Setting negative dependence for N random variables with copulas

Question

I would like to use normal copulas to set the dependence of N random variables. That is, I want the correlation $\rho$ to be equal between all variables. I am doing this with the copula package in R. This seems straightforward enough when $\rho$ is positive or when N = 1, however I get odd results when $\rho$ is negative and N > 1:

library(psych)
library(copula)

N = 6  #The number of random variables or dimensions 
m = ((N-1)*N)/2 #Number of correlated pairs 
correlations = rep(-.95, m) #Set all correlations to be the same value, in this case -0.95
listExp = list(rate = 2)
lists = list(listExp, listExp, listExp, listExp, listExp, listExp) #A list of exponential distributions that is N long

mv.NE = mvdc(normalCopula(correlations, dim = N, dispstr = "un"), rep("exp",N),lists) #create multivariate distribution 

x.samp = rMvdc(1000, mv.NE) #sample from multivariate distribution 
pairs.panels(x.samp) #plot results 

As you can see, the measured spearman's correlations are well below the set $\rho$ of 0.95. Does this arise from an error in this script, or is this a fundamental limitation of normal copulas? If its the latter, is there a different method I could use instead?

Answer 1

This is indeed a fundamental limitation and not only one of the normal copula. Your variables are exchangeable as identical margins of a normal copula with constant $\rho$. The correlation of $n$ exchangeable random variables each having variance $\sigma^2$ is bounded from below to be larger than $-\frac{\sigma^2}{n-1}.$ See the simple proof in this Wikipedia article