Variance vulnerability, background risks, and mean-variance preferences

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.