refinancing_07_10_03

NBER WORKING PAPER SERIES

OPTIMAL MORTGAGE REFINANCING:

A CLOSED FORM SOLUTION

Sumit Agarwal

John C. Driscoll

David Laibson

Working Paper 13487

http://www.nber.org/papers/w13487

NATIONAL BUREAU OF ECONOMIC RESEARCH

1050 Massachusetts Avenue

Cambridge, MA 02138

October 2007

We thank Michael Blank, Lauren Gaudino, Emir Kamenica, Nikolai Roussanov, Dan Tortorice, Tim

Murphy, Kenneth Weinstein and Eric Zwick for excellent research assistance. We are particularly

grateful to Fan Zhang who introduced us to Lambert?s W-function, which is needed to express our

implicit solution for the refinancing differential as a closed form equation. We also thank Brent Ambrose,

Ronel Elul, Xavier Gabaix, Bert Higgins, Erik Hurst, Michael LaCour-Little, Jim Papadonis, Sheridan

Titman, David Weil, and participants at seminars at the NBER Summer Institute and Johns Hopkins

for helpful comments. Laibson acknowledges support from the NIA (P01AG005842) and the NSF

(0527516). Earlier versions of this paper with additional results circulated under the titles "When Should

Borrowers Refinance Their Mortgages?" and "Mortgage Refinancing¸˛for Distracted Consumers."

The views expressed in this paper do not necessarily reflect the views of the Federal Reserve Board,

the Federal Reserve Bank of Chicago, or the National Bureau of Economic Research.

of text, not to exceed two paragraphs, may be quoted without explicit permission provided that full

credit, including © notice, is given to the source.

Optimal Mortgage Refinancing: A Closed Form Solution

Sumit Agarwal, John C. Driscoll, and David I. Laibson

NBER Working Paper No. 13487

October 2007

JEL No. G11,G21

ABSTRACT

We derive the first closed-form optimal refinancing rule: Refinance when the current mortgage

interest rate falls below the original rate by at least

[

N + W (! exp (!N))] .

In this fo

rmula W (.) is the Lambert W -function,

2 (D + 8) ,

R =

6/M ,

N = 1+R (D + 8)

! J )

D is the real discount rate, 8 is the expected real rate of exogenous mortgage repayment, F is the

standard deviation of the mortgage rate, 6/M is the ratio of the tax-adjusted refinancing cost and

the remaining mortgage value, and J is the marginal tax rate. This expression is derived by

solving a tractable class of refinancing problems. Our quantitative results closely match those

reported by researchers using numerical methods.

Sumit Agarwal

Financial Economist

Federal Reserve Bank of Chicago

230 South LaSalle Street

Chicago, IL 60604-1413

[email protected]

John C. Driscoll

Mail Stop 85

Federal Reserve Board

20th and Constitution Ave., NW

Washington, DC 20551

[email protected]

David Laibson

Department of Economics

Littauer M-14

Harvard University

Cambridge, MA 02138

and NBER

[email protected]

Optimal Mortgage Refinancing: A Closed Form Solution 3

1. Introduction

Hou sehold s in the US hold $23 trillion in real estate assets.

Almo st all home buy ers

obtain mortga ges and the total valu e of these mortgages is $10 trillion, exceeding the

value of US governm ent deb t. Decision s about mortgage reﬁnancing are among the

most important decisions th at households make.

Borrow ers reﬁnance mortgages to c h ange the size of their mortgage and/or to tak e

advantag e of lo wer borrowing rates. Many authors have calculated the optimal reﬁ-

nancing diﬀeren tial wh en the hou sehold is not m otivated by e quity extraction consid-

erations: Dunn and McConnell (1981a, 1981b); Dunn and Spatt (2005); Hendershott

and van Order (1987); Chen and Ling (1989); Follain, Scott and Yang (1992); Yang

and Maris (1993); Stanton (1995); Longstaﬀ (2004); and Deng and Quigley (2006).

A t the optimal diﬀerential, the NPV of the in terest sa ved equals the sum of reﬁnanc-

ing costs and the diﬀerence between an old ‘in the money’ reﬁnancing option that is

givenupandanew‘outofthemoney’reﬁnancing option that is acquired.

The actual behavior of mortg age holders often diﬀers from the predictions of

the optim al reﬁnancing model. In the 1980s and 1990s— when mortgage interests

rates generally fell — man y borro wers failed to reﬁnance despite holding options that

were deeply in the mone y (Giliberto and Thibodeau, 1989, Gr een and LaCou r-L ittle

(1999) and Deng and Quigley, 2006). On the other hand, Chang and Yavas (2006)

ha v e noted that ov er one-third of the borro wers reﬁnanced too e arly during the period

1996-2003.

Flow of Funds Accounts of the United States, Board of Governors of the Federal Reserve System,

June, 2007.

Dickinson and Heuson (1994) and Kau and Keenan (1995) provide extensive surveys of the

reﬁnancing literature. See Campbell (2006) for a broader discussion of the importance of studying

mortgage decisions by households.

Many other papers document and attempt to explain the puzzling behavior of mortgage hold-

Optimal Mortgage Refinancing: A Closed Form Solution 4

Anom alous reﬁnancing beha vior may be partially due to the complexit y of the

problem . Previous academ ic research has deriv ed the optimal diﬀerential as the

implicit numerical solution of a system of partial diﬀerential equations. Such option-

valueproblemsmaybediﬃcu lt to understand, or, in practice, solv e, for man y bor-

ro wers and their advisers. For instance, we analyze a sample of leading sources

of ﬁnancial advice and ﬁnd that none of these books and w eb sites ac knowledge or

discuss the (option) value of waiting for in terest rates to fall further. Instead these

advisory services discuss a “break-even” net presen t value rule: only reﬁnance if the

present value of the interest saving s is greater than or equal to the closing cost.

In the current paper, we derive a closed-form optimal reﬁnancing rule. We be-

gin our analysis b y iden tify in g an analy tically tractable class of mortg ag e reﬁnancing

problem s. We assum e that the real mo rtgag e interest rate and inﬂation follo w Brown-

ian motion, and the mortgage is structured so that its real value remains constan t

until an endogenou s reﬁnancing even t or an exogenous Poisson repaymen t even t.

The P oisson parameter can be calibrated to capture the combined eﬀects of mo v-

ing eve nts, principal repayment, and inﬂation-driv en depreciation of the mortgage

obligation. We derive a closed form solution for the optimal reﬁnancing threshold.

The optimal reﬁnancing solution depends on the discount factor, closing costs,

mortgage size, the marginal tax rate, the standard deviation of the inno vation in

the mortgag e in terest rate, and the P oisson rate of exogenous real repaym ent. For

calibrated c hoices of these param eters, the optimal reﬁnancing diﬀerentials w e deriv e

ers, including: Green and Shoven, (1986); Schw artz and Torous (1989,1992, 1993); Giliberto and

Thibodeau, (1989); Richard and Roll, (1989); Archer and Ling (1993); Stanton (1995); Archer, Ling

and McGill (1996); Hakim (1997); LaCour-Little (1999); Bennett, Peac h and Peristiani (2000, 2001);

Hurst (1999); Downing, Stanton and Wallace (2001); and Hurst and Staﬀord (2004).

Optimal Mortgage Refinancing: A Closed Form Solution 5

range typica lly from 100 to 200 basis poin ts. We comp are our interest rate diﬀeren-

tials with those computed by Chen and Ling (1989 ), wh o do not make our simplifying

assumptions. We ﬁn d that the two approaches generate recomm enda tions that diﬀer

by fewe r than 10 basis poin ts.

Many authors hav e called for greater attention to norm ative economic analysis

(e.g. Miller 1991 and Camp bell 2006 ). Ou r research follows this prescriptive line

of research. We solv e an optimal mortgage reﬁnancing problem . However, on its

o w n this is a redundant conceptual con tribu tion since other author s hav e numerically

solv ed mortga ge reﬁnancing problems. Ou r key con tribu tion is the derivation of a

closed-fo rm mortgag e ﬁnancing rule that has three good properties. It is easy to

verify. Itiseasytoimplement. Itisaccurateinthesensethatitmatchesoptimal

reﬁnancing diﬀerentials published b y other authors who do not make our simp lifying

assumptions.

We pro vid e two analytic solutions: a closed form exact solution — which appears

in the abstract — and a closed form second-order approximation , which w e refer to

as the square root rule. The closed form exact solution can be im plem ented on a

calculator that can make calls to Lam bert’s W -function (a little-kno w n but easily

comp utab le function that has only been actively studied in the past 20 y ea rs). By

contrast, our square root rule can be implemented with an y hand-held calculator. We

ﬁnd that this square root rule lies w ithin 10 to 30 basis points of the exact solution.

The paper has the follo w ing organization . Section 2 describes and solv es the mort-

gage reﬁnan cing problem. Section 3 analyzes our reﬁnancing result quantitativ ely and

comp ar es our results to the quantitativ e ﬁndings of other researc hers. Section 4 doc-

uments the advice of ﬁnancial planners, and derives the w elfare loss from following

Optimal Mortgage Refinancing: A Closed Form Solution 6

the net present value rule. Section 5 concludes.

2. The Model

In this section, we present a tractable contin uous-time model of mortgage reﬁnancing.

The ﬁrst subsection introduces the assumption s and notation. The next subsection

summ arizes the argument of the proof and reports the k ey results.

2.1. Notation and key assumptions.

The real in terest rate and the inﬂation rate. We assume that the real

interest rate, r, and inﬂation rate, π, jointly follo w Brownian motio n. Formally,

dr = σ

(1)

dπ = σ

, (2)

where dz represents Brownian incremen ts, and cov(dr, dπ)=σ

rπ

dt. Hence the nom-

inal interest rate, i = r + π follo ws a continu ous-tim e random walk. Li, Pearson ,

and Poteshman (2004) argue that the nom inal in ter est rate is w ell-approximated by a

random w alk, and that estimates showin g mean rev ersio n are biased.

The random

walk assum pt ions allo w us to considerably sim plify the analy sis. Chen and Ling

(1989), Follain, Scott and Yang (1992) and Yang and Maris (1993) also assume that

the nominal interest rate follow s a random walk. How ever, other authors assume

that interest rates are mean rever ting (e.g. Stan ton , 1995 and Downing, Stan ton, and

Wallace, 2005).

The in terest rate on a mortgage is ﬁxed at the time the mortgage is issued. Our

See Hamilton (1994) for a general discussion of the diﬃculties of distinguishing unit-root and

trend-stationary stochastic processes.

Optimal Mortgage Refinancing: A Closed Form Solution 7

analysis focuses on the gap between the curren t nominal in terest rate, i = r + π, and

the “mortgage rate,” i

= r

+ π

, which is the nominal interest rate at the time the

mortg age w a s issued. Let x represent the diﬀerence between the current nominal

in ter est rate and the mortgag e rate: x ≡ i − i

.Thisimpliesthat

dx =

+ σ

+2σ

rπ

dz (3)

= σdz, (4)

where σ ≡

+ σ

+2σ

rπ

The mortgage con tract. To elimin ate a state variable, w e counterfactually

assume that mortgage payments are structured so that the real value of the mort-

gage, M, remains ﬁxed until an exogenous and discrete mortgage repa ym ent event.

These repa ym ent ev ents follow a Poisson arrival process. Excluding these discrete

repa yment events, the continuous ﬂow of real mortgage repaym ent is given by

real ﬂo w of mortgage payments =(r

+ π

− π)M (5)

=(i

− π)M. (6)

In a standard mortgage con tract, the real value of a mortgage obligation declines

for three diﬀerentreasons: repaymentoftheentireprincipalatthetimeofarelocation

(or death), contracted nominal principal repaymen ts, and inﬂationary depreciation

of the real value of the mortgage. We capture all of these eﬀects when w e calibrate

the exogenous arrival rate of a mortgage repa ym ent eve nt.

We assume that the mortgage is exogenously repaid with hazard rate λ.In

Optimal Mortgage Refinancing: A Closed Form Solution 8

our calibration section, w e sho w how to choose a value of λ that simultaneously

captures all three c ha nnels of repa ym ent: relocation, nominal principal repa y m ent,

and inﬂation. Hence λ should be thought of as the expected exogenous rate of decline

in the real value of the mortgage.

Reﬁnancing. T he mortgage holder can reﬁnance his or her mortgage at real

(tax-adjusted) cost κ(M). These costs include points and any other explicit or imp licit

transactions costs (e.g. la w yers fees, mortgag e insurance, personal time). We deﬁne

κ(M) to represent the net presen t value of these costs, netting out all allow able tax

deductions generated by future deductions of amortized reﬁnancing points. For a

consum er who itemizes (and takes account of all allo wable deductions), the form u la

for κ(M) is provided in appendix A.

Our analysis translates costs and beneﬁts in to units of “discounted dollars of in-

terest pa ym ents.” Since κ(M) represen ts the tax-adjusted net present value of closing

costs, κ(M) needs to be adjusted so that the model recognizes that one unit of κ is

econom ically equal to

1−τ

dollars of curren t (fully and immedia tely tax-ded uc tible)

in ter est paym ents, where τ is the marginal tax rate of the household. Hence, w e

multiply κ(M) by

1−τ

and w ork with the normalized reﬁnancing cost

C(M)=

κ(M)

1 − τ

If a consum er does not itemize, set τ =0for both the calculation of κ(M) and

the calculation of C(M).

A borrower who itemizes is allowed to make the following deduction. If N is the term of the

mortgage, then the borrower can deduct

of the points paid for N years. If the mortgage is

reﬁnanced or otherwise prepaid, the borrower may deduct the remainder of the points at that time.

Appendix A derives a formula for κ(M).

Optimal Mortgage Refinancing: A Closed Form Solution 9

Optim ization problem. Mortgage holders pic k the reﬁnancing policy that

minimizes the expected NPV of their real interest payments, applying a ﬁxed dis-

count rate, ρ. We also assume that mortga ge holders are risk neutral.

Summing up these considerations, the consumer minimizes the expected value

of her real mortgage pa y me nts. Let value function V (r

,r,π

,π,M), represent the

expected v alue of her real mortgage payments. More formally, the instantanteous

Bellm an Equation for this problem is giv en b y

ρV =(r

+ π

− π)M + λM − λV +

E [dV ]

=(r

+ π

− π + λ)M − λV +

∂

∂r

∂

∂π

+ σ

rπ

∂

∂r∂π

This Bellman equ ation can be derived with a standard application of stochas-

tic calculus and Ito’s Lemma . First-order partial derivatives do not appear in this

expression, since r and π ha v e no drift.

At an endogenous reﬁnancing event, the mortgage holder exchanges V (r

,r,π

,π,M)

for V (r, r, π, π, M )+C(M). Hence, at an optimal reﬁnancing event value matching

will imply that

V (r

,r,π

,π,M)=V (r, r, π, π, M )+C(M).

Given our assum ption s, an optimizing mortg age holder pic ks a reﬁnancing rule

that minimizes the discounted v alue of her mortgage paym en ts. In other w ords, she

picks a reﬁnancing rule that minimizes V.

We next show that the second-order partial diﬀerential equation that c h aracterizes

V can be simpliﬁed .

Optimal Mortgage Refinancing: A Closed Form Solution 10

2.2. Our main result. Since M is a constant, we can partial M out of the

problem. This leaves four state variables: r

,r,π

,π.

The ﬁrst step in the proof decomposes the value function V (r

,r,π

,π). We de-

ﬁne Z to be the discoun ted value of expected future paym ents conditional on the

restriction that reﬁnancing is disallo wed. The Bellman Equation for Z is given by,

ρZ(r

,r,π

,π)=(r

+ π

− π + λ)M − λZ(r

,r,π

,π)+

E [dZ ]

It can be conﬁrm ed that the solution for Z is

Z(r

,r,π

,π)=

+ π

− π + λ) M

ρ + λ

. (7)

It follow s that Z can be reduced to a function of the state variable −x + r, which is

equal to r

+ π

− π.

We decomp ose V, by deﬁning R as

R(r

,r,π

,π) ≡ Z(r

,r,π

,π) − V (r

,r,π

,π). (8)

The function R represen ts the option value of being able to reﬁnance. R can be

expressed as a function of one state variable:

x = i − i

= r + π − r

− π

This one-variable simpliﬁcation can be deriv ed with the follow ing “replication” lemma.

Optimal Mortgage Refinancing: A Closed Form Solution 11

Lemm a 1. Replication.

R(r

,r,π

,π)=R(r

+ ∆,r+ ∆,π

,π) (9)

= R(r

,r,π

+ ∆,π+ ∆) (10)

= R(r

,r+ ∆,π

+ ∆,π) (11)

Proof: Consider an agent in state (r

+ ∆,r+ ∆,π

,π). Let this agent replicate

the reﬁna ncing strategy of an agen t in state (r

,r,π

,π). In other w ords, reﬁnance

after every sequence of innovation s in the Ito processes that w ould make the agen t

who started at (r

,r,π

,π) reﬁnan ce. So the agent in state (r

+ ∆,r+ ∆,π

,π) will

generate reﬁnan cing cho ices valued at V (r

,r,π

,π)+

∆M

ρ+λ

. Hence,

V (r

+ ∆,r+ ∆,π

,π) ≤ V (r

,r,π

,π)+

∆M

ρ + λ

Likewise, we have

V (r

,r,π

,π) ≤ V (r

+ ∆,r+ ∆,π

,π) −

∆M

ρ + λ

Combining these tw o inequ alities, and substituting equatio ns (7) and (8), yields equa-

tion 9. We now repeat this type of argument for other cases. By replication,

V (r

,r,π

+ ∆,π+ ∆) ≤ V (r

,r,π

,π)

V (r

,r,π

,π) ≤ V (r

,r,π

+ ∆,π+ ∆).

Optimal Mortgage Refinancing: A Closed Form Solution 12

Combining these t wo inequalities, w e ha ve equation 10. By replication ,

V (r

,r+ ∆,π

+ ∆,π) ≤ V (r

,r,π

,π)+

∆M

ρ + λ

V (r

,r,π

,π) ≤ V (r

+ ∆,r,π

,π+ ∆) −

∆M

ρ + λ

Before reﬁnan cing, the perturbed agen t pays ∆ more (the inﬂation rate at which the

perturbed agent borrowed is π

+∆ rather than π

). After reﬁnancing, the perturbed

agen t pa ys ∆ more (the real interest rate at which the perturbed agent reﬁnances is

r + ∆ rather than r). Combining the two inequa lities, we ha ve equation 11. ¤

The Lemm a implies that these equalities hold ev eryw h ere in the state space:

∂R

∂r

∂R

∂π

= −

∂R

∂r

= −

∂R

∂π

This in turn implies that R(r

,r,π

,π) can be rewritten as R(x).

We will sho w that the solution of R can be expressed as a second-order ordinary

diﬀerential equation with three unknowns: t wo constan ts in the diﬀerential equation

and one free boundary. To solve for these three unkno wns w e need three boundary

conditions. We will exploit a value matching constraint that links R the instan t

before reﬁnancing at x = x

∗

and the instant after reﬁnancing (when x =0).

R(x

∗

)=R(0) − C(M) −

∗

ρ + λ

We will also exploit smooth pasting at the reﬁnancing boundary.

∗

)=−

ρ + λ

Optimal Mortgage Refinancing: A Closed Form Solution 13

Finally, lim

x→∞

R(x)=0, since the option value of reﬁnancing vanishes as the in terest

diﬀerential gets arbitrarily large. See Lemm a 5 in Appendix B for a derivation of

the ﬁrst tw o boundary conditions.

The follow ing theorem characterizes the optimal threshold, x

∗

, and the v alue

functions. T h e threshold rule is expressed in x, the diﬀerence bet ween the curren t

nomina l in terest rate, i, and the nom inal in terest rate of the mortgage, i

Theorem 2. Reﬁnance when

i − i

≤ x

∗

≡

[φ + W (−exp (−φ))] . (12)

where W(.) is the Lambert W -function,

ψ =

2(ρ + λ)

φ =1+ψ (ρ + λ)

κ/M

(1 − τ)

When x>x

∗

the v alue function is

V (r

,r,π

,π)=−Ke

−ψx

− π + λ) M

ρ + λ

, (13)

where K

is given b y

K =

ψx

∗

ψ(ρ + λ)

. (14)

The option value of being able to reﬁnance is Ke

−ψx

when x>x

∗

K has an equivalent solution, K = −

−ψx

∗

− 1

−1

∗

ρ+λ

+ C(M)

Optimal Mortgage Refinancing: A Closed Form Solution 14

Proof: We can express V as

V (x, r)=

(−x + r + λ) M

ρ + λ

− R(x).

Using Ito’s Lemma, derive a contin uous time Bellman Equation for V :

ρV =(−x + r) M +

∂

∂x

+ λ (M − V ) . (15)

Substitu ting for V yields

(ρ + λ)

(−x + r + λ) M

ρ + λ

− R

=(−x + r + λ) M −

This simplifes to

(ρ + λ)R =

. (16)

The original value function V has been eliminated from the analysis, as has the

variable r. The option value R(x) has a solution of the form R(x)=Ke

−ψx

,with

exponen t

ψ =

2(ρ + λ)

We pick ψ>0 to satisfy the limiting boundar y condition (lim

∗

→∞

R(x

∗

)=0). The

remaining t wo parameters, K and x

∗

, solv e the system of equations deriv ed from the

value matching and smooth pasting conditions.

−ψx

∗

= K − C(M) −

∗

ρ + λ

(17)

−ψKe

−ψx

∗

= −

ρ + λ

. (18)

Optimal Mortgage Refinancing: A Closed Form Solution 15

We use the smooth pasting condition to solve for K and substitute it back into the

value matc hing condition. Hence,

K =

ψx

∗

ρ + λ

, (19)

yielding

ρ + λ

ψx

∗

ρ + λ

− C(M) −

∗

ρ + λ

, (20)

Mu ltiplyin g through b y the in verse of the left hand side yield s:

ψx

∗

− ψx

∗

=1+

C(M)

ψ(ρ + λ) (21)

Set k =1+

C(M)

ψ(ρ+λ) in Lemm a 6 (Appendix B) to yield the closed form expression

for x

∗

in the statem ent of the theorem.¤

The Lambert W function, whic h appears in the solution, is the inverse function

of f (x)=xe

. Hence, z = W (z)e

W (z)

. Although its origins can be traced to

Johann Lam bert and Leonhard Euler in the 18th cen tury, the function has only been

extensiv ely examined in the past 20 years. It has since been show n to be useful in

solvin g a wide variety of problem s in applied math e matics, and is built in to a n u mber

of comm on matherm atical programm ing packages, including Maple, Mathematica

and Ma tlab. For more information on the function and its uses, see Corless, Gonnet,

Hare, Jeﬀrey and Kn uth (1996) and Ha yes (2005).

We also study an additional threshold value at which the reduction in the NPV

of future in terest pa ym ents (assuming no more reﬁnan cing ) is exactly oﬀset b y the

cost of reﬁnancing, C(M). WerefertothisastheNPVbreak-eventhreshold.

Follain and Tzang (1988) also deriv e this diﬀeren tial. They note that, since it ignores the option

Optimal Mortgage Refinancing: A Closed Form Solution 16

Deﬁnition 3. The NPV break-even threshold, x

NPV

, is deﬁned as

−

NPV

ρ + λ

= C(M). (22)

In tu itively, the NPV break-even threshold is the poin t at which the expected

in ter est paymen ts saved from an immediate and ﬁnal reﬁnancing,

−xM

ρ+λ

,exactlyoﬀset

the tax-adjusted cost of reﬁnancing, C(M).

2.3. Second-order expansion. Our closed form (exact) solution for the optimal

reﬁnancing diﬀerential requires calls to the Lambert W -function. We also provide

an alternative solution that can be implemented on a hand-held calculator that does

not output the Lambert W-function.

The proof of the main theorem derives an implicit solution for x

∗

(equation 21),

which can be written as

f(x

∗

)=e

ψx

∗

− ψx

∗

− 1 − ψ(ρ + λ)

C(M)

A second-order Taylor series approximation to f(x

∗

) at x

∗

=0is given by:

f(x

∗

) ≈ f(0) + f

(0)x

∗

(0)x

∗2

= −ψ(ρ + λ)

C(M)

+0· x

∗

∗2

Setting this to zero and solving for x

∗

(picking the negative root) yields an approxi-

mation that we refer to as the square root rule,

to reﬁnance, this diﬀerential represents a lo wer bound to the reﬁnancing decision; they also note

that calculating the option value is complicated.

Optimal Mortgage Refinancing: A Closed Form Solution 17

∗

≈−

σκ

M (1 − τ)

2(ρ + λ).

We evaluate the practical accuracy of this app roximation in the calibration section

below.

We also evaluate a third-order approximation, which is given b y an implicit

cubic equation.

3. Calibration

We begin by illustrating the model’s predictions for the optimal threshold v alue x

∗

We n u m er ically solve equation (12) — the exact solution of the optimal reﬁnancing

problem — with ty pical values of parameters ρ, τ , κ(M), σ,andλ. We also provide

a web calculator

which readers can use to evaluate an y calibration of in terest.

For our ﬁrst illustrative analysis, we choose a 5% real discount rate, ρ =0.05. We

assume a 28% marginal tax rate, τ =0.28.

We assum e transaction s costs of 1 poin t

and $2000; κ(M) is giv en by the form ula in Appendix A (e.g. κ(M)=0.01M +2000 if

τ =0).Theﬁxed cost ($2000) reﬂects a range of fees including inspection costs, title

insurance, la w yers fees, ﬁling c h ar ges, and non-pecuniary costs lik e time.

Using

Not all of the limit properties of the second-order approximation match those of the exact

solution. In particular, as the standard deviation of the mortgage rate, σ, goes to zero, the second-

order approximation also goes to zero, while the exact solution goes to the NPV threshold. Because

of this, at low values of σ, the NPV threshold is a better approximation to the optimal threshold

than is the second-order approximation. Since the NPV threshold is also easily calculable, a

b etter reﬁnancing rule than simply using the second-order approximation alone is to reﬁnance when

x<min

−

C(M )

2(ρ + λ), −(ρ + λ)

C(M )

Higher order approximations provide greater accuracy at the cost of greater computational

complexit y. In our view, only the second-order approximation is of signiﬁcant interest due to its

ease of calculation.

http://www.n ber.org/mortgage-reﬁnance-calculator

In the 2007 tax code, the 28% marginal tax rate applies to joint ﬁlings for households with joint

income between $128,500 and $195,850, and to ﬁlings for single households with income between

$77,100 and $160,850.

See Federal Reserve Board and Oﬃce of Thrift Supervision (1996), Caplin, Freeman and Tracy

Optimal Mortgage Refinancing: A Closed Form Solution 18

historical data, we estimate that the annu alized standard deviation of the mortgage

interest rate is σ =0.0109.

Finally, we nee d to calibrate λ, the expected real repayment rate of the mortgage.

We need to calculate the value of λ that corresponds to a realistic mortgage con tract

— one in which there are three forms of repa y m ent: ﬁrst, a probability of exogenous

repa yment (due to a relocation); second, principal paymen ts that reduce the real

value of the mortgage; third, inﬂation that reduces the real value of the mortgage.

Formally, consider a household with a mortgage with a contemporaneous real (annual)

mortgage paym ent of p, rema ining principle M, an original nomin al in terest rate of

,andaμ hazard of relocation (implying that

is the expected time until the next

mov e). We’ll consider an enviro n m ent with current inﬂation π. For this mortgage,

the expected (ﬂow) value of the exogenous decline in the real mortgage obligation is

μM +(p − i

M)+πM.

The term in paren th eses corresponds to contracted principa l repa yment. The last

term represen ts inﬂation eroding the real value of the mortgage. Using this form ula,

we can calibrate the value of λ.

λ = μ +

− i

+ π

= μ +

nominal

− i

+ π

(1997), Danforth (1999), Lacour-Little (2000), and Chang and Yavas (2006) for data on transactions

costs.

The standard deviation for monthly diﬀerences of the Freddie Mac 30-year mortgage rate from

April 1971 to February 2004 is 0.00315, implying an annualized standard deviation of σ =

√

12 ×

0.00315 = 0.0109. By comparison, taking annual diﬀerences yields an average standard deviation of

σ =0.0144. These results are consistent with our decision to model in terest rate innovations as iid.

Optimal Mortgage Refinancing: A Closed Form Solution 19

In practice it will be easier for househo lds to use the latter “nominal” version of the

form ula since households kno w p

nominal

and M

nominal

, the nominal analogs of p and

We can also solve for the k ey terms in the equations above using formulae for a

standard ﬁxed rate mortgage. In this case, the calibration for λ is

λ = μ +

exp [i

Γ] − 1

+ π.

where Γ is the remaining life (in y e ars) of the mor tgage . See Appendix C for this

derivation.

Assum e that the household has a 10% chan ce of mo ving per y ear, so μ =10%,

and the expected dur ation of stayin g the house is 10 year s. Assum e that i

=0.06,

π =0.03, and Γ = 25 years, then, λ =0.147.

Table 1 reports the optimal reﬁnancing diﬀerentials calculated with our model for

the calibration summarized abo ve. We report the exact optimal rule, the second-

and third-order approximations to the optimal rule, and the (suboptimal) net present

value rule. We calculate the reﬁnancing diﬀerentials for mortg ag e sizes (M)of

$1,000,000, $500,000, $250,000 and $100,000.

This calibration is only an approximation, since the calibration formula will c h ange over time

(whereas λ is constant in the model from section 2).

Hayre, Chaudhary and Young (2000) estimate that 5 to 7 percent of single-family homes turn

over per year.

Optimal Mortgage Refinancing: A Closed Form Solution 20

Table 1: Reﬁnancing diﬀeren tials

inbasispointsbysolutionmethod

Mortgage

Exact optimum 2nd order 3rd order NPV rule

$1,000,000 107 97 109 27

$500,000 118 106 121 33

$250,000 139 123 145 44

$100,000 193 163 211 76

The optimal reﬁnancing threshold increases as mortgage size decreases, since in-

terest sa vin gs from reﬁnancing scale proportionately with mortgage size but part of

the reﬁnancing cost is ﬁx ed ($2000). The second-order approxim ation deviates by 10

to 30 basis points from the exact optimum . The third-order approximation deviates

b y only 2 to 18 basis points from the exact optimum. The NPV rule, b y con trast,

deviates by 80 to 117 basis points from the exact optimum .

Table 2 presents results for the six diﬀerent marginal tax rates that were in eﬀect

under the tax code in 2006:

Table 2: Optimal reﬁnancing diﬀerentials

in basis poin ts by marginal tax rate τ

Mortgage

0% 10% 15% 25% 28% 33% 35%

$1,000,000 99 101 103 106 107 109 110

$500,000 108 111 113 117 118 121 122

$250,000 124 129 131 137 139 143 145

$100,000 166 174 178 189 193 199 202

Optimal Mortgage Refinancing: A Closed Form Solution 21

The optimal diﬀerentials rise as the margina l tax rate rises, since interest pa ym ents

are tax deductible but reﬁnancing costs are not.

Table 3 reports the consequences of varying, λ, the expected real rate of repa y-

ment.

We consider cases in which the expected time to the next move is 5 y ears

(μ =0.20), 10 years (μ =0.10), and 15 years (μ =0.066), corresponding to values

for λ of .247, .147, and .114, respectively.

Table 3: Optimal reﬁnancing diﬀerentials

in basis poin ts by expected real rate of repaym ent λ

Mortgage

0.114 0.147 0.247

$1,000,000 101 107 122

$500,000 112 118 136

$250,000 131 139 161

$100,000 180 193 227

As expected, a higher hazard rate of prepa yment raises the optimal in ter est rate

diﬀerential, since the eﬀective amount of time o ver which the lowe r interest savings

will be realized is smalle r.

Table 4 reports the optimal diﬀerential assuming a reﬁnancing cost of only $1000,

which is of interest because of the wider a vailability of lo w-cost reﬁnancings. For

comp arison , w e also report the diﬀeren tials predicted by the NPV rule at a reﬁnancing

cost of $1000.

Unless otherwise speciﬁed, we now return to our earlier assumption of a marginal tax rate of

28%.

Optimal Mortgage Refinancing: A Closed Form Solution 22

Table 4: Optimal reﬁnancing diﬀerentials

in basis points by fee size

C(M)

Mortgage

$2000 + 0.01M $1000 $1000,NPV

$1,000,000 107 32 2

$500,000 118 45 4

$250,000 139 66 7

$100,000 193 108 18

Reducing the costs substantially reduces the optimal interest rate diﬀeren tia ls.

The diﬀerentials implied by the NPV rule also decline.

4. Comparison with Chen and Ling (1989)

We now compare the reﬁnancing diﬀeren tials implied by our model and those reported

b y Chen and Ling (1989). Chen and Ling calculate optimal diﬀerentials for a model

in which the log one-period nominal in terest rate follow s a random walk, the time of

exogenous prepa ym ent (or the expected holding period) is kno w n with certainty, and

the real mortg ag e principle is allo wed to decline over time because of inﬂation and

contin uous principle repa ymen t. C hen and Ling use numerical methods to solve the

resulting system of partial diﬀerential equations.

In contrast to their analysis, we make a simplifying assumption that allo w s us to

obtain an analytic solution to a closely related mortgag e reﬁnancing problem.

explained abo ve, w e assum e that the m ortg age is structured so that its real value

In one way, our paper adds greater realism when compared with previous work. We account

for the diﬀerential tax treatment of mortgage interest pa yments and reﬁnancing costs. Reﬁnancing

costs are not tax deductible (unlike the closing costs on an originating mortgage).

Optimal Mortgage Refinancing: A Closed Form Solution 23

rema ins constant. This allows us to a void trackin g a c ha nging value of time to

matu rity and a c h an ging remaining mortgage balance. In contrast to our approach ,

Chen and Ling’s model directly incoporates the eﬀects of principal repa yment and

the ﬁnite life of the mortgage con tract.

To bring our model into line with theirs, our parameter λ is calibrated to capture

the joint eﬀects of mo ving, principal repa ymen ts, and inﬂation. Hen ce, λ is set to

capture the three wa ys that the expected real value of the mortgag e declines over

time.

To calibrate our model to match the set-up in Chen and Ling, we set λ =0.173

to accoun t for (1) an 8 y ear expected holding period (

=8, so μ =0.125); (2) a

long-run inﬂation forecast (in 1989) of 4% (π =0.04); and (3) a principal repaym ent

rate of 0.8% at the beginning of a 30-year mortgage. We set the discoun t rate to

be 4%, ρ =0.04, matching Chen and Ling’s assumption of an 8% nominal interest

rate. Chen and Ling’s random w alk assumption for the log short-term interest rate

allows us to compute the implied standard deviation for the 30-y ear mortgage rate

(see Appendix D). We calculate an implied standard deviation for the innovations of

the 30-y ear mortgage rate of σ =0.012. Finally, to match the analysis of Chen and

Ling we assume a zero marginal tax rate.

Che n and Ling’s baseline calculations exclude the possibility of subsequent reﬁ-

nancings. But their analysis enables us to compute the additional points that w ould

be necessary to buy a new reﬁnancing option when the original mortgage is reﬁnanced.

There are two such cases that are analyzed in Chen and Ling.

With a reﬁnancing cost of 2 points (without a new option to reﬁnance), 2.24

We take results from the middle columns of their table 2. For consistency with our framework,

we consider cases from Chen and Ling in which the interest rate process has no drift.

Optimal Mortgage Refinancing: A Closed Form Solution 24

additional poin ts are c harged to purchase the right to reﬁnance again,

implying

total points of 4.24. For this case, Chen and Ling calculate an optimal reﬁnancing

diﬀerential of 228 basis points, while we calculate an optimal reﬁnancing diﬀerential

of 218 basis poin ts, a diﬀer ence of 10 basis point s.

With a reﬁnancing cost of 4 points (without a new option to reﬁnance), 1.51

additional points will be c harged to purchase the righ t to reﬁnance again,

implying

total points of 5.51 points. For this case, Chen and Ling calculate an optimal

reﬁnancing diﬀerential of 256 basis poin ts, while we calculate an optimal diﬀerential

of 255 basis poin ts, a diﬀer ence of 1 basis poin t.

5. Financial advice

Hou sehold s considering reﬁnancing use many diﬀerent sources of advice, including

mortgage brok ers, ﬁnancial planners, ﬁnancial advise books, and websites. In this

section, we describe the reﬁnancing rules recommended b y 25 leading books and

websites. We ﬁnd that none of the sources of ﬁnancial advice in our sample provide

a calculation of the optimal reﬁnancing diﬀerential. Instead, the advisory services

in our samp le oﬀer the break-even NPV rule as the only theoretical benchm ar k.

Most of the advice boils down to the follo w ing necessary condition for reﬁnancing —

only reﬁnance if you can recoup the closing costs of reﬁnancing in reduced interest

pa ymen ts.

First, we sampled books that were on top-ten sales lists at the Amazon and Barnes

& Nob le web sites (see the web appendix for a detailed description of our sampling

See panel 1, column 3, in Table 1 of Chen and Ling.

The second order approximations yield reﬁnancing diﬀerentials of 182 and 207, diﬀering from

Chen and Ling’s values by 46 and 48 basis points. These results reﬂect the general deterioration of

the approximation as reﬁnancing costs become very large.

Optimal Mortgage Refinancing: A Closed Form Solution 25

method and ﬁndings

). Of the 15 unique books in our sample, 13 pro vided a break-

even calculation of some sort. Most of the 15 books also provided some rules of

th umb (e.g. ‘wait for an interest diﬀerential of 200 basis points,’ or ‘only reﬁnance if

you can recoup the closing costs within 18 months’).

For websites, w e en tered the wo rds mortgage reﬁnanc in g advice in to Google and

examin ed the top twelv e sites whic h oﬀered information on reﬁnancing. Two of these

sites suggest a ﬁxed interest-rate diﬀerential of one-and -a-ha lf to two percent and

recomm end reﬁnancing only if the borro wer plans to stay in the house for at least

three to ﬁv e years. One of the sites pro vides a mon thly savings calculator, wh ile sev en

of the sites pro vide a reﬁnan cing calculator based on the NPV break-even criterion.

Theremainingthreesitesdidnotprovideareﬁn an cing calculator but still recomm end

break-even calculations.

None of the 15 books and 10 w eb sites in our sample discuss (or quantitativ ely

analyze) the value of waiting due to the possibilit y that interest rates migh t con tinue

to decline.

Finally, market data also sho ws that man y households did reﬁnance too close to

the NPV break-ev en rule during the last 15 years; see, for example, Ya vas and Chan g

(2006) and (Agarwal, Dri scoll and Laibson 2004).

How suboptimal is the NPV rule?. To measure the suboptimality of the

NPV rule, we consider an agent that starts life with state variable x =0(a new mort-

gage). We calculate the expected cost of using an arbitrary reﬁnancing diﬀeren tial,

, instead of using the optimal reﬁnancing rule speciﬁed in Theorem 2.

Proposition 4. The expected discoun ted Lo ss as a fraction of the mortgage size

http://www.n ber.org/mortgage-reﬁnance-calculator/appendix.py

Optimal Mortgage Refinancing: A Closed Form Solution 26

from using an arbitrary heuristic rule instead of using the optimal rule is giv en b y

Loss

C(M)

∗

ρ+λ

1 − e

−ψx

∗

−

C(M)

ρ+λ

1 − e

−ψx

(23)

ψx

∗

ψ(ρ + λ)

−

C(M)

ρ+λ

1 − e

−ψx

. (24)

where x

is the heuristic threshold rule. This implies that the expected discoun ted

Loss as a fraction of the mortgage size from using the suboptimal NPV rule instead

of using the optimal rule is given by

Loss

ψx

∗

ψ(ρ + λ)

. (25)

Proof : The loss is equal to the diﬀerence between the value function associated

with the optimal rule and the value function associated with the alterna tive rule. The

value function for the optima l rule is given in the statement of the main theorem.

Since the in terest paymen t term is the sam e for both the optim al and suboptimal

rules, the diﬀerence in value functions will be equal to the diﬀerence in option value

expressions. For both the suboptimal and appro x imate rules, the value matching

condition still applies, but with x

∗

replaced with the suboptimal diﬀerentials speciﬁed

by the alternative rule, x

Follow ing the line of argumen t in the proof of our main theorem, the option value

function, R(x), has a solution of the form R(x)=Ke

−ψx

. The parameter K is

deriv ed from the value matc hing condition,

−ψx

= K − C(M) −

ρ + λ

, (26)

Optimal Mortgage Refinancing: A Closed Form Solution 27

implying

K =

C(M)+

ρ+λ

1 − e

−ψx

. (27)

So the diﬀerence in value functions is giv en by

Loss

C(M)

∗

ρ+λ

1 − e

−ψx

∗

−

C(M)

ρ+λ

1 − e

−ψx

ψx

∗

ψ(ρ + λ)

−

C(M)

ρ+λ

1 − e

−ψx

Note that x

= x

NPV

implies that

C(M)

ρ+λ

=0, and hence

Loss

C(M)

∗

ρ+λ

1 − e

−ψx

∗

−ψx

ψ(x

∗

−x)

ψ(ρ + λ)

Set x =0, to reﬂect the perspectiv e of an agent with a newly issued mortgag e. ¤

Note that the loss from follow ing the NPV rule is equal to the option value of the

ability to reﬁna nce, evaluated for a new mortgage . By ignoring the existence of the

option value, the NPV rule creates a loss equal in size to the option value.

Using the same calibration assumptions that were used in section 3, we calculate

the economic losses of using the NPV rule and the second order rule instead of the

exactly optimal rule.

Optimal Mortgage Refinancing: A Closed Form Solution 28

Table 5: Expected losses in discounted dollars

from using the NPV and appro ximate rules

Mortgage

Loss (NPV rule) Loss (square root rule)

$1,000,000 $163,235 $15,253

$500,000 $86,955 $9,459

$250,000 $49,066 $7,020

$100,000 $26,479 $6,406

Tab le 6: Expected losses as a percent of mo rtgage face value

from using the NPV and appro ximate rules

Mortgage

Loss (NPV rule) Loss (square root rule)

$1,000 ,0 00 16.3% 1.5%

$500,000 17.4% 1.9%

$250,000 19.6% 2.8%

$100,000 26.8% 6.4%

Other rules of thumb. Som e advisers also refer to a rule of th umb in whic h

borro wers are encouraged to reﬁnan ce when the inter est rate has dropped b y 200 basis

points. We ha ve also heard more recen tly of a revised 100 basis poin t rule of thumb.

Both rules generally, though not alw a ys, imply reﬁnancings at bigger diﬀerentials than

those implied b y the NPV rule. How ever, our sim u lation s, show that the optimal

reﬁnancing diﬀeren tia l can vary quite substantially by expected holding period and

reﬁnancing cost, among other parameters. Hence a “one size ﬁts all” rule will lea d

to substantial w elfa re losses.

Optimal Mortgage Refinancing: A Closed Form Solution 29

6. Conclusion

Mortgage reﬁnancing is an importan t ﬁna ncial decision. Ma ny papers ha ve solv e d

for the optimal reﬁnan cing rule, whic h has been previously calculated by n umerically

solving a system of partial diﬀeren tial equations. Such numerical analysis is beyond

the capabilities of many borrowers and their advisers.

Indeed, w e sho w that leading ﬁnancia l advisers do not discuss (form ally or infor-

mally) option value considerations. Advisors t ypically discuss the net present value

rule: reﬁnanceonlyifthenetpresentvalueoftheinterestsavedisatleastasgreat

as the direct cost of reﬁnan cing. Comp ared to the optimal reﬁnancing rule, the NPV

rule generates expected discoun ted losses of over $85,000 on a $500,000 mortgage.

We solve an analytically tractable model of m ortga ge reﬁnancing. Our model

departs from existing analyses by making simplifying assumptions, but we sho w that

these simplifyin g assum p tions do not make a large diﬀerence to the results. We ﬁnd

that our closed-form calculations v er y closely match the nu m er ical results of Chen

and Ling (1989).

Our derived reﬁnancing rule tak es the following form: Reﬁnance when the current

mortg ag e inter est rate falls belo w the origina l mortgage int erest rate b y at least

[φ + W (−exp (−φ))] ,

where W(.) is the Lambert W -function,

ψ =

2(ρ + λ)

Optimal Mortgage Refinancing: A Closed Form Solution 30

φ =1+ψ (ρ + λ)

κ/M

(1 − τ)

ρ is the real discount rate, λ is the expected real rate of exogenous mortgage repay-

men t (including the eﬀects of mo ving, principal repaym ent, and inﬂation), σ is the

ann ual standard deviation of the mortgage rate, κ/M is the ratio of the tax-adjusted

reﬁnancing cost and the remaining value of the mortgage, and τ is the marginal tax

rate.

All of these variables are easy to calibrate, including λ. This variable can be

calibrated with the annual probability of relocating (μ), the ratio of total mortgage

paymen ts to the remaining value of the mortgage

nominal

, the initial mor tgage

interest rate (i

), and the current inﬂation rate (π):

λ = μ +

nominal

− i

+ π.

Equivalen tly, λ can be calculated b y using the remain ing ye ars left un til the mor tgag e

is fully repaid (Γ):

λ = μ +

exp(i

Γ) − 1

+ π.

We analyze both the exact solution of our mortgage reﬁnancing problem (above)

and a useful approximation to that solution. We sho w that a second-order Ta ylor

expansio n yields a square-root rule for optimal reﬁnancing: Reﬁnance when the

current mortg ag e in ter est rate falls below the original mortga ge interest rate b y at

least

σκ/M

(1 − τ)

2(ρ + λ).

Optimal Mortgage Refinancing: A Closed Form Solution 31

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Optimal Mortgage Refinancing: A Closed Form Solution 35

Appendix A: Partial deductibiity of points

Let κ(M)=F + fM,whereF denotes the ﬁxed cost of reﬁnancing and 100 × f

is the n umber of points. The expected arrival rate of a full deduc tib ility eve nt — a

mov e or a subsequent reﬁnancing — is θ.Attimet, the probability that such a full

deductibility ev ent has not y et occurred is e

−θt

. Thelikelihoodthatsuchanevent

occurs at date t is θe

−θt

Assume the term of the new mortgage is for N years. Each year, borro wers are

allowed to deduct amoun t

from their incom e, producing a tax reduction of

τfM

At the tim e of a full deductib ility even t, borro wers im m e diat ely deduct all of the

remaining undeducted points — i.e. they reduce their taxes by τfM

N−T

The real value of the deduction declines at the rate of inﬂation. H ence, the

payments are discoun ted eﬀectiv ely at the real discount rate r = ρ + π.

The present value of these tax beneﬁts is then:

−θt

−(ρ+π)t

τfM

dt +

θe

−θt

−(ρ+π)t

(τfM)

1 −

dt.

Using integ ratio n by parts, this simpliﬁes to

τfM

θ + ρ + π

∙µ

1 − e

−(θ+ρ+π)N

¶µ

ρ + π

θ + ρ + π

+ θ

Hence, total reﬁnancing costs κ(M) are given by

κ(M)=F + fM

∙

1 −

θ + ρ + π

∙µ

1 − e

−(θ+ρ+π)N

¶µ

ρ + π

θ + ρ + π

+ θ

¸¸

where κ(M) is deﬁned as the present value of the cost of reﬁnancing, net of future tax

beneﬁts. To calibrate this formula, set θ ' μ +0.10, where μ is the hazard rate of

mo ving and 0.10 is the (appro ximate) hazard rate of future reﬁnancing. The actual

hazard rate of reﬁnancing will be time-varying.

Optimal Mortgage Refinancing: A Closed Form Solution 36

Appendix B: Two lemmas

Lemm a 5. The boundary conditions for R are given b y

R(x

∗

)=R(0) − C(M) −

∗

ρ + λ

∗

)=−

ρ + λ

lim

x→∞

R(x)=0

Proof: We deriv e these from the boundary conditions on V. The value matching

and smooth pasting conditions at reﬁnancing boundary x

∗

are:

V (r

,r,π

,π)=V (r, r, π, π)+C(M).

ρ + λ

∂V (r

,r,π

,π)

∂r

Since V (r

,r,π

,π)=Z(r

,r,π

,π) − R(x

∗

), substitution into the ﬁrst equation im-

plies

Z(r

,r,π

,π) − R(x

∗

)=Z(r, r, π, π) − R(0) + C(M).

Rear rangin g the expression and simplifying the Z terms yields

R(x

∗

)=R(0) − C(M)+

− π + λ) M

ρ + λ

−

(r + λ) M

ρ + λ

= R(0) − C(M) −

∗

ρ + λ

The value matching equation states that the value of the program just before reﬁnanc-

ing, V (r

,r,π

,π), equals the sum of the value of the program just after reﬁnancing

and the cost of reﬁnancing, V (r, r, π, π)+C(M).

Changes in the interest rate (belo w the reﬁnancing poin t) do not change the

option value terms since the consumer is going to instantaneously reﬁnance an yw ay.

So a rise in the in terest rate only increases the NPV of future interest paymen ts.

This diﬀerential property must be con tinuous at the boundary (“smooth pasting”),

so R

∗

)=−

ρ+λ

The asymptotic boundary condition (for R)is

lim

∗

→∞

R(x

∗

)=0.

As the diﬀerence between the curren t nominal in terest rate and the original rate on

the mortgage grows beyond bound, the value of reﬁnancing goes to zero. ¤

Optimal Mortgage Refinancing: A Closed Form Solution 37

Lemm a 6.

If W is the Lambert W -function, then

− y = k

iﬀ

y = −k − W (−e

−k

Proof: Lambert’s W is the inverse function of f (x)=xe

, so

z = W(z)e

W (z)

Let z = −e

−k

, then

−k

= −W(−e

−k

W (−e

−k

)

Divide by e

W (−e

−k

)

and add k to yield

[

−k−W (−e

−k

)

]

−

−k − W (−e

−k

)

= k.

Hence, y = −k − W (−e

−k

) is the solution to e

− y = k. ¤

Appendix C: Formula for λ

Assum e that a mortgage is c h aracterized by a constant nominal payment, p, with a

nomina l in terest rate i

. The remaining nominal principal, N, is given by

N = −p + i

The boundary conditions are N(0) = N

and N(T)=0. The solution to this

diﬀerential equation is

N(t)=

−

exp(i

t).

Exploiting the boundary condition at T, we have

0=N(T )

−

exp(i

T ).

We are grateful to Fan Zhang for pointing this result out to us.

Optimal Mortgage Refinancing: A Closed Form Solution 38

This implies that the nominal pa ymen t stream is given by,

p =

1 − exp(−i

T )

We can also show that

N(t)

1 −

exp(i

1 − exp(i

[t − T ])

1 − exp(−i

T )

Hence,

N(t)

1 − exp(−i

T )

N(t)

1 − exp(i

[t − T ])

Sotherateofrealrepaymentatdatet is

λ = μ +

− i

+ π

= μ +

1 − exp(−i

Γ)

− i

+ π

= μ +

exp(i

Γ) − 1

+ π.

where μ is the hazard of moving, Γ is the n umber of remaining y ears on the mortgage,

and π is the current inﬂation rate.

Appendix D: Standard deviation calculations

Chen and Ling (1989)’s assumptions Chen and Ling (1989) assume that the

short rate x

follows the binomial process:

t+1

= 

t+1

where:



t+1

Uw/probπ

Dw/prob1 − π

Optimal Mortgage Refinancing: A Closed Form Solution 39

W ith constan t π, over time the logarithm of this ratio will follow a binomial distrib-

ution with an N-period mean of

μ = N [π ln (U)+(1− π)ln(D)]

and variance

= N

(ln (U) − ln (D))

π (1 − π)

The above expressions for μ and σ can be jointly solv ed for values of U and D in

terms of μ, σ, π,andN:

U =exp

σ (1 − π)

Nπ(1 − π)

and D =exp

−

σπ

Nπ(1 − π)

As N →∞, this log binomial distribution approac hes a log normal distribution.

Che n and Ling use this log normal approximation to calibrate values of μ and σ

from monthly data on three-month Treasury bills. They choose values for μ of -0.02,

0, and 0.02 and for σ of 5%, 15% and 25%.

They use the local expectations h ypothesis to compute the values of other securi-

ties as needed.

Current paper’s assumptions We assume that the 30-year m ortgage rate M

follows a driftless Brownian motion. We calibrate the varian ce with monthly data

on the ﬁrst diﬀerence of Freddie Mac’s 30-y ear mortgage rate series from 1971-2004,

ﬁnd ing an annu alized value of 0.000119.

Im plica tion s of Chen and Ling’s assum p tion s for the curre nt paper We

can use the log version of the expectations hypothesis to approximate the yield of

longer-term securities from Chen and Ling’s short-rate assum ptions.

For an y security of term s, the log yield of that security approximately satisﬁes:

ln x

+lnx

t+1

+lnx

t+2

+ ···+lnx

t+s−1

In each case, the superscript denotes the term of the security. Hence the log yield on

an s-period securit y is the a verage of the expected log yields on the future sequence

of s one-period securities.

Under Chen and Ling’s assumptions,

ln x

t+1

=lnx

+ln

t+1

Optimal Mortgage Refinancing: A Closed Form Solution 40

where

ln 

t+1

⎧

⎨

⎩

σ(1−π)

√

Nπ(1−π)

w/prob π

−

σπ

√

Nπ(1−π)

w/prob 1 − π

Hence:

ln x

t+i

=lnx

+ln

t+i−1

+ln

t+i−2

+ ···+ln

t+1

=lnx

j=1

ln 

t+j

and

ln x

t+i

= E

ln x

j=1

ln 

t+j

=lnx

j=1

ln 

t+j

Using the assum ptions above about how ln 

t+1

evolves,

ln 

t+j

= π

σ (1 − π)

Nπ(1 − π)

+(1− π)

−

σπ

Nπ(1 − π)

Th us:

ln x

t+i

=lnx

j=1

=lnx

+ i

This implies:

ln x

···+

ln x

+(s − 1)

´´

=lnx

s−1

k=1

=lnx

s(s − 1)

=lnx

(s − 1)

Th us the ﬁrst diﬀerence of the lev el of the yield is:

t+1

− x

= e

(s−1)

t+1

− x

≡ K

t+1

− x

Optimal Mortgage Refinancing: A Closed Form Solution 41

Deﬁne ∆x

t+1

≡ x

t+1

− x

.Then

∆x

t+1

= E

t+1

− x

= KE

t+1

− x

= KE

((

t+1

− 1)x

)

= Kx

π exp

σ (1 − π)

Nπ(1 − π)

+(1− π)exp

−

σπ

Nπ(1 − π)

and:

Var

∆x

t+1

= Var

t+1

− x

= K

Var

((

t+1

− 1)x

)

= K

Var

(

t+1

)

= K

π(1 − π)exp

ÃÃ

σ (1 − π)

Nπ(1 − π)

− exp

−

σπ

Nπ(1 − π)

Assum e π =

. N =12. Although Chen and Ling assume sev eral diﬀerent

values of μ, for our o w n speciﬁcation we assume lack of drift. Setting μ =0and

π =

implies K =1and simpliﬁes the expr essions for the conditional m ean and

variance considerably:

∆x

t+1

= x

exp

√

+exp−

√

− 1

Var

∆x

t+1

exp

2σ

√

+exp−

2σ

√

−

Note that, given the absence of drift, these expressions do not depend on the term s

of the securit y.

Chen and Ling start their short rate at x

=0.08. For the values of σ = {0.05, 0.15, 0.25}

assumed b y Chen and Ling, the corresponding mean , variance, and standard devia-

tion for the 30-y ear mortgage rate, annualized, are then:

σ Mean Variance Standard Deviation

0.05 0.0001 0.000016 0.0040

0.15 0.0009 0.000144 0.0120

0.25 0.0025 0.000401 0.0200