Mean-risk models using two risk measures: a multi-objective approach

Abstract

This paper proposes a model for portfolio optimization, in which distributions are characterized and compared on the basis of three statistics: the expected value, the variance and the CVaR at a specified confidence level. The problem is multi-objective and transformed into a single objective problem in which variance is minimized while constraints are imposed on the expected value and CVaR. In the case of discrete random variables, the problem is a quadratic program. The mean-variance (mean-CVaR) efficient solutions that are not dominated with respect to CVaR (variance) are particular efficient solutions of the proposed model. In addition, the model has efficient solutions that are discarded by both mean-variance and mean-CVaR models, although they may improve the return distribution. The model is tested on real data drawn from the FTSE 100 index. An analysis of the return distribution of the chosen portfolios is presented.

Keywords:

• Portfolio selection
• Multi-objective optimisation
• Risk measures
• Conditional Value at Risk

Notes

†There may be a situation when several mean-CVaR efficient portfolios have the same mean return and the same (optimal) CVaR, but different variances. Only the portfolio with the minimal variance is efficient in the proposed model. The same discussion applies for mean-variance efficient portfolios. We reconsider the issue in section 4.4.

†This is not necessarily the same as ‘the expected value of losses exceeding VaR at confidence level α’, as it is defined in earlier papers on CVaR. The two definitions lead to the same results when the distribution of the random variable under consideration is continuous, but differences may appear when the considered distribution has discontinuities---see Acerbi and Tasche (2002), and Rockafellar and Uryasev (2002) for more details.

†As stated in Proposition 1, CVaR is a convex function of x. Variance is also a convex function of x, since the variance-covariance matrix is positive semi-definite. The expected value is a linear function of x.

†If additionally there is the assumption of unique optimal solutions of (P3) when some of the weights are zero, then only the non-negativity condition is required for w ₁, w ₂ and w ₃.

†d _max is also equal to the highest expected return of the component assets in the portfolio selection problem.

†The CVaR level of is the arithmetic mean of the CVaR levels of and . Similarly, the CVaR level of is the arithmetic mean of the CVaR levels of and .

†For example, multiple optimal solutions of (P2) may have the same variance, the same expected return but different CVaRs; only the one with the lowest CVaR is Pareto efficient in (MVC).

‡In case there are several mean-variance efficient portfolios with expected return d, with different CVaR levels, only the portfolio with the lowest CVaR is efficient in the (MVC) model; its CVaR level is denoted by .