Strategic Financial Management · Equity and Bond Valuation and Evaluation of Performance

Risk-Adjusted Performance Measures: Sharpe, Treynor and Jensen

Updated 11 October 2026 · Fact-checked

Risk-adjusted performance measures judge a portfolio by the return it earned for each unit of risk taken, not by raw return. Sharpe divides excess return by total risk (standard deviation), Treynor by beta, and Jensen's alpha is the return above the CAPM-expected return. Compute excess return first, then apply the formula and rank.

Understand Risk-Adjusted Performance Measures

A portfolio that earned 18% is not automatically better than one that earned 14%. The first may have taken far more risk. Risk-adjusted measures fix this by comparing the excess return (portfolio return minus the risk-free rate) with the risk taken to earn it.

The Sharpe ratio uses total risk, measured by standard deviation. It suits an investor whose portfolio is the whole of their wealth, or when you compare portfolios that are not fully diversified. The Treynor ratio uses only systematic risk, measured by beta. It suits a well-diversified portfolio that is one part of a larger holding, because unsystematic risk is assumed to be diversified away.

Jensen's alpha works differently. It takes the return CAPM says the portfolio should have earned for its beta, and subtracts that from the actual return. A positive alpha means the manager added value; a negative alpha means the manager destroyed value. Alpha is in percentage points, not a ratio.

M-squared (Modigliani) restates Sharpe in percentage terms. You lever or de-lever the portfolio with the risk-free asset until its standard deviation equals the market's, then compare its return with the market return. The information ratio compares active return over a benchmark with the tracking error, which is the standard deviation of that active return. It tells you how consistently a manager beats the benchmark.

Sharpe, Treynor and M-squared give the same ranking only when portfolios are compared on the same basis. Rankings can differ between Sharpe and Treynor if the portfolios are not well diversified. Jensen's alpha can rank differently because it ignores the scale of beta.

Key rules to remember

Sharpe ratio: Sharpe = (Rp − Rf) ÷ σp; Rp is portfolio return, Rf risk-free rate, σp standard deviation of the portfolio. Higher is better.
Treynor ratio: Treynor = (Rp − Rf) ÷ βp; Uses beta. Meaningful for comparison when beta is positive. Higher is better.
Jensen's alpha: α = Rp − [Rf + βp × (Rm − Rf)]; Positive alpha means outperformance relative to CAPM.
M-squared: M² = Rf + Sharpe of portfolio × σm; M² difference = M² − Rm; σm is the market (benchmark) standard deviation. Positive difference means beating the market on a risk-matched basis.
Information ratio: IR = (Rp − Rb) ÷ Tracking error; Rb is benchmark return. Tracking error is the standard deviation of (Rp − Rb).
Market Sharpe and Treynor benchmarks: Market Sharpe = (Rm − Rf) ÷ σm; Market Treynor = Rm − Rf; Market beta is 1, so the market Treynor equals its excess return. Compare portfolio ratios with these.

How to solve Risk-Adjusted Performance Measures questions

Use the same sequence for every numerical, whichever measure the question names.

1List the given data: Rp, Rf, Rm, σ, β for each portfolio and for the market. Check all are in the same period (annual or monthly).
2Compute excess return Rp − Rf for each portfolio. If the question gives a series of returns, take the average first.
3Pick the measure the question asks for. If it says total risk or standard deviation, use Sharpe. If it says beta or systematic risk, use Treynor or alpha.
4Calculate the measure for every portfolio and for the market, if the market data is given.
5Rank the portfolios from highest to lowest. For alpha, a positive value means outperformance.
6Compare with the market benchmark and state a clear conclusion in one or two lines.
7If the rankings differ between measures, explain why: Sharpe penalises total risk, Treynor only systematic risk.

Quickest way: Excess return table method

When to use it: Use when the question gives two or three portfolios and asks for several measures or a ranking.

Draw a small table with columns: Rp, Rf, excess, σ, β.
Fill the excess return column first, because every ratio uses it.
Divide the excess by σ for Sharpe and by β for Treynor in the same pass.
For alpha, compute the CAPM return Rf + β(Rm − Rf) once per row and subtract from Rp.
Rank, then write one line of conclusion for each measure.

Common mistakes in Risk-Adjusted Performance Measures

Dividing the portfolio return by risk without subtracting the risk-free rate.
Students remember 'return per unit of risk' and skip the excess return.
Fix: Always write Rp − Rf as the first line of working.
Using beta in the Sharpe ratio or standard deviation in the Treynor ratio.
The two formulas look alike and the names are easy to swap.
Fix: Remember: Sharpe goes with Standard deviation (both start with S); Treynor goes with beta.
Mixing percentages and decimals, for example 12 − 0.06.
Data is given in different forms in the same question.
Fix: Convert everything to percentages or everything to decimals before calculating.
Reading alpha as a ratio and ranking it like Sharpe.
All the measures are called 'ratios' loosely.
Fix: Treat alpha as an excess return over CAPM. State whether it is positive or negative and what that says about the manager.
Concluding that the highest raw return is the best portfolio.
The conclusion line is written from habit, not from the computed ratios.
Fix: Base the ranking only on the measure computed and mention the risk taken.
Using the market return minus risk-free as the beta in Treynor comparison.
Confusion over the market benchmark.
Fix: Market beta is 1, so market Treynor is Rm − Rf. Compare the portfolio Treynor with that number.

Worked examples

Example 1

The risk-free rate is 6% and the market return is 12% with a standard deviation of 10%. Portfolio A returned 15% with σ = 12% and β = 1.2. Portfolio B returned 13% with σ = 8% and β = 0.9. Compute Sharpe, Treynor and Jensen's alpha for both, and rank them.

Show the solution

Excess return: A = 15 − 6 = 9%. B = 13 − 6 = 7%.
Sharpe: A = 9 ÷ 12 = 0.75. B = 7 ÷ 8 = 0.875.
Treynor: A = 9 ÷ 1.2 = 7.5. B = 7 ÷ 0.9 = 7.78 (approx).
CAPM return: A = 6 + 1.2 × 6 = 13.2%. B = 6 + 0.9 × 6 = 11.4%.
Alpha: A = 15 − 13.2 = 1.8%. B = 13 − 11.4 = 1.6%.
Market Sharpe = 6 ÷ 10 = 0.6 and market Treynor = 6. Both portfolios beat the market on both ratios.
Ranking: Sharpe gives B above A. Treynor gives B above A. Alpha gives A above B.

Answer: Sharpe: A 0.75, B 0.875. Treynor: A 7.5, B 7.78. Alpha: A 1.8%, B 1.6%. B ranks first on Sharpe and Treynor because it earned more per unit of risk; A has the higher alpha because it carries a higher beta and still beat the CAPM return by more. Both beat the market.

Example 2

Market return is 11%, market standard deviation 9%, risk-free rate 5%. A fund earned 14% with a standard deviation of 12%. Compute the fund's M-squared and say whether it beat the market on a risk-adjusted basis.

Show the solution

Fund Sharpe = (14 − 5) ÷ 12 = 0.75.
M-squared return = Rf + Sharpe × σm = 5 + 0.75 × 9 = 11.75%.
Check: levering down the fund to a 9% standard deviation means holding 9/12 = 75% in the fund and 25% in the risk-free asset. Return = 0.75 × 14 + 0.25 × 5 = 10.5 + 1.25 = 11.75%.
Difference from market = 11.75 − 11 = 0.75%.

Answer: M-squared = 11.75%, which is 0.75 percentage points above the market return of 11%. The fund beat the market on a risk-adjusted basis.

Exam tips

Write the formula and excess return before substituting. Method marks are awarded even if the arithmetic slips.
If the question gives no market data, you can still compute Sharpe and Treynor, but not alpha. Check what is available before choosing a measure.
End every numerical with a one-line ranking and recommendation. Examiners look for the conclusion.
For theory, be ready to explain why Sharpe and Treynor can rank differently: one uses total risk, the other only systematic risk.
In MCQs, check whether the question gives variance instead of standard deviation. Take the square root first.

Practice questions from Equity and Bond Valuation and Evaluation of Performance

Risk-Adjusted Performance Measures in other exams

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

Risk-Adjusted Performance Measures: frequently asked questions

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

Both measure excess return per unit of risk. Sharpe uses total risk (standard deviation), while Treynor uses only systematic risk (beta). Use Sharpe for undiversified portfolios and Treynor for well-diversified ones.

How do I calculate Jensen's alpha?

First find the CAPM return: 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.

What does M-squared tell you that Sharpe does not?

M-squared converts the risk-adjusted result into a return figure that you can compare directly with the market return. It gives the same ranking as Sharpe but is easier to interpret.

What is the information ratio?

It is the average return above a benchmark divided by the tracking error, which is the standard deviation of that excess return. A higher value shows more consistent outperformance.