Optimizing portfolio tail measures: Asymptotics and efficient simulation optimization

We consider a portfolio allocation problem where the objective function is a tail event such as probability of large portfolio losses. The dependence between assets is captured through multi-factor linear model. We address this optimization problem using two broad approaches. We show that a suitably scaled asymptotic of the probability of large losses can be developed that is a simple convex function of the allocated resources. Thus, asymptotically, portfolio allocation problem is approximated by a convex programming problem whose solution is easily computed and provides significant managerial insight. We then solve the original problem using sample average simulation optimization. Since rare events are involved, naive simulation may perform poorly. To remedy this, we introduce change-of-variable based importance sampling technique and develop a single change of measure that asymptotically optimally estimates tail probabilities across the entire space of feasible allocations.