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). By embedding this time-inconsistent problem into a family of expected utility maximization problems with a piecewise linear utility function, we solve the problem analytically. In contrast to the case of a complete, continuous-time market, the mean-CVaR efficient frontier in this generally incomplete, discrete-time setting is a straight line in the mean-CVaR plane and there is in particular a constant trade-off between risk and return. The cumulated amount invested in the risky assets under the optimal strategy is of a V -shaped pattern as a function of the current wealth. We further solve an inverse investment problem, where we investigate how mean-CVaR preferences need to adapt such that the pre-commited optimal strategy remains optimal at any point in time. Our result shows that, although conceptually distinct, a pre-commited mean-CVaR investor behaves like a naive mean-CVaR investor with a time-increasing confidence level for the CVaR, who revises her 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.