The task is the following:

Given is $Z_1,…Z_{50}$ different hypothetical assets.

Each $Z_k sim t_3$ with standard deviation $sigma=0.01$ and $tau(Z_i,Z_k)=0.4$ for $jneq k$.

I want to simulate from the distribution $(Z_1,…,Z_{50})$ assuming that it has a Gaussian Copula with $t_3$ margins.

For what I understand, from the kendals tau $tau$ I can estimate the correlation parameter $rho=sin(pitau/2)=0.5878$. From this I can determina the covariance matrix when I do want to do the simulation. Here is the code:

`require(mvtnorm) N <- 1000 tau <- 0.4 correlation <- sin(pi*tau/2) std <- 0.01 S <- matrix((std^2)*correlation,50,50 for (i in 1:50) { S[i,i] <- std^2 } gauss <- rmvt(N, sigma = S, df = 3) U_norm <- pnorm(gauss) Z_gauss <- qt(U_norm,df=3) plot(Z_gauss[,1],Z_gauss[,2]) `

However, this method is not correct. I cannot see any plot that resembles a gaussian copula. Any thoughts, corrections?

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#### Best Answer

There's an R package called "copula" that will let you do exactly this.

The process goes:

Specify a copula

Specify the population distribution, including whatever marginals you want. From the documentation: "A user-defined distribution, for example, fancy, can be used as margin provided that

`dfancy`

,`pfancy`

, and`qfancy`

are available."Generate samples from that multivariate distribution.

For you, you would specify a Gaussian copula in step 1 and then say that you want t-distributed marginals in step 2.

`# Step 1 # my_copula <- normalCopula(0.8) # Step 2 # my_population <- mvdc(my_copula, c("t","t"),list(t=3,t=3)) # Step 3 # my_sample <- rMvdc(1000,my_population) `

**Caveat**: I don't have access to this package right now, so I can't swear that this will compile, though it gives the gist of what to do.

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