Question

Solve this mathematically, don't use excel!

A sinking fund is set up to accumulate $70,000. The fund pays j1 = 15%. Annual deposits of $5,000 are made to the fund.

(a) How much interest does the fund earn during the year between the 5th and 6th deposits?

(b) How many $5,000 deposits are required?

(c) What is the size of the final smaller deposit? (Explain your answer).

Answer 1

We calculate the sinking fund balances year on year. We start with the opening deposit. The balance is then grown by interest earned on opening balances and additional deposits. For next year again we add deposits and repeat the process till $70,000 is collected

Using the above process we calculate sinking fund balances year on year

Year	Opening Balance	Deposits	Interest Earned	Closing Balance
1	-	5,000.00	750.00	5,750.00
2	5,750.00	5,000.00	1,612.50	12,362.50
3	12,362.50	5,000.00	2,604.38	19,966.88
4	19,966.88	5,000.00	3,745.03	28,711.91
5	28,711.91	5,000.00	5,056.79	38,768.69
6	38,768.69	5,000.00	6,565.30	50,334.00
7	50,334.00	5,000.00	8,300.10	63,634.10
8	63,634.10	0.00	9,545.11	73,179.21

a) Interest earned between year 5 &6 is 5056 as per the calculations above.

b) 7 $5,000 deposits are required as per the calculations.

c) No additional deposit is required in year 8 as interest for year 8 is enough for the balance to cross $ 70,000/

However, if we just want the balance to reach $ 70,000 at end of year 8, we might not make a deposit of $5000 at year 7 and make a lower deposit.

So we need to calculate at what amount if interest is earned for 2 years our balances reach 70,000 after year 8.

i.e X * (1+15%)^2 = 70,000

i.e X = 52,930

Hence additional deposit in year 7 = 52,930- 50,334 = 2596.05

Hence revised calculation sinking fund is as below:

Year	Opening Balance	Deposits	Interest Earned	Closing Balance
1	-	5,000.00	750.00	5,750.00
2	5,750.00	5,000.00	1,612.50	12,362.50
3	12,362.50	5,000.00	2,604.38	19,966.88
4	19,966.88	5,000.00	3,745.03	28,711.91
5	28,711.91	5,000.00	5,056.79	38,768.69
6	38,768.69	5,000.00	6,565.30	50,334.00
7	50,334.00	2,596.05	7,939.51	60,869.55
8	60,869.55		9,130.43	69,999.99

Hence final smaller deposit would be 2596.05