4.1 Rating Transition Probabilities

In this section, the way from raw data to estimated rating transition probabilities is described. First, migration events of the same kind are counted. The resulting migration counts are transformed into migration rates, which are used as estimates for the unknown transition probabilities. These estimates are complemented with estimated standard errors for two cases, for independence and for a special correlation structure.


4.1.1 From Credit Events to Migration Counts

We assume that credits or credit obligors are rated in $ d$ categories ranging from 1, the best rating category, to the category $ d$ containing defaulted credits. The raw data consist of a collection of migration events. The $ n$ observed migration events form a $ n \times 2$ matrix with rows

$\displaystyle (e_{i1},e_{i2}) \in \{1,\ldots,d-1\} \times \{1,\ldots,d\}, \quad i = 1,\ldots, n. $

Thereby, $ e_{i1}$ characterizes the rating of $ i$-th credit at the beginning and $ e_{i2}$ the rating at the end of the risk horizon, which is usually one year. Subsequently, migration events of the same kind are aggregated in a $ (d-1) \times d$ matrix $ \mathbf{C}$ of migration counts, where the generic element

$\displaystyle c_{jk} \stackrel{\mathrm{def}}{=}\sum_{i =1}^n \boldsymbol{1}\{ (e_{i1},e_{i2}) = (j,k) \} $

is the number of migration events from $ j$ to $ k$. Clearly, their total sum is

$\displaystyle \sum_{j=1}^{d-1} \sum_{k=1}^{d} c_{jk} = n. $


4.1.2 Estimating Rating Transition Probabilities

We assume that each observation $ e_{i2}$ is a realization of a random variable $ \tilde e_{i2}$ with conditional probability distribution

$\displaystyle p_{jk} = \textrm{P}(\tilde e_{i2} = k \vert \tilde e_{i1} = j), \quad \sum_{k=1}^d p_{jk} = 1, $

where $ p_{jk}$ is the probability that a credit migrates from an initial rating $ j$ to rating $ k$. These probabilities are the so called rating transition (or migration) probabilities. Note that the indicator variable $ \boldsymbol{1}\{\tilde
e_{i2}=k\}$ conditional on $ \tilde e_{i1}=j$ is a Bernoulli distributed random variable with success parameter $ p_{jk}$,

$\displaystyle \boldsymbol{1}\{\tilde e_{i2}=k \} \; \vert \; \tilde e_{i1}=j \sim \textrm{Ber}(p_{jk}).$ (4.1)

In order to estimate these rating transition probabilities we define the number of migrations starting from rating $ j$ as

$\displaystyle n_j \stackrel{\mathrm{def}}{=}\sum_{k=1}^{d} c_{jk}, \quad j = 1, \ldots, d-1$ (4.2)

and assume $ n_j > 0$ for $ j = 1, \ldots, d-1$. Thus, $ (n_1,\ldots,n_{d-1}) $ is the composition of the portfolio at the beginning of the period and

$\displaystyle \left( \sum_{j=1}^{d-1} c_{j1},\ldots,\sum_{j=1}^{d-1} c_{jd} \right)$ (4.3)

is the composition of the portfolio at the end of the period, where the last element is the number of defaulted credits. The observed migration rate from $ j$ to $ k$,

$\displaystyle \hat p_{jk} \stackrel{\mathrm{def}}{=}\frac{c_{jk}}{n_j},$ (4.4)

is the natural estimate of the unknown transition probability $ p_{jk}$.

If the migration events are independent, i. e., the variables $ \tilde e_{12},\ldots,\tilde e_{n2}$ are stochastically independent, $ c_{jk}$ is the observed value of the binomially distributed random variable

$\displaystyle \tilde c_{jk} \sim {\rm B}(n_j,p_{jk}), $

and therefore the standard deviation of $ \hat p_{jk}$ is

$\displaystyle \sigma_{jk} = \sqrt{ \frac{p_{jk}(1-p_{jk}) }{n_j}}, $

which may be estimated by

$\displaystyle \hat\sigma_{jk} = \sqrt{ \frac{\hat p_{jk}(1-\hat p_{jk}) }{n_j}}.$ (4.5)

The estimated standard errors must be carefully interpreted, because they are based on the assumption of independence.


4.1.3 Dependent Migrations

The case of dependent rating migrations raises new problems. In this context, $ \tilde c_{jk}$ is distributed as sum of $ n_j$ correlated Bernoulli variables, see (4.1), indicating for each credit with initial rating $ j$ a migration to $ k$ by 1. If these Bernoulli variables are pairwise correlated with correlation $ \rho_{jk}$, then the variance $ \sigma^2_{jk}$ of the unbiased estimator $ \hat p_{jk}$ for $ p_{jk}$ is (Huschens and Locarek-Junge; 2000, p. 44)

$\displaystyle \sigma^2_{jk} = \frac{ p_{jk}(1-p_{jk}) }{ n_j} +
\frac{n_j-1}{n_j}\rho_{jk} p_{jk}(1-p_{jk}).
$

The limit

$\displaystyle \lim_{n_j\to\infty} \sigma^2_{jk} = \rho_{jk} p_{jk}(1-p_{jk})
$

shows that the sequence $ \hat p_{jk}$ does not obey a law of large numbers for $ \rho_{jk} > 0$. Generally, the failing of convergence in quadratic mean does not imply the failing of convergence in probability. But in this case all moments of higher order exist since the random variable $ \hat p_{jk}$ is bounded and so the convergence in probability implies the convergence in quadratic mean. For $ \rho_{jk}=0$ the law of large numbers holds. Negative correlations can only be obtained for finite $ n_j$. The lower boundary for the correlation is given by $ \rho_{jk} \geq - \frac{1}{n_j-1}$, which converges to zero when the number of credits $ n_j$ grows to infinity.

The law of large numbers fails also if the correlations are different with either a common positive lower bound, or non vanishing positive average correlation or constant correlation blocks with positive correlations in each block (Finger; 1998, p. 5). This failing of the law of large numbers may not surprise a time series statistician, who is familiar with mixing conditions to ensure mean ergodicity of stochastic processes (Davidson; 1994, chapter 14). In statistical words, in the case of non-zero correlation the relative frequency is not a consistent estimator of the Bernoulli parameter.

The parameters $ \rho_{jk}$ may be modeled in consistent way in the framework of a threshold normal model with a single parameter $ \rho $ (Gupton et al.; 1997; Kim; 1999; Basel Committee on Banking Supervision; 2001). This model specifies a special dependence structure based on a standard multinormal distribution for a vector $ (R_1,
\ldots,R_n)$ with equicorrelation matrix (Mardia et al.; 1979, p. 461), where $ R_i$ ( $ i=1, \ldots,n$) is the standardized asset return and $ n$ is the number of obligors. The parameter $ \rho > 0$ may be interpreted as a mean asset return correlation. In this model each pair of variables $ (X,Y)=(R_{i},R_{i'})$ with $ i,i'=1,\ldots,n$ and $ i \neq i'$ is bivariate normally distributed with density function

$\displaystyle \varphi(x,y;\rho) = \frac{1}{2\pi\sqrt{1-\rho^2}} \exp \left(
-\frac{x^2 -2\rho xy + y^2}{2(1-\rho^2)} \right).
$

The probability $ \textrm{P}[(X,Y) \in (a,b)^2]$ is given by

$\displaystyle \beta(a,b;\rho) = \int_a^b \int_a^b \varphi(x,y;\rho) \; dx \; dy.$ (4.6)

Thresholds for rating $ j$ are derived from $ p_{j1},
\ldots,p_{j,d-1}$ by

$\displaystyle z_{j0} \stackrel{\mathrm{def}}{=}-\infty,\, z_{j1} \stackrel{\mat...
...{=}\Phi^{-1}(p_{j1}+p_{j2}),\ldots, z_{jd} \stackrel{\mathrm{def}}{=}
+\infty,
$

where $ \Phi$ is the distribution function of the standardized normal distribution and $ \Phi^{-1}$ it's inverse. Each credit in category $ j$ is characterized by a normally distributed variable $ Z$ which determines the migration events by

$\displaystyle p_{jk} = \textrm{P}(Z \in (z_{j,k-1},z_{jk})) = \Phi(z_{jk}) - \Phi(z_{j,k-1}). $

The simultaneous transition probabilities of two credits $ i$ and $ i'$ from category $ j$ to $ k$ are given by

$\displaystyle p_{jj:kk} = \textrm{P}(\tilde e_{i2} = \tilde e_{i'2} = k \vert \tilde e_{i1}
= \tilde e_{i'1} = j)= \beta(z_{j,k-1},z_{jk};\rho),
$

i.e., the probability of simultaneous default is

$\displaystyle p_{jj:dd} = \beta(z_{j,d-1},z_{jd};\rho). $

For a detailed example see Saunders (1999, pp. 122-125). In the special case of independence we have $ p_{jj:kk} = p_{jk}^2$. Defining a migration from $ j$ to $ k$ as success we obtain correlated Bernoulli variables with common success parameter $ p_{jk}$, with probability $ p_{jj:kk}$ of a simultaneous success, and with the migration correlation

$\displaystyle \rho_{jk} = \frac{p_{jj:kk} - p^2_{jk} }{p_{jk}(1 -p_{jk})}. $

Note that $ \rho_{jk}=0$ if $ \rho=0$.

Given $ \rho \geq 0$ we can estimate the migration correlation $ \rho_{jk} \geq 0$ by the restricted Maximum-Likelihood estimator

$\displaystyle \hat\rho_{jk} = \max \left\{0;\frac{\beta(\hat z_{j,k-1}, \hat z_{jk}; \rho) - \hat p_{jk}^2 }{\hat p_{jk}(1 - \hat p_{jk})} \right\}$ (4.7)

with

$\displaystyle \hat z_{jk} = \Phi^{-1}\left(\sum_{i=1}^{k} \hat p_{ji}\right).$ (4.8)

The estimate

$\displaystyle \hat\sigma_{jk} = \sqrt{ \frac{\hat p_{jk}(1-\hat p_{jk}) }{n_j} + \frac{n_j-1}{n_j}\hat\rho_{jk} \hat p_{jk}(1- \hat p_{jk})}$ (4.9)

of the standard deviation

$\displaystyle \sigma_{jk} = \sqrt{ \frac{p_{jk}(1- p_{jk}) }{n_j} +
\frac{n_j-1}{n_j} \rho_{jk} p_{jk}(1- p_{jk})}
$

is used. The estimator in (4.9) generalizes (4.5), which results in the special case $ \rho=0$.


4.1.4 Computation and Quantlets


counts = 9656 VaRRatMigCount (d, e)
computes migration counts from migration events

The quantlet 9659 VaRRatMigCount can be used to compute migration counts from migration events, where d is the number of categories including default and e is the $ n \times 2$ data matrix containing $ n$ migration events. The result is assigned to the variable counts, which is the $ (d-1) \times d$ matrix of migration counts.

9663 XFGRatMig1.xpl


b = 9675 VaRRatMigRate (c, rho, s)
computes migration rates and related estimated standard errors

The quantlet 9678 VaRRatMigRate computes migration rates and related estimated standard errors for $ m$ periods from an input matrix of migration counts and a given correlation parameter. Here, c is a $ (d-1) \times d \times m$ array of $ m$-period migration counts and rho is a non-negative correlation parameter as used in (4.6). For rho $ =0$ the independent case is computed.

The calculation uses stochastic integration in order to determine the probability $ \beta $ from (4.6). The accuracy of the applied Monte Carlo procedure is controlled by the input parameter s. For $ s>0$ the sample size is at least $ n \ge
(2s)^{-2}$. This guarantees that the user-specified value s is an upper bound for the standard deviation of the Monte Carlo estimator for $ \beta $. Note that with increasing accuracy (i. e. decreasing s) the computational effort increases proportional to $ n$.

The result is assigned to the variable b, which is a list containing:

The matrices b.nstart and b.nend have components given by (4.2) and (4.3). The matrices b.etp, b.emc, and b.esd contain the $ \hat p_{jk}$, $ \hat\rho_{jk}$, and $ \hat\sigma_{jk}$ from (4.4), (4.7), and (4.9) for $ j = 1, \ldots, d-1$ and $ k= 1,\ldots, d$. The estimates $ \hat\rho_{jk}$ are given only for $ \hat p_{jk} > 0$. The matrix b.etv contains the $ \hat z_{jk}$ from (4.8) for $ j,k = 1,\ldots,d-1$. Note that $ z_{j0} = -\infty$ and $ z_{jd} = +\infty$.

9682 XFGRatMig2.xpl