Single and multifactor models for investment returns: formula sheet

Key formulas

Market model

R_i = α_i + β_i R_M + e_i

E(e_i) = 0, Cov(e_i, R_M) = 0, Cov(e_i, e_j) = 0 for i ≠ j.

Expected return

E(R_i) = α_i + β_i E(R_M)

Take expectations of the model; the residual has mean zero.

Beta

β_i = Cov(R_i, R_M) ÷ Var(R_M) = ρ_iM σ_i ÷ σ_M

This is also the slope of the regression of R_i on R_M.

Alpha

α_i = E(R_i) − β_i E(R_M)

The intercept of the regression line.

Variance of a security

σ_i² = β_i² σ_M² + σ_ei²

Systematic part plus specific part.

Covariance between two securities

Cov(R_i, R_j) = β_i β_j σ_M²

Holds for i ≠ j because residuals are uncorrelated.

Portfolio beta

β_P = Σ w_i β_i

Weights w_i sum to 1.

Portfolio variance

σ_P² = β_P² σ_M² + Σ w_i² σ_ei²

Specific risk falls as the portfolio is diversified.

Multifactor return model

R_i = a_i + b_i1 F_1 + b_i2 F_2 + ... + b_iK F_K + e_i

F_k are factors, b_ik sensitivities, e_i specific return with E(e_i) = 0, uncorrelated with the factors and with other securities' specific terms.

APT expected return

E(R_i) = r_f + b_i1 λ_1 + b_i2 λ_2 + ... + b_iK λ_K

λ_k is the risk premium per unit of sensitivity to factor k. Here r_f is the return on a risk-free asset (sensitivity zero to all factors).

Factor premium as excess return

λ_k = E(R_P,k) − r_f

R_P,k is a portfolio with sensitivity 1 to factor k and 0 to all others. This is the usual way to read premiums from data.

Portfolio sensitivity

b_Pk = Σ w_i b_ik

Weights w_i sum to 1. Sensitivities are weighted averages, so portfolio factor exposure is linear in the weights.

Variance of a return (uncorrelated factors)

Var(R_i) = Σ b_ik² Var(F_k) + Var(e_i)

Valid only if factors are mutually uncorrelated. With correlated factors, add 2 b_ij b_ik Cov(F_j, F_k) terms.

Covariance between two securities

Cov(R_i, R_j) = Σ_k Σ_l b_ik b_jl Cov(F_k, F_l)

Assumes specific returns are uncorrelated. With uncorrelated factors it reduces to Σ b_ik b_jk Var(F_k).

CAPM as a one-factor case

E(R_i) = r_f + β_i [E(R_M) − r_f]

Here the single factor is the market and λ_1 = E(R_M) − r_f.

General multifactor model

R_i = a_i + b_i1 F_1 + b_i2 F_2 + ... + b_ik F_k + e_i

F_j are common factors, b_ij are sensitivities, e_i is specific return with zero mean, uncorrelated with the factors and with other securities' e.

Expected return

E(R_i) = a_i + b_i1 E(F_1) + ... + b_ik E(F_k)

Follows because E(e_i) = 0. If factors are defined with zero mean, E(R_i) = a_i.

Variance of return

Var(R_i) = Σ_j Σ_l b_ij b_il Cov(F_j, F_l) + Var(e_i)

If factors are uncorrelated this reduces to Σ_j b_ij² Var(F_j) + Var(e_i).

Covariance between two securities

Cov(R_i, R_m) = Σ_j Σ_l b_ij b_ml Cov(F_j, F_l)

The specific returns add nothing because they are uncorrelated across securities. This is the parameter saving.

Fama-French three-factor model

R_i − R_f = α_i + β_i (R_M − R_f) + s_i SMB + h_i HML + e_i

SMB = return on small-cap minus large-cap portfolio. HML = return on high minus low book-to-market portfolio. R_f is the risk-free rate.

Portfolio sensitivity

b_pj = Σ_i w_i b_ij

Portfolio loadings are the weighted average of the security loadings, with weights w_i summing to 1.

Single index model

R_i = α_i + β_i R_M + ε_i

E(ε_i) = 0, Cov(ε_i, R_M) = 0, Cov(ε_i, ε_j) = 0 for i ≠ j.

Variance of a security (single factor)

σ_i² = β_i² σ_M² + σ_εi²

Systematic plus specific variance.

Covariance of two securities (single factor)

Cov(R_i, R_j) = β_i β_j σ_M² (i ≠ j)

Holds only if residuals are uncorrelated.

Portfolio beta

β_P = Σ w_i β_i

Weights w_i sum to 1. Same rule applies to each factor loading.

Portfolio variance (single factor)

σ_P² = β_P² σ_M² + Σ w_i² σ_εi²

The specific term shrinks as the portfolio becomes diversified.

Multifactor expected return (APT form)

E(R_i) = r_f + b_i1 λ_1 + b_i2 λ_2 + ... + b_ik λ_k

λ_j is the risk premium for factor j.

Attribution of return

R_P − R_B = Σ (b_Pj − b_Bj) F_j + (α_P − α_B)

Active return = factor-tilt effects plus selection effect (benchmark B).

Quick revision

• Single factor model: Rᵢ = aᵢ + bᵢ·I + cᵢ, with cᵢ having zero mean.
• The market model uses a market index return as the factor.
• bᵢ measures sensitivity to the factor; it is the slope in a regression on the factor.
• Total variance of a security = bᵢ²·Var(I) + Var(cᵢ), as the error is uncorrelated with the factor.
• Covariance between two securities i and j = bᵢ·bⱼ·Var(I) when errors are uncorrelated.
• Systematic risk comes from the factors and cannot be diversified away; specific risk can be.
• A portfolio's sensitivity is the weighted average of its securities' sensitivities.
• Multifactor model adds further factors, each with its own sensitivity.
• APT is based on no-arbitrage, not on investor preferences.
• Three types: macroeconomic, fundamental and statistical factor models.
• Limitations: factors may be wrongly chosen, sensitivities may change over time, and past data may not predict the future.
• Always state your assumptions before computing.

Common mistakes

• Saying deterministic models are wrong or useless. Fix: State that deterministic models suit simple projections, sensitivity tests and communication. Choose by purpose.
• Confusing stochastic with statistical. Fix: Stochastic vs deterministic is about randomness in the model. Statistical vs economic is about how the model is built.
• Dividing covariance by the standard deviation of the market instead of its variance. Fix: Beta = Cov(R_i, R_M) ÷ σ_M². Square the standard deviation first.
• Forgetting to square beta in the systematic variance. Fix: Variance scales with the square of the multiplier: systematic variance = β²σ_M².
• Using total factor returns instead of risk premiums in the APT formula. Fix: Check that each λ_k is an excess return. If given a factor portfolio's total return, subtract r_f first.
• Saying APT needs the market portfolio or mean-variance investors. Fix: State that APT rests on a factor structure, well-diversified portfolios and no arbitrage. It does not need an equilibrium or a market portfolio.
• Calling Fama-French a macroeconomic model because it has a market factor. Fix: Classify by the extra factors. SMB and HML come from company characteristics (size, book-to-market), so it is fundamental-type.
• Saying statistical factors always have a clear economic meaning. Fix: State that statistical factors are extracted from data and may be hard to interpret. This is their main limitation.
• Adding specific variances without squaring weights. Fix: Specific variance of a portfolio is Σ w_i² σ_εi², not Σ w_i σ_εi².
• Using Cov(R_i, R_j) = β_iβ_jσ_M² when residuals are correlated. Fix: State that residuals are uncorrelated. If not, add the residual covariance term.

Exam tips

• Define each term in one line before comparing. Many marks go for clear definitions.
• Always tie your answer to the stated purpose. Generic lists score less.
• For comparison questions, give both advantages and disadvantages of each side.
• In MCQs, check whether the option talks about randomness (stochastic) or construction (statistical or economic).
• Mention calibration and parameter uncertainty when asked to evaluate a model.
• Write the model and its assumptions first. Examiners give marks for stating that residuals are uncorrelated with the market and with each other.
• Always check whether the question gives a standard deviation or a variance before you compute beta.
• In written answers, explain what the result means: systematic risk cannot be diversified away, specific risk can.