Using Copulas to Model Sums of Dependent Tail Risks

Summary

The document poses a problem about finding a high quantile for the sum of two dependent random variables. Each variable is modeled with an extreme value distribution, and their dependence is represented by a Gumbel copula. The question highlights a common distinction: a copula describes joint dependence, while the desired result is a quantile of the sum.

No answer or calculation method is included, so the document does not establish how to obtain the requested sum threshold. It also leaves key modeling details unresolved, such as the precise definition of the joint probability region and how uncertainty in the fitted marginal distributions affects the result. Its value is therefore mainly as a clear formulation of a dependence and aggregation problem, rather than as a usable estimation procedure.

Key ideas

• The target is a quantile of the sum of two dependent random variables.
• The example combines extreme value marginals with dependence represented by a Gumbel copula.
• A copula models dependence, but the document does not explain how to derive a quantile for the sum.
• The probability statement for a pair of values needs a precise definition before it can be calculated.

Tags

# Convolution of Dependent Random Variables with Copulas


# Convolution of Dependent Random Variables with Copulas












Lets say I have 2 different observations which are fitted to a parametric distribution. And lets say that they are dependent and can be modeled by one of the copulas. I want to calculate “a value” that is sum of “a possible value from the first distribution” and “a possible value from the second distribution” which their jointly exist with %99 probability.

Lets say I have 1000 observations that are distributed between (100 ~ 400) and fitted to an Extreme Value Distribution and another observation also EVT distributed between (200 ~ 600). Assume that their maximals are dependent and modeled by a Gumbel copula.

What I want to fetch that for example for %99 of chance, their sum will be 1050 (even their maximum observations were 400 and 600 which would yield to 1000). Since they are fitted to an EVT distribution, there is a possibility to be observed higher values than 400 and 600 in the near future.

From what I understand, Copula can give me the joint probability, but how can I tell the copula that give me a couple which their jointly happening probability would be 0.99 ? On normal parametric distributions we can do that by using Inverse CDF function, but copulas do not have inverse CDF or etc. I'm basically stuck.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.