Beneficial changes in dependence structures and two-moment decision models

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Abstract

We analyze the comparative static effects of beneficial changes in the dependence structure between risks. In a quasi-linear decision model with an endogenous risk and a dependent background risk, a mean–variance decision maker will choose a lower level of risky activities upon an increase in the coefficient of correlation of the risks if, and only if, the elasticity of risk aversion is larger than −0.5. For elliptical distributions, the elasticity condition is equivalent to relative prudence being smaller than 1.

Introduction

The dependence structure between risks and changes thereof have recently found great attention among economists. For financial and insurance economics, e.g., the propagation of financial crises, international repercussions, correlated and contagious risks are just a few keywords that indicate the relevance of dependences between risks. A large amount of research has been devoted to measuring and ranking bundled risks in the presence of multivariate dependences (see, e.g., [7]). However, relatively little work has been done on the behavioural consequences of changes in the dependence structure between risks: How do rational decision makers adjust their risk portfolios when dependence structures change? Can conditions on preferences and/or risk distributions be given such that the responses of rational decision makers always go into a certain direction?

This paper sheds some light on this issue. We investigate changes in the dependence structure between an endogenous risk and a background risk in a quasi-linear decision model [1], [9]. Quasi-linear decision models encompass portfolio selection [10], the output decision of a competitive firm under price uncertainty [20], liquidity preference [21], or the co-insurance problem [12], [17], [22]. Characterizations of comparative static changes in the location, scale and dependence parameters in the quasi-linear model are employed by [6]. In the present paper, we start our analysis in the mean–variance framework, where the dependence structure is naturally captured by Pearson’s coefficient of correlation between the risks. In contrast to [6] where the correlation coefficient is held fixed but the covariance can vary, here we aim at analyzing the comparative statics of changes in the correlation coefficient. It is shown that a higher correlation between the two risks leads to less risk-taking if and only if the decision maker’s risk aversion with respect to the riskiness of his final wealth is larger than −0.5 (Proposition 1).

Focussing on the often-discussed case of elliptically distributed risks renders the mean–variance approach equivalent to the expected-utility approach [2]. Within this distributional class, the coefficient of correlation is indeed an appropriate measure for the strength of association between risks in the sense of precedence in the concordance order (see [16]). Moreover, the elasticity condition on risk aversion turns out to be equivalent to the condition that the index of relative prudence does not exceed one in the expected-utility framework (Proposition 2). We observe that a higher degree of concordance between the loss and background wealth induces a second-order dominance shift in total wealth and, thus, constitutes a beneficial utility change for all risk-averters (Proposition 3). Combining all this gives: If risks are elliptically distributed, a decision maker will respond to a beneficial change in the concordance of risks with lower risk-taking if, and only if, his degree of relative prudence is smaller than 1 (Proposition 4).

Our analysis adds to a small amount of literature on the comparative statics of changes of dependence structures in economic models. [8] seems to be the pioneering paper in the expected-utility framework; its discussion of increases in correlation shows that the third derivative of von-Neumann–Morgenstern utility functions matters (which is related to prudence). [11] searches for restrictions on the set of changes in beliefs (rather than on the types of preferences) that give rise to unambiguous effects of an increased dependence. Closer relations (to be detailed below) exist to [22] which shows that relative prudence being smaller than one is a sufficient condition such that beneficial changes in copulas lead to higher risk-taking. Finally, [6] points out that the same condition also characterizes the comparative static effects of changes in the covariance between an endogenous and a background risk.

The rest of this paper is structured as follows: Section 2 presents the decision problem and its comparative statics with respect to increased correlation in a mean–variance framework. Section 3 discusses the relationship between mean–variance and expected utility framework, focussing on changes in the dependence structure. Section 4 revisits the results obtained in Section 2 against this backdrop. Section 5 concludes.

Section snippets

A quasi-linear decision model

Preferences. A decision maker’s preferences over his final wealth Z are represented by the two-parameter utility function U=R+×RR,U=U(vZ,μZ), where vZ and μZ denote, respectively, the variance and mean of random final wealth Z. Similarly, for any random variable A, the variance and mean will be denoted by vA and μA, respectively.

Throughout the paper we assume that U is an at least twice differentiable function with Uv(vZ,μZ)<0,Uμ(vZ,μZ)>0 for all (vZ,μZ)R+×R. Subscripts attached to functions

Expected utility, elliptical distributions and stochastic orders

Expected utility and elliptical distributions. Ranking lotteries over wealth via the expected von-Neumann/Morgenstern (vNM) utility of wealth or via a preference function that depends on the mean and the variance of wealth can be done in a mutually consistent way if wealth is elliptically distributed. In our linear framework, this will happen whenever the random variables themselves have a joint elliptical distribution. Recall that a two-dimensional random vector (X,W) has a bivariate

Comparative statics revisited

Let us return to the comparative statics of the quasi-linear model of Section 2. We can now apply Theorem 1 to changes in the dependence structure. By item (iii) of Theorem 1 we, first, derive that a reduction in ρXW enhances the decision maker’s utility.

Proposition 3

Consider two bivariate elliptically distributed random vectors (X0,W0) and (X1,W1) with the same marginals and the correlation coefficients ρ0 and ρ1 , respectively. For i=0,1 , let UimaxqU[vZi(q),μZi(q)] , where the Zi are defined, after

Concluding remarks

Recalling that the risk elasticity of risk aversion is defined as εv=vZααv, it is obvious that αv>0 is a sufficient condition for an agent to increase his risky activities upon a beneficial change in the dependence structure. The condition αv>0 is known as variance vulnerability [5] and captures the comparative static effect that an agent optimally reduces his risky activity in response to an increase in the variance of an independent background risk [4]. Interestingly, with decreasing

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