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(2p)
Note that we have employed same random deviate w uniformly taken from [0,1] in both expressions. This is due to the fact that (A) default rates in the active group and the passive group are perfectly correlated and (B) perfectly correlated uniform deviates are identical.
Therefore, the total default rate in the entire pool is a weighting of two perfectly correlated default rates taken from regular Vasicek distributions differing in parameters:
(3)
This is the final formula allowing for generating a random sample of default rates for an APD pool. Getting an explicit PDF requires resolving this expression for w, which does not seem possible due to its transcendent nature.
Using the Model
Early this year ( http://www.ad-co.com/newsletter/2006/Mar06/Consult.htm), we introduced a new approach to assessing market-implied default distribution in large pools of sub-prime borrowers. It is based on running our usual OAS valuation through a grid of default rates; the analysis is done for a set of ABS tranches taken from the same deal. The default distribution is then found to approximate observed market prices of all bonds involved into this analysis. We considered a few forms of this distribution and included the plain Vasicek and the APD Vasicek as the most attractive functional choices.
Arguably, the APD version should work with greater accuracy simply because it has more parameters to fit. The regular Vasicek has only two parameters to find, p and r. The active-passive version has four parameters – two for each of population groups - even after fixing the population variable y . The extended model, however, seems to be advantageous for several other less trivial reasons.
