A Comparison of Risk Measures for Portfolio Optimization with Cardinality Constraints

Abstract

Ever since the introduction of Markowitz’s classical quadratic programming problem, transforming portfolio optimization into a linear programming (LP) problem has drawn much attention from researchers and practitioners, given the tractability of LP. However, using non-linear risk measures and including real features and frictions may pose a challenge. In this paper, 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 from 2010 to mid-2020. 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 higher Sharpe ratio, the Expected Loss portfolios have higher absolute returns, Sortino and STARR ratios, and the Expected Shortfall portfolios have presented lower Beta coefficients than the benchmark and other risk-based portfolios. All portfolios present significant alpha after adjusting for several risk factors. The main results hold for subperiod analysis, different cardinalities, and other rebalancing periods. Our results show that superior performance can be achieved with simple linearized optimization models with lower market exposure measure by the CAPM beta.

Keywords: cardinality constraints, coherent risk measures, Risk, linear portfolio optimization, EVaR