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Talking Credit - November 2006Active-Passive Vasicek The Active-Passive Decomposition (APD) model of mortgage collateral is an essential structural feature of the AD&Co prepayment model and valuation system. It was born to kill two birds with one stone: to model the burnout and to enable a highly efficient backward induction for pricing fixed-rate pass-throughs. This approach will prove handy once again when we spark the new AD&Co default model. It turns out that as a pool burns out from the prepayment stand-point, the default rates tend to grow. The APD structure easily explains this by assuming that, between the two collateral groups, the passive borrowers have lower credit and are prone to default. This modeling step would naturally explain why the passive borrowers prepay slower in the first place. This article introduces the idea of active-passive split to improve the famous Vasicek model. O. Vasicek’s original works, published by KMV in 1989 and 1991, have solved the problem of finding probability distribution of default rates in a very large pool given default probability of individual loans and correlation between individual assets (explained by a common factor such as a home price index). This correlation is the key to understanding the model; if loan defaults occurred independently, the entire distribution would collapse into one point due to the infinite diversification . The Vasicek Model and Random Sampling of Default Rates Let p denote expected default rate (identical for every loan), r denotes correlation between asset’s values. Then, Vasicek shows that cumulative probability distribution function for default rate in a very large pool is
where, as usual, N stands for cumulative standard normal distribution.
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(1)