IV ExpertWeek 10 • Lesson 29Duration: 55 min

COP Correlation & Copulas

When markets crash, everything crashes together

Learning Objectives

•Understand how correlations change during crises
•Learn what copulas are and why they matter for portfolio risk
•Build realistic joint risk models
•See the correlation breakdown during market stress

Explain Like I'm 5

In calm markets, EURUSD and GBPUSD might be 60% correlated. During a crisis, suddenly they're 95% correlated — everything moves together. Your "diversified" portfolio isn't diversified when you need it most. Copulas model this — they capture how relationships between assets change during extremes.

Think of It This Way

In good times, your friends have different opinions. In a crisis, everyone panics the same way. Copulas model this "crisis behavior" — how relationships strengthen under stress. For portfolios: diversification disappears exactly when you need it most.

1Correlation Isn't Constant

The biggest mistake in portfolio risk: assuming correlations are stable. They're not. Correlation between assets typically: • Increases during market stress (everyone sells everything) • Is lower during calm, trending periods • Can shift dramatically in minutes during flash events Practical impact: if you're long EURUSD, GBPUSD, and AUDUSD simultaneously, you're really just long "not-USD" three times over. One dollar move hits all three. The fixes: • Cluster-based position limits (max N positions per correlated group) • Correlation monitoring (reduce exposure when correlations spike) • Regime-conditional risk that adapts to crisis conditions

2Normal vs Crisis Correlations

During normal conditions, asset classes have moderate correlations — some positive, some near zero. During crises, everything spikes toward 1.0. Your "diversified" five-asset portfolio becomes one concentrated bet. If you sized positions assuming normal correlations, you're holding 3–5× more risk than you think.

Asset Correlations: Normal vs Crisis

3What Copulas Do

Copulas separate the dependence structure between variables from their individual distributions. Each asset can have its own fat-tailed distribution (modeled by EVT). The copula captures how they move together, especially in the tails. Common copulas for finance: Gaussian copula: assumes normal dependence. Fails in tails. The 2008 crisis was partly caused by overreliance on Gaussian copula for CDO pricing. Student-t copula: allows tail dependence. Assets become more correlated during extremes. Much more realistic. Clayton copula: strong lower-tail dependence. Good for crash modeling — when one thing crashes, everything crashes together. For trading system risk, Student-t copula is the standard because it captures "in a crash, everything drops together" — which Gaussian copula misses entirely.

4The 2008 Copula Disaster

In 2000, David Li published a paper using Gaussian copula to model default correlations in CDOs. Banks loved it: mathematically clean, made risk calculations tractable, said widespread simultaneous defaults were nearly impossible. What it missed: • Tail dependence (Gaussian copula has λ = 0 — literally assumes no tail dependence) • Correlation regime shifts • Systemic risk where defaults cascade When the housing market turned, defaults happened simultaneously. The model said this was essentially impossible. Trillions of dollars in "safe" CDOs turned toxic overnight. Lesson: the wrong dependence model can be catastrophic. Always use copulas with tail dependence for financial applications. Salmon, F. (2009). "Recipe for Disaster: The Formula That Killed Wall Street." Wired.

5Portfolio Heat — Practical Monitoring

Track your portfolio "heat" — correlation-adjusted exposure — in real time. When heat spikes, you're overexposed to correlated risk. This shows portfolio heat over a month. The spikes are moments when multiple correlated positions were active simultaneously. The system should flag these and either close the weakest position or block new correlated trades. Simple metric, prevents blowups. If you have 5 correlated forex positions, your real risk is much higher than 5× individual risk.

Portfolio Heat (Correlation-Adjusted Exposure)

Key Formulas

Pearson Correlation

Linear correlation between X and Y. Ranges from -1 to +1. Standard measure but misses non-linear and tail dependencies.

Tail Dependence Coefficient

Probability that Y crashes given X crashes. Gaussian copula: λ=0. Student-t copula: λ>0. For finance, λ>0 is reality.

Hands-On Code

Normal vs Crisis Correlations

python

import numpy as np

def correlation_regimes(returns_a, returns_b, threshold_pct=10):
    """Compare correlation in normal vs crisis periods."""
    thresh_a = np.percentile(returns_a, threshold_pct)
    thresh_b = np.percentile(returns_b, threshold_pct)
    
    crisis = (returns_a < thresh_a) | (returns_b < thresh_b)
    
    corr_normal = np.corrcoef(returns_a[~crisis], returns_b[~crisis])[0,1]
    corr_crisis = np.corrcoef(returns_a[crisis], returns_b[crisis])[0,1]
    
    print(f"Normal correlation: {corr_normal:.3f}")
    print(f"Crisis correlation: {corr_crisis:.3f}")
    print(f"Increase: {corr_crisis - corr_normal:+.3f}")
    print(f"Your 'diversified' portfolio is {corr_crisis/corr_normal:.1f}x")
    print(f"more correlated during crises")

Crisis correlations are consistently higher than normal. This is empirical fact. Any risk model using fixed correlations underestimates joint tail risk.

Knowledge Check

Q1.Why did the Gaussian copula contribute to the 2008 crisis?

Assignment

Compute correlation between two currency pairs during "normal" and "crisis" periods (worst 10% of returns). Verify that crisis correlation is higher. Discuss what this means for position sizing with multiple correlated positions.

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