Linear Risk Tolerance and Mean-Variance Utility Functions

Transcription

1 Linear Risk Tolerance and Mean-Variance Utility Functions Andreas Wagener Department of Economics University of Vienna Hohenstaufengasse 9 00 Vienna, Austria andreas.wagener@univie.ac.at Abstract: The concept of linear risk tolerance is transferred from the expected utility framework to the two-parameter, mean-variance approach. We show how the requirement of a hyperbolical Arrow-Pratt index translates from the EU-approach into a condition on the marginal rate of substitution between return and risk in the two-parameter approach. As a spin-off from this translation, we derive a specific class of functional forms for twoparameter utility functions. JEL-classification: D8, D2. Keywords: Linear risk tolerance, Two-parameter preferences, HARA preferences. This version: July 24, 2004

2 Introduction Modelling preferences over lotteries by functions that only depend on the first and second moments of the probability distribution is very common and widely applied, both in economic theory and in practice. Mean-variance preferences reflect, in a very clear and straightforward way, the basic trade-off between higher return and higher riskiness that underlies many economic choices under uncertainty. However, how exactly should one model this trade-off? Given that there are zillions of functional forms in two variables that are increasing in one but decreasing in the other argument, what would a reasonable parametrization look like? Surprisingly little research has been devoted to this question in the context of two-parameter preferences. Things are much different for the cousin (or, as some would say, superior) approach of the mean-variance framework, namely the expected-utility (EU) approach. There, preferences over lotteries are represented by the expected value of a von-neumann-morgenstern utility index which is defined over the lottery outcomes. Several functional types of utility functions are widely applied in this setting, Think, e.g., of negative exponential functions, logarithmic functions, power functions etc. Such functional forms are not chosen ad hoc. Rather, each of them can be associated with certain behavioural attitudes towards risk which have been widely analysed and discussed in the literature. E.g., power and logfunctions are known to represent the risk attitude of constant relative risk-aversion which, on the behavioural side, translates into a unit elasticity of the demand for a risky asset with respect to changes in wealth. Even since its (re-)invention in Markowitz (952), mean-variance (or (µ, σ))-analysis has been recognized as deficient relative to the EU framework (in particular, see Markowitz, 959, pp ): The mean-variance approach is only consistent with the EU approach for quadratic utility functions (Baron, 977) or if all random variables are jointly elliptically distributed (Chamberlain, 983). As shown by Owen and Rabinovitch (983), the latter condition holds if all attainable distributions differ only by location and scale parameters. As has been convincingly argued by Meyer (987), Sinn (983) and others, this location-scale framework covers a wide range of economic decision problems. These insights recently have led to a partial rehabilitation of the mean-variance approach. When (µ, σ)- and EU-approach are perfect substitutes, a number of formal correspondences between them can be identified, relating to absolute and relative risk aversion, See Markowitz (999) for a stroll through the long-standing and venerable history of mean-variance considerations.

3 absolute and relative prudence, risk vulnerability, standardness, properness etc. (see, e.g., Meyer, 987; Lajeri and Nielsen, 2000; Lajeri-Chaherli, 2002, 2003; Eichner and Wagener, 2003b, 2004). This paper adds to these results by establishing a formal correspondence between linear risk tolerance (HARA-preferences) in the EU-setting and in the meanvariance framework. Linear risk tolerance is a helpful preference assumption in models of capital markets: If investors have homogeneous preferences, face equal investment opportunities and the capital market is complete, then the economy aggregates whenever the wealth coefficient in risk tolerance (i.e., the inverse of the Arrow-Partt index of risk aversion) is constant: The mean wealth of an economy entails enough information to predict the behaviour of aggregate variables and, in particular, asset prices in the equilibrium do not depend on the distribution of wealth. Even though recent empirical research suggests that risk tolerance is strictly concave rather tan linear in wealth (see, e.g., Guiso and Paiella, 200), the HARA case still serves as an important benchmark. As the HARA-class of preferences encompasses a wide range of different risk attitudes, our correspondence result for EU- and two-parameter approach generalizes several of the findings in the literature just mentioned. Moreover, we utilize our result which formally comes as a partial differential equation to specify that two-parameter utility functions with linear risk-tolerance must basically come as additive power functions of mean and risk. We then show that employing such functional forms in a simple decision problem under risk (portfolio choice with one risky and one riskfree asset) indeed yields comparative statics as one would expect them to be with linear risk tolerance. 2 Mean-Variance Preference Functionals As the mean-variance approach is popular in portfolio theory and its applications (as perceived by theorists), quite a number of specific functional forms for two-parameter preferences have been proposed and used. One observation from an anecdotal survey is that most of these specific functional forms are chosen, not without consideration, but without much reference to potentially underlying risk attitudes or underlying behaioural traits. Quite common is the quadratic form where utility U is quadratic and additively separable in the mean (µ) and the standard deviation (σ): U(µ, σ) = µ c (µ 2 + σ 2 ) () with c > 0. The motivation for this specific stems from the widespread (mis-)belief that EU- and mean-variance approach are only consistent in the case of quadratic von- 2

4 Neumann-Morgenstern utility functions. 2 Since quadratic vnm-functions suffer from serious defects, functional approaches as in () are more capable of discrediting, rather than promoting, the two-parameter approach. Dating back to Freund (956), mean-variance preferences are sometimes represented as linearly increasing in the mean and linearly decreasing in the variance (see, e.g., Chopra and Ziemba, 993): 3 U(µ, σ) = µ b 2 σ2 (2) with b > 0. This approach is motivated from portfolio choice problems where investors with exponential vnm-utility face returns to their activities that are jointly normally distributed. Under these conditions, maximizing the expected utility is equivalent to maximizing a functional form as in (2). While exponential vnm-functions have a not entirely unplausible underpinning in terms of risk attitudes (namely, constant absolute risk aversion), use of linear-quadratic functions as in (2) is valid only for Gaussian distributions which renders (2) a highly specific case. Antedating portfolio theory, Roy (952) proposes a two-parameter function of type U(µ, σ) = µ a σ where a > 0. Use of this specification is motivated by an explicit safety-first -criterion. The attempt to minimize the probability that the outcome of some risky choice falls below some exogenously given disaster level (represented by a) can, in the absence of full knowlegde of the underlying probability distribution, be approximated by Chebyshev s Inequality. Transformed into a maximization problem, this gives rise to (3). An exception to the more-or-less general neglect of references to measures of risk-attitudes in the discussion of two-parameter preference functionals can be found in Saha (997). Based on equivalences between two-parameter and EU-approach characterizations that were derived in Meyer (987) and which are partly outlined in Section 3 below, he empirically estimates parameters for the following functional specification: (3) U(µ, σ) = µ a σ b. (4) By appropriate choices of parameters a > 0 and b R, the function(4) is capable to exhibit different types of risk attitudes. E.g., decreasing, constant, and increasing absolute 2 If u(y) = (a b y) 2 is a quadratic vnm-function then, for all distributions of y, and up to innocuous normalizations of coefficients, EU(y) = µ c (µ 2 + σ 2 ) where µ = Ey and σ 2 = E[(y µ) 2 ]. 3 Such a specification may also result for quadratic vnm-utility functions. Steinbach (See, e.g., 200). 3

5 risk aversion are reflected as a >, a =, and a <, while decreasing, constant, and increasing relative risk aversion are reflected as a > b, a = b, and a < b. Below we will argue that functions of type (4) come quite close to reflecting the property of linear risk tolerance (HARA). More surprisingly, even the behavioural implications of (3) fit in the class of HARA preferences. 3 Notation and Preliminaries Before presenting our results, some notation and preliminaries are indispensable. Consider a choice set Y of random variables (lotteries) that have support in a (possibly unbounded) interval Y of the real line and that only differ from one another by location and scale parameters. I.e., if X is the random variable obtained by normalization of an arbitrary element of Y, then any Y Y is equal in distribution to µ y +σ y X, where µ y and σ y are the mean and the standard deviation of the respective Y. By M := {(µ, σ) R R + Y Y : (µ Y, σ Y ) = (µ, σ)} we denote the set of all possible (µ, σ)-pairs that can be obtained for Y Y. We assume that M is a convex set. Let u : R R be a von-neumann-morgenstern (vnm) utility index; for simplicity we assume that u is a smooth function. Then the expected utility from the lottery Y can be written in terms of the mean and the standard deviation of Y as: Eu(y) = u(µ Y + σ Y x)df (x) =: U(µ Y, σ Y ) (5) Y where F is the distribution function of X. Recall that the mean and the standard deviation of X are, respectively, zero and one by construction. It is evident from (5) that u(y) is increasing for all y Y if and only if U(µ, σ) is increasing in µ for all (µ, σ) M. As shown by Meyer (987), risk aversion in u(y) (i.e., u (y) < 0) is represented by U(µ, σ) being decreasing in σ. Denoting partial derivates by subscripts, we thus get that U µ (µ, σ) > 0 > U σ (µ, σ) (µ, σ) M u (y) > 0 > u (y) y Y. (6) Moreover, Meyer (987) also shows that u (y) < 0 is equivalent to U(µ, σ) being strictly concave. I.e., u (y) < 0 is equivalent for U µµ (µ, σ) < 0, U σσ (µ, σ) < 0, and U µµ (µ, σ)u σσ (µ, σ) U µσ (µ, σ) 2 > 0 (7) to hold separately and jointly for all (µ, σ) M. 4

6 Given two-parameter preferences U, let us define by α(µ, σ) = U σ(µ, σ) U µ (µ, σ) the marginal rate of substitution between risk and return in the two-parameter framework. It is well-known that α(µ y, σ y ) is the counterpart to the Arrow-Pratt measure of absolute risk aversion A(y) := u (y)/u (y) in the EU-framework. In particular, α and A share the same sign. Moreover, A (y) > < 0 y Y α(µ, σ) µ (y A(y)) > < 0 y Y α(λ µ, λ σ) λ > 0 (µ, σ) M; (8) < > 0 (µ, σ) M, λ > 0; (9) < see Meyer (987). In this paper, the cases of constant absolute risk aversion (i.e., A (y) = 0) and of constant relative risk aversion (i.e., (ya(y)) = 0) are of particular interest. 4 HARA Preferences A vnm-utility function is said to exhibit hyberbolical absolute risk aversion (for short: belongs to the HARA-class) if there exist constants a 0 and b R such that u (y) u (y) = a + b y for all y such that (a + b y) Y. Defining risk tolerance as the inverse of the Arrow- Pratt measure of risk aversion, HARA-preferences are sometimes said to exhibit linear risk tolerance. Condition (0) implies the following functional forms: (a + b y) b ( ) if b 0, b b b u(y) = u H (y) := a e ( y/a) if b = 0 ln(a + y) if b =. For a = 0 we, thus, get the case of constant relative risk aversion (CRRA) where b measures the degree of relative risk aversion. For b = 0 the case of constant absolute risk aversion (CARA) is obtained where a then measures the degree of absolute risk aversion. For b =, the utility function is quadratic. We will now derive the two-paramter counterpart of HARA-preferences. For (µ, σ) M, a 0, and b R define: G(µ, σ; a, b) := (a + b µ) α µ (µ, σ) + b σ α σ (µ, σ). (0) 5

7 Proposition. Let a 0 and b R. Then the following are equivalent: G(µ, σ; a, b) = 0 for all (µ, σ) M; A(y) = a + b y Proof: Calculate: for all y Y. G(µ, σ; a, b) = = (U µ ) [(a + b µ) (U µu 2 µσ U µµ U σ ) + b σ (U µ U σσ U µσ U σ )] [ ( = ( u df (x)) (a + b µ) u df (x) xu df (x) u df (x) ( 2 ) ] +b σ u df (x) x 2 u df (x) xu df (x) xu df (x) [ ( u u = (a + b µ) x u u df df u u ) u u df df u x u df df ( +b σ x 2 u u u u df df u u ) x u u df df u ] x u df df ) xu df (x) = (a + b µ) (E G (x A(y)) E G x E G A(y)) + b σ ( E G (x 2 A(y)) E G x E G (xa(y)) ) = (a + b µ) Cov G (x, A(y)) + b σ Cov G (x, xa(y)) = Cov G (x, (a + b (µ + σx)) A(y)) = Cov G (x, (a + b y) A(y)) where the argument of u is always (µ + σx) = y. The first equation comes from differentiating α with respect to µ and σ. In the second we used (5). The third follows from taking the expression ( u df ) 2 into the integrals. To obtain the fourth line, we used the distribution function G defined by dg = (u / u df )df ; E G denotes the expectation operator with respect to G. The fifth line is by definition of the covariance and the sixth follows from the additivity properties of the covariance. If we want to final expression to always be zero, (a + by) A(y) must not depend on x or, equivalently, is constant in y (recall that y = µ + σx). However, then u(y) must be a HARA-function as in (0). Observe two special cases of Proposition : Constant absolute risk aversion: By setting b = 0, we obviously obtain the equalitycase in (8): G(µ, σ; a, 0) = a α(µ, σ) where the value of a is irrelevant. 6

8 Constant relative risk aversion: By setting a = 0, the equality-case in (9) emerges. To see this, verify that for all λ is equivalent to ( U ) σ(λµ, λσ) = 0 λ U µ (λµ, λσ) (U µ ) 2 [U µ (µu µσ + σu σσ ) U σ (µu µµ + σu µσ )] = 0 to hold for all (µ, σ). This can be re-arranged to yield G(µ, σ; 0, b) = 0; note that b in fact is irrelevant here. 5 Functional Forms of HARA Preferences In the expected-utility framework, HARA-functions come in the forms listed in (0). Propostion allows us to derive explicit (µ, σ)-functions that exhibit HARA: Proposition 2. Let a 0 and b R. Then the following are equivalent: A(y) = a + b y for all y Y. There exists a positive constant d, d 2 such that U(µ, σ) equals { [ ] } U H (µ, σ) = exp b d (a + bµ) b d2 (bσ) b. () Proof: From Proposition we know that HARA-functions must solve the condition G(µ, σ; a, b) = 0 in the two-parameter framework. This constitutes a partial differential equation for U(µ, σ), (a + bµ) (U µ U σµ U σ U µµ ) + bσ (U µ U σσ U σ U µσ ) = 0, (2) which can be solved by a separation of variables. Using the approach U(µ, σ) = h(µ) g(σ), we break (2) into the following ordinary differential equations: (a + bµ) h (µ) 2 h(µ)h (µ) h(µ)h (µ) bσ g (σ) 2 g(σ)g (σ) g(σ)g (σ) = c = c 7

9 where c is a common constant. Without loss of generality, we can set c =. 4 irrelevant multiplicative constants, these equations have the solutions { } { d h(µ) = exp (a + bµ) b and g(σ) = exp d 2b b b σ } b. Up to where d, d 2 > 0 due to the desired monotonicity properties in µ and σ. Multiplying these functions then yields U H as defined in (). Hence, HARA-functions in the two-paramter framework come as exponential functions where the exponent consists of the difference of two power terms. The first term is increasing in the mean and corresponds to the expression in the vnm-function of the HARA-type (0). The second term is increasing in the standard deviation, reflecting risk-aversion. Seen independently from the EU-approach, (µ, σ)-utility functions are ordinal concepts: If U(µ, σ) represents preferences over some set of lotteries like M and Γ : R R is a strictly increasing function, then Ũ(µ, σ) = Γ(U(µ, σ)) represents preferences over M in the same manner. However, if we want to interpret (as we do), two-parameter preferences as a specific representation of the EU-framework in the sense of (5), then risk aversion implies that transformations of (µ, σ)-utility functions must be restricted to such that preserve both monotonicity (6) and concavity (7). We state this formally in 5 Lemma Suppose that in the location-scale framework, if U(µ, σ) represents preferences over M in the same way as function u(y) does over Y and that u(y) exhibts risk aversion. Let Γ : R R be a strictly increasing and strictly concave function. Then, the function Ũ(µ, σ) := Γ(U(µ, σ)) also represents preferences over M in the same way as function u(y) does over Y. Proof: Let Γ > 0 > Γ and U(µ, σ) be strictly concave (and, therefore, strictly quasiconcave). It is then easy to show that Ũ(µ, σ) exibits Ũµ(µ, σ) > 0, Ũ σ (µ, σ) < 0, Ũ µµ (µ, σ) < 0, Ũσσ(µ, σ) < 0 and Ũµµ(µ, σ)ũσσ(µ, σ) Ũµσ(µ, σ) 2 > 0. 4 Otherwise, use variables ã := a/c and b := b/c. It can be shown that in order for U to be concave, c must be positive. 5 Note that this does not contradict the well-known fact that vnm-utility functions may only undergo increasing linear transformations. In (5), preferences over lotteries would be a preserved by considering Γ(Eu(y)) rather than Eu(y) for all strictly increasing functions Γ; the VNM-function u(y) is not affected by this. 8

10 Taking the logarithm of () leads to the following admissable preference representation of HARA-functions: Ũ H (µ, σ) = b [ ] (a + bµ) b d (bσ) b where d = d 2 /d > 0. Observe that Ũ H (µ, 0) = u H (µ) which trivially represents the no-uncertainty case from (5). (3) The special cases of constant relative risk aversion, constant absolute risk aversion, or quadratic utility emerge for b = 0, a = 0, and b =, respectively. 6 Applications 6. Behavioural Implications For utility function () (and, naturally, also for (3)), the marginal rate of substitution between µ and σ is α H (µ, σ) = d ( ) /b ( ) /b bσ σ = d (4) a + bµ a/b + µ which is positive (we set d = d 2 /d > 0). One easily verifies that this indeed satisfies G(µ, σ; a, b) = 0. Moreover, this is can easily be used in comparative static exercises. Consider, e.g., the standard portfolio model with one safe and one risky asset. An investor is endowed with initial welth w > 0 which has to be divided between a riskfree asset with a zero rate of return (where zero is an innocuous but convenient normalization) and a risky asset with a random return s. Denoting by q the amount invested in the risky asset, final wealth is given by y = w + q s. Suppose that s has cumulative distribution F ; then µ s := E F s and σ s := E F [(s µ s ) 2 ] are the mean return and the standard deviation of s. In terms of the two-parameter framework, the investor would choose q as to maximize U(µ y, σ y ) where µ y = E F y = w + q µ s and σ y = E F [(y µ y ) 2 ] = q σ s. The firstorder condition for an optimal choice q then simply is α(w + q µ s, q σ s ) = σ y /µ s. With HARA-preferences (), this becomes: ( σ s q a/b + w + µ s q ) /b = d σ s µ s. Given that w does not appear on the RHS, the bracketed expression on the LHS must not depend on w either. This implies that the optimal investment strategy can be written as q = β (a + b w) 9

11 where β does not depend on w. This is exactly (and, given our modelling approach, not very surprisingly) the linear investment strategy that emerges from the EU-framework under linear risk-tolerance. In particular, if b = 0 investment is wealth-independent (CARA) while for a = 0, the wealth elasticity of q is constant at one (CRRA). 6.2 Related Functions It is quite obvious from () and especially from (3) that the functional form (4) suggested by Saha (997) is closely related to the functional form identified in our Propositions and 2 to be the (µ, σ)-equivalent to linear risk-tolerance. Less obvious perhaps, there also exists a relation to Roy s function (3). Observe that for this function, the marginal rate of substituion is given through: 6 α R (µ, σ) = µ a σ which is the same as for the quadratic case (b = ) in (4). Hence, the safety-first principle can (at least in the form proposed by Roy (952)) be interpreted as consistent with the assumption of linear risk tolerance. 7 Conclusion In this note, we transfer the concept of linear risk tolerance to the mean-variance framework. We show how the requirement of a hyperbolical Arrow-Pratt index translates from the EU-approach into a condition on the marginal rate of substitution between return and risk in the two-parameter approach. A spin-off from this translation is a partial differential equation which can be utilized to derive a functional class that might be employed in parametric models using the mean-variance approach. As a simple example of a portfolio problem demonstrates, the behavioural implications are indeed fully equivalent to those derived in the EU-framework. 6 This also provides a nice interpretation of the safety-first principle underlying which motivated Roy (952): Maximizing utility is equivalent to maximizing risk-aversion. 0

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