Mean–variance, mean–VaR, and mean–CVaR models for portfolio selection with background risk

This paper extends (Jiang et al. in J Bank Finance 34:3055–3060, 2010; Guo in Risk Manag 20(1):77–94, 2018) and others by investigating the impact of background risk on an investor’s portfolio choice in the mean–VaR, mean–CVaR, and mean–variance framework, and analyzes the characterization of the mean–variance, mean–VaR, and mean–CVaR boundaries and efficient frontiers in the presence of background risk. We derive the conditions that the portfolios lie on the mean–variance, mean–VaR, and mean–CVaR boundaries with and without background risk. We show that the MV, VaR, and CVaR boundaries depend on the covariance vector between the returns of the risky assets and that of the background asset and also the variance of the return of the background asset. We develop properties on MV, mean–VaR, and mean–CVaR efficient frontiers. In addition, we establish some new properties for the case with a risk-free security, extend our work to the non-normality situation, and examine the economic implication of the mean–VaR/CVaR model.