Pricing Default Risk in Mortgages

This paper examines the valuation of fixed-rate mortgages and the pricing of insurance against default on such mortgages. Both the mortgage and the insurance are treated as compound European put options. A put is the right, but not the obligation, to turn over an asset to another party for a specified payment, and being a European put indicates that this can only occur at a specified expiration date. The mortgage contract, and hence the insurance on it, fit into a European option framework because no rational borrower would ever choose to default until a payment is due. Mortgages are compound options in nature because at each payment data prior to the last one, the borrower either defaults or purchases a new option to default at the next payment date by making the scheduled payment. Since the current value of the mortgage is affected by options to default in the future, the problem is solved working backwards in time with the value of later options feeding into the earlier ones, so that the process builds on itself in a recursive fashion. Using familiar arguments from option-pricing theory, the value of any of the assets in the model is expressed as the solution to a partial differential equation, where the terms of the contract yield the appropriate terminal conditions. Standard numerical procedures are then used to produce the value of the mortgage and the insurance under various economic conditions. The simulations indicate that the prime determinants of the value of the assets considered are the volatility of the house price and the volatility of the spot interest rate. Sensitivity tests show that changing either of these parameters affects the results substantially more than any of the other parameters examined. The paper completely analyzes the default option and insurance against default on the mortgage. It is one part of a complete model of fixed-rate mortgages that would allow for both prepayment and default and treat the interaction of the two options. The general approach outlined in this paper can be used to develop such a model as well as to value any mortgage-related security. In light of the increasing variety and the complexity of such instruments in the market today, the presentation of our approach to these valuation problems is perhaps the most important contribution of the paper.     

Prepayment Behavior of Dutch Mortgagors: An Empirical Analysis

The suboptimal exercise of the prepayment option in a mortgage is relevant for mortgage pricing and the management of a mortgage portfolio. Construction of an accurate prepayment model requires quantification of driving factors such as seasoning, seasonality, refinance incentive and burnout. We focus on Dutch mortgages but also discuss the Dutch market in a European setting. Within the euro-denominated MBS market, the Dutch market is often referred to as the benchmark market. In our application we include typical Dutch market and contract characteristics such as the annual penalty-free prepayment of 10 to 20% of the original loan amount. We use loan-level historical data on mortgages originated between January 1989 and June 1999 to estimate separate models for two popular redemption types: savings mortgages and interest-only mortgages. In both models we allow for suboptimal prepayment behavior. The results clearly indicate that prepayment rates depend on interest rates and the age of the mortgage contract. Moreover, we find that burnout is an important element in describing the prepayment behavior of Dutch mortgagors.     

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.     

Optimal Mortgage Refinancing with Stochastic Interest Rates

The purpose of this paper is to develop a dynamic model of mortgage refinancing in a contingent claim framework that simultaneously solves for the borrower's optimal mortgage refinancing strategy, the value of the refinancing call option, the value of the mortgage liability to the borrower, and the market (lender) value of the fixed-rate contract. We also calculate the minimum differential between the contract rate on the existing mortgage and the current interest rate that is required to trigger an optimal mortgage refinancing.     

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.     

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.     

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.     

Mortgage Refinancing with Asymmetric Information

Information asymmetry exists between the lender and the borrower regarding the holding period of the mortgaged real estate; the lender does not know how long the borrower plans to own the house. This information asymmetry allows the cost of obtaining a mortgage to deviate from its value to the borrower. As a result, the exercise price of the option to refinance becomes the cost to the borrower of obtaining a new mortgage instead of the outstanding balance of the existing mortgage as used in previous models. The option to refinance is a sequential option; after the borrower refinances, a new option is obtained to refinance again in the future. A mortgage refinancing model is developed taking information asymmetry and sequential refinancing into account. The model is used to solve for (1) the value to the borrower of a callable mortgage and (2) the minimum interest rate differential between the contract rate of the existing mortgage and the market interest rate needed to justify refinancing.     

Mortgage Pricing: Some Provisional Empirical Results

We test some of the qualitative properties of mortgage pricing models. The models use option pricing techniques, focusing on prepayment as a call option. They imply a quite nonlinear relationship between mortgage price and coupon, interest rates and volatility. We test for both the first and second derivatives of the effects of these variables using data on Ginnie Mae mortgage backed securities. We find that the model is largely supported by the data.     

Mortgage Borrower Repayment Behavior: A Microeconomic Analysis with Canadian Adjustable Rate Mortgage Data

In this paper we estimate a model of mortgage borrower behavior using micro-level data on Canadian borrowers with rollover mortgages—a form of adjustable-rate mortgage. Our results suggest that the probability of default rises with a decrease in housing equity and an increase in the mortgage contract rate; however the size of these changes is relatively small. They also show that partial prepayment is sensitive to fluctuations in the rates of return from investing in housing versus other assets. For the United States experience, our results suggest that, relative to fixed-rate mortgage borrowers, adjustable-rate mortgage borrowers are more likely to default and less likely to prepay.     

An Arbitrage-Free Estimate of Prepayment Option Prices in Fixed-Rate GNMA Mortgage-Backed Securities

In an efficient market, the no-arbitrage condition implies that the price difference between any two assets must be the market value of all differences in their cash flows. We use this logic to deduce the price of the prepayment option embedded in fixed-rate Government National Mortgage Association (GNMA) mortgage-backed securities. The option price equals the difference between an observed GNMA price and the cost of a synthetic, nonprepayable GNMA constructed from the least expensive portfolio of Treasury securities that exactly replicates the promised GNMA cash flow stream, assuming prepayment is precluded. We regress the option prices on variables found significant in previous prepayment studies, finding that five key regressors explain more than 90% of the prepayment option value in pooled time-series cross-sectional analysis. We also show that the time value of the prepayment option calculated by our method displays a pattern similar to that produced by the Black-Scholes (1973) option pricing model. An additional empirical result is the existence of negative option prices and negative time value of the option prices. We attribute these to the fact that homeowners sometimes exercise their prepayment options when they are out-of-the-money, and to refinancing transaction costs. Our method is independent of assumptions regarding interest rate processes and the homeowner's prepayment behavior, and it provides a benchmark for testing theoretical prepayment models.     

Explaining Refinancing Decisions Using Microdata

This paper develops a model which explains how mortgage-rate movements, transactions costs, changes in borrower income and house value, personal financial opportunities and the prepayment option embedded in fixed-rate mortgages affect a financially flexible borrower's decision to refinance an existing loan while retaining the underlying home. Broadening the focus of previous analytical work, the model explains why households with similar mortgage loans may react differently as financial market conditions change. It contains definitive empirical predictions that are supported by an analysis of a choice-based sample of individual loan transactions. Results suggest that refinancings are motivated both by movements in the level of interest rates and by borrowers' desires to alter their capital structures in the face of changing income and housing wealth.     

UK Fixed Rate Repayment Mortgage and Mortgage Indemnity Valuation

We use a mean-reverting interest rate model and a lognormal house price diffusion model to evaluate British fixed rate repayment mortgage contracts with (embedded) default and prepayment options. The model also provides values for capped mortgage indemnity guarantees and the corresponding (residual) lender's coinsurance. Since the partial differential equation incorporating the general features of these mortgage contracts does not have a closed-form solution, an explicit finite difference method is used for the valuation (and sensitivity) results, with solution improvements to deal with error bounds. Then we provide graphical representations of each mortgage component as a function of house prices and interest rate levels, along with interpretations of the analysis. We calculate precisely the lender's (residual) exposure to house price risk, given the borrower's options, house and interest rate uncertainty, and customary mortgage indemnity insurance for high loan/collateral ratio mortgages.     

Efficiency in the Mortgage Market: The Borrower's Perspective

This paper uses mortgage history data from the Federal Home Loan Mortgage Corporation to analyze the prepayment behavior of homeowners and to test whether borrowers exercise their prepayment options in a manner consistent with contingent claims models. A variety of hazard models are estimated from individual data on more than 6000 mortgages issued during the 1976–1980 period. In these models, it is clear that the extent to which the prepayment option is "in the money" has a strong effect on behavior. However, it is less clear that the option is exercised quite as ruthlessly as the theory predicts.     

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.     

Implied Mortgage Refinancing Thresholds

The optimal prepayment model asserts that rational homeowners will refinance if they can reduce the current value of their liabilities by an amount greater than the refinancing threshold, defined as the cost of carrying the transaction plus the time value of the embedded call option. To compute the notional value of the refinancing threshold, researchers have traditionally relied on discrete- or continuous-time option-pricing models. Using a unique loan level database that links homeowner attributes with property and loan characteristics, this study proposes an alternative approach for estimating the implied value of the refinancing threshold. This empirical method enables us to measure the minimum interest-rate differential needed to justify refinancing conditional on the borrower's creditworthiness, loan-to-value ratio and other observable characteristics.     

Fixed Come Hell or High Water? Selection and Prepayment of Fixed‐Rate Mortgages Outside the United States

We examine the decision to prepay a fixed‐rate mortgage in the United Kingdom, Canada, Ireland, Australia and New Zealand. These countries are characterized by having substantial fees which are associated with breaking a fixed‐rate mortgage. We develop a model which allows for fluctuations both in banks’ wholesale rates and credit spreads. We find that households can achieve economically significant benefits both from following an optimal prepayment strategy contingent on the break fee used by their bank and also by selection of fixed interest rate term and (where available) break fee structure.     

ARM Wrestling: Valuing Adjustable Rate Mortgages Indexed to the Eleventh District Cost of Funds

This article analyzes adjustable rate mortgages (ARMs) based on the Eleventh District Cost of Funds Index (EDCOFI). The behavior of EDCOFI was examined over the period 1981–1993. Adjustments in this index lag substantially behind term structure fluctuations. Also, the seasonality and days-in-the-month effects noted by previous authors are really symptoms of a January effect . A finite difference valuation algorithm was developed which accounts for all usual ARM contractual features, in addition to the dynamics of EDCOFI. This pricing algorithm allows us to determine endogenously the optimal prepayment strategy for mortgage holders, and hence the value of their prepayment options. The dynamics of EDCOFI give significant value to this option, typically around 0.5% of the remaining principal on the loan.     

On Option-Pricing Models in Real Estate: A Critique

This paper considers the problems peculiar to options on real estate, because of the special set of institutional factors influencing real estate markets. It is intended to serve as a reply to Johnson and Wofford [15] as well as provide an overall critique of option-pricing models in a real estate context. Our major point is that a variety of real estate decisions, such as the abandonment decision, the option to refinance, or the option to exercise a contingent real estate purchase contract, may be modeled using option-pricing techniques. However, both the theoretical and institutional aspects of real estate markets must be taken into account in both developing and applying option models in a real estate context.     

Regressive Interest Rate Expectations and Mortgage Instrument Choice in the United Kingdom Housing Market

The paper considers the choice of mortgage instrument when the rate of interest is fixed for a short duration, with reversion to a variable (bullet) rate mortgage contract. The research is the first direct test for regressive interest rate expectations using United Kingdom data while testing for wealth and portfolio effects. The econometric modeling uses a variety of nonparametric and parametric techniques to control for classification error in the dependent variable. There is evidence that households adopt regressive interest rate expectations. The lack of statistical significance of wealth and portfolio effects confirms the short run cash flow perspective of United Kingdom mortgage choices.