FRM Exam Part II · Portfolio Performance Evaluation

Risk-Adjusted Performance Measures for FRM Part II

Updated 11 October 2026 · Fact-checked

Risk-adjusted performance measures compare a portfolio's return with the risk taken to earn it. Sharpe uses total risk, Treynor uses beta, Jensen's alpha is excess return over CAPM, M-squared restates Sharpe in return units, information ratio uses tracking error, and Sortino uses downside deviation. Compute each, then rank.

Understand Risk-Adjusted Performance Measures

Raw return tells you little. A fund that earned 12% with wild swings is not necessarily better than one that earned 10% with calm. Risk-adjusted measures put return and risk on one scale so you can compare managers fairly.

The measures split into two groups. Absolute measures compare a portfolio with the risk-free rate: Sharpe, Treynor, Jensen's alpha, M-squared and Sortino. Relative measures compare a portfolio with a benchmark: the information ratio. The choice of risk measure matters. Sharpe divides by total risk (standard deviation), so it penalises both systematic and diversifiable risk. Treynor and Jensen use beta, so they only count systematic risk.

Use the right one for the setting. If the portfolio is your investor's whole wealth, total risk matters, so use Sharpe. If it is one slice of a well-diversified pool, only beta matters, so Treynor or Jensen fits better. For an active manager judged against an index, use the information ratio. If downside losses matter more than upside swings, use Sortino.

Sharpe and Treynor give ratios, so they only rank portfolios. Jensen's alpha and M-squared give answers in return terms (percent), so you can read them directly as outperformance. M-squared works by levering or de-levering the portfolio with the risk-free asset until its volatility matches the market's, then comparing returns.

All of these rely on historical data and assumptions such as the CAPM and roughly symmetric returns. Rankings can change with the measure, the period and the benchmark.

Key formulas to remember

Sharpe ratio: SR = (Rp − Rf) ÷ σp; Rp = portfolio return, Rf = risk-free rate, σp = standard deviation of portfolio returns. Uses total risk.
Treynor ratio: TR = (Rp − Rf) ÷ βp; Uses systematic risk. Only meaningful for positive beta; compare among diversified portfolios.
Jensen's alpha: αp = Rp − [Rf + βp × (Rm − Rf)]; Excess return over the CAPM-required return. Positive alpha means outperformance for the beta taken.
M-squared (Modigliani): M² = Rf + SRp × σm, and M² difference = M² − Rm = (SRp − SRm) × σm; σm = market (benchmark) volatility. Result is a return, directly comparable with the benchmark return.
Information ratio: IR = (Rp − Rb) ÷ TE, where TE = standard deviation of (Rp − Rb); Rb = benchmark return. TE = tracking error. Numerator is active return.
Sortino ratio: Sortino = (Rp − MAR) ÷ DD; MAR = minimum acceptable return (often Rf or a target). DD = downside deviation, using only returns below MAR.
Link between Sharpe and Treynor: TR = SR × (σp ÷ βp); Rankings agree only if σp ÷ βp is similar, that is, if portfolios have similar diversification.

How to solve Risk-Adjusted Performance Measures questions

Use this routine for any question on risk-adjusted performance. It keeps you from mixing total and systematic risk.

1Read what the question gives: returns, risk-free rate, standard deviation, beta, benchmark return, tracking error or downside deviation.
2Decide whether the portfolio is the investor's whole wealth (total risk, Sharpe/M-squared) or part of a diversified pool (beta, Treynor/Jensen), or is judged against a benchmark (information ratio).
3Compute excess return first: Rp − Rf, or Rp − Rb for the information ratio.
4Divide by the correct risk measure, or for Jensen's alpha subtract the CAPM expected return.
5For M-squared, compute the Sharpe ratio, multiply by the market volatility, add Rf, then compare with Rm.
6Rank portfolios, or state the sign of alpha or M-squared difference. Higher is better for every measure here.
7Add a one-line interpretation and note any caveat, such as negative beta, skewed returns or short data.

Quickest way: Excess return over the right denominator

When to use it: Use when four options are close and you need the answer in under two minutes.

Write Rp − Rf at once and keep it as one number.
Match the denominator to the name: σ for Sharpe, β for Treynor, tracking error for IR, downside deviation for Sortino.
For M-squared, skip building the leveraged portfolio: compute (SRp − SRm) × σm and see if it is positive.
For Jensen, compute the CAPM return in one line, then subtract.
Check the sign and size against the options and eliminate any that use the wrong risk measure.

Common mistakes in Risk-Adjusted Performance Measures

Using standard deviation in the Treynor ratio or beta in the Sharpe ratio.
The two ratios look alike and both start with excess return.
Fix: Remember S for Sharpe, S for Standard deviation (total risk). T for Treynor, T for beta as the sensitivity to the market.
Forgetting to subtract the risk-free rate in the numerator.
Students use the headline return given in the question.
Fix: Always write Rp − Rf as step one. Only the information ratio uses Rp − Rb.
Treating M-squared as a ratio instead of a return.
It is built from the Sharpe ratio, so students leave it unitless.
Fix: Multiply the Sharpe ratio by σm and add Rf. The answer is a percentage return that you compare with Rm.
Using the standard deviation of the portfolio in the information ratio denominator.
Confusion between portfolio risk and tracking error.
Fix: Tracking error is the standard deviation of the active return (Rp − Rb), not of Rp.
Calculating downside deviation over only the negative returns, or dividing by the wrong count.
Students ignore that the deviation is measured from MAR across all observations.
Fix: Square the shortfalls below MAR (zero for returns above), average over all observations as the question specifies, then take the square root.
Ranking by Treynor or Jensen when beta is negative or portfolios are not diversified.
The formulas still produce a number.
Fix: State that beta-based measures need diversified portfolios, and that a negative beta reverses the ratio's meaning.

Worked examples

Example 1

Portfolio A returned 14% with standard deviation 20% and beta 1.2. Portfolio B returned 11% with standard deviation 12% and beta 0.8. The risk-free rate is 3%, the market return is 10% and market volatility is 15%. Compute the Sharpe ratio, Treynor ratio and Jensen's alpha of each and say which is better on each measure.

Show the solution

Excess returns: A = 14% − 3% = 11%. B = 11% − 3% = 8%.
Sharpe: A = 11 ÷ 20 = 0.55. B = 8 ÷ 12 = 0.667.
Treynor: A = 11 ÷ 1.2 = 9.17%. B = 8 ÷ 0.8 = 10.00%.
CAPM return: A = 3 + 1.2 × (10 − 3) = 11.4%. B = 3 + 0.8 × 7 = 8.6%.
Jensen's alpha: A = 14 − 11.4 = 2.6%. B = 11 − 8.6 = 2.4%.
Compare: B is better on Sharpe and Treynor. A is slightly better on Jensen's alpha.

Answer: Sharpe: A 0.55, B 0.667. Treynor: A 9.17%, B 10.00%. Jensen's alpha: A 2.6%, B 2.4%. B wins on Sharpe and Treynor; A has the higher alpha.

Example 2

A fund returned 9% with volatility 10%. The risk-free rate is 2%. The market index returned 8% with volatility 12%. Compute the fund's M-squared and say whether it beat the market on a risk-adjusted basis.

Show the solution

Sharpe ratio of the fund = (9 − 2) ÷ 10 = 0.70.
Sharpe ratio of the market = (8 − 2) ÷ 12 = 0.50.
M-squared = Rf + SRp × σm = 2% + 0.70 × 12% = 2% + 8.4% = 10.4%.
Check with the difference: (0.70 − 0.50) × 12% = 2.4%, and 10.4% − 8% = 2.4%.
The fund, levered or de-levered to 12% volatility, would earn 10.4% against the market's 8%.

Answer: M-squared = 10.4%, which is 2.4 percentage points above the market's 8%, so the fund beat the market on a risk-adjusted basis.

Exam tips

Questions often give both σ and β. Read the name of the measure carefully and pick the matching one.
When two measures give different rankings, expect a conceptual question: the answer usually turns on total versus systematic risk or on diversification.
For the information ratio, the key numbers are active return and tracking error. Ignore the risk-free rate unless the benchmark is cash.
Sortino questions test why it differs from Sharpe: it penalises only downside deviation, so it suits asymmetric or option-like returns.
Keep all inputs in the same units and periodicity. If the data are monthly, annualise consistently before comparing.

Practice questions from Portfolio Performance Evaluation

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 divide excess return over the risk-free rate by a risk measure. Sharpe uses total risk (standard deviation), while Treynor uses systematic risk (beta). Use Sharpe for a stand-alone portfolio and Treynor when the portfolio is one part of a diversified holding.

How do I calculate Jensen's alpha and M-squared?

For Jensen's alpha, compute the CAPM return Rf + β(Rm − Rf) and subtract it from the portfolio return. For M-squared, multiply the portfolio Sharpe ratio by the market volatility and add Rf. Compare the result with the market return.

How is the Sortino ratio different from the Sharpe ratio?

Sortino replaces standard deviation with downside deviation and replaces the risk-free rate with a minimum acceptable return. It ignores upside volatility, so it does not punish a manager for large gains. It is more useful when returns are skewed.

How do I interpret the information ratio?

It shows active return earned per unit of tracking error. A higher value means the manager earns more excess return over the benchmark for each unit of active risk. It measures consistency of skill against a benchmark, not absolute performance.