Capital Asset Pricing Model (CAPM): formula sheet

Key formulas

Portfolio expected return

E(Rp) = Σ wᵢ E(Rᵢ)

Weights wᵢ sum to 1. Short sales give negative weights.

Two-asset portfolio variance

Var(Rp) = w₁²σ₁² + w₂²σ₂² + 2w₁w₂Cov(R₁,R₂)

Cov(R₁,R₂) = ρ₁₂σ₁σ₂.

General portfolio variance

Var(Rp) = ΣᵢΣⱼ wᵢwⱼ Cov(Rᵢ,Rⱼ)

The sum includes i = j terms, which are the variances.

Correlation

ρ₁₂ = Cov(R₁,R₂) ÷ (σ₁σ₂)

Always between -1 and +1.

Minimum variance weight (two assets)

w₁ = (σ₂² − σ₁₂) ÷ (σ₁² + σ₂² − 2σ₁₂)

σ₁₂ is the covariance. w₂ = 1 − w₁. Weight on asset 1 is found by setting d Var/d w₁ = 0.

Quadratic utility approach (mean-variance utility)

U = E(R) − (λ ÷ 2) Var(R)

One common form. λ > 0 is the risk aversion coefficient. Larger λ means more risk averse.

Equally weighted n-asset diversification

Var(Rp) = σ̄²÷n + ((n−1)÷n) × average covariance

As n grows, variance tends to the average covariance: the systematic part.

Portfolio of risk-free asset and risky portfolio P

E(R) = (1 − w) r_f + w E(R_P); σ = |w| σ_P

w is the proportion in P. w > 1 means borrowing at r_f to invest more in P.

Capital Market Line

E(R_P) = r_f + [(E(R_M) − r_f) ÷ σ_M] × σ_P

Applies to efficient portfolios only. Intercept is r_f; slope is the market price of risk.

Slope of the CML (Sharpe ratio of the market)

(E(R_M) − r_f) ÷ σ_M

Extra expected return per unit of standard deviation.

Market portfolio weights

w_i = market value of asset i ÷ total market value of all risky assets

All investors hold the risky assets in these proportions under CAPM.

Standard deviation needed for a target return

σ_P = [E(R_P) − r_f] ÷ slope

Rearranged CML. Use it when the question gives a target return.

CAPM equation

E(Ri) = Rf + βi × (E(Rm) − Rf)

Gives the required (equilibrium) expected return. E(Rm) − Rf is the market risk premium.

Beta definition

βi = Cov(Ri, Rm) ÷ Var(Rm) = ρim × σi ÷ σm

ρim is the correlation between security i and the market. σ is standard deviation.

Portfolio beta

βp = Σ wi × βi

Weights wi are proportions of portfolio value and sum to 1. Beta is a weighted average.

Alpha

αi = E(Ri) − [Rf + βi(E(Rm) − Rf)]

Positive alpha: above the SML, underpriced. Negative alpha: below the SML, overpriced.

Systematic and specific risk

σi² = βi² × σm² + σ²(εi)

Holds under the single-factor (market model) set-up. The first term is systematic, the second is specific risk.

Price implied by required return

Fair price today = E(expected end-of-period value) ÷ (1 + required return)

Use for a single period. Compare with the market price to judge over- or underpricing.

CAPM

E(Ri) = Rf + βi [E(RM) − Rf]

Beta is Cov(Ri, RM) ÷ Var(RM). Only systematic risk earns a premium.

Single index model

Ri = αi + βi RM + εi

E(εi) = 0 and εi is uncorrelated with RM and with other εj. This is a return-generating model, not an equilibrium model.

Variance under single index model

Var(Ri) = βi² σM² + σεi²

Systematic part plus specific part.

Covariance under single index model

Cov(Ri, Rj) = βi βj σM²

Holds for i ≠ j. It cuts the number of inputs needed for a portfolio.

Multi-factor model

Ri = ai + bi1 F1 + bi2 F2 + … + bik Fk + εi

The bij are factor sensitivities. Factors can be macroeconomic or based on firm characteristics.

APT expected return

E(Ri) = λ0 + bi1 λ1 + bi2 λ2 + … + bik λk

λ0 is the return on a zero-sensitivity asset, often Rf. Each λj is the risk premium for factor j.

Test regression (cross-section)

R̄i = γ0 + γ1 β̂i + ηi

CAPM predicts γ0 = Rf and γ1 = E(RM) − Rf. Extra variables should have zero coefficients.

Quick revision

• Portfolio expected return: E(Rₚ) = Σ wᵢE(Rᵢ), with weights summing to 1.
• Two-asset variance: σₚ² = w₁²σ₁² + w₂²σ₂² + 2w₁w₂Cov(R₁, R₂).
• Correlation: ρ = Cov(R₁, R₂) ÷ (σ₁σ₂), and it lies between −1 and +1.
• The efficient frontier holds portfolios with the highest return for a given risk.
• The CML: E(Rₚ) = r_f + [E(R_M) − r_f] ÷ σ_M × σₚ, for efficient portfolios.
• The SML: E(Rᵢ) = r_f + βᵢ[E(R_M) − r_f], for any asset or portfolio.
• Beta: βᵢ = Cov(Rᵢ, R_M) ÷ Var(R_M). The market has beta 1 and the risk-free asset has beta 0.
• Only systematic risk is priced. Specific risk is removed by diversification.
• An asset above the SML is underpriced (positive alpha). One below is overpriced.
• The market risk premium is E(R_M) − r_f.
• Key assumptions: mean-variance investors, common expectations, one period, borrowing and lending at r_f, no taxes or transaction costs.
• Main limits: unrealistic assumptions, unobservable market portfolio, and weak empirical support for beta alone.

Common mistakes

• Averaging standard deviations to get portfolio risk. Fix: Use the variance formula with the covariance term. The weighted average of standard deviations is correct only when ρ = +1.
• Leaving out the factor 2 on the covariance term. Fix: Always write 2w₁w₂Cov(R₁,R₂) in the two-asset formula.
• Applying the CML to individual securities or inefficient portfolios. Fix: Remember: the CML uses σ and holds for efficient portfolios only. Individual assets lie below it. Use the SML with beta for any asset.
• Using beta on the x-axis of the CML. Fix: CML axes are standard deviation (x) and expected return (y). SML axes are beta (x) and expected return (y).
• Using E(Rm) instead of the market risk premium E(Rm) − Rf when multiplying by beta. Fix: Always compute the premium on its own line first, then multiply by beta, then add Rf.
• Dividing Cov(Ri, Rm) by the security's variance or by σm instead of Var(Rm). Fix: Beta is the regression slope of Ri on Rm, so the denominator is the variance of the market. Check units: covariance ÷ variance gives a pure number.
• Saying Roll's critique proves CAPM is false. Fix: Say it shows CAPM cannot be tested conclusively, because the true market portfolio is unobservable and any test is a joint test with the proxy.
• Treating the single index model and CAPM as the same. Fix: CAPM is an equilibrium model for expected returns. The single index model describes how returns are generated and includes an alpha and a random specific term.

Exam tips

• Show the formula first, then substitute. Markers give method marks even if arithmetic slips.
• State assumptions explicitly when asked about the theory: single period, rational risk-averse investors, mean and variance only, no frictions, common beliefs.
• For descriptive parts, distinguish systematic from unsystematic risk and say which one diversification removes.
• Sketch the frontier with axes labelled: standard deviation on the horizontal, expected return on the vertical. Mark the minimum variance point.
• In Paper B or R questions, state the covariance matrix and weights vector clearly, and compute variance as wᵀΣw.
• For assumption questions, give 5 or 6 assumptions and say what each one delivers. Marks usually go for the link, not the list.
• In a CML sketch, label the axes (σ and expected return), the intercept r_f, the point M and the slope. Show that the line continues beyond M for borrowing.
• Compare CML and SML in one line each when asked: CML is for efficient portfolios and uses σ; SML is for all assets and uses beta.