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Mil
118
THE
AFTER-TAX
MORTGAGE
REFINANCING
MODEL:
CLOSED-FORM
/'
SOLUTION
AND
ANALYSIS
(.._)
Richard
A.
Fo
ll
owill
Linda L. Johnson
Introduction
The
use
of
a before-tax, cash-flO\v decision model rather than an after-
tax, cash-flow decision model to evaluate a mortgage loan
refinancing deci-
sion may lead investors to make either subopti
mal
or
wealth-reducing deci-
sions. This paper expands the before-tax mortgage refinancing decision
model
to both open and closed-forms
of
an after-tax decision model. A compari-
son
of
net present valu
es
derived from both the before-tax and after-tax de-
cision models shows how the two procedures may yield conflicting
recommendations for refinancing a mortgage even under condiuons
of
cer-
tainty. Graphical analysis highlight the net present
val
ue comparisons and
the sensitivity
of
net present value to chan
ges
in marginal tax rates and time
remaining on the mortgage loan.
Lastly, important 1mplicauons
of
the after-
tax refinan
ci
ng model are summarized and discu sed.
Except for the complexity
of
monthly principal amortization, the decision
"hether
to refinance a mortgage
1s
similar
to
the bond refunding decision.
Unlike the bond refunding deci 10n,
,vh1ch
has received extensi\e coverage
in
the
fi
nance literature (Ang, 1975 and 1978; Bowlin, 1966; Emer
y,
1978;
Kalotay, 1978; Livingston, 1980; Ofer and Taggart, 1980; Siegel, 1984; Sir-
mans and Jaffe,
1988;
Yaw
Ill
and Ander<,on, 1977;
Z1ese
and Taylor, 1977),
most finance and real estate in\estment textbooks
d1~cuss
the mortgage
refinancing decision in either a cursory manner
or
fai
l to present a model
of
the decision process (Allen, 1989; Brueggeman, Fisher and Stone, I 989;
Den111s,
1989; Epley and Millar, 1991; irmans and Jaffe, 1988; Sirota, 1989;
Unger and Melicher, 1989; Wiedemer, 1990). For example, the Brueggeman,
Fisher, and Stone
text (pp. 448-450), a widel, adopted real estate finance
text,
o
nl
, addresses the refinancing decision from a before-tax, discounted
cash flow perspective.
In the literature, H
endersho11,
Hu and
Yilla111
(I 983) consider only the after-
tax benefib
of
interest savings, ignoring the substanual changes in the size
and timing
of
incremental payments on principal and, therefore, misspecify
the beneficial net cash
fl
ows
of
refinancing. Fo
ll
a
in
and Tzang ( 1988) and
Fo
ll
O
\\
i
ll
and Johnson ( 1989) properl) specify the net after-tax cash flows
but present only open-form versions
of
the mortgage refinan
ci
ng model.
Th
is
paper more completely a nal
yzes
the mortgage r
ef
ina n
ci
ng d
eci
ion
by developing a
cl
o ed-fo
rm
decision model that
ex
plic
it
ly
considers taxe .
Exampl
es
are prese nted
wh
ic
h contrast the before-tax and
af
ter-tax models
and illustrate that re
li
ance on the curre
nt
pre-tax methodology can e
li
c
it
de-
cisions w
hi
ch reduce wealth.
The
After
-T
ax
Mortgag
e Re
financin
g
Model
From
a before-tax perspective, the net present value
of
refinancing a
mort-
gage
is
relatively simple
to
calculate as follows:
NPV
n
-CF
0
+ !
t=l
(l+k)t
(I)
where PV is
the
net present value
of
replacing
the
existing
mortgage,
CF
0
is
the present value
of
the
initial cash flow required
to
refinance, Pc
is
the
current
mortgage
payment,
and
P,
is
the new
mortgage
payment
for a
replacement mortgage with a term equal
to
the
remaining
term
of
the
c
ur
-
r
ent
mortga
ge.
The
appropriate
discount r
ate
is
denoted
as
k,
and
n
is
the
number
of
mortgage
payments remaining.
A
more
co
rr
ect
approach
to
modeling the refinancing decision considers
after-tax cash flows, recognizing the tax deductibility
of
interest paid
on
both
the current
and
replacement
loan
. Additionally,
the
model
should
recognize
that
the relevant
holding
period
may
be less
than
the
entire
remaining
term
of
the
mortgage.
The
current
mortgage
payment
i
comp
ri
sed
of
two
components,
a pay-
ment
on
the loan
balance
and
an interest
payment.
(2)
where
!\B
e.,
= the
portion
of
the t
1
h
payment
dire
cted
toward
reducing the
current
mortgage
loan
balan
ce;
l
e.,
= the interest
payment
based
on
the balance
of
the
current
mortga
ge
loan;
and
t = I , 2, 3,
...
, n.
The
amount
by which the
current
mort
gage
loan
bala
nce is redu
ced
by the
tth
pa
yment
1s
given by
(3)
where ic
is
the
current
mortgage
interest r
ate.
S
ub
st
ituting
e
qu
ation
(3)
into
equation
(2)
and
rearranging
th
e terms give
an
expression for the interest
co
mponent
of
the
t
th
c
urrent
mort
gage
payment,
2
(4)
Similarly, each payment on the proposed replacement mortgage
is
com-
prised
of
an interest payment component and a debt reduction component:
(5)
where tiB,.
1
is the portion
of
the
(lh
payment directed toward reducing the
replacement mortgage loan balance, and
1,
,
1
is
the
in
terest payment based
on the balance
of
the replacement mortgage loan.
The debt reduction component
of
the
(lh
payment
of
the replacement loan
is
defined by
tiB
,.
1
=
P,
(6)
( I
+i,)
n-1+ 1
where i, is the replacement mortgage interest rate.
The
interest payment component
of
the
t1
h payment on the replacement
mortgage loan
1s
1
,,
1
= P, -
P,
(7)
(I+
i,
)n
-1
+ I
Changing the pre-tax methodology
of
equation
(I)
to renect the deducti-
biltty
of
interest payments yields the follo,1.ing open-form, after-tax, net
pre ent value model:
m
-Cr
0
+ 1
t=I
(1
,,
1
- 1,.,
)(
I
-T)
+
(tiB
,
-tiB
,,
1
)
'
( I +
J..)
1
(8)
where
Ti~
the marginal tax rate and m represents the number
of
paymencs
remaining m the relevant holding period.I
For computauonal .
1mpli
c
11
y,
the model assumes a precomputed pre ent
value amount
of
refinancing co ts,
CF
0
,
a
nd
that a mortgage having n pay-
ments remaining
is
repla
ce
d by an n-payment mortgage. The relevant hold-
ing period, m, however, may be less than the full term
of
the mortgage (i.e.,
m n).
The after-tax, open-form model depicted by equati
on
(8)
cannot
be
com-
puted by discounting a constant annuity payment as in the before-tax model
given by equation
(I).
Becau
se
the principal amortization and intere
lex-
pen
se
portions change with each payment, the solution
10
the after-tax, open-
form model given by equation
(8)
can require a lar
ge
number
of
repetitive
3
calculations. The after-tax refinancing model
of
equation (8) can be used
more efficiently if
it
is
expressed as a closed-form equation that can be solved
with a hand-held calculator.
The closed-form
of
equation (8)
is:
-CF
o +
(1-T)(P
c
-P,)
[
(l+k)
m
-J]
+
(I
+k)
mk
TP,[(I
+ k)m-(1 +i,)m]
(I+ 1,)n(l +
k)m(k-1
,)
TPc[(l
+k)
m
-(1
+()
m]
(I
+!J"(l
+k)m(k-iJ
(9)
All
parameter~ are
as
previous!) defined, and the discount rate, k,
is
not equal
10
i, or
i,
Complete development
of
the after-tax, clo\ed-form refinancing
de-
cision
model
given
b:r
equation (9)
1s
presented
in
the Appendix.
When
k equals 1 , the after-tax, clo\ed-form equation
I\
\IP\=
-CF
> +
(1-T)(P
,-
P,) [
(11-k)m-I
(1-t-k)mk
_
TP
,[(l
+k)
m
-(J
+i
,)n]
(I+
i,)
0
(
I+
k)"'(k - i,)
+
mTP
,
(l+k)
n+
I
When
k equab
i,.
the after-tax,
clo~ed
-form equauon
1\
(10)
(1
+
k)rn-1
l\lP\
=
-CF
+
(1-T)(P
c-
P,) [ I +
(I
+k)mk
TPJ(l
+
k)rn-(1
+1
)m]
(I +l.}n(l
+k)nt(k-IJ
mTP,
(11)
(1
+k)n
+I
'\
et Pr
ese
nt
\ al
ue
Com
pa
ri
so
n,
A comparison
of
net
pre~ent
values generated
by
the pre-lax model and
1he
af!er-tax
model
for
a h:rpotheucal refinancing example
1llustra1es
that ignoring
the
ta,
deduc11b1ht,
of
interest payments can
result
in
an erroneom dem1on. The
example
u~es
a
30
year, monthly payment mortgage loan
with
an original balance
of
$100,000, a current interN rate
of
J0<r'o,
and
fixed
monthly payment\
of
$877 .57
.2
A ne\\ interest rate
of
8<r'o
can be obtained
by
refinancing at
an
after-
tax pre ent value cost
of
$4,000 imposed for points, processing
fees,
appraisal
fees,
and title search costs.
We
arbitrarily
use
a discount rate,
k,
equal to the ne\\ interest rate
of
807o,
1
and
assume
1hat
the
relevant
holding
period
is
equal 10 t
he
fu
U term
of
the
rep
l
ace-
4
Both
before-tax and
af
ter-tax net present values are positive
when
T
is
.4
5
and the remaining term,
mor
n,
is
150 months. Such a
low
after-tax
net
prese
nt
value
as
$47,
however, may correctly
inOuence
a borrower to defer refinancing
with
the expectation that rates
will
decline funher. Vie\\ed \\ithin the context
of
option
pricing
theory,
refi
nancing a mongage involves exercising an option
to refinance
with
an
exercise
price
equal to
CF
0
•
Such an option
is
m-the-mon
ey
when
PV
is
positive. A borrower choosing to exercise this option
receive~
both
the
PV
of
refinancing and another
less
valuable out-of-the-money option to
refinance.
4
The optimal time to refinance a mongage docs not occur before the
combined
value
of
the after-tax
PV
and the ne\\ option
exceeds
the value of
the current, in-the-money option to refinance. Thus, a
positive
P\
docs not
ncces
arily warrant refinancmg.
In
our example, a borrower
who
refinances on
the
bcbis
of the before-tax
net
present value of
$4,068
at
150
months remaining
may
have
been
induced
to
exercise
his
refinancing opuon prematurel
y.
Table I shows that both before-tax and after-ta\
net
present values are
po~i
-
ti\e
for loam
\\
ith at least
150
months remaming. Conversely, for loans
with
90
months or les, remammg, both before-ta\ and after-tax
net
present
values
arc negative,
indi
cating that refinancing
is
not advisable
Figure I prO\idcs a graphical representation
of
the net
prescnr
values
of
the
mongage refinancing example depicted
by
Table I Figure
la
clearl}
illu
strates
the
positive, non-linear relationship bemeen net present \alue and time remain-
ing
on the mongage loan. It also
shO\\S
that
net
present \alues increase
\1ith
the holding period, but at a decreasing rate.
For
this
example, Figure
la
indicates that for a tax rate
of
45,
an erroneous
decision
to refinance
1s
reached
by
using a before-tax decision model
(T
=
0)
\I
hen
the remaining monj?age term
1s
greater than
90
months but
less
than
150
months.
An
erroneous
dcc1s1on
to
refinance
1s
more
likely
when a before-tax
de-
cbion
model
i,
used
by
borrowers
in
higher tax brackets.
Figure
lb
shows
the relation,hip
between
tax rates and
net
present
values
gener-
ated
by
equa11on
(
11)
for
a mongage loan
with
various remammg term,. The
graph indicates that
use
of
a before-tax model routine(;
yields
up\\ardly biased
net
present
values.
The relationship
between
net
present values and tax rates
ts
linear and in\er
se;
for any
given
n. 1
0
•
1 . and
k.
net
present value declines
b}
a constant amount for equal percentage
mcreases
m the tax rate. Moreover, the
rate
of
decline
in
net
present value increases as
11me
remaining on the mongage
loan
increcbes.
This result implies that the pre-tax cash now
dec1s1on
model
produces
the greatest errors m net present value
for
borrowers m high tax brack-
ets
with
long expected holding periods.
\\'hilc the refinancing example may
be
modified
by
changing the amount
of
the original mongage, the payment frequency, the term
of
the
rep
lacement mon-
gage
and holding period, the discount rate, the cost to refinance, and the current
and
rep
lacement mongage interest rates, the behavior
of
the
resu
lt
ing
net present
val
ues
ts
consistent
with
that depicted by Table I and
Fi
gu
res
I a and I b. T
hu
s,
an
analysis
of
net present val
ue
r
esu
lt
s for not only t
hi
s exam
pl
e, but for other
mongage loan r
efi
nan
ci
ng exa
mp
les
as
we
ll,
i
nd
ica
t
es
that the u
se
of
a befor
e-
tax refinancing model can eas
il
y
lea
d to an incor
rect
r
efi
nan
ci
ng
dec
ision.
6
S
umm
ar)
Although
ir
is
more
appropriare
to
analyze after-tax
ca~h
now~
relevant
10
mongage
refinancing,
most
textbooks
st
ill
present a refinancing
decision
mode
l
thar
analyzes
pre-tax
cash
now
s.
This
paper develops both an open-form
and
a
closed-form
after-tax
decision
model
and compares before-tax and after
-tax
net
present
values
for
a
hyporherical
refinancing example. Specilic
insrances
are
identified
that
demonstrate
rhat
rhe
use
of
a before-tax
dec1s1on
model
may
lead
to
incorrect
refinancing
decisions.
Graphical
analy
sis
funher shows that
the
differ-
ence
between
before-tax
and
after-tax
net
pre ent values and, therefore,
the
pos-
s
ibility
of
an
erroneous
decision,
increases
wirh
tax rates and the
relinancing
holding
period.
Appenfa
The
closed-form,
after-rax,
refinancing
model
i~
developed b)
irutially
recog-
nizing
that
the
present
value
of
the after-tax difference berwecn cur
rent
and
replacement
mongage
loan
payments
is
rn
2
t=l
(l
,_,
-I
,_.)
(1-T)
+
(liB
,.
1
-tiB
,.
1
)
(IH)c
Sub
tituting
equations
(3), (4), (6). and (7) into expression
(A
I)
yields
(Al)
P,
1)(1
-
T)+
pc
(~+~
--
()-l-1
)·
1
•+I
__
()
_+_
1,)_"·_•_~
which
reduce
10
m
2
(=)
or
(P
,
-P,)(1-T)
+
(l+k)'
(IH
..
)•
m m
(P
,
-P
,)(
1-T)
2
(IH:)
c + TP, 2
[=!
t=
I
m
TP
,
-TP
, 2
_____
(A2)
I=
I
(I+
i,)n-c+
I(
)+
k)t
8
The
closed-form for
the
first series
of
expression (A2), available from any basic
financial management text,
is
cPc-P,)(1-T)
(I
+k)"'
-
(I
+k)mk
(A3)
The
closed-form
of
the second series
of
expression (A2)
is
found by a straight-
forward application
of
the following relationship that gives the closed-form so-
lution for a j-term geometric series having an initial term, a,
and
a common ratio,
r.
j-1
L
ar'
= a
t=O
1
-rJ
1
-r
For
exposition the second series
of
expression (A2} i expanded as follows:
m
L
t=I
+
...
+
TP
c
TP
c
T Pc
+
_____
_
(l+iJn(I+k)
(l+lc)n
-t
(I+k)1
The
common
ratio for the sene
is
(I+
iJI(
I+
k); its closed-form
is
m
TP
, L
t=I
(l+t
c
}"
1+l
(l+k)t
TPcl(l+k)m
- (l +1Jm]
(I +i.:)"(I
+k)m(k-i,)
TP,
(I
+iJn(I
+k)
l-[(l+1J
(l+k)]m
I
-(I
+iJ
l (I +
k)
(A4)
A ,,rnilar procedure produce the closed-form olution
of
the third ene
of
e-.-
pres ion (A2}.
m
TP,
L
t=
I
(I
+i,)n
-t+
l( J
+k)t
9
TP,((I +
k)m
-
(I+
i,
)m]
k i i, (AS)
( I +
i,)"( I +
k)m(k
+ i,)
Summing
(A3),
(A
4), and
(AS)
yie
lds the closed-form equivalent to the
series
expressed
by
(Al). Substituting into equation
(8)
yield
the closed-form, after-
tax solution model, equation (9),
PV
= -
CFo
+
(1-l)(P
c
-P,)
(I+
k)m-
I ) + TPJ(l
+k)m-(1
+iJm]
(I
H)mk
(I +~)n(t
+k)m(k-iJ
TP,[(l
H)m-(1
+i,)m)
(I
+i
,
)n((
+k)m(k-i,)
E
ndn
ot
es
(A9)
1
The calculation
of
the
present
value
cash
flows
to
refinance,
CF
0
, should also
recognize
the tax deductibility
of
refinancing points which are amortized over
the
life
of
the ne\\ loan, unamortized points at the end
of
the holding period,
m, and prepayment penalties. The
initial
payment,
CF
0
,
is
calculated
as
follows:
M
CF
0
= P -
P·T
1(l+K)
- 1 -
LP-T(l+K)'t
+
PP(l-T)
+ FC
t=l
\\here
p
UP
:-.-1
T=
pp=
FC =
K =
points charged to refinance,
unamortized pomt ,
the
ne\\
mortgage term
in
years,
the holding period
in
years,
the marginal tax rate,
the deductible
prepayment penalty
other non-deductible financing
com,
and
the discount rate expressed
an effective annual rate.
2Monthly payment mortgage
rate~
are
generall}
stated as annual interest rates
compounded monthly.
J
Bond
refunding lnerature
extensively
discusses the appropriate discount rate
10
use
.
AJ1crnat1\e~
include the market rate, an after-tax market rate, or the cur-
rent
rate.
4
A discussion
of
the refinancing decision as an
op11on
pricing problem
is
provided by
Siegel,
(1984).
Referen
ces
Allen, Roger H. Re
al
E.s
tat
e
lm
es
tm
e
nt
S
trate
g
)'
. 3rd ed. Cincinnati: South-
Western Publishing Co., 1
989.
10
Ang, James, S.
"The
Two Faces
of
Bond Refunding," The
Jo
urn
al or Fm
ance
(June
19
75):869-874.
Ang, James S.
"The
Two
Faces
of
Bond
Refunding: Reply,"
Th
e Journal or
Finance (March
1978):354-356.
Bowlin, Oswald
D.
"The
Refunding
Decision:
Another Special Case
in
Capital
Budgeting," The Journal or
Finance
(March
1966):55-68.
Brueggeman,
William
B., Jeffrey
D.
Fisher and
Leo
D.
Stone.
Real
Estate
Fmance
.
8th ed. Homewood, Illinois: Richard
D.
Irwin,
In
c.,
1989.
Dennis, Marshall W. Res
id
ential Mortgage
Lending
. 2nd ed. Englewood Cliffs,
J: Prentice-Hall, Inc.,
1989.
Emery, Douglas
R.
"Overlapping Interest in Bond Refunding: A Reconsidera-
tion," Financial Management (Summer
1978):
19-20.
Epley, Donald
R.
and James
A.
Millar.
Basi
c
Real
Estate
Finance
and
Imest-
me
nu.
. 3rd ed.
ew
York: John
Wile}
& Sons, 1
991.
Follain,
Jam~
R.
and Dah-Nein Tzang. "Interest Rate Differential and
Refinanc-
ing
a Home Mortgage," The Appraisal Journal (April
1988):243-251.
Followtll, Richard
A.
and Linda
L.
Johnson.
"Taxe&
and Mortgage Refinanc-
ing,"
1l1e
Appraisal Journal (April
1989):
197
-
206.
Hendershott, P., S. Hu and K. Villani.
"The
Economic
of
1ortgage Termina-
tions: Implications for Mortgage Lenders and Mortgage Terms,"
Housin
g
Finan
ce
Re
,i
en
(April
1983):
127-142.
Kalotay, A.
J.
"On
the Advanced Refunding
of
Discounted Debt,"
Financial
Management (Summer
1978):
14-18.
Llvingsion,
Miles
. "Bond Refunding
Recon&idcrcd:
Comment,"
Th
e Journal or
Finance (March
1980):191
-
195.
Ofer,
A.
R.
and
R.
A. Taggart. "Bond Refunding Recon\idered: Reply,"
Th
e
Journal of Finance (March
1980):
196
-
200.
Siegel,
Jeremy J. "The Mortgage Refinancing
Dec1
10n," H
ous
in
g
Finance
Re,ien
(January
1984):91-97.
Sirmans, C. F. and Austin J. Jaffe.
Th
e
Co
mplet
e
Real
~
lat
e lm~tme
nt
Hand
-
book
. 4th ed. Englewood Cliffs, J: Prentice-Hall, Inc.,
1988.
11
I
•
Sirota, David.
Es.sentials
of
Real
E'.M
ate finan
ce.
5th ed. Chicago: Real Estate
Education Co.,
1989.
Unger, Maurice A. and Ronald
W.
Melicher. R
ea
l
E.s
tate f
in
ance. 3rd ed. Cin-
cinnati: South-Western Publishing Co.,
1989.
Wiedemer,
John P. R
ea
l
E'.Ma
tc finan
ce.
6th
ed.
Englewood Cliffs, J: Premice-
Hall, Inc.,
1990.
Yawitz,
Je s
B.
and James
A.
Anderson
"The
Effect
of
Bond Refunding on
Shareholder Wealth,"
Th
e Jo
urn
al of
fi
nance {December
1977):
1738-1746.
Ziese, Charles H. and Roger
K.
Taylor "Advance Refunding· A Practitioner's
Perspec11ve,"
f
in
ancial Management {Summer
1977):73-76
Richard A.
Follow11l
I an
A~s1stant
Professor
of
Finance, John A
\\
alker Col-
lege
of
Business
at Appalachian State Umversny. Linda
L.
Johnson
1s
an
As-
oc1ate
Prof~
or
of
finance, College
of
Busine~s
at Um\er~ny
of
South Flonda
at Fort
Myer .
12