Scenario generation for single-period portfolio selection problems with tail risk measures: coping with high dimensions and integer variables

摘要

In this paper we propose a problem-driven scenario generation approach to thesingle-period portfolio selection problem which use tail risk measures such asconditional value-at-risk. Tail risk measures are useful for quantifyingpotential losses in worst cases. However, for scenario-based problems these areproblematic: because the value of a tail risk measure only depends on a smallsubset of the support of the distribution of asset returns, traditionalscenario based methods, which spread scenarios evenly across the whole supportof the distribution, yield very unstable solutions unless we use a very largenumber of scenarios. The proposed approach works by prioritizing theconstruction of scenarios in the areas of a probability distribution whichcorrespond to the tail losses of feasible portfolios. The proposed approach can be applied to difficult instances of the portfolioselection problem characterized by high-dimensions, non-ellipticaldistributions of asset returns, and the presence of integer variables. It isalso observed that the methodology works better as the feasible set ofportfolios becomes more constrained. Based on this fact, a heuristic algorithmbased on the sample average approximation method is proposed. This algorithmworks by adding artificial constraints to the problem which are graduallytightened, allowing one to telescope onto high quality solutions.