The MEFAK methodology: Multi-year one-factor default model with temporal correlation

In the DZ BANK Group, the MEFAK test is the central tool for checking the calibration of a rating system. It is a method for simulating default figures to appropriately determine acceptance bands, which can also take correlations into account. As a further development of the well-known binomial test with asset correlation (Basel one-factor model), the MEFAK test takes into account the complete, multi-year data basis and can also map time-dependent correlations using an additional parameter. The distributions of default figures are not only simulated at portfolio level, but also at rating class and annual level. In this way, it is possible to assess the calibration simultaneously and at the same time differentiated across years and rating classes.

The correlation parameters required for the application of the MEFAK test must be determined in advance. To estimate the correlation parameters, the basic structure of the MEFAK methodology is inverted and Bayesian methods are used.

MEFAK Test

The MEFAK test is best explained using the graphical representation in Fig. 01. The parameters are represented as nodes or boxes. By connecting the parameters with arrows, the relationships – whether stochastic (red arrows) or deterministic (black arrows) – can be easily read off, with the colors of the boxes indicating the type of parameters involved.

Specifically, the blue fields represent the specified latent variables of the test, in this case the average probabilities of default specified by the rating ${PD}_{kj}$ per rating class $k$ and year $j$ and the two correlation parameters $ρ$ (asset correlation) and $τ$ (temporal correlation).

The observed variables (green field) are the number of obligors under default risk, $N_{kj}\$ per rating class and year.

The systematic factor (hereinafter also referred to as the “business cycle”) $C_j$ per year and the conditional probability of default co-determined by it ${PD}_{kj}^c$ per rating and year are not observed as latent variables (red fields).

The number of defaults $D_{kj}$ per rating class and year, as well as other error measures dependent on the number of defaults $E$ are dependent random variables (yellow fields) whose distribution is simulated.

The individual variables are specified below. The systematic factor $\vec{C}$ is called $J$-dimensional normally distributed random variable, where $J$ denotes the number of existing years, with the probability density.

Equation 01:

and the temporal covariance matrix

The following applies to the conditional PD

Equation 02:

analogous to the well-known Basel one-factor model, where $\Phi$ denotes the distribution function of the standard normal distribution.

For a given conditional probability of default and a known number of cases, the defaults are binomially distributed

Equation 03:

Thus, given the estimated PDs $PD=\left\{{PD}_{kj}|1\le k\le K,1\le j\le J\right\}$ of the rating system, the number of obligors $\ N=\left\{N_{kj}|1\le k\le K,1\le j\le J\right\}$ and correlation parameters $\rho$ and $\tau$, the number of defaults $D=\left\{D_{kj}|1\le k\le K,1\le j\le J\right\}$ are simulated under the null hypothesis that the estimated PDs correctly predict the probability of default. In the MEFAK test, the empirical default numbers are then compared with the distribution of the simulated default numbers under the null hypothesis, as is usual in calibration tests.

A distribution is generated as output (here in Fig. 02 the distribution of the default rate), which graphically shows where the empirical value (pink line) lies in relation to the acceptance ranges (highest density interval, HDI for short).

As the default figures are simulated simultaneously for all years, taking into account the correlation parameters, the calibration can be tested globally. At the same time, however, it is also possible to test each year and each rating class individually, as the defaults are sampled per year and rating class.

Fig. 03 contains a differentiated and compressed representation of the HDI bands and default rates from Figure 2, calculated at an annual level.

The MEFAK test is a generalization of established tests. On the one hand, the MEFAK test corresponds to a binomial test if the asset correlation is set to zero. On the other hand, the Basel one-factor model exists when the time correlation is zero. Due to the simultaneous simulation of default figures in all rating classes over the years, the rating class-specific PDs are included in the simulation of default figures at portfolio level rather than PDs averaged over the entire portfolio.

Correlation parameter estimation

In order to obtain suitable correlation parameters for the MEFAK test, the MEFAK methodology is also used when estimating the parameters. For this purpose, the MEFAK methodology must be “inverted” using Bayesian methods.

From the obligor and default figures observed $N_{kj}\$and $D_{kj}$ the goal is to find the most probable parameters $\rho$ and $\tau$ which fit the empirical data.

This means that from the model likelihood defined in the forward direction of the form $p\left(B\middle| A\right)$represented by a directed connection $A\rightarrow B$ the A posteriori distribution of $A$ conditional on all available observable data $B$ so $p\left(A\middle| B\right)\$calculated using the Bayesian formula:

Equation 04:

In contrast to the significance test, the default data $D$ are assumed to be observed and for the variables $\rho,\tau$ whose A posteriori distribution is determined.

For the approximation of $p(A|B)$ Gibbs sampling is used, a standard method of Bayesian statistics, which is an MCMC algorithm and represents an established special case of the Metropolis-Hastings algorithm. The asymptotic method of Gibbs sampling has become established for approximations of the A posteriori distribution in order to avoid the numerically complex and unstable estimation of the probability $p(B)$, i.e. the integral in the denominator from equation 4.

For the empirical default probabilities, the long-term average of the default rate is used for each rating class so that the PDs of the rating system are not included in the correlation parameter estimate.

The correlation parameters must be estimated specifically for each portfolio and each rating system, as they must reflect both the correlation behavior and the rating philosophy of the rating system. It has been shown that the level of the correlation parameters and the rating philosophy are closely related. For rating systems with a point-in-time character, the acceptance bands only become wider when the MEFAK test is carried out (and the correlation parameters are set to zero) due to the migration of customers to other rating classes. The correlation parameters determined using the MEFAK method are therefore small for point-in-time procedures, as all information about the migrations is taken into account in the estimation (by the $N_{kj}$) per year and rating class. In a rating system with a through-the-cycle philosophy, on the other hand, there is no or hardly any migration of customers to other rating classes, so that the cyclical fluctuations in default figures in the course of economic development must be carried out via the correlation parameters or the adjustment of default probabilities using conditional PD. Fig. 04 shows the relationship between the correlation parameters and the rating philosophy schematically.

Conclusion

The term MEFAK methodology includes the calibration test (MEFAK test) and the correlation parameter estimation required for the application of the MEFAK test. The MEFAK test has many advantages over other tests. As the simulation of the default figures is carried out simultaneously for all annual slices and rating classes, the calibration of the rating system is assessed much more robustly than on the basis of one-year tests or tests that have to be carried out for each rating class. Nevertheless, it is possible to test each year and each rating class individually. The correlation of default figures is differentiated according to intra-year and inter-year correlation via the asset and temporal correlation parameters. In addition to the default rate, other error measures dependent on the number of defaults can also be tested. Another advantage is that the output is a distribution that graphically shows exactly where the empirical value lies in relation to the acceptance ranges.

Finally, the inverted MEFAK methodology ensures that the estimation of the correlation parameters is determined taking into account the rating class information and not at overall portfolio level. This determines the correlation parameters suitable for the rating system and the rating philosophy.