Optimal prepayment and default rules for mortgage-backed securities

We study the optimal stopping problems embedded in a typical mortgage. Despite a possible non-rational behaviour of the typical borrower of a mortgage, such problems are worth to be solved for the lender to hedge against the prepayment risk, and because many mortgage-backed securities pricing models incorporate this suboptimality via a so-called prepayment function which can depend, at time t, on whether the prepayment is optimal or not. We state the prepayment problem in the context of the optimal stopping theory and present an algorithm to solve the problem via weak convergence of computationally simple trees. Numerical results in the case of the Vasicek model and of the CIR model are also presented. The procedure is extended to the case when both the prepayment as well as the default are possible: in this case, we present a new method of building two-dimensional computationally simple trees, and we apply it to the optimal stopping problem.     

A two-stage segment and prediction model for mortgage prepayment prediction and management

Predicting the risk of mortgage prepayments has been the focus of many studies over the past three decades. Most of these works have used single prediction models, such as logistic regressions and survival models, to seek the key influencing factors. From the point of view of customer relationship management (CRM), a two-stage model (i.e., the segment and prediction model) is proposed for analyzing the risk of mortgage prepayment in this research. In the first stage, random forests are used to segment mortgagors into different groups; then, a proportional hazard model is constructed to predict the prepayment time of the mortgagors in the second stage. The results indicate that the two-stage model predicts mortgage prepayment more accurately than the single-stage model (non-segmentation model).     

Optimal prepayment of Dutch mortgages*

Valuation of the prepayment option in Dutch mortgages is complicated. In the Netherlands, mortgagors are not allowed to prepay the full mortgage loan without a compensating penalty. Only a limited amount of the initial mortgage loan can be prepaid penalty‐free. We introduce a general model formulation for the valuation of limited callable mortgages, based on binomial trees. This model can be used for determining both the optimal prepayment strategy and the value of embedded prepayment options. For some mortgage types the prepayment option can be valued exactly, whereas other types require approximative methods for efficient valuation. The heuristic we propose here determines the prepayment option value efficiently and accurately for general mortgage types.     

Intensity-based framework and penalty formulation of optimal stopping problems

Financial derivatives commonly contain premature termination clauses, which are embedded rights held by the holder or writer. Well known examples of these stopping rights include the early exercise right in American options, the callable right in callable securities and the prepayment right in mortgage loans. In this paper, we show how to model the mortgagor's prepayment in mortgage loans and the issuer's call in the American warrant as an event risk using the intensity based approach, where the propensity of prepayment or calling is modeled by the intensity of a Poisson process. We illustrate that the corresponding pricing formulation resembles the penalty approximation approach commonly used in the solution of the linear complementarity formulation of an optimal stopping problem. We obtain several theoretical results on the prepayment strategies of mortgage loans and calling policies of American warrants. We also propose robust second order accurate numerical schemes for solving the penalty formulation of an optimal stopping problem.     

个人住房按揭贷款提前偿付风险研究

住房按揭贷款在国内银行的资产中正占据着越来越重要的地位,与此同时,按揭贷款的提前偿付行为也困扰着银行业的经营。本文分析了影响住房按揭贷款提前还款的主要因素,尝试建立一个符合我国目前状况的能够预测提前还款模型框架,同时提出我国商业银行个人住房按揭贷款提前还款风险管理的一些策略建议。     

Mortgagor Motivations in Prepayments for Adjustable Rate Mortgages

This paper provides the first rigorous analysis of residential adjustable mortgage prepayment using individual ARM mortgage data in Singapore. The prepayment rate for residential mortgages is low and is dominated more by macroeconomic factors than mortgage–specific factors. Specifically, the prepayment rate is increasing in residential property prices, but decreasing in income as proxied by GDP and volatility in mortgage rates. There is weak evidence to suggest that prepayment is increasing in the borrower's age, mortgage rate hikes, cash–availability variables and sentiments of the stock market, and decreasing in the price premium over valuation, payment–to–income ratio, loan–to–value ratio, loan term and floor level of the property.     

Mortgage default in an estimated model of the U.S. housing market

This paper models the housing sector, mortgages and endogenous default in a DSGE setting with nominal and real rigidities. We use data for the period 1981–2006 to estimate our model using Bayesian techniques. We analyze how an increase in risk in the mortgage market raises the default rate and spreads to the rest of the economy, creating a recession. In our model two shocks are well suited to replicate the subprime crisis and the Great Recession: the mortgage risk shock and the housing demand shock. Next we use our estimated model to evaluate a policy that reduces the principal of underwater mortgages. This policy is successful in stabilizing the mortgage market and makes all agents better off.     

Structural prepayment risk behavior of the underlying mortgages for residential mortgage life insurance in a developing market

Since 1998 all residential mortgages in China have been adjustable rate mortgages (ARMs). However, the borrower’s motivation for prepayment is different from that in the US or other developed mortgage markets. In the US, mortgage insurance plays an imperative role in covering some of the risk typically faced by housing finance institutions. However, China’s residential mortgage life insurance (RMLI) market is in its infancy. It offers the insured mortgagor a life-insurance death benefit, arising from only illness or accident, settling the insured’s outstanding residential mortgage balance. Prepayments of some RMLIs’ underlying mortgages are observed, leading to a premature termination of both the residential mortgage and the insurance commitment to settle the outstanding mortgage balance even though the insured has not yet passed away. Because such prepayments significantly influence the pricing of the RMLI, it is imperative to know more about the prepayment rate of occurrence and the prepayment characteristics of the underlying residential mortgages in terms of observable macro economic factors, loan specific factors and borrower specific characteristics. Hence, this study investigates the prepayment risk behavior of the underlying mortgages for RMLIs, utilizing a pilot study of 1000 Shanghai residential mortgagors who took up RMLIs between January 1999 and December 2003. This study uses the Cox proportional hazard model to investigate RMLI-mortgage prepayment risk behavior. The resultant hazard rate is dependent on four primary factors: combined monthly income of the co-borrowers, growth in the gross domestic product, number of co-borrowers and initial loan-to-value ratio.     

Mortgage Valuation with Optimal Intertemporal Refinancing Strategies

This article extends previous work on mortgage valuation in two ways. First, I identify the prepayment boundary by solving for the borrower's optimal prepayment strategy over the expected tenure in the house. Previous work has treated the prepayment decision as a one-time decision, not as an element of a multiperiod strategy. Second, the model incorporates borrower heterogeneity in terms of expected tenure in the house. The results show that the optimal refinancing strategy differs significantly from a sequence of one-time decisions. A borrower following the optimal strategy is less aggressive in refinancing and pays more interest and less transaction costs than does a borrower following a myopic strategy. Estimated mortgage values are higher and interest rate sensitivity is lower when compared to values calculated using the traditional approach.     

Valuation of fixed and variable rate mortgages: binomial tree versus analytical approximations

The recombining binomial tree approach, which has been initiated by Cox et?al. (J Financ Econ 7: 229?C263, 1979) and extended to arbitrary diffusion models by Nelson and Ramaswamy (Rev Financ Stud 3(3): 393?C430, 1990) and Hull and White (J Financ Quant Anal 25: 87?C100, 1990a), is applied to the simultaneous evaluation of price and Greeks for the amortized fixed and variable rate mortgage prepayment option. We consider the simplified binomial tree approximation to arbitrary diffusion processes by Costabile and Massabo (J Deriv 17(3): 65?C85, 2010) and analyze its numerical applicability to the mortgage valuation problem for some Vasicek and CIR-like interest rate models. For fixed rates and binomial trees with about thousand steps, we obtain very good results. For the Vasicek model, we also compare the closed-form analytical approximation of the callable fixed rate mortgage price by Xie (IAENG Int J Appl Math 39(1): 9, 2009) with its binomial tree counterpart. With respect to the binomial tree values one observes a systematic underestimation (overestimation) of the callable mortgage price (prepayment option price) analytical approximation. This numerical discrepancy increases at longer maturities and becomes impractical for a valuable estimation of the prepayment option price.     

Some Truths about Ostriches: Who Doesn't Prepay Their Mortgages and Why They Don't

Prepayment estimation is essential in forecasting expected mortgage cash flow patterns. Accordingly, mortgage and mortgage-backed security prices are highly dependent on prepayment assumptions. Yet borrower prepayment behavior appears to be highly irrational, in the sense that many borrowers prepay their mortgages when it is not optimal to do so and fail to prepay their mortgages when the prepayment option is substantially in the money. In this paper, we explore the latter phenomenon, using a large data set of loans originated during the relatively high-rate 1980s that have failed to prepay by year-end 1996. As a control group, we examine similar loans that did prepay during the refinancing boom of 1993. By coupling Case–Shiller house price index information at the zip-code level, we can analyze the effect of broader housing market trends, especially housing appreciation, on borrower prepayment behavior. Although housing prices did decline significantly during the late 1980s and early 1990s in many areas of the United States, we find evidence that only about 25% of non-refinancing households might have been constrained by declining collateral values. Household demographic characteristics may provide some further explanation for this apparently non-optimizing behavior; however, we do not have a complete explanation.     

Credit rationing, race, and the mortgage market

This study applies microdata from the 1983 Survey of Consumer Finances to evaluate the effects of borrower race and default risk in mortgage lending. The empirical analysis is based on a probit model of whether borrowers obtain FHA or conventional mortgages; the former are fully insured and are characterized by easier downpayment constraints, but are typically more expensive. Hence, households borrowing through the FHA will tend to be credit constrained in the conventional market. Results of the analysis indicate that variables which proxy lender concerns about default risk and cost have an important effect on the type of loan borrowers obtain. Empirical estimates also suggest that minority households are significantly less likely to obtain conventional financing than whites, even after controlling for various proxies of default risk. These results suggest that race effects in mortgage lending may persist for reasons unrelated to borrower default risk.     

Shared appreciation mortgages: Lessons from the UK

The recent rise in shared appreciation mortgage (SAM) availability motivates careful consideration of underlying borrower incentives. The lender’s share of appreciation in SAMs (share) is essentially a dynamic prepayment penalty imposed on the borrower. However, the borrower faces a moral hazard due to his ability to affect the penalty by reducing maintenance. We adapt a competing risks mortgage-pricing model to calculate SAM theoretical equilibrium rates. Our borrower possesses rational expectations of both the house price market and interest rates. Our simulation results may help explain the lack of secondary market interest for the UK SAMs containing extreme contract terms.     

Reconciling theory and empirics on the role of unemployment in mortgage default

Empirical models of mortgage default typically find that the influence of unemployment is negligible compared to other well known risk factors such as high borrower leverage or low borrower FICO scores. This is at odds with theory, which assigns a critical role to unemployment in the decision to stop payment on a mortgage. We help reconcile this divergence by employing a novel empirical strategy involving simulated unemployment histories to measure the severity of attenuation bias in loan-level estimations of default risk due to a borrower becoming unemployed. Attenuation bias results because individual data on unemployment status is unobserved, requiring that a market-wide unemployment rate be used as a proxy. Attenuation is extreme, with our results suggesting that the use of an aggregate unemployment rate in lieu of actual borrower unemployment status results in default risk from a borrower becoming unemployed being underestimated by a factor more than 100. In addition, our analysis indicates that adding the unemployment rate as a proxy for the missing borrower-specific unemployment indicator does not improve the accuracy of the estimated model over the specification without the proxy variable included. Hence, aggregate portfolio-level risk estimates for mortgage guarantors such as FHA also are not improved.These views represent those of the authors and not necessarily those of the Federal Reserve Bank of New York or the Federal Reserve System. This is a revised version of a paper that previously circulated under the title “Unemployment and Unobserved Credit Risk in the FHA Single Family Mortgage Insurance Fund (NBER Working Paper No. 18880). John Grigsby provided excellent research assistance. We appreciate the helpful comments of Andrew Haughwout, Wilbert van der Klaauw, the editor (Stuart Rosenthal) and referees, but remain responsible for any errors.     

Using a decision tree to analyze mortgage borrower decision behavior and values concerning interest rates and house prices

In reality, mortgage borrowers are more seriously concerned with the current mortgage boundary (i.e. option exercise) value than with the current option value (i.e. expected present value of the prospective option exercise value). Hence, by combining a simulation framework and a decision tree, the terminations of mortgage behavior can be classified forward but not backward as by the binomial lattice. After simulating 5000 projections for both Taiwan house prices and interest rates, as well as computing for current mortgage boundary values obtained by modifying Ambrose and Buttimer ( 2000 ) to step through the mortgage decision tree, the result shows that the prepayment is affected by rising interest rate volatility. Moreover, the delinquency and the reinstatement are affected by both rising interest rate and house price volatilities. However, due to the cost of delinquency and credit penalties, the foreclosure could not compete over the reinstatement when house prices and interest rates are in a high‐volatility situation. The reinstatement is encouraging for the borrowers. Copyright © 2010 John Wiley & Sons, Ltd.     

An analysis of default risk in the Home Equity Conversion Mortgage (HECM) program

While reverse mortgages are intended as a tool to enable financial security for older homeowners, in 2014, nearly 12 percent of reverse mortgage borrowers in the federally insured Home Equity Conversion Mortgage (HECM) program were in default on their property taxes or homeowners insurance. Unlike the traditional mortgage market, there were no risk-based underwriting guidelines for HECMs through 2014. In response to the relatively high default rate, a variety of policy responses were implemented, including establishing underwriting guidelines. However, there is a lack of data and analysis to inform such criteria. Our analysis follows 30,000 seniors counseled for reverse mortgages between 2006 and 2011. The data includes comprehensive financial and credit report attributes, not typically available in analyses of reverse mortgage borrowers. Using a bivariate probit model that accounts for selection, we estimate the likelihood of tax and insurance default. Financial characteristics that increase default risk include the percentage of funds withdrawn in the first month of the loan, a lower credit score, higher property tax to income ratio, low or no unused revolving credit, and a history of being past due on mortgage payments or having a tax lien on the property. Our estimate of the elasticity of default with respect to credit scores is similar to that for closed-end home equity loans, but higher than that for HELOCs. We simulate the effects of alternative underwriting criteria and policy changes on the probability of take-up and default. Reductions in the default rate with a minimal effect on participation can be achieved by requiring that participants with low credit scores set aside some of their HECM funds for future property tax and insurance payments, a form of escrowing.     

Modified duration and convexity with semiannual compounding

Both duration and convexity are a function of the curvilinear bond price: yield relationship. While duration measures the slope of the price:yield curve at a given yield-to-maturity, convexity measures the change in duration at this yield-to-maturity. Three shortcomings exist in the presentation of bond price volatility in financial education. First, modified duration and convexity should be used together as measures of bond price volatility. Second, these measures of bond price volatility should properly reflect semiannual compounding. Third, simple linear models for modified duration and convexity should be generally adopted in financial education literature.     

Equilibrium in collateralized asset markets: Credit contractions and negative equity loans

We address a general equilibrium model with collateralized debt, credit contractions, and financial market segmentation. Restrictions on credit access make borrower’s optimal payment strategies–coupon payment, prepayment, and default–sensitive to idiosyncratic factors, even though the only payment enforcement is the seizure of collateral guarantees. We prove equilibrium existence, characterize optimal borrower’s payment strategies, and provide a numerical example illustrating our main results. A remarkable feature of our model is that it rationalizes the prevalence of negative equity non-recourse loans.     

Deficiency Judgments and Borrower Maintenance: Theory and Evidence

It is well known that when property rights to an asset are divided, individual right holders may not have adequate incentives to invest in proper maintenance. In this paper, weexamine how mortgage laws affect the nature of the lender's claim to the house, and howthat claim in turn affects the incentives of borrowers to invest in home maintenance. Thespecific law that we examine concerns the right of lenders to pursue a borrower's nonhous-ingwealth in the event of default if the value of the house is less than the mortgage balance. Most states allow lenders to collect such “deficiency judgments,” while others either prohibit, or make it difficult to obtain them. The theoretical model developed inthis paper predicts that borrowers will maintain at a higher rate when lenders are allowedto seek deficiency judgments. Intuitively, when borrowers' nonhousing wealth is at risk,they have an incentive to invest more in maintenance in order to reduce the likelihood that the value of the property will fall below the mortgage balance. We attempt to measure this effect using data on household maintenance obtained from the American Housing Survey along with information on differences in mortgage laws across states. We estimate a three-equation simultaneous system relating maintenance expenditures, house value, and mortgage rates. The results provide confirmation that variation in mortgage laws affect homeowner maintenance in the manner predicted by the theory.