This document discusses amortization and sinking funds for repaying loans. Amortization involves making regular installment payments that divide each payment into interest and principal portions, with the outstanding principal decreasing with each payment. A sinking fund allows keeping the loan principal constant while accumulating a separate fund with deposits and interest to repay the principal in a lump sum later. Examples are provided for calculating payment amounts under each method. Yield rates equalize the value of payment streams for investors.

1. AMORTIZATION
AND
SINKING FUNDS

2. Amortization
• Amortization method: repay a loan by means of installment
payments at periodic intervals
• This is an example of annuity
• We already know how to calculate the amount of each
payment
• Our goal: find the outstanding principal
• Two methods to compute it:
–prospective
–retrospective

3. Two Methods
• Prospective method:
outstanding principal at any point in time is equal to
the present value at that date of all remaining
payments
• Retrospective method:
outstanding principal is equal to the original principal
accumulated to that point in time minus the
accumulated value of all payments previously made
• Note: of course, this two methods are equivalent.
However, sometimes one is more convenient than the
other

4. Examples
•(prospective) A loan is being paid off with payments
of 500 at the end of each year for the next 10 years. If
i = .14, find the outstanding principal, P, immediately
after the payment at the end of year 6.
•(retrospective) A 7000 loan is being paid of with
payments of 1000 at the end of each year for as long
as necessary, plus a smaller payment one year after
the last regular payment. If i = 0.11 and the first
payment is due one year after the loan is taken out,
find the outstanding principal, P, immediately after
the 9th payment.

5. One more example…
•(Different frequency) John takes out 50,000 mortgage at
12.5 % convertible semi-annually. He pays off the
mortgage with monthly payments for 20 years, the first
one is due one month after the mortgage is taken out.
Immediately after his 60th payment, John renegotiates
the loan. He agrees to repay the remainder of the
mortgage by making an immediate cash payment of
10,000 and repaying the balance by means of monthly
payments for ten years at 11% convertible semi-annually.
Find the amount of his new payment.

7. Amortization Schedule
• Goal: divide each payment (of annuity) into two parts – interest and principal
• Amortization schedule – table, containing the following columns:
• payments
• interest part of a payment
• principal part of a payment
• outstanding principal

8. Duration Payment Interest Principal
Repaid
Outstanding
Principal
0 5000.00
1 1387.05 600.00 787.05 4212.95
2 1387.05 505.55 881.50 3331.45
3 1387.05 399.77 987.28 2344.17
4 1387.05 281.30 1105.75 1238.42
5 1387.05 148.61 1238.44 0
Amortization schedule:
Example: 5000 at 12 % per year repaid by 5 annual payments

9. Example
t - 1 t
Payment
X
Outstanding principal
P
Interest earned during
interval (t-1,t) is iP
Therefore interest portion
of payment X is iP
and principal portion is
X - iP
A 1000 loan is repaid by annual payments of 150, plus a
smaller final payment. If i = .11, and the first payment is made
one year after the time of the loan, find the amount of
principal and interest contained in the third payment
Recall: in practical problems, the outstanding principal P can
be found by prospective or retrospective methods

10. General method
0 1 t
2 n
1 1 1 1
…..
an|
present value = outst.
principal at 0
…..
an-t|
outstanding principal at t
interest portion of (t+1)-st payment = i a n-t| = 1 – vn-t
t+1
1
principal portion of (t+1)-st payment = 1 – (1 – vn-t ) = vn-t
If each payment is X then
interest part of kth payment = X (1 – vn-k+1 )
principal part of kth payment = X∙vn-k+1

11. Example
•A loan of 5000 at 12% per year is to be
repaid by 5 annual payments, the first due
one year hence. Construct an amortization
schedule

12. General rules to obtain an amortization schedule
I. Take the entry from “Outs. Principal” of the previous row, multiply it by i, and
enter the result in “Interest”
II. “Payment” – “Interest” = “Principal Repaid”
III. “Outs. Principal” of prev. row - “Principal Repaid” = “Outs. Principal”
IV. Continue
Duration Payment Interest Principal
Repaid
Outstanding
Principal
0 5000.00
1 1387.05 600.00 787.05 4212.95
2 1387.05 505.55 881.50 3331.45
3 1387.05 399.77 987.28 2344.17
4 1387.05 281.30 1105.75 1238.42
5 1387.05 148.61 1238.44 0
i = 12 %

13. Example
• A 1000 loan is repaid by annual payments of 150, plus a smaller final
payment. The first payment is made one year after the time of the loan
and i = .11. Construct an amortization schedule
Duration Payment Interest Principal Repaid Outstanding Principal
0 1000
1 150 110.00 40.00 960.00
2
3
4
5
6
7
8
9
10
11
12

14. Sinking Funds
•Alternative way to repay a loan – sinking fund
method:
•Pay interest as it comes due keeping the
amount of the loan (i.e. outstanding
principal) constant
•Repay the principal by a single
lump-sum payment at some point in the
future

15. 0 1 2 n
interest
iL iL iL
…..
Loan L
lump-sum payment L
• Lump-sum payment L is accumulated by periodic deposits into a
separate fund, called the sinking fund
• Sinking fund has rate of interest j usually different from (and
usually smaller than) i
• If (and only if) j is greater than i then sinking fund method is better
(for borrower) than amortization method

16. Examples
• John borrows 15,000 at 17% effective annually. He agrees to pay the interest
annually, and to build up a sinking fund which will repay the loan at the end of 15
years. If the sinking fund accumulates at 12% annually, find
• the annual interest payment
• the annual sinking fund payment
• his total annual outlay
• the annual amortization payment which would pay off this loan in 15 years
• Helen wishes to borrow 7000. One lender offers a loan in which the principal is to be
repaid at the end of 5 years. In the mean-time, interest at 11% effective is to be paid
on the loan, and the borrower is to accumulate her principal by means of annual
payments into a sinking fund earning 8% effective. Another lender offers a loan for 5
years in which the amortization method will be used to repay the loan, with the first
of the annual payments due in one year. Find the rate of interest, i, that this second
lender can charge in order that Helen finds the two offers equally attractive.

17. Yield Rates
• Investor:
• makes a number of payments at various points in time
• receives other payments in return
• There is (at least) one rate of interest for which the value of his
expenditures will equal the value of the payments he received (at
the same point in time)
• This rate is called the yield rate he earns on his investment
• In other words, yield rate is the rate of interest which makes two
sequences of payments equivalent
• Note: to determine yield rate of a certain investor, we should
consider only payments made directly to, or directly by, this investor

18. Examples
•Herman borrows 5000 from George and agrees to
repay it in 10 equal annual instalments at 11%,
with first payment due in one year. After 4 years,
George sells his right to future payments to Ruth,
at a price which will yield Ruth 12% effective
• Find the price Ruth pays.
• Find George’s overall yield rate.

19. •At what yield rate are payments of 500 now and
600 at the end of 2 years equivalent to a
payment of 1098 at the end of 1 year?
•Henri buys a 15-year annuity with a present
value of 5000 at 9% at a price which will allow
him to accumulate a 15-year sinking fund to
replace his capital at 7%, and will produce an
overall yield rate of 10%. Find the purchase price
of the annuity.