2.3 Sinking Funds2.3 Sinking Funds

Sinking Funds

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

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

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

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 = iAA +isA

(1+ is)n −1

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

€

=Php141,917.12

2.3 Sinking Funds2.3 Sinking Funds

Period Beginning amount

Interest earned

Regular deposit

Ending amount

2.3 Sinking Funds2.3 Sinking Funds

Period Beginning amount

Interest earned

Regular deposit

Ending amount

1

2

3

4

5

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

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

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

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

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

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?

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

€

=Php640,630.44

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

⎡

⎣ ⎢

⎤

⎦ ⎥

€

=640,630.441.033 −1

.03

⎡

⎣ ⎢

⎤

⎦ ⎥

€

=Php1,980,124.63

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)

€

=Php246,151.91

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

€

=Php89,258.76

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 monthsb) 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?

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

€

=Php45,378.20

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

b) the loan is repaid through a sinking fund earning at 15% compounded semi-annually.

€

C = iAA +isA

(1+ is)n −1

€

=(.065)(500,000) + (.075)(500,000)

1.07520 −1

€

=Php44,046.10

€

The borrower saves Php1,332.10.

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 = iAA +isA

(1+ is)n −1

€

=(.01)(80,000) +.1312( )(80,000)

1+ .1312( )

48−1

€

=Php2,079.53