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Mor
tgag
e M
ath
Wha
t is
PV o
f $10
00 p
er m
onth
for 1
5 m
onth
s pl
us $
10,0
00 p
aid
15 m
onth
s fro
m
now
at 1
0% n
omin
al a
nnua
l int
eres
t?
= (1
4.04
5)10
00 +
(0.8
830)
1000
0 =
$14,
045
+ $8
,830
= (P
VIFA
.008
33,1
5)*P
MT
+ (
PVIF
.008
33,1
5)*F
V
()
()15
1512
10.1
000
,10$
1210.
100
0,1$
1210.
100
0,1$
875
,22$
++
++
++
=L
(With
cal
cula
tor s
et to
pm
ts a
t “EN
D”o
f per
iods
, an
d P/
YR=1
2…)
Mor
tgag
e M
ath
Key
s:D
CF
Key
s:15
---->
N
key
10---
-> I
/YR
key
10---
-> I
/YR
key
0 --
--> C
Fj k
ey10
00---
-> P
MT
key
1000
---->
CFj
key
1000
0----
> F
V k
ey14
---->
Nj
key
PV---
-> -
22,8
7511
000-
--->C
Fj k
eyN
PV---
-> 2
2,87
5
How
the
Cal
cula
tor "
Mor
tgag
e M
ath"
K
eys
Wor
k. .
.Th
e fiv
e "m
ortg
age
mat
h" k
eys
on y
our
calc
ulat
or (N
,I,PV
,PM
T,FV
) sol
ve:
()
()
()N
Nr
FVr
PMT
rPM
Tr
PMT
PV+
++
++
++
++
−=
11
11
02
L
or:0
= -P
V +
(PVI
FAr,N
)*PM
T +
(P
VIF r,N
)*FV
whe
re:
r = i
/ m,
whe
re:
i = N
omin
al a
nnua
l int
eres
t rat
e m
= N
umbe
r of p
aym
ent
perio
ds p
er y
ear (
mP/
YR).
Exam
ple:
10
%, 2
0-yr
fully
-am
ortiz
ing
mor
tgag
e w
ith p
aym
ents
of
$10
00/m
onth
.Th
e ca
lcul
ator
sol
ves
the
follo
win
g eq
uatio
n fo
r PV:
The
resu
lt is
: PV
= 10
3625
.
()
()
()24
024
02
0083
3.1
000
833
.110
0000
833
.110
0000
833
.11000
0+
++
++
−=
LPV
THE
BA
SIC
RU
LES
OF
CA
LCU
LATI
NG
LO
AN
PA
YMEN
TS &
BA
LAN
CES
Let: P
= In
itial
Con
tract
Prin
cipa
l (Lo
an B
alan
ce a
t tim
e ze
ro, w
hen
mon
ey is
bor
row
ed)
r t=
Con
tract
Inte
rest
rate
(per
pay
men
t per
iod,
e.g
., =i
/m) a
pplic
able
for p
aym
ent i
n Pe
riod
"t“IE
t=
Inte
rest
por
tion
of p
aym
ent i
n Pe
riod
"t“PP
t=
Prin
cipa
l pai
d do
wn
("am
ortiz
ed")
in th
e Pe
riod
"t" p
aym
ent
OLB
t=
Out
stan
ding
loan
bal
ance
afte
rthe
Per
iod
"t"
paym
ent h
as b
een
mad
ePM
T t=
Amou
nt o
f the
loan
pay
men
t in
Perio
d "t“
THE
FOU
R B
ASI
C R
ULE
S:1)
IEt
=r t(
OLB
t-1)
2)PP
t=
PMT t
–IE
t
3)O
LBt
=O
LBt-1
-PP
tEq
uiva
lent
to P
V of
rem
aini
ng lo
an p
aym
ents
4)O
LB0
= P
Know
how
to u
se th
ese
rule
s so
that
you
can
ca
lcul
ate
paym
ent s
ched
ule,
inte
rest
, prin
cipa
l, an
d ou
tsta
ndin
g ba
lanc
e af
ter e
ach
paym
ent,
for
any
type
of l
oan
that
can
be
drea
med
up!
APP
LIC
ATI
ON
OF
THE
FOU
R R
ULE
S TO
SP
ECIF
IC L
OA
N T
YPES
1)Fi
xed-
Rat
e lo
ans
(FR
Ms)
:Th
e co
ntra
ct in
tere
st ra
te is
con
stan
t th
roug
hout
the
life
of th
e lo
an:
r t=r,
all t
.
2) C
onst
ant-P
aym
ent l
oans
(CPM
s):
The
paym
ent i
s co
nsta
nt th
roug
hout
the
life
of th
e lo
an:
PMT t
=PM
T, a
ll t.
3) C
onst
ant-A
mor
tizat
ion
loan
s (C
AMs)
:Th
e pr
inci
pal a
mor
tizat
ion
is c
onst
ant t
hrou
ghou
t the
lif
e of
the
loan
: PP
t=PP
, all
t.
4) F
ully
-Am
ortiz
ing
loan
s:In
itial
con
tract
prin
cipa
l is
fully
pai
d of
f by
mat
urity
of
loan
:ΣP
P t=P
over
all
t=1,
…,N
.
5) P
artia
lly-A
mor
tizin
g lo
ans:
Loan
prin
cipa
l not
fully
pai
d do
wn
by d
ue d
ate
of
loan
:ΣP
P t<P,
so
OLB
Nm
ust b
e pa
id a
s “b
allo
on”a
t mat
urity
.
6) In
tere
st-O
nly
loan
s:Th
e pr
inci
pal i
s no
t pai
d do
wn
until
the
end:
PMT t=
IEt,
all t
(equ
ival
ently
:OLB
t=P,
all
t, an
d in
cal
cula
tor e
quat
ion:
FV
= -P
V).
7) G
radu
ated
Pay
men
t loa
ns (G
PMs)
:Th
e in
itial
pay
men
t is
low
, usu
ally
initi
al P
MT 1
< IE
1, so
OLB
at f
irst g
row
s ov
er ti
me
(“neg
ativ
e am
ortiz
atio
n”),
follo
wed
by
high
er p
aym
ents
sc
hedu
led
late
r in
the
life
of th
e lo
an.
8) A
djus
tabl
e-R
ate
loan
s (A
RM
s):
The
cont
ract
inte
rest
rate
var
ies
over
tim
e (r t
not c
onst
ant,
not k
now
n fo
r cer
tain
in
adva
nce,
loan
pay
men
t sch
edul
es &
ex
pect
ed y
ield
s m
ust b
e ba
sed
on
assu
mpt
ions
abo
ut fu
ture
inte
rest
rate
s).
Cla
ssic
al F
ixed
-Rat
e M
ortg
age
The
“cla
ssic
al”m
ortg
age
is b
oth
FRM
&
CPM
:PM
T =
P/(P
VIFA
r,N) =
P /
[(1 –
1/(1
+r)N
)/r]
$60,
000,
12%
, 30-
year
CPM
...
MO
NTH
BEG
. BA
L.IN
TER
EST
PMT
PRIN
END
BA
L.
1$6
0,00
0.00
$6
00.0
0 $6
17.1
7 $1
7.17
$5
9,98
2.83
2$5
9,98
2.83
$5
99.8
3 $6
17.1
7 $1
7.34
$5
9,96
5.49
3$5
9,96
5.49
$5
99.6
5 $6
17.1
7 $1
7.51
$5
9,94
7.98
Usi
ng Y
our C
alcu
lato
r
1)C
alcu
late
Loa
n Pa
ymen
ts:
Exam
ple:
$100
,000
30-
year
10%
mor
tgag
e w
ith m
onth
ly p
aym
ents
:36
0----
> N
10---
-> I
/YR
1000
00---
-> P
V0
---->
FV
PMT-
--->
-877
.57
2) C
alcu
late
Loa
n A
mou
nt
(Affo
rdab
ility
):Ex
ampl
e:Yo
u ca
n af
ford
$50
0/m
onth
pa
ymen
ts o
n 30
-yea
r, 10
% m
ortg
age:
360-
--->
N10
---->
I/Y
R50
0----
> P
MT
0----
> F
VPV
---->
-56
,975
.41
= Am
t you
can
bor
row
.
•If
you
can
bor
row
er 8
0% o
f hou
se v
alue
, ho
w m
uch
can
you
affo
rd to
pur
chas
e?–
Purc
hase
Pric
e =
$56,
975
/ 0.8
0–
Purc
hase
Pric
e =
$71,
218
3) C
alcu
late
Out
stan
ding
Loa
n B
alan
ce:
Exam
ple:
Wha
t is
the
rem
aini
ng b
alan
ce o
n $1
00,0
00, 1
0%, 3
0-ye
ar, m
onth
ly-p
aym
ent l
oan
afte
r 5 y
ears
(afte
r 60
paym
ents
hav
e be
en
mad
e)?
Firs
t get
loan
term
s in
the
regi
ster
s:36
0----
> N
10---
-> I
/YR
1000
00---
-> P
V0-
--->
FV
PMT-
--->
-877
.57
Then
cal
cula
te re
mai
ning
bal
ance
eith
er w
ay b
elow
:N
---->
60
N---
-> 3
00FV
---->
-96
,574
.32
PV---
-> 9
6,57
4.32
4) C
alcu
late
pay
men
ts &
bal
loon
on
part
ially
am
ortiz
ing
loan
:Sa
me
as (3
) abo
ve.
5) C
alcu
late
the
paym
ents
on
an in
tere
st-
only
loan
:Ex
ampl
e:A
$100
,000
inte
rest
-onl
y 10
% lo
an w
ith
mon
thly
pay
men
ts:
N
can
be a
nyth
ing,
10---
> I/Y
R,
1000
00---
> PV
, -1
0000
0--->
FV,
PMT
--->
-833
.33
6) M
eet a
fford
abili
ty c
onst
rain
t by
trad
ing
off
paym
ent a
mou
nt w
ith a
mor
tizat
ion
rate
:Ex
ampl
e:G
o ba
ck to
exa
mpl
e #2
on
the
prev
ious
pa
ge. T
he a
fford
abilit
y co
nstra
int w
as a
$50
0/m
o pa
ymen
t lim
it. S
uppo
se th
e $5
6,97
5 w
hich
can
be
borro
wed
at 1
0% w
ith a
30-
year
am
ortiz
atio
n sc
hedu
le fa
lls s
hort
of w
hat t
he b
orro
wer
nee
ds.
How
muc
h sl
ower
am
ortiz
atio
n ra
te w
ould
ena
ble
the
borro
wer
to o
btai
n $5
8,00
0?
Ente
r: I/YR
= 1
0, P
V =
-580
00, P
MT
= 50
0, F
V =
0,
Com
pute
: N =
410
. Th
us, t
he a
mor
tizat
ion
rate
wou
ld h
ave
to b
e 41
0 m
onth
s, o
r 34
year
s.
Not
e:Th
is d
oes
not m
ean
loan
wou
ld h
ave
to
have
a 3
4-ye
ar m
atur
ity, i
t cou
ld s
till b
e a
30-
year
par
tially
-am
ortiz
ing
loan
, with
bal
loon
of
$20
,325
due
afte
r 30
year
s.
7) D
eter
min
ing
prin
cipa
l & in
tere
st
com
pone
nts
of p
aym
ents
:Ex
ampl
e:Fo
r the
$10
0,00
0, 3
0-ye
ar, 1
0% m
ortg
age
in p
robl
em #
1 on
the
prev
ious
pag
e, b
reak
out
the
com
pone
nts
of th
e 12
pay
men
ts n
umbe
ring
50
thro
ugh
61.
In th
e H
P-10
B, a
fter e
nter
ing
the
loan
as
in p
robl
em #
1,
ente
r:50
, IN
PUT,
61,
AM
OR
T, =
$9,
696.
06in
t, =
$834
.80
prin
, =$
96,5
01 O
LB61
.To
get
the
corre
spon
ding
val
ues
for t
he s
ubse
quen
t ca
lend
ar y
ear,
pres
s AM
OR
T ag
ain,
to g
et: =
$9,
608.
65in
t, =
$922
.21
prin
, =$9
5,57
9 O
LB73
.(O
ther
bus
ines
s ca
lcul
ator
s ca
n do
this
too.
)
Loan
Yie
lds
and
Mor
tgag
e Va
luat
ion
Loan
Yie
ld =
Effe
ctiv
e In
tere
st R
ate
Yiel
d =
IRR
of l
oan
Rec
all:
IRR
bas
ed o
n ca
sh fl
ows.
Usi
ng c
alcu
lato
r equ
atio
n:
()
()
()N
Nr
FVr
PMT
rPM
Tr
PMT
PV+
++
++
++
++
−=
11
11
02
L
Let: PV
= C
F 0PM
T= C
F t,
t=1,
2,...
,N-1
PMT
+ FV
= C
F NN
= H
oldi
ng P
erio
dw
here
:CF j
repr
esen
ts a
ctua
l cas
h flo
w a
t en
d of
per
iod
"j".
Then
, by
the
defin
ition
of "
r" in
the
equa
tion
abov
e, w
e ha
ve:
()
()N
N rC
Fr
CF
rC
FC
FN
PV+
++
++
++
−=
=1
11
02
21
0L
(bea
ring
in m
ind
that
:
Expr
esse
d in
nom
inal
per
ann
um te
rms
(i=m
r, w
here
m=P
/YR
), w
e ca
n th
us fi
nd th
e yi
eld
by
com
putin
g th
e I/Y
R, p
rovi
ded
the
valu
es in
the
N, P
V, P
MT,
and
FV
regi
ster
s eq
ual t
he
appr
opria
te a
ctua
l cas
h flo
w a
nd h
oldi
ng p
erio
d va
lues
.
()
()
()
()N
NN
NN
rC
Fr
FVPM
Tr
FVr
PMT
+=
++
=+
++
11
11
In 2
ndar
ym
kt, l
oans
are
pric
ed s
o th
eir y
ield
s eq
ual t
he “m
kt’s
requ
ired
yiel
d”(li
ke e
xpec
ted
tota
l ret
urn,
E(r)
=rf+
RP,
from
bef
ore)
.
At th
e tim
e w
hen
a lo
an is
orig
inat
ed (p
rimar
y m
arke
t), th
e lo
an y
ield
is u
sual
ly
appr
oxim
atel
yeq
ual t
o its
con
tract
inte
rest
ra
te. (
But n
ot e
xact
ly…
)
The
tric
ky p
arti
n lo
an y
ield
cal
cula
tion:
(a
)The
hol
ding
per
iod
over
whi
ch w
e w
ish
to
calc
ulat
e th
e yi
eld
may
not
equ
al th
e m
atur
ity o
f th
e lo
an (e
.g.,
if th
e lo
an w
ill be
pai
d of
f ear
ly, s
o N
may
not
be
the
orig
inal
mat
urity
of t
he lo
an):
N
≠m
atur
ity ;
(b)T
he a
ctua
l tim
e-ze
ro p
rese
nt c
ash
flow
of t
he
loan
may
not
equ
al th
e in
itial
con
tract
prin
cipa
l on
the
loan
(e.g
., if
ther
e ar
e "p
oint
s" o
r oth
er
clos
ing
cost
s th
at c
ause
the
cash
flow
dis
burs
ed
by th
e le
nder
and
/or t
he c
ash
flow
rece
ived
by
the
borro
wer
to n
ot e
qual
the
cont
ract
prin
cipa
l on
the
loan
, P):
CF 0
≠P
;
(c)T
he a
ctua
l liq
uida
ting
paym
ent t
hat p
ays
off t
he
loan
at t
he e
nd o
f the
pre
sum
ed h
oldi
ng p
erio
d m
ay n
ot e
xact
ly e
qual
the
outs
tand
ing
loan
ba
lanc
e at
that
tim
e (e
.g.,
if th
ere
is a
"pre
paym
ent
pena
lty" f
or p
ayin
g of
f the
loan
ear
ly, t
hen
the
borro
wer
mus
t pay
mor
e th
an th
e lo
an b
alan
ce, s
o FV
is th
en d
iffer
ent f
rom
OLB
): C
F N≠
PMT+
OLB
N
So w
e m
ust m
ake
sure
that
the
amou
nts
in th
e N
, PV,
and
FV
regi
ster
s re
flect
the
actu
al
cash
flow
s…
Exam
ple
•$2
00,0
00 m
ortg
age,
30-
year
mat
urity
, m
onth
ly p
aym
ents
•10
% a
nnua
l int
eres
t•
The
loan
has
“2
poin
ts”
–(‘
disc
ount
poi
nts’
or p
repa
id in
tere
st)
•A
lso
a 3
poin
t pre
paym
ent p
enal
ty th
roug
h en
d of
5th
year
.
•W
hat i
s yi
eld
(“effe
ctiv
e in
tere
st ra
te”)
assu
min
g ho
ldin
g pe
riod
of 4
yea
rs (i
.e.,
borro
wer
will
pay
loan
off
afte
r 48
mon
ths)
?•
Brea
k th
is p
robl
em in
to 3
ste
ps:
(1)C
ompu
te th
e lo
an c
ash
flow
s us
ing
the
cont
ract
va
lues
of t
he p
aram
eter
s(N
=360
, I=1
0%, P
V=20
0000
, FV=
0, C
ompu
te
PMT=
$175
5.14
); (2
)Alte
r the
am
ount
s in
the
regi
ster
s to
refle
ct th
e ac
tual
cas
h flo
ws;
(3
)Com
pute
yie
ld.
•(Y
ou m
ust d
o th
ese
step
s in
this
ord
er.)
Step
1) 36
0----
> N
10---
-> I
/YR
2000
00---
-> P
V0
---->
FV
PMT-
--->
-17
55.1
4
Step
2) 48
---->
NFV
---->
-19
4804
X
1.0
3 =
-20
0,64
9 --
--> F
V19
6000
---->
PV
Step
3) I/Y
R---
-> 1
1.22
%
Expe
cted
yie
ld (l
ike
E(r)
or “g
oing
-in IR
R”)
is 1
1.22
%, e
ven
thou
gh “c
ontra
ctua
l in
tere
st ra
te”o
n th
e lo
an is
onl
y 10
%.
(Whe
n cl
osin
g co
sts
and
prep
aym
ent
pena
lties
are
quo
ted
in "p
oint
s", y
ou d
o no
t ne
ed to
kno
w th
e am
ount
of t
he lo
an to
find
its
yie
ld.)
Gen
eral
rule
to c
alcu
late
yie
ld:
Cha
nge
the
amou
nt in
the
PV R
egis
ter
last
, (ju
st p
rior t
o co
mpu
ting
the
yiel
d).
Equi
vale
nt s
olut
ion
to p
revi
ous
prob
lem
:U
se C
F ke
ys in
stea
d of
mor
tgag
e m
ath
keys
…19
6000
---->
CFj
key
-175
5.14
---->
CFj
key
47
---->
Nj
key
-202
404
---->
CFj
key
IRR
---->
11.
22%
Usi
ng M
arke
t Yie
lds t
o Va
lue
Mor
tgag
es(N
ote:
Thi
s is
per
form
ing
a D
CF
NPV
an
alys
is o
f the
loan
as
an in
vest
men
t, fin
ding
wha
t pric
e ca
n be
pai
d fo
r the
lo
an s
o th
e de
al is
NPV
=0. M
arke
t’s
requ
ired
yiel
d is
“r”,
the
oppo
rtuni
ty c
ost
of c
apita
l for
the
loan
.)
Exam
ple
•$1
00,0
00 m
ortg
age,
30-
year
, 10%
, 3
poin
ts p
repa
ymen
t pen
alty
bef
ore
5 ye
ars.
•Ex
pect
ed ti
me
until
bor
row
er p
repa
ys
loan
= 4
yea
rs.
•H
ow m
uch
is th
e lo
an w
orth
toda
y if
the
mar
ket y
ield
is 1
1.00
%?
Step
1)
360-
-->N
, 10
--->I
/YR
, 10
0000
--->P
V,
0--->
FV,
Com
pute
PM
T--->
-877
.57.
Step
2)
48---
>N,
FV---
> -9
7,40
2 * 1
.03
= -1
00,3
24 --
->FV
.St
ep 3
) I/Y
R---
->11
.00%
.St
ep 4
) PV
---->
98,
697.
The
loan
is w
orth
$98
,697
. (W
atch
out
for o
rder
of s
teps
. Cas
h flo
ws
first
, the
n in
put
the
mar
ket y
ield
, the
n co
mpu
te th
e lo
an v
alue
as
the
PV.)
Det
erm
inin
g re
quire
d “d
isco
unt
poin
ts”
(or “
orig
inat
ion
fee”
):To
avo
id le
nder
doi
ng N
PV <
0 d
eal i
n m
akin
g lo
an, w
e ne
ed:
(100
,000
-98
,697
) / 1
00,0
00
=
1.3
0%
=
1.30
poi
nts