Extreme Returns in a Shortfall Risk Framework

One of the most important 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. We propose a Monte Carlo simulation approach in order to solve an asset allocation problem with shortfall constraint, and to evaluate the exact portfolio risk-level when managers assume a misspecified tails behaviour. In particular, in a stochastic optimisation problem, we assume that returns are generated by a multivariate Student-t distribution, while in reality returns come from a multivariate distribution where each marginal is a Student-t but with different degrees of freedom. This method allows us to value the effective risk for managers. In the specific case analysed, it is also interesting to observe that a multivariate density resulting from different Student-t marginal distributions produces a shortfall probability and a shortfall return level that can be approximated adequately by assuming a multivariate Student-t with a common degree of freedom in the optimisation problem. The present simulation based approach could be an important instrument for investors who need a qualitative assessment of the reliability and sensitivity of their investment strategies when their models are potentially misspecified.