CFA Level I Exam · The Capital Asset Pricing Model, Market Model, and Other Factor-Based Equity Models

Risk-Adjusted Return Measures: Sharpe, Treynor, M-Squared, Alpha

Updated 7 October 2026 · Fact-checked

Risk-adjusted return measures compare portfolios after allowing for risk. Sharpe = (Rp − Rf) ÷ σp, Treynor = (Rp − Rf) ÷ βp, M² = Rf + (Rp − Rf)(σm ÷ σp) − Rm, Jensen's alpha = Rp − [Rf + β(Rm − Rf)], and information ratio = active return ÷ tracking error. Pick the measure by risk type.

Understand Performance Evaluation and Risk-Adjusted Return Measures

Raw return is a poor way to judge a manager. A portfolio that earned 12% with high risk may be worse than one that earned 9% with low risk. Risk-adjusted measures put returns on a common footing so you can compare them.

The first question is: what risk matters? If the portfolio is your whole wealth, total risk (standard deviation) matters. Use the Sharpe ratio or M-squared. If the portfolio is one part of a well-diversified larger portfolio, only systematic risk (beta) matters. Use the Treynor ratio or Jensen's alpha.

The Sharpe ratio is excess return per unit of total risk. It is the slope of the capital allocation line. M-squared restates Sharpe in percentage-return terms: it levers or de-levers the portfolio with the risk-free asset until its standard deviation matches the market's, then shows the return. It ranks portfolios the same way as Sharpe, but the answer is easier to read.

The Treynor ratio is excess return per unit of beta. Jensen's alpha is the return above what CAPM predicts for the portfolio's beta. Positive alpha means the portfolio beat its required return. The information ratio divides active return (portfolio minus benchmark) by tracking error, the standard deviation of those active returns. It measures how consistently a manager adds value over the benchmark.

Sharpe, Treynor and M-squared let you compare portfolios per unit of risk. Jensen's alpha is an absolute measure, not a ratio: it is the return in excess of the CAPM-required return. It does adjust for beta, because the required return includes beta times the market premium. But it is not expressed per unit of risk, so it is less suited than Treynor to ranking portfolios with very different betas.

Key formulas to remember

Sharpe ratio: SR = (Rp − Rf) ÷ σp; Uses total risk. Higher is better. Rf is the risk-free rate.
Treynor ratio: TR = (Rp − Rf) ÷ βp; Uses systematic risk. Can be misleading if beta is negative or near zero.
M-squared (M²): M² = Rf + (Rp − Rf) × (σm ÷ σp) − Rm; Equivalent to Rp* − Rm, where Rp* is the return of the portfolio rescaled to market volatility. Positive means it beat the market on a risk-matched basis.
Jensen's alpha: αp = Rp − [Rf + βp(Rm − Rf)]; Return above the CAPM-required return. Positive means outperformance.
Information ratio: IR = (Rp − Rb) ÷ tracking error; Tracking error is the standard deviation of (Rp − Rb) over time. Active return is the numerator.
Choosing the measure: Total risk: Sharpe, M². Systematic risk: Treynor, Jensen's alpha. Versus benchmark: information ratio.; Sharpe and M² give the same ranking of portfolios.

How to solve Performance Evaluation and Risk-Adjusted Return Measures questions

Use this method for any question on risk-adjusted performance.

1Identify what the question asks: a calculation, a ranking, or which measure suits the situation.
2Write down Rp, Rf, Rm, σp, σm, β and the benchmark return as given. Convert percentages to decimals only at the end if needed.
3Decide the risk type. Whole portfolio or total risk points to Sharpe or M². Part of a diversified portfolio points to Treynor or Jensen's alpha. A benchmark-relative mandate points to the information ratio.
4Compute the numerator first: excess return over Rf, or active return over the benchmark.
5Divide by the right denominator (σp, βp or tracking error), or subtract the CAPM required return for alpha.
6For M², scale the excess return by σm ÷ σp, add Rf, then subtract Rm.
7Check the sign and size. A positive alpha or M² means outperformance. Compare ratios only for the same measure.
8Eliminate options that use the wrong denominator or forget to subtract Rf.

Quickest way: Excess return first, then divide

When to use it: Use for numerical questions when time is short, which is about 90 seconds per question.

Subtract Rf from the portfolio return straight away. Most errors come from skipping this.
Look at the denominator word: standard deviation means Sharpe, beta means Treynor, tracking error means information ratio.
For alpha, compute required return Rf + β(Rm − Rf) and subtract it from Rp.
For M², compute Sharpe for the portfolio, multiply by σm, add Rf, and subtract Rm.
Pick the matching option. Options are listed smallest to largest, so check that your answer fits the order.

Common mistakes in Performance Evaluation and Risk-Adjusted Return Measures

Forgetting to subtract the risk-free rate in the numerator.
Return divided by risk feels like the natural ratio.
Fix: Sharpe and Treynor always use excess return, Rp − Rf. Write it first.
Using standard deviation in the Treynor ratio or beta in the Sharpe ratio.
The two ratios look similar and the names blur together.
Fix: Remember: Sharpe uses Standard deviation (S for S); Treynor uses beta (the systematic risk measure from CAPM).
Calling M² a ratio.
It is built from the Sharpe ratio, so it seems like a ratio too.
Fix: M² is a return difference in percentage points: the rescaled portfolio return minus the market return.
Using the wrong measure for the situation.
Candidates compute a number without asking how the portfolio is used.
Fix: Standalone or total-risk view: Sharpe or M². Part of a diversified portfolio: Treynor or alpha.
Using total return difference as the information ratio numerator over standard deviation of the portfolio.
Tracking error is confused with portfolio volatility.
Fix: Tracking error is the standard deviation of the active returns (portfolio minus benchmark), not of the portfolio.
Computing alpha with the wrong beta or using Rm instead of Rm − Rf.
The CAPM formula is applied from memory.
Fix: Required return = Rf + β × (Rm − Rf). Compute it separately, then subtract from Rp.

Worked examples

Example 1

Portfolio X returned 11% with a standard deviation of 15% and a beta of 1.20. The risk-free rate is 3% and the market returned 8% with a standard deviation of 10%. What is Portfolio X's Jensen's alpha: A) 0%, B) 2%, C) 4%?

Show the solution

Required return = Rf + β(Rm − Rf) = 3% + 1.20 × (8% − 3%).
1.20 × 5% = 6%, so required return = 9%.
Alpha = Rp − required return = 11% − 9% = 2%.

Answer: B) 2%

Example 2

Using the same data as above (Rp = 11%, σp = 15%, Rf = 3%, Rm = 8%, σm = 10%), what is the M-squared of Portfolio X: A) −0.67%, B) 0.33%, C) 1.33%?

Show the solution

Sharpe ratio of X = (11% − 3%) ÷ 15% = 0.5333.
Scale to market risk: risk-matched return = 3% + 0.5333 × 10% = 3% + 5.333% = 8.333%.
M² = 8.333% − 8% = 0.333%, about 0.33%.
Check: the market's Sharpe ratio is (8% − 3%) ÷ 10% = 0.50, below X's 0.5333, so M² is positive.

Answer: B) 0.33%

Exam tips

Questions are three-option MCQs with no penalty for guessing, so always eliminate the option that comes from skipping Rf or using the wrong denominator.
Expect conceptual items asking which measure fits a given investor view. Link total risk to Sharpe and M², and systematic risk to Treynor and alpha.
Sharpe and M² rank portfolios identically. Treynor and Sharpe rankings can differ when portfolios have different amounts of unsystematic (diversifiable) risk relative to their total risk.
On a BA II Plus, compute step by step with the memory keys: for Sharpe, (11 − 3) ÷ 15 =. Keep full precision until the end.
Know that the information ratio uses tracking error, and a higher value means more consistent active return per unit of active risk.

Practice questions from The Capital Asset Pricing Model, Market Model, and Other Factor-Based Equity Models

Performance Evaluation and Risk-Adjusted Return Measures in other exams

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

Performance Evaluation and Risk-Adjusted Return Measures: frequently asked questions

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

Both use excess return in the numerator. Sharpe divides by standard deviation, which is total risk. Treynor divides by beta, which is systematic risk only. Use Sharpe for standalone portfolios and Treynor for a portfolio that is one part of a diversified whole.

How do you calculate Jensen's alpha?

Find the CAPM required return: Rf + β(Rm − Rf). Subtract it from the portfolio's actual return. A positive result means the portfolio earned more than its beta justified.

What is the M-squared formula for CFA Level I?

M² = Rf + (Rp − Rf) × (σm ÷ σp) − Rm. You scale the portfolio's excess return to the market's volatility, add the risk-free rate, and subtract the market return. The answer is a percentage return difference.

What is tracking error in the information ratio?

Tracking error is the standard deviation of the differences between portfolio returns and benchmark returns. The information ratio is average active return divided by this figure. It shows how much active return the manager earned per unit of active risk.