Computational study of a chance constrained portfolio selection problem

We study approximations of chance constrained problems. In particular, we consider the Sample Average Approximation (SAA) approach and discuss convergence properties of the resulting problem. A method for constructing bounds for the optimal value of the considered problem is discussed and we suggest how one should tune the underlying parameters to obtain a good approximation of the true problem. We apply these methods to a linear portfolio selection problem with returns following a multivariate lognormal distribution. In addition to the SAA, we also analyze the Scenario Approximation approach, which can be regarded as a special case of the SAA method. Our computational results indicate the scenario approximation method gives gives a conservative approximation to the original problem. Interpreting the chance constraint as a Value-at-Risk constraint, we consider another approximation replacing it by the Conditional Value-at-Risk constraint. Finally, we discuss a method to approximate a sum of lognormals that allows us to find a closed expression for the chance constrained problem and compute an efficient frontier for the lognormal case.