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. We also suppose that the decision maker generally distorts the objective probabilities. 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 variance utility of Tobin and Markowitz, but also the mean-semivariance 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) 38 14 10 39,Valeri.Zakamouline@uia.no
†University of Agder, Faculty of Economics, Service Box 422, 4604 Kristiansand, Norway, Tel.: (+47) 38 14 15 31,Steen.Koekebakker@uia.no
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 ana-lyzed 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. Mea-surement 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 Morgen-stern, 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 Kah-neman and Tversky (1979) and Tversky and KahKah-neman (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
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 one-sided 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 uncertainty1). In contrast, our analysis shows that a decision maker
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 high2(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 equities3. 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 Sortino4 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.
2Previously, 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.
3See, 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.
4The Sortino ratio is a reward-to-semivariability type ratio that has been used by many researchers. Some
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)wealthW. In this section we assume that the decision maker has a von Neumann-Morgenstern utility function which is defined over wealth as a single functionU(W). The decision maker’s objective is to maximize the expected utility of wealth,E[U(W)], whereE[.] is the expectation operator.
We suppose that the utility functionU is increasing in wealth and is a differentiable func-tion. Then, taking a Taylor series expansion around some (deterministic) W0, the expected utility of the decision maker can be written as
E[U(W)] = ∞ X n=0 1 n!U (n)(W 0)E[(W −W0)n].
Our intension now is to keep the terms up to the second derivative ofU and disregard5 the terms with higher derivatives ofU. This gives us
E[U(W)]≈U(W0) +U(1)(W0)E[(W −W0)] +12U(2)(W0)E[(W −W0)2].
Since a utility function is unique up to a positive linear transformation andU(1) >0, then a convenient form of equivalent expected utility is
E[Ue(W)] =E · U(W)−U(W0) U(1)(W0) ¸ =E[(W −W0)]− 12γE[(W −W0)2], (1) where γ =−U(2)(W0) U(1)(W0) (2)
is the Arrow-Pratt measure of risk aversion. The equivalent expected utility (1) can be inter-preted as the mean-variance utility since the termE[(W −W0)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 −W0)−21γ(W −W0)2. (3)
5This 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 distri-bution belongs to the family of “compact” or “small risk” distridistri-butions (see Samuelson (1970)); the aggregate risk is small compared with the wealth (see Tsiang (1972)).
Consider now a risk averse decision maker with deterministic wealth W0. 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 thatE[x] = 0, then
E[U(W0+x)]< U(W0). (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(W0+x)] =U(W0+C(x)) =U(W0+E[x]−π(x)). (5)
This says that the decision maker is indifferent between receivingxand receiving a non-random amount ofC(x) =E[x]−π(x). Observe that for a fair gambleC(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 = W0 +x combined with the assumption of risk aversion (4) gives the following expression6 for the risk premium and certainty equivalent
π(x) =−C(x) = 1
2γVar[x]. (6)
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)
whereµ and σ are, respectively, the mean and volatility of the risky asset return per unit of time, andε is some (normalized) stochastic variable such thatE[ε] = 0 and Var[ε] = 1. The
return on the risk-free asset over the same time interval equals
r =ρ∆t, (8)
whereρ 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 WI
and he investsain the risky asset and, consequently, WI−ain the risk-free asset. Thus, the
investor’s wealth after ∆tis
W =a(x−r) +WI(1 +r). (9) The investor’s objective is to chooseato maximize the expected utility
E[U∗(W)] = max
a E[U(W)]. (10)
Observe that there is some ambiguity in the choice of the level of wealth W0 around which we perform the Taylor series expansion. A reasonable choice is W0 =WI(1 +r). With this
choiceW0 does not depend on aand the resulting risk measure (for example, a
p
E[(x−r)2]) exhibits the homogeneity property7 ina.
If ∆tis 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 2γa
2E[(x−r)2].
The first-order condition of optimality ofagives
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
7In 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 amountain the risky asset, the measure that assesses the investment risk should be a homogeneous function ina.
for the optimal value ofa, we obtain that the maximum expected utility of the investor is given by E[U∗(W)] = 1 2γ E[x−r]2 E[(x−r)2].
Note that for any investor the higher the value of EE[([xx−−rr])22], the higher the maximum expected
utility irrespective of the value ofγ. Thus, the value of
SR= ¯ ¯ ¯ ¯ ¯ E[x−r] p E[(x−r)2] ¯ ¯ ¯ ¯ ¯, (12)
which is nothing else than the absolute value of the Sharpe ratio8 (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 pointW0 and
U(W) = U+(W) if W ≥W0, U−(W) if W < W0.
This means that above the reference point the utility function is given byU+, whereas below
the reference point the utility function is given byU−. The continuity assumption gives
U−(W0) =U+(W0) =U(W0). (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 toU+ 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)(W0) = lim
h→0−
U−(W0)−U−(W0+h)
h .
Similarly, we denote the first-order right-sided derivative9
U+(1)(W0) = lim
h→0+
U+(W0+h)−U+(W0)
h .
The higher-order one-sided derivatives ofU at the reference point are denoted in the similar manner byU−(2)(W0), U+(2)(W0), etc.
Assumption 5. We suppose that the decision maker generally distorts the objective probabil-ity distribution. More formally, suppose that the cumulativeobjectiveprobability distribution of a gambleX is given byFX. Then the expected payoff of the gamble in the objective world
is
E[X] =
Z ∞
−∞
xdFX(x).
Observe that the integral above is a Lebesgue-Stieltjes integral which is defined for both dis-crete and continuous distributions. We denote by QX 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
ED[X] =
Z ∞
−∞
xdQX(x).
9To be more precise, we need to denote the left derivative asU(1)
− (W0−) and the right derivative asU
(1) + (W0+),
Note thatDdoes not denote some particular probability measure, butdenotes 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 QX the cumulative distorted probability distribution function of the complementary10 gamble X=−X. Observe that generally
ED[X]6=E[X],
This says that the expected value ofX under distortion of probabilities is generally different from the expected value ofX under the objective probabilities. This holds true for all other moments of distribution ofX. Moreover,
ED[−X]6=−ED[X].
This says that under the same rule of probability distortion the expected value of a complemen-tary gamble−X does not generally equal the expected value of a gambleX with the opposite sign. Some illustrations are provided in the online supplement.
Assumption 6. IfW =W0+X whereX is some gamble, then we suppose that
Z ∞
−∞
U(W0)dQX =U(W0). (14)
This is true when either the sum of all distorted probabilities is equal to 1 (that is,R−∞∞ dQX =
1) or the utility function is zero at the reference point (that is, as in PT/CPTU(W0) = 0). Definition 1 (The measure of loss aversion). The measure of loss aversion is given by
λ= U (1) − (W0)
U+(1)(W0)
. (15)
This measure of loss aversion was proposed by Bernartzi and Thaler (1995) and formalized by K¨obberling and Wakker (2005). Observe that if the decision maker does not exhibit loss
10By a complementary gamble X =−X we mean a gamble which is obtained by changing the signs of the
aversion, thenλ = 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 ofλis 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)(W0)
U+(1)(W0)
,
andthe measure of risk aversion in the domain of losses
γ− =−
U−(2)(W0)
U−(1)(W0)
.
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, ifγ+ <0, then the utility function for gains is convex which means that the decision maker is risk seeking in the domain of gains. Finally, ifγ+= 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 W0, a lower partial moment of order n under the distortion of probability is given by
LP MnD(W, W0) = (−1)n
Z W0
−∞
(w−W0)ndQW(w).
The coefficient (−1)n is chosen to bring our definition of a lower partial moment in
correspon-dence with the definition of Fishburn (1977). An upper partial moment of ordern under the distortion of probability is given by
U P MnD(W, W0) =
Z ∞
W0
(w−W0)ndQW(w).
The lower and upper partial moments of orderncomputed in the objective world are denoted byLP Mn(W, W0) andU P Mn(W, W0) respectively. See the online supplement that illustrates the computation of partial moments with and without the probability distortion.
Definition 4 (Performance Measure). By a performance measure we mean a score (num-ber/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 wealthW and reference pointW0. The decision maker generally distorts the objective probability distribution ofW such that the cumulative distorted probability distribution function isQW. The decision maker’s expected generalized
behavioral utility is, therefore, given by
ED[U(W)] = Z W0 −∞ U−(w)dQW(w) + Z ∞ W0 U+(w)dQW(w).
We apply Taylor series expansions forU−(w) and U+(w) aroundW0 which yields
ED[U(W)] = Z W0 −∞ à ∞ X n=0 1 n!U (n) − (W0)(w−W0)n ! dQW(w) + Z ∞ W0 Ã∞ X n=0 1 n!U (n) + (W0)(w−W0)n ! dQW(w) = Z ∞ −∞ U(W0)dQW(w) + ∞ X n=1 1 n!U (n) − (W0) Z W0 −∞ (w−W0)ndQW(w) + ∞ X n=1 1 n!U (n) + (W0) Z ∞ W0 (w−W0)ndQW(w) =U(W0) + ∞ X n=1 1 n!U (n) − (W0)(−1)nLP MnD(W, W0) + ∞ X n=1 1 n!U (n) + (W0)U P MnD(W, W0), (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 ofW belongs to the family of small risk distributions, then the equivalent expected utility can be written as the following mean-partial moments utility
ED[Ue(W)] =ED[(W −W0)]−(λ−1)LP M1D(W, W0) −1 2 ¡ λγ−LP M2D(W, W0) +γ+U P M2D(W, W0)¢, (17)
which means that we can assume the following quadratic form for the utility function11 U U(W) = (W −W0)−12γ+(W −W0)2 if W ≥W0, λ¡(W −W0)− 1 2γ−(W −W0)2 ¢ if W < W0. (18)
Proof. If utility functions U− and U+ are at most quadratic, then U−(n)(W0) = 0 and
U+(n)(W0) = 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 MnD(W, W0) and
U P MnD(W, W0),n≥3, are of smaller order than the second partial moments. To demonstrate this, suppose thatW =W0+xandx=tεwhereεis some random variable, and lettbe suffi-ciently small. Hence, if we keep only the first and the second partial moments of distribution, the expected utility is
ED[U(W)] =U(W0)+ 2 X n=1 1 n!U (n) − (W0)(−1)nLP MnD(W, W0)+ 2 X n=1 1 n!U (n) + (W0)U P MnD(W, W0).
Since the utility function is unique up to a positive linear transformation and U+(1)(W0) >0, an equivalent expected utility can be given by
E[Ue(W)] =E " U(W)−U(W0) U+(1)(W0) # =−U (1) − (W0) U+(1)(W0)LP M D 1 (W, W0) +U P M1D(W, W0) +1 2 Ã U−(2)(W0) U−(1)(W0) U−(1)(W0) U+(1)(W0) LP M2D(W, W0) + U+(2)(W0) U+(1)(W0) U P M2D(W, W0) ! .
To arrive at (17) recall Definitions 1 and 2 and note thatU P M1D(W, W0)−LP M1D(W, W0) =
ED[W −W0]. 2
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 W0 are equal. Hence, for a von Neumann-Morgenstern utility function λ = 1 and
γ−=γ+=γ. Finally note thatLP M2(W, W0) +U P M2(W, W0) =E[(W −W0)2]. 2
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[Ue(W)] =E[(W −W0)]−1
2γ−LP M2(W, W0).
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 functionλcontrols the loss aversion,γ+ controls the concavity/convexity of utility for gains, whereasγ− 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 arisk aversedecision maker equipped with the generalized behav-ioral 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
We want to derive the expressions for the certainty equivalent,CD(x), and the risk premium,
πD(x), which are now defined by the following indifference condition
ED[U(W0+x)] =U¡W0+CD(x)¢=U¡W0+ED[x]−πD(x)¢. (19)
Theorem 4. If either utility functionsU− andU+ 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
CD(x) = 1 λ µ U P M1D(x,0)−1 2γ+U P M D 2 (x,0) ¶ − µ LP M1D(x,0) +1 2γ−LP M D 2 (x,0) ¶ , (20) πD(x) = λ−1 λ ¡ ED[x] +LP M1D(x,0)¢+ 1 2 ³γ + λ U P M D 2 (x,0) +γ−LP M2D(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 =W0+x is given by ED[U(W0+x)] =U P M1D(x,0)− 1 2γ+U P M D 2 (x,0)−λ µ LP M1D(x,0) + 1 2γ−LP M D 2 (x,0) ¶ . (22) For a risk averse decision maker the certainty equivalent of a fair (in the objective world) gamble is negative. Consequently,
U¡W0+CD(x) ¢ =λ µ CD(x)− 1 2γ− ¡ CD(x)¢2 ¶ . (23)
Since the risk is small, ¡CD(x)¢2 ¿ CD(x) which means that the term with ¡CD(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 useπD(x) =ED[x]−CD(x). 2
Corollary 5. In the EUT framework, the risk premium (21) reduces to the Arrow-Pratt risk premium (6).
Proof. For a von Neumann-Morgenstern utility functionλ= 1 andγ−=γ+=γ. This gives π(x) = 1 2(γLP M2(x,0) +γU P M2(x,0)) = 1 2γVar[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 ofx and the second lower partial moment ofxrespectively. 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 in-creases. Conversely, the risk premium increases as the decision maker’s loss aversion inin-creases.
Proof. The first-order derivatives of the certainty equivalent and risk premium with respect toλ ∂CD(x) ∂λ =− ∂πD(x) ∂λ =− 1 λ2 µ U P M1D(x,0)−1 2γ+U P M D 2 (x,0) ¶ .
Consider the sign of
B=U P M1D(x,0)−1
2γ+U P M
D
Note that U P MD
1 (x,0)>0 and U P M2D(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)< γ+1 to ensure that the utility function is increasing12 for all outcomes ofx. Therefore B = Z 1 γ+ 0 µ y−1 2γ+y 2 ¶ dQx(y)>0,
sincey−12γ+y2 >0 for ally < γ+1 . Finally we obtain
∂CD(x)
∂λ =−
∂πD(x)
∂λ <0.
2
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 toγ+ ∂CD(x) ∂γ+ =− ∂πD(x) ∂γ+ =− 1 2λU P M D 2 (x,0)<0, sinceU P M2D(x,0)>0. 2
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 toγ− ∂CD(x) ∂γ− =− ∂πD(x) ∂γ− =− 1 2LP M D 2 (x,0)<0, sinceLP MD 2 (x,0)>0. 2
12To motivate for this, consider the generalized quadratic utility function (18). If γ+>0, then the function U(W) = (W−W0)−1
2γ+(W−W0)
2is increasing for 0< W−W0< 1
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 ∆tare given by equations (7) and (8) respectively. The investor’s initial wealth isWI and the investor’s objective is to maximizethe 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 area≥0, his future wealth is given by
W =a(r−x) +WI(1 +r).
Note that in both cases the value ofais non-negative. This is necessary to be able to distinguish between the probability distortion ofx−r andr−x.
Before attacking the optimal capital allocation problem, we need to choose a suitable reference pointW0 to which gains and losses are compared. One possible reference point is the “status quo”, that is, the investor’s initial wealthWI. Unfortunately, with this choice it is not
possible to arrive at a closed-form solution for the optimal capital allocation problem unless
WI = 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
W0 = WI(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
the mutual fund separation; (4) This choice is also justified if we require that all partial moments should exhibit the homogeneity property ina (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 amounta, as measured by downside deviation, equals
a
q
LP MD
2 (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
ED[x−r]−(λ−1)LP M1D(x−r,0)>0 (25)
and
λγ−LP M2D(x−r,0) +γ+U P M2D(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 MD
1 (x−r,0)
λγ−LP M2D(x−r,0) +γ+U P M2D(x−r,0)
, (27)
and the investor’s maximum equivalent expected utility from the buy-and-hold strategy is given by EBHD [U∗(W)] = 1 2 ¡ ED[x−r]−(λ−1)LP MD 1 (x−r,0) ¢2 λγ−LP M2D(x−r,0) +γ+U P M2D(x−r,0) . (28) If ED[r−x]−(λ−1)LP M1D(r−x,0)>0 (29) and λγ−LP M2D(r−x,0) +γ+U P M2D(r−x,0)>0, (30)
then the optimal amount of the risky asset that should be sold short is given by
a= ED[r−x]−(λ−1)LP M1D(r−x,0)
λγ−LP M2D(r−x,0) +γ+U P M2D(r−x,0)
and the investor’s maximum equivalent expected utility from the short selling strategy is given by ESSD [U∗(W)] = 1 2 ¡ ED[r−x]−(λ−1)LP M1D(r−x,0)¢2 λγ−LP M2D(r−x,0) +γ+U P M2D(r−x,0) . (32)
If all the conditions (25), (26), (29), and (30) are satisfied, then the maximization problem has two local maxima. In this case the optimal policy is the one which gives the highest expected utility (either buy-and-hold or sell short).
If neither (25) nor (29) is satisfied, then the investor’s optimal policy is to avoid the risky asset and invest all in the risk-free asset.
Proof. If either utility functionsU−andU+are at most quadratic in wealth or the investment risk is small, then, according to Theorem 1, the investor’s preferences can be represented by the generalized quadratic utility (18).
First, we consider the buy-and-hold strategy. In this case the investor’s expected utility is given by EBHD [U(W)] =a¡ED[x−r]−(λ−1)LP M1D(x−r,0)¢ −1 2a 2¡λγ −LP M2D(x−r,0) +γ+U P M2D(x−r,0) ¢ , (33)
where the expectation and the partial moments are computed using the cumulative distorted probability distribution functionQx−r. Observe that the investor’s expected utility (33) is a
quadratic function ina. To guarantee the existence of a local maximum (interior solution), the investor’s expected utility should be concave ina, which means that condition (26) should be satisfied. The first-order condition of optimality ofagives the solution (27). Since we assume thata≥0, the solution for the optimala is a valid solution only if condition (25) is satisfied. If not, the optimal solution isa= 0. Finally, inserting expression (27) for the optimal value of
ainto (33), we obtain the solution for the investor’s maximum expected utility (28).
Second, we consider the short selling strategy. In this case the investor’s expected utility is given by ESSD [U(W)] =a¡ED[r−x]−(λ−1)LP M1D(r−x,0)¢ −1 2a 2¡λγ −LP M2D(r−x,0) +γ+U P M2D(r−x,0) ¢ , (34)
where the expectation and the partial moments are computed using the cumulative distorted probability distribution functionQr−x. Observe again that the investor’s expected utility (34)
is a quadratic function in a. To guarantee the existence of a local maximum, the investor’s expected utility should be concave in a, which means that condition (30) should be satisfied. The first-order condition of optimality ofagives the solution (31). Since we assume thata≥0, the solution for the optimalais a valid solution only if condition (29) is satisfied. Otherwise, the solution isa= 0. Finally, inserting expression (31) for the optimal value ofainto (34), we obtain the solution for the investor’s maximum expected utility (32). 2
Remark 2. Observe that if, for example, condition (25) is satisfied whereas condition (26) is not, then the highera the higher the investor’s expected utility. In other words, in this case the investor is willing to borrow an infinite amount at the risk-free interest rate to invest in the risky asset. In this case the optimal capital allocation problem does not have a solution. For example, conditions (26) and (30) are violated if the investor appreciates uncertainty in losses,γ−<0, but is neutral to uncertainty in gains,γ+= 0. The other example when these conditions are violated occurs when γ− = γ+ = 0, that is, the investor is neutral to both uncertainties. In the latter case the investor’s utility can be represented by so-called “bilinear” utility . For bilinear utility the solution of the optimal capital allocation problem generally does not exist. This was noted by Sharpe (1998).
Corollary 10. In the EUT framework, the solution for the optimal amount that should be invested in the risky asset (or sold short) reduces to the Arrow’s solution given by (11).
Proof. This follows from the fact that for a von Neumann-Morgenstern utility functionλ= 1 andγ−=γ+=γ. 2
Remark 3. Observe from Arrow’s solution (11) that in the EUT framework it is always optimal to undertake a risky investment whenE[x]6=r. If E[x]> r, it is optimal for the investor to buy some amount of the risky asset, whereas ifE[x]< r, it is optimal for the investor to sell short some amount of the risky asset. By contrast, in our generalized framework the investor avoids the risky asset if
In particular, the investor does not invest in the risky asset if the perceived risk premium is not high enough. In other words, the perceived risk premium provided by the risky asset must be rather high to induce the investor to undertake a risky investment. Note that the investor might avoid the risky asset due to loss aversion and/or some particular probability distortion. For example, the higher the investor’s loss aversion, the higher must be the risk premium in order to induce the investor to undertake a risky investment. It is easy to show that without any probability distortion aloss averse investor does not investsif
1
λLP M1(x, r)< U P M1(x, r)< λLP M1(x, r).
However, some particular distortion of probabilities might cause the investor’s avoidance of the risky asset even in case of no loss aversion. In this case we must have
ED[x−r]<0 and ED[r−x]<0.
This may occur with a pessimistic probability distortion. In the online supplement we provide an example which demonstrates how a probability distortion might cause either avoidance of the risky asset or the existence of two local maxima in the optimal capital allocation problem. Observe that this result may help explain why many investors do not invest in equities. 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.
Next we present a brief comparative static analysis of expressions (27) and (31) for the optimal amount invested in the risky asset,a.
Corollary 11. The optimal amount that should be invested in the risky asset (or sold short) decreases as the investor’s risk aversion in the domain of gains increases.
derivative ofawith respect to γ+ is ∂a ∂γ+ =− ¡ ED[x−r]−(λ−1)LP MD 1 (x−r,0) ¢ U P MD 2 (r−x,0) ¡ λγ−LP M2D(r−x,0) +γ+U P M2D(r−x,0) ¢2 <0.
If it is optimal for the investor to sell short the risky asset, then the first-order derivative ofa
with respect toγ+ is ∂a ∂γ+ =− ¡ ED[r−x]−(λ−1)LP MD 1 (r−x,0) ¢ U P MD 2 (r−x,0) ¡ λγ−LP MD 2 (r−x,0) +γ+U P M2D(r−x,0) ¢2 <0. 2
Corollary 12. The optimal amount that should be invested in the risky asset (or sold short) decreases as the investor’s risk aversion in the domain of losses increases.
Proof. If it is optimal for the investor to buy-and-hold the risky asset, then the first-order derivative ofawith respect to γ− is
∂a ∂γ− =− λ¡ED[x−r]−(λ−1)LP M1D(x−r,0)¢LP M2D(r−x,0) ¡ λγ−LP M2D(r−x,0) +γ+U P M2D(r−x,0) ¢2 <0.
If it is optimal for the investor to sell short the risky asset, then the first-order derivative ofa
with respect toγ− is ∂a ∂γ− =− λ¡ED[r−x]−(λ−1)LP MD 1 (r−x,0) ¢ LP MD 2 (r−x,0) ¡ λγ−LP M2D(r−x,0) +γ+U P M2D(r−x,0) ¢2 <0. 2
When it comes to the dependence of the optimal amount that should be invested in the risky asset (or sold short) on the loss aversion parameterλ, the computation of the first-order derivative ofa, given by (27), with respect to λgives
∂a ∂λ =− A×LP MD 1 (x−r,0) +γ−B×LP M2D(r−x,0) ¡ λγ−LP M2D(r−x,0) +γ+U P M2D(r−x,0) ¢2 , where A=λγ−LP M2D(r−x,0) +γ+U P M2D(r−x,0), B =ED[x−r]−(λ−1)LP M1D(x−r,0).
Since A > 0 (due to (26)) and B > 0 (due to (25)), the sign of ∂a∂λ is obviously negative if
γ− ≥ 0. However, if γ− < 0, then the sign of ∂a∂λ might be positive. The latter means that when the investor appreciates uncertainty in losses, then an increase in loss aversion might increase the optimal amount invested in the risky asset. This is obviously counter-intuitive. We believe that the explanation of this paradox lies in the fact that the general quadratic utility function (18) is not always increasing for all values ofW −W0. This might result in a number of paradoxes similar to those for the standard quadratic utility function. Note that if
γ− <0, then by condition (26) we must have γ+ >0. In this case the generalized quadratic utility is increasing only in the range γ1
− < W −W0 < 1
γ+. For example, ifW −W0 decreases
below γ1
−, then the investor’s utility begins to increase which is not sensible. Generally, when the generalized quadratic utility function is not increasing for all W −W0, we can stumble upon some paradoxes. Some of these paradoxes will be due to the violation of the first-order stochastic dominance principle.
7
Performance Measurement in the Generalized Framework
In this section we discuss performance measurement in our generalized framework. First of all, note the following property of a performance measure:Proposition 1. A positive linear transformation of a performance measure produces an equiv-alent performance measure.
Proof. Suppose thatP MAis the performance measure of asset A andP MBis the performance measure of asset B. Define performance measuresP M0
A=cP MA+dand P MB0 =cP MB+d
for any realcand dsuch thatc >0. ThenP M0
A andP MB0 are equivalent to P MA and P MB
in the sense that they produce equal ranking of the assets. That is, ifP MA> P MB, then also
P M0
A> P MB0 . 2
Theorem 13. If conditions (26) and (30) in Theorem 9 are satisfied, then the performance measure of an asset might by given by
where P MBH = E D[x−r]−(λ−1)LP MD 1 (x−r,0) q λγ−LP M2D(x−r,0) +γ+U P M2D(x−r,0)
is the performance measure associated with the buy-and-hold strategy, and
P MSS = q ED[r−x]−(λ−1)LP M1D(r−x,0)
λγ−LP M2D(r−x,0) +γ+U P M2D(r−x,0)
is the performance measure associated with the short selling strategy.
Proof. According to Definition 4, a performance measure of an asset is related to the level of the investor’s expected utility associated with the investment in this asset. According to Theorem 9, if it is optimal to buy-and-hold the risky asset, then the investor’s maximum expected utility isEBHD [U∗(W)] = 12P MBH2 such that P MBH >0 (observe that numerator in
P MBH is positive due to (25)). In this case P MBH > P MSS because: if there exists a local
maximum for the short selling strategy as well, then ED
BH[U∗(W)]> ESSD [U∗(W)] = 12P MSS2
(since we assume that the buy-and-hold strategy gives higher expected utility) andP MSS >0; if there is no local maximum for the short selling strategy, thenP MSS ≤0.
Similarly, if it is optimal to sell short the risky asset, then the investor’s maximum ex-pected utility isED
SS[U∗(W)] = 12P MSS2 such that P MSS >0. In this caseP MSS > P MBH
because: if there exists a local maximum for the buy-and-hold strategy, then ED
SS[U∗(W)]>
ED
BH[U∗(W)] = 21P MBH2 and P MBH >0; if there exists no local maximum for the
buy-and-hold strategy, thenP MBH ≤0.
Finally, if it is optimal to avoid the risky asset, then the investor’s maximum expected utility is zero. In this caseP MBH ≤0 andP MSS ≤0. 2
Corollary 14. In the EUT framework, the performance measure (35) reduces to a performance measure that is equivalent to the Sharpe ratio (12).
Proof. For a von Neumann-Morgenstern utility function λ = 1 and γ− = γ+ = γ. The conditions (26) and (30) imply thatγ >0. This gives
P M = √1 γ ¯ ¯ ¯ ¯ ¯ E[x−r] p E[(x−r)2] ¯ ¯ ¯ ¯ ¯.
per-formance measure andγ >0, the performance measure P M is equivalent to the Sharpe ratio (12). 2
Observe that the computation of the Sharpe ratio does not require the knowledge of the investor’s coefficient of absolute risk aversion γ. By contrast, to compute the performance measure P M one generally needs to define the values of λ, γ−, and γ+ (and, possibly, the rule of distortion of the objective probability distribution). This means thatthe performance measure P M is not unique for all investors, but rather an individual performance measure. That is, investors with different preferences (different degrees of loss aversion and risk aversions in the domains of losses and gains) might rank differently the same set of risky assets.
Corollary 15. To compute a performance measure that is equivalent to P M given by (35), one needs to define maximum two parameters that describe the investor’s preferences.
Proof. We consider only the case when it is optimal to buy-and-hold the risky asset. In this case the performance measure is given by
P MBH = E D[x−r]−(λ−1)LP MD 1 (x−r,0) q λγ−LP M2D(x−r,0) +γ+U P M2D(x−r,0) .
Observe now that ifγ+ >0, then an equivalent performance measure is given by
P MBH0 = qED[x−r]−(λ−1)LP M1D(x−r,0)
λθLP MD
2 (x−r,0) +U P M2D(x−r,0)
, (36)
where θ = γ−
γ+. Note that to compute P MBH0 we need to define only two parameters that
describe the investor’s preferences: λ and θ, where the latter is the relation between the investor’s risk aversions in the domain of losses and the domain of gains. Similarly, ifγ−>0, then an equivalent performance measure is given by
P MBH0 = qED[x−r]−(λ−1)LP M1D(x−r,0)
λLP MD
2 (x−r,0) +θU P M2D(x−r,0)
, (37)
where θ = γ+γ
performance measure is given by
P MBH0 = ED[x−rq]−(λ−1)LP M1D(x−r,0)
U P MD
2 (x−r,0)
.
Ifγ+= 0 (in this caseγ−>0 due to condition (26)), then an equivalent performance measure is given by
P MBH0 = ED[x−rq]−(λ−1)LP M1D(x−r,0)
LP MD
2 (x−r,0)
. (38)
Observe that in the last two cases we need to define only one parameter,λ. 2
It is widely known that the Sharpe ratio is a meaningful measure of portfolio performance when the risk can be adequately measured by variance, for example, when returns are normally distributed. The literature on performance evaluation that tries to take into account non-normality of return distributions is a vast one. There have been proposed dozens of alternative performance measures. Unfortunately, most of these alternative performance measures lack a solid theoretical underpinning. However, the following performance measure can be justified by our analysis:
Corollary 16. If λ = 1, γ+ = 0, and the investor does not distort the objective probability distribution, then the performance measure P M given by (35) is equivalent to the following Sortino13 ratio (see Sortino and Price (1994))
DSR = pE[x−r]
LP M2(x, r)
.
Proof. Start with the equivalent performance measure (38), remove the probability distortion and setλ= 1. 2
8
Impact of the Decision Maker’s Skewness Preferences on
Risk Measurement
At the end of this paper we would like to discuss briefly the impact of the decision maker’s skewness preferences on risk measurement in our generalized framework. Recall that according
13To the best of the authors’ knowledge, Ang and Chua (1979) were the first to employ this particular form
to the mean-partial moments utility (17) the decision maker distinguishes between three sources of risk. Suppose that W = W0+x and x is a pure risk such that E[x] = 0 and there is no probability distortion. Then the total risk ofx is given by
Risk[x] =π(x) = λ−1 λ LP M1(x,0) + 1 2 ³ γ−LP M2(x,0) +γλ+U P M2(x,0) ´ . (39)
Observe that if γ− = γ+ = γ and λ = 1 as in EUT, then LP M2(x,0) and U P M2(x,0) have equal weights in the computation of risk which means that the risk is proportional to variance, in particular, Risk[x] = 12γVar[x]. That is, within EUT it is variance that has a first-order impact on risk measurement. By contrast, in many of the alternative theories generally γ− 6=γ+ and λ > 1 so that LP M2(x,0) and U P M2(x,0) have different weights in the computation of risk. However, if the probability distribution of x is symmetrical, then
LP M2(x,0) =U P M2(x,0) = 12Var[x] and the risk is again proportional to variance,
Risk[x] = λ−1 λ LP M1(x,0) + 1 4 ³ γ−+ γ+ λ ´ Var[x].
If, on the other hand, the probability distribution ofxisnotsymmetrical, then LP M2(x,0)6=
U P M2(x,0). In particular, if the distribution of x is skewed to the left then LP M2(x,0)>
U P M2(x,0), whereas if the distribution of x is skewed to the right then LP M2(x,0) <
U P M2(x,0). In this case skewness has a first-order impact on risk measurement when the value of λis notably different from 1 and/or the values ofγ− and γ+ are noticeably different. If, for example, γ− À γ+ then the main source of risk is the term with LP M2(x,0) which reflects the level of negative skewness. By contrast, ifγ−¿γ+then the main source of risk is the term withU P M2(x,0) which reflects the level of positive skewness. In other words, if the decision maker’s degrees of risk aversions below and above the reference point are substantially different, then, depending on the signs and the values ofγ− andγ+, either the downside or the upside part of variance is a more proper risk measure than variance. Which part is actually a proper risk measure? The answer depends on the theory being used. Whereas in most mod-els14the decision maker is more risk averse in the domain of losses than in the domain of gains and, therefore, the downside part of variance might be a proper risk measure, in PT/CPT the
14The examples are: the mean-semivariance utility of Markowitz (1959), the utility function of Markowitz
