Increases in skewness and three-moment preferences

Transcription

1 Increases in skewness and three-moment preferences Thomas Eichner a and Andreas Wagener b a) Department of Economics, University of Hagen, Universitätsstr. 41, Hagen, Germany. thomas.eichner@fernuni-hagen.de b) Institute of Social Policy, University of Hannover, Königsworther Plat 1, Hannover, Germany. wagener@sopo.uni-hannover.de Abstract: We call an agent skewness affine if and only if his marginal willingness to accept a risk increases when the distribution of the risk becomes more skewed to the right. Skewness affinity is shown to be equivalent to the marginal rate of substitution between mean and variance of wealth being decreasing in the skewness. This property allows us to characterie the comparative static effect of increases in the skewness in quasi-linear decision problems. Over domains of skewnesscomparable lotteries skewness affinity is equivalent to the von Neumann-Morgenstern utility index of relative temperance being smaller than three. JEL-classification: Keywords: D81 mean, variance, skewness, skewness affinity.

2 1 Introduction It appears to be well established empirically that, when ranking risky prospects over final consumption or wealth, individuals prefer distributions that are more positively skewed to such that are lesser so (Golec and Tamarkin 1998; Garrett and Sobel 1999; Bhattacharya and Garrett 2008). A basic intuition is that an increase in skewness involves a smaller probability for low (or large negative) returns. This trait is economically highly relevant, given that many distributions in the financial or actuarial realm are skewed rather than symmetric (think, e.g., of stock and bond returns, catastrophe insurance, credit default risks, options etc.). Consequently, a number of researchers (e.g., Kraus and Litenberger 1976; Lane 2000; Konno et al. 2003; Konno and Suuki 2007; Prakash et al. 2003) argue that moments of order three or above must not be neglected in choices under risk. Empirical evidence indeed suggests that the skewness of a lottery s distribution does not only affect an agent s well-being but also his risk taking behavior. E.g., Chunhachinda et al. (1997) point out that skewness (and possibly also other moments of higher order) affect investor s portfolio decisions. Patton (2004) shows that negative skewness in return distributions make risk-averse investors less aggressive (i.e., more risk averse) in their portfolio decisions, 1 as compared to symmetric distributions. The present paper analyes the effect of an increase in skewness on the willingness to take risks when an agent s preferences depend on the mean, the variance and the skewness of final wealth only henceforth denoted as three-moment preferences. In addition to a direct preference for higher degrees of (positive) skewness, we also capture behavioral responses to changes in skewness. An agent is called skewness affine if and only if his willingness to accept a risk rises when the distribution of the risk becomes more skewed to the right. We characterie skewness affinity in several different ways: in terms of comparative risk aversion, as a property of the risk premium and with the help of the marginal rate of substitution between mean and variance of final wealth. For quasi-linear decision models (which encompass a wide range of economic problems from co-insurance demand to portfolio selection) we show that skewness affinity is both necessary and sufficient for agents to increase risk-taking upon an increase in the skewness. 1 Kraus and Litenberger (1976) and Harvey and Siddique (2000) also demonstrate the relevance of skewness preferences in the capital asset pricing model. 1

3 While outnumbered by mean-variance analyses, three-parameter approaches have quite some history both in theoretical research and in applied works in economics and finance. Originally, mean-variance-skewness preferences have been derived in the standard expected utility (EU) framework with a cubic von-neumann-morgenstern-utility function (Levy, 1969) or from a third-order Taylor series approximation to a true utility function. These approaches are mainly motivated by their simplicity, which comes at some cost in generality. Consistency of three-moment approach and expected utility can only be achieved either by restricting utility functions or the random variables distributions. Chiu (2010) developed a far less limiting condition for the compatibility of both approaches, labelled skewness comparability. It requires that all lotteries from which an agent can choose differ by at most the first three moments of their distributions. Chiu (2010) argues that a wide range of economic decision problems satisfies the condition of skewness comparability. We employ that condition in our paper when analying quasi-linear decision problems. The rest of the paper is organied as follows: Section 2 introduces three-moment preferences as an approach which stands on its own. Section 3 characteries skewness affinity in terms of comparative risk aversion, risk premium, marginal rate of substitution between mean and variance and as a comparative static result in quasi-linear decision models. Since threeparameter and EU approach partially overlap, it is natural to ask for the precise translation of skewness affinity into the EU approach. Provided that three-moment preferences and expected utility approach are consistent, Section 4 shows that skewness affinity corresponds to the index of relative temperance being smaller than three. Section 5 provides some extensions and concludes. 2 Preferences Consider a decision maker whose preferences over random final wealth Y are represented by a three-parameter utility function U = R R + R R, (μ, v, m) U(μ, v, m). (1) In (1) μ R, v R +,and (y μ) 3 m := df (y) R v 3/2 2

4 denote, respectively, the mean, the variance and the third standardied central moment of the distribution F (y) of final wealth. The third moment m provides information on the skewness of the distribution. Throughout the paper we assume that U is a C 2 function with the following properties with respect to μ and v (subscripts to functions denote partial derivatives): U µ (μ, v, m) > 0 for all (μ, v, m) R R + R; (2a) U v (μ, v, m) < 0 for all (μ, v, m) R R + R; (2b) for all m : U(μ, v, m) is strictly quasi-concave in (μ, v) R R +. (2c) These properties are standard for mean-variance preferences; here we transfer them to a threeparameter framework. Condition (2a) is a non-satiation property and condition (2b) reflects risk aversion. Together, (2a) and (2b) imply that (μ, v)-indifference curves are upward-sloped. Quasi-concavity (2c) ensures that they are convex. Denote by α(μ, v, m) := U v(μ, v, m) U µ (μ, v, m) > 0 (3) the marginal rate of substitution between v and μ (i.e., the slope of an indifference curve of U in (μ, v)-space). α is the mean-variance analogue of the Arrow-Pratt measure of absolute risk aversion (Epstein 1985; Ormiston and Schlee 2001). For given (μ, v, m), we implicitly define the risk premium φ(m, k) for a change in riskiness (represented by a change of variance) as U(μ + φ(m, k),v+ k, m) =U(μ, v, m). (4) From (2a) and (2b), φ(m, k) is positive if and only if k>0. Preferences U are said to exhibit a higher degree of risk aversion for brevity, are more risk averse thanpreferencesū if U dislikes all lotteries (with a given skewness) that Ū dislikes, but not vice versa. Formally, this means that, for all v, h 0andallμ, k, andm, Ū(μ + k, v + h, m) Ū(μ, v, m) = U(μ + k, v + h, m) U(μ, v, m). (5) For mean-variance preferences greater risk aversion is equivalent to the marginal rate of substitution between μ and v to be uniformly larger for U than for Ū (Lajeri-Chaherli 2003). Since m does not vary in (5), this also translates to three-parameter preferences. Hence, (5) is tantamount to: α(μ, v, m) ᾱ(μ, v, m) for all (μ, v, m). (6) 3

5 3 Skewness and risk aversion In this section, we introduce the notions of skewness preference, skewness affinity and their relationship to comparative risk aversion and economic behaviour in a decision problem. Skewness preference and skewness affinity A direct preference for skewness in (1) is represented by the assumption U m (μ, v, m) > 0 (7) (see, e.g., Chiu 2010). Here, we propose a new concept, called skewness affinity. Formally, an agent with (μ, v, m)-preferences is said to be skewness affine if his marginal willingness to accept a risk rises if the distribution of random wealth becomes more skewed to the right. Formally, the utility function U exhibits skewness affinity whenever U(μ + k, v + h, m + l) U(μ, v, m + l) = U(μ + k, v + h, m) U(μ, v, m) (8) for all h 0, k, andl 0. According to (8), all risks that are rejected at a certain level of skewness, m + l should also be rejected at the lower level of skewness, m. In view of (5) an alternative interpretation of (8) is that U(μ, v, m) ismoreriskaversethanu(μ, v, m + l). Quasi-linear decision problems Following Bigelow and Menees (1995) we consider a class of decision problems where final wealth is given by y(q) =q x + f(q). (9) In (9), q 0 is a one-dimensional and non-negative decision variable, f(q) is a concave function with f(0) = 0, and x is a random variable. The class of quasi-linear models (9) covers, among other economic problems, the co-insurance problem (Hadar and Seo 1992 and Tibiletti 1995), output decision of competitive firms under price uncertainty (Sandmo 1971) and standard portfolio choice with a safe and a risky asset (Fishburn and Porter 1976). 4

6 In view of (1) and (9), an agent s optimiation problem can be written as max q U(μ y (q),v y (q),m y (q)) where μ y (q) = qμ x + f(q), v y (q) = q 2 v x, The optimal decision q > 0 is characteried by the first-order condition 2 m y (q) = m x. (10) A := (μ x + f (q )) 2q v x α(μ y,v y,m y )=0. (11) Observe that the decision maker cannot affect the skewness m y of final wealth by varying his risk-taking q. Nevertheless, as shown below, exogenous changes in the skewness affect the agent s decision. Relationships Skewness affinity, risk aversion, and the comparative statics of the decision problem (10) are related through Proposition 1. The following statements are equivalent: (i) the utility function U(μ, v, m) exhibits skewness affinity; (ii) the measure of absolute risk aversion α(μ, v, m) is decreasing in m; (iii) U(μ, v, m) is more risk averse than U m (μ, v, m); (iv) the risk premium φ(m, k) is decreasing in m; (v) the optimal level of risk taking q is increasing in m x. Proof: Let l 0. Due to (5) and (6), U(μ, v, m) ismoreriskaversethanu(μ, v, m + l) iff α(μ, v, m) α(μ, v, m + l) 0, (12) whichinturnisequivalenttoα m (μ, v, m) 0. Next, verify that α m (μ, v, m) := U mvu µ U v U mµ U 2 µ 0 U mv U mµ U v U µ (13) 2 The second-order condition A q < 0 is satisfied since U is quasi-concave in (µ y,v y)andf(q) isconcave. 5

7 which proves that U(μ, v, m) ismoreriskaversethan U m (μ, v, m) (cf. item (iii)). Now, observe that φ k φ = U v(μ + φ, v + k, m) U µ (μ + φ, v + k, m), (14a) m = U m(μ, v, m) U m (μ + φ, v + k, m). (14b) U µ (μ + φ, v + k, m) Accordingto(14b) φ/ m 0iffU m (μ, v, m) U m (μ + φ, v + k, m) 0. This difference is ero for k = 0. It is non-positive for all k>0iff U mµ φ k U mv 0. (15) Inserting (14a) proves that (15) is equivalent to α m (μ, v, m) 0. Finally, implicit differentiation of (11) establishes sign q m x = sign α m (μ, v, m). (16) Empirically, a higher skewness seems to induce higher risk-taking (see, e.g., Patton 2004). Proposition 1 then tells us that investor preferences exhibit skewness affinity. Equivalently, the marginal rate of substitution α(μ, v, m) between mean and variance, which measures the compensation that an agent requires to accept a higher risk (variance), is reduced when the skewness of the risk distribution is increased. 4 Expected utility The quasi-linear decision problem analyed in the previous section satisfies the location-scale property (Meyer 1987), which requires that the decision variable and the uncertainty interact in a linear way. As a consequence, changes in the decision variable only affect mean and variance of random final wealth (in particular, they leave skewness unaffected). In terms of the decision problem (10), for all q the random variables [y(q) μ y (q)]/ v y (q) are equal in distribution (linear distribution class). Meyer (1987) showed that with the location-scale property, expected utility and mean-variance preferences are compatible. Specifically, given a von Neumann-Morgenstern 6

8 (vnm) utility function u : R R, y u(y), the expected utility from lottery y(q) canbe written as E F u(y(q)) = ( ) u μ y (q)+ v y (q) df () =:ŨF (μ y (q),v y (q)), (17) where is a standardied random variable with support [, ], distribution F, E F () =0and Var F () =1. Ifu(y) is strictly increasing and concave, then ŨF satisfies properties (2a) to (2c) with respect to (μ y,v y ). Chiu (2010) calls two standardied 3 distributions F and G skewness-comparable if the change from F to G or its reverse constitute an increase in downside risk àlamenees et al. (1980). The distribution with the lower downside risk (F,say)is then more skewed to the right and has a higher (standardied) third moment: m F >m G. As a consequence, if F and G are skewness comparable and standardied, we can compare ŨF and ŨG in (17) by means of a three-parameter function U(μ y,v y,m). In particular, if more right-skewed distributions are preferred, then Ũ F (μ y,v y ) > ŨG(μ y,v y ) U(μ y,v y,m F ) >U(μ y,v y,m G ), or U m (μ y,v y,m) > 0, as in (7). Menees et al. (1980) coined the notion of downside risk aversion to capture the idea that higher skewness is welcome. Their Theorem 2 shows that U m > 0holds for all (μ y,v y,m) if and only if u (y) > 0 for all y. 4 Applying the technique by Menees at al. (1980), we are in a position to characterie skewness affinity as introduced above in terms of the EU approach. 5 For that purpose, define the measure of relative temperance for vnm function u(y) by T (y) := y u(4) (y) u (y). (18) 3 See Chiu (2010, Definition 5) for general probability distributions. 4 Already Scott and Horvath (1980) observed that investors with non-increasing absolute risk aversion (which implies that u > 0) exhibit skewness preference. 5 Applying the same technique to U µ(µ y,v y,m F )= we obtain that u (µ y + v y)df () and U µµ(µ y,v y,m F )= (i) U µm(µ y,v y,m F ) 0 if and only if u (4) (y) < 0; and (ii) U µµm(µ y,v y,m F ) 0 if and only if u (5) (y) > 0. u (µ y + v y)df (), 7

9 Proposition 2. A downside risk-averse agent is skewness affine if and only if T (y) 3 for all y. Proof: From Proposition 1, we can rephrase skewness affinity as: m F >m G = α(μ y,v y,m F ) α(μ y,v y,m G ) for all (μ y,v y ). With distributions F and G being skewness comparable and E F () =E G () =0 and Var F () =Var G () = 1, the property m G <m F is equivalent to G exhibiting higher downside risk than F.Nowverifythat: α(μ y,v y,m F ) α(μ y,v y,m G ) U v (μ y,v y,m F )U µ (μ y,v y,m G ) U v (μ y,v y,m G )U µ (μ y,v y,m F ) 0 U v (μ y,v y,m F )[U µ (μ y,v y,m G ) U µ (μ y,v y,m F )] U µ (μ y,v y,m F )[U v (μ y,v y,m G ) U v (μ y,v y,m F )] 0. Using (17) we then obtain (the argument of u is always y = μ y + v y ): α(μ y,v y,m F ) α(μ y,v y,m G ) [ ] [ ] U v (μ y,v y,m F ) u d(g() F ()) U µ (μ y,v y,m F ) 2 u d(g() F ()) 0 v y [ t ] U v (μ y,v y,m F ) v yu 3 (4) [F (r) G(r)]drdtd [ U µ (μ y,v y,m F ) vy 2 (3u + v y u (4) ) t ] [F (r) G(r)]drdtd 0. (19) The final line comes from integrating by parts three times the square-bracketed integrals in the line before and by observing that [F () G()]d = [F (r) G(r)]drd =0; this is due to the fact that E F = E G and Var F = Var G. Moreover, from G exhibiting more downside risk than F, t [F (r) G(r)]drdt 0 for all [, ] and strictly negative for some (Menees et al., 1980). Now recall that U µ (μ y,v y,m F ) > 0 >U v (μ y,v y,m F ). If we want (19) to hold for all distributions we must, thus, have 2α(μ y,v y,m F )v y u (4) +3u + v y u (4) 0. (20) 8

10 With downside risk-aversion, inequality (20) can be rearranged to ( 2α(μ y,v y,m F )v y + v y ) u (4) (μ y + v y ) u (μ y + v y ) 3. (21) Next, we substitute μ y + v y = y in (21) to obtain (y a) u(4) (y) u (y) 3, (22) where a := μ y 2α(μ y,v y,m F )v y. It remains to show that a 0. Since skewness affinity is equivalent to q / m x (see Proposition 1(i) and (v)) we can rewrite the first-order condition (11) as a = f(q)+f (q)q which is non-negative due to f(0) = 0 and f (q) 0. a 0inturn ensures that skewness affinity (see also (22)) prevails if and only if T (y) 3 for all y. An increase in skewness constitutes an increase in downside risk. For linear decision problems, Honda (1985) and Poulsson et al. (2003) pointed out that behavioral responses to downside risk shifts indeed match with relative temperance being smaller than three when agents are expected utility maximiers. Proposition 2 shows how this property directly translates to three-moments preferences. 5 Extensions and conclusion Under the condition of skewness comparability àlachiu (2010), an increase in skewness is equivalent to an increase in downside risk, or employing the terminology in Ekern (1980), an increase in third-degree risk over a random variable y, ify dominates the random variable ỹ via third-order stochastic dominance and the first two moments of the distribution of y and ỹ coincide. Our analysis has revealed that an increase in downside risk makes an agent less risk-averse (i.e., increases risk taking) if and only if he is skewness affine. The results of the previous section can be extended to higher-order risk changes: For n 3, define (y μ) n m n := df (y) v n/2 as the n th standardied central moment of F and suppose the decision maker s preferences are represented by a n-moment utility function U(μ, v, m 3,...,m n ). Then an agent can be called 9

11 k th -degree risk affine (for 3 k n) whenever m k > m k = α(μ, v, m 3,...,m k,...,m n ) α(μ, v, m 3,..., m k,...,m n ). In quasi-linear decision problems all central moments of degree higher than two are unaffected by the decision variable. Hence, in these problems increases in the k th -degree risk will induce an agent take more risks if and only if preferences exhibit k th -degree risk affinity. Presupposing that n-moment and EU approach are compatible, k th -degree risk affinity then translates into the condition that the k th -degree of relative risk aversion, yu k+1 (y)/u k (y), does nowhere exceed k. Interestingly, conditions on the k th -degree of relative risk aversion also serve to characterie the comparative statics of k th -degree increases in multiplicative risks (see Eeckhoudt and Schlesinger, 2008). On a more fundamental level, preference concepts that rely on higher degrees of relative risk aversion can be traced back to notions of multiplicative risk apportionment (Eeckhoudt et al., 2009; Wang and Li, 2010). 6 These concepts are related to our analysis as they also capture trade offs between different moments of lotteries. E.g., Eeckhoudt et al. (2009, p. 6) argue that relative prudence (i.e., second-degree relative risk aversion) being larger than two indicates that, when different risk apportionments with equal expected wealth have to be compared, a higher variance can be compensated for by a lower skewness which reflects a specific form of skewness aversion. In this paper we dealt with skewness affinity, which measures the willingness of individuals to accept risks of different skewness, rather than with direct preferences for higher or lower skewness. As could be expected, the comparative statics of risk aversion with respect to skewness involve higher degrees of relative risk aversion than are sufficient for direct utility comparisons relative temperance (third degree relative risk aversion) rather than relative prudence (second degree). However, the precise links still remain to be explored. It should be emphasied that the decision problems studied in this paper possess the location-scale property: the agent s chocies do not affect the skewness of the distribution he faces. Investigating the comparative statics of increases in skewness in decision problems where higher moments of final wealth depend on the agent s decisions as in quasi-linear models with 6 We thank a referee for pointing out this possible connection to us. 10

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