FRM Part II · FRM Exam Part II

Factors: formula sheet

Full chapter guide

Key formulas

CAPM expected return: E(Ri) = Rf + βi × [E(Rm) − Rf]; Only market beta is priced. [E(Rm) − Rf] is the market risk premium.
Beta: βi = Cov(Ri, Rm) ÷ Var(Rm) = ρ(i,m) × σi ÷ σm; Measures systematic risk relative to the market.
Jensen's alpha: αi = Ri − [Rf + βi × (Rm − Rf)]; Realized excess return not explained by market beta.
Multifactor model: E(Ri) = Rf + βi1 × λ1 + βi2 × λ2 + … + βik × λk; Each λ is the risk premium of factor k. βik is the exposure to that factor.
Factor return decomposition: Ri − Rf = α + Σ βik × Fk + εi; Fk are factor returns. εi is idiosyncratic and diversifiable.
CAPM (one factor): E(Ri) − Rf = βi × [E(Rm) − Rf]; Baseline. The Fama-French models extend this with more factors.
Fama-French three-factor regression: Ri − Rf = α + β_MKT(Rm − Rf) + β_SMB × SMB + β_HML × HML + ε; α is the intercept. ε is the residual with mean zero.
Fama-French five-factor regression: Ri − Rf = α + β_MKT(Rm − Rf) + β_SMB × SMB + β_HML × HML + β_RMW × RMW + β_CMA × CMA + ε; RMW = robust minus weak profitability. CMA = conservative minus aggressive investment.
Expected excess return from factors: E(Ri) − Rf = Σ βk × λk; λk is the risk premium of factor k. Without alpha, expected return is the sum of loading times premium.
Alpha: α = actual average excess return − Σ βk × (average factor return); Test significance with t = α ÷ standard error of α.
Variance decomposition: Var(Ri) = Σ Σ βj βk Cov(Fj, Fk) + Var(ε); First term is systematic. Var(ε) is idiosyncratic. R² = systematic ÷ total.
Portfolio loading: β_p = Σ wi × βi; Loadings are weighted averages of asset loadings by portfolio weights.
Factor premium: Factor return = Return(long high-exposure stocks) − Return(short low-exposure stocks); Long-short portfolio; for value, long high book-to-market minus short low book-to-market.
Multifactor return model: Rᵢ − R_f = α + β_MKT × (R_M − R_f) + β_V × F_value + β_M × F_mom + … + ε; Alpha is the part not explained by the factors. Loadings measure exposure.
Momentum signal: Cumulative return from month t−12 to t−2 (skip month t−1); Skipping the latest month avoids short-term reversal.
Sharpe ratio of a factor: SR = (mean factor return) ÷ (standard deviation of factor return); Annualize mean by ×12 and volatility by ×√12 for monthly data.
Low-beta bet: Beta-neutral BAB: long low-beta ÷ β_L, short high-beta ÷ β_H; Leverage the low-beta leg and de-leverage the high-beta leg so the net beta is zero.
Active (factor tilt) weight: Active weight = portfolio weight − benchmark weight; Sum of active weights is zero for a fully invested long-only portfolio versus its benchmark.
Tracking error: TE = standard deviation of (Rp − Rb); Long-only factor tilts usually have positive tracking error; it grows with the size of the tilt.
Information ratio: IR = (Rp − Rb) ÷ TE; Use average active return over tracking error, both annualised on the same basis.
Long-short factor return: Factor return = R(long leg) − R(short leg); Subtract financing and borrowing costs to get a net return.
Net factor premium: Net premium = gross premium − trading costs − financing and shorting costs − fees; Turnover × cost per trade is the main trading cost driver.
Composite score (integration): Score = Σ wᵢ × zᵢ; zᵢ is the standardised score on factor i and wᵢ its weight; rank stocks on the combined score.
Factor model: Rp = α + Σ βk × Fk + ε; βk is exposure to factor k, Fk is factor return, ε is the specific return.
Portfolio exposure: βp,k = Σ wi × βi,k; Weighted average of asset exposures. Linear in weights.
Systematic variance: σ²sys = β′ Σf β; Σf is the factor covariance matrix. With one factor: β² × σ²F.
Total variance: σ²p = β′ Σf β + Σ wi² σ²ε,i; Assumes specific returns are uncorrelated across assets and with factors.
Factor contribution to risk: Contribution of k = βk × Cov(Fk, Rp,sys) ÷ σ²p (as a % of variance); Contributions sum to the systematic share of variance.
Active exposure: Active βk = βp,k − βb,k; Portfolio minus benchmark exposure.
Factor attribution: Active return = Σ (Active βk × Fk) + active specific return; Residual is alpha, not necessarily skill.
R-squared: R² = σ²sys ÷ σ²p; Share of variance explained by the factors.

Quick revision

CAPM: E(Ri) = Rf + βi × [E(Rm) − Rf].
Beta = Cov(Ri, Rm) ÷ Var(Rm).
Alpha is return in excess of what the factor exposures explain.
Multifactor: E(R) = Rf + Σ βk × λk, where λk is the factor premium.
Total variance = systematic (factor) variance + specific variance, when residuals are uncorrelated with factors.
Fama-French three factors: market, size (SMB) and value (HML).
Value buys cheap stocks on fundamentals; momentum buys recent winners and sells losers.
Momentum can suffer sharp crashes after market reversals.
Low volatility strategies often carry hidden sector and interest rate exposure.
Smart beta changes the weighting rule but usually stays long-only, so it keeps market beta.
Factor returns are time-varying and can be crowded; diversification across factors helps but correlations rise in stress.
Attribution contribution = exposure × factor return; what is left over is specific return.

Common mistakes

Using total return instead of excess return for factor premiums. Fix: Confirm whether a premium is already net of Rf. Add Rf only once.
Saying the CAPM rewards total volatility. Fix: Only market beta is priced in the CAPM. Idiosyncratic risk is diversifiable and unpriced.
Treating alpha as the same as total excess return Fix: Alpha is only the part left after subtracting factor loadings times factor returns. Always subtract the factor-explained return first.
Reading a negative HML loading as poor performance Fix: A negative HML loading means a growth tilt, not a loss. Performance depends on the loading times the factor return.
Saying momentum crashes happen in market falls only Fix: Crashes typically occur in sharp rebounds after a prolonged decline, when losers, which are high-beta, surge and hurt the short leg.
Treating every factor premium as risk compensation Fix: Distinguish risk-based, behavioral and structural explanations. Value and size are more commonly attributed to risk, and momentum and low volatility to behavior and constraints, but the evidence is debated for all of them.
Saying long-only factor portfolios have no market exposure Fix: Long-only keeps market beta; only a beta-neutral long-short construction strips it out.
Treating smart beta as passive and costless Fix: Smart beta has active rule choices, tracking error, higher turnover and capacity limits than cap-weighted indices.
Adding factor volatilities instead of using covariances Fix: Risk adds through variance and covariance. Use β′Σβ, and take the square root only at the end.
Using total exposure instead of active exposure in attribution Fix: For a benchmark-relative question, always compute portfolio β minus benchmark β first.

Exam tips

Read whether the question asks for a CAPM answer or a multifactor answer. The model changes the alpha.
Expect conceptual questions that ask why a factor earns a premium. Match the story to risk-based, behavioral or structural.
Check that premiums are excess returns before adding the risk-free rate.
Remember that idiosyncratic risk is never priced in either model.
Know the sign-to-tilt mapping for every loading. Most conceptual questions test only this.
Always check whether returns are excess returns and whether the units are monthly or annual.
Alpha is a residual. If a question gives actual return and loadings, compute the factor return first.
For macro factors, link the factor to the bad state it represents (low growth, high inflation, high volatility) and the premium for bearing it.