Selected Payout Products of the Old-Age Pension Saving Scheme 3031238486, 9783031238482

Table of contents :
Preface
Acknowledgements
Contents
Part I Introduction and Preliminaries
1 Preliminaries
1.1 Basic Probabilities
1.2 Yield Curves
1.3 Administrative and Collection Costs
References
2 Model of Longevity
2.1 Model of Adult Age Longevity
2.2 Lee-Carter Model Implementation
2.3 Old-Age Longevity
2.4 Implementation of Kannistö's Model
2.5 Selection Factor
References
Part II Selected Payout Products
3 Selected Life Old-Age Pension Products
3.1 Selected Products
3.2 Basic Notations
3.3 Product 1
3.4 Product 2
3.5 Product 3
3.6 Product 4
3.7 Product 5
References
4 Analysis of the Products
4.1 Loading Assessment
4.1.1 Calculation of Loading Components
4.1.2 Effective Premium
4.1.3 Monthly Pension with Respect to the Retirement Age
4.2 Notes on Product 3
References
5 Test of Profitability
5.1 Test of Profitability Basis
5.2 Test of Profitability Process
5.2.1 Profitability Measures
5.3 Profitability Measures of Product 1
5.3.1 Sensitivity of Input Parameters
Reference
Part III Conclusions and Recommendations
6 Research Questions and Hypotheses
6.1 Research Questions
6.2 Research Hypotheses
6.3 Discussion About the Attitude to Risk and Overall Long-Term Savings
References
A Life Tables
B Procedure for Determining of the Effective Premium of Product 1
C Forward Rate for Continuous Compounding
Index

Citation preview

SpringerBriefs in Statistics Jana Špirková · Igor Kollár · Gábor Szűcs · Pavel Zimmermann

Selected Payout Products of the Old-Age Pension Saving Scheme

SpringerBriefs in Statistics

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Jana Špirková • Igor Kollár • Gábor Sz˝ucs • Pavel Zimmermann

Selected Payout Products of the Old-Age Pension Saving Scheme

Jana Špirková Faculty of Economics Matej Bel University in Banská Bystrica Banská Bystrica, Slovakia

Igor Kollár Faculty of Economics Matej Bel University in Banská Bystrica Banská Bystrica, Slovakia

Gábor Sz˝ucs Faculty of Mathematics, Physics and Informatics Comenius University in Bratislava Bratislava, Slovakia

Pavel Zimmermann Faculty of Informatics and Statistics Prague University of Economics and Business Prague, Czech Republic

ISSN 2191-544X ISSN 2191-5458 (electronic) SpringerBriefs in Statistics ISBN 978-3-031-23848-2 ISBN 978-3-031-23849-9 (eBook) https://doi.org/10.1007/978-3-031-23849-9 Copyright for Figure 4.2: © European Central Bank, Frankfurt am Main, Germany. The figure may be obtained free of charge through the website of the European Central Bank. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

To Peter, Veronika and Michal. J.Š. To Alena, Martin and Tomáš. I.K. To Zuzana and Beáta. G.Sz. To Markéta, Klára and Viktor. P.Z.

Preface

This monograph is intended for anyone interested in the amount of pension of old age that can be expected from pension savings of old age or voluntary pension savings. It is primarily intended for students of actuarial and financial mathematics and future economists. It deals with and analyses the so-called payout phase of old-age pension savings. This means that it examines the amount of the monthly lifetime pension that the beneficiary can expect from the amount saved. This saved amount represents a single premium for lifetime pension insurance. The monograph analyses five selected pension savings products. In addition to determining and analysing the basic lifetime monthly pension that the beneficiary will receive as long as he/she is alive, it also offers analyses of other pensions, which include benefits for survivors or authorized persons, in the event that the beneficiary dies. For example, selected products incorporate the risk for insurance companies that if a beneficiary dies during the first seven years of the beneficiary’s pension plan, a lump sum of not yet paid monthly pensions that would be paid to the beneficiary for seven years if he/she were alive will be paid to survivors or authorized persons. There are also other risks built into the products, e.g. the risk that a pension will be paid to survivors or an authorized person for either one or two years after the beneficiary’s death. It also analyses a product that includes a geometrically growing pension annuity. This work provides a perspective on the so-called effective premium, that is, that part of the accumulated amount that is intended directly for permanent monthly pensions. This effective premium is obtained by applying various risk loadings and other loadings that the actuary must use to prudently determine the amount of monthly pension. The aforementioned loadings include Kannistö’s model with selection, Kannistö’s model with selection and with 25% shock, market risk of an interest rate, and expense loadings. The work also offers profitability testing as a method that allows for the change of individual input parameters so that the monthly permanent pension is determined correctly and prudently. It also analyses the sensitivity of changes in individual input

vii

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Preface

parameters to the level of profitability rates, such as the net present value of cash flows, the profit margin and the internal rate of return. It answers questions concerning the effectiveness of pension savings and oldage insurance and takes a position on various hypotheses. The basic question being: “How many years would the beneficiary have to live in order to be paid the amount he/she invested in such insurance?” The fundamental data resources are the mortality tables of the Slovak Republic processed using the Kannistö’s model with selection with respect to the specific insurance pool of clients entering into such an insurance relationship. It is very important to talk about pensions. Not only about pensions in general, but especially about their amounts. People usually think that the entire amount saved will be spent on monthly payments. However, financial literacy should clarify that insurance companies are for-profit joint stock companies, and, in addition, actuaries need to calculate pensions prudently due to increasing life expectancy and uncertain interest rates, but also due to the insurance company’s investment costs and running an insurance company. It is essential that future pensioners know what the term effective premium means, what is its amount in relation to the money invested, and at least what yield curve should be used in calculating pensions. The vast majority of people have almost no knowledge of financial and actuarial mathematics, and this must change. Banská Bystrica, Slovakia Banská Bystrica, Slovakia Bratislava, Slovakia Prague, Czech Republic July 2022

Jana Špirková Igor Kollár Gábor Sz˝ucs Pavel Zimmermann

Acknowledgements

Our thanks go to the Slovak Scientific Grant Agency under projects VEGA no. 1/0150/21, VEGA no. 1/0760/22, and to the Slovak Research and Development Agency under project APVV-20-0311. We express our great gratitude to Mr. Martin Janeˇcek from Tools4F for his willingness to discuss and consult on our work.

ix

Contents

Part I

Introduction and Preliminaries

1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 1.1 Basic Probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 1.2 Yield Curves .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 1.3 Administrative and Collection Costs . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . .

5 5 6 7 7

2 Model of Longevity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 2.1 Model of Adult Age Longevity.. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 2.2 Lee-Carter Model Implementation . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 2.3 Old-Age Longevity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 2.4 Implementation of Kannistö’s Model . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 2.5 Selection Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . .

9 9 10 11 13 14 15

Part II

Selected Payout Products

3 Selected Life Old-Age Pension Products . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 3.1 Selected Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 3.2 Basic Notations.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 3.3 Product 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 3.4 Product 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 3.5 Product 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 3.6 Product 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 3.7 Product 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . .

19 19 21 21 23 25 27 29 30

4 Analysis of the Products .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 4.1 Loading Assessment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 4.1.1 Calculation of Loading Components . . . . . .. . . . . . . . . . . . . . . . . . . . . 4.1.2 Effective Premium . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 4.1.3 Monthly Pension with Respect to the Retirement Age.. . . . . . . .

31 31 33 34 36 xi

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Contents

4.2 Notes on Product 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 38 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 41 5 Test of Profitability.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 5.1 Test of Profitability Basis . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 5.2 Test of Profitability Process . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 5.2.1 Profitability Measures . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 5.3 Profitability Measures of Product 1 . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 5.3.1 Sensitivity of Input Parameters.. . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . Part III

43 43 44 47 48 49 51

Conclusions and Recommendations

6 Research Questions and Hypotheses .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 6.1 Research Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 6.2 Research Hypotheses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 6.3 Discussion About the Attitude to Risk and Overall Long-Term Savings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . .

55 55 57 57 58

A Life Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 59 B Procedure for Determining of the Effective Premium of Product 1 .. . . . 63 C Forward Rate for Continuous Compounding. . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 67 Index . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 69

Part I

Introduction and Preliminaries

Almost every day and from all sides, we hear how we should think about our future and how we should save for our retirement from a young age. Of course, it is a question of financial literacy and our attitude to the risk of investing in various financial instruments or real estate, precious metals, or other valuables, such as rare paintings. National governments are working by law to strengthen and stabilize old-age pension savings, whether mandatory or voluntary. Savers can invest their funds in various investment funds and savings schemes. However, the question arises as to whether investment funds can value their clients’ finances to be positively valued and exceed annual inflation. In most cases, these are very risky investments. It is obvious that risk-averse clients prefer to invest in risk-free bonds. However, it is well known that at present risk-free bond yields have long been negative or very low positive and do not even exceed inflation. In our monograph, the entire savings phase is skipped, and it focuses on determining the amount of pension based on the money already deposited. The amount of future pensions from the money saved in this way is interesting both for the young generation and for the generation of people who will retire in the foreseeable future. The monograph discusses the payout phase of the old-age pension saving, socalled effective premium, which represents the amount that will be paid in the form of monthly pensions after taking risk loadings, costs, and expenses into account. It also gives profitability testing, which is currently one of the most current methods of determining not only the amount of insurance premiums but also the amount of pensions. The monograph represents a summary of our actuarial analysis on the products of the payout phase of the old-age pension savings scheme, and it is inspired by the products that are offered as part of old-age pension savings in Slovakia. However, these products are not narrowly specific to Slovakia, and we can consider them as a relatively general offer of products within the old-age pension insurance anywhere in the world.

2

I Introduction and Preliminaries

The aim of the monograph is to model and analyse five products of the payout phase of the old-age pension scheme, as permanent old-age pension annuities. Our topic is specific in that we and the public have only limited access to the methodologies of real insurance companies by which actuaries calculate the amount of annuities in old-age pension savings. Our study provides insight into what happens to the savings accumulated by the savers after their relocation to the payout phase, i.e., after buying a life pension annuity from a life insurance company. The work gives insights into actuarial calculations and methodologies that allow them to determine the amount of permanent monthly pension annuities which follow from the old-age pension insurance. It provides information and procedures for beginner actuaries, as well as for the professional and lay public. The aim of this work is to get answers to the following research questions and to take a position on the hypotheses. Q1: What in our understanding represents the so-called effective premiums, i.e., in other words, what amount of the saved 10,000 euros will be paid out in the form of pensions? Q2: What is the lower limit of the monthly pension annuity from the accumulated sum of 10,000 euros of the individual products to observe the prudent behaviour of the insurance company? Q3: For how many years should a pensioner receive a monthly pension annuity to get back at least 10,000 euros? Q4: To what extent does the requirement of a 7-year annuity (84 monthly annuities) in Product 2 affects the amount of the monthly annuity in comparison with Product 1 (classical monthly permanent pension annuity)? H1: Life insurance companies consume more than 25% of pensioner savings, which means that the whole life annuity paid out is less than 75% of the savings. H2: Actual life expectancy for 62-year-old pensioner is 20.02 years. In the case that 62-year-old person lives for another 20 years, the whole saved amount will return to the person in the form of monthly pension payments. Even though our modelled products do not adhere in full detail to all the guidelines and recommendations of the EIOPA,1 the ESMA,2 and the European Union, we strive for a correct actuarial approach. As an actuarial basis, we use mortality tables and their adjustment according to the Lee-Carter model of longevity and Kannistö’s model with selection. We apply the methodology designed by the Austrian Actuarial Association, which was inspired by the methodology designed by the German Association of Actuaries. Our model also includes the returns from investment funds that are modelled by the Svensson yield, whose parameters are published on a daily basis by the European Central Bank (ECB). Moreover, it meets the requirements of the Council Directive 2004/113/EC of 13 December

1 2

European Insurance and Occupational Pensions Authority, https://www.eiopa.europa.eu/. European Securities and Markets Authority, https://www.esma.europa.eu/.

I Introduction and Preliminaries

3

2004 implementing the principle of equal treatment between men and women in the access to and supply of goods and services, and Directive 2009/138/EC of the European Parliament and of the Council of 25 November 2009 on the taking up and pursuit of the business of Insurance and Reinsurance (Solvency II). The monograph is structured as follows. Part I—Introduction and Preliminaries give basic information about our analysis regarding modelling permanent life pension annuities. Chapter 1—Preliminaries give basic actuarial concepts and notation as building blocks of actuarial modelling, that is, the relevant survival and mortality probabilities, the yield curve of the ECB, and mention the administrative and collection costs that are built into the actuarial models. Chapter 2—Model of Longevity brings a detailed view on the modelling of adult age longevity with special stress on the Lee-Carter model, Kannistö’s model and the so-called selection factor, while the population of pensioners who purchase a life pension has a significantly lower probability of death than the general population. Part II—Selected Payout Products provide a description of the selected products and the reasons why we chose these ones. Chapter 3—Selected Life Old-age Pension Products describe five selected products and their specifics and give an actuarial modelling of the products, which could be used to offer old-age pension products anywhere in the world. Among the products is the basic permanent monthly pension, without any benefits for survivors, as well as products which, in addition to the basic permanent pension, also include benefits for survivors or authorized persons in the event of the pensioner’s death. Chapter 4—Analysis of the Products gives an analysis of Product 1 and a relevant discussion regarding loading components, the so-called effective premium, and the amount of monthly pensions. Chapter 5—The Profitability Test gives a test of the profitability of Product 1 and the corresponding profitability measures. Part III— Conclusions and recommendations answer research questions and provide attitudes towards established hypotheses. Finally, the Appendix contains life tables, the procedure for determining the effective premium of the Product 1 and determining a formula for the forward rate of continuous compounding.

Chapter 1

Preliminaries

Abstract This chapter gives probabilities related to actuarial modelling, namely relevant survival and mortality probabilities, the so-called deferred mortality probability, i.e., a probability that death occurs in some interval following a deferred period. Because our research considers monthly paid pension annuities, we consider the most common fractional age assumptions using linear interpolation between integer ages and the uniform distribution of deaths. The Svensson yield curve, which is published daily on the European Central Bank website, is used as a model for the development of the risk-free interest rate. In our modelling, we assume standard costs that are published in the actuarial literature so that we comply with the requirements of a prudent business. In particular, we consider initial and administrative costs and legitimate expenditures.

1.1 Basic Probabilities The basic building blocks in modelling of all life insurance products are the relevant survival and probabilities, which are given as follows: • • •

px —the probability that an individual at age x survives at least to age .x + t. qx —the probability that an individual at age x dies before age .x + t. .r|t qx —the probability that an individual at age x survives r years and then dies in subsequent t years, that is, between ages .x + r and .x + r + t, see formula (1.1). .t .t

Probability .r|t qx is also called probability of deferred mortality [2] because it is the probability that death occurs in some interval after a deferred period. It can be calculated by formula .r|t

qx = r px − r+t px .

(1.1)

Because pensions from the second pillar pension savings are paid monthly immediately, we consider the fractional age assumption using linear interpolation. Linear interpolation between integer ages and the assumption of a uniform distribution of

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 J. Špirková et al., Selected Payout Products of the Old-Age Pension Saving Scheme, SpringerBriefs in Statistics, https://doi.org/10.1007/978-3-031-23849-9_1

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6

1 Preliminaries

deaths is the most common fractional age assumption, [2]. It can be formulated as follows. Definition 1.1 ([2]) For an integer x, provided the uniform distribution of deaths in every age interval .[x, x + 1[, and for .0 ≤ s < 1, assume that .s

qx = s × qx .

(1.2)

1.2 Yield Curves We can see interesting approaches to modelling the effect of the interest rate on the level of pensions in the literature. Very interesting approaches can be found in [5] and [6]. In our research we use the yield curve of the ECB because insurance companies invest premiums mainly in bonds issued by euro area countries. Furthermore, we decided to use the Svensson yield curve because it is available online on the ECB website [4, 7]. A yield curve is a representation of the relationship between market remuneration rates and the remaining time to maturity of debt securities. It can also be described as the term structure of interest rates. The euro area yield curve shows separately AAA-rated euro area central government bonds and all central government bonds of the euro area (including AAA-rated). It is updated every TARGET business day at noon, 12:00 Central European Time [4]. In this part, we give the basic formula of the Svensson yield curve according to [3]. For more information, see [1] and [7]. Definition 1.2 The Svensson yield curve is given by

Rt ∗ (z) = β0t ∗ + β1t ∗ ×

.

⎡ ×⎣

  1 − exp − τ z ∗ 1t

z

τ1t ∗

  1 − exp − τ z ∗ z

1t

τ1t ∗

⎡ + β3t ∗ × ⎣

+ β2t ∗

⎤   z ⎦ − exp − τ1t ∗

  1 − exp − τ z ∗ z

τ2t ∗

2t

⎤   z ⎦ − exp − , τ2t ∗

(1.3)

where • .Rt ∗ (z)—yield from a bond investment with continuous compounding (% p.a.), at the reference time .t ∗ • z—term to maturity , .z ∈]0, Tmax ]

References

7

• .Tmax —maximum term to maturity • .β0t ∗ , β1t ∗ , β2t ∗ , β3t ∗ , τ1t ∗ , τ2t ∗ —parameters of the Svensson yield curve, where .β0t ∗ , .τ1t ∗ , and .τ2t ∗ must be positive Yield curve (1.3) in Definition 1.2 is a parametric model which specifies a functional form for the spot interest rate and consists of four components. The parameters of this model can be interpreted as follows [1]. The parameter .β0t ∗ is the asymptotic long-term value, for very long maturities of .Rt ∗ (z); .β1t ∗ is the spread between the long-term and the short-term, and therefore .β0t ∗ + β1t ∗ is equal to the short-term rate (the rate at zero maturity). Furthermore, .τ1t ∗ specifies the position of the first hump or U-shape; .β2t ∗ determines the magnitude and direction of the hump. The parameter .β3t ∗ is analogous to .β2t ∗ and .τ2t ∗ can be interpreted as determining the magnitude and direction of the second hump or the U-shaped shape.

1.3 Administrative and Collection Costs In our models, we generally take into account the requirements under which insurance, reinsurance companies, and branches of foreign insurance and reinsurance companies should operate prudently with regard to taking into account and mitigating the risks to which they are exposed and, in addition, not endangering the interests of their clients’ impact on their financial situation In addition, according to the recommendations of the European Commission, the amount of pensions should be determined on the basis of reasonable actuarial assumptions so that insurance companies or branches of foreign insurance companies fulfil all their obligations and create an adequate amount of technical provisions. In our modelling, we assume standard costs that are published in the actuarial literature so that we comply with the requirements of a prudent business. We especially consider initial and administrative costs and the insurer’s legitimate expenditures. The specific costs and expenditures that we consider are described in Chap. 3, Sect. 3.2.

References 1. Z. Aljinovi´c, T. Poklepovi´c, K. Katalini´c, Best fit model for yield curve estimation. Croat. Oper. Res. Rev. 3(1), 28–40 (2012) 2. D.C.M. Dickson, M.R. Hardy, H.R. Waters, Actuarial Mathematics for Life Contingent Risks (Cambridge University Press, 2013) 3. European Central Bank, Technical notes. https://www.ecb.europa.eu/stats/financial_markets_ and_interest_rates/euro_area_yield_curves/html/technical_notes.pdf, 2019. Online; Accessed 6 Jan 2020

8

1 Preliminaries

4. European Central Bank, Euro area yield curves. https://www.ecb.europa.eu/stats/financial_ markets_and_interest_rates/euro_area_yield_curves/html/index.en.html, 6 Jan 2020. Online; Accessed on 6 Jan 2020 5. I. Melicherˇcík, G. Sz˝ucs, I. Vilˇcek, Investment strategies in the funded pillar of the Slovak ˇ pension system. J. Econ./Ekonomický Casopis 63(2), 133 (2015) 6. J. Mihalechová, M. Bilíková, The impact of interest rate on the amount of pensions of II. In Slovak: Vplyv úrokovej miery na výku dôchodkov z II. piliera. Econ. Inf. 13(2), (2015) 7. L.E.O. Svensson, Estimating and interpreting forward interest rates: Sweden 1992–1994. Technical report, National bureau of economic research (1994)

Chapter 2

Model of Longevity

Abstract An integral part of our model is the modelling of longevity as an essential part of the Solvency Capital Requirement. It is necessary to forecast future values of the probability of death and survival for each age in each future year. Our forecasts follow the methodology designed by the Austrian Actuarial Association published in Kainhofer et al. (2006, The new Austrian annuity valuation table AVÖ 2005R), which was inspired by the methodology designed by the German Association of Actuaries published in Pasdika et al. (2005, Coping with longevity: The new German annuity valuation table DAV 2004R). The approach is divided into two main parts: adult age modelling and old-age modelling. Each age interval is forecast with a different model and with different assumptions, as the number of observations available at each age is different.

2.1 Model of Adult Age Longevity For the adult age interval, the Lee-Carter model is one of the most popular models currently used by commercial insurance companies, [11]. The model is a semiparametric model with age- and time-specific parameters. Definition 2.1 ([11]) Lee-Carter model is defined by μx,t = exp(αx + βx × κt + x,t ), x ∈ X, t ∈ T,

.

(2.1)

where • .μx,t —the force of mortality of a person at age x in year t. • .αx —age-specific model parameters that do not depend on time. • .βx —age-specific model parameters that represent how rapidly or slowly mortality at each age varies when the general level of mortality changes. • .κt —time-varying index (independent of age), reflecting the general level of mortality, • .x,t —random noise (error term) with zero mean value and variance .σ 2 for .x ∈ X and .t ∈ T, where .X is the set of ages and .T is the set of times. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 J. Špirková et al., Selected Payout Products of the Old-Age Pension Saving Scheme, SpringerBriefs in Statistics, https://doi.org/10.1007/978-3-031-23849-9_2

9

10

2 Model of Longevity

Several parameter estimation techniques are applied for this model. Originally, the singular value decomposition algorithm was applied. Later, likelihood techniques appeared [2]. To forecast mortality in the future, the trend of the time-varying index .kt must be captured and forecast. The usual specification of the model is the linear autoregressive model ARIMA(1,1,0) with constant. Note that there are several alternative models to the Lee-Carter model. For example, the Cairns-Blake-Dowd (CBD) model was proposed in [4]. There are also several extensions and modifications to the original Lee-Carter model. An overview is presented, for example, in [18].

2.2 Lee-Carter Model Implementation The data used in our prediction were downloaded from the Human Mortality Database, [6]. The original data are published by the Statistical Office of the Slovak Republic, [12]. Due to the major trend change connected with the fall of the iron curtain, we only used data from the year 1992, which we consider to be representative. The last year available at the time of forecasting was 2017. As we consider a separate old-age model, we only use ages up to 85 years. There are several implementations of the Lee-Carter model for the estimation of parameters and the prediction of future mortality rates. For example, R software [15] has the Demography package [8] or StMoMo [18]. The second one we used in our analysis. Our estimates of the parameters are shown in Fig. 2.1. As stated in [9], the linear trend of mortality decrease has been stable for decades now, “However, a perpetual extrapolation with a high constant trend leads to vanishing death probabilities and thus unreasonably high life expectancy in the far future. Although this does not pose a problem for daily use of the annuity valuation table, it is an inherent weakness of a model with constant trend. To avoid this problem, we thus propose a long-term trend reduction, which leads to a reasonable limiting life expectancy and a non-vanishing limiting death probability in the limit .t → ∞”. The trend reduction factor is not based on data. It is an expert belief introduced to adjust the long-term behaviour of the model towards a more realistic expectation. We take over this adjustment and reduce the trend by the same time-dependent factor as in [9] as follows Z(t) =

.

1+



1 t −1992 t1/2

,

(2.2)

where the half-time .t1/2 is set to 100 years. This means that we assume that after t1/2 = 100 years, the trend is reduced by 50%. The trend reduction factor applied to the forecasts of .μx,t is shown in Fig. 2.2.

.

2.3 Old-Age Longevity

11

Fig. 2.1 The Lee-Carter model parameter estimation

2.3 Old-Age Longevity Due to the lack of data or even non-existence of data, mortality in very high ages is usually modelled separately from mortality in adult ages. For (very) high age, popular models are formulated as a one-dimensional function of age. Due to the fact that in very high ages, observations are scarce or do not (yet) exist at all, mortality is usually estimated using some extrapolation (in the age dimension) of mortality from common age to very high age. There are many parametric functions suggested based on a variety of assumptions. First models were not always specified only for very high ages and not specifically for extrapolation but rather to describe the growth of mortality with age (“mortality law”). They can be dated already to the nineteenth century by Benjamin Gompertz who assumed an exponential increase of the force of mortality with increasing age, [5]. Later this model was modified by William Makeham, who added an additional parameter to the model. With more data available in older ages, the exponential increase appears to be too fast for some authors, and other models were developed. In [10] a modification of the Gompertz–Makeham model was suggested. One of the most popular alternatives to exponential models are models based on logistic functions. Such a specification occurs, for example, in Beard’s model [1], Thatcher’s model [16], or Kannistö’s model [17]. But many other specifications

2 Model of Longevity

0.5

0.6

0.7

Z(t)

0.8

0.9

1.0

12

0

20

40

60

80

100

80

85

0.04 0.02

μ62+t

0.06

0.08

t

65

70

75

Age Fig. 2.2 Force of mortality .μ62+t,t , .t = 1, 2, . . . , 25 with trend reduction (solid line) and without trend reduction (dashed line)

may be found. An overview is provided in [3] or [13], which is a manual for an R package MortalityLaws, which is a consistent implementation of many old-age models. Models often coincide in the age interval in which data are available and diverge for ages for extrapolation to be performed. Therefore, it is difficult to select an appropriate model based on observed data, and more expert judgment on oldage development is required. As both [14] as well as [19] support a logistic type of models and the Human Mortality Database applies the Kannistö’s model [19] to smooth old-age data, we decided to assume the Kannistö’s specification with timevarying parameters μ(ηx,t ) =

.

1 , 1 + exp(−ηx,t )

(2.3)

2.4 Implementation of Kannistö’s Model

13

where ηx,t = at + bt x.

(2.4)

.

The parameters .at , .bt are discussed in the following section.

2.4 Implementation of Kannistö’s Model

1.006 1.004

ât

1.008

The parameters .at and .bt are first fitted independently at each historical time for .t = 1992, 1993, . . . , 2017 using observations from the age interval .x = 62, 63, . . . , 95 years. Time series of parameter estimates are fitted independently with ARIMA(1,1,0) models with constant, and forecasts are performed for the future years. For the parameter .at , the logarithmic transformation was applied. The same reduction factor (2.2) is applied to the future linear trend for both parameters. The resulting fitted force of mortality as a function of age is shown in Fig. 2.3.

2020

2030

2040

2050

2060

2050

2060

˸

0.110

bt

0.115

0.120

t

2020

2030

2040

t Fig. 2.3 Forecasts of the parameters .at and .bt with trend reduction (solid line) and without trend reduction (dashed line)

14

2 Model of Longevity

The package R MortalityLaws [13] was used to fit the Kannistö’s model for each year observed. The R package forecast [7] was used to perform ARIMA forecasts. Life tables can be found in the Appendix A.

2.5 Selection Factor Typically, for facultative pension programs, strong selection factors impact mortality, and the subgroup of annuitants has different mortality than the entire population. Publicly available data for such phenomena are very scarce. We take over the selection factors used in [9]. The selection factors are based on data pooled by Gen Re and the Munich Re Group of more than 20 German insurance companies. The selection factors used in [9] are expressed as an analytical function

s(x) =

.

⎧ f1 ⎪ ⎪ ⎪ ⎨ f1 − (f2 − f1 ) × ⎪ f2 + (1 − f2 ) × ⎪ ⎪ ⎩ 1

x−c1 c2 −c1 (x−c2 )2 (c3 −c2 )2

for x ≤ c1 for c1 ≤ x ≤ c2 for c2 ≤ x ≤ c3

,

(2.5)

for x ≥ c3

where the constants .f1 , f2 , f3 and .c1 , c2 , c3 are in Table 2.1. Function (2.5) is plotted in Fig. 2.4. As we do not work with separate mortality tables for males and females, we need to estimate a unisex selection factor. We calculate it as the weighted average of the selection factors for males and females, where the weights are at each age point the population exposure, see the Appendix. Table 2.1 Constants used in function (2.5)

Sex Males Females

.f1

.c1

.f2

.c2

.c3

0.8 0.8

40 40

0.51 0.55

60 60

100 100

Source: [9]

References

15

Fig. 2.4 The selection factors expressed as analytic functions (2.5)

References 1. R.E. Beard, Appendix: Note on some mathematical mortality models, in Ciba Foundation Symposium-The Lifespan of Animals (Colloquia on Ageing), Volume 5 (Wiley Online Library, 1959), pp. 302–311 2. N. Brouhns, M. Denuit, J.K. Vermunt, A Poisson log-bilinear regression approach to the construction of projected lifetables. Insur. Math. Econ. 31(3), 373–393 (2002) 3. B. Burcin, K. Tesárková, L. Šídlo, The most used balancing methods a extrapolation of the mortality curve and their application to the Czech population. Revue for research on population development. In Czech: Nejpoužívanejší metody vyrovnávání a extrapolace krivky úmrtnosti a jejich aplikace na ceskou populaci. Revue pro výzkum populaˇcního vývoje 52, 77–89 (2010) 4. A.J.G. Cairns, D. Blake, K. Dowd, Pricing death: Frameworks for the valuation and securitization of mortality risk. Astin Bulletin 36(1), 79 (2006) 5. B. Gompertz, On the nature of the function expressive of the law of human mortality, and on a new mode of determining the value of life contingencies. Roy. Soc. Lond. Philos. Trans. I 115, 513–583 (1825) 6. Human Mortality Database, University of California, Berkeley (USA), and Max Planck Institute for Demographic Research (Germany), 2019. https://www.mortality.org. Online; Accessed 10 Sept 2019 7. R.J. Hyndman, Y. Khandakar, Automatic time series forecasting: the forecast package for R. J. Stat. Softw. 26(3), 1–22 (2008). http://www.jstatsoft.org/article/view/v027i03 8. R.J. Hyndman, H. Booth, L. Tickle, J. Maindonald, Demography: Forecasting mortality, fertility, migration and population data (2019). https://CRAN.R-project.org/package=demography. R package version 1.22 9. R. Kainhofer, M. Predota, U. Schmock, The new Austrian annuity valuation table AVÖ 2005R, (2006). http://www.avoe.at/pdf/mitteilungen/H13_w3.pdf 10. F. Koschin, How high is the intensity of mortality at the end of human life. In Czech: Jak vysoká je intenzita úmrtnosti na konci lidského života. Demografie 41(2), 105–119 (1999) 11. R.D. Lee, L.R. Carter, Modelling and Forecasting U.S. Mortality (1992). http://pagesperso. univ-brest.fr/~ailliot/doc_cours/M1EURIA/regression/leecarter.pdf. Online; Accessed 2 Oct 2019

16

2 Model of Longevity

12. Mortality Tables, Statistical Office of the Slovak Republic (2019). https://www.statistics.sk. Online; Accessed 10 Sept 2019 13. M.D. Pascariu, MortalityLaws: Parametric mortality models, life tables and HMD (2019). https://CRAN.R-project.org/package=MortalityLaws. R package version 1.8.0 14. U. Pasdika, J. Wolff, G. Re, M. Life, Coping with longevity: The new German annuity valuation table DAV 2004 R, in The living to 100 and beyond symposium, Orlando Florida, vol. 2, p. 6 (2005) 15. R Core Team, R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria (2018). https://www.R-project.org/ 16. A.R. Thatcher, The long-term pattern of adult mortality and the highest attained age. J. R. Stat. Soc, A (Stat. Soc.) 162(1), 5–43 (1999) 17. A.R. Thatcher, V. Kannisto, J.W. Vaupel, et al., The force of mortality at ages 80 to 120, vol. 22 (Odense University Press Odense, 1998) 18. A.M. Villegas, V.K. Kaishev, P. Millossovich, StMoMo: An R package for stochastic mortality modeling. J. Stat. Softw. 84(3), 1–38 (2018). https://doi.org/10.18637/jss.v084.i03 19. J.R. Wilmoth, K. Andreev, D. Jdanov, D.A. Glei, C. Boe, M. Bubenheim, D. Philipov, V. Shkolnikov, P. Vachon, Methods protocol for the human mortality database [version 31/05/2007] (2007). http://mortality.org

Part II

Selected Payout Products

This part offers one of the most important ideas of the entire monograph, namely the description of the five products of the retirement pension phase payout.1 The first of the products examined is Product 1, which represents a very basic permanent immediate monthly pension annuity that the pensioner will receive as long as he/she is alive. Of course, there are future pensioners who care that their spouses or even children remain financially secure after their own death. Therefore, we also offer other products that have built-in, the so-called guaranteed or survivors’ pension. Here, we were inspired by the range of products offered by Act of the National Council of the Slovak Republic No. 43/2004 Coll. on the Old-Age Pension Scheme, as amended (“Act 43/2004 Coll.”), in Article 46. This law lists a product that is referred to in this monograph as Product 2. This product, similar to Product 1, offers a monthly permanent pension in advance. However, in addition, this product also includes the feature that if the pensioner dies before the end of the seventh year from the beginning of the pension, a lump sum in the amount of pensions not yet paid will be paid to the survivor or beneficiary. This means that if, for example, a pensioner entered the pension on 1 January at the age of 62 and died on 17 September of the same year, a lump sum of .(84 − 8) = 76 monthly pensions will be paid to the survivors at the end of the month of his death. This lump sum will be reduced by the necessary costs associated with its payment by the insurance company. We remind you that if a pensioner will receive a pension for more than seven years, he will receive a pension as long as he is alive, but the survivors will no longer have any right to the lump sum of not yet paid pensions. Another product is Product 3 which includes a permanent monthly annuity with a geometrically rising annuity by j % yearly and also includes the payment of a lump sum of not yet paid monthly annuities in the case of the beneficiary’s death during the period of the first seven years of pension payment to an authorized person or inheritors. In this case, it is very essential to determine to what extent it is

1

My cousin Rudolf says that he considers a product to be a loaf of bread but not a pension ©. J.Š.

18

II Selected Payout Products

appropriate and meaningful to set the amount of geometric progression quotient j to make the yield of financial flows positive. You can read more about this issue in Chap. 4, Sect. 4.2. You will also find two other products, namely Product 4 and Product 5, which are Product 2 and Product 3, respectively, extended by survivors benefits with a payment period of either one or two years. Whether the pensioner receives a pension from Product 2 or Product 3, after his death, the surviving or authorized person will receive a monthly pension in the amount of the last paid pension for one or two years, depending on how the pension is embodied in the policy. The rapidly developing insurance market headed by EIOPA came up with a novelty called the pan-European pension product (PEPP). The pan-European personal pension product is a voluntary personal pension scheme that will complement existing public and occupational pension systems, as well as national private pension schemes. The pan-European personal pension products are regulated by Regulation (EU) 2019/1238 of the European Parliament and of the Council of 20 June 2019 on a pan-European Personal Pension Product. These products are designed primarily for those who work outside of their home country. The European Commission sees great potential in the fact that only 27% of Europeans aged 25–59 years have their free funds in pension products. So far, only the savings phase of this product is being prepared, but in the future, our modelled products can also be the basis for the payout phase.

Chapter 3

Selected Life Old-Age Pension Products

Abstract This product range includes, like the first, the basic product of pension insurance—a permanent monthly pension. However, it also contains products in which a permanent pension is extended by a guarantee of the payment of pensions during the first seven years of payment of the pension, either to the policyholder of the pension or to his survivors in the event of his/her death. These hitherto mentioned pensions are extended in other products by a survivor’s pension for one or two years with a geometrically rising annuity by a certain percentage yearly. This offer of products is also inspired by the offer of pension products, which are listed in Act 43/2004 Coll. This product range is general and can be a basis for other products that could be offered not only within the European Union. The calculation of the amount of pensions in this chapter is made on the basis of classical actuarial modelling and forms the basis for profitability testing in Chap. 5.

3.1 Selected Products We offer five products of the old-age pension scheme. In all products, we assume that if the beneficiary dies before the payment of the first pension, the survivors will be reimbursed a premium reduced by the necessary costs associated with the termination of the insurance relationship. We have already researched all the products mentioned to some extent in our previous papers [1, 2] and [3]. The individual products are described in more details as follows: • Product 1—includes permanent monthly annuity and does not include raising of the pension and does not include survivors’ benefits. • Product 2—includes permanent monthly annuity and the payment of a lump sum equal to not yet paid monthly annuities in the case of the beneficiary’s death during the period of the first seven years of pension payment and does not include raising of the pension and does not include survivors’ benefits. • Product 3—includes permanent monthly annuity with (geometrically) raising the annuity by j % yearly, also includes the payment of a lump sum, which is the sum of not yet paid monthly annuities in the case of the beneficiary’s death © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 J. Špirková et al., Selected Payout Products of the Old-Age Pension Saving Scheme, SpringerBriefs in Statistics, https://doi.org/10.1007/978-3-031-23849-9_3

19

20

3 Selected Life Old-Age Pension Products

during the period of the first seven years of pension payment and does not include survivors’ benefits). • Product 4—includes permanent monthly annuity, the payment of a lump sum equal to not yet paid monthly annuities in the case of the beneficiary’s death during the period of the first seven years of pension payment, and includes survivors’ benefits with a payment period of either one or two years. This product is actually Product 2 extended by survivors’ benefits with a payment period of either one or two years. • Product 5—includes permanent monthly annuity with (geometrically) raising annuity by j % yearly, includes the payment of a lump sum which is the sum of not yet paid monthly annuities in the case of the beneficiary’s death during the period of the first seven years of pension payment, and includes survivors’ benefits with a payment period of either one or two years. This product is an extension of Product 3 by survivors’ benefits with a payment period of either one or two years. If it is agreed in a pension insurance contract to raise the old-age annuity, the increase shall be carried out each year on the same date on which the insurer’s obligation to perform the contract arises and by the percentage rate applicable as at the submission date. • Inheritance in the payout phase In case the beneficiary of whole life old-age annuities dies during the first seven years of payment, the rest of the sum to be paid during this period will be paid to a person designated by the saver in the old-age pension agreement. If there is no such person, the amount designated for the payment shall become the subject of inheritance proceedings. • Survivor’s benefits The saver can also arrange survivor’s benefits in their old-age pension agreement. In case of death of the beneficiary of whole life old-age annuities, the life insurance company will pay the survivor (widow, widower, orphan) a survivor’s pension during the agreed period, which is one or two years. We would like to emphasize that the calculation of the amount of pensions in this chapter is made on the basis of classical actuarial modelling and forms the basis for profit testing in Chap. 5 and therefore does not include all the financial flows that we consider in the profit testing. This is exactly the situation when a program has to calculate certain parameters for us, and it is necessary to enter the approximate input values of the parameters that we expect. In addition, we also offer the classical calculation of the amount of pensions, because we think that every future actuary should also know the classical actuarial modelling using the expected present value of selected cash flows.

3.3 Product 1

21

3.2 Basic Notations Firstly, we give the basic notations which are as follows: • S—accumulated sum, single premium in monetary units (m.u.), we use the euro throughout the text. • .P (z)—discounting factor using a yield .R(z) expressed in % p.a. given by (1.3) from a bond investment with continuous compounding,   R(z) × z . .P (z) = exp − (3.1) 100% • .P ∗ (z)—discounting factor using a constant technical interest rate i expressed in % p.a., yearly compounding,  P ∗ (z) = 1 +

.

i 100%

−z

,

(3.2)

• x—retirement age, in years, • j —geometric progression quotient as a % p.a., • .ω—maximum age to which a person can live to see (regarding used life tables: .ω = 110 years, • .α1 —initial costs as a % from the first yearly annuity, • .α2 —initial costs as an absolute amount (in m.u.) independent on an accumulated sum, • .δ1 —the insurer’s legitimate fees as a % from the amount of the lump sum premium related to the payment of this amount in cash or its transfer to a noneuro area country in a case of client’s death, • .δ2 —the insurer’s legitimate fees as a % from the amount of the expected present value of the sum of not yet paid monthly annuities in the case of the beneficiary death during the period of the first seven years of pension payment, • .β—administrative costs as a % p.m. from technical provisions on whole life immediate or on whole life immediate geometrically increasing annuity, respectively. Remark 3.1 The discounting factor in all formulas is identified as .P (z), but in specific calculations, we use .P ∗ (z), depending on whether we count with discounting factor using a yield .R(z) or with a constant interest rate i.

3.3 Product 1 Product 1 includes purely only permanent monthly annuity. This means that the insurer shall pay the beneficiaries of these annuities until their death. When the beneficiary dies before the date on which the insurer’s obligation to perform the

22

3 Selected Life Old-Age Pension Products

pension insurance contract arises, the amount of the lump sum premium paid under the terms of this contract, reduced by the amount of the insurer’s legitimate expenditures related to the payment of the amount in cash or its transfer to a noneuro area country. Subsequently, after the death of the beneficiary, survivors are not entitled to any benefits nor other financial compensation. Equivalence equation with the selected financial flows at the time of closing the policy agreement is as follows:  S = 12 × MPx × ax(12) + A1x:1/12 × S × 1 −

.

+ 12 × MPx ×

δ1  δ1 + A1x:1/12 × S × 100% 100%

α1 β +α2 +12 × MPx × × 100% 100%

12×(ω−x+1)  t =1

t 12 |

ax(12). (3.3)

On the left-hand side is an accumulated sum S which represents a single premium of the product. On the right-hand side are the following financial flows, sequentially: 1. The value .12 × MPx × ax(12) is the expected present value of whole life monthly immediate annuity .MPx , where 12×(ω−x+1) 

ax(12) =

.

t =1

1 × 12

t 12

px × P

 t  12

(3.4)

is the expected present value of whole life immediate benefits in the amount of 1 12 m.u., 12 times per year, under condition that the client is alive. 2. The second expression .

 A1x:1/12 × S × 1 −

.

δ1  , 100%

(3.5)

1 12

(3.6)

where A1x:1/12 =

.

1 12

qx × P

is the expected present value of benefit in the amount of 1 m.u. in the case of beneficiary’s death during the first month of retirement. 3. The value A1x:1/12 × S ×

.

δ1 100%

(3.7)

expresses expenditures of the insurance company associated with a quick withdrawal of finances from investment funds.

3.4 Product 2

23

In this case, we assume the insurer’s legitimate expenditures are just equal to the insurer’s costs associated with a quick withdrawal of finances from investment funds. 4. The expression 12 × MPx ×

.

α1 100%

(3.8)

gives a lump sum of the initial costs from the first yearly annuity. 5. Initial costs denoted by .α2 represent initial costs as an absolute amount in m.u. 6. The expression 12 × MPx ×

.

β × 100%

12×(ω−x+1)  t =1

t 12 |

ax(12)

(3.9)

is the expected present value of administrative costs, where the expected present value of deferred annuities for .t = 1, 2, . . . , 12 × (ω − x + 1) is given by (12) . t | ax 12

1 × = 12

12×(ω−x+1)  r=t

r+1 12

px × P

r + 1 12

12×(ω−x+1) 

hence the total sum of provisions is as follows .

t =1

t 12 |

(3.10)

ax(12).

Directly from described equivalence equation (3.3) we get formula on the calculation of the monthly pension annuity of this product in the form   S × 1 − A1x:1/12 − α2

MPx = 12 ×



ax(12)

 12×(ω−x+1)  β α1 (12) t ax + × + | 12 100% 100% t =1

.

(3.11)

3.4 Product 2 Product 2 includes permanent monthly annuity and, moreover, the payment of a lump sum equal to not yet paid monthly annuities in the case of the beneficiary’s death during the period of the first seven years of pension payment. As in Product 1, when the beneficiary dies before the date of the insurer’s obligation to perform the pension insurance contract arises, the amount of the lump sum premium paid under the terms of this contract, reduced by the amount of the insurer’s legitimate fees .δ1 related to the payment of the amount in cash or its transfer to a country not in the euro area.

24

3 Selected Life Old-Age Pension Products

As we mentioned above, survivors have the right to a lump sum equal to the monthly annuities not yet paid in the case of the beneficiary’s death during the period of the first seven years of pension payment, after a deduction of the amount of the insurer’s legitimate fees .δ2 . The equivalence equation with financial flows at the time of closing the policy agreement is as follows: δ1  δ1 + A1x:1/12 × S × 100% 100% 

 S = 12 × MPx × ax(12) + A1x:1/12 × S × 1 −

.

 1  + 12 × MPx × MA(12) x:7 × 1 − 1  + 12 × MPx × MA(12) x:7 × + 12 × MPx ×

δ2 100%

δ2 100%

(3.12)

α1 β + α2 + 12 × MPx × × 100% 100%

12×(ω−x+1)  t =1

t 12 |

ax(12).

On the left-hand side of equality, (3.12) is an accumulated sum S that represents a single premium of the product. On the right-hand side are sequentially following financial flows: 1. The value .12×MPx ×ax(12) is the expected present value of whole life immediate monthly annuity .MPx , where .ax(12)  is given by  (3.4). δ1 δ1 1 2. The expressions .Ax:1/12 × S × 1 − , .A1x:1/12 and .A1x:1/12 × S × 100% 100% have the same meaning as in Product 1, see formulas (3.5), (3.6) and (3.7). 3. The expression δ2  , 100%

(3.13)

t + 1 12

(3.14)

 1  12 × MPx × MA(12) x:7 × 1 −

.

where 

.

MA(12)

1 x:7

=

83  84 − t t =1

12

×

t 1 12 | 12

qx × P

represents the expected present value of the sum of the monthly annuities not yet 1 paid in the amount of . 12 m.u. in the case of the beneficiary’s death during the period of the first seven years of pension payment. 4. The expression 1  12 × MPx × MA(12) x:7 ×

.

δ2 100%

(3.15)

3.5 Product 3

25

gives fees of the insurance company associated with a quick withdrawal of finances from investment funds. Once again, we assume that the insurer’s legitimate expenditures are just equal to the insurer’s expenditures associated with a quick withdrawal of finances from investment funds. 5. The expression 12 × MPx ×

.

α1 100%

(3.16)

gives lump sum initial costs from the first yearly annuity. 6. Initial costs are denoted by .α2 as an absolute amount in monetary units. 7. The expression 12 × MPx ×

.

β × 100%

12×(ω−x+1)  t =1

t 12 |

ax(12)

(3.17)

is the expected present value of administrative costs, where the expected present value of deferred annuities for .t = 1, 2, . . . , 12 × (ω − x + 1) is given by (3.10). Directly, from the equation of equivalence (3.12) we get the formula for the calculation of the monthly pension annuity of this product. It is given by   S × 1 − A1x:1/12 − α2

MPx = 12 ×



ax(12)

. 12×(ω−x+1)  1  α1 β (12) (12) t ax + MA × + + x:7 12 | 100% 100% t =1 (3.18)

Remark 3.2 In our model, we could consider the expenditures of the insurance company associated with a quick withdrawal of finances from investment funds other than the legitimate expenditures of the insurer related to the payment of the amount in cash or its transfer to a non-euro area country. These expenditures do not affect the monthly annuity, but only the amount of lump sum which will be paid out in the case of the beneficiary’s death.

3.5 Product 3 Product 3 offers a permanent monthly annuity that includes raising the pension but does not include survivors’ benefits. This means that the permanent monthly immediate annuity increases geometrically by .j % per year. Furthermore, the product also includes the payment of a lump sum equal to geometrically increasing monthly annuities not yet paid in the case of the beneficiary’s death during the period of the first seven years of pension payment, reduced by .δ2 fees.

26

3 Selected Life Old-Age Pension Products

The basic equivalence equation represents the expected present values of all cash flows related to the basic monthly pension annuity .MPx which is geometrically increasing by .j % per year. Hence, we have an equivalence equation as follows:   (12) S = 12 × MPx × I˜a +A1x:1/12 × S × 1 −

.

x

1  + 12 × MPx × I MA(12)

x:7

 1 + 12 × MPx × I MA(12)

x:7

+ 12 × MPx ×

  δ2 × 1− 100% ×

 δ1 δ1 +A1x:1/12 × S × 100% 100% (3.19)

δ2 100%

α1 β +α2 +12 × MPx × × 100% 100%

12(ω−x+1)  t 12

t =1

   I˜a (12).  x

1. As we know the accumulated sum S represents a single premium of this product.  (12) 2. On the right-hand side the value .12 × MPx × I˜a is the expected present x

value of the permanent monthly immediate annuity geometrically increasing by .j % per year, where r 12  ω−x−1  (12)   j 1 ˜ × 1+ . Ia = × x 12 100 % r=0

t =1

 12×r+t 12

px × P

12 × r + t 12



(3.20) is the expected present value of the whole life immediate benefits of .1/12 m.u., 12 times per year, geometrically increasing by .j % per year, conditional upon the beneficiary’s life.   δ1 δ1 1 3. The expressions .Ax:1/12 × S × 1 − , .A1x:1/12 and .A1x:1/12 × S × 100% 100% have the same meaning as in Product 1, see formulas  and (3.7).  (3.5), (3.6)  1 δ2 (12) 4. The expression .12 × MPx × I MA × 1− represents the x:7 100% expected present value of benefits that are intended for the heirs or an authorized person if a beneficiary dies during the first seven years, reduced by the necessary fees .δ2 , where  1 (12) . I MA

x:7

 × 1+

=

j 100%

6  11  r=0 t =1

r

+

12 − t ×P 12×r+t 1 qx × 12 | 12 12

6 12×r   r=1 k=1

k 1 12 | 12

 qx × 1 +



j 100%

12 × r + t + 1 12

r

 ×P

k+1 12





(3.21)

3.6 Product 4

5.

6. 7. 8.

27

represents the expected present value of the sum of the not yet paid monthly annuity in the amount of .1/12 m. u. geometrically increasing by j % per year, in the case of the beneficiary dies during the period of the first seven years of pension payment.  1 δ2 The expression .12 × MPx × I MA(12) x:7 × expresses expenditures 100% of the insurance company associated with a quick withdrawal of finances from investment funds. Even in this case, we assume that the expenditures associated with withdrawing money are the same as the expenditures associated with paying the amount to the survivors or an authorized person. α1 The expression .12 × MPx × represents lump sum initial costs from the 100% first yearly annuity benefit. We denote initial costs as an absolute amount in monetary units by .α2 .   12(ω−x+1)  β  ˜ (12) represents the × The last expression .12 × MPx × t  Ia x 100% 12 t =1 expected present value of administrative costs, where the expected present value of deferred annuities for .t = 1, 2, . . . , 12 × (ω − x + 1) is given by  12 12 r   (12)  (12) 1   j  ˜ ˜ . t  Ia = Ia − × I (t+1−s− 12 × r) × 1 + x x 12 100% 12 r=0 s=1   12 × r + s , × 12×r+s px × P (3.22) 12 12 t−1

where  I (z) =

.

1 for z > 0, 0 otherwise.

From equivalence equation (3.19) we get a formula of the amount of monthly pension annuity .MPx of this product as follows: .MPx



= 12 ×



I˜a

(12) x

  S × 1 − A1x:1/12 − α2 12(ω−x+1) 1   α1 β + + I MA(12) x:7 + × 100% 100% t=1



 I˜a t  12

(12)

.

x

(3.23)

3.6 Product 4 The Product 4 includes a permanent monthly annuity and also other financial flows as Product 2 and moreover includes survivors’ benefits with a payment period of one or two years. This means that if a beneficiary dies, survivors or heirs will be

28

3 Selected Life Old-Age Pension Products

one more year or two more years to receive a monthly pension the same as getting a beneficiary. That means that equivalence equation (3.12) is extended to equivalence equation (3.24) by the last line. Therefore, the equivalence equation corresponding to this product is as follows: δ1  δ1 + A1x:1/12 × S × 100% 100% 

 S = 12 × MPx × ax(12) + A1x:1/12 × S × 1 −

.

 1  + 12 × MPx × MA(12) x:7 × 1 − 1  + 12 × MPx × MA(12) x:7 × + 12 × MPx ×

δ2 100%

δ2 100%

(3.24)

α1 β + α2 + 12 × MPx × × 100% 100%

+ 12 × MPx × a¨ (12) × n

1 12 |

A(12) + 12 × MPx × x (12)

1. The expression .12 × MPx × a¨ n (12)

a¨ n

.

=

×

1 12 |

(12)

Ax

12×(ω−x+1)  t =1

β × a¨ (12) × n 100%

t 12 |

ax(12)

1 12 |

A(12) x .

for .n = 1 or .n = 2, where

P (n) − 1 1 , ×   12 P 1 − 1 12

(3.25)

is the net present value of a certain monthly annuities in the amount of .1/12 m.u. over one or two years, and

. 1 12 |

A(12) = x



12(ω−x)+11  t =0

t 1 12 | 12

qx+ 1 × P 12

t +1 12

 (3.26)

is the expected present value of the benefit in the amount of 1 m.u. in the case of the pensioner’s death after the first month of retirement and is the expected present value of 12 monthly annuities that will be paid in the case of the beneficiary’s death. (12) β 2. The expression .12 × MPx × 100% × a¨ (12) is the expected present value 1 Ax n × 12 | of administrative costs, which are linked to the payment of a certain monthly pension for one or two years. Directly, from equivalence equation (3.24) we get the formula for the calculation of the monthly pension annuity of this product. It is given by

MPx =

  S × 1 − A1x:1/12 − α2 D4

,

(3.27)

3.7 Product 5

29

where  D4 = 12 × ax(12) +

.

+

 1 α1 + MA(12) + a¨ (12) × n 100% x:7

β × 100%

 12×(ω−x+1)  t =1

t 12 |

1 12 |

ax(12) + a¨ (12) × n

A(12) x

1 12 |

⎞  ⎠. A(12) x

3.7 Product 5 The Product 5 includes a permanent monthly annuity and also other financial flows such as Product 3 and moreover includes survivors’ benefits with a payment period of one or two years. This means that if a beneficiary dies, survivors or heirs will be one more year or two more years to receive a monthly pension the same as getting a beneficiary. This means that equivalence equation (3.19) is extended to equivalence equation (3.28) by the last line. For this product, it is given by   (12) S = 12 × MPx × I˜a +A1x:1/12 × S × 1 −

.

x

1  + 12 × MPx × I MA(12)

 × 1−

 1 + 12 × MPx × I MA(12)

×

x:7

x:7

+ 12 × MPx ×

δ2 100%



 δ1 δ1 +A1x:1/12 × S × 100% 100%

δ2 100%

(3.28)

α1 β + α2 + 12 × MPx × × 100% 100%

+ 12 × MPx × a¨ (12) × I nhpx(12) + 12 × MPx × n (12)

12(ω−x+1)  t =1

t 12

   I˜a (12)  x

β × a¨ (12) × I nhpx(12) . n 100%

(12)

1. The expression .12 × MPx × a¨ n × I nhpx is the expected present value of certain monthly annuities over one or two years in the amount of the annuity the pensioner received in the month of death. 2. The expression (12) .I nhpx

11 ω−x−1    = 1+ t =0

r=0

j 100%

r

× r px ν r ×

t 1 12 | 12

qx+r × ν

t+1 12

1

− 1 qx × ν 12 12

(3.29)

30

3 Selected Life Old-Age Pension Products

represents the expected present value of the permanent monthly benefits in the case of the pensioner’s death. β 3. The expression .12 × MPx × 100% × a¨ (12) × I nhpx(12) is the expected present n value of the administrative costs which are paid during payout of the survivors’ benefits. Then the formula for the calculation of the monthly pension of this product is given by

MPx =

.

  S × 1 − A1x:1/12 − α2 D5

,

(3.30)

where   (12) I˜a .D5 = 12 × + x

β × + 100%

1  α1 + I MA(12) x:7 100%

 12(ω−x+1)  t =1

t 12

⎞   (12) (12)  I˜a + a¨ n × I nhpx(12) ⎠ .  x

References 1. J. Špirková, I. Kollár, M. Spišiaková, Valuation of the second pillar pension products in Slovakia, in Proceedings of the 20th International Scientific Conference AMSE-Applications of Mathematics and Statistics in Economics, pp. 441–452 (2017) 2. J. Špirková, I. Kollár, G. Sz˝ucs, A payout product with increasing payments in the oldage pension saving scheme in Slovakia, in Proceedings of the 21th International Scientific Conference AMSE-Applications of Mathematics and Statistics in Economics, pp. 1–11 (2018) 3. J. Špirková, G. Sz˝ucs, I. Kollár, Detailed view of a payout product of the old-age pension ˇ saving scheme in Slovakia. J. Econ./Ekonomický Casopis 6(7), 287–306 (2019)

Chapter 4

Analysis of the Products

Abstract This chapter gives several numbers, comparisons, tables, and graphical representations, which give us a picture not only individual products but also their comparison. It offers a reflection not only on pension savings and insurance but also on how to secure yourself for the retirement period. In addition, it also analyses the amount of pensions from individual products, which provides an opportunity to think about long-term savings, in general. Moreover, the chapter discusses the socalled effective premium, which is intended for the payment of permanent pension annuities.

4.1 Loading Assessment For modelling, we used the Microsoft Excel 2016 system and the R software with demography package, [5, 6]. Furthermore, we were inspired by scientific research [1] and the psychological assessment of this problem in [2]. As the basic building blocks of our calculation and loading assessment are as follows: • The Lee-Carter model of longevity, Kannistö’s model and selection effects of retirement age without shock and with shock 25%, see Fig. 4.1. • Linear interpolation between integer ages and the uniform distribution of deaths assumption, see Sect. 1.1 and especially Definition 1.1. • The Svensson yield curve, which may be obtained free of charge from the European Central Bank website, on 6 January 2020, for AAA-rated bonds and all bonds [4], see Fig. 4.2. From this figure, it is possible to see negative yields for bonds with a term to maturity of approximately 7 years for all bonds and 15 years for AAA-rated bonds. • Initial costs, administrative costs and insurer’s legitimate expenditures as are stated in Sect. 3.2. • In our work, we offer a general model of pension calculation for x-aged individuals, but as a model situation, we offer an analysis for a beneficiary aged 62 years. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 J. Špirková et al., Selected Payout Products of the Old-Age Pension Saving Scheme, SpringerBriefs in Statistics, https://doi.org/10.1007/978-3-031-23849-9_4

31

32

4 Analysis of the Products

Fig. 4.1 Kannistö’s model and selection effects of retirement ages

Fig. 4.2 Svensson Yield Curves of the European Central Bank, 6 January 2020. © European Central Bank, Frankfurt am Main, Germany

All loadings are included in the insurance costs. These amounts cover the operating costs of the insurance company. To determine the correct amount of the monthly pension annuity, we start by using formula (3.11) to calculate the amount of the pension annuity from Product 1 without considering the costs of our virtual insurance company. Furthermore, we assume risk-free interest rates published by

4.1 Loading Assessment

33

the ECB [4] as “all bonds”, Kannistö’s model with selection and the accumulated sum as a single premium of .10, 000 euros. We receive the monthly pension annuity .MP62 = 35.90 euros.

4.1.1 Calculation of Loading Components In this subsection, we discuss the process of determining the loadings related to the pension annuity. Specifically, we focus on longevity, the specific composition of the insurance strain, market risk of the interest rate, profit of an insurance company, and expense loading. The percentage distribution of the effective premium and individual loadings of Product 1 for a 62-year-old person with the accumulated sum .S = 10, 000 euros are illustrated in Figs. 4.3 and 4.4. For more information, see Appendix B. 1. Calculation of the first loading component: the risk loading—shock If we use the same assumptions as in the previous case, but instead of Kannistö’s model with selection, we use Kannistö’s model with selection and 25% shock, we obtain the monthly pension annuity with the risk loading as follows: .MP62 (Risk loading 1) = 33.50 euros. 2. Calculation of the second loading component: the risk loading—market risk of an interest rate Applying the same assumptions as in the previous case, but considering the yield curve of AAA-rated bonds, the monthly pension annuity with the risk loading is .MP62 (Risk loading 2) = 29.90 euros.

Fig. 4.3 Effective premium and proportion of individual loadings using Kannistö’s model with shock loading, Product 1, 62-year-old client and .S = 10, 000 euros

34

4 Analysis of the Products

Fig. 4.4 Effective premium and proportion of individual loadings using Kannistö’s model without shock loading, Product 1, 62-year-old client and .S = 10, 000 euros

3. Calculation of the third loading component: profit loading The profit loading is defined as follows: the insurer will account for half of the .α1 costs and the initial costs .α2 , and both costs are charged at the time of signing the life insurance contract. The insurance company will satisfy the shareholders of the joint-stock company. Therefore, the insurance company expects to be able to invest only in AAA-rated bonds considering Kannistö’s model with selection and 25% shock. Moreover, we apply illustrative .α1 = 20% from the first yearly annuity (half of this goes into the profit loading, half is used to cover the initial costs of the insurance company). Moreover, we add the initial costs .α2 = 200 euros. If we use the same assumptions as in the previous case and if we also add the costs above, the monthly pension annuity with the loading is .MP62 (Loading 3) = 29.20 euros. 4. Calculation of the fourth loading component: expense loading For the determination of the expense loading, we assume illustrative administrative costs .β = 0.1%, half of .α1 costs, i.e., 10%. Next, we assume yields of AAA-rated bonds, considering the above-mentioned Kannistö’s model with selection and 25% shock. The monthly pension annuity with respect to our assumptions with Loading 4 is as follows .MP62 (Loading 4) = 24.61 euros.

4.1.2 Effective Premium Based on the loadings mentioned above, we can calculate the so-called Effective premium. For 62-year-olds, Effective premium is 68.55%, see Table 4.1.

4.1 Loading Assessment

35

Table 4.1 Monthly pension annuities and distribution of Effective premium and loadings according to retirement age with .S = 10, 000 euros of Product 1

Retirement age x (years) Monthly pension (euros) Effective premium (%) Shock loading (%) Market risk of the interest rate loading (%) Profit loading (%) Expense loading (%)

62 24.61 68.55 6.70 10.01

65 27.45 69.72 7.51 8.78

70 33.57 71.37 9.07 6.87

1.96 12.78

2.00 11.99

2.07 10.62

Table 4.2 Monthly pensions, loadings, and Effective premium for all products using Kannistö’s model with shock

Product 1 Product 2 Product 3 Product 4 (.n = 1) Product 4 (.n = 2) Product 5 (.n = 1) Product 5 (.n = 2)

Monthly Pension

Shock Loading

euros 24.61 24.55 20.88 23.89 23.25 20.25 19.65

% 6.70 6.57 7.49 6.25 5.94 7.17 6.87

Market Risk of the Interest Rate Loading % 10.01 10.00 10.81 10.41 10.81 11.27 11.71

Profit Loading

Expense Loading

Effective Loading

% 1.96 1.96 1.88 1.95 1.94 1.87 1.86

% 12.78 12.77 13.19 12.42 12.08 12.77 12.38

% 68.55 68.70 66.63 68.97 69.23 66.92 67.19

The Effective premium, as well as the share of individual loadings, varies depending on the retirement age. Table 4.1 offers the amount of the monthly pension annuity and the distribution of the effective premium and risk loadings, as well as profit and expense loadings, for retirement ages 62, 65, and 70 years. Note that while with increasing age values of the loading market risk of the interest rate and expense loading are falling, shock loading, and profit loading are rising. Remark 4.1 We made a similar consideration in the paper [9] with only a different probability of death, and the Svensson yield curve from 7 December 2017 was used. It is interesting to observe how these values affect the value of the future monthly pension. Figures 4.3 and 4.4 show the proportion of the amount of effective premium and individual loadings for Product 1, for a client with a retirement age of 62 years and an accumulated sum .S = 10, 000 euros. While Fig. 4.3 also includes shock loading, Fig. 4.4 does not consider it. Tables 4.2 and 4.3 offer individual values of the monthly pensions, loadings, and effective premiums of the examined products for the 62-year-old client and an accumulated sum of .10, 000 euros. It is very interesting to observe the development of individual values if we load Kannistö’s model with selection and then the same model with 25% shock.

36

4 Analysis of the Products

Table 4.3 Monthly pensions, loadings, and Effective premium for all products using Kannistö’s model without shock Monthly Pension Product 1 Product 2 Product 3 Product 4 (.n = 1) Product 4 (.n = 2) Product 5 (.n = 1) Product 5 (.n = 2)

euros 27.00 26.90 23.18 26.09 25.32 22.41 21.68

Market Risk of the Interest Rate Loading % 9.61 9.58 10.40 9.97 10.35 10.85 11.27

Profit Loading

Expense Loading

Effective Loading

% 2.15 2.15 2.09 2.13 2.11 2.07 2.05

% 13.04 12.49 13.54 12.56 12.15 13.03 12.55

% 75.19 76.79 73.96 75.33 75.40 74.05 74.13

If we use the Kannistö’s model with a selection without 25% shock, then the monthly pension or the first monthly pension, respectively, under all examined products would be higher by an average of 2.21 euros. In terms of the effective premium, this would increase by an average of 6.95 percentage points. Other loadings differ by only a few tenths of a percentage point on average.

4.1.3 Monthly Pension with Respect to the Retirement Age Tables 4.4 and 4.5 offer individual values of monthly pensions with respect to retirement age and the accumulated sum .10, 000 euros. It is very interesting to observe the development of the amount of monthly pensions and, of course, also in view of the fact that they offer other benefits than just a whole life monthly pension. We recall that in Product 3, where we consider the annuity increasing yearly with a geometric progression quotient of 1% p.a. considering notes to this product in the Sect. 4.2. When choosing from these products, the beneficiary’s family situation is also essential. It is up to him whether he/she prefers a slightly higher pension or a lower one, which, however, guarantees the payment of survivors’ pensions for a period of one or two years. If we look at these values from the insurance company, it is a question of whether the insurance company will apply a 25% shock to the mortality model or not. This is also related to profitability testing, which we discuss in Chap. 5. Tables 4.6 and 4.7 illustrate the monthly pensions for all examined products using a constant technical interest rate of .0.00% p.a. and the accumulated sum of the amount of .10, 000 euros. We consider these values to represent the lower limit of the amount of monthly pension annuities.

4.1 Loading Assessment

37

Table 4.4 Monthly pensions (in euros) using Kannistö’s model with shock and all mentioned loadings age x 62 63 64 65 66 67 68 69 70

Product 1 24.61 25.50 26.45 27.45 28.52 29.66 30.87 32.17 33.57

Product 2 24.55 25.43 26.36 27.34 28.39 29.50 30.68 31.93 33.27

Product 3 20.88 21.73 22.63 23.59 24.61 25.69 26.84 28.07 29.38

Product 4 = 1) 23.89 24.71 25.58 26.51 27.48 28.52 29.61 30.77 32.01

.(n

Product 4 = 2) 23.25 24.03 24.85 25.72 26.63 27.59 28.61 29.69 30.83

.(n

Product 5 = 1) 20.25 21.05 21.90 22.80 23.75 24.76 25.83 26.97 28.18

.(n

Product 5 = 2) 19.65 20.41 21.21 22.05 22.94 23.89 24.89 25.95 27.07

.(n

Table 4.5 Monthly pensions (in euros) using Kannistö’s model without shock and all mentioned loadings age x 62 63 64 65 66 67 68 69 70

Product 1 27.00 28.04 29.16 30.35 31.63 32.99 34.45 36.01 37.70

Product 2 26.90 27.93 29.02 30.18 31.41 32.72 34.12 35.61 37.20

Product 3 23.18 24.18 25.24 26.37 27.58 28.86 30.24 31.70 33.26

Product 4 = 1) 26.09 27.05 28.07 29.15 30.29 31.50 32.79 34.16 35.61

.(n

Product 4 = 2) 25.32 26.22 27.17 28.18 29.24 30.36 31.55 32.81 34.14

.(n

Product 5 = 1) 22.41 23.34 24.34 25.39 26.51 27.70 28.96 30.31 31.74

.(n

Product 5 = 2) 21.68 22.56 23.49 24.47 25.51 26.62 27.79 29.03 30.35

.(n

Table 4.6 Monthly pensions (in euros) using Kannistö’s model with shock, all mentioned loadings and .i = 0.00% p.a. age x 62 63 64 65 66 67 68 69 70

Product 1 24.15 25.08 26.07 27.12 28.24 29.43 30.70 32.05 33.50

Product 2 24.09 25.01 25.98 27.01 28.11 29.37 30.50 31.81 33.21

Product 3 20.39 21.28 22.22 23.22 24.28 25.41 26.61 27.89 29.25

Product 4 = 1) 23.41 24.27 25.19 26.15 27.18 28.26 29.41 30.63 31.92

.(n

Product 4 = 2) 22.76 23.57 24.44 25.34 26.30 27.32 28.39 29.52 30.72

.(n

Product 5 = 1) 19.79 20.63 21.51 22.45 23.44 24.49 25.61 26.80 28.06

.(n

Product 5 = 2) 19.13 19.92 20.76 21.64 22.57 23.56 24.61 25.72 26.89

.(n

38

4 Analysis of the Products

Table 4.7 Monthly pensions (in euros) using Kannistö’s model without shock, all mentioned loadings and .i = 0.00% p.a. age x 62 63 64 65 66 67 68 69 70

Product 1 26.66 27.75 28.92 30.16 31.49 32.91 34.44 36.07 37.82

Product 2 26.57 27.64 28.78 29.99 31.28 32.65 34.11 35.67 37.32

Product 3 22.80 23.84 24.95 26.13 27.39 28.73 30.16 31.68 33.31

Product 4 = 1) 25.73 26.74 27.80 28.93 30.13 31.40 32.74 34.17 35.69

.(n

Product 4 = 2) 24.94 25.89 26.89 27.94 29.06 30.23 31.48 32.80 34.20

.(n

Product 5 = 1) 22.01 22.98 24.02 25.12 26.29 27.53 28.85 30.26 31.75

.(n

Product 5 = 2) 21.26 22.18 23.16 24.18 25.27 26.43 27.66 28.95 30.33

.(n

4.2 Notes on Product 3 Product 3 is a monthly paid annuity that increases yearly in geometric progression. Determining the geometric progression quotient is a crucial task, especially when we consider the calculation of future cash flows based on a market-consistent valuation using risk-free bond yield curves. In this part, we point out the necessity of a mutual relationship between the riskfree bond yields and the yearly increase rate of pension payments. We were inspired by [3] and studied this problem in [8]. If we study an accumulated factor with progression quotient and a discounting factor on the basis of the Svensson yield curve from formula (3.20), we get .

 1+

j 100%

r

 ×P

12 × r + t 12  

⎧ ⎨ R 12×r+t 12 × exp − ⎩ 100%



 = 1+

r j 100% ⎫   12 × r + t ⎬ × . ⎭ 12

After performing a technical modification, we get the expression .

 exp

1



R 12×r+t 12 100%





  × 12×r+t 12

j 1+ 100%



(4.1)

r

that represents the actual discount factor for individual financial flows. From a mathematical point of view, we can really consider a positive capitalization if the

4.2 Notes on Product 3

39

Fig. 4.5 A comparison of the right-hand and left-hand sides of inequality (4.2)

denominator of (4.1) is greater than one. Thus, this condition is met if R .



12×r+t 12

100%



 ×

12 × r + t 12



 > r × ln 1 +

j 100%

 (4.2)

for all .r = 0, 1, 2, . . . , ω − x − 1 and .t = 1, 2, . . . , 12 and a quotient .j ≥ 0. However, we can investigate the impact of higher progression quotients. Figure 4.5 illustrates the amount of the right-hand side of inequality (4.2) for the geometric progression quotients gradually for .j = (1, 2, 3)% and the amount of the left-hand side of the yields of the Svensson yield curve.1 Because in our next study we consider the age of retirement of 62, we illustrate a retirement time ranging from 1 to 49 years. In Fig. 4.5 you can see how the values of the left- and right-hand sides of the inequality (4.2) develop, so we can decide which value to use. This will decide whether to use positive or negative capitalization. Based on the individual amounts presented in Fig. 4.5, we can consider for positive yield the maximum progression quotient .j = 1%. In Table 4.8 there are listed increasing monthly pension annuities for 62-aged pensioner starting with the basic monthly pension in the amount of 23.18 euros, gradually increasing by 1% per year up to his 110 years in the amount of 37.37 euros if the pensioner will be alive. In the second column, the monthly pension values are calculated using the Kannistö’s model selected tables with shock. It is also very interesting to study the impact of a higher progression quotient on the amount of monthly annuities. This situation is illustrated in Figs. 4.6 and 4.7. 1

The Svensson yield curve on 6 January 2020, for AAA-rated bonds (European Central Bank, 2020).

40

4 Analysis of the Products

Table 4.8 The amounts of the monthly pension annuities of Product 3 geometrically increasing with .j = 1% p.a. for 62-aged beneficiary, .S = 10, 000 euros

Payment year 1 2 3 .. .. 19 29 39 44 49

Monthly pension of Product 3 without shock (euros) Basic pension 23.18 23.41 23.65 .. .. 27.73 30.63 33.83 35.56 37.37

Monthly pension of Product 3 using shock (euros) Basic pension 20.88 21.09 21.30 .. .. 24.98 27.59 30.47 32.03 33.66

Fig. 4.6 Monthly pension annuities geometrically increasing by yearly progression quotient .j % p.a. for a 62-aged beneficiary with the accumulated sum .10, 000 euros using Kannistö’s model selected tables without shock

The monthly pension annuities are illustrated geometrically increasing gradually by j = (1, 2, 3)%. Furthermore, there is also an illustrated monthly pension annuity with .j = 0% that represents the monthly pension of Product 2, for more details, see [7]. Figures 4.6 and 4.7 show how pension annuities evolve according to the progression quotient. A beneficiary can expect the same pension annuity as from Product 2 approximately after 17 or 19 years, respectively.

.

References

41

Fig. 4.7 Monthly pension annuities geometrically increasing by yearly progression quotient .j % p.a. for a 62-aged beneficiary with the accumulated sum .10, 000 euros using Kannistö’s model selected tables with shock

Although for the first seventeen or nineteen years he or she will have a lower pension, in higher ages he can have a much higher pension if the pensioner will be alive. This is a more philosophical or psychological problem than a mathematical one.

References 1. H. Albrecher, P. Embrechts, D. Filipovi´c, G.W. Harrison, P. Koch, S. Loisel, P. Vanini, J. Wagner, Old-age provision: past, present, future. Eur. Actuar. J. 6(2), 287–306 (2016) 2. V. Baˇcová, L. Kostoviˇcová, Too far away to care about? Predicting psychological preparedness ˇ for retirement financial planning among young employed adults 1. J. Econ./Ekonomický Casopis 66(1), 43–63 (2018) 3. D.C.M. Dickson, M.R. Hardy, H.R. Waters, Actuarial mathematics for life contingent risks (Cambridge University Press, 2013) 4. European Central Bank, Euro area yield curves. https://www.ecb.europa.eu/stats/financial_ markets_and_interest_rates/euro_area_yield_curves/html/index.en.html, 6 Jan 2020. Online; Accessed 6 Jan 2020 5. R.J. Hyndman, H. Booth, L. Tickle, J. Maindonald, Demography: Forecasting mortality, fertility, migration and population data (2019). https://CRAN.R-project.org/package=demography. R package version 1.22 6. R Core Team, R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria (2018). https://www.R-project.org/ 7. J. Špirková, I. Kollár, M. Spišiaková, Valuation of the second pillar pension products in Slovakia, in Proceedings of the 20th International Scientific Conference AMSE-Applications of Mathematics and Statistics in Economics, pp. 441–452 (2017)

42

4 Analysis of the Products

8. J. Špirková, I. Kollár, G. Sz˝ucs, A payout product with increasing payments in the oldage pension saving scheme in Slovakia, in Proceedings of the 21th International Scientific Conference AMSE-Applications of Mathematics and Statistics in Economics, pp. 1–11 (2018) 9. J. Špirková, G. Sz˝ucs, I. Kollár, Detailed view of a payout product of the old-age pension saving ˇ scheme in Slovakia. J. Econ./Ekonomický Casopis 6(7), 287–306 (2019)

Chapter 5

Test of Profitability

Abstract The profitability test is a method to evaluate all products, not just financial and insurance. Within pension products, the objective of the profitability test is to determine not only the optimal immediate monthly old-age pension annuity but also the general analysis of the profitability of the product examined. Usually, an actuary proceeds as follows: determine the amount of the monthly pension using classical actuarial methods and subsequently determine the basis, i.e., all assumptions of the test of profitability, for example, expected mortality rates, expenses, and interest rates. The actuary selects a suitable sample of contracts, usually, the ones that are assumed to best represent the future real contracts portfolio, which he/she will value, based on input values, such as retirement age, insurance period, saved amount as a single premium for permanent or temporary pension, respectively, and other cash flows related to the product. If the expected profitability is less than required, then the actuary returns to the first step and adjusts the amount of the monthly pension annuity and other input assumptions. Model individual cash flows until he/she receives the required profitability. The final step is to test the sensitivity of the expected profitability with respect to the variation of the change in the test input values of a portfolio of contracts and the underlying assumptions.

5.1 Test of Profitability Basis An insurance company as a joint stock company is interested in the profitability of individual products. The process by which the profitability of individual insurance products is determined is usually called the test of profitability. Within classical actuarial mathematics, we can determine the amount of a pension annuity by classical actuarial modelling based on the equivalence equations and the relationships derived from them, which we used in Chap. 3. In general, these equivalence equations may not strictly include all cash flows as are used in the test of profitability. The whole process of the test may be iterative because premiums, fees, risk-free interest rates, inflation, or other assumptions are revised. In order to achieve the required profitability, it is necessary to examine in detail the impact of individual input values on all expected financial cash flows. The individual input values can be changed, © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 J. Špirková et al., Selected Payout Products of the Old-Age Pension Saving Scheme, SpringerBriefs in Statistics, https://doi.org/10.1007/978-3-031-23849-9_5

43

44

5 Test of Profitability

Table 5.1 Test of profitability basis S (euros) .10,000

x (years) 62

.MP62

(euros) 27.00

I nf (% p.a.) 2.00

IC (euros) 150.00

AMF (% p.a.) 0.20

.REabs

(euros) 2.00

increased, or decreased by the actuary, according to the current or foreseeable situation on the financial markets. It can even change the calculated amount of the pension, which is also the input value to the whole process. The amount of the pension that we have set can be adjusted now. The whole process of profitability aims to determine the amount of pension that will provide the insurance company with the required profitability. As part of our testing, we do not go into the full details required by the insurance company in practice, but we focus on the basic cash flows that we want to explain the essence of the whole process. We would like to point out that the individual input values and financial flows in actuarial modelling and profitability testing may be different and may also differ in the designation. We analyse the cash flows for Product 1 at discrete months throughout the term from 62 to 110 years, hence from 1 to 588 months. Moreover, we consider fixed and variable assumptions, a so-called test of profitability basis, which is listed in Table 5.1. The fixed assumptions are: • • • •

Accumulated amount .S = 10,000 euros Retirement age .x = 62 years Kannistö’s survival model without shock Svensson yield curve of the European Central Bank, AAA-rated, 6 January, 2020

Depending on what profitability rates we would like to achieve, respectively, as the financial market develops in real life, into our analysis, we enter the socalled variable assumptions. Based on these variable assumptions, we can analyse the product, examine the sensitivity of the impact of individual input values on profitability measures. They are as follows (initial values), see Table 5.1: • • • • •

Whole life immediate monthly pension annuity .MP62 Inflation I nf Initial costs I C incurred at the beginning of the insurance contract Asset management fee AMF as a percentage of the technical provision Renewal expenses .REabs in absolute terms

5.2 Test of Profitability Process In this section, we explain all cash flows in the deterministic test of the profitability process. The monthly pension annuity of Product 1 is calculated according to formula (3.11), where only cash flows are included within the classical actuarial

5.2 Test of Profitability Process

45

survival model. However, using the profitability testing, we can set the amount of the monthly pension annuity using profitability measures that an insurance company expects. We consider a 62-year-old beneficiary who is offered an immediate monthly pension annuity of the entire life. We assume a maximum life expectancy of 110 years. Therefore, the profitability test is explained using the cash flows in individual months t; .t = 1, 2, . . . , 588, for 49 years. The individual cash flows are recorded in Table 5.2. The description of individual financial flows in the 14 columns of Table 5.2 is as follows. 1st column: x—age of the beneficiary from the beginning of the considered retirement age 62 to the end of 110 years. 2nd column: t; .t = 1, 2, . . . , 588—discrete 588 months during the insurance period. 3rd column: . t p62 —the probability that individual at age 62 survives at least to 12 t .  age .62 + 12 t 4th column: .Rt ∗ 12 —spot yield of the Svensson yield curve, .t ∗ is January 6, 2020. t  5th column: .F Rt ∗ 12 —corresponding forward yields given by F Rt ∗

.

 t   t  t − 1 = t × Rt ∗ − (t − 1) × R t ∗ , 12 12 12

see Appendix C. 6th column: .t V62 =

t 12

MP62  t p62 ×P 12

×

588  r=t

r 12

p62 × P



r 12

 ; .t = 1, 2, . . . , 588—

technical provision at time t. We remind you that these technical provisions (2) are used to calculate renewal expenses .REt −1 , but are not included in the product’s cash flows. (1) 7th column: .REt −1 —renewal expenses which are calculated as a time value of renewal value .REabs , i.e.,  I nf  t−1 12 (1) REt −1 = REabs × 1 + . 100 %

.

(5.1)

 588 MP62 + REt(1) r  −1 r p62 × P t  × 8th = 12 ; .t = 1, 2, . . . , 588— 12 t p62 × P r=t 12 12 gross technical provision at time t which includes technical provisions on monthly annuities and provisions on renewal expenses .REt(1) −1 , too. We remind you that these technical provisions are used to calculate renewal expenses (2) .RE t −1 , but are not included in the product’s cash flows. 

B column: .t V62

12

. t

3

p62

0.999740041 0.999480082 0.999220123 0.998960164 0.998700205 0.998440246 0.998180287 0.997920328 0.997660369 0.997400410 0.997140451 0.996880492 0.996576799 0.996273106 0.995969413 0.995665720 .. .. .. .. .. .. 110 586 0.000708102 110 587 0.000667614 110 588 0.000627125

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

t

x

62 62 62 62 62 62 62 62 62 62 62 62 63 63 63 63

2

1



t 12



−0.58392336 −0.59083172 −0.59729719 −0.60333261 −0.60895052 −0.61416316 −0.61898252 −0.62342030 −0.62748792 −0.63119656 −0.63455712 −0.63758025 −0.64027636 −0.64265561 −0.64472789 −0.64650288 .. .. 0.41025318 0.41059665 0.41093896

.Rt ∗

4  t 12



−0.58392336 −0.59774008 −0.61022813 −0.62143887 −0.63142216 −0.64022636 −0.64789868 −0.65448476 −0.66002888 −0.66457432 −0.66816272 −0.67083468 −0.67262968 −0.67358586 −0.67373981 −0.67312773 .. .. 0.61187343 0.61187007 0.61187493

.F Rt ∗

5

Table 5.2 Test of profitability process of Product 1

8316.03 8287.06 8258.01 8228.89 8199.71 8170.47 8141.20 8111.89 8082.55 8053.18 8023.81 7994.42 7965.38 7936.35 7907.32 7791.37 .. .. 76.33 52.35 27.00

.t−1 V62

6 8932.03 8901.93 8871.73 8841.46 8811.11 8780.71 8750.25 8719.75 8689.21 8536.97 8506.74 8476.51 8446.29 8416.09 8385.91 8355.76 .. .. .. .. 5.25 91.18 5.26 62.55 5.27 32.27

B .t−1 V62

(1) .REt−1

2.00 2.00 2.01 2.01 2.01 2.02 2.02 2.02 2.03 2.03 2.03 2.04 2.04 2.04 2.05 2.05

8

7 27.00 27.00 27.00 27.00 27.00 27.00 27.00 27.00 27.00 27.00 27.00 27.00 27.00 27.00 27.00 27.00

.MP62t

10 .St−1

11 .I Ct−1

12 .It

13 .CF Vt

14

.CF St

15

10,000.00 150.00 −4.35 9815.17 9812.61 00.00 0.00 −4.43 −34.92 −34.90 00.00 0.00 −4.51 −35.00 −34.97 00.00 0.00 −4.58 −35.06 −35.02 00.00 0.00 −4.64 −35.12 −35.07 00.00 0.00 −4.68 −35.16 −35.11 00.00 0.00 −4.72 −35.20 −35.14 00.00 0.00 −4.75 −35.23 −35.16 00.00 0.00 −4.78 −35.25 −35.17 00.00 0.00 −4.78 −30.27 00.00 0.00 −4.79 −35.27 −35.18 00.00 0.00 −4.80 −35.27 −35.17 00.00 0.00 −4.80 −35.27 −35.16 00.00 0.00 −4.80 −35.27 −35.15 00.00 0.00 −4.79 −35.26 −35.13 00.00 0.00 −4.77 −35.24 −35.10 .. .. .. .. .. .. .. .. .. .. .. .. .. .. 0.02 27.00 00.00 0.00 −32.22 −0.02 −0.02 0.01 27.00 00.00 0.00 −32.24 −0.02 −0.02 0.01 27.00 00.00 0.00 −32.26 −0.02 −0.02 1.49 1.48 1.48 1.47 1.47 1.46 1.46 1.45 1.45 1.44 1.44 1.43 1.43 1.42 1.42 1.41

(2) .REt−1

9

46 5 Test of Profitability

5.2 Test of Profitability Process

47

(2)

9th column: .REt −1 —renewal expenses which represent asset management fees form the technical provisions from column 8, i.e., REt(2) −1 =

.

AMF B × t −1 V62 . 12 × 100%

(5.2)

10th column: .MP62t —monthly pension annuities payable at the end of the tth month. 11th column: .St −1 —premiums at time .t − 1, it means that the accumulated sum .S = 10,000 euros as a single premium payable to the insurance company at the beginning of the first month of insurance contract and others are equal to .0.00 euros. 12th column: .I Ct −1 —initial costs. .I Ct −1 for .t = 1 spent by the insurance company at the beginning of the first month of insurance contract. For other t corresponding cash flows are .0.00 euros. 13th column: .It —an interest at the end of t-month using the Svensson yield curve, i.e.,    FR ∗ 1 t B × − 1 × t −1 V62 It = exp . 100% 12

.

(5.3)

14th column: .CF Vt —cash flow vector which includes all flows at the end of tth month, i.e., (1)

(2)

CF Vt = St −1 − I Ct −1 − REt −1 − REt −1 − MP62t + It .

.

(5.4)

15th column: .CF St —cash flow signature is given by CF St = CF Vt ×

.

t 12

p62

(5.5)

and represents the expected value of individual cash flows. We now have all the cash flows we need to complete the test of profitability.

5.2.1 Profitability Measures We give the following profitability measures: net present value, profit margin, and internal rate of return, [1]. 1. The net present value is the value of the difference between the expected net present value of the cash flows that come to the insurance company (cash in) and the expected net present value of the flows that leave (cash out).

48

5 Test of Profitability

The Net Present Value NP V (euros) in this context is given by NP V =

12×(ω−x+1) 

.

CF St × P

t =1

 t  , 12

(5.6)

where • .CF  St is cash flow signature given by (5.5) and t • .P 12 is the corresponding discount factor (3.1) using the Svensson yield curve. 2. The Profit Margin P M (%) is given by PM =

.

NP V × 100%, S

(5.7)

i.e., the profit margin is expressed as a percentage of the net present value of expected cash flows to the present value of premiums received. Since in our research of pension products, we consider only a single premium in the amount of the saved amount S, we relate the net present value to this saved amount. 3. The Internal Rate of Return I RR (% p.a.) is the interest rate for which equation 12×(ω−x+1)  .

t =1

CF St × P

 t  =0 12

(5.8)

is satisfied. It is thus a constant interest rate that determines the net present value at zero.

5.3 Profitability Measures of Product 1 Based on data from Table 5.2, we get the following values of individual profitability measures (5.6), (5.7), and (5.8). Every insurance company expects that if it introduces a new product, all profitability rates will be at the required level. 1. The net present value (5.6) is in the amount of .NP V = 527.45 euros. With our set of basic profitability values, the net present value is positive, which is good. But even here it is important to know what value is required, especially in comparison with the expected present value of the premium. This percentage ratio is expressed in the following measure. 2. The profit margin (5.7) is at the level of .P M = 5.27%. This value is positive, but again, it depends on whether the insurance company demands higher profitability or not.

5.3 Profitability Measures of Product 1

49

3. The internal rate of return (5.8) is .I RR = −0.32% p.a. This interest rate represents a constant interest rate, which will ensure the net present value (5.6) at the level of 0 euros. It is interesting to observe the development of this value, given that we used the Svensson yield curve in the modelling, which acquires both positive and negative values.

5.3.1 Sensitivity of Input Parameters Table 5.3 gives profitability measures in the settings of input values from Table 5.1 with the changing value of the monthly pension .MP62 . During the initial phase of profitability testing, we set the amount of monthly pension at the level of 27 euros. However, if we reduced or increased the monthly pension annuity by 1 euro while maintaining all other values of profitability basis from Table 5.1, any such increase or decrease would cause a decrease or increase in the profit margin P M by 3.15 percentage points. Given the single premium that we consider in the amount of .10,000 euros, the value of the net present value NP V would also change to a level of 314.29 euros. The internal rate of return I RR will decrease or increase on average by a percentage point of .0.23. It is obvious that the change in the amount of monthly pension has a very significant effect on profitability and, therefore, it is essential to set this value sensitively. Table 5.4 gives profitability measures in the settings of the input values of Table 5.1 with the changing value of the asset management fee AMF . It can be seen that if AMF decreases by .0.05 percentage point, then P M will increase by an average of .0.65 percentage point and vice versa. The net present value NP V will change to a level of 64.80 euros. Regarding I RR, it will increase (decrease) by an average of .0.05 percentage point. However, it should be noted that this value is determined by the companies that manage the assets. Table 5.5 offers profitability measures in the setting of input values from Table 5.1 with the changing value of the renewal expenses .REabs (euros). If renewal expenses .REabs are changed by .0.50 euros, P M will change by approximately .2.10 percentage point. If renewal expenses were at the level of 2 euros or less, P M would increase on average by 2.10%. However, if renewal expenses are at the level of 2 euros and more, P M would fall to the level of 2.10% Table 5.3 Profitability measures of Product 1 using Kannistö’s model without shock with changing levels of the monthly annuity .MP62

(euros) NP V (euros) P M (%) I RR (% p.a.)

25.00 1156.03 11.56 −0.79

26.00 841.74 8.42 −0.55

27.00 527.45 5.27 −0.32

28.00 213.216 2.13 0.09

29.00 −101.14 −1.01 0.14

50

5 Test of Profitability

Table 5.4 Profitability measures of Product 1 using Kannistö’s model without shock with changing levels of the asset management fee AMF (% p.a.) NP V (euros) P M (%) I RR (% p.a.)

0.10 657.06 6.57 −0.41

0.15 592.25 5.92 −0.37

0.20 527.45 5.97 −0.32

0.25 462.64 4.63 −0.27

0.30 397.84 3.98 −0.23

Table 5.5 Profitability measures of Product 1 using Kannistö’s model without shock with changing levels of the renewal expenses .REabs

(euros) NP V (euros) P M (%) I RR (% p.a.)

1.00 946.90 9.47 −0.64

1.50 737.17 7.37 −0.48

2.00 527.45 5.27 −0.32

2.50 317.72 3.18 −0.16

3.00 108.00 1.08 −0.01

Table 5.6 Profitability measures of Product 1 using Kannistö’s model without shock with changing levels of the initial costs

I C (euros) NP V (euros) P M (%) I RR (% p.a.)

100.00 577.46 5.77 −0.35

125.00 552.45 5.52 −0.34

150.00 527.45 5.27 −0.32

175.00 502.44 5.02 −0.30

200.00 477.44 4.77 −0.28

Table 5.7 Profitability measures of Product 1 using Kannistö’s model without shock with changing levels of the inflation

I nf (% p.a.) NP V (euros) P M (%) I RR (% p.a.)

1.00 643.03 6.43 −0.41

1.50 588.15 5.88 −0.37

2.00 527.45 5.27 −0.32

2.50 460.20 4.60 −0.27

3.00 385.61 3.86 −0.21

and less. The net present value NP V would behave in the same way. This means that the value of .REabs is very sensitive and would jeopardize profitability. Values in Table 5.6 give profitability measures in the settings of the input values of Table 5.1 with the changing value of the initial costs I C. During the profitability testing, we set the value of 150 euros as the basis for the initial costs. However, if we increase or decrease this value by 25 euros, the profit margin would decrease or increase by approximately .0.25 percentage points. Similarly, there is no large change in other profitability measures. This means that a change in the value of initial costs does not have a major impact on the change in profitability. However, it is necessary to take into account the need to set these costs at a realistic level. Table 5.7 we give profitability measures in the settings of input values from Table 5.1 with the changing value of inflation I nf . Although inflation is usually around 2%, we can look at what will cause it to change in our tests. Inflation enters our testing as the time value of the base value of renewal expenses at the level of 2 euros. If inflation was to decrease (increase) by 0.50 percentage points, the profit margin would increase (decrease) by an average of .0.64 percentage points. The net present value would change by an average of .64.36 euros and the internal rate of return by .0.05 percentage points.

Reference

51

Reference 1. D.C.M. Dickson, M.R. Hardy, H.R. Waters, Actuarial mathematics for life contingent risks (Cambridge University Press, 2013)

Part III

Conclusions and Recommendations

We come to the conclusion of our reflection and analyses. It is appropriate to take a position on our results. Therefore, in this part, we answer the research questions set by us and take a position on individual hypotheses. Again, we recall the concept of the effective premium as part of the amount of money saved that will be paid out in the form of pensions. We answer the question, what is the lower limit of the monthly pension annuity from the accumulated sum of 10,000 euros of the individual products in order to observe the prudent behaviour of the insurance company, and for how many years should a beneficiary receive a monthly pension annuity to get back at least 10,000 euros. We also answer the question of how the requirement of a 7-year annuity in Product 2 affects the amount of the monthly annuity compared to Product 1. Moreover, we take a position on the established hypotheses concerning what part of the saved accumulated sum is consumed by life insurance companies or whether in the case that a 62-year-old beneficiary lives for another 20 years, the whole saved amount will return to the beneficiary in the form of monthly pension payments.

Chapter 6

Research Questions and Hypotheses

It is not important how high a pension we will receive, but how long.

Abstract In this chapter, we give the research questions and hypotheses that we set at the beginning of our analysis of the old-age pension annuities, and, of course, we answer them, and we also take a position on established hypotheses. In addition, we also offer a short, perhaps more philosophical discussion of the attitude to risk and overall long-term savings.

6.1 Research Questions Q1: What in our understanding represents the so-called effective premiums, i.e., in other words, what amount of the saved 10,000 euros will be paid out in the form of pensions? Q2: What is the lower limit of the monthly pension annuity from the accumulated sum of 10,000 euros of the individual products to observe the prudent behaviour of the insurance company? Q3: For how many years should a pensioner receive a monthly pension annuity to get back at least 10,000 euros? Q4: To what extent does the requirement of a 7-year annuity (84 monthly annuities) in Product 2 affect the amount of the monthly annuity compared to Product 1 (classical monthly permanent pension annuity)? The answers to the research questions are made in the same sequence as established here. Reply to Q1:

We consider as the effective premium that part of the saved amount which is intended for the payment of monthly pensions, after loading calculation of premiums, all considered risk loadings as well as profit and expense loadings.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 J. Špirková et al., Selected Payout Products of the Old-Age Pension Saving Scheme, SpringerBriefs in Statistics, https://doi.org/10.1007/978-3-031-23849-9_6

55

56

Reply to Q2:

Reply to Q3:

Reply to Q4:

6 Research Questions and Hypotheses

The effective premium from the accumulated sum S = 10,000 euros using shock loading for Product 1, is 68.55%, that is, 6855 euros, and it is illustrated on Fig. 4.3. In Fig. 4.4 the effective premium and percentage distribution of all considered loadings without shock loading for Product 1 are shown. In this case, the effective premium is at the level of 75.19%, i.e., 7519 euros. This difference represents 6.64 percentage points, that is, 664 euros. We also draw attention to the Table 4.1 where we present the distribution of all considered loadings and effective premiums based on the age at which the client retires and begins to receive a pension. In Tables 4.2 and 4.3 are given all effective premiums and all loadings of the examined products for the beneficiary of 62 years of age. We consider the lower limit of monthly pension annuities to be the values, which are calculated using a constant technical interest rate of 0% p.a. (instead of the Svensson yield curve on January 6, 2020). The values of these lower limits of monthly pensions are written in Tables 4.7 and 4.6. However, it is interesting to compare the values in Tables 4.7 and 4.5 (both calculated using life tables without shock) for ages 69 and 70, where we can consider the values of Table 4.5 as the lower limits, because these numbers are smaller than in Table 4.7. This situation is also caused by the development of the probability of death in older ages. a. Consider beneficiary of Product 1 and 62 years of age. To return at least the sum of 10,000 euros to the beneficiary, he/she would have to receive the monthly pension for at least 34 or 31 years, respectively, see Tables 4.4 and 4.5. b. If he/she started to receive a pension at the age of 70, he/she would have to receive it for at least 25 or 22 years and 2 months, respectively. In general, we can say that, on average, the beneficiary would have to live for at least 95 or 92 years. c. For the remaining products, the situation is very similar to that Product 1. However, we only recall that these products have “the advantages” that if the beneficiary dies during the first 7 years of receiving the pension, an insurance company will pay out a lump sum equal to not yet paid monthly annuities to the heirs or authorized person. In addition, Products 4 and 5 also guarantee the payment of pension annuities for one or two years to the heirs or authorized person in the event that the beneficiary dies. Please observe Tables 4.4 and 4.5, again. The difference in pensions from Products 1 and 2 is only a few cents, so it is really up to the serious consideration of the future pensioner which product to choose. If the pensioner has a spouse, it is definitely worth considering Product 2.

6.3 Discussion About the Attitude to Risk and Overall Long-Term Savings

57

6.2 Research Hypotheses We want to take a position on the following hypotheses: H1: Life insurance companies consume more than 25% of pensioner savings, which means that the whole life annuity paid out is less than 75% of the savings. H2: The actual life expectancy for the 62-year-old pensioner is 20.02 years. If a 62year-old person lives for another 20 years, the whole saved amount will return to the person in the form of monthly pension payments. Rejection or acceptance of hypotheses: To H1: If we consider the determination of the effective premium using Kannistö’s model with shock, we accept the assumption of hypothesis H1, see Table 4.2. But, if we assume calculation of the effective premium using Kannistö’s model without shock, we accept the assumption of hypothesis H1 only for Product 5. We reject this hypothesis for Products 1 to 4. For more information, see Table 4.3. To H2: If we consider a 62-year-old pensioner and monthly pension annuities in the amounts which are given in Tables 4.4 and 4.5, in order to get the .10,000 euros investment back, he/she would have to get a pension for at least 31 years or 34 years, so in this case, we would reject the hypothesis H2. We also discuss some of the previous research questions and hypotheses in the paper [3] but under different input conditions. It could certainly be interesting to compare the answers to the set questions, or to accept or reject our hypotheses. Based on other input parameters, we also examined individual products in our introductory work [1, 2], which relate to the products that we analyse in detail in our monograph.

6.3 Discussion About the Attitude to Risk and Overall Long-Term Savings We have come to the conclusion of our reflections and analyses. We answered the research questions and took a position on the hypotheses. However, our attitude towards retirement savings also depends on our attitude toward risk. We often get the question whether it would not be a better idea to save up “on the pillow”, as our grandmothers and great grandmothers did, or whether to trust the pension fund to really increase the value of our saved money at least so that it covers inflation or, of course, in the best-case scenario, increase the value at a much higher percentage beyond inflation. However, will it really be like that? Our analysis does not suggest that future pensions from our modelled products will be so high.

58

6 Research Questions and Hypotheses

However, there is also a philosophical question of whether we are able to save for retirement ourselves, whether we would not be tempted to spend the saved money at a convenient time on something tempting and current. There are certain individuals who could do this, but there are also those who live for the moment and do not think of how they will live in the future. Therefore, if the state offers retirement savings products and regular monthly benefits in the future, then it is definitely very good. So far, we have thought about retirement savings relatively conservatively. However, there is also the possibility of investing in pension funds, which are not conservative, but focus on investing in stocks that are more profitable but also riskier. However, this requires a targeted, almost daily monitoring of stock market development and the overall situation in the financial markets. Another excellent way to secure finances for the retirement period is, in our opinion, the purchase of real estate, which can also provide regular income from rent and its possible sale in the future. It is also suitable for buying rare images or precious metals that are easily turned into liquidity.

References 1. J. Špirková, I. Kollár, M. Spišiaková, Valuation of the second pillar pension products in Slovakia, in Proceedings of the 20th International Scientific Conference AMSE-Applications of Mathematics and Statistics in Economics, pp. 441–452 (2017) 2. J. Špirková, I. Kollár, G. Sz˝ucs, A payout product with increasing payments in the oldage pension saving scheme in Slovakia. in Proceedings of the 21th International Scientific Conference AMSE-Applications of Mathematics and Statistics in Economics, pp. 1–11 (2018) 3. J. Špirková, G. Sz˝ucs, I. Kollár, Detailed view of a payout product of the old-age pension saving ˇ scheme in Slovakia. J. Econ./Ekonomický Casopis 6(7), 287–306 (2019)

Appendix A

Life Tables

See Table A.1.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 J. Špirková et al., Selected Payout Products of the Old-Age Pension Saving Scheme, SpringerBriefs in Statistics, https://doi.org/10.1007/978-3-031-23849-9

59

Entry age x 62 63 64 65 66 67 68 69 70 71 72 73 74 75 .. .. 98 99 100 101 102

Kannistö’s model death probability at age x ∗ .qx 0.0058765199 0.0068698546 0.0079201065 0.0090625161 0.0103410046 0.0117219802 0.0132588872 0.0149336014 0.0167853893 0.0188141706 0.0210504717 0.0235065287 0.0262102603 0.0291823646 .. .. 0.2620878948 0.2837423176 0.3065120961 0.3303397226 0.3551478087

Selection factor males .s(x)M 0.511225 0.512756 0.514900 0.517656 0.521025 0.525006 0.529600 0.534806 0.540625 0.547056 0.554100 0.561756 0.570025 0.578906 .. .. 0.952225 0.975806 1.000000 1.000000 1.000000

Table A.1 Life tables using Kannistö’s model with selection Selection factor females .s(x)F 0.551125 0.552531 0.554500 0.557031 0.560125 0.563781 0.568000 0.572781 0.578125 0.584031 0.590500 0.597531 0.605125 0.613281 .. .. 0.956125 0.977781 1.000000 1.000000 1.000000 Weights males 0.691950 0.466676 0.461355 0.454986 0.450064 0.444991 0.439488 0.432507 0.423109 0.413160 0.402057 0.395824 0.389358 0.381786 .. .. 0.207430 0.199928 0.192426 0.184923 0.177421

.wx

Selection factor unisex .s(x) 0.532404 0.533969 0.536230 0.539116 0.542527 0.546527 0.551124 0.556357 0.562258 0.568755 0.575865 0.583371 0.591459 0.600157 .. .. 0.955316 0.977386 1.000000 1.000000 1.000000

Kannistö’s model selected death probability at age x ∗ .qx = s(x) × qx 0.0031195084 0.0036557193 0.0042302273 0.0048636772 0.0055813711 0.0063689740 0.0072590566 0.0082466810 0.0093589600 0.0106006135 0.0119955335 0.0135531019 0.0153008865 0.0172609283 .. .. 0.2202544230 0.2414529045 0.2639903847 0.2813204597 0.2989301887

60 A Life Tables

103 104 105 106 107 108 109 110

0.3808389448 0.4072963002 0.4343850262 0.4619544753 0.4898411937 0.5178725821 0.5458710636 0.5736585531

1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000

1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000

0.169919 0.162417 0.154915 0.147412 0.139910 0.132408 0.124906 0.117403

1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000

0.3167120722 0.3345530110 0.3523371597 0.3699489778 0.3872763092 0.4042133121 0.4206630758 0.4365397860

A Life Tables 61

Appendix B

Procedure for Determining of the Effective Premium of Product 1

In the following part, we describe the procedure for determining the effective premium and individual risk loadings. As an example, we use formula (3.11) for the calculation of the monthly premium from Product 1. Specific values can be compared with the values published in Sects. 4.1 and 4.1.1. For better orientation in the text, we will use the initial calculation of the monthly pension with item numbering 0 and the determination of risk loadings is numbered as in Sect. 4.1.1. 0. We assume risk-free interest rates published by the ECB as “all bonds”, Kannistö’s model of life tables with a selection from Table A.1 and the accumulated sum as a single premium S euros. The initial and administrative costs .α1 , .α2 and .β are equal to zero. Using formula (3.11) we obtain the first amount of the monthly premium and we will label it .(0)MPx . Now, we are going to start to determine individual risk loadings and, in the end, also the effective premium. 1. Calculation of the first component of the risk loading—shock (Risk loading 1) Again, we calculate the monthly pension using formula (3.11), but in the basic assumptions we will use life tables determined by Kannistö’s model with a selection from Table A.1, but loaded by shock 25%, and other assumptions remain the same. This is how we get the monthly premium, and we denote it by .(1)MPx . We express the accumulated sum S from formula (3.11). If we substitute .(1)MPx into this formula, we get the amount, which we denote by (12) .S1 . In all calculations we use the original values of .ax and .A1x:1/12 . Please, observe Table B.1. Then Risk loading 1 in absolute term is given by Risk loading 1abs = S − S1

.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 J. Špirková et al., Selected Payout Products of the Old-Age Pension Saving Scheme, SpringerBriefs in Statistics, https://doi.org/10.1007/978-3-031-23849-9

(B.1)

63

64

B Procedure for Determining of the Effective Premium of Product 1

Table B.1 Monthly pension annuities using the assumptions in the procedure for determining the effective premium

(12)

k

.(k)MP62

.a62

.A

0 1 2 3 4

35.90 33.50 29.90 29.20 24.61

23.2050221998

0.0002600749

1 62:1/12

and in relative term by Risk loading 1 =

.

S − S1 × 100%. S

(B.2)

2. Calculation of the second component of the risk loading—market risk of the interest rate (Risk loading 2) Now, in the calculation of .MPx , we use the same assumptions as in the previous case, but instead of risk-free interest rates all bonds we apply interest rates for AAA-rated bonds. Substituting .(2)MPx into the formula for S we get the amount .S2 . Hence, Risk loading 2 is given by Risk loading 2abs = S1 − S2

.

(B.3)

and in relative term by Risk loading 2 =

.

S1 − S2 × 100%. S

(B.4)

3. Calculation of the third component of the loading—profit loading (Loading 3) In this step, we use the same assumptions as in item 2 and moreover, we apply α1 . 2 % from the first yearly annuity. Furthermore, we add the initial costs in the amount of .α2 = 200 euros. Using these assumptions, we obtain again new monthly pension annuity .(3)MPx and we calculate “new” sum .S3 . Then, Loading 3 is given by Loading 3abs = S2 − S3

.

(B.5)

and in relative term by Loading 3 =

.

S2 − S3 × 100%. S

(B.6)

B Procedure for Determining of the Effective Premium of Product 1 Table B.2 Monthly pension annuities using the assumptions in the procedure for determining of the (risk) loadings and Effective premium

S .S1 .S2 .S3 .S4

10,000.00 9330.32 8329.30 8133.49 6855.35

669.68 1001.02 195.81 1278.14

100.00% 6.70% 10.01% 1.96% 12.78% 68.55%

65

Risk loading 1 Risk loading 2 Loading 3 Loading 4 Effective premium

4. Calculation of the fourth component of the loading—expense loading (Loading 4) For the determination of the expense loading, we assume illustrative administrative costs .β%, half of .α1 costs. Next, we assume yields of AAA-rated bonds, considering the mentioned Kannistö’s model mentioned with selection and 25% shock. The monthly pension annuity with respect to our assumptions is denoted by .(4)MPx and the corresponding sum .S4 . By similar way we get Loading 4 as follows: Loading 4abs = S3 − S4

.

(B.7)

and in relative term by Loading 4 =

.

S3 − S4 × 100%. S

(B.8)

Finally, we get formula for the calculation of the effective premium EP which is given by EP = 100% − Risk loading 1 − Risk loading 2 − Loading 3 − Loading 4. (B.9)

.

For illustration, we give a calculation of the above-mentioned loadings and effective premium for the beneficiary of Product 1 and 62 years of age with .S = 10,000 euros (Table B.2). Remark B.1 Very similar, using formulas (3.18)–(3.30) in Chap. 3 to calculate monthly pension annuities from other products, we can determine risk loadings and effective premiums. We can also expand the set of relevant loadings so that the overall analysis is consistent not only with the prudent behaviour of insurance companies but also with all the requirements for the professional work of the actuary.

Appendix C

Forward Rate for Continuous Compounding

We used the European Central Bank yield curve in our study, whose yields are modelled continuous compounding as spot yields,  t using   t .R t ∗ p.a., the monthly forward yields .F R t ∗ 12 ; .t = 1, 2, . . ., are determined 12 based on equation           t  t − 1 t − 1 t 1 t × exp F R t ∗ = exp R t ∗ . exp R t ∗ × × × 12 12 12 12 12 12 (C.1) From equality (C.1) we get the formula for the calculation of the forward rate as follows,

.

F Rt ∗

.

 t   t  t − 1 = t × Rt ∗ − (t − 1) × R t ∗ . 12 12 12

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 J. Špirková et al., Selected Payout Products of the Old-Age Pension Saving Scheme, SpringerBriefs in Statistics, https://doi.org/10.1007/978-3-031-23849-9

(C.2)

67

Index

A Age adult, 9 maximum, 21 old, 11 retirement, 21 Annuity, 35, 47 immediate, 5 increasing, 21 permanent, 19 Asset management fee, 44 Assumption fractional age, 6 uniform distribution, 6 B Bond AAA-rated, 6, 31 all bonds, 6, 31 risk-free, 1 C Cash flows, 43, 44 Costs, 7 administrative, 21, 31, 34 initial, 21, 31, 34, 44 Curve Svensson, 39 D Database Human Mortality, 10 Statistical Office of the Slovak Republic, 10

E European Central Bank (ECB), 6 Expenditures legitimate, 21, 31

F Factor discounting, 21, 38 reduction, 12 selection, 14 Forecasts, 10

H Hypothesis, 2, 57

I Inflation, 44 Interest rate of return, 48

L Lee-Carter parameters, 9 Life expectancy, 10 Life tables, 60 Loading, 35 components, 33 expense, 34 market risk of the interest rate, 33 profit, 34 shock, 31, 33

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 J. Špirková et al., Selected Payout Products of the Old-Age Pension Saving Scheme, SpringerBriefs in Statistics, https://doi.org/10.1007/978-3-031-23849-9

69

70

Index

Longevity adult-age, 9 old-age, 11

Q Quotient geometric, 21, 39

M Maturity, 6 Model ARIMA, 10 Beard’s, 11 Cairns-Blake-Dowd, 10 Gompertz, 11 Kannistö’s, 11, 31 Lee-Carter, 9, 10, 31 Makeham, 11 Thatcher’s, 11 Modelling actuarial, 20 Mortality, 5

R Renewal expenses, 44 Research question, 2, 55

N Net present value, 48 P Parameters Kannistö’s, 12, 13 Lee-Carter, 9, 10 Svensson, 7 Premium effective, 33, 35, 36, 63 single, 21 Probability, 5 deferred, 5 survival, 45 Product 1, 19, 21, 35, 44 Product 2, 19, 23, 35 Product 3, 20, 25, 35, 38 Product 4, 20, 27, 35 Product 5, 20, 29, 35 Profitability measures internal rate of return, 48 net present value, 47 profit margin, 48 Profit margin, 48

S Sensitivity asset management fee, 50 inflation, 50 initial costs, 50 monthly annuity, 49 renewal expenses, 50 Sum accumulated, 21

T Term to maturity, 7 Test of profitability, 43 basis, 44 process, 45, 47

U Uniform distribution, 6

Y Yield forward, 45, 67 spot, 6, 45, 67 Svensson, 39 Yield curve parameters, 7 Svensson, 6, 7, 21, 31, 35, 38