CFA Level II Exam · Using Multifactor Models

Factor Model Return Attribution and Risk Decomposition

Updated 7 October 2026 · Fact-checked

Active return is the portfolio return minus the benchmark return. A factor model splits it into factor return (active factor tilts times factor returns) and security selection (the leftover). Active risk squared splits into active factor risk and active specific risk. Compute each piece, then add.

Understand Factor Model Return Attribution and Risk Decomposition

A multifactor model says a portfolio's return comes from exposure to a few systematic factors, plus a part unique to the individual securities. You measure exposure with factor sensitivities (betas). Each portfolio has its own sensitivities, and so does its benchmark.

The key idea is to compare the portfolio with the benchmark, not with zero. The difference in sensitivity to a factor is the active factor tilt: portfolio sensitivity minus benchmark sensitivity. If the portfolio has a value sensitivity of 0.40 and the benchmark has 0.10, the tilt is +0.30.

Return attribution splits active return into two parts. Factor return is the sum over factors of (active tilt × factor return). It is the reward or cost of your factor bets. Security selection is what remains: active return minus factor return. It comes from picking securities that do better or worse than their factor exposures predict.

Risk decomposition works the same way, but with variances. Total active risk squared (also called tracking risk squared) is active factor risk plus active specific risk. Active factor risk is the variance of the active return explained by the factor tilts, calculated from the tilts and the factor variances and covariances. The tilts themselves are fixed exposures, so they have no variance. Active specific risk comes from the differences in security weights versus the benchmark, times each security's specific variance. You add variances, never standard deviations.

Key formulas to remember

Active return: Active return = Rp − Rb; Portfolio return minus benchmark return over the same period.
Active factor tilt: Tilt_k = βp,k − βb,k; Calculate it for each factor k. A positive tilt means the portfolio is more exposed than the benchmark.
Factor return (attribution): Factor return = Σ (βp,k − βb,k) × λk; λk is the return of factor k in the period. Use the factor return as given, not a premium assumed from memory.
Security selection: Security selection = Active return − Factor return; It is a residual. It equals Σ (wp,i − wb,i) × εi, where εi is each security's specific return, when the portfolio and benchmark betas are the weighted-average security betas.
Active risk: Active risk = √(Active risk squared); Also called tracking risk or tracking error. Take the square root last.
Active risk squared: Active risk squared = Active factor risk + Active specific risk; Variances add. Standard deviations do not.
Active factor risk (one factor): Active factor risk = (βp − βb)² × σ²factor; With several factors, you also need the covariances between the factors. The exam usually gives the variance-covariance data or the result.
Active specific risk: Active specific risk = Σ (wp,i − wb,i)² × σ²ε,i; Applies when security specific returns are uncorrelated. Weights are active weights versus the benchmark.
Share of risk: % from factors = Active factor risk ÷ Active risk squared; Use variance, not standard deviation, as the denominator.

How to solve Factor Model Return Attribution and Risk Decomposition questions

Use this order for any attribution or risk decomposition item. Pull the data from the vignette exhibit before you calculate.

1Identify what is asked: active return, factor return, security selection, active risk, or the share of risk from factors.
2Find the portfolio and benchmark sensitivities for each factor in the exhibit. Check that the factor list is the same for both.
3Compute the active tilt for each factor: portfolio beta minus benchmark beta. Keep the sign.
4Multiply each tilt by that factor's return for the period and add them. This is the factor return.
5Get active return as Rp − Rb. Subtract the factor return to find security selection.
6For risk, find active factor risk (variance of tilts) and active specific risk. Add them to get active risk squared.
7Take the square root only at the end to get active risk. Compute percentages on variances.
8Check signs and units. Convert percentages consistently and sanity check that the parts add to the total.

Quickest way: Tilt, multiply, subtract, add variances

When to use it: Use when the vignette gives betas for the portfolio and benchmark, factor returns, and either risk components or variances.

Write tilts in one line, such as +0.30, −0.20.
Multiply each by its factor return in your head or on the calculator, then sum for factor return.
Subtract factor return from active return to get selection. Do not recompute from security data.
For risk, square the factor-risk and specific-risk numbers only if they are given as standard deviations.
Add the two variances, then take √ once. Eliminate options that add standard deviations.

Common mistakes in Factor Model Return Attribution and Risk Decomposition

Using the portfolio's beta instead of the active tilt (beta difference).
Students recall the single-portfolio return formula and forget that attribution is relative to the benchmark.
Fix: Always subtract benchmark sensitivity first. Write the tilts before touching factor returns.
Adding standard deviations of factor risk and specific risk to get active risk.
The two numbers look like risks that should add, but only variances are additive here.
Fix: Square both, add, then take the square root. Check the answer is less than the sum of the two standard deviations.
Calling the whole active return security selection, or the whole factor return skill.
The labels are mixed up, or the residual idea is missed.
Fix: Factor return comes from tilts. Selection is the residual after you subtract it from active return.
Computing the share of risk with standard deviations instead of variances.
Active risk is quoted as a percentage, so it feels natural to divide those numbers.
Fix: Divide active factor variance by active risk squared. Square standard deviations first if needed.
Dropping the sign of a negative tilt.
Rushing through the multiplication, especially when the factor return is also negative.
Fix: Keep signs in brackets. A negative tilt times a negative factor return adds to return.
Treating specific risk as active risk because it is called idiosyncratic.
Students confuse total specific risk with active specific risk, which uses active weights.
Fix: Check whether the question asks about active risk (relative to the benchmark). If so, use active weights.

Worked examples

Example 1

A portfolio and its benchmark are described by two factors. Portfolio sensitivities: value 0.60, momentum 0.20. Benchmark sensitivities: value 0.30, momentum 0.50. Factor returns for the year: value 4.0%, momentum 2.0%. Portfolio return 9.5%, benchmark return 7.0%. (1) What is the active return? (2) What is the factor return? (3) What is the security selection contribution?

Show the solution

Active return = 9.5% − 7.0% = 2.5%.
Tilts: value 0.60 − 0.30 = +0.30; momentum 0.20 − 0.50 = −0.30.
Factor return = 0.30 × 4.0% + (−0.30) × 2.0% = 1.2% − 0.6% = 0.6%.
Security selection = 2.5% − 0.6% = 1.9%.

Answer: Active return 2.5%; factor return 0.6%; security selection 1.9%.

Example 2

For a portfolio versus its benchmark, a risk report shows active factor risk of 4.0 (in %², variance terms) and active specific risk of 5.0 (%²). (1) What is active risk squared? (2) What is active risk? (3) What share of active risk squared comes from factors?

Show the solution

Active risk squared = 4.0 + 5.0 = 9.0 (%²).
Active risk = √9.0 = 3.0%.
Factor share = 4.0 ÷ 9.0 = 0.444, or 44.4%.

Answer: Active risk squared 9.0 %²; active risk 3.0%; factors explain about 44.4% of active variance.

Exam tips

Read the exhibit headings carefully. Decide whether numbers are variances or standard deviations before you add anything.
Expect the vignette to give both portfolio and benchmark betas. The tilt is the first calculation, so do it before the question tempts you to shortcut.
If a question asks which part of active return reflects stock picking, the answer is the residual after the factor return, not the factor bets.
In conceptual questions, link a large share of active variance from factors to a portfolio that is a factor bet rather than stock selection.
You do not lose marks for wrong answers, so eliminate options that sum standard deviations and choose from what remains.

Factor Model Return Attribution and Risk Decomposition in other exams

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

Factor Model Return Attribution and Risk Decomposition: frequently asked questions

What is the difference between active return and factor return?

Active return is the portfolio return minus the benchmark return. Factor return is the part of it explained by active factor tilts times factor returns. The remainder is security selection.

How do I calculate active risk with a factor model?

Find active factor risk from the tilts and factor variances, and active specific risk from active weights and specific variances. Add the two variances to get active risk squared, then take the square root.

What is the difference between factor risk and idiosyncratic risk?

Factor risk comes from exposure to systematic factors shared across securities. Idiosyncratic, or specific, risk is unique to each security and is not explained by the factors. In active risk, both are measured relative to the benchmark.

Why can't I add standard deviations to get active risk?

Standard deviations do not add unless the components are perfectly correlated. Variances add when components are uncorrelated, so you add variances and then take the square root.