Lecture Notes in Economics and Mathematical Systems Founding Editors: M. Beckmann H.P. Künzi Managing Editors: Prof. Dr...

Author:
Rainer Brosch

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Lecture Notes in Economics and Mathematical Systems Founding Editors: M. Beckmann H.P. Künzi Managing Editors: Prof. Dr. G. Fandel Fachbereich Wirtschaftswissenschaften Fernuniversität Hagen Feithstr. 140/AVZ II, 58084 Hagen, Germany Prof. Dr. W. Trockel Institut für Mathematische Wirtschaftsforschung (IMW) Universität Bielefeld Universitätsstr. 25, 33615 Bielefeld, Germany Editorial Board: A. Basile, A. Drexl, H. Dawid, K. Inderfurth, W. Kürsten

611

Rainer Brosch

Portfolios of Real Options

Dr. Rainer Brosch Berlin, Germany [email protected]

ISBN 978-3-540-78298-8

e-ISBN 978-3-540-78299-5

DOI 10.1007/978-3-540-78299-5 Lecture Notes in Economics and Mathematical Systems ISSN 0075-8442 Library of Congress Control Number: 2008923931 © 2008 Springer-Verlag Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permissions for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, 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. Production: le-tex Jelonek, Schmidt & Vöckler GbR, Leipzig Cover design: WMX Design GmbH, Heidelberg Printed on acid-free paper 987654321 springer.com

To my sister Christina

Foreword

Valuing portfolios of options embedded in investment decisions is arguably one of the most important and challenging problems in real options and corporate finance in general. Although the problem is common and vitally important in the value creation process of almost any corporation, it has not yet been satisfactorily addressed. It is key for any corporation facing strategic resource allocation decisions, be it a pharmaceutical firm valuing and managing its pipeline of drugs, a telecom company having to select a set of technological alternatives, a venture capital or private equity firm investing in a portfolio of ventures, or any company allocating resources. Portfolios of real options typically interact such that the value of the whole differs from the sum of the separate parts. Thus one must address and value the particular configuration of options embedded in a specific situation, taking into account the configuration of other options already present in the portfolio, which in turn depends on the correlation structure among the various underlying assets and the strategic dependencies among the options themselves (e.g., mutual exclusivity, strategic additivity, compoundness, complementarity etc.). In that sense, optimal decisions also depend on past option exercise decisions by management and organizational capabilities put in place in the past. The optimal decision today as to whether to continue operating in the current mode or following the current strategy versus switching to the best among a set of alternative modes or strategies depends in part on the asymmetric costs of switching strategies and potentially the costs and probability of switching back in the future. Such asymmetric costs of switching and re-switching contribute to the breakdown of the standard value additivity. That is, the overall problem is path dependent and history matters! In a practical setting, organizations oftentimes face internal budgetary, human capital and other organizational constraints. From a portfolio perspective, any investment or financial resource commitment explicitly com-

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Foreword

petes with everything else. As a result, current commitments of financial, managerial or engineering resources or projects may need to be reduced or abandoned altogether to free up resources that can be devoted to more attractive future investment opportunities. The implications of this sort of options portfolio analysis can be non-conventional and far reaching. These aspects represent modeling challenges that the thesis of Mr. Brosch addresses by a tractable model solving important managerial decision problems. In particular, it tackles the portfolio problem head on within an overall budget constraint context in which interdependencies among optional decisions at each point in time and dynamically over time are explicitly considered. His thesis sets up a general portfolio valuation framework as a multi-dimensional real options problem involving path-dependent investment and dis-investment decisions and changes in the characteristics of the underlying assets. A key aspect of his model is that he explicitly links dynamic (dis-) investment decisions with a global dynamic budget constraint that is both pivotal for deriving the optimal operating policy while it remains relevant in practical applications. Mr. Brosch formulates his model as a constrained stochastic dynamic program that he subsequently solves. He then employs extensive numerical analysis and provides sensitivity results to investigate various portfolio effects. He confirms hypothesized deviations from value additivity and that the management of portfolios of real options is a complex task that requires careful, detailed, situation-specific analysis in order to properly capture the value that can be distilled from managing portfolios of real options. His model helps provide a general structure capable of handling these real-life complexities, both for the design of an optimal portfolio as well as for the execution or optimal exercise of the embedded options. The model properly reflects the inherent complexity of the underlying real-life constrained problem and the non-additivity and path-dependent challenges embedded therein. The framework developed by Mr. Brosch makes an important theoretical contribution in addressing this important and intellectually challenging portfolio problem, while at the same time it can be of significant value to practicing managers in facing this admittedly complex and difficult practical task of evaluating, managing and optimally exercising the set of interdependent corporate real options. Arnd Huchzermeier Vallendar, Germany

Lenos Trigeorgis Nicosia, Cyprus

Preface

Ever since I was a little boy, I was fascinated by investment decisions – not with financial assets in mind, but trucks. The bigger, the better. However, once it occurred to me that beyond sheer size, equipment with a broader application range may turn out to be more desirable. Many years later, taking a Master’s degree in Finance at Frankfurt University, I got hold of a copy of Lenos Trigeorgis’s book “Real Options”. It was all there: investment and flexibility, which is at the core of real options. However, I didn’t understand how this would work for portfolios. So when I met Lenos at the Real Options Conference and asked, he just smiled: “An interesting field – let’s work on it!” Now the work is ready and I hope it can be instrumental to managers and academics alike when dealing with portfolios of real options. It was submitted as doctoral thesis at WHU, Otto Beisheim School of Management, chaired by Professor Dr. Arnd Huchzermeier and Professor Lenos Trigeorgis, Ph.D. I would like to thank Arnd Huchzermeier for his guidance and the inspiring environment he creates at the Department of Production Management, including my opportunity to teach and to visit Stanford University. I am grateful to Lenos Trigeorgis who was always close despite the distance, by email or unforgettable meetings, and added both his great esprit and expertise. I am thankful to Stefan Spinler, for always being happy to discuss and help. Furthermore my thank goes to a friend at a Hightech Company, preferring to remain anonymous, who greatly contributed by discussing the applications. I thank my colleagues at the department for making my time at WHU an enjoyable one. Also I am grateful to Christian Artmann and Rolf Hellermann for tremendous intellectual and personal support, and for being a pleasure to be with far beyond research. Finally, I thank The Boston Consulting Group for financial support.

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Preface

The most thankful I am to my parents, Arnold and Barbara, and my godmother Margret, who created many growth options for my education and my future. To my sister Christina and Peter, who are always with me. And to Jens and Thomas, for being the great friends they are. From my first thoughts about flexibility to my dissertation it has been a challenging but inspiring and rewarding journey. The journey goes on and I am looking forward to it. Berlin, January 2008

Rainer Brosch

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Research Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Contribution and Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 3 3 5

2

Portfolio Approach to Real Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Motivation of the Portfolio Approach to Real Options . . . . . . . 2.1.1 Real Options and Financial Options . . . . . . . . . . . . . . . . . 2.1.2 Definition of Portfolios of Real Options . . . . . . . . . . . . . . 2.1.3 Difference Between Financial Portfolios and Portfolios of Real Options . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Implications for Modeling Approach . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Portfolio Aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Management of Portfolios of Real Options . . . . . . . . . . . 2.2.3 Translation into Model Features . . . . . . . . . . . . . . . . . . . . .

7 7 7 11

3

Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Financial Portfolio Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Mean–Variance Portfolio Analysis . . . . . . . . . . . . . . . . . . . 3.1.1.1 Traditional Markowitz Portfolio Selection . . . . 3.1.1.2 Capital Asset Pricing Model . . . . . . . . . . . . . . . . 3.1.1.3 Intertemporal Capital Asset Pricing Model and Consumption . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1.4 Arbitrage Pricing Theory . . . . . . . . . . . . . . . . . . . 3.1.1.5 Conclusion on Mean–Variance Portfolio Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Advanced Financial Portfolio Analysis . . . . . . . . . . . . . . . 3.1.2.1 Portfolio Analysis Considering Higher Moments of Distributions . . . . . . . . . . . . . . . . . . .

12 13 14 15 16 21 21 22 22 25 26 27 27 28 28

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3.1.2.2

Allowing for Frictions in the Management of Portfolios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2.3 Impact of Combining Financial Assets with Financial Options . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2.4 Portfolio Implications for Risk Management . 3.1.2.5 Conclusion on Advanced Financial Portfolio Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Corporate Capital Budgeting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Portfolio Perspective in Corporate Capital Budgeting . 3.2.3 Interdependencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.4 Conclusion on Corporate Capital Budgeting . . . . . . . . . 3.3 Financial and Real Options Theory . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Foundations of Real Options . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Real Options Analysis in a Portfolio Context . . . . . . . . . 3.3.3 Detailed Review of Related Papers . . . . . . . . . . . . . . . . . . 3.3.3.1 Kulatilaka (1988) . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3.2 Kulatilaka and Trigeorgis (1994) . . . . . . . . . . . . . 3.3.4 Conclusion on Financial and Real Options Theory . . . . 4

Valuation Model for Portfolios of Real Options . . . . . . . . . . . . . . . . 4.1 Model Features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 General Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Objective Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.3 Managerial Actions Permitted . . . . . . . . . . . . . . . . . . . . . . . 4.1.4 Risk Preference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.5 Stochastic Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.6 Volatility and Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.7 Inter– and Intra–Project Options Interactions . . . . . . . . . 4.1.8 Global Dynamic Budgets . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.9 Synergies and Operational Constraints . . . . . . . . . . . . . . . 4.1.10 Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.11 Diversification as a Passive Approach to Risk Management . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.12 Path–Dependencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.13 Modeling Approach and Solution . . . . . . . . . . . . . . . . . . . 4.1.14 Scalability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Binomial Option Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Stochastic Processes and System State Space . . . . . . . . . 4.2.2 Binomial Option Pricing Model with One Underlying Asset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 Binomial Option Pricing Model with Two Underlying Assets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

29 29 31 31 32 32 33 34 36 36 36 37 40 40 42 44 45 45 45 46 46 47 48 48 49 50 52 52 53 53 54 56 56 56 57 58

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5

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4.3

Real Options Portfolio Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Switching Between Modes . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Switching Formulation for One Underlying Asset . . . . 4.3.3 Switching Formulation for Two Underlying Assets . . . 4.3.4 Stochastic Mixed–Integer Program Formulation . . . . . . 4.3.4.1 One Underlying Asset . . . . . . . . . . . . . . . . . . . . . . 4.3.4.2 Two Underlying Assets . . . . . . . . . . . . . . . . . . . . . 4.3.4.3 n–Dimensional Model Formulation . . . . . . . . . . 4.4 Model Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Alternative Probability Formulation . . . . . . . . . . . . . . . . . 4.4.2 Change of the Underlying Stochastic Processes . . . . . . . 4.4.3 Explicit Modeling of Synergies . . . . . . . . . . . . . . . . . . . . . . 4.4.4 Operating and Other Constraints . . . . . . . . . . . . . . . . . . . . 4.4.5 Terminal Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.6 Further Generalizations of the Model Formulation . . . . 4.5 Model Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.1 Applicability of Risk–Neutral Valuation . . . . . . . . . . . . . . 4.5.2 Numerical Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.2.1 Optimality of Solutions and Linearization . . . . 4.5.2.2 Dimensionality of the Problem . . . . . . . . . . . . . . 4.5.2.3 Increasing Efficiency of the Implementation . . 4.5.3 Monte Carlo Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61 61 63 64 65 66 71 75 78 78 80 81 82 82 83 83 83 84 84 86 87 90

Numerical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Numerical Analysis for Two Underlying Assets . . . . . . . . . . . . . 5.1.1 Numerical Application Setting for Two Underlying Assets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.2 Overall Portfolio Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.3 Budget Effects and Other Constraints . . . . . . . . . . . . . . . . 5.1.3.1 One Portfolio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.3.2 Relative Attractiveness of Two Portfolios . . . . 5.1.3.3 Generalization of Budget Effects . . . . . . . . . . . . . 5.1.4 Volatility Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.5 Correlation Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.6 Starting Value of the Underlying Assets Effects . . . . . . . 5.1.7 Path–Dependency Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.8 Deviations from Value Additivity . . . . . . . . . . . . . . . . . . . 5.2 Extension of the Numerical Analysis to Three Underlying Assets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Numerical Application Setting for Three Underlying Assets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Correlation and Risk–Neutral Probabilities . . . . . . . . . . 5.2.3 Base Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

93 94 94 97 98 100 103 105 106 109 111 112 114 115 115 117 119

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5.2.4 Change in Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.5 Change in Budget . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.6 Change in Volatility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Numerical Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

122 123 123 127

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Summary of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Managerial Implications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Further Applications and Future Research . . . . . . . . . . . . . . . . . .

131 131 135 136

5.3 6

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

List of Abbreviations, Variables, and Functions

αi ai

− → a

APT ai0

− →0 a

β(t, s) B B1

Starting value of stochastic process i Entering operating mode for underlying asset i, ai = 1, . . . , Ai Vector of entering modes of underlying assets Arbitrage Pricing Theory Leaving operating mode for underlying asset i, ai0 = 1, . . . , Ai0 Vector of leaving modes of underlying assets New funds available at time t in scenario s Budget level Initial budget available at time t = 1, B1 e R ≥ 0

BARON© Branch And Reduce Optimization Navigator (optimizer) BMW Bayerische Motoren Werke ci (·, ai0 ) Cash flow function for operating underlying asset i in mode ai0 CAPM Capital Asset Pricing Model CCAPM Consumption–Based Capital Asset Pricing Model cf. confer (compare) Cplex© Optimizer by ILOG, Inc. CRR Cox-Ross-Rubinstein di Proportional down move of the price of asset i in the binomial model DCF Discounted Cash Flow E (·) Expected value

XVI

e.g.

List of Abbreviations, Variables, and Functions

exempli gratia (for example) e Base of the natural logarithm Eq. Equation et al. et alii (and others) f., ff. following GAMS© General Algebraic Modeling System (optimization platform) i.e. id est (that is to say) i Underlying asset, i = 1, . . . , n I (·) Investment or switching costs (I < 0 possible) ICAPM Intertemporal Capital Asset Pricing Model l Auxiliary index L Auxiliary variable [ x mod y] Modulo (remainder on integer division of x by y) MP Market portfolio max[·] Maximum operator NPV Net Present Value φ Indicator function for movement of underlying assets p., pp. page(s) p Risk–neutral probability for up-jump pk Risk–neutral probability for jump in the multinomial lattice, k = 1, . . . , 2n Q Large number ρij Coefficient of correlation between underlying assets i and j rf Risk-free interest rate

R(·) R&D RO σi s θi t ∆t

ui

V (·) VaR X

Binary switching variable Research and Development Real Options Standard deviation of underlying asset i Scenario indicator, st = 1, . . . , S(t) Stochastic value of underlying asset i Time indicator, t = 1, . . . , T Length of (infinitesimally small) time interval Proportional up move of the price of asset i in the binomial model Value of option or portfolio Value at Risk Exercise price of a financial option

1 Introduction

Corporate investment opportunities can be viewed as options, or more precisely, “real options”(Myers 1977). An option gives its holder the right, but not the obligation, to take a specific action in the future (Amram and Kulatilaka 1999, p. 5). This right is valuable if there is uncertainty about the future development of an underlying variable. Due to the discretionary decision right involved, options are asymmetric contracts: if and only if option exercise is valuable, the decision maker will exercise and realize value, otherwise, the option will expire worthless (Trigeorgis 1996, p. 4). In order to distinguish corporate investment decisions, or options, from financial options, it is helpful to use the term “real options” if the underlying asset of an option is a real asset (Dixit and Pindyck 1994, p. 7). With a close analogy to financial options, corporate decision making is naturally discretionary and asymmetric because management has future decision rights about the use of corporate resources, such as financial resources or assets in place. These future decisions are made contingent on how future uncertainty resolves. While it may be reasonable to invest today based on available information, it may be wiser to expand or abandon the investment if the environment should develop more or less favorably. Real options theory provides a rigorous modeling framework allowing to assess the flexibility inherent in real investment decisions, and to derive optimal investment policies. In that sense, real options theory enables to assess corporate investment problems that could not be fully captured before (cf. Trigeorgis and Mason 1987, p. 16). Therefore, it is insightful to apply well established concepts from financial option pricing to corporate finance, which is done increasingly in practice (Graham and Harvey 2001). Although the analogy between financial and real options is a close one, there is a major difference. While financial options are derivative instruments in the sense that the underlying asset and other options are typically unaffected by a particular option exercise, this is generally not the case for

2

1 Introduction

real options. Exercising a put option on BMW shares does not have any effect on the company itself or other BMW options because it is just the use of a separate financial distribution right. Exercising a put option on a BMW plant is fundamentally different in nature: if the plant is sold or abandoned, all subsequent real options linked to this plant are forgone. An intuitive yet more passive example for a capacity management perspective in the automotive industry is provided by Fleischmann et al. (2006) who discuss BMW’s management of global production sites through a strategic planning model. The interdependency between real options on the same underlying asset is well known academically, but the BMW example also gives some intuition about the complexity involved, making practical implementation of real options in a rigorous portfolio context challenging (cf. Trigeorgis 1993a). Additional complexity stems from the realization that the underlying variables can be affected through active management by the firm as managers can influence the risk profiles of their firms. When multiple real options on multiple underlying assets are considered, the interactions in the company’s portfolio increase. This aspect has not been adequately covered in the real options literature, but is of high relevance both for academia and in practical applications (cf. Triantis 2005, p. 15). The practical relevance of such a portfolio problem can be be illustrated through Toyota’s challenge of actively managing the demand chain of two product lines in the Japanese and North American markets, facing both demand risk and exchange rate risk, by aligning the capacity of its global manufacturing plants, inventory networks, and distribution channels (Lee et al. 2005). Closely related, Huchzermeier and Cohen (1996) consider a global manufacturing network under exchange rate risk with switching options between different manufacturing strategies contingent on exchange rate realizations (for a detailed review, see Kouvelis et al. 2006). From a strategic management perspective, a primary objective is to design the firm’s strategic portfolio and dynamically manage this portfolio. Managing a portfolio of real options is what strategy is all about. A business strategy is more like a series of options than a series of static cash flows (cf. Luehrman 1998, p. 90). More broadly, managing a portfolio of real options is both strategic and operational. In this context, strategy refers to the choice of assets, whereas operational refers to the optimal investment policy for a given set of assets. Both strategic and operational real options management need to account for interaction effects. The interdependencies between real options must be properly taken into account to fully capture realistic investment decision problems when companies are faced with multiple investment or disinvestment opportunities under budget or other constraints (Trigeorgis 2005, p. 50). Budget constraints deserve special attention since exercising options implicitly assumes that

1.2 Contribution and Main Results

3

funds are readily available. In practice this may not be the case either because of internal (divisional) guidelines or limited access to capital markets (cf. Brealey and Myers 1996, pp. 101 ff.). Constraints may also result from logical aspects of the investment program, such as mutual exclusivity among projects (Brown and Davis 1998). While constraints are not considered in standard financial option pricing models, they affect optimal decision making if the existence of binding constraints significantly influences the optimal investment strategy. For instance, it may sometimes be desirable to sell a faltering or even a profitable business unit to free up (capital) resources for another investment that is even more profitable.

1.1 Research Questions As a consequence of the possible interdependencies between investment decisions from a real options point of view, valuing and managing portfolios of real options is a complex task. The goal of this dissertation is to identify the key aspects involved in managing portfolios of real options and to provide a rigorous real option valuation framework enabling to optimize portfolio decisions quantitatively. The present dissertation develops a modeling approach for dynamic investment problems where limited resources are allocated to interacting risky projects over time, subject to constraints. It sheds light on the following research questions: • Which aspects need to be considered for the analysis of portfolios of real options? • How can the relevant aspects be captured quantitatively in a suitable modeling approach for the valuation of portfolios of real options? • Which numerical evidence and structural results characterize portfolios of real options? To our knowledge, a focused discussion of these research questions has been lacking in the literature so far. This dissertation aims at closing the gap, providing a framework that makes it possible to optimize investment decision problems that could not be handled before.

1.2 Contribution and Main Results Our formulation and solution of a general model of portfolios of real options provides a quantitative approach to valuing specific investment problems. At the same time, it makes it possible to draw general conclusions about the

4

1 Introduction

behavior of portfolios of real options. It confirms the earlier results provided in Trigeorgis (1993a) and extends these to more complex settings. Specifically, our contribution is twofold. First, we develop a rigorous real options valuation framework enabling to optimize such portfolio decisions involving dynamic investment problems with limited resources allocated to interacting risky projects over time, subject to various constraints. We set up our portfolio valuation framework as a multi–dimensional real options problem involving path–dependent (dis–) investment decisions. Our approach can handle compoundness of multiple options on multiple underlying assets, correlation and changes in the characteristics of the underlyings, investment synergies, asymmetric switching costs, learning effects, a dynamic budget (with or without borrowing) and other types of operational and financial constraints. One of the advantages is that this approach can account for changes in the properties of the underlying variable, e.g., changes in the volatility, or more generally, changes of the underlying stochastic processes. This captures explicitly the option to switch the underlying asset, e.g., as a function of available information at time t. Due to its flexibility, it can handle an extensive range of portfolio problems. The model is formulated as a stochastic program that we solve by an algorithmic solution implementation on a commercially available programming platform. Second, we use our solution implementation to numerically provide new insights into portfolio effects. These insights have not been presented before in the context of real options. Especially, we find that budget constraints influence portfolio value in a complex way, potentially causing the optimal investment policy to change profoundly for relatively small changes in the availability of funds. As a consequence, for different levels of the overall budget, different portfolios may be optimal. Given the existence of binding budget constraints, volatility has an ambiguous effect on portfolio value. While for some settings volatility increases portfolio value, as usually is the case for financial options, it can also have an adverse effect on portfolio value for other settings. Both of these effects are new in that they had not been identified in the context of portfolios of real options before. Many of the discussed effects present counter examples to prevailing thinking in financial option pricing and thus generate new insights. We find that: • Option values typically do not add up and tend to be sub–additive, such that it is not possible to value real options in isolation and consider the value of portfolios of real options as a sum of the parts. • Adding to the non–additivity of real options, budget constraints play a major role because they introduce a step–function of portfolio value. Portfolio optimization makes it possible to countervail the adverse budget effects, resulting in a adapted investment and disinvestment strategy.

1.3 Structure

5

• Volatility has an ambiguous effect on portfolio value. Increasing the volatility of an underlying asset can increase the value of a portfolio of real options related to this asset, but in fact can also reduce the value of a portfolio of real options. • Portfolio values of constrained portfolio problems can be decreasing in the correlation. Therefore, correlation in the underlying assets and in the cash flows can harm portfolio value. • Portfolio values are increasing in the underlying assets initial values if the problem is a net investment problem, i.e., it shows a net payoff profile equivalent to the payoff of a call option. The discussion of these effects shows the complexity inherent to managing portfolios of real options and the challenges involved. Different sources of interaction and resulting ambiguity in parameters are such that the dynamics of a specific investment problem cannot be specified without formal analysis, but are specific to the investment problem at hand. The proposed model provides a framework that can help to meet these challenges. By enhancing corporate investment decision making through dynamic portfolio analysis, it fills a gap in a key area for both theory and practice.

1.3 Structure The structure of the subsequent chapters is as follows. Chap. 2 motivates the portfolio perspective on real options. For this, it introduces the concept of real options and defines portfolios of real options, which are unlike portfolios of financial assets or financial options. The notion of portfolio aspects is clarified. Based on the presented understanding of portfolios of real options, the issue of valuing and managing such portfolios is highlighted. With this background, an overview of the chosen modeling approach is given. Chap. 3 reviews the relevant strands of the literature which cover financial portfolio theory, corporate capital budgeting, and financial and real options theory. This review identifies the key concepts that underly the following analyses. This sets the stage for the core of the dissertation which is the development of the general model formulation in Chap. 4. The model development starts with a discussion of the key model features that are to be included. Then, the model is built up successively by increasing the scope of the model in stages. It begins by introducing a binomial option pricing model with one time dimension and one underlying asset dimension. Then, this is expanded to a multinomial option pricing model with two underlying asset dimensions. Based on this, a real options portfolio model based on a switching concept is introduced, first for one underlying assets, then

6

1 Introduction

for two underlying assets, and finally generalized to n underlying assets. The chapter concludes with a discussion of the model features and possible extensions. Chap. 5 presents comparative statics of the presented model, based on the numerical analysis of specific investment setting examples. It discusses the impact of portfolio aspects stemming from constraints, volatility effects, correlation effects, and starting values effects. For each of these, an example is provided which illustrates the specific interest of the proposed portfolio analysis. It will be shown that the management of portfolios of real options navigates in a complex system state space where most of the value drivers are ambiguous. It is therefore not possible to specify ex ante, i.e., before a specific portfolio optimization, whether a change in a certain parameter will increase or decrease portfolio value. This runs counter to the basic intuition for standard financial options. Finally, Chap. 6 summarizes the main results, discusses key contributions alongside with possible limitations, and gives an outlook on further applications and future research.

2 Portfolio Approach to Real Options

A portfolio perspective on real options makes it possible to gain insights into the interplay of real options that could not be captured in a stand– alone analysis. It is well established in financial theory that financial assets must be valued from a portfolio perspective (cf., e.g., Constantinides 1989). A review of financial portfolio theory is provided in Sec. 3.1. Similarly to financial portfolios, portfolios of real options are affected by numerous aspects, which will be developed in the following. In this chapter, the portfolio perspective will be introduced in detail, and then linked to the ensuing general model that is developed subsequently.

2.1 Motivation of the Portfolio Approach to Real Options The motivation for considering real options jointly in a portfolio context derives from the specifics of real options. This is why this motivation is based on a focused introduction of the general concept of real options, and the definition of the resulting portfolio problem. 2.1.1 Real Options and Financial Options The term “real options” was coined by Myers (1977) who stated that “real options [...] are opportunities to purchase real assets on possibly favorable terms” (Myers 1977, p. 163). A more precise definition is found in Sick (1995, p. 631) who defines a real option as “the flexibility a manager has for making decisions about real assets. These decisions can involve adoption, abandonment, exchange of one asset for another or modification of the operating characteristics of an existing asset”. An introduction to real options can be found in Dixit and Pindyck (1994), Trigeorgis (1996), Amram and Kulatilaka (1999), Copeland and Antikarov (2001), and Smit and Trigeorgis (2004).

8

2 Portfolio Approach to Real Options

Recent comprehensive overviews of the growing body of real options literature are given in Lander and Pinches (1998), Baecker and Hommel (2004), and Trigeorgis (2005). A financial option gives the holder the right but not the obligation to buy or sell a financial asset, e.g., stock traded on financial markets, under specified terms. These terms define the period of time over which the option can be exercised, in exchange of the exercise price. A call option gives the right to acquire the underlying asset, a put option gives the right to sell the underlying assets. An American option can be exercised at any time before and including the expiration date, whereas a European option can only be exercised on the expiration date (cf. Luenberger 1998, pp. 319 ff.). The value of an option stems from the riskiness of the underlying assets. As compared to a long position in the underlying asset, the option provides insurance against losses (cf. McDonald 2003, pp. 44 ff.). It enables to protect from the downside potential of the underlying asset, while benefiting from the upside potential. This asymmetry between upside and downside is visualized in Fig. 2.1. At maturity, if the value of the underlying asset is below the exercise price, the option is not exercised and yields zero payoff. Otherwise, the option is exercised, with unlimited upside potential. The dotted line represents the value of the call option before maturity, which is always higher than the value of the call option at maturity (cf. Brealey and Myers 1996, p. 569). In order to distinguish these two, the value of an option at maturity is called “intrinsic value”, whereas the difference between intrinsic value and current value is called “time value” of the option (cf. Hull 2003, p. 154).

Call Option value

don't exercise (at maturity)

exercise (at maturity)

option value before maturity

"intrinsic value" of the option

exercise price

Value of underlying asset

Fig. 2.1. Payoff structure of a call option

As a consequence, the modeling of the riskiness of the underlying asset is key for valuing an option on this asset. This riskiness is captured by modeling the stochastic process followed by the price of the underlying asset (cf. Hull 2003, pp. 216 ff.). Asset prices are commonly assumed to follow a ran-

2.1 Motivation of the Portfolio Approach to Real Options

9

dom walk, i.e., price changes occur randomly (cf. Luenberger 1998, p. 308 f.). In continuous time, the random walk of a stock price is frequently modeled as a geometric Brownian motion (cf. Dixit and Pindyck 1994, p. 72), and in discrete time it is frequently modeled as a multiplicative binomial process (cf. McDonald 2003, pp. 341 ff.). Both stochastic processes obtain (the binomial process in the limit as time increments become infinitesimally small) lognormally distributed stock prices, i.e., the natural logarithm of stock price changes are normally distributed (cf. Luenberger 1998, p. 309). Real options are similar to financial options in that they “give the holder the right, but not the obligation, to take an action in the future”(Amram and Kulatilaka 1999, p. 5). The major difference lies in the nature of the underlying assets. Real options are written on real assets which are productive capacity creating streams of income. As opposed to this, financial options are written on financial assets which are claims to the income generated by real assets. Therefore, while financial assets only define distribution rights of a given value, real assets can create economic value (Bodie et al. 1999, p. 3). The term real options stresses both the methodological analogy to financial options and the fundamental difference in the underlying assets. Typically, the interpretation of investment decisions as real options is based on the following analogies to financial options: The current value of the stock corresponds to the (gross) present value of expected cash flows; the exercise price to the investment cost; time to expiration to the time until the opportunity disappears; stock value uncertainty to project value uncertainty; and the riskless rate is identical for both (cf. e.g. Trigeorgis 1988, p. 149). A real option that requires to pay the exercise price to receive some underlying asset corresponds to a call option, whereas a real options enabling to receive the exercise price in exchange for sacrificing some underlying asset is a put option. Throughout this dissertation, we can use the terms value and price interchangeably, which is common in the real options literature (cf., e.g., Copeland and Antikarov 2001, p. 6). This is because in the absence of mispricing or other distortions, these two coincide. The application of financial option pricing theory to real option valuation is subject to possible limitations of the transferability, such as incomplete markets, non–traded underlying assets, or shared options. Representing a large body of literature, a detailed discussion of these issues can be found, e.g., in Trigeorgis (1996), Smit and Trigeorgis (2004), and Smith and Nau (1995). Different types of real options are distinguished in the literature. A detailed overview can be found in Micalizzi and Trigeorgis (1999) and Lander and Pinches (1998, p. 540, Table 2). Trigeorgis (1996, pp. 2 f., Table 1.1) distinguishes the following types of real options:

10

2 Portfolio Approach to Real Options

• Option to defer investment. The investment decision can be postponed, making it possible to benefit from the resolution of uncertainty during the lifetime of the option. • Option to default during staged construction. For investment projects that take time to build, it is possible to abandon the project during construction based on the resolution of uncertainty during construction. This is desirable if the remaining investment costs are not covered by the expected future value of continuing the project as originally planned. • Option to expand. If the underlying asset and thus the project develops more favorable than initially expected, a possible increase of capacity realizes higher value than continuing in base scale. • Option to contract. As opposed to the option to expand, the option to contract enables to reduce the scale of operations and thus save costs if the underlying asset develops unfavorably. • Option to shut down and restart operations. This option enables to stop operating temporarily, with the perspective of reopening in a later period. This is desirable if variable costs exceed operating revenues by more than the costs incurred for shutting down and reopening. • Option to abandon. It is possible to scrap the project or sell it off, which is desirable if the salvage value, e.g., a resale value, is higher than the value of continuing operations. • Option to switch use. Production input factors or the product output can be changed, e.g., as a function of prices or demand. This makes it possible to choose the strategy which generates highest profit net of switching costs. • Growth options. An early investment opens up new investment opportunities. Huchzermeier and Loch (2001) extend this taxonomy by the option to improve. During projects it is possible to take corrective action, such as adding resources, in order to improve the expected performance of the product. While this approach captures the fact that managers can change the performance state, it cannot cope with changes in the underlying, i.e., the option to switch underlyings. The current dissertation takes a general view on real options by interpreting each discretionary decision right as a switch between modes, as proposed by Kulatilaka and Marcus (1988) and Kulatilaka and Trigeorgis (1994). Their concept enables interpreting real option exercise decisions as switching decisions among operating modes, which is the most general formulation of real options, having commonly analyzed real options as special cases.

2.1 Motivation of the Portfolio Approach to Real Options

11

2.1.2 Definition of Portfolios of Real Options “A portfolio is a particular combination of assets in question” (Neftci 2000, p. 17). In the present discussion, the assets in question include real assets and real options written on these assets. For ease of discussion it is convenient to label sources influencing the value of the portfolio as “portfolio aspects”. On the asset level, it is self–evident to assume that corporate decision makers will have a vast opportunity set, enabling them to invest in a myriad of real assets and possibly to allocate limited resources between these assets. Similarly, from a financial portfolio perspective, it is natural to consider a set of underlying assets simultaneously (see Sec. 3.1). On the real options level, it has been shown by Trigeorgis (1993a) that options on the same underlying assets interact, requiring a simultaneous valuation of all real options written on the same underlying asset. If there are two financial options, e.g., to acquire stock of Bayer AG and BMW AG, the values of these financial options are additive because the underlying assets and the terms of the options are independent from one another. As opposed to this, consider BMW who has a European option to abandon an existing plant and the subsequent European option to expand this plant. In options terms, the European option to abandon corresponds to a European put option, while the European option to expand corresponds to a European call option. Since both affect the same underlying asset, these real options interact. On the one hand, the value of the put is less than in isolation because it is written on the package of underlying asset and associated call option. An option is always valuable before expiration, hence the package is more valuable than the underlying asset in isolation. Since a put option value decreases with increasing value of the underlying asset, the option to abandon becomes less valuable due to the subsequent option to expand the plant. This is intuitive because exercising the put forfeits the subsequent option, such that investors may hesitate to exercise the put in order to preserve the future call option. On the other hand, the value of the subsequent call is reduced through the existence of the prior put, because with some probability it may not be possible to exercise the call, given prior exercise of the put. This prevents the call from unfolding its full value potential and translates into a lower call option value. Since both effects occur simultaneously, it is not possible to value put and call in isolation. The decision about exercising the first option needs to explicitly take into account the existence of the subsequent option. This relationship is structurally akin to the valuation of compound options (cf. Geske 1979). Hence, the arising effect can conveniently be labelled as com-

12

2 Portfolio Approach to Real Options

poundness. Specifically, Trigeorgis (1993a) defines interactions between real options written on the same underlying asset as “intra–project compoundness”. Following the same logic, an analogous effect is identified for several, interdependent underlying assets which he denotes as “inter–project compoundness” (Trigeorgis 1996, pp. 132 f.). Both inter–project and intra–project compoundness must be considered in the context of portfolios of real options. Further, both real options and real asset may be subject to constraints. Real assets can exhibit, for example, operating (technical or logical) or financial relationships. These can lead to mutually exclusive assets or strictly complementary assets which require one another. This affects the possibility of joint exercise of bundles of real options on different underlying assets. Likewise, the existence of constrained resources, e.g., funds available, influences the feasibility of joint option exercise. Therefore, portfolios of real options here are defined as combinations of multiple risky assets and multiple real options written on these assets subject to constraints. Cases with only one underlying asset, or one real option, are special portfolio cases that reduce the scope of portfolio analysis dramatically. In order to seize all possible portfolio effects, it is important to analyze multiple underlying assets with multiple real options simultaneously. The usual “laboratory” setting for real options analysis with one underlying asset and one real option does not provide a structure capable of handling realistic decision problems. It is thus prone to ignore key portfolio effects with possibly substantial impact on (optimal) option exercise. 2.1.3 Difference Between Financial Portfolios and Portfolios of Real Options The discussion of portfolios of real options evidently stands on a large body of literature on financial portfolios. This literature will be reviewed in detail in Chap. 3. For the motivation of the portfolio perspective chosen here, it is important to clarify the major differences between portfolios of real options and portfolios of financial options. Financial portfolio theory dates back to Markowitz (1952) who established the concept of mean–variance analysis. At its core, it is based on the concept of diversification. Diversification is about minimizing the variance of the return of a portfolio for a given mean of the return (or maximizing the return for a given variance). This can be achieved by increasing the number of assets included in the portfolio and in the limit by holding all available assets. The only relevant risk in this perspective is the covariance risk of each asset with the (market) portfolio. In essence, this implies a passive attitude towards risk because it exclusively consists of diversifying the risk over as many assets as possible.

2.2 Implications for Modeling Approach

13

The portfolio approach to real options is fundamentally different from this passive perspective on risk. For portfolios of real options, the variance of returns and the resulting distributions of values of the underlying assets are considered instead of covariance with a (market) portfolio. Real options make it possible to capitalize on desirable developments of risky underlying assets while protecting from undesirable movements. For instance, a real option to expand capacity is only exercised if the value of the underlying asset is higher than the investment costs. This implies an active attitude towards risk which consists of exercising real options as a function of the resolution of uncertainty. Therefore, in portfolios of real options, the attitude towards risk is different and diversification cannot be desirable. For financial portfolios, closely connected to the passive attitude towards risk, value additivity of the elements of the portfolio holds (cf. Brealey and Myers 1996, p. 19). Since a financial option derives its value directly from the value of the underlying financial asset, the same holds for financial options. Value additivity also holds for financial options. In contrast, a consequence of the more active real options approach is that value additivity among the elements of portfolios of real options generally breaks down (see Sec. 2.1.1). This is why it is not insightful to include financial assets and financial options in the analysis of portfolios of real options. Financial assets by definition are independent from other assets, therefore the values of the financial assets and the portfolio of real options are also independent from one another and thus do add up. Alternatively, if the portfolio of real options does have an impact on the financial asset, it can only do so by affecting the real asset underlying the financial asset. However, this is not a special case because in fact a real asset is considered, not a financial asset. Beyond this perspective, it can be insightful to consider interactions between real options and financial flexibility. This issue is discussed by Trigeorgis (1993b) where the idea is not to add an additional (financial) asset to the portfolio, but to assess the possible value of financial flexibility that can result from operating flexibility. In this context, Huchzermeier and Cohen (1996) discuss operational and financial hedging strategies. A review of this literature, citing Huchzermeier and Cohen (1996) as the key reference in the context of operational hedging, can be found in Van Mieghem (2003); Kouvelis et al. (2006). This discussion of interactions between financial and operational flexibility is beyond the scope of this dissertation.

2.2 Implications for Modeling Approach Based on the above developed specifics of portfolios of real options, a consistent modeling approach is required. In order to derive the implication of the specifics of real options for our modeling approach, in the following we

14

2 Portfolio Approach to Real Options

will highlight portfolio aspects, their consequences on how portfolios of real options need to be managed, and how this translates into our model. 2.2.1 Portfolio Aspects Understanding portfolios of real options as combinations of multiple risky assets and multiple real options written on these assets subject to constraints provides a rich structure for assessing real decision problems. In this structure, the overall portfolio value is influenced by the dynamic interaction of all elements of the portfolio. This portfolio perspective on real options is a fundamental way to properly handle interdependencies between decisions over time. A suitable perspective on portfolios of real options needs to include all relevant aspects that influence the value of such a portfolio. From the definition of portfolios of real options, it follows directly that portfolio aspects can be attributed to the real assets involved, the real options involved, or constraints. Budget constraints are of special importance because they can have a considerable limiting impact and require a detailed modeling of the investment dynamics. Moreover, the ensuing budget levels over time are state– and path–dependent. Based on these considerations, portfolio aspects can be categorized as follows (cf. Brosch 2001): • Interactions on the real options level: On the real options level, intra–project compoundness and inter–project compoundness can be distinguished. While the former is due to interdependencies of several real options written on the same underlying asset, the latter is connected to interdependencies of several real options and several underlying assets. For inter–project compoundness, the correlation between the underlying assets has to be modeled explicitly. • Interactions on the real asset level: On the real asset level, direct qualitative interactions and indirect qualitative interactions can be distinguished. Direct interactions are those which are inseparably connected to the underlying real assets, such as physical properties or operating synergies. Indirect interactions have their origin outside the strict asset level and are due to constraints, most prominently budget constraints. Both are qualitative in that they are not merely stochastic in nature, but result from the properties of projects or the specific background of the company (that would be different for another company). Due to the simultaneous nature of the interactions involved, it is not possible to isolate value impacts for separate portfolio aspects. Their effects need to be assessed jointly, in order to capture their interplay. This is why they require a simultaneous modeling approach, very much like compound option pricing. Specifically, in order to handle these interactions, existing

2.2 Implications for Modeling Approach

15

constraints can only be incorporated via a simultaneous modeling approach. Thus, the model is formulated as one stochastic optimization problem subject to constraints. Interactions are captured through the interplay of constraints as well as state– and path–dependency of investment decisions and cash flows. 2.2.2 Management of Portfolios of Real Options Based on this broad understanding of portfolios of real options, optimal management of portfolios of real options requires capturing all relevant portfolio aspects simultaneously in order to maximize overall portfolio value. Specifically, two dimensions of managing portfolios of real options can be distinguished: 1. Portfolio design: creating the optimal portfolio with maximum value, supposing optimal future exercise. 2. Portfolio execution: exercising existing real options optimally in order to realize the full value potential of real options. On the one hand, portfolio design analyzes which assets and which options to include in the portfolio of real options. This decision can be based on the proposed model, by choosing the portfolio which yields the highest value for the portfolio of real options. In order to assess different portfolio designs, a valuation needs to be carried out for each possible alternative. This process is depicted in Fig. 2.2. By choosing the optimal portfolio design, the decision maker in fact defines the company’s strategy, because this defines the way the company is planning to create value in the future. This approach helps to quantify the impact of new investment proposals and more generally ideas form a portfolio perspective, which in options terms corresponds to assessing new real options and their interactions with the portfolio. Apart from assessing new real options, management may be able to influence the underlying assets or the embedded real options, such as modifying the volatility involved (cf. Damisch 2002, pp. 365 ff.). The desirability of these alternatives can also be assessed as different design options. At its core, the design of a portfolio of real options is about the choice of the underlying assets to be included in the portfolio, and thus, strategic. This deserves special attention, because it will be shown that conventional wisdom may not hold true for portfolios of real options. For instance, it is not always desirable to increase volatility to enhance value (see Sec. 5). Hence the portfolio design can be supported by the proposed model. On the other hand, portfolio execution is about optimally managing the existing design of a portfolio of real options. The value created through real options implicitly assumes that real options are exercised at the optimal

16

2 Portfolio Approach to Real Options Start

(Re-) Design portfolio

Value portfolio

Alternatives?

yes

no Select optimal portfolio

End

Fig. 2.2. Portfolio design process

point in time. Therefore, management needs to know when this optimal point in time is. This information is provided together with the optimal portfolio value, because the optimization result contains the optimal contingent exercise policy. Thus, the proposed model gives clear management recommendations about which options should be exercised, and when, suggesting to exercise real options as in the optimal policy. Generally, the portfolio design defines the basis for the portfolio execution because the latter is about optimally using an existing structure which is defined in the first. At the same time, the portfolio execution can provide unanticipated new information which opens up new portfolio alternatives for the portfolio design. An example for this is research on a specific drug which, by chance, provides insights about another new drug that had not been considered before. Thus both dimensions of managing portfolios of real options are interrelated. 2.2.3 Translation into Model Features The motivation for considering portfolios of real options has not only clarified the understanding of portfolios of real options, but at the same time has also stated the requirements for the model formulation. A suitable model formulation needs to incorporate the above introduced elements. Before the model is derived in detail in Chap. 4, an overview of the model features is given below. The model formulation will incorporate the following features: 1. Multiple correlated underlying assets.

2.2 Implications for Modeling Approach

17

2. 3. 4. 5. 6.

Multiple real options embedded in the underlying assets. State–dependent cash flows leading to path–dependencies. intra–project compoundness. Inter–project compoundness. Direct qualitative interactions, i.e., operational (non–financial) constraints. 7. Indirect qualitative interactions, i.e., state–dependent budget constraints with carry–over of funds (including borrowing). 8. Choice of the underlyings. These features are incorporated into the model formulation as follows: 1. General model formulation: The case of portfolios of real options is formulated as one stochastic mixed–integer program that captures path– dependent and multi–dimensional real options settings. 2. Simultaneous, i.e., forward–backward looking, approach: The backward looking element consists of choosing the optimal investment strategy which maximizes both current cash flow and expected value of future cash flows. This is a standard approach in financial option pricing (cf. Hull 2003, pp. 392 ff.). This is complemented by a forward looking element which captures the sequence of decisions up to a certain point in time that meet the budget constraints on that path. This is not typical for financial options. Both are considered simultaneously in order to capture their interdependencies. 3. Model implementation and solution: The model is solved by an algorithmic solution implementation, using the GAMS modeling platform, which can readily be applied to practical decision problems. The proposed modeling approach provides a rigorous and flexible structure that can cope with more realistic and rich decision problems, deriving an optimal dynamic sequence of investment and disinvestment decisions from a portfolio perspective. It is very flexible in incorporating specific investment problems with different kinds of constraints and changes in the underlyings. Especially, the global dynamic budget depends on the sequence of decisions which depends on the paths taken by the joint stochastic processes of the underlying assets. This is achieved by a path–dependent, forward–backward looking approach that keeps track of past decisions and the expected future effect of current decisions. For a visualization of this approach, see Fig. 2.3. To our knowledge, the formulation of such a model for portfolios of real options has not been provided in the literature so far. In order to account for these features, this dissertation follows a new direction for the quantitative assessment of real options. The model interprets the portfolio problem as an optimization problem with the objective of maximizing portfolio value subject to “constraints”. On the one hand, these

18

2 Portfolio Approach to Real Options Path-dependent investment decisions

1 2 3 1 2 3

1 2 3

1 2 3

up

up down up

down 1 2 3

1 2 3

down

1 2 3

1 2 3

1 2 3

1 2 3

1 2 3

1 2 3

1 2 3

1 2 3

Path-dependent joint stochastic processes

Forward: Available Budget

Backward: Expected Value

Forward-backward looking algorithm incorporating dynamic global budgets

Fig. 2.3. Modeling approach

constraints model portfolio aspects which are automatically understood as constraints, such as budget constraints. On the other hand, other features are also captured by modeling these as balance equations, such as the specification of the processes of the underlying assets, the determination of available funds, or the calculation of expected values. The dynamic nature of the problem is incorporated through constraints that are either backward– or forward–looking, but either way dynamic. This makes it possible to handle simultaneous interaction effects and to define the problem in a formulation that can incorporate operating and financial constraints. Moreover, this approach makes it possible to solve the problem with commercially available solvers. The model implementation reverts to commercially available software packages, i.e., the optimization platform GAMS© (cf. Bussieck and Meeraus 2004), together with suitable optimizers such as BARON© and Cplex© . This implementation makes it possible to benefit from advanced optimization techniques that have been tested and adapted continuously and enables to separate the problem implementation from the actual optimization algorithms. Consequently, the model becomes verifiable and portable between applications.

2.2 Implications for Modeling Approach

19

Based on this understanding, our approach to portfolios of real options will be positioned in the relevant literature next. This will provide the basis for our ensuing model development.

3 Literature Review

This section reviews the key concepts provided in the relevant literature, drawing from different strands of the literature which will be discussed below. The structure of the literature review is as follows. First, key concepts of financial portfolio theory are discussed in Sec. 3.1. It introduces the basic concepts of portfolio analysis and then leads on to more advanced portfolio approaches. The goal of this discussion is twofold. On the one hand, it clarifies the positioning of our model in the portfolio literature. On the other hand, it shows that no suitable portfolio approach for our problem is available in the financial literature. Second, corporate capital budgeting is reviewed in Sec. 3.2. Special attention is paid to the treatment of interdependencies between projects, from which we will draw in our model. Third, the core of the relevant real options literature is examined in Sec. 3.3. This discussion identifies prior work that is most akin to our approach, and retains two references that have influenced our approach substantially. Therefore, these two are presented in more detail

3.1 Financial Portfolio Theory The seminal work of Markowitz (1952) marked the beginning of modern finance theory (Constantinides and Malliaris 1995). Today, concepts derived from modern portfolio theory are the foundation of most (if not all) financial analysis, directly or indirectly. For example, the risk-adjusted discount rate in a Discounted Cash Flow (DCF) analysis is motivated by the portfolio concept of diversification. Also, a key element of derivatives pricing is the construction of a dynamic self-financing replicating portfolio. An overview of the key concepts of portfolio theory can be found in Markowitz (1991), Constantinides and Malliaris (1995), and Sharpe (2000). More details including empirical tests can be found in Elton et al. (2003) and Rudolf (1994).

22

3 Literature Review

In the following discussion of financial portfolios, first the basic concepts of mean–variance portfolio analysis are discussed (Sec. 3.1.1). Second, more advanced portfolio models are presented (Sec. 3.1.2). 3.1.1 Mean–Variance Portfolio Analysis 3.1.1.1 Traditional Markowitz Portfolio Selection Markowitz (1952) established a relationship between risk and expected return in a portfolio context. This approach identifies the optimal portfolio configuration for a risk-averse investor in a one-period decision problem with the objective to allocate a given amount of money optimally to a portfolio of risky assets. The investor is assumed to consider “expected return a desirable thing and variance of return as an undesirable thing” (Markowitz 1952, p. 77). Based on this perspective on expected return and variance of return, the approach by Markowitz (1952) established what is called “mean– variance analysis”. Assuming non–satiation, always preferring more to less, it derives an efficient frontier in the mean–variance space which contains all portfolios that are not dominated by other portfolios, i.e., no portfolio yields a higher return for the same variance or the same return with less variance. More specifically, the set of efficient stock portfolios is obtained as follows (for the following exposition, cf. Rudolf 1994 and Constantinides and Malliaris 1995). x is a m-column vector whose components xi denote the weight of asset i in the portfolio, i = 1, . . . , m. Given that all available funds P must be allocated, the portfolio weights must add up to 1, i.e., i xi = 1. 1 is a m-column vector of ones, superscript T denotes the transpose of a vector or a matrix. R is a m-column vector whose components Ri denote the mean return of asset i. Further, V is the m × m covariance matrix of the returns of the assets, where the components are σjk , j, k = 1, . . . , m. It is assumed that V is nonsingular, i.e., there is no redundant asset and therefore, none of the return patterns of one asset can be replicated by a portfolio of the other assets. Given that variances of risky portfolios are strictly positive, V is positive definite, i.e., x T Vx > 0. The mean returns and covariances are endogenous to the model and assumed to be known. Finally, the variance of the portfolio is denoted as σP2 and the return of the portfolio is R P . With this notation, the portfolio selection problem can be formalized as follows: min σP2 = x T Vx x

(3.1)

subject to xT R = RP T

x 1=1

(3.2) (3.3)

3.1 Financial Portfolio Theory

23

In other words, this formulation minimizes portfolio variance by solving for the optimal portfolio weights (Eq. 3.1), for each given level of portfolio return (Eq. 3.3) where all funds are invested (Eq. 3.3). x T Vx is convex because V is positive definite; the constraints are linear and thus define a convex set. Therefore, the problem has a unique solution, which can be obtained through solving the Lagrangian function, L, with the Lagrangian factors λ1 and λ2 , as follows: L = x T Vx − λ1 (x T R − R P ) − λ2 (x T 1 − 1)

(3.4)

Solving this equation yields the vector of optimal portfolio weights that minimize portfolio variance for a given mean return, as follows: x = V −1 R

1

R T V −1 R R T V −1 1 −1 R P 1 R T V −1 1 1 T V −1 1

(3.5)

In turn, replacing this result in Eq. 3.1 gives the variance of a minimum variance portfolio, for a given return R P , as follows: σP2 = R P

1

R T V −1 R R T V −1 1 −1 R P 1 R T V −1 1 1 T V −1 1

(3.6)

This solution provides the minimum variance portfolio for a given R P , which is expressed by a parabola. In mean-variance space, i.e., by inverting the function, the mean-variance efficient minimum variance portfolios can be represented by a convex line, starting with the global minimum variance portfolio which is the portfolio with the lowest possible variance for any return. This relationship is graphically represented in Fig. 3.1. All portfolios inside the hyperbola are feasible; portfolios on the solid line are minimum variance efficient portfolios, because they yield the highest expected return for a given variance of return. Therefore, the efficient portfolios dominate all other portfolios including the minimum variance inefficient portfolios. The investor’s choice then consists of choosing that portfolio from the efficient frontier which maximizes her one-period utility. Evidently, this approach is sufficient if either returns are normally distributed (symmetrical), or if the investor has a quadratic utility function. Otherwise, the obtained results are an approximation only and the approximation quality depends on the deviation from the real distribution and utility function (cf. Markowitz 1990, pp. 6 ff.). One key insight of the mean–variance approach is the concept of diversification. Diversification can reduce risk without necessarily adversely affecting expected portfolio return. This leads to the conclusion that it is not the individual (total) risk of an asset that matters, but its contribution

24

3 Literature Review Expected Return Minimum Variance Efficient Portfolios Feasible Portfolios

Global Minimum Variance Portfolio

Feasible Portfolios Minimum Variance Inefficient Portfolios

Variance of Return

Fig. 3.1. Mean–variance efficient portfolios

to the risk of the entire portfolio. In turn, this risk contribution for a well diversified investor is captured by covariance risk, i.e., the correlation of the individual asset with the overall portfolio: σP2 =

X j

x2j σj2 +

XX j

x j xk ρ jk σj σk ,

(3.7)

k6= j

where x j are portfolio weights, σp2 denotes the variance of the return of the overall portfolio, j, k represent individual assets, and ρ jk σj σk is the covariance of their returns, expressed through the coefficient of correlation, ρ jk . With sufficient diversification and thus sufficiently small portfolio weights, the variance of each asset lose significance and σP2 approaches the average covariances between assets (interpreting k 6= j in Eq. 3.7 as a weighted average) (cf. Rubinstein 2002, p. 2). Thus, securities cannot be valued in isolation, but must be considered as a group. While Markowitz (1952) is renowned for introducing the concept of mean–variance analysis and its graphical representation, more rigorous formulations are given in subsequent work (see Markowitz 1991; Mertens 2004). Roy (1952) independently developed an alternative portfolio approach based on minimizing the probability of missing a required minimum return, the so–called “shortfall risk”. For normally distributed returns, this approach is equivalent to Markowitz (1952). Recently, the Markowitz portfolio selection has been extended to a multi–period portfolio selection by Li and Ng (2000). The outlined Markowitz portfolio selection provides the understanding of portfolios that underlies our model. At the same time, our approach fundamentally differs from this portfolio perspective in that it considers active management of the underlying assets, making it possible to influence the return pattern of the assets as endogenous in our model.

3.1 Financial Portfolio Theory

25

3.1.1.2 Capital Asset Pricing Model Related to the optimal portfolio choice of an investor, the (two–moment) Capital Asset Pricing Model (CAPM) developed by Sharpe (1964), Lintner (1965), and Mossin (1966) provided an equilibrium model for the pricing of risky assets.1 The CAPM follows logically from the Markowitz mean– variance portfolio theory. In its basic version, it assumes risk–free borrowing and lending at a given single interest rate, no transaction costs, shared common expectations of all market participants, and perfect capital markets. By an equilibrium argument, it is shown that each investor will hold the market portfolio, MP, regardless of her risk attitude, and will adjust her portfolio to her risk aversion preference by lending or borrowing. The capital asset pricing model is given by: h i (3.8) E (ri ) − r f = β i E (r MP ) − r f where βi =

σiMP . 2 σMP

(3.9)

This relationship states that the expected excess rate of return of asset i (above the risk–free rate, r f ) is a linear function of β i and the market risk premium (expected excess rate of return of the market portfolio MP over r f ). β i is the covariance of asset i with MP, normalized by the variance of MP. This implies that risk which is uncorrelated with the market is irrelevant in the pricing equation because it can be reduced by diversification such that a rational investor would not demand extra compensation for bearing that risk. This risk is usually referred to as nonsystematic, idiosyncratic, or specific risk. In contrast, covariance risk is referred to as systematic risk. From a portfolio pricing perspective it is not the total risk measured by the variance of the return of an individual asset that matters, but only the systematic risk measured by its covariance with the market. The above introduced basic CAPM is widely applied in practice, primarily because of its simplicity. The empirical validity of the CAPM has been widely discussed and challenged, but this discussion is not key for the purposes of our discussion since it will not explicitly refer to the CAPM. An overview of this discussion as well as key empirical results can be found in Fama (1976); Alexander and Francis (1986), “Roll’s critique” (Roll 1977), and an answer to Roll’s critique in Kandel and Stambaugh (1995).

1

For the following exposition, see Luenberger (1998, p. 173ff).

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3 Literature Review

3.1.1.3 Intertemporal Capital Asset Pricing Model and Consumption Both the traditional Markowitz portfolio selection problem and the CAPM are static models in the sense that they consider a single–period decision problem (cf. Constantinides and Malliaris 1995). Only under certain assumptions, especially when preferences and future investment opportunity sets are not state dependent, the solution for the single–period portfolio problem also solves the multi–period portfolio problem (cf. Fama 1970; Rudolf 1994; Brennan and Xia 2006). Relaxing these assumptions, the CAPM has been extended to the Intertemporal Capital Asset Pricing Model (ICAPM) by Merton (1973a). The ICAPM considers “consumer-investors” who maximize the expected utility of lifetime consumption (Merton 1993, p. 475). Therefore, the portfolio decisions over time take into account the relation between current period returns and returns that will be available in the future. Given that the investment opportunity set might shift over time, investors with a multi– period investment horizon need to consider ways of hedging against unfavorable shifts, to protect long–term wealth or consumption. Consequently, equilibrium expected returns depend both on the covariance of returns with the current return of the market portfolio, and information about future returns of the market portfolio Cochrane (cf. 2001, p. 172). For example, if the current return of an asset is negatively correlated with shifts in the investment opportunity set, it can serve as a hedge against negative future developments, thus requiring a higher equilibrium price for this asset through the hedging demand (cf. Merton 1993, p. 479). A more detailed discussion of the ICAPM can be found in Merton (1993) and Duffie (2001). The ICAPM models marginal utility not in terms of consumption, but in terms of other directly observably variables instead, i.e., current wealth as well as conditional distributions of asset returns, given shifts in the investment opportunity set (cf. Cochrane 2001, pp. 46 f., p. 166). In that sense, it is a special form of a consumption–based model (cf. Cochrane 2001, p. 170). Lucas (1978) and Breeden (1979) consider investment and consumption in an intertemporal Consumption–Based Capital Asset Pricing Model (CCAPM). For this, Breeden (1979) explicitly models the joint probability distribution of consumption and excess asset returns. In this setting, the expected excess return of an asset is proportional to the asset’s covariance with consumption, i.e., proportional to the asset’s consumption beta (cf. Constantinides 1989, p. 6). As a result, each asset’s risk is completely specified by the covariance of its return with the change in aggregate consumption (cf. Merton 1993, p. 520). A more detailed discussion of the CCAPM can be found in Merton (1993), Duffie (2003) and Campbell (2003).

3.1 Financial Portfolio Theory

27

3.1.1.4 Arbitrage Pricing Theory The CAPM provides a rather restrictive pricing model. Multi–factor models have been subsequently developed to explain returns of assets using a number of desired factors.2 The CAPM can be viewed as a special case of a single factor model, with the single factor being the return on the market. Factors can be grouped as external (e.g., gross national product), extracted factors (e.g., industry returns), and firm characteristics (e.g., price to earnings ratio). A prominent extension is the Arbitrage Pricing Theory (APT) which is more general than the CAPM, proposed by Ross (1976). It only assumes non–satiation and that the universe of investment alternatives is large, but is still based on the concept of diversification. By an arbitrage argument, it has been argued that a factor price of risk is derived for each factor, which can then be used for pricing securities. Multi–factor models can increase the empirical validity of the single–factor CAPM (Fama and French 1992, 1993, 1995, 1996). Likewise to the CAPM, the APT can be extended to a multi–period setting (cf. e.g. Ohlson and Garman 1980; Constantinides 1989). 3.1.1.5 Conclusion on Mean–Variance Portfolio Analysis Mean–variance analysis provides a simple structure for portfolio selection and asset pricing. Therefore, it conceptually sets the stage for any portfolio analysis and is also the foundation of the analysis of portfolios of real options. While originally set as single–period problems, portfolio analysis has been extended to multi–period problems successively. However, the assumptions which base the analysis on perfect capital markets in a mean–variance world are too restrictive for our problem. In the context of portfolios of real options, two groups of assumptions need to be relaxed. First, options are asymmetric contracts and thus can hardly be captured by a mean–variance perspective. Option asymmetries enhance skewness of the underlying distributions (Trigeorgis 2005), such that higher moments of the underlying distributions matter. Additionally, volatility dependence on project value may introduce fat tails and kurtosis (Sears and Trennepohl 1983). As a consequence, portfolio theory for higher moments, beyond µ and σ, becomes relevant. Second, portfolios of real options must be analyzed in the context of frictions, such as budget and operating constraints (Trigeorgis 2005). Since these reservations concern single–period portfolio models, they also apply for multi–period portfolio problems. This is why more advanced portfolio models are investigated next, which model higher moments of distributions and allow for frictions. 2

For the following exposition, see Luenberger (1998, p. 198ff).

28

3 Literature Review

3.1.2 Advanced Financial Portfolio Analysis 3.1.2.1 Portfolio Analysis Considering Higher Moments of Distributions As noted, Markowitz portfolio theory considers only the mean and variance of the distribution of returns of the underlying assets (i.e., only the first two moments). This is valid if the investor’s utility function is quadratic, or if the distribution can be sufficiently characterized by the first two moments, which holds true for the normal distribution. The assumption of quadratic utility functions has been criticized (Arditti 1967; Levy 1969), while the assumption of normality is not sufficient in the context of portfolios of real options (cf. Trigeorgis 2005). Consequently, higher moments in financial portfolio theory must also be considered. Generally, the r th univariate moment about the origin of a continuous random variable X is given by r

E( X ) =

Z

+∞ −∞

xr f ( x )dx,

(3.10)

with f ( x ) being the probability density function of X (cf. Ross 2003). While r can be any positive integer, the first four moments are most intuitive to characterize a distribution. The first moment is the mean, the second moment the variance (spread of outcome), the third moment is the skewness (measuring asymmetry), and the fourth moment is the kurtosis (measuring the probability mass on the tails of a distribution). For jointly distributed random variables, additional multivariate co–moments describe the multivariate distribution, such as co–skewness and co–kurtosis, analogously to the covariance. A recent review of portfolio optimization for higher moments can be found in Guse (2005). Early approaches analyzing the first three moments can be found in Arditti (1967), Samuelson (1970), and Jean (1971). Kraus and Litzenberger (1976) formulated a three–moment CAPM rigorously, with a more recent analysis of the three–moment CAPM found in de Athayde and Flores (2000) and Jurczenko and Maillet (2001). Fang and Lai (1997) extended the CAPM to four moments, while Jondeau and Rockinger (2003) estimated the first four moments and co– moments for a set of daily stock–index and foreign–exchange returns. While it is possible to consider even higher moments, the additional insights from better approximating the real underlying distribution is usually decreasing and the additional computational effort grows considerably (cf. Chung et al. 2004, who consider up to 10 moments).

3.1 Financial Portfolio Theory

29

3.1.2.2 Allowing for Frictions in the Management of Portfolios One direction for making the portfolio theory more realistic has been to allow for more complexity in either the number of factors or the number of moments considered. These approaches are typically set in a model with frictionless markets, involving no restriction on completeness of markets, continuous trading, no transaction costs, and no restrictions on short sales. Another strand of literature aims at making portfolio models more realistic by explicitly modeling specific frictions. A comprehensive overview of portfolio optimization under constraints and transaction costs can be found in Korn (1997). If rebalancing of a portfolio is costly, portfolio optimization must take such transaction costs into account. The bottom line effect is to hold less assets and trade less often, which counters the effect of diversification and continuous rebalancing. Key references analyzing the impact of transaction costs on the optimal portfolio include Brennan (1975), Constantinides (1979), Davis and Norman (1990), Li et al. (2000), Akian et al. (2001), and Sass (2005). More recently, Best and Hlouskova (2005) provide an algorithm reducing numerical expense for portfolio optimization. Going beyond pure transaction costs, it may sometimes be desirable to limit the number of portfolio positions, as in Jacob (1974) and Levy (1978). Li et al. (2001) modeled frictions by including transaction costs together with restrictions on short sales and taxes. The review of portfolio models for higher moments and models considering frictions is separated from one another because frictions have hardly been studied jointly with higher moments in the distributions of returns. Exceptions are Wang and Xia (2002) and Liu et al. (2003) who simplify their problem by piecewise linear approximations of the skewness. 3.1.2.3 Impact of Combining Financial Assets with Financial Options Up to this point, we have not yet considered financial options. Financial options are derivatives, i.e., redundant, in the sense that their cash flows can be replicated by directly holding positions in the corresponding underlying assets. An example is a call option on a share, whose payoff can be replicated by holding the share and borrowing a specific amount of money by shorting riskless bonds. The call option corresponds to a levered position in the underlying asset and actually can be viewed as some kind of portfolio because its payoff is identical to the payoff of an equivalent portfolio. In turn, the key concept of derivatives pricing is self–financing replication in arbitrage–free markets. If the call option yields a payoff identical to the dynamically rebalanced portfolio of share and shorted bond in all states of nature, it must have the same price. This assumes that markets are dynamically complete and in equilibrium, with correct prices for all assets. Options

30

3 Literature Review

can be “home–made” in that if a desired option is not available on the market, it can be replicated by holding the equivalent position in the underlying asset and riskless bonds. Based on this perspective, it is less insightful to develop a portfolio theory for portfolios composed of financial options alone. For plain–vanilla options, option values may be correlated as a function of the correlation of the underlying assets, but may otherwise be “strategically” independent. By this we mean that the option payoffs are independent from one another, and that the optimal exercise strategy for each option can be obtained by considering each option in isolation from the other options. A similar argument holds for more complex financial options written on several assets where the option price again depends on the correlation among the underlying assets, but is otherwise independent of other options. The same applies for options on other options, since this is the special case where the underlying asset in fact is an option. With this reservation, we retain that financial option values follow directly from the equilibrium prices of the corresponding underlying assets. Moreover, exercising a financial option typically does not affect the underlying asset. It is known that a portfolio of options is at least as valuable as an option on the corresponding portfolio of underlying assets because the portfolio of options gives more discretionary decision rights. Moreover, for option pricing, total risk (volatility) matters and standard option prices typically are increasing with higher volatility. Therefore, any diversification in the risk of the portfolio and reduction of volatility reduces option values (cf. Merton 1973b). This illustrates that the notion of risk for portfolios of options is fundamentally different from the understanding of risk in a standard portfolio perspective. According to Trigeorgis (2005), risk is not necessarily something bad that must be avoided, as posited by Markowitz, but a window of opportunity for capitalizing on upside potential. These different notions of risk need to be accommodated jointly when portfolios of assets and real options are considered simultaneously. The portfolio effects of options must be considered with care, as they modify the return pattern of the portfolio. For example, portfolio insurance is achieved by adding put options to a portfolio, locking in some minimum return at the expense of upside potential (Bookstaber and Clarke 1983, 1984) to provide a (partial) hedge. Besides hedging, options may be used for speculation, for saving on transaction costs, for helping complete an otherwise incomplete market, or for increasing liquidity when the markets for underlying securities are less liquid. These aspects require more advanced valuation techniques (cf. Korn and Trautmann 1999), but they do not represent a new theory as such. The assets are still priced in market equilibrium and options are still derivative securities, whose prices depend on the prices of

3.1 Financial Portfolio Theory

31

the underlying assets. Adding options to a portfolio of assets enables to design a desired payoff structure. 3.1.2.4 Portfolio Implications for Risk Management Risk management aims at altering the risk profile of a firm in order to protect its profitability (McDonald 2003, p. 89). An overview over risk management is found in Dempster (2002). One of the most widely used concepts in corporate risk management, “Value–at–Risk” (VaR), is reviewed in Duffie and Pan (1997). VaR quantifies the total risk of a portfolio, measuring the maximum loss that can materialize, with a given probability, over a defined time horizon (cf. Hull 2003, pp. 346 ff.). Risk management is closely connected to portfolio management because the risk of the portfolio has to be well understood before it can be controlled in a desired way. Also, portfolio management can be implemented via VaR, by using VaR as the measure of risk instead of volatility (cf. Alexander and Baptista 2004). In the context of this dissertation, risk management is not a relevant field. It represents an application of the portfolio concepts discussed so far, not a new theory in itself. Further, the underlying perspective of minimizing risk is fundamentally different from the real options perspective in portfolios of real options. The former considers risk as an undesirable thing, the latter capitalizes on the desirable effects of risk while reducing the corresponding downside. Thus, a rigorous portfolio approach to real options can ultimately provide further insights for risk management (cf. Trigeorgis 2005, pp. 36 f.) because real options affect the risk exposure of a company. The existence of real options provides a means of limiting downside risk. Consequently, the calculation of a risk exposure needs to take into account the effect of real options before it is insightful to manage that risk from a risk management perspective. In short, the analysis of portfolios of real options can shed light on risk management, not the other way round. 3.1.2.5 Conclusion on Advanced Financial Portfolio Analysis Advanced portfolio analysis is better able to handle complex problems with higher moments. Complex portfolio models have also been formulated for markets with frictions, such as to capture transaction costs. Still, the current state of research sheds very little light on portfolios of assets with higher moments and frictions. Yet a suitable model for portfolios of real options would need to go even further than that, and also include options. Incorporating options into a portfolio increases complexity because options enhance asymmetry. Even comparatively simple options like a compound option (cf. Geske 1979) or an option on the maximum or minimum of several

32

3 Literature Review

assets (cf. Stulz 1982; Johnson 1987) present challenges because the exercise region cannot be determined by simple arguments anymore (Broadie and Detemple 1997, p. 242). Further, the pricing of American, path–dependent or multi–asset options typically must resort to numerical methods since closed–form solutions may not be feasible (cf. Broadie and Detemple 2004, p. 1163). In the light of our portfolio approach, as defined in Sec. 2, the just presented review of the relevant financial portfolio literature has shown that no portfolio model suitable for the desired analysis is readily available.

3.2 Corporate Capital Budgeting 3.2.1 Overview “Capital budgeting is the decision area in financial management that establishes goals and criteria for investing resources in long–term projects” (Clark et al. 1984, p. 1). A synthesis of the capital budgeting literature can be found in Weingartner (1977) and more recently in Peterson and Fabozzi (2002) and Dayanada (2002). While the above mentioned definition is clearly based on a financial management perspective, the analysis of investments can also take place from an operations management point of view (for an overview, see Van Mieghem 2003). In this strand of literature, more emphasis is laid on detailed modeling of processing networks, including operating considerations such as production planning, inventory management, and supply chain management. We will not focus on this field here since the present dissertation is set in a real options and thus financial management perspective. A first model for harmonizing firm investment outlays and limited capital resources was provided by Dean (1951). It derives the optimal capital budget as the intersection of a capital demand and a capital supply curve in a one–period setting. Following this perspective, Lorie and Savage (1955) identify the three key problems in investment choice as the choice of profitable investments, forecasting cash flows, and rationing capital. Charnes et al. (1959) proposed a valuation approach based on (linear) programming which was widely applied subsequently. Key references for solving the capital budgeting problem by mathematical programming include Baumol and Quandt (1965), Weingartner (1966b, a, 1967), Arditti (1967), Tuttle and Litzenberger (1968), Hillier (1969), Nemhauser and Ullmann (1969), and Elton (1970). The relevant features considered in these papers, especially logical relationships and synergies among projects, will also be considered in our model, which we capture through constraints and correlated cash flows.

3.2 Corporate Capital Budgeting

33

While these authors mostly focused on implementation issues of the programming approach, attention subsequently shifted towards a harmonization of the capital budgeting decision with the portfolio concepts just recently developed for capital markets. In this context, the work of Markowitz (1952) as well as Sharpe (1964), Lintner (1965), and Mossin (1966) was guiding. It provided helpful insights as to what cost of capital should be used in the assessment of investment projects and in how far a portfolio of projects differs from stand–alone analysis. Key references discussing an explicit link between the CAPM and corporate investment decisions include Weston (1973), Carter (1974), Spies (1974), Nielsen (1975), Thompson (1976), and Myers and Turnbull (1977). While also based on an equilibrium framework, the approach chosen for the present dissertation will be substantially different from these papers, in that it is based on a risk–neutral valuation framework (see Sec. 4.1.4). Another approach to traditional capital budgeting is represented by the integer knapsack problem, e.g., in Dantzig (1957) and Cord (1964). In the context of capital budgeting, an integer knapsack problem is a combinatorial problem where the value of a selection of possible projects (e.g., items in the knapsack) is maximized, subject to budget constraints (e.g., the size of the knapsack) and to integer constraints for the projects. Knapsack problems provide a rich framework for handling capacity restrictions in decision analysis and have been widely extended in the last 50 years. A thorough review of the different types of knapsack models can be found in Kellerer et al. (cf., e.g., 2004). Still there is no readily available knapsack problem that can handle the portfolio problem considered here. Our model takes into account multiple correlated processes and dynamic interactions between decisions, at each point in time and over time. Such a stochastic knapsack model has not yet been presented in the literature. An alternative strand of literature stresses the flexibility inherent to investment management, and for this resorts to decision tree analysis (cf. e.g. Magee 1964a, b; Hespos and Strassmann 1965; Hax 1985). The relationship between real options and decision tree analysis has largely been discussed elsewhere (cf., e.g., Smith and Nau 1995; Trigeorgis and Mason 1987; Trigeorgis 1996). In the light of our discussion, decision tree analysis does not provide additional insights beyond real options analysis and therefore is not discussed further. 3.2.2 Portfolio Perspective in Corporate Capital Budgeting Investment decisions of companies are inherently concerned with portfolio considerations (cf. Weingartner 1966a). Companies typically are faced with investment program decisions where a set of investment projects, or an investment program, is chosen out of a larger set of investment opportunities.

34

3 Literature Review

This program can be considered as a portfolio. Further, since capital assets usually have a multi–period lifetime, the investment program has to optimally fit with the assets already in place, which together constitute the corporate asset portfolio. It is relatively straightforward to apply the concepts from standard (mean–variance) financial portfolio theory to corporate capital budgeting (see e.g. Hax 1985, pp. 133 ff.). Each investment project is assessed based on the risk it bears and the return commensurate with that risk. From a portfolio perspective, the relevant risk is not the total risk but the marginal systematic risk the individual project adds to the existing (market) portfolio, such that only the covariance risk with the (market) portfolio matters. In this context, corporate diversification is redundant (Brealey and Myers 1996, p. 166). With this background, companies should only care about the systematic risk of a project and can assess projects based on some equilibrium model such as the CAPM. As a function of project betas, the required rates of return are known. Each project that has a positive NPV is value creating, since its return is above the return demanded by capital markets for that type of systematic risk. The portfolio perspective on capital budgeting is very convenient because it allows to assess projects as if they were stand– alone, such that project values are additive. In this sense, pricing a real asset is analogous to pricing a financial asset (cf. Damodaran 1996, pp. 484 ff.). 3.2.3 Interdependencies The portfolio perspective on real assets requires more thorough analysis when projects cannot be assessed independently from one another in the CAPM sense (Peirson and Bird 1981, pp. 177 f.). Projects may be interdependent for a range of reasons (for this classification, see Betge (1995) and Brosch (2001)): • Direct qualitative interactions which have their origin in the investment plan as such, or which result from interactions with investments already undertaken and still generating cash flows. Direct qualitative interactions do not result from stochastic relationships, but from physical properties of the projects. This is why they cannot be avoided in any way comparable to diversification. Examples are mutual exclusive projects (=substitutive), projects requiring one another (=complementary), or synergies (positive or negative). • Indirect qualitative interactions which result from constraints beyond the investment plan. They are indirect because they are not inseparably connected to the investment opportunities considered; qualitative because they do not result from stochastic relationships and cannot be avoided

3.2 Corporate Capital Budgeting

35

by diversification. These constraints are connected to the firm making investment decisions. Most importantly, indirect qualitative interactions stem from binding capital constraints, i.e., capital rationing. Recently, Meier et al. (2001) explicitly discuss a portfolio of real options subject to a capital expenditure constraint and propose two model formulations for portfolios of real options. The first formulation is a standard knapsack problem, maximizing the value of the portfolio subject to a capital constraint. The value of the portfolio is calculated as the sum of the values of the options in the portfolio. Therefore, this model follows closely the traditional capital budgeting by replacing the NPVs of projects with their option values. Real option values do not interact and the model is static. In a second integer programming formulation, they provide a dynamic approach which models investment decisions over time within a binomial framework. All projects are related to the same underlying stochastic variable, but are independent from one another. Further, only call options are considered which can be postponed indefinitely. Once an option is exercised, there is no further option available. Therefore, the options are stand-alone options that do not interact. The budget constraint is static in the sense that it gives a fixed upper limit on outlays which must not be exceeded at any point in time. Their paper shares the conceptual origin with the present dissertation, in that it focuses on portfolios of real options, for limited funds available. As opposed to their approach, we go further into detail with regards to modeling options interactions, path–dependencies, dynamic budget constraints, and other constraints. In addition, our focus differs. While their paper lays emphasis on providing an efficient algorithm specific to the problem, our focus is more on the general model formulation. While interactions such as synergies were discussed extensively in the mathematical programming literature (see Sec. 3.2.1), they may be debated in the context of modern corporate finance. First, the presence of direct qualitative interactions violates the additivity of NPVs, which derives from the underlying mean–variance portfolio concept based on systematic risk. While this view is debatable (cf., e.g., Stulz 1999), it does not disallow the possibility of, e.g., synergies. NPVs add up if (and only if) the only portfolio aspect is diversification and markets are perfect and complete. Other interdependencies, e.g. due to budget constraints, can cause deviations from additivity. Second, in realistic investment decision settings there may exist “soft” or “hard” rationing of capital (cf. Brealey and Myers 1996, pp. 101 ff.). Soft rationing implies that although capital markets are perfect, the firm imposes an internal constraint voluntarily as a means of financial control. Hard rationing implies some sort of market imperfections. These imperfections may be caused by capital markets in a strict sense, or may result from the com-

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3 Literature Review

pany’s limited access to capital markets. Either way, as long as the ultimate investor is not hindered in her access to well–functioning capital markets, the existence of capital constraints does not by itself invalidate the NPV approach. An extensive review of literature on financing constraints is found in Hubbard (1998). Recently, Mello and Parsons (2000) analyze the impact of financing constraints on operating and hedging policies of a firm which is already invested. They show that the constraint effectively reduces the firm’s alternatives. Boyle and Guthrie (2003) focus on the effect of financing constraints on the initial decision to invest. Almeida et al. (2004) empirically test the effect of financial constraints on investment, proxied by the holding of cash reserves. 3.2.4 Conclusion on Corporate Capital Budgeting The literature on corporate capital budgeting has been concerned with both portfolio considerations and the modeling of interdependencies. Nevertheless, the traditional approach is oftentimes reduced to calculating NPVs under very restrictive assumptions, reducing the assessment of a specific project to its beta with the market. The present dissertation considers interactions jointly with the interactions stemming from correlation of real options and underlying assets alongside with asymmetries introduced by option exercise decisions. In doing so it unites elements taken from the traditional capital budgeting literature with more recent approaches based on contingent claims analysis.

3.3 Financial and Real Options Theory 3.3.1 Foundations of Real Options Concepts derived for financial options have provided a rich arsenal for valuing real options at the same time because real option valuation is based on concepts from financial option pricing. Cornerstones of real options pricing therefore are Black and Scholes (1973) and Merton (1973b) who were first to provide a closed–form solution for financial options, as well as Cox et al. (1979) who introduced an equivalent discrete binomial valuation approach that has proved to be very powerful in handling complex options settings. While these seminal contributions deal with single options in the first place, following work has focused on problems involving several options or several underlying assets which can be viewed as special cases of a portfolio. For example, an option on the maximum of two assets can be viewed as a portfolio investment problem with the opportunity to invest in two different projects which are mutually exclusive. This direction is particularly

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37

relevant in the light of our analysis here. Margrabe (1978) analyzes an option to exchange one asset for another while Stulz (1982) considers an option on the maximum or the minimum of two underlying assets, later generalized by Johnson (1987) and Boyle and Tse (1990) to several underlying assets. Laamanen (2000) models the related option on the best subset of multiple underlying assets. On a related strand of research, Geske (1979) provides a closed–form solution for two options which are compound, i.e., an option on an option. These two features are combined by Carr (1988) who values a sequential exchange option, i.e., an option on an option to exchange one asset for another. These concepts have also been widely applied to real option valuation problems, though mostly focusing on options analyzed in isolation. While the analysis of stand–alone options provided much insight for the understanding of flexibility, its usefulness is limited from a portfolio point of view. Exceptions are those papers explicitly taking into account the interaction of real options and real assets. Brennan and Schwartz (1985) analyze a natural resource project with options to shut down and restart or abandon. McDonald and Siegel (1985) consider an option to temporarily shut down a plant. The interconnection of separate options through a shared underlying asset makes it essential to value these options in combination in a compound option–like model.3 While these authors do take into account interactions, Trigeorgis (1993a) addresses options interactions explicitly. He points out that the existence of subsequent options may increase the value of the underlying asset for earlier options, whereas the possibility of exercising earlier options affects the value of the underlying asset for subsequent options. As a consequence, in the case of multiple options written on the same underlying asset, option values typically do not add up. Trigeorgis (1993a) discusses general properties concerning the nature of options interactions. Kulatilaka (1995) confirms the results of Trigeorgis (1993a) and explicitly considers the effect of additional options on the critical boundaries for option exercise, i.e., the exercise schedules. 3.3.2 Real Options Analysis in a Portfolio Context The above cited references have set the stage for real options analysis in a portfolio context. They have made clear that the analysis of interacting real options are of special interest because interactions have to be modeled explicitly. Following this line of argument, several authors have shed light on 3

Kulatilaka (1993) values a dual–fuel steam boiler where switching between different combustibles is costly. While it is desirable to switch between oil and gas as a function of the relative prices, the switching is subject to switching costs, implicitly making the problem a nested compound option problem.

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3 Literature Review

settings with interacting options. Following Trigeorgis (1996, pp. 132 f.), interactions between options on the same underlying are designated as “intra– project”, while interactions embracing several underlying assets are designated as “inter–project” interactions. The field of Research and Development (R&D), in particular, has paid special attention to options interactions. Due to the exploratory nature of R&D projects which are typically multi–stage, early projects can generate insights about future projects. Moreover, in an R&D pipeline many projects are typically undertaken but only few make it to become a marketable product, such that the bundle of projects must be assessed from a portfolio point of view. Problems involving inter–project interactions are thus particularly relevant. In this context, Childs et al. (1998) consider a setting where two projects may be run in parallel during the development phase, but only one may be implemented, i.e., they are mutually exclusive. They find that sequential development is desirable when project values are highly correlated, require large capital outlays, are short term, and have low volatility. Likewise, Childs and Triantis (1999) as well as Lint and Pennings (2002) consider the parallel development of two R&D projects that are mutually exclusive. Gustafsson and Salo (2005) focus on project selection subject to budget constraints, which is achieved by explicitly modeling the decision maker’s terminal utility. Denardo et al. (2004) propose a stochastic search algorithm motivated by an R&D application with a sequential compound decision process. Attention has also been paid to learning during the R&D process (cf. e.g. Koussis et al. 2003). Huchzermeier and Loch (2001) model a compound R&D decision problem where management has the option to take corrective action as a means of managing the risk involved in R&D projects which may stem from different sources (e.g., market payoffs and project schedules). Vassolo et al. (2004) find for the biotechnology sector that mutually competitive, correlated projects tend to be sub–additive, but that the sharing of resources among firms may create positive spill–overs resulting in super– additivity. A review of R&D specific investment problems is found in Chien (2002) and Kavadias and Loch (2004). In the context of exploration, Smith and Thompson (2005) show that learning between correlated oil exploration activities may invalidate the conventional wisdom of diversification, in that for a risk–averse investor it may be desirable to invest into highly correlated prospects. Dias (2006) confirms that in oil exploration, positive correlation is a desirable feature, as opposed to financial options intuition, due to learning and synergies. In other related work, Brown and Davis (1998) model an investment choice problem where a first project can be followed up by one of two mutually exclusive projects. Rose (1998) and Bowe and Lee (2004) consider the intra–project interactions of options on infrastructure projects, following

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39

the framework presented in Trigeorgis (1993a). Triantis and Hodder (1990) analyze a flexible production system that can switch the production mix among two products over time. Kogut and Kulatilaka (1994) and Huchzermeier and Cohen (1996) consider a global manufacturing network under exchange rate risk, with switching options between different manufacturing strategies contingent on exchange rate realizations. Wang and de Neufville (2004) value options inherent in the design of large physical systems, such as hydropower stations, modeled as path–dependent options. Their approach is related in that they model the option valuation problem as a stochastic mixed–integer program that can be implemented on commercially available optimization platforms. This approach will also be followed in the present dissertation in order to solve the formulated portfolio model. Martzoukos et al. (2003) model a path–dependent investment problems that is essentially a switching model subject to stochastic switching costs. In order to keep track of all possible paths and possible early exercise on each path, they propose a programming algorithm labelled as “exhaustive search” (cf. Martzoukos et al. 2003, p. 9). In the context of supply chain management, Kamrad and Siddique (2004) model the interactions between a producer’s and a supplier’s investment decisions, considering exchange rate uncertainty, supplier–switching options, order–quantity flexibility, profit sharing, and supplier reaction options. They consider discretized exchange rate processes as sources of uncertainty and solve the problem numerically through a backward recursion in dynamic programming. The mentioned references dealt with complex problems involving interactions between real options, either intra–project or inter–project. However, they did not take an explicit portfolio perspective on real options. Most explicit in presenting a portfolio perspective on real options, yet only qualitative, is Luehrman (1998). He chooses a gardening metaphor where the firm is behaving like a gardener who only picks tomatoes (= exercise options) which are “ripe and perfect”(= at the optimal time). There are good and bad gardeners, active and passive ones. A good gardener knows which tomatoes to pick, which ones to leave yet a little while, and which ones to pick even if not yet fully ripe, in order to prevent the squirrels (=the competitors4 ) from stealing them. While Luehrman (1998) is notable for stressing the intuition, he does not shed light on how interactions between options can be coped with in a rigorous valuation framework. Trigeorgis (1988) and Trigeorgis and Kasanen (1991) consider a portfolio perspective more explicitly by considering compound synergistic effects between parallel projects. Kester (1984, 1993) analyzed inter–project dependence with special attention to growth options, while Kasanen and Trigeor4

The author of the present dissertation was not aware of the fact that squirrels like tomatoes, but believes that this is only secondary to the argument.

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3 Literature Review

gis (1993) and Kasanen (1993) put more emphasis on modeling synergies. Mauer and Ott (1995) analyze replacement decisions as follow–up projects. 3.3.3 Detailed Review of Related Papers Beyond the so far reviewed references, two publications have particularly influenced the present dissertation: Kulatilaka (1988) and Kulatilaka and Trigeorgis (1994). In order to position the following analysis, these two reference will be discussed in more detail in what follows. 3.3.3.1 Kulatilaka (1988) In the paper “Valuing the Flexibility of Flexible Manufacturing Systems” Kulatilaka (1988) analyzes the case of a flexible manufacturing system, in which a flexible technology is one that can be operated in alternative modes. This perspective on modes inspired the setup of our model and therefore is presented in detail. At the same time it is noted that our modeling technique based on binomial option pricing is fundamentally different from the transition probability, i.e., Markov chain, approach in Kulatilaka (1988). For consistency the paper is not represented in its original notation, but we present the relevant elements with the variable definitions used in the following model development. Further, some key concepts are introduced somewhat differently in order to stress the elements employed later on. Especially, the view on entering and leaving modes is introduced here. This is not part of the original paper but helps to elucidate the design of the model. Nevertheless, despite these suitable modifications, the original content of the paper remains unchanged. In this paper, operating modes are interpreted in a very general way with the examples of different routings within a production system, different sets of inputs used in manufacturing the product, or different outputs. Also, modes can characterize waiting to invest, shutting down, and abandoning of projects. Operating modes are designated as a, a = 1, . . . , A. Further, it is insightful to distinguish another operating mode, a0 , a0 = 1, . . . , A0 . Mode a is the mode in which the technology is before a switch, a0 is the result of this switch, from a to a0 . Therefore it is convenient to define a as the entering mode and a0 as the leaving mode. The flexible technology is exposed to some uncertain variable, θk , which represents the uncertainty stemming from, e.g., demand, price, or exchange rate. θk follows some stochastic process which determines possible states in which θk can realize and the corresponding transition probabilities between states. More precisely, θk can realize N different states, i.e., k = 1, · · · , N. Further, the transition probability is given by pij , specifying the transition

3.3 Financial and Real Options Theory

41

probability that θ (t + 1) is in state i given that θ (t) was in state j. This discrete system state space can be interpreted in two ways. It may be viewed as an approximation of a problem which is continuous in reality. Alternatively, the discrete nature can be “given exogenously by contracting arrangements or other organizational constraints” (Kulatilaka 1988, p. 252). Associated with each production mode a0 is a cash flow function c(θ, a0 ) which is determined by the realizations of the stochastic process of the value of the underlying asset. Let us assume three cash flow functions, so we have for a0 = 1, 2, 3: c(θ, 1) = 0

(3.11)

c(θ, 2) = −0.6 + 2θ

(3.12)

c(θ, 3) = −0.1 + θ

(3.13)

Note that c(θ, 1) is the mode for waiting to invest. These parameters are chosen such that for low realizations of θ, i.e., θ < 0.5, c(θ, 3) > c(θ, 2) while for θ > 0.5, c(θ, 2) > c(θ, 3). Switching from any mode a, to any mode a0 (a and a0 can be identical) incurs a switching costs, I ( a, a0 ). Periods are counted by t, t = 1, . . . , T. The objective is to maximize the net present value, V (t, a). This present value is calculated iteratively using stochastic dynamic programming, with three simultaneous equations, at any state k in period t. For each entering mode, a, the optimal leaving mode, a0 is solved for. For the last period, the value of the system is given by the maximum of the cash flows resulting from the different leaving modes a0 : (3.14) V ( T, a) = max c(θ T , a0 ) − I ( a, a0 ) a0

In the last but one period, T − 1, future realizations of θ are unknown. This is why for intermediate periods the optimal value is obtained by finding the leaving mode that maximizes both current cash flow net of switching costs and the expected value for the following period which is entered in the leaving mode of the previous period. This expected value of cash flows, ET −1 [V ( T, a0 )], is calculated as the probability-weighted sum of all V for possible realizations of θ in the following period, given a0 . It is discounted by the risk–adjusted one–period discount factor, φ. Note that the expected value is calculated in each period, looking forward to the following period. This is indicated by the subscript in the expected value operator. Based on this, the value for the last but one period is as follows: n o V ( T − 1, a) = max c(θ T −1 , a0 ) − I ( a, a0 ) + φE T −1 [V ( T, a0 )] a0

(3.15)

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3 Literature Review

Generalizing this relationship establishes the stochastic dynamic programming equations: n o V (t, a) = max c(θt , a0 ) − I ( a, a0 ) + φEt [V (t + 1, a0 )] a0

a = 1, 2, 3 ; t = 1, . . . , T − 1.

(3.16)

Proceeding in a stochastic dynamic programming fashion, this problem can be solved by backward recursion. Folding back, the expected value in each period includes the optimal switching decisions and resulting cash flows from all subsequent decisions. This perspective on flexibility in investment decisions is particularly insightful because of the switching concept introduced. It can interpret any investment and disinvestment decision as a switching decision and thus is very general. This is why the present dissertation follows this switching perspective. At the same time, it substantially expands the model by linking it to a real options context with an explicit link to capital markets considering multiple underlying assets, and most of all, by allowing for constraints and a dynamic budget. These additional effects are such that the model in its core can stay with a similar switching concept, but requires a completely different, forward–backward looking modeling approach. 3.3.3.2 Kulatilaka and Trigeorgis (1994) While Kulatilaka (1988) has presented the interpretation of investment decision as switches, he did not stress the link to capital markets and thus real options theory. This specific link to real options theory is what the paper “The general flexibility to switch: real options revisited” by Kulatilaka and Trigeorgis (1994) is about. Following the reasoning of Kulatilaka (1988), they discuss the case of a car manufacturer. Cars can be produced with two alternative technologies and two corresponding cash flow functions. The net cash flows for each technology depend on the oil price. One technology has high fixed costs and low oil consumption thus low variable (oil) costs, such that it is attractive in periods where oil prices are high. The other technology has low fixed costs but high variable (oil) costs. They show that the possibility to switch between these two technologies, given switching costs, can boost value beyond the stand-alone value of these technologies. From a modeling standpoint, the paper is particularly insightful because it connects the switching concept to a real option valuation approach, based on the binomial model. It is based on two simultaneous option lattice trees, i.e., for each technology one. Both binomial lattice trees are incorporated into one stochastic dynamic program, which incorporates switching between corresponding nodes in the above described backward recursion.

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Within this setting, they further discuss the impact of the nature of switching costs on the optimal investment strategy and distinguish two cases: 1. No switching costs. Since switching between technologies is costless, the entering mode is irrelevant for the optimal operating mode to be chosen. A switching decision affects the current cash flow, but is irrelevant for all subsequent decisions. Thus, the flexibility to switch can be calculated in a straightforward manner. Specifically, it can be calculated as the present value of cash flows of one base technology, adding to this the values of European options to switch to the other technology at each decision point. Obviously, the same result is obtained, no matter which technology is taken as the base technology. 2. Switching costs. If switching costs exist, the entering mode affects the switching decision and thus has to be taken into account. This requires the two simultaneous option lattice trees, and corresponds to the case analyzed in Kulatilaka (1988). The case of switching costs is particularly interesting because it introduces a compound interaction. From a real options point of view, each switch, or exercise decision, creates a specific series of nested new options. This is analogous to a compound option and invalidates option value additivity. Interestingly, this may introduce a hysteresis range, or inactivity range in the optimal operating policy. This means that even if an immediate switch would create a positive net cash flow, it may be optimal to stay in the original mode and wait for stronger changes in the underlying variable. This is because the underlying variable must change sufficiently not only with respect to current cash flows, but also to future cash flows. This dynamic perspective takes into account the probability and costs of re-switching back in later stages; the costs of switching back and forth can be higher than the cash flows realized. Further, it may be desirable to wait and maintain options to switch later if the leaving mode is absorbing in the sense that further switches are prohibitively costly, or even impossible. The authors generalize the switching concept, showing that known real options can be valued as special cases of the switching concept. Specifically, they define suitable interpretations of switching modes for these options.5 This generalization underlines the large applicability of the switching concept. Kulatilaka and Trigeorgis (1994) is guiding for the approach presented here for two reasons. First, it completes the switching concept with a binomial real option pricing approach. The binomial switching concept is at the 5

They distinguish the following options: option to defer; to expand; to contract; to temporarily shut down and restart operations; to abandon for salvage value; to default during construction (or time-to-build option).

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3 Literature Review

core of our model. Second, the generalization of the switching concept to known types of real options explicitly connects our model to the existing body of real options literature where typically a real options terminology is used. Due to the generality of our switching concept, it contains known types of real options as special cases. 3.3.4 Conclusion on Financial and Real Options Theory The body of literature on real options has grown large, considering a wide range of problem settings that can only be fully captured with the help of contingent claims analysis. Nevertheless, the rather limited analysis of portfolios of real options in the literature so far has been motivated by specific decision problems where compoundness and interactions arose because of special features of the decision problem, such as the parallel or sequential development strategies of R&D projects, or investment in capacity. While some authors have focused more explicitly on interaction effects among options, such as involving growth options or synergies between projects, a more systematic treatment of the main portfolio effects, including budget constraints, has been lacking so far. We aim to close this gap with the general portfolio model framework discussed next. In order to do so, the work by Kulatilaka (1988) and Kulatilaka and Trigeorgis (1994) provides a starting point that we will build on and further extend.

4 Valuation Model for Portfolios of Real Options

This chapter contains the formulation of a valuation model for the portfolio problem defined in Sec. 2.1.2 where portfolios of real options are understood as a combination of multiple underlying assets and multiple real options written on these assets, subject to constraints. The model description is organized as follows: First, the described portfolio aspects are translated into model features that the general model formulation will take into account. Then, the binomial model is introduced as the starting point for our proposed model, which is then expanded successively to n underlying assets.

4.1 Model Features In order to ground the development of the proposed model, the key features that determine the model are presented in what follows. The description of these key features covers the objective function, the managerial actions permitted, the attitude towards risk, the modeling of the underlying stochastic processes, volatility and correlation, inter– and intra–project interactions, global dynamic budgets, synergies and operational constraints, learning, diversification, path–dependencies, as well as a discussion of the solution approach and the scalability of the model. 4.1.1 General Assumptions It has been widely discussed whether the real options approach is suitable for practical decision problems. Most importantly, the applicability of real options theory depends on whether the assumptions typically agreed on for financial assets and financial options hold for real options as well. The development of the portfolio model is based on a set of assumptions which

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4 Valuation Model for Portfolios of Real Options

are outlined here. These assumptions are standard in the context of real options (cf., e.g., Dixit and Pindyck 1994; Trigeorgis 1996; Copeland and Antikarov 2001). These assumptions are: • Capital markets are perfect and complete.1 Therefore, a spanning portfolio is available for each investment opportunity considered. • The risk–free interest rate is known, constant, and identical for all maturities. • The values of the underlying assets follow discrete binomial processes. • The underlying assets do not distribute any dividends. • Investments are at least in part irreversible, unless corresponding real options exist. • The decision maker has discretionary decision rights that can be interpreted as real options. • The universe of all available options can be specified exhaustively. • Options are proprietary options. There are no agency conflicts. • All input parameters necessary for the purpose of option valuation are unambiguous and known. 4.1.2 Objective Function The decision maker’s goal is to maximize shareholder value, and hence, market value of an all–equity financed firm (cf. Copeland et al. 1994, Part I). In the light of the present analysis, this translates into the objective of maximizing portfolio value, i.e., the combined value of the bundle of all underlying assets and all real options embedded in these underlying assets. Either this portfolio comprises the whole scope of the company, or it is perfectly independent from other ventures and can therefore be analyzed separately, contributing directly to the market value of the firm. The portfolio value is calculated as the expected value of the optimal exercise policy of the portfolio of real options, dynamically over all managerial actions permitted and stochastic realizations across time. This optimal exercise policy at each point in time corresponds to what is called “investment program” in the context of traditional capital budgeting. 4.1.3 Managerial Actions Permitted The decision maker’s actions determine the utilization of the underlying assets in question, by investing and disinvesting. For each underlying asset, i.e., investment prospect, the manager chooses the best available operating 1

This implies continuous trading on frictionless markets, no taxes, no restrictions on short sales, full divisibility of assets, and arbitrage–free markets (cf. Hull 2003, p. 242).

4.1 Model Features

47

mode out of the set of permissible modes. Without loss of generality, representative operating modes in the following model are: 1. Mode 1 = not invested. 2. Mode 2 = invested in low capacity. 3. Mode 3 = invested in high capacity. This translates into the following possible switches between modes, if permissible: • • • • •

1 → 1 : Stay uninvested, i.e., operate with zero capacity. 1 → 2, 1 → 3 : invest in low capacity, high capacity respectively. 2 → 3 : Expand capacity from low capacity to high capacity. 3 → 2 : Scale down capacity from high capacity to low capacity. 3 → 1, , 2 → 1 : Reduce capacity from high capacity to zero capacity or from low capacity to zero capacity. Depending on whether subsequent options to reinvest exist, this either represents an option to abandon (if switching back is not permitted), or an option to shut down and reopen (if switching back is permitted).

Switching between operating modes corresponds to a call or put option exercise at each decision point within the limits of permissible switches. Switching involves investment costs (switching costs) which can be positive or negative. Investments and disinvestments are treated similarly in the following, in that a disinvestment is an investment with negative sign. For instance, a European call option allows to switch from mode “not invested” to mode “invested” at maturity, whereas a European put option enables the reverse switch. 4.1.4 Risk Preference The expected value is calculated as a “risk–neutral” expected value. It is risk–neutral in the sense that it is independent from risk attitudes, such that a valuation can proceed as if the decision maker were risk–neutral (cf. Cox and Ross 1976; Duffie 2001). In other words, the decision maker can have any risk preference because it does not need to be modeled explicitly. Technically speaking, the risk–neutral probabilities are an equivalent martingale measure, transforming the actual real–world probabilities (see Harrison and Kreps 1979; Karatzas and Shreve 1998, pp. 33 f.; Paul and Baschnagel 1999, pp. 152 ff.). To avoid confusion, some authors prefer the term “synthetic” probabilities, since the latter differ from the probabilities for the actual occurrence of true states of the world (cf. Neftci 2000, pp. 21-23). The synthetic probabilities would coincide with the real probabilities only if all decision makers truly were risk–neutral; if not so, moving from the synthetic world

48

4 Valuation Model for Portfolios of Real Options

to the real world changes both expected returns on stock prices and risk– adjusted discount rates, in such a way that these changes cancel out each other and do not affect the result (cf. Hull 2003, p. 245). Risk–neutral valuation assumes arbitrage–free, frictionless, complete markets (Black and Scholes 1973; Luenberger 1998). Especially, frictionless markets make no restrictions on short sales. This assumption will be discussed in more depth in the context of capital constraints. 4.1.5 Stochastic Processes The proposed model is formulated as a stochastic program where the underlying assets follow stochastic diffusion processes. At its core, the problem formulation is based on a time– and state–discrete binomial lattice tree approach. The basic idea is that in each period it is assumed that the value of an underlying asset, θ, moves either up with probability p, or down with probability q = 1 − p. The most widely used approach is the CRR model, proposed by Cox et al. (1979), which is followed in the proposed model (see Sec. 4.2.2). 4.1.6 Volatility and Correlation The volatility of the value of the underlying assets, σi , is a key valuation parameter for any option and can be constant, i.e., stationary, or changing between periods. The present model will only consider the values of underlying assets as sources of uncertainty and therefore always assumes stationarity in the volatilities. Considering more than one underlying asset means that it is generally not sufficient to model the stochastic movement of each underlying asset separately; the joint movement of the prices of the underlying assets must also be taken into account (cf., e.g., Stulz 1982). However, by the flexible structure of our model we permit investment switches between alternative processes. In particular, this captures the case that the firm changes its risk exposure through investments. We model the values of the underlying assets with their respective volatilities and correlations as explicit sources of uncertainty. The values of the underlying assets are transformed into a cash flow measure, contingent on the scenario and operating mode, through a company specific cash flow function. On the one hand, this implies that the correlation of the underlying assets and the correlation of the cash flows need to be distinguished because they do not necessarily coincide. On the other hand, both are well–specified functions to our model. The former is determined directly through specifying the coefficient of correlation, ρij ,2 ; the latter is determined through 2

The joint movement of two random variables, X and Y, with finite variance can cov( X,Y )

be expressed by the coefficient of correlation, ρ XY = σ( X )σ(Y ) , with −1 ≤ ρ XY ≤ 1

4.1 Model Features

49

specifying the cash flow function. Therefore, all relevant information can be gained through ρij . Further, the model allows for any desired functional form of ci (θi (t, s), ai0 ), which defines the transformation of the correlation of values of the underlying assets into the correlation of the cash flows. An intuitive cash flow function, in analogy to Kulatilaka (1988), is of the form ci (θ, a0 ) = f a0 + v a0 θ,

with f e R, v e R > 0.

(4.1)

It bears the intuition with f negative, that f represents fixed costs, while v is the higher the lower the variable costs. For this class of affine functions, it is obvious that the correlation is unaffected by the transformation, such that the correlation of the underlyings and the correlation of the cash flows coincide. Therefore, without loss of generality in the following a cash flow function of the form c(θ, a0 ) = v a0 θ is used, and discussed in the numerical analysis. Extensions to other types of cash flow functions are straightforward, e.g., different cash flow functions for different states or paths. This would imply that cash flows can have a correlation different from the correlation of the underlying assets, especially, the sign of the correlation can be reversed. Nevertheless, such cash flow functions are still deterministic, hence all information is contained in ρij and correlation effects are sufficiently captured by discussing ρij alone. 4.1.7 Inter– and Intra–Project Options Interactions The analogy between financial and real options is greatest when modeling a simple, stand alone investment decision as a European call option. Yet when considering multiple real options on the same underlying asset, the valuation has to take into account the specific nature of real options. It has been shown by Trigeorgis (1993a) and Kulatilaka (1995) that real options written on the same underlying asset usually interact in the sense that value additivity does not hold. A valuation of these interacting real options must be carried out by valuing the bundle of real options and underlying asset as a whole, which is similar in structure to valuing a compound option. Intuitively speaking, this is because options on the same real assets are linked through this asset. By exercising any option the underlying asset is and as a special case ρ XY = 0 if X and Y are independent. For any variance– covariance matrix to be consistent, it must be positive semidefinite (cf. Jacod and Protter 2000, pp. 87 f.), which also extends to consistency conditions for coefficients of correlation (cf. Hull 2003, pp. 387 f.). Only in the special case where the underlying assets evolve independently from one another, the joint movement is not of special interest because it is trivial: given stochastic independence, i.e., ρ XY = 0, Prob( X ∧ Y ) = Prob( XA) × Prob(Y ).

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4 Valuation Model for Portfolios of Real Options

affected and with it, all other options tied to it. An extreme case is the option to abandon: by exercising it, all subsequent options are foregone. On the one hand, the put exercise decision has to take the existence of subsequent options into account because these have a strictly positive value (Merton 1973b) which is forfeited. On the other hand, the value of subsequent options is affected by the possibility that the put may be exercised. This interaction requires a valuation algorithm which is different from plain–vanilla financial option pricing (Trigeorgis 1991, 1993a; Brosch 2001). Trigeorgis (1993a) provides numerical analysis of options interactions, proving that values of real options are generally not additive and typically interactions are negative. Further, he derives that the nature of interactions generally depends on type, separation, degree of being in the money, and order of real options. This analysis finally shows that some usual options properties are preserved, e.g., sensitivity to time to maturity or volatility. Evidently, non–additivity is amplified in the case of multiple underlying assets. This holds true for financial options as well as real options. While there is a large body of literature on “rainbow” options (cf., e.g., Johnson 1987; Stulz 1982; Carr 1988; Boyle and Tse 1990), inter–project compoundness for real options has been analyzed little (cf., e.g., Childs and Triantis 1999; Lint and Pennings 2002). The model which is developed here will include both inter– and intra– project compoundness with multiple interacting real options written on multiple underlying assets. This requires valuing the portfolio of real options and underlying assets as a whole, in a compound option valuation logic. 4.1.8 Global Dynamic Budgets On the one hand, direct qualitative interactions are inseparably connected to the investment plan (Betge 1995). On the other hand, interactions or constraints may derive from indirect qualitative interactions (see Sec. 3.2.3). While any constraint is possible, e.g., a bottleneck in skilled engineers, we consider capital constraints as a representative indirect qualitative constraint for two reasons: First, for its big practical relevance that has only recently attracted attention (cf., e.g., Boyle and Guthrie 2003). Second, for being the most challenging constraint in that other constraints can be modeled as a special case. Financial option valuation models usually assume no capital scarcity: if an investment has a positive NPV, it is always possible to borrow money to finance this investment. In corporate finance, it is not always given that capital is unconstrained; this may be due to restricted access to capital markets or due to deliberately imposed rules and procedures. A compelling exam-

4.1 Model Features

51

β(t,s)

B(t-1,s-,a)

V(t,s,a)

B(t,s,a')

I(t,a,a') Fig. 4.1. Dynamic budgets depending on time, scenarios, and actions

ple for limited access to capital markets are startup companies which are typically facing significant financing challenges. Capital constraints in the general case introduce an additional dynamic aspect into real options portfolio management. Money that is not spent in one period will be available in the next period. This is depicted in Fig. 4.1, for one underlying asset. First, B(t − 1, s− , a) is the budget that is remaining from period t − 1 and thus passed on to period t. It depends on the switching decision at t − 1 and s− , a, which coming from period t − 1 is the entering mode for period t. s− is the predecessor of s, as defined by the binomial lattice. β(t, s) is the additional budget allowance that adds to the funds available at t; it is exogenous and depends on time and state. Moving forward in time by one period, the entering mode at time t is the leaving mode from t − 1, such that at t switching takes place from mode a to mode a0 . Therefore, B(t − 1, s− , a) + β(t, s) is the amount of funds available at time t before investment decisions at time t. The switch at time t realizes value V (t, s, a), incurring switching costs I (t, a, a0 ). For the budget level this implies that funds are increased by money freed up through disinvestment decisions and decreased by money spent for investment decisions. I (t, a, a0 ) depends on time and decisions, but is independent from s. These together determine the amount B(t, s, a0 ) that is remaining after the switching decision at t. This establishes the budget balance equation as follows: B(t, s, a0 ) = B(t − 1, s− , a) + β(t, s) − I (t, a, a0 ) where t s, s− a, a0 β(t, s) I

time indicator, t = 1, . . . , T, Scenario indicator, with s− = predecessor of s, Operating mode, a = 1, . . . , A and a0 = 1, . . . , A0 , New funds available at time t in scenario s, Investment or switching costs.

(4.2)

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4 Valuation Model for Portfolios of Real Options

This capital constraint is global in the sense that all investments compete for the same overall limited resources. It is dynamic in that money not invested today is available tomorrow and money freed up by disinvestments can be reinvested elsewhere. It is naturally path–dependent because it is a function of discretionary investment decisions along the path there, captured through state variables. This self–financing aspect yet has to be distinguished from the self–financing strategy in derivatives pricing, used for assembling a (dynamic) replicating portfolio (cf. Merton 1993, pp. 528 ff.), or hedging a given portfolio (cf. McDonald 2003, pp. 407 ff.). Constructing a replicating portfolio is a pricing technique, mimicking a known payoff structure through contracts with known prices. As opposed to this, we consider self–financing strategies in order to derive an optimal allocation of a limited amount of financial resources, i.e., the payoff structure is not known but obtained through our model in the first place. By bounding the budget variable, the nature of the budget constraint is set. If B(t, s, a0 ) > 0, a minimum liquidity level is required; if B(t, s, a0 ) ≥ 0, no borrowing is permitted; if B(t, s, a0 ) is bounded below by a negative real number, borrowing up to a defined level is permitted; if it is unbounded, capital availability is not restricted. 4.1.9 Synergies and Operational Constraints Beyond the correlation among cash flows, an investment into real assets may be related to direct qualitative interactions (see 2.1.2), involving physical or technical properties of the underlying assets. These interactions may come in different forms: partial if there are positive or negative synergies in operating projects simultaneously; binary if projects are mutual exclusive or depend on one another. Further, there may be other types of qualitative or logical interactions such as precedence constraints (Weingartner 1966a, 1977; Nemhauser and Ullmann 1969). The model can incorporate any type of qualitative interactions, including synergies. 4.1.10 Learning Closely related are learning effects. Learning goes beyond purely statistical relationships between underlying assets. Sometimes learning may be a side effect of option exercise, occasionally it may be the main goal of option exercise, e.g., a patent race in R&D. The presented model incorporates learning effects by considering staged investments, contingent cost structures, and path–dependencies.3 3

Papers that focus specially on learning effects include Koussis et al. (2003) and Smith and Thompson (2005).

4.1 Model Features

53

4.1.11 Diversification as a Passive Approach to Risk Management In financial portfolio management, diversification is key. Some authors even go further by claiming: “Portfolio theory is the analysis of the real world phenomenon diversification.” (Constantinides and Malliaris 1995). Since the seminal paper by Markowitz (1952), diversification has become a cornerstone of finance, including the CAPM. The argument of diversification is compelling in the context of financial markets where non–systematic risk can be avoided just by trading financial assets; for this reason, it can be assumed that any rational investor will be diversified. While diversification may be desirable in some specific contexts, overall portfolio risk minimization is not a legitimate objective for flexible investment decisions by companies (cf. Brealey and Myers 1996, p. 166). On arbitrage–free markets, companies must take risks in order to earn a return above the risk–free rate. The ultimate investor who will be diversified can then balance her portfolio as a function of her personal risk preferences and adjust the portfolio continuously. What is more, the perspective on risk, as measured by volatility and correlation, is different in a real options portfolio approach. Here a much more active approach towards risk is needed because options make it possible to capitalize on the upside and truncate the downside, depending on the nature of options. By designing an optimal portfolio of real options, risk is managed actively by an optimal sequence of option exercise decisions. If risk were minimized, this would typically adversely effect option values. Therefore, diversification is not considered as an objective in the management of portfolios of real options. To the contrary, the cardinality of the set of real options and stochastic processes, respectively, provides diversified inputs to the investment problem. For example, optimal management of a global manufacturing network through switching between sites requires some diversification between sites, e.g., sites in different countries with different exchange rates. With this notion of diversification as an input, active management of a portfolio of real options is achieved by optimal investment strategies in the given structure. When the structure of possible portfolio configurations is to be determined, diversification is relevant, but not at all with the goal of reducing risk, but instead in view of providing the optimal risk structure for future real options portfolio value (see Sec. 2.2.2). 4.1.12 Path–Dependencies One key feature of the presented model is that it considers path–dependencies. Path–dependencies arise for three reasons. First, a decision made in one period determines the basis for decisions made in the following periods. For a stand-alone analysis, this does not have any consequence because once

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4 Valuation Model for Portfolios of Real Options

an option is exercised, the decision problem ends. In our setting, switches back and forth are possible, which means that it models a compound option problem: With every switch made, new future switches are possible, contingent on the path of earlier switches. Given this logical connection between adjacent decisions, for any current decision the sequence of earlier decisions is relevant. Second, the dynamic global budget introduces an additional source of path–dependencies because on each path, the funds available are a function of all decisions made up to this point, which depend on the realizations of the underlying assets. In other words, depending on the path taken and the corresponding optimal investment decisions on that path, the availability of funds for further investment decisions is determined. Third, path–dependencies result from the interaction between underlying assets, which is closely linked to their joint stochastic movements. Especially, this source of interaction is closely connected to the budget constraint. If assets compete for shared funds, investment into each asset depends on investments on the other assets and the resulting funds available. Due to these sources of path–dependencies, the model is path–dependent no matter whether the underlying assets themselves are path-dependent or not. In the proposed model, path–dependent underlying assets can be integrated in a straightforward way, but would not change the portfolio problem. The system state space already models a path–dependent problem with a suitable tracking of the system state space, such that path–dependent stochastic processes of the underlying assets are straightforward to include in that structure if desired. It is noted that the distinction of different sources of path–dependencies is only helpful for the conceptual understanding of the path–dependent nature of the problem. From a modeling standpoint, it is not possible to separate the effects of different sources of path–dependencies because all sources act simultaneously and jointly. 4.1.13 Modeling Approach and Solution The general real options portfolio problem with budget constraints and path–dependency is formulated as a stochastic mixed–integer program using risk–neutral probabilities. The program recurs to mixed–integer decision variables to take account of option exercise decisions. Especially, it resides to binary variables which turn one if and only if an option is exercised. This makes the problem non-linear because the binary variables “activate” the operating modes which determine the value generated in these modes, in the current and all subsequent periods. This naturally introduces an asymmetry in switching choices and hence nonlinearity. Further, it is stochastic in

4.1 Model Features

55

modeling the stochastic processes of the underlying assets and their interdependencies. These interdependencies dynamically capture the interaction effects of decisions over time. The program formalizes a simultaneous forward– and backward–looking procedure. On the one hand, a backward–looking element consists of the calculation of expected values in a rolling-back fashion which is well know from standard option pricing theory, by a backward recursion. On the other hand, a forward recursion is required to cope with path–dependencies. This can best be expressed by the budget balance equation: money left over from one period can be invested in the following period, for the states following that specific path. What is special here is that both the backward– and the forward–looking element have to be integrated simultaneously. This can be achieved by formulating the problem as a simultaneous program. This is made possible through an appropriate modeling of the binary variables which combine both the backward– and the forward–looking element via an appropriate mapping of the system state space. Given the complexity of the problem, it is evident that a solution cannot be obtained closed–form. At the same time, commercially available solvers have been developed for decades and today provide a heavily researched and powerful arsenal for solving problems with high dimensionality. Numerically solving the problem proceeds in a two-staged process: 1. The general formulation is given in its “implicit” form where all equations are indexed over time, scenarios, and actions. This implicit form needs to be expanded to its “explicit” form where for each combination of indices just one equation is generated. This is implemented using a modeling language, such as the “General Algebraic Modeling System” (GAMS© ), which we will use here. Both versions, implicit and explicit, are equivalent. Intuitively speaking, while the implicit version gives an aggregated view of the problem, the explicit version expands the problem and hence builds up the landscape in which the optimization takes places. 2. The explicit problem is then passed on to a suitable solver. Given nonlinearity, it is important to ensure that an optimum is not only local but global. This can be ascertained by using a global solver, such as BARON© which offers the advantage of being able to prove global optimality of the solution (Tawarmalani and Sahinidis 2002a, b). For our problem, we have reformulated the numerical implementation as an equivalent linear problem (see Sec. 4.5.2.1), such that it is possible to reside to linear solvers for mixed-integer programs, such as CPLEX© . This offers the advantage of much faster solutions and avoids the problem of local optima (for a discussion, see Sec. 4.5.2.1).

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4 Valuation Model for Portfolios of Real Options

4.1.14 Scalability Theoretically, there is no bound on model size, since the model can readily be formulated for any number of underlying assets, real options, and constraints. Numerically, the solution depends on computing power. Given the exponential growth of the problem size, practically speaking, the size of solvable decision problems cannot easily be generalized. After presenting the general model formulation, routes for making the solution more efficient will be indicated.

4.2 Binomial Option Pricing So far, we have presented the features of the model in an overview, aiming to introduce the nature of the presented model. With this background, the stochastic modeling and the valuation approach of the model will be presented in what follows. A core element of the model is binomial option pricing, which will be developed step by step. The basic binomial option pricing is presented, which is then expanded to two underlying assets. 4.2.1 Stochastic Processes and System State Space The stochastic processes of the underlying assets are assumed as discrete binomial processes. First, the CRR model is presented due to its widespread use. It offers many advantages, especially that the discretization of the system state space is very intuitive. Also, the distribution of the stock price converges to the log–normal distribution in the limit, when ∆t → 0 (Kwok 1998, pp. 199 ff.). This is standard for financial options pricing where log– normal processes and specifically geometric Brownian motions are widely used to model stock prices (Duffie 2001, p. 88). Moreover, in a binomial lattice tree the central node at time 2∆t is the same as in the initial node because d = u1 and hence u × d = 1. This is particularly convenient in the calculation of “options Greeks” (Hull 2003, p. 407). However, this comes at the disadvantage that the probabilities may turn negative when time steps are not sufficiently small, which also is the case in the extension to many underlying assets (Boyle 1988; Boyle et al. 1989). In the presented model, it is neither important to converge to a log– normal distribution of the underlying assets, nor to calculate options Greeks. Therefore, it is not necessary to consider small time steps and it is helpful to derive risk–neutral probabilities in a way that they cannot turn negative. The most prominent alternative binomial approach that ensures non– negativity is provided by Jarrow and Rudd (1983).

4.2 Binomial Option Pricing

57

The binomial model is most known in its standard version which can be represented by binomial lattice trees. One dimension represents time (usually the horizontal dimension), the other dimension represents the up and down movements of the underlying asset. The binomial model can readily be extended to multiple underlying assets, with one additional dimension for each additional asset. For example, a binomial lattice tree for two underlying assets has three dimensions: one time dimension and two asset dimensions. This three–dimensional character can be represented graphically by a pyramid. To simplify exposition and tracking, we adopt a few conventions. All variables are indexed first by time t (t = 1, . . . , T ), then by scenario s (st = 1, . . . , S(t)), and then – if necessary – by actions. The combinations of all relevant state indicators are defined as the “system state space”. Scenarios are counted from high to low scenarios. For example, the option value in the one underlying case in period t=2 for scenario “up” is V (2, 1) and for scenario “down” V (2, 2) (see Fig. 4.2). By “scenario” we mean chance realizations through the lattice, at each point in time. This distinguishes a scenario from the more common term “state” which in our convention contains all relevant information at a given point in time. As opposed to this, depending on their focus of attention some authors find it more convenient to call a path through the tree a scenario (e.g., cf. Høyland and Wallace 2001). 4.2.2 Binomial Option Pricing Model with One Underlying Asset The reference model for our model development is the Cox–Ross–Rubinstein (CRR) version of the binomial model (cf. for the derivation of the CRR model Cox et al. 1979). It is introduced here in the notation needed for the following development of the model, which differs from the usual representation. In that sense, the model is presented here mainly in order to introduce the notation while the content of the CRR model is unchanged. The binomial model assumes that the value of the underlying asset, θ, follows a stationary time–discrete multiplicative binomial process, with up factor u and down factor d. Up und down are specified as a function of the volatility of the underlying asset, σ, such that in the limit the process converges to a log–normal distribution of the return of the underlying asset, i.e.: √ 1 (4.3) u = eσ ∆t , and d = . u These up and down factors determine the stochastic process of the value of the underlying asset, θ. Due to the recombining nature of the tree, the system state space grows in t (t = 1, . . . , T ) and s (st = 1, . . . , t) as follows:

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4 Valuation Model for Portfolios of Real Options

Fig. 4.2. Recombining binomial lattice tree for one underlying asset

θ (t, s) = θ (1, 1) × ut−s ds−1 .

(4.4)

The development of θ (t, s) can be represented by a binomial lattice tree where scenarios are indexed from the up scenario to the down scenario, as given in Fig. 4.2). In this framework, it is possible to assign risk–neutral probabilities p and 1 − p such that no arbitrage opportunities exist. In each period, the up scenario occurs with probability p and the down scenario with probability 1 − p where r f is the continuously compounded risk–free rate of return: p=

er f ∆t − d . u−d

(4.5)

This provides the basis for valuing an option on the underlying asset, which can be exemplified for a European option as follows. First, the boundary condition on option exercise at t = T is imposed. Supposing for example a European call option with exercise price X, we have the terminal value condition: (4.6) V ( T, s) = max[θ ( T, s) − X, 0], s = 1, . . . , T. Valuation in the binomial model proceeds in a backward recursion, as a discounted probability–weighted expected value: V (t − 1, s) =

V (t, s) × p + V (t, s + 1) × (1 − p) er f ∆t t = 2, . . . , T ; s = 1, . . . , t.

(4.7)

The value of the option corresponds to V (1, 1). 4.2.3 Binomial Option Pricing Model with Two Underlying Assets The presented model so far was only able to cope with one underlying asset. In what follows, the case of two underlying assets is considered.

4.2 Binomial Option Pricing

59

Fig. 4.3. Recombining binomial lattice tree for two underlying assets, with ρ12 6= 0

First assume a coefficient of correlation of zero, i.e., ρ12 = 0. Looking at two underlying assets and their respective values, θ1 (t, s) and θ2 (t, s), a tree for each underlying asset can be built up just as before, with σ1 , u1 , d1 as well as σ2 , u2 , d2 . With recombiningness preserved, the system state space for each underlying asset grows in t = 1, . . . , T and s = 1, . . . , t, so that overall it grows in 2t: θ1 (t, s) = θ1 (1, 1) × u1t−s d1s−1 ,

(4.8)

θ2 (1, 1) × u2t−s d2s−1 .

(4.9)

θ2 (t, s) =

Second, assume ρ12 6= 0. In this case, the two separate trees must be considered simultaneously. Here it makes a difference whether asset 1 moves up and asset 2 moves down, as opposed to asset 1 moves down and asset 2 moves up. This generates a binomial lattice tree which can be represented graphically by a pyramid (see Fig. 4.3). For valuation purposes, the backward recursion can proceed stepwise through the pyramid, in a way identical to the approach for two dimensions. The only difference is that each node has four successors, with respective risk–neutral probabilities. Obviously the system state space grows faster than in the one asset case. It is given by t = 1, . . . , T and s = 1, . . . , t2 . The recombining nature of the tree is preserved in the sense that, e.g., for asset 1 it does still not matter whether it first moves up and then down, or the other way round, so that ud = du. However, scenarios grow with order t2 instead of order 2t.

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The effect of joint movement of the underlying assets can be taken into account by corresponding risk–neutral probabilities. Following Boyle et al. (1989), these probabilities can be specified as follows (see Fig. 4.4): " !# √ r f − 12 σ22 r f − 12 σ12 1 + 1 + ρ + ∆t , p1 = 4 σ1 σ2 for asset 1=up, asset 2=up, " √ 1 p2 = 1 − ρ + ∆t 4

(4.10)

r f − 12 σ22 r f − 12 σ12 − σ1 σ2

!# ,

for asset 1=up, asset 2=down,

(4.11)

" !# √ r f − 12 σ12 r f − 12 σ22 1 p3 = + 1 − ρ + ∆t − , 4 σ1 σ2 for asset 1=down, asset 2=up,

(4.12)

" !# √ r f − 12 σ12 r f − 12 σ22 1 p4 = − 1 + ρ + ∆t − , 4 σ1 σ2 for asset 1=down, asset 2=down.

(4.13)

This derivation of the risk–neutral probabilities has been generalized to n underlying assets by Boyle et al. (1989) and can therefore be used to capture the joint movements of multiple underlying assets in a risk–neutral valuation framework. As mentioned before, a numerical disadvantage of this approach, just as in the CRR two–dimensional case, √ is that the probabilities may turn negative when ∆t is large, i.e., σ ≤| r ∆t | (cf. Hull 2003, p. 407). This can only be avoided by increasing the number of time steps at the expense of increased computing time. With the help of this lattice tree, it is possible to value an option in the above introduced backward recursion fashion, by stochastic dynamic programming. In order to stay within the above introduced notation (see Fig. 4.2), we preserve the intuition of two combined binomial trees by inp3

d1;u2

u1;u2

p1

p4

d1;d2

u1;d2

p2

Fig. 4.4. Three–dimensional risk–neutral probabilities

4.3 Real Options Portfolio Model

61

dexing the scenarios separately, i.e., s1 for asset 1 and s2 for and asset 2. For a European option on the maximum of 2 underlying assets, this gives: V ( T, (s1 , s2 )) = max[θ1 ( T, s1 ) − X, θ2 ( T, s2 ) − X, 0] s1 = 1, . . . , T ; s2 = 1, . . . , T.

(4.14)

Valuation in the binomial model proceeds in a backward recursion, as a discounted probability–weighted expected value: n V (t − 1, (s1 , s2 )) = e−r f ∆t × V (t, (s1 , s2 )) × p1 + V (t, (s1 , s2 + 1)) × p2 o + V (t, (s1 + 1, s2 )) × p3 + V (t, (s1 + 1, s2 + 1)) × p4 t = 2, . . . , T ; s1 , s2 = 1, . . . , t.

(4.15)

The value of the European option on the maximum of two assets corresponds to V (1, (1, 1)). While only the model for 2 underlying assets can conveniently be represented graphically, the approach can be generalized in a straightforward manner to more than two underlying assets, e.g., as specified by Boyle et al. (1989), or in a log–transformed version that offers computational advantages, by Gamba and Trigeorgis (2001).

4.3 Real Options Portfolio Model Up to this point, basic concepts of option pricing within the CRR binomial model have been discussed. The discussion has not primarily focused on real options as such, but has been applicable to both financial and real option pricing. These basic concepts set the stage for the following development of a general model for valuing portfolios of real options, subject to constraints. 4.3.1 Switching Between Modes The core of the proposed model is a switching algorithm, based on a general framework of switching among nodes described in Kulatilaka (1988) and Kulatilaka and Trigeorgis (1994). At any time, the firm can either continue operating in the current mode for one more period, receiving a short–term payoff (i.e., current cash flow) plus the long–term option value from optimal future switching, or switch immediately to a new mode by incurring specified switching costs. For each tuple (t, s) and each underlying asset i, an entering mode ai (t, s) = 1, . . . , Ai (t, s) ∀(t, s) and a leaving mode ai0 (t, s) = 1, . . . , Ai0 (t, s) ∀(t, s) are distinguished. Switching takes place from mode ai (t, s) to mode

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ai0 (t, s) where staying in the same mode is represented by ai = ai0 , if permitted. Without loss of generality, we assume that ai (t, s) = ai ∀(t, s) and ai0 (t, s) = ai0 ∀(t, s), i.e., the set of permitted operating modes is not state– dependent. Considering one underlying asset, switches from entering mode a may involve: • a0 = 1: waiting to invest, • a0 = 2: operate with low capacity, • a0 = 3: operate with high capacity. Associated with each switch is a cash flow realization, c(θ (t, s), a0 ). In each period, depending on the leaving operating mode a0 , a cash flow is realized. This cash flow can be any function of the value of the underlying asset, θ (t, s). Without loss of generality, the cash flow function can be represented by a positive scalar, drawing on an analogy to levels of capacity: • • • •

c(θ (t, s), a0 ) = 0: not invested, 0 < c(θ (t, s), a0 ) < 1: invested with low capacity, c(θ (t, s), a0 ) = 1: invested with base–scale capacity, c(θ (t, s), a0 ) > 1: invested with expanded capacity.

Switching between any mode a and a0 involves a time dependent switching costs, I (t, a, a0 ), ∀ a, a0 . The switching costs represent investment outlays, or in options terms, exercise prices. Without loss of generality, we can assume that staying in the same operating mode is costless (the respective costs are already taken into account in the cash flow). Examples of basic switching costs are as follows: • I (t, a, a0 ) = ∞: no switching possible,4 • I (t, a, a0 ) = 0 : staying in the same mode (or, exceptionally, costless switching), • I (t, a, a0 ) > 0 : call option, i.e., investment (utilization of resources), • I (t, a, a0 ) < 0 : put option, i.e., disinvestment (freeing up of resources). The switching algorithm is very flexible. It can handle any structure of time–dependent symmetric or asymmetric switching costs, any cash flow payoff function and any options combinations in terms of type, sequence and option maturity. The switching algorithm is designed to model a bundle of real options that can be interpreted as a joint complex, compound option (Trigeorgis 1996, pp. 185 ff.). Since each switching possibility represents a possible real option exercise and the switching costs matrix introduces an asymmetry, the valuation cannot be achieved by valuing each real option separately and adding up the separate values. Deviations from value 4

In an implementation, this is incorporated by using a sufficiently large number for I (t, a, a0 ).

4.3 Real Options Portfolio Model

63

additivity make the valuation and interpretation of incremental option contributions challenging (Trigeorgis 1993a). 4.3.2 Switching Formulation for One Underlying Asset Following this switching perspective, we can formalize the valuation algorithm as a backward recursion in a stochastic dynamic programming fashion. Intuitively speaking, this program integrates a double iteration: First, it moves backward in time, applying the Bellman principle of optimality. Second, at each point in time, it iterates over all entering modes, each time choosing the optimal leaving mode. The value of the underlying asset, θ (t, s), follows a binomial process with probabilities p for an up movement and probability (1 − p) for a down movement, as introduced above for the two–dimensional CRR framework. Each decision is based on both the current cash flow per period which is realized after the switch, as a function of θ (t, s) and a0 , as well as the expected value of future net cash flows, starting from the derived leaving mode. This can be incorporated into the following stochastic dynamic program based on backward recursion, which is in analogy with Trigeorgis (1996, p. 185): V ( T, s, a) = max c(θ ( T, s), a0 ) − I ( T, a, a0 ) a0

s = 1, . . . , T ; a = 1, . . . , A V (t − 1, s, a) = max c(θ (t − 1, s), a0 ) − I (t − 1, a, a0 )

(4.16)

a0

+

where t s a a0 V c(·, a0 ) θ I p rf

V (t, s, a0 ) × p + V (t, s + 1, a0 ) × (1 − p) er f ∆t t = 2, . . . , T ; s = 1, . . . , t ; a = 1, . . . , A.

(4.17)

Time indicator, t = 1, . . . , T, Scenario indicator, s = 1, . . . , t, Entering operating mode, a = 1, . . . , A, Leaving operating mode, a0 = 1, . . . , A0 , Value of portfolio, Cash flow function for operation in mode a0 , a0 = 1, . . . , A0 , Stochastic value of underlying asset, Investment or switching costs, Risk–neutral probability for up–jump, Risk–free rate of return.

At each node of the system state space, for each entering mode, the stochastic dynamic program determines the optimal operating mode. This decision

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takes into account the switching costs contingent on the entering mode, the payoff in that specific period contingent on switching costs, as well as the expected future value given the leaving mode. This expected value represents the value of all future switching decisions with the leaving mode as the new starting point. Overall, the logic sequence is as follows: i) observe the realization of the underlying stochastic variable; ii) switch optimally; and iii) realize the cash flow in this period for the leaving mode. The leaving mode will then be the entering mode for the following period. This stochastic dynamic programming formulation takes into account any structure of switching costs, including the non–symmetric case. Therefore in some nodes, it may be optimal to stay in the same mode in order to prevent a future costly switching back to the earlier mode. This introduces a hysteresis effect: even though immediate switching may appear profitable from a short–term perspective, for dynamic long–term considerations it can be optimal to remain in the original mode (Dixit and Pindyck 1994; Trigeorgis 1996). 4.3.3 Switching Formulation for Two Underlying Assets The above introduced switching model can be extended towards two underlying assets in a straightforward manner. In this case, there are two underlying assets with their respective values, θ1 (t, s1 ) and θ2 (t, s2 ); four risk–neutral probabilities p1 , p2 , p3 , p4 ; entering modes for each underlying asset, a1 and a2 ; leaving modes for each underlying asset, a10 and a20 ; and a switching costs matrix for each underlying asset, I1 (t, a1 , a10 ) and I2 (t, a2 , a20 ). For each underlying asset, a binomial lattice tree can be constructed as in Fig. 4.2. Both underlying assets have to be considered simultaneously, as has been shown for the example of the option on the maximum of two assets. The dynamic program for the switching model therefore becomes:

4.3 Real Options Portfolio Model

65

V ( T, (s1 , s2 ), ( a1 , a2 )) = max c1 (θ1 ( T, s1 ), a10 ) − I1 ( T, a1 , a10 ) a10

+ max c2 (θ2 ( T, s2 ), a20 ) − I2 ( T, a2 , a20 ) a20

s1 , s2 = 1, . . . , T ; (4.18) a1 = 1, . . . , A1 ; a2 = 1, . . . , A2 0 0 V (t − 1, (s1 , s2 ), ( a1 , a2 )) = max c1 (θ1 (t − 1, s1 ), a1 ) − I1 (t − 1, a1 , a1 ) a10

+ max c2 (θ2 (t − 1, s2 ), a20 ) − I2 (t − 1, a2 , a20 ) a20

+ + + +

−r f ∆t

× (V (t, (s1 , s2 ), ( a10 , a20 )) × p1 V (t, (s1 , s2 + 1), ( a10 , a20 )) × p2 V (t, (s1 + 1, s2 ), ( a10 , a20 )) × p3 V (t, (s1 + 1, s2 + 1), ( a10 , a20 )) × p4 ) ] , t = 2, . . . , T ; s1 , s2 = 1, . . . , t ; a1 = 1, . . . , A1 ; a2 = 1, . . . , A2 . e

(4.19)

Time indicator, t = 1, . . . , T, Scenario indicator for underlying asset i = 1, 2, si = 1, . . . , t, Entering operating mode for underlying asset i = 1, 2, ai ai = 1, . . . , Ai , Leaving operating mode for underlying asset i = 1, 2, ai0 ai0 = 1, . . . , Ai0 , V Value of portfolio, ci (·, ai0 ) Cash flow function for operating asset i = 1, 2 in mode ai0 , ai0 = 1, . . . , Ai0 , Stochastic value of underlying asset i = 1, 2, θi Investment or switching costs for underlying asset i = 1, 2, Ii p Risk–neutral probability, Risk–free rate of return. rf

where t si

4.3.4 Stochastic Mixed–Integer Program Formulation The above derived switching model is a stochastic dynamic programming model that derives option values iteratively. One key characteristic is that it does not consider capital scarcity and that it can be based on a recombining tree structure which allows to value portfolios of real options in a straight backward recursion. With a dynamic budget, this logic cannot be applied. The tree of an underlying asset is no longer recombining, because the available budget depends on the realized path and the switching decisions along this path. For

66

4 Valuation Model for Portfolios of Real Options

Fig. 4.5. Non–recombining binomial lattice tree for one underlying asset

a two–dimensional, non–recombining tree, the scenarios grow to s = 2(t−1) instead of s = t. Likewise, for a three–dimensional, non–recombining tree, the scenarios grow exponentially to s = 22(t−1) , instead of s = t2 . This additional source of dynamics is hard to integrate into a recursion in a dynamic programming fashion. Alternatively, this can be modeled as a stochastic mixed–integer program for a non–recombining lattice tree. This way, the structure of the decision problem is translated into a set of constraints that must be satisfied simultaneously in the optimal solution. In the following, we will present such a model formulation in successive steps. First, we will present the necessary building blocks with the help of a problem with one underlying asset. Second, we develop the changes which are necessary to take into account an additional underlying asset, i.e., two underlying assets in total. Finally, this is generalized to a model formulation for n underlying assets. 4.3.4.1 One Underlying Asset First, the model formulation for one underlying asset is presented. Taking into account budget dynamics, the problem becomes path–dependent, no matter whether the underlying process itself is recombining. Therefore, the scenarios of the stochastic process have to account for non– recombiningness. This is represented graphically in Fig. 4.5, following the indexing logic of scenarios, from up to down scenarios as introduced before. In addition to time and scenario, the system state space integrates the entering operating mode, a, and leaving operating mode, a0 . Decisions are modeled by a binary switching variable, R, which is state dependent, hence R(t, s, a, a0 ). The objective function maximizes the overall portfolio value, V (1, 1, 1), over all decision switching variables, R(1,1,1,a’). The program derives the

4.3 Real Options Portfolio Model

67

maximum expected portfolio value at t = 1, and the corresponding optimal dynamic strategy (Eq. 4.20). The constraints are described in blocks. First, the dynamic movements of the underlying asset are modeled by its starting value, α, (Eq. 4.22) and its development across time (Eq. 4.22), spanning the underlying binomial lattice tree.5 The modulo function, [mod 2], serves to model the non– recombining structure of the binomial lattice and ensures the correct linkage between parent node to child node. This can be achieved by drawing from our design that we count states from up to down states and that the non– recombining lattice tree grows with order t2 . Based on this, the property of the modulo function can be implied, because the remainder on integer division of states by the basis 2 is such that it yields either a 1 for the up state successor node or a 0 for the down state successor node. This feature can be exploited for adding over the successor nodes in Eq. 4.22, because exactly one factor is 1 while the other is 0. This formalizes the linkage visualized in Fig. 4.5. Next, the core of the set of constraints is provided by the switching value function, V (t, s, a), for intermediate periods in Eq. 4.23 and for the terminal period in Eq. 4.24. Switching from mode a to mode a0 involves switching costs I (t, a, a0 ). Once a switch is made, cash flow c(θ (t, s), a0 ) is realized, given by the cash flow function c(·, a0 ) of the underlying asset. For all periods t except for the terminal period t = T, the leaving mode in a period t also serves as the entering mode for the following period t + 1, allowing to determine an expected value for the leaving mode in all subsequent periods. This expected value implicitly takes into account optimal switching in all subsequent periods. In this sense, switching incorporates a backward recursive element. Optimal switching is determined by the binary variable R(t, s, a, a0 ). As ensured by Eq. 4.25, the system is always exactly in one (and only one) operating mode, i.e., it is not possible to make several successive switches in a single period.6 Capital rationing is accommodated through the global dynamic budget constraint, B(t, s, a0 ), given by Eq. 4.26 for the first period (t = 1) and Eq. 4.27 for all subsequent periods. B(t, s, a0 ) represents the budget remaining after a switching decision. The first period is special in that it specifies 5

6

In financial option pricing applications, the starting value of the underlying asset typically is represented by the variable S0 , or S(0). Since our underlying variables are not “stock”, we prefer to use α instead. The fact that there is always exactly one switch is only technical. In a specific decision problem it may be possible to realize successive switches immediately. This can be taken into account by introducing a separate mode for the decision to realize immediately successive switches involving several modes, making this decision alternative technically a single, additional mode.

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4 Valuation Model for Portfolios of Real Options

the initial funds available, B1 , with B1 e R ≥ 0. In subsequent periods, this is replaced by the funds remaining from the prior period. Moreover, there is an allocation of a new budget per period, β(t, s), with β(t, s) e R ≥ 0. The above budget constraint is dynamic in that it incorporates both investment and disinvestment decisions, the latter with I (t, a, a0 ) < 0. Money that is freed up by disinvesting in one period is available for investments in subsequent periods. It is noted that cash flows from operating the asset cannot be reinvested here. If no borrowing is permitted, we have B(t, s, a0 )e R ≥ 0; otherwise, borrowing can be partially relaxed by allowing a certain level of financing or can be fully relaxed by setting B(t, s, a0 )e R. One element of the formulation, which shows in the value function as well as in the budget constraint, merits special attention: the interconnection of decisions across periods, i.e., the compoundness of decisions. Each decision sets the stage for the subsequent decisions and determines which future decisions are possible. The formulation is such that the switch made in one period is the basis for the switch in the subsequent period. This ensures consistency over time and enforces non–anticipativity.7 This applies both forward and backward looking in time. Since the dynamic budget constraint is forward looking, consistency is ensured for B(t, s, a0 ) when moving from t − 1 to t since the leaving mode at t − 1 is set to be the entering mode at t. Conversely, the value function V (t, s, a) incorporates the backward– looking feature of the program, moving from t + 1 to t. Given a switching decision at t, the entering mode at t + 1 is fixed to be the leaving mode for t. Further, some comments on equations that are technically required are in place. In Eq. 4.23, R(t, s, a, a0 ) expresses the optimal switch in a way that exactly one binary variable turns 1. Also, R(t, s, a, a0 ) is linked to the following period by multiplying it by V (t + 1, ·). This linkage also ensures consistency, i.e., that the leaving mode from t is the entering mode at t + 1. However, this can only work when the term in brackets is different from zero. Otherwise, the optimization can set any (but exactly) one of the binary variables to 1 because it is multiplied with zero and thus does not impose any restriction on the binary variables because it has no impact on the optimum. This special case is obtained in states where it is optimal to remain uninvested. Consequently, this does not affect the optimal portfolio value because it has zero value impact, but it can falsify the vector of binary variables and consequently give meaningless investment strategies. To avoid this, Eq. 4.28, Eq. 4.29 and Eq. 4.30 are used to ensure that the switches are always consistent over time. The core of the consistency constraints is Eq. 4.29. It ensures 7

Non–anticipativity refers to the fact that each decision can only be made with the information available at this time. Through the implicit tree structure underlying the program, this non–anticipativity is satisfied (Birge and Louveaux 1997).

4.3 Real Options Portfolio Model

69

that the leaving mode for any binary variable that turns 1 is indeed the entering mode for both following scenarios. This formulation exploits that there are always two successors and that for each successor there is exactly one binary that turns 1, making it possible to add over binaries and divide by 2. Finally, Eq. 4.31 simply ensures the binary nature of the switching decision variable, R(t, s, a, a0 ). This modeling logic is combined in the model formulation for one underlying asset as follows: max

R(1,1,1,a0 )

V (1, 1, 1)

(4.20)

subject to θ (1, 1) = α

(4.21)

√ s+1 ) × eσ ∆t [s mod 2] 2 √ s + θ (t − 1, ) × e−σ ∆t [(s + 1) mod 2] 2 t = 2, . . . , T ; s = 1, . . . , 2(t−1)

θ (t, s) = θ (t − 1,

(4.22)

0

V (t, s, a) =

A X

R(t, s, a, a0 ) × c(θ (t, s), a0 ) − I (t, a, a0 )

a 0 =1

+

V (t + 1, 2s − 1, a0 ) × p + V (t + 1, 2s, a0 ) × (1 − p)

er f ∆t t = 1, . . . , T − 1 ; s = 1, . . . , 2(t−1) ; ∀ a

(4.23)

0

V ( T, s, a) =

A X

R( T, s, a, a0 ) × c(θ ( T, s), a0 ) − I ( T, a, a0 )

a 0 =1

s = 1, . . . , 2(T −1) ; ∀ a

(4.24)

0

A X A X a =1

R(t, s, a, a0 ) = 1

a 0 =1

∀t , s = 1, . . . , 2(t−1)

(4.25)

B(t, s, a0 ) = R(t, s, a, a0 ) × B1 + β(t, s) − I (t, a, a0 ) t = 1 ; s = 1 ; a = 1 ; ∀ a0

(4.26)

70

4 Valuation Model for Portfolios of Real Options B(t, s, a0 ) =

A X

n R(t, s, a, a0 ) × β(t, s) − I (t, a, a0 )

a =1

s+1 , a) × [s mod 2] 2 o s + B(t − 1, , a) × [(s + 1) mod 2] 2 t = 2, . . . , T ; s = 1, . . . , 2(t−1) ; ∀ a0

+ B(t − 1,

(4.27)

0

L(t, s, a) =

A X

R(t, s, a, a0 )

a 0 =1

t = 2, . . . , T ; s = 1, . . . , 2(t−1) ; ∀ a R(t, s, a, a0 ) =

×

(4.28)

0 1 2 × R ( t, s, a, a ) A0 X 1 X

L(t + 1, 2s − l, a0 )

a 0 =1 l =0

t = 2, . . . , T ; s = 1, . . . , 2(t−1) ; ∀ a ; ∀ a0

(4.29)

0

A X

R(t, s, a, a0 ) = 1

a 0 =1

t=1; s=1; a=1

(4.30)

R(t, s, a, a0 ) ∈ {0; 1}

∀t ; ∀s ; ∀ a ; ∀ a0 .

(4.31)

4.3 Real Options Portfolio Model

71

where t s a a0 R V θ α c(·, a0 )

Time indicator, Scenario indicator, Entering operating mode, a = 1, . . . , A, Leaving operating mode, a0 = 1, . . . , A0 Binary switching variable, Value of portfolio, Stochastic value of underlying asset, Starting value of underlying asset, Cash flow function for operating in mode a0 , a0 = 1, . . . , A0 , I Investment or switching costs, p Risk–neutral probability, Risk–free rate of return, rf B Budget level, β(t, s) New funds available at time t in scenario s, Initial budget available at time t = 1, B1 e R ≥ 0, B1 L, l Auxiliary variable and index.

4.3.4.2 Two Underlying Assets The formulation as a nonlinear stochastic program for multiple underlying assets can be achieved by combining the elements introduced for a multinomial valuation procedure, with a simultaneous switching approach similar to the one presented for one underlying asset. More explicitly, the approach consists of introducing a stochastic variable θi for the value of each underlying asset i. Spanning up the multinomial lattice tree can still be achieved by using the modulo function, with [mod 2n ], alongside a starting value for each underlying asset, αi . Entering and leaving modes for each underlying asset are distinguished, i.e., ai and ai0 , as well as cash flow functions for each underlying asset, ci (·, ai0 ). The risk–neutral probabilities are adapted to the number of underlying assets, taking into account their respective joint movements. Boyle et al. (1989) have provided a formulation of risk–neutral probabilities for n underlying assets. Since the global dynamic budget is shared by all investment decisions for all underlying assets, the budget constraint incorporates investment decisions from a global portfolio perspective where it may be desirable to free up money by disinvesting some underlying assets in order to invest in other more profitable underlying assets. In the case of i = 2 underlying assets, the binomial processes can still be represented by a pyramid, as depicted in Fig. 4.3. Yet, with the dynamic budget constraint, the pyramid becomes non-recombining since it does matter whether a node is reached from the “up” or the “down” path. This path dependent peculiarity can be translated into an indexing logic as depicted in Fig. 4.6, following the indexing logic of joint scenarios from the up to

72

4 Valuation Model for Portfolios of Real Options

the down states. Using this indexing logic, the model formulation for i = 2 underlying assets is as follows: s=3

θ(2,3) d1;u2

θ(2,1)

s=1

θ(2,2)

s=2

u1;u2 θ(1,1)

s=4

θ(2,4)

u1;d2

d1;d2

Fig. 4.6. Indexing in a three–dimensional lattice tree

In a straightforward manner, the formulation for two underlying assets is then given by: max

R(1,1,(1,1),( a10 ,a20 ))

V (1, 1, (1, 1))

(4.32)

subject to θ1 (1, 1) = α1

(4.33)

θ2 (1, 1) = α2

(4.34)

θ1 (t − 1, θ (t − 1, 1 θ1 (t, s) = θ 1 ( t − 1, θ1 (t − 1,

√ s +3 σ1 ∆t if 4 )×e √ s +2 σ1 ∆t ) × e if 4 √ s +1 − σ ∆t 1 if 4 )×e √ s −σ1 ∆t ) × e if 4 2( t −1)

[s mod 4] = 1, [s mod 4] = 2, [s mod 4] = 3, [s mod 4] = 0

t = 2, . . . , T ; s = 1, . . . , 2 θ2 (t − 1, θ (t − 1, 2 θ2 (t, s) = θ 2 ( t − 1, θ2 (t − 1,

√ s +3 σ2 ∆t if 4 )×e √ s +2 −σ2 ∆t if ) × e 4 √ s +1 σ2 ∆t if 4 )×e √ s −σ2 ∆t ) × e if 4 2( t −1)

t = 2, . . . , T ; s = 1, . . . , 2

(4.35)

[s mod 4] = 1, [s mod 4] = 2, [s mod 4] = 3, [s mod 4] = 0 (4.36)

4.3 Real Options Portfolio Model 0

V (t, s, ( a1 , a2 )) =

73

0

A1 A2 X X

R(t, s, ( a1 , a2 ), ( a10 , a20 )) × (c1 (θ1 (t, s), a10 )

a10 =1 a20 =1

− I1 (t, a1 , a10 )) + (c2 (θ2 (t, s), a20 ) − I2 (t, a2 , a20 )) h + e−r f ∆t × V (t + 1, 4s − 3, ( a10 , a20 )) × p1 + V (t + 1, 4s − 2, ( a10 , a20 )) × p2 + V (t + 1, 4s − 1, ( a10 , a20 )) × p3 i + V (t + 1, 4s, ( a10 , a20 )) × p4 t = 1, . . . , T − 1 ; s = 1, . . . , 22(t−1) ; ∀( a1 , a2 ) 0

V ( T, s, ( a1 , a2 )) =

(4.37)

0

A1 A2 X X

R( T, s, ( a1 , a2 ), ( a10 , a20 )) × (c1 (θ1 ( T, s), a10 )

a10 =1 a20 =1

− I1 ( T, a1 , a10 )) + (c2 (θ2 ( T, s), a20 ) − I2 ( T, a2 , a20 )) ] s = 1, . . . , 22(T −1) ; ∀( a1 , a2 ) 0

(4.38)

0

A1 A2 A1 X A2 X X X

R (t, s, ( a1 , a2 ), ( a10 , a20 )) = 1

a1 =1 a2 =1 a10 =1 a20 =1

∀t ; s = 1, . . . , 22(t−1)

(4.39)

B(t, s, ( a10 , a20 )) = R(t, s, ( a1 , a2 ), ( a10 , a20 )) × B1 + β(t, s) − I1 (t, a1 , a10 ) − I2 (t, a2 , a20 ) t = 1 ; s = 1 ; a1 = a2 = 1 ; ∀( a10 , a20 ) B(t, s, ( a10 , a20 )) =

A1 X A2 X

(4.40)

R(t, s, ( a1 , a2 ), ( a10 , a20 )) × [ β(t, s)

a1 =1 a2 =1

B(t − 1, B(t − 1, + B(t − 1, B(t − 1,

mod 4] = 1,

t = 2, . . . , T ; s = 1, . . . , 2

; ∀( a10 , a20 )

s +3 4 , ( a1 , a2 )) if [ s s +2 4 , ( a1 , a2 )) if [ s s +1 4 , ( a1 , a2 )) if [ s s if [s 4 , ( a1 , a2 )) 0 0 − I1 (t, a1 , a1 ) − I2 (t, a2 , a2 ) ] 2( t −1)

mod 4] = 2, mod 4] = 3, mod 4] = 0

(4.41)

74

4 Valuation Model for Portfolios of Real Options 0

L(t, s, ( a1 , a2 )) =

0

A1 A2 X X

R(t, s, ( a1 , a2 ), ( a10 , a20 ))

a10 =1 a20 =1

t = 2, . . . , T ; s = 1, . . . , 22(t−1) ; ∀( a1 , a2 ) R(t, s, ( a1 , a2 ), ( a10 , a20 )) =

×

(4.42)

0 0 1 4 × R ( t, s, ( a1 , a2 ), ( a1 , a2 )) A10 A20 3

X XX a10 =1 a20 =1

L(t + 1, 4s − l, ( a10 , a20 ))

l =0

t = 2, . . . , T ; s = 1, . . . , 22(t−1) ;

∀( a1 , a2 ) ; ∀( a10 , a20 ) 0

(4.43)

0

A1 A2 X X

R(t, s, ( a1 , a2 ), ( a10 , a20 )) = 1

a10 =1 a20 =1

t = 1 ; s = 1 ; a1 = a2 = 1

(4.44)

R(t, s, ( a1 , a2 ), ( a10 , a20 )) ∈ {0; 1}

∀t ; ∀s ; ∀( a1 , a2 ) ; ∀( a10 , a20 ).

where t s ai

(4.45)

Time indicator, Scenario indicator Entering operating mode, underlying asset i = 1, 2, ai = 1, . . . , Ai , Leaving operating mode, underlying asset i = 1, 2, ai0 ai0 = 1, . . . , Ai0 , R Binary switching variable, V Value of portfolio, Stochastic value of underlying asset i = 1, 2, θi Starting value of underlying asset i = 1, 2, αi ci (·, ai0 ) Cash flow function for operating underlying asset i = 1, 2 in mode ai0 , ai0 = 1, . . . , Ai0 , Investment or switching costs for underlying asset i = 1, 2, Ii p Risk–neutral probability, Risk–free rate of return, rf B Budget level, β(t, s) New funds available at time t in scenario s, Initial budget available at time t = 1, B1 e R ≥ 0, B1 L, l Auxiliary variable and index.

4.3 Real Options Portfolio Model

75

4.3.4.3 n–Dimensional Model Formulation The model formulation for one underlying asset has established the modeling logic behind portfolios of real options. Especially it has introduced the switching concept, embedded into a program that can take into account both forward– and backward–looking elements. The model formulation for 2 underlying assets has then demonstrated how the model needs to be modified in order to handle a second underlying dimension. This has introduced the consideration of the joint stochastic processes of the underlying assets, while all investments are financed through a joint budget. The model formulation for n underlying assets is structurally similar to the formulation for two underlying assets and is composed of identical building blocks. With this background, it is possible to extend the model to i > 2 underlying assets in a straightforward manner. The indexing logic of the system state space needs to take into account the spanning defined by n underlying assets. Therefore, scenarios for θi (t, s) expand with s = 1, . . . , (2n )(t−1) = 2n(t−1) . This number expresses the total number of scenarios as a function of time. This implies that each node branches into exactly 2n successor nodes. For example, a tree representing the joint movement of i = 3 underlying assets branches into 23 = 8 nodes per time step. The risk–neutral probabilities for n underlying assets, based on a multidimensional extension of the Cox–Ross–Rubinstein (CRR) binomial (cf., e.g., Cox et al. 1979), are given in Boyle et al. (1989). For the general formulation, it is useful to define the following indicators: ( 1 if i moves up in scenario s, φi (s) = (4.46) −1 if i moves down in scenario s. 1 if i, j move in the same direction in scenario s, (4.47) φij (s) = −1 if i, j move in the opposite direction in scenario s. In analogy with CRR, the up and down movements are specified as ui = and di = u1 . The general formulation for the risk–neutral probability i in scenario s, pk , is then given by: √ eσi ∆t

" n # n √ X r f − 12 σi2 1 X pk = n φ (s)ρij + ∆t φi (s) 2 i,j=1 ij σi i< j

k = 1, . . . , 2n ,

i =1

(4.48)

76

4 Valuation Model for Portfolios of Real Options

√

where ∆t is the length of the time step, r f is the risk–free interest rate, and σ is the standard deviation of asset i returns. This formulation does not guarantee positive probabilities and so must be checked in each application. Further, the core of the model formulation is based on vectors for en→ a as the vector of entering modes of all tering and leaving modes, i.e., − − →0 underlying assets and a as the vector of leaving modes of all underlying assets: − → − → a = ( a1 , a2 , . . . , an ) , and a0 = ( a10 , a20 , . . . , a0n ). (4.49)

− → What is more, the vector of ones is given by 1 . Based on this, all variables can be defined over the corresponding vectors, instead of the more tedious expression through all elements of these vectors. For example, the binary variable R is defined over the system state − → → space as R(t, s, − a , a0 ). With these modifications, the model can be specified for n underlying assets in close analogy to the case of 2 underlying assets: − → max− V (1, 1, 1 ) →

(4.50)

− → R(1,1, 1 , a0 )

subject to θi (1, 1) = αi i = 1, . . . , n

θi (t, s) =

(4.51)

√ s + 2n − 1 φi (s)σi ∆t θ ( t − 1, ) × e n i 2 √ s + 2n − 2 φi (s)σi ∆t θi (t − 1, 2n ) × e ... √ θi (t − 1, s2+n1 ) × eφi (s)σi ∆t √ θ (t − 1, s ) × eφi (s)σi ∆t i 2n

if

[s mod 2n ] = 1,

if

[s mod 2n ] = 2,

if

[s mod 2n ] = 2n − 1,

if

[s mod 2n ] = 0

i = 1, . . . , n ; t = 2, . . . , T ; s = 1, . . . , 2n(t−1) 0

→ a) = V (t, s, −

A1 X a10 =1

0

···

An X

− → → R(t, s, − a , a0 ) ×

a0n =1

ci (θi (t, s), ai0 ) − Ii (t, ai , ai0 ) )

i =1 n

+ e−r f ∆t ×

X n

(4.52)

2 hX

i − → V (t + 1, 2n (s − 1) + k, a0 ) × pk

k =1

→ i = 1, . . . , n ; t = 1, . . . , T − 1 ; s = 1, . . . , 2n(t−1) ; ∀− a

(4.53)

4.3 Real Options Portfolio Model 0

77

0

A1 An n hX X X − → → → a) = ··· R( T, s, − a , a0 ) × V ( T, s, − ci (θi (t, s), ai0 ) − Ii (t, ai , ai0 ) a10 =1

a0n =1

i =1

i = 1, . . . , n ; s = 1, . . . , 2 A1 X

0

...

A1 An X X

0

···

an =1 a10 =1

a1 =1

An X

n ( T −1)

→ ; ∀− a

− → → R(t, s, − a , a0 ) = 1

a0n =1

∀t ; s = 1, . . . , 2n(t−1) − → − → → B(t, s, a0 ) = R(t, s, − a , a0 ) " ×

(4.54)

B1 + β(t, s) −

n X

(4.55)

# Ii (t, ai , ai0 )

i =1

− → − → → a = 1 ; ∀ a0 i = 1, . . . , n ; t = 1 ; s = 1 ; −

(4.56)

A1 An X X − → − → → B(t, s, a0 ) = ··· R(t, s, − a , a0 ) × [ β(t, s) a1 =1

a n =1

n → B(t − 1, s+22n −1 , − a) n − → s + 2 − 2 B ( t − 1, , a) 2n + ··· → B(t − 1, s2+n1 , − a) → s − B(t − 1, 2n , a )

−

n X

if [s mod 2n ] = 1, if [s mod 2n ] = 2, if [s mod 2n ] = 2n − 1, if [s mod 2n ] = 0

Ii (t, ai , ai0 ) ]

i =1

− → i = 1, . . . , n ; t = 2, . . . , T ; s = 1, . . . , 2n(t−1) ; ∀ a0 0

(4.57)

0

A1 An X X − → → → a) = ··· R(t, s, − a , a0 ) L(t, s, − a10 =1

a0n =1

→ t = 2, . . . , T ; s = 1, . . . , 2n(t−1) ; ∀− a − → → a , a0 ) = R(t, s, − ×

1 2n × A10

X a10 =1

(4.58)

− → → R(t, s, − a , a0 )

···

0

n

A n 2 −1 X X

− → L(t + 1, 2n s − l, a0 )

a0n =1 l =0

− → → t = 2, . . . , T ; s = 1, . . . , 2n(t−1) ; ∀− a ; ∀ a0

(4.59)

78

4 Valuation Model for Portfolios of Real Options 0

A1 X

0

···

a10 =1

An X

− → → R(t, s, − a , a0 ) = 1

a0n =1

− → → a = 1 t = 1 ; s = 1 ;−

(4.60)

− → → a , a0 ) ∈ {0; 1} R(t, s, − − → → ∀t ; ∀s ; ∀− a ; ∀ a0 ,

where t s i ai

− → a ai0

(4.61)

Time indicator, Scenario indicator, Underlying asset i = 1, . . . , n, Entering operating mode for underlying asset i = 1, . . . , n, ai = 1, . . . , Ai , Vector of entering modes of underlying assets, Leaving operating mode for underlying asset i = 1, . . . , n, ai0 = 1, . . . , Ai0 ,

− →0 a Vector of leaving modes of underlying assets, R Binary switching variable, V Value of portfolio, Stochastic value of underlying asset i = 1, . . . , n, θi Starting value of underlying asset i = 1, . . . , n, αi U p and down movement of underlying asset i = 1, . . . , n, φi ci (·, ai0 ) Cash flow function for operating und. asset i in mode ai0 , i = 1, . . . , n ; ai0 = 1, . . . , Ai0 , Investment or switching costs for und. asset i = 1, . . . , n, Ii p Risk–neutral probability, Risk–free rate of return, rf B Budget level, β(t, s) New funds available at time t in scenario s, Initial budget available at time t = 1, B1 e R ≥ 0, B1 L, l Auxiliary variable and index.

4.4 Model Extensions Up to this point, the core general model has been presented. This model already incorporates the key portfolio aspects of portfolios of real options. Some further possible model details will be given in the next section. 4.4.1 Alternative Probability Formulation The three–dimensional model has been based on risk–neutral probabilities as formulated by Boyle et al. (1989). On the one hand, this formulation of-

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79

fers the advantage that it is general enough to control for any number of underlying assets. Therefore, the above introduced model can be used for any number of underlying assets. On the other hand, as discussed before, it has the disadvantage that the probabilities may turn negative for large time steps. In financial option pricing where asset prices are typically modeled as log–normal processes, this disadvantage is hardly relevant because either way time steps are chosen such that the log–normal distribution is approximated well enough by the discrete distribution. In practice, usually 30 time steps per year are regarded as sufficient to obtain a good approximation (Hull 2003, p. 397). In our model, it is assumed that the values of the underlying assets follow a multivariate binomial distribution. Following this assumption, it is sufficient to consider less time steps, i.e., at least as many time steps as there are decision points. Hence it is important to ensure non– negativity of probabilities because otherwise arbitrage possibilities would exist, which would invalidate risk–neutral valuation. An approach ensuring non–negativity was first suggested in the context of interest rate trees by Hull and White (1994) and then applied by Hull (2003, pp. 473 f.) for a binomial lattice tree built up according to Jarrow and Rudd (1983, pp. 183 ff.). Instead of imposing the assumption u = 1d as in CRR, the probabilities in the two–dimensional case are set to p = 1 − p = 0.5. For the three– dimensional case, based on the coefficient of correlation, ρ, this gives risk– neutral probabilities with: p1 = p2 = p3 = p4 =

1 4 1 4 1 4 1 4

[1 + ρ ] ,

for asset 1=up, asset 2=up,

(4.62)

[1 − ρ ] ,

for asset 1=up, asset 2=down,

(4.63)

[1 − ρ ] ,

for asset 1=down, asset 2=up,

(4.64)

[1 + ρ ] ,

for asset 1=down, asset 2=down.

(4.65)

In the limit as the number of time steps approaches infinity, these probabilities are the same as are obtained for the formulation by Boyle et al. (1989). By modifying the jump factors the alternative formulation makes it possible to use the probabilities that in a CRR setting are only obtained in the limit. Since the degree of freedom has been used for fixing the probabilities, the up and down factors must be calculated separately as follows: u = e (r f − σ

2 /2)∆t + σ

√ ∆t

, and

d = e (r f − σ

2 /2)∆t − σ

√ ∆t

.

(4.66)

With these u and d factors, the values of the underlying assets do not hit the central node at each time 2∆t any more (see Sec. 4.2.1). This is less convenient for calculating options Greeks, but since this is not key for the presented model, the advantage of always non–negative probabilities clearly prevails. For this convenience, in our numerical analysis in Chap. 5 we will

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4 Valuation Model for Portfolios of Real Options

use the formulation by Jarrow and Rudd (1983) for two underlying assets (see Sec. 5.1. However, given that this formulation is only available for two underlying assets, we will come back to the original formulation by Boyle et al. (1989) for our discussion of three underlying assets (see Sec. 5.2). 4.4.2 Change of the Underlying Stochastic Processes As discussed before, our model is based on a switching concept that allows to switch between operating modes of different assets. Up to this point, we assumed that the stochastic processes of the underlying assets remain unchanged. Nevertheless, an investment policy may result in an alteration of an underlying asset, and thus, the stochastic process. Especially, the investment policy may alter the risk profile of an asset, and thus, its volatility. This is depicted in Fig. 4.7, where an investment decision at some point in time shifts the stochastic process of an asset upwards, and lowers its variability. If a change of the underlying process occurs, all future real options will be based on this change.

Fig. 4.7. Change of the underlying asset

Our model can readily incorporate this alteration of the underlying assets, by tying the stochastic processes to the decision variable. Based on Eq. 4.52, this gives θi (t, s, ai0 ) instead of θi (t, s). Further, it is conceivable that the stochastic process is affected by decisions about all underlying assets, − → which would give the expression θi (t, s, a0 ). By altering the underlying stochastic process, the multinomial grid on which our model is based would shift accordingly, as visualized in Fig. 4.8.

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81

Fig. 4.8. Change of the underlying asset in the multinomial grid

4.4.3 Explicit Modeling of Synergies In the traditional capital budgeting literature, a typical way of taking synergies into account is to model bundles of synergistic projects as new projects. For example, two projects that exhibit synergies among each other can be modeled as three projects, i.e., project 1, project 2, and a synthetic project 3 which consists of doing project 1 and 2 together. This technically assumes that these three projects are mutually exclusive because project 3 incorporates the case where both projects are undertaken simultaneously (cf. e.g. Hax 1985, pp. 97 f.). In that sense the modeling of synergies does not require any modification of the model and does not merit special attention. Synergies just imply additional projects. Nevertheless it should be pointed out that, while leaving the structure of the model unchanged synergies can be handled through the binary switching variable R because it already controls for the necessary information. Following this approach, synergies can be modeled explicitly by making switching costs state–dependent, by making the cash flow function state– dependent, or both. State dependent switching costs for asset i = 1 would mean changing I1 (t, a1 , a10 ) to I1 (t, a1 , a10 , a20 ) if the switching costs of underlying asset i = 1 depends on the operating mode of the underlying asset i = 2; moreover the switching costs may also depend on where the switch was made from, yielding I1 (t, ( a1 , a2 ), ( a10 , a20 )). Other cash flow functions can also be specified in a straightforward manner. The cash flow function so far was only dependent on the leaving mode, e.g., c1 (θ1 (t, s), a10 ). To take synergies into account, a cash flow function can be specified dependent on combinations of operating modes, e.g.,

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4 Valuation Model for Portfolios of Real Options

c1 (θ1 (t, s), ( a10 , a20 )). In general, the advantage of our mathematical programming formulation is that it keeps track of all variables throughout the system − → → state space and thus can incorporate any desirable function of (t, s, − a , a 0 ). All alternatives can be implemented readily without affecting the model setup. This is why they are not analyzed in more detail here. 4.4.4 Operating and Other Constraints Operating constraints have not been formulated so far. In order to include operating constraints, corresponding equations need to be added to the model formulation, by the help of the binary variables R. For the example of two underlying assets, if in period t = 2 it is not possible to simultaneously operate asset 1 in mode 1 and asset 2 in mode 3, i.e., these modes are mutually exclusive in all scenarios and independent on the entering mode, the additional constraint is: R(t = 2, s, ( a1 , a2 ), ( a10 = 1, a20 = 3)) = 0 , s = 1, . . . , 4 ; ∀( a1 , a2 ).

(4.67)

This example shows the flexibility of the approach which can handle path– and state–dependent constraints. Further, this formulation makes it possible to link option pricing to traditional capital budgeting and the known formulation of “project interrelationships” constraints (cf. Clark et al. 1984, pp. 330 ff.). Similarly, any precedence constraint can be handled. Depending on the specific modeling context, it may be desirable to include further constraints, such as maximum losses, minimum profits, or VaR constraints per period (cf. Kleindorfer and Li 2005). All of these can be readily accommodated. 4.4.5 Terminal Values The model formulation considers all periods in which cash flows are realized explicitly. This means that there are no further cash flows beyond period t = T. Alternatively, it is possible to include a terminal value in the assessment of a project, representing the aggregate cash flow in all periods beyond the explicit planning horizon, e.g., calculated as a perpetuity. The terminal value can represent a large share of the overall value (cf. Copeland et al. 1994, chapter 9). Our model is formulated without including terminal values, making the implicit assumption that terminal values are zero. This is done in order to avoid that a large fraction of value comes from decisions and cash flows that are not modeled explicitly because this would dilute the portfolio effects

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83

which derive from explicit option exercise strategies. If desired, terminal → a ). values can be included conveniently in the equation for V ( T, s, − 4.4.6 Further Generalizations of the Model Formulation Generally, the modeling approach provides a very flexible structure that can readily be customized to specific applications. This stems from two sources: 1. Controlling of decisions via binary variables: The indexing of the system state space is such that the binary variables keep track of the sequence of decisions and hence, their dynamic interaction. The binary variables can serve as reference points to any feature related to desired or permissible actions. 2. Discrete programming approach: In this structure, it is straightforward to modify the set and nature of constraints, as well as any input to the problem. For instance, the cash flow functions could be replaced; β(t, s) could be made endogenous by linking the level of additional funds to other endogenous variables (e.g., cumulated profits over time) instead of being exogenous; or different stochastic processes (e.g., mean–reverting processes) could be used. Adapting the problem setup leaves the structure of the problem unchanged. Therefore, changes can be made in a straightforward way in order to asses different problems.

4.5 Model Discussion Up to this point, we have presented a general formulation for portfolios of real options, and have discussed possible extensions of this formulation. In the following, we will discuss features of this model formulation. This discussion will cover the applicability of risk–neutral valuation, numerical efficiency, and a positioning with regards to Monte Carlo simulation. 4.5.1 Applicability of Risk–Neutral Valuation Risk–neutral valuation assumes no restrictions on short sales and hence unrestricted borrowing. Restrictions on borrowing are assumed only for the investing firm, whereas capital markets are assumed to be frictionless, both for the ultimate investor and for the underlying stochastic variables. This logic is founded by two elements. First, it is still a sufficient goal of the investment strategy to maximize net present value, calculated as the value of the portfolio of real options. If one were to consider the capital constraint as inconsistent with risk–neutral valuation, the same objection

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4 Valuation Model for Portfolios of Real Options

would make it impossible to calculate even a NPV and make it hard to obtain any valuation results at all. We assume that the ultimate investor has unrestricted access to perfect capital markets and can choose her portfolio as a function of her personal preferences (Brealey and Myers 1996, pp. 105), such that risk–neutral valuation is validated. Second, given frictionless markets in the first place, all replication arguments on the level of the underlying stochastic variables and the derivation of risk–neutral probabilities still hold. The boundary between capital markets in general, and firm–specific investment decisions, is the transformation of θi (t, s) into a cash flow by the function ci (θi (t, s), ai0 ). The dynamics of the underlying stochastic processes are unaffected by this. While the firm has financing constraints, capital markets can be assumed to be frictionless. This assumption is typically made in recent work focusing on financing constraints (cf., e.g., Mello and Parsons 2000; Boyle and Guthrie 2003). Based on these arguments, the presented binomial model gives unique equivalent martingale measures, such that any contingent claim can be priced by risk–neutral valuation (Bingham and Kiesel 1998, pp. 105 ff.). 4.5.2 Numerical Efficiency The pricing of complex financial options typically requires the use of numerical methods which can result in a “tremendous computational challenge” (p. 1163 Broadie and Detemple 2004). Not surprisingly, the numerical solution of the presented model for portfolios of real options encounters the same challenges. This makes it worthwhile to discuss the efficiency of the numerical solution approach. The discussion of the numerical efficiency covers the optimality of solutions, the dimensionality of the problem, and possible ways for increasing the efficiency of solution algorithms. 4.5.2.1 Optimality of Solutions and Linearization The problem is formulated as a stochastic mixed–integer program which can be solved by commercially available implementations of suitable algorithms. This is why it is insightful to discuss the quality of possible solutions. The nonlinearity of the model merits special attention. It arises because the binary decision variable, R, is multiplied with the optimal portfolio value, V, introducing non-linear equations of the type R × V. These equations are inseparably tied to the general formulation because they are a core feature of the asymmetry stemming from the optionality involved. The decision maker has discretionary decision rights, which may or may not be exercised. Exercising a real option, which is equivalent to switching, is

4.5 Model Discussion

85

represented by a binary variable R = 1. Therefore, the nonlinearity is unavoidable in the general formulation. As opposed to this, the implementation of the problem in GAMS© can be done through an equivalent, linear version of the problem. This can be achieved by replacing the nonlinear equations by sets of linear inequalities, based on the upper and lower bounds of the variables involved. Schematically, assume that the binary variable R is multiplied by some function of V, f (V ), which yields the variable X. Further assume that bounds on X can be inferred with upper bound Xup and lower bound Xlo . Also, let there be a sufficiently large number Q. Then, the given nonlinearity of the form X = R × f (V )

(4.68)

can be substituted by the system of linear inequalities X ≤ f (V ) + Q × (1 − R )

(4.69)

X ≥ f (V ) − Q × (1 − R )

(4.70)

X ≤ Xup × R

(4.71)

X ≥ Xlo × R.

(4.72)

In order to improve numerical efficiency, we use this linear reformulation in our implementation for the numerical analysis. The logic is as follows: On the one hand, for R = 0 (i.e., option not exercised), the non-linear equation yields X = 0. The same is achieved by the set of linear equations. Due to Q, the first two equations are not binding whereas the last two equations yield X ≤ 0 and X ≥ 0 thus X = 0. On the other hand, for R = 1 (i.e., option exercised), the nonlinear equation yields X = f (V ). In this case, the last two equations do not turn out binding. The first two equations yield X ≤ f (V ) and X ≥ f (V ) thus X = f (V ). All in all, the set of inequalities can represent the nonlinearity. It is important to note that the interplay of these inequalities still expresses a nonlinear problem which can only be avoided technically in the implementation. The character of the problem remains nonlinear. Based on this reformulation, for the purpose of the discussion of numerical solutions, the problem can be considered as linear. This leads to a discussion of the mixed-integer character of the linearized problem. Integer problems are much harder to solve than continuous problems (cf. Williams 1993, chapter 8), such that it is much more expensive both in memory and computing time. Given that there are no general efficient methods available, this field is actively researched (cf. Birge and Louveaux 1997, p. 253). As the mixed-integer formulation is unavoidable for the valuation of portfolios of real options, the challenges represented by this class of problems also apply here.

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4 Valuation Model for Portfolios of Real Options

More precisely, mixed-integer programs are non-convex and thus, a local optimum typically cannot be guaranteed to be a global optimum (cf. Kall and Wallace 1994, pp. 2 f.). This means that it is not necessarily clear whether an optimal solution actually is the best solution. It may be the best solution locally, but there may be another local optimum which gives a higher value of the objective function. Problems of this kind are also referred to as “global optimization” (Kall and Wallace 1994, p. 2). For these, generally speaking, the proof of global optimality is difficult. In the light of the absence of general proofs of global optimality, recent developments in solver techniques become very important. Specifically, so-called global solvers have emerged and have actively been researched lately (cf., e.g., Horst and Pardalos 1995; Dzemyda et al. 2002). For example, the solver BARON© has been demonstrated to be very powerful, based on a branch– and–bound approach (Tawarmalani and Sahinidis 2002a, b; Ahmed et al. 2004). One key contribution of this solver is its ability to check global optimality numerically with high precision. In our numerical analyses, we applied this solver to the nonlinear problem, observing a high reliability but relatively slow computation time. In some cases, it is possible to prove global optimality in a very simple way. If in the relaxed mixed-integer problem the binary decision variables turn out exactly 0 or 1, the solution must be globally optimal. This is because the relaxed problem is linear thus convex, and thereby each local optimum is a global optimum (Kall and Wallace 1994, Lemma 1.1). This idea can be combined with a branch–and–bound technique in order to ensure integrality of the binary variables, i.e., branch–and–bound using linear programming relaxations (Nemhauser and Wolsey 1988, Chapter II.4.2). If the branch–and–bound algorithm terminates with an optimal solution, it is a global optimum. This requires that the solver does not terminate before its algorithmic termination, which can be the case if a given limit is reached, e.g., the maximum number of iterations, or memory. In the context of the above discussed computational burden of a mixed–integer algorithm, the bottleneck is memory. Concluding, if the algorithm provides a solution, we can rely on it to be a global optimum. Whether it gives a solution mainly depends on the dimension of the problem, which leads us to the discussion of the dimensionality of the problem. 4.5.2.2 Dimensionality of the Problem The size of the model can be characterized by the number of equations in the explicit version of the program. The number of equations grows exponentially in the number of periods and quadratic in the number of modes per underlying asset, i.e., (2n )(t−1) . Since the problem is path–dependent and thus non–recombining, the system state space grows exponentially. For

4.5 Model Discussion

87

example, two underlying assets make the system state space grow by factor 22 = 4, i.e., in each period a node in the tree branches into four successors. Further, the model grows quadratically in the modes per underlying asset since in the general case switches from each mode to each other mode are permissible. More precisely, for a portfolio problem with two underlying assets, the number of equations is a function of the number of operating modes for both underlying assets, A1 and A2 , as well as of the number of periods, T. For a portfolio with A1 > 1, A2 > 1, and T > 1, the number of equations in our implementation is given by: T h iX 1 + 3( A1 )2 ( A2 )2 + 1 22(t−1) + ( A1 )2 ( A2 )2

(4.73)

t =1

≈ 3( A1 )2 ( A2 )2

T X

22(t−1) + ( A1 )2 ( A2 )2

(4.74)

t =1

For A1 = A2 , this dimensionality is depicted in Fig. 4.9. It is noted that the scale of the vertical axis is logarithmic, which captures the exponential growth of the problem. In the implementation, this growth in the system state space translates directly into high memory requirements for generating the problem. Given the fact that mixed-integer problems are computationally very expensive, the high dimension of the system state space is very demanding in terms of memory and processor performance. Nevertheless, the linearized problem for 2 underlying assets, 3 modes each, and 5 periods takes a Pentium IV-M Laptop (CPU 1.7GHz RAM 512MB) 2 minutes to solve to optimality. In this light, it is vital to implement the problem as efficiently as possible. This is why possible routes for implementing the problem more efficiently will be pointed out in what follows. 4.5.2.3 Increasing Efficiency of the Implementation In order to increase the efficiency of the implementation, two directions can be followed. The first direction aims at reducing the original dimension of the problem before the solver is applied. The second, given the dimension of the problem, is concerned with making the optimization algorithm more efficient. Following the first direction, the efficiency of the implementation is closely tied to the modeling of the system state space. The system state space is generated exhaustively, i.e., it is built up for all combinations of periods, scenarios, and actions. It does not eliminate regions in the system state space which are irrelevant to the solutions, i.e., regions which will

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4 Valuation Model for Portfolios of Real Options

1.0E+10 1.0E+09 1.0E+08 1.0E+07 1.0E+06 Number of equations 1.0E+05 1.0E+04 1.0E+03 1.0E+02

10 8

1.0E+01 6

1.0E+00 2

3

4

Periods

4 5

6

Modes per underlying asset

7

8

9

10

2

Fig. 4.9. Dimensionality for two underlying assets

never be anywhere near an optimum. This translates into high memory requirements when generating the model. Once the problem is generated, the solver needs to solve a correspondingly big problem, with sparse matrices for the binary variables. The binary variables are sparse because at each time in each scenario exactly one binary variable turns one, while all other binary variables there turn zero. The solver then eliminates as many irrelevant regions as possible. This is why it would be desirable to generate a model that is reduced in the first place. For instance, the size could substantially be reduced by gridding the system state space adaptively to the problem, in the fashion of an adaptive mesh approach known from derivatives pricing (Figlewski and Gao 1999). The basic idea is to build up a lattice tree that is finer in relevant regions and coarser in other regions. Along the same lines, it is conceivable to distinguish time periods and decision points. This would increase numerical efficiency when decision points are much fewer than points in time, because it would reduce the problem size. Each additional decision point increases the number of paths that can be followed over time, because it increases the number of possible combinations of decisions over time. As opposed to this, allowing for additional time periods that do not increase the number of decision points leaves the problem size unchanged since it only requires additional discounting operations. In our numerical example, this consideration would not materialize because we assume that there are decision points at each point in time.

4.5 Model Discussion

89

Another approach to reducing the dimension of the problem can be achieved by replacing the original system state space by a smaller one that gives a sufficient representation of the original system state space, in that its solution is not much different from the original problem. Such an approach is proposed by Casey and Sen (2005). Likewise, Zenios and Shtilman (1993) provide a method for generating subsets of an exponentially growing lattice tree, representing the whole lattice tree at a given level of accuracy. Following the second direction, different ways of making the optimization algorithm more efficient are conceivable. In our implementation, we considered implementation and solution as independent steps. This leaves some efficiency unexploited because a general formulation can be implemented in many different but equivalent ways. Alternative implementations can be assessed as a function of their efficiency with the solution algorithm and the most efficient implementation can be retained. As an example, in our implementation we have products of sums of variables. For numerical purposes, we checked whether a product disaggregation would increase efficiency (cf. Tawarmalani et al. 2002), but this did not prove helpful. Another possibility of better handling a complex problem is to combine a sequence of several optimization runs. A first run may eliminate large regions of the system state space quickly, which, making subsequent and tighter optimization runs more efficient. Also, there is a tradeoff between using commercially available solvers and tailor–made algorithms for specific problems. The former make it possible to benefit from very powerful engines that are continuously researched, tested, and improved. Most importantly, the results are perfectly reliable. The latter make it possible to further exploit the specific structure of a problem, but it can be very time consuming and expensive to show that the algorithm is robust. This is the case for the above described analysis of portfolios of real options in Meier et al. (2001). A more recent example for a tailor–made algorithm developed for a capacity planning problem, can be found in Huh and Roundy (2005). Probably, a good compromise between the two is to customize a commercially available solver specific to the problem. It has to be noted that the goal of our implementation was to illustrate the general model formulation. If the focus lies more on the efficiency of the implementation, it may be desirable to investigate alternative implementation structures. For example, for the given model formulation, it may be possible to avoid the growth of the system state space through an implementation as a network design problem (cf. Nemhauser and Wolsey 1988). Finally, a wide range of numerical techniques has been employed in the context of financial derivatives (cf. Broadie and Detemple 2004), and continues to be actively researched (cf., e.g., Broadie and Yamamoto 2005). It is conceivable to complement the solution approach used for our prob-

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4 Valuation Model for Portfolios of Real Options

lem with elements provided in this strand of research. For instance, Kishimoto (2004) present a discrete-time numerical procedure for pricing path– dependent options, labelled as the “extended tree” method. The tree is extended by a “supplementary variable” which keeps track of the relevant information associated with the path through the tree. The method is illustrated by applications to collateralized mortgage obligations as well as American average option. Wang (2006) explores dimension reduction techniques for high–dimensional problems such as the pricing of path– dependent options and bond valuation. Interestingly, while the proposed dimension reduction techniques can achieve substantial dimension reduction in some settings, in other settings they may even increase the effective dimension. While the path–dependency is in close analogy to our model, the problems are different enough to exclude a straightforward adaptation. Like these examples, analogies to the model presented here can be tracked in other derivative pricing settings. 4.5.3 Monte Carlo Simulation With growing importance of complex financial derivatives, numerical methods have been used increasingly. Boyle (1977) was first to suggest Monte Carlo Simulation for estimating security prices. The approach consists of the following steps (cf. Boyle and Broadie 1997; Lehoczky 1997): 1. Simulate sample paths of the underlying variables over the relevant time horizon with respect to the risk–neutral measure. 2. Evaluate the discounted cash flows from the derivative, on each sample path. 3. Average the discounted cash flows over sample paths. This procedure shows that the approach is basically a forward–looking one. The advantage of this approach is that it can be used to price derivatives that only depend on the final values as well as derivatives that are path–dependent. The drawbacks of Monte Carlo simulation are that it typically is computationally very expensive and that it cannot easily handle cases with early exercise opportunities because these introduce a backward looking element (Hull 2003, p. 411). The dimension reduction techniques discussed in Wang (2006), for specific settings, can alleviate the computational burden of Monte Carlo simulation approaches. Early exercise is immanent to American–style options. For these, cash flows depend on both the price paths of the underlying assets and the decisions of the owner (Boyle and Broadie 1997, p. 1309), which invalidate the forward–looking nature of simulation. Only recently, simulation approaches that are able to handle American–style derivatives have been proposed

4.5 Model Discussion

91

(cf., e.g., Longstaff and Schwartz 2001; Andersen and Broadie 2004). Essentially, some iterative procedure needs to be added to the simulation which can incorporate the impact of exercise decisions. This can happen as a nesting of simulations and optimizations for the optimal decisions. Evidently, these simulation approaches preserve and further increase the drawback of computational burden required for obtaining reliable results. In the context of the model presented here, simulation does not represent a modeling alternative. The dynamics of the underlying assets are modeled in a straightforward way, through discrete binomial processes. These could alternatively be replaced by simulation runs of the underlying assets, especially, when many underlying assets are considered. Yet, the core of the model and source of its complexity is the interaction of decisions and resulting level of budgets over time. These would still need to be incorporated in a simulation approach, along the lines of pricing American–style options through simulation. Thus, using simulation would leave this programming core unchanged and hence would face the same challenges. One could think that at least simulation could be beneficial when considering many underlying assets and their joint stochastic processes which can readily be simulated. Yet, the challenge of the presented model which lies in the huge system state space does not result from the underlying assets in the first place. In fact, the underlying stand–alone binomial processes are recombining. The system state space becomes only bigger because it has to capture the path–dependencies in decisions. Again, this would remain unchanged even when simulating the processes of the underlying assets.

5 Numerical Analysis

In the previous chapter, a general model formulation for portfolios of real options has been proposed. This model provides a framework that incorporates the identified specific aspects of portfolios of real options and thus helps to optimally manage portfolios of real options. Up to this point, we have provided a general valuation structure but have not yet shed any light on the structural properties of such portfolios. This is why in this section we will exploit the impact of the main drivers of portfolio value and their impact on the optimal investment strategy. Keys drivers of the value of portfolios of real options are budget effects, volatility effects, correlation effects, and starting value effects. Embracing all of these effects are path–dependencies which simultaneously result from the interdependency of the budget constraint and of option exercise decisions. It will be shown that these value drivers can cause investment strategies that are puzzling in the first place, e.g., they imply shutting down an asset that is profitable. Nevertheless, the optimal investment strategy can be related, with more thorough analysis, to financial intuition and prior work on complex financial options. As to be expected, these drivers affect portfolio value in a complex way. Budget constraints establish sub–additivity of options value and may require to move in and out of assets in order to realize the highest value. This can cause disinvestment decisions that from a naïve point of view seem to be value destroying but which make it possible to undertake or help finance even more profitable investments for different assets and later periods with a positive net effect on portfolio value. Closely connected to budget effects are volatility effects. A basic understanding from standard option pricing would justify expecting options portfolio values to increase in the volatility of the underlying assets. For portfolios of real options, the impact of increasing volatility may be reversed. Typically, options benefit from higher volatility because they allow to capitalize on the upside while protecting from the downside of an increased variability in the out-

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5 Numerical Analysis

comes. This may not materialize when the interplay of volatility and budget constraints prevents from capitalizing on the upside. Correlation and starting values can cause different effects, depending on the structure of the investment problem. Intuitively, the encountered effects can be attributed to the net character of the investment problem, i.e., whether it tends to be an investment problem with some disinvestment options or the other way round. Thus, all key drivers of the value of portfolios of real options can have an ambiguous impact on the value of the portfolio. Beyond this, an additional source of ambiguity results from the fact that these drivers all interact simultaneously with varying relative magnitudes, making it impossible to derive value effects in general. This ambiguity underlies any practical application. As a consequence, each investment problem needs to be solved specifically, which is achieved by the presented model. Knowing that in different settings the relative impact of different drivers will vary, the net effect only becomes evident after the valuation has been carried out. In the following, a representative investment problem is analyzed in order to give evidence of the ambiguity of key value drivers. This problem illustrates that much care must be dedicated to portfolio effects because these can substantially affect valuation results and hence optimal decision making. Specifically, for each value driver an example is provided that illustrates effects which can be connected to financial intuition only with detailed analysis. In this sense, pinpointing portfolio effects by example shows in how far these effects can only be captured by a thorough portfolio analysis. Most of these effects present counter examples to prevailing thinking in financial option pricing and thus generate new insights. The numerical example analyzed here is stylized in the sense that it does not represent an actual decision problem. It is purposely simplified because this makes it possible to isolate separate portfolio effects and keep the intuition of the underlying dynamics. Given that the model formulation is much more general, much more complex settings are readily solvable. Here, the focus lies on the basic structure of the portfolio setting. Nevertheless, the numerical application has been developed in cooperation with an anonymous high–tech company analyzing their manufacturing sites. Thus, while the problem is stylized and made anonymous, the general structure of the presented problem is real.

5.1 Numerical Analysis for Two Underlying Assets 5.1.1 Numerical Application Setting for Two Underlying Assets In the following, the basic structure of the investment problem is presented which will be analyzed to highlight portfolio features. We give a short

5.1 Numerical Analysis for Two Underlying Assets

95

Table 5.1. Capacity factors for two underlying assets

1 2 3 Plant 1 = Asset 1 0 1.0 1.4 Plant 2 = Asset 2 0 2.4 3.2 overview of the investment context and production process before we define the base case investment parameters. The production process of the considered high–tech company consists of two major steps: first a basic production process that is identical for all products, second a finishing process that is specific to the desired applications. The same production technology is used in different plants on different sites. The installment of a plant is naturally linked to expansion options because there typically is a “ramp up” of capacity in several capacity steps. This is feasible because a plant is designed modularly, with several production lines. Two partially complementary standards are produced, which are both exposed to highly volatile markets. The analyzed example is as follows. The company may invest in two plants, one for each standard, which are labelled as plant 1 and plant 2, respectively. For both we assume a base capacity of 100,000 units output per month. Plant 1 is less technologically intensive because it is a much more common technology. This is why it is possible to use this plant also for other products. Plant 2 is a new technology and hence much more specific, such that there is no alternative use for it. In that sense, it is a risky venture for the company. The free cash flows realized in both plants follow Jarrow–Rudd stochastic processes (see Sec. 4.4.1). Each investment installment is instantaneous and two decision points are considered. This short decision period makes it possible to keep track of the system state space and the interdependent decisions made at each state. For a more real–life decision problem a longer period would have to be modeled and a state–dependent residual value should be included. Here it is sufficient to highlight the dynamics of decisions over time, which can be shown in a short period. Both plants can be operated in 3 operating modes which determine capacity as a factor of base capacity. Plant 1, i.e., asset 1 can be operated at a capacity level of 0, 1.0 and 1.4 of base capacity, respectively; plant 2, i.e., asset 2 can be operated at a capacity level of 0, 2.4 and 3.2 of base capacity, respectively (see Table 5.1). The investment cycle starts in mode 1 “not invested”, i.e., zero capacity for both assets. Thus, for each asset there are 2 European options in period 1. In the second period, each asset can enter in exactly 1 out of 3 modes and can either stay in the same mode or switch to 1 out of 2 alternative modes. Thus, each mode can switch to two alternative modes, establishing

96

5 Numerical Analysis

3 × 2 = 6 European options for each asset. Adding up, there are 2 + 6 = 8 European options for each asset and 2 × 8 = 16 European option for the entire decision problem. By the help of this example it is also insightful to illustrate the number of resulting paths which are a consequence of investment decisions and stochastic realizations of the underlying assets. In the first period, each underlying asset enters in mode ai = 1, and can be switched to all ai0 = 1, 2, 3. Therefore, there are 32 = 9 possible combinations of investment decisions. After the switch, the binomial lattice tree branches into four successor nodes, at t = 2. Based on the entering mode, in each of the 4 nodes, each asset can again choose among ai0 , such that there are again 9 possible combinations of decisions. Combining investment decisions and stochastic realizations, for n = 2 underlying assets, Ai0 = 3 leaving modes, and time horizon T = 2, the number of paths through the tree is given by: Y i

Ai0 ∗ [2n

Y

Ai0 ](T −1) = 9 ∗ [4 ∗ 9]1 = 324

paths.

(5.1)

i

Switching costs represent investment cash outflow (positive sign) and disinvestment cash inflow (negative sign). Non–permissible switches are represented by a prohibitively large number (here, 10, 000); staying in the same mode is always costless. Switching costs are assumed constant over time.1 The structure of the switching costs is such that there are synergies to scale in building a higher capacity for asset 2. Thus, it is cheaper to move from no capacity to large capacity instead of passing via intermediate capacity (see Table 5.3). Together with the capacity levels in the respective modes, it becomes clear that asset 1 is designed smaller than asset 2, which also translates into lower investment outlays. On the other hand, given the economies of scale in investing, the investment outlays for asset 2 in relation to capacity are more attractive, but the involved risk is also higher because there are no options to scale down or abandon asset 2. While asset 1 is less technologically intensive and can be used for several alternative products, asset 2 is totally specific and therefore a bigger gamble. The company may invest in one out of the two assets, both assets, or not at all, and adjust the investments over time. For doing so, its shareholders have scheduled a specific budget for future periods, as an organizational measure of controlling their specific risk exposure in their overall portfo1

Constant switching costs make the options American since the exercise price is independent of the exercise date and exercising in period 1 can be viewed as early exercise. It is preferable though to keep the concept of European options in mind here because the assumption of constant switching costs is only made for ease of discussion. Therefore, the general case in the spirit of the presented model are European options.

5.1 Numerical Analysis for Two Underlying Assets

97

Table 5.2. Switching costs for asset 1 (two underlying assets)

Asset 1 1 2 3 1 0 30 40 2 -20 0 8 3 10,000 10,000 0 Table 5.3. Switching costs for asset 2 (two underlying assets)

Asset 2 1 2 3 1 0 45 50 2 10,000 0 20 3 10,000 10,000 0 lios, which must not be exceeded. Within these limits the company has been delegated full discretion how to best invest these funds. Thus, for both periods, a constant budget allowance is available. Unused budget allowance from period 1 remains with the investors but may be drawn in period 2 (bearing no interest though) and money freed up by disinvesting may be plowed back immediately. Further, we assume that there are no relevant interactions beyond the considered business. With this, the investment problem constitutes a portfolio of real options with two underlying assets and 16 real options. 5.1.2 Overall Portfolio Effects Due to the flexibility of the proposed approach, even the “simple” case of a two–period, two–asset, three–mode problem can have a large number of parameter combinations. Each case is fully specified by 45 parameters: 2 underlying processes with starting values and volatilities each; a joint coefficient of correlation; an initial budget endowment and 2 budget allowances; 9 switching costs for each period and asset; a risk–free interest rate.2 In the following numerical analysis, this will be simplified by fixing some degrees of freedom: no initial funds are available; budget allowance and switching costs are constant over time; switching costs and risk–free rate are fixed. Thus, the following numerical analysis is fully specified by variations of 6 parameters leading to 2, 880 cases,3 each one solved for the optimal portfolio value and the implicit contingent optimal exercise policy, through one optimization run. The following parameter variations are considered: 2 3

This makes a total of 2 × 2 + 1 + 3 + 2 × (2 × 9) + 1 = 45 parameters. For each underlying asset, 3 starting values and 2 volatilities; 2 coefficients of correlation; 40 levels of budget allowance per period, therefore (3 ∗ 2)2 ∗ 2 ∗ 40 = 2, 880 cases.

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5 Numerical Analysis

• Underlying processes for each underlying asset: starting values α1 , α2 = 25, 35, 45; volatilities σ1 , σ2 = 0.3, 0.7; coefficient of correlation ρ = −0.6, 0.7. • Budget constraint: budget allowance is constant per period, i.e., β(1) = β(2) = β, β = 14, 16, . . . , 92, i.e., varying between 14 and 92 in 39 steps of size 2. No initial funds are available, i.e., B1 = 0. For T = 2, the joint representation of the behavior of the two-period optimization model is given in Fig. 5.1. It depicts overall portfolio value as a function of budget allowance per period for different portfolios. Each line in the landscape represents a portfolio and each portfolio is specified by a joint combination of parameters within the above specified ranges (72 series in total). It gives an intuitive understanding of the sensitivity of the overall system which in some ranges can be relatively stable (flat regions) whereas in other regions it is more sensitive (rugged regions). It must be noted though that this representation provides limited insight since there is no logical relationship between the different series. Further, the joint effects are too rich to be clearly captured, such that it is preferable to identify them separately in a comparative–static analysis, in order to pinpoint specific aspects in this complex rugged landscape. Thus, in the following the single aspects will be analyzed separately with suitable graphical representations. Specifically, budget effects, volatility effects, and correlation effects will be exposed in detail. 5.1.3 Budget Effects and Other Constraints One main point of interest in the present discussion are budget effects. From a theoretical standpoint it is compelling to consider budget constraints in the optimization because it is intuitive that they will affect the global optimum significantly. In stark contrast to this consideration, budget constraints are oftentimes neglected. This can only be correct if valuation results are independent from budgets and do add up. We can show by example that this is generally not the case. The intuition behind the importance of considering budget constraints is the following: a binding budget constraint means that the optimization cannot achieve the optimum that would be obtained in the unconstrained case. Therefore, a binding budget constraint immediately implies deviations from value additivity, as compared to the unconstrained case. An analog to the case of two underlying assets with binding budget constraints can be found in the financial option on the maximum of two underlying assets (Stulz 1982) if the budget is such that only one asset can be acquired. In financial option pricing, it is evident that the value of the option on the maximum of two underlying assets can only correspond to the sum of two

5.1 Numerical Analysis for Two Underlying Assets

99

350 300 250

Portfolio value

200 150 100 50 0

86 68 50

Portfolio specifications

32 14

Budget allowance per period

Fig. 5.1. Portfolio landscape

options on the respective underlying assets if one option is approximately worthless. In the analyzed portfolios of real options, the portfolio value is a step– function of the budget level. This is depicted in Fig. 5.2 which visualizes a selection out of the whole range of data as presented in Fig. 5.1. It shows 9 different portfolio configurations of the above introduced investment problem where each portfolio is represented by a separate line. Each line gives the overall portfolio value created on the ordinate for different levels of budget allowance per period on the abscissa. Since the budget level is a constraint, changes in portfolio value as a function of budget allowance imply a change in the corresponding optimal investment strategy. Whenever a binding budget level hits a new “relevant” budget level from below, portfolio value increases. A relevant budget level here is defined as a budget level which enables to implement a different, more valuable investment strategy which requires higher investment outlays. The value effects in some regions are higher than in other regions and small changes in budget level can have a considerable impact on the portfolio value and the respective investment strategy. Overall, this shows that, over this range, there are no diminishing returns to investing more funds into projects.

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5 Numerical Analysis

Fig. 5.2. Portfolio value as a step–function of budget allowance Table 5.4. Portfolio highlighting budget effects

Portfolio α1 α2 σ1 σ2 ρ Portfolio 6 45 35 0.7 0.3 0.7 Portfolio 7 25 45 0.7 0.7 −0.6 For the discussion of budget effects, the effects for one portfolio separately can be distinguished from the effects when comparing two portfolios. 5.1.3.1 One Portfolio As a starting point, it is insightful to consider the budget effects for one portfolio separately, e.g., portfolio 6 as specified in Table 5.4. In some regions, value is independent of the available budget. This case does not require further analysis since it is trivial in that the optimal strategy remains unchanged. Second, in other cases, value increases with increasing funds available but to a rather small degree, e.g., for increasing β from 36 to 38. Third, value can substantially increase in other cases, e.g., for increasing β from 48 to 50. Investment strategies for portfolio 6 can be found in Fig. 5.3, on the left hand side for a budget allowance of β = 36, on the right hand side for β = 38. Each table gives the time and scenario specific optimal investment decisions for each of the two underlying assets. In each box, the first number indicates the entering mode while the second number indicates the leaving mode. For instance, (1 → 2) indicates that the asset enters the respective

5.1 Numerical Analysis for Two Underlying Assets

101

Fig. 5.3. Small change in portfolio value

scenario in mode 1, (not invested) and in that scenario switches to mode 2, investing into low capacity. The box on the left hand sides gives switching decisions for asset 1, the one on the right hand side switching decisions for asset 2. Changes in the optimal strategy are highlighted with a bold frame. While for β = 36 in period 2 it is optimal to disinvest asset 1 and install large capacity for asset 2, β = 38 makes it possible in period 2 to keep operating asset 1 and to invest in low capacity for asset 2, resulting in an increase of the portfolio value from 101.5 to 110.2. This is because two opposing effects occur in parallel: The new investment strategy sacrifices value realized by asset 2 in that it switches to low capacity instead of high capacity, but in return can avoid to disinvest asset 1. The relatively small value impact shows the positive net effect which is a consequence of a substantial modification of the optimal strategy, involving parallel investment and disinvestment. Changing β from 48 to 50 has a much bigger effect, with value boosted from 155.2 to 204.6 (see Fig. 5.4). Here, the additional budget allowance makes it possible to invest more profitably, i.e., returns do not diminish with increasing investment. It is optimal to install high capacity for asset 2 in the first period and then remain in that mode, instead of first installing low capacity and then expanding capacity (which can only happen in states where enough money is available). This makes it possible to realize cash flows earlier and to benefit from synergies of scale in the investment costs. Since no value needs to be sacrificed, the net effect is just the additional investment value and hence much bigger. Structurally, this case here is much more straightforward than the case of β = 38 because here the additional money simply is used without substantial impact on the overall strategy. The more complex case requires more detailed analysis and the optimization can partially overcome the adverse effect of the constraint by readjusting the overall strategy to a maximum; the system “breathes”. The second case is simpler because it just implies waiting for more money and investing that money once it is available, where it creates highest value, but otherwise leaving the investment strategy unaffected.

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5 Numerical Analysis

Fig. 5.4. Substantial change in portfolio value

In the presented example, the budget allowance per period was assumed to be constant over time, i.e., in each period the same amount of budget allowance is added to the funds available. The general model formulation allows for varying budget allowances per period. Following this direction further, it is also conceivable to allow for state–dependent budget allowances, especially as some function of past cash flows or expected cash flows in the future. This may help to model specific decision settings but does not shed further light on the discussion of the effects of the budget constraint on portfolio value. This is because the budget equation either way is path– dependent as it is a function of investment decisions which in turn are state– and path–dependent. What matters in the discussion of budget effects is this path–dependency. Adding more detail to the modeling of the budget equation clearly changes the actual results of a specific investment problem because it modifies the funds available at each state. Conversely, the structure of the budget dynamics remains unchanged. Thus, the effects encountered in the analyzed example are general in that they would also arise for state–dependent budget allowances and further possible extensions. Result: For the behavior of one portfolio at a time, it is possible to distinguish three categories of effects4 that stem from an increase in budget allowance: 1. Value and optimal strategy are unaffected. The additional money available does not permit to implement a new strategy with higher value. 2. Value increases to some degree and the optimal strategy is typically modified substantially. The gain in value is obtained from simultaneous disinvesting (at some opportunity cost) and investing. 4

These categories are combinations of a) whether the portfolio value changes (yes/no), and b) whether the optimal exercise strategy changes (yes/no). However, we only consider three instead of four categories, because a different optimal exercise strategy typically translates into a change of value. If not so, this would be the special case of equivalent alternative strategies, which is not insightful for the discussion of portfolio effects.

5.1 Numerical Analysis for Two Underlying Assets Portf. 7 β = 44 s=1

t=1

1→2

t=2

t=1

2→3

1→3

s=1

1→1

2→3

2→3

1→3

s=2

1→3

2→2

s=3

2→1

1→3

s=3

1→1

2→3

2→1

1→3

s=4

1→1

2→2

Asset 2

s=4

Value = 122.2

Portf. 6 β = 44 s=1

t=1

1→2

Asset 1

1→2

Asset 2

+ 59 %

t=2

Value = 194.8

Portf. 6 β = 46

t=1

2→3

1→3

s=1

1→3

2→2

2→3

1→3

s=2

1→3

2→2

s=3

2→1

1→3

s=3

1→1

2→3

2→1

1→3

s=4

1→1

2→3

Asset 2

s=4

Value = 134.7

1→1

t=2

s=2 Asset 1

1→1

1→1

t=2

s=2 Asset 1

1→1

Portf. 7 β = 46

103

Asset 1

+ 15 %

1→2

Asset 2

Value = 155.2

Fig. 5.5. Comparison of optimal investment strategies

3. Value increases substantially and the optimal strategy is typically modified in part only. The additional funds available enable to undertake a new investment. 5.1.3.2 Relative Attractiveness of Two Portfolios Besides the effect of level shifts in the portfolio value of one portfolio, an increase in available money can reverse the relative attractiveness of two portfolios, i.e., the graphs in Fig. 5.2 can intersect. To see exactly why this happens, it is insightful to consider two examples, portfolio 6 and portfolio 7, as given in Table 5.4. These two settings can be considered as representative because they cover important classes of possible parameter specifications. They cover one case each for positive and negative correlation, reversed starting values for the underlying assets, as well as low and high volatilities. The analysis of their joint behavior can best be achieved by discussing their respective investment strategies (as given in Fig. 5.5). As a reference case, consider portfolio 7 for β = 44. In this case, it is optimal in the first period to invest in asset 1 with low capacity. Then, in the second period, invest in asset 2 with high capacity, and depending on the scenario either scale up asset 1 (in the up scenarios) or disinvest (in the down scenarios). This strategy yields an overall portfolio value of 122.2. As opposed to this, by just allowing for an additional budget allowance of 2

104

5 Numerical Analysis

units per period, i.e., β = 46, the optimal investment strategy is completely reconfigured. In this case, it is optimal to move into asset 2 with low capacity, in the first period. Then, in the second period, the strategy becomes more elaborate. Asset 2 remains operating in low capacity in scenario 2 and 4, which is only made possible through the corresponding strategy for asset 1. In scenario 2, i.e., in scenario (u1 , d2 ), asset 1 gets invested in high capacity and asset 2 continues operating in low capacity. As opposed to this, in scenario 1, i.e., scenario (u1 , u2 ), asset 1 remains uninvested whereas asset 2 is scaled up. It is apparent that the decision on asset 1 would be to invest in large capacity in the case of unrestricted funds. However, since money is short, it is more attractive to spend the limited money on asset 2, which is true except for scenario 2. Overall, with the increased budget level much less money is invested in asset 1 and it is feasible to invest earlier into the relatively more attractive asset 2. Figuratively speaking, asset 1 steps back for asset 2. The portfolio value significantly benefits from this shifted strategy with an increase of roughly 60%. This example has presented a case where money is channelled to one asset at the disadvantage of another asset which would receive the funds if the other asset were not more profitable. More profitable means that it must be profitable as such and overcompensate for the opportunity cost in the not invested or disinvested asset. Generalizing, this effect is encountered when the relative attractiveness of investments is decisive for the allocation of limited resources between assets. In the presented example, the interaction between assets took place in the same period where a profitable opportunity was not seized. Structurally, the same effect can be obtained if an already operating asset gets disinvested to free up the necessary resources. Both disinvesting an existing asset and not seizing an investment opportunity can happen in one period or between periods. An investment may not be undertaken in one period in order to save funds for investment in a later period (keeping the options open); or a disinvestment may be realized earlier in order to seize a put option before it expires. These effects are different versions of the effect demonstrated in portfolio 7, yet structurally they are the same. Further, the relative behavior of portfolio 7 in comparison to portfolio 6 is insightful. for For β = 44, portfolio 6 yields a portfolio value of 134.7, as opposed to a value of 122.2 for portfolio 7. These values are realized through an identical investment strategy as in portfolio 7, which is feasible for both portfolios since both have an identical budget allowance of β = 44, but differ in value due to the differences on the portfolio parameters. Remarkably, for β = 46 the investment strategy substantially changes, both compared to the lower budget level and to portfolio 7 for the same budget level. For portfolio 6 and β = 46, it is optimal in the first period to remain uninvested in asset 1 and to install low capacity for asset 2, like for portfolio 7. Unlike portfolio

5.1 Numerical Analysis for Two Underlying Assets

105

7, in the second period in scenario 1 it is optimal to install high capacity for asset 1 and stay in low capacity for asset 2, whereas in scenario 4 it is optimal to scale up capacity for asset 2. The different investment strategy is due to the different relative attractiveness of the investment opportunities since asset 1 is much more valuable and asset 2 is less valuable than in portfolio 7 (see Fig. 5.4), while the switching costs matrix is identical. This is why there is not as much upside potential in changing the investment strategy; value sacrificed by not investing into asset 1 is still overcompensated by investing into asset 2 in the first period, but not as clearly. Also, given the different relative attractiveness of asset 1 and 2 in portfolio 6, the investment strategy in period 2 is different from the strategy in portfolio 7, which is explained by the switching costs involved. The increase in value is still substantial and the investment strategy completely changes, yielding an increase in portfolio value to 155.2, or 15%. Compared to the budget effect for portfolio 7 with a shift in portfolio value of some 60%, the effect on portfolio 6 is much less pronounced. A slight relaxation of the budget constraint not only results in a fully reconfigured optimal investment strategy and higher values for both portfolios, but also in a net effect where the relative attractiveness of the two portfolios changes. Graphically, this translates into intersecting lines in Fig. 5.2. This intensifies the observation that budget constraints must be considered when comparing different possible portfolios. It is not straightforward to judge which portfolio would be preferable because it all depends on the budget, and more so, is highly sensitive to it. For less funds available portfolio 6 is more attractive, whereas with little more funds available portfolio 7 takes the lead. Result: For the budget effects on the optimal investment strategy and corresponding portfolio value, the following properties can be summarized: 1. Binding budget constraints have a significant impact on the optimal investment strategy, clearly deviating from value additivity. 2. The portfolio value is a nonlinear step–function of budget level. 3. The investment decision is highly reactive to budget levels: small changes in the budget constraint can result in a considerable adjustment of the optimal investment strategy, and have a substantial impact on portfolio value. 4. The relative attractiveness of different portfolios can be reversed for different available budget levels. 5.1.3.3 Generalization of Budget Effects The above discussion has demonstrated that binding budget constraints play an important role in the investment decision process. They need to

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5 Numerical Analysis

be considered in all dimensions of real options portfolio management, i.e., both in the design and in the execution of portfolios. Budget constraints have been exposed in detail as representative for any other type of constraint that can be included in the model. This is for two reasons: First, budget constraints appear as key to any investment process which at its core is about spending money that must be available. Second, budget constraints as balance equations represent a modeling challenge that requires special attention. Obviously, any constraint will have an impact specific to the actual decision problem, but structurally the effect will be the same as for budget constraints: Binding constraints imply deviations from the unconstrained optimum and the optimization ensures to obtain the best possible result given the constraints. Relaxing constraints even slightly can have a huge impact on the optimal strategy and portfolio value, stressing the potential of a simultaneous optimization. Result: Binding budget constraints reduce the value of portfolios of real options, but the negative effect can be minimized by a suitable investment strategy. Such an investment strategy, can be sophisticated in the sense that it involves investment and disinvestment decisions in parallel, realizing a positive value effect. If constraints exist in the real world investment problem, ignoring them in the investment analysis forgoes some of the value potential for a (by luck) basically correct investment strategy. In other cases, this can lead to a plainly sub–optimal investment strategy. 5.1.4 Volatility Effects Closely related to the discussed budget effects, it is insightful to analyze the effect of changes of volatility on portfolio value. Increasing volatility in the proposed model can have two main effects: value increasing in volatility and value decreasing in volatility. First, option value increasing in the volatility is a common property of standard options, both puts and calls (Black and Scholes 1973 and Merton 1973b, Theorem 8). In our numerical investigation, we have confirmed this behavior for the discussed portfolio of real options. Therefore, this behavior of the model is straightforward and will not be discussed in detail here. Second, as opposed to these first two effects, option value decreasing in the volatility is an interesting case that merits special attention. This effect has first been shown by Stulz (1982) for European options on the minimum or the maximum of two assets. He shows that the effect of an increase in volatility of the price of one underlying asset is ambiguous and can also result in decreasing option value. The budget constraint can introduce, to some degree, a choice between assets that compete for the joint funds available. As a consequence, the case of two underlying assets is structurally comparable to a call on the maximum of two underlying assets.

5.1 Numerical Analysis for Two Underlying Assets

107

Table 5.5. Portfolio with value decreasing in volatility

Portfolio Portfolio 3Portfolio 3 Portfolio 3+

α1 45 45 45

α2 25 25 25

σ1 0.3 0.3 0.7

σ2 ρ β Portfolio value 0.3 0.7 32 73.6 0.7 0.7 32 85.0 0.7 0.7 32 78.7

As a special case, value can also be independent from volatility, but this is due to the model setup only. The possibility of value independent from volatility is implied by assuming discrete processes for the underlying assets. With discrete processes, option values also become non-smooth in the underlying assets. A change in the volatility only affects the exercise strategy and thus the option value if it makes the underlying asset move beyond a given exercise price (cf. Cox et al. 1979). This introduces discontinuity. The choice of a discrete framework was motivated by the resulting flexibility for the modeling of options and constraints (see Sec. 4.2.1). Thus, the effect is by design and does not deserve special attention, because it would vanish in a continuous model setup. Further, for exotic financial options, this same property can be observed, e.g., for an up–and–out barrier call option (see Merton 1973b, pp. 177 f.). This option is structured like a plain vanilla call option with the difference that it becomes worthless if the underlying asset rises above a given barrier, throughout the lifetime of the option. This is why a barrier option is path–dependent: the value of the option at expiry depends on the path taken by the underlying asset. On all paths that have hit the barrier at least once throughout the lifetime of the option, the option has zero value since it is foregone (Kwok 1998, pp. 246 ff.). In a recent application, Wong and Kwok (2003) provide examples for a barrier option where increasing volatility translates into a lower option value because the chance of hitting the barrier increases and more probability is on cases where the option becomes worthless. Similar to the results by Stulz (1982), barrier options are complex contracts because they are ambiguous in the sensitivity to parameter variations (Wilmott 1998, pp. 189-201). The effect of increasing and decreasing value can best be shown by the example of portfolio 3, portfolio 3-, and portfolio 3+, as given in Table 5.5. First, increasing the volatility of asset 2, σ2 , from 0.3 to 0.7 increases portfolio value by 15.5%, from 73.6 to 85.0. As discussed before, this case is expected and does not merit special attention. More interestingly, increasing the volatility of asset 1, σ1 , from 0.3 to 0.7 reduces portfolio value by 7.4%, dropping from 85.0 to 78.7. This case deserves a more careful discussion. Overall, the effect is due to a binding budget constraint, which is apparent in the optimal operating strategy, as depicted in Fig. 5.6. For portfolio 3, in period 1 it is optimal to install low capacity for asset 1 and stay unin-

108 Portf. 3 σ1= 0.3 s=1

5 Numerical Analysis t=1

1→2

t=2

t=1

2→1

1→3

s=1

2→1

1→3

2→3

1→1

s=2

2→3

1→1

s=3

2→1

1→3

s=3

2→1

1→3

2→3

1→1

s=4

2→1

1→1

Asset 2

s=4

Value = 85.0

1→2

t=2

s=2 Asset 1

1→1

Portf. 3+ σ1= 0.7

Asset 1

1→1

Asset 2

- 7.4 %

Value = 78.7

Fig. 5.6. Portfolio value decreasing in σ1

vested for asset 2. In period 2, it is optimal to invest in high capacity of asset 3 in scenarios 1 and 3, i.e., when the value of asset 2 develops favorably. This is only made possible through disinvesting asset 1 in these scenarios; in the other scenarios it is optimal to expand asset 1. In other words, asset 1 is profitable both in its respective up and the down scenarios, but needs to be disinvested in order to provide the funds necessary to realize the yet more profitabel investment in asset 2, in scenarios 1 and 3. Increasing the volatility of asset 1, in portfolio 3+, naturally results in a higher variability of the outcomes of asset 1. Usually, a standard option can profit from the increased variability by capitalizing on the upside while protecting from the downside; basically this is why standard option values are increasing in volatility. This cannot happen in the presented portfolio 3+. On the upside, in period 2 in scenario 1, the portfolio cannot exploit the higher value of asset 1 because it is still optimal to disinvest asset 1 in order to finance asset 2. Reversely on the downside, in period 2 and scenario 4, the portfolio is fully hit by the reduced value of asset 1. While in that scenario it is optimal to scale up asset 1 for lower volatility, it is optimal to disinvest for higher volatility. The additional value realized in period 2 in scenario 2 for asset 1 cannot overcompensate for the loss in value, such that the overall net effect is negative. The money freed up by disinvesting here is assumed to be constant, therefore independent of time and scenario, in analogy to the deterministic exercise price of a standard put option. Allowing for state–dependent disinvestment payoffs, which is a possible extension of the proposed model, can further increase this effect. The analogy to a barrier option consists of the analogous existence of an adverse effect on value. For the barrier option, the existence of the barrier introduces some probability that the option cannot be exercised even if it expires in the money, depending on the path taken by the underlying. Just the same holds true for a budget constrained portfolio of real options. Here the existence of the budget is such that on certain paths the full (unrestricted) value potential cannot be exploited and portfolio optimization can only partly countervail the adverse effect on value. At the same time, the

5.1 Numerical Analysis for Two Underlying Assets

109

analogy between barrier options and portfolios of real options is intuitive but cannot shed light on the valuation of portfolios of real options because they are concerned with valuation issues that are fundamentally different. The balance equation of the budget, path–dependencies of decisions, as well as interactions and different types of constraints are key for portfolios of real options, but irrelevant for barrier options. Reversely, the valuation of barrier options within a discrete valuation framework requires special attention to the discretization of the system state space relative to the barrier, such that nodes lie on the barrier (Hull 2003, pp. 468 ff.). This is not an issue for portfolios of real options because the budget level is independent from the underlying assets. Result: The values of portfolios of real options can be either decreasing or increasing in the volatility of the underlying assets. This attaches special importance to the assessment of volatility and risk in portfolios of real options. Contrary to financial asset portfolio management, adding new assets or options to the portfolio and by this changing the overall risk profile can have a positive or a negative effect on portfolio value. 5.1.5 Correlation Effects The present analysis deals with both correlation in the underlying assets and correlation in the cash flows. As argued in Sec. 4.1.6 and without loss of generality, it is sufficient to focus on the effect of the correlation of the underlying assets on the values of the portfolios of real options. Considering correlation effects can best be positioned by pointing out that the investment problem analyzed here has structural parallels to a call option on the maximum of two assets, as discussed before. For a European call option, Stulz (1982) shows that option value is decreasing in the correlation: the higher the coefficient of correlation, the less valuable the option. In our numerical analysis, the same effect is found. This result is mainly driven by the binding nature of the budget constraint, and the payoff function. In settings without a binding budget constraint, it is possible to observe positive correlation effects on portfolio value. For example, Dias (2006) recently discuss correlation in oil exploration, without budget constraints. In this setting, correlation tends to have a positive effect because learning about one asset helps to optimally manage other assets. Further, correlation makes it possible to capitalize further on the upside potential for all assets, leading to higher overall returns the higher the correlation. It is noted though that this positive correlation effect can only be encountered when these assets do not compete for limited funds. For our setting, we can show a negative correlation effect by the example of portfolio 3 (see Table 5.5), for β = 32. The optimal exercise strategy for this portfolio is depicted in Fig. 5.7, on the top. Below, the state–dependent cash

110

5 Numerical Analysis Portf. 3 β = 32 s=1

t=1

1→2

t=2

2→1

1→3

s=2

2→3

1→1

s=3

2→1

1→3

2→3

1→1

Asset 1

1→1

Asset 2

s=4

Portf. 3 β = 32

t=1

t=2

Prob.

Prob.

ρ = -0.6

ρ = 0.7

20.0

82.6

0.100

0.425

s=2

77.5

0.0

0.400

0.075

s=3

20.0

82.6

0.400

0.075

38.9

0.0

0.100

0.425

s=1

s=4

15.0

0.0

Asset 1

Asset 2

Value = 97.0

-12.3 %

Value = 85.0

Fig. 5.7. Portfolio value decreasing in ρ

flows for the optimal exercise strategy are given. Next to it, for both levels of ρ and the corresponding risk–neutral probabilities, the optimal value is given. In the presented case, increasing ρ from −0.6 to 0.7 does not affect the optimal exercise policy, and therefore, not the state–dependent cash flows but the corresponding portfolio value. This is due to the fact that the risk– neutral probabilities change. While ρ = −0.6 puts much probability on scenarios 2 and 3, this changes for ρ = 0.7 where most probability is on scenario 1 and 4. Intuitively speaking, scenario 1 and scenario 4 are scenarios where both assets behave in parallel, such that for a higher correlation these scenarios increase in probability and reversely so for scenarios 2 and 3. Since scenarios 2 and 3 jointly yield higher cash flows than scenarios 1 and 4, reducing their respective probabilities decreases portfolio value. This observation confirms the intuition. If both assets compete for a binding budget constraint, scenarios where both assets develop in the same direction suffer most from the constraint. For example, in the unconstrained case in scenario 1 it would be optimal to expand asset 1. In the constrained case, the opposite is optimal: asset 1 is disinvested to finance asset 2. Putting more probability on this case reduces overall portfolio value because the adverse budget effect has more weight. Just as for financial options, it is also possible that a change in the correlation has no value effect at all. For a financial option on the maximum of two assets, the correlation impact depends on the relative being in or out

5.1 Numerical Analysis for Two Underlying Assets

111

of the money of the option with regard to the separate underlying assets. As an intuitive example, if the exercise price is high enough that one of the assets will never be acquired, i.e., one asset is dominated, the option can be priced independently from the dominated asset because the correlation in the probabilities cancels out. Supposing asset 2 is dominated and using the formulation given in Sec. 4.2.3, the risk–neutral probabilities for asset 1 give: " !# √ r f − 12 σ12 1 1 + ∆t (5.2) p( asset1, up) = p1 + p2 = 2 σ1 " !# √ r f − 12 σ12 1 1 − ∆t (5.3) p( asset1, down) = p3 + p4 = 2 σ1 Also in the limit, the correlation cancels out (cf. Boyle et al. 1989, p. 246): √lim p1 ∆t→0

+ p2 =

1 2

(5.4)

√lim p3 ∆t→0

+ p4 =

1 2

(5.5)

In our investment setting, a similar effect is encountered when one asset is too expensive to be ever acquired. Nevertheless, this is just a special case; for investment decisions that are not as extreme, a negative effect of correlation is to be expected. In an investment setting as is discussed here, the budget constraint is such that the competition between projects for limited funds makes it desirable to look for reduced correlation, or even negative correlation.It is important to note that this is not in order to diversify, but on the contrary to increase the range of outcomes in order to capitalize on the upside potential while being protected from the downside. Result: In the presence of budgets constraints, the values of portfolios of real options can be decreasing in the correlation. In other settings, when there are synergies or learning effects between assets in the absence of budget constraints, the values of portfolios of real options can be increasing in the correlation. 5.1.6 Starting Value of the Underlying Assets Effects Further to the so far discussed effects, it is evident that the starting values of the underlying assets have an important impact on portfolio value. For the presented problem, the value of the portfolios of real options is increasing in the starting value of the underlying assets. This can be visualized by the

112

5 Numerical Analysis

Fig. 5.8. Portfolio value increasing in α1 and α2

example of portfolio 6 (Table 5.4), for β = 38, and varying α1 and α2 (see Fig. 5.8). It is noted that in the specific case, the behavior is nearly linear, i.e., a change in the starting value linearly translates into a change in the portfolio value. This is not the point to be made here though because it is not a general result. A counter–example is a case where only by increasing the starting value, a specific option is exercised. This would result in a kinked curve known as a standard payoff–profile of a call option at maturity. If the price of the underlying asset is higher than the option exercise price, the option value is a linear function of the underlying, otherwise its value is zero. Result: This further confirms that the presented problem has the net payoff profile of a call option. This is intuitive, as the decision problem starts in a not invested mode and is therefore mainly an investment problem with disinvestment options, not the other way round. 5.1.7 Path–Dependency Effects All of the discussed effects are inseparably connected to the path– dependencies involved. Decisions about one of the underlying assets depend on the path taken by the other assets and the decisions made until this point since the budget is a path–dependent function of earlier decisions. This can be illustrated by a decision problem for three decision points, as given in Table 5.6. It follows the same structure as the examples discussed before, with the difference that there is one more decision point, i.e., T = 3. The budget starts with an initial endowment, B1 , of 0 and additional allowances per period, β, of 32.

5.1 Numerical Analysis for Two Underlying Assets

113

Table 5.6. Portfolio highlighting path–dependency for T=3

α1 α2 σ1 σ2 ρ B1 β Portfolio value 45 25 0.7 0.7 0.7 0 32 178.5 Asset 2, t = 3 t=1

A s s e t 1

t=2

u2u2

u2d2

d2u2

d2d2

u1u1

3→3

3→3

3→3

3→3

u1d1

3→3

3→3

3→3

3→3

d1u1

1→2

1→2

2→3

2→3

d1d1

1→1

1→1

2→1

2→1

t=3

u1

u1

d1

d1 u1 d1

Fig. 5.9. Path–dependency of underlying asset 1

The path–dependency underlying this investment problem can be visualized in Fig. 5.9. On the left hand side, it visualizes the jump process for asset 1 and T = 3. Here, the path–dependency of asset 1 becomes apparent up the nodes for (up, down) and (down, up) do not coincide. On in the fact that down the right hand side, the figure gives an overview of the investment decisions made for asset 1 at t = 3 which are indicated in the 4 × 4 = 16 boxes on the right. The corresponding investment decisions for asset 2 are omitted here because they would not clarify the path–dependency further. Here the goal is to show how asset 1 is affected by the joint movement with asset 2, which is captured through the scenarios for asset 2. On the horizontal of the right hand side, the different scenarios for asset 2 are specified alongside. Asset 2 is also path–dependent since (up, down) needs to be distinguished from (down, up). In Fig. 5.9, path–dependency for asset 1 can be observed in three ways: 1. Path–dependency arising from decisions over time. The sequence of decisions before period t = 3 and especially the decision made at t = 2 determine the operating mode in which asset 1 enters period 3. Precisely, the entering mode at t = 3 must be identical to the leaving mode at t = 2. This mode is the starting point for a possible switch and hence has an impact on which leaving mode is optimal from there. This source of path–dependency is visible in all 16 boxes because all possible entering modes, a1 = 1, 2, 3, appear. 2. Path–dependency arising from the stochastic process of asset 1. Following the stochastic process of asset 1, the scenarios of (up,down) and (down, up) need to be distinguished. They correspond to the second and the third line of the boxes. Fixing the second asset to (up, up), it

114

5 Numerical Analysis

becomes clear that although the scenario of asset 2 is the same, the two lines present different operating strategies (see boxes in bold frame). Thus, it is not possible to model asset 1 as recombining, which establishes the second source of path–dependency. 3. Path–dependency arising from the interplay of assets. Before, it was insightful to fix the scenario of asset 2. Reversely, it is insightful to fix the scenario of asset 1 to (down, up). Looking at the different scenarios for asset 2 reveals different strategies (see dotted boxes). Since the scenario for asset 1 is fixed, these different strategies result from the development of asset 2 and the respective optimal investment strategies. The strategy for asset 1 is different for the scenarios (up, down) and (down, up) of asset 2, thus another source of path–dependency. This path–dependency for the decisions made about asset 1 is driven by the path–dependency of the other asset. Result: This example has visualized the path–dependencies which are at the core of the proposed model. It simultaneously stems from the path– dependencies in decisions over time in the development of each underlying asset and in the interplay of the underlying assets. It should be noted though that this distinction is helpful to fully grasp the degree of path–dependency, but that it is not very rigorous. Given that these drivers occur simultaneously, rigorously speaking, there is only one joint path–dependency. 5.1.8 Deviations from Value Additivity The above discussion of value drivers has repeatedly recurred to the fact that value additivity does not hold for portfolios of real options and that option values are typically sub–additive. In order to stress that nature, the reasons behind this are reviewed to conclude the numerical analysis. First, budget constraints introduce deviations from value additivity which was the key insight from Fig. 5.2. This is because a binding constraint, by definition, requires the optimal strategy to deviate from the unconstrained optimum. Second, the switching concept subject to switching costs represents a compound option valuation approach, in which each option exercise, or switching decision, gives birth to new options, contingent on this switch. This also, by definition, establishes deviations from value– additivity. Third, it was shown that an increase in volatility, contrary to common understanding of financial options, can harm portfolio value. Fourth, an increase in correlation for an investment problem that is a net call option reduces portfolio value. These effects occur simultaneously, such that it is not possible to attribute the effect to a single source. Overall, these considerations confirm the expectation that options do not add up, as shown by Trigeorgis (1993a).

5.2 Extension of the Numerical Analysis to Three Underlying Assets

115

For completeness, the sub–additivity can be shown, e.g., for portfolio 3. In the constrained case with β = 32, the portfolio of real options has a value of 85, as given in Table 5.5 and Fig. 5.6. Relieving the budget constraint, it would be optimal to install large capacity for both assets immediately, yielding a portfolio value of 194.6. For completeness, it is noted that options can also be super–additive. Beyond the reasoning given in Trigeorgis (1993a, pp. 8 f.), in the proposed model super–additivity may result from synergies in operating projects together. Result: Given the impact of budget constraints on the optimal policy, typically sub–additivity is to be expected for portfolios of real options. Therefore, in practice the negative effect of options interactions is likely to be more prevalent (cf. Trigeorgis 1993a, p. 19).

5.2 Extension of the Numerical Analysis to Three Underlying Assets Up to this point, the numerical analysis has shed light on the relevant portfolio aspects encountered in portfolios consisting of two underlying assets. This analysis has not yet addressed the joint stochastic movements of several assets, which is considered in what follows. 5.2.1 Numerical Application Setting for Three Underlying Assets The discussion of three underlying assets will focus on the additional portfolio aspects which go beyond the features discussed for two underlying assets. For this, we will present a specific portfolio configuration which will highlight the respective portfolio features successively. The intuition behind the numerical example for two plants in 5.1.1 can be extended by assuming that the additional underlying asset would be located in another country, thus operating with a different cost structure, and especially, with a different correlation structure. In the numerical analysis for three underlying assets, investment decisions are equivalent to switching decisions between 3 operating modes, for each underlying asset, as depicted in Table 5.7. Compared to the before discussed numerical setting for two underlying assets, asset 2 can be considered as an “intermediate” asset, with capacity factors situated between the otherwise unchanged capacity factors of the other two assets (cf. Sec. 5.1.1). Switching between these capacity factors is subject to the corresponding switching costs matrices (see Table 5.8, 5.9, and 5.10), which are assumed

116

5 Numerical Analysis Table 5.7. Capacity factors for three underlying assets

1 2 3 Asset 1 0 1.0 1.4 Asset 2 0 1.4 1.8 Asset 3 0 2.4 3.2 Table 5.8. Switching costs for asset 1 (three underlying assets)

Asset 1 1 2 3 1 0 30 40 2 10,000 0 8 3 10,000 10,000 0 Table 5.9. Switching costs for asset 2 (three underlying assets)

Asset 2 1 2 3 1 0 25 40 2 -25 0 20 3 -40 15 0 Table 5.10. Switching costs for asset 3 (three underlying assets)

Asset 3 1 2 3 1 0 45 50 2 10,000 0 20 3 10,000 10,000 0 constant over time. Compared with the numerical setting for two underlying assets, the newly introduced intermediate asset 2 incorporates a flexible technology, with lowest investment costs and scaling down possible between all operating modes. The switching costs for the other two assets are identical to the structure analyzed for two underlying assets, with the exception that for asset 1, switching back from mode 2 to mode 1 is not permissible (cf. Sec. 5.1.1). As before, all investment decisions are subject to a dynamic capital constraint. Here we assume B1 = 18, and an allocation of new budget which is constant over time and scenario, β(t, s) = β. In the presented cases, we will discuss two cases for β, i.e., β = 10 and β = 20. With these parameters as the common basis for all portfolios that will be discussed in the following, we can outline the specific portfolio settings. First, a base case is presented for three underlying assets with three different initial values of the underlying assets, i.e., α1 = 20, α2 = 40 and α3 = 50. While asset 1 and asset 3 exhibit relatively low volatility with σ1 = σ3 = 0.3,

5.2 Extension of the Numerical Analysis to Three Underlying Assets

117

Table 5.11. Portfolio examples with 3 underlying assets

Portfolio Base case Correlation Budget Volatility

α1 20 20 20 20

α2 40 40 40 40

α3 50 50 50 50

σ1 0.3 0.3 0.3 0.3

σ2 0.5 0.5 0.5 0.5

σ3 0.3 0.3 0.3 0.7

ρ12 0.8 -0.6 -0.6 -0.6

ρ13 0.8 0.6 0.6 0.6

ρ23 0.8 -0.4 -0.4 -0.4

B1 18 18 18 18

β 10 10 20 20

Value 189.2 205.7 361.3 373.3

asset 2 is more volatile, with σ2 = 0.5. Further, the values of the underlying assets develop in a highly correlated way, with ρ12 = ρ13 = ρ23 = 0.8 (see Table 5.11). This parameter setting is in line with the setting presented for two underlying assets (cf. Sec. 5.1.1). Second, with all else being equal, we focus on correlation effects by changing the correlation structure to ρ12 = −0.6, ρ13 = 0.6, and ρ23 = −0.4. This structure is internally consistent, as formulated in Eq. 5.12. While asset 1 and asset 2 are negatively correlated, asset 1 and asset 3 are positively correlated, which limits the degree to which asset 2 and asset 3 can be negatively correlated in a consistent way. Overall, by this setting we obtain a substantially expanded range of correlation, from negative to positive (see Table 5.11). Third, with all else being equal, we show the impact of an increased availability of funds, by increasing β = 10 to β = 20 (see Table 5.11). Finally, with all else being equal, we highlight volatility effects, by increasing the volatility of asset 2, from σ2 = 0.5 to σ2 = 0.7, thus making asset 2 “riskier” (see Table 5.11). The numerical settings will be discussed for T = 3. Therefore, for n = 3, Ai0 = 3, and T = 3, the number of paths through the tree is given by (cf. also Sec. 5.1.1): Y Y Ai0 ∗ [2n Ai0 ](T −1) = 27 ∗ [8 ∗ 27]2 = 1, 259, 712 paths. (5.6) i

i

With these specifications, the numerical results for three underlying assets will be discussed next. 5.2.2 Correlation and Risk–Neutral Probabilities The joint behavior of n underlying assets is captured by the correlation structure between these underlying assets, which can be represented in a n × n matrix given in Table 5.12, where i, j = 1, . . . , n. This matrix gives the correlation of each pair ij out of all n assets and is thus quadratic. Further, it holds that

118

5 Numerical Analysis Table 5.12. Correlation structure

1 2 3 .. . i

1 ρ11 ρ21 ρ31

2 ρ12 ρ22 ρ32 ..

ρi1

ρi2

3 ρ13 ρ23 ρ33

··· ··· ··· ···

. ρi3

···

ρij = ρ ji

j ρ1j ρ2j ρ3j .. . ρij

, and

ρii = 1.

(5.7) (5.8)

While for two underlying assets there is only one correlation, for three underlying assets correlations for all pairs of underlying assets are to be considered. Therefore, the case for n = 3 captures the general structure of the correlation matrix and can thus be analyzed as representative for n underlying assets. Increasing the number of underlying assets for n > 3 leaves the structure of the correlation matrix unchanged and only increases its columns and rows. Thus, for the case of n = 3 invoking Eq. 5.7 and Eq. 5.8, we have to consider the following three correlations: ρ12

correlation between assets 1 and 2,

(5.9)

ρ13

correlation between assets 1 and 3,

(5.10)

ρ23

correlation between assets 2 and 3.

(5.11)

Correlations need to be internally consistent. If the interplay of asset 1 with asset 2 and 3 is defined, consequently the interplay between assets 2 and 3 loses degrees of freedom because of the given joint relation with asset 1. This consistency is exogenous to our model, i.e., it must be satisfied in the input data to our model, but leaves the model formulation as such unaffected. For three underlying assets (cf. Hull 2003, p. 388), the condition for a matrix of correlations to be internally consistent is: ρ212 + ρ213 + ρ223 − 2ρ12 ρ13 ρ23 ≤ 1.

(5.12)

This correlation structure enters the model through a suitable risk– neutral probability measure which assigns scenarios to joint realizations of the stochastic processes of the underlying assets. For three underlying as-

5.2 Extension of the Numerical Analysis to Three Underlying Assets

119

Table 5.13. Scenarios and risk–neutral probabilities for three underlying assets

Scenario s=1 s=2 s=3 s=4 s=5 s=6 s=7 s=8

Asset 1 up up up up down down down down

Asset 2 up up down down up up down down

Asset 3 Probability up p1 down p2 up p3 down p4 up p5 down p6 up p7 down p8

sets, this translates into a branching of 8 successor nodes from each parent node, as depicted in Table 5.13. As discussed in Sec. 4.4.1 and Sec. 4.2.1, the risk–neutral probabilities for more than two underlying assets need to be formulated for a CRR specification of the binomial model, which we apply here following Boyle et al. (1989). This approach is equivalent to the one proposed by Kamrad and Ritchken (1991) for the stretch parameter used herein set to 1. This specifies the risk–neutral probabilities, as given in Table 5.14 for 3 underlying assets. Based on these definitions, we will investigate the joint behavior of three underlying assets in a portfolio of real options setting next. 5.2.3 Base Case The base case, as introduced above, considers the parameters as summarized in Table 5.15. The optimal strategy is represented in 5.10. For each underlying asset, it represents the entering mode, and the leaving mode, at each state. For instance, at t = 3 in scenario s = 43, it is optimal to switch asset 2 from mode 2 to mode 1, i.e., it is optimal to shut down. In scenario s = 43, the stochastic processes for the underlying assets 1, 2, and 3, have developed with (down, up), (up, down), and (down, up), respectively (cf. Table 5.13). In that setting, it is optimal to install low capacity for asset 2 at t = 1, which is the only feasible investment decision given the budget constraint. At t = 2, the operating strategy remains unchanged, continuing to operate asset 2. At t = 3, the strategy becomes more refined, expressing the competition between assets for limited resources available. For instance, at t = 3 and s = 1, asset 2 is shut down in order to free funds for asset 3, which dominates asset 2. As opposed to this, in s = 2 it is optimal to expand capacity for asset 2 and keep asset 3 uninvested. This is intuitive because in both scenarios

120

5 Numerical Analysis t=1 Asset 1 Asset 2 Asset 3

1→1

1→2

1→1

t=2 Asset 1 Asset 2 Asset 3

t=3 Asset 1 Asset 2 Asset 3

s

1→1

2→2

1→1

1→1 1→1 1→1 1→1 1→1 1→1 1→1 1→1

2→1 2→3 2→1 2→1 2→1 2→3 2→1 2→1

1→2 1→1 1→2 1→2 1→2 1→1 1→2 1→2

s=1 s=2 s=3 s=4 s=5 s=6 s=7 s=8

1→1

2→2

1→1

1→1 1→1 1→1 1→1 1→1 1→1 1→1 1→1

2→3 2→3 2→1 2→2 2→3 2→3 2→1 2→2

1→1 1→1 1→2 1→1 1→1 1→1 1→2 1→1

s=9 s=10 s=11 s=12 s=13 s=14 s=15 s=16

1→1

2→2

1→1

1→1 1→1 1→1 1→1 1→1 1→1 1→1 1→1

2→1 2→1 2→1 2→1 2→1 2→1 2→1 2→1

1→2 1→2 1→2 1→2 1→2 1→2 1→2 1→2

s=17 s=18 s=19 s=20 s=21 s=22 s=23 s=24

1→1

2→2

1→1

1→1 1→1 1→1 1→1 1→1 1→1 1→1 1→1

2→1 2→2 2→1 2→1 2→1 2→2 2→1 2→1

1→2 1→1 1→2 1→2 1→2 1→1 1→2 1→2

s=25 s=26 s=27 s=28 s=29 s=30 s=31 s=32

1→1

2→2

1→1

1→1 1→1 1→1 1→1 1→1 1→1 1→1 1→1

2→1 2→3 2→1 2→1 2→1 2→3 2→1 2→1

1→2 1→1 1→2 1→2 1→2 1→1 1→2 1→2

s=33 s=34 s=35 s=36 s=37 s=38 s=39 s=40

1→1

2→2

1→1

1→1 1→1 1→1 1→1 1→1 1→1 1→1 1→1

2→3 2→3 2→1 2→2 2→3 2→3 2→1 2→2

1→1 1→1 1→2 1→1 1→1 1→1 1→2 1→1

s=41 s=42 s=43 s=44 s=45 s=46 s=47 s=48

1→1

2→2

1→1

1→1 1→1 1→1 1→1 1→1 1→1 1→1 1→1

2→1 2→1 2→1 2→1 2→1 2→1 2→1 2→1

1→2 1→2 1→2 1→2 1→2 1→2 1→2 1→2

s=49 s=50 s=51 s=52 s=53 s=54 s=55 s=56

1→1

2→2

1→1

1→1 1→1 1→1 1→1 1→1 1→1 1→1 1→1

2→1 2→2 2→1 2→1 2→1 2→2 2→1 2→1

1→2 1→1 1→2 1→2 1→2 1→1 1→2 1→2

s=57 s=58 s=59 s=60 s=61 s=62 s=63 s=64

Fig. 5.10. Optimal strategy for three underlying assets - base case

5.2 Extension of the Numerical Analysis to Three Underlying Assets

121

Table 5.14. Specification of risk–neutral probabilities for n = 3 assets (i = 1, 2, 3) in 8 scenarios

p1 18

√ r − 1 σ2 1 + ρ12 + ρ13 + ρ23 + ∆t + f σ 2 1 +

r f − 12 σ22 σ2

p2 18

√ r − 1 σ2 1 + ρ12 − ρ13 − ρ23 + ∆t + f σ 2 1 +

1 8

+

r f − 12 σ32 σ3

r f − 12 σ22 σ2

−

r f − 12 σ32 σ3

√ r − 1 σ2 1 − ρ12 + ρ13 − ρ23 + ∆t + f σ 2 1 −

r f − 12 σ22 σ2

r f − 12 σ32 σ3

+

√ r − 1 σ2 p4 18 1 − ρ12 − ρ13 + ρ23 + ∆t + f σ 2 1 −

r f − 12 σ22 σ2

r f − 12 σ32 σ3

−

√ r − 1 σ2 p5 18 1 − ρ12 − ρ13 + ρ23 + ∆t − f σ 2 1 +

r f − 12 σ22 σ2

r f − 12 σ32 σ3

+

√ r − 1 σ2 1 − ρ12 + ρ13 − ρ23 + ∆t − f σ 2 1 +

r f − 12 σ22 σ2

r f − 12 σ32 σ3

−

√ r − 1 σ2 1 p7 8 1 + ρ12 − ρ13 − ρ23 + ∆t − f σ 2 1 −

r f − 12 σ22 σ2

+

r f − 12 σ32 σ3

√ r − 1 σ2 1 + ρ12 + ρ13 + ρ23 + ∆t − f σ 2 1 −

r f − 12 σ22 σ2

−

r f − 12 σ32 σ3

p3

1

1

1

1

1

p6

1 8

1

1

p8 18

1

Table 5.15. Three underlying assets – base case

Portfolio α1 α2 α3 σ1 σ2 σ3 ρ12 ρ13 ρ23 B1 β Value Base case 20 40 50 0.3 0.5 0.3 0.8 0.8 0.8 18 10 189.2 the value of the underlying asset is the same for asset 2 (up, up), whereas asset 3 jumps up in s = 1 but down in s = 2. The competition for funds is also apparent in scenarios s = 3 and s = 4, where asset 2 has jumped (up, down). Here asset 3 gets invested no matter whether it moves up or down, because it faces no competition any more from asset 2, due to the unfavorable development of the latter. In the other states at t = 3, an analogous pattern can be observed. Asset 1 never gets invested, because given the relative switching costs, expansion factors, and funds available, it is dominated by the other two assets. Assets 2 and 3 compete for the funds available, and contingent on the relative joint stochastic realizations and the path taken there, the one or the other prevails. Given that the operating policy is not reconfigured at t = 2 and thus the operating policy is identical in all states, path dependency does not materialize here. This will change in 5.2.6.

122

5 Numerical Analysis Stochastic Movement s s=1 s=2 s=3 s=4 s=5 s=6 s=7 s=8

Asset 1 Asset 2 Asset 3

up up up up down down down down

up up down down up up down down

up down up down up down up down

Probabilities Base case

Correlation

0.410 0.006 0.048 0.044 0.006 0.002 0.044 0.440

0.060 0.006 0.348 0.094 0.056 0.302 0.044 0.090

Fig. 5.11. Optimal strategy for three underlying assets - changed correlation

Result: The behavior of the base case for three underlying assets is consistent with the discussion for two underlying assets, in Sec. 5.1.1. The optimal strategy realizes an overall portfolio value of 189.2. 5.2.4 Change in Correlation Building up on the base case, we change the correlation structure, with all else being equal, as summarized in Table 5.16. While before all three assets were highly positively correlated, asset 1 is negatively correlated with asset 2 but still positively correlated with asset 3. Asset 2 and asset 3 are also negatively correlated, which satisfies the consistency of the correlation structure. Table 5.16. Three underlying assets – changed correlation

Portfolio α1 α2 α3 σ1 σ2 σ3 ρ12 ρ13 ρ23 B1 β Value Correlation 20 40 50 0.3 0.5 0.3 -0.6 0.6 -0.4 18 10 205.7

The correlation enters the model through the risk–neutral probabilities, as discussed in 5.1.5 and applied to three underlying assets in Sec. 5.2.2. This implies that the correlation does not affect the optimal exercise policy in each specific scenario, but affects the probabilities attributed to that scenario. For the discussed setting, the change in correlation shifts the weights away from scenarios where the negatively correlated assets behave in a likewise manner, and increases the relative weight for scenarios where the negatively correlated assets develop in an opposing direction. For instance, the probability for all assets to jointly jump up, i.e., p1 , drops from 41% to a mere 6%, as depicted in Fig. 5.11.

5.2 Extension of the Numerical Analysis to Three Underlying Assets

123

Result: The findings obtained for two underlying assets are confirmed. In the specific case, we show that reducing the correlation among three underlying assets can increase portfolio value, from 189.2 to 205.7. This again confirms the financial intuition that the overall investment problem follows the structure of a call option (cf. Sec. 5.1.5). As noted before, in other settings a positive correlation effect can exist. 5.2.5 Change in Budget Next, we will develop the problem by allowing for more funds available, by increasing β from 10 to 20 and all else being equal. This case is summarized in Table 5.17. Table 5.17. Three underlying assets – increased budget allowance

Portfolio α1 α2 α3 σ1 σ2 σ3 ρ12 ρ13 ρ23 B1 β Value Budget 20 40 50 0.3 0.5 0.3 -0.6 0.6 -0.4 18 20 361.3

The optimal strategy for this case is depicted in Fig. 5.12. Here, the change in the optimal policy, as compared to the cases discussed before, is indicated by bold boxes. As discussed in the base case (cf. Sec. 5.2.3), asset 3 is more attractive than the other two assets. Here the budget constraint is such that at t = 1, funds are not sufficient for investment in asset 3, such that asset 2 steps in. What changes is that at t = 2, it is possible to install full capacity for asset 3, requiring to free funds by shutting down asset 2. At t = 3, it is optimal to keep on operating asset 3 at full capacity. Further, funds are sufficient to reinvest into asset 2, at low capacity, which is optimal in all scenarios where asset 2 has followed at least one up-jump. Result: The findings obtained for two underlying assets are confirmed, in that a sufficiently large increase in the budget allowance makes it possible to implement a substantially reconfigured operating strategy, realizing higher value. In our case, the portfolio value increases from 205.7 to 361.3. It is noted though that the optimal strategy does not exhibit a wide range of switching decisions, due to the relative attractiveness of asset 3. This will fundamentally change in the following numerical setting. 5.2.6 Change in Volatility Next, the numerical example is modified by considering a higher volatility for asset 3, increased from 0.3 to 0.7, and all else being equal. This case is summarized in Table 5.18.

124

5 Numerical Analysis t=1 Asset 1 Asset 2 Asset 3

1→1

1→2

1→1

t=2 Asset 1 Asset 2 Asset 3

t=3 Asset 1 Asset 2 Asset 3

s

1→1

2→1

1→3

1→1 1→1 1→1 1→1 1→1 1→1 1→1 1→1

1→2 1→2 1→2 1→2 1→2 1→2 1→2 1→2

3→3 3→3 3→3 3→3 3→3 3→3 3→3 3→3

s=1 s=2 s=3 s=4 s=5 s=6 s=7 s=8

1→1

2→1

1→3

1→1 1→1 1→1 1→1 1→1 1→1 1→1 1→1

1→2 1→2 1→2 1→2 1→2 1→2 1→2 1→2

3→3 3→3 3→3 3→3 3→3 3→3 3→3 3→3

s=9 s=10 s=11 s=12 s=13 s=14 s=15 s=16

1→1

2→1

1→3

1→1 1→1 1→1 1→1 1→1 1→1 1→1 1→1

1→2 1→2 1→1 1→1 1→2 1→2 1→1 1→1

3→3 3→3 3→3 3→3 3→3 3→3 3→3 3→3

s=17 s=18 s=19 s=20 s=21 s=22 s=23 s=24

1→1

2→1

1→3

1→1 1→1 1→1 1→1 1→1 1→1 1→1 1→1

1→2 1→2 1→1 1→1 1→2 1→2 1→1 1→1

3→3 3→3 3→3 3→3 3→3 3→3 3→3 3→3

s=25 s=26 s=27 s=28 s=29 s=30 s=31 s=32

1→1

2→1

1→3

1→1 1→1 1→1 1→1 1→1 1→1 1→1 1→1

1→2 1→2 1→2 1→2 1→2 1→2 1→2 1→2

3→3 3→3 3→3 3→3 3→3 3→3 3→3 3→3

s=33 s=34 s=35 s=36 s=37 s=38 s=39 s=40

1→1

2→1

1→3

1→1 1→1 1→1 1→1 1→1 1→1 1→1 1→1

1→2 1→2 1→2 1→2 1→2 1→2 1→2 1→2

3→3 3→3 3→3 3→3 3→3 3→3 3→3 3→3

s=41 s=42 s=43 s=44 s=45 s=46 s=47 s=48

1→1

2→1

1→3

1→1 1→1 1→1 1→1 1→1 1→1 1→1 1→1

1→2 1→2 1→1 1→1 1→2 1→2 1→1 1→1

3→3 3→3 3→3 3→3 3→3 3→3 3→3 3→3

s=49 s=50 s=51 s=52 s=53 s=54 s=55 s=56

1→1

2→1

1→3

1→1 1→1 1→1 1→1 1→1 1→1 1→1 1→1

1→2 1→2 1→1 1→1 1→2 1→2 1→1 1→1

3→3 3→3 3→3 3→3 3→3 3→3 3→3 3→3

s=57 s=58 s=59 s=60 s=61 s=62 s=63 s=64

Fig. 5.12. Optimal strategy for three underlying assets - increased budget allowance

5.2 Extension of the Numerical Analysis to Three Underlying Assets

125

Table 5.18. Three underlying assets – increased budget allowance and increased volatility

Portfolio α1 α2 α3 σ1 σ2 σ3 ρ12 ρ13 ρ23 B1 β Value Volatility 20 40 50 0.3 0.5 0.7 -0.6 0.6 -0.4 18 20 373.3 The optimal strategy for this case is depicted in Fig. 5.13. Again, the change in the optimal policy is indicated by bold boxes. By increasing the volatility of asset 3, the optimal investment at t = 2 is more differentiated. When both asset 2 and asset 3 move up, it is still optimal to shut down asset 2 and invest in full capacity for asset 3. What changes here is that in states where asset 2 moves up and asset 3 moves down, it is more attractive to expand asset 2 and remain uninvested for asset 3, because the latter is adversely affected by the down movement, to a higher extent than before. This is the case for s = 2 and s = 6, where s = 6 is endowed with a relatively high probability, given the correlation structure (cf. Sec. 5.2.4). Given that the strategy only changes in the successor scenarios of these scenarios, it is insightful to consider scenarios s = 9, . . . , 16 and s = 41, . . . , 48, at t = 3. The starting situation for all of these scenarios is the same: asset 2 is operating at full capacity, while asset 3 is still uninvested; asset 2 has followed an up jump while asset 3 has suffered from a down jump. First, consider scenarios s = 41, . . . , 48. Capacity for asset 3 is installed whenever it follows an up jump at t = 3, which is the case in scenarios s = 41, 43, 45, 47. The size of the capacity expansion depends on the relative development of asset 2. On the one hand, when asset 2 moves down, as is the case in scenarios s = 43, 47, it is optimal to install full capacity for asset 3, which is financed by a shut down of asset 2. On the other hand, when asset 2 moves up, as is the case in scenarios s = 41, 45, it is optimal to scale down asset 2 from high to low capacity, and install medium capacity for asset 3. We note that asset 1 does not affect the optimal strategy, since it is never invested. Second, consider scenarios s = 9, . . . , 16. The situation is different here, because asset 1 here follows an up jump, instead of a down jump in scenarios s = 41, . . . , 48. Thus, in scenarios s = 9, . . . , 12, asset 1 has developed with (up, up), i.e., has turned as valuable as possible. In scenario s = 10, it is optimal to install low capacity for asset 1. A comparison with scenario s = 42 shows that the investment into asset 1 leaves the operating strategy for the other assets unaffected. This is different for scenario s = 12, as compared to scenario s = 44. In scenario s = 12, the relative attractiveness of asset 1 causes a scaling down of asset 2, which here was hit by a down movement instead of moving up in scenario s = 10. The funds available through scaling down asset 2 make it possible to install high capacity for asset 1 here.

126

5 Numerical Analysis t=1 Asset 1 Asset 2 Asset 3

1→1

1→2

1→1

t=2 Asset 1 Asset 2 Asset 3

t=3 Asset 1 Asset 2 Asset 3

s

1→1

2→1

1→3

1→1 1→1 1→1 1→1 1→1 1→1 1→1 1→1

1→2 1→2 1→2 1→2 1→2 1→2 1→2 1→2

3→3 3→3 3→3 3→3 3→3 3→3 3→3 3→3

s=1 s=2 s=3 s=4 s=5 s=6 s=7 s=8

1→1

2→3

1→1

1→1 1→2 1→1 1→3 1→1 1→1 1→1 1→1

3→2 3→3 3→1 3→2 3→2 3→3 3→1 3→3

1→2 1→1 1→3 1→1 1→2 1→1 1→3 1→1

s=9 s=10 s=11 s=12 s=13 s=14 s=15 s=16

1→1

2→1

1→3

1→1 1→1 1→1 1→1 1→1 1→1 1→1 1→1

1→2 1→2 1→1 1→1 1→2 1→2 1→1 1→1

3→3 3→3 3→3 3→3 3→3 3→3 3→3 3→3

s=17 s=18 s=19 s=20 s=21 s=22 s=23 s=24

1→1

2→1

1→3

1→1 1→1 1→1 1→1 1→1 1→1 1→1 1→1

1→2 1→2 1→1 1→1 1→2 1→2 1→1 1→1

3→3 3→3 3→3 3→3 3→3 3→3 3→3 3→3

s=25 s=26 s=27 s=28 s=29 s=30 s=31 s=32

1→1

2→1

1→3

1→1 1→1 1→1 1→1 1→1 1→1 1→1 1→1

1→2 1→2 1→2 1→2 1→2 1→2 1→2 1→2

3→3 3→3 3→3 3→3 3→3 3→3 3→3 3→3

s=33 s=34 s=35 s=36 s=37 s=38 s=39 s=40

1→1

2→3

1→1

1→1 1→1 1→1 1→1 1→1 1→1 1→1 1→1

3→2 3→3 3→1 3→3 3→2 3→3 3→1 3→3

1→2 1→1 1→3 1→1 1→2 1→1 1→3 1→1

s=41 s=42 s=43 s=44 s=45 s=46 s=47 s=48

1→1

2→1

1→3

1→1 1→1 1→1 1→1 1→1 1→1 1→1 1→1

1→2 1→2 1→1 1→1 1→2 1→2 1→1 1→1

3→3 3→3 3→3 3→3 3→3 3→3 3→3 3→3

s=49 s=50 s=51 s=52 s=53 s=54 s=55 s=56

1→1

2→1

1→3

1→1 1→1 1→1 1→1 1→1 1→1 1→1 1→1

1→2 1→2 1→1 1→1 1→2 1→2 1→1 1→1

3→3 3→3 3→3 3→3 3→3 3→3 3→3 3→3

s=57 s=58 s=59 s=60 s=61 s=62 s=63 s=64

Fig. 5.13. Optimal strategy for three underlying assets - increased budget allowance and increased volatility

5.3 Numerical Summary Stochastic Movement

127

Optimal Strategy

Asset 1 t=2 t=3

Asset 2 t=2 t=3

s=12

up

up

up

down

down

down

1→3

3→2

1→1

s=26

up

up

down

up

down

down

1→1

1→2

3→3

s

Asset 3 t=2 t=3

Asset 1 Asset 1 Asset 1 t=3 t=3 t=3

Fig. 5.14. Path dependency for three underlying assets

At the same time, putting together these elements illustrates the path– dependencies underlying the portfolio problem. For instance, the optimal strategy differs completely in scenarios s = 12 and s = 26. The only difference here is that in scenario s = 12, asset 2 moves first up then down, and reversely, in scenario s = 26, first down then up. This feature is depicted in Fig. 5.14. Result: The value of the portfolio of real options increases from 361.3 to 373.3. It is observed that the optimal policy has changed significantly. Especially, while we only changed the volatility of one underlying asset, the optimal policy changed for all underlying assets. This underlines the results that all investment decisions interact, and must be assessed simultaneously, considering all investment opportunities jointly.

5.3 Numerical Summary The goal of the numerical analysis was to shed light on the effects encountered in the general model formulation. It was based on a stylized investment problem. The numerical analysis has demonstrated that the assessment of portfolios of real options must pay special attention to budget constraints, volatility effects, and correlation effects. Similarities to complex financial options have been identified, corroborating the results obtained through the presented model. While sometimes not straightforward, the encountered effects could always be explained by financial intuition. The complexity of the joint effects was made accessible through a comparative– static analysis of one parameter at a time. It should be noted though that all parameters interact dynamically and thus must be assessed simultaneously. This is achieved by the proposed model. The overall behavior of the presented investment problem resembles a general pattern of a call option. This is intuitive since it is an investment problem that starts in a not invested mode and corresponding options to invest. More precisely, the problem exhibits analogies to a call option on the maximum of two underlying assets. Another analogy could be drawn

128

5 Numerical Analysis

to barrier options. Both analogies confirm the financial intuition behind the proposed model. While constraints have been discussed for budgets, the proposed model can incorporate any number and type of additional constraints. One example are investments that are mutually exclusive (see Sec. 4.4.4). The effect is very much like the effect of a binding budget constraint: both result in deviations from the optimal unrestricted policy. The constraint prevents from making an investment that is profitable as such and that would be realized in the unrestricted case. This is why it is sufficient here to explicitly discuss budget constraints only, representing other types of constraints. The effects on portfolio value are structurally the same, but budget constraints require the biggest modeling challenge and thus provide the most insights. Result: Concluding, we find that portfolios of real options are subject to the following portfolio effects: 1. Option values typically do not add up and tend to be sub–additive, such that it is not possible to value real options in isolation and consider the value of portfolios of real options as a sum of the parts. 2. Adding to the non–additivity of real options, budget constraints play a major role because they introduce a step–function of portfolio value. Portfolio optimization makes it possible to countervail the adverse budget effects, resulting in a adapted investment and disinvestment strategy. 3. Volatility has an ambiguous effect on portfolio value. Increasing the volatility of an underlying asset can increase the value of a portfolio of real options related to this asset, but in fact can also reduce the value of a portfolio of real options. 4. Portfolio values of portfolio problems, subject to binding budget constraints, can be decreasing in the correlation. Therefore, correlation in the underlying assets and in the cash flows can harm portfolio value. In other portfolio settings without binding budget constraints, correlation can have a positive effect. 5. Portfolio values are increasing in the underlying assets initial values if the problem is a net investment problem, i.e., it shows a net payoff profile equivalent to the payoff of a call option. The main differences between a stand–alone perspective on real options and a portfolio perspective on real options are related to volatility effects, correlation effects, and budget effects, as summarized in Table 5.19. All of these lead to deviations from value additivity, making it impossible to value a portfolio of real options as sum of the parts. Based on these observations, the management of portfolios of real options is a complex task. The proposed model provides a structure capable of handling these complexities. It can be used for two goals: First, designing

5.3 Numerical Summary

129

Table 5.19. Stand–alone vs. portfolio perspective on real options ...lets the option value An increase in ... Stand-alone perspective Volatility

Portfolio perspective

increase

increase / decrease

Correlation

–

(increase /) decrease

Funds available

–

increase as a step–function

an optimal portfolio and second, exercising existing real options optimally. Both can be achieved by solving the model repeatedly for scenarios with different bundles of real options. In the light of portfolio design, options are desirable in the portfolio if (and only if) they increase overall portfolio value. As to exercising existing real options, the model provides clear managerial recommendations since a contingent exercise policy is implicit in the optimal value. Along the same lines, the model can be used to assess different levels of budgets available as a basis for strategic discussion of suitable levels of budget allowances. Presenting the ambiguous nature of portfolio interactions in the context of real options has demonstrated the complexities the model can handle and thus its contribution. While the numerical analysis took place in a rather simple setting to pinpoint specific effects clearly and to keep a sound intuition of the encountered effects, other cases are readily solvable. The flexible nature of the discrete modeling approach as well as the use of advanced optimization algorithms provide a rich framework for handling challenging investment problems. These become less intuitive with increasing complexity since it is harder to extract single effects from an interacting system. Nevertheless, this only affects the specific valuation results but leaves the structural results discussed here unchanged. Concluding, the proposed model formulation provides a powerful approach for the valuation and the management of portfolios of real options, taking into account significant portfolio effects that are otherwise hardly controllable.

6 Conclusion

The main contribution of this dissertation is to provide a rigorous valuation framework for portfolios of real options handling several important portfolio features that have not been covered so far in the literature. Our model is set up as a multi–dimensional real options analysis problem based on n underlying assets and their joint behavior along with embedded real options in a path–dependent setting. The modeling of the underlying assets is flexible and allows for changes in the characteristics of these assets, possibly at the discretion of management. It also has the flexibility to handle any kind of (financial) constraints and desired operational policies as well as investment synergies, asymmetric and time–dependent switching costs and learning effects. It can handle a wide range of intra– and inter–project compoundness of multiple options on multiple underlying assets. The model links dynamic (dis–) investment decisions with a global dynamic budget constraint that is both pivotal for the optimal operating policy and relevant in practical applications. Our portfolio approach incorporates features which are not only relevant for practical applications, but which also represent significant modeling challenges when considered in combination. Through our model implementation, we not only obtained a solution to this complex portfolio problem but also gained new insights into the behavior of portfolios of real options through sensitivity analysis. After summarizing the results, an outlook on managerial implications as well as on further applications and future research, is provided.

6.1 Summary of Results As an introduction, Chap. 1 motivated the portfolio perspective on real options. It is natural to assume that companies have discretionary decision rights about the operation of many physical assets. This introduces an asym-

132

6 Conclusion

metry in decision making which establishes a strong analogy between corporate investment decisions and financial options. Consequently, the application of financial option pricing theory to corporate investment decisions can provide valuable insights. At the same time, the key difference to financial options lies in the underlying asset which is real. The term real options stresses both analogy and difference. While the analogy is a close one, the real nature of the underlying assets gives rise to specific interaction effects. It has been shown that real options on the same underlying asset inherently interact, such that they cannot be valued independently from one another, but need to be modeled in a compound option type approach. If this is true for one underlying asset, it is all the more true for several underlying assets, especially when constraints are being considered. To capture all relevant aspects, portfolios of real options are defined as a combination of multiple underlying assets and multiple real options written on these assets, subject to constraints. Managing such portfolios of real options envisions two dimensions: strategic portfolio design and operational portfolio execution. The proposed model is capable of handling both. In this context, strategic refers to the choice of assets, whereas operational refers to the optimal investment policy. A discussion of the relevant literature, in Chap. 3, has related the present dissertation to prior work and has assessed different strands of literature in order to collect necessary elements for the ensuing model. Traditional mean–variance portfolio analysis has been examined first. It is based on the concept of diversification where risk as measured by volatility is minimized. For a real options analysis, this perspective is not valid since real options provide a way of handling risk other than by statistical diversification. Real options enable to capitalize on the upside of uncertainty and at the same time to truncate on the downside. Moreover, the asymmetry involved in real options fundamentally affects the distribution of returns such that it cannot be described sufficiently by its mean and variance. More advanced portfolio analysis was reviewed next. While the complexity of existing models goes far beyond the traditional approach, nevertheless they can only shed very little light on portfolios of real options, most of all due to the dynamic interaction effects between real options in a portfolio context. Likewise, a discussion of portfolio models including frictions, portfolios with financial options, and risk management evidenced that the models existing in the literature do not provide suitable starting points for the analysis of portfolios of real options. None can readily incorporate the specifics of portfolios of real options. In a next step, the literature on corporate capital budgeting was reviewed from a portfolio point of view. Of particular interest was the analysis of interactions in the mathematical programming literature, which could be gen-

6.1 Summary of Results

133

eralized to direct and indirect qualitative interactions. These interactions were later adopted in the model. Finally, the main basis of the present dissertation is the literature on financial and real options. As for any real options analysis, analogies to financial option pricing could be drawn, in our case to rainbow options on multiple underlying assets such as the option on the maximum of several underlying assets, as well as exotic options such as compound options. This lead to discussing the body of real options literature, covering a wide range of applications. A suitable perspective on real options as a real options switching algorithm could be identified in a general framework of switching among modes. Nevertheless, the rather limited analysis of portfolios of real options in the literature so far has been motivated by specific decision problems where compoundness and interactions arose because of special features of the decision problem. A systematic treatment of the main portfolio effects, including budget constraints, has been lacking so far. This discussion has set the stage for the core of the present dissertation, which is the real options portfolio model provided in Chap. 4. For a sound representation, it started with a description of the relevant model features. These include the objective function, the managerial actions permitted, the attitude towards risk, the modeling of the underlying stochastic processes, volatility and correlation, inter– and intra–project interactions, global dynamic budgets, synergies and operational constraints, learning, diversification, path–dependencies, as well as a discussion of the solution approach and the scalability of the model. A key feature of the model is the dynamic global budget equation which has the form of a balance equation. Funds that are not spent can be invested in later periods and funds freed up from disinvestment can be plowed back. This is completed by an initial endowment in funds and an additional allocation of new funds per period. Together with the interdependencies of investment decisions over time, this gave rise to path–dependencies. The interdependency between decisions across time makes it necessary to calculate mode–specific expected values in a backward–looking perspective. At the same time, the budget balance requires a forward–looking perspective. These interact simultaneously, which is impossible to model through standard dynamic programming. This is why we proposed a general formulation as a stochastic mixed–integer program that captures path–dependent and multi–dimensional real options settings. The program can be solved numerically, which we did by an implementation under GAMS© and the help of the commercial solvers BARON© and Cplex© . The mixed–integer nature of the problem and the exponential growth of the system state space present a challenging numerical problem. We could alleviate the complexity by linearizing the problem implementation which makes it possible to ensure global optimality. Nevertheless, the

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6 Conclusion

challenge remains in the curse of dimensionality of such a problem. This is why possible ways for improving numerical efficiency were discussed. This general development and discussion of the model was then illustrated, in Chap. 5, with the help of a numerical application. The complex landscape that is spanned by portfolios of real options was made accessible by a comparative–static analysis, tracing each effect by a specific (counter) example. This allowed to provide new insights into portfolio effects which were made accessible through the discussion of state–dependent investment decisions for specific examples. The revealed effects provide new insights for portfolios of real options and are contrary to effects one may infer from plain–vanilla option features. At the same time, all effects could be traced back to financial intuition. The discussion of value drivers for portfolios of real options focused on budget effects, volatility effects, correlation effects, starting value effects, as well as path–dependencies and deviations from additivity. First, a binding budget constraint plays an important role for understanding the optimal management of portfolios of real options. Binding budget constraints reduce the value of portfolios of real options, in a non–trivial way in that the optimal policy countervails the adverse effects. For cases involving little sensitivity of portfolio value with regards to changing budget constraints, fundamentally modified investment policies may be needed since parallel switching in and out of assets takes place. Reversely, stronger sensitivities may be simpler to track because they may include an additional investment while leaving the operating policy otherwise unchanged. Moreover, it is possible that different portfolios are optimal for different budget levels and that the relative attractiveness of portfolios are reversed. Generally, portfolios show a high sensitivity to changes in the budget constraint. Thus, it is clearly inadequate to abstract from constraints if they exist in the real world. Ignoring them in the investment analysis forgoes some of the value potential of portfolios of real options, or in the extreme case, can lead to plainly sub–optimal investment strategies. It invalidates the abstraction from constraints typically made in corporate finance. Second, a non-monotonic effect of volatility under budget constraints with disinvestment options was shown. This means that an increase of volatility can reduce portfolio value. Interestingly, this effect is ambiguous in that an increase in volatility can either increase or decrease portfolio value. In general, it cannot be specified without formal analysis whether a change in volatility will raise or reduce the value of a portfolio. This runs counter to standard features of financial options. Third, for an investment problem that has basically a call option structure, correlation of the underlying assets can harm portfolio value. In other terms, it is desirable to look for less correlated, or even negatively correlated ventures. It was noted, though, that in other settings without budget con-

6.2 Managerial Implications

135

straints, correlation can have a positive portfolio effect. Further, the call option structure translates into portfolio value increasing the more valuable the underlying assets are, as measured by their starting values. All of these features contribute to the fact that portfolios of real options generally show deviations from value-additivity and tend to be sub–additive. Based on these observations, the management of portfolios of real options is a complex task and requires detailed analysis in order to fully unfold the value that can be distilled from portfolios of real options. The proposed model provides a structure capable of handling these complexities, both for the design of an optimal portfolio and for the execution of existing real options. Hopefully this can contribute to both theory and practice. Based on these results, key managerial implications and room for further applications and future research will be discussed in the following.

6.2 Managerial Implications From a managerial point of view, the real options approach has often been criticized for its apparent complexity. Based on financial options pricing, it inherently models asymmetric decisions and relies on distributions of returns instead of point estimates. Analyzing options portfolios presumes a sound understanding of option pricing techniques, and hence involves a strong modeling aspect. For this reason, it is not trivial to gain acceptance of real option valuation models in practice, particularly when they involve portfolios of options. Naturally, it is important to discuss and structure the option valuation model with line managers. However, it is generally hard to communicate the valuation framework in detail for two reasons: it presumes a detailed understanding of the underlying option valuation concepts, and the nature of risk and uncertainty for portfolios of real options is different from that of stocks or primitive underlying assets. Nonetheless, the complexity of our model unavoidably reflects the inherent complexity of the underlying problem and the challenges embedded therein. Our model also has a key advantage. In view of the flexibility it provides for modeling specific investment problems, it can tackle actual decision problems in a tailor-made fashion. This should be helpful in making the essence of the model more transparent in the light of the specific application at hand. With a sound understanding of the investment structure, a line manager will be more inclined to accept the model. If the decision problem itself exhibits strong optionality and flexibility that can be well captured by the model, the model will be more compelling and more easily accepted. In our discussions with practitioners we experienced that the model was well received, as they were more involved with a concrete application.

136

6 Conclusion

The most convincing argument for using the model is that it can significantly improve corporate portfolio management. The model application and sensitivity analysis of results made it clear that managing portfolios of real options is a complex task with many interacting effects. In dynamic environments with a high degree of flexibility and uncertainty, practitioners understand perfectly well that it is unwise to ignore these interactions and that these need to be embraced proactively. Our model can be of significant value to managers and academics alike in facing this admittedly complex and difficult task.

6.3 Further Applications and Future Research The present dissertation has introduced a framework that can serve as a stepping stone for future work on portfolios of real options. Focusing on the setup of the model, some insightful questions are left for future research. These can best be pointed out by distinguishing possible ensuing questions in theory and in practice. From an academic standpoint, it would be valuable to enhance the mathematical efficiency of the numerical implementation. This dissertation focuses on formulating the general portfolio problem and provides a feasible numerical implementation that can solve the problem in view of illustrating the relevant portfolio problem and the impact of different value drivers. It did not focus on the efficiency of the implementation. The discussion of possible ways of improving the implementation showed further potential of a more efficient implementation, which is beyond the scope of this dissertation. In that sense, the goal of the implementation was chiefly conceptual. When solving real-life cases, increasing the efficiency is vital in a setting that is naturally prone the “curse of dimensionality”. Increasing the number of underlying assets, the number of options, the horizon, and the number of time steps evidently makes the model more realistic, yet increases the dimensionality of the problem. Second, the modeling approach chosen here and its implementation – based on an optimization platform that is more common in engineering than in finance – has hardly been applied to real options. It is possible to extend the presented model into many of the directions that were excluded by assumption in this dissertation. Especially, an extension towards real options and games, agency conflicts, incomplete markets and fuzziness of input parameters, and non-stationarity of parameters appears as promising. The flexibility of the model and its implementation via an optimization platform are such that, presumably, many additional effects can be modeled. From a practical standpoint, first it would be desirable to further adapt the model to different types of industries, i.e., tailor it to the specific needs

6.3 Further Applications and Future Research

137

and aspects that are most relevant. This would establish a toolbox readily available for different types of problems. Second, it would be desirable to apply the model to different cases in order to check the insights obtained and the reliability of the model. Third, once the model is confirmed it would have to become more user friendly with a convenient user interface before it can be used on a regular basis. Finally, the nature of the model is naturally quantitative and goes into deep modeling detail. At the same time, the problem of managing portfolios of real options, at its core, is fundamentally strategic. It doubtlessly can be put to good use in strategic portfolio management. For this, it would be necessary to integrate the model consistently into frameworks that are currently used for the assessment of strategic portfolios.

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Index

applications manufacturing, 2, 39, 40, 53, 94 R&D, 38, 52 supply chain management, 2, 32, 39 budget constraint balance equation, 18, 51, 55, 106, 109, 133 hard rationing, 35 modeling, 52, 54 soft rationing, 35 capital budgeting decision tree analysis, 33 definition, 32 interactions, 34 knapsack, 33 mathematical programming, 32 portfolio perspective, 33, 34 synergies, 35 financial options American Option, 8 barrier option, 107, 128 call option, 8 compound option, 11, 14, 31, 37, 43, 49, 50, 62, 114, 132, 133 definition, 2, 8 European option, 8 option on the maximum, 31, 36, 61, 64, 98, 109, 110, 127, 133 financial portfolio theory APT, 27

CAPM, 25 diversification, 23 efficient frontier, 23 frictions, 29 higher moments, 28 ICAPM, 26 Markowitz, 22 minimum variance portfolio, 23 portfolio selection, 22 interactions inter–project, 12, 14, 17, 38, 39, 50, 131, 133 intra–project, 12, 14, 17, 38, 50, 131, 133 model formulation change of underlying, 4, 10, 17, 80, 131 correlation, 14, 16, 24, 30, 32, 33, 38, 48, 59, 109, 114, 117, 118, 122, 125, 128 forward–backward looking, 17, 42, 55, 68, 75, 133 lattice tree, 42, 48, 51, 56, 59, 64, 66, 71, 79, 88 overview, 16 path dependency, 14, 15, 17, 39, 52, 53, 66, 86, 90, 91, 93, 107, 112, 127, 133 risk–neutral valuation, 33, 47, 54, 56, 64, 71, 75, 78, 83, 110, 118, 122

152

Index

stochastic process, 9, 40, 48, 54, 56, 66, 75, 80, 83, 84, 95, 133 stochastic program, 4, 17, 39, 41, 48, 54, 60, 63, 66, 71, 75, 84, 133 model implementation efficiency, 85, 87, 136 GAMS, 17, 55, 85, 133 linearization, 85, 134 optimality, 55, 63, 86, 87, 133 option pricing redundant, 29 replication, 21, 22, 29, 52, 84 self–financing, 29, 52 portfolios of real options definition, 12

design, 15 execution, 15 portfolio aspects, 14 portfolio management, 6, 15, 51, 53, 106, 109, 128, 134–137 real options definition, 1, 7 interdependencies, 2 parameters, 9 types, 9 recursion, 39, 55, 58, 59, 63, 65 strategic management, 2, 15, 30, 129, 132, 137 utility, 23, 26, 28

List of Tables

5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 5.11 5.12 5.13 5.14 5.15 5.16 5.17 5.18 5.19

Capacity factors for two underlying assets . . . . . . . . . . . . . . . . . . . Switching costs for asset 1 (two underlying assets) . . . . . . . . . . . Switching costs for asset 2 (two underlying assets) . . . . . . . . . . . Portfolio highlighting budget effects . . . . . . . . . . . . . . . . . . . . . . . . Portfolio with value decreasing in volatility . . . . . . . . . . . . . . . . . Portfolio highlighting path–dependency for T=3 . . . . . . . . . . . . . Capacity factors for three underlying assets . . . . . . . . . . . . . . . . . Switching costs for asset 1 (three underlying assets) . . . . . . . . . . Switching costs for asset 2 (three underlying assets) . . . . . . . . . . Switching costs for asset 3 (three underlying assets) . . . . . . . . . . Portfolio examples with 3 underlying assets . . . . . . . . . . . . . . . . . Correlation structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Scenarios and risk–neutral probabilities for three underlying assets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Specification of risk–neutral probabilities for n = 3 assets (i = 1, 2, 3) in 8 scenarios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Three underlying assets – base case . . . . . . . . . . . . . . . . . . . . . . . . . Three underlying assets – changed correlation . . . . . . . . . . . . . . . Three underlying assets – increased budget allowance . . . . . . . Three underlying assets – increased budget allowance and increased volatility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stand–alone vs. portfolio perspective on real options . . . . . . . . .

95 97 97 100 107 113 116 116 116 116 117 118 119 121 121 122 123 125 129

List of Figures

2.1 2.2 2.3

Payoff structure of a call option . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Portfolio design process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Modeling approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8 16 18

3.1

Mean–variance efficient portfolios . . . . . . . . . . . . . . . . . . . . . . . . . .

24

4.1 4.2 4.3

Dynamic budgets depending on time, scenarios, and actions . Recombining binomial lattice tree for one underlying asset . . . Recombining binomial lattice tree for two underlying assets, with ρ12 6= 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Three–dimensional risk–neutral probabilities . . . . . . . . . . . . . . . . Non–recombining binomial lattice tree for one underlying asset Indexing in a three–dimensional lattice tree . . . . . . . . . . . . . . . . . . Change of the underlying asset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Change of the underlying asset in the multinomial grid . . . . . . Dimensionality for two underlying assets . . . . . . . . . . . . . . . . . . .

51 58

Portfolio landscape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Portfolio value as a step–function of budget allowance . . . . . . . Small change in portfolio value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Substantial change in portfolio value . . . . . . . . . . . . . . . . . . . . . . . . Comparison of optimal investment strategies . . . . . . . . . . . . . . . . Portfolio value decreasing in σ1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Portfolio value decreasing in ρ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Portfolio value increasing in α1 and α2 . . . . . . . . . . . . . . . . . . . . . . Path–dependency of underlying asset 1 . . . . . . . . . . . . . . . . . . . . . Optimal strategy for three underlying assets - base case . . . . . . Optimal strategy for three underlying assets - changed correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

99 100 101 102 103 108 110 112 113 120

4.4 4.5 4.6 4.7 4.8 4.9 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 5.11

59 60 66 72 80 81 88

122

156

List of Figures

5.12 Optimal strategy for three underlying assets - increased budget allowance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 5.13 Optimal strategy for three underlying assets - increased budget allowance and increased volatility . . . . . . . . . . . . . . . . . . . 126 5.14 Path dependency for three underlying assets . . . . . . . . . . . . . . . . 127

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Portfolios of Real Options

Dr. Rainer Brosch Berlin, Germany [email protected]

ISBN 978-3-540-78298-8

e-ISBN 978-3-540-78299-5

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To my sister Christina

Foreword

Valuing portfolios of options embedded in investment decisions is arguably one of the most important and challenging problems in real options and corporate finance in general. Although the problem is common and vitally important in the value creation process of almost any corporation, it has not yet been satisfactorily addressed. It is key for any corporation facing strategic resource allocation decisions, be it a pharmaceutical firm valuing and managing its pipeline of drugs, a telecom company having to select a set of technological alternatives, a venture capital or private equity firm investing in a portfolio of ventures, or any company allocating resources. Portfolios of real options typically interact such that the value of the whole differs from the sum of the separate parts. Thus one must address and value the particular configuration of options embedded in a specific situation, taking into account the configuration of other options already present in the portfolio, which in turn depends on the correlation structure among the various underlying assets and the strategic dependencies among the options themselves (e.g., mutual exclusivity, strategic additivity, compoundness, complementarity etc.). In that sense, optimal decisions also depend on past option exercise decisions by management and organizational capabilities put in place in the past. The optimal decision today as to whether to continue operating in the current mode or following the current strategy versus switching to the best among a set of alternative modes or strategies depends in part on the asymmetric costs of switching strategies and potentially the costs and probability of switching back in the future. Such asymmetric costs of switching and re-switching contribute to the breakdown of the standard value additivity. That is, the overall problem is path dependent and history matters! In a practical setting, organizations oftentimes face internal budgetary, human capital and other organizational constraints. From a portfolio perspective, any investment or financial resource commitment explicitly com-

VIII

Foreword

petes with everything else. As a result, current commitments of financial, managerial or engineering resources or projects may need to be reduced or abandoned altogether to free up resources that can be devoted to more attractive future investment opportunities. The implications of this sort of options portfolio analysis can be non-conventional and far reaching. These aspects represent modeling challenges that the thesis of Mr. Brosch addresses by a tractable model solving important managerial decision problems. In particular, it tackles the portfolio problem head on within an overall budget constraint context in which interdependencies among optional decisions at each point in time and dynamically over time are explicitly considered. His thesis sets up a general portfolio valuation framework as a multi-dimensional real options problem involving path-dependent investment and dis-investment decisions and changes in the characteristics of the underlying assets. A key aspect of his model is that he explicitly links dynamic (dis-) investment decisions with a global dynamic budget constraint that is both pivotal for deriving the optimal operating policy while it remains relevant in practical applications. Mr. Brosch formulates his model as a constrained stochastic dynamic program that he subsequently solves. He then employs extensive numerical analysis and provides sensitivity results to investigate various portfolio effects. He confirms hypothesized deviations from value additivity and that the management of portfolios of real options is a complex task that requires careful, detailed, situation-specific analysis in order to properly capture the value that can be distilled from managing portfolios of real options. His model helps provide a general structure capable of handling these real-life complexities, both for the design of an optimal portfolio as well as for the execution or optimal exercise of the embedded options. The model properly reflects the inherent complexity of the underlying real-life constrained problem and the non-additivity and path-dependent challenges embedded therein. The framework developed by Mr. Brosch makes an important theoretical contribution in addressing this important and intellectually challenging portfolio problem, while at the same time it can be of significant value to practicing managers in facing this admittedly complex and difficult practical task of evaluating, managing and optimally exercising the set of interdependent corporate real options. Arnd Huchzermeier Vallendar, Germany

Lenos Trigeorgis Nicosia, Cyprus

Preface

Ever since I was a little boy, I was fascinated by investment decisions – not with financial assets in mind, but trucks. The bigger, the better. However, once it occurred to me that beyond sheer size, equipment with a broader application range may turn out to be more desirable. Many years later, taking a Master’s degree in Finance at Frankfurt University, I got hold of a copy of Lenos Trigeorgis’s book “Real Options”. It was all there: investment and flexibility, which is at the core of real options. However, I didn’t understand how this would work for portfolios. So when I met Lenos at the Real Options Conference and asked, he just smiled: “An interesting field – let’s work on it!” Now the work is ready and I hope it can be instrumental to managers and academics alike when dealing with portfolios of real options. It was submitted as doctoral thesis at WHU, Otto Beisheim School of Management, chaired by Professor Dr. Arnd Huchzermeier and Professor Lenos Trigeorgis, Ph.D. I would like to thank Arnd Huchzermeier for his guidance and the inspiring environment he creates at the Department of Production Management, including my opportunity to teach and to visit Stanford University. I am grateful to Lenos Trigeorgis who was always close despite the distance, by email or unforgettable meetings, and added both his great esprit and expertise. I am thankful to Stefan Spinler, for always being happy to discuss and help. Furthermore my thank goes to a friend at a Hightech Company, preferring to remain anonymous, who greatly contributed by discussing the applications. I thank my colleagues at the department for making my time at WHU an enjoyable one. Also I am grateful to Christian Artmann and Rolf Hellermann for tremendous intellectual and personal support, and for being a pleasure to be with far beyond research. Finally, I thank The Boston Consulting Group for financial support.

X

Preface

The most thankful I am to my parents, Arnold and Barbara, and my godmother Margret, who created many growth options for my education and my future. To my sister Christina and Peter, who are always with me. And to Jens and Thomas, for being the great friends they are. From my first thoughts about flexibility to my dissertation it has been a challenging but inspiring and rewarding journey. The journey goes on and I am looking forward to it. Berlin, January 2008

Rainer Brosch

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Research Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Contribution and Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 3 3 5

2

Portfolio Approach to Real Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Motivation of the Portfolio Approach to Real Options . . . . . . . 2.1.1 Real Options and Financial Options . . . . . . . . . . . . . . . . . 2.1.2 Definition of Portfolios of Real Options . . . . . . . . . . . . . . 2.1.3 Difference Between Financial Portfolios and Portfolios of Real Options . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Implications for Modeling Approach . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Portfolio Aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Management of Portfolios of Real Options . . . . . . . . . . . 2.2.3 Translation into Model Features . . . . . . . . . . . . . . . . . . . . .

7 7 7 11

3

Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Financial Portfolio Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Mean–Variance Portfolio Analysis . . . . . . . . . . . . . . . . . . . 3.1.1.1 Traditional Markowitz Portfolio Selection . . . . 3.1.1.2 Capital Asset Pricing Model . . . . . . . . . . . . . . . . 3.1.1.3 Intertemporal Capital Asset Pricing Model and Consumption . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1.4 Arbitrage Pricing Theory . . . . . . . . . . . . . . . . . . . 3.1.1.5 Conclusion on Mean–Variance Portfolio Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Advanced Financial Portfolio Analysis . . . . . . . . . . . . . . . 3.1.2.1 Portfolio Analysis Considering Higher Moments of Distributions . . . . . . . . . . . . . . . . . . .

12 13 14 15 16 21 21 22 22 25 26 27 27 28 28

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3.1.2.2

Allowing for Frictions in the Management of Portfolios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2.3 Impact of Combining Financial Assets with Financial Options . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2.4 Portfolio Implications for Risk Management . 3.1.2.5 Conclusion on Advanced Financial Portfolio Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Corporate Capital Budgeting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Portfolio Perspective in Corporate Capital Budgeting . 3.2.3 Interdependencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.4 Conclusion on Corporate Capital Budgeting . . . . . . . . . 3.3 Financial and Real Options Theory . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Foundations of Real Options . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Real Options Analysis in a Portfolio Context . . . . . . . . . 3.3.3 Detailed Review of Related Papers . . . . . . . . . . . . . . . . . . 3.3.3.1 Kulatilaka (1988) . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3.2 Kulatilaka and Trigeorgis (1994) . . . . . . . . . . . . . 3.3.4 Conclusion on Financial and Real Options Theory . . . . 4

Valuation Model for Portfolios of Real Options . . . . . . . . . . . . . . . . 4.1 Model Features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 General Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Objective Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.3 Managerial Actions Permitted . . . . . . . . . . . . . . . . . . . . . . . 4.1.4 Risk Preference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.5 Stochastic Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.6 Volatility and Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.7 Inter– and Intra–Project Options Interactions . . . . . . . . . 4.1.8 Global Dynamic Budgets . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.9 Synergies and Operational Constraints . . . . . . . . . . . . . . . 4.1.10 Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.11 Diversification as a Passive Approach to Risk Management . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.12 Path–Dependencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.13 Modeling Approach and Solution . . . . . . . . . . . . . . . . . . . 4.1.14 Scalability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Binomial Option Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Stochastic Processes and System State Space . . . . . . . . . 4.2.2 Binomial Option Pricing Model with One Underlying Asset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 Binomial Option Pricing Model with Two Underlying Assets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

29 29 31 31 32 32 33 34 36 36 36 37 40 40 42 44 45 45 45 46 46 47 48 48 49 50 52 52 53 53 54 56 56 56 57 58

Contents

5

XIII

4.3

Real Options Portfolio Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Switching Between Modes . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Switching Formulation for One Underlying Asset . . . . 4.3.3 Switching Formulation for Two Underlying Assets . . . 4.3.4 Stochastic Mixed–Integer Program Formulation . . . . . . 4.3.4.1 One Underlying Asset . . . . . . . . . . . . . . . . . . . . . . 4.3.4.2 Two Underlying Assets . . . . . . . . . . . . . . . . . . . . . 4.3.4.3 n–Dimensional Model Formulation . . . . . . . . . . 4.4 Model Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Alternative Probability Formulation . . . . . . . . . . . . . . . . . 4.4.2 Change of the Underlying Stochastic Processes . . . . . . . 4.4.3 Explicit Modeling of Synergies . . . . . . . . . . . . . . . . . . . . . . 4.4.4 Operating and Other Constraints . . . . . . . . . . . . . . . . . . . . 4.4.5 Terminal Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.6 Further Generalizations of the Model Formulation . . . . 4.5 Model Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.1 Applicability of Risk–Neutral Valuation . . . . . . . . . . . . . . 4.5.2 Numerical Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.2.1 Optimality of Solutions and Linearization . . . . 4.5.2.2 Dimensionality of the Problem . . . . . . . . . . . . . . 4.5.2.3 Increasing Efficiency of the Implementation . . 4.5.3 Monte Carlo Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61 61 63 64 65 66 71 75 78 78 80 81 82 82 83 83 83 84 84 86 87 90

Numerical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Numerical Analysis for Two Underlying Assets . . . . . . . . . . . . . 5.1.1 Numerical Application Setting for Two Underlying Assets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.2 Overall Portfolio Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.3 Budget Effects and Other Constraints . . . . . . . . . . . . . . . . 5.1.3.1 One Portfolio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.3.2 Relative Attractiveness of Two Portfolios . . . . 5.1.3.3 Generalization of Budget Effects . . . . . . . . . . . . . 5.1.4 Volatility Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.5 Correlation Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.6 Starting Value of the Underlying Assets Effects . . . . . . . 5.1.7 Path–Dependency Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.8 Deviations from Value Additivity . . . . . . . . . . . . . . . . . . . 5.2 Extension of the Numerical Analysis to Three Underlying Assets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Numerical Application Setting for Three Underlying Assets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Correlation and Risk–Neutral Probabilities . . . . . . . . . . 5.2.3 Base Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

93 94 94 97 98 100 103 105 106 109 111 112 114 115 115 117 119

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Contents

5.2.4 Change in Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.5 Change in Budget . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.6 Change in Volatility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Numerical Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

122 123 123 127

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Summary of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Managerial Implications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Further Applications and Future Research . . . . . . . . . . . . . . . . . .

131 131 135 136

5.3 6

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

List of Abbreviations, Variables, and Functions

αi ai

− → a

APT ai0

− →0 a

β(t, s) B B1

Starting value of stochastic process i Entering operating mode for underlying asset i, ai = 1, . . . , Ai Vector of entering modes of underlying assets Arbitrage Pricing Theory Leaving operating mode for underlying asset i, ai0 = 1, . . . , Ai0 Vector of leaving modes of underlying assets New funds available at time t in scenario s Budget level Initial budget available at time t = 1, B1 e R ≥ 0

BARON© Branch And Reduce Optimization Navigator (optimizer) BMW Bayerische Motoren Werke ci (·, ai0 ) Cash flow function for operating underlying asset i in mode ai0 CAPM Capital Asset Pricing Model CCAPM Consumption–Based Capital Asset Pricing Model cf. confer (compare) Cplex© Optimizer by ILOG, Inc. CRR Cox-Ross-Rubinstein di Proportional down move of the price of asset i in the binomial model DCF Discounted Cash Flow E (·) Expected value

XVI

e.g.

List of Abbreviations, Variables, and Functions

exempli gratia (for example) e Base of the natural logarithm Eq. Equation et al. et alii (and others) f., ff. following GAMS© General Algebraic Modeling System (optimization platform) i.e. id est (that is to say) i Underlying asset, i = 1, . . . , n I (·) Investment or switching costs (I < 0 possible) ICAPM Intertemporal Capital Asset Pricing Model l Auxiliary index L Auxiliary variable [ x mod y] Modulo (remainder on integer division of x by y) MP Market portfolio max[·] Maximum operator NPV Net Present Value φ Indicator function for movement of underlying assets p., pp. page(s) p Risk–neutral probability for up-jump pk Risk–neutral probability for jump in the multinomial lattice, k = 1, . . . , 2n Q Large number ρij Coefficient of correlation between underlying assets i and j rf Risk-free interest rate

R(·) R&D RO σi s θi t ∆t

ui

V (·) VaR X

Binary switching variable Research and Development Real Options Standard deviation of underlying asset i Scenario indicator, st = 1, . . . , S(t) Stochastic value of underlying asset i Time indicator, t = 1, . . . , T Length of (infinitesimally small) time interval Proportional up move of the price of asset i in the binomial model Value of option or portfolio Value at Risk Exercise price of a financial option

1 Introduction

Corporate investment opportunities can be viewed as options, or more precisely, “real options”(Myers 1977). An option gives its holder the right, but not the obligation, to take a specific action in the future (Amram and Kulatilaka 1999, p. 5). This right is valuable if there is uncertainty about the future development of an underlying variable. Due to the discretionary decision right involved, options are asymmetric contracts: if and only if option exercise is valuable, the decision maker will exercise and realize value, otherwise, the option will expire worthless (Trigeorgis 1996, p. 4). In order to distinguish corporate investment decisions, or options, from financial options, it is helpful to use the term “real options” if the underlying asset of an option is a real asset (Dixit and Pindyck 1994, p. 7). With a close analogy to financial options, corporate decision making is naturally discretionary and asymmetric because management has future decision rights about the use of corporate resources, such as financial resources or assets in place. These future decisions are made contingent on how future uncertainty resolves. While it may be reasonable to invest today based on available information, it may be wiser to expand or abandon the investment if the environment should develop more or less favorably. Real options theory provides a rigorous modeling framework allowing to assess the flexibility inherent in real investment decisions, and to derive optimal investment policies. In that sense, real options theory enables to assess corporate investment problems that could not be fully captured before (cf. Trigeorgis and Mason 1987, p. 16). Therefore, it is insightful to apply well established concepts from financial option pricing to corporate finance, which is done increasingly in practice (Graham and Harvey 2001). Although the analogy between financial and real options is a close one, there is a major difference. While financial options are derivative instruments in the sense that the underlying asset and other options are typically unaffected by a particular option exercise, this is generally not the case for

2

1 Introduction

real options. Exercising a put option on BMW shares does not have any effect on the company itself or other BMW options because it is just the use of a separate financial distribution right. Exercising a put option on a BMW plant is fundamentally different in nature: if the plant is sold or abandoned, all subsequent real options linked to this plant are forgone. An intuitive yet more passive example for a capacity management perspective in the automotive industry is provided by Fleischmann et al. (2006) who discuss BMW’s management of global production sites through a strategic planning model. The interdependency between real options on the same underlying asset is well known academically, but the BMW example also gives some intuition about the complexity involved, making practical implementation of real options in a rigorous portfolio context challenging (cf. Trigeorgis 1993a). Additional complexity stems from the realization that the underlying variables can be affected through active management by the firm as managers can influence the risk profiles of their firms. When multiple real options on multiple underlying assets are considered, the interactions in the company’s portfolio increase. This aspect has not been adequately covered in the real options literature, but is of high relevance both for academia and in practical applications (cf. Triantis 2005, p. 15). The practical relevance of such a portfolio problem can be be illustrated through Toyota’s challenge of actively managing the demand chain of two product lines in the Japanese and North American markets, facing both demand risk and exchange rate risk, by aligning the capacity of its global manufacturing plants, inventory networks, and distribution channels (Lee et al. 2005). Closely related, Huchzermeier and Cohen (1996) consider a global manufacturing network under exchange rate risk with switching options between different manufacturing strategies contingent on exchange rate realizations (for a detailed review, see Kouvelis et al. 2006). From a strategic management perspective, a primary objective is to design the firm’s strategic portfolio and dynamically manage this portfolio. Managing a portfolio of real options is what strategy is all about. A business strategy is more like a series of options than a series of static cash flows (cf. Luehrman 1998, p. 90). More broadly, managing a portfolio of real options is both strategic and operational. In this context, strategy refers to the choice of assets, whereas operational refers to the optimal investment policy for a given set of assets. Both strategic and operational real options management need to account for interaction effects. The interdependencies between real options must be properly taken into account to fully capture realistic investment decision problems when companies are faced with multiple investment or disinvestment opportunities under budget or other constraints (Trigeorgis 2005, p. 50). Budget constraints deserve special attention since exercising options implicitly assumes that

1.2 Contribution and Main Results

3

funds are readily available. In practice this may not be the case either because of internal (divisional) guidelines or limited access to capital markets (cf. Brealey and Myers 1996, pp. 101 ff.). Constraints may also result from logical aspects of the investment program, such as mutual exclusivity among projects (Brown and Davis 1998). While constraints are not considered in standard financial option pricing models, they affect optimal decision making if the existence of binding constraints significantly influences the optimal investment strategy. For instance, it may sometimes be desirable to sell a faltering or even a profitable business unit to free up (capital) resources for another investment that is even more profitable.

1.1 Research Questions As a consequence of the possible interdependencies between investment decisions from a real options point of view, valuing and managing portfolios of real options is a complex task. The goal of this dissertation is to identify the key aspects involved in managing portfolios of real options and to provide a rigorous real option valuation framework enabling to optimize portfolio decisions quantitatively. The present dissertation develops a modeling approach for dynamic investment problems where limited resources are allocated to interacting risky projects over time, subject to constraints. It sheds light on the following research questions: • Which aspects need to be considered for the analysis of portfolios of real options? • How can the relevant aspects be captured quantitatively in a suitable modeling approach for the valuation of portfolios of real options? • Which numerical evidence and structural results characterize portfolios of real options? To our knowledge, a focused discussion of these research questions has been lacking in the literature so far. This dissertation aims at closing the gap, providing a framework that makes it possible to optimize investment decision problems that could not be handled before.

1.2 Contribution and Main Results Our formulation and solution of a general model of portfolios of real options provides a quantitative approach to valuing specific investment problems. At the same time, it makes it possible to draw general conclusions about the

4

1 Introduction

behavior of portfolios of real options. It confirms the earlier results provided in Trigeorgis (1993a) and extends these to more complex settings. Specifically, our contribution is twofold. First, we develop a rigorous real options valuation framework enabling to optimize such portfolio decisions involving dynamic investment problems with limited resources allocated to interacting risky projects over time, subject to various constraints. We set up our portfolio valuation framework as a multi–dimensional real options problem involving path–dependent (dis–) investment decisions. Our approach can handle compoundness of multiple options on multiple underlying assets, correlation and changes in the characteristics of the underlyings, investment synergies, asymmetric switching costs, learning effects, a dynamic budget (with or without borrowing) and other types of operational and financial constraints. One of the advantages is that this approach can account for changes in the properties of the underlying variable, e.g., changes in the volatility, or more generally, changes of the underlying stochastic processes. This captures explicitly the option to switch the underlying asset, e.g., as a function of available information at time t. Due to its flexibility, it can handle an extensive range of portfolio problems. The model is formulated as a stochastic program that we solve by an algorithmic solution implementation on a commercially available programming platform. Second, we use our solution implementation to numerically provide new insights into portfolio effects. These insights have not been presented before in the context of real options. Especially, we find that budget constraints influence portfolio value in a complex way, potentially causing the optimal investment policy to change profoundly for relatively small changes in the availability of funds. As a consequence, for different levels of the overall budget, different portfolios may be optimal. Given the existence of binding budget constraints, volatility has an ambiguous effect on portfolio value. While for some settings volatility increases portfolio value, as usually is the case for financial options, it can also have an adverse effect on portfolio value for other settings. Both of these effects are new in that they had not been identified in the context of portfolios of real options before. Many of the discussed effects present counter examples to prevailing thinking in financial option pricing and thus generate new insights. We find that: • Option values typically do not add up and tend to be sub–additive, such that it is not possible to value real options in isolation and consider the value of portfolios of real options as a sum of the parts. • Adding to the non–additivity of real options, budget constraints play a major role because they introduce a step–function of portfolio value. Portfolio optimization makes it possible to countervail the adverse budget effects, resulting in a adapted investment and disinvestment strategy.

1.3 Structure

5

• Volatility has an ambiguous effect on portfolio value. Increasing the volatility of an underlying asset can increase the value of a portfolio of real options related to this asset, but in fact can also reduce the value of a portfolio of real options. • Portfolio values of constrained portfolio problems can be decreasing in the correlation. Therefore, correlation in the underlying assets and in the cash flows can harm portfolio value. • Portfolio values are increasing in the underlying assets initial values if the problem is a net investment problem, i.e., it shows a net payoff profile equivalent to the payoff of a call option. The discussion of these effects shows the complexity inherent to managing portfolios of real options and the challenges involved. Different sources of interaction and resulting ambiguity in parameters are such that the dynamics of a specific investment problem cannot be specified without formal analysis, but are specific to the investment problem at hand. The proposed model provides a framework that can help to meet these challenges. By enhancing corporate investment decision making through dynamic portfolio analysis, it fills a gap in a key area for both theory and practice.

1.3 Structure The structure of the subsequent chapters is as follows. Chap. 2 motivates the portfolio perspective on real options. For this, it introduces the concept of real options and defines portfolios of real options, which are unlike portfolios of financial assets or financial options. The notion of portfolio aspects is clarified. Based on the presented understanding of portfolios of real options, the issue of valuing and managing such portfolios is highlighted. With this background, an overview of the chosen modeling approach is given. Chap. 3 reviews the relevant strands of the literature which cover financial portfolio theory, corporate capital budgeting, and financial and real options theory. This review identifies the key concepts that underly the following analyses. This sets the stage for the core of the dissertation which is the development of the general model formulation in Chap. 4. The model development starts with a discussion of the key model features that are to be included. Then, the model is built up successively by increasing the scope of the model in stages. It begins by introducing a binomial option pricing model with one time dimension and one underlying asset dimension. Then, this is expanded to a multinomial option pricing model with two underlying asset dimensions. Based on this, a real options portfolio model based on a switching concept is introduced, first for one underlying assets, then

6

1 Introduction

for two underlying assets, and finally generalized to n underlying assets. The chapter concludes with a discussion of the model features and possible extensions. Chap. 5 presents comparative statics of the presented model, based on the numerical analysis of specific investment setting examples. It discusses the impact of portfolio aspects stemming from constraints, volatility effects, correlation effects, and starting values effects. For each of these, an example is provided which illustrates the specific interest of the proposed portfolio analysis. It will be shown that the management of portfolios of real options navigates in a complex system state space where most of the value drivers are ambiguous. It is therefore not possible to specify ex ante, i.e., before a specific portfolio optimization, whether a change in a certain parameter will increase or decrease portfolio value. This runs counter to the basic intuition for standard financial options. Finally, Chap. 6 summarizes the main results, discusses key contributions alongside with possible limitations, and gives an outlook on further applications and future research.

2 Portfolio Approach to Real Options

A portfolio perspective on real options makes it possible to gain insights into the interplay of real options that could not be captured in a stand– alone analysis. It is well established in financial theory that financial assets must be valued from a portfolio perspective (cf., e.g., Constantinides 1989). A review of financial portfolio theory is provided in Sec. 3.1. Similarly to financial portfolios, portfolios of real options are affected by numerous aspects, which will be developed in the following. In this chapter, the portfolio perspective will be introduced in detail, and then linked to the ensuing general model that is developed subsequently.

2.1 Motivation of the Portfolio Approach to Real Options The motivation for considering real options jointly in a portfolio context derives from the specifics of real options. This is why this motivation is based on a focused introduction of the general concept of real options, and the definition of the resulting portfolio problem. 2.1.1 Real Options and Financial Options The term “real options” was coined by Myers (1977) who stated that “real options [...] are opportunities to purchase real assets on possibly favorable terms” (Myers 1977, p. 163). A more precise definition is found in Sick (1995, p. 631) who defines a real option as “the flexibility a manager has for making decisions about real assets. These decisions can involve adoption, abandonment, exchange of one asset for another or modification of the operating characteristics of an existing asset”. An introduction to real options can be found in Dixit and Pindyck (1994), Trigeorgis (1996), Amram and Kulatilaka (1999), Copeland and Antikarov (2001), and Smit and Trigeorgis (2004).

8

2 Portfolio Approach to Real Options

Recent comprehensive overviews of the growing body of real options literature are given in Lander and Pinches (1998), Baecker and Hommel (2004), and Trigeorgis (2005). A financial option gives the holder the right but not the obligation to buy or sell a financial asset, e.g., stock traded on financial markets, under specified terms. These terms define the period of time over which the option can be exercised, in exchange of the exercise price. A call option gives the right to acquire the underlying asset, a put option gives the right to sell the underlying assets. An American option can be exercised at any time before and including the expiration date, whereas a European option can only be exercised on the expiration date (cf. Luenberger 1998, pp. 319 ff.). The value of an option stems from the riskiness of the underlying assets. As compared to a long position in the underlying asset, the option provides insurance against losses (cf. McDonald 2003, pp. 44 ff.). It enables to protect from the downside potential of the underlying asset, while benefiting from the upside potential. This asymmetry between upside and downside is visualized in Fig. 2.1. At maturity, if the value of the underlying asset is below the exercise price, the option is not exercised and yields zero payoff. Otherwise, the option is exercised, with unlimited upside potential. The dotted line represents the value of the call option before maturity, which is always higher than the value of the call option at maturity (cf. Brealey and Myers 1996, p. 569). In order to distinguish these two, the value of an option at maturity is called “intrinsic value”, whereas the difference between intrinsic value and current value is called “time value” of the option (cf. Hull 2003, p. 154).

Call Option value

don't exercise (at maturity)

exercise (at maturity)

option value before maturity

"intrinsic value" of the option

exercise price

Value of underlying asset

Fig. 2.1. Payoff structure of a call option

As a consequence, the modeling of the riskiness of the underlying asset is key for valuing an option on this asset. This riskiness is captured by modeling the stochastic process followed by the price of the underlying asset (cf. Hull 2003, pp. 216 ff.). Asset prices are commonly assumed to follow a ran-

2.1 Motivation of the Portfolio Approach to Real Options

9

dom walk, i.e., price changes occur randomly (cf. Luenberger 1998, p. 308 f.). In continuous time, the random walk of a stock price is frequently modeled as a geometric Brownian motion (cf. Dixit and Pindyck 1994, p. 72), and in discrete time it is frequently modeled as a multiplicative binomial process (cf. McDonald 2003, pp. 341 ff.). Both stochastic processes obtain (the binomial process in the limit as time increments become infinitesimally small) lognormally distributed stock prices, i.e., the natural logarithm of stock price changes are normally distributed (cf. Luenberger 1998, p. 309). Real options are similar to financial options in that they “give the holder the right, but not the obligation, to take an action in the future”(Amram and Kulatilaka 1999, p. 5). The major difference lies in the nature of the underlying assets. Real options are written on real assets which are productive capacity creating streams of income. As opposed to this, financial options are written on financial assets which are claims to the income generated by real assets. Therefore, while financial assets only define distribution rights of a given value, real assets can create economic value (Bodie et al. 1999, p. 3). The term real options stresses both the methodological analogy to financial options and the fundamental difference in the underlying assets. Typically, the interpretation of investment decisions as real options is based on the following analogies to financial options: The current value of the stock corresponds to the (gross) present value of expected cash flows; the exercise price to the investment cost; time to expiration to the time until the opportunity disappears; stock value uncertainty to project value uncertainty; and the riskless rate is identical for both (cf. e.g. Trigeorgis 1988, p. 149). A real option that requires to pay the exercise price to receive some underlying asset corresponds to a call option, whereas a real options enabling to receive the exercise price in exchange for sacrificing some underlying asset is a put option. Throughout this dissertation, we can use the terms value and price interchangeably, which is common in the real options literature (cf., e.g., Copeland and Antikarov 2001, p. 6). This is because in the absence of mispricing or other distortions, these two coincide. The application of financial option pricing theory to real option valuation is subject to possible limitations of the transferability, such as incomplete markets, non–traded underlying assets, or shared options. Representing a large body of literature, a detailed discussion of these issues can be found, e.g., in Trigeorgis (1996), Smit and Trigeorgis (2004), and Smith and Nau (1995). Different types of real options are distinguished in the literature. A detailed overview can be found in Micalizzi and Trigeorgis (1999) and Lander and Pinches (1998, p. 540, Table 2). Trigeorgis (1996, pp. 2 f., Table 1.1) distinguishes the following types of real options:

10

2 Portfolio Approach to Real Options

• Option to defer investment. The investment decision can be postponed, making it possible to benefit from the resolution of uncertainty during the lifetime of the option. • Option to default during staged construction. For investment projects that take time to build, it is possible to abandon the project during construction based on the resolution of uncertainty during construction. This is desirable if the remaining investment costs are not covered by the expected future value of continuing the project as originally planned. • Option to expand. If the underlying asset and thus the project develops more favorable than initially expected, a possible increase of capacity realizes higher value than continuing in base scale. • Option to contract. As opposed to the option to expand, the option to contract enables to reduce the scale of operations and thus save costs if the underlying asset develops unfavorably. • Option to shut down and restart operations. This option enables to stop operating temporarily, with the perspective of reopening in a later period. This is desirable if variable costs exceed operating revenues by more than the costs incurred for shutting down and reopening. • Option to abandon. It is possible to scrap the project or sell it off, which is desirable if the salvage value, e.g., a resale value, is higher than the value of continuing operations. • Option to switch use. Production input factors or the product output can be changed, e.g., as a function of prices or demand. This makes it possible to choose the strategy which generates highest profit net of switching costs. • Growth options. An early investment opens up new investment opportunities. Huchzermeier and Loch (2001) extend this taxonomy by the option to improve. During projects it is possible to take corrective action, such as adding resources, in order to improve the expected performance of the product. While this approach captures the fact that managers can change the performance state, it cannot cope with changes in the underlying, i.e., the option to switch underlyings. The current dissertation takes a general view on real options by interpreting each discretionary decision right as a switch between modes, as proposed by Kulatilaka and Marcus (1988) and Kulatilaka and Trigeorgis (1994). Their concept enables interpreting real option exercise decisions as switching decisions among operating modes, which is the most general formulation of real options, having commonly analyzed real options as special cases.

2.1 Motivation of the Portfolio Approach to Real Options

11

2.1.2 Definition of Portfolios of Real Options “A portfolio is a particular combination of assets in question” (Neftci 2000, p. 17). In the present discussion, the assets in question include real assets and real options written on these assets. For ease of discussion it is convenient to label sources influencing the value of the portfolio as “portfolio aspects”. On the asset level, it is self–evident to assume that corporate decision makers will have a vast opportunity set, enabling them to invest in a myriad of real assets and possibly to allocate limited resources between these assets. Similarly, from a financial portfolio perspective, it is natural to consider a set of underlying assets simultaneously (see Sec. 3.1). On the real options level, it has been shown by Trigeorgis (1993a) that options on the same underlying assets interact, requiring a simultaneous valuation of all real options written on the same underlying asset. If there are two financial options, e.g., to acquire stock of Bayer AG and BMW AG, the values of these financial options are additive because the underlying assets and the terms of the options are independent from one another. As opposed to this, consider BMW who has a European option to abandon an existing plant and the subsequent European option to expand this plant. In options terms, the European option to abandon corresponds to a European put option, while the European option to expand corresponds to a European call option. Since both affect the same underlying asset, these real options interact. On the one hand, the value of the put is less than in isolation because it is written on the package of underlying asset and associated call option. An option is always valuable before expiration, hence the package is more valuable than the underlying asset in isolation. Since a put option value decreases with increasing value of the underlying asset, the option to abandon becomes less valuable due to the subsequent option to expand the plant. This is intuitive because exercising the put forfeits the subsequent option, such that investors may hesitate to exercise the put in order to preserve the future call option. On the other hand, the value of the subsequent call is reduced through the existence of the prior put, because with some probability it may not be possible to exercise the call, given prior exercise of the put. This prevents the call from unfolding its full value potential and translates into a lower call option value. Since both effects occur simultaneously, it is not possible to value put and call in isolation. The decision about exercising the first option needs to explicitly take into account the existence of the subsequent option. This relationship is structurally akin to the valuation of compound options (cf. Geske 1979). Hence, the arising effect can conveniently be labelled as com-

12

2 Portfolio Approach to Real Options

poundness. Specifically, Trigeorgis (1993a) defines interactions between real options written on the same underlying asset as “intra–project compoundness”. Following the same logic, an analogous effect is identified for several, interdependent underlying assets which he denotes as “inter–project compoundness” (Trigeorgis 1996, pp. 132 f.). Both inter–project and intra–project compoundness must be considered in the context of portfolios of real options. Further, both real options and real asset may be subject to constraints. Real assets can exhibit, for example, operating (technical or logical) or financial relationships. These can lead to mutually exclusive assets or strictly complementary assets which require one another. This affects the possibility of joint exercise of bundles of real options on different underlying assets. Likewise, the existence of constrained resources, e.g., funds available, influences the feasibility of joint option exercise. Therefore, portfolios of real options here are defined as combinations of multiple risky assets and multiple real options written on these assets subject to constraints. Cases with only one underlying asset, or one real option, are special portfolio cases that reduce the scope of portfolio analysis dramatically. In order to seize all possible portfolio effects, it is important to analyze multiple underlying assets with multiple real options simultaneously. The usual “laboratory” setting for real options analysis with one underlying asset and one real option does not provide a structure capable of handling realistic decision problems. It is thus prone to ignore key portfolio effects with possibly substantial impact on (optimal) option exercise. 2.1.3 Difference Between Financial Portfolios and Portfolios of Real Options The discussion of portfolios of real options evidently stands on a large body of literature on financial portfolios. This literature will be reviewed in detail in Chap. 3. For the motivation of the portfolio perspective chosen here, it is important to clarify the major differences between portfolios of real options and portfolios of financial options. Financial portfolio theory dates back to Markowitz (1952) who established the concept of mean–variance analysis. At its core, it is based on the concept of diversification. Diversification is about minimizing the variance of the return of a portfolio for a given mean of the return (or maximizing the return for a given variance). This can be achieved by increasing the number of assets included in the portfolio and in the limit by holding all available assets. The only relevant risk in this perspective is the covariance risk of each asset with the (market) portfolio. In essence, this implies a passive attitude towards risk because it exclusively consists of diversifying the risk over as many assets as possible.

2.2 Implications for Modeling Approach

13

The portfolio approach to real options is fundamentally different from this passive perspective on risk. For portfolios of real options, the variance of returns and the resulting distributions of values of the underlying assets are considered instead of covariance with a (market) portfolio. Real options make it possible to capitalize on desirable developments of risky underlying assets while protecting from undesirable movements. For instance, a real option to expand capacity is only exercised if the value of the underlying asset is higher than the investment costs. This implies an active attitude towards risk which consists of exercising real options as a function of the resolution of uncertainty. Therefore, in portfolios of real options, the attitude towards risk is different and diversification cannot be desirable. For financial portfolios, closely connected to the passive attitude towards risk, value additivity of the elements of the portfolio holds (cf. Brealey and Myers 1996, p. 19). Since a financial option derives its value directly from the value of the underlying financial asset, the same holds for financial options. Value additivity also holds for financial options. In contrast, a consequence of the more active real options approach is that value additivity among the elements of portfolios of real options generally breaks down (see Sec. 2.1.1). This is why it is not insightful to include financial assets and financial options in the analysis of portfolios of real options. Financial assets by definition are independent from other assets, therefore the values of the financial assets and the portfolio of real options are also independent from one another and thus do add up. Alternatively, if the portfolio of real options does have an impact on the financial asset, it can only do so by affecting the real asset underlying the financial asset. However, this is not a special case because in fact a real asset is considered, not a financial asset. Beyond this perspective, it can be insightful to consider interactions between real options and financial flexibility. This issue is discussed by Trigeorgis (1993b) where the idea is not to add an additional (financial) asset to the portfolio, but to assess the possible value of financial flexibility that can result from operating flexibility. In this context, Huchzermeier and Cohen (1996) discuss operational and financial hedging strategies. A review of this literature, citing Huchzermeier and Cohen (1996) as the key reference in the context of operational hedging, can be found in Van Mieghem (2003); Kouvelis et al. (2006). This discussion of interactions between financial and operational flexibility is beyond the scope of this dissertation.

2.2 Implications for Modeling Approach Based on the above developed specifics of portfolios of real options, a consistent modeling approach is required. In order to derive the implication of the specifics of real options for our modeling approach, in the following we

14

2 Portfolio Approach to Real Options

will highlight portfolio aspects, their consequences on how portfolios of real options need to be managed, and how this translates into our model. 2.2.1 Portfolio Aspects Understanding portfolios of real options as combinations of multiple risky assets and multiple real options written on these assets subject to constraints provides a rich structure for assessing real decision problems. In this structure, the overall portfolio value is influenced by the dynamic interaction of all elements of the portfolio. This portfolio perspective on real options is a fundamental way to properly handle interdependencies between decisions over time. A suitable perspective on portfolios of real options needs to include all relevant aspects that influence the value of such a portfolio. From the definition of portfolios of real options, it follows directly that portfolio aspects can be attributed to the real assets involved, the real options involved, or constraints. Budget constraints are of special importance because they can have a considerable limiting impact and require a detailed modeling of the investment dynamics. Moreover, the ensuing budget levels over time are state– and path–dependent. Based on these considerations, portfolio aspects can be categorized as follows (cf. Brosch 2001): • Interactions on the real options level: On the real options level, intra–project compoundness and inter–project compoundness can be distinguished. While the former is due to interdependencies of several real options written on the same underlying asset, the latter is connected to interdependencies of several real options and several underlying assets. For inter–project compoundness, the correlation between the underlying assets has to be modeled explicitly. • Interactions on the real asset level: On the real asset level, direct qualitative interactions and indirect qualitative interactions can be distinguished. Direct interactions are those which are inseparably connected to the underlying real assets, such as physical properties or operating synergies. Indirect interactions have their origin outside the strict asset level and are due to constraints, most prominently budget constraints. Both are qualitative in that they are not merely stochastic in nature, but result from the properties of projects or the specific background of the company (that would be different for another company). Due to the simultaneous nature of the interactions involved, it is not possible to isolate value impacts for separate portfolio aspects. Their effects need to be assessed jointly, in order to capture their interplay. This is why they require a simultaneous modeling approach, very much like compound option pricing. Specifically, in order to handle these interactions, existing

2.2 Implications for Modeling Approach

15

constraints can only be incorporated via a simultaneous modeling approach. Thus, the model is formulated as one stochastic optimization problem subject to constraints. Interactions are captured through the interplay of constraints as well as state– and path–dependency of investment decisions and cash flows. 2.2.2 Management of Portfolios of Real Options Based on this broad understanding of portfolios of real options, optimal management of portfolios of real options requires capturing all relevant portfolio aspects simultaneously in order to maximize overall portfolio value. Specifically, two dimensions of managing portfolios of real options can be distinguished: 1. Portfolio design: creating the optimal portfolio with maximum value, supposing optimal future exercise. 2. Portfolio execution: exercising existing real options optimally in order to realize the full value potential of real options. On the one hand, portfolio design analyzes which assets and which options to include in the portfolio of real options. This decision can be based on the proposed model, by choosing the portfolio which yields the highest value for the portfolio of real options. In order to assess different portfolio designs, a valuation needs to be carried out for each possible alternative. This process is depicted in Fig. 2.2. By choosing the optimal portfolio design, the decision maker in fact defines the company’s strategy, because this defines the way the company is planning to create value in the future. This approach helps to quantify the impact of new investment proposals and more generally ideas form a portfolio perspective, which in options terms corresponds to assessing new real options and their interactions with the portfolio. Apart from assessing new real options, management may be able to influence the underlying assets or the embedded real options, such as modifying the volatility involved (cf. Damisch 2002, pp. 365 ff.). The desirability of these alternatives can also be assessed as different design options. At its core, the design of a portfolio of real options is about the choice of the underlying assets to be included in the portfolio, and thus, strategic. This deserves special attention, because it will be shown that conventional wisdom may not hold true for portfolios of real options. For instance, it is not always desirable to increase volatility to enhance value (see Sec. 5). Hence the portfolio design can be supported by the proposed model. On the other hand, portfolio execution is about optimally managing the existing design of a portfolio of real options. The value created through real options implicitly assumes that real options are exercised at the optimal

16

2 Portfolio Approach to Real Options Start

(Re-) Design portfolio

Value portfolio

Alternatives?

yes

no Select optimal portfolio

End

Fig. 2.2. Portfolio design process

point in time. Therefore, management needs to know when this optimal point in time is. This information is provided together with the optimal portfolio value, because the optimization result contains the optimal contingent exercise policy. Thus, the proposed model gives clear management recommendations about which options should be exercised, and when, suggesting to exercise real options as in the optimal policy. Generally, the portfolio design defines the basis for the portfolio execution because the latter is about optimally using an existing structure which is defined in the first. At the same time, the portfolio execution can provide unanticipated new information which opens up new portfolio alternatives for the portfolio design. An example for this is research on a specific drug which, by chance, provides insights about another new drug that had not been considered before. Thus both dimensions of managing portfolios of real options are interrelated. 2.2.3 Translation into Model Features The motivation for considering portfolios of real options has not only clarified the understanding of portfolios of real options, but at the same time has also stated the requirements for the model formulation. A suitable model formulation needs to incorporate the above introduced elements. Before the model is derived in detail in Chap. 4, an overview of the model features is given below. The model formulation will incorporate the following features: 1. Multiple correlated underlying assets.

2.2 Implications for Modeling Approach

17

2. 3. 4. 5. 6.

Multiple real options embedded in the underlying assets. State–dependent cash flows leading to path–dependencies. intra–project compoundness. Inter–project compoundness. Direct qualitative interactions, i.e., operational (non–financial) constraints. 7. Indirect qualitative interactions, i.e., state–dependent budget constraints with carry–over of funds (including borrowing). 8. Choice of the underlyings. These features are incorporated into the model formulation as follows: 1. General model formulation: The case of portfolios of real options is formulated as one stochastic mixed–integer program that captures path– dependent and multi–dimensional real options settings. 2. Simultaneous, i.e., forward–backward looking, approach: The backward looking element consists of choosing the optimal investment strategy which maximizes both current cash flow and expected value of future cash flows. This is a standard approach in financial option pricing (cf. Hull 2003, pp. 392 ff.). This is complemented by a forward looking element which captures the sequence of decisions up to a certain point in time that meet the budget constraints on that path. This is not typical for financial options. Both are considered simultaneously in order to capture their interdependencies. 3. Model implementation and solution: The model is solved by an algorithmic solution implementation, using the GAMS modeling platform, which can readily be applied to practical decision problems. The proposed modeling approach provides a rigorous and flexible structure that can cope with more realistic and rich decision problems, deriving an optimal dynamic sequence of investment and disinvestment decisions from a portfolio perspective. It is very flexible in incorporating specific investment problems with different kinds of constraints and changes in the underlyings. Especially, the global dynamic budget depends on the sequence of decisions which depends on the paths taken by the joint stochastic processes of the underlying assets. This is achieved by a path–dependent, forward–backward looking approach that keeps track of past decisions and the expected future effect of current decisions. For a visualization of this approach, see Fig. 2.3. To our knowledge, the formulation of such a model for portfolios of real options has not been provided in the literature so far. In order to account for these features, this dissertation follows a new direction for the quantitative assessment of real options. The model interprets the portfolio problem as an optimization problem with the objective of maximizing portfolio value subject to “constraints”. On the one hand, these

18

2 Portfolio Approach to Real Options Path-dependent investment decisions

1 2 3 1 2 3

1 2 3

1 2 3

up

up down up

down 1 2 3

1 2 3

down

1 2 3

1 2 3

1 2 3

1 2 3

1 2 3

1 2 3

1 2 3

1 2 3

Path-dependent joint stochastic processes

Forward: Available Budget

Backward: Expected Value

Forward-backward looking algorithm incorporating dynamic global budgets

Fig. 2.3. Modeling approach

constraints model portfolio aspects which are automatically understood as constraints, such as budget constraints. On the other hand, other features are also captured by modeling these as balance equations, such as the specification of the processes of the underlying assets, the determination of available funds, or the calculation of expected values. The dynamic nature of the problem is incorporated through constraints that are either backward– or forward–looking, but either way dynamic. This makes it possible to handle simultaneous interaction effects and to define the problem in a formulation that can incorporate operating and financial constraints. Moreover, this approach makes it possible to solve the problem with commercially available solvers. The model implementation reverts to commercially available software packages, i.e., the optimization platform GAMS© (cf. Bussieck and Meeraus 2004), together with suitable optimizers such as BARON© and Cplex© . This implementation makes it possible to benefit from advanced optimization techniques that have been tested and adapted continuously and enables to separate the problem implementation from the actual optimization algorithms. Consequently, the model becomes verifiable and portable between applications.

2.2 Implications for Modeling Approach

19

Based on this understanding, our approach to portfolios of real options will be positioned in the relevant literature next. This will provide the basis for our ensuing model development.

3 Literature Review

This section reviews the key concepts provided in the relevant literature, drawing from different strands of the literature which will be discussed below. The structure of the literature review is as follows. First, key concepts of financial portfolio theory are discussed in Sec. 3.1. It introduces the basic concepts of portfolio analysis and then leads on to more advanced portfolio approaches. The goal of this discussion is twofold. On the one hand, it clarifies the positioning of our model in the portfolio literature. On the other hand, it shows that no suitable portfolio approach for our problem is available in the financial literature. Second, corporate capital budgeting is reviewed in Sec. 3.2. Special attention is paid to the treatment of interdependencies between projects, from which we will draw in our model. Third, the core of the relevant real options literature is examined in Sec. 3.3. This discussion identifies prior work that is most akin to our approach, and retains two references that have influenced our approach substantially. Therefore, these two are presented in more detail

3.1 Financial Portfolio Theory The seminal work of Markowitz (1952) marked the beginning of modern finance theory (Constantinides and Malliaris 1995). Today, concepts derived from modern portfolio theory are the foundation of most (if not all) financial analysis, directly or indirectly. For example, the risk-adjusted discount rate in a Discounted Cash Flow (DCF) analysis is motivated by the portfolio concept of diversification. Also, a key element of derivatives pricing is the construction of a dynamic self-financing replicating portfolio. An overview of the key concepts of portfolio theory can be found in Markowitz (1991), Constantinides and Malliaris (1995), and Sharpe (2000). More details including empirical tests can be found in Elton et al. (2003) and Rudolf (1994).

22

3 Literature Review

In the following discussion of financial portfolios, first the basic concepts of mean–variance portfolio analysis are discussed (Sec. 3.1.1). Second, more advanced portfolio models are presented (Sec. 3.1.2). 3.1.1 Mean–Variance Portfolio Analysis 3.1.1.1 Traditional Markowitz Portfolio Selection Markowitz (1952) established a relationship between risk and expected return in a portfolio context. This approach identifies the optimal portfolio configuration for a risk-averse investor in a one-period decision problem with the objective to allocate a given amount of money optimally to a portfolio of risky assets. The investor is assumed to consider “expected return a desirable thing and variance of return as an undesirable thing” (Markowitz 1952, p. 77). Based on this perspective on expected return and variance of return, the approach by Markowitz (1952) established what is called “mean– variance analysis”. Assuming non–satiation, always preferring more to less, it derives an efficient frontier in the mean–variance space which contains all portfolios that are not dominated by other portfolios, i.e., no portfolio yields a higher return for the same variance or the same return with less variance. More specifically, the set of efficient stock portfolios is obtained as follows (for the following exposition, cf. Rudolf 1994 and Constantinides and Malliaris 1995). x is a m-column vector whose components xi denote the weight of asset i in the portfolio, i = 1, . . . , m. Given that all available funds P must be allocated, the portfolio weights must add up to 1, i.e., i xi = 1. 1 is a m-column vector of ones, superscript T denotes the transpose of a vector or a matrix. R is a m-column vector whose components Ri denote the mean return of asset i. Further, V is the m × m covariance matrix of the returns of the assets, where the components are σjk , j, k = 1, . . . , m. It is assumed that V is nonsingular, i.e., there is no redundant asset and therefore, none of the return patterns of one asset can be replicated by a portfolio of the other assets. Given that variances of risky portfolios are strictly positive, V is positive definite, i.e., x T Vx > 0. The mean returns and covariances are endogenous to the model and assumed to be known. Finally, the variance of the portfolio is denoted as σP2 and the return of the portfolio is R P . With this notation, the portfolio selection problem can be formalized as follows: min σP2 = x T Vx x

(3.1)

subject to xT R = RP T

x 1=1

(3.2) (3.3)

3.1 Financial Portfolio Theory

23

In other words, this formulation minimizes portfolio variance by solving for the optimal portfolio weights (Eq. 3.1), for each given level of portfolio return (Eq. 3.3) where all funds are invested (Eq. 3.3). x T Vx is convex because V is positive definite; the constraints are linear and thus define a convex set. Therefore, the problem has a unique solution, which can be obtained through solving the Lagrangian function, L, with the Lagrangian factors λ1 and λ2 , as follows: L = x T Vx − λ1 (x T R − R P ) − λ2 (x T 1 − 1)

(3.4)

Solving this equation yields the vector of optimal portfolio weights that minimize portfolio variance for a given mean return, as follows: x = V −1 R

1

R T V −1 R R T V −1 1 −1 R P 1 R T V −1 1 1 T V −1 1

(3.5)

In turn, replacing this result in Eq. 3.1 gives the variance of a minimum variance portfolio, for a given return R P , as follows: σP2 = R P

1

R T V −1 R R T V −1 1 −1 R P 1 R T V −1 1 1 T V −1 1

(3.6)

This solution provides the minimum variance portfolio for a given R P , which is expressed by a parabola. In mean-variance space, i.e., by inverting the function, the mean-variance efficient minimum variance portfolios can be represented by a convex line, starting with the global minimum variance portfolio which is the portfolio with the lowest possible variance for any return. This relationship is graphically represented in Fig. 3.1. All portfolios inside the hyperbola are feasible; portfolios on the solid line are minimum variance efficient portfolios, because they yield the highest expected return for a given variance of return. Therefore, the efficient portfolios dominate all other portfolios including the minimum variance inefficient portfolios. The investor’s choice then consists of choosing that portfolio from the efficient frontier which maximizes her one-period utility. Evidently, this approach is sufficient if either returns are normally distributed (symmetrical), or if the investor has a quadratic utility function. Otherwise, the obtained results are an approximation only and the approximation quality depends on the deviation from the real distribution and utility function (cf. Markowitz 1990, pp. 6 ff.). One key insight of the mean–variance approach is the concept of diversification. Diversification can reduce risk without necessarily adversely affecting expected portfolio return. This leads to the conclusion that it is not the individual (total) risk of an asset that matters, but its contribution

24

3 Literature Review Expected Return Minimum Variance Efficient Portfolios Feasible Portfolios

Global Minimum Variance Portfolio

Feasible Portfolios Minimum Variance Inefficient Portfolios

Variance of Return

Fig. 3.1. Mean–variance efficient portfolios

to the risk of the entire portfolio. In turn, this risk contribution for a well diversified investor is captured by covariance risk, i.e., the correlation of the individual asset with the overall portfolio: σP2 =

X j

x2j σj2 +

XX j

x j xk ρ jk σj σk ,

(3.7)

k6= j

where x j are portfolio weights, σp2 denotes the variance of the return of the overall portfolio, j, k represent individual assets, and ρ jk σj σk is the covariance of their returns, expressed through the coefficient of correlation, ρ jk . With sufficient diversification and thus sufficiently small portfolio weights, the variance of each asset lose significance and σP2 approaches the average covariances between assets (interpreting k 6= j in Eq. 3.7 as a weighted average) (cf. Rubinstein 2002, p. 2). Thus, securities cannot be valued in isolation, but must be considered as a group. While Markowitz (1952) is renowned for introducing the concept of mean–variance analysis and its graphical representation, more rigorous formulations are given in subsequent work (see Markowitz 1991; Mertens 2004). Roy (1952) independently developed an alternative portfolio approach based on minimizing the probability of missing a required minimum return, the so–called “shortfall risk”. For normally distributed returns, this approach is equivalent to Markowitz (1952). Recently, the Markowitz portfolio selection has been extended to a multi–period portfolio selection by Li and Ng (2000). The outlined Markowitz portfolio selection provides the understanding of portfolios that underlies our model. At the same time, our approach fundamentally differs from this portfolio perspective in that it considers active management of the underlying assets, making it possible to influence the return pattern of the assets as endogenous in our model.

3.1 Financial Portfolio Theory

25

3.1.1.2 Capital Asset Pricing Model Related to the optimal portfolio choice of an investor, the (two–moment) Capital Asset Pricing Model (CAPM) developed by Sharpe (1964), Lintner (1965), and Mossin (1966) provided an equilibrium model for the pricing of risky assets.1 The CAPM follows logically from the Markowitz mean– variance portfolio theory. In its basic version, it assumes risk–free borrowing and lending at a given single interest rate, no transaction costs, shared common expectations of all market participants, and perfect capital markets. By an equilibrium argument, it is shown that each investor will hold the market portfolio, MP, regardless of her risk attitude, and will adjust her portfolio to her risk aversion preference by lending or borrowing. The capital asset pricing model is given by: h i (3.8) E (ri ) − r f = β i E (r MP ) − r f where βi =

σiMP . 2 σMP

(3.9)

This relationship states that the expected excess rate of return of asset i (above the risk–free rate, r f ) is a linear function of β i and the market risk premium (expected excess rate of return of the market portfolio MP over r f ). β i is the covariance of asset i with MP, normalized by the variance of MP. This implies that risk which is uncorrelated with the market is irrelevant in the pricing equation because it can be reduced by diversification such that a rational investor would not demand extra compensation for bearing that risk. This risk is usually referred to as nonsystematic, idiosyncratic, or specific risk. In contrast, covariance risk is referred to as systematic risk. From a portfolio pricing perspective it is not the total risk measured by the variance of the return of an individual asset that matters, but only the systematic risk measured by its covariance with the market. The above introduced basic CAPM is widely applied in practice, primarily because of its simplicity. The empirical validity of the CAPM has been widely discussed and challenged, but this discussion is not key for the purposes of our discussion since it will not explicitly refer to the CAPM. An overview of this discussion as well as key empirical results can be found in Fama (1976); Alexander and Francis (1986), “Roll’s critique” (Roll 1977), and an answer to Roll’s critique in Kandel and Stambaugh (1995).

1

For the following exposition, see Luenberger (1998, p. 173ff).

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3 Literature Review

3.1.1.3 Intertemporal Capital Asset Pricing Model and Consumption Both the traditional Markowitz portfolio selection problem and the CAPM are static models in the sense that they consider a single–period decision problem (cf. Constantinides and Malliaris 1995). Only under certain assumptions, especially when preferences and future investment opportunity sets are not state dependent, the solution for the single–period portfolio problem also solves the multi–period portfolio problem (cf. Fama 1970; Rudolf 1994; Brennan and Xia 2006). Relaxing these assumptions, the CAPM has been extended to the Intertemporal Capital Asset Pricing Model (ICAPM) by Merton (1973a). The ICAPM considers “consumer-investors” who maximize the expected utility of lifetime consumption (Merton 1993, p. 475). Therefore, the portfolio decisions over time take into account the relation between current period returns and returns that will be available in the future. Given that the investment opportunity set might shift over time, investors with a multi– period investment horizon need to consider ways of hedging against unfavorable shifts, to protect long–term wealth or consumption. Consequently, equilibrium expected returns depend both on the covariance of returns with the current return of the market portfolio, and information about future returns of the market portfolio Cochrane (cf. 2001, p. 172). For example, if the current return of an asset is negatively correlated with shifts in the investment opportunity set, it can serve as a hedge against negative future developments, thus requiring a higher equilibrium price for this asset through the hedging demand (cf. Merton 1993, p. 479). A more detailed discussion of the ICAPM can be found in Merton (1993) and Duffie (2001). The ICAPM models marginal utility not in terms of consumption, but in terms of other directly observably variables instead, i.e., current wealth as well as conditional distributions of asset returns, given shifts in the investment opportunity set (cf. Cochrane 2001, pp. 46 f., p. 166). In that sense, it is a special form of a consumption–based model (cf. Cochrane 2001, p. 170). Lucas (1978) and Breeden (1979) consider investment and consumption in an intertemporal Consumption–Based Capital Asset Pricing Model (CCAPM). For this, Breeden (1979) explicitly models the joint probability distribution of consumption and excess asset returns. In this setting, the expected excess return of an asset is proportional to the asset’s covariance with consumption, i.e., proportional to the asset’s consumption beta (cf. Constantinides 1989, p. 6). As a result, each asset’s risk is completely specified by the covariance of its return with the change in aggregate consumption (cf. Merton 1993, p. 520). A more detailed discussion of the CCAPM can be found in Merton (1993), Duffie (2003) and Campbell (2003).

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3.1.1.4 Arbitrage Pricing Theory The CAPM provides a rather restrictive pricing model. Multi–factor models have been subsequently developed to explain returns of assets using a number of desired factors.2 The CAPM can be viewed as a special case of a single factor model, with the single factor being the return on the market. Factors can be grouped as external (e.g., gross national product), extracted factors (e.g., industry returns), and firm characteristics (e.g., price to earnings ratio). A prominent extension is the Arbitrage Pricing Theory (APT) which is more general than the CAPM, proposed by Ross (1976). It only assumes non–satiation and that the universe of investment alternatives is large, but is still based on the concept of diversification. By an arbitrage argument, it has been argued that a factor price of risk is derived for each factor, which can then be used for pricing securities. Multi–factor models can increase the empirical validity of the single–factor CAPM (Fama and French 1992, 1993, 1995, 1996). Likewise to the CAPM, the APT can be extended to a multi–period setting (cf. e.g. Ohlson and Garman 1980; Constantinides 1989). 3.1.1.5 Conclusion on Mean–Variance Portfolio Analysis Mean–variance analysis provides a simple structure for portfolio selection and asset pricing. Therefore, it conceptually sets the stage for any portfolio analysis and is also the foundation of the analysis of portfolios of real options. While originally set as single–period problems, portfolio analysis has been extended to multi–period problems successively. However, the assumptions which base the analysis on perfect capital markets in a mean–variance world are too restrictive for our problem. In the context of portfolios of real options, two groups of assumptions need to be relaxed. First, options are asymmetric contracts and thus can hardly be captured by a mean–variance perspective. Option asymmetries enhance skewness of the underlying distributions (Trigeorgis 2005), such that higher moments of the underlying distributions matter. Additionally, volatility dependence on project value may introduce fat tails and kurtosis (Sears and Trennepohl 1983). As a consequence, portfolio theory for higher moments, beyond µ and σ, becomes relevant. Second, portfolios of real options must be analyzed in the context of frictions, such as budget and operating constraints (Trigeorgis 2005). Since these reservations concern single–period portfolio models, they also apply for multi–period portfolio problems. This is why more advanced portfolio models are investigated next, which model higher moments of distributions and allow for frictions. 2

For the following exposition, see Luenberger (1998, p. 198ff).

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3 Literature Review

3.1.2 Advanced Financial Portfolio Analysis 3.1.2.1 Portfolio Analysis Considering Higher Moments of Distributions As noted, Markowitz portfolio theory considers only the mean and variance of the distribution of returns of the underlying assets (i.e., only the first two moments). This is valid if the investor’s utility function is quadratic, or if the distribution can be sufficiently characterized by the first two moments, which holds true for the normal distribution. The assumption of quadratic utility functions has been criticized (Arditti 1967; Levy 1969), while the assumption of normality is not sufficient in the context of portfolios of real options (cf. Trigeorgis 2005). Consequently, higher moments in financial portfolio theory must also be considered. Generally, the r th univariate moment about the origin of a continuous random variable X is given by r

E( X ) =

Z

+∞ −∞

xr f ( x )dx,

(3.10)

with f ( x ) being the probability density function of X (cf. Ross 2003). While r can be any positive integer, the first four moments are most intuitive to characterize a distribution. The first moment is the mean, the second moment the variance (spread of outcome), the third moment is the skewness (measuring asymmetry), and the fourth moment is the kurtosis (measuring the probability mass on the tails of a distribution). For jointly distributed random variables, additional multivariate co–moments describe the multivariate distribution, such as co–skewness and co–kurtosis, analogously to the covariance. A recent review of portfolio optimization for higher moments can be found in Guse (2005). Early approaches analyzing the first three moments can be found in Arditti (1967), Samuelson (1970), and Jean (1971). Kraus and Litzenberger (1976) formulated a three–moment CAPM rigorously, with a more recent analysis of the three–moment CAPM found in de Athayde and Flores (2000) and Jurczenko and Maillet (2001). Fang and Lai (1997) extended the CAPM to four moments, while Jondeau and Rockinger (2003) estimated the first four moments and co– moments for a set of daily stock–index and foreign–exchange returns. While it is possible to consider even higher moments, the additional insights from better approximating the real underlying distribution is usually decreasing and the additional computational effort grows considerably (cf. Chung et al. 2004, who consider up to 10 moments).

3.1 Financial Portfolio Theory

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3.1.2.2 Allowing for Frictions in the Management of Portfolios One direction for making the portfolio theory more realistic has been to allow for more complexity in either the number of factors or the number of moments considered. These approaches are typically set in a model with frictionless markets, involving no restriction on completeness of markets, continuous trading, no transaction costs, and no restrictions on short sales. Another strand of literature aims at making portfolio models more realistic by explicitly modeling specific frictions. A comprehensive overview of portfolio optimization under constraints and transaction costs can be found in Korn (1997). If rebalancing of a portfolio is costly, portfolio optimization must take such transaction costs into account. The bottom line effect is to hold less assets and trade less often, which counters the effect of diversification and continuous rebalancing. Key references analyzing the impact of transaction costs on the optimal portfolio include Brennan (1975), Constantinides (1979), Davis and Norman (1990), Li et al. (2000), Akian et al. (2001), and Sass (2005). More recently, Best and Hlouskova (2005) provide an algorithm reducing numerical expense for portfolio optimization. Going beyond pure transaction costs, it may sometimes be desirable to limit the number of portfolio positions, as in Jacob (1974) and Levy (1978). Li et al. (2001) modeled frictions by including transaction costs together with restrictions on short sales and taxes. The review of portfolio models for higher moments and models considering frictions is separated from one another because frictions have hardly been studied jointly with higher moments in the distributions of returns. Exceptions are Wang and Xia (2002) and Liu et al. (2003) who simplify their problem by piecewise linear approximations of the skewness. 3.1.2.3 Impact of Combining Financial Assets with Financial Options Up to this point, we have not yet considered financial options. Financial options are derivatives, i.e., redundant, in the sense that their cash flows can be replicated by directly holding positions in the corresponding underlying assets. An example is a call option on a share, whose payoff can be replicated by holding the share and borrowing a specific amount of money by shorting riskless bonds. The call option corresponds to a levered position in the underlying asset and actually can be viewed as some kind of portfolio because its payoff is identical to the payoff of an equivalent portfolio. In turn, the key concept of derivatives pricing is self–financing replication in arbitrage–free markets. If the call option yields a payoff identical to the dynamically rebalanced portfolio of share and shorted bond in all states of nature, it must have the same price. This assumes that markets are dynamically complete and in equilibrium, with correct prices for all assets. Options

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3 Literature Review

can be “home–made” in that if a desired option is not available on the market, it can be replicated by holding the equivalent position in the underlying asset and riskless bonds. Based on this perspective, it is less insightful to develop a portfolio theory for portfolios composed of financial options alone. For plain–vanilla options, option values may be correlated as a function of the correlation of the underlying assets, but may otherwise be “strategically” independent. By this we mean that the option payoffs are independent from one another, and that the optimal exercise strategy for each option can be obtained by considering each option in isolation from the other options. A similar argument holds for more complex financial options written on several assets where the option price again depends on the correlation among the underlying assets, but is otherwise independent of other options. The same applies for options on other options, since this is the special case where the underlying asset in fact is an option. With this reservation, we retain that financial option values follow directly from the equilibrium prices of the corresponding underlying assets. Moreover, exercising a financial option typically does not affect the underlying asset. It is known that a portfolio of options is at least as valuable as an option on the corresponding portfolio of underlying assets because the portfolio of options gives more discretionary decision rights. Moreover, for option pricing, total risk (volatility) matters and standard option prices typically are increasing with higher volatility. Therefore, any diversification in the risk of the portfolio and reduction of volatility reduces option values (cf. Merton 1973b). This illustrates that the notion of risk for portfolios of options is fundamentally different from the understanding of risk in a standard portfolio perspective. According to Trigeorgis (2005), risk is not necessarily something bad that must be avoided, as posited by Markowitz, but a window of opportunity for capitalizing on upside potential. These different notions of risk need to be accommodated jointly when portfolios of assets and real options are considered simultaneously. The portfolio effects of options must be considered with care, as they modify the return pattern of the portfolio. For example, portfolio insurance is achieved by adding put options to a portfolio, locking in some minimum return at the expense of upside potential (Bookstaber and Clarke 1983, 1984) to provide a (partial) hedge. Besides hedging, options may be used for speculation, for saving on transaction costs, for helping complete an otherwise incomplete market, or for increasing liquidity when the markets for underlying securities are less liquid. These aspects require more advanced valuation techniques (cf. Korn and Trautmann 1999), but they do not represent a new theory as such. The assets are still priced in market equilibrium and options are still derivative securities, whose prices depend on the prices of

3.1 Financial Portfolio Theory

31

the underlying assets. Adding options to a portfolio of assets enables to design a desired payoff structure. 3.1.2.4 Portfolio Implications for Risk Management Risk management aims at altering the risk profile of a firm in order to protect its profitability (McDonald 2003, p. 89). An overview over risk management is found in Dempster (2002). One of the most widely used concepts in corporate risk management, “Value–at–Risk” (VaR), is reviewed in Duffie and Pan (1997). VaR quantifies the total risk of a portfolio, measuring the maximum loss that can materialize, with a given probability, over a defined time horizon (cf. Hull 2003, pp. 346 ff.). Risk management is closely connected to portfolio management because the risk of the portfolio has to be well understood before it can be controlled in a desired way. Also, portfolio management can be implemented via VaR, by using VaR as the measure of risk instead of volatility (cf. Alexander and Baptista 2004). In the context of this dissertation, risk management is not a relevant field. It represents an application of the portfolio concepts discussed so far, not a new theory in itself. Further, the underlying perspective of minimizing risk is fundamentally different from the real options perspective in portfolios of real options. The former considers risk as an undesirable thing, the latter capitalizes on the desirable effects of risk while reducing the corresponding downside. Thus, a rigorous portfolio approach to real options can ultimately provide further insights for risk management (cf. Trigeorgis 2005, pp. 36 f.) because real options affect the risk exposure of a company. The existence of real options provides a means of limiting downside risk. Consequently, the calculation of a risk exposure needs to take into account the effect of real options before it is insightful to manage that risk from a risk management perspective. In short, the analysis of portfolios of real options can shed light on risk management, not the other way round. 3.1.2.5 Conclusion on Advanced Financial Portfolio Analysis Advanced portfolio analysis is better able to handle complex problems with higher moments. Complex portfolio models have also been formulated for markets with frictions, such as to capture transaction costs. Still, the current state of research sheds very little light on portfolios of assets with higher moments and frictions. Yet a suitable model for portfolios of real options would need to go even further than that, and also include options. Incorporating options into a portfolio increases complexity because options enhance asymmetry. Even comparatively simple options like a compound option (cf. Geske 1979) or an option on the maximum or minimum of several

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3 Literature Review

assets (cf. Stulz 1982; Johnson 1987) present challenges because the exercise region cannot be determined by simple arguments anymore (Broadie and Detemple 1997, p. 242). Further, the pricing of American, path–dependent or multi–asset options typically must resort to numerical methods since closed–form solutions may not be feasible (cf. Broadie and Detemple 2004, p. 1163). In the light of our portfolio approach, as defined in Sec. 2, the just presented review of the relevant financial portfolio literature has shown that no portfolio model suitable for the desired analysis is readily available.

3.2 Corporate Capital Budgeting 3.2.1 Overview “Capital budgeting is the decision area in financial management that establishes goals and criteria for investing resources in long–term projects” (Clark et al. 1984, p. 1). A synthesis of the capital budgeting literature can be found in Weingartner (1977) and more recently in Peterson and Fabozzi (2002) and Dayanada (2002). While the above mentioned definition is clearly based on a financial management perspective, the analysis of investments can also take place from an operations management point of view (for an overview, see Van Mieghem 2003). In this strand of literature, more emphasis is laid on detailed modeling of processing networks, including operating considerations such as production planning, inventory management, and supply chain management. We will not focus on this field here since the present dissertation is set in a real options and thus financial management perspective. A first model for harmonizing firm investment outlays and limited capital resources was provided by Dean (1951). It derives the optimal capital budget as the intersection of a capital demand and a capital supply curve in a one–period setting. Following this perspective, Lorie and Savage (1955) identify the three key problems in investment choice as the choice of profitable investments, forecasting cash flows, and rationing capital. Charnes et al. (1959) proposed a valuation approach based on (linear) programming which was widely applied subsequently. Key references for solving the capital budgeting problem by mathematical programming include Baumol and Quandt (1965), Weingartner (1966b, a, 1967), Arditti (1967), Tuttle and Litzenberger (1968), Hillier (1969), Nemhauser and Ullmann (1969), and Elton (1970). The relevant features considered in these papers, especially logical relationships and synergies among projects, will also be considered in our model, which we capture through constraints and correlated cash flows.

3.2 Corporate Capital Budgeting

33

While these authors mostly focused on implementation issues of the programming approach, attention subsequently shifted towards a harmonization of the capital budgeting decision with the portfolio concepts just recently developed for capital markets. In this context, the work of Markowitz (1952) as well as Sharpe (1964), Lintner (1965), and Mossin (1966) was guiding. It provided helpful insights as to what cost of capital should be used in the assessment of investment projects and in how far a portfolio of projects differs from stand–alone analysis. Key references discussing an explicit link between the CAPM and corporate investment decisions include Weston (1973), Carter (1974), Spies (1974), Nielsen (1975), Thompson (1976), and Myers and Turnbull (1977). While also based on an equilibrium framework, the approach chosen for the present dissertation will be substantially different from these papers, in that it is based on a risk–neutral valuation framework (see Sec. 4.1.4). Another approach to traditional capital budgeting is represented by the integer knapsack problem, e.g., in Dantzig (1957) and Cord (1964). In the context of capital budgeting, an integer knapsack problem is a combinatorial problem where the value of a selection of possible projects (e.g., items in the knapsack) is maximized, subject to budget constraints (e.g., the size of the knapsack) and to integer constraints for the projects. Knapsack problems provide a rich framework for handling capacity restrictions in decision analysis and have been widely extended in the last 50 years. A thorough review of the different types of knapsack models can be found in Kellerer et al. (cf., e.g., 2004). Still there is no readily available knapsack problem that can handle the portfolio problem considered here. Our model takes into account multiple correlated processes and dynamic interactions between decisions, at each point in time and over time. Such a stochastic knapsack model has not yet been presented in the literature. An alternative strand of literature stresses the flexibility inherent to investment management, and for this resorts to decision tree analysis (cf. e.g. Magee 1964a, b; Hespos and Strassmann 1965; Hax 1985). The relationship between real options and decision tree analysis has largely been discussed elsewhere (cf., e.g., Smith and Nau 1995; Trigeorgis and Mason 1987; Trigeorgis 1996). In the light of our discussion, decision tree analysis does not provide additional insights beyond real options analysis and therefore is not discussed further. 3.2.2 Portfolio Perspective in Corporate Capital Budgeting Investment decisions of companies are inherently concerned with portfolio considerations (cf. Weingartner 1966a). Companies typically are faced with investment program decisions where a set of investment projects, or an investment program, is chosen out of a larger set of investment opportunities.

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3 Literature Review

This program can be considered as a portfolio. Further, since capital assets usually have a multi–period lifetime, the investment program has to optimally fit with the assets already in place, which together constitute the corporate asset portfolio. It is relatively straightforward to apply the concepts from standard (mean–variance) financial portfolio theory to corporate capital budgeting (see e.g. Hax 1985, pp. 133 ff.). Each investment project is assessed based on the risk it bears and the return commensurate with that risk. From a portfolio perspective, the relevant risk is not the total risk but the marginal systematic risk the individual project adds to the existing (market) portfolio, such that only the covariance risk with the (market) portfolio matters. In this context, corporate diversification is redundant (Brealey and Myers 1996, p. 166). With this background, companies should only care about the systematic risk of a project and can assess projects based on some equilibrium model such as the CAPM. As a function of project betas, the required rates of return are known. Each project that has a positive NPV is value creating, since its return is above the return demanded by capital markets for that type of systematic risk. The portfolio perspective on capital budgeting is very convenient because it allows to assess projects as if they were stand– alone, such that project values are additive. In this sense, pricing a real asset is analogous to pricing a financial asset (cf. Damodaran 1996, pp. 484 ff.). 3.2.3 Interdependencies The portfolio perspective on real assets requires more thorough analysis when projects cannot be assessed independently from one another in the CAPM sense (Peirson and Bird 1981, pp. 177 f.). Projects may be interdependent for a range of reasons (for this classification, see Betge (1995) and Brosch (2001)): • Direct qualitative interactions which have their origin in the investment plan as such, or which result from interactions with investments already undertaken and still generating cash flows. Direct qualitative interactions do not result from stochastic relationships, but from physical properties of the projects. This is why they cannot be avoided in any way comparable to diversification. Examples are mutual exclusive projects (=substitutive), projects requiring one another (=complementary), or synergies (positive or negative). • Indirect qualitative interactions which result from constraints beyond the investment plan. They are indirect because they are not inseparably connected to the investment opportunities considered; qualitative because they do not result from stochastic relationships and cannot be avoided

3.2 Corporate Capital Budgeting

35

by diversification. These constraints are connected to the firm making investment decisions. Most importantly, indirect qualitative interactions stem from binding capital constraints, i.e., capital rationing. Recently, Meier et al. (2001) explicitly discuss a portfolio of real options subject to a capital expenditure constraint and propose two model formulations for portfolios of real options. The first formulation is a standard knapsack problem, maximizing the value of the portfolio subject to a capital constraint. The value of the portfolio is calculated as the sum of the values of the options in the portfolio. Therefore, this model follows closely the traditional capital budgeting by replacing the NPVs of projects with their option values. Real option values do not interact and the model is static. In a second integer programming formulation, they provide a dynamic approach which models investment decisions over time within a binomial framework. All projects are related to the same underlying stochastic variable, but are independent from one another. Further, only call options are considered which can be postponed indefinitely. Once an option is exercised, there is no further option available. Therefore, the options are stand-alone options that do not interact. The budget constraint is static in the sense that it gives a fixed upper limit on outlays which must not be exceeded at any point in time. Their paper shares the conceptual origin with the present dissertation, in that it focuses on portfolios of real options, for limited funds available. As opposed to their approach, we go further into detail with regards to modeling options interactions, path–dependencies, dynamic budget constraints, and other constraints. In addition, our focus differs. While their paper lays emphasis on providing an efficient algorithm specific to the problem, our focus is more on the general model formulation. While interactions such as synergies were discussed extensively in the mathematical programming literature (see Sec. 3.2.1), they may be debated in the context of modern corporate finance. First, the presence of direct qualitative interactions violates the additivity of NPVs, which derives from the underlying mean–variance portfolio concept based on systematic risk. While this view is debatable (cf., e.g., Stulz 1999), it does not disallow the possibility of, e.g., synergies. NPVs add up if (and only if) the only portfolio aspect is diversification and markets are perfect and complete. Other interdependencies, e.g. due to budget constraints, can cause deviations from additivity. Second, in realistic investment decision settings there may exist “soft” or “hard” rationing of capital (cf. Brealey and Myers 1996, pp. 101 ff.). Soft rationing implies that although capital markets are perfect, the firm imposes an internal constraint voluntarily as a means of financial control. Hard rationing implies some sort of market imperfections. These imperfections may be caused by capital markets in a strict sense, or may result from the com-

36

3 Literature Review

pany’s limited access to capital markets. Either way, as long as the ultimate investor is not hindered in her access to well–functioning capital markets, the existence of capital constraints does not by itself invalidate the NPV approach. An extensive review of literature on financing constraints is found in Hubbard (1998). Recently, Mello and Parsons (2000) analyze the impact of financing constraints on operating and hedging policies of a firm which is already invested. They show that the constraint effectively reduces the firm’s alternatives. Boyle and Guthrie (2003) focus on the effect of financing constraints on the initial decision to invest. Almeida et al. (2004) empirically test the effect of financial constraints on investment, proxied by the holding of cash reserves. 3.2.4 Conclusion on Corporate Capital Budgeting The literature on corporate capital budgeting has been concerned with both portfolio considerations and the modeling of interdependencies. Nevertheless, the traditional approach is oftentimes reduced to calculating NPVs under very restrictive assumptions, reducing the assessment of a specific project to its beta with the market. The present dissertation considers interactions jointly with the interactions stemming from correlation of real options and underlying assets alongside with asymmetries introduced by option exercise decisions. In doing so it unites elements taken from the traditional capital budgeting literature with more recent approaches based on contingent claims analysis.

3.3 Financial and Real Options Theory 3.3.1 Foundations of Real Options Concepts derived for financial options have provided a rich arsenal for valuing real options at the same time because real option valuation is based on concepts from financial option pricing. Cornerstones of real options pricing therefore are Black and Scholes (1973) and Merton (1973b) who were first to provide a closed–form solution for financial options, as well as Cox et al. (1979) who introduced an equivalent discrete binomial valuation approach that has proved to be very powerful in handling complex options settings. While these seminal contributions deal with single options in the first place, following work has focused on problems involving several options or several underlying assets which can be viewed as special cases of a portfolio. For example, an option on the maximum of two assets can be viewed as a portfolio investment problem with the opportunity to invest in two different projects which are mutually exclusive. This direction is particularly

3.3 Financial and Real Options Theory

37

relevant in the light of our analysis here. Margrabe (1978) analyzes an option to exchange one asset for another while Stulz (1982) considers an option on the maximum or the minimum of two underlying assets, later generalized by Johnson (1987) and Boyle and Tse (1990) to several underlying assets. Laamanen (2000) models the related option on the best subset of multiple underlying assets. On a related strand of research, Geske (1979) provides a closed–form solution for two options which are compound, i.e., an option on an option. These two features are combined by Carr (1988) who values a sequential exchange option, i.e., an option on an option to exchange one asset for another. These concepts have also been widely applied to real option valuation problems, though mostly focusing on options analyzed in isolation. While the analysis of stand–alone options provided much insight for the understanding of flexibility, its usefulness is limited from a portfolio point of view. Exceptions are those papers explicitly taking into account the interaction of real options and real assets. Brennan and Schwartz (1985) analyze a natural resource project with options to shut down and restart or abandon. McDonald and Siegel (1985) consider an option to temporarily shut down a plant. The interconnection of separate options through a shared underlying asset makes it essential to value these options in combination in a compound option–like model.3 While these authors do take into account interactions, Trigeorgis (1993a) addresses options interactions explicitly. He points out that the existence of subsequent options may increase the value of the underlying asset for earlier options, whereas the possibility of exercising earlier options affects the value of the underlying asset for subsequent options. As a consequence, in the case of multiple options written on the same underlying asset, option values typically do not add up. Trigeorgis (1993a) discusses general properties concerning the nature of options interactions. Kulatilaka (1995) confirms the results of Trigeorgis (1993a) and explicitly considers the effect of additional options on the critical boundaries for option exercise, i.e., the exercise schedules. 3.3.2 Real Options Analysis in a Portfolio Context The above cited references have set the stage for real options analysis in a portfolio context. They have made clear that the analysis of interacting real options are of special interest because interactions have to be modeled explicitly. Following this line of argument, several authors have shed light on 3

Kulatilaka (1993) values a dual–fuel steam boiler where switching between different combustibles is costly. While it is desirable to switch between oil and gas as a function of the relative prices, the switching is subject to switching costs, implicitly making the problem a nested compound option problem.

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3 Literature Review

settings with interacting options. Following Trigeorgis (1996, pp. 132 f.), interactions between options on the same underlying are designated as “intra– project”, while interactions embracing several underlying assets are designated as “inter–project” interactions. The field of Research and Development (R&D), in particular, has paid special attention to options interactions. Due to the exploratory nature of R&D projects which are typically multi–stage, early projects can generate insights about future projects. Moreover, in an R&D pipeline many projects are typically undertaken but only few make it to become a marketable product, such that the bundle of projects must be assessed from a portfolio point of view. Problems involving inter–project interactions are thus particularly relevant. In this context, Childs et al. (1998) consider a setting where two projects may be run in parallel during the development phase, but only one may be implemented, i.e., they are mutually exclusive. They find that sequential development is desirable when project values are highly correlated, require large capital outlays, are short term, and have low volatility. Likewise, Childs and Triantis (1999) as well as Lint and Pennings (2002) consider the parallel development of two R&D projects that are mutually exclusive. Gustafsson and Salo (2005) focus on project selection subject to budget constraints, which is achieved by explicitly modeling the decision maker’s terminal utility. Denardo et al. (2004) propose a stochastic search algorithm motivated by an R&D application with a sequential compound decision process. Attention has also been paid to learning during the R&D process (cf. e.g. Koussis et al. 2003). Huchzermeier and Loch (2001) model a compound R&D decision problem where management has the option to take corrective action as a means of managing the risk involved in R&D projects which may stem from different sources (e.g., market payoffs and project schedules). Vassolo et al. (2004) find for the biotechnology sector that mutually competitive, correlated projects tend to be sub–additive, but that the sharing of resources among firms may create positive spill–overs resulting in super– additivity. A review of R&D specific investment problems is found in Chien (2002) and Kavadias and Loch (2004). In the context of exploration, Smith and Thompson (2005) show that learning between correlated oil exploration activities may invalidate the conventional wisdom of diversification, in that for a risk–averse investor it may be desirable to invest into highly correlated prospects. Dias (2006) confirms that in oil exploration, positive correlation is a desirable feature, as opposed to financial options intuition, due to learning and synergies. In other related work, Brown and Davis (1998) model an investment choice problem where a first project can be followed up by one of two mutually exclusive projects. Rose (1998) and Bowe and Lee (2004) consider the intra–project interactions of options on infrastructure projects, following

3.3 Financial and Real Options Theory

39

the framework presented in Trigeorgis (1993a). Triantis and Hodder (1990) analyze a flexible production system that can switch the production mix among two products over time. Kogut and Kulatilaka (1994) and Huchzermeier and Cohen (1996) consider a global manufacturing network under exchange rate risk, with switching options between different manufacturing strategies contingent on exchange rate realizations. Wang and de Neufville (2004) value options inherent in the design of large physical systems, such as hydropower stations, modeled as path–dependent options. Their approach is related in that they model the option valuation problem as a stochastic mixed–integer program that can be implemented on commercially available optimization platforms. This approach will also be followed in the present dissertation in order to solve the formulated portfolio model. Martzoukos et al. (2003) model a path–dependent investment problems that is essentially a switching model subject to stochastic switching costs. In order to keep track of all possible paths and possible early exercise on each path, they propose a programming algorithm labelled as “exhaustive search” (cf. Martzoukos et al. 2003, p. 9). In the context of supply chain management, Kamrad and Siddique (2004) model the interactions between a producer’s and a supplier’s investment decisions, considering exchange rate uncertainty, supplier–switching options, order–quantity flexibility, profit sharing, and supplier reaction options. They consider discretized exchange rate processes as sources of uncertainty and solve the problem numerically through a backward recursion in dynamic programming. The mentioned references dealt with complex problems involving interactions between real options, either intra–project or inter–project. However, they did not take an explicit portfolio perspective on real options. Most explicit in presenting a portfolio perspective on real options, yet only qualitative, is Luehrman (1998). He chooses a gardening metaphor where the firm is behaving like a gardener who only picks tomatoes (= exercise options) which are “ripe and perfect”(= at the optimal time). There are good and bad gardeners, active and passive ones. A good gardener knows which tomatoes to pick, which ones to leave yet a little while, and which ones to pick even if not yet fully ripe, in order to prevent the squirrels (=the competitors4 ) from stealing them. While Luehrman (1998) is notable for stressing the intuition, he does not shed light on how interactions between options can be coped with in a rigorous valuation framework. Trigeorgis (1988) and Trigeorgis and Kasanen (1991) consider a portfolio perspective more explicitly by considering compound synergistic effects between parallel projects. Kester (1984, 1993) analyzed inter–project dependence with special attention to growth options, while Kasanen and Trigeor4

The author of the present dissertation was not aware of the fact that squirrels like tomatoes, but believes that this is only secondary to the argument.

40

3 Literature Review

gis (1993) and Kasanen (1993) put more emphasis on modeling synergies. Mauer and Ott (1995) analyze replacement decisions as follow–up projects. 3.3.3 Detailed Review of Related Papers Beyond the so far reviewed references, two publications have particularly influenced the present dissertation: Kulatilaka (1988) and Kulatilaka and Trigeorgis (1994). In order to position the following analysis, these two reference will be discussed in more detail in what follows. 3.3.3.1 Kulatilaka (1988) In the paper “Valuing the Flexibility of Flexible Manufacturing Systems” Kulatilaka (1988) analyzes the case of a flexible manufacturing system, in which a flexible technology is one that can be operated in alternative modes. This perspective on modes inspired the setup of our model and therefore is presented in detail. At the same time it is noted that our modeling technique based on binomial option pricing is fundamentally different from the transition probability, i.e., Markov chain, approach in Kulatilaka (1988). For consistency the paper is not represented in its original notation, but we present the relevant elements with the variable definitions used in the following model development. Further, some key concepts are introduced somewhat differently in order to stress the elements employed later on. Especially, the view on entering and leaving modes is introduced here. This is not part of the original paper but helps to elucidate the design of the model. Nevertheless, despite these suitable modifications, the original content of the paper remains unchanged. In this paper, operating modes are interpreted in a very general way with the examples of different routings within a production system, different sets of inputs used in manufacturing the product, or different outputs. Also, modes can characterize waiting to invest, shutting down, and abandoning of projects. Operating modes are designated as a, a = 1, . . . , A. Further, it is insightful to distinguish another operating mode, a0 , a0 = 1, . . . , A0 . Mode a is the mode in which the technology is before a switch, a0 is the result of this switch, from a to a0 . Therefore it is convenient to define a as the entering mode and a0 as the leaving mode. The flexible technology is exposed to some uncertain variable, θk , which represents the uncertainty stemming from, e.g., demand, price, or exchange rate. θk follows some stochastic process which determines possible states in which θk can realize and the corresponding transition probabilities between states. More precisely, θk can realize N different states, i.e., k = 1, · · · , N. Further, the transition probability is given by pij , specifying the transition

3.3 Financial and Real Options Theory

41

probability that θ (t + 1) is in state i given that θ (t) was in state j. This discrete system state space can be interpreted in two ways. It may be viewed as an approximation of a problem which is continuous in reality. Alternatively, the discrete nature can be “given exogenously by contracting arrangements or other organizational constraints” (Kulatilaka 1988, p. 252). Associated with each production mode a0 is a cash flow function c(θ, a0 ) which is determined by the realizations of the stochastic process of the value of the underlying asset. Let us assume three cash flow functions, so we have for a0 = 1, 2, 3: c(θ, 1) = 0

(3.11)

c(θ, 2) = −0.6 + 2θ

(3.12)

c(θ, 3) = −0.1 + θ

(3.13)

Note that c(θ, 1) is the mode for waiting to invest. These parameters are chosen such that for low realizations of θ, i.e., θ < 0.5, c(θ, 3) > c(θ, 2) while for θ > 0.5, c(θ, 2) > c(θ, 3). Switching from any mode a, to any mode a0 (a and a0 can be identical) incurs a switching costs, I ( a, a0 ). Periods are counted by t, t = 1, . . . , T. The objective is to maximize the net present value, V (t, a). This present value is calculated iteratively using stochastic dynamic programming, with three simultaneous equations, at any state k in period t. For each entering mode, a, the optimal leaving mode, a0 is solved for. For the last period, the value of the system is given by the maximum of the cash flows resulting from the different leaving modes a0 : (3.14) V ( T, a) = max c(θ T , a0 ) − I ( a, a0 ) a0

In the last but one period, T − 1, future realizations of θ are unknown. This is why for intermediate periods the optimal value is obtained by finding the leaving mode that maximizes both current cash flow net of switching costs and the expected value for the following period which is entered in the leaving mode of the previous period. This expected value of cash flows, ET −1 [V ( T, a0 )], is calculated as the probability-weighted sum of all V for possible realizations of θ in the following period, given a0 . It is discounted by the risk–adjusted one–period discount factor, φ. Note that the expected value is calculated in each period, looking forward to the following period. This is indicated by the subscript in the expected value operator. Based on this, the value for the last but one period is as follows: n o V ( T − 1, a) = max c(θ T −1 , a0 ) − I ( a, a0 ) + φE T −1 [V ( T, a0 )] a0

(3.15)

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3 Literature Review

Generalizing this relationship establishes the stochastic dynamic programming equations: n o V (t, a) = max c(θt , a0 ) − I ( a, a0 ) + φEt [V (t + 1, a0 )] a0

a = 1, 2, 3 ; t = 1, . . . , T − 1.

(3.16)

Proceeding in a stochastic dynamic programming fashion, this problem can be solved by backward recursion. Folding back, the expected value in each period includes the optimal switching decisions and resulting cash flows from all subsequent decisions. This perspective on flexibility in investment decisions is particularly insightful because of the switching concept introduced. It can interpret any investment and disinvestment decision as a switching decision and thus is very general. This is why the present dissertation follows this switching perspective. At the same time, it substantially expands the model by linking it to a real options context with an explicit link to capital markets considering multiple underlying assets, and most of all, by allowing for constraints and a dynamic budget. These additional effects are such that the model in its core can stay with a similar switching concept, but requires a completely different, forward–backward looking modeling approach. 3.3.3.2 Kulatilaka and Trigeorgis (1994) While Kulatilaka (1988) has presented the interpretation of investment decision as switches, he did not stress the link to capital markets and thus real options theory. This specific link to real options theory is what the paper “The general flexibility to switch: real options revisited” by Kulatilaka and Trigeorgis (1994) is about. Following the reasoning of Kulatilaka (1988), they discuss the case of a car manufacturer. Cars can be produced with two alternative technologies and two corresponding cash flow functions. The net cash flows for each technology depend on the oil price. One technology has high fixed costs and low oil consumption thus low variable (oil) costs, such that it is attractive in periods where oil prices are high. The other technology has low fixed costs but high variable (oil) costs. They show that the possibility to switch between these two technologies, given switching costs, can boost value beyond the stand-alone value of these technologies. From a modeling standpoint, the paper is particularly insightful because it connects the switching concept to a real option valuation approach, based on the binomial model. It is based on two simultaneous option lattice trees, i.e., for each technology one. Both binomial lattice trees are incorporated into one stochastic dynamic program, which incorporates switching between corresponding nodes in the above described backward recursion.

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Within this setting, they further discuss the impact of the nature of switching costs on the optimal investment strategy and distinguish two cases: 1. No switching costs. Since switching between technologies is costless, the entering mode is irrelevant for the optimal operating mode to be chosen. A switching decision affects the current cash flow, but is irrelevant for all subsequent decisions. Thus, the flexibility to switch can be calculated in a straightforward manner. Specifically, it can be calculated as the present value of cash flows of one base technology, adding to this the values of European options to switch to the other technology at each decision point. Obviously, the same result is obtained, no matter which technology is taken as the base technology. 2. Switching costs. If switching costs exist, the entering mode affects the switching decision and thus has to be taken into account. This requires the two simultaneous option lattice trees, and corresponds to the case analyzed in Kulatilaka (1988). The case of switching costs is particularly interesting because it introduces a compound interaction. From a real options point of view, each switch, or exercise decision, creates a specific series of nested new options. This is analogous to a compound option and invalidates option value additivity. Interestingly, this may introduce a hysteresis range, or inactivity range in the optimal operating policy. This means that even if an immediate switch would create a positive net cash flow, it may be optimal to stay in the original mode and wait for stronger changes in the underlying variable. This is because the underlying variable must change sufficiently not only with respect to current cash flows, but also to future cash flows. This dynamic perspective takes into account the probability and costs of re-switching back in later stages; the costs of switching back and forth can be higher than the cash flows realized. Further, it may be desirable to wait and maintain options to switch later if the leaving mode is absorbing in the sense that further switches are prohibitively costly, or even impossible. The authors generalize the switching concept, showing that known real options can be valued as special cases of the switching concept. Specifically, they define suitable interpretations of switching modes for these options.5 This generalization underlines the large applicability of the switching concept. Kulatilaka and Trigeorgis (1994) is guiding for the approach presented here for two reasons. First, it completes the switching concept with a binomial real option pricing approach. The binomial switching concept is at the 5

They distinguish the following options: option to defer; to expand; to contract; to temporarily shut down and restart operations; to abandon for salvage value; to default during construction (or time-to-build option).

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3 Literature Review

core of our model. Second, the generalization of the switching concept to known types of real options explicitly connects our model to the existing body of real options literature where typically a real options terminology is used. Due to the generality of our switching concept, it contains known types of real options as special cases. 3.3.4 Conclusion on Financial and Real Options Theory The body of literature on real options has grown large, considering a wide range of problem settings that can only be fully captured with the help of contingent claims analysis. Nevertheless, the rather limited analysis of portfolios of real options in the literature so far has been motivated by specific decision problems where compoundness and interactions arose because of special features of the decision problem, such as the parallel or sequential development strategies of R&D projects, or investment in capacity. While some authors have focused more explicitly on interaction effects among options, such as involving growth options or synergies between projects, a more systematic treatment of the main portfolio effects, including budget constraints, has been lacking so far. We aim to close this gap with the general portfolio model framework discussed next. In order to do so, the work by Kulatilaka (1988) and Kulatilaka and Trigeorgis (1994) provides a starting point that we will build on and further extend.

4 Valuation Model for Portfolios of Real Options

This chapter contains the formulation of a valuation model for the portfolio problem defined in Sec. 2.1.2 where portfolios of real options are understood as a combination of multiple underlying assets and multiple real options written on these assets, subject to constraints. The model description is organized as follows: First, the described portfolio aspects are translated into model features that the general model formulation will take into account. Then, the binomial model is introduced as the starting point for our proposed model, which is then expanded successively to n underlying assets.

4.1 Model Features In order to ground the development of the proposed model, the key features that determine the model are presented in what follows. The description of these key features covers the objective function, the managerial actions permitted, the attitude towards risk, the modeling of the underlying stochastic processes, volatility and correlation, inter– and intra–project interactions, global dynamic budgets, synergies and operational constraints, learning, diversification, path–dependencies, as well as a discussion of the solution approach and the scalability of the model. 4.1.1 General Assumptions It has been widely discussed whether the real options approach is suitable for practical decision problems. Most importantly, the applicability of real options theory depends on whether the assumptions typically agreed on for financial assets and financial options hold for real options as well. The development of the portfolio model is based on a set of assumptions which

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4 Valuation Model for Portfolios of Real Options

are outlined here. These assumptions are standard in the context of real options (cf., e.g., Dixit and Pindyck 1994; Trigeorgis 1996; Copeland and Antikarov 2001). These assumptions are: • Capital markets are perfect and complete.1 Therefore, a spanning portfolio is available for each investment opportunity considered. • The risk–free interest rate is known, constant, and identical for all maturities. • The values of the underlying assets follow discrete binomial processes. • The underlying assets do not distribute any dividends. • Investments are at least in part irreversible, unless corresponding real options exist. • The decision maker has discretionary decision rights that can be interpreted as real options. • The universe of all available options can be specified exhaustively. • Options are proprietary options. There are no agency conflicts. • All input parameters necessary for the purpose of option valuation are unambiguous and known. 4.1.2 Objective Function The decision maker’s goal is to maximize shareholder value, and hence, market value of an all–equity financed firm (cf. Copeland et al. 1994, Part I). In the light of the present analysis, this translates into the objective of maximizing portfolio value, i.e., the combined value of the bundle of all underlying assets and all real options embedded in these underlying assets. Either this portfolio comprises the whole scope of the company, or it is perfectly independent from other ventures and can therefore be analyzed separately, contributing directly to the market value of the firm. The portfolio value is calculated as the expected value of the optimal exercise policy of the portfolio of real options, dynamically over all managerial actions permitted and stochastic realizations across time. This optimal exercise policy at each point in time corresponds to what is called “investment program” in the context of traditional capital budgeting. 4.1.3 Managerial Actions Permitted The decision maker’s actions determine the utilization of the underlying assets in question, by investing and disinvesting. For each underlying asset, i.e., investment prospect, the manager chooses the best available operating 1

This implies continuous trading on frictionless markets, no taxes, no restrictions on short sales, full divisibility of assets, and arbitrage–free markets (cf. Hull 2003, p. 242).

4.1 Model Features

47

mode out of the set of permissible modes. Without loss of generality, representative operating modes in the following model are: 1. Mode 1 = not invested. 2. Mode 2 = invested in low capacity. 3. Mode 3 = invested in high capacity. This translates into the following possible switches between modes, if permissible: • • • • •

1 → 1 : Stay uninvested, i.e., operate with zero capacity. 1 → 2, 1 → 3 : invest in low capacity, high capacity respectively. 2 → 3 : Expand capacity from low capacity to high capacity. 3 → 2 : Scale down capacity from high capacity to low capacity. 3 → 1, , 2 → 1 : Reduce capacity from high capacity to zero capacity or from low capacity to zero capacity. Depending on whether subsequent options to reinvest exist, this either represents an option to abandon (if switching back is not permitted), or an option to shut down and reopen (if switching back is permitted).

Switching between operating modes corresponds to a call or put option exercise at each decision point within the limits of permissible switches. Switching involves investment costs (switching costs) which can be positive or negative. Investments and disinvestments are treated similarly in the following, in that a disinvestment is an investment with negative sign. For instance, a European call option allows to switch from mode “not invested” to mode “invested” at maturity, whereas a European put option enables the reverse switch. 4.1.4 Risk Preference The expected value is calculated as a “risk–neutral” expected value. It is risk–neutral in the sense that it is independent from risk attitudes, such that a valuation can proceed as if the decision maker were risk–neutral (cf. Cox and Ross 1976; Duffie 2001). In other words, the decision maker can have any risk preference because it does not need to be modeled explicitly. Technically speaking, the risk–neutral probabilities are an equivalent martingale measure, transforming the actual real–world probabilities (see Harrison and Kreps 1979; Karatzas and Shreve 1998, pp. 33 f.; Paul and Baschnagel 1999, pp. 152 ff.). To avoid confusion, some authors prefer the term “synthetic” probabilities, since the latter differ from the probabilities for the actual occurrence of true states of the world (cf. Neftci 2000, pp. 21-23). The synthetic probabilities would coincide with the real probabilities only if all decision makers truly were risk–neutral; if not so, moving from the synthetic world

48

4 Valuation Model for Portfolios of Real Options

to the real world changes both expected returns on stock prices and risk– adjusted discount rates, in such a way that these changes cancel out each other and do not affect the result (cf. Hull 2003, p. 245). Risk–neutral valuation assumes arbitrage–free, frictionless, complete markets (Black and Scholes 1973; Luenberger 1998). Especially, frictionless markets make no restrictions on short sales. This assumption will be discussed in more depth in the context of capital constraints. 4.1.5 Stochastic Processes The proposed model is formulated as a stochastic program where the underlying assets follow stochastic diffusion processes. At its core, the problem formulation is based on a time– and state–discrete binomial lattice tree approach. The basic idea is that in each period it is assumed that the value of an underlying asset, θ, moves either up with probability p, or down with probability q = 1 − p. The most widely used approach is the CRR model, proposed by Cox et al. (1979), which is followed in the proposed model (see Sec. 4.2.2). 4.1.6 Volatility and Correlation The volatility of the value of the underlying assets, σi , is a key valuation parameter for any option and can be constant, i.e., stationary, or changing between periods. The present model will only consider the values of underlying assets as sources of uncertainty and therefore always assumes stationarity in the volatilities. Considering more than one underlying asset means that it is generally not sufficient to model the stochastic movement of each underlying asset separately; the joint movement of the prices of the underlying assets must also be taken into account (cf., e.g., Stulz 1982). However, by the flexible structure of our model we permit investment switches between alternative processes. In particular, this captures the case that the firm changes its risk exposure through investments. We model the values of the underlying assets with their respective volatilities and correlations as explicit sources of uncertainty. The values of the underlying assets are transformed into a cash flow measure, contingent on the scenario and operating mode, through a company specific cash flow function. On the one hand, this implies that the correlation of the underlying assets and the correlation of the cash flows need to be distinguished because they do not necessarily coincide. On the other hand, both are well–specified functions to our model. The former is determined directly through specifying the coefficient of correlation, ρij ,2 ; the latter is determined through 2

The joint movement of two random variables, X and Y, with finite variance can cov( X,Y )

be expressed by the coefficient of correlation, ρ XY = σ( X )σ(Y ) , with −1 ≤ ρ XY ≤ 1

4.1 Model Features

49

specifying the cash flow function. Therefore, all relevant information can be gained through ρij . Further, the model allows for any desired functional form of ci (θi (t, s), ai0 ), which defines the transformation of the correlation of values of the underlying assets into the correlation of the cash flows. An intuitive cash flow function, in analogy to Kulatilaka (1988), is of the form ci (θ, a0 ) = f a0 + v a0 θ,

with f e R, v e R > 0.

(4.1)

It bears the intuition with f negative, that f represents fixed costs, while v is the higher the lower the variable costs. For this class of affine functions, it is obvious that the correlation is unaffected by the transformation, such that the correlation of the underlyings and the correlation of the cash flows coincide. Therefore, without loss of generality in the following a cash flow function of the form c(θ, a0 ) = v a0 θ is used, and discussed in the numerical analysis. Extensions to other types of cash flow functions are straightforward, e.g., different cash flow functions for different states or paths. This would imply that cash flows can have a correlation different from the correlation of the underlying assets, especially, the sign of the correlation can be reversed. Nevertheless, such cash flow functions are still deterministic, hence all information is contained in ρij and correlation effects are sufficiently captured by discussing ρij alone. 4.1.7 Inter– and Intra–Project Options Interactions The analogy between financial and real options is greatest when modeling a simple, stand alone investment decision as a European call option. Yet when considering multiple real options on the same underlying asset, the valuation has to take into account the specific nature of real options. It has been shown by Trigeorgis (1993a) and Kulatilaka (1995) that real options written on the same underlying asset usually interact in the sense that value additivity does not hold. A valuation of these interacting real options must be carried out by valuing the bundle of real options and underlying asset as a whole, which is similar in structure to valuing a compound option. Intuitively speaking, this is because options on the same real assets are linked through this asset. By exercising any option the underlying asset is and as a special case ρ XY = 0 if X and Y are independent. For any variance– covariance matrix to be consistent, it must be positive semidefinite (cf. Jacod and Protter 2000, pp. 87 f.), which also extends to consistency conditions for coefficients of correlation (cf. Hull 2003, pp. 387 f.). Only in the special case where the underlying assets evolve independently from one another, the joint movement is not of special interest because it is trivial: given stochastic independence, i.e., ρ XY = 0, Prob( X ∧ Y ) = Prob( XA) × Prob(Y ).

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4 Valuation Model for Portfolios of Real Options

affected and with it, all other options tied to it. An extreme case is the option to abandon: by exercising it, all subsequent options are foregone. On the one hand, the put exercise decision has to take the existence of subsequent options into account because these have a strictly positive value (Merton 1973b) which is forfeited. On the other hand, the value of subsequent options is affected by the possibility that the put may be exercised. This interaction requires a valuation algorithm which is different from plain–vanilla financial option pricing (Trigeorgis 1991, 1993a; Brosch 2001). Trigeorgis (1993a) provides numerical analysis of options interactions, proving that values of real options are generally not additive and typically interactions are negative. Further, he derives that the nature of interactions generally depends on type, separation, degree of being in the money, and order of real options. This analysis finally shows that some usual options properties are preserved, e.g., sensitivity to time to maturity or volatility. Evidently, non–additivity is amplified in the case of multiple underlying assets. This holds true for financial options as well as real options. While there is a large body of literature on “rainbow” options (cf., e.g., Johnson 1987; Stulz 1982; Carr 1988; Boyle and Tse 1990), inter–project compoundness for real options has been analyzed little (cf., e.g., Childs and Triantis 1999; Lint and Pennings 2002). The model which is developed here will include both inter– and intra– project compoundness with multiple interacting real options written on multiple underlying assets. This requires valuing the portfolio of real options and underlying assets as a whole, in a compound option valuation logic. 4.1.8 Global Dynamic Budgets On the one hand, direct qualitative interactions are inseparably connected to the investment plan (Betge 1995). On the other hand, interactions or constraints may derive from indirect qualitative interactions (see Sec. 3.2.3). While any constraint is possible, e.g., a bottleneck in skilled engineers, we consider capital constraints as a representative indirect qualitative constraint for two reasons: First, for its big practical relevance that has only recently attracted attention (cf., e.g., Boyle and Guthrie 2003). Second, for being the most challenging constraint in that other constraints can be modeled as a special case. Financial option valuation models usually assume no capital scarcity: if an investment has a positive NPV, it is always possible to borrow money to finance this investment. In corporate finance, it is not always given that capital is unconstrained; this may be due to restricted access to capital markets or due to deliberately imposed rules and procedures. A compelling exam-

4.1 Model Features

51

β(t,s)

B(t-1,s-,a)

V(t,s,a)

B(t,s,a')

I(t,a,a') Fig. 4.1. Dynamic budgets depending on time, scenarios, and actions

ple for limited access to capital markets are startup companies which are typically facing significant financing challenges. Capital constraints in the general case introduce an additional dynamic aspect into real options portfolio management. Money that is not spent in one period will be available in the next period. This is depicted in Fig. 4.1, for one underlying asset. First, B(t − 1, s− , a) is the budget that is remaining from period t − 1 and thus passed on to period t. It depends on the switching decision at t − 1 and s− , a, which coming from period t − 1 is the entering mode for period t. s− is the predecessor of s, as defined by the binomial lattice. β(t, s) is the additional budget allowance that adds to the funds available at t; it is exogenous and depends on time and state. Moving forward in time by one period, the entering mode at time t is the leaving mode from t − 1, such that at t switching takes place from mode a to mode a0 . Therefore, B(t − 1, s− , a) + β(t, s) is the amount of funds available at time t before investment decisions at time t. The switch at time t realizes value V (t, s, a), incurring switching costs I (t, a, a0 ). For the budget level this implies that funds are increased by money freed up through disinvestment decisions and decreased by money spent for investment decisions. I (t, a, a0 ) depends on time and decisions, but is independent from s. These together determine the amount B(t, s, a0 ) that is remaining after the switching decision at t. This establishes the budget balance equation as follows: B(t, s, a0 ) = B(t − 1, s− , a) + β(t, s) − I (t, a, a0 ) where t s, s− a, a0 β(t, s) I

time indicator, t = 1, . . . , T, Scenario indicator, with s− = predecessor of s, Operating mode, a = 1, . . . , A and a0 = 1, . . . , A0 , New funds available at time t in scenario s, Investment or switching costs.

(4.2)

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4 Valuation Model for Portfolios of Real Options

This capital constraint is global in the sense that all investments compete for the same overall limited resources. It is dynamic in that money not invested today is available tomorrow and money freed up by disinvestments can be reinvested elsewhere. It is naturally path–dependent because it is a function of discretionary investment decisions along the path there, captured through state variables. This self–financing aspect yet has to be distinguished from the self–financing strategy in derivatives pricing, used for assembling a (dynamic) replicating portfolio (cf. Merton 1993, pp. 528 ff.), or hedging a given portfolio (cf. McDonald 2003, pp. 407 ff.). Constructing a replicating portfolio is a pricing technique, mimicking a known payoff structure through contracts with known prices. As opposed to this, we consider self–financing strategies in order to derive an optimal allocation of a limited amount of financial resources, i.e., the payoff structure is not known but obtained through our model in the first place. By bounding the budget variable, the nature of the budget constraint is set. If B(t, s, a0 ) > 0, a minimum liquidity level is required; if B(t, s, a0 ) ≥ 0, no borrowing is permitted; if B(t, s, a0 ) is bounded below by a negative real number, borrowing up to a defined level is permitted; if it is unbounded, capital availability is not restricted. 4.1.9 Synergies and Operational Constraints Beyond the correlation among cash flows, an investment into real assets may be related to direct qualitative interactions (see 2.1.2), involving physical or technical properties of the underlying assets. These interactions may come in different forms: partial if there are positive or negative synergies in operating projects simultaneously; binary if projects are mutual exclusive or depend on one another. Further, there may be other types of qualitative or logical interactions such as precedence constraints (Weingartner 1966a, 1977; Nemhauser and Ullmann 1969). The model can incorporate any type of qualitative interactions, including synergies. 4.1.10 Learning Closely related are learning effects. Learning goes beyond purely statistical relationships between underlying assets. Sometimes learning may be a side effect of option exercise, occasionally it may be the main goal of option exercise, e.g., a patent race in R&D. The presented model incorporates learning effects by considering staged investments, contingent cost structures, and path–dependencies.3 3

Papers that focus specially on learning effects include Koussis et al. (2003) and Smith and Thompson (2005).

4.1 Model Features

53

4.1.11 Diversification as a Passive Approach to Risk Management In financial portfolio management, diversification is key. Some authors even go further by claiming: “Portfolio theory is the analysis of the real world phenomenon diversification.” (Constantinides and Malliaris 1995). Since the seminal paper by Markowitz (1952), diversification has become a cornerstone of finance, including the CAPM. The argument of diversification is compelling in the context of financial markets where non–systematic risk can be avoided just by trading financial assets; for this reason, it can be assumed that any rational investor will be diversified. While diversification may be desirable in some specific contexts, overall portfolio risk minimization is not a legitimate objective for flexible investment decisions by companies (cf. Brealey and Myers 1996, p. 166). On arbitrage–free markets, companies must take risks in order to earn a return above the risk–free rate. The ultimate investor who will be diversified can then balance her portfolio as a function of her personal risk preferences and adjust the portfolio continuously. What is more, the perspective on risk, as measured by volatility and correlation, is different in a real options portfolio approach. Here a much more active approach towards risk is needed because options make it possible to capitalize on the upside and truncate the downside, depending on the nature of options. By designing an optimal portfolio of real options, risk is managed actively by an optimal sequence of option exercise decisions. If risk were minimized, this would typically adversely effect option values. Therefore, diversification is not considered as an objective in the management of portfolios of real options. To the contrary, the cardinality of the set of real options and stochastic processes, respectively, provides diversified inputs to the investment problem. For example, optimal management of a global manufacturing network through switching between sites requires some diversification between sites, e.g., sites in different countries with different exchange rates. With this notion of diversification as an input, active management of a portfolio of real options is achieved by optimal investment strategies in the given structure. When the structure of possible portfolio configurations is to be determined, diversification is relevant, but not at all with the goal of reducing risk, but instead in view of providing the optimal risk structure for future real options portfolio value (see Sec. 2.2.2). 4.1.12 Path–Dependencies One key feature of the presented model is that it considers path–dependencies. Path–dependencies arise for three reasons. First, a decision made in one period determines the basis for decisions made in the following periods. For a stand-alone analysis, this does not have any consequence because once

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an option is exercised, the decision problem ends. In our setting, switches back and forth are possible, which means that it models a compound option problem: With every switch made, new future switches are possible, contingent on the path of earlier switches. Given this logical connection between adjacent decisions, for any current decision the sequence of earlier decisions is relevant. Second, the dynamic global budget introduces an additional source of path–dependencies because on each path, the funds available are a function of all decisions made up to this point, which depend on the realizations of the underlying assets. In other words, depending on the path taken and the corresponding optimal investment decisions on that path, the availability of funds for further investment decisions is determined. Third, path–dependencies result from the interaction between underlying assets, which is closely linked to their joint stochastic movements. Especially, this source of interaction is closely connected to the budget constraint. If assets compete for shared funds, investment into each asset depends on investments on the other assets and the resulting funds available. Due to these sources of path–dependencies, the model is path–dependent no matter whether the underlying assets themselves are path-dependent or not. In the proposed model, path–dependent underlying assets can be integrated in a straightforward way, but would not change the portfolio problem. The system state space already models a path–dependent problem with a suitable tracking of the system state space, such that path–dependent stochastic processes of the underlying assets are straightforward to include in that structure if desired. It is noted that the distinction of different sources of path–dependencies is only helpful for the conceptual understanding of the path–dependent nature of the problem. From a modeling standpoint, it is not possible to separate the effects of different sources of path–dependencies because all sources act simultaneously and jointly. 4.1.13 Modeling Approach and Solution The general real options portfolio problem with budget constraints and path–dependency is formulated as a stochastic mixed–integer program using risk–neutral probabilities. The program recurs to mixed–integer decision variables to take account of option exercise decisions. Especially, it resides to binary variables which turn one if and only if an option is exercised. This makes the problem non-linear because the binary variables “activate” the operating modes which determine the value generated in these modes, in the current and all subsequent periods. This naturally introduces an asymmetry in switching choices and hence nonlinearity. Further, it is stochastic in

4.1 Model Features

55

modeling the stochastic processes of the underlying assets and their interdependencies. These interdependencies dynamically capture the interaction effects of decisions over time. The program formalizes a simultaneous forward– and backward–looking procedure. On the one hand, a backward–looking element consists of the calculation of expected values in a rolling-back fashion which is well know from standard option pricing theory, by a backward recursion. On the other hand, a forward recursion is required to cope with path–dependencies. This can best be expressed by the budget balance equation: money left over from one period can be invested in the following period, for the states following that specific path. What is special here is that both the backward– and the forward–looking element have to be integrated simultaneously. This can be achieved by formulating the problem as a simultaneous program. This is made possible through an appropriate modeling of the binary variables which combine both the backward– and the forward–looking element via an appropriate mapping of the system state space. Given the complexity of the problem, it is evident that a solution cannot be obtained closed–form. At the same time, commercially available solvers have been developed for decades and today provide a heavily researched and powerful arsenal for solving problems with high dimensionality. Numerically solving the problem proceeds in a two-staged process: 1. The general formulation is given in its “implicit” form where all equations are indexed over time, scenarios, and actions. This implicit form needs to be expanded to its “explicit” form where for each combination of indices just one equation is generated. This is implemented using a modeling language, such as the “General Algebraic Modeling System” (GAMS© ), which we will use here. Both versions, implicit and explicit, are equivalent. Intuitively speaking, while the implicit version gives an aggregated view of the problem, the explicit version expands the problem and hence builds up the landscape in which the optimization takes places. 2. The explicit problem is then passed on to a suitable solver. Given nonlinearity, it is important to ensure that an optimum is not only local but global. This can be ascertained by using a global solver, such as BARON© which offers the advantage of being able to prove global optimality of the solution (Tawarmalani and Sahinidis 2002a, b). For our problem, we have reformulated the numerical implementation as an equivalent linear problem (see Sec. 4.5.2.1), such that it is possible to reside to linear solvers for mixed-integer programs, such as CPLEX© . This offers the advantage of much faster solutions and avoids the problem of local optima (for a discussion, see Sec. 4.5.2.1).

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4.1.14 Scalability Theoretically, there is no bound on model size, since the model can readily be formulated for any number of underlying assets, real options, and constraints. Numerically, the solution depends on computing power. Given the exponential growth of the problem size, practically speaking, the size of solvable decision problems cannot easily be generalized. After presenting the general model formulation, routes for making the solution more efficient will be indicated.

4.2 Binomial Option Pricing So far, we have presented the features of the model in an overview, aiming to introduce the nature of the presented model. With this background, the stochastic modeling and the valuation approach of the model will be presented in what follows. A core element of the model is binomial option pricing, which will be developed step by step. The basic binomial option pricing is presented, which is then expanded to two underlying assets. 4.2.1 Stochastic Processes and System State Space The stochastic processes of the underlying assets are assumed as discrete binomial processes. First, the CRR model is presented due to its widespread use. It offers many advantages, especially that the discretization of the system state space is very intuitive. Also, the distribution of the stock price converges to the log–normal distribution in the limit, when ∆t → 0 (Kwok 1998, pp. 199 ff.). This is standard for financial options pricing where log– normal processes and specifically geometric Brownian motions are widely used to model stock prices (Duffie 2001, p. 88). Moreover, in a binomial lattice tree the central node at time 2∆t is the same as in the initial node because d = u1 and hence u × d = 1. This is particularly convenient in the calculation of “options Greeks” (Hull 2003, p. 407). However, this comes at the disadvantage that the probabilities may turn negative when time steps are not sufficiently small, which also is the case in the extension to many underlying assets (Boyle 1988; Boyle et al. 1989). In the presented model, it is neither important to converge to a log– normal distribution of the underlying assets, nor to calculate options Greeks. Therefore, it is not necessary to consider small time steps and it is helpful to derive risk–neutral probabilities in a way that they cannot turn negative. The most prominent alternative binomial approach that ensures non– negativity is provided by Jarrow and Rudd (1983).

4.2 Binomial Option Pricing

57

The binomial model is most known in its standard version which can be represented by binomial lattice trees. One dimension represents time (usually the horizontal dimension), the other dimension represents the up and down movements of the underlying asset. The binomial model can readily be extended to multiple underlying assets, with one additional dimension for each additional asset. For example, a binomial lattice tree for two underlying assets has three dimensions: one time dimension and two asset dimensions. This three–dimensional character can be represented graphically by a pyramid. To simplify exposition and tracking, we adopt a few conventions. All variables are indexed first by time t (t = 1, . . . , T ), then by scenario s (st = 1, . . . , S(t)), and then – if necessary – by actions. The combinations of all relevant state indicators are defined as the “system state space”. Scenarios are counted from high to low scenarios. For example, the option value in the one underlying case in period t=2 for scenario “up” is V (2, 1) and for scenario “down” V (2, 2) (see Fig. 4.2). By “scenario” we mean chance realizations through the lattice, at each point in time. This distinguishes a scenario from the more common term “state” which in our convention contains all relevant information at a given point in time. As opposed to this, depending on their focus of attention some authors find it more convenient to call a path through the tree a scenario (e.g., cf. Høyland and Wallace 2001). 4.2.2 Binomial Option Pricing Model with One Underlying Asset The reference model for our model development is the Cox–Ross–Rubinstein (CRR) version of the binomial model (cf. for the derivation of the CRR model Cox et al. 1979). It is introduced here in the notation needed for the following development of the model, which differs from the usual representation. In that sense, the model is presented here mainly in order to introduce the notation while the content of the CRR model is unchanged. The binomial model assumes that the value of the underlying asset, θ, follows a stationary time–discrete multiplicative binomial process, with up factor u and down factor d. Up und down are specified as a function of the volatility of the underlying asset, σ, such that in the limit the process converges to a log–normal distribution of the return of the underlying asset, i.e.: √ 1 (4.3) u = eσ ∆t , and d = . u These up and down factors determine the stochastic process of the value of the underlying asset, θ. Due to the recombining nature of the tree, the system state space grows in t (t = 1, . . . , T ) and s (st = 1, . . . , t) as follows:

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Fig. 4.2. Recombining binomial lattice tree for one underlying asset

θ (t, s) = θ (1, 1) × ut−s ds−1 .

(4.4)

The development of θ (t, s) can be represented by a binomial lattice tree where scenarios are indexed from the up scenario to the down scenario, as given in Fig. 4.2). In this framework, it is possible to assign risk–neutral probabilities p and 1 − p such that no arbitrage opportunities exist. In each period, the up scenario occurs with probability p and the down scenario with probability 1 − p where r f is the continuously compounded risk–free rate of return: p=

er f ∆t − d . u−d

(4.5)

This provides the basis for valuing an option on the underlying asset, which can be exemplified for a European option as follows. First, the boundary condition on option exercise at t = T is imposed. Supposing for example a European call option with exercise price X, we have the terminal value condition: (4.6) V ( T, s) = max[θ ( T, s) − X, 0], s = 1, . . . , T. Valuation in the binomial model proceeds in a backward recursion, as a discounted probability–weighted expected value: V (t − 1, s) =

V (t, s) × p + V (t, s + 1) × (1 − p) er f ∆t t = 2, . . . , T ; s = 1, . . . , t.

(4.7)

The value of the option corresponds to V (1, 1). 4.2.3 Binomial Option Pricing Model with Two Underlying Assets The presented model so far was only able to cope with one underlying asset. In what follows, the case of two underlying assets is considered.

4.2 Binomial Option Pricing

59

Fig. 4.3. Recombining binomial lattice tree for two underlying assets, with ρ12 6= 0

First assume a coefficient of correlation of zero, i.e., ρ12 = 0. Looking at two underlying assets and their respective values, θ1 (t, s) and θ2 (t, s), a tree for each underlying asset can be built up just as before, with σ1 , u1 , d1 as well as σ2 , u2 , d2 . With recombiningness preserved, the system state space for each underlying asset grows in t = 1, . . . , T and s = 1, . . . , t, so that overall it grows in 2t: θ1 (t, s) = θ1 (1, 1) × u1t−s d1s−1 ,

(4.8)

θ2 (1, 1) × u2t−s d2s−1 .

(4.9)

θ2 (t, s) =

Second, assume ρ12 6= 0. In this case, the two separate trees must be considered simultaneously. Here it makes a difference whether asset 1 moves up and asset 2 moves down, as opposed to asset 1 moves down and asset 2 moves up. This generates a binomial lattice tree which can be represented graphically by a pyramid (see Fig. 4.3). For valuation purposes, the backward recursion can proceed stepwise through the pyramid, in a way identical to the approach for two dimensions. The only difference is that each node has four successors, with respective risk–neutral probabilities. Obviously the system state space grows faster than in the one asset case. It is given by t = 1, . . . , T and s = 1, . . . , t2 . The recombining nature of the tree is preserved in the sense that, e.g., for asset 1 it does still not matter whether it first moves up and then down, or the other way round, so that ud = du. However, scenarios grow with order t2 instead of order 2t.

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The effect of joint movement of the underlying assets can be taken into account by corresponding risk–neutral probabilities. Following Boyle et al. (1989), these probabilities can be specified as follows (see Fig. 4.4): " !# √ r f − 12 σ22 r f − 12 σ12 1 + 1 + ρ + ∆t , p1 = 4 σ1 σ2 for asset 1=up, asset 2=up, " √ 1 p2 = 1 − ρ + ∆t 4

(4.10)

r f − 12 σ22 r f − 12 σ12 − σ1 σ2

!# ,

for asset 1=up, asset 2=down,

(4.11)

" !# √ r f − 12 σ12 r f − 12 σ22 1 p3 = + 1 − ρ + ∆t − , 4 σ1 σ2 for asset 1=down, asset 2=up,

(4.12)

" !# √ r f − 12 σ12 r f − 12 σ22 1 p4 = − 1 + ρ + ∆t − , 4 σ1 σ2 for asset 1=down, asset 2=down.

(4.13)

This derivation of the risk–neutral probabilities has been generalized to n underlying assets by Boyle et al. (1989) and can therefore be used to capture the joint movements of multiple underlying assets in a risk–neutral valuation framework. As mentioned before, a numerical disadvantage of this approach, just as in the CRR two–dimensional case, √ is that the probabilities may turn negative when ∆t is large, i.e., σ ≤| r ∆t | (cf. Hull 2003, p. 407). This can only be avoided by increasing the number of time steps at the expense of increased computing time. With the help of this lattice tree, it is possible to value an option in the above introduced backward recursion fashion, by stochastic dynamic programming. In order to stay within the above introduced notation (see Fig. 4.2), we preserve the intuition of two combined binomial trees by inp3

d1;u2

u1;u2

p1

p4

d1;d2

u1;d2

p2

Fig. 4.4. Three–dimensional risk–neutral probabilities

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61

dexing the scenarios separately, i.e., s1 for asset 1 and s2 for and asset 2. For a European option on the maximum of 2 underlying assets, this gives: V ( T, (s1 , s2 )) = max[θ1 ( T, s1 ) − X, θ2 ( T, s2 ) − X, 0] s1 = 1, . . . , T ; s2 = 1, . . . , T.

(4.14)

Valuation in the binomial model proceeds in a backward recursion, as a discounted probability–weighted expected value: n V (t − 1, (s1 , s2 )) = e−r f ∆t × V (t, (s1 , s2 )) × p1 + V (t, (s1 , s2 + 1)) × p2 o + V (t, (s1 + 1, s2 )) × p3 + V (t, (s1 + 1, s2 + 1)) × p4 t = 2, . . . , T ; s1 , s2 = 1, . . . , t.

(4.15)

The value of the European option on the maximum of two assets corresponds to V (1, (1, 1)). While only the model for 2 underlying assets can conveniently be represented graphically, the approach can be generalized in a straightforward manner to more than two underlying assets, e.g., as specified by Boyle et al. (1989), or in a log–transformed version that offers computational advantages, by Gamba and Trigeorgis (2001).

4.3 Real Options Portfolio Model Up to this point, basic concepts of option pricing within the CRR binomial model have been discussed. The discussion has not primarily focused on real options as such, but has been applicable to both financial and real option pricing. These basic concepts set the stage for the following development of a general model for valuing portfolios of real options, subject to constraints. 4.3.1 Switching Between Modes The core of the proposed model is a switching algorithm, based on a general framework of switching among nodes described in Kulatilaka (1988) and Kulatilaka and Trigeorgis (1994). At any time, the firm can either continue operating in the current mode for one more period, receiving a short–term payoff (i.e., current cash flow) plus the long–term option value from optimal future switching, or switch immediately to a new mode by incurring specified switching costs. For each tuple (t, s) and each underlying asset i, an entering mode ai (t, s) = 1, . . . , Ai (t, s) ∀(t, s) and a leaving mode ai0 (t, s) = 1, . . . , Ai0 (t, s) ∀(t, s) are distinguished. Switching takes place from mode ai (t, s) to mode

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ai0 (t, s) where staying in the same mode is represented by ai = ai0 , if permitted. Without loss of generality, we assume that ai (t, s) = ai ∀(t, s) and ai0 (t, s) = ai0 ∀(t, s), i.e., the set of permitted operating modes is not state– dependent. Considering one underlying asset, switches from entering mode a may involve: • a0 = 1: waiting to invest, • a0 = 2: operate with low capacity, • a0 = 3: operate with high capacity. Associated with each switch is a cash flow realization, c(θ (t, s), a0 ). In each period, depending on the leaving operating mode a0 , a cash flow is realized. This cash flow can be any function of the value of the underlying asset, θ (t, s). Without loss of generality, the cash flow function can be represented by a positive scalar, drawing on an analogy to levels of capacity: • • • •

c(θ (t, s), a0 ) = 0: not invested, 0 < c(θ (t, s), a0 ) < 1: invested with low capacity, c(θ (t, s), a0 ) = 1: invested with base–scale capacity, c(θ (t, s), a0 ) > 1: invested with expanded capacity.

Switching between any mode a and a0 involves a time dependent switching costs, I (t, a, a0 ), ∀ a, a0 . The switching costs represent investment outlays, or in options terms, exercise prices. Without loss of generality, we can assume that staying in the same operating mode is costless (the respective costs are already taken into account in the cash flow). Examples of basic switching costs are as follows: • I (t, a, a0 ) = ∞: no switching possible,4 • I (t, a, a0 ) = 0 : staying in the same mode (or, exceptionally, costless switching), • I (t, a, a0 ) > 0 : call option, i.e., investment (utilization of resources), • I (t, a, a0 ) < 0 : put option, i.e., disinvestment (freeing up of resources). The switching algorithm is very flexible. It can handle any structure of time–dependent symmetric or asymmetric switching costs, any cash flow payoff function and any options combinations in terms of type, sequence and option maturity. The switching algorithm is designed to model a bundle of real options that can be interpreted as a joint complex, compound option (Trigeorgis 1996, pp. 185 ff.). Since each switching possibility represents a possible real option exercise and the switching costs matrix introduces an asymmetry, the valuation cannot be achieved by valuing each real option separately and adding up the separate values. Deviations from value 4

In an implementation, this is incorporated by using a sufficiently large number for I (t, a, a0 ).

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63

additivity make the valuation and interpretation of incremental option contributions challenging (Trigeorgis 1993a). 4.3.2 Switching Formulation for One Underlying Asset Following this switching perspective, we can formalize the valuation algorithm as a backward recursion in a stochastic dynamic programming fashion. Intuitively speaking, this program integrates a double iteration: First, it moves backward in time, applying the Bellman principle of optimality. Second, at each point in time, it iterates over all entering modes, each time choosing the optimal leaving mode. The value of the underlying asset, θ (t, s), follows a binomial process with probabilities p for an up movement and probability (1 − p) for a down movement, as introduced above for the two–dimensional CRR framework. Each decision is based on both the current cash flow per period which is realized after the switch, as a function of θ (t, s) and a0 , as well as the expected value of future net cash flows, starting from the derived leaving mode. This can be incorporated into the following stochastic dynamic program based on backward recursion, which is in analogy with Trigeorgis (1996, p. 185): V ( T, s, a) = max c(θ ( T, s), a0 ) − I ( T, a, a0 ) a0

s = 1, . . . , T ; a = 1, . . . , A V (t − 1, s, a) = max c(θ (t − 1, s), a0 ) − I (t − 1, a, a0 )

(4.16)

a0

+

where t s a a0 V c(·, a0 ) θ I p rf

V (t, s, a0 ) × p + V (t, s + 1, a0 ) × (1 − p) er f ∆t t = 2, . . . , T ; s = 1, . . . , t ; a = 1, . . . , A.

(4.17)

Time indicator, t = 1, . . . , T, Scenario indicator, s = 1, . . . , t, Entering operating mode, a = 1, . . . , A, Leaving operating mode, a0 = 1, . . . , A0 , Value of portfolio, Cash flow function for operation in mode a0 , a0 = 1, . . . , A0 , Stochastic value of underlying asset, Investment or switching costs, Risk–neutral probability for up–jump, Risk–free rate of return.

At each node of the system state space, for each entering mode, the stochastic dynamic program determines the optimal operating mode. This decision

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takes into account the switching costs contingent on the entering mode, the payoff in that specific period contingent on switching costs, as well as the expected future value given the leaving mode. This expected value represents the value of all future switching decisions with the leaving mode as the new starting point. Overall, the logic sequence is as follows: i) observe the realization of the underlying stochastic variable; ii) switch optimally; and iii) realize the cash flow in this period for the leaving mode. The leaving mode will then be the entering mode for the following period. This stochastic dynamic programming formulation takes into account any structure of switching costs, including the non–symmetric case. Therefore in some nodes, it may be optimal to stay in the same mode in order to prevent a future costly switching back to the earlier mode. This introduces a hysteresis effect: even though immediate switching may appear profitable from a short–term perspective, for dynamic long–term considerations it can be optimal to remain in the original mode (Dixit and Pindyck 1994; Trigeorgis 1996). 4.3.3 Switching Formulation for Two Underlying Assets The above introduced switching model can be extended towards two underlying assets in a straightforward manner. In this case, there are two underlying assets with their respective values, θ1 (t, s1 ) and θ2 (t, s2 ); four risk–neutral probabilities p1 , p2 , p3 , p4 ; entering modes for each underlying asset, a1 and a2 ; leaving modes for each underlying asset, a10 and a20 ; and a switching costs matrix for each underlying asset, I1 (t, a1 , a10 ) and I2 (t, a2 , a20 ). For each underlying asset, a binomial lattice tree can be constructed as in Fig. 4.2. Both underlying assets have to be considered simultaneously, as has been shown for the example of the option on the maximum of two assets. The dynamic program for the switching model therefore becomes:

4.3 Real Options Portfolio Model

65

V ( T, (s1 , s2 ), ( a1 , a2 )) = max c1 (θ1 ( T, s1 ), a10 ) − I1 ( T, a1 , a10 ) a10

+ max c2 (θ2 ( T, s2 ), a20 ) − I2 ( T, a2 , a20 ) a20

s1 , s2 = 1, . . . , T ; (4.18) a1 = 1, . . . , A1 ; a2 = 1, . . . , A2 0 0 V (t − 1, (s1 , s2 ), ( a1 , a2 )) = max c1 (θ1 (t − 1, s1 ), a1 ) − I1 (t − 1, a1 , a1 ) a10

+ max c2 (θ2 (t − 1, s2 ), a20 ) − I2 (t − 1, a2 , a20 ) a20

+ + + +

−r f ∆t

× (V (t, (s1 , s2 ), ( a10 , a20 )) × p1 V (t, (s1 , s2 + 1), ( a10 , a20 )) × p2 V (t, (s1 + 1, s2 ), ( a10 , a20 )) × p3 V (t, (s1 + 1, s2 + 1), ( a10 , a20 )) × p4 ) ] , t = 2, . . . , T ; s1 , s2 = 1, . . . , t ; a1 = 1, . . . , A1 ; a2 = 1, . . . , A2 . e

(4.19)

Time indicator, t = 1, . . . , T, Scenario indicator for underlying asset i = 1, 2, si = 1, . . . , t, Entering operating mode for underlying asset i = 1, 2, ai ai = 1, . . . , Ai , Leaving operating mode for underlying asset i = 1, 2, ai0 ai0 = 1, . . . , Ai0 , V Value of portfolio, ci (·, ai0 ) Cash flow function for operating asset i = 1, 2 in mode ai0 , ai0 = 1, . . . , Ai0 , Stochastic value of underlying asset i = 1, 2, θi Investment or switching costs for underlying asset i = 1, 2, Ii p Risk–neutral probability, Risk–free rate of return. rf

where t si

4.3.4 Stochastic Mixed–Integer Program Formulation The above derived switching model is a stochastic dynamic programming model that derives option values iteratively. One key characteristic is that it does not consider capital scarcity and that it can be based on a recombining tree structure which allows to value portfolios of real options in a straight backward recursion. With a dynamic budget, this logic cannot be applied. The tree of an underlying asset is no longer recombining, because the available budget depends on the realized path and the switching decisions along this path. For

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Fig. 4.5. Non–recombining binomial lattice tree for one underlying asset

a two–dimensional, non–recombining tree, the scenarios grow to s = 2(t−1) instead of s = t. Likewise, for a three–dimensional, non–recombining tree, the scenarios grow exponentially to s = 22(t−1) , instead of s = t2 . This additional source of dynamics is hard to integrate into a recursion in a dynamic programming fashion. Alternatively, this can be modeled as a stochastic mixed–integer program for a non–recombining lattice tree. This way, the structure of the decision problem is translated into a set of constraints that must be satisfied simultaneously in the optimal solution. In the following, we will present such a model formulation in successive steps. First, we will present the necessary building blocks with the help of a problem with one underlying asset. Second, we develop the changes which are necessary to take into account an additional underlying asset, i.e., two underlying assets in total. Finally, this is generalized to a model formulation for n underlying assets. 4.3.4.1 One Underlying Asset First, the model formulation for one underlying asset is presented. Taking into account budget dynamics, the problem becomes path–dependent, no matter whether the underlying process itself is recombining. Therefore, the scenarios of the stochastic process have to account for non– recombiningness. This is represented graphically in Fig. 4.5, following the indexing logic of scenarios, from up to down scenarios as introduced before. In addition to time and scenario, the system state space integrates the entering operating mode, a, and leaving operating mode, a0 . Decisions are modeled by a binary switching variable, R, which is state dependent, hence R(t, s, a, a0 ). The objective function maximizes the overall portfolio value, V (1, 1, 1), over all decision switching variables, R(1,1,1,a’). The program derives the

4.3 Real Options Portfolio Model

67

maximum expected portfolio value at t = 1, and the corresponding optimal dynamic strategy (Eq. 4.20). The constraints are described in blocks. First, the dynamic movements of the underlying asset are modeled by its starting value, α, (Eq. 4.22) and its development across time (Eq. 4.22), spanning the underlying binomial lattice tree.5 The modulo function, [mod 2], serves to model the non– recombining structure of the binomial lattice and ensures the correct linkage between parent node to child node. This can be achieved by drawing from our design that we count states from up to down states and that the non– recombining lattice tree grows with order t2 . Based on this, the property of the modulo function can be implied, because the remainder on integer division of states by the basis 2 is such that it yields either a 1 for the up state successor node or a 0 for the down state successor node. This feature can be exploited for adding over the successor nodes in Eq. 4.22, because exactly one factor is 1 while the other is 0. This formalizes the linkage visualized in Fig. 4.5. Next, the core of the set of constraints is provided by the switching value function, V (t, s, a), for intermediate periods in Eq. 4.23 and for the terminal period in Eq. 4.24. Switching from mode a to mode a0 involves switching costs I (t, a, a0 ). Once a switch is made, cash flow c(θ (t, s), a0 ) is realized, given by the cash flow function c(·, a0 ) of the underlying asset. For all periods t except for the terminal period t = T, the leaving mode in a period t also serves as the entering mode for the following period t + 1, allowing to determine an expected value for the leaving mode in all subsequent periods. This expected value implicitly takes into account optimal switching in all subsequent periods. In this sense, switching incorporates a backward recursive element. Optimal switching is determined by the binary variable R(t, s, a, a0 ). As ensured by Eq. 4.25, the system is always exactly in one (and only one) operating mode, i.e., it is not possible to make several successive switches in a single period.6 Capital rationing is accommodated through the global dynamic budget constraint, B(t, s, a0 ), given by Eq. 4.26 for the first period (t = 1) and Eq. 4.27 for all subsequent periods. B(t, s, a0 ) represents the budget remaining after a switching decision. The first period is special in that it specifies 5

6

In financial option pricing applications, the starting value of the underlying asset typically is represented by the variable S0 , or S(0). Since our underlying variables are not “stock”, we prefer to use α instead. The fact that there is always exactly one switch is only technical. In a specific decision problem it may be possible to realize successive switches immediately. This can be taken into account by introducing a separate mode for the decision to realize immediately successive switches involving several modes, making this decision alternative technically a single, additional mode.

68

4 Valuation Model for Portfolios of Real Options

the initial funds available, B1 , with B1 e R ≥ 0. In subsequent periods, this is replaced by the funds remaining from the prior period. Moreover, there is an allocation of a new budget per period, β(t, s), with β(t, s) e R ≥ 0. The above budget constraint is dynamic in that it incorporates both investment and disinvestment decisions, the latter with I (t, a, a0 ) < 0. Money that is freed up by disinvesting in one period is available for investments in subsequent periods. It is noted that cash flows from operating the asset cannot be reinvested here. If no borrowing is permitted, we have B(t, s, a0 )e R ≥ 0; otherwise, borrowing can be partially relaxed by allowing a certain level of financing or can be fully relaxed by setting B(t, s, a0 )e R. One element of the formulation, which shows in the value function as well as in the budget constraint, merits special attention: the interconnection of decisions across periods, i.e., the compoundness of decisions. Each decision sets the stage for the subsequent decisions and determines which future decisions are possible. The formulation is such that the switch made in one period is the basis for the switch in the subsequent period. This ensures consistency over time and enforces non–anticipativity.7 This applies both forward and backward looking in time. Since the dynamic budget constraint is forward looking, consistency is ensured for B(t, s, a0 ) when moving from t − 1 to t since the leaving mode at t − 1 is set to be the entering mode at t. Conversely, the value function V (t, s, a) incorporates the backward– looking feature of the program, moving from t + 1 to t. Given a switching decision at t, the entering mode at t + 1 is fixed to be the leaving mode for t. Further, some comments on equations that are technically required are in place. In Eq. 4.23, R(t, s, a, a0 ) expresses the optimal switch in a way that exactly one binary variable turns 1. Also, R(t, s, a, a0 ) is linked to the following period by multiplying it by V (t + 1, ·). This linkage also ensures consistency, i.e., that the leaving mode from t is the entering mode at t + 1. However, this can only work when the term in brackets is different from zero. Otherwise, the optimization can set any (but exactly) one of the binary variables to 1 because it is multiplied with zero and thus does not impose any restriction on the binary variables because it has no impact on the optimum. This special case is obtained in states where it is optimal to remain uninvested. Consequently, this does not affect the optimal portfolio value because it has zero value impact, but it can falsify the vector of binary variables and consequently give meaningless investment strategies. To avoid this, Eq. 4.28, Eq. 4.29 and Eq. 4.30 are used to ensure that the switches are always consistent over time. The core of the consistency constraints is Eq. 4.29. It ensures 7

Non–anticipativity refers to the fact that each decision can only be made with the information available at this time. Through the implicit tree structure underlying the program, this non–anticipativity is satisfied (Birge and Louveaux 1997).

4.3 Real Options Portfolio Model

69

that the leaving mode for any binary variable that turns 1 is indeed the entering mode for both following scenarios. This formulation exploits that there are always two successors and that for each successor there is exactly one binary that turns 1, making it possible to add over binaries and divide by 2. Finally, Eq. 4.31 simply ensures the binary nature of the switching decision variable, R(t, s, a, a0 ). This modeling logic is combined in the model formulation for one underlying asset as follows: max

R(1,1,1,a0 )

V (1, 1, 1)

(4.20)

subject to θ (1, 1) = α

(4.21)

√ s+1 ) × eσ ∆t [s mod 2] 2 √ s + θ (t − 1, ) × e−σ ∆t [(s + 1) mod 2] 2 t = 2, . . . , T ; s = 1, . . . , 2(t−1)

θ (t, s) = θ (t − 1,

(4.22)

0

V (t, s, a) =

A X

R(t, s, a, a0 ) × c(θ (t, s), a0 ) − I (t, a, a0 )

a 0 =1

+

V (t + 1, 2s − 1, a0 ) × p + V (t + 1, 2s, a0 ) × (1 − p)

er f ∆t t = 1, . . . , T − 1 ; s = 1, . . . , 2(t−1) ; ∀ a

(4.23)

0

V ( T, s, a) =

A X

R( T, s, a, a0 ) × c(θ ( T, s), a0 ) − I ( T, a, a0 )

a 0 =1

s = 1, . . . , 2(T −1) ; ∀ a

(4.24)

0

A X A X a =1

R(t, s, a, a0 ) = 1

a 0 =1

∀t , s = 1, . . . , 2(t−1)

(4.25)

B(t, s, a0 ) = R(t, s, a, a0 ) × B1 + β(t, s) − I (t, a, a0 ) t = 1 ; s = 1 ; a = 1 ; ∀ a0

(4.26)

70

4 Valuation Model for Portfolios of Real Options B(t, s, a0 ) =

A X

n R(t, s, a, a0 ) × β(t, s) − I (t, a, a0 )

a =1

s+1 , a) × [s mod 2] 2 o s + B(t − 1, , a) × [(s + 1) mod 2] 2 t = 2, . . . , T ; s = 1, . . . , 2(t−1) ; ∀ a0

+ B(t − 1,

(4.27)

0

L(t, s, a) =

A X

R(t, s, a, a0 )

a 0 =1

t = 2, . . . , T ; s = 1, . . . , 2(t−1) ; ∀ a R(t, s, a, a0 ) =

×

(4.28)

0 1 2 × R ( t, s, a, a ) A0 X 1 X

L(t + 1, 2s − l, a0 )

a 0 =1 l =0

t = 2, . . . , T ; s = 1, . . . , 2(t−1) ; ∀ a ; ∀ a0

(4.29)

0

A X

R(t, s, a, a0 ) = 1

a 0 =1

t=1; s=1; a=1

(4.30)

R(t, s, a, a0 ) ∈ {0; 1}

∀t ; ∀s ; ∀ a ; ∀ a0 .

(4.31)

4.3 Real Options Portfolio Model

71

where t s a a0 R V θ α c(·, a0 )

Time indicator, Scenario indicator, Entering operating mode, a = 1, . . . , A, Leaving operating mode, a0 = 1, . . . , A0 Binary switching variable, Value of portfolio, Stochastic value of underlying asset, Starting value of underlying asset, Cash flow function for operating in mode a0 , a0 = 1, . . . , A0 , I Investment or switching costs, p Risk–neutral probability, Risk–free rate of return, rf B Budget level, β(t, s) New funds available at time t in scenario s, Initial budget available at time t = 1, B1 e R ≥ 0, B1 L, l Auxiliary variable and index.

4.3.4.2 Two Underlying Assets The formulation as a nonlinear stochastic program for multiple underlying assets can be achieved by combining the elements introduced for a multinomial valuation procedure, with a simultaneous switching approach similar to the one presented for one underlying asset. More explicitly, the approach consists of introducing a stochastic variable θi for the value of each underlying asset i. Spanning up the multinomial lattice tree can still be achieved by using the modulo function, with [mod 2n ], alongside a starting value for each underlying asset, αi . Entering and leaving modes for each underlying asset are distinguished, i.e., ai and ai0 , as well as cash flow functions for each underlying asset, ci (·, ai0 ). The risk–neutral probabilities are adapted to the number of underlying assets, taking into account their respective joint movements. Boyle et al. (1989) have provided a formulation of risk–neutral probabilities for n underlying assets. Since the global dynamic budget is shared by all investment decisions for all underlying assets, the budget constraint incorporates investment decisions from a global portfolio perspective where it may be desirable to free up money by disinvesting some underlying assets in order to invest in other more profitable underlying assets. In the case of i = 2 underlying assets, the binomial processes can still be represented by a pyramid, as depicted in Fig. 4.3. Yet, with the dynamic budget constraint, the pyramid becomes non-recombining since it does matter whether a node is reached from the “up” or the “down” path. This path dependent peculiarity can be translated into an indexing logic as depicted in Fig. 4.6, following the indexing logic of joint scenarios from the up to

72

4 Valuation Model for Portfolios of Real Options

the down states. Using this indexing logic, the model formulation for i = 2 underlying assets is as follows: s=3

θ(2,3) d1;u2

θ(2,1)

s=1

θ(2,2)

s=2

u1;u2 θ(1,1)

s=4

θ(2,4)

u1;d2

d1;d2

Fig. 4.6. Indexing in a three–dimensional lattice tree

In a straightforward manner, the formulation for two underlying assets is then given by: max

R(1,1,(1,1),( a10 ,a20 ))

V (1, 1, (1, 1))

(4.32)

subject to θ1 (1, 1) = α1

(4.33)

θ2 (1, 1) = α2

(4.34)

θ1 (t − 1, θ (t − 1, 1 θ1 (t, s) = θ 1 ( t − 1, θ1 (t − 1,

√ s +3 σ1 ∆t if 4 )×e √ s +2 σ1 ∆t ) × e if 4 √ s +1 − σ ∆t 1 if 4 )×e √ s −σ1 ∆t ) × e if 4 2( t −1)

[s mod 4] = 1, [s mod 4] = 2, [s mod 4] = 3, [s mod 4] = 0

t = 2, . . . , T ; s = 1, . . . , 2 θ2 (t − 1, θ (t − 1, 2 θ2 (t, s) = θ 2 ( t − 1, θ2 (t − 1,

√ s +3 σ2 ∆t if 4 )×e √ s +2 −σ2 ∆t if ) × e 4 √ s +1 σ2 ∆t if 4 )×e √ s −σ2 ∆t ) × e if 4 2( t −1)

t = 2, . . . , T ; s = 1, . . . , 2

(4.35)

[s mod 4] = 1, [s mod 4] = 2, [s mod 4] = 3, [s mod 4] = 0 (4.36)

4.3 Real Options Portfolio Model 0

V (t, s, ( a1 , a2 )) =

73

0

A1 A2 X X

R(t, s, ( a1 , a2 ), ( a10 , a20 )) × (c1 (θ1 (t, s), a10 )

a10 =1 a20 =1

− I1 (t, a1 , a10 )) + (c2 (θ2 (t, s), a20 ) − I2 (t, a2 , a20 )) h + e−r f ∆t × V (t + 1, 4s − 3, ( a10 , a20 )) × p1 + V (t + 1, 4s − 2, ( a10 , a20 )) × p2 + V (t + 1, 4s − 1, ( a10 , a20 )) × p3 i + V (t + 1, 4s, ( a10 , a20 )) × p4 t = 1, . . . , T − 1 ; s = 1, . . . , 22(t−1) ; ∀( a1 , a2 ) 0

V ( T, s, ( a1 , a2 )) =

(4.37)

0

A1 A2 X X

R( T, s, ( a1 , a2 ), ( a10 , a20 )) × (c1 (θ1 ( T, s), a10 )

a10 =1 a20 =1

− I1 ( T, a1 , a10 )) + (c2 (θ2 ( T, s), a20 ) − I2 ( T, a2 , a20 )) ] s = 1, . . . , 22(T −1) ; ∀( a1 , a2 ) 0

(4.38)

0

A1 A2 A1 X A2 X X X

R (t, s, ( a1 , a2 ), ( a10 , a20 )) = 1

a1 =1 a2 =1 a10 =1 a20 =1

∀t ; s = 1, . . . , 22(t−1)

(4.39)

B(t, s, ( a10 , a20 )) = R(t, s, ( a1 , a2 ), ( a10 , a20 )) × B1 + β(t, s) − I1 (t, a1 , a10 ) − I2 (t, a2 , a20 ) t = 1 ; s = 1 ; a1 = a2 = 1 ; ∀( a10 , a20 ) B(t, s, ( a10 , a20 )) =

A1 X A2 X

(4.40)

R(t, s, ( a1 , a2 ), ( a10 , a20 )) × [ β(t, s)

a1 =1 a2 =1

B(t − 1, B(t − 1, + B(t − 1, B(t − 1,

mod 4] = 1,

t = 2, . . . , T ; s = 1, . . . , 2

; ∀( a10 , a20 )

s +3 4 , ( a1 , a2 )) if [ s s +2 4 , ( a1 , a2 )) if [ s s +1 4 , ( a1 , a2 )) if [ s s if [s 4 , ( a1 , a2 )) 0 0 − I1 (t, a1 , a1 ) − I2 (t, a2 , a2 ) ] 2( t −1)

mod 4] = 2, mod 4] = 3, mod 4] = 0

(4.41)

74

4 Valuation Model for Portfolios of Real Options 0

L(t, s, ( a1 , a2 )) =

0

A1 A2 X X

R(t, s, ( a1 , a2 ), ( a10 , a20 ))

a10 =1 a20 =1

t = 2, . . . , T ; s = 1, . . . , 22(t−1) ; ∀( a1 , a2 ) R(t, s, ( a1 , a2 ), ( a10 , a20 )) =

×

(4.42)

0 0 1 4 × R ( t, s, ( a1 , a2 ), ( a1 , a2 )) A10 A20 3

X XX a10 =1 a20 =1

L(t + 1, 4s − l, ( a10 , a20 ))

l =0

t = 2, . . . , T ; s = 1, . . . , 22(t−1) ;

∀( a1 , a2 ) ; ∀( a10 , a20 ) 0

(4.43)

0

A1 A2 X X

R(t, s, ( a1 , a2 ), ( a10 , a20 )) = 1

a10 =1 a20 =1

t = 1 ; s = 1 ; a1 = a2 = 1

(4.44)

R(t, s, ( a1 , a2 ), ( a10 , a20 )) ∈ {0; 1}

∀t ; ∀s ; ∀( a1 , a2 ) ; ∀( a10 , a20 ).

where t s ai

(4.45)

Time indicator, Scenario indicator Entering operating mode, underlying asset i = 1, 2, ai = 1, . . . , Ai , Leaving operating mode, underlying asset i = 1, 2, ai0 ai0 = 1, . . . , Ai0 , R Binary switching variable, V Value of portfolio, Stochastic value of underlying asset i = 1, 2, θi Starting value of underlying asset i = 1, 2, αi ci (·, ai0 ) Cash flow function for operating underlying asset i = 1, 2 in mode ai0 , ai0 = 1, . . . , Ai0 , Investment or switching costs for underlying asset i = 1, 2, Ii p Risk–neutral probability, Risk–free rate of return, rf B Budget level, β(t, s) New funds available at time t in scenario s, Initial budget available at time t = 1, B1 e R ≥ 0, B1 L, l Auxiliary variable and index.

4.3 Real Options Portfolio Model

75

4.3.4.3 n–Dimensional Model Formulation The model formulation for one underlying asset has established the modeling logic behind portfolios of real options. Especially it has introduced the switching concept, embedded into a program that can take into account both forward– and backward–looking elements. The model formulation for 2 underlying assets has then demonstrated how the model needs to be modified in order to handle a second underlying dimension. This has introduced the consideration of the joint stochastic processes of the underlying assets, while all investments are financed through a joint budget. The model formulation for n underlying assets is structurally similar to the formulation for two underlying assets and is composed of identical building blocks. With this background, it is possible to extend the model to i > 2 underlying assets in a straightforward manner. The indexing logic of the system state space needs to take into account the spanning defined by n underlying assets. Therefore, scenarios for θi (t, s) expand with s = 1, . . . , (2n )(t−1) = 2n(t−1) . This number expresses the total number of scenarios as a function of time. This implies that each node branches into exactly 2n successor nodes. For example, a tree representing the joint movement of i = 3 underlying assets branches into 23 = 8 nodes per time step. The risk–neutral probabilities for n underlying assets, based on a multidimensional extension of the Cox–Ross–Rubinstein (CRR) binomial (cf., e.g., Cox et al. 1979), are given in Boyle et al. (1989). For the general formulation, it is useful to define the following indicators: ( 1 if i moves up in scenario s, φi (s) = (4.46) −1 if i moves down in scenario s. 1 if i, j move in the same direction in scenario s, (4.47) φij (s) = −1 if i, j move in the opposite direction in scenario s. In analogy with CRR, the up and down movements are specified as ui = and di = u1 . The general formulation for the risk–neutral probability i in scenario s, pk , is then given by: √ eσi ∆t

" n # n √ X r f − 12 σi2 1 X pk = n φ (s)ρij + ∆t φi (s) 2 i,j=1 ij σi i< j

k = 1, . . . , 2n ,

i =1

(4.48)

76

4 Valuation Model for Portfolios of Real Options

√

where ∆t is the length of the time step, r f is the risk–free interest rate, and σ is the standard deviation of asset i returns. This formulation does not guarantee positive probabilities and so must be checked in each application. Further, the core of the model formulation is based on vectors for en→ a as the vector of entering modes of all tering and leaving modes, i.e., − − →0 underlying assets and a as the vector of leaving modes of all underlying assets: − → − → a = ( a1 , a2 , . . . , an ) , and a0 = ( a10 , a20 , . . . , a0n ). (4.49)

− → What is more, the vector of ones is given by 1 . Based on this, all variables can be defined over the corresponding vectors, instead of the more tedious expression through all elements of these vectors. For example, the binary variable R is defined over the system state − → → space as R(t, s, − a , a0 ). With these modifications, the model can be specified for n underlying assets in close analogy to the case of 2 underlying assets: − → max− V (1, 1, 1 ) →

(4.50)

− → R(1,1, 1 , a0 )

subject to θi (1, 1) = αi i = 1, . . . , n

θi (t, s) =

(4.51)

√ s + 2n − 1 φi (s)σi ∆t θ ( t − 1, ) × e n i 2 √ s + 2n − 2 φi (s)σi ∆t θi (t − 1, 2n ) × e ... √ θi (t − 1, s2+n1 ) × eφi (s)σi ∆t √ θ (t − 1, s ) × eφi (s)σi ∆t i 2n

if

[s mod 2n ] = 1,

if

[s mod 2n ] = 2,

if

[s mod 2n ] = 2n − 1,

if

[s mod 2n ] = 0

i = 1, . . . , n ; t = 2, . . . , T ; s = 1, . . . , 2n(t−1) 0

→ a) = V (t, s, −

A1 X a10 =1

0

···

An X

− → → R(t, s, − a , a0 ) ×

a0n =1

ci (θi (t, s), ai0 ) − Ii (t, ai , ai0 ) )

i =1 n

+ e−r f ∆t ×

X n

(4.52)

2 hX

i − → V (t + 1, 2n (s − 1) + k, a0 ) × pk

k =1

→ i = 1, . . . , n ; t = 1, . . . , T − 1 ; s = 1, . . . , 2n(t−1) ; ∀− a

(4.53)

4.3 Real Options Portfolio Model 0

77

0

A1 An n hX X X − → → → a) = ··· R( T, s, − a , a0 ) × V ( T, s, − ci (θi (t, s), ai0 ) − Ii (t, ai , ai0 ) a10 =1

a0n =1

i =1

i = 1, . . . , n ; s = 1, . . . , 2 A1 X

0

...

A1 An X X

0

···

an =1 a10 =1

a1 =1

An X

n ( T −1)

→ ; ∀− a

− → → R(t, s, − a , a0 ) = 1

a0n =1

∀t ; s = 1, . . . , 2n(t−1) − → − → → B(t, s, a0 ) = R(t, s, − a , a0 ) " ×

(4.54)

B1 + β(t, s) −

n X

(4.55)

# Ii (t, ai , ai0 )

i =1

− → − → → a = 1 ; ∀ a0 i = 1, . . . , n ; t = 1 ; s = 1 ; −

(4.56)

A1 An X X − → − → → B(t, s, a0 ) = ··· R(t, s, − a , a0 ) × [ β(t, s) a1 =1

a n =1

n → B(t − 1, s+22n −1 , − a) n − → s + 2 − 2 B ( t − 1, , a) 2n + ··· → B(t − 1, s2+n1 , − a) → s − B(t − 1, 2n , a )

−

n X

if [s mod 2n ] = 1, if [s mod 2n ] = 2, if [s mod 2n ] = 2n − 1, if [s mod 2n ] = 0

Ii (t, ai , ai0 ) ]

i =1

− → i = 1, . . . , n ; t = 2, . . . , T ; s = 1, . . . , 2n(t−1) ; ∀ a0 0

(4.57)

0

A1 An X X − → → → a) = ··· R(t, s, − a , a0 ) L(t, s, − a10 =1

a0n =1

→ t = 2, . . . , T ; s = 1, . . . , 2n(t−1) ; ∀− a − → → a , a0 ) = R(t, s, − ×

1 2n × A10

X a10 =1

(4.58)

− → → R(t, s, − a , a0 )

···

0

n

A n 2 −1 X X

− → L(t + 1, 2n s − l, a0 )

a0n =1 l =0

− → → t = 2, . . . , T ; s = 1, . . . , 2n(t−1) ; ∀− a ; ∀ a0

(4.59)

78

4 Valuation Model for Portfolios of Real Options 0

A1 X

0

···

a10 =1

An X

− → → R(t, s, − a , a0 ) = 1

a0n =1

− → → a = 1 t = 1 ; s = 1 ;−

(4.60)

− → → a , a0 ) ∈ {0; 1} R(t, s, − − → → ∀t ; ∀s ; ∀− a ; ∀ a0 ,

where t s i ai

− → a ai0

(4.61)

Time indicator, Scenario indicator, Underlying asset i = 1, . . . , n, Entering operating mode for underlying asset i = 1, . . . , n, ai = 1, . . . , Ai , Vector of entering modes of underlying assets, Leaving operating mode for underlying asset i = 1, . . . , n, ai0 = 1, . . . , Ai0 ,

− →0 a Vector of leaving modes of underlying assets, R Binary switching variable, V Value of portfolio, Stochastic value of underlying asset i = 1, . . . , n, θi Starting value of underlying asset i = 1, . . . , n, αi U p and down movement of underlying asset i = 1, . . . , n, φi ci (·, ai0 ) Cash flow function for operating und. asset i in mode ai0 , i = 1, . . . , n ; ai0 = 1, . . . , Ai0 , Investment or switching costs for und. asset i = 1, . . . , n, Ii p Risk–neutral probability, Risk–free rate of return, rf B Budget level, β(t, s) New funds available at time t in scenario s, Initial budget available at time t = 1, B1 e R ≥ 0, B1 L, l Auxiliary variable and index.

4.4 Model Extensions Up to this point, the core general model has been presented. This model already incorporates the key portfolio aspects of portfolios of real options. Some further possible model details will be given in the next section. 4.4.1 Alternative Probability Formulation The three–dimensional model has been based on risk–neutral probabilities as formulated by Boyle et al. (1989). On the one hand, this formulation of-

4.4 Model Extensions

79

fers the advantage that it is general enough to control for any number of underlying assets. Therefore, the above introduced model can be used for any number of underlying assets. On the other hand, as discussed before, it has the disadvantage that the probabilities may turn negative for large time steps. In financial option pricing where asset prices are typically modeled as log–normal processes, this disadvantage is hardly relevant because either way time steps are chosen such that the log–normal distribution is approximated well enough by the discrete distribution. In practice, usually 30 time steps per year are regarded as sufficient to obtain a good approximation (Hull 2003, p. 397). In our model, it is assumed that the values of the underlying assets follow a multivariate binomial distribution. Following this assumption, it is sufficient to consider less time steps, i.e., at least as many time steps as there are decision points. Hence it is important to ensure non– negativity of probabilities because otherwise arbitrage possibilities would exist, which would invalidate risk–neutral valuation. An approach ensuring non–negativity was first suggested in the context of interest rate trees by Hull and White (1994) and then applied by Hull (2003, pp. 473 f.) for a binomial lattice tree built up according to Jarrow and Rudd (1983, pp. 183 ff.). Instead of imposing the assumption u = 1d as in CRR, the probabilities in the two–dimensional case are set to p = 1 − p = 0.5. For the three– dimensional case, based on the coefficient of correlation, ρ, this gives risk– neutral probabilities with: p1 = p2 = p3 = p4 =

1 4 1 4 1 4 1 4

[1 + ρ ] ,

for asset 1=up, asset 2=up,

(4.62)

[1 − ρ ] ,

for asset 1=up, asset 2=down,

(4.63)

[1 − ρ ] ,

for asset 1=down, asset 2=up,

(4.64)

[1 + ρ ] ,

for asset 1=down, asset 2=down.

(4.65)

In the limit as the number of time steps approaches infinity, these probabilities are the same as are obtained for the formulation by Boyle et al. (1989). By modifying the jump factors the alternative formulation makes it possible to use the probabilities that in a CRR setting are only obtained in the limit. Since the degree of freedom has been used for fixing the probabilities, the up and down factors must be calculated separately as follows: u = e (r f − σ

2 /2)∆t + σ

√ ∆t

, and

d = e (r f − σ

2 /2)∆t − σ

√ ∆t

.

(4.66)

With these u and d factors, the values of the underlying assets do not hit the central node at each time 2∆t any more (see Sec. 4.2.1). This is less convenient for calculating options Greeks, but since this is not key for the presented model, the advantage of always non–negative probabilities clearly prevails. For this convenience, in our numerical analysis in Chap. 5 we will

80

4 Valuation Model for Portfolios of Real Options

use the formulation by Jarrow and Rudd (1983) for two underlying assets (see Sec. 5.1. However, given that this formulation is only available for two underlying assets, we will come back to the original formulation by Boyle et al. (1989) for our discussion of three underlying assets (see Sec. 5.2). 4.4.2 Change of the Underlying Stochastic Processes As discussed before, our model is based on a switching concept that allows to switch between operating modes of different assets. Up to this point, we assumed that the stochastic processes of the underlying assets remain unchanged. Nevertheless, an investment policy may result in an alteration of an underlying asset, and thus, the stochastic process. Especially, the investment policy may alter the risk profile of an asset, and thus, its volatility. This is depicted in Fig. 4.7, where an investment decision at some point in time shifts the stochastic process of an asset upwards, and lowers its variability. If a change of the underlying process occurs, all future real options will be based on this change.

Fig. 4.7. Change of the underlying asset

Our model can readily incorporate this alteration of the underlying assets, by tying the stochastic processes to the decision variable. Based on Eq. 4.52, this gives θi (t, s, ai0 ) instead of θi (t, s). Further, it is conceivable that the stochastic process is affected by decisions about all underlying assets, − → which would give the expression θi (t, s, a0 ). By altering the underlying stochastic process, the multinomial grid on which our model is based would shift accordingly, as visualized in Fig. 4.8.

4.4 Model Extensions

81

Fig. 4.8. Change of the underlying asset in the multinomial grid

4.4.3 Explicit Modeling of Synergies In the traditional capital budgeting literature, a typical way of taking synergies into account is to model bundles of synergistic projects as new projects. For example, two projects that exhibit synergies among each other can be modeled as three projects, i.e., project 1, project 2, and a synthetic project 3 which consists of doing project 1 and 2 together. This technically assumes that these three projects are mutually exclusive because project 3 incorporates the case where both projects are undertaken simultaneously (cf. e.g. Hax 1985, pp. 97 f.). In that sense the modeling of synergies does not require any modification of the model and does not merit special attention. Synergies just imply additional projects. Nevertheless it should be pointed out that, while leaving the structure of the model unchanged synergies can be handled through the binary switching variable R because it already controls for the necessary information. Following this approach, synergies can be modeled explicitly by making switching costs state–dependent, by making the cash flow function state– dependent, or both. State dependent switching costs for asset i = 1 would mean changing I1 (t, a1 , a10 ) to I1 (t, a1 , a10 , a20 ) if the switching costs of underlying asset i = 1 depends on the operating mode of the underlying asset i = 2; moreover the switching costs may also depend on where the switch was made from, yielding I1 (t, ( a1 , a2 ), ( a10 , a20 )). Other cash flow functions can also be specified in a straightforward manner. The cash flow function so far was only dependent on the leaving mode, e.g., c1 (θ1 (t, s), a10 ). To take synergies into account, a cash flow function can be specified dependent on combinations of operating modes, e.g.,

82

4 Valuation Model for Portfolios of Real Options

c1 (θ1 (t, s), ( a10 , a20 )). In general, the advantage of our mathematical programming formulation is that it keeps track of all variables throughout the system − → → state space and thus can incorporate any desirable function of (t, s, − a , a 0 ). All alternatives can be implemented readily without affecting the model setup. This is why they are not analyzed in more detail here. 4.4.4 Operating and Other Constraints Operating constraints have not been formulated so far. In order to include operating constraints, corresponding equations need to be added to the model formulation, by the help of the binary variables R. For the example of two underlying assets, if in period t = 2 it is not possible to simultaneously operate asset 1 in mode 1 and asset 2 in mode 3, i.e., these modes are mutually exclusive in all scenarios and independent on the entering mode, the additional constraint is: R(t = 2, s, ( a1 , a2 ), ( a10 = 1, a20 = 3)) = 0 , s = 1, . . . , 4 ; ∀( a1 , a2 ).

(4.67)

This example shows the flexibility of the approach which can handle path– and state–dependent constraints. Further, this formulation makes it possible to link option pricing to traditional capital budgeting and the known formulation of “project interrelationships” constraints (cf. Clark et al. 1984, pp. 330 ff.). Similarly, any precedence constraint can be handled. Depending on the specific modeling context, it may be desirable to include further constraints, such as maximum losses, minimum profits, or VaR constraints per period (cf. Kleindorfer and Li 2005). All of these can be readily accommodated. 4.4.5 Terminal Values The model formulation considers all periods in which cash flows are realized explicitly. This means that there are no further cash flows beyond period t = T. Alternatively, it is possible to include a terminal value in the assessment of a project, representing the aggregate cash flow in all periods beyond the explicit planning horizon, e.g., calculated as a perpetuity. The terminal value can represent a large share of the overall value (cf. Copeland et al. 1994, chapter 9). Our model is formulated without including terminal values, making the implicit assumption that terminal values are zero. This is done in order to avoid that a large fraction of value comes from decisions and cash flows that are not modeled explicitly because this would dilute the portfolio effects

4.5 Model Discussion

83

which derive from explicit option exercise strategies. If desired, terminal → a ). values can be included conveniently in the equation for V ( T, s, − 4.4.6 Further Generalizations of the Model Formulation Generally, the modeling approach provides a very flexible structure that can readily be customized to specific applications. This stems from two sources: 1. Controlling of decisions via binary variables: The indexing of the system state space is such that the binary variables keep track of the sequence of decisions and hence, their dynamic interaction. The binary variables can serve as reference points to any feature related to desired or permissible actions. 2. Discrete programming approach: In this structure, it is straightforward to modify the set and nature of constraints, as well as any input to the problem. For instance, the cash flow functions could be replaced; β(t, s) could be made endogenous by linking the level of additional funds to other endogenous variables (e.g., cumulated profits over time) instead of being exogenous; or different stochastic processes (e.g., mean–reverting processes) could be used. Adapting the problem setup leaves the structure of the problem unchanged. Therefore, changes can be made in a straightforward way in order to asses different problems.

4.5 Model Discussion Up to this point, we have presented a general formulation for portfolios of real options, and have discussed possible extensions of this formulation. In the following, we will discuss features of this model formulation. This discussion will cover the applicability of risk–neutral valuation, numerical efficiency, and a positioning with regards to Monte Carlo simulation. 4.5.1 Applicability of Risk–Neutral Valuation Risk–neutral valuation assumes no restrictions on short sales and hence unrestricted borrowing. Restrictions on borrowing are assumed only for the investing firm, whereas capital markets are assumed to be frictionless, both for the ultimate investor and for the underlying stochastic variables. This logic is founded by two elements. First, it is still a sufficient goal of the investment strategy to maximize net present value, calculated as the value of the portfolio of real options. If one were to consider the capital constraint as inconsistent with risk–neutral valuation, the same objection

84

4 Valuation Model for Portfolios of Real Options

would make it impossible to calculate even a NPV and make it hard to obtain any valuation results at all. We assume that the ultimate investor has unrestricted access to perfect capital markets and can choose her portfolio as a function of her personal preferences (Brealey and Myers 1996, pp. 105), such that risk–neutral valuation is validated. Second, given frictionless markets in the first place, all replication arguments on the level of the underlying stochastic variables and the derivation of risk–neutral probabilities still hold. The boundary between capital markets in general, and firm–specific investment decisions, is the transformation of θi (t, s) into a cash flow by the function ci (θi (t, s), ai0 ). The dynamics of the underlying stochastic processes are unaffected by this. While the firm has financing constraints, capital markets can be assumed to be frictionless. This assumption is typically made in recent work focusing on financing constraints (cf., e.g., Mello and Parsons 2000; Boyle and Guthrie 2003). Based on these arguments, the presented binomial model gives unique equivalent martingale measures, such that any contingent claim can be priced by risk–neutral valuation (Bingham and Kiesel 1998, pp. 105 ff.). 4.5.2 Numerical Efficiency The pricing of complex financial options typically requires the use of numerical methods which can result in a “tremendous computational challenge” (p. 1163 Broadie and Detemple 2004). Not surprisingly, the numerical solution of the presented model for portfolios of real options encounters the same challenges. This makes it worthwhile to discuss the efficiency of the numerical solution approach. The discussion of the numerical efficiency covers the optimality of solutions, the dimensionality of the problem, and possible ways for increasing the efficiency of solution algorithms. 4.5.2.1 Optimality of Solutions and Linearization The problem is formulated as a stochastic mixed–integer program which can be solved by commercially available implementations of suitable algorithms. This is why it is insightful to discuss the quality of possible solutions. The nonlinearity of the model merits special attention. It arises because the binary decision variable, R, is multiplied with the optimal portfolio value, V, introducing non-linear equations of the type R × V. These equations are inseparably tied to the general formulation because they are a core feature of the asymmetry stemming from the optionality involved. The decision maker has discretionary decision rights, which may or may not be exercised. Exercising a real option, which is equivalent to switching, is

4.5 Model Discussion

85

represented by a binary variable R = 1. Therefore, the nonlinearity is unavoidable in the general formulation. As opposed to this, the implementation of the problem in GAMS© can be done through an equivalent, linear version of the problem. This can be achieved by replacing the nonlinear equations by sets of linear inequalities, based on the upper and lower bounds of the variables involved. Schematically, assume that the binary variable R is multiplied by some function of V, f (V ), which yields the variable X. Further assume that bounds on X can be inferred with upper bound Xup and lower bound Xlo . Also, let there be a sufficiently large number Q. Then, the given nonlinearity of the form X = R × f (V )

(4.68)

can be substituted by the system of linear inequalities X ≤ f (V ) + Q × (1 − R )

(4.69)

X ≥ f (V ) − Q × (1 − R )

(4.70)

X ≤ Xup × R

(4.71)

X ≥ Xlo × R.

(4.72)

In order to improve numerical efficiency, we use this linear reformulation in our implementation for the numerical analysis. The logic is as follows: On the one hand, for R = 0 (i.e., option not exercised), the non-linear equation yields X = 0. The same is achieved by the set of linear equations. Due to Q, the first two equations are not binding whereas the last two equations yield X ≤ 0 and X ≥ 0 thus X = 0. On the other hand, for R = 1 (i.e., option exercised), the nonlinear equation yields X = f (V ). In this case, the last two equations do not turn out binding. The first two equations yield X ≤ f (V ) and X ≥ f (V ) thus X = f (V ). All in all, the set of inequalities can represent the nonlinearity. It is important to note that the interplay of these inequalities still expresses a nonlinear problem which can only be avoided technically in the implementation. The character of the problem remains nonlinear. Based on this reformulation, for the purpose of the discussion of numerical solutions, the problem can be considered as linear. This leads to a discussion of the mixed-integer character of the linearized problem. Integer problems are much harder to solve than continuous problems (cf. Williams 1993, chapter 8), such that it is much more expensive both in memory and computing time. Given that there are no general efficient methods available, this field is actively researched (cf. Birge and Louveaux 1997, p. 253). As the mixed-integer formulation is unavoidable for the valuation of portfolios of real options, the challenges represented by this class of problems also apply here.

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4 Valuation Model for Portfolios of Real Options

More precisely, mixed-integer programs are non-convex and thus, a local optimum typically cannot be guaranteed to be a global optimum (cf. Kall and Wallace 1994, pp. 2 f.). This means that it is not necessarily clear whether an optimal solution actually is the best solution. It may be the best solution locally, but there may be another local optimum which gives a higher value of the objective function. Problems of this kind are also referred to as “global optimization” (Kall and Wallace 1994, p. 2). For these, generally speaking, the proof of global optimality is difficult. In the light of the absence of general proofs of global optimality, recent developments in solver techniques become very important. Specifically, so-called global solvers have emerged and have actively been researched lately (cf., e.g., Horst and Pardalos 1995; Dzemyda et al. 2002). For example, the solver BARON© has been demonstrated to be very powerful, based on a branch– and–bound approach (Tawarmalani and Sahinidis 2002a, b; Ahmed et al. 2004). One key contribution of this solver is its ability to check global optimality numerically with high precision. In our numerical analyses, we applied this solver to the nonlinear problem, observing a high reliability but relatively slow computation time. In some cases, it is possible to prove global optimality in a very simple way. If in the relaxed mixed-integer problem the binary decision variables turn out exactly 0 or 1, the solution must be globally optimal. This is because the relaxed problem is linear thus convex, and thereby each local optimum is a global optimum (Kall and Wallace 1994, Lemma 1.1). This idea can be combined with a branch–and–bound technique in order to ensure integrality of the binary variables, i.e., branch–and–bound using linear programming relaxations (Nemhauser and Wolsey 1988, Chapter II.4.2). If the branch–and–bound algorithm terminates with an optimal solution, it is a global optimum. This requires that the solver does not terminate before its algorithmic termination, which can be the case if a given limit is reached, e.g., the maximum number of iterations, or memory. In the context of the above discussed computational burden of a mixed–integer algorithm, the bottleneck is memory. Concluding, if the algorithm provides a solution, we can rely on it to be a global optimum. Whether it gives a solution mainly depends on the dimension of the problem, which leads us to the discussion of the dimensionality of the problem. 4.5.2.2 Dimensionality of the Problem The size of the model can be characterized by the number of equations in the explicit version of the program. The number of equations grows exponentially in the number of periods and quadratic in the number of modes per underlying asset, i.e., (2n )(t−1) . Since the problem is path–dependent and thus non–recombining, the system state space grows exponentially. For

4.5 Model Discussion

87

example, two underlying assets make the system state space grow by factor 22 = 4, i.e., in each period a node in the tree branches into four successors. Further, the model grows quadratically in the modes per underlying asset since in the general case switches from each mode to each other mode are permissible. More precisely, for a portfolio problem with two underlying assets, the number of equations is a function of the number of operating modes for both underlying assets, A1 and A2 , as well as of the number of periods, T. For a portfolio with A1 > 1, A2 > 1, and T > 1, the number of equations in our implementation is given by: T h iX 1 + 3( A1 )2 ( A2 )2 + 1 22(t−1) + ( A1 )2 ( A2 )2

(4.73)

t =1

≈ 3( A1 )2 ( A2 )2

T X

22(t−1) + ( A1 )2 ( A2 )2

(4.74)

t =1

For A1 = A2 , this dimensionality is depicted in Fig. 4.9. It is noted that the scale of the vertical axis is logarithmic, which captures the exponential growth of the problem. In the implementation, this growth in the system state space translates directly into high memory requirements for generating the problem. Given the fact that mixed-integer problems are computationally very expensive, the high dimension of the system state space is very demanding in terms of memory and processor performance. Nevertheless, the linearized problem for 2 underlying assets, 3 modes each, and 5 periods takes a Pentium IV-M Laptop (CPU 1.7GHz RAM 512MB) 2 minutes to solve to optimality. In this light, it is vital to implement the problem as efficiently as possible. This is why possible routes for implementing the problem more efficiently will be pointed out in what follows. 4.5.2.3 Increasing Efficiency of the Implementation In order to increase the efficiency of the implementation, two directions can be followed. The first direction aims at reducing the original dimension of the problem before the solver is applied. The second, given the dimension of the problem, is concerned with making the optimization algorithm more efficient. Following the first direction, the efficiency of the implementation is closely tied to the modeling of the system state space. The system state space is generated exhaustively, i.e., it is built up for all combinations of periods, scenarios, and actions. It does not eliminate regions in the system state space which are irrelevant to the solutions, i.e., regions which will

88

4 Valuation Model for Portfolios of Real Options

1.0E+10 1.0E+09 1.0E+08 1.0E+07 1.0E+06 Number of equations 1.0E+05 1.0E+04 1.0E+03 1.0E+02

10 8

1.0E+01 6

1.0E+00 2

3

4

Periods

4 5

6

Modes per underlying asset

7

8

9

10

2

Fig. 4.9. Dimensionality for two underlying assets

never be anywhere near an optimum. This translates into high memory requirements when generating the model. Once the problem is generated, the solver needs to solve a correspondingly big problem, with sparse matrices for the binary variables. The binary variables are sparse because at each time in each scenario exactly one binary variable turns one, while all other binary variables there turn zero. The solver then eliminates as many irrelevant regions as possible. This is why it would be desirable to generate a model that is reduced in the first place. For instance, the size could substantially be reduced by gridding the system state space adaptively to the problem, in the fashion of an adaptive mesh approach known from derivatives pricing (Figlewski and Gao 1999). The basic idea is to build up a lattice tree that is finer in relevant regions and coarser in other regions. Along the same lines, it is conceivable to distinguish time periods and decision points. This would increase numerical efficiency when decision points are much fewer than points in time, because it would reduce the problem size. Each additional decision point increases the number of paths that can be followed over time, because it increases the number of possible combinations of decisions over time. As opposed to this, allowing for additional time periods that do not increase the number of decision points leaves the problem size unchanged since it only requires additional discounting operations. In our numerical example, this consideration would not materialize because we assume that there are decision points at each point in time.

4.5 Model Discussion

89

Another approach to reducing the dimension of the problem can be achieved by replacing the original system state space by a smaller one that gives a sufficient representation of the original system state space, in that its solution is not much different from the original problem. Such an approach is proposed by Casey and Sen (2005). Likewise, Zenios and Shtilman (1993) provide a method for generating subsets of an exponentially growing lattice tree, representing the whole lattice tree at a given level of accuracy. Following the second direction, different ways of making the optimization algorithm more efficient are conceivable. In our implementation, we considered implementation and solution as independent steps. This leaves some efficiency unexploited because a general formulation can be implemented in many different but equivalent ways. Alternative implementations can be assessed as a function of their efficiency with the solution algorithm and the most efficient implementation can be retained. As an example, in our implementation we have products of sums of variables. For numerical purposes, we checked whether a product disaggregation would increase efficiency (cf. Tawarmalani et al. 2002), but this did not prove helpful. Another possibility of better handling a complex problem is to combine a sequence of several optimization runs. A first run may eliminate large regions of the system state space quickly, which, making subsequent and tighter optimization runs more efficient. Also, there is a tradeoff between using commercially available solvers and tailor–made algorithms for specific problems. The former make it possible to benefit from very powerful engines that are continuously researched, tested, and improved. Most importantly, the results are perfectly reliable. The latter make it possible to further exploit the specific structure of a problem, but it can be very time consuming and expensive to show that the algorithm is robust. This is the case for the above described analysis of portfolios of real options in Meier et al. (2001). A more recent example for a tailor–made algorithm developed for a capacity planning problem, can be found in Huh and Roundy (2005). Probably, a good compromise between the two is to customize a commercially available solver specific to the problem. It has to be noted that the goal of our implementation was to illustrate the general model formulation. If the focus lies more on the efficiency of the implementation, it may be desirable to investigate alternative implementation structures. For example, for the given model formulation, it may be possible to avoid the growth of the system state space through an implementation as a network design problem (cf. Nemhauser and Wolsey 1988). Finally, a wide range of numerical techniques has been employed in the context of financial derivatives (cf. Broadie and Detemple 2004), and continues to be actively researched (cf., e.g., Broadie and Yamamoto 2005). It is conceivable to complement the solution approach used for our prob-

90

4 Valuation Model for Portfolios of Real Options

lem with elements provided in this strand of research. For instance, Kishimoto (2004) present a discrete-time numerical procedure for pricing path– dependent options, labelled as the “extended tree” method. The tree is extended by a “supplementary variable” which keeps track of the relevant information associated with the path through the tree. The method is illustrated by applications to collateralized mortgage obligations as well as American average option. Wang (2006) explores dimension reduction techniques for high–dimensional problems such as the pricing of path– dependent options and bond valuation. Interestingly, while the proposed dimension reduction techniques can achieve substantial dimension reduction in some settings, in other settings they may even increase the effective dimension. While the path–dependency is in close analogy to our model, the problems are different enough to exclude a straightforward adaptation. Like these examples, analogies to the model presented here can be tracked in other derivative pricing settings. 4.5.3 Monte Carlo Simulation With growing importance of complex financial derivatives, numerical methods have been used increasingly. Boyle (1977) was first to suggest Monte Carlo Simulation for estimating security prices. The approach consists of the following steps (cf. Boyle and Broadie 1997; Lehoczky 1997): 1. Simulate sample paths of the underlying variables over the relevant time horizon with respect to the risk–neutral measure. 2. Evaluate the discounted cash flows from the derivative, on each sample path. 3. Average the discounted cash flows over sample paths. This procedure shows that the approach is basically a forward–looking one. The advantage of this approach is that it can be used to price derivatives that only depend on the final values as well as derivatives that are path–dependent. The drawbacks of Monte Carlo simulation are that it typically is computationally very expensive and that it cannot easily handle cases with early exercise opportunities because these introduce a backward looking element (Hull 2003, p. 411). The dimension reduction techniques discussed in Wang (2006), for specific settings, can alleviate the computational burden of Monte Carlo simulation approaches. Early exercise is immanent to American–style options. For these, cash flows depend on both the price paths of the underlying assets and the decisions of the owner (Boyle and Broadie 1997, p. 1309), which invalidate the forward–looking nature of simulation. Only recently, simulation approaches that are able to handle American–style derivatives have been proposed

4.5 Model Discussion

91

(cf., e.g., Longstaff and Schwartz 2001; Andersen and Broadie 2004). Essentially, some iterative procedure needs to be added to the simulation which can incorporate the impact of exercise decisions. This can happen as a nesting of simulations and optimizations for the optimal decisions. Evidently, these simulation approaches preserve and further increase the drawback of computational burden required for obtaining reliable results. In the context of the model presented here, simulation does not represent a modeling alternative. The dynamics of the underlying assets are modeled in a straightforward way, through discrete binomial processes. These could alternatively be replaced by simulation runs of the underlying assets, especially, when many underlying assets are considered. Yet, the core of the model and source of its complexity is the interaction of decisions and resulting level of budgets over time. These would still need to be incorporated in a simulation approach, along the lines of pricing American–style options through simulation. Thus, using simulation would leave this programming core unchanged and hence would face the same challenges. One could think that at least simulation could be beneficial when considering many underlying assets and their joint stochastic processes which can readily be simulated. Yet, the challenge of the presented model which lies in the huge system state space does not result from the underlying assets in the first place. In fact, the underlying stand–alone binomial processes are recombining. The system state space becomes only bigger because it has to capture the path–dependencies in decisions. Again, this would remain unchanged even when simulating the processes of the underlying assets.

5 Numerical Analysis

In the previous chapter, a general model formulation for portfolios of real options has been proposed. This model provides a framework that incorporates the identified specific aspects of portfolios of real options and thus helps to optimally manage portfolios of real options. Up to this point, we have provided a general valuation structure but have not yet shed any light on the structural properties of such portfolios. This is why in this section we will exploit the impact of the main drivers of portfolio value and their impact on the optimal investment strategy. Keys drivers of the value of portfolios of real options are budget effects, volatility effects, correlation effects, and starting value effects. Embracing all of these effects are path–dependencies which simultaneously result from the interdependency of the budget constraint and of option exercise decisions. It will be shown that these value drivers can cause investment strategies that are puzzling in the first place, e.g., they imply shutting down an asset that is profitable. Nevertheless, the optimal investment strategy can be related, with more thorough analysis, to financial intuition and prior work on complex financial options. As to be expected, these drivers affect portfolio value in a complex way. Budget constraints establish sub–additivity of options value and may require to move in and out of assets in order to realize the highest value. This can cause disinvestment decisions that from a naïve point of view seem to be value destroying but which make it possible to undertake or help finance even more profitable investments for different assets and later periods with a positive net effect on portfolio value. Closely connected to budget effects are volatility effects. A basic understanding from standard option pricing would justify expecting options portfolio values to increase in the volatility of the underlying assets. For portfolios of real options, the impact of increasing volatility may be reversed. Typically, options benefit from higher volatility because they allow to capitalize on the upside while protecting from the downside of an increased variability in the out-

94

5 Numerical Analysis

comes. This may not materialize when the interplay of volatility and budget constraints prevents from capitalizing on the upside. Correlation and starting values can cause different effects, depending on the structure of the investment problem. Intuitively, the encountered effects can be attributed to the net character of the investment problem, i.e., whether it tends to be an investment problem with some disinvestment options or the other way round. Thus, all key drivers of the value of portfolios of real options can have an ambiguous impact on the value of the portfolio. Beyond this, an additional source of ambiguity results from the fact that these drivers all interact simultaneously with varying relative magnitudes, making it impossible to derive value effects in general. This ambiguity underlies any practical application. As a consequence, each investment problem needs to be solved specifically, which is achieved by the presented model. Knowing that in different settings the relative impact of different drivers will vary, the net effect only becomes evident after the valuation has been carried out. In the following, a representative investment problem is analyzed in order to give evidence of the ambiguity of key value drivers. This problem illustrates that much care must be dedicated to portfolio effects because these can substantially affect valuation results and hence optimal decision making. Specifically, for each value driver an example is provided that illustrates effects which can be connected to financial intuition only with detailed analysis. In this sense, pinpointing portfolio effects by example shows in how far these effects can only be captured by a thorough portfolio analysis. Most of these effects present counter examples to prevailing thinking in financial option pricing and thus generate new insights. The numerical example analyzed here is stylized in the sense that it does not represent an actual decision problem. It is purposely simplified because this makes it possible to isolate separate portfolio effects and keep the intuition of the underlying dynamics. Given that the model formulation is much more general, much more complex settings are readily solvable. Here, the focus lies on the basic structure of the portfolio setting. Nevertheless, the numerical application has been developed in cooperation with an anonymous high–tech company analyzing their manufacturing sites. Thus, while the problem is stylized and made anonymous, the general structure of the presented problem is real.

5.1 Numerical Analysis for Two Underlying Assets 5.1.1 Numerical Application Setting for Two Underlying Assets In the following, the basic structure of the investment problem is presented which will be analyzed to highlight portfolio features. We give a short

5.1 Numerical Analysis for Two Underlying Assets

95

Table 5.1. Capacity factors for two underlying assets

1 2 3 Plant 1 = Asset 1 0 1.0 1.4 Plant 2 = Asset 2 0 2.4 3.2 overview of the investment context and production process before we define the base case investment parameters. The production process of the considered high–tech company consists of two major steps: first a basic production process that is identical for all products, second a finishing process that is specific to the desired applications. The same production technology is used in different plants on different sites. The installment of a plant is naturally linked to expansion options because there typically is a “ramp up” of capacity in several capacity steps. This is feasible because a plant is designed modularly, with several production lines. Two partially complementary standards are produced, which are both exposed to highly volatile markets. The analyzed example is as follows. The company may invest in two plants, one for each standard, which are labelled as plant 1 and plant 2, respectively. For both we assume a base capacity of 100,000 units output per month. Plant 1 is less technologically intensive because it is a much more common technology. This is why it is possible to use this plant also for other products. Plant 2 is a new technology and hence much more specific, such that there is no alternative use for it. In that sense, it is a risky venture for the company. The free cash flows realized in both plants follow Jarrow–Rudd stochastic processes (see Sec. 4.4.1). Each investment installment is instantaneous and two decision points are considered. This short decision period makes it possible to keep track of the system state space and the interdependent decisions made at each state. For a more real–life decision problem a longer period would have to be modeled and a state–dependent residual value should be included. Here it is sufficient to highlight the dynamics of decisions over time, which can be shown in a short period. Both plants can be operated in 3 operating modes which determine capacity as a factor of base capacity. Plant 1, i.e., asset 1 can be operated at a capacity level of 0, 1.0 and 1.4 of base capacity, respectively; plant 2, i.e., asset 2 can be operated at a capacity level of 0, 2.4 and 3.2 of base capacity, respectively (see Table 5.1). The investment cycle starts in mode 1 “not invested”, i.e., zero capacity for both assets. Thus, for each asset there are 2 European options in period 1. In the second period, each asset can enter in exactly 1 out of 3 modes and can either stay in the same mode or switch to 1 out of 2 alternative modes. Thus, each mode can switch to two alternative modes, establishing

96

5 Numerical Analysis

3 × 2 = 6 European options for each asset. Adding up, there are 2 + 6 = 8 European options for each asset and 2 × 8 = 16 European option for the entire decision problem. By the help of this example it is also insightful to illustrate the number of resulting paths which are a consequence of investment decisions and stochastic realizations of the underlying assets. In the first period, each underlying asset enters in mode ai = 1, and can be switched to all ai0 = 1, 2, 3. Therefore, there are 32 = 9 possible combinations of investment decisions. After the switch, the binomial lattice tree branches into four successor nodes, at t = 2. Based on the entering mode, in each of the 4 nodes, each asset can again choose among ai0 , such that there are again 9 possible combinations of decisions. Combining investment decisions and stochastic realizations, for n = 2 underlying assets, Ai0 = 3 leaving modes, and time horizon T = 2, the number of paths through the tree is given by: Y i

Ai0 ∗ [2n

Y

Ai0 ](T −1) = 9 ∗ [4 ∗ 9]1 = 324

paths.

(5.1)

i

Switching costs represent investment cash outflow (positive sign) and disinvestment cash inflow (negative sign). Non–permissible switches are represented by a prohibitively large number (here, 10, 000); staying in the same mode is always costless. Switching costs are assumed constant over time.1 The structure of the switching costs is such that there are synergies to scale in building a higher capacity for asset 2. Thus, it is cheaper to move from no capacity to large capacity instead of passing via intermediate capacity (see Table 5.3). Together with the capacity levels in the respective modes, it becomes clear that asset 1 is designed smaller than asset 2, which also translates into lower investment outlays. On the other hand, given the economies of scale in investing, the investment outlays for asset 2 in relation to capacity are more attractive, but the involved risk is also higher because there are no options to scale down or abandon asset 2. While asset 1 is less technologically intensive and can be used for several alternative products, asset 2 is totally specific and therefore a bigger gamble. The company may invest in one out of the two assets, both assets, or not at all, and adjust the investments over time. For doing so, its shareholders have scheduled a specific budget for future periods, as an organizational measure of controlling their specific risk exposure in their overall portfo1

Constant switching costs make the options American since the exercise price is independent of the exercise date and exercising in period 1 can be viewed as early exercise. It is preferable though to keep the concept of European options in mind here because the assumption of constant switching costs is only made for ease of discussion. Therefore, the general case in the spirit of the presented model are European options.

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97

Table 5.2. Switching costs for asset 1 (two underlying assets)

Asset 1 1 2 3 1 0 30 40 2 -20 0 8 3 10,000 10,000 0 Table 5.3. Switching costs for asset 2 (two underlying assets)

Asset 2 1 2 3 1 0 45 50 2 10,000 0 20 3 10,000 10,000 0 lios, which must not be exceeded. Within these limits the company has been delegated full discretion how to best invest these funds. Thus, for both periods, a constant budget allowance is available. Unused budget allowance from period 1 remains with the investors but may be drawn in period 2 (bearing no interest though) and money freed up by disinvesting may be plowed back immediately. Further, we assume that there are no relevant interactions beyond the considered business. With this, the investment problem constitutes a portfolio of real options with two underlying assets and 16 real options. 5.1.2 Overall Portfolio Effects Due to the flexibility of the proposed approach, even the “simple” case of a two–period, two–asset, three–mode problem can have a large number of parameter combinations. Each case is fully specified by 45 parameters: 2 underlying processes with starting values and volatilities each; a joint coefficient of correlation; an initial budget endowment and 2 budget allowances; 9 switching costs for each period and asset; a risk–free interest rate.2 In the following numerical analysis, this will be simplified by fixing some degrees of freedom: no initial funds are available; budget allowance and switching costs are constant over time; switching costs and risk–free rate are fixed. Thus, the following numerical analysis is fully specified by variations of 6 parameters leading to 2, 880 cases,3 each one solved for the optimal portfolio value and the implicit contingent optimal exercise policy, through one optimization run. The following parameter variations are considered: 2 3

This makes a total of 2 × 2 + 1 + 3 + 2 × (2 × 9) + 1 = 45 parameters. For each underlying asset, 3 starting values and 2 volatilities; 2 coefficients of correlation; 40 levels of budget allowance per period, therefore (3 ∗ 2)2 ∗ 2 ∗ 40 = 2, 880 cases.

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5 Numerical Analysis

• Underlying processes for each underlying asset: starting values α1 , α2 = 25, 35, 45; volatilities σ1 , σ2 = 0.3, 0.7; coefficient of correlation ρ = −0.6, 0.7. • Budget constraint: budget allowance is constant per period, i.e., β(1) = β(2) = β, β = 14, 16, . . . , 92, i.e., varying between 14 and 92 in 39 steps of size 2. No initial funds are available, i.e., B1 = 0. For T = 2, the joint representation of the behavior of the two-period optimization model is given in Fig. 5.1. It depicts overall portfolio value as a function of budget allowance per period for different portfolios. Each line in the landscape represents a portfolio and each portfolio is specified by a joint combination of parameters within the above specified ranges (72 series in total). It gives an intuitive understanding of the sensitivity of the overall system which in some ranges can be relatively stable (flat regions) whereas in other regions it is more sensitive (rugged regions). It must be noted though that this representation provides limited insight since there is no logical relationship between the different series. Further, the joint effects are too rich to be clearly captured, such that it is preferable to identify them separately in a comparative–static analysis, in order to pinpoint specific aspects in this complex rugged landscape. Thus, in the following the single aspects will be analyzed separately with suitable graphical representations. Specifically, budget effects, volatility effects, and correlation effects will be exposed in detail. 5.1.3 Budget Effects and Other Constraints One main point of interest in the present discussion are budget effects. From a theoretical standpoint it is compelling to consider budget constraints in the optimization because it is intuitive that they will affect the global optimum significantly. In stark contrast to this consideration, budget constraints are oftentimes neglected. This can only be correct if valuation results are independent from budgets and do add up. We can show by example that this is generally not the case. The intuition behind the importance of considering budget constraints is the following: a binding budget constraint means that the optimization cannot achieve the optimum that would be obtained in the unconstrained case. Therefore, a binding budget constraint immediately implies deviations from value additivity, as compared to the unconstrained case. An analog to the case of two underlying assets with binding budget constraints can be found in the financial option on the maximum of two underlying assets (Stulz 1982) if the budget is such that only one asset can be acquired. In financial option pricing, it is evident that the value of the option on the maximum of two underlying assets can only correspond to the sum of two

5.1 Numerical Analysis for Two Underlying Assets

99

350 300 250

Portfolio value

200 150 100 50 0

86 68 50

Portfolio specifications

32 14

Budget allowance per period

Fig. 5.1. Portfolio landscape

options on the respective underlying assets if one option is approximately worthless. In the analyzed portfolios of real options, the portfolio value is a step– function of the budget level. This is depicted in Fig. 5.2 which visualizes a selection out of the whole range of data as presented in Fig. 5.1. It shows 9 different portfolio configurations of the above introduced investment problem where each portfolio is represented by a separate line. Each line gives the overall portfolio value created on the ordinate for different levels of budget allowance per period on the abscissa. Since the budget level is a constraint, changes in portfolio value as a function of budget allowance imply a change in the corresponding optimal investment strategy. Whenever a binding budget level hits a new “relevant” budget level from below, portfolio value increases. A relevant budget level here is defined as a budget level which enables to implement a different, more valuable investment strategy which requires higher investment outlays. The value effects in some regions are higher than in other regions and small changes in budget level can have a considerable impact on the portfolio value and the respective investment strategy. Overall, this shows that, over this range, there are no diminishing returns to investing more funds into projects.

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5 Numerical Analysis

Fig. 5.2. Portfolio value as a step–function of budget allowance Table 5.4. Portfolio highlighting budget effects

Portfolio α1 α2 σ1 σ2 ρ Portfolio 6 45 35 0.7 0.3 0.7 Portfolio 7 25 45 0.7 0.7 −0.6 For the discussion of budget effects, the effects for one portfolio separately can be distinguished from the effects when comparing two portfolios. 5.1.3.1 One Portfolio As a starting point, it is insightful to consider the budget effects for one portfolio separately, e.g., portfolio 6 as specified in Table 5.4. In some regions, value is independent of the available budget. This case does not require further analysis since it is trivial in that the optimal strategy remains unchanged. Second, in other cases, value increases with increasing funds available but to a rather small degree, e.g., for increasing β from 36 to 38. Third, value can substantially increase in other cases, e.g., for increasing β from 48 to 50. Investment strategies for portfolio 6 can be found in Fig. 5.3, on the left hand side for a budget allowance of β = 36, on the right hand side for β = 38. Each table gives the time and scenario specific optimal investment decisions for each of the two underlying assets. In each box, the first number indicates the entering mode while the second number indicates the leaving mode. For instance, (1 → 2) indicates that the asset enters the respective

5.1 Numerical Analysis for Two Underlying Assets

101

Fig. 5.3. Small change in portfolio value

scenario in mode 1, (not invested) and in that scenario switches to mode 2, investing into low capacity. The box on the left hand sides gives switching decisions for asset 1, the one on the right hand side switching decisions for asset 2. Changes in the optimal strategy are highlighted with a bold frame. While for β = 36 in period 2 it is optimal to disinvest asset 1 and install large capacity for asset 2, β = 38 makes it possible in period 2 to keep operating asset 1 and to invest in low capacity for asset 2, resulting in an increase of the portfolio value from 101.5 to 110.2. This is because two opposing effects occur in parallel: The new investment strategy sacrifices value realized by asset 2 in that it switches to low capacity instead of high capacity, but in return can avoid to disinvest asset 1. The relatively small value impact shows the positive net effect which is a consequence of a substantial modification of the optimal strategy, involving parallel investment and disinvestment. Changing β from 48 to 50 has a much bigger effect, with value boosted from 155.2 to 204.6 (see Fig. 5.4). Here, the additional budget allowance makes it possible to invest more profitably, i.e., returns do not diminish with increasing investment. It is optimal to install high capacity for asset 2 in the first period and then remain in that mode, instead of first installing low capacity and then expanding capacity (which can only happen in states where enough money is available). This makes it possible to realize cash flows earlier and to benefit from synergies of scale in the investment costs. Since no value needs to be sacrificed, the net effect is just the additional investment value and hence much bigger. Structurally, this case here is much more straightforward than the case of β = 38 because here the additional money simply is used without substantial impact on the overall strategy. The more complex case requires more detailed analysis and the optimization can partially overcome the adverse effect of the constraint by readjusting the overall strategy to a maximum; the system “breathes”. The second case is simpler because it just implies waiting for more money and investing that money once it is available, where it creates highest value, but otherwise leaving the investment strategy unaffected.

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5 Numerical Analysis

Fig. 5.4. Substantial change in portfolio value

In the presented example, the budget allowance per period was assumed to be constant over time, i.e., in each period the same amount of budget allowance is added to the funds available. The general model formulation allows for varying budget allowances per period. Following this direction further, it is also conceivable to allow for state–dependent budget allowances, especially as some function of past cash flows or expected cash flows in the future. This may help to model specific decision settings but does not shed further light on the discussion of the effects of the budget constraint on portfolio value. This is because the budget equation either way is path– dependent as it is a function of investment decisions which in turn are state– and path–dependent. What matters in the discussion of budget effects is this path–dependency. Adding more detail to the modeling of the budget equation clearly changes the actual results of a specific investment problem because it modifies the funds available at each state. Conversely, the structure of the budget dynamics remains unchanged. Thus, the effects encountered in the analyzed example are general in that they would also arise for state–dependent budget allowances and further possible extensions. Result: For the behavior of one portfolio at a time, it is possible to distinguish three categories of effects4 that stem from an increase in budget allowance: 1. Value and optimal strategy are unaffected. The additional money available does not permit to implement a new strategy with higher value. 2. Value increases to some degree and the optimal strategy is typically modified substantially. The gain in value is obtained from simultaneous disinvesting (at some opportunity cost) and investing. 4

These categories are combinations of a) whether the portfolio value changes (yes/no), and b) whether the optimal exercise strategy changes (yes/no). However, we only consider three instead of four categories, because a different optimal exercise strategy typically translates into a change of value. If not so, this would be the special case of equivalent alternative strategies, which is not insightful for the discussion of portfolio effects.

5.1 Numerical Analysis for Two Underlying Assets Portf. 7 β = 44 s=1

t=1

1→2

t=2

t=1

2→3

1→3

s=1

1→1

2→3

2→3

1→3

s=2

1→3

2→2

s=3

2→1

1→3

s=3

1→1

2→3

2→1

1→3

s=4

1→1

2→2

Asset 2

s=4

Value = 122.2

Portf. 6 β = 44 s=1

t=1

1→2

Asset 1

1→2

Asset 2

+ 59 %

t=2

Value = 194.8

Portf. 6 β = 46

t=1

2→3

1→3

s=1

1→3

2→2

2→3

1→3

s=2

1→3

2→2

s=3

2→1

1→3

s=3

1→1

2→3

2→1

1→3

s=4

1→1

2→3

Asset 2

s=4

Value = 134.7

1→1

t=2

s=2 Asset 1

1→1

1→1

t=2

s=2 Asset 1

1→1

Portf. 7 β = 46

103

Asset 1

+ 15 %

1→2

Asset 2

Value = 155.2

Fig. 5.5. Comparison of optimal investment strategies

3. Value increases substantially and the optimal strategy is typically modified in part only. The additional funds available enable to undertake a new investment. 5.1.3.2 Relative Attractiveness of Two Portfolios Besides the effect of level shifts in the portfolio value of one portfolio, an increase in available money can reverse the relative attractiveness of two portfolios, i.e., the graphs in Fig. 5.2 can intersect. To see exactly why this happens, it is insightful to consider two examples, portfolio 6 and portfolio 7, as given in Table 5.4. These two settings can be considered as representative because they cover important classes of possible parameter specifications. They cover one case each for positive and negative correlation, reversed starting values for the underlying assets, as well as low and high volatilities. The analysis of their joint behavior can best be achieved by discussing their respective investment strategies (as given in Fig. 5.5). As a reference case, consider portfolio 7 for β = 44. In this case, it is optimal in the first period to invest in asset 1 with low capacity. Then, in the second period, invest in asset 2 with high capacity, and depending on the scenario either scale up asset 1 (in the up scenarios) or disinvest (in the down scenarios). This strategy yields an overall portfolio value of 122.2. As opposed to this, by just allowing for an additional budget allowance of 2

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5 Numerical Analysis

units per period, i.e., β = 46, the optimal investment strategy is completely reconfigured. In this case, it is optimal to move into asset 2 with low capacity, in the first period. Then, in the second period, the strategy becomes more elaborate. Asset 2 remains operating in low capacity in scenario 2 and 4, which is only made possible through the corresponding strategy for asset 1. In scenario 2, i.e., in scenario (u1 , d2 ), asset 1 gets invested in high capacity and asset 2 continues operating in low capacity. As opposed to this, in scenario 1, i.e., scenario (u1 , u2 ), asset 1 remains uninvested whereas asset 2 is scaled up. It is apparent that the decision on asset 1 would be to invest in large capacity in the case of unrestricted funds. However, since money is short, it is more attractive to spend the limited money on asset 2, which is true except for scenario 2. Overall, with the increased budget level much less money is invested in asset 1 and it is feasible to invest earlier into the relatively more attractive asset 2. Figuratively speaking, asset 1 steps back for asset 2. The portfolio value significantly benefits from this shifted strategy with an increase of roughly 60%. This example has presented a case where money is channelled to one asset at the disadvantage of another asset which would receive the funds if the other asset were not more profitable. More profitable means that it must be profitable as such and overcompensate for the opportunity cost in the not invested or disinvested asset. Generalizing, this effect is encountered when the relative attractiveness of investments is decisive for the allocation of limited resources between assets. In the presented example, the interaction between assets took place in the same period where a profitable opportunity was not seized. Structurally, the same effect can be obtained if an already operating asset gets disinvested to free up the necessary resources. Both disinvesting an existing asset and not seizing an investment opportunity can happen in one period or between periods. An investment may not be undertaken in one period in order to save funds for investment in a later period (keeping the options open); or a disinvestment may be realized earlier in order to seize a put option before it expires. These effects are different versions of the effect demonstrated in portfolio 7, yet structurally they are the same. Further, the relative behavior of portfolio 7 in comparison to portfolio 6 is insightful. for For β = 44, portfolio 6 yields a portfolio value of 134.7, as opposed to a value of 122.2 for portfolio 7. These values are realized through an identical investment strategy as in portfolio 7, which is feasible for both portfolios since both have an identical budget allowance of β = 44, but differ in value due to the differences on the portfolio parameters. Remarkably, for β = 46 the investment strategy substantially changes, both compared to the lower budget level and to portfolio 7 for the same budget level. For portfolio 6 and β = 46, it is optimal in the first period to remain uninvested in asset 1 and to install low capacity for asset 2, like for portfolio 7. Unlike portfolio

5.1 Numerical Analysis for Two Underlying Assets

105

7, in the second period in scenario 1 it is optimal to install high capacity for asset 1 and stay in low capacity for asset 2, whereas in scenario 4 it is optimal to scale up capacity for asset 2. The different investment strategy is due to the different relative attractiveness of the investment opportunities since asset 1 is much more valuable and asset 2 is less valuable than in portfolio 7 (see Fig. 5.4), while the switching costs matrix is identical. This is why there is not as much upside potential in changing the investment strategy; value sacrificed by not investing into asset 1 is still overcompensated by investing into asset 2 in the first period, but not as clearly. Also, given the different relative attractiveness of asset 1 and 2 in portfolio 6, the investment strategy in period 2 is different from the strategy in portfolio 7, which is explained by the switching costs involved. The increase in value is still substantial and the investment strategy completely changes, yielding an increase in portfolio value to 155.2, or 15%. Compared to the budget effect for portfolio 7 with a shift in portfolio value of some 60%, the effect on portfolio 6 is much less pronounced. A slight relaxation of the budget constraint not only results in a fully reconfigured optimal investment strategy and higher values for both portfolios, but also in a net effect where the relative attractiveness of the two portfolios changes. Graphically, this translates into intersecting lines in Fig. 5.2. This intensifies the observation that budget constraints must be considered when comparing different possible portfolios. It is not straightforward to judge which portfolio would be preferable because it all depends on the budget, and more so, is highly sensitive to it. For less funds available portfolio 6 is more attractive, whereas with little more funds available portfolio 7 takes the lead. Result: For the budget effects on the optimal investment strategy and corresponding portfolio value, the following properties can be summarized: 1. Binding budget constraints have a significant impact on the optimal investment strategy, clearly deviating from value additivity. 2. The portfolio value is a nonlinear step–function of budget level. 3. The investment decision is highly reactive to budget levels: small changes in the budget constraint can result in a considerable adjustment of the optimal investment strategy, and have a substantial impact on portfolio value. 4. The relative attractiveness of different portfolios can be reversed for different available budget levels. 5.1.3.3 Generalization of Budget Effects The above discussion has demonstrated that binding budget constraints play an important role in the investment decision process. They need to

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5 Numerical Analysis

be considered in all dimensions of real options portfolio management, i.e., both in the design and in the execution of portfolios. Budget constraints have been exposed in detail as representative for any other type of constraint that can be included in the model. This is for two reasons: First, budget constraints appear as key to any investment process which at its core is about spending money that must be available. Second, budget constraints as balance equations represent a modeling challenge that requires special attention. Obviously, any constraint will have an impact specific to the actual decision problem, but structurally the effect will be the same as for budget constraints: Binding constraints imply deviations from the unconstrained optimum and the optimization ensures to obtain the best possible result given the constraints. Relaxing constraints even slightly can have a huge impact on the optimal strategy and portfolio value, stressing the potential of a simultaneous optimization. Result: Binding budget constraints reduce the value of portfolios of real options, but the negative effect can be minimized by a suitable investment strategy. Such an investment strategy, can be sophisticated in the sense that it involves investment and disinvestment decisions in parallel, realizing a positive value effect. If constraints exist in the real world investment problem, ignoring them in the investment analysis forgoes some of the value potential for a (by luck) basically correct investment strategy. In other cases, this can lead to a plainly sub–optimal investment strategy. 5.1.4 Volatility Effects Closely related to the discussed budget effects, it is insightful to analyze the effect of changes of volatility on portfolio value. Increasing volatility in the proposed model can have two main effects: value increasing in volatility and value decreasing in volatility. First, option value increasing in the volatility is a common property of standard options, both puts and calls (Black and Scholes 1973 and Merton 1973b, Theorem 8). In our numerical investigation, we have confirmed this behavior for the discussed portfolio of real options. Therefore, this behavior of the model is straightforward and will not be discussed in detail here. Second, as opposed to these first two effects, option value decreasing in the volatility is an interesting case that merits special attention. This effect has first been shown by Stulz (1982) for European options on the minimum or the maximum of two assets. He shows that the effect of an increase in volatility of the price of one underlying asset is ambiguous and can also result in decreasing option value. The budget constraint can introduce, to some degree, a choice between assets that compete for the joint funds available. As a consequence, the case of two underlying assets is structurally comparable to a call on the maximum of two underlying assets.

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107

Table 5.5. Portfolio with value decreasing in volatility

Portfolio Portfolio 3Portfolio 3 Portfolio 3+

α1 45 45 45

α2 25 25 25

σ1 0.3 0.3 0.7

σ2 ρ β Portfolio value 0.3 0.7 32 73.6 0.7 0.7 32 85.0 0.7 0.7 32 78.7

As a special case, value can also be independent from volatility, but this is due to the model setup only. The possibility of value independent from volatility is implied by assuming discrete processes for the underlying assets. With discrete processes, option values also become non-smooth in the underlying assets. A change in the volatility only affects the exercise strategy and thus the option value if it makes the underlying asset move beyond a given exercise price (cf. Cox et al. 1979). This introduces discontinuity. The choice of a discrete framework was motivated by the resulting flexibility for the modeling of options and constraints (see Sec. 4.2.1). Thus, the effect is by design and does not deserve special attention, because it would vanish in a continuous model setup. Further, for exotic financial options, this same property can be observed, e.g., for an up–and–out barrier call option (see Merton 1973b, pp. 177 f.). This option is structured like a plain vanilla call option with the difference that it becomes worthless if the underlying asset rises above a given barrier, throughout the lifetime of the option. This is why a barrier option is path–dependent: the value of the option at expiry depends on the path taken by the underlying asset. On all paths that have hit the barrier at least once throughout the lifetime of the option, the option has zero value since it is foregone (Kwok 1998, pp. 246 ff.). In a recent application, Wong and Kwok (2003) provide examples for a barrier option where increasing volatility translates into a lower option value because the chance of hitting the barrier increases and more probability is on cases where the option becomes worthless. Similar to the results by Stulz (1982), barrier options are complex contracts because they are ambiguous in the sensitivity to parameter variations (Wilmott 1998, pp. 189-201). The effect of increasing and decreasing value can best be shown by the example of portfolio 3, portfolio 3-, and portfolio 3+, as given in Table 5.5. First, increasing the volatility of asset 2, σ2 , from 0.3 to 0.7 increases portfolio value by 15.5%, from 73.6 to 85.0. As discussed before, this case is expected and does not merit special attention. More interestingly, increasing the volatility of asset 1, σ1 , from 0.3 to 0.7 reduces portfolio value by 7.4%, dropping from 85.0 to 78.7. This case deserves a more careful discussion. Overall, the effect is due to a binding budget constraint, which is apparent in the optimal operating strategy, as depicted in Fig. 5.6. For portfolio 3, in period 1 it is optimal to install low capacity for asset 1 and stay unin-

108 Portf. 3 σ1= 0.3 s=1

5 Numerical Analysis t=1

1→2

t=2

t=1

2→1

1→3

s=1

2→1

1→3

2→3

1→1

s=2

2→3

1→1

s=3

2→1

1→3

s=3

2→1

1→3

2→3

1→1

s=4

2→1

1→1

Asset 2

s=4

Value = 85.0

1→2

t=2

s=2 Asset 1

1→1

Portf. 3+ σ1= 0.7

Asset 1

1→1

Asset 2

- 7.4 %

Value = 78.7

Fig. 5.6. Portfolio value decreasing in σ1

vested for asset 2. In period 2, it is optimal to invest in high capacity of asset 3 in scenarios 1 and 3, i.e., when the value of asset 2 develops favorably. This is only made possible through disinvesting asset 1 in these scenarios; in the other scenarios it is optimal to expand asset 1. In other words, asset 1 is profitable both in its respective up and the down scenarios, but needs to be disinvested in order to provide the funds necessary to realize the yet more profitabel investment in asset 2, in scenarios 1 and 3. Increasing the volatility of asset 1, in portfolio 3+, naturally results in a higher variability of the outcomes of asset 1. Usually, a standard option can profit from the increased variability by capitalizing on the upside while protecting from the downside; basically this is why standard option values are increasing in volatility. This cannot happen in the presented portfolio 3+. On the upside, in period 2 in scenario 1, the portfolio cannot exploit the higher value of asset 1 because it is still optimal to disinvest asset 1 in order to finance asset 2. Reversely on the downside, in period 2 and scenario 4, the portfolio is fully hit by the reduced value of asset 1. While in that scenario it is optimal to scale up asset 1 for lower volatility, it is optimal to disinvest for higher volatility. The additional value realized in period 2 in scenario 2 for asset 1 cannot overcompensate for the loss in value, such that the overall net effect is negative. The money freed up by disinvesting here is assumed to be constant, therefore independent of time and scenario, in analogy to the deterministic exercise price of a standard put option. Allowing for state–dependent disinvestment payoffs, which is a possible extension of the proposed model, can further increase this effect. The analogy to a barrier option consists of the analogous existence of an adverse effect on value. For the barrier option, the existence of the barrier introduces some probability that the option cannot be exercised even if it expires in the money, depending on the path taken by the underlying. Just the same holds true for a budget constrained portfolio of real options. Here the existence of the budget is such that on certain paths the full (unrestricted) value potential cannot be exploited and portfolio optimization can only partly countervail the adverse effect on value. At the same time, the

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109

analogy between barrier options and portfolios of real options is intuitive but cannot shed light on the valuation of portfolios of real options because they are concerned with valuation issues that are fundamentally different. The balance equation of the budget, path–dependencies of decisions, as well as interactions and different types of constraints are key for portfolios of real options, but irrelevant for barrier options. Reversely, the valuation of barrier options within a discrete valuation framework requires special attention to the discretization of the system state space relative to the barrier, such that nodes lie on the barrier (Hull 2003, pp. 468 ff.). This is not an issue for portfolios of real options because the budget level is independent from the underlying assets. Result: The values of portfolios of real options can be either decreasing or increasing in the volatility of the underlying assets. This attaches special importance to the assessment of volatility and risk in portfolios of real options. Contrary to financial asset portfolio management, adding new assets or options to the portfolio and by this changing the overall risk profile can have a positive or a negative effect on portfolio value. 5.1.5 Correlation Effects The present analysis deals with both correlation in the underlying assets and correlation in the cash flows. As argued in Sec. 4.1.6 and without loss of generality, it is sufficient to focus on the effect of the correlation of the underlying assets on the values of the portfolios of real options. Considering correlation effects can best be positioned by pointing out that the investment problem analyzed here has structural parallels to a call option on the maximum of two assets, as discussed before. For a European call option, Stulz (1982) shows that option value is decreasing in the correlation: the higher the coefficient of correlation, the less valuable the option. In our numerical analysis, the same effect is found. This result is mainly driven by the binding nature of the budget constraint, and the payoff function. In settings without a binding budget constraint, it is possible to observe positive correlation effects on portfolio value. For example, Dias (2006) recently discuss correlation in oil exploration, without budget constraints. In this setting, correlation tends to have a positive effect because learning about one asset helps to optimally manage other assets. Further, correlation makes it possible to capitalize further on the upside potential for all assets, leading to higher overall returns the higher the correlation. It is noted though that this positive correlation effect can only be encountered when these assets do not compete for limited funds. For our setting, we can show a negative correlation effect by the example of portfolio 3 (see Table 5.5), for β = 32. The optimal exercise strategy for this portfolio is depicted in Fig. 5.7, on the top. Below, the state–dependent cash

110

5 Numerical Analysis Portf. 3 β = 32 s=1

t=1

1→2

t=2

2→1

1→3

s=2

2→3

1→1

s=3

2→1

1→3

2→3

1→1

Asset 1

1→1

Asset 2

s=4

Portf. 3 β = 32

t=1

t=2

Prob.

Prob.

ρ = -0.6

ρ = 0.7

20.0

82.6

0.100

0.425

s=2

77.5

0.0

0.400

0.075

s=3

20.0

82.6

0.400

0.075

38.9

0.0

0.100

0.425

s=1

s=4

15.0

0.0

Asset 1

Asset 2

Value = 97.0

-12.3 %

Value = 85.0

Fig. 5.7. Portfolio value decreasing in ρ

flows for the optimal exercise strategy are given. Next to it, for both levels of ρ and the corresponding risk–neutral probabilities, the optimal value is given. In the presented case, increasing ρ from −0.6 to 0.7 does not affect the optimal exercise policy, and therefore, not the state–dependent cash flows but the corresponding portfolio value. This is due to the fact that the risk– neutral probabilities change. While ρ = −0.6 puts much probability on scenarios 2 and 3, this changes for ρ = 0.7 where most probability is on scenario 1 and 4. Intuitively speaking, scenario 1 and scenario 4 are scenarios where both assets behave in parallel, such that for a higher correlation these scenarios increase in probability and reversely so for scenarios 2 and 3. Since scenarios 2 and 3 jointly yield higher cash flows than scenarios 1 and 4, reducing their respective probabilities decreases portfolio value. This observation confirms the intuition. If both assets compete for a binding budget constraint, scenarios where both assets develop in the same direction suffer most from the constraint. For example, in the unconstrained case in scenario 1 it would be optimal to expand asset 1. In the constrained case, the opposite is optimal: asset 1 is disinvested to finance asset 2. Putting more probability on this case reduces overall portfolio value because the adverse budget effect has more weight. Just as for financial options, it is also possible that a change in the correlation has no value effect at all. For a financial option on the maximum of two assets, the correlation impact depends on the relative being in or out

5.1 Numerical Analysis for Two Underlying Assets

111

of the money of the option with regard to the separate underlying assets. As an intuitive example, if the exercise price is high enough that one of the assets will never be acquired, i.e., one asset is dominated, the option can be priced independently from the dominated asset because the correlation in the probabilities cancels out. Supposing asset 2 is dominated and using the formulation given in Sec. 4.2.3, the risk–neutral probabilities for asset 1 give: " !# √ r f − 12 σ12 1 1 + ∆t (5.2) p( asset1, up) = p1 + p2 = 2 σ1 " !# √ r f − 12 σ12 1 1 − ∆t (5.3) p( asset1, down) = p3 + p4 = 2 σ1 Also in the limit, the correlation cancels out (cf. Boyle et al. 1989, p. 246): √lim p1 ∆t→0

+ p2 =

1 2

(5.4)

√lim p3 ∆t→0

+ p4 =

1 2

(5.5)

In our investment setting, a similar effect is encountered when one asset is too expensive to be ever acquired. Nevertheless, this is just a special case; for investment decisions that are not as extreme, a negative effect of correlation is to be expected. In an investment setting as is discussed here, the budget constraint is such that the competition between projects for limited funds makes it desirable to look for reduced correlation, or even negative correlation.It is important to note that this is not in order to diversify, but on the contrary to increase the range of outcomes in order to capitalize on the upside potential while being protected from the downside. Result: In the presence of budgets constraints, the values of portfolios of real options can be decreasing in the correlation. In other settings, when there are synergies or learning effects between assets in the absence of budget constraints, the values of portfolios of real options can be increasing in the correlation. 5.1.6 Starting Value of the Underlying Assets Effects Further to the so far discussed effects, it is evident that the starting values of the underlying assets have an important impact on portfolio value. For the presented problem, the value of the portfolios of real options is increasing in the starting value of the underlying assets. This can be visualized by the

112

5 Numerical Analysis

Fig. 5.8. Portfolio value increasing in α1 and α2

example of portfolio 6 (Table 5.4), for β = 38, and varying α1 and α2 (see Fig. 5.8). It is noted that in the specific case, the behavior is nearly linear, i.e., a change in the starting value linearly translates into a change in the portfolio value. This is not the point to be made here though because it is not a general result. A counter–example is a case where only by increasing the starting value, a specific option is exercised. This would result in a kinked curve known as a standard payoff–profile of a call option at maturity. If the price of the underlying asset is higher than the option exercise price, the option value is a linear function of the underlying, otherwise its value is zero. Result: This further confirms that the presented problem has the net payoff profile of a call option. This is intuitive, as the decision problem starts in a not invested mode and is therefore mainly an investment problem with disinvestment options, not the other way round. 5.1.7 Path–Dependency Effects All of the discussed effects are inseparably connected to the path– dependencies involved. Decisions about one of the underlying assets depend on the path taken by the other assets and the decisions made until this point since the budget is a path–dependent function of earlier decisions. This can be illustrated by a decision problem for three decision points, as given in Table 5.6. It follows the same structure as the examples discussed before, with the difference that there is one more decision point, i.e., T = 3. The budget starts with an initial endowment, B1 , of 0 and additional allowances per period, β, of 32.

5.1 Numerical Analysis for Two Underlying Assets

113

Table 5.6. Portfolio highlighting path–dependency for T=3

α1 α2 σ1 σ2 ρ B1 β Portfolio value 45 25 0.7 0.7 0.7 0 32 178.5 Asset 2, t = 3 t=1

A s s e t 1

t=2

u2u2

u2d2

d2u2

d2d2

u1u1

3→3

3→3

3→3

3→3

u1d1

3→3

3→3

3→3

3→3

d1u1

1→2

1→2

2→3

2→3

d1d1

1→1

1→1

2→1

2→1

t=3

u1

u1

d1

d1 u1 d1

Fig. 5.9. Path–dependency of underlying asset 1

The path–dependency underlying this investment problem can be visualized in Fig. 5.9. On the left hand side, it visualizes the jump process for asset 1 and T = 3. Here, the path–dependency of asset 1 becomes apparent up the nodes for (up, down) and (down, up) do not coincide. On in the fact that down the right hand side, the figure gives an overview of the investment decisions made for asset 1 at t = 3 which are indicated in the 4 × 4 = 16 boxes on the right. The corresponding investment decisions for asset 2 are omitted here because they would not clarify the path–dependency further. Here the goal is to show how asset 1 is affected by the joint movement with asset 2, which is captured through the scenarios for asset 2. On the horizontal of the right hand side, the different scenarios for asset 2 are specified alongside. Asset 2 is also path–dependent since (up, down) needs to be distinguished from (down, up). In Fig. 5.9, path–dependency for asset 1 can be observed in three ways: 1. Path–dependency arising from decisions over time. The sequence of decisions before period t = 3 and especially the decision made at t = 2 determine the operating mode in which asset 1 enters period 3. Precisely, the entering mode at t = 3 must be identical to the leaving mode at t = 2. This mode is the starting point for a possible switch and hence has an impact on which leaving mode is optimal from there. This source of path–dependency is visible in all 16 boxes because all possible entering modes, a1 = 1, 2, 3, appear. 2. Path–dependency arising from the stochastic process of asset 1. Following the stochastic process of asset 1, the scenarios of (up,down) and (down, up) need to be distinguished. They correspond to the second and the third line of the boxes. Fixing the second asset to (up, up), it

114

5 Numerical Analysis

becomes clear that although the scenario of asset 2 is the same, the two lines present different operating strategies (see boxes in bold frame). Thus, it is not possible to model asset 1 as recombining, which establishes the second source of path–dependency. 3. Path–dependency arising from the interplay of assets. Before, it was insightful to fix the scenario of asset 2. Reversely, it is insightful to fix the scenario of asset 1 to (down, up). Looking at the different scenarios for asset 2 reveals different strategies (see dotted boxes). Since the scenario for asset 1 is fixed, these different strategies result from the development of asset 2 and the respective optimal investment strategies. The strategy for asset 1 is different for the scenarios (up, down) and (down, up) of asset 2, thus another source of path–dependency. This path–dependency for the decisions made about asset 1 is driven by the path–dependency of the other asset. Result: This example has visualized the path–dependencies which are at the core of the proposed model. It simultaneously stems from the path– dependencies in decisions over time in the development of each underlying asset and in the interplay of the underlying assets. It should be noted though that this distinction is helpful to fully grasp the degree of path–dependency, but that it is not very rigorous. Given that these drivers occur simultaneously, rigorously speaking, there is only one joint path–dependency. 5.1.8 Deviations from Value Additivity The above discussion of value drivers has repeatedly recurred to the fact that value additivity does not hold for portfolios of real options and that option values are typically sub–additive. In order to stress that nature, the reasons behind this are reviewed to conclude the numerical analysis. First, budget constraints introduce deviations from value additivity which was the key insight from Fig. 5.2. This is because a binding constraint, by definition, requires the optimal strategy to deviate from the unconstrained optimum. Second, the switching concept subject to switching costs represents a compound option valuation approach, in which each option exercise, or switching decision, gives birth to new options, contingent on this switch. This also, by definition, establishes deviations from value– additivity. Third, it was shown that an increase in volatility, contrary to common understanding of financial options, can harm portfolio value. Fourth, an increase in correlation for an investment problem that is a net call option reduces portfolio value. These effects occur simultaneously, such that it is not possible to attribute the effect to a single source. Overall, these considerations confirm the expectation that options do not add up, as shown by Trigeorgis (1993a).

5.2 Extension of the Numerical Analysis to Three Underlying Assets

115

For completeness, the sub–additivity can be shown, e.g., for portfolio 3. In the constrained case with β = 32, the portfolio of real options has a value of 85, as given in Table 5.5 and Fig. 5.6. Relieving the budget constraint, it would be optimal to install large capacity for both assets immediately, yielding a portfolio value of 194.6. For completeness, it is noted that options can also be super–additive. Beyond the reasoning given in Trigeorgis (1993a, pp. 8 f.), in the proposed model super–additivity may result from synergies in operating projects together. Result: Given the impact of budget constraints on the optimal policy, typically sub–additivity is to be expected for portfolios of real options. Therefore, in practice the negative effect of options interactions is likely to be more prevalent (cf. Trigeorgis 1993a, p. 19).

5.2 Extension of the Numerical Analysis to Three Underlying Assets Up to this point, the numerical analysis has shed light on the relevant portfolio aspects encountered in portfolios consisting of two underlying assets. This analysis has not yet addressed the joint stochastic movements of several assets, which is considered in what follows. 5.2.1 Numerical Application Setting for Three Underlying Assets The discussion of three underlying assets will focus on the additional portfolio aspects which go beyond the features discussed for two underlying assets. For this, we will present a specific portfolio configuration which will highlight the respective portfolio features successively. The intuition behind the numerical example for two plants in 5.1.1 can be extended by assuming that the additional underlying asset would be located in another country, thus operating with a different cost structure, and especially, with a different correlation structure. In the numerical analysis for three underlying assets, investment decisions are equivalent to switching decisions between 3 operating modes, for each underlying asset, as depicted in Table 5.7. Compared to the before discussed numerical setting for two underlying assets, asset 2 can be considered as an “intermediate” asset, with capacity factors situated between the otherwise unchanged capacity factors of the other two assets (cf. Sec. 5.1.1). Switching between these capacity factors is subject to the corresponding switching costs matrices (see Table 5.8, 5.9, and 5.10), which are assumed

116

5 Numerical Analysis Table 5.7. Capacity factors for three underlying assets

1 2 3 Asset 1 0 1.0 1.4 Asset 2 0 1.4 1.8 Asset 3 0 2.4 3.2 Table 5.8. Switching costs for asset 1 (three underlying assets)

Asset 1 1 2 3 1 0 30 40 2 10,000 0 8 3 10,000 10,000 0 Table 5.9. Switching costs for asset 2 (three underlying assets)

Asset 2 1 2 3 1 0 25 40 2 -25 0 20 3 -40 15 0 Table 5.10. Switching costs for asset 3 (three underlying assets)

Asset 3 1 2 3 1 0 45 50 2 10,000 0 20 3 10,000 10,000 0 constant over time. Compared with the numerical setting for two underlying assets, the newly introduced intermediate asset 2 incorporates a flexible technology, with lowest investment costs and scaling down possible between all operating modes. The switching costs for the other two assets are identical to the structure analyzed for two underlying assets, with the exception that for asset 1, switching back from mode 2 to mode 1 is not permissible (cf. Sec. 5.1.1). As before, all investment decisions are subject to a dynamic capital constraint. Here we assume B1 = 18, and an allocation of new budget which is constant over time and scenario, β(t, s) = β. In the presented cases, we will discuss two cases for β, i.e., β = 10 and β = 20. With these parameters as the common basis for all portfolios that will be discussed in the following, we can outline the specific portfolio settings. First, a base case is presented for three underlying assets with three different initial values of the underlying assets, i.e., α1 = 20, α2 = 40 and α3 = 50. While asset 1 and asset 3 exhibit relatively low volatility with σ1 = σ3 = 0.3,

5.2 Extension of the Numerical Analysis to Three Underlying Assets

117

Table 5.11. Portfolio examples with 3 underlying assets

Portfolio Base case Correlation Budget Volatility

α1 20 20 20 20

α2 40 40 40 40

α3 50 50 50 50

σ1 0.3 0.3 0.3 0.3

σ2 0.5 0.5 0.5 0.5

σ3 0.3 0.3 0.3 0.7

ρ12 0.8 -0.6 -0.6 -0.6

ρ13 0.8 0.6 0.6 0.6

ρ23 0.8 -0.4 -0.4 -0.4

B1 18 18 18 18

β 10 10 20 20

Value 189.2 205.7 361.3 373.3

asset 2 is more volatile, with σ2 = 0.5. Further, the values of the underlying assets develop in a highly correlated way, with ρ12 = ρ13 = ρ23 = 0.8 (see Table 5.11). This parameter setting is in line with the setting presented for two underlying assets (cf. Sec. 5.1.1). Second, with all else being equal, we focus on correlation effects by changing the correlation structure to ρ12 = −0.6, ρ13 = 0.6, and ρ23 = −0.4. This structure is internally consistent, as formulated in Eq. 5.12. While asset 1 and asset 2 are negatively correlated, asset 1 and asset 3 are positively correlated, which limits the degree to which asset 2 and asset 3 can be negatively correlated in a consistent way. Overall, by this setting we obtain a substantially expanded range of correlation, from negative to positive (see Table 5.11). Third, with all else being equal, we show the impact of an increased availability of funds, by increasing β = 10 to β = 20 (see Table 5.11). Finally, with all else being equal, we highlight volatility effects, by increasing the volatility of asset 2, from σ2 = 0.5 to σ2 = 0.7, thus making asset 2 “riskier” (see Table 5.11). The numerical settings will be discussed for T = 3. Therefore, for n = 3, Ai0 = 3, and T = 3, the number of paths through the tree is given by (cf. also Sec. 5.1.1): Y Y Ai0 ∗ [2n Ai0 ](T −1) = 27 ∗ [8 ∗ 27]2 = 1, 259, 712 paths. (5.6) i

i

With these specifications, the numerical results for three underlying assets will be discussed next. 5.2.2 Correlation and Risk–Neutral Probabilities The joint behavior of n underlying assets is captured by the correlation structure between these underlying assets, which can be represented in a n × n matrix given in Table 5.12, where i, j = 1, . . . , n. This matrix gives the correlation of each pair ij out of all n assets and is thus quadratic. Further, it holds that

118

5 Numerical Analysis Table 5.12. Correlation structure

1 2 3 .. . i

1 ρ11 ρ21 ρ31

2 ρ12 ρ22 ρ32 ..

ρi1

ρi2

3 ρ13 ρ23 ρ33

··· ··· ··· ···

. ρi3

···

ρij = ρ ji

j ρ1j ρ2j ρ3j .. . ρij

, and

ρii = 1.

(5.7) (5.8)

While for two underlying assets there is only one correlation, for three underlying assets correlations for all pairs of underlying assets are to be considered. Therefore, the case for n = 3 captures the general structure of the correlation matrix and can thus be analyzed as representative for n underlying assets. Increasing the number of underlying assets for n > 3 leaves the structure of the correlation matrix unchanged and only increases its columns and rows. Thus, for the case of n = 3 invoking Eq. 5.7 and Eq. 5.8, we have to consider the following three correlations: ρ12

correlation between assets 1 and 2,

(5.9)

ρ13

correlation between assets 1 and 3,

(5.10)

ρ23

correlation between assets 2 and 3.

(5.11)

Correlations need to be internally consistent. If the interplay of asset 1 with asset 2 and 3 is defined, consequently the interplay between assets 2 and 3 loses degrees of freedom because of the given joint relation with asset 1. This consistency is exogenous to our model, i.e., it must be satisfied in the input data to our model, but leaves the model formulation as such unaffected. For three underlying assets (cf. Hull 2003, p. 388), the condition for a matrix of correlations to be internally consistent is: ρ212 + ρ213 + ρ223 − 2ρ12 ρ13 ρ23 ≤ 1.

(5.12)

This correlation structure enters the model through a suitable risk– neutral probability measure which assigns scenarios to joint realizations of the stochastic processes of the underlying assets. For three underlying as-

5.2 Extension of the Numerical Analysis to Three Underlying Assets

119

Table 5.13. Scenarios and risk–neutral probabilities for three underlying assets

Scenario s=1 s=2 s=3 s=4 s=5 s=6 s=7 s=8

Asset 1 up up up up down down down down

Asset 2 up up down down up up down down

Asset 3 Probability up p1 down p2 up p3 down p4 up p5 down p6 up p7 down p8

sets, this translates into a branching of 8 successor nodes from each parent node, as depicted in Table 5.13. As discussed in Sec. 4.4.1 and Sec. 4.2.1, the risk–neutral probabilities for more than two underlying assets need to be formulated for a CRR specification of the binomial model, which we apply here following Boyle et al. (1989). This approach is equivalent to the one proposed by Kamrad and Ritchken (1991) for the stretch parameter used herein set to 1. This specifies the risk–neutral probabilities, as given in Table 5.14 for 3 underlying assets. Based on these definitions, we will investigate the joint behavior of three underlying assets in a portfolio of real options setting next. 5.2.3 Base Case The base case, as introduced above, considers the parameters as summarized in Table 5.15. The optimal strategy is represented in 5.10. For each underlying asset, it represents the entering mode, and the leaving mode, at each state. For instance, at t = 3 in scenario s = 43, it is optimal to switch asset 2 from mode 2 to mode 1, i.e., it is optimal to shut down. In scenario s = 43, the stochastic processes for the underlying assets 1, 2, and 3, have developed with (down, up), (up, down), and (down, up), respectively (cf. Table 5.13). In that setting, it is optimal to install low capacity for asset 2 at t = 1, which is the only feasible investment decision given the budget constraint. At t = 2, the operating strategy remains unchanged, continuing to operate asset 2. At t = 3, the strategy becomes more refined, expressing the competition between assets for limited resources available. For instance, at t = 3 and s = 1, asset 2 is shut down in order to free funds for asset 3, which dominates asset 2. As opposed to this, in s = 2 it is optimal to expand capacity for asset 2 and keep asset 3 uninvested. This is intuitive because in both scenarios

120

5 Numerical Analysis t=1 Asset 1 Asset 2 Asset 3

1→1

1→2

1→1

t=2 Asset 1 Asset 2 Asset 3

t=3 Asset 1 Asset 2 Asset 3

s

1→1

2→2

1→1

1→1 1→1 1→1 1→1 1→1 1→1 1→1 1→1

2→1 2→3 2→1 2→1 2→1 2→3 2→1 2→1

1→2 1→1 1→2 1→2 1→2 1→1 1→2 1→2

s=1 s=2 s=3 s=4 s=5 s=6 s=7 s=8

1→1

2→2

1→1

1→1 1→1 1→1 1→1 1→1 1→1 1→1 1→1

2→3 2→3 2→1 2→2 2→3 2→3 2→1 2→2

1→1 1→1 1→2 1→1 1→1 1→1 1→2 1→1

s=9 s=10 s=11 s=12 s=13 s=14 s=15 s=16

1→1

2→2

1→1

1→1 1→1 1→1 1→1 1→1 1→1 1→1 1→1

2→1 2→1 2→1 2→1 2→1 2→1 2→1 2→1

1→2 1→2 1→2 1→2 1→2 1→2 1→2 1→2

s=17 s=18 s=19 s=20 s=21 s=22 s=23 s=24

1→1

2→2

1→1

1→1 1→1 1→1 1→1 1→1 1→1 1→1 1→1

2→1 2→2 2→1 2→1 2→1 2→2 2→1 2→1

1→2 1→1 1→2 1→2 1→2 1→1 1→2 1→2

s=25 s=26 s=27 s=28 s=29 s=30 s=31 s=32

1→1

2→2

1→1

1→1 1→1 1→1 1→1 1→1 1→1 1→1 1→1

2→1 2→3 2→1 2→1 2→1 2→3 2→1 2→1

1→2 1→1 1→2 1→2 1→2 1→1 1→2 1→2

s=33 s=34 s=35 s=36 s=37 s=38 s=39 s=40

1→1

2→2

1→1

1→1 1→1 1→1 1→1 1→1 1→1 1→1 1→1

2→3 2→3 2→1 2→2 2→3 2→3 2→1 2→2

1→1 1→1 1→2 1→1 1→1 1→1 1→2 1→1

s=41 s=42 s=43 s=44 s=45 s=46 s=47 s=48

1→1

2→2

1→1

1→1 1→1 1→1 1→1 1→1 1→1 1→1 1→1

2→1 2→1 2→1 2→1 2→1 2→1 2→1 2→1

1→2 1→2 1→2 1→2 1→2 1→2 1→2 1→2

s=49 s=50 s=51 s=52 s=53 s=54 s=55 s=56

1→1

2→2

1→1

1→1 1→1 1→1 1→1 1→1 1→1 1→1 1→1

2→1 2→2 2→1 2→1 2→1 2→2 2→1 2→1

1→2 1→1 1→2 1→2 1→2 1→1 1→2 1→2

s=57 s=58 s=59 s=60 s=61 s=62 s=63 s=64

Fig. 5.10. Optimal strategy for three underlying assets - base case

5.2 Extension of the Numerical Analysis to Three Underlying Assets

121

Table 5.14. Specification of risk–neutral probabilities for n = 3 assets (i = 1, 2, 3) in 8 scenarios

p1 18

√ r − 1 σ2 1 + ρ12 + ρ13 + ρ23 + ∆t + f σ 2 1 +

r f − 12 σ22 σ2

p2 18

√ r − 1 σ2 1 + ρ12 − ρ13 − ρ23 + ∆t + f σ 2 1 +

1 8

+

r f − 12 σ32 σ3

r f − 12 σ22 σ2

−

r f − 12 σ32 σ3

√ r − 1 σ2 1 − ρ12 + ρ13 − ρ23 + ∆t + f σ 2 1 −

r f − 12 σ22 σ2

r f − 12 σ32 σ3

+

√ r − 1 σ2 p4 18 1 − ρ12 − ρ13 + ρ23 + ∆t + f σ 2 1 −

r f − 12 σ22 σ2

r f − 12 σ32 σ3

−

√ r − 1 σ2 p5 18 1 − ρ12 − ρ13 + ρ23 + ∆t − f σ 2 1 +

r f − 12 σ22 σ2

r f − 12 σ32 σ3

+

√ r − 1 σ2 1 − ρ12 + ρ13 − ρ23 + ∆t − f σ 2 1 +

r f − 12 σ22 σ2

r f − 12 σ32 σ3

−

√ r − 1 σ2 1 p7 8 1 + ρ12 − ρ13 − ρ23 + ∆t − f σ 2 1 −

r f − 12 σ22 σ2

+

r f − 12 σ32 σ3

√ r − 1 σ2 1 + ρ12 + ρ13 + ρ23 + ∆t − f σ 2 1 −

r f − 12 σ22 σ2

−

r f − 12 σ32 σ3

p3

1

1

1

1

1

p6

1 8

1

1

p8 18

1

Table 5.15. Three underlying assets – base case

Portfolio α1 α2 α3 σ1 σ2 σ3 ρ12 ρ13 ρ23 B1 β Value Base case 20 40 50 0.3 0.5 0.3 0.8 0.8 0.8 18 10 189.2 the value of the underlying asset is the same for asset 2 (up, up), whereas asset 3 jumps up in s = 1 but down in s = 2. The competition for funds is also apparent in scenarios s = 3 and s = 4, where asset 2 has jumped (up, down). Here asset 3 gets invested no matter whether it moves up or down, because it faces no competition any more from asset 2, due to the unfavorable development of the latter. In the other states at t = 3, an analogous pattern can be observed. Asset 1 never gets invested, because given the relative switching costs, expansion factors, and funds available, it is dominated by the other two assets. Assets 2 and 3 compete for the funds available, and contingent on the relative joint stochastic realizations and the path taken there, the one or the other prevails. Given that the operating policy is not reconfigured at t = 2 and thus the operating policy is identical in all states, path dependency does not materialize here. This will change in 5.2.6.

122

5 Numerical Analysis Stochastic Movement s s=1 s=2 s=3 s=4 s=5 s=6 s=7 s=8

Asset 1 Asset 2 Asset 3

up up up up down down down down

up up down down up up down down

up down up down up down up down

Probabilities Base case

Correlation

0.410 0.006 0.048 0.044 0.006 0.002 0.044 0.440

0.060 0.006 0.348 0.094 0.056 0.302 0.044 0.090

Fig. 5.11. Optimal strategy for three underlying assets - changed correlation

Result: The behavior of the base case for three underlying assets is consistent with the discussion for two underlying assets, in Sec. 5.1.1. The optimal strategy realizes an overall portfolio value of 189.2. 5.2.4 Change in Correlation Building up on the base case, we change the correlation structure, with all else being equal, as summarized in Table 5.16. While before all three assets were highly positively correlated, asset 1 is negatively correlated with asset 2 but still positively correlated with asset 3. Asset 2 and asset 3 are also negatively correlated, which satisfies the consistency of the correlation structure. Table 5.16. Three underlying assets – changed correlation

Portfolio α1 α2 α3 σ1 σ2 σ3 ρ12 ρ13 ρ23 B1 β Value Correlation 20 40 50 0.3 0.5 0.3 -0.6 0.6 -0.4 18 10 205.7

The correlation enters the model through the risk–neutral probabilities, as discussed in 5.1.5 and applied to three underlying assets in Sec. 5.2.2. This implies that the correlation does not affect the optimal exercise policy in each specific scenario, but affects the probabilities attributed to that scenario. For the discussed setting, the change in correlation shifts the weights away from scenarios where the negatively correlated assets behave in a likewise manner, and increases the relative weight for scenarios where the negatively correlated assets develop in an opposing direction. For instance, the probability for all assets to jointly jump up, i.e., p1 , drops from 41% to a mere 6%, as depicted in Fig. 5.11.

5.2 Extension of the Numerical Analysis to Three Underlying Assets

123

Result: The findings obtained for two underlying assets are confirmed. In the specific case, we show that reducing the correlation among three underlying assets can increase portfolio value, from 189.2 to 205.7. This again confirms the financial intuition that the overall investment problem follows the structure of a call option (cf. Sec. 5.1.5). As noted before, in other settings a positive correlation effect can exist. 5.2.5 Change in Budget Next, we will develop the problem by allowing for more funds available, by increasing β from 10 to 20 and all else being equal. This case is summarized in Table 5.17. Table 5.17. Three underlying assets – increased budget allowance

Portfolio α1 α2 α3 σ1 σ2 σ3 ρ12 ρ13 ρ23 B1 β Value Budget 20 40 50 0.3 0.5 0.3 -0.6 0.6 -0.4 18 20 361.3

The optimal strategy for this case is depicted in Fig. 5.12. Here, the change in the optimal policy, as compared to the cases discussed before, is indicated by bold boxes. As discussed in the base case (cf. Sec. 5.2.3), asset 3 is more attractive than the other two assets. Here the budget constraint is such that at t = 1, funds are not sufficient for investment in asset 3, such that asset 2 steps in. What changes is that at t = 2, it is possible to install full capacity for asset 3, requiring to free funds by shutting down asset 2. At t = 3, it is optimal to keep on operating asset 3 at full capacity. Further, funds are sufficient to reinvest into asset 2, at low capacity, which is optimal in all scenarios where asset 2 has followed at least one up-jump. Result: The findings obtained for two underlying assets are confirmed, in that a sufficiently large increase in the budget allowance makes it possible to implement a substantially reconfigured operating strategy, realizing higher value. In our case, the portfolio value increases from 205.7 to 361.3. It is noted though that the optimal strategy does not exhibit a wide range of switching decisions, due to the relative attractiveness of asset 3. This will fundamentally change in the following numerical setting. 5.2.6 Change in Volatility Next, the numerical example is modified by considering a higher volatility for asset 3, increased from 0.3 to 0.7, and all else being equal. This case is summarized in Table 5.18.

124

5 Numerical Analysis t=1 Asset 1 Asset 2 Asset 3

1→1

1→2

1→1

t=2 Asset 1 Asset 2 Asset 3

t=3 Asset 1 Asset 2 Asset 3

s

1→1

2→1

1→3

1→1 1→1 1→1 1→1 1→1 1→1 1→1 1→1

1→2 1→2 1→2 1→2 1→2 1→2 1→2 1→2

3→3 3→3 3→3 3→3 3→3 3→3 3→3 3→3

s=1 s=2 s=3 s=4 s=5 s=6 s=7 s=8

1→1

2→1

1→3

1→1 1→1 1→1 1→1 1→1 1→1 1→1 1→1

1→2 1→2 1→2 1→2 1→2 1→2 1→2 1→2

3→3 3→3 3→3 3→3 3→3 3→3 3→3 3→3

s=9 s=10 s=11 s=12 s=13 s=14 s=15 s=16

1→1

2→1

1→3

1→1 1→1 1→1 1→1 1→1 1→1 1→1 1→1

1→2 1→2 1→1 1→1 1→2 1→2 1→1 1→1

3→3 3→3 3→3 3→3 3→3 3→3 3→3 3→3

s=17 s=18 s=19 s=20 s=21 s=22 s=23 s=24

1→1

2→1

1→3

1→1 1→1 1→1 1→1 1→1 1→1 1→1 1→1

1→2 1→2 1→1 1→1 1→2 1→2 1→1 1→1

3→3 3→3 3→3 3→3 3→3 3→3 3→3 3→3

s=25 s=26 s=27 s=28 s=29 s=30 s=31 s=32

1→1

2→1

1→3

1→1 1→1 1→1 1→1 1→1 1→1 1→1 1→1

1→2 1→2 1→2 1→2 1→2 1→2 1→2 1→2

3→3 3→3 3→3 3→3 3→3 3→3 3→3 3→3

s=33 s=34 s=35 s=36 s=37 s=38 s=39 s=40

1→1

2→1

1→3

1→1 1→1 1→1 1→1 1→1 1→1 1→1 1→1

1→2 1→2 1→2 1→2 1→2 1→2 1→2 1→2

3→3 3→3 3→3 3→3 3→3 3→3 3→3 3→3

s=41 s=42 s=43 s=44 s=45 s=46 s=47 s=48

1→1

2→1

1→3

1→1 1→1 1→1 1→1 1→1 1→1 1→1 1→1

1→2 1→2 1→1 1→1 1→2 1→2 1→1 1→1

3→3 3→3 3→3 3→3 3→3 3→3 3→3 3→3

s=49 s=50 s=51 s=52 s=53 s=54 s=55 s=56

1→1

2→1

1→3

1→1 1→1 1→1 1→1 1→1 1→1 1→1 1→1

1→2 1→2 1→1 1→1 1→2 1→2 1→1 1→1

3→3 3→3 3→3 3→3 3→3 3→3 3→3 3→3

s=57 s=58 s=59 s=60 s=61 s=62 s=63 s=64

Fig. 5.12. Optimal strategy for three underlying assets - increased budget allowance

5.2 Extension of the Numerical Analysis to Three Underlying Assets

125

Table 5.18. Three underlying assets – increased budget allowance and increased volatility

Portfolio α1 α2 α3 σ1 σ2 σ3 ρ12 ρ13 ρ23 B1 β Value Volatility 20 40 50 0.3 0.5 0.7 -0.6 0.6 -0.4 18 20 373.3 The optimal strategy for this case is depicted in Fig. 5.13. Again, the change in the optimal policy is indicated by bold boxes. By increasing the volatility of asset 3, the optimal investment at t = 2 is more differentiated. When both asset 2 and asset 3 move up, it is still optimal to shut down asset 2 and invest in full capacity for asset 3. What changes here is that in states where asset 2 moves up and asset 3 moves down, it is more attractive to expand asset 2 and remain uninvested for asset 3, because the latter is adversely affected by the down movement, to a higher extent than before. This is the case for s = 2 and s = 6, where s = 6 is endowed with a relatively high probability, given the correlation structure (cf. Sec. 5.2.4). Given that the strategy only changes in the successor scenarios of these scenarios, it is insightful to consider scenarios s = 9, . . . , 16 and s = 41, . . . , 48, at t = 3. The starting situation for all of these scenarios is the same: asset 2 is operating at full capacity, while asset 3 is still uninvested; asset 2 has followed an up jump while asset 3 has suffered from a down jump. First, consider scenarios s = 41, . . . , 48. Capacity for asset 3 is installed whenever it follows an up jump at t = 3, which is the case in scenarios s = 41, 43, 45, 47. The size of the capacity expansion depends on the relative development of asset 2. On the one hand, when asset 2 moves down, as is the case in scenarios s = 43, 47, it is optimal to install full capacity for asset 3, which is financed by a shut down of asset 2. On the other hand, when asset 2 moves up, as is the case in scenarios s = 41, 45, it is optimal to scale down asset 2 from high to low capacity, and install medium capacity for asset 3. We note that asset 1 does not affect the optimal strategy, since it is never invested. Second, consider scenarios s = 9, . . . , 16. The situation is different here, because asset 1 here follows an up jump, instead of a down jump in scenarios s = 41, . . . , 48. Thus, in scenarios s = 9, . . . , 12, asset 1 has developed with (up, up), i.e., has turned as valuable as possible. In scenario s = 10, it is optimal to install low capacity for asset 1. A comparison with scenario s = 42 shows that the investment into asset 1 leaves the operating strategy for the other assets unaffected. This is different for scenario s = 12, as compared to scenario s = 44. In scenario s = 12, the relative attractiveness of asset 1 causes a scaling down of asset 2, which here was hit by a down movement instead of moving up in scenario s = 10. The funds available through scaling down asset 2 make it possible to install high capacity for asset 1 here.

126

5 Numerical Analysis t=1 Asset 1 Asset 2 Asset 3

1→1

1→2

1→1

t=2 Asset 1 Asset 2 Asset 3

t=3 Asset 1 Asset 2 Asset 3

s

1→1

2→1

1→3

1→1 1→1 1→1 1→1 1→1 1→1 1→1 1→1

1→2 1→2 1→2 1→2 1→2 1→2 1→2 1→2

3→3 3→3 3→3 3→3 3→3 3→3 3→3 3→3

s=1 s=2 s=3 s=4 s=5 s=6 s=7 s=8

1→1

2→3

1→1

1→1 1→2 1→1 1→3 1→1 1→1 1→1 1→1

3→2 3→3 3→1 3→2 3→2 3→3 3→1 3→3

1→2 1→1 1→3 1→1 1→2 1→1 1→3 1→1

s=9 s=10 s=11 s=12 s=13 s=14 s=15 s=16

1→1

2→1

1→3

1→1 1→1 1→1 1→1 1→1 1→1 1→1 1→1

1→2 1→2 1→1 1→1 1→2 1→2 1→1 1→1

3→3 3→3 3→3 3→3 3→3 3→3 3→3 3→3

s=17 s=18 s=19 s=20 s=21 s=22 s=23 s=24

1→1

2→1

1→3

1→1 1→1 1→1 1→1 1→1 1→1 1→1 1→1

1→2 1→2 1→1 1→1 1→2 1→2 1→1 1→1

3→3 3→3 3→3 3→3 3→3 3→3 3→3 3→3

s=25 s=26 s=27 s=28 s=29 s=30 s=31 s=32

1→1

2→1

1→3

1→1 1→1 1→1 1→1 1→1 1→1 1→1 1→1

1→2 1→2 1→2 1→2 1→2 1→2 1→2 1→2

3→3 3→3 3→3 3→3 3→3 3→3 3→3 3→3

s=33 s=34 s=35 s=36 s=37 s=38 s=39 s=40

1→1

2→3

1→1

1→1 1→1 1→1 1→1 1→1 1→1 1→1 1→1

3→2 3→3 3→1 3→3 3→2 3→3 3→1 3→3

1→2 1→1 1→3 1→1 1→2 1→1 1→3 1→1

s=41 s=42 s=43 s=44 s=45 s=46 s=47 s=48

1→1

2→1

1→3

1→1 1→1 1→1 1→1 1→1 1→1 1→1 1→1

1→2 1→2 1→1 1→1 1→2 1→2 1→1 1→1

3→3 3→3 3→3 3→3 3→3 3→3 3→3 3→3

s=49 s=50 s=51 s=52 s=53 s=54 s=55 s=56

1→1

2→1

1→3

1→1 1→1 1→1 1→1 1→1 1→1 1→1 1→1

1→2 1→2 1→1 1→1 1→2 1→2 1→1 1→1

3→3 3→3 3→3 3→3 3→3 3→3 3→3 3→3

s=57 s=58 s=59 s=60 s=61 s=62 s=63 s=64

Fig. 5.13. Optimal strategy for three underlying assets - increased budget allowance and increased volatility

5.3 Numerical Summary Stochastic Movement

127

Optimal Strategy

Asset 1 t=2 t=3

Asset 2 t=2 t=3

s=12

up

up

up

down

down

down

1→3

3→2

1→1

s=26

up

up

down

up

down

down

1→1

1→2

3→3

s

Asset 3 t=2 t=3

Asset 1 Asset 1 Asset 1 t=3 t=3 t=3

Fig. 5.14. Path dependency for three underlying assets

At the same time, putting together these elements illustrates the path– dependencies underlying the portfolio problem. For instance, the optimal strategy differs completely in scenarios s = 12 and s = 26. The only difference here is that in scenario s = 12, asset 2 moves first up then down, and reversely, in scenario s = 26, first down then up. This feature is depicted in Fig. 5.14. Result: The value of the portfolio of real options increases from 361.3 to 373.3. It is observed that the optimal policy has changed significantly. Especially, while we only changed the volatility of one underlying asset, the optimal policy changed for all underlying assets. This underlines the results that all investment decisions interact, and must be assessed simultaneously, considering all investment opportunities jointly.

5.3 Numerical Summary The goal of the numerical analysis was to shed light on the effects encountered in the general model formulation. It was based on a stylized investment problem. The numerical analysis has demonstrated that the assessment of portfolios of real options must pay special attention to budget constraints, volatility effects, and correlation effects. Similarities to complex financial options have been identified, corroborating the results obtained through the presented model. While sometimes not straightforward, the encountered effects could always be explained by financial intuition. The complexity of the joint effects was made accessible through a comparative– static analysis of one parameter at a time. It should be noted though that all parameters interact dynamically and thus must be assessed simultaneously. This is achieved by the proposed model. The overall behavior of the presented investment problem resembles a general pattern of a call option. This is intuitive since it is an investment problem that starts in a not invested mode and corresponding options to invest. More precisely, the problem exhibits analogies to a call option on the maximum of two underlying assets. Another analogy could be drawn

128

5 Numerical Analysis

to barrier options. Both analogies confirm the financial intuition behind the proposed model. While constraints have been discussed for budgets, the proposed model can incorporate any number and type of additional constraints. One example are investments that are mutually exclusive (see Sec. 4.4.4). The effect is very much like the effect of a binding budget constraint: both result in deviations from the optimal unrestricted policy. The constraint prevents from making an investment that is profitable as such and that would be realized in the unrestricted case. This is why it is sufficient here to explicitly discuss budget constraints only, representing other types of constraints. The effects on portfolio value are structurally the same, but budget constraints require the biggest modeling challenge and thus provide the most insights. Result: Concluding, we find that portfolios of real options are subject to the following portfolio effects: 1. Option values typically do not add up and tend to be sub–additive, such that it is not possible to value real options in isolation and consider the value of portfolios of real options as a sum of the parts. 2. Adding to the non–additivity of real options, budget constraints play a major role because they introduce a step–function of portfolio value. Portfolio optimization makes it possible to countervail the adverse budget effects, resulting in a adapted investment and disinvestment strategy. 3. Volatility has an ambiguous effect on portfolio value. Increasing the volatility of an underlying asset can increase the value of a portfolio of real options related to this asset, but in fact can also reduce the value of a portfolio of real options. 4. Portfolio values of portfolio problems, subject to binding budget constraints, can be decreasing in the correlation. Therefore, correlation in the underlying assets and in the cash flows can harm portfolio value. In other portfolio settings without binding budget constraints, correlation can have a positive effect. 5. Portfolio values are increasing in the underlying assets initial values if the problem is a net investment problem, i.e., it shows a net payoff profile equivalent to the payoff of a call option. The main differences between a stand–alone perspective on real options and a portfolio perspective on real options are related to volatility effects, correlation effects, and budget effects, as summarized in Table 5.19. All of these lead to deviations from value additivity, making it impossible to value a portfolio of real options as sum of the parts. Based on these observations, the management of portfolios of real options is a complex task. The proposed model provides a structure capable of handling these complexities. It can be used for two goals: First, designing

5.3 Numerical Summary

129

Table 5.19. Stand–alone vs. portfolio perspective on real options ...lets the option value An increase in ... Stand-alone perspective Volatility

Portfolio perspective

increase

increase / decrease

Correlation

–

(increase /) decrease

Funds available

–

increase as a step–function

an optimal portfolio and second, exercising existing real options optimally. Both can be achieved by solving the model repeatedly for scenarios with different bundles of real options. In the light of portfolio design, options are desirable in the portfolio if (and only if) they increase overall portfolio value. As to exercising existing real options, the model provides clear managerial recommendations since a contingent exercise policy is implicit in the optimal value. Along the same lines, the model can be used to assess different levels of budgets available as a basis for strategic discussion of suitable levels of budget allowances. Presenting the ambiguous nature of portfolio interactions in the context of real options has demonstrated the complexities the model can handle and thus its contribution. While the numerical analysis took place in a rather simple setting to pinpoint specific effects clearly and to keep a sound intuition of the encountered effects, other cases are readily solvable. The flexible nature of the discrete modeling approach as well as the use of advanced optimization algorithms provide a rich framework for handling challenging investment problems. These become less intuitive with increasing complexity since it is harder to extract single effects from an interacting system. Nevertheless, this only affects the specific valuation results but leaves the structural results discussed here unchanged. Concluding, the proposed model formulation provides a powerful approach for the valuation and the management of portfolios of real options, taking into account significant portfolio effects that are otherwise hardly controllable.

6 Conclusion

The main contribution of this dissertation is to provide a rigorous valuation framework for portfolios of real options handling several important portfolio features that have not been covered so far in the literature. Our model is set up as a multi–dimensional real options analysis problem based on n underlying assets and their joint behavior along with embedded real options in a path–dependent setting. The modeling of the underlying assets is flexible and allows for changes in the characteristics of these assets, possibly at the discretion of management. It also has the flexibility to handle any kind of (financial) constraints and desired operational policies as well as investment synergies, asymmetric and time–dependent switching costs and learning effects. It can handle a wide range of intra– and inter–project compoundness of multiple options on multiple underlying assets. The model links dynamic (dis–) investment decisions with a global dynamic budget constraint that is both pivotal for the optimal operating policy and relevant in practical applications. Our portfolio approach incorporates features which are not only relevant for practical applications, but which also represent significant modeling challenges when considered in combination. Through our model implementation, we not only obtained a solution to this complex portfolio problem but also gained new insights into the behavior of portfolios of real options through sensitivity analysis. After summarizing the results, an outlook on managerial implications as well as on further applications and future research, is provided.

6.1 Summary of Results As an introduction, Chap. 1 motivated the portfolio perspective on real options. It is natural to assume that companies have discretionary decision rights about the operation of many physical assets. This introduces an asym-

132

6 Conclusion

metry in decision making which establishes a strong analogy between corporate investment decisions and financial options. Consequently, the application of financial option pricing theory to corporate investment decisions can provide valuable insights. At the same time, the key difference to financial options lies in the underlying asset which is real. The term real options stresses both analogy and difference. While the analogy is a close one, the real nature of the underlying assets gives rise to specific interaction effects. It has been shown that real options on the same underlying asset inherently interact, such that they cannot be valued independently from one another, but need to be modeled in a compound option type approach. If this is true for one underlying asset, it is all the more true for several underlying assets, especially when constraints are being considered. To capture all relevant aspects, portfolios of real options are defined as a combination of multiple underlying assets and multiple real options written on these assets, subject to constraints. Managing such portfolios of real options envisions two dimensions: strategic portfolio design and operational portfolio execution. The proposed model is capable of handling both. In this context, strategic refers to the choice of assets, whereas operational refers to the optimal investment policy. A discussion of the relevant literature, in Chap. 3, has related the present dissertation to prior work and has assessed different strands of literature in order to collect necessary elements for the ensuing model. Traditional mean–variance portfolio analysis has been examined first. It is based on the concept of diversification where risk as measured by volatility is minimized. For a real options analysis, this perspective is not valid since real options provide a way of handling risk other than by statistical diversification. Real options enable to capitalize on the upside of uncertainty and at the same time to truncate on the downside. Moreover, the asymmetry involved in real options fundamentally affects the distribution of returns such that it cannot be described sufficiently by its mean and variance. More advanced portfolio analysis was reviewed next. While the complexity of existing models goes far beyond the traditional approach, nevertheless they can only shed very little light on portfolios of real options, most of all due to the dynamic interaction effects between real options in a portfolio context. Likewise, a discussion of portfolio models including frictions, portfolios with financial options, and risk management evidenced that the models existing in the literature do not provide suitable starting points for the analysis of portfolios of real options. None can readily incorporate the specifics of portfolios of real options. In a next step, the literature on corporate capital budgeting was reviewed from a portfolio point of view. Of particular interest was the analysis of interactions in the mathematical programming literature, which could be gen-

6.1 Summary of Results

133

eralized to direct and indirect qualitative interactions. These interactions were later adopted in the model. Finally, the main basis of the present dissertation is the literature on financial and real options. As for any real options analysis, analogies to financial option pricing could be drawn, in our case to rainbow options on multiple underlying assets such as the option on the maximum of several underlying assets, as well as exotic options such as compound options. This lead to discussing the body of real options literature, covering a wide range of applications. A suitable perspective on real options as a real options switching algorithm could be identified in a general framework of switching among modes. Nevertheless, the rather limited analysis of portfolios of real options in the literature so far has been motivated by specific decision problems where compoundness and interactions arose because of special features of the decision problem. A systematic treatment of the main portfolio effects, including budget constraints, has been lacking so far. This discussion has set the stage for the core of the present dissertation, which is the real options portfolio model provided in Chap. 4. For a sound representation, it started with a description of the relevant model features. These include the objective function, the managerial actions permitted, the attitude towards risk, the modeling of the underlying stochastic processes, volatility and correlation, inter– and intra–project interactions, global dynamic budgets, synergies and operational constraints, learning, diversification, path–dependencies, as well as a discussion of the solution approach and the scalability of the model. A key feature of the model is the dynamic global budget equation which has the form of a balance equation. Funds that are not spent can be invested in later periods and funds freed up from disinvestment can be plowed back. This is completed by an initial endowment in funds and an additional allocation of new funds per period. Together with the interdependencies of investment decisions over time, this gave rise to path–dependencies. The interdependency between decisions across time makes it necessary to calculate mode–specific expected values in a backward–looking perspective. At the same time, the budget balance requires a forward–looking perspective. These interact simultaneously, which is impossible to model through standard dynamic programming. This is why we proposed a general formulation as a stochastic mixed–integer program that captures path–dependent and multi–dimensional real options settings. The program can be solved numerically, which we did by an implementation under GAMS© and the help of the commercial solvers BARON© and Cplex© . The mixed–integer nature of the problem and the exponential growth of the system state space present a challenging numerical problem. We could alleviate the complexity by linearizing the problem implementation which makes it possible to ensure global optimality. Nevertheless, the

134

6 Conclusion

challenge remains in the curse of dimensionality of such a problem. This is why possible ways for improving numerical efficiency were discussed. This general development and discussion of the model was then illustrated, in Chap. 5, with the help of a numerical application. The complex landscape that is spanned by portfolios of real options was made accessible by a comparative–static analysis, tracing each effect by a specific (counter) example. This allowed to provide new insights into portfolio effects which were made accessible through the discussion of state–dependent investment decisions for specific examples. The revealed effects provide new insights for portfolios of real options and are contrary to effects one may infer from plain–vanilla option features. At the same time, all effects could be traced back to financial intuition. The discussion of value drivers for portfolios of real options focused on budget effects, volatility effects, correlation effects, starting value effects, as well as path–dependencies and deviations from additivity. First, a binding budget constraint plays an important role for understanding the optimal management of portfolios of real options. Binding budget constraints reduce the value of portfolios of real options, in a non–trivial way in that the optimal policy countervails the adverse effects. For cases involving little sensitivity of portfolio value with regards to changing budget constraints, fundamentally modified investment policies may be needed since parallel switching in and out of assets takes place. Reversely, stronger sensitivities may be simpler to track because they may include an additional investment while leaving the operating policy otherwise unchanged. Moreover, it is possible that different portfolios are optimal for different budget levels and that the relative attractiveness of portfolios are reversed. Generally, portfolios show a high sensitivity to changes in the budget constraint. Thus, it is clearly inadequate to abstract from constraints if they exist in the real world. Ignoring them in the investment analysis forgoes some of the value potential of portfolios of real options, or in the extreme case, can lead to plainly sub–optimal investment strategies. It invalidates the abstraction from constraints typically made in corporate finance. Second, a non-monotonic effect of volatility under budget constraints with disinvestment options was shown. This means that an increase of volatility can reduce portfolio value. Interestingly, this effect is ambiguous in that an increase in volatility can either increase or decrease portfolio value. In general, it cannot be specified without formal analysis whether a change in volatility will raise or reduce the value of a portfolio. This runs counter to standard features of financial options. Third, for an investment problem that has basically a call option structure, correlation of the underlying assets can harm portfolio value. In other terms, it is desirable to look for less correlated, or even negatively correlated ventures. It was noted, though, that in other settings without budget con-

6.2 Managerial Implications

135

straints, correlation can have a positive portfolio effect. Further, the call option structure translates into portfolio value increasing the more valuable the underlying assets are, as measured by their starting values. All of these features contribute to the fact that portfolios of real options generally show deviations from value-additivity and tend to be sub–additive. Based on these observations, the management of portfolios of real options is a complex task and requires detailed analysis in order to fully unfold the value that can be distilled from portfolios of real options. The proposed model provides a structure capable of handling these complexities, both for the design of an optimal portfolio and for the execution of existing real options. Hopefully this can contribute to both theory and practice. Based on these results, key managerial implications and room for further applications and future research will be discussed in the following.

6.2 Managerial Implications From a managerial point of view, the real options approach has often been criticized for its apparent complexity. Based on financial options pricing, it inherently models asymmetric decisions and relies on distributions of returns instead of point estimates. Analyzing options portfolios presumes a sound understanding of option pricing techniques, and hence involves a strong modeling aspect. For this reason, it is not trivial to gain acceptance of real option valuation models in practice, particularly when they involve portfolios of options. Naturally, it is important to discuss and structure the option valuation model with line managers. However, it is generally hard to communicate the valuation framework in detail for two reasons: it presumes a detailed understanding of the underlying option valuation concepts, and the nature of risk and uncertainty for portfolios of real options is different from that of stocks or primitive underlying assets. Nonetheless, the complexity of our model unavoidably reflects the inherent complexity of the underlying problem and the challenges embedded therein. Our model also has a key advantage. In view of the flexibility it provides for modeling specific investment problems, it can tackle actual decision problems in a tailor-made fashion. This should be helpful in making the essence of the model more transparent in the light of the specific application at hand. With a sound understanding of the investment structure, a line manager will be more inclined to accept the model. If the decision problem itself exhibits strong optionality and flexibility that can be well captured by the model, the model will be more compelling and more easily accepted. In our discussions with practitioners we experienced that the model was well received, as they were more involved with a concrete application.

136

6 Conclusion

The most convincing argument for using the model is that it can significantly improve corporate portfolio management. The model application and sensitivity analysis of results made it clear that managing portfolios of real options is a complex task with many interacting effects. In dynamic environments with a high degree of flexibility and uncertainty, practitioners understand perfectly well that it is unwise to ignore these interactions and that these need to be embraced proactively. Our model can be of significant value to managers and academics alike in facing this admittedly complex and difficult task.

6.3 Further Applications and Future Research The present dissertation has introduced a framework that can serve as a stepping stone for future work on portfolios of real options. Focusing on the setup of the model, some insightful questions are left for future research. These can best be pointed out by distinguishing possible ensuing questions in theory and in practice. From an academic standpoint, it would be valuable to enhance the mathematical efficiency of the numerical implementation. This dissertation focuses on formulating the general portfolio problem and provides a feasible numerical implementation that can solve the problem in view of illustrating the relevant portfolio problem and the impact of different value drivers. It did not focus on the efficiency of the implementation. The discussion of possible ways of improving the implementation showed further potential of a more efficient implementation, which is beyond the scope of this dissertation. In that sense, the goal of the implementation was chiefly conceptual. When solving real-life cases, increasing the efficiency is vital in a setting that is naturally prone the “curse of dimensionality”. Increasing the number of underlying assets, the number of options, the horizon, and the number of time steps evidently makes the model more realistic, yet increases the dimensionality of the problem. Second, the modeling approach chosen here and its implementation – based on an optimization platform that is more common in engineering than in finance – has hardly been applied to real options. It is possible to extend the presented model into many of the directions that were excluded by assumption in this dissertation. Especially, an extension towards real options and games, agency conflicts, incomplete markets and fuzziness of input parameters, and non-stationarity of parameters appears as promising. The flexibility of the model and its implementation via an optimization platform are such that, presumably, many additional effects can be modeled. From a practical standpoint, first it would be desirable to further adapt the model to different types of industries, i.e., tailor it to the specific needs

6.3 Further Applications and Future Research

137

and aspects that are most relevant. This would establish a toolbox readily available for different types of problems. Second, it would be desirable to apply the model to different cases in order to check the insights obtained and the reliability of the model. Third, once the model is confirmed it would have to become more user friendly with a convenient user interface before it can be used on a regular basis. Finally, the nature of the model is naturally quantitative and goes into deep modeling detail. At the same time, the problem of managing portfolios of real options, at its core, is fundamentally strategic. It doubtlessly can be put to good use in strategic portfolio management. For this, it would be necessary to integrate the model consistently into frameworks that are currently used for the assessment of strategic portfolios.

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Index

applications manufacturing, 2, 39, 40, 53, 94 R&D, 38, 52 supply chain management, 2, 32, 39 budget constraint balance equation, 18, 51, 55, 106, 109, 133 hard rationing, 35 modeling, 52, 54 soft rationing, 35 capital budgeting decision tree analysis, 33 definition, 32 interactions, 34 knapsack, 33 mathematical programming, 32 portfolio perspective, 33, 34 synergies, 35 financial options American Option, 8 barrier option, 107, 128 call option, 8 compound option, 11, 14, 31, 37, 43, 49, 50, 62, 114, 132, 133 definition, 2, 8 European option, 8 option on the maximum, 31, 36, 61, 64, 98, 109, 110, 127, 133 financial portfolio theory APT, 27

CAPM, 25 diversification, 23 efficient frontier, 23 frictions, 29 higher moments, 28 ICAPM, 26 Markowitz, 22 minimum variance portfolio, 23 portfolio selection, 22 interactions inter–project, 12, 14, 17, 38, 39, 50, 131, 133 intra–project, 12, 14, 17, 38, 50, 131, 133 model formulation change of underlying, 4, 10, 17, 80, 131 correlation, 14, 16, 24, 30, 32, 33, 38, 48, 59, 109, 114, 117, 118, 122, 125, 128 forward–backward looking, 17, 42, 55, 68, 75, 133 lattice tree, 42, 48, 51, 56, 59, 64, 66, 71, 79, 88 overview, 16 path dependency, 14, 15, 17, 39, 52, 53, 66, 86, 90, 91, 93, 107, 112, 127, 133 risk–neutral valuation, 33, 47, 54, 56, 64, 71, 75, 78, 83, 110, 118, 122

152

Index

stochastic process, 9, 40, 48, 54, 56, 66, 75, 80, 83, 84, 95, 133 stochastic program, 4, 17, 39, 41, 48, 54, 60, 63, 66, 71, 75, 84, 133 model implementation efficiency, 85, 87, 136 GAMS, 17, 55, 85, 133 linearization, 85, 134 optimality, 55, 63, 86, 87, 133 option pricing redundant, 29 replication, 21, 22, 29, 52, 84 self–financing, 29, 52 portfolios of real options definition, 12

design, 15 execution, 15 portfolio aspects, 14 portfolio management, 6, 15, 51, 53, 106, 109, 128, 134–137 real options definition, 1, 7 interdependencies, 2 parameters, 9 types, 9 recursion, 39, 55, 58, 59, 63, 65 strategic management, 2, 15, 30, 129, 132, 137 utility, 23, 26, 28

List of Tables

5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 5.11 5.12 5.13 5.14 5.15 5.16 5.17 5.18 5.19

Capacity factors for two underlying assets . . . . . . . . . . . . . . . . . . . Switching costs for asset 1 (two underlying assets) . . . . . . . . . . . Switching costs for asset 2 (two underlying assets) . . . . . . . . . . . Portfolio highlighting budget effects . . . . . . . . . . . . . . . . . . . . . . . . Portfolio with value decreasing in volatility . . . . . . . . . . . . . . . . . Portfolio highlighting path–dependency for T=3 . . . . . . . . . . . . . Capacity factors for three underlying assets . . . . . . . . . . . . . . . . . Switching costs for asset 1 (three underlying assets) . . . . . . . . . . Switching costs for asset 2 (three underlying assets) . . . . . . . . . . Switching costs for asset 3 (three underlying assets) . . . . . . . . . . Portfolio examples with 3 underlying assets . . . . . . . . . . . . . . . . . Correlation structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Scenarios and risk–neutral probabilities for three underlying assets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Specification of risk–neutral probabilities for n = 3 assets (i = 1, 2, 3) in 8 scenarios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Three underlying assets – base case . . . . . . . . . . . . . . . . . . . . . . . . . Three underlying assets – changed correlation . . . . . . . . . . . . . . . Three underlying assets – increased budget allowance . . . . . . . Three underlying assets – increased budget allowance and increased volatility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stand–alone vs. portfolio perspective on real options . . . . . . . . .

95 97 97 100 107 113 116 116 116 116 117 118 119 121 121 122 123 125 129

List of Figures

2.1 2.2 2.3

Payoff structure of a call option . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Portfolio design process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Modeling approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8 16 18

3.1

Mean–variance efficient portfolios . . . . . . . . . . . . . . . . . . . . . . . . . .

24

4.1 4.2 4.3

Dynamic budgets depending on time, scenarios, and actions . Recombining binomial lattice tree for one underlying asset . . . Recombining binomial lattice tree for two underlying assets, with ρ12 6= 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Three–dimensional risk–neutral probabilities . . . . . . . . . . . . . . . . Non–recombining binomial lattice tree for one underlying asset Indexing in a three–dimensional lattice tree . . . . . . . . . . . . . . . . . . Change of the underlying asset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Change of the underlying asset in the multinomial grid . . . . . . Dimensionality for two underlying assets . . . . . . . . . . . . . . . . . . .

51 58

Portfolio landscape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Portfolio value as a step–function of budget allowance . . . . . . . Small change in portfolio value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Substantial change in portfolio value . . . . . . . . . . . . . . . . . . . . . . . . Comparison of optimal investment strategies . . . . . . . . . . . . . . . . Portfolio value decreasing in σ1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Portfolio value decreasing in ρ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Portfolio value increasing in α1 and α2 . . . . . . . . . . . . . . . . . . . . . . Path–dependency of underlying asset 1 . . . . . . . . . . . . . . . . . . . . . Optimal strategy for three underlying assets - base case . . . . . . Optimal strategy for three underlying assets - changed correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

99 100 101 102 103 108 110 112 113 120

4.4 4.5 4.6 4.7 4.8 4.9 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 5.11

59 60 66 72 80 81 88

122

156

List of Figures

5.12 Optimal strategy for three underlying assets - increased budget allowance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 5.13 Optimal strategy for three underlying assets - increased budget allowance and increased volatility . . . . . . . . . . . . . . . . . . . 126 5.14 Path dependency for three underlying assets . . . . . . . . . . . . . . . . 127

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