# Does Value-at-Risk have any mathematical equivalence to copulas?

Portfolio Value-at-Risk estimated using the copula approach often just means generating artificial data sampled from a parametric copula('s joint multivariate distribution) as a model fit over the real data, and then estimating VaR from this artificial data,

but is there any actual equivalence or connection between the copula and Value-at-Risk measure? In other words, is VaR actually equivalent or mathematically linked to copula somehow, using integrals and probabilities?

From what I know, VaR just measures the quantiles in the tails of joint distributions, whereas copula is an estimate or fit of the entire joint distribution, not just tails. Why would anyone have thought this would be a good idea in the first place knowing that these concepts operate over two distinct regions?

• VaR does not even measure the tails; it just measures a quantile in the tails -- the best of the worst $\alpha$% days. Sep 14 '20 at 3:13
• thanks, edited it Sep 14 '20 at 3:17

Your confusion stems from you confusing several aspects of VaR and copulas. Note first that Portfolio Value at Risk measures the value at risk of a portfolio. This means the total loss of your portfolio is the sum of losses from single assets, instruments, entities, lines of business ... whatever

$$S = \sum_{i=1}^n L_i.$$

Now what causes a large total loss? There are basically two causes:

• one of the $$L_i$$ is particularly large
• many of the $$L_i$$ are jointly bad.

The first point is handled by the fat tails (or not) of your marginal loss distributions, the second point relates to the tendency of losses occurring together, i.e. the joint loss distribution.

Copulas enter the picture for the second point: You might have innocuous marginal distributions, meaning if you are in trouble you loose some limited amount of money, but a very heavy joint tail dependence, meaning if one deal goes bad all others are in bad shape as well.

You can test the difference yourself: Calculate the risk of a "well diversified" CDO (= a large portfolio or Bernoulli variables) based on a Gaussian copula (which your friendly investment banker suggests) or based on a Clayton copula calibrated to produce serious tail dependency.

You'll notice the difference!

This is especially so since a Bernoulli variable will not have fat tails, so the first point will not matter and total portfolio loss rests entirely on the copula. Under these circumstances it does not even matter a lot what particular risk measure you choose, be it VaR, ES or any other.