Workers' Compensation Insurance: Claim Costs, Prices, and Regulation [1 ed.] 0792391705, 9780792391708, 9780585325309

The first four papers in this volume address benefit system policy matters, and the last ten papers address the pricing,

406 9 16MB

English Pages 384 [367] Year 1992

Report DMCA / Copyright

DOWNLOAD FILE

Polecaj historie

Workers' Compensation Insurance: Claim Costs, Prices, and Regulation  [1 ed.]
 0792391705, 9780792391708, 9780585325309

Citation preview

Workers' Compensation Insurance: Claim Costs, Prices, and Regulation

Huebner International Series on Risk, Insurance, and Economic Security J. David Cummins, Editor The Wharton School, University of Pennsylvania, Philadelphia, Pennsylvania, USA Series Advisors: Dr. Phelim P. Boyle University of Waterloo, Canada Dr. Jean Lemaire University of Pennsylvania, USA Professor Akihiko Tsuboi Kagawa University, Japan Dr. Richard Zeckhauser Harvard University, USA Other books in the series: Cummins, J. David; Smith, Barry D.; Vance, R. Nell; VanDerhei, Jack L.: Risk Classification in Life Insurance Mintel, Judith: Insurance Rate Litigation Cummins, J. David: Strategic Planning and Modeling in Property-Liability Insurance Lemaire, Jean: Automobile Insurance: Actuarial Models Rushing, William.: Social Functions and Economic Aspects of Health Insurance Cummins, J. David and Harrington, Scott E.: Fair Rate of Return in PropertyLiability Insurance Appel, David and Borba, Philip S.: Workers Compensation Insurance Pricing Cummins, J. David and Derrig, Richard A.: Classical Insurance Solvency Theory Borba, Philip S. and Appel, David: Benefits, Costs, and Cycles in Workers Compensation Cummins, J. David and Derrig, Richard A.: Financial Models of Insurance Solvency Williams, C. Arthur: An International Comparison of Workers' Compensation Cummins, J. David and Derrig, Richard A.: Managing the Insolvency Risk of Insurance Companies Dionne, Georges: Contributions to Insurance Economics Dionne, Georges: Foundations of Insurance Economics--Readings in Economics and Finance Klugman, Stuart: Bayesian Statistics in Actuarial Science with Emphasis on Credibility The objective of the series is to publish original research and advanced textbooks dealing with all major aspects of risk bearing and economic security. The emphasis is on books that will be of interest to an international audience. Interdisciplinary topics as well as those from traditional disciplines such as economics, risk and insurance, and actuarial science are within the scope of the series. The goal is to provide an outlet for imaginative approaches to problems in both the theory and practice of risk and economic security.

Workers' Compensation Insurance: Claim Costs, Prices, and Regulation

Edited by David Durbin Milliman & Robertson, Inc. and Philip S. Borba Milliman & Robertson, Inc.

Kluwer Academic Publishers Boston/Dordrecht/London

Distributors for North/~merlca: Kluwer Academic Publishers 101 Philip Drive Assinippi Park Norwell, Massachusetts 02061 USA Dlstrlbutore for all other countries: Kluwer Academic Publishers Group Distribution Centre Post Office Box 322 3300 AH Dordrecht, THE NETHERLANDS

Library of Congress Cataloging-in-Publication Data Workers' compensation insurance: claim costs, prices, and regulation/edited by David Durbin and Philip S. Borba. p. cm.--(Huebner international series on risk, insurance, and economic security) Papers first presented at the 7tb and 8th Conferences on Economic Issues in Workers' Compensation, sponsored by the National Council on Compensation Insurance. Includes bibliographical references and index. ISBN 0-7923-9170-5 (alk. paper) 1. Workers' compensation--United States--Costs. 2. Workers' compensation--United States--Rates and tables. I. Durbin, David L. IL Borba, Philip S. III. Series. HD7103.65.U6W664 1992 368.4' 101 '0973--dc20 92-22576 CIP Copyright ~) 1993 by Kluwer Academic Publishers All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, mechanical, photo-copying, recording, or otherwise, without the prior written permission of the publisher, Kluwer Academic Publishers, 101 Philip Drive, Assinippi Park, Norwell, Massachusetts 02061. Printed on acid-free paper.

Printed in the United States of America

Contents

Preface List of Contributors

vii xvii

1

The Determination of Workers' Compensation Benefit Levels Patricia M. Danzon 2

Workers' Compensation Costs and Heterogeneous Claims Richard J. Butler and John D. Worrall

25

3

The Transition from Temporary Total to Permanent Partial Disability: A Longitudinal Analysis John D. Worrall, David Durbin, David Appel and Richard J. Butler

51

4

The Transition from Temporary to Permanent Disability: Evidence from New York State Terry Thomason

67

5

Capital Flows and Underwriting Cycles in Liability Insurance David J. Cummins and Patricia M. Danzon

97

6

Self Insurance in Workers' Compensation Richard J. Butler and John D. Worrall 7 On the Use of Option Pricing Models for Insurance Rate Regulation Neil A. Doherty and James R. Garven

129

147

vi

CONTENTS

8

Some Caveats for the Use of Forecasting Models for Assessing Rates of Return in Workers' Compensation A.S. Paulson, R.L. Boylan and L.T. Lim

171

9

Leverage, Interest Rates, and Workers' Compensation Survival John D. Worrall, Richard J. Butler, David Durbin and David Appel 10 Predicting Insurance Insolvency Using Generalized Qualitative Response Models James B. McDonald 11 Firm Characteristics and Workers' Compensation Claims Incidence Allan Ho Hunt, Rochelle V. Habeck and Michael J. Leahy 12 The Impact of Experience-Rating on Employer Behavior: The Case of Washington State James Chelius and Robert S. Smith

187

223

243

293

13

"Good Days" and "Bad Days": Risk Management Decisions When Effectiveness is Unreliable Harris Schlesinger

307

14

Economic Consequences of Third-Party Actions for Workplace Injuries Joan T. Schmit

329

Index

341

Preface

The articles in this volume were first presented at the Seventh and Eighth Conferences on Economic Issues in Workers' Compensation sponsored by the National Council on Compensation Insurance. A principal objective of the Conference series has been for workers' compensation insurance researchers to apply state-of-the-art research methodologies to policy questions of interest to the workers' compensation insurance community. This community is a rather diverse group--it includes employers, insurers, injured workers, regulators, and legislators, as well as those who service or represent these groups (e.g., physicians, rehabilitation specialists, labor unions). Despite this diversity and the variety of agendas, the Conference series continues to address many important policy questions. Readers familiar with the Conference series and the four previously published volumes should notice an evolution in terms of the topics addressed in this volume. In the earlier conferences, the topics were more often concerned with the underlying causes of the tremendous increase in workers' compensation benefit payments. In the present volume, however, only four of the fourteen chapters directly concern workers' compensation insurance benefits, while the other ten concern the pricing of workers compensation insurance. This is not to suggest that workers' compensation cost increases have abated. In 1989, workers' compensation incurred losses exceeded $45 billion to continue the annual double-digit cost increases. Two explanations can be offered for the somewhat altered focus of this volume. First, despite the continued increase in prices, the financial results for the workers' compensation insurance line continue to be poor. Second, the number of property-casualty insurer insolvencies increased dramatically in the latter half of the 1980s. vii

viii

PREFACE

With regard to the first explanation, virtually all those familiar with workers' compensation insurance will agree that 1) this insurance coverage has become much more expensive to employers, and 2) despite increasing prices, operating results have not improved. Regarding the second explanation, although insolvency is often the consequence of poor management--in property-casualty insurance as well as other industries--insolvencies present special problems for the property-casualty insurance industry. In most instances, the unfunded liabilities of a property-casualty insolvency must be paid from assessments levied on solvent insurers; thus, the mistakes of the few are borne by the successes of the many. Furthermore, more than the case in other markets, a property-casualty insolvency creates a discomforting public relations reaction. Unlike most products, an insurance policyholder may not receive the full value of the product for some time--for a long-tail liability claim, payments many not be complete for a number of years. An insolvency causes a loss of trust that is directed not only toward the insolvent insurer but also to all insurers. Similar to the problems in the Savings and Loan banking industry, the financial stability of the property-casualty insurance industry has economy-wide ramifications. The ability to determine prices that are neither inadequate, excessive, nor unfairly discriminatory becomes more than just an adage. The financial tools at the disposal of insurance regulators for both determining fair prices and keeping track of insurer financial health are thus worthy of critical examination. The first four chapters in this volume address benefit system policy matters, and the last ten address the pricing, regulation, and potential insolvency of workers' compensation insurance. Within each general area, the chapters are arranged so that the first chapters address broad issues of workers compensation benefits and prices; while the later chapters address issues which are more specific in nature. The first four chapters address: • the determinants of the level of workers' compensation benefit levels (Danzon), • the determinants of the shape and location of a loss distribution (Butler and Worrall), and • the factors that affect the propensity of temporary total disabilities to become permanent disabilities (Worrall, Durbin, Appel and Butler; Thomason). The ten chapters that concern workers compensation insurance pricing address:

PREFACE

ix

• explaining the flow of capital to the property-casualty insurers over the underwriting cycle (Cummins and Danzon), • the determinants of self insurance (Butler and Worrall), • models for pricing insurance products (Doherty and Garven; Paulson, Boylan and Lim; Worrall, Butler, Durbin and Appel), • predicting insurer insolvencies (McDonald), • explaining differences in loss experience across firms in the same industry (Hunt, Habeck and Leahy), • the incentives of an experience rating program on small employers (Chelius and Smith), • the effectiveness of loss control activities on insurance prices (Schlesinger), and • the effect that third-party actions brought by injured workers against product manufacturers might have on workplace safety (Schmit). In the first chapter, Danzon investigates the relationship between the characteristics of a workforce and the workers' compensation insurance benefit system. With heterogeneous preferences among workers, and a common benefit structure, the conventional assumption that compensating wage differentials are individually fair may not be correct. The empirical results suggest: that maximum weekly benefits are higher when the proportion of workers in low-income families is high; that a large proportion are employed in small firms; and that there is a high proportion of workers employed in the agriculture and service sections. Legislators, regulators, and insurers frequently inquire into the determinants of the losses for a given workers' compensation insurance benefit system. Butler and Worrall extend previous research by distinguishing between two forms of injured-worker heterogeneity in the empirical analysis of loss distributions. One form concerns factors that are unobservable (e.g., physical endowment, attitude, administration of claims), while a second form concerns observable characteristics (e.g., age, benefit award size, wages). They find that by allowing both the shape and location of various loss distributions to be affected by the observable heterogeneity, they are better able to fit the long-tail that has been observed for workers' compensation costs. In essence, they conclude that the distributional effects of policy changes are complex and that estimates of the cost impact of such changes need to consider impacts along the entire loss distribution. The two chapters by Worrall, Durbin, Appel and Butler and Thomason narrow the investigation of workers' compensation insurance benefits to a costly feature of most benefit systems: compensating workers with permanent disabilities. Permanent disabilities account for approximately

X

PREFACE

5 percent to 7 percent of all workers' compensation claims but between 40 percent and 50 percent of all incurred benefit costs. Except in the obvious cases of dismemberment, or death, most workers' compensation claims for lost worktime benefits are initially classified as temporary total disabilities. Over time, most workers fully recover from a lost worktime disability and return to work. The more seriously disabled workers, however, achieve only a partial recovery and are reassigned to a permanent disability claim status. Although the severity of the disability is an important factor affecting the assignment to a permanent disability status, economic incentives can also impart a significant influence. The Worrall, Durbin, Appel and Butler and Thomason chapters offer the first systematic investigation of the factors that influence the transition from temporary total claim status to permanent partial status. Worrall et al. point out that understanding the determinants of this transition is important for two reasons. First, increases in permanent disability benefits increase the incentives for workers to obtain a permanent disability status, which increases the proportion of these claims and alters the distribution of losses among the different claim statuses. Second, insurer case reserves should be more accurate with an improved understanding of which injured workers are most likely to become permanent disability beneficiaries. Worrall et al. investigate the temporary-permanent transition among Massachusetts and Illinois injured workers, and Thomason does the same among New York injured workers. Both empirical investigations find that the benefit structure is an important determinant of permanent disability claim status, that is, the higher the permanent disability benefits, the higher the propensity for a permanent disability claim status. Thomason also found that reduced employment opportunities increase the propensity of permanent disability status, while greater uncertainty over these benefits--measured by the variability of the permanent disability awards--reduces this propensity. The remaining ten chapters in this volume address various topics that directly or indirectly concern the pricing of workers' compensation insurance. Cummins and Danzon review two existing and competing theories that have been used to explain the movement of insurance prices and capital flows over an underwriting cycle: the perfect capital markets theory and the capacity constraint theory. They also present two additional explanations. One alternative explanation concerns informational asymmetries between buyers and sellers. This line of thinking argues that insurers can manage reductions in demand caused by a price increase by reducing policy limits. The reduced policy

PREFACE

xi

limits constrain the insurer's losses, which improves its profit levels. A second alternative explanation begins with the hypothesis that insurers prefer internal capital sources (from retained earnings) to external capital. This preference for internal capital is positively correlated to the uncertainty caused by inadequate reserves following an adverse loss experience. Under those conditions, capital markets expect a premium over the normal cost of capital. To avoid paying this premium, insurers may increase prices that lead to larger retained earnings and thus avoid the need to raise capital from external sources. Butler and Worrall offer another view regarding fluctuations in the demand for insurance. As an alternative to price increases, individual firms have incentives to retain more control over losses. The result may be an increase in firms that self insure. They investigate the determinants of self insurance. Using an expected utility framework, where controls for firm size distribution are included, they use aggregate state data for Census of Manufacturers years and find that self insurance in workers' compensation also increases with increases in larger size firms and in interest rates. The next two chapters address a difficult problem, presented by the substantial volatility of insurance prices over an underwriting cycle: the setting of insurance prices through a regulatory rate-setting procedure. Doherty and Garven summarize the two principal approaches for using rate-of-return models from insurance pricing: discounted cash flow models and asset pricing models. Examples of the discounted cash flow models include the internal rate of return approach frequently used by NCCI and the Myers-Cohn model that has been used in Massachusetts automobile and workers' compensation insurance rate proceedings. Examples of asset pricing models include the capital asset pricing model and arbitrage pricing models. In this chapter, Doherty and Garven also apply the option pricing model to the problem of insurance rate regulation. An implicit solution for the fair insurance price is derived by setting the value of the shareholders' option equal to the initial surplus. They argue that the treatment of ruin probability and tax shields in the options pricing model is an improvement over the discounted cash flow and asset pricing models. However, they also recognize that the complexity of the options pricing model is a limitation for use in a regulatory rate-setting environment. Paulson, Boylan and Lim note that workers' compensation insurance pricing models must capture the expected underwriting results, the investment of policyholder and equityholder funds, as well as the regulatory process. They note that an insurance pricing model must be able to

xii

PREFACE

capture and be adaptable to external shocks--such as the effect of the October 19, 1987 stock market crash on investment values and the Tax Reform Act of 1986. Paulson, Boylan, and Lim also note that insurance pricing is inherently more risky for an individual insurer than for the statewide average, which is the basis for regulated price setting, because of a larger potential fluctuation in the individual insurer's loss distribution. Through empirical evaluation, they illustrate the relationship between the rate of return and firm size for various premium-to-surplus and liabilities-to-surplus ratios. They also enumerate the inherent modelling problems incorporating e x a n t e informational asymmetries. Worrall, Butler, Durbin, and Appel address the effects of external shocks on workers' compensation insurance prices from a different but related perspective. They consider the implications on individual insurer loss experience and insolvency potential of alternative financial market and leverage assumptions in insurance pricing models. Using a simulation approach and employing assumptions occasionally presented in regulatory rate-setting proceedings, they demonstrate that, under plausible scenarios, many insurers could potentially become insolvent. The Paulson, Boylan and Lim and Worrall, Butler, Durbin and Appel chapters discuss conditions that might give rise to an insurer insolvency. McDonald also addresses the problem of insurer insolvency, but from the perspective of identifying financially troubled insurers with information on past insolvencies. McDonald uses the characteristics (e.g., selected IRIS measures) of insurers that became insolvent during the 1980s and a random sample of solvent insurers to test the predictive power of different estimating procedures. McDonald estimated the probability of insolvency using conventional estimating procedures and a set of alternative methods that provide greater flexibility for data that may not meet the restrictive assumptions implicitly underlying conventional approaches. McDonald found the selected IRIS and other characteristics useful for predicting insurer insolvencies. More important, the alternative tests McDonald introduces improved the predictive power of insurer insolvency. Using information on insurers and McDonald's alternative prediction procedures should improve the ability of regulators and others to detect financially troubled insurers. It was noted above that the Paulson, Boylan and Lim and Worrall, Butler, Durbin and Appel chapters discuss the problems of extrapolating statewide, from "typical insurer" characteristics, the operating results of all insurers. The issue is the greater variability and, hence, uncertainty of individual insurer operating results and the problems this can create for rate regulation. Hunt, Habeck and Leahy extend this investigation by

PREFACE

xiii

examining the variability of loss experiences among employers within the same industry. The principal objective of their investigation was to determine whether certain characteristics differentiate low-claimsincidence employers from high-incidence employers in the same industry. Hunt, Habeck and Leahy found tremendous variability in claims incidence rates within 2-digit SIC groups. In some cases, the claim incidence among the high-incidence employers was ten times the incidence among the lowincidence employers. Further analysis found, as might be expected, that the specific industry type had the largest influence on claims incidence, and that claim incidence was lower for employers with more than 500 employees and higher for employers with facilities in several counties. Hunt, Habeck and Leahy also reported the findings from a survey of low-and high-incidence employers. The last three chapters narrow the issue of workers' compensation insurance pricing to its most basic unit: designing employer-specific pricing programs that better tailor an employers' premiums to their own loss experience. These last chapters concern the incentives of an experience rating program on small employers, the effectiveness of loss control activities on insurance prices, and the effect that third-party actions brought by injured workers against product manufacturers might have on workplace safety. A primary objective of an experience rating program is to present employers the prospect of reducing the insurance premium if losses can be reduced. In every state, employers above a certain premium threshold must participate in an experience rating program. Generally speaking, these programs use up to three years of the employer's past experience. This experience is weighted with the class-wide experience, with the weight an increasing function of employer size. The experience rating program for Washington state employers however is distinctive. The threshold for participation in the Washington program is lower than other states; thus, a larger proportion of small businesses in Washington have their workers' compensation insurance premium tied to their past experience. All other factors the same, it should be expected that injury rates among small employers in Washington would be lower than the injury rates among small employers in other states. Chelius and Smith compared the lost worktime injury rates of small employers in Washington to the injury rates of small employers in other states and find no evidence to conclude that the Washington experience rating program reduced the injury rates of small employers. Chelius and Smith caution, however, that it is still premature to conclude that an experience rating program does not reduce injury rates for small

xiv

PREFACE

employers. First, there may be a lag between an employer's introduction of an injury-reducing program and its effect on premium. Further, the effects of such a program may be muted, at least initially, given the inclusion of three years of information in the premium modification calculation. Second, the premium modification calculation may be too complicated for many employers to appreciate the potential premium reduction prospects. Third, the benefits of a reduction in workers' compensation insurance premium may be too small for many employers to undertake the costs of an injury-reducing program. Finally, there may be a failure on the part of insurers to adequately market the premium-reducing potential of the experience rating programs. Broadly speaking, employers can improve their losses through activities that either reduce the frequency of accidents or reduce the severity of a given loss. Collectively, these loss prevention and loss reduction activities are referred to as loss control activities; however, loss control programs occasionally fail. Schlesinger analyzes the interactions between loss control activities and insurance prices and finds that loss reduction activities and insurance prices are not always substitutes. For most situations, an increase in the price of insurance should encourage the undertaking of more loss control activities. Yet, if the uncertainty of the loss control activities is taken into consideration, that is, loss control may have a "bad" day, Schlesinger finds that risk management activities and insurance prices may be complements. This finding complicates models of risk management and may help explain the difference between empirical evidence and the existing theoretical models. The last chapter in this volume concerns the importance of productrisk information in allocating the costs of workers' compensation insurance. While workers' compensation insurance provides for partial replacement of lost earnings and full reimbursement of reasonable medical charges, Schmit also points out that injured workers can sue product manufacturers or health care providers for negligence. The availability of these third-party actions affect workers, employers, and manufacturers safety incentives. If markets are competitive, and perfect information exists, then the full cost of accident liability will be captured in the product prices paid by employers. The more usual situation, however, is that information is imperfect, which gives rise to an ambiguous effect for third-party actions arising from workplace accidents. Cognitive dissonance by employees will tend to reduce the compensating wage differentials that otherwise would be demanded for the total job risk. Alternatively, the knowledge that third-party actions can be brought, may cause employees to demand a

PREFACE

xv

less-than-optimal level of safety. Or, employers may not have complete (or understate the) information concerning the risk of a product used by its employers and, therefore, provide a less-than-optimal level of safety. Using a model of imperfect information, Schmit concludes that thirdparty actions may be undesirable if workers' compensation benefits provide an optimal level of compensation. If employers or employees misperceive risk, however, some form of product liability may be desirable. Suboptimal levels of safety may still remain because of the incomplete risk information. In closing, we wish to thank the National Council on Compensation Insurance for its financial support of the Conference series and this volume. We also wish to express our gratitude and the gratitude of the authors to the anonymous referee, who provided many helpful comments.

List of Contributors

David Appel Milliman & Robertson, Inc. 2 Pennsylvania Plaza, #1552 New York, NY 10001 R.L. Boylan Rensselaer Polytechnic Institute Troy, NY 12181 Professor Richard J. Butler Department of Economics Brigham Young University Provo, UT 84601 Professor James Chelius Institute of Management and Labor Relations Rutgers University P.O. 231 New Brunswick, NJ 08903 Professor J. David Cummins Department of Insurance The Wharton School University of Pennsylvania Philadelphia, PA 19104-6366

Professor Patricia A. Danzon Health Care Systems and Insurance Departments The Wharton School University of Pennsylvania Philadelphia, PA 19104 Professor Neil Doherty The Wharton School The University of Pennsylvania 3641 Locust Walk Philadelphia, PA 19104 David Durbin Milliman & Robertson, Inc. 2 Pennsylvania Plaza, #1552 New York, NY 10001 Professor Rochelle V. Habeck University of Washington Seattle, WA Dr. Alan A. Hunt The W.E. Upjohn Institute 300 S. Westnedge Avenue Kalamazoo, MI 49907

xvii

xviii Professor Michael J. Leahy Michigan State University East Lansing, MI 48824 L.Y. Lim Rensselaer Polytechnic Institute Troy, NY 12181 Professor James B. McDonald Department of Economics Brigham Young University Provo, UT 846404 Professor A1 S. Paulson Department of Operations Research Rensselaer Polytechnic Institute Troy, NY 12181 Professor Harris Schlesinger Department of Finance University of Alabama Box 870224 Tuscaloosa, AL 35487

LIST OF CONTRIBUTORS

Professor Joan T. Schmit University of Wisconsin-Madison 1155 Observatory Drive Madison, WI 53706 Professor Robert S. Smith New York State School of Industrial and Labor Relations Cornell University Ithaca, NY 14853 Professor Terry Thomason McGill University Faculty of Management 1001 Sherbrooke Street West Montreal, Quebec Canada H3A1G5 Professor John D. Worrall Bureau of Economic Research and Department of Economics Rutgers University New Brunswick, NJ 08903

THE DETERMINATION OF WORKERS' COMPENSATION BENEFIT LEVELS* Patricia M. Danzon

Introduction The purpose of this chapter is to analyze the incidence of the cost of workers' compensation (WC) benefits and to examine whose preferences are reflected in the benefit levels chosen. The standard assumption of economists is that the costs of WC benefits are borne by workers in the form of a compensating wage differential. Several empirical studies appear to confirm this hypothesis. Dorsey and Walzer (1983) conclude that for nonunion workers there is a fully offsetting wage differential for compensation benefits, although for union members they find a significantly positive correlation between wages and benefit levels.

* I would like to thank several individuals and institutions for assistance in this research. The Center for Risk and Insurance at the University of Pennsylvania provided financial support. The National Council of Compensation Insurance and Jack Worrall provided data. I have greatly benefitted from conversations with David Appel, Philip Borba, David Cummins, David Durbin, Mark Pauly and Jack Worrall. Dong Han Chang provided valuable research assistance.

2

WORKERS' COMPENSATIONINSURANCE

Viscusi and Moore (1987) conclude that "the observed rate at which workers are willing to trade off base wage rates for higher levels of compensation greatly exceeds the actuarial rate of trade-off, even taking into account administrative costs. These results suggest that current benefit levels are suboptimal, provided that one abstracts from moral hazard considerations. ''1 It would be surprising to find suboptimal benefit levels, since that would imply failure to maximize worker utility, which in turn implies that labor costs to employers are not minimized. However, this estimate of a more than one-for-one trade-off may be biased. 2 And as Feldstein (1973) points out in the context of the Social Security payroll tax, a finding that net wages fall by the amount of the tax does not necessarily imply that the incidence of the tax is on labor, if either labor or capital is in imperfectly elastic supply. Similarly, a finding that wages fall by the cost of WC benefits would not necessarily imply that the incidence of the cost is on labor or that workers receive their preferred level of benefits. The policy-oriented literature has also tended to conclude that WC benefit levels are inadequate. These criticisms were particularly strong prior to the reforms of the 1970s. The National Commission on State Workmen's Compensation Laws (hereafter National Commission) concluded: In general, workmen's compensation programs provide cash benefits that are inadequate. The majority of disabled beneficiaries receive less than two-thirds of the lost wages. 3 In most States, the most a beneficiary may receive, "the maximum weekly benefit", is less than the poverty level of income for a family of four. Payments are inequitable as well as inadequate. Benefits differ widely from State to State. Within States, high-wage workers, if disabled, receive a smaller proportion of their lost earning than do low-wage earners because they are limited by the ceiling of the maximum weekly benefits. The National Commission also noted that many states limit the duration or the total amount of cash payments for permanent total disabilities. In 1972, fifteen states limited the duration of benefits for permanent total disabilities to ten years and eleven states limited the amount payable to $25,000, which was less than the average U.S. worker earned in four years. 4 Payment of relatively more generous benefits for routine minor injuries than for less frequent serious injuries seems to conflict with basic principles of optimal insurance coverage. Benefit levels have increased substantially since the National Commission made its recommendations (see table 1-1). 5 Maximum benefits have been increased, although generally not to the recommended 200

THE DETERMINATION OF WORKERS' COMPENSATION BENEFIT LEVELS

3

percent of the average weekly wage. Most states have not adopted the Commission's recommendation that the replacement rate for temporary total disability be 80 percent of the worker's spendable weekly earnings. There remains substantial variation across states. Burton and Krueger (1986) report that interstate differences widened between 1972 and 1978. How far the increase in benefits during the 1970s reflects higher preferred levels of benefits in the face of changed economic circumstances, and how far it was a reluctant response to the threat of Federal intervention in the event of failure to comply with at least the major recommendations remains an unanswered question. Implicit in the criticisms that WC benefit levels are inadequate is the presumption that benefits do not reflect the preferences of workers. Determining whose preferences are reflected in WC benefit levels is essential to evaluating these criticisms. Another reason for studying the determinants of WC benefit levels is that if benefits do in fact reflect workers' preferences and if these preferences for compensation are correlated with wages or injury rates, then estimates of compensating wage differentials that treat benefit levels as exogenous are subject to simultaneous equations bias. 6 Specifically, if preferred replacement rates are inversely related to wage levels, then estimates of compensating wage differentials for WC benefits will be upward biased (in absolute value). Standard theory of the demand for insurance implies that for any individual worker the optimal mix. of cash wages and compensation benefits depends on the potential wage, the probability of injury, the loading charge, risk preferences, and on tax rates on cash wages. It may also depend on the degree of experience rating and on moral hazard. However, because the structure of WC benefits is determined at the state level, the level of compensation cannot be tailored to the preferences of each worker or even the workers in each firm, as is the case with private health and disability insurance. The WC benefit structure is effectively a public good for the workers in each state. The outcome therefore reflects the filtering of the preferences of workers and possibly of other interest groups through the collective decision-making process. The structure of this chapter is as follows. Section one presents a simple model of individual demand for benefits. Section two discusses alternative models of the collective choice of benefit levels and examines circumstances in which interest groups other than workers may have a stake in and influence the outcome. Section three describes the data and estimation issues. Section four reports empirical estimates of the determinants of benefit levels over the period 1965 to 1985.

4

WORKERS' COMPENSATION INSURANCE

The Optimal Structure of Benefits Consider first the level of cash benefits that a cost minimizing employer would choose to provide if benefits could be tailored to the preferences of each individual worker or if all workers within the firm are homogeneous in all relevant respects. In each period there is a probability p that the worker suffers an injury that results in total loss of wages for the remainder of the period. Initially, assume that the level of benefits has no effect on the probability of injury or duration of benefits. Further, assume that the firm is perfectly experience rated, and that workers are fully informed about the risks of injury and about benefits. The optimal level cash wages (W) and benefits (K) is selected to maximize the worker's utility, subject to the constraint that the expected cost of the total compensation package is equal to the potential wage with zero benefits (we). Assuming profit maximization, W e is also the value of marginal product of labor: Max L = (1 - p ) U I [ W ( 1 W,K,rn

-

-

t)]

+ pUo[K ] + m [ W e

--

(1 - p ) W - p(1 + h)K]

(1)

where t = p = W = K = h = We =

the worker's tax rate on cash income the probability of injury cash wages compensation if injured loading charge on WC insurance, h / > 0 potential wage with zero benefits = value of marginal product m i s a Lagrange multiplier

Subscripts 0, 1 denote the states of injury and no injury, respectively. First order conditions for W and K yield: UI'(1 - t) --- m U0'(1 + h) -1 = m or

u0'

= (1 - t)(1 + h)

(2) (3)

(4)

U1 '

This has the familiar interpretation that if h = 0 and t = 0, and utility is state-independent, optimal compensation is full compensation. 7 If h = 0 and t > 0 and utility is state-independent, then optimal compensation would be more than full compensation: K* > W*. But this ignores the effects of moral hazard that would impose the additional constraint K* ~ W*.

THE DETERMINATION OF WORKERS' COMPENSATION BENEFIT LEVELS

5

To determine how the optimal replacement rate changes in response to changes in the potential wage or skill level, the risk of injury, the tax rate and loading charge, the first order conditions are differentiated with respect to W e, p, t, and h. It is shown in the Appendix that an increase in the potential wage W e increases both cash wages and benefits. Ignoring taxes, the optimal replacement rate increases (decreases) if there is increasing (decreasing) absolute risk aversion, assuming state independent utility: 8 OK/OW e

UI"(1 - 02(1 + h)

OWlOW e

Uo"

=

U1"(1 - 0 2 / U 1 ' Uo"lUo'

> o

(5)

If this expression is less than 1, the optimal replacement rate declines as W e increases. With a progressive tax rate, decreasing absolute risk aversion is necessary but not sufficient condition for O ( K / W ) / O W e < 1. With state-dependent utility, these conditions do not necessarily hold. An increase in the risk of injury (p) decreases both the optimal cash wage and benefits. The effect on the replacement rate again depends on the tax rate and on whether absolute risk aversion is increasing, constant or decreasing, as in equation 5. 9 An increase in the loading charge lowers the optimal replacement rate. Thus, assuming that administrative costs are higher in small firms, workers in small firms would prefer lower benefits. An increase in the tax rate on cash income increases the optimal replacement rate. The results so far must be modified to take account of moral hazard and imperfect experience rating. As argued by Butler and Worrall (1983) and others, higher benefit levels may induce employee moral hazard in the form of higher claim rates and longer duration of injuries. Although large firms are self-insured or self-rated, perfect experience rating at the firm level may not be sufficient to eliminate moral hazard at the level of the individual worker. Optimal benefits are lower in the presence of claimant moral hazard. 1° It is sometimes argued that for a given replacement rate claimant moral hazard is more severe at higher wage levels, because a given replacement rate provides a higher absolute level of "discretionary income" over and above the expenses of basic living. In that case, optimal replacement rates would be lower for higher wage workers, ceteris paribus. The optimal replacement rate is thus affected by several factors which vary with wage level: risk aversion, tax rates, and moral hazard. The net effect is uncertain a priori, but is likely to be negative if there is decreasing absolute risk aversion and if moral hazard increases at

6

WORKERS' COMPENSATION INSURANCE

higher wage levels. If moral hazard requires the constraint K < W(1 - t) then progressive tax rates alone imply [O(K/W)/OW] < O. Relaxing the assumption of perfect experience rating implies that each firm, and hence its workers, face less than the full cost of their losses. If benefit decisions were made at the firm level, each firm would have incentives to provide more generous benefits than their workers would prefer if they bore the full cost. However as will be argued below, when benefits are collectively chosen, imperfect experience rating at the firm level need not imply above-optimal levels of benefits.

Collective Choice of Benefits Because the benefit structure is set by statute, it is effectively a public good for all covered workers in the state, although the value that individual workers place on the common benefit structure may differ. The collective choice of WC benefits at the state level differs in important respects for the collective choice of group health insurance benefits or workplace safety at the level of the individual firm. 11 When noncash job attributes are determined at the level of the individual finn, market equilibrium involves the self-selection or matching of workers and firms. If there is a sufficient number of firms, in equilibrium workers sort themselves into homogeneous groups, such that within each firm all workers receive their preferred mix of cash and noncash job attributes.12 But given the small number of states and other factors affecting location decisions, the presumption of sorting into homogeneous groups is not plausible. A realistic model of the collective choice of WC benefits must address the question of filtering of heterogeneous preferences. However, to demonstrate some of the effects of state-level as opposed to firm-level choice, let us consider first the case of within-state homogeneity.

Within-State Homogeneity Assume that within each state employers face the same production function of safety and supply price of insurance, and that all firms are perfectly experience rated and are price takers in product markets. Workers are identical with respect to potential wage or skill (we), nonwage income and risk preferences. However, across states there are differences in safety production functions, skill levels, nonwage income, loading charges and tax rates.

THE D E T E R M I N A T I O N OF W O R K E R S ' COMPENSATION BENEFIT LEVELS

7

Given these assumptions, there is a unique utility-maximizing benefit structure that is simultaneously optimal for all workers in each state and so would receive unanimous support. Since each worker receives his or her preferred level of benefits, the incidence of benefit costs is fully on workers. 13 Differences in benefit levels across states reflect differences in opportunities. Specifically, across states the replacement rate is expected to be positively related to average firm size TM and negatively related to skill levels because of progressive tax rates--reinforced if there is decreasing absolute risk aversion or if moral hazard increases with income for a given replacement rate. Even with imperfect experience rating for the individual firm, provided that there is experience rating in the aggregate (costs are fully borne by workers in the state), the benefit level that is collectively chosen should be optimal--given the level of moral hazard. 15 Thus, when firms are imperfectly experience rated, collective choice of benefits is superior to noncooperative, firm-specific choice of benefits (assuming homogeneous preferences): collective choice eliminates the illusion of a free lunch for workers in each firm. Some of the costs of WC benefits can be shifted from workers through the preferential tax treatment of benefits and, in some states, by insurance arrangements that may permit hidden subsidies. Where the incidence of benefit costs is not fully on workers, benefit levels are likely to be above optimal if cost-shifting is to out-of-state taxpayers, through the exemption from federal income and payroll tax. Similarly, the potential for substituting Social Security Disability (SSDI) benefits for WC benefits creates incentives for workers in each state to set low benefits for total disability, since the payroll taxes that finance SSDI are paid largely by out-of-state taxpayers. 16 The probability of qualifying for SSDI is likely to increase with the duration of the disability. In that case, limits on duration of WC benefits for total disabilities or dollar caps may be an optimal strategy, particularly in low wage states since SSDI replacement rates are higher for low wage workers. At age sixty-two workers become eligible for the old age component of Social Security and at sixty-five qualify for Medicare; these programs would further reduce the demand for indefinite WC benefits for total disability. Thus, dollar or duration limits on WC benefits for permanent total disability may be Consistent with standard theory of (privately) optimal insurance, given federal funding for SSDI. The 1969 Federal Coal Mine Health and Safety Act, which established the Black Lung program, also creates the potential for shifting costs outof-state. Under this program, federal payments are excess to the state payments. Payments from federal sources are inversely related to benefits

8

WORKERS' COMPENSATION INSURANCE

received under the state-run WC program. Thus, this creates an incentive for states with large mining interests to reduce benefit levels, to the extent disbursements from federal funds are not assessed back to coal operators in the state.

Within-state Heterogeneity More realistically, within each state there is heterogeneity of firms and workers that results in differences in preferred benefit levels. Some industries face higher costs of producing safety. Small firms face diseconomies of small scale in producing safety and administering insurance, and are not fully self-rated because their experience is not fully credible. Workers differ in potential wage, nonwage income and tax rates and, hence, in preferred replacement rates, even if they faced similar supply prices. Thus, within any state workers are expected to differ with respect to preferred benefit levels, both within firms and across firms. The statutory benefit structure provides for a uniform replacement rate for temporary total disabilities. However, in most states the maximum benefit implies a sharply declining replacement rate for higher wage workers, and the minimum benefit implies a higher replacement rate for the lowest wage workers. For permanent-partial disabilities scheduled benefits provide for little variation in absolute compensation based on preinjury earnings, implying much lower replacement rates for high wage earners. Increasingly, these simple schedules are being replaced by formulae that incorporate some measure of wage loss. Nevertheless, it remains true that the benefit structure is to a large degree a public good for all workers in a state. Given heterogeneity of worker preferences there is no unique benefit structure that is optimal for all workers within the state. The benefit structure that is chosen will reflect a compromise that results from the collective decision-making process, given the incidence of the cost of WC benefits, which determines the stakes to the various parties and costs of participating in the political process. Even if the incidence is on workers in the aggregate, individual employers would have an indirect stake in the outcome, since by influencing the outcome to more closely match the interests of their workers, they may reduce their own total labor costs. Median Worker Model.

By analogy with the median voter model of the political process in a democracy, 17 one simple hypothesis is that the benefit structure reflects the preferences of the worker with median pref-

THE D E T E R M I N A T I O N OF W O R K E R S ' COMPENSATION BENEFIT LEVELS

9

erences. If income and risk level are the major determinants of differences in preferred benefits, then interstate variation in benefit levels should reflect interstate differences in median income and risk levels. This model requires the assumptions that each worker pays a compensating differential equal to his or her expected WC benefits (such that no other parties have a stake in the outcome), and that the costs of trading votes are prohibitive or that all workers have equal marginal net benefits (benefits minus costs) of participating in the decision-making process. These assumptions are unlikely to be met because of the operation of labor markets. Any worker for whom statutory benefits exceed his preferred level would not be willing to pay a fully compensating wage offset if he could get his preferred mix of wages and benefits in another state or in the uncovered sector; similarly, any worker who gets less than his preferred level of benefits and can get his preferred level elsewhere may require some additional compensation. Thus, mandatory benefits impose a tax on workers for whom the mandatory benefit levels are nonoptimal. The incidence of the tax depends on elasticities of supply of different types of labor and other factors of production; elasticities of demand for final products; factor intensities and elasticities of substitution between inputs; and the relative size of the uncovered sector. A general equilibrium model of the incidence of mandatory benefits has been developed elsewhere (Danzon, 1989) in the context of mandated employment-based health insurance. The relevant results are summarized here. In general, mandating benefits imposes a tax per worker equal to the difference between the cost to the employer and the worker's valuation of the benefits. The net-of-benefits demand curve shifts down by the costs of the benefits, P = (1 + h)pK. For workers who prefer less than the mandated benefit level, the supply curve shifts down by the reduction in the reservation cash wage, vP, where v ~< 1 is the worker's valuation of benefits, UI,/[Uw(1 - t)(1 + h)]. The difference between the demand and the supply shifts is the tax, T = P[1 - v]. For workers whose preferred level of benefits exceeds the mandated level, the supply shift is uncertain. If the mandatory benefits simply displace voluntary private insurance and supplementation beyond the mandated minimum is feasible, there is no tax. There may still be a positive tax if the supply price of private insurance exceeds the supply price of WC benefits (for example, because of adverse selection or diseconomies of smaller scale in the provision of private disability and medical insurance) and if they can get their preferred benefit mix in another state. Assuming that private insurance is a reasonable substitute for WC benefits, WC benefits impose taxes primarily from mandating higher than preferred benefit levels.

10

W O R K E R S ' COMPENSATION INSURANCE

For any benefit level the effective tax rate, measured as a percentage increase in cost per hour of labor, will vary across firms. It will be higher in firms with a relatively high loading charge or high cost of safety (small firms), and in firms whose workers face low tax rates on cash income or have low risk aversion. The incidence of a tax that differs across firms may be analyzed using Mieskowski's (1972) general equilibrium analysis of the incidence of the local property tax on reproducible capital, when the tax rate varies across jurisdictions. 18 Consider three factors of production: L (low wage workers) for whom mandated benefits exceed the optimal level K; H (high wage workers) for whom mandated benefits are either optimal or simply displace private insurance at equal cost; and capital F, which includes potentially immobile factors such as land and some physical and human capital. Total supplies of all factors are assumed to be fixed in aggregate. L is assumed to be mobile across firms, but H and F may be imperfectly mobile. If the tax is uniform on all units of L, the full incidence is on L. But if the tax rate differs across units of L because the supply price of safety or insurance differs across firms, and if L is perfectly mobile across firms, then L in high cost firms will not bear the cost differential in these firms since wages will be equalized in all employments. Wages of L fall by the average tax per hour. Deviations from the mean tax may be shifted forward to consumers, if demand facing highly taxed firms is not perfectly elastic--for example, if they produce nontraded goods and services, such as retail trade and construction. There may also be some backward shifting to other imperfectly mobile factors of production in the high cost firms. Of course to the extent that L in high cost firms is imperfectly mobile, it will bear (part of) the cost differential of providing the mandated benefits. This analysis has several implications for collective choice of WC benefit levels. Workers with a relatively low demand for benefits will vote for Kl. But the H workers who would prefer higher benefits, if they bore only the costs of their own benefits, will not necessarily vote for higher benefits, Kh, because they may bear some of the costs of imposing K > KI on L. Type H workers are less likely to oppose Kl if supplementary private insurance is a good substitute for higher WC benefits. Even if private insurance is an imperfect substitute, H are less likely to oppose K~ if type L workers could get their preferred benefits in another state and are mobile. Thus, where workers who prefer relatively low benefits are mobile, any one state will be reluctant to be the first to raise benefits above the level preferred by type L workers. The existence of an uncovered sector of employment within the state

THE DETERMINATION OF WORKERS' COMPENSATION BENEFIT LEVELS 11

has the same effect in constraining benefit levels in the covered sector to the level preferred by workers who prefer low benefits. Thus, one factor contributing to the increase in benefit levels in the 1970s may be the simultaneous extension of WC coverage to sectors of the economy that were previously not covered and that included many low wage workers who would probably have a low willingness to pay for benefits. 19 The fact that employers lobby over the determination of WC benefits does not necessarily imply that the incidence of costs is on capital or that employers are opposing their workers interests, defined as lobbying for benefits below the level their workers would be willing to pay for. An employer may lobby on behalf of his workers if this reduces the cost of representing their interests, since a cost-minimizing employer has an incentive to provide any noncash job attribute that is valued at more than cost by his workers. On the other hand, to the extent the collectively chosen level exceeds ~he level that an employer's workers are willing to pay for, the incidence cf the excess cost may be partly on immobile factors within the firm, so employers are expected to lobby to prevent benefits exceeding the level their workers are willing to pay for. In particular, the owners of imperfectly mobile factors in small firms, who face relatively high costs of providing any given level of benefits, will oppose mandated benefits that exceed the level preferred by their workers. Benefit levels a~e expected to be higher in states where final product demand for domestically produced goods is relatively inelastic, since inelastic product demand facilitates the pass-through of excess costs to consumers. Consumers are likely to face high costs and relatively small per capita benefits from becoming informed about WC benefit choices, compared to other factors of production that bear excess costs when final product demand is perfectly elastic. Thus, benefits are expected to be higher in states with a relatively large share of value added in goods and services for which out-of-state substitutes are imperfect, such as retail trade, services, and possibly construction. Previous studies indicate that unions do influence the operation of the WC system. Dorsey and Waltzer (1983) find a positive association between benefits and wages for unionized workers, rather than the expected negative compensating wage differential that was found for nonunionized workers. Butler and Worrall (1983) find a positive association between unionization and WC claim rates. Both of these estimates could be influenced by simultaneous equations bias if unions affect choice of benefit levels. Unions may affect WC benefit choice in several ways. If individual Unions.

12

WORKERS' COMPENSATION INSURANCE

workers face high costs of information about job risk or about benefit levels, they would choose nonoptimal benefit levels. 2° If imperfect information implies underestimation of job risk, and if unions reduce the costs of information, benefit levels would be higher in heavily unionized states. Unions may also reduce the costs of representing workers interests in the political process. However, as noted above, employers also have an incentive to provide this service so it does not necessarily follow that worker preferences are better represented in heavily unionized states. Some models of union decision making imply that representation through unions may affect the relative weight given to preferences of different workers. In the context of group health insurance, Goldstein and Pauly (1976) suggest that union choice may reflect the preferences of the median worker or of more senior, higher paid workers, rather than the marginal worker whose preferences are expected to dominate in nonunionized firms. However, this ignores the issue of who bears the cost of forcing other marginal workers away from their preferred level of benefits. The general equilibrium model outlined above suggests that unions may influence the level and incidence of costs of mandating aboveoptimal benefit levels for some workers. If a union raises the overall level of compensation of labor and controls the wage structure for different types of labor, then workers who receive more than their preferred level of benefits would be willing to pay the required compensating differential, since total compensation could still exceed their best alternative in nonunionized employment. Another way by which unions may raise the collectively chosen benefit level is by simultaneously raising benefits in adjacent states. By eliminating low benefit alternatives in other states, workers who prefer lower benefits are forced to bear the costs of aboveoptimal benefits; the incidence of excess costs on other factors of production, including workers who prefer higher benefits, is thereby reduced. Finally, in industries where unions negotiate labor compensation on an industry-wide basis, the demand facing any firm is less elastic than in a nonunionized industry because all firms' costs increase simultaneously. The more inelastic the final product demand, the greater the share of excess costs that can be passed on to consumers. Thus, in general the level of benefits in a state is expected to be positively related to the degree of unionization. However the effect of unionization may differ across industries and states, depending on their penetration within and across firms in an industry and across states.

High Versus Low Wage Workers.

High and low wage workers may have different preferred replacement rates for several reasons. First, as argued

THE DETERMINATION OF WORKERS' COMPENSATION BENEFIT LEVELS

13

above, the optimal replacement rate is likely to be lower for higher paid workers. This may explain the use of minimum and maximum benefits, although the wide range in maximum benefits across states would remain a puzzle, if workers voted only for their individually preferred benefit levels. Second, the rating of WC premiums on the basis of total payroll may yield a lower expected return for high wage workers. 21 The expected compensation per dollar of payroll is less for high wage than for low wage workers both because of the maximum benefit cap and because higher paid workers tend to be in less risky jobs and so have lower frequency of injury. Even if experience rating modifications to manual rates yield perfect experience rating at the firm level, if individual compensating wage differentials do not correctly reflect the lower expected loss per dollar of payroll for higher paid workers, then high paid workers face a lower expected return on WC premiums (or a higher effective loading charge) and would prefer lower benefit, ceteris paribus. Third, it is often argued that minimum and maximum benefits reflect a socially preferred redistribution of income from high paid to low paid workers. In practice, attempts at redistribution through the benefit structure may be undone if competitive labor markets force actuarially fair compensating differentials at the level of the individual worker. Assuming that some redistribution is possible or at least is attempted, a voluntary demand for redistribution would suggest that minimum benefits would be higher where the ratio of low wage to high wage workers is low, since the effective tax price per high wage workers varies inversely with the ratio of subsidy recipients to donors. But, if redistribution is involuntary, then the minimum benefits will be positively related to the ratio of low wage to high wage workers. Minimum benefits are also expected to be higher in states where WC coverage is more extensive, if the imposition of minimum benefits tends to raise wage levels for workers who are substitutes for those workers who receive more than the preferred level of benefits. 22 Small Firms. Workers in small firms face a higher loading charge and so are expected to prefer lower benefits, if perfectly experience rated. This preference for lower benefits is reinforced if small firms as a group bear their own costs, but imperfect experience rating at the firm level induces moral hazard. However, evidence in Harrington (1987) that loss ratios are negatively related to firm size (average payroll) suggests that small firms as a group do not fully bear their own costs. To the extent the insurance costs of small firms are subsidized, workers in small firms would be expected to prefer high benefits.

14

W O R K E R S ' COMPENSATION INSURANCE

A subsidy to small firms could be effected either directly through the rating structure in the primary market or through residual market plans. In the absence of a subsidized state fund, the costs of the subsidy are borne by firms or workers in firms that are not large enough to self insure. Their opposition is expected to depend on the per capita subsidy cost to workers in medium and large firms, T = [(1 - b ) K * Ns]/N1, where K is the benefit level, b is the fraction borne by small firms, Ns is the number of workers in small firms, and NI is the number of workers in larger insured firms. NI is thus N - Ns - Ni, where Ni is the number of workers in self-insured firms. 23 The potential for subsidy to small firms is inversely related to the fraction of the market that is large enough to self-insure and is lower in states that permit group self-insurance. The potential for subsidy to small firms may be greater in states with a monopoly or competitive state fund, to the extent state funds are subsidized from other sources. In summary, benefit levels are predicted to be negatively related to the percent of workers in a state that are in small firms. The effect should be less negative in states that establish obstacles to self-insurance, such as not permitting group self-insurance. The effect may be less negative in states that have a state fund, particularly a monopoly state fund. The prediction is uncertain because taxpayers may block the use of state funds to subsidize WC benefits.

Young Versus Older Workers.

Older workers may prefer shorter duration of benefits for permanent disabilities than younger workers, if older workers are more likely to qualify for SSDI benefits, or if Social Security payments that commence at age sixty-five provide adequate replacement rates. A second reason for including a measure of the age distribution in the empirical analysis is that, in theory, the demand for permanent disability benefits is a function of permanent income. The empirical measures of the current wage distribution in a state are a downward-biased measure of the distribution of permanent income in states with a relatively young population, because age-earnings profiles are positively sloped. Conclusion. Theory implies that there will be a compensating wage differential equal to the cost of WC benefits if firms employ labor to the point where the wage (gross of benefit costs) is equal to the value of marginal product. However, an empirical finding of a dollar-for-dollar tradeoff between WC benefits and wages would not necessarily imply that the incidence of benefit costs is on workers, if the supply of labor to the

THE DETERMINATION OF WORKERS' COMPENSATION BENEFIT LEVELS 15

covered sector in each state is not perfectly inelastic. A one-for-one tradeoff would also not necessarily imply that each worker receives his preferred level of benefits. In fact, with heterogeneous preferences but a common statutory benefit structure there is no benefit level that is simultaneously optimal for all workers. Workers for whom statutory benefits exceed their preferred level will not pay for the excess if they can get their preferred mix elsewhere. The costs of excess benefits may be shifted forward to consumers if product demand is imperfectly elastic. However, for traded goods the incidence of excess costs will be on less mobile factors of production, including owners of land, owner-operators of small firms, and possibly other workers. Workers with high-preferred benefit levels will be less willing to bear any excess costs and hence are less likely to vote for high WC benefits if they can supplement low WC benefits with private insurance. If private insurance were a perfect substitute, statutory WC benefits would reflect the lowest common denominator of preferences. 24 But it would be inappropriate to derive any normative conclusion that benefits are inadequate or suboptimal.

Empirical Evidence Table 1-1 shows trends between 1960 and 1985 in the mean, minimum, and maximum replacement rates for the sample of roughly thirty-eight states for which the National Council collects data. Changes in the nominal replacement rates were modest over this period. The mean increased from 0.64 in 1960 to 0.68 in 1985, while the minimum increased from 0.55 to 0.60 and the maximum increased from 0.70 to 0.80. Table 1-1 also shows the ratio of the minimum and maximum weekly benefit relative to the average weekly wage. Because of the minimum and maximum benefit constraints, the replacement rate actually received by many workers differs from the nominal replacement rate. The mean ratio of maximum benefit to average weekly wage was less than 0.5 during the 1960s, but increased sharply after 1972, reaching 0.81 in 1985. Thus, prior to the changes of the 1970s, workers earning more than half the average wage in most states actually received less than the nominal replacement rate. The ratio of minimum benefit to average weekly wage has increased in some states but decreased in others, with little change in the mean. The multivariate analysis reported here is confined to the maximum weekly benefit for temporary total and permanent total disabilities. As shown in table 1-1, the maximum weekly benefit determines the effective

16

WORKERS' COMPENSATION INSURANCE

Table 1-1. Trends in Workers' Compensation Benefits for Temporary Total Disability, 1960 to 1985 Replacement Rate. Replacement Rate

Maximum Benefit a

Minimum Benefit a

Year

Mean

Min

Max

Mean

Min

Max

Mean

Min

Max

1960 1961 1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973 1974 1975 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985

0.639 0.642 0.642 0.642 0.642 0.643 0.643 0.645 0.645 0.645 0.646 0.648 0.644 0.656 0.674 0.674 0.674 0.674 0.674 0.674 0.672 0.676 0.676 0.678 0.679 0.679

0.550 0.550 0.500 0.550 0.550 0.550 0.550 0.550 0.550 0.550 0.550 0.550 0.550 0.550 0.600 0.600 0.600 0.600 0.600 0.600 0.600 0,600 0.600 0.666 0.600 0.600

0.700 0.750 0.750 0.750 0.750 0.750 0.750 0.750 0.750 0.750 0.750 0.750 0.700 0.800 0.800 0.800 0.800 0.800 0.800 0.800 0.800 0.800 0.800 0.800 0.800 0.800

0.475 0.447 0.447 0.435 0.449 0.433 0.454 0.450 0.444 0.449 0.462 0.451 0.457 0.480 0.510 0.533 0.621 0.638 0.673 0.673 0.675 0.729 0.771 0.763 0.803 0.807

0.316 0.307 0.308 0.296 0.329 0.322 0.312 0.298 0.273 0.269 0.304 0.300 0.301 0.289 0.281 0.303 0.292 0.267 0.355 0.363 0.343 0.363 0.351 0.316 0.346 0.357

0.622 0,573 0.662 0.628 0.828 0.757 1.119 1.017 0.932 0.871 0.828 0.785 0.701 0.853 1.006 1.257 1.500 1.598 1.620 1.678 1.556 2.004 2.051 2.028 2.403 2.321

0.169 0,159 0.156 0.150 0.157 0.152 0.151 0.148 0.140 0.138 0.140 0.136 0.135 0.145 0.145 0.143 0.154 0.154 0.153 0.146 0.139 0.139 0.139 0.138 0.149 0.150

0.067 0.060 0.059 0.055 0.059 0.058 0.058 0.055 0.051 0.048 0.047 0.044 0.041 0.041 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

0.285 0.277 0.293 0.284 0.270 0.261 0.261 0.251 0.249 0.243 0.267 0.785 0.336 0.344 0.462 0.395 0.428 0.399 0.438 0.397 0.352 0.370 0.376 0.388 0.431 0.416

aRelative to state average weekly wage. Based on 37 states. Sources: National Council of Compensation Insurance.

r e p l a c e m e n t rate for w o r k e r s at the average wage and above, and there is significant variation across states. Analysis of the nominal r e p l a c e m e n t rate and the m i n i m u m benefit yielded very few significant results, plausibly because of the lack of variation in these variables. 25 Table 1 - 2 lists e x p l a n a t o r y variables with definitions, m e a n s and standard deviations. W A G E is the log of average weekly earnings of private nonagricultural production workers wage, converted to 1985

THE DETERMINATION OF WORKERS' COMPENSATION BENEHT LEVELS 17 Table 1-2.

Variable Definitions, Means, Standard Deviations.

Variable

Mean

Deviation

Definition

REPL MAXR MINR WAGE (log) POOR RICH SMALL UNION MANUF AGRIC MINING CONSTR SERVICE EDUC > 12 COMPFD GRPSELF MIDAGE PERSELF PERSTATE

66.36 0.58 0.15 5.94 21.57 15.91 16.92 18.05 16.65 0.81 1.36 5.33 20.85 59.99 0.19 0.09 19.69 13.70 7.15

4.24 0.26 0.09 0.15 6.10 6.54 2.19 7.61 7.51 0.57 2.10 1.24 4.43 13.07 0.40 0.28 1.32 6.59 15.19

Replacement rate Maximum benefit % average weekly wage Minimum benefit % average weekly wage Average weekly wage % Low income families % High income families % Employees in firms ~ 12 years schooling Competitive state fund. Dummy Group self insurance. Dummy % Population 45-64 % WC payments by self-insured firms % WC payments from state and federal funds

dollars using the consumer price index. 26 POOR and RICH measure the percent of families with income below or above, respectively, a low and a high income threshold. 27 SMALL is the percent of workers in establishments of twenty or fewer employees. GRPSELF is a dummy variable indicating states that permit firms to group together to self-insure. PERSELF is the percent of total WC payments paid by self-insured firms. COMPFD is a dummy variable indicating states with a competitive state fund. PERSTATE is the percent of total payments paid by state funds or from federal sources. Table 1-3 reports OLS regression equations for the log of the maximum weekly benefit (MAX) (in constant 1985 dollars) for approximately thirty-seven states in 1970, 1975, 1980, and 1985. 28 MAX is a public good for all workers with wages above the threshold at which MAX becomes a binding constraint on the effective replacement rate. For lower wage workers it should be irrelevant if compensating wage differentials are individually fair and general equilibrium effects are irrelevant. The significant negative effect of the percent of low-income families (POOR) is consistent with the hypothesis that general equilibrium effects

18

W O R K E R S ' COMPENSATION INSURANCE

Table 1-3. Maximum Weekly Cash Benefit (log) (1970, 1975, 1980, 1985). Variable Intercept Wage (LOG) Injury Rate POOR a SMALL a UNION a MANUF a AGRIC a MINING a CONST a SERVICES a EDUC > 12 D75 D80 D85 R2 n = 146

Coefficient

t-Statistic

4.724 0.083 -0.000 -0.022 -0.047 0.003 0.004 0.280 0.007 0.026 0.034 0.007 0.083 0.196 0.153 0.739

3.52 0.41 -0.04 -3.15 -3.38 0.60 0.76 6.38 0.36 -0.96 3.56 1.16 1.30 2.09 1.14

a Measured as percent.

matter. It suggests that some of the incidence of the cost of WC benefits for higher wage workers falls on lower wage workers, presumably because W C wage loss benefits are not valued at cost by high wage workers. Benefits are negatively related to the percent of workers in establishments of twenty, or fewer, workers ( S M A L L ) , which is consistent with a negative price elasticity of demand in response to the higher loading charge in small firms. The significant positive coefficient of the percent of workers in agriculture and services could reflect the higher cost of private supplementation in these industries, as evidenced by the fact that a relatively large proportion of workers in these industries lack private health insurance. D u m m y variables 1975, 1980, and 1985 are positive although not highly significant. This indicates that benefits have increased over time due to factors not measured here. It is not possible to distinguish the effect of the threat of federal intervention following the National Commission from other influences that have led to rising benefits in other social insurance programs and tort awards over this period. The income elasticity ( W A G E ) is insignificantly different from zero, as

THE DETERMINATION OF WORKERS' COMPENSATION BENEFIT LEVELS

19

are the effect of the average injury rate, unionization, and other industry mix variables. Measures of the structure of insurance--the percent of payments made by self-insured firms and state funds, and dummy variables indicating group self insurance and competitive state funds-were generally not significant and are excluded from the regressions reported here.

Conclusions This analysis has important implications for interpreting the choice and "adequacy" of WC benefit levels. The theory implies that with heterogeneous preferences and a common benefit structure, the standard assumption that compensating wage differentials are individually actuarially fair may not be correct. Given the public good nature of the benefit structure and the possibility of supplementation, it may be optimal to choose the benefit level that reflects the lowest common denominator of preferences in order to avoid deadweight losses that result from imposing benefits above the preferred level. The empirical analysis of determinants of the maximum weekly benefit, which was a binding constraint for the majority of workers during this period, provides some evidence consistent with the model. However, some results are not consistent. A useful next step in understanding the determinants of WC benefit levels would be to extend the model to include other factors, such as size of the uncovered sector, benefit levels in other states, and structure of insurance regulation.

Appendix A The firm chooses W and K to maximize the worker's utility, subject to the constraint that total compensation is equal to the value of marginal product, We: Max L = (1 - p)U1 [W(1 - t)] + p U o ( K ) + m [ W e W,K,,,, -- p(1 + h)K]

--

(1

--

p)W

(A1)

First order conditions for a maximum are: Ul'(1-t)-m W e --

=0

Uo'/(1 + h) - m = 0 (1 -- p ) W - p(1 + h ) K = 0

(A2) (A3)

20

WORKERS' COMPENSATION INSURANCE

Changes in p The effect of a change in p can be determined using Cramer's rule:

OK

UI"(10 - 02

00

-(1-1+ h)

-(1 - p)

- W + (1 + h ) K

0

Op

D (1 + h)K][-UI"(1 - t)z(1 + h)]

= [W-

Since D and the expression in the second brackets are both positive if OK

utility is indeed maximized, -~p ~ as W ~ (1 + h) K. Similarly, OW

OW

= [(1 + h ) K - W ] U o " / D ~ 0 as (1 + h ) K ,~ W.

Op OK

Since-0--fr ° - and -~p clearly cannot both be positive, it follows that both are non-positive. A change in W e alters K and W as follows:

ow

U~"(Io - t)2

O0

-(1-1+ h)

-(1 -p)

-1

o

OW e

D

= -[U1"(1 - 02(1 + h)l/D > 0

OW

0 0

0 Uo"

-1 - ( 1 + h)

--1

- p ( l + h)

0

OW ~

D = -Uo"/D > 0

A change in 14x and in p have similar effects on the ratio, K / W : OK/OW ~

OK/Op

ow/ow e

oW/Op

_

UI"(1 - t ) 2 / U 1 Uo"/Uo'

'

after substituting from first order condition. A change in the loading charge, h, has the following effects: OW Oh = rap(1 + h) - pKUo" > O,

THE D E T E R M I N A T I O N OF WORKERS' COMPENSATION BENEFIT LEVELS

OK Oh

-

-m(1

21

- p ) + p g ( 1 + h)(1 - t)2fm " < 0

T h u s an increase in h lowers the optimal r e p l a c e m e n t rate.

Notes 1. Viscusi and Moore (1987), 25. 2. In the equation W = a + b ( B / W ) , the coefficient b is ( W O B / O W - B ) / W 2. The variation in B is small relative t o t h e variation in W, because of maximum benefits. Within any one state OB/OW = 0 for all workers for whom W > MAX/REPL, where M A X is the maximum benefit and REPL is the replacement rate. This may include over half the workers in a state (see table 1-1). Much of the variation in B / W may therefore reflect variation in W. 3. The Commission notes that the majority of states had a maximum benefit of two-thirds of the average weekly wage in 1940 but had subsequently not kept pace with rising wage levels. Thus, it is possible that the lower levels observed in 1970 reflected an unintended failure to keep pace with unanticipated inflation. However, with individual benefits set at the lesser of the maximum benefit or two-thirds of the individual's wage, all workers earning more than the state average wage would have replacement rates of less than two-thirds. 4. National Commission, 19. 5. Krueger and Burton (1983, 26) estimate that in thirty-one states in 1972, temporary total, permanent total and fatal benefits would have been about 40 percent higher if the eight essential benefit recommendation Of the Commission had been in effect. By 1978, replacing actual statutory benefits with the recommended benefits would have increased costs only 6.7 percent on average. However, Burton and Krueger (1986, 112 and 199) report that interstate differences had widened between 1972 and 1978. They conclude that this evidence provides support for the Commission's case for federal minimum standards for workers' compensation. 6. The possibility of simultaneous equations bias in estimates of compensating differentials is noted by Leigh (1985). He does not develop a formal theory pf, the determinants of W C benefits, but suggests that legislators may be influenced by accident rates, political ideology, and unionization. He finds that introducing state dummy variables as a rough control for endogeneity of benefits reduces estimates of compensating differentials. 7. This condition for optimal compensation when wages are taxed is derived in Viscusi and Moore (1987). 8. Eq (5) is obtained by applying the implicit function theorem to eq (4). 9. See Appendix. 10. Where the benefit structure contains features designed to control moral hazard, such as a long waiting period and retroactive period, or a rehabilitation program, optimal replacement rates would be higher than in the absence of such features. 11. See, for example, Thaler and Rosen (1976) for the choice of job risk; Goldstein and Pauly (1976) and Danzon (1989) for analysis of the choice of group health insurance. 12. This is analogous to the Tiebout model of the choice of local and public goods and services. 13. If workers value benefits at cost, the demand and supply schedules shift down by

22

W O R K E R S ' COMPENSATION INSURANCE

the same amount. Cash wages fall by the cost of benefits but the level of employment is unaffected. 14. Typically, the manual premium per employer includes a flat expense constant to cover costs of issuing a policy. Until recently, there was also a flat loss constant charge, intended to "compensate for the generally inferior record of small businesses" Burton and Krueger (1986, 133). This generates a higher percent load for small firms. 15. There are two potential sources of moral hazard that may be affected by the level of benefits; worker moral hazard in filing claims and employer moral hazard in preventing injuries. Assuming that workers are fully informed about job risk and transactions costs are zero, compensating wage differentials for job risk should provide optimal employer incentives for injury reduction, while benefit levels are optimally designed to control claimant moral hazard. 16. Since SSDI replacement rates were substantially increased in the 1970s, the substitution-of-SSDI hypothesis would predict a smaller increase in WC benefits for permanent total disability than in benefits for temporary or permanent partial disability. 17. See, for example, Borcherding and Deacon (1972); Bergstrom and Goodman (1973). 18. Courant (1977) shows that the Mieskowski analysis may be only approximately correct. 19. The extension of mandated benefits to previously uncovered sectors could actually increase wages of workers in previously covered sectors if some workers who value the benefits below cost withdraw from the labor market or move out of state. See Danzon (1989). 20. See Spence (1977) for an analysis of the effect of consumer misperceptions on the market equilibrium choice of product safety and injury compensation. 21. For many years most states applied a payroll limitation, that is, a cap on the amount of an employee's weekly earnings used to compute the insurance premium. Most states dropped this limit in 1974 to 1975; by 1983 only Texas had a payroll limit ($300). Burton and Krueger (1986, 116). 22. This is analogous to the hypothesis that higher wage workers may benefit from the imposition of the minimum wage. 23. Strictly, the incentive of firms to self-insure is directly related to the magnitude of the tax, given that other necessary conditions are satisfied, so N~ is endogenous. 24. In a market subject to adverse selection, it may be Pareto improving to mandate a low level of benefits and permit private supplementation by those who prefer higher benefits (Pauly, 1976; Dahlby, 1983). Low WC benefits could be interpreted in this light, since workers desiring higher benefits can supplement by purchasing private disability insurance. However, since the statutory minimum WC benefit applies only to work-related injuries, it does not suffice to eliminate all informational asymmetries in the private disability insurance market. It therefore cannot provide a first best solution. 25. Roughly three-fourths of all claims are for temporary total disabilities, but they account for only one-fifth of cash benefits (Price, 1984). The formula for permanent total disability payments is often similar, but the expected value depends on the limits on the duration of benefits or dollar ceilings. The formula for permanent partial benefits is more complex. 26. The median wage is not available. The log of median family income was tried but had slightly lower explanatory power than WAGE. 27. The Census used different breakpoints in the income distribution for the 1960, 1970, and 1980 censuses. Values were chosen to yield roughly the same percentage of the income distribution in each year.

THE DETERMINATION OF WORKERS' COMPENSATION BENEFIT LEVELS 23 28. Because benefit levels remain stable for several years at a time, including the intervening years, does not necessarily add information but did increase the serial correlation significantly.

References Bergstrom, Theodore and Robert Goodman. (1973). "Private Demands for Public Goods." American Economic Review 63:280-296. Borcherding, Thomas and Robert Deacon. (1972). "The Demand for the Services of Non-Federal Governments." American Economic Review 62:891-901. Burton, John F., Jr. and Alan B. Krueger. (1986). "Interstate Variations in the Employers' Costs of Workers' Compensation, with Particular Reference to Connecticut, New Jersey, and New York." In James Chelius (ed.). Current Issues in Workers Compensation. W.E. Upjohn Institute for Employment Research. Butler, Richard J. and John D. Worrall. (1983). "Workers' Compensation: Benefit and Injury Claims Rates in the Seventies." The Review of Economics and Statistics 65(4) :580- 599. Courant, Paul N. (1977). "A General Equilibrium Model of Heterogeneous Local Property Taxes." Journal of Public Economics 8:313-327. Dahlby, B.G. (1981). "Adverse Selection and Pareto Improvements Through Compulsory Insurance." Public Choice 37:547-558. Danzon, Patricia M. (1989). "Mandated Employment-Based Health Insurance: Incidence and Efficiency Effects." LDI Discussion Paper No. 66. Leonard Davis Institute of Health Economics. University of Pennsylvania. Dorsey, Stuart and Norman Walzer. (1983). "Workers' Compensation, Job Hazards, and Wages." Industrial and Labor Relations Review 36:642-654. Feldstein, Martin S. (1972). "The Incidence of the Social Security Payroll Tax: Comment." American Economic Review 62:735-738. Goldstein, G.S. and M.V. Pauly. (1976). "Group Health Insurance as a Local Public Good." In R. Rosett (ed.) The Role of Health Insurance in the Health Services Sector. New York: NBER. Harrington, Scott E. (1987). "Cross-Subsidation and the Economics of Regulation: Theory and Evidence from Automobile Insurance." Working Paper No. 87-12. Center for Research on Risk Insurance. University of Pennsylvania. Krueger, Alan B. and John F. Burton. (1983). "Interstate Differences in the Employers' Costs of Workers Compensation: Magnitudes, Causes, and Cures." Leigh, Paul J. (1985). "Analysis of Workers' Compensation Using Data on Individuals." Industrial Relations 24:247-256. Mieskowski, P. (1972). "The Property Tax: An Excise Tax or a Profits' Tax?", Journal of Public Economics: 73-96.

24

WORKERS' COMPENSATION INSURANCE

Price, Daniel N. (1984). "Workers' Compensation: Coverage, Benefits, and Costs, 1982." Social Security Bulletin 47:7-46. Spence, Michael. (1977). "Consumer Misperceptions, Product Failure, and Producer Liability." Review of Economic Studies 44:561-572. Thaler, Richard and Sherwin Rosen. (1976). "The Value of Saving a Life: Evidence from the Labor Market." In N. Terleckyz (ed.) Household Production and Consumption. NEBR Studies in Income and Wealth No. 40. New York: Columbia University Press. The National Commission on State Workmen's Compensation Laws. (1972) Report of the National Commission on State Workmen's Compensation Laws. Washington: U.S. Government Printing Office. Viscusi, W. Kip and Michael J. Moore. (1987). "Workers' Compensation: Wage Effects Benefit Inadequacies, and the Value of Health Losses." The Review of Economics and Statistics LXIX:249-261.

WORKERS' COMPENSATION COSTS AND HETEROGENEOUS CLAIMS Richard J. Butler* John D. Worrall

High Frequency Versus Long Duration The distribution of Workers Compensation claims are known to have thick tails so that even though claims are of relatively short duration a substantial amount of the indemnity losses are accounted for by claims that are very long in their duration. The former types of claims can be reasonably characterized as "high frequency/short duration" claims, whereas the later are "low frequency/long duration" claims. Duration and frequency are, of course, highly correlated with the type of claim. Temporary Total claims arising from lacerations, strains, and sprains are likely to be relatively high frequency events of short duration so that the claim indemnity costs are small per individual claim. In contrast,

*Prepared for the National Council on Compensation Insurance's Conference on Worker' Compensation Issues, Nov. 13, 1987. We wish to thank the National Council on Compensation Insurance for providing the data used in this study, and the College of Family, Home and Social Sciences at Brigham Young University for computer support.

25

26

W O R K E R S ' COMPENSATION INSURANCE

a permanent partial spinal cord injury is a relatively low frequency, but long duration, type of claim whose costs per claim are inevitably very large. It's important to determine the relative importance of each type of claim for a number of reasons. High frequency/short duration events are likely to be claim types that are more susceptible to risk reduction efforts where the number of claims are reduced through safety programs. They may also be better candidates for risk retention than those claims that are infrequent but very expensive to a firm. On the other hand, changes in federal programs that cover the long-term disabled, such as Disability Insurance under the Social Security system, the introduction of a Negative Income Tax, or any other income transfer program, would certainly affect those in the "low frequency/long duration" claim group but not those with "high frequency/short duration" claims. Differences in claim type is not the only form of heterogeneity that determines whether a claim will likely be a high frequency or a long duration claim. Within any given claim type, the duration of claims will also vary so that whether most of the indemnity losses are in the form of the frequent short term claims or the relatively infrequent longer term claims is an empirical question. This study looks at the relative importance of losses arising from high versus low frequency claims for only one class of claims: the low-back temporary total. We further restrict the type of heterogeneity we allow by examining a random sample of male workers whose claims arose in 1985.1 We consider how two forms of heterogeneity affect the distribution of losses. The first type we characterize here as intrinsic heterogeneity. These differences arise because each claim type involves workers with different (unobservable) physical endowments, different attitudes toward work and claimant status, differences in administrative stringency, etc. We allow for such differences by introducing both parametric and nonparametric mixing of our models of claim duration. These models and their results are presented in the next section. The second type of heterogeneity, that associated with observable differences in the attributes of the claim, is given in section three. This is perhaps the more interesting form of heterogeneity since it allows us to predict how, for example, changes in the age of the claimant, size of indemnity benefit, or the wage, affect the breakdown of low versus high frequency claims. Some concluding observations are drawn in the last section.

WORKERS' COMPENSATION COSTS AND HETEROGENEOUS

CLAIMS

27

Gamma Models of Duration with Intrinsic Heterogeneity By estimating the distribution of time spent on a claim we can calculate both the number of claims that are short as well as the proportion of all losses that result from very long duration claims. We will lety y be the random variable given the time spent on a Workers Compensation claim and write the probability density of these nonwork spells as f(ylO, a) and the distribution function as F(YIO, a). o is the location parameter of the distribution and the vector a consists of the shape and scale parameters. We call f(ylO, a) the "structural" distribution since it usually embodies all the testable hypotheses of economic theories. 2 We distinguish between the location parameter and other parameters of the distribution because we assume that any intrinsic heterogeneity simply shifts the location parameter of the structural distribution while leaving the other parameters unaffected. In other words, we assume that all claimants have the same structural distribution for time spent on a claim except for possible differences in their location parameters. Differences in the location parameters capture the intrinsic (unobservable) heterogeneity. The Structural Duration Distribution Without Heterogeneity is simply a generalized gamma distribution where the location parameter 0 is assumed to be a degenerate distribution with all its probability mass at point b. In other words, 0 is a constant, equal b, and the contribution to the likelihood function of a completed spell of joblessness due to workplace injury is given by

f(ylO, a) = GG(yl0 = b, a, p)

=

ayap-lexp(-(y/b)a)/bapF(p)

(1)

where F(p) is the gamma function with parameter p. Special cases of this distribution include the following distributions Gamma = GG(yl0 = b, a, = 1, p) = yp-lexp(-(y/b))/bPF(p) Weibull = GG(yl0 = b, a, p = 1) = ay"-lexp(-(y/b)a)/b a (2) Exponential = GG(yl0 = b, a = 1, p = 1) = exp(-(y/b))/b To include Parametric Heterogeneity controls into the model we relax the assumption that the distribution of 0 (the location parameter of structural distribution, f(ylo, a) is degenerate, and assume rather that 0 is randomly distributed across spells of Workers Compensation (i.e., we treat 0 as a random effect across individual workers that represents intrinsic heterogeneity). In particular we assume that 0 is a member of the inverse generalized gamma distribution. Denoting the distribution of heterogeneity as g(01q~), we have

28

W O R K E R S ' COMPENSATION INSURANCE

g(014~) = IGG(01b, a, q) = aO-'~q-lexp(-(O/b)-a)/b-aqF(q)

(3)

Exactly as with equation (2) above, by setting a or q to 1 we get the inverse gamma (Weibull and exponential distributions as special cases). Given equations (1) and (3), a general model of jobless spells (in the presence of parametric heterogeneity) is

h(ylc~, a) = f f(ylO, a)g(Oldp)dO

(4)

which for the class of probability density functions we consider here becomes GB2(ylb, a, p, q) =

f GG(ylO,a, p)IGG(Olb, a, q)ao

= ayaP-1/bapB(p, q)(1 + (y/b)a) p+q

(5)

where B(p, q) denotes the beta function with parameters p and q. The GB2 is the generalized beta distribution (of the second kind), and since it can be shown to include the generalized gamma (GG) distribution as a limiting case, it is flexible enough to include all of the distributions considered in this chapter as special or limiting cases (see McDonald (1984) for details). 3 By setting the parameters a, p, or q to 1, several different specifications of the parametric forms for the structural and heterogeneity distributions are possible as indicated in table 2-1. The relationships indicated in the left hand column of table 2-1 (as well as in equation (2)) imply, for example, that if there is no unobservable heterogeneity present, then a test for the appropriateness of the Weibull distribution as a description of jobless spells would be possible by comparing the fitted log-likelihood of it against that of a generalized gamma distribution. On the other hand, if the Weibull were known to be the appropriate structural distribution, then a test for heterogeneity would consist of comparing fitted log-likelihood values of Weibull and BR12 (Burr type 12, also known as the Singh-Maddala distribution in economics) distributions as indicated moving horizontally in table 2-1 (the third line up from the bottom). Other nested tests for the appropriate structural or heterogeneity distributions are easily read off of table 2-1. To include Nonparametric Heterogeneity controls for intrinsic differences in claims, we let b be an unspecified distribution with a finite number of support points; in other words, the location parameter simply takes on an unspecified number of values. The number of distinct values of the location parameter, which we write as 6i, as well as the probability mass of each of those values, is determined empirically by increasing the

WORKERS' COMPENSATION COSTS AND HETEROGENEOUS CLAIMS Table 2-1.

Summaryof Parametric Models.

Structural Distribution

Heterogeneity Distribution 1

Generalized Gamma

29

Estimated Distribution 2

tl~o=B

Generalized Gamma ayap-lexp(- (y /b )a)/baPF(p )

Inverse Weibull

Burr 3 (Gen. Loglogistic) apya-1/bap(1 + (y/b)a) p+I

Inverse Generalized Gamma

Generalized Beta type 2 ayap-x/baPB(p, q)(1 + (y/b)") p+q

qlo=p

Gamma yP- %xp( - (y/b))/bPF(p)

Inverse Gamma

Beta type 2 (Gen. Pareto) yP-1/bPB(p, q)(1 + (y/b)) p+q

0o=~

Weibull ay a-lexp(-(y/b)a)/b a

Inverse Weibull

Fisk aya-1/b"(1 + (y/b)a) 2

Inverse Generalized Gamma

Burr-12 (Singh-Maddala) aqy"-l/ba(1 + (y/b)") q+l

Do=#

Exponential exp((y/b))/b

Inverse Gamma

Lomax (Pareto) q/(1 + (y/b))q+lb

a=l

Gamma p=l p = 1 Weibull

a=l

Exponential

1Note that q~o=adenotes the degenerate distribution with probability mass at 0 = fl, 2The probability density functions of the respective distributions are given in the far right hand column.

number of support points until additional points are no longer statistically significant. For example, for a completed or uncensored spell of joblessness the likelihood function with three support points in the nonparametric heterogeneity distribution, is in general given by L = f ( y l g l , a)P1 +

f(yl~2,

a)P2 +

f(yl~3, a)P3

(6)

30

W O R K E R S ' COMPENSATION INSURANCE

Hence, two support points amount to a random intercept in which the intercept takes a value of 61 with probability P~ and a value of 62 with probability P2. The explicit log-likelihood function for an exponential distribution of jobless spells and 3 support points is, for uncensored observations, given as In L = '~ log((exp(--(y/61))/61)Pl + (exp(-(y/62))/62)P2 i

+ (exp(-(y/63))/63)P3)

(7)

In the sense that the number, value, and probabilities of the support points are not specified in advance of the empirical estimation, this type of intrinsic heterogeneity control is "nonparmetric." And since these parameters can be estimated along with the other parameters in equations (6) or (7) by maximizing the log likelihood function, their large sample properties are known. The tests for appropriate structure were made by comparing the loglikelihood values of the various distributions using a sample of claims from the NCCI's Detailed Claim information file. This data base is a random sample of all indemnity claims made against private insurance carriers for work injury or occupational disease under the Workers' Compensation Insurance Program. Each individual observation includes: a continuous age variable, sex,

preinjury wage, injury date, report date, employment status at claim filing (primarily employed, unemployed, on strike), type of claim, indemnity benefits, the body part injured (or the occupational disease), and the nature of the injury (spain, strain, fracture, rupture, etc.). In order to minimize the impact of unobservable heterogeneity due to differences in the type of claim, we drew a random sample of low-back claims for 1,783 male workers in 1985. The impact of adding intrinsic heterogeneity, both in its nonparametric and parametric forms, is apparent in table 2-2. T h e generalized gamma fits significantly better than any of the other three "structural" distributions as can be seen by comparing the log-likelihood values on the first

WORKERS'

COMPENSATION

Table 2 - 2 .

COSTS AND

HETEROGENEOUS

CLAIMS

31

Intrinsic Heterogeneity in Claims: Mixture Models.

Generalized Gamma Structure

a

No Mixture (GG)

0.471

b-1 0.417

b-2 --

p

--

6.589

Inverse Weibull

1.242

4.455

. --

Mix (Burr3) Inverse GG Mix

-2.151

(1.000) 7.568

. --

.

(GB2) 2 Support Point

-0.826

(1.000) 2.616

. 15.668

.

(GG-2) 3 Support Point (GG-3)

-0.792 --

(0.684) 2.128 (0.665)

(0.316) 10.231 (0.262)

--

16.167

--

--

1.560

--

(1.000)

.

b-3 .

. 4.624

.

.

. 1.631

.

.

. 4.159

.

----

. . . 8.99E6 4.527 (0.073) --

q

loglikelihood (minus)

--

5170.825

--

5096.469

0.441

5093.223

--

5089.594

. ---

5089.465 --

--

5225.899

Gama Structure No Mixture (GA) Inverse Gamma Mix (B2) 2 Support Points (GA-2) 3 Support Points (GA-3)

(1.000) 1.937 (1.000) 4.683

--

--

7.119

27.413

--

3.118

--

5090.154

(0.669) 4.578 (0.652)

(0.331) 22.529 (0.282)

729.463 (0.066)

3.152

--

5089.964

29.974 (1.000) 19.733 (1.000) 8.961 (1.000) 15.135

.

.

.

.

5257.198

.

.

.

.

5140.037

--

--

--

91.195

--

--

--

5104.460

2.120

(0.657) 14.954 (0.599)

(0.343) 54.217 (0.270)

8.5Ell (0.131)

--

--

5097.169

--

30.723

.

----

0.879

5141.910

Weibull Structure No Mixture (WEIB) Inverse Weibull (FISK) Inverse GG (Burrl2) 2 Support Points (WEIB-2) 3 Support Points (WEIB-3)

1.224 1.828 2.846 1.926

0.310

5094.309

Exponential Structure No Mixture (EXP) Inverse Gamma (LOMAX) 2 Support Points (EXP-2) 3 Support Points (EXP-3)

----

(1.000) 532.257 (1.000) 26.076 (0.928) 26.333 (0.927)

-2.39E6 (0.072) 12.48E7 (0.059)

.

. ---8.19E8 (0.014)

.

5292.718 --

17.797

5292.099

----

----

5283.428 5283.405

32

WORKERS' COMPENSATION INSURANCE

row of each grouping. In every case, except for the Lomax distribution (which is the parametric "mixed" distribution for the exponential), all forms of heterogeneity control are sufficient improvements over the models with no heterogeneity controls. Twice the difference in the loglikelihood functions for the GB2 and G G distributions is 155 (take twice the difference between 5,170.825 and 5,093.223), a statistically significant value for a chi-square statistic with one degree of freedom. The test using nonparametric forms of intrinsic heterogeneity is similar: a chisquare value of 162 with two degrees of freedom. In all of the G G models the a and b parameters increased, and the p parameter decreased once intrinsic heterogeneity across individuals was allowed in the specification. Similar systematic changes in the structural coefficients of the gamma and Weibull distributions are also present in table 2 - 2 , although the form of heterogeneity control significantly affects the magnitude of the parameter shifts. One of the reasons that fit of the mixed distributions is better than the fit of the (GG) structural distribution is that the tails of the G G distributions (no heterogeneity) aren't thick enough. But while the thicktailed parametric distributions may fit the data better, the very thickness of the tails frequently implies that many of the higher order moments of the distribution are undefined. For the GB2 distribution, for example, the h th moment of distribution exists only if it is less than the product of the parameters q and a, that is, only if q* a > h. A check of the estimated parameters in table 2 - 2 show that not even the first moment is defined for GB2, B2 and Burrl2 distributions. Since computing the losses associated with various claim durations depends crucially on the first moment, this makes accounting for cost effects of different lengths of claims difficult. Two remedies for undefined moments are possible: either use the nonparametric distributions for the cost accounting, or find a parametric distribution for which the moments are defined and suitably close in fit to the more general GB2 distribution. We pursue this latter course in the next section by allowing key parameters of the distribution to vary between individuals on the basis of the attributes of the claims. We close this section with a brief discussion of how to use our nonparameteric distributions for simple cost accounting purposes. If we put aside the question of the number of parameters for a moment, it seems that the nonparametric distributions seem to fit the data about as well as the parametric heterogeneity controls: the GG-2 fits as well as the GB2 distribution, the GA-2 fits as well as the Beta distribution, and WEIB-3 fits better than the Fisk and almost as well as the Burrl2. Since

WORKERS' COMPENSATION COSTS AND HETEROGENEOUS CLAIMS

33

these nonparametric mixtures are just linear combinations of Generalized Gamma (for which all moments are defined) distributions, they allow one to analyze higher order momenets even when the parametric mixtures don't have the appropriate moments defined. To understand how duration affects Workers' Compensation losses we use the "incomplete moments" of the duration distribution. The incomplete moments for the distributions given in table 2 - 1 are convenient to work with since they all have a closure property which facilitates their calculation. 4 The H t h incomplete moment for the density function is defined as (':yhf(y)dy h th incomplete moment = J0 E - - ~

(8)

So in terms of the moments of the duration distribution, setting h = 0 yields the number of claims with duration less than x weeks in duration (the h = 0 incomplete moment is just the distribution function), whereas h = 1 yields the proportion of total losses accounted for by those with claim duration of x weeks or less. These moments provide important information about insurance risks: if 50 percent of the claimants (the value of the 0 th moment is 0.5) account for 40 percent of the total losses (the value of the 1st moment is 0.4) is a skewed distribution, then reducing the number of high frequency-shorter duration claims could significantly affect the insurance losses. On the other hand, if the first 50 percent of claimants only account for 10 percent of total losses, then a efficacious strategy would focus on the risk associated with the relatively few long-term claims. To see how estimates of these moments are effected by intrinsic heterogeneity we turn to figure 2 - 1 (as well as the numeric results in table 2-3). The density functions of the G G distribution given in the first row of table 2 - 2 , as well as the heterogeneity induced distributions from the third (GB2) and fourth (GG-2) rows are pictured in figure 2-1. While all three distributions peak at around twelve weeks, the implied frequency of claims is higher at the peak when controlling for heterogeneity and, though the heterogeneity distributions intially fall off more quickly from the mode than the G G distribution, they approach the asymptote much more slowly. In particular, the form of G G distribution forces the tails to be too thin for the data so that it provides a relatively poor fit to the data. The GG-2 has a better likelihood value than the GB2, perhaps because it allows for even thicker tails than the GB2 in this application. But while the GG-2 has thicker tails than the GB2, the moments are defined for the GG-2 but not for the GB2. Values of those moments for

34

WORKERS' COMPENSATION INSURANCE PDF-GG 0.040 0.038 o .036 0.034 0.032 . 0.0300.028 " 0.026 0.024 ' 0.022 • 0.020



0.018 -

\

\

0.016 " 0.014 " 0.012 " 0.010 0.008 0.006-

0.004 0.002 0.000 0

20

40

60

60

1~

120

140

160

160

200

Y

Figure 2-1. Comparing the GG (long-dash); GG-2 (short-dash); and GB2 (solid line) density functions. different weeks of duration are given in table 2 - 3 . From the first three rows it is clear that the distribution function for the GB2 and GG-2 are very similar to each other, and quite different than the GG distribution. In particular, between the twenty-fifth and fiftieth week of duration there is a reversal in the implied pattern of claims between the heterogeneityinduced distributions and the GG distribution. This difference is also clearly reflected in figure 1. But while h = 0 moments of the GG-2 and GB2 are similar, the h = 1 moments cannot be compared for our heterogeneity distributions because they are not defined in the GB2 case. Hence, all statements we make about the relative cost importance of different durations of claims must be made on the basis of the GG-2 results when compared to the GG results. Differences in the shape of the distributions are even more remarkable: the GG indicates that only 20

WORKERS' COMPENSATION COSTS AND HETEROGENEOUS CLAIMS

Table 2-3.

35

Comparing the Incomplete Moments for GG, GB2 and GG-2.

Fraction of Claimants With Claims Duration Less Than (in weeks): Distribution GG GB2 GG-2

5 0.066 0.052 0.054

10 0.210 0.236 0.225

15 0.359 0.403 0.400

20 0.488 0.519 0.528

25 0.594 0.600 0.610

50 0.871 0.785 0.756

100 0.983 0.888 0.889

200 0.999 0.941 0.986

Fraction o f Total Losses Accounted for by Those with claim Duration Less Than: GG GB2* GG-2

5 0.008 NA 0.012

10 0.048 NA 0.098

15 0.117 NA 0.240

20 0.200 NA 0.382

25 0.288 NA 0.495

50 0.648 NA 0.701

100 0.923 NA 0.817

200 0.995 NA 0.963

** Relative Claim Costs Accounts f o r Claims Falling in the Respective Claim

Intervals GG GB2 GG-2

0-5 0.123 NA 0.237

5-10 0.280 NA 0.502

10-15 0.461 NA 0.809

15-20 0.645 NA 0.108

20-25 0.829 NA 0.367

25-50 0.298 NA 0.417

50-100 0.457 NA 0.877

100-200 0.603 NA 0.494

* Incomplete first moment is undefined since a*q < 1. ** Computed as (First Moment (t) - First Moment (t - 1))/ (Distribution (t) - Distribution (t - 1)).

percent of all costs are due from claims less than 20 weeks, while for the GG-2 distribution that number is nearly doubled. Short term claims appear relatively much more important here when we control for heterogeneity (the GG-2 results) than when we make no such controls (the GB2 results). On the other hand, the GG-2 indicates that claims longer than two years are also relatively more important is suggested by the GG distribution: the later implies that less than 10 percent of all claim losses arise from such claims (i.e., 1-0.924), while the former implies that nearly 20 percent of all costs are due to these long duration claims (i.e., 1-0.818). The most notable difference is in the one to two year duration category: with no heterogeneity correction the implied share of total costs for these claims is nearly 30 percent, a number that falls to only 10 percent once one allows for heterogeneous claims.

Controlling for Observed Heterogeneity Differences

Heterogeneity Controls for Observable Characteristics by allowing the parameters of the distributions in table 2 - 1 to vary as a linear function of those observable characteristics so that

We introduce

36

WORKERS' COMPENSATION INSURANCE

ai = exp(X/r) b i = exp(Xifl)

Pi = exp(Xip)

q; = exp(Xi~)

(9)

where Xi is the vector of the workers' age, wage, compensation benefits, and employment status, and r, fl, p, ~ are (vectors of) parametric weights assumed to be constant from worker to worker. From the workers' perspective, the relative cost of staying on a claim increases with the expected wage (the probability of having a job when the claim is over times the wage offer of that job) and decreases with increases in indemnity benefits. This suggests that the duration of a claim should decrease as the wage offer or the probability of a wage offer increases and increase with increases in the reservation wage. Proxies for these variables are readily available. One can approximate the probability of a wage offer with a measure of the expected probability that an employee will be able to return to his previous employer. In our empirical work reported below, this probability is captured by an employment status variable (see also Butler and Worrall, 1985) that equals one if the claimant was a regular employee at the time of the claim and zero otherwise. When multiplied by the wage, this constitutes the expected wage offer--though we include them as separate variables to allow for the possibility that they may differentially affect the decision to end the claim. Hence, if one regressed the logarithm of duration on these variables, one would expect a negative relationship between the probability of receiving a wage offer or the preinjury wage and the length of duration in the Workers' Compensation. Benefits should increase the expected duration as it lowers the opportunity cost of a nonwork spell. We control for vintage and life cycle effects with the claimants age at the time of injury. We hypothesize that as one gets older two effects tend to increase claim duration: the older one is the more slowly one recuperates from a job injury, and the older one is (for a given wage) the lower is the present value of lost market skills and, hence, the lower is the incentive to return to work quickly. The results from estimating this model is given in table 2-4. We also tried to similarly parameterize the GB2 and GG-2 mixture distributions, but internal software constraints did not allow us to estimate these more complicated functions. But as can be seen from table 2 - 2 , the Burr3 and Burrl2 distributions fit the data about as well as the GB2 (the Burrl2 is not significantly different from the GB2 at the 5 percent, and the Burr3 is not significantly different at the 1 percent). That they do not differ

WORKERS' COMPENSATION COSTS AND HETEROGENEOUS CLAIMS Table 2-4.

Parameter a

b

p

q

loglikelihood

37

Observable Heterogeneity in the Burr3 and Burr12 Distributions.

Heterogeneity Due to:

Burr3 (std. error)

Burrl 2 (std. error)

aO (constant) al (ln wages) a2 (ln benefits) a3 (employ) a4 (ln age) b0 (constant) bl (In wages) b2 (ln benefits) b3 (employ) b4 (ln age) p0 (constant) pl (ln wages) p2 (ln benefits) p3 (employ) p4 (ln age) q0 (constant) ql (ln wages) q2 (ln benefits) q3 (employ) q4 (ln age)

0.295 (3.48) -0.279 (0.54) 0.205 (0.75) 0.433 (2.08) -0.002 (0.54) 4.077 (24.93) -0.283 (0.92) -0.267 (2.40) -0.438 (14.96) 0.195 (3.31) -0.904 (19.59) -0.094 (1.16) 0.516 (2.01) 0.307 (8.90) 0.040 (3.16) ----------5079.72

0.174 (6.27) -0.375 (1.12) 0.443 (0.94) 0.412 (3.87) 0.084 (1.37) 3.190 (10.46) -0.063 (0.66) -0.177 (0.96) -0.279 (8.57) 0.161 (1.13) ----------0.454 (13.46) 0.261 (1.55) -0.529 (1.55) 0.265 (9.59) -0.174 (2.19) 5078.52

markedly in shape from each other is clear also from figure 2 - 2 that plots all three fitted functions for our sample. But because the Burr3 a n d Burrl2 distributions have closed forms-for both the density and distribution functions, they are relatively easy to e s t i m a t e - - e v e n with the complex parameterization given in equation (9) and presented in table 2 - 4 . Since these distributions seem to fit the data as well as the more complex (but presently inestimable) GB2 and GG-2 functions better than those with fewer parameters (the Beta2, Lomax, and structural distributions), we focus our discussion on results using these two distributions. Notice that the parameterization of the a, b, p, q parameters provides a significantly better fit in the statistical sense. Previous studies of duration have restricted observed heterogeneity to shifting only the location parameter b, hence, we have no bench mark with which to compare our results. As was the case in previous analyses of the duration of Workers' Compensation claims (Worrall et al., 1987),

38

WORKERS' COMPENSATION INSURANCE PDF-GB2

0.040 0.038 0.036 0.034 0.032 0.030 0.028 0.026 0.024 0.022

-

0.020

-

0.018

-

0.016 .~ 0.014, 0.012 0.010. 0.008 -" 0.006 0.004 0.002 0.~O 0

20

40

60

80

100

120

140

160

180

200

Y

Figure 2-2. Comparing the BR3 (long-dash); BR12 (shorl-dash); and GB2 (solid line) density functions.

the A G E variable is seen to increase b, while WAGE and EMPLOY decrease it as expected. BENEFITs have an unexpected negative impact on the location of the distribution since a negative coefficient implies the mean duration falls as benefits increase. However, since the a, p, q parameters are now also functions of these claim characteristics, we have to judge the effect of BENEFITS or any other variable on the basis of how it effects the entire distribution. To do this we have simulated the effect of increasing each of the variables by 50 percent from their mean value 5 on the shape of the distribution. The densities of these stimulated changes are given in figures 2 - 3 and 2 - 4 , with the corresponding moments recorded in table 2 - 5 .

WORKERS' COMPENSATION COSTS AND HETEROGENEOUS CLAIMS

39

BR3-MEAN

0.039 0.038 0.036 0.(Y34 0.032 0.030

i

0.026 . 0.026 0.024 •

,

0,022 0.020 0.018 0.016 0.014 0.012 0.010 0.008 0.006 0.004

0.002 0.000

0

20

40

60

80

100

120

140

160

180

200

Y

Figure 2-3a. Burr3 results: distribution of means (solid line); and with 50% higher wages (dashed line).

In figures 2 - 3 a and 2 - 4 a we observe that higher W A G E S appear to shift the distribution to the left, while very slightly increasing the thickness of the right tail. The shift to the left is consistent with the hypothesis that higher wages represent greater opportunity costs of being on a Workers' Compensation claim. Workers respond when wages are higher by decreasing the length of a claim. The quantitative impact of these responses are detailed in table 2 - 5 . The first line of each group in table 2 - 5 is the baseline measure, with subsequent lines representing the effects of a 50 percent increase in the relevant variable. For example, the Burr3 distribution indicates that a 50 percent increase in wages above the average increases the proportion of

40

WORKERS'

COMPENSATION

INSURANCE

BR3-MEAN 0.044 It 0.042 0.040

I I I I

0.038 0.036 0.034 0.032 0.030 0.028 0.026 0.024 0.022 0.020 0.018 0.016 0.014 0.012 0.010 0.0080.006 0.004 0.002

0.000 i

i

1

i

i

i

r

20

40

60

80

100

120

140

i

160

i

180

i

200

Figure 2-3b. Burr3 results: Distribution of means (solid line); and with 50% higher benefits (dashed line).

claims that are five weeks or less from about 5.4 percent to 7.6 percent (compare the first columns in the first two lines). Because higher wages induce more short-term claims the total losses associated with claims of five weeks or less increases from 3.6 percent to 4.9 percent (see the first column in line eleven and twelve from the top). But also, because of the slightly fatter right tail associated with higher benefits, beyond about the twelth week the proportion of claims actually begins to fall as wages increase (again, compare the eleventh and twelth lines). The Burrl2 results show a similar shift without the thickening of the right tail.

WORKERS' COMPENSATION COSTS AND HETEROGENEOUS CLAIMS

41

BR3-MEAN

0.038 0.036 0.004 0.002 , I

0.030, ~\1

o.028. ! iI i

0.026 ,

I

1

0.024 0.022 , 0.020 0,018

.

0.016 0.014. 0.012 • 0.010, 0.008



0.006, i 0.004. 0.002 o.ooo i 0

20

40

60

80

10O

120

140

160

180

200

Y

Figure 2-3c. Burr3 results: Distribution of means (solid line); and with 25% lower employment probability (dashed line).

It is also clear from figures 2-3c and 2-4c that the EMPLOY variable, the other part of the expected wage, effects the distribution of losses differently from the W A G E variable since in figures 2 - 3 c and 2-4c we are lowering the employment probability, while in figures 2 - 3 a and 2-4a we are increasing the value of the wage variable. The leftward shift in the distribution due to an increase in EMPLOY is barely discernible, and the mode rises as EMPLOY increases and the right tail thins rather than thickens, as it does for the W A G E variable. Because of their differential impact on the distribution, interpreting the multiplicative effect of W A G E and EMPLOY as an expected wage effect is problematic: they

42

WORKERS' COMPENSATION INSURANCE BR3-MEAN

0.039 0.038 0.036 0.034. 0.032 0.030 0.028 ' 0.026 0.024 0.0~. 0.020,

0.0180.016 0.014 0.012 0,0100.008 0.006. 0.004 0.002. 1 0.000,

20

40

60

80

100

120

140

160

180

200

Y

Figure 2-3d. Burra results: Distribution of means (solid line); and with 50% higher age (dahsed line).

both tend to shift the distribution to the left, but while the right tail thins as EMPLOY increases, it thickens as W A G E increases. As is apparent from figures 2 - 3 b and 2 - 4 b the effect of an increase in BENEFITS is nearly symmetric to that of wages: the mode of the distribution increases rather than decreases; the distribution shifts to the right rather than to the left; and the right tail becomes somewhat thinner instead of thicker. That benefits should work in the opposite direction from wages is expected on the basis of standard theories of search. That they should effect not only the location of the mean, but the thickness of the tail, has not been noted before. Again, the thinning of the right tail is more pronounced for the Burrl2 distribution than for the Burr3 dis-

WORKERS' COMPENSATION COSTS AND HETEROGENEOUS CLAIMS

43

BRI2MEA~

0041~ 0.040~ 0038! 0.036~ 0.0341

o.o32~ 0,030~ 0.02B~

002G~ 0.024~

oo~2~ 0.020~ 0 0~8~ 0.016~ 0.014: 0 012~ O.OlOO.OOBi 0 0061 0.004~

o.oo2~ 0.000~ I

2O

'

I

b

40

60

I

BO

I

I

I

IO0

~20

14o

i

~Gq

I

1~0

'

I

200

Y

Figure 2-4a. Burr12 results: Distribution of means (solid line); and with 50% higher wages (dashed line).

tribution. The result is that there are relatively fewer small claims, but many more claims in the ten- to thirty-week category. Losses reflect this pattern: a smaller fraction of losses accruing to the short term claims, but relatively higher losses explained by longer-termed claims. The reason for the offsetting changes in the tails of the distribution is known generally as the "selectivity bias" problem in econometrics. In labor economics, it is well known that an increase in nonwage income affects females' wage income in two ways: it affects labor supply incentives and hence wages directly, but it also changes the composition of women choosing to work. This latter impact is the "sample selection" effect since the composition of women working, and hence the number of women

44

WORKERS' COMPENSATION INSURANCE BR12MEAN

0.050

0.045

0.040

0.035¸

0.030

0.025

0,020.

0.01.5.

0.010.

0.005,

0.0Q0.

i

i

I

I

20

40

60

80

'

I

I

I

I

I

I

1~

1~

1~

1~

1~

200

Y

Figure 2-4b. Burr12 results: Distribution of means (solid line); and with 50% higher benefits (dashed line).

for whom wages are observed, is also effected by things like nonwage income. Frequency effects duration. The same thing is happening here: benefits differentially increase small and moderately sized claims. This increase in frequency of claims affects the proportion of claims at the mode of the distribution accounting for the "thinning" of the right tail. The effect of the A G E variable on the distribution of losses, on the other hand, is not at all surprizing. Figures 2 - 3 d and 2 - 4 d indicate that as workers grow older the distribution of losses shifts to the right, and the right tail thickens as well. Both of these trends suggest that as the current generation of baby boomers mature and the working population ages,

WORKERS' COMPENSATION COSTS AND HETEROGENEOUS CLAIMS

45

BR12MEAN 0.041 0,040

0,038 0.036 0.034 0,032 0,030 0.028 0.026 0.024 0.022 0.020 0.018 0,016 0.014 0.012

o.010 0.008 0.006.

XN

0.004.

xN x

0.002 0.1300. '

i

20

I

40

I

60

'

I

80

}

i

100

I

120

'

I

140

'

1

180

"

I

180

'

I

200

Y

Figure 2-4c. Burr12 results: Distribution of means (solid line); and with 25% lower employment probability (dashed line).

there may be profound changes in the distribution of losses associated with Workers' Compensation claims. Note that the number of claims remaining increases at every level of duration: at the duration mode of fifteen weeks, for example, the fraction of completed claims falls from 40 percent to roughly 35 percent, and the total Workers' Compensation losses that they account for falls from 35 percent to 30 percent. Since similar differences are observed at all levels of duration, the total impact of an aging population on the cost of Workers' Compensation will be substantial indeed.

46

WORKERS' COMPENSATION INSURANCE BR12MEAN 0.041 0.040 0.038 0.036 0.034 0.032 0.030 , 0.028 0.026, 0.024 ' 0.022 0.020 0.018 0.016. 0.014 0.012 • 0.010 • 0.008 0.006. t 0.004. 0.002. 0.000. I

I

i

i

I

20

40

60

80

100

'

I

I

120

140

i

160

I

180

i

200

Figure 2-4d. Burr12 results: Distribution of means (solid line); and with 50% higher age (dashed line).

Conclusions We have introduced two general forms of heterogeneity in the estimation of Workers' Compensation claims in order to more adequately characterize the relative cost importance of short and long term claims. In the second section we characterized as "intrinsic" heterogeneity those shifts in the distribution (specifically in the location parameter of the distribution) that are uncorrelated with the observable characteristics of the claim. We allowed for such shifts in the distributions by "mixing" on the location parameter. Since such mixtures tend to thicken the right tail, the

WORKERS' COMPENSATION COSTS AND HETEROGENEOUS CLAIMS

47

Table 2-5. Incomplete Moments When Wages, Benefits, Employment Probability and Age Increase. Distribution Fraction of Claimants, With Claims Duration Less Than (in weeks): Burr3 5 10 I5 20 25 50 100 200 means 0.053 0.237 0.399 0.516 0.600 0.798 0.907 0.958 +wage 0.075 0.249 0.389 0.492 0.567 0.756 0.873 0.936 +benefit 0.043 0.246 0.431 0.559 0.648 0.841 0.934 0.973 -empl.prob. 0.046 0.191 0.326 0.432 0.512 0.723 0.857 0.930 +age 0.038 0.196 0.351 0.469 0.557 0.772 0.893 0.952 Burrl2 means 0.050 0.234 0.407 0.525 0.781 0.880 0.934 0.937 +wage 0.078 0.264 0.419 0.525 0.598 0.769 0.869 0.926 +benefit 0.036 0.234 0.419 0.537 0.614 0.783 0.878 0.932 -empl.prob. 0.046 0.193 0.339 0.446 0.523 0.709 0.825 0.895 +age 0.034 0.189 0.356 0.476 0.560 0.750 0.860 0.921 * Fraction of Total Losses Burr3 means 0.035 +wage 0.048 +benefit 0.031 -empl.prob. 0.027 +age 0.024 ** Relative Claim Intervals: Burr3 means +wages +benefits -empl.prob. +age

Accounted for by Claims Duration Less than (in weeks): 0.193 0.196 0.214 0.144 0.156

0.350 0.331 0.396 0.269 0.303

0.469 0.435 0.528 0.374 0.422

0.558 0.514 0.620 0.457 0.513

0.773 0.721 0.826 0.684 0.744

0.894 0.853 0.927 0.835 0.879

0.953 0.926 0.970 0.918 0.946

Costs Accounted for by Claims Falling in the Respective Claim

0.659 0.641 0.730 0.591 0.633

0.860 0.851 0.899 0.805 0.835

0.969 0.962 0.986 0.927 0.949

1.021 1.017 1.025 1.991 1.005

1.051 1.050 1.045 1.03 1.038

1.085 1.091 1.067 1.078 1.075

1.116 1.131 1.086 1.125 1.110

1.131 1.151 1.094 1.149 1.127

*Incomplete First Moment is undefined for the Burrl2 since at the mean values of the parameters a * q < 1. ** Computed as (First Moment (t) - First Moment (t - 1))/ (Distribution (t) - Distribution (t - 1)).

requisite m o m e n t s of the distribution may not exist. W e suggested that " n o n p a r a m e t r i c " mixtures m a y provide useful models of "intrinsic" heterogeneity since they allow for thick tails and (for the class considered here at least) always have their m o m e n t s defined. Indeed, we provide an empirical example where the nonparametric mixtures actually fit the data

48

WORKERS' COMPENSATION INSURANCE

better (in a log-likelihood sense) than the parametric mixtures (i.e., the GG-2 versus GB2 comparison in section two). In section three we discussed the ways in which observable claim characteristics may be introduced into the analysis. Where previous studies of jobless (unemployment as well as Workers' Compensation) durations let observable heterogeneity change only the location parameter, this is the first analysis to extend the possibility that observable claim differences may effect the other parameters as well. We find that changes in the other parameters appear to significantly affect the shape of the distribution, 6 and hence are important to include in empirical research where the entire shape of the loss distribution is important. To facilitate the qualitative impact the observable characteristics had on losses, we used the incomplete moments of the distribution and simulated the effect that changing claim characteristics had on these moments. While the A G E and E M P L O Y variables shifted the distribution of losses in the way that we expected, the W A G E and BENEFITS variables had some surprising effects on the shape of the distribution. Though W A G E and ENEFITS shifted the mean of the distribution in the way that our simple theory suggested, the tail of the distribution thickened in ways that tended to offset the shift of the mean. Higher values of the W A G E variable tend to shift the distribution to the left so that the average duration and the cost of Workers' Compensation claims is lowered, at the same time the right tail thickens so that average duration and costs of claims tend to increase. The reason for this shift is simple: benefits differentially increase small and moderately sized claims. This increase in frequency of claims affects the proportion of claims at the mode of the distribution accounting for the "thinning" of the right tail. This sort of "selectivity bias" effect is well known in many fields of research: in labor economics, we know that labor force participation rates ("frequency") differentially affects the wage distribution ("duration"). Whether similar offsetting effects occur in other samples is certainly an empirical question that we hope to pursue in the future, perhaps with general models that control for the selectivity bias effect. Whatever the direction that this future research takes us, it is obvious that to address the question of how benefits, employees' age, etc. affect the distribution of claims, we must employ models that allow these characteristics to affect more than just the mean location parameter.

WORKERS' COMPENSATION COSTS AND H E T E R O G E N E O U S CLAIMS

49

Notes 1. The restriction to male workers eliminates heterogeneity in claims arising from physical differences associated with gender, although Worrall, Appel and Butler (1986) provide evidence that such differences are generally not statistically significant with respect to Workers Compensation losses. The restriction to 1985 minimizes temporal differences in cyclical economic activity. 2. The analogy between "structural" equations in standard simultaneous equations systems and our use here of the term "structural" equation is not exact but it is useful. The "reduced" form of a simultaneous equation system has its analogue here in equation (8) below, in either case (here or in simultaneous equation case) the reduced form is only informative about economic hypotheses when elements of the structural equation can be inferred from it. 3. McDonald and Butler (1987) include these and other more general mixtures of distributions, although their application is different (aggregate unemployment data) and they do not consider nonparametric heterogeneity in their models. See also Venter (1984) for an insightful discussion of the generalized gamma and beta distributions in an actuarial context. 4. See Butler and McDonald (1986) and references cited there for applications of incomplete moments to income inequality. Not only do the incomplete moments satisfy the requirements for a distribution function, they have the same functional form as the distribution function for the distributions we consider in this chapter. Hence, if one has the computer program to calculate the GB2 distribution, then the incomplete moments are trivially calculated by modifying the progra m slightly: simply replace the p parameter for the GB2 distribution with p+h/a and q with q-h/a for the h 'h moment. A similar result holds for the GG distribution by replacing p with p+h/a. This is summarized as follows (with the necessary code in the SAS computer program to calculate the distributions and their moments):

Distribution function

h m Incomplete Moment GG(y; a, b, p+(h/a)) [probgam((y/b)**a, (p+h/a))] GB2(y; a, b, p+(h/a), q-(h/a)) [probf( ( (q-h/a)/(p+h/a) )*(y/b )**a, 2*(p+h/a), 2*(q-h/a))l

GG(y; a, b, p)

[probgam( (y/b )** a, p)] GB2 (y; a, b, p, q,)

[probf((q/p)*(y/b)**a, 2*p, 2*q)]

5. Since the mean of the employ variable is 0.984 (virtually everyone in the sample was a regular employee at the time that the claim was filed), for this variable only we stimulated what would happen if it decreased by 24 percent by replacing the mean value with 0.734 and then examining the implied moments. 6. The specifications in table 2-5 can be simplified in the usual way if we restrict changes in the form of the distribution to the location parameter alone. When fit with our sample, such a restriction yields the following results: Parameters a b0 bl (wage) b2 (benf) b3 (empl)

Burr 3 (std, error) 1.225 1.497 -0.122 0.80 -0.489

(0.24) (2.47) (0.40) (0.46) (0.72)

Burr12 (std, error) 2.899 2.328 -0.097 0.029 -0.545

(0.40) (1.81) (0.42) (0.49) (0.71)

WORKERS' COMPENSATION INSURANCE

50 Parameters b4 (age) p q log-likelihood

Burr 3 (std, error) 0.224 (0.41) 4.531 (1.54) -5088.686

Burr12 (std, error) 0.229 (0.41) -0.306 (0.66) 5085.869

By comparing these likelihood values with those in tables 2-2 and 2-4, we observe that the WAGE, BENEFIT, EMPLOY and AGE variables are jointly significant at the 1 percent level in explaining shifts in the location parameter (compare the above likelihood values with those in table 2-2), and that these same variables are also significant at the 5 percent level for the Burr3 and the 10 percent level for the Burrl2 in explaining changes in the a and p or q parameters. More significiantly, all of the shift variables (bl, b2, b3, and b4) in these simplified specifications have the expected sign and would unambiguously shift the distribution in the indicated direction. However, as we see in tables 2-3 and 2-4 when shifts in the p and q parameters are also allowed, the impact of these shifts is mitigated by changes in the tail.

References Butler, Richard J. and John D. Worrall. (1985). " W o r k Injury Compensation and the Duration of Nonwork Spells." Economic Journal 714-724. Butler, Richard J. and James B. McDonald. (1986). "Trends in Unemployment Duration D a t a . " Review of Economics and Statistics 68:545-557. Heckman, James J. and Burton Singer. (1984). " A Method for Minimizing the Impact of Distributional Assumptions in Econometric Models for Duration D a t a . " Econometrica 52:271-320. Hogg, Robert V. and Stuart G. Klugman. (1985). " O n the Estimation of LongTailed Skewed Distributions with Actuarial D a t a . " Journal of Econometrics 23:91-102. McDonald, James B. (1984). "Some Generalized Functions for the Size Distribution of Income." Econometrica 52:647-663. McDonald, James B. and Richard J. Butler. (1987). "Generalized Mixture Distributions with an Application to Unemployment Insurance." Reivew of Economics and Statistics 69:232-240. Ralston, M.L. and R.I. Jennrich (1979). " D U D , A D e r i v a t i v e - - F r e e Algorithm for Nonlinear Least Squares." Technometrics 1:7-14. Trussel, James and Toni Richards. (1985) "Correcting for Unmeasured Heterogeneity in Hazard Models Using the Heckman-Singer Procedure."

Sociological Methodology. Venter, Gary S. (1984). "Transformed Beta and G a m m a Distributions." 1984 Proceedings of the Casualty and Actuarial Society LXX: 156-1923. Worrall, John D., Richard J. Butler, Philip Borba, and David Durbin. (1987). "Estimating the Exit Rate from Workers Compensation: New Hazard Rate Estimates." Unpublished manuscript (revised October 1987).

3

THE TRANSITION FROM TEMPORARY TOTAL TO PERMANENT PARTIAL DISABILITY: A LONGITUDINAL ANALYSIS John D. Worrall* David Durbin** David Appel** Richard J. Butler***

Introduction There are more than 8 million workplace injuries per year compensated by private insurers or competitive state funds under the various workers compensation programs. Of these only 5 percent are sufficiently severe to result in permanent disability or the death of the injured worker, yet this subsample accounts for some 75 percent of all incurred claims costs. Identifying the more severe injuries is of substantial importance both in the estimating the costs of the workers compensation system, as well as in the effort to contain those costs. Unfortunately, identification of permanently disabling injuries is not

A paper prepared for the 7th annual NCCI seminar on economic issues in workers' compensation, The Wharton School, University of Pennsylvania. *Rutgers University, **The National Council on Compensation Insurance, and *** Brigham Young University. Any opinions expressed in this paper are our own and are not necessarily shared by our sponsoring organizations. We thank the national council on compensation for research support and rich derrig for research suggestions.

51

52

W O R K E R S ' COMPENSATION INSURANCE

necessarily obvious when claims initially arise. Consider compensable injury arising in a given year that initially prohibits the claimant from working, initially it is a "total" disability. Within months the ultimate medical condition of the claimants become obvious, and the majority of these claims will be injuries of short duration in which the worker will fully recover. Such claims are "Temporary Total" disability claims. However, some of these claims will ultimately be found to be permanently disabling: those in which the worker can return to gainful employment are "Permanent Partial" injuries; those injuries sufficiently severe so as to prevent them from returning to work are "Permanent Total" injuries. In our sample at least 30 percent of all claims that initially begin as temporary total disabilities that remain open six months after injury are ultimately classified as permanent disabilities. Since these claims have costs that far exceed those of the less severe claim type, insurers and employers will have a significant interest in their early and accurate recognition. In this chapter we shall attempt to retrieve point estimates of the determinants of the transition from temporary total to permanent partial disability status. There is evidence that the ultimate disposition of claims is a function of the level of the expected indemnity benefit that can, and does, vary with claim type. (See Butler, 1983; Butler and Worrall, 1983; Butler and Worrall, 1985; Chelius, 1982, 1983; Fenn, 1981; Ruser, 1985; Worrall and Butler, 1985; and Worrall and Appel, 1982). The empirical evidence that exists on the impact of the structure of Workers' Compensation benefits on the percentage distribution of claim types is either aggregate time series (Worrall and Appel, 1982) or pooled cross-section time series (Butler, 1983; Butler and Worrall, 1983; Chelius, 1982, 1983; and Ruser, 1985) at the state level. Another group of studies have found that the benefit structure also increases the length of claims (Butler and Worrall, 1985; Fenn, 1981; Worrall and Butler, 1985; Worrall, Butler, Borba, and Durbin, 1988). No one yet, however, has examined how benefits affect the transition from temporary total to permanent partial disability status. In this chapter we use longitudinal data on claimant status to begin to fill that research void. The motivation for the research is twofold. We would like to be able to determine which individual claims classified as temporary totals at first report will ultimately be classified as permanent disabilities; and we would like to determine if the probability of such classification varies with program structure. The research question is one of more than academic curiosity. It has important implications for practitioners in workers compensation. At the rate hearing level, it addresses the question: Will

THE TRANSITION FROM TEMPORARY TOTAL

53

increases in benefits lead to more permanent claims, and if so how many? At the insurance carrier level, it addresses the question: How do we reserve an individual claim that is still open? (Or what do we put up in bulk for temporary totals that are still open?) Estimates of severity (or claim length) will also be improved. The duration of many temporary total claims is sufficiently great that even in fairly lengthy longitudinal data-sets some claims will not have closed and may still be made a permanent disability claim. These are "open" or "right censored" claims. Since these claims might ultimately be classified as permanent disability, the point estimates of durations of times in temporary total claimant status may be "biased" in the sense that we would be mixing some claims that will ultimately be permanent partial or permanent total disabilities with claims that are "purely" temporary total claims. One of our research goals is to purge these censored (open) temporary total claims of those likely to make the transition to permanent disabilities in order to produce better estimates of the duration of temporary total spells. We can attempt to do this in two ways. First, the sample of open temporary total claims may be may literally purged of claims that have estimated transit probabilities greater than some given cutoff (say 50 percent, or more generally, a cutoff that is optimal by some criterion). This method would, unfortunately, eliminate some true temporary total claims from our sample (i.e., those predicted transits that do not actually occur). On the one hand it may appear to do little harm since such claims are, in some sense, most like permanent disabilities (and the least likely candidates for a "search" model). However, since these true temporary totals may differ systematically from temporary totals in the sample, their elimination could affect the estimates of the population parameters of the distributions of temporary total claims. A second, alternative approach is to enter the individual estimated transit probability as a covariate in the duration model. This would enable us to control for transit probability without eliminating true temporary totals from the sample. In effect, we would be trying to control for the potential sample selection of long temporary total claims into permanent disability status by using the predicted probabilities. There is a simple linear approximation for a process, sample selection, that in theory can always be captured by sufficient expansions in the probability of transition (see Butler and Worrall, 1983 for an application of this principle to claim frequency estimation). In addition to improving the point estimates of the parameters of the temporary total duration distribution, the technique may help in the

54

WORKERS' COMPENSATION INSURANCE

monitoring of individual claims. For example, the State of Maine has an incurred development factor of 2.25 for indemnity claims--that is, the ultimate claims costs are 225 percent of the initial estimates; ultimate claim costs exceed initial estimates by 125 percent. Even in this extreme case the overwhelming majority of claims will be properly reserved, but 5 percent to 10 percent of the claims will have much higher development factors more on the order of 10.0 to 20.0.1 These claims can present tremendous problems because they may be seriously under reserved. Insurers cannot simply load this development across all claims, as risks would balk as they saw their experience rating modifications affected. That is, individual insureds would not want their o w n experience rating to suffer as a result of reserving for s o m e o n e else's serious claim. This problem is basically an information problem. E x ante, those temporary total claims that will transit to permanent claim status are not known. If the standard techniques (logit, probit, discriminant function) enable a carrier to predict which individual open temporary total claim is likely to transit to permanent partial status, they may prove valuable as both a monitoring device and a vehicle for making decisions on renewal business. Our plan for this chapter is as follows. First, we present the theoretical models and data employed in the analysis. We then discuss the results of our model estimations, and we conclude with thoughts for future research.

Model and Data We are using a random sample of temporary total disability claims from Massachusetts and Illinois 2 arising during 1982. The data is from the Detailed Claim Call of the National Council on Compensation Insurance. The sample was designed to yield 1,200 permanent partial claims per state and year with the states in the sample comprising approximately 40 percent of countrywide workers compensation premium. Information on unresolved (open) claims is collected at six, eighteen, thirty, forth-two and fifty-four months. Not all sampled claims will have five reports. For example, if a claim closes in five months, it would only have a First Report. If a claim closed in seven months, it would only have two reports, a First Report at six months and a Second Report at eighteen months (or, at the carriers discretion, at closure). Not all claims are closed. Although we are basically

THE TRANSITION FROM TEMPORARY TOTAL

55

using 1982 accident year data, a small number of claims classified as temporary totals at First Report are still open at Fifth Report. The Detailed Claim Call is a rich data source. The sample provides the following data on each claimant: time on a claim, claim type and status, wage, indemnity benefit, attorney involvement, time spent in hospital, age, marital status, sex, body part injured and the nature of the injury. These variables were used in the studies of indemnity claims cited above, as well as in a study of medical utilization (Worrall, Appel and Butler, 1987). Two states were selected for our initial look at the determinants of permanent partial disability, Massachusetts and Illinois. Massachusetts was chosen based on previous (unpublished) work that found the distributions of temporary total spells to be inordinately long. This was the case whether sophisticated modelling techniques were used to estimate the parameters of the distribution, or simply if the mean durations were compared across the Detailed Claim Call states. Illinois was selected due to previous success in modelling the durations of temporary total spells in that state (Worrall et al., 1988); thus, it will provide a comparison with the point estimates from the Massachusetts transition model. An advantage of estimating the models within each state is that such a technique may eliminate some heterogeneity (for example, economic conditions may vary radically by state), and it provides strong controls for program administration. A disadvantage is that estimation within a single state makes the identification of the two main variables of interest--the wage and benefit variables--more problematic. Although a claimant's wage and workers' compensation benefit are not perfectly collinear, the replacement rate is fixed within a state for all claimants over the range of the minimum to maximum benefit level. 3 Although part of the claim experience is due to stochastic influences beyond the control of the worker, part is--we believe--subject to economic incentives associated with observable traits. This part can be described in a simple search framework in which the claimant returns to work when the expected value of doing so (measured by his wage) is greater than the expected value of staying on the claim (measured by his benefit), holding other demographic variables and type of injury constant. This simple theory predicts that the probability of remaining on a claim (and hence the claim duration) will increase as benefits increase and decrease as wages decrease. Hence, our basic model is: Prob y = f(ln w, In age, male, lawyer, In hosp, married, In ben, injury type).

56 Where / n w In age male lawyer In hosp married In ben injury type

WORKERS' COMPENSATION INSURANCE

= = = = = = =

log wage log age l if male l if attorney involvement log hospital days4 l if married log benefits a vector of injury types: head & neck; upper extremities; trunk; lower extremities; low back; and all others (the basis)

We have estimated this model using maximum likelihood techniques to retrieve point estimates of the Logit model. 5 For comparison purposes, we also estimated discriminant functions for our data, however these would provide inconsistent estimates (see Maddala, 83). As the results using both techniques were qualitatively similar, we present only the maximum likelihood results below. 6 The model was estimated in the two states selecting the sample of all temporary total claims that were open at First report. The basic model assigns a value of l to those temporary total claims that closed permanent partial between First and Fifth report. Although we have censoring, we are treating the estimates from our basic model as First to Ultimate (by "ultimate" we mean the ultimate disposition of the claim into temporary or permanent disability). The estimates are probably slightly biased because forty of the initial 4,636 temporary total claims were still open at Fifth report in our Massachusetts data. 7 We suspect that most of these claims will ultimately be classified as permanent. If the censored claims are passed through the survivor function, the likelihood function would be weighted more heavily in favor of no transit. As we are treating the 40 censored claims as if they are complete in the sample, our results can be treated as First to Fifth transit probabilities (or slightly biased First to Ultimate). Point estimates of the probability of transit during the intervals from: First to Second, Second to Third, Third to Fourth, and Fourth to Fifth are also retrieved. These estimates are presented in the Appendix tables. Although the primary interest is in predicting which open temporary total claims will be closed as permanent claims, we have also estimated the model for all claims closed at First report. The determinants of permanent partial closure at First report are estimated because we believe that there will be factors that will enable one to predict permanent partial status early in the life of a claim. Even beyond the obvious cases like amputa-

57

THE TRANSITION FROM T E M P O R A R Y T O T A L

tion, for example, those seriously injured and hospitalized for extensive stays (holding constant the injury type) are more likely to be identified as permanent disabilities early in the claims process. These claims may close more rapidly, leaving more problematic claims open in the sample. The Logit results for claims closed at First report are also presented in the Appendix tables. We hypothesize that the signs of the key variables will be as follows:

In wage 0, lawyer In hosp

>0, >0,

In age male married injury type

>0 >0 9 ?

Wage: Our research on duration leads us to believe that higher wage workers return to work more rapidly than low wage workers. As the money measure of the opportunity cost of time is higher, we predict that high wage workers will be less likely to extend stays on claims and fight for permanent partial awards. 8 Benefits: The effect of higher benefits should be the reverse of the wage effect. That is, all else the same, higher benefits should provide incentives to prolong the disability spell. Lawyer: We predict permanent partial closure is more likely, given attorney involvement, because the financial incentives, and expected attorney rewards, are greater for permanent claims. Hospital Days: The longer a hospital stay, the more serious an injury and likely a permanent claim. This may be attenuated since the more medical care one receives, ceteris paribus, the less likely is residual impairment. Age: We predict older workers will take longer to heal, and, given the same force in trauma, will be more like to suffer permanent injury. This likelihood may be offset by a self-selection process as well; older workers who remain in strenuous and dangerous jobs may be stronger and in better health than some younger workers who have not sorted out of dangerous jobs. Male: The conventional wisdom is that males historically have been holding the more dangerous jobs. Not only are they injured more frequently, but also more seriously. Married: We have no strong priors on marital status. Married claimants may have a working spouse, which would enable them to extend their stays on claims and, through a tax effect, raise the dollar value of benefits. However, marital status may imply financial responsibility and the need for speedy return to work. It may also affect risk bearing or be a signal about one's willingness to bear risk.

58

WORKERS' COMPENSATION INSURANCE

Injury Type: The probability of permanent partial claim status will vary with the type of injury especially since more serious injuries are more likely to be categorized as a permanent disability at first report (e.g., loss of limb). In addition, certain other claim types which are slow to develop and difficult to monitor (e.g., low back injuries), may also be likely to develop permanent status. We use a set of dummy variables to control for injury type, the excluded category being "all other injuries," that account for about 5 percent of the claims.

Results

Table 3 - 1 contains the point estimates and their standard errors for the Logit model estimated on a First to Fifth report basis. There is some support for our economic hypothesis, with the case being stronger in the Massachusetts data. 9 In both cases the model is significant at the 0.01 level with Chi Squares (not reported here) well beyond those required to reject the null of a zero Beta vector. In both cases the wage and benefit variables have the hypothesized signs, although only the wage variable in the Massachusetts data is significant at the 0.05 level. The Massachusetts benefit variable is not quite significant at the 0.10 level. As the Massachusetts results are stronger than those for Illinois, we will limit the rest of our discussion to the Massachusetts results. Attorney involvement has the hypothesized sign and is significant at the 0.01 level; however, hospital days not only does not have the hypothesized sign, but it is negative and significantly different from zero at the 0.01 level. At first blush it seems strange that longer hospital stays should be associated with lower probabilities of transit from temporary total to permanent partial status between the First and Fifth reports. But an inspection of the point estimates for claims closed at First report is quite revealing. Hospital days has a powerful impact on early claim closure as a permanent partial claim (with an asymptotic t statistic over 10.0). It seems that those claimants who do not close by the First report (six months) and receive more hospital care show a greater likelihood of ultimately closing as temporary totals. This may be closely related to the fact that this set of claimants, who may tend to have more soft tissue rather than loss of limb type of injuries, may in fact respond to additional medical treatment. Closure as a permanent partial at First report is also strongly associated with being male, and with having an attorney involved in the claim. Suffering a low back injury or a head injury is associated with a lower likelihood of closure as a permanent partial at First report. Low back

THE TRANSITION FROM TEMPORARY TOTAL

59

Table 3-1. 1982 Claim Yransistion: First to Fifth Report Maximum Likelihood Legit Parameter Estimates (Asymptotic Standard Errors in Parentheses). Variable

Massachusetts

Illinois

Intercept

- 1.491 (0.947) -0.631" (0.315) 0.169 (0.184) 0.187 (0.156) 0.311" (0.126) -0.074" (0.019) 0.107 (0.138) 0.548 (0.347) 0.361 (0.373) 0.759* (0.290) 0.850" (0.319) 0.372 (0.301) 0.108 (0.292) 1351

0.894 (1.053) -0.324 (0.320) -0.120 (0.216) 0.269 (0.191) -0.026 (0.137) -0.067" (0.021) 0.060 (0.161) 0.132 (0.349) 0.268 (0.336) 0.510" (0.260) -0.120 (0.320) 0.528"* (0.271) 0.085 (0.269) 929

Lnwage Lnage Male Lawyer Lnhosp Married Lnben Head & Neck Upextr Trunk Lowextr Lowback N * Significant at 5%. ** Significant at t0%.

claims may be slow to develop and have very long durations in the Massachusetts data. In addition, these claims may be more difficult to monitor. In future work, we intend to partition claims data to ascertain whether the determinants of permanent disability classification vary across injury type, testing whether soft tissue injuries are more responsive to economic incentive. Both the low back and head and neck injury variables were significant at the usual levels. Interestingly, the wage and benefit variables are not significant for claims closed at First report and

60

WORKERS' COMPENSATION INSURANCE

they have the wrong sign. This has some intuitive appeal because claims that are clearly permanent partial early on in the claims process may be less responsive to economic incentive. Returning to our estimates for transit between the First and Fifth reports, age and male have the hypothesized signs but they are not significant at even the 0.10 level. Given an open status at First report, each of the injury types included in our logistic regression had a higher probability of closure as a permanent disability than the base case (all other injuries), but only upper extremity and trunk injures were significant (both at the 0.01 level). We also retrieved point estimates on transits between First and Second, Second and Third, Third and Fourth, and Fourth and Fifth Reports. 1° These estimates are presented in the Appendix tables. We treated these transits as independent events, although they are clearly not. 11 As the sample sizes are smaller and the difficulties in identifying the wage and benefit effects likely to be greater, we didn't expect the model to perform as well when applied to shorter time frames. Nonetheless, the wage and benefit effects are significant at the 0.10 level or better in the First to Second, and in the Third to Fourth model. The Model Chi Square is significant at the 0.10 level in all but the Third to Fourth report case. 12 As measures of the performance of our models, we will compare our forecast of transits to permanent partial, at both the individual and aggregate level, with the actual transits. These results are presented in table 3 - 2 below. Suppose our decision rule is that any individual observation with a predicted transit probability greater than 0.36 will be predicted to be a permanent partial. 13 In the Massachusetts data, 461 of 1,351 claims had predicted transit probabilities greater than 0.36. The percentage of predicted transits that actually occurred is referred to as Sensitivity. The Sensitivity of the Massachusetts model was 41.7 percent as 186 of the 461 claims that were predicted to be transits actually were. On the other hand, Specificity of our model is defined as its ability to accurately predict temporary total closure. We predicted 890 temporary total closures, with 630 or 69.6 percent of these actually closing as temporary totals. In the Massachusetts data our total Correct forecast was 60.4 percent (630 correct temporary total and 186 correct permanent partial predictions in a sample of 1,351 claims). The Sensitivity, Specificity and Correct measures given above imply that we predicted transits (False Positive Rate = 59.7 percent) and temporary total closures (False Negative Rate = 29.2 percent) that did not occur. The estimated Sensitivity, Specificity and Correct rates given above essentially are ex post in the sense that we have used the same sample

THE TRANSITION FROM TEMPORARY TOTAL Table 3-2.

61

Massachusetts 1982 Claim Transistion Legit Analysis Goodness

of Fit. Threshold Probability Goodness o f Fit

Chi-Square Correct Prediction (%) Specificity (%) Sensitivity (%) Predicted/Actual Ratio

Classification Table Predicted Negative Positive

Negative True Positive Total

630 260 890

275 186 461

Probability = 0.36

0.36

0.50

49.95" 60.4 69.6 41.7 461/446

49.95" 67.8 97.3 7.8 59/446

Classification Table Predicted Negative Positive

Total

905 Negative 446 True Positive 1,351 Total

881 411 1,292

24 35 59

Total

905 446 1,351

Probability = 0.50

to derive point estimates of the population parameters and check the performance of the model. There are several ways to explicitly test the performance of the model including the use of the parameter estimates to predict transit on another sample of claims. The point estimates from the 1982 accident year data are used to predict transits to permanent disability status in the 1983 accident year data. This not only provides a truer standard to assess the model but also enables us to determine if our point estimates can be used to reserve claims arising in future years. 14 The results of our forecast outside the sample period are presented in table 3-2. With a predicted probability of 0.38 we forecast 241 permanent disabilities in the accident year data. There were actually 225 that occurred. This forecast has Sensitivity of 28.4 percent. However, 95 of 101 claims that were open at Fourth report had no Fifth report. We suspect that many of these open claims with no Fifth report will actually close as permanent disabilities. If this is the case, both the Sensitivity and Correct measures could improve. For instance, if one third of the 95 claims currently classified as temporary totals were actually permanent disabilities and correctly predicted as such, the model Sensitivity would rise to

62

WORKERS' COMPENSATION INSURANCE

Table 3-3. Massachusetts 1983 Predicted Claim Yransistion Legit Analysis Goodness of Fit.

Threshold Probability Goodness of Fit Correct Prediction (%) Specificity (%) Sensitivity (%) Predicted/Actual Ratio

0.36

0.38

58.95 68.8 30.7 271/225

61.2 72.6 28.4 241/225

Table of Migrate by PRED6 Migrate PRED6

Table of Migrate by PRED8 Migrate PRED8

Frequency percent row PCT col PCT 0

Frequency percent row PCT col PCT 0

1

Total

0 445 51.03 68.78 74.04 156 17.89 69.33 25.96 601 68.92

1 202 23.17 31.22 74.54 69 7.91 30.67 25.46 271 31.08

Probability = 0.36

Total 647 74.20

225 25.80

1

872 100.00

Total

0 470 53.90 72.64 74.48 161 18.46 71.56 25.52 631 72.36

1 177 20.30 27.36 73.44 64 7.34 28.44 26.56 241 27.64

Total 647 74.20

225 25.80

872 100.00

Probability = 0.38

42.7 percent (96 of 225 claims). This is considerably better than a random assignment based on a predicted transit rate, which would have a Sensitivity of approximately 11 percent. We are also interested in the responsiveness of claims to the structure of the program. As a measure of the wage elasticity we have increased each open t e m p o r a r y total claimant's wage by 10 percent, holding benefit constant, and calculated the percentage decrease in claims that would be predicted as p e r m a n e n t disabilities. For example, suppose a claim in our sample had a predicted transit probability of 0.40. If it is found that the predicted probability for some claims drop below 0.36 when wages are increased by 10 percent, and all other variables are held constant, these claims enter the elasticity calculation. The benefit elasticity is also

63

THE TRANSITION FROM T E M P O R A R Y T O T A L

Table 3-4. Massachusetts Claim Transition Logit Elasticity Estimates*. Variable

Raw (%) Chance

Wages Benefits

-47 (-10.2) 68 (14.75)

*The elasticity is evaluated for a 10% change in the specific variable all other variables held constant.

calculated, that is, we increased benefits by 10 percent holding all other variables constant and calculated the increase in expected permanent disability claims. These result are presented in table 3-4. Increasing each individual's wage by 10 percent results in a prediction of 414 permanent disability claims, a decrease of 47 predicted permanent disabilities. This decrease of 47 claims is a decrease of 10.2 percent and results in am implied elasticity of 1.02. Increasing each individual's benefit by 10 percent results in an increase of 68 predicted permanent disabilities, an increase of 14.75 percent, and an implied elasticity of 1.47. These elasticity estimates computed using individual data are reasonably close to the elasticity estimates obtained using aggregate pooled cross section time series data at the state level.

Conclusions Although this is just the first step in an attempt to predict permanent partial closure for open temporary total claims, we believe the technique has merit. The evidence indicates that the structure of benefits is a determinant of permanent partial closure. The elasticities presented in table 3 - 4 are revealing: increases in real benefits may lead to sizable increases in permanent disability claims. The specification of our basic model m u s t be refined. We appear to have the ability to forecast with some accuracy outside the sample period, but we view the duration and classification of claims to be a simultaneous problem. A natural extension of the analysis is to determine if better duration estimates and heterogeneity controls can be produced using Logit as a first stage procedure. Since the use of this technique as a reserving device may be problematic, companies might apply it as a claims monitoring device, and regulatory agencies might apply it as a guide to understanding why development factors can get so large. Rating organizations and regulators might

WORKERS' COMPENSATION INSURANCE

64

consider the technique as o n e aid in measuring the likely response of claimants to benefit increases. As this chapter is the first step in our research plan, there are many questions to be answered. Does the predicted transit rate vary across states and across accident years? Are the point estimates on the individual coefficients stable? Do monitoring costs matter (are soft tissue claims more responsive to benefit changes)? Does the nature of the injury, given body part, matter? Should the specification of the model change from

Appendix A: Illinois Claim Transitions Maximum Likelihood Legit Parameter Estimates (Asymptotic Standard Errors in Parentheses). Variable

1st to 2nd

2nd to 3rd

3rd to 4th

Intercept

0.162 (1.895) -0.274 (0.585) -0.123 (0.390) 0.137 (0.354) 1.062" (0.289) 0.033 (0.038) O.128 (0.289) 0.287 (0.537) -0.031 (0.574) -0.128 (0.433) - 0.747 (0.563) 0.046 (0.470) - 1.120" (0.473)

- 6.324 (4.184) - 1.984 (1.507) -0.365 (0.831) -0.172 (0.671 ) 1.265" (0.491) -0.060 (0.071) 1.186"* (0.642) 3,162" * (1.829) 2.325* (1.181) 2.125" (0.851) 2.251 * (1.072) 2.356" (0.974) 1.266 (0.812)

- 0.301 (11.286) 4.126 (3.719) - 1.768 (2.011) - 1.695 (1.619) 2.447"* (1.401) -0.293"* (0.161) -0.715 (1.409) -3.403 (3.431 ) 2.214 (2.370) 2.509 (1.743) - 6.733 (19.954) 1.572 ( 1.658) 0.009 (1.387)

333

106

37

Lnwage Lnage Male Lawyer Lnhosp Married Lnben Head & Neck Upextr Trunk Lowextr Lowback N

*Significant at 5 percent. **Significantat 10 percent.

65

THE TRANSITION FROM TEMPORARY TOTAL

Appendix A: Massachusetts Claim Transitions Maximum Likelihood Legit Parameter Estimates (Asymptotic Standard Errors in Parentheses). Variable

1st to 2nd

2nd to 3rd

3rd to 4th

4th to 5th

Intercept

- 0.160 (1.332) - 1.241 * (0.544) 0.003 (0.250) 0.093 (0.210) 0.473" (0.182) -0.016 (0.026) -0.181 (0.187) 1.175" (0.611) -0.738 (0.598) 0.270 (0.424) 0.146 (0.471) - 0.068 (0.439) -0.124 (0.422)

3.890* * (2.338) 0.380 (0.610) -0.548 (0.447) 0.196 (0.344) 0.144 (0.290) 0.097" (0.043) -0.020 (0.331) - 1.180" * (0.735) -0.130 (1.086) 0.925 (0.826) 0.724 (0.869) 0.816 (0.826) 1.068 (0.802)

- 3.656 (4.168) -3.252* (1.303) 0.893 (0.691) -0.258 (0.486) -0.181 (0.442) -0.005 (0.066) -0.527 (0.550) 3.362* (1.489) -0.894 (1.366) 0.432 (0.948) 0.394 (0.998) 0.865 (0.940) 0.062 (0.886)

- 1.537 (5.027) -0.195 (1.458) -0.520 (0.770) -0.092 (0.583) -0.060 (0.499) 0.040 (0.077) 0.235 (0.577) 0.656 (1.613) 0.752 (1.170) 0.262 (1.033) 3.523' (1.441) 0.513 (1.021) 1.301 (0.946)

Lnwage Lnage Male Lawyer Lnhosp Married Lnben Head & Neck Upextr Trunk Lowextr Lowback N

762

292

160

90

*Significant at 5 percent. **Significant at 10 percent.

report to report? W e do not have all the answers, but we are encouraged that we will eventually be able to answer some of the questions.

Notes 1. In comparing statewide development factors with individual firm development factors, two caveats should be borne in mind. a) The statewide development factor may

66

W O R K E R S ' COMPENSATION INSURANCE

include the effects of changes in bulk (or IBNR) reserves. This could cause some incompatibility with development factors for individual firms, b) The statewide development factors are obtained from insurance company financial data, while the information for experience rating is obtained from individual risk information. 2. Benefits for permanent disabilities, as well as temporary total disabilities, are at least as high in Illinois as they are in Massachusetts for any given wage (both minimums and maximums are higher in Illinois, and the waiting period is only three days instead of the five days in Massachusetts). Though we don't expect that it will have a large impact on the results, the retroactive period in Massachusetts is six days, whereas it is two weeks in Illinois. Otherwise, the two states seem to be administrately similar, at least with respect to employees' incentives. 3. In future work we shall pool across states. As mentioned above the sample weights vary across states. We have not written maximum likelihood routines to account for the varying sample weights. 4. We set hospital days to 0.01 for those claimants with zero hospital days. 5. We used the SAS PROC LOGIST for our Logit routines and SAS PROC DISCRIM for our discriminant function routines. 6. The discriminant function results are available from the authors on request. 7. Only four temporary total claims were open at Fifth Report in the Illinois data. 8. This may be offset by the higher compensating differentials required to compensate workers more likely to suffer permanent injuries. 9. In both the Massachusetts and Illinois data we had some claims with a Second or Third report and no First Report. We also had some claims that had their claims numbers changed by the carriers reporting to the National Council on Compensation Insurance. We are attempting to create First reports for the former to pick up observations for our First to Second Logits. We are attempting to match records, in the latter cases, to avoid duplication. 10. There were not enough observations to estimate the model for closures between the Fourth and Fifth reports in Illinois (n = 20). 11. Butler and Moffitt have considered the case of Logit estimation with panel data (Butler and Moffit, 1982). 12. We also estimated all of the models with duration included as a right hand side variable. As duration is a function of the vector of right hand side variables as well, we knew we would have bias problems. The inclusion of the duration variable had very little effect on the sign or size of the wage and benefit variables. We believe this is due to the highly nonlinear relationship between the covariate vector and duration. 13. The actual transit rate observed in the Massachusetts data between the First and Fifth reports was 0.33. This transit percentage, 446 of 1,351 temporary total claims open at First report that were ultimately classified as permanent partial, the 40 temporary total claims open at Fifth report. Counting those 40 as permanent claims would have raised the transit percentage to 0.36. 14. We also intend to derive the point estimates for different accident years to see if they are stable.

References B u t l e r , J.S. a n d R o b e r t Moffitt. (1982). " A C o m p u t a t i o n a l l y Q u a d r a t u r e P r o c e d u r e for t h e O n e - F a c t o r M u l t i n o m i a l P r o b i t Econometrica 5 0 : 7 6 1 - 6 4 .

Efficient Model."

THE TRANSITION FROM TEMPORARY TO PERMANENT DISABILITY

67

Butler, Richard J. (1983). "Age and Injury Rate Response to Shifting Levels of Workers' Compensation." In John D. Worrall (ed.) Safety and the Workforce: Incentives and Disincentives in Workers' Compensation. Ithaca, NY: ILR Press, 61-86. Butler, Richard J. and John D. Worrall. (1983). "Workers' Compensation: Benefit and Injury Claims Rates in the Seventies." Review of Economics and Statistics LXV(4):580-589. Butler, Richard J. and John D. Worrall. (1985). "Work Injury Compensation and the Duration of Nonwork Spells." Economic Journal 95:714-724. Butler, Richard J. and John D. Worrall. (1987). "Gamma Duration Models With Heterogeneity." Paper Presented at the Risk Theory Seminar, University of Texas, Austin (April). Chelius, James R. (1983). "The Incentive to Prevent Injuries." In John D. Worrall (ed.) Safety and the Workforce: Incentives and Disincentives in Workers' Compensation. Ithaca, NY: ILR Press, 154-160. Chelius, James R. (1982). "The Influence of Workers' Compensation on Safety Incentive." Industrial and Labor Relations Review 35:235-242. Ehrenberg, Ronald G. (1984). "Workers' Compensation, Wages and the Risk of Injury." A paper presented at the Conference on New Perspectives on Workers' Compensation, ILR School, Cornell University. Ithaca, NY (October 16). Fenn, Paul. (1981). "Sickness Duration Residual Disability and Income Replacement: An Empirical Analysis." Economic Journal: 158-173. Maddala, G.S. (1983). Limited Dependent and Qualitative Variables in Econometrics. NY: Cambridge University Press. Ruser, John W. (1985). "Workers' Compensation Insurance, Experience Rating, and Occupational Injuries." Rand Journal of Economics 16(4):487-503. Worrall, John D., Richard J. Butler, Philip S. Borba, and David Durbin. (1988). "Estimating Exit Rates from Workers Compensation: New Hazard Rate Estimates." Unpublished manuscript. Worrall, John D. and Butler, Richard J. (1985). "Workers' Compensation: Benefits and Duration of Claims." In John D. Worrall and David Appel (eds.), Workers' Compensation Benefits: Adequacy, Equity and Efficiency. Ithaca, NY: ILR Press, 57-70. Worrall, John D. and Appel, David. (1982). "The Wage Replacement Rate and Benefit Utilization in Workers' Compensation Insurance." The Journal of Risk and Insurance XLIX (3):361-371. Worrall, John D., David Appel, and Richard J. Butler. (1987). "Sex, Marital Status, and Medical Utilization by Injured Workers." Journal of Risk and Insurance: 27-44.

This page intentionally blank

THE TRANSITION FROM TEMPORARY TO PERMANENT DISABILITY: EVIDENCE FROM NEW YORK STATE* Terry Thomason

Workers' compensation is a social insurance program providing cash benefits, medical care, and rehabilitation services to workers disabled by work-related injuries or diseases. Workers' compensation is one of the largest social programs in the United States. In 1984 almost 82 million workers were covered by workers' compensation programs, which paid over 19 billion dollars in benefits (Nelson, 1988). In terms of benefits paid, workers' compensation is larger than the Social Security Disability Income program, which paid approximately 18 billion dollars in benefits in 1984, and the several state and federal unemployment insurance programs, which paid about 15 billion in benefits that same year (Nelson, 1988).

*The writing of this paper was funded by the National Council of Compensation Insurers. For their helpful comments, I wish to thank John F. Burton, Jr., Ronald Ehrenberg, Stewart Schwab, an anonymous reviewer, and the participants at the NCCI's Eighth Annual Seminar on Economic Issues in Workers' Compensation. Special thanks to William P. Currington who provided the data used in my analyses. This data set was created at the University of Arkansas for research supported by the National Science Foundation (Grant No. SES-8107426).

69

70

WORKERS' COMPENSATION INSURANCE

Workers' compensation programs provide several benefits to injured workers. First, claimants are reimbursed for all medical expenses related to their injuries. Additionally, a variety of cash benefits are paid, including survivor benefits to members of the claimant's immediate family if the worker suffers a fatal injury as well as indemnity benefits for claimants who are unable to work as a result of their injuries. Four types of indemnity benefits are provided, which may be categorized according to the permanency and the severity of the disability. These are: 1) temporary total, 2) temporary partial, 3) permanent total, and 4) permanent partial disability benefits. Generally, any worker who is disabled as the result of an injury will receive temporary partial or temporary total disability benefits. ~ Temporary benefits do not normally begin immediately following the worker's injury; there is typically a waiting period of three to seven days before the injured worker may begin to collect benefits. 2 Once begun, temporary benefits are paid until the claimant reaches the point of maximum medical improvement. Maximum medical improvement is that point where further medical treatment is not expected to result in any substantial improvement in the claimant's physical condition. Maximum medical improvement occurs sometime after the date of injury, often as much as two or three years afterwards. In most cases the worker is no longer disabled once he or she reaches this point. If so, temporary benefits cease and the claimant returns to work. For example, the worker may suffer an arm fracture that prevents him (or her) from working until the fracture heals. Once healed, however, the worker suffers no permanent consequence and is able to return to work without loss of earning capacity. This person would receive temporary benefits for the period between the date of injury and the time that his or her fracture heals but would not receive any permanent disability benefits. A worker who continues, however, to suffer some permanent consequence as the result of his or her injury is eligible for permanent disability benefits. For example, a worker whose arm is amputated as the result of an industrial accident suffers a permanent consequence and i's entitled to permanent compensation benefits. Permanent disability benefits may be total or partial depending on the degree of disability. While a variety of methods have been used to provide permanent partial disability benefits to injured workers, two broad categories may be identified: an ex ante payment system in which indemnity benefits are fixed in advance of any wage-loss due to permanent disability and wageloss (or ex post) benefits, which are based on the claimant's current, actual loss of earnings due to disability. Ex ante compensation: By this method, the compensation system

THE TRANSITION F R O M T E M P O R A R Y TO P E R M A N E N T DISABILITY

71

determines the amount of benefits to which the claimant is entitled at the time of maximum medical improvement. Various proxies are used as substitutes for work disability. For example, workers who suffer a 100 percent loss (or loss of use) of a hand in New York are entitled to thirtytwo weeks of permanent partial benefits. 3 These benefits are paid to all workers suffering such an impairment regardless of the actual disability that the worker may experience. Most states have adopted this approach, in some form, to the compensation of permanent partial disabilities. Wage-loss benefits: The wage-loss approach endeavors to compensate claimants for actual wage loss experienced after the date of maximum medical improvement. Following this date, the system compares the claimant's preinjury wage income with his or her post-injury income to determine wage loss. Benefit levels are set in relation to the amount of wage loss experienced so that if there is no wage loss, no benefits will be received. The claimant will collect compensation benefits for as long as he or she experiences wage loss, up until the age of normal retirement. Should the worker's wage-loss experience change, then the level of benefits is adjusted. If, for example, the claimant should recover to the point where he or she is earning wages that are equivalent to preinjury real wages (thus protecting the worker against inflation), then that worker will cease receiving benefits. 4 Data presented in table 4-1 reveal that a substantial proportion of monies paid as indemnity benefits is paid as compensation for a permanent partial disability. These data indicate that over 65 percent of cash benefits paid in 1982 went to claimants with permanent partial disabilities. (As a proportion of the total number of compensation cases, permanent partial claims were not nearly as significant, representing only 23.8 percent of the total). In contrast, permanent total benefits accounted for only 3.5 percent of the indemnity benefits paid in 1982. A further break-down between major and minor partial disabilities 5 indicates that a little over a quarter of these permanent partial disability cases (or 7 percent of the total number of cases paying indemnity benefits) accounted for almost 43 percent of the total cost. Furthermore, it would appear that the cost of permanent partial claims, and in particular, the cost of major permanent partial disabilities, is increasing at a proportionately faster rate than the cost of indemnity payments generally. In 1958 major permanent partial disability benefits accounted for less than 25 percent of all cash benefits. As noted, this share had grown to almost 43 percent by 1982. Despite the growing significance of permanent partial disabilities as an element of workers' compensation costs, the compensation of permanent partial disabilities remains problematic. Permanent partial compensation

72

WORKERS' COMPENSATIONINSURANCE

Table 4-1. Number and Cost of Permanent Partial Disability Cases in the United States, Selected Years, 1958-1982.

Year

All Cases with Indemnity Benefits

1958 1968 1973 1978 1982

716 896 1,031 1,392 1,260

1958 1968 1973 1978 1982

591 1,287 1,786 4,042 4,861

Major Permanent Partial Number or Cost

Percent of Total

Minor Permanent Partial Number or Cost

Number of Claims 21 2.9% 39 4.4% 42 4.1% 72 5.1% 83 6.6%

Percent of Total

All Permanent Partial Number or Cost

Percent of Total

(Thousands)

167 218 213 260 218

23.3% 24.3% 20.7% 18.7% 17.3%

Cost of Claims ($ Millions) 144 24.4% 209 35.4% 391 30.4% 448 34.8% 559 31.3% 560 31.4% 1,536 38.0% 1,043 25.8% 2,079 42.8% 1,078 22.2%

188 257 255 332 300

26.2% 28.7% 24.8% 23.8% 23.8%

354 839 1,119 2,579 3,157

59.9% 65.2% 62.6% 63.8% 65.0%

Source: National Council on Compensation Insurance, "Countrywide Workers' Compensation Experience IncludingCertain CompetitiveState Funds--lst Report Basis."

was described by the National Commission (1972, 67) as "the most controversial and complex aspect of workmen's compensation." Although no specific recommendations were made, the Commission realized that a "bold and substantial reform of permanent partial benefits is necessary. Indeed there is no more pressing and fundamental issue confronting workmen's compensation." Despite the urgency of this call, reform efforts that followed the publication of the Commission's report centered on improvements in coverage and benefit levels, leaving untouched the problems of permanent partial disability (Berkowitz and Burton, 1987). One impact of these improvements, however, was a dramatic increase in the overall cost of workers' compensation. This cost increase elicited legislative concern about the permanent partial problem, since permanent partial disabilities account for such a large share of overall costs (Berkowitz and Burton, 1987). These concerns have led to revision or attempted revision of the statutory law in a number of jurisdictions, including Florida, which has revamped its system of permanent partial benefits in wholesale fashion.

THE TRANSITION FROM TEMPORARY TO PERMANENT DISABILITY

73

Considerable academic interest has been devoted to the study of work injury rates in recent years. On the other hand, little attention has been given to the factors that influence the injured worker's transition from temporary to permanent disability. The purpose of this chapter is to help remedy this deficiency. The problem is discussed in four parts. The first section presents the pertinent theory and empirical literature. Next, relevant aspects of the New York workers' compensation program are described. Hypotheses based on theory from the first section are presented in the third part of the chapter, as well as a description of the data set and methodology used to test those hypotheses. Finally, the results of these analyses are presented along with conclusions and their implications for public policy.

Theory and Literature The economic model predicting the injured worker's transition from temporary to permanent disability is similar to the more familiar and well-studied model that predicts the likelihood of a work injury. 6 Both processes (the transition from temporary to permanent disability and the occupational injury) are simultaneously determined by the behaviors of two actors. In the case of occupational injuries, the two actors are the worker and the employer. With respect to the compensation claimant's transition from temporary to permanent disability, the parties are the claimant and the insurance carrier liable for the claim. In occupational injury models, the worker compares the expected utility of the injured state with the expected utility of the noninjured state. As the relative expected utility of the injured state increases, the worker will be more likely to engage in behaviors that will result in the worker being classified as "injured." As the injured-state utility increases, workers may be expected to report injuries that previously would have gone unreported or fraudulently report injuries where no injury occurred. Additionally, as injury utility increases, workers may be expected to more frequently engage in risky activities on the job. Conversely, when considering the adoption of "safe" practices or technology the employer will assess the relative cost of these practices and technology against the cost of the work injuries that would result if those practices are not adopted. The transition from temporary to permanent disability is similarly determined. In this case, the workers' compensation claimant compares his or her expected utility in the permanently disabled state with the expected utility of the nondisabled state. The greater the utility of the

74

WORKERS' COMPENSATIONINSURANCE

disability, the more likely the claimant will be to engage in behaviors that positively affect the probability of transition. These behaviors include the pursuit of the claim through hearings and appeals, etc. Conversely, the higher the costs of disability, the greater will be the insurer's efforts to prevent the claimant's transition to permanent disability. To better understand this phenomenon we may construct a model of the processes by which the claimant decides whether or not to pursue a permanent disability claim. First, we note that the claimant's income if he or she is classified as permanently disabled is: YD = E(B) + WD -- C(WD, Z).

(la)

If the claimant is not categorized as permanently disabled, his or her income is: (lb)

YN = WN

where E ( B ) is the expected value of permanent partial compensation benefits, WD is the claimant's post-injury wage income if he or she is categorized as permanently disabled; WN is the claimant's post-injury wage income if he or she is not categorized as disabled; and C(WD, Z ) is the cost of filing a permanent disability claim. We may further assume that workers who claim to be disabled may expect to work less than claimants who do not, so that WD = rWN, where 0 < r < 1. The designation of benefits as an expected value indicates that the level of permanent disability benefits is uncertain. Filing costs include two components: 1) out-of-pocket expenses such as legal fees, and 2) the opportunity costs of time spent in pursing the claim. Out-of-pocket expenses are designated as Z, while opportunity costs are a function of the claimant's post-injury wage income. Note that OC/OWD > O, OC/OZ > O. The claimant's willingness to pursue a permanent disability claim depends on the utility of the disabled state relative to the nondisabled state. Let I be an index function representing this willingness, such that: I = U(YD) - U(YN)

(2)

Taking a Taylor expansion of the claimant's utility in the disabled state around the utility of the nondisabled state we have: U(YD) = U(YN) + U ' ( Y N ) ( Y D -- YN) +

U"(YN) t v 2 ~" D -- YN) 2

(3)

where higher order terms have been suppressed. Substituting equation (3) into equation (2) and dividing by U', we have:

THE TRANSITION F R O M T E M P O R A R Y TO P E R M A N E N T DISABILITY

I

U"

U-'-7 = (YD -- YN) + ~ - ~ 7 ( Y D -- Y N ) 2

75 (4)

where U' and U" are evaluated at YN. U " / U ' is the Pratt-Arrow measure of absolute risk aversion. This measure is positive if the claimant is risk seeking, zero if the claimant is risk neutral, and negative if the claimant is risk averse. Equation (4) indicates that the claimant's propensity to engage in disability-seeking behaviors is a function of both the relative income in the disabled and nondisabled states and the claimant's attitudes toward risk. Risk in this model arises from the uncertainty of benefits in the disabled state. Compensation claimants do not know, a priori, whether they will be successful in establishing a claim or, if successful, the level of benefits that they will receive. If claimants are risk averse, then they will be willing to pay a premium in order to avoid this risk; risk-averse claimants will be willing to accept a certain benefit payment that is less than E ( B ) + (r - 1)Wu - C ( W D , Z ) . It may be demonstrated that this risk premium is equal to (1/2)Va 2, where a 2 is the variance of compensation benefits and V is the Pratt-Arrow measure of absolute risk aversion (Deaton and Muellbauer, 1980). If we assume that at the margin E ( Y D -Y N ) ---- E ( B ) + (r - 1 ) W N -- C ( d W N , Z ) = O, then (YD - Y N ) 2 = 6r2. To the extent that the probability of a permanent disability classification depends on the claimant's claim-seeking behavior, which is, in turn, dependent on the relative utility of the disabled and nondisabled states, we may express the probability of a permanent disability (p) as a function of the variables on the left-hand side of equation (6). Substituting (1/2)Va 2 for ( 1 / 2 ) ( U " / U ' ) ( Y D -- Yu) 2, we have: p = g [ E ( B ) + (r - 1 ) W N -- C ( d W N , Z ) + (1/2)Vo-2]

(7)

Equation (7) indicates that the probability of a permanent disability is positively related to the level of compensation benefits, but negatively related to the magnitude of claim-filing costs, as well as the variability of permanent disability benefits--assuming the claimant is risk averse. As mentioned, the factors influencing occupational injury rates are well researched. In particular, empirical research has examined the effect of workers' compensation benefit levels on the likelihood of a work injury. Theoretically, the impact of this variable is indeterminate since a change in benefit levels will have opposing effects on worker and employer behavior. Higher compensation benefits will increase the claimant's perceived utility of the injured state. The claimant will be more likely to report injuries and to engage in risky behaviors at work. By

76

WORKERS' COMPENSATION INSURANCE

itself, this should cause the occupational injury rate or the likelihood of an injury to increase. On the other hand, higher benefits will raise the employer's cost of work injuries relative to the cost of "safe" technology and work practices. The employer will be more likely to adopt safer technology and practices, which should, ceteris paribus, cause the injury rate to decline. Theory does not yield an unambiguous prediction concerning the impact of a change in compensation benefits on the likelihood of an occupational injury. However, estimation of reduced form injury rate equations that include benefit levels as an independent variable has, in most cases, revealed that higher benefits are associated with higher injury rates (Chelius, 1973, 1974, 1977, 1982, and 1983; Worrall and Appel, 1982; Butler and Worrall, 1983; Butler, 1983; Ruser, 1985; and Krueger, 1988). 7 It is important to note that with the exception of the Krueger (1988) study, which used logit regressions to predict the probability of an injury, these studies employ data aggregated by industry rather than individual claims data. Use of such micro-level data offers several advantages over aggregate-level data. As Krueger (1988) has noted, individual claims data allow the researcher to explore the impact of claimant characteristics, such as age and sex, and to more exactly measure the effect of benefit level variation on the behavior of injured workers. To date, there is little empirical research examining the transition from temporary to permanent disability. Recently, Worrall, Durbin, Appel, and Butler (1987) analyzed a longitudinal sample of individual claims data from Massachusetts and Illinois, finding that increased compensation benefits are positively related to the probability that a claim will close as a permanent partial disability. Additionally, a few studies have investigated the influence of benefit levels on the severity of a claimant's injury. For example, Chelius (1982) found a significant inverse relationship between benefit levels and the number of lost-time workdays per injury. He explained these results as indicating that employers are primarily concerned with major injuries of long duration and respond to higher benefits by acting to reduce the duration of the claimant's injury. A study with similar data, yielded the opposite relationship; the number of lost workdays was directly related to benefit levels (Chelius, 1983). Additionally, Worrall and Appel (1982) found higher benefits to be associated with an higher costs and frequency of indemnity benefit claims relative to claims where only medical benefits were paid, indicating that claim severity is directly related to benefit levels. Furthermore, two papers by Richard Butler and John Worrall (Worrall and Butler, 1985; and Butler and Worrall, 1985) found that the duration of temporary total benefits is

THE TRANSITION FROM TEMPORARY TO PERMANENT DISABILITY

77

positively related to the level of temporary total benefits. In summary, these studies (with the exception of Chelius (1983)) suggest that both the severity of a workers' compensation claimant's injury, in general, and the probability of a permanent disability, in particular, are positively related to the size of compensation benefits. This suggests that the claimant's response to increased benefits dominates the employer's or insurer's response.

The New York System As discussed, previous research indicates that the probability that a employee will suffer an occupational injury is positively related to the level of worker's compensation benefits. Furthermore, a review of the literature indicates that higher benefit levels may also be related to more severe injuries. Two fundamentally different approaches to the compensation of permanent partial disability are used in New York, depending on the body part injured. In this section the two approaches are described and data are presented concerning their relative cost and frequency in New York. Finally, the economically relevant differences between these two approaches are discussed.

Scheduled Injuries For injuries involving the total or partial physical loss, or loss of use of an extremity (such as a toe, hand, leg, etc.), loss of sight in one eye or hearing in one or both ears, benefits are established by statute and are called scheduled injuries. Compensation for permanent partial scheduled injuries is determined ex ante; a worker who suffers a scheduled injury is automatically entitled to a specific duration of benefits, depending upon the body part injured and the severity of the injury. Thus, a claimant with a 10 percent permanent disability of the arm will receive 31.2 weeks of permanent partial disability benefits, since the maximum duration for the loss of an arm according to the schedule is 312 weeks. The size of the weekly benefit payment is 66.67 percent of the worker's average weekly earnings before the date of injury (subject to minimum and maximum amounts). Average weekly earnings are computed on the basis of all work income made during the previous year, including overtime and earnings from a second job. s

78

WORKERS' COMPENSATION INSURANCE

Nonscheduled Injuries Injuries other than scheduled injuries (termed "nonscheduled" injuries) are compensated using what may be called a "wage-loss" approach. The weekly benefit amount is computed as 66.67 percent of the difference between the worker's preinjury earnings and the worker's actual earnings after his or her medical condition has stabilized (the maximum medical improvement date), subject to a minimum and a maximum amount. For those workers with no actual earnings on the maximum medical improvement date, a percentage disability (loss of earning capacity) is determined (for example, 25 percent); the weekly benefit is 66.67 percent of the loss of earning capacity times preinjury earnings (subject to minimum and maximum amounts). For nonscheduled permanent partial disabilities, the duration of benefits is not limited by statute. Rather, benefits continue as long as the worker is experiencing lost earnings Table 4 - 2 . Number of Cases and Costs of Compensation for Scheduled and Nonscheduled Permanent Partial Disability, as a Percentage of All Cases, Selected Years, 1960-1983. Number Year 1960 1965 1970 1971 1972 1973 1974 1975 1976 1977 1978 1979 1980 1981 1982 1983

of C a s e s 34.1 38.9 37.9 37.7 37.6 36.3 37.2 37.3 35.8 35.4 36.2 36.4 35.6 35.4 36.4 34.8

Amount of

Compensation 35.0 34.8 35.7 37.6 34.2 32.7 32.0 30.0 26.8 24.9 25.8 23.9 24.0 24.7 24.2 23.0

Number

of C a s e s 3.0 2.6 2.6 2.4 3.0 3.0 2.8 3.1 3.6 3.9 3.7 3.9 3.8 3.8 4.2 4.4

Amount of Compensation 32.2 34.8 34.5 33.4 38.7 40.9 41.5 44.7 51.7 54.3 54.3 54.6 52.2 50.9 52.4 52.1

Source: Compensated Cases Closed, State of New York Workers' Compensation Board, 1960, 1965-1983, inclusive. The figures for 1983 are preliminary and subject to change.

THE TRANSITION FROM T E M P O R A R Y TO P E R M A N E N T DISABILITY

79

due to the injury. Benefit payments, therefore, may extend over the remainder of the claimant's life.

The New York Experience Table 4 - 2 provides currently available information on the numbers and costs of scheduled and nonscheduled permanent partial disability cases for New York state during the 1960 to 1983 period. 9 The numbers refer to cases closed in the particular years and the costs refer to the past and anticipated future benefit payments for these closed cases. 1° Scheduled permanent partial disabilities have consistently accounted for about 35 percent to 38 percent of all cases throughout the period, while nonscheduled permanent partial disabilities have accounted for only 2.5 percent to 4.5 percent of all cases. (The bulk of the cases in New York involve temporary disability benefits only.) Scheduled permanent partial disability benefits accounted from 25 percent to 35 percent of all cash benefits. In contrast, the share of cash benefits accounted for by nonscheduled permanent partial disability benefits increased from approximately 35 percent (from 1960 to 1971) to over half of all expenditures in the 1976 to 1983 period--a striking result given that such cases represent only about four percent of all cases.

Scheduled Versus Nonscheduled Compensation These two methods of compensating permanent partial disabilities offer substantially different incentives to the workers' compensation claimant. As noted, weekly benefits are a function of the claimant's preinjury weekly wage. To the extent that post-injury wages are related to preinjury wages, the higher the claimant's weekly benefits, the greater his or her post-injury wages. Recall that the nonscheduled injury claimant will receive benefits only by forgoing post-injury wage income. We would expect to find, then, that the relationship between permanent partial benefits and the probability of observing a permanent disability is attenuated for nonscheduled injury claimants. Higher weekly benefits imply higher post injury wages; higher benefits imply greater claimant utility in both the disabled and nondisabled states. Since permanent partial disability compensation for scheduled injuries is not dependent on post-

80

WORKERS' COMPENSATION INSURANCE

injury income, higher benefits will increase the utility in the disabled state but not in the nondisabled state. To illustrate the economic differences between scheduled and nonscheduled permanent partial compensation, I assume that the claimant's post-injury wage is related to his or her preinjury wage, or WN = aWnds, where ds represents the duration of the scheduled benefits, a (0 < a < 1) is an inverse measure of the severity of the claimant's injury, and W/-/is the claimant's preinjury wage. For scheduled-injury claimants, compensation benefits are a function of the claimant's preinjury wage such that B = bWH, where b is the proportion of wages replaced by compensation benefits and 0 < b < 1. Because compensation benefits are subject to minimums and maximums, b is a nonlinear function of the worker's preinjury wage or b = b(WH) where Ob/OWn < O. 11 Substituting into equation (7) and taking the first derivative, we have:

Op [ Ob OC ] OVI~ = bds + -~--~HWI-lds + (r -- 1)a - ~--~ads Op Og

(g)

and

~Op= WHds~-[g

(9)

While equation (9) indicates that an increase in b will result in an increased likelihood that the claimant will pursue a permanent disability classification, equation (8) does not yield an unambigous prediction concerning the impact of a change in preinjury wage income on permanent disability probability, since the terms (Ob/OWn)WI-I, (r - 1)a, and (OC/OWn)ads enter the equation negatively, while b enters positively. For nonscheduled claimants in New York, the benefit formula is B = b(Wn - Wo) = b(1 - a)WHdu, where du is the duration of nonscheduled benefits. ~2 Substituting into equation (7) and taking first derivatives we have: =

OWn

[{bdN+~W--~HW,dN}

_

a) + ( r - 1)a

OC .lOp

-O-ff-~Haaj-~g(10)

and

OPob= (1 - a)WrlduoO-~pg

(11)

As was the case for scheduled injuries, the higher the replacement rate, the more likely the claimant will be to develop a permanent disability.

THE TRANSITION FROM T E M P O R A R Y TO P E R M A N E N T DISABILITY

81

Also, the effect of a change in the claimant's preinjury wage is ambiguous, as it was for scheduled injuries. If we assume that dN ---- ds 13 and compare equations (8) and (10), we may conclude that the impact of preinjury wages on the transition from temporary to permanent disability for scheduled injuries is greater than its impact on the transition to permanent disability for nonscheduled injuries. Similarly, the effect of variation of the replacement rate on transition probability is also greater for scheduled than for nonscheduled injuries, as an examination of equations (9) and (11) indicates.

Data Sources and Methodology Data Sources and Sampling Procedures Empirical investigation of the hypotheses postulated in the previous section was implemented through the estimation of reduced form equations predicting the probability that a compensation claimant would develop a permanent disability. Data were provided by the New York Workers' Compensation Board's Closed Case Tapes. These tapes are created each year by the New York Workers' Compensation Board and contain information for each worker's compensation claim closed during the previous calendar year. A stratified random sampling procedure was used to construct the data set used in this study. Observations on 35,902 worker's compensation claims from the New York Closed Case tapes were selected from a universe consisting of a subset of all claims that resulted from injuries occurring during the period from 1971 to 1977 and that were closed prior to 1983.14 Claims in this subset were restricted to those in which the claimant experienced a permanent partial or a temporary disability. Excluded from the sampling universe were injuries that resulted in the claimant's death, a permanent total disability, or a facial disfigurement. Stratification was based on the extent of the claimant's injury. 15 An attempt was made to: 1) draw an equal number of permanent partial and temporary disability claimants and 2) draw an equal number of nonscheduled and scheduled permanent partial claimants. In other words, permanent disabilities and, in particular, nonscheduled permanent disabilities were oversampled, while temporary disabilities were undersampled. 16

82

WORKERS' COMPENSATION INSURANCE

Estimating Equation The dependent variable used in these analyses is a 0-1 variable indicating whether the claim was categorized as a permanent disability or as a temporary disability on the date that the claim was closed (designated as PERM in the following tables, 1 -- permanent disability). The expected utility model presented earlier hypothesizes that three factors influence the claimant's propensity to pursue a permanent disability claim: 1) the disability benefits provided by workers' compensation programs; 2) claim-filing costs incurred by the claimant, including out-ofpocket expenses and opportunity costs; and 3) variability in permanent disability compensation benefits and the claimant's aversion to risk. The discussion around equations (8) through (11) indicated that permanent compensation benefits in New York are determined by three factors: 1) the claimant's preinjury wage (Wn), 2) the replacement rate (b), and 3) the severity of the claimant's injury (1 - a). Preinjury wages are measured by W A G E . 17 Two measures were employed to capture variation in the replacement rate: TFBEN, the weekly temporary total benefit payment as determined by the claimant's preinjury wage, and PPBEN, the weekly permanent partial benefit payment. TTBEN and PPBEN are expected to be positively related to the probability of permanent disability, although a stronger relationship is expected for scheduled than for nonscheduled injuries for both variables. The permanent disability model assumes that increased injury severity increases nonscheduled benefit levels by reducing the claimant's postinjury wage income, is The model also predicts that injury severity will be negatively related to claimant opportunity costs for both types of injuries. Additionally, we may expect injury severity will affect the objective criteria determining permanent disability, independent of its effect on economic incentives; more severely injured claimants are likely to have medical symptoms indicative of a permanent disability than are less seriously injured claimants. Two variables are included in the equation to capture injury severity effects. The principle measure is LOST, the number of weeks the claimant collected temporary disability benefits. AGE, the claimant's age at the time of injury, may also capture some of the effects of injury severity since older workers recuperate more slowly from injuries, ceteris paribus. Both LOST and A G E are expected to be positively related to the probability of permanent disability. Unfortunately, the data set did not contain an adequate measure of the claimant's out-of-pocket claim-filing costs; however, UE, the average annual unemployment rate for the region and year in which the claim was

THE TRANSITION FROM TEMPORARY TO PERMANENT DISABILITY

83

closed, 19 served as a proxy for claimant opportunity costs. 2° The lower the unemployment rate, the greater the costs of pursuing a claim and the lower the probability of permanent disability. Additionally, since nonscheduled benefits are dependent, in part, on the post-injury wage, UE is hypothesized to have a greater impact on the probability of permanent disability for nonscheduled than for scheduled injuries. As noted, if the claimant is risk averse, the probability of permanent disability will be negatively related to variability in permanent disability compensation benefits. Since compensation benefits are a function of the 1) the claimant's preinjury weekly wage and 2) the severity of the claimant's injury as rated by the Workers' Compensation Board, claimant uncertainty with respect to benefits may arise from either or both of these two sources. For scheduled injury claimants, the injury severity influences the duration of benefits, while injury severity determines wage-loss and, hence, the size of the weekly benefit payment for nonscheduled benefits. Since there is usually little question concerning the claimant's preinjury wage rate, uncertainty arises from the claimant's lack of prior knowledge concerning injury severity or, at least, the compensation board's evaluation of injury severity and their concomitant determination of benefits. This uncertainty is measured by A W D V A R , the variance in permanent compensation benefits by type of injury and body part. 21 A W D V A R is expected to be negatively related to permanent disability probability. Finally, the estimating equation included a number of control variables: dummies designating the claimant's gender (SEX, Female = 1) as well as the claimant's occupation (OCC) and industry (IND) at the time of injury. Finally, INJTYPE identified the claim as to whether or not it is typically classified by the workers' compensation board as nonscheduled or scheduled (INJTYPE, 1 = nonscheduled injury). Unweighted sample means as well as means weighted to reflect characteristics of the population and disaggregated by type of injury, are presented in tables 4-3 to 4-5. The estimating equation may be written: {~ Yi

=

if y ' i > 0 otherwise

where

Y*i = fllSEXi + fl2AGEi

flaPPBENi + flsTTBENi + fl6WAGEi + fl7AWDVARi + flaUEi + fl3LOSTi + 18

+ fl9INJTYPEi + ~ j=lO

23

fljlNDij + ~ k=19

flkOfCik '1- ~,i

Table 4-3. Weighted and Unweighted Sample Deviations*uAII Injuries. Variables PERM SEX AGE LOST PPBEN TrBEN WAGE UE AWDVAR N PERM SEX AGE LOST PPBEN TTBEN WAGE UE AWDVAR

All Claimants

Permanently Disabled

Panel A: Unweighted 0.53477 -(0.49880) 0.23188 0.23345 (0.42204) (0.42304) 42.19 45.62 (14.90) (14.18) 16.32 25.67 (31.89) (40.66) 81.05 82.09 (15.31) (14.51) 95.05 96.86 (24.36) (23.72) 181.61 190.72 (84.86) (89.78) 7.9702 8.0975 (1.8154) (1.7284) 62,349 62,526 (51.767) (54.383) 35,092 18,766 0.23046 (0.42113) 0.22510 (0.41765) 39.16 (14.87) 6.89 (13.61) 80.06 (16.00) 93.31 (24.85) 173.67 (79.86) 7.8475 (1.8907) 52,717 (49,421)

Panel B: Weighted --

*Standard deviations within parentheses.

0.20854 (0.40626) 42.18 (14.88) 11.27 (23.39) 80.70 (15.64) 94.47 (24.60) 182.15 (86.67) 7.9261 (1.8574) 21,232 (37,784)

Means and Standard Non-Permanently Disabled -0.23006 (0.42089) 38.25 (14.74) 5.58 (8.32) 79.87 (16.10) 92.97 (24.92) 171.13 (77.53) 7.8239 (1.9000) 62,147 (48.587) 16,326 -0.23006 (0.42087) 38.25 (14.74) 5.58 (8.32) 79.87 (16.10) 92.97 (24.92) 171.13 (77.52) 7.8239 (1.9000) 62,147 (48,586)

Table 4-4. Weighted and Unweighted Sample Deviations*--Nonscheduled Injuries. Variables

PERM SEX AGE LOST PPBEN TTBEN WAGE UE AWDVAR N PERM SEX AGE LOST PPBEN TTBEN WAGE UE AWDVAR

All Claimants

Permanently Disabled

Panel A: Unweighted 0.51618 -(0.49975) 0.23468 0.24771 (0.42381) (0.43170) 44.26 48.53 (14.36) (12.80) 21.85 36.20 (36.87) (46.12) 82.43 83.45 (14.08) (13.14) 97.42 99.20 (23.24) (22.52) 189.23 199.18 (85.60) (91.79) 8.0449 8.2389 (1.7656) (1.6098) 100,400 105,126 (31,049) (31,042) 20,483 10,573 0.08824 (0.28364) 0.22290 (0.41619) 40.39 (14.63) 8.32 (15.82) 81.44 (14.88) 95.72 (23.81) 180.06 (78.52) 7.8652 (1.8843) 94,406 (31,850)

Panel B: Weighted --

* Standard deviations within parentheses.

0.24472 (0.42992) 47.47 (13.74) 26.73 (39.61) 82.56 (14.09) 97.81 (23.25) 194.93 (90.84) 8.1457 (1.7311) 84;577 (43,960)

Means and Standard Non-permanently Disabled

-0.22079 (0.41480) 39.71 (14.53) 6.54 (9.31) 81.33 (14.95) 95.52 (23.85) 178.62 (77.07) 7.8380 (1.8964) 95,357 (30,255) 9,910 -0.22079 (0.41478) 39.71 (14.53) 6.54 (9.31) 81.33 (14.95) 95.52 (23.85) 178.62 (77.07) 7.8380 (1.8963) 95,357 (30,254)

Table 4-5. Weighted and Unweighted Sample Means and Standard Deviations*--Scheduled Injuries. Variables PERM SEX AGE LOST PPB EN T-rBEN WAGE UE AWDVAR N PERM SEX AGE LOST PPBEN T/BEN WAGE UE AWDVAR

All Claimants

Permanently Disabled

Panel A: Unweighted 0.56082 -(0.49630) 0.22794 0.21506 (0.41952) (0.41089) 39.29 41.86 (15.17) (14.97) 8.57 12.07 (20.83) (26.74) 79.13 80.33 (16.69) (15.94) 91.72 93.84 (25.47) (24.86) 170.92 179.81 (82.64) (85.90) 7.8654 7.9150 (1.8780) (1.8547) 8,900 7,551 (14,464) (12,888) 14,609 8,193 0.37987 (0.48535) 0.22742 (0.41916) 37.86 (15.00) 5.39 (10.60) 78.60 (16.98) 90.78 (25.66) 166.96 (80.71 ) 7.8289 (1.8973) 8,923 (14,105)

Panel B: Weighted --

* Standard deviations within parentheses.

0.19971 (0.39978) 40.89 (14.86) 7.5017 (15.01) 80.24 (15.97) 93.65 (24.85) 179.04 (85.34) 7.8725 (1.8831) 5,775 (9,294)

Non-Permanently Disabled -0.24439 (0.42976) 36.00 (14.78) 4.09 (6.21) 77.60 (17.49) 89.02 (25.99) 159.57 (76.81 ) 7.8021 (1.9056) 10,851 (16,067) 6,416 -0.24439 (0.42972) 36.00 (14.78) 4.09 (6.21) 77.60 (17.49) 89.02 (25.99) 159.57 (76.80) 7.8021 (1.9055) 10,851 (16,066)

THE TRANSITION FROM TEMPORARY TO PERMANENT DISABILITY

87

where y, is an indicator variable that is equal to 1 if the claim was closed as a permanent disability and equal to 0 if the claim was closed as a temporary disability and ei is a random error term.

Statistical Methodology Logit regressions were used to estimate the impact of these independent variables on the probability that the claimant will develop a permanent disability. Since the data were a stratified random sample of the universe of claims and since the stratification criteria were related to the dependent choice variable, the data were weighted to avoid choice-based sampling bias problems (Manski and Lerman, 1977).

Results Table 4 - 6 presents results from logit equations predicting whether New York workers' compensation claimants will develop a permanent partial disability. Coefficients presented in these equations represent partial derivatives of the probability that the claimant will become permanently disabled given a unit change in the independent variable. 22 With the exception of A W D V A R , these coefficients have been increased by a factor of 100 so that they measure the percentage change in the probability of a permanent disability associated with a unit change in the independent variable. The coefficient for A W D V A R has been increased by a factor of 100,000. Seven different specifications are presented in this table, designated as Roman numerals I to VII. The results from table 6 are generally consistent with the predictions made in the previous section. As predicted, the greater the severity of the claimant's injury, as indicated by LOST, the greater the probability that the claimant will develop a permanent partial disability. Similarly, the likelihood of a permanent disability is positively related to the claimant's age (AGE). As indicated earlier, injury severity is expected to be positively related to the claimant's age. Table 4 - 6 also indicates that claimants exercise some discretion with respect to the transition from temporary to permanent disability. For example, these results show that an increase in the unemployment rate (UE) is associated with an increased probability that the claimant will develop a permanent disability. The more pessimistic the claimant's labour market alternatives, as indicated by higher regional unemployment

88

WORKERS' COMPENSATION INSURANCE

Table 4-6. Legit Coefficients Predicting Transition Permanent Disability,* All Injuries (N = 35,092). Variables SEX

AGE LOST PPBEN T-I'BEN WAGE UE AWDVAR INJTY PE CHI-SQ Variables SEX

I - 1.08251 (2.04) 0.32404 (24.20) 0.95018 (39.48) 0.07842 (2.23) -0.06684 (2.49) 0.03019 (7.34) 0.39667 (3.80) -0.36417 (38.45) - 8.11126 (10.49) 11,673

H - 1.51792 (2.64) 0.36457 (25.19) 1.02978 (39.44) 0.01430 (0.39) 0.41867 (1.70) --

0.42013 (3.71) -0.39507 (38.42) - 8.81043 (10.49) 11,618

III - 1.08032 (1.88) 0.35018 (24.11) 1.03146 (39.48) 0.00065 (0.04) --

0.02686 (7.13) 0.40553 (3.60) -0.39572 (38.48) - 8.80593 (10.48) 11,666

TTBEN

V - 1.64365 (2.89) 0.36772 (25.61) 1.02960 (39.43) 0.07213 (5.07) --

WAGE

--

0.05069 (5.34) --

0.43879 (3.89) -0.39472 (38.40) -8.81127 (10.49) 11,615

0.42074 (3.71) -0.39509 (38.42) -8.81145 (10.49) 11,618

AGE LOST PPBEN

from Temporary to IV - 1.14505 (1.99) 0.35051 (24.13) 1.03181 (39.49) --

-0.01463 (1.10) 0.03027 (7.01) 0.43320 (3.83) -0.39551 (38.45) - 8.81192 (10.49) 11,668

VI - 1.50804 (2.63) 0.36416 (25.23) 1.02984 (39.44) --

VII - 1.08200 (1.89) 0.35018 (24.11) 1.03147 (39.48) --

-0.02694 0.40636 (3.68) -0.39571 (38.49) -8.08615 (10.48) 11,667

(8.75) UE AWDVAR INJTYPE

CHI-SQ

*Absolute t-ratioswithin parentheses.

T H E TRANSITION F R O M T E M P O R A R Y TO P E R M A N E N T DISABILITY

89

rates, the more likely it is that the claimant will develop a permanent disability. Additionally, these results support the hypotheses that claimants are risk averse and that the greater the uncertainty regarding the size of the permanent disability award (as measured by the proxy AWDVAR), the less likely the claimant is to develop a permanent disability. This finding further implies that there are costs to the claimant associated with attempting to make the transition from temporary to permanent status. Regression results with respect to the size of the weekly permanent disability benefit payment (PPBEN), the temporary disability benefit (TFBEN), and the claimant's preinjury weekly wage (WAGE) are less satisfying. While W A G E bears a consistent, significant, and positive relationship with the probability of permanent disability in all equations, the coefficients for PPBEN and TTBEN vary depending on the model specification. PPBEN is positively related to the probability of permanent disability in every equation, but this relationship is only statistically significant in equations I and V. The coefficient for TTBEN is negative in equations that include W A G E (I and IV), while it is positive in equations that do not include WAGE. The instability of the estimates for PPBEN and TTBEN may be attributed to the high degree of collinearity among these two variables and WAGE. Nevertheless, if we assume that the model that includes both the W A G E measure as well as the two benefit variables (equation I) is correctly specified, the result for PPBEN supports the theoretical model. As permanent partial benefits increase so do the chances that the claimant will be classified as permanently disabled, as hypothesized. The negative coefficient for TTBEN is more difficult to explain. It is possible that these results are attributable to the influence of temporary compensation benefits on the duration of temporary disability. As indicated earlier, prior research has found that higher temporary benefits induce longer spells of temporary disability (Butler and Worrall, 1985; and Worrall and Butler, 1985). We may assume that the longer the period between injury and case closing, the greater the improvement in the claimant's medical condition. It is possible that higher temporary benefits are extending temporary disability spells to the point where the claimant's medical condition has improved, such that the claimant will not be classified as permanently disabled. SEX is consistently negatively related to the probability of a permanent disability indicating that women are less likely to develop permanent disabilities than are men. One explanation for this result is that female claimants, who are generally in less hazardous industries and occupations than male claimants, suffer less serious injuries than their male counterparts. Interestingly claimant gender appears to be related to the wage and

90

WORKERS' COMPENSATION INSURANCE

benefit variables, as revealed by a comparison of the SEX coefficients across specifications. Female claimants may be expecting to receive lower compensation benefits than similarly situated males. Finally, the results for INJTYPE indicate that scheduled injuries are more likely to develop into permanent disabilities than are nonscheduled injuries. These results are consistent with the incidence of permanent disability by injury type. As may be seen from the 1983 data presented in table 4-2, scheduled permanent partial disabilities occur approximately eight times more frequently than do nonscheduled permanent partial disabilities. 23 Tables 4 - 7 and 4 - 8 present separate logit regressions for scheduled and nonscheduled injuries predicting the probability that a claim will be classified as permanently disabled. Nonscheduled injury regressions are presented in table 4-7, while scheduled injury regressions are displayed in table 4-8. As before, coefficients reported in this table are partial derivatives of the probability that the claim will be classified a permanent disability, given a unit change in the independent variable. Likelihood ratio tests comparing the models presented in tables 4-7 and 4 - 8 with the models presented in table 4 - 6 reject the hypothesis that the slope coefficients in the regression equations are identical for scheduled and nonscheduled injuries. A comparison of the nonscheduled and scheduled logit equations in tables 4 - 7 and 4 - 8 reveals a number of differences between the processes determining the transition from temporary to permanent disability for the two injury types. Significantly, the relative magnitudes of the coefficients in the scheduled and nonscheduled injury equations indicate that the nonscheduled-injury claimant's response to W A G E is attenuated when compared with the response exhibited by a claimant with a scheduled injury, as predicted by the theoretical model. Additionally, the unemployment rate has a greater impact on the probability that a nonscheduled injury will become a permanent disability than it does on the probability of a scheduled permanent disability. This is also as predicted by theory, given that the unemployment rate provides a measure of the claimant's post-injury wage. On the other hand, the results for PPBEN and T-I'BEN do not support the model. While not substantially different, PPBEN appears to have a greater effect on the probability of permanent disability for nonscheduled injuries than for scheduled injuries. Similarly, TTBEN appears to have a smaller effect on scheduled permanent partial disabilities than it does on nonscheduled permanent disabilities. Collinearity among these two variables and W A G E may be adversely affecting these estimates. The process determining a scheduled permanent disability differs from

THE TRANSITION FROM TEMPORARY TO PERMANENT DISABILITY

91

Table 4-7. kogit Coefficients Predicting Transition from Temporary Permanent Disability,* Nonscheduled Injuries (N = 20,483).

to

Variables

SEX AGE LOST PPBEN TTBEN WAGE UE AWDVAR CHI-SQ

I

H

III

3.17778 (4.76) 0.28537 (16.52) 0.62109 (31.96) 0.11359 (2.39) -0.08425

2.92136 (4.39) 0.29•63 (16.95) 0.62258 (31.99) 0.06788 (1.48) -0.01478

3.27716 (4.91) 0.28331 (16.42) 0.62116 (31.95) 0.00715 (0.34)

(2.50)

(o.5o)

0.01937 (4.29) 0.71174 (5.37) -0.17173 (25.52) 2,819

--

AGE LOST PPBEN TTBEN WAGE UE AWDVAR CHI-SQ

-

0.01389 (3.49) 0.68680 (5.19) - O.17210 (25.57) 2,813

0.71470 (5.38) -0.17224 (25.57) 2,801

Variables

SEX

-

/V 3.19629 (4.78) 0.28220 (16.41) 0.62149 (31.96)

-0.01190 (0.80) 0.01693 (3.84) 0.71687 (5.41) -0.17176 (25.51) 2,814

V

VI

VII

2.87351 (4.48) 0.28213 (16.99) 0.60411 (31.99) 0.04513 (2.62) --

2.95287 (4.44) 0.28908 (16.90) 0.62272 (31.99) --

3.25570 (4.90) 0.28286 (16.45) 0.62127 (31.96) --

0.02523 (2.24) --

--

-

-

0.68787 (5.36) -0.16721 (25.58) 2,801

* Absolute t-ratios within parentheses.

0.71791 (5.41) -0.17222 (25.56) 2,799

0.01462 (4.37) 0.69533 (5.35) -0.17201 (25.57) 2,813

92

WORKERS' COMPENSATION INSURANCE

Table 4-8. Logit Coefficients Predicting Transition Permanent Disability,* Scheduled Injuries (N = 14,609). Variables SEX AGE LOST PPBEN "I-TBEN WAGE UE AWDVAR CHI-SQ Variables SEX AGE L O ST PPBEN TTBEN WAGE UE AWDVAR CHI-SQ

1 - 3.87423 (4.21) 0.46656 (19.69) 1.48001 (24.97) 0.07329 (1.21) -0.06531 (1.37) 0.05260 (6.72) 0.22444 (1.22) - 1.41489 (32.81) 2,972

H - 4.36756 (4.74) 0.48678 (20.62) 1.46756 (24.82) -0.03684 (0.63) 0.11626 (2.93) -0.20142 (1.10) - 1.40724 (32.67) 2,925

V - 4.73179 (5.18) 0.49605 (21.18) 1.46481 (24.77) 0.12118 (5.39)

from Temporary

III - 3.78775 (4.12) 0.46521 (19.65) 1.47988 (24.98) 0.00007 (0.003) -0.04663 (7.23) 0.20193 (1.10) - 1.41547 (32.81) 2,970

to

IV - 3.84338 (4.17) 0.46566 (19.66) 1.47996 (24.98)

-0.01393 (0.63) 0.05007 (6.65) 0.22684 (1.24) - 1.41506 (32.80) 2,970

VI - 4.39645 (4.78) 0.48777 (20.71) 1.46726 (24.81 ) --

VII - 3.78787 (4.13) 0.46521 (19.65) 1.47970 (24.98) ---

--

0.09316 (6.10) --

0.25267 (1.38) - 1.40360 (32.63) 2,916

0.19952 (1.09) - 1.40696 (32.67) 2,924

-

-

*Absolute t-ratioswithin parentheses.

-0.04664 (9.00) 0.20202 (1.13) - 1.41544 (32.82) 2,970

THE TRANSITION FROM TEMPORARY TO PERMANENT DISABILITY

93

that determining a nonscheduled permanent disability in other ways. Specifically, while injured female workers, who suffer nonscheduled injuries are more likely than their male counterparts to be classified as permanently disabled, female claimants with scheduled injuries are less likely than males to become permanently disabled. In addition, the variability in the permanent disability award (AWDVAR) had a much greater impact on scheduled injury transition probability than on the nonscheduled transition probability. Finally and less obviously, the resuits from tables 4 - 7 or 4 - 8 indicate that the severity of the claimant's injury as proxied by A G E and WKSLOST has a greater impact on the probability that a scheduled injury claimant will develop a permanent disability than it does on the probability of permanent disability for nonscheduled claimants.

Conclusions Two conclusions may be drawn from the previous analyses. First, the evidence suggests that the claimant exercises some discretion with respect to the pursuit of a permanent disability claim and is thus able to affect the probability that he or she will be categorized as permanently disabled. The higher the payoff from a permanent disability classification, in the form of higher compensation benefits, the greater the likelihood that the claimant will become permanently disabled. On the other hand, as the costs of pursuing a compensation claim increase, in the form of higher opportunity costs (as proxied by lower unemployment rates) and greater uncertainty (as proxied by greater variability in the permanent disability award), the probability of a permanent disability declines. Second, the results of this study demonstrate that there are marked differences between the processes determining the transition from temporary to permanent disability for claimants with scheduled and claimants with nonscheduled injuries. Unfortunately, these results offer only equivocal support for the hypothesis that the effect of wage-loss compensation on the probability of permanent disability is attenuated relative to the effect of e x a n t e compensation. While the claimant's preinjury wage rate has the hypothesized differential effect on the probability of permanent disability for scheduled and nonscheduled injuries, the results for benefit levels contradict the theoretical model. These conflicting results are confounded by a high degree of collinearity between benefit levels and the wage rate. In addition, any inference concerning differences in the effects of wage

94

W O R K E R S ' COMPENSATION INSURANCE

loss and e x a n t e compensation based on this data set must be tempered with the recognition that the New York data confound body part and compensation method. Wage-loss benefits are predominantly paid to workers suffering injuries to the back or head, while e x a n t e benefits are paid to workers suffering injury to an extremity, the eye, or one or both ears. Differences (or the lack of difference) between nonscheduled and scheduled injuries observed by this study could be the result of differences in the body part injured rather than the method in which the injured workers are compensated. These weaknesses imply the need for future research. A definitive test of the hypothesized relationship between the compensation method and the probability of permanent disability awaits further research. This research will require data from different jurisdictions to allow for variation between benefit levels and the claimant's preinjury wage and to disentangle the effects of body part and compensation method.

Notes 1. Partial benefits are paid to those claimants who retain some wage-earning capacity, while total benefits are paid to workers who are completely disabled and unable to work. 2. If the claimant is disabled for an extended period, he or she will receive payment for the waiting period. The date that the worker qualifies for waiting period benefits is known as the retroactive date and usually occurs fourteen to twenty-one days after the date of injury. 3. This is typically in addition to whatever temporary total benefits the claimant may have collected. 4. These are characteristics of an ideal wage-loss system (Burton, 1983). Those jurisdictions that have actually adopted the "wage-loss" approach do not necessarily share each of these characteristics. For example, Florida does not provide lifetime compensation to workers with lifetime wage loss. At most, injured workers in Florida receive benefits for slightly more than ten years. Additionally, wage-loss systems generally do not provide full protection against inflation. 5. The breakdown between major and minor permanent partial is based on the total cash benefits paid in the case and varies both across jurisdictions and over time. 6. Due to space limitations, this discussion is heuristic rather than exhaustive. For a more complete presentation of the economic theory of work injuries, see Chelius (1977) or Darling-Hammond and Kniesner (1980). 7. A few studies have, however, detected a negative relationship between injury rates and workers' compensation benefits (Burton, 1983; and Welland, 1987). 8. In addition, New York subtracts from the permanent partial benefit duration the number of weeks of temporary benefits received by the claimant. This period, termed the scheduled healing period, is also subject to certain maximums, defined by the body part injured. The maximum for the arm is thirty-two weeks. Expanding on the example used in the text, if our injured worker had received ten weeks of temporary benefits, then he/she would receive 31.2 - 10 = 21.2 weeks of permanent partial disability benefits. A worker

THE TRANSITION FROM T E M P O R A R Y TO PERMANENT DISABILITY

95

with the same loss who had collected thirty-two weeks of temporary disability benefits would receive no permanent partial compensation. 9. Unfortunately, comparable nationwide statistics are not available. 10. Cases closed in any particular year include cases involving accidents from prior years, e.g., 1980 closings include claims arising from accidents that occurred in 1979, 1978, etc. An examination of the distribution of claims arising from accidents that occured in 1964 and that closed prior to 1983 reveals that 23.65 percent of these claims closed within one year of the date of injury, 82.17 percent within two years, and 98.68 percent within five years. For more detailed information concerning the distribution of the lag between injury and closing for New York compensation claims, see Thomason (1989). 11. In New York, for claimants earning less than the statutory minimum and more than the statutory maximum, b is 66.67 percent. For workers earning a wage greater than the maximum or less than the minimum, the replacement rate declines as the preinjury wage increases. 12. As previously noted, the duration of wage-loss benefits is dependent on the claimant's post-injury earnings so that benefits cease any time the claimant's post-injury earnings exceed preinjury earnings. It is recognized that while benefit duration is theoreticaly indefinite, the effective duration is often limited. 13. This assumption is plausible if compensation claimants heavily discount future income, i.e., if claimants have a strong time preference for immediate income relative to future income or if claimants believe that future wage income will eventually exceed preinjury earnings. 14. The original Closed Case data were sorted by year of closing. These data were resorted by year of injury by Professo, William Currington of the University of Arkansas. 15. Observations that were missing data for any of the independent variables in the regression equation (see below) were also excluded from the analyses. This was particularly significant for the variable measuring the claimant's preinjury wage. Approximately 15 percent of observations in the data set were missing data for this variable. 16. Throughout this chapter I have categorized claims as "scheduled" or "nonscheduled" based on the body part injured. All claims involving injuries to the eye, ear, leg, foot, toe, arm, hand, finger, or thumb were identified as "scheduled" injuries. Claims resulting from injuries to other body parts or from injuries to multiple body parts were identified as "nonscheduled" injuries. This classification scheme is not identical to that used by the New York Workers' Compensation Board, since that agency only classifies injuries where the claimant has been categorized as having a permanent partial disability; temporary disabilities are not categorized as "scheduled" or "nonscheduled" by the Board. Additionally, the classification scheme used in this chapter was applied to permanent partial disability claims, regardless of the Workers' Compensation Board's categorization of the claim. For example, an arm injury that had been designated as a nonscheduled permanent partial disability by the Board would be classified as a scheduled injury for purposes of my analysis. 17. As mentioned in note 15, a large proportion of the observations in the data set were missing data for the wage variable. These observations were eliminated from the sample analyzed in this study. To determine whether the elimination of these observations had an adverse affect on the regression estimates reported below, these regressions were also estimated with a sample that included the deleted observations, with sample means substituted for missing W A G E values. The results for these regressions did not differ substantially from equations reported in tables 4 - 6 to 4 - 8 below. 18. Injury severity is also positively related to scheduled benefits through its effect on benefit duration.

96

WORKERS' COMPENSATION INSURANCE

19. Compensation claims may be filed in any one of six district offices of the New York Workers' Compensation Board. The districts served by these offices are more or less coextensive with the Statistical Metropolitan Areas (SMAs) used by the Department of Labor to report employment statistics. Unemployment rates for these SMAs were used in the analyses (New York Division of the Budget, 1978 and 1986). 20. The model also predicts that preinjury wages and injury severity will be similarly related to claimant opportunity costs. 21. AWDVAR was calculated by pooling all permanent disability claims (total and partial) in the population, classifying these claims by body part and nature of injury, and then calculating the variance in permanent disability benefits awarded to each claimant categorized as permanently disabled for each "body part-injury nature" cell. These benefits were scaled by the weekly permanent disability payment (PPBEN) to eliminate variation due to weekly benefits. Individual claims were assigned AWDVAR scores based upon the nature of the claimant's injury and the body part affected. 22. These probability derivatives were computed according to the following formula:

N

0t) OX~

exp(fliXij)

=j~l [1 ; ~ i j ) ]

2ffi

N

where fli (i = 1, k) are the maximum likelihood estimates corresponding to the k variables used to predict settlement probability, j = 1, N are the N claims in the data set. 23. 34.8 percent of the total number of permanent partial claims were scheduled claims, while only 4.4 percent were nonscheduled claims.

References Berkowitz, Monroe and John F. Burton, Jr. (1987). Permanent Disability Benefits in Workers' Compensation. Kalamazoo MI: The W . E . Upjohn Institute for Employment Research. Burton, John F., Jr. (1983). "Compensation for Permanent Partial Disabilities." In John D. Worrall (ed.), Safety and the Workforce: Incentives and Disincentives in Workers' Compensation. Ithaca, NY: I L R Press. Butler, Richard J. (1983). "Wage and Injury Rate Response to Shifting Levels of Workers' Compensation." In John D. Worrall (ed.), Safety and the Workforce: Incentives and Disincentives in Workers' Compensation. Ithaca, NY: I L R Press. Butler, Richard J. and John D. Worrall. (1983). " W o r k e r s ' Compensation: Benefit and Injury Claims Rates in the 1970s." Review of Economics and Statistics 65:580-89. Butler, Richard J. and John D. Worrall. (1985). " W o r k Injury Compensation and the Duration of Nonwork Spells." Economic Journal 5:714-724. Chelius, James R. (1973). " A n Empirical Analysis of Safety Regulation." In Monroe Berkowitz (ed.), Supplemental Studies for the National Commission on State Workmen's Compensation Laws, Volume 3. Washington, DC: Government Printing Office.

UNDERWRITING CYCLES IN LIABILITY INSURANCE

97

Chelius, James R. (1974). "The Control of Industrial Accidents: Economic Theory and Empirical Evidence." Law and Contemporary Problems 38:700729. Chelius, James R. (1977). Worklace Safety and Health: The Role of Workers' Compensation. Washington, DC: American Enterprise Institute. Chelius, James R. (1982). "The Influence of Workers' Compensation on Safety Incentives." Industrial and Labor Relations Review 35:235-242. Chelius, James R. (1983). "Workers' Compensation and the Incentive to Prevent Injuries." In John D. Worrall (ed.), Safety and the Workforce: Incentives and Disincentives in Workers' Compensation. Ithaca, NY: ILR Press. Darling-Hammond, Linda and Thomas J. Kneisner. (1980). The Law and Economics of Workers' Compensation. Santa Monica, CA: The Institute for Civil Justice, The Rand Corporation. Deaton, Angus and John Muellbauer. (1980). Economics and Consumer Behavior. Cambridge: Cambridge University Press. Krueger, Alan B. (1988). "Moral Hazard in Workers' Compensation Insurance." Mimeo. Princeton University. Manski, Charles and Lerman S. (1977). "The Estimation of Choice Probabilities from Choice Based Samples." Econometrica 45:1977-1988. National Commission on State Workmen's Compensation Laws. (1972). The Report of the National Commission on State Workmen's Compensation Laws. Washington, DC: Government Printing Office. Nelson, William J., Jr. (1988). "Workers' Compensation: Coverage, Benefits, and Costs, 1985." Social Security Bulletin 51:4-9. Ruser, John W. (1985). "Workers' Compensation Insurance, Experience-Rating, and Occuptational Injuries." Rand Journal of Economics 16:487-503. Thomason, Terry L. (1989). "The Compensation of Permanent Partial Disability in New York State: An Examination of Wage-Loss and Ex Ante Workers' Compensation Benefits." Doctoral Dissertation. Cornell University. Welland, Deborah A. (1986). "Workers' Compensation Liability Changes and the Distribution of Injury Claims." Journal of Risk and Insurance 53:662-678. Worrall, John D. and David Appel. (1982). "The Wage Replacement Rate and Benefit Utilization in Workers' Compensation Insurance." Journal of Risk and Insurance 49:361-371. Worrall, John D. and Richard J. Butler. (1985). "Benefits and Claim Duration." In John D. Worrall and David Appel (eds.), Workers' Compensation Benefits. Ithaca, NY: ILR Press. Worrall, John D., David Durbin, David Appel, and Richard J. Butler. (1987). "The Transition from Temporary Total to Permanent Partial Disability: A Longitudinal Analysis." Mimeo. Rutgers University.

This page intentionally blank

CAPITAL FLOWS AND UNDERWRITING CYCLES IN LIABILITY INSURANCE J. David Cummins Patricia M. Danzon

Introduction Reported underwriting profits in property-liability insurance are characterized by significant cyclical fluctuations. The pricing pattern usually identified as the underwriting cycle is portrayed in figure 5-1, which shows the all-lines combined ratio for the period 1951 to 1987. Venezian (1985) identified the cycle as a second-order autoregressive process. The cycle in figure 5-1 is statistically significant and approximately six years in length. The presence of significant autoregression in prices and profits would not seem to be characteristic of a rational market (see, e.g., Abel and Mishkin, 1983). In property-liability insurance, however, cycles are not necessarily inconsistent with rationality. Cummins and Outreville (1987) have shown that cycles can occur in property-liability insurance markets even if prices reflect rational expectations. They demonstrate that intervention factors, such as policy renewal lags, information lags and regulatory delays, can create autoregressive patterns very similar to those observed in actual insurance markets. Even though intervention factors appear to be largely responsible for the general autoregressive pattern in reported underwriting profits, they 99

100

WORKERS' COMPENSATION INSURANCE COMBINED RATIO 1.2 1.15 1.1 1.05 1 0.95. i~-~rl

i

i

i

i

i

i

[

i

i

i

i

i

i

i

i

i

i

i

i

i

i

i

i

i

i

i

i

i

i

i

i

i

0.9 1951 1954 1957 1960 1963 1966 1969 1972 1975 1978 1981 1984 1987 ' ALL LINES

Figure 5 - 1 .

Combined ratio: all lines.

do not appear capable of explaining the severe price and availability problems that have plagued liability insurance markets intermittently since the mid-1970s. The problems in the liability market are often described as "cycles," although many observers confuse them with the rational-autoregressive process explained by Cummins and Outreville. We prefer to distinguish between the two phenomena, while pointing out areas in which they interact. Any explanation of price and availability problems must be based upon a theory of price formation and market equilibrium. The two theories of insurance pricing that have gained the most prominence can be termed the perfect capital markets hypothesis and the capacity constraint hypothesis. The former approach relies on concepts from financial economics to explain insurance pricing. The literature in this area is extensive. Among the most prominent papers are those by Kraus and Ross (1982) and Myers and Cohn (1987). In efficient markets theory, insurance is priced as a contingent claim (debt instrument). Capital is freely available as long as insurance is priced to yield the market equilibrium rate of return for securities of comparable risk. Other than regulatory meddling, the efficient markets approach typically does not envision scenarios in which insurance would be unavailable or "unfairly" priced. The insurance markets are not expected to spend significant amounts of time out of equilibrium. In contrast, the capacity constraint hypothesis, advanced by Winter (1988) and Gron (1989), postulates that capital does not always flow

UNDERWRITING CYCLES IN LIABILITY INSURANCE

101

freely between the capital markets and the insurance industry. The primary rationale is the argument that external capital is more costly than internal capital (retained earnings). Thus, if the industry sustains a negative shock to capital, insurers do not fully and immediately replenish capital from external sources but rather wait for retained earnings to close the gap. During the period of disequilibrium, the supply price rises and buyers compete for scarce capacity. Likewise, positive shocks to capital create periods of excess capacity. Insurers do not distribute all of the "surplus-surplus" as dividends but prefer to maintain "slack" capacity to help cushion the next negative capital shock. In this model, disequilibria are common and not necessarily short-lived. The efficient markets approach, at least in its essentials, does not offer a completely satisfying explanation for price and availability problems in insurance markets. The capacity constraint hypothesis is promising but also suffers from limitations, as explained below. One objective of this chapter is to evaluate the perfect markets and capacity constraint hypotheses and to formulate testable h'ypotheses based on the two theories. We also extend the existing theories by offering two new hypotheses that may help to explain insurer behavior during price and availability crises: 1) changes in price and coverage amounts may represent rational behavior for insurers facing asymmetric information with regard to the risk rather than the expected value of policyholder loss distributions--thus, it may not be necessary to resort to capital market imperfections to explain nonlinear pricing; 1 2) investor uncertainty about the adequacy of reserves may explain insurer preferences for internal rather than external capital. Thus, information asymmetries between investors and insurers may be partly responsible for any capacity constraints that may exist in insurance markets. We consider this a more satisfying explanation for capacity constraints than the dividend tax explanation proposed by Winter (see below). The chapter begins with an empirical overview of the market for liability insurance. The results indicate that most of the increase in losses during the 1984 to 1985 liability crisis represented changes in insurer expectations with regard to future claim payments (reserves), rather than actual increases in paid claims. Premium increases based on changes in expectations would be fully consistent with the perfect capital markets model. However, changes in liability prices during the crisis appear to be larger than would be predicted from changes in expectations alone (see also Harrington, 1988). Although changes in the systematic risk of underwriting and federal tax rates could explain this discrepancy within the context of the perfect markets model, it is likely that other factors

102

WORKERS' COMPENSATIONINSURANCE

such as information asymmetries with respect to reserves also play an important role. In future papers, we will report the results of empirical tests of these hypotheses.

The Liability Market: An Empirical Overview As in other property-liability lines, profits in general liability insurance exhibit a cyclical pattern. In recent years, however, the cyclicality in liability insurance has been much more pronounced than in other lines. This is shown in figure 5-2, which graphs the combined ratios for general liability and medical malpractice, as well as the all-lines combined ratio, for the period 1975 to 1987. As explained below, the pronounced cyclicality in liability profits is partly, but not totally, attributable to changes in interest rates and loss expectations. Profit cyclicality is associated with sharp premium increases in liability insurance markets. Net premiums written for general liability increased at over 70 percent per year between 1984 and 1986 (Hafrington, 1988). If demand is not totally inelastic, this would imply an even larger percentage increase in premium rates. Rate hikes of several hundred percent have been reported for some coverages. A recent survey by the U.S. General Accounting Office (GAO) shows a median increase between 1985 and 1986 of 54 percent for comprehensive general liability (CGL) COMBINED RATIO 1.6

1.4

j,~_

\'\

,~/~J

0.8 1975

i 1976

L 1977

t 1978

ALL LINES

Figure 5-2.

J 1979

i 1980

I

= 1981

GEN LIAB

Liability combined ratios.

'\,,,

i ~ 1982 1983

~

i 1984

~ 1985

GL ÷ MED MAL

1986

1987

103

UNDERWRITING CYCLES IN LIABILITY INSURANCE

primary coverage, 214 percent for CGL excess coverage, for large corporations responding to the survey. These sharp rate increases follow several years of flat or negative growth. General liability premiums and losses for the period 1975 to 1987 are portrayed in figure 5-3, along with GNP. Losses parallel GNP until 1983 and do not exhibit a cyclical pattern. Premiums are cyclical but do not depart significantly from losses and GNP between 1975 and 1984. Beginning in 1984, losses begin a very steep upward climb, increasing by more than 200 percent between 1983 and 1987. Premiums follow losses, and the historical linkage with GNP is severed. Although the aggregates shown in figure 5-3 suggest overcharging followed by undercharging, this comparison can be misleading because reported losses are not discounted, while premiums reflect present values. The relationship between "economic" and reported premiums and losses is discussed below. It is interesting to attempt to identify the source of the marked increase in incurred losses that is shown in figure 5-3. Incurred losses reported at the end of an accident year consist of three components: 1) paid claims arising from coverage provided during that accident year, 2) reserves for claims arising out of that accident year, and 3) an adjustment (the loss reserve adjustment) for errors in loss estimates for prior accident years. Conventional economic pricing theory implies that only the first two components are relevant for setting future prices, although information revealed by the reserve adjustment also would play a role. In INDEX 1975 - 1.0

/ij

/,/~/~

4

,/ j

2

~

0

1975 1976 1977 1978 1979 1980 LOSSES

Figure 5-3.

~

1981 1982 1983 1984 1985 1986 1987 PREMIUMS

General liability V. GNP.

[]

GNP

104

WORKERS' COMPENSATION INSURANCE Thousands

5

0

t980

t981

1982

1983

AY LOSSES

Figure 5-4.

1984

1985

1986

t987

/'iiiiiii~ADJ PRIOR YEAR

Components of reported losses.

addition to its information content, the loss reserve adjustment affects key financial statistics such as leverage and equity, which also have an impact on price. The reserve adjustment can be substantial, as shown in figure 5 - 4 which portrays the adjustment as a proportion of total incurred losses. The adjustments in the 1980s range from 5 percent to 20 percent of incurred losses. These results suggest that part of the loss acceleration shown in figure 5 - 3 was attributable to revaluations of claims for prior years. However, a substantial acceleration is still present after the loss reserve adjustment has been removed from the data. In setting premiums, insurers form expectations with regard to future losses and interest rates. Expectations about future (reserved) losses rather than trends in claim payment were primarily responsible for increases in reported losses from 1984 to 1987. This is shown in figure 5-5, which plots indices of first-year paid claims and reserves for accident years 1980 to 1987. 2 Figure 5 - 5 reveals that paid claims did not accelerate during the latter half of the period but maintained a steady upward trend. The acceleration took place entirely in the reserves. It should be noted that the reserves shown here do n o t include the loss reserve adjustment for prior years. Plots of paid and reserved claims for the second, third, and fourth development years show the same result: the claims acceleration of 1984 to

105

UNDERWRITING CYCLES IN LIABILITY INSURANCE INDEX tgSO

-

1,0 s

j~

2,5'-

1980

1981

t982

1983

- - - - PAID C L A I M S

Figure 5-5.

1984 I

1985

1986

1987

RESERVE

Loss growth: paid V. reserve first development year.

1987 was due to expectations about future claims (reserves) rather than changes in claim payments. Models of Insurance Pricing As mentioned above, the two insurance pricing models with the most relevance for explaining price and availability problems are the perfect capital markets model and the capacity constraint model. This section briefly explains and evaluates these models and specifies their testable implications. The Perfect Capital Markets Model Financial pricing models based on capital market theory are emerging as the prevailing ratemaking methodology in property-liability insurance. 3 Various models have been proposed, utilizing the capital asset pricing model (CAPM), arbitrage pricing theory (APT), and capital budgeting theory. Reviews are presented in D'Arcy and Doherty (1988) and Cummins (1990), while specific models are discussed in Kraus and Ross (1982), Myers and Cohn (1987), Cummins (1988b), and other sources. Financial pricing (perfect capital market) models formulate prices in terms of present values of expected losses and expenses. For exam-

106

WORKERS' COMPENSATION INSURANCE

pie, the Myers-Cohn model obtains the price as the solution of the following equation: PV(Premiums) -- PV(Expected Losses) + PV(Expenses) + PV(Taxes)

(1) where P V ( . ) is the present value operator. Thus, policyholders pay for the anticipated losses, expenses, and taxes arising from the insurance transaction. To derive the testable implications of the perfect capital markets approach, it is helpful to consider a simple discounted cash flow model of insurance pricing. Our model focuses on inflationary expectations, which were not dealt with explicitly by Myers and Cohn. It abstracts from the tax issue in order to conserve on notation. However, taxes are incorporated into our testable hypotheses through the Myers-Cohn formulation. Our discounted cash flow model is the following: P = tY.Ctpt/R t

(2)

where the summation is indexed from t -- 1 to t = N, and L = total losses, valued at the price level at the beginning of the accident year, C = (1 + c), where c is expected claims inflation, p, = the proportion of total claims paid during development year t, and R = (1 + r), where r is the nominal interest rate. To simplify the discussion, the price expressed by equation (2) is both risk-neutral and tax-neutral. In most of the discussion, prices will be normalized to base-period losses, so that L in equation (2) is set equal to 1. An important related concept is the accounting loss ratio, defined as follows: Q = LZCtpt/P

(3)

The accounting loss ratio expresses undiscounted losses as a ratio to premiums. It is important because the loss ratio or its inverse has been widely used as a proxy for price and because conventional analyses of underwriting cycles focus on this statistic. This ratio differs from the economic loss ratio, which is the present market value of losses divided by the loss component of the premium actually charged. A significant issue is the extent to which price fluctuations in insurance can be traced to changes in interest rates (see Harrington, 1988) and inflationary expectations. Kraus and Ross (1982) argue that the impact of

UNDERWRITING CYCLES IN LIABILITY INSURANCE

107

interest rate changes on premiums will be minimal as long as the Fisher effect prevails and insurance inflation does not depart significantly from general inflation. The Fisher hypothesis is that R = (1 + r) = (1 + i) (1 + rr), where i is the inflation rate and rr is the real rate of interest. If i = c,-it is clear from (2) that prices will equal losses at the base year (accident year) price level times the sum of the payout proportions discounted at the real rate. Under these assumptions, substantial changes in premiums would not be expected in response to changes in interest rates. However, changes in the loss ratio could be much larger, since nominal inflation appears directly in the numerator of equation (3). The impact of interest rates and inflation on premiums can be expressed in terms of elasticities. The elasticity of premiums with respect to claims inflation is given by equation (4): Epc = epc(1 - E R c ) = [(ZtCtPt/Rt)/p][1 - (dR/dC)(C/R)]

(4)

where Epc = the elasticity of premiums with respect to claims inflation, C, epc = the partial elasticity of premiums with respect to C, 4 and ERC = the elasticity of nominal interest with respect to C.

Under the Fisher hypothesis, ERC = 1, and Epc = 0, that is, premiums do not respond to changes in inflation as long as expected inflation is offset by nominal interest. In order for premiums to change significantly in response to inflation, insurance inflation must diverge from general inflation. The elasticity of the accounting loss ratio with respect to insurance inflation can be expected to be higher than Epc. This elasticity is: EQC = (EtCtpt/EC'p,) - Epc

(5)

where EOC = the elasticity of the loss ratio with respect to C. The first term on the right-hand-side of (5) is the elasticity of undiscounted losses. This term is likely to be quite large, for example, in the neighborhood of 5 for general liability in the 1980s. Thus, if the Fisher effect prevails, the loss ratio elasticity will be quite high. If claims inflation departs from general inflation, loss ratio elasticity is reduced because both premiums and undiscounted losses respond to changes in C. The discounted cash flow model (equation (2)) can be used to analyze pricing behavior in liability insurance markets (see Harrington, 1988, for a previous application of this approach). We use estimated liability payout proportions for general liability and estimated yield curves to develop predicted liability insurance prices for the period 1980 to 1987. Accident year incurred losses (exclusive of adjustments for prior years)

108

WORKERS' COMPENSATION INSURANCE Thousands 16 14 12

1o

8

i ~,

198(}

1981

t982

...... ACTUAL PW

Figure 5-6.

1983

1984

ACTUAL PE

-~

t985

t888

1987

MODEL= iNiTIAL INC

GL premiums: predicted V. actual.

are used as the best available estimate of undiscounted losses at the time prices are set. 5 The results are presented in figure 5-6, which plots predicted (model) prices against actual premiums written and premiums earned. The figure reveals that model prices were above actual prices from 1982 through 1985 and below actual prices in 1986 and 1987. Figure 5 - 6 suggests systematic underpricing and overpricing that would be inconsistent with the efficient markets model, at least in its pure form. One must be cautious in drawing conclusions on the basis of this analysis, however, because of the risk and tax-neutrality of equation (2). It is possible that changes in underwriting risk and taxes could account for the patterns observed in the figure. The Tax Reform Act of 1986 significantly increased the effective Federal income tax rates for property-liability insurers. If one assumes a tax burden of 20 percent of investment income in 1986 and 1987, the model prices coincide with premiums earned in these two years. 6 An increase in the systematic risk of underwriting also would increase predicted prices. There is no obvious explanation for the apparent underpricing from 1982 to 1985.

The Capacity Constraint Model This discussion of the capacity constraint model focuses on the version of the model presented in Winter (1988). Winter adopts the insurance

109

UNDERWRITING CYCLES IN LIABILITY INSURANCE

industry's definition of capacity, that is, capacity = net worth (policyholders' surplus). As mentioned above, a key premise of his model is that capital does not flow freely into and out of the insurance sector. This is attributed to the hypothesized higher costs of external capital in comparison with internal capital (retained earnings). Various explanations have been offered for the higher costs of external capital including the dividend tax, which makes it costly to withdraw capital (Winter, 1988) and information asymmetries between managers and suppliers of capital (e.g., Myers and Majluf, 1984). A second critical assumption in Winter's model is that risks in liability insurance markets are not independent. A "limited liability" constraint is assumed, such that the probability of ruin is essentially zero. 7 With limited liability, dependence among risks, and restricted capital flows, shocks to capital can occur that lead to an inelastic supply of insurance. Steep price increases result, and the market clears at a higher price and lower quantity than those characterizing the "normal" equilibrium. Dependence among risks implies that prices exceed the expected value of loss (by a covariance term). On average, this difference between pure premiums and expected losses provides retained earnings that restore capacity, and prices fall to normal levels. Formulated more precisely, Winter's version of the capacity constraint model postulates an equilibrium that satisfies the following set of equations: Xt+l = [St + (P, - pt)Qt](1 + r - k)

(6)

P Q + S > pI4Q

(7)

Pt = Max[P*, p .

P* = E ( p ) +

(8)

- St/Qt]

Cov(q,+l,

p,)/E(q,+l)

(9)

q(SM) = 1 - t

(10)

q(S,,,) = 1

(11)

q(X,) = E [ q ( X t + I ) X t + , ] / X t ( 1

+ r)

(12)

where Xt = surplus at the beginning of period t, before the firm makes its decision with regard to paying a dividend or issuing equity, St = surplus at the beginning of period t, after the dividendequity payments have been made, Pt = the loss probability in period t, Pt = the premium per dollar of coverage in period t,

110

WORKERS' COMPENSATION

INSURANCE

the quantity of insurance offered by the firm in period t, PH = the maximum loss probability that could possibly occur in period t, P* = the equilibrium premium in period t, qt = Tobin's q function, which maps surplus into firm value, such that firm value is the present value of future dividends, t = the tax rate on dividend payments, SM = the value of surplus that must be reached before a dividend payment will be made, Sm = the value of surplus that must be reached before new equity will be issued, where Sm < SM. Equation (6) specifies the relationship between end-of-period equity, beginning of period equity (St), and underwriting profits, (Pt - p t ) O t . Equation (7) is the limited liability constraint, while equations (8) and (9) specify the price that will prevail in the market. Equations (10) and (11) are boundary conditions, indicating respectively the values of equity at which dividends will be paid and new equity issued. A typical scenario envisioned by the capacity constraint theory is the following: Assume that the level of capacity in the market is relatively high. Thus, the supply curve is elastic over a reasonably wide range of surplus, due to the dividend tax that would be incurred in order to get capital out of the insurance sector. An adverse shock to losses occurs, in the form of higher pt. 8 I f surplus is sufficiently depleted by these shocks, the limited liability constraint becomes binding, moving the market into the inelastic segment of the supply curve. Price rises and the market reaches a new equilibrium at a lower quantity of insurance. The new equilibrium prevails until surplus becomes so low that q(Sm) = 1, triggering the issuance of new equity, or until surplus is replenished through retained earnings. Ot

=

Testable Implications The perfect capital markets model suggests the following testable hypotheses: 1. There are no significant impediments to capital flows into or out of the insurance industry. Thus, if capital is depleted by adverse loss shocks, new funds will be readily available, preventing the type of high-price, low-quantity equilibrium predicted by the capacity constraint model. 2. A direct relationship exists between the (relative) level of capital and the price of insurance because of the tax component of equation (1). An extension of the basic model of equation (1) to consider insurer default risk provides another rationale for a direct relationship between

UNDERWRITING CYCLES IN LIABILITY INSURANCE

111

relative capital and price (Cummins, 1988b). Insurers with higher relative capital issue safer policies and thus command higher market prices. 3. Prices are directly related to the amount of Federal corporate income tax paid by insurers. 4. Prices are responsive to the relationship between nominal interest rates and claims inflation rates in accordance with the elasticity relationships specified above. 5. Prices are directly related to the covariability of underwriting results and securities market return factors. Covariability of underwriting returns among risks is not relevant to capital flows as long as underwriting risk is diversifiable through the capital markets. The principal implications of the capacity constraint model are as follows: 1. Capital inflows and outflows occur only at the boundary points of the valuation function defined by equations (10) and (11). Thus, a shock to capital does not necessarily result in an immediate capital flow. 9 2. Prices are not unbiased predictors of losses. In particular, the price (underwriting profit) is inversely related to the level of surplus relative to its "normal" level. 3. Covariability among risks is important even if underwriting returns are uncorrelated with securities market pricing factors. The covariability between losses and the valuation function (q) for the insurer's equity enters the pricing equation even if insurance risk is unsystematic with respect to the securities market. This is a direct result of the capacity constraints due to costly external equity. Since there are impediments to the inflow of new capital, existing capital is more valuable if it is low relative to the demand for insurance. Underwriting losses deplete capital and thus increase the value of the remaining capital, resulting in higher premiums. Implications (1) and (2) of the capacity constraint model are directly opposite to those of the perfect capital markets model. Thus, future research focusing on tests of these implications can provide an indication of the relative performance of the two models in describing insurance markets. The hypothesized impact of covariability in underwriting returns also differs between the two models. This hypothesis is more difficult to test, given the available data.

Critique of the Models The perfect capital markets model has proven to be a powerful tool for analyzing financial markets. However, the model has some limitations,

112

WORKERS' COMPENSATION INSURANCE

which we address below in developing our new hypotheses about the liability insurance market. The pure form of this model does not provide an explanation for capacity constraints and availability problems. One potential problem is the reliance of the model on the assumption of perfect information. As discussed below, information asymmetries in product and capital markets may provide an explanation for some of the phenomena observed during a liability crisis. At first glance, the capacity constraint model appears to provide an explanation for observed behavior in liability insurance markets. However, empirical tests presented in Winter (1988) indicate that the model fails to provide an empirical explanation for the most recent liability crisis. Specifically, Winter found that relative surplus levels are directly, rather than inversely, related to the economic price of insurance during the 1980s. 1° The capacity constraint theory predicts an inverse relationship. Winter's version of the capacity constraint model has several potential weaknesses that may explain its poor performance with regard to the liability crisis of the 1980s. The limited liability constraint, for example, does not reflect reality. There is a vast actuarial literature dealing with probabilities of ruin for property-liability insurers (see Beard, Pentikainen, and Pesonen (1984) for a review). This literature makes clear that any insurer could be faced with claims larger than its resources. If the capacity constraint model provided a consistent empirical explanation for liability crises, one might be able to argue that limited liability is a reasonable abstract representation of a "small" ruin probability. However, it is more likely that the price of insurance reflects a "demand for solvency" and that equilibrium, non-zero ruin probabilities, exist which affect pricing behavior. Another possible problem with the capacity constraint hypothesis is that capital impediments may not exist or that they may exist in a different form than postulated by the model. Contrary to Winter's contention (1988, 6), the reason for and precise nature of capital impediments may matter in developing empirical predictions. By treating the industry as a single aggregate firm, Winter's version of the capacity constraint model abstracts from the possibility of different types of firms--old firms that have unfunded liabilities and new firms that do not. For purposes of discussion, define the triple (P, Q, K) as the price, quantity of insurance, and quantity of capital that yield "normal" profits in the insurance market. For a single homogeneous firm it may be value-maximizing to set price above P and wait to accumulate retained earnings, rather than issue new equity that would then be trapped. But

UNDERWRITING CYCLES IN LIABILITY INSURANCE

113

this would seem to provide a profitable opportunity for new entrants, who should be willing to offer new policies at P. Thus, if entry were low cost and if it were costless for policyholders to switch to the new firms, the potential for entry would prevent established firms raising prices above P to restore K. Thus, Winter's model seems to require either homogeneity of old firms and no possibility of entry or a definition of equilibrium that requires extreme farsightedness on the part of potential entrants.

Nonlinear Pricing and Insurance Market Equilibrium In this section we formulate two new hypotheses regarding insurer behavior during market crises. A crisis is defined as a period of increased prices accompanied by restrictions on coverage. Although prices tend to increase during a crisis, they do not rise sufficiently to eliminate coverage restrictions. We suggest two related reasons for this. First, raising prices may exacerbate adverse selection. If policyholders are heterogeneous, raising P may induce more good risks than bad risks to drop out of the pool. Thus, an increase in the premium may reduce net revenue, even if aggregate demand is inelastic, because the revenue gain is more than offset by an increase in expected loss per policyholder. Adverse selection may also increase insurer risk. Reducing limits of coverage at the same time as raising premiums dominates (in terms of value maximization) a strategy of adjustment solely through a premium increase. The second factor that may prevent prices rising sufficiently to eliminate coverage restrictions is the possibility of entry by new firms. As noted above, Winter's model of the industry as an aggregate firm obscures differences between old firms, which have incurred a capital shock due to an unanticipated increase in loss liabilities, and new firms that do not have this unfunded liability. 11 If it were costless for new firms to enter and for old policyholders to switch, old firms could not raise premiums above the long run equilibrium level P. The following section sketches the adverse selection argument.12 This model postulates an insurance market where adverse selection derives from heterogeneity in the dispersion rather than the mean loss of individual policyholders. We show that upper limits of coverage can induce positive selection, whereas deductibles could induce adverse selection. The adverse selection discussion is followed by an analysis of market adjustment to an unanticipated increase in liabilities on prior policies that depletes capital below K.

114

WORKERS' COMPENSATION INSURANCE

Adverse Selection With Deductibles and Policy Limits Assume that the firm maximizes an objective function V(R, z), where R is net revenue and z is the insurer's risk index (presently unspecified), where Vn > O, VRR ~< 0, Vz < 0, and Vz~ < 0. Insurer risk z depends upon the shape of the insurer's loss distribution. The risk index z also depends inversely on capital at the start of the period (K0). Thus, the index impounds both the loss distribution facing the firm and its ability to satisfy its loss obligations. The adverse selection model depends upon the assumption that insurers behave as if they were risk averse. Recent research on the theory of the firm has provided cogent arguments to support this hypothesis. As pointed out by Greenwald and Stiglitz (1987, 1990), firm behavior will appear to be risk averse if managers are subject to agency arrangements that tie their compensation to firm performance but are subject to large penalties in the event of firm bankruptcy. The bankruptcy penalties include loss of professional reputation as well as the loss of both fungible and nonfungible equity interests in the firm. These conditions seem to fit insurers particularly well because of the prevalence of mutuals, the highly specialized nature of human capital in the insurance industry, and the regulatory emphasis on solvency monitoring. Assume that information in the insurance market is asymmetric. Insureds know their expected losses and their risk characteristics. Insurers know only the expected loss for each individual. Risks are categorized into classes such that all individuals in a class have the same mean but are heterogeneous with respect to dispersion, that is, members of a class are distinguished by mean-preserving spreads. The insurer knows the expected loss within each class of risks but cannot identify the risk characteristics of individual insureds. Insureds are assumed to be willing to pay more than the expected loss for insurance. This may be because they are expected utility decision makers (e.g., consumers) or due to agency costs, bonding, or other factors (e.g., corporate buyers). A typical insurer has a pool of N risks in a given class, with average loss E(x). Initially, assume that a common insurance contract is offered to all policyholders within the class, at a premium that must break even. The policy may offer unlimited coverage at a fixed price per dollar of coverage, or may have either a deductible (D), an upper limit (M), or both. To show the effects of within-class heterogeneity in risk, assume that there are two types of policyholder, H and L, whose loss distributions

115

U N D E R W R I T I N G CYCLES IN LIABILITY INSURANCE PROBABiLiTY

o.2 0

fl 10

D

2o

M

30

40

LOSS AMOUNT LOW SIGMA

Figure 5-7.

HIGH SIGMA

Mean preserving spreads,

differ by a mean-preserving spread. Define the following two distribution functions with the same mean: Fh = F(x; Sh) = high risk distribution function El = F(x; st ) = low risk distribution function where sj, j = h, 1, is an index of dispersion. Also define: D = deductible M = policy limit The distribution functions are graphed in figure 5-7. An important result used in the following analysis is that the mean of the distribution is the area above the distribution function (Feller, 1971, 150-151). Likewise, integration by parts can be used to show that imposing deductibles and policy limits subdivides the expected value proportionally to the areas above the distribution function bounded by vertical lines at D and M. For example, the expected value of a policy with a deductible but no policy limit is the area above the distribution function and to the right of the vertical line at D (see figure 5-7).

116

W O R K E R S ' COMPENSATION INSURANCE

Effects of Deductibles.

Let EH>o = expected value of t t R loss above deductible EL>o = expected value of LR loss above deductible We know that EH>o > EL>D because the deductible cuts off more of the expected value for the low risks. It can also be shown that:

dEH>D/dD

=

-[I

-

Fh(D)] >

-[i

-

Ft(D)] = dEL>D/dD

(13)

Thus, imposing a deductible (given P fixed or adjusted to reflect the change in expected loss for the class as a whole) imposes a higher cost on low risks than on high risks. The effects of deductibles can be summarized as follows: 1. 2. 3.

Introducing or raising D (with a fair adjustment in P) will cause low risks to drop out disproportionately relative to high risks. Raising the premium for a fixed deductible will cause low risks to drop out because EL>D < EH>D. Deductibles will not be favored by insurers in lines where information asymmetries of this type are high. There are of course other reasons for using deductibles, such as control of moral hazard. In that case, there is a tradeoff for the insurer between controlling moral hazard and controlling adverse selection.

Effects of Upper Limits. The relationship between the expected value of the high and low risks above a policy limit (M) is similar to that above a deductible, that is, EH>~t > EL>M. However, the selection effect is reversed for the upper limit because the expectations represent the amounts not covered. Unlike deductibles, upper limits are more likely to cause high risks to drop out, assuming that high risks are expected-value decision makers. If they are expected utility maximizers, the effect may hold afortiori. 13 The effects of changes in premiums and upper limits, assuming that D = 0, are as follows: 1. Introducing (or lowering) an upper limit, with actuarially fair adjustment in P, tends to induce positive selection since high risks receive less in expected value from the insurance, compared to low risks: dEH>M/dM = - [ 1 - Fh(m)] < - [ 1 - F,(M)I = dEL>M/M

(14)

2. If the class mean increases, raising P with fixed M > 0 and D = 0 will impose higher costs on high risks than on low risks. Thus, raising the premium should improve the pool and reduce the risk to the insurer. The adverse selection at work here affects the dispersion rather than

UNDERWRITING CYCLES IN LIABILITY INSURANCE

117

the mean of the insurer's loss distribution. This would not affect the contracts offered if the insurer is indifferent to risk. If insurers are averse to risk, reducing the policy limit may be an optimal response to an increase in risk in the form of a mean preserving spread in the policyholders' loss distributions. Even with within-class homogeneity, if insurers are averse to risk, upper limits might still be the optimal policy design: 1. Upper limits tend to reduce the ruin probability disproportionately to the amount of expected loss they eliminate, because most of the variance and skewness in the loss distribution is created by the upper tail. 2. Upper limits may be necessary in order for certain risks to be insurable, Many insurance loss distributions have been shown to have infinite moments (e.g., Cummins, et al. 1990). Such risks are uninsurable because by definition there is no finite expected value and thus no risk-neutral premium. Pooling may be inoperable for infinite moment distributions. 14 Underwriting Crises. Berger and Cummins (1992) investigate the effects of this type of informational asymmetry on market equilibrium. As in the classical Rothschild-Stiglitz model, a pooling equilibrium is unlikely to exist. Multiple types of separating equilibria can occur. For continuous loss distributions, four types of separating equilibria are identified, that is, equilibria where: 1) low risks receive "optimal" coverage and high risks are restricted; 2) high risks are required to buy higher than optimal amounts of coverage; 3) high risks receive optimal coverage and low risks are restricted; and 4) both high and low risks receive optimal coverage. The type of equilibrium that occurs depends upon the degree of the mean-preserving-spread and on both policyholder and insurer risk aversion. An underwriting crisis scenario can be generated in this model, with lower coverage accompanied by higher prices. This type of crisis requires an increase in insurer "risk aversion." Such an increase could be triggered by factors such as adverse shocks to surplus. Increases in loss-distribution means which are not accompanied by increases in insurer risk aversion lead to increases in price but usually not to decreases in coverage.

Market Adjustment to Unanticipated Surplus Shocks Consider two types of firms, old firms and new entrants. New firms could include established firms that are new or expanding in a particular line of

118

WORKERS' COMPENSATIONINSURANCE

insurance. Firms of each type are assumed to be homogeneous so that each group can be represented by a single firm. Initially, the old firm is the sole supplier. The demand for insurance is expressed by the inverse demand function:

Qa = f(P, M, x, z)

(15)

where P is the premium rate per dollar of coverage, x is the expected loss per policyholder, M is the upper limit on coverage, z is a measure of the insurer's risk of insolvency: dQd/dP < O, dQd/dM < O, dQJdx > O,

dQd/dZ < O. Supply is a function of P, given the mean expected loss per policyholder x and the upper limit M. Supply is inversely related to the insurer's cost of capital r(z, Ko, w), which is a function of the insurer's risk z, its initial capital (K0), and the risk of inadequacy of its existing liabilities (w):

Qs = g(P, x, M, r(z, Ko, w))

(16)

with dQs/dP > O, dQJdr < 0, and dQs/dz < 0. The supply price of new capital r is directly related to z and w and inversely related to K0. The initial equilibrium price and quantity of insurance and level of capital are denoted P, Q, and K. We do not analyze the determinants of K, but simply assume 0 < K < 0o. Now assume that losses on policies written in year t-1 exceed anticipated levels. Loss reserves for prior years must be increased, such that capital falls to K' which is less than K. The question is: what is the optimal adjustment of price, policy limits, and new equity? The firm could raise new equity to make up the shortfall, K-K'. But if potential suppliers of new equity are still uncertain whether loss reserves for prior years are adequate, the supply price of new equity will be r' > r when the premium is set at P. Recall that P implies an expected rate of return of r when capital equals K and the quantity sold equals Q. But if there is still some risk that loss reserves for prior policy years may prove inadequate, the rate of return required by new equity will include a markup over r to reflect the risk that the new equity will be used to pay for unfunded liabilities on prior policies. Thus, the short-run supply curve of new equity is imperfectly elastic, at least when the market is uncertain about reserve adequacy. 15 The hypothesis that the true value of losses on prior policies may remain uncertain and that this may raise the supply price of new equity to old firms is particularly relevant to the insurance "crisis" of the 1980s. There remains great uncertainty about the ultimate liability of the insurance industry, both in the aggregate and for individual insurers,

UNDERWRITING CYCLES IN LIABILITY INSURANCE

119

for both asbestos-related losses (personal and property damage) and hazardous waste clean-up under Superfund. In both cases, the uncertainty derives in large part from conflicting judicial interpretations of insurance contracts. In the case of asbestos, the issue whether liability accrues during the plaintiff's exposure or at time of manifestation of the injury. In the case of Superfund, uncertainty derives both from joint and several liability and from conflicting rulings concerning the pollution exclusion in general liability policies. If uncertainty about prior losses raises the supply price of new equity, such that it is optimal for the old firm to restore K through retained earnings, there remains a question of the optimal rate of accumulation of retained earnings. Equivalently, the question is what is the optimal path of adjustment of P. The optimal path of adjustment depends on the adverse selection effects of raising prices, the price elasticity of demand, and the feasibility of entry. We consider two possible extremes: no entry and free entry. These define bounds within which the actual path of adjustment will fall. 1. No Entry. Several scenarios could lead to market conditions similar to those observed in an insurance crisis. One illustrative example is portrayed in figure 5-8. In the figure, the initial supply and demand

PRICE \"

S'

f,f J j f j JJ" jJ'J f/j,J f

"\ p*

,,

J

,i

jf~J

\ \

// / f

y,//"

\ j~

D Q~ Q

QUANTITY

Figure 5-8.

Insurance crisis.

S

120

WORKERS' COMPENSATION INSURANCE

curves for insurance are denoted by D and S, respectively, and the initial equilibrium occurs at P and Q. Assume there has been a one-time shock to capital that causes a shift in the supply curve from S to S'. Also assume for the moment that the capital shock does not affect the position of the demand curve, if entry is not feasible, the new equilibrium will occur at price P' > P and quantity Q' < Q, even though P' is an "unfair" price, that is, a price higher than would be implied by the perfect markets model. The difference between P' and the fair price P is the loading that policyholders have to pay to restore capital from K' to K. The amount of the supply price increase and hence the optimal path of adjustment depends on several factors, including the severity of the shortfall in K, the elasticity of demand, and the loss of goodwill resulting from charging unfair premiums. The path will also depend upon the length of time before entry becomes feasible, that is, it is not assumed that the insurer can charge economically unfair premiums indefinitely. Of course, it is also likely that the shock to capital will result in a shift in the demand curve. Policyholders are willing to pay less for insurance policies from companies with higher default risk, so the demand curve is likely to shift to the left. This would mitigate the price increase but aggravate the decline in quantity sold. However, in the presence of guaranty funds, it is possible that the demand curve would not shift very much. In addition, the factors that caused the shock could lead to an increase in demand for insurance that could offset the decline due to higher ruin probabilities. This could occur without necessarily affecting the equilibrium supply price of capital r, if the shock to capital were caused by an increase in unsystematic risk. Two points are worth noting here. First, in this model unanticipated losses on prior policies are recouped by a charge to the current cohort of policyholders--either a one-time assessment or a markup that generates retained earnings over time. Second, in order for the insurer to implement the equivalent of an assessment through a sharp premium hike, entry must be delayed for at least the duration of the contract, if entry is delayed for a year, the necessary premium surcharge can be spread over a year of coverage. This will imply a smaller percentage increase in P than if the same total levy had to be concentrated into the cost of a six month contract. Thus, the percentage increase in P, for a given shortfall in K, will be inversely related to the length of the insurance contract. If it is low cost for policyholders to cancel during a policy term, then the critical determinant of the percentage change in P is the delay in entry, rather than the policy term.

U N D E R W R I T I N G CYCLES IN LIABILITY INSURANCE

121

2. Free Entry. A new firm that is not encumbered with uncertain losses on prior policies faces a cost of capital of r and may be able to offer coverage at P, provided that it faces the same loss distribution on new policies as established firms and this distribution is known. Thus, if entry and exit are costless and it is costless for policyholders to switch, old firms will be unable to raise their premium above P. In practice, the financial costs of entry appear to be quite small. But imperfect information about the loss distribution faced by new entrants may raise the supply price of capital above r and possibly above r'. There are two sources of uncertainty for new firms. First, the loss distribution for the market as a whole may be uncertain. This is plausible in the case where the initial shock to capital was caused by an unanticipated loss on prior policies, and it is uncertain whether this change is permanent or transitory. If managers of old firms have more precise estimates of the parameters of the new loss distribution, based on their own experience, new entrants face greater risk, and this acts as a barrier to entry (defined as a cost faced by new entrants that is not faced by incumbents). Second, new entrants may face greater adverse selection risk than old firms, in part because of the risk-reduction strategies of old firms. If old firms reduce their exposure by lowering M, rather than simply raising P, they will tend to retain the good risks and the less risky layers of coverage. More realistically, old firms do not have to rely solely on selfselection but are likely to have acquired information from experience that enables them to select out the good risks. On the other hand, positive selection by old firms will be imperfect to the extent that unobservable heterogeneity in risk aversion is negatively correlated with unobservable differences in inherent risk. In this case lowering M may force out the risk averse good risks along with the less risk averse bad risks. For third-party liability insurance, a negative correlation between risk aversion and intrinsic riskiness is not implausible since inherent riskiness is to some degree a choice of the insured: risk tolerant individuals may be attracted to activities that are intrinsically risky or may take fewer precautions to reduce risk. In summary, suppliers of new equity to old insurers face three sources of risk: 1) inadequacy of prior loss reserves, 2) parameter uncertainty for the prospective loss distribution and nonindependence, and 3) sampling or selection risk. Suppliers of equity to new entrants face only the two latter sources of risk. Both of these types of risk may be greater for the new entrant than for the old firm. But this may be more than offset by the fact that new entrants are not encumbered with uncertainty about the

122

WORKERS' COMPENSATION INSURANCE

adequacy of loss reserves on prior policies. It seems plausible that the firm-specific information needed to assess the adequacy of loss reserves of individual old firms will be less precise than the information on the loss distribution for the market as a whole, since the former involves sampling risk as well as parameter uncertainty. In that case, old firms are at a disadvantage relative to new entrants. The optimal strategies for old and new firms also depend on the costs of switching for policyholders. If switching is costless, then, as soon as entry occurs, policyholders would cancel policies with old firms and switch to new entrants, constraining the post-entry price to P. But in markets with asymmetric information, switching is likely to be costly. Insurers who face a risk of adverse selection will acquire information through initial underwriting of new applicants and through experience accumulated over time. Each time a policyholder switches, these costs are incurred again and, in competitive markets, will be reflected in the premium charged. Further, if established insurers use acquired information to ration coverage in periods of relatively high insurer risk, an insured who switches from his old insurer to a new entrant when the old insurer raises the price destroys the stock of information that may affect his access to a stable supply of insurance in the future.

Conclusions Recent research on underwriting cycles has focused on two models of insurance pricing: the perfect markets model of corporate finance and the capacity constraint model. The perfect markets model states that insurance prices reflect the discounted value of the cash flows arising from insurance transactions. The principal cash flows are losses, expenses, and corporate income taxes. The discount rate reflects the systematic risk of underwriting and may reflect insurer investment and default risk. The pure form of the model recognizes no impediments to capital flows into and out of the insurance industry. In contrast, sticky capital flows play a major role in the capacity constraint model. This model states that insurers prefer internal to external capital either due to the tax effects of dividend payments or to information asymmetries between managers and investors. According to this model, insurers will be reluctant to raise new captial externally following a market shock. A capacity constraint develops and insurance prices are bid up as buyers compete for scarce capacity. Prices in excess of expected values lead to the accumulation of retained earnings,

U N D E R W R I T I N G CYCLES IN LIABILITY INSURANCE

123

eventually leading to slack capacity. Slack is accumulated because insurers recognize that it will be difficult to raise capital externally following a shock. Prices decline as slack accumulates leading to a so-called "soft market." The perfect markets and capacity constraint models have different implications with regard to the relationship of capacity and price. The perfect markets model implies that the price of insurance will be directly related to the amount of capital maintained by an insurer relative to its obligations. The reason is the reduced risk of default. Thus, when capital falls, the perfect markets model predicts a drop in price. The capacity constraint model predicts the opposite, that is, an increase in price following an adverse shock to capital. This paper has provided an evaluation of the two models and proposed two new hypotheses that may help to explain insurer behavior during market crises. The empirical analysis shows that the increase in reported liability losses during the 1984-1985 crisis was primarily due to expectations (reserve increases) rather than increases in loss payments. Insurers strengthened reserves on existing policies, as well as reporting a large increase in incurred losses on policies issued in 1984 and 1985. When taken in conjunction with the declines in interest rates during the period, the increase in loss expectations explains most but not all of the increase in premiums during this period. Part of the discrepancy can be explained by the increase in Federal tax rates for property-liability insurers that were anticipated in the Tax Reform Act of 1986. However, the difference appears too large to be fully explained by the discounted cash flow model alone. One of the most puzzling aspects of the underwriting cycle is nonlinear pricing, that is, increases !in price that exceed increases in the expected value of loss. Anecdotally, nonlinear pricing often takes the form of a price increase accompanied by a mandated reduction in coverage. We propose a new explanation for nonlinear pricing, that is, the possible existence of information asymmetries between buyers and sellers with regard to risk rather than (or perhaps in addition to) expected value. If policyholders have the same mean but differ in risk, then reducing the policy limit penalizes high-risk policyholders more than low-risk policyholders, thus inducing favorable selection. Insurers may be able to manage the reductions in demand accompanying a price increase by coupling the price change with a change in limits. Our second contribution is to provide a possible explanation for insurer preferences for internal over external capital. We hypothesize that uncertainty regarding the adequacy of reserves following an adverse loss

124

W O R K E R S ' COMPENSATION INSURANCE

shock may lead capital markets to charge a markup to insurers over the true cost of capital. Instead of penalizing existing stockholders by raising capital on adverse terms, insurers may attempt to charge new policyholders higher premiums in order to recoup the capital deficit through retained earnings. Limitations on entry such as switch costs and information asymmetries between old and new firms may be present that permit insurers to recover at least part of the capital deficit from incoming cohorts of policyholders. Our future research will further develop and empirically test these hypotheses.

Notes 1. Nonlinear pricing refers to price increases in excess of increases in the expected value of loss. 2. For each accident year, figure 5 - 5 shows the amount of claims paid during the accident year (e.g., claims paid during 1981 that were attributable to coverage provided in 1981) and the amount reserved for that year's coverage at the end of the year. The indices do not reflect reserve development after the end of the accident year in order to focus on the information that was available for ratemaking at the end of each year. 3. While it is clearly an exaggeration to state that all financial pricing models are based on the assumption of "perfect" capital markets, the models are grouped under this heading for purposes of discussion. The models typically assume that markets are perfect in the sense that all relevant information is instantaneously and costlessly available and that adjustments of prices between equilibria in response to changes in parameters is quite rapid. 4. The partial elasticity epc is the Macaulay duration of the premium P, familiar from analyses of bond price sensitivity in the financial literature. 5. Actually, reported accident year losses may not reflect the company's best estimate of losses at the end of the accident year. Companies often misstate incurred losses for purposes of financial reporting and tax management. See Cummins and Grace (1988). These misstatements usually show up primarily in the incurred but not reported (IBNR) reserve, which is included in the reported accident year losses used in our estimates. Harrington (1988) made an attempt to use developed losses, in addition to accident year incurred losses, in estimating predicted prices. The problem with this approach is that developed losses do not mature until several years after the accident year. In addition, the losses as ultimately developed reflect information that would not have been available in the company's best estimate at the end of the accident year. 6. Since the Tax Reform Act was not fully implemented until 1987, one would have to hypothesize that the market was anticipating its effects in order to explain 1986 prices in terms of a tax adjustment. This hypothesis may have some justification in view of the fact that the tax implications of writing long-tail lines of insurance span a considerable period of time. 7. Winter (1988) indicates that this assumption would be justified if consumers are infinitely risk averse or if implicit ex post bankruptcy costs are present. 8. The parameter Pt is determined by a random draw each period and then becomes the loss probability for all liability risks.

UNDERWRITING CYCLES IN LIABILITY INSURANCE

125

9. It is interesting to note that the region within which the trap occurs was reduced by the Tax Reform Act of 1986. Thus, the trap hypothesis would imply that capital flows should be more responsive at present than they have been historically. 10. The economic price is the ratio of premiums to the discounted value of losses. Actually, Winter's tests used the economic loss ratio (the reciprocal of the economic price) as the dependent variable. We have reinterpreted his results in terms of the economic price. 11. Because of its long payout tail, liability insurance in susceptible to shocks, such as changes in the legal system, that increase unpaid claims (reserves) from prior years as well as expectations regarding future claims. Thus, there is likely to be positive covariability between leverage ratios and estimated losses on future policies that can exacerbate price and quantity adjustments. 12. The implications of the adverse selection argument are explored in detail in Berger and Cummins (1992). 13. Assuming identical utility functions, the utility loss from an upper limit for H exceeds the utility loss for L by more than the difference in expected loss. 14. The hypothesis that some liability risks have become so unpredictable that their loss distributions have infinite moments might explain the long-term lack of availability of pollution coverage but probably cannot explain temporary shortages of excess coverage without an additional assumption that the uncertainty was reduced over time as a result of learning about the new loss distribution. With inelastic capital supply, it is also possible that the existing capital is fully sold-off to lower risk policyholders. 15. This explanation requires the existence of information asymmetries. If all market participants have the same information, homogeneous expectations will be present and the usual theory would suggest that a market price will exist that is fair to all participants. Thus, unless the insurer possesses information that the market does not have, availability problems would be difficult to explain using this argument. Infinite moment distributions and inter-generational covariances are a more likely explanation for some of the "uninsurable" risks of the 1980s.

References Abel, Andrew and Frederic Mishkin. (1983). " A n Integrated View of Tests of Rationality, Market Efficiency, and the Short-Run Neutrality of Monetary Policy." Journal of Monetary Economics 11:3-24. Beard, R.E., T. Pentikainen, and E. Pesonen. (1984). Risk Theory (New York: Chapman and Hall). Berger, A. Lawrence, and J. David Cummins. (1992). "Adverse Selection and Equilibrium In Liability Insurance Markets." Journal of Risk and Uncertainty 5:273- 288. Cummins, J. David, et al. (1990). "Applications of the GB2 Family of Probability Distributions In Collective Risk Theory." Insurance: Mathematics and Economics 9:257-272. Cummins, J. David. (1990). "Asset Pricing Models and Insurance Ratemaking." Astin Bulletin 20:125-166. Cummins, J. David, and Elizabeth Grace. (1992). "The Demand for Tax-Exempt

126

WORKERS' COMPENSATION INSURANCE

Bonds By Financial Institutions: The Case of Property-Liability Insurers." Working paper. University of Pennsylvania. Cummins, J. David, and Francois Outreville. (1987). "An International Analysis of Underwriting Cycles." Journal of Risk and Insurance. Cummins, J. David. (1988a). "Risk-Based Premiums for Insurance Guaranty Funds." Journal of Finance. Cummins, J. David. (1988b). "Capital Structure and Fair Profits In PropertyLiability Insurance." Working paper. University of Pennsylvania. Danzon, Patricia. (1988). "Medical Malpractice Liability." In R.E. Litan and C. Winston (eds.) Liability: Perspectives and Policy. Washington, DC: The Brookings Institution. D'Arcy, Stephen, and Neil A. Doherty. (1988). Financial Theory of Insurance Pricing. Philadelphia, PA: S.S. Huebner Foundation. D'Arcy, Stephen, and James R. Garven. (1988). "Financial Pricing Models of Property-Liability Insurance: An Empirical Evaluation." Working paper. Department of Finance; The Pennsylvania State University. Derrig, A. Richard. (1985). "The Effect of Federal Income Taxes on Investment Income in Property-Liability Ratemaking." Working paper. Massachusetts Rating Bureau, Boston, MA. Doherty, Neil A. and James R. Garven. (1986). "Price Regulation in PropertyLiability Insurance: A Contingent Claims Approach." Journal of Finance 41:1031-1050. Doherty, Neil A. and H.B. Kang. (1988). "Interest Rates and Insurance Price Cycles." Journal of Banking and Finance 12:199-214. Feller, William. (1971). An Introduction to Probability Theory and Its Applications, (Vol. 2) New York: John Wiley. Greenwald, Bruce C. and Joseph E. Stiglitz. (1987). "Financial Market Imperfections and Business Cycles." NBER Working Paper No. 2494. - - ( 1 9 9 0 ) . "Asymmetric Information and the New Theory of the Firm: Financial Constraints and Risk Behavior." American Economic Review 80:160-165. Gron, Anne. (1989). "Capacity Constraints and Cycles In Property-Casualty Insurance Markets." Working paper. Department of Economics, MIT Cambridge, MA. Harrington, Scott E. (1988). "Prices and Profits In the Liability Insurance Market." In R.E. Litan and C. Winston (eds.) Liability: Perspectives and Policy. Washington, DC: The Brookings Institution. Kraus, Alan and Stephen Ross. (1982). "The Determination of Fair Profits for the Property-Liability Insurance Firm." Journal of Finance 33:1015-1028. Myers, Stewart and Richard Cohn. (1987). "Insurance Rate Regulation and the Capital Asset Pricing Model." In J.D. Cummins and S.E. Harrington (eds.) Fair Rate of Return In Property-Liability Insurance. Norwell, MA: Kluwer Academic Publishers. Myers, Stewart and Nicholas Majluf. (1984). "Corporate Financing and Investment Decisions When Firms Have Information That Investors Do Not Have." Journal of Financial Economics 13:187-221.

UNDERWRITING CYCLES IN LIABILITY INSURANCE

127

Rothschild, Michael and Joseph Stiglitz. (1970). "Increasing Risk: I.A. Definition." Journal of Economic Theory 2:225-243. Venezian, Emilio. (1985). "Ratemaking Methods and Profit Cycles In PropertyLiability Insurance." Journal of Risk and Insurance: 477-500. Winter, Ralph. (1988). "The Liability Crisis of 1984-1986 And the Economics of Competitive Insurance Markets." Working Paper. Yale Law School and the University of Toronto.

This page intentionally blank

SELF INSURANCE IN WORKERS' COMPENSATION Richard J. Butler* John D. Worrall

Increasing Self Insurance in Workers' Compensation Employers can fulfill their obligations to provide for workers' compensation coverage by purchasing insurance from a private insurance carrier, or from an insurance fund run by the state or by self-insuring. Eighteen states have state funds. Twelve of these compete with privace insurance carriers for business and are usually referred to as competitive state funds. The other states have exclusive state funds, and private insurance carriers are not permitted to sell workers compensation insurance in those states. Firms that self-insure can usually do so by posting a bond or deposit of securities with the state industrial commission. Most firms are too small to meet the requirements of state law for self-insurance. Group self-

* We wish to thank Eric Eide for competent research help and the College of Family, Home and Social Studies for research support. Useful comments were received from David Appel, Neil Doherty, and Alan Krueger on an earlier draft. Some data help from the NCCI is also greatly appreciated. The opinions expressed here are, however, our own.

129

130

WORKERS' COMPENSATION INSURANCE

insurance is permitted in some states, and firms, usually in the same industry, can jointly self-insure. From 1946 to 1970, losses covered by self-insurance payments have generally bracketed between 12 percent and 15 percent of total losses and have shown fairly slow fluctuations in the aggregate. Since 1970 there has been a more pronounced trend in the proportion of self-insurance, falling from 0.143 of total losses in 1971 to 0.116 in 1973, and then rising monotonically to 0.146 by 1978. Interestingly, this rise in the mid-1970s coincided with the publication and dissemination of the report by the National Commission on State Workmen's Compensation Laws. Understanding why firms choose to self-insure is important both as an example of how firms make choices under uncertainty and as a means for obtaining "clean" estimates of such things as benefit elasticities and loss forecasts. Self-insurance complicates the analysis of the workers compensation program because the self-insuring firm is not a typical insurance risk--it is invariably larger than other firms and often is safer. As more firms self-insure, the observed values of the dependent variables in the sample will be systematically affected whenever the sample information comes from insurance claim data. ~ Systematic truncation of the dependent variable leads to what is called the "sample selection" problem. Consider a (employee) size distribution of firms which is used to generate estimates of how changes in the structure of benefits affect losses in the private insurance market in the upcoming years. Due to economies of scale in the provision of safety, let's assume the largest firms are also the safest. Because they are large they are also most likely to be experience rated, bearing the costs of the claim experience they generate. The moral hazard problem faced by manually rated firms, namely that their individual loss experience is unrelated to their insurance premium, is absent for the larger firms. Because of manual rating, small firms may have less incentive to monitor the utilization of benefits by their employees. Employee incentives are larger for small firms, and the response of the employees from these smaller firms to benefit changes will likely be larger than for similar employees in the larger firms. 2 As self-insurance increases and the size distribution of firms becomes truncated, as the larger firms withdraw from the insurance market, then the estimated benefit elasticities may increase. So unless one controls for the selective nature of the sample, estimated benefit elasticities may well be increasing over time as the larger firms increasingly self-insure. This can also lead to bias forecasts of cost increases for rate-making purposes. If large and safer risks withdraw themselves from the market, then rates based on historical experience may be inadequately low.

SELF INSURANCE IN WORKERS' COMPENSATION

131

Predictions on how changes in the age distribution affect losses can similarly be skewed by the selective withdrawal of firms from the private market into the pool of self-insurers. Larger firms tend to have more experienced, older workers. Again, if safety economies are important and larger firms have lower losses per worker, then the truncation of these large firms by increased self-insurance is likely to decrease the age of the workers in the sample, while simultaneously increasing the average loss experience. The effect of age on losses will be changed by this truncation. Knowing why firms in the aggregate self-insure is essential not only because it allows us to predict the size of the private insurance market (or the size of a state monopolistic fund), but it is also essential for making adjustments for sample selection bias in the estimation of many other facets of the workers' compensation system, where the only sample data comes from the insurance industry. Since this has usually been the case, the model and estimates here provide a useful starting point for future analyses. Ours is, to the best of our knowledge, the first attempt to estimate the determinants of self insurance for the workers compensation system.

Self Insurance and Market Insurance We use expected utility theory to model the uncertainty associated with workplace injuries and disease. Firms are modeled as being risk averse, 3 and are willing to forgo some amount of income in order to reduce the uncertainty associated with compensation claims. Firms have two ways to insure: they can buy insurance in the marketplace (i.e., from a private carrier or from a state fund) at a going rate, or they can self-insure their compensation losses. The insurance market is assumed to be competitive with insurance available at actuarially fair rates. This is not central to any of the results below, but is a useful pedagogical assumption: workers' compensation requires that firms be "fully" insured for their liabilities, a result that also obtains when insurance is sold to risk averse firms at fair rates. We also assume that all firms can trade off current income in order to prevent future losses. To do so is to "self-insure." Self-insurance can be accomplished by adding machine guards, increasing safety training, carefully placing employees in appropriately safe lines of work, etc. Choosing to self insure instead of purchasing market insurance depends on which is cheaper for the firm. Again, for pedagogical reasons we can assume that this tradeoff is fixed to each firm, but that some firms are

132

WORKERS' COMPENSATION INSURANCE

more efficient in providing self-insurance in the sense that they give up relatively less income when protecting themselves against future losses. Except for this difference in their ability to self-insure, firms in a given state and year are assumed to be otherwise identical with respect to insurance incentives--facing the same insurance rates, the same potential losses, and having the same risk preferences. Data we use in this analysis is aggregated to the state level. Our theoretical model has been similarly simplified to accommodate this level of aggregation. Throughout our analysis, we will assume that there is one type of claim that occurs, with resulting liability L and a fixed probability p. While the firms' direct liability, L, for an injury can be changed either by "self-insuring", or by buying market insurance, the probability of a loss is assumed constant. In figure 6 - 1 all firms are assumed to have the same income and losses, with the vertical axis representing the level of income, N, when there is no accident, which is 1 - p percent of the time. The horizontal axis gives the income of the firm when an accident occurs. When no investment in insurance is made, all firms will be at point E in figure 6-1. The curve running through E that is convex to the origin is the "self-insuring" curve which represents the marginal firms' incentive to self-insure. For although each firm has constant costs of self-insuring (and, hence, a linear tradeoff between income in the no-accident state and the accident state), these marginal costs of self insuring vary across firms. 4 The slope of the curve indicates how much income must be forgone in a no-accident state in order to achieve a given level of protection from losses. Hence, the slope indicates the cost of self-insuring to the marginal firm. Those firms that are most efficient at self-insuring are, by definition, able to get the greatest amount of loss prevention for given expenditure of current income. These firms are represented by that part of the "selfinsuring" curve located nearest to point E in figure 6-1; these are the last firms to buy market insurance. Firms with progressively worse terms of trade are indicated by the steeper slope as we move southeast along the curve away from point E. Hence, those firms nearest the 45 degree line buy insurance first. Our assumption of actuarially fair prices is sufficient to guarantee full market or full self-insurance for liability, though these are in fact mandated by law. 5 With risk-averse firms buying insurance at fair market rates, all firms will try to locate on the 45 degree line emanating from the origin, since along this line income is the same whether or not a claim occurs. Figure 6 - 1 , which represents the aggregation of firms with differing self-insuring capabilities, shows the breakdown of firms into

133

S E L F I N S U R A N C E IN W O R K E R S ' C O M P E N S A T I O N N

A

:" N - L set f"~

insurers

Figure 6-1.

mar ket insurers

Theoretical model.

market and self-insuring groups. But care must be taken in interpreting this diagram: by assumption all firms locate at point E with the self

insuring curve only serving to differentiate firms by their marginal costs of self insurance. When the market insurance line, AB, is placed so as to be just tangent to the self-insuring line, at point S, then the incentives to self-insure become obvious. Firms self-insure when the costs of doing so is less than the costs of buying market insurance. All firms located to the left of point S will choose to self-insure. Similarly, all those located to the right of point S will choose to buy market insurance, since market insurance is cheaper than self insurance (the slope of the self-insuring line for these firms is steeper than the slope of the market insurance line). We use the model implicit in figure 6 - 1 to derive our self-insurance estimation equation. We assume that a firm chooses to self-insure when the utility of doing so is greater than the utility of insuring in the market. The utility of income is equivalent whether or not one self-insures, though we indicate the source of insurance on the utility functions with i and s subscripts, for market and self insurance respectively. Let Y

WORKERS' COMPENSATION INSURANCE

134

be uncertain income in the face of job risk, with Cm the unit cost of market insurance and Cs be the unit cost of self-insuring. A firm chooses self-insurance when it maximizes utility:

Us(Y - Cs) > U i ( Y - Cm)

or

U s ( Y - Cs) - U i ( Y - Cm) >

0

(l)

Ui(Y - Cm) is the same for all firms under the assumptions of our model, but Us(Y - C~) varies between firms since the cost of self insuring varies. Because of the variation in C~, we approximate U~(Y - Cs) with a second order Taylor series expansion around C~ = Cm as follows: Us(Y-

Cs) = U s ( Y -

Cm) + U;[C m -

Cs] -Jr

(U;'/2)[Cm - Cs] 2 -- R

(2)

where R is the remainder term in the Taylor Series expansion, and the first and second derivatives of the utility function are evaluated at the Y -- C m level of income. Since Us(Y - C m ) = U i ( Y -- C m ) , substituting equation (2) into (1), and dividing through by the marginal utility of income from self-insurance yields the following condition for self insurance: [Crn -

Cs] --I- (Us'/(2* U s ) ) [ C m -

Cs] 2 -

R/U s > 0

(3)

The probability of self insuring is just

Prob{R/Uj < [ C m - Cs] = Prob{self-insuring}

-J¢- (Us'/(2

* Us))[C m

-

Cs] 2} (4)

If R is distributed across firms as a standard normal random variable, then equation (4) becomes a probit analysis. And if R follows a logistic distribution, then one can use a logit model for equation (4). If one had data on self-insurance at the firm level and proxies for the associated costs of insuring, Cs, then estimating equation (4) would be straight forward since C,,, is constant in a given state. Good proxies for the cost of self-insuring are the size of the firm and the interest rate. The size of firm is an obvious proxy for the Cm - C~ term for two reasons: there is a legal restriction against small firms self-insuring (apparently the law of large numbers reduces the uncertainty of the average type of claim), and there may be safety economies associated with large firms that are not fully priced out in the market so that Cm > Cs. We think that the interest rate may also affect the cost of self-insurance if the higher realized return on reserves invested by the insurance company is not returned to the firm in terms of a lowered rate premium, Cm. If competition between carriers forces the rates charged to reflect these gains, then the interest rate will be statistically insignificant in our empirical analysis. If the market doesn't do this (either because the market isn't fully competitive, or more likely, because capital markets are not perfect

135

SELF INSURANCE IN W O R K E R S ' COMPENSATION

in the sense that borrowing and lending rates differ), then we expect rising interest rates will increase the probability of self insuring. When one aggregates across firms in a given state, then the appropriate proxies for Cm - Cs is the firm size distribution and the market rate of interest. Proxies for the (U"/(2 * U'~))[Cm - Cs]2 term are also readily available. We assume that the term Us'~(2 * U ' ) is a linear function of income as follows U~'/(2 * U~) = ao + a l BUS

(5)

where BUS is the level of business income in the state. The left hand side of equation (5) is a measure of "absolute risk aversion," which decreases with business income if al < 0, and increases if al > 0. When one aggregates over all firms, the [Cm - Cs]2 term becomes the aggregate sample variance in the self-insurance costs between firms. We assume that this variance is proportional to the variance in the size distribution of firms as follows: [Cm -

(6)

Cs] z = 6o'2

with 6 > 0. Hence, the second term in equation (3), multiplying equation (5) and (6) together is written as follows (U"/(2* U j ) ) [ C m

--

Cs] 2 = aoaa 2

+

a l a ( O -2 *

BUS)

(7)

Add to these terms the interest rate and the size distribution of firms for the " C m -- Cs" term and we get for equation (4) Probability{self-insurance} = Prob{R < fl0 + fll FIRM SIZE q-/?2 INTEREST + fl3cr 2

+/?4(0-2 * BUS)} (8)

The expected pattern of signs on the coefficients are fl~ > 0, f12 >/ 0, /73 1> 0, and/?4 > 0 (/?4 = a,a) if absolute risk aversion is increasing in income) and/?4 < 0 if it's decreasing in income. Measuring Firm Size Effects on Self Insurance The data that we use to estimate the model is a cross-section of states during the manufacturing census years of 1954, 1958, 1963, 1967, 1972, 1977, and 1982. The restriction to the census years is necessary in order to get the size distribution of manufacturers--though, of course, we would prefer the size distribution of all firms in the state. The dependent variable is the proportion of dollars paid from self-insuring firms in a given

136

WORKERS' COMPENSATION INSURANCE

Table 6-1. OLS Regression Results Using Alternative Definitions of "Large-" and "Medium-" Sized Firms (absolute t-statistics).

Independent Variables intercept medium ~ large~ medium 2 large 2 variance variance*value group interest rate year effects R2

0.080 (6.83) -0.140 (0.91) 3.180 (1.14) --2.5E-7 (1.13) 3.9E-11 (5.52) 0.036 (3.51) 0.004 (3.31) NO 0.300

0.128 (8.70) -0.021 (0.13) 1.444 (0.51) --3.8E-7 (1.71) 4.2E-11 (5.93) 0.035 (5.93) -YES 0.335

1.55 (6.25) --

0.215 (7.48) --

--

--

-0.389 (3.42) 0.352 (1.34) 5.4E-7 (4.74) 4.9E-11 (6.47) 0.36 (3.61) 0.005 (3.88) NO 0.323

-0.382 (3.35) 0.423 (1.63) 5.5E-7 (4.89) 5.1E-11 (6.86) 0.036 (3.66) -YES 0.358

Notes: 312 observations were employed in these regressions (cross sections from 1954, 1958, 1963, 1967, 1972, 1977, and 1982). medium I = proportion of firms with 100 to 999 employees. large ~ = proportion of firms with 1,000 or more employees. mediumz = proportion of firms with 20 to 250 or more employees. large 2 = proportion of firms with 250 or more employees.

state a n d c o m e s f r o m various issues of the Social Security Bulletin, in an article written a n n u a l l y by D a n Price. T h o u g h o u r discussion o f the results will f r e q u e n t l y p r o c e e d as if the self-insurance is for n u m b e r s o f firms, r a t h e r t h a n dollars o f insurance costs, it is i m p o r t a n t to r e m e m b e r that we can o n l y m e a s u r e the latter. Business i n c o m e is p r o x i e d by real valud a d d e d by m a n u f a c t u r e r s . T h e interest rate data c o m e s f r o m the 1987 Economic Report of the President. T h e m e a n and v a r i a n c e for the size distribution o f firms was calculated f r o m the p u b l i s h e d ( g r o u p e d ) d a t a by a s s u m i n g that all o b s e r v a t i o n s were u n i f o r m l y distributed within each cell. This simplified the calculation o f the variance. H o w e v e r , the p r o b l e m o f c o n d e n s i n g the i m p a c t o f the size distribution o f firms d o w n into a few p a r a m e t e r s remains. T w o

SELF INSURANCE IN WORKERS' COMPENSATION

137

approaches were tried, with results using the first, the "aggregation" approach, presented in table 6-1. In our first attempt at collapsing the impact of the firm size distribution on self-insuring, we aggregated the ten firm size groupings ("cells") into three variables: the proportion of small, the proportion of medium, and the proportion of large-sized firms. The "proportion medium" and the "proportion large" sized firms were included as unconstrained regressors to measure the firm size distribution, where we expected the "proportion large" coefficient to be positive and larger than the "proportion medium" coefficient. Although the signs of the coefficients were as expected in table 6-1, we noted that the coefficients on the firm size variables were extremely sensitive to various groupings of the firm distribution cells. Note, for example, the difference in magnitude and statistical significance in the firm size variables between the first two columns and the last two columns. The estimated self insurance elasticity with respect to the medium sized firm i s - 0 . 0 1 in column 2, and -0.74 in column 4. Similar instabilities in the coefficients (and their implied elasticities) were detected for other groupings and for the various other probability models. Though these estimates give qualitative support to the theory, they do not shed much light on the quantitative effect of firm size on the probability of self-insurance. "Aggregating" the firm size distribution by arbitrarily putting cells together simply does not yield very robust results in this example. An alternative method of summarizing the effect of the firm size distribution is to constrain the coefficients of the individual firm size cell to vary smoothly according to a quadratic equation and then to estimate the terms in the quadratic. This is a variant of the Almon distributed lag procedure, which we call "Hedonic Distribution" approach. To see the difference between the "hedonic" approach and the previous "aggregation" approach consider the ten cells of firm size that we have in the census of manufacturers' number of firms by employment size tables. The first cell in that table, with the greatest number of firms, records the number of firms with one to four employees; the second cell has five to nine; the third has ten to nineteen, etc. Lets indicate these proportions with the variables P1, P 2 , - . . P10. Where the "aggregate" approach would collapse these probabilities into more aggregated groups (such as P6 + P7 + /°8, and P9 + P10), the "hedonic" approach uses all ten cells, but assumes that their associated coefficients vary smoothly from cell to cell according to some quadratic formula. The original model is Proportion Self = ~IP1 + ~b2P2 4- ~b3e3 + ~4P4 q- (o5P5 + ~b6P6 + q~7P7 + ~bsP8 + ~9P9 + q51oPlo

(9)

138

W O R K E R S ' COMPENSATION INSURANCE

but now we define the quadratic coefficients, ~bl - ~b3, in terms of the ~bs as ~j+l

(10)

= I/JO + ~ l j + ~zj 2

Substituting equation (10) into (9), we get Proportion self = ~0Z0 + ~'1Z1 + ~'2Z2 where J-1

J-1

]-1

Zoi ~- 2 (Pj+I/P), Zli = ~ j(Pj+I{P), Z2i = E j2(ej+,/p) j=o

j=o

j=o

(11)

(J = l0 in this example). The hedonic distribution approach not only smooths the effect of the estimated ~bs in equation (9), reducing the number of parameters left to estimate, but also makes possible various tests about pattern of influence that firm size has on the probability of self insuring. 6 For example, setting ~1 = ~2 = 0 and comparing the fit with the unrestricted model using standard F-tests gives a test of the hypothesis that the coefficients are equivalent, that all of the size groups have an equal chance of self-insuring. Excluding the ~'2 term and comparing a model with ~u0 and ~'1 against a model with ~'0 alone is a test of equality (the ~Ul = 0 model) of self-insurance probability across firm size groups against a model of differential incentives to self-insure (if Iffl > 0, then larger firms have higher proabilities, and if ~'1 < 0, then larger firms have relatively lower probabilities than small firms of self-insuring). Finally, models including and excluding the ~u2 term provide tests of the importance of concavity (~'2 < 0) and convexity (~'2 > 0) of the impact of firm size on self-insuring probabilities.

Empirical Results The effect on self-insurance not only of firm size, but also the effects of the interest rate, group self-insurance and value-added on the probability of self-insuring are given in table 6-2. Several models were estimated to check for the robustness of the results to alternative specifications. This included estimating each of the models with year specific effects (in which case, the interest rate has to be deleted since it is our only year-specific regressor), running weighted least squares to adjust for heteroskedasticity in the models, and trying alternative forms for the dependent variable. 7 As indicated in the last two lines of each of the models, we also tested for structural shift in the coefficients after 1972. Many analyses of workers'

SELF INSURANCE

o

139

IN W O R K E R S ' C O M P E N S A T I O N

"p.

UJ 0

i ~ i I

I

v

v

~

~

~

N .~_

~q

~

0Z

¢¢~ l ~

~',

rq

r~ C~ p-.

4~

g ~ ~ , ~ ~ . ~ ~ , ~ ~,da ~ - ~ ---,~ 5

~q

c~

e--

p... e~

.___ e-

7-

N

rm ffl

rr

~ ~ ¢" ~ ~ ' 0 0 ~ "~- " ~ ~ ' ~ ~ " ~ ~ ' ~ "-""~ ~. ,q ~ ~. oR. ~. ~. ~. ~. ~. ~ ~ ~ ~. Z z

~~c~

oo

e--

¢o3 i-

N

140

WORKERS' COMPENSATION INSURANCE

c~

I~

~

c~

m

~

.

.

~

~

. ~ ~ 1

~Z 0

t'~" ""~:)

(,.q

to) ,r-,

~ ' 0 0

o

~o,

~.

k..

c~ ~D

Z~ I--

SELF INSURANCE IN WORKERS' COMPENSATION

141

compensation have found 1972 to be a watershed year, in the sense that the relatively large benefit increases adopted after the report of National Commission on State Workmen's Compensation Laws seem to change incentives under the system. We worried that the post-1973 period might see an increased movement towards self-insurance among the largest firms because of these changes in incentives. Although in all specifications the shift in the Z1 and Z2 variables indicate that the largest firms, indeed, more frequently choose to self-insure after 1972 than before, these shifts in the coefficients were statistically insignificant as indicated in the last two lines at the bottom of the table 6-2. 8 Generally, we find that a 10 percent increase in the interest rate from 5 percent to 5.5 percent leads to about a 3 percent increase in the proportion choosing to self-insure, from about 14 percent to about 14.5 percent (the implied elasticity for the interest rate ranges from about 0.2 to about 0.4 in table 6-2). This is a significant finding, which suggests that firms in our sample may have felt like the interest income on reserves earned by private carriers or state funds were not being fully returned to them in the form of lower premiums; hence, firms were more likely to self insure when the interest rate was higher. This is an interesting and plausible result, which must remain tentative given the limited numbers of years of data that we use in this analysis. We trust the estimates using the year effects much more than those with just the interest rate, since the interest rate is just a particular linear combination of these year effects. Because of this we focus our discussion on the specifications that include these year effects. States that allow group self-insurance 9 have only about an additional 1 percent or 2 percent of their firms choosing to self insure. This effect is generally significant in all of the specifications, although it is quantitatively smaller when we control for heteroskedasticity in the errors than when we just use ordinary least squares regression. The variance terms, reflecting a sort of "risk aversion" in the aggregate, were more puzzling to us. While we expected the variance term by itself to have a positive coefficient, we also expected the interaction of it with output to be negative. A negative coefficient in our specification would be interpreted as indication that absolute risk aversion falls with "business" income, when in fact the positive estimate indicates that it is increasing with income (as proxied by the real value of value added).1° The main variables of focus in our analysis were the implied patterns of firm size effects. Note the relative robustness of these results across the different probability specifications employed in our analysis. Both the Z1 and Z2 variables are statistically significant, indicating that the incentive

142

WORKERS'

COMPENSATION

INSURANCE

e0

I11

II

I l l

IIII

n II

00

ddddooooddZ~ I I I I I I I

°_

oo~ooo~oooZZ

oooooo~oo~Z~

e-

I1,1

I

I

I

I

I

I

I

I

o00 I

I

d d d d d d d d d d ~ I I I I I I I I

I

I

ce-

0

ooooooooooZZ

o

Iit

IIII

rr~ I

I

~E

oooooooooo;~ I

I

I

I

I

I

II

dddcSdddcSdd~ I

m

I

II

o o o o ~ o o ~ o o Z > I

~

~

~

~

0

o

o o o o o o o o o o z z I

~ o ~ o ~ o o ~ Z ~ I l r l l l l

~ o ~ o o

o~ooooooooZZ I

I

I

I

I

I

I

I

I

I

" 5 ®e -

i_~ MI tu~

o

0 °.

~

~

~

~

0

0

0

°~

~ ~ , ~ , ~ , 0

0

©

,~+~

SELF INSURANCE IN WORKERS' COMPENSATION

143

to self-insure is not uniform across firm size groups--indeed, it appears to be a convex function of firm size with larger firms having an increasingly larger estimated firm size effect (that is, Z2 is significantly positive in all of the specifications). However, because of the differences in functional form between these alternative specifications, the implied elasticity of self-insuring with respect to each of the firm size groups need not be convex and, indeed, is not generally convex as indicated by the implied elasticities that are given in table 6 - 3 for the different sized groups. 11 The pattern there is generally of a negative elasticity for the smallest firms, with increasing use of market insurance for the medium-sized firms up through firms with about 50 employees. Then there seems to be decreasing use of market insurance and increasing use of self-insurance with 500 or more employees (based on the results with year effects). For all of these estimates, the "partial" nature of the estimated elasticity must be understood--here, we are holding constant all other firm size proportions and then imagining a 1 percent increase in the ith firm size group when there must be, in fact, offsetting changes in those other proportions (in firm size group j = i). This helps to explain the preponderance of negative elasticities in table 6-3. Although we don't address this issue directly with these estimates, the relatively small elasticities at the largest firm level suggests that it is not the growth of the largest firms that has been responsible for recent shifts in the amount of self-insurance in the market, but the shifting of firms among the various firm size classes that best accounts for these trends.

Notes 1. Studies based on data from the insurance industry (which doesn't include selfinsurers) such as that used in Butler and Worrall (1983), or Worrall and Appel (1982), will suffer from the sample selection basis discussed here. Studies based on noninsurance sources, such as data in the Current Population Survey (used in Krueger, 1988) or Bureau of Labor Statistics (used in Ruser, 1985), would not suffer from these selectivity effects. Note that Butler and Worrall (1983) explicitly recognize the sample selection problem and make the appropriate statistical adjustments in their study. Worrall and Appel (1982) avoid sample selection problems by examining trends from a state in which there is no self-insurance allowed (Texas). 2. Empirical evidence for this has been found in Ruser (1984), Butler and Worrall (1988), and Worrall and Butler (1988). They find a significant negative coefficient estimated for a benefit/firm size interaction variable using very different techniques and samples. 3. Either they are privately held, small in size, or they are unable to diversify all of the financial risks associated with workplace disease or injuries. Given the recent trends-including the unexpected increase in stress claims, changes in what are considered

144

W O R K E R S ' COMPENSATION INSURANCE

occupationally related injuries and diseases, the uncertainities of ratemaking, and the unpredictabilities of benefit increases--the assumption that firms are risk averse seems quite reasonable. 4. We are simplifying incentives under Workers' Compensation enormously. In particular, Victor (1982, 1984) has shown with simulations that the largest firms may have a financial incentive not to self-insure, even when it would seem profitable to do so. Cash flows over time may make it advisable for some large firms to finance their liability with insurance purchased from private carriers. 5. Though whether the requirements for self-insurance, which exhibit incredibly large variations from state to state, are sufficient to guarantee "full" self-insurance is a difficult question that we do not address in this chapter. The adequacy of the self-insurance laws is a separate (though possibly related) issue from our focus: namely, developing a model of when firms choose to buy market insurance and when they choose to self-insure. 6. One can also impose endpoint restrictions when appropriate, see Almon (1965). Note that in our variant of the procedure we completely circumvent one of the major criticisms of the Almon technique, namely, the specification of the "lag length" (the number of previous periods' coefficients to be smoothed for the analysis). Here, the "number of lags" is unambiguously defined as the number of firm size groups. Note also that, unlike the standard application of the Almon procedure, we lose none of our data points in the analysis. 7. Other types of tests were also made on the data. In the tables we report the results using the three month Treasury Bill rate--estimates using the three year Treasury security rates were virtually identical with those reported here. The heteroskedasticity adjustments were those suggested in Maddala (1983, 29-30). Most significantly, we have excluded state specific effects in the analysis. State "effects" are highly correlated with the firm size distribution: relatively large firms appear to cluster in specific states. Hence, adding state specific dummy variables removes much of the inherent differences in the firm size distribution, and has a substantial impact on the estimated parameters of the firm size distribution implying some rather unreasonable patterns of firm size effects. Hence, we exclude state specific effects in the results reported here. 8. There were, in fact, only two significant shifts in the regressors after the 1973 period. The intercept fell slightly, and the effect of the interest rates increased. In the model with all of the dummy variable interaction terms, the effect of the interest rate was negative before 1972, and significantly positive afterwards; the positive shift far outweighing the slight negative effect before 1972. This result is robust across all the specifications reported in this chapter. This shift is probably due to the increase in benefits associated with recommendations of the National Commission on State Workmen's Compensation Laws that benefits be substantially increased. Their report recommended changes that led to an increase in firms' overall liability under Workers Compensation; as a proportion of payroll, the costs of the program about doubled in the decade following the report. With these increased liabilities and associated reserves, firms increasingly found it profitable to self insure in periods were the interest rate was higher. 9. These data were provided to us from NCCI and are for those states that allowed selfinsurance as of 1980, according to the U.S. Chamber of Commerce's Analysis of Workers' Compensation Laws. Data for earlier years were not available. Hence our "group" coefficient is probably best interpreted as a sentiment variable--that those states that did not have this law earlier at least had a sentiment for passage of the law that led to its existence later in the sample period. Excluding the "group" variable from the analysis had no impact on any of the quantitative results of the other regressors. Self-insurance for individual firms is only excluded in Texas (which was dropped from our sample).

SELF INSURANCE IN WORKERS' COMPENSATION

145

10. The estimated positive effect is unexpected. Neil Doherty has suggested that the positive coefficient may represent an "Options Pricing" effect. Larger variances represent more chances not only for large gains but larger losses as well. However, the losses are truncated by the limited liability of most firms (losses are bounded below by the value of their assets), and so the "option" of self insuring is more valuable when the variance is higher. Hence, more firms self insure as the variance increases. 11. These elasticities were all calculated at the sample means of all of the variables. In particular, for f(P) = fl~PFiwhere P is the proportion self insuring, and PF~is the proportion of firms in the ith size group (so that ill. is the coefficient implied by the polynomial as indicated in equation (10) in the text), the following elasticity formulas were used:

dependent variable P

LOG(P)

LOG(P/(1 - e)) PROBIT

elasticityformula PFi *fli/P PF, * Bi

fie*(1 - P)*PF~

gp(.)PF~*flJP

where ~b(.) is the density function for the standard normal evaluated at the sample means of the specification.

References Butler, Richard J. and John D. Worrall. (1988). "Labor Market Theory and the Distribution of Workers' Compensation Costs." In Appel and Borba, (eds.), Workers' Compensation Insurance Pricing. Boston, MA: Kluwer Academic Publishers. Butler, Richard J. (i983). "Wage and Injury Rate Response to Shifting Levels of Workers' Compensation." In John D. Worrall (ed.) Safety and the Work Force: Incentives and Disincentives in Workers' Compensation, Ithaca, NY: I L R Press. Butler, Richard J. and John D. Worrall. (1983). "Workers' Compensation: Benefit and Injury Claim Rates in the Seventies." Review of Economics and Statistics 65(4):580-589. Krueger, Alan B. (1988). "Moral Hazard in Workers' Compensation Insurance." Unpublished manuscript. Princeton University. Maddala, G.S. (1983). Limited Dependent and Qualitative Variables in Econometrics Cambridge: Cambridge University Press. Ruser, John W. (1985). "Workers' Compensation Insurance, Experience Rating, and Occupational Injuries." Rand Journal of Economics 16(4):487-503. Social Security Bulletin, various issues. Victor, Richard B. (1985). "Experience Rating and Workplace Safety." In John D. Worrall and David Appel (eds.) Workers' Compensation Benefits: Adequacy, Equity, and Efficiency. Ithaca, NY: I L R Press. Victor, Richard B., Linda Cohen, and Charles Phelps. (1982). Workers' Com-

pensation and Workplace Safety: Employer Response to Financial Incentives. Santa Monica CA: Rand Corporation.

146

--WORKERS' COMPENSATION INSURANCE

Worrall, John D. and Richard J. Butler. (1988). "Experience Rating Matters." In Appel and Borba (eds.) Workers' Compensation Insurance Pricing. Boston, MA: Kluwer Academic Publishers. Worrall, John D. and David Appel. (1982). "The Wage Replacement Rate and Benefit Utilization in Workers' Compensation Insurance." Journal of Risk and Insurance 49(3):361-371.

7

ON THE USE OF OPTION PRICING MODELS FOR INSURANCE RATE REGULATION Neil A. Doherty* James R. Garven**

Introduction In the past decade, insurance regulatory authorities have witnessed the introduction of various economic or quasi-economic models for estimating the "fair" rate of return on equity. These models succeeded earlier rules of thumb such as the target of 5 percent underwriting profit. The economic model that has received most attention is the Capital Asset Pricing Model (CAPM). However, this model has serious flaws arising from its stylized treatment of taxes, its failure to consider the possibility of ruin, and the difficulties associated with measuring underwriting betas. More recently, models based upon option pricing theory have been applied to insurance pricing in an attempt to address at least some of the flaws of the CAPM. In this chapter, we discuss the use of option-based

*Professor, The Wharton School, University of Pennsylvania, and **Assistant Professor, College and Graduate School of Business, The University of Texas at Austin respectively. We would like to thank David Appel and Steve D'Arcy for their useful comments. Of course, we are solely responsible for any remaining errors.

147

148

WORKERS' COMPENSATIONINSURANCE

models for insurance pricing and will summarize two recent applications by the authors. We will then present specimen applications of these models using workers' compensation data. For comparison, we show comparable results for the CAPM.

Asset Pricing Models and Discounted Cashflow Models Much of the early regulation of insurance prices in the United States appears to have been based on rules of thumb concerning the appropriate level of underwriting profit. This approach does not merit serious discussion here for it was not based on any apparent economic theory. Moreover, the use of positive target ratios for underwriting profit conveys the unnerving implication that policyholders are supposed to pay the insurer for the use of their funds. Such a negative rate of interest is obviously incompatible with the fair return objective. The history of price regulation was summarized recently by the National Association of Insurance Commissioners in their examination of the role of investment income. This report acknowledges the obvious inadequacy of the earlier rule of thumb underwriting profit approach and considers more modern methods that reflect the investment income of the insurer. Again, we give little attention to several of the methods reported that have no clear economic foundations. Instead, we concentrate on methods that attempt to estimate the level of insurance prices that will, through the intermediation process, deliver competitive rates of return to equityholders. The main forms of such models are the Discounted Cash Flow (DCF) models and the Asset Pricing models. These forms are not necessarily mutually exclusive and our treatment here is not intended to be exhaustive. We merely wish to comment on the main features of these approaches.

Discounted Cashflow Models The NCCI Internal Rate of Return Model.

The most primitive DCF model that attempts to derive a competitive price is the Internal Rate of Return (IRR) model as used by the NCCI. The term "primitive" is not meant in a derogatory sense; rather, it conveys that the approach makes minimal use of the competing theorems of financial economics. The NCCI model simply identifies the cashflows to and from the equityholders and from these estimates the internal rate of return on equity. The generality of this approach is both its strength and its weakness. The

ON THE USE OF OPTION PRICING MODELS

149

weakness of this model lies in its failure to specify important economic relationships, for example: 1.

2. 3. 4. 5.

The expected investment income is estimated externally and simply plugged into the model. There is no internal mechanism to ensure that the assumed investment return is competitive or that any specific constraints on investment income for the insurer are incorporated. There is no internal guidance for when some cashflows should be realized. An example is the underwriting profit or loss. Taxes are not modelled internally but are simple imposed upon pretax cashflows. The model offers no competitive return on equity with which to compare the internal rate of return. The calculated IRR's may turn out to irrelevant if the solution has multiple roots.

The fact that the NCCI structure gives little theoretical guidance also lends generality. For example, the calculated IRR can be compared with an estimated equilibrium rate of return estimated from any desired asset pricing model (e.g., the Sharpe/Lintner/Mossin CAPM, Merton's multiperiod CAPM, or Ross's Arbitrage Pricing Model). Another example lies in the possibilities for external modelling of imperfections and constraints on the insurer's investment behavior. Whatever modelling process used, the estimated investment cashflows can simply be inserted into the multiperiod structure to estimate the internal rate of return on equity. The Myers/Cohn Net Present Value Model. While the NCCI model explicitly calculates the internal rate of return on equity for any postulated insurance price, the Myers/Cohn model derives an insurance price that implicitly yields a competitive expected rate of return on equity. In doing so, the Myers and Cohn model makes use of the separability properties of the insurer's principal cashflows. With pricing models such as the CAPM, the component cashflows of a firm may be valued separately under certain conditions. For example, with proportional taxes and zero probability of bankruptcy, the investment and underwriting cashflows are separable in this way. If the investment activities of the firm take place under competitive capital market conditions, their impact on the expected return to shareholders of the insurance firm will be neutral and can be ignored. This implies that the competitive insurance premium may b e calculated simply by discounting the expected cashflows under the insurance contract. The gain from making this assumption is considerable. The

150

WORKERS' COMPENSATION INSURANCE

problems of trial and error calculation, multiple roots, and arbitrary timing of the underwriting profit are avoided. Moreover, the estimated fair insurance price is independent of the particular investment strategy adopted by the insurer. Whether these gains are worth the accepted restrictions must ultimately be considered an empirical matter. Thus, it is not possible to say whether the Myers and Cohn model is superior to the NCCI approach. The Myers/Cohn model, like that of the NCCI, does not avoid a specific choice among asset pricing models. Whereas the NCCI approach required a hurdle rate with which to compare the IRR, so the Myers/ Cohn model requires an appropriate discount rate. An obvious choice, used by the authors, is the CAPM, but other choices also could have been made. It is a glaring inconsistency of these models that, while they both have multiperiod structures, they have both relied on single period asset pricing models to provide discount rates.

Asset Pricing Models Capital Asset Pricing Models. The single period CAPM has been used by many researchers to value an insurance policy or an insurance portfolio (see Haugen and Kronke, 1971; Quirin and Waters, 1975; Cooper, 1974; Biger and Kahane, 1978; Fairley, 1979; and Hill, 1979). This application of the CAPM to insurance pricing also has made use of the separability property of internal cashflows. Thus, the fair insurance price may be derived independently of the investment strategy of the insurer. Again, the models do not ignore investment income, rather they simply assume that it accrues at a competitive expected rate of return and, therefore, the particulars of the portfolio investment strategy are irrelevant. The basic proposition of the CAPM is that capital assets are priced according to their level of systematic risk as measured by beta. Beta is the standardized covariance of the return on the asset and the return on the market portfolio; it measures the degree to which the risk of the asset is not amenable to simple diversification within the investor's portfolio. Thus, the equilibrium expected rate of return for a capital asset i is E(rg) = rf + fli[E(rm) - rf],

where rf = the risk free rate of interest; fli = beta of the i th asset; E ( r m ) = the expected rate of return on the market portfolio.

(1)

ON T H E USE OF OPTION PRICING MODELS

151

The equity of an insurance company will be priced in the same way as all other capital assets under this model. Thus, using equation (1) to derive the expected rate of return on insurance stocks, and recognizing the financial structure of the property-liability insurer, we can derive the expected rate of underwriting profit required to deliver a competitive expected return on equity. The solution, E(ru), is E(ru) = - k r f + flu[E(rm) - rf],

(2)

where flu = the underwriting beta; k --- the funds generating coefficient. A comment is in order on a couple of the terms in equation (2). The underwriting beta is the standardized covariance of the underwriting profit and the return on the market portfolio. It is a direct analogue of the betas on stocks or other capital assets. The funds generating coefficient is a correction for the fact that this is a single period model. With longtailed lines the typical premium dollar is held a considerable period (perhaps several years) before it is paid out as a claim. This implies that the insurer will earn several years of investment income from each dollar of premium. This feature clearly reduces the amount of expected underwriting profit required to deliver a fair expected rate of return on equity. Thus k may be thought of as the average time delay between receipt of premium and the payment of claims. This feature of the model is a somewhat crude correction for the application of a single period model to a multiperiod problem. However, it will be recalled that the DCF models also had their own crude adaption in the use of a single period discount factor to what appeared to be a multiperiod structure. This CAPM insurance pricing model can be (and has been) extended to include the effects of taxes and imperfections in investment performance (see Hill and Modigliani, 1987; and D'Arcy and Doherty, 1988). Despite these extensions, the CPM insurance pricing model has several serious shortcomings: .

2.

As mentioned, the model is crude adaption of a single period model to a multiperiod problem. The model only copes with proportional taxes. In reality, taxes are not proportional. This is extremely important for the insurer since the tax shields on underwriting losses in different lines compete with each other to shelter the taxable investment income. This feature alone is sufficient to break down the additive properties of the insurer's internal cashflows. 1

152

WORKERS' COMPENSATIONINSURANCE

.

The model also assumes away any probability of default. With a nonzero bankruptcy probability and bankruptcy costs, the additive properties of the CAPM also would fail. . Another problem with the model is of a practical nature, the underwriting betas are extremely difficult, if not impossible, to calculate with any degree of accuracy. The two methods, backing out market betas or direct calculation of accounting betas, are fraught with all manner of pitfalls and arbitrary assumptions. The resulting calculations turn out to be quite unstable and subject to extremely wide confidence intervals (see Cummins and Harrington (1985) for a discussion of these issues).

The Arbitrage Pricing Model. Brief mention is also given here of the arbitrage pricing model (APM) as applied to fair insurance pricing by Kraus and Ross (1982). To our knowledge, this has not been applied in rate hearings. Perhaps this is due to its complexity or to the lack of consensus among financial economists as to whether the APM has succeeded the CAPM as the asset pricing paradigm of choice. Like the CAPM, the APM predicts that the capital market prices only undiversifiable risk. However, the APM allows for the existence of several underlying factors (not just the market factor) in the determination of equilibrium rates of return. The strength of the model lies in its generality and its specification of the arbitrage process that leads to an equilibrium in the capital market. Furthermore, the CAPM can under certain assumptions be characterized as a special case of the APM. However, the APM's most glaring weakness lies in its omission of any economic theory that will permit any e x a n t e identification of precisely what the priced factors are. Thus, the researcher is often left to use capital market data both to identify the factors and to test the pricing relationships. The application of the APM to insurance pricing by Kraus and Ross has a major advantage over the models cited above in that it is truly a multiperiod model. Their model specifies a continuous time stochastic process by which the insurer's cashflows are generated. This advantage is potentially important. However, their derivation comes at great cost in terms of application. They do not specify the all important tax relationships, nor is it clear that nonproportional taxes could be incorporated into this model. Like the CAPM, the probability of ruin is also ignored. Finally, the application of the APM to insurance pricing encounters the severe practical problems of measuring betas (factor loadings) that plagued the CAPM. Consequently, arbitrage pricing theory does not address many of the important regulatory problems.

ON THE USE OF OPTION PRICING MODELS

153

Other Models The four models outlined above are not exhaustive; moreover, they have been summarized in an overly simplistic manner that is not altogether flattering. Many of their authors, as well as other researchers (e.g., Derrig, 1987), recognize their shortcomings and have adopted ad hoc solutions to address these problems. But the point is that the solutions are ad hoc and are not an integral part of the models. One is left without any guarantee of internal consistency. The four models were presented as "ideal" types that serve to classify modern approaches to insurance price regulation. The other main class of pricing models that recently has been introduced to insurance pricing is the option class, which is the subject of the remainder of this chapter.

Option Pricing Models in Insurance Some General Comments on Options The option pricing model has recently been used by a number of researchers to value insurance policies. In fact, the idea of using option pricing to value insurance policies is not particularly new. For example, a number of economists have noted that financial guarantees, such as federal deposit insurance and pension insurance, can be modelled as put options, but only recently has the concept been applied directly to property liability insurance and to ratemaking in particular. To our knowledge, the main applications of option pricing techniques to the pricing of property liability insurance are those of Cummins (1988), Derrig (1989), and Doherty and Garven (1986). Cummins derives a sophisticated continuous time/jump process model for valuing the liabilities of solvency guarantee schemes. Although this model is not explicitly applied to the issue of rate regulation, his method could in fact be adapted for this purpose. Derrig's objectives are somewhat different. He is concerned with the estimation of solvency probabilities and premium loadings. Doherty and Garven are directly concerned with derivation of the competitive or fair insurance price and, for this reason, the following summarizes this approach. A typical option contract endows its holder with the privilege to either buy or sell a particular asset at a given price within a specified period of time. It is not an obligation to buy or sell but a choice that may be exercised at the option of the holder. Call options derive value from the possibility that the underlying asset can be purchased at some point in

154

WORKERS' COMPENSATION INSURANCE

time for a price that is less than the market price, thus securing a profit to the holder. Similarly, put options derive value from the possibility that the underlying asset can be sold at some point in time for a price that exceeds its market price. Options are now routinely traded on financial assets both for speculative and for hedging purposes; but this level of trading activity is relatively new, following the establishment of the principle exchanges. One of the important factors in stimulating this activity was the derivation of a class of useable option pricing models. The breakthrough is generally attributed to Black and Scholes (1973). Their invaluable insight was to incorporate the arbitrage mechanism into the pricing model. In principle, it is possible for the holder of an option to construct a hedge portfolio comprising both the option and the underlying asset on which the option is written. If this portfolio is constructed to yield a riskless payoff, then the portfolio must earn the risk free rate of return. Since one knows the total value of portfolio and the prices of each of the components except the option, the value of the option is a simple matter of arithmetic. There are complications since the relative components of the riskless hedge are constantly changing, and the hedge must be continuously maintained. However, by using this framework, Black and Scholes were able to derive closed form solutions for call and put option values. This model has subsequently been successfully adapted to the analysis of contingent claim assets in general. Other researchers have also extended option pricing theory to the case of discrete time models which will discussed later. To understand the application of options to insurance one must look a little more closely at the source of value from holding an option. For sake of example, consider the case of a European call option. The holder of a European call option is endowed with the right to buy a security at a future date for a price agreed upon now. The future date is known as the maturity date, and the agreed price is the exercise, or striking, price. To clarify the example, we will insert values. Suppose the current price of the underlying stock, A0, is $95, the striking price, X, is $100 and the maturity date is six months from now. When the option is purchased, the parties do not know what the price of the stock will be at maturity. The unknown maturity price is denoted A1. If the price at maturity is less than the exercise price of $100, the holder of the option would allow the option to expire worthless, since it would not be rational to purchase an asset for a price in excess of its market value. But if the price at maturity exceeds the exercise price of $100, the holder will find it worthwhile to exercise his option and purchase the stock at a price less than the market price. The difference between the market price at maturity and

ON THE USE OF OPTION PRICING MODELS

$

155

MAX[AI-X,O]

X

Figure 7-1.

TERMINAL VALUE OF UNDERLYING ASSET (At)

Terminal payoff on call option.

the exercise price is gain to the holder. Thus, the holder is in the enviable position of holding a security that yields nonnegative payoffs at maturity, namely, there is only upside potential. Such a "no lose" position has value. This is the value of the option and in an efficient market the option would trade for this value. The payoff to the option described in the previous paragraph are depicted in figure 7-1. The payoff to the holder can be presented alternatively as Terminal Payoff on Call Option = [A1 - X \u

if A1 > X; otherwise.

(3)

Use of Options to Value Insurance Portfolios Now consider the property liability insurance firm. It is easily shown that the payments to the major stakeholders of the insurance firm resemble option payoffs. We will show how option pricing theory can be used to value these payoffs. First, we simply identify the principal cashflow to and from the insurance firm. Imagine that the insurance firm is set up at one period of time (e.g., at the beginning of the year) and runs for one period

156

WORKERS' COMPENSATIONINSURANCE

(e.g., one year) at which time all liabilities are discharged or reserved. At the beginning of the period, the insurer receives surplus (equity) and premiums and pays its marketing and production expenses. Thus, the opening cash flow is Y0 = So + e0,

(4)

where So = the initial surplus; Po = the premiums (net of expenses). At the end of the period, allowing for the accumulation of investment income at a rate ri, the insurer will have the following amount from which to discharge liabilities to the various claimholders, }I1 = So + Po + (So + kPo)ri,

(5)

where k is the funds-generating coefficient as used in the CAPM above. The value Y~ is allocated first to the policyholders to discharge policy claims, second to pay taxes, and the residual is available to the shareholders. For simplicity, we assume that Y1 is nonnegative. Thus, the policyholders' claim, H~, is Hi = MIN[L, Y,].

(6)

The tax claim reflects the statutory tax rate, z, the fact that only a proportion, 0, of investment income is taxable, 2 that any underwriting loss can be used to shield tax on investment income, namely, T1 = MAX[z(0( Y1 - Yo) + Po - L), 0].

(7)

These payoffs have the characteristics of options. This is seen clearly in figure 7-2. For example, the tax claim is simply z times a call option written on the taxable income. The striking price is the level of income at which tax liabilities are perfectly offset by tax writeoffs. The policyholders' claim is the value of the firm's total cashflow minus a call option. The striking price for this call is the face value of losses, L. One unique feature of both these claims is that their striking prices are random. The valuation model we use can easily cope with this feature. Finally, as residual claimholders, the shareholders have the value of the firm minus the policyholders' claim minus the tax call. The present values of the claims of the IRS (To), the policyholders (H0) and the shareholders (Ve) can be presented in the following manner: To = zC[0(Y, - Yo) + e0; L],

(8)

Ho = V(Y1) - C[Y1; L], and

(9)

157

ON THE USE OF OPTION PRICING MODELS

$

payoff to shareholders payoff to policyholders

L

o IRS

/ |

I

L Figure 7-2.

TS

Terminal Value Y1

Payoffs to insurance claimholders.

Ve = C[YI; L] - ~C[O(Y1 - Yo) + P0; L] = C1 - re2,

(10)

where V(}11) is the current market value of cash flow Y1 and C[A; B] is the current market value of a European call option written on an asset with a terminal value of A and exercise price of B. It is worthwhile noting that the sum (To + Ho + Ve) is equal to V(Y1).

Deriving the "Fair" Price for Insurance Equations (8), (9) and (10) show the present values of the claims of the respective parties. The regulatory problem is to price the insurance policies so that the shareholders receive a "fair" rate of return on the equity investment in the insurance firm. Following the precendent set in the Hope Natural Gas decision, we assume that a "fair" return is that which would be earned in a competitive capital market. Such a return would be made for investors if the present value of their future payoff is equal to the value of the capital they invest in the firm, that is, Ve

=

C[YI(P~);

= c~

-

~c~

LI = So.

zC[O(YI(P~) - Yo(P~).) + P~; L] (11)

158

WORKERS' COMPENSATION INSURANCE

This is an implicit solution to the "fair" insurance price. The values of the two calls, C1 and (72, depend among other things on the premiums charged to policyholders. The premiums clearly affect values of the underlying asset on which the two calls are written. Thus, the solution requires that a level of premiums, P~, be chosen so that equation (11) is satisfied. The remaining task required to make calculations of the fair insurance price is to specify how the two options are valued. There are several frameworks that can be chosen. Broadly speaking, we can choose between the continuous time framework originally developed by Black and Scholes or the discrete time framework developed by Rubinstein (1976), Brennan (1979), and Stapleton and Subrahmanyam (1984). Other possibilities include the mixed continuous jump process used by Cummins (1988) for valuing insurance solvency guarantees. We will limit our presentation to discrete time models, noting that the advantages of such an approach to valuation are particularly useful in valuing nontraded claims such as taxable income. 3 The solutions we present are those we derived in a recent paper (Doherty and Garven, 1986). Two solutions are given. The first assumes that the value of the asset portfolio held by the insurer is jointly and normally distributed with aggregate wealth, and that the representative investor displays constant absolute risk aversion. The second solution assumes joint lognormal distribution and constant relative risk aversion. Numerical simulations will be presented for both solutions.

Option Pricing Solutions to the Fair Rate of Return Next, we present the normal and lognormal CAPM and option pricing solutions to the fair rate of return as derived in Doherty and Garven (1986). Assuming normally distributed claims costs and investment returns in conjunction with constant absolute risk aversion, the CAPM solution to the fair premium and rate of return on underwriting is given by equation (12) as

e~=

E(L) (1

-

where E(r*) = [Pg - E(L)]/Pg _ (1-Or) . ( (1 ~) krr + (VJPo)

(12)

E(r*))'

(12a) Or )

rf + 2COV(r.,

rm), and

ON THE USE OF OPTION PRICING MODELS

159

2 = the market price of risk = [E(rm) - rrl/o-2m. Hill and Modigliani derive a comparable expression for E(r*) using the Sharpe/Lintner/Mossin CAPM, and a similar relationship is derived by Fairley. The corresponding option model is given by equation (13) as V e

=

(1 + rf)-l[E(X)N[l~(X)/ax] - rI~(W)N[E(W)/aw] + axn[E(X)/ax] - zawn[~.(W)/aw]],

(13)

where /~(.) = the certainty-equivalent expectation operator; 4 /~(X) = /~(Y1) - I~(L) = So + (So + kPo)rf + Po - /~(L);

= (So + kPo)2a~ + a2L -- 2(S0 + k P o ) C O V ( L , ri); I~(W) = O(So + kPo)rf + Po - /~(L); -~ (So + kP)020262 + 6 2 2(S0 + k P o ) 0 C O V ( L , ri); (7 2w N[.] n[.]

-- the standard normal distribution function; = the standard normal density function.

Although the option model does not provide a closed form solution for P~, we know from our previous analysis that there exists a value of P~ which satisfies the fair return criterion implied by equation (11). Furthermore, P~ may be translated into the fair underwriting profit rate E(r*) by the routine solution of E(r*) - P~ -p jE(L)

(14)

Next, we present the lognormal CAPM and option pricing solutions to the fair rate of return on underwriting. Assuming lognormally distributed claims costs and investment returns in conjunction with constant relative risk aversion, the CAPM solution to the fair premium and rate of return on underwriting is given by equation (15) as P•

E(L) (1 - E(r*))'

(15)

where E(r*) = [P~ - E(L)]/P~ = 1 + [1

(1 - 0 z ) . Or -(1 -- --Q K r f - ( V J P o ) ~ r f ]

exp{ ~,COV[lnL, lnRm]}, and ~u

= the market price of risk

(15a)

160

WORKERS' COMPENSATION INSURANCE E(lnRm)

-- [ n R f

R,,,

VAR(lnRm) = 1 + r,,,;

Rr

= 1 + ry.

1

2'

The corresponding option pricing model is given by equation (16) as V e =

VUN(d U) - zV~N(d~) - R?XPo[N(dU) - rN(dT)],

(16)

where VoV.004 the contemporaneous value of the claim U = V Y -- E L + R ? l p o

ni

V• d~=

= SO +

Rfleo(2 + krf):-

vL;

(L)

RT'E(L ) e x p { - ~COV[InL, lnRm]} ; In (vU/Po) + lnRf + ~ / 2 (7u

d~= d ~ - au; O'u~ the standard deviation of the natural logarithm of U [~ + ~ - 2COV(lnY, lnL)l'a; the standard deviation of the natural logarithm of Y1; the standard deviation of the natural logarithm of L; v r = the contemporaneous value of the claim T

R71[0(So + kPo)rf + 2Po] - VoL; dr= In (Vr/Po) + ln R£ + trot~2 t7t

dr

= cl

ot

= the standard deviation of the natural logarithm of T = [~ay + ~ - 2COV(ln[O( Y, - Y0)], lnL)]la; = the standard deviation of the natural logarithm of O( Y1 - Ii0).

(70Ay

-

As in the case of the normal option pricing model, the lognormal model can be solved numerically in order to determine implicit values for P~ and E(r*). Simulations of the Fair Price for Insurance Using the Option Model and the CAPM In this section we provide a numerical illustration that provides points of comparison between the alternative option-based models presented here

161

ON THE USE OF OPTION PRICING MODELS -U%

C

-12%

,~

-13%

~

-11%

j ~ J j

s

f

C °~ ~

"0 C "~

.i'

-15%"

S

J -16%

f r" J

-17%

$0.25

$0.50

I .-m- CAPM Figure 7 - 3 .

$0.75 $1.00 Initial Equity (Dollars) ~

OPM (Normal)

~

$1.50

$2.00

OPM (Lognormal) I

Vary level of initial equity.

0% .5% ¸

E -10% tr.

.15%

~

.20%.

IT. -25%' C

•~

...30%.

~ Q

-35%':

"0

C

-40%':

-...,.

..45%" I -50%''

1.00

0.50

2.oo

3.oo

4.00

s.oo

Funds Generating Coefficient

I -mFigure 7 - 4 .

CAPM

OPM (Normal)

Vary funds generating coefficient.

~

OPM (Lognormal) I

4.oo

162

WORKERS'

-10%

COMPENSATION

INSURANCE

/.._------~

-15%

.c •,i

-2O%

IIC

.~%

~

-3o%

\

m rr

-35%

t-

-45%

\

\

G

C

-50%

\

-55% 2O%

0%

\ 6O%

4O%

Standard Deviation of Investment Return

[-mFigure 7-5.

OPM (Normal)

C~=M

~

OPM (Lognormal) I

Vary investment standard deviation.

-10%

-IS%

CC 0

i~

-25%

e-

~ -3~ 0 "0 C

;:)

-35%

-40%

So.2s

$0.50

Q.75

1.00

Z.O0

;L50

Standard Deviation of ;laims Costs

I- m Figure 7-6.

CAPM

- ~ - OPU ( ~ l J

Vary claims costs standard deviation.

] i

ON THE

USE OF OPTION

PRICING

163

MODELS

-5%

t.~

-10%

rr

q)

-15%

tic c "r. Q '13 e.

-25%

-30%

9%

7%

5%

Riskless

OPM (Normal)

Figure 7-7.

13%

11%

Rate of Interest ~

OPM (Lognorma|) I i

Vary riskless rate of interest.

-10%' -U%

E -I

-12%.

n,"

-13%

~

-14%

J

(II fie" -15% c ~ 4) "0

S

S

sj f

J

-16% -17%

,1t

Y

-10% .20% ¸ 0.00

0.20

Y 0.60

0.40

0.80

Theta OPM

Figure 7-8.

Vary tax parameter theta.

(Normal)

-gK-- OPM (Lognormal)

J

I

Table 7-1.

Model Parameterization: The Base Case.

Initial Equity (So) Funds-Generating Coefficient (k) Standard Deviation of Investment Returns (tri) Expected Claims Costs (E(L)) Standard Deviation of Claims Costs (at) Correlation Between Investment Returns/Claims Costs Riskless Rate of Interest (rf) Statutory Tax Rate (7) Tax Adjustment Parameter (0) Beta of Investment Portfolio (fli) Expected Return on the Market (E(rm)) Standard Deviation of Market Return (am)

1.00 2.00 0.0427 1.80 0.142 0.114 0.07 0.34 0.60 0.20 0.15 0.2137

(PiL)

Table 7-2. Effects of Variations in Model Parameters Upon the Equilibrium Rate of Return on Underwriting.

CAPM So

E(ru)

0.25 -0.1653 0.50 -0.1619 0.75 -0.1584 1.00 -0.1550 1.50 -0.1481 2.00 -0.1414

Panel A: Effects of Variations in Initial Equity (So). OPM (Normal) OPM (Lognormal) E(ru) -0.1409 -0.1352 -0.1338 -0.1324 -0.1292 -0.1256

P(default) P(no tax) 0.0469 0.0013 0.0000 0.0000 0.0000 0.0000

0.6161 0.5614 0.5239 0.4876 0.4187 0.3566

E(ru) -0.1381 -0.1301 -0.1284 -0.1268 -0.1234 -0.1197

P(default) P(no tax) 0.0873 0.0053 0.0002 0.0000 0.0000 0.0000

0.6430 0.5884 0.5575 0.5287 0.4741 0.4241

Panel B: Effects of Variations in the Funds Generating Coefficient (k). CAPM OPM (Normal) OPM (Lognormal) k 0.50 1.00 2.00 3.00 4.00 5.00 6.00

E(ru)

E(ru)

-0.0298 - 0.0715 -0.1550 -0.2384 - 0.3218 -0.4052 -0.4887

-0.0223 - 0.0596 -0.1324 -0.2036 - 0.2741 -0.3443 -0.4144

P(default) P(no tax) 0.0000 0.0000 0.0000 0.0000 0.0000 0.0002 0.0007

0.3449 0.4005 0.4875 0.5453 0.5836 0.6098 0.6287

E(ru) -0.0167 - 0.0538 -0.1268 -0.1988 - 0.2704 -0.3416 -0.4127

P(default) P(no tax) 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0001

0.3371 0.4089 0.5287 0.6189 0.6877 0.7416 0.7852

Table 7-2.

(Continued).

Panel C: Effects of Variations in the Standard Deviation of Investment Returns (~i) CAPM OPM (Normal) OPM (Lognormal) ~i

0.00 0.20 0.40 0.60

E(ru)

E(r~)

-0.1550 -0.1550 -0.1550 -0.1550

-0.1311 -0.1389 -0.2949 -0.5582

P(default) P(no tax) 0.0000 0.0846 0.2712 0.3730

0.4797 0.5039 0.5941 0.6555

E(ru) -0.1323 -0.1052 -0.1404 -0.2042

P(default) P(no tax) 0.0000 0.0606 0.2530 0.3868

0.5518 0.5364 0.6197 0.6883

Panel D: Effects of Variations in the Standard Deviation of Claims Costs ((Yi) CAPM OPM (Normal) OPM (Lognormal) ~L

E(r~)

0.25 -0.1550 0.50 -0.1550 0.75 -0.1550 1.00 -0.1550 1.50 -0.1550 2.00 -0.1550

E(ru) -0.1261 -0.1107 -0.1057 -0.1177 -0.1950 -0.3555

P(default) P(no tax) 0.0000 0.0103 0.0560 0.1193 0.2402 0.3359

0.4770 0.4685 0.4746 0.4886 0.5232 0.5566

E(ru) -0.1182 -0.1044 -0.1115 -0.1347 -0.1987 -0.2633

P(default) P(no tax) 0.0011 0.0495 0.1469 0.2399 0.3778 0.4680

0.5176 0.5326 0.5721 0.6140 0.6835 0.7331

Panel E: Effects of Variations in the Riskless Rate of Interest (rf). CAPM OPM (Normal) OPM (Lognormal) r~ 0.05 0.07 0.09 0.11 0.13

E(ru)

E(ru)

-0.1111 -0.1550 -0.1986 -0.2419 -0.2849

-0.0913 -0.1324 -0.1733 -0.2140 -0.2546

P(default) P(no tax) 0.0000 0.0000 0.0000 0.0000 0.0000

0.3370 0.4875 0.6316 0.7528 0.8442

E(ru) -0.0854 -0.1268 -0.1680 -0.2091 -0.2500

P(default) P(no tax) 0.0000 0.0000 0.0000 0.0000 0.0000

0.4304 0.5287 0.6212 0.7037 0.7738

Panel F: Effects of Variations in the Tax Parameter Theta (0). CAPM OPM (Normal) OPM (Lognormal) 0 0.00 0.20 0.40 0.60 0.80 1.00

E(ru)

E(ru)

-0.1977 -0.1835 -0.1692 -0.1550 -0.1407 -0.1265

-0.1430 -0.1418 -0.1387 -0.1324 -0.1224 -0.1095

P(default) P(no tax) 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.9436 0.8573 0.6923 0.4875 0.3122 0.1940

E(ru) -0.1394 -0.1381 -0.1342 -0.1270 -0.1166 -0.1037

P(default) P(no tax) 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.9602 0.8585 0.6975 0.5286 0.3869 0.2807

166

WORKERS' COMPENSATION INSURANCE

and the CAPM model. The option-based models were solved iteratively from equations (13) and (16), whereas the normal and lognormal CAPM models were solved from equations (12a) and (15a) after substituting for P0 from equations (12) and (15). 5 The solutions were derived from a set of parameters presented in table 7-1, which are intended as a crude representation of a typical workers compensation insurance business. Table 7 - 2 and figures 7 - 3 to 7 - 8 show the rates of underwriting profit required to deliver a competitive rate of return on equity over different ranges of values for the model parameters. Furthermore, we also show the implied probabilities of insolvency and tax shield redundancy for the option-based models in table 7-2. The points of interest include the following. In general, the optionbased models provide higher rates of underwriting profit than the CAPM. The most useful comparison is between the normal CAPM results and those produced under the joint normal/CARA option pricing model. Since the distributional assumptions are comparable, the differences in fair underwriting profit are explained by the attention paid in our optionpricing model to the probabilities of insolvency and redundant tax shields. Another point of interest is between the insolvency and tax shield redundancy probabilities produced by the two option models. It is well known in the actuarial literature that estimates of the probability mass in the extreme tail of a fitted distribution are highly sensitive to the functional form used, a fact borne out by our simulation results. Both normal and lognormal density functions have been used to describe the insurer's aggregate loss distribution (see Cummins and Nye, 1980), as well as other distributions. Since the probabilities of insolvency and tax shield redundancy are at issue, some prudent curve fitting should influence the choice of regulatory model.

Conclusions and Discussion of the Relative Merits of this Model The option-pricing model developed in this chapter works on straightforward basic principles. The insurance firm must discharge a sequence of liabilities to policyholders, the tax authorities and to its shareholders. The value of these respective claims rests on the value of the insurer's assets. With no assets, the claims would have no value. The option model of insurance pricing presented here values the various claims as options written on the insurer's asset portfolio. The virtue of this approach is that the insurer's asset, being primarily financial securities, are easily valued

ON THE USE OF OPTION PRICING MODELS

167

and that models to derive the value of contingent claims on these assets are readily available. The fair price for insurance is derived by chosing the premium such that the present value of the shareholders' claim is equal to the value of their equity (surplus) investment in the firm. To conclude, we will compare the features of the option-pricing model presented here with earlier models. The option model has three main advantages over earlier models. First, it explicitly models the prospect of ruin and will yield an estimate of the ruin probability implicit in the calculation of the fair insurance price. Second, the model explicitly models the affects of tax shields, and this can have a major impact on the result. The use of proportional taxes in earlier models tends to overstate tax liabilities; the option model appropriately considers the effects of tax shields. A third advantage is that underwriting betas do not have to be directly calculated. Instead, we require an estimate of the covariance between the assessed value of losses and the market value of the insurer's asset portfolio. While there will be some error in providing such an estimate, the method avoids the use of accounting underwriting betas as a proxy for market betas, which is the usual solution with CAPM. We feel that these advantages are significant. The principal disadvantages are three, although we feel the second and the third of these are shared with most earlier models. First, the model is more complex and difficult to understand than the DCF models or the CAPM though not necessarily more difficult than the APM. Given an adversarial setting for the rate regulation process, it will presumably take more effort, and an eloquent advocate, to secure the adoption of this model. The second disadvantage is that this model applies most readily to the single line firm since it is difficult to allocate tax shields and wealth transfers due to insolvency across lines. Other models appear to solve this problem by treating the cashflows in different lines as separable, by treating taxes as proportionate, and ignoring ruin. But this solution is simply turning a blind eye to the real problem. The option model faces the problem and correctly models taxes and ruin but, as yet, can only offer arbitrary solutions to the multiline case. All models face the problems of allocating surplus and expenses across lines. The third problem is that the option-pricing model is a single period model that has crudely recognized the multiperiod nature of price setting by use of the funds generating coefficient, k; but the DCF and CAPM models also faced this problem and had no superior resolution. The only model to bypass this problem was the Kraus/Ross APM; but their model has yet to be used, perhaps because of its complexity and because of the other problems it shares with the CAPM approach to rate regulation.

168

WORKERS' COMPENSATION INSURANCE

T h u s , on b a l a n c e we speculate that, while n o t perfect, the o p t i o n pricing a p p r o a c h will p r o b a b l y d o a b e t t e r j o b o f a p p r o x i m a t i n g fair insurance prices t h a n p r e v i o u s models. 6 M o r e o v e r , the o p t i o n pricing m o d e l p r o v i d e s a m o r e p r o m i s i n g f r a m e w o r k for addressing the r e m a i n i n g unsolved problems.

Notes 1. It may be noted that this objection rests upon the absence of perfect markets for tax arbitrage. 2. This reflects the corporate shield for dividend income and the ability to invest in tax exempt securities. 3. For an elaboration of the advantages of discrete time over continuous time models for valuing nontraded claims such as taxable income, see the excellent discussion by Brennan (1979). 4. Mathematically, the calculation of a certainty-equivalent expectation involves integrating the product of a random variable and its "risk neutral" density function. As shown by Brennan (1979) and Stapleton and Subrahmanyam (1984), the location parameter of a risk neutral density function is chosen so that the expected value of the random variable is its certainty equivalent. In the case of a multivariate risk neutral density function, the same result holds for the location parameters of the marginal distributions. 5. Since the results obtained with the lognormal CAPM do not differ materially from the results obtained with the normal CAPM, only the latter model's results are presented here. 6. See D'Arcy and Garven (1990) for corroborating empirical evidence.

References Biger, N. and Y. Kahane. (1978). "Risk Considerations in Insurance Ratemaking." Journal of Risk and Insurance 45:121-132. Black, F. and M. Scholes. (1973). "The Pricing of Options and Corporate Liabilities." Journal of Political Economy 81:637-659. Brennan, M.J. (1979). "The Pricing of Contingent Claims in Discrete Time Models." Journal of Finance 34:53-68. Cooper, R. (1974). "Investment Return and Property-Liability Insurance Ratemaking." Homewood, IL: Richard D. Irwin. Cummins, J.D. (1988). "Risk Based Premiums for Insurance Guaranty Funds." Journal of Finance 43:823-839. Cummins, J.D. and S. Harrington. (1985). "Property-Liability Insurance Rate Regulation: Estimation of Underwriting Betas Using Quarterly Profit Data." Journal of Risk and Insurance 52:16-43. Cummins, J.D. and D.J. Nye. (1980). "The Stochastic Characteristics of Property-Liability Insurance Underwriting Profits." Journal of Risk and Insurance 48:61-77.

ON THE USE OF OPTION PRICING MODELS

169

D'Arcy, S. and N.A. Doherty. (1988). The Financial Theory of Pricing PropertyLiability Insurance Contracts. Homewood, IL: Richard D. Irwin. D'Arcy, S. and J.R. Garven. (1990). "Property-Liability Insurance Pricing Models: An Empirical Evaluation." Journal of Risk and Insurance (forthcoming). Derrig, R.A. (1989). "Solvency Levels and Risk Loadings Appropriate for Fully Guaranteed Insurance Contracts." In Derrig and Cummins (eds.) Financial Models of Insurance Solvency. Boston, MA: Kluwer-Nijhoff. Derrig, R.A. (1987). "The Use of Investment Income in Massachusetts Private Passenger Automobile and Worker's Compensation Ratemaking." in Cummins and Harrington (eds.) Fair Rate of Return in Property and Liability Insurance. Boston, MA: Kluwer-Nijhoff. Doherty, N.A. and J.R. Garven. (1986). "Price Regulation in Property Liability Insurance: A Contingent Claims Approach." Journal of Finance 41:10311050. Fairley, W. (1979). "Investment Income and Profit Margins in Property-Liability Insurance: Theory and Empirical Results." Bell Journal of Economics 10:192210. Haugen R.A. and C.O. Kronke. (1971). "Rate Regulation and the Cost of Capital in the Insurance Industry." Journal of Financial and Quantitative Analysis 6:1283-1305. Hill, R. (1979). "Profit Regulation in Property-Liability Insurance." Bell Journal of Economics 10:172-191. Hill, R.D. and F. Modigliani. (1987). The Massachusetts Model of Profit Regulation in Non-Life Insurance: An Appraisal and Extensions." In Cummins and Harrington (eds.) Fair Rate of Return in Property and Liability Insurance. Boston, MA: Kluwer-Nijhoff. Kraus, A. and S.A. Ross. (1982). "The Determination of Fair Profits for the Property-Liability Insurance Firm." Journal of Finance 37:1015-1028. Lintner, J. (1965). "The Valuation of Risk Assets and the Selection of Risk Investments in Stock Portfolios and Capital Budgets." Review of Economics and Statistics 47:13-37. Merton, R. (1973). "An Intertemporal Capital Asset Pricing Model." Econometrica 41. Mossin, J. (1966). Equilibrium in a Capital Asset Market." Econometrica 34:768783. Myers, S.C. and R.A. Cohn. (1987). "A Discounted Cash Flow Approach to Property-Liability Insurance Rate Regulation." In Cummins and Harrington (eds.) Fair Rate of Return in Property and Liability Insurance. Boston, MA: Kluwer-Nijhoff. Quirin, G.D. and W.R. Waters. (1975). "Market Efficiency and the Cost of Capital: The Strange Case of Fire and Casualty Insurance Companies." Journal of Finance 30:427-450. Rubinstein, M. (1976). "The Valuation of Uncertain Income Streams and the Pricing of Options." Bell Journal of Economics 7:407-425.

170

WORKERS' COMPENSATION INSURANCE

Sharpe, W.F. (1964). "Capital Asset Prices: A Theory of Market Equilibrium Under Conditions of Risk." Journal of Finance 19:425-442. Stapleton, R.C. and M.G. Subrahmanyam. (1984). "The Valuation of Multivariate Contingent Claims in Discrete Time Models." Journal of Finance 39:207-228.

8

SOME CAVEATS FOR THE USE OF FORECATING MODELS FOR ASSESSING RATES OF RETURN IN WORKERS' COMPENSATION* A.S. Paulson R.L. Boylan L.T. Lim Introduction A model, in the most general sense, is a simplified representation of reality, as it is, or as it is to come to be. The reality of interest may be a phenomenon, a system, a process, a living thing, that is, virtually anything. The simplification embodied in the representation may range from slight to extensive, from simple to highly sophisticated. All models are incorrect in some way; some models are useful; all, even good and correct models, are capable of being used in an inappropriate way. Complex economic models that have not been appropriately validated, preferably in at least two ways, are particularly dangerous. Our discussion and results will be concerned exclusively with models that are constructed to determine the prospective rate of return for the workers' compensation line of an insurance business, although our

* We are indebted to the Massachusetts Rating Bureau for providing data concerning the workers' compensation line and in particular to Dr. Richard A. Derrig for much stimulating discussion concerning the modelling process.

171

172

W O R K E R S ' COMPENSATION INSURANCE

comments will be much more generally applicable. Since our focus is on prospective rates of return, we are concerned with models of reality as it is to come to be, that is, forecasting models. The subject is currently of great interest and importance but is also one of considerable difficulty. Some of the difficulty associated with forecasting models stems from the nature of forecasting itself; even the near future is uncertain and subject to some variation from expectations, while the remote future is very uncertain and subject to marked variation from expectations. Deviations from expectations will be due both to ordinary fluctuations and to fluctuations induced by influential factors that are hitherto unknown or thought to be of no consequence. There are also many further difficulties associated with producing tractable but realistic representations of the very complex environment in which the workers' compensation line of insurance resides. Indeed, the determination of the appropriate degree of detail and structure that must be included in any to-be-useful forecasting modeling endeavor must be cognizant of the influence of known but nonquantifiable factors, because eventually such nonquantifiable or qualitative factors produce a quantitative effect on rates of return. A general discussion, as above, of forecasting models for specifying or assessing rates of return is useful to fix some ideas but is more useful when complemented with a discussion of specific aspects of surplus, loss flows, expense flows, capital market returns, the timing of flows, etc. In future work we shall examine in great detail the problems of producing an estimate of the future late of return for the workers' compensation line of insurance and also provide several recommendations and strategies for dealing with these problems.

Some General Considerations This chapter is an outgrowth of extensive efforts over the past five years to develop stochastic cash flow models to provide prospective premium prices, rates of return and capital surplus requirements, among other things, given accounting cashflows of loss and loss adjustment expense payment patterns, other expense payment patterns, investment portfolio structure and capital market returns. These models are a dynamic representation of future cash flows, incorporating federal tax ramifications (including the 1986 Tax Law), capital structure, etc., and thus represent a statutory-regulatory firm. The current presentation of the model is necessarily brief for this article, but incorporates a reasonable variety of decision points in order to explore many aspects of rate of return ques-

ASSESSING RATES OF RETURN IN WORKERS' COMPENSATION

173

tions. These stochastic cashflow models are the focal point of this work because of our belief that these are the only type of models that can be made sufficiently detailed and dynamic to capture the character of the environment in which the workers' compensation line of insurance resides, on both an individual firm and a statewide (because of regulatory considerations) or industry-wide basis. Even though our investigation is centered on stochastic cashflow models, a variety of additional models were developed or reproduced in order to compare the forecast results of one model vis-a-vis the others. It is difficult to obtain a sound understanding of the workings and results of a complex model that represents a complex reality without making use of the perspective of different models and extreme checks of model behavior. Accordingly, in conjunction with stochastic cashflow models, at the minimum a relatively gross-in-character model should always be constructed to provide a check of the more complex model. Further, since we are effectively concerned with forecasts and their results, and since the cash flow or other detailed models possess both micro-economic and macro-economic characteristics that are meant to match up ultimately with observed macro-economic results, it is necessary to match the results of the forecasts with actual realization. Historical results often provide or should provide the basis for model validation and evaluation. If this cannot be done, then an accounting information system should be set up to accomplish this. Certainly, individual firms will often do this or should do this, but so also should the regulatory process. Even the best short-term economic forecasts are often wrong and long-term economic forecasts fare even more poorly, so forecasts should be calibrated against actual results to ensure that systematically low or high deviations do not become the rule. There is a nonobvious added complication here, however, in that accounting lag and regulatory lag may, in the setting of premiums, destabilize the return process and induce a cyclical effect (Cummins and Outreville, 1987). At this point we expostulate some conclusions concerning the utilization of models for forecasting rate of return on the workers' compensation line of insurance. • Forecasts cannot be proved to be either correct or incorrect. • Construction of forecasting models to help determine necessary prices and "fair" rates of return can be useful. But it is necessary in the modeling process to account for and build in validation and calibration capabilities for both individual firms and for the regulation process. If firms allow for validation and calibration, so also should regulators, if models are to be used to set rates. It is not appropriate in modeling to

174

WORKERS' COMPENSATION INSURANCE

be completely dependent on their validity or truth, especially if there are reasonable ways to reduce the degree of dependence. Prospective models must be linked to the past through the accounting process. • Micro-economic models can be useful, but their usefulness may be heavily overstated if external macro-economic checks against their conclusions are not possible or are not effected. • Too much summarization, or too much detailing, makes for poor and possibly dangerous modeling. Comparison of at least two distinct models for the same forecast allows for an assessment of internal consistency of assumptions and results. This type of exercise also allows for assessing the sensitivity of model outputs to assumptions. It is well known that the future need not be like the past. This is the case in at least two ways. First, the operating environment may be evolving. We may see more rapid or deeper fluctuations in the future than had been the case in the past. The behavioral characteristics of the workers' compensation environment, the fluctuations in losses, or the fluctuations in capital markets may show big variations in a short period of time. An example of such fluctuations is evidenced by the extreme drop in equity and bond values on October 19, 1987. There exist a few quantitative ways of dealing with such fluctuations and we shall briefly discuss these subsequently. Second, the operating environment may be completely changed with no counterpart in the past. The Tax Law of 1986 provides an example of a totally new environment. A major question concerning the effect of the 1986 Tax Law was the magnitude of the effect of the Alternative Minimum Tax (AMT) on profitability. Figures 8 - 1 and 8 - 2 provide the results of two separate cashflow models in determining the loss plus expense to premium ratio, (L + E)/P, that must be maintained to earn the same internal rate of return (IRR) before and after the implementation of the AMT. Both models generate expected time path models and account for the potential for fluctuations in capital markets through the vehicle of a capital asset pricing model approach. Figure 8 - 1 provides the internal rate of return as a function of the loss plus expense to premium ratio under the modelling assumption that surplus is held to support potential liabilities until liabilities are satisfied. As liabilities are satisfied on a quarterly basis, surplus is returned to the firm in the proportion two to one. Flow of expense patterns, payout patterns, indeed, all cash flow patterns have been obtained from empirical studies by the Massachusetts Rating Bureau. Taxes are determined in accordance with the 1986 Tax Reform Act provisions, including the AMT. Underwriting losses are assumed to be deterministic, not stochastic, for figures 8 - 1 and 8-2. These results

175

ASSESSING RATES OF R E T U R N IN W O R K E R S ' COMPENSATION 22 21.8 21.6 21.4 21.2 21 20.8 et-

20.6

n,"

20.4

IRR= 20.34%

20.2 2O 19.8 t9.6

19.4 19.2

i

1.08

i

i

1.1 2

1.10

i

1.14

(L*E)/P []

Without AM]"

+

With AMT

Figure 8 - 1 . Increase in profit provision needed to retain same level of profitability as expected for previous year.

constitute a special case of the Paulson and Dixit (1988) stochastic cashflow models. Figure 8 - 2 provides the internal rate of return as a function of the loss plus expense to premium ratio under the modeling assumption that the surplus is returned in a block at the end of the accounting year and is not held to support the potential liabilities. The assumptions underlying figure 8 - 2 are the same as those of figure 8 - 1 in all other particulars. The results of figure 8 - 2 can be used as a check against the validity of the model underlying figure 8 - 1 and also to determine the sensitivity of the I R R to the surplus return assumption. In spite of the major difference in the surplus return assumption, there is a systematic consistency between the results. Both models indicate that the figure 8 - 1 and figure 8 - 2 firms must reduce the (L + E)/P ratio to roughly the same level to achieve the same IRR. In each case, the effect of the AMT is to produce roughly a 1 percent decrease in the IRR. The influence of A M T on the I R R is also a function of the investment portfolio. The results of figures 8 - 1 and 8 - 2 reflect the investment portfolio given in Table 1. The asset proportions given in table 8 - 1 represent an industry average

176

WORKERS' COMPENSATION INSURANCE 33.4 33.2 33 32.8 32.6 32.4 32.2

r,,

32

he,

31.8 IRR=31.54 %

31.6 31.4 31.2 31 30.8 30.6

I

1.08

1.12

t.10 []

1.1/,

(L.E)/P Without A M I

+

With AMT

Figure 8-2. Increase in profit provision needed to retain same level of profitability as expected for previous year.

Table 8-1. Assumed Industry-Wide Structure of Investment Portfolio and Expected Returns.

Category U.S. Gov't Bonds Other Taxable Bonds Tax Exempt Bonds Stocks: Dividends Capital Gains Other Assets Total

Expected Return on Assets %

% of Total Income

Tax Rates (%)

Taxes as % of Income

8.13 8.49 8.09

13.33 15.42 29.33

34.0 34.0 22.0

4.53 5.24 6.46

2.2

4.91 11.84 18.55

11.16 26.91 3.80

6.8 34.0 27.16

0.75 9.15 1.03

100.0

10.74

100.00

% of Total Assets 17.6 19.5 36.3 24.4

27.16

ASSESSING RATES OF R E T U R N IN W O R K E R S ' COMPENSATION

177

for the beginning of 1987. A M T results will change as the relative proportion of assets held by an "average" firm changes. The A M T results also depend on the rates of return assumed for the assets change: The rates of return used for this study are those quoted in table 8 - 1 and are average rates of return for the beginning of 1987. A variety of different models were used to forecast the influence of AMT on rates of return for workers' compensation. The results concerning rates of return vary markedly as a function of the assumptions, even without the stochastic influences due to underwriting losses. Unfortunately, there is no possibility of using historical data against which to calibrate any model. A final note in this regard is that once the rules have been specified, the firms that underwrite workers' compensation will have begun to evolve so as to reduce any negative impact of the 1986 Tax Law and the AMT.

The Underwriting Component of Rate of Return Models The complex environment in which workers' compensation resides permits the production of a wide variety of models for a rate of return forecast. These forecasts can be useful for product pricing, that is, determination of premiums, for regulatory consideration and for loss control considerations. The modeling must address two components: the underwriting side of the workers' compensation business, and the investment of funds that results from the flow of premiums and from the supporting surplus. The regulatory process, a third and very important component, influences both the underwriting and the investment side and is influenced in turn by these components. The initial formulation of formal forecasting models will usually incorporate the underwriting and investment components. Additional layers of complexity are added at subsequent stages of the modeling process. We first consider the underwriting side of the business. In order to ultimately forecast losses to be experienced in underwriting a cohort of business, say from January 1, 1988 to December 31, 1988, we will first forecast the total losses that will be incurred. This can be done in several ways, depending on the context in which the forecast is being made. For a company, consider the situation in which the total losses are restricted to those of a single state. The company can base the forecast on an internally developed historical series of total losses, or the company can develop the forecast from an internally developed historical series of frequency of losses and severity of losses.

178

W O R K E R S ' COMPENSATION INSURANCE

Total losses may also be forecast from a historical series of total losses at the statewide level, even though there may be a one or two year lag in the availability of the information. The use of statewide workers' compensation information provides at least an additional check on the consistency of forecasts. A forecast of statewide total losses will involve an error of the forecast. The error inherent in any forecast has the net effect of spreading, or rescaling the total loss distribution while leaving the mean of the distribution invariant. The total loss distribution of the forecast may be summarized in terms of certain moments and momentrelated quantities. A summarization of this type is not only convenient but also retains nearly all of the useful information concerning total loss fluctuations. The error of the forecast will be more important for an analysis of fluctuations at the statewide level than will the distribution of total losses. At the individual underwriting firm level, the total loss distribution will be important in the description or representation of loss experience, possibly more so than the error of the forecast. The forecast error is usually represented as an additive normal (Gaussian) random variable. For an individual firm writing workers' compensation, the coefficient of variation of the total loss distribution can be large, (for example, greater than unity), depending on the size of the firm or, equivalently, on the fraction of the state-wide losses the firm experiences. The relatively smaller the firm, as a fraction of the state-wide total, the relatively larger the fluctuations about the mean of the firm's total loss distribution. The following are summary statistics for estimated indemnity costs at a statewide level of Permanent Partial Disability Claims for Workers' Compensation, 1979-1980, as reported in 1982: Number of Claims Total Amount Mean Mode Median

3785 $90.4 mil. $23,900 $10,000 $20,000

Maximum Standard Deviation, s Coeff. of Variation, c.v. Coeff. of Skewness, c.s. Coeff. of Kurtosis, c.k.

$360,000 $16,900 0.71 6.50 93.80

Let s, c.v., c.s., c.k. denote the standard deviation, the coefficient of variation, the coefficient of skewness, the coefficient of kurtosis, respectively for an individual loss. Let c. v (S), c.s. (S), c.k. (S) denote the coefficient of variation, coefficient of skewness, coefficient of kurtosis, respectively for a sum of m losses. Thus, a firm that holds 10 percent of the workers' compensation market can expect to experience about m =

179

ASSESSING RATES OF R E T U R N IN W O R K E R S ' COMPENSATION

400 (approximately 10 percent of 3,785 claims) losses. For this firm, we expect to experience a mean loss of (400) ($23,900) and (Abramowitz and Stegun, 935) c.v.(S)

= m -1/2 c . v . ,

c.v.(S)

= 0.036,

c.s.(S)

= m -1/2 c . s . ,

c.s.(S)

= 0.325,

c.k.(S)

c.k.(S)

= m -1 c.k.,

= 0.235.

Thus, in this relatively benign situation the total loss distribution will be moderately skewed to the right with a fairly thin tail. For the total workers' compensation line, a much less homogenous situation, it would be common to find coefficients of variation in excess of 50, coefficients of skewness in excess of 200, and coefficients of kurtosis in excess of 100,000. A firm that averages 5,000 claims per year, for the specifications immediately above, that is, m = 5,000, 50

C.V.

(5,000)1/2

0.71,

c.s.(S)=2.83,

c.k.(S)

20.00,

a situation characterized by very substantial fluctuations on an expected basis. If the state-wide total number of claims is 100,000, then the total loss distribution state-wide will have a 50 c.v.(S)

-

(100,000)1/2 = 0.16,

c.s.(S)

= 0.63,

c.k.(S)

= 1.00.

Since the coefficient of variation is the dominant factor in determining the magnitude of potential fluctuations, the example above clearly illustrates the fact that fluctuations about the mean total loss is much less at the state-wide level than at the individual firm level. Accordingly, if statewide characteristics are used to set rates of return, the risk to any firm in the state is much understated and may not be priced properly, since the state enjoys a level of concentration with corresponding fluctuation control not shared by any individual firm. This set-up suggests the evaluation of the workers' compensation line of business may be regarded as a sort of option pricing problem, under non-standard conditions. The non-Gaussian character of the fluctuations is perhaps the most important of the nonstandard conditions. Given a known empirical distribution, the fluctuations in random samples drawn from that distribution may at least be partially characterized by the variance or standard deviation of that distribution. In a forecasting setting that distribution is known only from past data, and the future need not completely reflect the past. If n distinct and independent past observations are available, then an adjustment to

180

WORKERS' COMPENSATIONINSURANCE

account for the possibility of greater fluctuations in a future one-year period is usually made by inflating the variance o-2 in accordance with

By making this modification, stochastic cash flow simulations will account for the uncertainty in moving to the future and ensure that this element of risk will be partially priced. Alternate modifications can be developed for forecasts more than one year in the future (Box and Jenkins, 1984; see also Draper and Smith for an alternate illustration in terms of extrapolation and regression analysis). Interestingly, but expected from standard financial theory, if profit margins are set at the state-wide level, stochastic cash flow simulations indicate that the understated risk borne by firms smaller than 100 percent of the state-wide total is compensated by these firms achieving a higher internal rate of return (see figure 8-3). But as figures 8 - 4 , 8 - 5 , and 8 - 6 show, the fluctuation in the internal rate of return is now much larger and the magnitude of the fluctuations increase with decreasing size of the o O c~ a6

o

o uS cn

S/•ize

of f i r m =2.5%

o o0 o

~oo ~LO or" n," o ~e6 -.4" O.

C~

O.

= --------~_

--m ~oo'/.

o

c50.0 0.5 1.0 1.5 . .2.0 . 2.5 . 3.0 3.5 2.0 ' 1..5 5.0 PREHIUM-SURPLUS RATIO Figure 8 - 3 . Average IRRs at various P/S levels. Number of coborts at risk = 1,000. The size of the firm figure represents the size of an individual firm as a fraction of a statewide firm (i.e., 100%).

ASSESSING RATES OF RETURN IN WORKERS' COMPENSATION

181

C:)

L¢)

~2

.12] /

~

f i n -~

m.

(,i

-'iN . . . . . []

Lr3

(2)

0.0

Figure 8-4.

0.5

1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 PREMIUM-SURPLUS RATIO

Average IRRs, +I standard deviation at various P/S levels.

N u m b e r of cohorts at risk = 1,000. Size of firm = 1 0 0 % .

/,/I

E]

// // (IC rr

C)

""Eg- . . . . E] i

0.0

Figure 8-5.

0.5

i

1.0

i

i

i

i

i

i

,

1.5 2.0 2.5 3.0 3.5 4.0 Z,.5 5.0 PREMIUM-SURPLUS RATIO

Average IRRs, _ I standard deviation at various P/S levels.

N u m b e r of cohorts at risk = 1,000. Size of firm = 1 0 % .

O

/ / / / ~~

O Q °

O OO

n~" n,-

O c.O

o o i

i

"N

o m

i

0.0

0.5

1.0

i

i

1.5 2.0 2.5 3.0 PREMIUM-SURPLUS

3,5 4.0 RATIO

4.5

5.0

Figure 8 - 6 . Average IRRs, _+1 standard deviation at various P/S levels. Number of cohorts at risk = 1,000. Size of firm = 2.5%. o ¢,3

o

n,-

~o Q

o Q o.,.=

%

1

0.7

0.8 0.9 1.0 LIABILITIES-PREMIUM

I. I 1.2 RATIO

1.3

Figure 8 - 7 . Average IRRs, +1 standard deviation at various k = (L + E ) / P levels. Number of cohorts at risk = 100. Size of firm = 100%.

ASSESSING RATES OF R E T U R N IN W O R K E R S ' COMPENSATION (:3 O'i P,I

183

\

C~

O

(:3

\\\\ \\\\

O

O E: re"

\\\ \\\\

Q

O

06

0.

u~

O

Q J

1

0.7

i

i

0.8 0.9 1.0 1.1 1.2 LIABILITIES-PREMIUM RATIO

1.3

Figure 8-8. Average IRRs, +1 standard deviation at various k = (L + E ) / P levels. Number of cohorts at risk = 100. Size of firm = 10%.

firm, namely, the state-wide firm. The probability of capital impairment and probability of ruin rapidly increases with decreasing firm size. These fluctuations are based on the assumption that the coefficient of variation for the total loss distribution is 0.3, the total loss distribution is symmetric, and the fluctuations inherent in capital market returns are implicitly accounted for. The influence of size and internal rate of return is also depicted in figures 8 - 7 and 8 - 8 . Again, it is shown that smaller firms will have a larger expected IRR but also larger variability.

T h e Investment C o m p o n e n t of Rate of Return Models

Investment income on various instruments as measured by annual returns exhibits marked volatility or variability over the years. This is clearly shown in table 8 - 2 , taken from Ibbotson and Sinquefield (1982). From table 8 - 2 we see that the coefficient of variation associated with an investment in large common stocks is approximately 2, the coefficient

184 Table 8-2.

WORKERS' COMPENSATIONINSURANCE Investment Total Annual Returns, 1926 to 1981.

Series

Common Stock Long Term Corporate Bonds Long Term Government Bonds U.S. Treasury Bills Small Stocks Inflation

Arithmetic Mean

Standard Deviation

11.4%

21.9%

3.7%

5.6%

3.1% 3.1% 18.1% 3.1%

5.7% 3.1% 37.3% 5.1%

of variation associated with an investment in long-term corporate bonds is greater than unity, etc. Accordingly, we should expect considerable variability on returns, on assets, or portfolios of assets, held for the purpose of producing income on both a forecast and on a realized basis, whether the return period is quarterly or annually. For example, consider the portfolio structure, portfolio proportions, and the anticipated returns as given in table 8-1. This "average" portfolio was used in determining the influence of investment income on rate of return for workers' compensation for 1987 to 1988. The time path of returns is, on average, upward, but the fluctuations about this average time path can be very considerable, as judged by the coefficient of variation of the individual instruments. The fluctuations to be experienced in portfolio returns need to be considered in conjunction with the fluctuations to be experienced in loss and loss adjustment expense experience. On a future expectation basis, models for setting premiums and rates of return show that an expected return time path can vary substantially, both upward and downward. The use of models that focus on the expected time paths of underwriting experience and investment experience, even though purportedly allowing for fluctuations, will not correctly account for fluctuations, since the underwriting sides and the investment sides of the workers' compensation line fluctuate jointly as a unit, and not independently. Further, expected time path models are not capable of representing the influence of nonlinearities, boundaries, and constraints on the evolving and dynamic time paths which produce the ultimate returns. Callable bonds and correlations between stock and bond returns provide examples of nonlinearities and constraints on returns that may come into play when fluctuations in financial markets are sum-

ASSESSING RATES OF RETURN IN WORKERS' COMPENSATION

185

marized in an explicit form. To successfully capture the constraints on forecasting rates of return imposed by financial instruments, such as a callable bond, it is necessary that a stochastic cash flow model have the capability of simulating yield curves for treasuries. If bonds are callable, yields and durations become random variables. Finally, returns on a portfolio of instruments will not vary independently, the usual assumption of many expected time path models.

Some Final Conclusions The models that we have discussed treat the workers' compensation line as a completely isolated entity. Clearly, this is in no way a reflection of the workers' compensation environment. Because of this isolation, it is not clear whether the severe "end-effects" of start-up and close-down produce any distortions in forecasts of rates of return vis-h-vis actual realizations. This seems to be a very difficult issue to sort out without having dedicated and extensive accounting information systems in place. In general, it may be necessary to impose extensive accounting information systems at the state-wide and individual firm levels to make effective use of models for forecasting and setting profit margins. With an extensive information system in place, regulatory entities could utilize forecasting models of rates of return and make adjustments based on prior experience, say over a three to five year period.

References Abramowitz, M. and I. Stegun. (1964). Handbook of Mathematical Functions, Tables and Graphs. Washington, DC: U.S. Government Printing Office. Box, G.E.P. and G. Jenkins. (1984). Times Series Analysis: Forecasting and Control. San Francisco CA: Holden Day. Cummins, J.D. and J.F. Outreville. (1987). "An International Analysis of Underwriting Cycles in Property-liability Insurance." The Journal of Risk and Insurance: 246-262. Doherty, N.A. and J.R. Garven. (1986). "Price Regulation in Property-liability Insurance: A contingent Claims Approach. The Journal of Finance: 10111030. Draper, N. and H. Smith. (1985). Applied Regression Analysis. New York: Wiley. Ibbotson, R. and R. Sinquefield. (1982). Stocks, Bonds, Bills, and Inflation: The Past, and the Future. Charlottesville: Financial Analysis Research Foundation.

186

WORKERS' COMPENSATION INSURANCE

Paulson, A.S. and R.V.S. Dixit. (1988). "Cash Flow Simulation Models for Premium and Surplus Analysis." In J.D. Cummins and R.A. Derrig (eds.) Financial Models of Insurance Solvency. Norwell, MA: Kluwer Academic Publishers. Paulson, A.S. and R.V.S. Dixit. (1988). "Some General Approaches to Computing Total Loss Distributions and the Probability of Ruin. In J.D. Cummins and R.A. Derrig (eds.), Financial Models of Insurance Solvency. Norwell, MA: Kluwer Academic Publishers. Pentikainen, T. and J. Rantala. (1982). Solvency of Insurers and Equalization Reserves, (Vol. I and II). Helsinki: Insurance Publishing Company.

9

LEVERAGE, INTEREST RATES, AND WORKERS' COMPENSATION SURVIVAL John D. Worrall Richard J. Butler David Durbin David Appel

Introduction Potential insolvency has always played an important role in insurance regulation. In fact, concerns about the public interest are such that the insurance industry is required to establish guarantee funds so that, in the event a particular company is unable to meet its obligations to policyholders, the insureds would still be covered. Even so, the number and magnitude of insurance carrier insolvencies have risen dramatically in the past few years. Probably the most serious was the Mission Insurance Company in 1987. The original cost was estimated at $520 million, which has since grown to be in excess of $1.5 billion (New York Times, November 15, 1988). In addition, the size of the guarantee funds has grown eleven-fold in just the past four years to approximately $917 million in 1987. While regulators are clearly concerned with solvency, they also are required to approve prices (in the regulated lines) that are "not excessive." Thus, they are asked to perform a political balancing act that has potentially severe ramifications. In order to reduce the profit and contingency loadings that are proposed in rate proceedings, regulators 187

188

WORKERS' COMPENSATION INSURANCE

may impose unrealistic assumptions about financial conditions that will prevail in insurance markets. This becomes especially important for lines of insurance, such as workers compensation, where liability payments may stretch out over a number of years. Indeed, assumptions on leverage, interest rates, and the cost of capital can be critical in determining the final price of insurance in regulatory proceedings. In such proceedings, the government is, in effect, mandating a price ceiling for contracts written over a future time interval. If this constraint is binding, both nominal and real insurance prices and the quality and quantity of insurance services provided can be affected. In the absence of government regulation, the market would determine the optimal price of insurance, as well as the rates of profit, leverage, and insolvency. Prices could adjust relatively rapidly to changing market conditions, particularly to changes in the nominal rates of return on financial placements. Consider a set of contracts written to cover the time period t to t + 1. If there is perfect certainty with respect to claim frequency and severity, as well as the time of claim occurrence and payment dates, an insurer can match the duration of financial placements with the duration of liabilities, that is, an insurer could immunize the liabilities by matching them to financial placements. In such a world, the insurer knows e x a n t e the rate of return generated on its book of business at various prices. With perfect certainty and perfect capital markets, leverage loses its meaning, surplus its purpose, and insurance our interest. Actual results in real financial markets have varied considerably from expected results, both in the aggregate and across firms in the industry. The underwriting margins postulated e x a n t e in rate hearings have generally not been achieved. Insurers are, in fact, uncertain about the frequency, severity, and time payment of claims to be made against contracts written over a future time interval. And no one knows with perfect certainty what set of interest rates will prevail over future time intervals. Yet even in the face of this experience, regulators in some jurisdictions continue to adopt optimistic assumptions in order to reduce insurance prices. In this chapter, we shall consider the implications of these regulatory constraints for potential insolvency or impairment of firms' capital base or surplus. In particular, we examine whether extreme regulatory assumptions increase insolvency risk by examining the distribution of loss ratios of actual state data, and by simulating the effects of the regulatory constraints estimate the probability of sustaining sufficient losses to exhaust surplus. In the next section, we describe a method by which the actual data are adjusted to yield "transformed" loss ratios. We then

WORKERS' COMPENSATION SURVIVAL

189

characterize the distribution of these transformed loss ratios and estimate the survivor function (the complement of the distribution function) at alternative values of the transformed loss ratio. These results are used in the final section to estimate the probabilities of insurer insolvency given the imposition of severe regulatory assumptions.

The Internal Rate of Return and Accident Year Loss Ratio Adjustments In property-casualty ratemaking for regulated lines of insurance there is (often ?) a provision for profit and contingencies. This provision can depend upon, among other things, leverage, the post tax rates of return on assets, and the equity cost of capital. The higher the leverage ceteris paribus the greater the risk of insolvency, but the higher the potential rate of return that can be earned on the insurance transaction. Consider an insurer with a premium to surplus ratio of 2 to 1, no investment income, and a 34 percent corporate tax rate. For such a firm, a 50 percent underwriting loss would cause serious surplus impairment (i.e., its surplus would fall 66 percent). Another insurer with a premium to surplus ratio of 3 to 1 having the same underwriting experience would be virtually wiped out. This relationship can be expressed in terms of the amount of surplus remaining as a function of underwriting experience, investment experience, tax rates and leverage, that is, SR = ([1 - ( L R + ER)](1 - 7",) + 1}1(1 - Ti)} * ( P / S ) + 1

(1)

where S R is surplus remaining in percentage terms, L R is the pure loss ratio, E R is the expense ratio, Tu is the corporate tax rate, I Y is investment income, Ti is the tax rate on investments, and P/S is the premium to surplus ratio. As noted above, it is our intention to estimate insolvency probabilities under alternative regulatory regimes. For example, we can compare the probability of insolvency or surplus impairment under the traditional workers compensation 2.5 percent profit and contingency factor versus the factor that would be in the rates given extreme regulatory assumptions. However, the specific value of the profit and contingency factor is conditional on the specific input assumptions imposed by the regulator. In the work below, we investigate the implications of extreme assumptions on leverage, investment yields and the cost of capital. It should be noted that while we term these assumptions extreme, each one has been

190

WORKERS' COMPENSATION INSURANCE

implicitly or explicitly imposed in regulatory proceedings in recent years. Thus, our work holds more than academic interest to insurers writing regulated lines of business in a number of states.

The Internal Rate of Return Model The major portion of this chapter is devoted to estimating the probability of achieving a loss ratio sufficient to produce insolvency after consideration of tax credits (assuming underwriting losses) and investment income. By necessity, this must be estimated assuming a "stand alone" workers compensation carrier writing business in a single state. While this is admittedly an uncommon situation, it is not unheard of in the industry, nor does it deviate from characterizations used by regulators in setting rates. In order to estimate this probability under the extreme assumptions referred to above, we must analyze a distribution of loss ratios that would have ocurred under such regulatory constraint. It is impossible to do this using observed data inasmuch as the observed loss ratios in most jurisdications have been generated under a 2.5 percent profit and contingency loading. Thus, we must transform the observed data to reflect the loss ratios that would have occurred had the regulatory imposed unrealistic assumptions (discussed below), the effect of which produce substantial negative profit and contingency factors. The internal rate of return model (IRR) is the basis for the loss ratio transformation. The I R R model has been used by regulators in some states to determine the profit factor in the rates consistent with a target rate of return on equity. It can be used, of course, to estimate the profit load that will achieve the required cost of capital under any assumptions. In our case, we estimate that profit factor that will produce a 13 percent return on equity assuming the carrier writes at a 3 to 1 premium to surplus ratio and can earn 9 percent after tax on its investment portfolio. We discuss each of these assumptions briefly below.

Leverage The observed writings to surplus ratio for the Property Casualty industry has traditionally been around 2 to 1. At the same time, the loss reserve to surplus ratios have, in recent years, hovered at roughly the same value. This occurs because property casualty insurance, in the aggregate, is a mix of long and short tailed lines; the long tailed lines tend to produce high levels of reserves relative to premiums, while the short tailed lines

WORKERS' COMPENSATION SURVIVAL

191

produce low reserve to premium ratios. When aggregated across all lines, both the reserve and premium to surplus ratios are roughly the same. Notice that a constant premium to surplus ratio implies significantly different reserve to surplus levels across lines. For example, workers compensation has a reserve to premium ratio of approximately 1.9; thus, assuming it is written at a premium to surplus of 2 implies a reserve to surplus ratio of 3.8 ( R / P * P / S = R/S). Contrast this with a line like automobile physical damage that has a very short tail, and thus produces a reserve to premium ratio that is very low (about 1/8 in 1987). Such a line produces a reserve to surplus ratio of 2/8 under the same premium to surplus assumption as above. The internal rate of return models assumes that surplus is allocated by line proportional to reserves. Therefore, we must solve by iteration for the reserve to surplus ratio that is consistent with any specified premium to surplus ratio. While an often articulated standard is a premium to surplus ratio of 2, suppose that regulators impose a 3 to 1 premium to surplus constraint. The impact of changing leverage on the I R R and, hence, on insurer insolvency probabilities can be examined.

Investment Yields The yields on investments that insurers can earn, if all loss and premium flow expectations are met and the term structure of interest rates is constant, may be calculated by using the aggregate loss payout pattern for a specific state. The idea is that insurers can match the duration of the loss payouts with the duration of investments, to immunize their cash flows. However, since ratemaking (and consequently the IRR) are prospective analyses, prospective yields on various securities are needed. We compute the yields on investments assuming a crude form of immunization, maturity matching. Thus, we assume that insurers purchase a diversified portfolio of securities that have terms equal to the average length of loss payout. The rate of return for this portfolio can be calculated by applying current yields by type of financial instrument to their percentage distribution in the portfolio and the adjusting for their current tax rates.

Cost of Capital The primary application of the I R R model is in investment decision making: the firm should accept all projects for which the I R R exceeds the

192

WORKERS' COMPENSATION INSURANCE

cost of capital. However, in ratemaking the concern of regulators is to set prices that allow a reasonable or fair rate of return. If rates of return on insurance operations are not sufficient to induce investment, the continuing viability of such operations will be compromised. Usually, the I R R model is used to solve for the rate of return to investing in the insurance operation. However, by combining the assumptions on leverage and investment yields, the I R R model can be solved for the profit and contingency factor consistent with a selected target rate of return. This backwards (iterative) process involves using leverage assumptions (assuming different rates of growth for the business) until the real premium to surplus ratio and allowable yield to return (IRR) are observed. We are, therefore, able to combine the regulatory constraints together with a final assumption that regulators allow (at least implicitly) a profit and contingency factor designed to generate, say, a yield to maturity of 13.0 percent on the workers compensation business, to determine what would happen to the target loss ratio, ceteris paribus. Equation (1) can then be evaluated to determine the surplus impairment and thus the probability of insolvency.

An Example To illustrate the above framework, and how we may transform individual firm loss ratios to simulate the effects of the extreme regulatory constraints, the internal rate of return model has been run for workers compensation operations in the state of Alabama employing the net expense provisions contained in the latest rate filing. The Alabama provisions are: Target Loss Ratio State Taxes General Expenses Production Profit and Contingency Factor

75.0 5.8 6.2 10.5 2.5

The Internal Rate of Return algorithm was solved iteratively to find the reserve to surplus ratio that generates a premium to surplus ratio of 3 to 1, and under this scenario, determine the profit and contingency factor

WORKERS'

COMPENSATION

193

SURVIVAL

rr

0 O-

tO

= 0

r~ ~0

tX

kU v

E .0

~6 k 0 0 t-

x O9 ttO

tO_

E 0

0 tO

~ o ~ ~= . ~

t-i

. O 0 ~ O ~

0

0

0

"'~

.... °

t-

.o_ Z

I O~ °_ r X

U.I

0 f.

L; . . . .

194

WORKERS' COMPENSATION INSURANCE

nX

0

cO

a

exl

eX ILl

E

o0

e-

t-

cO e-

~o exl

~e

WORKERS'

e',l

t'-.

e~

c',l

I

I---

COMPENSATION

SURVIVAL

195

196

WORKERS' COMPENSATION INSURANCE

that would prevail given the regulatory constraints of a 13 percent total return on the business and a 9 percent post-tax return on assets. For comparison to the regulatory constraints, exhibit 9 - 1 contains for the time period May 1986 to February 1988, the weighted (by the loss payout pattern) treasury yields for the Alabama workers compensation business. The yields ranged from 3.83 percent to 5.45 percent. Similarly, the post-tax rate of return for the diversified portfolio held by insurers doing business in Alabama ranged from 5.64 percent to 7.20 percent over the same time period (see exhibit 9-2). If the workers compensation business were experiencing no growth, a profit and contingency factor of - 1 0 . 2 would have generated a 13 percent internal rate of return, given the net expense provisions, 9 percent return to assets, and 3 to 1 writings to surplus ratio. If the workers' compensation business were growing at 10 percent, a profit and contingency of - 9 . 7 2 would have yielded a 13 percent internal rate of return under the same assumptions. Using the 10 percent growth scenario for illustrative purposes, note that if the -9.72 profit and contingency were imposed, our target loss ratio would have increased from 75.0 to 87.22.1 The ratio of the two (87.22/75.0), 1.1629, can be applied to the distribution of insurers' loss ratios, given an assumption that the observed loss ratios would have been independent of this adjustment. This is an heroic assumption because insurers could change their underwriting standards, adjust their expenses, and make portfolio adjustments. For example, insurers could have taken on a more risky portfolio to offset the change in the regulatory framework. This would clearly increase the danger of insolvency, while tightening underwriting standards would reduce it. Insurers could cede more of the business (but they would pay a higher price to do so). However, as a first-order approximation of the upper bound of the effect of the regulatory constraints, we have multiplied the loss ratio of each insurer (at the group level) by the scalar, 1.1629, as our first step in adjusting the observed distribution of loss ratios. The second step taken to adjust the observed loss ratio distribution was to add to each observed loss ratio the average state wide dividend, 3.28. We did not use individual company dividend data in our first attempt, but may in subsequent revisions. We have dividends as a percentage of net premium so we have taken both discounts and deviations into account. Using the state-wide average requires an assumption that dividends do not vary with experience. While this is obviously not the case, we believe that a part of dividends is a cost of doing business that must be paid. In addition, dividends are probably a function of lagged experience, so that if the regulatory scenario changed

WORKERS' COMPENSATION SURVIVAL

197

in year t, dividends could be paid on the basis of previous business. Under some circumstances, the payment of dividends could exacerbate solvency problems. If a firm had terrible experience during year t, it might pay dividends to retain business in order to generate cash flow to pay current loss payments.

The Distribution of Loss Ratios In exhibit 9 - 3 , we present the distribution of adjusted loss ratios for the State of Alabama. Under this scenario, fifteen firms would have had loss ratios over 150, as compared with nine firms when a 2.5 profit and contingency loading was in the rates (see exhibit 9 - 4 for the unadjusted loss ratio distribution). Similarly, seven firms would have had loss ratios over 170, before the imposition of the extreme regulatory constraints, and ten firms would have had such loss ratios after their imposition. The loss ratio distributions presented in exhibits 9 - 3 and 9 - 4 are accident year loss ratios. We have attempted to use parametric techniques to fit the distribution of loss ratios in order to calculate the probability of being at or above a critical loss ratio. Although we have had some success in the past fitting size of loss distributions with the Singh-Maddala (Beta P or Burrl2), and Generalized Beta of the Second Kind distributions, we have no strong priors on the distribution of loss ratios. We have fit the Weibull, Singh-Maddala and the Generalized Beta type 2:2 Estimated Distribution

Weibull Singh-Maddala GB2

aya-lexp( - (y/b)a)/b a aqya-1/ba(1 + (y/b)a) q+l ayaP-1/b~PB(p, q)(1 + (y/b)~) p+q

for Alabama using the D U D option 3 of P R O C NLIN in SAS version 5.16. Parameter Estimate a

Weibull Singh-Maddala GB2

1.1457 1.1459 12.6130

b

1.2560 870212 0.8777

q

p

In likeli -111.2826

5092193 0.1587

0.2172

-72.4608

198

WORKERS'

~4

tr l.I

II .£1

II

t~

.o t~

._> t'-

t6 t~

e't"- t¢3 t"q

O t~

tr

0 ..~



,?, CO CO O~ v-,-

E

t~ ..O

~6 I O'~ iX

Ill

Z

~ d ~

COMPENSATION

INSURANCE

199

WORKERS' COMPENSATION S U R V I V A L

~

~

M

~

" ~

.

.

.

.

.

.

.

200

WORKERS' COMPENSATION I N S U R A N C E

0 ~ o o o 5. .

--I

II e-i .~ al

> .t

ill ._> t-

_>, ¢-

o rr 0 .--I

® >-

E o

"10 0

GO

E ¢11 .I:1

.

~

I ._ tX

ll.I

~ ~ Z

WORKERS' COMPENSATION SURVIVAL

201

202

WORKERS' COMPENSATIONINSURANCE

Although the GB2 fits better than the Weibull or Singh-Maddala (Chi Square of 77.65 > 3.84 with 1 degree of freedom), as the improvement in the Log Likelihood (111.2826 to 72.4608) indicates, the GB2 does not pass a strong goodness of fit test (that is we reject the null hypothesis that the distribution is indeed GB2). We have generated the deciles of the GB2 with the estimated parameters and tested predicted versus observed with a Chi Square test. The results for Alabama are as follows: YHA T

EXPECT

OBSERVE

DIFFER

CUM

0.405 0.564 0.685 0.789 0.890 1.007 1.166 1.430 2.020 >2.02

10 10 10 10 10 10 10 10 10 10

6 7 11 19 14 14 8 5 9 7

4 3 1 9 4 4 2 5 1 3

5 11 14 20 34 52 71 80 91 100

N = 100

CHI SQUARE = 17.8

Where YHAT is the adjusted loss ratio value that we entered into the cumulative distribution function to generate the 10th, 2 0 t h , . . . , 90th percentile; EXPECT is the number of observations expected in a decile (it is a coincidence that our sample size was 100); OBSERVE is the actual number of firms we observe between the loss ratios defining the deciles; D I F F E R is the difference between the actual number of firms observed and the number predicted. In exhibit 9-5, we present the value of the G B2 survivor function (1 - CDF) for each transformed loss ratio we observed in the data. Although the GB2 did not pass the goodness of fit test, it fit the tails reasonably well when compared with the Weibull or Singh-Maddala. For example, if we use the deciles generated from the Weibull distribution and compare the predicted tails to the observed tails (e.g., the first three and last three deciles), we see the following:

203

WORKERS' COMPENSATION SURVIVAL YHAT

WEIB PRED

ACTUAL

GB2 PREDICTED

0.176

10

2

1.7

0.339 0.511 1.900 2.600 >2.60

10 10 10 10 10

4 6 7 3 5

5.2 9.4 7.4 5.3 6.3

In order to present a clearer picture of the goodness of fit, figure 9 - 1 contains a plot of the observed cumulative distribution, which we have labeled OBSPER to indicate that it represents the observed percentiles in the transformed loss ratio distribution, versus the GB2 cumulative distribution, which we have labeled CUMPER. In order to save space we have not printed the plot for the 3 loss ratios that exceeded 4.00. If the GB2 fit the data perfectly, a plot of the cumulative distributions should lie along a 45 degree line. As indicated in figure 9 - 1 , the GB2 fits both tails reasonably well. In figure 9 - 2 , we present plots of the cumulative distribution

OBSPER

1.0

AA AA A

0.9

A A AA AA 0.8

AA

A A

AAAAA

0.7

~ AA 0.6

AA A ~A ~~BABA BAA

0.5

0.4

03

A

A

02 AB A

A

A A

0.1

0.0

,

i

i

i

,

i

,

,

,

i

0

0

0

0

0

0

0

0

0

0

0

0

5

0

5

0

5

0

5

0

5

0

I



0

0

0

0

0

0

0

0

0

1

5

0

5

0

5

0

5

0

5

0

CUMPER

Figure 9 - 1 . OBS, etc.

SAS. Plot of O B S P E R * C U M P E R .

Legend: A = 1 0 B S ,

B = 2

204

WORKERS' COMPENSATION INSURANCE

o5 o'1

u'i

t"" X ILl

a~

WORKERS' COMPENSATION SURVIVAL

205

206

WORKERS' COMPENSATION INSURANCE

u5 O) . m

°_ t" X t.l.I

WORKERS' COMPENSATION SURVIVAL

207

208

WORKERS' COMPENSATION INSURANCE

CUMPER

1,1 1.0

0.9 AAA B

0.8 0.7

5

AA

A

A

AA

A

j0~A A

B A C CA AA 0 C E

0.6

0.5 0,4

E

D5

B

E D D[

0.3

/

0.2 BA AA A

0.1

0.0

AA 0.00

A

A AA

0.25

0.50

0.75

1.00

1.25

1.50

1 . 7 5 2.00

2.25

2.50

2.75

3.00

3.25

3.50

3.75

400

TLR NOTE:

3 0 B S HAD MISSING VALUES OR WERE OUT OF RANGE

Figure 9-2.

SAS. Plot of COMPER*TLR. Legend: A = 1 0 B S , B = 2 0 B S ,

etc.

that we actually observed in the data and the cumulative distribution predicted by the GB2. Exhibit 9 - 6 presents the results of the calculations described above, and addresses the issue of primary interest--the probability of insolvency under extreme regulatory constraints. Specifically, it contains an evaluation of equation (1), using actual Alabama data, and shows the amount of surplus that would remain under a range of assumed loss ratios. Also shown are the probabilities of observing loss ratios greater than or equal to the value indicated based on the estimated GB2 survivor function. Thus, at adjusted loss ratios of 1.50 and greater, surplus is completely exhausted; the probability of observing firms with such loss ratios is 16.7 percent. While this work is clearly in its infancy, it addresses an issue of increasing significance within the insurance industry. As regulators try to cope with the myriad of demands from their many constituencies, reductions in the profit and contingency margin in the rates can become an attractive means of holding price increases down. This does not come at

209

WORKERS' COMPENSATION SURVIVAL

Exhibit 9-6. Surplus Remaining: Simulation Analysis Assuming a 3 to 1 Premium to Surplus Ratio. OBS 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28

LR

SURVIV

ER

TU

IY

TI

SURPR

1.041 1.064 1.074 1.121 1.127 1.146 1.149 1.206 1.260 1.300 1.336 1.419 1.460 1.561 1.563 1.584 1.590 1.657 1.771 1.835 1.906 2.165 2.256 2.994 3.244 4.140 6.668 15.528

0.375 0.359 0.352 0.324 0.321 0.310 0.308 0.280 0.257 0.249 0.228 0.202 0.191 0.167 0.167 0.162 0.161 0.148 0.130 0.121 0.112 0.087 0.080 0.045 0.039 0.024 0.009 0.002

0.225 0.225 0.225 0.225 0.225 0.225 0.225 0.225 0.225 0.225 0.225 0.225 0.225 0.225 0.225 0.225 0.225 0.225 0.225 0.225 0.225 0.225 0.225 0.225 0.225 0.225 0.225 0.225

0.34 0.34 0.34 0.34 0.34 0.34 0.34 0.34 0.34 0.34 0.34 0.34 0.34 0.34 0.34 0.34 0.34 0.34 0.34 0.34 0.34 0.34 0.34 0.34 0.34 0.34 0.34 0.34

0.1717 0.1717 0.1717 0.1717 0.1717 0.1717 0.1717 0.1717 0.1717 0.1717 0.1717 0.1717 0.1717 0.1717 0.1717 0.1717 0.1717 0.1717 0.1717 0.1717 0.1717 0.1717 0.1717 0.1717 0.1717 0.1717 0.1717 0.1717

0.1796 0.1796 0.1796 0.1796 0.1796 0.1796 0.1796 0.1796 0.1796 0.1796 0.1796 0.1796 0.1796 0.1796 0.1796 0.1796 0.i796 0.1796 0.1796 0.1796 0.1796 0.1796 0.1796 0.1796 0.1796 0.1796 0.1796 0.1796

0.896 0.850 0.831 0.738 0.726 0.688 0.682 0.569 0.462 0.383 0.312 0.147 0.066 -0.134 -0.138 -0.179 -0.191 -0.324 -0.549 -0.676 -0.817 -1.330 -1.510 -2.971 -3.466 -5.240 -10.246 -27.788

Notes: LR = Loss Ratio SURVIV = Survival Probability ER = Expense Ratio TU = Corporate Tax Rate IY = Investment Income TI = Investment Tax Rate SURPR = Surplus Remaining

210

WORKERS' COMPENSATION INSURANCE

zero cost, as our research clearly demonstrates. A l t h o u g h the model is crude, it indicates that arbitrary assumptions that justify reduced profit margins also increase the probability of insolvency. Such assumptions should be carefully considered before implementation.

Notes 1. The negative P & C does not have to been achieved and insurers can earn amounts below the allowable for a number of reasons. See Derrig (1987) for an example from the Massachusetts experience. 2. see Venter (1983), McDonald (1984), Hogg and Klugman (1985), and McDonald and Butler (1987) for discussion and derivation of these and related distributions. 3. A nonderivative method (see Ralston and Jennrich, 1979).

References Derrig, Richard. (1987). "The Massachusetts Experience." In D. Cummins and S. Harrington (eds.) Fair Rate of Return in Property Liability Insurance, Kluwer Academic Press. Hogg, Robert V. and Stuart G. Klugman. (1985). "On the Estimation of LongTailed Skewed Distributions with Actuarial Data." Journal of Econometrics. McDonald, James B. (1984). "Some Generalized Functions for the Size Distributions of Income." Econometrica. McDonald, James B. and Richard J. Butler. (1987). "Generalized Mixture Distributions with an Application to Unemployment Insurance." Review of

Economics and Statistics. Ralston, M.L. and R.I. Jennrich. (1979). "DUD, A Derivative Free Algorithm for Nonlinear Least Squares." Technometrics. Venter, Gary S. "The Transformed Beta and Gamma Distributions." Proceedings

of the Casualty Actuarial Society.

211

WORKERS' COMPENSATION SURVIVAL

Appendix of Tables Table 9-1. Assumptions. National Council on Compensation Insurance Internal Rate of Return Analysis State of Alabama. Item loss ratio commissions other expenses state premium taxes tax 1 tax 2 tax 3 profit and contingency deviations and sched, rtgs (if any) dividends to policyholders investment income pretax return on assets investment income tax rate lost loss deduction posttax return on assets reserve to surplus ratio premiums written collected premium

Percent 87.22 7.00 10.50 4.00 1.00 0.00 9.72 2.00 3.37 10.97 1.75 0.22 9.00 4.44 1,000,000.00 980,000.00

212

WORKERS' COMPENSATION INSURANCE (/)

{.o~

E

_= (~

rr

rr

E

0

(I) (-E ' 0. = E (1) ~L

E Q E 0 (.Q

~J eQ

N Z

o r--

I I I

t~

0

e,i I

I.-- (/)

l i l l

213

WORKERS' COMPENSATION SURVIVAL

O

O

O

O

O ~

O

O

O

O

O

O

O

O

O

O

O

O

O

O

O

O

O

O

O

O

O

O

O

O

O

O

O

O

O

O

O

O

O

~

O

O

O

O

O

O

O

O

~

O O

O

O

O

O

O

O

O

O

~

O

O

O

~

O

O

O

O

O

O

O

214

WORKERS' COMPENSATION

t-

-I O9

t"

tO t~ tO t--

E O tO ¢O

tO

z

0~ (1) tO

tr

E ._~ E o O_

¢-

I -.I

O9

l.J_

Oo

III

I g.

I I I I

~

~

~

INSURANCE

WORKERS' COMPENSATION S U R V I V A L

215

216

WORKERS' COMPENSATION INSURANCE

t~

E t--

t-

t~ t"

t'q t'-

._o tt) O.

E O

tO t.-~ 0

tO Z tO ,~==,

0 e-

e,-

~,,,-

~'~

I

Table 9-4b. Net Cash Flow from Underwriting. National Council on Compensation Insurance Internal Rate of Return Analysis State of Alabama.

(1) time from -1.00 -0.75 -0.50 -0.25 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00 3.25 3.50 3.75 4.00 4.25 4.50 4.75 5.00 6.00 7.00 8.00 9.00 10.00 11.00 12.00 13.00 14.00 15.00 16.00 17.00 18.00 19.00 20.00 21.00 22.00 23.00

premium interval flows net of to reserves : -0.75 : -0.50 : -0.25 : 0.00 : 0.25 : 0.50 : 0.75 : 1.00 : 1.25 : 1.50 : 1.75 : 2.00 : 2.25 : 2.50 : 2.75 : 3.00 : 3.25 : 3.50 : 3.75 : 4.00 : 4.25 : 4.50 : 4.75 : 5.00 : 6.00 : 7.00 : 8.00 : 9.00 : 10.00 : 11.00 : 12.00 : 13.00 : 14.00 : 25.00 : 16.00 : 17.00 : 18.00 : 19.00 : 20.00 : 21.00 : 22.00 : 23.00 : 24.00

0.00 0.00 0.00 0.00 3,368.75 10,106.25 16,843.75 23,581.25 23,581.25 16,843.75 10,106.25 3,368.75 -9,212.00 588.00 -1,176.00 980.00 1,274.00 -196.00 -294.00 1,078.00 2,254.00 2,156.00 1,666.00 784.00 98.00 0.00 0.00 0.00 0.00 • 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

(2)

(3)

(4)

(5)

tax credits

expenses

dividends

net cash flow from underwrtg

1,052.75 1,052.75 1,052.75 1,052.75 -8,401.18 -8,401.18 -8,401.18 -8,401.18 9,752.71 9,752.71 9,752.71 9,752.71 4,910.68 4,910.68 4,910.68 4,910.68 1,311.84 1,311.84 1,311.84 1,311.84 862.99 862.99 862.99 862.99 2,576.62 2,085.15 1,756.39 1,478.21 1,220.54 929.46 666.54 429.22 185.26 56.18 45.49 36.84 29.83 24.15 19.56 15.84 12.83 10.39 9.17

171.50 695.80 2,839.55 8,678.39 18,580.80 26,178.25 29,176.31 28,179.16 47,087.53 15,729.98 10,845.41 5,825.36 24,830.75 126.91 -82.83 68.60 598.78 -13.72 -20.58 75.46 250.88 150.92 116.62 54.88 349.86 4.90 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 8,256.50 8,256.50 8,256.50 8,256.50 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

881.25 356.95 -1,786.80 -7,625.64 -23,613.23 -24,473.18 -20,733.74 -12,999.09 -13,753.57 2,609.98 757.05 -960.40 -37,388.57 5,371.77 3,817.00 5,822.08 1,987.06 1,129.56 1,038.42 2,314.38 2,866.11 2,868.07 2,412.37 1,592.11 2,324.76 2,080.25 1,756.39 1,478.21 1,220.54 929.46 666~54 429.22 185.26 56.18 45.49 36.84 29.83 24.15 19.56 15.84 12.83 10.39 9.17

Table 9-5. Derivation of Funds in Surplus Account. National Council on Compensation Insurance Internal Rate of Return Analysis State of Alabama.

time from -1.00 -0.75 -0.50 -0.25 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75

2.00 2.25 2.50 2.75 3.00 3.25 3.50 3.75 4.00 4.25 4.50 4.75 5.00 6.00 7.00 8.00 9.00 10.00 11.00 12.00 13.00 14.00 15.00 16.00 17.00 18.00 19.00 20.00 21.00 22.00 23.00

interval to : -0.75 : -0.50 : -0.25 0.00 0.25 0.50 0.75 : 1.00 : 1.25 : 1.50 : 1.75 : 2.00 : 2.25 : 2.50 : 2.75 : 3.00 : 3.25 : 3.50 : 3.75 : 4.00 : 4.25 : 4.50 : 4.75 : 5.00 : 6.00 : 7.00 : 8.00 : 9.00 : 10.00 : 11.00 : 12.00 : 13.00 : 14.00 : 15.00 : 16.00 : 17.00 : 18.00 : 19.00 : 20.00 : 21.00 : 22.00 : 23.00 : 24.00

(1)

(2)

(3)

loss and loss adj. reserves

unearned premium reserves

admitted agents balances

0.00 0.00 0.00 0.00 15,655.99 74,224.22 175,704.69 320,097.40 433,265.35 491,920.80 496,063.75 445,694.20 403,392.50 361,090.80 318,789.10 276,487.40 252,283.85 228,080.30 203,876.75 179,673.20 165,718.00 151,762.80 137,807.60 123,852.40 120.363.60 90 708.80 76 753.60 63 670.60 51 024.98 40 784.89 32 492.72 25 777.94 20,340.49 15,937.38 12,371.85 9,484.58 7,146.54 5,253.26 3,720.13 2,478.64 1,473.31 659.23 0.00

0.00 0.00 0.00 0.00 214,375.00 367,500.00 459,375.00 490,000.00 275,625.00 122,500.00 30,625.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

0.00 0.00 0.00 -1,862.00 199,038.00 340,158.00 432,376.00 480,004.00 292,530.00 154,056.00 62,034.00 19,012.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

(4)

O)

cash level

funds in surplus account

0.00 0.00 0.00 0.00 0.00 0.00 1,862.00 0.00 30,992.99 3,529.46 101,566.22 16,733.00 202,703.69 39,610.60 330,093.40 72,162.27 416,360.35 97,674.68 460,364.80 110,897.88 464,654,75 111,831.86 426,682.20 110,476.62 403,392.50 90,940.19 361,090.80 81,403.76 318,789.10 71,867.33 276,487.40 62,330.90 252,283.85 56,874.49 228,080.30 51,418.08 203,876.75 45,961.66 179,673.20 40,505.25 165,718.00 37,359.21 151,762.80 34,213.17 137,807.60 31,067.14 123,852.40 27,921.10 120,363.60 27,134.59 90,708.80 20,449.25 76,753.60 17,303.21 63,670.60 14,353.80 51,024.98 11,502.99 40,784.89 9,194.48 32,492.72 7,325.11 25,777.94 5,811.34 20,340.49 4,585.53 15,937.38 3,592.90 12,371.85 2,789.09 9,484.58 2,138.19 7,146.54 1,611.11 5,253.26 1,184.29 3,720.13 838.66 2,478.64 558.78 1,473.31 332.14 659.23 148.62 0.00 0.00

WORKERS' COMPENSATION SURVIVAL

219

,= -'-,i

rr" O

II]1

I

f12 E e--

t----t t--to .i.--, e'~t.

E

O

o

I I I I I

co {.0

o I

~

0 ,4,-.,

z 0 Q t--

p.

f.,.

I I I I I

o .-~

z~

~

~

I

(-.

III

I I I I

I

I

~

I

I

~

I

I

I

I

I

I

I

I

220

WORKERS' COMPENSATION I N S U R A N C E

I

I

I

I

~

~ I I I I I

-6 t-" tO

~6 O'~

t~

°~

I

I

~

~

I

~

I

~

~

i

i

i

i

I

National Council on Compensation Insurance Internal Rate of Return Analysis State of Alabama Calculations

Table 9 - 3 (1) (2) (3) (4) (5) (6) (7) (8)

Table 9 - 2 , col. (1) * Table 9 - 1 , item (11). Cumulative prem. written (post-dev.) - col. (1). Estimate based upon premium collection pattern. Col. (2)-col. (3). Table 9-1, item (10) * Table 9-1, item (1). * cumulative loss incurrence pattern. Cumulative prem. written (post-dev.) - cumulative prem. earned (post-dev). col. (1) + col. (2) - col. (5) - col. (6). col. (7) current period - col. (7) previous period. Table 9 - 4 a

(4),(5) 0.5 * Table 9-1, item 1 * Table 9-1, item (10) *accident year loss payout pattern. (6),(7) Period-by-period change in discounted accident year reserve, i.e. change in accident year reserve level * reserve discount factor: factors are calculated per tax reform act of 1986, and 1987 IRS discount rate. (8) - 0 . 3 4 * (col. (1) - (col. (2) * 0.8) - col. (3) - col. (4) - col. (5) - col. (6) - col. (7)). Table 9 - 4 b (1) Table 9-3, col. (8). (2) Table 9-4a, col. (8) (split quarterly for time -1.00 to time 5.00). (3) Table 9 - 1 , item (11) * (Table 9-1, item 2 * Table 9-2, col. (1) + Table 9-1, item 3 * Table 9-2, col. (3) + Table 9-1, item 4a * Table 9 - 2 , col. (4) + Table 9-1, item 4b * Table 9-2, col. (5) + Table 9-1, item 4c * Table 9 - 2 , col. (6)). (4) Table 9-1, item (11) * Table 9-1, item 7 * Table 9-2, col. (7). (5) Col. (1) + col. (2) - col. (3) - col. (4). Table 9 - 5 (1) (2) (3) (4) (5)

Table 9 - 3 , col. (5) - cumulative losses paid out as of end of period. Table 9-3, col. (6). Table 9 - 3 , col. (4). Col. (1) + col. (2) - col. (3). Col. (1)/Table 9-1, item 9. Table 9 - 6

(1) Table 9-4b, col. (5). (2) Table 9-5, (col. (4) current period + col. (4) previous period)/2 * Table 9-1, item ba. (3) Table 9-5, (col. (4) current period + col. (4) previous period)/2 * Table 9-1, item bb + bc. (4) Table 9 - 5 , col. (5), previous period - col. (5), current period. (5) Table 9 - 5 , (col. (5) current period + col. (5) previous period)/2 * Table 9-1, item ba. (6) Table 9-5, (col. (5) current period + col. (5) previous period)/2 * Table 9-1, item bb + bc. (7) Col. (1) + col. (2) + col. (3) + col. (4) + col. (5) + col. (6).

This page intentionally blank

10

PREDICTING INSURANCE INSOLVENCY USING GENERALIZED QUALITATIVE RESPONSE MODELS* James B. McDonald

Introduction The general problem of corporate or business failure is very important and can generate significant losses to creditors and stockholders. Models of predicting business failure have been studied by numerous authors including Aharony (1980), Altman (1968), Beaver (1967), Dambolena and Khoury (1980), Deakin (1972), and Zavgren (1983). The problem of predicting insolvency of insurance companies has become an important issue for the National Association of Insurance Commissioners as well as state and federal legislators. More than 130 property and casualty insurance companies have failed during the last ten years. Daenzer (1984)

* Prepared for the Eighth Annual Seminar on Economic Issues in Workers Compensation, Wharton School, University of Pennsylvania, November 18, 1988. The author appreciates David Appel and the National Council on Compensation Insurance for encouragement and for providing the data used in this study. David Durbin's assistance and interpretation of the data, as well as the excellent research assistance of Mont Timmins and Steve White, are greatly acknowledged. The comments of a referee are appreciated.

223

224

WORKERS' COMPENSATION INSURANCE

indicates that the total statutory underwriting loss for the period 1979 to 1983 is $33.7 billion. In an interesting article entitled "Is 'A-Plus' Really a Passing Grade?" Denenberg (1967) examined Best's ratings for size and financial strength for the six years preceding insolvency and found that the ratings were useful in predicting solvency status. These ratings attempt to summarize many factors, such as quality of underwriting, management, adequacy of reserves, investment quality and a financial rating. This has provided a useful, but not a perfect, guideline of an insurance companies financial health. Predicting insolvency continues to attract a great deal of attention in the academic and professional literature. BarNiv and Hershbarger (1987) and Shaked (1985) discuss some approaches used in life insurance and provide a brief review of the literature. Some recent research and additional relevant references to the insolvency problem for propertycasualty companies can be found in Ambrose and Seward (1988), BarNiv and Smith (1987), Harrington and Nelson (1986), Pinches and Trieschmann (1974, 1977) and Trieschmann and Pinches (1973). The most common models used in the analysis are based on predicting binary dependent variables. Linear probability models and multiple discriminant analysis (MDA) have been widely used in this context. Pinches and Trieschmann (1974) used MDA to determine which financial ratios were most correlated with insolvency of property-liability companies. Other studies have been done by Harmelink (1974), who uses MDA to predict declines in Best's letter ratings, and Harrington and Nelson (1986), who use insurer characteristics to predict premium-to-surplus ratios. Ambrose and Seward (1988) applied MDA to compare the predictive ability of Insurance Regulatory Information System (IRIS) ratios and some other portfolio characteristics relative to the explanatory power of the Best's letter ratings. Qualitative response models provide another approach to these problems. Amemiya (1981) has written an excellent survey of qualitative response models. He states that "one of the most important developments in econometrics in the past ten years has occurred in the area of qualitative response models". These models were used in areas of biological research long before they were used in economics, (e.g., see Bliss (1935), Berkson (1944) and Finney (1952)). Probit and logit models are the most popular qualitative response models. Recent economic applications of qualitative response models include studies of labor force participation, union membership, housing decisions, voting and legislation, occupational choice, as well as providing models for individual and corporate bankruptcy. BarNiv and Hershbarger (1987) use the logit

P R E D I C T I N G INSURANCE INSOLVENCY

225

model in predicting insolvent life insurance companies. Ohlson (1980) and Zemijewski (1984) suggest the use of qualitative response models in predicting financial distress. The purpose of this chapter is to develop a generalized statistical model that will help identify insurance companies having financial problems and may be headed towards insolvency. Linear probability models, multiple discriminant analysis, probit and logit models are each based upon particular assumptions that may or may not be satisfied in a given application. These models will be reviewed in the second section and a generalization of the probit and logit model considered. The third section contains an application of these models to the problem of forecasting insolvency of property-casualty companies.

Limited Dependent Variable Models and Classification The Models The discussion will be focused on models for predicting whether a company is going to be insolvent, but the form of the model is the same for any binary choice problem. The linear probability (LPM) model is defined by Yt = Xtb + e,

(1)

where Y, = 1 for an insolvent company and 0 otherwise; X, is a Kxl vector of explanatory variables that are helpful in predicting insolvency or solvency; e, denotes a random disturbance. The values of Xtb are sometimes referred to as "scores" and are used in classifying the individual companies with "high" values (relative to some threshold) leading to a prediction of insolvency. The residuals in equation (1) will be nonnormal and heteroskedastic. Thus, ordinary least squares estimators may still be unbiased but will be neither efficient or normally distributed. Weighted or generalized least squares provides only an incomplete approach to the problem. It does solve the heteroskedasticity problem (Goldberger, 1964; and Judge et al., 1985). Predictions obtained from the LPM that have been adjusted for heteroskedasticity need not be between 0 and 1. Multiple discriminant Analysis (MDA) has been widely used in insolvency models for the insurance industry (e.g., Ambrose and Seward, 1988). MDA assumes that the vectors iV, are distributed as two multivariate normals, one for the solvent companies and another for the insolvent companies. It is also assumed that the two multivariate normals

226

W O R K E R S ' COMPENSATION INSURANCE

have the same variance-covariance matrix with different mean vectors (/11, /~2). The methodology is based upon estimating a discriminant function of the form

Zt = Xtg

(2)

that selects the estimator of the vector g to maximize the variance between groups relative to the variance within groups. The scores Zt again provide the basis for predicting insolvency. The estimators of g are proportional to E-l(gl f12), as well as to the OLS slope coefficients in equation (1) (Cooley and Lohnes, 1962). These estimators are "optimal" if the assumptions are satisfied, that is, normality of the explanatory variables, equal covariance matrices with different means. The assumptions of both symmetry and normality are often violated in applications involving economic and demographic variables, (e.g., Bookstaber and McDonald, 1987; and McDonald, 1984). Pinches and Trieschmann (1977) test and reject the validity or normality and equal variance-covariance assumptions underlying the method of MDA in an insurance insolvency model. Ohlson (1980) and others suggested using qualitative response models, in particular a logit model, to reduce problems associated with violations of the assumptions underlying MDA which are common in distress prediction models. See Zemijewski (1984) for a discussion of estimation biases in such applications. -

Qualitative Response models.

LPM and MDA models are both based upon "scores" that are linear functions of the explanatory variables. Qualitative response models provide an alternative approach based on nonlinear functions of the explanatory variables. These models take the following form

Pr(Yt = 1/X,) = ~ J_

fXtb(s; O)ds oo

= F(X,b; O)

(3)

where f ( ) and F( ) denote the statistical density and cumulative distribution functions and 0 denotes the parameters of the statistical distribution. Since F(Xtb; O) is a cumulative distribution function, the dependent variable Yt is bounded between zero and one. The literature has primarily focused upon two special cases of equation (3), the logit and probit models, Judge et al. (1985) and McFadden (1974). The probit model is based upon selecting f(s; O) to be the standard normal, N(s; O, 1) and the logit model corresponds to using the logistics density

PREDICTING INSURANCE INSOLVENCY

227

function f(s) = e-S/(1 + e - S ) 2. Maximum likelihood estimation of the parameters in (3) involves the cumulative distribution function. The logit model has a closed form distribution, F(s) = 1/(1 + e -s) that simplifies estimation; whereas, the distribution for the normal involves an infinite series, a confluent hypergeometric series. However, accurate approximations to the cumulative normal distribution are now available and make the probit analysis a viable procedure. Amemiya (1981) cites several references and suggests that it is often difficult to discriminate between the probit and logit in practice, unless a large number of observations are available, or if the data are concentrated in the tails. Both the probit and logit models implicitly assume the estimated scores are symmetrically distributed about zero.

Generalized Qualitative Response Models. We consider a variation of (3) that retains the desirable property of predictions being bounded between 0 and 1, but does not impose symmetry on the distribution of the scores. The generalization also includes both the probit and logit models as special cases. This specification is based on the density EGB2(s; a, fl, p, q) = esap/[flapB(p, q)(1 + (eS/fl)a)p+q] --00 .< S < oo (4) where a, fl, p and q are positive parameters. This density is based upon a generalized beta of the second type (GB2) and is symmetric about zero if p = q and not otherwise. The logit model is a special case of equation (4) where a = fl = p = q = 1. Similarly, the probit model is also a special case of (4) obtained by taking the limit of (4) in (3) as the parameter a approaches zero and q grows indefinitely large in a particular way (see McDonald, 1984). The log-likelihood function for applications of qualitative response models is given by N

1 = Z [Y, lne(Xtb; O) + (1 - Y,) In(1 - F(X,; 0))]

(5)

t=l

and provides the basis for maximum likelihood estimators which are obtained by maximizing equation (5) with respect to the independent "score" or regression parameters, b, and distributional parameters a, fl, p and q. The parameters a and b appear as a product in the log-likelihood function based on the general specification in (4) and can not estimated simultaneously. Except in the limiting case, such as with the probit model, where a approaches 0, the parameter a can be assumed to be equal to 1 with no loss of generality. The special cases of p = 1 and of

228

WORKERS' COMPENSATION INSURANCE

q = 1 correspond to the Burr12 or Burr3 distributions, respectively (McDonald, 1984). These are both associated with closed distributional forms and facilitate maximum likelihood estimation. We will work with special cases of (4) that we will call the Lomit (a = fl = p = 1) and the Burrit (a = fl = q = 1) models. These models, respectively, are based upon distributions known as the Lomax and a special case of the Burr type three distributions in the same way as the logit model is based on the logistics model. The qualitative response model based on the EGB2 given in (4) includes both the probit and logit models as special or limiting cases. It also includes two qualitative response models (the Lomit and Burrit models) that are generalizations of the logit model. This nested structure of the models, combined with maximum likelihood estimation, lends itself to statistical tests based upon the likelihood ratio test. These models will be used for the empirical work in section three. The more general a model the better it should do relative to the log-likelihood value. However, this does not imply an improvement in predictive ability, but the increased flexibility of the distribution certainly provides that potential. The generalizations permit, but do not impose symmetry on the underlying density function.

Application to Predicting Insolvency The insolvent companies studied in this chapter are those reported in the Best's Insurance Reports (Property and Casualty) as having been declared insolvent by their state insurance commissioner during 1983 to 1987 and for which two years of data were available. The last years data available before insolvency is denoted with a t and the second to last year as (t - 1). Pinches and Trieschmann (1974) suggest that the (t - 1) data may be the more helpful for regulation due to time lags involved with the availability of financial reports as well as possible "window dressing" associated with the most recent data. Data on 58 insolvent firms and a random sample of 254 solvent firms were available. The analysis in this chapter is based on 23 independent variables that collectively yield a high degree of predictability for the one and two year ahead predictions. These variables are a combination of some explanatory variables used in BarNiv and Hershbarger (1987) and Ambrose and Seward (1988). Stepwise regression and M D A were used to identify important factors identifying insolvent firms. Different variables appeared to play key roles for predicting one period ahead than for two period predictions. The two

229

PREDICTING INSURANCE INSOLVENCY Table 10-1.

Variable IRIS: 11

12 16 17 R1 R3 R12 R16

Variable Definitions.

Definitions Change in Capital and Surplus Net Gain to Total Income Real Estate to Capital and Surplus Investments in Affiliate to Capital and Surplus Net Premiums written/Surplus (P/S) Percent Change in Net Premiums Written (•9) Net investment Income/Invested Assets (/4) Stated Liabilities/Invested Assets

SIZE: SURPLUS PREMIUMS ASSETS GROWTH: Ln(GRS) Log of Growth of Surplus LEVERAGE RATIO: A/S Total Assets/Surplus CEDED PREMIUM: M9: Change in Reinsurance Ceded/Total Surplus INVESTMENT: R20 Admitted Assets-Invested Assets/Admitted Assets EXPENSE: R24 Expense Incurred/Written Premium R26 Investment Expense Incurred/Invested Assets R30 Loss Adjustment Expense/Losses Incurred RESERVES: R22 Excess Loss Reserves/(Losses Incurred + Loss Adjustment) MISCELLANEOUS: R28 (Losses Incurred + Loss Adjustment Expense)/Net Premiums Earned R32 Liability Premiums/Net Premiums Written R40 Net Income/Surplus R42 Percent change in (Net Income/Surplus) different sets of variables were combined into a single common set of regressors for the alternative qualitative response and MDA models considered in this chapter. Thus, the focus of this chapter is to compare the relative predictive performance of several alternative econometric models, rather than identifying a minimal number of variables to be used in such an analysis. Table 10-1 summarizes the definition and character-

230

WORKERS' COMPENSATION INSURANCE

istics of the variables used. The Ri's in table 10-1 correspond to the same variables used by Ambrose and Seward and the remaining variables were considered by BarNiv and Hershbarger. The IRIS ratios are tests developed by the National Association of Insurance Commission called the Insurance Regulatory Information System (IRIS) or early warning system. These ratios are constructed from the companies annual statements and collectively are supposed to be leading indicators of financial distress. Numerous papers have been written on the validity of these ratios as providing an early warning, (see Bloom, 1988; and Thornton and Meador, 1977). The other variables measure the size or scale of the firm (surplus, total premiums and total admitted assets), the log of the growth of surplus to measure growth, the ratio of total assets to surplus as a leverage ratio and various other variables and ratios to reflect investments, expenses, reserves, reinsurance and other financial characteristics of the firm. In addition to the papers by Ambrose and Seward (1988) and Barniv and Hershbarger (1987) see Booth (1983), Kahane (1978); Kahane et al. (1986) and the various papers by Pinches and Trieschmann. The formal definition of these variables expressed in terms of the entries from Best's data base are given in Appendix A. Requiring two years of data on all of the explanatory variables (including some variables defined in terms of lagged variables) in table 10-1 further reduced the sample to 35 insolvent firms and 177 solvent firms. A list of the firms used in the analysis is included in Appendix B. The sample data characteristics are summarized in table 10-2. The means and standard deviations permit some simple univariate comparisons between the solvent and insolvent firms for one and two year lags. MDA, probit, logit and several generalizations of probit and logit were estimated. The MDA models were estimated using SAS. The qualitative response models were estimated using maximum likelihood estimation procedures based on equations (4) and (5). In each case the sample consisted of 212 observations (35 insolvent and 177 solvent companies). Many of the previous empirical studies have used a sample consisting of an equal number of solvent and insolvent companies. There are inherent biases in the parameters when the fraction of solvent and insolvent companies in the sample differs from that of the population. Manski and Lerman (1977) have outlined an approach using a weighted likelihood function that circumvents this problem. The impact of these weighting schemes are being investigated by the author, but are not reported in this chapter. Separate models were estimated for predicting solvency status for one and two periods ahead using the 22 explanatory variables. This is the approach adopted by Barniv and Hershbarger (1987).

231

PREDICTING INSURANCE INSOLVENCY

~.~v~-~r..4

~ . ~ ' ~ , - ~ , ~ . ~ 1 ~.

~ j ~ - ~

t-"

t< ~ e.i t"

t"

E ¢-~ .m

O e" t~ X 1.13

.m

L" t~

E E --'1 e,i O

t~

I---

~,,.,q ~

~"~-~

I~ -

~-~..~

232

W O R K E R S ' COMPENSATION INSURANCE

Table 10-3.

Likelihood Ratio Values.

Data Set EGB2 vs. Logit EGB2 vs. Probit

Lag 1

Lag 2

4.2 4.2

6.4 5.4

The relative performance of the alternative models can be compared in a number of ways. A natural comparison for the nested models is to use a likelihood ratio test to check for significant improvements in moving to a more general formulation. The likelihood ratio test is obtained by calculating twice the difference of the maximized log-likelihood values and has an asymptotic Chi-square distribution. Table 10-3 reports the values of the likelihood ratio statistics obtained by comparing the logit and probit models with the GB2 qualitative response model. The likelihood ratio statistics for these hypotheses are asymptotically chi square with two degrees of freedom and have 0.1 and 0.05 critical values of 3.84 and 5.99. Thus, the qualitative response model based on the EGB2 distribution provides a statistically significant improvement in fit relative to the logit and probit model at the 10 percent level for both data sets (Lag 1 and 2). However, the improvement is only statistically significant at the 0.05 level for the logit model in the two period lag model. Statistical significance of the likelihood ratio test does not necessarily imply increased predictive ability and does not permit a comparison of the previously mentioned qualitative response models with the MDA procedures. Recall that the probit and logit models are based upon symmetric distributions and that (4) is symmetric for p = q. The estimated (p, q) for the lag one and lag two data sets are (0.0043, 26,041) and (0.0063, 451) respectively suggest non symmetry. A comparison of the relative mean values of the explanatory variables from table 10-2 provides useful, but incomplete and possibly misleading information about "causal" factors associated with insolvency. The signs of the "regression coefficients" in the qualitative response models disentangle the interrelationships between the explanatory variables and indicate important information about solvent and insolvent firms when the impact of other contributing factors is controlled for. This type of analysis was not the primary objective of the chapter and the collinearity and the use of t statistics from nonlinear qualitative response models makes such comparisons somewhat questionable. However, a few qualitative relationships appear to be rather consistent in the estimated

233

PREDICTING INSURANCE INSOLVENCY Table 10-4.

Prediction Results.

One Period Lag:

Models: MDA Logit Probit Lomit Burrit EGB2

Insolvent (35) number correct

Solvent (177) number correct

Percent Correct

23 27 27 27 29 29

171 173 173 173 171 171

91.5 94.3 94.3 94.3 94.3 94.3

Insolvent (35) number correct

Solvent (177) number correct

Percent Correct

17 31 31 31 31 31

171 172 173 173 174 174

88.7 95.8 96.2 96.2 96.7 96.7

Two Period lag:

Models: MDA Logit Probit Lomit Burrit EGB2

models. Insolvent firms (or more accurately increased probability of insolvency) tended to be associated with higher levels of the IRIS variables (I1, 16, 17, R1, R3 and R12), other things equal. For example, many studies have found that distressed firms often attempt to generate a positive cashflow by writing new business. Thus, higher values of R1 may be correlated with insolvent firms; however, as Bloom (1988) points out "There are many good reasons why a company would want to increase surplus. To suggest that such action is improper is irresponsible." The "regression coefficients" for size as measured by the level of total premiums or surplus are negative indicating larger companies are less likely to become insolvent. The signs of the estimated regression coefficients also tend to suggest that solvent firms have larger leverage ratios as compared to insolvent firms. The relative predictive ability of the MDA and various qualitative response models yields a method of comparing non nested binary choice models. Table 10-4 summarizes the results of the various estimated models for one and two period predictions. The results are not as dramatic as desired, but some conclusions appear valid. The qualitative

234

WORKERS' COMPENSATIONINSURANCE

Table 10-5.

0.3 0.4 0.5 0.6 0.7 0.8 0.9

Predictive Performance (Number and % correct).

Sol

Logit Insol

%

Sol

Probit Insol

%

Sol

EGB2 lnsol

%

176 174 172 171 167 167 162

24 29 31 32 33 34 35

94.3 95.8 95.8 95.8 94.3 94.8 92.9

176 174 173 171 168 167 163

26 29 31 31 33 34 35

95.3 95.8 96.2 95.3 94.8 94.8 93.4

174 174 174 174 172 169 168

27 31 31 33 34 35 35

94.8 96.7 96.7 97.6 97.2 96.2 95.8

response models appear to be better able to predict both the solvent and insolvent companies than MDA. This may be explained in part to the observation by Pinches and Trieschmann (1977) of nonnormality and unequal variance-covariance matrices. The qualitative response models are also seen to yield similar results with the more general forms having marginally better predictive ability. It is interesting to note that the two period lag model appears to more accurately predict insolvency than the one period lag model. This might reflect the "window dressing" effect discussed in Pinches and Trieschmann. The prediction results in table 10-4 are based upon a threshold (Z*) such that F(Z*; 0) = 0.5. If X t b < Z* solvency will be predicted and insolvency otherwise. The use of a 0.5 threshold implicitly assumes equal costs of misclas-sification. A marginal increase in predictive ability associated with the generalized forms is seen. The importance of this improvement depends upon the costs and benefits of correct classification as well as costs of misclassification. McDonald and Clarke (1992) investigate these tradeoffs in a different context. The predictive performance depends upon the fractile of the value of the selected threshold. The results reported in table 10-4 correspond to using a decision criterion based on F(Z*; 0) = 0.5. The predictive performance and misclassification errors associated with solvent and insolvent companies will change as Z* is altered. A sensitivity analysis of the predictive performance of the various estimated two period lag models is presented in table 10-5 for different fractiles. While the results for the logit, probit and EGB2 based model are fairly close, the EGB2 appears to have a greater ability to "separate" both the solvent and insolvent firms. The maximum percent correct is achieved for the probit and logit at a threshold corresponding to 0.5 and for the EGB2 at 0.6; however, the differences are not large. If the costs of misclassification are the same,

PREDICTING INSURANCE INSOLVENCY

235

the decision criterion would be to maximize the percent of correct classifications. For the case of unequal costs of misclassification, the optimal decision need not correspond to the maximum of correct classifications, but would be based on expected profits or costs of misclassification. The possible tradeoffs are easily visualized from the results in table 10-5.

Conclusions The problem of predicting insurance company insolvency has been considered. Various multivariate procedures have been investigated and found to provide helpful guidelines to this problem. The IRIS variables tended to have the predicted impact upon insolvency as did other variables measuring size and rate of growth. Multiple discriminant analysis has been a widely used procedure. Qualitative response models such as probit or logit are shown to provide better predictions of both solvency and insolvency than obtained from multiple discriminant analysis. The generalizations of probit and logit models provide statistical improvements and some improvement in predicting insolvencies. Additional areas which need to be investigated further include the impact of biases resulting from samples that are not representative of the population, as well as gains that might be achieved incorporating relative costs differentials of misclassifying solvent and insolvent companies. Additional efforts to reduce the number of variables involved and to build a single model for use in predicting insolvency for different time periods would also be of interest.

Appendix A: Variable Definitions VARNAME= 11 I2

I6

17

VAR F O R M U L A VAR DESCRIPTION = VAR94/VAR80 Percentage Change in Surplus as regards policyholders = (VAR79 - Lagl[VAR79])/VAR79 Change in Net Income after taxes and dividends to Net Income after taxes and dividends = (VAR10 + V A R l l ) / V A R 9 5 Real Estate to Surplus as regards policyholders = (VAR150 + VAR152 + VAR154)/VAR95 Bonds, Preferred and Common Stocks to Surplus as regards policyholders

236 SURPLUS PREMIUM ASSETS A/S P/S L/S Ln(GRS)

R1 R3 R12

R16

R20

R22

R24

R26

R28

R30

WORKERS' COMPENSATION INSURANCE

= VAR95 Surplus as regards policyholders -- VAR302 Net Premiums Written = VAR28 Total Admitted Assets = VAR28/VAR95 Total Admitted Assets to Surplus as regards policyholders = VAR302/VAR95 Net Premiums Written to Surplus as regards policyholders = VAR51/VAR95 Total Liabilities to Surplus as regards policyholders = Ln(VAR95/VAR80) Natural Logarithm of Growth in Surplus as regards policyholders = VAR302/VAR95 Net Premiums Written to Surplus as regards policyholders = (VAR302 - LagI[VAR302])/Lagl[VAR302] Percentage Change in Net Premiums Written = VAR70/(VAR16 + VAR24) Net Investment Gain or Loss to Cash, Accrued Interest and Invested Assets = VAR51/(VAR16 + VAR24) Total Liabilities to Cash, Accrued Interest and Invested Assets = (VAR28 - VAR16 - VAR24)/VAR28 Total Admitted Assets less Cash, Accrued Interest and Invested Assets to Total Admitted Assets = VAR43/(VAR62 + VAR63) Excess of Statutory Reserves Over Statement Reserves to Losses Incurred and Loss Expenses Incurred -- (VAR63 + VAR65)/VAR302 Loss Expenses Incurred and Aggregate Write-ins for Underwriting Deductions to Net Premiums Written = VAR470/(VAR16 + VAR24) Total Investment Expenses Paid to Cash, Accrued Interest and Invested Assets = (VAR62 + VAR63)/VAR239 Losses Incurred and Loss Expenses Incurred to Total Premiums Earned During Year = VAR63/VAR62 Loss Expenses Incurred to Losses Incurred

PREDICTING INSURANCE INSOLVENCY R32

R40

R42

M9

237

= (VAR278 + VAR279 + VAR280 + VAR281 + VAR283 + VAR284 + VAR289 + VAR290 + VAR291 + VAR293 + VAR298)/VAR302 Net Farm owners' multiple peril, Homeowners' multiple peril, Cormercial multiple peril, Ocean marine, Aggregate write-ins for other, Medical malpractice, Workers' compensation, Other liability, Auto liability, Aircraft (all perils) and Boiler & machinery Premiums Written to Net Premiums Written = VAR79/VAR95 Net Income after taxes and dividends to Surplus as regards policyholders = (VAR79/VAR95 Lagl[VAR79]/LagI[VAR95])/ (Lagl[VAR79]/LagI[VAR95]) Percentage Change in Net Income after taxes and dividends to Surplus as regards policyholders = (VARl125 - LagI[VARl125])/VAR95 Change in Total Reinsurance Ceded to Surplus as regards policyholders

Appendix B Insolvent Insurers Beacon Insurance Cal-Farm Ins Carriers Ins Glacier Gen Assur Ideal Mutual Ins Integrity Ins Iowa National Mutual North-West Ins Southwestern Ins Quality Ins RGAF Underwriters Professional Mutual

Amer Consumer Ins Transit Casualty Early American Ins Heritage Ins Comm Standard Ins Southwestern Nat Homeland Ins Aspen Indemnity Oklahoma Ins Logist Guaranty Ins Pacific Amer Ins Great Global Assur

Eastern Ind (Md) Consumers Ins Gr Texas Fire and Gas Merchants & Mfrs Ins Golden West Ins Exch Pine Top Ins Columbus Ins Independent Indem Carriers Casualty Allied Fidelity Ins S & H Ins

Solvent Insurers Amer Bankers Ins Angelina Casualty Co Barnstable Cnty Mut

Amer Road Ins Co Audubon Ins Co Beacon National Ins

Nat Continental Ins Avemco Ins Co Calif Comp & Fire

238 Canal Ins Co Cumberland Mut Fire Farm Bur Mut of Ida Home & Auto Ins Co Mercury Casualty Co Mut Ins Burlington Rock River Ins Co United Services Auto Amer Hallmark Ins Tx Amer Freedom Ins Co Sagamore Ins Co United Financial Cas Southern Ins Co Ins Co of State PA Argonaut Ins Co Republic Mutual Ins Comm Union Ins Atlanta Intern Ins Employers Cas Co New England Reins Cp Grange Ins Assn Dependable Ins Co Country Casualty Ins Ina of Texas Amer Motorists Ins Maryland Casualty Co Green Mountain Ins Nationwide Mut Ins Peerless Ins Co Progressive Mutual Security Ins Co Hart Charter Oak Fire Ins Mt Vernon Fire Ins Western Fire Ins Co Amer Empire Ins EBI Ins Co (OR) Nat Merit Ins Co Lincoln Nat Reins Co Paramount Ins Co (NY) Amer Resources Ins Gerling Amer Ins Co Western Employer Ins Western World Ins Co Eastern Shore of Va

WORKERS' COMPENSATION INSURANCE Colonial Ins Co Cal Employers Reins Corp Harbor Ins Co Prudential-Lmi Comml Mid-continent Cas Co Nat Farmers Un P & C Stonewall Ins Co Wausau Unders Ins Co Penn Mfrs Idemnity Mieo Ins Co Progressive Nrthwstn Alfa Mutual Ins Co Selective Ins of SC Cambridge Mut Fire Argonaut-Northwest Sun Ins Co of NY Aetna Fire Undrs Ins Continental Cas Co Northeastern Ins Co Foremost Ins Co Amer National Fire Hanover Ins Co Calif Union Ins Co Pacific Employers Ins Liberty Mutual Fire Green Mountain Ins Shelter Genl Ins Co Unigard Security Ins Farmers Auto Ins Asn Western Alliance Ins Universal Scour (TN) Union Mut Fire Ins United States Liab Consolidated Ins Asn Hartford Ins of SE MIC General Ins Corp Safeco Ins Co of Ill Lincoln Nat Hlth & Cas Fremont Reins Co Columbia Mut Cas Co Admiral Ins CNA Casualty of Cal Calmutual Ins Co Nat Lloyds Ins Co

Concord General Mut Farm & City Ins Co Hawaiian Ins & Guar Merchants Prop Ins Motorists Mutual Ins Penn Mfrs Assn Ins Transport Indemn Co Alfa General Ins CP Narragansett Bay Ins Cumberland Ins Co Voyager Protection Alfa Mutual Fire Ins Selective Ins of SE Merrimack Mut Fire Amer Select Ins Co North Amer Co P & C Boston Old Colony Northern Assurance Fireman's Ins of DC Kansas City Fire & Mar First Ins Co of Haw Hartford A c c & Indem Ins Co of N Amer Chicago Ins Co Universal Undrs Ins Maryland Casualty Co Millers Casualty Ins Oregon Auto Ins Co Preferred Risk Mut General Ins America Transamerica Premier Prairie State Farmrs Virginia F B Mut Ins Comstock Ins Co Allianz Undrs Ins Co Zurich Reins Co NY Georgia Amer Ins Co CNA Lloyd's of Texas Penn Casualty Ins Metropolitan Gen Ins Gerling Global Reins Mut Assur Co Western Mutual Ins Alfa Mutual General

PREDICTING INSURANCE INSOLVENCY Ins Co of Greater NY Columbia Casualty Co Scor Reins Co Aetna Reins Co Prudential Gen Ins Professionals Ins Co Oklahoma Surety Co USAA Casualty Ins Co Global Surety & Ins Reliance Ins Co Ill Progressive American Commerce Ins Co

Congregation Ins Co Sheffield Ins Co (RI) Republic-Vang Reins Pohjola America Rein Calif Cas General Amerisure Ins Co Westfield National Amer Family Home Ins Cornhusker Casualty Queen City Indemnity Design Professionals Northfield Ins Co

239 CNA Casualty Co Wisconsin Emplrs Cas Signet Reins Co Fenix de Puerto Rico Voyager Prop & Cas Nat Farmers Un Stand Century Indemnity Co Kemper Reins Co Safeco National Ins Beaver Ins Co Atlanta Casualty Co Amer Deposit Ins

References Aharony, J., C.P. Jones and I. Swary. (1980). "An Analysis of Risk and Return Characteristics of Corporate Bankruptcy Using Capital Market Data." Journal of Finance 35:1001-1016. Altman, E.I. (1968). "Financial Ratios, Discriminant Analysis, and the Prediction of Corporate Bankruptcy." Journal of Finance 23:589-609. Ambrose, J.M. and J.A. Seward. (1988). "Best's Ratings, Financial Ratios and Prior Probabilities in Insolvency Prediction." The Journal of Risk and Insurance 55:229-244. Amemiya, T. (1981). "Qualitative Response Models: A Survey." Journal of Economic Literature 19:1483-1536. BarNiv, Ran and Robert A. Hershbarger. (1987). "Classifying Financial Distress in the Life Insurance Industry." Mimeographed Manuscript. Mississippi State University (September). BarNiv, R. and M. L. Smith. (1987). "Underwriting, Investment and Solvency." Journal of Insurance Regulation 409-428. Beaver, W.H. (1967). "Financial Ratios as Predictors of Failure," Empirical Research in Accounting: Selected Studies. Journal of Accounting Research, Supplement to Vol. 5:71-102. Berkson, J. (1949). "Application of the Logistic Function to Bio-assay", Journal of the American Statistical Association 39:357-365. Best's Insurance Reports--Property and Casualty Edition. Oldwick, NJ: A.M. Best (various issues, annual). Bliss, C.I. (1935). "The Calculation of the Dosage-Mortality Curve." Annals of Applied Biology 22:135-67. Bookstaber, R. and J.B. McDonald. (1987). "A General Distribution for Describing Security Price Returns." The Journal of Business 60:401-424. Bloom, Thomas S. (1988). "Passing the Test: Regulators Need New Way to Measure Strength of Insurers." Business Insurance March:27-28.

240

WORKERS' COMPENSATIONINSURANCE

Booth, P.J. (1983). "Decomposition Measures and the Prediction of Financial Failure." Journal of Business Finance and Accounting 10(1):67-82. Cooley, W.W. and P.R. Lohnes. (1962). Multivariate Procedures for the Behavioral Sciences. New York: Wiley. Daenzer, B.J. (1984). "Insurers Solvency: Determining Which Companies Can Survive the Storm." Risk Management July:22-30. Dambolena, I.G. and S.J. Khoury. (1980). "Ratio Stability and Corporate Failure," Journal of Finance 35:1017-1026. Deakin, E. (1972) "A Discriminant Analysis of Predictors of Business Failure." Journal of Accounting Research 10(1):167-179. Denenberg, H. (1967). "Is 'A-Plus' Really a Passing Grade?" Journal of Risk and Insurance 34(3):371-384. Finney, P.J. (1952). Probit Analysis, Cambridge: Cambridge University Press. Goldberger, A. (1964). Econometric Theory, New York: Wiley. Harmelink, P.J. (1974). "Prediction of Best's General Policyholder Ratings." Journal of Risk and Insurance 41(4):621-632. Harrington, S.E. and J.M. Nelson. (1986). "A Regression-Based Methodology for Solvency Surveillance in the Property-Liability Insurance Industry." The Journal of Risk and Insurance 53(4):583-605. Judge, G.G., W.E. Griffiths, R.C. Hill, H. Lutkepohl and T. Lee. (1985). The Theory and Practice of Econometrics. New York: Wiley. Kahane, Y. (1978). "Solidity, Leverage and Regulation of Insurance Companies." The Geneva Papers on Risk and Insurance December:3-19. Kahane. Y., C.S. Tapiero and L. Jacque. (1986). "Concepts and Trends in the study of Insurer's Solvency." Paper presented at the International Conference on Insurance Solvency, Wharton School, (June). Manski, C.F. and S.R. Lerman. (1977). "The Estimation of Choice Based Samples." Econometrica 45:1977-1988. McDonald, J. (1984). "Some Generalized Functions for the Size Distribution of Income," Econometrica 52:647-663. McDonald, J. and D.G. Clarke. (1992). "Generalized Logit Models with an Application to Consumer Credit Behavior." Presented at the International Symposium on Forecasting, Amsterdam (1988). Journal of Economic and Business 44:47-62. McFadden, D. (1974). "Conditional Logit Analysis of Qualitative Choice Behavior." In Frontiers in Econometrics P. Zaremka (ed.). New York: Academic Press. Ohlson, J.A. (1980). "Financial Ratios and the Probabilistic Prediction of Bankruptcy." Journal of Accounting Research 18:109-131. Pinches, G.E. and J.S. Trieschmann. (1977). "Discriminant Analysis, Classification Results and Financially Distressed P-L Insurers." The Journal of Risk and Insurance 44:289-298. Pinches, G.E. and J.S. Trieschmann. (1974). "The Efficiency of Alternative Models for Solvency Surveillance in the Insurance Industry." Journal of Risk and Insurance 41(4):563-577.

PREDICTING INSURANCE INSOLVENCY

241

Pinches, G.E. and J.S. Trieschmann. (1973). "A Multivariate Model for Predicting Financially Distressed Property-Liability Insurers." Journal of Risk and Insurance 40(3):327-338. Shaked, L. (1985). "Measuring Prospective Probabilities of Insolvency: An Application to the Life Insurance Industry." The Journal of Risk and Insurance 52:59-80. Thornton, J.H. and J.W. Meador. (1977). "Comments on the Validity of the NAIC Early Warning System for Predicting Failures Among P-L Insurance Companies." CPCU Annuals September:191-211. Trieschmann, J.S. and G.E. Pinches. "A Multivariate Model for Predicting Financially Distressed P1-L Insurers." The Journal of Risk and Insurance 40:327-338. Zavgren, C.V. (1983). "The Prediction of Corporate Failure: The State of the Art." Journal of Accounting Literature 2:1- 38. Zemijewski, M.E. (1984). "Methodological Issues Related to the Estimation of Financial Distress Prediction Models." Journal of Accounting Research 22:59-82.

This page intentionally blank

1

FIRM CHARACTERISTICS AND WORKERS' COMPENSATION CLAIMS INCIDENCE H. Allan Hunt Rochelle V. Habeck Michael J. Leahy

Introduction The research effort that underlies this chapter began with an intriguing discovery, namely, wide differences in litigation rates among the workers' compensation case (WC) populations in different cities in Michigan that were demonstrated in a previous Upjohn Institute study (Hunt, 1981). The question of the possible WC cost impact of such differences became a policy issue in Michigan during the battle for the location of the new GM Saturn facility. Workers' compensation costs in Michigan had risen significantly over the twenty years from 1958 to 1978, to the point that WC costs were a very significant negative element in the Michigan business climate (Burton, Hunt, and Krueger, 1985). If lower WC litigation rates in southwest Michigan could offset some of the cost disadvantage to a Michigan Saturn location, the Michigan Department of Commerce wanted to know about it. More directly, Ed Welch, Director of the Bureau of Workers' Disability Compensation of the Michigan Department of Labor, was searching for a way to demonstrate that there were policies and techniques that could help employers to lower WC costs and reduce workers' suffering at the same time. The Michigan statute had been amended in 1980, 1981, and 1984 with resulting substantial cost savings for employers and fairly 243

244

W O R K E R S ' COMPENSATION INSURANCE

small benefit reductions for workers. However, there were real questions about whether additional cost reduction was politically feasible, since there was no obvious way to reduce costs further without significant benefit reductions. This research project grew out of these policy interests under the sponsorship of the Bureau of Workers' Disability Compensation, in partnership with the Upjohn Institute and Michigan State University. The project drew upon the disability management and rehabilitation perspectives of researchers at Michigan State University in order to broaden the scope of the research effort. The work of these researchers in the relatively new field of disability management seemed to offer some degree of employer control over WC costs and improved employment outcomes for injured workers (Tate, Habeck and Galvin, 1986). Berkowitz (1985) has pointed out that work disability is a socioeconomic phenomenon that must be understood in the context of the disabled person's total situation. Persons with the same apparent medical condition will have very different disability outcomes that cannot be explained by their diagnosis alone. In fact, disability is influenced by a complex set of factors, including such job-related items as income replacement benefit levels, alternative wages, types of work available to the person, and the relationship between the mental and physical requirements of the job and the residual capacities of the person. All of these factors potentially influence ability or incentive to work and can be expected to help determine a given individuals reaction to a particular physical limitation. Similarly, the impact of the firm environment on the onset and duration of work disability must also be considered. Internal characteristics of firms such as job hazards, company size, and unionization have been related to hazard abatement and disability cost containment actions (Sims, 1988). Further, more subtle characteristics such as the attitudinal environment and human resource orientation of firms have been implicated as impacting employee health and claims incidence (Rosen, 1986). Berkowitz (1985) cited general factors that lie within the control of employers and can be addressed at the firm or plant level to deal with the disability problem. Employers can 1) prevent some disabilities from occurring through safety and health measures and access to proper care for limitations that do occur; 2) provide early intervention to disabilities of gradual onset by noting patterns of absenteeism, benefit use, and grievance rates; 3) know the mental and physical requirements of their jobs in order to appropriately place workers and assist health care and rehabilitation providers with accommodation or placement of impaired workers; and 4) address collective bargaining issues and create a transfer policy that allows

W O R K E R S ' COMPENSATION CLAIMS INCIDENCE

245

temporary or modified work assignments with minimal disruption of productivity. In fact, Askey (1988) argues that over 50 percent of prevailing workers' compensation costs are directly attributable to a company's internal practices in response to injured workers and their claims. Rising disability costs have clearly increased the incentives for employers to develop their capacity to effectively manage disability factors that are within their control. Although only very limited empirical information is available, a growing body of descriptive accounts has created an acknowledged set of principles to guide employer initiatives. The concept of disability management has emerged as an important component of effective employer action. The concept entails a comprehensive, systematic, goal-oriented, employer based approach to manage the occurrence and outcomes of disability. The overall aim of disability management is to minimize the impact of disability on the individual worker and the workforce as a whole--thus improving workers' quality of life, enhancing productivity, and reducing the costs of injury and illness for the company and for society (Tate, Habeck, and Galvin, 1986). Some of these factors were conceptualized and refined for this research effort on the basis of a three year study of disability management initiatives and outcomes among Michigan employers conducted by Michigan State University (Munrowd and Habeck, 1987). The goals of this research effort, then, were first, to substantiate the variability of claims incidence among employers within the same industry and regulatory environment; and second, to explore the relationship among organizational characteristics of firms with their claims incidence status. Ultimately, the research seeks to test the assumption that some significant portion of the variability in workers' compensation experience among employers is due to organizational factors and practices that are at least partially within the control of the employer. The purpose of this specific chapter is to identify the central organizational characteristics that differentiate employers with low WC claims incidence from those with high WC claims incidence in order to help identify and guide employer actions that may favorably impact their own workers' compensation experience.

The Bivariate Analysis of Claims Incidence This section will present a bivariate analysis of the relationship between selected organizational characteristics and the incidence of closed WC

246

W O R K E R S ' COMPENSATION INSURANCE

claims among Michigan firms in 1986. Later, a multivariate analysis will be presented which considers the impact of all factors simultaneously. The bivariate analysis of claims incidence among Michigan employers utilizes a matched sample of 5,568 firms representative of all employers with more than fifty employees. 1 The independent variables to be examined include: 1) location, by county in which the injury occurred; 2) firm size, as indicated by the level of employment in the second quarter of 1986; and 3) industry of the employer, based on the product produced or the arena of economic activity of the firm. Table 11-1 shows the distribution of closed claim rates by location within the state of Michigan. The individual closed claim rates of the finns are collapsed into seven classes to represent the entire continuous distribution. The grouping of counties for analysis is explained in Appendix A. The table presents the cross tabulation of the dependent variable, closed claims rate, by the independent variable, location within the state. It also reports the means and standard deviations for each category. The chi-square statistic provides a test of the hypothesis that these two variables are unrelated, given the sample size and the distributions of the two variables. For table 11-1, the chi-square statistic is statistically significant at the five percent level. The highest level of closed claims incidence occurs among the statewide (i.e., multiplant) firms. The lowest closed claims rates are observed in the Southern tier of counties, the Northern sector, and the Central section of Michigan respectively. Detroit and the Western counties fall in the middle. Treating the multiplant firms as a group in the analysis of claims incidence by location could bias the results on location. For example, it is clear that the big three auto producers would be included in the statewide group. Thus their unique WC experience will be attributed to the statewide group rather than to the specific locations of the plants. If the individual installations of these large firms are more like one another than they are like their physical neighbors in workers' compensation experience, this will cause no problem. It is possible that treating these firms as a group could tend to minimize the significance of physical location as an influence on WC claims incidence, particularly if these are the largest employers in the state. Table 11-2 presents the bivariate analysis of closed claim rates by employer size, as measured by the number of employees in 1986. Again, the relationship is very highly significant in a statistical sense, and the means show a distinct pattern across employer size categories. The incidence of claims declines with size until the very largest category. Study of the individual cell patterns reveals a number of interesting items.

247

WORKERS' C O M P E N S A T I O N CLAIMS I N C I D E N C E

tt'3 ¢"q

tO

O

0 0 ._1

rr E

e-,

0

°i

.ff II e~

Q

rq

-ff

~6

o6

~

,--,

248

WORKERS' COMPENSATION INSURANCE 0

~

o o

~A

-~

~.

o.

,~ ~ -

~.

~



o, ~-

. ~ °°

~

~

~

~ '~



O

0

E LLI

t'q

T--

I-

~4

v

~

"~

'~

~

"~

=

WORKERS' COMPENSATION CLAIMS INCIDENCE

249

First, it is clear that the larger the firm, the less likely they are to fall within the highest closed claims category, over eleven claims per 100 workers. There is a clear step function across the bottom row of the table. As one proceeds across to larger employers, the percentage of firms experiencing these very high closed claim rates declines steadily, so that firms with more than 1,000 employees have less than one-half the chance of experiencing more than eleven claims per 100 employees. On the other end of the distribution, less than one claim per 100 employees, the relationship is not so clear. The table shows that small employers (50 to 99 employees) are less likely to experience these extremely low levels of closed claims incidence. However, the extreme gap between the smallest firms and all other size categories probably reflects the systematic exclusion of employers with zero closed claims in 1986 from the analysis. An employer with 50 to 99 employees who had a true claims incidence of 1.0 claim per 100 employees per year would have a predicted value of 0.5 to 0.99 claims. But they would not have appeared in the sample at all unless they actually closed one claim in 1986. Nevertheless, it seems clear that closed claim rates differ substantially among Michigan firms. Table 11-3 presents the bivariate analysis of closed claim rates by industry group of the employer. This independent variable can be expected to have a relatively powerful impact on claim rates, as evidenced by the fact that the insurance industry WC rate classification system is largely industry, or process, based. The data in table 11-3 confirm this presumption; they show that industry does have a very significant impact on closed claims incidence. The range goes from retail trade and service industries, which have closed claim rates of under 2.5 per 100 employees, to construction, transportation and utilities, and public administration, which all have over five closed claims per 100 employees. 2 Thus, it is clear that average rates of claims incidence vary substantially by industry, even at this level of aggregation. One important technical detail to note is that the precision of measurement for industry as represented in table 11-3 is not very good. The insurance industry uses about 550 classifications, of which 250 to 300 are in rather common usage, to represent the variety of exposure for all employers. Using eight industrial groups to represent the variety of circumstances encountered in the real world is not sufficient, but it is still indicative of the power of industry in determining WC claims incidence. It is clear that industry is a very important determinant of closed claims rates. Still, the major impression created by table 11-3 is of the variety of

250

WORKERS' COMPENSATION I N S U R A N C E

O

"O t" t"-i

t'-I

t'-I

~

..O

tO

tr

E t,.) "O ¢/)

oc

o ¢..)

¢6 I

t~

I-.--

"~

~q

,4

,6

06

,-

I

I

I

I

I

WORKERS' COMPENSATION CLAIMS INCIDENCE

251

experience among Michigan employers. Even within the service sector, where over 30 percent of firms experience less than one claim per 100 employees, there are another five percent of firms who have more than seven claims per 100 employees. Furthermore, this contrast would be even greater if it was possible to include the firms that had zero claims in 1986. Because of the variety of closed claim rates shown in table 11-3 and the fear that this could be due more to grouping dissimilar firms than to fundamentally different claims experiences, it was decided to do a more precisely focussed analysis. This analysis will be presented graphically and will show the distribution of closed WC claims in 1986 for more detailed industry groups than would be feasible in tabular form.

Graphical Analysis of Claims Incidence by Industry This section provides an overview of closed claims experience by industry. Each industry presented for analysis is represented by a figure, depicting the actual range of closed claims incidence among the individual firms in that industry in 1986. The point of the analysis is to explore the degree of concordance in WC experience among employers in the same basic line of business. From the matched database of 5,568 firms, there were a total of seventy-five industries (2-digit SIC level) with one or more firms (employing more than fifty people in Michigan as of the second quarter of 1986) who closed at least one WC claim in 1986. There were twenty-nine industries that had at least fifty such firms; table 11-4 lists these industries according to their rank ordered average closed claim rate. Local and Long Distance Trucking (SIC 42) is the industry with the highest average closed claim rate in Michigan among those industries with at least fifty firms in our matched sample. This industry experienced an average of nearly seven closed claims per 100 employees in 1986. Rubber and Miscellaneous Products (SIC 30) was close behind with a mean of 6.3 closed claims per 100 employees. Next comes Primary Metals Manufacturing (SIC 33) with a mean of just under 6.0 claims per 100 from a sample of 113 firms. On the other end of the distribution, the Educational Services industry (SIC 82) had the lowest average with only 1.1 closed claims per 100 employees in 1986. Retailers (SIC 59) were next lowest with 1.6 closed claims per 100 employees. These numbers are not surprising, nor are the ranking of industries. Generally, white-collar and service industries are

252

WORKERS' COMPENSATION INSURANCE

Table 11-4.

SIC 42 30 33 34 16 17 37 20 15 51 91 26 25 35 80 50 28 73 36 27 70 54 79 89 55 83 58 59 82

Closed Claims Rate by Industry (2 Digit SIC).

Industry

N

Mean

Standard Deviation

Coefficient of Variation

Trucking, Local & Long Distance Rubber & Miscellaneous Primary Metals Fabricated Metals Construction Special Trade Transportation Equipment Food Production General Contractors Wholesale Trade-Nondurable Executive, Legislative Paper & Allied Products Furniture Production Machinery Production Health Services Wholesale-Durable Goods Chemical Production Business Services Electric & Electronic Printing Trade Hotels & Other Lodging Food Stores Amusement & Recreation Misc. Other Services Automotive Dealers Residential & Social Services Drinking & Eating Places Retailers Educational Services

115 183 113 382 53 195 168 114 63 192 194 69 59 352 339 192 53 241 78 82 70 163 57 51 221 84 370 73 393

6.82 6.29 5.95 5.37 5.23 5.21 5.07 4.79 4.67 4.33 4.03 4.00 3.90 3.85 3.38 3.37 2.93 2.91 2.84 2.54 2.50 2.42 2.32 2.26 2.25 2.22 2.11 1.61 1.13

4.09 10.65 3.96 4.12 5.37 3.48 4.14 3.70 3.02 3.95 3.98 2.45 2.54 3.57 3.74 3.30 3.21 3.38 2.03 1.84 1.64 1.87 1.61 3.71 1.33 1.56 1.66 1.15 3.42

59.99 169.30 66.61 76.79 102.79 66.73 81.61 77.16 64.77 91.21 98.81 61.10 65.30 92.68 110.66 97.93 109.35 116.34 71.56 72.29 65.64 77.38 69.66 163.81 59.01 70.21 78.45 71.78 303.59

n e a r the b o t t o m of the list while construction and heavy manufacturing industries are n e a r the top. W h a t is s o m e w h a t surprising is the degree of variation in claim rates within each industry, and this will be m o r e effectively d e m o n s t r a t e d by the individual industry distributions. Figures 1 1 - 1 through 1 1 - 3 represent durable goods manufacturing industries, historically the heart of the Michigan e c o n o m y . A total of 183 firms manufacturing R u b b e r and Miscellaneous Plastic Products (SIC 30), 113 firms f r o m the

WORKERS'

COMPENSATION

253

CLAIMS INCIDENCE

NUMBER OF FIRMS 601

50 //

40 30

//2 /// /// / / I /// ///

//

//.o

//

///

20 / / I

/ /

//

//t / / I

10

/L

(1

1- 2,99

F// /// F.-J ¢// /// /// i// /~-,s //J /// /// //7 ///

/ / / /

¢'//

3-4.99

5-6.99

'7/ 7-8.99

9-I0.99

)11

CLAIMS RATE PER 400 EMPLOYEES Figure 1 1 - 1 , Rubber & Miscellaneous Plastics Products SIC-30. N u m b e r of firms = 183 M e a n claim rate = 6.29 per 100 employees Coefficient of variation = 1 6 9 . 3 0

Primary Metal Industries (SIC 33), and 382 firms producing Fabricated Metal Products (SIC 34), are represented in figures 11-1 through 11-3 respectively. Mean values for these industries range from 5.37 to 6.29 closed claims per 100 employees. Coefficients of variation range from 66.61 to 169.30, revealing very substantial variation in WC claims experience among firms in the same two-digit industries. It is not at all unusual among this group of industries to see substantial numbers of firms at very extreme values, particularly in primary metals where over 12 percent of sample firms had closed claim rates of 11.0 per 100 employees or greater in 1986. For rubber products and fabricated metals, about 8 percent of sample firms reached these levels. Yet, these industries also show substantial numbers of firms with less than 3.0 closed claims per 100 employees. The next three figures are for service industries. Figure 11-4 portrays the closed claim records for 241 firms in Business Services (SIC 73) and figure 11-5 is for 339 firms in the Health Services (SIC 80) industry. Figure 11-6 rounds out this group with Social Services (SIC 83). These service firms generally experience very low WC claims incidence, with from

254

WORKERS' COMPENSATION

INSURANCE

NUMBER OF FIRMS 30 /// /// /// /// /// r/I f/v #//

25

r// /// ///

20

10 ¢// ¢'//

ff~

1-2.99

(t

///

///

/// F// f//

/// f// ///

/// ///

///

r//

//I

f// #// //s /// /f/ //I /// /// f/J

/// //I ~// ri/ //I //P

//I

/// e'// w//

//# ///

//]

3-4.99

5-6.99

7-8.99

///

//1

I/I ill I/1 111 i11 wll I/I i/-i

t/1

f/J //I t'//

I/I //J f/J I/1 i11 I/I

9-t0.99

)H

CLAIMS RATE PER 100 EMPLOYEES

Figure 1 1 - 2 . Primary Metal Industries SLC-33. Number of firms = 113 Mean claim rate = 5.95 per 100 employees Coefficient of variation = 66.61

NUMBER OF FIRMS 120 t00 /J/ /// ///

///

80 //, /// /// f// //., f// /// f// //1 f//

60 40

/// F//

20

///

//. //., //., /// //t /// /// //1 /// f// /// /// //J ///

"I // sI

rll r/j

,// (I

D

1-2.99 5-g.99 3-4.99 7-8.99 CLAIMS RATE PER t00 EMPLOYEES

Figure 1 1 - 3 . Fabricated Metal Products SIC-34. Number of firms = 382 Mean claim rate = 5.37 per 100 employees Coefficient of variation = 76.79

9-I0.99

)11

255

WORKERS' COMPENSATION CLAIMS INCIDENCE NUMBER OF FIRMS 120: ,.'// /.// / / / f / / / / / / / / t'.// / / / /.// /'.// / / t

100 80 60 40 20

D (1

f / / /J/ //J /.'/ //J / / t / / / / / / //-/

F'-/A

I / /

1-2.99

3-4.99

5-6.99

7-8.99

9-I0,99

CLAIMS RATE PER t00 EMPLOYEES

Figure 1 1 - 4 . Business Services SIC-73. Number of firms = 241 Mean claim rate = 2.91 per 100 employees Coefficient of variation = 116.34 NUMBER OF FIRMS 180 160

t " .,,- / i / / r / I / / / / / i

140

f / /

t20

i / /

100

r/., r/,, t'/,, / / /

80

r/1 r / / //,,

60

/./1 f / / //., / / I

40

/ / I / / t / / I / / / i / / //,, / / / / / / //J

20

1,000 Employees Employees Employees 152 17.57 713 82.43 865 10.06

39 10.48 333 89.52 372 4.33

14 5.83 226 94.17 240 2.79

Total 3031 35.25 5568 64.75 8599 100.00

Chi-square = 558.85*** with 4 degrees of freedom.

The final data issue was the decision to exclude all firms with less than fifty employees. This primarily reflected concern over unreliable sample estimates due to the problem of estimating infrequent events with a one year slice-in-time sample mentioned earlier. The smaller firms also proved to be considerably harder to match between the two databases and the quality of the match was thought to be important in establishing the face validity of the overall analysis.

The Adequacy of the Match The first difficulty encountered with the match was that the ES-202 tape showed there were 154,882 Michigan firms paying unemployment insurance taxes in the middle of 1986, while Bureau records only showed closed WC claims against 19,250 employers. The discrepancy in size between the two databases is due to the fact that many employers do not typically close any WC claims in a particular year, especially smaller employers or those in sectors with low accident frequency. Table A l l - 2 shows the gross match rates achieved by firm size category. As might be predicted, the adequacy of the match increases with the size of firm. For employers with more than 1,000 employees in 1986, over 94 percent were successfully matched with their WC closed claims record. On the other hand, for firms with fifty to ninety-nine employees, only about 54 percent were successfully matched. Of course, this could easily be due to the fact that these smaller employers did not have any closed cases in 1986. Overall, the match was "successful" for 65 percent of all firms in the population that employed fifty persons or more. Confining our attention just to those firms with 500 employees or more in table A l l - 2 , 91.3 percent of these firms were successfully matched

WORKERS'

COMPENSATION

CLAIMS

281

INCIDENCE

MATCH ST

No

log(3a.~g~)

~

\

I

I

I

_

\

WE

No

Chl-equor(~

~0~3~.2 X)5

~ 37.497... wffh 5 d/gtOOS Ol froodom

w,

/

?~3 (sT.ssx) r -

r---'

.,,,, )

17 -' 0. It is interesting to note that the default probability, q, affects the insurance price. This may show up in an indirect manner. The insurer is likely to publish a price of P ( a ) = (] + ?)paL, where ~ is the nominal loading factor. The insurer does not officially interject default risk into its pricing formula; but if the default probability is known (or a subjective one is postulated), then 2 can be easily related to ~ via the equivalence of P ( a ) in both cases: = 2 - (1 + ~)q.

(3)

Thus, the true ("effective") loading 2 is larger than the nominal loading y, since the nominal loading fails to adjust for the discrepancy between the probability of a claim and the probability of being fully indemnified. Maximizing equation (1) is straightforward, and yields a first-order condition 2 of V ' ( a ) = - P ' ( a ) [EU'] + p(1 - q ) L U ' ( W 2 ) = 0.

(4)

where E denotes the expectation operator and where I define W1-A -P W2- A - PW3-A -P-L.

L + aL

The first term in equation (4) is the marginal cost of increasing coverage, weighted by expected marginal utility. It is thus the expected marginal reduction in utility due to a higher insurance premium for a higher a. The second term in equation (4) is the expected increased indemnity due to a higher a, weighted by marginal utility in the state of the world in which the indemnity is paid. Thus, equation (4) merely equates mar-

314

WORKERS' COMPENSATION INSURANCE

ginal benefits and marginal costs, in terms of utility, for increasing the level of coverage. Dividing through by p(1 - q)L, it follows from equation (4) that te'(W2) - t~U'(Wl) - CU'(W3) = 0,

(5)

where B ---

(1 + 2)(1 - p ) 1 -

(1 + ).)p(1 -

q)

and

C =

(1 + 1 -

2)pq

(1 + 2 ) p ( 1

The left-hand side of (5) clearly has the same sign as

-

q)"

V'(a).

If 2 = q = 0, so that insurance premiums are actuarially fair and there is no possibility of insolvency, it follows trivially from (5) that full coverage, a* = 1, will be purchased. This result is well known in the insurance literature and first appears in papers by Mossin (1968) and Smith (1968), who also show that less than full coverage is optimal when 2 > 0 and q = 0. If 2 = 0 but q is nonzero, then less than full coverage will be optimal. This is easily seen by noting that B + C = 1 in this case; thus U' (1412) is a weighted average of U'(W1) and U'(W3). For a more complete analysis of the level of coverage purchased, the reader is referred to Doherty and Schlesinger (1990). When q > 0, less than full coverage will be purchased, even at a "fair" price with 2 = 0. In this case, what happens when the degree of risk aversion increases? This is easily seen by considering a more risk averse set of preferences, which I will assume are strongly more risk averse in the sense of Ross (1981). This stronger measure of risk aversion may be appropriate since there are now actually two sources or risk, namely, the original risk of loss and the added risk of insurer insolvency. Note that this added risk is very real and if the firm puts in a claim to an insolvent insurer, it will actually end up worse off than it would be absent any insurance since it will lose premium dollars in addition to suffering the loss. The Ross measure of risk aversion is designed to analyze behavior under multiple sources of risk. An example showing why the usual Arrow-Pratt measure of risk aversion is insufficient with multiple risks is given in Turnbull (1983). Applying Ross' main theorem, any set of strongly-more-risk-averse preferences can be written as bU(W) + G(W), where b is a positive constant and G is a decreasing, strictly concave function, that is, G ' < 0 and G" < 0. Note that the addition of G makes utility more concave (risk averse) in a particular sense. Ross (1981) shows how his measure is stronger than the Arrow-Pratt measure.

315

" G O O D DAYS" AND " B A D D A Y S "

Considering the left-hand side of equation (5) using the strongly-morerisk-averse preferences and evaluating this expression at a = 1, I obtain: G'(W2) - [BG'(W~) + CG'(W3)].

(6)

The expression in (6) can be either positive or negative. In the case where there is no insolvency risk (q = 0), C = 0 and it follows trivially, since in this case B > 1 when 2 > 0, that the expression in (6) is positive and hence more insurance should be purchased. However, when q > 0, it is possible that either more or less insurance will be purchased with a higher degree of risk aversion. Since "strongly more risk averse" in the sense of Ross implies more risk averse in the usual Arrow-Pratt sense as well, this ambiguity holds for "more-risk-averse" preferences in general. In order to see that the ambiguity is not vacuously satisfied, I consider the following example.

Example 1: Let preferences be given by the utility function U(W) = - e x p ( - f l W ) , with fl > 0. This is the well-known constant-absolute-risk-aversion utility, which was applied by Cozzolino (1978) and by Freifelder (1979) in examining risk management decisions by non-insurance and by insurance firms respectively. The parameter fl denotes the degree of absolute risk aversion, which is calculated in general as - U " ( W ) / U ' ( W ) ; for the assumed utility function, fl is constant over all values of W. I use the following parameter values in my calculations: A = 4, L = 2, p = 0.2

and

2=0.3.

I consider three different probabilities of insolvency, q = 0, q = 0.1 and q = 0.25; and I let fl vary from fl = 0.5 to fl = 3.5. Computer results are presented in table 13-2 and shown graphically in figure 13-1. For the case where there is no chance of default, q = 0, we see that coverage is monotonic increasing in the degree of risk aversion. However, when there is a 10 percent or a 25 percent chance of insolvency, we see that the level of coverage decreases for high enough levels of risk aversion. In particular, a is decreasing in fl for fl > 1.25 when q = 0.1, and for fl > 1.0 when q = 0.25.

316

WORKERS' COMPENSATION INSURANCE

Table 13-2.

Optimal Insurance.

Risk Aversion

0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5 2.75 3 3.25 3.5

q = 0

q = 0.1

q = 0.25

0.659674194063 0.773116129375 0.829837097031 0.863869677625 0.886558064688 0.902764055447 0.914918548516 0.924372043125 0.931934838813 0.938122580739 0.943279032344 0.947642183702 0.951382027723

0.628458657247 0.725338672529 0.762347156832 0.771335423126 0.762438823112 0.740298830342 0.708361927882 0.67009320238 0.62888968508 0.587577813729 0.54809139978 0.511490300177 0.478179975705

0.586954323354 0.66461582441 0.68209339132 0.671101601552 0.643520953903 0.606366090968 0.564862766716 0.522856513428 0.48281449698 0.446021024272 0.4129268729 0.383489960763 0.357424329668

1.0

~

0.9

'

'

'

~

-~q--0

0.8

~

0.7

g 0.6 "6 ~ 0.5

- " ' " - a..,.

0.4

" ' - - ' ' . . . . ,,..,_..q=O. I "~

- - c ~ ...oq=0.25

0.3

0

1

2

3

Degree of risk aversion

Figure 13-1.

Loss Reduction I now consider the same model with two modifications. First, I no longer allow for the purchase of insurance; and second, I now allow for investment in loss reduction. T o this end, let x denote the level of investment in loss reduction and let c(x) d e n o t e the cost of x, where c(0) = 0, c' > 0 and c" >I O. The size of the loss is affected by the level of loss reduction and is given by L ( x ) , where L ' < 0. Thus, the size of the loss decreases with investment in loss reduction. I assume, however, that the reduction in loss severity is achieved only if the loss reduction turns out to be

317

" G O O D D A Y S " AND " B A D DAYS"

effective ex post. If the loss reduction is ineffective, I consider only the simplest case where the loss then remains at L ( 0 ) - - t h e same loss as would have occurred absent any investment in loss reduction. Of course, in this circumstance, the cost of the loss reduction, c(x), would also be lost. As in the previous section, I let q denote the probability of failure for the risk-management tool under consideration. The valuation function for a particular level of loss reduction is given as follows: V ( x ) = (1 - p ) U [ A - c(x)] + p(1 - q ) U [ Z - c(x)] - L ( x ) + p q U [ A - c(x) - L(0)].

(7)

Choosing x to maximize (7) obtains the first-order condition, V'(x) = -c'(x)[EU'] -p(1

- q ) L ' ( x ) [ U ' ( W 2 ) ] = O,

where W1 = - A - c(x) w2 =-A

- c(x) -

L(x)

W3 - A - c(x) - L(O).

The second-order condition for a maximization is easily verified. The first-order condition in equation (8), equates the expected marginal utility cost of increasing the investment in loss reduction, c ' ( x ) [ E U ' ] , with the expected marginal utility benefit of a lower loss severity when the loss reduction is effective, p(1 - q ) L ' ( x ) [ U ' ( W 2 ) ] . Let x* denote the "optimal" level of loss reduction, that is, the value of x satisfying equation (8). I now consider a strongly-more-risk-averse valuation function in the sense of Ross, and evaluate the left-hand side of equation (8) at x*. This obtains -c'(x)[EG']

- p(1 - q ) L ' ( x ) [ G ' ( W a ) I ,

(9)

where G is as defined in the preceeding section. The conditions on G are not strong enough to sign the expression in (9), and hence the optimal level of loss reduction might be either higher or lower than x*. Note that when the loss reduction is perfectly reliable, that is, q = 0, then (9) reduces to -c'(x)(1 - p ) [ G ' ( W a ) ] - [c'(x) + L ' ( x ) ] p G ' ( w 2 ) .

(10)

From the first order conditions for the case where q = O, it follows that (1 - p ) c ' ( x ) > - p [ c ' ( x ) + L'(x)]. It then follows in (10) that more "positive weight" is put on the G'(W1) term. Since both G' terms are

318 Table 13-3. Risk Aversion

0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5 2.75 3 3.25 3.5

WORKERS' COMPENSATION INSURANCE Optimal Loss Reduction. q = 0

q = 0.1

q = 0.25

-0.13083460829 0.032399861307 0.152088508609 0.245055275696 0.319765975802 0.381110748833 0.432185003605 0.47508336873 0.511299478129 0.541945894783 0.567883391627 0.589800797092 0.608266033126

-0.18950360326 -0.028006366568 0.084492990165 0.164890418494 0.22129348194 0.25847809014 0.280074581973 0.289471396625 0.289968626687 0.284546731965 0.275607506011 0.264882436526 0.253504688042

-0.283156732344 -0.120729081181 -0.013413187606 0.057880498529 0.103254730121 0.129687001155 0.142778407419 0.147070076083 0.145999582879 0.141948814277 0.136438480418 0.130373726654 0.124259751707

negative and G ' ( W I ) < G'(Wz), it follows that the expression in (10) must be positive. H e n c e , x* will be higher with strictly-more-risk-averse preferences in the case where q = 0. In fact, this is true for Arrow-Pratt risk aversion, in addition to the Ross risk aversion considered here, as was shown by Dionne and E e c k h o u d t (1985). T o show that the case where q > 0 can indeed lead to a negative relationship between risk aversion and the level of loss reduction investment, I consider the following example, which is similar to Example 1.

Example 2: This example is similar to Example 1, but with no insurance. I still assume A -- 4 and p = 0.2, with L(0) = 2. I now assume that loss reduction will reduce the size of the loss, when it is effective, to L ( x ) = 2e -x. The cost function that I use is c ( x ) = x . I once again let fl vary from fl = 0.5 to fl = 3.5 and I consider three alternative probabilities of default: q = 0, q = 0.1 and q = 0.25. Computer calculations appear in table 1 3 - 3 and are illustrated graphically in figure 13-2. H e r e we see once again that d e m a n d for the riskm a n a g e m e n t tool under consideration, in this case loss reduction, is only monotonic increasing for the case where the tool is perfectly reliable, q =

319

,'GOOD DAYS" AND "BAD DAYS" i

0.7 0.6

I

i

,

i

i

i

i

t

i

i

q=0

0.5 0.4 0.3 0,2 0.1

f:5.*L.

0,0 -0.1 -0.2 -0.3

-"

_n(/,,

:

:

~I

i

q=O.2S

~ - -

,

,

i

1

i

i

i

!

2

J

;

i

i

i

3

Degree of risk aversion

Figure 13-2• 0. For the cases where q > 0, the level of loss reduction is decreasing in the degree of risk aversion for high enough values of ft.

Loss Prevention Here I consider the case in which the firm can invest in loss prevention, rather than in either insurance or loss reduction. The size of a loss is fixed at L, but the probability of loss is now assumed to be given by the function p ( y ) , where p' < 0 and p" > O, and y denotes the level of loss prevention. Of course, p ( y ) is bounded from below by zero. The cost of loss prevention is assumed to be given by the function c(y), where c' > 0 and c" t> O. I once again interject nonreliability risk by assuming that the loss prevention fails to perform properly with probability q. In this case, I assume that the probability of loss remains at p(O). The firm's valuation function can now be written as V(y)

where

= (1 - ~ ) U [ A - c(y)l + f i U [ A - c ( y ) - L], fi -

qp(O) +

(11)

(! - q ) p ( y ) .

Choosing the level of loss prevention to maximize V ( y ) requires V'(y) -- - c ' ( y ) [ E U ' ]

+ (1 -

q)p'(y)[U(W2)

-

U(W1)] -- O,

(12)

where W 2 =- A W1 - A -

c(y) c(y)

L.

Equation (12) equates the expected marginal utility cost of increasing y with the marginal utility benefit of shifting probability weight from W2

320

WORKERS' COMPENSATION INSURANCE

Table 13-4.

Optimal Loss Prevention.

Risk Aversion

q = 0

q = 0.1

q = 0.2

0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00 3.25 3.50

0.409 0.458 0.510 0.564 0.621 0.679 0.739 0.799 0.860 0.920 0.980 1.037 1.092

0.372 0.405 0.433 0.454 0.466 0.466 0.457 0.439 0.414 0.384 0.349 0.310 0.266

0.333 0.353 0.366 0.369 0.362 0.347 0.324 0.296 0.264 0.229 0.191 0.150 O.106

to W1. Let y* denote the optimal level of loss prevention, satisfying equation (12). U n d e r strongly-more-risk-averse preferences, evaluated at y*, we obtain V'(y*) = -c'(y)[EG']

+ (1 - q ) p ' ( y ) [ G ( W 2 )

-

G(W1)].

(13)

Conditions on G are not strong enough to sign (13), even in the case where q = 0. Further discussion of the q = 0 case can be found in Briys and Schlesinger (1990). Example 3 illustrates that the optimal level of loss prevention might be either higher or lower with an increase in risk aversion.

Example3: Utility is as in the previous examples, with initial wealth still A = 4. I assume that p(0) = 0.5 and that p ( y ) = 0 . 5 e - y , with c ( y ) = y. I consider three alternative probabilities of ineffectiveness for loss protection: q = 0, q = 0.1 and q = 0.2. C o m p u t e r calculations are shown in table 1 3 - 4 with the same data plotted graphically in figure 13-3. In table 1 3 - 4 , we see that loss prevention is monotonic increasing in the degree of risk aversion only for the case where loss prevention is perfectly reliable, q = 0. For cases where there is some chance that loss

"GOOD DAYS'AND"BAD

321

DAYS"

1.2

q=0

1.0 ~

0.8

8 0.6 0

0.4

"''"

..q

-'----_.,....

0.2

0.0

~ ~q~O. 2 I

|

I

1

Figure 1 3 - 3 .

0 1

I

I

I

I

I

I

2

I

I

I

I

I

3

Degree of risk aversion

prevention may fail to operate, we see that the level of loss prevention is decreasing in the degree of risk aversion, for high enough levels of ft. I should m e n t i o n that the case where q = 0 does not always lead to a monotonic relationship between loss prevention. An example to the contrary was provided by Dionne and Eeckhoudt (1985), and was also discussed, albeit heuristically, by Ehrlich and Becker (1972).

Combination of Risk-Management Tools Thus far, I have only considered the use of the key tools of risk management used in isolation. Of course, such tools are used most effectively in combination with one another. Naturally, the ambiguity with regards to the relationship between risk aversion and the level of risk management used extends from the case of using one tool alone to using combinations of tools. My focus here is a bit different. It is to consider the interaction between risk-management tools, and whether or not these tools might be economic substitutes or complements. For the sake of tractability, I consider first the use of only two of the tools, insurance and loss reduction. For now, I assume that insurance is perfectly reliable and that only the loss reduction may fail to perform properly. Let q denote the probability of such a failure. The valuation function can now be written as

322

WORKERS' COMPENSATION INSURANCE

V(a, x) = (1 - p ) U ( W 1 ) + p(1 - q)U(W2) + pqU(W3),

(14)

where W1 =-A - c(x) - P ( a ) W2 = A - c(x) - P(a) - L(x) + a L ( x ) W3 - A - c(x) - P ( a ) - L(O) + aL(O) P ( a ) - (1 + 2)ap[qL(O) + (1 - q)L(x)].

The firm must simultaneously choose both the level of insurance and the level of loss reduction to maximize V(a, x). This yields the first-order conditions Va =- - e ' ( a ) [ E V ' l

+ p{(1 - q ) L ( x ) U ' ( W z ) + qL(O)U'(W3)} = 0

(15)

and Vx - - [ c ' ( x ) + (1 + 2)p(1 - q ) L ' ( x ) ] E U ' + p(1 - q)[(1 - ct)L'(x)]U'(W2) = 0. (16) Equation (15) implies that the marginal utility cost of a higher premium for a higher a equals the expected marginal utility benefit of receiving a larger share of the total loss (either L(0) or L(x)) as an indemnity. The first term in equation (16) considers the marginal utility cost of increasing the investment in loss reduction, x. Note that, in addition to the direct marginal cost c'(x), there is a term which reflects the insurance premium reduction for reducing the size of the loss (at least when the loss reduction works). Thus, the first term in (16) reflects the net cost of increasing x. The second term in (16) is the marginal utility benefit of having a lower loss severity when the loss reduction is effective. The optimal x* equates these marginal costs and benefits. I now examine whether loss reduction and insurance are substitutes or complements by varying the "price" of insurance (which I consider as the pure premium plus the loading per dollar of coverage, 1 + 2). The comparative static analysis is too complicated to present the general case. However, the following example shows that insurance and loss reduction can be either substitutes or complements.

Example 4: In this example, the utility function is the same as in previous examples, but with ,8 fixed at ,8 = 0.3. I also assume that A = 4, L(0) = 2, L(x) = 2e -x, as in example 2. The cost function is assumed to be c(x) = 0.2x,

323

"GOOD DAYS" AND "BAD DAYS" Table 13-5.

Insurance and Loss Reduction.

Price of Insurance

Loss Reduction

Insurance

1 1.05 1.1 1.2 1.23 1.25 1.27 1.3 1.35

0.916290731874 0.959405510929 0.989394261722 1.013934263703 1.012820467029 1.010015073169 1.005586774475 0.995941272806 0.971978288421

1 0.841182161531 0.690528512556 0.406919057278 0.324836683631 0.270553801395 0.216501548787 0.135631678695 0.000555040450

1.02 1.01 1.00 O.gO .~ 0.gQ i 0.97

i

i

l

0.96 0.95 0.94 0.93 0,92 0.91

i

0.1

Figure 13-4.

i

0.2

I

i

I

i

,

i

J

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Level of insurance

and the probabilities of loss and of loss reduction's ineffectiveness are assumed to be p = 0.4 and q = 0.375, respectively. I let the insurance loading vary from 2 = 0 to 2 = 0.35. Computer calculations of the optimal levels of insurance and loss reduction are shown in table 13-5 and illustrated graphically in figure 13-4. We see that, as the price of insurance rises, the firm will decrease its purchase of insurance. For relatively low insurance prices, we see a substitution of more loss reduction to replace the reduction in insurance protection. However, for relatively high insurance prices (loadings above approximately 20 percent in this example), we see that the level of loss reduction also falls as insurance prices rise. Thus, insurance and loss reduction are seen to be substitutes at low insurance prices, but complements at higher prices. The above analysis considers only the case of the joint use of insurance

324

W O R K E R S ' COMPENSATION INSURANCE

and loss reduction when insurance is fully reliable. It should be apparent that interjecting the nonreliability of insurance will only add to the ambiguity. The case of insurance and loss reduction has been examined when both tools are perfectly reliable by Ehrlich and Becker (1972), who show that these two services are always substitutes. Example 4 shows that the Ehrlich and Becker result no longer holds when the tools are not fully reliable. Ehrlich and Becker also showed that insurance and loss prevention are not always substitutes, in the case of full reliability, and may be complements. The addition of nonreliability to the model maintains this ambiguity. Combining loss reduction and loss prevention, or using all three tools, there is no guarantee that any two of the tools are either substitutes or complements with one another. The fact that loss-control tools and insurance are not necessarily substitutes implies that it is possible to desire a higher optimal investment in loss control when insurance levels are increased, a notion that is in direct contrast to the usual implied assumption of possible moral hazard. When insurance and loss control are complements, we have what Colwell and Wu (1988) call "moral imperative"--a higher level of insurance actually leads to behavior to improve the loss distribution.

Concluding Comments I have considered a basic model of a firm's investment in risk management, when the possibility exists for the risk-management tools not to perform as promised. I label this the "nonreliability problem." On a "good day," the risk manager might see his or her well calculated strategy come into fruition. A potential fire might be extinguished immediately by an effective sprinkler system, for example. However, there is always the chance of a "bad day," where everything just seems to go wrong. The sprinkler doesn't activate. The fireproof walls prove to be defective. The sprinkler finally activates after the fire is extinguished, causing water damage on top of the fire damage. The insurer eventually fails to pay the ensuing claim due to its own insolvency. Since the risk management tools are themselves risky, it is perhaps not surprising to find that, with a greater concern for pure (i.e., downside) risk as measured via the degree of risk avesion, a firm may elect to use less of these "risky" riskmanagement tools. When considering the joint use of these tools, I found that they might

"GOOD DAYS" AND "BAD DAYS"

325

be complements as well as substitutes. In one sense, this says that we cannot use much of the existing theory to predict behavior. Indeed, normative models of risk-management decision making become hardpressed to yield many definite conclusions, and what we are left with are case-by-case decisions that cannot be generalized very easily. While this may distract from the elegance of normative models of choice behavior, it may help to explain why empirical evidence does not always agree with the inferences made from existing theoretical models.

Appendix A I present here a brief argument concerning risk-averse behavior by a firm. For small firms, large firms that are closely held, or publicly held firms with an individual as a majority shareholder, it might not be difficult to imagine why risk-averse behavior would prevail. For large publicly held firms that have a manager with much power, this might also be understandable. However, publicly held firms serve shareholder interests and shareholders do not usually concern themselves with firm-specific risks. Since shareholders have an opportunity to diversify their stock holdings, only market risk is important if we stay within the confines of the well-entrenched Capital Asset Pricing Model (CAPM). Several papers have addressed this issue recently, mostly in asking the question: why do publicly held corporations buy insurance? (See, for example, Mayers and Smith, 1982). These papers provide several reasons why the firm insures, many of which focus on the many claimants on a firms assets. In addition to the obvious shareholders and bondholders, the firm's customers, suppliers, workers, landlord and other various parties also have claims on the firm. The firm-specific risk may affect the nature of many of these claims. For example, consider two firms identical in every way except that one firm is fully insured and against potential loss and one is not. If the loss might jeopardize the firm's survival and its ability to make payroll, workers at the uninsured firm may demand a higher salary, one that includes a risk premium for the payroll risk. Since workers cannot readily diversify this risk, we might see the lower wages translated into a higher expected cash flow (since wage costs are reduced) for the insured firm. Of course, if insurance premiums include an expense loading, the cash flow will be reduced, on average, by the amount of the expense loading. 3 If insurance premiums are actuarially fair, we might expect the cash flow to be a function of the level of insurance coverage. Let Z(a)

326

W O R K E R S ' COMPENSATION INSURANCE

denote the expected cash flow when insurance is purchased at level ct. I assume Z'(a) > 0 for a < 1, and Z'(a) < 0 for a > 1. Thus, expected cash flow is at its highest when a -- 1, (i.e., with full coverage). I also assume that cashflow returns to higher insurance levels are diminishing, so that Z"(a) < O. Other claimants on the firm, such as suppliers of raw materials, may be willing to allow the firm to have more favorable trade-credit terms, for example, net sixty rather than net ten, when the firm is insured. Banks might also be willing to supply more funds at better rates to insured firms. Such phenomena will tend to reduce the weighted average cost of capital (WACC) for the insured firm. Let k(a) denote the WACC and assume that k'(a) < 0 for a < 1, and k'(a) > 0 for a > 1, so that WACC is lowest a fully-insured firm. I further assume k"(a) > 0. Assuming a zero-growth firm, although only minor changes are necessary to relax this assumption, the value of the firm is given by

V(a) = Z(a)/k(a).

(A1)

It is easy to show that firm value is maximized when a = 1. In other words, full insurance is optimal at an actuarial-fair price. If we include an expense loading, similar to that in equation (2) but with q = 0 for the moment, the net expected cash flow will be Z(a) a2E(L). In this case, Z(a) is maximized at some value ao < 1. If we now consider the value of the firm,

V(a) = [Z(a) - a2E(L)]/k(a),

(A2)

it is apparent that V(a) is maximized at some value of a between a = a0 and a = 1, say a*. Thus, less than full coverage is optimal for 2 > 0. This result, together with the previous one, are key results in the insurance literature, stemming from Mossin (1968) and Smith (1968). Note how the above scenario implies a type of pseudo-risk-averse behavior by the corporation. In reality, cashflow and the WACC are sensitive to changes in the level of insurance. Indeed, looking at gains and losses in firm valuation due to changes in discounted cash flow which are due to a, there is a concave valuation of the cashflows. This concavity of firm valuation yields behavior that is identical to that which would obtain, if discounted cash flows remained unchanged (except for the insurance premium and indemnity) and the firm used a concave (i.e., risk-averse) valuation of the risky cash flows. For the reason presented above, as well as others (see Cozzolino, 1978), it may not be inappropriate to use a risk-averse valuation of risky

"GOOD DAYS" AND "BAD DAYS"

327

cash flows to a corporation. A m o r e extensive evaluation of this a p p r o a c h is currently being investigated by B e n - A r a b and Schlesinger (1990).

Notes 1. See, for example, Industrial Risk Insurers (1986) or Factory Mutual Engineering (1983). 2. The second-order condition is satisfied, which is easily verified by calculating V"(a). 3. Note that the pure premium has a net effect of zero on expected cash flow, since we pay the pure premium as part of the insurance premium but expect to receive it back in indemnity payments.

References Ben-Arab, M. and H. Schlesinger. (1990). "Corporate Purchase of Insurance." Unpublished manuscript. Briys, E. and H. Schlesinger. (1990). "Risk Aversion and the Propensities for Self-Insurance and Self-Protection." Southern Economic Journal 57:13151324. Briys, E., H. Schlesinger and J.M. Schulenburg. (1991). "The Reliability of Risk Management: Market Insurance, Self-Insurance and Self-Protection Reconsidered." Geneva Papers on Risk and Insurance Theory 16:45-59. Colwell, P. and C. Wu. (1988). "Moral Hazard and Moral Imperative." Journal of Risk and Insurance 55:101-117. Cozzolino, J. (1978). "A Method for the Evaluation of Retained Risk." Journal of Risk and Insurance 45:449-471. Dionne, G. and L. Eeckhoudt. (1985). "Self-Insurance, Self-Protection and Increased Risk Aversion." Economics Letters 17:39-42. Doherty, N. and H. Schlesinger. (1990). "Rational Insurance Purchasing: Consideration of Contract Nonperformance." Quarterly Journal of Economics 105:243-253. Ehrlich, I. and G. Becker. (1972). "Market Insurance, Self-Insurance, and SelfProtection." Journal of Political Economy 80:623-647. Factory Mutual System. (1983). The Handbook of Property Conservation. Norwood, MA: Factory Mutual Engineering Corporation. Friedfelder, L. (1979). "Exponential Utility Theory Ratemaking: An Alternative Ratemaking Approach." Journal of Risk and Insurance 46:515-530. Industrial Risk Insurers. (1986). Overview: A Total Management Program for Loss Prevention and Control. Hartford, CT: Industrial Risk Insurers. Mayers, D. and C.W. Smith. (1982). "On the Corporate Demand for Insurance." Journal of Business 55:281-296.

328

WORKERS' COMPENSATION INSURANCE

Mossin, J. (1968). "Aspects of Rational Insurance Purchasing." Journal of Political Economy 76:553-568. Ross, S. (1983). "Some Stronger Measures of Risk Aversion in the Small and in the Large with Applications." Econometrica 49:621-638. Schlesinger, H. and J.M. Schulenburg. (1987). "Risk Aversion and the Purchase of Risky Insurance." Journal of Economics, 47:309-314. Smith, V. (1968). "Optimal Insurance Coverage." Journal of Political Economy 76:68-77. Turnbull, S.M. (1983). "Additional Aspects of Rational Insurance Purchasing." Journal of Business 56:217-229.

14

ECONOMIC CONSEQUENCES OF THIRDPARTY ACTIONS FOR WORKPLACE INJURIES Joan T. Schmit

Introduction Safety has been an issue of importance to business since long before the industrial revolution. The advent of industry, however, brought with it both a concentration of exposure to loss (many people injured simultaneously) and a complexity of loss potential through such environmental factors as use of synthetic materials. Demand for safety rose significantly as a result. Most attention to safety, however, was focused on workplace hazards. Not until after the 1965 Restatement of Torts (Second) did product safety become a prominent issue. Since then, and especially following the Report of the National Commission on Product Safety (NCPS), the economics of (product) safety has been a topic given substantial treatment in economic, legal, and insurance literature. Much of the existing literature analyzes the impact on efficient resource allocation of regulated safety standards, mandated compensation awards, and various liability schemes. In this article, the question of what effect third-party actions in work-related accidents have on safety is considered. The NCPS concluded that lack of consumer information necessitated a shift in the assignment of financial responsibility for product-related accidents from consumer to producer, that is, strict producer liability. Oi (1973) responded to the NCPS with perhaps the seminal economic 329

330

WORKERS' COMPENSATION INSURANCE

analysis of product safety. Oi concluded that strict producer liability could yield less safety than exist under negligence or strict consumer liability doctrines. Since Oi's work, the analysis has been expanded to consider the economics of liability rules, of which many studies exist (e.g., Brown, 1973; Diamond, 1974), noncompetitive markets (Schlesinger, 1985), risk associated with the outcome of costly litigation (Simon, 1981), the impact of the liability insurance market (Shavell, 1982), and consumer misperception of safety (Spence, 1977), among other circumstances. In general, a conflict seems to arise in encouraging safety while compensating victims. As stated by Spence: . . . liability of the producer to the consumer has a dual function. It is an adjustment in the incentives for the producer to supply reliability and in this respect is like a fine. It is also nonvoluntary insurance for the consumer. The optimal fine and the optimal level of insurance will not necessarily be the same. (561) Spence concluded that the fine should be made in two parts, one to the victim-consumer to compensate the victim appropriately, and the other to the state to set liability at proper levels for deterrence. His conclusions coincide with those of Chelius (1982), holding that optimality can be achieved, but not through the channels currently available. Rea (1981) employed Spence's model of imperfect information to analyze the impact of workers' compensation and occupational safety legislation. He did so through the employee's demand function for wages, safety, and disability insurance. Rea concluded that the mandatory insurance required by workers' compensation could work to reduce safety. One element absent from Rea's analysis is the existence of third-party suits. Given the importance of work-related accidents on products liability costs, 1 and the prominence in tort reform proposals of some relief from that liability, 2 thrid-party suits appear deserving of consideration. Third-party suits arise in workers' compensation when an employee is injured on the job due to some action of another party. For example, an employee may be injured while driving a truck that has defective brakes. Workers' compensation benefits are available to the employee because of the work-relatedness of the injury. The employee may choose also to sue the truck manufacturer in a products liability case. Workers contracting disease from exposure to toxic chemicals or materials (such as asbestos) similarly may seek workers' compensation benefits and also sue the chemical manufacturer for negligent warning or some other form of

ECONOMIC CONSEQUENCES OF THIRD-PARTY ACTIONS

331

liability. Likewise, an employee injured due to negligent medical treatment of a work-related injury may sue the medical care provider in a malpractice suit. In this article, the models of Spence and Rea are modified to include the impact of third-party actions by employees and employers against product manufacturers. 3 Because the focus is on the circumstances involving product-related hazards, the other situations in which employees are injured are not considered. The results closely follow those of Rea. Specifically, because third-party suits increase the employees' "disability insurance" coverage, workers are less willing to trade wages for safety. Yet, because the employers' marginal cost of an accident could rise or fall depending on the extent to which information is imperfectly conveyed and on whether or not subrogation is permitted, firms might be more or less willing to buy safety.

The Model 4

In the work environment, two states of nature exist. Either the employee is injured (here only product-related injuries are considered) or the employee is not injured. The probability of not being injured [a(d, s, p)] is a function of the employee's safety precautions (d), the employer's safety precautions (s), and the product manufacturer's safety precautions (p). Each is costly. If an employee is not injured, utility (U) is received from wages (w). Contrarily, when injured, an employee receives utility (V) from a products liability (or any third-party) award (y).5 Ex ante, the expected utility of the employee in either state is improved by undertaking safety precautions. As in Rea's model, the labor supply is assumed fixed and utility is concave increasing. The expected utility is:

a(d, s, p)U(w, d) + (1 - a(d, s, p))V(y, d)

(1)

Strict products liability is assumed so that when an accident occurs from use of a defective product, the third-party suit is successful. In addition, the third-party judgment (y) is assumed to exceed the workers' compensation award (m) to which the worker is entitled. Rea pointed out that the employer's profit equation can be affected by the costs and benefits of safety precautions in a number of ways. He simplified the analysis by considering labor as the only factor of production. 6 In addition, Rea assumed that precautions affected the fixed costs

332

W O R K E R S ' COMPENSATION INSURANCE

of hiring an employee, but were independent of the level of employment. These assumptions are included here. The employer's labor costs are comprised of wages (w) to nondisabled employees, disability benefits to disabled employees adjusted for effects of subrogation against product manufactures and the resulting increase in product costs (D), and a fixed cost (C(s)) per employee for safety other than the purchase of work products. Competition for employees is taken to yield an expected marginal product of labor equal to the expected marginal cost of labor: a(d, s, p)Z(a) = w + [(1 - a(d, s, p))/a(d, s,

p)](D) 4 C(s)/a(d, s, p) (2)

where Z(a) equals the marginal revenue product of labor, Z' < 0; C', C" > 0 . The producer's production function is not considered. Rather, increased production costs associated with increased liability are factored into the analysis through the employer's purchase price of work products. Producer safety (p) will therefore be dropped, assumed to be incorporated in both the employer's costs for disability (D) and efforts toward safety (s). This treatment appears appropriate because of the lack of a direct contract between product manufacturer and product user, as well as the fact that the analysis can be made under simplified circumstances without extreme assumptions. The topic is given further consideration later in the paper. Also considered is the importance of administrative costs, such as litigation expenses in tort actions. Administrative costs can be sufficiently large to outweigh the benefits of determining exact fault for a given loss.

Optimal Conditions Perfect Information The optimal levels of safety and compensation are achieved by maximizing the expected utility of each worker (denoted G) with respect to m (y here), d, and s, constrained by the equality between marginal product of labor and marginal cost of labor. The purpose of this chapter is to compare the optimum calculated under an assumption of accurate estimation of risk with the optimum calculated under as assumption of misestimation of risk. If competitive markets including perfect information exist, the full cost

ECONOMIC CONSEQUENCES OF THIRD-PARTY ACTIONS

333

of accident liability will be placed on employers through product prices, that is, D = y. Hence, G is given by

G = a(d, s)U((Z(a) - [(1 - a)/a]y - C/a), d) + (1 - a(d, s))V(y, d) (3) The first order conditions are

Uw = U~

(4)

-[aUa + (1 - a)Va] = aa[U - V] + [ad(y + C)/a]Uw + aU~Z'ad (5) C' = [as(U- V)]/Uw + [as(y + C)]/a + aZ'as

(6)

Equation (4) dictates that the marginal utility derived from wages equals the marginal utility of a third-party tort award; that is, marginal utilities are equated between both states of the world. According to equation (5), a worker will choose a level of safety such that the expected marginal cost of safety equals the marginal reduction in uncompensated loss (aa[U - V]), plus the marginal reduction in the employer's workers' compensation and safety costs

(aa(y + C)U~) a

- , plus the declining

marginal productivity of labor (aUwZ'aa). Employers, in turn, choose safety levels such that the marginal cost of safety equals the marginal benefits. Benefits are represented by

- v)) plus the the marginal reduction in uncompensated loss \(as(UU~ reduction in workers' compensation costs and nonproductive job slots

(as(C+ y).), both offset by the dectine in marginal productivity (aZ'as). Of particular importance here is that the wage workers are willing to forgo in return for greater safety is (U - V)/Uw. Given that U - V is smaller when third-party actions are available than when only workers' compensation is obtainable (because m is less than y), workers are less willing to buy safety under these circumstances than under Rea's assumptions. On the other hand, employers are willing to buy more safety because the marginal cost of labor is greater when product prices reflect third-party actions. If perfect information and perfect markets are assumed, the parties will bargain for optimal levels of wages, safety, and compensation (Rea, 1981). The circumstances when the assumptions of perfect information and perfect markets do not hold, therefore, provide the impetus for studies such as this one.

334

WORKERS' COMPENSATION INSURANCE

Imperfect Information As done elsewhere, an element of imperfect information is incorporated in the model by use of a perceived probability of no injury distinct from the true probability. The worker, therefore, views a ( d , s) as r ( d , s). 7 The employer and empolyee maximize

r(d, s)U(w, d) + [1 - r(d, s)V(y, d)]

(7)

with the employer constrained by Z(a)

-

w -

[(1 - a ) / a ] y - C ( s ) / a = 0

(8)

The first order conditions are (together with (8) and d held constantS): rUw/[(1 rs(U -

r)Vy] = a/(1 - a)

V) = rUw(C'/a)

- a s [ ( y + C ) / a 2] - a s Z ' a

(9) (10)

Employee misperception of risk could induce reduced safety because the willingness to substitute wages for safety is reduced. As before, this likelihood is affected by third-party actions in that U - V is smaller, while the employer's marginal cost of an accident is larger than when such actions are not present. Imperfect information, however, leads to an ambiguous effect of thirdparty actions over workers' compensation benefits only. An employee who underestimates risk will not demand sufficient wages to account for the reduced utility of workers' compensation benefits, which are less than the full loss. This same employee, when finding third party actions available, will continue to demand less safety than optimal, and of course less safety than without these extra awards. The employer, however, will pay increased product prices for the producers' liabilities to employees. If producers properly estimate risk, the full cost (possibly greater than the reduction in employee's demand for safety) is passed on to the employer, inducing increased safety. The net result of third-party actions, therefore, is ambiguous. It depends on the extent to which workers misperceive the impact of expenditures on safety (rs), as well as the extent to which workers misperceive the level of risk. If rs is far below as, workers are unwilling to buy optimal safety, with or without third-party actions. Legislated levels of compensation, as a result, may be appropriate. Danzon (1987) argues that when workers are perfectly informed no need exists to place liability on employers because the employment contract will yield optimal levels of wages, safety, and compensation for injuries. If workers are imperfectly informed, however, suboptimal levels of safety implicitly demanded through compensating wage differentials

ECONOMIC CONSEQUENCES OF THIRD-PARTYACTIONS

335

may be corrected through a rule of employer liability. The optimal award under a rule of employer liability is the amount of compensation fully-informed workers would have been willing to purchase. Danzon (1984) and Viscusi and Evans (1990) argue that the optimal level of compensation is well represented by workers' compensation benefits. Danzon further postulates that people typically are unwilling to purchase compensation for pain and suffering, which may be available in tort awards but not in workers' compensation awards. Under this theory, then, third-party actions ought not be allowed. One potential problem with Danzon's argument is the possible discord generated in the minds of workers when the disparity between tort awards and workers' compensation awards is observed. Such discord appears to be a major factor in attempts by workers to abrogate the exclusivity of workers' compensation. Merely to eliminate the option for third-party actions, therefore, could yield significant problems in the workers' compensation systems, including excessive litigation costs.

Imperfect Product Markets Of greater interest in this piece is to discuss the effect of a product market that fails to account perfectly for the workers' compensation costs. That is, employers and product manufacturers do not have perfect information in their contractual setting. If manufacturers and employers have perfect information (even if employees do not), then they will internalize the product manufacturers' contribution to workers' losses through product prices. If, however, employers underestimate the risks associated with use of various products, they may not demand sufficient safety. This result is obtained by using the same type of analysis as applied to employee misperception of risk. The marginal benefit of another dollar of safety spent by the manufacturer may be perceived as having lower value than the extra dollar in price. The employer having such a perception would be unwilling to purchase the additional safety. If the fully-informed employer would purchase the additional safety, to do otherwise is suboptimal. A potential source of improvement is to permit employer subrogation against the manufacturer for the workers' compensation costs. As used above and discussed by Danzon (1987), strict liability placed in a setting of misperceived risk can, in theory, correct for the market imperfection. If the subrogation right is part of a third-party action available to the employee, including awards for general damages, however, Danzon

336

WORKERS' COMPENSATION INSURANCE

would argue that the ultimate payment to the employee exceeds the optimum. To correct for this excess, the subrogation action could be available solely to the employer. How much the employer ought to be able to recover is the amount of protection the employer would have purchased under full information. One might postulate that value to be nearly equal to or equal to the amount that the employee would have demanded. Thus, as is possible today in all but three states, 9 for the employer to recoup from the third party the value of workers' compensation payments may be appropriate. A key concern considered in this discussion is whether or not assessment of workers' compensation costs on manufacturers will yield optimal levels of safety and also optimal amounts of compensation. One alternative to the present liability system discussed by a number of authors (Danzon, 1987; Spence, 1977, among others) and mentioned previously, is to utilize a two-tiered system of compensation. One tier would be the amount by which the plaintiff is compensated, set at a level that would yield appropriate demand for safety by plaintiffs. To reduce moral hazard the compensation would be less than full. The second tier would be an additional fine against the defendant, intended to place responsibility for the full value of the loss on a negligent party and thus encourage safety. A system of government fines has been suggested as a means of encouraging proper levels of safety by all parties. The ability to split responsibilities accurately, of course, is unclear. Yet, to permit subrogation by the employer is to add another layer of administrative and legal costs to the system. Proving causation against a product manufacturer (which is required even in strict liability) typically is more difficult than proving work-relatedness. Third-party actions by necessity increase noncompensation costs. Whether or not the additional costs are well spent depends upon the degree of disparity between the employer's perception of risk (and the impact of safety) and the true level of risk (and the true impact of safety). Assuming that most employers can obtain reasonably adequate information about product safety, this disparity may be minor. The benefit of permitting tort actions against product manufacturers, therefore, may be small, especially if it is determined that workers' compensation benefits are the appropriate payments to employees. Providing an opportunity to sue third parties may also yield counteractions by those third parties against employers. These suits are termed "actions over," and are more likely to occur when employees can receive tort awards than when they cannot. As presented earlier, if the thirdparty action is limited to the employer's subrogation rights, they may be

ECONOMIC CONSEQUENCES OF THIRD-PARTY ACTIONS

337

appropriate. In the current situation, however, tort awards are available to employees. And actions over have received some favorable attention by courts despite the fact that, in essence, they destroy the exclusivity of workers' compensation programs. One method of eliminating this additional layer of administrative costs is to incorporate a theory of contributory negligence in the subrogation action against the manufacturer. As pointed out by Danzon (1987) (referencing a wealth of literature), where both parties have some control over safety, a rule of strict liability with contributory negligence can result in optimal loss prevention. To permit subrogation without contributory negligence defenses may be too beneficial to the employer to yield optimal safety levels, and in turn, may encourage manufacturers to bring actions over against employers. Eliminating the employee's action reduces this possibility. The actions-over phenomenon is an example of how two competing liability systems can cause undue administrative wastes, and demonstrates the need to coordinate them.

Conclusions Product safety is an important issue, both for use in and out of the workplace. Previous studies have shown that regulation of compensation for work-related accidents (worker's compensation legislation) can affect workplace safety (often adversely). In this chapter, the impact of third-party actions for work-related injuries and/or illnesses is considered. The general impact of third-party actions can be assessed only after making assumptions concerning the degree, if any, to which information is imperfect, and whether or not employers are given legal rights to receive indemnity from third parties. It can be stated unambiguously, however, that employees will demand less safety when they have available third-party actions than when they do not because their compensation for work-related injuries/illnesses is greater with third-party actions. Whether or not this will yield suboptimal levels of safety depends upon the counter-balancing effect of increased labor costs reflected in higher work-product prices and more frequent workers' compensation payments. If workers' compensation benefits approximate optimal levels of compensation, as argued by Danzon (1984) and Viscusi and Evans (1990) third-party actions by employees are undesirable. Where employers as well as employees misperceive risks, however, lack of producer liability may yield suboptimal levels of safety. Subrogation rights by employers against manufacturers may correct for this error, but does so at increased

338

WORKERS' COMPENSATION INSURANCE

administrative costs and incentives for third-party actions over. At the very least, a liability rule including defenses of contributory negligence by the employer, therefore, appears needed. Alternatively, if employer misperception of risk is slight, complete elimination of third-party actions, including subrogation rights (to eliminate the administrative costs) may be appropriate. What appears clear is that further study of the relative risk perceptions of employers, employees, and producers is needed to assess various liability rules available to these parties. The time for such research appears ripe as greater right-to-know laws are enacted and as producer liability in the workplace (consider asbestos and chemical manufacturers) increases exponentially.

Note 1. The 1977 Insurance Services Office Product Liability Closed Claim Survey produced data indicating that "workers injured on the job are involved in 11 percent of product liability incidents resulting in claim payments. However, these incidents account for 42 percent of total bodily injury payments" ("Highlights," 5). 2. For example, S.44, introduced to the 98th Congress of the United States, devotes Section 10 to the "Effect of Workers' Compensation Benefits." 3. The analysis could apply equally well to other third parties, such as medical care providers. 4. The notation follows directly from that utilized by Rea. 5. The allocation of liability (y) between the employer and the third party is an issue considered below. An assumption is made, however, that an employee does not receive both workers' compensation and a tort award. Thus, either the producer pays the full amount (y), or the employer pays the workers' compensation award (m) and the third party is obligated to contribute the remainder (y - m). 6. The reader might be confused by the thought of labor as the sole factor of production, while a capital good is the object of the third-party action. To alleviate such confusion, consider the capital good, not as a factor of production, but rather as an element of the work environment. Employers can affect employees' safety through their choice of capital good. 7. Employees are assumed to underestimate risk. Such an assumption is consistent with evidence recently provided by Dickens (1985): Finally, once workers are on the job, there will probably be a tendency for them to view their jobs as being safer than they actually are. Employers are assumed to estimate risk perfectly for this part of the analysis. 8. The assumption of constant d is to ignore employee moral hazard. In essence, the assumption is a conservative move. Furthermore, one could consider d as reflected in r. 9. The three states are Georgia, Ohio, and West Virginia (Larson, 1988). Oklahoma and Minnesota disallow subrogation in special circumstances.

ECONOMIC CONSEQUENCES OF THIRD-PARTY ACTIONS

339

References Brown, John Prather. (1973) "Toward an Economic Theory of Liability." Journal of Legal Studies 2:323-349. Chelius, R. James. (1982). "The Influence ofr Workers' Compensation on Safety Incentives." Industrial and Labor Relations 35:235-242. Danzon, M. Patricia (1987). "Compensation for Occupational Disease: Evaluating the Options." Journal of Risk and Insurance 54:263-282. (1984). "Tort Reform and the Role of Government in Private Insurance Markets." Journal of Legal Studies 13:517-549. Diamond, Peter. (1974). "Single Activity Accidents." Journal of Legal Studies 3:107-164. Dickens, T. William (1985). "Occupational Safety and Health and "Irrational" Behavior: A Preliminary Analysis." Workers' Compensation Benefits: Adequacy, Equity and Efficiency, In D. John Worrall and David Appel (eds.) Ithaca, NY: ILR Press, 19-40. Larson, Arthur. (1988). Workmen's Compensation Laws, §71.21. New York: Mathew Bender. Oi, Y. Walter (1973). "The Economics of Product Safety." The Bell Journal of Economics and Management Science 4:3-28. Rea, A. Samuel, Jr. (1981). "Workmen's Compensation and Occupational Safety Under Imperfect Information." American Economic Review 71:80-93. Schlesinger, Harris. (1985). "Product Safety for a Monopolist Under Strict Liability." Insurance: Mathematics and Economics 4:253-263. Shavell, Steven. (1982) "On Liability and Insurance." The Bell Journal of Economics 13:120-132. Simon, Marilyn. (1981). "Imperfect Information, Costly Litigation, and Product Liability." The Bell Journal of Economics 12:171-184. Spence, Michael. (1977) "Consumer Misperception, Product Failure and Producer Liability." Review of Economic Studies 64:561-572. Viscusi, W.K. and W. Evans. (1990). "Utility Functions that Depend on Health Status: Estimates and Economic Implications." American Economic Review 80:353-374.

-

-

.

This page intentionally blank

Index

Abel, A., 99 Abramowitz, M., 179 Absenteeism, claims incidence and, 244, 271,274, 275, s e e a l s o Lost work days Accounting loss ratios, 106 Actions over suits, 336 Adverse selection, 113, 114-115, 116-117, 121,122 Age benefit levels and, 14 claims incidence and, 269-270, 275 heterogeneous claims and, 26, 36, 38, 44-45, 48 self insurance and, 131 temporary/permanent disability transition and, 57, 60, 82,87, 93 Aggregate pooled cross-section time series, 63 Aggregate time series, 52 Aharony, J., 223 Alabama IRR in, 192-197, 211,212-221 loss ratios in, 198-201,208 Almon distributed lag procedure, 137 Alternative minimum tax (AMT), 174, 175-177 Altman, E.I., 223 Ambrose, J.M., 224, 225,228, 230 Amemiya, T., 224,227 Amputation, 56-57, 58, 70, 77 AMT, s e e Alternative minimum tax APM, s e e Arbitrage pricing model Appel, D., viii, ix, x, xii, 49n. 1, 52, 55, 76, 143n.1,297

Arbitrage pricing model (APM), xi, 105, 149, 152, 167 Arrow-Pratt measure of risk aversion, 75, 314, 315,318 Asbestos injuries, 119,330 Askey, D.E., 245 Asset pricing models, xi, 148, 150-152, s e e a l s o specific types Assets to surplus ratios, 230 Attorney involvement, disability transition and, 57, 58 Automobile manufacturing industry, 246, 257, 258, 259 Autoregressive patterns, 99-100 B2, s e e Beta distributions of 2nd kind Back injuries, 58-59, 94 Bankruptcy probability, s e e Ruin probability BarNiv, R., 224,228, 230 Beard, R.E., 112 Beaver, W.H., 223 Becker, G., 321,324 Ben-Arab, M., 312, 327 Benefit levels, viii, 1-21 claims incidence and, 244 experience-rating and, 297, 298, 304 heterogeneous claims and, 26, 36, 38, 42-43, 48 optimal structure of, 4-6 temporary/permanent disability transition and, 55, 57, 60, 62-63, 71-73, 75-77, 78, 83, 89-90, 93, 94 within-state heterogeneity in, 8-15

341

342 within-state homogeneity in, 6-8 Berger, A.L., 117, 125n.12 Bergstrom, T., 22n. 17 Berkowitz, M., 72, 244 Berkson, J., 224 Best's Insurance Reports, 224, 228, 230 Beta distributions, in option pricing models, 147, 150-151,152, 167, s e e a l s o Beta distributions of 2nd kind; Generalized beta distributions of 2nd kind Beta distributions of 2nd kind (B2), of heterogeneity, 32, 37 Biger, N., 150 Binary choice models, 224,233 Bivariate analysis, of claims incidence, 245-251,271 Black, F., 154, 158 Black Lung program, 7-8 Bliss, C.I., 224 Bloom, T.S., 230, 233 Bonds, 184-185 Bookstaber, R., 226 Booth, P.J., 230 Borba, P.S., 52 Borcherding, T., 22n. 17 Box, G.E.P., 180 Boylan, R.L., ix, xi, xii BR, s e e Burr distributions Brennan, M.J., 158, 168n.3 Briys, E., 312,320 Brown, J.P., 330 Bureau of Workers' Disability Compensation (BWDC), Michigan, 243,244, 275-278, 280, 282 Burr3 (BR3) distributions in generalized qualitative response models, 228 of heterogeneity, 36-37, 39-40, 42-43 Burrl2 (BR12) distributions in generalized qualitative response models, 228 of heterogeneity, 28, 32, 36-37, 42 in loss ratios, 197-202 Burrit models, 228 Burton, J.F., Jr., 3, 21n.5, 22n.14, 72, 94n.4, 243 Business services, 253,255 Butler, J.S., 66n.ll

INDEX Butler, R.J., viii, ix, x, xi, xii, 5, 11, 36, 49n.1, 52, 53, 55, 76, 89, 143n.1,297, 298 BWDC, s e e Bureau of Workers' Disability Compensation, Michigan Call options, 153-155, 156-157, 158 Capacity constraint hypothesis, x, 100-101, 105, 108-110, 122-123 critique of, 112-113 testable implications of, 111 Capital, cost of, 188, 189, 191-192,326 Capital asset pricing model (CAPM), xi, 147, 150-152, 167, 174 of capital flows, 105 fair rate of return and, 158-166 Merton's multiperiod, 149 portfolio valuing with, 156 in risk management, 325 Sharpe/Lintner/Mossin, 149, 159 Capital budgeting theory, 105 Capital flows, ix, x-xi, 99-124 capacity constraint hypothesis of, s e e Capacity constraint hypothesis nonlinear pricing and, 113-122, 123 perfect capital markets hypothesis of, s e e Perfect capital markets hypothesis CAPM, s e e Capital asset pricing model Chelius, J.R., ix, xiii, 52, 76, 77, 94n.6, 297, 330 Chi-square statistics in claims incidence, 246, 265,289, 290 in generalized qualitative response models, 232 in loss ratio distributions, 202 in temporary/permanent disability transition, 58, 60 Claim-filing costs, 74, 82-83, 93 Claims incidence, 243-290 bivariate analysis of, 245-251,271 BWDC data on, s e e Bureau of Workers' Disability Compensation, Michigan graphical analysis of, 251-258 MESC data on, s e e Michigan Employment Security Commission data regression analysis of, 259-264 survey of, 264-273 Upjobn Institute data on, s e e Upjohn

INDEX Institute Clarke, D.G., 234 Closed Case Tapes, 81 Cohn, R., 100, 105 Colwell, P., 324 COMPMAST, 275-276, 291n.6 Construction industry, 249, 252, 264,283 Continuous time/jump process models, 153, 158 Contributory negligence, 337, 338 Cooley, W.W., 226 Cooper, R., 150 Courant, P.N., 22n.18 Cozzolino, J., 315,326 Cummins, D.J., ix, x, 99, 100, 105,111, 117, 124n.5,125n.12, 152, 153, 158, 166, 173 Currington, W., 95n.14 Daenzer, B.J., 223 Dahlby, B.G., 22n.24 Dambolena, I.G., 223 Danzon, P.M., viii, ix, x, 9, 21n.11, 22n.19, 334,335-336, 337 D'Arcy, S., 105,151,168n.6 Darling-Hammond, L., 94n.6 DCF model, see Discounted cashflow model Deacon, R., 22n.17 Deakin, E., 223 Deaton, A., 75 Deductibles, 116-122 adverse selection and, 113, 114-115, 116-117 Deneber, H., 224 Derrig, R., 153,210n.I Detailed Claim Call, 30, 54, 55 Detroit, claims incidence in, 246,259, 261, 262, 263,290 Diamond, P., 330 Dickens, T.W., 338n.7 Dionne, G., 318, 321 Discounted cashflow (DCF) model, xi, 106, 107, 148-150, 151,167 Discrete time framework models, 158 Disfigurement, 81 Dividends, 196-197 Dividend taxes, 101,109, 122 Dixit, R.V.S., 175

343 Doherty, N.A., ix, xi, 105,145n.10, 151, 153,158, 314 Dorsey, S., 1, 11 Draper, N., 180 Duration of claims frequency vs., 25-26, 48 gamma models of, 27-35 Durbin, D., viii, ix, x, xii, 52, 76 Economic loss ratios, 106 Economic Report o f the President, 136

Education, claims incidence and, 269 Educational services, 251,264 Eeckhoudt, L., 318,321 Ehrlich, I., 321,324 ES-202 data, 278, 280, 287,291n.8 Evans, W., 335,337 Ex ante compensation, xii, 70-71, 77, 93, 94 Excess loss components, 294-296 Expected-utility theory, 131,312, 313 Experience-rating, xiii-xiv, 293-305 empirical test of, 299-304 formula used in, 294-296 research on, 297-298 Exponential distributions, of heterogeneity, 28 Ex post benefits, 70 External capital, xi, 101,109, 122, 123-124 Extremity injuries, 94, see also Amputation Fairley, W., 150, 159 Fair rate of return, in option pricing models, 147,158-166, 167 Fatal injuries, 70, 81 Federal Coal Mine Health and Safety Act, 7-8 Feldstein, M.S., 2 Feller, W., 114 Finney, P.J., 224 Firm size, see also Small firms claims incidence and, 244, 246-249, 259, 263,273 experience-rating and, 297-298 forecasting models and, 183 self insurance and, 130, 135-138, 139140, 142 Fisher effect, 107 Fisk distributions, of heterogeneity, 32

344 Florida, disability transition in, 72 Food production industry, 264, 284, 288, 289 Forecasting models, 171-185 investment component of, 183-185 underwriting in, 174, 177-183 Freifelder, L., 315 Frequency of claims, duration vs., 25-26, 48 F-tests, 138,261 Galvin, D.E., 244, 245 Gamma distributions, of heterogeneity, 27-35, s e e a l s o Generalized gamma distributions; Generalized gamma distributions of 2nd kind Garven, J.R., ix, xi, 153, 158, 168n.6 Gender claims incidence and, 268, 274 heterogeneous claims and, 43-44 temporary/permanent disability transition and, 57, 60, 83, 89-90, 93 General Accounting Office survey, 102 General equilibrium models, of benefit levels, 9, 10, 12, 17-18 Generalized beta distributions of 2nd kind (GB2), s e e a l s o Beta distributions; Beta distributions of 2nd kind of heterogeneity, 28, 32-35, 36-37, 48 in insolvency prediction, 227, 228, 234-235 in loss ratios, 197-208 Generalized gamma distributions (GG), of heterogeneity, 27-28, 30, 32-35, s e e a l s o Gamma distributions Generalized gamma distributions of 2nd kind (GG-2), of heterogeneity, 32-35, 36-37, 48 Generalized qualitative response models, 223-239 applications of, 228-235 GG, s e e Generalized gamma distributions Goldberger, A., 225 Goldstein, G.S., 12, 21n.ll Goodman, R., 22n. 17 Grace, E., 124n.5 Grand Rapids, claims incidence in, 290 Greenwald, B.C., 114

INDEX Grievance rates, claims incidence and, 244, 271,274, 275 Gron, A., 100 Gross-in-character models, 173 Guaranty funds, s e e Solvency guarantees Habeck, R.V., ix, xii-xiii, 244,245 Harmelink, P.J., 224 Harrington, S.E., 13, 101,102, 106, 107, 152, 224 Haugen, R.A., 150 Hazardous waste clean-up, 119 Head injuries, 58-59, 94 Health insurance, 6, 9 Health services, 253,255,256, 264,284, 285,288 Hearing loss, 77, 94 Heavy manufacturing industries, 252 Hedonic distribution approach, in self insurance, 137-138, 139-140 Hershbarger, R.A., 224,228, 230 Heterogeneous claims, 25-48, 55 controlling for observed differences in, 35-45 frequency vs. duration in, 25-26, 48 intrinsic, s e e Intrinsic heterogeneity nonparametric, 28-30, 32-35, 47-48 parametric, 27-28, 29, 32 Hill, R., 150, 151,159 Hogg, R.V., 210n.2 Hope Natural Gas decision, 157 Hospital stays, disability transition and, 57, 58 Hunt, A.H., ix, xii-xiii, 243,291n.1 Ibbotson, R., 183 Illinois, disability transition in, x, 54, 55, 58, 64, 76 Imperfect information, xv, 334-335 Imperfect product markets, 335-337 Income taxes, 7, 108, 111 negative, 26 Industrial machinery/equipment industries, 263 Industry groups claims incidence and, 246,249-258, 259, 263-264,283,284-287 experience-rating and, 300

INDEX temporary/permanent disability transition and, 83 Inflation, 71,106-107 Injuries, 73 benefit levels and, 3, 5, 75-76,297, 298 experience-rating and, xiii-xiv, 297,298, 303,304 third-party actions for, s e e Third-party actions Insolvency, vii-viii, xii, 118, 166, 187-220, s e e a l s o Ruin probability in generalized qualitative response models, s e e Generalized qualitative response models loss ratio adjustments and, 189-197 loss ratio distributions and, 197-210 risk management and, 309, 311,312-313, 314-315 Insolvent insurers, 237 Insurance, s e e a l s o specific types benefit levels and, 15 in risk management, 307, 308,311-315, 316, 318, 319,321-324,325-327 self insurance vs., 129, 131-135 Insurance Regulatory Information System (IRIS), xii, 224, 230, 233,235,311 Interest rates, 191,298 capital flows and, 106-107 negative, 148 self insurance and, 134-135, 138, 141 Internal capital, xi, 101,122, 123-124 Internal rate of return (IRR), xi, 211, 212-221 in forecasting models, s e e Forecasting models loss ratio adjustments and, 189-197 in option pricing models, 148-149, 150 Intrinsic heterogeneity, 26, 46-48 gamma models of duration with, 27-35 Investments, 148, 183-185,191 IRIS, s e e Insurance Regulatory Information System Jennrich, R.I., 210n.3 Judge, G.G., 225,226 Kahane, Y., 150, 230 Khoury, S.J., 223

345 Klugman, S.G., 210n.2 Kniesner, T.J., 94n.6 Kraus, A., 100, 105,106, 152, 167 Kronke, C.O., 150 Krueger, A.B., 3, 21n.5, 22n.14, 76, 143n. 1,243 Larson, A., 338n.9 Leahy, M.J., ix, xii-xiii Leigh, P.J., 21n.6 Lerman, S.R., 87,230 Leverage, 190-191,192 Liabilities-to-surplus ratios, xii Lim, L.Y., ix, xi, xii Limited dependent variable models, 225-228 Limited liability constraints, 109,110 Linear probability models (LPM), 224, 225, 226 Loading charges, benefit levels and, 5, 6, 10, 13 Location, claims incidence and, 246, 247, 259-263,264 Logit models insolvency predicted with, 224, 225, 226-227,230, 232,235 of self insurance, 134 of temporary/permanent disability transition, 56, 58, 59, 61, 62, 63, 64, 65, 76, 87, 90, 91, 92 Lohnes, P.R., 226 Lomax distributions in generalized qualitative response models, 228 of heterogeneity, 32, 37 Lomit models, 228 Loss control, xiii, 307-308,309-310 Loss distribution, 121,122 Loss prevention, 307, 308,309,310, 319-321 Loss ratios, 106, 107 adjustments in, 189-197 distribution of, 197-210 transformed, 188-189 Loss reduction, 307,308,309, 310, 316319, 321-324 Loss reserves, 103-104, 121,122, 190 Lost work days, s e e a l s o Absenteeism

346 benefit levels and, 76 claims incidence and, 271-273,274, 275 experience-rating and, 299-300, 303, 304 LPM, s e e Linear probability models Maddala, G.S., 56, 144n.7 Maine, disability transition in, 54 Majluf, N., 109 Manski, C.F., 87, 230 Manufacturing industries, 252-253,264, 283,284 Marginal utility of risk management, 313-314,319-320, 322 of third-party actions, 333 Marital status, disability transition and, 57 Market equilibrium, nonlinear pricing and, 113-122 Massachusetts, disability transition in, x, 54, 55, 56, 58, 59, 60, 65, 76 Massachusetts Rating Bureau, 174 Maximum medical improvement, 70 Mayers, D., 312, 325 McCaffrey, P.D., 293 McDonald, J.B., ix, xii, 28, 49n.3,210n.2, 226, 227, 228,234 McFadden, D., 226 MDA, s e e Multiple discriminant analysis Meador, J.W., 230 Median worker model, 8-11 Medical malpractice insurance, 102 Medicare, 7 Meiskowski, P., 10 Merton's multiperiod capital asset pricing model (CAPM), 149 MESC, s e e Michigan Employment Security Commission Metal manufacturing industries, 251,253, 254, 256-258, 264, 284, 288 Michigan, claims incidence in, 243-290 Michigan Employment Security Commission (MESC) data, 278-280, 282 Michigan State University, 244, 245 Minority workers, claims incidence in, 268-269, 274 Mishkin, F., 99 Mission Insurance Company, 187 Modigliani, F., 151,159

INDEX Moffitt, R., 66n.ll Moore, J.M., 2, 21n.1 Moral hazards benefit levels and, 4, 5-6, 7 deductibles and, 116 risk management and, 307, 308, 312 self insurance and, 130 third-party actions and, 336 Mossin, J., 314,326 Muellbauer, J., 75 Multiperiod models, 149, 151,152, 167 Multiple discriminant analysis (MDA), insolvency predicted with, 224, 225, 225-226, 228-229,230, 232, 233,234, 235 Multivariate analysis, of benefit levels, 15 Munrowd, D.A., 245 Myers, S., 100, 105,109 Myers-Cohn model, xi, 106, 149-150

NAIC, s e e National Association of Insurance Commissioners National Association of Insurance Commissioners (NAIC), 148,223,230, 311 National Commission on Product Safety (NCPS), 329-330 National Commission on State Workmen's Compensation Laws, 2, 3, 18, 72, 130, 141,304 National Council on Compensation Insurance (NCCI), xi, 15, 30, 298 IRR of, 148-149, 150, 211,212-221 NCCI, s e e National Council on Compensation Insurance NCPS, s e e National Commission on Product Safety Negative income taxes, 26 Negative interest rates, 148 Nelson, J.M., 224 Nelson, W.J., Jr., 69 New York, disability transition in, x, 69-94 Nonlinear pricing, 113-122, 123 Nonparametric heterogeneity, 28-30, 32-35, 47-48 Nonscheduled injuries, 78-81, 90-93 Nye, D.J., 166

INDEX Occupation, disability transition and, 83 Occupational Safety and Health Act, 293 Occupational Safety and Health Administration (OSHA), 266,271, 273,274, 275, 299 Ohlson, J.A., 225,226 Oi, Y.W., 329-330 Opportunity costs, disability transition and, 83, 93 Option pricing models, xi, 147-168, s e e also specific types advantages vs. disadvantages of, 166-168 fair insurance price derived with, 153, 157-158 fair rate of return determined with, 147, 158-166, 167 portfolio valuing with, 154, 155-157 OSHA, see Occupational Safety and Health Administration Outreville, F., 99, 100,173 Overtime, claims incidence and, 268, 274 Parametric heterogeneity, 27-28, 29, 32 Partial disabilities, s e e also Temporary/ permanent disability transition permanent, 26, 51-65, 70, 71-72, 74, 77-81, 82, 87, 178 temporary, 70 Part-time employment, claims incidence and, 267-268, 274 Paulson, A.S., ix, xi, xii, 175 Pauly, M.V., 12, 21n.ll, 22n.24 Payroll taxes, 7 Pentikainen, T., 112 Perfect capital markets hypothesis, x, 100, 101,105-108, 122, 123 critique of, 111-112 testable implications in, 110-111 Perfect information, 332-333,335 Permanent disabilities, 14, 69-94, s e e also Temporary/permanent disability transition partial, 26, 51-65, 70, 71-72, 74, 77-81, 82, 87, 178 total, 7, 70, 71 Pesonen, E., 112 Pinches, G.E., 224, 226, 228,230, 234 Plating & polishing industries, 258

347 Pooled cross-section time series, 52, 63 Pooling, 117 Portfolios forecasting models for, 172 option pricing models for, 154, 155-157 Pratt-Arrow measures of risk aversion, s e e Arrow-Pratt measures of risk aversion Premiums determinants of, 294-296 risk management and, 322, 325 Premium-to-surplus ratios, xii, 224 Price, D., 22n.25,136 Primary loss components, 294-296 Probit models insolvency predicted with, 224, 225, 226-227, 230, 232, 235 of self insurance, 134 Progressive taxes, 5, 6, 7 Property-casualty insurance, s e e Propertyliability insurance Property-liabilityinsurance, vii-viii, 151, 190-191,225,311, s e e also Capital flows Property taxes, 10 Proportional taxes, 151 Public administration, 249, 283 Put options, 155 Qualitative response models, 226-227, see also Generalized qualitative response models Quirin, G.D., 150 Ralston, M.L., 210n.3 Rate of return, 188, see also Fair rate of return; Internal rate of return Rate regulation, xi, 187-189, 192,208, s e e also Option pricing models Rea, A.S., Jr., 330, 331-332, 333 Regression analysis, of claims incidence, 259-264 Restatement of Torts, 329 Retail sector, 249,251,283 Retained earnings, xi, 110, 119, 122, 124 Risk aversion Arrow-Pratt measure of, 75,314, 315,318 benefit levels and, 5, 7, 10 capital flows and, 114, 117, 121

348 risk management and, 312, 314-315,318, 320, 321,325,326 self insurance and, 131,132, 135,141 temporary/permanent disability transition and, 75, 82, 83, 89 Risk management, 307-327 combination of tools used in, 321-324 insurance in, see under Insurance loss control in, see Loss control loss prevention in, see Loss prevention loss reduction in, see Loss reduction nonreliability problem in, 308-312, 324 Rosen, R.H., 244 Rosen, S., 21n.ll Ross, S., 100, 105, 106, 149, 167, 314, 315, 317, 318 Rotating shifts, claims incidence and, 268, 274 Rothschild-Stiglitz model, 117 Rubber products industry, 251,252 Rubinstein, M., 158 Ruin probability, see also Insolvency adverse selection and, 114 in forecasting models, 183 in option pricing models, xi, 149, 152, 166, 167 Ruser, J., 52, 76, 143n.1,297, 298 Safety measures benefit levels and, 6 claims incidence and, 244 self insurance and, 131 temporary/permanent disability transition and, 76 Sample selection effect, 43 Saturn facility, 243 Savings and Loan industry, viii Scheduled injuries, 77, 78, 79-81, 90-93 Schlesinger, H., ix, xiv, 312, 314, 320, 327, 330 Schmit, J.T., ix, xiv, xv Scholes, M., 154, 158 Schulenburg, J.M., 312 Screw machine industry, 257, 258 Selection risk, 121 Selectivity bias, 43 Self insurance, ix, xi, 129-143 benefit levels and, 5, 14, 17, 19

INDEX firm size and, 130, 135-138,139-140, 142 increase in, 129-131 market insurance vs., 129,131-135 Service sector, 249, 251-252, 283,284 Seward, J.A., 224, 225,228, 230 Shaked, L., 224 Sharpe/Lintner/Mossin capital asset pricing model (CAPM), 149, 159 Shavell, S., 330 Sight loss~ 77, 94 Simon, M., 330 Sims, R.H., 244 Singh-Maddala distributions, see Burrl2 distributions Single period models, 151,167 Sinquefield, R., 183 Small firms, see also Firm size benefit levels in, 8, 10, 11, 13-14, 17, 18 experience-rating in, see Experiencerating Smith, C.W., 312,325 Smith, H., 180 Smith, M.L., 224 Smith, R.S., ix, xiii, 297 Smith, V., 314, 326 S o c i a l Security B u l l e t i n , 136 Social Security Disability (SSDI), 7, 14, 22n.16, 26, 69 Social Security program, 2, 7, 14 Social services, 253,256 Solvency guarantees, 120, 153, 158 Solvent insurers, 237-239 Spence, M., 22n.20, 330, 331,336 SSDI, see Social Security Disability Stapleton, R.C., 158, 168n.4 State funds benefit levels and, 14, 17, 19 self insurance vs., 129, 131 Stegun, l., 179 Stiglitz, J.E., 114 Stochastic cash flow models, 173, 175, 180 Stock market crash of 1987, xii Stocks, 184-185 Structural duration distribution without heterogeneity, 27, 37 Subrahmanyam, M.G., 158, 168n.4 Superfund, 119 Surplus shocks, 117-122

INDEX Tate, D.G., 244, 245 Tax credits, 190 Taxes, s e e also specific types benefit levels and, 5, 6, 7, 8, 9-10, 13 capital flows and, 101,106, 108, 109, 110, 111,122, 123 forecasting models and, 172, 174, 175177 heterogeneous claims and, 26 loss ratio adjustments and, 189, 190 option pricing models and, 149, 151,152, 156, 166, 167 Tax Reform Act of 1986, xii, 108,123, 124n.6, 125n.9, 172, 174, 177 Tax shields, xi, 166, 167 Temporary disabilities, 69-94, s e e also Temporary/permanent disability transition partial, 70 total, 25, 51-65, 70, 76-77, 82 Temporary/permanent disability transition, viii, ix-x, 51-65, 69-94 age and, 57, 60, 82, 87, 93 attorney involvement and, 57, 58 benefit levels and, see under Benefit levels gender and, 57, 60, 83, 89-90, 93 hospital stays and, 57, 58 marital status and, 57 nonscheduled injuries and, 78-81, 90-93 scheduled injuries and, 77, 78, 79-81, 90-93 theory and literature on, 73-77 unemployment rate and, 83, 87-89, 90, 93 wages and, see under Wages Thaler, R., 21n.ll Third-party actions, xiii, xiv, 329-338 imperfect information and, xv, 334-335 imperfect product markets and, 335-337 perfect information and, 332-333,335 Thomason, T., viii, x Thornton, J.H., 230 Tiebout model, 21n.12 Time path models, 184 Tort actions, Third-party actions Total disabilities, see also Temporary/ permanent disability transition permanent, 7, 70, 71

349 temporary, 25, 51-65, 70, 76-77, 82 Transfer policies, Claims incidence and, 244-245 Transformed loss ratios, 188-189 Transportation industry, 249, 264,283,284, 285,286, 289 Trieschmann, J.S., 224, 226, 228,230, 234 Trucking industry, 251,263 T tests, 264, 265,268,269, 270 Turnover, claims incidence and, 270, 271, 274,275 Two period lag model, 232, 234 Underwriting capital flows and, see Capital flows in forecasting models, 174, 177-183 Unemployment rate, disability transition and, 83, 87-89, 90, 93 Unions, 224 benefit levels and, 1, 11-12 claims incidence and, 244,267, 274 Upjohn Institute, 243,244, 278,283-290 Utilities industry, 249 Venezian, E., 99 Venter, G.S., 49n.3,210n.2 Victor, R.B., 144n.4 Viscusi, W.K., 2, 21n.1,335,337 Wages benefit levels and, 1, 3, 4, 5, 8, 9, 11, 12-13, 14, 17, 18-19 claims incidence and, 244, 268, 274 heterogeneous claims and, 26, 36, 38, 39-40, 41-42, 48 temporary/permanent disability transition and, 55, 57, 58, 59-60, 62-63, 82, 83, 89-90 third-party actions and, 332, 333,334 Walzer, N., 1, 11 Washington state, experience-rating in, xiii- xiv, 293-305 Waters, W.R., 150 Weibull distributions of heterogeneity, 28, 32 of loss ratios, 197-202 Weighted average cost of capital, 326 Welch, E., 243

350

Welland, D.A., 94n.7 Winter, R., 100, 108-109, 112, 113, 124n.7, 125n. 10 Women, see Gender Worra/l, J.D., viii, ix, x, xi, xii, 5, 11, 36, 37, 49n.1, 52, 53, 55, 76, 89, 143n.1, 297, 298

INDEX

Wu, C., 324 X2

test, 264

Zavgren, C.V., 223 Zemijewski, M.E., 225, 226