Performance evaluation: risk-adjusted return
Raw return is misleading — risk-adjusted measures (Sharpe, Treynor, Jensen's alpha) reveal whether a fund truly beat the market.
Each measure charges the fund's excess return Rp − Rf for the risk it took: Sharpe per unit of total risk σp, Treynor per unit of beta βp, and Jensen's alpha as the gap above the CAPM line. A fund above the line out-earns its systematic risk; one below it lagged.
Fund A Jensen's alpha at β = 1.10+1.4%
Sharpe (total risk) 0.44Treynor (per β) 0.07Jensen's α +1.4%
Market Sharpe 0.40Fund A vs market (Sharpe) beats
Fund A
Mean return R_p12.0%
Total volatility σ_p18.0%
Beta β_p1.10
Market & risk-free
Risk-free rate R_f4.0%
Market return R_m10.0%
Market volatility σ_m15.0%
Try this
Fund A returns 12.0%, but the CAPM only requires 10.6% for its β = 1.10 (R_f 4.0% plus β times the 6.0% premium). Positive alpha of +1.4% puts it above the SML: it genuinely beat the market line.
Total vs systematic risk. Sharpe 0.44 divides excess return by total σ; the market Sharpe is 0.40, so Fund A beats the market for a fully diversified investor. Treynor 0.07 divides by β instead, the right lens when the fund is one slice of a larger diversified book.
Discuss: a fund tops the league table on raw return yet has the lowest alpha of its peers. What did it do to earn the headline number, and which measure — Sharpe or Treynor — would you trust if it were the investor's only holding versus one fund among many?