The probability of fixed- and adjustable-rate mortgage termination

This article analyzes mortgage terminations using a national individual loan data set for the 1986–1992 period. The standard option-choice-theoretic framework is supplemented with variables to proxy for non-option-related termination determinants. Separate multinominal logit models are estimated for three mortgage types: 30-year FRMs, 15-year FRMs, and 30-year ARMs. The results indicate substantial differences in the response of the mortgage types to variables included in the model. FRM15 prepayments are the most responsive to prepayment option values; FRM30 prepayments are less responsive to option values and are dirven by local area housing market and economic conditions; ARM prepayment rates are higher but defeult rates are lower relative to the FRMs. A noteworthy finding is that teaser discounts reduce the likelihood of ARM defaults.     

The valuation at origination of fixed-rate mortgages with default and prepayment

This paper develops a model to rationally price fixed-rate mortgages, using the arbitrage principles of option pricing theory. The paper incorporates amortization, prepayment and default in valuing the mortgage. Having completely specified the model, numerical procedures value the different features of the mortgage contract under a variety of economic conditions. The necessity of having both the interest rate and the house price as explanatory variables, due to the interaction of default and prepayment, is demonstrated. The numerical solutions presented center around mortgage pricing at origination. Thus, variations in the equilibrium contract rate are examined for differing economic conditions and changes in the contract. Finally, by presenting a complete model, the paper yields insights for the existence of common institutional practices.     

Derivative mortgage securities: An overview

Derivative mortgage securities have proliferated since planned amortization and floating rate CMO classes were introduced in late 1986. Other recently created derivative securities include reverse floaters and deep-discount bonds of CMOs, CMO residuals, and stripped and senior/subordinated passthroughs. These securities, which are derived from fixed-rate mortgages, were created to meet investor demands for maturity certainly, interest rate and prepayment hedging, and enhanced credit. The rapid growth of derivative securities reflects expansion of the investor base for fixed-rate mortgages. It also suggests that these mortgages will continue to be a viable housing finance instrument in a volatile interest rate environment. For the future, the increased creation of derivative securities will make the secondary mortgage market more efficient, facilitating the funding of fixed-rate mortgage originations.The substance of this paper was originally written in late 1987 and many of the specific data reflect that time period.     

A Dynamic Analysis of Fixed- and Adjustable-Rate Mortgage Terminations

This paper provides a side-by-side comparison of loan-level statistical models for fixed- and adjustable-rate mortgages. Multinomial logit models for quarterly conditional probabilities of default and prepayment are estimated. We find that the estimated impacts of embedded option values for prepayment and default are generally quite similar across both FRM and ARM loans, providing additional empirical support for the basic predictions of the options theory. We also find that differences in estimates of conditional probabilities of prepayment and default associated with mortgage age, origination period, original LTV, and relative loan size, indicate the continued significance of these other economic and demographic factors for empirical models of mortgage terminations.     

An Analysis of Mortgage Termination Risks: A Shared Frailty Approach with MSA-Level Random Effects

Investigating the residential mortgage defaults and prepayments has been the subject of research for the past three decades. The literature on mortgage default and prepayment is often used to inform credit risk policies and asset pricing strategies. This literature has evolved from the use of logistic regressions to the use of survival and frailty models that control for unobserved heterogeneity. In this paper, we apply a shared-frailty survival model to analyze the mortgage termination risks. In particular, we investigate whether mortgages originated in the same Metropolitan Statistical Area (MSA) share common unobserved factors and how these factors affect the mortgage termination risks. The paper demonstrates that MSA-level frailty, together with other risk factors, has significant effects on the probability of mortgage terminations risks.     

Fixed-Rate Endowment Mortgage and Mortgage Indemnity Valuation

We use a contingent claims framework for valuing the the default and prepayment embedded options in certain British fixed-rate endowment mortgages, with a (capped) mortgage indemnity guarantee (MIG). This methodology provides a template for the borrower, lender, and insurer to compare mortgage terms, including the fairness of contract rates, arrangement fees, prepayment penalties, any MIG premiums required, and co-insurance exposure. With empirical inputs, this model may eventually be useful as a mark-to-value proxy for all parties, as expected parameters change (especially interest rate and house price levels, and expected future volatilities), for purposes of determining valued added accounting, appropriate reserves, and indeed for setting premiums and business drivers. Fixed-rate endowment mortgages differ from fixed-rate repayment mortgages primarily because, in the event of early termination, the amount owed by the borrower is a function of the evolution of the term structure of interest rates, whereas for a repayment mortgage it is pre-determined. We compare endownment and repayment mortgages for different levels of loan-to-value ratios, interest rate and house price volatilities.     

An Early Assessment of Residential Mortgage Performance in China

The residential mortgage market becomes a financial engine for the booming residential housing development and sustained economic growth in China. Our study provides the first rigorous empirical analysis on the earlier performance of residential mortgage market in China based on a unique micro dataset of mortgage loan history collected from a major residential mortgage lender in China. We found that while the option theory fails to explain prepayment and default behavior in the residential mortgage market in China, other non-option theory related financial economic factors play major roles in determining the prepayment and default risks in China. We also found that borrower’s characteristics are significant in determining prepayment behavior, hence may be used as an effective tool for screening potential high risk borrowers in the loan origination process. Adopting a risk-based pricing in residential mortgage lending in China can improve the efficiency of the market, and enhance the credit availability to the most needed households, i.e., the younger households, blue-collar workers, lower income households, and help them become homeowners.     

Credit History and the Performance of Prime and Nonprime Mortgages

Although nonprime lending has experienced steady or even explosive growth over the last decade very little is known about the performance characteristics of these mortgages. Using data from national secondary market institutions, this paper estimates a competing risks proportional hazard model, which includes unobserved heterogeneity. The analysis examines the performance of 30-year fixed rate owner occupied home purchase mortgages from February 1995 to the end of 1999 and compares nonprime and prime loan default and prepayment behavior. Nonprime loans are identified by mortgage interest rates that are substantially higher than the prevailing prime rate. Results indicate that nonprime mortgages differ significantly from prime mortgages: they have different risk characteristics at origination; they default at elevated levels; and they respond differently to the incentives to prepay and default. For instance, nonprime mortgages are less responsive to how much the option to call the mortgage or refinance is in the money and this effect is magnified for mortgages with low credit scores. Tests also reveal that default rates are less responsive to homeowner equity when credit scores are included in the specification.     

Loss given default of high loan-to-value residential mortgages

This paper studies loss given default using a large set of historical loan-level default and recovery data of high loan-to-value residential mortgages from several private mortgage insurance companies. We show that loss given default can largely be explained by various characteristics associated with the loan, the underlying property, and the default, foreclosure, and settlement process. We find that the current loan-to-value ratio is the single most important determinant. More importantly, mortgage loss severity in distressed housing markets is significantly higher than under normal housing market conditions. These findings have important policy implications for several key issues in Basel II implementation.     

Valuation of Residential Mortgage-Backed Securities with Default Risk Using an Intensity-Based Approach

This paper develops a valuation model for fixed-rate mortgages, mortgage pools, and residential mortgage-backed securities (RMBS's) using an intensity-based approach. This model incorporates full prepayment, partial prepayment, and default in valuing a mortgage. Full prepayment is further classified into “refinancing” and “sale of a house” depending on the reason. The time of occurrence of each of these three types of prepayment and default is modeled as the first jump time of a Cox process. Under these conditions, the valuation formula for a mortgage as well as a partial differential equation (PDE) that the mortgage value satisfies is provided. As for implementation of the model, the short-term riskless interest rate and the house price are adopted as state variables. Each intensity process is specified in a manner that allows a jump in intensity depending on the state variables and the borrower's incentive for prepayment or default. Through such specifications, it is shown that our model has characteristics similar to some structural models in previous literature. As for the numerical method for valuation, we propose a simple backward induction technique on a tree instead of the commonly used Monte Carlo method. Additionally, the method for estimating the model is discussed, and the results of numerical simulations are reported.This paper represents the view of the author and does note necessarily the views of the Mitsubishi UFJ Securities Co., Ltd. or members of its staff.     

Valuing mortgage loan servicing

We develop an intertemporal model for valuing mortgage loan servicing contracts. The model includes a stochastic short-term interest rate and realized inflation rate which jointly determine the current mortgage coupon rate, the mortgagor's prepayment decision, the servicer's future net cash flows, and the rate at which to discount these future cash flows. Several potential uses of the model for institutions that service mortgages and trade servicing portfolios are illustrated by the application of the model to servicing fixed-rate mortgages and adjustable-rate mortgages. The model also is applicable to regulatory issues and to the servicing of other types of loans.The authors gratefully acknowledge support from the Federal Home Loan Bank Board, Washington, D.C., the Purdue Research Foundation, West Lafayette, Indiana, and the Richard D. Irwin Foundation, Homewood, Illinois.     

The impact of the agencies on conventional fixed-rate mortgage yields

Between the early 1980s and 1986, the share of new conforming (under $153,000 in 1986) conventional fixed-rate mortgages (FRMs) that went into Fannie Mae and Freddie Mac mortgage pools increased from under 5% to over 50%. The impact of these agencies moving from negligible participants to dominant players in this market is investigated in this study by an analysis of yields on 4,900 loans closed in California during May–June 1978 and 1,800 closed in May–June 1986.Our analysis indicates that the loan rate depends on the loan-to-value ratio, the loan size, and, in 1986, whether the loan is far above, just above, or below the conforming loan limit. Rates on loans far above the conforming loan limit exceed those on otherwise comparable loans below the limit by 30 basis points and those on loans destined to exceed the limit within a year by 15 basis points. That is, the expanded agency securitization of conforming FRMs has significantly lowered the rates on both conforming loans and loans somewhat above the conforming limit (27% of nonconforming loans in 1986) relative to what they would otherwise have been.Earlier versions of this paper were presented at an OSU Finance Seminar and the Midyear Meetings of the American Real Estate and Urban Economics Association. We acknowledge the financial support of the Garn Institute.     

Omitted Mobility Characteristics and Property Market Dynamics: Application to Mortgage Termination

Property market dynamics depend on changes in long run equilibrium and on impediments to adjustment towards equilibrium. Mortgage termination due to mobility, default and refinancing provides a lens for evaluating property market adjustments. The borrower’s decision to move as an adjustment mechanism in the property market is associated with utility-maximizing decisions to either prepay or default on the mortgage. The optimal choice between these two termination events may depend on unobserved propensities related to change in income, job location, or family size, and substantial inertial forces including search costs, neighborhood change and attachment to an area. We propose a method for modeling variables determining the impact of mobility on mortgage terminations with imperfect household and loan level data. Since omitted variables contribute to moving decisions and therefore to mortgage prepayment and default decisions, utility functions for sale and default are correlated through these unobservable variables; thus, the IIA assumption of the widely used Multinomial Logit Model (MNL) is violated. Under such circumstances, econometric theory suggests that the Nested Logit Model (NMNL) is a better choice, which obviates the limitation of MNL by allowing correlation in unobserved factors across alternatives. Using loan level micro data, we find empirical evidence showing significant correlation between sale and default due to omitted borrower mobility characteristics. Our simulations find that NMNL out performs MNL in out-of-sample prediction. Improved predictions of moves and defaults are applicable to micro and macro analysis of the housing market system.     

Leverage and Mortgage Foreclosures

In this paper, we deduce the default and prepayment characteristics of mortgages by examining the actual behaviors of a large set of conforming fixed rate mortgages tracked over time. Employing reduced form pricing techniques, we are then able to fully value such mortgages, and so determine the cost as well as the probability of default for any particular mortgage. The analysis reveals the levels of foreclosures that can be expected when loans are leveraged at the high loan–to–value ratios characteristic of recent years.     

Mortgage Termination: An Empirical Hazard Model with a Stochastic Term Structure

Pricing for mortgage and mortgage-backed securities is complicated due to the stochastic and interdependent nature of prepayment and default risks. This paper presents a unified economic model of the contingent claims and competing risks of mortgage termination by prepayment and default. I adopt a proportional hazard framework to analyze these competing and interdependent risks in a model with time-varying covariates. The paper incorporates a stochastic interest rate model into the hazard function for prepayment. The empirical results reported in the paper provide new evidence about the ruthlessness of default and prepayment behavior and the sensitivity of these decisions to demographic as well as financial phenomena. The results also illustrate that evaluating the interest rate contingent claims with a stochastic term structure has effects on predicting not only the mortgage prepayment behavior but also the mortgage default behavior.     

An Improved Fixed-Rate Mortgage Valuation Methodology with Interacting Prepayment and Default Options

This paper considers in detail a realistic mortgage valuation model (including the potential for early prepayment and the risk of default), based on stochastic house-price and interest-rate models. As well as the development of a highly accurate numerical scheme to tackle the resulting partial differential equations, this paper also exploits singular perturbation theory (a mathematically rigorous procedure, based on the idea of the smallness of the volatilities), whereby mortgage valuation can be accurately approximated by very simple closed-form solutions. Determination of equilibrium contract rates, previously requiring many computational hours is reduced to just a few seconds, rendering this a highly useful portfolio management tool; these approximations compare favorably with the full numerical solutions. The method is of wide applicability in US or other mortgage markets and is demonstrated for UK fixed-rate mortgages, including insurance and coinsurance.     

Housing-Price Cycles and Prepayment Rates of U.S. Mortgage Pools

Empirical mortgage prepayment models generally have trouble explaining differences in mortgage-prepayment speeds among pools with similar interest rates on the underlying mortgages. In this article, we model some of the sources of termination heterogeneity across mortgage pools, particularly the role of regional variations in housing prices in generating atypical prepayment speeds. Using a sample of Freddie Mac mortgage pools from 1991 to 1998, we compare two classes of empirical models: a rational option-pricing model using a backward-solving pricing algorithm and an empirical hazard model. In both empirical estimation strategies, we find evidence that differences in house-price dynamics across regions are an important source of between-pool heterogeneity. This finding is then shown to be robust to alternative ways of parameterizing pool heterogeneity in mortgage termination models.     

The Effects of Purchases of Mortgages and Securitization by Government Sponsored Enterprises on Mortgage Yield Spreads and Volatility

In this paper, we investigate the effects of GSE (government sponsored enterprise) activities on mortgage yield spreads and volatility. Using various regression procedures (i.e., vector error correction (VEC) and GARCH models) and controlling for default and prepayment risk, we find that securitizations and purchases of mortgages by GSEs reduce mortgage yield spreads and volatility. In particular, we find that the yield spread between conforming and 10-year constant maturity treasury (CMT) rates decreases by 8.0 bp per $1billion increase in the level of GSE securitizations. Similarly, if GSEs increase mortgage purchases, the yield spread decreases 10.5 bp per $1billion increase of purchases. In addition, we hypothesize and find that GSE activities have a spillover effect to the non-conforming mortgage market; via investor substitutions, GSE purchases and securitizations of conforming loans reduce non-conforming loan rates. Thus, the measured influence of GSE activities is biased downward when measured using the spread of non-conforming loans over conforming loan rates. We also find that purchases of mortgages by GSEs significantly reduce mortgage yield volatility. In sum, our findings show that GSE activities reduce and stabilize mortgage market rates.     

Fixed‐Rate Mortgages,Labor Markets,and Efficiency

The paper studies the effects of mortgage choices between fixed‐rate mortgages (FRMs) and adjustable‐rate mortgages (ARMs) on labor market efficiency. FRMs provide insurance for risk‐averse borrowers in the sense that they pay the same rate over time and are not subject to uncertain spot market rates. FRMs, however, discourage borrowers from moving to other regions despite better employment opportunities, as they terminate the FRM contracts in order to move and their new loan interests may be higher. As FRM‐borrowers do not move to other regions due to the interest lock‐ins, entrepreneurs in other regions lose the potential surpluses from productive matches that would have occurred between borrowers‐workers and entrepreneurs. Borrowers ignore this negative externality they impose on the entrepreneurs when choosing their mortgages, and too many borrowers choose FRMs relative to the efficient level. If FRMs are eliminated and ARM‐insurance (that protects ARM‐borrowers against uncertain adjustable interest rates) is created, it will improve efficiency. The paper also assesses quantitatively the welfare effects of eliminating FRMs and providing ARM‐insurance.     

A Model of Mortgage Default

In this paper, we solve a dynamic model of households' mortgage decisions incorporating labor income, house price, inflation, and interest rate risk. Using a zero‐profit condition for mortgage lenders, we solve for equilibrium mortgage rates given borrower characteristics and optimal decisions. The model quantifies the effects of adjustable versus fixed mortgage rates, loan‐to‐value ratios, and mortgage affordability measures on mortgage premia and default. Mortgage selection by heterogeneous borrowers helps explain the higher default rates on adjustable‐rate mortgages during the recent U.S. housing downturn, and the variation in mortgage premia with the level of interest rates.