Risk Aversion and Portfolio Selection in a Continuous-Time Model

Abstract

The comparative statics of the optimal portfolios across individuals is carried out for the Black-Scholes market model. It turns out that the indirect utility functions inherit the order of risk aversion (in the Arrow-Pratt sense) from the von Neumann-Morgenstern utility functions, and therefore, a more risk-averse agent would invest less wealth (in absolute value) in the risky asset. Introduction.Portfolio selection is one of the classical problems in the economics of uncertainty. The optimal portfolios depend on agents' characters (preference and wealth level) and on the market's structure (the risk-free return, the return and risk of the risky assets). Various agents would have different allocations of wealth between the risk-free asset and the risky assets, due to the differences in preference and/or the differences in wealth level. The comparative statics of the optimal portfolios w.r.t. (with respect to) preference and/or wealth level was first carried out by Arrow [1] and Pratt [18] for a static model with a risk-free asset and a risky asset. For this model, if the excess return of the risky asset is positive, then (i) the more risk-averse an agent is, the less wealth is invested in the risky asset; and (ii) if an agent displays decreasing absolute (relative) risk aversion, then the amount (proportion) of wealth invested in the risky asset is increasing in the total wealth.Since then, decades have passed. Except for some specific cases such as constant absolute (relative) risk aversion, in which the solutions can be explicitly worked out, few works have been reported for dynamic models, as far as we know, until Borell [3]. For a continuous-time complete market model, where the risky assets price process follows a joint geometric Brownian motion, and for an agent who consumes only at the terminal time, Borell [3] analyzed the changes of the optimal portfolios across the wealth levels. The similar conclusions that hold for the static models have been obtained there by showing that the indirect utility function inherits the decreasing absolute (relative) risk aversion from the vNM (von Neumann-Morgenstern) utility function. 1 The purpose of this paper is to investigate how the agents' preferences impact the optimal portfolios for the Black-Scholes market model. Here we compare the optimal portfolios across individuals instead of across wealth levels. As a result (see

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