FRM Part II · FRM Exam Part II

Factor Theory: formula sheet

Full chapter guide

Key formulas

CAPM: E(Ri) = Rf + βi × [E(Rm) − Rf]; βi = Cov(Ri, Rm) ÷ Var(Rm). Only market (systematic) risk is priced.
Multifactor / APT expected return: E(Ri) = Rf + βi1 × λ1 + βi2 × λ2 + … + βik × λk; λ is the risk premium for each factor. Betas are exposures to each factor.
Factor return model: Ri = αi + βi1 F1 + … + βik Fk + εi; εi is idiosyncratic and assumed uncorrelated with the factors.
Alpha: α = actual return − expected return from factor model; Measured in excess of Rf. Positive alpha means return above what factor exposure explains.
Portfolio beta: βp = Σ wi × βi; Weights are portfolio weights. Betas add linearly.
Systematic variance: Var(Ri) = β² × Var(F) + Var(ε) (single factor); The first term is systematic risk. The second is diversifiable.
Multi-factor expected return: E(Rᵢ) = R_f + β_i1 × λ₁ + β_i2 × λ₂ + … + β_iK × λ_K; λ is the risk premium per unit of exposure to each factor. R_f is the risk-free rate. The result is an expected total return.
Excess return form: E(Rᵢ) − R_f = Σ β_ik × λ_k; The risk premium of an asset is the sum of exposure times factor premium across all factors.
Realised return in a macro factor model: Rᵢ = E(Rᵢ) + Σ β_ik × F_k + εᵢ; F_k is the unexpected surprise in factor k (actual minus expected). εᵢ is idiosyncratic and diversifiable.
Factor variance of a portfolio: σ²_p = Σ Σ β_pj β_pk Cov(F_j, F_k) + σ²_ε; With uncorrelated factors, systematic variance = Σ β_pk² × σ²(F_k). Add residual variance for total variance.
Real versus nominal rate (approximation): Nominal rate ≈ real rate + expected inflation + inflation risk premium; The inflation risk premium compensates nominal bond holders for uncertainty about inflation.
Single-factor case (CAPM): E(Rᵢ) = R_f + β_i × (E(R_m) − R_f); The market is one factor. Macro models extend this to several economic factors.
Fama-French three-factor model: Rᵢ − R_f = αᵢ + βᵢ(R_m − R_f) + sᵢ·SMB + hᵢ·HML + εᵢ; s and h are loadings on size and value. Alpha is the return left unexplained.
SMB: SMB = return of small-cap portfolio − return of large-cap portfolio; Positive SMB means small caps outperformed large caps in that period.
HML: HML = return of high book-to-market (value) portfolio − return of low book-to-market (growth) portfolio; Positive HML means value beat growth.
Four-factor (Carhart) model: Rᵢ − R_f = αᵢ + βᵢ(R_m − R_f) + sᵢ·SMB + hᵢ·HML + mᵢ·MOM + εᵢ; MOM is the winners-minus-losers return.
Expected excess return from factors: E[Rᵢ] − R_f = βᵢ·λ_m + sᵢ·λ_SMB + hᵢ·λ_HML; λ is the factor premium; multiply each loading by its premium and add.
Active return: Active return = Portfolio return − Benchmark return; Active weights sum to zero for a fully invested long-only portfolio.
Tracking error: TE = standard deviation of active returns; Annualise by multiplying periodic TE by √(periods per year).
Information ratio: IR = Expected active return ÷ Tracking error; Measures active return earned per unit of benchmark-relative risk.
Multi-factor model: R = α + β₁F₁ + β₂F₂ + … + ε; The β values are factor exposures. Active exposure is portfolio β minus benchmark β.
Long-short factor return: Factor return = Return of top group − Return of bottom group; Dollar-neutral, so market beta is close to zero.
Active risk from factor tilts: TE² ≈ Σ(active exposure² × factor variance) + covariance terms + residual variance; Correlated factors need the covariance terms.
Factor model: Rp − Rf = α + β1 × F1 + β2 × F2 + … + βk × Fk + ε; Rp − Rf is the portfolio excess return. Fk are factor returns. ε is the residual, with mean zero and uncorrelated with the factors.
Single-factor variance decomposition: σp² = β² × σF² + σε²; Systematic variance = β² × σF². Idiosyncratic variance = σε². Use variances, never add volatilities.
Multi-factor systematic variance: σsys² = Σ βi² σi² + 2 Σ βi βj ρij σi σj (i < j); The covariance term drops out only when the factors are uncorrelated. Total variance = σsys² + σε².
R-squared: R² = σsys² ÷ σp² = 1 − σε² ÷ σp²; Share of total variance explained by the factors. The idiosyncratic share is 1 − R².
Return attribution: Factor contribution = βk × Fk; α = (Rp − Rf) − Σ βk × Fk; Alpha is the unexplained part of excess return. Keep the sign of each beta, since short exposures subtract.
t-statistic for alpha: t = α̂ ÷ SE(α̂); An absolute value above roughly 2 is usually read as significant at about the 5% level. A large alpha with a large standard error may be noise.
Information ratio: IR = α ÷ σε; Alpha per unit of residual (idiosyncratic) risk, with both on the same annual basis.
Annualising monthly figures: Return × 12; volatility × √12; variance × 12; Apply to monthly data, assuming returns are independent over time.
Portfolio return from factors: Rp = α + Σ βk × Fk + ε; βk is exposure to factor k; Fk is the factor return. Asset classes are described by their βs.
Portfolio factor exposure: βp,k = Σ wi × βi,k; Weights times each asset's loading, summed across holdings. Use this to find hidden concentration.
Two-factor portfolio variance: σp² = w1²σ1² + w2²σ2² + 2 w1 w2 ρ σ1 σ2; Lower ρ between factors means more diversification benefit. If ρ = 1 there is none.
Sharpe ratio: SR = (E[R] − Rf) ÷ σ; Used to compare factor premiums per unit of risk.

Quick revision

A factor is a systematic source of return and risk that is shared across many assets.
Multifactor model: return = alpha + Σ (beta × factor return) + residual.
Alpha is return not explained by the factors in the model; change the model and alpha changes.
Risk premium is compensation for bearing a risk that investors do not want to hold, or a persistent behavioural or structural effect.
Macro factors include economic growth, inflation and interest rates; assets differ in their exposure to each.
Style factors such as value, size and momentum are defined from asset characteristics, not from the economy directly.
Momentum can reverse sharply, so it carries crash risk.
Long-only smart beta carries the market beta as well as the factor tilt; long-short isolates the factor but needs leverage and shorting.
Factor risk contribution depends on exposure, factor volatility and correlations, not on exposure alone.
Specific risk is the part of risk left after factor risk; diversification reduces it.
Factors can be correlated and can become crowded, which reduces diversification when stress hits.
Diversifying across factors helps because factor premiums do not all go bad at the same time, though correlations can rise in a crisis.

Common mistakes

Using the market return instead of the market risk premium in CAPM. Fix: Always compute E(Rm) − Rf first, then multiply by beta and add Rf.
Saying total volatility is priced under CAPM. Fix: Only beta, the non-diversifiable part, earns a premium. Idiosyncratic risk does not.
Using the level of inflation or growth instead of the surprise to compute a realised return effect. Fix: Subtract the expected value first. Only actual minus expected enters the realised return equation.
Forgetting to add the risk-free rate to the sum of β × λ. Fix: Read whether the question asks for expected return or risk premium. Expected return needs R_f.
Reading a negative HML loading as 'poor performance'. Fix: A negative loading means growth-like exposure. Performance depends on whether HML was positive or negative in the period.
Confusing SMB direction. Fix: SMB is small minus big. Positive SMB means small caps beat large caps.
Calling smart beta the same as active management. Fix: Smart beta is rules-based, transparent and systematic. Active management relies on discretionary views and is usually costlier.
Assuming a long-short factor portfolio has no risk. Fix: It still has factor risk, crash risk, leverage, borrowing costs and crowding risk.
Adding volatilities instead of variances Fix: Square each volatility, add the variances, then take the square root. For example, 18% and 6% give √(324 + 36) ≈ 18.97%, not 24%.
Using total returns instead of excess returns Fix: Subtract the risk-free rate from the portfolio return before comparing it with factor-explained return. Factor returns for long-short factors are already excess-style returns.

Exam tips

Read whether the question gives a market return or a premium. This is the most common trap.
Know the CAPM versus APT contrast: CAPM has one factor and strong investor assumptions, APT has multiple unspecified factors and relies on no-arbitrage.
Expect interpretation questions: which risk is priced, and why diversifiable risk earns nothing.
When alpha appears, check whether it is measured against one factor or several.
Keep units consistent, excess returns with excess returns.
Read whether the question gives levels or surprises. Only surprises drive realised returns.
Expect conceptual questions on why hedges earn lower premiums and why only systematic risk is priced.
Watch signs. Negative betas and negative premiums multiply to a positive contribution.