Parameter estimation of default portfolios using the Merton model and Phase transition

TL;DR

This paper explores phase transitions in default probability estimation using the Merton model, revealing slow convergence issues linked to power-law decay of correlations, impacting parameter estimation accuracy.

Contribution

It introduces a phase transition analysis within the Merton model framework, highlighting the effects of power-law decay on default probability estimation and convergence behavior.

Findings

Phase transition occurs when temporal correlation decays by power law.

PD estimator converges slowly when the power index is less than one.

Empirical data suggests PD has long memory, complicating estimation.

Abstract

We discuss the parameter estimation of the probability of default (PD), the correlation between the obligors, and a phase transition. In our previous work, we studied the problem using the beta-binomial distribution. A non-equilibrium phase transition with an order parameter occurs when the temporal correlation decays by power law. In this article, we adopt the Merton model, which uses an asset correlation as the default correlation, and find that a phase transition occurs when the temporal correlation decays by power law. When the power index is less than one, the PD estimator converges slowly. Thus, it is difficult to estimate PD with limited historical data. Conversely, when the power index is greater than one, the convergence speed is inversely proportional to the number of samples. We investigate the empirical default data history of several rating agencies. The estimated power index is in the slow convergence range when we use long history data. This suggests that PD could have a long memory and that it is difficult to estimate parameters due to slow convergence.