5.4 Multivariate Random Variables

Multivariate Random Variables

Market, credit, and operational risks move together. QA–4 develops joint distributions, dependence measures, and portfolio variance—the bridge from single-name math to book-level risk.

Joint Probability Matrices

For discrete RVs (X, Y), the joint PMF p(x, y) = P(X = x, Y = y) can be written as a matrix with rows indexed by x and columns by y. Entries are nonnegative and sum to 1.

Worked joint matrix

Let X = 1 if equity book loses money today, else 0. Let Y = 1 if credit book loses money, else 0.

So P(X=1, Y=1) = 0.20. Both books lose money on 20% of days in this toy history.

Marginals and Conditionals

Marginal of X: p_X(x) = Σ_y p(x, y). From the table, P(X=1) = 0.40. Marginal of Y: P(Y=1) = 0.35.

Conditional of Y given X=x: p(y|x) = p(x, y) / p_X(x). Given equity lost money (X=1):

P(Y=1|X=1) = 0.20 / 0.40 = 0.50, P(Y=0|X=1) = 0.20 / 0.40 = 0.50.

Unconditionally P(Y=1) = 0.35, so equity losses raise the chance of credit losses from 35% to 50%—dependence is visible without computing covariance yet.

For continuous vectors, joint PDFs f(x, y) replace matrices; marginals integrate; conditionals are f(x, y)/f_X(x).

Covariance, Correlation, and Independence

Covariance: Cov(X, Y) = E[(X − μ_X)(Y − μ_Y)] = E[XY] − μ_X μ_Y.

Correlation: ρ(X, Y) = Cov(X, Y) / (σ_X σ_Y), defined when σ_X, σ_Y > 0. Always −1 ≤ ρ ≤ 1. Correlation is invariant to positive scale changes of each variable separately; Cov(aX, bY) = a b Cov(X, Y).

Independence means the joint factors: p(x, y) = p_X(x)p_Y(y) for all x, y (or f(x, y) = f_X f_Y). Independence ⇒ Cov = 0 (when variances exist), but zero covariance does not imply independence in general. Classic counterexamples exist with nonlinear dependence. Special case: if (X, Y) are bivariate normal, then uncorrelated ⇔ independent.

Worked covariance on the 0–1 matrix

E[X] = 0.40, E[Y] = 0.35, E[XY] = P(X=1, Y=1) = 0.20. Cov(X, Y) = 0.20 − 0.40·0.35 = 0.20 − 0.14 = 0.06. Var(X) = 0.40·0.60 = 0.24, Var(Y) = 0.35·0.65 = 0.2275. ρ = 0.06 / √(0.24·0.2275) ≈ 0.06 / 0.2337 ≈ 0.257.

Positive correlation matches the conditional probability increase computed earlier.

Linear Transforms of Vectors

If Z = aX + bY + c, then E[Z] = aE[X] + bE[Y] + c and

Var(Z) = a²Var(X) + b²Var(Y) + 2ab Cov(X, Y).

For a portfolio return R_p = Σᵢ wᵢ Rᵢ,

Var(R_p) = Σᵢ wᵢ² σᵢ² + Σᵢ≠ⱼ wᵢ wⱼ Cov(Rᵢ, Rⱼ) = wᵀ Σ w

in matrix form, where Σ is the covariance matrix. Diversification works when covariances are small or negative; in stress, correlations often rise and Var(R_p) jumps even if weights are unchanged.

Worked two-asset variance

σ₁ = 20%, σ₂ = 30%, ρ = 0.4, w₁ = w₂ = 0.5. Var(R_p) = (0.5)²(0.20)² + (0.5)²(0.30)² + 2(0.5)(0.5)(0.4)(0.20)(0.30) = 0.25·0.04 + 0.25·0.09 + 0.5·0.4·0.06 = 0.010 + 0.0225 + 0.012 = 0.0445. σ_p = √0.0445 ≈ 21.1%.

Equal-weight average of standalone vols would be 25%; diversification cuts vol to ~21% here. If ρ → 1, σ_p → 0.5·20% + 0.5·30% = 25%. If ρ → 0, σ_p → √(0.010 + 0.0225) ≈ 18.0%.

Conditional Expectation

The conditional expectation E[Y|X = x] is the mean of the conditional distribution of Y given X = x. As a random variable, E[Y|X] is a function of X. It is the minimum mean-squared-error predictor of Y among functions of X. Tower property: E[E[Y|X]] = E[Y].

In the binary example, E[Y|X=1] = 0.50 and E[Y|X=0] = P(Y=1|X=0) = 0.15/0.60 = 0.25. Knowing X updates the expected credit-loss indicator.

Risk uses: forecasting losses given macro factors, expected exposure given market moves, and regression as linear conditional-mean modeling (later QA readings).

i.i.d. Sums

If X₁, …, Xₙ are i.i.d. with mean μ and variance σ², the sum Sₙ = Σᵢ Xᵢ satisfies E[Sₙ] = nμ and Var(Sₙ) = nσ², so Std(Sₙ) = σ√n. The sample mean X̄ = Sₙ/n has E[X̄] = μ and Std(X̄) = σ/√n—the √n risk-reduction law for independent risks.

Independence is essential for Var(Sₙ) = nσ². With pairwise correlation ρ among every pair (equal-correlation model),

Var(Sₙ) = nσ² + n(n−1)ρ σ² = nσ²[1 + (n−1)ρ],

which grows like n²ρ σ² when ρ > 0 is fixed—systematic risk that diversification cannot kill. That formula is why “many small independent loans” and “many loans tied to one factor” produce different capital answers.

Worked i.i.d. versus correlated

One-period losses on n = 100 exposures, each with σ = $1m.

• Independent: Std(S) = 1·√100 = $10m.
• Equal ρ = 0.2: Std(S) = √{100[1 + 99·0.2]} = √{100·20.8} = √2080 ≈ $45.6m.

Correlation multiplies portfolio risk far more than adding names reduces it.

Synthesis for Exam Day

Read joint tables carefully (marginals vs joints). Do not equate ρ = 0 with independence unless normality (or another listed special case) is given. For portfolio questions, write wᵀΣw or the two-asset expansion explicitly. Use conditional probabilities and E[Y|X] when information is revealed. For aggregation, ask whether risks are i.i.d. or factor-linked before applying √n rules.