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Chapter 49 MBS VALUATION AND PREPAYMENTS C.H. TED HONG, Beyond Bond Inc., USA WEN-CHING WANG, Robeco Investment Management, USA Abstract This paper not only provides a comparison of recent models in the valuation of mortgage-backed securities but also proposes an integrated model that addresses important issues of path-dependence, exogenous prepayment, transaction costs, mortgagors’ heterogeneity, and the housing devaluation effect. Recent research can be categorized into two frameworks: empirical and theoretical option pricing. Purely empirically derived models often consider estimation of the prepayment model and pricing of the mortgage-backed security as distinct problems, and thus preclude explanation and prediction for the price behavior of the security. Some earlier theoretical models regard mortgage-backed securities as default-free callable bonds, prohibiting the mortgagors from exercising the default (put) option, and therefore induce bias on the pricing of mortgagebacked securities. Other earlier models assume homogeneity of mortgagors and consequently fail to address important issues of premium burnout effect and the path-dependence problem. The model proposed is a two-factor model in which the housing price process is incorporated to account for the effect of mortgagor’s default and to capture the impact of housing devaluation. Default is correctly modeled in terms of its actual payoff through a guarantee to the investors of the security such that the discrepancy is eliminated by assuming mortgage securities as either default-free or unin- sured. Housing prices have been rising at unsustainable rates nation wide, especially along the coasts, suggesting a possible substantial weakening in house appreciation at some point in the future. The effect of housing devaluation is specifically modeled by considering the possibility that the mortgagor might be restrained from prepayment even if interest rates make it advantageous to refinance. Mortgagors’ heterogeneity and the separation of exogenous and endogenous prepayments are explicitly handled in the model. Heterogeneity is incorporated by introducing heterogeneous refinancing transaction costs. The inclusion of heterogeneous transaction costs not only captures premium burnout effect but also solves the path-dependence problem. Finally, the model separates exogenous prepayment from endogenous prepayment, and estimates their distinct magnitudes from observed prepayment data. This construction provides a better understanding for these two important components of prepayment behavior. The generalized method of moments is proposed and can be employed to produce appropriate parameter estimates. Keywords: MBS valuation; option pricing theory; exogenous and endogenous prepayments; housing devaluation effect; devaluation trap; transaction costs of refinancing and default; generalized method of moments; path dependency; premium burnout effect; heterogeneity 730 ENCYCLOPEDIA OF FINANCE 49.1. Introduction The main objective of this paper is to gain a better understanding of the valuation of mortgage-backed securities. Mortgage-backed securities have attracted unprecedented investor interest over the last decade, spurring tremendous growth in the market for this important financial instrument. There are over $7.7 trillion worth of residential mortgage loans outstanding, an amount far exceeding the size of the corporate debt market. Approximately $5.1 trillion worth of securitized mortgage-backed securities and CMOs are outstanding, and well over $1.8 trillion new mortgage-backed securities and whole loans pools are issued each year for the past three years.1 Mortgage-backed securities are extensively held by every class of institutional investor, including commercial banks, saving institutions, insurance companies, mutual funds, and pension plans. An in-depth study of the valuation of mortgagebacked securities is of interest to financial economists because mortgage-backed securities have unique characteristics that are distinct from other contingent claims, such as monthly amortization, negative convexity, premium burnout, and pathdependence. This paper examines recent developments in the area of valuing mortgage-backed securities and proposes a model that accommodates these factors affecting the price of mortgage-backed securities. The core issue in valuing mortgage-backed securities is the modeling of the prepayment behavior of mortgagors in the pool backing the security. Continuous-time option pricing methodology has been a popular method in the mortgagebacked securities valuation because of the obvious parallel between the call option and the right of a mortgagor to prepay. In order to model the mortgagors’ prepayment behavior more realistically, recent theoretical models have added modifications to the original stock option pricing theory framework. The first of these modifications broadly accounts for prepayment due to reasons exogenous to financial consideration, such as mov- ing and job changes. The second group of modification addresses transaction costs. The third considers heterogeneity among mortgagors, and the fourth group discusses the separation of exogenous prepayment and endogenous prepayment. The observation that homeowners clearly do not prepay as objectively as option pricing models imply has motivated many researchers to add prepayment functions that allow prepayments for reasons that are exogenous to purely financial considerations. Such research includes the work of Dunn and McConnell (1981a,b), and Brennan and Schwartz (1985), and most of the prepayment functions have been arbitrary. The main drawback of adding an arbitrary prepayment function is that it does not aid in the identification of the factors responsible for prepayment behavior. Identifying these factors would go a long way toward enhancing the explanatory power of the model. Applying the option pricing theory to the valuation of residential mortgage-backed securities, one can see a departure from the perfect market assumption when homeowners face transaction costs upon refinancing or defaulting. For this reason, Dunn and Spatt (1986) and Timmis (1985) add homogenous refinancing transaction costs in their models to adjust the prepayment speeds from those implied in the frictionless economic environment. Kau et al. (1993) also add the transaction cost of default in their modeling of the probability of default for residential mortgages. Addressing mortgagors’ heterogeneity is a more complex matter. Many earlier models assumed homogeneity among mortgagors to avoid complexity in the pricing process. However, the assumption of mortgagors’ homogeneity fails to address the issue of premium burnout which is an important empirical effect of homeowner heterogeneity. And this assumption also results in a pathdependent problem when numerically solving the optimal refinancing strategies backwards. The premium burnout effect is the tendency of prepayments from premium pools to slow down over time, with all else held constant. If a large number MBS VALUATION AND PREPAYMENTS of mortgagors have already prepaid, those remaining are likely to have a relatively low probability of prepaying. Conversely, the smaller the number of previous prepayments, the higher the probability of prepaying by the remaining mortgagors. The aforementioned path-dependent problem occurs because any mortgage pool contains a group of mortgagors who behave differently in their prepayment decisions: these mortgagors differ in their willingness or ability to prepay their loans under favorable circumstances. As a result, without knowing either the type of mortgagor or the entire path of interest rates from origination, backward optimization is not applicable because there is no way of knowing whether the earlier prepayment exercise is optimal. Johnston and Van Drunen (1988), and Davidson et al. (1988) improve on the homogenous transaction cost model by introducing heterogeneous transaction models. They assume that different homeowners face different levels of refinancing transaction costs. In addition to the ability to capture the premium burnout, the inclusion of heterogeneous transaction costs also solves the path-dependent problem encountered when pooling individual mortgagors, who behave differently in their prepayment decisions. Another common problem in existing models is the lack of differentiation between exogenous prepayment and endogenous prepayment. This lack of distinction between the two thereby precludes explanation of the interrelation between these important behavioral components. Endogenous prepayment refers to any prepayment decision that occurs in response to changes in underlying economic processes, such as the interest rate. Stanton (1990) incorporates an endogenous decision parameter that enables separate estimations of endogenous prepayment and prepayment for exogenous reasons. As a result, the explanatory power of the model is improved. In addition to the inclusion of the previously discussed modifications, our model introduces two adjustments. One is the treatment of mortgagors’ right to default in the content of mortgage-backed securities valu- 731 ation. And the other is the impact of the housing prices on prepayment behavior. Although default has been modeled as a put option in the models of residential mortgages or commercial mortgage-backed securities, many earlier models have not incorporated it in the valuation of residential mortgage-backed securities. This is because government agency guarantees lead to the perception that securities are defaultfree. Default should be taken into consideration because there is a payoff difference between a guaranteed mortgage-backed security and a default-free security. The payoff from a guarantee in the event of default is the par amount rather than the market value of the security, thus producing an asymmetric return for investors. In modeling default, we expand previous default-free models into a default-risky model in which the housing price process is included as a second-state variable. Default is explicitly modeled in terms of its actual payoff through a guarantee to the investors of the residential mortgage-backed security. This is in contrast to models for individual mortgages or commercial mortgages in which mortgages are neither insured nor guaranteed. Consequently, the payoff in the event of default in these cases is the value of the house. By correctly modeling the effect of default, our model reduces the discrepancy from assuming mortgage-backed securities as either default-free or uninsured. The housing price process is incorporated in the model not only to account for the effect of default on security price, but also to determine its impact on the prepayment behavior of mortgagors. The effect of housing prices on prepayment is specifically modeled by considering the possibility that the mortgagor might be restrained from prepaying even if interest rates make it advantageous to refinance. This is because housing prices have fallen to the extent that the mortgagor is no longer qualified for refinancing. The model we propose not only captures the fundamental characteristics of the mortgagors’ prepayment behavior but it also combines parametric heterogeneity and variability of the decision 732 ENCYCLOPEDIA OF FINANCE parameter to the extent that our model can come closer than previous models in describing empirical prepayment behaviors. 49.2. The Model The central issue in valuing mortgage-backed securities is the treatment of prepayment uncertainty. The valuation model of mortgage-backed securities proposed here is based on the continuous-time option pricing methodology. This methodology treats the right of a mortgagor to prepay as a call option and the right to default as a put option. Modifications to the assumption of perfect capital markets and the principle that borrowers act to minimize the market cost of their mortgages are required to portray mortgagors’ actual prepayment behavior in a more realistic manner. According to Dunn and McConnell (1981) and Brennan and Schwartz (1985), we allow mortgagors to prepay for reasons exogenous to purely financial considerations. In contrast to their models that assume arbitrary exogenous prepayment functions, our model utilizes the proportional hazard function and can be estimated from observable prepayment data. To account for the fact that homeowners face transaction costs when they prepay or default on their mortgages, we follow Johnston and Van Drunen (1988). Consequently, we add heterogeneous refinancing transaction costs in our models to adjust the prepayment speeds from those implied in the frictionless economic environment. Following Kau et al. (1993), we also add the transaction cost of default in modeling the effect of default. Default has been modeled as a put option in the valuation of residential mortgages or commercial mortgage-backed securities. However, many models have not incorporated default in the valuation of residential mortgage-backed securities because government agency guarantees lead to the perception that securities are default-free. Moreover, there is a significant difference between the payoff of a guaranteed mortgage-backed security and that of a default-free security. The payoff from insurance in the event of default is the par amount rather than the market value of the security, producing an asymmetric return for investors. Kau and associates (1992) develop a two-factor model for both prepayment and default only in the context of evaluating individual mortgages, where mortgages are considered as uninsured. As discussed in the Chapter 49, the payoff from uninsured mortgages is the value of the house when the mortgage is defaulted. In our model, the payoff to the investor from default is explicitly modeled as insured mortgages. This eliminates the potential bias in the pricing of mortgage-backed securities. A significant relationship between observed prepayment and housing prices data pointed out by Richard (1991) leads us a final adjustment of the two-factor model. The housing price process is brought in not only to account for the effect of default on security price, but also to determine its restraining effect on mortgagors’ refinancing decisions. Figure 49.1 outlines these differences between one-and two-factor models and the innovations presented in this study. In the one-factor model, the prepayment decision responds to the level of interest rates. The two-factor model adds two additional termination outcomes that follow from the level of housing prices. At very low housing prices, the mortgagors may default regardless of the interest rate in order to cut their losses. Finally, the mortgagor might be restrained from prepaying even if interest rates make it advantageous to refinance. This occurs when the housing prices fall to the extent that the new loan cannot cover the costs of refinancing. In addition to capturing these fundamental characteristics of the mortgagor’s termination behavior, this model aggregates the underlying pool of mortgages according to the heterogeneity of transaction costs. And it is the specification of heterogeneous transaction costs that also solves the path-dependent problem displayed by pooled mortgages. The following first section pertains to the modeling of termination decisions affected by exogenous MBS VALUATION AND PREPAYMENTS One-factor model r > r* No prepayment r ≤r * Prepayment 733 H(t ) > Hdn No prepayment from low interest rates H(t ) ≤ Hdn Default H(t ) ≥ H * Prepayment from low interest-rate H * > (Ht ) > Hdn Restrained from prepayment H(t ) ≤ Hdn Default r >r * Two-factor model r ≤r * r* H* Hdn Figure 49.1. adjustments introduced by our model critical value for interest rate motivated prepayment housing price upper limit restraining mortgagor from refinancing housing price upper limit of default Model trees and endogenous factors, housing prices, and transaction costs. The later section introduces our model, which is a two-factor pricing framework that provides exact security prices given underlying interest rate and housing prices processes, and precludes arbitrage opportunities. 49.2.1. Modeling Issues p(t) ¼ p, t
0 and p > 0: Pr(Exogeneous prepayment in(t, t þ dt)j No prepayment prior to t) dt!0 : dt (49:1) There are numerous parametric methods used in the analysis of duration data and in the modeling of aging or failure processes. We use the exponential distribution in the model for its simplicity. The (49:2) The probability that an individual has not prepaid for exogenous reasons until time t is given by the survival function S(t), S(t) ¼ ep(t) ¼ ept , t
0 49.2.1.1. Exogenous Prepayment In practice, exogenous reasons for termination include factors such as relocation, death, divorce, or natural disasters. Exogenous prepayments are also known as turnover prepayments. A hazard function is used to model exogenous prepayment as follows: p(t) ¼ limþ distribution is characterized by the constant hazard function (49:3) 49.2.1.2. Endogenous Termination A mortgage is terminated when mortgagors either prepay or default on their mortgages. Any termination which affects the cash flows passed through to the investors will have an impact on the price of the mortgage-backed securities. Throughout the model, endogenous termination is defined as any rational termination decision that occurs in response to underlying economic processes rather than personal considerations. We assume that mortgagors maximize their current wealth, or equivalently, minimize their liabilities. Mortgagors’ liabilities can be thought of as 734 ENCYCLOPEDIA OF FINANCE composed of three parts. The first part consists of owing the scheduled streams of cash flows associated with the mortgage. The second part constitutes their option to prepay at any time, which is equivalent to possessing a call option. And the third part consists of mortgagor’s option to default, which functions as a put option. Option pricing theory is, therefore, an appropriate method for determining the value of mortgagors’ mortgage liability. A model of mortgage pricing should incorporate both refinancing transaction costs and default transaction costs in order to more accurately portray the decision-making processes of mortgagors. Although including transaction costs causes the resulting termination strategy to deviate from the perfect market assumption, the strategy still remains rational. In order to derive the magnitude of endogenously determined termination, we follow Stanton (1990) and introduce r, which measures the frequency of mortgagors’ termination decisions. The time between successive decision points is described as an exponential distribution. If we let Ti be one such decision point, and Tiþ1 the next, then Pr(Tiþ1  Ti > t) ¼ ert (49:4) If mortgagors are continually re-evaluating their decisions, then the parameter r takes on a value of infinity. If mortgagors never make endogenous termination decisions and only terminate for exogenous reasons, then r takes on a value of zero. If r takes on a value between these limits, then this signifies that decisions are made at discrete times, separated on average by 1=r. Given this specification, the magnitude of endogenized termination can be estimated and studied. The contribution of this device is to separate the magnitude of endogenized termination from that of exogenous termination. It also serves to help understand the actual termination behavior of mortgagors. Without this specification, it would be difficult to know the proportion of termination from endogenous optimization decisions and the proportion due to exogenous factors. Utilizing the definitions from Sections 49.2.1.1 and 49.2.1.2, we notice that the optimal exercise strategy immediately leads to a statistical representation of the time to terminate for a single mortgagor. If termination is due exclusively to exogenous factors, then the termination rate is p and the survival function is defined as in Equation (49.4). When termination occurs for endogenous reasons, the probability that the mortgagor terminates in a small time interval, dt, is the probability that the mortgagor neither prepays for exogenous reasons nor makes a rational exercise decision during this period. This survival function can be approximated by
S(t) ¼ epdt  erdt ¼ e(pþr)dt epdt if endogenous termination : if no endogenous termination (49:5) 49.2.1.3. Transaction Costs and Aggregation of Heterogeneous Mortgages The cash flows that accrue to the investor of a mortgage-backed security are not determined by the termination behavior of a single mortgagor, but by that of many mortgagors within a pool. To cope with the path-dependent problem caused by the heterogeneity within a pool of mortgages, we assume that the different refinancing transaction costs each mortgagor faces is the only source of heterogeneity. Although the costs of initiating a loan vary among different types of mortgages, some of the most common costs borrowers face include credit report, appraisal, survey charges, title and recording fees, proration of taxes or assessments, hazard insurance, and discount points. The transaction costs of individual mortgagors are drawn from a univariate discrete distribution, which allows for underlying heterogeneity in the valuation of the mortgage-backed security. A better way to choose the underlying distribution that represents this heterogeneity would be to look at summary statistics of transaction costs actually incurred by mortgagors when they refinanced. A discrete rectangular distribution is chosen for MBS VALUATION AND PREPAYMENTS its simplicity and the task of determining which distribution improves the fit is left for future research. The value of the security is equal to the expected value of the pool of mortgages weighted by the proportions of different refinancing transaction cost categories. Suppose that each Xi (the refinancing transaction costs faced by mortgagor i) is drawn from a discrete rectangular, or uniform distribution Pr (x ¼ a þ ih) ¼ M 1 , i ¼ 1, . . . , M (49:6) Various standard forms are in use. For this application, we set a ¼ 0, h ¼ RM 1 , so that the values taken by x are RM 1 , 2RM 1 , . . . , R. The upper bound R of the transaction cost is set at 10 percent. The distribution for the transaction costs is then defined as:   0:1 Pr x ¼ i (49:7) ¼ M 1 , i ¼ 1, . . . , M M In principle, given any initial distribution of transaction costs, it is possible to value a mortgage-backed security backed by a heterogeneous pool of mortgages in a manner similar to the valuation of a single mortgage. If the value of individual mortgages is known, then the value of the pool is the sum of these individual values. When the value of individual mortgages is not known, but a distribution of transaction costs is generated that accounts for heterogeneity, the expected value of a pool of mortgages is the sum of the transaction cost groups times the probability of their occurrence in the pool. Recall from Section 49.1.2 that for a given transaction cost Xi and state of the world, if any mortgagor finds it optimal to terminate, the hazard rate is the sum of the exogenous prepayment rate, p, and the endogenized termination rate, r. If it is not optimal to terminate, the hazard rate falls back to the background exogenous prepayment rate p. Models that neither permit the estimation of r nor consider exogenous factors in the prepay- 735 ment decision imply that r ¼ 1 and p ¼ 0, and the single-transaction cost level predicts that all mortgages will prepay simultaneously. Adding heterogeneous transaction costs addresses the problem of path dependence, however, keeping the same parameter values still does not permit hesitation in the prepayment decision. Although prepayment rates fluctuate, in reality, they do tend to move fairly smoothly. The effect of setting r to a value other than 1 is to permit a delay even when it is optimal to prepay. And prepayment need not occur at all if interest rates or housing prices change such that it is no longer optimal. The actual value of r determines how fast this drop occurs. Thus, combining parametric heterogeneity and variability of the parameter r would allow the model to come closer than previous rational models to describe empirical prepayment behavior. 49.2.2. A Model for Pricing Mortgage-Backed Securities 49.2.2.1. Termination Decision of a Single Mortgagor The following is a model of rational prepayment behavior of mortgages that extends the rational prepayment models of Stanton (1990) and Kau and associates (1993). Mortgagors may terminate their mortgages for endogenous financial reasons that include interest rates and housing prices, or for exogenous reasons. They also face transaction costs, which are used to differentiate mortgagors and solve the path-dependent problem. Mortgagors choose the strategy that minimizes the market value of the mortgage liability. The following assumptions are employed: 1. Trading takes place continuously and there are no taxes or informational asymmetries. 2. The term structure is fully specified by the instantaneous riskless rate r(t). Its dynamics are given by. pffiffi (49:8) dr ¼ k( mr  r)dt þ sr rdzr 736 ENCYCLOPEDIA OF FINANCE 3. The process to capture the housing price is assumed to follow a Constant Elasticity of Variance (CEV) diffusion process dH ¼ mH Hdt þ sH H g=2 dzH , (49:9) where mH , sH > 0, 0 < g < 2, and {zH (t), t
0} is a standard Wiener Process, which may be correlated with the process {zr (t), t
0}. When g ¼ 2, the process is lognormal. The underlying state variables in the model are the interest rate r(t) and the housing price H(t). By applying the arbitrage argument, the value of the ith mortgage liability V i (r,H, t) satisfies the following partial differential equation: pffiffi 1 2 i i sr rVrr þ rsr sH rH g=2 VrH 2 1 i þ s2H H g VHH þ [k(mr  r)  lr]Vri 2 þ rHVHi þ Vti  rV i ¼ 0, (49:10) where lr represents factor risk premium. The value of the mortgage liability is also required to satisfy the following boundary conditions: 1. At maturity T, the value of a monthly amortization bond is equal to the monthly payment: V i (r, H, T) ¼ MP 2. As r approaches infinity, the payoff of the underlying mortgage bond approaches zero: lim V i (r, H, t) ¼ 0 r!1 Figure 49.2 summarizes the remaining conditions, which establish the boundaries of the various circumstances affecting the termination decision. 3. At any time t, the mortgage value satisfies the following conditions: Let V i (r, H, tþ ) ¼ V i (r, H, t þ 1) þ MP, then 8 i V (r, H, tþ ) > > > > < V i (r,H, t) ¼ U(t) > > > > : U(t) if Vi if U if H(t) > Hdn and U(t)(1 þ Xi ) > (r , H, tþ ) if continued H(t) > Hdn and V i (r,H,tþ )
(t ) (1 + Xi ) if refinanced H(t)  Hdn if defaulted where U(t) is the principal remaining at time t. Hdn is the boundary of default, defined as the housing price times the cost of default, or Hdn ¼ (V i r, H, tþ )=(1 þ d):Xi is the prepayment transaction costs for individual i and d is the transaction cost of default for all individuals. This boundary condition defines the default and refinancing regions in Figure 49.2. When housing prices fall so low that they are exceeded by the default cost-adjusted mortgage value, the mortgagor will exercise their put option by defaulting. The refinancing region describes a situation in which H(t) : Continuance region H * < H( t ) : Termination region Hdn < H(t) < Hup (refinacing region) r * : v(r *,H,t) = u/(t)(1+xi) (contiuance region) H* H * : u(t)(1+xi)-LTV *H(t)=V(r,H,t)-u(t) Hdn < H < H * Hdn: Hdn = V(r,H,t)/(1+d) (devaluation trap) Hdn H(t) < Hdn (default region) r* V(r,h,t) > u(t)(1+xi) Figure 49.2. Diagram of boundary conditions r(t) V(r,H,t) < u(t)(1+xi) MBS VALUATION AND PREPAYMENTS the interest rate falls to the point where the mortgage value is greater than the refinancing cost-adjusted unpaid principal. In this case, the mortgagor exercises the call option by refinancing their loan. The value of the mortgage liability takes on the value of unpaid principal U(t) unadjusted by transaction costs (1 þ Xi ), because the refinancing costs are collected by the third party who services the mortgage. 4. To improve on the previous model, we have included the effect of housing prices on the termination decision Working back one month at a time, we can value the ith mortgage liability V i (r, H, t) by solving Equation (49.10), given boundary condition 1 through 4. Given p and r, we can also calculate the probability that the mortgage is terminated in month t. Denote Pe the probability of termination if only exogenous prepayment occurs. Denote Pr the probability of termination if it is endogenous conditions that lead to a decision to terminate in month t. According to the survival function Equation (49.5), these termination probabilities are given by Pr ¼ 1  e(pþr)=12 if endogenous termination V i (r, H, t) ¼ V i (r, H, tþ )ifH  > H(t) > Hdn and V i (r, H, tþ )
U(t) (1 þ Xi) if restrained, where LTV is the loan-to-value ratio and H  is determined at U(t) þ (1 þ Xi)  LTV  H(t) ¼ V i (r, H, t)  U(t): 737 (49:11) This condition encompasses the devaluation trap. The devaluation trap occurs when housing prices fall between H  and Hdn, where the costs of refinancing exceed its benefits. The mortgagor will be unable to refinance their loan, even though interest rates are advantageous, because they will have to pay the difference out of their pocket. And since the housing price remains above the default threshold, the mortgagor continues the mortgage. The present value of costs is determined by the left-hand side of Equation (49.11), that is the difference between the unpaid principal plus refinancing transaction cost and the new loan amount, which is the housing price times the loan-to-value ratio. The benefit of refinancing is given by the right-hand side of Equation (49.11), i.e. the mortgage value minus the unpaid principal. The role of the loan-tovalue ratio is important in determining the size of the devaluation trap. The higher loan-to-value ratios result in decreases in the range of the devaluation trap. Pe ¼ 1  ep=12 if no endogenous termination We can now calculate the expected value of a single mortgage liability. That is 8 (1  Pr )V i (r, H, tþ ) þ Pr U(t) > > < (if endogenous termination) V i (r, H, t) ¼ > (1  Pe )i (r, H, tþ ) þ Pe U(t) > : (if no endogenous termination) 49.2.2.2. Valuation of a Pool of Mortgages To determine the value of the mortgage-backed security at any time t, as mentioned above, we can simply take the expected value of pooled mortgage liabilities V (r, H, t) ¼ M X V i (r, H, t)  P(Xi ¼ x) i¼1 (49:12) x 2 (0, 0:1] 49.3. Estimation A model for valuing mortgage-backed securities was described that permits the determination of the security’s price for given parameter values describing exogenous and endogenous factors that contribute to the termination decision. The next logical step would be to estimate these parameter values from prepayment data. In this sector, the generalized method-of-moment technique is 738 ENCYCLOPEDIA OF FINANCE proposed for the estimation, where the termination probability at any given time t is required for equating the population and sample moments. In order to accomplish this, we must determine the model in terms of the probability rather than in terms of the dollar value of the security. 49.3.1. Determination of the Expected Termination Probability produces a change in drift in the underlying diffusions. Consequently, one must substitute the risk-adjusted processes for the actual stochastic processes in Equations (49.8) and (49.9), which in this case are dr ¼ (kmr  (k þ l)r)dt þ sr dz̃r (49:14) and dH ¼ rHdt þ sH dz̃H : (49:15) In addition to equating the population and sample moments when the generalized method-of-moment technique is employed for the estimation, the calculation of termination probability is useful because it can also be utilized to determine the expected cash flows for any other mortgage-related securities, such as collateralized mortgage obligations. We first restate the procedure for determining the price in order to provide a comparison to the procedure for determing termination probability. When the housing price process is transformed to its risk-adjusted form, the actual required rate of return on the house mH drops out of the equation. Therefore mH does not influence the mortgage and default option values. We know that the mortgage value V i (r, H, t) satisfies the partial differential equation specified in Equation (49.10). And thus, with the appropriate terminal and boundary conditions, the value of the mortgage is determined by solving this partial differential equation (PDE) backwards in time. 49.3.1.1. Procedure for Determining the Security Price In this model, the uncertain economic environment a homeowner faces is described by two variables: the interest rate and the housing price. The term structure of the interest rate is assumed to be generated from the stochastic process described in Equation (49.8) and the process of the housing prices is represented in Equation (49.9). Assuming perfect capital markets, the present value V i (r, H, t) of the mortgage contract at time t is of the form 3 2 T Ð 6  r(t)dt
i 7 (49:13) V i (r, H, t) ¼ Ẽt 4e t V (T)5, 49.3.1.2. Deriving the Expected Termination Probability of Mortgage i In order to implement the parameter estimation, we are now concerned with the actual occurrence of termination instead of the dollar value of the mortgage. We begin the derivation of termination probability with the following definition: i where V
(T) is the terminal value of the mortgage liability at expiration date T. This equation states that the value of the mortgage is equivalent to the discounted-expected-terminal payoff under the riskneutral measure. By Girsanov’s theorem, under certain circumstances, the change in measure merely Pi (r, H, t) ¼ Pr( (r(t), H(t),t) 2 termination region of mortgage i, for some t > t, given(r(t), H(t), t) ¼ (r, H, t)) (16) where (r, H, t) are the interest rate and housing price at current time t, while Pi (r, H, t) is the probability that termination ever occurs beyond the current situation. The general theory of stochastic processes allows that such a probability satisfies the Kolmogorov backward equation pffiffi 1 2 i 1 sr rPrr þ rsr sH rH g=2 PirH þ s2H H g PiHH 2 2 i i i þ k(mr  r)Pr þ mH PH þ Pt ¼ 0 (17)