Portforlios of real options

exempli gratia for GAMS© General Algebraic Modeling System ICAPM Intertemporal Capital Asset Pricing Model max[·] Maximum operator NPV Net Present Value φ Indicator function for correla-

and Mathematical Systems

Rainer Brosch

Portfolios of Real Options

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To my sister Christina

Valuing portfolios of options embedded in investment decisions is arguablyone of the most important and challenging problems in real options andcorporate finance in general Although the problem is common and vitallyimportant in the value creation process of almost any corporation, it hasnot yet been satisfactorily addressed It is key for any corporation facingstrategic resource allocation decisions, be it a pharmaceutical firm valuingand managing its pipeline of drugs, a telecom company having to select

a set of technological alternatives, a venture capital or private equity firminvesting in a portfolio of ventures, or any company allocating resources.Portfolios of real options typically interact such that the value of thewhole differs from the sum of the separate parts Thus one must addressand value the particular configuration of options embedded in a specificsituation, taking into account the configuration of other options alreadypresent in the portfolio, which in turn depends on the correlation struc-ture among the various underlying assets and the strategic dependenciesamong the options themselves (e.g., mutual exclusivity, strategic additiv-ity, compoundness, complementarity etc.) In that sense, optimal decisionsalso depend on past option exercise decisions by management and organi-zational capabilities put in place in the past The optimal decision today as

to whether to continue operating in the current mode or following the rent strategy versus switching to the best among a set of alternative modes

cur-or strategies depends in part on the asymmetric costs of switching gies and potentially the costs and probability of switching back in the fu-ture Such asymmetric costs of switching and re-switching contribute to thebreakdown of the standard value additivity That is, the overall problem ispath dependent and history matters!

strate-In a practical setting, organizations oftentimes face internal budgetary,human capital and other organizational constraints From a portfolio per-spective, any investment or financial resource commitment explicitly com-

Mr Brosch formulates his model as a constrained stochastic dynamicprogram that he subsequently solves He then employs extensive numericalanalysis and provides sensitivity results to investigate various portfolio ef-fects He confirms hypothesized deviations from value additivity and thatthe management of portfolios of real options is a complex task that requirescareful, detailed, situation-specific analysis in order to properly capture thevalue that can be distilled from managing portfolios of real options Hismodel helps provide a general structure capable of handling these real-lifecomplexities, both for the design of an optimal portfolio as well as for theexecution or optimal exercise of the embedded options The model prop-erly reflects the inherent complexity of the underlying real-life constrainedproblem and the non-additivity and path-dependent challenges embeddedtherein.

The framework developed by Mr Brosch makes an important cal contribution in addressing this important and intellectually challengingportfolio problem, while at the same time it can be of significant value topracticing managers in facing this admittedly complex and difficult practicaltask of evaluating, managing and optimally exercising the set of interdepen-dent corporate real options

Ever since I was a little boy, I was fascinated by investment decisions – notwith financial assets in mind, but trucks The bigger, the better However,once it occurred to me that beyond sheer size, equipment with a broaderapplication range may turn out to be more desirable Many years later, tak-ing a Master’s degree in Finance at Frankfurt University, I got hold of acopy of Lenos Trigeorgis’s book “Real Options” It was all there: investmentand flexibility, which is at the core of real options However, I didn’t under-stand how this would work for portfolios So when I met Lenos at the RealOptions Conference and asked, he just smiled: “An interesting field – let’swork on it!”

Now the work is ready and I hope it can be instrumental to managersand academics alike when dealing with portfolios of real options It wassubmitted as doctoral thesis at WHU, Otto Beisheim School of Manage-ment, chaired by Professor Dr Arnd Huchzermeier and Professor LenosTrigeorgis, Ph.D

I would like to thank Arnd Huchzermeier for his guidance and the spiring environment he creates at the Department of Production Manage-ment, including my opportunity to teach and to visit Stanford University I

in-am grateful to Lenos Trigeorgis who was always close despite the distance,

by email or unforgettable meetings, and added both his great esprit and pertise I am thankful to Stefan Spinler, for always being happy to discussand help Furthermore my thank goes to a friend at a Hightech Company,preferring to remain anonymous, who greatly contributed by discussing theapplications I thank my colleagues at the department for making my time

ex-at WHU an enjoyable one Also I am grex-ateful to Christian Artmann andRolf Hellermann for tremendous intellectual and personal support, and forbeing a pleasure to be with far beyond research Finally, I thank The BostonConsulting Group for financial support

X Preface

The most thankful I am to my parents, Arnold and Barbara, and mygodmother Margret, who created many growth options for my educationand 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 achallenging but inspiring and rewarding journey The journey goes on and

I am looking forward to it

1 Introduction 1

1.1 Research Questions 3

1.2 Contribution and Main Results 3

1.3 Structure 5

2 Portfolio Approach to Real Options 7

2.1 Motivation of the Portfolio Approach to Real Options 7

2.1.1 Real Options and Financial Options 7

2.1.2 Definition of Portfolios of Real Options 11

2.1.3 Difference Between Financial Portfolios and Portfolios of Real Options 12

2.2 Implications for Modeling Approach 13

2.2.1 Portfolio Aspects 14

2.2.2 Management of Portfolios of Real Options 15

2.2.3 Translation into Model Features 16

3 Literature Review 21

3.1 Financial Portfolio Theory 21

3.1.1 Mean–Variance Portfolio Analysis 22

3.1.1.1 Traditional Markowitz Portfolio Selection 22

3.1.1.2 Capital Asset Pricing Model 25

3.1.1.3 Intertemporal Capital Asset Pricing Model and Consumption 26

3.1.1.4 Arbitrage Pricing Theory 27

3.1.1.5 Conclusion on Mean–Variance Portfolio Analysis 27

3.1.2 Advanced Financial Portfolio Analysis 28

3.1.2.1 Portfolio Analysis Considering Higher Moments of Distributions 28

XII Contents

3.1.2.2 Allowing for Frictions in the Management of

Portfolios 29

3.1.2.3 Impact of Combining Financial Assets with Financial Options 29

3.1.2.4 Portfolio Implications for Risk Management 31 3.1.2.5 Conclusion on Advanced Financial Portfolio Analysis 31

3.2 Corporate Capital Budgeting 32

3.2.1 Overview 32

3.2.2 Portfolio Perspective in Corporate Capital Budgeting 33 3.2.3 Interdependencies 34

3.2.4 Conclusion on Corporate Capital Budgeting 36

3.3 Financial and Real Options Theory 36

3.3.1 Foundations of Real Options 36

3.3.2 Real Options Analysis in a Portfolio Context 37

3.3.3 Detailed Review of Related Papers 40

3.3.3.1 Kulatilaka (1988) 40

3.3.3.2 Kulatilaka and Trigeorgis (1994) 42

3.3.4 Conclusion on Financial and Real Options Theory 44

4 Valuation Model for Portfolios of Real Options 45

4.1 Model Features 45

4.1.1 General Assumptions 45

4.1.2 Objective Function 46

4.1.3 Managerial Actions Permitted 46

4.1.4 Risk Preference 47

4.1.5 Stochastic Processes 48

4.1.6 Volatility and Correlation 48

4.1.7 Inter– and Intra–Project Options Interactions 49

4.1.8 Global Dynamic Budgets 50

4.1.9 Synergies and Operational Constraints 52

4.1.10 Learning 52

4.1.11 Diversification as a Passive Approach to Risk Management 53

4.1.12 Path–Dependencies 53

4.1.13 Modeling Approach and Solution 54

4.1.14 Scalability 56

4.2 Binomial Option Pricing 56

4.2.1 Stochastic Processes and System State Space 56

4.2.2 Binomial Option Pricing Model with One Underlying Asset 57

4.2.3 Binomial Option Pricing Model with Two Underlying Assets 58

4.3 Real Options Portfolio Model 61

4.3.1 Switching Between Modes 61

4.3.2 Switching Formulation for One Underlying Asset 63

4.3.3 Switching Formulation for Two Underlying Assets 64

4.3.4 Stochastic Mixed–Integer Program Formulation 65

4.3.4.1 One Underlying Asset 66

4.3.4.2 Two Underlying Assets 71

4.3.4.3 n–Dimensional Model Formulation 75

4.4 Model Extensions 78

4.4.1 Alternative Probability Formulation 78

4.4.2 Change of the Underlying Stochastic Processes 80

4.4.3 Explicit Modeling of Synergies 81

4.4.4 Operating and Other Constraints 82

4.4.5 Terminal Values 82

4.4.6 Further Generalizations of the Model Formulation 83

4.5 Model Discussion 83

4.5.1 Applicability of Risk–Neutral Valuation 83

4.5.2 Numerical Efficiency 84

4.5.2.1 Optimality of Solutions and Linearization 84

4.5.2.2 Dimensionality of the Problem 86

4.5.2.3 Increasing Efficiency of the Implementation 87

4.5.3 Monte Carlo Simulation 90

5 Numerical Analysis 93

5.1 Numerical Analysis for Two Underlying Assets 94

5.1.1 Numerical Application Setting for Two Underlying Assets 94

5.1.2 Overall Portfolio Effects 97

5.1.3 Budget Effects and Other Constraints 98

5.1.3.1 One Portfolio 100

5.1.3.2 Relative Attractiveness of Two Portfolios 103

5.1.3.3 Generalization of Budget Effects 105

5.1.4 Volatility Effects 106

5.1.5 Correlation Effects 109

5.1.6 Starting Value of the Underlying Assets Effects 111

5.1.7 Path–Dependency Effects 112

5.1.8 Deviations from Value Additivity 114

5.2 Extension of the Numerical Analysis to Three Underlying Assets 115

5.2.1 Numerical Application Setting for Three Underlying Assets 115

5.2.2 Correlation and Risk–Neutral Probabilities 117

5.2.3 Base Case 119

XIV Contents

5.2.4 Change in Correlation 122

5.2.5 Change in Budget 123

5.2.6 Change in Volatility 123

5.3 Numerical Summary 127

6 Conclusion 131

6.1 Summary of Results 131

6.2 Managerial Implications 135

6.3 Further Applications and Future Research 136

References 139

Index 151

αi Starting value of

Naviga-Werke

ci(·, a0i) Cash flow function for

operating underlyingasset i in mode a0iCAPM Capital Asset Pricing

ModelCCAPM Consumption–Based

Capital Asset PricingModel

cf confer (compare)Cplex© Optimizer by ILOG,

DCF Discounted Cash Flow

E (·) Expected value

XVI List of Abbreviations, Variables, and Functions

e.g exempli gratia (for

GAMS© General Algebraic

Modeling System

ICAPM Intertemporal Capital

Asset Pricing Model

max[·] Maximum operator

NPV Net Present Value

φ Indicator function for

correla-tion between

underly-ing assets i and j

rf Risk-free interest rate

R(·) Binary switching

vari-ableR&D Research and Develop-

X Exercise price of a

financial option

Corporate investment opportunities can be viewed as options, or more cisely, “real options”(Myers 1977) An option gives its holder the right, butnot the obligation, to take a specific action in the future (Amram and Ku-latilaka 1999, p 5) This right is valuable if there is uncertainty about thefuture development of an underlying variable Due to the discretionary de-cision right involved, options are asymmetric contracts: if and only if optionexercise is valuable, the decision maker will exercise and realize value, oth-erwise, the option will expire worthless (Trigeorgis 1996, p 4) In order todistinguish corporate investment decisions, or options, from financial op-tions, it is helpful to use the term “real options” if the underlying asset of

pre-an option is a real asset (Dixit pre-and Pindyck 1994, p 7)

With a close analogy to financial options, corporate decision making isnaturally discretionary and asymmetric because management has future de-cision rights about the use of corporate resources, such as financial resources

or assets in place These future decisions are made contingent on how futureuncertainty resolves While it may be reasonable to invest today based onavailable information, it may be wiser to expand or abandon the investment

if the environment should develop more or less favorably Real options ory provides a rigorous modeling framework allowing to assess the flexibil-ity inherent in real investment decisions, and to derive optimal investmentpolicies In that sense, real options theory enables to assess corporate in-vestment problems that could not be fully captured before (cf Trigeorgisand Mason 1987, p 16) Therefore, it is insightful to apply well establishedconcepts from financial option pricing to corporate finance, which is doneincreasingly in practice (Graham and Harvey 2001)

the-Although the analogy between financial and real options is a close one,there is a major difference While financial options are derivative instru-ments in the sense that the underlying asset and other options are typicallyunaffected by a particular option exercise, this is generally not the case for

The interdependency between real options on the same underlying set is well known academically, but the BMW example also gives some in-tuition about the complexity involved, making practical implementation ofreal options in a rigorous portfolio context challenging (cf Trigeorgis 1993a).Additional complexity stems from the realization that the underlying vari-ables can be affected through active management by the firm as managerscan influence the risk profiles of their firms.

as-When multiple real options on multiple underlying assets are ered, the interactions in the company’s portfolio increase This aspect hasnot been adequately covered in the real options literature, but is of highrelevance both for academia and in practical applications (cf Triantis 2005,

consid-p 15) The practical relevance of such a portfolio problem can be be trated through Toyota’s challenge of actively managing the demand chain

illus-of two product lines in the Japanese and North American markets, facingboth demand risk and exchange rate risk, by aligning the capacity of itsglobal 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 switchingoptions between different manufacturing strategies contingent on exchangerate realizations (for a detailed review, see Kouvelis et al 2006)

From a strategic management perspective, a primary objective is to sign the firm’s strategic portfolio and dynamically manage this portfolio.Managing a portfolio of real options is what strategy is all about A busi-ness strategy is more like a series of options than a series of static cashflows (cf Luehrman 1998, p 90) More broadly, managing a portfolio of realoptions is both strategic and operational In this context, strategy refers tothe choice of assets, whereas operational refers to the optimal investmentpolicy for a given set of assets Both strategic and operational real optionsmanagement need to account for interaction effects

de-The interdependencies between real options must be properly taken intoaccount to fully capture realistic investment decision problems when com-panies are faced with multiple investment or disinvestment opportunitiesunder budget or other constraints (Trigeorgis 2005, p 50) Budget constraintsdeserve special attention since exercising options implicitly assumes that

funds are readily available In practice this may not be the case either cause of internal (divisional) guidelines or limited access to capital mar-kets (cf Brealey and Myers 1996, pp 101 ff.) Constraints may also resultfrom logical aspects of the investment program, such as mutual exclusivityamong projects (Brown and Davis 1998) While constraints are not consid-ered in standard financial option pricing models, they affect optimal deci-sion making if the existence of binding constraints significantly influencesthe optimal investment strategy For instance, it may sometimes be desir-able to sell a faltering or even a profitable business unit to free up (capital)resources for another investment that is even more profitable.

iden-The present dissertation develops a modeling approach for dynamic vestment problems where limited resources are allocated to interacting riskyprojects over time, subject to constraints It sheds light on the following re-search questions:

in-• Which aspects need to be considered for the analysis of portfolios of realoptions?

• How can the relevant aspects be captured quantitatively in a suitablemodeling 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 hasbeen lacking in the literature so far This dissertation aims at closing thegap, providing a framework that makes it possible to optimize investmentdecision 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 optionsprovides 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 realoptions valuation framework enabling to optimize such portfolio decisionsinvolving dynamic investment problems with limited resources allocated

to interacting risky projects over time, subject to various constraints Weset up our portfolio valuation framework as a multi–dimensional real op-tions problem involving path–dependent (dis–) investment decisions Ourapproach can handle compoundness of multiple options on multiple under-lying assets, correlation and changes in the characteristics of the underly-ings, investment synergies, asymmetric switching costs, learning effects, adynamic budget (with or without borrowing) and other types of operationaland financial constraints One of the advantages is that this approach can ac-count for changes in the properties of the underlying variable, e.g., changes

in the volatility, or more generally, changes of the underlying stochasticprocesses 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 mulated as a stochastic program that we solve by an algorithmic solutionimplementation on a commercially available programming platform.Second, we use our solution implementation to numerically provide newinsights into portfolio effects These insights have not been presented before

for-in the context of real options Especially, we ffor-ind that budget constrafor-intsinfluence portfolio value in a complex way, potentially causing the optimalinvestment policy to change profoundly for relatively small changes in theavailability of funds As a consequence, for different levels of the overallbudget, different portfolios may be optimal Given the existence of bindingbudget constraints, volatility has an ambiguous effect on portfolio value.While for some settings volatility increases portfolio value, as usually is thecase for financial options, it can also have an adverse effect on portfoliovalue for other settings Both of these effects are new in that they had notbeen identified in the context of portfolios of real options before Many

of the discussed effects present counter examples to prevailing thinking infinancial option pricing and thus generate new insights We find that:

• Option values typically do not add up and tend to be sub–additive, suchthat it is not possible to value real options in isolation and consider thevalue 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 bud-get effects, resulting in a adapted investment and disinvestment strategy

• Volatility has an ambiguous effect on portfolio value Increasing thevolatility 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 inthe correlation Therefore, correlation in the underlying assets and in thecash flows can harm portfolio value

• Portfolio values are increasing in the underlying assets initial values ifthe problem is a net investment problem, i.e., it shows a net payoff profileequivalent to the payoff of a call option

The discussion of these effects shows the complexity inherent to ing portfolios of real options and the challenges involved Different sources

manag-of interaction and resulting ambiguity in parameters are such that the namics of a specific investment problem cannot be specified without formalanalysis, but are specific to the investment problem at hand The proposedmodel provides a framework that can help to meet these challenges By en-hancing corporate investment decision making through dynamic portfolioanalysis, it fills a gap in a key area for both theory and practice

dy-1.3 Structure

The structure of the subsequent chapters is as follows Chap 2 motivates theportfolio perspective on real options For this, it introduces the concept ofreal options and defines portfolios of real options, which are unlike portfo-lios of financial assets or financial options The notion of portfolio aspects isclarified Based on the presented understanding of portfolios of real options,the issue of valuing and managing such portfolios is highlighted With thisbackground, an overview of the chosen modeling approach is given.Chap 3 reviews the relevant strands of the literature which cover fi-nancial portfolio theory, corporate capital budgeting, and financial and realoptions theory This review identifies the key concepts that underly the fol-lowing analyses

This sets the stage for the core of the dissertation which is the opment of the general model formulation in Chap 4 The model devel-opment starts with a discussion of the key model features that are to beincluded Then, the model is built up successively by increasing the scope

devel-of the model in stages It begins by introducing a binomial option pricingmodel with one time dimension and one underlying asset dimension Then,this is expanded to a multinomial option pricing model with two underly-ing 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 possibleextensions

Chap 5 presents comparative statics of the presented model, based onthe numerical analysis of specific investment setting examples It discussesthe 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 portfolioanalysis It will be shown that the management of portfolios of real optionsnavigates in a complex system state space where most of the value driversare ambiguous It is therefore not possible to specify ex ante, i.e., before aspecific portfolio optimization, whether a change in a certain parameter willincrease or decrease portfolio value This runs counter to the basic intuitionfor standard financial options

Finally, Chap 6 summarizes the main results, discusses key tions alongside with possible limitations, and gives an outlook on furtherapplications and future research

contribu-Portfolio Approach to Real Options

A portfolio perspective on real options makes it possible to gain insightsinto the interplay of real options that could not be captured in a stand–alone analysis It is well established in financial theory that financial assetsmust be valued from a portfolio perspective (cf., e.g., Constantinides 1989)

A review of financial portfolio theory is provided in Sec 3.1 Similarly tofinancial portfolios, portfolios of real options are affected by numerous as-pects, which will be developed in the following In this chapter, the portfolioperspective will be introduced in detail, and then linked to the ensuing gen-eral 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 contextderives from the specifics of real options This is why this motivation isbased on a focused introduction of the general concept of real options, andthe 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 “realoptions [ ] are opportunities to purchase real assets on possibly favorableterms” (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 makingdecisions about real assets These decisions can involve adoption, abandon-ment, exchange of one asset for another or modification of the operatingcharacteristics of an existing asset” An introduction to real options can befound in Dixit and Pindyck (1994), Trigeorgis (1996), Amram and Kulati-laka (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 ature are given in Lander and Pinches (1998), Baecker and Hommel (2004),and Trigeorgis (2005)

liter-A financial option gives the holder the right but not the obligation tobuy or sell a financial asset, e.g., stock traded on financial markets, underspecified terms These terms define the period of time over which the optioncan be exercised, in exchange of the exercise price A call option gives theright to acquire the underlying asset, a put option gives the right to sell theunderlying assets An American option can be exercised at any time beforeand including the expiration date, whereas a European option can only beexercised 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 insuranceagainst losses (cf McDonald 2003, pp 44 ff.) It enables to protect from thedownside potential of the underlying asset, while benefiting from the up-side potential This asymmetry between upside and downside is visualized

in Fig 2.1 At maturity, if the value of the underlying asset is below theexercise price, the option is not exercised and yields zero payoff Otherwise,the option is exercised, with unlimited upside potential The dotted line rep-resents the value of the call option before maturity, which is always higherthan 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 valueand current value is called “time value” of the option (cf Hull 2003, p 154)

Value of underlying asset

"intrinsic value"

of the option

don't exercise (at maturity) (at maturity)exercise

Call Option value

option value before maturity

exercise price

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 eling 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-

mod-dom walk, i.e., price changes occur ranmod-domly (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 indiscrete time it is frequently modeled as a multiplicative binomial process(cf McDonald 2003, pp 341 ff.) Both stochastic processes obtain (the bino-mial process in the limit as time increments become infinitesimally small)lognormally distributed stock prices, i.e., the natural logarithm of stock pricechanges are normally distributed (cf Luenberger 1998, p 309)

Real options are similar to financial options in that they “give the holderthe right, but not the obligation, to take an action in the future”(Amram andKulatilaka 1999, p 5) The major difference lies in the nature of the under-lying assets Real options are written on real assets which are productivecapacity creating streams of income As opposed to this, financial optionsare written on financial assets which are claims to the income generated byreal 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 financialoptions and the fundamental difference in the underlying assets

Typically, the interpretation of investment decisions as real options isbased on the following analogies to financial options: The current value ofthe stock corresponds to the (gross) present value of expected cash flows;the exercise price to the investment cost; time to expiration to the time untilthe opportunity disappears; stock value uncertainty to project value uncer-tainty; 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 someunderlying asset corresponds to a call option, whereas a real options en-abling to receive the exercise price in exchange for sacrificing some under-lying asset is a put option

Throughout this dissertation, we can use the terms value and priceinterchangeably, which is common in the real options literature (cf., e.g.,Copeland and Antikarov 2001, p 6) This is because in the absence of mis-pricing 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 incompletemarkets, non–traded underlying assets, or shared options Representing alarge 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 tailed overview can be found in Micalizzi and Trigeorgis (1999) and Landerand Pinches (1998, p 540, Table 2) Trigeorgis (1996, pp 2 f., Table 1.1) dis-tinguishes the following types of real options:

de-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 duringthe lifetime of the option

• Option to default during staged construction For investment projectsthat take time to build, it is possible to abandon the project duringconstruction based on the resolution of uncertainty during construction.This is desirable if the remaining investment costs are not covered by theexpected future value of continuing the project as originally planned

• Option to expand If the underlying asset and thus the project developsmore favorable than initially expected, a possible increase of capacityrealizes higher value than continuing in base scale

• Option to contract As opposed to the option to expand, the option tocontract enables to reduce the scale of operations and thus save costs ifthe underlying asset develops unfavorably

• Option to shut down and restart operations This option enables to stopoperating temporarily, with the perspective of reopening in a later pe-riod This is desirable if variable costs exceed operating revenues bymore 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 thevalue of continuing operations

• Option to switch use Production input factors or the product outputcan be changed, e.g., as a function of prices or demand This makes

it possible to choose the strategy which generates highest profit net ofswitching costs

• Growth options An early investment opens up new investment tunities

oppor-Huchzermeier and Loch (2001) extend this taxonomy by the option toimprove During projects it is possible to take corrective action, such asadding resources, in order to improve the expected performance of theproduct While this approach captures the fact that managers can changethe 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 preting each discretionary decision right as a switch between modes, asproposed by Kulatilaka and Marcus (1988) and Kulatilaka and Trigeorgis(1994) Their concept enables interpreting real option exercise decisions asswitching decisions among operating modes, which is the most general for-mulation of real options, having commonly analyzed real options as specialcases

inter-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 assetsand real options written on these assets For ease of discussion it is con-venient to label sources influencing the value of the portfolio as “portfolioaspects”

On the asset level, it is self–evident to assume that corporate decisionmakers 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 aset of underlying assets simultaneously (see Sec 3.1)

On the real options level, it has been shown by Trigeorgis (1993a) thatoptions on the same underlying assets interact, requiring a simultaneousvaluation of all real options written on the same underlying asset If thereare two financial options, e.g., to acquire stock of Bayer AG and BMW AG,the values of these financial options are additive because the underlyingassets and the terms of the options are independent from one another Asopposed to this, consider BMW who has a European option to abandon anexisting plant and the subsequent European option to expand this plant Inoptions terms, the European option to abandon corresponds to a Europeanput option, while the European option to expand corresponds to a Europeancall option Since both affect the same underlying asset, these real optionsinteract

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 morevaluable than the underlying asset in isolation Since a put option value de-creases with increasing value of the underlying asset, the option to abandonbecomes 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 thefuture call option

On the other hand, the value of the subsequent call is reduced throughthe existence of the prior put, because with some probability it may not bepossible to exercise the call, given prior exercise of the put This preventsthe call from unfolding its full value potential and translates into a lowercall option value

Since both effects occur simultaneously, it is not possible to value putand call in isolation The decision about exercising the first option needs

to explicitly take into account the existence of the subsequent option Thisrelationship 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 realoptions written on the same underlying asset as “intra–project compound-ness” Following the same logic, an analogous effect is identified for several,interdependent underlying assets which he denotes as “inter–project com-poundness” (Trigeorgis 1996, pp 132 f.) Both inter–project and intra–projectcompoundness must be considered in the context of portfolios of real op-tions

Further, both real options and real asset may be subject to constraints.Real assets can exhibit, for example, operating (technical or logical) or fi-nancial relationships These can lead to mutually exclusive assets or strictlycomplementary assets which require one another This affects the possibil-ity of joint exercise of bundles of real options on different underlying assets.Likewise, the existence of constrained resources, e.g., funds available, influ-ences the feasibility of joint option exercise

Therefore, portfolios of real options here are defined as combinations ofmultiple risky assets and multiple real options written on these assets subject toconstraints Cases with only one underlying asset, or one real option, arespecial portfolio cases that reduce the scope of portfolio analysis dramati-cally In order to seize all possible portfolio effects, it is important to analyzemultiple underlying assets with multiple real options simultaneously Theusual “laboratory” setting for real options analysis with one underlying as-set and one real option does not provide a structure capable of handlingrealistic decision problems It is thus prone to ignore key portfolio effectswith 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 isimportant to clarify the major differences between portfolios of real optionsand portfolios of financial options

Financial portfolio theory dates back to Markowitz (1952) who lished the concept of mean–variance analysis At its core, it is based on theconcept of diversification Diversification is about minimizing the variance

estab-of the return estab-of a portfolio for a given mean estab-of the return (or maximizing thereturn 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 availableassets The only relevant risk in this perspective is the covariance risk ofeach asset with the (market) portfolio In essence, this implies a passive at-titude towards risk because it exclusively consists of diversifying the riskover as many assets as possible

The portfolio approach to real options is fundamentally different fromthis passive perspective on risk For portfolios of real options, the variance

of returns and the resulting distributions of values of the underlying assetsare considered instead of covariance with a (market) portfolio Real optionsmake it possible to capitalize on desirable developments of risky underlyingassets while protecting from undesirable movements For instance, a realoption to expand capacity is only exercised if the value of the underlyingasset is higher than the investment costs This implies an active attitudetowards risk which consists of exercising real options as a function of theresolution of uncertainty Therefore, in portfolios of real options, the attitudetowards risk is different and diversification cannot be desirable

For financial portfolios, closely connected to the passive attitude towardsrisk, value additivity of the elements of the portfolio holds (cf Brealey andMyers 1996, p 19) Since a financial option derives its value directly fromthe value of the underlying financial asset, the same holds for financial op-tions Value additivity also holds for financial options In contrast, a con-sequence of the more active real options approach is that value additivityamong the elements of portfolios of real options generally breaks down (seeSec 2.1.1) This is why it is not insightful to include financial assets andfinancial options in the analysis of portfolios of real options Financial as-sets by definition are independent from other assets, therefore the values

of the financial assets and the portfolio of real options are also dent from one another and thus do add up Alternatively, if the portfolio

indepen-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 isnot a special case because in fact a real asset is considered, not a financialasset Beyond this perspective, it can be insightful to consider interactionsbetween real options and financial flexibility This issue is discussed by Tri-georgis (1993b) where the idea is not to add an additional (financial) asset

to the portfolio, but to assess the possible value of financial flexibility thatcan result from operating flexibility In this context, Huchzermeier and Co-hen (1996) discuss operational and financial hedging strategies A review ofthis 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 andoperational 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 sistent modeling approach is required In order to derive the implication ofthe specifics of real options for our modeling approach, in the following we

con-14 2 Portfolio Approach to Real Options

will highlight portfolio aspects, their consequences on how portfolios of realoptions 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 riskyassets and multiple real options written on these assets subject to constraintsprovides a rich structure for assessing real decision problems In this struc-ture, the overall portfolio value is influenced by the dynamic interaction ofall elements of the portfolio This portfolio perspective on real options is afundamental way to properly handle interdependencies between decisionsover time

A suitable perspective on portfolios of real options needs to include allrelevant aspects that influence the value of such a portfolio From the defi-nition of portfolios of real options, it follows directly that portfolio aspectscan be attributed to the real assets involved, the real options involved, orconstraints Budget constraints are of special importance because they canhave a considerable limiting impact and require a detailed modeling of theinvestment dynamics Moreover, the ensuing budget levels over time arestate– and path–dependent Based on these considerations, portfolio aspectscan be categorized as follows (cf Brosch 2001):

• Interactions on the real options level: On the real options level, intra–projectcompoundness and inter–project compoundness can be distinguished.While the former is due to interdependencies of several real optionswritten on the same underlying asset, the latter is connected to inter-dependencies of several real options and several underlying assets Forinter–project compoundness, the correlation between the underlying as-sets has to be modeled explicitly

• Interactions on the real asset level: On the real asset level, direct qualitativeinteractions and indirect qualitative interactions can be distinguished.Direct interactions are those which are inseparably connected to the un-derlying real assets, such as physical properties or operating synergies.Indirect interactions have their origin outside the strict asset level andare due to constraints, most prominently budget constraints Both arequalitative in that they are not merely stochastic in nature, but resultfrom the properties of projects or the specific background of the com-pany (that would be different for another company)

Due to the simultaneous nature of the interactions involved, it is notpossible to isolate value impacts for separate portfolio aspects Their effectsneed to be assessed jointly, in order to capture their interplay This is whythey require a simultaneous modeling approach, very much like compoundoption pricing Specifically, in order to handle these interactions, existing

constraints can only be incorporated via a simultaneous modeling approach.Thus, the model is formulated as one stochastic optimization problem sub-ject to constraints Interactions are captured through the interplay of con-straints as well as state– and path–dependency of investment decisions andcash flows.

2.2.2 Management of Portfolios of Real Options

Based on this broad understanding of portfolios of real options, optimalmanagement of portfolios of real options requires capturing all relevantportfolio aspects simultaneously in order to maximize overall portfoliovalue Specifically, two dimensions of managing portfolios of real optionscan be distinguished:

1 Portfolio design: creating the optimal portfolio with maximum value, posing optimal future exercise

sup-2 Portfolio execution: exercising existing real options optimally in order torealize the full value potential of real options

On the one hand, portfolio design analyzes which assets and which tions to include in the portfolio of real options This decision can be based

op-on the proposed model, by choosing the portfolio which yields the highestvalue for the portfolio of real options In order to assess different portfoliodesigns, a valuation needs to be carried out for each possible alternative.This process is depicted in Fig 2.2 By choosing the optimal portfolio de-sign, the decision maker in fact defines the company’s strategy, because thisdefines the way the company is planning to create value in the future Thisapproach helps to quantify the impact of new investment proposals andmore generally ideas form a portfolio perspective, which in options termscorresponds to assessing new real options and their interactions with theportfolio

Apart from assessing new real options, management may be able to fluence the underlying assets or the embedded real options, such as modi-fying the volatility involved (cf Damisch 2002, pp 365 ff.) The desirability

in-of these alternatives can also be assessed as different design options Atits core, the design of a portfolio of real options is about the choice of theunderlying assets to be included in the portfolio, and thus, strategic Thisdeserves special attention, because it will be shown that conventional wis-dom may not hold true for portfolios of real options For instance, it is notalways desirable to increase volatility to enhance value (see Sec 5) Hencethe portfolio design can be supported by the proposed model

On the other hand, portfolio execution is about optimally managing theexisting design of a portfolio of real options The value created through realoptions implicitly assumes that real options are exercised at the optimal

16 2 Portfolio Approach to Real Options

Alternatives?

(Re-) Design portfolio Start

Value portfolio

no yes

Select optimal portfolio End

Fig 2.2.Portfolio design process

point in time Therefore, management needs to know when this optimalpoint in time is This information is provided together with the optimalportfolio value, because the optimization result contains the optimal con-tingent exercise policy Thus, the proposed model gives clear managementrecommendations about which options should be exercised, and when, sug-gesting to exercise real options as in the optimal policy

Generally, the portfolio design defines the basis for the portfolio tion because the latter is about optimally using an existing structure which

execu-is defined in the first At the same time, the portfolio execution can provideunanticipated new information which opens up new portfolio alternativesfor the portfolio design An example for this is research on a specific drugwhich, by chance, provides insights about another new drug that had notbeen considered before Thus both dimensions of managing portfolios ofreal options are interrelated

2.2.3 Translation into Model Features

The motivation for considering portfolios of real options has not only fied the understanding of portfolios of real options, but at the same time hasalso stated the requirements for the model formulation A suitable modelformulation needs to incorporate the above introduced elements Before themodel is derived in detail in Chap 4, an overview of the model features isgiven below The model formulation will incorporate the following features:

clari-1 Multiple correlated underlying assets

2 Multiple real options embedded in the underlying assets.

3 State–dependent cash flows leading to path–dependencies

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 mulated as one stochastic mixed–integer program that captures path–dependent and multi–dimensional real options settings

for-2 Simultaneous, i.e., forward–backward looking, approach: The backwardlooking element consists of choosing the optimal investment strategywhich maximizes both current cash flow and expected value of futurecash flows This is a standard approach in financial option pricing (cf.Hull 2003, pp 392 ff.) This is complemented by a forward looking el-ement which captures the sequence of decisions up to a certain point

in time that meet the budget constraints on that path This is not cal for financial options Both are considered simultaneously in order tocapture their interdependencies

typi-3 Model implementation and solution: The model is solved by an rithmic solution implementation, using the GAMS modeling platform,which can readily be applied to practical decision problems

algo-The proposed modeling approach provides a rigorous and flexible ture that can cope with more realistic and rich decision problems, deriv-ing an optimal dynamic sequence of investment and disinvestment deci-sions from a portfolio perspective It is very flexible in incorporating spe-cific investment problems with different kinds of constraints and changes inthe underlyings Especially, the global dynamic budget depends on the se-quence of decisions which depends on the paths taken by the joint stochasticprocesses of the underlying assets This is achieved by a path–dependent,forward–backward looking approach that keeps track of past decisions andthe expected future effect of current decisions For a visualization of thisapproach, see Fig 2.3 To our knowledge, the formulation of such a modelfor portfolios of real options has not been provided in the literature so far

struc-In order to account for these features, this dissertation follows a newdirection for the quantitative assessment of real options The model inter-prets the portfolio problem as an optimization problem with the objective ofmaximizing portfolio value subject to “constraints” On the one hand, these

18 2 Portfolio Approach to Real Options

Path-dependent joint stochastic processes Path-dependent investment decisions

1 2 3 1 3 2

1 2 3 1 3 2

1 2 3

1 2 3 1 2 3

1 2 3

1 2 3 2 3 1

1 2 3 2 3 1

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 asconstraints, such as budget constraints On the other hand, other featuresare also captured by modeling these as balance equations, such as the speci-fication of the processes of the underlying assets, the determination of avail-able funds, or the calculation of expected values The dynamic nature of theproblem is incorporated through constraints that are either backward– orforward–looking, but either way dynamic This makes it possible to handlesimultaneous interaction effects and to define the problem in a formulationthat can incorporate operating and financial constraints Moreover, this ap-proach makes it possible to solve the problem with commercially availablesolvers

The model implementation reverts to commercially available softwarepackages, i.e., the optimization platform GAMS© (cf Bussieck and Meer-aus 2004), together with suitable optimizers such as BARON© and Cplex©.This implementation makes it possible to benefit from advanced optimiza-tion techniques that have been tested and adapted continuously and enables

to separate the problem implementation from the actual optimization rithms Consequently, the model becomes verifiable and portable betweenapplications

algo-Based on this understanding, our approach to portfolios of real optionswill be positioned in the relevant literature next This will provide the basisfor our ensuing model development.

Literature Review

This section reviews the key concepts provided in the relevant literature,drawing from different strands of the literature which will be discussedbelow

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 basicconcepts of portfolio analysis and then leads on to more advanced portfo-lio approaches The goal of this discussion is twofold On the one hand,

it clarifies the positioning of our model in the portfolio literature On theother hand, it shows that no suitable portfolio approach for our problem isavailable in the financial literature Second, corporate capital budgeting isreviewed in Sec 3.2 Special attention is paid to the treatment of interdepen-dencies between projects, from which we will draw in our model Third, thecore of the relevant real options literature is examined in Sec 3.3 This dis-cussion identifies prior work that is most akin to our approach, and retainstwo 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 modernfinance theory (Constantinides and Malliaris 1995) Today, concepts derivedfrom modern portfolio theory are the foundation of most (if not all) finan-cial analysis, directly or indirectly For example, the risk-adjusted discountrate in a Discounted Cash Flow (DCF) analysis is motivated by the portfolioconcept of diversification Also, a key element of derivatives pricing is theconstruction 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 includ-ing empirical tests can be found in Elton et al (2003) and Rudolf (1994)

In the following discussion of financial portfolios, first the basic concepts

of mean–variance portfolio analysis are discussed (Sec 3.1.1) Second, moreadvanced 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 turn in a portfolio context This approach identifies the optimal portfolioconfiguration for a risk-averse investor in a one-period decision problemwith the objective to allocate a given amount of money optimally to a port-folio of risky assets The investor is assumed to consider “expected return adesirable thing and variance of return as an undesirable thing” (Markowitz

re-1952, p 77) Based on this perspective on expected return and variance ofreturn, 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 allportfolios 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 fol-lows (for the following exposition, cf Rudolf 1994 and Constantinides and

Malliaris 1995) x is a m-column vector whose components xi denote theweight of asset i in the portfolio, i=1, , m Given that all available fundsmust be allocated, the portfolio weights must add up to 1, i.e.,P

ixi = 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

as-sumed 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 ofthe other assets Given that variances of risky portfolios are strictly positive,

V is positive definite, i.e., xTVx>0 The mean returns and covariances areendogenous 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 RP Withthis notation, the portfolio selection problem can be formalized as follows:

3.1 Financial Portfolio Theory 23

In other words, this formulation minimizes portfolio variance by solvingfor 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) xTVx is convex

be-cause V is positive definite; the constraints are linear and thus define a

con-vex 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=xTVx−λ1(xTR−RP) −λ2(xT1−1) (3.4)Solving this equation yields the vector of optimal portfolio weights thatminimize portfolio variance for a given mean return, as follows:

x=V−1 R 1 RTV−1R RTV−11

RTV−11 1TV−11

−1

 RP1



(3.6)

This solution provides the minimum variance portfolio for a given RP,which is expressed by a parabola In mean-variance space, i.e., by invertingthe function, the mean-variance efficient minimum variance portfolios can

be represented by a convex line, starting with the global minimum varianceportfolio which is the portfolio with the lowest possible variance for anyreturn This relationship is graphically represented in Fig 3.1 All portfoliosinside the hyperbola are feasible; portfolios on the solid line are minimumvariance efficient portfolios, because they yield the highest expected returnfor a given variance of return Therefore, the efficient portfolios dominateall other portfolios including the minimum variance inefficient portfolios.The investor’s choice then consists of choosing that portfolio from theefficient frontier which maximizes her one-period utility Evidently, this ap-proach is sufficient if either returns are normally distributed (symmetrical),

or if the investor has a quadratic utility function Otherwise, the obtained sults are an approximation only and the approximation quality depends onthe deviation from the real distribution and utility function (cf Markowitz

re-1990, pp 6 ff.)

One key insight of the mean–variance approach is the concept of versification Diversification can reduce risk without necessarily adverselyaffecting expected portfolio return This leads to the conclusion that it isnot the individual (total) risk of an asset that matters, but its contribution

Return

Variance of Return

Minimum Variance Efficient Portfolios Feasible Portfolios

Global Minimum Variance Portfolio Feasible Portfolios

Minimum Variance Inefficient Portfolios

Fig 3.1.Mean–variance efficient portfolios

to the risk of the entire portfolio In turn, this risk contribution for a welldiversified investor is captured by covariance risk, i.e., the correlation of theindividual asset with the overall portfolio:

where xj are portfolio weights, σ2 denotes the variance of the return of the

overall portfolio, j, k represent individual assets, and ρjkσjσk is the

covari-ance of their returns, expressed through the coefficient of correlation, ρjk With sufficient diversification and thus sufficiently small portfolio weights,

the variance of each asset lose significance and σP2 approaches the averagecovariances between assets (interpreting k6=j in Eq 3.7 as a weighted aver-age) (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 ofmean–variance analysis and its graphical representation, more rigorous for-mulations are given in subsequent work (see Markowitz 1991; Mertens2004) Roy (1952) independently developed an alternative portfolio ap-proach based on minimizing the probability of missing a required minimumreturn, the so–called “shortfall risk” For normally distributed returns, thisapproach is equivalent to Markowitz (1952) Recently, the Markowitz port-folio selection has been extended to a multi–period portfolio selection by Liand Ng (2000)

The outlined Markowitz portfolio selection provides the understanding

of portfolios that underlies our model At the same time, our approach damentally differs from this portfolio perspective in that it considers activemanagement of the underlying assets, making it possible to influence thereturn pattern of the assets as endogenous in our model

fun-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 borrowingand lending at a given single interest rate, no transaction costs, shared com-mon expectations of all market participants, and perfect capital markets By

an equilibrium argument, it is shown that each investor will hold the marketportfolio, MP, regardless of her risk attitude, and will adjust her portfolio

to her risk aversion preference by lending or borrowing The capital assetpricing model is given by:

E (ri) −rf =βi

h

E (rMP) −rfi (3.8)where

βi= σiMP

This relationship states that the expected excess rate of return of asset i(above the risk–free rate, rf) is a linear function of βi and the market riskpremium (expected excess rate of return of the market portfolio MP over rf)

βi 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 suchthat a rational investor would not demand extra compensation for bearingthat risk This risk is usually referred to as nonsystematic, idiosyncratic, orspecific risk In contrast, covariance risk is referred to as systematic risk.From a portfolio pricing perspective it is not the total risk measured bythe variance of the return of an individual asset that matters, but only thesystematic risk measured by its covariance with the market

The above introduced basic CAPM is widely applied in practice, ily because of its simplicity The empirical validity of the CAPM has beenwidely discussed and challenged, but this discussion is not key for the pur-poses of our discussion since it will not explicitly refer to the CAPM Anoverview of this discussion as well as key empirical results can be found

primar-in Fama (1976); Alexander and Francis (1986), “Roll’s critique” (Roll 1977),and an answer to Roll’s critique in Kandel and Stambaugh (1995)

1For the following exposition, see Luenberger (1998, p 173ff)

3.1.1.3 Intertemporal Capital Asset Pricing Model and Consumption

Both the traditional Markowitz portfolio selection problem and the CAPMare static models in the sense that they consider a single–period decisionproblem (cf Constantinides and Malliaris 1995) Only under certain as-sumptions, especially when preferences and future investment opportu-nity sets are not state dependent, the solution for the single–period port-folio problem also solves the multi–period portfolio problem (cf Fama1970; Rudolf 1994; Brennan and Xia 2006) Relaxing these assumptions,the CAPM has been extended to the Intertemporal Capital Asset PricingModel (ICAPM) by Merton (1973a)

The ICAPM considers “consumer-investors” who maximize the pected utility of lifetime consumption (Merton 1993, p 475) Therefore, theportfolio decisions over time take into account the relation between currentperiod returns and returns that will be available in the future Given that theinvestment opportunity set might shift over time, investors with a multi–period investment horizon need to consider ways of hedging against unfa-vorable shifts, to protect long–term wealth or consumption Consequently,equilibrium expected returns depend both on the covariance of returns withthe current return of the market portfolio, and information about future re-turns of the market portfolio Cochrane (cf 2001, p 172) For example, if thecurrent return of an asset is negatively correlated with shifts in the invest-ment opportunity set, it can serve as a hedge against negative future devel-opments, thus requiring a higher equilibrium price for this asset throughthe hedging demand (cf Merton 1993, p 479) A more detailed discussion

ex-of the ICAPM can be found in Merton (1993) and Duffie (2001)

The ICAPM models marginal utility not in terms of consumption, but interms of other directly observably variables instead, i.e., current wealth aswell as conditional distributions of asset returns, given shifts in the invest-ment opportunity set (cf Cochrane 2001, pp 46 f., p 166) In that sense, it is aspecial form of a consumption–based model (cf Cochrane 2001, p 170) Lu-cas (1978) and Breeden (1979) consider investment and consumption in anintertemporal 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 excessreturn of an asset is proportional to the asset’s covariance with consump-tion, 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 variance of its return with the change in aggregate consumption (cf Merton

co-1993, p 520) A more detailed discussion of the CCAPM can be found inMerton (1993), Duffie (2003) and Campbell (2003)