Copulas & dependence

Every portfolio decision rests on a model of how assets move together. The number almost everyone uses — the Pearson correlation — measures only linear co-movement and is blind to exactly the dependence that destroys portfolios: joint crashes. This chapter separates the two questions a joint distribution answers (how does each asset behave alone, and how are they wired together), states Sklar’s theorem that makes the separation exact, and shows why a Gaussian dependence model has zero tail dependence — the mathematical reason diversification evaporates precisely when you need it.

Where Pearson correlation fails#

Pearson’s $\rho$ is the covariance normalised to $[-1,1]$:

It measures the strength of the best linear fit between $X$ and $Y$ — nothing more. Three failure modes matter at the desk:

• Nonlinearity. If $Y=X^2$ with $X$ symmetric around zero, then $\rho =0$exactly — perfect deterministic dependence, zero measured correlation. Like Anscombe’s quartet for regression, radically different joint scatters can share one correlation number.
• Non-ellipticity. Interpreting $\rho$as “the” dependence is only justified for elliptical distributions (multivariate normal, multivariate t). Real return pairs are not elliptical: their dependence is asymmetric, stronger on the downside.
• Tail blindness. $\rho$ is dominated by the mass of ordinary days. Two assets can have modest full-sample correlation and still crash together with near certainty — the statistic barely notices, because crash days are rare in the average it computes.

Rank correlations: Kendall and Spearman#

A first repair is to measure dependence on ranks rather than values. Kendall’s tau asks: for a random pair of observations, are they concordant (both up or both down together) or discordant?

Spearman’s rho is the Pearson correlation of the ranks. Both are invariant under any strictly increasing transform of either variable: replace $X$ with $\ln X$ or $e^X$ and $\tau$ does not move, because ranks do not move. That invariance is the tell: rank correlations depend only on the dependence structure, not on the marginal distributions — they are properties of the copula, defined next.

import edgekit as ek

rets = prices.pct_change().dropna()          # DataFrame, one column per strategy

tau = ek.dependence.kendall_matrix(rets)     # rank-based dependence, robust to marginals
mst = ek.dependence.corr_mst(rets)           # minimum spanning tree on d = sqrt(2(1 - rho))
for a, b, dist in mst:
    print(f"{a} -- {b}: {dist:.3f}")         # the backbone of the correlation structure

Sklar’s theorem — marginals versus wiring#

The central result of dependence modelling says the split we have been gesturing at is exact. A copula is a joint CDF on $[0,1{]}^d$ whose marginals are uniform — a pure dependence structure with the individual distributions stripped out.

Gaussian versus t-copula#

The two workhorse copulas both come from elliptical families. The Gaussian copula is the dependence structure of a multivariate normal: transform correlated normals through the normal CDF, $U_i=\Phi (Z_i)$. The t-copula does the same with a multivariate Student-t: divide the normals by a shared $\sqrt{{\chi }_{\nu }^2/\nu }$ draw first. That shared divisor is the whole story — when the chi-square draw is small, every component is inflated at once, manufacturing joint extremes that the Gaussian cannot produce.

Tail dependence — the killer fact#

The property that separates them has a name. Lower tail dependence is the probability that one asset is in its worst $q$-fraction of outcomes given the other is:

with ${\lambda }_U$ defined symmetrically in the upper corner. For the Gaussian copula with any correlation $\rho <1$:

Zero. Not small — zero. Under Gaussian dependence, conditional on one asset having an extreme day, the probability the other one does too vanishes as you go further into the tail. The t-copula, by contrast, has strictly positive tail dependence for every $\rho >-1$: the closed form is $\lambda =2t_{\nu +1}(-\sqrt{(\nu +1)(1-\rho )/(1+\rho )})$, where $t_{\nu +1}$ is the Student-t CDF — increasing in $\rho$, decreasing in the degrees of freedom $\nu$, and positive even at $\rho =0$.

Worked scenario — two strategies, calm correlation 0.3#

Take two strategies whose daily returns show Pearson correlation $\rho =0.3$ over a calm sample. You allocate half to each, expecting crash days to partially offset. How often do both land in their worst 5% of days at the same time?

import edgekit as ek

rets = strat_returns.dropna()                          # DataFrame: two strategy return columns

# Fit both wirings to the same data (correlation from Kendall tau: rho = sin(pi/2 * tau))
gauss = ek.dependence.fit_copula(rets, family="gaussian")
tcop = ek.dependence.fit_copula(rets, family="t", df=4.0)

# Empirical tail dependence in the historical sample
td = ek.dependence.tail_dependence(rets.iloc[:, 0], rets.iloc[:, 1], q=0.05)
print(td)                                              # {"lower": ..., "upper": ...}

# Simulate 100k joint days under each wiring and compare co-crash rates
u_g = ek.dependence.sample_copula(gauss, n=100_000)    # uniforms in (0,1)
u_t = ek.dependence.sample_copula(tcop, n=100_000)
co_crash_g = ((u_g < 0.05).all(axis=1)).mean()
co_crash_t = ((u_t < 0.05).all(axis=1)).mean()
print(f"co-crash 5%: gaussian {co_crash_g:.4f}, t {co_crash_t:.4f}")

The workflow generalises: fit marginals however you like (empirical, fat-tailed), fit the copula on ranks, then stress the portfolio under t-wiring even if calm-period statistics look Gaussian. The copula samples are uniforms — push them through each strategy’s inverse marginal CDF to get joint return scenarios for the Monte-Carlo engine.

Assumptions versus reality#

Next: copulas describe how assets crash together; the marginals still need a model of how extreme a single crash can be. That is the domain of extreme value theory — the statistics of maxima and tails, where the Gaussian is not just inaccurate but structurally wrong. Extreme value theory.