Global maximal Sharpe ratios for active portfolios

Abstract

ABSTRACT Active fund performance is assessed relative to specified benchmarks. Fund managers aim to generate portfolio returns higher than the benchmarks while adhering to constraints such as tracking errors (TE) and asset weight restrictions. For each TE, the maximum active portfolio return and associated absolute portfolio risk traces out a hyperbolic TE frontier in risk/return space. The locus of all possible active portfolio coordinates for a given level of TE traces out a closed, distorted ellipse in risk/return space. Attempts to optimize active portfolio performance at a given TE include the maximum active return portfolio, the maximum active return portfolio with the same absolute risk as the benchmark and the tangent portfolio (maximal risk adjusted return or Sharpe ratio (SR)). Using the maximum SR approach, the locus of maximal, active portfolio SRs for different TEs is roughly parabolic with a clearly defined maximum at a unique TE. We derive a solution for identifying these global maximum SR portfolios and find that these occur at feasible TEs (i.e. – within the range of most active portfolios) but only if inter-asset correlations are relatively low (). For asset correlations , maximal SRs occur at unrealistically high TEs ().