An Empirical Investigation of Alternative Contingent Claims Models for Pricing Residential Mortgages

Journal of Real Estate Finance and Economics, 17:2, 139±162 (1998)

# 1998 Kluwer Academic Publishers, Boston. Manufactured in The Netherlands.

An Empirical Investigation of Alternative ContingentClaims Models for Pricing Residential Mortgages

AMITAVA CHATTERJEE

Department of Economics and Finance, Fayetteville State University, 1200 Murchison Road, Fayetteville,NC 23801-4298

ROBERT O. EDMISTER

Department of Economics and Finance, The University of Mississippi, University, MS 38677

GAY B. HATFIELD

Department of Economics and Finance, The University of Mississippi, University, MS 38677

Abstract

Researchers have employed option pricing techniques to analyze mortgage ®nancing and valuation. Alternative

models (one-, two-, and three-variable models) employing different variables (short- and long-term interest rates

and building value) have been designed to price mortgage securities. No prior research has addressed the question

of whether the pricing accuracy of these contingent claims models improves as states increase or whether

contingent claims models' valuation abilities generate reasonable estimates of primary mortgage market prices.

The articles investigates the relative ef®ciency of each of these alternative mortgage valuation models in

predicting primary market mortgage values. Our results show that a two-variable model (short rate and building

value) is the most ef®cient. Valuation results indicate a positive pricing spread between the primary market and

the theoretically estimated value.

Key Words: mortgages, mortgage valuation, residential mortgages

With the development of collateralized mortgage securities and mortgage pass-through

securities, the secondary mortgage market has expanded dramatically during the past

decade. The amount of outstanding mortgage debt in the United States is actually greater

than the amount of corporate debt. Consequently, ®nancial analysts have become

concerned about this segment of the capital markets. Following the seminal work on

pricing contingent claims in capital markets (Black and Scholes, 1973; Merton, 1973;

Brennan and Schwartz, 1977), researchers have employed option pricing techniques to

analyze ®nancing and valuation. As (Kau, Keenan, Muller, and Epperson, 1992, p. 279)

point out, ``The provisions of a standard ®xed-rate mortgage (FRM) rival those of any

other ®nancial instrument in complexity.''

This complexity results, in large part, because a mortgage contract can be canceled by

the borrower for two reasonsÐprepayment or default. Borrowers may prepay a mortgage

for multiple reasons. Mortgages may be paid off early when interest rates are declining so

that the borrower may re®nance at a lower rate of interest. Non®nancial reasons, such as

moving or divorce, can also result in prepayment of a mortgage. Kau, Keenan, and Kim

(1994) note that default occurs due to the action of an individual agent (the borrower).

Kau, Keenan, Muller, and Epperson (1992) state that a borrower will choose to default on

his mortgage contract only if the market value of the loan is greater than the value of the

house. In summary, prepayment and default options are `èssential characteristics'' of a

mortgage (Kau, Keenan, Muller, and Epperson, 1995).

Research that has utilized option pricing theory to price mortgage securities has taken

several directions. The resulting models have differed from each other according to

inclusion or omission of diverse state variables (short- and long-term interest rates and

building value) and prepayment and default options. Dunn and McConnell (1981b)

designed a one-variable model and employed an instantaneous risk-free rate as the state

variable for pricing default-free, mortgage-backed GNMA securites.1 Using a similar

approach, Cunningham and Hendershott (1984) examined the borrower's default option,

while Buser and Hendershott (1984) considered the prepayment option. Neither study

included both default and prepayment.

The one-variable model was enlarged to a two-variable model that extended in two

separate directions. One dual-variable model employs both short- and long-term interest

rates as the two-state variables to explain the term structure for valuing a default-free,

prepayable mortgage. Brennan and Schwartz (1982, 1983, 1985) adopted this framework

to analyze the pricing of GNMA mortgage-backed securities. Other notable research in

this same vein includes studies by Buser, Hendershott, and Sanders (1990), Schwartz and

Torous (1989a, 1989b, 1991), and McConnell and Singh (1993, 1994).

The alternative two-variable model includes the short-rate as one variable but replaces

the long-rate variable with building value (to capture the default character). This two-

variable model has been utilized by Foster and Van Order (1984, 1985), Epperson, Kau,

Keenan, and Muller (1985), Titman and Torous (1989), Kau, Keenan, Muller, and

Epperson (1987, 1990a, 1990b, 1992, 1993), and Schwartz and Torous (1992).2

This study further recognizes a three-state variable model3 that includes both

prepayment and default options. Drawing from the previous two-state variable models,

the three-state variables employed in the model are short-term interest rates, long-term

interest rates, and building value.

To summarize, alternative models have been designed to price mortgage securities. No

prior research addresses the question of whether the pricing accuracy of contingent claims

models improves as states increase or whether contingent claims models' valuation

abilities generate reasonable estimates of primary mortgage market prices. Further, no

prior research has employed the backward-solving methodology on a three-variable

model. In addition, no previous research includes estimates based on primary mortgage

data for purposes of valuation and comparison.4 The purpose of this study is to compare

individual contingent claims mortgage valuation models to determine the relative

ef®ciency of each model in predicting primary market mortgage values.

The rest of the article is organized in the following manner. Section 1 reviews previous

literature concerning model comparisons and describes our research design. Individual

contingent claims models for pricing residential mortgages are presented in section 2, and

the data and valuation methodology follow in section 3. Section 4 presents valuation

summaries and hypothesis test results, while conclusions regarding the optimal number of

model states and primary market pricing are given in section 5.

140 CHATTERJEE ET AL.

1. Previous Literature and Research Design

Several studies exist that compare one- and two-variable models' valuations for mortgages

priced in the secondary mortgage market. Dunn and McConnell (1981a) compared a

single-variable contingent claims model against two discrete time models to value GNMA

mortgage-backed securities. The results indicated that, when term structure is rising,

discrete time models compute prices that are less than the prices generated by contingent

claims models. The reverse was observed for a falling term structure. Brennan and

Schwartz (1985) analyzed the outcome of employing the Black±Scholes option pricing

model, the Dunn±McConnell one-variable model, and a two-interest-rate model to price

GNMA mortgage-backed securities. Their ®ndings indicated that, relative to the two-

variable model, single-variable models understate the yield differential between GNMA

securities and T-bonds and the value of the prepayment options. Buser, Hendershott, and

Sanders (1990) concluded that both one-interest-rate and two-interest-rate models produce

adequate and ef®cient results. Gilliberto and Ling (1992) employed both one-variable

(short-rate) and two-variable (short-rate and building value) models to price residential

mortgages and found that both models provide unbiased predictions of actual data.

This study tests four contingent claims models for pricing residential mortgages in order

to analyze the relative ef®ciency of each model in predicting mortgage values in the

primary market. The ®rst model to be tested will be the two-variable model, which

includes both the short interest rate and building value (Kau, Keenan, Muller, and

Epperson, 1987, 1990a, 1990b, 1992, 1993). The results of this model are compared to the

one-variable model (risk-free rate), the alternative two-variable model (the short- and

long-term interest rates), and the three-variable model (short- and long-term interest rates

and building value). To examine the ef®ciency of different models in predicting residential

mortgage prices, both parametric and nonparametric tests are employed.

2. Contingent Claims Models

Each of the four contingent-claims models employed in this article includes one or more of

three-state variables, which are as follows: r, the instantaneous risk-free short-term interest

rate (the spot rate); l, the yield on a bond with in®nite maturity (the consol rate); and B, the

value of the mortgaged building. The four models and their variables are as follows:5

SPOT-CONSOL-VALUE model: a three-variable model (spot and consol rate, and

building value),

SPOT-CONSOL model: a two-variable model (spot and consol rate),

SPOT-VALUE model: a two-variable model (spot rate and building value), and

SPOT model: a one-variable model (spot rate).

All the mortgages in this study include prepayment and default options, and therefore,

the values of these options must be included in the model. The necessary terminal

and boundary conditions ( prepayment and default options) to solve respective partial

differential equations (PDEs) will be described following the models' speci®cations.

ALTERNATIVE CONTINGENT CLAIMS MODELS FOR PRICING RESIDENTIAL MORTGAGES 141

2.1. SPOT-CONSOL-VALUE Model

The term structure of interest rates is summarized by two-state variablesÐr, the spot rate,

and l, the consol rate. The stochastic processes are described as follows:

dr � �ar � br�lÿ r��dt� srrdzr; �1�

dl � �al � blr � cll�dt� slldzl; �2�

dzrdzl � rrldt: �3�

Here, dzr and dzl are standardized Weiner processes, rrl is the instantaneous correlation

coef®cient between r and l, sr and sl are respective standard deviations, and a's, b's, and

c's are coef®cients for any risk premium and speed-of-adjustment processes. These

speci®cations imply that the scale of unanticipated changes in both r and l are

proportional to their current values. Also, r reverts to the current value of l, which varies

stochastically over time.

The state variable B summarizes all information about the values of building collateral.

The stochastic process follows a lognormal diffusion process, which is described as

dB � �aÿ b0�Bdt� sBBdzB; �4�

dzrdzB � rrBdt: �5�

Here, a is the instantaneous total expected rate of return to the asset, and b0 is the

continuous payout rate generated by the building. Accordingly, �aÿ b0� is the

instantaneous mean rate of appreciation in property value, dzB is a standardized

Weiner process, and rBr is the instantaneous correlation coef®cient between r and B. As

demonstrated by Titman and Torous (1989), the instantaneous correlation between the

building value process and the long-term yield process is assumed to be zero.

The value of the three assets (the default-free bond, the building, and the mortgage) can

be determined as functions of the three-state variablesÐr, l, and B. The value of the

default-free bond is generated by two interest rates, while the value of the mortgage is

generated by two interest-rate processes and the building-value process. Further, the

default-free bond and the mortgage are combined into a portfolio that is instantaneously

risk free. As illustrated by Cox, Ingersoll, and Ross (1985a, 1985b), this no-arbitrage

condition ensures that the value of any derivative asset, such as a mortgage, can be

derived from a partial differential equation (PDE) with appropriate boundary and

initial conditions.

The valuation equation contains three market prices of risk for r, l, and B. However, as

Brennan and Schwartz (1982, 1985) noted, the no-arbitrage condition ensures that the

market price of risk for long-term yield, l, and the building value, B, can be expressed in

terms of their derivatives with respect to r, l, and B. Black and Scholes (1973) showed that

the value of a stock or its risk price is unnecessary to price an option on the stock and that


the market price of risk cannot be eliminated because the instantaneous security is not a

traded asset. The market price of consol rate risk, Ll, and the building value risk, LB are

expressed as

Ll � ��al � blr � cll�=lÿ �s2l � lÿ r��=sl; �6�

LB � �aÿ r�=�sBB�: �7�

After substituting equations (3), (5), (6), and (7) into equations (1), (2), and (4), and

applying Ito's lemma, the valuation equation (which must be satis®ed by the mortgage

value V), is expressed as

12s2

r r2Vrr � srslrlrrlVrl � 12s2

l l2Vll � srsBrBrrBVrB � 12s2

BB2VBB

� �ar � br�lÿ r� ÿ Lrsrr�Vr � l�s2l � lÿ r�Vl � �r ÿ b0�BVB

� Vt ÿ rV � m � 0: �8�

Here, Lr is the market price of risk associated with r, m is the continuous rate of mortgage

payment, and V � V�r; l;B; t� is the mortgage value �t 2 �0; T��. Subscripted V's are

respective partial derivatives. With appropriate terminal and boundary conditions, the

valuation equation (8) represents the three-state variable valuation model, and it provides

the value of the mortgage as a function of the three-state variables.

2.2. SPOT-CONSOL Model

The SPOT-CONSOL model demonstrates that the mortgage return can be described by

two-state variablesÐr and l. Substituting equations (3) and (6) into equations (1) and (2)

and applying Ito's lemma results in the valuation equation (which must be satis®ed by the

mortgage value V), which is expressed as

12s2

r r2Vrr � srslrlrrlVrl � 12s2

l l2Vll � �ar � br�lÿ r� ÿ Lrsrr�Vr

� l�s2l � lÿ r�Vl � Vt ÿ rV � m � 0; �9�

where the mortgage value is V � V�r; l; t�. This model is a special case of the SPOT-

CONSOL-VALUE model when srsBrrBrBVrB � 0; 12s2

BB2VBB � 0; and �r ÿ b0�BVB � 0

in equation (8). The solution to equation (9) yields the mortgage value as a function of

two-state variables.

2.3. SPOT-VALUE Model

The SPOT-VALUE model demonstrates that the return of a mortgage can be described by

two-state variablesÐr and B. Unlike the SPOT-CONSOL model, the SPOT-VALUE


model assumes that the term structure of interest rates can be summarized by a single-state

variable r. Interest rates are assumed to follow a mean reverting processÐthat is, r reverts

to its steady-state mean. The stochastic process is given as

dr � �ar � br�iÿ r��dt� srrdzr: �10�

Here, br is the speed of adjustment parameter, and i is the steady state mean of the

process (long-term mean of r). After substituting equations (5) and (7) into equations (4)

and (10) and applying Ito's lemma, the valuation equation (which must be satis®ed by the

mortgage value V), can be expressed as

12s2

r r2Vrr � srsBrBrrBVrB � 12s2

BB2VBB � �ar � br�iÿ r� ÿ Lrsrr�Vr

� �r ÿ b0�BVB � Vt ÿ rV � m � 0: �11�

Here, V � V�r;B; t� represents the value of the mortgage. This model is a special case of

the SPOT-CONSOL-VALUE model when

srslrrlrlVrl � 0; 12s2

l l2Vll � 0; l�s2l � lÿ r�Vl � 0;

and l � i � constant in equation (8). The solution to equation (11) yields the

mortgage value as a function of two-state variables.

2.4. SPOT Model

This model demonstrates that the return on a mortgage can be described by a single-state

variable r. The corresponding valuation equation that satis®es the mortgage value V is

derived from equation (10) and, using Ito's lemma, can be expressed as

12s2

r r2Vrr � �ar � br�iÿ r� ÿ Lrsrr�Vr � Vt ÿ rV � m � 0: �12�

Here, V � V�r; t� is the mortgage value. This model is a special case of the three previous

models if srsBrrBrBVrB � 0; 12s2

BB2VBB � 0; and �r ÿ b0�BVB � 0 in equation (11) or

srslrrlrlVrl � 0; 12s2

l l2Vll � 0; l�s2l � lÿ r�Vl � 0, and l � i � constant in equation (9).

The solution of equation (12) yields the value of the mortgage as a function of a single-

state variable.

2.5. Default and Prepayment Options

As the residential mortgages in this study include default and prepayment options, for

accurate mortgage pricing, the values of these options must be incorporated in the model.

These options are speci®ed by the terminal and boundary conditions to solve respective


partial differential equations (PDEs). The framework for our boundary conditions follows

the arguments of Kau, Keenan, Muller, and Epperson (1987).

Default occurs when the mortgage value exceeds the value of the mortgaged building.

As Kau and Keenan (1995, p. 225) point out, ``Given that one can enjoy the service or

rental ¯ow of a house until a payment is due, one will not rationally default except at

payment dates.'' Accordingly, at each payment date k, the default option value is evaluated

as follows:

DEFk � DEFk�1ifXn

j�k

Mj ÿ DEFk�1 ÿ PREk�15Bk �no default�

�Xn

j�k

Mj ÿ Bk otherwise �default�; �13�

where, DEFk is the default option value at period k, Mj is the monthly mortgage payment,

PREk�1 is the prepayment option value at period k � 1, and Bk is the building value at

period k. If the building value is greater than the mortgage value at period k, the default

option is carried unexercised to the next period; otherwise, the difference between the

unpaid balance and the building value represents the value of the default option.

Prepayment occurs when the loan's contract rate exceeds the prevailing re®nancing

rate. Unlike the default option, the prepayment option may be exercised between any two

payment dates. Thus, at each payment date k, the prepayment option can take two

positions, which are

PREk � 0 if �default� � PREk�1 otherwise �no default�: �14�

At payment date k, if default occurs, the prepayment option will have no value.

Otherwise, it will be carried unexercised to the next period. If prepayment occurs at any

time t, the prepayment value option is described as

PREt �Xn

j�t

Mj ÿ �1� ckftÿ �k ÿ 1�g�Ukÿ18 k ÿ 15t � k: �15�

Pnj�t Mj represents the total promised mortgage payment at period t, ck is the contract

rate at period k, Ukÿ1 is the unpaid principal at period k ÿ 1, and ckftÿ �k ÿ 1�gUkÿ1 is

the accrued interest on the unpaid balance for holding the mortgage during period k ÿ 1 to

t. The difference between the promised mortgage payment and the unpaid principal at

period t represents the value of the prepayment option.

Last, the terminal boundary condition is expressed as follows:

V�r; l;B; T� � 0; �16�

where t 2 �0; T�. The time span �0; T� denotes the term-to-maturity of each mortgage.

Equation (16) states that the mortgage is fully amortized at the maturity date. Subject to

the terminal and boundary conditions, all four PDEs can now be solved.


3. Data and Methodology

3.1. Data

Detailed mortgage data were collected from mortgage originators (savings and loans in

three states). A survey6 was sent to 149 institutions located in Mississippi, Arkansas, and

Louisiana. Institutions in RTC conservatorship as of August 1, 1990 were not included in

the population.7 Mortgages quali®ed for the survey if they met the following criteria:

* First mortgage loans granted to individuals for owner-occupied residential proper-

ties.* Loans booked on one of twenty-two business dates, randomly selected8 from each

quarter in even years beginning in 1980 and ending in mid-1990.

All loans meeting the above quali®cations were selected for the survey.9 The booking

dates represent 1% of all business dates during the period; thus, each loan in the sample

corresponds to approximately 100 loans originated.

The survey generated mortgage loan responses from twenty-nine institutions, each

response representing one mortgage loan package. The mortgages had a total original loan

amount of $23,070,609 and a total current balance of $17,334,211; the average original

amount borrowed was $59,004, and the average amount currently owed was $44,333. The

summary statistics (number of loans booked, average loan-to-value (LTV) ratio, average

contract rates, and average points for twenty-two dates) are provided in table 1.

The largest number of loans booked was in 1988, followed by 1986, 1984, 1980, 1990,

and 1982. With the exception of a single date (November 17, 1988), the average LTV

ratios are generally between 70 and 80%. For all loans pooled together, the average LTV

ratio is 0.74, though a number of individual mortgages report high LTVs.10 Titman and

Torous (1989) and Kau, Keenan, Muller, and Epperson (1992) observed that the value of

the default option in the presence of the prepayment option will have little impact on

mortgage valuation if the LTV ratio is lower than 0.80. Other research suggests that low

LTV ratios are important for default option valuation. Brennan and Schwartz (1985)

suggest that model valuation will be sensitive to building value. Building value and the

accompanying LTV ratio may be viewed as an indirect indicator of mortgage default

probability.

During the sample period, the average contract rates vary directly with the T-bill and

T-bond rates. Average points, however, fail to show any systematic pattern.

The spot rate, consol rate, and building value parameters are proxied as follows:

* The yield-to-maturity on one-month certi®cates of deposit proxies the spot rate;* The coupon yield on a thirty-year maturity T-bond proxies the consol rate; and* The average new building value in the mid-south region proxies the building value.

Monthly CD rates and T-bond rates are obtained from the Federal Reserve Bulletin, while

monthly average building values are obtained from Current Construction Reports.


Table 1. Statistical summary of all loans booked on different dates.

Dates Loans Booked Average Loan-to-Value Ratio Average Interest Rate Average Points

01/16/80 7 0.6867 10.8786 2.2857

(0.1864) (1.5411) (1.359)

06/05/80 5 0.7542 12.05 1

(0.1621) (2.0025) (0.8367)

07/24/80 11 0.7854 11.5375 2.1

(0.1735) (0.5064) (1.3565)

12/10/80 7 0.7026 12.2321 3.2143

(0.1744) (1.5533) (1.8489)

09/27/82 3 0.7388 15.4667 2.00

(0.0405) (0.4497) (1.4142)

11/26/82 5 0.721 14.05 1.5

(0.1217) (1.1522) (1.0243)

03/02/84 18 0.7343 12.0476 2.1765

(0.1815) (1.0834) (1.4583)

05/02/84 15 0.6827 12.7747 1.0667

(0.2185) (0.9439) (0.717)

08/30/84 21 0.7603 12.2553 1.8095

(0.1139) (1.2158) (1.5026)

12/05/84 13 0.7909 12.5312 1.3846

(0.1274) (0.9776) (0.9869)

03/17/86 24 0.7823 10.1156 2.0312

(0.1168) (0.7279) (1.1852)

06/19/86 27 0.7215 9.8982 2.0555

(0.1963) (0.8601) (1.5206)

09/02/86 13 0.7919 9.75 1.9615

(0.1019) (0.8506) (0.8009)

10/28/86 36 0.7865 9.5632 1.8184

(0.1643) (0.8008) (1.073)

01/04/88 4 0.783 8.625 1.25

(0.062) (0.4714) (1.4142)

04/25/88 26 0.779 9.19 1.4615

(0.1793) (0.6145) (1.1145)

09/30/88 79 0.768 9.6668 1.5367

(0.1847) (1.5011) (1.0441)

11/17/88 30 0.801 9.6633 1.375

(0.275) (0.8314) (1.2242)

02/07/90 14 0.7022 9.9714 1.00

(0.2297) (1.3125) (0.7216)

05/22/90 8 0.7462 9.6719 1.4219

(0.1021) (1.0559) (0.7764)

Total 374 0.7444 10.2322 1.6581

Note: Standard deviations are in parentheses.


3.2. Valuation Methodology

We solve PDEs in two steps. First, the parameters of the three-state variable processes are

estimated. These estimated values are then plugged back into different PDEs. Second, we

numerically solve the equations by the explicit ®nite difference methods.

We extend the methodology of Brennan and Schwartz (1982) for estimating the

stochastic processes. All state variable stochastic processes are replaced by discrete

approximations. Equations (1), (2), (4), and (10) are approximated as11

�rt ÿ rtÿ1�=rtÿ1 � ar=rrÿ1 � br��ltÿ1 ÿ rtÿ1�=rtÿ1� � ert; �17�

�lt ÿ ltÿ1�=ltÿ1 � al=ltÿ1 � bl�rtÿ1=ltÿ1 � cl � elt; �18�

�Bt ÿ Btÿ1�=Btÿ1 � aBrtÿ1 ÿ b0 � eBt; �19�

�rt ÿ rtÿ1�=rtÿ1 � ar=rtÿ1 � br��9:75ÿ rtÿ1�=rtÿ1� � ert: �20�

Here, t represents the current period, and 9.75 is the steady-state mean of r (the eleven-

year mean of monthly CD rates). The market price of the risk of r minimizes the price-

prediction errors between primary market prices and model-generated prices. These

equations are estimated by an iterative Aitken procedure, which yields the maximum

likelihood estimator. The estimated values of these parameters are discussed in the next

section.

The explicit ®nite difference method used here extends to methodology suggested by

Hull and White (1990, 1993) to the three-variable case. The numerical solution of a PDE

with more than a one-state variable is possible with two assumptions: (1) the instantaneous

standard deviation of each variable is constant, and (2) the variables are uncorrelated.

These assumptions are resolved by two sets of transformations.

Three state variables r, l, and B are transformed into three new state variables (Y1;Y2, and Y3) so that the instantaneous standard deviations are constant. The three

corresponding processes are described as follows:

dY1 � q1dt� k1dz1; �21�

dY2 � q2dt� k2dz2; �22�

dY3 � q3dt� k3dz3: �23�

The expressions for q's and k's are provided in the appendix. In multiple-variable model

situations, Weiner processes dz1, dz2, and dz3 are instantaneously correlated to each other.

More speci®cally, the correlation between dz1 and dz2 represents rrl, and the correlation

between dz1 and dz3 represents rrB. The correlation between dz2 and dz3 is assumed to be

zero in the model speci®cation. Therefore, three variables Y1, Y2, and Y3 are transformed


into four new state variables f1, f2, f3, and f4, which are instantaneously uncorrelated.

The four corresponding processes can be described as follows:

df1 � �k2q1 � k1q2�dt� k1k2

p�2�1� rrl��dz4; �24�

df2 � �k2q1 ÿ k1q2�dt� k1k2

p�2�1ÿ rrl��dz5; �25�

df3 � �k3q1 � k1q3�dt� k1k3

p�2�1� rrB��dz6; �26�

df4 � �k3q1 ÿ k1q3�dt� k1k3

p�2�1ÿ rrB��dz7: �27�

These new variables are independent of each other; therefore, their possible

unconditional movements (the corresponding PDEs) can be found individually. The

PDE for any of the new variables is expressed as

Viÿ1;j � f1=�1� rDt�gfPj;jÿ1Vi;jÿ1 � Pj;jVj;j � Pj;j�1Vi;j�1 �Mig: �28�

Here, I denotes a drift in time interval, j denotes a drift in the variable, Mi is the

continuous rate of mortgage payment at I, Dt is the unit time interval, V denotes the value

of the mortgage, and P denotes the corresponding probability. In this backward-solving

method, for fully amortized loans (when the mortgage value will be equal to zero), the

valuation starts at the maturity date T. The value of the mortgage at time t can be obtained

by repeatedly working backward from time T to time t in steps of Dt. For each variable,

the value of the mortgage at time I ÿ 1 is associated with three possible unconditional

movements: jÿ 1, j, and j� 1 at time I. Therefore, the PDE corresponding to any

variable can be modeled by using a one-dimensional lattice with three branches coming

out of each node.

The PDEs for more than one variable are now easier to form as all four new variables are

independent of one another. The SPOT-CONSOL-VALUE model contains the new

variables f1, f2, f3, and f4. The probability of reaching any given point is the product of

unconditional probabilities associated with corresponding movements from f1 to f4. We

represent the SPOT-CONSOL-VALUE model with a four-dimensional lattice (a product

of four unrelated one-dimensional lattices). The value of the mortgage at time I ÿ 1 is

associated with eighty-one alternative values of the mortgage at time I (34 branches at

each node).

The two-variable models can be estimated in the same fashion. The SPOT-CONSOL

model is estimated by using f1 and f2: The SPOT-VALUE model is estimated by using f3

and f4: We represent these two models with two-dimensional lattices (a product of two

one-dimensional lattices), and, for each model, the value of the mortgage at time I ÿ 1 is

associated with nine alternative values of the mortgage at time I (32 branches at each

node).

Because of the absence of any correlation terms, the SPOT model is estimated by using

the ®rst set of transformations or the new state variable Y1 corresponding to three

probabilities. This model is represented by using a one-dimensional lattice, and the value


of the mortgage at time I ÿ 1 is associated with three alternative values of the mortgage at

time I.

4. Results

4.1. Parameter Estimates

The estimated parameter values were entered into valuation equations to compare the

prices of various mortgage contracts. The parameter estimates for the individual models

are given in table 2.

The estimated values of br are positive but statistically insigni®cant for all four models.

This is consistent with the assumption of the short-rate process as a mean reverting

process. For the SPOT and SPOT-VALUE models, the positive b1 values imply that the

short-rate regresses slowly toward its long-term mean over time. For the SPOT-CONSOL

and SPOT-CONSOL-VALUE models, the positive b values imply that the short rate will

have a tendency to regress towards the current value of the long rate. This ®nding supports

that of Brennan and Schwartz (1982, 1985).

The estimated values of ar are predominantly negative for the SPOT and SPOT-VALUE

models. In both models, the absolute values of ar are approximately 9.6 times the values of

br. This illustrates that the change in the short rate at r � 0 is positive for all current rvalues, as long as the consol rate exceeds 9.6%. The mean of the short rate (the long-term

mean of r) is 9.75%; therefore, the possibility of any misspeci®cation of the short-rate

process is unfounded. The estimated values of ar are positive for both the SPOT-CONSOL

and SPOT-CONSOL-VALUE models. This implies that the change in the short rate at

r � 0 is positive for all current values of r and l. This ®nding is also consistent with that of

Brennan and Schwartz (1982, 1985). The estimated values of the volatility parameter, sr

for all models are as expected. The D-W statistic for all models shows values less than 2;

therefore, the presence of positive autocorrelation is not indicated.

In the SPOT-CONSOL and SPOT-CONSOL-VALUE models, the absolute values of al

are approximately equal to the absolute values of bl; however, the signs of bl and cl are

both negative. This result contradicts the ®ndings of Brennan and Schwartz (1982, 1985);

however, our study period covers January 1980 through May 1990, a period of extremely

volatile interest rates. The Brennan and Schwartz studies are based on data from the two

previous decades. The estimated values of sl are as expected. The estimated values of the

correlation parameter rrl, however, show strong positive correlation between two

processes. The D-W statistic for both models shows values less than the lower bound of the

critical value, thus suggesting the presence of positive autocorrelation and a possible

omission of relevant state variables.

In the SPOT-VALUE and SPOT-CONSOL-VALUE models, the estimated values of the

payout rate b0 are negative. This ®nding, though contrary to popular intuition, does not

refute the assumption of lognormality of the building value process. The negative values

may be a result of speci®c regional factors. The estimated values of the volatility

parameter, sB, conform to observations in Titman and Torous (1989). The estimated


values of the correlation parameter, rrB, are observed as positive. This is contrary to the

popular belief that interest rates and building value movements are inversely related;

however, the values of rrB are relatively smaller than the implied correlation parameter

values observed by Titman and Torous (1989). The positive correlation coef®cient

indicates some form of mispricing in the building value process. The D-W statistics for

both models show values greater than 2 and less than the lower bound of the critical value,

indicating the presence of negative autocorrelation and the possible omission of relevant

state variables.

4.2. Valuation Results

Table 3 reports the mortgage valuation results for each of the four models. For each date

the loans are booked, results are shown for the average model price per $100 of primary

mortgage value for all mortgages in the sample and for two subgroups, short-term loans

(one to ®fteen years) and long-term loans (sixteen to thirty years).

We ®nd that the average price predicted by each model is more than $100 for all four

models and for all dates. The average positive spread is lowest for the SPOT model,

Table 2. Parameter estimates for different models.

Different Models

Parameters Spot Spot-Consol Spot-Value Spot-Consol-Value

ar ÿ 0.1795 0.0001 ÿ 0.1815 0.0002

(ÿ 0.5902) (0.1505) (ÿ 0.5967) (0.1551)

br 0.0187 0.0531 0.0189 0.0528

(0.5929) (1.0822) (0.5994) (1.0789)

sr 0.0919 0.0915 0.0919 0.0915

DW-statistic 1.4623 1.4519 1.4621 1.4521

al 0.0023 0.0023

(1.4003) (1.4021)

bl ÿ 0.0023 ÿ 0.0023

(ÿ 0.1189) (ÿ 0.1166)

cl ÿ 0.0177 ÿ 0.0178

(ÿ 0.6617) (ÿ 0.6643)

sl 0.0392 0.0392

DW-statistic 1.2274 1.2274

rrl 0.5234 0.5234

b0 ÿ 0.0055 ÿ 0.0051

(ÿ 0.1172) (ÿ 0.1087)

sB 0.1604 0.1604

DW-statistic 2.9763 2.9762

rrB 0.0164 0.0158

Note: t-statistics are in parentheses.


Table 3. All loans valuation results.

Average Model prices per $100 of Actual Mortgage Values

Dates

Years-to-

Maturity

(in years) Spot

Spot-

Consol

Spot-

Value

Spot-Consol-

Value

01/16/80 01±30 112.33 112.83 113.09 112.97

15±30 109.77 110.19 110.41 110.30

01±15 127.70 128.63 129.19 128.98

06/05/80 01±30 115.25 116.75 118.49 118.18

15±30 108.27 108.80 109.11 108.94

01±15 143.18 148.56 156.00 155.13

07/24/80 01±30 113.17 113.80 114.03 113.90

15±30 110.22 110.75 110.91 110.82

01±15 124.95 126.00 126.50 126.22

12/10/80 01±30 108.97 109.55 109.81 109.67

15±30 108.97 109.55 109.81 109.67

01±15 0.00 0.00 0.00 0.00

09/27/82 01±30 114.97 115.99 116.27 116.15

15±30 108.15 108.80 109.01 108.94

01±15 118.37 119.59 119.90 119.75

11/26/82 01±30 110.66 111.29 111.55 111.41

15±30 107.72 108.30 108.53 108.41

01±15 122.43 123.26 123.60 123.41

03/02/84 01±30 113.44 114.00 114.29 114.13

15±30 109.41 109.90 110.13 110.01

01±15 126.54 127.32 127.84 127.56

05/02/84 01±30 116.33 116.94 117.12 117.25

15±30 110.10 110.69 111.01 111.19

01±15 121.01 121.63 121.70 121.81

08/30/84 01±30 110.25 110.67 110.90 110.85

15±30 108.75 109.14 109.36 109.33

01±15 119.22 119.81 120.16 119.98

12/05/84 01±30 110.28 110.77 111.23 111.11

15±30 108.70 109.14 109.60 109.49

01±15 129.18 130.32 130.83 130.55

03/17/86 01±30 111.01 111.36 111.56 111.46

15±30 109.63 109.99 110.21 110.09

01±15 113.59 113.91 114.11 114.01

06/19/86 01±30 113.36 113.86 114.15 113.98

15±30 110.31 110.70 110.91 110.78

01±15 115.40 115.97 116.31 116.12

09/02/86 01±30 113.67 114.06 114.37 114.22

15±30 108.73 109.22 109.47 109.34

01±15 116.75 117.08 117.44 117.28

10/28/86 01±30 111.85 112.75 112.59 112.39

15±30 109.92 110.32 110.67 110.42

01±15 114.16 115.63 114.87 114.74

01/04/88 01±30 109.72 110.19 110.47 110.36

15±30 109.08 109.59 109.85 109.76

01±15 111.65 111.99 112.30 112.15


followed by the SPOT-CONSOL, SPOT-CONSOL-VALUE and SPOT-VALUE models;

however, no signi®cant pattern in the spread is observed over different dates.

The investigation of average model prices reveals that the average prices for short-term

loans are systematically higher than for long-term loans. This ®nding is valid for all

models and all dates, thus extending the research of Titman and Torous (1989). Our

observation strongly suggests an inverse relationship between the average model price (as

well as the average spread amount) and the term-to-maturity of a mortgage. Roll (1994)

observed that the explanatory power of contingent claims models greatly increases with

longer-term maturities. In the context of mortgage valuation models, the increase in

explanatory power reduces the pricing spread, further implying that the differences

between average model prices are more pronounced for short-term loans.

Unlike an accounting approach to transactions cost estimation (margin points,

origination fees, prepayment penalty clauses, and past-due penalties), the valuation

procedure implicitly estimates transaction costs. Several factors justify the existence of a

systematic pricing spread. Brennan and Schwartz (1985) and Titman and Torous (1989)

observe that model valuation is more sensitive to l (the consol rate) and B (building value)

than to r (the spot rate). The zero correlation between the long-rate and the building value

Table 3. (continued)

Average Model prices per $100 of Actual Mortgage Values

Dates

Years-to-

Maturity

(in years) Spot

Spot-

Consol

Spot-

Value

Spot-Consol-

Value

04/25/88 01±30 112.59 113.07 113.32 113.19

15±30 109.01 109.41 109.60 109.50

01±15 117.96 118.56 118.88 118.72

09/30/88 01±30 113.21 113.76 113.98 114.03

15±30 108.98 109.40 109.61 109.52

01±15 118.91 119.62 119.85 120.09

11/17/88 01±30 113.56 114.17 114.44 114.30

15±30 108.69 109.19 109.45 109.31

01±15 119.13 119.85 120.15 120.00

02/07/90 01±30 115.53 116.19 116.49 116.41

15±30 107.55 107.93 108.15 108.04

01±15 118.72 119.49 119.83 119.76

05/22/90 01±30 113.49 113.97 114.23 114.09

15±30 108.98 109.35 109.58 109.46

01±15 119.28 119.90 120.21 120.06

Total 01±30 112.78 113.35 113.59 113.50

15±30 109.17 109.61 109.86 109.75

01±15 118.25 119.02 119.24 119.19


(rlB) in the SPOT-CONSOL-VALUE model is, perhaps, the most important contributing

factor to the spread. Furthermore, the high implied long-term mean of r in the SPOT and

SPOT-VALUE models, and the frequently higher long-rate values (higher than short-rate

values) in the SPOT-CONSOL model and SPOT-CONSOL-VALUE model can be thought

of as contributing factors. This observation is supported by both Brennan and Schwartz

(1985) and Titman and Torous (1989). Another contributing factor to the spread might be

the possible mispricing of the building value process parameters.

4.3. Statistical Test Results

Two parametric tests (t and likelihood-ratio) and two nonparametric tests (Mann±Whitney

and Kruskal±Wallis) were performed to test the ef®ciency of each model in predicting

primary market values and the relative ef®ciency of individual models. As our valuation

results indicate an inverse relationship between the spread amount and term-to-maturity,

these tests are also applied not only to the total sample of loans but also to subgroups of

long-term and short-term loans. Test results are provided in table 4 (total sample of loans),

table 5 (long-term loans), and table 6 (short-term loans).

The t-test analysis for the equality of two means between actual market primary value

and model estimated value for all four models fails to reject the null hypothesis; therefore,

the results suggest that all models are equally ef®cient. The likelihood-ratio test results for

all possible restricted±unrestricted pairs show that the likelihood ratios for all pairs are

statistically insigni®cant for all samples (total, long-term, and short-term).

The Mann±Whitney test for the equality of two medians between the actual market

primary value and all four models' estimated values reveals that the null hypothesis is

rejected for the total sample of mortgage loans (table 4) and short-term loans (table 6);

while it is rejected for the subsample of long-term loans (table 5). The inef®ciency in

pricing for the total loan sample is attributed to the inclusion of short-term loans in the

sample. The test between the SPOT-CONSOL-VALUE model and the other models shows

that all models are equally ef®cient (or inef®cient) irrespective of the term-to-maturity of

the loans.

The Kruskal±Wallis test among the actual mortgage values for the primary market and

four estimated model values indicates that the null hypothesis of the equality among

all medians is rejected for the total sample (table 4) but is accepted for both long-term

(table 5) and short-term (table 6) loans. This result suggests that these models are ef®cient

for each category; however, for pooled data, they are inef®cient.

In summary, the test results strongly indicate that, on average, all four models are

ef®cient in predicting actual primary mortgage prices. Ef®ciency is more pronounced for

long-term loans than for short-term loans. These results are empirically supported by

Buser, Hendershott, and Sanders (1990) and Gilliberto and Ling (1992).

4.4. Regression Results

The presence of the positive pricing spread amount12 between the model estimates and the

actual mortgage values calls for further investigation. Regression analysis was selected


as the methodology to test the importance of various mortgage-related factors. The depen-

dent variable is the spread amount.13 Five mortgage-related variables are considered as

possible explanatory variables. They are as follows: short-term interest rate (SHORT), long-

term interest rate (LONG), building value (BUILDING), loan-to-value ration (LTV), and

initial contract rate (CONTRACT). The results of the regressions are presented in table 7.

The intercept terms for all four models are statistically signi®cant and positive. The

short-term rate process coef®cients are statistically signi®cant and negative for all four

Table 4. Statistical results for all loans: comparison among different models.

Models for Comparison

Statistical Procedures SPT SPTCON SPTVAL SPTCONVALa

T-test

Dependent variable ORIGa

T-values ÿ 1.18 ÿ 1.22 ÿ 1.24 ÿ 1.23

Dependent variable SPTCONVALa

T-values 0.06 0.01 ÿ 0.01

Mann±Whitney test


Statistic T 58898.0 58621.0 58493.0 58588.0

z-value ÿ 2.3542** ÿ 2.4526** ÿ 2.4981** ÿ 2.4642**


Statistic T 65017.0 65313.0 65271.0

z-value ÿ 0.1795 ÿ 0.0743 ÿ 0.892

Likelihood ratio test

Unrestricted model SPTCONVALa

Likelihood ratio L* 1.0019 1.0016 0.9998

Chi-square value ÿ 0.0024 ÿ 0.0032 ÿ 0.0039

Unrestricted model SPTCONa

Likelihood ratio L* 0.9996

Chi-square value 0.0008

Unrestricted model SPTVALa


Chi-square value ÿ 0.0028

Kruskal±Wallis test

Comparative basis: ORIG, SPT,

SPTCON, SPTVAL, SPTCONVALa

Chi-square value 9.6031**

Comparative basis: SPT, SPTCON,

SPTVAL, SPTCONVALa


aVariable description:

ORIG: actual mortgage values,

SPT: SPOT model mortgage values,

SPTCON: SPOT-CONSOL model mortgage values,

SPTVAL: SPOT-VALUE model mortgage values,

SPTCONVAL: SPOT-CONSOL-VALUE model mortgage values.

**Signi®cant at the 5% level.


models. Negative coef®cients imply that the higher the short-term rate, the lower the

pricing spread. Since the short-term rate re¯ects current market volatility, low short rates

re¯ect both the borrower's and lender's perception of future rate increases, which would

result in a higher pricing spread. This study employed a backward-solving valuation

method for mortgages with ten to thirty years until maturity, and with this valuation

methodology, long-term economic conditions are more important to the lenders. The

results show that the long-term interest-rate coef®cients are negative and statistically

Table 5. Statistical results for long term (15 to 30 years) loans: comparison among different models.



T-test


T-values ÿ 0.75 ÿ 0.78 ÿ 0.79 ÿ 0.79


T-values 0.04 0.01 ÿ 0.01

Mann±Whitney test


Statistic T 21527.0 21434.0 21370.0 21414.5

z-value ÿ 1.6989 ÿ 1.7696 ÿ 1.8183 ÿ 1.7844


Statistic T 23520.0 23621.5 23622.5

z-value ÿ 0.1841 ÿ 0.1068 ÿ 0.1062




Chi-square value ÿ 0.0086 0.0016 0.0011












SPTVAL, SPTCONVALa










insigni®cant for all four models. Consequently, the short-term rate process alone appears

to be suf®cient to capture the current and future interest-rate volatility.

These results raise the question as to why the long-term rate is not statistically

signi®cant while the short-term rate is signi®cant. A possible answer could be the variance

of the long-term rate. The calculated standard deviation for both rates was quite similar

(0.53 and 0.56); consequently, a wider variance does not explain the differences.

Building value and the LTV ratio can be viewed as the indirect indicators of the default

Table 6. Statistical results for short-term (1 to 15 years) loans: comparison among different models.



T-test


T-values ÿ 1.55 ÿ 1.60 ÿ 1.61 ÿ 1.60


T-values 0.05 0.00 ÿ 0.01

Mann±Whitney test


Statistic T 8958.0 8900.0 8865.0 8892.0

z-value ÿ 1.9953** ÿ 2.0774** ÿ 2.1269** ÿ 2.0887**


Statistic T 10244.0 10302.0 10290.0

z-value ÿ 0.1755 ÿ 0.0934 ÿ 0.1104




Chi-square value 0.0035 ÿ 0.0066 ÿ 0.0014












SPTVAL, SPTCONVALa










probability of a mortgage. Both the building value and the LTV ratio coef®cients are

decidedly statistically signi®cant and positive for each model, thus indicating that the

higher the building value and/or the LTV ratio, the greater the default risk and,

correspondingly, the pricing spread.

The coef®cients for the initial contract rate are statistically signi®cant for only three of

the four models (SPOT, SPOT-CONSOL, and SPOT-VALUE). The positive coef®cients

imply that the higher the initial contract rate, the greater the prepayment risk and,

concomitantly, the pricing spread. Most important, the absence of any signi®cant

coef®cient for the long-term rate process suggests that the short-term rate and building

value are adequate economic variables to value a mortgage (the R2 values are more than

0.70 for all four models).

5. Conclusion

This study has examined four individual contingent claims mortgage valuation modelsÐ

SPOT, SPOT-CONSOL, SPOT-VALUE, and SPOT-CONSOL-VALUEÐto determine the

relative ef®ciency of each in predicting mortgage values from primary mortgage market

data. Each model includes one or more of three state variablesÐthe spot rate, the consol

rate, and the value of the mortgaged building. All of the mortgages in the sample included

both prepayment and default options. The values of these options are incorporated in the

model as the necessary terminal and boundary conditions to solve respective partial

Table 7. The effects of various mortgage indicators on the relative pricing spread.

Comparison for Different Models

Variablea Spot Spot-Consol Spot-Value Spot-Consol-Value

INTERCEPT 1.2261 1.3499 1.3935 1.4241

(4.034)** (4.531)** (4.791)** (4.789)**

SHORT ÿ 0.2577 ÿ 0.2648 ÿ 0.2576 ÿ 0.2591

(ÿ 2.028)** (ÿ 2.125)** (ÿ 2.118)** (ÿ 2.083)**

LONG ÿ 0.1124 ÿ 0.0869 ÿ 0.0905 ÿ 0.0619

(ÿ 0.583) (ÿ 0.459) (ÿ 0.491) (ÿ 0.328)

BUILDING 0.6941 0.6854 0.6796 0.6755

(28.358)** (28.569)** (29.012)** (28.200)**

LTV 0.3405 0.3318 0.3478 0.3546

(6.721)** (6.681)** (7.175)** (7.151)**

CONTRACT 0.2435 0.2307 0.2475 0.2244

(2.037)** (1.969)** (2.164)** (1.918)

aVariable description

SHORT: The natural logarithm of the short rates,

LONG: The natural logarithm of the long rates,

BUILDING: The natural logarithm of the building values,

LTV: The natural logarithm of the LTV ratios,

CONTRACT: The natural logarithm of the contract rates.



differential equations. A backward-solving valuation method was employed. The data

came from survey responses (one mortgage loan package) from twenty-nine savings and

loans in three southern states.

All four models demonstrate ef®ciency in predicting primary mortgage prices. Model

ef®ciency is more pronounced for long-term than for short-term mortgages. For each

model's pricing estimate, a systematic positive spread exists between the model's

estimated price and the actual primary market price. The average amount of spread is

lowest for the SPOT model, followed by the SPOT-CONSOL model, SPOT-CONSOL-

VALUE model, and SPOT-VALUE model; however, differences in the spread amounts are

not statistically signi®cant. All models show an inverse relationship between term-to-

maturity and the spread amount of a mortgage. Long-term loans produce lower spread

amounts than short-term loans.

Regression analysis was performed to attempt to determine the cause of the positive

pricing spread. Each model included the three state variables cited above plus the loan-to-

value ratio and the initial contract as explanatory variables. The empirical results indicated

that the long-term interest rate was not statistically signi®cant. Three variables, the short-

term rate, the building value, and the loan-to-value ratio were positively statistically

signi®cant. Both building value and the loan-to-value ratio are indirect indicators of

default probability, and therefore, the higher the building value and loan-to-value ratio, the

greater the default risk, and accordingly, the pricing spread. The short-term rate was

signi®cant but negative in sign, thus implying that the higher current interest rates are, the

lower the pricing spread.

The models and results presented in this article suggest that the SPOT-VALUE model is

the superior model for mortgage valuations. While all four models price ef®ciently, the

regression results suggest that the SPOT-VALUE model is the most ef®cient. The single

variable SPOT model does not include building value, which is highly statistically

signi®cant in alternative models. Both the SPOT-CONSOL-VALUE and SPOT-CONSOL

models include a variable (the long rate) that is not signi®cant. Accordingly, the two

variable SPOT-VALUE model prices ef®ciently and includes what (Kau, Keenan, Muller,

and Epperson, 1995) note are the `èssential characteristics'' of a mortgage ( prepayment

and default options). Our empirical ®ndings suggest that a simple but powerful model

(SPOT-VALUE) can ef®ciently value actual primary mortgage data.

Appendix

The expressions for q's and k's are

q1�r; t� � dY1=dt� �ar � br�lÿ r� ÿ Lrsrr��dY1=dr� � 12s2

r r2�d2Y1=dr2�q2�l; t� � dY2=dt� �s2

l � lÿ r�l�dY2=dl� � 12s2

l l2�d2Y2=dl2�q3�B; t� � dY3=dt� �r ÿ b0�B�dY3=dB� � 1

2s2

BB2�d2Y3=dB2�k1 � �dY1=dr�srr

k2 � �dY2=dl�sll

k3 � �dY3=dB�sBB:


Acknowledgments

Gay Hat®eld received grant support from the School of Business at The University of

Mississippi, University, MS.

Notes

1. This was the ®rst study to utilize an instantaneous risk-free rate as the sole variable to price GNMA securities.

2. For a comprehensive review of the contingent claims mortgage valuation literature, see Hendershott and Van

Order (1987) and Kau and Keenan (1995).

3. For a complete discussion of the three-state variable model (model speci®cation and estimation), see

Chatterjee, Edmister, and Hat®eld (1995).

4. Hendershott and Van Order (1987) and Kau, Keenan, Muller and Epperson (1992) employ simulations, not

actual data.

5. For ease of substituting in equations, the three-state-variable model is presented ®rst, followed by both two-

state models, and then the single-state model.

6. The survey design emphasized consistency and ¯exibility. The objective was to ensure consistent responses

despite different accounting systems and application forms at separate institutions. To improve feasibility and

provide a consistent interpretation of responses, data items were extracted primarily from original mortgage

documents by researchers rather than respondents.

7. Including failed thrift institutions would bias the study in one way, while excluding them would result in a

separate bias.

8. A random number table was employed to select random dates and months. The table had 100 lines and 14

columns. Matching began with the ®rst line and column (cell(1,1)), and the last two digits of that number was

used to select months. The rule was as follows: if the two digit number is between 01 and 03, then select;

otherwise, go to the next number in the second line, ®rst column (cell (2,1)), and so on. When numbers in the

®rst column were ®nished, the same procedure was started from the ®rst line, second column (cell (1,2)). The

number between 01 and 03 indicated the month in each quarter. For example, month 02 from the third quarter

in 1984 would be August 1984. After twenty-two random months were selected, twenty-two random days

were selected in a similar fashion.

9. No loans were booked on two dates (January 12, 1982 and April 27, 1982) by the responding institutions.

10. During the 1980s, a period of highly volatile interest rates, savings and loan institutions avoided issuing

®xed-rate loans with high LTV ratios.

11. In the estimation process, the instantaneous rate of return on building �a� in equation (19) is approximated as

a fraction of the instantaneous (spot) interest rate �aBrtÿ1�: Cunningham and Hendershott (1984) and

Hendershott and Van Order (1987) showed that the building value appreciation rate �aÿ b0� exactly equals to

�r ÿ b0� in a taxless world, and the end results hold even in a world with taxes.

12. The spread is the amount for every $100 of original loan amount (the actual value) for each mortgage.

13. The spread amount for each observation was subjected to a natural-log transformation so that normal

distribution could be assumed for subsequent regression analyses. For a discussion, see Theerathorn, Bos,

Fetherston, and Chuwonganant (1995).

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Documents

An Empirical Investigation of Alternative Contingent Claims Models for Pricing Residential Mortgages