Question

Starting today, a series of annual deposits is made into an account offering 8 percent compounded annually. The first deposit is $2,000. The second deposit is 2 percent smaller than the first, that is, $2,000 (0.98)^1. The third deposit is 2 percent smaller than the second, that is, $2,000 (0.98)^2. This pattern of declining deposits, with each deposit being 2 percent smaller than the previous one, continues until the last deposit is made at the end of year 40, that is, 40 years from today. What will be the balance in the account at the end of year 40, immediately after the last deposit is made?

Answer 1

Future Value of each year deposit made for 40 years are as follows:

Year	Deposits	FV= PV (1+r)^n
1	2000.000000	40230.595364
2	1960.000000	36505.540238
3	1920.800000	33125.397623
4	1882.384000	30058.231177
5	1844.736320	27275.061623
6	1807.841594	24749.592955
7	1771.684762	22457.963977
8	1736.251066	20378.522868
9	1701.526045	18491.622603
10	1667.495524	16779.435325
11	1634.145614	15225.783906
12	1601.462701	13815.989100
13	1569.433447	12536.730850
14	1538.044779	11375.922438
15	1507.283883	10322.596286
16	1477.138205	9366.800334
17	1447.595441	8499.504006
18	1418.643532	7712.512895
19	1390.270662	6998.391330
20	1362.465248	6350.392133
21	1335.215944	5762.392862
22	1308.511625	5228.837967
23	1282.341392	4744.686303
24	1256.694564	4305.363497
25	1231.560673	3906.718729
26	1206.929460	3544.985514
27	1182.790870	3216.746114
28	1159.135053	2918.899252
29	1135.952352	2648.630802
30	1113.233305	2403.387210
31	1090.968639	2180.851357
32	1069.149266	1978.920676
33	1047.766281	1795.687280
34	1026.810955	1629.419939
35	1006.274736	1478.547723
36	986.149241	1341.645156
37	966.426256	1217.418752
38	947.097731	1104.694794
39	928.155777	1002.408239
40	909.592661	909.592661
	55429.959605	425576.421857

Balance in the account at the end of year 40, immediately after the last deposit is made will be $425,576.421857

It is assumed that today is end of 1^st year, therefore until end of 40^th year 40 annual deposit would be made.

Deposit:

First year end deposit is $2000.

Every year 2% declined from 2^nd year onwards on previous year deposit was made.

Therefore 1^st year end $2000 deposited then, 2^nd year end it will be deposited 98% of 1^st year’s deposits i.e., $2000 X 0.98= $1960,

Similarly 3^rd year end deposit will be 98% of 2^nd year’s deposit i.e., $1960 X 0.98 =$1920.80 and so on continue until 40^th year.

Future Value at end of 40^th year immediately after last deposit made:

Balance in the account at the end of 40^th year = FV for 1^st year deposit + FV for 2^nd year deposit + …… + FV for 40^th year deposit

Balance in the account at the end of 40^th year = (FV = PV (1+r)^39) + (FV = PV (1+r)^38)+…….. + (FV = PV (1+r)^0)

Where

FV= Future value

PV= Present value