This document discusses sinking funds. Some key points: - A sinking fund is an accumulation scheme where periodic deposits are made to build up a target amount over time including interest earned. - Formulas are provided to calculate the periodic deposit amount, amount in the fund after k deposits, interest earned, and more. - Examples demonstrate calculating annual deposits to a sinking fund to reach $800k in 5 years, quarterly deposits to reach $15M in 4.5 years, and comparing costs of loan amortization vs. repayment via sinking fund.

1. 2.3  Sinking Funds2.3  Sinking Funds
Sinking Funds

2. 2.3  Sinking Funds2.3  Sinking Funds
• Sinking fund – a fund-accumulation scheme
wherein the amount is generated by making
periodic deposits
• In amortization problems, we usually want to
find the following values:
– Periodic deposit
– Amount in the fund after any kth
deposit
– Interest earned in any period
– Increase in the fund in any period

3. 2.3  Sinking Funds2.3  Sinking Funds
• Sinking fund schedule – a table which shows
how a target amount is completely attained
through periodic deposits as well as interests
these deposits earn in the process
• Sinking fund method of paying off a debt –
happens when the debtor pays interest
periodically and pays the principal in one
lump-sum payment at the end of the term
from a sinking fund

4. 2.3  Sinking Funds2.3  Sinking Funds
Formulas:
• Amount in the fund after k deposits
• Periodic deposit
Sk = D
(1+is)k
−1
is
⎡
⎣
⎢
⎤
⎦
⎥
D =
Skis
(1+is)k
−1

5. 2.3  Sinking Funds2.3  Sinking Funds
Formulas:
• Interest earned
• Principal repayment
• Periodic cost (when a sinking fund is used to
pay off a debt)
IEk = Sk−1is = D (1+is)k−1
−1[ ]
INCk = Sk − Sk−1 = D(1+is)k−1
C = iA A +
isA
(1+is)n
−1

6. 2.3  Sinking Funds2.3  Sinking Funds
1. In order to have Php800,000 in 5 years, Shane
deposits an amount each year in a sinking
fund earning 6% effective. Find the annual
deposit and construct the sinking fund
schedule.
D =
Skis
(1+is)k
−1
=
(800,000)(.06)
1.065
−1
= Πηπ141,917.12

7. 2.3  Sinking Funds2.3  Sinking Funds
Period Beginning
amount
Interest
earned
Regular
deposit
Ending
amount

8. 2.3  Sinking Funds2.3  Sinking Funds
Period Beginning
amount
Interest
earned
Regular
deposit
Ending
amount
1
2
3
4
5

9. 2.3  Sinking Funds2.3  Sinking Funds
Period Beginning
amount
Interest
earned
Regular
deposit
Ending
amount
1 0 0 141,917.12 141,917.12
2
3
4
5

10. 2.3  Sinking Funds2.3  Sinking Funds
Period Beginning
amount
Interest
earned
Regular
deposit
Ending
amount
1 0 0 141,917.12 141,917.12
2 141,917.12 8,515.03 141,917.12 292,349.27
3
4
5

11. 2.3  Sinking Funds2.3  Sinking Funds
Period Beginning
amount
Interest
earned
Regular
deposit
Ending
amount
1 0 0 141,917.12 141,917.12
2 141,917.12 8,515.03 141,917.12 292,349.27
3 292,349.27 17,540.96 141,917.12 451,807.35
4
5

12. 2.3  Sinking Funds2.3  Sinking Funds
Period Beginning
amount
Interest
earned
Regular
deposit
Ending
amount
1 0 0 141,917.12 141,917.12
2 141,917.12 8,515.03 141,917.12 292,349.27
3 292,349.27 17,540.96 141,917.12 451,807.35
4 451,807.35 27,108.44 141,917.12 620,832.91
5

13. 2.3  Sinking Funds2.3  Sinking Funds
Period Beginning
amount
Interest
earned
Regular
deposit
Ending
amount
1 0 0 141,917.12 141,917.12
2 141,917.12 8,515.03 141,917.12 292,349.27
3 292,349.27 17,540.96 141,917.12 451,807.35
4 451,807.35 27,108.44 141,917.12 620,832.91
5 620,832.91 37,249.97 141,917.12 800,000

14. 2.3  Sinking Funds2.3  Sinking Funds
3. To raise money for office space expansion, a
small business operator estimates that
Php15M will be needed in 4 ½ years. He
projects that a certain amount must be
invested every 3 months in a fund which earns
interest at 12% converted quarterly.
a) How much is the quarterly investment?
b) How much will be in the fund after the 3rd
deposit?
c) How much interest is earned on the 3rd year?

15. 2.3  Sinking Funds2.3  Sinking Funds
3. To raise money for office space expansion, a
small business operator estimates that
Php15M will be needed in 4 ½ years. He
projects that a certain amount must be
invested every 3 months in a fund which earns
interest at 12% converted quarterly.
a) How much is the quarterly investment?
D =
Skis
(1+is)k
−1
=
(15,000,000)(.03)
1.0318
−1
= Πηπ640,630.44

16. 2.3  Sinking Funds2.3  Sinking Funds
3. To raise money for office space expansion, a
small business operator estimates that
Php15M will be needed in 4 ½ years. He
projects that a certain amount must be
invested every 3 months in a fund which earns
interest at 12% converted quarterly.
b) How much will be in the fund after the 3rd
deposit?
S3 = D
(1+is)3
−1
is
⎡
⎣
⎢
⎤
⎦
⎥ = 640,630.44
1.033
−1
.03
⎡
⎣
⎢
⎤
⎦
⎥= Πηπ1,980,124.63

17. 2.3  Sinking Funds2.3  Sinking Funds
3. To raise money for office space expansion, a
small business operator estimates that
Php15M will be needed in 4 ½ years. He
projects that a certain amount must be
invested every 3 months in a fund which earns
interest at 12% converted quarterly.
c) How much interest is earned on the 3rd year?
IE12 = 640,630.44(1.0311
−1) = Πηπ246,151.91

18. 2.3  Sinking Funds2.3  Sinking Funds
5. Caloi and Santi want to pool their money in
order to have P1M in 5 years. They will place
semi-annual deposits in a fund that earns 5%
compounded semi-annually. How much each
of these deposits should be?
D =
Skis
(1+is)k
−1
=
(1,000,000)(.025)
1.02510
−1
= Πηπ89,258.76

19. 2.3  Sinking Funds2.3  Sinking Funds
9. A Php500,000 loan at 13% interest rate
payable semi-annually is to be repaid in 10
years. Find the semi-annual expense if
a) the loan is to be amortized every 6 months
b) the loan is repaid through a sinking fund
earning at 15% compounded semi-annually.
How much does the borrower save semi-
annually by choosing the cheaper method?

20. 2.3  Sinking Funds2.3  Sinking Funds
9. A Php500,000 loan at 13% interest rate
payable semi-annually is to be repaid in 10
years. Find the semi-annual expense if
a) the loan is to be amortized every 6 months.
R =
(500,000)(.065)
1−1.065−20
= Πηπ45,378.20

21. 2.3  Sinking Funds2.3  Sinking Funds
9. A Php500,000 loan at 13% interest rate
payable semi-annually is to be repaid in 10
years. Find the semi-annual expense if
b) the loan is repaid through a sinking fund
earning at 15% compounded semi-annually.
C = iA A +
isA
(1+is)n
−1
= (.065)(500,000) + (.075)(500,000)
1.07520
−1
= Πηπ44,046.10
Τηε βορροωερ σαϖεσ Πηπ1,332.10.

22. 2.3  Sinking Funds2.3  Sinking Funds
11. George borrows Php80,000, with a plan to
repay the whole amount in 4 years. Between
now and that time, he has to make monthly
interest payments at 12% compounded
monthly. The Php80,000 will be repaid by
making deposits each month in a sinking fund
that earns 13% compounded monthly. How
much is the monthly cost of the loan?
C = iA A +
isA
(1+is)n
−1
= (.01)(80,000) +
.13
12( )(80,000)
1+ .13
12( )
48
−1
= Πηπ2,079.53