Copulas and Sklar's theorem: modelling dependence without touching the margins

CS2 · Risk Modelling & Survival Analysis · 13 min read

The problem correlation cannot solve

Two lines of business. Each has a loss distribution you are happy with. What is the probability that both have a bad year at the same time?

The natural reflex is to reach for a correlation coefficient. It is not enough, and the reason is structural rather than technical. A correlation is a single number. The dependence between two random variables is a whole two-dimensional function. One number cannot carry it, and two pairs of variables with identical margins and identical correlation can have completely different probabilities of failing together.

That is not a mathematical curiosity. It is the difference between a capital model that holds up and one that does not, because the quantity a capital model needs is precisely the probability of simultaneous extremes — and that probability is almost entirely invisible to a correlation coefficient. Linear correlation also has two specific defects worth naming: it is not defined at all when variances are infinite, which heavy-tailed claim distributions often are, and it is not invariant under non-linear transformations, so the correlation of two losses and the correlation of their logarithms are different numbers describing the same relationship.

A copula is the object that carries the whole of the dependence, separately from the margins.

The idea: work in probabilities, not in money

Take a loss X with continuous distribution function F. The transformed variable U = F(X) is uniform on (0, 1) — this is the probability integral transform, and it is the one piece of machinery the topic needs. Whatever units X was in, U is a percentile.

Do that to both losses and you have a pair (U, V) of uniforms. The margins have been standardised away; whatever is left is the dependence, and nothing else. The joint distribution function of that pair is the copula.

In words: the probability that the first loss is below its u-th percentile and the second is below its v-th, at the same time. A function of two percentiles, returning a probability, with no reference to any currency or scale.

A copula is therefore just a joint distribution function whose margins are uniform. It satisfies C(u, 1) = u and C(1, v) = v — the margin conditions, worth checking on any candidate — equals 0 if either argument is 0, and is increasing in a two-dimensional sense.

Sklar’s theorem

This is the result the whole topic rests on, and it says two things.

Every joint distribution contains a copula. For any joint distribution function H with margins F and G, there is a copula C with

and if the margins are continuous, that copula is unique.

And the construction runs the other way. Take any copula and any two margins you like, and the formula above defines a valid joint distribution with exactly those margins.

In words: margins and dependence are separable, completely and without loss. You can model each loss on its own, choose a dependence structure independently, and bolt them together.

That second half is what makes copulas a working tool rather than a theorem. You are not restricted to joint families that happen to exist in closed form. Lognormal losses in one line, a GPD tail in another, and a dependence structure chosen because it has the tail behaviour you believe in — Sklar's theorem says the result is a legitimate joint distribution. Before copulas, that combination was simply unavailable.

The fundamental copulas, and the bounds

Three copulas are the reference points against which every other is read.

Independence: C(u, v) = uv. Nothing to see.

Comonotonic (perfect positive dependence): C(u, v) = min(u, v). The two losses are increasing functions of one another; the same percentile in both, always.

Countermonotonic (perfect negative dependence): C(u, v) = max(u + v − 1, 0). One is a decreasing function of the other.

Those last two are not just examples. They are limits — the Fréchet-Hoeffding bounds — and no copula can go outside them:

This is the cheapest check in the topic. Any copula value you compute must sit between those two numbers, and at u = v = 0.9 that means between 0.8 and 0.9. An answer outside the bounds is wrong before you look at the algebra.

Tail dependence: the property that matters

Here is the question a capital model actually asks: given that one line has had a one-in-a-hundred year, how likely is it that the other has too? Push that to the limit and you have the coefficient of upper tail dependence:

In words: far out in the tail, the chance that the second loss is also extreme given that the first one is.

A positive λU means extremes genuinely arrive together, however far out you look. Zero means that far enough into the tail, the two become effectively independent — each may be bad, but not together.

And here is the fact that makes this more than a definition. The Gaussian copula has λU = 0 for every correlation below 1. Fit a Gaussian copula with a correlation of 0.9 and you have a model which says that sufficiently extreme losses never coincide. Correlation of 0.9 looks like a prudent, conservative assumption. On the question a capital model exists to answer, it is a statement of asymptotic independence. Choosing a copula family is therefore a decision about tail behaviour, and it cannot be deferred to the fitting.

A lower tail dependence coefficient λL is defined the same way at the bottom end, and the two need not be equal — which is itself a reason a one-number summary cannot do the job.

The named families, by what their tails do

Gaussian: symmetric, no tail dependence at either end, one parameter. The default, and the default is the problem.

Student’s t: like the Gaussian but with a degrees-of-freedom parameter, and it has tail dependence at both ends, increasing as the degrees of freedom fall. Often the smallest honest change to a Gaussian model.

Archimedean copulas are built from a single decreasing generator function φ:

One function of one variable generates the whole dependence structure, which makes the family easy to work with and easy to extend to more dimensions. Three members matter here. Gumbel has upper tail dependence and none in the lower tail — the natural choice when losses spike together but good years are unrelated. Clayton is the mirror image, with lower tail dependence only. Frank has neither, but allows negative dependence, which the other two do not.

Where marks are lost

Confusing C(u, v) with a joint tail probability. C(u, v) is the probability that both are below their percentiles. The probability that both are above is 1 − u − v + C(u, v), from inclusion-exclusion. The two are different numbers and the second is the one a capital calculation wants.

Forgetting to check the Fréchet bounds. Ten seconds, catches most algebra slips.

Assuming a copula value exceeds the independence value. It does for positive dependence and it does not for negative. Compare against uv deliberately rather than assuming.

Treating the copula parameter as a correlation. Gumbel's θ is 1 at independence and increases without limit; Clayton's is 0 at independence. Neither is on a −1 to 1 scale and neither equals a correlation coefficient. Quote the dependence measure the question asks for — Kendall's tau and Spearman's rho are the ones that depend on the copula alone and not on the margins, which is exactly why they are preferred to linear correlation here.

Reading tail dependence off a correlation. They are unrelated. A Gaussian copula at correlation 0.99 still has λU = 0.

Questions students ask

If the margins are modelled separately, what does the copula actually change?

Nothing about either loss on its own. Every marginal probability, percentile and mean is untouched by the choice of copula — that is Sklar's theorem. What changes is every statement involving both at once: the distribution of the sum, the probability of two simultaneous large losses, and therefore the capital. Two models with identical margins can differ by a wide margin in aggregate capital on the strength of the copula alone.

Why are Kendall’s tau and Spearman’s rho preferred to ordinary correlation?

Because they are properties of the copula and not of the margins. Both are computed from ranks, so they survive any increasing transformation of either variable — the tau of two losses equals the tau of their logarithms. Linear correlation does not have that property and is not even defined when a variance is infinite, which for heavy-tailed claims is a live possibility rather than an edge case.

Is the Gaussian copula wrong?

It is a model, and like any model it is right about some things and wrong about others. It is wrong about joint extremes, specifically and always, and joint extremes are usually the reason an actuary reached for a copula. If what you need is a convenient dependence structure for the body of the distribution it is perfectly serviceable. If you are setting capital, say what its tail behaviour implies and choose deliberately.

Does this extend beyond two dimensions?

Sklar's theorem holds in any number of dimensions, and the Gaussian and t copulas extend naturally through a correlation matrix. Archimedean copulas extend too, but a single generator imposes a symmetric, exchangeable dependence on all the variables at once, which is restrictive in practice; hierarchical constructions exist for that reason. Two dimensions is where the ideas are learned.

A worked example

An insurer writes two lines of business. The marginal loss distribution of each has been modelled separately and need not be named: everything below works in percentiles, which is the whole point of a copula. The dependence is modelled with a Gumbel copula with θ = 2. The parameter is invented for this example.

Step one: the probability that both lines have a good year

Take u = v = 0.9: each line's loss is below its own 90th percentile.

Take the same value by the shorter route, which is exact. Along the diagonal [2(−ln u)θ]1/θ = 21/θ(−ln u), so C(u, u) = u21/θ with no square root to take:

0.8615674 against 0.8615672. The second is the right one, and the gap is instructive rather than embarrassing: the first route squares a small number, adds, then takes a square root, and a square root magnifies the rounding error in what you feed it. Where a formula offers an algebraically exact shortcut, it is also usually the numerically better one. Everything below uses 0.8615672.

Check the bounds and the reference point

Inside the bounds, and above independence — which is correct, since Gumbel with θ > 1 is a positive dependence structure. Had the answer come out below 0.81, or outside 0.8 to 0.9, there would be an error to find.

Step two: the probability that both have a bad year

This is the number the capital model wants, and it is not the copula value. By inclusion-exclusion:

Under independence the same calculation gives

which is of course 0.1 × 0.1, as it must be — a second route to the independence figure and a check that the inclusion-exclusion has been written correctly.

0.0616 against 0.0100. Both lines having a one-in-ten year together is 6.16 times more likely under this copula than under independence. The margins are identical in both calculations. Every single-line figure — every percentile, every expected loss, every reserve for one line considered alone — is unchanged. Only the joint statement moved, and it moved by a factor of six.

Step three: the upper tail dependence coefficient

For the Gumbel copula the closed form is

And the same number from its definition

Rather than trust the formula, evaluate the limit. Along the diagonal the Gumbel copula simplifies, because [2(−ln u)θ]1/θ = 21/θ(−ln u):

Now push u towards 1 in (1 − 2u + C(u, u))/(1 − u):

The numerical limit walks down to the closed form — 0.5887211, 0.586079, 0.58582, 0.5858, against 0.5857864 — so two independent routes reach the same number. Verify a row by hand: 0.9991.4142136 = exp(1.4142136 × ln 0.999) = exp(−0.0014149) = 0.998586079, and (1 − 1.998 + 0.998586079)/0.001 = 0.000586079/0.001 = 0.586079.

Notice that the last column loses a digit on every row, and that is not sloppiness in the table — it is the arithmetic. The numerator 1 − 2u + C(u, u) subtracts numbers that agree in their leading digits, so the leading digits cancel and only what was left at the bottom survives. At u = 0.99999 the numerator is 0.000005858: nine printed digits of C have become four usable ones. Push one row further and there is nothing left to divide. This is why the closed form matters. 2 − 21/θ is exact at the limit itself, where the numerical route has already run out of precision — and the same cancellation is waiting in any capital model that computes a joint tail probability by subtracting two numbers close to 1.

What 0.586 means in words: however far into the tail you go, given that one line has had an extreme year there remains roughly a 59% chance the other has too. That probability does not fade away. It is a property of the dependence structure and it survives to the very end of the tail.

Step four: what a Gaussian copula would have said

Fit a Gaussian copula to the same two lines, at any correlation short of 1, and the same limit is exactly zero. Far enough into the tail, the Gaussian model says the two lines stop failing together altogether.

At u = 0.9 the two models would look similar enough; a Gaussian copula with a high correlation would also put C(0.9, 0.9) somewhere near 0.86. The difference is invisible where the data is and decisive where the capital is. That is the single most important sentence on this page, and it is why "we used a copula" is not a statement about prudence — the family carries the assumption, and the assumption is about the tail.

And a Clayton copula, for contrast

Inside the Fréchet bounds and above independence, so Clayton at θ = 2 is also positively dependent — but less so at the top than Gumbel's 0.8616. Its upper tail dependence is zero, and its lower tail dependence is 2−1/θ = 2−0.5 = 0.7071068. It is the mirror image: a model in which good years arrive together and bad years do not.

Three copulas, one pair of margins, the same percentiles throughout. C(0.9, 0.9) comes out as 0.8100 under independence, 0.8250 under Clayton, 0.8616 under Gumbel — and the upper tail dependence as 0, 0 and 0.5858. The spread in the first row is small and the spread in the second is everything, which is the whole case for choosing a copula on its tail behaviour rather than on how well it fits the middle.

Where this fits in CS2

This page explains one idea from CS2 Risk Modelling & Survival Analysis. The full CS2 syllabus, with what each part is examined on, sits on the course page, and past-paper style mock exams with full solutions are where you find out whether the idea has stuck under time pressure.

Other CS2 topics explained

• The survival function and the hazard rate, explained
• The Kaplan-Meier estimator, and why censoring needs it
• The Cox proportional hazards model, and what a hazard ratio means
• Markov chains: transition matrices, Chapman-Kolmogorov and the stationary distribution
• Central exposed to risk, and the two mortality rates it produces
• Bühlmann credibility: how much to trust one risk's own experience
• Extreme value theory: block maxima, thresholds and the generalised Pareto distribution

Stuck on a term rather than the idea? The actuarial glossary defines the notation used above. If something here is wrong or unclear, tell us — these pages get corrected.