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How much can the option value to refinance explain the premium
associated with an assumable loan?
Jyh-Bang Jou*
Department of Economics and Finance, Massey University Albany Campus,
New Zealand
Graduate Institute of National Development, National Taiwan University, Taiwan
Tan (Charlene) Lee*
Department of Finance, Auckland University of Technology, New Zealand
*
Correspondence to: Jyh-Bang Jou, Department of Economics and Finance, Massey University Albany
Campus, New Zealand
Tel: 64-9-4140800 ext. 9429
Fax: 64-9-4418156
E-mail: J.B.Jou@massey.ac.nz
1
How much can the option value to refinance explain the premium
associated with an assumable loan?
Abstract
An assumable loan that carries a lower-than-market rate of interest has value because
of its advantageous rate. An investor who employs a traditional loan can, however,
avoid the downside risk of interest rates because the investor has the option to
refinance the loan when future market interest rates decrease. This option value can
explain why only one-third to two-thirds of the premium associated with assumption
financing is capitalized into the home selling prices. The premium associated with the
assumable loan will be eroded more by the option value to refinance when the term of
the loan lasts longer, when the interest rate becomes more volatile, and when the
spread between the assumable rate and the market rate shrinks.
Key Words: assumable loan, downside risk, interest rate, option value
2
How much can the option value to refinance explain the premium
associated with an assumable loan?
I. Introduction
In a period of high inflation, an assumable loan that carries a lower-than-market
rate of interest offers an option for a homebuyer, who also has the other option to
apply for a traditional loan that carries a market rate of interest. In a low inflation
period, a loan that carries a fixed market rate of interest may become an assumable
loan in the future once inflation floods. Since the assumable loan has value, the
buyer of a house would be willing to pay more than he/she would if there were no
assumable loans. Garcia (1972) argued that we should use the cash-equivalence
method to value the assumable loan.1 As such, the additional amount that a buyer
would be willing to pay should be equal to the premium associated with the
assumable loan; i.e., the difference between the present value of the loan discounted at
the assuming rate and that discounted at the market rate.2
Empirical studies that test Garcia’s hypothesis typically find, however, that the
value of the assumable loan only partly transfers to the home transaction price. For
example, Sirmans, Smith, and Sirmans (1983) concluded that only one-third of the
premium associated with assumption financing was capitalized, while Clauretie (1984)
found that 64% was capitalized into home selling prices.3 Clauretie and Sirmans
1
Some papers argue that the terms of financing would affect the transaction price of homes. For
example, both Lusht and Hanz (1994) and Asabere, Hoffman, and Mehdian (1992) found that
properties in all cash transactions sold for more than 10% discount relative to transactions involving
financing terms.
2
Guntermann (1978) argued that selling prices of homes financed with assumed loans should be lower
than selling prices of homes with conventional financing. As Sirmans, Smith, and Sirmans (1983)
suggest, this argument is wrong, as the mortgage, having once occurred, is always a liability rather than
as asset to the homebuyer.
3
Two exceptions are Rosen (1982), which found that capitalization rates of the cash equivalence
premium were greater than 100%, and Gaines, Ingram, and Gregory (1983), which found no premium
paid by the assumable loan.
3
(2006, chapter 7) summarized the literature and offered the following reasons to
explain a reduction in the value of assumable loans: (i) Buyers value only the payment
savings that would accrue during their expected value in the house (Sirmans et al.,
1983); (ii) the after-tax savings of an assumable loan is less than the before-tax
savings (Clauretie, 1982); (iii) buyers and sellers could not accurately quantify the
present value of good financing (Friedman, 1984); (iv) the cash-equivalence
technique overvalues the assumption financing, because it abstracts from the
loan-to-value ratio (Sunderman, Cannaday, and Colwell, 1990); (v) most properties
sold with assumption financing involve a second and junior loan (Dale-Johnson,
Findlay, Schwartz, and Kapplin, 1985); and (iv) future interest rates may decrease,
which increases the value of a traditional loan, thus reducing the value of an
assumable loan (Ferreira and Sirmans, 1987). Among those reasons, Clauretie and
Sirmans (2006) argued that the last one is the most convincing. This is because an
investor who employs a traditional loan can avoid the downside risk of interest rates
by refinancing the loan, while an investor who carries an assumable loan usually does
not have this option unless future interest rates drop below the assuming rate.
It is unfortunate that both Ferreira and Sirmans (1987) and Clauretie and Sirmans
(2006) only offer a numerical example to demonstrate that the traditional loan accrues
some value when the market interest rate goes down by 1%. The contribution of this
article is to employ an option pricing technique to exactly quantify the proportion of
the premium associated with an assumable loan, which is accounted for by the option
value to refinance embedded in the traditional loan. To this end, we will abstract
from the transaction costs associated with refinancing and will also assume that the
market interest rate evolves stochastically, as addressed in Cox, Ingersoll, and Ross
(1985). As such, we are able to investigate how the following factors affect the
option value to refinance: (i) The reversion speed of interest rate movements; (ii) the
4
steady state rate of interest; (iii) the volatility of the interest rate; (iv) the spread
between the assuming rate and the current market rate; and (v) the term of the loan.
The remaining sections are organized as follows. The next section first offers
an option pricing model to calculate the value of a traditional loan that is able to
refinance over time and then compares this value with the value of an assumable loan.
Section III employs plausible parameter values to investigate the comparative-statics
results mentioned above. The last section concludes and offers directions for future
research.
II. The Model
We assume that the market interest rate at instant t , r (t ) , follows Cox et al.
(1985):
dr (t ) = a (b − r (t ))dt + c r (t )dZ (t ) , a > 0 , b > 0 , c > 0 . (1)
The term Z (t ) represents a Wiener process, and a , b , and c are parameters.
The equation states that, on average, the interest rate converges toward value b .
The rate of the convergence is governed by the value of a and the volatility of the
interest rate is c r (t ) . Note that Equation (1) lacks an interest-rate risk premium,
because we assume that this premium has been absorbed by parameters a and b
(Cox, Ross and Rubenstein, 1979).
The movement in the interest rate r (t ) enables us to calculate the value of the
option to refinance embedded in a traditional loan. For ease of exposition we
assume that a buyer of a traditional loan can costlessly refinance the loan when the
interest rate decreases in the future. Incorporating the transaction costs associated
with refinancing will naturally erode the value of the traditional loan. Suppose that
V (r (t ), t ) denotes the value of the traditional loan with a payment P (r (t ), t ) at
5
instant t . This value function satisfies the partial differential equation:
c2 ∂ 2V (⋅) ∂V (⋅) ∂V (⋅)
r (t ) + a (b − r (t )) + − rV (⋅) + P (⋅) = 0. (2)
2 ∂r (t ) 2 ∂r (t ) ∂t
Suppose that t0 is the current date and T is the term of the loan. Two
constraints are associated with this loan.
Suppose that ε is an infinitesimal positive number, then:
(i) V (r (t0 + T + ε ), t0 + T + ε ) = 0, where t0 + T is the maturity date; and, (3)
(ii) assuming that τ = t + ε , where t0 + T > t > t0 , then
V (r (τ ),τ ) = V (r (t ),τ ) , if r (τ ) > r (t ) (4)
Condition (3) captures the requirement that the balance of the tradition loan is equal to
zero at the date immediately after the loan matures. Condition (4) captures the
option to refinance. That is, when the future interest rate goes down, then an
investor purchasing a traditional loan will refinance the loan through borrowing at a
lower rate. The investor will, however, still be charged at the contract rate if the
future interest rate goes up. We are, however, unable to derive a closed-form
solution for V (r (t ), t ) , because the option to refinance contains numerous compound
options, as addressed in Geske (1979).
Ferreira and Sirmans (1987) argued that a valuation formula capable of
measuring the assumable loan value has not been developed, mainly due to the
following three reasons: (i) The unknown time the potential buyer intends to stay in
house; (ii) the market interest rate is a random variable; and (iii) the assumable loan
contains compound option values. We can tackle the first problem by assuming
either that the potential buyer intends to stay in house in the same period as the term
of the loan, or that the buyer could capture the value of remaining payment savings
even if he/she stays shorter than the term of the loan. We have already tackled the
second problem by assuming that the interest rate follows the stochastic movement
6
shown in Equation (1).4 We can then use the backward pricing method to calculate
the option value to refinance. More precisely, we will tackle the third problem by
employing the modified explicit finite difference method developed by Hull and
White (1990) to approximate the value of V (r (t ), t ) .5 For ease of exposition, we
will normalize the monthly payment for both assumable and traditional loans to one
unit: i.e., P (r (t ), t ) = 1 at a certain day in each month when payment is due.
c 1
Let φ (t ) = r (t ) . Define ( ) 2 Δt / (Δφ ) 2 = , where Δt is a small change in
2 3
the time interval, and Δφ is a small change in φ (t ) . Hull and White (1990) then
show that:
φmin ≤ φ (t ) ≤ φmax , (5)
where (
φmin = − β + β 2 + 4α1α 2 / 2α 2 ) , ( )
φmax = β + β 2 + 4α1α 2 / 2α 2 ,
Δφ 4ab − c 2 a
β= , α1 = , and α 2 = .
2Δt 8 2
The values of φ (t ) considered on the grid for the explicit finite difference
method are φ0 , φ1 , ..., φ j , K , φn , where φ0 is the largest multiple of Δφ less
than φmin , φ j = φ0 + j Δφ , and n is the smallest integer, such that φn ≥ φmax . We
also choose Δφ , such that φ0 plus some multiple of Δφ equals the current value of
φ (t0 ) . Similarly, the values of t considered in the grid are t0 , t1 , ..., ti , K ,
t0 + T , where ti = t0 + iΔt .
The modified explicit finite difference method can be used to value V (φ (t ), t ) .
4
As Cox et al. (1985) indicated, the interest rate implied by this movement has the following
empirically relevant properties: (i) Negative interest rates are precluded; (ii) if the interest rate
reaches zero, it can subsequently become positive; (iii) the absolute variance of the interest rate
increases when the interest rate itself increases; and (iv) there is a steady state distribution for the
interest rate.
5
The modified explicit finite difference method is more suitable for our purpose than either the
implicit finite difference method, which must impose more boundary conditions (Brennan and
Schwartz, 1978), or the binomial method, which only allows the interest rate to either go up, or down
(see, e.g., Hilliard, Kau and Slawson, 1998).
7
Define Vi , j as the value of a tradition loan at the (i, j ) node. We also assume that
the loan matures at time t0 + (m − 1)Δt ; i.e., T = (m − 1)Δt , such that the loan is
worthless at date t0 + mΔt . Equivalently, this implies that boundary condition (3)
can be written as Vm, j = 0 for all j . That is, the traditional loan is worthless
immediately after the maturity date. We then use Figure 1 to explain how we grid
time period t and the state variable φ (t ) to find Vi , j .
The partial derivatives of V , with respect to φ at node (i − 1, j ) , are
approximated as follows:
∂V Vi , j +1 − Vi , j −1
= (6)
∂φ 2Δφ
∂ 2V Vi , j +1 + Vi , j −1 − 2Vi , j
= (7)
∂φ 2 (Δφ ) 2
and the time derivative is approximated as
∂V Vi , j − Vi , j −1
= (8)
∂t Δt
Since the short-term interest is φ (t ) 2 , in the absence of the option to refinance, we
can calculate the value of the loan as follows (see, Hull and White, 1990):
1 1 2 1
Vi , j = [ Vi +1, j −1 + Vi +1, j + Vi +1, j +1 ] (9a)
1 + φ 2 Δt
j 6 3 6
or
1 1 2 1
Vi , j = [ Vi +1, j −1 + Vi +1, j + Vi +1, j +1 + 1] (9b)
1 + φ 2 Δt 6
j 3 6
for j = 1, 2,K , n − 1 , while
1 1 2 1
Vi ,0 = [ Vi +1,0 + Vi +1,1 + Vi +1,2 ] (10a)
1 + φ0 Δt
2 6 3 6
1 1 2 1
or = [ Vi +1,0 + Vi +1,1 + Vi +1,2 + 1] (10b)
1 + φ0 Δt
2 6 3 6
8
1 1 2 1
and Vi ,n = [ Vi +1,n− 2 + Vi +1,n−1 + Vi +1,n ] (11a)
1 + φn Δt 6
2 3 6
1 1 2 1
or Vi ,n = [ Vi +1,n −2 + Vi +1,n−1 + Vi +1,n + 1] (11b)
1 + φn Δt 6
2 3 6
Suppose that we divide every month into d intervals. Then Equations (9b),
n
(10b), and (11b) will be applied to i = kd , where k = 1, 2,K , , while Equations (9a),
d
(10a), and (11a) will be applied to i ≠ kd . Furthermore, since the loan will be due
T years later, we can define m = (12 × T × d ) + 1 , such that at any node ( i, m − 1 ) the
loan matures and, thus, the final payment P (⋅) = $1 is paid.
To capture the restriction of Condition (4), we employ the following procedures.
Consider the interest rate and the value of the traditional loan at the (i, j ) node,
where 1 ≤ i ≤ m − 1 and 1 ≤ j ≤ n − 1 . As shown by Figure 2, one period later, the
interest rate will jump up to the (i + 1, j + 1) node, remain unchanged at the (i + 1, j )
node, or jump down to the (i + 1, j − 1) node. At these nodes, the value of the
traditional loan will jump down to Vi +1, j +1 , remain unchanged as Vi +1, j , or jump up
to Vi +1, j −1 , respectively, since the value of the loan is inversely related to the interest
rate. As an investor purchasing the traditional loan has the option, but not the
obligation, to refinance the loan, he/she will thus refinance the loan once the interest
rate goes down, but not refinance it when the interest rate goes up. In other words,
when the interest rate increases from φ j to φ j +1 , then the investor will not adjust the
loan, such that the value of the loan remains at Vi +1, j , rather than decreases to
Vi +1, j +1 . As a result, we need to replace Vi +1, j +1 with Vi +1, j in Equations (9a) and
(9b). That is, we must impose the restriction that the value of the loan when the
interest rate goes up one period later remains equal to that when the interest rate
remains the same one period later. Similarly, we also need to replace Vi +1,2 with
9
Vi +1,1 in Equations (10a) and (10b) and Vi +1,n with Vi +1,n−1 in Equations (11a) and
(11b). By contrast, when the interest rate decreases from φ j to φ j −1 , then the
investor will refinance the loan such that the value of the loan increases from Vi +1, j
to Vi +1, j −1 .
<Insert Figures 1 and 2 here>
Let r denote the interest rate for the assumable loan, such that φ = r . Let
φ = φ0 + g Δφ and let φl = φ0 + l Δφ denote the current value of φ (t0 ) . We only
need to consider φ (t ) in the region between φ and φl , because both the traditional
and assumable loans can be refinanced if the future market interest rate is below the
assumable rate; i.e., if φ (t ) < φ . By contrast, when φ < φ (t ) < φl , only the investor
who employs the traditional loan would like to refinance the loan. As such, we can
use V0l to denote the present value of the traditional loan at the current time t 0
with the interest rate equal to φl2 , such that the investor who employs the traditional
loan can always refinance the loan when the future interest rate decreases.
Suppose that we start from t = t0 . Consider a loan that matures at T years;
i.e., t = t0 + T . If the loan is discounted at the current market rate, r (t0 ) ( = φ (t0 ) 2 ),
then the cash equivalence value of this loan, denoted by PVM , is given as:
12×T
1
PVM = ∑ r (t )
(12)
i =1 (1 + 0 )i
12
If the loan is discounted at the assumable rate, r ( = φ 2 ), then the cash equivalence
value of this loan, denoted by PVA , is given as:
10
12×T
1
PVA = ∑ r
(13)
i =1 (1 + )i
12
The present value of the payment savings; i.e., the premium associated with the
assumable loan; is thus equal to PVA − PVM . Under perfect capital market
conditions, Brueckner (1983) argued that homebuyers should be indifferent between
an assumable and new conventional loan if the premium paid by the assumable loan is
equal to PVA − PVM . However, Ferreira and Sirmans (1987) argued that
homebuyers should prefer a new conventional loan when market interest rates are
expected to decrease, since they will then have the option value to refinance, which is
given by V0l − PVM in our framework. As a result, the proportion of the premium
associated with the assumable loan that is accounted for by this option value is given
as:
V0l − PVM
RR = (14)
PVA − PVM
III. Numerical Examples
We choose a set of plausible parameter values to investigate the issue at hand.
The following benchmark parameter values resemble those set by Cox et al. (1985)
and Hilliard et al. (1998): (i) The current interest rate is equal to 10% per year, i.e.,
r (t0 ) = 0.1 ; (ii) the reversion speed of the interest rate per year is equal to 25%, i.e.,
a = 0.25 ; (iii) the long-run steady state of interest rate is equal to 7.5% per year, i.e.,
b = 0.075 ; (iv) the parameter c is set equal to 0.1, such that the volatility of the
interest rate evaluated at its current rate is equal to c r (t0 ) = 31.6% per year; and (v)
the assumable rate of interest is equal to 5% per year, i.e., r = 0.05 .
Given these benchmark parameter values, Table 1 shows the results for PVM ,
PVA , V0l , and RR for different loan terms (T ) , including 5, 10, 15, 20, 25, and 30
11
years. Table 1 indicates that all PVM , PVA , V0l , and RR increase with the term
of the loan. In other words, the present value of a traditional loan ( PVM ), the
present value of an assumable loan ( PVA ), and the expected present value of a loan
with the option to refinance ( V0l ) will all be more valuable when the loan lasts longer,
because the buyer of the loan then incurs more payments. The ratio of the option
value to refinance over the premium associated with the assumable loan ( RR ) also
increases with T . More precisely, the option value to refinance accounts for 30.4%
of the premium for the assumable loan, when the loan matures 5 years later. The
ratio then increases to 47.3%, 62.9%, and 68.7%, respectively, when the loan matures
10, 20, and 30 years later. This positive relation is due to the fact that the market
rate of interest follows a Wiener process, such that it is more likely to reach the level
of the assumable rate if it is allowed to travel longer. As a result, the premium
associated with the assumable loan is more likely to erode as the loan term lengthens.
Table 2 presents the results of RR for changes of the standard deviation c
over the region (5%, 15%), the assumable rate r over the region (2.5%, 7.5%), and
the current market interest rate r (t0 ) over the region (7.5%, 12.5%), holding the
other parameters at their benchmark values. The table shows that the premium
associated with the assumable loan will be eroded more by the option value to
refinance if the interest rate is more volatile ( c increases), if the spread between the
current market interest rate and the assumable rate shrinks (either r increases, or
r (t0 ) decreases). The intuition behind these results is as follows. As the interest
rate becomes more volatile, the market rate of interest is more likely to touch the level
of the assumable rate in the future. Similarly, as the spread between the current
market interest rate and the assumable rate shrinks, in the future it is also more likely
for the market rate of interest to touch the level of the assumable rate.
Take the example of a loan that matures 15 years later. We consider the case
12
where the standard deviation, c , increases from 5% to 15%. Table 2 then shows
that the option value to refinance accounts for 34.0% of the premium associated with
the assumable loan when the interest rate volatility is equal to 22.4% per year
( 0.05 0.1 ), such that 66.0% ( 100% − 34.0% ) of the premium associated with the
assumable loan is capitalized. By contrast, when the interest rate volatility is equal
to 38.7% per year ( 0.15 0.1 ), then the option value to refinance accounts for 69.3%
of the premium associated with the assumable loan, such that only 30.7%
( 100% − 69.3% ) of the premium associated with the assumable loan is capitalized.
These results fit well into the empirical results, indicating that only one-third to
two-thirds of the premium associated with the assumable loan transfers to the
transaction price of housing (Clauretie and Sirmans, 2006).
Table 2 does not report how changes of the reversion speed ( a ), nor the long-run
interest rate ( b ) affect RR , because RR is independent of these changes. The
reason is due to the fact that changes in a , or b , only affect the value of φmin and
φmax , but not the value of Δt . Given that φmin is much smaller than the
2
assumable rate, r , and φmax is much larger than the current interest rate, r (t0 ) , in
2
our analysis, and given that only the region between the assumable rate and the
current market interest rate is relevant to the calculation of V0l , changes in a and
b thus do not affect RR , since they don’t change the value of V0l .
<Insert Tables 1 and 2 here>
IV. Conclusion
This article has employed the option pricing model to calculate the proportion of
the premium associated with the assumable loan that is accounted for by the option
value to refinance embedded in a traditional loan. This option to refinance is more
valuable especially when the term of both loans last longer, when the interest rate is
13
more volatile, and when the spread between the current market interest rate and the
assumable rate shrinks. We also find that this option value can account for one-third
to two-thirds of the premium associated with the assumable loan, which is in line with
findings in the existing empirical studies.
This article makes several simplified assumptions, which may be relaxed in the
future. First, we may also allow stochastic movements of housing prices (see, e.g.,
Ambrose and Buttimer, 2000; Hilliard et al., 1998; Kau et al., 1994; Leung and
Sirmans, 1990), in addition to stochastic movements of spot interest rates. We can
then simultaneously value the option to refinance, as well as the option to default.
Second, we may include other factors, such as the loan-to-value ratio, tax rules, or
transaction costs (see, e.g., Kau, Keenan and Kim, 1993; Stanton, 1995), into our
framework and then examine how these factors affect the results of this research.
14
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17
φ
φn
φmax Δφ
φn−1
φl
φ1
φmin Δφ
φ0 t
t0 t1 ti tm−1 tm
Δt Δt
Figure 1: The grid method for finding Vi , j .
φ j +1 Vi +1, j −1
φj φj Vi , j Vi +1, j
φ j −1 Vi +1, j +1
(a) (b)
Figure 2: The motion of φ j and Vi , j . When the buyer has the option to
refinance, we must replace Vi +1, j +1 by Vi +1, j .
18
Table 1: The present values of an assumable loan, the present value of a
traditional loan without the option to refinance, the expected present
value of the traditional loan with the option to refinance, and the value
of the option to refinance as a percentage of the premium associated
with the assumable loan Unit: $
Variation in T (years)
5 10 15 20 25 30
(1) PVA 47.06 75.67 93.06 103.6 110.0 114.0
(2) PVM 52.99 94.28 126.5 151.5 171.1 186.3
(3) V0l 48.86 84.48 112.2 133.8 150.6 163.7
(3) − (1)
(4) RR = (%) 30.4 47.3 57.3 62.9 66.4 68.7
(2) − (1)
Note:
1. PVA , PVM , V0l , and RR denote the present values of an assumable loans,
the present value of a traditional loan without the option to refinance, the
expected present value of a traditional loan with the option to refinance, and the
value of the option to refinance as a percentage of the premium associated with
the assumable loan, respectively.
2. The benchmark parameter values are as follows: a = 25% per year; b = 7.5%
per year; c = 10% per year; r = 5% per year; and r (t0 ) = 10% per year.
Also, T , a , b , c r (t0 ) , r , and r (t0 ) denote the term of the loan, the
reversion speed, the long-run interest rate, the volatility of interest rate, the
assuming rate of interest, and the current market rate of interest, respectively.
We divide each month into ten intervals; i.e., d = 10 . Our simulation results
converge rapidly when d ≥ 5 .
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Table 2: Value of the option to refinance as a percentage of the premium
associated with the assumable loan, RR Unit: %
Variation in Variation in T (years)
c (%) 5 10 15 20 25 30
5 17.8 25.9 34.0 40.4 44.9 48.1
7.5 23.9 37.3 47.6 53.9 58.0 60.7
10 30.4 47.3 57.3 62.9 66.4 68.7
12.5 36.9 55.3 64.3 69.3 72.3 74.2
15 42.7 61.1 69.3 73.7 76.3 78.0
Variation in
r (%)
2.5 19.3 29.9 39.0 45.3 49.5 52.4
3.75 24.1 37.8 48.0 54.2 58.1 60.8
5 30.4 47.3 57.3 62.9 66.4 68.7
6.25 41.7 59.7 68.2 72.7 75.4 77.2
7.5 60.3 74.2 79.9 82.8 84.5 85.6
Variation in
r (t0 ) (%)
7.5 52.7 69.6 76.9 80.7 83.0 84.6
8.75 37.9 56.7 66.0 71.0 74.2 76.2
10.0 30.4 47.3 57.3 62.9 66.4 68.7
11.25 25.8 40.2 50.1 55.9 59.5 61.9
12.5 21.9 33.9 43.1 48.8 52.4 54.8
Note: Same as in Table 1.
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