APT and multi-factor models

Ross's Arbitrage Pricing Theory (1976) generalised CAPM in two important ways. First, it allowed multiple sources of priced risk. Second, it dispensed with the mean-variance optimisation assumption — APT's logic is built on the absence of arbitrage, not on investor preferences.

The APT model

R_i = α_i + β_i,1 F_1 + β_i,2 F_2 + ... + β_i,k F_k + ε_i

R_i is the return on asset i. F_j are k common factors (with zero mean). β_i,j is the exposure of asset i to factor j. ε_i is the idiosyncratic, mean-zero residual uncorrelated across assets. The factors can be macroeconomic (inflation, output, term spread) or statistical (principal components).

The pricing relation

If markets admit no arbitrage and idiosyncratic risk can be diversified away, then expected returns are a linear combination of factor risk premia:

E[R_i] = r_f + β_i,1 λ_1 + β_i,2 λ_2 + ... + β_i,k λ_k

λ_j is the risk premium for factor j — the expected reward per unit of factor exposure. CAPM is the special case k = 1 with F_1 = market excess return and λ_1 = market risk premium.

The arbitrage argument

Suppose two assets have identical factor exposures (β_i,j = β_j,j for all j) but different expected returns. Construct a portfolio long the higher-return asset and short the lower-return asset, weighted to eliminate idiosyncratic risk by diversification. The result is a riskless position with positive expected return — an arbitrage. Absence of arbitrage rules out the spread, hence the pricing equation must hold.

Choosing factors

• Macroeconomic factors (Chen-Roll-Ross 1986): industrial production, unanticipated inflation, term spread, default spread.
• Statistical factors: principal components of the return covariance.
• Style / characteristic factors (Fama-French): size, value, momentum, profitability, investment.
• Currency, commodity, and credit factors for fixed-income and global portfolios.

Estimation procedure (Fama-MacBeth 1973)

1. First-stage: time-series regressions of each asset's return on the factors. Obtain β̂_i,j for each (asset, factor) pair.
2. Second-stage: cross-sectional regressions of returns each period on the estimated betas. Obtain estimated factor premia λ̂_j,t.
3. Time-series average of λ̂_j,t gives the factor risk premium estimate; standard errors from the time-series variation (Newey-West correction is standard).

Mimicking portfolios

For each factor, construct a portfolio whose return mimics the factor's movements. Fama-French SMB (small minus big), HML (high B/M minus low B/M) are precisely such mimicking portfolios. Once factors are mimicking portfolios, factor risk premia are directly observable as the portfolios' expected returns.

Strengths and weaknesses of APT

• Flexible: any factor structure consistent with the data can be tested.
• No mean-variance optimisation assumption.
• Identifiability problem: which factors? How many? Hard to test exhaustively.
• Sensitivity: small changes in factor specification produce different estimated premia.

Modern multi-factor models

Fama-French 3-factor (1992), 4-factor with momentum (Carhart 1997), 5-factor (Fama-French 2015 adding profitability and investment), q-factor (Hou-Xue-Zhang 2015) — all are APT-style multi-factor models. They dominate academic and practitioner-level cross-sectional asset pricing.