Benchmark Mean-Variance Optimization

Abstract Benchmarks arise naturally in many asset management contexts. For example, an equity manager’s performance is typically evaluated relative to the return and tracking risk of an index such as the S&P 500. Tactical asset allocation performance is generally measured relative to a return index. Investment policy asset allocation may be associated with funding an appropriate return liability. This chapter addresses MV optimization relative to a return index or benchmark. The benchmark redefines risk in terms of the return of an investment-relevant objective. Mathematically, benchmark-relative optimization is MV optimization with a linear equality constraint representing the benchmark. Benchmark-relative optimization without linear inequality constraints, such as sign constraints, is subject to the same estimation error investment limitations as described in Jobson and Korkie (1980, 1981) and discussed in Chapter 4. While MV optimization relative to a benchmark may reduce instability at low risk, the statistical significance of investment information may be diminished.