Learning Module 5
Portfolio Mathematics
Key Outcomes Summary & Practice Problems
What you must be able to do
Curriculum Year: 2026
Calculate and interpret the expected value, variance, standard deviation, covariances, and correlations of portfolio returns.
Calculate and interpret the covariance and correlation of portfolio returns using a joint probability function for returns
Define shortfall risk, calculate the safety-first ratio, and identify an optimal portfolio using Roy's safety-first criterion.
1 · Portfolio Expected Return & Variance
Weighted average of component expected returns:
E(Rp) = w₁E(R₁) + w₂E(R₂) + … + wₙE(Rₙ) = Σᵢ wᵢE(Rᵢ)
Where wᵢ = proportion of portfolio in asset i (weights sum to 1).
Example: 50% S&P 500 (13%), 25% US Bonds (6%), 25% MSCI EAFE (15%)
E(Rp) = 0.50(13) + 0.25(6) + 0.25(15) = 11.75%
Probability-weighted average of deviation cross-products:
Cov(Rᵢ,Rⱼ) = E[(Rᵢ − ERᵢ)(Rⱼ − ERⱼ)]
Alternatively: Cov(Rᵢ,Rⱼ) = ρᵢⱼ × σᵢ × σⱼ
Sign interpretation:
Positive → assets tend to move in the same direction
Zero → returns are unrelated (linear)
Negative → assets tend to move in opposite directions
Own covariance: Cov(R,R) = σ²(R) (variance is a special case)
σ²(Rp) = w₁²σ²(R₁) + w₂²σ²(R₂) + 2w₁w₂Cov(R₁,R₂)
Substituting Cov = ρσ₁σ₂:
σ²(Rp) = w₁²σ₁² + w₂²σ₂² + 2w₁w₂ρ₁₂σ₁σ₂
σ²(Rp) = w₁²σ₁² + w₂²σ₂² + w₃²σ₃²
+ 2w₁w₂Cov(R₁,R₂) + 2w₁w₃Cov(R₁,R₃) + 2w₂w₃Cov(R₂,R₃)
General n-asset formula:
σ²(Rp) = Σᵢ Σⱼ wᵢwⱼCov(Rᵢ,Rⱼ)
For n assets: n variance terms + n(n−1)/2 unique covariance terms
(5 assets → 5 variances + 10 covariances; 20 assets → 20 + 190)
Standardised covariance — dimensionless, bounded [−1, +1]:
ρ(Rᵢ,Rⱼ) = Cov(Rᵢ,Rⱼ) / [σ(Rᵢ) × σ(Rⱼ)]
ρ = +1 → perfect positive linear relationship
ρ = 0 → no linear relationship (may still have nonlinear)
ρ = −1 → perfect negative (inverse) linear relationship
Closeness to 0 indicates weakness; distance from 0 indicates strength.
ρ = −0.24 is WEAKER than ρ = 0.33 (both closer to 0 vs −0.67).
The covariance matrix organises all inputs for portfolio variance. Diagonal = variances (bold); off-diagonal = covariances. It is symmetric — upper triangle mirrors lower triangle.
S&P 500 | US LT Bonds | MSCI EAFE | |
|---|---|---|---|
S&P 500 | 400 | 45 | 189 |
US LT Bonds | 45 | 81 | 38 |
MSCI EAFE | 189 | 38 | 441 |
Units: %² (e.g. 400 = 20%² → σ = 20%). Green diagonal = variances. Tan off-diagonal = covariances.
Diversification insight: As the number of assets in a portfolio increases, covariances dominate the portfolio variance calculation (380 covariance terms vs 20 variances for a 20-asset portfolio). The key insight of Modern Portfolio Theory: negative or low covariances reduce portfolio risk without reducing expected return — the diversification benefit. As correlation between two assets approaches +1, diversification benefits disappear.
n assets → n(n−1)/2 unique covariances: 2 assets = 1 covariance; 5 assets = 10; 10 assets = 45; 20 assets = 190.
Cov(R,R) = σ²(R): a random variable's covariance with itself is its own variance. The covariance matrix diagonal contains variances.
Portfolio variance ≠ weighted average of individual variances unless all assets are perfectly correlated (ρ=1). Covariance terms always matter.
If covariances are zero (uncorrelated assets): σ²(Rp) = Σwᵢ²σᵢ², which is strictly less than the variance of any single asset. Pure diversification benefit.
2 · Forecasting Covariance via Joint Probability Function
When we can specify the joint probability function P(Rₐ, R_b) — giving the probability of each combination of returns — we can compute covariance directly from scenario analysis.
Cov(Rₐ, R_b) = Σᵢ Σⱼ P(Rₐᵢ, R_bⱼ)(Rₐᵢ − ERₐ)(R_bⱼ − ER_b)
Steps:
1. Identify all scenario combinations and their joint probabilities
2. Compute E(Rₐ) and E(R_b) from the marginal distributions
3. For each scenario: multiply (deviation of Rₐ)(deviation of R_b)(joint prob)
4. Sum all probability-weighted cross-products → Cov(Rₐ, R_b)
BankCorp / NewBank example: Three scenarios (good/average/poor banking conditions) with joint probabilities 0.20, 0.50, 0.30.
E(BankCorp) = 14%; E(NewBank) = 15%
Good: (25−14)(20−15) = 11×5 = 55; × 0.20 = +11
Average: (12−14)(16−15) = −2×1 = −2; × 0.50 = −1
Poor: (10−14)(10−15) = −4×−5 = 20; × 0.30 = +6
Cov(Rₐ, R_b) = 11 − 1 + 6 = 16
Independence condition: X and Y are independent if and only if P(X,Y) = P(X)×P(Y) for all values. Independence is stronger than uncorrelatedness (ρ=0).
Product rule for uncorrelated variables: E(XY) = E(X)×E(Y) when X and Y are uncorrelated. Useful for computing expected values of products (e.g. revenue = price × quantity).
Diagonal joint probability matrices (like BankCorp/NewBank) occur when returns are perfectly positively associated — only one outcome per scenario combination has non-zero probability.
3 · Portfolio Risk Measures: Roy's Safety-First Criterion
Shortfall risk is the probability that a portfolio return falls below a minimum acceptable level R_L over a given horizon. Roy's safety-first criterion selects the portfolio that minimises this probability — equivalently, maximises the safety-first ratio.
SFRatio = [E(Rp) − R_L] / σp
Where:
E(Rp) = expected portfolio return
R_L = threshold / minimum acceptable return
σp = portfolio standard deviation
Roy's criterion: choose the portfolio with the highest SFRatio
P(Rp < R_L) = Normal(−SFRatio) ← lower is better
Special case: when R_L = Rf (risk-free rate), SFRatio = Sharpe ratio
Implementation — two steps:
1. Calculate SFRatio for each candidate portfolio: SFRatio = [E(Rp) − R_L] / σp
2. Choose the portfolio with the highest SFRatio (= lowest probability of shortfall)
Example: Client with CAD800,000 needs CAD30,000 → R_L = 30,000/800,000 = 3.75%
(25−3.75)/27 = 0.787
P(shortfall) ≈ 21.6%
(11−3.75)/8 = 0.906
P(shortfall) ≈ 18.2%
(14−3.75)/20 = 0.513
P(shortfall) ≈ 30.4%
Shortfall risk examples: defined benefit pension assets falling below liabilities; a client needing to withdraw funds without invading principal; minimum return required to meet regulatory requirements.
SFRatio vs Sharpe ratio: when R_L = risk-free rate, SFRatio = Sharpe ratio. The Sharpe ratio therefore minimises the probability that portfolio return falls below the risk-free rate.
Higher SFRatio always preferred: a higher SFRatio means the mean return is farther above the threshold in standard-deviation units → lower probability of shortfall → lower tail risk.
Normality assumption required: the exact probability calculation P(Rp < R_L) = N(−SFRatio) assumes normally distributed returns. In practice this is an approximation.
Intuition: a portfolio with high E(R) but also high σ may have a worse SFRatio than a lower-return, lower-volatility portfolio. Allocation B above (11% return, 8% σ) beats Allocation A (25% return, 27% σ) on this criterion.