FINC3017 · Investments And Portfolio Management
Anomalies, APT & Multi-Factor Models
Anomalies, APT & Multi-Factor Models (Week 7) is the empirical answer to the CAPM's failures. You catalogue the documented return anomalies — size, value, momentum, post-earnings drift, low-beta, profitability, investment — and then build models that price them. Arbitrage Pricing Theory allows several unspecified factors; the Fama-French three-factor model adds size (SMB) and value (HML) to the market, Carhart adds momentum, and the five-factor model adds profitability (RMW) and investment (CMA). The recurring caution is the 'factor zoo': with enough data-mining, spurious factors appear, so a credible factor needs an economic story and out-of-sample evidence.
What this chapter covers
- 01The main anomalies: size, value (high book-to-market), momentum, PEAD, low-beta, profitability, investment
- 02Arbitrage Pricing Theory: E[R_i] = λ₀ + β_i¹λ₁ + … + β_iⁿλₙ (factors unspecified)
- 03Fama-French 3-factor: market + SMB (size) + HML (value)
- 04Carhart 4-factor: adds momentum (MOM)
- 05Fama-French 5-factor: adds RMW (profitability) and CMA (investment)
- 06Reading factor loadings: a non-zero alpha means the model is incomplete
- 07Risk-based vs behavioural explanations of each anomaly
- 08Factor-zoo cautions: data-mining, overfitting and publication bias
Expected return from a Fama-French three-factor model
- 1 markWrite the FF3 model in expected-return form: E[R] = rf + β_M·MRP + β_S·SMB + β_H·HML.
- 3 marksCompute each factor contribution: market 1.1 × 6% = 6.6%, size 0.6 × 2% = 1.2%, value 0.4 × 3% = 1.2%.
- 1 markAdd to the risk-free rate: E[R] = 3% + 6.6% + 1.2% + 1.2% = 12.0%.
- 1 markInterpret the loadings: positive SMB and HML loadings mean the stock behaves like a smaller, value (high book-to-market) firm, so it earns size and value premiums on top of its market exposure.
Key terms
- Arbitrage Pricing Theory (APT)
- A multi-factor pricing model, E[R_i] = λ₀ + Σβ_i^kλ_k, that says expected return is a linear function of several systematic factor exposures. Unlike the CAPM it does not specify what the factors are or require a market portfolio, resting instead on no-arbitrage.
- SMB and HML factors
- The Fama-French size and value factors. SMB (small minus big) is the return of small-cap over large-cap stocks; HML (high minus low) is the return of high book-to-market (value) over low (growth) stocks. Loadings on them capture exposure to the size and value premiums.
- Momentum (the Carhart factor)
- The tendency of recent winners (past 3-12 month returns) to keep outperforming recent losers, documented by Jegadeesh-Titman. Carhart adds it as a fourth factor (MOM) to the FF3 model; it is the hardest anomaly to explain on a purely risk basis.
- Fama-French 5-factor model
- The FF3 model plus a profitability factor (RMW, robust minus weak operating profitability) and an investment factor (CMA, conservative minus aggressive asset growth). It absorbs more cross-sectional return variation, though momentum is still notably absent.
- Factor zoo
- The proliferation of hundreds of published 'factors', many of which are statistical flukes from data-mining and publication bias. The warning is to demand a clear economic rationale and out-of-sample survival before trusting any newly discovered factor.
Anomalies, APT & Multi-Factor Models FAQ
What is the difference between the APT and a specific model like Fama-French?
The APT is a general framework that says expected returns are linear in some set of systematic factors, but it stays silent on what those factors are. Fama-French (and Carhart, FF5) are concrete implementations that name the factors — market, size, value, momentum, profitability, investment — and supply the data to estimate the premiums, so they are testable special cases of the APT idea.
Are anomalies explained by risk or by behaviour?
Both stories compete and the exam expects you to know each. The risk-based view says an anomaly's premium is compensation for a hidden systematic risk (so it persists), while the behavioural view attributes it to investor mistakes like under-reaction (momentum) or over-extrapolation (value). Momentum is especially hard to rationalise as risk, which keeps the debate alive.
What does a non-zero alpha in a factor model tell you?
It signals the model has not fully captured the cross-section of returns — there is return the included factors cannot explain. Historically this drove the progression from CAPM to FF3 to Carhart to FF5: each added factor was introduced to shrink the alphas left by the previous model. A persistent alpha therefore points to either genuine skill or a still-missing factor.
Exam move
Be able to compute a multi-factor expected return as a sum of loading-times-premium terms, and to read what a set of loadings implies about a stock's style. On the conceptual side, memorise which factors each model adds (FF3 → Carhart → FF5) and rehearse name-the-anomaly with a one-line risk-versus-behavioural explanation, since these are high-frequency MCQ targets.
