On the distribution of the sum of dependent standard normally distributed random variables using copulas

The distribution function of the sum $Z$ of two standard normally distributed random variables $X$ and $Y$ is computed with the concept of copulas to model the dependency between $X$ and $Y$. By using implicit copulas such as the Gauss- or t-copula as well as Archimedean Copulas such as the Clayton-, Gumbel- or Frank-copula, a wide variety of different dependencies can be covered. For each of these copulas an analytical closed form expression for the corresponding joint probability density function $f_{X,Y}$ is derived. We apply a numerical approximation algorithm in Matlab to evaluate the resulting double integral for the cumulative distribution function $F_Z$. Our results demonstrate, that there are significant differencies amongst the various copulas concerning $F_Z$. This is particularly true for the higher quantiles (e.g. $0.95, 0.99$), where deviations of more than $10\%$ have been noticed.