Multiple Risks and Mean-Variance Preferences

Multiple Risks and Mean-Variance Preferences

Thomas Eichnera and Andreas Wagenerb

a) Department of Economics, University of Bielefeld, Universit¨atsstr. 25, 33615 Bielefeld, Germany. E-mail: [email protected] b) Institute of Social Policy, University of Hannover, Koenigsgworther Platz 1, 30167 Hannover, Germany. E-mail: [email protected]

Subject classification: Area of Review:

Decision analysis: Risk

Decision analysis

Abstract: We analyze comparative static effects under uncertainty when a decision maker has mean-variance preferences and faces a generic, quasi-linear decision problem with both an endogenous risk and a background risk. In terms of mean-variance preferences, we fully characterize the effects of changes in the location, scale, and concordance parameters of the stochastic environment on optimal risk-taking. Presupposing compatibility between mean-variance and expected-utility approach we then translate these mean-variance properties into their analogues for von-Neumann-Morgenstern utility functions.

2

1

Introduction

Comparative statics for decision problems with uncertainty over monetary outcomes have since long found the interest of economists. Most of the numerous studies in that field operate within the expected utility (EU) framework, the standard (but not entirely undisputed) approach to rational choice under risk. The analyses encompass scenarios with single as well as with multiple risks, and they set up generic decision models (Feder, 1977; Dionne et al., 1993; Bigelow and Menezes, 1995) as well as specific applications for portfolio selection (Fishburn and Porter, 1976), firm behaviour (Sandmo, 1971), insurance demand (Meyer, 1992; Hadar and Seo, 1992), hiring under uncertainty (Feder, 1977), and many others. Most prominent among the various alternatives to the EU-approach is mean-variance analysis 1 which represents preferences over lotteries as functions of the mean and the variance or standard deviation of final wealth. Since its (re-)invention2 by Markowitz (1952) and Tobin (1958), meanvariance analysis has been highly popular both in economic theory (notably in the theory of finance) and in practical applications. Its main merits are expository simplicity and an intuitive interpretation: all effects can be couched in terms of risk and returns, and mean-variance models remain two-dimensional even with multi-dimensional risks or choice variables. In recent years mean-variance analysis has also re-gained some attention among theorists (Lajeri, 1995; Lajeri and Nielsen, 1995, 2000; Wagener, 2002; Eichner and Wagener, 2003a; Mathews, 2004). Yet, in spite of the widespread use of mean-variance analysis in theory and practice, comparative static analyses in that framework are remarkably scant, both in absolute number and relative to the EU-paradigm. They include an early graphical analysis of the behaviour of the competitive firm under uncertainty by Hawawini (1978), a formal analysis of the portfolio problem with a riskless and a risky asset by Lajeri-Chaherli (2003) and the analysis of linear risk tolerance in Wagener (2005). Eichner and Wagener (2003a) employ the notion of variance vulnerability to characterize the effects of changing an independent background risk in a generic decision problem. Lajeri-Chaherli (2002, 2005) investigates the comparative statics of different risk premia, employing risk-attitudes such as properness and standardness. In the standard 1

This approach comes under different labels: two-parameter approach, mean-standard deviation approach,

(µ, σ)-approach etc. 2 Markowitz (1999) provides a historical account of mean-variance analysis.

1

portfolio problem augmented by an independent background risk, Eichner (2007) shows that the properties of properness and standardness also suffice for the comparative static effect of increases in background risk. In this paper, we aim at a systematic and comprehensive analysis of behavioural responses in the mean-variance framework. For a generic decision problem with two risks we characterize the comparative-static effects for the full set of distribution parameters (i.e., for the means, variances, and the covariance of the two random variables). We trace back all comparative static effects to simple and intuitive sensitivity or curvature properties of the marginal rates of substitution between risk and return (i.e., of the willingness-to-pay for risk reductions). This is the first contribution of this paper. Generally, mean-variance and EU-approach constitute distinct frameworks for decision making under risk (Ormiston and Schlee, 2001). The former is largely deemed deficient relative to the latter (which derives much of its appeal from the underlying von-Neumann-Morgenstern [vNM] axiomatization). However, in certain contexts, EU- and mean-variance approach are closely related. In fact, they are perfect substitutes if (and only if) all attainable lotteries can, after normalizing them by their means, be completely ordered by the Rothschild-Stiglitz criterion for increases in risk (Bigelow, 1993). This concept of normalized risk comparability generalizes the location-scale property which requires that attainable lotteries differ from one another only by the mean and standard deviation of final wealth (Meyer, 1987; Sinn, 1983). In particular, it applies when a single choice variable and a one-dimensional risk interact in a linear way — as it happens in a wide range of important economic applications, encompassing the problems of co-insurance, hedging, portfolio choice, liquidity demand, or output choices under uncertainty.3 For set-ups that possess the location-scale property, comparative-static effects in the meanvariance framework have equivalent counterparts in terms of expected utility and vNM utility indexes. Changes of means, variances and covariance correspond to first-order stochastic dominance shifts, mean-preserving contractions, or changes in the concordance structure between random variables. For the EU-framework, characterizations of the corresponding comparative 3

For multi-dimensional risks, the location-scale property is satisfied whenever final monetary outcomes result

from a linear combination of risks that jointly possess an elliptically symmetric distribution (Chamberlain, 1983; Owen and Rabinovitch, 1983). For more on this, see Section 5.

2

statics are well-known from the literature (see Section 5 for an extensive discussion). We provide “translations” of these characterizations in terms of the mean-variance approach. This is the second contribution of this paper. Naturally, characterizations of comparative-static effects derived in a mean-variance setup remain, once appropriately translated, valid also in the EU-framework on that subset of decision problems where the two approaches are perfect substitutes. We exploit this observation to derive new insights for the EU-framework via the detour of mean-variance analysis. E.g., comparative statics of changes in background risks have so far only partly been analyzed within the EUframework proper. In particular, only little is known about the comparative statics of adding (or, more generally, of increasing) dependent background risks.4 By contrast, the mean-variance characterizations we establish for changes in background risks do not require any restriction on the dependence structures between endogenous and background risks or on the presence or absence of initial background uncertainty. Within the (reasonably general) location-scale framework, the full (and yet simple) characterization of comparative statics for changes in dependent risks, thus, generates new insights also for models of decision making using the EU-approach. This is a third contribution of our paper. We proceed as follows: Section 2 sets up a generic decision problem with mean-variance preferences. In Section 3 we characterize, in terms of restrictions on preferences, comparative static effects for changes in each of the parameters of the risky environment. Section 4 clarifies the relation between EU- and mean-variance framework within the location-scale property and translates the conditions obtained in Section 3 into their EU-analogues. Section 5 embeds these results into the existing literature on the EU-approach. Section 6 concludes. 4

Lajeri-Chaherli (2002) and Eichner (2005) are exceptions. While Lajeri-Chaherli (2002) analyzes the addition

of a dependent risk on equivalent and compensating risk premia, Eichner (2005) studies the portfolio problem with dependent background risk and points out that properness is sufficient for the comparative static effect of increases in background risk.

3

2

Set-up

An individual decision maker chooses among different lotteries over his final wealth y. His preferences for lotteries are represented by a two-parameter utility function V : R+ × R → R,

V = V (σy , µy ),

(1)

where σy ∈ R+ and µy ∈ R denote, respectively, the standard deviation and the expected value of random final wealth y, calculated for an underlying probability distribution (that will most often be suppressed in our notation). Generally, for a random variable  we shall denote its expected value by µ and its standard deviation by σ . The preference functional V (σy , µy ) is assumed to be at least four times continuously differentiable, increasing in µy (non-satiation), decreasing in σy (risk aversion), and to exhibit strictly convex indifference curves in (σy , µy )-space. Denoting partial derivatives with subscripts, we, thus, require that the following properties are satisfied for all (σy , µy ) ∈ R++ × R:5 Vµ (σy , µy ) > 0;

(2a)

Vσ (σy , µy ) < 0;

(2b)

V (σy , µy ) is strictly quasi-concave in (σy , µy ).

(2c)

We denote by m(σy , µy ) := −

Vσ (σy , µy ) Vµ (σy , µy )

the marginal rate of substitution (MRS) between σy and µy . Lajeri and Nielsen (1995, 2000) and Ormiston and Schlee (2001) identify m(σy , µy ) as the two-parameter analogue of the ArrowPratt concept of absolute risk aversion. Following Tobin (1958, p. 13) and Sinn (1983, pp. 112f.) we assume that (σy , µy )-indifference curves enter the µy -axis with slope zero, representing risk neutrality with respect to very small risks: m(0, µy ) = Vσ (0, µy ) = 0 5

∀µy ∈ R.

As is shown in Eichner (2007, Proposition 3), quasi-concavity (2c) requires, for all (σy , µy ), −Vσ2 · Vµµ + 2Vµ · Vσ · Vµσ − Vµ2 · Vσσ > 0.

4

(2d)

We consider a class of decision problems where final wealth y is a function of a one-dimensional and non-negative choice variable q ≥ 0. In particular, we assume the following quasi-linear form of Feder (1977)’s general approach: y(q) = x · q + f (q) + z.

(3)

Both x and z are random variables. As the impact of q on the distribution of final wealth y directly works through x, we call x the endogenous risk; the “passive” random variable z will be referred to as the background risk. The interaction between the endogenous risk x and the choice variable q is linear. The function f (q) is assumed to be smooth and concave in q, and to satisfy f (0) = 0. This implies that f 0 (q) · q − f (q) ≥ 0 for all q ≥ 0. Without loss of generality, we assume that the expected value of z is non-negative.6 The dependence structure of endogenous risk x and background risk z is captured by their covariance σxz := Cov(x, z), and the degree of their linear dependence is measured by the Pearson coefficient of correlation ρ := σxz /(σx · σz ) ∈ (−1, 1). Denote by Θ := {(µx , µz , σx , σz , σxz ) |µx ∈ R, µz , σx , σz ∈ R+ , −σx σz ≤ σxz ≤ σx σz } the set of possible parameters; a typical element of Θ will be denoted by θ. Given (3), the expected value and the standard deviation of final wealth are given by: µy (q) = µx · q + f (q) + µz p σy (q) = σx2 · q 2 + σz2 + 2 · q · σxz .

(4) (5)

By an appropriate interpretation or re-definition of variables, specification (3) captures a wide array of standard economic choice problems under risk: portfolio choice, saving under uncertainty, insurance demand, output choices of competitive firms under uncertainty etc. To illustrate, consider the following Example: Portfolio choice with two assets.

An individual has one unit of wealth which

has to be divided on two assets with (random) net returns a and b. Denote by µc and σc the 6

Alternatively, we could replace z by a random variable z˜ + µz where z˜ has expected value zero and include

µz in f as µz = f (0). This would be notationally slightly more inconvenient without affecting any results.

5

mean and standard deviations of returns c = a, b and by σab their covariance. With pa and pb as the amounts that go, respectively, to assets a and b, final wealth y is given by y(q) = pa · a + pb · b = qx + z where we set q := pb = 1 − pa and z := a. Moreover, x := b − a measures the difference between the returns of the assets (and f (q) ≡ 0).7 Obviously, µx = µb − µa , σx2 = σb2 + σa2 − 2σab , and σxz = σab − σa2 (which is zero if and only if return a is certain). Clearly, µy (q) = µx · q + µz = pa · µa + pb · µb , q p σy (q) = σx2 · q 2 + σz2 + 2 · q · σxz = p2a · σa2 + p2b · σb2 + 2 · pa · pb · σab .

(6a) (6b)

Observe that this problem is structurally identical to a portfolio problem with one risky asset, one safe asset, and some (possibly uncertain background) wealth (see Fishburn and Porter, 1976; Eichner, 2007, on that problem).

Turning back to the general case, the decision maker solves the following problem: max V (σy (q), µy (q)) q≥0

subject to (4) and (5).

(7)

Restricting attention to interior solutions of this program, the optimal level q ∗ = q ∗ (θ) > 0, where θ ∈ Θ, is then determined by the first-order condition Φ (q ∗ , θ) = 0

(8)

where we set Φ (q, θ) := =

∂µy ∂σy (q) · Vµ (σy (q), µy (q)) + · Vσ (σy (q), µy (q)) ∂q ∂q   ∂σy (q) µx + f 0 (q) · Vµ (σy (q), µy (q)) + · Vσ (σy (q), µy (q)). ∂q

The second-order condition for a maximum8 is satisfied due to the quasi-concavity of V (σy , µy ) in (σy , µy ) and the concavity of f (q) in q. 7 8

The reason for the redundany in notation will (hopefully) become clear below. After some rearrangements, the second-order condition can be written as  2   2 −Vσ2 Vµµ + 2Vµ Vσ Vµσ − Vµ2 Vσσ ∂σy (q ∗ ) σx2 σz2 − σxz 00 ∗ Φq (q ∗ , θ) = − + f (q )V + < 0. µ ∂q Vµ2 σy (q ∗ )

6

We will henceforth assume that q denotes a risky activity in the sense that it marginally increases the standard deviation of final wealth at its optimal level q ∗ : ∂σy (q ∗ ) σ 2 · q ∗ + σxz = x > 0. ∂q σy (q ∗ )

(9)

From (8) this obviously implies that, at the optimal level, activity q marginally increases expected final wealth, too: µx + f 0 (q ∗ ) > 0. We wish to ensure that increases in σx or σz indeed constitute increases in risk, i.e., they lead to an increase in the standard deviation of final wealth. This obviously holds whenever we keep the dependence structure between x and z fixed upon a change in σx or σz ; from (5), ∂σy (q ∗ ) >0 ∂σx

and

∂σy (q ∗ ) >0 ∂σz

hold for all q ∗ = q ∗ (θ). In addition we require that increases in the standard deviations of x and z constitute increases in risks also if we keep their correlation fixed but allow their covariance to vary. Rewriting the standard deviation of final wealth as σ ˜y (q) =

p

σx2 · q 2 + σz2 + 2 · q · ρ · σx · σz ,

we require that ∂σ ˜y (q ∗ ) ≥0 ∂σx

and

∂σ ˜y (q ∗ ) > 0. ∂σz

The first of these conditions is equivalent to (9), i.e., to the fact that q is a risky activity. The second condition requires σz + q ∗ ρσx > 0 or σz2 + q ∗ · σxz > 0.

(10)

This automatically holds whenever x and z are non-negatively correlated; for negative values of σxz , condition (10) constitutes an assumption on its own. Example (ctd.):

To illustrate these assumptions, we return to the two-asset portfolio example

from above. It is straightforward to show that (9) and (10) are together equivalent to: pb · σa2 + pa · σab > pa · σb2 + pb · σab > 0

7

(where pa = 1 − q ∗ and pb = q ∗ have to be evaluated at their optimal levels). From (6b), this is tantamount to: ∂σy ∂σy > 0. > ∂pb ∂pa Hence, both investments pa and pb are risky in the sense that independently increasing either leads to higher overall risk. However, activity pb is (in the optimum) riskier than activity pa – which, by the constraint that pb = q and pa = 1 − q must add up to unity, implies that q itself is a risk-increasing activity. Hence, conditions (9) and (10) have quite simple interpretations.

To summarize, we consider quasi-linear choice problems of type (3) where increases in the standard deviations of the endogenous or of the background risk always lead to an increase in overall risk (i.e., where (9) and (10) hold).

3

Comparative Statics

We are interested in how optimal risk-taking responds to changes in the stochastic parameters of the decision problem. In this section we provide complete characterizations of the comparative statics of q ∗ (θ) with respect to all components of θ.

3.1

Changes in the Background Risk

Our first result deals with the comparative statics for changes in the distribution of the background risk z: Proposition 1. a) An agent will, for all θ ∈ Θ, increase q ∗ (θ) upon an increase in µz if and only if mµ (σy , µy ) < 0 for all (σy , µy ). b) An agent will, for all θ ∈ Θ, decrease q ∗ (θ) upon an increase in σz if and only if mσ (σy , µy ) > 0 and mσσ (σy , µy ) > 0 for all (σy , µy ). Proof: The first-order condition (8) can be rewritten as   ∂σy (q ∗ ) ∗ 0 ∗ ∗ ∗ Φ(q , θ) = Vµ · µx + f (q ) − · m(σy , µy ) = 0. ∂q 8

(11)

a) Implicit differentiation of (11) with respect to µz yields   Vµ ∂σy (q ∗ ) ∂q ∗ =− · mµ (σy∗ , µ∗y ) · − ∂µz Φq (q ∗ , θ) ∂q which is positive if and only if mµ (σy∗ , µ∗y ) < 0 for all (σy∗ , µ∗y ) (recall assumption (9)). b) Implicit differentiation of (11) with respect to σz yields, after some rearrangements,   m(σy∗ , µ∗y ) Vµ ∂σy (q ∗ ) σz ∂q ∗ ∗ ∗ · ∗· − mσ (σy , µy ) =− · ∂σz Φq (q ∗ , θ) ∂q σy σy∗

(12)

which is negative if and only if m(σy∗ , µ∗y ) − mσ (σy∗ , µ∗y ) < 0 σy

(13)

for all (σy∗ , µ∗y ). Obviously, this necessitates mσ (σy∗ , µ∗y ) > 0. Given this and (2d), condition (13) is equivalent to mσσ (σy∗ , µ∗y ) > 0.



As shown by Lajeri and Nielsen (1995) and Ormiston and Schlee (2001), the slope of (σy , µy )indifference curves with respect to µy determines whether absolute risk aversion increases or decreases. Formally, the more appealing property of decreasing absolute risk aversion (DARA) is represented in the two-parameter framework by mµ (σy , µy ) < 0

∀(σy , µy ) ∈ R+ × R.

(14)

Eichner and Wagener (2003a) show that convexity of the slope of (σy , µy )-indifference curves with respect to σy , i.e., ∀(σy , µy ) ∈ R+ × R,

mσσ (σy , µy ) > 0

(15)

together with mσ (σy , µy ) > 0, generally characterizes the comparative static effect that individuals behave in a more-risk averse way when they are confronted with an increase in an independent background risk. Inspired by the concept of “risk vulnerability” in the expected-utility framework, Eichner and Wagener (2003a) refer to property (15) as variance vulnerability.9 9

Note that mσ (σy , µy ) > 0 and the convexity of the indifference curves (2c) together imply that mµ (σy , µy ) < 0

(see also Lajeri-Chaherli, 2003, p. 90). Hence, DARA is a necessary condition for variance vulnerability. Variance

9

In short, Proposition 1 conveys that the individual behaves in a less risk-averse way when facing an increase in the expected value of the (possibly degenerate) background risk or a decrease in its variability if his preferences exhibit DARA (mµ (σy , µy ) < 0) or, respectively, variance vulnerability (mσ (σy , µy ) > 0 and mσσ (σy , µy ) > 0). For these observations, the distribution parameters µx and σx of the direct risk and the dependence structure between endogenous and background risk do not play any role.10

3.2

Changes in the Endogenous Risk

We now turn to the comparative statics for the distribution parameters of x and for the covariance. For (σy , µy ) ∈ R++ × R, define by εµ (σy , µy ) :=

∂m(σy , µy ) µy · ∂µy m(σy , µy )

(16)

the elasticity of the marginal rate of substitution between return and risk with respect to µy . Similarly, let εσ (σy , µy ) :=

∂m(σy , µy ) σy · ∂σy m(σy , µy )

(17)

denote the elasticity of the marginal rate of substitution between return and risk with respect to σy . With a grain of salt, the marginal rate of substitution between return and risk reflects the decision maker’s degree of risk aversion; its elasticities then measure how sensitive risk aversion is with respect to changes in return and risk. The following result shows that these sensitivity measures provide crucial information also in comparative static analyses: Proposition 2. vulnerability is related to two other concepts in the mean-variance approach, standardness and properness: LajeriChaherli (2002) shows that proper risk aversion is equivalent to the mean-variance utility function being quasiconcave and displaying DARA. Lajeri-Chaherli (2005) shows that standard risk aversion is equivalent to the mean variance utility function displaying DARA and decreasing absolute prudence. Eichner (2005, 2007) shows that properness and standardness, repectively, are both sufficient for variance vulnerability. 10 Conditions (11) and (12) admit Proposition 1 to be phrased more generally as follows: The decision maker will always respond to a change in µz [or σz ] with a change in q ∗ (θ) that is in opposite direction to ∂σy (q ∗ )/∂q if and only if mµ (σy , µy ) < 0 [or mσ (σy , µy ) > 0 and mσσ (σy , µy ) > 0]. Thus, Proposition 1 not only applies to risk-taking (in the sense of (9)) but also to risk-avoidance (e.g., in the contexts of insurance or hedging).

10

a) An agent will, for all θ ∈ Θ, increase q ∗ (θ) upon an increase in µx if and only if εµ (σy , µy ) < 1 for all (σy , µy ). b) An agent will, for all θ ∈ Θ, decrease q ∗ (θ) upon an increase in σx if and only if εσ (σy , µy ) > −1 for all (σy , µy ). c) An agent will, for all θ ∈ Θ, decrease q ∗ (θ) upon an increase in σxy if and only if εσ (σy , µy ) > 0 for all (σy , µy ). Proof: For k ≥ 0, verify that: 

εµ (σy , µy ) ≤ k

⇐⇒

εσ (σy , µy ) ≥ −k

⇐⇒

 Vµ ≥0 k · Vµ + µy Vµµ − Vσµ Vσ   Vσ k · Vσ + σy Vσσ − Vσµ ≤0 Vµ

(18) (19)

Implicit differentiation of (8) with respect to µx , σx , and σxy yields the following comparative statics: ∂q ∗ ∂µx

∂q ∗ ∂σx

∂q ∗ ∂σxz

  ∂σy∗ 1 ∗ ∂µy = − · Vµ + q · · Vµµ + q · Vσµ Φq (q ∗ , θ) ∂q ∂q    Vµ Vµ K(q ∗ ) ∗ = − · + µy · Vµµ − · Vσµ , Φq (q ∗ , θ) K(q ∗ ) Vσ   q ∗ · σx ∗ ∗ ∂σy ∗ ∂µy · (2 − A(q )) · Vσ + q · · Vσσ + q · · Vµσ = − Φq (q ∗ , θ) · σy∗ ∂q ∂q    q ∗ · σx · A(q ∗ ) 2 ∗ ∗ Vσ = − · − 1 · Vσ + σy · Vσσ − σy · · Vµσ , Φq (q ∗ , θ) · σy∗ A(q ∗ ) Vµ   1 ∗ ∗ ∂µy ∗ ∗ = − · (1 − A(q )) Vσ + q · · Vµσ + σy · A(q ) · Vσσ Φq (q ∗ , θ) · σy∗ ∂q    A(q ∗ ) 1 ∗ ∗ Vσ = − · − 1 Vσ + σy · Vσσ − σy · · Vµσ . Φq (q ∗ , θ) · σy∗ A(q ∗ ) Vµ

(20a)

(20b)

(20c)

Here we used the FOC (8) and we defined: K(q) :=

q ∂µy · µy ∂q

and A(q) =

q ∂σy · . σy ∂q

(21)

1 . K(q ∗ )

(22)

From (18) and (20a) we get ∂q ∗ ≥0 ∂µx Observe that K(q) =

qµx +qf 0 (q) qµx +µz +f (q)

⇐⇒

εµ (µ∗y , σy∗ ) ≤

∈ [0, 1] since µz ≥ 0 and f (q) is concave with f (0) = 0.

Independently of q ∗ , K will take the value of 1 if µz = 0 and f (q) is linear. Hence, we get ∂q ∗ ≥0 ∂µx

for all θ

⇐⇒ 11

εµ (µ∗y , σy∗ ) ≤ 1.

From (19) and (20b) we get ∂q ∗ ≤0 ∂σx

⇐⇒

εσ (µ∗y , σy∗ ) ≥ 1 −

2 . A(q ∗ )

(23)

⇐⇒

εσ (µ∗y , σy∗ ) ≥ 1 −

1 . A(q ∗ )

(24)

Similarly, from (19) and (20c) we get ∂q ∗ ≤0 ∂σxz

From (9), A(q) > 0 for all q. We now show that A(q) cannot exceed 1. Rewriting A(q) as A(q) =

σx2 · q 2 + q · σxz σz2 + q · σxz = 1 − σy2 σy2

(25)

leads, from (9) and (10) to A(q) ∈ [0, 1] for all q ≥ 0 (in particular, for q ∗ ). Moreover, independently of q ∗ , the case A = 0 can be arbitrarily closely approximated by letting σx approach zero, and A = 1 will hold when z is non-random (σz = σxz = 0). This provided, we obtain from (23) and (24) ∂q ∗ ≤ 0 for all θ ∂σx ∂q ∗ ≤ 0 for all θ ∂σxz

⇐⇒

εσ (µ∗y , σy∗ ) ≥ −1

⇐⇒

εσ (µ∗y , σy∗ ) ≥ 0. 

Proposition 2 fully characterizes the comparative statics for changes in the distribution of random variable X in terms of elasticities of risk aversion. Intuition for these elasticity conditions can be gained via a graphical representation, inspired by Hawawini (1978). Consider the (σy , µy )-plane. Rewrite the first-order condition (8) as: m(σy∗ , µ∗y ) =

µx + f 0 (q ∗ ) ∂σy (q ∗ ) ∂q

,

(26)

This defines a point of tangency where the left-hand side measures the slope of a (σy , µy )indifference curve while the right-hand side reflects the slope of the efficiency frontier, formally defined through the locus {σy (q), µy (q) | q ≥ 0}. Consider first a small increase in µx . Graphically, this will induce the decision maker to increase q (or, equivalently here, to opt for a higher σy ), if the change in the slope of the indifference curve that it induces is smaller than the change in the slope of the efficiency frontier, i.e., if mµ q ∗ < 1/(∂σy (q ∗ )/∂q) = 12

m . µx + f 0 (q ∗ )

(27)

Observe here that the left-hand side is (locally) proportional to mµ while the left-hand side is (locally) proportional to m. Hence, if a higher µx is supposed to increase the decision maker’s activity q, then risk aversion must, in relative terms, not increase too strongly upon increases in expected wealth. Since we wish this to always hold, we must consider the “hardest case”, which occurs when µz = 0 and f is linear (i.e. when µy = qµx + f (q) = qµx + f 0 (q)). This then leads to the elasticity condition εµ < 1 in (27). In an analogous fashion, increases in σx will lead to a reduction of q (and hence to lower expected wealth µy ), if the slope of the indifference curve increases more strongly in σx than the slope of the efficiency frontier. Formally, we have mσ

∂σy ∂ 2 σy /∂q∂σx µx + f 0 (q ∗ ) ∂ 2 σy >− · = −m . ∂σx (∂σy /∂q)2 ∂q∂σx ∂σy /∂q

(28)

Hence, if a larger σx is supposed to decrease the decision maker’s activity q, then risk aversion must, in relative terms, not decrease too strongly upon increases in risk. The “hardest” case here occurs when z is non-random (σz = σxz = 0 and σy = qσx ). Then (28) simplifies to εσ > −1. The same intuition holds for changes in the covariance. To summarize, the comparative statics of parameter changes depend on how sensitively the decision maker’s risk aversion, i.e., her willingness-to-pay for additional risks, responds to changes in expected wealth and wealth risk (measured by mµ and mσ ). Sensitivity is not measured in absolute terms, but relative to changes of the efficiency frontier which, at an optimum, is proportional to the (initial) value of risk aversion. This then naturally gives rise to elasticity conditions for risk aversion.

3.3

A Parametric Example

Let us illustrate Propositions 1 and 2 and their strenght by a parametric example. We apply the following specific utility function, which is due to Saha (1997): V (σy , µy ) = µνy − σyγ .

(29)

The monotonicity and quasi-concavity properties (2a) to (2d) require ν > 1 and γ > 1, which we henceforth assume. From m(σy , µy ) =

γ γ−1 1−ν ·σ · µy ν y 13

we obtain:11 γ(1 − ν) γ−1 −ν · σy · µy < 0 ν γ(1 − γ)(2 − γ) γ−3 1−ν mσσ (σy , µy ) = · σy · µy > 0 ν

⇐⇒

ν>1

(30a)

⇐⇒

γ>2

(30b)

εµ (σy , µy ) = 1 − ν < 1

⇐⇒

ν>0

(30c)

εσ (σy , µy ) = γ − 1 > k

⇐⇒

γ > k + 1.

(30d)

mµ (σy , µy ) =

Propositions 1 and 2 then predict the following: Corollary 1. Suppose that two-parameter preferences are given by (29). Then: a) an increase in µz will always induce higher risk-taking if and only if ν > 1; b) an increase in σz will always lead to lower risk-taking if and only if γ > 2; c) an increase in µx will always induce higher risk-taking if and only if ν > 0; d) an increase in σx will always reduce risk-taking if and only if γ > 0; e) an increase in σxz will always reduce risk-taking if and only if γ > 1. To see Corollary 1 in action in a specific decision problem, we once more take up the portfolio example introduced in Section 2. Using (6a) and (6b), utility maximization for (29) necessitates: 0=

∂V ∂q

= νµx µν−1 − γσyγ−2 (σxz + qσx2 ) y = νµx (µx q + µz )ν−1 − γ · (σxz + qσx2 ) · (q 2 σx2 + σz2 + 2qσxz )γ/2−1 =: Φ.

Parallel to items a) through e) in Corollary 1 we then get: a) Comparative statics with respect to µz : ∂q > 0 ⇐⇒ Φµz > 0 ⇐⇒ ν(ν − 1)µx (µx q + µz )ν−2 > 0 ∂µz — which is satisfied if and only if ν > 1. 11

Conditions (30a) and (30c) always hold since we assume ν > 1. The same applies to some of the conditions

presented below.

14

b) Comparative statics with respect to σz : ∂q < 0 ⇐⇒ Φσz < 0 ⇐⇒ (2 − γ)γ(σxz + qσx2 )σz σyγ−4 < 0 ∂σz — which is satisfied if and only if γ > 2. c) Comparative statics with respect to µx : ∂q > 0 ⇐⇒ Φµx > 0 ⇐⇒ ν(µx qν + µz )(µx q + µz )ν−2 > 0 ∂µx — which always holds if, as assumed, ν > 0. d) Comparative statics with respect to σx : ∂q 0 and σz2 + qσxz > 0. Hence, γ > 0 is sufficient for ∂q/∂σx < 0. To see necessity set σxz = 0 and let σz converge to zero. e) Comparative statics with respect to σxz : ∂q 0 and σz2 + qσxz > 0. Hence, γ > 1 is sufficient for ∂q/∂σxz < 0. To see necessity set σxz = 0 and let σz converge to zero. Obviously, the calculations for these comparative statics (though they still can be solved explicitly) are much more tedious than just “looking up” the general conditions provided in Propositions 1 and 2. This is also true for applications different from portfolio choice.

4

Mean-Variance Preferences and EU-Approach

Under certain conditions, the mean-variance approach and expected-utility (EU) approach are perfect substitutes. The most important and useful of these conditions for economic applications is the so-called location-scale property (Meyer, 1987; Sinn, 1983). The location-scale 15

condition requires that all feasible distributions differ only by location and scale parameters. In univariate settings the location-scale condition is satisfied if the decision variable and the sources of uncertainty interact in a linear way. In multivariate settings with linear interaction between the decision variable and the sources of risks the location-scale condition holds if the risks are jointly elliptically distributed (Chamberlain, 1983; Owen and Rabinovitch, 1983). Many standard problems in decision making under risk possess this feature – including those already mentioned above: portfolio choice, co-insurance, liquidity demand, output choices of competitive firms under uncertainty, etc. (also see the references cited in Section 1). Formally, the location-scale assumption requires that the set Y of lotteries that can be obtained by the agent’s choices only contains random variables whose distributions differ from one another by location and scale parameters (i.e., by means and standard deviations). Consider such a class of random variables y ∈ Y and assume that the union of their supports is a (possibly unbounded) interval Y of the real line. Then let s be the random variable obtained by normalization of one arbitrary y ∈ Y. The location-scale property is then satisfied if all y ∈ Y are equal in distribution to µy +σy s where µy and σy denote the mean and the standard deviation of the respective y ∈ Y. By M := {(σ, µ) ∈ R+ × R|∃y ∈ Y : (σy , µy ) = (σ, µ)} we denote the set of all possible (σ, µ)-pairs that can be obtained from y ∈ Y.12 Given a von-Neumann/Morgenstern (vNM) utility index u : R → R, the expected utility from a lottery y ∈ Y can be written in terms of the mean and the standard deviation of y: Z Eu(y) =

s

u(µy + σy s)dH(s) =: V (σy , µy ).

(31)

s

The interval (s, s) with s < s is the (not necessarily finite) support of s and H is its distribution. As shown by Meyer (1987), identity (31) generates various relations between EU- and twoparameter preferences. In particular, u0 (y) > 0 ∀y ∈ Y 12

⇐⇒

Vµ (σy , µy ) > 0

∀(σy , µy ) ∈ M;

(32a)

A formal relation between the set-up here and the decision problem in Section 2 is as follows: The set Y

contains all lotteries over final wealth y that can, for some parameter vector θ ∈ Θ, be obtained by an appropriate choice of q > 0. Then, M = {(σ, µ)|∃θ ∈ Θ : (σy (q), µy (q)) = (σ, µ) for some q ≥ 0}.

16

u00 (y) < 0 ∀y ∈ Y

⇐⇒

Vσ (σy , µy ) < 0

u0 (y) > 0, u00 (y) < 0 ∀y ∈ Y

=⇒

V (σy , µy ) is quasi-concave on M;

Vσ (0, µy )

=

0

∀(σy , µy ) ∈ M;

∀(0, µy ) ∈ M.

(32b) (32c) (32d)

These four relations match with assumptions (2a) to (2d) which we previously imposed on V . For the vNM index u(y) define A(y) := −

u00 (y) u0 (y)

as the Arrow-Pratt index of absolute risk aversion. As shown by Meyer (1987), A0 (y) < 0 ∀y ∈ Y

⇐⇒

mµ (σy , µy ) < 0

∀(σy , µy ) ∈ M.

(33)

Lajeri and Nielsen (1995, 2000) and Wagener (2002) show that also the property of prudence (convex marginal utility) has a simple two-parameter analogue: u000 (y) > 0 ∀y ∈ Y

⇐⇒

Vµσ (σy , µy ) > 0

∀(σy , µy ) ∈ M.

(34)

For z, b ∈ R such that z + b ∈ Y, we denote by RP (z, b) := −z ·

u00 (z + b) u0 (z + b)

and PP (z, b) := −z ·

u000 (z + b) u00 (z + b)

the index of partial relative risk aversion and the index of partial relative prudence, respectively. The following result clarifies the relationship between the elasticities of risk aversion and the measures RP and PP : Proposition 3. If u0 (y) > 0 > u00 (y) for all y ∈ Y the following statements are true for β ∈ {1, 2}: εµ (σy , µy ) ≤ 1

∀(σy , µy ) ∈ M

⇐⇒

RP (z, b) ≤ 1

∀(z, b) s.t. z + b ∈ Y,

εσ (σy , µy ) ≥ 1 − β

∀(σy , µy ) ∈ M

⇐⇒

PP (z, b) ≤ β

∀(z, b) s.t. z + b ∈ Y.

The proof of Proposition 3, which works through two lemmas, is relegated to the Appendix. Obviously, the index of partial relative risk aversion, RP (z, b), being smaller than one for all (z, b) is equivalent to the (normal) index of relative risk aversion, −zu00 (z)/u0 (z), being smaller than one (Hanson and Menezes, 1968). Similarly, if PP (z, b) < β for all (z, b), then −zu000 (z)/u00 (z) < β, too. Hence, from Proposition 3 we can identify 17

• the condition εµ (σy , µy ) < 1 in the two-parameter framework with the property that the index of relative risk aversion is smaller than one in the EU-framework; and • the condition εσ (σy , µy ) > −1 [0] in the two-parameter framework with the property that the index of relative prudence is smaller than two [one] in the EU-framework.

5

Translating Propositions 1 and 2

Against the backdrop of Section 4, Propositions 1 and 2 can be interpreted as two-parameter analogues for several well-known theorems on comparative statics in the EU-framework. The parameter changes considered in the two-parameter framework are closely connected to firstorder stochastic dominance (FSD) shifts and mean-preserving contractions (MPC) that are widely discussed in the EU-framework. Specifically, suppose that (x, z) are (jointly and separately) elliptically symmetrically distributed with means (µx , µz ), variance-covariance structure (σx , σz , σxz ), and some generating function (of no further interest here). • An increase in µx , leaving all other parameters unchanged, constitutes a FSD shift of x; similarly, a ceteris-paribus increase of µz is a FSD shift of z. • A decrease in σx , leaving all other parameters unchanged, constitutes a MPC of x; similarly, a ceteris-paribus decrease of σz is a MPC of z. • For elliptical distributions, the Pearson coefficient of correlation ρ is a copula-based measure of concordance (or association) between random variables (see Embrechts et al., 2002). Recalling σxz = ρσx σz , increasing the covariance while leaving σx and σz unchanged means that the random variables x and z become more concordant. Using these features, we can now verify that Propositions 1 and 2 indeed translate some wellknown results from the EU-framework to the two-parameter set-up. In addition, the meanvariance framework allows for some characterizations of comparative statics that are simpler or impossible to obtain in the more general EU-framework.

18

5.1

Changes in µz (Proposition 1a)

An increase in the expected value of a background risk is, if that risk is degenerate, tantamount to an increase in deterministic wealth. For that case, Proposition 1a just confirms the standard DARA-effect that wealthier people engage in higher risk-taking (Pratt, 1964; Lajari and Nielsen, 1995). Moreover, in the mean-variance framework, an increase in µz corresponds to a FSD shift in the distribution of the background risk. For independent background risks, such shifts have been investigated in the EU-framework by Eeckhoudt et al. (1996). They find that DARA in the sense of Ross (1981)13 — which is a stronger notion than Arrow-Pratt-DARA — is necessary and sufficient to make individuals behave in a less risk averse way when the background uncertainty ameliorates in the FSD-sense. DARA in the Arrow-Pratt sense is only a sufficient condition. In the less rich mean-variance framework, DARA in the Arrow-Pratt sense turns out to be both sufficient and necessary. Moreover, while Eeckhoudt et al. (1996)’s finding only applies when endogenous and background risk are independently distributed, Proposition 1a holds for an arbitrary dependence structure between the two risks.

5.2

Changes in σz (Proposition 1b)

The relation between an increase in σz and an increase (or the addition) of a background risk in the EU-framework has been studied in Eichner and Wagener (2003a). They show that variance vulnerability (i.e., mσ (σ, µ) > 0 and mσσ (σ, µ) > 0) is closely related to, and in various cases, coincides with concepts of risk vulnerability and related properties of VNM-utility functions. In the EU-framework, these properties capture the tempering effects of background risks, i.e., the idea that individuals reduce their risk-taking when confronted with increases in independent background uncertainty (see, e.g., Eeckhoudt et al., 1996; Gollier and Pratt, 1996).14 Let us (temporarily) assume that endogenous and background risk are independently distributed. For elliptically symmetric distributions this is equivalent to assuming that both risks are jointly and separately normally distributed (Fang et al., 1990, Theorem 4.11). Chipman (1973, Theo13

DARA in the sense of Ross (1981) means that there exists a scalar λ > 0 such that −u000 (y + c)/u00 (y + c) ≥

λ ≥ −u00 (y + c0 )/u0 (y + c0 ) for all y + c and y + c0 such that u is defined. 14 For the EU-framework, Eeckhoudt et al. (1996) establish the necessity of DARA for individuals to behave in a more risk-averse way when background risks get larger. This corresponds to the observation, made in Section 3, that mµ (σy , µy ) < 0 (i.e., the mean-variance analogue for DARA) is a precondition for variance vulnerability.

19

rem 1) has shown that mean-variance preferences under normality obey the differential equation Vσ (σy , µy ) = σy · Vµµ (σy , µy )

(35)

for all (σy , µy ). Using (35) and presupposing DARA, condition (13) can be rewritten as follows: m(σy , µy ) − mσ (σy , µy ) < 0 σy

⇐⇒

Vµµµµ (σy , µy ) · Vµ (σy , µy ) − Vµµ (σy , µy ) · Vµµµ (σy , µy ) < 0

⇐⇒

−

Eu0000 (y) Eu00 (y) > − Eu000 (y) Eu0 (y)

(36)

for all y = µy + σy s. Here, we used identity (31) and the fact that DARA requires u000 (y) > 0. Condition (36) necessitates

−

u0000 (y) u00 (y) > − u000 (y) u0 (y)

(37)

for all y. In conjunction with DARA, (37) is the property of local risk vulnerability which Gollier and Pratt (1996) show to be necessary and sufficient for the addition to an endogenous risk of a small, independent, and unfair background risk to increase risk aversion. However, (36) is stronger than (37) as the latter condition also captures increases of pre-existing background risks. Eichner and Wagener (2003a) show that Kimball (1993)’s property of standardness (i.e., both −u00 (y)/u0 (y) and −u000 (y)/u00 (y) are decreasing in y) is sufficient for (36) to hold.15 Standardness, however, is a refinement of local risk vulnerability (Gollier and Pratt, 1996). Departing from normality allows for stochastic dependence between background and endogenous risk. For that case, Eichner and Wagener (2003a) show that, if mean-variance and EU-approach are perfect substitutes, the property mσσ (σy , µy ) > 0 necessitates temperance, i.e., the underlying vNM utility index u to have a positive fourth derivative. As proved by Kimball (1993), Eeckhoudt et al. (1996), or Gollier and Pratt (1996), temperance is one of the necessary conditions in the EU-framework to make agents reduce their risk-taking when independent background risks increase (in certain specific senses). As expected, this property is relevant also for more general dependence structures between direct and background risk. 15

It should be mentioned that Lajeri-Chaherli (2005)’s standardness is compatible to Kimball (1993)’s stan-

dardness, while Lajeri-Chaherli’s (2002)’s properness is compatible to Pratt and Zeckhauser (1987)’s properness only if risks are normally distributed.

20

The condition in Proposition 1b, i.e., that m(σy , µy ) is increasing and convex in σy , is a single and quite simple requirement for increases in background risks (of whatever dependence with respect to the endogenous risk) to induce more risk-averse behaviour; no similar condition has yet been derived in the EU-framework. In fact, analyses of the impact of dependent background risks are still scarce. In a study on optimal financial risk-taking in the presence of return-correlated background uncertainty over income, Elmendorff and Kimball (2000) derive standardness as a sufficient condition for an increase in background risk to lead to a reduction in financial risk-bearing whenever background and return risks are non-negatively correlated; they exclude, however, negative correlations. In a similar framework, Arrondel and Calvo-Pardo (2002) analyse the effects of adding correlated background risks to a problem of portfolio choice. They observe that, in order to generate a tempering effect on risk-taking by adding small background risks, local risk vulnerability is no longer a necessary [sufficient] condition when background and endogenous risk are positively [negatively] correlated — which stands in marked contrast to the observation made by Gollier and Pratt (1996) for independent risks. In total, for the EU-framework no set of preference restrictions has so far been established that characterizes behavioural responses to the addition (not to speak of the increase of) dependent background risks. Against that backdrop, the simple mean-variance conditions in Proposition 1b are quite remarkable in capturing all sorts of correlation. To repeat, the price to be paid for this is the restriction to decision problems satisfying the location-scale property.

5.3

Changes in µx (Proposition 2a)

An increase in µx constitutes a FSD-shift in the endogenous risk, and behavioural responses to such shifts in the EU-framework have been studied intensively. Meyer (1992) shows that with independence between background risk and endogenous risk a FSD shift in the latter leads to an increase in the risky activity whenever relative risk aversion does not exceed unity (i.e., [yu0 (y)]0 > 0). This extends the observations previously made for the single-risk case by Fishburn and Porter (1976) and Cheng et al. (1987) to the multi-risk framework. Hadar and Seo (1992) and Tibiletti (1995) show that the characterization by Meyer (1992) generalizes to the case of dependent risks (if the FSD shift leaves the dependence structure between endogenous and

21

background risk unaffected). Against the background of Proposition 3, Proposition 2a then is the one-by-one translation of all these results to the two-parameter framework.

5.4

Changes in σx (Proposition 2b)

A decrease in σx constitutes a MPC of the endogenous risk. The behavioural responses to such changes in the EU-framework have, for the single-risk case, been studied by Diamond and Stiglitz (1974) or Bigelow and Menezes (1995), for the multiple-risk case with an independent background risk by Meyer (1992), and for multiple risks with dependent background uncertainty by Hadar and Seo (1992) and Tibiletti (1995). These studies share the result that a MPC in the endogenous risk leads to an increase in risk taking whenever individuals are prudent (u000 (y) > 0) and their index of relative prudence does not exceed two (i.e., [yu0 (y)]00 < 0). Against the background of Proposition 3, Proposition 2b together with (34) then is the one-byone translation of this observation to the two-parameter framework.

5.5

Changes in σxz (Proposition 2c)

Not much work has so far been devoted to the analysis of shifts in the dependence structure. In an EU-framework with two dependent risks, Tibiletti (1995, Theorem 4) shows that if the dependence structure between the two random variables changes such that the risks become more concordant (leaving the marginal distributions of the single risks unchanged), individuals will behave in a more risk-averse way if they are prudent and their index of relative prudence falls short of unity. Again, Proposition 2 (in its part c) together with (34) provides the twoparameter analogue. However, in the mean-variance case we obtain an equivalence result while Tibiletti’s theorem “only” provides a set of sufficient conditions, albeit for a more general class of decision problems.

6

Concluding remarks

In a generic decision model for an individual with mean-variance preferences, we (re-)examined the comparative statics with two types of risk: an “endogenous” risk, the size of which is

22

affected by the agent’s choices, and a “background risk” which is unalterable, but affects decision making through additional riskiness and possible correlation with the endogenous risk. Given the different nature of these risks, the behavioural responses to their changes are driven by different properties of the agent’s preferences. We provide full characterizations for changes in every single parameter of the stochastic environment. For the background risk, DARA (in the mean-variance sense) and variance vulnerability constitute necessary and sufficient conditions for the plausible properties that the individual engages in higher risk-taking when the background risk undergoes a FSD shift or a mean-preserving contraction. Comparative-statics for the endogenous risk are determined by the elasticities of risk aversion with respect to mean and variance. The latter also determines responses to changes in the dependence structure that make endogenous and background risk more concordant. Presupposing substitutability between mean-variance and EU-specification, we then translate our results, relating the mean-variance concepts of DARA, variance vulnerability, and the properties of the elasticities of risk aversion to, respectively, DARA (in the Arrow-Pratt sense), risk vulnerability and its subcategories, and the magnitudes of the indexes of relative risk aversion and relative prudence. Rephrased in that way, our results resonate well-known observations from the EU-framework, demonstrating that mean-variance analysis can indeed handle comparative static problems in a similar fashion as the EU-approach. This narrows the gap between the theoretical analyses of mean-variance and EU-approach considerably. An attractive feature of the conditions we derive for the mean-variance approach is their simple interpretation: all conditions are phrased in terms of the marginal rate of substitution between the risk and the expected value of wealth, i.e., in terms of the decision maker’s (marginal) willingness-to-pay for changes in risk. The monotonicity and elasticity properties of this willingness-to-pay may, for some readers, be more accessible and intuitive than the (possibly equivalent) conditions on higher-order derivates of utility functions and their composites that emerge from the EU-approach.

23

Appendix: Proof of Proposition 3 The proof will use functions F G, defined for (σy , µy ) ∈ R++ × R, α ∈ [0, 1] and β > 0 by: F (σy , µy , α) := Vµ (σy , µy ) + µy · Vµµ (σy , µy ) + α · σy · Vσµ (σy , µy ), G(σy , µy , α, β) := (β − α) · Vσ (σy , µy ) + µy · Vµσ (σy , µy ) + α · σy · Vσσ (σy , µy ). We first show that Proposition 2 can be equivalently couched in terms of F and G: Lemma 1. For parts b) and c), suppose that Vµσ (σy , µy ) ≥ 0. a) An agent will, for all θ ∈ Θ, increase q ∗ (θ) upon an increase in µx if and only if F (σy , µy , α) > 0 for all (σy , µy ) and for all α ∈ [0, 1]. b) An agent will, for all θ ∈ Θ, decrease q ∗ (θ) upon an increase in σx if and only if G(σy , µy , α, 2) < 0 for all (σy , µy ) and for all α ∈ [0, 1]. c) An agent will, for all θ ∈ Θ, decrease q ∗ (θ) upon an increase in σxy if and only if G(σy , µy , α, 1) < 0 for all (σy , µy ) and for all α ∈ [0, 1]. Proof: Conditions (20a), (20b), and (20c) can be expressed as follows: ∂q ∗ ∂µx ∂q ∗ ∂σx ∂q ∗ ∂σxz

  1 · F (σy∗ , µ∗y , A(q ∗ )) − µy (1 − K) · Vµµ , ∗ Φq (q , θ)   q ∗ · σx = − · G(σy∗ , µ∗y , A(q ∗ ), 2) − µy (1 − K) · Vµσ , ∗ ∗ Φq (q , θ) · σy   1 ∗ ∗ ∗ = − · G(σ , µ , A(q ), 1) − µ (1 − K) · V , y µσ y y Φq (q ∗ , θ) · σy∗ = −

where K and A were defined in (21). Recall from the proof of Proposition 2 that, independently of one another, K and A can vary between zero and one. Hence, F (σy , µy , α) > 0, G(σy , µy , α, 2) < 0 and G(σy , µy , α, 1) < 0 for all α ∈ [0, 1] are both necessary and sufficient to generate, respectively, ∂q ∗ /∂µx > 0, ∂q ∗ /∂σx < 0 and ∂q ∗ /∂σxy < 0 (observe that the terms •

including K always have the “right” sign.)

Lemma 2. If u0 (y) > 0 > u00 (y) for all y ∈ Y the following statements are true for all β > 0: F (σy , µy , α) ≥ 0

∀(σy , µy , α) ∈ M × [0, 1]

⇐⇒

RP (z, b) ≤ 1

∀(z, b) s.t. z + b ∈ Y,

G(σy , µy , α, β) ≤ 0

∀(σy , µy , α) ∈ M × [0, 1]

⇐⇒

PP (z, b) ≤ β

∀(z, b) s.t. z + b ∈ Y.

24

Proof: a) Using the derivatives of V (σy , µy ) yields (observe that y = µy + σy s): s  00 u (y)

Z F

=

u0 (y)

s



· (µy + ασy s) + 1 u0 (µy + σy s)dH(s)

s  00 u ((µy + ασy s) + (1 − α)σy s) u0 ((µy + ασy s) + (1 − α)σy s) s

Z =



· (µy + ασy s) + 1 u0 (µy + σy s)dH(s).

Since u0 (µy + σy s) > 0 for all µy , σy and s, the integral is positive [negative] for all (σy , µy , α) if and only if −z ·

u00 (z + b) < [>] 1 u0 (z + b)

where we set z = µy + ασy s and b = (1 − α)σy s. b) Using the derivatives of V (σy , µy ) one gets Z G =

s

  s · (β − α) · u0 (µy + σy s) + (µy + ασy s) · u00 (µy + σy s) sdH(s)

s

Z = −σy

s

000

00


u (µy + σy s) · (µy + ασy s) + β · u (µy + σy s)

Z

s

 tdH(t) ds,

s

s

where the second line follows from integration by parts. Since E(s) = 0, the inner integral is always negative. Hence, for u00 < 0, the entire integral is always positive [negative] if and only if −z ·

u000 (z + b) < [>] β u00 (z + b) •

where z and b are defined as above. Proof of Proposition 3: Putting together Proposition 2 and the two lemmas, we get: εµ (σy , µy ) ≤ 1 for all (µy , σy )

⇐⇒

∂q ∗ ≥ 0 for all θ ∈ Θ ∂µx

⇐⇒

F (σy , µy , α) ≥ 0

⇐⇒

RP (z, b) ≤ 1

∀(σy , µy , α) ∈ M × [0, 1]

∀(z, b) s.t. z + b ∈ Y,

which was to be shown. In a similar fashion one proceeds to deal with εσ (σy , µy ). Here, we have to add that PP (z, b) > 0 requires u000 (y) > 0 for which Vµσ (σy , µy ) > 0 is the mean-variance analogue (see (34)).

 25

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