#### Date of Award

12-1-2016

#### Degree Type

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

#### Department

Economics

#### Advisor(s)

Jan Ondrich

#### Keywords

A class of discrete transformation survival models, Multinomial logit model, Sueyoshi proportional hazards model, Termination risks of Single-family mortgages, The Competing Risks Model, The Competing Risks of Prepayment and Default

#### Subject Categories

Social and Behavioral Sciences

#### Abstract

This study contains three chapters. Since the subprime crisis, it has become increasingly important to understand the competing risks of prepayment and default on the single-family mortgage market. This research studies the economic factors that affect the competing risks of prepayment and default in locations where the aggregate of the prepayment risk and the default risk are simultaneously high.

Chapter 1 outlines the analysis based on thirty-year, fixed-rate, single-family mortgages in five Metropolitan Statistical Areas (MSAs): Phoenix, Miami, Tampa, Detroit and Las Vegas. These MSAs are chosen from a sample Single-family Loan-Level Dataset constructed by Freddie Mac based on high simultaneous prepayment and default rates. The results are estimated by a discrete time competing risks model based on restricted multinomial logit. Two different combinations of dependent variables are used to make the analysis more comprehensive. The first combination is prepayment and default and the second is prepayment and 90-days-delinquency.

The indicator for a prepayment penalty and the value of the call option are used to evaluate the prepayment risk and the best three combinations (using the Bayesian information criterion) of explanatory variables – the value of negative equity, a negative equity dummy, and original loan-to-value ratio – together with the unemployment rate are used to evaluate the default risk. Because of the ambiguous effect of the credit score on the prepayment decision discussed in the previous literature, two estimations to explain the effect of the credit score on the prepayment and default are run. One estimation considers how the credit score affects both termination risks and the other considers how the credit score affects only default. The effect of the debt-to-income ratio on the prepayment and default is also tested.

By controlling other mortgage characteristics, such as loan size and loan age, the brief summary of results shows us the prepayment penalty has a negative effect on the prepayment decision. However, its effect in Detroit is insignificantly positive. The value of the call option has a significantly positive impact on the prepayment in each MSA and its effect is the strongest among all of the explanatory variables. The unemployment rate has a positive effect on default/90-days-delinquency. Moreover, negative equity, the negative equity dummy and original loan-to-value are all positively related to the default/90-days-delinquency decision. As expected, in most MSAs, the credit score has a strongly positive effect on the prepayment; comparatively, it has a strong negative impact on the default. When using the credit score only in the default risk and not in the prepayment risk, the effect of the credit score is still significantly negative, but the coefficients decrease slightly. Moreover, debt-to-income does not appear to affect prepayment; however, it has a positive relationship with the default/90-days-delinquency in most MSAs. A Wald test is constructed to test the equality of the coefficients among five MSAs and the results of this test support the argument that prepayment and default risks are heavily influenced by local characteristics.

In earlier studies, the proportional hazards model becomes a popular method to analyze the single (prepayment or default) risk of mortgages. This model has been developed into a competing risks model and been widely used to analyze the prepayment and default risks simultaneously in later studies. However, the process of constructing the competing risks model is ambiguous. In Chapter 2, this study clearly presents the calculation process of this model based on the proportional hazards model using Sueyoshi’s method and then implements the model to analyze the termination risks of single-family mortgages in Phoenix.

Ding, Tian, Yu and Guo (2012) construct a new model based on a class of transformation survival models to analyze the risk of bankruptcy and they argue that that the proportional hazards model is not the best model to analyze this risk. Therefore, a question is raised by this argument: whether the proportional hazards model is the best model to analyze the default/prepayment risk of single-family mortgages? A new competing risks model based on a class of discrete transformation survival models is constructed in Chapter 2 and it is used to analyze the termination risks of the single-family mortgages in Phoenix. The model is controlled by the transformation parameters 𝑐𝑝 (for the prepayment risk) and 𝑐𝑑 (for the default risk). When 𝑐𝑝=0 and 𝑐𝑑=0, it is the competing risks model based on proportional hazards, and when 𝑐𝑝>0and 𝑐𝑑>0, its framework is changed according to the value of 𝑐𝑝 and 𝑐𝑑.

The results show that the proportional hazards framework is the best model to estimate the prepayment risk, but it is not the best model to estimate the default/90-days-delinquency risk. The results of both models support the important arguments made in Chapter 1. Comparing the coefficients estimated by three competing risks models, the coefficients estimated by the model based on the Sueyoshi proportional hazards are insignificantly distinguishable from those estimated by the model based on the multinomial logit. Moreover, the coefficients estimated by the model based on a class of transformation survival models are significantly different from those estimated by the other two models.

Unobserved heterogeneity is an important component that should be considered in the modeling process, even though it is not commonly involved in the analysis of the termination risks of the mortgages. In Chapter 3, this study uses latent classes to control unobserved heterogeneity of two different groups of borrowers and constructs three competing risks models based on the multinomial logit, the proportional hazards model and a class of transformation survival models. The models allow the coefficients of the explanatory variables to be different between two groups of borrowers by keeping the baseline the same (the coefficients of the loan age splines are the same between two groups of borrowers). The models are used to analyze the competing risks of prepayment and default/90-days-delinquency of the single-family mortgages in Phoenix and the estimated average conditional hazard for prepayment, default and 90-days-delinquency are compared with those estimated by models that do not control for unobserved heterogeneity. The results show that when the loan age is between 120 and 165 months, models that do not control for unobserved heterogeneity highly overpredict the prepayment hazard. In the average conditional default and 90-days-delinquency hazard, models that do not control for unobserved heterogeneity overpredict the average conditional hazard compared with models that control for unobserved heterogeneity when the loan age is between around 49 and 94 months.

Another question answered in this study is that if housing prices did not boom and bust since 2004, what would the average conditional default hazard and the average conditional 90-days-delinquency hazard be? This paper constructs a simulation process by assuming that the housing price remains the same since September 2004 and compares the simulated conditional hazards with those estimated based on the real trend of the housing price. The results show that, in the case when the housing price changes across time, the average conditional hazard dramatically increases from around age month 11 and reaches the maximum hazard at around age month 54, and then sharply decreases until around age month 93. This dramatic change of the average conditional hazard disappears in the case when the housing price is assumed to be unchanged after September 2004. The simulated average conditional hazard slowly increases from age month 1 up to age month 169 with an average rate of increase of 3.72 percent for default hazard and 1.80 percent for 90-days-delinquency hazard. The average difference of the conditional hazard is approximately 0.21 percent between the two cases.

#### Access

Open Access

#### Recommended Citation

An, Ran, "The Competing Risks of Prepayment and Default on the Single-Family Mortgage Market" (2016). *Dissertations - ALL*. 556.

https://surface.syr.edu/etd/556