The Geneva Papers on Risk and Insurance Theory, 29: 137 144, 2004 c 2004 The Geneva Association Portfolio Selection with Quadratic Utility Revisited TIMOTHY MATHEWS tmathews@csun.edu Department of Economics, California State University-Northridge, 18111 Nordhoff St., Northridge, CA 91330-8374 Received December 10, 2002; Revised May 15, 2003 Abstract Considering a simple portfolio selection problem by agents with quadratic utility, an apparently counterintuitive outcome results. When such a choice is over two assets that can be ordered in terms of riskiness, an agent that is more risk averse may optimally invest a larger portion of wealth in the riskier asset. It is shown that such an outcome is not counterintuitive, since for the portfolios from which agents optimally choose, a larger proportion of investment in the riskier asset leads to a less risky portfolio. Key words: portfolio selection, choice under uncertainty JEL Classification No.: G11, D8 1. Introduction Ross [1981] criticized the standard definition of more risk averse by highlighting some apparently counterintuitive choices made by agents that can be ordered in terms of degree of risk aversion. One of Ross examples showed that in a simple portfolio selection problem, an individual with a higher degree of absolute risk aversion may choose to invest more in a riskier asset (using his new definition of an asset being riskier). 1 Ross proposed a stronger concept of more risk averse and showed that such a counterintuitive outcome cannot occur when his new concept of more risk averse is applied (again, using his new definition of an asset being riskier). Hadar and Seo [1990] examine the implications of Ross definition of more risk averse in situations in which a choice is over two assets that can be compared using the traditional definition of riskier. They show that Ross stronger definition of more risk averse may not necessarily rule out the counterintuitive outcomes it was intended to. One of the examples considered by Hadar and Seo is a simple portfolio selection problem in which an agent must decide upon the proportion of wealth to invest in each of two risky assets. Comparing the optimal choice of agents with quadratic Bernoulli utility functions, they arrive at the seemingly counterintuitive conclusion that the more risk averse agent optimally chooses to invest more in the riskier asset. It is argued here that this outcome, which was labelled counterintuitive by both Ross and Hadar and Seo, is not so counterintuitive after all.
138 MATHEWS 2. Simple portfolio selection problem Consider an economic agent making a decision under uncertainty. The level of utility from a payoff of x is obtained by evaluating the Bernoulli utility function, u( ), at x. Given a distribution of payoffs F(x), the utility of the agent is given by the von-neumann- Morgenstern utility function U(F(x)) = u(x) df(x). Suppose an agent must choose optimal proportions of a fixed amount of wealth to invest in two risky assets X and Y. Further, suppose that Y is riskier than X in that X second order stochastic dominates Y and has a higher mean than Y. Hadar and Seo consider such a choice by agents with Bernoulli utility functions of the form u i (w) = w k i w 2 where k i > 0 and 1 2k i exceeds the largest possible realization of w. Considering two agents A and B, A is more risk averse than B in the Arrow-Pratt sense if u A (x) B (x) u A (x) u u B for all x (with strict inequality almost everywhere). In the current context, this (x) is satisfied if and only if k A > k B. A is more risk averse than B as defined by Ross if for all x and y there exists λ>0such that u A (x) u B (x) λ u A (y) u B (y). For agents with different preferences, this condition is again satisfied in the current context if and only if k A > k B. Let α denote the proportion invested in asset Y. That is, the portfolio of an agent is given by the random variable W α = αy + (1 α)x with α [0, 1]. Denote the value of α characterizing the optimal portfolio of agent i by αi. 3. A counterintuitive outcome? As a function of α, the von-neumann-morgenstern utility for agent i is U i (α) = E(W α ) k i E ( W 2 α ). Supposing that each agent optimally chooses to invest a positive amount in each asset, the optimal portfolio choices for agents A and B are such that and U A (α A ) = E(Y X) 2k A E(WA (Y X)) = 0 U B (α B ) = E(Y X) 2k B E(WB (Y X)) = 0 where W A = α A Y + (1 α A )X and W B = α B Y + (1 α B )X.
PORTFOLIO SELECTION WITH QUADRATIC UTILITY REVISITED 139 As shown by Hadar and Seo, these conditions imply α A >α B. That is, they arrive at the apparently counterintuitive result that the more risk averse agent will choose to invest more in the riskier asset. 2 Thus, Hadar and Seo illustrate that Ross definition of more risk averse is not strong enough to rule out such a seemingly counterintuitive result when the standard definition of riskier is used. It has been shown by Rothschild and Stiglitz [1970] that the preferences of an expected utility maximizing agent over different distributions of wealth cannot always be consistently stated as preferences over mean and variance alone. This can only be done if restrictive assumptions are made about either the Bernoulli utility function of the agent or the specific class of distributions from which the agent must choose. Borch [1969] and Baron [1977] have shown that for any arbitrary payoff distribution, preferences can be stated over mean and variance alone only if an agent has a quadratic Bernoulli utility function. Others have attempted to show consistency of mean-variance analysis and expected utility analysis for restricted classes of random variables. Meyer [1987] proves such consistency for situations in which all risks under consideration belong to a common two-parameter family, the elements of which differ from one another by only location and scale. It should be noted that the class of random variables resulting from the portfolio selection problem considered here (with two risky assets) need not satisfy the location and scale parameter condition identified by Meyer. 3 Bigelow [1993] provides a necessary and sufficient condition for consistency of mean-variance analysis and expected utility analysis. Bigelow defines a class of random variables to be normalized risk comparable if all members of the class can be ordered in terms of riskiness (in the traditional sense of Rothschild and Stiglitz) after being normalized to have zero mean. Preferences resulting from an arbitrary Bernoulli utility function are consistent with preferences over mean and variance if and only if the class of random variables under consideration is normalized risk comparable. It again should be noted that the portfolio selection problem considered here (with two risky assets) need not give rise to a class of distributions which is normalized risk comparable. 4 Mathews [2002] proposes a definition of more risk averse for situations in which preferences can be stated over mean and variance alone (in which case variance completely indexes riskiness). This definition leads to an intuitive restriction on indifference curves in (σ 2,µ) space. Letting S i (σ 2,µ) denote the slope of the indifference curve of agent i at the point (σ 2,µ), agent A is defined to be more risk averse than agent B if and only if S A (σ 2,µ) > S B (σ 2,µ) for all (σ 2,µ). The notion of more risk averse described by Mathews corresponds to the notion of more variance averse which is discussed by Lajeri and Nielsen [2000] and Lajeri-Chaherli [2002]. Comparing the optimal choices by two agents from any common set of risky alternatives, Mathews shows that (when his proposed definition can be applied) an agent that is more risk averse cannot optimally choose an alternative characterized by a higher level of risk. For agents with quadratic Bernoulli utility functions, the notion of more risk averse proposed by Mathews is not only applicable, but corresponds to the traditional Arrow-Pratt notion of more risk averse. 5 The outcome of the portfolio selection problem considered here and the observation by Mathews that a more risk averse agent cannot optimally choose a higher level of risk initially appear to be at odds with each other. However, as more light is shed upon the matter, it is
140 MATHEWS clear that not only is there no contradiction, but further, the observation by Hadar and Seo does not conflict with intuition. 4. Further inspection of portfolio choice Foranagent i with u i (w) = w k i w 2 we have U i (α) = µ α k i µ 2 α k iσα 2 where µ α is the expected value and σα 2 is the variance of W α.asaresult, and U i µ α = 1 2k i µ α > 0 U i σ 2 α = k i < 0. The first inequality follows from the assumption that 2k i exceeds the largest possible realization of w; the second inequality follows from the assumption that k i > 0. Thus, an increase in expected payoff leads to an increase in von-neumann-morgenstern utility, while an increase in variance (an increase in risk) leads to a decrease in von-neumann-morgenstern utility. Let µ Y denote the expected value of Y and µ X denote the expected value of X. Since µ X >µ Y, µ α = αµ Y + (1 α)µ X is decreasing in α, asillustrated in figure 1. Letting µ B denote the expected value of WB and µ A denote the expected value of W A, α B <α A implies µ B >µ A. Let σb 2 denote the variance of W B and σ A 2 denote the variance of W A.Itmust be that σb 2 >σ2 A.Inorder to see this, suppose that the converse is true. If σ B 2 σ 2 A and µ B >µ A, then any agent with a quadratic Bernoulli utility function would prefer WB to WA, contradicting the optimality of α A for agent A. Asaresult, σ B 2 >σ2 A, implying that agent A chooses a portfolio that both agents view as less risky than the portfolio that B chooses. This outcome accords with intuition: the agent that is more risk averse chooses a portfolio which is less risky. Direct inspection of σα 2 shows that over the entire range of α from which agents might optimally choose, the riskiness of a portfolio decreases as α increases. Letting σy 2 denote the variance of Y, σx 2 denote the variance of X, and Cov(Y, X) denote the covariance between Y and X, σ 2 α = α2 σ 2 Y + 2α(1 α) Cov(Y, X) + (1 α)2 σ 2 X. For this expression α = 2{ ασy 2 + (1 2α) Cov(Y, X) (1 α)σ X 2 } = 2 { αvar(y X) + Cov(Y, X) σx} 2. σ 2 α 1
PORTFOLIO SELECTION WITH QUADRATIC UTILITY REVISITED 141 Figure 1. Mean and variance of portfolio. Var(Y X) isalways positive. Assume Cov(Y, X) σx 2 < 0, so that agents always wish to diversify. As a result, for small values of α the second term dominates this expression, leading to σ2 α α < 0. Since σ Y 2 >σ2 X, σ α 2 is as illustrated in figure 1.6 Thus, as α is increased from zero to one, σα 2 decreases up to some level ᾱ and then increases beyond ᾱ. Since U i (α) isincreasing in µ α and decreasing in σα 2, the optimal choice of α cannot exceed ᾱ. That is, the optimal portfolio of any such agent is characterized by α (0, ᾱ]. This can also be seen by illustrating the resulting values of (σα 2,µ α)in(σ 2,µ) space, as in figure 2. Different values of α lead to different portfolios along the locus from X = W 0 to Y = W 1. From the observations thus far, this locus is positively sloped at points induced by α [0, ᾱ) and negatively sloped at points induced by α (ᾱ, 1]. Further, as α is increased from zero to ᾱ (that is, as we move along the locus from X = W 0 to Wᾱ), the slope of the locus increases. Clearly the optimal portfolio of any agent must lie on this locus somewhere between X = W 0 and Wᾱ, corresponding to α (0, ᾱ). When preferences can be stated over mean and variance alone, the shape of indifference curves has been characterized: by Meyer and by Lajeri and Nielsen (when all risks under consideration belong to a common two-parameter family) and by Lajeri-Chaherli (when
142 MATHEWS Figure 2. Optimal choice of agent A and agent B. such preferences do not necessarily arise from an expected utility framework). From each of these studies, it follows that at any point in (σ 2,µ) space, S i (σ 2,µ)isgreater for agents that are more risk averse in the traditional Arrow-Pratt sense. 7 It is however important to recognize that the portfolio selection problem considered here does not necessarily give rise to a class of random variables belonging to a two-parameter family. Thus, when analyzing such a choice for an expected utility maximizing agent, the aforementioned references do not provide an immediate characterization of the shape of indifference curves in (σ 2,µ) space. Foranagent with u i (w) = w k i w 2, the slope of an indifference curve in (σ 2,µ) space is S i (σ 2,µ) = k i 1 2k i µ > 0. Considering two agents A and B such that k A > k B,atany point in (σ 2,µ) space the indifference curve of A is steeper than the indifference curve of B. These insights follow directly from observations made by Baron. From here it immediately follows that 0 <α B <α A < ᾱ. The optimal choice of each agent is illustrated in (σ 2,µ) space in figure 2. Since σα 2 decreases as α is increased over the range of α from which agents will choose, optimal portfolios characterized by larger portions of the riskier asset are in fact less risky portfolios. Thus, we arrive at the intuitive result that a more risk averse agent chooses a portfolio which all agents view as less risky. 5. Conclusion One of the examples presented by Hadar and Seo when examining the implications of Ross definition of more risk averse was a simple portfolio selection problem in which an
PORTFOLIO SELECTION WITH QUADRATIC UTILITY REVISITED 143 agent must decide upon the proportion of a fixed amount of wealth to invest in two risky assets. Comparing the optimal choice by agents with quadratic Bernoulli utility functions, they arrive at the seemingly counterintuitive conclusion that the more risk averse agent will choose to invest more in the riskier asset. It has been argued here that this outcome does not conflict with intuition, by pointing out that the portfolio chosen by the agent that is more risk averse is viewed as a less risky portfolio by both agents. Acknowledgment Iwould like to thank Tom Lee, Qihong Liu, and Soiliou Namoro for helpful comments. Notes 1. Ross states that an asset Y offering a higher return than an asset X is riskier if E(Y X x) 0 for all x X. 2. Lajeri-Chaherli [2002] considers a portfolio selection problem with one riskless and one risky asset (in which there are no constraints on borrowing or short sales) in a mean-variance framework, and arrives at the intuitive conclusion that a more risk averse agent will hold fewer shares of the risky asset. 3. As noted by Meyer, a portfolio selection problem of this nature over one riskless and one risky asset would lead to a class of random variables satisfying his location and scale condition. 4. Thus, in order to legitimately analyze any such portfolio selection problem in an expected utility framework as a choice over mean and variance, attention must be restricted to agents with quadratic Bernoulli utility functions. 5. When agents have non-quadratic Bernoulli utility functions the notion of more risk averse defined by Mathews need not apply, even when preferences are over mean and variance alone. Mathews provides an example of a class of random variables which is normalized risk comparable for which agents with differing degrees of constant absolute risk aversion are not ordered by the proposed definition. An implication of this observation is that for such a class of random variables, there exist points in (σ 2,µ) space for which an agent with a higher degree of constant absolute risk aversion has a flatter indifference curve. 6. If Cov(Y, X) σx 2 0 (so that it never pays to diversify), then σ α 2 would not be as illustrated in figure 1, but would rather be increasing in α for all α [0, 1]. 7. A characterization of the shape of indifference curves in (σ 2,µ) space is also provided by Mathews (when preferences arise from an expected utility framework) in which, as previously noted, one agent is defined as being more risk averse than another precisely when S i (σ 2,µ)isgreater at every point in (σ 2,µ) space. References BARON, D.P. [1977]: On the Utility Theoretic Foundations of Mean-Variance Analysis, Journal of Finance, 32, 1683 1697. BIGELOW, J.P. [1993]: Consistency of Mean-Variance Analysis and Expected Utility Analysis: A Complete Characterization, Economics Letters, 43, 187 192. BORCH, K. [1969]: A Note on Uncertainty and Indifference Curves, Review of Economic Studies, 36, 1 4. HADAR, J. and SEO, T.K. [1990]: Ross Measure of Risk Aversion and Portfolio Selection, Journal of Risk and Uncertainty, 3,93 99. LAJERI, F. and NIELSEN, L.T. [2000]: Parametric Characterizations of Risk Aversion and Prudence, Economic Theory, 15, 469 476. LAJERI-CHAHERLI, F. [2002]: Partial Derivatives, Comparative Risk Behavior and Concavity of Utility Functions, forthcoming in Mathematical Social Sciences. MATHEWS, T. [2002]: The Meaning of More Risk Averse when Preferences are over Mean and Variance, forthcoming in The Manchester School (volume 43, issue 1, January 2005).
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