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The Bayesian Approach to Default Risk: A Guide Michael Jacobs, Jr. Credit Risk Analysis Division Office of the Comptroller of the Currency Nicholas M. Kiefer Cornell University Departments of Economics and Statistical Science March 2010 Forthcoming: “Rethinking Risk Measurement & Reporting”, Risk Books, Ed. Klaus Blocker The views expressed herein are those of the authors and do not necessarily represent the views of the Office of the Comptroller of the Currency or the Department of the Treasury.
The formula for the conditional PD (4) still holds, but we don’t get the Vasicek distribution of the default rate (5) and (6)-(6.1) becomes this without the Vasicek distributed default rate:
Now the unconditional distribution is given by the T-dimensional integration as the likelihood now can’t be broken up the period-by-period (8):
Where is the joint-density of a zero-mean random variable following and AR(1) process
While Model 1 is a very simple example of a Generalized Linear Model - GLMs (McCullagh and Nelder, 1989), Models II &III are Generalized Linear Mixed Models - GLMMs), a parametric mixture (McNeil and Wendin, 2007; Kiefer, 2009)
We address the boundary problem, that K has a larger support by p ME , using the reflection technique (Schuster, 1985):
For asset correlation in Models 2 & 3, B2 recommends a value of about 0.20 for this segment, so due to little expert information on this, we choose a Beta(12.6, 50.4) prior centered at to 0.20
With even less guidance on the autocorrelation in Model 3, other than from asset pricing literature that is likely to be positive, we chose a uniform prior in [-1,1], with the B2 value of 0 as its mean