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CFA Level I Exam · Portfolio Risk and Return: Part II

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

Updated 7 October 2026

Risk-adjusted performance measures compare portfolios after allowing for risk. Sharpe ratio divides excess return by total risk (standard deviation). Treynor divides excess return by beta. M-squared restates Sharpe in return units. Jensen's alpha is return above what CAPM predicts. Compute each from the formula, then rank the portfolios.

Understand Risk-Adjusted Performance Measures

Raw return is a poor way to compare managers. A portfolio that earned 12% by taking large risk is not clearly better than one that earned 10% with little risk. Risk-adjusted measures put returns on a common footing.

The measures split into two families. Total-risk measures use standard deviation: the Sharpe ratio and M-squared. Use them when the portfolio is your whole investment, or when you compare undiversified portfolios. Systematic-risk measures use beta: the Treynor ratio and Jensen's alpha. Use them when the portfolio is one part of a well-diversified larger portfolio, because only systematic risk is rewarded in that case.

Sharpe and Treynor are ratios: excess return per unit of risk. A higher number is better, but the number alone has no unit you can read as a return. M-squared fixes this. It scales your portfolio with risk-free borrowing or lending until its standard deviation equals the benchmark's, then reports the return difference. The answer is in percent, so it is easy to read.

Jensen's alpha is the actual return minus the return CAPM says the portfolio should earn for its beta. Positive alpha means the manager added value beyond the risk taken. Alpha is in return units, but it does not rank portfolios with different betas fairly, since it is not scaled by beta.

The measures can rank portfolios differently. This happens because they use different risk measures. If a portfolio is fully diversified, its total risk is mostly systematic, and Sharpe and Treynor rankings tend to agree.

Key formulas to remember

Sharpe ratio
Sharpe = (Rp − Rf) ÷ σp
Rp is portfolio return, Rf the risk-free rate, σp the portfolio standard deviation. Higher is better.
Treynor ratio
Treynor = (Rp − Rf) ÷ βp
Uses beta. Gives a meaningful result for positive beta; be careful interpreting it with negative or tiny beta.
M-squared (M²)
M² = (Sharpe_p − Sharpe_m) × σm = Rp* − Rm, where Rp* = Rf + (Rp − Rf) × (σm ÷ σp)
σm is the benchmark (market) standard deviation. Positive M² means the portfolio beat the benchmark at equal risk.
Jensen's alpha
α = Rp − [Rf + βp × (Rm − Rf)]
Rm is the market or benchmark return. Positive alpha means outperformance relative to CAPM.
Which measure when
Sharpe and M²: total risk. Treynor and Jensen's alpha: systematic risk (beta).
Match the measure to whether the portfolio is standalone or part of a diversified portfolio.

How to solve Risk-Adjusted Performance Measures questions

Use this routine for any question on risk-adjusted performance. It works whether the question asks for a ratio, a ranking or an interpretation.

  1. 1Read what is given: portfolio return, risk-free rate, standard deviation, beta and market return. Note which risk measure is supplied.
  2. 2Decide which measure the question asks for, or which fits the situation: standalone portfolio means Sharpe or M²; part of a diversified portfolio means Treynor or alpha.
  3. 3Convert all returns to the same basis, usually percent per year, and compute excess return Rp − Rf.
  4. 4Apply the formula. For M², first find the scaled return Rp* using σm ÷ σp, then subtract Rm.
  5. 5Check the sign and size. Sharpe and Treynor should be positive when excess return is positive; alpha is positive only if the return beats the CAPM-required return.
  6. 6Compare across portfolios: higher ratio is better. Then match to the three options, listed smallest to largest, and eliminate the two that fail your check.

Quickest way: Rank by ratio, then sanity-check

When to use it: Use when you must compare two or three portfolios quickly and options are close.

  1. Write excess return for each portfolio in one line.
  2. Divide each by its σ (Sharpe) or β (Treynor) in your head or on the calculator. Do not round until the end.
  3. For M², remember M² = (Sharpe difference) × σm. If the portfolio's Sharpe is higher than the benchmark's, M² must be positive.
  4. For alpha, compute the CAPM required return first; alpha is the leftover.
  5. Calculator: on the BA II Plus, put the excess return in parentheses before dividing. For a portfolio return of 12%, a risk-free rate of 3% and a standard deviation of 15%, key ( 12 − 3 ) ÷ 15 =. The display shows 0.6. The BA II Plus uses chain calculation by default, so keying 12 − 3 ÷ 15 without parentheses would be evaluated left to right as (12 − 3) ÷ 15 = 0.6. A calculator that follows the algebraic order of operations would instead give 12 − 0.2 = 11.8. Always use parentheses so the result does not depend on the calculator mode.

Common mistakes in Risk-Adjusted Performance Measures

  • Using total return instead of excess return in the numerator.

    The risk-free rate is often given as a side detail and gets ignored.

    Fix: Always write Rp − Rf first. Never divide raw return by risk.

  • Using beta in the Sharpe ratio or standard deviation in the Treynor ratio.

    Both ratios look alike and the names blur.

    Fix: Remember: Sharpe uses σ (total risk); Treynor uses β (systematic risk).

  • Computing M² as the scaled return Rp* without subtracting the benchmark return.

    Students stop after the leverage or de-leverage step.

    Fix: M² = Rp* − Rm. The scaled return is only a middle step.

  • Treating a higher Jensen's alpha as always better across portfolios with different betas.

    Alpha is in return units, so it looks directly comparable.

    Fix: Alpha is not adjusted for the size of beta. Use Treynor to rank when betas differ, and say so if asked for limitations.

  • Using the wrong measure for the situation.

    Students memorise formulas but skip the context in the question stem.

    Fix: Standalone or undiversified portfolio: Sharpe or M². One piece of a diversified portfolio: Treynor or alpha.

Worked examples

Example 1

The risk-free rate is 3%. Portfolio X earned 11% with a standard deviation of 16% and a beta of 1.2. Portfolio Y earned 9% with a standard deviation of 10% and a beta of 0.8. Which portfolio has the higher Sharpe ratio, and what are the Sharpe values?

Show the solution
  1. Excess return X = 11 − 3 = 8%. Excess return Y = 9 − 3 = 6%.
  2. Sharpe X = 8 ÷ 16 = 0.50.
  3. Sharpe Y = 6 ÷ 10 = 0.60.
  4. Y has the higher Sharpe ratio, so Y earned more excess return per unit of total risk.

Answer: Portfolio Y has the higher Sharpe ratio: 0.60 versus 0.50 for X.

Example 2

The risk-free rate is 2%. The market return is 8% with a standard deviation of 12%. A portfolio returned 10% with a standard deviation of 16% and a beta of 1.25. Calculate M² and Jensen's alpha.

Show the solution
  1. Scale the portfolio to the market's risk: weight in portfolio = σm ÷ σp = 12 ÷ 16 = 0.75, rest in the risk-free asset.
  2. Rp* = 2 + (10 − 2) × 0.75 = 2 + 6 = 8%.
  3. M² = Rp* − Rm = 8 − 8 = 0.00%.
  4. Check with Sharpe: portfolio = 8 ÷ 16 = 0.50; market = 6 ÷ 12 = 0.50. Equal, so M² = 0, which agrees.
  5. CAPM required return = 2 + 1.25 × (8 − 2) = 2 + 7.5 = 9.5%.
  6. Alpha = 10 − 9.5 = 0.5%.

Answer: M² is 0.00% (same total-risk performance as the market) and Jensen's alpha is +0.5%. The two measures can point different ways because one uses total risk and the other beta.

Exam tips

  • Check the wording: 'well-diversified' or 'one of several managers' points to Treynor or alpha; 'sole investment' points to Sharpe or M².
  • M² is positive exactly when the portfolio's Sharpe ratio exceeds the benchmark's Sharpe ratio. If the sign of M² and the Sharpe comparison disagree, you made an arithmetic slip.
  • Options are listed smallest to largest. Estimate first (for example 8 ÷ 16 is 0.5), then pick the matching option instead of guessing.
  • Do not round Rp* or ratios mid-calculation. Round only the final answer.
  • If the question asks about a limitation, state it: Sharpe assumes standard deviation captures risk, Treynor and alpha depend on the benchmark and a valid CAPM.

Practice questions from Portfolio Risk and Return: Part II

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 divides by standard deviation, which is total risk. Treynor divides by beta, which is systematic risk only. Use Sharpe for a standalone portfolio and Treynor for a portfolio that is part of a diversified one.

How do I calculate the M-squared measure?

Scale your portfolio with the risk-free asset so its standard deviation equals the benchmark's. The scaled return is Rf + (Rp − Rf) × (σm ÷ σp). Subtract the benchmark return from it. The result is M² in percent.

What is the Jensen's alpha formula?

Alpha = Rp − [Rf + β × (Rm − Rf)]. The term in brackets is the return CAPM requires for the portfolio's beta. A positive alpha means the portfolio earned more than that required return.

Can these measures rank portfolios differently?

Yes. Sharpe and M² use total risk, while Treynor and alpha use beta. A portfolio with a lot of non-systematic risk can look good on Treynor but poor on Sharpe. The rankings tend to agree when the portfolios are well diversified.