FRM Exam Part II · Portfolio Risk: Analytical Methods

Factor Models and Systematic Risk Decomposition Explained

Updated 11 October 2026 · Fact-checked

A factor model explains asset returns using exposures (betas) to common risk factors plus a unique residual. Portfolio variance splits into systematic risk, driven by factors, and idiosyncratic risk, which diversifies away. To solve questions, compute portfolio betas as weighted averages, then add factor variance and residual variance.

Understand Factor Models and Systematic Risk Decomposition

A factor model says that an asset's return comes from a few common drivers plus something unique to that asset. In the single-factor form, R_i = α_i + β_i × F + ε_i. F is the factor return, β_i is the asset's sensitivity to it, and ε_i is the idiosyncratic (specific) return with mean zero.

The model lets you split risk into two parts. Systematic risk comes from the factors. You cannot remove it by holding more assets, because all assets respond to the same factors. Idiosyncratic risk comes from ε. It is unrelated across assets, so it shrinks as you add names.

In a multi-factor model, R_i = α_i + β_i1 F1 + β_i2 F2 + ... + ε_i. Factors can be macroeconomic (inflation, GDP growth, rates) or style-based (value, size, momentum). Each beta is the change in the asset's return for a one-unit change in that factor, holding the other factors fixed.

Portfolio betas are simple. The portfolio's beta to each factor is the weighted average of the asset betas. Portfolio systematic variance depends on those betas and on the factor covariance matrix. Idiosyncratic variance is the weighted sum of squared weights times residual variances, assuming residuals are uncorrelated.

Diversification works through the residual term. With n equally weighted assets, each with residual variance σ_ε², the portfolio residual variance is σ_ε² ÷ n. It falls toward zero, but the systematic part stays. The same logic applies to VaR: lower correlation between positions lowers portfolio VaR, and VaR is below the sum of stand-alone VaRs unless correlation is 1.

Key formulas to remember

Single-factor model: R_i = α_i + β_i × F + ε_i; ε_i has mean zero and is uncorrelated with F and with other assets' residuals.
Single-factor variance of an asset: σ_i² = β_i² × σ_F² + σ_εi²; First term is systematic, second is idiosyncratic.
Portfolio beta: β_p = Σ w_i × β_i; Applies to each factor separately. Weights include shorts as negative values.
Portfolio variance, single factor: σ_p² = β_p² × σ_F² + Σ w_i² × σ_εi²; Assumes uncorrelated residuals.
Multi-factor portfolio variance: σ_p² = β_p' Σ_F β_p + Σ w_i² × σ_εi²; Σ_F is the factor covariance matrix. With two factors the systematic part is β1²σ1² + β2²σ2² + 2β1β2ρσ1σ2.
Systematic share of risk (R²): R² = β² σ_F² ÷ σ²; Idiosyncratic share is 1 − R².
Equal-weight idiosyncratic variance: σ_ε²(p) = σ_ε² ÷ n; Holds for n equally weighted assets with equal, uncorrelated residual variance.
Two-asset portfolio VaR: VaR_p = √(VaR_1² + VaR_2² + 2ρ × VaR_1 × VaR_2); Normal returns, same confidence level, zero mean. Equals VaR_1 + VaR_2 only when ρ = 1.

How to solve Factor Models and Systematic Risk Decomposition questions

Use this order for any factor-model or decomposition question.

1Identify the model: single or multi-factor, and what each beta and factor volatility refers to.
2List portfolio weights, including negatives for shorts, and check they are in the units the question wants (weights or amounts).
3Compute portfolio betas for each factor as the weighted average of asset betas.
4Compute systematic variance: β_p² σ_F² for one factor, or the full β' Σ_F β with covariance terms for several.
5Compute idiosyncratic variance as Σ w_i² σ_εi², assuming residuals are uncorrelated unless told otherwise.
6Add the two for total variance, then take the square root for volatility.
7If asked for VaR, multiply volatility by the z-score (1.645 at 95%, 2.326 at 99%) and by portfolio value, over the stated horizon.
8Interpret: state the systematic share and what diversification can and cannot remove.

Quickest way: Variance bookkeeping in three lines

When to use it: Use it when the question gives betas, factor volatility and residual volatilities and asks for total risk or the systematic share.

Write β_p, then systematic variance β_p² σ_F².
Write idiosyncratic variance Σ w² σ_ε² and add to get total.
Compute systematic ÷ total for the share, and check that the answer options differ enough that rounding will not matter.

Common mistakes in Factor Models and Systematic Risk Decomposition

Adding volatilities instead of variances when combining systematic and idiosyncratic risk.
Volatility is the number most often quoted, so it feels natural to add.
Fix: Convert to variance, add, then take the square root.
Forgetting to square the weights in the idiosyncratic term.
Beta uses weights linearly, so students reuse the same pattern.
Fix: Betas are linear in w. Residual variance uses w² because residuals are uncorrelated.
Ignoring factor correlation in a multi-factor model.
Students treat factors as independent to save time.
Fix: Include 2β1β2ρσ1σ2 unless the question says factors are uncorrelated.
Claiming diversification removes all risk.
Overgeneralising from the idiosyncratic result.
Fix: Only idiosyncratic risk diversifies. Systematic risk remains, so variance floors at β_p² σ_F².
Assuming portfolio VaR equals the sum of individual VaRs.
It is true only at perfect correlation.
Fix: Use the correlation formula. Sum of VaRs is an upper bound for normal returns with ρ ≤ 1.
Treating a short position's beta with the wrong sign.
Weights are entered as positive amounts.
Fix: Use a negative weight for shorts so it reduces β_p.

Worked examples

Example 1

A portfolio holds 60% in asset A (β = 1.2, residual volatility 10%) and 40% in asset B (β = 0.8, residual volatility 15%). The market factor has volatility 18%. Residuals are uncorrelated. Find the portfolio volatility.

Show the solution

β_p = 0.6 × 1.2 + 0.4 × 0.8 = 0.72 + 0.32 = 1.04.
Systematic variance = 1.04² × 0.18² = 1.0816 × 0.0324 = 0.035044.
Idiosyncratic variance = 0.6² × 0.10² + 0.4² × 0.15² = 0.36 × 0.01 + 0.16 × 0.0225 = 0.0036 + 0.0036 = 0.0072.
Total variance = 0.035044 + 0.0072 = 0.042244.
Volatility = √0.042244 ≈ 0.2055.

Answer: Portfolio volatility is about 20.5%. Systematic risk is roughly 83% of total variance.

Example 2

Two positions have stand-alone 1-day 99% VaRs of $4 million and $3 million. Returns are normal and the correlation is 0.5. What is the portfolio VaR, and how much does diversification save?

Show the solution

VaR_p² = 4² + 3² + 2 × 0.5 × 4 × 3 = 16 + 9 + 12 = 37.
VaR_p = √37 ≈ 6.08 million.
Undiversified sum = 4 + 3 = 7 million.
Diversification benefit = 7 − 6.08 ≈ 0.92 million.

Answer: Portfolio VaR is about $6.08 million, a diversification benefit of about $0.92 million versus the $7 million sum.

Exam tips

Check whether the question gives volatilities or variances before computing. Mixing them is the commonest slip.
If a statement says diversification eliminates systematic risk, it is wrong. Look for this distractor.
For multi-factor betas, compute each factor's portfolio beta separately before combining.
Interpret results in words: systematic share, effect of lowering correlation, or what a higher beta does to VaR.
Eliminate options with the wrong order of magnitude first, then compute only what is needed.

Practice questions from Portfolio Risk: Analytical Methods

Factor Models and Systematic 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.

CFA Level IIFactor Model Return Attribution and Risk Decomposition

Factor Models and Systematic Risk Decomposition: frequently asked questions

What is the difference between systematic and idiosyncratic risk?

Systematic risk comes from common factors that affect many assets, so diversification cannot remove it. Idiosyncratic risk is specific to one asset and falls as you add uncorrelated holdings.

How do I calculate portfolio beta in a multi-factor model?

Take the weighted average of asset betas separately for each factor. Shorts carry negative weights. You then use these portfolio betas with the factor covariance matrix to get systematic variance.

How does correlation affect portfolio VaR?

Lower correlation gives a lower portfolio VaR for the same stand-alone VaRs. At a correlation of 1, portfolio VaR equals the sum of the individual VaRs for normal returns.

Why does idiosyncratic risk disappear in large portfolios?

With uncorrelated residuals, each weight shrinks as the number of assets grows, so the sum of squared weights times residual variance goes toward zero. Systematic variance stays.