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I would like to get your opinion on the following topic:

I am comparing the behaviour of Gaussian and Student-t Copulas.

I employ the follwing procedure:

  1. Simulate N=100,000 samples from a Student Copula with 3 DoF, Student Copula with 100 DoF and a Gaussian Copula.The dimension of the copulas is 3.
  2. I then transform these samples via a Student-t Distribution with 3 DoF.
  3. Finally I compute the empirical quantile function of the sum of the marginals and look at the tail of the distribution between 99% and 99.9%. For people familiar with the domain of finance, this is very close to be computing the VaR.

The following graphs are obtained with a Correlation of 0.1. enter image description here

The following graphs are obtained with a Correlation of 0.9. enter image description here

From the above graph I conclude that the higher the correlation the lesser the impact of the copulas. How can this be explained mathematically?

Below the code I used (Copula package)

seed1 <- runif(1,0,100000)

n<- 100000
cor <- c(0.9, 0.9,0.9)
t.cop <- tCopula(cor, dim = 3, dispstr = "un",df = 3)
t.cop100 <- tCopula(cor, dim = 3, dispstr = "un",df = 100)
n.cop <- normalCopula(cor, dim = 3, dispstr = "un")
set.seed(seed1)
tCop <- rCopula(n, t.cop)
set.seed(seed1)
tCop100 <- rCopula(n, t.cop100)
set.seed(seed1)
nCop <- rCopula(n, n.cop)

StudentN <- qt(nCop,3)
StudentT <- qt(tCop,3)
StudentT100 <- qt(tCop100,3)

# StudentN <- qnorm(nCop)
# StudentT <- qnorm(tCop)
# StudentT100 <- qnorm(tCop100)

Seq_L <- seq(0.99,0.999,0.00001)
plot(Seq_L,quantile(rowSums(StudentT),Seq_L), type="l", col="red")
lines(Seq_L,quantile(rowSums(StudentN),Seq_L), type="l", col="blue")
lines(Seq_L,quantile(rowSums(StudentT100),Seq_L), type="l", col="green")

nam <- c("Student Copula with 3 DoF", "Gaussian Copula", "Studentr Copula with 100 DoF")
legend('topleft',  nam, 
       lty=1, col=c('red', 'blue', 'green'), bty='n', cex=.75)
title("Comparison of 3 Copulas with correlation: 0.9")
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The first graph with $\rho=0.1$ is straightforward. The t-copula presents more tail dependence than the gaussian copula. Hence, when you look at the tail, there is more probability mass in the case of a student copula. When the degree of freedom increases, you converge to the gaussian copula which explains why the 100df is close to the gaussian. In case $\rho=0.9$, the difference is very small because all these copulas converge to the comonotonic copula obtained when $\rho=1$.

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