Library/CFA (Chartered Financial Analyst)/Quantitative Methods — CFA Program Curriculum, 2026 • LEVEL I • VOLUME 1/Learning Module 5 Portfolio Mathematics

Learning Module 5 Portfolio Mathematics

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Portfolio expected return and variance (definitions and formulas)5 min

This chapter develops core portfolio mathematics used to measure expected return and risk. Portfolio expected return equals the sum of asset expected returns weighted by portfolio weights: E(Rp) = sum wi E(Ri). Portfolio variance quantifies investment risk and depends on both individual asset variances and covariances between assets: Var(Rp) = sum_i sum_j wi wj Cov(Ri,Rj). Covariance is the expected product of deviations from means, Cov(Ri,Rj) = E[(Ri - E(Ri))(Rj - E(Rj))], and can be estimated from joint probability functions by summing probability-weighted deviation cross-products. Correlation standardizes covariance by asset standard deviations: rho_ij = Cov(Ri,Rj)/(sigma_i sigma_j), bounded between -1 and +1; correlation captures the linear association and is unitless. A covariance matrix (or variance-covariance matrix) summarizes variances on the diagonal and covariances off-diagonal; with n assets there are n(n-1)/2 distinct covariances to estimate. Portfolio variance expands into variance terms and twice each unique covariance multiplied by corresponding weights: for three assets Var(Rp)=w1^2 Var1 + w2^2 Var2 + w3^2 Var3 + 2w1w2 Cov12 + 2w1w3 Cov13 + 2w2w3 Cov23. Diversification benefits arise when covariances are low or negative: as the number of assets increases with limited correlation, portfolio variance can fall substantially relative to average individual variance. The chapter illustrates computing covariance and correlation from a given numerical covariance matrix and from a joint probability distribution of returns (example: BankCorp and NewBank returns with three states).

Key Points

E(Rp) is the weighted average of asset expected returns.
Var(Rp) depends on asset variances and covariances: Var(Rp)=sum_i sum_j wi wj Cov(Ri,Rj).
Covariance measures joint deviation; correlation standardizes covariance between -1 and +1.
A covariance matrix concisely contains variances (diagonal) and covariances (off-diagonal).
There are n(n-1)/2 distinct covariances for n assets.

Covariance from joint probabilities and independence5 min

Independence implies P(X,Y)=P(X)P(Y) and uncorrelatedness implies E(XY)=E(X)E(Y) for independent variables. The chapter shows how to compute covariance directly from a joint probability table by calculating deviations from expected values, forming the cross-products, weighting by joint probabilities, and summing. Example tables with three states compute expected returns for each asset, deviations, products, and the final covariance. It reiterates that covariance can be positive, negative, or zero, and that sign indicates average same-side or opposite-side deviations from means, respectively. Forecasting covariances for many assets can be challenging because the number to estimate increases as n(n-1)/2, and small-sample noise can make estimated covariances unreliable without smoothing or modeling.

Key Points

Covariance can be computed from joint probability distributions by summing probability-weighted deviation cross-products.
Independent variables have joint probability equal to product of marginals and yield E(XY)=E(X)E(Y).
Covariance sign interprets co-movement (positive = same-side on average; negative = opposite-side).
Estimating many covariances is data intensive and can be noisy.

Correlation, diversification, and examples6 min

Correlation is obtained by dividing covariance by the product of standard deviations: rho_ij = Cov(Ri,Rj)/(sigma_i sigma_j). The chapter provides numeric examples showing how correlation can differ across asset classes even when covariances or variances differ in absolute terms. It illustrates a three-asset portfolio example (S&P 500, long-term corporate bonds, MSCI EAFE) with weights and a covariance matrix, computes expected portfolio return, portfolio variance, and standard deviation, and compares the variance if covariances were zero or negative to show diversification benefits. Another example explores a two-asset portfolio where varying weights produce a frontier of expected return versus standard deviation. The text shows how increasing correlation toward +1 reduces diversification benefits, and counts distinct covariance terms to emphasize dimensionality: n(n-1)/2 distinct covariances and n variances.

Key Points

Correlation standardizes covariance making comparisons easy.
Numerical examples demonstrate portfolio variance computation and diversification benefits.
Increasing correlation reduces diversification benefits.
Portfolio optimization depends critically on covariances, not just individual variances.

Safety-first rules and practical applications6 min

The chapter applies the normal distribution to safety-first rules focusing on shortfall risk: the probability portfolio return falls below a threshold RL. Roy's safety-first ratio is SFratio = (E(Rp) - RL)/sigma_p; under normality, choosing a portfolio that maximizes SFratio minimizes P(Rp < RL). If RL is the risk-free rate, the safety-first ratio equals the Sharpe ratio, linking mean-variance analysis and shortfall minimization. Examples calculate SFratio for multiple allocations given a client withdrawal threshold and use the normal distribution to compute the probability of shortfall (Normal(-SFratio)). The chapter also notes practical considerations: use of VaR and scenario analysis for tail risk, and that small input changes can change rankings under safety-first. Practice problems reinforce solving for expected recovery, covariances from joint distributions, portfolio variance from covariance matrices, and safety-first comparisons.

Key Points

Shortfall risk is P(Rp < RL); Roy's safety-first criterion chooses portfolio that minimizes shortfall probability.
SFratio = (E(Rp)-RL)/sigma_p; maximizing SFratio minimizes P(Rp < RL) under normality.
When RL = risk-free rate, SFratio equals the Sharpe ratio.
Safety-first applications include portfolio choice given a withdrawal or liability threshold.

Questions

Question 1

Given three assets with weights 0.4, 0.3, and 0.3, individual variances 100, 81, and 144 (in % squared) and covariances Cov12=20, Cov13=10, Cov23=30 (all in % squared), what is the portfolio variance (in % squared)?

Learning Module 5 Portfolio Mathematics

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