4.3 Multi-Period Transitions

In the multi-period case, transitions in credit ratings are also characterized by rating transition matrices. The $ m$-period transition matrix is labeled $ \mathbf{P}^{(m)}$. Its generic element $ p_{jk}^{(m)}$ gives the rating transition probability from rating $ j$ to $ k$ over the $ m \geq 1$ periods. For the sake of simplicity the one-period transition matrix $ \mathbf{P}^{(1)}$ is shortly denoted by $ \mathbf{P}$ in the following. This transition matrix is considered to be of type $ d \times d$. The last row contains $ (0,0, \ldots ,0,1)$ expressing the absorbing default state. Multi-period transition matrices can be constructed from one-period transition matrices under the assumption of the Markov property.


4.3.1 Time Homogeneous Markov Chain

Let $ \{X(t)\}_{t \geq 0}$ be a discrete-time stochastic process with countable state space. It is called a first-order Markov chain if

\begin{multline}
\textrm{P}\left[( X(t+1) = x(t+1) \vert X(t) = x(t), \ldots, X(...
... \\
= \textrm{P}\left[ X(t+1) = x(t+1) \vert X(t) = x(t) \right]
\end{multline}

whenever both sides are well-defined. Further, the process is called a homogeneous first-order Markov chain if the right-hand side of (4.16) is independent of $ t$ (Brémaud; 1999).

Transferred to rating transitions, homogeneity and the Markov property imply constant one-period transition matrices $ \mathbf{P}$ independent of the time $ t$, i. e.  $ \mathbf{P}$ obeys time-stability. Then the one-period $ d \times d$ transition matrix $ \mathbf P$ contains the non-negative rating transition probabilities

$\displaystyle p_{jk} = \textrm{P}(X(t+1)=k\vert X(t)=j). $

They fulfill the conditions

$\displaystyle \sum_{k=1}^d p_{jk} = 1 $

and

$\displaystyle (p_{d1}, p_{d2}, \ldots ,p_{dd}) = ( 0,\ldots,0,1). $

The latter reflects the absorbing boundary of the transition matrix $ \mathbf{P}$.

The two-period transition matrix is then calculated by ordinary matrix multiplication, $ \mathbf P^{(2)} = \mathbf P\mathbf P$. Qualitatively, the composition of the portfolio after one period undergoes the same transitions again. Extended for $ m$ periods this reads as

$\displaystyle \mathbf P^{(m)} = \mathbf P^{(m-1)} \mathbf P = \mathbf{P}^m$    

with non-negative elements

$\displaystyle p_{jk}^{(m)}=\sum_{i=1}^d p_{ji}^{(m-1)}p_{ik}. $

The recursive scheme can also be applied for non-homogeneous transitions, i.e. for one-period transition matrices being not equal, which is the general case.


4.3.2 Bootstrapping Markov Chains

The one-period transition matrix $ \mathbf{P}$ is unknown and must be estimated. The estimator $ \mathbf{\hat{P}}$ is associated with estimation errors which consequently influence the estimated multi-period transition matrices. The traditional approach to quantify this influence turns out to be tedious since it is difficult to obtain the distribution of $ (\hat{\textbf{P}} -\textbf{P})$, which could characterize the estimation errors. Furthermore, the distribution of $ (\hat{\textbf{P}}^{(m)} -\textbf{P}^{(m)})$, with

$\displaystyle \hat{\textbf{P}}^{(m)}\stackrel{\mathrm{def}}{=}\hat{\textbf{P}}^m,$ (4.16)

has to be discussed in order to address the sensitivity of the estimated transition matrix in the multi-period case. It might be more promising to apply resampling methods like the bootstrap combined with Monte Carlo sampling. For a representative review of resampling techniques see Efron and Tibshirani (1993) and Shao and Tu (1995), for bootstrapping Markov chains see Athreya and Fuh (1992) and Härdle et al. (2001).

Assuming a homogeneous first-order Markov chain $ \{X(t)\}_{t \geq 0}$, the rating transitions are generated from the unknown transition matrix $ \textbf{P}$. In the spirit of the bootstrap method, the unknown transition matrix $ \textbf{P}$ is substituted by the estimated transition matrix $ \hat{\textbf{P}}$, containing transition rates. This then allows to draw a bootstrap sample from the multinomial distribution assuming independent rating migrations,

$\displaystyle (\tilde c_{j1}^*,\ldots, \tilde c_{jd}^*) \sim {\rm Mult}(n_j; \hat{p}_{j1},\ldots, \hat{p}_{jd}),$ (4.17)

for all initial rating categories $ j = 1, \ldots, d-1$. Here, $ \tilde
c_{jk}^*$ denotes the bootstrap random variable of migration counts from $ j$ to $ k$ in one period and $ \hat{p}_{jk}$ is the estimated one-period transition probability (transition rate) from $ j$ to $ k$.

Then the bootstrap sample $ \{c_{jk}^*\}_{j=1, \ldots,d-1,k=1,
\ldots, d}$ is used to estimate a bootstrap transition matrix $ \hat{\textbf{P}}^*$ with generic elements $ \hat{p}_{jk}^*$ according

$\displaystyle \hat{p}_{jk}^* = \frac{c_{jk}^*}{n_j}.$ (4.18)

Obviously, defaulted credits can not upgrade. Therefore, the bootstrap is not necessary for obtaining the last row of $ \hat{\textbf{P}}^*$, which is $ (\hat{p}_{d1}^*,\ldots,\hat{p}_{dd}^*) = (0,\ldots,0,1)$. Then matrix multiplication gives the $ m$-period transition matrix estimated from the bootstrap sample,

$\displaystyle \hat{\textbf{P}}^{*(m)}= \hat{\textbf{P}}^{*m}, $

with generic elements $ \hat{p}_{jk}^{*(m)}$.

We can now access the distribution of $ \hat{\textbf{P}}^{*(m)}$ by Monte Carlo sampling, e. g.  $ B$ samples are drawn and labeled $ \hat{\textbf{P}}^{*(m)}_b$ for $ b=1, \ldots, B$. Then the distribution of $ \hat{\textbf{P}}^{*(m)}$ estimates the distribution of $ \hat{\textbf{P}}^{(m)}$. This is justified since the consistency of this bootstrap estimator has been proven by Basawa et al. (1990). In order to characterize the distribution of $ \hat{\textbf{P}}^{*(m)}$, the standard deviation $ \textrm{Std}\left(\hat{p}_{jk}^{*(m)}\right)$ which is the bootstrap estimator of $ \textrm{Std}\left(\hat{p}_{jk}^{(m)}\right)$, is estimated by

$\displaystyle \widehat{\textrm{Std}}\left(\hat{p}_{jk}^{*(m)}\right) = \sqrt{\f...
...t{p}^{*(m)}_{jk,b} -\hat{\textrm{E}}\left(\hat{p}_{jk}^{*(m)}\right) \right]^2}$ (4.19)

with

$\displaystyle \hat{\textrm{E}}\left(\hat{p}_{jk}^{*(m)}\right)=\frac{1}{B} \sum_{b=1}^B \hat{p}^{*(m)}_{jk,b}$    

for all $ j = 1, \ldots, d-1$ and $ k= 1,\ldots, d$. Here, $ \hat{p}_{jk,b}^{*(m)}$ is the generic element of the $ b$-th $ m$-period bootstrap sample $ \hat{\textbf{P}}^{*(m)}_b$. So (4.20) estimates the unknown standard deviation of the $ m$-period transition rate $ \textrm{Std}\left(\hat{p}_{jk}^{(m)}\right)$ using $ B$ Monte Carlo samples.


4.3.3 Computation and Quantlets

For time homogeneity, the $ m$-period rating transition matrices are obtained by the quantlet 10774 XFGRatMig5.xpl (q = XFGRatMig5(p, m)). It computes all $ t=1,2,\ldots,\texttt{m}$ multi-period transition matrices given the one-period $ d \times d$ matrix p. Note that the output q is a $ d\times d
\times m$ array, which can be directly visualized by 10777 XFGRatMig6.xpl (XFGRatMig6(q)) returning a graphical output. To visualize $ t$-period transition matrices each with $ d^2$ elements for $ t= 1,\ldots,m$, we plot $ d^2$ aggregated values

$\displaystyle j - 1 + \sum_{l=1}^k p_{jl}^{(t)}, \quad j,k = 1,\ldots, d$ (4.20)

for all $ t= 1,\ldots,m$ periods simultaneously.

Figure: Example for 10781 XFGRatMig6.xpl :
\includegraphics[width=1.42\defpicwidth]{RM02.ps} Aggregated values of multi-period transition matrices.

A typical example is shown in Figure 4.2 for the one-year transition matrix given in Nickell et al. (2000, p. 208), which uses Moody's unsecured bond ratings between 31/12/1970 and 31/12/1997. According (4.21), aggregated values are plotted for $ t=1, \ldots ,
10$. Thereby, the transition matrix is condensed for simplicity to $ 4 \times 4$ with only 4 basic rating categories, see the example in Section 4.2.4. Again, the last category stands for defaulted credits. Estimation errors are neglected in Figure 4.2.


out = 10791 VaRRatMigRateM (counts, m, B)
bootstraps $ m$-period transition probabilities

Bootstrapping is performed by the quantlet 10794 VaRRatMigRateM . It takes as input counts, the $ (d-1) \times d$ matrix of migration counts, from which the bootstrap sample is generated. Further, m denotes the number of periods and B the number of generated bootstrap samples. The result is assigned to the variable out, which is a list of the following output:

The components of the matrices out.btm are calculated according (4.18) and (4.19). The matrices out.etm and out.stm have components given by (4.17) and (4.20).


4.3.4 Rating Transitions of German Bank Borrowers

In the following the bootstrapping is illustrated in an example. As estimator $ \hat{\textbf{P}}$ we use the $ 7 \times 7$ rating transition matrix of small and medium-sized German bank borrowers from Machauer and Weber (1998, p. 1375), shown in Table 4.1. The data cover the period from January 1992 to December 1996.


Table 4.1: German rating transition matrix ($ d=7$) and the number of migrations starting from rating $ j=1, \ldots ,d$
  To $ k$  
From $ j$ 1 2 3 4 5 6 Default $ n_j$
1 0.51 0.40 0.09 0.00 0.00 0.00 0.00 35
2 0.08 0.62 0.19 0.08 0.02 0.01 0.00 103
3 0.00 0.08 0.69 0.17 0.06 0.00 0.00 226
4 0.01 0.01 0.10 0.64 0.21 0.03 0.00 222
5 0.00 0.01 0.02 0.19 0.66 0.12 0.00 137
6 0.00 0.00 0.00 0.02 0.16 0.70 0.12 58
Default 0.00 0.00 0.00 0.00 0.00 0.00 1.00 0


With the quantlet 10920 VaRRatMigRateM the $ m$-period transition probabilities are estimated by $ \hat p_{jk}^{(m)}$ and the bootstrap estimators of their standard deviations are calculated. This calculations are done for 1, 5 and 10 periods and $ \texttt{B}=1000$ Monte Carlo steps. A part of the resulting output is summarized in Table 4.2, only default probabilities are considered. Note that the probabilities in Table 4.1 are rounded and the following computations are based on integer migration counts $ c_{jk}
\approx n_j p_{jk}$.

10924 XFGRatMig7.xpl


Table 4.2: Estimated $ m$-period default probabilities and the bootstrap estimator of their standard deviations for $ m=1,5,10$ periods
             
From $ j$ $ \hat{p}_{jd}^{(1)}$ $ \widehat{Std}\left(\hat{p}^{*(1)}_{jd}\right)$ $ \hat{p}_{jd}^{(5)}$ $ \widehat{Std}\left(\hat{p}^{*(5)}_{jd}\right)$ $ \hat{p}_{jd}^{(10)}$ $ \widehat{Std}\left(\hat{p}^{*(10)}_{jd}\right)$
             
1 0.00 0.000 0.004 0.003 0.037 0.015
2 0.00 0.000 0.011 0.007 0.057 0.022
3 0.00 0.000 0.012 0.005 0.070 0.025
4 0.00 0.000 0.038 0.015 0.122 0.041
5 0.00 0.000 0.079 0.031 0.181 0.061
6 0.12 0.042 0.354 0.106 0.465 0.123



4.3.5 Portfolio Migration

Based on the techniques presented in the last sections we can now tackle the problem of portfolio migration, i. e. we can assess the distribution of $ n(t)$ credits over the $ d$ rating categories and its evolution over periods $ t \in \{1, \ldots m\}$. Here, a stationary transition matrix $ \mathbf{P}$ is assumed. The randomly changing number of credits in category $ j$ at time $ t$ is labeled by $ \tilde n_j(t)$ and allows to define non-negative portfolio weights

$\displaystyle \tilde{w}_j(t) \stackrel{\mathrm{def}}{=}\frac{ \tilde n_j(t)}{n(t)}, \quad j=1, \ldots , d,$    

which are also random variables. They can be related to migration counts $ \tilde c_{jk}(t)$ of period $ t$ by

$\displaystyle \tilde w_k(t+1) = \frac{1}{n(t)} \sum_{j=1}^d \tilde c_{jk}(t)$ (4.21)

counting all migrations going from any category to the rating category $ k$. Given the weights $ \tilde w_j(t) = w_j(t)$ at $ t$, the migration counts $ \tilde c_{jk}(t)$ are binomially distributed

$\displaystyle \tilde c_{jk}(t)\vert\tilde w_j(t) = w_j(t) \sim \textrm{B} \left( n(t) \, w_j(t),p_{jk} \right).$ (4.22)

The non-negative weights are aggregated in a row vector

$\displaystyle \tilde w(t) = (\tilde w_1(t), \ldots, \tilde w_{d}(t)) $

and sum up to one

$\displaystyle \sum_{j=1}^{d} w_j(t) = 1.$    

In the case of independent rating migrations, the expected portfolio weights at $ t+1$ given the weights at $ t$ result from (4.22) and (4.23) as

$\displaystyle \textrm{E}[ \tilde w(t+1)\vert\tilde{w}(t)=w(t) ] = w(t) \mathbf{P} $

and the conditional covariance matrix $ V[\tilde
w(t+1)\vert\tilde{w}(t)=w(t) ]$ has elements

$\displaystyle v_{kl}\stackrel{\mathrm{def}}{=}\left\{ \begin{array}{lll} \frac{...
...c{1}{n(t)} \sum_{j=1}^d w_j(t) p_{jk}p_{jl} & & k\neq l. \\ \end{array} \right.$ (4.23)

For $ m$ periods the multi-period transition matrix $ \mathbf{P}^{(m)}=\mathbf{P}^m$ has to be used, see Section 4.3.1. Hence, (4.22) and (4.23) are modified to

$\displaystyle \tilde w_k(t+m) = \frac{1}{n(t)} \sum_{j=1}^d \tilde c_{jk}^{(m)}(t)$    

and

$\displaystyle \tilde c_{jk}^{(m)}(t)\vert\tilde w_j(t) = w_j(t) \sim \textrm{B} \left( n(t) \, w_j(t),p_{jk}^{(m)} \right).$    

Here, $ c_{jk}^{(m)}(t)$ denotes the number of credits migrating from $ j$ to $ k$ over $ m$ periods starting in $ t$. The conditional mean of the portfolio weights is now given by

$\displaystyle \textrm{E}[ \tilde w(t+m)\vert\tilde{w}(t)=w(t) ] = w(t) \mathbf{P}^{(m)}$    

and the elements of the conditional covariance matrix $ V[\tilde
w(t+m)\vert\tilde{w}(t)=w(t)]$ result by replacing $ p_{jk}$ and $ p_{jl}$ in (4.24) by $ p_{jk}^{(m)}$ and $ p_{jl}^{(m)}$.