5.3 Dependence modelling

To formalize the ratings-based approach, we characterize each exposure $ j \in \{1, \ldots, n\}$ by a four-dimensional stochastic vector

$\displaystyle (S_j,k_j,l_j,\pi(j,k_j,l_j)), $

where for obligor $ j$
(1)
$ S_j$ is the driving stochastic process for defaults and rating migrations,
(2)
$ k_j,l_j$ represent the initial and end-of-period rating category,
(3)
$ \pi(.)$ represents the credit loss (end-of-period exposure value).
In this context $ S_j$ (which is, with reference to the Merton model, often interpreted as a proxy of the obligor's underlying equity) is used to obtain the end-of-period state of the obligor. If we assume $ N$ rating classes, we obtain cut-off points $ -\infty = z_{k,0},z_{k,1},z_{k,2},\ldots,z_{k,N-1}, z_{k,N} = \infty$ using the matrix of transition probabilities together with a distributional assumption on $ S_j$. Then, obligor $ j$ changes from rating $ k$ to rating $ l$ if the variable $ S_j$ falls in the range $ [z_{k,l-1},z_{kl}]$. Our default-mode framework implies two rating classes, default resp. no-default, labeled as $ 1$ resp. 0 (and thus only a single cut-off point obtained from the probability of default). Furthermore, interpreting $ \pi( \bullet )$ as the individual loss function, $ \pi(j,0,0)=0$ (no default) and according to our zero recovery assumption $ \pi(j,0,1)=1$. To illustrate the methodology we plot in Figure 5.1 two simulated drivers $ S_1$ and $ S_2$ together with the corresponding cut-off points $ z_{1,1}$ and $ z_{2,1}$.

Figure: Two simulated driver $ S_j$ and the corresponding cut-off points for default. 11497 XFGSCP01.xpl
\includegraphics[width=1.3\defpicwidth]{SCPfig1.ps}


5.3.1 Factor modelling

In a typical credit portfolio model dependencies of individual obligors are modelled via dependencies of the underlying latent variables $ S=(S_1,\ldots,S_n)^\top $. In the typical portfolio analysis the vector $ S$ is embedded in a factor model, which allows for easy analysis of correlation, the typical measure of dependence. One assumes that the underlying variables $ S_j$ are driven by a vector of common factors. Typically, this vector is assumed to be normally distributed (see e.g. JP Morgan (1997)). Thus, with $ Z \sim \textrm{N}(0,\Sigma)$ a $ p$-dimensional normal vector and $ \epsilon=(\epsilon_1,\ldots,\epsilon_n)^\top $ independent normally distributed random variables, independent also from $ Z$, we define

$\displaystyle S_j = \sum_{i=1}^p a_{ji}Z_i +\sigma_j \epsilon_j, \;\; j=1,\ldots n.$ (5.1)

Here $ a_{ji}$ describes the exposure of obligor $ j$ to factor $ i$, i.e. the so-called factor loading, and $ \sigma_j$ is the volatility of the idiosyncratic risk contribution. In such a framework one can easily interfere default correlation from the correlation of the underlying drivers $ S_j$. To do so, we define default indicators

$\displaystyle Y_j=\boldsymbol{1}(S_j \leq D_j),
$

where $ D_j$ is the cut-off point for default of obligor $ j$. The individual default probabilities are

$\displaystyle \pi_j=\textrm{P}(Y_j=1)=\textrm{P}(S_j \leq D_j),$

and the joint default probability is

$\displaystyle \pi_{ij}=\textrm{P}(Y_i=1,Y_j=1)=\textrm{P}(S_i\leq D_i,S_j \leq D_j).$

If we denote by $ \rho_{ij}=Corr(S_i,S_j)$ the correlation of the underlying latent variables and by $ \rho^D_{ij}=Corr(Y_i,Y_j)$ the default correlation of obligors $ i$ and $ j$, then we obtain for the default correlation the simple formula

$\displaystyle \rho^D_{ij}= \frac{\pi_{ij}-\pi_i\pi_j}{\sqrt{\pi_i\pi_j(1-\pi_i)(1-\pi_j)}}.$ (5.2)

Under the assumption that $ (S_i,S_j)$ are bivariate normal, we obtain for the joint default probability

$\displaystyle \pi_{ij}=\int_{-\infty}^{D_i}\int_{-\infty}^{D_j}
\varphi(u,v;\rho_{ij})dudv,
$

where $ \varphi(u,v;\rho)$ is bivariate normal density with correlation coefficient $ \rho $. Thus, asset (factor) correlation influences default correlation by entering in joint default probability. Within the Gaussian framework we can easily evaluate the above quantities, see (5.1).

Table 5.1: Effect of asset correlation on default correlation
Asset correlation Default correlation
   
*[-2pt] 0.1 0.0094
0.2 0.0241
0.3 0.0461


We see, that under our modelling assumption default correlation is of an order of magnitude smaller than asset correlation (which is also supported by empirical evidence).


5.3.2 Copula modelling

As an alternative approach to the factor assumption, we can model each of the underlying variables independently and subsequently use a copula to generate the dependence structure. (For basic facts on copulae we refer the reader to Chapter 2 and the references given there.)

So, suppose we have specified the individual distributions $ F_j$ of the variables $ S_j$ and a copula $ C$ for the dependence structure. Then, for any subgroup of obligors $ \{j_1,\ldots,j_m\}$, we have for the joint default probability

\begin{displaymath}
\begin{array}{ll}\displaystyle
&\displaystyle
\textrm{P}\lef...
...{F_{j_1}(D_{j_1}),\ldots,
F_{j_m}(D_{j_m})\right\},
\end{array}\end{displaymath}

where we denote by $ C_{j_1,\ldots,j_m}$ the $ m$-dimensional margin of $ C$. In particular, the joint default probability of two obligors is now

$\displaystyle \pi_{ij}=C_{i,j}\left\{F_{i}(D_{i}),F_{j}(D_{j})\right\}.
$

To study the effect of different copulae on default correlation, we use the following examples of copulae (further details on these copulae can be found in Embrechts et al. (2001)).
1.
Gaussian copula:

$\displaystyle C_R^{Gauss}(u)=\Phi^n_R(\Phi^{-1}(u_1),\ldots,\Phi^{-1}(u_n)).
$

Here $ \Phi^n_R$ denotes the joint distribution function of the $ n$-variate normal with linear correlation matrix $ R$, and $ \Phi^{-1}$ the inverse of the distribution function of the univariate standard normal.
2.
$ t$-copula:

$\displaystyle C^t_{\nu,R}(u)=t^n_{\nu,R}(t_\nu^{-1}(u_1),\ldots,t_{\nu}^{-1}(u_n)),
$

where $ t^n_{\nu,R}$ denotes the distribution function of an $ n$-variate $ t$-distributed random vector with parameter $ \nu>2$ and linear correlation matrix $ R$. Furthermore, $ t_\nu$ is the univariate $ t$-distribution function with parameter $ \nu$.
3.
Gumbel copula:

$\displaystyle C_\theta^{Gumbel}(u)=\exp\left\{-[(-\log u_1)^\theta + \ldots +(-\log
u_n)^\theta]^{1/\theta}\right\},
$

where $ \theta \in [1,\infty)$. This class of copulae is a sub-class of the class of Archimedean copulae. Furthermore, Gumbel copulae have applications in multivariate extreme-value theory.
In Table 5.2 joint default probabilities of two obligors are reported using three types of obligors with individual default probabilities roughly corresponding to rating classes A,B,C. We assume that underlying variables $ S$ are univariate normally distributed and model the joint dependence structure using the above copulae.


Table 5.2: Copulae and default probabilities
Copula Default probability
  class A ( $ \times 10^{-6}$) class B ( $ \times 10^{-4}$) class C ( $ \times 10^{-4}$)
       
*[-2pt] Gaussian 6.89 3.38 52.45
$ C^t_{10}$ 46.55 7.88 71.03
$ C^t_4$ 134.80 15.35 97.96
Gumbel, $ C_2$ 57.20 14.84 144.56
Gumbel, $ C_4$ 270.60 41.84 283.67


The computation shows that $ t$ and Gumbel copulae have higher joint default probabilities than the Gaussian copula (with obvious implication for default correlation, see equation (5.2)). To explain the reason for this we need the concept of tail dependence:

DEFINITION 5.1   Let $ X$ and $ Y$ be continuous random variables with distribution functions $ F$ and $ G$. The coefficient of upper tail dependence of $ X$ and $ Y$ is

$\displaystyle \lim_{u \rightarrow 1} \textrm{P}[Y > G^{-1}(u) \vert X >F^{-1}(u)]= \lambda_U$ (5.3)

provided that the limit $ \lambda_U\in[0,1]$ exists. If $ \lambda_U
\in (0,1]$, $ X$ and $ Y$ are said to be asymptotically dependent in the upper tail; if $ \lambda_U=0$, $ X$ and $ Y$ are said to be asymptotically independent in the upper tail.

For continuous distributions $ F$ and $ G$ one can replace (5.3) by a version involving the bivariate copula directly:

$\displaystyle \lim_{u\rightarrow 1} \frac{1-2u+C(u,u)}{1-u} = \lambda_U.$ (5.4)

Lower tail dependence, which is more relevant to our current purpose, is defined in a similar way. Indeed, if

$\displaystyle \lim_{u\rightarrow 0} \frac{C(u,u)}{u} = \lambda_L$ (5.5)

exists, then $ C$ exhibits lower tail dependence if $ \lambda_L\in
(0,1]$, and lower tail independence if $ \lambda_L=0$.

It can be shown that random variables linked by Gaussian copulae have no tail-dependence, while the use of $ t_\nu$ and the Gumbel copulae results in tail-dependence. In fact, in case of the $ t_\nu$ copula, we have increasing tail dependence with decreasing parameter $ \nu$, while for the Gumbel family tail dependence increases with increasing parameter $ \theta $.