Analyses of Mortgage-Backed Securities Based on Unobservable Prepayment Cost Processes

We propose a prepayment model of mortgage based on a structural approach in order to analyze prepayment risk of mortgage-backed securities (MBS). We introduce a continuous process named prepayment cost process. Specifically, each mortgager's prepayment time is defined by the first time when her or his prepayment cost process falls below zero, but prepayment cost processes are supposed to be unobservable in the market. We also introduce a risk unique to each loan pool of mortgages, called a loan pool risk (LPR), and we regard LPR as a systematic risk other than interest rate. Using the model, we discuss the conditional distribution of prepayment times and a risk-neutral valuation of pass-through MBS. It is shown that each mortgager's conditional non-prepayment probability and the posterior distribution of LPR play quite important roles in our study.This research is partially supported by Grant-in-Aid for Young Scientists (B) No. 16710108 from the Ministry of Education, Culture, Sports, Science and Technology.     

Market Risk of Mortgage-Backed Securities with Consistent Measures

Due to the complex prepayment behavior, mortgage contracts and their derivatives are generally priced using Monte Carlo simulations. The typical approach used by the industry, which involves simulating interest rates under the risk-neutral measure and applying a physically measured prepayment function, is subject to the problem of internal inconsistency. This is the first paper that directly investigates the potential impact of this issue. Following the general equilibrium setting by Cox, Ingersoll and Ross, we incorporate the market risk price parameter to derive the physical interest rate process from an observed yield curve. This allows us to model mortgage values under the consistent physical measures of interest rates and prepayment functions. By analyzing a default-free Ginnie Mae MBS, we find that the mixed measures lead to slower prepayment rate estimates and overpriced mortgage securities by approximately 5%. Further, there can be substantial biases in the duration and convexity measures depending on market condition and the particular security of interest. The internal inconsistency also leads to biased predictions of both expected and stressed returns for different investment horizons. Depending on the particular security, the bias in expected and stressed returns can be either positive or negative. These biases in risk estimates can introduce misallocation of risk-based capital and/or failure in hedging the market risk of a mortgage-related portfolio.

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.     

Semi-analytical MBS Pricing

This paper presents a multi-factor valuation model for fixed-rate callable mortgage backed securities (MBS). The model yields semi-analytic solutions for the value of MBS in the sense that the MBS value is found by solving a system of ordinary differential equations. Instead of modelling the conditional prepayment rate (CPR), as is customary, the pool size is the primary modelling object. It is shown that the value of a single MBS payment due at time t _n can be found by computing two expectations of the pool size at time t _n–1 and t _n respectively. This is a general result independent of any interest rate model. However, if the pool size is specified in a way that makes the expectations solvable using transform methods, semi-analytic pricing formulas are achieved. The affine and quadratic pricing frameworks are combined to get flexible and sophisticated prepayment functions. We show that the model has no problem of generating negative convexity as the spot rate falls, and still be close to a similar non-callable bond when the spot rate rises.     

Prepayment and the Valuation of Mortgage-Backed Securities

This paper puts forward a valuation framework for mortgage-backed securities. Rather than imposing an optimal, value-minimizing call condition, we assume that at each point in time there exists a probability of prepaying; this conditional probability depends upon the prevailing state of the economy. To implement our valuation procedure, we use maximum-likelihood techniques to estimate a prepayment function in light of recent aggregate GNMA prepayment experience. By integrating this empirical prepayment function into our valuation framework, we provide a complete model to value mortgage-backed securities.     

Limits of Arbitrage: Theory and Evidence from the Mortgage-Backed Securities Market

“Limits of Arbitrage” theories hypothesize that the marginal investor in a particular asset market is a specialized arbitrageur rather than a diversified representative investor. We examine the mortgage‐backed securities (MBS) market in this light. We show that the risk of homeowner prepayment, which is a wash in the aggregate, is priced in the MBS market. The covariance of prepayment risk with aggregate wealth implies the wrong sign to match the observed prices of prepayment risk. The price of risk is better explained by a kernel based on MBS market‐wide specific risk, consistent with the specialized arbitrageur hypothesis.     

Microfoundations of a mortgage prepayment function

This article focuses on the following question: how much of an interest rate decline is needed to justify refinancing a typical home mortgage? Modern option pricing theory is used to answer the question; this theory indicates that the answer depends upon several factors, which include the volatility of interest rates and the expected holding period of the borrower. The analysis suggests that the commonly espoused “rule of thumb” refinance if the interest rate declines by 200 basis points — is a fair approximation to the more precisely derived differential for many households. We also construct the prepayment behavior of a pool of mortgages in which the expected holding periods of the borrowers in the pool vary. The prepayment behavior of this simulated pool is used to generate a series of empirically testable hypotheses regarding the likely shape of an actual prepayment function and its determinants. Finally, actual prepayment data are used to estimate a hazard function that explains prepayment behavior. We find that the estimated model understates prepayment behavior relative to that predicted by the simulation model, which suggests that the simple option pricing model is not adequate to explain aggregate prepayment behavior.     

Mortgage convexity

Most home mortgages in the United States are fixed-rate loans with an embedded prepayment option. When long-term rates decline, the effective duration of mortgage-backed securities (MBS) falls due to heightened refinancing expectations. I show that these changes in MBS duration function as large-scale shocks to the quantity of interest rate risk that must be borne by professional bond investors. I develop a simple model in which the risk tolerance of bond investors is limited in the short run, so these fluctuations in MBS duration generate significant variation in bond risk premia. Specifically, bond risk premia are high when aggregate MBS duration is high. The model offers an explanation for why long-term rates could appear to be excessively sensitive to movements in short rates and explains how changes in MBS duration act as a positive-feedback mechanism that amplifies interest rate volatility. I find strong support for these predictions in the time series of US government bond returns.     

Private information and incentives: Implications for mortgage contract terms and pricing

We examine the linkage in equilibrium among (1) contract design; (2) expected prepayment and default likelihoods; and (3) the pricing of mortgage contracts by focusing upon the effects of the borrower's private information at the time of contracting. We examine the implications of these perspectives upon the empirical analysis of prepayment behavior and use the framework to examine the predominance of long-term mortgage contracts in the United States. We consider examples that explore the trade-offs between fixed and adjustable rate instruments, assumable and due-on-sale loans, and contract interest rates and initial discounts (points).     

The refinancing threshold pricing model: An economic approach to valuing MBS

Investigation of MBS prepayment data indicates that mortgagors have different interest rate levels, or thresholds, at which they exercise their option to prepay their mortgage. In order to properly value an MBS with heterogeneous mortgagors, Merrill Lynch has developed the Refinancing Threshold Pricing Model (RTP). The RTP model focuses on the refinancing decision of the mortgagor when pricing the mortgage pool. The model divides each pool into groups of mortgagors who share similar refinancing costs. Using market data, the RTP model endogenously determines both the implied costs that mortgagors face, as well as the proportion of the MBS pool in each refinancing cost group. In addition to determining pool value, the RTP model also calculates MBS duration, dP/dY and convexity. Comparison between RTP model values and actual market data reveals a strong correlation. The RTP has a wide range of applications, including valuing 15-year and 30-year conventional MBS; pricing interest-only (IO)/principal-only (PO) derivative MBS; determining new versus seasoned MBS price spreads; and valuing specific MBS pools.The information set forth was obtained from sources we believe reliable, but we do not guarantee its accuracy. Neither the information, nor any opinion expressed constitutes a solicitation by us for the purchase or sale of any securities or commodities. Merrill Lynch, Pierce, Fenner & Smith, Inc. or its affiliates may have either a long or short position in, and may buy and sell for its own account or the accounts of others, these securities.     

Unobservable heterogeneity and rational learning: Pool-specific versus generic mortgage-backed security prices

Previous mortgage prepayment and valuation models assume that two mortgage pools with the same observable characteristics should behave indistinguishably. However, even pools with apparently identical characteristics often exhibit markedly different prepayment behavior. The sources of this heterogeneity may be unobservable, but we can infer information about a pool from its prepayment behavior over time. This article develops a methodology for using this information to calculate pool-specific mortgage-backed security prices. Knowledge of these prices is important both for portfolio valuation and for determining the cheapest pool to deliver when selling mortgagebacked securities. We find that unobservable heterogeneity between mortgage pools is statistically significant, and that pool-specific prices, calculated for a sample of outwardly identical mortgage pools between 1983 and 1989, may differ greatly from any single representative price.     

Duration and the price behavior of fixed-rate level payment mortgages: An analytical investigation

In this research we use a continuous payment formula for duration to examine the price behavior of a fixed-rate level payment mortgage. In the case where the mortgage is held to maturity, duration increases monotonically as term-to-maturity increases, regardless of changes in the market rate of interest. In the case where the mortgage is prepaid prior to maturity, there exists a unique market interest rate below which duration is a monotonically increasing function of time of prepayment, but above which duration has a global maximum at some time of prepayment prior to the term-to-maturity.     

Rational Prepayments and the Valuation of Collateralized Mortgage Obligations

This article presents a procedure for evaluating collateralized mortgage obligation (CMO) tranches. The solution procedure is in the spirit of a dynamic programming problem in which an individual mortgagor's decision to prepay is the feedback control variable―the mortgagor seeks to minimize the value of the mortgage subject to refinancing costs. We employ a two-step procedure to solve this dynamic programming problem. The first step uses an implicit finite difference backward solution procedure to determine the “optimal” prepayment boundary for a class of mortgagors, each of whom confronts the same proportional refinancing cost. This step is repeated for several different classes of mortgagors that differ in the level of refinancing costs that they confront. The outcome of this first step is a series of prepayment boundaries―one set of boundaries for each level of refinancing costs (i.e., one set of boundaries for each refinancing cost category of mortgagors). In the second step, the prepayment boundaries determined in the first step are used in conjunction with Monte Carlo simulation to value the CMO tranches. The essence of the second step is that when the simulated interest rate hits the boundary for a particular class, it triggers a prepayment scenario for that class of mortgagors. We conduct extensive sensitivity analysis to determine the robustness of this approach (and our solution procedure) to alternative single-factor models of the term structure of interest rates and to alternative specifications of the distribution of refinancing cost levels confronted by mortgagors. The sensitivity analysis indicates that CMO tranche valuation is not particularly sensitive to alternative models of the term structure so long as the models are consistent with the current yield curve, but, even when alternative specifications of the refinancing cost categories generate nearly identical values for the collateral underlying the CMO (i.e., the generic mortgage-backed securities), the resulting tranche values can differ widely between the two specifications. The results point out the importance of accurate estimation of the distribution of refinancing costs when the rational valuation model is used for the analysis of CMO tranches.     

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.     

Pricing mortgage insurance contracts under housing price cycles with jump risk: evidence from the U.K. housing market

Previous studies have investigated the determinants of housing price cycles in the housing market; however, we observed the phenomenon of housing price jumps in the 2007 subprime crisis. This paper presents a discussion on the housing price cycle and abnormal price jumps to describe the behavior of housing prices in the United Kingdom. The empirical results show that the impact factors of housing cycles are market risk and the switching factor. Furthermore, the impact factors of jump risks include the bursting of the housing bubble and financial crises. Therefore, in this paper, we employ the Markov switching model with jump risks to value the MI contracts and analyze the influences of housing price cycles, jump risks, risks of market interest rate, and the prepayment risks on MI premiums. The results of sensitivity analysis show that more volatile housing price index returns, as well as longer periods of higher volatility in housing prices, raise MI premiums. Moreover, the MI premium is positively related to the absolute value of the average jump amplitude and the shock frequency of abnormal events. There is the tradeoff between the market interest rate and the prepayment risk. The influences of market interest rate are different on MI premium with/without prepayment risks.     

The valuation and securitization of commercial and multifamily mortgages

This paper develops a pricing model capable of accurately valuing commercial mortgages and their mortgage backed securities (MBS). It is the first example in the general literature on mortgage pricing in which the valuation of an MBS is explicitly tied to that of the underlying mortgages, making possible a comparison of the performance of the loans and the securities they back. We have shown that while there are similarities between mortgages and their mortgage backed securities, they act in different ways. In general, it turns out that despite being the more passive asset, the mortgage backed security exhibits the more complicated behavior.     

SECONDARY MORTGAGE MARKET PURCHASE COMMITMENT YIELDS

We provide empirical evidence that quoted secondary market mortgage yields conform to the predictions of option theory. We compare Fannie Mae and Freddie Mac origination yields offered in the secondary market from 1985 to 2003 with the predictions of a two‐state binomial mortgage option valuation model. Our two‐state approach considers a mean‐reverting interest rate process as well as a stochastic housing price. Using predictions from option simulations, we find strong links between market practice and mortgage option prepayment and default factors over time. We also find cross‐sectional differences that are consistent with the institutional structure of the markets.     

Time-Varying Mortgage Prepayment Penalties

Recent empirical findings reveal that prepayment decisions of commercial property owners are slower than predicted by the pure options-pricing model (OPM). Borrower decisions appearing slow, however, may be quite rational when prepayment penalties of a time-varying nature are incorporated into the OPM. This article uses a competing risks OPM, adjusted for each of four different categories of prepayment penalties, to analyze borrower prepayment behavior. We find the value of delaying prepayment is often higher for mortgages with declining-rate penalties than for mortgages with static-rate penalties, frequently requiring a substantially higher interest rate spread to trigger a refinance. Multifamily loan prepayment records reveal the type of prepayment pattern that the adjusted OPM indicate should occur, reducing the gap between empirical findings and theoretical predictions. The results have implications for the specification of regressions fit to historical data, for the pricing of newly originated commercial mortgages, and for pricing in the single-family market where prepayment penalties are reemerging.     

Anatomy of an ARM: The Interest-Rate Risk of Adjustable-Rate Mortgages

This article analyzes the dynamics of the commonly used indices for adjustable rate mortgages and systematically compares the effects of their time-series properties on the interest-rate sensitivity of adjustable-rate mortgages. Our ARM valuation methodology allows us simultaneously to capture the effects of index dynamics, discrete coupon adjustment, mortgage prepayment, and both lifetime and periodic caps and floors. We can, moreover, either calculate an optimal prepayment strategy for mortgage holders or use an empirical prepayment function. We find that the different dynamics of the major ARM indices lead to significant variation in the interest-rate sensitivities of loans based on different indices. We also find that changing assumptions about contract features, such as loan caps and coupon reset frequency, has a significant, and in some cases unexpected, impact on our results.