Minimising value‐at‐risk in a portfolio optimisation problem using a multi‐objective genetic algorithm

In this paper, we develop a general framework for market risk optimisation that focuses on VaR. The reason for this choice is the complexity and problems associated with risk return optimisation (non‐convex and non‐differential objective function). Our purpose is to obtain VaR efficient frontiers using a multi‐objective genetic algorithm (GA) and to show the potential utility of the algorithm to obtain efficient portfolios when the risk measure does not allow calculating an optimal solution. Furthermore, we measure differences between VaR efficient frontiers and variance efficient frontiers in VaR‐return space and we evaluate out‐sample capacity of portfolios on both bullish and bearish markets. The results indicate the reliability of VaR‐efficient portfolios on both bullish and bearish markets and a significant improvement over Markowitz efficient portfolios in the VaR‐return space. The improvement decreases as the portfolios level of risk increases. In this particular case, efficient portfolios do not depend on the risk measure minimised.