More on properness: The case of mean-variance preferences

This paper focuses on the situations where individuals with mean-variance preferences add independent risks to an already risky situation. Pratt and Zeckhauser (Econometrica, 55, 143-154, 1987) define a concept called proper risk aversion in the expected utility framework to describe the situation where an undesirable risk can never be made desirable by the presence of an independent undesirable risk. The assumption of mean-variance preferences allows us to study proper risk aversion in an intuitive manner. The paper presents an economic interpretation for the quasi-concavity of a utility function derived over mean and variance. The main result of the paper says that quasi-concavity plus decreasing risk aversion is equivalent to proper risk aversion.