Multivariate Risk Premia With a Stochastic Objective

Abstract

where u( ) is a utility function; x a vector of decision variables; e a vector of random variables and the maximisation is subject to a linear constraint, with x usually having the interpretation of a quantity vector and p a price vector. Most existing approaches to comparing different probability distributions of e use the ordering of these distributions by Eu(x, e) and often proceed by exploring risk premia defined by vectors 0 which satisfy Eu(x, e) = u(x, E + 0). The argument proceeds by relating the sign of 0 to risk attitudes and, perhaps, characterising choices between distributions by properties of 0. However there are two disadvantages here. First for problems where constrained maximisation is the essence of the problem as in all the examples cited above the interesting choice between distributions is that after maximisation has taken place. In other words one wants simple characteristics of preferences which determine whether the maximum attainable expected utility is greater or less from one distribution than another, given the constraints. There is relatively little interest in the ordering of distributions by Eu(x, e) alone. Secondly as both Epstein and Duncan (1977) vividly show risk premia or certainty equivalents defined in terms of quantities of commodities are not unique: in which good or combination of goods should the risk premium be measured? In a natural sense there is a whole subset of an n dimensional space which are suitable candidates for a certainty equivalent measure. The purpose of the present paper is to argue that, at least for discrete distributions, a much more natural measure of risk premium of a distribution which also provides an ordering of distributions by maximum attainable expected utility is given by minimum cost. Suppose that we are faced with the problem

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