Risk models

~8 min read

Interview occasional

Why it matters

A risk model tells you what the book is exposed to and how large a move you should expect. It is not an alpha model. Interviews use it to catch a specific mistake: calling a portfolio “idiosyncratic” when most of its variance is still industry, market, or a style factor. You should be able to write the one-line decomposition and say what each piece means.

Core concepts

Factor risk plus specific risk. The standard linear model for asset returns is r ≈ Xf + e, where X holds exposures (industry dummies, beta, size, value, momentum), f holds factor returns, and e holds specific returns. Portfolio variance is then approximately:

h′Vh = h′X F X′h + h′D h

h is the vector of weights, F is the factor-covariance matrix, and D is the specific-variance matrix (often diagonal). The first term is common-factor risk. The second is specific risk. If you cannot point at which term dominates, you do not know why the book moves.
Exposures are not the same object as alpha scores. A risk exposure to value can be a characteristic loading used only to estimate variance. An alpha score is a forecast of expected return. Using the risk model as your alpha model, or vice versa, confuses a covariance story with a forecast. Teams often use a richer risk model than the three factors they cite in the alpha write-up. That is normal. Disclose both.
Sample covariance is a bad risk model on its own. With hundreds of names and a modest window, the sample covariance is noisy and often not a stable input to an optimizer. Structured factor models (the XFX′ + D form) and shrinkage (Ledoit–Wolf is the textbook version) exist because of that. An interview answer of “I used the sample covariance of 500 stocks over 60 days” should make you flinch.
“Market neutral” is a statement about one exposure. Net market beta near zero can sit next to a large industry exposure or a large size exposure. Look at the vector X′h, not a single beta. If one industry is half the factor variance, the book is an industry bet that happens to be dollar neutral.
Specific risk is where concentration hides. Equal industry weights can still put specific risk in one name. The diagonal (or block) specific term is how a max-name discussion shows up in the variance, not just in a weight cap. See Portfolio construction.
Risk models are estimated, so they are stale and wrong in crises. Factor volatilities jump, correlations move toward one, and specific risk is not independent across names when a real event hits a sector. A risk model is a baseline for sizing and for attributing P&L. It is not a guarantee that a 1-standard-deviation band contains the next month.
Attribution closes the loop. After the fact, split P&L into factor P&L (exposures times realized factor returns) and specific P&L. If the money came from the factor piece, you ran a factor portfolio. If it came from specific P&L and the specific piece is a few names, you got lucky or concentrated. Either way you learned something the Sharpe did not say.

Mental model

portfolio variance
    = common factor variance (exposures × factor cov × exposures)
    + specific variance (name-level, mostly)

P&L after the fact
    = factor P&L + specific P&L

The risk model is how you stop describing every up month as “alpha.”

Interview questions

Write the decomposition of portfolio variance in a factor risk model and say what you learn if the first term is 90% of the total. Answer: h′Vh ≈ h′X F X′h + h′D h. If the first term is most of the variance, the book’s risk is common-factor exposure, not a pile of independent residuals. You are being paid (or hurt) for factor bets. That may be intentional. It is not a specific-risk alpha story.
Why is the sample covariance of raw stock returns a poor thing to drop straight into an optimizer? Answer: Too many parameters for the length of the sample, so weights chase noise and the matrix can be ill-conditioned. A factor structure or shrinkage cuts the effective number of parameters. Mention that the means are usually even noisier than the covariance if the question turns into full mean-variance.
You hedged market beta to zero and the daily residual is still obviously one sector. What exposure did the hedge leave in X′h? Answer: Industry (or another non-market factor). Market beta is one row of exposures. Sector weights can be large with a zero market beta. Add industry factors to the risk model or constrain sector active weights, then recompute factor variance.
How can a risk model and an alpha model disagree without someone being “wrong”? Answer: They answer different questions. The alpha model forecasts expected residual or total return. The risk model forecasts covariance. A name can have a positive alpha score and a large specific variance; the portfolio step decides whether the return per unit of that variance is worth it. Conflict is a sizing decision, not a data error.
A calm-sample risk model says this book should move about 20 bps a day. It moves 150 bps on an event day. Is the model refuted? Answer: Not by one day. Correlations and volatilities change in stress, so a model estimated in a calm window will understate event days. The operational response is a stress scenario or a higher risk estimate in the tails, not a claim that factor models “don’t work” because one day broke the band.

Watch

Quantopian Lecture Series: Risk Factor Exposure — Quantopian. The lecture title on YouTube is misspelled (“Expsosure”); it is their factor-risk-exposure session.
Quantopian Lecture Series: Fundamental Factor Models — Quantopian. Same family of models, from the characteristic side. Pair with the exposure lecture.