CFA Level II · CFA Level II Exam

Using Multifactor Models: formula sheet

Full chapter guide

Key formulas

Single-factor return-generating equation: Ri = E(Ri) + βi,1 F1 + εi; F1 is the factor surprise (actual minus expected). εi is the asset-specific return with mean zero.
Multifactor return-generating equation: Ri = E(Ri) + βi,1 F1 + βi,2 F2 + ... + βi,k Fk + εi; One surprise term per factor. Betas are sensitivities to each factor.
APT expected return (multifactor): E(Rp) = RF + λ1 βp,1 + λ2 βp,2 + ... + λk βp,k; λj is the risk premium for factor j. It is the expected return on a portfolio with beta 1 to factor j and 0 to all others, minus RF.
APT expected return (single factor): E(Rp) = RF + λ1 βp,1; Same form as CAPM when the factor is the market and λ1 = E(RM) − RF.
Portfolio beta: βp,k = Σ wi βi,k; Weights times asset betas, taken factor by factor. It is a weighted average.
Mispricing check: Alpha = Expected (forecast) return − APT required return; Positive means underpriced (buy). Negative means overpriced (sell or short).
Macroeconomic factor model: R_i = a_i + b_i1 × F1 + b_i2 × F2 + … + b_ik × Fk + ε_i; F are surprises in macro variables (actual − expected). a_i is the expected return. b are sensitivities. ε_i is the asset-specific return.
Factor surprise: Surprise = Actual value − Expected value; Only the surprise enters the model. The expected part is already in a_i.
Fundamental factor model: R_i = a_i + b_i1 × F1 + … + b_ik × Fk + ε_i; Same form, but the sensitivities b_i are the standardized attributes of the company (for example standardized book-to-market) and the factor returns F are estimated by cross-sectional regression.
Fama-French three-factor model: R_i − R_f = a_i + b_mkt × (R_m − R_f) + b_SMB × SMB + b_HML × HML + ε_i; SMB is small minus big. HML is high minus low book-to-market. The factors are returns on long-short portfolios formed on company attributes.
Carhart four-factor model: Fama-French three factors + b_WML × WML; WML is winners minus losers, the momentum factor.
APT / factor risk premium expected return: E(R_i) = R_f + b_i1 × λ1 + … + b_ik × λk; λ is the risk premium of each factor and b are the sensitivities. Use it when the question gives the risk-free rate and factor risk premiums. Do not confuse it with the macro model above, where a_i is the expected return and F are surprises.
Active return: Active return = Rp − Rb; Portfolio return minus benchmark return over the same period.
Active factor tilt: Tilt_k = βp,k − βb,k; Calculate it for each factor k. A positive tilt means the portfolio is more exposed than the benchmark.
Factor return (attribution): Factor return = Σ (βp,k − βb,k) × λk; λk is the return of factor k in the period. Use the factor return as given, not a premium assumed from memory.
Security selection: Security selection = Active return − Factor return; It is a residual. It equals Σ (wp,i − wb,i) × εi, where εi is each security's specific return, when the portfolio and benchmark betas are the weighted-average security betas.
Active risk: Active risk = √(Active risk squared); Also called tracking risk or tracking error. Take the square root last.
Active risk squared: Active risk squared = Active factor risk + Active specific risk; Variances add. Standard deviations do not.
Active factor risk (one factor): Active factor risk = (βp − βb)² × σ²factor; With several factors, you also need the covariances between the factors. The exam usually gives the variance-covariance data or the result.
Active specific risk: Active specific risk = Σ (wp,i − wb,i)² × σ²ε,i; Applies when security specific returns are uncorrelated. Weights are active weights versus the benchmark.
Share of risk: % from factors = Active factor risk ÷ Active risk squared; Use variance, not standard deviation, as the denominator.
Sharpe ratio: SR = (Rp − Rf) ÷ σp; Uses total standard deviation. Use the same period for return and risk.
Active return: Rp − Rb = Σ[(wp,k − wb,k) × factor return k] + security selection; The sum is the factor tilt return. The remainder is selection. Here w means factor sensitivity.
Active risk (tracking risk): σ(Rp − Rb) = standard deviation of (Rp − Rb); Also called tracking error. It is not the portfolio's standard deviation.
Information ratio: IR = (Rp − Rb) ÷ σ(Rp − Rb); Mean active return divided by active risk. Compare this with the benchmark, not with Rf.
Active risk decomposition: Active risk² = active factor risk + active specific risk; Add variances, not standard deviations.
Maximum Sharpe ratio: SR*² = SRB² + IR²; Holds when active positions are optimally sized. So SR* = √(SRB² + IR²).
Optimal active risk: σA* = (IR ÷ SRB) × σB; Scales active risk to the IR and to the benchmark's reward per unit of risk.
Portfolio factor sensitivity: b_p,k = Σ w_i × b_i,k; Weighted average of asset sensitivities to factor k. Weights sum to 1 unless a cash or hedge position is stated.
Factor model return: R_i = a_i + b_i1 F1 + b_i2 F2 + … + b_ik Fk + ε_i; F are factor surprises or factor returns as defined in the vignette. ε is the specific return.
Pure factor portfolio: b_p,k = 1 for the chosen factor and b_p,j = 0 for every other factor j; Expected return is the risk-free rate plus that factor's risk premium in an APT-style setting.
Tracking portfolio condition: b_tracking,k = b_target,k for every factor k; Solve the weights from the matching equations plus weights summing to 1.
Active return: R_active = R_p − R_B; Portfolio return minus benchmark return.
Active factor tilt: Tilt_k = b_p,k − b_B,k; Positive means overweight to that factor relative to the benchmark.
Active return from factors: Σ (b_p,k − b_B,k) × F_k + active specific return; Splits active return into factor tilts and security selection.
Active risk (tracking error): Active risk = √(Active factor risk + Active specific risk); Variances add, not standard deviations. Active factor risk and active specific risk are variances.
Information ratio: IR = Active return ÷ Active risk; Active return per unit of tracking error.

Quick revision

APT: E(Rp) = Rf + λ1·β1 + λ2·β2 + ... where λ is the factor risk premium and β the sensitivity.
APT rests on no arbitrage, so expected returns must reflect factor exposures.
Macroeconomic models use surprises in economic variables as factors.
Fundamental models use company attributes such as size or value characteristics.
Statistical models extract factors from return data, so they can be hard to interpret.
Active return = portfolio return − benchmark return.
Factor return contribution = (portfolio sensitivity − benchmark sensitivity) × factor return, summed over factors.
Active return not explained by factors is the security selection part.
Active risk splits into active factor risk and active specific risk.
Sharpe ratio = (Rp − Rf) ÷ σp, using total risk.
Information ratio = active return ÷ active risk (tracking error).
A tracking portfolio replicates a target set of factor exposures; smart beta tilts toward factors using rules-based indexes.

Common mistakes

Using the full factor return instead of the factor risk premium. Fix: Read the exhibit label. If it is a return that includes the risk-free rate, subtract RF first. If it says premium, use it as is.
Saying APT requires the market portfolio to be efficient. Fix: Remember that CAPM needs the market portfolio to be mean-variance efficient. APT needs only a factor model and no arbitrage among well-diversified portfolios.
Plugging actual GDP growth or inflation into the model instead of the surprise. Fix: Subtract the expected value first. Only the difference enters a macro model. The expected part is already in the intercept.
Calling a model with factors like book-to-market and size a macroeconomic model. Fix: Ask whether the factor is a company attribute. If yes, it is fundamental.
Using the portfolio's beta instead of the active tilt (beta difference). Fix: Always subtract benchmark sensitivity first. Write the tilts before touching factor returns.
Adding standard deviations of factor risk and specific risk to get active risk. Fix: Square both, add, then take the square root. Check the answer is less than the sum of the two standard deviations.
Using portfolio standard deviation as the denominator of the information ratio. Fix: IR always uses active risk, the standard deviation of portfolio minus benchmark returns. Sharpe uses total standard deviation.
Subtracting the risk-free rate in the IR numerator. Fix: The IR numerator is portfolio return minus benchmark return. Rf appears only in Sharpe.
Treating a factor portfolio as having zero betas everywhere. Fix: A factor portfolio has beta 1 on the target factor and 0 on all others. Check both conditions.
Adding standard deviations to get active risk. Fix: Add variances: active factor risk plus active specific risk, then take the square root.

Exam tips

Read each exhibit label carefully. Premium versus expected factor return decides whether you subtract the risk-free rate.
For conceptual questions, link each assumption to its role: factor model, diversification removing specific risk, and no arbitrage producing the pricing line.
Keep the sign of each premium. A negative premium paired with a positive beta lowers required return.
For compare-and-contrast items, remember APT allows many factors and does not name them, while CAPM has one factor, the market.
Check your arithmetic by recomputing the sum once. In an item set, an error on required return often carries into later questions.
Look for the word surprise. If the vignette gives actual and expected values, expect a calculation step before the model is applied.
Classification questions usually hinge on what the factors are. Read the factor list before reading anything else.
Know one strength and one weakness for each type: macro is intuitive but surprises are hard to measure; fundamental is easy to interpret; statistical fits well but has unclear factors.