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:
- 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.
- I then transform these samples via a Student-t Distribution with 3 DoF.
- 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.
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")