A comparison of risk measures for portfolio optimization with cardinality constraints

Abstract

We solve the optimization problem of minimum portfolio risk for seven measures using linear programming under cardinality constraints. The risk measures used are Expected Loss, Expected Loss Deviation, Expected Shortfall, Shortfall Deviation Risk, Expectile Value at Risk, Deviation Expectile Value at Risk, and Maximum Loss. We assess the out-of-sample performance of seven risk-optimized portfolios with a maximum size of 20 assets for S&P 100 components. After subtracting transaction costs, the Expected Loss Deviation portfolios have shown superior performance in terms of diversification and risk, the Maximum Loss portfolios have presented a higher Sharpe ratio (1.098 against 0.990 for the benchmark), the Expected Loss portfolios have higher absolute returns (660%), Sortino and STARR ratios. Expected Shortfall portfolios have presented the lowest Beta coefficients (0.616), and all portfolios returned lower Betas than the benchmark. All portfolios present significant annual alpha of 25% after adjusting for several risk factors. Our results show that superior performance can be achieved with simple linearized optimization models with lower market exposure measures to the CAPM beta.