CFA Level II Exam · Using Multifactor Models

Sharpe Ratio and Information Ratio in Multifactor Models

Updated 7 October 2026 · Fact-checked

The Sharpe ratio is excess return over the risk-free rate divided by total risk. The information ratio is active return over a benchmark divided by active risk. In a multifactor model, you split active return into factor tilts and selection, then choose exposures that maximise the Sharpe ratio, where SR² = benchmark SR² + IR².

Understand Risk-Adjusted Performance: Sharpe and Information Ratios

Return alone does not tell you whether a manager did well. You need to know how much risk was taken to earn it. Two ratios do this job in the multifactor chapter.

The Sharpe ratio measures reward per unit of total risk. You subtract the risk-free rate from the portfolio return and divide by the portfolio's standard deviation. Use it to judge a whole portfolio, or to compare the benchmark with the portfolio.

The information ratio (IR) measures reward per unit of active risk. Active return is portfolio return minus benchmark return. Active risk (tracking risk) is the standard deviation of that active return. A manager can have a high Sharpe ratio just because the market rose, while the IR shows whether the manager added value over the benchmark.

A multifactor model helps explain where active return came from. Active return = the sum of (portfolio factor sensitivity − benchmark factor sensitivity) × factor return, plus security selection. The first part is the factor tilt return. The second is the part the factors do not explain. Active risk likewise splits into active factor risk and active specific (selection) risk.

The final idea is optimal exposure. You want the portfolio with the highest Sharpe ratio. Taking active positions adds Sharpe only through the IR: the maximum Sharpe ratio squared equals the benchmark Sharpe ratio squared plus the IR squared. A higher IR lets you reach a higher Sharpe ratio. The best level of active risk is proportional to the IR.

Key formulas to remember

Sharpe ratio: SR = (Rp − Rf) ÷ σp; Uses total standard deviation. Use the same period for return and risk.
Active return: Rp − Rb = Σ[(wp,k − wb,k) × factor return k] + security selection; The sum is the factor tilt return. The remainder is selection. Here w means factor sensitivity.
Active risk (tracking risk): σ(Rp − Rb) = standard deviation of (Rp − Rb); Also called tracking error. It is not the portfolio's standard deviation.
Information ratio: IR = (Rp − Rb) ÷ σ(Rp − Rb); Mean active return divided by active risk. Compare this with the benchmark, not with Rf.
Active risk decomposition: Active risk² = active factor risk + active specific risk; Add variances, not standard deviations.
Maximum Sharpe ratio: SR*² = SRB² + IR²; Holds when active positions are optimally sized. So SR* = √(SRB² + IR²).
Optimal active risk: σA* = (IR ÷ SRB) × σB; Scales active risk to the IR and to the benchmark's reward per unit of risk.

How to solve Risk-Adjusted Performance: Sharpe and Information Ratios questions

Use the same sequence for any vignette on risk-adjusted performance or optimal factor exposure.

1Read the question first and decide whether it asks about total risk (Sharpe) or active risk (IR).
2Find the data in the vignette or exhibit: portfolio return, benchmark return, risk-free rate, standard deviation and tracking risk. Check the periods match.
3For Sharpe, subtract Rf from the return, then divide by standard deviation. For IR, subtract the benchmark return, then divide by tracking risk.
4If factor data are given, compute active factor return as (portfolio sensitivity − benchmark sensitivity) × factor return for each factor, then add them up. Subtract the total from active return to get selection.
5If active risk has two parts, add the variances (squares), then take the square root.
6For optimal exposure, compute SRB, then use SR*² = SRB² + IR² and σA* = (IR ÷ SRB) × σB.
7Check the answer is sensible: a higher Sharpe ratio or IR is better, and the optimal Sharpe ratio must be at least the benchmark Sharpe ratio.

Quickest way: Identify the denominator, then plug in

When to use it: Use when the vignette gives clean returns and risk numbers and you have about two minutes per question.

Ask: excess over Rf or over the benchmark? That tells you Sharpe or IR.
Write numerator and denominator on scratch paper before calculating.
Keep returns and risks in percent, and divide directly, for example 1.5 ÷ 4.
For optimal Sharpe, square both ratios, add them, then take the square root.
Eliminate any answer where the optimal Sharpe is below the benchmark Sharpe.

Common mistakes in Risk-Adjusted Performance: Sharpe and Information Ratios

Using portfolio standard deviation as the denominator of the information ratio.
Both ratios are called risk-adjusted, so the denominators get mixed up.
Fix: IR always uses active risk, the standard deviation of portfolio minus benchmark returns. Sharpe uses total standard deviation.
Subtracting the risk-free rate in the IR numerator.
Students carry over the Sharpe formula automatically.
Fix: The IR numerator is portfolio return minus benchmark return. Rf appears only in Sharpe.
Adding active factor risk and active specific risk as standard deviations.
Students forget that only variances add.
Fix: Square each, add, then take the square root.
Using the factor sensitivity itself instead of the difference from the benchmark.
Students compute total factor return, not active factor return.
Fix: Subtract the benchmark sensitivity first, then multiply by the factor return.
Writing SR* = SRB + IR.
It looks like a simple add-on.
Fix: The relationship is in squares: SR*² = SRB² + IR². Take the square root at the end.
Treating a high IR as proof of skill without checking the source.
Students stop at the ratio.
Fix: Decompose active return. If most of it comes from factor tilts, it is a factor bet, not security selection.

Worked examples

Example 1

A portfolio returned 9.5% with a standard deviation of 14%. Its benchmark returned 8.0% with a standard deviation of 12%. The risk-free rate is 3.0% and tracking risk is 4.0%. (1) Compute the portfolio Sharpe ratio. (2) Compute the information ratio. (3) Compute the maximum Sharpe ratio from optimally sized active positions, and the optimal active risk.

Show the solution

Portfolio Sharpe = (9.5 − 3.0) ÷ 14 = 6.5 ÷ 14 = 0.4643.
Active return = 9.5 − 8.0 = 1.5%. IR = 1.5 ÷ 4.0 = 0.375.
Benchmark Sharpe = (8.0 − 3.0) ÷ 12 = 0.4167.
SR*² = 0.4167² + 0.375² = 0.1736 + 0.1406 = 0.3142. SR* = 0.5605.
Optimal active risk = (0.375 ÷ 0.4167) × 12% = 0.9 × 12% = 10.8%.

Answer: (1) Sharpe ≈ 0.46. (2) IR = 0.375. (3) Maximum Sharpe ≈ 0.56 with optimal active risk of 10.8%.

Example 2

A fund and its benchmark have these sensitivities: value factor 0.6 (fund) vs 0.2 (benchmark); size factor 0.1 (fund) vs 0.3 (benchmark). Factor returns for the period: value 3%, size 2%. The fund beat the benchmark by 1.5% with tracking risk of 3.5%. (1) Compute the active factor return. (2) Compute the security selection return. (3) Compute the information ratio.

Show the solution

Value tilt = (0.6 − 0.2) × 3% = 0.4 × 3% = 1.2%.
Size tilt = (0.1 − 0.3) × 2% = −0.2 × 2% = −0.4%.
Active factor return = 1.2% + (−0.4%) = 0.8%.
Selection = total active return − factor return = 1.5% − 0.8% = 0.7%.
IR = 1.5 ÷ 3.5 = 0.4286.

Answer: (1) Active factor return = 0.8%. (2) Security selection = 0.7%. (3) IR ≈ 0.43.

Exam tips

Read the vignette for the word 'tracking risk' or 'active risk'. That number is the IR denominator and is easy to confuse with the portfolio standard deviation.
Expect a question that asks which source of active return is larger: factor tilts or selection. Do the decomposition with differences in sensitivities.
When asked to improve the Sharpe ratio, link it to the IR: only a positive IR raises the maximum Sharpe ratio above the benchmark's.
Do the squares carefully and keep four decimals until the final step, then round to the precision the options use.
If an exhibit shows several managers, compute the ratio for each one. Do not rank by return alone.

Risk-Adjusted Performance: Sharpe and Information Ratios 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: Sharpe and Information Ratios: frequently asked questions

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

The Sharpe ratio divides return above the risk-free rate by total standard deviation. The information ratio divides return above the benchmark by active risk. Sharpe judges the whole portfolio, and IR judges the value added by active management.

How do I calculate the information ratio from active return?

Subtract the benchmark return from the portfolio return to get active return. Then divide by the standard deviation of that active return, also called tracking risk. For example, 1.5% active return and 4% tracking risk gives an IR of 0.375.

How does a multifactor model give the maximum Sharpe ratio?

The model separates active return into factor tilts and selection, and active risk into factor and specific risk. With optimally sized active positions, the maximum Sharpe ratio satisfies SR*² = SRB² + IR². A higher IR therefore means a higher achievable Sharpe ratio.

Do I add active factor risk and active specific risk directly?

No. Add them as variances: active risk² = active factor risk + active specific risk. Then take the square root to get active risk.