Factor Models, APT and Factor Investing

The CAPM rests on heavy economic assumptions. This lesson reaches the same kind of risk-return relationship by two looser routes: a factor model that lets many common forces drive returns, and arbitrage pricing theory (APT), derived from the single principle that there is no free lunch. We then meet the Fama-French factors, the smart-beta investing they spawned, and how factor models attribute performance — including unpicking Buffett's alpha.

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Factor models Arbitrage & APT The APT equation CAPM vs APT Fama-French 3-factor Factor investing Style & attribution Buffett's alpha

A factor model generalises the single-index model: instead of one market factor, returns are driven by \(K\) common factors — which can be the market, macro variables, or empirical regularities.

Factor models

Each factor loading \(b_{ik}\) is the asset's sensitivity to factor \(k\) (it behaves like beta); \(\varepsilon_i\) is zero-mean firm-specific noise, uncorrelated with the factors and other assets' noise. The second form reads return as mean + Σ(loading × factor surprise) + noise. A portfolio's loading on each factor is, like portfolio beta, the weighted average of constituent loadings; and in a well-diversified portfolio the firm-specific noise washes out, leaving only the common factors as sources of randomness.

Arbitrage and APT

Arbitrage is a strategy generating a riskless profit with zero net investment (e.g. IBM at \$160 on NYSE and \$162 on NASDAQ — buy one, sell the other). In an efficient market, persistent arbitrage should not exist. APT uses far weaker assumptions than CAPM: (1) enough assets to diversify, (2) no arbitrage, (3) the factor model is accurate.

The derivation: build a well-diversified portfolio (noise ≈ 0) whose net factor loading is zero. Its return is then certain, so by no-arbitrage it must equal \(r_f\). Working this through for any pair of assets forces a common ratio of excess return to loading — the factor's risk premium \(\lambda\).

The APT pricing equation

The factor risk premium \(\lambda_k\) is the fair excess return per unit of loading on factor \(k\). When a factor is the return of a tradable portfolio, \(\lambda_k=E(F_k)-r_f\); with a single market factor, this collapses to the CAPM security market line — hence the name multifactor SML. (If a factor is not a tradable asset, its \(\lambda\) may differ.)

CAPM vs APT

The Fama-French three-factor model

Empirically the market factor alone leaves systematic patterns unexplained. Fama-French add two:

SMB ("small minus big") is the return of small-cap over large-cap stocks; HML ("high minus low") is value (high book-to-market) over growth. They are built by sorting stocks into six size × book-to-market groups: \(\text{SMB}=\tfrac{1}{3}(r_{S/H}+r_{S/M}+r_{S/L})-\tfrac{1}{3}(r_{B/H}+r_{B/M}+r_{B/L})\) and \(\text{HML}=\tfrac{1}{2}(r_{S/H}+r_{B/H})-\tfrac{1}{2}(r_{S/L}+r_{B/L})\). Because SMB and HML are zero-investment long-short portfolios, no \(r_f\) is subtracted from their expected returns. Estimate by multivariate regression: the intercept \(\alpha\) should be zero if the model holds, and the loadings reveal style (high positive SMB → small-cap tilt; high HML → value tilt). Other popular factors: momentum (UMD), quality (QMJ), low-beta (BAB) — data free on Kenneth French's site and AQR.

Factor investing ("smart beta")

Traditional allocation targets asset classes or sectors; factor investing instead targets the factors with strong academic evidence of historical risk-adjusted excess returns — value, size, momentum, quality, low-volatility, liquidity. "Smart-beta" strategies harvest these premia through systematic, rules-based processes — a hybrid of active and passive, at lower cost than active funds, now widely packaged as ETFs. But factor returns are cyclical over the short run, so factor investing is genuinely risky — the premia can disappear for years.

Style analysis and performance attribution

Factor loadings summarise a fund's style (its exposures), useful for mimicking, risk management and attribution. Extending Jensen's alpha to a multifactor benchmark:

Buffett's alpha

Frazzini et al. (2018) regressed Berkshire Hathaway's public-stock returns on six factors (market, SMB, HML, UMD, BAB, QMJ). The result: significant loadings on HML (value), BAB (bet-against-beta / low-beta) and QMJ (quality) — Buffett buys cheap, safe, high-quality stocks. Controlling for these factors drove the portfolio's alpha from 5.8% down to a statistically insignificant 0.3% — the factors almost entirely explain his returns. In principle one could mimic the style by factor-tilting a diversified portfolio to match Berkshire's loadings — though, as ever, treat such results critically.