The Arbitrage Pricing Theory and Multifactor Models of Risk and Return: formula sheet

Key formulas

Multifactor return model

R = E(R) + β1·F1 + β2·F2 + … + βk·Fk + e

F are surprises in each factor (zero mean). e is firm-specific return with zero mean, uncorrelated with the factors.

Expected return (excess return form)

E(R) = Rf + β1·λ1 + β2·λ2 + … + βk·λk

λ is the risk premium on factor portfolio j, meaning expected return on a portfolio with beta 1 on factor j and 0 on the others, minus Rf.

Single-factor model

R = E(R) + β·F + e

The market model is a special case when F is the market surprise.

Portfolio beta

βp,j = Σ wi · βi,j

Portfolio factor beta is the weighted average of asset betas. Weights sum to 1.

Systematic variance (uncorrelated factors)

σ² = β1²σ1² + β2²σ2² + … + σe²

Add covariance terms 2·β1·β2·Cov(F1,F2) if factors are correlated.

Multifactor return model

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

F are factor surprises with mean zero. ε is firm-specific risk, uncorrelated with the factors.

APT expected return (K factors)

E(Ri) = Rf + βi1 λ1 + βi2 λ2 + ... + βiK λK

λk is the risk premium on factor k, often the expected return of a factor portfolio with beta 1 on it minus Rf.

Single-factor APT

E(Rp) = Rf + βp λ

Applies to well-diversified portfolios. All must lie on one line.

Portfolio beta

βp = Σ wi βi

Weights sum to 1. Use the same for each factor separately.

Arbitrage profit for a mispriced portfolio

Profit = E(R) of the long portfolio − E(R) of the short replicating portfolio (identical betas), per unit invested

Go long the portfolio with the higher return and short a mix of assets with the same betas on every factor. Multiply the return difference by the amount invested. The profit is approximately riskless only when both sides are well diversified.

Alpha

α = actual expected return − APT required return

Non-zero alpha for a well-diversified portfolio signals arbitrage.

Single-factor return model

r = E(r) + β × F + e

F is the surprise in the factor (mean zero). e is firm-specific noise with mean zero.

Firm-specific risk of an equally weighted portfolio

σ²(e_P) = σ²(e) ÷ n, so σ(e_P) = σ(e) ÷ √n

Assumes firm-specific shocks are uncorrelated and all stocks have the same σ(e). Falls toward zero as n rises.

Total variance of a portfolio (one factor)

σ²_P = β_P² × σ²_F + σ²(e_P)

For a well-diversified portfolio, σ²(e_P) ≈ 0, so σ_P ≈ |β_P| × σ_F.

Portfolio beta

β_P = Σ wᵢ βᵢ

Weights sum to 1. Beta is a weighted average.

One-factor APT expected return

E(r_P) = rf + β_P × [E(r_F) − rf]

Holds for well-diversified portfolios. Applies to individual assets only approximately.

Multifactor APT expected return

E(r_P) = rf + β₁ × RP₁ + β₂ × RP₂ + … + β_k × RP_k

Each RP is the premium of a factor portfolio with beta 1 on that factor and 0 on the others.

Alpha (mispricing)

α = E(r) − [rf + Σ βⱼ RPⱼ]

Non-zero alpha on a well-diversified portfolio signals an arbitrage opportunity.

CAPM

E(Ri) = Rf + βi × [E(Rm) − Rf]

One factor. βi = Cov(Ri, Rm) ÷ Var(Rm). Market risk premium is E(Rm) − Rf.

APT return-generating process

Ri = E(Ri) + βi1F1 + βi2F2 + ... + βiKFK + ei

Factors F have zero expected value. ei is firm-specific risk, uncorrelated with the factors.

APT pricing equation

E(Ri) = Rf + βi1λ1 + βi2λ2 + ... + βiKλK

λk is the risk premium per unit of exposure to factor k, a factor portfolio's expected return minus Rf.

CAPM as special case of APT

K = 1, F1 = Rm − E(Rm), λ1 = E(Rm) − Rf

The APT equation then becomes the CAPM.

Alpha under a factor model

αi = E(Ri) − [Rf + Σ βik λk]

Non-zero alpha means mispricing relative to the model.

Fama-French three-factor model (expected return)

E(Ri) = Rf + βi,M × [E(RM) − Rf] + βi,S × E(SMB) + βi,H × E(HML)

Rf is the risk-free rate. Each beta multiplies its own factor premium. The market premium is RM minus Rf; SMB and HML are already spreads.

Regression form

Ri − Rf = αi + βi,M (RM − Rf) + βi,S SMB + βi,H HML + εi

Use excess returns on the left. Alpha is the intercept. Under the model, expected alpha is zero.

SMB

SMB = Return(small-cap portfolio) − Return(large-cap portfolio)

Positive when small caps outperform large caps.

HML

HML = Return(high book-to-market) − Return(low book-to-market)

Positive when value outperforms growth.

Carhart four-factor model

E(Ri) = Rf + βi,M × [E(RM) − Rf] + βi,S × E(SMB) + βi,H × E(HML) + βi,MOM × E(MOM)

MOM is winners minus losers. Same method, one more term.

Multifactor return model

Ri = αi + βi1·F1 + βi2·F2 + … + βik·Fk + εi

F are factor surprises or factor returns, ε is the firm-specific error with mean zero. Check whether the question gives factors as returns or as surprises.

Expected return (factor model)

E(Ri) = Rf + βi1·λ1 + βi2·λ2 + … + βik·λk

λk is the risk premium for factor k. This is the fair return given the betas.

Alpha

α = E(R) − [Rf + Σ βk·λk]

Use the realised return instead of E(R) for a realised (ex-post) alpha. Alpha is in return units, for example %.

Portfolio factor beta

βp,k = Σ wi·βi,k

Weights can be negative for short positions. Weights are on the portfolio value, so they need not sum to 1 when hedging with futures-style positions.

Zero-exposure hedge condition

βp,k + Σ hj·βj,k = 0 for every factor k

hj is the position in hedge instrument j (negative means short). You need one equation per factor, so solve the simultaneous equations.

Targeted exposure

βp,k + Σ hj·βj,k = target βk

Use this to tilt a portfolio to a chosen beta rather than to zero.

Quick revision

• A multifactor model splits return into factor-driven parts and a firm-specific residual.
• A factor beta measures the sensitivity of an asset's return to a one-unit move in that factor.
• APT: E(R) = Rf + Σ βi × λi, where λi is the risk premium for factor i.
• APT rests on no-arbitrage: two assets with the same factor exposures must offer the same expected return.
• In a well-diversified portfolio, firm-specific risk is diversified away, so only factor risk is priced.
• The CAPM is a single-factor special case, with the market as the factor.
• The APT does not name its factors, while the CAPM specifies the market portfolio.
• Fama-French three factors: market, size (small minus big) and value (high minus low book-to-market).
• Momentum is a common extension, adding a winners-minus-losers factor.
• Alpha = actual return minus the return expected from factor exposures.
• To hedge a factor, take a position that offsets the portfolio's beta to it, sized by beta times portfolio value.
• Portfolio factor beta is the weighted average of the individual asset betas.

Common mistakes

• Leaving out the risk-free rate in expected return Fix: Always start from Rf unless the question asks for the excess return.
• Using the factor level instead of the surprise Fix: Subtract the expected value first. Only the unexpected part enters the return equation.
• Forgetting to add the risk-free rate when λ is a premium over Rf. Fix: Read the wording. If λ is a risk premium, add Rf. If it is the total expected factor-portfolio return, subtract Rf first.
• Treating APT as requiring the market portfolio or mean-variance investors. Fix: Remember APT needs only a factor structure, diversification and no arbitrage. A single-factor APT with the market as the factor resembles CAPM, but CAPM rests on mean-variance assumptions and the market portfolio.
• Adding firm-specific risk to the expected return calculation. Fix: Only factor betas are priced. Firm-specific risk can be diversified away, so it carries no premium.
• Using the factor portfolio's total return as the premium. Fix: Subtract rf first. The risk premium is the excess return over the risk-free rate.
• Saying APT tells you which factors to use. Fix: APT gives the structure but not the factors. You choose them (macro, statistical or fundamental).
• Believing APT needs the market portfolio. Fix: APT needs no market portfolio. The market can be one of the factors, but it is not required.
• Using the market return instead of the market premium. Fix: Always compute RM − Rf first. SMB, HML and MOM are already spreads, so they are not reduced by Rf.
• Subtracting Rf from SMB, HML or MOM. Fix: These are long-short returns. Use them as given.

Exam tips

• Check whether the question gives factor risk premiums or total factor returns. If total, subtract Rf first.
• When a portfolio is described, average the betas by weight before using the formula. It is faster than computing each asset's return.
• Questions often test the idea that only systematic factors are priced. Firm-specific risk earns no premium in a well-diversified portfolio.
• Watch for a single-factor versus multifactor comparison: the multifactor model captures more sources of systematic risk but needs more estimates.
• Expect numerical questions where you compute Rf + Σ βλ. Practise doing this in under a minute.
• Know the contrast with CAPM: APT is multi-factor, needs no market portfolio, and is derived from no-arbitrage.
• Questions often say the portfolio is well diversified. That phrase signals that only factor risk matters.
• In arbitrage questions, check that betas match on both sides before choosing the long and short legs.