Performance Appraisal
Performance appraisal asks a different question from attribution: not where the excess return came from, but whether the manager demonstrated skill once the risk taken to earn it is accounted for. Level I tests the formulas, when each measure is appropriate, and the diagnostics that tell you when a ratio is meaningless.
The core risk-adjusted measures
- Sharpe ratio = (R_p − R_f) ÷ σ_p. Uses total risk, so it suits a portfolio that is the investor's whole holding. Higher is better.
- Treynor ratio = (R_p − R_f) ÷ β_p. Uses systematic risk only, so it suits a well-diversified sub-portfolio judged within a larger fund.
- Jensen's alpha = R_p − [R_f + β_p(R_m − R_f)]. A positive alpha means the return exceeded what CAPM required for the beta taken.
- Information ratio = active return ÷ tracking error (standard deviation of active returns). It measures active risk efficiency.
- M² scales the portfolio's return to the market's risk level: R_f + Sharpe_p × σ_m, then subtracts R_m.
- Sortino, capture ratios and maximum drawdown address downside risk, which symmetric standard deviation treats as no worse than upside.
Benchmarks and style
A valid benchmark is specified in advance, investable, measurable, unambiguous, reflective of the manager's style, and known to the manager. Returns-based style analysis regresses portfolio returns on asset-class indices; the coefficients are the implied style exposures, and R² tells you how much of the return variation the style explains. Low R² means the style model does not describe the portfolio - and Treynor or alpha built on a mismatched beta is unreliable.
Worked example
R_p = 12%, R_f = 3%, σ_p = 18%, β_p = 1.2, R_m = 9%. Sharpe = (12 − 3) ÷ 18 = 0.50. Treynor = 9 ÷ 1.2 = 7.5%. Required CAPM return = 3 + 1.2(9 − 3) = 10.2%, so Jensen's alpha = 12 − 10.2 = +1.8%. If tracking error is 6%, the information ratio is 3 ÷ 6 = 0.50.
Traps that cost marks
- Using beta to rank a portfolio that is not diversified - the Sharpe ratio is the correct tool there.
- Forgetting to subtract R_f in Jensen's alpha or subtracting actual market return instead of the beta-adjusted requirement.
- Reading a high alpha as skill when R² is low or the benchmark is not investable.
- Confusing appraisal (risk-adjusted judgement of skill) with attribution (decomposition of the return).