3.2 APT and Multifactor Models

From One Factor to Many

The CAPM is powerful because it is simple: one systematic factor—the market—explains expected returns. Empirically, however, average returns also line up with characteristics such as firm size, value versus growth, momentum, and various macroeconomic surprises. Arbitrage Pricing Theory (APT) and related multifactor models give FRM candidates a rigorous way to price those additional dimensions of risk without requiring every CAPM assumption.

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APT versus CAPM

APT does not claim that every individual security’s alpha must be exactly zero if idiosyncratic risk remains. The sharp prediction is that well-diversified portfolios with the same factor betas must offer the same expected return; otherwise arbitrageurs can long the cheap mix and short the rich mix until prices adjust.

Factor Model Representation

A linear factor model for returns is:

Ri = E(Ri) + βi1 F1 + βi2 F2 + ··· + βik Fk + εi

where Fj are factor surprises (mean zero if factors are demeaned), βij are factor betas (factor loadings), and εi is idiosyncratic residual with E(εi)=0, uncorrelated with the factors and (ideally) with other assets’ residuals.

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Single-Factor APT Expected Return

If a single factor carries risk premium RP1 (the expected excess return earned for bearing one unit of that factor), then for a well-diversified portfolio or in exact APT form:

E(Ri) = Rf + βi1 RP1

When the single factor is the market excess return and RP1 = E(Rm)−Rf, this collapses to CAPM algebra—even though the economic story differs.

Worked Single-Factor Example

Rf = 4%. A macroeconomic “industrial production growth surprise” factor has premium RP1 = 5%. Portfolio A has βA1 = 1.4.

E(RA) = 4% + 1.4 × 5% = 11%

If A’s forecast mean return is only 9%, APT flags a negative alpha of about 2% for diversified holders of that exposure—suggesting A is rich relative to its factor risk.

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Multifactor Expected Returns

With k factors expressed as risk premia RPj:

E(Ri) = Rf + βi1 RP1 + βi2 RP2 + ··· + βik RPk

If factors are specified as excess returns on tradable portfolios (as in Fama–French), the same linear structure holds and Rf plus the sum of beta-weighted factor premia again delivers expected return.

Worked Two-Factor Example

Rf = 3%. Factor 1 (inflation surprise) premium = 2%. Factor 2 (credit-spread widening) premium = 4%. Stock Z loadings: βZ1 = 0.5, βZ2 = 1.25.

E(RZ) = 3% + 0.5(2%) + 1.25(4%) = 3% + 1% + 5% = 9%

Suppose a second stock Y has the same betas but an analyst forecast of 11%. In a pure APT world with diversified portfolios, that gap is an arbitrage signal: overweight Y / underweight Z (or construct a zero-beta long-short book) until expected returns align with betas.

Interpreting Factor Betas

Factor betas are estimated with time-series regressions of asset excess returns on factor realizations. FRM questions often hand you the betas and premia and ask only for the linear combination.

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Hedging Factor Exposures

Factor hedging means offsetting unwanted betas while retaining desired ones (or retaining alpha). If a portfolio has βp,1 = 0.80 to an oil-price factor and the manager wants oil beta near zero, she can short a package of oil-sensitive instruments (or oil futures) with combined oil beta 0.80. Net oil beta becomes approximately zero; other factor betas change only to the extent the hedge instruments load on them—so hedges should be as factor-pure as practicable.

Hedge Ratio Sketch

If one unit of hedge instrument H has oil beta βH,oil and the portfolio notional is Vp with oil beta βp,oil, the hedge notional VH satisfies:

βp,oil Vp + βH,oil VH ≈ 0 ⇒ VH ≈ −(βp,oil / βH,oil) × Vp

Worked Hedge Example

Portfolio value $100 million, inflation beta 0.60. Inflation-linked hedge note has inflation beta 1.20 per unit notional. Required short notional:

VH = −(0.60/1.20) × $100m = −$50m

After the hedge, inflation beta ≈ 0. If the inflation risk premium was 3%, removing 0.60 units of exposure also removes about 0.60 × 3% = 1.8% from APT-required return—useful when comparing residual expected performance.

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Fama–French Three-Factor Model

Eugene Fama and Kenneth French augment the market factor with two empirically motivated equity factors:

1. Market excess return: Rm − Rf
2. SMB (Small Minus Big): return of small-cap portfolios minus large-cap portfolios
3. HML (High Minus Low): return of high book-to-market (value) portfolios minus low book-to-market (growth) portfolios

The time-series regression is:

Ri − Rf = αi + βi,m(Rm − Rf) + βi,s SMB + βi,h HML + εi

Expected excess return in the corresponding pricing equation (setting αi = 0 if the model prices perfectly) is:

E(Ri) − Rf = βi,m E(Rm−Rf) + βi,s E(SMB) + βi,h E(HML)

Worked Fama–French Example

Rf = 2%, E(Rm−Rf) = 6%, E(SMB) = 2.5%, E(HML) = 3%. A small-value stock has βm = 1.10, βs = 0.80, βh = 0.70.

E(R) = 2% + 1.10(6%) + 0.80(2.5%) + 0.70(3%) E(R) = 2% + 6.6% + 2.0% + 2.1% = 12.7%

A large-growth stock with βm = 0.95, βs = −0.40, βh = −0.50 has:

E(R) = 2% + 0.95(6%) + (−0.40)(2.5%) + (−0.50)(3%) = 2% + 5.7% − 1.0% − 1.5% = 5.2%

The model therefore rationalizes why small-value names can require substantially higher expected returns than large-growth names even after market beta is considered.

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Practical Uses for Risk Managers

• Risk decomposition: attribute P&L and VaR-like sensitivities to market, size, value, rates, credit, FX, and commodity factors.
• Mandate design: constrain factor betas (for example, market-neutral, size-neutral) rather than only tracking-error versus a cap-weighted index.
• Stress testing: shock factor realizations and map through betas to portfolio losses—more interpretable than shocking every security price independently.
• Performance evaluation: multifactor alpha is residual return after all named factor premia; a “positive CAPM alpha” can disappear once SMB/HML are included.

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Exam Pitfalls

• Treating APT as requiring the true market portfolio to be observable.
• Adding factor premia without multiplying by betas.
• Confusing factor surprises in the return-generating process with factor risk premia in the expected-return equation.
• Claiming APT guarantees zero alpha for every concentrated single stock with large εi.

If you can move fluently between CAPM’s single beta and APT’s beta vector—and compute the linear required return—you are ready for the multifactor items in Foundations and for later valuation/risk-model applications.