FRM Exam Part II · Factor Theory

Factor Risk Decomposition and Performance Attribution

Q: How do I decompose portfolio risk into factor risk?

Regress portfolio excess returns on the factor returns to get betas and a residual. Systematic variance comes from the betas and the factor variances (and correlations). Idiosyncratic variance is the residual variance, and the two add up to total variance.

Q: What is the difference between alpha and factor beta?

Factor beta is the return you get from exposure to a systematic factor, which you could in principle get cheaply through factor portfolios. Alpha is the return left after removing all factor exposures. It only counts as evidence of skill if it is statistically significant and no relevant factor is missing.

Q: How do I read a factor model regression in the exam?

The intercept is alpha, each slope is the exposure to that factor, and R² is the share of variance explained by the factors. Check the t-statistics to see which estimates are reliable. A significant alpha with low R² points to stock-specific return and risk.

Q: Why can't I just add systematic and idiosyncratic volatility?

Volatility is a square root, so it does not add across independent sources. Variance does add when the residual is uncorrelated with the factors. Square each volatility, add them, then take the square root.

Updated 11 October 2026 · Fact-checked

Factor risk decomposition splits portfolio variance into systematic variance from factor exposures (betas) and idiosyncratic variance from the regression residual. Performance attribution multiplies each beta by its factor return; what is left over is alpha. Use excess returns, add variances rather than volatilities, and test alpha with a t-statistic.

Understand Factor Risk Decomposition and Performance Attribution

A factor model says a portfolio's excess return comes from a few common drivers, such as the market, value, momentum or interest rates, plus a part that is unique to the portfolio. You estimate it by regressing the portfolio's excess returns on the factor returns. The slope coefficients are the factor exposures (betas), the intercept is alpha, and the residual is the idiosyncratic return.

Risk splits the same way. Systematic risk is the variance that the factors explain. Idiosyncratic risk (also called specific or residual risk) is the variance they do not explain. If the residual is uncorrelated with the factors, total variance is exactly the sum of the two. Variances add. Volatilities do not.

Performance attribution does the same job for return. Each factor's contribution is your beta times that factor's return over the period. Add the contributions to get the factor-explained return. Subtract that from the portfolio's excess return and you have alpha, the part not explained by the factor exposures.

The key interpretation point is that a high return is not skill by itself. A manager who earned 11% may simply have held a high-beta portfolio in a rising market. Alpha only counts after factor exposures are removed, and it only counts as evidence of skill if it is statistically distinguishable from zero. A high R² means most variance is systematic. A low R² means idiosyncratic risk dominates, which is typical of concentrated stock-picking portfolios.

Key formulas to remember

Factor model: Rp − Rf = α + β1 × F1 + β2 × F2 + … + βk × Fk + ε; Rp − Rf is the portfolio excess return. Fk are factor returns. ε is the residual, with mean zero and uncorrelated with the factors.
Single-factor variance decomposition: σp² = β² × σF² + σε²; Systematic variance = β² × σF². Idiosyncratic variance = σε². Use variances, never add volatilities.
Multi-factor systematic variance: σsys² = Σ βi² σi² + 2 Σ βi βj ρij σi σj (i < j); The covariance term drops out only when the factors are uncorrelated. Total variance = σsys² + σε².
R-squared: R² = σsys² ÷ σp² = 1 − σε² ÷ σp²; Share of total variance explained by the factors. The idiosyncratic share is 1 − R².
Return attribution: Factor contribution = βk × Fk; α = (Rp − Rf) − Σ βk × Fk; Alpha is the unexplained part of excess return. Keep the sign of each beta, since short exposures subtract.
t-statistic for alpha: t = α̂ ÷ SE(α̂); An absolute value above roughly 2 is usually read as significant at about the 5% level. A large alpha with a large standard error may be noise.
Information ratio: IR = α ÷ σε; Alpha per unit of residual (idiosyncratic) risk, with both on the same annual basis.
Annualising monthly figures: Return × 12; volatility × √12; variance × 12; Apply to monthly data, assuming returns are independent over time.

How to solve Factor Risk Decomposition and Performance Attribution questions

Use this order for any question on factor risk or attribution. It keeps the units and the logic straight.

1Check that returns are excess returns (over the risk-free rate) and that all numbers use the same period, such as annual.
2Write down the betas, factor returns, factor volatilities, factor correlations and residual volatility from the question.
3For risk: compute systematic variance (β² × σ² for each factor, plus covariance terms if factors are correlated).
4Add the idiosyncratic variance (residual volatility squared) to get total variance. Take the square root only at the end.
5Compute R² as systematic variance ÷ total variance, and the idiosyncratic share as 1 − R².
6For return: multiply each beta by its factor return, keeping signs, and sum to get the factor-explained return.
7Subtract the factor-explained return from the portfolio's excess return to get alpha. Divide by its standard error for the t-statistic.
8Interpret: is alpha significant, which factor drives risk, and is the portfolio mostly factor exposure or stock-specific?

Quickest way: Variance table for uncorrelated factors

When to use it: Use when the question says factors are uncorrelated or gives a single factor, and the options are numerically distinct.

Compute each factor's variance contribution as (β × σ)², and square the residual volatility.
Add the contributions. This is the total variance, and the share of each one is its own part ÷ total.
Estimate the volatility with a quick square root. Check it falls between the largest single component and the sum of components.
For attribution, compute β × factor return per line, sum, then subtract from the excess return to get alpha.

Common mistakes in Factor Risk Decomposition and Performance Attribution

Adding volatilities instead of variances
Systematic and idiosyncratic risk are both quoted as percentages, so adding them feels natural.
Fix: Square each volatility, add the variances, then take the square root. For example, 18% and 6% give √(324 + 36) ≈ 18.97%, not 24%.
Using total returns instead of excess returns
The question gives a portfolio return, and the risk-free rate is easy to forget.
Fix: Subtract the risk-free rate from the portfolio return before comparing it with factor-explained return. Factor returns for long-short factors are already excess-style returns.
Treating alpha as skill without checking significance
A positive intercept looks like outperformance.
Fix: Compute t = α̂ ÷ SE(α̂). If the t-statistic is small, the alpha is within estimation noise. Also ask whether an omitted factor could explain it.
Ignoring the sign of a negative beta
Candidates multiply magnitudes and forget the exposure is short.
Fix: Keep the sign. A beta of −0.2 on a factor that returned 5% contributes −1.0%, which lowers factor-explained return and raises alpha.
Assuming factor variances simply add when factors are correlated
The single-factor formula is memorised and applied to a multi-factor case.
Fix: Include 2 × βi × βj × ρij × σi × σj. Only drop it when the question states the factors are uncorrelated.
Confusing R² with the share of return explained
Both are described as 'explained by the factors'.
Fix: R² measures the share of variance explained. Return attribution is a separate sum of beta × factor return, and the two are not the same number.

Worked examples

Example 1

A portfolio's monthly excess returns are regressed on the market excess return. Annualised results: market beta 1.2, market volatility 15%, residual volatility 6%. Assume the residual is uncorrelated with the market. What share of total portfolio variance is systematic? (A) 60% (B) 75% (C) 90% (D) 95%

Show the solution

Systematic volatility = β × σM = 1.2 × 15% = 18%. Systematic variance = 18² = 324 (in %²).
Idiosyncratic variance = 6² = 36.
Total variance = 324 + 36 = 360. Total volatility = √360 ≈ 18.97%.
Systematic share = 324 ÷ 360 = 0.90, so R² = 90%. The idiosyncratic share is 10%.

Answer: (C) 90%. Total volatility is about 18.97%.

Example 2

Over a year a portfolio earned an excess return of 11.0%. Its regression exposures and the factor returns were: market beta 1.1 (market excess return 8%), value beta 0.4 (value factor return 3%), momentum beta −0.2 (momentum factor return 5%). The standard error of the estimated alpha is 0.8%. Which statement is correct? (A) Alpha 2.0%, t = 2.5, significant at 5% (B) Alpha 2.0%, t = 1.25, not significant (C) Alpha 9.0%, t = 11.25, significant (D) Alpha 3.0%, t = 3.75, significant

Show the solution

Market contribution = 1.1 × 8% = 8.8%.
Value contribution = 0.4 × 3% = 1.2%.
Momentum contribution = −0.2 × 5% = −1.0%.
Factor-explained return = 8.8% + 1.2% − 1.0% = 9.0%.
Alpha = 11.0% − 9.0% = 2.0%.
t-statistic = 2.0 ÷ 0.8 = 2.5, which is above about 1.96, so alpha is significant at the 5% level.

Answer: (A) Alpha is 2.0% with t = 2.5, significant at 5%. Most of the return, 9.0 of the 11.0 percentage points, came from factor exposures.

Exam tips

Read what the question asks for: variance share, volatility, factor contribution or alpha. Many wrong options are correct numbers for a different quantity, such as variance when volatility was asked.
Check the units before any arithmetic. If the data is monthly and the answer is annual, convert returns by ×12 and volatility by ×√12.
When a regression table is given, read the intercept as alpha, slopes as betas and R² as the systematic variance share. Then check the t-statistic before you call alpha skill.
Expect interpretation items: high R² means factor exposure drives risk, low R² means stock-specific risk, and alpha that disappears after adding a factor was factor beta in disguise.
If a result looks too good, such as alpha larger than the total return or R² above 100%, you have probably dropped a sign or mixed total and excess returns.

Practice questions from Factor Theory