redeemed with a 216-payment annuity-immediate so that the balance is
216
e
0
05
12
=
2,639.82
×
142
241
=
$
375,490
7

By the retrospective method, the balance is
400,000
μ
1 +
0
05
12
¶
24
−
2,639
24
e
0
05
12
=
441,976.53
−
2,639.82
×
25
186
=
$
375,490
Hence, the two methods give the same answer. After the increase in the
rate of interest, if the loan is to be repaid within the same period, the
revised monthly installment is
375,490
216
e
0
055
12
=
375,490
136
927
=
$
2,742.26
so that the increase in installment is $102.44.
Let
be the remaining
number of installments if the amount of installment remained unchanged.
Thus,
e
0
055
12
=
216
e
0
05
12
= 142
241
8

from which we solve for
= 230
88
≈
231
.
Thus, it takes 15 months
more to pay back the loan.
2
Example 5.2:
A housing loan is to be repaid with a 15-year monthly
annuity-immediate of $2,000 at a nominal rate of 6% per year. After 20
payments, the borrower requests for the installments to be stopped for
12 months.
Calculate the revised installment when the borrower starts
to pay back again, so that the loan period remains unchanged.
What
is the di
ff
erence in the interest paid due to the temporary stoppage of
installments?
Solution:
The loan is to be repaid over
15
×
12 = 180
payments. After
20 payments, the loan still has 160 installments to be paid.
Using the
prospective method, the balance of the loan after 20 payments is
2,000
×
160
e
0
005
= $
219,910
9

Note that if we calculate the loan balance using the retrospective method,
we need to compute the original loan amount. The full calculation using
the retrospective method is
2,000
180
e
0
005
(1
005)
20
−
2,000
20
e
0
005
=
2,000
(130
934
−
20
979)
=
$
219,910
Due to the delay in payments, the loan balance 12 months after the 20th
payment is
219,910
(1
005)
12
= $
233,473
which has to be repaid with a 148-payment annuity-immediate.
Hence,
the revised installment is
233,473
148
e
0
005
=
233,473
104
401
=
$
2,236.31
10

The di
ff
erence in the interest paid is
2,236.31
×
148
−
2,000
×
160 = $
10,973
2
Example 5.3:
A man borrows a housing loan of $500,000 from Bank
A to be repaid by monthly installments over 20 years at nominal rate of
interest of 4% per year. After 24 installments Bank B o
ff
ers the man a loan
at rate of interest of 3.5% to be repaid over the same period. However,
if the man wants to re-
fi
nance the loan he has to pay Bank A a penalty
equal to 1.5% of the outstanding balance. If there are no other re-
fi
nancing
costs, should the man re-
fi
nance the loan?
11

Solution:
The monthly installment paid to Bank A is
500,000
240
e
0
04
12
=
500,000
165
02
=
$
3,029.94
The outstanding balance after paying the 24th installment is
3,029.94
216
e
0
04
12
=
3,029.94
×
153
80
=
$
466,004
If the man re-
fi
nances with Bank B, he needs to borrow
466,004
×
1
015 = $
472,994
so that the monthly installment is
472,994
216
e
0
035
12
=
472,994
160
09
=
$
2,954.56
12

As this is less than the installments of $3,029.94 he pays to Bank A, he
should re-
fi
nance.
2
13

5.2 Amortization
•
If a loan is repaid by the amortization method, each installment is
fi
rst used to o
ff
set the interest incurred since the last payment.

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