How much can the option value to refinance

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: [email protected] 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 ) ∂t ∂r (t )

Suppose that t0 is the current date and T is the term of the loan.

(2) 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 , where

β=

(

(5)

)

φmin = − β + β 2 + 4α1α 2 / 2α 2

,

(

)

φmax = β + β 2 + 4α1α 2 / 2α 2

,

a Δφ 4ab − c 2 , and α 2 = . , α1 = 2Δt 2 8 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 = 2Δφ ∂φ ∂ 2V ∂φ 2

=

(6)

Vi , j +1 + Vi , j −1 − 2Vi , j

(7)

(Δφ ) 2

and the time derivative is approximated as ∂V Vi , j − Vi , j −1 = ∂t Δt

(8)

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): Vi , j =

1 1 + φ 2j Δt

1 2 1 [ Vi +1, j −1 + Vi +1, j + Vi +1, j +1 ] 6 3 6

(9a)

or Vi , j =

1

1 2 1 [ V + V + Vi +1, j +1 + 1] i + 1, j − 1 i + 1, j 3 6 1 + φ 2j Δt 6

(9b)

for j = 1, 2,K , n − 1 , while Vi ,0 =

or

=

1 1 + φ02 Δt 1 1 + φ02 Δt

1 2 1 [ Vi +1,0 + Vi +1,1 + Vi +1,2 ] 6 3 6

(10a)

1 2 1 [ Vi +1,0 + Vi +1,1 + Vi +1,2 + 1] 6 3 6

(10b)

8   

and Vi ,n = or

Vi ,n =

1

1 2 1 V + V + Vi +1,n ] [ i + 1, n − 2 i + 1, n − 1 3 6 1 + φn2 Δt 6 1

1 2 1 V + V + Vi +1,n + 1] [ i + 1, n − 2 i + 1, n − 1 3 6 1 + φn2 Δt 6

(11a) (11b)

Suppose that we divide every month into d intervals. Then Equations (9b), (10b), and (11b) will be applied to i = kd , where k = 1, 2,K ,

n , 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 . 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: PVM =

12×T

∑ i =1

1 r (t ) (1 + 0 )i 12

(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   

PVA =

12×T

∑ i =1

1 r (1 + )i 12

(13)

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: RR =

V0l − PVM PVA − PVM

(14)

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 .

2 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

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 .

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   

References

Ambrose, B.W. and R.J. Buttimer. 2000. Embedded Options in the Mortgage Contract. Journal of Real Estate Finance and Economics, 21(2): 95-111. Asabere, P.K., F.E. Hoffman and S. Mehdian. 1992. The Prime Effects of Cash Versus Mortgage Transactions. Journal of the American Real Estate and Urban Economics Association 20: 141-153. Brennan, M.J. and E.S. Schwartz. 1978. Finite Difference Method and Jump Processes Arising in the Pricing of Contingent Claims. Journal of Financial and quantitative Analysis 13: 461-474. Brueckner, J.K. 1983. Creative Financing and House Prices: A Theoretical Inquiry into the Capitalization Issue. Paper presented at the meeting of the American Real Estate and Urban Economics Association. Clauretie, T.M. 1982. How much is an assumable loan worth? Real Estate Review 12: 52-56. Clauretie, T.M. 1984. Capitalization of Seller Supplied Financing: Implications for assessors. Property Tax Journal 3: 229-238. Clauretie, T.M. and G.S. Sirmans. 2006. Real Estate Finance: Theory and Practice, 5th edition, Thomson South-Western. Cox, J.C., J.E. Ingersoll and S.A. Ross. 1985. A Theory of the Term Structure of Interest Rates. Econometrica 53: 385-407. Cox, J.C., S.A. Ross and M. Rubinstein. 1979. Option Pricing: a Simplified Approach. Journal of Financial Economics 7: 229-264. Dale-Johnson, D., M.C. Findlay, A.L. Schwartz Jr. and S.D. Kapplin. 1985. Valuation and Efficiency in the Market for Creatively Financed Houses. AREUEA Journal 13: 388-403. 15   

Ferreira, E.J. and G.S. Sirmans. 1987. Interest Rate Changes, Transactions Costs, and Assumable Loan Values. Journal of Real Estate Research 2: 29-49. Friedman, J. 1984. Cash Equivalence: Market Knowledge and Judgments. The Appraisal Journal 52: 129-132. Gaines, J.P., F.J. Ingram and C.W. Gregory. 1983. Conventional Financing Effects on Residential Market Prices During Periods of Increasing Interest Rates. Presented at the annual meeting of the American Real Estate and Urban Economics Association. Garcia, K. 1972. Sales Prices and Cash Equivalents. The Appraisal Journal 40: 7-16. Geske, R. 1979. The Valuation of Compound Options. Journal of Financial Economics 7: 63-81. Guntermann, K.L. 1978. Adjusting Comparable Sales Data for the Effects of FHA Financing. The Real Estate Appraiser 44: 12-18. Hilliard, J.E., J.B. Kau and V.C. Slawson, Jr. 1998. Valuing Prepayment and Default in a Fixed-Rate Mortgage: a Bivariate Binomial Options Pricing Technique. Journal of Real Estate Economics 26(3): 431-468. Hull, J.C. and A. White. 1990. Valuing Derivative Securities Using the Explicit Finite Difference Method. Journal of Financial and Quantitative Analysis 25: 87-100. Kau, J.B., D.C. Keenan and T. Kim. 1993. Transaction Costs, Suboptimal Termination, and Default Probabilities. Journal of American Real Estate and Urban Economics Association 21(3): 247-263. Kau, J.B., D.C. Keenan, W.J. Muller, III and J. Epperson. 1994. The Value at Origination of Fixed-Rate Mortgages with Default and Prepayment. Journal of Real Estate Finance and Economics 11(1): 5-36. Leung, W.K. and C.F. Sirmans. 1990. A Lattice Approach to Pricing Fixed-Rate Mortgages with Default and Prepayment Options. Journal of the American Real Estate and Urban Economics Association 18(1): 91-104. 16   

Lusht, K.M. and J.A. Hanz. 1994. Some Further Evidence on the Price of Mortgage Contingency Clauses. Journal of Real Estate Research 9: 213-218. Rosen, K.T. 1982. Creative Financing and House Prices: A Study of Capitalization Effects. Working Paper. Center for Real Estate and Urban Economics, University of California at Berkeley. Sirmans, G.S., S.D. Smith and C.F. Sirmans. 1983. Assumption Financing and Selling Prices of Single-Family Homes. Journal of Financial and Quantitative Analysis, 18: 307-318. Stanton, R. 1995. Rational Prepayment and the Valuation of Mortgage-Backed Securities. Review of Financial Studies 8(3): 677-708. Sunderman, M.A., R.E. Cannaday and P.F. Colwell. 1990. The value of Mortgage Assumptions: An Empirical Test. Journal of Real Estate Research 5: 247-257.

17   

φ  φn   φmax   φn−1  

Δφ  

φl  

φ1   φmin   φ0  

Δφ  

t0  

t1  

ti

tm−1

Δt  

t 

tm Δt

Figure 1: The grid method for finding Vi , j .

φ j +1   φj  

Vi +1, j −1  

φj  

Vi , j

φ j −1  

Vi +1, j +1  

(b)

(a)

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   

Vi +1, j  

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

30.4

47.3

57.3

62.9

66.4

68.7

(4) RR =

(3) − (1) (%) (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 . 19 

 

Table 2: Value of the option to refinance as a percentage of the premium Unit: %

associated with the assumable loan, RR Variation in T (years)

Variation in 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

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

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

Variation in r (%)

Variation in r (t0 ) (%)

Note: Same as in Table 1.

20