A Generalization of the Mean-Variance Analysis

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1 A Generalization of the Mean-Variance Analysis Valeri Zakamouline and Steen Koekebakker This revision: May 30, 2008 Abstract In this paper we consider a decision maker whose utility function has a kink at the reference point with different functions below and above the reference point. suppose that the decision maker generally distorts the objective probabilities. We also First we show that the expected utility function of this decision maker can be approximated by a function of mean and partial moments of distribution. This mean-partial moments utility generalizes not only the mean-variance utility of Tobin and Markowitz, but also the meansemivariance utility of Markowitz. Then, in the spirit of Arrow and Pratt, we derive an expression for a risk premium when risk is small. Our analysis shows that a decision maker in this framework exhibits three types of aversions: aversion to loss, aversion to uncertainty in gains, and aversion to uncertainty in losses. Finally we present the solution to the optimal capital allocation problem and derive an expression for a portfolio performance measure. This performance measure generalizes the Sharpe and Sortino ratios. We demonstrate that in this framework the decision maker s skewness preferences have first-order impact on risk measurement even if the risk is small. Key words: mean-variance utility, quadratic utility, mean-semivariance utility, risk aversion, loss aversion, risk measure, probability distortion, partial moments of distribution, risk premium, optimal capital allocation, portfolio performance evaluation, Sharpe ratio. JEL classification: D81, G11. Corresponding author, University of Agder, Faculty of Economics, Service Box 422, 4604 Kristiansand, Norway, Tel.: (+47) , University of Agder, Faculty of Economics, Service Box 422, 4604 Kristiansand, Norway, Tel.: (+47) , 1

2 1 Introduction Expected Utility Theory (EUT) of von Neumann and Morgenstern has long been the main workhorse of modern financial theory. A von Neumann and Morgenstern s utility function is defined over the decision maker s wealth. The properties of a von Neumann and Morgenstern s utility function have been studied in every detail. The concept of risk aversion was analyzed by Friedman and Savage (1948) and Markowitz (1952). They showed that the realistic assumption of diminishing marginal utility of wealth explains why people are risk averse. Measurement of risk aversion was developed by Pratt (1964) and Arrow (1971). These authors analyzed the risk premium for small risks and introduced a measure which is widely known now as the Arrow-Pratt measure of risk aversion. The celebrated modern portfolio theory of Markowitz and the use of the mean-variance utility function can be justified by approximating a von Neumann and Morgenstern s utility function by a function of mean and variance, see, for example, Samuelson (1970), Tsiang (1972), and Levy and Markowitz (1979). In this sense, the use of the Sharpe ratio (see Sharpe (1966)) as a measure of performance evaluation of risky portfolios is also well justified. However, not very long ago after EUT was formulated by von Neumann and Morgenstern, questions were raised about its value as a descriptive model of choice under uncertainty. Influential experimental studies have shown the inability of EUT to explain many observed phenomena and reinforced the need to rethink much of the theory. An enormous amount of theoretical effort has been devoted towards developing alternatives to EUT. Many of the alternative models of choice under uncertainty can predict correctly individual choices, even in the cases in which EUT is violated, by assuming that the decision maker s utility has a reference point and/or the decision maker distorts the objective probabilities. Most prominent examples of these types of models are (for a brief description, see the online supplement): the utility function of Markowitz (1952); the mean-lower partial moment model of Fishburn (1977) and Bawa (1978); Prospect Theory/Cumulative Prospect Theory (PT/CPT) of Kahneman and Tversky (1979) and Tversky and Kahneman (1992); Disappointment Theory (DT) of Bell (1985) and Loomes and Sugden (1986); Anticipated Utility Theory/Rank Dependent Expected Utility (AUT/RDEU) of Quiggin (1982); etc. Even though the models where the utility function has a reference point and/or the objective probabilities are distorted have been 2

3 known for quite a while, very little is known about general implications of these models for decision making except for the prescription that the decision maker should maximize expected utility. The goal of this paper is to provide some general insights into this broad class of models of choice under uncertainty. In this paper we consider a decision maker with a generalized behavioral utility function that has a reference point. We assume that the decision maker regards the outcomes below the reference points as losses, while the outcomes above the reference point as gains. We suppose that the behavioral utility generally has a kink at the reference point and different functions below and above the reference point. We require only that the behavioral utility function is continuous and increasing in wealth and has at least the first and the second onesided derivatives at the reference point. Moreover, we also suppose that the decision maker generally distorts the objective probability distribution. Note that a utility function in EUT is a particular case of our generalized behavioral utility function, so that our generalized framework encompasses EUT as well. The first contribution of this paper is to provide an approximation analysis of the expected generalized behavioral utility function. Our analysis shows that the expected generalized behavioral utility function can be approximated by a function of mean and partial moments of distribution. The mean-partial moments utility generalizes not only the mean-variance utility of Tobin and Markowitz, but also the mean-semivariance utility mentioned by Markowitz (1959) and discussed further by many others. The mean-partial moments utility appears to have reasonable computational possibilities (in, for example, the optimal portfolio choice problem) as well as a great degree of flexibility in modelling different risk preferences of a decision maker. The second contribution of this paper is, in the spirit of Pratt (1964) and Arrow (1971), to derive an expression for a risk premium when risk is small. We show that the mean-partial moments utility allows for a much richer and detailed characterization of a risk premium. It is well known that for a decision maker with the mean-variance utility the only source of risk is variance. Moreover, the risk attitude of this decision maker is completely described by a single measure widely known as the Arrow-Pratt measure of risk aversion (which is, in fact, the aversion to uncertainty 1 ). In contrast, our analysis shows that a decision maker 1 Here by uncertainty we mean actually a deviation measure of uncertainty which assesses the dispersion of 3

4 with a generalized behavioral utility distinguishes between three sources of risk: expected loss, uncertainty in losses, and uncertainty in gains. Consequently, a decision maker in our framework exhibits three types of aversions: aversion to loss, aversion to uncertainty in gains, and aversion to uncertainty in losses. Loss aversion leads to different weights of losses and gains in the expression for the risk premium. The third contribution of this paper is to generalize the famous Arrow s solution (see Arrow (1971), page 102) to the optimal capital allocation problem of an investor. In this setting the investor s objective is to allocate optimally the wealth between a risk-free and a risky asset. It is widely known that a mean-variance utility maximizer will always want to allocate some wealth to the risky asset if the risk premium is non-zero, no matter how small it might be. By contrast, we show that an investor with the generalized behavioral utility function will want to allocate some wealth to the risky asset only when the perceived risk premium is sufficiently high 2 (how high depends on the level of loss aversion and the degree of probability distortion). Otherwise, if the perceived risk premium is small, the investor avoids the risky asset and invests all wealth in the risk-free asset. This result may help explain why many investors do not invest in equities 3. This result may also help explain the equity premium puzzle discovered by Mehra and Prescott (1985). One possible explanation is presented by Bernartzi and Thaler (1995) who showed that loss aversion can explain why stock returns are too high relative to bond returns. Another explanation suggested by our analysis is that some particular types of probability distortion (for example, pessimistic distortion) may cause the observed equity premium puzzle. Our fourth contribution is to derive an expression for the portfolio performance measure of an investor with the generalized behavioral utility. This measure generalizes the Sharpe ratio (see Sharpe (1966)) and the Sortino 4 ratio (see Sortino and Price (1994)). As compared to either the Sharpe or Sortino ratio where the investor s risk preferences seemingly disappear, the computation of the generalized performance measure usually requires the knowledge of distribution. 2 Previously, this was briefly mentioned by Segal and Spivak (1990). Using a simple one-period binomial model with no probability distortion, Gomes (2005) has also shown that loss aversion causes avoidance of risky asset when the risk premium is small. 3 See, for example, Agnew, Balduzzi, and Annika (2003) who report that about 48% of participants of retirement accounts do not invest in equities. This behavior clearly contradicts EUT. 4 The Sortino ratio is a reward-to-semivariability type ratio that has been used by many researchers. Some examples are Klemkosky (1973), Ang and Chua (1979), and Ziemba (2005). 4

5 the investor s risk preferences. This means that this performance measure is not unique for all investors, but rather an individual performance measure. The explanation for this is the fact that in our generalized framework an investor distinguishes between several sources of risk. Since each investor may exhibit different preferences to each source of risk, investors with different preferences might rank differently the same set of risky assets. The fifth contribution of this paper is to provide some new insights on risk measurement. In modern financial theory most often one uses variance as a risk measure. This is because in the EUT framework variance has the first-order impact on risk measurement, at least when the risk is small. We demonstrate that in many alternative theories variance is generally not a proper risk measure even if the risk is small. In these theories it is the decision maker s skewness preferences that have first-order impact on risk measurement. This is the consequence of the presence of loss aversion and the fact that the decision maker s degrees of risk aversions below and above the reference point might be substantially different. In the latter case if probability distributions are not symmetrical, then, depending on the signs and the values of skewness preferences, either the downside or the upside part of variance is a more proper risk measure than variance. The rest of the paper is organized as follows. In Section 2 we briefly review the justification of the mean-variance analysis and present the results we want to generalize. In Section 3 we present assumptions, definitions, and notation that we will use in our generalized framework. In Section 4 we perform the approximation analysis and generalize the mean-variance utility and the quadratic utility. In Section 5 we generalize the Arrow-Pratt risk premium. In Section 6 we analyze the optimal capital allocation problem. In Section 7 we derive the expression for a portfolio performance measure. In Section 8 we discuss briefly the impact of the decision maker s skewness preferences on risk measurement in our generalized framework. Section 9 concludes the paper. As a supporting document this paper has an extensive online supplement with lots of illustrations and additional information. 2 Expected Utility Theory and the Mean-Variance Analysis In this section we present a brief justification of the mean-variance analysis as well as some of its most important results. Throughout the paper we consider a decision maker with (random) 5

6 wealth W. In this section we assume that the decision maker has a von Neumann-Morgenstern utility function which is defined over wealth as a single function U(W ). The decision maker s objective is to maximize the expected utility of wealth, E[U(W )], where E[.] is the expectation operator. We suppose that the utility function U is increasing in wealth and is a differentiable function. Then, taking a Taylor series expansion around some (deterministic) W 0, the expected utility of the decision maker can be written as E[U(W )] = n=0 1 n! U (n) (W 0 )E[(W W 0 ) n ]. Our intension now is to keep the terms up to the second derivative of U and disregard 5 the terms with higher derivatives of U. This gives us E[U(W )] U(W 0 ) + U (1) (W 0 )E[(W W 0 )] U (2) (W 0 )E[(W W 0 ) 2 ]. Since a utility function is unique up to a positive linear transformation and U (1) > 0, then a convenient form of equivalent expected utility is [ ] U(W ) E[Ũ(W )] = E U(W0 ) U (1) = E[(W W 0 )] 1 (W 0 ) 2 γe[(w W 0) 2 ], (1) where γ = U (2) (W 0 ) U (1) (W 0 ) (2) is the Arrow-Pratt measure of risk aversion. The equivalent expected utility (1) can be interpreted as the mean-variance utility since the term E[(W W 0 ) 2 ] is proportional to variance. Observe that we can arrive at the same expression for the expected utility as (1) if we assume that the utility function of the decision maker is quadratic U(W ) = (W W 0 ) 1 2 γ(w W 0) 2. (3) 5 This can be justified when at least one of the following conditions is satisfied: the investor utility function is quadratic (see Tobin (1969)); the probability distribution is normal (see Tobin (1969)); the probability distribution belongs to the family of compact or small risk distributions (see Samuelson (1970)); the aggregate risk is small compared with the wealth (see Tsiang (1972)). 6

7 Consider now a risk averse decision maker with deterministic wealth W 0. According to Arrow (1971) (page 90) a risk averter is defined as one who, starting from a position of certainty, is unwilling to take a bet which is actuarially fair. More formally, if x is a fair gamble such that E[x] = 0, then E[U(W 0 + x)] < U(W 0 ). (4) Beginning from the landmark papers of Arrow and Pratt, it became a standard in economics to characterize the risk aversion in terms of either a certainty equivalent, C(x), or a risk premium, π(x), which are defined by the following indifference condition E[U(W 0 + x)] = U(W 0 + C(x)) = U(W 0 + E[x] π(x)). (5) This says that the decision maker is indifferent between receiving x and receiving a non-random amount of C(x) = E[x] π(x). Observe that for a fair gamble C(x) = π(x). If the risk of a fair gamble is small, then the decision maker s preferences can be approximated by quadratic utility. The use of quadratic utility (3) in the indifference condition (5) for W = W 0 + x combined with the assumption of risk aversion (4) gives the following expression 6 for the risk premium and certainty equivalent π(x) = C(x) = 1 γvar[x]. (6) 2 This is the famous result of Pratt (1964). Now consider the optimal capital allocation problem of an investor with quadratic utility function. The investor wants to allocate the wealth between a risk-free and a risky asset. The return on the risky asset over a small time interval t is x = µ t + σ t ε, (7) where µ and σ are, respectively, the mean and volatility of the risky asset return per unit of time, and ε is some (normalized) stochastic variable such that E[ε] = 0 and Var[ε] = 1. The 6 We need also to disregard the term with C 2 (x). This is valid since we assume that the risk is small. 7

8 return on the risk-free asset over the same time interval equals r = ρ t, (8) where ρ is the risk-free interest rate per unit of time. We assume that the risky asset can be either bought or sold short without any limitations and the risk-free rate of return is the same for both borrowing and lending. We further assume that the investor s initial wealth is W I and he invests a in the risky asset and, consequently, W I a in the risk-free asset. Thus, the investor s wealth after t is W = a(x r) + W I (1 + r). (9) The investor s objective is to choose a to maximize the expected utility E[U (W )] = max E[U(W )]. (10) a Observe that there is some ambiguity in the choice of the level of wealth W 0 around which we perform the Taylor series expansion. A reasonable choice is W 0 = W I (1 + r). With this choice W 0 does not depend on a and the resulting risk measure (for example, a E[(x r) 2 ]) exhibits the homogeneity property 7 in a. If t is rather small, then the risk is small and the use of quadratic utility is well justified. Consequently, using quadratic utility (3) in the investor s objective function (10) we arrive to the following maximization problem E[U (W )] = max a ae[x r] 1 2 γa2 E[(x r) 2 ]. The first-order condition of optimality of a gives a = 1 γ E[x r] E[(x r) 2 ], (11) which is the famous Arrow s solution (see Arrow (1971), page 102). Finally, using the expression 7 In the landmark paper of Artzner, Delbaen, Eber, and Heath (1999), the authors argue that a sensible risk measure should satisfy several properties. One of these properties is the homogeneity property. This says that if one invests the amount a in the risky asset, the measure that assesses the investment risk should be a homogeneous function in a. 8

9 for the optimal value of a, we obtain that the maximum expected utility of the investor is given by E[U (W )] = 1 E[x r] 2 2γ E[(x r) 2 ]. Note that for any investor the higher the value of utility irrespective of the value of γ. Thus, the value of E[x r] 2, the higher the maximum expected E[(x r) 2 ] E[x r] SR = E[(x r) 2 ], (12) which is nothing else than the absolute value of the Sharpe ratio 8 (see Sharpe (1966)), can be used as the ranking statistics in the performance measurement of risky assets. Note that the Sharpe ratio is usually presented based on the assumption that either E[x r] > 0 or short sales are restricted. Here we also allow for profitable short selling strategies as well. Therefore we compute the absolute value of the Sharpe ratio in order to properly measure the performance. 3 Assumptions, Definitions, and Notation The purpose of this paper is to generalize the mean-variance analysis and some of its important results presented in the preceding section. Before proceeding to the analysis, in this section we would like to present the assumptions, definitions, and notation. Assumption 1. We suppose that the decision maker s utility function is continuous and increasing in wealth. Assumption 2. We suppose that the decision maker s utility function generally has a kink at the reference point W 0 and U + (W ) if W W 0, U(W ) = U (W ) if W < W 0. This means that above the reference point the utility function is given by U +, whereas below 8 Observe that E[(x r) 2 ] = Var[x] + (E[x] r) 2 Var[x] for a small t. 9

10 the reference point the utility function is given by U. The continuity assumption gives U (W 0 ) = U + (W 0 ) = U(W 0 ). (13) Assumption 3. The decision maker regards the outcomes below the reference points as losses, while the outcomes above the reference point as gains. Consequently, we will refer to U as the utility function for losses, whereas to U + as the utility function for gains. Assumption 4. We suppose that the left and right derivatives of U at the reference point exist and finite. We denote the first-order left-sided derivative U (1) (W 0) = lim h 0 U (W 0 ) U (W 0 + h). h Similarly, we denote the first-order right-sided derivative 9 U (1) + (W 0) = lim h 0+ U + (W 0 + h) U + (W 0 ). h The higher-order one-sided derivatives of U at the reference point are denoted in the similar manner by U (2) (W 0), U (2) + (W 0), etc. Assumption 5. We suppose that the decision maker generally distorts the objective probability distribution. More formally, suppose that the cumulative objective probability distribution of a gamble X is given by F X. Then the expected payoff of the gamble in the objective world is E[X] = xdf X (x). Observe that the integral above is a Lebesgue-Stieltjes integral which is defined for both discrete and continuous distributions. We denote by Q X the cumulative distorted probability distribution function of X. Under the distortion of probabilities, the expected payoff of the gamble for the decision maker is given by E D [X] = xdq X (x). 9 To be more precise, we need to denote the left derivative as U (1) (1) (W0 ) and the right derivative as U + (W0+), but this would enlarge the notation. 10

11 Note that D does not denote some particular probability measure, but denotes some particular probability distortion rule. Since the probability distortion may depend either on the cumulative objective distribution function or on whether the outcomes of X are interpreted as losses or gains, we need to distinguish between the probability distortion of X and X. For this purpose we denote by Q X the cumulative distorted probability distribution function of the complementary 10 gamble X = X. Observe that generally E D [X] E[X], This says that the expected value of X under distortion of probabilities is generally different from the expected value of X under the objective probabilities. This holds true for all other moments of distribution of X. Moreover, E D [ X] E D [X]. This says that under the same rule of probability distortion the expected value of a complementary gamble X does not generally equal the expected value of a gamble X with the opposite sign. Some illustrations are provided in the online supplement. Assumption 6. If W = W 0 + X where X is some gamble, then we suppose that U(W 0 )dq X = U(W 0 ). (14) This is true when either the sum of all distorted probabilities is equal to 1 (that is, dq X = 1) or the utility function is zero at the reference point (that is, as in PT/CPT U(W 0 ) = 0). Definition 1 (The measure of loss aversion). The measure of loss aversion is given by λ = U (1) (W 0) U (1) + (W 0). (15) This measure of loss aversion was proposed by Bernartzi and Thaler (1995) and formalized by Köbberling and Wakker (2005). Observe that if the decision maker does not exhibit loss 10 By a complementary gamble X = X we mean a gamble which is obtained by changing the signs of the outcomes of gamble X. 11

12 aversion, then λ = 1. Loss aversion implies λ > 1. Conversely, loss seeking behavior implies λ < 1. Finally note that since the first-order derivatives of U are positive (follows from Assumption 1), the value of λ is also positive, that is, λ > 0. Definition 2 (Two measures of risk aversion). Recall the measure of risk aversion (2) that was introduced by Arrow and Pratt. Since the utility function of the decision maker generally has different functions for losses and gains, we introduce: the measure of risk aversion in the domain of gains γ + = U (2) + (W 0) U (1) + (W 0), and the measure of risk aversion in the domain of losses γ = U (2) (W 0) U (1) (W 0). If, for example, γ + > 0, then the utility function for gains is concave which means that the decision maker is risk averse in the domain of gains. By contrast, if γ + < 0, then the utility function for gains is convex which means that the decision maker is risk seeking in the domain of gains. Finally, if γ + = 0, then the decision maker is risk neutral in the domain of gains. Definition 3 (Lower and Upper Partial Moments). If the decision maker s random wealth is W and the reference point is W 0, a lower partial moment of order n under the distortion of probability is given by W0 LP Mn D (W, W 0 ) = ( 1) n (w W 0 ) n dq W (w). The coefficient ( 1) n is chosen to bring our definition of a lower partial moment in correspondence with the definition of Fishburn (1977). An upper partial moment of order n under the distortion of probability is given by UP M D n (W, W 0 ) = W 0 (w W 0 ) n dq W (w). The lower and upper partial moments of order n computed in the objective world are denoted by LP M n (W, W 0 ) and UP M n (W, W 0 ) respectively. See the online supplement that illustrates the computation of partial moments with and without the probability distortion. 12

13 Definition 4 (Performance Measure). By a performance measure we mean a score (number/value) attached to each financial asset. A performance measure is related to the level of expected utility provided by the asset. That is, the higher the performance measure of an asset, the higher level of expected utility the asset provides. 4 Mean-Partial Moments Utility and Generalized Quadratic Utility The purpose of this section is to generalize the mean-variance utility (1) and quadratic utility (3). We consider a decision maker with random wealth W and reference point W 0. The decision maker generally distorts the objective probability distribution of W such that the cumulative distorted probability distribution function is Q W. The decision maker s expected generalized behavioral utility is, therefore, given by E D [U(W )] = W0 U (w)dq W (w) + U + (w)dq W (w). W 0 We apply Taylor series expansions for U (w) and U + (w) around W 0 which yields ( W0 ) E D 1 [U(W )] = n! U (n) (W 0 )(w W 0 ) n dq W (w) + n=0 W 0 1 W0 = U(W 0 )dq W (w) + n! U (n) (W 0 ) (w W 0 ) n dq W (w) + = U(W 0 ) + n=1 n=1 1 n! U (n) (W 0 )( 1) n LP M D n (W, W 0 ) + n=1 ( n=0 ) 1 n! U (n) + (W 0 )(w W 0 ) n dq W (w) 1 n! U (n) + (W 0 ) (w W 0 ) n dq W (w) W 0 n=1 1 n! U (n) + (W 0 )UP M D n (W, W 0 ), (16) supposing that the Taylor series converge and the integrals exist (also recall Assumption 6). Theorem 1. If either utility functions U and U + are at most quadratic in wealth or the probability distribution of W belongs to the family of small risk distributions, then the equivalent expected utility can be written as the following mean-partial moments utility E D [Ũ(W )] = ED [(W W 0 )] (λ 1)LP M D 1 (W, W 0 ) 1 2 ( λγ LP M D 2 (W, W 0 ) + γ + UP M D 2 (W, W 0 ) ), (17) 13

14 which means that we can assume the following quadratic form for the utility function 11 U (W W 0 ) 1 2 U(W ) = γ +(W W 0 ) 2 if W W 0, λ ( (W W 0 ) 1 2 γ (W W 0 ) 2) if W < W 0. (18) Proof. If utility functions U and U + are at most quadratic, then U (n) (W 0) = 0 and U (n) + (W 0) = 0 for n 3. If the probability distribution belongs to the family of small risk distributions, then we can assume that all the terms in (16) with LP M D n (W, W 0 ) and UP M D n (W, W 0 ), n 3, are of smaller order than the second partial moments. To demonstrate this, suppose that W = W 0 + x and x = tε where ε is some random variable, and let t be sufficiently small. Hence, if we keep only the first and the second partial moments of distribution, the expected utility is E D [U(W )] = U(W 0 )+ 2 n=1 1 n! U (n) (W 0)( 1) n LP M D n (W, W 0 )+ 2 n=1 1 n! U (n) + (W 0)UP M D n (W, W 0 ). Since the utility function is unique up to a positive linear transformation and U (1) + (W 0) > 0, an equivalent expected utility can be given by [ ] E[Ũ(W )] = E U(W ) U(W 0 ) U (1) + (W 0) ( (2) U (W 0) U (1) (W 0) U (1) (W 0) U (1) = U (1) (W 0) U (1) + (W 0) LP M 1 D (W, W 0 ) + UP M1 D (W, W 0 ) (2) + (W ) 0) U (1) + (W 0) UP M 2 D (W, W 0 ). + (W 0) LP M D 2 (W, W 0 ) + U To arrive at (17) recall Definitions 1 and 2 and note that UP M D 1 (W, W 0) LP M D 1 (W, W 0) = E D [W W 0 ]. Corollary 2. In the EUT framework, the mean-partial moments utility (17) reduces to the mean-variance utility (1), and the generalized quadratic utility (18) reduces to the quadratic utility (3). Proof. For a von Neumann-Morgenstern utility function the left and right derivatives of U at point W 0 are equal. Hence, for a von Neumann-Morgenstern utility function λ = 1 and γ = γ + = γ. Finally note that LP M 2 (W, W 0 ) + UP M 2 (W, W 0 ) = E[(W W 0 ) 2 ]. 11 This utility function can be denoted as piecewise-quadratic utility. 14

15 Corollary 3. If λ = 1, γ + = 0, and there is no probability distortion, then the mean-partial moments utility (17) reduces to the mean-semivariance utility of Markowitz E[Ũ(W )] = E[(W W 0)] 1 2 γ LP M 2 (W, W 0 ). Remark 1. Observe that the mean-semivariance utility is a particular case of the mean-partial moments utility when the decision maker exhibits no loss aversion, risk aversion in the domain of losses, and risk neutrality in the domain of gains. Note that for a decision maker with the mean-variance utility there is only one source of risk, namely, the variance which measures the total uncertainty (or dispersion). In contrast, a decision maker with the mean-partial moments utility generally distinguishes between three sources of risk: the lower partial moment of order 1 which is related to the expected loss, the lower partial moment of order 2 which is related to the uncertainty in losses, and the upper partial moment of order 2 which is related to the uncertainty in gains. Note also that in the mean-partial moments utility the measure of risk aversion in the domain of losses is scaled up by the measure of loss aversion. This allows us to say that a decision maker with loss aversion puts more weight on the uncertainty in losses than on the uncertainty in gains. Observe that the generalized quadratic utility function (18) is very flexible with regard to the possibility of modelling different preferences of a decision maker. In this function λ controls the loss aversion, γ + controls the concavity/convexity of utility for gains, whereas γ controls the concavity/convexity of utility for losses. Some possible shapes of this utility function are presented in the online supplement. 5 Generalization of the Arrow-Pratt Risk Premium In this section we consider a risk averse decision maker equipped with the generalized behavioral utility function and who generally distorts the objective probability distribution. Recall that we say that the decision maker is risk averse if a fair (in the objective world) gamble decreases the utility of the decision maker. More formally, for a fair gamble x such that E[x] = 0 E D [U(W 0 + x)] < U(W 0 ). 15

16 We want to derive the expressions for the certainty equivalent, C D (x), and the risk premium, π D (x), which are now defined by the following indifference condition E D [U(W 0 + x)] = U ( W 0 + C D (x) ) = U ( W 0 + E D [x] π D (x) ). (19) Theorem 4. If either utility functions U and U + are at most quadratic in wealth or the risk of a fair gamble is small, then for a risk averse decision maker the certainty equivalent and the risk premium of the fair gamble are given by C D (x) = 1 λ ( UP M1 D (x, 0) 1 ) ( 2 γ +UP M2 D (x, 0) LP M1 D (x, 0) + 1 ) 2 γ LP M2 D (x, 0), (20) π D (x) = λ 1 λ ( E D [x] + LP M D 1 (x, 0) ) ( γ+ λ UP M D 2 (x, 0) + γ LP M D 2 (x, 0)). (21) Proof. If either utility functions U and U + are at most quadratic in wealth or the risk of a fair gamble is small, then, according to Theorem 1, the decision maker s preferences can be represented by the generalized quadratic utility (18). Consequently, the expected utility of W = W 0 + x is given by E D [U(W 0 + x)] = UP M1 D (x, 0) 1 ( 2 γ +UP M2 D (x, 0) λ LP M1 D (x, 0) + 1 ) 2 γ LP M2 D (x, 0). For a risk averse decision maker the certainty equivalent of a fair (in the objective world) gamble is negative. Consequently, (22) U ( W 0 + C D (x) ) ( = λ C D (x) 1 2 γ ( C D (x) ) ) 2. (23) Since the risk is small, ( C D (x) ) 2 C D (x) which means that the term with ( C D (x) ) 2 can be disregarded. Thus, the use of (22) and (23) in the indifference condition (19) gives the expression for the certainty equivalent (20). To derive the expression for the risk premium, we use π D (x) = E D [x] C D (x). Corollary 5. In the EUT framework, the risk premium (21) reduces to the Arrow-Pratt risk premium (6). 16

17 Proof. For a von Neumann-Morgenstern utility function λ = 1 and γ = γ + = γ. This gives π(x) = 1 2 (γlp M 2(x, 0) + γup M 2 (x, 0)) = 1 γvar[x]. (24) 2 Observe that since a decision maker with a von Neumann-Morgenstern utility exhibits no loss aversion, both (infinitesimal) losses and gains are treated similarly (as it is clearly seen from equation (24)). Consequently, the Arrow-Pratt measure of risk aversion is really a measure of aversion to variance, or a measure of aversion to uncertainty (or dispersion) of the probability distribution. In other words, when risk is small, a decision maker with a von Neumann-Morgenstern utility exhibits only aversion to uncertainty. In addition, the risk premium is fully characterized by a measure of uncertainty aversion and the variance, which is a measure of uncertainty. A utility function with loss aversion and different functions for losses and gains allows a much richer and detailed characterization of risk aversion. According to equation (20) a decision maker exhibits three different types of aversions: aversion to loss, aversion to uncertainty in gains, and aversion to uncertainty in losses. The loss is measured by the expected loss, and the uncertainties in gains and losses are measured by the second upper partial moment of x and the second lower partial moment of x respectively. Observe that losses and gains have different weights in the computation of the risk premium. In particular, for a loss averse decision maker, losses are λ times more important than gains. Finally, a brief comparative static analysis of the expressions for the certainty equivalent and the risk premium is presented by means of the following corollaries. Corollary 6. The certainty equivalent decreases as the decision maker s loss aversion increases. Conversely, the risk premium increases as the decision maker s loss aversion increases. Proof. The first-order derivatives of the certainty equivalent and risk premium with respect to λ C D (x) λ = πd (x) λ = 1 (UP λ 2 M1 D (x, 0) 1 ) 2 γ +UP M2 D (x, 0). Consider the sign of B = UP M D 1 (x, 0) 1 2 γ +UP M D 2 (x, 0). 17

18 Note that UP M D 1 (x, 0) > 0 and UP M D 2 (x, 0) > 0. Obviously, if γ + 0, then the sign of B is positive. However, even if γ + > 0 the sign of B is positive since in this case we need to impose the upper limit max(x) < 1 γ + outcomes of x. Therefore B = 1 γ + 0 to ensure that the utility function is increasing 12 for all (y 12 γ +y 2 ) dq x (y) > 0, since y 1 2 γ +y 2 > 0 for all y < 1 γ +. Finally we obtain C D (x) λ = πd (x) λ < 0. Corollary 7. The certainty equivalent decreases as the decision maker s risk aversion in the domain of gains increases. Conversely, the risk premium increases as the decision maker s risk aversion in the domain of gains increases. Proof. The first-order derivatives of the certainty equivalent and risk premium with respect to γ + since UP M D 2 (x, 0) > 0. C D (x) γ + = πd (x) γ + = 1 2λ UP M D 2 (x, 0) < 0, Corollary 8. The certainty equivalent decreases as the decision maker s risk aversion in the domain of losses increases. Conversely, the risk premium increases as the decision maker s risk aversion in the domain of losses increases. Proof. The first-order derivatives of the certainty equivalent and risk premium with respect to γ since LP M D 2 (x, 0) > 0. C D (x) γ = πd (x) γ = 1 2 LP M D 2 (x, 0) < 0, 12 To motivate for this, consider the generalized quadratic utility function (18). If γ + > 0, then the function U(W ) = (W W 0) 1 2 γ+(w W0)2 is increasing for 0 < W W 0 < 1 γ +. 18

19 6 Optimal Capital Allocation in the Generalized Framework The set up of the problem is the same as that described in Section 2. That is, we consider an investor who wants to allocate the wealth between a risk-free and a risky asset. The returns on the risky and the risk-free assets over a small time interval t are given by equations (7) and (8) respectively. The investor s initial wealth is W I and the investor s objective is to maximize the expected utility of his future wealth. Since the resulting expression for the investor s expected utility depends on whether the investor buys and holds or sells short the risky asset, we need to consider these two cases separately. In particular, if the investor uses the amount a 0 to buy the risky asset, his future wealth is given by equation (9). If the investor sells short the risky asset such that the proceedings are a 0, his future wealth is given by W = a(r x) + W I (1 + r). Note that in both cases the value of a is non-negative. This is necessary to be able to distinguish between the probability distortion of x r and r x. Before attacking the optimal capital allocation problem, we need to choose a suitable reference point W 0 to which gains and losses are compared. One possible reference point is the status quo, that is, the investor s initial wealth W I. Unfortunately, with this choice it is not possible to arrive at a closed-form solution for the optimal capital allocation problem unless W I = 0. However, according to Markowitz (1952), Kahneman and Tversky (1979), Loomes and Sugden (1986), and others, the investor s initial wealth does not need to be the reference point. Following Barberis, Huang, and Santos (2001) we assume that the reference point is W 0 = W I (1 + r). This is the investor s initial wealth scaled up by the risk-free rate. This choice of reference point is sensible due to many reasons: (1) This level of wealth serves as a benchmark wealth. The idea here is that the investor is likely to be disappointed if the risky asset provides a return below the risk-free rate of return; (2) With this reference point the performance measure does not depend on the investor s wealth (see the subsequent section). That is, the investor s utility of returns is independent of wealth; (3) De Giorgi, Hens, and Mayer (2006) show that with this choice of reference point a prospect theory investor satisfies 19

20 the mutual fund separation; (4) This choice is also justified if we require that all partial moments should exhibit the homogeneity property in a (for explanation, see footnote 7). For example, if the investor is equipped with the Fisburn s mean-lower partial moment of order 2 utility, then the risk of investing the amount a, as measured by downside deviation, equals a LP M2 D (x r, 0), which seems reasonable. Having decided on the investor s reference wealth, we are ready to state and prove the following theorem. Theorem 9. If either utility functions U and U + are at most quadratic in wealth or the investment risk is small, then the investor s optimal capital allocation problem has the following solution: If E D [x r] (λ 1)LP M1 D (x r, 0) > 0 (25) and λγ LP M D 2 (x r, 0) + γ + UP M D 2 (x r, 0) > 0, (26) then the optimal amount invested in the risky asset is given by a = E D [x r] (λ 1)LP M1 D (x r, 0) λγ LP M2 D(x r, 0) + γ +UP M2 D (27) (x r, 0), and the investor s maximum equivalent expected utility from the buy-and-hold strategy is given by EBH[U D (W )] = 1 ( E D [x r] (λ 1)LP M1 D(x r, 0)) 2 2 λγ LP M2 D(x r, 0) + γ +UP M2 D (28) (x r, 0). If E D [r x] (λ 1)LP M D 1 (r x, 0) > 0 (29) and λγ LP M D 2 (r x, 0) + γ + UP M D 2 (r x, 0) > 0, (30) then the optimal amount of the risky asset that should be sold short is given by a = E D [r x] (λ 1)LP M1 D (r x, 0) λγ LP M2 D(r x, 0) + γ +UP M2 D (31) (r x, 0), 20

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