Phase transition in the Bayesian estimation of the default portfolio

The probability of default (PD) estimation is an important process for financial institutions. The difficulty of the estimation depends on the correlations between borrowers. In this paper, we introduce a hierarchical Bayesian estimation method using the beta binomial distribution, and consider a multi-year case with a temporal correlation. A phase transition occurs when the temporal correlation decays by power decay. When the power index is less than one, the PD estimator does not converge. It is difficult to estimate the PD with the limited historical data. Conversely, when the power index is greater than one, the convergence is the same as that of the binomial distribution. We provide a condition for the estimation of the PD and discuss the universality class of the phase transition. We investigate the empirical default data history of rating agencies, and their Fourier transformations to confirm the correlation decay equation. The power spectrum of the decay history seems to be 1/f of the fluctuations that correspond to long memory. But the estimated power index is much greater than one. If we collect adequate historical data, the parameters can be estimated correctly.