Strategic Financial Management · Portfolio Theory and Practice

Sharpe, Treynor and Jensen Portfolio Performance Measures

Updated 11 October 2026 · Fact-checked

Portfolio performance evaluation ranks portfolios by return earned per unit of risk. Sharpe divides excess return over the risk-free rate by standard deviation. Treynor divides it by beta. Jensen's alpha is actual return minus the CAPM required return. Fama's decomposition splits excess return into risk-taking, diversification and net selectivity. Compute, compare, then recommend.

Understand Portfolio Performance Evaluation Measures

Raw return is a poor way to judge a fund manager. A manager who earned 15% by taking very high risk may be worse than one who earned 12% with low risk. Performance measures adjust return for risk so that portfolios can be compared fairly.

All three main measures start from excess return: the portfolio return minus the risk-free return. This is the reward for bearing risk. The measures differ in how they define risk.

Sharpe ratio uses total risk (standard deviation). Use it when the portfolio is your whole investment, or when it is not fully diversified. Treynor ratio uses systematic risk (beta). Use it when the portfolio is one part of a larger, well-diversified holding, because only market risk matters then. Jensen's alpha is an absolute figure: the extra return over what CAPM says the portfolio should have earned for its beta. Positive alpha means the manager added value.

Sharpe and Treynor give ratios, so you rank portfolios by the higher value. Jensen gives a percentage, so you rank by the higher alpha. The rankings can differ, because the risk measures differ. A portfolio with a lot of unsystematic risk looks worse on Sharpe than on Treynor.

Fama's decomposition goes further. It breaks the excess return into the part earned by taking market risk, the part linked to diversification, and the part due to the manager's selectivity (skill). It tells you why a portfolio performed as it did, not just how well.

Key rules to remember

Sharpe ratio: Sharpe = (Rp − Rf) ÷ σp; Rp = portfolio return, Rf = risk-free rate, σp = standard deviation of the portfolio. Higher is better.
Treynor ratio: Treynor = (Rp − Rf) ÷ βp; βp = portfolio beta. Higher is better. Market portfolio has Treynor = Rm − Rf because its beta is 1.
Jensen's alpha: α = Rp − [Rf + βp × (Rm − Rf)]; Positive alpha means outperformance, zero means as expected, negative means underperformance.
Fama: overall excess return: Rp − Rf = (Rp − Rσ) + (Rσ − Rβ) + (Rβ − Rf); Net selectivity + diversification + risk-taking (beta) component.
Fama: benchmark returns: Rβ = Rf + βp × (Rm − Rf); Rσ = Rf + (σp ÷ σm) × (Rm − Rf); Rβ is the CAPM return for the beta. Rσ is the return on the capital market line at the portfolio's total risk.
Fama: components: Selectivity = Rp − Rβ; Net selectivity = Rp − Rσ; Diversification = Rσ − Rβ; Selectivity equals Jensen's alpha. It equals net selectivity plus diversification.

How to solve Portfolio Performance Evaluation Measures questions

Use this method for any performance-evaluation question. Work in a small table, one column per portfolio.

1List the given data: Rp, σp, βp for each portfolio, plus Rf, Rm and σm for the market.
2Compute excess return Rp − Rf for every portfolio.
3Compute Sharpe (excess ÷ σ) and Treynor (excess ÷ β) for each portfolio. Also compute them for the market if asked to compare with it.
4Compute the CAPM required return Rf + β(Rm − Rf) and then alpha = Rp − required return.
5Rank the portfolios separately on each measure. State the ranking clearly.
6If Fama is asked, compute Rβ and Rσ, then net selectivity, diversification and risk-taking, and check that they add to Rp − Rf.
7Give a recommendation in one or two lines. Explain why rankings differ and which measure suits the investor's situation.

Quickest way: One-table shortcut

When to use it: When the question gives three or four portfolios and asks for all three measures and a ranking, and time is tight.

Compute Rp − Rf once for each portfolio and reuse it in all three measures.
Compute Rm − Rf once. Required return = Rf + β × (Rm − Rf).
Fill the table row by row: Sharpe, Treynor, Alpha. Keep two decimals.
Rank with 1, 2, 3 beside each figure so the answer is visible to the examiner.
Write one line of comment: Sharpe for undiversified portfolios, Treynor and Jensen for diversified ones.

Common mistakes in Portfolio Performance Evaluation Measures

Forgetting to subtract the risk-free rate in the numerator.
Students remember the ratio as return divided by risk.
Fix: Always write (Rp − Rf) first. Without it the measure is not a risk premium.
Using beta in the Sharpe ratio or standard deviation in the Treynor ratio.
The two formulas look alike and the names get mixed up.
Fix: Remember: Sharpe = standard deviation (total risk), Treynor = beta (systematic risk).
Computing alpha as Rp − Rm.
Students compare the portfolio with the market directly, ignoring beta.
Fix: Alpha is Rp minus the CAPM return, Rf + β(Rm − Rf). Adjust the market premium for beta first.
Expecting all three measures to give the same ranking.
Students assume one answer must be wrong when rankings differ.
Fix: Different risk measures can legitimately change the ranking. Report each ranking and explain the reason.
Using σp ÷ σm wrongly in Fama's decomposition, or using β where σ is needed.
Rβ and Rσ look similar.
Fix: Rβ uses β and Rσ uses σp ÷ σm. Check that net selectivity + diversification + risk-taking equals Rp − Rf.
Giving numbers without a conclusion.
Students treat it as pure calculation.
Fix: End with a recommendation: which portfolio performed best and on what basis.

Worked examples

Example 1

Risk-free rate is 6% and market return is 12% with standard deviation 10%. Three portfolios: A has return 14%, standard deviation 12%, beta 1.1. B has return 12%, standard deviation 8%, beta 0.8. C has return 15%, standard deviation 15%, beta 1.5. Compute Sharpe, Treynor and Jensen's alpha for each and rank them.

Show the solution

Market premium Rm − Rf = 12 − 6 = 6%.
Excess returns: A = 14 − 6 = 8%; B = 12 − 6 = 6%; C = 15 − 6 = 9%.
Sharpe: A = 8 ÷ 12 = 0.667; B = 6 ÷ 8 = 0.750; C = 9 ÷ 15 = 0.600.
Treynor: A = 8 ÷ 1.1 = 7.27; B = 6 ÷ 0.8 = 7.50; C = 9 ÷ 1.5 = 6.00.
Required returns: A = 6 + 1.1 × 6 = 12.6%; B = 6 + 0.8 × 6 = 10.8%; C = 6 + 1.5 × 6 = 15.0%.
Alpha: A = 14 − 12.6 = 1.4%; B = 12 − 10.8 = 1.2%; C = 15 − 15 = 0%.
Rankings: Sharpe B, A, C. Treynor B, A, C. Jensen A, B, C.

Answer: Sharpe: A 0.667, B 0.750, C 0.600. Treynor: A 7.27, B 7.50, C 6.00. Alpha: A 1.4%, B 1.2%, C 0%. B is best on Sharpe and Treynor; A is best on alpha. C only earned what its high beta required, so it added no value.

Example 2

Using the data for Portfolio A in the previous example (Rf 6%, Rm 12%, σm 10%, Rp 14%, σp 12%, β 1.1), decompose its excess return using Fama's method.

Show the solution

Total excess return = Rp − Rf = 14 − 6 = 8%.
Rβ = 6 + 1.1 × 6 = 12.6%.
Rσ = 6 + (12 ÷ 10) × 6 = 6 + 7.2 = 13.2%.
Risk-taking component = Rβ − Rf = 12.6 − 6 = 6.6%.
Diversification = Rσ − Rβ = 13.2 − 12.6 = 0.6%.
Net selectivity = Rp − Rσ = 14 − 13.2 = 0.8%.
Check: 6.6 + 0.6 + 0.8 = 8.0%. Selectivity = 0.6 + 0.8 = 1.4%, which equals Jensen's alpha.

Answer: Of the 8% excess return, 6.6% comes from market risk, 0.6% from diversification and 0.8% is net selectivity. Total selectivity is 1.4%, equal to alpha. The manager added value, but most of the return came from bearing market risk.

Exam tips

Draw the table first. Examiners give marks for each measure and for the ranking, so show every column.
Carry at least two decimals in Sharpe and Treynor. Ranking can flip if you round too early.
Always add a short comment on why rankings differ and which measure fits the investor. Case-based MCQs often test this reasoning.
In Fama's decomposition, show the check that the three components add up to Rp − Rf.
Read whether the question gives standard deviation or beta. If only one is given, the question is pointing to the matching measure.

Practice questions from Portfolio Theory and Practice

Portfolio Performance Evaluation Measures in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Portfolio Performance Evaluation Measures: frequently asked questions

What is the difference between Sharpe and Treynor ratio?

Both divide excess return by risk. Sharpe uses standard deviation, which is total risk. Treynor uses beta, which is only systematic risk. Use Sharpe for an undiversified portfolio and Treynor for a well-diversified one.

How do you calculate Jensen's alpha?

Find the required return from CAPM: Rf + β × (Rm − Rf). Then subtract it from the actual portfolio return. A positive result means the manager beat the return expected for the risk taken.

Why do Sharpe and Treynor sometimes rank portfolios differently?

A portfolio with a large share of unsystematic risk has high standard deviation but a modest beta. It looks weaker on Sharpe than on Treynor. The difference reflects how well diversified the portfolio is.

What does Fama's decomposition tell you that the other measures do not?

It explains the source of the excess return. It separates the return for taking market risk, the effect of diversification and the manager's net selectivity. This shows whether outperformance came from skill or just from higher risk.