On Relation Between Expected Regret and Conditional Value at Risk

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Abstract
The paper compares portfolio optimization approaches with expected regret and Conditional Value-at-Risk (CVaR) utility functions. The expected regret is defined as an average portfolio underperformance comparing to a fixed target of some benchmark portfolio. By definition, CVaR is the mean of the worst x% portfolio losses in a specified time period. CVaR is also called Mean Excess Loss or Expected Shortfall. Recently, it was demonstrated that the optimization of CVaR can be performed using linear programming. We formally prove that a portfolio, which minimizes CVaR, can be obtained by doing a sensitivity analysis 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 sensitivity analysis 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.

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