**Fixed-Rate Mortgages**

### James B. Kau; Donald C. Keenan; Walter J. Muller III; James F. Epperson

*The Journal of Business, Vol. 66, No. 4. (Oct., 1993), pp. 595-618.*

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**Donald C. Keenan **

*University of Georgia *

**Walter J. Muller Ill **

*NationsBank *

**James F. Epperson **

*University of Alabama in Huntsville *

### Option Theory and

### Floating-Rate Securities with a

### Comparison of Adjustable- and

### Fixed-Rate Mortgages

**I. Introduction **

Both the primary and the more newly developed secondary mortgage markets are of enormous size. The securitized portion of the single family mortgage markets alone is over 1.4 trillion dol- lars, which is roughly half the size of all market- able Treasury debt outstanding. Mortgage bor- rowing almost exclusively takes the form of either fixed-rate mortgages (FRMs) or adjust-able-rate mortgages (ARMs), with ARMs having varied between 20% and 60% of the mortgage market over the last decade (Hu 1992). The de- velopment of the primary market in ARMS has led to increased interest in the securitization of ARMs. Yet the lack of experience with ARMs and the complexity of the instrument itself have slowed efforts to develop the market for ARM securities. For example, in the past both the Fed- eral Home Loan Mortgage Corporation and the Federal National Mortgage Association have vir- tually ignored the ARM market, securitizing less

*(Journal of Business, 1993, vol. 66, no. 4) *

*O *1993 by The University of Chicago. All rights reserved.
0021-939819316604-0005$01**.50 **

This article demon- strates how to value floating-rate securities, in particular adjust- able-rate mortgages (ARMs), in the pres- ence of default. The problem is not a straightforward one since endogenous termi- nation (default and pre- payment) necessitates solution by backward procedures, but caps on the floating rate ,

then create path de- pendencies. The solu- tion is to introduce an artificial state variable, the past contract rate, in addition to the natu- ral stochastic variables, the interest and the house price process. With this technique, a numerical investiga- tion of the properties of defaultable ARMs is provided. In all cases, a comparison is made with standard fixed- rate mortgages.

than 5% of total ARM originations (Houston, Sa-Aada, and Shilling 1991). Therefore, a model thoroughly analyzing the features of an ARM could increase the pace of securitization and contribute toward a secondary market for ARMs as fully developed as the one for FRMs.

### '

Adjustable-rate mortgages invariably have caps and floors on the mortgage contract rate. The analytical difficulty with valuing such in- struments is that they are path dependent, since future coupon rates depend on whether past interest rates caused the caps or floors to bind. In Kau et al. (1990a), we show how to overcome these difficulties for the case where the term structure is the only source of uncertainty. This method leads to the analytical valuation of default-free

### ARMS.^

However, empirical studies (e.g., Cunningham and Capone 1990) indi- cate that ARMS have significant default rates; thus the failure to in- clude default in the valuation of ARMS may be considered a serious omission. Consequently, this article introduces techniques to value ARMs where default is permitted.Since FRMs can be treated as degenerate ARMs whose caps and floors coincide, the model presented here is ideal for a direct compari- son of the two types of mortgages. The introduction of default as an option is critical in comparing the behavior of FRMs and ARMs. Un- less the option to default and the option to prepay are both considered, a borrower's behavior cannot be correctly described even qualita-tively. Thus, previous attempts to compare ARMs and FRMs are suspect.

While we focus on the specific features of an ARM, the techniques presented in this article are quite robust. In fact, ARMS-having op-tions to prepay and default coupled with periodic and lifetimes caps-

are one of the most complicated assets available. Many of the securi- ties in the $4 trillion derivative interest rate markets have similar features and are easily valued using our methods. Some examples are floating-rate notes, swaps, caps, floors, and indexed amortizing '

In Section 11, we set out the basic valuation model and then explain 1. By now a number of authors have developed models using backward-pricing tech- niques to value fixed-rate instruments that incorporate endogenous prepayment (Bren- nan and Schwartz 1979; Dunn and McConnell 1981) and default (Kau et al. 1987, 1990b, 1992; Titman and Torous 1989).

2. Not all adjustable-rate mortgages may be exactly solved by this particular proce- dure we follow; i.e., they may depend on considerations beyond the current and future term structure and the past contract rate. For instance, mortgages that set their contract rate according to a cost-of-funds index (COFI) have additional path dependencies due to the lagged structure of COFI, which then calls for additional state variables.

**3. **The difficulties with accurately describing floating-rate instruments are pointed out
by Ramaswamy and Sundaresan (1986), who finesse the issue by selecting a distributed
lag variable to represent the effect of past interest-rate behavior. While ingenious, their
procedure does not describe any actual floating-rate instruments in an exact manner.
Kau et al. (1990a) show that it is possible with a single auxiliary state variable to exactly
model a significant class of default-free floating-rate instruments.

how the introduction of an auxiliary variable eliminates the path- dependency problem, first for the value of payments, where the effect of the housing asset is absent, and then for default and prepayment, where the house value enters the option to terminate. In Section 111, we provide numerical results valuing the various components of a mortgage, including insurance against default. In all cases, comparison is made between results for adjustable-rate mortgages and those for fixed-rate mortgages.

**11. ** **The Model **

**A . The Economic Environment **

The very notion of an adjustable-rate mortgage requires that the term structure be accounted for. It is assumed that this is specified by the spot interest rate r(t), where

Thus, the interest rate evolves according to a mean-reverting CIR sto-
**chastic process (Cox, Ingersoll, and Ross 1985b), where 0 is the **
steady-state spot rate and y is the speed of reversion. The deterministic
trend in interest rates indicated by the first term of (1) is continually
being disturbed by the second, stochastic term, where

*z, *

is a Wiener
*process and a, controls the variance of the disturbance.*

In order that default be considered, it is necessary to include the underlying asset, the house H(t). In standard fashion (Merton 1973), this asset is assumed to satisfy the lognormal process

Here p denotes the instantaneous total rate of return on the asset,
whereas **s **denotes the per unit service flow. The interpretation of the
second, disturbance term is similar to that for the interest rate process.
The two Wiener processes may in fact be correlated as

By standard results of contingent claims analysis (Cox, Ingersoll, and Ross 1985a, 1985b), the value of any derivative asset X(r,H,t) depending solely on the assumed state variables must satisfy the partial differential equation (PDE)~

4. The absence of an interest-rate risk premium in the PDE indicates either that the local expectations hypothesis holds (see Cox, Ingersoll, and Ross 1981) or that any such premium has been absorbed into the term structure parameters y and 8 (see Cox, Inger- soll, and Ross 1979).

The value of a contingent asset at its expiration is known from the
contractual specification, given the prevailing economic environment.
It is rather the value of the asset in the present that must be deter-
mined. This can be achieved by starting from the termination of the
contract and solving the PDE (4) back over time, to the contract's
origination. It is this specific backward procedure that we employ in
our numerical analysis (see the Appendix). To see that floating-rate
securities pose path-dependency problems for such backward ap-
proaches but that this problem may in fact be circumvented, we now
specify the details in the case of an adjustable mortgage contract with
default**.5 **

B. The Contract

We may assume without loss of generality that the contract rate for an ARM is adjusted at yearly intervals. To describc this procedure, we introduce the following notation:

n = the life of the mortgage in years;

~ ( i ,j ) = the j t h monthly payment date after the ith yearly
adjustment date, where 0 * 5 *i

*n, 0*

**5***j*

**5***12 (note that*

**5**~ ( i , 12) = ~ ( i

### +

1, 0));b(i) = the mortgage equivalent rate for a 1-year, default-free pure discount bond set on the ith adjustment date ~ ( i , 0); m = the margin;

y = the yearly cap and floor; c = the life of loan cap; and

a(i) = the contract rate set on the ith adjustment date.

In each year i, at adjustment date ~ ( i , O), a new contract rate is calculated according to the rule

a(i) = max{min[b(i)

### +

m, a(i - 1)### +

y, a(0)### +

c], a(i -1) -y).*(5)*

This indicates that the new contract rate a(i) consists of the yearly mortgage equivalent rate b(i) plus the margin m, insofar as this does not exceed the initial contract rate a(0) by more than the life-of-loan 5. If instead of using a backward procedure, one used a forward procedure such as a Monte Carlo method, there would be no path-dependency problem. This would work well for valuing promised payments when no termination is allowed, but forward meth- ods cannot handle endogenous termination. Since the point of the present work is to include such endogenous termination in particular default, backward methods were nec- essarily employed.

cap c, or deviate from that previous rate by more than the yearly cap y. The only departure from this occurs with teasers, which are applied in the first year. At the origin of the contract, we therefore have

where a is the teaser rate.

Given the contract rate a(i), the monthly payments M(i) over the coming year are calculated as if the mortgage is going to be a fixed-rate contract, one that over the remaining life of the loan completely amor- tizes the current unpaid balance U(i, 0) at the now prevailing contract rate a(i). This gives

where the unpaid balance after the payment at date ~ ( i , j ) is then

The value V of the mortgage itself will be broken into components
* A , D, and C, reflecting the amortizing payment, the default option, *
and the prepayment call option, respectively. Except as noted other-
wise, the mortgages we examine contain all three of these features.
That is, we have

A(r, t) = the value at time t of the amortizing loan payments that remain,

C(H, r, t) = the value at time t of the call option to prepay the loan, and

D(H, r, t) = the value at time t of the mortgage's default option. We occasionally speak of J = C

### +

D as the joint option to terminate the loan, and so we have the value of the mortgage being V = A -J= A - (C

### +

D). At payment dates only, a distinction must be made between the value of an asset, both immediately before and after the payment is made. This distinction will be indicated by the notations X-(H, r, t) and X + ( H , r, t), re~pectively.~If prepayment occurs, the borrower must prepay the current unpaid principal plus any accrued interest. This amount, the face value, is denoted

F(t) = (1

### +

a(i)[t -~ ( i , j)])U(i, j ) for ~ ( i , t j )*~ ( i ,j*

**5**5### +

1). (9)6. Parameters other than the state and time variables always appear after a semicolon. When there is little risk of confusion, variables not under discussion will be deleted from the notation.

One other asset, which is not part of the mortgage but depends on it, is insurance I . We denote this as

I(H, r, t) = the value at time t of insurance on the mortgage, and so the position of the lender on an insured mortgage may be written as L = V

### +

I . Insurance is a passive asset, in the sense that its payoffs occur as the result of the borrower acting to minimize the cost V of the mortgage without regard to the presence of insurance.C. Floating-Rate Payments

The promised payments A(r, t) do not depend on the house price

H and so do not reflect the full complexity involved in solving the path-dependency problem. Nonetheless, the main elements of the problem are already present. At termination of the loan, a final pay- ment M is made, so we have

The path-dependency problem occurs because the current contract rate a(n - 1) is not considered until the year's beginning is reached, and even then neither this contract rate nor the beginning balance U(n - 1, 0) are known. Indeed, they depend on yet earlier contract rates for which the caps may have bound.

The problem with the unpaid balance may be easily disposed of, though, by observing that the value of promised payments is homoge- nous of degree one in U, and so we may normalize the current unpaid balance to unity. When later we desire to change this unpaid principal by some proportion, we need merely change the value of A in that same proportion. Notationally, it is convenient to delete U whenever its value is taken to be unity.

On the other hand, the unknown contract rate a(n - 1) is treated by introducing it as an auxiliary state variable, so that effectively all its possible values are considered. To see that this device succeeds, we follow through the valuation procedure.

The PDE (4) may be solved backward using terminal condition (lo), until month's beginning, when a payment is due. In general, if we are at such a payment date ~ ( i , j ) , j # 0 or 12, then we have solved for At[r, ~ ( i , 0); a(i)] and we may then write the terminal condition to begin the succeeding month as

A-[r, t; a(i)l = A t [r, t; a(i)]

### +

M[i; a(i)](11) for t = ~ ( i ,j ) , j # 0 or 12.

This merely says that the value of the promised payments changes at a due date by the amount of the payment then being made.

At year's beginning, time ~ ( i , 0) = ~ ( i-1, 12), one wants to treat the succeeding yearly valuation procedure in the same manner as pre- viously. Thus, the new auxiliary variable a(i - 1) must be introduced and the subsequent unpaid balance U(i - 1, 0) must be set to unity. To introduce a(i - I), one must be able to drop a(i) as an auxiliary variable. This can in fact be done, since the current interest rate r [ ~ ( i , O)] and any assumed value a(i - 1) now determine the old contract rate a(i) via its definition (5). (Recall that since all values of a(i) have been considered, the desired one is available.) Finally, the value of A-[r, ~ ( i , 0); a(i)] is adjusted to correspond to the value at which the unpaid balance U(i, 0) has been reset. This latter value is now dictated through (8) by the assumed value of the new contract rate a(i - 1) and the normalization of the new unpaid balance U(i - 1, 0) to unity.

Thus, at adjustment dates, we have the terminal condition A-[r, t; a(i - I)] = A t [ r , t;a(i)]U(i, 0 ) + M[i - 1;a(i - I)]

(12) fort = ~ ( i ,0) = ~ ( i- 1, 12),

where a(i) is determined by r and a(i- 1) together, while both U(i, 0)
and M(i - 1) are determined by a ( i - 1) and U(i - 1, 0) = 1.
With the specification of natural boundary conditions, the valuation
* procedure for A is closed and one may obtain A(r(O), O), the value at *
origination t = 0 of the promised payments per dollar loan.

D. Solution of the Path-dependency Problem with Default

The valuations of prepayment C and default D are complicated by the influence of the house price H . While the house price has no direct effect on the payoff to the prepayment option, it does have a direct effect on the default option. Since exercise of the latter option renders the former one valueless, this means that the house price does indi- rectly affect the prepayment option. Clearly, these two options cannot be considered separately. We must instead consider the entire mort- gage V = A

### -

J , where J = C### + D

is the joint option to terminate the mortgage. From the actions of the borrower to minimize V, the conditions on default D and hence prepayment C may then be inferred.The value of the mortgage at termination is

V-[H, r, ~ ( n

### -

1,12); a(n - I)] = min{M[n - 1; a(n - I)], H ) (13) since the lender winds up with either the final payment M or a defaulted house H. By the same reasoning, the general terminal condition for payment dates other than adjustment dates isV-[H, r, t; a(i - I)] = min{Vt(H, r, t)

### +

M[i; a(i - I)], H)(14) f o r t = ~ ( i ,j ) , j # 0 or 12.

The only complication arises at an adjustment date, when the unpaid
principal U(i, 0) for the just completed year is to be recalibrated. The
term U(i, 0) must comply with the requirement that not it but rather
the previous year's unpaid principal U(i - 1, 0) is now to be set to
* unity. The problem is that, unlike payments A , neither prepayment C *
nor default D will be homogenous in U, since they both depend on the
house price H . That is, doubling the unpaid principal doubles the value
of payments, but it does not of itself double the value of prepayment

**or default on a house of a given value. However, while neither C nor**D are homogenous in * U *itself, each is homogenous in both H and U
together. That is, the value of default becomes twice as great when
the loan becomes twice the amount and the house becomes twice as
valuable. In essence, you then just have two copies of the original
house and loan. Since you make the identical decision in each case,
the result is simply that the stakes become twice as high.

In view of this discussion, the correct transition at an adjustment date is

*V - [H, r, t*;a(i - I)] = min

### {

**V+ [u:o),**-r,

*t ;*a(i)

## I

~ ( i ,0) ,151### +

M i - 1 a### -

1 , H## I

fort = ~ ( i ,0) = ~ ( i- 1, 12), where a(i) is determined by a(i - 1) and r, while both U(i, 0) and M(i - 1) are determined by a(i - 1) and U(i - 1, 0) = 1.It is important to observe that the technique used here to incorporate
house price into the valuation technique is fundamentally different
from the basic technique that suffices to value the promised payments
* A . That basic technique, which extends to include prepayment on a *
default-free mortgage, in no way depends on the precise form of the
interest rate process. However, the method used here to incorporate
house prices works because the lognormal form assumed for the under-
lying asset H is a proportionate process, with the property that the
rate of appreciation is independent of the size of the house. While
specific, processes of this type are the ones generally assumed in the
finance literature when an underlying asset is to be modelede7

Once boundary conditions are introduced, the solution technique for
V[H(O), r(O), 0] is closed. Note that in addition to the natural boundary
conditions applying when r or H take on extreme values, there is also
a free boundary formed by the decision to prepay. This boundary can
be described by standard "high contact" conditions, as in Merton
7. Since **does not actually appear in the valuation eq. (2), its value need not be **
independent of H or r ; hence, the class of permitted stochastic processes is actually
much wider than the lognormal ones.

(1973). (See the Appendix for the procedure employed here.) The rea- son that there is no corresponding free boundary for default is that financially rational default will always be put off until a payment is due. Thus, the conditions for default are entirely absorbed into the terminal conditions at each month's end.

The value of default is characterized by the general terminal condi- tion

D-(~(i,j)) =

D'[T(~, j)] if V- [ ~ ( i , j)] = V' [ ~ ( i , j)]

### +

M(i) (no default) A - [ ~ ( i , j ) ]-H i f V - [ ~ ( i , j ) ]= H (default)If default occurs, the borrower essentially gives up the house H for the promised payments A-

### .

On the other hand, if the payment at time ~ ( i ,j ) is made, and so default does not occur, then the value D of default becomes merely its value in the future. Since the technique here for adjusting unpaid balances and carrying the auxiliary variable a(i) is the same as that elaborated in the discussion of the full mortgage V, we avoid further exposition of this procedure. Note that with the specification of the previous components, one can now infer the value of prepayment as*= V - A*

**C**### -

D.The only remaining asset to be considered is insurance. Here the general terminal condition is

I-[T(i, j)] =

* 1' * [ ~ ( i ,j)] if V- [ ~ ( i , j)] = V+ [ ~ ( i , j)]

### +

*M ( i )*(no default) max (0, min { F - [ ~ ( i , j)] -H,

**C$**FP[7(i,j)]) (default)

if V-[~(i, j)] = H

If default occurs, the insurer makes good the lender's shortfall up to
some fraction of **C$ **of the face value. Note that the lender's shortfall
*is defined in terms of the unpaid balance plus interest F , whereas the *
borrower's default decision is driven by the promised payments A. If
default does not occur at that payment date, then the value of insur-
ance becomes merely its value against future default.

E. Mortgages a t Origination

A final condition that may be imposed on a newly issued contract is that the value of the contract to the lender should balance with the actual amount lent. Such arbitrage reasoning leads to

where *6 *is the amount of points deducted from the loan. (Recall that
we have expressed assets to par, so that the value of the loan is unity .)

Here, for definiteness, we have assumed that the insurance is pur- chased up front by the borrower.' Note that while insurance may not be strictly part of the mortgage, it does affect the margin m selected when the contract rate is drawn up. See the Appendix for further description of the numerical methods that yield the results presented below from the model just discussed.

**111. ** **Numerical Results **

* A. Changes in the Economic Environment *
1. Description of Results

Table 1 considers two adjustable-rate mortgages, both with 2% yearly and 5% lifetime caps. The first contract involves an 80% loan-to-value ratio, while the second involves a 90% one. Included throughout the table is the case where the mortgage is fixed rather than adjustable. All values of the mortgages are stated to par, that is, as a percent of the loan amount.

All parameters describing the economic environment and the con-
tract are recorded in the table itself. The parameter values used were
chosen to be typical of those reported in the literaturee9 The term
structure is upward sloping, since the initial interest rate r(0) was taken
**to be 8%, while the steady-state rate 0 was set at 10%. A base volatility **
of 10% was selected for the term structure volatility IT,, while the

housing price volatility u, was put at 15%. At these volatilities, a
margin m, and hence an initial contract rate a(O), were calculated for
each of the two base cases. These margins satisfy the arbitrage condi-
tion (18) that characterizes any newly originated loans. (Thus, given
the assumed **11/2 **points on the loan, the values of these mortgages

balance at 98.5% of par value.) Part A of the table then provides varia- tions in volatilities around the base situations, with the margin being held fixed in each case. Part B of the table provides similar variations in the initial interest rate r(0) and house value H(0). All these variations should be interpreted as occurring just after origination of the contract since they are not allowed to affect the margin or the initial contract rate.

8. Figure 2 exhibits the balancing contract rate without insurance for that particular economic environment. Note that this contract rate is also the appropriate one in the case where insurance is thought of as being paid as part of the monthly mortgage pay- ment since the contract rate (and points) then cover all benefits the borrower receives and it is irrelevant whether or not the lender purchases insurance.

9. Titman and Torous (1989) examine what the reasonable parameter values should be for the interest rate and the house processes, in the context of commercial real estate. Also see the estimates in Case and Shiller (1989).

**A. Partials of Volatilities u,,, u, **

- - - **d **

*o r *

- -

-Inatlt~al Value of Payments Default Insurance Prepayment Mortgage

Contract * A * D I

**C***Rate - -- - -- --*

**L****-arj**10% 5% 10% 15% 5% 10% 15% 5% 10% 15% 5% 10% 15% 5% 10% - - 15% -80% loan- to-value ratatlo 10% ARM 103 69 07 04 5 25 98 42 FRM 106 92 07 03 8 43 98 45 15% ARM 8 34 106 87 103 69 100 60 45 44 47 28 32 37 6 9 1 5 0 8 4 32 99 80 98 50 96 19 FRM 1017 10624 10692 10778 07 37 46 04 19 19 6 18 8 25 11 16 100 04 98 50 96 35 m 159 39 20% ARM 103 69 90 72 4 95 98 56 FRM 106 92 8 1 46 8 00 98 58 90% loan- to-value ratlo 10% ARM 102 71 95 62 4 05 98 33 FRM 105 90 1 04 45 7 04 98 26 15% ARM 8 1 7 10586 10271 9962 2 4 6 2 1 7 2 0 2 1 6 3 1 6 9 1 7 2 5 1 6 3 7 2 3 22 99 88 98 50 96 10 FRM 10 00 105 22 105 90 I06 75 2 11 2 46 2 79 1 3 3 1 30 1 17 3 98 6 23 8 83 100 46 98 50 96 30 m 142 64 20% ARM 102 71 3 14 2 61 3 64 98 54

*FRM 105 90 3 64 2 13 5 70 98 70*

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Inspection of the table confirms one's expectations that the larger default values are associated with higher loan-to-value ratios, higher house price volatilities, and lower house prices. What may not be so obvious is just how much difference it makes to the value of default whether the mortgage features an 80% or a 90% loan-to-value ratio. In the latter case, default approaches prepayment in value. Indeed, as results reported later in table 2 show, if teasers are dropped so that prepayment loses some of its value, then a mortgage otherwise identi- cal to the 90% loan-to-value base case has a value of default exceeding that of prepayment. Notice also from table 1 that the value of default for an ARM is typically comparable to that of an FRM, indicating in a very rough way that the default experiences of the two type mortgages should be similar. Since it is the term structure that is the primary source of whatever differences do arise between the two type mort- gages, we now explore its effects in finer detail.

2. Interest Rates and Their Volatility

*The effect o n payments. Examining the effects of the interest rate *
volatility a,, the most striking feature is the opposite effect on prom-
ised payments A between the case of an FRM and that of an ARM.
In the case of an FRM, Jensen's inequality assures that the fixed pay-
ments increase in market value with a rise in volatility a,. However,
in the case of an ARM, the payments are not fixed, and so the same
reasoning does not apply.

One may think of upward caps as an option the borrower may exer-
cise against an ideal ARM, whose value would otherwise be indepen-
dent of the term structure. As the volatility **a, **increases, so does the
value of this option to the borrower, thus lowering the cost of promised
payments A. There is, of course, an opposite effect due to floors, but
the situation is not symmetric. Whereas there are downward yearly,
floors counteracting the upward caps, there is not a downward lifetime
floor. Of greater consequence, however, is the asymmetric effect of
the yearly caps attributable to the chosen upward term structure. Since
interest rates are expected to trend upward, the caps are more likely
to bind than the floors.

*The effect o n prepayment. Teasers exercise an enormous influence *
on the value of prepayment, as seen later in table 2. Across the board,
prepayment increases significantly with introduction of teasers, with
the relative effects being larger for the less tightly capped ARMS. The
modest teaser rate of 1'12% we employ in our base run nearly triples
the value of prepayment over what it would be in the absence of teasers
(from 1.26 to 3.72% par value for the 90% loan-to-value contract). To
compensate for the initial teaser rate, the lender requires a higher
margin than otherwise (via eq. [18]) and, hence, a higher contract
rate once the effects of the teaser wear off. This in turn encourages

considerable prepayment after the first year. The net effect of this is to lower the effective life of the loan, which then serves to diminish the influence of the term structure volatility a,.

As table 1 reveals, the effects of interest rate volatility on pre-
payment are quite complex. At low volatilities, the high margin that
is appropriate with teasers and higher volatilities instead becomes ex-
cessive, and so prepayment becomes inevitable, given the likely course
of interest rates. Thus, at first, increases in term structure volatility,
and the alternative interest rates scenarios which thereby gain promi-
nence, can as a whole only lower the incidence of prepayment. Eventu-
ally, though, as one would expect, the incidence of prepayment begins
to rise with increases in volatility a,. Note, however, because the im-
mediate payoff to prepayment, A(r, t) - F(t), is being diminished by
the previously described effect of volatility a, on payments A, the
value of prepayment actually starts to rise well after the incidence of
prepayment begins turning up against increases in volatility **a,. **

The effect on default and insurance. For a satisfactory description of the term structure's impact on default itself, one needs to recognize the presence of prepayment. In the absence of prepayment, default, whose payoff is A(r, t) - H(t), would be strictly declining in value

against the interest rate. However, since actual prepayment makes default worthless, the presence of this alternative form of termina- tion causes default to lose much of its value at low interest rates (see fig. 1).

The overall behavior of insurance is similar to that of default. The only major difference arises because the payoff to insurance, F(t)

-H, is unaffected by r, unlike the payoff to default, A(r, t)

### -

H . While the present value of insurance is being discounted directly by r, it is not being discounted indirectly through A, as is default. Because of*this, changes in insurance's value I actually give a better indication of*changes in the occurrences of default than do changes in the value D

of default itself.

As discussed, beginning from a low volatility, increases in a, initially decrease the incidence of prepayment. This then serves to increase the incidence of the alternative to prepayment, default. This can be observed in the behavior of the value of insurance, the more accurate indicator of the incidence of default. However, as table 1 shows, the value of default itself does not turn up against a, until much higher volatilities because of the depressing effect volatility a, has on the immediate payoff to default, A(r) - H. At high enough volatilities, though (from 10% to 15% in the case of no teasers and a 90% loan-to-value ratio), insurance begins to turn down against a,, as the resurging incidence of prepayment starts to depress the incidence of default.

The effect on the entire contract. Notice that the interest rate vola- tility a, has the same effect on mortgage value V whether one considers

**Value to par **
**as a percent **

**default D(r,H(O)) **

### 1

**and ** **for a 90% loan to "due ratio **
**,(~nsurance ** **I(r,H (0)) **

**default D(r,H(O)) **

I I I I I r

FIG.1.-Default values at origination against the spot interest rate. All pa- rameter values are the same as those listed in table l .

an ARM or an FRM. Therefore, this volatility also has the same effect
on the initial contract rates required to balance these two types of
contracts. This is despite the fact that the effects of interest rate volatil-
* ity on payments A are opposite between an ARM and an FRM. Given *
the termination pattern, the effects of volatility

**a,**on payments

**A **

are
largely canceled in the value of the mortgage V = *- C - D by the'*

**A*** additional presence of A in the payoffs to prepayment *C and default

D. Thus, the overall effect of volatility a, on the contract is dictated
by the change in the termination pattern. Since a rise in volatility **a, **

tends to increase the frequency of termination, the value of the mort- gage falls, whichever contract is considered. This is, of course, the phenomenon of "negative convexity" due to termination, an issue of much concern in fixed-income analysis.

**3. ** Housing Prices and Their Volatility

The effect on an ARM of housing prices and their volatility is straight- forward. In all cases, the effects are the same as for an FRM. This should not be surprising, since the distinctions that separate an adjust- able from a fixed-rate mortgage do not deal as much with house prices

directly affecting the prospects for default as with interest rates indi- rectly affecting these prospects.1°

We do not explicitly discuss the effect of the loan-to-value ratio on
insurance, default, or prepayment since they are all as one would ex-
* pect. It might not seem, though, that payments A should fall with *
increases in the loan-to-value ratio, when these are being expressed to
par. Notice, however, that the contract rate has fallen. The reason for
the latter fall is that this contract is being insured by an up-front insur-
ance payment. An increased loan-to-value ratio increases default, of
course, but it also decreases prepayment. Since default is insured, it
is mainly the effect on prepayment that needs to be covered by the
contract rate, and so the rate required falls."

*B. * *Comparison of Different Contracts *

1. Distinguishing Characteristics of ARMS

The motivation for an ARM is to adjust payments to the prevailing
market and so to insulate the contract's value from interest rate varia-
tions. This diminishes the role of prepayment. As a result, the initial
* contract rate and hence the value of payments A for an adjustable-rate *
mortgage can be significantly lower than for a fixed-rate mortgage.12 If
in a flat-yield-curve environment a lender were to charge a fixed con-
tract rate near the market interest rate, he would lose money whenever
interest rates rose significantly. The lender would not, however, gain
correspondingly when interest rates fell, since borrowers would avoid
the losses to themselves by prepaying. The contract rate for an
adjustable-rate mortgage, on the other hand, need not be substantially
above the market rate. The lessened chance of prepayment for an
ARM will exactly compensate the lender for accepting the lower con-
tract rate that in part results in this lessened prepayment.

The above reasoning is, of course, attenuated by the presence of
* 10. Clearly, payments A are affected by neither house price nor its volatility. How- *
ever, a decrease in house price H or an increase in its volatility

*can substantially raise the value of default, particularly at high loan-to-value ratios. Insurance responds in a similar manner. Finally, since prepayment and default serve as substitutes, an increase in the prospect of default due to a fall in house price H or a rise in volatility*

**u~**(Iwill then cause prepayment C to fall in value. ,

11. We also made a series of runs varying p, the coefficient correlating term structure
and house price volatility. With a balancing contract, and the situation the same as the
90% loan-to-value base case, save for a correlation coefficient * p *of -lo%, the results
are a(0) = 8.17% and m = 142.6 basis points, with A = 102.71%, D = 2.17%, I =
1.59%, and C = 1.87% of par value, respectively, whereas for a correlation coefficient

*of lo%, the results are a(0) = 8.16% and m = 141.3 basis points, with A = 102.62%. D = 1.62%, I = 1.43% and C = 3.93% of par value, respectively.*

**p**12. Without teasers, the margin for an ARM can become negative, as shown in table 2, for the case of a 90% loan-to-value contract with no caps. The margin m is then seen to be -16.8 basis points. Note that the assumed 1'12 points on the loan also make the margin smaller than it would otherwise be.

teasers. To be compensated for such teasers, the lender will require a higher contract rate than otherwise after the first year. Caps also tend to undo the distinctive features of an ARM and restore properties of a fixed-rate mortgage. One would nevertheless expect default D for a capped ARM to be relatively immune from interest rate fluctuations, compared to the case of an FRM. This is borne out by the results in the default column of table 1, particularly for changes in the volatility

**a,.**In fact, this stability causes default at the lower volatilities **a,**to be
of noticeably greater value for an ARM than for an FRM.

2. Insurance

Insurance for an ARM is typically much closer in value to default than
is the case with an FRM. The reason why default D for an FRM is
*more valuable than insurance I is not because of the limit to coverage *

### +.

With continuous price movements, the home owner typically de- faults well before this consideration comes into play. The difference in value between default and insurance occurs when drops in the inter-*price then drops below the face value of the loan, default may be of considerable value. However, the resulting insurance payouts need not be large, since insurance coverage is based on the face value F of*

**est rate drive up the cost of the promised payments A . If the house****the loan, not the present value of promised payments A.13 With an**adjustable-rate mortgage, on the other hand, the effects of interest rates are reduced, so default occurs almost entirely due to housing

*F deviate compar- atively little for an ARM approaching default, the values of default D*

**price movements. Since payments A and face value**and insurance I also deviate by only a small amount.

**3. ** FRMs versus ARMs

To gain some sense of how FRMs and ARMs differ in the large, in,
figure 2 we have graphed both sorts of contracts against the initial
contract rate *a(0).*These particular results have no teasers and use an
interest rate volatility of **5%.14 *** Since, the initial rate r ( O ) , like other *
parameters, is being held fixed here, variations in the initial contract
rate of an ARM arise solely from equivalent changes in the margin

*m.*

13. Of course, downward variations in the value of payments would create instances
for an FRM where default would not occur or would have less payoff than an ARM.
These instances, however, are small in value compared to the value contributed to FRM
* default by upward movements in A . The reason one does not exercise an option as soon *
as it is in the money is the loss of the future value of exercise. Without prepayment,
however, much of the future value of termination is absent. With no major upward

*it becomes of immediate value. The payoffs will consequently be small in value com- pared to the large payoffs possible with an FRM.*

**movements in A to give future default value, default on an ARM will occur soon after**14. The only other deviation from the base runs was that the service flow was taken to be 5%.

**Value to par **
**as a percent **

**1°1 **

### 1

**a1 **: **ARM with insurance, default, and prepayment **
**a2:ARM with prepayment **

**a3: ARM with default and prepayment **

**f l **: **FRM with insurance, default, and prepayment **
**f2: FRM with prepayment **

**f3: FRM with default and prepayment **

**92 ** **Contract **

**rate **

FIG.2.-Fixed-rate mortgage and adjustable-rate mortgage values at origina- tion against the initial contract rate. Note that, for an ARM, a(0) = b(0)

### +

*m.*All variations in the ARM'S contract rate here are with respect to the margin

*(The value of b(0) is*

**m.****8.246%.) All parameter values are the same**

as in table 1 for a loan-to-value ratio = 90%, except **a,.**= *5%, ***s **= * 5%, *and
there are no teasers (a = 0).

For both fixed and adjustable mortgages, we have graphed contracts with and without default, and in the former case, with and without insurance. Since insurance I is a passive claim, for any contract rate

*a(0) *the value of I there can be seen by taking the difference between'

the vertical heights of the graphs with and without insurance. No simi- lar procedure gives the value of default, however, since 'on adding default the value of prepayment changes.

In the absence of prepayment, the value of default would be greater
with an FRM than with the corresponding ARM, since large upward
* variations in the value of payments A with an FRM would induce large *
default payoffs.15 However, in the presence of prepayment, default for
an ARM has more value than for an FRM, at least in the absence of
teasers. Even with the modest interest rate volatilities selected for the

*and the face value*

**graphs, the resulting variations between the payment values A ( r , t )***F ( t )*of an FRM mean that prepayment is much greater for an FRM than an ARM. This likelihood of prepayment 15.

*default nor insurance are likely to be of value, so there is little difference between them to offset the low interest rate situations.*

**On the other hand, when A goes down due to high interest rates, then neither**crowds out the possibility of default for an FRM, so now it is the ARM that has the greater value of default.

With the FRM, prepayment is a much more immediate prospect than for an ARM that lacks teasers; any substantial rise in the contract rate induces such prepayment. In figure 2, this is indicated by the relatively small distance between the contract rate where the graph of the full FRM contract balances, that is, where it first crosses the ordi- nate axis, and the contract rate where the FRM first prepays, that is, where the graph merges with the ordinate axis. On the other hand, for the ARM, prepayment is a more remote possibility so it takes a substantially greater increase in the margin before default loses all its value and prepayment is induced.

4. Caps

Table 2 presents trade-offs between the tightness of caps and the mar- gins required to balance the contract. This is done for contracts both with and without teasers. The tighter the caps, the more in general the contract resembles a fixed-rate mortgage. On the other hand, the wider the caps, the more the contract resembles an ideal ARM, fully insu- lated in real value from interest rate variations. Naturally, the FRM has the highest initial contract rate, whereas the uncapped ARM has the lowest.

Notice that with 1% yearly caps there is very little difference be- tween the ARM that has a lifetime cap and the one without such a lifetime cap. Except for an offsetting difference in the value of insur- ance, the margin for an ARM with an upward cap must never be above the margin of the corresponding contract without such a cap. The fact that the difference is so slight in this particular instance indicates that there is little additional benefit to the borrower from being guaranteed a lifetime cap, given that the yearly cap is already so tight. This differ- ence between a contract with a lifetime cap and one without such a cap becomes much more significant when the yearly caps are loosened to 2%.

One other interesting feature of the results in table 2 is that default and insurance basically follow opposite patterns as caps tighten. We have already commented that insurance, rather than the value of de- fault, is the surer indication of the likelihood of default. As caps are tightened, the possibility of prepayment comes to the fore, making the likelihood of default more remote, hence insurance less valuable. Nonetheless, default tends to increase in value with tighter caps, fol- lowing exactly the pattern of payments A. On those occasions where it continues to occur, the payoff to default becomes greater, since tighter caps increase the uppermost values that payments A may as- sume. Note that prepayment, whose payoff is also in terms of A, re- sponds in the same manner as does default to changes in caps.

**Life of ** **Initial **

**Loan ** **Contract ** **Margin ** **Value of **

**Annual Cap ** **Cap ** **Rate ** **in Basis Points ** **Payments ** **Default ** **Insurance ** **Prepayment **

**Y ** **c ** **a ****m ****A ****D ****I ****C **

**Fixed-rate **
**mortgage **

NOTE.-Results without parentheses are for a I%% teaser; results with parentheses are without teasers.

* All results are to par value for a 15-year loan: a rising-term structure (8% spot interest rate r(0); 10% steady-state rate **(0)); **25% adjustment parameter **y;**0% correlation
coefficient **p; **8%% service flow **s; **10% and 15% interest rate and house volatilities **u, **and **u,, **respectively; 1'12% points 6; 25% insurance coverage **4;**and a 90% loan-to-value
ratio.

**IV, Conclusion **

By including default, this article extends previous work on pricing floating-rate instruments. The procedure for eliminating path depen- dency due to floating-rate payments is to introduce an auxiliary vari- able to carry relevant past information. It is shown here that this proce- dure continues to apply when the asset price explaining default joins the interest rate as a state variable. The solution procedure is used to provide a detailed analysis of adjustable-rate mortgages. A comparison is made between these and fixed-rate mortgages with default. It is noted that the solution procedure is also applicable to index-amortizing notes and other floating-rate instruments not related to mortgages.

Since the point of an adjustable-rate mortgage is to keep the present value of payments near the face value of the loan, prepayment loses much of its value going from an FRM to an ARM that has no teasers. In such cases, default is affected in the same manner. The resulting drop in terminations allows lower contract rates than with FRMs, which then further lowers the value of termination. On the other hand, teasers on an ARM restore much. of the value of prepayment. This then requires higher margins, hence, contract rates similar to those of an FRM, once the temporary effect of the teasers wears off. This, in turn, results in default values similar to those of an FRM. The dramatic consequences of teaser rates on prepayment, and thus on margins and default, seem to have been largely undiscussed in the literature.

The most unusual feature found in our study is the opposite effect of interest rate volatility on the value of payments, when comparing an FRM to an ARM. This in turn affects the behavior of prepayment and default, whose payoffs are expressed in terms of the value of payments. It is because of these differing volatility effects that the value of default for an ARM can noticeably exceed that for an FRM when interest rates have little volatility. Much of this differing effect of volatilities disappears, however, when looking at the aggregate value of a mortgage. This follows since the effects of interest rate volatility on the payoffs to the various components of the mortgage tend to cancel one another. The remaining effect is primarily one of "negative convexity," due to the increased terminations at higher in- terest rate volatilities.

The effects of tightening both yearly and lifetime caps were exam- ined across adjustable-rate mortgages. As would be expected, a pro- gressive tightening of caps makes an adjustable-rate mortgage more nearly resemble a fixed-rate one. Insurance, whose payoff does not reflect the value of payments, fell in value as caps were tightened due to substitution into prepayment. On the other hand, the values of de- fault and prepayment moved in concert with the value of payments, since this value enters into their payoff. Finally, it was found that in

the presence of tight yearly caps, lifetime caps have little additional impact.

Default and prepayment are, in principle, decisions endogenously
determined as part of the valuation of a mortgage, since the value of
the mortgage depends on their likely occurrence, and their occurrence
depends on the cost of the mortgage. It has been frequently pointed
out, though, that much prepayment is induced by considerations be-
yond the actual cost of the mortgage itself, and indeed, much effort
has been expended in modeling such "suboptimal" prepayment. It
may turn out empirically that one also wants to consider "suboptimal"
default, though it is anticipated that default in response to a desire to
move is a much more remote possibility than is prepayment. More
likely is the opposite occurrence, where "transaction costs" are asso-
ciated with default, and so default does not occur when strict minimiza-
tion of the cost of the mortgage would indicate that it should. The
point, however, is that "suboptimal" default and particularly transac-
tion costs represent modifications to be built into the basic valuation
model of an adjustable-rate mortgage with default, and it is only with
the methods described here that one has such a valuation model.16
**Appendix **

**Numerical Methods **

**The PDE equation ( 4 ) was solved using a standard two-state explicit finite *** difference method, which was applied to (r, H ) space after it had been com- *
pacted into a unit square under a nonlinear transformation and then uniformly
discretized. The specific natures of the lognormal house process and the CIR
mean-reverting interest rate process provided natural boundary conditions at
extreme house values and interest rates. The explicit need for free boundary
conditions delineating where prepayment first occurs was avoided by

**ex-tending the domain of the PDE equation ( 4 ) over the prepayment region to***originally of the form,*

**include the entire unit square [(r, H ) space]. Thus, the valuation equation,**V = Fotherwise,

where * 2 *is the second-order linear differential operator in equation

**(4),**became

once it was observed that

16. For a discussion of transaction costs and default with FRMs, see Kau, Keenan, and Kim (1994, in press).

### - -

- when V = F, and by this means, the PDE was defined for all*r*and

*H ,*in the manner of Berger, Ciment, and Rogers

*(1975).*A spatial grid of size

*.02*was used to discretize the

*(r, H )*unit square, and thus

*66*time steps a month were used to assure numerical stability. A grid of

*24*points was used to span the range of possible contract rate. Simple interpolation is used to calculate values lying between grid points. Once the above procedure was completed backward over time through each subsequent month for a given margin, a secant method was then used to iteratively locate those margins which balanced the contract, according to equation

*(18).*Accuracy is estimated to be within tens of dollars on a

*$100,000*principal, with most inaccuracy due to the interpolation on the auxiliary contract rate variable, and to a lesser extent, due to allowed error in balancing

*(18).*Runs were done on an IBM RISC

*6000*minicomputer.

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