Joint Distributions and Independence

Single random variables live in isolation. Finance does not. A portfolio is a weighted sum of correlated returns. A credit portfolio’s loss depends on how defaults cluster. An interest rate model couples multiple maturities. To handle any of this, we need the theory of joint distributions.

This article formalizes the joint behavior of multiple random variables, defines independence precisely, introduces the covariance and correlation structures that drive portfolio theory, and builds toward the single most important multivariate distribution in finance: the multivariate normal. It also previews copulas --- the tool for modeling dependence structures that go beyond linear correlation.

The article builds on expectation-variance-and-mgfs for the moment concepts (covariance, correlation) and connects forward to conditional-expectation where we condition one variable on another.

Key Topics

• Joint PMF and PDF: $f_{X,Y}(x,y)$, the full specification of two (or more) random variables
• Marginal distributions: recovering $f_X(x)=\int f_{X,Y}(x,y)dy$ from the joint
• Independence: $X\perp Y$ iff $f_{X,Y}(x,y)=f_X(x)f_Y(y)$ for all $x,y$. Independence implies zero correlation but not vice versa
• Covariance and correlation: ${\rho }_{X,Y}=\frac{\text{Cov}(X,Y)}{{\sigma }_X{\sigma }_Y}$, $-1\le \rho \le 1$
• Correlation matrices: positive semi-definiteness, eigenvalue decomposition, what happens when your estimated correlation matrix is not PSD
• The multivariate normal distribution: $\mathbf{X}\sim N(\mathbf{\mu },\mathbf{\Sigma })$, density, marginals are normal, conditionals are normal, uncorrelated implies independent (unique among distributions)
• Copulas (introduction): separating marginal distributions from dependence structure, Sklar’s theorem, Gaussian copula and its role in the 2008 crisis (Li’s 2000 paper)

Finance Connections

• Portfolio variance: $\text{Var}(R_p)={\mathbf{w}}^{\top }\mathbf{\Sigma }\mathbf{w}$. Markowitz (1952) optimization lives entirely in this framework. Diversification works because $\rho <1$
• Correlation breakdown in crises: correlations spike toward 1 in market stress, exactly when diversification is needed most. This is a failure of the multivariate normal assumption
• Credit risk: the Vasicek model assumes latent normal variables driving defaults; the Gaussian copula model for CDO pricing uses joint normality to model default dependence. Its limitations were a contributing factor to the 2008 financial crisis
• Risk management: VaR and CVaR for portfolios require joint distribution of returns, not just marginals

Prerequisites

• expectation-variance-and-mgfs

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This article is a roadmap --- content to be developed in future sessions.