CAPM and Factor Models: From One Beta to Many

Factor models answer one question: of the return this asset produced, how much was compensation for exposures anyone could have bought, and how much was something else? Everything from CAPM to a modern 200-factor risk model is that question at increasing resolution.

CAPM: the one-factor case

The Capital Asset Pricing Model says an asset's expected excess return is proportional to its beta to the market:

$$E[R_i] - r_f = \beta_i \left(E[R_m] - r_f\right), \qquad \beta_i = \frac{\operatorname{Cov}(R_i, R_m)}{\operatorname{Var}(R_m)}$$

Note that beta is literally an OLS regression slope — regress the asset's excess returns on the market's and $\beta$ is the coefficient. The economic argument: idiosyncratic risk diversifies away (the $1/n$ term in portfolio math), so the market only pays for the risk that doesn't — covariance with the aggregate. An asset that pays off in crashes (negative beta) is insurance, and rationally earns less than the risk-free rate.

Beta intuition worth having ready: $\beta = 1.5$ means the stock moves 1.5% for a 1% market move on average; total risk splits as $\sigma_i^2 = \beta_i^2\sigma_m^2 + \sigma_\epsilon^2$ (systematic + idiosyncratic); and beta of a portfolio is the weighted average of its constituents' betas.

Alpha is the regression intercept: return unexplained by market exposure,

$$\alpha_i = E[R_i] - r_f - \beta_i(E[R_m] - r_f).$$

Under the CAPM, every alpha is zero. The entire active-management industry is a bet that this is false.

The empirical failures that built factor investing

Tested on data, CAPM fails in systematic, repeatable ways — and each failure became a "factor":

• Size (SMB): small-cap stocks earned more than their betas justified.
• Value (HML): high book-to-market ("cheap") stocks likewise.
• Momentum (UMD): past 12-month winners kept winning over the next months.
• Low-beta anomaly: low-beta stocks earned more per unit of beta than high-beta — arguably because leverage-constrained investors overpay for high-beta stocks as embedded leverage.

Fama and French packaged the first two into the three-factor model:

$$R_i - r_f = \alpha_i + \beta_i^{mkt}\, MKT + \beta_i^{smb}\, SMB + \beta_i^{hml}\, HML + \epsilon_i$$

where SMB and HML are returns of long-short portfolios (long small/cheap, short big/expensive). The framework is the same regression with more right-hand-side variables; "alpha" now means return unexplained by all included factors — a moving target that shrinks as the factor zoo grows. A strategy pitched as alpha in 1990 (buy cheap small-caps) is beta today. That migration — yesterday's alpha becomes today's factor — is the central sociological fact of quant equity.

How practitioners actually use this

• Risk decomposition: a risk model (Barra-style) with dozens of factors — industries, countries, value, momentum, vol, quality — decomposes any portfolio's variance into factor bets plus idiosyncratic residual. A portfolio manager discovers they're unintentionally short momentum before momentum rips.
• Hedging: to isolate a stock-specific view, short out the factor exposures ($\beta$ dollars of market, industry baskets, etc.), leaving the residual — this is the "market-neutral" in market-neutral funds.
• Performance attribution: did the fund earn its return from skill or from static factor tilts an ETF replicates at 10bps? Regress the track record on factor returns; the intercept is the honest answer.

The interview version

"Your friend's fund returned 20% last year. Impressed?" The factor-model answer: can't say without exposures — if it ran beta 1.5 in a year the market did 15%, the CAPM-implied return was ~22% and the alpha is negative. Then the standard follow-ups: "how would you estimate beta?" (regress excess returns, mind the estimation window and that beta drifts), and "the regression gives $\alpha = 3\%$ with standard error 2% — is the manager skilled?" ($t \approx 1.5$: statistically indistinguishable from luck — and most track records are).