Portfolio & Asset Pricingintermediate

Performance Evaluation

Raw returns are misleading because a high return may simply reflect high risk, so we use risk-adjusted performance measures. The Sharpe ratio divides excess return by total risk and judges a stand-alone portfolio. The Treynor ratio divides excess return by beta and judges a portfolio held as part of a larger diversified pool. Jensen’s alpha is the return earned above the CAPM benchmark, the cleanest gauge of manager skill. The choice hinges on the risk that matters. Use total risk when the portfolio is the whole investment, and use beta-based measures when it is one slice of a diversified whole.

Try it yourself

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 R_p − R_f for the risk it took: Sharpe per unit of total risk σ_p, Treynor per unit of beta β_p, 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%

Compare a second fund (B)

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?

Why it matters

Beating the market means little if you took on far more risk to do it, so every serious scorecard divides reward by the risk borne. Sharpe uses total volatility because it asks how the portfolio does on its own, while Treynor uses beta because it asks how the portfolio contributes to an already diversified book where only systematic risk survives. Jensen’s alpha goes a step further and asks whether the manager beat the exact return the CAPM demanded for that beta, so a positive alpha is genuine value added. The unifying question is always the same. What risk is actually being borne, total or systematic, and is the return worth it.

Formulas

Sharpe ratio

S = \dfrac{E(R_p) - R_f}{\sigma_p}

Excess return per unit of total risk. Best when the portfolio is the investor’s entire risky holding.

Treynor ratio

T = \dfrac{E(R_p) - R_f}{\beta_p}

Excess return per unit of systematic risk. Best when the portfolio is one component of a larger diversified pool.

Jensen alpha

\alpha_p = R_p - \left[\,R_f + \beta_p\!\left(R_m - R_f\right)\right]

Return above the CAPM benchmark for the portfolio beta. A positive

\alpha_p

signals outperformance net of systematic risk.

Worked examples

Scenario

A fund returns 14% with $\sigma=20\%$ and $\beta=1.2$ . The risk-free rate is 4% and the market returns 11%. Compute its Sharpe ratio, Treynor ratio, and Jensen alpha.

Solution

Sharpe is $(14\%-4\%)/20\%=0.50$ . Treynor is $(14\%-4\%)/1.2\approx 8.3\%$ . The CAPM benchmark is $R_f+1.2(11\%-4\%)=12.4\%$ , so Jensen alpha is $\alpha=14\%-12.4\%=1.6\%$ . The positive alpha says the fund beat its risk-adjusted benchmark by 1.6 percentage points.

Scenario

Two funds have the same Sharpe ratio but fund B has the higher Treynor ratio. What does that reveal?

Solution

Equal Sharpe means equal reward per unit of total risk, but a higher Treynor for B means more reward per unit of systematic risk. So B carries relatively more diversifiable risk that inflates its total volatility. Inside a diversified portfolio, where only beta matters, fund B is the better addition.

Common mistakes

✗A higher raw return always signals better performance. Raw return ignores risk, so a fund may simply have taken more risk, which is why risk-adjusted measures are required.
✗The Sharpe and Treynor ratios always rank funds the same way. They agree only for fully diversified portfolios, since Sharpe penalises total risk while Treynor penalises only systematic risk.
✗A positive return guarantees a positive Jensen alpha. Alpha is measured against the CAPM benchmark, so a fund can earn a solid positive return yet have negative alpha if it failed to beat that benchmark.
✗The Sharpe ratio is the right measure for a single sleeve of a larger portfolio. For a component of a diversified pool the beta-based Treynor or alpha is appropriate, because only systematic risk is added at the margin.

Revision bullets

•Risk-adjusted measures divide reward by the risk borne
•Sharpe uses total risk, best for a stand-alone portfolio
•Treynor uses beta, best for one sleeve of a diversified pool
•Jensen alpha is the return above the CAPM benchmark, a skill gauge
•Choose the measure by whether total or systematic risk is relevant

Quick check

When a portfolio is held as one component of a larger, well-diversified pool, the most appropriate performance measure is

Jensen’s alpha measures

Connected topics

Active vs Passive Return & Risk Capital Allocation CAPM & SML Factor Models

Sources

Brailsford, Heaney & Bilson (2015), Ch. 10
Brailsford, T., Heaney, R., & Bilson, C. Investments: Concepts and Applications. 5th ed. Cengage Learning Australia, 2015.
Defines the Sharpe ratio, Treynor ratio, and Jensen alpha and when each applies.
Bodie, Kane & Marcus (2021), Ch. 24
Bodie, Z., Kane, A., & Marcus, A. J. Investments. 12th ed. McGraw-Hill Education, 2021.
Reference treatment of portfolio performance evaluation and risk-adjusted measures.

How to cite this page

Dr. Phil's Quant Lab. (2026). Performance Evaluation. Derivatives Atlas. https://phucnguyenvan.com/concept/im-performance-evaluation

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