There are different ways to measure portfolio risk, and one such way is with copulas. Join us to discover why using copulas is a great alternative, and let’s put it into practice.

Only one of the two sets of points below follows a bivariate normal distribution, and yet both come from pairs of normal marginal distributions, such that the linear correlation coefficient between pairs is the same in both examples: 0.9.

Both sets present differences that show that they do not follow the same multivariate distribution; it’s sufficient to see the differences in the extreme values.

With this simple glance we can draw a conclusion: even if the risk factors are individually normal or assumed under another known distribution, the portfolio risk may be greater than we believe it to be. It’s therefore necessary to adequately reflect the dependence between individual factors so as not to incur large losses.

The graph on the left corresponds to a normal multivariate distribution, and the one on the right has been generated from what is known as a copula (in particular, the more well-known Clayton’s copula), a concept that we present in the post.

We are proposing copulas as an alternative measure to calculate portfolio risk for several reasons:

• The concept models the dependence of variables from the ‘perfect world’, that is, the Gaussian world. As we have seen, we cannot assume that a multivariate distribution is normal, although its marginal ones are. A quick look at surrounding literature in this field will show that this assumption is erroneously extended. On the other hand, assuming individual normality per asset is not a correct assumption either. With copulas, it’s not necessary to assume a specific distribution for yields; neither marginal (for assets) nor multivariate (for portfolios).
• We were able to model the dependence of variables outside the strictly linear. It’s important to remember that the correlation coefficient reflects the linear dependence of two variables, and is a measure that falls short to describe the relationship between variables. In particular, it’s very affected by atypical data, something so important in the measurement of financial risk. The correlation identification has been wrongly extended to any type of dependency or causality. For there to be correlation, the variance of the random variables must exist and therefore, this measure excludes variables with thick tails.

Now, what’s a copula?

A copula is a multivariate distribution whose marginal distributions are uniform in the interval (0,1). Thanks to the Sklar’s Theorem, it’s known that for any multivariate distribution function F, there always exists a copula C, such that:

F(x₁,x₂, … , x_n) = C(U₁(x₁),U₂(x₂), … , U_n(x_n) )

…where the U variables indicate uniform distributions.

Where’s the advantage?

Any random variable can be mapped into a uniform variable through its cumulative distribution. We can therefore see that a copula is a useful tool for simulating multivariate random variables with given marginal distributions, and not just the classic, known distributions.

It’s only necessary to simulate uniform random variables (which come from the projection of the individual’s marginal distributions) with dependencies determined by the chosen copulas. Notably, using the copulas of Gumbel, Frank and Clayton to model the extreme value tails.

Example

We will estimate the weekly VaR of 2 shares at the 1% confidence level using the Gumbel, Frank and Clayton copulas.

We can say that the estimated VaR of two stocks is the pair (a, b) if:

F(a,b) = C(F₁(a), F₂(b)) = 0.99

The idea is to estimate two values for two actions while somehow picking up the dependency between them. Simply:

1. Take the yields of the actions in a window, map them into a uniform distribution and with them calculate the necessary parameters for each of the known copulas (actually, the only parameter used in the 3 copulas in this post is the well-known Kendall Rank).
2. The estimated values are only probabilities, so we have to look for the pair that have returned a probability of approximately 0.01. (A previous analysis has shown that if we take a window of 52 weeks, we find that value).
3. Vector (“denormalised”) is the estimated VaR for the two shares at that time.

In particular, we will validate this prediction by evaluating weekly profitability from the time of estimation. If the performance for both shares does not exceed the estimated VaR, the prediction is met. In turn, we have also calculated the individual VaR of the two shares, taking the 1st percentile of yields in each window. The latter method is successful for two shares if, in the same estimation, neither of the two VaR estimates for the two assets is exceeded in the following week.

Two shares have been chosen from very different markets with the idea of setting an uncorrelated scenario; one from Japanese market, and the other from the US.

Estimates have been made weekly and the percentage of times that both shares have predicted a pair of loss levels that have not been exceeded, have been recorded for all methods:

Therefore, the use of copulas has not improved the prediction that, at the individual level, takes the historical percentile individually per share.

The possibility to go deeper into this subject is left open. Rather than lean directly on certain copulas, I suggest analysing copula that can be adjusted more to a pair of series. That is, the study of a priori in which copula can best model the dependence of variables and then, once the choice of copula is made, measure its predictive capacity.

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