Long-Short Portfolio Optimisation in the Presence of Discrete Asset Choice Constraints and Two Risk Measures

This paper considers long-short portfolio optimization in the presence of two risk measures: variance and Conditional Value at Risk (CVaR) and asset choice constraints of (i) buy, sell and holding thresholds (ii) cardinality restrictions on the number of stocks to be held in the portfolio. The mean-variance-CVaR model improves upon the classical mean-variance model by controlling both the variance and CVaR of the resulting return distribution. Our long-short extension to the mean-variance-CVaR model incorporates many financial institutions' practices in respect of the short decisions. We highlight that introducing short selling leads to superior choice of portfolios, with higher expected return and much lower risk exposures, as characterized by CVaR and variance. We further analyze the effects of applying buy and sell thresholds and cardinality restrictions on the number of stocks. Such constraints are of practical importance but make the efficient frontier discontinuous. When stocks' returns are represented as discrete random variables, the formulation leads to a Quadratic Mixed Integer Program (QMIP). We conclude that the long-short model with cardinality constraint is superior to the long only model even without cardinality constraint. The models are tested on real data drawn from the FTSE 100 index