Correlation Risk and Copulas

In 2007 a portfolio of senior CDO tranches was priced under Gaussian copula correlations near 0.30. By autumn 2008 those correlations approached 1.0 across nearly every reference name. The same model, the same portfolio, gave a VaR of single-digit millions before the breakdown and triple-digit millions after. The exam tests whether you understand correlation as a number that moves under stress, not a fixed parameter.

Correlation risk shows up across every desk:

• Trading book: A long-short portfolio's VaR depends on the correlation between the legs. If correlation rises, the hedge stops working.
• Credit book: Default correlation drives CDO and basket-credit pricing. Senior tranches are short correlation; equity tranches are long.
• Counterparty risk: Wrong-way risk is correlation between counterparty default and portfolio exposure. Right-way risk is the reverse.
• Multi-asset options: Quantos, basket options, best-of/worst-of structures all have correlation embedded in their prices.

Correlation is not a stable parameter. It moves with regime, sector concentration, leverage, and liquidity conditions. Risk management treats it as a stochastic variable, not a constant.

Common mistakes

• Treating correlation as a stable parameter. Correlations move with regime. A model calibrated to a calm period systematically understates stress-period correlation and stress-period VaR. Trap: a question with calm-period historical correlation 0.30 and asks for stress VaR using the same number: the right answer overrides with a stress correlation (typically 0.7+).
• Using Gaussian copula for tail-dependent assets. Gaussian copula has zero tail dependence regardless of correlation. For equity portfolios with empirical joint extreme losses, Student-t or Clayton copulas are required. Trap: a question describes a portfolio with documented joint tail losses; the Gaussian copula is a distractor.
• Confusing correlation with dependence. Correlation captures linear dependence; copulas capture full dependence including tail behavior. Two distributions can have correlation 0 but still be highly dependent (e.g., $Y=X^2$ has zero correlation but perfect dependence). Trap: a question reports $\rho =0$ and asks if the assets are independent: not necessarily; check the copula.

Bottom line

• Financial correlation risk is the chance asset correlations move adversely; correlations rise toward 1 in stress and diversification benefits evaporate exactly when capital is most needed.
• Empirical correlation properties: equity correlations are mean-reverting (daily speed below 0.05), regime-shifting, and right-skewed; equity fits Johnson SU, bonds mixed normal, default correlations state-dependent.
• Copula functions separate marginal distributions from joint dependence; Sklar's theorem decomposes any joint distribution into marginals plus a copula.
• Tail dependence by family: Gaussian copula has zero tail dependence, Student-t is symmetric, Clayton is lower-tail, Gumbel is upper-tail.

Exam shortcut

When a question describes a portfolio with empirical joint extreme losses, the right copula is Student-t or Clayton, not Gaussian. When a question asks about post-2008 CDO model failures, the Gaussian copula's zero tail dependence is the answer, not the correlation parameter itself. Stress correlation tests should use $\rho$ values near 0.7-0.9, not historical averages.

The full lesson (about 3,239 words, 22 min read) adds 2 worked examples, all 6 common mistakes, a self-check, free in the app.

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