Portfolio Allocation and Asset Demand with MeanVariance Preferences


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1 Portfolio Allocation and Asset Demand with MeanVariance Preferences Thomas Eichner a and Andreas Wagener b a) Department of Economics, University of Bielefeld, Universitätsstr. 25, Bielefeld, Germany. b) Institute of Social Policy, University of Hannover, Königsgworther Platz 1, Hannover, Germany. Abstract: We analyze the comparative static effects of changes in the means, the standard deviations and the covariance of asset returns in a standard portfolio selection problem when investors have mean variance preferences. Simple and intuitive characterizations in terms of the elasticity of risk aversion are provided. JELclassification: Keywords: D81 Mean, Variance, Elasticity of risk aversion.
2 1 Introduction The theory of portfolio selection, originating from the works of Markowitz (1952) and Tobin (1958) and underlying the capital asset pricing model (CAPM), is based on the idea that investors, when choosing between assets with uncertain prospects, trade off (undesirable) risks and (lookedfor) returns. In its most popular form, this idea is formalized by meanvariance (MV) or, equivalently, meanstandard deviation utility functions. Due to its expository simplicity and its ease of interpretation, MV analysis 1 is widely used in the theory and practice of finance and economics. Given that (asset) demand functions make up a main ingredient in models of (capital) markets (see Nielsen, 1990), it is quite remarkable that so far not much is known about the comparative statics of problems of portfolio choice in a MV framework. As Ormiston and Schlee (2001, p. 850) note: [L]ittle work has been done on the following basic question: How does a riskaverse investor s holdings of the risky asset, or optimal level of risktaking, change in response to a change in the return distribution? Interest in this question, however, dates at least back to Tobin (1958) s graphical analysis which highlights the ambiguity of the properties of asset demand functions. The reason why nothing much of generality can be said about the comparative statics of portfolio choices is the ubiquitous conflict of income and substitution effects (ibidem, p. 79). 2 While the decomposition into income and substitution effects clearly provides many insights into the motives that drive the properties of asset demand, it does not answer the potentially more relevant question for the types of risk preferences that lead to clear properties of asset demand functions, i.e., that decide which of the sides dominates in Tobin s ubiquitous conflict. To the best of our knowledge only Ormiston and Schlee (2001) and LajeriChaherli (2003) deal with these issues in a MV framework. For socalled standard portfolio problems (one safe and one risky asset), Ormiston and Schlee (2001) and LajeriChaherli (2003) derive sufficient conditions (basically, the analogue to the EUconcept of decreasing absolute risk aversion) 1 In portfolio analyses, meanvariance analysis is often confined either to the search for meanvariance efficient portfolios or to the (implausible) case of separable utility functions that are linear in the expected value and the variance of final wealth. For this paper, meanvariance analysis refers to the (constrained) maximization of utility functions that depend on the first two moments of the wealth distribution. 2 Sometimes these effects are referred to as indirect and direct effects, respectively.
3 such that the demand for the risky asset increases in its expected rate of return and decreases in its variance. 3 In this paper we continue the study of the comparative static properties of the MV model. Specifically, we characterize the properties of asset demand functions in a systematic fashion. In a classical portfolio problem with two (potentially) risky assets, we investigate the effects of changes in the means, standard deviations and covariances of the asset returns on the portfolio choices of investors with MV preferences. We are interested in restrictions on preferences that always (i.e., for all combination of random parameters) give rise to unambiguous properties of asset demand functions. As the key primitives we identify the elasticities of risk aversion with respect to the mean and the variance of final wealth (i.e., the relative responsiveness of the marginal rate of substitution between risk and return). These elasticities which are in principle empirically estimable enable us to provide simple and intuitively appealing characterizations for comparative static effects. Relative to the standard approach of decision making under risk, the expectedutility (EU) approach, MV analysis is often regarded as deficient. Indeed, the MV and the EU approach are mutually compatible only under specific restrictions on the choice problem (Meyer, 1987; Bigelow, 1993) or on distribution classes (Chamberlain, 1983). For the (not implausible) cases where problems of portfolio choice meet these restrictions (see Meyer and Rasche, 1992), MV analysis constitutes a perfect substitute for the EU approach. Generally, however, MV analysis should be interpreted as a genuine and insightful approach to decision making under risk that can stand on its own (Nielsen, 1990; Ormiston and Schlee, 2001). As such, it has experienced a recent renaissance in the theoretical literature (see Eichner, 2008, for an overview). Portfolio choices and their properties have been extensively studied in the EU framework. Since Fishburn and Porter (1976) s observation that risk aversion alone does not suffice to have investors in a standard portfolio problem increase their demand for a risky asset when the returns to that asset undergo a firstorder stochastically dominant (FSD) shift, the search for restrictions on preferences or although this is not our focus on the class of distributional changes that ensure certain intuitively appealing behavioural responses has produced numerous results. Presupposing that the EU and the MV approach are compatible, the MV elasticity conditions derived in this paper can naturally be related to these results. We shall refer to the 3 LajeriChaherli (2003) also discusses the comparative statics for a twoperiod problem with precautionary saving.
4 various findings later in this paper; here, we just wish to emphasize that each of our findings for the MV approach has a onetoone analogue in the EUframework. Section 2 sets up the framework for the analysis. Section 3 contains and discusses the results, which are proven in the Appendix. Section 4 briefly concludes. 2 Portfolio Allocation with Two Risky Assets 2.1 Preferences Consider a decision maker whose preferences over wealth lotteries are represented by a twoparameter utility function U : R + R R, (σ, µ) U(σ, µ) where σ R + and µ R denote, respectively, the standard deviation and the expected value of random wealth. Throughout the paper we assume that U is a C 2 function with the following standard properties 4 (σ, µ) > 0, (σ, µ) < 0, U(σ, µ) is strictly quasiconcave (1a) (1b) (1c) for all (σ, µ) R + R. Equations (1a) and (1b) reflect risk aversion and imply that indifference curves in (σ, µ)space are upwardsloped. Condition (1c) ensures that indifference curves of U are strictly convex in (σ, µ)space. Denote by α(σ, µ) = (σ, µ) (σ, µ) (2) the marginal rate of substitution between σ and µ, which is the MV analogue of the ArrowPratt measure of absolute risk aversion (Lajeri and Nielsen 2000; Ormiston and Schlee 2001). From α we derive ε µ (σ, µ) := α(σ, µ) µ µ α(σ, µ) (3) as the elasticity of risk aversion with respect to µ, and ε σ (σ, µ) := α(σ, µ) σ σ α(σ, µ) (4) 4 Subscripts denote partial derivatives. These properties are also imposed, e.g., in Meyer (1987), Nielsen (1990), or Ormiston and Schlee (2001).
5 as the elasticity of risk aversion with respect to σ. For k R + verify that 2.2 The Decision Problem ( ) ε µ (σ, µ) k k + µ y µ µ ε σ (σ, µ) k k + σ y ( σ µ In the portfolio problem at hand the investor s random final wealth Y is given by 0; (5) ) 0. (6) Y = qx + (1 q)z, (7) where X and Z are random variables representing the return from allocating all of wealth to asset X or Z, respectively, and q is the proportion of wealth the agent chooses to allocate to asset X. Denoting by µ w and σ w, respectively, the mean and standard deviation of a random variable w, the investor is assumed to choose q to maximize U(σ y, µ y ) from final wealth, taking the distributions of X and Z as given. Formally: max 0 q 1 U(σ y (q), µ y (q)) s.t. σ y (q) = q 2 σ 2 x + (1 q) 2 σ 2 z + 2q(1 q)σ xz, µ y (q) = qµ x + (1 q)µ z, (8) where the dependence structure of the risks X and Z is captured by their covariance σ xz. For sake of reference, we first consider the case µ x = µ z. Then the minimumvariance portfolio will be chosen. We assume that parameters σ x, σ z and σ xz are such that the minimumvariance portfolio requires investment in both assets. This will happen whenever σ y (0)/ q < 0 < σ y (1)/ q or, equivalently, if σ xz < min{σ 2 x, σ 2 z} (9) (Wright, 1987). Assumption (9) allows for positively correlated asset returns. It also implies that σ xz < 0.5(σ 2 x + σ 2 z) which ensures that σ(q) is strictly convex in q. With (9), the minimumvariance portfolio is then uniquely given by q = σ 2 z σ xz σ 2 x + σ 2 z 2σ xz (0, 1); (10) it will always allocate more than 50% to the asset with the lower variance: q 1/2 σ x σ z. (11)
6 Let us now discuss the general case of assets with unequal means. Specifically, but without loss of generality, suppose that µ x > µ z. Condition (9) ensures that it is optimal to invest some money in asset X: q is optimally positive. We shall henceforth assume that it is not optimal to invest everything in asset X (diversification). I.e., q is optimally smaller than one (condition (9) is necessary for this. 5 The optimal q (0, 1) then satisfies the FOC for problem (8): where φ := (µ x µ z ) (σ y (q ), µ y (q )) + σ y(q ) q (σ y (q ), µ y (q )) = 0, (12) σ y (q ) q = q σx 2 (1 q )σz 2 + (1 2q )σ xz σ y (q. (13) ) Due to the quasiconcavity of U and the convexity of σ y (q), this q is unique. 6 The FOC (12) can only hold if investment q in asset X is a risky activity in the sense that, at its optimal level q, it marginally increases the standard deviation of final wealth: σ y (q ) q > 0 (14) Obviously, if investment in X is riskincreasing, investment in Z must be riskreducing at q. In addition, we assume that increases in the standard deviations σ x and σ z constitute increases in risks if we keep the correlation of X and Z fixed, but allow their covariance to vary. Rewriting the standard deviation of final wealth as σ y (q) = q 2 σ 2 x + (1 q) 2 σ 2 z + 2q(1 q)ρσ x σ z, (15) where ρ is the Pearson coefficient of correlation between the returns to X and Z, we require that σ y (q ) σ x > 0 and σ y(q ) σ z > 0. (16) The optimal portfolio allocation q varies with the distribution parameters (µ x, µ z, σ x, σ z, σ xz ). We restrict attention to parameter combinations that satisfy µ x > µ z, conditions (9), (16) and lead to q < 1. In the propositions to follow the word always should be understood to encompass all parameter combinations with the aforementioned properties. 5 These assumptions also preclude existence problems for optimal meanvariance portfolios, as discussed in Nielsen (1990). 6 At any solution to (12), the secondorder derivative is negative due to the monotonicity and quasiconcavity ) 2 properties of U and due to the convexity of σ y(q): φ q(q ) = 1 ( ) U 2 σ µ + 2µ UµU 2 σσ + 2 σ y q 2 < 0. ( σy q U 2 µ
7 3 Comparative Statics We now consider three types of shifts in the distribution of assets, namely increases of mean returns, increases of standard deviations and changes of the dependence structure. 3.1 Changes in the RiskIncreasing Asset Our first result deals with the comparative statics for changes in the distribution of asset X. 7 Proposition 1. Suppose the investment in asset X is a risky activity (µ x > µ z ). (i) An investor will always increase the optimal share of asset X in the portfolio in response to an increase in the mean return of asset X if and only if ε µ (σ y, µ y ) < 1 for all (σ y, µ y ). (ii) An investor will always decrease the optimal share of asset X in the portfolio in response to an increase in the standard deviation of asset X if and only if ε σ (σ y, µ y ) > 1 for all (σ y, µ y ). The comparative statics of the riskincreasing asset in Proposition 1 are driven by the elasticity of risk aversion. A straightforward intuition behind these elasticity conditions can be gathered from a geometric interpretation. Recall that the firstorder condition (12) requires the strictly convex (σ, µ)indifference curve to be tangent to the upwardsloped efficiency frontier: 8 α(σ y (q ), µ y (q )) = µ x µ z σ y. (17) A small increase in µ x will lead to a higher investment in X (or, equivalently here, to a higher overall risk σ y ) if the change in the slope of the indifference curve upon an increase in µ x (which is proportional to α µ ) is smaller than the change in the slope of the efficiency frontier (which is locally proportional to α), i.e., if α µ q < 1/( σ y / ) = α/(µ x µ z ). As we wish this to always hold, we have to consider the worst case, which occurs when µ z = 0 (i.e., when µ y = qµ x ). This leads to the elasticity condition ε µ < 1: If an increase in the expected return of X is supposed to always trigger an increase in the exposure to X, then risk aversion must not increase too strongly in µ y, relative to its initial value. Similarly, increasing σ x will lead to lower exposure to X (and, thus, to a lower expected wealth µ y ) if the slope of the indifference curve reacts more strongly to an increase in σ x than 7 Proofs for all propositions are in the Appendix. 8 The efficiency frontier is given through the set {(σ y(q), µ y(q)) 0 q 1}.
8 the slope of the efficiency frontier (which locally is equal to α). Formally, α σ σ y σ x > 2 σ y q σ x µ x µ z ( σ y / q) 2 = σ y / q σ x α 2. σ y / q The hardest case here is that Z is a safe asset (σ y = qσ x ), which directly gives the elasticity condition ε σ > 1. Risk aversion must not deteriorate too elastically upon increases in risk. The same idea is also behind the results to follow: The effect of a parameter change depends on how the reaction of risk aversion (measured via α µ or α σ ) relates to the change in the slope of the efficiency frontier (which, in an optimum, is locally proportional to the value of risk aversion α). This naturally gives rise to elasticity conditions on risk aversion. Whenever meanvariance and expectedutility (EU) approach are compatible, the elasticity properties of risk aversion can be translated into properties of the vonneumannmorgenstern utility functions that underly the equivalent EUrepresentation. Eichner and Wagener (2008, Proposition 3) show that ε µ < 1 is equivalent to the index of relative risk aversion being smaller than one, and ε σ > 1 β is equivalent to the index of relative prudence being smaller than β in the EU framework. Hence, setting β = 2, Proposition 1 corresponds to the results obtained by Hadar and Seo (1990) for the case of independent asset returns and by Meyer and Ormiston (1994) for correlated asset returns. 3.2 Changes in the RiskReducing Asset Next, we turn to the comparative statics for the distribution parameters of asset Z. Proposition 2. Suppose the investment in asset X is a risky activity (µ x > µ z ). An investor will always decrease the optimal share of asset X (and, thus, increase the optimal share of asset Z) in the portfolio in response to an increase in the mean return of asset Z if and only if ε µ (σ y, µ y ) > 0 for all (σ y, µ y ). An investor will always increase the optimal share of asset X in the portfolio in response to an increase in the standard deviation of asset Z if and only if ε σ (σ y, µ y ) < 1 for all (σ y, µ y ). Given that α is positive, ε µ > 0 is equivalent to α µ > 0, which reflects increasing absolute risk aversion (IARA; see Ormiston and Schlee, 2001). The first part of Proposition 2 then conveys that assets X and Z are substitutes (i.e., the investor will always reduce her exposure to a riskincreasing asset if the mean return to a riskreducing asset increases) if and only if the investor s
9 preferences display IARA. For σ z = 0, this result is in perfect harmony with Fishburn and Porter (1976) s corresponding observation for the standard portfolio problem (one safe and one risky asset) in the EUframework, where IARA is found as a sufficient and necessary condition such that investors will always invest more in the safe asset when the rate of return of that asset rises. IARA (α µ > 0) also is a necessary condition if an investor should always increase his exposure to X upon an increase in the riskiness of the other asset. To see this, observe that ε σ < 1 requires that α σ < 0. With strictly convex (σ y, µ y )indifference curves (i.e., for α σ +αα µ > 0), this, in turn, necessitates α µ > 0. 9 Given that Proposition 2 rests on the empirically implausible concept of IARA, the following result checks for conditions on preferences such that q µ z > 0 and q σ z We obtain < 0, respectively. Proposition 3. Suppose the investment in asset X is a risky activity (µ x > µ z ). (i) An investor will always increase the optimal share of asset X in the portfolio in response to an increase in the mean return of asset Z if and only if ε µ (σ y, µ y ) for all (σ y, µ y ). (ii) An investor will always decrease the optimal share of asset X in the portfolio in response to an increase in the standard deviation of asset Z if and only if ε σ (σ y, µ y ) for all (σ y, µ y ). Although Proposition 3 characterizes the comparative static effects in terms of the elasticities of risk aversion, the result has to be classified as an impossibility result, since meanvariance utility functions with the property ε µ (σ y, µ y ) [ε σ (σ y, µ y ) ] for all (σ y, µ y ) do not exist. In view of Proposition 2 and 3, we conclude that under empirically plausible assumptions on preferences such as decreasing absolute risk aversion comparative statics for beneficial changes in the return of a hedge asset must necessarily remain ambiguous. There is no hope that Tobin s ubiquitous conflict between income and substitution can be generally resolved. 3.3 Changes in the Dependence Structure Finally, we turn to the comparative statics of the dependence structure captured by the covariance. 9 To our knowledge, no EUanalogue for this observation exists.
10 Proposition 4. Assume that the investment in asset X is a risky activity (µ x > µ z ). (i) Suppose the optimal share of asset X satisfies q < 1 2. Then the investor will always10 decrease the optimal share of asset X in the portfolio upon an increase in the covariance if ε σ (σ y, µ y ) > 1 for all (σ y, µ y ). (ii) Suppose the optimal share of asset X satisfies q > 1 2. Then the investor will always increase the optimal share of asset X in the portfolio upon an increase in the covariance if ε σ (σ y, µ y ) < 1 for all (σ y, µ y ). According to Proposition 4, it is crucial for the comparative statics of the covariance whether ε σ is larger or smaller than one. To better understand this, it is useful to employ concept of variance vulnerability, due to Eichner and Wagener (2003). An agent is said to be variance vulnerable if she reduces her risky activity in response to an increase in an exogenous variability of her wealth; such increases can be caused by the increase in a genuine background risk or, as here, by an increase in the covariance of the risks the agent faces (observe that σ y (q)/ σ xz > 0). A MV utility function can be shown to exhibit variance vulnerability if and only if α σσ > 0. If, as assumed in Tobin (1958) and Ormiston and Schlee (2001), α(0, µ y ) = 0 for all µ y, it can be shown that 11 ε σ 1 α + σα σ 0 α σσ 0. (18) To coin a term, let us call an agent with ε σ < 1 varianceaffine. An increase in σ xz has two effects: First, for any given q, it makes the portfolio riskier. Second, if q > 1/2 [alternatively, if q < 1/2] it decreases [increases] the marginal impact of q on the portfolio risk: 2 σ 2 y(q) σ xz q = 1 2q. Variancevulnerable [varianceaffine] agents would respond to an exogenous increase in the riskiness of their portfolio by a lower [higher] q (since σ y (q)/ q > 0). However, such a change in q can, in combination with the increase in σ xz, impact on the riskiness of the portfolio in an undesirable way unless q < 1/2 in case of the variancevulnerable agent or q > 1/2 in the case of the varianceaffine agent. 10 I.e., for all distribution parameters that lead to q < 1/2. Similarly, in (ii) always refers to all distribution parameters that lead to q > 1/2. 11 The second equivalence in (18) is proven in Eichner and Wagener (2003).
11 Starting from q < 1/2 [alternatively, q > 1/2], a reduction [an increase] of q means a move towards less diversification. Proposition 4, thus, provides conditions such that the investor responds to an increased correlation of asset returns with a more specialized portfolio. With this interpretation, Proposition 4 is very much in the spirit of the results obtained by Epstein and Tanny (1980) for the portfolio problem in the EUframework (although that analysis only deals with nonpositively correlated asset returns). Epstein and Tanny (1980, p. 25) show that less negatively correlated asset returns lead (in our notation and terminology) to a higher level of q if and only if the index of relative prudence exceeds (1 2q ). I.e., if q > 1/2 and the index of relative prudence is positive, then / σ xz > 0. Conversely, if q < 1/2 and the index of relative prudence is negative, then / σ xz < 0. Using the correspondence (mentioned earlier) found by Eichner and Wagener (2008, Proposition 3) that ε σ > 1 β in the meanvariance framework is equivalent to the index of relative prudence being smaller than β in the EU framework, Epstein s and Tanny s observation is identical to Proposition 4 (set β = 0). We should stress, however, that Proposition 4 also covers the case of positively correlated asset returns (as long as (9) is satisfied). 4 Concluding Remarks For a standard portfolio selection problem when investors have MV preferences, we analyzed the comparative static effects of changes in the stochastic parameters of asset returns. The key determinants for the properties of asset demand functions turn out to be elasticities of risk aversion with respect to the mean and the standard deviation of final wealth. Given that these elasticities can be empirically retrieved, our theoretical results are testable. E.g., Saha (1997) proposes a MV utility function of type V (σ, µ) = µ β σ γ (such that ε µ = β 1 and ε σ = 1 γ) and estimates (in a model of firm behaviour, not portfolio choice) that β range between 1.86 and 3.81 and γ between 1.96 to Using these estimates (purely for illustrative purposes) in Proposition 1 (with its threshold levels ε µ = 1 and ε σ = 1) would leave the comparative statics of changes in parameters of asset X empirically generally unclear. The portfolio problem analysed is an archetypal scenario out of a large class of linear choice problems (encompassing insurance demand, output choices with random prices etc.). Our approach and its results are, mutatis mutandis, applicable in many more MV settings than just portfolio choice problems.
12 References Bigelow, J.P. (1993). Consistency of meanvariance analysis and expected utility analysis. Economics Letters 43, Chamberlain, G. (1983). A characterization of the distributions that imply meanvariance utility functions. Journal of Economic Theory 29, Eichner, T. and A. Wagener (2008). Multiple risks and mean variance preferences. Discussion paper. Eichner, T. and A. Wagener (2003). Variance vulnerability, background risk and mean variance preferences. General Papers of Risk and Insurance Theory 28, Eichner, T. (2008). Mean variance vulnerability. Management Science 54, Epstein, L.G. and S. M. Tanny (1980). Increasing generalized correlation: a definition and some economic consequences. Canadian Journal of Economics 13, Fishburn, P.C., and R. B. Porter (1976). Optimal portfolios with one safe and one risky asset: effects of changes in rate of return and risk. Management Science 22, Hadar J. and T.K. Seo (1990). The effect of shifts in a return distribution on optimal portfolios. International Economic Review 31, Lajeri, F. and L.T. Nielsen (2000). Parametric characterizations of risk aversion and prudence. Economic Theory 15, LajeriChaherli, F. (2003). Partial derivatives, comparative risk behavior and concavity of utility functions. Mathematical Social Sciences 46, Markowitz, H. (1952). Portfolio selection. Journal of Finance 7, Meyer, J. (1987), Twomoment decision models and expected utility maximization. American Economic Review 77, Meyer, J. and MB. Ormiston (1994). The effect on optimal portfolios of changing the return to a risky asset: The case of dependent risky returns. International Economic Review 35,
13 Meyer, J. and R. H. Rasche (1992). Sufficient conditions for expected utility to imply meanstandard deviation rankings: Empirical evidence concerning the location and scale condition. Economic Journal 102, Nielsen, L.T. (1990). Existence of equilibrium in CAPM. Journal of Economic Theory 52, Ormiston, M.B. and E. Schlee (2001). Mean variance preferences and investor behavior. Economic Journal 111, Saha, A. (1997). Risk preference estimation in the nonlinear standard deviation approach. Economic Inquiry 35, Sharpe, W.F. (1964). Capital asset prices: A theory of market equilibrium under conditions of risk. Journal of Finance 19, Tobin, J. (1958). Liquidity preference as behavior towards risk. Review of Economic Studies 25, Wright, R. (1987). Expectation dependence of random variables, with an application in portfolio theory. Theory and Decision 22,
14 Appendix Proof of Proposition 1. (i) Implicit differentiation of (12) with respect to µ x yields = 1 [ µ x φ q (q ) = K 1(q ) φ q (q ) + q µ y q µ + q σ ] y q µ [ ( Uµ K 1 (q ) + µ y µ µ )]. (19) In (19) we used the firstorder condition (12) and defined K 1 (q) := q µ y µ y q = 1 µ z µ x. (20) Activity q being risky ensures via the first order condition (12) and (20) that K 1 (q) [0, 1]. In view of (5) and (19) we get 0 ε µ (σ y, µ y ) sup K 1 (q) = 1. µ x q (0,1) (ii) Implicit differentiation of (12) with respect to σ x yields = 1 [ 2 σ y σ x φ q (q ) = 1 φ q (q ) + σ ( y µy q σ x σ x q σ + σ )] y q σ [ 2 σ y + σ ( y σ y σ µ q σ x σ x q )]. (21) With the help of σ y σ x = q2 σ x σ y, and 2 σ y = qσ ( x 2 σ y q σ x σ y q q σ y ) = σ y σ x 1 q [2 A 1(q)], where expression (21) can be rearranged to: [ = q σ x σ x φ q (q )σ y = q σ x A 1 (q ) φ q (q )σ y A 1 (q) := q σ y σ y q = 1 (1 q)σ2 z + qσ xz, (22) σ 2 y [2 A 1 (q )] + σ y A 1 (q ) [[ ] ( 2 A 1 (q ) 1 + σ y ( σ µ σ µ )] )]. (23) Activity q being risky (i.e., σy(q ) q > 0) and σ y (q ) σ z > 0 (1 q)σ 2 z + qσ xz > 0
15 ensures A 1 (q) [0, 1]. In view of (6) and (23) we get σ x 0 ε σ (σ y, µ y ) 1 sup q (0,1) 2 A 1 (q) = 1. (24) Proof of Proposition 2. (i) Implicit differentiation of (12) with respect to µ z yields µ z = 1 φ q (q ) = K 2(q ) φ q (q ) [ + (1 q ) µ y [ Uµ K 2 (q ) + µ y In (25) we again used the FOC and defined q µ + (1 q ) σ ] y q µ ( µ µ )]. (25) (1 q) µ y K 2 (q) := µ y q = 1 µ x. (26) µ z Here, K 2 < 0, K 2 /φ q > 0, sup K 2 (q) = 0, and K 2 (q) is unbounded from below. Expression (25) is negative for all parameters if and only if ε µ (σ y, µ y ) > 0 for all (σ y, µ y ). (ii) Implicit differentiation of (12) with respect to σ z yields Here we put = 1 [ 2 σ y σ z φ q (q + σ y σ y ) q σ z σ z q = (1 [ q )σ z φ q (q )σ y = (1 q )σ z A 2 (q ) φ q (q )σ y A 2 (q) := ( σ µ )] ( )] [2 A 2 (q )] + σ y A 2 (q ) σ µ [[ ] ( 2 A 2 (q ) 1 + σ y σ µ (1 q) σ y σ y q = 1 qσ2 x + (1 q)σ xz σ 2 y )]. (27). (28) Assuming that q is a risky activity, i.e. σy(q ) q > 0, and σ y (q ) σ x > 0 qσ 2 x + (1 q)σ xz > 0 leads to 12 A 2 (q) (, 0] and 2/A 2 (q) 1 (, 1]. In view of (6) expression (27) is positive for all parameter values if and only if ε σ (σ y, µ y ) < 1 for all (σ y, µ y ). 12 If q converges to zero, which is implied by the FOC if σ x is large (possibly infinite) and if α is bounded from above, then lim q 0 A 2(q) = which proves that A 2(q) is not bounded from below.
16 Proof of Proposition 3. Part (i) follows from (25) and part (ii) from (27). Proof of Proposition 4. Implicit differentiation of (12) with respect to σ xz yields, after similar rearrangements as above: where = 1 [ 2 σ y σ xz φ q (q ) σ xz q + σ y σ y σ xz σy q = σ y σ y φ q (q ) 2 σ y σ xz q σ y σ y σ xz q σ xz ( µy q σ + σ )] y q σ + σ y ( σ µ = σy σ y σ xz q σ y φ q (q ) [G(q )α + σ y α σ ], (29) ) G(q) := σ y 2 σ y σ xz q σ y σ y σ xz q = σ y 1 2q q(1 q) σy q 1. (30) Since σy σ xz = q(1 q)/σ y is positive, σ xz < 0 G(q )α + σ y α σ > 0 ε σ (σ y, µ y ) > G(q ). Observe that we get G(q) 1 if and only if q 1/2. For q > 1/2, we get G(q) (, 1); the case G will occur if q is close to the minimumvariance portfolio q, i.e., when µ x is close to µ z. Hence, q 1/2 ε σ (σ y, µ y ) > 1 = q σ xz < 0 q > 1/2 ε σ (σ y, µ y ) < 1 = q σ xz > 0. (31a) (31b)
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