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Estimating Continuous Time Transition Matrices From Discretely Observed Data

by Yasunari Inamura of the Bank of Japan

April 2006

Abstract: A common problem in credit risk management is the estimation of probabilities of rare default events in high investment grades, when sufficient default data are not available. In addressing this issue, increasing attention has been paid to the use of continuous time Markov chains for modeling transition matrices. This approach incorporates the possibility of successive downgrades leading to defaulting in such a way that a very slight probability of default can be captured. In banking applications, however, the approach faces a problem with data limitations, since it requires continuously observed rating data to estimate intensities for transition matrices. In reality, the data frequency of internal rating systems for individual banks is either annual or bi-annual. To make the approach more applicable, the estimation methodology based on discretely observed rating data needs to be examined from a practical perspective. Against this background, the paper discusses and compares the small sample performances of the five estimation methods designed for discrete time observations -- diagonal adjustment, weighted adjustment, quasi-optimization approach, expectation maximization algorithm and Markov chain Monte Carlo (MCMC) estimation -- by measuring differences in default probabilities of investment grades and several matrix norms. Monte Carlo experiments reveal that the MCMC gives the most accurate finite-sample performance, both in terms of the estimated default probabilities and the matrix norms. Moreover, a case study to examine the impact on the loss distribution of a hypothetical investment grade portfolio shows that differences in these estimation methods have the potential to yield significantly different estimates of economic capital.

JEL Classification: C13, G21.

Keywords: Default probability, LDPs, Markov chains, Infinitesimal generator matrix.

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