How Gaussian and Student-t Copulas Affect Portfolio Tail Risk
Summary
The document compares Gaussian and Student-t copulas in a three-variable portfolio setup. It describes simulating copula samples, transforming each uniform marginal through a Student-t distribution, summing the resulting values, and examining empirical quantiles in the upper tail as a VaR-like measure. The question focuses on why differences among the copulas appear smaller when the specified correlation is high.
The response attributes the stronger tail outcomes at low correlation to the Student-t copula's greater tail dependence. It also notes that increasing the Student-t copula's degrees of freedom brings its behavior closer to the Gaussian case. At high correlation, all three dependence structures approach comonotonicity, so their differences diminish. This is a qualitative explanation supported by the stated copula properties; the document gives no formal derivation or numerical comparison beyond the described simulation procedure. Its conclusions depend on the chosen marginal distribution, dimension, correlation structure, and tail range.
Key ideas
- Student-t copulas can exhibit greater tail dependence than Gaussian copulas.
- A Student-t copula with more degrees of freedom approaches Gaussian copula behavior.
- At high correlation, dependence structures approach comonotonicity and differences shrink.
- The described tail-quantile comparison depends on both the copula and the chosen marginal distribution.
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Full text
# Gaussian vs Student Copula applied to finance
# Gaussian vs Student Copula applied to finance
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.
The following graphs are obtained with a Correlation of 0.1.
The following graphs are obtained with a Correlation of 0.9.
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")
```
## Answer by Wiles01 (score 2)
https://quant.stackexchange.com/a/33615
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 $\rho=1$.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.