Active Portfolio Management with Benchmarking: Adding a Value-at-Risk Constraint

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Abstract
We investigate the impact of adding a Value-at-Risk (VaR) constraint to the problem of an active manager who aims to beat the return on a benchmark by a given percentage. In doing so, we use the mean-tracking error variance (TEV) model examined by Roll (1992). We show that portfolios on the constrained mean-TEV boundary still exhibit three-fund separation, but the weights of the three funds when the constraint binds differ from those in Roll's model. Furthermore, we show that an appropriately chosen VaR constraint leads to the selection of a portfolio that dominates the portfolio chosen in its absence and the benchmark according to both mean-variance and mean-VaR criteria. Hence, the constraint mitigates the well-known problem that when an active manager seeks to track (or outperform) a benchmark, he or she may select a particularly inefficient portfolio.

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