CFA Level I · CFA Level I Exam

Portfolio Risk and Return: Part I: formula sheet

Full chapter guide

Key formulas

Holding period return: HPR = (P₁ − P₀ + D) ÷ P₀ = (P₁ + D) ÷ P₀ − 1; D is income received during the period. Reinvested income needs care.
Multi-period HPR: HPR = (1 + R₁)(1 + R₂)…(1 + Rₙ) − 1; This is also the time-weighted cumulative return when R values are sub-period returns.
Arithmetic mean return: R̄ = (R₁ + R₂ + … + Rₙ) ÷ n; Best single-period forecast. Overstates compound growth when returns vary.
Geometric mean return: R_G = [(1 + R₁)(1 + R₂)…(1 + Rₙ)]^(1/n) − 1; Use for compound growth over past periods. Always ≤ arithmetic mean.
Annualizing a return: Annual return = (1 + HPR)^(365 ÷ days) − 1; For a period shorter than a year, this compounds up. For longer periods, the exponent is below 1.
Money-weighted return: Σ CFₜ ÷ (1 + IRR)ᵗ = 0; Treat deposits as outflows and ending value as an inflow. Solve with the calculator.
Real return (exact): (1 + real) = (1 + nominal) ÷ (1 + inflation); Approximation: real ≈ nominal − inflation. Use exact when options differ closely.
Net return: Net return ≈ gross return − fees and expenses; Fees are usually a percentage of assets. Net is what the investor keeps.
Utility of a portfolio (mean-variance): U = E(r) − 0.5 × A × σ²; E(r) and σ² in decimals. A is the risk-aversion coefficient. A > 0 risk averse, A = 0 risk neutral, A < 0 risk seeking.
Utility of the risk-free asset: U = Rf; Variance is zero, so utility equals the risk-free rate.
Variance and standard deviation: σ = √σ²; Standard deviation is in the same units as return, so it is easier to compare.
Risk preference rule: Risk averse: needs higher E(r) for higher σ; risk neutral: ranks by E(r) only; risk seeking: may accept lower E(r) for higher σ; Compare assets at equal expected return first.
Sample variance: s² = Σ(Rᵢ − R̄)² ÷ (n − 1); Use n − 1 when the data are a sample. For a whole population use σ² = Σ(Rᵢ − μ)² ÷ N.
Standard deviation: s = √s²; Same units as returns. Never average variances and call it a standard deviation.
Variance with probabilities: σ² = Σ P(i) × [Rᵢ − E(R)]²; Use when you are given scenarios with probabilities. E(R) = Σ P(i) × Rᵢ.
Sample covariance: Cov(A,B) = Σ(RA,ᵢ − R̄A)(RB,ᵢ − R̄B) ÷ (n − 1); With probabilities, use Σ P(i)(RA,ᵢ − E(RA))(RB,ᵢ − E(RB)). Cov(A,A) equals the variance of A.
Correlation: ρ(A,B) = Cov(A,B) ÷ (σA × σB); Rearranged: Cov(A,B) = ρ × σA × σB. Always between -1 and +1.
Two-asset portfolio variance: σp² = wA²σA² + wB²σB² + 2wAwB ρ σA σB; The last term equals 2wAwB Cov(A,B). Portfolio standard deviation is the square root of this.
Portfolio expected return: E(Rp) = Σ wᵢ × E(Rᵢ); Weights sum to 1. Use decimals or percentages consistently.
Covariance from correlation: Cov(A,B) = ρ(A,B) × σA × σB; Rearrange to get ρ = Cov ÷ (σA × σB). Covariance of an asset with itself is its variance.
Two-asset portfolio variance: σp² = w1²σ1² + w2²σ2² + 2 w1 w2 ρ12 σ1 σ2; Take the square root for standard deviation. Do not square-root before adding.
General portfolio variance: σp² = Σᵢ Σⱼ wᵢ wⱼ Cov(i,j); Sum over every cell of the covariance matrix. Off-diagonal pairs appear twice, so each pair carries a factor of 2.
Number of distinct covariances: n(n − 1) ÷ 2; For n assets. Three assets need three covariances, four need six.
Perfect positive correlation (ρ = +1): σp = w1σ1 + w2σ2; Only in this case is portfolio standard deviation the weighted average of the standard deviations (long positions).
Perfect negative correlation (ρ = −1): σp = |w1σ1 − w2σ2|; Risk can fall to zero if w1σ1 = w2σ2.
Covariance from correlation: Cov(1,2) = ρ12 × σ1 × σ2; Use this when the question gives correlation and standard deviations, so you can plug covariance into the variance or weights formula.
Two-asset portfolio variance: σp² = w1²σ1² + w2²σ2² + 2 w1 w2 ρ12 σ1 σ2; Portfolio standard deviation is the square root. Weights are squared in the first two terms. Do not forget the final square root.
Perfect positive correlation (ρ = +1): σp = w1σ1 + w2σ2; The only case where portfolio risk equals the weighted average of the standard deviations. There is no diversification benefit.
Perfect negative correlation (ρ = −1): σp = |w1σ1 − w2σ2|; Risk can be reduced to zero by choosing w1 = σ2 ÷ (σ1 + σ2).
Two-asset minimum-variance weight: w1 = (σ2² − Cov12) ÷ (σ1² + σ2² − 2 Cov12), and w2 = 1 − w1; With ρ = 0 this simplifies to w1 = σ2² ÷ (σ1² + σ2²). The lower-risk asset gets the larger weight.
Risk decomposition: Total risk = Systematic risk + Nonsystematic risk; Diversification removes nonsystematic risk only. Investors are not rewarded for bearing risk that can be diversified away.
Efficient frontier definition: Efficient portfolios: maximum expected return for a given σ (or minimum σ for a given return); It starts at the global minimum-variance portfolio and runs upward and to the right.
Expected return of the mix: E(Rc) = Rf + w × [E(Rp) − Rf]; w is the weight in the risky portfolio. Same as w × E(Rp) + (1 − w) × Rf.
Standard deviation of the mix: σc = w × σp; Works because the risk-free asset has zero standard deviation and zero correlation with P.
Capital allocation line: E(Rc) = Rf + [(E(Rp) − Rf) ÷ σp] × σc; Straight line. Intercept is Rf, slope is the Sharpe ratio.
Sharpe ratio: Sharpe = (E(Rp) − Rf) ÷ σp; Uses total risk (standard deviation). Higher is better.
Weight for a target risk: w = σtarget ÷ σp; Use to find how much to hold in P for a chosen standard deviation.
Optimal risky portfolio: Choose the portfolio with the highest Sharpe ratio; It is the tangency point of the CAL with the efficient frontier of risky assets.
Utility of a portfolio: U = E(R) − 0.5 × A × σ²; A is the risk-aversion coefficient. Higher A means more risk averse. The investor picks the point on the CAL with the highest U.

Quick revision

HPR = (ending value − beginning value + income) ÷ beginning value.
The geometric mean is never above the arithmetic mean, and they are equal only when all returns are identical.
Variance is the average squared deviation. Standard deviation is its square root.
Covariance shows the direction of co-movement. Correlation = Cov(A,B) ÷ (σA × σB) and lies between −1 and +1.
Portfolio expected return is the weighted average of asset expected returns.
Two-asset variance = wA²σA² + wB²σB² + 2wAwB·Cov(A,B).
Portfolio risk is not a weighted average of standard deviations unless correlation is +1.
Lower correlation gives more diversification benefit, and perfect −1 can bring risk to zero at some weights.
Risk-averse investors want more return for more risk and prefer less risk at equal return.
The efficient frontier holds portfolios with the highest return for each level of risk.
The CAL joins the risk-free asset and the optimal risky portfolio, and its slope is the Sharpe ratio. If investors share the same expectations (homogeneous expectations), every investor holds the same tangency risky portfolio.
The optimal portfolio is where the investor's highest indifference curve touches the CAL.

Common mistakes

Averaging returns with the arithmetic mean when asked for compound growth. Fix: If the question asks for the average annual growth of an investment over past years, use the geometric mean.
Calculating real return as nominal minus inflation when the options are close. Fix: Use (1 + nominal) ÷ (1 + inflation) − 1 whenever the answer choices are within a few basis points.
Using percentages instead of decimals in the utility formula Fix: Convert to decimals first: 20% is 0.20, variance 0.04.
Forgetting the 0.5 in U = E(r) − 0.5Aσ² Fix: Write the full formula at the top of your working every time.
Dividing by n instead of n − 1 for sample data. Fix: Check for the words sample or population. Default to n − 1 for a list of historical returns unless told otherwise. Read Sx versus σx carefully on the calculator.
Reporting variance when the question asks for standard deviation, or the reverse. Fix: Underline the requested measure. Variance is in %² and standard deviation is in %.
Taking the weighted average of standard deviations as portfolio risk Fix: Use the variance formula with the covariance term. The weighted average is correct only when ρ = +1.
Forgetting the factor of 2 on the covariance term Fix: Always write 2 × w1 × w2 × Cov. In a matrix, either use both cells or use one cell with a factor of 2.
Averaging standard deviations to get portfolio risk Fix: Use the variance formula with the correlation term. The weighted average of σs is correct only when ρ = +1.
Saying diversification can eliminate all risk Fix: Diversification removes nonsystematic risk only. Systematic risk remains, apart from the special case of ρ = −1 between two assets.

Exam tips

Questions are three-option and standalone, so first apply quick sanity checks: geometric ≤ arithmetic, real < nominal when inflation is positive.
When flows occur mid-period, expect a time-weighted calculation that needs sub-period returns. Compute the value just before each flow.
If two options differ by a few basis points, use the exact real return formula, not the subtraction shortcut.
Learn the story: if a large inflow arrives just before strong returns, the money-weighted return exceeds the time-weighted return. If the inflow arrives just before weak returns, the money-weighted return falls below it. Time-weighted return is unaffected by the timing of flows. Conceptual questions often test this.
Budget about 90 seconds per question. Chain growth factors on the calculator in one pass rather than computing each period separately.
Utility questions are short calculations. Do them with decimals and check that the answer is plausible: utility should be below E(r) for a risk-averse investor.
When two options have the same expected return, the preference question is really a risk question. Decide using the three definitions.
For asset class statements, reject options with words like always or guaranteed.