FRM Part II · FRM Exam Part II

Alpha (and the Low-Risk Anomaly): formula sheet

Full chapter guide

Key formulas

CAPM expected return: E(Ri) = Rf + βi × [E(Rm) − Rf]; Beta is the only risk that earns a premium under CAPM. (Rm − Rf) is the market risk premium.
Jensen's alpha: α = Rp − [Rf + βp × (Rm − Rf)]; Use realised returns for Rp and Rm over the same period. Positive means outperformance versus CAPM.
Single-index regression: Rp − Rf = α + β × (Rm − Rf) + ε; The intercept is alpha. The slope is beta. ε is the residual with mean zero.
Beta: β = Cov(Rp, Rm) ÷ Var(Rm) = ρ × σp ÷ σm; ρ is the correlation between portfolio and market returns.
Benchmark-relative alpha: Active return = Rp − Rb; This is not Jensen's alpha unless the benchmark beta is 1 and the benchmark is the market. Adjust for beta first if betas differ.
Systematic variance share: β² × σm² ÷ σp² = ρ²; This is R² of the regression. The rest of variance is residual risk.
Single-factor (CAPM) alpha: α = (Rp − Rf) − β × (Rm − Rf); Also called Jensen's alpha. Use average returns over the same period for all terms.
Multi-factor regression: Rp − Rf = α + β1F1 + β2F2 + ... + βkFk + ε; α is the intercept. Factors can be market, size, value, momentum and others.
Active return: Active return = Rp − Rb; Rb is the benchmark return. Simple but depends entirely on the benchmark chosen.
Tracking error: TE = standard deviation of (Rp − Rb); Annualise by multiplying a monthly figure by √12.
Information ratio: IR = α ÷ ω, or (Rp − Rb) ÷ TE; ω is residual (active) risk. Compare the same horizon and annualisation for both numerator and denominator.
t-statistic of alpha: t = α̂ ÷ standard error of α̂; Roughly, |t| above about 2 suggests alpha is significant at the 5% level.
CAPM expected return (security market line): E(Ri) = Rf + βi × [E(Rm) − Rf]; The theoretical benchmark. Slope of the SML is the market risk premium.
Alpha (Jensen's alpha): α = Ri − [Rf + βi × (Rm − Rf)]; Use realised or average returns. Positive alpha means the stock plots above the SML.
Sharpe ratio: Sharpe = (Ri − Rf) ÷ σi; Return per unit of total volatility. The anomaly says this falls as volatility rises.
Treynor ratio: Treynor = (Ri − Rf) ÷ βi; Return per unit of beta. Flat SML means low-beta stocks have a higher Treynor ratio.
Beta: β = ρ(i, m) × σi ÷ σm; Low beta can come from low volatility, low correlation, or both.
Beta-neutral BAB portfolio: BAB return = (1 ÷ βL) × (RL − Rf) − (1 ÷ βH) × (RH − Rf); Long the low-beta leg scaled up, short the high-beta leg scaled down. Both legs have beta of 1, so net beta is zero.
CAPM expected return: E(Ri) = Rf + βi × [E(Rm) − Rf]; The benchmark line. The anomaly says the actual line is flatter than this.
Jensen's alpha: αi = Ri − [Rf + βi × (Rm − Rf)]; Positive for low-beta stocks and negative for high-beta stocks under the anomaly.
Sharpe ratio: SR = (Rp − Rf) ÷ σp; Low-risk portfolios tend to show higher Sharpe ratios than high-risk ones.
Mechanism chain (rule): Constraint or preference → extra demand for high-risk stocks → higher price → lower expected return; Apply the same chain to leverage constraints, lottery preferences and benchmarking.
Beta of a portfolio: β_p = Σ w_i × β_i; Weights include leverage and shorts. Short positions carry negative weights.
BAB leverage on the long leg: Long leverage = 1 ÷ β_L; Scales the low-beta basket to beta of 1. Needs borrowing, so the leveraged amount is financed at the funding rate.
BAB short leg scaling: Short size = 1 ÷ β_H; The high-beta basket is scaled down to beta of 1 before shorting.
BAB portfolio return: r_BAB = (1 ÷ β_L)(r_L − r_f) − (1 ÷ β_H)(r_H − r_f); Net beta is zero by construction. The risk-free terms matter because the long leg is financed.
Minimum variance weights (fully invested, no constraints): w = Σ⁻¹ 1 ÷ (1ᵀ Σ⁻¹ 1); Σ is the covariance matrix. Weights depend on estimated covariances, so estimation error matters.
Two-asset minimum variance weight in asset 1: w₁ = (σ₂² − ρσ₁σ₂) ÷ (σ₁² + σ₂² − 2ρσ₁σ₂); Valid when short selling is allowed. Weights can be negative if correlation is high.

Quick revision

CAPM: E(Ri) = Rf + βi × (E(Rm) − Rf).
Alpha = actual return − CAPM required return, so it is a gap, not a total return.
Positive alpha means return above what beta explains; it does not by itself prove skill.
Beta = Cov(Ri, Rm) ÷ Var(Rm), the sensitivity to market movements.
Alpha measured against too few factors may be a hidden factor exposure.
The low-risk anomaly: low-beta and low-volatility assets have earned higher risk-adjusted returns than CAPM predicts.
The security market line predicts a rising return with beta; the empirical line is flatter.
Leverage constraints push some investors toward high-beta stocks, which can lift their prices and lower their returns.
Benchmarked managers and lottery-like preferences are other proposed explanations.
Betting-against-beta style strategies go long low-beta and short high-beta assets, scaled so each leg has beta near one.
Leverage, funding costs, turnover and trading costs can erode the gain from exploiting the anomaly.
Always state the method, the number and the interpretation in your working.

Common mistakes

Computing alpha as Rp − Rm. Fix: Always adjust for beta. Use Rp − [Rf + β(Rm − Rf)]. Rp − Rm is only an active return.
Forgetting to subtract the risk-free rate inside the beta term. Fix: Multiply beta by (Rm − Rf), then add Rf.
Treating a factor premium as alpha Fix: Ask whether the return is compensation for a known factor. If the model includes that factor, it is beta, not alpha.
Dividing by total volatility to get the information ratio Fix: Information ratio uses tracking error or residual risk. Sharpe uses excess return over the risk-free rate and total volatility.
Saying low-beta stocks earn higher raw returns than high-beta stocks. Fix: The claim is about risk-adjusted return and alpha. High-beta stocks may still earn more in raw terms, just not enough for their risk.
Describing the flat SML as having a negative slope. Fix: Flatter means a smaller slope than the market risk premium. The slope is still usually positive.
Saying leverage constraints make investors buy low-beta stocks. Fix: Constrained investors still want higher returns. They buy high-beta stocks instead of leveraging, which overprices high beta.
Treating lottery preference as an institutional constraint. Fix: Lottery preference is a behavioural taste for skewed payoffs. Leverage limits and benchmarking are institutional.
Treating minimum variance and low beta as the same thing. Fix: Low beta ranks on market sensitivity alone. Minimum variance uses full covariances and can hold higher-beta stocks that diversify.
Forgetting the risk-free rate in BAB returns. Fix: Subtract r_f from each leg before scaling. Leverage is financed at the funding rate.

Exam tips

Write the CAPM line first, then subtract. Most wrong options come from skipping beta.
If a question mentions a low-beta or low-volatility portfolio with positive alpha, think about the low-risk anomaly and factor exposure.
Check whether returns given are total or excess. If excess, do not subtract Rf again.
Questions on interpretation often test that alpha depends on the chosen model or benchmark.
Know that the regression intercept is alpha and the slope is beta.
Always read which model defines alpha. The answer often changes when a factor is added.
Information ratio questions nearly always use tracking error or residual risk in the denominator, not total volatility.
Watch for options that confuse alpha with active return. Active return ignores beta.