Sinking fund is when a specified amount of money is needed at a specified future date, it is a good practice to accumulate systematically a fund by means of equal periodic deposits.
Sinking funds are used to pay off debts, to redeem bond issues, to replace worn-out equipment, to buy new equipment, or in one of the depreciation methods. Since the amount needed in the sinking fund, the time the amount is needed and the interest rate that the fund earns are known, we have an annuity problem in which the size of the payment, the sinking-fund deposit, is to be determined. A schedule showing how a sinking fund accumulates to the desired amount is called a sinking-fund schedule.
Sinking fund schedule for a loan of $1000 repaid over 4 years at 8%
Year
Interest paid
Sinking fund
Interest earned
Amount in
Net amount of
deposit
on sinking fund
sinking fund
loan
0
1000.00
1
80.0
221.92
0
221.92
778.08
2
80.0
221.92
17.75
461.59
538.41
3
80.0
221.92
36.93
720.44
279.56
4
80.0
221.92
57.64
1000
0
An eight-storey condominium apartment building consists of 146 twobedroom apartment units of equal size. The Board of Directors of the Homeowners’ Association estimated that the building will need new carpeting in the halls at a cost of $25 800 in 5 years. Assuming that the association can invest their money at j12 = 8%, what should be the monthly sinking-fund assessment per unit? Solution The sinking-fund deposits form an ordinary simple annuity with S = 25 800, i = 2%, n = 60. We calculate the total monthly sinking-fund deposit R=25800S60|66.67%=$351.13
Per-unit assessment should be 351.13146=$2.405
In this case, a two-step procedure can be followed: Find the rate of interest, convertible at the same frequency as payments are made, that is equivalent to the giver rate of interest. E.g: +
đ?‘›
=
+
đ?‘› đ?‘›
Using this new rate of interest, construct the amortization schedule using the techniques developed previously. *The frequency of the following may differ. • Interest payments on the loan • Sinking fund deposits • Interest conversion period on the sinking fund
A borrower takes out a loan of $2000 for 2 years. Construct a sinking fund schedule if the lender receives 10% effective on the loan and if the borrower replaces the amount of the loan with semiannual deposits in a sinking fund earning 8% convertible quarterly. Year
Interest paid
Sinking fund deposit
Interest earned on sinking fund
Amount in
Net amount
sinking fund
of loan
0 Âź
0
0
0
0
2000.00
½
0
470.70
0
470.70
1529.30
ž
0
0
9.41
480.11
1519.89
1
200
470.70
9.60
960.41
1039.59
1Âź
0
0
19.21
979.62
1020.38
1½
0
470.70
19.59
1469.91
530.09
1ž
0
0
29.4
1499.31
500.69
2
200
470.70
29.99
2000.00
0
1. The interest is paid at the end of every year due to the annual effective rate charged which is 10%. đ?‘° = đ?’Š . đ?‘Šđ?&#x;Ž 200 = (0.1)2000.00
2.The sinking fund deposits on the other hand is given 8% interest rate quarterly but is required to be deposited semiannually +
=
= .
+
.
So, we used j in this equation: đ?‘›
+ .
−
đ??ˇ= á . =470.70 (rounded off)
The interest earned on sinking fund is quarterly with interest rate of 8% therefore; j = 0.02 = 0.02Ă— . =9.414 Amount in sinking fund (1/2year ) = D(1/4year) +D(1/2year) +đ??źđ?‘†đ??š (1/2year) So, the net amount of loan or outstanding loan balance can be calculated: đ?‘?
đ??ľđ?‘Ą = đ?‘Žđ?‘›âˆ’đ?‘Ą đ??ľđ?‘Ąđ?‘&#x; = đ?‘Žđ?‘› +
đ??ľ =đ??ľ −đ?‘Ž
đ?‘Ą
−
đ?‘Ą
đ?‘Ś đ?‘Ž
Notes; 
If a loan is being paid by the amortization method, it is possible that the borrower repays the loan with installments which are not level.

In this case, we consider more general pattern of variation.

We will assume that the interest conversion period and the payment period are equal and coincide.
A loan is to be repaid with n periodic installments are R1, R2, …., Rn formula:
Assume that the varying payments by the borrower are R1, R2, …., Rn and that i ≠j. Loan is denoted as L. Then, the sinking fund deposit for the tth period is Rt-iL. The accumulated value of sinking fund at the end of n period must be L. So, we have
OR
A loan of RM39999.85 is to be repaid by payments at the end of each quarter for eight years. Each payment is 2% higher than its predecessor. The loan is made at nominal rate of discount of 4% payable quarterly. Find the balance just after twentieth payment, the amount of interest in the twentieth payment and the amount of principal in twentieth payment.
.
=[
= =
. .
.
= 1081.099917
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