Predicting the Credit Cycle with an Autoregressive Model

Credit default events show cross sectional as well as serial correlation. While the latter is often neglected by current credit risk models, this work incorporates both types of dependence. A Bernoulli mixture model is considered, where in each rating grade the probit of the stochastic Bernoulli parameter follows an autoregressive stationary process with exogenous variables. The model parameters are estimated for a large retail portfolio. Exemplarily, prediction intervals of the default probabilities of the best and worst nondefault rating grade are given and predicted credit portfolio loss distributions are plotted in contrast to the unconditional loss distributions. 1 Motivation One of the main problems in credit risk modeling is that the observations available for the estimation of model parameters are dependent over time. However, time series in the credit risk industry are often so short that calibrations are accomplished under the assumption of independence. Whereas this negligence could be accepted for a model fit, forecasts of default probabilities need to take the credit cycle into account. A parsimonious parameterization is required due to the scarce data situation. This paper examines what can be achieved using stationary autoregressive processes with an additional exogenous variable in order to incorporate the credit cycle. 2 Notations and assumptions In the following survey a homogeneous portfolio with nt obligors belonging to the same rating grade is considered over time periods t = 1, . . . , T . Let At,i denote the default-indicator variable of the i-th obligor in time period t, which can take a value of one in the case of default or zero otherwise. The defaults are modeled within a Bernoulli mixture model, where at the beginning of time period t, the stochastic default probability π̃t takes a number πt ∈ ]0, 1[ . The variables At,i for i = 1, 2, . . . , nt are assumed to be conditionally independent for given realizations of the stochastic default probabilities π̃t, π̃t−1, . . . and for given realizations of macroeconomic impact variables Vt−1, Vt−2, . . .. Thus, their conditional distribution is a Bernoulli distribution, At,i ∣