Econ 251
Fall 2001
Mortgage Prepayments and Valuation
1
Mortgage Arithmetic
A mortgage is a promise backed by a house as collateral. There are records of mort
gages as far back as 1000 BC in Babylon.
1.1
Amortizing Mortgages
Mortgages since the depression of the 1930s amortize. Unlike the old mortgages (or
bonds), which pay a small coupon every period and a large
fi
xed sum at maturity, an
amortizing (
fi
xed rate) mortgage makes the same level payment every period. After
the depression of the 1930s, in which many farmers defaulted on their loans just
before the big
fi
nal payment came due, the amortizing loan became common.
Mortgages had originally been structured like bonds, with a coupon payment
covering the interest and then the big principal payment at maturity, because that
made it easy to
fi
gure out how much a homeowner who wanted get out of his loan
should pay. That is an important consideration, since nobody would ever borrow on
the condition that he never move, and if the homeowner moves and sells his house, he
no longer has the collateral to back the remainder of the loan. Under the old ”bullet”
mortgage, it was easy to
fi
gure out what to charge the homeowner. No matter when
he sold, he would have to make the next period payment and then give back the loan
original loan amount (say $100). When the payment is not directly broken up into
interest payments and a
fi
nal principal payment, it is not as easy to
fi
gure out what
the homeowner should pay. But it is not so di
ﬃ
cult either.
1.1.1
Computing the Level Payments
If the mortgage rate is
m
, and the maturity of the mortgage is
T
periods, then the
level payment
x
needed on a dollar loan of
B
0
is
fi
gured out to solve the equation:
B
0
=
T
X
t
=1
x
1
(1 +
m
)
t
From our formula for annuities, this means that
x
=
mB
[1
−
1
(1+
m
)
T
]
Thus when
T
=
∞
,
the mortgage pays
x
=
m
per period on a loan of
B
= $1
. If
T
= 30
and
m
= 7%
and
B
= 1
, then (by our doubling rule) (1.07)
30
= 8
,
and so
x
≈
8
7
m.
1
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A mortgage convention
Typically, mortgage payments come monthly, and the
mortgage rate is quoted at the monthly rate times 12. Thus an 8% mortgage means
that
T
= 360
and
m
=
8
12
%
.
For simplicity (i.e., to
fi
t everything into an excel spread
sheet) we will suppose that
T
= 30
years and
m
is an annual rate, and that all
payments come at the end of the year.
1.1.2
Remaining Balance
Just after the level payment
x
at any time
n
, the remaining balance is
B
n
=
T
−
n
X
t
=1
x
1
(1 +
m
)
t
For example, suppose
B
= 100
, and
m
= 7%
.
Then the
fi
rst few years of level
payments and balances work out to
8
.
06
8
.
06
8
.
06
8
.
06
8
.
06
8
.
06
100
98
.
94
97
.
81
96
.
60
95
.
30
93
.
91
92
.
43
1.2
Principal and Interest
In keeping with the original form of the mortgage, which divided the payments into
interest and principal, we can do the same with an amortizing mortgage. Notice that
by construction, for every
n
,
B
n
=
1
1 +
m
[
x
+
B
n
+1
]
Hence
B
n
[1 +
m
] =
x
+
B
n
+1
x
=
payment
=
interest + principal paydown
=
mB
n
+ (
B
n
−
B
n
+1
)
The division of the payment
x
between the interest payment
mB
n
and the principal
paydown amortizing amount
(
x
−
mB
n
) = (
B
n
−
B
n
+1
)
starts o
ff
mostly interest and
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 '09
 GEANAKOPLOS,JOHN
 Interest Rates, Interest

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