Most real-world finance problems are multivariate. A portfolio's risk depends not just on individual asset distributions but on how they move together. Joint distributions, correlation, and copulas are how we formalise dependence — and the cleanest illustration of why the standard tools quietly broke during the global financial crisis.
Joint, marginal, conditional
For two random variables X, Y: the joint PDF is f(x, y); the marginal PDFs are obtained by integrating out the other variable; the conditional PDF is f(x | y) = f(x, y) / f_Y(y). Independence: f(x, y) = f_X(x) f_Y(y) for all x, y.
Covariance and correlation
Cov(X, Y) = E[(X - E[X])(Y - E[Y])] = E[XY] - E[X]E[Y]ρ(X, Y) = Cov(X, Y) / (σ_X σ_Y), ρ ∈ [-1, 1]
Correlation is a unit-free measure of linear association. It is bounded by ±1 (Cauchy-Schwarz). Sign and magnitude have intuitive meaning: ρ near ±1 means almost-perfect linear relationship; ρ near 0 means no linear relationship.
Three failure modes of correlation
(1) Non-linear dependence: Y = X² gives ρ ≈ 0 even though X determines Y. (2) Asymmetric tails: two assets can be uncorrelated in normal markets but jointly extreme during crashes — exactly when you care. (3) Sample variability: with 252 days, a sample correlation has standard error roughly 1/√252 ≈ 6%. Reporting 'correlation rose from 0.30 to 0.35' is usually noise.
The multivariate normal — and its limitations
X ~ N(μ, Σ) is the canonical multivariate distribution. Properties: every marginal is normal; every linear combination is normal; conditional distributions are normal; Σ is the full description of dependence.
Limitation: the MVN forces all pairwise dependence to be Gaussian, which means no tail dependence. Two normally-distributed assets, however highly correlated, become asymptotically independent in the tails. Real financial returns don't behave that way.
Copulas — separating margins from dependence
Sklar's theorem (1959): every joint distribution F(x, y) can be written F(x, y) = C(F_X(x), F_Y(y)), where C is a copula — a joint CDF on the unit square with uniform marginals. The copula carries all the dependence; the marginals are separate.
- Gaussian copula: derived from the MVN. Used for risk modelling before 2008. Symmetric tail behaviour — same dependence in upper and lower extremes.
- Student-t copula: derived from MVN with t scaling. Has symmetric tail dependence even when correlation is moderate.
- Clayton copula: lower tail dependence, upper tail independence. Models joint crashes well; not joint booms.
- Gumbel copula: upper tail dependence, lower tail independence. Models joint booms.
Li's 2000 paper and the Gaussian copula crisis
David Li's 2000 paper proposed the Gaussian copula for pricing CDOs. It worked when defaults are mildly correlated and the tail was not the binding case. In 2007-2008, when housing collapsed everywhere simultaneously, the Gaussian copula's zero tail dependence catastrophically underpriced default correlation. The model didn't lie; the people using it didn't understand its tail assumption.
Kendall's tau and Spearman's rho
Rank-based dependence measures. Robust to monotonic transformations of the data (taking log returns vs simple returns doesn't change them). For elliptical distributions including MVN, related to Pearson ρ by: τ = (2/π) arcsin(ρ) and ρ_S ≈ (6/π) arcsin(ρ/2). For non-elliptical data, they capture dependence Pearson ρ misses.
Estimating Σ in practice
Sample covariance is unbiased but high-variance when T is not much larger than n. Production fixes: Ledoit-Wolf shrinkage (covered in Linear Algebra Module 12), factor-model covariance (Σ = BΩBᵀ + D for factor loadings B, factor covariance Ω, idio variance D), exponentially-weighted covariance (EWMA — RiskMetrics standard).
Exercise
Two stocks A and B have daily volatilities 2% each and correlation +0.5. (1) Compute the volatility of an equal-weighted portfolio. (2) Now suppose the correlation jumps to +0.9 during a crisis. By how much does portfolio volatility increase? (3) Comment on the diversification illusion this exposes.