I am trying to get my head around the impact of different dependence structures (copulas) on the risk (quantiles) of a sum of dependent random variables (with arbitrary marginals).
In a multivariate normally distributed setting (Gauss copula + Gauss marginals) everything works (for me) intuitively. Increasing the correlation between the variables under joint normality for example also increases the 99% quantile of the sum of the random variables.
However, things become quite unintuitive (for me at least) when deviating from normality (either with the dependence structure or the marginals). For exmaple, if I use a Gauss-Copula on Pareto marginals increasing the dependence results in lower 99% quantiles than for the case with a zero correlation parameter. I am aware that the Gauss Copula is unable to capture tail-dependence but I still find it hard why the uncorrelated case produces higher quantiles than the correlated one. For less heavy-tailed marginals (such as a student-t) the effect is more similar to the Gaussian marginal case.
Any hint on why an increase in dependence of the Gauss copula results in an decrease of (upper-tail) quantiles would be appreciated (and/or good resources with regard to the impact of dependence structures on diversification / risk quantiles).
Thanks / Best
Best Answer
Say you have two random variables $X$ and $Y$ and their sum $S=X+Y$. Then the variance of $S$ is $$Var[S] = Var[X] + Var[Y] + 2Cov(X,Y) = Var[X] + Var[Y] + 2\rho \sqrt{Var[X]}\sqrt{Var[Y]}$$ where $\rho$ is the correlation between $X$ and $Y$. From this you can conclude that increased correlation will lead to increased variance. Put simply, you have discovered that increased variance will not always lead to increased (tail) quantiles.
Actually, this is only incidentally related to $S$ being a sum or the structure of dependence between $X$ and $Y$. The pertinent fact is that only for a few distributions of $S$ there is a tight connection between Variance and tail behaviour. A well known class with this property are the elliptical distributions, normal and student-t among them. For those Variance determines quantiles. See proposition 6.1.3 in Embrechts, Frey, Mc Neil, "Quantitative Risk Management" which is the go-to source for these kind of questions.
In general the connection between Variance and tail quantiles is much looser and you have freedom to wiggle both around. I did explain in some detail the connection between Expected Shortfall and Variance in this answer.
For exactly this reason Variance is quite useless or even misleading as a risk measure for non-elliptical distributions. Furthermore, correlation is of very limited use to express dependence in theses cases.