Question

Ten months from today, you plan to make the first of several quarterly deposits into an account paying an APR of 4% with monthly compounding. After your first deposit, subsequent deposits will grow by 1% each. After your final deposit three years and seven months from today, you will simply let interest accrue on the account. Set up the calculations needed to determine your first and your final deposit if you want the balance in your account to equal $100,000 four years from today.

Answer 1

Suppose it is 1st Jan 2020 today and the after 4 years, the date will be 31st Dec 2023
Balance is account on 31st Dec 2023	$100,000
First deposit is made on 1st Nov 2020
Last deposit is made on 1st Aug 2023
No deposit is made during Aug to Dec 2023 i.e. during 5 months
APR	4%
Monthly interest rate	0.333333%
Value as at 1st Aug 2023 after making the last deposit	$98,349.87
Let the first deposit be D1
As stated in the question, every subsequent deposit will increase by 1%
Therefore, D2 = D1 + 1%
Hence, D12 = D11 + 1%
Also, interest shall accrue on these deposits
Now, we shall compute the future value of $100 deposit made on 1st Nov 2020 and then we would use value derived to find out the actual monthly deposit

S.No.	Deposit amount	Number of months till last deposit	Monthly interest rate	Interest @ 4% APR	Future value
1	100.00	33.00	0.33333%	11.61	111.61
2	101.00	30.00	0.33333%	10.60	111.60
3	102.01	27.00	0.33333%	9.59	111.60
4	103.03	24.00	0.33333%	8.57	111.60
5	104.06	21.00	0.33333%	7.53	111.59
6	105.10	18.00	0.33333%	6.49	111.59
7	106.15	15.00	0.33333%	5.43	111.59
8	107.21	12.00	0.33333%	4.37	111.58
9	108.29	9.00	0.33333%	3.29	111.58
10	109.37	6.00	0.33333%	2.21	111.57
11	110.46	3.00	0.33333%	1.11	111.57
12	111.57	-	0.33333%	-	111.57
				TOTAL	1,339.05

Now, we have arrived at a future value of $100 deposit. Actual deposit = $ (98,349.57/1339.05)*100 = $ 7,344.78

And similarly, value of last deposit = $8,194.33