Variance Vulnerability, Background Risks, and Mean-Variance Preferences

The Geneva Papers on Risk and Insurance Theory, 28: 173–184, 2003 c 2003 The Geneva Association
Variance Vulnerability, Background Risks, and Mean-Variance Preferences THOMAS EICHNER VWL IV, FB 5, University of Siegen, Hölderlinstr. 3, 57068 Siegen, Germany ANDREAS WAGENER Department of Economics, University of Vienna, Hohenstaufengasse 9, 1010 Vienna, Austria Abstract An agent with two-parameter, mean-variance preferences is called variance vulnerable if an increase in the variance of an exogenous, independent background risk induces the agent to choose a lower level of risky activities. Variance vulnerability resembles the notion of risk vulnerability in the expected utility (EU) framework. First, we characterize variance vulnerability in terms of two-parameter utility functions. Second, we identify the multivariate normal as the only distribution such that EU- and two-parameter approach are compatible when independent background risks prevail. Third, presupposing normality, we show that—analogously to risk vulnerability— temperance is a necessary, and standardness and convex risk aversion are sufficient conditions for variance vulnerability. Key words: mean-variance preferences, background risk, variance vulnerability JEL Classification No.: D81, D21 1. Introduction In this paper we investigate the effect of an increase in a mean-zero background risk on an agent’s willingness to take another independent risk when the agent has two-parameter, mean-standard deviation preferences. While this question has quite found extensive coverage in the expected-utility (EU) approach (see, among others, Gollier and Pratt [1996]; Kimball [1993]; Pratt and Zeckhauser [1987]), it has so far not been fully analyzed in the context of mean-standard deviation preferences. Lajeri-Chaherli [2002] recently studied the addition of independent undesirable risks to an already risky situation and transferred the concept of proper risk aversion, originally due to Pratt and Zeckhauser [1987], to the two-parameter framework. Our focus here is on increases in background risks. Inspired by Gollier and Pratt [1996]’s notion of risk vulnerability, we propose and characterize the concept of variance vulnerability to formally capture the idea that an agent reduces his risky activities when being confronted with the increase in the variance of an independent background risk. Two-parameter, mean-standard deviation analysis has a long and productive history both in theoretical research and in applied work on decision making under uncertainty. Due to 174 EICHNER AND WAGENER its simplicity and ease of interpretation it continues to be used although it is well-known to be consistent with EU rankings only under specific circumstances. The least restrictive and, for economic applications, most useful condition for compatibility of both approaches is the location-scale assumption under which all lotteries an agent might choose from differ only in location and scale parameters (Sinn [1983]; Meyer [1987]). This assumption will be satisfied whenever the interaction between the agent’s choices and a univariate source of uncertainty is linear, as it is, e.g., the case in Sandmo [1971]’s analysis of a competitive firm under price uncertainty, Fishburn and Porter [1976]’s portfolio choice problem with one risky and one risk-free asset and Ehrlich and Becker [1972]’s study of insurance demand. As two-parameter and the EU approach ought, in general, to be regarded as two distinct models of choice under risk (Ormiston and Schlee [2001], we first develop the concept of variance vulnerability in a pure two-parameter setting without reference to the EU approach (Sections 2 through 4). We derive necessary and sufficient conditions on mean-standard deviation utility functions such that variance vulnerability prevails and background risks have a tempering effect on risk taking (Proposition 1). Since two-parameter and EU approach at least partially overlap it is, then, natural to ask for the precise connection between them in a setting with independent background risks. We elaborate on this in Section 5. We start from Chamberlain [1983]’s observation that EUand two-parameter approach are compatible in multivariate settings if and only if risks are jointly elliptically symmetric distributed. Under this distributional assumption, all economic models where the interactions between an agent’s choices and the multivariate sources of uncertainty are linear still satisfy the location-scale assumption à la Sinn [1983] and Meyer [1987]. Among the elliptically symmetric distributions, however, the multivariate normal is the only one that allows for stochastic independence of risks—as it is required in the generic background risk problem. Hence, we conclude that EU- and two-parameter approach are compatible in settings with independent background risks if and only if stochastics are Gaussian (Proposition 3). Presupposing normality, we find in Section 6 that temperance (Kimball [1992]) is a necessary condition, and that Kimball [1993]’s property of standardness (i.e., the combination of decreasing absolute risk aversion and decreasing absolute prudence) or convexity of risk aversion (Gollier and Pratt [1996]) are sufficient restrictions on risk preferences to display variance vulnerability (Propositions 4 and 5). The same restrictions emerge as, respectively, necessary and sufficient conditions for risk vulnerability in the original setting by Gollier and Pratt [1996]. Thus, variance vulnerability and risk vulnerability are closely related concepts. At this point it is worth emphasizing that a distinction should be made between adding a (new) background risk and increasing a (pre-existing) background risk. While most analyses in the EU-framework (with the exception of Eeckhoudt, Gollier and Schlesinger [1996]) deal with the addition of a background risk and risk vulnerability applies to this setting only, mean-variance preferences allow for a simple way to capture and characterize increases in the background risk. However, in case where mean-variance approach and EU-approach are perfect substitutes, our necessary and sufficient conditions are coupled with a loss of generality, namely with the restriction to multivariate normal random variables. VARIANCE VULNERABILITY, BACKGROUND RISKS, AND PREFERENCES 2. 175 Preferences Individual preferences over lotteries are represented by a two-parameter preference function V : R+ × R → R, V = V (σ y , µ y ), (1) where y denotes random final wealth and σ y and µ y are, respectively, the standard deviation and the expected value of y. We assume that the function V is at least four times continuously differentiable, increasing in µ y , decreasing in σ y (risk aversion), and strictly concave (such that (σ y , µ y )-indifference curves are strictly convex). Denoting partial derivatives with subscripts, we thus require Vµ (σ y , µ y ) > 0; (2a) Vσ (σ y , µ y ) < 0; (2b) Vµµ (σ (...truncated)