CFA Level I · CFA Level I Exam

The Capital Asset Pricing Model, Market Model, and Other Factor-Based Equity Models: formula sheet

Full chapter guide

Key formulas

CAPM required return: E(Ri) = Rf + βi × [E(Rm) − Rf]; [E(Rm) − Rf] is the market risk premium. This is the SML equation.
Beta: βi = Cov(Ri, Rm) ÷ Var(Rm) = ρi,m × σi ÷ σm; Beta of the market is 1. Beta of the risk-free asset is 0.
Portfolio beta: βp = Σ wi × βi; Weights are market-value weights and sum to 1 (include the risk-free asset at beta 0).
Capital market line: E(Rp) = Rf + [(E(Rm) − Rf) ÷ σm] × σp; Uses total risk σ and applies to efficient portfolios only.
Mispricing test (alpha): α = E(R) − required return from CAPM; α > 0: undervalued, above the SML. α < 0: overvalued, below the SML.
Market model: Ri = αi + βi Rm + ei; Regression of security return on market return. Intercept is alpha, slope is beta, e is the residual with expected value zero.
Beta from covariance: βi = Cov(Ri, Rm) ÷ Var(Rm); Same as the OLS slope. Use the same sample or population basis for numerator and denominator.
Beta from correlation: βi = ρ(i,m) × σi ÷ σm; Useful when you are given correlation and standard deviations.
Adjusted beta (Blume-type): Adjusted β = (2/3) × raw β + (1/3) × 1.0; Pulls raw beta toward 1. Weights of 2/3 and 1/3 are the commonly used ones.
Alpha from regression: αi = mean(Ri) − βi × mean(Rm); The regression line passes through the sample means.
CAPM expected return: E(Ri) = Rf + βi × [E(Rm) − Rf]; Use beta from the market model to find required return or cost of equity.
Systematic variance share: R² = βi² σm² ÷ σi²; For a simple regression, R² equals the squared correlation between security and market.
APT expected return: E(Rᵢ) = Rf + λ₁βᵢ₁ + λ₂βᵢ₂ + … + λₖβᵢₖ; λ is the risk premium for each factor. β is the asset's sensitivity to that factor. If a question gives the expected return on a factor portfolio rather than a premium, subtract Rf to get λ, then compute E(R) = Rf + Σβλ, adding Rf only once.
Macroeconomic factor model return: Rᵢ = E(Rᵢ) + βᵢ₁F₁ + βᵢ₂F₂ + … + βᵢₖFₖ + εᵢ; F is the surprise in each factor (actual minus expected). ε is the asset-specific return, with mean zero.
Single-factor special case: E(Rᵢ) = Rf + βᵢ × λ; With the market as the only factor this has the same form as CAPM.
Portfolio factor sensitivity: β_p,k = Σ wᵢ × βᵢ,k; A portfolio's sensitivity to a factor is the weighted average of the asset sensitivities.
Fama-French three-factor model: E(Ri) − Rf = βmkt × [E(Rm) − Rf] + βSMB × E(SMB) + βHML × E(HML); Add Rf to the right side to get expected return. Factor premiums are the expected returns of the long-short portfolios.
Fama-French five-factor model: E(Ri) − Rf = βmkt × [E(Rm) − Rf] + βSMB × E(SMB) + βHML × E(HML) + βRMW × E(RMW) + βCMA × E(CMA); RMW is profitability (robust minus weak). CMA is investment (conservative minus aggressive).
Carhart four-factor model: E(Ri) − Rf = βmkt × [E(Rm) − Rf] + βSMB × E(SMB) + βHML × E(HML) + βWML × E(WML); WML is winners minus losers, the momentum factor.
Factor definitions: SMB = small-cap return − large-cap return; HML = high B/M return − low B/M return; High book-to-market means value stocks; low means growth stocks.
CAPM as a special case: E(Ri) = Rf + βi × [E(Rm) − Rf]; Single factor. Multifactor models reduce to this if all other loadings are zero.
Sharpe ratio: SR = (Rp − Rf) ÷ σp; Uses total risk. Higher is better. Rf is the risk-free rate.
Treynor ratio: TR = (Rp − Rf) ÷ βp; Uses systematic risk. Can be misleading if beta is negative or near zero.
M-squared (M²): M² = Rf + (Rp − Rf) × (σm ÷ σp) − Rm; Equivalent to Rp* − Rm, where Rp* is the return of the portfolio rescaled to market volatility. Positive means it beat the market on a risk-matched basis.
Jensen's alpha: αp = Rp − [Rf + βp(Rm − Rf)]; Return above the CAPM-required return. Positive means outperformance.
Information ratio: IR = (Rp − Rb) ÷ tracking error; Tracking error is the standard deviation of (Rp − Rb) over time. Active return is the numerator.
Choosing the measure: Total risk: Sharpe, M². Systematic risk: Treynor, Jensen's alpha. Versus benchmark: information ratio.; Sharpe and M² give the same ranking of portfolios.
CAPM cost of equity: r(e) = R(f) + β × [E(R_m) − R(f)]; E(R_m) − R(f) is the equity risk premium. Use the premium, not the market return, in the bracket.
Multifactor cost of equity: r(e) = R(f) + β1 × λ1 + β2 × λ2 + … + βk × λk; Each λ is the risk premium for that factor. Each β is the sensitivity to that factor.
Fama-French style factor set: r(e) = R(f) + β_mkt × λ_mkt + β_size × λ_size + β_value × λ_value; Size is small minus big. Value is high minus low book-to-market. Carhart adds momentum. Use the size and value premiums as given. Subtract R(f) only if a total market return is given instead of the market premium.
Active return: Active return = R(p) − R(benchmark); Can also be split into factor-driven return and residual (selection) return.
Active return by factor exposure: Active return = Σ (β_p,i − β_b,i) × F_i + residual; Differences in factor sensitivity between portfolio and benchmark, times factor returns.

Quick revision

CAPM: E(Ri) = Rf + βi × [E(Rm) − Rf].
The security market line plots expected return against beta; the market risk premium is E(Rm) − Rf.
Beta measures systematic risk only; CAPM gives no reward for unsystematic risk.
A security plotting above the SML is undervalued; below it is overvalued.
Market model: Ri = α + βi × Rm + error; beta is the regression slope.
APT uses several systematic factors and assumes no arbitrage opportunities persist.
Fama-French adds size and value factors to the market factor; Carhart adds momentum.
Sharpe ratio = (Rp − Rf) ÷ σp, using total risk.
Treynor ratio = (Rp − Rf) ÷ βp, using systematic risk.
Jensen's alpha = Rp − [Rf + βp × (Rm − Rf)]; it is an absolute return, not a ratio.
Use Sharpe for a standalone or whole portfolio. Use Treynor and Jensen's alpha when the portfolio is one component of a larger diversified portfolio.
Cost of equity from CAPM is the required return used to discount equity cash flows.

Common mistakes

Multiplying beta by the market return instead of the market risk premium. Fix: Always subtract Rf from Rm first. Required return = Rf + β(Rm − Rf).
Confusing the SML with the CML. Fix: SML: x-axis is beta, applies to any asset. CML: x-axis is standard deviation, applies only to efficient portfolios.
Dividing covariance by the security's variance instead of the market's variance. Fix: Beta measures sensitivity to the market, so the market's variance always goes in the denominator.
Treating the market model and CAPM as the same thing. Fix: The market model is a regression describing historical returns with a free intercept. CAPM is an equilibrium model for expected returns, with no alpha.
Using the factor's actual value instead of its surprise in a macroeconomic factor model Fix: Subtract the expected value first. Only the unexpected part moves the return away from the expected return.
Forgetting to add the risk-free rate Fix: The sum of factor terms is the risk premium. Expected return = Rf + that sum.
Forgetting to add the risk-free rate at the end. Fix: Ask whether the question wants expected return or excess return. If expected return, add Rf.
Treating HML as a high-growth factor. Fix: High means high book-to-market, which is value. A positive HML loading signals a value tilt; negative signals growth.
Forgetting to subtract the risk-free rate in the numerator. Fix: Sharpe and Treynor always use excess return, Rp − Rf. Write it first.
Using standard deviation in the Treynor ratio or beta in the Sharpe ratio. Fix: Remember: Sharpe uses Standard deviation (S for S); Treynor uses beta (the systematic risk measure from CAPM).

Exam tips

Questions often hide beta inside correlation and standard deviations. Compute beta first.
Know the CAPM assumptions as a list, and expect a question asking which assumption is unrealistic or what happens if it is relaxed.
For mispricing items, compare expected and required returns only. Do not worry about the price level.
Use logic to eliminate: for 0 ≤ beta < 1 the required return lies between Rf and the market return; beta of 0 means required return equals Rf; a negative beta means required return below Rf.
For SML versus CML items, check the axis: beta means SML, standard deviation means CML.
Read whether the question asks for raw or adjusted beta before calculating. The wrong one is usually an option.
Adjusted beta always sits between raw beta and 1. Use this to remove two options fast.
Know the conceptual contrast: market model is a regression with an intercept, CAPM is an equilibrium model about expected returns.