Commercial mortgage prepayments under heterogeneous prepayment penalty structures

Journal of Real Estate Research, The, Jul-Sep 2003 by Fu, Qiang, LaCour-Little, Michael, Vandell, Kerry D

Cox's partial likelihood method was selected to test the propositions embodied in the model specification above, focusing on the impact of various prepayment penalties on the prepayment hazard. This makes Cox's partial likelihood method especially desirable, since the estimates of the [beta] coefficients can be recovered without assuming a particular functional form for baseline hazard. Follain, Ondrich and Sinha (1997) attempt to extend the Cox model into a competing risks framework that recognizes the existence of possible failure due to both prepayment and default. However, since our data contains few defaults, the competing risk model estimates would be difficult, if not impossible, to estimate, especially when we are examining the timing, as well as the incidence, of events. Accordingly, we treat the small number of defaults as censored. Other research on prepayments in the single-family market has shown that when there are only a very small number of defaults, obtaining statistically significant results is unlikely (Ambrose and LaCour-Little, 2001).8

To recover the baseline hazard, a two-step limited information maximum likelihood method is applied. In the first step, the Cox's partial likelihood is performed to obtain the estimated ... coefficients. In the second step, the baseline hazard is expressed as a function of mortgage age:

where ... is the estimated [beta] coefficients from the first step partial likelihood procedure. Notice that by substituting [beta] with ..., the term exp...) is now a known quantity. The baseline hazard can be estimated using the maximum likelihood method. The log likelihood function is:

where I(Prepayment) is an indicator variable equaling 1 if the mortgage was prepaid in month j and 0 otherwise. Murphy and Topel (1985) show that the maximum likelihood estimators at the second step are consistent and asymptotically normal.

The seasoning ramp for multifamily mortgages has been little studied.9 Lacking a particular shape to the hazard function, the baseline hazard is modeled as a polynomial of mortgage age, allowing flexibility in functional forms:

This two-step estimation method can take full advantage of the desirable features of Cox's partial likelihood method. Computational simplicity is a further attraction.

As previously discussed, many argue that default and prepayment probabilities be jointly determined and thus be estimated simultaneously under a competing risk framework (e.g., Deng, 1995; and Deng, Quigley and Van Order, 2000). Those models assume a correlation structure in the error terms of prepayment and default equations. Computations are considerably more complicated. In our data set, default risk is very low (there are only 36 out of 1,165 loans defaulted during the study period). To address the competing risk issue, the variables influencing default (such as LTV, regional dummies) were incorporated into the prepayment equation, instead of modeling the competing risk explicitly through adding a default equation and a correlated error structure.