FRM Exam Part II · Factors

Factor Risk, Performance Attribution and Diversification Explained

Updated 11 October 2026 · Fact-checked

Factor risk decomposition splits a portfolio's return and variance into exposures (betas) times factor returns, plus a stock-specific residual. Attribution multiplies each active exposure by that factor's return. Solve questions by finding exposures, applying factor returns, separating systematic from idiosyncratic risk, then interpreting the result.

Understand Factor Risk, Performance Attribution and Diversification

A factor model says an asset's return comes from a few common drivers plus a part unique to that asset. Write it as R = α + β1·F1 + β2·F2 + … + ε. Each β is the factor exposure (loading). Each F is the factor return. ε is the idiosyncratic (specific) return, assumed uncorrelated with the factors.

Portfolio exposures are the weighted average of asset exposures. If you hold 60% of an asset with a value beta of 0.5 and 40% of one with a value beta of -0.25, your portfolio value beta is 0.6 × 0.5 + 0.4 × (-0.25) = 0.20. Exposures add linearly. Risk does not, because factors are correlated.

For risk, total variance = systematic variance + idiosyncratic variance. Systematic variance is β′Σβ, where Σ is the factor covariance matrix. Each factor's contribution to risk is its exposure times its marginal contribution (the covariance of the factor with the portfolio). These contributions sum to total systematic variance. A factor can have a small exposure and still dominate risk if it is volatile or highly correlated with other factors.

For performance attribution, you compare the portfolio with a benchmark. Active exposure = portfolio beta minus benchmark beta. Factor contribution to active return = active exposure × factor return. What is left over is alpha, or specific return. Attribution tells you what was earned, not whether it was skill. Earning a value premium through a value tilt is factor return, not stock-picking.

Diversification across factors works because factor premia are not perfectly correlated, so combining value, momentum, quality and low volatility can lower volatility. It is weaker in crises, when correlations rise. Factor timing tries to vary exposures using valuation spreads, momentum or the economic cycle. Evidence is mixed, and signals are noisy, so treat success as hard to prove. Crowding occurs when many investors hold the same factor tilts. Crowded factors can suffer sharp drawdowns when positions are unwound together, especially with leverage and low liquidity.

Key formulas to remember

Factor model: Rp = α + Σ βk × Fk + ε; βk is exposure to factor k, Fk is factor return, ε is the specific return.
Portfolio exposure: βp,k = Σ wi × βi,k; Weighted average of asset exposures. Linear in weights.
Systematic variance: σ²sys = β′ Σf β; Σf is the factor covariance matrix. With one factor: β² × σ²F.
Total variance: σ²p = β′ Σf β + Σ wi² σ²ε,i; Assumes specific returns are uncorrelated across assets and with factors.
Factor contribution to risk: Contribution of k = βk × Cov(Fk, Rp,sys) ÷ σ²p (as a % of variance); Contributions sum to the systematic share of variance.
Active exposure: Active βk = βp,k − βb,k; Portfolio minus benchmark exposure.
Factor attribution: Active return = Σ (Active βk × Fk) + active specific return; Residual is alpha, not necessarily skill.
R-squared: R² = σ²sys ÷ σ²p; Share of variance explained by the factors.

How to solve Factor Risk, Performance Attribution and Diversification questions

Use this order for most factor risk and attribution questions.

1Identify what is asked: risk decomposition, attribution, diversification, timing or crowding.
2List exposures. Compute portfolio exposures as weighted averages. For active questions, subtract the benchmark exposures.
3For attribution, multiply each exposure by the matching factor return. Add the results and compare with the actual return to find the residual.
4For risk, compute systematic variance (β²σ² for one factor, or β′Σβ for several). Add idiosyncratic variance for total variance. Take the square root for volatility.
5Compute shares: systematic ÷ total, or each factor's contribution ÷ total.
6Check units and signs. Variance is not volatility. Negative exposure with a positive factor return lowers return.
7Interpret in one line: what drives the result, and is it factor return or skill, or concentrated or diversified.
8For conceptual items, test each option against the conditions: correlations in stress, signal noise in timing, leverage and liquidity in crowding.

Quickest way: Exposure times factor return, then variance split

When to use it: Numerical MCQs with one or two factors and a short data table.

Compute exposure × factor return for each factor and sum.
Subtract from the actual return to get the residual.
For variance, compute β² × σ²F, add specific variance, then divide.
Eliminate options that confuse variance with volatility or use total instead of active exposure.

Common mistakes in Factor Risk, Performance Attribution and Diversification

Adding factor volatilities instead of using covariances
Return contributions add, so students assume risk does too.
Fix: Risk adds through variance and covariance. Use β′Σβ, and take the square root only at the end.
Using total exposure instead of active exposure in attribution
The benchmark row is skipped under time pressure.
Fix: For a benchmark-relative question, always compute portfolio β minus benchmark β first.
Treating the attribution residual as proof of skill
Alpha is the leftover, so it looks like manager contribution.
Fix: Residual also includes omitted factors and model error. Say it is unexplained by the chosen factors.
Assuming factor diversification always holds
Low average correlations are remembered, not their rise in stress.
Fix: State that diversification benefits are reduced when correlations rise in crises.
Claiming factor timing reliably adds value
Valuation spreads and momentum look like clear signals.
Fix: Say evidence is mixed, signals are noisy and costs matter. Do not call timing proven.
Ignoring leverage and liquidity in crowding
Crowding is seen only as a valuation issue.
Fix: Link crowding to simultaneous deleveraging, fire sales and sharp drawdowns when many hold the same tilt.

Worked examples

Example 1

A portfolio has betas of 1.10 to market, 0.40 to value and -0.20 to momentum. The benchmark has betas of 1.00, 0.00 and 0.00. Factor returns for the period were market 5.0%, value 2.0%, momentum 4.0%. The portfolio outperformed the benchmark by 1.5%. How much of the active return is unexplained by these factors?

Show the solution

Active betas: market 0.10, value 0.40, momentum -0.20.
Market contribution: 0.10 × 5.0% = 0.50%.
Value contribution: 0.40 × 2.0% = 0.80%.
Momentum contribution: -0.20 × 4.0% = -0.80%.
Factor total: 0.50% + 0.80% - 0.80% = 0.50%.
Residual: 1.5% - 0.50% = 1.00%.

Answer: The unexplained (specific) active return is 1.00%.

Example 2

A portfolio has a single-factor beta of 1.2. The factor volatility is 15% a year. Idiosyncratic volatility of the portfolio is 6% a year. What share of total variance is systematic, and what is total volatility?

Show the solution

Systematic variance = 1.2² × 0.15² = 1.44 × 0.0225 = 0.0324.
Idiosyncratic variance = 0.06² = 0.0036.
Total variance = 0.0324 + 0.0036 = 0.0360.
Systematic share = 0.0324 ÷ 0.0360 = 0.90, or 90%.
Total volatility = √0.0360 = 0.18974, about 18.97%.

Answer: 90% of variance is systematic; total volatility is about 18.97% a year.

Exam tips

Read whether the question asks for variance, volatility or percentage of variance. Options often include all three.
For attribution, check whether the return is total or active before choosing exposures.
On conceptual items, prefer cautious wording: factor timing is hard, diversification weakens in stress, crowding raises drawdown risk.
Link crowding to leverage, liquidity and common positioning, not just to popularity of a factor.
Sanity-check signs: a negative exposure to a factor with a positive return reduces return.

Practice questions from Factors

Factor Risk, Performance Attribution and Diversification in other exams

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

Factor Risk, Performance Attribution and Diversification: frequently asked questions

What is factor risk decomposition?

It splits portfolio variance into the part explained by factor exposures and the part that is stock-specific. You can also assign the systematic part to each factor. This shows which exposures actually drive risk.

How do you do factor-based performance attribution?

Compute active exposures versus the benchmark, multiply each by the factor return, and sum. The difference from actual active return is the specific or unexplained part. Use the same period for exposures and returns.

Can factor timing work?

Possibly in some cases, but evidence is mixed. Signals such as valuation spreads are noisy, and trading costs reduce gains. For the exam, treat timing as difficult and uncertain, not a dependable source of return.

Why does factor crowding cause drawdowns?

When many investors hold the same factor tilt, a shock can trigger common selling. Leverage and limited liquidity make this worse. Prices then move against everyone at once, causing sharp losses.