A Lattice Approach to Pricing Fixed-Rate Mortgages with Default and Prepayment Options

Existing models on the pricing of default and prepayment options in fixed-rate mortgages either use numerical methds or they do not consider refinancing or other transaction costs involved in default and prepayment. We provide in this paper an application of the Boyle [1] lattice model to price secured debt with two risky assets. This model is simple, efficient and capable of considering the major types of transaction costs involved in prepayment and default. Using our model, we estimate the option values under a range of assumptions about the underlying parameters. We also provide some comparisons of the lattice model estimates to other models in the literature.     

Mortgage Prepayment and Default Decisions: A Poisson Regression Approach

This paper uses an extensive and geographically dispersed sample of single-family fixed rate mortgages to assess the prepayment and default behavior of individual homeowners. We make use of Poisson regression to efficiently estimate the parameters of a proportional hazards model for prepayment and default decisions. Poisson regression for grouped survival data has several advantages over partial likelihood methods. First, when dealing with time-dependent covar-iates, it is considerably more efficient in terms of computations. Second, it is possible to estimate full-hazard models which include, for example, functions of time as well as multiple time scales (i.e., age of the loan and calendar time), in a much more straightforward manner than partial likelihood methods for un-grouped data. Third, Poisson regression can be used to estimate non-proportional hazards models such as additive excess risk specifications. Taken together, our data and estimation methodology allow us to obtain a better understanding of the economic factors underlying prepayment and default decisions.     

An Analysis of the Ex Ante Probabilities of Mortgage Prepayment and Default

Observed mortgage prepayment and default rates have been far different than the ruthless option exercise rates predicted by contingent claims models of mortgage pricing. The discrepancies have been attributed to both the competing-risk nature of prepayment and default and to transactions costs. This paper tries a different means of reconciliation. We introduce a third stochastic process, household income, into the usual pricing model that includes only the spot interest rate and the house price. The presence of income allows considering consumption-theoretic determinants of termination. The role of mortgage underwriting rules in restricting optimal prepayment is also explicitly modeled. Numerical ex ante prepayment and default rates based on the theoretical model come much closer to historical experience.     

The Termination of Commercial Mortgage Contracts through Prepayment and Default: A Proportional Hazard Approach with Competing Risks

This article examines the factors driving the borrower's decision to terminate commercial mortgage contracts with the lender through either prepayment or default. Using loan–level data, we estimate prepayment and default functions in a proportional hazard framework with competing risks, allowing us to account for unobserved heterogeneity. Under a strict definition of mortgage default, we do not find evidence to support the existence of unobserved heterogeneity. However, when the definition of mortgage default is relaxed, we do find some evidence of two distinctive borrower groups. Our results suggest that the values of implicit put and call options drive default and prepayment actions in a nonlinear and interactive fashion. Prepayment and default risks are found to be convex in the intrinsic value of call and put options, respectively. Consistent with the joint nature of the two underlying options, high value of the put/call option is found to significantly reduce the call/put risk since the borrower forfeits both options by exercising one. Variables that proxy for cash flow and credit conditions as well as ex post bargaining powers are also found to have significant influence upon the borrower's mortgage termination decision.     

Prepayment Penalty as a Screening Mechanism for Default and Prepayment Risks

The objective of this article is to offer a theoretical model of asymmetric information to analyze the screening role of prepayment penalty. We consider both default risk and prepayment risk. What makes the role of prepayment penalty interesting and more complicated is that a borrower's contract choice could send conflicting signals to the lender about that borrower's default and prepayment risk type. This is different from earlier theoretical models of mortgage choice under asymmetric information where a certain aspect of the contract (e.g., discount points or loan‐to‐value ratio) is explored as a screening mechanism for a single risk dimension (default risk or prepayment risk) of the borrower type. We show the existence of separating equilibria where different default and prepayment risk types choose contracts with different combinations of prepayment penalty and interest rate. For certain parameter combinations, the model also generates a pooling equilibrium where all borrower types obtain the same contract. Our analysis could offer a partial explanation for the observation that contracts with prepayment penalties are a lot less prevalent than contracts with points.     

Mortgage Default and Prepayment Risks among Moderate‐ and Low‐Income Households

Using a unique sample of community reinvestment loans, we study the propensity of very low‐income households to terminate a mortgage and compare it to the outcomes for low‐income and moderate‐income households. The results indicate that, even within moderate‐ and low‐income segments, lower or very low income is associated with higher default and lower prepayment probabilities. In addition, depending on how low the borrower's income is, classic determinants of loan termination such as credit scores, the amount of equity in the home and local labor market conditions can have different impacts on default and prepayment probabilities.     

Elective Mortgage Prepayment: Termination and Curtailment

Mortgage-prepayment risk underlies the structuring of mortgage-backed derivative securities, such as tranched real estate mortgage investment conduits. This prepayment comes either from mortgage termination or from curtailment, where the borrower retains the existing mortgage and prepays a portion. There are differences in cash flows from the two types of prepayment. In termination, the loan disappears from a pool, and the scheduled payment to investors in the pool is reduced. In curtailment, the loan survives, and the scheduled payment is unchanged but the term is reduced. There are implications for structuring mortgages and derivative securities. The prepayment decision is embedded in an in-tertemporal household utility maximization framework where choices are made between refinancing, making the regular payment, default or curtailment. Empirical results are presented for Government National Mortgage Association (GNMA) pools, and an algorithm is presented that separates the termination and curtailment components, facilitating the development of derivative securities.     

Optimal Sharing of Interest‐Rate Risk in Mortgage Contracts: The Effects of Potential Prepayment and Default

Much of the literature on the economics of mortgage markets has studied the fixed vs. adjustable‐rate mortgage choice made by individual borrowers. However, to decide if the outcome of such a choice is efficient or approximately so, it is necessary to explore the question of optimal risk‐sharing in mortgage contracts. But because only a small literature has studied this question, more research is clearly warranted. The present article helps fill this gap by developing a simplified version of Arvan and Brueckner's model, using it to characterize optimal contracts in the absence of mortgage termination, and then exploring how termination via prepayment or default affects optimal risk‐sharing. The broad conclusion of the analysis is that potential mortgage termination makes higher risk exposure for borrowers optimal.     

A Note on Mortgage Risk: Default vs. Loss Rates

This note reexamines the role of the loan-to-value ratio on mortgage risk. Whereas previous studies have focused on the default rate as a function of this term, this study considers the additional effect on the loss rate of defaulted loans. Because the dollar loss per amount originated is the product of the default rate and the loss rate on defaulted loans, the impact of the loan-to-value ratio on both the default and loss rates is crucial to explaining the impact of the loan-to-value ratio on mortgage risk. I find that both rates are significantly positively related to loan-to-value ratio and that the loss rate accounts for between 13% and 20% of total loan-to-value impact.     

Handing Over the Keys: A Perspective on Mortgage Default Research

This paper is the text of the 1992 Presidential Address for the American Real Estate and Urban Economics Association. A comparative evaluation of mortgage default research finds that both the residential and commercial markets evolved from informal underwriting rules, to formalized (though unvalidated) ratios and rules of thumb, to early risk ratings based upon empirical evidence, to gener-alizable econometric models of default, to option-based pricing models. The commercial market lagged the residential market by about 10 to 20 years at first but is now only about five years behind. The survey finds that research and progress in understanding mortgage credit risk has been precipitated by a public policy need or mandate, data availability, and adequate technology. The absence of any one of these factors has hindered progress in the past. Finally, six emerging issues in default research are identified and discussed: (1) the degree of "ruth-lessness" with which default is exercised, (2) loan recourse, (3) the magnitude and timing of revenues and losses associated with default, (4) loan modification, (5) default in a portfolio context, and (6) leasehold default. Progress in these areas will enhance the efficiency of both the residential and commercial markets.     

FHA Terminations: A Prelude to Rational Mortgage Pricing

Recent models of pricing mortgages and/or mortgage insurance have used option-pricing models as their framework. The focus is usually on default, which is viewed as a put option (to sell the house to the lender in exchange for the mortgage) and prepayment, which is viewed as a call option (to buy the mortgage from the lender). Analysis then uses techniques like those used to price options in capital markets. Unfortunately, homeowners do not seem to exercise their option as quickly as do traders in organized markets. We estimate prepayment and default functions, which are meant to be a first step in developing modified, option-based models of mortgage pricing.     

Call Options and Liquidation in Commercial Mortgage Financing

This paper models a commercial real estate project where a wealth-constrained manager uses outside debt financing to purchase a project who's return depends on future economic conditions and the manager's investment in the project. It is shown that it is inefficient to finance the project with callable debt. This prediction is consistent with observed real estate financing practice. I also model the outcome of financial distress allowing for (1) debt forgiveness, (2) equity in exchange for debt forgiveness, and (3) foreclosure. The model motivates (1) why commercial real estate loans are often foreclosed, and (2) why lenders foreclose assets at fire-sale prices.     

Regional Economic Stability and Mortgage Default Risk in the Netherlands

This paper investigates the relationship between regional economic diversification and stability, and residential mortgage default risk in the Netherlands. To describe and measure regional economic diversity and stability, methods from both the regional economics and the industrial economics literature are used. All measures are based on regional employment characteristics. Mortgage default rates were obtained from a database of the population of insured mortgage defaults in the Netherlands from 1983 through 1990. To test the relationship between the measures and mortgage default risk, cross sectional Seemingly Unrelated Regression was used. The paper concludes that the employed measures explain regional mortgage default rates to a significant extent, and that stability measures outperform diversity measures.     

Pricing Mortgage-Backed Securities: Integrating Optimal Call and Empirical Models of Prepayment

Residential mortgage borrowers frequently appear to behave suboptimally with respect to their mortgage prepayment options. Many borrowers fail to exercise even well-into-the-money options while others prepay when the call option is out-of-the-money. To account for these apparently suboptimal prepayments, the recent trend in mortgage-backed securities research has been away from optimal call valuation models, in which the decision to exercise is determined endoge-nously, in favor of models in which prepayment behavior is exogenously specified based on empirical estimation. This paper develops a rational model of mortgage prepayment which incorporates both types of "non-optimal" prepayment and retains endogenous call. This enables the model to disentangle and compare the separate effects of the interest rate call, impeded by transaction costs, and of non-interest-rate driven prepayment. In addition, by recognizing heterogenous borrower transaction costs, the model presents a way to account more precisely for the varying prepayment lags associated with well-into-the-money call options and to account for the phenomenon of "burnout" within a mortgage pool. The paper includes an empirical test of the unbiasedness of the integrated pricing model by comparing simulated prices from our theoretical model to observed prices on traded Fannie Mae and GNMA securities.     

Spatial Heterogeneity in Mortgage Terminations by Refinance, Sale and Default

This article investigates the impact of spatially correlated unobservable variables on the refinancing, selling and default decisions of mortgage borrowers. Virtually the entire mortgage literature acknowledges that borrower-specific characteristics, such as culture, education or access to information, play an important role in mortgage termination decisions. While we do not observe these variables directly, we note that borrowers of similar background tend to cluster together in neighborhoods. We estimate a competing risks hazard model with random effects using a three-stage maximum likelihood estimation approach. We utilize the space-varying coefficient method to modify the covariance structure according to the spatial distribution of the observations. Beyond a significant improvement of the model performance, this yields a number of insightful implications for mortgage termination behavior. For instance, borrowers of the affluent "West Side" of Los Angeles County both refinance and move at a higher rate than predicted by the standard maximum likelihood estimation method. At the same time, borrowers from some lower-valued neighborhoods tend to stay longer than expected with their mortgages and properties.     

A Range-Based Finite Difference Approach to Valuing Options in Markov Switching Regimes*

Recognizing the inflexibility inherent in standard capital budgeting analysis, recent research has provided new insights using a real options framework. This paper uses the explicit finite difference approach to value real options. However, instead of assuming a constant return and volatility term, we assume that these variables are sensitive to changes in the economic environment. Accordingly, we adapt our approach to incorporate a Markov switching feature. Further, we recognize that some of the modeling assumptions can be violated in a practical application. Therefore, we recommend using range based estimates of the real option value, as opposed to a point estimate.     

Hedging Forward Mortgage Loan Commitments: The Option of Futures and a Future for Options

The purpose of this paper is to analyze theoretically the problem of how mortgage lending institutions assume the interest rate risks inherent in the granting of fixed-rate mortgage loan commitments.
Two approaches to hedge this risk are analyzed. First, the use of the GNMA futures market is evaluated from the standpoint of how it might be used to hedge against mortgage commitment risks. Secondly, the use of an appropriate pricing model—the Black-Scholes option pricing model—is offered as a proxy for establishing the market value of a fixed-rate mortgage commitment. This model is extended and empirically estimated for several hypothetical environments.
The paper demonstrates a basic flaw in the GNMA futures market as a hedge against mortgage commitments. Once this is established, the use of the options pricing approach is offered as a more rational approach for hedging these risks.     

Privatized Default Risk and Real Estate Recessions: The U.K. Mortgage Market

A mortgage pricing model is developed when a borrower goes through a series of distress states, including delinquency, long-term nonpayment and ultimate default. These steps are sequential, and depend on prices and alternatives faced by the borrower. The multistate default model is applied to the mortgage market in the United Kingdom. As a byproduct, a pricing structure for the U.K. endowment mortgage, which combines a good and a life insurance policy, is developed. Income and liquidity constraints are shown to affect the decision to keep a mortgage current in different states of distress. Solvent borrowers may thus keep their mortgages current, even when equity is negative.     

Managing Mortgage Interest-Rate Risks in Forward, Futures, and Options Markets

This article reviews the Housing Commission's perspective and recommendations on management of interest-rate risks in housing finance, and considers the relative advantages of various techniques by which institutions on the supply side of mortgage markets can absorb or shift such risks. It is argued that exchange-based options can provide a more reliable way than cash forward contracting for originators or purchasers of mortgages to manage commitment-period risk, but that commitment fees charged household borrowers should not fully correspond to premiums for put options "traded" on the exchanges. It also is argued that exchange-based futures can provide a more effective and economical way than asset-liability maturity matching in cash markets for thrift institutions to manage portfolio interest-rate risks; in particular, futures trading can permit these institution to meet the maturity preferences of liquidity-conscious creditors and risk-averse borrowers, to reduce the risk associated with unexpected shifts of the yield curve, and to maintain a higher degree of asset quality. The capacity of futures markets to handle large-scale hedging by mortgage market participants will depend upon heavy participation by highly leveraged speculators who are willing to take long positions without the receipt of substantial risk premiums from hedgers.     

Industrial Real Estate Mortgage Default Experience of the New York State Job Development Authority Second Loan Program: A Preliminary Investigation

This paper explores the loan loss experience of a public industrial lending authority, employing contemporaneous borrower net equity as a link to mortgage loan default. The relationship between default, net equity and bankruptcy is tested on a small longitudinal data set of loans using nonparametric statistics and a proportional hazard model. Results show that negative net equity and firm bankruptcy are strongly associated with default among the study population. Further, the borrowers studied did not exercise the put option promptly, suggesting potential benefits from monitoring net equity one year or more prior to default.