On Relation Between Expected Regret and Conditional Value at Risk

Abstract

The paper compares portfolio optimization approaches with expected regret and Conditional Value-at-Risk (CVaR) performance functions. The expected regret is defined as an average portfolio underperformance comparing to a fixed target or some benchmark portfolio. For continuous distributions, CVaR is defined as the expected loss exceeding α-Value-at Risk (VaR), i.e., the mean of the worst (1-α) 100% losses in a specified time period. However, generally, CVaR is the weighted average of VaR and losses exceeding VaR. Optimization of CVaR can be performed using linear programming. We formally prove that a portfolio with a continuous loss distribution, which minimizes CVaR, can be obtained by doing a line search with respect to the threshold in the expected regret. An optimal portfolio in CVaR sense is also optimal in the expected regret sense for some threshold in the regret function. The inverse statement is also valid, i.e., if a portfolio minimizes the expected regret, this portfolio can be found by doing a line search with respect to the CVaR confidence level. A portfolio, optimal in expected regret sense, is also optimal in CVaR sense for some confidence level. The relation of the expected regret and CVaR minimization approaches is explained with a numerical example.