Portfolio Optimization Under Limited Value at Risk

In this paper we examine the problem of optimizing portfolios under limited downside risk. The portfolios risk exposure is measured assuming that the portfolio manager is averse to portfolio values falling below a given benchmark. We apply a downside risk approach using shortfall constraints, a framework that is quite well justified in the literature. As a special benchmark we choose the value at risk (VaR) since this is the probably most important benchmark in measuring the downside risk exposure of a portfolio. We approximate the portfolios distribution function and develop a mixed-integer optimization problem that finds portfolios with limited VaR and a given minimum expected final value (FV). We also derive a larger scale mixed-integer program that maximizes the expected FV of a portfolio under limited VaR. A case study shows the practical usefulness of the procedures developed by applying them to optimize portfolio protection against downside risk.