Gaussian vs t-Copula: Capturing Joint Crashes

The claim we test: a correlation number tells you how two assets co-move on average — it says nothing about whether they crash together. The Gaussian copula assumes joint extremes are vanishingly rare (zero tail dependence); the t-copula assumes crashes cluster. Using two sector ETFs — Financials (XLF) and Technology (XLK) — through the GFC, COVID and 2022, we measure which model matches how markets actually behave on the worst days.

python

import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
import seaborn as sns
from scipy import stats

np.random.seed(42)
plt.rcParams["figure.dpi"] = 110
sns.set_style("whitegrid")

1.The disagreement

A copula separates how assets co-move from how each behaves alone. Two copulas can share the same correlation $ρ$ yet disagree completely about the tails:

Gaussian copula: as moves get more extreme, the dependence fades. Joint crashes are essentially coincidences. Lower-tail dependence $λ$ = 0, always.
t-copula: dependence persists into the extremes — crashes arrive together. $λ > 0$ , controlled by the degrees of freedom $ν$ (smaller $ν$ = fatter joint tails).

This isn't academic. The Gaussian copula priced the CDOs of 2008 — and its zero-tail-dependence assumption is precisely what failed when housing defaults arrived together.

2.The contenders

	Gaussian copula	t-copula
Parameters	$ρ$	$ρ, ν$ (degrees of freedom)
Lower-tail dependence $λ$	0 — always	$2 t_{ν + 1} (- \frac{( ν + 1 ) ( 1 - ρ )}{1 + ρ}) > 0$
Joint crashes	Coincidences	Expected
As $ν \to \infty$	—	Converges to Gaussian

The t-copula contains the Gaussian as a limit — so if the data prefers small $ν$ , that's the data voting for tail dependence.

3.The test

3.1Data: XLF vs XLK through three crises

python

TICKERS = ["XLF", "XLK"]
START, END = "2007-01-01", "2024-12-31"

def load_prices(tickers, start, end):
    """Adjusted-close prices: yfinance first, Stooq as fallback."""
    try:
        import yfinance as yf
        df = yf.download(tickers, start=start, end=end, auto_adjust=True, progress=False)["Close"]
        if not df.empty:
            return df[tickers].dropna()
    except Exception as exc:
        print(f"yfinance failed ({exc}); trying Stooq…")
    cols = {}
    for t in tickers:
        url = f"https://stooq.com/q/d/l/?s={t.lower()}.us&i=d"
        cols[t] = pd.read_csv(url, parse_dates=["Date"], index_col="Date")["Close"].rename(t)
    return pd.concat(cols, axis=1).loc[start:end].dropna()

px = load_prices(TICKERS, START, END)
rets = px.pct_change().dropna()
n = len(rets)
print(f"{n} trading days, {px.index[0].date()} → {px.index[-1].date()}")
print(f"Linear correlation: {rets['XLF'].corr(rets['XLK']):.2f}")

3.2Strip the marginals: pseudo-observations

Copulas work on ranks. Converting each series to its empirical percentile removes the marginal distributions and leaves only the dependence structure.

python

u = rets.rank() / (n + 1)          # pseudo-observations in (0,1)

fig, axes = plt.subplots(1, 2, figsize=(11, 4.6))
axes[0].scatter(rets["XLF"], rets["XLK"], s=4, alpha=0.4, color="steelblue")
axes[0].set_title("Raw daily returns"); axes[0].set_xlabel("XLF"); axes[0].set_ylabel("XLK")
axes[1].scatter(u["XLF"], u["XLK"], s=4, alpha=0.4, color="darkorange")
axes[1].set_title("Pseudo-observations (the copula's view)")
axes[1].set_xlabel("XLF percentile"); axes[1].set_ylabel("XLK percentile")
plt.tight_layout(); plt.show()

print("Note the crowding in the corners of the right plot — especially bottom-left (joint crashes).")

3.3Fit both copulas

Gaussian: correlation of normal scores. t-copula: same $ρ$ (via Kendall's $τ$ ), with $ν$ chosen by maximum likelihood.

python

# --- Gaussian copula fit ---
z = stats.norm.ppf(u.values)
rho_g = np.corrcoef(z.T)[0, 1]

# --- t-copula fit: rho from Kendall's tau, nu by profile likelihood ---
tau = stats.kendalltau(u["XLF"], u["XLK"]).statistic
rho_t = np.sin(np.pi * tau / 2)

def t_copula_loglik(nu, u, rho):
    x = stats.t.ppf(u, df=nu)
    cov = np.array([[1, rho], [rho, 1]])
    num = stats.multivariate_t(loc=[0, 0], shape=cov, df=nu).logpdf(x)
    den = stats.t.logpdf(x, df=nu).sum(axis=1)
    return np.sum(num - den)

nus = np.arange(2, 31)
lls = [t_copula_loglik(v, u.values, rho_t) for v in nus]
nu_hat = nus[int(np.argmax(lls))]

fig, ax = plt.subplots(figsize=(9, 3.8))
ax.plot(nus, lls, "o-", color="steelblue")
ax.axvline(nu_hat, color="crimson", ls="--", label=f"MLE: ν = {nu_hat}")
ax.set_xlabel("Degrees of freedom ν"); ax.set_ylabel("Copula log-likelihood")
ax.set_title("The data votes: profile likelihood over ν"); ax.legend()
plt.tight_layout(); plt.show()

print(f"Gaussian copula: rho = {rho_g:.3f}")
print(f"t-copula:        rho = {rho_t:.3f}, nu = {nu_hat}")
print("\nSmall ν = strong tail dependence. ν above ~30 would mean 'basically Gaussian'.")

4.The scoreboard

4.1Tail dependence: the headline number

Probability that one ETF is in its worst q% given the other is — as q → 0.

python

# closed-form lower-tail dependence of the t-copula
lam_t = 2 * stats.t.cdf(-np.sqrt((nu_hat + 1) * (1 - rho_t) / (1 + rho_t)), df=nu_hat + 1)
print(f"Lower-tail dependence λ — Gaussian: 0.000 (by construction)")
print(f"Lower-tail dependence λ — t-copula: {lam_t:.3f}")

4.2Empirical test: joint crash frequency

Count the days both ETFs landed in their worst 5% (and 1%) simultaneously — then ask each fitted copula how many such days it predicts over the same sample size.

python

def simulate_gaussian(rho, size):
    cov = np.array([[1, rho], [rho, 1]])
    z = np.random.multivariate_normal([0, 0], cov, size)
    return stats.norm.cdf(z)

def simulate_t(rho, nu, size):
    cov = np.array([[1, rho], [rho, 1]])
    g = np.random.multivariate_normal([0, 0], cov, size)
    chi = np.random.chisquare(nu, size) / nu
    x = g / np.sqrt(chi)[:, None]
    return stats.t.cdf(x, df=nu)

N_SIM = 2_000_000
u_g = simulate_gaussian(rho_g, N_SIM)
u_t = simulate_t(rho_t, nu_hat, N_SIM)

print(f"{'q':>4} {'empirical':>10} {'Gaussian':>10} {'t-copula':>10}   (joint-crash days per sample of {n})")
for q in [0.05, 0.01]:
    emp = float(((u['XLF'] < q) & (u['XLK'] < q)).mean())
    pg  = float(((u_g[:,0] < q) & (u_g[:,1] < q)).mean())
    pt  = float(((u_t[:,0] < q) & (u_t[:,1] < q)).mean())
    print(f"{q:>4.0%} {emp*n:>10.1f} {pg*n:>10.1f} {pt*n:>10.1f}")

python

# visual: simulated pseudo-observations vs reality, tails highlighted
fig, axes = plt.subplots(1, 3, figsize=(13.5, 4.4))
sets = [(u.values, "Real data (XLF/XLK)", "black"),
        (u_g[:len(u)], f"Gaussian (ρ={rho_g:.2f})", "steelblue"),
        (u_t[:len(u)], f"t-copula (ρ={rho_t:.2f}, ν={nu_hat})", "crimson")]
for ax, (dat, title, c) in zip(axes, sets):
    ax.scatter(dat[:, 0], dat[:, 1], s=3, alpha=0.35, color=c)
    ax.axvline(0.05, color="gray", ls=":", lw=0.8); ax.axhline(0.05, color="gray", ls=":", lw=0.8)
    joint = ((dat[:, 0] < 0.05) & (dat[:, 1] < 0.05)).sum()
    ax.set_title(f"{title}\njoint 5% tail: {joint} days")
    ax.set_xlabel("XLF percentile")
axes[0].set_ylabel("XLK percentile")
plt.tight_layout(); plt.show()

5.Verdict

The t-copula wins where it counts — the corner where portfolios die. Typical result on this pair: the Gaussian copula underpredicts joint 1%-tail days by a large factor, while the t-copula lands close to the observed count. The likelihood agrees: the fitted $ν$ is small, which is the data explicitly rejecting the Gaussian limit.

Why it matters:

Risk models built on Gaussian dependence understate exactly the events that cause maximum damage — simultaneous crashes across holdings. Your 99% “diversified” VaR is too optimistic in the only scenario you bought diversification for.
The 2008 lesson, quantified — the Gaussian copula's $λ = 0$ is the mathematical form of “national housing markets can't all fall together.” They did. Any dependence model with zero tail dependence repeats that mistake in new clothing.

The honest caveats:

The t-copula is symmetric— it implies joint booms are as likely as joint crashes. Equity data usually shows asymmetric dependence (crashes cluster more than rallies). A skewed-t or Clayton copula captures that; the t is the pragmatic middle ground.
One $ν$ for all seasons — dependence itself shifts with regimes (see our 60/40 case study). Time-varying copulas exist but multiply complexity.

Where to go next: the Copulas & Tail Dependence tutorial for foundations; CVaR / Expected Shortfall for the risk measure that consumes these joint-tail probabilities; and the CDO story in our curriculum's credit module for the trillion-dollar version of this comparison.