Real Option Analysis and Climate Change: A New Framework for Environmental Policy Analysis [1st ed.] 978-3-030-12060-3;978-3-030-12061-0

Table of contents :
Front Matter ....Pages i-xiii
Prolegomena: What Does Real Option Analysis Bring to Climate Change Policy (Benoit Morel)....Pages 1-8
Toward a General Theory of Real Options (Benoit Morel)....Pages 9-33
Real Option Analysis: Work in Progress, in Need of Progress (Benoit Morel)....Pages 35-49
Extreme Events, Cat Bonds, ROA in the Context of Fat Tail Distributions, and the Weitzman Effect (Benoit Morel)....Pages 51-67
Global CO2 Emission Mitigation Through the Looking Glass of ROA (Benoit Morel)....Pages 69-84
Internationalization of the Response: The Example of the REDD Credits (Benoit Morel)....Pages 85-94
Prioritizing the Investments Needed to Avoid the Unmanageable (Mitigation) and to Manage the Unavoidable (Adaptation) (Benoit Morel)....Pages 95-115
Unanswered Questions About Uncertainty, Information, and Investment Decisions (Benoit Morel)....Pages 117-128
Back Matter ....Pages 129-160

Citation preview

Springer Climate

Benoit Morel

Real Option Analysis and Climate Change A New Framework for Environmental Policy Analysis

Springer Climate Series Editor John Dodson, Chinese Academy of Sciences, Institute of Earth Environment, Xian, Shaanxi, China

Springer Climate is an interdisciplinary book series dedicated to climate research. This includes climatology, climate change impacts, climate change management, climate change policy, regional climate studies, climate monitoring and modeling, palaeoclimatology etc. The series publishes high quality research for scientists, researchers, students and policy makers. An author/editor questionnaire, instructions for authors and a book proposal form can be obtained from the Publishing Editor. Now indexed in Scopus1 !

More information about this series at http://www.springer.com/series/11741

Benoit Morel

Real Option Analysis and Climate Change A New Framework for Environmental Policy Analysis

Benoit Morel Carnegie Mellon University Pittsburgh, PA, USA

Note: In this book, ROA refers to real option analysis, not to return on assets or other uses of that acronym. ISSN 2352-0698 ISSN 2352-0701 (electronic) Springer Climate ISBN 978-3-030-12060-3 ISBN 978-3-030-12061-0 (eBook) https://doi.org/10.1007/978-3-030-12061-0 © Springer Nature Switzerland AG 2020 This work is subject to copyright. All rights are reserved 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, express 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 Alexandra, without whom that book would not exist

Preface

For the better of ten years I taught a course on real options at Carnegie Mellon University. The goal was to show that real option analysis (ROA) had the potential to fill a serious gap in environmental policy by providing a powerful tool to deal with large uncertainty. Most papers on the subject start with the observation that such tools do not exist. Net present value (NPV) is blatantly limited as it deals with expected values, i.e. averages of distributions. But it is insensitive to the details of the uncertainty. The “neo-classical approach” is an awkward framework for quantitative analysis of uncertainty, especially if and when the uncertainty is large. But ROA in its present form is also an awkward tool to support investment decisions under uncertainty. It dawned on me progressively that ROA could be approached differently from what is the case. It is treated as an extension of financial option theory to be applied to corporate investments and eventually to all investment under uncertainty. The first extension is relatively solid, but the second extension has been done in such a way that the result was this ugly duckling where the Black– Scholes formula shows up in context where it does not belong and the uncertainty does not reflect the problem at hand. Worse, ROA degenerated in a set of half baked recipes where several different approaches are used in parallel, with little conceptual commonality. It dawned on me that it was not necessarily that difficult to improve the situation by realizing that ROA is a stand-alone paradigm and financial options are a special case, where the uncertainty stems from the dynamics of change of tradable assets. The narrative of this book is about that. In the first chapters, the fundamentals of option theory are revisited by coming back to the origin, i.e. the Ph.D. thesis of Gaston Bachelier. The mathematical framework that comes out is applied to the existing field of real options, partially to reinforce its strong points but also to point to some of its weaknesses. In fact this book is at times quite critical of the present state of ROA for its lack of mathematical and conceptual rigour. The chapter about extreme events and catbonds (Chap. 4) makes the transition to the second part of the book. Extreme events provide an example where the vii

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Preface

uncertainty is described by fat-tailed distributions, and the estimation of the value of catbonds, something that earlier versions of ROA did not know how to deal with (let alone NPV). The next chapters are about environmental policy, mitigation and adaptation. The idea is to illustrate the kind of insights that ROA provides. Each chapter brings its set of surprising results. The intended message is that ROA is best used not as a set of recipes but as a powerful tool to be used intelligently. In Appendix A, I tried as diplomatically as possible but firmly to point to a shameful confusion commonly made between two kinds of “real options”. The only feature those two forms of real options have in common is that they deal with investments under uncertainty. One kind of real options has to do with finding the optimal conditions for investments under uncertainty. The other kind of real options are those who were inspired originally from the Black–Scholes contribution to financial options. They are a quantification of the effect of uncertainty on the value of investments. Both are mathematically and conceptually different. This book is about the Black–Scholes inspired real options. Appendix B is for those who want to see more concretely how the mathematical apparatus developed in the book can be used in practice. In other words, I tried to write the book, I wished existed when I started teaching about ROA. Pittsburgh, PA

Benoit Morel

Contents

1

2

Prolegomena: What Does Real Option Analysis Bring to Climate Change Policy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Introducing the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 The IPCC Reports . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Dealing with the Uncertainty and the Limitations of Existing Mathematical Models . . . . . . . . . . . . . . . . . 1.2.2 Cost Benefit Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Limitations of Net Present Value . . . . . . . . . . . . . . . . . 1.2.4 Prospects Offered by Real Options . . . . . . . . . . . . . . . . 1.3 Risk and Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . .

1 1 2

. . . . . .

3 5 5 6 6 7

Toward a General Theory of Real Options . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 The Impact of Black–Scholes on Real Options . . . . . . . . 2.1.2 Black–Scholes as a Black Swan . . . . . . . . . . . . . . . . . . . 2.2 What Are “Options”? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Value of Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Risk-Neutral Valuation of Options . . . . . . . . . . . . . . . . . 2.3 The Role of the “Corporate Culture” in the Genesis of Real Option Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Financial Options as a Particular Case of Real Options . . . . . . . . 2.5 Prospects of ROA in Environmental Policy . . . . . . . . . . . . . . . . 2.6 Foundations of Option Theory . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.1 Bachelier’s “Théorie de la Spéculation” . . . . . . . . . . . . . 2.6.2 The Emergence of Real Options . . . . . . . . . . . . . . . . . . . 2.7 The Mathematical Foundations of ROA . . . . . . . . . . . . . . . . . . 2.7.1 Mathematical Framework . . . . . . . . . . . . . . . . . . . . . . . 2.7.2 The Value of Exchanging Two Risky Assets . . . . . . . . . 2.7.3 First-Degree Homogeneity . . . . . . . . . . . . . . . . . . . . . . .

9 10 10 11 12 12 13 13 14 15 15 15 17 21 21 21 22 ix

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2.7.4 2.7.5

Sum and Ratio Distributions . . . . . . . . . . . . . . . . . . . . . Risk Neutrality Mathematically and Its Effect on the Valuation of Options . . . . . . . . . . . . . . . . . . . . . . 2.7.6 The Black–Scholes Formula . . . . . . . . . . . . . . . . . . . . . 2.7.7 Black–Scholes Cannot Be Applied Everywhere, but There Is Life in Real Option Theory Outside of Black–Scholes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.8 When Can Risk Neutrality Be Invoked? Portfolio Argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.9 How Does One Recognize When Risk Neutrality Applies? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 Toward a Real Options Theory . . . . . . . . . . . . . . . . . . . . . . . . . 2.8.1 The Basic Structure of a Real Option Valuation . . . . . . . 2.8.2 Can We Speak of a Theory of Real Options? . . . . . . . . . 2.8.3 What Do Theories Do? . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

4

Real Option Analysis: Work in Progress, in Need of Progress . . . . . 3.1 Net Present Value Compared with Real Options . . . . . . . . . . . . 3.2 Classical Real Options Success Stories . . . . . . . . . . . . . . . . . . . 3.2.1 Strategic Investments . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Compound Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 The “License to Kill” Problem . . . . . . . . . . . . . . . . . . . . 3.2.4 Flexibility in the Design of a Project . . . . . . . . . . . . . . . 3.2.5 The Value of Abandoning a Project . . . . . . . . . . . . . . . . 3.2.6 Growth Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 When ROA Raises Serious Legitimate Questions . . . . . . . . . . . . 3.3.1 The Borison Controversy . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 A Call for More Theoretical Precision: The Georgetown Challenge . . . . . . . . . . . . . . . . . . . . . . 3.3.3 A “Solution” of the Problem of Borison that Meets the Georgetown Criteria . . . . . . . . . . . . . . . . . . . . . . . . 3.3.4 The McDonald-Siegel Model . . . . . . . . . . . . . . . . . . . . . 3.4 Are Real Options Speculative Instruments? Hedging Instruments? Decision Tools? Risk Management Tools? Are They Truly Derivatives? Does ROA Somehow Relate to Bond Valuation? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Thales of Miletus: The First Instantiation of Option as Speculative Instrument . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Nature of Real Options . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Extreme Events, Cat Bonds, ROA in the Context of Fat Tail Distributions, and the Weitzman Effect . . . . . . . . . . . . . . . . . . . . . 4.1 Setting the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Catbonds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Extreme Events and Power Law Distributions . . . . . . . .

. . . .

23 25 26

27 28 29 30 30 31 32 33 35 35 36 36 37 37 39 40 41 42 42 45 45 47

48 48 48 49 51 51 51 52

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4.2

56 56 57 61 61 62 67

The Weitzman Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Conditional Probability for Triggering Events . . . . . . . . . 4.2.2 The Weitzman Effect Mathematically . . . . . . . . . . . . . . . 4.3 Catbonds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Catbonds in Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Catbonds in Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

6

7

Global CO2 Emission Mitigation Through the Looking Glass of ROA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 ROA and Climate Change: Setting Up the Problem . . . . . . . . . . 5.2.1 Irreducible Uncertainty on Temperature Increase . . . . . . . 5.2.2 Building the Mathematical Expression of the ROA Value of Mitigation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.3 The ROA Value of a Climate Change Mitigation Policy Is Infinite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.4 Deconstructing the Source of the Infinity . . . . . . . . . . . . 5.2.5 The Weitzman Effect Applies Here on Steroids . . . . . . . . 5.3 Extracting Finite Numbers Out of Infinities . . . . . . . . . . . . . . . . 5.3.1 Probability of Exceedance and the Importance of the Parameter ΔT0 in Eq. 5.1 . . . . . . . . . . . . . . . . . . . 5.3.2 Environmental Value at Risk (EVaR) . . . . . . . . . . . . . . . 5.3.3 Relative Entropy and the Kullback–Leibler Divergence . . 5.4 A Cheap and Easy Way to Improve the Situation? . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

69 69 70 70 74 74 76 77 78 78 80 81 82 83

Internationalization of the Response: The Example of the REDD Credits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Importance of Forests and REDD Credits . . . . . . . . . . . . . . . . 6.1.1 REDD Credits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 What REDD Credits Could Be in Theory . . . . . . . . . . . . . . . . 6.2.1 Modeling the Uncertainties . . . . . . . . . . . . . . . . . . . . . 6.2.2 Framing the Quantification of the Transaction . . . . . . . . 6.3 Complicating Factors: Alternative Use, Ecosystem Services, Biodiversity, etc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Valuing Ecosystem Services . . . . . . . . . . . . . . . . . . . . 6.3.2 Biodiversity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.3 Weisbrodian Perspective . . . . . . . . . . . . . . . . . . . . . . . 6.4 REDD Credits in Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . .

85 85 86 87 87 88

. . . . . .

91 91 92 92 93 93

Prioritizing the Investments Needed to Avoid the Unmanageable (Mitigation) and to Manage the Unavoidable (Adaptation) . . . . . . 7.1 Adaptation and Mitigation Are Different Problems . . . . . . . . . . 7.2 Financing Adaptation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 The Context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2 Where ROA Comes In . . . . . . . . . . . . . . . . . . . . . . . .

. . . . .

95 96 96 96 98

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7.3

Financing Mitigation and the Transition into a Green Economy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 Corporate Culture and Green Investments . . . . . . . . . . . . 7.3.2 Risk Profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.3 Co-benefits of Green Investments . . . . . . . . . . . . . . . . . . 7.3.4 Financing Innovation . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Climate Bonds, Green Bonds, and the Financing of Green Investments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.1 Green Bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

Unanswered Questions About Uncertainty, Information, and Investment Decisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Climate Change Decision-Making . . . . . . . . . . . . . . . . . . . . . . . 8.2 Thinking in Probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 The Intrusion of Subjectivity as a Legitimate Contributor of Collective Decision-Making . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 The Scarcity of Information Conundrum . . . . . . . . . . . . . . . . . . 8.5 Can ROA Make a Difference in Climate Change Policy? . . . . . . 8.6 Ugliness, Thersites, and Climate Change . . . . . . . . . . . . . . . . . . 8.7 Oil Painting and ROA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

104 104 105 105 108 112 113 114 117 117 119 120 122 123 124 127 128

A

Optimizing the Conditions of Investments Under Uncertainty: “Real Option” Can Mean Different Things that Should Not Be Confused . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 A.1 McDonald and Siegel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 A.2 Linkage with the Black–Scholes-Based Approach to ROA . . . . . 131

B

ROA and Climate Change in Practice . . . . . . . . . . . . . . . . . . . . . . . B.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.2 The Basics of ROA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.2.1 Effect of First-Degree Homogeneity . . . . . . . . . . . . . . . B.3 Climate Change Policy Choices . . . . . . . . . . . . . . . . . . . . . . . . B.3.1 Discussion of Report Recommendations . . . . . . . . . . . . B.3.2 Deforestation, Reforestation, Afforestation . . . . . . . . . . . B.3.3 Vulnerability to Extreme Events . . . . . . . . . . . . . . . . . . B.3.4 Promoting Energy Production Less Carbon Intensive . . . B.4 France as Example or Case Study on How High Level Recommendations Speak to Environmental Policy . . . . . . . . . . . B.4.1 France’s Policy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.4.2 Contours of an ROA-Based Optimal Strategy . . . . . . . . . B.5 Transports: Electric Vehicles as case Study . . . . . . . . . . . . . . . . B.5.1 About France and Electric Cars . . . . . . . . . . . . . . . . . . . B.5.2 The ROA Value of Electric Vehicles (EV) . . . . . . . . . . .

133 133 134 135 135 138 139 139 140 141 142 145 146 146 147

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B.6

B.7

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B.5.3 A Few Vital Statistics to Anchor the ROA Estimate . . . B.5.4 ROA of Moving Toward Electric Vehicles in France . . B.5.5 What ROA Has to Say About Electric Cars in France . . Combining Climate Change Adaptation with Disaster Risk Management and Sustainable Development is a Common Recommendation, Because They Are Often Linked. Can One Quantify Those Linkages? . . . . . . . . . . . . . . . . . . . . . . . . . . . B.6.1 ROA and Linked Investments, a Few Random Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.6.2 How to Compute Ratio, Product, Sum, and Difference Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . What ROA Is and Is Not . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. 147 . 148 . 150

. 151 . 152 . 154 . 155

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

Chapter 1

Prolegomena: What Does Real Option Analysis Bring to Climate Change Policy

Abstract The word prolegomena is defined as “a critical or discursive introduction to a book.” This is what it means here but with a twist. The German philosopher Immanuel Kant (1724–1804) wrote Prolegomena zu einer jeden künftigen Metaphysik, die als Wissenschaft wird auftreten können, or Prolegomena to Any Future Metaphysics That Will Be Able to Present Itself as a Science. In that book, Kant in a rather polemic way explained what he wanted to accomplish by writing a previous book, Critique of Pure Reason, which many think is his masterpiece but was poorly received at the time.

1.1

Introducing the Problem

Similarly, the premise of this book is a bit polemic, in that it is critical of the evolution of real option analysis (ROA). It begins with the rather banal observation that climate change policy suffers due to the inadequacy of existing policy tools. Whereas the science of climate change is quite advanced and still progressing, the related economics and policy lag far behind. This is illustrated in the so-called AR5 IPCC report. Every few years, the Intergovernmental Panel on Climate Change (IPCC) produces an Assessment Report (AR). AR5—which came out in 2013–2014—is the fifth, and for the time being (2017) the most recent, such report. Adaptation and mitigation are the two components of the response to climate change. Although they are conceptually completely different, they tend to be treated together. They differ in the sense that mitigation is a collective effort of all nations to control the atmospheric content of green house gases (GHG), whereas adaptation refers to the steps each country takes individually to mitigate the effects of climate change. Each nation has an interest in what other nations do in mitigation as it potentially affects them. Mitigation is an instantiation of the problem of the management of the commons, along with the lines of the late Nobel laureate in economy, Elinor Ostrom that is, it is based on the interaction of many actors with ecosystems. Here, it is through multilateral international agreements. Adaptation is ultimately a national problem, and in most cases what individual nations do does not affect the other nations. The fact that there is an overlap between © Springer Nature Switzerland AG 2020 B. Morel, Real Option Analysis and Climate Change, Springer Climate, https://doi.org/10.1007/978-3-030-12061-0_1

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1 Prolegomena: What Does Real Option Analysis Bring to Climate Change Policy

adaptation and mitigation, in the sense that some mitigation measures can contribute to adaptation or vice versa, does not change the fundamental difference between the two. The IPCC report treats adaptation and mitigation separately. Adaptation is treated in the WGII report, whereas mitigation is the subject of the WGIII report. The IPCC reports are huge endeavors, involving thousands of the best experts on all aspects of climate change. The list of authors reads like a Who’s Who in that field. As reports, they are the ultimate authority on the topic of climate change. The reports are divided in several sub-reports produced by different Working Groups (WG). WGI focuses on the “physical science basis” of climate change: WGII focuses on “impact, adaptation and vulnerability”; and WGIII focuses on “mitigation of climate change.” Finally, there is a “synthesis report.” Each sub-report is extremely long, and it is difficult to believe that anybody would go through them. The WGII report on adaptation is close to 1800 pages, while the WGIII report has no less than 1454 pages. They are longer than the Bible, and in one sense better, in that they leave much less room for conflicting interpretations. Furthermore, they have far more authors, and we know their names. Still, treating them as a Bible may not be the best idea. They are still work in progress.

1.2

The IPCC Reports

The reports are basically a review of all contributions to climate change. At times, they go to such great lengths to avoid omissions that the reader gets the sense of going through a laundry list of contributions, connected by very thin thread. As a result, they provide a thorough overview of the present state of the art. However, especially in climate change policy, the “state of the art” is somewhat scattered, and trying to create the impression of greater coherence would be a distortion. The executive summary of the WGIII report reads: “Climate change mitigation can be framed as a risk management exercise. It may provide large opportunities to humankind, but will also be associated with risks and uncertainties. [. . .] The intent of the report is to facilitate an integrated and inclusive deliberation of alternative climate policy goals and the different possible means to achieve them (e. g., technologies, policies, institutional settings). It does so through informing the policymakers and general public about the practical implications of alternative policy options, i.e., their associated costs and benefits, risks and trade-offs.” In practice, the report’s findings or conclusions do not easily translate into policy recommendations, and a lot is left to the policymakers. If the goal is to provide policymakers with everything they need to make informed, wise, or optimal climate change decisions, the reports fall seriously short. The few policymakers who have the attention span to go through the reports are left in a jungle of unsolved issues, the most prominent of which are the uncertainties. Mitigation policy has a long-term dimension (the benefit will be felt in a distant future) and a short-term dimension (the cost starts as soon as the policy starts). Both are mired in uncertainty. And a discount term, whose value is notoriously

1.2 The IPCC Reports

3

controversial, has to enter. Its effect is to make the benefits that belong to the longterm future smaller without affecting the cost, which have to be born in the short term.

1.2.1

Dealing with the Uncertainty and the Limitations of Existing Mathematical Models

Furthermore, uncertainty is not static. It changes with experience and additional knowledge: knowledge about the success of the mitigation policy as measured by the evolution of the atmospheric load of CO2; knowledge of how fast the transition to a green economy is going; and knowledge of how the environment responds to climate change. Mitigation policy is a path-dependent process finding its way in a thick fog of uncertainties. Whether in the parlance of Manne and Richels (1991), it is best framed as “learn and act” or “act and learn” or a combination of both, one of the best ways to get some insight into the possible futures is based on scenarios. This technique is used to predict the trajectory of hurricanes. Many scenarios are run, and the results give the best guess on the distribution of probability as to where the hurricane is heading. In the case of hurricanes, the laws of physics involved are well known. Still, the dynamics of airflows is so complicated that the best computer programs reach their limits and the simulation has to be updated as soon as more is known. When it comes to predicting the future of the world in the context of mitigation, the knowledge base is much smaller, but the number of variables is not. The tools used to base the scenarios, so-called integrated assessment models (IAM), tend to be models like the Manne and Richels Global 2100 (1991) or the Nordhaus DICE (Nordhaus 1993), among others. Those models couple the economy to the climate change policy. They model the economy with the Ramsey macroeconomic model, which assumes an economy made of capital and labor. Capital and labor enter in the production function (proxy for the GDP). The fundamental assumption of that model is that there is a natural mechanism that maximizes an expected utility build from the consumption. This model plays a central role in macroeconomics. Manne and Richels extended this model to make investment in diverse forms of energy more visible. This has become a model for an optimized economic management of energy policy. DICE and the models inspired by it introduce an environmental damage function, which negatively affects the GDP. From a mathematical point of view, the Ramsey model is an optimal control problem where the consumption is the control variable and the capital is the dynamic variable. Mitigation adds a new control variable: the level of mitigation. The level of mitigation comes at a cost of the economy. The problem is determining the optimal cost level. This construct builds a model for an optimal mitigation policy. IAM have known limitations: one is that the result depends strongly on the choice of environmental damage function. Nobody in their right mind knows what a

4

1 Prolegomena: What Does Real Option Analysis Bring to Climate Change Policy

damage function should be if the temperature increases by more than 2  C or 3  C, except that the damage is probably severe. Optimal control problems by nature need to allow the variables take those kinds of values. As a result, those models give predictions even for situations where the increase of global temperature exceeds 10  C, a world we cannot begin to imagine. In a nutshell, their damage function stinks when it matters. A second well-known limitation is that being economic models, IAM have a discount rate. Considering that the time period is far in the future and the potential damage becomes serious only on the distant horizon, its impact on the result is small as it is discounted over many years. One result is the embarrassingly small cost of mitigation predicted by those models. More fundamentally, the impression is that what really matters—the long-term environmental damage—plays a very small role. This has significant implications for mitigation policy and, for example, the value of hedging or “buying greenhouse insurance” (Manne and Richels 1991). In the words of Lorenz et al. (2012), “Common integrated assessment models (IAM) of climate change suggest uncertainty has little effect because the marginal risk premium in these models is small.” A third, less-discussed limitation is how it deals with the possibility of an environmental catastrophe. The fat tail behind this possibility originates in the relation between the atmospheric load of GHG and its translation into an increase of temperature (cf Chap. 5). In principle, this could be approached in those models. The atmospheric load of GHG is a dynamic variable. The damage function is expressed as a function of the increase in temperature associated with the atmospheric load of GHG. In principle, it should be possible to translate within those models any value of the atmospheric load of GHG into a probability distribution function (PDF) with fat tail for the increase of temperature. But this is easier said than done. Those models are based on a dynamic representation of climate change. To operationalize the idea of including fat tails in those IAMs, one would need to be able to generate the fat tail dynamically. This would necessitate an understanding of how a fat tail in the PDF for the temperature increase progressively develops as a result of climate change. One way to do this is for every value of the increase of GHG atmospheric load to systematically sample the PDF of the corresponding ΔT. That distribution has a growing fat tail. But the fat tail is due to low-probability events. The logistic of generating such an effect in the context of IAM seems hopeless and is probably not worth the effort. In other words, IAM are not good instruments to discuss the effect of uncertainty. One thing is certain: uncertainty is the mother of the unsolved problems in climate change. The word uncertainty appears on average once per page in the 1454 pages of the AR5 WGIII report, and it inspires all sorts of comments and developments. Many of them are biodegradable. The most portent comments are those one would reserve for an untamed shrew.

1.2 The IPCC Reports

1.2.2

5

Cost Benefit Analysis

According to the same report, CBA offers a lot of promises, if it were not for the damage caused by the fact that uncertainty is so out of control. When explaining the limitations of CBA, the report invariably mentions the problem with low-probability catastrophic events, i.e., fat tail distributions of effects. CBA in its present form does not handle those situations well. In fact, CBA in its present stage has the same problem as Net Present Value (NPV). It is blind to the details of the “shape” of the uncertainty. When one sets a limit of 2  C or 1.5  C for the “tolerable” increase of global average temperature, it is implicit (but not stated with that level of precision) that the atmospheric concentration of GHG at its peak should yield a PDF for the increase of average temperature whose average is 2  C or 1.5  C. There is a large uncertainty as to how a given atmospheric load of GHG translates into a global average for the surface temperature. With a low but non-zero probability, it could be significantly above the average. In fact, that probability cannot be made even negligible. When this uncertainty is expressed as a PDF, that PDF has a fat tail (Chap. 5). All the AR5 IPCC WGIII has to offer to policymakers is, “Climate policy may be informed by a consideration of a diverse array of risks and uncertainties, some of which are difficult to measure, notably events that are of low probability but which would have a significant impact if they occur.” It is up to the policymakers to transform this pumpkin of a statement into a royal coach appropriate for the beautiful princess that their policy initiatives are expected to be. The authors of the IPCC reports are the best and the brightest in the field. If the reports fall short in important aspects of climate change policy, it is because the state of the art is not where one would like it to be, something difficult to cover in a report like that.

1.2.3

Limitations of Net Present Value

When it comes to have a better grasp of the effect of the uncertainty, NPV-based approaches are inferior because they are not sensitive to the “shape” of the PDF, expressing the uncertainty only to the average value. This is where real option analysis has the potential to make a difference. Real options are the expected benefit from an investment under uncertainty. Also, the cost can be known only with uncertainty. As shown in this book, the uncertainty can be any uncertainty, as long as it can be expressed as a PDF. Not all uncertainties do but most can. If real options were playing a much more central role in climate change policy, the translation of uncertainties into PDF would be far more advanced. Today there is no such need as there is no receiving structure for such PDFs, because neither NPV nor CBA or IAM qualify. Why then is real options not used more in the context of climate change policy? The answer offered here is that the field of real options evolved in such a manner that

6

1 Prolegomena: What Does Real Option Analysis Bring to Climate Change Policy

it lost its way and its early promises did not materialize. The point of this book is that by revisiting the fundamentals of real options, one can recover or discover its potentials, which have not been exploited.

1.2.4

Prospects Offered by Real Options

Real options in a sense are closely related to financial options. But too much has been made of this apparent proximity. They also have differences, which are important here. Financial options are in fact only a subset of a larger world, the world of real options. Financial options are a special case of real options where the uncertainty can be modeled by a stochastic process like a geometric Brownian motion. Concepts like risk neutrality apply because financial options can be traded. They are speculative instruments. In general, real options cannot be traded. They represent the value (estimated benefit) of an investment. Financial options can be purchased. But in most cases, real options cannot be purchased. Their value is indicative. Risk neutrality applies in rare cases in real options. For example, it applies when measuring the value of a corporate investment, where the investment is part of a strategic corporate plan intended to generate a predetermined return. This rarely occurs in the context of environmental policy. In environmental policy, the uncertainty rarely if ever proceeds from a stochastic process. Uncertainty can have all types of origins. As long as it can be expressed as a PDF, real option analysis is possible. But it proceeds somewhat differently from the way people approach real option valuation. Compared with what has been the tradition in real options so far, much more mathematical rigor and precision is needed to make real option analysis work as a stand-alone paradigm. For example, risk neutrality applies sometimes, but not always. Other mathematical considerations—which were ignored or treated as technicalities—are in fact very important, for example, “first-degree homogeneity.” No practitioner of real option analysis seems to have realized that first-degree homogeneity does not always apply and this has consequences for the valuation of the option. Am I suggesting that the field of real option progressively sank into some kind of intellectual morass? I will let the reader decide. The goal of this book is to show that real option analysis can be framed in such a way that it speaks to all aspects of climate change policy.

1.3

Risk and Uncertainty

Have no respect whatsoever for authority: forget who said it and instead look at what it starts with, where it ends up and ask yourself: is it reasonable? Richard Feynman

References

7

Economists seem to agree that a distinction should be made between risk and uncertainty (Knight 1921). It is not obvious that they agree on how to define the difference. F. Knight suggests that “we speak of the ‘risk’ of a loss, [and of] the ‘uncertainty’ of a gain”. Still, there is a consensus that for an investor to take a risk, he must have something to lose. In the context of climate change investment, when the risk is environmental damage, the situation differs from commercial investments. Can we say that the uncertainty is in the benefit of mitigation or adaptation measures, and the risk is measured in environmental damage? Many investments pertaining to the response to climate change share the same risk and uncertainty characteristics as commercial investments, such as investments in electric cars, solar power, or carbon capture and storage. But when the risk is environmental damage, there is potentially a lot to lose by not making those investments. In other words, environmental policy involves risk of a completely different nature from the risk economists have in mind. Uncertainty plays a central role in climate change policy. Uncertainty is based on the fact that we do not have a perfect view of the future. This is true in all contexts. But the factors behind our inability to predict the future and the implications of this are significantly different for economic investments and climate change. In the case of climate change, the implications of uncertainty can be ominous. Uncertainty means that the outcome at best can only be predicted probabilistically, by a PDF. If one repeated the same investment many times, one would reproduce the PDF representing the uncertainty. As long as the uncertainty is not very large, the outcome can be approximated by its expected value in the spirit of NPV. But when the uncertainty becomes large, because, for example, the PDF of outcomes has a fat tail, the situation is qualitatively different. It is a case where quantity affects quality, a situation popularized by the German philosopher Georg Hegel (1812). When the uncertainty is very large, what works for lower levels of uncertainty no longer applies (Courtney et al. 1997). The degree of uncertainty encountered in environmental policy is rarely present in commercial investments. Furthermore, in corporate investors typically avoid situations with high levels of uncertainty. This option does not exist in climate change. It is imperative to confront a situation mired in a high level of uncertainty and develop a way to make this uncertainty speak to the policy choices. This begins with quantifying the effect of the uncertainty, if and when possible. This is what real option analysis is about.

References Courtney, Hugh, Jane Kirkland Patrick Viguerie, (1997), Strategy under uncertainty, Harvard Business Review November–December 1997 Issue, https://hbr.org/1997/11/strategy-underuncertainty

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1 Prolegomena: What Does Real Option Analysis Bring to Climate Change Policy

Hegel, G.: Wissenschaft der Logik (1812), translated as Science of Logic, http://www.inkwells.org/ index_htm_files/hegel.pdf Knight, F.: Risk, Uncertainty and Profit. Schaffner and Marx, Boston (1921). http://www.econlib. org/library/Knight/knRUP.html Lorenz, A., Kriegler, E., Held, H., Schmidt, M.G.W.: Climate Change Economics. 03(01), (2012) Manne, A.S., Richels, R.G.: Buying Greenhouse Insurance. MIT, Cambridge (1991). See also: Manne A. S., and R. G. Richels (1991). Energy Policy 19, 543–552 Nordhaus, W.D.: Rolling the “DICE”: an optimal transition path for controlling greenhouse gases. Resour. Energy Econ. 15, 27–50 (1993). See also: Nordhaus W. D. (1994). Managing the Global Commons: The Economics of Climate Change. MIT Press, Cambridge, MA

Chapter 2

Toward a General Theory of Real Options

You can’t cross the sea merely by standing and staring at the water — Rabindranath Tagore (https://www.brainyquote.com/ quotes/rabindranath_tagore_383735).

Abstract This chapter reviews history of financial option from its origin with Bachelier and continuing with the contributions of Black–Scholes and Merton. The origin of the concept of real option (S. Myers) is also discussed. The relation (conceptual and mathematical) between financial and real option, as well as the concept of risk neutrality and its relevance for real options, is discussed ad nauseam. In the process, a mathematical framework for real option analysis (ROA) is developed. This chapter is somewhat math-intensive. Of particular importance for the rest of the book are the discussions of first-degree homogeneity and risk neutrality and their mathematical implications. Without the Black–Scholes formula, ROA would probably not exist. It was Black–Scholes who inspired Stewart Myers to introduce the concept of real options in his study of the value of a firm. He emphasized the importance of growth options in the valuation of a firm, and he called those options “real options.” As a result, not only does ROA have its roots in the culture of corporate investments, but it also grew in that cobweb. The downside is that ROA is seen as a mere extension of financial options, when in fact it should be the opposite: financial options being the particularization to the world of finance of a broader concept, real options. When it comes to climate change policy, this distinction is fundamental, because it is what makes ROA applicable there.

© Springer Nature Switzerland AG 2020 B. Morel, Real Option Analysis and Climate Change, Springer Climate, https://doi.org/10.1007/978-3-030-12061-0_2

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2.1

2 Toward a General Theory of Real Options

Introduction

The goal of this book is to show that ROA can have a much broader range of applications than it has today, using climate change policy as example. Those who have tried to apply real option theory to climate change could not fail to notice that it is as if they tried to make a paradigm apply outside its natural field of application. The question becomes: is it possible to apply ROA to climate change policy? And if yes, to what kind of restricted use? This book aims to prove that there is no hard limit for applying ROA, hence the idea of building a “general theory of real options.” J.M. Keynes (1936) justifies the title of his famous book, The General Theory of Employment, Interest and Money, by placing the emphasis on the prefix “General.” In his words: “The object of such a title is to contrast the character of my arguments and conclusions with those of the classical theory of the subject, upon which I was brought up and which dominates the economic thought, both practical and theoretical, of the governing and academic classes of this generation, as it has for a hundred years past. I shall argue that the postulates of the classical theory are applicable to a special case only and not to the general case, the situation which it assumes being a limiting point of the possible positions of equilibrium. Moreover, the characteristics of the special case assumed by the classical theory happen not to be those of the economic society in which we actually live, with the result that its teaching is misleading and disastrous if we attempt to apply it to the facts of experience.” This is exactly, but in a much more modest way, what this book contends when it comes to real options. . . Does the field of real options need a “general theory”? What could that add? When you look at that question from the perspective of how to generalize the scope of application of ROA, the answer becomes obvious. The need is obvious, but the question is how to do it.

2.1.1

The Impact of Black–Scholes on Real Options

After the 1973 seminal papers of Black, Scholes and Merton, financial option theory embarked on a spectacular trajectory. It was like a testimony to human ingenuity, a revolution in the way humans approach finance. But when used as a template to develop ROA as an extension or spin-off of financial options outside the confines of finance, the majesty of Black–Scholes turned out to be seriously stifling. It is not the way the “practitioners” of ROA would put it. Instead they found solace in finding any possible excuse to stay as close as possible to Black–Scholes. That meant using the formula as often as possible (without necessarily a lot of justifications), assuming risk neutrality whether or not it could be justified, and assuming that any form of algorithm used in the context of investment under uncertainty is an instantiation of “real option.” In the latter case, optimal timing of an investment is considered an instantiation of ROA even if (as shown in the Appendix of this book) this is a completely different problem (conceptually, mathematically, and otherwise). Over

2.1 Introduction

11

time, ROA developed different ways to estimate the real option value of an investment without clear conceptual reasons to justify that these apparently completely different algorithms were equivalent ways to estimate this value. In fact they were not and were yielding different results. This no doubt contributed to weakening the appeal of an approach to investment under uncertainty. Instead of being “not wrong but nearly right” (as Lowenstein said about Merton and Scholes in his book on the fiasco of Long-Term Management Fund in 1997), the proponents of ROA did a good job of projecting the impression that they had hardly any clue as to the actual value of an investment under uncertainty. It progressively became an intellectual swamp. Needless to precise that the world of real options did not replicate the splendor of Black–Scholes, otherwise this book would have no raison d’être. The fact that ROA slowly dug its own hole does not mean that the achievements of real options are nonexistent and their whole history has been a complete failure. But compared with what it could or should have been, ROA is an “ugly duckling.”

2.1.2

Black–Scholes as a Black Swan

Nassim Taleb made “Black Swan” a buzzword (Taleb 2007). He asserts: “What we call here a Black Swan (and capitalize it) is an event with the following three attributes. First, it is an outlier, as it lies outside the realm of regular expectations, because nothing in the past can convincingly point to its possibility. Second, it carries an extreme ‘impact.’ Third, in spite of its outlier status, human nature makes us concoct explanations for its occurrence after the fact, making it explainable and predictable.” The Black–Scholes formula meets all three criteria of Nassim Taleb, so it qualifies as a Black Swan. This is an observation that may not be music to the ears of Nassim Taleb, as he forcefully denounced that formula, among other things, as being responsible for the 2008 financial crisis. Black–Scholes has been a milestone in the history of financial options and a nemesis for real options. A lot of the flaws and problems ROA has run into have to do with its inability to contain the impact of Black–Scholes by applying its formula and conceptual framework where it does not belong. Instead, it should have taken a much cooler look at how Black–Scholes is derived and based ROA on the same level of mathematical precision and rigor. This is the subject of the next chapter and the rest of this book. Black in Black–Scholes refers not to a color but to Fischer Black, a distinguished economist. Black in “Black Swan” has a more colorful origin. The expression “Black Swan” goes back centuries to a poem—which by today’s standards would be deemed politically incorrect—by the Latin humorist Juvenal who lived in the first century AD. In a poem entitled Why marry?, he compared a perfect wife to “a rare bird on this earth, as rare as is a black swan.”1 The next verse reads: “Who could

1

This is the translation by Creekmore (1963): “rara avis in terris nigroque simillima cygno.”

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2 Toward a General Theory of Real Options

endure a wife who has all the virtues known?” Because of the rarity of Black Swans, very few have been given the opportunity to make the control experiment.

2.2

What Are “Options”?

The conventional wisdom is that: “Options at a cost give a right (as opposed to an obligation) to do something. In the context of financial options, the right is to buy or sell a good (like a share) at a certain price (exercise price) sometime in the future (exercise time).” This characterization of options applies well to financial options, but when it comes to real options in general, it is more often wrong than true. One is never careful enough with conventional wisdom. “Conventional wisdom” is an oxymoron because by nature wisdom is not conventional. For good reasons, the economist John Kenneth Galbraith used that expression in a derisive way. A much better characterization of an option would be the value of an option measures the expected benefit that the purchaser of the option can reap by owning it. In some contexts such as financial options, this is the value of the right to exercise the option. But in the context of an investment under uncertainty, it is the potential benefit of making the investment. In the case of financial option, you exercise the right of exercising it only if it benefits you. In the case of an investment, the value of the option is a virtual number: it gives an estimate of the expected benefit of the investment. The option in that case is never purchased, but the investment makes sense only if the expected benefit offsets its cost. The real option in that case is a decision variable, not an instrument of speculation as is the case with financial options.

2.2.1

Value of Options

The conventional wisdom: options are “derivatives,” i.e., their value is determined by how an underlying asset evolves with time. Typically the underlying asset (a share) follows a stochastic process. In the seminal case of Black–Scholes, that process is a geometric Brownian motion. Real options need not be derivatives and in general are not derivatives. Their value reflects an uncertainty, which in general does not derive from a stochastic process. The concept of options as derivatives is a special case, which occurs mostly in the narrow confines of finance. If the value of an investment depends on the future value of oil or gas or palm oil or sugar cane, i.e., on the value of a commodity whose price tends to follow a stochastic process, the real option value of that investment is a “derivative.” But in general, the uncertainty associated with an investment has an origin, which is not translatable into a stochastic process. For example, the uncertainty regarding the increase of surface temperature associated with a given atmospheric load of GHG has to do with the complexity of physics and cannot be expressed as a stochastic process.

2.3 The Role of the “Corporate Culture” in the Genesis of Real Option Analysis

2.2.2

13

Risk-Neutral Valuation of Options

This is one place where ROA erred badly. Conventional wisdom: what made Black–Scholes and the field of option theory so unique is the observation by Fischer Black, Myron Scholes, and Robert Merton of the relevance of risk neutrality. The concept of risk neutrality has been a milestone in the history of financial option valuation. A popular mistake is to apply it where it does not belong and more generally to use the Black–Scholes formula in situations where the uncertainty does not proceed from a geometric Brownian motion and risk neutrality does not apply. How many practitioners of real options have bothered trying to understand what risk neutrality is, and when it applies? Risk neutrality is discussed ad nauseam in the next chapter and the rest of the book. It reflects a deep insight into the valuation of financial option, which has been attributed by Fischer Black and Myron Scholes to Robert Merton. It has everything to do with the fact that financial options can be traded and are instruments of speculation. It reflects the fact that through trading financial options will provide on average to speculators the market return, referred to as the “riskless rate.” This kind of situation emerges in real options only when the investment needs to have a predetermined return. This is often true for corporate investments, but definitely not the case for environmental investments.

2.3

The Role of the “Corporate Culture” in the Genesis of Real Option Analysis

Without Black–Scholes has been a milestone in the history of finance. And without it, real option analysis would probably not exist. Black–Scholes originated within the corporate culture. One of the starting points of Black and Scholes (Black and Scholes 1973) in their seminal paper, “The Pricing of Options and Corporate Liabilities,” was the observation that “corporate liabilities (i.e., warrants, common stock, corporate bonds, and debt) can be seen as a combination of options.” The reference to corporate liabilities clearly guided their intuition when they were deriving their famous formula, which is the foundation of what is today financial option theory. The “Black–Scholes” formula is used mostly to value purely financial instruments like European or American call options or European put options. Real options also grew within the same corporate culture. This may be why it went astray. As an intellectual construct, Black–Scholes was so impressive that it inspired Stewart Myers to explore the possibility of applying it outside of the narrow confines of speculative finances. It was in a paper entitled “Determinants of Corporate Borrowings”(Myers 1977) that Stewart Myers coined the expression “real options.” Central to the paper was the concept of “value of the firm,” which still is a subject of major research. The insight of Stewart Myers was that “the value of a firm as a going concern depends on its future investment strategy.” In other words, the firm is

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influenced by growth opportunities, or “growth options,” which he called “real options.” The paper of Stewart Myers appeared in 1977, 4 years after the Black– Scholes paper. They were all—with Robert Merton—at MIT at the time. The field of real options grew on that seminal contribution of Myers. Considering the context in which the concept was introduced, it is not surprising that corporate investments were the natural realm of application of ROA. What ROA developers failed to appreciate was that when extending further the realm of application of ROA, it was essential to revisit the fundamentals. The Black–Scholes formula can be applied when a few conditions are met, like the uncertainty proceeds from a geometric Brownian motion, risk neutrality, and first-degree homogeneity. None of those conditions apply universally. Many are aware of the concept of risk neutrality, even if they apply it more often than they should. But few seem to appreciate the importance of the assumption of first-degree homogeneity and those who do hide it very well.

2.4

Financial Options as a Particular Case of Real Options

Corporate investments were the first application of real options, and to a large extent, ROA has developed around that kind of application. It is also the area where ROA is closest to financial option theory. Real options have been treated as extensions of options outside of finances. But most problems where ROA should apply do not fit well in the intellectual construct associated with Black–Scholes. In its first incarnation, the concept of real options was an extension of financial options, but not a trivial extension. The value of the real options or growth options was the expected benefits a company could make through discretionary investments. Progressively the realm of application of ROA became broader to basically encompass any investment under uncertainty. A better approach would be to treat financial options as a special case of real options. Both are expected benefits, but financial options correspond to the case where the uncertainty proceeds from some stochastic process (geometric Brownian motion in the case of Black–Scholes). In real options, the uncertainty can have all sorts of origins and take a number of forms. Even if real options should and can be made a stand-alone paradigm (as shown in the next chapter)—and even if financial options are best seen as mere special cases of real options—it remains that financial option theory is much more developed and polished than real option theory. In developing the larger framework of ROA, one keeps having to wrestle with the ramifications of the sophistication of financial option theory. This permeates the entire field of ROA, all the way to areas sitting as far apart from corporate investments as mitigation policy.

2.6 Foundations of Option Theory

2.5

15

Prospects of ROA in Environmental Policy

The science of climate change still has some room for improvement, but it is significantly more advanced than its economics. Thanks to the reports produced by authoritative sources like the Intergovernmental Panel on Climate Change (IPCC), the world is becoming increasingly aware of the multiplicity and complexity of the impact of climate change. The response calls for all kinds of investments, private and public and small and large, in advanced economies as well as in nations with a development deficit and an adaptation deficit. We are still at the stage where the best reports provide laundry lists of things to do to obviate the negative impacts of climate change, but with little help on how to prioritize investments. “When everything is a priority, nothing is a priority.” Environmental policy is replete with uncertainties, some with fat tail, i.e., low probability but severe consequences, as when Hurricane Katrina hit New Orleans. The existing policy tools—like net present value (NPV)—tend to be based on expected values or expected utilities. The problem with these kinds of tools is that they see only averages. They are not sensitive to the size and shape of the uncertainties, let alone fat tails. That makes them poor instruments for climate change policy. ROA is much more sensitive than NPV to the details of uncertainty. That has been one of the major reasons for the appeal of ROA in investments under uncertainty. At one point, some mistakenly went as far as predicting that ROA would replace NPV as the main tool for investment decision. This did not happen. NPV and ROA are not exclusive of each other but often complementary to each other. Still, in the context of environmental policy, ROA is clearly superior. To be able to use option valuation in climate change policy, one first needs to be able to cope with uncertainties that do not proceed from a stochastic process. This is possible and the subject of the next chapter. Climate change policy is such a different world from corporate investments that, to make ROA speak to it all, the fundamentals of the theory have to be revisited. The practitioner should be able to recognize the anatomy and physiology of his or her problem when applying that paradigm. Not only should it be user-friendly; it should also be mathematically and conceptually transparent.

2.6 2.6.1

Foundations of Option Theory Bachelier’s “Théorie de la Spéculation”

The PhD thesis (Bachelier 1900) of Louis Bachelier (1870–1946), entitled “Théorie de la Spéculation,” is absolutely seminal for the theory of options. When Bachelier died in 1946, he had not achieved much notoriety. The fact that Bachelier was inspired to apply advanced math to finance at the time was an oddity. The part of Bachelier’s thesis that was best received was his contribution to the mathematics of stochastic processes. In 1900 this field was at a very primitive stage. Bachelier

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completed his thesis before Einstein had developed the theory of the “Brownian motion” in a paper published in 1905 (the same year he published his paper on special relativity). The goal of Einstein’s paper was to “explain” mathematically an experimental observation made in 1827 by the botanist Robert Brown that molecules were moving randomly. Bachelier in his thesis assumed that the prices of shares were undergoing a random walk, i.e., before the name existed, he assumed that shares were following a Brownian process. Today, it is more popular to assume that the prices of shares or commodity are following a geometric Brownian motion. This is equivalent to saying that their rate of change is following a Brownian motion.

2.6.1.1

Bachelier Rediscovered

A bit like what happened to Shakespeare or Mozart, it took decades before people realized the importance of Bachelier’s contribution. Allegedly the mathematical statistician Jimmie Savage accidentally stumbled upon one of Bachelier’s papers in the University of Chicago’s library sometime in the 1950s. He alerted, among others, Paul Samuelson from MIT, who could not find the paper but found Bachelier’s thesis in the MIT library (Bernstein 2005). The rest is history.

2.6.1.2

L’espèrance Mathématique du Spéculateur est Nulle

The seminal assumption of Bachelier’s PhD thesis, which captures the essence of the valuation of options, financial, real, and otherwise, is best expressed in its original French: “l’espèrance mathématique du spéculateur est nulle.” In English, this translates to the “mathematical expectation of a speculator is zero.” In other words, this means that the value of an option is equal to the expected gain or loss (depending on which side of the transaction one is on) of the speculator or investor. This is exactly what the Black–Scholes formula for financial options captures. It is also true for real options: their value is the expected benefit of an investment.

2.6.1.3

Black–Scholes

The Black–Scholes formula (Black and Scholes 1973) is a mega milestone in the history of options. It is used widely and earned the Nobel Prize in 1997 for Myron Scholes and Robert Merton (Fischer Black had passed away by then). The Black– Scholes formula gives the value of European call or put options in the case where the uncertainty on the value of the underlying asset proceeds from a particular stochastic process, a geometric Brownian motion. Bachelier, being the first to model the evolution of stock prices as a stochastic process, assumed that the probability distribution function (PDF) of their future values was a Gaussian distribution. That was in the spirit of the central limit theorem and amounted to assume that the prices follow a Brownian motion. The prices of assets, like shares or commodities, are better modeled by a geometric Brownian

2.6 Foundations of Option Theory

17

motion. As a result, the PDF of the values of shares in the future is a lognormal distribution. This is the distribution used in Black–Scholes. When it comes to real options, the uncertainty in general has a completely different origin, and in most cases, it cannot be described by a lognormal distribution. In that case, Black–Scholes does not apply. But the real option valuation is still possible.

2.6.1.4

Risk Neutrality: A Milestone in the History of Options

The Black–Scholes formula is a continuation of Bachelier’s theory, but with a major addition: the realization of the importance of the concept of risk neutrality in the context of financial options. In their seminal paper (Black and Scholes 1973), Fischer Black and Myron Scholes credit Robert Merton for having made them realize that because a financial option can be traded continuously as long as it is alive, the return it can provide to its owner is the market rate, also referred to as the “riskless rate,” independent of what happens to the value of the underlying asset. To show rigorously how that works, using what he called “Ockham’s razor,” Merton (1973) demonstrated that in a financial world where options are traded continuously and making money with them through arbitrage is not possible, the return on the option is the market rate. This is a very profound insight, which plays a central role in the whole construct of the “financial option theory.” The degree of depth, rigor, and sophistication that exudes from these papers has been inspirational for many of the subsequent developments in option theory. But it has not been replicated in the context of real options.

2.6.2

The Emergence of Real Options

The concept of real options dates back to 1977 (Myers 1977), i.e., it is almost as old as the Black–Scholes formula, which dates back to 1973. Still, ROA is far from being as mature and accomplished as the theory of financial options. After all these years, ROA is still a “work in progress.” Today, ROA gives the impression of being a set of artificial recipes bringing more opacity than transparency to investment decisions involving risks and uncertainty. One goal of this book is to put an end to that situation, and this chapter is at the core of that effort.

2.6.2.1

What Does the Value of an Option Measure: The Price of a Right or Something Else?

The seminal assumption of Louis Bachelier is that the value of a financial option is the expected gain of the purchaser of the option. In the same way, the value of a real option is the expected gain from an investment. But this is basically all that financial options and real options have in common.

18

2 Toward a General Theory of Real Options

A “European financial call option” gives its owner the right to purchase an asset at a certain price at a certain date. Contrary to what some think, real options do not in general operate that way. Stewart Myers coined the word “real option” to refer to the value of corporate investments within firms. Today, the word is used more broadly. The real option value of an investment measures the expected benefit, sometime in the future, of making the investment today. Making the investment is not paying for a right, but the value of the option helps one decide whether making the investment under uncertainty makes economic sense. In other words, in most cases a real option does not provide a right: it supports an investment decision. Rarely is the value of a real option actually paid. It is an abstract number, merely a reference to compare with the actual cost of an investment and help make investment decisions. There are circumstances where the value of a real option is related to purchasing a right. But the value of the real option is in general different from the cost of the right. This is illustrated with the following example: an investor is given the choice between making an investment decision immediately and delaying it for a certain amount of time, at a cost. Real options enter into this situation as they measure the expected benefit of waiting. It is only if this expected benefit is larger than the sum asked for that waiting makes economic sense. Paying the sum provides the investor with a right (to delay his investment decision) without the obligation to invest. The sum paid has the attribute of a financial option. But the real option value is a separate number. It is a reference number, not something to be paid but to be compared with the sum to pay. In the case of environmental decisions, ROA is used to decide whether an investment makes economic sense or which investment (if there is a choice) provides the larger expected benefit. ROA can be very useful in prioritizing investments. This is very different from an instrument, which provides a right without the obligation to do something.

2.6.2.2

Real Options as “Derivatives”

Options are also called “derivatives” as they “derive” from the dynamic of change of the value of underlying assets. Derivatives in general have to be treated with caution, as mistakes can have dire consequences. And mistakes are not so easy to avoid, as shown by the fiasco of Long-Term Capital Management. Formulas like Black– Scholes are not so easy to use in practice. Still, derivatives attract. According to global research (Chapman 2012), the amount invested in “Global Derivatives [is] at 1,200 Trillion US Dollars, about 20 Times the World Economy.” The fact that they are complex instruments that should be used with caution makes them similar to fire. One can easily get burnt, as happened with Long-Term Capital Management. On the other hand, like fire, used astutely, they can yield great results. The art is to get the best of them without being burnt. There are things one can do with them that no other financial instrument comes close to offering.

2.6 Foundations of Option Theory

19

Real options are not “derivatives” in general. Their value does not necessarily derive from the price of an asset. For example, the value of an investment in mitigation is measured by the environmental damage saved. The real option measuring the value of such investment is not a “derivative.” Other investments can be derivatives, if their value depends on the price of a commodity like oil or gas, for example. On a case-by-case basis, whether a real option has to be treated as a derivative or not makes a difference, conceptually, but also in the way it is used. The value of a real option is an expected benefit. Often this value is only indicative. The cost of the investment follows a different logic. The value of the real option is not the amount the investor should pay (unlike with financial options). The value of the real option has to be compared with the cost of the investment. If it is larger, the expected benefit offsets the cost, and the investment makes economic sense. The value of the real option is in fact an abstract number to help an investment decision. But there are investment situations—for example, in an oil or gas field—where it can make sense to delay the investment, at a cost. That price is a real option. Its value is a derivative. ROA in that case enters differently in the strategic or decision process from the way it does to mitigation investment. In other words, ROA is not a one-size-fits-all concept. Its use and implication is very much context dependent. But whatever the context, the value of the real option is always an expected benefit. The math of real option always boils down to estimate an expected benefit, i.e., it looks very similar for all real options. This is the subject of the rest of this chapter.

2.6.2.3

Real Options as Hedging Instruments

Options can be used to value of a hedge against risks. For example, car insurances can be construed as “American put options.” The premium paid by the insured can be interpreted as the value of a put option that the owner of the car exercises if and when his car has an accident, as long as this is before the expiration of the put option. When car insurances compute the value of the premium they charge, they implicitly make some kind of ROA. They have a good idea, based on experience and lots of data, of their risks and their associated costs. They therefore have some idea of the expected sums they will have to cover, and from there they compute their premiums to transform that into an expected profit. For the owner of the insurance, the premium he pays is the value of a hedge against the danger of losing the value of the car abruptly. Although few make the computation, this is typically a real option valuation, which is the reverse of the one the insurer does. The ultimate value of the premium is not strictly the real option value of the deal, but a bit more to let the insurer (the risk taker) make a profit. The value of the hedge in a sense is the extra cost paid to the risk taker. This interpretation of real options as hedging instruments can be applied widely, including in climate change policy. For example, the cost to the economy of a carbon tax can be treated as a premium of a hedge against some future environmental damage (Yohe et al. 2004).

20

2.6.2.4

2 Toward a General Theory of Real Options

Risk Neutrality and Real Options

Financial options can be traded: they are speculative instruments. In a speculative world where arbitrage is not an option, the revenue one can extract by trading speculative instruments is the same for all instruments. It is the market rate. In other words, the revenue an owner of an option can hope to get by trading it is what the market for speculative instruments offers: the market rate, the average return shared by all participants to that market. This is the basis of “risk neutrality.” The key observation is that this rate of return is independent of what happens to the value of the asset underlying the financial option. In other words, risk neutrality applies to financial options because they can be traded continuously and are speculative instruments. Real options are in general non-tradable, i.e., not speculative instruments. So the insight of Merton, which underlies “risk neutrality,” does not apply straightforwardly to real options. Still, the concept of risk neutrality extends to a certain but limited extent to real options. Depending on the context, risk neutrality can apply to corporate investments. More precisely, that could be the case in situations where corporate investments are part of a general strategic planning of a company and the investments have to fit into the planning for yearly returns decided by the company. In its calculation of the real option value of some corporate investments, a company can require that the investment must have a predetermined return, which serves as a basis for its real option valuation. That predetermined rate of return plays a similar role as the market rate in the risk-neutral valuation of financial options. In that case, the valuation is “risk neutral,” but “risk neutrality” means that a predetermined rate of return is used in the valuation. That rate of return is in general not market-based. ROA can also be used to estimate the value of investments where risk neutrality simply does not belong. For example, risk neutrality does not have a place in the valuation of the expected benefits of environmental investments like reductions of CO2 emissions. Still, a valuation of the real option is possible and a useful tool to “value” the investments and prioritize them.

2.6.2.5

How Risk Neutrality Enters in the Valuation of Options

Risk neutrality, when it applies, has subtle implications in the valuation of options. Risk neutrality means that in the computation of the “expected gain,” one has to use “risk-neutral” probabilities instead of the “physical probabilities” of the value of the underlying asset. In the case of Black–Scholes, the risk-neutral probabilities are fictitious probabilities, which reflect the fact that as long as the option is alive, the changes of its value are determined by market conditions, not by the changes of value of the underlying asset.

2.7 The Mathematical Foundations of ROA

2.7

21

The Mathematical Foundations of ROA

It is impossible to go very far in the field of options without using some math. The “Mathematical Giant” Israel Moiseevich Gelfand (1913–2009) has been quoted as saying (Frenkel 2013): “People think they don’t understand math, but it’s all about how you explain it to them. If you ask a drunkard what number is larger, 2/ 3 or 3/ 5, he won’t be able to tell you. But if you rephrase the question: what is better, 2 bottles of vodka for 3 people or 3 bottles of vodka for 5 people, he will tell you right away: 2 bottles for 3 people, of course.”

2.7.1

Mathematical Framework

Starting with financial option The value H(X, Y, T ) of a financial call option on a share X is the “discounted expected gain” made at the exercise time T, by exercising the option of purchasing the share at the exercise price Y. If the price of the share X is less the exercise price Y, the option is not exercised. Translated in mathematical terms, the value of the option is: H ðX, Y, T Þ ¼ e

ρ T

Z1 ðu  Y Þ Ψðu, X, Y, T Þ du

ð2:1Þ

Y

eρ T is a discount factor. Ψ(u, X, Y, T ) is a probability distribution for the values of X at time T. If the price of the share at the exercise time is less than the exercise price Y, the value of the option is zero. Otherwise, it is the expected benefit of buying the share at the exercise price Y, which is inferior to the actual price of the share at that moment. This is a direct instantiation of Bachelier’s assumption that the “mathematical expectation” of a speculator is zero: the speculator pays exactly what his expected gain is, with the risk of losing his original investment, but with the equal prospect to benefit from that investment. In the case of Black–Scholes, the probability distribution Ψ(u, X, Y, T ) is a “riskneutral” lognormal distribution. We discuss that later.

2.7.2

The Value of Exchanging Two Risky Assets

In the example of Black–Scholes, only one asset is “risky”: the value of the share X. As noticed by William Margrabe (Margrabe 1978), this kind of valuation can be used to measure the expected gain of exchanging, in the future, two assets whose values are known with uncertainty. The rationale behind computing the value of

22

2 Toward a General Theory of Real Options

exchanging two risky assets is that the same math can be used to estimate the ROA value of an investment whose benefit and cost are both known with uncertainty. Let X1 and X2 denote two “risky assets.” The expression H ðX 1 , X 2 , Y, T Þ ¼ e

ρ T

Z1 ðu  Y Þ Ψðu, X 1 , X 2 , T Þ du

ð2:2Þ

Y

can be interpreted as the discounted expected benefit of exchanging the two assets at time T. In that expression, Ψ(u, X1, X2, T ) describes the distribution comparing the values of X1 and X2. It can be either based on the difference or the ratio between the values of X1 and X2. Whether one should use the difference or the ratio makes a difference. This is where first-degree homogeneity comes in. If it applies, the ratio should be used, whereas if it does not apply, the difference should be used. We explain why later. In the case where Ψ(u, X1, X2, T) is the difference distribution, to compute the expected value of exchanging two assets, the expected “gain” of exchanging two assets is the expression: H ðX 1 , X 2 , T Þ ¼ e

ρ T

Z1 u Ψðu, X 1 , X 2 , T Þ du

ð2:3Þ

0

It is the cumulative probability that the value of the difference between the two variables is positive (u ¼ X1  X2  0). The result of Margrabe corresponds to the case where Ψ(u, X1, X2, T ) is the ratio distribution (in that case first-degree homogeneity applies). So to compute the expected value of a variable being larger than the other (here that means u ¼ XX 12  1), the expected gain of exchanging the two assets is: H ðX 1 , X 2 , T Þ ¼ e

ρ T

Z1 ðu  1Þ Ψðu, X 1 , X 2 , T Þ du

X2

ð2:4Þ

1

2.7.3

First-Degree Homogeneity

“First-degree homogeneity” is never discussed in the context of ROA and tends to be taken for granted. But first degree does not apply to all situations, and whether it applies or not makes a big difference in the valuation of the option. Applied to financial options, it means that the value of a call or put for k shares is k times the value for one share everything else being the same. Mathematically that means: H ðλX, λY, T Þ ¼ λH ðX, Y, T Þ

ð2:5Þ

2.7 The Mathematical Foundations of ROA

23

First-degree homogeneity is not a universal property for real options. An obvious example where first-degree homogeneity does not apply is the mitigation of GHG emissions. The value of the option in that case quantifies the benefit of mitigation. Doubling the cost of the policy will not reduce the level of emission by a factor two or divide the environmental damage by two. Whether or not first-degree homogeneity applies determines whether the probability distribution Ψ(u, X, Y, T), in the expression of options, is a ratio or a difference distribution. When first-degree homogeneity applies, the ratio distribution should be used. First-degree homogeneity means that the condition Eq. (2.5) is satisfied. One can see immediately that the expression for the option Eq. (2.4) satisfies that condition because the ratio distribution Ψ(u, X, Y, T) does not change when both variables X and Y are multiplied by the same factor, so the value of the option is multiplied by that factor as required by Eq. (2.5). But if first-degree homogeneity does not apply, the difference distribution should be used. As can be seen readily, Eq. (2.3) does not satisfy the condition Eq. (2.5).

2.7.4

Sum and Ratio Distributions

Since they play an important role in ROA valuation, we remind the reader how the difference and ratio distributions can be inferred from the knowledge of the original distributions for the variables. The ratio distribution of two distributions ψ 1(X1) and ψ 2(X2) is: Z1 Ψratio ðu, t Þ ¼

jX 2 jψ 1 ðuX 2 , t Þ ψ 2 ðX 2 , t Þ dX 2

ð2:6Þ

1

whereas the difference distribution is: Z1 Ψdiff ðu, t Þ ¼

ψ 1 ðu þ X 2 , t Þ ψ 2 ðX 2 , t ÞdX 2

ð2:7Þ

1

In Box 2.1, difference and ratio distributions are compared. The original distributions used are lognormal distributions, chosen because they appear often in financial options and also because the variables are necessarily positive. So their ratio takes only positive value, whereas their difference can take positive and negative values. That makes the comparison between the two situations more striking.

24

2 Toward a General Theory of Real Options

Box 2.1 Difference and Ratio Distributions Compared For illustration: Combining two Lognormal Distributions: with parameters μ ¼ 0.5, σ ¼ 1 and μ ¼ 1.5, σ ¼ 1, respectively:

0.4

0.3

0.2

0.1

5

10

15

20

5

10

Fig. 2.1 Original distributions

0.10

0.08

0.06

0.04

0.02

–10

–5

Fig. 2.2 Difference distribution

2.7 The Mathematical Foundations of ROA

25

0.25 0.20

0.15

0.10

0.05

5

10

15

20

Fig. 2.3 Ratio distribution

2.7.5

Risk Neutrality Mathematically and Its Effect on the Valuation of Options

Since risk neutrality was introduced in the context of Black–Scholes, we discuss it first in that context. Before the words were actually coined, the idea of “risk neutrality” appeared in the original paper of Black and Scholes (1973) in the following form: “The return of an hedged portfolio is completely independent of the change in the value of the stock. In fact the return on the hedged position becomes certain.” They added, “this was pointed to us by Robert Merton.” That means that Merton is the culprit, or hero, who started the risk neutrality saga, which is also the insight that made financial option theory what it is today. Merton was then a young faculty member at MIT, colleague of the likes of Fischer Black, Myron Scholes, and Stewart Myers, under the intellectual leadership of Samuelson. These are the founding heroes of modern option theory. Black–Scholes deals with assets whose values follow a geometric Brownian motion, i.e., their evolution is described by the following stochastic differential equation2: dX ¼ αX dt þ σX dz

ð2:8Þ

The solutions of stochastic differential equations are time-dependent PDFs. In the particular case of this geometric Brownian motion, the solution is the lognormal distribution:

2

“dz” is the “stochastic” term. By definition hdzi ¼ 0 and hdz2i ¼ dt.

26

2 Toward a General Theory of Real Options

2      2 3 X= σ2 ln þ α  X 0 2 t 7 1 6 ψ phys ðX, t Þ ¼ pffiffiffiffiffiffiffi exp4 5 σ2t σ 2πt

ð2:9Þ

ψ phys(X, t) is the “physical” PDF for the stochastic variable X. The distribution used in Black–Scholes is a slightly modified distribution, the “risk-neutral” distribution ψ RN(X, t). 2      2 3 X= σ2 ln þ r  X 0 2 t 7 1 6 ψ RN ðX, t Þ ¼ pffiffiffiffiffiffiffi exp4 5 σ2t σ 2πt

ð2:10Þ

The difference between the two distributions is that the parameter α is replaced by the “riskless rate” r. The parameter α is the “physical” average rate of change of the value of the stochastic variable X. The substitution by the “riskless rate” r reflects the risk neutrality. Translated in math, risk neutrality means that for the valuation it is as if the actual rate of change of the variable X instead of being its “physical” rate of change is the riskless rate. In other words, it is as if the stochastic differential equation for the variable X were not Eq. (2.8) but: dX ¼ rX dt þ σX dz

ð2:80 Þ

Those who are uncomfortable with that cursory way to express risk neutrality mathematically should be reassured that this substitution has a solid mathematical basis in the theory of Stochastic Processes and Martingales, in which the theorem of Girsanov plays an important role (Lalley course Chicago). This is one of the most solid results of that theory and giving more detail about it here would clutter rather than clarify the discussion. Good information about that can be found in many places. It remains that the physical distribution for the variable X does not enter in the valuation of the option. What enters is the “risk-neutral” probability, which technically is a fictitious probability. This is one of those moments where option theory becomes less intuitive if not slippery. . . Caution is advisable when invoking risk neutrality.

2.7.6

The Black–Scholes Formula

In the case of Black–Scholes, first-degree homogeneity applies. The value of an option on two shares is twice the value of an option on one share. So Eq. (2.4) is the appropriate formula for the option, and the distribution Ψ(u, X, Y, T) is a ratio distribution. Therefore the Black–Scholes formula for a call option can be written as:

2.7 The Mathematical Foundations of ROA

H call BS ðX, Y, T Þ

¼e

ρ T

27

Z1 RN ðu  1ÞΨratio ðu, X, Y, T Þdu

Y

ð2:11Þ

1

The corresponding formula for a put option is:

put H BS ðX, Y, T Þ

¼e

ρ T

Z1 RN ð1  uÞΨratio ðu, X, Y, T Þdu

Y

ð2:12Þ

0

Those two expressions are equivalent to the way the Black–Scholes formula is in general presented, for example, in the case of a call option: ρ T H call BS ðX, Y, T Þ ¼ XN ðd 1 Þ  YN ðd 2 Þe

ð2:13Þ

with:       1 X σ2 þ ρþ T d1 ¼ pffiffiffiffi ln Y 2 σ T pffiffiffiffi d2 ¼ d1  σ T 1 N ðdÞ ¼ pffiffiffiffiffi 2π

Zd

ez dz 2

ð2:14Þ ð2:15Þ ð2:16Þ

1

put H call BS ðX, Y, T Þ and H BS ðX, Y, T Þ do not depend on the parameter α. This is a consequence of risk neutrality.

2.7.7

Black–Scholes Cannot Be Applied Everywhere, but There Is Life in Real Option Theory Outside of Black– Scholes

To use Black–Scholes, a few conditions have to be met. The uncertainty must stem from a geometric Brownian motion, and both first-degree homogeneity and risk neutrality must apply. If any of these conditions are not met, it is simply incorrect to use Black–Scholes. In the realm of real options, it is relatively rare to have all those conditions met. So in most situations in ROA, Black–Scholes should not be used. There is a whole taxonomy of financial options (Hull 2014). They represent a vast world of hedging instruments against all sorts of financial risks. As instruments of speculations, they are rather complex and when used unwisely can lead to disasters. But even if the world of financial options is vast and the field of real options is much less advanced, the world of real options is also much larger. Instead of treating real options

28

2 Toward a General Theory of Real Options

as an extension of financial options, it is more appropriate to treat financial options as a special case of real options. For example, Black–Scholes could and should be seen as a particular case of ROA where the uncertainty comes from a geometric Brownian motion and both first-degree homogeneity and risk neutrality apply.

2.7.8

When Can Risk Neutrality Be Invoked? Portfolio Argument

There is a portfolio argument for “risk neutrality,” which can be enlightening when it is used appropriately in the context of real options or a good source of confusion otherwise. “Risk neutrality” in the context of finance means that it is possible to build a “hedged portfolio” Π containing a certain amount of shares X, together with the option H(X, Y, t), where Y is the exercise price and t the exercise time, such that the rate of change of the market value of the portfolio Π follows the market or riskless rate “r” and is indifferent to the changes of value of the underlying asset X. Mathematically this is a way to say: dΠ ¼ rΠ dt

ð2:17Þ

A change of value of the underlying asset ΔX will affect the value of H(X, Y, t) by If we assume that a change of value of ΔX does not affect the value of the portfolio, the portfolio must have the general form:

∂H ðX , Y , t Þ ΔX. ∂X

Π ¼ H ðX, Y, t Þ 

∂H ðX, Y, t Þ X ∂X

ð2:18Þ

One can verify that a change of value ΔX does not change the value of Π: ΔΠ ¼

∂H ðX, Y, t Þ ∂H ðX, Y, t Þ ΔX  ΔX ¼ 0 ∂X ∂X

ð2:19Þ

Using Ito’s lemma and Eq. (2.8) (that uses the fact that hdz2i ¼ dt): 2

dH ðX, Y, t Þ ¼

∂H ðX, Y, t Þ ∂H ðX, Y, t Þ σ 2 X 2 ∂ H ðX, Y, t Þ dt þ dX þ dt ∂t ∂X 2 ∂X 2

ð2:20Þ

Remembering that hdzi ¼ 0, Eq. (2.17) combined with Eqs. (2.18), (2.19), and (2.20) yields the “Black–Scholes equation”: 2

rH ðX, Y, t Þ ¼

∂H ðX, Y, t Þ ∂H ðX, Y, t Þ σ 2 X 2 ∂ H ðX, Y, t Þ þ rX þ ∂t ∂X 2 ∂X 2

ð2:21Þ

2.7 The Mathematical Foundations of ROA

29

whose solution, with appropriate boundary conditions (Black and Scholes 1973), is the Black–Scholes formula. This portfolio argument is enlightening in the sense that one can see mathematically how the parameter α, which appears in the stochastic differential equation for X (Eq. 2.8), disappears and is replaced by the riskless rate r in Eq. (2.21), as a result of the risk neutrality assumption, i.e., the combined effect of Eqs. (2.17), (2.18), (2.19), and (2.20). But this portfolio argument can also create confusion when applied to real options. A popular mechanism to justify risk neutrality in ROA is to construct a “replicating” portfolio—one that replicates the properties of the Black–Scholes portfolio. In the case of Black–Scholes, risk neutrality does not emerge by accident. A financial option is a speculative instrument, which can be used to hedge against the risk of owning shares. There is a whole literature (involving a certain element of controversy) on the management of hedged portfolios. In order for the portfolio to stay “risk neutral,” its composition should be continuously changed to reflect the X , Y , tÞ changes in the value of the “hedge ratio” ∂H ð∂X . The hedge ratio is called Δ in the financial world, and the process of making the portfolio as close as possible to a “hedged portfolio” is called Δ hedging. A replicating portfolio would be a portfolio designed in such a way that the real option mitigates the risk associated with the investment (the risky asset in this construct) to make the whole portfolio “riskless.” This may be possible at times, but certainly not always. It is dangerous or potentially wrong to try to build one replicating portfolio at all cost. It should appear naturally, not as a result of an artificial construct.

2.7.9

How Does One Recognize When Risk Neutrality Applies?

How does risk neutrality fit in the world of ROA? The answer is not straightforward. One reason that financial options are “risk neutral” is that they can be traded continuously in an open market. Real options are not tradable in general. Why then should risk neutrality apply to them? In the book they co-edited (Schwartz and Trigeorgis 2004), Eduardo Schwartz and Leon Trigeorgis stated: “[Several authors] have shown that in perfect market, the absence of arbitrage implies the existence of a probability distribution such that the equities are priced based on their discounted (at the risk-free rate) expected cash flows, where expectations are determined under this risk neutral or risk adjusted probability measure (also called equivalent martingale measure). If markets are complete and all risks can be hedged, these probabilities are unique. [...] If markets are not complete, these risk neutral distributions are not necessarily unique, but anyone of the probability distributions would determine the same market value of the security.[...] The critical advantage of working in this

30

2 Toward a General Theory of Real Options

risk neutral environment in which the relevant discount rate is the risk-free rate of interest is that is that it is a convenient environment for option pricing.” The value of a corporate investment, and of delaying it or of abandoning a project, can be estimated by ROA. In fact it is exactly for that kind of situation that ROA was originally developed. Clearly that kind of real option cannot be traded. Arbitrage does not enter. Still in many cases, corporate investments call for a “risk-neutral” valuation. The reason is that they belong to a “portfolio” of strategic investments for a firm. And these strategic investments have to meet the firm’s requirements of yearly returns. In the valuation of the real options associated with corporate investments, this rate of return plays the same role as the riskless rate in the context of financial options. In that (narrow) sense, the real options associated with those investments are “risk neutral” or more exactly justify a “risk-neutral” valuation. But the question is: how far can this kind of reasoning go in justifying using a riskneutral valuation in the ROA of investments under uncertainty? The answer is: not very far. We show abundantly in the rest of this book that risk neutrality is not always relevant in ROA, or more exactly in general it does not apply. But there are many situations where there are no clear-cut criteria to decide whether risk neutrality applies or not. The context sometimes matters. Whether risk neutrality applies should not be a subjective choice, but as we show in this book, there is an element of complexity. This is not the only element of complexity in ROA.

2.8 2.8.1

Toward a Real Options Theory The Basic Structure of a Real Option Valuation

Reduced to its bare bones, a real option valuation starts with two PDFs: ψ 1(X) and ψ 2(Y ) for two variables. X typically represents a “benefit,” and Y represents a cost. The real option H(X, Y, T ) represents the expected benefit that can be earned at time T by paying the cost Y. Two situations have to be distinguished depending on whether first-degree homogeneity is satisfied, i.e., whether: H ðλX, λY, T Þ ¼ λH ðX, Y, T Þ

ð2:5Þ

If this is the case, the general form of H(X, Y, T ) is: H ðX, Y, T Þ ¼ e

ρ T

Z1 ðu  1Þ Ψratio ðu, X, Y, T Þdu

Y

ð2:4Þ

1

The integral is over all the values of X larger than Y. Ψratio(u, X, Y, T) is the ratio distribution build from the two original distributions ψ 1(X) and ψ 2(Y ).

2.8 Toward a Real Options Theory

31

If first-degree homogeneity does not apply, the general form of H(X, Y, T ) is: H ðX, Y, T Þ ¼ e

ρ T

Z1 u Ψdiff ðu, X, Y, T Þdu

ð2:3Þ

0

The integral is still over the values of X larger than Y, but in this case, Ψdiff(u, X, Y, T) is the difference distribution build from the two original distributions ψ 1(X) and ψ 2(Y ). Equations (2.4) and (2.3) represent all the forms that a real option valuation can take, including financial options, whatever they may be. Black–Scholes, for example, corresponds to Eq. (2.4). In that case ψ 1(X) is a “risk-neutral” distribution and concretely a lognormal distribution (Eq. 2.10). ψ 2(Y ) is a trivial distribution because there is only one cost: the exercise price Y. “Risk neutrality” affects typically the PDF ψ 1(X): should it be the “physical” distribution or a modified distribution to reflect the risk neutrality?

2.8.2

Can We Speak of a Theory of Real Options?

One goal of this chapter is to show that real options are not mere extension of financial option. They are a stand-alone paradigm, and financial options are a special case of real options. To conduct an ROA, all one needs is an expression for ψ 1(X) and ψ 2(Y ) and to know whether the condition H(λX, λY, T) ¼ λH(X, Y, T) holds or not. The rest is only crunching numbers. It is noteworthy that the value of a real option H(X, Y, T ) is always positive. That does not mean that ROA always recommends that an investment should be made. The expected benefit, i.e., H(X, Y, T ), can be very small. An interesting situation is when the NPV of the same investment is negative. The question becomes whether the value of the real option H(X, Y, T ) plus the NPV build a positive number or not. If the sum is positive, the investment makes economic sense. In what kind of situation could the NPV be negative and H(X, Y, T ) + NPV > 0? This occurs typically if the PDF ψ 1(X) for the benefit of the investment has a fat tail. This affects positively the value of H(X, Y, T ). The NPV sees only the average value of the distribution. It is the same whether there is a fat tail or not in the distribution of the benefit. This example also shows that ROA is not necessarily a substitute for NPV valuation but a useful complement. But it also shows that NPV-based investment decisions can be suboptimal, because they do not factor the “shape” of the uncertainty. When one refers to “option theory,” one refers not only to the Black–Scholes formula but also to the role that this kind of valuation plays in financial decisions. It provides a mathematical insight into the spookiest aspect of the very complex

32

2 Toward a General Theory of Real Options

financial world: uncertainty and risk. As illustrated by the eventual fiasco of LongTerm Capital Management (Lowenstein 2000), uncertainty is not only mathematical. Countries (like Russia) can default and thereby invalidate the whole construct. The financial world does not have a monopoly on having problems with dealing with uncertainty. Uncertainty is amenable to a mathematical treatment up to a point. One challenge with ROA is to know how to make it speak to situations of complexity. It should not be used as a set of formula and the numbers coming out a substitute for thinking deeper. In that sense ROA fits well in the world of climate change policy. In that world uncertainty is arguably the biggest challenge. ROA has the potential to help the policy debate and facilitate some decisions because it will allow a much more quantitative treatment of uncertainty. But before we can speak of a “real options theory,” much more epistemological elaboration will have to take place.

2.8.3

What Do Theories Do?3

What is a theory? Coming back to the Greeks and in particular Plato and Aristotle, “theoria” meant speculation of contemplation. In practice (which Aristotle opposed to theory), the word theory can refer to something as trivial as “this is my theory of why this happened.” But the word theory means something completely different in “theory of relativity.” In that case, as in any case of epistemological interest, “theory” refers to a body of knowledge. The question here is: what is “options theory,” or whether real options could become a “theory.” Theories can be built in a variety of ways. Newton’s “theory of gravity” proceeds from an equation. That equation turned out to be only an approximation, and from it a lot of successful predictions and explanations of physical phenomena on earth or in space were made. In a sense, ROA also proceeds from one equation. Is that enough to make it a “theory” on par with Newton’s theory of gravity? Obviously not. The question is: what is missing? ROA would be a theory of what? Would it be a theory of how to have a quantitative approach to uncertainty or something to do with that? The word theory is sometimes used in ways that are epistemologically suspicious, for example, decision theory. It is particularly relevant here as ROA, from an epistemological point of view, has a lot in common with decision theory and even overlaps with it. As noticed by many, the opposition dear to Aristotle between theory and practice (“praxis”) does not stand scrutiny. At its best a theory has practical implications. This is definitely the case with ROA. In that case, the practical implications are easier to see than the theoretical component. This is what will have to be built for ROA to qualify as a “theory.” That theory will have to be built from the practice. Aristotle will be happy. . . 3

https://www.ssc.wisc.edu/~jpiliavi/357/theory.white.pdf

References

33

References Bachelier, L.: Théorie de la Spéculation. Annales Scientifiques de l’École Normale Supérieure 3 (17), 21–86 (1900) Bernstein, P.L.: Capital Ideas, pp. 22–23. Wiley, Hoboken, NJ (2005) Black, F., Scholes, M.: The pricing of options and corporate liabilities. J. Polit. Econ. 81, 641 (1973) Chapman, D.: Could global derivative market tip over?, May 25. MGI Securities (2012) Creekmore, H.: The Satires of Juvenal, Mentor Book, p. 34. New American Library, New York (1963) Frenkel, E.: Love and Math: The Heart of Hidden Reality. Basic, New York (2013) Hull, J.C.: Options, Futures and Other Derivatives. Pearson, Tokyo (2014) Keynes, J.M.: General Theory of Employment, Interest and Money. Palgrave MacMillan, London (1936) Lalley, S.: University of Chicago, Mathematical Finance Course, 2001, Lecture 10. http://galton. uchicago.edu/~lalley/Courses/390/ Lowenstein, R.: When Genius Failed: The Rise and Fall of Long Term Capital Management. Random House, New York (2000) Margrabe, W.: The value of an option to exchange one asset for another. J. Finance 33, 177–186 (1978) Merton, R.C.: Theory of rational option pricing. Bell J. Econ. Manag. Sci. 4, 141–183 (1973) Myers, S.: Determinants of corporate borrowing. J. Financ. Econ. 5, 147–175 (1977) Schwartz, E., Trigeorgis, L.: Real Options and Investment Under Uncertainty: Classical Readings and Recent Contributions. MIT Press, Cambridge (2004) Taleb, N.: The Black Swan. Penguin, London (2007) Yohe, G., Andronova, N., Schlesinger, M.: To hedge or not to hedge. Science 306, 416 (2004)

Chapter 3

Real Option Analysis: Work in Progress, in Need of Progress

There’s no sense in being precise when you don’t even know what you’re talking about John von Neumann (https://www.brainyquote.com/quotes/ john_von_neumann_137953)

Abstract This chapter is openly polemical. This book would not exist if real options analysis (ROA) had progressed the way other fields of knowledge do. It did not. That has everything to do with the stifling impact of Black-Scholes on ROA. The modern form of financial option theory started around 1973 with Black-Scholes. The concept of real option was introduced soon afterward in 1977. After all these years, ROA is nowhere as developed as financial option theory. The difference of degree of advancement between the two is so huge that it begs to be explained. This is the theme of this chapter.

3.1

Net Present Value Compared with Real Options

Net present value (NPV) is the most popular method to support investment decisions under uncertainties. NPV has the advantage of being simple to understand and use: typically the rule is that the investment makes economic sense as long as its NPV is positive. A major shortcoming of NPV is that it sees only the expected values and not the variance. In other words, it does not distinguish between large and small uncertainties. If NPV were the only tool available, basically no investment in drug research and development would take place, as most of the time, their NPV is negative. There are several other examples in addition to the drug industry. The advantage of ROA is that it is affected not only by the size of the uncertainty but also by its shape. The apparent superiority of ROA was so obvious that some went as far as predicting that ROA would substitute for NPV altogether in future investment decisions. This has not happened. One reason is because ROA went © Springer Nature Switzerland AG 2020 B. Morel, Real Option Analysis and Climate Change, Springer Climate, https://doi.org/10.1007/978-3-030-12061-0_3

35

36

3 Real Option Analysis: Work in Progress, in Need of Progress

astray, which is the subject of this chapter. Another is because, all things considered, ROA is not a total substitute to NPV. There are circumstances under which NPV is awkward—for example, fat tail situations. But when both yield useful information, they are powerful complements to each other.

3.2

Classical Real Options Success Stories

In this paragraph, we present superficially a few illustrative and well-known examples—borrowed from the literature—where ROA was shown to provide useful quantitative information helping the design of investment strategy, such as adding flexibility to project, the value of abandoning it, etc. The following paragraphs deal with some of the “problems” with ROA.

3.2.1

Strategic Investments

This example is borrowed from one of the many editions of the Brealey and Myers book (Brealey et al. 2011). It is about whether a company should invest in the first generation of a line of computers, knowing that the NPV of that investment is negative. The reason for not immediately abandoning the project is the possibility that at a certain time T in the future, this first investment will make possible a followup investment in the second generation of this line of computers. So in a nutshell, the question is whether the value of the option of investing in the second line of computers is large enough to offset the negative value of the NPV. The NPV is the discounted sum of the difference between expected revenue of an investment minus its cost. In continuous time, the sum can be written as (for simplicity we assume the cost to be a onetime value C1): ZT NPV ¼

eρt hX 1 ðt Þidt  C 1

ð3:1Þ

0

In this formula: Z1 hX 1 ðt Þi ¼

X 1 P1 ðX 1 , t ÞdX 1

ð3:2Þ

0

is the expected value of the revenue. It is time dependent. P1(X1, t) is the probability distribution function (PDF) of the revenue from the first project at time t in the future. The NPV does not depend on the details of the PDF. This is a major limitation. It

3.2 Classical Real Options Success Stories

37

sees only the average value of the variable X1, which represents the revenue. The uncertainty could be large or small, and the NPV is the same. The ROA values of the next project are: H ðT Þ ¼ e

ρT

Z1 ðX 2  C 2 ÞP2 ðX 2 , T ÞdX 2

ð3:3Þ

C2

It is the discounted expected gain from the second project. By design H is always positive, whereas the NPV can be positive or negative. The decision to invest in the first projects hinges on whether or not H(T ) + NPV > 0. Brealey and Myers cooked up their example, i.e., chose numbers for both projects such that the condition H(T ) + NPV > 0 was met. Their conclusion was that the investment in the first project made sense even if its NPV was negative. A slight change of numbers would have led to the opposite conclusion. This points to a typical problem with ROA: to what extent is it prudent to make an investment decision on the basis of numbers, which by nature are imprecise, when the outcome could be altered by a small change of any of them?

3.2.2

Compound Options

In this problem involving two consecutive projects, it may be tempting to try an approach based on “compound options” (Geske 1979), Compound options are options on options. The problem with them today is that they are mathematically intensive to the point of opacity and their extension to ROA is not developed.

3.2.3

The “License to Kill” Problem

The ROA value of investment is influenced by the shape of the uncertainty. Here, that means by the shape of the PDF P2(X2, t). If, for example, P2(X2, t) has a fat tail, the ROA value of the investment will be disproportionately large as can be inferred from Eq. 3.3. A fat tail means that with a small probability the return on the investment will be large. In other words, the presence of a fat tail in that distribution acts an incentive to invest, even if the probability of the large return is small. In the ROA world, the riskier the investment, the more valuable it is: this is what the “license to kill” problem is. This is illustrated in Box 3.1. This does not invalidate the whole approach. But it shows that ROA results should not be taken at face value without having a deeper understanding of where they come from. ROA is a powerful tool when used intelligently.

38

3 Real Option Analysis: Work in Progress, in Need of Progress 0.8

0.6

0.4

0.2

2

4

6

8

Fig. 3.1 Two PDF of returns from an investment, with negative NPV, same expected return but different ROA values

Box 3.1 The License to Kill Problem Tail effect: In that “cooked up” example, the economics of two different investments described by two different functions is compared. Both have exactly the same expected return. One is described by a normal distribution with parameters μ ¼ 1.87, σ ¼ 0.5, the other a lognormal distribution with parameters μ ¼ 0.5, σ ¼ 0.5. Both have exactly the same mean (1.87). The cost of the investment is assumed to be 2.1. They have therefore the same expected return NPV ¼ 1.87  2.1 ¼  0.23, which is negative. But the values of the expected gains are different: it is H(X, Y, t) ¼ 0.102 in the case of the normal distribution and H(X, Y, t) ¼ 0.286 in the lognormal case. In the first case, the investment does not make economic sense, because the value of H(X, Y, t) does not offset the negative value of NPV, in the second, it does. . . In this “analysis” we neglect the effect of the discount factor. Its effect is to complicate a bit, without changing the fundamental message (Fig. 3.1). ZT Z1 H¼ 0

eρt ðX  Cðt ÞÞθðX  Cðt ÞÞPðX, t ÞdXdt

0

A more realistic scenario: with a tail (most animals have tails. . .)

3.2 Classical Real Options Success Stories

39

0.4

Gas 0.3

oil

0.2

0.1

2

4

6

8

10

$

Fig. 3.2 Horizontal axis is the price (arbitrary scale). These are completely notional graphs without any attempt to be realistic. Just for illustration. Sometimes gas is more expensive than oil; sometimes the opposite is true

3.2.4

Flexibility in the Design of a Project

This example is borrowed from Amram and Kulatikala (1999). The problem is to choose between three boilers: one burning gas, one burning oil, and a third, which can do both. Each boiler comes at its price. The question is whether it is better to buy two separate boilers, one for gas and one for oil, or to rely on boilers that can do both but also are more expensive. The question is to figure what the difference in prices should be to make the dual boiler more advantageous. The source of the uncertainty in this problem is the prices of oil and gas. Their prices can be assumed to follow a geometric Brownian motion. Their PDFs therefore are lognormal distributions. A notional graph for their PDFs is shown in Fig. 3.2.1 The idea is that the dual boiler will switch depending on whether the price of gas or oil is higher. What would be the expected gain of being able to do that? This is a good ROA problem. First-degree homogeneity applies here: if both prices are doubled, the return on the dual boiler will be doubled too. So the valuation should use the ratio distribution of prices. In this problem the uncertainty stems from the prices of gas and oil, i.e., assets, which can be assumed to follow a geometric Brownian motion. The expected benefit of the dual boiler over the boiler burning only gas (or only oil) is given by:

1 These are completely notional graphs without any attempt to be realistic. Just for illustration. Sometimes gas is more expensive than oil; sometimes the opposite is true.

40

3 Real Option Analysis: Work in Progress, in Need of Progress

H ðX, Y, T Þ ¼ e

ρ T

Z1 ðu  1ÞΨratio ðu, X, Y, T Þdu

ð3:4Þ

1

In that formula, H(X, Y, T) measures the expected benefit of being able to choose continuously between oil and gas, whichever is cheaper at any given time, over the time horizon T as compared not to be able to change, because one has only one boiler. The decision to go for the flexible boiler or not hinges on whether that expected benefit is larger or smaller than the difference of price between buying specialized boilers or dual boilers. Does risk neutrality apply here? Whether it does or not makes a difference. As explained in Chap. 2, that changes the details of the ratio distribution. Instead of being driven by the changes in the price of gas and oil, those changes would become irrelevant. They would be replaced by a “riskless” rate. The question of risk neutrality here boils down (so to speak) to whether the fact that the prices of gas and oil have different dynamics matters or not. Intuitively, it seems that the ROA valuation of the choice should be influenced by the fact that the prices of gas and oil do not grow the same way. That would preclude a risk-neutral valuation. On the other hand, the investment in the boilers can be construed as a corporate investment, which has to fit in the predetermined rate of return the firm wants to have. That would make a case for a risk-neutral valuation, where the riskless rate would be that rate of return. That seems like an awfully artificial construct. Corporate culture often looks that way. . . The fact that the price of oil and gas have different dynamic of change is an external factor that will affect the operational benefits of the boilers. Even if the boiler’s decision falls in the category of corporate investments, risk neutrality is questionable in this case.

3.2.5

The Value of Abandoning a Project

This is another example of ROA inspired from one of the many editions of the Brealey et al. (2011) book on Corporate Finance. The question is whether it makes economic sense to buy a more expensive piece of equipment, knowing that if the project fails, it could be sold at a certain price Y, whereas the other cheaper equipment would be lost if the project fails. The value of abandoning this project can be modeled as a put option, where the exercise price is Y. If the exercise time is not known precisely, this would correspond to an American put option. American option differs from a European option in that an American option can be exercised at any moment before the option expires, whereas a European option can be exercised only when the option expires, i.e., at the exercise time. This is reminiscent of car insurance. Against a premium, the owner of the car can turn to his insurance when his car has a crash. The premium he has to pay can be construed as the value of an American put option.

3.2 Classical Real Options Success Stories

41

American put options are notoriously difficult to valuate. So as was the case in the book of Brealey and Myers, we assume that there is an exercise time and make it a European put option. The value of the “put” option to sell the equipment at time T at the price Y is: H ðX, Y, T Þ ¼ e

ρ T

Z1 RN ð1  uÞΨratio ðu, X, Y, T Þdu

Y

ð3:5Þ

0

X is the value of the project and Y the value of the equipment. In that formula RN Ψratio ðu, X, Y, T Þ is the probability distribution is the ratio distribution as first-degree homogeneity applies. Furthermore, the decision of continuing the project or abandoning it is a corporate decision, part of the investments strategy of the firm, and therefore a “risk-neutral” valuation is of essence. H(X, Y, T ) measures the discounted expected benefit of being able to sell the equipment at the price Y at time T. If that value is larger than the difference of price between the two pieces of equipment, it is worth buying the more expensive one. The additional cost for the equipment can be seen as a hedge against the risk of the failure of the project. Real option valuations often intersect with hedging.

3.2.6

Growth Options

Growth options were what Stewart Myers had in mind when he coined the words “real options.” (Myers 1977). The possibility that a firm will grow through investments adds to its value. Real option is used to quantify this added value. H(X, Y, T ) represents the expected increase of value of the firm if instead of its future value being represented by the probability distribution ψ ðY, T Þ, because of the investment, it is represented by another probability distribution ψ ðX, T Þ. The idea is to compare the two distributions to estimate the expected benefit of the investment. When the value of the “real option” H(X, Y, T ) exceeds the cost of the investment, the investment makes economic sense. Risk neutrality applies here because it would be a strategic investment for a company organizing itself around a predetermined return. Superficially, growth options resemble call options as they are a way to take a “long” position on the value of the firm. Still, they differ in fundamental ways from financial call options: unlike financial options, a growth option does not provide a right. It is an abstract number (nobody pays it) used to decide whether an investment decision makes economic sense. Unlike debts and assets, growth options are not tangibles. Still, they enter in the value of a firm.

42

3.3

3 Real Option Analysis: Work in Progress, in Need of Progress

When ROA Raises Serious Legitimate Questions

One problem is that ROA as a field does not project a sense of rigor, coherence, and precision. There are many parallel approaches, and it projects the impression of looking like a set of recipes that one can choose from and then apply blindly. Furthermore, ROA-based decisions seem to assume that the numbers produced by ROA should be believable to the third decimal, when all the inputs are often rough estimates. The Journal of Applied Corporate Finances (JACF) from time to time dedicates a whole issue on ROA. The 2005 issue on that topic (vol. 17, no 2) was particularly interesting in that regard.

3.3.1

The Borison Controversy

In that issue of JACF, Adam Borison began an article entitled “Real Options Analysis: where are the Emperor's clothes?”(Borison 2005), with the statement: “The state of the topic at the analytical level from a potential practitioner’s point of view is problematic. There is a great deal of agreement about the appeal of the underlying concepts. However, a variety of contradictory approaches have been suggested for implementing real options in practice. The assumptions underlying these different approaches and the conditions that are appropriate for their application are typically not spelled out. Where they are spelled out or can be inferred, they differ widely from approach to approach. The difficulties in implementing the approaches are rarely discussed, and the pros and cons of alternative approaches are not explained. This situation leaves potential practitioners in troubling circumstances. In principle, the concept seems valuable and appealing. But given the current state of practice, there is a good chance that one could either apply an unsound approach or make inappropriate use of a sound one. The result is not simply a lack of theoretical precision, but mistaken investment decisions and lost value.” In order to illustrate his point, he “cooked up” an investment decision problem: “The investment is the possible acquisition of an undeveloped natural gas field in the Western U.S. Uncertainty about the size of the business is represented by the amount of natural gas in the field. Proven reserves are currently (May) estimated by the firm at 100 billion cubic feet (BCF). Uncertainty about the profitability of the business is represented by the future price of natural gas. The current (May 2003) Henry Hub price of natural gas is $5.25 per thousand CF. The firm can purchase and develop the field now for $175 million, decide never to purchase and develop the field, or acquire an option for $20 million to purchase and develop the field in 2 years for $175 million. [. . .] The option will expire if not exercised at the end of 2 years. Over the 2-year period, the value of the field may change as uncertainty evolves regarding the amount and price of natural gas. [. . .] The cost of information collection is assumed to be included in the $20 million option price.”

3.3 When ROA Raises Serious Legitimate Questions

43

The idea is to compare two key numbers: the expected profit if the investment is made immediately and the expected profit if the investment is delayed by 2 years. What matters is whether the option of waiting is larger or smaller than the cost of waiting, $20 million, and the value of the investment is larger or smaller than its cost of $175 million. The controversy is on the valuation of the option of waiting and on the value of the option of investing.

3.3.1.1

Five Different ROA Valuations

Adam Borison used the fact (that was his bone of contention) that in ROA as it stands, there are five different possible ways to value the option of waiting: “classical,” “subjective,” “MAD,” (which stands for “Marketed Asset Disclaimer”) “revised classical,” and “integrated.” According to Borison’s calculations, they yield very different values for the option of waiting: $19 million, $75 million, $113 million, $300 million, and $145 million respectively. Translated into a decision, in one case the value of waiting is less than the cost; hence the decision is now or never. The other approaches pointed to the opposite conclusion. ROA was also used to estimate the value of the investment. In some cases it was above $175 million, and in some cases, it was less than that value. Hence by using these different approaches, Borison showed that it was possible to get any imaginable conclusion. Those who tried to invalidate Borison’s results did not question the legitimacy of using those five different methods. They merely fiddled with the numbers to show that in fact the different methods did not lead to such different values for the option of waiting. They find: $66.8 million, $70.3 million, $66.1 million, $68.7 million, and $58.4 million, respectively (Copeland and Antikarov 2005). The fact that they yield numbers not totally divergent does not conceal the uncomfortable fact that the five approaches use different assumptions and one has to fiddle with numbers (with little guidance) to make the analysis proceed. There is no attempt to explain why approaches that are conceptually so totally different were supposed to be quantitatively equivalent. This suggests that real options represent an uncomfortably loose paradigm—or is there a hidden equivalence somewhere between the different approaches?

3.3.1.2

When Black-Scholes Is Used

The two first approaches (“classical” and “subjective”) make explicit use of the Black-Scholes formula. They differ by the value of some parameters. As we saw in Chap. 2, to justify using Black-Scholes, a few conditions have to be met. The uncertainty has to result from a geometric Brownian motion, and furthermore firstdegree homogeneity and risk neutrality should apply. Each of these conditions deserves a discussion (which is completely absent). In this problem, the relevant uncertainties for computing the option of waiting have two origins: the price of gas and the amount of gas in the field. The former like most commodities can arguably

44

3 Real Option Analysis: Work in Progress, in Need of Progress

be modeled as a geometric Brownian motion. But not the latter. Using Black-Scholes ignores the effect of the second uncertainty. Borison uses a different value for the volatility in the two cases (25% and 30%). In ROA, when Black-Scholes is used, it is not uncommon to see the volatility being treated as an adjustable (subjective) parameter. Who can be comfortable with a subjective valuation proceeding that way?

3.3.1.3

Replicating Portfolio and Risk Neutrality

To justify using a risk-neutral valuation, a “replicating portfolio” is used in both approaches, i.e., a portfolio that mimics the riskless portfolio used by Black and Scholes in their derivation of their formula. In Black and Scholes, the portfolio was composed of an option together with a chosen amount of the risky asset, to build a “riskless” portfolio, i.e., a portfolio whose value changes at a “riskless” rate. Here the underlying risky asset is the value of the project, because apparently this is the only possibility. Then Black-Scholes is used, assuming a riskless rate of 3% and the cost of the project ($175 million) being the exercise price. This looks like a very artificial if not “brute force” replicating portfolio and the riskless rate a somewhat arbitrary number. How much of the original Black-Scholes construct can be found here?

3.3.1.4

The Other Approaches

The other approaches do not invoke any replicating portfolio or riskless rate. In fact the one key assumption of the “Marketed Asset Disclaimer” approach is that the replicating portfolio is not necessary. One major motivation to do without it is the difficulty of building one that makes sense. In the words of their inventors (Copeland and Antikarov 2001): “Where does one find the twin security?” Who can disagree with that? The last two approaches introduce a distinction between private and public risks. In the context of the “revised classical” approach, Borison argues that real options can be applied to only one of the two types of risks: “ROA should be used when investments are dominated by market-priced or public risks, and dynamic programming/decision analysis should be used when investments are dominated by corporate-specific or private risks.” We discuss the “dynamic programming/decision analysis” in the Appendix and show that it is very different conceptually and mathematically from ROA. We argue that both can be applied to public as well as private investments. The fifth approach, or “integrated approach,” acknowledges that most investment problems encountered in practice have both kinds of risk (private and public)—and the approach is designed to address that very situation. This approach was first described in depth in a 1995 article by James Smith and Robert Nau (1995) and was later discussed in a 1998 article by James Smith and Kevin McCardle (1998). Both articles refer specifically to their approach as the “integration” of option pricing and decision analysis, not as a ROA approach per se.

3.3 When ROA Raises Serious Legitimate Questions

3.3.2

45

A Call for More Theoretical Precision: The Georgetown Challenge

What Borison at the minimum has shown is that, as it stands, the field of real options is in need of more “theoretical precision.” That by itself would be enough to explain why ROA has not enjoyed its expected popularity. Frustration about real options inspired the so-called Georgetown Challenge (Copeland and Antikarov 2005). Here is the “challenge”: “To gain acceptance among academics and practitioners, real options methodology should: 1. 2. 3. 4. 5. 6. 7.

Intuitively dominate other decision-making methods Capture the reality of the problem Use math that everyone can understand Rule out the possibility of mispricing by eliminating arbitrage Be empirically testable Appropriately incorporate risk Use as much market information as possible”

The first and second requirements are emblematic of the discomfort inspired by ROA. People should recognize the problem just by looking at the formula giving the valuation. If that were the case, the two first requirements would not be problematic. The third requirement is problematic. Like statistics, ROA supposes some minimum mathematical ability. Paraphrasing Einstein, the math should be made as simple as possible, but not simpler. The other requirements are relevant for some applications of ROA, not necessarily all. Arbitrage, for example, is a requirement that can make sense only in certain contexts. But the fifth requirement (empirically testable) is unrealistic if not epistemologically invalid. By nature the predictions of ROA are probabilistic, and the same exact situation never occurs again.

3.3.3

A “Solution” of the Problem of Borison that Meets the Georgetown Criteria

Here is an attempt to approach the problem of Borison in a way that meets the Georgetown criteria: There are two sources of uncertainty: the price of gas and the size of the field. Both can be modeled separately. The price of the gas can be assumed to follow a geometric Brownian motion (this is the implicit assumption used in the different approaches discussed by Borison): dpg ¼ αg pg dt þ σ g pg dz

ð3:6Þ

46

3 Real Option Analysis: Work in Progress, in Need of Progress

  Implying that the physical distribution ψ phys pg , t for the gas price is the g lognormal distribution: 2      2 3 σg 2 pg = ln p0g  αg  2 t 7   1 6 ψ phys pg , t ¼ pffiffiffiffiffiffiffiffi exp4 5 g 2σ g 2 t σ g 2π t

ð3:7Þ

The corresponding “risk-neutral” distribution is: 2      2 3 σg 2 pg = ln  r  p0g   2 t 7 1 6 ψ gRN pg , t ¼ pffiffiffiffiffiffiffiffi exp4 5 2σ g 2 t σ g 2π t

ð3:8Þ

The only difference is that αg, the average rate of change of the price of gas, is replaced by the “riskless” rate r. We can discuss afterward which one is the most appropriate here. In absence of any additional information, one can model the uncertainty on the amount of gas by a normal distribution: 

ψ Rg



"  2 # Rg  R0g 1 pffiffiffi exp  ¼ σ Rg π 2σ Rg 2

ð3:9Þ

R0g is the estimated size of the reserve, and σ Rg represents the uncertainty on that estimate. Another distribution could be used if there was more information on the subject of the reserve. The expected revenue X from that field is built from the product pgRg. Its PDF ψ ðX, t Þ is built from the product distribution of the price and the reserve. This implies that the PDF for the revenue X from the investment can be written as: Z1 ΨðX, t Þ ¼

    1 X ψg , t ψ Rg Rg dRg Rg Rg

ð3:10Þ

0

And the expected gain of the investment, i.e., its ROA value, is: H ðY, T Þ ¼ e

ρ T

Z1 ðX  Y ÞΨðX, T ÞdX

ð3:11Þ

Y

Y is the cost of the investment $175 million. T is the timing of the investment, here T ¼ 0 or T ¼ 2 years. The idea is to compute the two values and see how the difference compares with the cost of waiting 2 years, i.e., $20 million.

3.3 When ROA Raises Serious Legitimate Questions

47

In this formula, we plug in the noncontroversial numbers used by Borison, p0g ¼ $2.25 million per billion of cubic feet, the estimated reserve R0g at 100 billion cubic feet (for a total value for the reserve of p0gR0g of $225 million), the average growth rate of the price of gas 5%, with a volatility of 25%, an uncertainty on the reserve of 10%, and a discount rate of 13%, and one gets that the value of waiting for 2 years is $62.1 million. It is $42.1 million above the $20 million asked to pay. In other words, waiting for 2 years would add $42.1 million in value to the project. But this has to be balanced with the fact that the $50 million profit on the project ($225 million to $175 million) has to be discounted over 2 years (~$38.6 million). When the decision has to be made, the expected net benefit of waiting for 2 years to make the investment is 38.6 + 42.1  50 ¼ $30.7 million. Had we assumed risk neutrality (there could be good argument to invoke it here), the computation would have been identical except that we would have replaced the average growth rate of the price of gas 5% by a riskless rate, for example, 3%. The value of the option would as a result be slightly smaller. The important part of this remark is that in this approach, we can easily make a “risk-neutral” valuation without making use of the Black Scholes formula. In real options valuation, it is not uncommon that risk neutrality is invoked to justify using Black-Scholes as if there was no other way to value an option.

3.3.4

The McDonald-Siegel Model

The phrase “real option” is also used to mean something completely different, leading to confusion. McDonald and Siegel have established the optimal conditions for a certain kind of investment under uncertainty. This is an important and extremely useful result. This result is often referred to as real options. That means that the words “real options” are used to mean two completely different things. The McDonald and Siegel problem is a problem of optimal control under uncertainty, i.e., a stochastic dynamic programming problem. The driving equation in the case of McDonald and Siegel is a Bellman equation. The control variable in that problem is the timing of the investment. ROA is not an optimization let alone an optimal control problem. Time is not a control variable but a boundary condition. The two approaches are fundamentally different mathematically and conceptually. The fact that the fundamental difference between them is not more widely recognized is an additional symptom of the conceptual pathos in which the field of real options is mired. This is the subject of the Appendix.

48

3.4

3.4.1

3 Real Option Analysis: Work in Progress, in Need of Progress

Are Real Options Speculative Instruments? Hedging Instruments? Decision Tools? Risk Management Tools? Are They Truly Derivatives? Does ROA Somehow Relate to Bond Valuation? Thales of Miletus: The First Instantiation of Option as Speculative Instrument

In discussions of options, a popular anecdote is the story of Thales of Miletus (624 BC, 546 BC) recorded by the Greek philosopher Aristotle. Thales, a pre-Socratic philosopher, was apparently interested in more than mere philosophical pursuits. Anticipating a bountiful olive harvest, he bought from the owners the right to lease their olive oil presses for the whole region. With a relatively small investment, he generated a huge profit for himself. This is often cited as the first recorded example of the use of real options. He computed the expected gain of his investment. Thales made a calculus, which can be construed as an instantiation of ROA. Thales was also a respected mathematician, known to like to use geometric arguments. In this investment, it is doubtful that he used the theorem he is known for. . . One implication of that story is that real options can be speculative instruments. So much so, that in some other countries, option trading is banned because it is considered gambling (Wisegeek).

3.4.2

Nature of Real Options

ROA is a powerful tool to maximize returns in situations of uncertainty. Risk management is one of the most promising applications of ROA. In an article entitled “Rethinking Risk Management” (Shultz 1996), one can read: “For one thing, large companies make far greater use of derivatives than small firms, even though small firms have more volatile cash flows, more restricted access to capital and thus presumably more reason to buy protection against financial trouble. Perhaps more puzzling however is that many companies appear to use risk management to pursue their goals other than reducing variance.” “Puzzling?” Managing risk does not necessarily mean reducing it. It can mean being clever about it. Risk offers opportunities. Those companies, which “appear to use risk management to pursue their goals other than reducing variance,” may be examples of that intelligent attitude with respect to risk. To the question are real options speculative instruments, hedging instruments, decision tools, risk management instruments, or derivatives, the answer is that they are all of the above. But not at the same time. It depends on the context. This is one of the themes of the rest of this book.

References

49

References Amram, M., Kulatikala, N.: Real options, managing strategic investments in an uncertain world. Harvard Business School Press, Boston, MA (1999) Borison, A.: Real options analysis: where are the emperor clothes? J Appl Corp Financ. 17(2), 17–32 (2005) Brealey, R.C., Myers, S.A., Allen, F.: Principles of corporate finance, 10th edn. McGrawhill, New York (2011) Copeland, T., Antikarov, V.: Real options: a practitioner’s guide. TEXERE, New York (2001) Copeland, T., Antikarov, V.: Real options: Meeting the Georgetown challenge. J. Appl. Corp. Financ. 17(2), 32–51 (2005) Geske, R.: The valuation of compound options. J. Financ. Econ. 7, 63–81 (1979) Myers, S.: Determinants of corporate borrowing. J. Financ. Econ. 5, 147–175 (1977) Stultz, R.M.: Rethinking risk management. J. Appl. Corp. Financ. 9(3), 8–25 (1996). https://doi. org/10.1111/j.1745-6622.1996.tb00295.x Smith, J., McCardle, K.: Valuing oil properties: integrating option pricing and decision analysis approaches. Oper. Res. 46(2), 198–217 (1998) Smith, J., Nau, R.: Valuing risky projects: option pricing theory and decision analysis. Manag. Sci. 41(5), 795–816 (1995) Wisegeek. What are futures. http://www.wisegeek.com/what-are-futures.htm

Chapter 4

Extreme Events, Cat Bonds, ROA in the Context of Fat Tail Distributions, and the Weitzman Effect

Abstract In this chapter the mathematical framework developed in Chap. 2 is used to apply ROA in an area for which extensions of Black-Scholes or NPV cannot be used: fat tail distributions, i.e., areas in distributions where extreme events reside. This chapter paves the way to the policy discussion of the response to climate change, where such distributions are pervasive.

4.1

Setting the Problem

Extreme events like hurricanes or earthquakes occur relatively rarely, but when they do, they can generate huge damages in real time. They live in the tail of fat tail distributions. Existing mechanisms to hedge against the economic consequences of that sort of contingency are not robust. The response to extreme events is about being able to withstand their impact. This can take several forms like taking preventive measures or acquire the means to survive the economic impact of such disasters by building adequate insurance protection. Insurers and “reinsurers” make a living by covering risks associated with extreme events and, judging by Berkshire Hathaway, a good living. But “global warming” may change all that (Schneider 2001). It is blamed for the apparent increase in intensity of hurricanes or typhoons. “Thinking in terms of risk management can help identify how and where reducing uncertainties could produce considerable social benefits. This includes exploring the high tail of the PDF” (Weitzman 2008).

4.1.1

Catbonds

To face extreme contingencies, insurers can try to share the financial risks with investors by issuing catastrophe bonds, catbonds for short. Catbonds are bonds with a significant probability of default. The fact that they may default is their raison d’être. They are risky by design. © Springer Nature Switzerland AG 2020 B. Morel, Real Option Analysis and Climate Change, Springer Climate, https://doi.org/10.1007/978-3-030-12061-0_4

51

52

4 Extreme Events, Cat Bonds, ROA in the Context of Fat Tail. . .

Catbonds have to be competitive with other bonds. To guarantee that the “expected gain” of the investor offsets his expected losses is not trivial. Investors who buy the bonds receive a regular return (coupon or dividend) as long as the bond is alive, i.e., as long as no “triggering event” covered by the bond occurs. If none occurs before the expiration of the bond, the investor recoups his investment at expiration. But a “triggering event” puts an end to the bond, and its principal is then used to pay for the damage. The investor loses what was left of his investment. That means that the interest on the bond must commensurate with the risk of default. Too large a risk of default means that the issuer of the bond will have to pay huge interest. That makes the bond expensive to the issuer. To reduce the cost to the issuer, the bond has to be such that the probability of default is not too high. That means that the bond should cover specific contingencies tailored around their probability of occurring. That limits the extent of coverage by individual bonds. If catbonds cover only limited contingencies, many different catbonds will be needed to cover the spectrum of all the contingencies of interest. A necessary prelude to the design of catbonds is the estimate of the expected losses covered by the bonds. When these losses proceed from fat tail distributions, the estimate of expected losses is out of the grasp of NPV. NPV calculations are based on first moments. NPV calculations are impossible if when distributions do not have first moments, as happen sometime with fat tail distributions. But even when the distribution has a first moment, the fact that the NPV is insensitive to the shape of the distribution makes them awkward tools for situations where so much is based on the shape of the distribution. On the other hand, the math developed in Chap. 2 can be used to estimate the expected loss due to extreme events.

4.1.2

Extreme Events and Power Law Distributions

Fat tail distributions tend to be a code word for power law distributions (Resnick 2000). Although the reason for that is not completely clarified, it has been observed empirically (Engelhard 2002) that the size distribution of the damages due to extreme events such as hurricanes, earthquakes, or nuclear accidents is a power law: ψ(X)~X
β. They are also called Pareto distributions. The exponent β is not universal. Its value varies with the type of disaster and with the location. The value of β matters a lot. Whether β > 2 or β  2, for example, makes a big difference.

4.1.2.1

A Few Properties of Power Law Distributions

To be a probability distribution, a power law distribution ψ(X)~X
β must satisfy: XÐmax ψ ðX Þ dX ¼ 1. This means that precautions have to be taken. If Xmin ¼ 0 and X min

β > 0, this condition is impossible to fulfill. The distribution must therefore have a

4.1 Setting the Problem

53

minimum value for the variable. That raises the issue of choosing a value for Xmin. Xmin has an interpretation: in the context of extreme events, power law distributions can be interpreted as the asymptotic part of the distribution of damages for large damages. Whatever the real distribution of damage is for small values of the damage, for large values of the damage, it can be approximated as a power law distribution. By choice or definition is Xmin where the tail begins. What about the other boundary Xmax? If β  1, Xmax cannot be infinite, because 1 Ð ψ ðX Þ dX ¼ 1. In most cases of interest β > 1, so in principle the value of Xmax is X min

not as much of a mathematical concern in general. But there are exceptions and there are other issues. In the context of financial damages, there is a natural limit Xmax to what insurers can afford to cover. What if the damage exceeds Xmax? Additional resources will have to be summoned up until the level of the “unmanageable” is reached. From a mathematical point of view, both Xmin and Xmax must be treated with some care. They break the scale invariance of the distribution. Power laws are scale invariant in the sense that all they say is that the probability that the damage is doubled is 2
β of the probability of the original damage. But the scale of the damage can be anything. Giving a value to Xmin automatically sets a scale. XÐmax ψ ðX Þ dX ¼ 1, the With all these caveats in mind, we can see that to have X min

probability distribution must be: ψ ðX, X min , X max Þ ¼ 

4.1.2.2

ðβ
1Þ X
β  X min 1
β
X max 1
β

ð4:1Þ

Hedging Against Extreme Events

The expected loss of an insurer covering damages between Xmin and Xmax is:

H ðX min , X max Þ ¼ e


ρ T

Xð max

ðX
X min Þ ψ ðX, X min , X max Þ dX

ð4:2Þ

X min

T is the expected amount of time before the “triggering event.” H(Xmin, Xmax) is also what at the minimum he has to recover through the insurance premium. For illustration, we use “The Most Storm-Exposed Country on Earth,” i.e., the Philippines (Brown 2013). In Fig. 4.1, we made a log-log plot for the size distribution of the ten most destructive typhoons in the period 1995–2011. The most devastating one was Haiyan in 2013. So it is not included. The plot points to a power law distribution with β ¼ 1.75.

4 Extreme Events, Cat Bonds, ROA in the Context of Fat Tail. . .

54 10.0 7.0 5.0

3.0 2.0 1.5

10

15

20

Fig. 4.1 Log-log plot of the data from a local newspaper (Bacani 2013), for the ten most destructive typhoons in the Philippine in the period 1947–2011. The horizontal axis represents the damage in billions of pesos. The vertical axis is the rank order of the typhoons by size, the largest being the rank 1

In that plot the destructiveness of the smallest was 5.086 billion pesos (local currency of the Philippines). That typhoon was called Reming and took place in December 2006. The cost of the most destructive was 27.195 billion pesos. It was called Pepeng and occurred in October 2009. Suppose the government of the Philippines wanted to hedge against the large ones by buying an insurance policy, how large would the “premium” be? We assume that the government is prepared to cope with any typhoon whose cost is less than 5 billion pesos (roughly $10 millions). That would define the Xmin. Xmax could be the cost of the largest typhoon which took place between 1995 and 2011, 30 billion pesos. As far as the frequency of “triggering” events is concerned, 10 major typhoons in 16 year, suggest that the expected time between large typhoons is roughly 1.6 years. Using Eq. 4.1 with β ¼ 1.75, Xmin ¼ 5, and Xmax ¼ 30, the PDF for typhoons is ψ ðX, X min , X max Þ ¼ 3:39 X
1:75

ð4:3Þ

Plugging Eq. 4.3 into Eq. 4.2, assuming a discount rate of 10% and expected time for a triggering event of 1.6 year, the expected loss that this insurance policy would cover would be:

H ðX min , X max Þ ¼ 3:39 e


1:6

3ð0

5

ðX
5Þ X
1:75 dX ¼ 0:38

ð4:4Þ

4.1 Setting the Problem

55

In words, the expected loss to be covered by the insurance would be 0.38 billion pesos (~$7.6 million) over 1.6 years. This does not seem so out of reach. But in this estimate, we did not include the far more devastating Haiyan typhoon of 2013, whose cost is estimated above $6 billion (~300 billion pesos and so on). The same calculation means that if the government wanted to be covered for up to a 300 billion pesos contingency, everything else being equal, the expected loss to be covered would be 1.7 billion pesos (~$35 million) over 1.6 years, i.e., significantly larger. Considering that climate change seems to translate into an increase in intensity of typhoons, prudence would be to prepare for larger contingencies. It is quite possible that climate change could lead to a change in the value of β. The expected losses depend strongly on the value β as can be seen from analyzing their general expression obtained by plugging Eq. 4.1 into Eq. 4.2: ( H ðX min , X max , T Þ ¼ e


ρ T

)   ðβ
1Þ X max 2
β
X min 2
β  
1 ð2
βÞ X min 1
β
X max 1
β

ð4:5Þ

If as is the case in the example of the Philippines 1 < β < 2, when Xmax ! 1, H (Xmin, Xmax, T ) grows like Xmax: H ðX min , X max ! 1, T Þ  e
ρ T

ðβ
1Þ X max 2
β ð2
βÞ X min 1
β

ð4:6Þ

For 1 < β < 2, there is no limit to the expected loss. This does not happen if β > 2. In that case when Xmax ¼ 1: H ðX min , X max ¼ 1, T Þ ¼ e


ρ T



ðβ
1ÞX min
1 β
2

 ð4:7Þ

Whether β < 2 or β > 2 makes a huge difference, β ¼ 2 is a kind of bifurcation value for the exponent. When β < 2, the distribution does not have a finite average.1 In the context of mitigation in the next chapter, we encounter distributions, which also do not have first and/or higher moments. Clearly the value of β plays a central role in the estimation of expected losses. But in general there is some uncertainty as what is the exact value of β . This uncertainty has a pernicious effect, which complicates the design of catbonds. We call this effect the “Weitzman effect” for the first to have detected it in the context of fat tail distributions (Weitzman 2012).

If β < 2, the distribution does not have even an average. And when 2  β < 3, the distribution has an average, but it does not have a variance. Both kinds of distributions have a mathematical raison d’être (Samorodnitsky Taqqu 1994). 1

4 Extreme Events, Cat Bonds, ROA in the Context of Fat Tail. . .

56

4.2

The Weitzman Effect

In a nutshell the Weitzman effect says that uncertainty in the value of the exponent β has a flattening effect on fat tails, i.e., it strengthens the importance of events deep in the tails. This has the potential to affect significantly the valuation of the expected losses due to extreme events. The derivation of the effect is a bit involved mathematically (and the paper of Weitzman is somewhat mathematically opaque), so we do it in as user-friendly way as possible. In the main text, we give a scenic tour of the derivation of Weitzman. The gory details of the derivation are reserved for Box 4.1 and 4.2.

4.2.1

Conditional Probability for Triggering Events

Here, fat tail distribution is the code word for power law distribution (Resnick 2000) and events in the tail, i.e., “triggering events” refer to events such that x  Xmin. So the cumulative conditional probability for triggering events is: Pβ ðxjx  X min Þ ¼

1 ð

probðτ  xjx  X min Þ dτ

ð4:8Þ

x

It is easy to see that with power law distributions: Pβ ðxjx  X min Þ ¼

  X min β x

ð4:9Þ

Pβ ðxjx  X min Þ is the probability that a triggering event is larger than x. Equation 4.9 shows that this probability decreases with x. But when x ¼ Xmin, this probability should be one Pβ ðX min Þ ¼ 1 as implied by Eq. 4.9. Any event larger than Xmin is by definition a triggering event. Following Weitzman, we introduce the “slimming rate,” which plays an important role in the derivation of his result: 

 X min Þ ∂x ∂log Pβ ðxjx  X min Þ ¼
λ β ð xÞ ¼
 Pβ ðxjx  X min Þ ∂x ∂ Pβ ðxjx

ð4:10Þ

In the case of power law distributions: λ β ð xÞ ¼


β x

ð4:11Þ

The slimming rate quantifies the speed at which the probability of events deep in the tail decreases. Weitzman observation is that if the exponent of the power law is

4.2 The Weitzman Effect

57

not known with certainty or belongs to a distribution, the slimming rate is affected: it decreases when x grows, i.e., the further one goes in the tail. From Eq. 4.10, it follows that: 2 Pβ ðxjx  X min Þ ¼ Exp4


1 ð

3 λβ ðzÞ dz5

ð4:12Þ

x

The first step toward the Weitzman effect is that if β has a distribution of values g(β), the cumulative conditional probability Pðxjx  X min Þ for an event x to be a triggering event is: Pðxjx  X min Þ ¼

1 ð

Pβ ðxjx  X min Þ gðβÞdβ

ð4:13Þ

0

In the case where the size distribution of events is a power law, Pðxjx  X min Þ can be written: Pðxjx  X min Þ ¼

1 ð

   x exp
βlog gðβÞdβ X min

ð4:14Þ

0

It is noteworthy (we use that observation in Box 4.2 to derive Eq. 4.19 from the  main text) Eq. 4.14 implies that

Pðxjx  X min Þ is the Laplace transform of g(β) where x the conjugate variable is log X min . Both Eq. 4.9(with Eq. 4.13) and Eq. 4.14 are different ways to say: Pðxjx  X min Þ ¼

1 ð

X min x

β

gðβÞdβ

ð4:15Þ

0

4.2.2

The Weitzman Effect Mathematically

When g(β) ¼ δ(β
β0): Pðxjx  X min Þ ¼

  X min β0 x

We recover the expression Eq. 4.9 for Pβ0 ðxjx  X min Þ, and therefore:

ð4:16Þ

4 Extreme Events, Cat Bonds, ROA in the Context of Fat Tail. . .

58

β λðxÞ ¼
0 x

ð4:17Þ

The Weitzman effect is the observation that when g(β) 6¼ δ(β
β0), i.e., g(β) is a nontrivial distribution. There is a flattening of the tail, i.e., β in λðxÞ ¼
βx becomes a function β(x) decreasing when x increases. There is a general argument for that shown in Box 4.1 and an illustrative example for that due to Weitzman shown in Box 4.2. of Weitzman, g(β) is a gamma distribution

In the illustrative example

 gðβÞ ¼ ba Γ1ðaÞβa
1 exp
βb . The effect on the shape of the tail is to change the cumulative probability of triggering events from the power law:   X min β0 x

ð4:18Þ

  
a x 1 þ blog X min

ð4:19Þ

Pðxjx  X min Þ ¼ Into: Pðxjx  X min Þ ¼

In the process the “slimming rate” went from: β λðxÞ ¼
0 x

ð4:20Þ

1 ab

 λ ð xÞ ¼
x 1 þ b log x X min

ð4:21Þ

to:

This means that as a function of x, the slimming rate is decreases by a logarithm term. What may look like a small effect on the “slimming” of the tail leads to a major difference in the behavior of the conditional probability for high x. This result applies only to power law fat tail distributions, which are also the most common and important fat tail distributions. The choice by Weitzman of a gamma distribution for g(β) was not innocent, even if not necessarily very realistic, because the math simplifies elegantly and the effect can be seen in all its glory. This is not the case for other distributions and even less if the distribution g(β) is poorly known. The only thing one knows is that the mere fact that g(β) 6¼ δ(β
β0) implies that the slimming rate λðxÞ ¼
βx has an additional x-dependence. Instead of being constant the numerator β becomes a decreasing function of x. This means that when the exponent β of the power law is not known with certainty, the events deep in the tails however rare they may be tend

4.2 The Weitzman Effect

59

to be more frequent than a superficial reading of the parameters controlling the tail may suggest. The deeper one goes in the tail the more pronounced the effect. This observation is relevant here, as illustrated by the graph Fig. 4.1. In the example of the typhoons in the Philippines, the size distribution of large typhoons is best described as a power law distribution, but the value exponent β is best described by a distribution around an “effective value.” So in the estimation of the expected losses associated with typhoons in the Philippines, one should be prepared for the fullness of the Weitzman effect. The fact that climate change seems to have something to do with the observed and documented increase of frequency of extreme events amplifies the importance of this profound insight, which has major implications for catbonds. Box 4.1 Weitzman Effect, the General Argument  min Þ The question is how the “slimming rate” λðxÞ ¼
∂log½PðxjxX is affected by the ∂x 1  

Ð  min Þ x fact that: ∂PðxjxX ¼
1x β exp
βlog Xmin gðβÞdβ with g(β) ¼ 6 δ(β
β0). ∂x 0

Following hWeitzman

i we exp
β0 log

ωðx, β0 Þ ¼ Ð1 1 Ð

h

exp
βlog

0

x X min

i

x X min

the

“distribution”

able

to

introduce

gðβ 0 Þ

,

to

be

gðβÞdβ

write,

ω(x, β0): λðxÞ ¼
1x

β ωðx, βÞdβ.

0

ω(x, β0) is a distribution in the sense that

1 Ð

ωðx, β0 Þdβ0 ¼ 1. The slimming

0 hβ i

rate has the general form: λðxÞ ¼
x ω . hβiω is the average value of the parameter β with respect to the measure ω(x, β0). When g(β) ¼ δ(β
β0), ω(x, β0) ¼ δ(β
β0) and we recover: λðxÞ ¼
βx0 . The question is: what happens to the slimming rate when g(β) 6¼ δ(β
β0)? Now the focus of the analysis shifts to the distribution ω(x, β0) and in particular on its x-dependence when g(β) 6¼ δ(β
β0). Its derivative with ½β
hβi  ∂ωðx, β0 Þ ¼
0 x ω ωðx, β0 Þ, with respect to x simplifies into: ∂x 1 Ð hβiω ¼ β ωðx, βÞ dβ. 0

This in turn implies that:

∂hβiω ∂x

This can also be written as: This shows that

∂hβiω ∂x

¼

∂hβiω ∂x

1 Ð

1 Ð

ðhβiω
βÞ β ωðx, βÞ dβ. x 0 0 

  ¼ 1x hβiω 2
β2 ω ¼
Var½ωxðx, βÞ. β

∂ωðx, βÞ ∂x

dβ ¼

 0. This is the Weitzman effect in its general form:

whatever the PDF g(β) may be, as long as it satisfies g(β) 6¼ δ(β
β0), λðxÞ hβiω x

¼


flattens with x. (continued)

4 Extreme Events, Cat Bonds, ROA in the Context of Fat Tail. . .

60

Box 4.1 (continued) Key in the derivation was the fact that the slimming rate is “controlled” by the average hβiw and that the derivative with respect to x of that average is ∂hβiω related to minus the variance of the distribution ω(x, β): ¼
Var½ωðx, βÞ. ∂x

x

Both of those properties are attributable to the fact that the starting distribution was a power law distribution. So this result is valid only with that kind of distributions.

Box 4.2 Weitzman’s Illustrative Example In this box, we reproduce an illustrative example of Weitzman. In that

example, g(β) is the gamma distribution: gðβÞ ¼ ba Γ1ðaÞ βa
1 exp
βb . We observed in the main text (Eq. 4.14) that Pðxjx  Xmin Þ is the Laplace x . This implies: transform of g(β) where the conjugate variable is log X min Pðxjx  X min Þ ¼



 1 þ blog

x


a :

X min

Whatever the values of the parameters a and b may be, for large values of the damage x, there is huge difference of behavior between the original  β conditional probability Pβ ðxjx  X min Þ ¼ X min and the “effective” one P x


a x ðxjx  X min Þ ¼ 1 þ blog X min when g(β) is gamma distribution. The definition of ω(x, β0) (Cf Box 4.1) applied to the gamma function yields:   a      x β0 x 1 ωðx, β0 Þ ¼ 1 þ b log blog β a
1 exp
þ1 X min X min ba Γ ð aÞ 0 b Implying: hβ iw ¼

a b

1þb log

x X min

 and λðxÞ ¼
1 x

a b 1þb log



. x X min

min Þ It is easy to check that: λðxÞ ¼
∂log½PðxjxX , i.e., Eq. 4.10 is satisfied


a ∂x x with Pðxjx  X min Þ ¼ 1 þ blog X min .

4.3 Catbonds

4.3

61

Catbonds

Catbonds are problematic financial instruments. There are quite a few moving parts, which have to work together for the whole scheme to function. The sums of money needed to cover properly the whole gamut of future natural disasters are potentially colossal. This system will work well only if it enjoys a huge participation. That means that catbonds must be made attractive to many investors. But investing in catbonds is somewhat similar to making a loan with a significant probability of default.

4.3.1

Catbonds in Theory

In a nutshell this is how a catbond works: if Pj is the “principal” the investor provides and p is the probability that there will be a “triggering”event during the life  of the n P 1
pÞ ι catbond, the expect return for the investor is R j ¼ P j ðð1þρ þ , which ÞN ð1þρÞk k¼1

can be written as (Cf Boxes 4.3, 4.4, and 4.5): " Rj ¼ Pj

ð 1
pÞ ð1 þ ρÞN

þ

ι ð1 þ ρÞn
1



ð1 þ ρÞn
1 ρ

# ð4:22Þ

N is the maximum number of annuities after which the bond expires, ρ is the discount rate, and ι is the interest rate of the bond. With a probability p, a triggering event will take place before the expiration. n is the expected number of annuities before a triggering event. A triggering event kills the bond. With a probability 1
p, n ¼ N. To be competitive against “safe” or riskless bonds, i.e., bonds with zero or low probability of default, the interest rate on catbonds should reflect their inherent risk, in other words ι has to be high (ι > > r, if r is a “riskless” rate). According to Swiss Re, the “Cat Bond Total Return Index” shows that they provided a return of about 10% in 2013. This compared well with the corresponding return of 2.5% from US 10-year treasury bonds. A way to make catbonds as cheap as possible is to design them in such a way that their probability p of “default” is small. But p cannot be put to zero, otherwise catbonds would lose their raison d’être. In practice, catbonds are designed to cover limited or specific risks. That implies that to cover a substantial spectrum of risks, one needs to issue many different kinds of catbonds. The bonds are financed through premiums paid by the insured parties. Issuers can issue many kinds of catbonds, each with a low probability of default to finance those, which are triggered. “Triggering” events are events whose “size” is larger than a threshold Xmin. Xmin determines also the minimum amount that should be covered. It must be fine-tuned in such a way that its probability p of occurring during the

62

4 Extreme Events, Cat Bonds, ROA in the Context of Fat Tail. . .

lifetime of the bonds is not too high but also in such a way that the insured party is prepared to pay an adequate premium to make the bond economically viable. This is not a set of constraints easy to meet (Cf Box 4.5). Insurers also have to put an upper limit Xmax to what they cover. This too is problematic (also for the insured) as it is not easy to limit the amount of damage that natural disasters can create. In the case of Katrina, for example, which devastated New Orleans in August 2005, it is difficult to imagine what kind of catbonds could have been set up to cover properly the whole damage. But had the vulnerability of New Orleans been detected, some precautionary measures may have limited the damage that Katrina made. In other words, catbonds could be part of a wider set of measures to face the threat of natural disasters. Catbonds are about insuring against events relatively rare, with large consequences, i.e., events in the tails of power law distribution. ROA is the only available tool to compute the expected losses. In Box 4.5, we conduct what can be construed only as a very superficial analysis inspired by ROA. This very superficial analysis uncovers a wealth of complications, most of them only alluded to, none addressed seriously. If there is a message from ROA, it is that catbonds are very complex financial instruments, with many moving parts. Still when it comes to get analytical power in this spooky world of low-probability events with large consequences, there is no substitute to ROA, even if it is not a silver bullet. ROA, among other things, helps clarifying how to choose the threshold for the triggering events Xmin and also how to deal with the maximum level of damage that the catbond should cover Xmax. Both are problematic. Xmin is related to the probability that a triggering event will take place during the lifetime of the bond. The choice of Xmax involves several considerations. In principle it is in the economic interest of the issuer to let Xmaxbe as large as possible. But there are practical considerations: the ability to finance very large contingencies if needed and the willingness of potential buyers of the insurance to pay to be covered to that extent. The parameter β, which controls the power law of the size distribution of the triggering events, plays a very important role throughout the analysis. Considering that this parameter is in general not very well known, this means that the Weitzman effect applies to catbonds and complicates further their already intricate design.

4.3.2

Catbonds in Practice

Although there are indications that things may be changing, in practice, catbonds so far have played a marginal role. For example, when the Tohoku earthquake hit in 2011 generating a Tsunami, which damaged heavily the Fukushima nuclear plant, catbonds played a negligible role. The total cost of the disaster is estimated between $200 billion and $300 billion. The cost to the insurance companies was estimated between $21 billion and $34 billion. Only one catbond of $300 million issued by the reinsurance company Munich Re was paid out. At that time, the sum covered by catbonds for the whole of Japan at the time was a miserly $1.7 billion, and they were

4.3 Catbonds

63

mostly designed to cover damage from earthquakes affecting the Tokyo metropolitan area where 40% of the economy is concentrated. After the traumatic experience with the typhoon Haiyan in 2013, the government of the Philippines apparently toyed with the idea of “issuing” catbonds to insure their infrastructures against large typhoons. But interestingly it chose not to. Preliminary estimates of the cost of Haiyan were around $6 billion, since the estimates rose to as high as $14 billion. For comparison the GDP of the Philippines is about $290 billion. The cost of Haiyan was significantly more than the average $4 billion that the Philippines lose every year to typhoons, but significantly less than the cost of the 2011 Tohoku earthquake. The Philippine government apparently decided that catbonds were not cost-efficient enough tools to hedge against extreme events like Haiyan, which can hit in many places. Each set of catbond would have to cover several billions USD. The Philippines being a developing country can find a more generous financial support in international institutions such as the World Bank. But, international institutions like the World Bank cannot take on themselves to finance the response to natural disasters. Catbonds seem to be the only financial instruments available to set up privately funded responses to that kind of threat. They are complex instruments, but that does not mean that the art of designing them cannot be perfected. Box 4.3 Designing a Cat Bond The first challenge in the design of catbonds is to make them attractive to investors. They are put in a situation to have to lend money with a significant probability of default. If Pj is the principal of the investors, their revenue is  n P 1
pÞ ι þ . R j ¼ P j ðð1þρ ÞN ð1þρÞk k¼1

ι is the interest rate, and n is the expected number of times the interest will be paid. n is related to the probability p. If N is the maximum number of annuities after which the bond expires, with a probability 1
p, n ¼ N. The total cost covered C by the P catbonds is the sum of the contributions of the individual investors: C ¼ P j . j h

i 1
pÞ ð1þρÞn
1 ι Rj can also be written:R j ¼ P j ðð1þρ . One approach N þ n
1 ρ Þ ð1þρÞ to gauge the attractiveness of the bonds is to estimate their Sharpe ratio, i.e., to scale their return by hcomparison to the

expected i revenue R0j of a riskless investment: R j ¼ P j

ð1
pÞ ð1þρÞN

þ ð1þρr Þn
1

ð1þρÞn
1 ρ

, where r is the “riskless

rate.” The probability p of a triggering event plays a central role. To a certain extent, it is a control variable as the issuers of catbonds can choose which risks they want to insure. With a probability 1
p, the bond will expire before a (continued)

4 Extreme Events, Cat Bonds, ROA in the Context of Fat Tail. . .

64

Box 4.3 (continued) triggering event takes place. The expected maximum return for the investor therefore is: " R j ¼ P j ð 1
pÞ

1 ð 1 þ ρÞ N

þ

ι ð1 þ ρÞN
1

ð 1 þ ρÞ N
1 ρ

!# :

This gives some idea of how the interest rates ι and r are related to make catbonds comparable to riskless bond. To have at least  an equal expected return would imply ð1
pÞ ι 

pρ ð1þρÞðð1þρÞN
1Þ

þ r . This is worse than

r simply ι  ð1
p Þ because the principal is also lost after a triggering event. The larger the p, the more pronounced the effect. This is one reason catbonds tend to be designed for very specific contingencies: to minimize the probability of a triggering event. But there is a limit to that: catbonds have to cover something, otherwise what would be their purpose. . .

Box 4.4 Catbonds, the Challenge The revenue Rj(x) of an investor in a catbond has a maximum value (when there is no triggering event): h

i ð1þρÞN
1 1 ι þ R j ðx  X min Þ ¼ P j ð1þρ ρ ÞN ð1þρÞN
1 If there is a triggering event, to simplify, we will assume that the bond dies automatically. That means that for x  Xmin h i : R j ðx  X min Þ ¼ P j

ι ð1þρÞn
1

ð1þρÞn
1 ρ

To get some grip on the variance on the revenue, we compare the two extreme situations: a triggering event at the onset of the bond, the revenue is Rjmin ¼ 0. At h the other extreme,

there isno i triggering event, and the revenue is:

R j max ¼ P j

1 ð1þρÞN

þ ð1þριÞN
1

ð1þρÞN
1 ρ

. This is the interval within which lies

the revenue for the investor. That suggests a variance for the revenue close to the whole value of the revenue. The answer to the question of what is the distribution of return for the investor between these two extremes is context dependent when it exists. If p is the probability that a triggering event takes place during the lifetime of the bond, the n expectedh revenue for the investor has i the h general form: i o Ret j ¼P j ð1
pÞ

1 þ ι ð1þρÞN ð1þρÞN
1

ð1þρÞN
1 ρ

þp

ι ð1þρÞn
1

ð1þρÞn
1 ρ


1

(continued)

4.3 Catbonds

65

Box 4.4 (continued) The Sharpe ratio (which is used to see whether the risk in an investment is adequately rewarded) is the difference Retj
R0j divided by the variance on the return on the risky asset. R0j is the riskless return. In the case of catbonds, the variance looks so large (even before adding the Weitzman effect) that it is dubious that catbonds could be made to have good Sharpe ratios. They should be made attractive otherwise.

Box 4.5 Catbonds Through the Looking Glass of ROA: The Issuer’s Perspective Using the fact that the size distribution of events is a power law distribution and introducing the time dimension: We assume in this Box that β > 2. β < 2 is obviously of interest but seriously more involved. A key quantity for the issuer is his expected loss. Assuming he plans to cover events whose sizes are between Xmin and Xmax, his expected loss is:   ðβ
1Þ X min
βþ2
X max
βþ2   : hX i ¼ ðβ
2Þ X min 1
β
X max 1
β In the limit Xmax ! 1 (which cannot be taken if β < 2), it takes the value: Þ ðβ
1Þ hX i ¼ ððβ
1 β
2Þ X min . Since in general Xmax > > Xmin, hX i ¼ ðβ
2Þ X min is a good estimate for the expected losses, even when Xmax is finite. One could deduce that the smaller the Xmin, the smaller the expected loss, and therefore there is an advantage in choosing Xminas small as possible. But that would forget that when the size distribution of events is a power law, if the threshold for triggering events is lowered, the probability of a triggering event is increased exponentially. Catbonds are designed to ensure that with a significant probability, there should be no triggering event during the lifetime of the bond. That sets a lower limit to Xmin, which is context dependent. The probability that a triggering event will exceed the “expected” loss, in


βþ1 Þ the limit Xmax ! 1 is: probðX  hX iÞ ¼ ððβ
1 . This is not a negligible β
2Þ number: if β ¼ 3, the probability is 25%. If β is smaller, the probability is higher. We revisit that issue later in this Box, when we discuss the catbond value at risk (CVaR). (continued)

66

4 Extreme Events, Cat Bonds, ROA in the Context of Fat Tail. . .

Box 4.5 (continued) The issuer has to put a limit to the value of Xmax. This is discussed in the next continuation of this Box 4.5. Catbonds through the looking glass of ROA: How to choose the maximum level of damage Xmax: If a catbond covers damages from a triggering event, from Xmin to Xmax, if the size of the event is Xmin  x  Xmax, the expected “saving” or gain for the issuer is the ROA-like quantity: XÐmax β
1Þ x
β ROA ¼ . Introducing ðX max
xÞψ ðxÞdx where ψ ðxÞ ¼ X ð1
β
X max 1
β Þ ð min X min   ðλβ
2
1Þ β
1Þ min λ ¼ XXmax , ROA becomes: ROA ¼ X max 1
λ λβ
1
1 ððβ
2 Þ . As usual we ð Þ assume β > 2. To see more clearly through this formula for ROA, we take the simplifying
X min . With the same assumption the example of β ¼ 3. Then ROA ¼ X max XX max max þX min . ROA increases linearly with expected losses of the issuer are hX i ¼ X2XminmaxþXX min max Xmax, i.e., faster than the expected loss, which asymptotically becomes 2 Xmin. It could seem that the larger the Xmax, the better. But ROA is a fake number. It is based on the assumption that the issuer can cover damages up until Xmax. In the context of this power law, the probability that the damage to cover will be Xmax decreases like Xmax
3. If Xmax is large, this probability is very small. But realistically the issuer has to think of what level of damage he can cover with a given probability. The value of his expected loss does not capture the fullness of his risk. He has to be able to foot the bill if asked to, even if there is a very small probability of that to occur. This is the whole point of catbonds: covering low-probability events of high size. There is more to the calculation than expected loss or expected saving. They may provide an economic estimate of the ROA value of the catbond for the issuer. But they do not solve the problem of having the financing available, i.e., having Xmaxsomewhere in reserve in case it is needed. A bond issuer to maximize the value of issuing the bond should push the value of Xmaxas high as possible. By covering many uncorrelated contingencies, he could exploit the fact that with a probability so high that it is almost a certainty, he will not have to face more than one at a time. Catbond value at risk (CVaR) Value at risk (VaR) is a powerful tool used in the context of risk management (McNeil et al. 2005), in particular when it comes to extreme events [“Black Swans” (Taleb 2007)]. This is exactly what catbonds are about. . . . So it is tempting to extend the concept of VaR to them, i.e., to use the “catbond value at risk” (CVar) as an additional tool to assess the risk associated with catbonds. (continued)

References

67

Box 4.5 (continued) Given a probability p, the CVaR associated with it is a value of the damage XÐmax ðx1
β
X max 1
β Þ ðβ
1Þ z
β x such that: p ¼ dz, i.e., p ¼ X 1
β
X 1
β , i.e., X min 1
β
X max 1
β Þ ð ð Þ min max x 1    1
β 1
β X min min
1 X max or x ¼ with x ¼ 1 þ p XXmax 1 fλβ
1 þpð1
λβ
1 Þgβ
1 min λ ¼ XXmax . In general λ < < 1, so x  X min is a good estimate for the CVaR. 1 pβ
1

This means that CVaR is basically determined by Xmin. Xmax plays a negligible role.


βþ1 Þ β
1Þ If p ¼ ððβ
1 , x ¼ ððβ
2 β
2Þ Þ X min , which the result we found earlier in this Box. To put some numbers, if β ¼ 3, the CVaR for the probability of 25% is as before the expected loss hXi ¼ 2 Xmin. Like VaR, CVaR is more useful for lower probabilities. For a probability of 1% ( p ¼ 0.01), the CVaR is x ¼ 10 Xmin. In other words, with a probability of 99%, the catbond will not have to cover more than 10 Xmin.

References Bacani, L.: Deadliest, most destructive cyclones in the Philippines. Philstar. http://www.philstar. com/headlines/2013/11/11/1255490/deadliest-most-destructive-cyclones-philippines (2013) Brown, S.: The Philippines is the most storm-exposed country on earth. Time (2013) Engelhard, J.D.: Risk Anal. (2), 369–381 (2002) McNeil, A.J., Frey, R., Embrechts, P.: Quantitative Risk Management. Princeton University Press, Princeton (2005) Resnick, S.I.: Heavy Tail Phenomena, Probabilistic and Statistical Modeling. Springer, New York (2000) Samorodnitsky, G., Taqqu, M.S.: Stable Non-Gaussian Random Processes, Stochastic Processes with Infinite Variance. Chapman & Hall, New York (1994) Schneider, S.H.: What is dangerous climate change? Nature. 411, 17–19 (2001) Taleb, N.: The Black Swan: The Impact of the Highly Improbable. Random House, New York (2007) Weitzman, M.: On modeling and interpreting the economics of catastrophic climate change. Harvard Preprint. http://www.economics.harvard.edu/faculty/weitzman/papers_weitzman (2008) Weitzman, M.: Precautionary tale about uncertain fat tail flattening, NBER, Working Paper 18144 (2012)

Chapter 5

Global CO2 Emission Mitigation Through the Looking Glass of ROA

What does culture want? To make infinity comprehensible, Umberto Eco (https://www.brainyquote.com/quotes/ umberto_eco_457208) I cannot help it. In spite of myself, infinity torments me, Alfred de Musset (https://www.brainyquote.com/quotes/ alfred_de_musset_169890)

Abstract Except for a few loonies, most agree that there is an imperative not to let the atmospheric load of CO2 grow indefinitely. “Doing nothing implies that risks are negligible. That position implies an absurd degree of certainty” [Wolf, Financial Times (27 Oct 2015)]. As some like to say, there is no planet B. The economics of climate change is far, far behind its science. Given the important role of uncertainty in climate change policy and the debilitating limitations of alternatives such as NPV or the “neoclassical” approach, i.e., the “integrated assessment models,” when it comes to uncertainty, the detour of ROA is unavoidable. The interface between ROA and climate change turns out to be rather explosive and reveals how deep the need for a response to climate change goes.

5.1

Introduction

ROA perspective here means estimating the expected benefit of global CO2 mitigation. The expected benefit is the difference between the expected environmental damages with and without mitigation. The estimate of the expected damages proceeds very much like in the previous chapter, which dealt with fat tail distributions. But here the fat tail distribution is generated by a best guess of what the actual uncertainty of the temperature increase is for a given load of atmospheric GHG. Then this increase of temperature is translated into environmental damage. Each of these steps has its share of problems. The result is an explosion or to make it more recreational a firework of unresolved issues. © Springer Nature Switzerland AG 2020 B. Morel, Real Option Analysis and Climate Change, Springer Climate, https://doi.org/10.1007/978-3-030-12061-0_5

69

70

5.2

5 Global CO2 Emission Mitigation Through the Looking Glass of ROA

ROA and Climate Change: Setting Up the Problem

In the nineteenth century, an English physicist John Tyndall (1820–1893) explained the importance of greenhouse gases (GHGs) for life on earth. Where it is not for the GHGs, the temperature of the earth would get so low at night that there would be no life. Svante Arrhenius (1859–1927), a Nobel Prize-winning Swedish scientist, pushed this observation one step further. When the industrial revolution was at an early stage, he predicted that one of its effects would be to increase the load of atmospheric GHGs, to the point that the temperature of the earth could increase dangerously. One hundred years later, we are nearing that point. Today’s consensus is that the temperature of the earth should not be allowed to increase by more than 2
C or even 1.5
C. In fact those numbers conceal a lot of complexity. They refer to a “global average.” Different areas are affected differently. The relation between atmospheric concentrations of CO2 and temperature is not precisely known. A lot of physical processes are involved. Local climate and its change have a lot to do with the specifics of the heat transfer from the equator to the pole, and the poles are warming faster than the equator. That dynamics perturbed by global warming and responsible for climate change involves sensitive physical phenomena not often discussed in the public debate such as baroclinic instabilities (Phillips 1954) and their role in atmospheric heat transfer (Stone 1978; Held 1978). They mediate the process of heat transfer, and a change in the boundary conditions means that local climatic conditions will be modified: this is part of what creates local climate change. This is just an example. The physics of climate is replete with such examples. In other words, the problem of climate change is far more complex than what the “experts” in the US Congress who call it a hoax can possibly appreciate. Reducing this entangled web of effects local and global to one number, the value for the average increase of global temperature does not get close to reflect the intricate complexity of climate change. But however aggregated it may be, temperature increase is a useful number as even US congressmen and policymakers can understand it. . .The problem is that temperature increase should be treated not only as a number but also as a distribution of numbers. It cannot be known with absolute certainty.

5.2.1

Irreducible Uncertainty on Temperature Increase

The dynamic of climate change involves the simultaneous effect of processes with very different time scales. Oceans, for example, trap heat for a very long time and release it very slowly. Their dynamic is so slow compared to the other processes that they make sure that no equilibrium is ever reached. When a system never reaches any equilibrium, it is in a permanent transient state. This is what scientists studying climate change have to contend with when they build their models, the so-called global circulation models. However detailed these models try to be, because of the

5.2 ROA and Climate Change: Setting Up the Problem

71

limits of what can be accomplished today with computer power, they are coarse and aggregated representations of the processes involved. Furthermore they use different assumptions. And as a result, they provide different pictures of what we thought were the same phenomena. There is so much variability between models that a meaningful is not possible. They deal with transient behaviors, with different boundary conditions. This is a good prescription to generate spurious discrepancies. This is one reason why the concept of “climate sensitivity” has been usefully introduced.

5.2.1.1

Climate Sensitivity

It is fundamental to keep in mind in all meaningful debates about climate change that climate sensitivity is an abstract number, useful for discussions but not representing any reality. It represents what the increase of global average temperature “at equilibrium” would be in case the atmospheric content of CO2 was doubled from what it was in the preindustrial time. The “doubling” is a convention. Useful too (and maybe more useful) is to ask what “at equilibrium” the global temperature would look like at another (lower) level of the atmospheric load of CO2. This kind of equilibrium situation is purely theoretical, since there is no hope that any equilibrium will ever be reached. However abstract this number may be, used intelligently, it is a useful proxy to estimate the effect of any atmospheric load of CO2 on the average global temperature and compare model predictions.

5.2.1.2

PDF for the Relation Between Atmospheric Load of CO2 and Global Temperature

Even if in practice the relation between atmospheric load of CO2 and global temperature is complex, an accepted rule of thumb is that the translation of a change in atmospheric concentration of CO2 into a temperature increase, ΔT0 has the general form (Baker and Roe 2007): ΔT0  Δ ln [CO2]. The uncertainty in the relation between atmospheric CO2 concentration and temperature is expressed as a probability distribution around ΔT0. The source of the uncertainty is the complexity of the so-called feedback: climate change changes the humidity content of the atmosphere. Under some circumstances, but not always, that translates in clouds. Clouds affect the albedo of the earth as well as the amount of heat trapped in the atmosphere. Trying to put all these effects together in an aggregated way, Roe and Baker eventually came out with the following mathematical formulation for that distribution of the temperature increase for a given CO2 atmospheric load:

5 Global CO2 Emission Mitigation Through the Looking Glass of ROA

72

2 ! 3 ~  ΔT 0 2 1  f 1 ΔT 0 1 ΔT 5 ψ T ðΔT Þ ¼ pffiffiffiffiffi exp4 2 σ f~ σ f~ 2π ΔT 2

ð5:1Þ

In absence of any feedback ( f ¼ 0), the increase of temperature associated with the atmospheric load of CO2 would be ΔT0. In this formula Eq. 5.1, f~ is the “average” 0 value of the “feedback” term. By definition f ¼ 1  ΔT ΔT . The feedback cannot be neglected, and its value cannot be known precisely. This is an irreducible uncertainty. In a very aggregated way, it captures the combined impact of all the processes that build the response of the earth to the greenhouse effect. Baker and Roe modeled the feedback by a Gaussian distribution with variance σ f and average f~. ψ T (ΔT) as given by Eq. 5.1 is the probability distribution of ΔT resulting from that set of assumptions. In the words of Baker and Roe, “Despite the enormous complexity of the climate system, the probability distribution of equilibrium climate sensitivity is well characterized by ψ T (ΔT), which reflects the straightforward, compounding effect of essentially linear feedbacks and depends on only the two parameters f~ and σ f.” Doubling the amount of atmospheric CO2 with respect to the preindustrial age
amounts to take ΔT0  1.2 C. Roe and Baker use f~ ¼ 0:7 and σ f ¼ 0.14. With those numbers, the general shape of the distribution of the increase of temperature ΔT is plotted in Fig. 5.1:

y(Δt) 0.30 0.25 0.20 0.15 0.10 0.05

2

4

6

8

10

12

14

ΔT

Fig. 5.1 General shape of the temperature distribution assumed by Baker and Roe (2007)

5.2 ROA and Climate Change: Setting Up the Problem

5.2.1.3

73

Relevance of Fat Tail Distributions

The distribution ψ T (ΔT ) has a fat tail. When ΔT 1, ψ T ðΔT Þ  ΔT1 2. This implies ! R1 ψ T ðΔT Þ dΔT ¼ 1 , it does not that even if it is a well-defined distribution 1   R1 have a finite first moment hΔT i ¼ ΔT ψ T ðΔT Þ dΔT ¼ 1 and even less finite 1

higher moments. The graph of Fig. 5.1 may be only “notional,” but it is not arbitrary although the consequences of not having a first moment are so drastic that one could be tempted to go back to how this expression was arrived at and question some fundamentals. That is not the point of that book. This book is about looking at those issues from a ROA perspective, taking for granted that accepted results have been accepted for good reasons. So we take this expression as a given and explore its implications for climate change policy from an ROA perspective. All distributions for climate sensitivity are fat tail distributions (Meinshausen et al. 2009). The fact that they do not have necessarily first moments means that the average temperature of the distribution does not necessarily exist. . .This means that it is technically incorrect or misleading to state that a given concentration of atmospheric CO2 would lead to a well-defined temperature increase, knowing that what that statement referred to is the expected temperature increase, i.e., the first moment of the distribution. There is an “irreducible” uncertainty on what the actual value of ΔT is for a given increase of CO2 atmospheric load. For a given atmospheric load, all one can estimate is the probability that the temperature increase will or will not exceed a certain value. As long as the distribution of temperature increase not only has a fat tail, in the sense that it does not have a first or second moment, the possibility of an environmental catastrophe is part of the possible futures (Weitzman 2011). When it does not have a first moment, basically when we speak of temperature increase, we do not know what we are speaking about. But when it has a first moment, i.e., a well-defined average or expected value, if it does not have a second moment (the variance is infinite), the possibility of an environmental catastrophe is still among the possible futures. There is no mathematical way out of that. Obviously, an interesting line of research is to clarify the physical origin of this mathematical fact. Can we through a clever monitoring of the evolution of global warming detect evidence that our climate is in the path of the “fat tail”? What sorts of early evidence could we have that this possibility is entering seriously the realm of the contingencies to contend with?

74

5.2.2

5 Global CO2 Emission Mitigation Through the Looking Glass of ROA

Building the Mathematical Expression of the ROA Value of Mitigation

The ROA value, i.e., the expected benefit of a mitigation policy, is measured by the amount of environmental damage saved. If we knew the relation between temperature increase and environmental damage, i.e., if we had a “damage function” we can believe in, we could build a PDF for the damage from the knowledge of the PDF for the temperature increase (Eq. 5.1) with and without mitigation, i.e., for the two corresponding atmospheric loads of CO2. Let ψ 1(X1, t) and ψ 2(X2, t) represent the PDF for the environmental damage with and without mitigation. We can use those distributions to compute the ROA value of mitigation. In this case, first-degree homogeneity does not apply: doubling the investment in mitigation will not reduce the damage by a factor two. So in this case, the ROA value of the mitigation policy will be based on the difference distribution Ψ(u, X1, X2, t) for X1 and X2 (Cf. Chap. 2). Ψ(u, X1, X2, t) is built from the distributions ψ 1(X1, t) and ψ 2(X2, t). The value of or expected benefit of mitigation is the value of exchanging the PDF of damage ψ 1(X1, t) with the new PDF of damage ψ 2(X2, t). It is an instantiation of the example of the value of exchanging two “risky” assets (in the sense of Margrabe) studied in Chap. 2. This means that the value of mitigation is: H ðX 1 , X 2 , t Þ ¼ e

ρ t

Z1 u Ψðu, X 1 , X 2 , t Þ du

ð5:2Þ

1

The mitigation policy makes economic sense if the expected benefit exceeds the cost of the policy (H(X1, X2, t)  C). This consideration is moot here because as we show in the next paragraph, in the case of CO2 emission mitigation, H(X1, X2, t) is infinite. This is the first salvo of the firework of unsolved issues in global mitigation policy.

5.2.3

The ROA Value of a Climate Change Mitigation Policy Is Infinite

We need a damage function to translate the distribution of temperature increase into a distribution of environmental damage. There is no such a thing as a good damage function, as nobody knows how an increase of temperature of 4
C or more translates into environmental damage. A popular damage function is (Nordhaus) X ¼ θ(ΔT)γ. θ is constant, which plays no role in this analysis, and the exponent γ is typically 2 but should be allow to grow with ΔT. Clearly this expression underestimates the environmental damage for large values of ΔT. The “effective” exponent γ should

5.2 ROA and Climate Change: Setting Up the Problem

75

grow when ΔT becomes larger, and when ΔT is large enough, the damage should be “catastrophic.” How does one capture that in a damage function? That kind of damage function should be used for small temperature increase, i.e., when we vaguely know what we are speaking about. But the problem is that we cannot by convention decide what the maximum temperature increase will be in the future. Somehow we must have a theoretical framework, which accommodates the possibility of an environmental catastrophe, since that possibility cannot be excluded. We are not yet there. But using a damage function which demonstrably underestimates the damage for large (but not so large) values of ΔT, we already get additional contributions to the firework of unresolved issues. To get the probability distribution of the damages X1 (with mitigation) and X2  1 (without mitigation), we just have to substitute ΔT ¼ Xθ γ in the probability distribution for the temperature ψ T (ΔT) (Eq. 5.1) and get: 2 3 θΔT !2 1  f~  X 1X1 0 1 ΔT X 1 0 1 5 ψ 1 ðX 1 Þ ¼ 3 pffiffiffiffiffi exp4 1 2 σ f~ γθ γ σ f~ 2π X 1 ðγ þ1Þ 1

As long as γ < 1,

R1

ð5:3Þ

ψ 1 ðX 1 ÞdX 1 ¼ 1, i.e., ψ 1(X1), is a legitimate PDF. But for

0

γ > 1, it has no finite average or higher moments. This is because when X1 ! 1, it 11

goes like X 1 γ . When it comes to compute overall damages, we are already in the realm of infinities. The expected level of damage before mitigation is infinite. The damage distribution ψ 2(X2) for the case without mitigation differs only by the value of ΔT X 1 ,0 . The two parameters ΔT X 1 ,0 and ΔT X 2 ,0 represent what the equilibrium temperature would be in the absence of feedback ( f ¼ 0), for the two atmospheric loads of carbon. Clearly ΔT X 1 ,0 > ΔT X 2 ,0 . In both cases, as long as γ > 1, the environmental damage is potentially infinite, i.e., an environmental catastrophe belongs to the possible futures, before and after mitigation. The difference is in the probability of such an outcome. . .What is the value of reducing the probability of a catastrophe? In other words, what is the ROA value of global mitigation? If γ > 1 the ROA Value of Mitigation Is Infinite The reason for the fact that the value of the option of mitigating is infinite when γ > 1 1 is that when the damage difference X ! 1, Ψðu, X 1 , X 2 , t Þ  uγ . So when γ > 1, the integral for H(X1, X2, t) ¼ 1. This is a consequence of the fat tail of the original distribution for the uncertainty on the temperature. It goes asymptotically (when ΔT ! 1) as ΔT 2. Because of the relation between damage and temperature 1 1 increase ΔT  X γ , the damage functions asymptotically behave like X γ 1 . This is 1 reflected in the properties of the difference function Ψðu, X, t Þ  uγ 1 , when R1 u ! 1. This means that the integrand in H ðX, Y, t Þ ¼ eρt ðu  Y ÞΨðu, X, t Þdu Y

behaves like uγ , when u ! 1, implying that the integral is infinite when γ  1. 1

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5 Global CO2 Emission Mitigation Through the Looking Glass of ROA

γ > 1 is a safe assumption here. It means that damage X grows faster than linearly with the increase of temperature. In the most popular models, γ ¼ 2. As we discussed earlier in this chapter, γ ¼ 2 gives unrealistically small values for the cost of environmental damage when ΔT gets larger than 3 or 4
C. It is safe to say that when speaking of potential environmental catastrophes, we are swimming deep in a soup of infinities. We are deep in the space in which Weitzman invited us in his series of papers (Weitzman 2008, 2012).

5.2.4

Deconstructing the Source of the Infinity

The reason why the ROA value of global mitigation here appears infinite is not only due to the choice of damage function. As we noted earlier, the distribution of temperature increase (Eq. 5.1) ψ T (ΔT ) has a fat tail, since when ΔT ! 1, ψ T ðΔT Þ  ΔT1 2 . Since any damage function would grow faster than linearly in ΔT, the fact that the damage function has a fat tail is a direct consequence and the fact the ROA of mitigation is infinite follows. The question is what is the origin and meaning of the fat tail of the distribution ψ T (ΔT )? In other words how did Roe and Baker (Baker and Roe 2007) come up with that expression for ψ T (ΔT )? Technically the atmospheric load of GHG does not appear explicitly in the expression for ψ T (ΔT ) in Eq. 5.1. Instead there is ΔT0 and a parameter f expressing the strength of the “feedback.” This is based on the fact that the physical effect behind the global warming is the increase in the radiation forcing ΔRf due to the 0 greenhouse 1 effect (Hansen et al. 2013). Typically σΔT0 ¼ κ0 ΔRf, oK

where κ 0  0:3 @W A is a physical constant measured empirically. The differ=m2 ence between ΔT0 and the actual value of the warming ΔT due to ΔRf is associated with the physical phenomena which are triggered by the radiation forcing. Those effects are captured by the “feedback” parameter f. In the words of Baker and Roe, “How do the uncertainties in the physical processes translate into an uncertainty in climate sensitivity? Explanations for the range of predictions of ΔT have focused on (1) uncertainties in our understanding of the individual physical processes (in particular, those associated with clouds), (2) complex interactions among the individual processes, and (3) the chaotic, turbulent nature of the climate system, which may give rise to thresholds, bifurcations, and other discontinuities, and which remains poorly understood on a theoretical level. We show here that the explanation is far more fundamental than any of these.” What Baker and Roe do by using a feedback term is to interpret the collective effect of all these processes as an emergent property, which cannot be reduced to the effect of the individual component.

5.2 ROA and Climate Change: Setting Up the Problem

77

The effect of the feedback enters in the following relation between the actual 0 value of ΔT and ΔT0: f ¼ 1  ΔT ΔT . If f ¼ 0, ΔT ¼ ΔT0 and 1 > f > 0, ΔT > ΔT0. So what do we know objectively about the parameter f ? Can we see wherefrom the fat tail of ψ T (ΔT ) comes, i.e., what is behind the fact that ψ T ðΔT Þ  ΔT1 2 when ΔT ! 1? In the climate system, 1 > f > 0 is a safe assumption. Not much more is known about f. So the assumption of Baker and Roe is that it has a Gaussian distribution of values with average f and variance σ f: "  2 # f  f 1 ψ f ð f Þ ¼ pffiffiffiffiffi exp  2σ f 2 σ f 2π

ð5:4Þ

The expression for ψ T (ΔT) is directly derived from Eq. 5.4 by writing it as: "   # 0  2 ΔT 0 1  ΔT df ðΔT Þ 1 ΔT  f ¼ pffiffiffiffiffi exp  ψ T ðΔT Þ ¼ ψ f ð f ðΔT ÞÞ d ΔT 2σ f 2 ΔT 2 σ f 2π

ð5:5Þ

Equation 5.5 is identical with Eq. 5.1. That means that the mathematical origin of ΔT Þ the fat tail, i.e., of the ΔT1 2 asymptotic dependence of ψ T(ΔT ), is the Jacobian d df ðΔT T0 ¼Δ . . . The “physical” origin is the combination of the assumptions that f ¼ 1 Δ T2 ΔT 0  ΔT with the assumption that 1 > f > 0 and that the values of f build a Gaussian distribution. They seem safe and rather innocent assumptions. But considering their implications, it would be nice to be able to anticipate them from first principles. The existence of the fat tail suggests the non-zero probability of an environmental catastrophe. This is due to the feedback. But note that even if the average value of the feedback f were small as well as the variance, the fat tail would still be there. The probability of events in the fat tail would be smaller, but still not zero. All this suggests that buried in the extreme complexity of the physical processes accompanying global warming, there is the possibility of some runaway process, which would make the temperature increase beyond the acceptable. This is plausible, but the way this conclusion is reached through this feedback argument is somewhat unsettling. It is as if that high-level argument had revealed the possibility of something sinister.

5.2.5

The Weitzman Effect Applies Here on Steroids

The “Weitzman effect” is explained in details in Chap. 4 in the context of fat tail (power law) distributions and catbonds. It is relevant here because the fat tail distribution of consequence here is the damage distribution. The damage distribution is what enters in ROA. Environmental damage X is related to the temperature by

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5 Global CO2 Emission Mitigation Through the Looking Glass of ROA

X ¼ θ (ΔT)γ . The exponent γ cannot be known precisely, except that it is larger than one (γ > 1) and grows when ΔT gets larger and larger. That means that the tail of the temperature distribution ΔT1 2 becomes the damage distribution 12γ . In fact, when one X

includes the effect of the change of variables, the actual asymptotic dependence in X of the damage distribution is 1γ1þ1 (Cf. Eq. 5.3). The fact that γ > 1 and its value is X

not known precisely means that the damage distribution is an example of a distribution with fat tail where the parameter controlling the tail is not known with precision. It falls squarely in the domain of application of the Weitzman effect. Basically the Weitzman effect says that if there is an uncertainty in the parameters controlling the tail of the distribution, the effect of the uncertainty is to strengthen the tail effect, i.e., increase the probability of large-scale events. Not only is there an uncertainty in the value of the parameter γ controlling the tail but also the effect of that uncertainty gets more and more dramatic when one goes deeper in the tail, because one can assume that γ increases with ΔT. This is a case of Weitzman effect on steroids. . . Considering that the value of mitigation is infinite in the first place, it emphasizes the conclusion of Weitzman that mitigation policy calls for approaches based on the precautionary principle. But invoking the precautionary principle cannot be the end of the story. It is a statement without solid quantitative content. Does that mean that we should approach mitigation the way urban pollution is approach: put in it what we can afford? That approach to pollution had its share of success, especially in rich countries. But there are quite a few large cities in the world, where in some places, it is almost dangerous to lie down too long: one may not get up again. . .Furthermore, one is not forced to live in an excessively polluted city, and pollution can be cleaned up over time. That option does not exist for global warming. There is no planet B to go to.

5.3

Extracting Finite Numbers Out of Infinities

Does the fact that the ROA value of mitigation is infinite preclude the possibility of finding finite numbers useful for mitigation policy? After all the 1965 Nobel Prizewinning physicists (Richard Feynman, Julian Schwinger, Shin’ichirō Tomonaga) perfected the art of extracting meaningful finite numbers out of infinities in quantum field theory.

5.3.1

Probability of Exceedance and the Importance of the Parameter ΔT0 in Eq. 5.1

For each value of the atmospheric load of CO2, there is a different PDF for the temperature increase. They all have the same asymptotic behavior when ΔT ! 1,

5.3 Extracting Finite Numbers Out of Infinities

79

Exceedance prob 1.0

0.8

0.6 0.4

0.2

0.5

1.0

1.5

ΔT0

Fig. 5.2 The probability of exceeding 2
C, as a function of ΔT0 with the formula of Baker and Roe. The horizontal axis is ΔT0 and the vertical axis gives the probability of exceedance

i.e., they go like ψ T ðΔT Þ  ΔT1 2 , but they differ on the value of one key parameter ΔT0. ΔT0 is what the increase of temperature at equilibrium would be, without
feedback. Baker and Roe used ΔT0 ¼ 1.2 C for climate sensitivity, which corresponds to doubling the atmospheric load of CO2 with respect to preindustrial time. With that value for ΔT0 (and the value of the feedback parameters f~ ¼ 0:7 and σ f ¼ 0.14), the probability that the temperature increase at equilibrium will exceed 2
C is above 90% (Cf. Fig. 5.2). To have a probability less than 5%, the value of ΔT0 should be about ten times smaller. That means that the atmospheric load should be dramatically smaller, but not ten times smaller as the relation between atmospheric load and value of ΔT0 is complicated. In fact it is so poorly understood that some prefer to use radiative forcing instead of atmospheric load of CO2. In the words of Hansen et al. (Hansen et al. 2013), “We usually discuss climate sensitivity in terms of a global mean temperature response to a 4 W/m2 CO2 [radiative] forcing. [. . . That] avoids the uncertainty in the exact magnitude of a doubled CO2 forcing estimate.” The best that mitigation can accomplish is to reduce further and further the probability of exceedance of 2
C. Because ψ T (ΔT ) has a fat tail, the probability of exceedance above any temperature cannot be reduced to zero. Fat tails can have very different origins and interpretations: when it comes to extreme events, they reflect a frequency distribution. When it comes to environmental damage, the cause is different: it comes from the fat tail of the temperature distribution. A compelling reason to use the probability of exceedance, instead of a value for temperature increase, is that as noticed earlier, the distribution ψ(ΔT ) does not have an average or more exactly its average is infinite:

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5 Global CO2 Emission Mitigation Through the Looking Glass of ROA

Z1 hΔT i ¼

ΔT ψ ðΔT ÞdΔT ¼ 1

ð5:6Þ

0

5.3.2

Environmental Value at Risk (EVaR)

Temperature increase is a phony measure of mitigation as at best that refers to the expected value hΔTi of a distribution ψ(ΔT), which does not have first moment. Instead it could be more consistent to use the probability of exceedance or to introduce an environmental equivalent of the value at risk. EVaR would be the value of the temperature increase ΔTEVaR, which corresponds to a predetermined probability of exceedance τ, i.e., prob[ΔT  ΔTEVaR] ¼ τ or equivalently R1 ψ ðΔT Þ dΔT. One could characterize a mitigation policy by the change of τ¼ X EVaR

EVaR it is expected to accomplish. The EVaR is very sensitive to the value of the parameters f~ and σ f, which control the distribution of values of the feedback. For illustration, the EVaR of τ ¼ 1% of the
PDF for temperature increase with ΔT0 ¼ 1.2 C is 40
C. That means that the temperature increase that has a probability of exceedance to 1% is 40
C! It is difficult to be very comfortable playing with formula, which generates such crazy results. For the record, if, instead of f~ ¼ 0:7 (the number chosen by Baker and Roe), we had taken f~ ¼ 0:3, everything else unchanged, the EVaR of τ ¼ 1% would have been 3.2
C. . . On the other hand, if one insists to keep f~ ¼ 0:7 and σ f ¼ 0.4 as values for those
parameters, to have 2
C as EVaR with τ ¼ 1%, one needs ΔT0 ¼ 0.06 C instead of
ΔT0 ¼ 1.2 C. This implies a rather drastic reduction of the atmospheric load of CO2. . . As mentioned before, the origin of the fat tail of the distribution ψ(ΔT ) (which 0 has a lot to do with those crazy numbers) is the relation f ¼ 1  ΔT ΔT . This looks like a solid assumption not so easy to relax. So it seems that if one wants to keep the basic structure of the Baker and Roe formula, tightening the estimate of the feedback seems the most attractive avenue. . . One advantage of introducing the concept of EVaR is that it connects mitigation policy to the world of financial risk management and in particular the value at risk (VaR) (McNeil et al. 2005). EVaR is a goal or constraint to be met in a context where economic forces, not that well understood, are at work. As Taleb et al. (Geman et al. 2015) suggest in the context of VaR, it is something that “can be constrained in a robust manner” in conjunction with the dynamics of economic change often poorly modeled by economic theories (Jaynes 1991). That dynamics reflects the laws of nature in the sense that they lead to the maximization of entropy compatible with constraints like

5.3 Extracting Finite Numbers Out of Infinities

81

VaR. “We equate the return distribution with the maximum entropy extension of constraints expressed as statistical expectations on the [..] tail behavior [. . .]. One strategy is to make such estimates and predictions under the most unpredictable circumstances consistent with the constraints. That is, use the maximum entropy extension (MEE) of the constraints.” This vision can be mapped on mitigation policy to provide an interesting framework to discuss it. With the absence of any mitigation effort, the entropy of climate change will naturally increase to its maximum, whether this is due to natural variability of anthropogenic effects. Mitigation policy is a way to introduce additional information or constraints in that system and thereby reduce its maximum entropy. The climate change system at any given time can be characterized by its EVaR or equivalently a distribution ψ(ΔT ). The corresponding entropy is: Z1 hðΔT 0 Þ ¼

ψ ðΔT Þ logðψ ðΔT ÞÞ dΔT

ð5:7Þ

1

The term ΔT0 is to remind the reader that mitigation means changing the value of ΔT0 to get a new EVaR. Mitigation can be interpreted as a transport in the space of distributions from one distribution ψ 1(ΔT ) for the temperature increase to another ψ 2(ΔT ) with a smaller EVaR. In the process, the maximum entropy of the system is reduced. This could provide some quantification of the effect of mitigation in the same spirit as Taleb et al. used maximum entropy extension of the constraints to estimate returns in portfolios.

5.3.3

Relative Entropy and the Kullback–Leibler Divergence

The fact that mitigation can be conceptualized as a transport in the space of probability distributions suggests that the mathematics of mitigation policy could benefit from the vast, hot, and fast-advancing world of optimal transport (Villani 2008). A natural place to start is the “Kullback–Leibler (KL) divergence” (Kullback and Leibler 1951). The KL divergence is a special case of the “f-divergence,” i.e., it is a tool used to discriminate between distributions. It is based on the observation that any distribution φ(x) has the information content:

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5 Global CO2 Emission Mitigation Through the Looking Glass of ROA

Z1 h ð φÞ ¼ 

φðxÞ logðφðxÞÞ dx

ð5:8Þ

1

It is the negative of its entropy. The KL “divergence” is defined as:

Z1 KL½ψ 1 , ψ 2  ¼

ψ 1 ðΔT Þ log 1

ψ 1 ðΔT Þ dΔT ψ 2 ðΔT Þ

ð5:9Þ

KL[ψ 1, ψ 2] is called KL “divergence” and not KL “distance”, because (among other reasons) as can be seen by its definition, it is not in ψ 1(ΔT ) and h symmetric i ψ 1 ðΔT Þ ψ 2(ΔT ). The KL divergence is the average value of log ψ ðΔT Þ over the distribution 2 h i ψ 1 ðΔT Þ ψ 1(ΔT ). log ψ ðΔT Þ is the information discriminating between the distributions 2

ψ 1(ΔT ) and ψ 2(ΔT ). Sometimes called the relative entropy, the KL divergence can be interpreted as the amount of information needed to go from the distribution ψ 1(ΔT ) to ψ 2(ΔT ). It has two attractive features here: first it is finite and second it provides a kind of proxy to quantify the effect of mitigation. Using the formulas for ψ 1(ΔT ) and ψ 2(ΔT ), we get: KL½ψ 1 , ψ 2  ¼ log

  ΔT 0,1 ΔT 0,1 pffiffiffiffiffi Ω  ΔT 0,2 4 σ f 3 2π

with: Z1 Ω¼ 1

1 ΔT 2

2 ! 3 2  2 ) ΔT 0,1 2 ~ ΔT 0,1 ΔT 0,2 1 1f  ΔT 5dΔT 1~f   1~f  exp4 2 ΔT ΔT σf

(

KL[ψ 1, ψ 2] is finite as seen in Fig. 5.3. The smaller ΔT0, 2 (i.e., the more drastic the mitigation), the larger KL[ψ 1, ψ 2].

5.4

A Cheap and Easy Way to Improve the Situation?

We end up this chapter by a remark, which may look flippant but which is based on real facts and may be illustrative of some of the deeper problems of mitigation policy. Before we try to use sophisticated tools, there are some basic issues which in a rational world would have been addressed and solved long ago. . .

References KL[

83 1,

2]

3.0 2.5 2.0 1.5 1.0 0.5

0.2

0.4

0.6

0.8

1.0

1.2

ΔT02


Fig. 5.3 Horizontal axis
is ΔT0, 2; vertical axis is KL[ψ 1, ψ 2]. We assumed ΔT0, 1 ¼ 1.2 C. As a result when ΔT0, 2 ¼ 1.2 C, KL[ψ 1, ψ 2] ¼ 0

The IMF (IMF 2015) calculated that “were the subsidies [the governments provide to support the use of fossil fuel] to be eliminated this year they would raise government revenues round the world by $2.9tn, cut global CO2 emissions by more than 20% and reduce the estimated 1m people who die premature deaths from air pollution by more than half. They would also raise ‘global economic welfare’ by $1.8tn, or 2.2% of GDP.” As Nick Stern says, “it shatters the myth that fossil fuels are cheap by showing just how huge their real costs are.”

References Baker, M.B., Roe, G.H.: Science. 318, 629–632 (2007) Geman, D., Geman, H., Taleb, N.N.: Tail risk constraints and maximum entropy, entropy. 17, 3724–3737 (2015) Hansen, J., Kharecha, P., Sato, M., Masson-Delmotte, V., Ackerman, F., et al.: Assessing “dangerous climate change”: required reduction of carbon emissions to protect young people, future generations and nature. PLoS One. 8(12), e81648 (2013). https://doi.org/10.1371/journal.pone. 0081648 Held, I.M.: The Vertical Scale of an unstable baroclinic wave and its importance for Eddy heat flux parameterization. J. Atmos. Sci. 35, 572 (1978). http://journals.ametsoc.org/doi/pdf/10.1175/ 1520-0469%281978%29035%3C0572%3ATVSOAU%3E2.0.CO%3B2 IMF: Climate, environment, and the IMF. http://www.imf.org/external/np/exr/facts/enviro.htm (2015) Jaynes, E.T.: How should we use entropy in economics? Saint John’s College, Cambridge (1991) Kullback, S., Leibler, R.A.: On information and sufficiency. Ann. Math. Stat. 22(1), 79–86 (1951) McNeil, A.J., Frey, R., Embrecht, P.: Quantitative Risk management, concepts, techniques and tools. Princeton University Press, Princeton (2005) Meinshausen, M., Meinshausen, N., Hare, W., Raper, S.C.B., Frieler, K., Knutti, R., Frame, D.J., Allen, M.R.: Nature. 458, 1158 (2009)

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Nordhaus, W.: The Dynamic Integrated Climate Economic model (DICE). http://www.econ.yale. edu/~nordhaus/homepage/documents/DICE_Manual_100413r1.pdf Phillips, N.A.: Tellus. 6, 273–286 (1954) Stone, P.H.: Baroclinic adjustment. J. Atmos. Sci. 35, 561 (1978). http://eaps4.mit.edu/research/ papers/Stone_1978.pdf Villani, C.: Optimal transport, old and new. Springer, Berlin (2008). http://cedricvillani.org/wpcontent/uploads/2012/08/preprint-1.pdf Weitzman, M.: On modeling and interpreting the economics of catastrophic climate change. 680 Harvard Preprint. http://www.economics.harvard.edu/faculty/weitzman/papers_weitzman (2008) Weitzman, M.: Rev. Environ. Econ. Policy. 5(2), 275–292 (2011) Weitzman, M.: Precautionary tale about uncertain fat tail flattening, NBER, Working Paper 18144 683 (2012)

Chapter 6

Internationalization of the Response: The Example of the REDD Credits

Trees are the earth’s endless effort to speak to the listening heaven, Rabindranath Tagore (https://www.brainyquote.com/quotes/ rabindranath_tagore_101654)

Abstract One can read in Chap. 11 of the WGIII AR5 IPCC report that REDD credits (REDD is for Reduced Emissions from Deforestation and Forest Degradation) “can represent a cost-effective option for mitigation with economic, social, and other environmental co-benefits, e.g., conservation of biodiversity and water resources” with the caveat that there is limited evidence for that and only medium agreement. In this chapter, using the ROA looking glass, we investigate some aspect of how to make REDD credits cost-effective and compare with what actually happens. Despite the apparent difference, there is more similarity between what the ROA approach would recommend and what takes place.

6.1

Importance of Forests and REDD Credits

It may be a platitude but still true that forests are influenced by climate and at the same time a contributor to climate. They are an important reservoir for carbon but, at the same time, can be a good emitter of carbon when they burn. Estimates of the contribution to GHG emissions coming from deforestation or forest degradation vary somewhat, but they tend to be around 15–20% of the total GHG emissions worldwide (comparable with the transport sector). Deforestation results in immediate release of the carbon stored in the trees as CO2 emissions (with small amounts of CO and CH4). According to the UN agency FAO, deforestation, mainly for the conversion of forests to agricultural land, continues at an alarming rate. For the period 1990–2005, it was approximately 13 million hectares per year. Most of the deforestation takes place in tropical forests. The preservation of tropical forests has become an important component of global climate change policy. In countries like

© Springer Nature Switzerland AG 2020 B. Morel, Real Option Analysis and Climate Change, Springer Climate, https://doi.org/10.1007/978-3-030-12061-0_6

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6 Internationalization of the Response: The Example of the REDD Credits

Brazil or Indonesia, deforestation has an economic purpose, such as logging or planting industrial crops (sugar cane or oil palm).

6.1.1

REDD Credits

REDD credits (originally introduced in 2005 at the UNFCCC COP11) are financial subsidies aimed at keeping and/or maintaining tropical forests. Tropical forests more often than not are in poor countries. If a developing nation is asked to refrain from an activity, which would have benefited its economy, the REDD subsidy should offset the economic loss. The way REDD credits work in practice is very suboptimal, and that may explain why they have not been very effective. Since REDD credits are a way to offset carbon emissions, REDD credits could be construed as a form of carbon credits. If there were a universal price for carbon (Parry et al. 2014), the quantification of REDD credits would be far less arbitrary. Still its implementation raises a variety of issues, like how they are distributed and how does one guarantee compliance. Despite the fact that there are reasons to be skeptical that such a thing as a universal price for carbon can be agreed upon in a world where there is such a high level of economic diversity, to establish a price for carbon is a stated goal of the post COP21 UNFCCC cycle of conferences. In the executive summary of Chap. 20 of the IPCC AR4 wg2 full report, one can read: “More than 100 estimates of the social cost of carbon are available. They run from US$
10 to US$+350 per ton of carbon. Peer-reviewed estimates have a mean value of US$43 per ton of carbon with a standard deviation of US$83 per ton.” Still REDD credits have the potential to add some cost.efficiency to global mitigation. This is because it is conceivable that at the same cost, more carbon could be saved from forests than from industrial activity. The carbon content of forests varies widely. If one assumes a price of US$43 per ton of carbon, a forest with 200 t of carbon per ha would theoretically justify $860 per ha of REDD subsidy in average. But the price of carbon comes with a large variance, and if it is decided by a market, it is bound to vary with time, as a commodity. How does that compare with the revenue the potential “deforester” would have made? Furthermore when computing the value of the forest, is the carbon content the only consideration? What about biodiversity, ecosystem services, or even tourism? Clearly if one wants to optimize the REDD credit system and make it as cost-efficient as possible, each forest should be treated a special case. In the next paragraph, we make an ROA estimate of how theoretically REDD credits could work. Then we compare with the way REDD credits actually work.

6.2 What REDD Credits Could Be in Theory

6.2

87

What REDD Credits Could Be in Theory

The receiver and provider of REDD credits follow different rationalities. Both face uncertainties in their economic assessment of the transaction. But the uncertainties they face are completely different in their nature and origins.

6.2.1

Modeling the Uncertainties

Key in the analysis are the probability distributions ψ X(X, t) and ψ Y (Y, t) for the benefit for the provider and the “loss” for the receiver. The former is basically measured as the amount of carbon “saved.” The second is the revenue that the deforester would have reaped over time from exploiting his forest. In the case of the benefit for the provider, the main source of uncertainty is the value of carbon. In the case of the deforester, the main source of uncertainty is the price of commodities like palm oil. The time dependence of the distributions ψ X(X, t) and ψ Y (Y, t) reflects how the uncertainty looks like at a time “t” in the future, at the time of the REDD credit transaction. The larger the “t” is, the larger the uncertainty (Cf Box 6.1). Box 6.1 Modeling the Uncertainties If the uncertainties here stem either from the price of carbon or the price of a commodity like palm oil, i.e., market-based prices decided, it is fair to model them as proceeding from a geometric Brownian motion, i.e., with the stochastic differential equation: dX ¼ α X dt þ σ X dz This assumes that the price of carbon is also determined through a market mechanism as was implicitly what the Kyoto Protocol called for. The “solution” of the equation is the time-dependent lognormal distribution: 2      2 3 X= σ2 ln þ α
X 0 2 t 7 1 6 ψ ðX, t Þ ¼ pffiffiffiffiffiffiffiffi exp4
5 σ2 t σ 2π t αt mean  The  of that distribution e grows with time as does its variance: 2 eσ t
1 e2α t . With α ¼ 0.1 and σ ¼ 0.5, the mean is 1.1 at t ¼ 1 and 1.6 at

(continued)

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6 Internationalization of the Response: The Example of the REDD Credits

Box 6.1 (continued) t ¼ 5. The variance goes from 0.34 to 6.8. When the volatility is high, the uncertainty grows very fast. 0.8

0.6

0.4

0.2

2

4

6

8

10

The time-dependent lognormal distribution for the price of carbon for t ¼ 1 and t ¼ 5, with X0 ¼ 1

6.2.2

Framing the Quantification of the Transaction

If ψ Y (Y, t) is the revenue at time t the receiver of the credit would have reaped from exploiting the forest, the expected revenue the deforester should be compensated for, Z1 at time t, is hY ðt Þi ¼ Y ψ Y ðY, t Þ dY. His revenue streams up until the horizon time ZT T is Y total ðT Þ ¼ 0

0

e
ρ t hY ðt Þidt. Ytotal(T ) measures the NPV value of what the

receiver “losses” by refraining to exploit his forest. The total expected “environmental benefit” Xtotal(T ) can be computed along the same lines as Ytotal(T ). Forests have different carbon content. That means that the value of Xtotal(T ) depends on the forest. If CREDD is the REDD credit, i.e., the amount of money actually exchanged between the provider and the deforester, what should its value be?

6.2 What REDD Credits Could Be in Theory

6.2.2.1

89

Valuing CREDD

The expected benefit for the receiver is the following expression:

H receiver ðC REDD , T Þ gain

¼e


ρ T

CZREDD

ðCREDD
Y Þ ψ Y ðY, T ÞdY

ð6:1Þ

0

It is controlled by the probability that he gets more through REDD credit than he would have exploiting his forest. From the perspective of the provider of REDD credit, the expected benefit is that the price of carbon could have been higher than what he paid for, i.e.: H provider ðC REDD , T Þ gain

¼e


ρ T

Z1 ðX
CREDD Þ ψ X ðX, T ÞdX

ð6:2Þ

C REDD

H receiver ðCREDD , T Þ and H provider ðCREDD , T Þ look very much like the value of a put gain gain option and call option, with exercise time T and exercise price CREDD (Chap. 2, Eqs. 2.11 and 2.12). CREDD is a parameter. Its value could be chosen so that the two expected benefits are equal. That would provide a “fair” or optimal value for the REDD credit. A much simpler way to evaluate CREDD for the provider would be to compare that value with the expected savings Xtotal(T ) it buys in carbon emission. For the receiver the comparison should be with respect to its “expected loss” Ytotal(T ). In that logic, one could look for a value of CREDD, which is halfway between Xtotal(T ) and Ytotal(T ). In Box 6.2 we show (on a cooked-up example) that the value of CREDD depends on what approach is chosen. H receiver ðC REDD , t Þ and H provider ðC REDD , t Þ are better gain gain measures of the expected gains for each party. A “fair” or optimal value for CREDD is when both are equal.

6.2.2.2

All Forests Are Not Equal

People tend to assume that CREDD should be the value of the carbon saved. No consideration is made of the economic impact of not deforesting, and the uncertainty on the price of carbon in the future does not enter in the calculation. In this ROA approach, CREDD is recomputed on a case-by-case basis, and it takes care of the specific of the forest and of the economic situation of the deforester. The carbon content of forests varies. For forest with low carbon content, CREDD needs to be small for the provider to make a profit. But if CREDD is small, the expected gain of the receiver is small too. Then the fair value of CREDD corresponds to a situation where the expected benefits are small. Whereas if the carbon content of

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6 Internationalization of the Response: The Example of the REDD Credits

the forest is high, the fair value of the REDD credit CREDD will be larger as will the expected benefits of the provider and receiver. That suggests that to maximize the cost efficiency of the REDD credit mechanism, the policy should focus on the forest which creates the largest expected benefit. Carbon content is not the only consideration. Ecosystem services as well as biodiversity, for example, are additional considerations. Box 6.2 Estimates for CREDD Cooked-up example: We chose ψ X(X, t) and ψ Y (Y, t) with the same average, i.e., we assumed hX(t)i ¼ hY(t)i. That implies that in the simpler scheme, CREDD ¼ hX(t)i ¼ hY(t)i. We did it on purpose because it makes the effect of the uncertainties on the different other quantities easier to appreciate.

0.4

0.3

0.2

0.1

2

4

6

8

10

The distributions ψ X(X) and ψ Y (Y ) are assumed to be the lognormal distributions with parameters μX ¼ 0.5, σ X ¼ 1.3 and μY ¼ 1.22, σ Y ¼ 0.5, respectively

This example has been “cooked up” so that hXi ¼ hYi ¼ 3.83819. If one takes CREDD ¼ hXi ¼ 3.83819, the expected gain of the provider is ðCREDD Þ ¼ 1:86, whereas the expected gain of the receiver is H provider gain receiver H gain ðCREDD Þ ¼ 0:76. If CREDD ¼ 5.02, both gains become equal at 1.59. What is the “fairest” choice?

6.3 Complicating Factors: Alternative Use, Ecosystem Services, Biodiversity, etc

6.3

91

Complicating Factors: Alternative Use, Ecosystem Services, Biodiversity, etc

One consideration that is never discussed is how the REDD credit is used. If the subsidy is invested in other lucrative activity, this could enter in the general calculation. And if it is to engage in an economic activity producing carbon, this would defeat the purpose of the subsidy. Another consideration when computing the amount of carbon saved is to investigate how the land would have been used after deforestation. If it were for agriculture (it probably would have in most cases), then the question is how much carbon that would have released in the atmosphere. Agriculture is a good source of GHG. That should be added to the amount of carbon saved by refraining from deforesting. But in addition, global warming creates an environment more conducive to forest fire, as happened in Borneo and Sumatra in 1997 and 1998. Not only they affected the air quality in the area, but they also were very good atmospheric CO2 contributors. More events like that belong clearly in the possible futures. Should that be a consideration when valuing REDD credits? Where does that stop? Probably it never stops. But many of these effects can be factored in a ROA estimate. The possibility of a forest fire, for example, can be treated like a natural disaster as discussed in Chap. 4. It is possible to quantify this effect. One obvious problem is that pushing the economic estimate to that limit of precision is far and above the head of the prevailing bureaucratic rationality. This is a binding constraint, which comes at a cost, an overhead due to the overwhelming power of bureaucracy. By contrast, the fact that the carbon content is not the only factor in the value of a forest is often mentioned. Biodiversity and ecosystem services play an important role. “Ecosystem services are the benefits people obtain from ecosystems. These include provisioning services such as food, water, timber, and fiber; regulating services that affect climate, floods, disease, wastes, and water quality; cultural services that provide recreational, aesthetic, and spiritual benefits; and supporting services such as soil formation, photosynthesis, and nutrient cycling. The human species, while buffered against environmental changes by culture and technology, is fundamentally dependent on the flow of ecosystem services” (MEA 2005).

6.3.1

Valuing Ecosystem Services

How to price ecosystem services is not obvious: the cost of losing ecosystem services is difficult to assess, and those losses are irreversible. A rational strategy is to hedge against those losses. Refraining from deforesting could be part of that hedging strategy (Fisher 2000) (Pindyck 2000) (Hanemann 1989). Some ecosystem services are threatened by climate change. Protecting those ecosystem services are part of the response to climate change. Adaptation and mitigation are very different

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6 Internationalization of the Response: The Example of the REDD Credits

problems (Cf Chap. 7). The protection of ecosystem services is one of a few instances where mitigation and adaptation intersect.

6.3.2

Biodiversity

And there is biodiversity. Biodiversity deals with the whole fabric of life on the planet. It is not affected only by climate change and it is not limited to forests. The “sixth great extinction” which is taking place now has more to do with economic and industrial activity than climate change. Unlike the previous five ones, this extinction is primarily manmade (Sachs 2015). The sixth great extinction affects biodiversity and its ecological “services.” The economics of biodiversity is a foggy subject (Helm and Hepburn 2014). Readers wanting to learn about that subject have to go through a barrage of papers, which with (not so) different words reiterate the same fundamental fact that although biodiversity has clearly a high economic value, that value is difficult, if not close to impossible to estimate.

6.3.3

Weisbrodian Perspective

Running the danger of going a bit too far in trying to give an option value to biodiversity and its protection, we make a small Weisbrodian digression. In 1964, Weisbrod used the word “option value” in the context of environmental policy (Weisbrod 1964). He took the example of a national park, which operates at a cost. In order to avoid closing the park, he suggested the possibility of enticing potential customers to pay to support the park. They would pay to keep open the option of visiting the park any time in the future. One problem is to estimate the value of that kind of “option.” Clearly the willingness to pay is the real yardstick. But what does it measure? Should it be assimilated to a risk premium, people are prepared to pay to hedge against the uncertainty on the future value of the park (Schmalensee 1972), or can it be interpreted as a reaction against the irreversibility of the decision of closing the park (Arrow 1974)? One way or the other, the value of the option as defined by Weisbrod is the price or premium that people accept to pay to have the (public) good available in the future. If public willingness to pay can be taken as a measure for the value of a public good, can that be applied to the valuation of biodiversity? If the protection of biodiversity is a way to maintain a certain level of ecological resilience, ecological resilience is a kind of a “Weisbrodian public good”. There should be an option value for it. Its interpretation could be that it represents the cost to ensure that these public goods will be available in the future. Its actual quantification would be based on a willingness to pay.

References

6.4

93

REDD Credits in Practice

REDD in its present incarnation is not so much a market-based system as it is part of development programs. To qualify under UNFCCC for REDD funds, the countries have to produce a REDD+ strategic plan. REDD+ adds to REDD a few more concerns: “sustainable forest management,” “conservation of forest,” and “enhancement of carbon sinks.” The REDD+ credits tend to be negotiated at the state level, based on national programs, i.e., they seem to follow a completely different rationality than any market mechanism. In fact they seem to be treated as component of sustainable development where mitigation and adaptation (potentially) synergize. The system whereby REDD+ credits are allocated from “proposals” by the receiver of the credit in fact potentially makes their assessment not so dissimilar from the ROA exercise earlier in this chapter. To justify the amount of REDD credits he seeks, the receiver makes an evaluation of the economic implication for him of not deforesting as well as an informed estimate of the amount of carbon saved in the process. His evaluation must pass the scrutiny of the donor. Between the two they make an evaluation of the project along the same lines as the ROA. Unfortunately, this is the way it should be, but not the way things are done in practice. REDD+ credits tend to be more influenced by political and bureaucratic considerations rather than by an ROA-like economic rationality. There is little to no effort to address the drivers of deforestation let alone maximize the cost efficiency of the policy, which has yielded few tangible results. In fact, deforestation in Indonesia, for example, is accelerating, despite REDD+. Why is that? One, there is no agreed upon price for carbon, since the European CO2 emission cap and trade system (the “Emission Trading Scheme” or ETS) collapsed as a result of the financial crisis of 2008. Without any carbon price reference, it is difficult to put a dollar value to the amount of carbon saved. Second, the negotiating partners tend to be governments, which more often than not have limited authority to implement their side of the deal. Hence there is a problem of monitoring of what happens to the forests. There is a talk after the Paris COP 21 to resurrect the ETS system in Europe. In its previous incarnation, forestry and agriculture were not included. Will they be this time? In any case, why not letting European authorities decide whether they prefer to pay their CO2 allowance in avoiding deforestation instead of at high cost reducing their emission? Arguments against more bang for the buck can only come from people for whom cost efficiency is a second priority, i.e., typically bureaucrats.

References Arrow, K.J.: The limits of organization. Norton, New York (1974) Fisher, A.: Introduction to special issue on irreversibility. Resour. Energy Econ. 22(3), 189–196 (2000)

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Hanemann, W.M.: Information and the concept of option value. J. Environ. Econ. Manag. 16(1), 23–27 (1989) Helm, D., Hepburn, C.: The economic analysis of biodiversity. In: Helm, D., Hepburn, C. (eds.) Nature in the Balance. Oxford University Press, Oxford (2014) Millennium Ecosystem Assessment. http://www.millenniumassessment.org/en/index.html (2005) Parry, I., Veung, C., Heine, D.: How much carbon pricing is in countries’ interests? The critical role of co-benefits. IMF working Paper (2014) Pindyck, R.: Irreversibilities and the timing of environmental policy. Resour. Energy Econ. 22(3), 233–259 (2000) Sachs, J.D.: The age of sustainable development. Columbia University Press, New York (2015) Schmalensee, R.: Option demand and consumer’s surplus: valuing price changes under uncertainty. Am. Econ. Rev. 62(5), 813–824 (1972) Weisbrod, B.A.: Collective-consumption services of individual-consumption goods. Q. J. Econ. 78 (3), 471–477 (1964)

Chapter 7

Prioritizing the Investments Needed to Avoid the Unmanageable (Mitigation) and to Manage the Unavoidable (Adaptation)

Men argue. Nature acts. –Voltaire (https://www.goodreads.com/quotes/445307-menargue-nature-acts) Tackling the immense challenge of climate change must involve both mitigation—to avoid the unmanageable—and adaptation—to manage the unavoidable—all while maintaining a focus on its social dimensions. –The World Bank (https://unfccc.int/sites/default/files/ worldbankgreenbondfactsheet.pdf)

Abstract According to the present “scientific evidence,” the “unmanageable” is part of our possible futures. Mitigation of CO2 emissions is an imperative. There are many sources of emissions of greenhouse gases (GHGs), and however global the mitigation effort may be, it has a strong local component. The rest of the world has a vested interest in mitigation efforts everywhere. It is a case of “act locally but think globally.” Mitigation involves also transitioning to a greener economy. Adaptation, on the other hand, is the purview of individual countries. Their needs and situations differ widely. Most countries have an adaptive capacity gap, including the USA (Preston et al. 2011). That was eloquently illustrated by the devastation that Hurricane Katrina brought to New Orleans in August 2005. Still, the gap is as a rule more pronounced in poor countries, which are also the least equipped to fill it. Furthermore, adaptation intersects with other concerns: “Climate change adds to the list of stressors that challenge our ability to achieve the ecologic, economic and social objectives that define sustainable development” (IPCC 2007). This chapter discusses the interface between real option analysis (ROA) and the response to climate change in general terms. The interface is rather complex and involves quite a few moving parts. In this chapter those issues are discussed in general. A more pedestrian introduction to how to operationalize ROA in the context climate change investments can be found in Appendix B.

© Springer Nature Switzerland AG 2020 B. Morel, Real Option Analysis and Climate Change, Springer Climate, https://doi.org/10.1007/978-3-030-12061-0_7

95

96

7.1

7

Prioritizing the Investments Needed to Avoid the Unmanageable. . .

Adaptation and Mitigation Are Different Problems

Characterizing the response to climate change as “avoiding the unmanageable and managing the unavoidable” (SEGCC 2007) has the advantage of capturing the imagination and of pointing to the important distinction between mitigation (avoiding the unmanageable) and adaptation (managing the unavoidable), although they sometimes tend be considered as part of the same package, because both are part to the “response to climate change.” The fact that some mitigation measures also contribute to adaptation is not an excuse to confuse them. Mitigation is a global effort to limit the emission of GHG. Adaptation for individual nations means addressing the impact of global change on their welfare and economy. Mitigation is about transitioning to a green economy. The focus is on reducing GHG emissions in all possible ways. Adaptation is about responding to the negative impact of climate change. All nations are affected, but very differently. And it takes many forms, including change of local climate, desertification, intensification of hurricanes or typhoons, protecting biodiversity in forest and oceans, facing the danger of sea level change, water management, and security issues raised through migration or a fight for resources. Mitigation and adaptation have something in common: the response involves many investments of all shapes and sizes. When it comes to adaptation, there is what N. Stern calls a “double inequity”: the countries that are the most vulnerable to climate change are also the least responsible for it. Furthermore, they tend to be countries with the most limited resources. ROA is appropriate to a subset of the investments needed: those that are substantial and involve a large uncertainty. Publications like the authoritative IPCC reports provide laundry lists of recommendations. The resources needed to fund those recommendations are in the trillions of dollars, far more than what realistically can be raised. A system of prioritization is needed, but no good ones exist. This is where ROA can be useful. By nature those investments are mired in uncertainty, making ROA potentially the only tool available to estimate their economic value. Another concern about the response to climate change is how to summon enough interest from the private sector to fund the response. ROA can help design climate bonds and green bonds.

7.2 7.2.1

Financing Adaptation The Context

Chapter 20 of the IPCC AR4 states: “Climate change leads to the degradation of forests, ecosystems, coastal area; it complicates the management of fresh water resources; it is responsible for the destruction of mangroves, the acidification of the oceans, the melting of the poles, mass extinctions and the shrinking of

7.2 Financing Adaptation

97

biodiversity; it represents an existential threat on some low lying islands; it is behind the spread of arid lands in areas where there is agriculture, leading to social disorder through environmental migrations and even potential wars on scarcer water resources. Furthermore, climate change acts also as an additional stressor in areas where there are sanitation problems like large cities in poor countries or areas exposed to natural disasters like typhoons or hurricanes. Climate change has also health effects and even can affect industry. It is difficult to identify any sector of the economy completely immune from climate change. The space within which climate change policy has to live includes also concerns about poverty reduction and sustainable development, a widely-held social and political goal, even though, implementation remains problematic” (IPCC 2007). Adaptation to climate change means addressing a large cocktail of entangled concerns including sustainable development. Furthermore, climate change does not affect all countries in the same way, and to make things worse, the communities or countries that are most affected tend to be the ones that can least afford an adaptation policy. The “climate vulnerable forum” (CVF) seeks to address that problem: “The CVF is a global partnership of countries that are disproportionately affected by the consequences of global warming, [. . .] that actively seek a firm and urgent resolution to a current intensification of climate change [because it represents] an existential threat to [those] nations, cultures and way of life.” Based on their Climate Change Vulnerability Index (CCVI), the company Maplecroft has determined that in 2014, Bangladesh, Guinea-Bissau, Sierra Leone, Haiti, South Sudan, Nigeria, Democratic Republic of Congo, Cambodia, the Philippines, and Ethiopia were the most vulnerable countries to climate change. In previous years, India, Madagascar, Nepal, and Mozambique were also deemed among the most vulnerable. Bangladesh tops the list of vulnerable countries. Today, Bangladesh loses 1% to 4% of its GDP annually to natural disasters. Infrastructure, sanitation, and urbanization projects are significantly more difficult to carry out due to continuous interference from natural disasters. Any major calamity—the kind Bangladesh experiences regularly—can destroy any socioeconomic gains from other programs. Nigeria produces oil and could soon be among the 20 largest economies in the world. But it is also very vulnerable to climate change, in particular its agriculture, which could suffer losses of about 30%. The fact that poor countries are the most vulnerable and the least equipped to pursue an aggressive adaptation policy has led the United Nations Framework for Climate Change (UNFCC) to launch an initiative whereby the least-developed nations can produce a National Adaptation Program for Action (NAPA) report, in which they can estimate the price of their immediate needs. There are about 38 NAPAs in the database.1 They are only about costs. Benefits are treated as selfevident, even when they differ widely in size and uncertainty. Bangladesh stated in

1 http://unfccc.int/adaptation/workstreams/national_adaptation_programmes_of_action/items/4583. php

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Prioritizing the Investments Needed to Avoid the Unmanageable. . .

its NAPA report: “Comprehensive estimates of adaptation costs and benefits are currently lacking.” Under those circumstances, how can one prioritize investments? In the IPCC AR4 report, one can read the sad but valid comment: “The low levels of funding available to LDCs for completing NAPAs likely placed inherent constraints on their ability to serve as robust platforms for long-term adaptation planning”(IPCC 2007).

7.2.2

Where ROA Comes In

In the words of Benjamin Preston et al. (2011), “Climate adaptation is fundamentally a process of social learning. Yet, in the absence of methods for evaluating adaptation, opportunities for learning are lost. [There is a need to develop a new methodology that] enables the development of more robust adaptation policy over time, through adaptive management.” Those writing the executive summary for policymakers in IPCC reports have perfected the art of presenting their recommendations in such a way that they poorly conceal what they actually are: a laundry list of vulnerabilities to address, without guidance for prioritization. No cost-benefit analysis of adaptation measures is possible without some quantitative estimates of impacts. Learning to estimate impacts and not limiting the analysis to vulnerability assessment is clearly a high priority if one wants to go forward. Prioritization is an important missing link where ROA can make a real difference. An ROA analysis of the risks of hurricanes in the Louisiana coastal region prior to Katrina would have shown a fat tail distribution of potential damage. No doubt, the fat tail would have pointed to the vulnerability of New Orleans and the value of reinforcing it against hurricanes. That did not take place. The rest is history. After the traumatic episode of Katrina in 2005, the famous Yale economist William Nordhaus asked (Nordhaus 2006): “What kinds of policies should be undertaken to cope with rising seas and the possibility of more intense hurricanes? Should cities like New Orleans be abandoned to return to salt marshes or ocean?” Or as Mario Draghi once said (with regard to the Euro), we should “do whatever it takes” to save cities like New Orleans. Large hurricanes destroying major cities, as occurred with Katrina, are deep in the tail of fat tail distributions. There is not much we can do to avoid that kind of hurricane. But there are things that can be done to limit the damage they might inflict. That kind of investment comes at a cost and will have to be assessed in comparison with other adaptation investments. Their costs can be estimated. Prioritizing such investments should not be on the basis of their costs only but also—and mostly—on the basis of their expected benefits. That requires being able to cope with an uncertainty whose distribution is a fat tail. That requires the detour of ROA. That net present value (NPV) is useless in those situations (although it is used) is obvious. The action is in the tail of distribution. NPV does not make any difference between distributions with or without tail.

7.2 Financing Adaptation

99

Extreme events are becoming a major concern in the context of climate change. IPCC dedicated a 594-page report to that subject, entitled “Special Report on Managing the Risks of Extreme Events and Disasters to Advance Climate Change Adaptation” (SREX 2012). The cause of the problem raised by extreme events is global warming. So trying to mitigate that could be construed as part of the measures to address extreme events. The problem is that the effect of mitigation of GHG emissions on extreme events will not be known for a very long time. Immediate measures have to be taken, which fall squarely in the area of adaptation. The IPCC report SREX is very good at projecting the impression that extreme events can take so many forms that confronting them is a daunting task. But in some cases, the complication can be an advantage. For example, the report states, “tropical cyclones can have very different impacts depending on where and when they make landfall.” This is an advantage in the sense that the expected benefit of adaptive measures against that kind of contingency is distributed very unequally, helping to identify where to concentrate the greater adaptation effort. Vulnerability assessment plays a central role in the design of adaptation policies. Sometimes many factors collude to build “extreme events”: for example, the combination of a heat wave and low humidity can trigger forest fires in some forests with old trees. Those fires can become completely out of control if a response system is not in place. In California, for example, some fires stress the response system. In much poorer countries, the actual fires could easily be much larger. But what makes the event “extreme” is the cost inflicted on the community. Arguably, the economic damage to poor communities may be smaller than to the rich communities that inhabit California in dollars terms, but as proportion of the GDP of the countries involved, the impact is often the opposite. This quantification ex ante is problematic. Simulating scenarios may provide some analytic power. But extreme events are by nature rare. That scenario exercise will have to explore remote corners of the space of the possible, which have very low probability. By nature it is difficult to generate those low probability events through random walk or Monte Carlo, i.e., tools that analysts like to use. An important fraction of the financing of adaptation is through insurance. The way insurance companies compute their premiums follows closely the logic of ROA. They need to have a good idea of what their expected losses are to decide their coverage. This is an area of adaptation where ROA is useful and in fact is used in practice, as is the case whenever one needs the expected value of a benefit or of a loss. This is what the ROA value is. When it comes to ROA, it is important to emphasize that ROA is not a simple set of recipes. It requires additional intelligence or judgment. One has to perfect the art of using ROA as an analytical tool. In Boxes 7.1 and 7.2, some complications are illustrated. In the example in Box 7.1, a choice has to be made between two policies with the same NPV but a different ROA value because of the details of the uncertainty. It is (if needed) an additional nail in the coffin of the use of NPV in such situations, but it also sheds light on the importance of understanding the effect of the shape of uncertainty in the interpretation of ROA results.

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Prioritizing the Investments Needed to Avoid the Unmanageable. . .

NPV can be a positive or negative number. That tells the difference in that approach between investments, which make economic sense or not. In the case of most environmental investments, the NPV is negative. Real options are always positive. That does not imply that from an ROA perspective, any investment makes economic sense. If the expected benefit of an investment is smaller than its cost, there is an issue. The comparison between the NPV and the ROA value can be a precious additional piece of information. One can argue that it is only if the ROA value (i.e., the expected benefit of the investment) offsets the negative value of the NPV that the investment makes economic sense. But responsible investors would put all these numbers in perspective before jumping to a decision. Intelligence and judgment help. Box 7.1 ROA Values of Two Investments with Same Expected Cost but Different Uncertainties In both cases we assume that the benefit of the policy is described by the same lognormal distribution with parameters μ ¼ 0.5, σ ¼ 0.5. So the expected benefit is 1.86. For the higher and lower uncertainty cost, we use two lognormal distribution with parameters μ ¼ 1.85, σ ¼ 0.5 (Fig. 7.1) and μ ¼ 1.5, σ ¼ 1 (Fig. 7.2), respectively. Both have the expected value, 7.39, i.e., significantly larger than the expected benefit. The two situations are described in the graphs Figs. 7.1 and 7.2. In both cases the ROA value of the policy is small, but they are significantly different. In the high uncertainty case, the ROA value is 0.018 (Fig. 7.3), whereas in the lower uncertainty case, the value is 0.077 (Fig. 7.4), roughly five times larger. . .

7.2 Financing Adaptation

101

0.5

0.4

0.3

0.2

0.1

5

10

15

20

10

15

20

Fig. 7.1 Distributions with higher uncertainty

0.5

0.4

0.3

0.2

0.1

5

Fig. 7.2 Distributions with lower uncertainty

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Prioritizing the Investments Needed to Avoid the Unmanageable. . .

0.12 0.10 0.08 0.06 0.04 0.02

–20

–10

10

20

10

20

Fig. 7.3 Difference distribution with higher uncertainty

0.14 0.12 0.10 0.08 0.06 0.04 0.02

–20

–10

Fig. 7.4 Difference distribution with lower uncertainty

7.2 Financing Adaptation

103

0.8

0.6

0.4

0.2

2

4

6

8

Fig. 7.5 Illustration of the tail effect, which is also behind the “license to kill effect” (Cf Fig. B3.1.1)

Box 7.2 Tail effect In that “cooked-up” example, the economics of two different investments described by two different functions is compared. Both have exactly the same expected return. One is described by a normal distribution with parameters μ ¼ 1.87, σ ¼ 0.5 and the other a lognormal distribution with parameters μ ¼ 0.5, σ ¼ 0.5. Both have exactly the same mean (1.87). The cost of the investment is assumed to be 2.1. They have the same NPV ¼ 1.87  2.1 ¼  0.23, which is negative. But their ROA values are different: it is H(X, Y, t) ¼ 0.104 in the case of the normal distribution and H(X, Y, t) ¼ 0.286 in the lognormal case. In the first case, the investment does not make economic sense, because the value of H(X, Y, t) does not offset the negative value of NPV, and in the second, it does. . . In this “analysis” we neglect the effect of the discount factor. Its effect is to complicate a bit, without changing the fundamental message (Fig. 7.5). ðT 1 ð H¼

eρt ðX  Cðt ÞÞθðX  Cðt ÞÞPðX, t ÞdXdt

0 0

A more realistic scenario: with a tail (A more realistic scenario. . . Box 7.2 points to a potential complication with the interpretation of an ROA value: the “license to kill effect.” The fact that the distribution entering in the expression for ROA has a fat tail for high values of the benefit means that the

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value of ROA is larger. This high value comes from the fact that with a low probability, the benefit can be large. Letting that consideration dominate the investment decision is a bit like gambling. It is important before basing a decision on the ROA value to have a clear idea of what is behind that value. This is one instance where ROA requires intelligence or judgment. Like fire it can burn those who use it without prudence. Out of fear of being burned, one could choose to do without fire altogether. But when used intelligently, fire is a major contributor to civilization. In the same way because ROA (as well as other derivatives) can lead to disastrous decisions, one can follow the advice of Warren Buffett and treat them as financial weapons of mass destruction and keep away from them. Or one can accept the challenge of learning to use them.

7.3

Financing Mitigation and the Transition into a Green Economy

Mitigation is about transitioning into a green economy. In practice, mitigation policy can take many forms, including regulations, carbon tax, investment in equipment like carbon capture and storage (CCS), producing energy with low carbon technology, and refraining from deforesting. If, as Nick Stern asserted, climate change is a case of market failure (Stern 2006), it belongs to the governments to do what the climate response requires. But without the participation of the private sector, there will not be enough resources available. As the businessman turned environmental activist, Tom Steyer says (WebTV 2014) somehow we should find ways to unleash the “magical power of business.” It is in the area of “green investments” that the involvement of the private sector is the most promising: there is money to be made.

7.3.1

Corporate Culture and Green Investments

“Green projects” refer to a hodgepodge of investments. The total cost of the needed “green projects” is not precisely known, except that it is in the trillions of dollars. To raise the trillions of dollars required, the private sector will have to be involved in a big way. Some green investments are attractive to the private sector but most are not. Renewable energy projects have traditionally not been considered smart capital investments but “a foolish use of corporate funds” (Schwartz 2011). Companies with high demands of energy tend to avoid renewable such as solar, wind, and biomass because they are seen as money-losing both in the short and the long term. But things are changing. There is a sense that climate change will not go away and that the future belongs to clean technologies. The movement away from fossil fuel means the possibility that large quantities of unused reserves of oil and gas will progressively lose value and even eventually become worthless and stay underground as “stranded assets.” “There could be as much as $7 trillion worth of fossil

7.3 Financing Mitigation and the Transition into a Green Economy

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fuels that will need to stay in the ground,” forcing investors to reconsider the structure of their long-term energy investments, along the line of “buy green sell stranded.” There are reasons why the private sector is not naturally excited by green investments. First, in many cases those investments benefit society as a whole but not the investor directly, and second, their “risk profile” does not make them attractive (cf Box 7.3). “Risk profile” refers to the relation between the expected return and the “riskiness” of the investment. The risk profile of green investments affects the conditions of their financing.

7.3.2

Risk Profile

“Risk profile” refers to the relation between the expected return and the “riskiness” of the investment. In the context of finances, this relation tends to be quantified by the “Capital Asset Price Model” (CAPM). CAPM is a Nobel Prize (1990) winning approach where the risk is measured by the volatility in the value of what controls the return of investments. This characterization of risk does not work in the context of environmental investments. On the other hand, clearly some green investments are riskier than others. One could associate a “Sharpe ratio” (based on their PDF of return) to those investments to “value” them, i.e., to assess whether their return is commensurate with their risk. The variance of the investment tends to reduce the Sharpe ratio. Investments with large variance are intrinsically riskier as the probability that their return will be small is larger. On the other hand, the probability of large returns is also larger. In Box 7.3, we illustrate this based on a contrived example. One message of Box 7.3 is that even when the expected return of risky investments can be reasonably good, the probability of such profits tends to be very small. The government can alleviate the concerns of the investors (at low cost) by a system of incentives. This is the subject of Box 7.4. For example, what the government can do is to invite investors to foot the cost of the investment. If the return is smaller than the cost, the government can guarantee the investors their early investment back (possibly with a small premium). But if the return happens to be above the cost, the investors will be allowed to collect the return completely only as long as it is less than a predetermined amount. If the return is above that amount, the government collects that excess of return. Governments have the luxury to offer that deal for many green investments. Even if the revenue of each individual investment has a small probability, summed over many investments, statistically the expected returns become a good predictor of the actual overall return. With this technique, the government can make money out of those investments.

7.3.3

Co-benefits of Green Investments

Green investments have co-benefits. The value of technologies such as solar, wind, and geothermal—and energy smart technologies such as power storage, fuel cells,

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and carbon capture—tends to be measured by the carbon saved, but it is not limited to that. They can also lead to the reduction of other pollutants, fostering economic development and improving adaptive capacity. In the long term, green investments may generate huge returns. The challenge is to quantify them and to identify who will be the beneficiaries. One should not expect private investors to factor those co-benefits as part of the return on their investment, if they do not benefit directly. But if the common good benefits, governments are natural candidates to fill that gap. Intelligent governments (a rare commodity) could calibrate additional incentives on green investments based on the expected additional benefit. This supposes being able to compare the additional benefit to the cost. This is a good ROA problem. Box 7.3 Risk Profiles Different “risk profiles” can have exactly the same expected benefit. Figure 7.6 is an example. Both curves are lognormal distributions: one with large variance ( f1(x)) with parameters (μ, σ) ¼ (0.9451) and one (small variance, f2(x)) with parameters (μ, σ) ¼ (0.05, 0.1). The parameters have been chosen to give very different return (risk) profiles. Neglecting discounting, the expected 1 Ð return xf 2 ðxÞdx of the investment with small variance is 1.06, whereas the 0

expected return

1 Ð

xf 1 ðxÞdx from the investment with large variance is 0.64.

0

But if the cost of both investments is K ¼ 0.875, both have exactly the same 1 1 Ð Ð expected benefit: ðx  K Þf 1 ðxÞdx ¼ ðx  K Þf 2 ðxÞdx ¼ 0:18. K

K

On the other hand, the probability that the return is larger than K is not the same in both cases. When K ¼ 0.875, the probability is 0.96 for the investment with small variance and 0.2 for the case with large variance. The fact that expected benefits are identical has to do with the tail of f1(x). If the cost of the investments is larger than 0.875, the investment with the larger variance has a higher expected benefit than the small variance investment, whereas if the cost is smaller, the opposite is true. For example, if the cost K ¼ 1, the expected benefits are 0.16 and 0.07, respectively, whereas if K ¼ 0.5, they are 0.29 and 0.55, respectively.

7.3 Financing Mitigation and the Transition into a Green Economy

107

3

2

1

0.5

1.0

1.5

2.0

2.5

3.0

Fig. 7.6 These are two different distributions with the same average, but different level of uncertainty

Box 7.4 Involving Private Investors at No Risk for Them and with Benefit for Governments We use the same two curves but they have a different interpretation. Let f1(x) represent the return of an investment. It is very risky and does not attract investors. But governments can make the investment more attractive by offering to the investors the following deal: we guarantee the investments in such a way that your return is described by the curve with small variance f2(x). The cost of the investment is assumed to be k1  0.83. If the actual return happens to be less than k1, the investor is fully reimbursed by the government. If the return happens to be larger than K1, the investor does not get reimbursed, and he cannot get more than k2  1.34. If the return is larger than K2, the government pockets the difference. Is that a good deal? The expected benefit Ðk2 for the investors is ðx  k1 Þf 2 ðxÞdx ¼ 0:216. This is a no-risk deal for the k1

investors, but what about the government? The expected loss for the governÐk1 ment is ðk1  xÞf 1 ðxÞdx ¼ 0:39. 0

But the government will have an expected gain too, if the return happens to 1 Ð be larger than k2 , ðx  k 2 Þf 1 ðxÞdx þ k1 ¼ 0:11 þ 0:83 ¼ 0:94. k2

The term k1 reflects the fact the government did not pay the original investment. The investors did. By offering that deal, the government has an expected benefit of 0.94 minus its expected loss 0.39, i.e., 0.55. (continued)

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Box 7.4 (continued) If the government had made the investment by itself, its expected benefit 1 Ð would have been ðx  k1 Þf 1 ðxÞdx ¼ 0:19. k1

By using the detours of private investors that way, the government pushes its expected benefit for 0.19 to 0.55. . . The main reason for that result is the fact that the government has to pay the cost of the investment k1 only if the return is less than the cost. If the government wants to finance some investment whose cost is k1 that way, it can choose an optimal value for k2. It is technically a free parameter here. But it is to be emphasized that this approach applies only on investment generating a return to the investors. Many environmental investments benefit society as a whole, so the investors have to be rewarded otherwise, like through bonds.

7.3.4

Financing Innovation

The response to climate change calls for a vast array of investments with completely different characteristics, risk levels, time scales, and lots of innovations, some revolutionary. Different types of innovations require different types of financing: venture capital (VC), stock market funding, funding from public agencies, or state investment banks. Some are more appropriate than others. VC wants quick return. It is not optimal for science-based research: “Science is not a Business. [. . .] This problem is being felt in the emerging clean-technology sector where venture capital is either absent or producing the quick in/out funding dynamic that results in bankruptcies such as that of solar panel company Solyndra in 2012” (Pisano 2006).

7.3.4.1

Green Innovations and “Patient Capital”

Innovation needs “patient capital.” The private sector is better known for “impatient” investments and “short-termism.” For the Italian economist Mariana Mazzucato, it is possible to go beyond this “market failure” framework through “mission oriented investments” (Mazzucato and Caetano 2015). The idea is to “adopt a market framework, where the public and private sector share the risks and the rewards.”

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109

This is not without precedent. Mission-oriented investments are behind major innovations in the USA, such as the development of transportation, electrification, public universities, aeronautics, and computers, to name a few. Important in this vision is the relationship between finance and innovation. Says Mazzucato (2013): “It was Joseph Schumpeter who first drew a strong connection between the innovation performance of an economy and the functioning of its credit and capital markets.” 7.3.4.2

Tesla and “Patient Investment”

Founded in 2003, Tesla allegedly makes the best electric cars (Forbes 2015). It competes with Chevrolet, Fiat, Ford, Kia, Mercedes, Mitsubishi, Nissan, Volkswagen, and the Smart Electric Drive, to name a few (Plugincars). It had yet to make a profit after 13 years of existence, during which time it “burnt” several billions of dollars of public money. Still, in 2017 the market capitalization of Tesla became larger than Ford and GM, companies that were selling many millions of cars per year, while Tesla was selling at best a few hundreds of thousands. Stewart Myers explained that what he called “growth options” in its original paper on real options (Myers 1977) (this is the ROA value) play an important role in the valuation of any firm. Tesla’s success with investors suggests that its ROA value is very large. The ROA value of the investments in electric cars by Tesla looks impossible to estimate, but the market capitalization of Tesla may be a good proxy. Box 7.5 Straw Man Calculation of the Value of a Bond The issuer can follow at least two different paths to value the bond he wants to finance his project with: the net present value (NPV) approach or the ROA approach. They yield different results. To a certain extent, they are complementary. But ROA is a much more powerful source of information. The NPVproject is the discounted sum of the difference between expected n P hX i iCi revenue of an investment minus its cost: NPV project ¼ . ð1þr Þi i¼0

In continuous time, the sum can be written as the integral: ðT

NPV project ¼ eρt ðhX ðt Þi  Cðt ÞÞdt 0

In that expression, C(t) is the cost the issuer will have to pay at time t. It is the sum of the actual cost of the investment at time t and the interest to be paid regularly to the purchaser of the bond.hX(t)i is the “expected revenue” at time t, i.e., the average value of X: hX ðT Þi ¼

1 Ð

XPðX, T ÞdX. The probability

0

(continued)

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Box 7.5 (continued) distribution P(X, t) is central to the analysis as it carries the information about the uncertainty on the benefit from the investment. The NPVproject of the project does not depend on the details of P(X, t). It only sees on the average value hX(t)i of the variable X. On the other hand, the real option value of the investment (its “expected benefit”) depends on the details of the probability distribution P(X, t) at the expiration of the bond, i.e., when t ¼ T: H¼e

ρT

1 ð

ðX  C tot ÞPðX, T ÞdX C tot

ÐT With C tot ¼ eρt C ðt Þdt is the total discounted cost of the investment up to 0

time T. The real option value H is by definition always positive, whereas NPVproject can be positive or negative. And H is sensitive to the shape of the probability distributionP(X, t), in the sense that the value of H depends on whether the distribution has a tail or not. If the distribution has a tail for high values of X, H is larger. This is a key feature of real options.

Box 7.6 How Does the Approach of Box 7.4 Compare with Issuing Bonds? The government could instead issue bonds, i.e., financial instruments with fixed return. The bonds have to cover the cost of the investment k1. The N P λ k1 over the N years before government will have to pay the interest ð1þrÞi i¼1

expiration of the bond (λ being an interest rate). This has to be compared to the 1 Ð 1 discounted expected benefit: ð1þr ðx  k1 Þf 1 ðxÞdx (here we cannot ignore ÞN k1

the effect of discounting). A necessary condition for the “bond” approach to be beneficiary is that 1 N Ð P λ k1 1 ðx  k1 Þ f 1 ðxÞdx > . But for investors to be interested in ð1þrÞi ð1þrÞN k1

bonds, the condition

i¼1

N P i¼1

λ k1 ð1þrÞi

>> k 1 must be met, i.e., the interest rate λ (continued)

7.3 Financing Mitigation and the Transition into a Green Economy

Box 7.6 (continued) 1 must satisfy λ > P N i¼1

111

. With N ¼ 10 (10 years bond) and a discount rate

1 ð1þr Þi

r ¼ 0.1, this means that λ must be larger than 16%. For governments to have a positive expected profit (N ¼ 10, r ¼ 0.1, 1 N Ð P λ k1 λ ¼ 0.16), ðx  k 1 Þf 1 ðxÞdx > ð1 þ r ÞN  2:56 k1 is needed. This ð1þrÞi i¼1

k1

implies: k1 < 0.18. Another scenario is when the government reimburses the principal at expiration and offers a much lower interest rate. If the interest λ is 5%, N ¼ 10 and r ¼ 0.1, the condition for profitability becomes   1 N Ð N P λ k1 ðx  k 1 Þ f 1 ðxÞdx > k 1 þ ð1 þ r Þ  1:8 k1, i.e., k1 < 0.24. ð1þrÞi

k1

i¼1

This is somewhat better than the other scenario, but still not that good, compared with the scenario of Box 7.2. Bonds require a large expected benefit (with respect to the cost of the investment) to be profitable. Environmental investments do not often generate large returns, and more often than not, the returns are to society as a whole, not to individual investors. When the returns of the investments do not come back to the investors, bonds (whatever form they take) are the only solution to attract investors. In Box 7.6, we revisit the problem of the design of bonds and what ROA has to offer.

Box 7.7 Bond Design Figure 7.7 illustrates a cooked-up example where the “difference distribution” P(X, T ) is a lognormal distribution with parameters μ ¼  1, σ ¼ 1.3. We assume that the cost of the investment Ctot ¼ 1, that the time horizon for the project is 10 years (T ¼ 10), and that the discount rate r ¼ 0.05. We also simplify things by assuming a two-time episode: cost at the beginning and benefit at the end: hX T i NPV project ¼ C þ ð1þr ¼ C þ e0:05 T hX ðT Þi and H ¼ e0:05 T ÞT 1 Ð ðX  1ÞPðX, T ÞdX: 1

In that “cooked-up” example, NPV ¼ 0.52  1 ¼  0.48, and the value of the real option is H(X, Y, t) ¼ 0.228. The expected gain from the investment does not offset the negative value of NPV, but thanks to the fat tail, it points to (continued)

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P(X,T)

1.5

1.0

0.5

1

2

3

4

Fig. 7.7 Example where P(X,T) is a lognormal distribution (Cf text in Box 7.7)

Box 7.7 (continued) a positive expected return of 228. So it makes economic sense to design the bond in such a way that the sum of the discounted interests over 10 years does 10 P 1 not exceed 0.228, i.e., κ ð1þr  0:228, implying that κ < 0.03. Since the Þi i¼1

cost is assumed to be 1, that means an interest rate of at most 3%. For a developing country with a weak currency, this is challenging. Ultimately, what matters is to have a competitive bond. This was only a hypothetical example. . . If the cost had been 0.7, instead of 1, the effect would have been NPV ¼  0.18, H(X, Y, t) ¼ 0.28, and κ < 0.036. The real option value would have offset the negative value of the NPV. Still the interest rate would have been 4%, i.e., quite low. Just to show that funding adaptation through bonds may not be so easy to developing countries. But if they have to offer 10% interest, there would be an expected loss of 10 P 1 0.49:0:1 ð1þr  0:28 ¼ 0:77  0:28 ¼ 0:49. Þi i¼1

When it comes to design bonds, ROA has much more to offer than NPV.

7.4

Climate Bonds, Green Bonds, and the Financing of Green Investments

Bonds seem a natural approach to attract private money to the financing of climate change policy.

7.4 Climate Bonds, Green Bonds, and the Financing of Green Investments

113

There are all sorts of bonds, although they have a few things in common. As is the case with loans, the issuer of a bond pays interest and reimburses the “principal” at the expiration of the bond. The difference with plain loans is that bonds are speculative instruments because they can be traded in “secondary markets.” From the speculators’ point of view, the value of the bonds varies, but unlike stocks, they provide “fixed” revenue. What changes is their yield (the ratio between the fixed return and the value of the bond). Climate bonds are in direct competition with a slew of other kinds of bonds. The bond market is huge. It is estimated at around $100 trillion, comparable to the world GDP (CIA World Factbook). Only a small fraction of that is needed to finance the response to climate change. Still, work has to be done to make climate bonds competitive enough to attract that fraction. And as Box 7.6 shows, this is a challenge. One immediate problem is to identify the issuers of the bond. The issuer of the bond has to pay an interest and eventually return the principal at the expiration of the bond. By then he hopes to have made enough return. When returns are very uncertain and far in the future, ROA may be a good way to estimate their expected benefit. However large the uncertainty, the fact that the return will be far in the future already limits the value of the bond (because of the discount rate). Basically only governments and some international institutions would be able to stomach that kind of investment. For investors on the other hand, those bonds would be a bit like treasury bonds, except that their purpose is not to finance a national debt, but a climate policy whose political popularity may not last as long as the bond. There is a whole taxonomy of climate bonds2, and among them, the “green bonds” have the potential to play a prominent role (Bonds & Climate Change 2014).

7.4.1

Green Bonds

Green bonds, in their first incarnation, were created by “an amendment to the America Jobs Creation Act of 2004, which was officially titled the ‘Brownfields Demonstration Program for Qualified Green Buildings and Sustainable Design Projects’.” Lisa Smith (2012) added “for those who prefer to have a little oxygen left at the end of their sentences, this has been shortened to ‘Green Bonds’.” After the green bonds issued by the US treasury in 2004, one of the earliest occurrences of such bonds may have been in May 2007, when the European Investment Bank issued more than €1 billion in “Climate Awareness Bonds.” The funds were for renewable energy projects as part of Europe’s commitment to produce 20% of its energy from renewable sources by 2020. In 2008, the World Bank started issuing green bonds to “cater a specific group investors who asked for a product with the same standards and safe credit quality as the regular World Bank regular bonds, but with a defined purpose.” This was in response to Scandinavian

2

https://www.climatebonds.net/standards/taxonomy

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pension funds interested in supporting activities that address mitigation and adaptation to climate. According to its August 2013 “fact sheet,” the World Bank had issued by then about $4 billion in green bonds (Reichelt 2013). That was part of the World Bank “Strategic Framework for Development and climate change to help stimulate and coordinate public and private sector activity in this area.” The size of the green bond markets is poorly known. In part, this is because it is not always easy to distinguish a “green bond” from another kind of climate bond. Using its own definition of what constitutes an investment in clean technology, the Climate Bonds Initiative estimates that in 2013, the market for climate-related bonds stood at $346 billion, but the market for green bonds specifically was just over $15 billion. Considering the size of the global bond market (estimated at around $100 trillion), there is considerable space for growth in climate and green bonds. But investors are quite aware of and sensitive to the fact that as of 2013, only about $35 billion of the $346 billion in climate-themed bonds “have yields of 3% or higher” (Trabish 2013). In Box 7.5, we compare the pros and cons for government financing of green investments, directly or by issuing bonds. The message is that bonds are much less “profitable” to the government than the investment incentive of Box 7.4. But this is not the whole story. Both Boxes 7.4 and 7.6 are based on the premise that the return of the investments goes to the investors. But many green investments benefit society as a whole. For those investments, there is no tangible short-term return to investors. In that case, to have the participation of private investors, there is no alternative to bonds. The difficulty is not to estimate the ROA value of such bonds, as it is to identify how to make that market as large as possible. In Appendix B, we review this problem in a pedestrian way.

References Bonds & Climate Change 2014: Report from the climate bond initiative. http://www.climatebonds. net/resources/publications/bonds-climate-change-2014 CIA World Facts Book: https://www.cia.gov/library/publications/the-world-factbook/ Forbes: Tesla’s business model highlights what the shift to electric means for the auto industry. http://www.forbes.com/sites/greatspeculations/2015/09/01/teslas-business-model-highlightswhat-the-shift-to-electric-means-for-the-auto-industry/#50b6a1715029 (2015). Accessed 1 Sept 2015 IPCC AR4: Climate change fourth assessment. Cambridge University Press, Cambridge (2007) Maplecroft.: https://maplecroft.com/ Mazzucato, M.: Financing creative destruction versus destructive creation. Ind. Corp. Chang. 22(4), 851–867 (2013) Mazzucato, M., Penna, C.C.R. (eds.): Mission oriented finance for innovation. Rowman Littlefield, London (2015) Myers, S.: Determinants of corporate borrowing. J. Financ. Econ. 5, 147–175 (1977) Nordhaus, W.: The economics of hurricanes in the United States. NBER Working Paper No. 12813, National Bureau of Economic Research (NBER), Cambridge, MA, USA, 46 pp. http://www. econ.yale.edu/~nordhaus/homepage/documents/hurr_083109.pdf (2006) Pisano, G.: Can science be a business? Harvard Business Review. 1 (2006, October) Plugincars.: http://www.plugincars.com/cars

References

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Preston, B.L., Westaway, R.M., Yuen, E.J.: Climate adaptation planning in practice. Mitig. Adapt. Strat. Glob. Chang. 16, 407–438 (2011). http://www.researchgate.net/profile/Benjamin_Pres ton/publication/226149604_Climate_adaptation_planning_in_practice_an_evaluation_of_adap tation_plans_from_three_developed_nations/links/00b7d530e7ee36cee1000000.pdf Reichelt, H.: World Bank green bonds surpass US$4 billion mark – reflections five years on, The World Bank. http://blogs.worldbank.org/climatechange/world-bank-green-bonds-surpass-us4billion-mark-reflections-five-years (2013). Accessed 28 Oct 2013 Schwartz, E.: Investing in the clean trillion: closing the clean energy investment gap executive summary is renewable energy a “good” Investment? MIT Technological Review. (2011). Accessed 6 Jan 2011 SEGCC: Confronting climate change: avoiding the unmanageable and managing the unavoidable. In: Bierbaum, R.M., J.P. Holdren, M.C. MacCracken, R.H. Moss, and P.H. Raven (eds.) Report prepared for the 15th session of the United Nations Commission on sustainable development by the United Nations-Sigma XI Scientific Expert Group on Climate Change (SEGCC), Sigma Xi, Research Triangle Park, NC, USA and United Nations Foundation, Washington, DC, USA, 144 pp., http://www.globalproblems-globalsolutions-files.org/unf_website/PDF/climate%20_ change_avoid_unmanagable_manage_unavoidable.pdf (2007) Smith, L.: Fixed returns to fix the planet. Investopedia. http://www.investopedia.com/articles/ bonds/07/green-bonds.asp (2012). Accessed 12 Aug 2012 SREX: Special report on managing the risks of extreme events and disasters to advance climate change adaptation (SREX), the applicability of rigorous community-based adaptations (CBAs) for evaluations of adaptation to climate variability and change may be limited (Handmer et al., 2012). https://www.ipcc.ch/pdf/special-reports/srex/SREX_Full_Report.pdf (2012) Stern, N.: The Stern review on the economics of climate change. (2006) Trabish, H. K.: New financing for renewables and energy efficiency promises GHG cuts. (2013). Accessed 26 Dec 2013 WebTV.: http://webtv.un.org/meetings-events/conferencessummits/3rd-international-conferenceon-financing-for-development-addis-ababa-ethiopia-13%E2%80%9316-july-2015/press-confer ences/watch/part-1-2014-investor-summit-on-climate-risk-financing-the-clean-energy-future/ 3059096647001 (2014)

Chapter 8

Unanswered Questions About Uncertainty, Information, and Investment Decisions

Prediction is very difficult, especially if it’s about the future. Niels Bohr (https://www.brainyquote.com/quotes/quotes/n/ nielsbohr130288.html) We cannot predict the future, but we can create it, Jim Collins (Collins, Jim, Hansen Morten, T., “Great by Choice,” www.harperscollins.com)

Abstract Books far too often come with answers only. They convey a warm feeling that everything is under control. This book is not in that tradition. Hopefully, it will inspire some to realize that the frontier of knowledge in ROA is only work in progress and at an early stage of progress. Unanswered questions are the prerequisites to change, progress, and even sometime scientific revolutions. This book is an invitation for those who think that ROA can make a difference to help making it go to the next level.

8.1

Climate Change Decision-Making

Climate change policy is an extreme case where the knowledge about the future is particularly important and at the same time particularly problematic. The policies rely excessively on our representation of the future provided by PDFs. Does the information PDFs carry, represent some form of “probabilistic knowledge” of the future and can climate change policy decision-making confidently rely on that form of knowledge? How solid are the epistemological foundations of climate change decision-making? Mitigation can be described as moving from a PDF of futures ψ 1(x) into another ψ 2(x). Both carry different levels of information. In Chap. 5 , the Kullback–Leibler “divergence” KL was used to measure the amount of information separating the two distributions ψ 1(x) and ψ 2(x) or alternatively the amount of information that mitigation should introduce to make the transition. The formula for the “divergence” was:

© Springer Nature Switzerland AG 2020 B. Morel, Real Option Analysis and Climate Change, Springer Climate, https://doi.org/10.1007/978-3-030-12061-0_8

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8 Unanswered Questions About Uncertainty, Information, and Investment Decisions

h i h i Ð KL ¼ ψ 1 ðxÞ log ψψ 1 ððxxÞÞ dx, in which log ψψ 1 ððxxÞÞ is the “relative” entropy between the 2

2

two distributions. As explained in Chap. 5, KL is the value of the relative information averaged over the original distribution, i.e. the amount of “information” that separates the two distributions. Information here is to be taken in the sense of Boltzmann or physics in general, i.e., the opposite of entropy. Kullback and Leibler (1951) derived their formula having in mind the discrimination between statistical distributions built from empirical data. This was based on the very Fisherian (from Sir Ronald Aylmer Fisher, 1890–1962) assumption that distributions elicited from data represent empirical information. By extension, any distribution can be assumed to carry some information. But when one details the make-up of different distributions, it becomes obvious that one is speaking of conceptually different kinds of information. PDFs used in climate change policy are not based on data; they are based on representations of the futures generated by simulations or scenarios, using our limited knowledge to the best of our ability to “predict” the future. Ð There is a consensus that I ¼  φ(x) log [φ(x)] dx has something to do with the “information” associated with the distribution φ(x). That does not imply that the Ð formula I ¼  φ(x) log [φ(x)] dx captures all the possible forms and aspects that information can take. From Boltzmann (1844–1906) to Tsallis (1943–), passing through Fisher (1890–1962) and Shannon (1916–2001) and more, the complementary concepts of entropy and information have had quite a ride. But through this brouhaha, some fundamentals survived. One is the seminal importance of PDFs. PDFs find their origin in models, equations, assumptions, simulations, or informed or wild guesses, and more, basically there is no limit. As one can read in IPCC AR5: “Individuals and organizations that link science with policy grapple with several different forms of uncertainty. These uncertainties include absence of prior agreement on framing of problems and ways to scientifically investigate them (paradigmatic uncertainty), lack of information or knowledge for characterizing phenomena (epistemic uncertainty), and incomplete or conflicting scientific findings (translational uncertainty).” A PDF is basically a tool, which summarizes and carries a lot of information. Whether the information carried by the PDF is reliable enough for momentous policy decisions depends also on the collateral information. Archeologists have perfected the art with one tooth to reconstruct the whole animal, tell you the color of its skin and eyes, at least so the story goes. . . When it comes to PDFs, we are nowhere close to that. Without the context, by just analyzing a PDF, it would be impossible to reconstruct the whole set of assumptions, inferences, and scientific speculations that led to it. This is part of the collateral information that is needed to be able to use that PDF knowingly. The use of PDFs in policy decision today is as much an art as a science?

8.2 Thinking in Probabilities

8.2

119

Thinking in Probabilities

In practice, investments are made only once, and their actual return will very rarely be exactly the same as their “expected” return. The fact that investment decisions under uncertainty are made on probabilistic grounds raises a wealth of unresolved epistemological and philosophical issues. Having a PDF for the return is vital for investment decision. Without PDFs for the return on investments, ROA and most other forms of quantitative analysis of the value of investments under uncertainty are basically impossible. We said that many times, but that is so true it is worth to be repeated: PDFs are problematic. They play a big role in investment decisions, and they often have spooky origins. In economics, all PDFs are not equal even if they enter in computations in similar ways. In the evaluation of the value of a put or call option in finance, for example, using the Black-Scholes (BS) formula, the PDF expressing the uncertainty reflects the dynamics of the value of the underlying variable a share, in that particular case referred to as a “risky asset.” In the case of BS, the assumption is that the value of the share follows a geometric Brownian motion whose parameters can be empirically estimated. When it comes to investments under uncertainty, the PDF used to estimate the expected benefits has a totally different grounding. It reflects the probabilities investors attribute to the different futures. Where do these probabilities come from and what do they mean is very context dependent. Weather forecasters use often the language of probabilities: 60% chance of precipitation, for example. That speaks to our intuition. But a bit like the numbers given by experts in climate change, that number cannot be “validated.” Those numbers express a degree of probability that something will happen. They have a completely different interpretation and quantitative implications from the BlackScholes lognormal distribution of values of the risky asset. The lognormal distribution of Black-Scholes is due to the intrinsic “randomness” of the evolution of the market price of the asset. Uncertainty can have completely different origins. Uncertainty due to randomness as in Black-Scholes is irreducible (it is impossible to predict the detailed fluctuations of the price of a share), but at the same time, it is “predictable” (all this takes place within the framework of a well-understood stochastic process, and more importantly over time with enough data, it is possible to confirm whether these predictions were “true” in a statistical sense). In that case, we know how to build the PDF of that uncertainty. Uncertainty due to ignorance is different. It may not be “irreducible” if it is possible to acquire additional knowledge. This is a point that did not escape analysts of climate change policy. This explains the interest in pricing the value of additional information (ROA used intelligently can help).

120

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8 Unanswered Questions About Uncertainty, Information, and Investment Decisions

The Intrusion of Subjectivity as a Legitimate Contributor of Collective Decision-Making

The issues associated with PDFs fed an intriguing controversy in the climate change establishment in the early 2000s, which has not evolved in a very promising way but which illustrates the damage that uncertainty can inflict on rationality. The most authoritative sources on that subject are the IPCC reports published every few years. The last report so far is the “Fifth Assessment Report” (AR5), which came out in 2014. In a Science policy forum, Reilly et al. (2001) deplored the fact that the “Third Assessment Report” (TAR), which appeared in 2001, did not provide PDFs for the uncertainties. They inspired a rebuttal by Allen et al. (Allen et al. 2001), which appeared in the same issue of Science. The main argument of Allen et al. to defend the IPCC report (of which they were leading authors. . .) was: “We should recall that the IPCC was under considerable pressure in 1990s to make a statement attributing observed climate changes to human influence because if they don’t, someone else will” (and, indeed, did). The IPCC is a cautious body, and if the evidence is not available in the peer-reviewed literature to support a statement, it will not make it, no matter how great the interest in that statement might be. In the end, this caution resulted in the attribution statement made in the Second Assessment Report having much more impact than if it had been made prematurely.” In other words, Allen et al. did not conclude that PDFs were not vital for policy, but they chose the bureaucratic prudent line that our present level of understanding of the economic uncertainty made IPCC report not the right venue to propose any. Even more bureaucratic was the way this controversy was reported in the “Ffourth Assessment Report” (AR4): In Chap. 2 by the Working Group III of the AR4 report, one can read: “Grübler and Nakicenovic (2001) and Allen et al. (2001) argued that good scientific arguments preclude determining objective probabilities or the likelihood that future events will occur. They explained why it was the unanimous view of the IPCC report’s lead authors that no method of assigning probabilities to a 100-year climate forecast was sufficiently widely accepted and documented to pass the review process. They underlined the difficulty of assigning reliable probabilities to social and economic trends in the latter half of the twenty-first century, the difficulty of obtaining consensus range for quintiles such as climate sensitivity, and the possibility of a non-linear geophysical response.” The implication is that climate change policy will have to live without PDFs. This would be a disaster for policy making. But importantly that was also a misrepresentation of what Allen et al. as well as Grübler and Nakicenovic said and through the next IPCC report sent the debate in a policy wasteland. The last sentence of Allen et al. was: “We hope the research community will develop a capacity for fully probabilistic 100-year climate forecasting over the coming years and commend the efforts of many groups working toward this goal. When it happens, the IPCC will report that development. But not before.” Grübler and Nakicenovic were holding against Reilly et al. their suggestion that if there was no better way to get a grip on the uncertainties, there was still the recourse

8.3 The Intrusion of Subjectivity as a Legitimate Contributor of. . .

121

of “expert elicitation” (they called it “spurious expert opinion”) opening the door to highly subjective approaches based on assumptions of future societal behavior. The follow-up IPCC report (AR5, WGIII Chap. 2) took the discussion in a direction confirming the worst fears of Grübler and Nakicenovic. In that report, one can read: “Compared to AR4, where judgment and choice were primarily framed in rationaleconomic terms, this chapter reviews the psychological and behavioral literature on perceptions and responses to risk and uncertainty.” Then it proceeds to state: “Laypersons tend to judge risks differently than experts. [. . .] Experts engage in more deliberative thinking than laypersons by utilizing scientific data to estimate the likelihood and consequences of climate change.[. . .]. Formalized expert judgment and elicitation processes improve the characterization of uncertainty for designing climate change strategies (high confidence).” This looks like saying that the wisdom of the experts is a valid substitute to a more “scientific approach” attempting to derive actual probabilities. Experts do not agree on everything. They have their own individual belief systems. The problem at hand is the design of an optimal mitigation policy. When one tries to merge points of view of experts, one can encounter the Dempster–Shafer problem, a genuine complication in “probabilistic logic.” The Dempster–Shafer “problem” (in the field of “probabilistic logic” this is called the Dempster–Shafer “theory”) refers to the observation that probabilistic logic can lead to counterintuitive results. A famous example is when two doctors are diagnosing a patient with huge headaches. One thinks it is a tumor with 99% probability or a meningitis with 1% probability. The other thinks that it is a concussion with 99% probability or a meningitis with 1% probability. Following the probabilistic logic rules to merge the two expert points of view, the patient is treated for a meningitis, something both experts agreed had a very low but non-zero probability to explain the symptoms. The alternative would have been no agreement and no treatment. No question the outcome is “counterintuitive.” In fact it does not inspire any comfort, especially if one is the patient. At this stage probabilistic logic is problematic. Centuries ago, Leibniz (1646–1716) stated: “J’ai dit plus d’une fois qu’il faudrait une nouvelle espèce de logique, qui traiterait des degrés de Probabilité.”1(quoted in (Keynes 1921)). We definitely are not yet there. In other words, expert elicitation if used in policy decisions is sending us in a dubious epistemological territory. Still, there are instances where it is necessary to live dangerously. Words are no substitute to numbers. But if purely scientific methods do not deliver enough useable information to build credible PDFs, and one cannot delay policy decisions, as the French saying goes: “Faute de grives, on mange des merles2.” In other words, when one is desperate, expert elicitation as unsavory as it is may be sometimes the only recourse. Luckily, not all investment

1

Leibniz was German, but he said that in French, which with Latin at the times were the language of Science. Translated in English: “I said more than once that we need a new kind of logic which would deal with different degrees of probabilities.” It does not sound as good as in French. 2 This translates as: If one cannot get thrushes, one has to eat blackbirds instead.

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decisions under uncertainty need that kind of detour through expert elicitation. This is a consideration for only a small fraction of them. Is it prudent or wise to base economic decisions on quantities as fickle as PDFs? PDFs introduce a level of mathematical rigor in the analysis. But as remarked by John Maynard Keynes (Keynes 1921), the world of probabilities is not as solid epistemologically as its deep reliance on mathematical rigor suggests. How often can we trust that a PDF is a faithful representation of the future? More often than not PDFs are merely guesses about the future. When it comes to fat tail distributions, there is a relatively low degree of probability that events in the tail will take place. But events lying in the tail tend to be more “significant” than the more probable ones. Where does that information appear? A possible approach is to exploit the fact that the significance of lowprobability event with high consequence is measured by expected losses or gain. This introduces a ranking system for the significance of events not solely based on their probability. But any quantification has a subjective component, from the price of a car to the value of a reserve of oil, let alone the value of biodiversity or of ecosystem services. By the time the mathematical machinery of the calculus of probability is put in use, the subjective elements have been disposed of and are buried in the PDFs somewhere in the information they carry. Depending on the context, one speaks of the “Fisher information,” “Rényi entropy,” “Shannon entropy,” “Wiener entropy,” and “Tsallis entropy,” to name a few. Does that imply that like in a candy store we can choose which one we prefer? Or is there a way to know which one is the most appropriate for our purpose?

8.4

The Scarcity of Information Conundrum

Mitigation is a case where we know enough to know that something should be done, but not enough to know exactly what and how fast. Risk managers are sometimes confronted to situations where there is no good risk quantification available but still an imperative to act. Those risk managers have to assume that some losses they do not know much about (“unknown unknowns3”) will happen. Those “unknown unknowns” cannot be captured by PDFs but still require some kind of intelligent response. Could fuzzy logic (Kosko 1990), imprecise probabilities (Walley 1991), and the like help? In fuzzy logic, decisions can be made with minimum information. A situation not uncommon in policy decisions dealing with mitigation or adaptation. At its most fundamental fuzzy logic is a system of decisions based on a consensus using several inputs. The system can behave well even when the inputs are not trustworthy or even available. Admittedly, this means that the inputs talk to a system pre-trained.

3

Donald Rumsfeld, US Secretary of Defense 1975–1977 and 2001–2006.

8.5 Can ROA Make a Difference in Climate Change Policy?

123

And the system can learn to improve on its rule-based response with time. This analogy could be used in the context of the policy decision process in mitigation policy, if there were reasons to believe that policymakers are also able to learn and improve as do well-programmed machines ... That would be good news as those systems perform well, unlike the present response system to climate change.

8.5

Can ROA Make a Difference in Climate Change Policy?

Science is the belief of the ignorance of experts Richard Feynman4

Could ROA be a game changer for climate change policy? Debates and recommendations about climate change policy are conspicuously weak in quantitative content. CBA is mentioned a lot but hardly operationalized, because cost and benefits are basically never quantified. Sources of funding are evoked, but amounts are not as is the prioritization of investments. Does that gap hamper progress? For sure, the answer is yes. But could ROA fill that gap? One indication that it could help is the fact that the inability to make convincing policy recommendations in the context of the kind of uncertainty in which climate change is mired is at the root of the weakness that some would call with good reasons, failing of climate change policy implementation. ROA is a dignified form of cost-benefit analysis. It is more sensitive to the shape of the uncertainty (or risk) than NPV. But is it a game changer? What ROA does is to give an estimate of the expected benefit associated with any policy measure. It provides a kind of quantitative landscape in which to allocate resources when choices have to be made. What effect would the existence of such a landscape have on the way mitigation and adaptation are approached today? Would the recommendations made be very different if they had to backed with some quantitative components provided by tools like ROA? Would the whole climate change response debate be different if every argument had to be justified more quantitatively? The “Copenhagen Consensus” recommends to refrain from immediate aggressive mitigation measure because they claim that there are higher priorities (Copenhagen 2012). Those recommendations were uttered by the like of Nobel Laureates in Economics Thomas Schelling, Finn Kydland, Vernon Smith, and Robert Mundell. The quantitative or technical analyses accompanying those recommendations were not on a par with what gave them their Nobel Prize. Considering the importance of the issue, ROA could help getting a reality check on their recommendations, i.e., whether the expected benefits of what they recommend are better than the alternative. Forcing naysayers to back up their claims with that kind of quantitative arguments could only improve the quality of the debate. It is tempting to quote

4

https://blog.jim.com/global-warming/science-is-the-belief-in-the-ignorance-of-experts/

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Christopher Hitchens: “That which can be asserted without evidence can be dismissed without evidence” (Hitchens, 2003). Climate change policy has a strong financial component. That component needs to be clarified. None of the tools used do a good job at helping estimate the cost and benefit of particular measure, not as good a job as ROA. This is because of the pervasive uncertainty. One important contribution of ROA could be to confront the uncertainties better than is done today, having in mind to quantify them, i.e., as far as possible to translate them into PDFs. ROA translates this knowledge into numbers policymakers can play or fight with. The landscape in which responses to climate change are discussed is devoid of significant financial content. Is it fair to say that the climate change policy debate suffers from a lack of quantitative inputs? Considering that the fact that resources are limited is probably the most binding constraint, this is definitely a debilitating situation. It shows when recommendations dwell mostly with who should take responsibility for what: international organizations or local, regional, national, and international institutions. Concrete, compelling, i.e., useful recommendations come with numbers and ways to identify priorities. The concept of “no regrets measures” has become a cop-out: by definition “no regret measures” cannot do harm and may do good. But can a policy restricted to no regret measures solve climate change? The short answer is no. Much more analytic power is needed. Policy recommendations could (and should) be accompanied with an estimate of their expected benefits compared to alternatives. Policy decisions should be based on informed choices. There are so many moving parts in developing the analytic power needed, that it would be ludicrous to hope that being able to estimate expected benefits in situations of uncertainty and complexity, what ROA is all about, could be a panacea. But the numbers that a systematic use of ROA would provide ROA could significantly improve the situation. The requirement of translating uncertainties into PDFs whenever possible, would force the policy analysts to explore deeper the anatomy and physiology of uncertainties. Organizing the policy discussion around the uncertainty would be an instantiation of the observation of the German philosopher Georg Hegel (Hegel 1812) mentioned in the prolegomena of this book that quantity affects quality. When the uncertainty is very large, it becomes an organizing factor for the analysis. Whereas, when it is small, it merely complicates somewhat the analysis as is the case with NPV.

8.6

Ugliness, Thersites, and Climate Change

Climate change has its deniers. When more than 90% of the scientists are on one side of the issue, they stick to the less than 10% left and proclaim that this scientific consensus is worthless: they know better. They write books like the Greatest Hoax

8.6 Ugliness, Thersites, and Climate Change

125

(Inhofe 2012) or use their presidential power to make the USA officially withdraw from COP 21. Both are biodegradable moves as few take them seriously. Still they make noise as does the so-called Copenhagen Consensus. They have some visibility but inspire no real respect. The rest of the world is forging ahead in pursuing mitigation and adaptation, maybe painfully or haphazardly but independently of them. The deniers do not have real policy impact. If they have one, it is the opposite of the one they seek. No American initiative had more a stimulating effect on the resolve of the world post COP 21 than the “withdrawal” of the USA officially announced by the president. One tangible effect has been that nations like China put their objections aside and choose instead to take a leading role in the international mitigation effort. . . The concern about climate change and the international initiative like COP 21 it inspired, although based on science, is mostly an attitude with respect to potential long-term dangers to a certain degree hypothetical. The uncertainty of everything related to climate change is such that it is difficult to claim that we know fully what will be the benefit in the long term of our mitigation effort. There is a lot of righteousness in our policy attitude with respect to climate change. Righteousness is not the same as the certainty of being right. The deniers have understood that. The deniers play the wise men in front of an international audience poo pooing them when they do not ignore them altogether. This is where the parallel with ugliness in art becomes enlightening. Climate change deniers are the equivalent of ugly figures in art. Art cannot be reduced to the pursuit of beauty; ugliness is part of it too as demonstrated by Goya, Dali, or Bosch to name a few. The role of ugliness in art may not seem central, but it is non-negligible (Gagnebin 1978). In the context of today, climate change deniers look like semi-lunatic figures, but something would be missing without them. Obviously, the climate change deniers will not easily accept the notion that they are merely grotesque figures in a debate they do not influence. Ugliness tends to be associated with being wrong. But there are some caveats. In its Iliad, Homer invented a character called Thersites. He was ugly in appearance because it went well with what he had to say. In the Iliad, Thersites claimed that Agamemnon was greedy and coward, as an excuse to not go to war against Troy (following the same line as Achilles). He was severely punished. Shakespeare used him as a dark character in some of his plays: Troilus and Cressida and Cymbeline. Goethe made him appear shortly in Faust (part II). He also appeared in some Plato’s dialogue, like the Gorgias and the Republic The German philosopher Hegel coined the word “Thersitism,” which could be applied to some climate change denier. Starting with words from Kenneth Burke (Burke, 1966): “If an audience is likely to feel that it is being crowded into a position, if there is any likelihood that the requirements of dramatic “efficiency” would lead to the blunt ignoring of a possible protest from at least some significant portion of the onlookers [there is case for taking a seriously controversial stand (like withdrawing from COP 21 when nobody in his/her right mind would advise such a politically costly and ultimately useless move)].”

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But Thersitism in spite of its association with ugliness, is not necessarily deprecatory if one believes this quote from Robert Graves: “To dissociate himself from Thersites’ sentiments, Homer presents him as bow-legged, bald, hump-backed, horrible-looking, and a general nuisance; but the speech and Odysseus’ brutal action stay on record.” Thersites was physically repulsive but fundamentally right in its stand, i.e., exactly the opposite of the US president and his fellow deniers. In their case, the ugliness is in their position. Why pay any cost to protect against a hypothetical outcome, which at best will take place after our death, even if it may be the end of civilization as we know it? There are so many more urgent things to do, like partying under the pretext of fighting diseases, poverty, and hunger. Defining ugliness is like trying to define beauty, humor, or pornography: difficult to define but one recognizes it when one sees it. The attitude of the deniers is definitely ugly even if they were right. In a situation of uncertainty, nobody knows anything for sure. As noted by Socrates, an ugly stand typically does not have much influence, because it does not convince. This applies to climate change deniers. So in what sense is the ugliness of the deniers useful? An obvious reason is related to the fact that climate change is swimming so deep in uncertainty that doing anything or not about it generates anxiety. How do we convince ourselves that the sacrifices we make for mitigation are justified? The contribution of the deniers is that we know that they try their best to invalidate the present efforts. So the fact that their recommendations are so pathetic, inspire confidence that contrarily to what they want us to believe, there are no obvious better alternatives to what we are doing. This is one function of ugliness: it makes the alternative look beautiful. This is not the only reason that makes the ugliness of the deniers worthwhile. Deniers are like the monsters of Goya: they are “esthetic curiosities” as the French poet and also art critique, Baudelaire pointed out (Baudelaire 1868). Those who are uncomfortable with the rhetoric around climate change may find solace in joining the deniers in the dustbin of their discontent. Imagine what masterpiece Goya would have produced painting that scene. He was born too early. The deniers are the climate change equivalent of the gargoyles in Gothic architecture, i.e. they have their usefulness (they part of the gutter system) mostly harmless because few take them seriously but a remainder that ugliness is here to stay. . . In one of his books, (Nietzche 1888), Nietzche states that “the wisest men have come to the same conclusion [. . .]. Could it be that wisdom appears on earth as a raven, inspired by a little scent of carrion?” Ask the climate change deniers or better ask yourself whether there is merit in that Nietzche’s remark and what that says about the climate change deniers. . .

8.7 Oil Painting and ROA

8.7

127

Oil Painting and ROA

Visitors to museums like the Museum of Uffizi in Florence or the Louvre in Paris tend to congregate around the most well-known paintings like Mona Lisa and like the sociologist Duncan Watts (Watts 2011) may “experience a sense of, well, disappointment upon laying eyes on the most famous painting in the world. To start with, it is amazingly small. [. . .] You expect to see something special [..] “The supreme example of perfection”, which causes viewers to “forget all our misgivings in admiration of perfect mastery. [. . .] As far as I could tell Mona Lisa looked like an amazing accomplishment of artistic talent but no more so than the four other Da Vinci paintings the Louvre5 has and on which people do not pay the slightest attention.” The 1434 wedding portrait of Giovanni Arnolfini and his wife6 by Jan Van Eyck does not have the same public attraction. But it has played a prominent role in the history of painting. It is one of the first instantiations of a technique based on oil paint developed by the painter.7 The painting may not strike immediately as very exciting, but the quality of the details makes all the difference.8 This painting got notoriety through because it raised the unsettling question of whether the bride was already pregnant or not.9 Maybe the quality of the details was excessive. Oil paint progressively displaced the egg yolk10-based “tempera.” Tempera was difficult to use because it dries fast. Oil paint dries much less fast and as a result makes the life of the painters. It is so to speak, less “tempera-mental.” But using tempera, Botticelli11 painted timeless masterpieces such as Primavera12 and the Birth of Venus.13 In a not totally dissimilar way, ROA may progressively displace NPV as the natural tool to assess the value of investment under uncertainty. It has not happened yet. But investors anxious to quantify the effect of uncertainty on the value of their investments cannot be comfortable with what NPV has to offer. This has been 5 Virgin on the Rocks, La belle Ferronnière, The Virgin and Child with Saint Anne, and Saint John the Baptist (who seems to make an obscene gesture with his finger, http://merovingio.c2rmf.cnrs.fr/ iipimage/showcase/StJohnTheBaptist/ 6 http://www.nationalgallery.org.uk/paintings/jan-van-eyck-the-arnolfini-portrait 7 http://www.cyberlipid.org/perox/oxid0011.htm 8 “Van Eyck was intensely interested in the effects of light: oil paint allowed him to depict it with great subtlety in this picture, notably on the gleaming brass chandelier,” http://www.nationalgallery. org.uk/paintings/jan-van-eyck-the-arnolfini-portrait 9 “His wife is not pregnant, as is often thought, but holding up her full-skirted dress in the contemporary fashion. Arnolfini was a member of a merchant family from Lucca living in Bruges. The couple is shown in a well-appointed interior.” http://www.nationalgallery.org.uk/paintings/janvan-eyck-the-arnolfini-portrait 10 http://www.britannica.com/art/tempera-painting 11 http://www.artble.com/artists/sandro_botticelli/more_information/style_and_technique 12 http://www.uffizi.org/artworks/la-primavera-allegory-of-spring-by-sandro-botticelli/ 13 http://www.uffizi.org/artworks/the-birth-of-venus-by-sandro-botticelli/

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known for decades. ROA seems the only alternative but it has not yet reached enough maturity to inspire confidence in financial decisions.

References Allen, M., Raper, S., Mitchell, J.: Uncertainty in the IPCC’s third assessment report. Science. 293 (July 20), 430–433 (2001) Baudelaire, Charles, (1868): Quelques Caricaturistes Étrangers, in Curiosités Ésthétiques. http:// www.artandpopularculture.com/Curiosit%C3%A9s_Esth%C3%A9tiques Burke, K.: Language as Symbolic Action. University of California Press, Berkeley (1966) Copenhagen Consensus 2012, Expert Panel Findings., http://www.copenhagenconsensus.com/ copenhagen-consensus-iii/outcome Gagnebin, M.: Fascination de la Laideur. L’ Age d’Homme, Lausanne (1978) Graves, R.: The Anger of Achilles: Homer’s Iliad. Doubleday, Garden City (1959). https://www. libertarianism.org/columns/ancient-greeces-legacy-liberty-counsel-thersites Grübler, A., Nakicenovic, N.: Nature. 412, 15 (2001) G. Hegel, Wissenschaft der Logik (1812), translated as Science of Logic, http://www.inkwells.org/ index_htm_files/hegel.pdf Hitchens, C., 2003, Slate 20 October 2003: “Mommy Dearest”, https://www.goodreads.com/ quotes/12042-that-which-can-be-asserted-without-evidence-can-be-dismissed Inhofe, J.: The Greatest Hoax: How the Global Warming Conspiracy Threatens your Future. WND Books, Washington, D.C. (2012) IPCC AR5: Climate Change Fourth Assessment. Cambridge University Press, Cambridge (2013). WGIII chapter 2 Keynes, J.M.: Treatise on Probability. Macmillan & Co, London (1921) Kosko, B.: Fuzzyness versus probability. Int. Journ. General Systems. 17, 211–240 (1990) Kullback, S., Leibler, R.A.: On information and sufficiency. Ann. Math. Stat. 22(1), 79–86 (1951) Nietzche, F., (1888) The Twilight of the Idols, or How One Philosophizes with a Hammer, http:// www.inp.uw.edu.pl/mdsie/Political_Thought/twilight-of-the-idols-friedrich-neitzsche.pdf Reilly, J., Stone, P.H., Forest, C.E., Webster, M.D., Jacoby, H.D., Prinn, R.G.: Science. 293, 430 (2001) Walley, P.: Statistical Reasoning with Imprecise Probabilities. Chapman and Hall, London (1991) Watts, D.: Everything is Obvious, How Common Sense Fails Us. Random House, New York (2011)

Appendix A Optimizing the Conditions of Investments Under Uncertainty: “Real Option” Can Mean Different Things that Should Not Be Confused

Abstract In this Appendix, we point to a major mistake and confusion pervasive in the world of real option. There are two COMPLETELY different things; both are called real options: one is the ROA which is the subject of this book and related somehow to Black–Scholes, and the other one is the McDonald–Siegel model which sets optimal conditions for investments under uncertainty. Failing to distinguish the two benefits neither ROA nor the problem of optimizing the conditions for investments under uncertainty.

A.1

McDonald and Siegel

In a paper entitled “The Value of Waiting to Invest” (McDonald and Siegel 1986), McDonald and Siegel proposed a model to find the optimal conditions of an investment under uncertainty. Although this approach differs profoundly from the ROA derived from financial option theory, this approach is also called “real option.” Some people call them “strategic” options as opposed to “real” options. This approach was made popular by a book by A. Dixit and R. Pindyck published in 1994 (Dixit Pindyck1994). The model of McDonald and Siegel assumes that the “value” of an investment follows a geometric Brownian motion (one limitation of this approach is that it is difficult to generalize to cases where the uncertainty on the value of the investment proceeds completely differently): dS ¼ ðρ  δÞ S dt þ σ S dz

ðA:1Þ

The drift is written as (ρ  δ) to facilitate the discussion which follows. ρ is a discount rate and δ a “free” parameter around which the discussion is based. The problem is to maximize the expected return of the investment, i.e., to maximize the expression: H ¼ eρ th(S  I)i. I is the cost of the investment. At © Springer Nature Switzerland AG 2020 B. Morel, Real Option Analysis and Climate Change, Springer Climate, https://doi.org/10.1007/978-3-030-12061-0

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A Optimizing the Conditions of Investments Under. . .

130

the outset, one can or should notice that this expression for H is different from the expression used in ROA. The expression to maximize is: H¼e

ρ t

1 ð

ðS  I Þ ΨðS, t Þ dt

ðA:2Þ

0

The expression giving the ROA value of the investment is: H ROA ¼ e

ρ t

1 ð

ðS  I Þ ΨðS, t Þ dt

ðA:3Þ

I

To a blind eye, they may look similar. But the fact that the second has a lower bound makes it different. HROA is the expected benefit from the investment. Its value plays a central role in ROA. The more fundamental difference is that ROA does not involve any maximization. In the McDonald–Siegel problem, H is the expression to maximize. What matters are the conditions that maximize H. The problem of McDonald and Siegel is a stochastic dynamic programming problem. The optimal conditions for the investment are solutions of the “Bellman” equation (dH ¼ 0): 2

ρ H þ

∂H σ 2 S2 ∂ H ðρ  δÞ S þ ¼0 ∂S 2 ∂S2

ðA:4Þ

This differential equation is in fact easy to solve. One can make the “ansatz”: H  Sβ, to get that the exponent β is the solution of a second-degree equation: 2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi3  2 2   2 1 4σ σ  ðρ  δÞ þ  ðρ  δÞ þ 2ρ σ 2 5 β¼ σ2 2 2

ðA:5Þ

There are three possibilities (now the advantage of introducing δ becomes clearer and will be even more so later): δ < 0: The value of the investment grows faster than the discount rate. The optimum policy is to wait forever. δ > ρ > 0: The value of the investment decreases with time (ρ  δ < 0). The best decision is now or never. Finally: ρ > δ > 0, the optimal policy is to wait that:

A Optimizing the Conditions of Investments Under. . .

S 

β I β1

131

ðA:6Þ

The time T where this condition is met is not known. Using stochastic calculus techniques (Karlin Taylor 1981), it can be estimated as well as the probability that this condition will eventually be met.

A.2

Linkage with the Black–Scholes-Based Approach to ROA

This approach differs fundamentally from the ROA described in this book. The Bellman equation is different from the Black–Scholes equation conceptually and their solution completely different mathematically. In ROA, like in financial options, there is a specific time horizon or exercise time. Here the time is a control variable in an optimization, and it is impossible to predict precisely when the investment should be made. Risk neutrality does not enter; otherwise the parameter δ would disappear. It is possible to somehow clarify the difference between the world of financial options and the world of investment under uncertainty (Pindyck 2008). It starts by creating the same risk free portfolio used in the derivation of Black–Scholes: Π¼H

∂H S ∂S

ðA:7Þ

In Black–Scholes, the next assumption is: dΠ ¼ dH 

∂H dS ¼ ρΠdt ∂S

ðA:8Þ

In the context of the McDonald–Siegel model, the next assumption is: dΠ ¼ dH 

∂H ∂H dS  δ S dt ∂S ∂S

ðA:9Þ

In the case of Black–Scholes δ ¼ 0. The additional term δ S∂H dt is an artificial ∂S way to connect with the Black–Scholes equation, i.e., to get rid of the parameter δ in Eq. A.9. δ is the difference between the return of the share, which we chose to write as ρ  δ and the riskless rate ρ. One reason for using ρ  δ as rate of change of the share is that it makes it more obvious that “risk neutrality” requires δ ¼ 0 .The excuse for ∂H dt is that it is a “payment” for the “short” position (Pindyck 2008). the term δ S ∂S When δ ¼ 0, i.e., in the case of Black–Scholes, in the context of this optimization, this would mean β ¼ 1 and S ¼ 1. But mixing the two approaches that way or in any way is the equivalent of making a mathematical ratatouille.

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A Optimizing the Conditions of Investments Under. . .

Clearly, ROA and McDonald–Siegel are totally different whatever way one looks at them. They can provide complementary perspectives but completely different perspectives. In the context of climate policy, having to wait that optimal conditions are met before investing can make a lot of sense. The question is to what extent can the result of McDonald and Siegel be generalized to situations relevant for climate change policy, i.e., where one can write a Bellman equation, because the dynamics of change of the relevant variables is known. There are a lot of uncertainties in climate change, many changing with time, but few if any driven by a geometric Brownian motion. There is an ROA version of the value of waiting before making an investment. It is computed completely differently. If the distribution of return is time dependent (like here), one can compute the value of waiting a fixed amount of time, by computing the difference of expected return at that time. This was done in Chap. 3 in a real option example. In that case the waiting time is fixed. It is not a control variable as here. References Dixit, A. K., Pyndyck, R. S.: Investments Under Uncertainty, Princeton, New Jersey (1994) Karlin, S., Taylor, H.M.: Second Course on Stochastic Processes, Academic Press (1981) McDonald, R., Siegel, D.: Quar. J. Econ. 101(4): 707–728 (1986) Pindyck, R.: Lectures on Real Options, Part II, 2008

Appendix B ROA and Climate Change in Practice

I’d better have questions that can’t be answered than answers that can’t be questioned. Richard Feynman1

Abstract Many of the IPCC guidelines or recommendations are either not easy to implement or platitudes or paradoxical like recommending to wait for a disaster to occur before acting. Examples: “Effective risk management generally involves a portfolio of actions to reduce and transfer risk and to respond to events and disasters, as opposed to a singular focus on any one action or type of action (high confidence).” “The more astute and effective the investments made in mitigation and adaptation, the less will be the suffering.” “Increases in exposure will result in higher direct economic losses from tropical cyclones. Losses will also depend on future changes in tropical cyclone frequency and intensity (high confidence)”. “Post-disaster recovery and reconstruction provide an opportunity for reducing weather- and climate-related disaster risk and for improving adaptive capacity (high agreement, robust evidence)”.

B.1

Introduction

Those reports are good at pointing to cheap measures that contribute to mitigation and/or improve adaptive capacity. There is no excuse not to implement those measures. But it is important to emphasize that in and of themselves those measures are not sufficient. The remaining problem is to decide which among the more

1 https://www.goodreads.com/quotes/1134331-i-would-rather-have-questions-that-can-t-beanswered-than

© Springer Nature Switzerland AG 2020 B. Morel, Real Option Analysis and Climate Change, Springer Climate, https://doi.org/10.1007/978-3-030-12061-0

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expensive measures should be given higher priorities. This is where ROA enters and does a better job than NPV. ROA is about the expected benefits of measures, individual or in combination, and comparing them to their costs. In general, the weakest part of the recommendations of those reports is how to prioritize the investments. There is so much emphasis on the risk management aspect of the response to climate change and its psychological dimension that one almost forgets that the deciding factor most of the time is more prosaic, i.e., cost or cost efficiency. Those reports are “high level,” i.e., they make broad recommendations whose implementations can be (and in general are) problematic. The translation of those recommendations into actual policy is the theme of this Appendix. More exactly this Appendix is about what ROA can contribute to this most important step.

B.2

The Basics of ROA

Real option is the expected benefit or loss in any transaction. One pays a price for a return, which is uncertain, i.e., described by a PDF (probability distribution function) ψ 1(X). The cost itself can be uncertain and could be described by another PDF: ψ 2(Y ). If the cost is fixed, the PDF is obviously trivial. But the interesting situations are when there is uncertainty in both the cost and the benefit. It is important to keep in mind that ψ 1(X) and ψ 2(Y ) do not need to be correlated in any form of way. Without the distributions ψ 1(X) and ψ 2(Y ), ROA does not proceed. They are vital. So later in this Appendix, we dwell about how to know what the PDFs ψ 1(X) and ψ 2(Y ) are or how to figure what they are. ROA, i.e., the expected benefit of doing one investment, boils down to ONE formula, the expression for the value of the real option H(X, Y, T ) of paying the cost Y to get the benefit X. T is the time at which the determination of whether the investment was worth it or not is made. In some contexts, it is called the exercise time. The general form of H(X, Y, T ), i.e., the formula to start from, is: H ðX, Y, T Þ ¼ e

ρT

1 ð

ðu  Y Þ Ψðu, X, Y, T Þdu

ðB:1Þ

Y

H(X, Y, T ) is the expected benefit of doing the investment whose cost is uncertain and described here as ψ 2(Y) for a benefit also uncertain and described here as ψ 1(X). C is the probability distribution that the benefit X is larger than the cost Y. It can be a X difference distribution (X > Y ) or a ratio distribution ( > 1). We explain later in Y this Appendix (as well as in Chap. 2) how such distributions can be computed from the original distributions ψ 1(X) andψ 2(Y ). As explained in too many details in Chap. 2, there are two situations: whether first-degree homogeneity, i.e., the relation H(λX, λY, T ) ¼ λH(X, Y, T ), applies or not.

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This is discussed in Chap. 2 ad nauseam. But it applies rarely in the context of environmental investment. When it does not apply, which is what happens most of the time in environmental problems, Ψ(u, X, Y, T) is the “difference distribution” build from ψ 1(X) and ψ 2(Y ). But when first-degree homogeneity applies as it does in most if not all cases in financial option, Ψ(u, X, Y, T ) is the ratio distribution build from ψ 1(X) and ψ 2(Y ). In both cases, the integral computing the expected benefit picks the values of X larger than Y. But in the first case (when first-degree homogeneity does not apply) that means that the difference has to be larger than zero; in the second case (when firstdegree homogeneity does apply) the ratio has to be larger than one.

B.2.1

Effect of First-Degree Homogeneity

If first-degree homogeneity applies, Eq. B.1 becomes by dividing by Y, i.e., introducing v ¼ Yu : H ðX, Y, T Þ ¼ e

ρT

1 ð

ðv  1Þ Ψratio ðv, X, Y, T Þdv

Y

ðB:2Þ

1

But if it does not apply, Eq. B.1 becomes, by subtracting with Y, H ðX, Y, T Þ ¼ e

ρT

1 ð

w Ψdiff ðw, X, Y, T Þdw 1

In those formulae, Ψratio(v, X, Y, T ) and Ψdiff(w, X, Y, T) are, respectively, the ratio and difference distributions for X and Y. How to get them knowing ψ 1(X) and ψ 2(Y ) is explained in Chap. 2. The less trivial point is how we get the distributions ψ 1(X) and ψ 2(Y) in the first place in the context of climate change policy choices. This is the subject of this Appendix.

B.3

Climate Change Policy Choices

Unabashedly or as an expression of respect for the importance of their work (IPCC 2014), the examples around which the discussion of this Appendix is built were borrowed from a couple of selected sources, namely, the report entitled “Confronting Climate Change: Avoiding the Unmanageable and Managing the Unavoidable”

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(SEGCC 2007), and the IPCC “Special Report on Managing the Risks of Extreme Events and Disasters to Advance Climate Change Adaptation” (SREX 2012). Recommendations tend to come as laundry lists of things to do, like: Avoiding the unmanageable and managing the unavoidable will require an immediate and major acceleration of efforts to both mitigate and adapt to climate change. The following are our recommendations for immediate attention by the United Nations (UN) system and governments worldwide. 1. Accelerate implementation of win-win solutions that can moderate climate change while also moving the world toward a more sustainable future energy path and making progress on attaining the MDGs (see Box ES.1). Key steps must include measures to: Improve efficiency in the transportation sector through measures such as vehicle efficiency standards, fuel taxes, and registration fees/rebates that favor purchase of efficient and alternative fuel vehicles, government procurement standards, and expansion and strengthening of public transportation and regional planning. Improve the design and efficiency of commercial and residential buildings through building codes, standards for equipment and appliances, incentives for property developers and landlords to build and manage properties efficiently, and financing for energy efficiency investments. Expand the use of biofuels, especially in the transportation sector, through energy portfolio standards and incentives to growers and consumers, with careful attention to environmental impacts, biodiversity concerns, and energy and water inputs. Promote reforestation, afforestation, and improved land-use practices in ways that enhance overall productivity and delivery of ecological services while simultaneously storing more carbon and reducing emissions of smoke and soot. Beginning immediately, design and deploy only coal-fired power plants that will be capable of cost-effective and environmentally sound retrofits for capture and sequestration of their carbon emissions. 2. Implement a new global policy framework for mitigation that results in significant emissions reductions, spurs development and deployment of clean energy technologies, and allocates burdens and benefits fairly. Such a framework needs to be in place before the end of the Kyoto Protocol’s first commitment period in 2012. 3. Develop strategies to adapt to ongoing and future changes in climate by integrating the implications of climate change into resource management and infrastructure development and by committing to help the poorest nations and most vulnerable communities cope with increasing climate change damages. Taking serious action to protect people, communities, and essential natural systems will involve commitments to: Undertake detailed regional assessments to identify important vulnerabilities and establish priorities for increasing the adaptive capacity of communities, infrastructure, and economic activities. For example, governments should commit to incorporate adaptation into local Agenda 21 action plans and national sustainable-development strategies. Develop technologies and adaptive-management and disaster-mitigation strategies for water resources, coastal infrastructure, human health, agriculture, and ecosystems/biodiversity, which are expected to be challenged in virtually every region of the globe, and define a new category of “environmental refugee” to better anticipate support requirements for those fleeing environmental disasters. Avoid new development on coastal land that is less than one meter above present high tide, as well as within high-risk areas such as floodplains. Ensure that the effects of climate change are considered in the design of protected areas and efforts to maintain biodiversity. Enhance early-warning systems to provide improved prediction of weather extremes, especially to the most vulnerable countries and regions.

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Bolster existing financial mechanisms (such as the Global Environment Facility)—and create additional ones—for helping the most vulnerable countries cope with unavoidable impacts, possibly using revenues generated from carbon pricing, as planned in the Adaptation Fund of the Clean Development Mechanism. Strengthen adaptation-relevant institutions and build capacity to respond to climate change at both national and international levels. The UN Commission on Sustainable Development (CSD) should request that the UN system evaluate the adequacy of, and improve coordination among, existing organizations such as the CSD, the Framework Convention on Climate Change, the World Health Organization, the Food and Agriculture Organization, the UN Refugee Agency, the World Bank, and others to more effectively support achievement of the MDGs and adaptation to climate change. 4. Create and rebuild cities to be climate resilient and GHG-friendly, taking advantage of the most advanced technologies and approaches for using land, freshwater, and marine, terrestrial, and energy resources. Crucial action items include the following elements: Modernize cities and plan land-use and transportation systems, including greater use of public transit, to reduce energy use and GHG intensity and increase the quality of life and economic success of a region’s inhabitants. Construct all new buildings using designs appropriate to local climate. Upgrade existing buildings to reduce energy demand and slow the need for additional power generation. Promote lifestyles, adaptations, and choices that require less energy and demand for nonrenewable resources. 5. Increase investments and cooperation in energy-technology innovation to develop the new systems and practices that are needed to avoid the most damaging consequences of climate change. Current levels of public and private investment in energy technology research, development, demonstration, and pre-commercial deployment are not even close to commensurate with the size of the challenge and the extent of the opportunities. We recommend that national governments and the UN system: Advocate and achieve a tripling to quadrupling of global public and private investments in energy-technology research, emphasizing energy efficiency in transportation, buildings, and the industrial sector; biofuels, solar, wind, and other renewable technologies; and advanced technologies for carbon capture and sequestration. Promote a comparable increase in public and private investments—with a particular emphasis on public–private partnerships—focused on demonstration and accelerated commercial deployment of energy technologies with large mitigation benefits. Use UN institutions and other specialized organizations to promote public–private partnerships that increase private sector financing for energy efficiency and renewable energy investments, drawing upon limited public resources to provide loan guarantees and interest rate buy-downs. Increase energy-technology research, development, and demonstration across the developing regions of the world. Potential options for achieving this goal include twinning arrangements between developed and developing countries and strengthening the network of regional centers for energy-technology research. Over the next two years, complete a study on how to better plan, finance, and deploy climate-friendly energy technologies using the resources of the UN and other international agencies such as the UN Development Programme, the World Bank, and the Global Environment Facility.

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B.3.1

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Discussion of Report Recommendations

Those “recommendations” were made in 2007 (SEGCC 2007). Ten years later, not many have been followed. Point 2 in particular has serious room for improvement, as well as any recommendation asking the UN to use its resources to foster some international initiatives, even if during that period the UN secretary general (Ban Ki Moon) dedicated a lot of his personal energy and prestige to environmental issues. But there is such a thing as bureaucratic inertia. In addition, the private sector is not only greedy; it has also common sense. It will act when it makes economic sense. That much cannot be said for the public sector. Private–public partnership requires from the public sector to project a sense that finally they got the message: make economic sense. Everything will be easier after that. The contribution of the taxpayer will be more rational, i.e., greater impact at lesser cost.

B.3.1.1

The ROA Journey from “Recommendations” to Policy

Of interest here is to see whether ROA can bring some policy clarity. The first distinction to make among the recommendations made in 2007 (SEGCC 2007) is between policy measures in the sense of regulation and policy measures involving investments. ROA is more relevant for the second. Here that means focusing on: “Transport: promoting greener technology and strengthening public transport. Promoting biofuel, while preserving biodiversity”. . . Housing: make buildings more energy efficient. “Promote energy production less carbon intensive”. The emphasis was on Carbon Capture and Sequestration in addition to renewables and quadrupling the level of public and private investments. “Promote reforestation, afforestation, and improved land-use practices in ways that enhance overall productivity and delivery of ecological services while simultaneously storing more carbon and reducing emissions of smoke and soot”. And working at making areas vulnerable to extreme weather, like coastal or arid areas, less vulnerable, by developing technologies and adaptive-management and disaster-mitigation strategies for water resources, coastal infrastructure, human health, agriculture, and ecosystems/biodiversity. Next comes the translation of these “recommendations” into actual investments. The “recommenders” leave it to the policy makers to go that extra mile. In fact it probably looks to them more like an extra thousand miles. As the Chinese proverb says (Lao Tzu 500 BCE), “A journey of a thousand miles begins with a single step (or under one’s foot, depending on the translation. The original version is: ).”

B ROA and Climate Change in Practice

B.3.1.2

139

The First Step

As far as possible you should not start a journey without knowing when you want to go; otherwise at best you end up in Rome, but more probably you will get lost. Here that means trying to take full advantage of what ROA has to offer. It starts with identifying PDFs for benefits or costs and make them speak to the policy decisions.

B.3.2

Deforestation, Reforestation, Afforestation

In Chap. 6, we dealt with the problem of deforestation and afforestation in the context of REDD credits. Any mitigation policy dealing with forests has its economic and other moving parts. Often, refraining from deforesting means not being able to engage in a lucrative activity. Most of the forests of interest for mitigation are located in tropical regions and in developing countries. A system of subsidies (like the REDD credits) is of essence. Forests are not equal. They differ by their carbon content, age, ecosystem services, and biodiversity content. In Chap. 6, we saw how ROA can help deciding the level of subsidies. We also noticed that even if ROA is hardly invoked in the context of REDD credits, in practice when countries make bids to get credits what they do is very related.

B.3.3

Vulnerability to Extreme Events

This problem is related to what Chap. 5 tried to accomplish. The problem is to deal with events lying deep in fat tails of distributions. On one hand, if the choice is between using NPV and ROA to get analytic power, this is a no-brainer. NPV simply fails to capture the effect of tails of distribution. ROA on the other hand does. So, in principle ROA is the only tool which would allow potentially to conduct a quantitative approach to value investments. In one of the reports quoted earlier in this Appendix, it was suggested that “postdisaster recovery and reconstruction provide an opportunity for reducing weatherand climate-related disaster risk and for improving adaptive capacity (high agreement, robust evidence).” This is one approach. Whether it worked well in New Orleans after Katrina is not obvious. Preventives measures have their merit too. The policy problem addressed here is the choice of prophylactic investments to reduce the damage from extreme events like Katrina. In practical terms, can ROA be used to estimate the reduction of damage that can be accomplished preventively, at a given cost? ROA is not a substitute to a good vulnerability assessment. What it does at best is to help factoring the findings of vulnerability assessments into an estimate of expected benefits of investments to address them. The “expected” part is the most

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difficult part. Extreme events are by nature rare. An event like Katrina occurs so rarely that some may question the wisdom to even bother. Had the hurricane hit tens of miles in any direction, the damage would have been significantly less serious. But as emphasized in Chap. 5, it is enough for the fat tail of the damage distribution to be known with uncertainty to have an enhancement of the effect of the tail (Weitzman effect). Hence, a plain ROA estimate in fact underestimates the value of the expected benefit of preventive measures. As a result, quite a bit of hedging has to be involved. Extreme events have multiple effects and many of them can be covered by insurances. There is something to be said to rely as much as possible on insurances. They may act for profit, but they also introduce an economic rationality in those situations of complexity. Furthermore, whether they do it consciously or not, the way they set up their insurance follows exactly the logic of ROA. So from an ROA perspective, insurances and their culture should be given as large role as possible to the response. There are additional remedial or preventive measures that can hardly be covered by insurances, like infrastructure investments. They have to be valued on a case-bycase basis. Their expected benefit includes the fact that they potentially reduce some of the damage that insurances may have to cover. ROA here is useful to focus on the expected benefits and itemize that contribute to it. (De-)forestation and extreme events were the easiest to handle in this Appendix, since both were discussed earlier in this book. But climate change response is not limited to them as proved by the fact that there are other recommendations, which also merit attention.

B.3.4

Promoting Energy Production Less Carbon Intensive

This is not a new debate, although it evolves with time, not necessarily expressing evidence of movement of a frontier of wisdom on the subject, but more as evidence of fads. Carbon capture and sequestration (CCS) for example at one point was prominent. Far before there was evidence that it was a dead end, the emphasis moved elsewhere, like renewables. Nuclear, which qualifies to be not carbon intensive, has a mixed record in that debate. The decision by some countries to move completely (like Germany, Switzerland) or partially (France, UK, Japan) away from it proceeded from a rationality, which may have economic merit, but that was not visible. There is an economic case to be made against nuclear: because of safety reasons, the capital cost to build a plant is always severely underestimated. People forget that phasing down a nuclear plant costs US$ billions of cleanup. The operating cost of a nuclear plant is not very high (compared with the alternative), but it has its own inertia. It is not possible to change fast the power output. Still, it does not produce any carbon and its safety record is not so bad in countries like France, for

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141

example, which has relied heavily (>70%) on that source of electricity. Russia may not be as good an example. James Hansen (one of the most thoughtful experts in climate change) made his discomfort with the present stand on that issue of Germany or Japan amply clear, even if earlier in his life he expressed skepticism about nuclear power (Vaidyanathan 2015). Facts may compel the countries would decide to forego the nuclear component, to revisit that stand. For the time being, this is the context in which the future of electricity production has to be approached. ROA can be used to judge the economic wisdom of the political choices of countries like Germany, Switzerland, Japan, or France to phase out as much as possible of their reliance on nuclear, apparently for no other reasons that this is popular. ROA is about crude economic rationality in the presence of uncertainty. ROA deals with objective uncertainty, not emotions or risk aversion. From an ROA perspective, the major question is what are the expected benefits of not using nuclear as compared to the alternatives? The alternatives are the renewables, i.e., wind, solar, hydropower, biomass, geothermic, tides, mills in rivers, hamsters or squirrels in tread mills, caressing cats to make them electrostatic, and a few more. Each has its merits and limits. Some couple more easily to the electric grid than others. Some scale better than others. The cost structure is different for each. Some are more reliable than others. The impact on the environment of each is also different. Some can be located only in specific regions: hydropower for example requires water with current. Solar works better during the day; does it make it more or less economically valuable in high latitude, where there is alternation of long days and short days during the year than close to the equator? Cloud cover is also a concern. And there are many other considerations. Clearly economic rationality in energy production is not best served by blank choices, i.e., without taking care of the specifics, as if it was a case of one size fits all.

B.4

France as Example or Case Study on How High Level Recommendations Speak to Environmental Policy

As illustration, it is instructive to see (superficially) what ROA has to say on the way France intends to undergo an energy transition toward a greener economic growth by de-emphasizing its reliance on nuclear power and broadening its energy portfolio. This move has been welcomed by IEA2: “France’s energy transition is vital for energy security.” Let us have a look.

2 https://www.iea.org/newsroom/news/2017/january/energy-policies-of-iea-countries-france-2016. html

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B.4.1

B ROA and Climate Change in Practice

France’s Policy

The new French energy policy was codified in the 2015 law: “La LOI no 2015-992 du 17 août 2015 relative à la transition énergétique pour la croissance verte.”3 In the words of IEA4 “the cornerstone of the transition, [is that] the role of renewable energy in the power mix is to be increased to 40% by 2030 (from its current share of 16.5%), and France aims to accelerate energy savings while preparing for the future, given its ageing nuclear fleet. The government sets the ambitious target of reducing the share of nuclear from 78% in 2015 to 50% by 2025. [. . .] To achieve a share of 50% by 2025, without increasing CO2 emissions, would require an increase of non-hydro renewable energies (wind, solar, biomass) from their current level of 6.8% to 34% within 10 years, assuming a stable electricity demand.” From an ROA standpoint, it is far from obvious that there is not a better way to manage or design the “energy transition” envisioned by the French. The detour of ROA here is a way to illustrate how ROA works. But it becomes clear that ROA here is an extension of common sense. Here is how an ROA approach might look like: 1. 2. 3. 4.

Get the ROA value of building new nuclear plants Get the ROA value of extending the life of existing plants Get the ROA value of developing renewables Because the latter may (will) not substitute in time for the nuclear, get the ROA value of alternative additions like gas plants 5. Combine the results and develop a realistic/optimal strategy. 6. Compare with the existing plan (optional or not advised to those inclined toward radicalization). To discuss the economics of power plants, “professionals” use the concept of “levelized cost of electricity (LCOE)”.5 “The LCOE represents the price that the electricity must fetch if the project is to break even (after taking account of all lifetime costs, inflation and the opportunity cost of capital through the application of a discount rate).” Each plant has a different LCOE. With enough data it is possible to gas renewable build a distribution of LCOE ψ nuclear ðX Þ, ψ LCOE ðX Þ for nuclear, LCOE ðX Þ, ψ LCOE renewables, and gas plants. In a comparative study of LCOE,6 numbers are given for the average of those distributions in the case of France. The numbers for 2014 are (assuming a discount rate of 7%): hLCOEnucleari  $80/MWh, hLCOEgasi  $100/

3

http://archive.wikiwix.com/cache/?url¼http%3A%2F%2Fwww.developpement-durable.gouv.fr %2FIMG%2Fpdf%2Fjoe_20150818_0189_0001_1_-2.pdf 4 http://www.iea.org/publications/freepublications/publication/Energy_Policies_of_IEA_Coun tries_France_2016_Review.pdf 5 http://www.world-nuclear.org/information-library/economic-aspects/economics-of-nuclearpower.aspx 6 http://www.world-nuclear.org/information-library/economic-aspects/economics-of-nuclearpower.aspx

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143

MWh, hLCOEwindi  $(110  210)/MWh, hLCOEsolari  $180/MWh. In the case of the wind, the difference is between onshore and offshore wind. gas renewable ψ nuclear ðX Þ, ψ LCOE ðX Þ are the cost distributions associated with LCOE ðX Þ, ψ LCOE these different technologies. To get the benefits these costs should be subtracted from the price of electricity. The price of electricity in the future can be described by the PDF:ψ elec(Y, T ). So the expected benefit at time T of using any of those technologies is: H j ðX, Y, T Þ ¼ e

ρT

1 ð

uΨ j ðu, CX, Y, T Þdu

ðB:3Þ

ψ elec ðX þ u, T Þψ j ðX ÞdX

ðB:4Þ

0

j ¼ nuclear, renewables, gas and: 1 ð

Ψ j ðu, X, Y, T Þ ¼ 1

is the difference distribution the price of electricity and the cost of producing it with the different technologies. Hj(X, Y, T ) is the expected benefit of using any of those technologies, i.e., the ROA value of those technologies. gas renewable The distributions ψ nuclear ðX Þ, and ψ LCOE ðX Þ are expected to change LCOE ðX Þ, ψ LCOE with time. Building new nuclear plants has turned out to be anything but a smooth exercise. In the USA (plant Vogtle, in Georgia,7 Watts Bar unit 2 in Tennessee8) as well as in France (Flamanville9), that kind of projects has generated massive initial capital cost overruns as well as costly delays.10 Those are (facetiously) pictured in Figs. B.1 and B.2. In Fig. B.1, ψ cost(X) is a fictional distribution for the initial capital cost of a new nuclear plant. In that fictional distribution, it is assumed that the actual cost turns out to more than five times the estimated cost. The reality is not as bad as that. ψ delay(X) in Fig. B.1 is also a fictional distribution for the delay before the start of a new nuclear plant. That kind of delays adds to the cost of the project and has to be factored in when planning new plants. One can argue that the recent delays and cost overruns have to do with the fact that the new plants were technologically different from the existing PWR. They were part of a learning curve. In an ROA analysis how would one use ψ delay(T ) or ψ cost(X) ? The answer is obvious for ψ cost(X) : you factor it in the cost function. When it comes to ψ delay(T ), the answer is a little bit (but not much) more involved. The return from the nuclear

7

https://www.reuters.com/article/us-usa-nuclearpower-vogtle/u-s-offers-vogtle-nuclear-plant-3-7billion-in-loan-guarantees-idUSKCN1C42BG 8 http://www.latimes.com/business/hiltzik/la-fi-hiltzik-nuclear-shutdown-20170508-story.html 9 http://www.world-nuclear-news.org/NN-EDF-confirms-Flamanville-EPR-start-up-schedule1207174.html 10 https://www.eia.gov/outlooks/capitalcost/pdf/updated_capcost.pdf

144 Fig. B.1 Typical distribution of cost of nuclear plant project

B ROA and Climate Change in Practice

cost(X )

Coverun

Cest

Fig. B.2 Typical time delay in the construction of nuclear plants

delay(T )

0

5yr

10yr

project will not start before the nuclear plant is up and running. This will happen not when it was planned but at a later time Tdelay. Hence the discounting factor used in the estimation of the ROA value should be T + Tdelay  and the expression for the ROA value H(X, Y, T ) should include the factor ψ delay T delay eρT delay in front. The whole ROA value of the project has to be discounted by the effect of the delay. This means that the LCOE associated with nuclear energy may be affected negatively. Also the cost efficiency of the technology of renewables is improving: their LCOE may improve (i.e., be lower). Finally, gas plant releases CO2. Even if they are weaker emitter than other fossil fuel-based plants, this carbon will generate an additional cost. In 2015, the share of gas in France electricity production was 3.5%.11 Is that “optimal”? Coal is “responsible” for 2.2% of electricity generation and decreasing. Carbon capture and sequestration (CCS) is not a relevant technology for France. The premises of the calculation of what is the best mix for France in the future should be revisited regularly as those changes go fast compared with the life expectancy of the different kinds of plants. The strategy should allow some flexibility or adaptiveness. What is best for France may not be for other countries. The LCOE for nuclear is lower in Korea than France but larger in the UK and USA. On the other hand, the LCOE for gas is significantly lower in the USA as is the LCOE for solar, for

11 http://www.iea.org/publications/freepublications/publication/Energy_Policies_of_IEA_Coun tries_France_2016_Review.pdf

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example. These disparities have natural explanations related to climate and access to raw material like gas and regulatory environment. In a changing technological environment, where electricity plays a central role, one should expect that the ideal mix of source of electricity should change too. Seen from today what should be the optimal mix, or optimal policy? One may ask how did the French Government got to numbers like 50% share of nuclear by 2025 and renewables will fill the gap. They look good (from a political correctness perspective), but are they the result of a thorough economic optimization? It is far from obvious that there was a lot of consideration of whether the same reduction in carbon release in electricity production could not have been achieved otherwise and in a more cost-efficient way. Whether these numbers stand scrutiny or not, they have become binding constraints. This is a typical case of “bounded rationality” (Simon 1990). Bounded rationality is a step above “irrationality,” i.e., what seems to inspire the recent German and Japanese new stand on nuclear energy in the context of CO2 emissions. In 2015, 53.96% of the electricity in Germany was based on coal, gas, or oil12— the lion share being the most generous in CO2 emission, i.e., coal. Japan does not fare much better as far as dependence on CO2 emitting fossil fuel is concerned.13 Under those conditions, why go against the least CO2 emitting form of electric power, whose technology has still more to offer than we already know (think small reactors, breeder reactors, or controlled fusion), if it is not for pure political reasons, the energy policy equivalent of “populism”?

B.4.2

Contours of an ROA-Based Optimal Strategy

The “plan de transition energétique” of France calls for a reduction of the share of the nuclear in electricity production to 50% by 2025 from 77.7% in 2015 and the gap in production being filled by renewables. By renewables in the case of France one means hydropower (9.7% of electricity generation in 2015), wind (3.8%), and solar (1.5%). There is not much possibility for increasing the production of electricity through hydropower in France. The bulk of the contribution of the new renewables will have to be from wind and solar. IEA in its 2016 review of France’s energy policy14 states in its recommendations: “The government of France should Implement the reformed renewable electricity support scheme in line with the pluriannual energy programming to ensure it reflects market signals and benchmarked technology costs, while addressing the need for transparency, long-term predictability and certainty to gain investors' confidence.”

12

https://www.statista.com/statistics/436880/share-of-electricity-from-oil-coal-gas-germany/ https://www.eia.gov/beta/international/analysis.cfm?iso¼JPN 14 http://www.iea.org/publications/freepublications/publication/Energy_Policies_of_IEA_Coun tries_France_2016_Review.pdf 13

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This is code word for one can be somewhat skeptical that the renewable part of your plan will deliver in a timely way. As the LCOEs suggest, the expected benefits, i.e., from ROA value of renewables, are not very competitive with nuclear. Reliance on what renewables can provide may turn out to be excessive. As compared the nuclear contribution is far less uncertain. A more progressive reduction of the nuclear contribution and substitution by renewables seems a much more prudent policy. In fact, if one wanted to push the use of ROA to its logical conclusion, one could use it to estimate the value of acquiring additional information about the renewables and organize the buildup of that sector around its uncertainty. This is left as an exercise to the readers.

B.5

Transports: Electric Vehicles as case Study

“Our personal vehicles are a major cause of global warming. Collectively, cars and trucks account for nearly one-fifth of all US emissions, emitting around 24 pounds of carbon dioxide and other global-warming gases for every gallon of gas. About five pounds comes from the extraction, production, and delivery of the fuel, while the great bulk of heat-trapping emissions—more than 19 pounds per gallon—comes right out of a car’s tailpipe.”15

B.5.1

About France and Electric Cars

One can read in the IEA review of France energy policy16: “Transport consumes on average one-third of total energy use in IEA member countries and the sector is the main emitter of greenhouse gases globally. EVs are more energy-efficient than internal combustion engine vehicles (ICV) and enable low-carbon emissions if the electricity is supplied from low-carbon energy sources. Furthermore, EVs can reduce local air pollution and noise levels in urban areas. For oil-importing countries with high reliance on fossil fuels in the transport sector, EVs can also provide improved energy security. [. . .] With nuclear electricity providing 77% of electricity, electrification of the transport sector will ensure an important part of the transformation and decarbonisation of the transport sector.” In the previous discussion about nuclear versus renewables in France, the assumption was that the consumption of electricity would not increase in the foreseeable future. If anything it would decrease. But what if electric vehicle

15

http://www.ucsusa.org/clean-vehicles/car-emissions-and-global-warming#.WiCGzoXBgSY http://www.iea.org/publications/freepublications/publication/Energy_Policies_of_IEA_Coun tries_France_2016_Review.pdf 16

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(EV) become the favored means of transport. What impact would that have on electricity demand?

B.5.2

The ROA Value of Electric Vehicles (EV)

Given the choice between an electric car and a gas fueled car, with identical properties when it comes to price, range, speed, and ease of refilling, any rational driver would choose the electric option. But today somehow, the impression is that to choose to drive electric cars involves quite a few concessions: range (not the kind of cars people today would be comfortable to cross the country with) and limited options for the refill. On the other hand, electricity cost in the USA is around 12 cents per kWh. That means that it costs a fraction of what gasoline costs to refill the car. That depends on the specific but the ratio is close to a factor 3.17

B.5.3

A Few Vital Statistics to Anchor the ROA Estimate

B.5.3.1

CO2 Reductions

Some advocate EVs as a way to reduce CO2 emissions. But that supposes that the production of electricity powering those vehicles does not emit CO2 in the first place. Unlike with most other countries, in the case of France, electricity tends to be mostly produced from nuclear and increasingly renewables, the movement from combustion engines to EV is a net reduction of CO2 emission. Although there is a large variance, it is said that in average cars emit about six tons of CO2 every year.18

B.5.3.2

EV Need for Electricity

Those are rough numbers. Still we take the Ford Focus EV as a basic example. Allegedly the electric energy it needs to travel 100 miles is 32 kWh.19 The same car to travel 10,000 miles would therefore require 3.2 MWh. Electric cars have an efficiency around 60%,20 i.e., they transform into mechanical energy 60% of the electricity they get. Hence such a car would require about 5.3 MWh. If 10,000 miles were the average distance such EVs travel in a year, each EV would require 5.3 MWh. Some would argue that 15,000 miles is a better estimate of the average

17

https://pluginamerica.org/how-much-does-it-cost-charge-electric-car/ https://www.cartalk.com/content/global-warming-and-your-car-0 19 https://www.edmunds.com/fuel-economy/decoding-electric-car-mpg.html 20 http://www.fueleconomy.gov/feg/evtech.shtml 18

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distance traveled by cars yearly. That would add 50% on all the other estimates. Instead of 5.3 MWh, individual cars would require about 8 MWh per year. The next question is how many cars have to receive that amount of electric energy. In 2015, the fleet of EV in France is 54,300. The needed electricity was about 0.29 TWh. At that time France was producing 563 TWh. It was not stressed by such demand. But the number of electric cars is expected to grow, maybe fast. In 2015, the French fleet of passenger cars was about 32 million.21 Assuming a reasonably deep penetration of EVs in France, i.e., something like 10 million electric passenger cars, the need of electricity would become 53 TWh. This is close to 10% of the present total production of electricity in France. This 10% becomes 15% if the cars traveled 15,000 miles in average per year.

B.5.3.3

How Many Power Plants Would Be Needed

The amount of electricity that a power plant generates depends on the specifics of its operation: amount of time and power level. France has 58 nuclear reactors. Together they supplied 437 TWh of electricity in 2015 (77% of the total electricity production). That means that in average each nuclear reactor contributed about 7.5 TWh. To support a fleet of 10 million EVs at least seven such reactors would be needed. The number would go to above ten in case cars in average traveled 15, 000 miles per year. Nuclear reactors tend to have a significantly higher yield (>> 1 GWe) than power plants using renewables (