CFA Level I · CFA Level I Exam

Portfolio Risk and Return: Part II: formula sheet

Full chapter guide

Key formulas

Expected return of the combined portfolio: E(Rc) = w × E(Rp) + (1 − w) × Rf = Rf + w × [E(Rp) − Rf]; w is the weight in the risky portfolio. w > 1 means borrowing at Rf.
Standard deviation of the combined portfolio: σc = w × σp; Holds because the risk-free asset has zero standard deviation and zero covariance with the risky portfolio. Use w as a positive number here.
CAL equation: E(Rc) = Rf + [(E(Rp) − Rf) ÷ σp] × σc; Slope is the Sharpe ratio of the risky portfolio. Intercept is Rf.
Weight needed for a target risk: w = σtarget ÷ σp; Use this to find leverage or lending for a given standard deviation.
Weight needed for a target return: w = [E(Rtarget) − Rf] ÷ [E(Rp) − Rf]; If w > 1, borrow; the risk-free weight (1 − w) is negative.
Sharpe ratio (CAL slope): Sharpe = [E(Rp) − Rf] ÷ σp; Same for every point on the CAL.
Capital market line: E(Rc) = Rf + [(E(RM) − Rf) ÷ σM] × σc; A CAL where the risky portfolio is the market portfolio M.
CML equation: E(Rp) = Rf + [(E(Rm) − Rf) ÷ σm] × σp; Applies to efficient portfolios made of the risk-free asset and the market portfolio. The slope is the market Sharpe ratio.
Slope of the CML: Slope = (E(Rm) − Rf) ÷ σm; Extra return per unit of total risk. It is the Sharpe ratio of the market portfolio.
Portfolio return with weight w in the market: E(Rp) = w × E(Rm) + (1 − w) × Rf; w > 1 means borrowing at Rf. w < 1 means lending.
Portfolio risk with the risk-free asset: σp = w × σm; The risk-free asset has zero standard deviation and zero covariance with the market, so risk scales with w.
SML (for contrast): E(Ri) = Rf + βi × (E(Rm) − Rf); Uses beta, not σ. Applies to any asset, efficient or not.
Total risk decomposition: Total risk = Systematic risk + Nonsystematic risk; Systematic risk cannot be diversified away. Nonsystematic risk can.
Market model: Ri = αi + βi × Rm + ei; ei is the firm-specific return with zero expected value and zero correlation with Rm.
Variance decomposition: σi² = βi² × σm² + σe²; Systematic variance is βi² × σm². Nonsystematic variance is σe². Variances add, standard deviations do not.
Proportion of systematic risk: R² = βi² × σm² ÷ σi² = ρ(i,m)²; R² is the share of total variance that is systematic.
Beta: βi = Cov(Ri, Rm) ÷ σm² = ρ(i,m) × σi ÷ σm; Measures sensitivity to market moves.
Equal-weighted portfolio variance: σp² = (1 ÷ n) × avg variance + ((n − 1) ÷ n) × avg covariance; As n grows, the first term shrinks toward zero and the average covariance remains.
Single-index (market) model: Ri = αi + βi·RM + ε; One factor. ε has expected value zero and is specific to the asset.
Beta from covariance: β = Cov(Ri, RM) ÷ Var(RM); Use the same type of variance and covariance (both sample or both population). Units cancel in the ratio.
Beta from correlation: β = ρ(i, M) × σi ÷ σM; Useful when you are given correlation and standard deviations.
Alpha (intercept): α = mean(Ri) − β × mean(RM); The regression line passes through the sample means.
R-squared: R² = ρ² = β²·σM² ÷ σi²; Share of asset variance explained by the market. Valid for a one-variable regression.
Systematic and nonsystematic variance: σi² = β²·σM² + σε²; Systematic part is β²·σM². Nonsystematic part is σε² = (1 − R²)·σi².
Adjusted beta (Blume-style): Adjusted β = (2/3) × raw β + (1/3) × 1.0; Pulls beta toward 1. Use only if the question gives these weights or asks for adjustment.
CAPM required return: E(Ri) = Rf + βi × [E(Rm) − Rf]; [E(Rm) − Rf] is the market risk premium. Use decimals or percentages consistently.
Beta: βi = Cov(Ri, Rm) ÷ Var(Rm) = ρ(i,m) × σi ÷ σm; Use whichever inputs the question gives. Remember Cov(Rm, Rm) = Var(Rm), so the market beta is 1.
Portfolio beta: βp = Σ wi × βi; Weights are market-value weights and sum to 1. Beta is a weighted average.
Alpha (Jensen's): α = expected (or actual) return − CAPM required return; Positive alpha means the security plots above the SML (undervalued). Negative means below (overvalued).
SML: Intercept = Rf; slope = E(Rm) − Rf; x-axis is beta. The SML is a line for all securities, not only efficient portfolios.
Utility of a portfolio (mean-variance): U = E(R) − 0.5 × A × σ²; A is the risk-aversion coefficient. Higher A means more risk-averse. Use σ² as a decimal, e.g. 20% gives 0.04.
Capital allocation line: E(Rp) = Rf + [(E(Rm) − Rf) ÷ σm] × σp; Slope is the Sharpe ratio of the risky portfolio. On the CML the risky portfolio is the market portfolio.
Two-asset mix with risk-free asset: E(Rp) = w × E(Rrisky) + (1 − w) × Rf; σp = w × σrisky; The risk-free asset has zero standard deviation, so its covariance with any risky asset is zero (correlation is not defined). w above 1 means borrowing.
Optimal weight in risky portfolio: w* = [E(Rrisky) − Rf] ÷ (A × σ²risky); Maximizes U on the CAL. Higher A gives smaller w*.
Tangency rule: Optimal point: indifference curve slope = frontier (or CAL) slope; Tangent to the highest feasible indifference curve.

Quick revision

CAL: E(Rp) = Rf + [(E(Ri) − Rf) ÷ σi] × σp, where Ri is the risky portfolio. Equivalently, E(Rp) = Rf + w × [E(Ri) − Rf], where w is the weight in the risky portfolio.
The CML uses the market portfolio and plots return against total risk (standard deviation).
CML slope = [E(Rm) − Rf] ÷ σm.
Total risk = systematic risk + nonsystematic risk.
Diversification removes nonsystematic risk; systematic risk remains and is priced.
Beta = Cov(Ri, Rm) ÷ Var(Rm).
CAPM: E(Ri) = Rf + βi × [E(Rm) − Rf].
The SML plots required return against beta; the market portfolio has a beta of 1.
A security above the SML is undervalued; below it is overvalued.
Sharpe ratio = (Rp − Rf) ÷ σp; Treynor ratio = (Rp − Rf) ÷ βp.
Jensen's alpha = Rp − [Rf + βp × (Rm − Rf)].
M² = Rp* − Rm, where Rp* = Rf + (Sharpe ratio of the portfolio × σm). Equivalently, M² = (Sharpe_p − Sharpe_m) × σm. It is the excess return over the market at equal risk.

Common mistakes

Using σc = w × σp with w as the risk-free weight. Fix: Always convert to the risky weight first: w = 1 − wf. Standard deviation scales with the risky weight only.
Calculating portfolio standard deviation as a weighted average including the risk-free asset's term or with a correlation. Fix: The risk-free asset has zero variance and zero covariance, so the formula collapses to σc = w × σp.
Using beta on the CML x-axis Fix: CML: σ on the x-axis, efficient portfolios only. SML: β on the x-axis, any asset.
Thinking riskier investors hold a different risky portfolio Fix: Everyone holds the same risky portfolio. Only the mix between it and Rf changes.
Saying a large portfolio has zero risk. Fix: Diversification removes only nonsystematic risk. Systematic risk remains.
Adding standard deviations instead of variances. Fix: Add variances, then take the square root of the total.
Dividing covariance by the asset's variance instead of the market's variance. Fix: Beta measures sensitivity to the market. The market is the independent variable, so Var(RM) goes in the denominator.
Using beta as a measure of total risk. Fix: Beta captures only systematic risk. Total risk needs σi, which includes nonsystematic variance.
Multiplying beta by the market return instead of the market risk premium Fix: Always subtract Rf first. Beta multiplies [E(Rm) − Rf].
Calling a stock with a high expected return undervalued without comparing to its required return Fix: Compare with the SML return for that beta. A high-beta stock needs a high return.

Exam tips

Numerical options run from smallest to largest. Compute w first, then return and risk, and match the value asked for.
Use the sense-check: lending keeps both measures below the risky portfolio's, borrowing pushes both above. This can eliminate two options in seconds.
Watch the wording: 'borrows 40% of his own capital' means w = 1.4. 'Invests 40% in the risk-free asset' means w = 0.6.
Conceptual items often ask what changes along a CAL. Answer: return and risk change, the Sharpe ratio does not.
If asked which CAL is better, pick the higher slope. No calculator keystrokes are needed beyond basic arithmetic.
If the question mentions total risk or standard deviation with efficient portfolios, use the CML. If it mentions beta, use the SML.
Remember the separation theorem as: one risky portfolio for all, the amount depends on risk aversion.
Weights above 1 signal borrowing. Use that to rule out options fast.