Finding the Distribution of a Copula-Dependent Sum
Summary
The document considers how to calculate the distribution of a sum of dependent risks when their joint dependence is specified with a copula. It asks whether Fourier methods or the mixture representation of Archimedean copulas can produce the result without Monte Carlo simulation. The response cautions that a copula does not, in general, make the sum-distribution problem easier: numerical integration may still be needed. It points to a survey chapter on calculating distributions of sums of risks and mentions a deterministic method designed for this task, alongside recursive and FFT approaches.
A second response gives a direct route when the joint density of the variables is known: transform the variables to their sum and one component, then integrate out that component to obtain the sum’s density. The document does not derive these methods, compare their accuracy or computational cost, or provide a worked example. It also reports no specific solution for Archimedean copulas, so the suggested avenues require further investigation for that case.
Key ideas
- A copula specifies dependence, but does not automatically simplify the distribution of a sum.
- When the joint density is available, a change of variables followed by integration yields the density of the sum.
- Numerical integration may be necessary in the general case.
- The discussion points to deterministic, recursive, and FFT methods as alternatives to Monte Carlo simulation.
- The document leaves the application of these methods to Archimedean copulas unresolved.
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# Copula models and the distribution of the sum of random variables without Monte Carlo
# Copula models and the distribution of the sum of random variables without Monte Carlo
There is a vast literature on copula modelling. Using copulas I can describe the joint law of two (and more) random variables $X$ and $Y$, i.e. $F_{X,Y}(x,y)$. Very often in risk management (credit risk, operational risk, insurance) the task is to model a sum $$Z=X+Y$$ and find its distribution $$F_Z(z) = F_{X+Y}(z)$$
I know several approaches that do not directly use copulas (e.g. commons shock models and mixed compound Poisson models) but how can I elegantly combine a copula model and a model for the sum (without Monte Carlo of course - otherwise it would be easy).
Is there some useful Fourier-transform approach? I had the feeling that in the case of Archimedian copulas there could be a chance (looking at this mixture representation as in e.g. in Embrechts, Frey, McNeil). Who has an idea? Are there any papers on this?
## Answer by Alexey Kalmykov (score 5, accepted)
https://quant.stackexchange.com/a/7279
In general setting this is quite a tough problem and it looks like just switching from regular multivariate probability to copulas doesn't make it easier. In general case you need to rely on numerical methods for integration.
There is a nice overview of the problem in Copula Theory and Its Applications: Proceedings of the Workshop Held in Warsaw, 25-26 September 2009, Part I, Section 5.3, "The Calculation of the Distribution of the Sum of Risks".
If you want to avoid Monte Carlo methods, you can look at a new deterministic method AEP, which was specially designed to tackle this problem. Recursive and FFT methods are compared here.
I haven't seen any work that deals with this problem specifically for Archimedean copulas.
## Answer by Julian Wergieluk (score 5)
https://quant.stackexchange.com/a/7099
If the density of $(X,Y)$ is known, then you may obtain the density of the sum $X+Y$ simply by applying the Jacobi's transformation formula, which describes the density of the transformed random variable $g(X,Y)$ for $g(x,y) = (x+y, x)$. Integrating out the $x$-component yields the density of $X+Y$. See Jacod/Protter Probability Essentials ch. 12 for details.
I am not sure, if this is an answer you are expecting. Perhaps you could provide us with some more specific description of your modelling task.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.