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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.

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