Stochastic Optimisation for Allocation Problem with Shortfall Risk Constraints

One of the crucial aspects in asset allocation problems is the assumption concerning the probability distribution of asset returns. Financial managers generally suppose normal distribution, even if extreme realizations usually have an higher frequency than in the Gaussian case. The aim of this paper is to propose a general Monte Carlo simulation approach able to solve an asset allocation problem with shortfall constraint, and to evaluate the exact portfolio risk-level when managers assume a misspecified return behaviour. We assume that returns are generated by a multivariate skewed Student-t distribution where each marginal can have different degrees of freedom. The stochastic optimization allows us to value the effective risk for managers. In the empirical application we consider a symmetric and heterogeneous case, and interestingly note that a multivariate Student-t with heterogeneous marginal distributions produces in the optimization problem a shortfall probability and a shortfall return level that can be adequately approximated by assuming a multivariate Student-t with common degrees of freedom. Thus, the proposed simulation-based approach could be an important instrument for investors who require a qualitative assessment of the reliability and sensitivity of their investment strategies in the case their models could be potentially misspecified.