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Investment under risk tolerance constraints and non-concave utility functions: implicit risks, incentives… Rodriguez-Mancilla, Jose Ramon 2007

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I N V E S T M E N T U N D E R RISK T O L E R A N C E C O N S T R A I N T S A N D N O N - C O N C A V E U T I L I T Y FUNCTIONS: IMPLICIT RISKS, INCENTIVES A N D O P T I M A L S T R A T E G I E S by JOSE R A M O N R O D R I G U E Z - M A N C I L L A B.Sc , Universidad Nacional Autonoma de Mexico, 1994 M.Sc., University of Minnesota, 1999 M . S c , The University of British Columbia, 2001 A THESIS S U B M I T T E D IN P A R T I A L F U L F I L L M E N T O F T H E R E Q U I R E M E N T S FOR T H E D E G R E E O F D O C T O R O F P H I L O S O P H Y in T H E F A C U L T Y OF G R A D U A T E STUDIES (Business Administration) T H E U N I V E R S I T Y OF BRITISH C O L U M B I A August 2007 © Jose Ramon Rodn'guez-Mancilla, 2007 Abstract The objective of this thesis is to contribute in the understanding of both the induced behavior and the underlying risks of a decision maker who is rewarded through option-like compensation schemes or who is subject to risk tolerance constraints. In the first part of the thesis we consider a risk averse investor who maximizes his expected utility subject to a risk tolerance constraint expressed in terms of the risk measure known as Conditional Value-at-Risk. We study some of the implicit risks associated with the optimal strategies followed by this investor. In particular, embedded probability measures are uncovered using duality theory and used to assess the probability of surpassing a loss threshold defined by the risk measure known as Value-at-Risk. Using one of these embedded probabilities, we derive a measure of the financial cost of hedging the loss exposure associated to the optimal strategies, and we show that, under certain assumptions, it is a coherent measure of risk. In the second part of the thesis, we analyze the investment decisions that managers undertake when they are paid with option-like compensation packages. We consider two particular cases: • We study the optimal risk taking strategies followed by a fund manager who is paid through a relatively general option-like compensation scheme. Our analysis is develo-ped in a continuous-time framework that permits to obtain explicit formulas. These are used first to analyze the incentives induced by this type of compensation schemes and, second, to establish criterions to determine appropriate parameter values for these compensation packages in order to induce specific manager's behaviors. • We consider a hedge fund manager who is paid through a simple option-like compensa-tion scheme and whose investment universe includes options. We analyze the nature of ii the optimal investment strategies followed by this manager. In particular, we establish explicit optimal conditions for option investments in terms of embedded martingale measures that are derived using duality theory. Our analysis in this case is developed in a discrete-time framework, which allows to consider incomplete markets and fat-tailed distributions -such as option return distributions- in a much simpler manner than in a continuous-time framework. iii Contents A b s t r a c t 1 1 Contents : i v L i s t of Figures v i i i Preface x i v Acknowledgements X 1 X D e d i c a t i o n x x -1 L i v i n g on the Edge: H o w r isky is it to operate at the l i m i t of the tolerated risk? 1 1.1 Introduction 2 1.2 Decision Space Framework 4 1.2.1 Probability Space 4 iv 1.2.2 Financial Market 5 1.3 Basic Model 8 1.4 Risk Management Modelling 8 1.5 Embedded Probabilities 13 1.6 Implicit Risks 27 1.7 Conclusions and Further Research 44 2 T h e D u a l i t y o f O p t i o n I n v e s t m e n t S t r a t e g i e s f o r H e d g e F u n d s 4 6 2.1 Introduction 47 2.2 Decision Space Framework 52 2.3 The Basic Problem 55 2.3.1 Buying Options 56 2.3.2 Selling Options 68 2.3.3 Buying and Selling Options 73 2.3.4 Feasibility of the Basic Problem 76 2.4 Other Utility Functions 79 2.5 Risk Incentives 85 2.5.1 Buying Options 86 v 2.5.2 Buying and Selling Options 90 2.6 Multiple Monitoring Dates 93 2.6.1 Two-Period Monitoring Case 94 2.6.2 Multiple Monitoring Dates 103 2.7 Advanced Models 106 2.7.1 Underperformance Risk Management 107 2.7.2 Option Risk Management 109 2.7.3 Other Extensions 113 2.8 Conclusions 115 3 Incentives and Design of Option-Like Compensation Schemes 117 3.1 Introduction 118 3.2 The Model 124 3.2.1 The Fund Value 124 3.2.2 The Manager Problem 125 3.3 Derived Utility Function Analysis 127 3.3.1 Motivation 128 3.3.2 Concavification Function Construction 129 vi 3.4 Optimal Solution 142 3.4.1 Optimal Fund Value 143 3.4.2 Optimal Risk Taking 148 3.5 Induced Incentives 151 3.5.1 Fund Value Incentives 151 3.5.2 Risk Incentives 153 3.6 Incentives Design 165 3.6.1 Terminal Fund Value Design 166 3.6.2 Risk Profile Design 172 3.6.3 An Example 177 3.7 Conclusions 182 F i n a l R e m a r k s 1 8 5 B i b l i o g r a p h y 1 8 7 A p p e n d i x 1 1 9 8 A p p e n d i x 2 1 9 9 A p p e n d i x 3 2 0 4 vii List of Figures 1.1 Convexity of ESH. This graph shows a density P (gray line) and a density P' (black line) such that the ESH, under P, coincides with the CVaR, under P ' , for a confidence level of a = 0.95 and where P' is obtained by distorting appropriately the tail of the P density 43 2.1 Optimal Strategy. This graph shows the optimal long positions in the risky security and the put options, as a percentage of the sum of the amount borrowed and the initial capital W0 88 2.2 Risk Aversion Effect. This graph shows the effect of the risk aversion level on the risk induced by the compensation scheme 90 2.3 Variable Fee Effect. This graph shows the effect of the variable fee per-centage on the risk induced by the compensation scheme, for different levels of risk aversion 91 2.4 Risk Aversion Effect: Buying versus Buying-and-Selling. This graph compares the effect of the risk aversion level on the risk induced by the com-pensation scheme, for the two sets of investment conditions considered in this section 93 viii 3.1 S h a p e o f U o 4 > . Th is is the graph of U o < / > when the following parameters are considered: X0 = 0.005, p = 0.01, q = 0.02, B = 1.04, K = 0.76 and 7 = 2 . Note that (U o </>) (0) = ( -1 ) (0.005" l ) = - 2 0 0 127 3.2 T w o - p i e c e s h a p e C o n c a v i f i c a t i o n o f U o<p. Th is is the graph of U o < j ) and its concavification for a case in which the concavification function has a two-piece shape. The parameters are: XQ = 0.005, p = 0.01, q = 0.02, B = 1.04, K = 0.76 and 7 = 2. U o cj) and its concavification function coincide, for this particular example, over the interval [2.0544, 00). The concavification function displayed in this graph corresponds to the derived ut i l i ty function of Figure 3.1 142 3.3 F o u r - p i e c e s h a p e C o n c a v i f i c a t i o n o f Uo<f>. This is the graph of U o c j ) and its concavification for a case in which the concavification function has a four-piece shape. The parameters are: X0 = 0.005, p = 0.01, q = 0.02, B = 1.04, K = 1.46 and 7 = 2. U o </> and its concavification function coincide, for this particular example, over the intervals [1.7611,2.0858] and [2.7391, 00) 143 3.4 O p t i m a l C h o i c e o f vf- T w o - p i e c e s h a p e c a s e . This is a graph that shows the opt imal terminal value as a function of the pricing kernel £T for a particular example. The parameters are: X o = 0.005, p = 0.01, q = 0.02, B = 1.04, K = 0.16, 7 = 2, V0 = 1, p = 0.04, a = 0.4, T = 1. The relevant tangency point in this case is V4 = 1.5838, which corresponds to the threshold value £ 0 = (1/A) (U o 4>) (V4) = 1.2462 153 ix 3.5 O p t i m a l C h o i c e o f V r - F o u r - p i e c e s h a p e c a s e . This is the graph of the optimal terminal value as a function of the pricing kernel £r- The parameters are: X0 = 0.005, p = 0.01, q = 0.02, B = 1.04, K = 1.26, 7 = 2, V0 = 1, /x = 0.04, a — 0.4, T = 1. The relevant tangency points are in this case Vy = 1.7611 and V3 = 2.5147, with the corresponding threshold values £1 = 1.0524 and & = 0.8144 154 3.6 O p t i m a l V o l a t i l i t y : S i n g l e - O p t i o n C a s e . This graph shows the optimal volatility strategy for the case of a single option (i.e. q = 0%) with the following parameters: X0 = 0.005, p = 0.01, B = 1.04, 7 = 2, V0 = 1, H = 0.04, a = 0.4, T = 1 155 3.7 C o m p e n s a t i o n S c h e m e S h a p e . This is a graph shows the shape of the compensation scheme for the single option case (q = 0%), and the double option case (q = 2%) 156 3.8 K E f fect (I). This graph shows the effect of parameter K on the optimal volatility decisions, for a fixed q ( = 2 %). The parameters used are: XQ = 0.005, p = 0.01, B = 1.04, 7 = 2, V0 = 1, \t = 0.04, a = 0.4, and T = 1. The values of K considered are 0, 0.5, and 1.0. The concavification function associated with these values has a two-piece shape 157 3.9 K Ef fect (II). This graph exemplifies the effect of parameter K on the op-timal volatility decisions, for a fixed q ( = 2 %). The parameters used are: X0 = 0.005, p = 0.01, B = 1.04, 7 - 2, V0 = 1, M = 0.04, a = 0.4, and T = 1. The values of i f considered are 1.5, 2.0, and 2.5. The concavification function associated with these values has a four-piece shape 158 x 3.10 M i n i m a l O p t i m a l V o l a t i l i t y : K E f f e c t (I). This graph exemplifies the effect of the parameter K on the minimal volatility decision, for a fixed q ( = 2 %). The solid line represents the minimal volatility decision, while the dashed lined corresponds to the difference between the tangent points V2 and V i , for each value of K considered. The concavification function has a four-piece shape if and only if V2 — V\ > 0 (Proposition 3.3.4). The parameters used are: X0 = 0.005, p = 0.01, B = 1.04, 7 = 2, V0 = 1, p. = 0.04, a = 0.4, and T = 1 159 3.11 M i n i m a l O p t i m a l V o l a t i l i t y : K E f f e c t (II). This graph exemplifies the effect of the parameter K on the minimal volatility decision, for a fixed q ( = 10 %). The solid line represents the minimal volatility decision, while the dashed lined corresponds to the difference between the tangent points V2 and Vi, for each value of K considered. The concavification function has a four-piece shape if and only if V2 — V\ > 0 (Proposition 3.3.4). The parameters used are: X0 = 0.005, p = 0.01, B = 1.04, 7 = 2, V0 = 1, p. = 0.04, a = 0.4, and T = 1 160 3.12 q E f fect . This graph illustrates the effect of parameter q on the optimal volatility decision, for a fixed K ( = 0.5). The parameters used are: XQ = 0.005, p = 0.01, B = 1.04, 7 = 2, V0 = 1, fx = 0.04, a = 0.4, and T = 1. The values of q considered are 0.5%, 1%, and 1.5% 161 3.13 M i n i m a l O p t i m a l V o l a t i l i t y : q E f f e c t . This graph illustrates the effect of parameter q on the optimal volatility decision, for a fixed K ( = 0.5). The parameters used are: X0 = 0.005, p = 0.01, B = 1.04, 7 = 2, V0 = 1, p = 0.04, a = 0.4, and T = 1 162 xi 3.14 M i t i g a t i o n of U n e x p e c t e d R i s k Profiles. This graph plots the maximal minimal volatility level for a given q. It shows that the unexpected risk profile observed for the single-option case illustrated in Figure 3.6 can be mitigated. The parameters used are: X0 = 0.005, p = 0.01, B = 1.04, K = 2.5, 7 = 2, V0 = l,n = 0.04, a = 0.4, T = 1, and T - t = 0.5 163 3.15 O p t i m a l V o l a t i l i t y Decis ion: S m a l l q a n d Large K. This graph shows the optimal volatility decisions for a relatively small q ( = 0.225 %) and a rel-atively large K ( = 3), and the corresponding single-option optimal volatility curve (q = 0%). The parameters used are: X0 = 0.005, p = 0.01, B = 1.04, 7 = 2, V0 = 1, fi = 0.04, a = 0.4, and T = 1 164 3.16 Sensi t iv i ty of V\ w i t h respect to p. This graph exemplifies the way VI changes as p changes. The parameters used are: X0 = 0.005, B = 1.04, and 7 = 2. For these parameters, and the range of values considered for p: condition (3.26) of Proposition 3.6.1 is satisfied and therefore ^ < 0 168 3.17 K*(q): This graph shows the minimal value of K required to assure that condition V2 — v\ > 0 is satisfied for each value of q, and given a set of parameters (X0,p,B). The parameters used are: X0 — 0.005, B = 1.04, and 7 = 2. The range of values of q considered is Q = [0.001,0.1]. 170 3.18 D e t e r m i n a t i o n of K*(q): This graph shows V2 as function of K, given risk aversion parameter 7 = 2 and the parameter values X0 = 0.005, p = 0.01, B = I, and q = 0.02. K*(q) is determined as the value at which V2 (K*(q)) equates VI, which in this case is equal to 1.7071. The terminal fund value is of the form {0} U (V4, oo) if and only if K 6 [0, K*(q)} 179 xu 3.19 O p t i m a l S e l e c t i o n o f K: This graph shows the objective function of Prob-lem (3.31) for K e [0,1]. The optimal value is reached at K** = 0.39. The optimal exercise price for the second option is then B + K** = 1.39 180 3.20 D e t e r m i n a t i o n o f q*(K): This graph shows Vi as function of q, given risk aversion parameter 7 = 2 and the parameter values X0 = 0.005, p = 0.01, B = 1, and K = 0.80. The point at which the function V2(q) crosses the value of Vi = 1.7071 is q*(K) = 0.0036. The terminal fund value is of the form {0} U (Vi, V2] U (V4, oo) if and only if q G [0, q*(K)} 182 3.21 O p t i m a l S e l e c t i o n o f q : This graph shows the objective function of Problem (3.32) for q e [0,0.02]. The optimal value is reached at q** = 0.0096 183 xiii Preface The development of the foundations of Portfolio Theory in the nineteen fifties by Harry Markowitz marked an important breakthrough in Economics and Finance. In his seminal pa-per "Portfolio Selection", published in 1952, Markowitz assembled for the first time a theory of choice under certainty that was capable of explaining the observed diversification behavior of investors in the financial markets. This theory was the first mathematical formulation of diversification (Rubinstein (2002)), yet simple enough to be understood and applied. There-fore, this seminal work, together with other early important contributions by Tobin (1958), Markowitz (1959), Sharpe (1964), Lintner (1965), Samuelson (1969), Hakasson (1970), Fama (1970), Merton (1969, 1973), among others, opened the door to a torrent of research that has extended, enriched and applied this theory in many ways, with no sign of diminishing its momentum yet. Each investment decision has an associated risk. Variance from the mean rate of return is the measure that Markowitz used to quantify and include risk in his basic portfolio model. He used it because variance captures, in a relatively simple manner, the interrelations among the different securities's returns included in the portfolio's investment universe. The degree of these return interrelations -measured by the covariances- was precisely what explained, under Markowitz's model, the investor's diversification behavior. However, as Markowitz himself recognized since the very beginning (Markowitz (1959)), variance has several inconveniences as a risk measure. For instance, variance does not distinguish between values that are above or below their expected mean, by the same amount. This is contrary to the notion that risk quantification should be more concerned about values below the average than about those above the average. Therefore, since the early days of portfolio theory there has been a per-manent interest of proposing other risk measures. Notable initial efforts in such issue include those done by Stone (1973) and Fishburn (1977), among others. At the beginning of the xiv nineties, J.P. Morgan popularized the use of the Value-at-Risk (VaR) measure, given its easy interpretation (Jorion (2000b)). However, this measure has serious drawbacks. Probably the most important one is that it can penalize diversification. That is, the risk of a portfolio constructed by investing our wealth equally among two financial positions can be strictly larger, as measured by VaR, than the average risk of investing our money in any of these two positions (e.g., Follmer and Scheid (2002)). This counterintuitive situation in which VaR might incur has led to the recent development of the concept of "Coherent Measures of Risk" (Artzner, et al. (1999)). This concept established a benchmark for differentiating between "good" and "bad" risk measures (Ortobelli, et al. (2005)) and motivated the definition of other related classes of risk measures such as: Expected Bounded Risk Measures (Rockafel-lar, et al. (2002b)), Spectral Measures of Risk (Acerbi (2002)), and Conditional Drawdown Risk Measures (Chekhlov et al (2005)). Nonetheless, it is important to emphasize that be-sides its desirable properties, the proper use of a risk measure depends importantly on the context where it is used. In particular, in Portfolio Theory, optimal investment decisions are typically consistent with the solution of an "expected utility maximization problem" (e.g., Ortobelli, et al. (2005)), where this consistency is usually characterized through the concept of stochastic dominance (Hadar and Rusell (1969), Rothschild and Stiglitz (1970)). Conditional Value-at-Risk 1 (CVaR), defined as the expected loss given that the loss has exceeded VaR (refer to Rockafellar and Uryasev (2002a) for a formal and general definition), is currently the most popular member of the class of coherent risk measures. This is mainly because of its closed relation with VaR, nowadays the most used risk measure in financial markets, but also significantly because of its nice mathematical properties. For instance, CVaR is -for general loss distributions- a convex function and, moreover, it can be deter-mined as the solution of a minimization convex problem (Rockafellar and Uryasev (2002a)). 1 For continuous loss distributions, CVaR is also known as Mean Excess Loss, Mean or Expected Shortfall (e.g., Acerbi and Tasche (2002)), or Tail-Value-at-Risk (Artzner, et al. (1999)). xv Therefore, its inclusion in portfolio optimization, either in the objective function or in the set of constraints, is rather appealing. For example, consider a typical portfolio problem in which the investor wants to minimize risk subject to an expected return restriction and, probably, to other linear constraints (e.g., non-short-selling constraints), where risk is mea-sure in terms of CVaR. Hence, since CVaR is convex, the problem has a unique solution and, given the linearity of the constraints, there exists many efficient algorithms to obtain the optimal solution (e.g., Bazaraa, et al. (1993)). Convexity of CVaR has been also used to derive equivalent formulations of generic portfolio problems that consider CVaR in the objective function or constraints (Krokhmal, et al. (2001)). Chapter 1 of this Ph.D. thesis considers a risk averse investor who maximizes his expected utility subject to a risk tolerance constraint expressed in terms of CVaR. This chapter studies some of the implicit risks associated with the optimal strategies followed by this investor. In particular, embedded probability measures are uncovered using duality theory and used to assess the probability of surpassing a loss threshold defined by the VaR measure. Using one of these embedded probabilities, we derive a measure of the financial cost of hedging the loss exposure associated to the optimal strategies, and we show that, under certain assumptions, it is a coherent measure of risk. The variety of financial problems addressed from Markowitz's seminal work has reached enormous proportions in both the academic and practitioner's arenas. Indeed, it is not presumptuous to say that Portfolio Theory laid the groundwork for the Mathematical Theo-ry of Finance. One of the major applications of Portfolio Theory is precisely the issue that motivated it in the first place: the understanding of investors' behavior. This issue has recently attracted the attention of both academics and finance professionals in relation to the compensation packages that many managers and executives receive nowadays. For instance, the interest in executive compensation has grown significantly in the last fifteen xvi years because of two main reasons. First, the escalation and recent decline of the average executive compensation amount. Second, the replacement of base salaries by stock options as the single largest component of executive compensation (Hall and Murphy (2003)). On the other hand, managerial compensation has captured the interest of many investors and academics, especially after the nearly catastrophic bankrupt of LTCM (Long Term Capital Management) hedge fund in 1998 which, according to the U.S. Federal Reserve (Jorion (2000a)), jeopardized the world economy. Hedge Fund managers, like some other managers and traders, are compensated with option-like schemes. Option-like compensation packages for both, executives and managers, are designed, in principle, to align interests between investors and executives or managers. Therefore, it is crucial to understand the induced incentives derived from these compensation mechanisms. Chapters 2 and 3 of this Ph.D. thesis analyze the investment decisions that managers un-dertake when they are paid with this type of compensation packages. Chapter 3 studies the optimal risk taking strategies followed by a fund manager who is paid through a relatively general option-like compensation scheme. This chapter is developed in a continuous-time framework that permits to obtain explicit formulas. These are used first to analyze the induced incentives by this type of compensation schemes and, second, to establish criterions to determine appropriate parameter values for these compensation packages in order to in-duce specific manager's behaviors. Chapter 2 focuses on the particular case of a hedge fund manager who is paid through a simple option-like compensation scheme and whose invest-ment universe includes options. This chapter analyzes the nature of the optimal investment strategies followed by this manager. In particular, it establishes explicit optimal conditions for option investments in terms of embedded martingale measures that are derived using duality theory. Given the inclusion of options in the manager's investment universe, Chap-ter 2 is developed in a discrete-time framework that allows to consider incomplete markets, xvii in which options are not necessarily replicable, and fat-tailed distributions -such as option return distributions- in a much simpler manner than in a continuous-time framework. Overall, the objective of this Ph.D. thesis is to contribute in the understanding of both the induced behavior and the underlying risks of a decision maker who is rewarded through option-like compensation schemes or who is subject to risk tolerance constraints. Although related, each chapter is intended to be self-contained. xviii Acknowledgements I would like to express my gratitude to Dr. William Ziemba, my thesis supervisor, for believing in me and taking me as his student; to Dr. Ali Lazrak and Dr. Ivar Ekeland, for their guidance as members of my supervisory committee; to Dr. Ulrich Hausmann, for introducing me into the fascinating topic of Mathematical Finance and for his support in critical moments of my graduate studies; to Samuel Alfaro-Desentis, for all his support during the conclusion of this thesis; to Dr. Gabriel Casillas-Olvera, for reading entirely and thoroughly this thesis and providing me with many useful comments; to the graduate students with whom I shared challenging situations during the difficult and uncertain times of my doctoral studies; and to all the people who in one way or another have contributed to the achievement of this goal. xix Dedication I dedicate this thesis to God, who has blessed me in so many ways; to my precious wife, who has given me her entire love and unconditional support, regardless of the personal and professional sacrifices that she has had to make; to my three wonderful kids, Maria, Sebastian and Isabel, who have brought so much meaning and happiness to my life; to my parents, whose love and support have never been apart from me regardless the distance and the challenging situations; to my brother and sister, who have always given me their love, respect and their unconditional support. xx Chapter 1 Living on the Edge: How risky is it to operate at the limit of the tolerated risk? i 1.1 Introduction As financial markets have grown in complexity and volume of operation, risk management has become an increasingly crucial issue. Therefore, different risk measures have been pro-posed to evaluate the risk of financial positions, particularly to evaluate the risk of positions associated with investment portfolios. For a long time the variance has been a standard to evaluate the risk of such portfolios. Therefore, it is a typical element included in models that look for optimal portfolios such as the mean-variance approach (Markowitz (1952)), that targets a portfolio with the best mean-variance tradeoff. However, the variance has several drawbacks. For instance, it has the disadvantage of not distinguising between losses and profits. To overcome such drawbacks, alternative risk measures have been proposed. The most popular of these, and indeed the standard in many financial markets worldwide, is Value at Risk (VaR). This risk measure evaluates the maximal loss of a financial position over a time horizon and with a certain level of confidence. Yet, while VaR penalizes only for the losses, it could also penalize for the diversification of positions. That is, the VaR measure could indicate a risk increment of a portfolio, even if the volatility of the portfolio is reduced through the diversification of its positions. Therefore, VaR is said to be an incoherent mea-sure of risk. Hence, alternative coherent risk measures, in the sense of Artzner, et al. (1999), have been proposed to be used instead of or as a complement to VaR. One of the measures within this class of coherent measures of risk is the one known as Conditional Value at Risk (CVaR), which is the expected loss in the event that the loss has surpassed the standard VaR threshold (Jorion (2000b)). This chapter studies the implicit risks of the strategies followed by an investor who maximizes the expected value of his final wealth, subject to budget and risk tolerance constraints expressed in terms of coherent risk measures such as CVaR. This chapter undercovers embedded probabilities that are implicit in the optimal strate-gies followed by the type of investor considered. Key relationships between these embedded 2 probabilities are obtained and used to evaluate the implicit risks associated with the optimal strategies whenever these involve operating at the limit of the tolerated risk. This evalu-ation is done, under each embedded probability, in terms of the probability assessment of surpassing a certain VaR threshold and, more importantly, in terms of a risk measure, in the sense of Follmer H. and Shied (2002), which is a coherent measure of risk (Artzner, et al. (1999)), under certain conditions. This chapter is organized as follows: Section 2 describes the decision space framework, while Section 3 specifies the basic model that constitutes the starting point for the analysis of this chapter. Section 4 develops the way in which risk management is considered and how it is incorporated into the basic model to derive the model over which the main analysis of this chapter is done. In Section 5, duality is applied to the model of Section 4 to undercover (three) embedded probabilities within this model, and then it is used again to establish one of the main results of the chapter (Theorem 1.5.1), which describes a general relationship between these embedded probabilities. Other interesting, and useful, consequences of The-orem 1.5.1 are also established in Section 5. Section 6 defines a measure that attempts to evaluate the financial cost of hedging the loss exposure associated to the optimal strategies followed by the type of investor considered in the model of Section 4. This measure, named Expected Shortfall Hedge, is proved to be a monetary measure of risk (Follmer Ft. and Shied (2002)) and, under certain conditions, a coherent measure of risk (Artzner, et al. (1999)). Section 7 concludes and proposes two main directions for further research. 3 1.2 Decision Space Framework 1.2.1 Probability Space The probability space uses a scenario tree structure that models all possible scenarios or states (represented by nodes of a tree) of the market over a finite number of discrete time periods t = 0, . . . , T. The scenario tree structure is such that every possible state is the consequence of a unique sequence (trajectory) of states (events). This is convenient for assigning probabilities to each of the tree scenario nodes. Assume that every node n E NT, where NT denotes the set of all nodes at time t, has a unique parent denoted by a(n) G Nt_x and a set of child nodes C(n) C NT+Y. Defining a probability measure (i.e. assigning probabilities to each node of the tree), say P, consists of assigning (strictly) positive weights pn to each leaf node n G NT such that Y^neNT Pn — 1 and then recursively computing the remaining node probabilities Pn= YI Pm , V n G i V t , V i = T - l , . . . , 0 . m€C(n) Let fi be the set of possible trajectories or sequences of events (from time 0 to the end of period T) in the scenario tree, then ( f i , P) defines a sample space. Every node n E NT has a unique history up to time t and a unique set of possible future trajectories. NT induces a unique set of histories up to time t, say Ft, and a partition of fi. The collection of sets {Ft}t=o,...,T s atisfy Ft c Ft+i for t = 0,. . . , T — 1. The triplet (fi, FT, P) forms a probability space. Hence, for a probability space (fi, F T , P), the conditional probability of state (event) m given that n occurs (m € C{ri)) is (^r) i a n d if {Xt}t 0^ T is a discrete stochastic process defined on (fi, FT, P), then E p[Xt\ = £n(EiVt XnPn] and E p[Xt+l\Ft\ = }ZmeC(n) ( ^ ) * m is random variable taking values over the nodes n G NT. 4 D e f i n i t i o n 1.2.1 ( M a r t i n g a l e s ) Let {Zt}t=Q T be a discrete stochastic process defined on (0.,FT,P). If there exists a probability measure Q such that zt = EQ[zt+1\Ft] , t = o , . . . , r - i then {Zt}t=0 T is called a martingale under Q, or simply a Q-martingale, and Q is called a martingale measure for the process {Zt}t=0 T . 1.2.2 Financial Market The market consists of / + 1 tradable securities i = 0, . . . , / whose prices at node n are Sn = ( 5 ° , . . . , S^). Assume that one of the securities, the numeraire, always has a strictly positive value and, without loss of generality, assume it is security 0. Define discount fac-tors (3n — -go V n € Nt, and the discounted price (relative to the numeraire) Zxn = /3nS^, Vi = 0,. . . , I where Z° = 1 V n 6 Nt, V t — 0, . . . , T. The market can be complete or incomplete. Let 6ln be the amount of security i held by the investor in state n € iV t. Thus, the portfolio value in state n € Nt is i Zn-0n = 'Y^Z]l-61n . t=0 We assume a class of investors that do not influence the prices of any security and trade at every time-step based on historical information up to time t. 5 Arbitrage Arbitrage refers to the opportunity of making a sure profit out of nothing (usually through the purchase and sale of assets). In our framework, arbitrage reduces to finding a portfolio with zero initial value whose terminal values, obtained through self-financing strategies, are nonnegative for any scenario and for which at least one of those is strictly positive and has a positive probability of occurring. D e f i n i t i o n 1.2.2 ( A r b i t r a g e ) There is arbitrage if there exists a strategy {Qn}n€Nt,0<t<T ' where 6n is the quantity of security i held in state n € Nt, such that z0-e0 = o Zn • {On - &a(n)) = 0 , V n G Nt , V t = 1, . . . , T. Zn • 0n > 0 , V n G NT , and P {Zm • 0m > 0} > 0 , for some m G NT • Assume that there is no arbitrage opportunity in our financial market. This assump-tion guarantees, under this framework, the existence of a martingale measure for {Zt}t=0 T (King (2002), Theorem 2.2). 6 Completeness Completeness of a financial market refers to the capability of replicating any sequence of payoffs. That is, a financial market is said to be complete if given a sequence of payoffs, it is always possible to construct a portfolio whose values coincide with such sequence of payoffs, for each possible scenario. Definition 1.2.3 (Completeness) The financial market is said to be complete if and only if given a sequence of payoffs {Cn}ne^t o<t<r > ihere exists a strategy where 6ln is the quantity of security i held in state n G Nt, such that Zn-6n = cn, VneNt, t = Ot...,T. A financial market is said to be incomplete if it is not complete. We do not make any assumption regarding the completeness of the market. Therefore, all the results developed in this chapter applied to both complete and incomplete markets. 7 1.3 Basic Model Consider the model Max g s.t. ne NT u(zn-en)P, (l.i) = / W o V n G Nt Vt = l , . . . , T where £/(•) is assumed to be a strictly increasing concave function and W 0 > 0 is the initial wealth. This model applies to several relevant cases including • self-financing strategies, when Ln = 0 for all n G Nt , V t = 1,..., T. This case is relevant to study, for example, the problem of finding an arbitrage (let Wo = 0 and U(x) be the identity function); and • the case of strategies that consider the payment of liabilities, when Ln > 0, and/or the inclusion of future endowments, when Ln < 0. An example of this case is the problem faced by the writer of a contingent claim who receives a payment of Fn (= Wo) monetary units an who is obliged to pay an amount of Fn (= Ln) in scenario n G Nt, for t = 1 , . . . ,T. 1.4 Risk Management Modelling Risk management is incorporated to Model (1.1) through the addition of a risk tolerance con-straint that specifies the maximum risk tolerated, where risk is measured in terms of CVaR. This risk constraint can be established by the investor or by a supervisory/regulatory institu-tion, such as a central bank. In the case that this constraint is established only by investor's risk concerns, it is natural to think that these risk concerns can be, in theory, included in 8 the investor's utility function. However, in practice, determining the precise form of the investor's utility function is rather difficult. Therefore, the inclusion of this risk constraint in Model (1.1) can be seen as a practical way of approximating the true investor's utility function. Conditional Value at Risk (CVaR) Let l(x, y) a loss function, where x represents a decision and y symbolizes the future (stochas-tic) values of relevant variables (e.g. interest rate). Then, the Conditional Value at Risk associated with the loss function / and a level of confidence a, which we denote by <fra(x), is defined as the mean of the a-tail distribution of the loss function l(x, •). This distribution is defined as and VaRa(x) is the Value at Risk associated with decision x at a confidence level a. Value at Risk (VaR) is a popular measure of risk which aids to define a threshold for the maximum loss associated with a loss function l(x,y). VaRa(x) is defined as where #(2,77) is the distribution of the loss function l(x,y), i.e. y(x,V) = P{l(x,y) <rj} , (1.2) VaRQ(x) = Inf {q\^{x,ri) > a} 9 where ^(x,r)) is defined in (1.2). Conditional Value at Risk (CVaR) is a measure of risk with several important properties not shared by VaR (Pflug (2000), Rockafellar and Uryasev (2000), Rockafellar and Uryasev (2002)). For instance, • if the loss function l(x,y) is convex (sublinear) with respect to x then Qa(x) is convex (sublinear), and • <£a(x) behaves continuously and has left and right derivatives with respect to the level of confidence a € (0,1). The convexity of a risk measure is an appealing property since it means that, under such risk measure, diversification does not increase the risk. Continuity, on the other hand, assures that slight changes on the level of confidence imply slight changes on the risk of the underlying position. These two properties are a direct consequence of the following useful way of expressing 4>a(x), * a(x) = Min „ FQ{x,ri) (1.3) where Fa(x,n) = TJ+^E [(l(x,y) - n)+] and (x)+ = max {x,0} (Rockafellar and Uryasev (2002)). Formula (1.3) is useful not only for modelling CVaR within optimal investment problems (as described below), but also for obtaining VaR (for the same level of confidence) simultaneously since VaRa(x) e Argmin , Fa(x,n) (Rockafellar and Uryasev (2002)). 10 Risk Tolerance Constraint The risk tolerance constraint that is considered for Model (1.1) is *a(6) < RT , (1.4) where RT is the maximum tolerated loss that the investor is expecting to face if the loss surpasses the Value at Risk threshold and whose amount could be directly established by the investor or indirectly set by a supervisory institution. RT is presumably, although not necessarily, chosen in such a way that it is not redundant with respect to the utility function £/(•), i.e. that restriction (1.4) effectively constraints the set of optimal solutions of Model (1.1). For instance, the investor could solve Model (11) to obtain an optimal strategy 6** with an associated conditional value at risk of <1>Q(#**). Hence, if <ba(9**) is not affordable by the investor and/or not allowed by a relevant supervisory institution, RT should be equal to a value strictly less than <E>a (#**). Constraint (1.4) is, by (1.3), equivalent to Min „ Fa(6,r)) < RT or, Fa(0, r))<RT , for some n e 3? i.e., 11 n + Y~E l(!(0>y) - ^ R T . f o r s o m e V € & (1.5) Constraint (1.5) reduces to V + Z") ~ - ^ T ' f o r s o m e ^ G ^' C1-6) which, within any optimization problem, can be modelled as * / + ( l ^ ) E „ e J V T * n P » < #T - s „ + J(0, jfe) - n < 0 i V n £ NT (1.7) -s n < 0 , V n G NT since (sn)neNr satisfying (1.7) and such that sn > (1(8, yn) — n)+ for some n 6 NT is never better than (sn)neNT, where sn = (1(0,yn) - n)+ V n € 7V T . Clearly, (s n ) n e W T satisfies (1.7), and hence (1.6), but also in a less restrictive manner, thus allowing for a broader set of feasible solutions and therefore leading to a better, or at least not worse, objective function value. Within the context of portfolio management, a particular relevant case for the loss func-tion /(•, •) is I (6, Zn) = -(Zn-9n- [30W0) for n G NT. The model is . s.t. Zo • 6o Zn • {On — 0a(n)) *7 "b ( l ^ ) 2~2neNT SnPn -sn-(Zn-On-p0W0)-r, <0 < / W o -f3nLn , V n e J V , , V t = l T . (1.8) RT , V n € NT , V n G NT 12 and whose dual (Zangwill (1969)) is Min a,x,y £ n € i v T (U(Zn • 9n) - U'(Zn • 0n)(Zn • 6n)) P n +(RT - p0W0)x + (yofoWo - £ t r = 1 £ n e / V t 3/.AA.) s.t. If (Zn-6n)pn + x n - y n = 0 , V n G NT VnZn - 2~2meC(n) VmZm = 0 , V Tl G NT, Q gs v* = o , . . . , r - 1 . x » ~ x te) ^ 0 , V n G JVy -x < 0 - « „ < 0 , V n G NT • where: 1. yo corresponds to the initial portfolio's value constraint (first restriction of the primal problem); 2. y = (yn)neJv t te{i .,T} ' s associated with the re-balancing conditions over the portfolio's value at each period (second set of restrictions of the primal problem), and 3. x = (x, {xn)neNT) corresponds to the risk tolerance constraint (specified by the third and the last two sets of restrictions of the primal problem). 1.5 Embedded Probabilities The dual problem contains embedded probability measures that yield different assessments of the probability of surpassing the VaR threshold. These probability measures and their relationship are described in the following propositions. 13 Proposition 1.5.1 (Embedded Probabilities) Let (0*,x*,y*) be an optimal solution of the dual problem (1.9), where x* = (x*, (x^)neNT) and y' = (l/n)n€Afc. t=0 T' T H E T L ' i) Q* = (ln)neNT' w/iere Qn = (jj. ) V n € NT, defines a martingale measure of {Zt}t=0 T in(Q,F). ii) If x* > 0, i.e. if the risk tolerance constraint is binding for the optimal solution, then P = (Pn)N<ENT, where pn = (j^j V n G NT, defines a probability measure on (Q,F). iii) P = (pn) € N , where pn = u ?^<'VP" . V n G N T , defines a probability measure in (n,F). Proof: i) From (1.9), y*n = If (Zn-9*n)pn + x*n £/(•) strictly increasing and x* > 0 implies y* > 0 V n G NT and hence, y$ = J2neNT Vn > 0-Therefore, 1) q* = | > 0 V n G NT ; . 2) 1 = 2~2neNT ( § ) = ZneNT C a n d 3) ^n = E m e c(n) ( § ) ^ m V n G i V t , V t = 0,..., T - 1.. That is, £ « ' [ Z t + 1 | F t ] = Z t , vt = o , . . . , r - i . 14 Hence, {Zt}t=0 T is a C}*-martingale. ii) By hypothesis and from (1.9), !) Pn = > 0 , V n € NT; and 2 ) 1 = £ „ e j v T (ff) = E„ eyv rPn- H e n c e - (Pn)neNT defines a probability measure in (Q, F). iii) U(-) strictly increasing implies U'(Zn • 9n)pn > 0 for all n € NT. Therefore, ' ^neNT E n e W l.t/ '(z„.e-) P n - 2^neNTPn • Q.E.D. Remark 1.5.1 Q* is not necessarily the unique martingale measure of {Zt}l=0 T , since our framework allows for incomplete markets. Indeed, for any feasible solution y of Problem (1.8), Q = (^) defines a martingale measure of {Zt}t=0 T . These three probability measures, Q*, P, and P, have special and, in general, different meanings. Q* is a martingale measure and hence linked to arbitrage pricing. P is a measure depending directly on the preferences of the investor and standard for models such as Model (1.1). P is shown later on to be a conditional distribution. We are particularly interested in comparing and interpreting the different assessments of the event of surpassing the VaR threshold, when using an optimal strategy, under Q*, P, 15 and P. We define the latter event in terms of the (primal) variables of Problem (1.8). Definition 1.5.1 Let (r)*,6*,s*) be an optimal solution of problem (1.8). Define A = {n G NT\s*n > 0} , i.e., A is the event of surpassing the VaR threshold. Before assesing the probability of A under Q*, P, and P, we state some basic relations between VaR and CVaR within the context of Problem (1.8) and describe precisely which conditional distribution P is. Proposition 1.5.2 Let (r/*,9*,s*) be an optimal solution of problem (1.8), and A as defined in 1.5.1. Then, i) If A = ® then <Pa (0*) = VaRa (9*). ii) IfA^tD then $ Q (9*) > VaRa (9*). Moreover, ifZneA(t^)<lthen $a {9*) = VaRa (9*) + Ep [s*T] (1.10) where P is any probability measure in (ft, F) that satisfies pn = ^ for all n G A. Proof: Observe that 16 Under the hypothesis EneA (l?^) < 1, it is clear that there exists a probability measure P such that pn = p„/(l — a), for all 1 n 6 A. Therefore, M A ' ) = (**) + £ „ < = * P n < = Vai** (0*) + E n e X P n ^ + E „ ^ P n < = VaRa(9*) + Ef'[sT\ . Q.E.D. The next two propositions show that, under certain conditions, P not only satisfies the conditions in Proposition 1.5.2 ii), and hence (1.10) holds under P, but also coincides with the ct-tail distribution of the loss function i(0,zn) = -{zn-en-powo). Proposition 1.5.3 If x* > 0, i.e. if the risk tolerance constraint is binding for the optimal solution, and A ^ 0 then i) $ a (0*) = RT = VaRa (0*) + Ep [s*T]. ii) If \A\ > 2 then (Z) (:::) — 1For instance, let pn = p„/{l — a) for all n € A and, if A ^ NT, for ne N T \ A . 17 Proof: i) From Proposition 1.5.1, P = (pn)neNT is a probability measure and thus it is satisfied I > < 1 • (1.11) From the dual problem (1.9) and complementarity, < = W - ^ ) V n e A , V 1 — a ' which, under the assumption x* > 0, implies x* 1 — a Therefore, from (1.11) and (1.12), 3> = E ( r ^ j £ i -Thus, the hypothesis of Proposition 1.5.2 ii) is satisfied and hence $Q (0*) = VaRa (0*) + Ep [s*T]. Finally, from the assumption x* > 0 and applying the property of complementarity to the corresponding primal restriction leads to $a (0*) = RT . 18 ii) Let n^rij e A. Then, from (1.12) pni = and p = -— I - a ' I - a Therefore, Pru = PJH Pnj Pn; Q.E.D. Proposition 1.5.4 If x* > 0, i.e. if the risk tolerance constraint is binding for the optimal solution, then P is the a-tail loss distribution. Proof: From Proposition 1.5.1, P is a probability measure, and from (1.12) P satisfies that pn = ^ V n € A. Let n € NT \ A. Hence, by definition of A, s* =0 . If pn > 0 then, by definition of P, x*n > 0 and thus, by the application of the complementarity property to the corresponding primal O = < = -(Zn-0*-0oWo)-ri'. = -(Zn-8'-aoWo)-VaRa(6*) Therefore, -(Zn-0n-(3oWo) = VaRa(e*) . 19 That is, pn > 0 for n G NT \ A implies that scenario n corresponds to the VaR scenario. Thus, for any other scenario n € NT \ A such that — (Zn • 0* — (3<}Wo) ^ VaRa (0*) it must be satisfied that pn = 0. Hence, Pn = < ( iine A. l - r ^ E ^ P n - ^ T f " ^ i f n € i V T \ A a n d - (Zn • 6*n - (30W0) = VaRa (6*). otherwise. Therefore, P is the a-tail loss distribution. Q.E.D. We now obtain a lower bound of the probability of surpassing VaR under P. This is useful to compare the assessment of the probability of such event under the different embedded probabilities. Corollary 1.5.1 If x* > 0, i.e. if the risk tolerance constraint is binding for the optimal solution, then P(A) > 1 PVaR (1-a) where pVaR = Pn and n € NT is such that — (Zn • 6*n — POWQ) = VaRa (0*). Proof: By definition of VaR, neNT\A 20 Hence, < P V a f l - ( E „ g , V T \ J 4 P " ) + 1 1—a — {T,neAP"\ , PVaR \ 1 - Q J ^  1 - a where the last equality follows from Proposition 1.5.4. Thus 1 < p{A) + ?™ (1.13) 1 — a v ' The result follows from (1.13). Q.E.D. The main result that establishes the relations among the embedded probabilities (Q*, P, and P) and P is presented in the following theorem. Theorem 1.5.1 (Relations among the embedded probabilities) 0 - s ) « ^ s ) ( f ^ ) - ™ Proof: From Problem (1.9) if (Zn • 6n) pn + x*n = y*n , V n E NT (1.15) and hence 21 U'(Zn-en)pn + x*n = y*n , V n e A Therefore, neA neA neA SO v y o / neA \yo/ n e A „ e A % B y complementarity, 1 — a Thus E< = (T^)^" = (T^) F ( A )' ( U 7 ) neA v ' n 6 X x 7 Hence, combining (1.16) and (1.17), leads to (s )£**-^ + (s ) (^)-£^™- <-> Now, from (1.15) and dual problem (1.9) 22 V0= Y.U'^-en)Pn+ E < = £ C / ' ( Z n - ^ ) p „ + x* neNT Thus, (1.19) Therefore, combining (1.18) and (1.19), V yJ [zneNT u' (zn • en)Pn)  + U J V - J ~ Q { ) which, by definition of P, implies Q.E.D. Whether the risk tolerance constraint is binding or not and/or whether the investor or decision maker is risk-neutral or not, Theorem 1.5.1 allows us to compare the embedded probabilities and P with each other. The following corollaries and lemmas establish these comparisons. C o r o l l a r y 1.5.2 If x* = 0, i.e. if the r isk tolerance constraint is not b i n d i n g , then Q*(A) = P(A). 23 Proof: The result is a direct consequence of applying condition x* = 0 to Equation (1.14). Q.E.D. C o r o l l a r y 1 . 5 . 3 IfU(x) = x for all x in its domain, i.e. if the i n v e s t o r i s r i s k - n e u t r a l , then P(A) = P(A) and hence, Proof: Observe that p(A) = ^ neAUT}?;'e:\Pn = = £ p » = p ( A ) where the second inequality is due to the risk-neutrality assumption on the investor prefer-ences. The result follows from substituting the above expression in Equation (1.14). Q.E.D. L e m m a 1 . 5 . 1 / / U(x) = x for all x in the domain of £/(•), i-e. if the i n v e s t o r i s r i s k -n e u t r a l , then Q*{A) > P(A). Moreover, the latter i n e q u a l i t y i s s t r i c t if x* > 0, i.e. if the r i s k t o l e r a n c e c o n s t r a i n t i s b i n d i n g . Proof: From Equation (1.20) Q*(A) = P ( A ) [ l - | + ^ = ^ ) h ( g ) t e ) " > P(A). 24 Finally, the latter inequality is strict if x* > 0. Q.E.D. C o r o l l a r y 1 . 5 . 4 / / x* > 0, i.e. if the r i s k t o l e r a n c e c o n s t r a i n t i s b i n d i n g for the optimal solution, then Proof: From Equation (1.12) pn = ^ V n 6 A. Therefore, Then, by Theorem 1.5.1 the result follows. Q.E.D. L e m m a 1 . 5 . 2 lfx*>0 ,i.e. if the r i s k t o l e r a n c e c o n s t r a i n t i s b i n d i n g for the optimal solution andpvaR < ( l - P{A)) (1 - a), then Q*(A) > P{A). Proof: From Corollaries 1.5.4 and 1.5.1, Q\A) = (l - P{A) + U\ P(A) > ( l - | ) ^ ) + ( | ) ( l - T ^ ) > Min{P(A),l-&i*} Hence, if condition pyaR < ( l — P(A)) (1 — a) is satisfied then 25 Min^P(A),l-^^j =P(A) . Therefore Q*{A) > P{A). Q.E.D. Corollary 1.5.5 IfU(x) = x for all x in its domain, i.e. if the investor is risk-neutral, and x* > 0, i. e. if the risk tolerance constraint is binding for the optimal solution, then ( 1 - 5 ) ^ ! ) ^ - < ™ -Proof: This follows from Corollaries 1.5.3 and 1.5.4. Q.E.D. In summary, if the investor is risk-neutral or the risk tolerance constraint is binding and the probability of losing an amount equal to VaR is small enough, then the probability of surpassing VaR is larger or equal under Q* than under P or P respectively. Otherwise, the probability of surpassing VaR under Q* is a convex combination of the corresponding probability assessments of P and P. Moreover, the probability of surpassing VaR is strictly larger under Q* than under P or P if the risk tolerance constraint is binding and, respectively, the investor is risk neutral or the probability of losing an amount equal to VaR is small enough. Therefore, if the risk tolerance constraint is binding, Q*(A) is in some sense a measure of how risky it is to operate at the limit of the tolerated risk. Indeed, Q*(A) has the interpretation 26 Q*(A) = EQ' [1A] (1.21) where 1 if to E A 0 otherwise Hence, since Q* is a martingale measure, Q*(A) can be interpreted as a no-arbitrage price of a binary option (Hull (2003)) that provides one monetary unit in the event of a loss greater than VaR. Therefore, if the market is complete, Q* (A) can be regarded as the cost of hedging one monetary unit of loss beyond the VaR threshold. The latter interpretation of Q*(A) serves as a motivation to define an implicit risk measure for the optimal strategies of Problem (1.8). This risk measure is described in the following section. 1.6 Implicit Risks The assessment of the probability of surpassing the VaR threshold under the different em-bedded probabilities provides us with an indication of the implicit risks associated with an optimal strategy for Problem (1.8). However, these embedded probabilities do not directly define measures of risk in the sense of quantifying the required capital to hedge for the im-plicit risks of such optimal strategy. From this perspective of capital requirement, a measure of risk is defined (Follmer and Shied (2002)) as follows: Definition 1.6.1 (Monetary Measure of Risk) Let p : X — > 3? be a mapping, where X is a class of the financial positions. Then, p is said to be a monetary measure of risk in X, if it satisfies the following two conditions: 27 • (Monotonicity) Let X,Y € X. Then, X <Y a.s. implies p(X) > p(Y). • (Translation Invariance) Let X G X and m S E Then, p(X + m) = p(X) - m . Within the framework of Problem (1.8), the class of financial positions that we consider is the set of the feasible investment strategies of (1.8) for different values of risk tolerance parameter RT in a relevant convex set S (e.g. S = [0, RT], where RT < RT0 = <f>a(0**) and 6** is an optimal strategy for Problem (1.1)). That is, we consider X = {6(RT)\RT e S and 6 is a feasible solution of Problem (1.8)} which, under the linearity of the restrictions in (1.8) and the convexity of S, is a convex set. VaR and CVaR are monetary measures of risk in the sense of Definition 1.6.1 (Follmer and Shied (2002)). Indeed, VaR and its multiples are in practice capital requirements of some sort for many financial institutions (Jorion (2000b)). However, VaR has the disadvantage of not providing information about the extent of the losses beyond the threshold amount that it defines. On the other hand, CVaR does quantify losses beyond the VaR threshold and hence a capital requirement equal to CVaR should cover, at least, for the expected shortfall beyond VaR. Nevertheless, CVaR could be much larger than VaR and so it could be very costly as a capital requirement. Nonetheless, in principle, there is no need to require for the whole expected shortfall, CVaR - VaR, but rather for the premium of an insurance, or a hedge, that will ensure for the payment of the expected shortfall in case the loss goes beyond the VaR threshold. In other words, it would be enough to require the price of a binary call option 28 whose underlying is the loss of the portfolio and which pays (CVaR - VaR) with a strike price equal to VaR. From observation (1.21), and denoting CVaR and VaR (as before) by $ a (0) and VaRa(6) respectively, for a feasible strategy 6, it follows that, under the assumption of completeness of the market, such binary option would cost Q*(A) ($ A (0) - VaRa(9)) . (1.22) Therefore, a natural capital requirement that takes into account losses beyond the VaR threshold is VaRa{6) + Q*(A) (Qa(9) - VaRa(6)) or equivalently, (1 - Q*(A)) VaRa(6) + Q*{A)<S>a(6) . (1.23) The latter measure, called the Expected Shortfall Hedge (ESH), is a monetary risk mea-sure (See Proposition 1.6.1). When the market is incomplete Q* is only one of several martingale measures of the discount price process {Zt}t=0 T . Hence, (1.22) is just one possible no-arbitrage price of a binary option that pays the expected shortfall loss beyond the VaR threshold. However, the election of a different martingale measure Q does not change most of the results that are next obtained for ESH. For instance, if other martingale measure Q is chosen, the associated expected shortfall hedge ESH®, i.e. expression (1.23) but with Q instead of Q*, will still 29 be a monetary measure of risk and will posses the basic properties described in Proposition 1.6.2. Nonetheless, it is important to emphasize that under the generality of allowing for incompleteness of the market, ESHa would not necessarily superhedge for the expected shortfall, i.e. it would not necessarily be enough to provide for a payoff of at least the expected shortfall in all possible scenarios. Therefore, in this case, the decision maker should consdider Q* and the other martingale measures in his analysis (see Follmer and Shied (2002) for a detailed treatment on this issue). P ropos i t i on 1.6.1 Let ESHa(6) = VaRa{6) + Q*(A) (*a(0) - VaRa{9)), for a E (0,1). Then, ESHa is a monetary measure of risk. Proof: Let 9U 92 E X. • Monotonicity: Assume that 9\ > 92 a.s.. Then, given that VaR and <&a are monetary measures of risk, it is satisfied that VaRaiOi) < VaRa(92) *a(0i) < $«(0 2 ) Since Q*{A) G [0,1], then (1 - Q*(A)) € [0,1] and so ESHa(9x) = ( l - Q * ( A ) ) V a i ? Q ( 0 1 ) + g*(A)$ Q (0 1 ) < (1 - Q"{A)) VaRa{92) + Q*(A)$a(82) = ESHa(92). • Translation Invariance: Let m £ S . Hence, ESHafa + m) = (l-Q*(A))VaRa(91+m) + Q*{A)<f>a{91 + m) = (1 - Q*(A)) (VaRa(91) - m) + Q*(A) ($a(0,) - m) = (1 - QT {A)) VaRa(9i) + Q'iA)*^) - m = ESHa(8i) - m , 30 where the second equality follows because VaR and CVaR are monetary measures of risk. Q.E.D. The essence of proving that ESH is a monetary risk measure relies on expressing it as a convex combination of VaR and CVaR. Indeed, from substituting VaR and CVaR by any other pair of monetary measures of risk in the proof of Proposition 1.6.1, it can be concluded that any convex combination of monetary measures of risk is also a monetary measure of risk. Therefore, are also monetary measures of risk though these measures do not share the hedging meaning of ESH. Moreover, the definition of ESH as a weighted average of VaR and CVaR also implies that ESH inherits certain properties shared by both VaR and CVaR. For instance, VaR and CVaR are positive homogeneous for any loss distribution and therefore, ESH is also positive homogeneous. Furthermore, although VaR is not a convex risk measure for any loss distribution, ESH is convex whenever VaR is convex. The latter claims are formalized in the next proposition. (1 - P(A)) VaRa(0) + P(A)$a{0) and 31 P r o p o s i t i o n 1 . 6 . 2 ( B a s i c P r o p e r t i e s o f E S H ) Let a G (0,1). Then, i) ESHa is positive homogeneous in X for all a G (0,1). ii) If VaRa is convex in X, for a G (0,1), then ESHa is convex in X. Proof: i) Let A > 0, a G (0,1) and 9 G X. Then, ESHa{\6) = (l-Q*(A))VaRa(X9) + Q*(A)$a(X9) = (1 - Q*(A)) [\VaRa{e)\ + Q*(A) [X$a(9)} = X[(l-Q*(A))VaRa(9) + Q*(A)<l>a(9)} = XESHa(9) where the second equality is due to the fact that VaR and CVaR are positive homogeneous (Follmer and Shied (2002)). ii) Since the loss function 1(9, ZN) = — (ZN • 9n — f30W0) for n G NT is convex in 9, then CVaRa is convex in 9 (Corollary 11 in Rockafellar and Uryasev (2002)). Therefore, ESHa(X9 + (l- X)9') = (l-Q*(A))VaRQ(\e + (l-X)9') + Q*(A)$a (X9 + (l-X)9') < (1 - Q*(A)) [XVaRa(9) + (1 - X)VaRa(9')] + Q*(A) [A$ a(0) + (1 - X)$a(9')] = A [(1 - QT (A)) VaRa(9) + Q*(A)$a(6)] +(1 - A) [(1 - Q*(A)) VaRa(9') + Q*(A)<S>a(9')] = XESHa(9) + (1 - X)ESHa(9') for any A G (0,1) and 9,9' G X. Hence, ESHa is convex. Q.E.D. 32 Proposition 1.6.2 states that ESH is, at least, as appealing as VaR in the sense that ESH is positive homogeneous, as is VaR, and convex whenever VaR is convex. Within the context of the optimal investment Problem (1.8), ESH is convex for a broader class of loss distributions than VaR. This is shown in the next proposition, for which we require first the following preliminary result: Lemma 1.6.1 Let a £ (0,1) and assume the following: 1. The investor of Problem (1.8) is risk neutral with a maximum tolerated risk equal to its initial wealth, i.e. RT = PQWQ. 2. Q = P. If the corresponding dual problem (1.9) has a solution (x,y) such that xn ^ 0 , VneNT , (1.24) and, > Pn , V n,ne NT (1.25) Pn then there exists an optimal solution (x*,y*) of Problem (1.9) that satisfies (1.26) for some (3, (3+ € [a, 1) and (3+ > p. 33 Proof: Let (x, y) be an optimal solution of Problem (1.9) that satisfies conditions (1.24) and (1.25), under assumptions 1 and 2. If (x,y) satisfies (1.26) then lemma's claim holds. Otherwise, we will construct, departing from (x,y), another solution (x, y) of Problem (1.9) that does satisfy (1.26) for some 3, 3+ G [a, 1) and 8+ > 3. Let 3 = a and 3+ G [a, 1). The idea is to scale x and y in such a way that the derived pair (x, y) is feasible, optimal and satisfies condition (1.26). That is, we find 5 > 0 and 7 „ > 0, for all n G NT, such that xn = 7„x n , yn = Syn , \/ n E NT , (1.27) is a solution of (1.9) and satisfies (1.26). To show this, we primarily focus on the first set of constraints of (1.9) and condition (1.26). The pair (x,y), as defined in (1.27), satisfies such conditions if and only if Pn + lnXn = <fyn , V U G NT (1.28) and, ^2neNT lnXn Syo a + 1 -a (1.29) . ( l - o + ) 2 where we recall that 3 has been chosen equal to a. The set of conditions defined by (1.28) and (1.29) form a system of \NT\ + 1 linear equations with \NT\ + 1 variables. This system always has a solution. For instance, let 34 m(/3+) EE l - + l Therefore, from (1.28) and (1.29), and provided that m(/3+) ^ 1, we obtain y 0 ( l -m( /3+) ) from which, using (1.28), _ 3 M - P n j / o ( l - m Q g + ) ) > ~ z „ y o ( l - r o ( / 3 + ) ) where S and jn are not necessarily positive unless the value of [3+ is chosen appropriately. Later on we establish conditions on fl+ under which m(/3+) ^ 1, and 6 and 7„'s are strictly positive. Therefore, (x, y), as defined in (1.27), satisfies by construction condition (1.26) and the first set of constraints of Problem (1.9). The vector y clearly satisfies the second set of constraints of Problem (1.9) and if we define x — ^ ^ xn neNT then, x satisfies the last three sets of constraints of Problem (1.9). Hence, the only constraint of Problem (1.9) that remains to be verified by (x,y) is Pn 35 V nE NT . (1.30) To prove (1.30) recall that, by assumption, x is optimal and thus it must be satisfied that l - a , V n G NT • Let n G NT- Therefore, 7n*n < m ^ t e ) it) 1 fe) We claim that assumption (1.25) implies that I n 7« > \xn J \_Syn-pn J f5a. ^ (M2k) for all n , n G Af T . If the latter inequality holds then 2~2neNT ( 1 % ) (~x) — E n e N r ( x ) ( y ^ ) = ( i f ) (?M) ^ n€NT Vn " (5) = ft) = I (1.31) where the last equality is due to the assumption Q = P. Hence, 36 < 1 Therefore, ^ » 6 " T ( W ) ( " ? ) To prove (1.31) it is enough to show that Syn-Pn \yn (1.32) since xn > 0, for all n € AV- Observe that dm{8+) d(3+ a ( l — cv+)2 and so, d(i-mV)) d/3+ (1 - m(/J+))J < 0 . Therefore, m(/3+) and 1_m1(j9+) are strictly decreasing functions for all /3 + € 5?. If /3 + is ( l - a + ) 2 , 2 restricted to the interval (1 — v X_J , 1 — (1 — c r ) ) then 0 < m(3+) a + i < 1 . 37 That is, the function m is bounded if 8+ € (1 - ^ , 1 - (1 - « + ) 2 ) - Thus, as 3+ approaches ^1 — ^ 7_a^) f r o m t n e r i § n t ' t n e u i m t of m(8+) exists and equals lim m(/?+) = 1 This implies that lim = — . T ^ T T = co . (1.33) F- -Q (1T-^)2) y0(l-m(/?+)) Therefore, On the other hand, assumption (1.25) implies Hm T2Vn~Pn =  V~ (1-34) - [j/nPfi - 2/nPn] > 0 , V Tl E NT • (1.35) 2 •0/3+ W + ) l t e - P f t ) That is, j(^+)"-Zp? is a monotone increasing function under assumption (1.25). Hence, (1.35) and (1.34) imply (1.32) (See Appendix 1). To complete the proof, we now establish appropriate conditions on 3+ to guarantee the nonnegativity of 5 and the 7„'s as well as the condition m(6+) ^ 1. Recall that 3+ E (1 - ( i i r ? ! > 1 - (1 - o + ) 2 ) C [a, 1) implies 0 < m(B+) < 1 and so 38 However, the nonnegativity of 5 does not ensure that the 7„'s are nonnegative too. Further restrictions on ft+ need to be imposed. Recall that 7 n = (Syn—pn)/xn, for all n 6 NT- Hence, by (1.33), P+ can be chosen ( ( l - a + ) 2 \ 2 sufficiently close to I 1 — v X_J 1, but above it and less than 1 — (1 — a+) , such that we assure that all the 7„'s are positive. That is, there exists P 6 I 1 — ^  X_J , 1 — (1 — a+)2 1 such that if 0+ € (l - ^~"+J ,p\ then Xrt Therefore, if 3+ e (l - ^ , P that ( x , y ) is also optimal. , then ( x , y ) is a feasible solution of (1.9). We now prove By assumption, the investor is risk neutral and the maximum tolerated risk is the initial wealth. Hence, the dual objective function of (1.9) associated with ( x , y ) reduces to VOPOWQ - £ £ VnPnLn t=l neNt Thus, since multiplying the objective function by a positive constant does not alter the point at which the optimal is reached, (x , y ) must be an optimal dual solution associated with 6. Q.E.D. 39 Under the assumptions of Lemma 1.6.1, there are multiple optimal solutions. Proposition 1.6.3 (Convexity of ESH) Suppose that the assumptions of Lemma 1.6.1 hold, and assume that the cardinality of the sample space \Q\ = \ N T \ is such that given 0\, 02 € X, A € (0,1) and a G (0,1), there exists a probability measure P', defined over Q, that satisfies VaRay (0) = VaRa,P(0) , for 0 = 0U02, X9i + (1 - \)02 $+p, (0) = $+p(0) ,fore = 61,92, A0i + (1 - X)02 Let (x*,y*) be a solution of the dual problem (1.9) that satisfies condition (1.26), and let ESH be the expected shortfall measure associated with this solution. Then, ESH is a convex measure of risk in X. Proof: By definition of ESH (see Equation (1.23)), ESHa(9) = (1 - Q*{A)) VaRa{0) + Q*(A)$a(9) , V 0 G X , (1.36) where Q* is defined as in Proposition 1.5.1. On the other hand, $ a can be expressed as (See Proposition 6 in Rockafellar and Uryasev (2002)) = ( ^ f £ ^ ) VaRa{0) + (^r~rP) *a (<?) , V 9 G X . (1.37) where *+(0) = Ep [1(9, y)\l(0, y) > VaRa(9)\ and a+ = V(9,VaRa(9)). Therefore, substi-tuting (1.37) into (1.36) leads to 40 ESHa{9) = -Q*(A) (l^^VaRa{9)+ Q*(A) ( T £ ) ] < W ) > V 0 e * . (1.38) The essence of the proof relies on showing that, under the conditions stated in the proposition, given 61, 92 G X and A G (0,1), there exist a probability measure P' and a level of confidence P such that 1. (T(A)(^) = V 0 G * , a n d VaRa,P{9) CVaR+P(9) = VaR0P>(9) = CVaR+0pl(9) where B+ = * p (9,VaRp(9)) and 9 G {0i,0 2 ,A0i + (1 - A)6>2}- If these two conditions are satisfied, then ESHa,P(0) = [l - Q*{A) (T^)] VaRa,P(0) = l ^ ) v a R P t P , ( e ) + { ^ ) ^ ( 0 ) for 9 = fl,,02, A0i + (1 - A)02 and, therefore, ESHaiP(\91 + (l-X)92) = ^PtP>(X91 + (l-X)92) < A * A J y ( 0 1 ) + ( l - A ) * A i y ( 0 2 ) = XESHa<P{9{) + (1 - X)ESHa<P(02) where the last inequality is due to the convexity of CVaR (See Corollary 11 in Rockafellar and Uryasev (2002)). Thus, given that 0i, 0 2, and A are arbitrary, we conclude that ESHa<P 41 is convex in X. We now prove claims 1 and 2 to complete the proof. 1. Lemma 1.6.1 assures the existence of (x*,y*). By assumption, (x*,y*) satisfies x ¥0 1 - a l-a+J \ l - 0 for some 3, 3+ 6 [a, 1) and 3+ > 8. From Lemma 1.5.1, + 1 - -a (1.39) Q*(A) = P(A) 1 + (S) (*)] = [i+(s) (i^ >; Substitution of (1.39) into (1.40) leads to Condition 1. (1.40) 2. Given 9i, 92 6 X, and A G (0,1) we need to show that there exists a probability measure P' = (pn)neNr such that (a) VaRa,P(9) = VaRaP>{9) , for 8 = 9U92, A0i + (1 - \)92. (b) $+ P (0) = , for 9 = 01,02^0! + (1 - A)02. Conditions (a) and (b) determine a system of equations for every triplet (9i,82,X). Therefore, if the sample space is large enough, as it is appropriately assumed in the statement of the Proposition, we can always find a set of values (p'n)neNT that define a probability measure which solves this system of equations. Q.E.D. The convexity of ESH relies on the idea that, under certain assumptions, the Expected Shortfall Hedge of a given strategy, with a pre-specified level of confidence, is equivalent to 42 Density induced by P' Density induced by P ESHQ95P = C V a R o p -/ VaRo.gs, p- — VaR 0 g5i P 'o.95,P -4 -3 -2 -1 0 1 2 3 4 S 6 Figure 1.1: Convexity of E S H . This graph shows a density P (gray line) and a density P' (black line) such that the ESH, under P, coincides with the CVaR, under P', for a confidence level of a = 0.95 and where P' is obtained by distorting appropriately the tail of the P density. the Conditional Value at Risk of the same strategy but under a different probability measure and, possibly, a different level of confidence. Although, in principle, the alternative proba-bility measures under which the ESH is a CVaR could be quite different from the original distribution, we should focus on those which tend to be close to the original distribution given that this distribution should contain important historical and expert information. In partic-ular, given that ESH and CVaR are risk measures for. extreme losses, we should concentrate on those alternative distributions that are as close as'possible to the original distribution for losses up to the VaR threshold and therefore, differ from the original distribution only on the tail area, (see Figure 1.1). From Propositions 1.6.2 and 1.6.3 we know that, under certain conditions, ESH is both convex and positive homogeneous. Therefore, ESH is a coherent measure of risk in the sense 43 of Artzner et al (1999). Next corollary states this fact. Corollary 1.6.1 (Coherence of ESH) Suppose that the assumptions of Proposition 1.6.3 hold. Then, ESH is a coherent measure of risk in X. Proof: A direct consequence of Propositions 1.6.2 and 1.6.3. Q.E.D. 1.7 Conclusions and Further Research The motivation for this research was to assess the implicit risks of the strategies followed by an investor who maximizes the expected value of his final wealth, subject to budget and risk tolerance constraints. A discrete-time framework that allows to consider incomplete markets and distributions other than the normal distribution, including the typically observed fat-tailed distributions, is used. Risk tolerance constraints are considered in terms of a maximum tolerated loss, which is measured using a coherent and stable measure of risk such as the Conditional Value at Risk (CVaR). Implicit Risks are uncovered in terms of the likelihood of surpassing the standard Value at Risk (VaR) threshold. For instance, three embedded probability measures are determined and compared with each other when assessing the prob-ability of surpassing the VaR threshold. One of these embedded probability measures is a martingale measure and, hence, it is used to quantify the cost of hedging losses beyond the VaR threshold. In fact, it is proved that if this hedging cost is visualized as a capital requirement, then the mapping that associates this financial cost to a given strategy is a monetary measure of risk. Moreover, this monetary measure of risk, which we denominate Expected Shortfall Hedge (ESH), is proved to be a coherent measure of risk for a broader class of loss distributions than those for which VaR is coherent. The other two embedded 44 probability measures are related to the particular utility function of the decision maker and the risk tolerance constraint. Finally, it is also proved that if the risk tolerance constraint is binding, then the probability of surpassing the VaR threshold and the cost of hedging losses beyond the VaR threshold are strictly larger under the embedded probabilities than under the natural probability measure. Therefore, this research particularly emphasizes the importance of considering these implicit risks, especially when the selected strategy operates at the limit of the tolerated risk. Further research can be done in at least two main directions. A first direction should be in the line of extending the results obtained in this chapter by considering other convex risk measures and/or other type of loss functions and/or dynamic risk measures. A second and broader direction should be based on the observation that the inclusion of a convex risk measure, such as CVaR, in a stochastic programming model induces another implicit risk measure, such as ESH. Therefore, a problem to address is whether, in general, the inclusion of convex risk measures in a stochastic programming model always induces implicit risk measures. The inverse problem is also of interest. That is, given a risk measure, is there an associated stochastic programming model that induces such risk measure? 45 Chapter 2 The Duality of Option Investment Strategies for Hedge Funds 46 2.1 Introduction Hedge Funds are private investment partnerships that attempt to obtain superior risk-adjusted returns in any market condition for their mostly wealthy investors. Although hedge funds have long existed1, it was until the collapse of the Long-Term Capital Management (LTCM) hedge fund in 1998 that the influence and role that these institutions play in the financial markets was fully realized (e.g. de Brouwer (2001)). The high-profile (four billion dollars) failure of L T C M jeopardized several large financial institutions and, according to the U.S. Federal Reserve, the world economy.(e.g. Jorion (2000a)). The L T C M event attracted immediately the attention of academics, who were interested in understanding well what actually happened and which was the real nature of these financial institutions. Early studies on hedge funds, such as Fung and Hsieh (1997), Eichengreen, et al. (1998), Fung and Hsieh (1999), Ackermann, et al. (1999) and Brown, et al. (1999), among the most important, were particularly difficult to realize since, at that time, information about hedge funds, both qualitative and quantitative, was not freely available for the general investment public. This is because hedge funds deal with sophisticated investors, such as wealthy and institutional investors, for which standard disclosure information regulations do not apply. Over the past six years, the number of hedge funds has doubled and the assets under management by the hedge fund industry have grown exponentially, being one of the main reasons of this phenomena the significant increment of asset allocation to hedge funds by institutional investors (Fung and Hsieh (2006)). 1The term hedge fund describes the "hedge" against risk that some of these partnerships aim for through their strategies. The first official hedge fund was founded in 1949 by Alfred Winslow Jones in the United States. For more on the history of hedge funds, see Eichengreen, et al. (1998). Early hedge funds, such as the Chest Fund at King's College (Cambridge) which was managed by J .M. Keynes from 1927 - 1945, are discussed in Ziemba (2003), and Ziemba (2007). 47 Academic research on hedge funds has been mainly focused on their performance, their strategies and the risk associated with such strategies. Performance of hedge funds is typi-cally measure in absolute terms with respect to the risk free rate and they normally use a wide variety of dynamic investment strategies. Hedge fund performance is perhaps the main research topic given the popular claim that hedge funds obtain superior risk-adjusted returns. To verify this claim, many related studies do some kind of empirical analysis based on a performance model, which is typically a generalization or an extension of Sharpe's Performance Model (Sharpe (1992)). Sharpe proposes a simple linear regression model to explain portfolio returns in terms of suitable chosen factor variables and finds that his model explains reasonably well mutual fund returns. However, Fung and Hsieh (1997), when studying the nature of the hedge fund strategies, apply Sharpe's model to a large database of hedge fund returns and observe that Sharpe's model does not explain satisfactorily these returns. Therefore, several extensions of Sharpe's model have been proposed for such matter. Most of these extensions are based on the work of Glosten and Jagannathan (1994), who propose a general performance model based on a contingent claim perspective and which, in practical terms, reduces to add call-option payoff terms to Sharpe's model. Recently, Agarwal and Naik (2004) extend a bit further Glosten and Jagannathan's model and propose to include also put-option payoff terms. They find that this model fits well observed hedge fund returns. Overall, performance studies conclude that hedge funds consistently outperform mutual funds but not standard market indices. Explaining the performance of hedge funds is intimately related to understanding their strategies. Therefore, studies that focus on performance also provide insights about their strategies and vice versa. For example, Agarwal and Naik (2004) conclude from the good fit of their performance model, the nonlinear option-like nature of hedge fund returns, which are characteristics that have been deduced by others that were primarily studying hedge 48 fund strategies (e.g. Fund and Hsieh (1997)). Describing the strategies followed by hedge funds is not an easy task. This is due to the lack of information, given the sui generis nature of hedge funds as financial entities for sophisticated investors, and the wide variety of strategies that hedge funds actually follow. However, there are some common characteristics that are well known about their strategies. For instance, most hedge funds use dynamic investment strategies that very often involve leverage and which have a low correlation with standard asset classes (Fung and Hsieh (1999, 2006)). Nonetheless, there exists great variety of investment styles. Consultants classify hedge funds according to self-described styles. Among the most common are Long/short equity funds, which are typically exposed to a long-short portfolio of equities with a long bias, Event driven funds, that specialize in trading corporate events, such as merger transactions or corporate restructuring, and Equity market neutral funds, which usually trade long-short portfolios of equities with little directional exposure to the stock market. Nevertheless, there are alternative classifications. For example, Fung and Hsieh (1997), using a principal component approach based on returns alone, provide a quantitative classification of hedge fund styles. One of the popular beliefs about hedge funds is that these financial institutions do not have systematic risk. There are few studies that address this issue. This is because hedge fund managers usually diversify their fund's performance across a variety of strategies. Hence, risk analysis has focused on specific hedge fund style strategies. For example, Fung and Hsieh (2001) study the risk associated with the strategy style known as trend following and Mitchell and Pulvino (2001) analyze risk associated with the risk-arbitrage strategy style. Recently, Agarwal and Naik (2004) have proposed a general approach to characterize the risk of any hedge fund strategy. These studies conclude that hedge fund strategies do have systematic risk, and moreover, that in some cases, such as equity-oriented hedge fund 49 strategies, payoffs resemble a short position in a put option on the market index. Therefore, risk management of hedge fund strategies is a crucial topic that needs to be addressed further. Instead of trying to describe or deduce the strategies that hedge funds actually follow, other academics have theoretically derived the strategies that hedge fund managers should implement, given some preference model framework, and analyze the risks associated with such strategies. Hedge fund managers are typically compensated with an incentive fee formed by a fixed compensation plus a variable payment that is equal to a predetermined percentage of the positive profits over a specified benchmark. In other words, hedge fund managers are compensated with a fixed compensation plus a percentage of the payoff of a call option, whose underlying in the fund value. Carpenter (2000) solves explicitly for the first time, in a continuous-time framework, the dynamic investment problem of a risk averse manager who is compensated with a call option on the assets he controls, such as a hedge fund manager. Carpenter finds that the optimal strategy implies that the compensation option ends either deep in or deep out of the money. That is, the hedge fund manager implements a policy that leads to a fund value that is either well above or well below the benchmark. Carpenter also finds that the volatility associated with the optimal strategy can be strictly below the volatility of the policy followed by the same manager if he were trading his own account. This latter feature has questioned, in general, the ability of this type of option-like mechanisms to induce specific risk profiles. This topic is one of the main issues discussed in Chapter Three of this Ph.D. Thesis. Cadenillas, et al. (2004) extend Carpenter's (2000) model, in a continuous-time frame-work, to study the case in which the manager chooses, in addition to the volatility of the portfolio of the assets that he controls, his level of effort. Carlson and Lazrak (2005) com-plement Cadenillas, et al. (2004) analysis by investigating the case in which the manager decides on the leverage instead of the effort level. Recently, Panageas and Westerfield (2006) 50 obtain, also in a continuous-time framework, the optimal portfolio choice of a hedge fund manager who is compensated with a high-water-mark contract, where the benchmark is the last recorded maximum fund value. Unlike previous related studies, they assume that the horizon time is indefinite or infinite. They find that under such assumption the optimal portfolio will place a constant fraction in a mean-variance portfolio and the rest in a riskless asset. This implies that even risk-neutral investors will not invest unboundedly in risky assets, contrary to what previous studies with finite horizons would imply (e.g. Carpenter (2000)). Therefore, they conclude that risk-seeking incentives of option-like compensation schemes rely more on the horizon than in the convexity of the compensation scheme. This chapter studies the nature of the optimal strategies followed by a hedge fund ma-nager who includes stock index options in his investment universe. Our main motivations are, on one hand, that payment benchmarks for hedge fund managers are typically established in reference to a certain stock index and, on the other hand, that previous studies on optimal strategies use probability frameworks that do not allow to consider the well known skewed and fat-tailed features of option return distributions. Therefore, we consider a discrete-time framework that, in contrast to typical used continuous-time frameworks, can be easily adapted to consider not only these type of distributions, but also incomplete markets. We consider a general risk averse hedge fund manager who is compensated with a fixed salary and predetermined percentage of the net profits with respect to a benchmark. Using duality theory, we obtain explicit theoretical conditions under which it is optimal for this manager to invest in stock index options. These conditions establish pricing thresholds for the stock index options, in terms of embedded martingale measures that are linked to the preferences of the hedge fund manager. We derive these optimal investment conditions for different benchmark policies and risk management considerations. The numerical valuation of these optimal conditions is relatively easy to obtain, given that all the models used involved 51 optimization problems with linear constraints, for which there are well established algorithms to solve them (e.g. Bazarra, et al. (1993)). We illustrate our results with some examples. This chapter is organized as follows: Section 2 describes the decision space framework used for our models. Section 3 presents our basic model, establishes explicit optimal invest-ment conditions for such model, and develops a detailed example that shows the application of these conditions. Section 4 extends these results for a broader class of utility functions. Section 5 illustrates the risk incentives induced by the manager's compensation scheme. Sec-tion 6 generalizes the results of Section 4 for multiple periods of fee payments and different policies to determine the benchmark. Section 7 considers more advanced models, and Section 8 concludes. 2.2 Decision Space Framework Our framework is based on King (2002) and uses three elements: a probability space, a financial market, and a class of investors. Probability Space The probability space uses a scenario tree structure that models all possible scenarios or states (represented by nodes of a tree) of the market over a finite number of discrete time periods t = 0 , . . . , T. The scenario tree structure is such that every possible state is the consequence of a unique sequence (trajectory) of states (events). That is, every node n G Nt, where Nt denotes the set of all nodes at time t, has a unique parent, denoted by a(n) G Nt-\, although with a set on possibly many child nodes, denoted by C(n) C J V t + 1 . This is convenient for assigning probabilities to each of the tree scenario nodes. Defining a probability measure P (i.e., assigning probabilities to each node of the tree) in this type of scenario tree consists of 52 assigning weights pn > 0 to each leaf node n G NT with J2neNT Vn — 1 and then recursively computing the remaining node probabilities via Pn= ^2 Pm, V n€ Nt, t = T-1,...,0. meC(n) Let ft be the set of possible trajectories or sequence of events (from time 0 to the end of period T) in the scenario tree, then (ft, P) defines a sample space. Every node n G Nt has a unique history up to time t and a unique set of possible future trajectories. There-fore, Nt induces a unique set of histories up to time t, say FT, and a partition of ft. The collection of sets { F T } ( = 0 ] T satisfy FT C Ft+i for t = 0,..., T - 1. The triplet (ft, FT, P) forms a probability space. For the probability space (Q.,FT,P), the conditional probability of state (event) m given that n occurs, where m G C ( n ) , is ( ^ ) , a n d if {^t}t=o,...,T *S A discrete stochastic process defined on our probability space, then EP [XT] — 2~2neNt XnPn and EP [XT+1\FT] = E m 6 C ( n ) ( ^ ) Martingales Definition 2.2.1 Let {Zt}t=0 T be a stochastic process defined in (ft, FT). / / there exists a probability measure Q such that Zt = Efi[Zt+1\Nt], t = 0 , . . . , T - l then the stochastic process {Zt}t=0 T is called a martingale under Q, and Q is called a martingale measure for the process {Zt}t=Q T . Martingales are useful in the financial context, for example, to determine if a price 53 process is fair in the sense that at any time the price equates its expected future value (under a probability measure Q), i.e., if the price process is a martingale (under Q). Financial Market and Investors The market has 1+1 tradable securities i = 0,..., I, whose prices at each node n are denoted by the vector Sn = ( 5 ° , . . . , S^)- We assume that one of the securities, the numeraire, has always a strictly positive value and without loss of generality we assume that it is security 0. This numeraire defines the discount factors Bn — (-^) and the discounted prices (relative to the numeraire) Zn = 3nSn for i = 0 , . . . , / . The price of the numeraire in any state equals one. The market may be either complete or incomplete. Let 6%n be the amount of security i held by the investor in state n G Nt. Thus, the portfolio value in state n G Nt is Zn • 6n = 2~2{=o %n' ®n-We assume that investors do not influence the security prices of any security and trade at every time-step based on information up to time t. Arbitrage Arbitrage is the opportunity of making a sure profit with no risk (usually through the purchase and sale of assets). Here, arbitrage is to find a portfolio with zero initial value whose terminal values, obtained through self-financing strategies, are nonnegative for all scenarios and at least one of those terminal values is strictly positive with a positive proba-bility. In other words, there is an arbitrage if there exists a sequence of portfolio holdings {9n}neNt, 0<t<T  S U c h t h a t 54 Z0 • 0o = 0 Zn • (<?n - 0a(n)) = 0 V n 6 i V » , V t = l l ...,T Z n • 0 n > 0 V n G iV T , and P {Zm • 6m > 0} > 0 for some m G NT We assume that there is no arbitrage in our financial market. This assumption guaran-tees the existence of martingale measures for {Zt}t=0 T (King (2002), Theorem 2.2), which are used to express optimal conditions for option investments for models presented below. 2.3 The Basic Problem We consider a hedge fund manager who controls assets with a current value of Wo and who receives compensation of a fixed fee / and an incentive fee of a percentage a of the positive profit, if any, of the hedge fund manager's portfolio with respect to a previously specified benchmark, usually based on a stock index, over a T-period horizon. The hedge fund manager can invest in any of / + 1 securities and can buy or sell European stock index options. Our basic problem is that faced by a hedge fund manager determining the investment strategy that maximizes his expected fee. We focus on two disjoint cases: either the hedge fund manager is only allowed to buy stock index options or he is only allowed to sell stock index options. We make the following assumptions: • The hedge fund manager only considers strategies that do not lead to negative values of his portfolio. Hence, Zn- 6n> 0, for all n G NT-• The manager can re-balance his portfolio periodically. However, there are no inflows or outflows of capital during [0,T]. hence, the hedge fund manager only considers 55 self-financing strategies that satisfy Zn • {On - ««(„)) = 0, for all n € NT, V t = 1,..., T. This restriction is relaxed in subsequent sections. • If the hedge fund manager takes a position in stock index options at time 0, then he maintains such position until time T. 2.3.1 Buying Options We consider the set of strategies in which the hedge fund manager is allowed to invest in any of the 7 + 1 securities and to buy (European) Options (vanilla Calls and Puts) whose underlying is a Stock Index, which is assumed to be the benchmark. The manager's objective is to maximize his expected fee (we study more complex objective functions in subsequent sections). The optimization investment problem, without the constant term / (Y2neNr PnPn), which does not influence the optimal solution, is Max 0,eo,s a [J2n&NT PnSnPn] s.t. Z0-80-rB0(e0-V0) = 60W0 Zn • ( 0 „ - 0a(n)) = 0, V n e W t , V t = 1 , . . . , T (2.1) Zn • 6n - 6nsn + Bn ( e 0 • Vn) = BnBn , V n € NT Zn • 6n > 0, sn > 0 , V n e NT eo > 0 , where e0 = [e£, e£ ] , V0 = [ C 0 ( l + tcc), P0(l + tep)], Vn = [Cn, Pn], 0 = [0,0], and 56 : The surplus over the benchmark in scenario n £ A ^ . (tf) : Quantity of Call (Put) options that are purchased at time 0. Co (Po) : Call (Put) price at time 0. Cn (Pn) : Call (Put) payoff in scenario n £ NT-tcc (tcT) : Transaction Costs when buying Calls (Puts) at time 0. (as a percentage of the option's value). W0 : Initial Value of the Hedge Fund Manager's portfolio. B n : Benchmark value in scenario n £ NT-The dual problem of (2.1), which provides insights into the nature of the optimal strate-gies for the hedge fund manager, is (see Appendix 2) M i n !/o,y,x,A,M,7,c,»?p VOPOWQ - Y^neNT xnPnBn s.t. (XPn + Hn ~ Xn = 0, V Tl £ NT A„ + xn - yn = 0, V n £ NT (ynZn - £ m e c ( n ) ymZm) = 0, V n 6 Nt, V t = 0,. . . , T - 1 (2.2) 2ZneNT XnPnQn ~ y0P0C0(l + tCC) + T]C = 0 T,n€NT XnPnPn ~ y ^ P ^ l + tcP) + TJp = 0 A„ > 0, pn > 0, V n £ NT Vc > 0, VP > 0 • The first set of restrictions corresponds to the positive profits, if any, over the Bench-mark. The second set of restrictions is associated with the final value of the hedge fund manager's portfolio. The third set of restrictions is consequence of the self-financing strate-gies constraint. The last two constraints correspond to the purchase of options. Observe from the first two sets of restrictions of the dual that Xn = + CtPn > CtPn > 0 , V U £ NT , 57 and yn = xn + A n > xn , V n 6 NT. Thus, 2/n > xn > apn > 0 . (2.3) The third set of restrictions of the dual problem and the fact that Z® = 1 for all n G Nt, V t = 1,..., T, imply that yn= I/m, V n e TYt, Vt = 0 , . . . , r -1 . (2.4) meC(n) Hence, Therefore, from (2.3) and (2.5), 2/o > £ xn • (2.6) These preliminary observations of the dual problem are key for its analysis and interpre-tation. A n a l y s i s a n d I n t e r p r e t a t i o n o f t h e D u a l P r o b l e m 1. Let J/Q ^ d {x*n)neNT b e the optimal values for the linear programming dual variables yo and {xn)n€Nr. These values represent, respectively, the marginal change of the optimal expected fee of the hedge fund manager with respect to the initial hedge fund portfolio's value Wo, and the possible benchmark values (Bn)neNT. 58 2. The last two restrictions of the dual problem are equivalent to Vo > T.neNr^){%)(l + tcc)^ (2.7) VO > J2neNTXn(%)(^)(l+tCP)~\ (2.8) The expressions on the right hand side of these two inequalities can be interpreted as a (discounted) weighted average return of the Stock Index Options. 3. From (2.6) - (2.8), it must be satisfied that yo > Max{yZneNTxn(%){%){l + tc^)-\ Therefore, the optimal value of yo, say y^, must satisfy yl = Max{En e Jv 7.<(|)(t)( 1 + * c C)- 1, where ( z n ) n ( E A r r are the optimal values of (xn)neNl.- Hence, if neNT or then, from 2, c a n &e interpreted as a (discounted) weighted average return of Stock Index Options. Otherwise, if 59 V0= Xn  ) (2.9) n€NT then, by (2.3) and (2.5), x*n = y*, for all n G N T , and hence, A* = 0, for all n € N T . Thus, by linear complementarity, ZN • 6n > 0 for all n G NT- That is, if condition (2.9) is satisfied, then the hedge fund manager aims for a strictly positive value of his security's portfolio for every possible final scenario. Therefore, given the no arbitrage assumption, the initial value of the security's portfolio, Z0-9Q, must be strictly positive. In other words, if condition (2.9) holds, then the hedge fund manager must invest in the security's portfolio. 4. The dual objective function can be expressed as can be interpreted, using 2 and 3, as a weighted average return over the remaining capital derived from subtracting the weighted average, neNT Therefore, the optimal dual value, 'n from the current value of the hedge fund manager's portfolio. 5. From equations (2.3)-(2.5) yo 60 defines a probability measure Q. Furthermore, the third set of restrictions of the dual problem (2.2) implies that {qn}neNT defines a martingale measure for the (discounted) price process {Zt}t=0 T . Moreover, if the final values of the portfolio (Zn-8n, n 6 NT), are unrestricted, then xn = yn for all n € NT (see the Appendix 2 for the proof), and, thus, the dual objective function is = yo ( /Wo - EQ \5TBT\) . (2.10) Hence, the optimal dual value, y*0 ( /Wo - E<3' [0TBT}) , can be interpreted as a weighted return over the remaining capital derived from sub-tracting the expected value of the benchmark (under Q*) at time T from the current value of the hedge fund manager's portfolio. 6. From equations (2.3) and (2.5), it follows that y0 > a. Therefore, based on the inter-pretations of I/Q argued in Items 4 and 5, the optimal hedge fund manager's expected fee must be at least a percent of the remaining capital derived from subtracting a weighted average value (expected value, under the no-bankruptcy assumption made in Item 5) of the benchmark from the current value of the hedge fund manager's portfolio. 7. Finally, equation (2.3) provides us with a set of lower bounds for the dual variables tin} neNT> and, hence, with a set of (scaled) lower bounds for the martingale probabil-ities {qn}n<zNT defined in Item 5. That is yo qn > apn , V n G NT-We summarize the previous analysis in Table 1. 61 Table 1 Expression Interpretation Bounds 1. Marginal change of the optimal expected fee with respect to Wo, 2. Weighted Average Return. Vo > a r* Marginal change of the optimal expected fee with respect to the benchmark value Bn, n G NT. Martingale Measure of the discounted price process (Zt)t=o T-y o " ( / W o - £ n 6 J v T ( f ) / U * » ) Weighted average return over the remaining capital derived from subtracting an average of the possible benchmark values at T from the current value of the hedge fund manager's portfolio a ( /Wo) -«E„ eyv T(S)^n Example 1 Description: Consider a static one-period horizon problem with two equally likely scenarios, n\ and n2, and a financial market with two securities: a bond (security 0, the numeraire) and a stock (security 1). We assume that the hedge fund manager receives, at time T, a variable incentive fee of 20% over the positive profit of the portfolio he controls, with respect to a benchmark B. This benchmark is established as the capital that would be accumulated if the current portfolio value W0 would be invested at a rate of return equal to the return gained by the stock index SI plus a fixed return n. That is, the benchmark B can take the two following possible values: 62 Bni = (JTJJ^ + KJWO , in scenario ni, and Bn2 = (j&h + KJWQ , in scenario n 2 , where SIo is the current value of the stock index and SInj represents the stock index value in scenario j, j = 1,2. Finally, we assume that there is an option market on the stock index SI. The basic problem is then M a x w (0.2) [pni SniPm "F Pn.2^712^712] S.t . 0° + Z\9\ + p0e$C0 + 0oepPo = f%W0 « - 0O°) + Z\x (0 N I -9$=0 K ~ 0°o) + Zi2 {9i2 -9l)=0 QNI + Zni9ni — 0NXSNI + 0ni€QCni + PTII^QPni = PniBni @n2 + Zn29n2 — PN2SN2 + / ? n 2 e 0 ^ » » 2 ~b Pn2e0 Pn2 = Pn2Bn2 (2.11) > 0, e £ > 0 , s n i > 0, s „ 2 > 0 . where /?nj. = (S° ) _ 1 = (S§(1 + r ) ) _ 1 , j = 1,2; r is the one-period (fixed) interest rate; Cnj (Pnj) is the Call (Put) payoff under scenario rij, j = 1,2; and C 0 (Po) is the current price of the Call (Put) option. We assume that the current price of the options on the Stock Index SI correspond to the no-arbitrage prices Co = ( i k ) [qCni + (1 - q)Cn2] , and Po = ^)[qPni + (l-q)Pn2] , 63 where „ _ y > - « . . Suppose that the current security prices are = 90 and = 10, and that W0 = 500, SIQ = 100, r = 5% and K = 2%, with the following scenarios: Scenario n\: Sni = (1 + r)Sg = 10.5, 5 n i = 108, and 5 / n i = 115. Therefore, BNI = + K^JWO = 585. Scenario n 2 : S? = (1 + rJSg = 10.5, Sn2 = 67.5, and SIn2 = 70. Therefore, Din Thus, and hence, considering an exercise price of 80 for both options, it holds that Co = 25.926 and P 0 = 2.116. 64 Solution: The optimal strategy for the hedge fund manager is 9°o* = -92.857 , e\>* = 14.444 <£'* = () , ep'* = 60.75 which means that the manager borrows cash, at a rate of interest r = 5%, takes a long position in the risky security S1, and buys put options. This leads to the expected variable fee 0.2 [PnisniPni + Bn2sn2pn2] = [(0) (|) + (247.501) (§)] = 2.35715 . The Dual Problem and its Interpretation: The optimal solution of the dual problem of (2.11) is W5 = 0.45, j/J = 0.3, x l = 0.3, y*2 = 0.15, x*2 = 0.1 . Then (0-45) [(**) - (^) ({j&}(585) + {&}(360))] (0.45) [50 - 44.762] 2.35715 0.2 [/^ m^mPm + Pn2Sn2Pn2] • Thus, the optimal expected variable fee for the hedge fund manager is approximately2 45% (= 2/Q) over the remaining capital derived from subtracting the expected (discounted) 2[(|) + (|) Pn2Bn,} = 44.762 * 48.571 = [(g) /?NIBM + (g) /?„25„2] = [pTBT\ 65 2/5 / W o benchmark value = 48.571 , from the current (discounted) hedge fund manager portfolio 80W0 = (1/10)500 = 50. Note that Q* does not coincide with the probability measure Q = (q, 1 — q), where q is defined in (2.12). Moreover, observe that PoPo = m?r) = ( l f e ) [ ^ ( 0 ) + ^ ( 1 0 ) ] = (|)(^1)(^1)(l+fcP)-1 + (|)(/3n2)(p,l2)(i+tepr\ • This latter condition is expected to hold since optimal purchase of (Put) Stock Index Options, i.e., e^ '* > 0, implies, by linear complementarity, PoPo = 2ZneNT(ll)(Pn)(Pn)(l+tCP)-l ^ 2ZneNT(i)(Pn)(Pn)(l + tcp)^ (2-13) = EV[0TPT}(l + tcp)-1. where the first inequality follows from (2.3). Therefore, from (2.13), we have proved P ropos i t i on 2.3.1 If e£* > 0, then /3 0C 0(1 + tcc) = }ZneNT (§) PnCn < Eft* [0TCT], where Q* = \ % \ . Analogously, if > 0 then, I yo ) neNT 0OPO(1 + tCP) = (~) PnPn < E®' [BTPT] . n e N T Wo/ 66 The optimal investment condition for the purchase of stock index options given in Propo-sition 2.3.1 can be restated as the Stock Index Options must not be overpriced. This is proved in Coro l l a ry 2.3.1 If eg'* > 0 then /3bCb(l + tcc) < MaxQeQ EQ [pTCT] , (A>P„(1 + tcp) < Max{QeQ} EQ [j3TPT]) where Q denotes the set of martingale measures o, '/ {Ct}t=o,...,T ({Pt}t=0i...tT) Proof: The result follows directly from Proposition 2.3.1 and the obvious fact that Within this framework King (2002, page 550) proves that the fair value of the contingent claim Vt at time T is where Vt is the contingent's claim price at time t, and Q denotes the set of martingale measures of { V t } t = 0 T . Hence, Corollary 2.3.1 states that it is optimal to purchase Stock Index Options if these are not overpriced. In summary, if the hedge fund manager is allowed to purchase Stock Index Options, then the optimal strategy leads to a percentage (of at least a) over the remaining capital derived MaxQeQ EQ [pTCT] > EQ' [/3TCT]. Q.E.D. MaxQeQ EQ [pTVT] 6 7 from subtracting the average value of the Benchmark from the current value of the hedge fund manager's portfolio and will include the purchase of options if these are not overpriced (with respect to the no-arbitrage price under Q*). 2.3.2 Selling Options We concentrate now on the set of strategies where the hedge fund manager is allowed to sell Stock Index Options. The manager's objective is again to maximize his expected variable fee. The optimal investment problem is e,eg, « [EneNr PnSnPn] s.t. Z0-00- eg(30Co(l + tcc) - ip/30P0(l + tcp) = / W o Zn • (On ~ 0a(n)) = 0 , V n G Nu V t = 1 , . . . , T (2.14) Zn • 0n ~ PnSn ~ ^ P n C n ~ tPf3nPn = PnBn , V U G NT Zn • 0n > 0, sn > 0 , V n G NT > 0, ep > 0 . where (ep) is the amount of Stock Index Call (Put) Options that are sold and Co (-Po) is the current (unitary) price of such Call (Put) options. As in the case of purchasing Stock Index Options, the dual of (2.14) provides insights into the structure of the optimal strategies for the hedge fund manager. The feasibility of (2.14) is discussed in Section 2.3.4). The dual (see Appendix 2 for its derivation) of problem (2.14) is 68 S.t. ®Pn + P-n ~ xn = 0, V n 6 NT K + xn-yn = 0, V n € NT (ynZn - EmeC(n) VmZm) = 0, V 71 € JVt, V * = 0 , . . . , T - 1 (2.15) yoPoCo(l + < c c ) - E / 3 n C n + 7?C = 0 2 / o / W l + i c p ) - E n 6 W 7 . XnPnPn + VP = 0 A„ > 0, > 0, V n € A^T Vc > 0, 77P > 0 . A n a l y s i s a n d I n t e r p r e t a t i o n o f t h e D u a l P r o b l e m The difference between the dual problems (2.2) and (2.15) is only in the last two restrictions. Therefore, we expect that most of the interpretations derived for purchasing options are valid for the case of selling options. For instance, interpretations 1, 4, 5, 6 and 7 of Section 2.3.1 apply exactly or almost exactly in the same manner. Interpretations 2 and 3 do not have a straight forward interpretation in this case although they do have the following corresponding counterparts: 1. (2') The last two restrictions of the dual problem are equivalent to (2.16) yo < E „ 6 ^ [ ( | ) ( c > ) ( l + tec)-1]^. yo < E n e A T T [ ( t ) ( t ) ( i + ^ H ^ . The right hand side of each of the previous restrictions can be interpreted as weighted average returns of Stock Index Options. 2. (4') Equations (2.16) and (2.6) imply that 69 xn < yo < Min{Y:n€NT{(%)(%)(l + tcc)-1 ^ [ ( * ) ( & ) ( l + tep)-1W-Therefore, optimality implies y* = J2n€NT xn- Hence, x*n = y* and, thus, A* = 0 for all n 6 NT- By linear complementarity, Hence, given the assumption of no arbitrage, the initial value of the security's portfolio, Zn • 9Q, must be strictly positive. That is, in the case of selling Stock Index Options, the optimal strategy is to always invest in the security's portfolio. The reason is that if the manager sells options, he must, at least partially,, hedge his position by investing in the security's portfolio; otherwise, he needs to invest in his attempt to beat the benchmark. Therefore, in any case, the hedge fund manager must invest in the security's portfolio. We now state necessary optimality conditions for the sale of options. P ropos i t i on 2.3.2 If eg'* > 0, then A,C0(1 + tcc) = EQ' [PTCT], where Q* = i.e., selling calls or puts on a Stock Index is optimal if these are fairly priced (under Q*). Proof: Assume that eg'* > 0. Then, by linear complementarity, zn • e*n > o , v n e NT . Analogously, if eP'* > 0 then, pnP0(l+tcP) = EQ*[PTPT] , (2.17) neNT 70 Optimality implies yl = YlneN x*n a n d , hence, x* = y* , V n € iVr- Therefore, neNT neNT Thus, combining equations (2.17) and (2.18) yields y^C^l + tcc) Hence, 0oCo(l + tcc) = T,neNT(ACn) (I) = [PTCT} • The case of selling put options is proved in an analogous manner. Q.E.D. In summary, if the hedge fund manager is allowed to sell Stock Index Options, then the optimal strategy leads to a percentage (of at least a %) over the remaining capital derived from subtracting the average value of the Benchmark from the current value of the hedge fund manager's portfolio. Moreover, the optimal strategy will always involve an investment in the security's portfolio and will include the sale of options only if these are fairly priced (with respect to the no-arbitrage price under Q*). Recall that in this case the optimal strategy for hedge fund managers involves an invest-ment in the security's portfolio as a way of hedging, at least partially, the associated risk with the sale of the Stock Index Option. Risk Management is discussed in Section 2.7. However, as motivation, we present a simple modification of model (2.14) that considers this issue. (2.18) — YLn€NT(PnCn)yn-71 Risk Management Suppose that the hedge fund manager, who is usually a major investor of the hedge fund, agrees with the investors to consider strategies that could reduce, at least in a proportion 6, the expected payoff of the Stock Index Options that he decides to sell. Therefore, the hedge fund manager considers the modified objective function Max ^ ^ Pn^nPn .n€NT -5 E Pn (CO • K.) Pn .n€NT (2.19) where 2^2neNT Pn (^ o • Vn)pn represents the expected payoff of the sold options and 5 6 [0,1] is the proportion of such payoff that is desired to be minimized. In this case, the manager will sell Stock Index Options if such options are overpriced. Proposition 2.3.3 proves that the overpricing of the Stock Index Options is a necessary optimal condition for the sale of such options and also states by how much these options must be overpriced. P ropos i t i on 2.3.3 Consider model (2.14) but with the objective function (2.19). Then, > 0 implies /?oC0(l + tcc) = E®" {(3tCT\ + (4) EP [6TCT] , Wo/ where Q* = \ % \ Proof: The addition of the term S [2~2neNr (^ o • Vn) pn] to the objective function of model (2.14) implies that the dual restriction, y0PoCo(l + tC°) - XnPnCn < 0 , neNr 72 is replaced by y 0A)Co(l + tcc) - Y^neNT xn(3nCn - $2~2neNT (PnCn)pn < 0. Therefore, e\g'* > 0 implies, by linear complementarity, A c 0 ( i + tcc) = £ (§) pncn + ( A ) n e N T Wo/ Wo/ [neNl Recall from the proof of Proposition 2.3.2 that optimality implies x* = y* V n E NT. Hence, This result shows not only how risk management considerations can be included but also how to extract relevant information through the use of the duality framework. 2.3.3 Buying and Selling Options The analysis of the two previous subsections allow us to study the nature of the optimal strategies for the entire basic problem in a straight forward manner. For instance, the opti-mization model for the basic problem is PoC0(l + tcc) = E n e ^ ( | ) ^ C 7 N + ( ^ ) [ E n e J V r / ? n C 7 n p , Q.E.D. Max e,eo,eo,s a s.t. Z0-00 + Po (e0 -Vo-io- V0) = /Wo zn• ( 0 „ - 0 „ ( „ ) ) = o, V n e i V t vt = i , . . . , r Zn • 0n ~ 0nSn + /?„ (to - C 0 ) " Vn = 0nBn , V 71 E NT Zn • 0n > 0, sn > 0 , V n E NT e0 > 0, e0 > 0 , (2.20) 73 where e0 =: [eg, eg] , e0 = [eg, e^ ] , Vn = [Cn, Pn] V n E NT , and Vb = [(1 + icc)C70, (1 + ic p )P 0 ] , V0 = [(1 + tcc)C0, (1 + itf The dual is Min yo<yiX yoPoW0 - 2~2n^NT xnPnBn s.t. &Pn -xn<0, V n E NT xn-yn<0, V n E NT [ynZn - 2~2mec(n) VmZm) = 0, V n € Nt, V t = 0, . . . , T - 1 J2neNT XnPnCn - y0/30C0(l + tCC) < 0 E n e / V r XnPnPn ~ B b f t W + tCP) < 0 y0poC0(i + tcc) - j : n e N r Xnpncn < o 0bA>Po(l + ftO - E „ e ^ r xnpnPn < 0. Comparing the dual of the basic problem with the dual problems (2.2) and (2.15), we observe that the dual problem (2.21) has only more restrictions than the other two dual problems. This follows because the model for the basic problem can be derived from any of the primal models (2.1) and (2.14) by simply adding terms into two of their set restrictions (the initial and the final set of restrictions). Therefore, necessary optimality conditions for the purchase and sale of Stock Index Options should be similar to those obtained in Propositions 2.3.1 and 2.3.2 since these conditions are derived from the application of linear complementarity to the dual restrictions shown in Proposition 2.3.4 i) Purchase Conditions: If eg'* > 0 (eg'* > dY then A)C0(1 + tc( •C) < EQ* [PTCT] , (A 10P0(l + tcp) <Eq'[(3TPT}) . 74 ii) Sale Conditions: If > 0 [e^'* > Oj, then 0oCo(l + tcc) < EQ' [fhCr] , {0OPo(l + tcp) < EQ' [0TPT]) . Proof: i) Assume e^ '* > 0. By linear complementarity, 0OCO(l+tCC)= V ( ^ ) P n C n , n e N r Wo/ but, x*n < y*n V n G NT. So 0OCO(1 + tcc) < E®' \p\CT\. ii) Assume e^ '* > 0. By linear complementarity, y*o0OCo(l+tCC)= J2(PnCnX • neNx Therefore, 0OCO(1 + tcc) = £ n e J V T (jf ). But, x*n < y*n , V n G NT. Hence, (%Co(l + tcc) < £ (0nCn) (£) = E®' [0TCT} . n e N r Wo/ In general, 0QCO(1 + tcc) ^ EQ' [0TCT], since feasibility of the dual problem implies yl = MaXnENT J E n e i V r X n > 2~2neNT X*n ( f ^ ) (c^) ' 2~2neNT X n ( f o ) ( l l ) } • Q.E.D. 75 2.3.4 Feasibility of the Basic Problem We now establish conditions for feasibility of the optimization problems (2.1), (2.14), and (2.20). Feasibility of Model (2.1): Model (2.1) is feasible if and only if the following inequality holds: / W o > Min e,eo Zo-90 + p0 (c0 • V0) s.t. zn• {en-ea{n)) = o, v n e wt, vt = I,..., r (2.22) Zn • 6n + 3neo • Vn = pnBn, V n £ NT e0 > 0, Zn • On > 0 V n € NT. Model (2.1) is feasible if and only if the current value of the hedge fund portfolio, 30W0, is no less than the minimal amount needed to replicate the benchmark through self-financing strategies derived from the investment in the security's portfolio and the purchase of Stock Index Options. The dual of the optimization problem on the right hand side of inequality (2.22) is Max X,Q E S.t. QnZn = 2~2meC(n) 9m^m, V 71 G Nt , V t = 1, . . . , T. qn>0, \/n€NT (2.23) <?o = 1 POVQ < J2neNT XnPnVn qn>xn>0, V n € NT • 7 6 The first three restrictions imply that Q = {qn}neN is a martingale measure of {Zt}t=0tT-Hence, recalling that Q represents the set of all the martingale measures of {Zt}t=0 then the dual problem (2.23) becomes Max x , q 2_jn(zNT Xn@nBn S.t. qn>xn>0, V n G NT PoV0 < £ n€NT xnPnVn tin}neNT =QCQ which is equivalent to Max x , q J2neNT QnPnBn s.t. {qn}neNT = QCQ Therefore, we have proved Proposition 2.3.5 Model (2.1) is feasible if and only if P0W0 > MaxQeQ [PTBT] s.t. (2.24) PoV0 < E® [PTVT] . That is, Model (2.1) is feasible if and only if the current value of the hedge fund manager's portfolio is no less than the maximum expected value of the (discounted) benchmark over all the martingale measures of the (discounted) price process {Zt}t=0>T under which the Stock Index Options are overpriced with respect to its current market value Vo. 77 The feasibility of Model (2.1) requires that the optimization problem on the right hand side of the inequality (2.24) is bounded. Therefore, feasibility of Model (2.1) demands the feasibility of Model (2.24) which needs the existence of a martingale measure of the (discounted) price process, say Q, such that 0OVO < EQ [0TVT]. Feasibility of Model (2.1) implies the existence of a martingale measure Q under which the current market value of the Stock Index Option, V0, is not greater than the corresponding no-arbitrage price under Q. This condition is consistent with the result of Proposition 2.3.1 in which the purchase of a Stock Index Options is optimal if its current price is not greater than the no-arbitrage price under the specific martingale measure Q* defined in Proposition 2.3.1. The analysis of the feasibility of Model (2.14) is completely analogous to the case of Model (2.1). The necessary and sufficient conditions for the feasibility of Model (2.14) are in P r o p o s i t i o n 2 . 3 . 6 Model is feasible if and only if / W o > MaxQeQ [PTBT] s.t. (2.25) /Wo > £ ° [PTVT] • Hence, Model (2.14) is feasible if and only if the current value of the hedge fund manager's portfolio is at least equal to the maximum expected value of the (discounted) benchmark over all the martingale measures of the (discounted) price process {Zt}t=Q T under which the Stock Index Options are underpriced with respect to the current market value Vo. As with Model (2.1), the feasibility of Model (2.14) requires the existence of a martin-gale measure Q of the discounted price process {Zt}t=0 T, under which the current (sale) 78 price of the current value of the Stock Index Option is not less than the corresponding no-arbitrage price under Q. This condition is consistent with the optimal sale condition stated in Proposition 2.3.2. The analysis of the feasibility of Model (2.20) is analogous and straightforward from the results established in Propositions 2.3.5 and 2.3.6. Proposition 2.3.7 Model (2.20) is feasible if and only if p0W0 > MaxQeQ Efi [PTBT] s.t. (2.26) PoV0 > E® [prVr] > (30V0. Therefore, the basic problem is feasible if and only if the current value of the hedge fund manager's portfolio is at least equal to the maximum expected value of the (discounted) benchmark over all the martingale measures of the (discounted) price process {Zt}t=0 T under which the corresponding no-arbitrage price of the Stock Index Options is within the bid-ask price range [Vo, Vb]. 2.4 Other Utility Functions We now extend the analysis of the previous section to the more general class of utility functions U formed by the set of functions that are twice differentiable, concave, and strictly increasing. This class of utility functions is standard to incorporate risk aversion (Pratt (1964)). Second order differentiability assures that [/(•) and [/'(•) are continuous, and, hence, if the feasible region is bounded, it guarantees that the maximum is achieved. 79 Let U(-) be a utility function belonging to the class of functions U. Consider the basic problem with Y U([f + asn}0n)pn neNr replacing / [J2neNr PnPn] + a [EneN r 0nSnPn) as the objective function. That is, the opti-mization problem is M a X O,e0,s EneJV r U(if + aSn] Pn) Pn S.t. Z 0 - 9 0 + 0o (e 0 • Vb - e 0 • V0) = 0OWO Zn • (0„ - 9a{n)) = 0 , V n £ JVt , V t = 1 T . (2.27) Zn • &n ~ PnSn + Pn (Co ~ e 0 ) • Vn = 0 n B n , V Tl £ NT Z n - 6 n > 0 , s n > 0 , V n £ N T eo > 0, e 0 > 0 The dual of this problem provides us with information about the optimal investment strategies. From standard nonlinear programming theory (e. g., Zangwill (1969), pp. 47 -52), the dual of (2.27) is Min yo,y,x [{yo0OWo - E r e € ^ . xn0nBn) + ZneNr {U ([/ + asn] Bn) - asnU' ([/ + asn] 0n)) pn] s.t. aPnU' ([/ + asn] 0n)-xn<O, V n 6 NT x n - V n < 0 , VneNT (VnZn - ZmeC(n)ymZm) = 0, V U G Nt , V t = 0, . . . , T - 1 . E „ e A r T xn0nVn - 0oVoyo < 0 - E n € i V T xnPnVn + A)Vfafo < 0 . (2.28) Except for the first restriction, the rest of the dual restrictions do not depend on the utility function U, therefore, most of the results and interpretations developed in Subsection 80 2.3.3 are preserved. For instance, y* still satisfies the condition, ^M-[£<'„S<I)(I)'S.<I)(§)}' ^ and, hence, J/Q is still interpreted as an average return. Other important results are also maintained. For example, qn = (^j ,Vn€NT, still defines a martingale measure of the (discounted) price process {Zt}t=Q T . Nonnega-tivity of yn is guaranteed since U is strictly increasing. For instance, the first two sets of restrictions of (2.28) imply that Vn > xn > apnU' ([/ + asn] pn) > 0, V Tl 6 NT . (2.30) The dual objective function is the sum of the dual objective function of the basic problem, VOPOWQ - Y XnPnBn , n€NR and the nonlinear term ^2neNT (U ([/ + a s « ] Z3") ~ OiSnU' ([/ + asn] Pn)) pn. The dual objective function of the the basic problem has a key role in the understanding of the optimal strategies for the hedge fund manager. It tells us that the manager should obtain the best return out of the remaining capital derived from subtracting the expected benchmark value, under a specific martingale measure Q*, from the hedge fund manager's portfolio value. Therefore, martingale measures are crucial for evaluating appropriately the 81 Benchmark. We inherit such importance of the martingale measures since the current dual objective function includes the dual objective function of the basic problem. Hence, it is important to understand the parameters that define such martingale measures. In particular, it is useful to comprehend the way the utility function U(•), the variable fee percentage a, and the natural or given probabilities (pn)n&Nr determine these martingale measures. Equation (2.30), a generalization of equation (2.3), provides understanding of the relationship between such parameters and the martingale measures. Equation (2.30) implies that yo qu > apnU' ({f + asn] Bn), V n G NT . (2.31) Hence, {aPnU'([f + asn]/3n)\ defines a set of (scaled) lower bounds for the martingale measures Q = {qn)neNT- The value of each of these lower bounds depends on a combination of three factors: a, (pn)neNT, and the marginal utility of hedge fund manager's compensation. This dependency of the lower bounds with respect to these three factors has the following characteristics • Each of these factors has, ceteris paribus, a monotonically increasing relationship with the corresponding lower bounds; that is, the higher the value of any of the factors, the higher the corresponding lower bound value. • The first two factors, a and (pn)neNT, do not depend on the particular strategy followed by the hedge fund manager. • Given a feasible strategy, two lower bounds for two different scenarios have the same value if they have equivalent tradeoff between natural (or given) probability and marginal 82 utility of the hedge fund manager's compensation fee. That is, given a feasible strategy (6, e, s), two scenarios n, and rij have the same lower bound if Pni U ([/ + asni This tradeoff condition implies, given the concavity of U, that the larger the surplus on a particular scenario is with respect to the surplus of an equivalent scenario, the higher the probability of occurrence of such a scenario. Although yoqn, for n € NT, does not necessarily equate its corresponding lower bound, the previous analysis describes the behavior of the forces that determine qn, n G NT- The optimal value y* is the best return from the remaining capital /?nWo — E®' [PTBT]-For the basic problem, y^ must be at least a. We now obtain the corresponding lower bound of J/Q f ° r the current case. Again, we use Equation (2.30) and that yo = 2~2neNT V™ IS also satisfied. Therefore, assuming that (9*,e*,s*) is an optimal solution, then y*>aJ2PnU'({f + asn}(3n) . n£NT The lower bound of depends, in general, on the marginal utility of the optimal surplus. If we consider a risk-neutral hedge fund manager, as in the basic problem case, this lower bound reduces to a. To conclude this section, we state necessary conditions under which it is optimal to invest in options. 83 P r o p o s i t i o n 2.4.1 (i) If > 0 ( e £ * > o), then 0oCo (1 + tcc) < El' [pTCT] , (0OPO (1 + tcp) < E^ \0TPT}) , and, if (ii) e^'* > 0 (eP'* > fj), then 0OCO (1 + tcc) < EQ' [0tCT] , {0oPo (1 + tcp) < E®' [0tPT]) , where Co (Co) and P0 (Po) denote the Call and Put bid (ask) prices, and Q* = {jj?} is a martingale measure for the process {Zt}t=Q T , that is, buying or selling options is optimal if these are not overpriced under Q*. Proof: (i) Assume e^ '* > 0- Then, by complementarity, Y xn0nCn = 0oCo{l + tcc)y* . neNr But y*n>xnVne NT, hence J2neNT Vl^Cn > EneNT <p\Cn > 0OCQ ( l + tcc) y*0. There-fore, EQ' [0TCT] = V qn0nCn = T ^0nCn > 0OCO ( l + tCC) . neNT neNT u The proofs of the purchase of Stock Index Options case and analogous. Q.E.D. 84 Proposition 2.4.2 proves that if the purchase and the sale of Stock Index Options are optimal in a simultaneous manner, then the sale and the purchase prices must be equal. P r o p o s i t i o n 2.4.2 (i) If eg'* > 0 and eg'* > 0, then CQ = C0. (ii) Analogously, if eg'* > 0 and eg'* > 0, then P0 = P0. Proof: (i) Assume eg'* > 0 and e^ '* > 0. Then, by complementarity, (30Co (1 + tcc) = <^Cn = pQCo (1 + tcc) . nENT Q.E.D. The latter result can also be obtained from the feasibility of (2.28) and the assumption % > V0. Feasibility of (2.28) implies PoVoyo < 2~2neNr xnPnVn < PoVoVo, which combined with Vo > Vo yields Vo = Vo-Next section solves Problem (2.27) for two particular examples to illustrate the risk incentives induced by the manager's compensation scheme, for a specific utility function. 2.5 Risk Incentives In the previous sections we have characterized, using duality theory, the nature of the optimal strategies followed by the hedge fund manager, when stock index options are included in his investment universe. For instance, we have proved that the purchase or sale of such options is optimal if these are, respectively, underpriced or not overpriced, and established explicit 85 pricing thresholds to determine the underpricing or overpricing of these options. Moreover, we have shown that these pricing thresholds depend on the variable fee percentage and the utility function that describes the preferences of the manager. In this section, we illustrate the connection of the variable fee (a) and the risk aversion level (7) of the manager with the optimal strategy, in terms of the risk incentives induced by the combination of a and 7. We do this in the following manner. We consider a specific risk averse hedge fund manager that solves Problem (2.27), for particular market and investment conditions, and measure the variability of his compensation with respect to its optimal expected value. The variability of the manager's optimal compensation is directly linked to the variability of the final value of the portfolio. Therefore, the higher the variability of the optimal manager's compensation, the riskier we consider the strategy that leads to such compensation. Our risk incentives study focuses on two particular cases. First, we analyze the situation in which the manager is only allowed to buy options. Second, we study the more general case in which the manager is permitted to buy and sell options. In both cases, we study the risk incentives induced for different degrees of risk aversion and distinct variable compensation fees. 2.5.1 Buying Options Consider the static one-period horizon problem described in Example 1 of Section 2.3.1, but we now suppose that the hedge fund manager is risk averse and his preferences are described by the utility function U(x) = x 1 - 7 1 - 7 86 where 7 > 0 represents his level of risk aversion and U belongs to the class of utility functions U, described in Section 2.4. That is, the hedge fund manager solves the problem s.t. 91 + Z\9\ + PotgCo + 0OepPQ = p0W0 K - e°o) + zni K - el) = o (e°2 - e°) + z\2 (e\2 -e*)=o &m + Z\x eni — 0ni sni + 3ni e 0 Cni + 0m eoPm = Pnt Bni ®ni "b Zn20n2 — 0n2Sn2 + 0n2€^Cn2 + /3n 2 £0 "^"2 = Pm^m o0ni + znieni>o, el2 + zn2e\2>o > 0, ep > 0 , sni ^ °> s„ 2 > 0 . where WQ = 500, and Variable Value at t = 0 Value in scenario n\ Value in scenario n2 S1 90 108 67.5 s° 10 10.5 10.5 SI 100 115 70 B - 585 360 C 25.926 35 0 P 2.116 0 10 where 0nj = ( S ^ . ) - 1 = (5°(1 + r)) \ j = 1,2; r = 5% is the one-period (fixed) interest rate, and the fixed salary / is assumed to be 0.5% of the current fund capital (Wo = 500), i.e. / = 2.5. We solved Problem (2.32) for different levels 7 of risk aversion and distinct values of the variable fee percentage a. For all the cases that we consider, the optimal investment strategy followed by the hedge fund manager is the same: 87 Optimal Investment Strategy: Long positions in the Risky Asset and Put Options 100.00% 99.00% 98.00% 97.00% a, 96.00% ut n I 95.00% u ^ 94.00% 93.00% 92.00% 91.00% 90.00% 8 . 3 4 % 7.51% 7.06% 6.79% 6.61% 6 4 r . 6.40% 6. . I3° . 6.26% -•IIP min | | | | | | | | | Wm • • Jw|§ * --•• i l l • —TT~ "IJtri ,. i': _ if©! IMP - ppls Ip sill - ":rf'f I f f f l l S i 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5 • Put Options • Risky Asset Risk Aversion Level Figure 2.1: Optimal Strategy. This graph shows the optimal long positions in the risky security and the put options, as a percentage of the sum of the amount borrowed and the initial capital WQ. • Borrow money from the bank, at an interest rate r, • Take a long position in the risky asset S1, and • Buy put options. This strategy is somewhat natural. This is because, on one hand, the expected returns of the risky asset and a long position in put options are both larger than the interest rate r and, on the other hand, the expected return of a long position in call options is significantly smaller than a long position in put options. Figure 2.1 shows the optimal long positions in the risky security and the put options, as a percentage of the sum of the amount borrowed and the initial capital WQ, for different levels of risk aversion. As it can be observed, although 88 both percentages are relatively stable across the levels of risk aversion considered, the higher the manager's risk aversion, the smaller the percentage invested in put options. In order to study the risk incentives induced by the compensation scheme as 7 and a vary, we use the standard deviation of the hedge fund manager's optimal compensation, where s* is the optimal surplus value for scenario i = 1,2 and M* = [/ + < ] Pn , + [/ + OiS*n2] pn2 , as a measure of risk. We analyze the effect of the risk aversion level by solving Problem (2.32) for different values of 7, given a fixed variable fee percentage a of 20%. Figure 2.2 shows the graph of CT*, as a function of 7, for the values of 7 considered. From this graph we can observe that the higher the risk aversion level, the smaller the risk incentives induced by the compensation scheme. Moreover, this behavior holds when a* (7) is scaled by the corresponding expected compensation /it*(7). This means, in particular, that the higher the risk aversion, the higher the expected compensation per unit of risk. In other words, as the manager's risk aversion increases, he tends to target for a more efficient mean-risk relationship. We study the impact of the variable fee percentage a in the same manner that we analyzed the effect of the risk aversion level. That is, we solved Problem (2.32) for different values of a, given a fixed risk aversion level 7. Figure 2.3 plots several curves of a*, as a function of a, for different values of 7. From these curves, we conclude that (i) managers with the same risk aversion will have higher risk incentives for higher variable fee percentages a, 89 Risk Incentives 20.5 19.5 18.5 17.5 16.5 15.5 14.5 13.5 12.5 11.5 10.5 9.5 8.5 7.5 6.5 5.5 4.5 3.5 \\ \\ \ \ \ \ \ \ \ \ \ \ \ \ \ x \ \ X \ \ ^ —. '— ^^•w """"" 60.00% O o "Jo *> .E 50.00% , ° ™ C 40.00% tn 0.5 0.75 1.25 1.5 1.75 Risk Aversion Level 2.25 2.5 - S t d . Dev. S td . Dev. / Mean Figure 2.2: Risk Aversion Effect. This graph shows the effect of the risk aversion level on the risk induced by the compensation scheme. and (ii) given a fixed a, the higher the the risk aversion level, the smaller the risk incentives induced by the compensation scheme. 2.5.2 Buying and Selling Options Consider the example developed in 2.5.1, but we now allow the manager to sell options too. Therefore, in this case the hedge fund manager solves for Problem (2.32) but using 9°0 + ZlX + f% (<£ - e£) Co + 6o {4 - ep) P0 = / W o as the initial budget constraint and 90 Risk incentives Figure 2.3: Variable Fee Effect. This graph shows the effect of the variable fee percentage on the risk induced by the compensation scheme, for different levels of risk aversion. ^ n i "f" Zni8ni BNISNI + Pni i^o ^) Cni + Bnx (tP — eP) Pni — BniBnx @U2 Zn2@n2 — BN2SN2 + Bn2 (CQ — C g ' ) Cn2 + Bn2 (t.P — £Q) Pn2 = Bn2Bn2 as the surplus constraints. We solved the modified optimization problem for different levels 7 of risk aversion and distinct values of the variable fee percentage a. For any combination of ( 7 , a) considered, the optimal strategy is the following: • Borrow money from the bank, at an interest rate r, • Take a long position in the risky asset S1, and 91 • Buy put options. • Sell call options. The unique difference between the case in which the only the purchase of options is allowed and the current case is that now the manager also sells call options. This is because (i) taking a long position in put options is significantly more profitable than taking a long position in call options, (ii) the sale of call options can be hedged by taking a long position in the risky asset, and (iii) the risky asset generates a higher expected return than the interest rate r. Therefore, the manager borrows extra money by selling call options, and uses this money to buy more put options and to take a longer position in the risky asset. The latter, in order to hedge the sale of call options. The risk incentives results are similar to those obtained in 2.5.1. That is, managers with the same risk aversion will have higher risk incentives for higher variable fee percentages a, and given a fixed a, the higher the the risk aversion level, the smaller the risk incentives induced by the compensation scheme. However, a* values are significantly higher than in previous case. These results are expected since the manager has now a broader set of investment possibilities, which implies that the optimal strategy leads to a higher expected compensation and thus, to a higher risk too. We compare the risk incentives results with those of 2.5.1 by scaling the standard de-viations of the manager's compensation by its corresponding mean. For instance, Figure 2.4 shows a*/ii* for different values of 7. It can be observed that, in relative terms, the manager takes higher risks when he is allowed to buy and sell options. Chapter 3 of this Ph.D. Thesis treats in depth, among other issues, the risk incentives induced by more general option-like compensation schemes. 92 Risk Incentives Comparison 80.00% i Risk Aversion Level | 'Buy only" - Buy and Sell | Figure 2.4: Risk Aversion Effect: Buying versus Buying-and-Selling. This graph compares the effect of the risk aversion level on the risk induced by the compensation scheme, for the two sets of investment conditions considered in this section. 2.6 Multiple Monitoring Dates All the models considered in the previous sections assume that the performance of the hedge fund manager is measured, or monitored, at the end of the planning horizon. In this section, we relax this assumption and study the nature of the optimal strategies when the hedge fund manager is faced with multiple dates in which his performance, and hence his compensation, is evaluated. It is shown that the nature of such optimal strategies depends on the mechanism or policy to determine the benchmark. We focus on two policies: Fixed or Stock Index Based Benchmarks and High Water Marks Benchmarks. The first policy refers to the common practice of determining the benchmark based on a preestablished fixed return (such as zero) or on a Stock Index (e.g., return obtained by S&P500 over the planning horizon). In the second policy the benchmark is the maximum historical value of the portfolio. We proceed 93 in two phases. In phase one, we study the case of two monitoring dates, and, then, we study the general case in phase two. 2.6.1 Two-Period Monitoring Case Consider two monitoring dates Ti and T 2 , where Ti < T 2 , in which the hedge fund portfolio's value is measured against its benchmark. Suppose that the hedge fund manager follows self-financing strategies between monitoring dates and that a proportion q (with q < 100%) of the profit over the benchmark is subtracted from the hedge fund portfolio's value at T\. This amount should cover at least the variable fee of the hedge fund manager, i.e. q > a, and it could possibly include some revenue payments to the investors. We further assume that the rebalancing of the investment positions does not occur at Ti but a period later, Ti + 1. Finally, we consider a utility function U that is time-additive, i . e. U (8Tl[f + asTl],pT2[f + asT2}) = XJX (/%[/ + asTl]) + U2 (pT2[f + as-*]) , where each £/; (i = 1,2) belongs to the class of functions U that is defined in the previous section. 94 Fixed or Stock Index Based Benchmarks We assume that the benchmark that applies is either fixed (e. g. 0 %) or based on a Stock Index (e. g. 2 % above a certain Stock Index return). The model in this case is Max e , £ 0 , s [}2neNTl Ui (if + asn)pn)pn + J2neNT2 ^ ([/ + Pn s.t. Z0-90 + 0O (e0 • Vo - e0 • V0) = 0OWO Zn • [9n - 9a{n)) = 0 , V n £ Nt V t = 1,..., Tx Zn'Gn- 0nS\ + f% (e0 - €0) ' K? = M , V 71 € NTl Zn- 0n + Pn (eri+i • Vri+i.n — ej^+i • V r 1 + i ) n ) — >» • *«(») + /?» (e0 - 6 0 ) • K°W ( ^ f 1 ) - QPns\[n) ( ^ ) ] = 0, V n E A T X l + 1 • (0n - <?«(n)) = 0 , V 71 € W t V t = TL + 2, . . . , T 2 2» • 6>n - « + /J n ( £ T i + 1 _ e T i + 1 ) . = ^B2 ; V n € i V T 2 Zn • 9n > 0, si > 0 , V n 6 iVr2 eo > 0, e0 > 0, eTl > 0, eTl > 0 , (2.33) where Vttn = [Ct,n, Pt,n] '• Purchase prices in scenario n, n E Nt. V \ n = [C t, n,Pf,n] : Sale prices in scenario n, n E Nt. V* = [Cn, P*} : Payoff in the event n, n E Nt, of an option purchased at time t. et = [ef,ef] : Amount of options purchased at time t, where t = 0,Ti + 1. et = [ef,ef] : Amount of options sold at time t, where t = T i , T 2 — 1. 95 The dual is + EneNTl (Ui ([/ + asUM - asnU[ ([/ + as*]/?n)) pn + >ZneNr2 (Ul ( [ / + CtsftPn) ~ « * ^ 2 ( [ / + Pn S.t. *PnU'L ([/ + as'M - 4 - ^ < 0 , V n e NTl xn-yn + yl<Q,\f n€ NTl (VnZn ~ EmeC{n) vM = 0, V n G 7Vt) V t = 0,..., Tx - 1 E n ^ - AVo«J + J2neNTl VIM < 0 (2-34) E n e ^ 2 * n / W + 1 - E n € A r T l + 1 y2n0nVTl+hn < 0 - E ^ , + E „ e ^ 1 + 1 ^ / W 1 + i , B < 0 ap„c/; ([/ + a S 2 ] & ) - x 2 < 0 ) v n € JVr2 ( & - E m e c ( n ) W = 0, VnGiV, , V i = T 1 ; . . . , T 2 - 1 , where y£ = £ m g C ( n ) y>m, V n G JVTl. The dual yields necessary conditions under which it is optimal to invest in options at t = 0 and t = 7\ + 1. Proposition 2.6.1 (Optimal Option Investment at t = 0) (i; 7/e|p* > 0 (e£* > o), then 0OCO < [0TlCTl] {0OPO < Efll [PTLPTL])-(ii) If&* > 0 (e£* > O), then 0OCO < [0TlCTl] {0OPO < [PT,PTI})-where Q\ = < 1 is a martingale measure for the process {Zt}t=0 T , and [ • ] denotes the expected value operator under Ql, and based (conditional) on the information at t = 0. 96 Proof: (i) Assume eg'* > 0. Then, by complementarity, A)Co(z70r= E {(<Y + MY) neNTl From the first two sets of dual restrictions yi > xl + yl > x\ + qy2n > apnU[ (/ + asn) > 0 , V n € J V n , where the strict inequality follows from the fact that £/(•) is strictly increasing and differen-tiable. From the third set of restrictions, Vo= E yn • n€NTl Hence, Q* — I T^Tr f i s a probability measure. Therefore, from the fourth set of dual )n<ENTl restrictions, we obtain The martingale property of Q\ comes from the third set of restrictions in (2.34). (ii) Assume e^ '* > 0. By complementarity f3oCo(y0Y= E (KY + (ylY) PnCn n€NTl 97 Therefore, by the properties of yl and yn discussed in (i), ^ E (jf£) ^Cl = [f3TlCTl] . Q.E.D. Proposition 2.6.2 (Optimal Option Investment at t = Ti + 1) (i) //e%*+1 > 0 (e£* + 1 > o), tfien Bfli [PT2C%+1] > E& [pTl+iCTl+1] , (E& [PT2P%+1] > [PT1+1PT1+I]) , and, if (ii) c^ +x > 0 > o)> ^ e n where Q2 = { 1 is o martingale measure for the process {Zt}t=Tl+1 T2, and y^ is J„ 6 JV T 2 defined as yl- E y" • F / ^ [ . ] denotes the expected value operator under Q2 and based (conditional) on the infor-mation at t = 0. Proof: (i) Assume e^i+i > 0- Then, by the application of the complementarity property to the fifth set of dual restrictions 98 n 6 N r 2 n€NTl+1 From second and third to last set of dual restrictions yl>x2n> apnU'2 (/ + as2n) > 0 , V n € NT2 , where the strict inequality follows from the assumption that U is strictly increasing and differentiate. Therefore, £ G£)*A.c?+1> £ (2-35) n€NT2 neNTl+i and — I T^V f defines a probability measure. Dividing both sides of (2.35) by (yl)* yields The martingale property of Q2 comes from the last set of restrictions in (2.34). (ii) The proof is analogous to (i). Q.E.D. Propositions 2.6.1 and 2.6.2 state that the purchase and sale of options at t = 0 and t = Ti + 1 is optimal if these options are not overpriced under Q{ and Q2, respectively. The conditions of Proposition 2.6.2 are a sort of forward-looking conditions. 99 In addition to the optimal investment conditions in terms of the embedded martingale measures Ql and Q*2, the dual problem (2.34) provides us with other insights. For instance, the inequalities vi > < + vl > < + wl > ^pnu[ ([/ + asl}pn) , v n G N T I , (2.36) and yl > x2n > apnU'2 ([/ + asl]f%) V n £ NT2 , (2.37) imply that yWn = yl > aPnU[ ([f + asn)0n) , V n G NTl = y2n > aPnU2 ({f + asl]3n) , V n G NT2 (2.38) This set of inequalities establishes (scaled) lower bounds for the martingale measures Q\ and Q2 which are of the same form as those in Section 2.4 for the single-period monitoring case. The probability value that the martingale measure Q* assigns to each scenario depends on the natural or given probability and the marginal utility under the utility function Ui, (i = 1, 2), of the profit over the benchmark for such scenario. In particular, observe that the Ql probability values at Ti do not depend on then the utility gained over the fee at monitoring date T 2 . Therefore, the Ql probabilities at Ti are, in a certain sense, independent of the future scenarios at T2. This does not occur for high water marks. At the beginning of the time horizon (t = 0) there is a preestablished benchmark B1 for the first monitoring date T i . For the second monitoring date T 2 , the corresponding benchmark, H i g h W a t e r M a r k s 100 B2, is determined through the policy of high water marks. Hence, B2 depend upon B1 and the (positive) profit obtained with respect to the benchmark, if any. Therefore, we add -(3nBn + (3nBn + (1 - a)(3nsn = 0, V n € NTl /3nB2n-pa{n)B2a{n) = 0, V n e i V t , V t = 7\ + 1,... , T 2 , ( 2 " 3 9 ) to formulation (2.33) to model such a benchmark determination policy. The first set of re-strictions in (2.39) models the benchmark update at time Ti by adding the adjusted profit (1 — ajs1 (after subtracting the incentive fee paid to the hedge fund manager) to the bench-mark B1. The second set of restrictions rolls over the updated benchmark value at Ti until time T 2 where the hedge fund portfolios's value is measured against its benchmark again. Then, (2.33) and (2.39) implies: • the definition of additional dual variables (/„) that satisfy In- E ' m = 0 , V n e / V 4 , t = Tu...,T2-l , (2.40) meC(n) -x2n + ln = 0 , V n E NT2 , (2.41) • the replacement of the first set of restrictions in (2.34) aPnU[ ([/ + asn]Pn) - x\ - qyl < 0 , V n € NTl by aPnU[ ([f + asn}Pn) -x\- qy2n + (1 - a)ln < 0 , V n € W T l , (2.42) 101 • and the addition of the term Y2neNT ^nPnBn to the dual objective function of (2.34). Restriction (2.42) expresses the optimal investment conditions in terms of implicit mar-tingale measures and defining (scaled) lower bounds for these measures. So <xPnU'L([f + asn]Pn) + (1 - a)ln <xn + qy2n . But, x\ + qyl < x\ + y\ < yn , V n G i V T l . The fact that U[(-) is strictly increasing, together with (2.40), (2.41), and x\ > apnU'2([f + as£]/3 n) yields Propositions 2.6.1 and 2.6.2 hold for the high water marks policy, although with different embedded martingale measures Ql and Ql; Ql has different lower bounds than the corres-ponding ones for the case of fixed benchmarks (see equation (2.38)). To see this, observe that if we apply Equation (2.40) repeatedly 0 < apnU'1([f + asn]8n) + (1 - a)ln , V n G NTl . Therefore, 0 < aPnU[{[f + asn]0n) + (1 - a)ln <x\ + qy2n < y\ . (2.43) Hence, Qi = {$o} n£jV T l defines a probability measure. meDT2 (n) 102 where DT2(n) is the set of descents of node n by time T2. Then, by equations (2.41) and (2.37), Y2 l m = YI x ™ - Y2 <*PmU'2{[f + as2m]pm) • meDR2(N) meDT2(n) meDT2(n) Therefore, ln > Emeo^fn) " P m ^ d / + " s l l ^ ) - Hence, using (2.43), Q\ is bounded from below by apn + (1 - cc) ]T apmU'2{[f + as2m]Pm) m€DT2 (n) (2.44) neNTl i.e. !/„•* = vWr? > aPnU'r ([/ + asln}Pn) + (1 - a) I £ " P ^ L f + ] , V n S i V r i . \meDT2 (n) / Therefore, the value of the embedded (martingale) probability measure Q\ for each scenario n £ depends on the combination of natural probabilities and marginal utilities in n at T i , and all its descendant scenarios by monitoring date T2. 2.6.2 Multiple Monitoring Dates We now extend the results to the multi-period monitoring case. Assume that there are m monitoring dates T i , . . . , T m (m > 2) where the portfolio's value is measured against its benchmark. As in the two-period case, we assume that the hedge fund manager follows self-financing strategies between monitoring dates and that a proportion q (with q < 100 %) of the profit over the benchmark is subtracted from the hedge fund portfolio's value at each 103 monitoring date Tj, except for the last one (i. e., j — 1,..., m — 1). The subtracted amount should cover at least the incentive fee of the hedge fund manager (i.e., q > oi). As in the two-period case, the re-balance of the investment positions does not occur at Tj but a period later at Tj + 1. The utility function over the entire horizon is assumed to be time-additive and each of its utility components belongs to the class of utility functions IA. Theorem 2.6.1 (i) If > 0, then EQ'^ PTJ+1C^\ > EQ'^ [0TJ+1CTj+i] and (ii) > 0, then EQ'^ IpT^C^] > EQ'^ [ 0 T J + L C T J + L ] , Q*+1 = I / ,y \ is a martingale measure, and E®i [ • ] is the expected value operator under Qj and based (conditionally) on the information att = 0. Proof: Only (i) is proved since (ii) is completely analogous. Fixed or Stock Index Based Benchmarks The corresponding dual restrictions are a p » ^ + i ([/ + *si+1]0n) - x{+1 - qyi+2 < 0 V n e NT. (2.45) xi+1 + yi+2-yn+1<0, VneNT, (2.46) n <Tj+l _ £ » j + 1 A l Cr i + i , „ < 0 (2.47) neNT n£NTj for j = 0, . . . , m — 2, and 104 <*PnUm ([f + c < ] / ? n ) - < < 0 , V n e 7Yrm , (2-48) * n - y J T < o , V n e i V T m , (2.49) E ( C ) / ? n C j - 1 + 1 - E « C 7 T m _ 1 + 1 , n < 0 (2.50) for Tm-i. Assume e '^+i > 0- Then, Case 1: j = 0,. . . , m — 2. Applying the complementarity property to (2.47) yields E ((Xi+1Y + (yn+2T) PnCTJ+1 = E ( 2 / n + 1 ) * / 5 n C T j + 1 , n . From (2.45) and (2.46) 0 < aPnU'j+1 ([/ + as„ + 1]/3 n) < xn+1 + W n + 2 < x„ + 1 + y£+2 < Hence, E ^ N ^ Vn+1PnC^+1 > £ „ £ ^ y n + 1 P n C T j + 1 , n , and, dividing by y ^ 1 = ^ y ^ 1 , Case 2: The proof is analogous to Case 1. The case of put options is proved in an analogous manner. 105 High Water Marks The dual restrictions for high water marks are the same as the two cases treated for fixed or stock index benchmarks except that (2.45) and (2.48) are replaced respectively by *PnU'j+1 ([/ + asi+1}pn) - xi+1 - qyi+2 + (1 - a)ln+1 < 0 , V n E NTj+1 , j = 0, . . . , m - 2 , and aPnU'm ([/ + as™]A,) - x™ + (1 - a)lm < 0, V n E NTm, where ln+1 satisfies ' n + 1 - £ m e c ( « ) 4 + 1 = 0 , VnENt t = Tj+1,... ,Tj+i-l, j = 0 , . . . , m - 2 , -xn+1 + Pn+l = 0 1 V n 6 % 1 , J = l , . . , m - l . The proof for this benchmark policy follows the same structure used for the case of fixed or stock index benchmarks. Q.E.D. 2.7 Advanced Models The models given here incorporate risk management features and other considerations that are not so far. These models include measures of underperformance (with respect to a bench-mark), risk management of options, and other risky factors such as short selling constraints. We focus on the single monitoring case and depart from the following basic set of constraints Z0 • 00 + fa (e0 • Vo - eo • Vo) = / W o , Zn • (On - 0a{n)) = 0 , V n e Nt , V t = 1,..., T , Zn-On- PnSn + Pn Of) - e0) • V„ = 8 n B n , V U E NT (2.51) Zn • 0n > 0, Sn > 0 , V Tl E NT , eo > 0, e0 > 0 . 106 2.7.1 Underperformance Risk Management The risk of underperforming the benchmark is a major issue for any hedge fund manager, whose performance and compensation fee is based on it. Therefore, appropriate risk measures of underperformance should be included in the objective function of the hedge fund manager. A n important class of measures in that sense are so called downside-risk aversion measures (Fishburn (1977)) that penalize the shortfall of the portfolio relative to a given benchmark BT- Within this class of downside-risk aversion measures, there is a subclass of measures that is widely applied and which is known as the class of lower partial moment measures (Bawa and Lindenberg (1977)). In this subclass, the penalization is done through partial (statistical) moments of the shortfall. That is, underperformance measures are of the form i t v ( B r ) = E [Max (BT - WT, 0)7] , where 7 > 0 and WT is the value of the portfolio at time T. The values of 7 determine the weight that the investor gives to small or large deviations. The larger the 7, the more the investor cares about larger deviations and vice versa. Popular values of 7 are 7 = 0, which defines the so called shortfall probability; 7 = 1, which is simply the expected shortfall; and 7 = 2, which defines the downside variance. Underperformance measures of the form of FCy(BT) may be combined with the goal of maximizing the hedge fund manager's compensation fee, using the objective function Maxg^s E[(f + asT)3T]-ARN(BTBT) , (2.52) where A is a nonnegative constant that determines the risk aversion of the hedge fund manager (towards underperformance) or equivalently, a tradeoff parameter between expected 107 profit and underperformance risk. The objective function (2.52) is Max 0,£)S E if + asn]PnPn - A ^ ( S * » / ? n ) 7 p n , (2.53) where S F n is nonnegative and represents the shortfall relative to the benchmark Bn in scenario n € NT, and must satisfy the following set of (modified) constraints Zn-9n- pnsn + pn (eo - eo) • + P„SFn = PnBn , V n £ NT , (2.54) plus nonnegativity restrictions, where A > a to assure feasibility. Consider the case of the expected shortfall as the measure of underperformance, i . e., 7 = 1. Then, SF; = Max (pnBn - [Zn -en + pn (eo - e0) K?] , 0) . since the objective function can be re-expressed as £ fPnPn +J2ia iZ" -0n + Pn OS ~ ej) K° - pnBn) P m + (« - A ) S F > n ] . neNT neNT Hence, if Zn • 0n + /?„ (e*. — ejj) > PnBn for some n 6 N T , then, given that (a — A) < 0, then S F * = 0. Otherwise, if Zn-0n + Pn (ej - e*,) Kf < PnBn, for some n € N T , the objective function is f^JPn +Y,[A {Zn • 9*n + p\ (e*0 - t0) V* - pnBn) pm + (a - A)slpn) . 108 Thus, (a- A) < 0, implies that s* = 0. Hence, by feasibility, SF* must satisfy that The inclusion of the set of variables (SFn)neNr extends the feasible region defined by (2.51) and adds the restrictions to the dual associated to (2.51) (see (2.28) in Section 2.4). The model formed by the objective function (2.53) subject to the constraints (2.51) (with the modified restriction (2.54)) possesses the same optimality conditions for option investment stated in Proposition 2.4.1. The addition of the dual restriction (2.55) gives us more structure; and, hence, more insights about the optimal solutions. For instance, if SF* > 0, then, by complementarity, xn = Apn. Analogously, s* > 0 implies x* = apn. Furthermore, from our previous analysis, snSF* = 0. Therefore, That is, the dual optimal variable x*n plays the role of an indicator function of profits and shortfalls for each scenario. 2.7.2 Option Risk Management Hedge fund managers should consider appropriate risk management for any trading strategy that is implemented in their pursuit of superior returns. We propose some risk measures for the case of strategies involving options and the way these measures could be included in our SF* = 6nBn — Zn-6*n~ Pn (e$ ~~ €S) Xn < Apn , V n e NT (2.55) 109 framework. We also study the implications of the inclusion of such measures on the nature of the optimal strategies. Buying options is less risky than selling options since the maximum loss is bounded. At worst, the premium paid for the option is lost if the option expires out of the money. Therefore, an appropriate risk measure for the purchase of an option is VWo) , (2.56) where ip is the probability that the option expires out of the money (ip can be computed once the scenarios have been set up and therefore it is known at the moment of solving the model), and Vn is the option's price (call or put). Hence, assuming homogeneity of the risk measure (i.e., a double position have double risk, see Artzner, et al. (1999)), a model that incorporates a tradeoff between profit and risk of purchased options in our framework is Max e>e0iS £ [/ + asn}(3nPn - eoIVWo)] (2.57) n£iV T subject to (2.51). The inclusion of enfa/K A^))] yields the following proposition: Propos i t i on 2.7.1 / / > 0, then 0OCO = }ZneNT PnCn < £°* \p\CT), where Q* = \ % \ . Analogously, if eP'* > 0 then, ^ = E (^) ^Pn < E®' [3TPT]. neNT V ^ y ° ' Proof: The proof is analogous to that of Proposition 2.3.1. The key is that the corresponding dual problem differs only in the last two constraints from (2.2). The last two constraints are 110 E„ ejv T xnBnCn - (yo + ip) P0C0(l + tcc) + r]C = 0 , EneNT XnPnPn ~ fob + A>P0(1 + tCP) + 7/p = 0 , Q.E.D. i.e., buying options in the current model is optimal if these are underpriced (under Q*). Selling options implies taking a position where losses are, in principle, unbounded and, hence, risk management is crucial. Hedging the risk of sold options is usually carried out through either the implementation of a portfolio of assets that replicates the payout of the option or the purchase of an option of the same characteristics (i.e., same expiry date and exercise price). The latter is partially implicit in the basic formulation (2.51) since it allows for buying and selling options of the same type although the model does not enforce the purchase of options if stock index options are sold. Perfect replication of the option's payoff is not always possible since our framework allows for incomplete markets. Therefore, it is imperative to include risk measures into the basic formulation. We addressed this problem for the basic problem in Section 2.3.2 by including Pn LneNr We now address the risk management for sold options in a broader sense. For instance, we consider a class of risk measures that satisfies two properties: homogeneity and a priori determination (i.e., once the scenarios have been set up, the risk measure can be determined). Two members of this class of measures are Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR) (see Mina and Y i Xiao (2001)). VaR is the maximum expected loss under a certain confidence level and CVaR is defined, given a certain level of confidence, as the expected loss given that the loss has surpassed the VaR of the corresponding confidence 111 level. Although, VaR is the most widely used risk measure in the financial context, CVaR has more appealing properties. For instance, Artzner, et al. (1999) prove that CVaR is a coherent risk measure (while VaR is not), Rockafellar and Uryasev (2000 and 2000a) prove that CVaR is convex and continuous, and Rockafellar, et al. (2000b) prove that CVaR is an expectation-bounded risk measure (while VaR is not). Other convex risk measures, which place more weight on losses, are discussed in Rockafellar and Ziemba (2000) and Cariho and Ziemba (1998). If VaR is computed at the confidence level of (1 — r/)%, then a tradeoff between profit and risk over sold options can be modelled via Max 0 , £ o , s £ [/ + asn}0nPn - e0(nVaR) (2.58) n€NT subject to (2.51) or Max e,eo,s £ [/ + oisn)0npn - e0 {nCVaR) (2.59) n€NT subject to (2.51) if CVaR is used. Application of duality leads to P ropos i t i on 2.7.2 Ife^'* > 0 (ep'* > 0J, then (i) under model (2.58J, PoCo = J2 (4) PnCn + (\) (nVaR) , If%P0 = £ (^) 0nPn + (\) (vVaR)) , n 6 J v T V 2 / o / Wo/ \ n e N r Wo/ Wo/ J (ii) under model (2.59), 112 Proof: Only (i) is proved since (ii) is completely analogous. Assume e0'* > 0. Then, by complementarity, yo(30C0 = Z)n€jv„, x*n(3nCn + r/Vai?. Therefore, 2.7.3 Other Extensions Restrictions on the positions of the portfolio None of the models studied before impose any restrictions on the asset positions within the portfolio. Nevertheless, such restrictions often appear in practice. For instance, liquidity and short-selling restrictions are common investment constraints that investors impose on the management of their portfolios. Such restrictions can be included into the set of constraints (2.51) through the addition of the constraints where bn € 5ft. Therefore, the dual restriction, ynZn — Y^meC(n) Vm.Zm = 0, for all n G Nt, and t = 0 , . . . , T — 1, is replaced by Q.E.D. 0n >k V n S Nt , t = 0 T ynZn - E y™-Zm - ^n = 0 , V nE Nt, t = 0 , . . . , T - 1 ; meC(n) 113 xn ~ Vn + A° < 0 , V n E NT ; A „ > 0 , Vn£Nt, t = o , . . . , r - i ; and the term — X } n 6 J V r ^ " ^ n * s a ^ded to the dual function. Although yn > 0 , V n E JVt, j <?„ = ^ }• is no longer a probability measure on (n,FT). Further Extensions The model (2.51) assumes that the hedge fund manager buys and sells options with the same payoff. A natural extension is to allow for buying and selling options with different payoffs and/or different times to maturity. For instance, the budget constraint is still of the form Z0 • 0o + A> (eo • Vb - e0 • Vo) - p0W0 , but now e0 = (e j , . . . , e^) and e0 = (4> • • • > e^)) where e30 (4) is the purchase (sale) price of the jth option (European calls or puts) for j = 1,..., N (j = 1,..., M). Self-financing restrictions are modelled using Zn • {On ~ 0a{n)) ~ 0n (c0 • Vn ~ A ' Vn) = 0, V Tl E Nt , where e0 (e0) is the vector of positions of the option purchased (sold) at time 0 and which matures at time t, and V£ (V*) is the vector of corresponding payoffs. The Benchmark 114 restriction is Zn-9n- Bnsn + pn (el • - el • VJ) = 0nBn, V n E NT . Duality can be applied to obtain insights on the structure of optimal investments as in this chapter. 2.8 Conclusions This chapter develops a duality framework for a variety of models of the problem faced by a hedge fund manager whose goal is to maximize his expected compensation fee using option investment strategies. This framework leads to explicit characterizations of the optimal strategies that contrast with many stochastic programming models used in finance that only provide numerical results. The application of the duality framework for the hedge fund manager's problem leads to the general, and expected, conclusion that the purchase or sale of Stock Index Options is optimal if these options are not overpriced or not underpriced respectively. However, our approach derives the pricing threshold that determines the overpricing or underpricing of such options, and the relation of this threshold with the parameters that characterize his compensation scheme. We obtain explicit relations between the predetermined percentage that the manager obtains from the profits over a specified benchmark and the embedded probability measure that defines the pricing threshold. Moreover, we generalize our results to a broad class of utility functions, risk management considerations, short selling restrictions, multiple monitoring dates, and different policies for determining the benchmark, such as fixed and high water marks schemes. 115 The framework has several advantages: First, it allows for a broad class of distributions including the commonly observed fat-tailed return price distributions. Second, it is applica-ble to both, incomplete and complete markets. Third, the mathematics used to construct this framework are simpler, and hence more accessible, than those in the continuous-time framework. Fourth, the implementation of this framework to establish optimality conditions is relatively simple. Our duality framework can be extended to consider more complex or general problems such as more general convex compensation schemes that include problems such as the com-pensation of a corporate manager that controls firm leverage or the compensation of a trader at a security firm. 116 Chapter 3 Incentives and Design of Option-Like Compensation Schemes 117 3.1 Introduction Two of the most significant changes in corporate compensation practices in the last fifteen years have been, on the one hand, the escalation and recent decline of the average executive compensation amount and, on the other hand, the replacement of base salaries by stock options as the single largest component of executive compensation (Hall and Murphy (2003)). For instance, since the early 1990s stock options have replaced base salaries as the single largest component of executive compensation in almost all industry sectors and, during the same period of time, firms related to the technology sector (computers, software, internet, telecommunications and networking) have had a pronounced increase in employee stock options (Hall and Liebman (1998), Murphy (1999)). The explosive growth in the use of options as a compensation can be roughly explained by two main factors. First, the belief that option-like compensation schemes are capable of aligning interests between executives, employees and shareholders while, at the same time, they can attract, motivate and retain high qualified executives and employees (e.g. DeFusco et al (1990)). Second, practical reasons for companies related with the conservation of cash and the reduction of reporting accounting expenses (e.g. Oyer (2004)). Regardless of its causes, the use of option-like compensation schemes has generated concerns about their design and the behavior they induce, as well as about the valuation of the options embedded in these payment schemes from the executives' and shareholders' point of view. The design and induced behavior of these option-like compensation schemes are inti-mately related. The compensation scheme design determines the induced behavior of share-holders, executives, and employees. Conversely, a desired induced behavior characterizes the required compensation schemes. 118 Since the early nineties, there has been an intensive research on the issue of induced be-havior of option-like compensation schemes (Murphy (1999)). Most of these studies support the idea that these compensation mechanisms can be effectively used to align interests and induce desired behaviors, such as the implementation of riskier strategies by an investment manager. However, recent studies by Carpenter (2000), Lewellen (2006), Ross (2004) and Braido and Ferreira (2006) conclude that this type of payment schemes do not necessarily imply the expected or desired behaviors, unless some assumptions on the executives' and employees' preferences, or on the specific form of the compensation scheme, are made. For instance, Carpenter considers a risk averse manager compensated with a call option on the assets he controls. Under certain assumptions, Carpenter obtains the manager's explicit optimal policy and finds that this type of option compensation does not necessarily lead to greater risk-taking, as it might be intuitively expected 1. Ross (2004) corroborates Carpen-ter's theoretical findings in a simpler static framework and proves that, among the class of convex compensation schemes, such as vanilla call options, there is no single scheme that would make all strictly-concave utility functions uniformly display lower^  (absolute) risk aver-sion in their domains. However, very recently, Braido and Ferreira (2006) find, in a static setting too, that for the particular case of a call option, there exist conditions on the stock price distribution -not considered by Ross (2004)- under which all strictly-concave managers prefer riskier projects to safer ones. Many of the ideas developed recently about induced risk incentives by option-like com-pensation schemes (e.g., Carpenter (2000), Ross (2004), Braido and Ferreira (2006)) have been applied to other important corporate issues such as optimal leverage decisions. For instance, Cadenillas, et al. (2004) and Carlson and Lazrak (2005) are two important, and complementary, studies that have analyzed the optimal leverage choice from the shareholder's 1Prom option pricing theory, it is well known that the higher the volatility, the higher the option value. Therefore, it is intuitive to think that a manager who holds a stock option as part of his compensation, will be encouraged to take higher levels of volatility in order to increase his expected compensation. 119 and the manager's point of view. Research on corporate compensation design has been importantly influenced by agency theory (Ross (1973)). This theory considers the problem of a person, the principal, who wants to induce another person, the agent, to take some action which is costly to the agent. The principal's problem consists in designing an incentive payment from the principal to the agent that induces the agent to take the best action from the principal's point of view. In the context of executive compensation the shareholders play the role of the principal, while the executives represent the agents. The majority of executive compensation design studies that are based on agency theory, visualize executive compensation as a remedy to the agency problem (e.g. Murphy (1999) and Core, et al. (2003) for surveys). However, there exist some recent studies that consider executive compensation not only as a remedy, but also as part of an agency problem itself (Bebchuk and Fried (2003)), arguing that managers usually have a certain degree of power or influence over the people (e.g. board of directors) that set their compensation payments. For the shareholders, writing compensation options represents an opportunity cost, while for the executives or employees these options are an expected benefit. However, the expected benefits usually do not equate the opportunity costs of issuing these options. In fact, these costs are often greater than the expected benefits, since compensation options typically have trading and hedging restrictions that an outside investor would not have. The magnitude of the difference between the expected benefits and the opportunity costs derived from these options is directly related to their induced behavior and, thereafter, their design. For in-stance, the smaller this magnitude is, the stronger the induced incentives are, and vice versa. Therefore, the design of these options is connected to the control of this magnitude. Determining the company's opportunity cost from issuing compensation options is less challenging than estimating their value for executives or employees, since they have trading 120 and hedging constraints than an outside investor would not have. In fact, the outside in-vestor's freedom usually fits into the model assumptions made by the existing frameworks to value options and therefore, opportunity costs can be usually estimated by using an appro-priate option-valuation model. Hedging constraints on compensation options imply that executives' and employees' va-luations depend on their particular preferences. Therefore, research efforts on the valuation of these options have mostly relied on utility-based frameworks with two types of approaches: a certainty equivalent approach, which determines the riskless cash compensation that the executive or employee would exchange for the option (e.g. Hall and Murphy (2000,2002) and Lambert, et al. (1991)), and an optimal strategy approach in which, given an objective function, the optimal exercising policy is obtained and used to value the option (e.g. Hudart (1994) and Marcus and Kulatilaka (1994)). Nevertheless, there are some studies that have addressed the valuation problem without considering an explicit preference-based model. For example, Carpenter (1998) shows that a simple extension of the American option (binomial) model is reasonable to value, in practice, these compensation options. Characterizing the optimal exercise policy of a compensation option is not only important for obtaining the subjective value of compensation options, but also for determining the hedging cost for the shareholders. This is because even though the outside investor has, in principle, no trading restrictions, the person who actually exercises the option is the executive or employee. Theory of optimal exercise of compensation options is still under development. Most part of the efforts on this direction are based on the intuition that, given the payoff risk and the undiversified condition of the executive or employee, the compensation option holder chooses an option exercise policy as a consequence of a greater utility maximization problem that includes other decisions, such as portfolio and consumption choice and managerial strategy. The majority of the papers under this perspective, make some kind of exogenous 121 assumptions about how non-option wealth is invested (e.g. Huddart (1994), Marcus and Kulatilaka (1994) and Carpenter (1998)) or about the exercise policy (e.g. Jennergren and Naslund (1993), Cvitanic, et al. (2004), and Hull and White (2004)). This chapter focuses on the design and induced behavior of option-like compensation schemes for a fund manager. It considers payment schemes composed by cash and two call options with the same underlying -the fund value- but different strike prices. There are two main goals: First, studying the induced incentives of this type of compensation package -fund value and risk incentives- and analyzing if these compensation schemes have the same unexpected theoretical features that have been documented for the single-option case. Second, finding criterions to determine appropriate parameter values of the considered compensation package to induce, as much as possible, specific behaviors. In order to achieve these goals, we develop our analysis in a continuous-time framework similar to the one used by Carpenter (2000). This framework presents two main advantages for our study. First, it assumes that the fund value is governed by a geometric Brownian motion that allow us to derive explicit formulas that facilitate our analysis. Second, it is easily adaptable to general convex compensations schemes, as the one treated in this chapter. We obtain the explicit optimal risk-taking strategy for an undiversified risk-averse ma-nager who maximizes the expected utility value of his compensation package. We find that including two distinct options in the compensation scheme does not eliminate the unex-pected risk-taking behavior observed for the single-option case, but it does provide us with a significant higher control on the manager's implied actions. In particular, we obtain that the sensitivity of optimal risk-taking choice with respect to fund value is roughly propor-tional to a stability factor, which involves only the compensation parameters. Therefore, the smaller the value of this factor, the less sensitive is the risk-taking decision to changes in the fund value. This result can be used, for example, by investors to decrease the incentives 122 to rebalance the fund's portfolio by appropriately choosing the compensation parameters. For instance, we consider in our analysis a situation in which investors desire to induce an strategy that leads to a range of fund values of the form {0} U (V, oo), where V is a mini-mal target value. This type of strategy is typically implied by a single-option compensation scheme (Carpenter (2000)) and can be induced by the double-option payment scheme con-sidered in this chapter. However, this strategy can involve, in both cases, drastic changes of optimal risk-taking decisions given small variations of the fund value, specially when the options are out-of-the money. This is costly in terms of transaction costs or investment irreversibility or lumpiness. Therefore, it is imperative to be able to control the possibility of such drastic changes. Although there exists a stability factor for the single-option case that could be used to control stability, this is hardly possible in practice since there are typical values for the corresponding compensation parameters. However, for the double-option case, the stability factor involves two extra parameters, namely the second-option exercise price and the relative weight of the second-option payoff in the compensation scheme, for which there are not such typical values. Hence, adding a second option provide to investors with two degrees of freedom for controlling stability. We propose specific criterions to determine optimal values for these extra parameters. Therefore, this stability factor, together with other results obtained in this chapter, provide incentives for investors to grant the manager additional options. Nevertheless, we also discover that the risk-taking optimal profile can change abruptly for a certain second option's threshold exercise price. This means that two managers with very similar option-like compensation packages can have different risk-taking profiles. In other words, a marginal change in the compensation structure may imply drastic changes in the risk-taking incentives and hence a policy implication of our work suggest that investors should pay attention to global compensation scheme of their managers. To our knowledge, 123 this is one of the few studies on how exercise price decisions may affect managerial risk-taking behavior (Hjortsh0j (2006)). This chapter is structured as follows: Section two describes the model. Section three analyzes the utility function derived from the manager's preferences over his compensation package. Section four develops closed-form expressions for the manager's optimal choice of terminal fund value and volatility. Section five illustrates different types of fund value and risk incentives that the proposed option-like compensation scheme is, in theory, able to induce. Section six proposes simple criterions, based on the theoretical results and analysis of the previous sections, to determine appropriate compensation parameter values in order to induce specific fund value or risk profiles. Section seven presents the conclusions. 3.2 The Model We consider the problem of an undiversified risk-averse manager who maximizes the expected utility value of his compensation package. We assume that the manager makes decisions regarding the risk level of the fund, given a compensation package. To address this problem we consider a continuous-time framework that is described in the following subsections. 3.2.1 The Fund Value We depart from the standard assumption that the value of a fund is governed by a geometric Brownian motion (Black and Scholes (1973), Merton (1974)) and consider the extended model dVt = nVtdt + avtVtdt + vtVtdzu (3.1) 124 where p and a are fixed parameters, (^t) t > 0 is an adapted process and (zt)t>0 is a standard Brownian motion. Model (3.1) is equivalent to the one used in Carpenter (2000) and a particular case of the one used in Cadenillas, et al. (2004). The parameter p is exogenous and represents the expected return due to the fund's momentum which cannot be affected by the manager. In an equilibrium setting, p, would be related to the prevailing interest free rate in the economy. The process {vt)t>o represents the volatility of the fund's value and it is assumed that it can be controlled by the manager and that it has an impact on the expected return of the fund. This impact is captured by the parameter a and so it can be interpreted as the expected return per unit of risk undertaken by the manager, as measured by the volatility. 3.2.2 The Manager Problem The manager is risk averse with utility function2 U(x) = ^ - , 7 > 0 , 1 - 7 where x is the compensation that the manager receives and 7 represents the manager's risk aversion, and who receives a compensation package formed by a fixed salary X0, a proportion p of a call option payoff on the fund's value with a a strike value B, that represents a benchmark payoff for the manager, and a proportion q of the payoff of another call option on the fund's value with a strike price B + K, where K > 0. Therefore, the total value of the compensation package is cb(V) = X0+ p(V - B)+ + q(V-B- K)+ 2This is the power utility function, the most popular member of the class of C R R A utility functions. 125 where (x)+ = max(x,0). We assume that the values B and K are preestablished by the investors. If we were in a firm value context, the parameter B might represent the leverage level of the firm, then the compensation package is composed by a fixed salary X0, a pro-portion p of the shares of the firm, and a proportion q of the payoff of a call option on the levered value of the firm. The goal of the manager is to maximize the expected derived utility of his compensation package at the end of the horizon time [0, T], by choosing the level of risk for the fund. That is, the objective of the manager is sup E (<KVr)) 1 - 7 1-7 (3.2) where the volatility process v is well behaved and is such that Vt > 0, 0 < t < T. It can be shown that the solution to Problem (3.2) is uniquely characterized (e.g. Karatzas et al (1987)) as the solution of the static problem sup E vT (<f>(Vr)) 1 - 7 1-7 (3.3) under the positivity constraint Vr > 0, i.e. under the assumption that the fund's value is always nonnegative, and the constraint E [ZTVT] < Vo, where 126 Shape of the Derived Utiity Function U " 0 Oi 1 1 1 1 1 1 1 r Value of the Fund Figure 3.1: Shape of U o (f>. This is the graph of U o <f> when the following parameters are considered: X0 = 0.005, p = 0.01, q = 0.02, B = 1.04, K = 0.76 and 7 = 2. Note that (U o 0) (0) = (-1) (0.005-1) = -200. £ t = exp |-pt - y t - azt j , for * € [0, T], (3.4) and which, in the case that p represents the (instantaneous) interest free rate, represents a state price density or a pricing kernel. 3.3 Derived Utility Function Analysis Problem (3.3) cannot be solved directly using standard theory since the derived utility func-tion U o d) = 0 1 _ 7 / (1 — 7) is not concave overall its domain (See Figure 3.1). This lack of concavity is expected up to some extent since the convex compensation schedule cj) is designed with the intend of inducing the manager to be less risk-averse and therefore with 127 the goal of making its utility function less concave. In order to solve Problem (3.3) we construct an equivalent problem where Uocb is replaced by an appropriate concave function which we refer to as the concavification function of U o f>. This section is devoted to the related analysis of the derived utility function U o <b in order to construct its concavification function. 3.3.1 Motivation Concavity of the concavification function of U o <b is required in order to apply standard methods while its appropriateness refers roughly to the property that its solution is also optimal for the original problem. Intuitively, one would expect that the concavification function of U o d>, say £/, is closed to it since U should reflect the essential set of preferences represented by the derived utility function U o <b, at least over a neighborhood of the optimal solution. Thus, in such sense, the construction of U entails to find the closest concave function to U o <b. This closeness to U o <b must be ruled by the referred appropriateness of U. That is, it must be guided by the premise of assuring that the optimal solution under U is also a solution of Problem (3.3). In other words, the optimal solution must not change if U o (b is replaced by U. One way of satisfying such premise is to restrict our search of U to the set of all the concave functions that dominate U ocb, i.e. to consider the set of functions G = {g : 5R+ —> 5? | g is concave and g(x) > (U o </>) (x) for all x > 0} . The reason to focus on this class of functions is the following: suppose that it exists g G G such that E[g(X*)} = E[(Uocb) (X*)} 128 (3.5) where X* is the optimal solution of Problem (3.3) when U o (fi is replaced by g. Then, E {(U o 0) (X*)} = E [g(X*)} > E [g(X)} > E[(U o «/>) (X)} for all X. Thus, X* must be an optimal solution of Problem (3.3). Therefore, the construc-tion of the concavification function JJ can be formulated as determining a concave function that dominates U o </> and that satisfies condition (3.5). This condition is precisely what we mean by the closest concave function to U o (fi. 3.3.2 Concavification Function Construction As previously motivated, the concavification function of U o (fi is roughly the closest concave function that dominates U o (fi. Using this idea, we provide a geometrical motivation for the construction of the concavification function U, derived from conditions established in terms of the absolute risk aversion induced by the compensation schedule (fi. Then, we state formal concavification results for the particular compensation scheme (fi that we consider. G e o m e t r i c a l M o t i v a t i o n To motivate the geometrical construction of the concavification function U, we assume that it is of the form (U o (fi) + h, where h is a function, and analyze the conditions that should be imposed on h to assure that U is the closest concave function that dominates U o 0. Dominance of U over U o (fi holds by simply imposing the restriction that h(x) > 0 for all x > 0. Concavity and optimal closeness of U require a deeper analysis. To perform such analysis we use the concept of absolute risk aversion (Pratt (1964)). We start off by assuming that h is twice differentiable over all its domain and that its first derivative does not vanish over the interval [0, B). Therefore, the induced absolute risk aversion of (U o (fi) + h can be 129 decomposed as follows: A (U°4>)+h (Uo<t>) +h {Uo<t>j+h' = < h (C/Q0) (U°4>)'+h' (Uo<t>) _ h (£/o</>)' (C/o0)' if x < B. if x G (B, B + K) U (B + K, 00) . Therefore, in the case that x < B, the induced absolute risk aversion of U, and thereafter its concavity, depends on the term Ah = ti'jx) ~h'{x) " So, U is concave for x < B if and only if Ah > 0. On the other hand, in the case that x G (B, B + K) U (B + K, 00) observe that A(uo4>)+h can be rewritten as F {A* + Au((b) where F = (U o <j>)' / \(U o <j>)' + ti + . Hence, if we assume that the concavification function U is within the class of (strictly) monotone utility functions, then F(x) > 0 for all x > 0. Therefore, the induced absolute risk aversion of U, and thereafter its concavity, depends on the expression A^ + Av(4>) cb - 1 + M<t>) {Uocb) Thus, for the case x G (B, B + K)U (B + K, 00) , the absolute risk aversion of U, and its concavity, depend on the interaction of three effects (Ross (2004)): a convexity effect, 130 represented by Af, a magnification effect, represented by Au(<f>) [(f) — l ] ; and a translation effect, represented by L4(/(</>) - 7-^-7] • Hence, U is concave in this case if and only if the translation effect is positive and it offsets both, the convexity and the magnification effects. That is, concavity of U requires that In summary, if the concavification function U is of the form (U o <j>) + h, then h must satisfy the following conditions: D o m i n a n c e ( D ) : • h(x) > 0, for all x > 0. M o n o t o n i c i t y ( M ) : • h'(x) > - [U o (j))' (x), for all x > B. C o n c a v i t y ( C ) : or equivalently h" < (Uo(f>)' {A^ + Autt) [</»'- 1] +M4)} 1. Ah(x) = > 0, for all x < B. 2. h"(x) <-(Uo (j))" (x), for all x € (B, B + K) U [B + K, 00 ) . 131 Therefore, intuitively, the function (U o </>) + h is the closest concave function that dom-inates U o <f> if h is the smallest function that satisfies the D, M and C conditions. If the D and M conditions were the only ones considered, h would simply be the zero function (i.e. h(x) = 0 for all x). However, the zero function does not satisfy the C con-ditions. For instance, under the assumption that h is twice differentiable, the zero function clearly does not satisfy the C.2 condition. Therefore, the C conditions are the key ones. We focus first on C.2 —h" > (U o <f>) , and then analyze conditions C . l , D, and M. In particular, we study the minimal conditions under which —ti satisfies C.2. That is, we analyze the condition Although we are interested on the fulfillment of (3.6) for x > B, we omit for the moment this range restriction and analyze the consequences of this condition. Equation (3.6) implies that h must be of the form That is, we need to add to U o cp what it is necessary to obtain a linear function, at least over the range of values in which U o <p is not concave. This makes sense considering that linear functions are minimal concave functions, since any linear function is both, concave and convex. We now analyze the range of values for b and m that would lead h to satisfy the D and M conditions. If h satisfies the D condition, then -ti' = (Uo (j)) (3.6) h(x) — b + mx — (U o (/)) (x) . b + mx > (U o <f>) (x) (3.7) 132 i.e., the linear function b + mx must dominate the derived utility function (U o cb) (x). It turns out that h must satisfy /t(0) = 0 in order to be optimal3. This implies b = (U o cb) (0), which means that this linear function must be anchored to the point (0, (U o cb) (0)). The inequality (3.7) cannot be satisfied in a strict manner for all x > 0 since that would imply the existence of a concave function that is closer to U o cb. Indeed, it exists x > B such that b + mx = (U o cb) (0) + mx = (U o <b) (x) . Therefore, b + mx must be the line that is tangent to the graph of U o <b at x that passes through the point (0, (U o <b) (0)). Thus, x. satisfies (U o <b) (0) = (Uo(j>) (x) + (U o <b)' (x)(0 - x) . and m = (U o (b) (x) > 0. Hence, note that h(x) = b + mx — (U o <b) (x) = (Uocb) (0) + \{U o <b)' (x)l x-{Uo<b) (x) satisfies h'(x) = (Uo </>)' (x) -(Uo <b) (x) >-{Uo <f>) (x) i.e., h satisfies the M condition. Moreover, note that 3If /i(0) > 0, it is always possible to construct a function h such that h(0) = 0, (U o <p) + h concave, and 0 < h(x) < h(x) for all x £ [0, B), given that U o <j> is flat over this interval. 133 Ah{x) h (x) ' h'(x) (U°<t>) (*) = 0 (U04) (x)-(t/o0) (x) for x > B. Therefore, h(x) = (U o <f>) (0) + {U o(b) (x)j x - (U o cb) (x) satisfies the D, M , and C conditions. However, note that for x > x, it is enough, and thus optimal, to set h(x) = 0 in order to satisfy the D, M , and the C conditions. So, h ( x ) = i (U°4>) ( ° ) + [(U ° 0)' ( £ ) J x-(Uo</>) (x) for rc < x, [ 0 otherwise That is, the geometrical construction of the concavification function entails drawing a line that is tangent to the graph of U o <b that passes through the point (0, (U o <b) (0)). The concavification function will be formed by this tangent line, for values less than or equal to the tangency point x, and by the derived utility function otherwise. We now formalize the construction of the concavification function for the type of com-pensation scheme (b that is considered in this chapter. Concavification Results The current analysis of the concavification function oiUocb begins from its conceptualization as the closest concave function that dominates U o cb. In the motivational part, the idea of closeness is mapped into a set of conditions that characterize the offsetting forces of the convexity effects induced by the compensation scheme cb. Therefore, closest within this context means to satisfy minimal conditions to offset these convexity effects, which lead to the described geometry of the concavification function construction. 134 Although, the previous motivational subsection provides us with a good understanding of the construction and shape of the concavification function, its concept has not been fully formalized yet. To formalize the concept of concavification function, we use the charac-terization of a concave function / in terms of its hypograph, defined by {(x,y)\y < f(x)}. For instance, a function / is concave if and only if its hypograph is a convex set (Zangwill (1969)). Therefore, using this characterization, the concavification function of / can be for-mally described as the function / whose hypograph is the smallest convex set that contains the hypograph of / . The smallest convex set that contains a given set is called the convex hull of such set. Hence, the definition of the concavification of / can be rephrased in the following manner: Definition 3.3.1 (Concavification) Let f be a real valued function on 3? and let Hf = {(x,y)\y < f(x)} be the hypograph of f. Let HJ be the convex hull of Hf. Then, if it exists a function f such that its hypograph coincides with the convex hull of Hf, i.e. if Hj = HJ, then f is said to be the concavification of f. Armed with the above definition, we formally characterize the concavification function for particular cases derived from the consideration of U o </> over special subsets of its domain and then combine appropriately such cases to define the concavification function of U o (f> overall its domain. To follow this strategy, we need to establish first the following preliminary results: Lemma 3.3.1 Let 7 > 0. Then, it exists Vi > B such that 135 £ 1 = (Xo+Py_-B))^ _ p V l { X o + p { V l _ B)y ,ifi±\,or log(Xo) = log((X 0 +p(Vi - 5)) - X o + g i _ B ) , i /7 = I-Proof: See Appendix 3. Lemma 3.3.2 Let 7 > 0. Then, it exists V 4 > B + K such that xjg_ = ^Xn+p{V4—B)+g{Vi—B—K))1~1 _ { p + q ) y4 [ X q + p { V l _ B) + q (V4 - B - K)r , if 7 ^ 1, o r log(Xo) = log ((Xo + p (Vi - B)) - X o + p ( V i i g ^ V 4 - B - K ) . i / 7 = 1. (3.9) Proof: See Appendix 3. 136 Lemma 3.3.3 Let 7 > 0. Then, there exist V2, V 3 G 5? such that satisfy the following system of equations: ( l - 7 ) ( X 0 + p ( V 2 - J B ) ) 1 - 7 = (i-1)(X0 + p(V3-B) + q(V3-B-K))1-'1 + (p + q) (X0 + p(V3-B) + q(V3-B-K))^ (V2 - V3) (3.10) and (1 - 7) (X0 + p(V3-B) + q(V3-B- K)) '-t = (i-y)'x0 + p(yi-B))1-1' + P(X0+p(V2-B))-'t(V3-V2) (3.11) « / 7 1, or l o g ( X 0 + p ( V 2 - B ) ) = l o g ( X o + p ( V 3 - B ) + ( 7 ( V 3 - J B - i i : ) ) + (p + q) (X0+p(V3 -B) + q(V3-B- K))~l (V2 - V3) (3.12) and \og(Xo +P(V3-B) + q(V3-B-K)) = log (X0 +p(V2 — B)) + p(Xo+p(V2-B))-1(V3-V2) ( 3 ' 1 3 ) if 7 = 1. Proof: See Appendix 3. The previous results establish, under certain assumptions, the existence of four tangency points over the graph of U o <f>. For instance, it is established the existence of 137 • the points (Vi, (U o (b) (Vi)) and (V4, (U o </>) (V4)) on the graph of £/o</> for which the co-rresponding tangent lines have the property of crossing through the point (0, (U o <b) (0)), and • the points (V2, (U o <b) (V2)) and (V3, (U o cb) (V3)) for which the line that passes through them is tangent to the graph of U o <b at both points. These tangency points are crucial for the construction of the concavification function of Uo<b. Next proposition proposes and characterizes formally, under certain assumptions, the concavification function of U o (b when its domain is restricted to the interval [0, B + K]. P r o p o s i t i o n 3 . 3 . 1 Assume 7 > 0, let v\ > B such that it satisfies equation (&.%), and define the function IfV\ < B + K, then U\ is the concavification function of the U o cb restricted to the domain Proof: See Appendix 3. The following proposition characterizes the concavification function of U o <b over the interval [B,oo). [0,B + K]. P r o p o s i t i o n 3 . 3 . 2 Lety > 0 and assume that V2 and V 3 satisfy equations (3AO) and (3A1), if'>y ^ 1, or equations (SA2) and (3A3) otherwise. Define the following function: 138 - . , J (Uo <b) (V2) + (v- V2) (U o <b) (V2) for v e (V2, V3] U2(v) = < I (U o(b) (v) otherwise. Then, U2 is the concavification function ofUocb over the interval \B,oo). Proof: See Appendix 3. So far, it has been proved that U\ and U2 are the concavification functions of U o <b restricted to [0,B + K] and [B, oo), respectively. We use these two functions to define a candidate function for being the concavification oiUocb overall its domain. The construction of this candidate function is based on the idea of combining U\ and U2 in such way that its combination possess, at least, the property of dominating the derived utility Uocb. Therefore, a natural candidate in such sense is the following function: U3(v) = Max jCMv), U2(v)j for all v > 0 . It is immediate that U3(v) >{Uo(b) (TJ) for all v > 0 given that both, U\ and U2, dominate /7o</>. However, U3 is in general not a concave function. Nevertheless, it turns out that if C/3 is concave then <73 must be the concavification function of U o cb. This fact is proved in the next proposition. P ropos i t i on 3.3.3 If U$ is concave, then U3 is the concavification ofU o f>. Proof: See Appendix 3. 139 There are two natural issues that follow from the previous result: 1. Finding conditions under which t/3 is concave, and 2. Proposing another candidate function in case that t/3 is not concave. The first issue is implicitly, although partially, addressed in the proof of Proposition 3.3.3 (See Appendix 3) where the condition V 2 > Vi is proved to be necessary from the assumption of concavity of t/3. The next proposition proves that this condition is also sufficient to assure that t/3 is concave. P ropos i t i on 3.3.4 t/3 is concave if and only if V2 > V i . Proof: See Appendix 3. If U3 is not concave, we need to look for another candidate function to be the concav-ification function of U o </>. Note that t/3 dominates U o f> and it is at least by pieces its concavification function. Therefore, the search for an alternative candidate function should depart in some way from t/3. A natural guess is to propose the concavification function of t/3 as a candidate function. It results that this is the right guess. This is proved in the next two propositions. P ropos i t i on 3.3.5 Assume that t/3 is not concave. Then, its concavification function is the function U defined as for v E [0, V4] for v > V4 (3.14) where V 4 satisfies the equation 140 (1 - 7 ) (X^) (1 - 7 ) (X0 +p(Vi -B) + q(Vi-B- K)f-V,(p + q) (X0+p(V4 -B) + q(VA-B- K)) - 7 (3.15) if •y y£ 1, and l o g ( X o ) = \og(X0+ p(Vi-B) + q(V4-B-K)) (3.16) X0+p(V4-B)+q(V4-B-K) if 7 = 1-Proof: See Appendix 3. P r o p o s i t i o n 3.3.6 U is the concavification ofUofi. Proof: See Appendix 3. S h a p e o f t h e c o n c a v i f i c a t i o n o f U o < f The concavification function oiUocb has two possible shapes. It has a four-piece shape if U$ is concave, and hence characterized by Us itself (See Figure 3.3). Otherwise, the concavification function has a two-piece shape characterized by the function U defined in (3.14) (See Figure 3.2). Therefore, the shape type of the concavification function oi U o <f> depends on the concavity of U3 and hence, it is characterized by Proposition 3.3.4. 141 Concavification function of U ° <t>: Two-piece shape case Oi i 1 1 1 1 1 1 1 r Value of the Fund Figure 3.2: T w o - p i e c e s h a p e C o n c a v i f i c a t i o n o f U o < f > . This is the graph of U o 4 > and its concavification for a case in which the concavification function has a two-piece shape. The parameters are: X0 = 0.005, p = 0.01, q = 0.02, B = 1.04, K = 0.76 and 7 = 2. U o (f> and its concavification function coincide, for this particular example, over the interval [2.0544, 00) . The concavification function displayed in this graph corresponds to the derived utility function of Figure 3.1. 3.4 Optimal Solution The main reason for the construction of the concavification function of U o </> is the definition of an alternative problem to (3.3), by replacing U o $ by its concavification function, for which standard techniques can be applied and whose solution is also a solution of (3.3). In this section we solve this alternative problem and demonstrate that its solution is optimal for (3.3) too 4 . 4Refer to Corollary 3.4.1. 142 Concavification function of U ° <J>: Four-piece shape case Oi 1 i 1 1 p 1 1 1 r Value of the Fund Figure 3.3: Four-piece shape Concavification of U o </>. This is the graph of U o cj) and its concavification for a case in which the concavification function has a four-piece shape. The parameters are: X0 = 0.005, p = 0.01, q = 0.02, B = 1.04, K = 1.46 and 7 = 2. U o (j) and its concavification function coincide, for this particular example, over the intervals [1.7611,2.0858] and [2.7391, oo). 3.4.1 Optimal Fund Value Consider the alternative problem s u p ^ E p V r ) J s.t. E[ZTVT}<V0 Vt>0 (3.17) where U is the concavification function of C/o c/>, and where ( £ t ) t e [ 0 T ] is the pricing kernel as defined in (3.4). The following proposition states explicitly the optimal terminal fund value of Problem (3.17). 143 Proposition 3.4.1 Let 7 > 0 and Vi and V2 as defined in Lemmas 3.3.1 and 3.3.3, respec-tively. Then, ifV2 > Vi and gip{\) = E [ f r / i , (Afr) l{v 2>/» 1(A ? r)>v 1}] + E [frh2 (Afr) l{h1(Ae7.)>vb}] < 0 0 ( 3 - 1 8 ) for all A > 0, where hi(y) ( ^ f 1 / 7 - * ° + ^ P+9 + 5 , i/ie optimal terminal fund value of Problem (3.17) is V* — vT — + B + P ( p ) ^ ^ ° l{p{X0+v(V2-B)}-< < Mr < P&o+piVi-B)]-"1} B + {p+l) K + ( p f e ) (p+i) ^ ^ ~ X0 l{XiT < p[x0+p(V2-B))-i} (3.19) where X is a nonnegative scalar that satisfies E[ZTVT-] = V0 (3.20) where VQ is the initial fund value. Otherwise, if g2p(X) = E [frh2 (Afr) l{fc2(A£T)>v4}] < 00 (3.21) for all X > 0, where V4 is defined as in Lemma 3.3.2, then the optimal terminal fund value is given by 144 V? = B + p + q K + p + q p + q 1 { A € T < (p+q)lX0+P(V4-B)+q(Vi-B-K)]-'} (3.22) where X is a nonnegative scalar that satisfies condition (3.20). Proof: Assumption (3.18) implies that g4p is continuous and strictly decreasing and hence, it assures that Equation (3.20) has a solution for some A > 0. Suppose that V2 > V i . Therefore, the concavification function of U o cb is (U o ch) (Vi) + (U o cb)' (Vi)(v - V i ) , i f v < Vx. U(v) = { (Uo<b)(v) , i f T ; G [ y 1 , V 2 ] U ( V 3 , o o ) . (U o <p) (V2) + (U o cb)' (V2)(v -V2) , i f v € (V2, V3}. By standard theory (Karatzas et al (1987) and Cox and Huang (1989)), (j'(V*) = Afr a.s. (3.23) where A is a nonnegative scalar that satisfies Equation (3.20). Therefore, condition (3.23) and definition of U implies U'(VT(u)) • p[X0+p(VX - fi)]"7 , if Afr(w) G C I (p + q) [Xo + p (V?(w) - B) + q (V?H - B - K)r , if A f r M G C2 p [X0 + p (V^(u) - -B)] - 7 , otherwise for all u G Vt for which equation (3.20) has a a solution, and where C i ' = {x\x = p[X0+p(V1-B)}-'1} C2 = {x\x<p[Xo+p(V2-B)}-1} • 145 Therefore, if A £ r M < p [X 0 + p (Vi - B)] 7 , for some w e d , then VT(u) = (U'y1 (\fr(u)) B + l v X , if e C 2 , if A £ r M € (Ci U C2) Note that if A£T(u;) G C i then Vy(a;) G (0, Vi] (£/' is not defined at w = 0), while if A£r(w) > p [X 0 + p (Vi - B ) ]~ 7 , then equation (3.20) does not have a solution for A > 0. We extend the definition of if to consider the latter two cases using the following mapping 5: u(VT(u)) = I p [ X 0 + p ( V i - B ) r 7 (p + q) [X0 +p(v — B) +q(v — B — K)]~J , i i A C , T ( W ; t u 2 [p [ X 0 + p (Vi - B)], oo) , if Afr (u>) G C73 , if A f r H G C, f £ w) G C  [ p [ X 0 + p f o - B ) ] - 7 , otherwise where C 3 = {a:|a; > p [ X 0 + p ( V i - B)] 7 } . Hence, condition (3.23) generalizes to 1 u'(V^) = A£r a.s. from which condition A£T(w) > p [ X 0 + p ( V i - B ) ] - 7 implies V£(u;) G [0, Vi]. Therefore, the optimal terminal fund value can be expressed as VT = + X n 1{P[X0+p(V2-B)]-> < Afr < pIXo+KVi-B)]"^} 1{\ir < p [ X 0 + p ( V 2 - B ) ] - T } 5Formally speaking, u is a subdifferential of (7 (e.g. Rockafellar (1970)). 146 The proof of expression (3.22) is similar. Q.E.D. Corollary 3.4.1 The optimal solution of Problem (3.17) is also optimal for Problem (3.3). Proof: By construction of the concavification function U it holds the condition E U(VT)\ >E[U(VT)] for VT such that E [frVr] < V0 VT>0 From Proposition 3.4.1 it is clear that E tf(V?) =E[U(V^)] where is the optimal solution of Problem (3.17). Therefore, it follows that Vf must be an optimal solution of Problem (3.3). Q.E.D. 147 3.4.2 Optimal Risk Taking We now explicitly characterize the manager's optimal risk-taking strategy. Proposition 3.4.2 Let 7 > 0 and V\ and V2 as defined in Lemmas 3.3.1 and 3.3.3, respec-tively. If V2 > V\, then the manager's optimal volatility choice for any t 6 [0, T) is given by v; = a ( l - 7 * ) + + + [( B P+Q f) e " M ( r " ° ] ( v f ) [n{dT-J^'l)) ~ a( l - 7*)A^(ciT_ t(m2,1)) (^B — e-At(T-t) j ^ j ^ " ( d r - t ( m i , l ) ) - n ( d r _ t ( m 2 , l ) ) (B - f) e-^-')] a( l - 7*) W{dT-t{mu 1)) - -A/" 1))] ( ^ ) [Vi* - - ) e-^N{dT.t{m2A))] (^ ) G(X0,p,B,q,Kn) w/iere 7* = 1 — 1/7, (p+<?)' G ( X 0 , p , B , ?,/<-, 7 ) = r n (d T -t(m 2 ,7*)) + ^fri [n(dT-t(muY)) ~ n (d T-t(m 2 ,7*))] (p + q)~r M(dr_t(m2,7*)) + P~7* W(dr-t(mi,7*)) - Af(d T - t (m 2 ,7*))] and du(-, •) zs the function defined by du(m,0) = log m — I — pu + a u I 0 — - / (a-v/w) . (3.24) 148 N~(-) and n(-) are the standard normal cumulative distribution and density, 7* = 1 — I/7, and A is the nonnegative scalar that satisfies Equation (3A7). Otherwise, if < V\, the manager's optimal volatility choice for any t G [0, T) is a ( l - 7 * ) + + (B - ^f) e - ^ - » ] ( i ) ["^-dgu.i)) - a ( l - 7 ^ Wr-«(m*, 1)) K - ( 5 - ) ^ ( * - « K D ) ] (*) U g ^ y ^ • Proof: Assume that V2 > Vi. Therefore, the opt imal fund value for any time t G [0,T] is given by (See Proposi t ion 3.7.1 in the Appendix 3) B _ Xp-qK p+q e - / 4 T - t ) j v - ( d r _ t ( m 2 | 1 ) ) + + + (p + qr* V-1*!'-1] e~("+2i^>*(T- V ( d T _ t ( m 2 , 7 * ) ) e - * r - t ) [ A f ( ( i T _ ( ( m i ) i ) ) _ A^ (c i T _ t (m 2 , 1 ) ) ] - ( M + = ^ ) 7 ' ( T -[Jv ' (dr_ t (m 1 , 7 *)) -Jv ' (dr_t(m 2 > 7*))] where mi V_ [Xo+piV-B)}-for i = 1,2. Therefore, 149 dv; + + + B - e-"<r-W (dT_ t(m 2, 1)) Q _ Xp-qK P+Q p Ar{dT-t(m2,l))e->lV-1)tidt e-^-^d [N~ (dr_ t (mi, 1)) - M {dr-t(™*, 1))] [ A A ( d T _ t ( m 1 , l ) ) - A r ( d T - t ( m 2 , l ) ) ] e - " ( T - t ) M + [(P + <?)~7* A 7 * " 1 * ; 7 * - 1 ] d ^"(^ a a ( , r*07^-V(dr_e(m2,r))^ + [ p - ^ A 7 * - ^ 7 * " 1 ] d ( - { ^ " - ^ y ^ [r^(dT.t(m1,Y))-^(dT-t(m2,'f))] By Ito's lemma, d(Af(dT.t(mi,P))) d (a'-1) and, [\n (dr_t(m i,)9))] dt + n(d r-t("i i,j9))rf(dr-t(m i, i0)) ,for % = 1,2, i . J d ( d r _ t ( m i , / ? ) ) = _ ^ p ^ , - s r % t + _ ^ ( [ o g fe] [ A j + p w _ ^ aJT-t '/x + f + log ( [^] [Xo + p (V- - i?)]"7) di + / T - t dzt for /? = 1,7* = 1 — I/7. Thus, the stochastic part of V7 is given by 150 B _ X0-qK p+q „-n(T-t) n(rfT_t(w2,l)) + e-fi(T-t) »(dr_t(mi,l)) _ n(dT-t(m2,l)) W(dr-t (ma, 7 * ) ) ( f ) + " ^ ^ ^ L ' J y ^ ( 0 ^ ( ^ , 7 * ) ) (a) + Mda^dmh2l r / 2 , \ ^ 7 ' v ^ t L ' J 7 ^ ( d r _ t ( m 2 , 7 * ) ) ( V ) + Z i i ^ m ^ ,7')) )) (3.25) On the other hand, Vt* must satisfy Therefore, tfVt* must equate expression (3.25), from which the result follows. Q.E.D. 3.5 Induced Incentives Based on the theoretical results of the previous section, we analyze and illustrate in this section the different fund value and risk incentive profiles that can be induced by the com-pensation scheme cf>. 3.5.1 Fund Value Incentives The optimal choice of terminal fund value implies a sort of gambling behavior of the manager. In the two-piece shape case (See Equation (3.22)), the solution implies an all-or-nothing 151 strategy. Either the manager is far out (Vf = 0) or well in the money (V? > V4 > B+K > B) for the two embedded options of the manager's compensation package. In the four-piece shape case (See Equation (3.19)), the optimal solution implies that the manager is either far out (Vf = 0) or well in the money for at least one of the two embedded options of his compensation package. That is, either Vf > V3 > B + K > B or V2 > Vf > V\ > B. This gambling behavior of the manager is up to some extent expected. The manager will get the same compensation, and thus the same utility, either if VT = 0 or 0 < Vr < B, although he will have to use part of his budget in the latter case. Hence, if the manager chooses to be out of the money, he will be as far out of the money as possible. On the other hand, if the manager decides to be in the money, he will target for firm values that will provide him a sufficiently small marginal utility. For instance, in the two-piece shape case, the manager aims for fund values with smaller marginal utility than the one provided by a fund value equal to V 4 , and characterized in terms of the pricing kernel fr (See Figure 3.4). The intuition behind the threshold value of V4 relies in the fact C (U o cb) (0) + (1 - 0 (U o cb) (V<) > (U o cb) (C • 0 + (1 - C) V4) , V C G (0,1) . That is, the weighted average utility of the zero and V4 payoffs, surpasses the utility of the weighted average of those payoffs. Therefore, any strategy that implies to target for fund values between zero and V4 could be dominated, in the sense of expected utility, by a strategy that takes either the value of zero, with an appropriate chosen probability V, or V4 with probability 1 — V (See Appendix 3). For the four-piece shape case, the manager targets for fund values with a marginal utility necessarily smaller than that associated to the fund value V i , but without aiming for fund values between V2 and V3. The reason for excluding [V2, V 3 ) is that the utility of any fund 152 Optimal Choice of V T (Two-piece Case) Figure 3.4: Optimal Choice of Vr'- Two-piece shape case. This is a graph that shows the optimal terminal value as a function of the pricing kernel £ T for a particular example. The parameters are: X0 = 0.005, p = 0.01, q = 0.02, B = 1.04, K = 0.16, 7 = 2, V0 = 1, p = 0.04, a = 0.4, T = 1. The relevant tangency point in this case is V4 = 1.5838, which corresponds to the threshold value £0 = (1/A) (U o (j)) (V4) = 1.2462. value between V2 and V3 is dominated by the corresponding weighted average of the utility values of V2 and V3. Therefore, in this case, the manager targets for fund values in the range (Vi, V2] or the range (V3, 0 0 ) . The election between these two ranges can be characterized in terms of the pricing kernel (See Figure 3.5). 3.5.2 Risk Incentives One of our primary goals is to explore the risk incentives induced by the compensation scheme 4> a n d analyze if these incentives present the same unexpected features documented by Carpenter (2000). In particular, we are interested in studying if cj> can induce the manager to take lower volatility levels than investors would like. In order to do such analysis, we 153 Figure 3.5: Optimal Choice of Vf: Four-piece shape case. This is the graph of the optimal terminal value as a function of the pricing kernel fr. The parameters are: X0 = 0.005, p = 0.01, q = 0.02, B = 1.04, K = 1.26, 7 = 2, V0 = 1, \i = 0.04, a = 0.4, T = 1. The relevant tangency points are in this case V\ = 1.7611 and V 3 = 2.5147, with the corresponding threshold values £ 1 = 1.0524 and £2 = 0.8144. consider first an example of the single option case (i.e. (7 = 0%) for which, contrary to what it is supposed to be designed for, the compensation scheme induces less risk (volatility) taking of the manager than if he were trading his own account (characterized by Merton's constant (0.2 in this case), see Figure 3.6). Then, we study the impact of adding the second option payoff into the compensation scheme. This analysis is divided in three parts. First, we analyze the effect of parameter K for a fixed q ^ 0. Second, we study the impact of q, for a fixed K. Third, we analyze the combined effect of K and q. 154 Optimal Volatility: Single Option Case 0.21 . Volatility (v() • Merton's constant Value of the Fund Figure 3.6: O p t i m a l V o l a t i l i t y : S i n g l e - O p t i o n C a s e . This graph shows the optimal volatility strategy for the case of a single option (i.e. q = 0%) with the following parameters: X0 = 0.005, p = 0.01, B = 1.04, 7 = 2, V0 = 1, p = 0.04, a = 0.4, T = 1. K E f f e c t We computed the optimal volatility curve, as a function of the fund value, for different values of K using the same parameters of the single-option example considered in Figure 3.6, except that we now set q equal to 2 %. We choose q — 2% to make our compensation scheme <fi more convex than with q — 0 % (See Figure 3.7), for fund values greater or equal than B + K. This should, intuitively, make more attractive for the manager to target for fund values greater or equal than B + K. Figures 3.8 and 3.9 show these volatility curves. A primer observation that we can derived from Figures 3.8 and 3.9 is that the compensa-tion scheme (fi shows, for q ^ 0, the same unexpected features observed for the single-option case studied by Carpenter (2000). For instance, all volatility curves lie below Merton's 155 Compensation scheme $ 0.071 1 1 — 1 1 1 1 — r q = 2 % q = 0 % 0 . 0 6 [ - / -0 . 0 5 - / 0 . 0 4 - / 0 . 0 3 - / ~ 0 . 0 2 -0 . 0 1 - ^ ^ ^ ^ 0I 1 1 1 1 1 1 1 0 0 . 5 1 1 . 5 2 2 . 5 3 3 . 5 4 Value of the Fund Figure 3.7: Compensat ion Scheme Shape. This is a graph shows the shape of the com-pensation scheme for the single option case (q = 0%), and the double option case (q = 2%). constant for most of the range of fund values considered. Other observations that can be made from Figures 3.8 and 3.9 are the following: 1. The minimal value of the optimal volatility curve for the case q = 0 % (around 0.16) is greater or equal than the minimal value associated with each of the volatility curves plotted. That is, for the particular example and values of K considered, the effect of the second option is inducing less volatility taking than with a single option (q — 0 %). 2. For values of K associated with the same concavification function shape (Figure 3.8 corresponds to a two-piece shape, while Figure 3.9 is associated with a four-piece shape), we observe that the larger the value of K, (a) the smaller the minimal value of the optimal volatility curve, and 156 Optimal Volatility: Effect of the parameter K for a given percentage q ( I ) Value of the Fund Figure 3.8: K Effect ( I ) . This graph shows the effect of parameter K on the optimal volatility decisions, for a fixed q ( = 2 %). The parameters used are: X0 = 0.005, p = 0.01, B = 1.04, 7 = 2, V0 = 1, p = 0.04, a = 0.4, and T = 1. The values of i f considered are 0, 0.5, and 1.0. The concavification function associated with these values has a two-piece shape. (b) the steeper the volatility curve for values smaller than the fund value at which the minimum volatility is reached, which we denote by V(K) (e.g. V ( l ) « 2.6). For fund values larger than V(K), the volatility curve has essentially the same type of curvature for all the values of K considered. 3. The minimal values of the optimal volatility curves that are shown in Figure 3.9, which correspond to values of K larger than one, are smaller than the corresponding minimal values showed in Figure 3.8, which corresponds to values of K smaller than one (see also Figure 3.10). That is, for the particular parameters considered, the minimal volatility value decreases as K increases, regardless of the shape of the concavification function. Unfortunately, this is not always true as it is shown in Figure 3.11. 157 Optimal Volatility: Effect of the parameter K for a given percentage q (II ) Value of the Fund Figure 3.9: K Effect (II). This graph exemplifies the effect of parameter K on the optimal volatility decisions, for a fixed q ( — 2%). The parameters used are: X0 = 0.005, p — 0.01, B = 1.04, 7 = 2, Vo = 1, p = 0.04, a = 0.4, and T = 1. The values of K considered are 1.5, 2.0, and 2.5. The concavification function associated with these values has a four-piece shape. Observation 2 can be explained in the following manner: the higher the second option exercise price, the smaller the manager's incentive to pursue higher risk-taking once he is in the money, but the higher the risk incentives if the manager is out of the money. Nevertheless, Observation 2 does not hold always. For instance, if q is relatively large with respect to p then, for sufficiently large values of K, it holds that the larger the value of K, the larger the minimal value of the optimal volatility curve (See Figure 3.11). This fact can be reconcile with Observation 2 by arguing that the manager will pursue higher risk-taking if the relative weight of the second option payoff with respect to the compensation scheme is sufficiently large, even in the case that both options are in the money. In addition to the piecewise monotonic behavior of the minimal optimal volatility as 158 Minimal Volaltility for different values of K 0.121 1 1 1 1 — I 1 5 Figure 3.10: Minimal Optimal Volatility: K Effect (I). This graph exemplifies the effect of the parameter K on the minimal volatility decision, for a fixed q ( = 2 %). The solid line represents the minimal volatility decision, while the dashed lined corresponds to the difference between the tangent points V2 and V i , for each value of K considered. The concavification function has a four-piece shape if and only if V2 — V\ > 0 (Proposition 3.3.4). The parameters used are: X0 = 0.005, p = 0.01, B = 1.04, 7 = 2, Vo = 1, p = 0.04, a = 0.4, and T = 1. a function of K, Figures 3.10 and 3.11 show an abrupt change of the minimal volatility decision at K s=a 1.05 and K w 1.38, respectively. Coincidentally, these values of K co-rrespond, respectively, to the threshold values, ceteris paribus, between a two-piece and a four-piece shape concavification function. The size and direction of this discontinuity (upward or downward) depend on the parameter q. This phenomena is relevant because of its policy implications: it means that it is possible that managers with similar compensation schemes behave in a different way. 159 Minimal Volaltility for different values of K 0.081 1 -i 1 1 11 0.041 ' 1 1 1 1 -1 0 0.5 1 1.5 2 2.5 K Figure 3.11: Min imal Optimal Volatility: K Effect (II). This graph exemplifies the effect of the parameter K on the minimal volatility decision, for a fixed q ( = 10 %). The solid line represents the minimal volatility decision, while the dashed lined corresponds to the difference between the tangent points V2 and Vi, for each value of K considered. The concavification function has a four-piece shape if and only if V2 — V\ > 0 (Proposition 3.3.4). The parameters used are: X0 = 0.005, p = 0.01, B = 1.04, 7 = 2, V0 = 1, p = 0.04, a = 0.4, and T = 1. q Effect In order to get a sense of the q effect, we computed the optimal volatility, as a function of the firm value, for different values of q. To make this numerical exercise comparable with those done for the analysis of the K effect, we use the same parameters of the single-option example considered in Figure 3.6, except that we now set K = 0.5. We plotted these optimal curves in Figure 3.12. For the particular case and values of q considered, the effects are analogous to those observed in the analysis of the K effect. For instance, 160 Optimal Volatility: Effect of parameter q for a given K 0,2-0 1 I 1 1 1 ' 2 4 6 8 10 12 14 Value of the Fund Figure 3.12: q Effect. This graph illustrates the effect of parameter q on the optimal volatility decision, for a fixed K ( = 0.5). The parameters used are: X0 = 0.005, p = 0.01, B = 1.04, 7 = 2, Vo = 1, p = 0.04, a = 0.4, and T = 1. The values of q considered are 0.5%, 1%, and 1.5%. • it is induced less risk-taking than with a single option (q = 0%), and • minimal volatility decisions decrease as q increases, although, contrary to what it is observed in the K effect case, such monotonic behavior is continuous over all the range of q values considered (see Figure 3.13). Combined Effect of K and q From the numerical analysis done previously, we conclude that the inclusion of a second option payoff in the compensation scheme does not eliminate the unexpected volatility levels that may arise in the single option-case. However, this analysis considers separately the effect of the two parameters that determine the second option, K and q. Therefore, a natural 161 Minimal Volaltility for different values of q 0.171 1 1 1 1 1 1—— 1 1 r I i i i i 1 1 1 1 1 1 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 q Figure 3.13: M i n i m a l O p t i m a l V o l a t i l i t y : q E f f e c t . This graph illustrates the effect of parameter q on the optimal volatility decision, for a fixed K ( = 0.5). The parameters used are: X0 = 0.005, p = 0.01, B = 1.04, 7 = 2, V 0 = 1, A* = 0.04, a = 0.4, and T = 1. question is if K and q can be chosen simultaneously in such way that the unexpected risk profiles -when investors want to induce higher risk-taking- can be mitigated, at least in some degree. This is indeed possible as it is exemplified in Figure 3.14. This figure shows that for K = 2.5 and relatively small values of q (approximately less than or equal to 0.15%), the minimal volatility value is higher than that associated to the single-option case (q = 0 %). Optimal volatility decisions can be more complex than those exemplified previously. For instance, if q is relatively small and K is relatively large the shape of the optimal volatility curve can take the form shown in Figure 3.15. In this case, the rationale of the manager is the following: • If both options are out of the money, i.e. V < B, the manager will take higher risk-162 Minimal Volaltility for different values of q 0.181 1 1 1 r 0.16 0.14 0.12 0.1 0.08 . | l l l l I 0 0.005 0.01 0.015 0.02 0.025 q Figure 3.14: Mitigation of Unexpected Risk Profiles. This graph plots the maximal minimal volatility level for a given q. It shows that the unexpected risk profile observed for the single-option case illustrated in Figure 3.6 can be mitigated. The parameters used are: X0 = 0.005, p = 0.01, B = 1.04, K = 2.5, 7 = 2, V0 = 1, ft = 0.04, a = 0.4, T = 1, and T - * = 0.5. taking than if he were trading his own account and, the farther the option is from being in the money, the higher the risk the manager will take in order to increase the possibility that the first option, with payoff (V — B)+, ends up in the money at time T. • If only the first option is in the money then, — if the fund value V is not sufficiently above the benchmark B (B < V < V* in Figure 3.15), then the higher the fund value, the lesser the risk the manager will take. The manager acts in this way to decrease the chance that the first option, which is already in the money, ends out of the money at time T. — if the fund value V is sufficiently above the benchmark B but still significantly 163 Figure 3.15: O p t i m a l Vo la t i l i t y Decis ion: Smal l q and Large K. This graph shows the optimal volatility decisions for a relatively small q (= 0.225 %) and a relatively large K ( = 3), and the corresponding single-option optimal volatility curve (q = 0%). The parameters used are: X0 = 0.005, p = 0.01, B = 1.04, 7 = 2, Vo = 1, p = 0.04, a = 0.4, and T = 1. below the other benchmark value B + K (V* < V < V** in Figure 3.15) then, the higher the fund value, the higher the risk the manager will take. That is, the manager will increase risk-taking as the fund value increases, once the first option is sufficiently deep in the money. The manager behaves this way to increase both, the expected first option payoff and the possibility that the second option, with payoff (V — B — K)+, ends in the money at time T. Note in Figure 3.15 that the difference between the volatility curves associated, respectively, with (q = 0.225%, K = 3) and q — 0%, captures the manager's risk appetite induced to increase the chance that the second option ends in the money. — if the fund value V is sufficiently closed to the benchmark value B + K 164 (V** < V < B + K in Figure 3.15) then, the higher the fund value, the lesser the risk the manager will take. In this situation, the second option is relatively near to be in the money and therefore, the closer is the fund value to B + K, the higher the incentive of the manager to guarantee that the second option ends up in the money at time T. • If both options are in the money then — if the fund value V is not sufficiently above the benchmark value B + K [B + K < V < V*** in Figure 3.15), then the higher the fund value, the lesser the risk the manager will take. In this case, given that the second option is not sufficiently deep in the money, the manager tries to reduce the possibility that the second option ends out of the money at time T. - if the fund value V is sufficiently above the benchmark B + K (V > V***), then the higher the fund value, the higher the risk the manager will take. That is, once both options are sufficiently deep in the money, the manager will increase risk-taking as fund value grows in order to increase both expected payoffs. The observations made in this section suggest that an appropriate selection of the second option parameters (q, K) allow investors to have a significant higher control on the manager's risk profile they want to incentive than with a single option. Next section analyzes this issue further. 3.6 Incentives Design This section is devoted to the design of terminal fund value and risk profiles. For this purpose, analytical results related to the sensitivity and stability of the induced profiles with respect 165 to the compensation parameters are obtained. Based on these results, specific criterions are proposed in order to induce particular fund value and risk profiles. 3.6.1 Terminal Fund Value Design The range of (optimal) terminal fund values depends on the shape of the concavification function of U o (p. In the two-piece shape case, the range is of the form {0} U (V4,oo). Otherwise, in the four-piece shape case, the range is of the form {0} U (Vi, V2] U (V3,oo). That is, in the two-piece shape case the manager follows an all-or-nothing strategy while in the four-piece shape case the manager targets for a less disperse range of fund values than in the two-piece case. Therefore, it is crucial for the investors, or whoever establishes the manager's compensation scheme, to understand the way in which the compensation parameters determine the shape of the concavification function. To do so, recall that this shape can be characterized in terms of the sign of the difference Vi — Vi (See Proposition 3.3.4). If Vi — Vi is negative we have a two-piece shape concavification function. Otherwise, we have four-piece shape concavification function. Four-piece shape concavification function In the case that investors want to induce a terminal fund value range of the form m u ^ . V a l u C V a . o o ) , it is important to characterize the set of compensation parameters values that guarantees condition Vi — V\ > 0, and their impact on Vi, V2 and V 3 . To accomplish these two goals, we proceed in the following manner: First, we analyze the effect on Vi from the set of parameters that determines it: XQ, p, and B. Second, we study the impact of q and K on V2, ceteris 166 paribus (X0,p,B). Third, we do analysis on the simultaneous effect of (X0,p, B,q, K) on 6 Vi and V2. Proposition 3.6.1 (Sensitivity of Vi with respect to X0, B, and p) Lety > Q,p € (0, and X0 > 0. Then, 1 25i > o 1- dB u -2. ^ < 0 i/and only if Vi > B [1 + { (p7Vi ) _ 1 ( X 0 + p ( V i - £))}] (3.26) 5. | ^ > 0 if and only if (Xo + p (Vi - B)) (Xo + p (Vi - . B ) ) - 7 Proof: See Appendix 3. Proposition 3.6.1 states on one hand, that Vi increases as B increases. On the other hand, changes in the fixed compensation X 0 or the proportion p could either derive in increments or decrements of Vi (See Figure 3.16). These results are useful to determine the impact, both qualitative and quantitative, of the decisions made over the compensation parameters X 0 , p, and B. In the following proposition we analyze the sensitivity of V2 with respect to the second option compensation parameters q and K. 6Vs is a linear transformation of V2 (see Equation (3.40)). Therefore V2 determines V3 and viceversa. 167 Sensitivity of V, with respect to p 0.001 0.002 0.003 0.004 0.005 Figure 3.16: Sensi t ivi ty of Vi w i th respect to p. This graph exemplifies the way Vi changes as p changes. The parameters used are: X0 — 0.005, B — 1.04, and 7 = 2. For these parameters, and the range of values considered for p, condition (3.26) of Proposition 3.6.1 is satisfied and therefore ^ < 0. P ropos i t ion 3.6.2 (Sensitivity of V 2 w i th respect to q and K) Letj > 0 andp G (0,1). Then, 1. | ^ < 0 if and only if j < 1. 2. ^ < 0 if and only if 1 - 7 1\ 5a / a \ d£ ~ (3) dq + {(3*) dq\ < 0 where a and 6 are defined in (3.39J in Appendix 3. Proof: See Appendix 3. 168 Proposition 3.6.2 states that V2 increases or decreases (proportionally) as K increases, depending upon if the risk aversion parameter 7 is, respectively, strictly greater or smaller than one. The sensitivity of V2 with respect to q is more complex. It depends on the risk aversion parameter 7 and the sign of For instance, assume that 7 < 1. Then, ^ < 0 implies that V2 increases as q increases. However, ^ > 0 does not imply that V2 decreases as q increases. A practical consequence of Propositions 3.6.1 and 3.6.2 for the design of the compensation scheme is the following: suppose manager's risk aversion parameter is 7 > 1 and that investors' first priority is to establish the threshold value V i . Thereafter, they select (Xo, p, B) accordingly, and for which Proposition 3.6.1 is helpful. In order to determine q and K, and given result one of Proposition 3.6.2 ( | ^ > 0 iff 7 > 1), it is useful to obtain, for each value of q, the minimal value of K that assures V2 — V\ > 0, which we denote by K*(q). Let Q be the set of values of q considered by the investors. Hence, the set of values {B + K* (q) \ q G Q} specifies the lower bound, and thereafter the range of values, of the second option exercise price that induces a four-piece shape concavification function for each value q considered (see Figure 3.17). Hence, if investors want to induce a four-piece shape concavification function, they choose q associated with the range of values of K they are more comfortable with. That is, they select the interval [K*(q), 00) they prefer the most. Finally, they choose K G [K* (q), 00) using some criterion. This two-phase procedure for establishing the compensation parameters is not only prac-tical but also useful to give a characterization7 of the set of compensation parameter values that guarantees V2 — V\ > 0. For instance, 7There axe other ways to characterize condition V2 — V\ > 0. For example, given (X0,p,B), we could determine a threshold value of q for each value of K, say q*(K), in the same spirit that K*(q) was defined. However, the determination of feasible intervals will not be as straight forward as in the case of K*(q) because the sign of depends on the risk aversion parameter 7 and the ^ (Result 2 in Proposition 3.6.2). 169 K*(q) 0 0.01 0.02 0.03 0.04 0.05 0.00 0.07 0.08 0.09 0.1 q Figure 3.17: K*(q): This graph shows the minimal value of K required to assure that condition V2 — V\ > 0 is satisfied for each value of q, and given a set of parameters (X0,p, B). The parameters used are: X0 = 0.005, B = 1.04, and 7 = 2. The range of values of q considered is Q = [0.001, 0.1]. {(X0,p,B,q,K)\V2-Vl>0}= (J {(X0,p, B,q, K)\ K > K*(q), q € {x0,p,B)m3+ Compensation parameters can also be chosen simultaneously to induce a specific terminal fund value profile. For example, suppose investors want to incentive a fund value range of the form {0} U (Vi, V2] U (V3, 00) and that they prefer the strategies that lead to a final fund value within the range (Vi, V2] rather than those strategies that lead to the range (V3,oo). Therefore, they would like to increase the length of the interval (Vi, V2] and the value of V 3 . In order to determine the required compensation parameters to induce this specific fund value incentive profile, investors can solve the following optimization problem: 170 Max Xo,p,B,q,K SVs + i l - S ) ^ - ^ ) s.t. ^327^ V2 - Vi > 0 (X0,p,B,q,K) E AC^+ where 6 and (1 — 8), for 5 6 [0,1], represent, respectively, the relative weight that investors assign to strategies that lead to fund values within the interval (V3, 00) and (Vi, V2\, where the set A specifies other restrictions that might apply to the compensation parameters, including nonnegativity conditions. Section 3.6.3 develops a numerical example for this criterion. T w o - p i e c e s h a p e c o n c a v i f i c a t i o n f u n c t i o n In the case that investors want to induce a terminal fund value of the form { 0 } U ( V 4 , 0 0 ) , there are two crucial issues at hand: the characterization of the set of compensation parameter values that implies V2 — V\ < 0 and their impact on V4. The following proposition establishes the sensitivity of V4 with respect to each of the compensation parameters. P r o p o s i t i o n 3.6.3 ( S e n s i t i v i t y o f V 4 ) Let 7 > 0 andp € (0,1). Then, 1 9¥±. > 0 1- dB ^ u -2. ^ < 0 if and only if V4 > B [1 + {((p + q)7V4)-1 (X0 +p(V4-B)+q(Vi-B-iv"))}] 3. ^ < 0 if and only if V4> (B + K) [1 + {((p + q) 7V4)- 1 ( X 0 + p (V 4 - B) + q (V4 - B - K))}] 171 dVj dK > 0. 5. dV4 dX0 > 0 if and only if > 1 + (P + a) TVi (X0 +p (V4 - B) + 9 (V 4 - B - X ) ) " 7 ( X 0 + p (V 4 - 5 ) + q (V4 - B - K)) Proof: The proof of all the results uses the same methodology in Proposition 3.6.1. The sensitivity of V4 with respect to the type of compensation parameter (proportions and exercise prices) is identical to that of Vi : V4 decreases, under certain conditions, as the value of the proportions p and q increase; V 4 increases as the option exercise prices B and B + K increase; and the sensitivity of V4 with respect to the fixed salary X0 depends on a certain condition that involves, among other quantities, the risk aversion parameter 7. Two important features of manager's risk profile for the investors are the range of the vola-tility levels that the manager can take and the sensitivity of the risk-taking decisions with respect to the firm values and the market conditions. Range of Volatility Values The range of volatility values is of the form (v, 0 0 ) , where Q.E .D. 3.6.2 Risk Profile Design u(P) = Minv>0 v*(V,P) , 172 V represents the fund value, and P = (X0,p, B, q, K) is the vector of compensation parame-ters. The analysis done in 3.5.2 illustrates that regardless the convexity of the compensation scheme 4>, v_ can be below the volatility level that the manager will take if he were trading his own account. However, it is also observed in 3.5.2 that if the compensation parameters are chosen appropriately, this undesired feature -for the investors that want to induce higher risk-taking- can be mitigated and even eliminated in some cases. For instance, given the compensation parameters (Xn,p, B,q), K can be chosen equal to K*(q), where K*(q) is defined as K*(q) = ArgMax K Min v v* (q, K, V) , where v* (q, K, V) denotes the optimal volatility choice given the parameter values (q, K)and the firm value V , in order to mitigate as much as possible the undesired low volatility effect for any firm value. In general, investors can set a minimum level ML and choose a vector of compensation parameters P* that solves the following optimization problem MinpeV [v{V)-ML]2 where the set V specifies particular restrictions that the vector of compensation parameters P must satisfy. Stability Drastic changes of optimal risk-taking decisions given small variations of the fund value and the market conditions are costly in terms of transactions costs or investment irreversibility 173 or lumpiness. Therefore, it is crucial for investors to induce stable risk-taking decisions. Stability could be indirectly achieved by controlling the range of volatility values. However, under the current model we only have control over the lower bound of the range of volatility values v_. Hence, it is necessary to study directly the sensitivity of the optimal volatility with respect to changes of the firm value V. In order to accomplish such analysis, we uss the following result. Lemma 3.6.1 (Sensitivity of volatility with respect to the fund value) Let 7 > 0 and Vi and V2 as defined in Lemmas 3.3.1 and 3.3.3, respectively. IfV2 > Vi, then dv* = ( ^ f ) e-^- ' ) ( ^ ) [ n ( d r ^ " 1 ) ) ~ M (4r-t(m2> 1)) {a (1 - 7*) + G(X0, p, B, q, K, 7)}" - (f) e _ M ( T _ t ) (T?) [ n ( d T ^ l ) ) - M (dT-t(rri2,1)) {a (1 - 7*) + G(X0,p, B, q, K, 7)}' - ^B-f)e-^T-^(^) [ w ( d ^ " 1 ) ) - M (dT-t(mi, 1)) {a (1 - 7*) + G(X0,p, B, q, K, 7)} where du(-,-) is the function defined by du(m,0) logm -pu + a2u 0 — - I (as/u) , JV(-) and n(-) are the standard normal cumulative distribution and density, 7* = 1 — I /7 , and A is the nonnegative scalar that satisfies Equation (ZA7). Otherwise, ifV2 < Vi, then dv* dvt ( B - j ^ > - » ( T - t ) /T-t N(dr. N(dT- [dr-t (m 4 ) 7*)) - n (aV_t (m4,1))] ( ^ ) + ( 5 - M ) e-dr-t) [a(l - 7 * ) ^ (dr_t (m4,1))] ( ^ ) Proof: Differentiate with respect to Vt the optimal volatility expressions obtained in Propo-sition 3.4.2. 174 Q.E.D. The explicit expressions obtained in the previous lemma show a direct, and useful, link between the compensation parameters and the sensitivity of the volatility with respect to which involves all the parameters that determine the compensation scheme <p, without con-sidering the risk aversion parameter 7. In other words, expression (3.28) is a scaling factor of o^-. Therefore, we can control the stability of v*t with respect to V by adjusting the absolute value of this factor. The smaller the absolute value of this factor, the more stable is v* with respect to V. Factor (3.28) can be used to establish conditions to maintain a stability level. For example, suppose that investors want to keep a stability level 8 of e. Hence, we can infer the value of any compensation parameter to maintain the stability level e. For instance, suppose that XQ, p, B, and q are given. Thus, the fund value. For instance, for the two-piece shape case (i.e., V2 < Vi) , observe that dv*t dVt can be factorized in terms of the expression (3.28) (3.29) assures 8< The value of e could be chosen to guarantee, for example, that du" dV <(3% V ~ where f3 is a predetermined percentage and V € 5R+. 175 B ( 3 . 3 0 ) p + q ) Moreover, we can deduce the required increment or reduction of a parameter value. For instance, from (3.29) we obtain f - ( ? > < * - « > - * i So, ^ > 0 if and only if pB > X0 + pe. That is, if the manager's fixed salary is small enough (i.e. Xo < pB — pe) then a larger value of q will require a larger value of K in order to preserve the same stability level e. This is consistent with intuition. If the manager's fixed salary is relatively small (i.e. X0 < pB — pe) and q is increased, the manager will be encouraged to take higher risks to get a better expected compensation. To mitigate an excessive increase in the volatility level, the exercise price of the second option, B + K, must be increased. In the four-piece shape case, there are two factors -instead of one- controlling the size of |p£r , that involve only compensation parameters. These factors are: X0 - qK X0 p+q p and B Xo P 176 Stability can be controlled by setting the value of the factors equal to some predetermined constants9, say ei and e2, and therefore establishing conditions and implicit relationships that compensation parameters must satisfy, as it is illustrated previously for the two-piece shape case. A particular and relevant case occurs when the constants ei and e2 are equal. This assumption implies that ^ is proportional to the single factor X0-qK X0 _ B X0 p + q p p from which we derive the implicit relation B - x ° - ^ = o. p + q Therefore, factor (3.28) is relevant for the stability of | ^ in any concavification case. For instance, condition (3.30), for e small, allows to control stability through a single factor for both cases. 3.6.3 An Example The goal of this section is to exemplify some of the theoretical results that are developed in the previous sections. We focus on the selection of the second option parameters q and K, given that in practice there exists a typical set of values for the first option parameters and the fixed salary. That is, we assume that the compensation parameters X0, p, and B are given, the risk aversion parameter 7 is known, and propose specific criterions to choose q and K. For instance, suppose that the initial or current fund value is Vn = 1, that the coefficient 9Using some criterion as that outlined in the precedent footnote to the previous one. 177 of risk aversion is 7 = 2, and the following compensation parameters Xo = 0.005, p = 0.01, and B — 1. That is, investors compensate the manager with 1% of the profits above the current firm's value and a fixed salary equivalent to 0.5% of the actual firm's value. We assume that investors decide first on the range type of terminal firm values and then on the stability of the optimal volatility decisions. To determine the type of terminal fund range, we.need the reference value V i , in this case equal 1 0 to 1.7071, and select q and K such that • V2 > V i , if investors want to induce a terminal fund value range of the form {0} U ( V i , V 2 ] U (Va.oo), • or V2 < Vi , if investors want to induce a range of the form {0} U ( V 4 , 0 0 ) We suppose, for simplicity, that q and K are not chosen simultaneously. Choosing q first Assume that investors set q = 2% and want to determine K. The election of K depends primarily on the type of terminal fund value range they want to induce. Given that 7 > 1, we know that V2 is an increasing function of K (Refer to Proposition 3.6.2). Therefore, it must exist a threshold value K*(q) such that K < K*(q) implies a terminal fund value of the form {0} U (V4 ,oo). Otherwise, if K > K*(q), the terminal fund value range is of the form {0} U (Vi, V2] U (V3, 0 0 ) . Under the given assumptions, K*(q) = 1.0305 (See Figure 3.18). Suppose that investors want to induce a range of the form {0} U (V4, 0 0 ) . How should they select K e [0, K*(q)]l There are certainly many ways to do such selection, depending upon what extra criterions investors would like to consider. To illustrate how this extra information could be used to determine a specific K, we consider the following situation: 10Refer to Lemma 3.3.1 for the characterization of Vi. 178 Graphical Determination of K*(q) Figure 3.18: Determina t ion of K*{q): This graph shows V2 as function of K, given risk aversion parameter 7 = 2 and the parameter values X0 = 0.005, p = 0.01, B = 1, and q = 0.02. K*(q) is determined as the value at which V2 (K*(q)) equates Vi , which in this case is equal to 1.7071. The terminal fund value is of the form {0} U (V4,oo) if and only if K€[0,K*(q)}. Assume that investors want to increase fund's value but without inducing the manager to follow a relatively unstable risk-taking policy. To determine the appropriate value of K, investors can solve the following optimization problem Max m o , K . { g ) ] (1-6)V4-8[B- \ q (3.31) V P + Q J where 1 — 5 and 5, for 5 € [0,1], are the relative weights that investors assess, respectively, to a minimum target fund value, if the manager decides to be in the money, and stability. Recall that it was shown in 3.6.2 that the term B — X°p+qqK is a key factor for controlling the stability of optimal risk-taking decisions. By solving Problem (3.31), investors try to choose 179 Optimal Selection of K 0.71 T 0.63 -I , 1 , , , 1 , 1 , 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 K Figure 3.19: Optimal Selection of K: This graph shows the objective function of Problem (3.31) for K € [0,1]. The optimal value is reached at K** = 0.39. The optimal exercise price for the second option is then B + K** = 1.39. the value of K that leads to the best tradeoff between the minimum target value V 4 and stability. The form of the objective function in (3.31) emulates the classical mean-variance approach of portfolio theory (Markowitz (1952)). The squared of the factor B — Xa~+qK can be directly related to the variability of the portfolio's return 1 1 . We solved Problem (3.31) for 5 = 35%. We obtained that the value of K that leads to n L e t V : Q —> 5R a random variable and G : K —> 5R a real valued function. Therefore, by Taylor's Theorem G(V(u)) « G{V0) + ( V » - Vo)G'(Vo) for u € O, and Vb £ SR. Hence, the variance of G(V), which we denote by VAR(G(V)), can be approximated in the following manner: VAR{G{V)) & [G'(y0)]2 VAR{V) If V represents the fund's value and G(V) the optimal volatility decision v* then, by the analysis developed in 3.6.2, JG'(Vo)] is proportional to [B - • 180 the best tradeoff is K** = 0.39 (See Figure 3.19). Hence, investors should set a percentage of q = 2% and an exercise price of B + K** = 1.39 for the second option included in the compensation scheme. In other words, investors should set a second benchmark, 39% higher than the current firm's value (Vo = 1), if the manager is compensated with 2% of the profits over such benchmark. Choosing K first Assume now that investors set K = 0.8 and want to determine q. The election of q depends primarily on the type of terminal fund value range they want to induce. Under the given parameters, V2 is a decreasing function of q (Refer to Proposition 3.6.2). Therefore, it exists a threshold value q*(K) such that if q < q*(K) then the fund value range is of the form {0} U (Vi, V2] U (V3, 00). Otherwise, if q > q*(K), the fund value range is of the form {0} (V4 ,oo). In this case, q*(K) = 0.0036 (See Figure 3.20). Suppose that investors want to induce a terminal fund value range of the form {0} U (Vi, V2] U (V3,oo). Thus, investors must select q from [0,q*(K)). Furthermore, assume that investors want to amplify the range (Vi, V2] as much as possible and, at the same time, to maximize the minimum target value V 3 . To define the value of q, investors can solve the following optimization problem: to a minimum target value V3 and the length of the range of fund values (Vi, V2]. Note that Problem (3.32) is a particular case of Problem (3.27) outline in 3.6.1. We solved Problem (3.32) for <5 = 55%. We obtained that the value of q that leads to (3.32) where S and for f5 G [0,1], are the relative weights that investors assess, respectively, 181 Graphical Determination of q*(K) 1.8 T : 1.5-1 , , , , , , , , , 1 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 q Figure 3.20: Determina t ion of q*(K): This graph shows V2 as function of q, given risk aversion parameter 7 = 2 and the parameter values X0 = 0.005, p = 0.01, B = 1, and K = 0.80. The point at which the function V2(q) crosses the value of Vi = 1.7071 is q*(K) = 0.0036. The terminal fund value is of the form {0} U (Vi, V2] U (V4, 00) if and only iiqE[0,q*(K)}. the best tradeoff between strategies that lead to terminal firm values in (Vi,!^] U (1/3,00) is q** = 0.18% (See Figure 3.21). That is, investors should compensate the manager with 0.18% of the profits over a benchmark set 80% above the current firm's value. 3.7 Conclusions The main motivation of this research is to contribute on the understanding of the induction and design of risk incentives derived from option-like compensation schemes. In particular, this study focus on the intriguing finding that option-like compensation schemes might ge-nerate unexpected risk-taking policies. For instance, single-option compensation schemes, 182 Optimal Selection of q 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01 q Figure 3.21: O p t i m a l Selection of q: This graph shows the objective function of Problem (3.32) for q <E [0,0.02]. The optimal value is reached at q** = 0.0096. which are in principle established to induce higher risk-taking, can imply that the manager actually takes less risk than if he were trading his own account. We studied the case of a compensation scheme formed by two options instead of one, as it is a common case in practice. Our expectation was to find out that two-option compensation schemes could alleviate, in some degree, the unexpected features of the single-option case given the extra degrees of freedom (exercise price and the proportion that the second option represents from the whole compensation package). We corroborated our expectation by obtaining the explicit optimal risk-taking policy and making a detailed analysis on the induction and design of such optimal policy. 183 The theoretical results obtained and the criterions proposed here on relation to the design of incentives for the type of option-like compensation schemes considered, should be useful in practice and a starting point for adapting these schemes to particular needs or proposing others. 184 Final Remarks The main motivation of this Ph.D. thesis is understanding the behavior of investors that have to operate under risk tolerance constraints or managers who are compensated through standard and relatively more sophisticated option-like compensation schemes. Our analysis is developed in two different model frameworks: a discrete-time framework, that can easily consider incomplete markets and fat tailed distributions, and under which we study investment under risk constraints and standard option-like compensations; and a continuous-time framework, in which we explore the optimal strategies and induced incentives of slightly more complex option-like compensation schemes than the typical single-option case observed in practice. The discrete-time analysis framework derives many important and useful insights about the nature of the optimal manager's strategies followed by the type of investors and managers considered. For instance, in the case of an investor subject to risk tolerance constraints, we uncover implicit probability measures that are used to assess the risk of implementing optimal strategies that involve to invest at the limit of the tolerated risk. For a manager or executive paid with a typical option-like compensation scheme, the optimal strategy leads to an expected gain that is proportional to the difference between the initial capital and the expected market value of replicating the benchmark. Moreover, if options are considered in the investment universe, we establish explicit optimal investment conditions for such instruments. The study realized in the continuous-time framework reveals that although compensation schemes formed by two different options might present the same counterintuitive behavior that the standard single-option case, these more complex compensation packages can be designed to mitigate this kind of behavior and to induce richer terminal firm value and risk taking profiles. We obtain several explicit theoretical results on the sensitivity of the terminal 185 firm value and the stability of the risk taking decisions with respect to the compensation parameters. Furthermore, we propose and illustrate specific criterions that combine some of these theoretical results to determine concrete firm value and risk profiles. The analysis done in this Ph.D. thesis should be useful to administration boards, that are responsible for establishing compensation packages and risk limits for investment managers, and to researchers who are interested in the induced behavior and the design of option-like compensation packages. The results can be extended to consider other monotone risk-averse utility functions and different convex risk measures. The discrete-time framework developed in this thesis should be particularly appealing, because of its relative simplicity and generality, to explore the induced behavior under other investment restrictions, while the concavification technique developed in the continuous-time framework, can be applied to more sophisticated option-like compensation schemes than the one studied in this research work. 186 Bibliography 1. 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(2003), The Stochastic Programming Approach to Asset Liability and Wealth Management, A I M R . 110. Ziemba, R.E.S. and Ziemba W.T. (2007), Scenarios for Risk Management and Global Investment Strategies, Wiley, in press. 197 Appendix 1 L e m m a 3.7.1 Let f : 9? —> 3? be a real-valued function that is monotone nondecreasing. Assume that lim x_ > a+ f(x) (limit of f as x approaches to a from the right) exists for some a 6 9ft and is equal to L. Then, f(x) > L for x > a. Proof: The proof is by contradiction. Assume that there exists x > a such that f(x) < L. Let e = EzMzl > o. Therefore, there must exist <5 > 0 such that if 0 < x — a < 6 then \f(x) -L\<e = (3.33) Let <5 = min (5,x — a). Hence, 0 < x — a < 5 implies x < x and (3.33). Therefore, then f(x) -L< f{x) -L - , which implies fix) > f{x) + ( ^ ^ ) > /(*) thus leading to a contradiction, since / is monotone nondecreasing and x < x. Therefore, f(x) > L for all x > a. Q.E.D. 198 A p p e n d i x 2 1. Derivation of the Dual Problem (2.2) The Lagrangian of (2.1) is L(0, ef?> eo°> s> Vo, y, x, A, H, rjc, nP) = Pn -y0(Z0 • 90 + egp0C0{\ + tcc) + ep0oPo(l + tcp) - p0W0) ~ Z)<Ll UneNt yn[Zn " {0n ~ # a (n ) ) ] + EneNT xn(Zn • 0n - (3nsn + egpnCn + ep(3nPn - pnBn) + Z ) „ 6 J V r *>n(Zn • 0n) + Y^n€NT HnPnSn + T]Ceg + T)Peg , where A„ > 0, ^ > 0, V n € NT , Vc > 0, T]P > 0, and — y0, —yn and xn are free variables since they correspond to equality constraints. The signs have been chosen to give later on a contextual interpretation. The Lagrangian can be rewritten as (see Item 3 of this Appendix) L{0, eg, eg, s, y0, y, x, A, /j,, T]C, VP) = yoPoW0 - }2neNT XnPnBn + J2neNT(aPn + Hn~ Xn)PnSn + £ n 6 j v T ( A „ + Xn - yn) (Zn • 0n) — X ] * = 0 Y2neNt (VnZn ~ X ^ m e C ( n ) 2 / m ^ m ) " On + e 0 EneJVr XnPnCn ~ (30C0(l + tCC)(y0) + 7)C} +«* En€JV T XnPnPn ~ P0P0(l + tCP)(y0) + TJp] The dual constraints require that the factors of the primal variables are zero, and the dual objective function comes from all the Lagrangian terms that do not involve primal variables. So, the dual is 199 Min sojr.x.A.M.TC.up 2/o/W 0 - E n € A f T XnPnBn S.t. ®Pn + Hn - Xn = 0, V n E NT K + x n - y n = 0, V n E NT (VnZn - Em€C(n) VmZm) = 0, V n € 7Y£, V < = 0, . . . , T - 1 E n € J V T Z r A C n - /3 0Co(l + tC C)(y 0) + VC = 0 E n e / V T Z r A ^ n ~ /?0-P 0 ( l + + VP = 0 An > 0, / J n > 0, VnENT Vc>0, T]P>0. Q.E.D. 2. Der iva t ion o f t h e D u a l P r o b l e m (2.14) The Lagrangian of (2.14) can be expressed as (see (3.34) and (3.35)) L(Q, eo , . s, 2/o, y, x, A, //, r/c, r/p) = 2/oA)Wo - E „ e / V r xnPnBn + T,neNT(aPn + Hn~ Xn)(3nSn + E n € J V 7 - ( A ™ + x n ~ yn)Zn • 0n > — E f = 0 lL/neNt(ynZn ~ LmeC(n) 2/m^m)^n +ef?MoC7 0(l + t C c ) - E - E n e J V 7 . ( A . C n ) P n + Vc] +eP[y0PoW + * C P ) - E n g / V r ^ / ^ n ^ n ~ E n e / V T ( / ^ " K + VP] • where A„ > 0, y L i „ > 0, V n E NT , Vc >0, VP> 0, and —yo) — yn and are free variables since they correspond to equality constraints. 200 The dual constraints require that the factors of the primal variables are zero; and the dual objective function comes from the Lagrangian terms that do not involve primal variables. The dual is S.t. apn + pn - xn = 0, V n€ NT , A n + xn - yn = 0, V n G NT , (VnZn ~ Y,meC(n) VmZm) = 0, V 71 G Nt, V t = 0, . . . , T - 1 , y0P0C0(l + tcC) - 23n€/VV XnPnCn ~ Y,neNT(PnCn)Pn + VC = 0 , yO0OPo(l + tCP) - YlneN-j- XnPnPn ~ Y.neNT(PnPn)Pn + VP = 0 , A„ > 0, pn > 0, VneNT , VCall > 0, VP > 0 • Q.E.D. 3. Equivalence between equations (3.34) and (3.35) The equivalence between equations (3.34) and (3.35) follows from showing that y0 (Zo • do) + £ f = l Yln€Nt Vn [Zn • (0n - 9a(n))} = ^2neNT Vn (Zn - #n) + 2~Zt=0 ^2neNt (jJnZn ~ ]CmeC(n) ymZrnj ' @n 201 Proof: 2/0 (Z0 • Oo) + EfLl J2neNt Vn [Zn ' {0n ~ o^(n))] = J2t=0 EneNt {Zn • 0n) ~ Et=l EneNt VnZn M 0a(n) — Y%=0 T,n€Nt ^ (Z* ' Gn) ~ J2n€Nt+l Vn (Zn ' 0<j(n)) + Y,n€NT Vn (Zn ' On) Observe that y i Vn {Zn • 0a(n)) = Y [ Y2 V m Z m ) 0n • neNt+i neNt \m€C(n) Hence, 2/0 (Z0 • Oo) + £f=l T,neNt Vn [Zn ' (0n - 0„(n))] = Et=0 Ene/Vt 2/™ ' ~~ EnGJVt (SmeC(n) 2/m^m^ " $n + En6A^T ^" ^™ ' = £t=0 En€JV"i (jJnZn ~ YlmeC(n) UmZm^ • #n + En6ArT J/n ' 0n) Q.E.D. 4. Proof of Equation (2.10) If ZN • 0n is unrestricted for n € NT, then the term 202 vanishes from the Lagrangian function of Problem (2.1) (see (3.34) in Item 1 of this Appendix); hence, ^2n€NT(Xn + xn - yn) (Zn • 6n) is replaced by ^ (xn - yn)Zn • 6n n£NT in the Lagrangian equation (3.35), and xn = yn for all n G NT. Thus, the dual objective function is yo \ PoW0 - (-) PnBn) = yo (/Wo - £ Q [3TBT]) . n€NT WO/ J Q.E.D. 203 A p p e n d i x 3 Proof of Lemma 3.3.1 Consider the function g : [B, oo) —» 3ft defined in the following manner: 5 * ~ j log (X0 +p(x — B)) — Xo+%x_B) - log(Xo) if 7 = 1-Clearly, this function is differentiable over its domain and hence it is also continuous. There-fore, it is enough to show that there exist x and x such that g(x) < 0 < g(x) and then apply the Intermediate Value Theorem to claim that there must exist Vi in the interior of the domain of g such that g(Vi) = 0, from which we can conclude (3.8). Note that g(B) = -pBXf < 0 . To prove that it exists x such that g(x) > 0, observe that the function g can be expressed, for the case of 7 ^ 1, in the following manner: g(x) = - ^ l + (X0 + p(x-B))-1 — 7 Xo 1 - 7 pB + 7 1 - 7 p{x-B) Thus, by applying L' Hopital's Theorem, it follows that lim (X0+p{x- B)Y ( I ^ ) + (T^><H = { 00 if 7 < 1. 0 if -y > 1. Thus, 204 { oo if 7 < 1. i f 7 > l . Note that —-j^— > 0 for 7 > 1. For the case of 7 = 1, observe that lim ' ^ +<x>\X0+p(x - B) Hence, lim g(x) = 00 x—>oo for 7 = 1. Therefore, for any 7 > 0, it must exist x such that g(x) > 0. Hence, by the Intermediate Value Theorem, it must exists Vi in the interior of the interval [B, x], so Vi > B, such that g(Vi) = 0 and thus it follows that Vi must satisfy equation (3.8). Q.E.D. P r o o f o f L e m m a 3.3.2 The proof is similar to the proof of Lemma 3.3.1. Q.E.D. P r o o f o f L e m m a 3.3.3 Let 7 > 0 and 7 ^ 1 . Then, the system formed by equations (3.10) and (3.11) imply that p (X0 + p (V2 - B))-1 = {p + q) (Xo + p (V3 -B) + q(V3-B- K))^ (3.37) 205 or equivalently, X0 + p(V2-B) = 1 + - (X0+p(V3-B) + q(V3-B-K)) (3.38) v; Solving for V2 from equation (3.38) leads to where: V2 •= BV3 + a 0 a = x 0 f ( i + ; ) " - i + (3.39) Thus, (3.40) Also, from equation (3.38), we deduce that (X0+p(V2-B)) 1 - 7 1 - 7 1 - A 7 1 - 7 1 - 7 i.e., (X 0 + p (V2 - AO) 1 " 7 = ^(Xp + p (14 - B) + g (Va - B - K))1^ _ ( g ^ 1 - 7 1 - 7 206 Therefore, using (3.37), (3.40) and (3.41), we can rewrite equation (3.10) in the following manner: { X 0 + p ( V 2 - B ) t 1 - 7 '1" ~{XQ+p(V2-B)f^ fi. 1 - 7 +p(X0 + p(V2-B))-'' l V2 p So, (X0 + p(V2-B)) 1 - 7 1-7 P. p(x0+P(v2-B)y Hence, (X0 + p(V2-B)) 1 - 7 P. V 2 [ l - i ) + ^ PJ P. Therefore, V2 P 1 -P 7 P 1 - 7 ( X 0 - p 5 ) from which Vo 1-7 ( X o - p B ) (3.42) Q.E.D. 207 P r o o f of P ropos i t ion 3.3.1 Vi exits by Lemma 3.3.1 and hence the function U\ is well defined. Denote the derived utility function U o cb restricted to the domain [0, B + K] by (U o <b) \ [Q,B+K]- We need to show that the hypograph oiU\, restricted to [0, B + K], is the convex hull of the hypograph of (U o cb) \ [QTB+K\- Let HQI and Hua), be, respectively, the hypographs of V\ and U o cb restricted to [0, B + K\. Thus, we have to prove the following conditions: 1. Hfj is a convex set. 2. Hjjx 3> B\jo<i>-3. If A D Huo</> and A is convex then A D Hfj . 1. By the construction of (7i, lf[ exists over the interval [0, B + K] and it is defined as f 0 for TJ G [0,Vi] U l ^ = j (<7o <b)" (TJ) otherwise. where (Uocb)" (TJ) = If (<Kv))<f(v) + U" (cb(v)) ( « ^ ' ( T J ) ) . (3.43) Note that the assumptions that U is concave and differentiable, together with the observation that the compensation scheme satisfies the condition d>"(v) = 0 for all TJ > 0, imply that (U o (/>)" (TJ) < 0 for all v > V i . Therefore, U'i(v) < 0 for all v > 0, and thus it follows that Ui must be concave over all its domain. Hence, by the characterization of a concave function in terms of its hypograph, HQ is a convex set. 208 2. By construction of U\, U\ dominates U o <b over the interval [0, B + K], that is U\(v) > (U o<b)(v) for all v G [0, B + K]. Therefore, using the definition of the hypograph of a function, it is clear that HQ 5 Huo<t>-3. Let A be a convex set such that A D HUo(t). We have to show that A I> H Q I . TO do so, we partition A, HfJi, and Huo<p in the following manner: A = Ai\J(A\Ai) , with AL = {{v,y) G A\v e [0, Vi]} H6l = H^ljfaXHlJ , with = {(v,y)€HOi\ve{0,V1}} HUo(t> = H^UiHu^XH^) , with Hh0<t> = {(v,y)eHUo+\ve[0,V1}} and where C\B denotes the set formed by all the elements of C that are not contained in B. That is, we partition each of these sets with respect to the tangency point Vi. The rationale of this partition is to prove A D HQI by pieces. First, we prove that (A \ A\) ~D {HQi \ H^^j and then we prove that A\ D HQ . . (AXA^D^XHiy. By the design of U\, U\ and U o cb coincide over the interval (Vi,oo). In particular, Ui(v) = (U o cb) (v) for v G [Vi, B + K]. Hence, it must be satisfied that \ H± = Huo<j> \ Hjjort>. Thus, it is clear that the assumption A D Huo<t> implies (A \ A\) D Observe first that the assumption A D Hu0<i> assures that Ai D H^^. Then, by construction of f/i, U\ dominates U o cb over the interval [0, Vi] and thus, it is satisfied that HQ 5 Hy0lp. Therefore, to prove A\ D HQ it is enough to show that Ai D \ Hy0(^j. To accomplish this task, we use the following strategy: choose an 209 arbitrary point in [HQ \ Hy0^ and show that this point can be expressed as a convex combination of two elements of A\. Then, we prove that Ai is a convex set and thus conclude that the arbitrary point chosen must belong to A\, by the convexity of A\. Let (vo, y0) G \ Hho^ a n a consider the line that passes through the points (0, (U o cj)) (0)) and (v0,y0). This line is described by the equation , ( ! l = ( J o W 0 ) t ( t M i ) < „ _ „ „ ) where its slope, by construction of Ui, satisfies ( g°- (^ ) ( 0 ))<(cw)> 1). Recall that Vi satisfies the following equation: (U o <j>) (Vi) = (U o <f>) (0) + Vi (U o cj))' (Vi) . Therefore, (£W ) (Vi) = (t/o^)(O) + Vi(/7o0)'(Vi) > (Uocfi)(0) + vo(UocP)' (Vx) > (UoJ)) (0) + v 0 ( y ° - ^ 0 ) ) = IM = yo Thus, we deduce that I must cross the graph of U o <fi at some value v* < Vi. That is, it exists (v*,y*) such that v* < Vi and l(v*) = (U o(p)(v*). Clearly, (v*,y*) G HUo<j,. Therefore, the point (v0, yo) belongs to the line segment between the points (0, (U o cj)) (0)) and (v*,y*). Hence, it must exist A G [0,1] such that 210 K yo) = A(0, (U o cb) (0)) + (1 - X)(v*, y*) . Now, we claim that Ai is a convex set. To prove such claim let (v, y), (v, y) G A\ and 77 G (0,1). Observe that it must be satisfied the condition nv + (1 — r])v < Vi since v and v are, by definition of A\, less than or equal to V i . On other side, AL C A and A convex implies that n(v,y) + (1 - 7 7 ) (v,y) G A . Therefore, by definition of A\, r](v,y) + (1 - 7 7 ) (v,y) G Ax and so, A\ is a convex set. To finish the proof of (3), recall that (0, (U o cb) (0)), (v*,y*) G H i / 0 ^ C A, and use the convexity of A\ to conclude that (v0,yo) G and thus, since (uo,yo) is arbitrary, Therefore, from conditions (1),(2), and (3), we conclude that is the smallest convex set that contains to HUo<p. Hence, by definition 3.3.1, U\ must be the convavification of U o cb over the interval [0, B + K]. Q.E.D. 211 P r o o f of P ropos i t ion 3.3.2 By Lemma 3.3.3, V2 and V3 exist and therefore, U2 is well defined. Denote the derived utility function U ocb restricted to [B, oo) by (U o <b) \{B,OO)- The goal is to show that the hypograph of U2, restricted to [JB, oo), is the convex hull of the hypograph of (U o cb) \[B,OO)- L e t HQ2 and Huo<t> be, respectively, the hypographs of U2\[B,OO) and (U o <b) \ [B,OO)- Thus, we need to prove the following conditions: 1. H(j is a convex set. 2. HQ2 D Huo(j>-3. If A D HUo<j, and A is a convex set then A 2 HQ2. 1. By the construction of U2, U2 exists over the interval [B, 00) and it is defined as 0 for v E (V2,V3}. (U o (b)" (v) for all v > 0 such that v <£ {(V2, V3] U {0} U {B}} where (U o <£)' (v) = if (4>(v)) cb"(v) + U" (<f>(v)) (cb'ivtf < 0 The assumptions that U is concave and differentiable, together with the observation that the compensation scheme satisfies condition <b"(v) = 0 for all v > 0, imply that (U o (b) (v) < 0 for all TJ > 0 such that TJ {(V2, V3] U {0} U {B}}. Therefore, (U o <f>)" (TJ) < 0 for all TJ > 0 such that TJ 7^  B. Thus, U2 must be a concave function. Hence, by the characterization of a concave function in terms of its hypograph, HQ2 must be a convex set. 212 2. By the construction of U2, U2 dominates U o <f> over the interval [B, oo), that is U2(v) > (U o (j)) (v) for all v G [B, oo). Therefore, using the definition of hypograph of a function, we obtain that HQ 3 Huo<j>-3. Let A be a convex set such that A D Hu0<p- We need to show that A D H Q 2 . TO accomplish that, partition A, H Q 2 , and Hu0<t> in the following manner: {(v,y)eA\v e(V2,V3}} {(v,y)eH02\ve(V2,V3]} {(v,y) G HUo<p\v e (V2,V3}} and where C \ B denotes the set formed by all the elements of C that are not contained in B. That is, we partition each of these sets with respect to the tangency points V2 and V3. The rationale of this partition is to prove A D by pieces. First, to prove (A\A\) D (jJ(j2 \ > a n a 1 then, to prove that A\ D HQI. By design of U2, U2 coincides with U o <f> over the set [0,V2) U (V^oo). That is, (Uod))(v) = U2(v) for all v G [0, V2) U (V3,oo). Hence, it must be satisfied that HQ2 \ Hfji = HUo(t> \ H\j0(j). Thus, it is clear that the assumption A D HUo<p implies ( A \ A 1 ) D [ H 0 2 \ H 0 , ) . The assumption A D Huolf, assures that Ai D Hy^. Then, by construction of U2, U2 dominates U o<fi over all its domain. Hence, it is satisfied that HQI D Hy^. Therefore, 213 A = Hu2 Huo(j> Ax\J(A\Ai) . with Ai ^ 2 U ( ^ 2 \ ^ 2 ) , with Hi HhM{Hu^\H}j^) , with Hh^ to prove Ai I) it is enough to prove that Ai ~D \ ^rjo^J • To do so, we follow the next strategy: choose an arbitrary point in \ Hjj0lp^ and prove that this point can be expresses as a convex combination of two elements of the set A\. Then, prove that Ai is a convex set and thus conclude that the arbitrary point chosen must belong to Ai, by the convexity of Ai. Let (vn,yo) € ^H^\ H}jot^j and consider the line that passes through the points (V2, (U o (b) (V2)) and (urj,2/o)- This line is described by the equation i M - ( i r ° « W + ( » - ( g ° * ) W ) ) ( . - » 4 ) V V0-V2 j where its slope, by construction of U2, satisfies V V0-V2 J Recall that V3 satisfies the equation (U o <b) (V3) = (U o cb) (V2) + (V3 - V2) (U o <b)' (V3) Therefore, (Uo(b)(V3) = (Uo<h)(V2) + (V3-V2)(Uo<bj (V3) > (Uo<b)(V2) + (v0-V2)(Uo<b)'(V3) > (Uo<h)(v2) + (v0-v2) = l(v0) = yo Thus, we deduce that I must cross the graph of U o cb at some value v* < V3. That is, it exists (v*,y*) such that v* < V3 and l(v*) = (U ocb)(v*). Clearly, (v*,y*) € 214 Huo<f>- Therefore, the point (vo,yo) belongs to the line segment between the points (V2, (U o <p) (V2)) and (v*, y*). Hence, it must exists A G [0,1] such that (vo,yo) = A(V 2 , (U o <t>) (V2)) + (1 - \)(v\y*) . We claim that Ai is a convex set. To prove this claim, let (v, y),(v,y) G A\ and rj G (0,1). By definition oi Ai, v and v are less than or equal to V3, and greater than V2. Hence, V 2 < T)V + (1 - T])V < V 3 . On the other hand, Ai C A and A convex implies that V(v,y) + (1 - rj)(v,y) G A . Therefore, by definition of A\, rj(v,y) + (1 — r/)(v,y) £ Ai, and so, A\ is convex. To end the proof of (3), recall that (V 2, (17 o d>) (V2)), (v*, y*) G HUoij) C Ax and use the convexity of A\ to conclude that (vo,yo) G A\. Thus, since (un,2/o) i s arbitrary, we deduce that A± D Hy . So, from conditions (1),(2), and (3) we conclude that HQ2 is the smallest convex set that contains to H{j0^. Hence, by definition 3.3.1, U2 must be the concavification of U o </> over the interval [B, 00) . Q.E.D. 215 Proof of Proposition 3.3.3 Let HUs and HUo<i> be, respectively, the hypographs of U3 and U o d>. We need to show that HUs is the convex hull of HUo(f,. By construction of U3, U3(v) > (U o d>) (v) for all v > 0 and hence, HQ3 2 HUo<t>. On the other hand, concavity of U3 implies that HUs is a convex set. Therefore, the only issue that remains to be proved is that Hv is the smallest set that satisfies the latter two properties. To prove this, the strategy is the following: show that the assumption of concavity of U3 implies that V2 > Vi and hence, deduce that U3 coincides with Ui over the interval [0, V2], and with U2 over the interval (V2,oo). Therefore, Hfr = Hf, | U Hr, , and Hn , D Hf,, = 0 where HQ^ is the hypograph of Ui restricted to [0, V2], and H Q 2 ^ ^ represents the hypograph of U2 restricted to (V2, oo). Finally, use the fact that t/i l^vy must be the concavi-fication function of U o d> restricted to [0, V2], while U2\(v2,oo) is the concavification function of U o cj) restricted to (V2,oo) to deduce that any convex set A such that A D HJJ0^ must satisfy that A D HUs. We now prove this strategy in detail: By construction of U3, the following conditions are satisfied: • V3 > Vi. U3(V3) = (Uo<f>)(V3) • C/3(Vi) = (UofliVi) ' • U3 is twice differentiable and, - (c/o^)'(V 2) = (<7o«/,),(V3) , 216 — under the assumption of concavity, U3(v) < 0 for all v > 0. Thus, (U o <j>)' (V2) = (U o <(,)' (V3) = U3(V3) < c73(Vi) = (U o </>)' (Vi) . Therefore, (Uocfi)' (V2) < (/7o«/,)'(Vi) which, by the concavity of U o cp over the interval [B, B + K], implies that V2 > V\. Hence, by the definition of U\ and U2, UM = U2(v) , f o r v G [VI, V 2 ], Ui(v) > U2{v) , for7j<Vi U2(v) > Ui(v) , {orv>V2 Therefore, by the construction of U3, ( UAv) for v < V2 « - { « , ( . ) fcr^H ( 3 ' 4 4 ) So, Hfr =Hfl, DH,-,, andHrj, n H0. = 0 (3.45) From Propositions 3.3.1 and 3.3.2 we know that Ui and U2 are, respectively, the concavifi-cation functions of U o <f> restricted to [0, B + K) and [5, oo). Thus, given that V2 > Vi, it 217 is clear that C/i|[o,vi] must be the concavification function of U o cb restricted to [0, V2], while U2\(v2,oo) is the concavification function of U o cb restricted to (V2, 00). Let A be a convex set such that A D HUo<i> and partition A and HUo(p in the following manner: A = ^ I J ^ U i ) , A i = {(v,y) <E A\v € [0,V2]} HUo<t> = H^UiHuoAHbo^) , = {(v,y)eHUo(t>\ve[0,V2}} where C\B denotes the set of all the elements of C that ar not in B. Thus, by definition of this partition, Ax D and (A \ A{) D {HUo<t> \ H1^) Hence, since £/i|[o,v2] and U2\[v2,oo) are, respectively, the concavification functions of U o cb over [0, V2] and (V2,00), it must be satisfied then that A, D Hf,, and (A \ Ax) Z> Hf,, Therefore, by (3.45), A D HQ^. Q.E.D. Proof of Proposition 3.3.4 The necessity of condition V2 > Vi is proved in the first part of the proof of the previous proposition. 218 We now prove the sufficiency of condition V2 > V i . This condition implies, by the definition of Ui and U2, that Utiv) = U2(v) , for v e [VUV2], UM > U2(v) , for7j<Vi, U2(v) > Ui(v) , iorv>V2. Therefore, by construction of t/3, Ui(v) for v< V2 U2(v) for v > V2 Hence, since Ui and U2 are concave, twice differentiable and U[(V2) = U2(V2), then U'3\v) < 0 for all v > 0, and so we conclude that U3 must be concave. Q.E.D. P r o o f of P ropos i t ion 3.3.5 The proof is completely analogous to the proof in Proposition 3.3.1 except that in the current case V 4 , instead of V i , plays the role of the pivot to partition the domain of <73. Conditions (3.15) and (3.16) are simply the explicit statement that the point (0, (U o d>) (0)) belongs to the tangent line of the graph of (U o d>) |[jg+jc,oo) at ( V 4 , (U o d>) ( V 4 ) ) . That is, V 4 must satisfy (U o <f>) (0) = (Uo<f>) ( V 4 ) + (0 - VA) (U o </>)' (V4) Q.E.D. 219 Proof of Proposition 3.3.6 If U3 is concave then, by Definition 3.3.1, U(v) = U3(v) for all v > 0, and hence, by Proposition 3.3.3, U is the concavification of U o <b. Otherwise, assume that f/3 is not concave and let HQ be the hypograph of U. By construction, U is concave and satisfies U{v) > U3(v) > (U o cb) (v) for all v > 0. Therefore, HQ is a convex set that contains to Huotj,. Hence, what remains to be shown is that HQ is the smallest set with the latter two properties. Let A be a convex set such that A D Huo^- We need to prove that A D HQ. It is enough to show that A Z> HQ3 given that U is the concavification of t/3. To prove A D HQ3, we decompose A and HUo^ into A = Ai U A2 HUo4> = H}jo4> U i/£o0 where: Ai = { > , y ) e A\ve [0,B + K]} A2 = {(v,y) G A\v e [B,oo)} Hb* = {(v,y)eHu^\ve[0,B + K}} Hue* = {(", y) e HUo<l>\v e [B, oo)} Thus, A D HUo<f, implies Ax 5 and A2 D Hy^. Therefore, given that Ui and U2 are, respectively, the concavification functions of U o <b restricted to [0,5 + K] and [B, 00), it must be satisfied that Ai D H}-r and A2 D H% , where Ul U2 HlY = {(v,y)€HOi\v€[0,B + K}} Hi = {(v,y)EH02\ve[B,^)} 220 Hence, Ax U A2 2 H± U H ^ . Moreover, by construction of Ux and U2 and the definition of U3 , Thus, A = Ai U A2 D % . Therefore, since (7 is the concavification function of U3, A 5 HQ must be satisfied. Q.E.D. P r o o f of P ropos i t i on 3.6.1: By construction, Vi satisfies Vi > B and (see Lemma 3.3.1) 0 = (U o cb) (Vi) -(Uo(f>) (0) - Vi (U o </>)' (Vi) . (3.46) The results are obtained by deriving the previous equation with respect to B, p, and X0 and then solving of the corresponding partial derivative, as it is shown next: 1. Deriving (3.46) with respect to B leads to 0 = (X0+p(Vi-B)rp[^-l}+p^Vi(X0+p(V1-B)r-1[^-l] - P(x0+p(Vi-B)r[^} or equivalently, 221 0 = p dVi dB + p21V1(X0+p(V1-B)y1 dB - 1 dB from which <3Vj 95 1 + ( X 0 + p (Vi - 73)) ( p 7 V i ) _ 1 > 0 . 2. Deriving (3.46) with respect to p leads to 0 = ( X o + M V i - B ) ) - 7 p ^ + ( V i - i ? ) + FYVi (Xo + p(V1- B)r~l [p9^ + (Vi - B) - V i ( X 0 + p ( V i - 5 ) P - p ( X o + p ( V i - £ ) p dVr dp or equivalently, 0 = -B + p 7 V i (X 0 + p (Vi - /J))" 1 <9Vi Hence, r3Vj _B[1 + {(PJVI)'1 (XQ +p(Vj - B))}] - Vj dp p from which the result follows. 3. Deriving (3.46) with respect to X 0 leads to 0 = ( X 0 + p ( V i - B ) ) - 1+p dV1 dX0 + p 7 V i ( X 0 + p ( V i - 5 ) ) -7-1 1 +P dX0 • p ( X 0 + p ( V i - £ ) ) - dX0 222 or equivalently, 0 = ( p 2 7 V i ) ( * o + p ( V i - £ ) ) + pyV^Xo+piVi-B))-1 - l dVi ex0 + 1 X~1 (X0+p{V1-B))-< Thus, from which result (3) follows. L(x 0 + p(vi-s))-l - p 7 V i ( X o + p ( V i - / 3 ) ) - i P r o o f of P ropos i t i on 3.6.2: 1. Deriving (3.42) with respect to K leads to Q.E.D. p 2 7 dV2 dK V ( l - 7 ) / ? (l - 1\ d°L + ( °L \ ii. \ L (3) dK ^ \02 J dK where a and B are defined in (3.39). Therefore, the sign of ^ depends on the expres-sion From (3.39) p 2 7 ( 1 - 7 ) / ? B) 3K + \32) dK da dK 90. dK SB- = 0 223 Therefore, dK \{\ \ p) w V p) which does not depend on K. Thus, V2 is a linear function with respect to K. Moreover, from (3.39), (1 - 1/8) < 0 if and only if 7 < 1. Hence, f £ < 0 if and only if 7 < 1. 2. Deriving (3.42) with respect to q leads to where a and /? are defined in (3.39). The result follows from the above equation. dV2_( P21 \ dq \{l-l)8) Q.E.D. Lemma 3.7.2 Let X ~ N(p,a2) be defined in a probability space (Vt,J:,TJ) and let ACQ. Then, E [ e ^ l { X e A ) ] = e ^ ^ 2 p ( x e A ) where X ~ N(p + 8a2, a2). Proof: Just note that 8x-^{x-p)2 = g ^ ' - f r ^ a x ^ ) = ~^(x2-2x(p + Pa2) + p2) = -2^((x-(p + pa2))2-0a2(2p + 0a^ = -^{x-{p + Po2))2 + pP+\a2p2 . 224 ) Hence, E[ef*xU = = ert+^Pp {X G A) where X ~ N (p + 0a2, a2) Q.E.D. Lemma 3.7.3 (Characterization of A) IfV2 > Vi, the parameter X must satisfy the equa-tion V o = ^B-^e-»TW(dT(rnul))-M(dT(rn2,l))} + (*) ( £ ) 7 * ~ V ( M + " ^ r T K ( r f r ( m i , f ) ) - ^ r K , 7 ' ) ) ] + ^(dr(m2,7*)) where du(-, •) is £/ie function defined by du(m, 0) = log m — I — pu + a u \ 0 — -J\f(-) is the standard normal cumulative distribution, 7* = 1 — I /7 , and (3.47) (3.48) mi JL [Xo + piVi-B)]-"* , fori = 1,2, 225 and where Vi and V2 are defined in Lemmas 3.3.1 and 3.3.3, respectively. Otherwise, A must satisfy the equation V« = [B- 2[% !f}e-*TM(dT(m1,l)) 4(A)M«"(wJV3N*(».n) + where m 4 p + q [ X o + p ( V 4 - £ ) + < z ( V 4 - £ - / 0 ] - 7 and V 4 is defined in Lemma 3.3.2. Proof: Assume that V2 > V i . Therefore, condition (3.20) is equivalent to V0 = + + B *f E frlSp[x0+p{V2-B)}-' < \(,T < P [ X o + p ( V 1 - B ) ] - T } ( p ) ( p ) ^ ^ P I X O + P ^ - B ) ] - 1 < A? T < p [ X o + p ( V x - B ) ] - ^ } S ~ ^ ^ T 1 { A ? t < p [ X 0 + p ( V 2 - B ) ] - 7 } + (,P+9 J \p+q J E fa 1{PIX0+P(V2-B)}-'> < A? T < p[X0+p(V1-B)]-'} By Lemma 3.7.2 . (3.49) (3.50) E ^ l { l o g ^ } ] = e - H 4 ) M » ] ^ ( l e A ) (3.51) where X ~ AT ( - /x + ) (1 - / ? ) T , a 2 r ) . The result follows from applying (3.51) to (3.50). 226 The proof of condition (3.49) is similar. Q.E.D. P r o p o s i t i o n 3.7.1 ( O p t i m a l F i r m V a l u e f o r a n y t i m e t G (0, Tj) Let 7 > 0 and Vi and V2 as defined in Lemmas 3.3.1 and 3.3.3, respectively. If V2 > V\, then the optimal fund value for any t € [0, T) is B ~ I e - " ( T - t W ( r i T - t ( m 2 , 1 ) ) + + + l £ 7 * - l -(/i+"a(12"7*))>y(r-« M(dr-t (m 2,7*)) e-„(r-t) [ /V(d T _ 4 ( m i , 1)) - /V(d T - t(m 2 ,1))] l c 7 * - l r ( ^ ) , ' ( r - [ T V ( r i r ^ K , 7 * ) ) - /V( r i T _ t (m 2 ,7*) ) ] where du(-,f is defined by expression (3A8), Tfli V A6 [Xo+piVi-B)}^ , fori = 1,2, A/"(-) is the standard normal cumulative distribution, 7* = 1 — I /7 , X is a nonnegative scalar that satisfies Equation (3A7). Otherwise, the optimal fund value for any t € [0,T) is given by V* = r> Xg-qK (P+qr A 7 * - I £ e-xF-'Widr-timA,!)) V J N(dT-t(m4,i*)) - l £ 7 * - l where 227 P + Q [X0+p(V4- B) + q(VA- B - K)r , V4 is defined as in Lemma 3.3.2, and X is a nonnegative scalar that satisfies Equation (3A9). Proof: Assume that V2 > Vi. Note that (f*V t*) t > 0 is a martingale. Therefore, for all t £ [0, T] and where Et [ • ] is the expected value operator with information up to time t. Hence, the above expression and (3.19) lead to v* = i£U&vy] B-X* p E 7 * - l (if)  1{p[X0+p(V2-B)r~< < A f r < p[XQ+p{V1-B)]-<} + (P) (P) ^ L E [ ( > ) 1{P[X0+P(V2-B)}-~> < A f r < p[X0+p(V1-B)}-'<} (tf) 1 { M T < p[X0+p(V2-B)]-~i} D _ Xp-qK P+q \p+qj \p+qj E (7*-l) a'-lE ( f f ) 1{KT < p[X0+p(V2-B)]-<} By Lemma 3.7.2 (3.52) E (3.53) where X ~ N ( - //+ (1 -/?) T > 2 r ) . The result follows from applying (3.53) to (3.52). The proof for the case V2 < Vi is similar. Q.E.D. 228 Observation made in 3.5.1 Any strategy that leads to fund values between 0 and V4 can be characterized in terms of a density function g(x), for x £ (0, V 4), that specifies the likelihood of choosing the intermediate value x. By construction, (U o cj>) (x) < X(x) (U o cfi) (0) + (1 - \{x)) (U o cj>) (V4) where \(x) £ (0,1) and it satisfies x = X(x) • 0 + (1 - X(x)) VA. Therefore, /;4 (U o 4>) (x)dg(x) < [J0V4 X(x)dg(x)\ (U o cf>) (0) + [l - fQv* X(x)dg(x)\ (U o cj>) (V4) = V(Uo<j>) (0) + [1 - V] (U o cj)) (V^ where V = JQ4 X(x)dg(x). Clearly, V £ [0,1]. Hence, following a strategy that takes on the payoff of zero with probability V and the payoff of V 4 , with probability 1 — V is better, in the sense of expected utility, than any strategy that takes on intermediate values between zero and V4. 229 

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