Discrete-time mean-CVaR portfolio selection and time-consistency induced term structure of the CVaR

Abstract

We investigate a discrete-time mean-risk portfolio selection problem, where risk is measured by the conditional value-at-risk (CVaR). A substantial challenge is the combination of a time-inconsistent objective with an incomplete and dynamic model for the financial market. We are able to solve this problem analytically by embedding the original, time-inconsistent problem into a family of time-consistent expected utility maximization problems with a piecewise linear utility function. The optimal investment strategy is a fully adaptive feedback policy and the cumulated amount invested in the risky assets is of a characteristic V-shaped pattern as a function of the current wealth. For the incomplete, discrete-time market considered herein, the mean-CVaR efficient frontier is a straight line in the mean-CVaR plane and thus economically meaningful. This contrasts the complete, continuous-time setting where the mean-CVaR efficient frontier is degenerate or does not exist at all. We further solve an inverse investment problem, where we investigate how mean-CVaR preferences need to adapt in order for the pre-committed optimal strategy to remain optimal at any point in time. Our result shows that a pre-committed mean-CVaR investor behaves like a naive mean-CVaR investor with a time-increasing confidence level for the CVaR, who revises the investment decision at every point in time. Finally, an empirical application of our results suggests that risk measured by the CVaR might help to understand the long-standing equity premium puzzle.