Some covariance inequalities for non-monotonic functions with applications to mean-variance indifference curves and bank hedging

Abstract

In several problems of decision-making under uncertainty, it is necessary to study the sign of the covariance between marginal utilities. All of the works that study the covariance signs are based on Chebyschev’s integral inequality. However, this inequality requires that both functions be monotonic. There are many cases, originated basically by new alternative theories, which assume that the marginal utilities of interest are non-monotonic. Thus, we cannot use Chebyschev’s result as it relies on monotonic functions. In this article, I derive some new covariance inequalities for utility functions which have non-monotonic marginal utilities. I also apply the theoretical results to two problems in economics: First, I study some properties of the indiference curve in the mean-variance space for Prospect Theory and for Markowitz utility functions. Second, I analyze the asset hedging policies of a bank that behaves as predicted by Prospect Theory.