Triple-objective models for portfolio optimisation with symmetric and percentile risk measures

The purpose of this paper is to compare three different triple‐criteria portfolio optimisation models. The first model is constructed with the use of percentile risk measure value‐at‐risk and solved by mixed integer programming. The second one is constructed with the use of percentile risk measure conditional value‐at‐risk and solved by linear programming. The third one is constructed with the use of a symmetric measure of risk ‐ variance of return ‐ as in the Markowitz portfolio and solved by quadratic programming. Cardinality constraints are formulated in all models that limit the number of assets selected in the portfolio. Computational experiments are conducted for triple‐criteria portfolio stock exchange investments. The results obtained prove that the triple‐objective portfolio optimisation models with value‐at‐risk and conditional value‐at‐risk could be used to shape the distribution of portfolio returns. The decision maker can assess the value of portfolio return, the risk level and number of assets chosen in the portfolio, and can decide how to invest in a real life situation comparing with ideal (optimal) portfolio solutions. The proposed scenario‐based portfolio optimisation problems under uncertainty, formulated as a triple‐objective linear, mixed integer or quadratic program are solved using commercially available software (AMPL/CPLEX) for mathematical programming.