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Sections 6.3, 6.4

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... 6.1 on page 170 of the textbook displays the format for an amortization schedule. ... an amortization schedule and a sinking fund schedule for a loan of ... – PowerPoint PPT presentation

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Title: Sections 6.3, 6.4


1
Sections 6.3, 6.4
When a loan is being repaid with the amortization
method, each payment is partially a repayment of
principal and partially a payment of interest.
Determining the amount of each for a payment can
be important (for income tax purposes, for
example).
Table 6.1 on page 170 of the textbook displays
the format for an amortization schedule. The
entries are for a loan of at interest rate
i per period being repaid with payments of 1 at
the end of each period for n periods. Observe
each of the following from this table 1. 2.
a n
At the end of period 1, the interest paid is i
1 vn, the principal repaid is vn, and the
outstanding loan balance is .
a n
a n1
At the end of period k, the interest paid is i
1 vnk1, the principal repaid is
vnk1, and the outstanding loan balance is Bkp
.
a nk1
a nk
2
3. 4.
Sum of Principal Repayments Original Amount of
Loan
Sum of Interest Payments Sum of Total
Payments Sum of Principal Repayments
3
A 5000 loan is being repaid by payments of X at
the end of each half year for as long as
necessary until a smaller final payment is made.
The nominal rate of interest convertible
semiannually is 14. (a) If X 400, find the
principal and the interest in the sixth payment.
(Note it is not necessary to calculate any of
the formulas in the amortization table.) The
outstanding balance at the beginning of the sixth
half-year is
B5r
5000(1.07)5 400
7012.76 400(5.7507)
4712.48
s 5
The interest in the sixth payment is
(0.07)(4712.48)
329.87
The principal in the sixth payment is
400 329.87
70.13
4
(b) If X 350, find the principal in the sixth
payment, and interpret this.
The outstanding balance at the beginning of the
sixth half-year is
B5r
5000(1.07)5 350
7012.76 350(5.7507)
5000.01
s 5
If X ? 350, then the loan will never be paid off,
because every payment will count only toward
principal.
5
Jones borrows 20,000 from Smith and agrees to
repay the loan with equal quarterly installments
of principal and interest at 10 convertible
quarterly over eight years. At the end of three
years, Smith sells the right to receive future
payments to Collins at a price which produces a
yield rate of 12 convertible quarterly for
Smith. Find the total amount of interest
received (a) by Collins, and (b) by Smith.
20000
20000 21.8492
915.37
(a) Each quarterly payment by Jones is
a 32 0.025
Total payments by Jones to Collins over the last
five years are
(20)915.37
18307.40
After three years, the price Collins pays to
Smith is
915.37
a 20 0.03
915.37(14.8775)
13618.42
Total amount of interest received by Collins is
18307.40 13618.42
4688.98
(b) Total payments by Jones to Smith over the
first three years are
(12)915.37
10984.44
6
Jones borrows 20,000 from Smith and agrees to
repay the loan with equal quarterly installments
of principal and interest at 10 convertible
quarterly over eight years. At the end of three
years, Smith sells the right to receive future
payments to Collins at a price which produces a
yield rate of 12 convertible quarterly for
Smith. Find the total amount of interest
received (a) by Collins, and (b) by Smith.
(b) Total payments by Jones to Smith over the
first three years are
(12)915.37
10984.44
After three years, the outstanding loan balance is
915.37
a 20 0.025
915.37(15.5892)
14269.89
Recall that after three years, the price Collins
pays to Smith is
915.37
915.37(14.8775)
13618.42
a 20 0.03
Total amount of interest received by Smith is
13618.42 10984.44 20000
4602.86
7
An amount is invested at an annual effective rate
of interest i which is exactly sufficient to pay
1 at the end of each year for n years. In the
first year, the fund earns rate i and 1 is paid
at the end of the year. However, in the second
year, the fund earns rate j i. If X is the
revised payment which could be made at the end of
years 2 to n, then find X assuming that (a) the
rate reverts back to i again after this one year,
From the amortization table, the balance after
one year is , and therefore the balance
after two years must be
a n1 i
(1 j) X .
a n1 i
However, after two years, the balance must be
equal to the present value of all future
payments. Consequently, we have that
(1 j) X
X
a n1 i
a n2 i
(1 ) (1 j)
X
a n1 i
a n2 i
(1 i) (1 j)
X
1 j X 1 i
a n1 i
a n1 i
8
An amount is invested at an annual effective rate
of interest i which is exactly sufficient to pay
1 at the end of each year for n years. In the
first year, the fund earns rate i and 1 is paid
at the end of the year. However, in the second
year, the fund earns rate j i. If X is the
revised payment which could be made at the end of
years 2 to n, then find X assuming that
(b) the rate earned remains at j for the rest of
the n-year period.
From the amortization table, the balance after
one year is , and therefore the balance
after two years must be
a n1 i
(1 j) X .
a n1 i
However, after two years, the balance must be
equal to the present value of all future
payments. Consequently, we have that
(1 j) X
X
a n1 i
a n2 j
(1 ) (1 j)
X
a n1 i
a n2 j
(1 j) (1 j)
X
X
a n1 i
a n1 j
a n1 i
a n1 j
9
Instead of repaying a loan in installments by the
amortization method, a borrower can accumulate a
fund which will exactly repay the loan in one
lump sum at the end of a specified period of
time. This fund is called a sinking fund. It is
generally required that the borrower periodically
pay interest on the loan, sometimes referred to
as a service.
Consider a loan of amount 1 repaid over n periods.
1
With the amortization method, the payment each
period is .
a n
With the sinking fund method, the payment at the
end of each period is the sum of the periodic
sinking fund deposit necessary to accumulate the
amount of the loan at the end of n periods and
the amount of the interest paid on the loan each
period
1 i
s n
1
1 i
Recall that , which implies that if the
rate of interest paid on the loan equals the rate
of interest earned on the sinking fund, then the
sinking fund method and the amortization method
are equivalent.
a n
s n
10
Table 6.2 on page 171 of the textbook and Table
6.3 on page 177 of the textbook are respectively
an amortization schedule and a sinking fund
schedule for a loan of 1000 repaid over 4 years
at 8.
Suppose i the rate of interest paid on the loan
is not necessarily equal to j the rate of
interest earned on the sinking fund. (Usually j

the present value of an annuity which pays 1 at
the end of each period for n periods with i and j
as previously defined.
a n ij
11
Consider a loan of 1 with level payments for n
periods. Each payment must be , but each of
these payments must pay interest rate i on the
loan and provide for a sinking deposit which will
accumulate at rate j to the amount of the loan at
the end of n periods. Consequently, we have
1
a n ij
1
1 i

a n ij
s n j
1
1 j i

a n ij
a n j
1

1 (i j)
a n ij
a n j
a n j

1 (i j)
a n ij
a n j
12
The sinking fund schedule in Table 6.3 on page
177 in the textbook is for i j 0.08.
Consider how this schedule would change if i
0.10.
13
Abernathy wishes to borrow 5000. Lender Barnaby
offers a loan in which the principal is to be
repaid at the end of six years. In the meantime,
12 effective is to be paid on the loan and
Abernathy is to accumulate the amount necessary
to repay the 5000 by means of annual deposits in
a sinking fund earning 9 effective. Lender
Cromwell offers a loan for six years in which
Abernathy repays the loan by the amortization
method. Find which offer would be better for
Abernathy to take if Cromwell charges an
effective rate of (a) 10 and (b) 12.
With the sinking fund offer from Barnaby, the
annual payment is
5000
1 0.12
1 0.12 7.5233

5000
5000
1264.60
a 6 0.120.09
s 6 0.09
With the amortization offer from Cromwell, the
annual payment at effective rate
In either case, the amortization offer which
involves smaller payments is from
5000
5000 4.3553
10 is
1148.03
a 6 0.10
5000
5000 4.1114
12 is
1216.13
Cromwell.
a 6 0.12
14
With the sinking fund method, the borrower is not
only paying i per unit borrowed but is investing
in a sinking fund on which interest is being
sacrificed at rate i j per period.
Roughly speaking, the average balance in the
sinking fund is 1/2 per unit borrowed (i.e., half
the original loan amount).
Consequently, the average rate interest is being
sacrificed is approximately (i j)/2.
The total interest cost per unit borrowed, which
would be the interest charged in the amortization
method in order for payments to be equal to those
in the sinking fund method, is approximately i
(i j)/2.
In the example just completed, we have i (i
j)/2 0.12 (0.03)/2
0.135
Consider Example 6.7 on page 181 of the textbook.
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