5.4 Simulations

The purpose here is to generate portfolios with given marginals (normal) and the above copulae. We focus on the Gaussian and $ t$-copula case.


5.4.1 Random sample generation

For the generation of an $ n$-variate Normal with linear correlation matrix $ R$, $ (x_1,\ldots,x_n)^\top \sim \textrm{N}(0,R)$, we apply the quantlet 11906 gennorm . To obtain realizations from a Gaussian copula we simply have to transform the marginals:

To generate random variates from the $ t$-copula $ C^t_{\nu,R}$ we recall that if the random vector $ X$ admits the stochastic representation

$\displaystyle X = \mu + \sqrt{\frac{\nu}{Z}}Y \; \; \;\textrm{ (in distribution)},$ (5.6)

with $ \mu \in \mathbb{R}^n$, $ Z \sim \chi_\nu^2$ and $ Y\sim
\textrm{N}(0,\Sigma)$, where $ Z$ and $ Y$ are independent, then $ X$ is $ t_\nu$ distributed with mean $ \mu$ and covariance matrix $ \frac{\nu}{\nu-2}\Sigma$. Here we assume as above, that $ \nu>2$. While the stochastic representation (5.6) is still valid, the interpretation of the parameters has to change for $ \nu \leq 2$. Thus, the following algorithm can be used (this is Algorithm 5.2 in Embrechts et al. (2001)): Having obtained the $ t$-copula $ C_{\nu,R}^t$, we only need to replace the $ u_i$ with $ \Phi^{-1}(u_i)$ in order to have a multivariate distribution with $ t$-copula and normal marginals.

The implementation of these algorithms in XploRe is very straightforward. Indeed, using the quantlet 11920 normal we can generate normally distributed random variables. Naturally all the distribution functions needed are also implemented, 11923 cdfn , 11926 cdft etc.


5.4.2 Portfolio results

We simulate standard portfolios of size $ 500$ with all obligors belonging to one rating class. We use three rating classes, named A,B,C with default probabilities $ 0.005, 0.05, 0.15$ roughly corresponding to default probabilities from standard rating classes, Ong (1999), p. 77.

For our first simulation exercise we assume that the underlying variables $ S_j$ are normally distributed within a single factor framework, i.e. $ p=1$ in (5.1). The factor loadings $ a_{j1}$ in (5.1) are constant and chosen so that the correlation for the underlying latent variables $ S_j$ is $ \rho=0.2$, which is a standard baseline value for credit portfolio simulations, Kiesel et al. (1999). To generate different degrees of tail correlation, we link the individual assets together using a Gaussian, a $ t_{10}$ and a $ t_4$-copula as implemented in 12168 VaRcredN and 12171 VaRcredTcop .


out = 12186 VaRcredN (d, p, rho, opt)
simulates the default distribution for a portfolio of d homogeneous obligors assuming a Gaussian copula.
out = 12189 VaRcredTcop (d, p, rho, df, opt)
simulates the default distribution for a portfolio of d homogeneous obligors assuming a $ t$-copula with df degrees of freedom.

The default driver $ S_j$ are normal for all obligors $ j$ in both quantlets. p denotes the default probability $ \pi_j$ of an individual obligor and rho is the asset correlation $ \rho $. opt is an optional 12192 list parameter consisting of opt.alpha, the significance level for VaR estimation and opt.nsimu, the number of simulations. Both quantlets return a 12195 list containing the mean, the variance and the opt.alpha-quantile of the portfolio default distribution.


Table: Effect of different copulae 12198 XFGSCP02.xpl
    VaR
Portfolio Copula $ \alpha=0.95$ $ \alpha=0.99$
A Normal 10 22
  $ t_{10}$ 14 49
  $ t_4$ 10 71
B Normal 77 119
  $ t_{10}$ 95 178
  $ t_4$ 121 219
C Normal 182 240
  $ t_{10}$ 198 268
  $ t_4$ 223 306


The most striking observation from Table 5.3 is the effect tail-dependence has on the high quantiles of highly-rated portfolios: the $ 99\%$-quantile for the $ t_4$-copula is more than $ 3$-times larger than the corresponding quantile for the Gaussian copula. The same effect can be observed for lower rated portfolios although not quite with a similar magnitude.

To assess the effects of increased correlation within parts of the portfolio, we change the factor loading within parts of our portfolio. We assume a second factor, i.e. $ p=2$ in (5.1), for a sub-portfolio of $ 100$ obligors increasing the correlation of the latent variables $ S_j$ within the sub-portfolio to $ 0.5$. In the simulation below, the quantlets 12201 VaRcredN2 and 12204 VaRcredTcop2 are used.


out = 12217 VaRcredN2 (d1, d2, p, rho1, rho2, opt)
simulates the default distribution for a portfolio consisting of two homogeneous subportfolios using a Gaussian copula.
out = 12220 VaRcredTcop2 (d1, d2, p, rho1, rho2, df, opt)
simulates the default distribution for a portfolio consisting of two homogeneous subportfolios using a $ t$-copula with df degrees of freedom.

The number of obligors in the first (second) subportfolio is d1 (d2). rho1 (rho2) is the asset correlation generated by the first (second) factor. The other parameters correspond to the parameters in 12223 VaRcredN and 12226 VaRcredTcop .

Such a correlation cluster might be generated by a sector or regional exposure for a real portfolio. Again, degrees of tail correlation are generated by using a Gaussian, a $ t_{10}$ and a $ t_4$-copula.

Table: Effect of correlation cluster 12229 XFGSCP03.xpl
    VaR
Portfolio Copula $ \alpha=0.95$ $ \alpha=0.99$
A Normal 10 61
  $ t_{10}$ 9 61
  $ t_4$ 5 60
B Normal 161 318
  $ t_{10}$ 157 344
  $ t_4$ 176 360
C Normal 338 421
  $ t_{10}$ 342 426
  $ t_4$ 350 432


As expected the results in Table 5.4 show a slight increase in the quantiles due to the increased correlation within the portfolio. However, comparing the two tables we see that the sensitivity of the portfolio loss quantiles is far higher with regard to the underlying copula - and its corresponding tail dependence - than to the correlation within the portfolio.

Our simulation results indicate that the degree of tail dependence of the underlying copula plays a major role as a credit risk characteristicum. Thus, while analysis of the driving factors for the underlying variables (obligor equity, macroeconomic variables, ..) remains an important aspect in modelling credit risky portfolio, the copula linking the underlying variables together is of crucial importance especially for portfolios of highly rated obligors.