Variance stochastic orders

Suppose that the decision-maker is uncertain about the variance of the payoff of a gamble, and that this uncertainty comes from not knowing the number of zero-mean i.i.d. risks attached to the gamble. In this context, we show that any n-th degree increase in this variance risk reduces expected utility if and only if the sign of the 2n-th derivative of the utility function u is (−1)n+1. Moreover, increasing the statistical concordance between the mean payoff of the gamble and the n-th degree riskiness of its variance reduces expected utility if and only if the sign of the (2n+1)-th derivative of u is (−1)n+1. These results generalize the theory of risk apportionment developed by Eeckhoudt and Schlesinger (2006). They are useful to better understand the impact on asset prices of stochastic volatility, which has recently been shown to be a crucial ingredient to solve the classical puzzles of asset pricing theory.