FRM Exam Part I · Modern Portfolio Theory (MPT) and the Capital Asset Pricing Model (CAPM)

Sharpe, Treynor, Jensen and Information Ratio Explained

Updated 11 October 2026 · Fact-checked

Risk-adjusted performance measures scale a portfolio's excess return by a risk number. Sharpe divides by total risk (standard deviation), Treynor by systematic risk (beta), and Jensen's alpha is the return above the CAPM prediction. Information ratio divides active return over a benchmark by tracking error. Compute, then rank.

Understand Performance Measures: Sharpe, Treynor, Jensen, Information Ratio

A higher return means little if it came from taking more risk. Performance measures fix this by asking: how much return did you earn for each unit of risk? Each measure uses a different definition of risk, so each answers a slightly different question.

The Sharpe ratio uses total risk, the standard deviation of portfolio returns. It is the slope of the line from the risk-free asset to your portfolio, so it is the natural measure when the portfolio is your whole investment, or when you compare stand-alone funds. The Treynor ratio uses beta, so it only counts market (systematic) risk. It suits a portfolio that is one piece of a larger, well-diversified holding, where the fund's own specific risk is diversified away.

Jensen's alpha works in return units, not ratios. You use CAPM to find the return the portfolio should have earned given its beta, then subtract that from the actual return. Positive alpha means outperformance after adjusting for beta. Alpha does not penalise specific risk and is not scaled by beta, so a high-beta fund and a low-beta fund with the same alpha are not equally good in every sense.

The information ratio judges an active manager against a benchmark. The numerator is active return (portfolio minus benchmark). The denominator is tracking error, the standard deviation of that active return. It tells you how consistently the manager beats the benchmark per unit of active risk.

Rankings can disagree. Sharpe and Treynor agree only when portfolios are well diversified, since then total risk is almost all systematic. Alpha needs a benchmark market return and a risk-free rate, so it depends on the CAPM being a fair model.

Key formulas to remember

Sharpe ratio: Sharpe = (Rp − Rf) ÷ σp; Rp is portfolio return, Rf the risk-free rate, σp the standard deviation of portfolio returns. Use for total risk.
Treynor ratio: Treynor = (Rp − Rf) ÷ βp; Result is in return units per unit of beta, for example 8% per 1.0 of beta. Use for systematic risk.
Jensen's alpha: α = Rp − [Rf + βp × (Rm − Rf)]; Rm is the market return. The bracket is the CAPM expected return. Positive alpha means outperformance.
Tracking error: TE = σ(Rp − Rb); Standard deviation of the active returns, where Rb is the benchmark return. It is not the standard deviation of the portfolio itself.
Information ratio: IR = (Rp − Rb) ÷ TE; Numerator is the average active return. Use the same period basis (annual with annual) for both parts.
Treynor-Black (appraisal) ratio: Appraisal ratio = α ÷ σ(ε); Alpha divided by residual (non-systematic) risk from the regression. It is the information ratio form where the benchmark is the CAPM-implied return.
M-squared (Modigliani): M² = Rf + Sharpe_p × σm; M² difference = M² − Rm; Sharpe restated as a return by leveraging or de-leveraging the portfolio to the market's standard deviation σm.

How to solve Performance Measures: Sharpe, Treynor, Jensen, Information Ratio questions

Use this routine for any question on risk-adjusted performance. Identify what the question asks, pick the matching risk measure, then compute carefully in consistent units.

1Write down every input given: Rp, Rf, Rm, Rb, σp, βp, tracking error. Convert percentages to decimals or keep all as percentages, not a mix.
2Decide the risk concept. Total risk or stand-alone portfolio points to Sharpe. Diversified portfolio inside a larger holding points to Treynor. Return versus CAPM line points to Jensen. Return versus a benchmark points to information ratio.
3Compute the excess return first: Rp − Rf for Sharpe and Treynor, Rp − Rb for the information ratio.
4For Jensen's alpha, compute the CAPM required return Rf + β(Rm − Rf), then subtract it from Rp.
5Divide by the correct risk number. Check you used σp for Sharpe, β for Treynor, and tracking error (not σp) for the information ratio.
6If tracking error must be found from a series, compute active returns each period, then their sample standard deviation (divide by n − 1 unless told otherwise).
7Compare or rank. Higher is better for all four. Remember rankings by different measures can differ, and state which is appropriate for the context.
8Sanity check: Sharpe of a typical equity portfolio is well below 2, and Jensen's alpha should be small relative to Rp. A wildly large value usually means a unit slip.

Quickest way: Match the measure to the risk, then divide

When to use it: Use when a multiple-choice question gives you the inputs directly and the time per question is about two and a half minutes.

Scan the options. If they are ratios like 0.50 or 0.65, it is Sharpe or the information ratio. If they are in percent units per beta, it is Treynor. If they are a plain percent difference, it is alpha.
Do the excess return in your head or on the calculator, then one division.
For alpha, compute Rf + β(Rm − Rf) first and only then subtract it from Rp.
Check the keyword: 'benchmark' or 'active' means tracking error. 'Well-diversified' means Treynor or Jensen. 'Total risk' or 'standalone' means Sharpe.
Eliminate options that flip the sign or use σ in place of β. That usually leaves one answer.

Common mistakes in Performance Measures: Sharpe, Treynor, Jensen, Information Ratio

Using portfolio standard deviation as the denominator of the information ratio.
Sharpe and the information ratio look alike, so students carry the same denominator over.
Fix: The information ratio always divides active return by the standard deviation of active return (tracking error), not by σp.
Forgetting to subtract the risk-free rate in Sharpe or Treynor.
Students divide the plain return by risk because it feels simpler.
Fix: Write 'Rp − Rf' as the first line every time. Only the information ratio uses a benchmark instead of Rf.
Computing Jensen's alpha as Rp − Rm.
Students treat the market as the benchmark and ignore beta.
Fix: Alpha is Rp minus the CAPM-implied return Rf + β(Rm − Rf). Rp − Rm is the active return, a different quantity.
Choosing Sharpe for a fund that is a small part of a diversified portfolio.
Sharpe is the best-known measure, so students default to it.
Fix: When the fund's specific risk is diversified away by other holdings, beta is the relevant risk, so Treynor or alpha fits better. Use Sharpe for stand-alone or total-risk cases.
Assuming Sharpe and Treynor always give the same ranking.
Both use excess return, so it feels as if the order must match.
Fix: They agree only when portfolios are well diversified so that total risk is mostly systematic. A fund with large specific risk can rank high on Treynor but low on Sharpe.
Mixing frequencies, such as monthly tracking error with an annual active return.
Data are given in different periods and students skip the conversion.
Fix: Put both numerator and denominator on the same basis. Annualise monthly standard deviation by multiplying by √12 and monthly mean by 12 when told to use simple scaling.

Worked examples

Example 1

Risk-free rate is 3%, market return is 8%. Portfolio A returned 12% with standard deviation 18% and beta 1.2. Portfolio B returned 9.5% with standard deviation 10% and beta 0.8. Which statement is correct? (a) A ranks higher on Sharpe and Treynor. (b) B ranks higher on Sharpe and Treynor, A ranks higher on Jensen's alpha. (c) B ranks higher on all three measures. (d) A ranks higher on all three measures.

Show the solution

Sharpe A = (12 − 3) ÷ 18 = 9 ÷ 18 = 0.50.
Sharpe B = (9.5 − 3) ÷ 10 = 6.5 ÷ 10 = 0.65.
Treynor A = (12 − 3) ÷ 1.2 = 7.50%.
Treynor B = (9.5 − 3) ÷ 0.8 = 6.5 ÷ 0.8 = 8.125%.
CAPM return A = 3 + 1.2 × (8 − 3) = 3 + 6 = 9%. Alpha A = 12 − 9 = 3.0%.
CAPM return B = 3 + 0.8 × 5 = 7%. Alpha B = 9.5 − 7 = 2.5%.
B is higher on Sharpe (0.65 vs 0.50) and Treynor (8.125% vs 7.50%). A is higher on alpha (3.0% vs 2.5%).

Answer: (b). B ranks higher on Sharpe and Treynor, while A has the higher Jensen's alpha.

Example 2

A fund's annual returns over four years exceeded its benchmark by 2%, −1%, 3% and 0%. Using the sample standard deviation, what is the information ratio?

Show the solution

Mean active return = (2 − 1 + 3 + 0) ÷ 4 = 4 ÷ 4 = 1.0%.
Deviations from the mean: 1, −2, 2, −1.
Squared deviations: 1, 4, 4, 1. Sum = 10.
Sample variance = 10 ÷ (4 − 1) = 3.333. Tracking error = √3.333 = 1.826%.
Information ratio = 1.0 ÷ 1.826 = 0.548.

Answer: Information ratio ≈ 0.55 (using the population standard deviation, 1.581%, you would get 0.63, so use n − 1 unless told otherwise).

Exam tips

Read the denominator word carefully: 'standard deviation' means Sharpe, 'beta' means Treynor, 'tracking error' means information ratio.
Expect questions that give two portfolios and ask which measure ranks them differently. Compute all measures for both and compare, since a single measure rarely settles it.
For alpha questions, always compute the CAPM required return as a separate line. Most wrong answers skip beta.
Check whether returns are given as annual or monthly before dividing. If the question requires annualising, apply it to both numerator and denominator consistently.
Know the 'when to use' logic in words: Sharpe for total risk or whole portfolio, Treynor for one component of a diversified portfolio, alpha and information ratio for assessing active management.

Practice questions from Modern Portfolio Theory (MPT) and the Capital Asset Pricing Model (CAPM)

Performance Measures: Sharpe, Treynor, Jensen, Information Ratio in other exams

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

Performance Measures: Sharpe, Treynor, Jensen, Information Ratio: frequently asked questions

What is the difference between the Sharpe ratio and the Treynor ratio?

Both divide excess return over the risk-free rate by a risk number. Sharpe uses total risk (standard deviation), while Treynor uses systematic risk (beta). They give the same ranking only when the portfolios are well diversified.

Which performance measure should I use for a diversified portfolio?

If the portfolio is well diversified, or is one part of a larger diversified holding, specific risk is largely gone, so Treynor ratio or Jensen's alpha is appropriate. Sharpe is better for stand-alone portfolios or when total risk matters.

How do I calculate Jensen's alpha?

First find the CAPM expected return: Rf + β × (Rm − Rf). Then subtract it from the portfolio's actual return. A positive result means the portfolio beat what its beta required.

How is the information ratio related to tracking error?

Tracking error is the standard deviation of the portfolio's returns minus the benchmark's returns. The information ratio is the average active return divided by that tracking error, so it measures active return per unit of active risk.

Can the Sharpe ratio be negative?

Yes. If the portfolio return is below the risk-free rate, the excess return is negative and so is the ratio. Be careful ranking negative Sharpe ratios, because a higher risk number then makes the ratio look less negative.