Proper Risk Aversion in Presence of Multiple Sources of Risk

Abstract

A risk-averse agent finds each of two independent risky assets undesirable. If he is imposed to take one, will he continue to evaluate the other undesirable one? In other words, can an undesirable lottery be made desirable by the presence of another undesirable lottery? In the negative case, the agent is said to have a proper risk aversion. This notion has been introduced by Pratt and Zeckhauser (1987) and a weaker version has recently been proposed by Gollier and Pratt (1993). Families of utility functions displaying the properness and weak properness properties for all couples of undesirable lotteries have been found (Kimball (1993), Gollier and Pratt (1993, 1995)). The aim of this note is to face the problem from a different point of view. Conditions to guarantee properness and weak properness properties are no longer stated for all couples of undesirable risks, but are formulated by means of a characterisation of the utility function (in terms of the coefficients of risk-aversion, of prudence and temperance) and by imposing restrictions on the marginal distributions of the risks involved. An extended notion has also been proposed for non-independent risks and sufficient conditions for properness are provided. Proper risk aversion should have important consequences in applications. For example, properness guarantees that the demand in insurance and in other forms of hedging does not decrease as the number of independent unfavourable risks increases.