Question

After graduation, you start your job with a salary of $70,000 per year (paid annually). Starting the second year, your salary increases by a rate of 3% per year through year 30 and then you retire. Determine the amount available in your retirement account when you retire if you put 12% of salary in the retirement account on an annual basis. Suppose the interest rate on the retirement account is 5% per year.

Answer 1

Salary earned in 1st year = 70,000. This will grow by 3% every year for the next 30 years and so salary in year 2 = 70,000*1.03 = 72,100. Salary in year 3 = 72,100*1.03 = 74,263. This pattern will continue till the 30th year when salary will become $164,959.59

Now 12% of salary every year is put in retirement account. So amount put in year 1 = 70,000*12% = $8400. In year 2 = 72,100*12% = $8652 and so on. These numbers are shown below:

Year	Salary	Amount put in retirement account
1	70,000.00	8,400.00
2	72,100.00	8,652.00
3	74,263.00	8,911.56
4	76,490.89	9,178.91
5	78,785.62	9,454.27
6	81,149.19	9,737.90
7	83,583.66	10,030.04
8	86,091.17	10,330.94
9	88,673.91	10,640.87
10	91,334.12	10,960.09
11	94,074.15	11,288.90
12	96,896.37	11,627.56
13	99,803.26	11,976.39
14	102,797.36	12,335.68
15	105,881.28	12,705.75
16	109,057.72	13,086.93
17	112,329.45	13,479.53
18	115,699.33	13,883.92
19	119,170.31	14,300.44
20	122,745.42	14,729.45
21	126,427.79	15,171.33
22	130,220.62	15,626.47
23	134,127.24	16,095.27
24	138,151.06	16,578.13
25	142,295.59	17,075.47
26	146,564.46	17,587.73
27	150,961.39	18,115.37
28	155,490.23	18,658.83
29	160,154.94	19,218.59
30	164,959.59	19,795.15

Now the amount available in retirement account at the end of the 30th year can be computed using the FVIF (future value interest factor) for 5% for these amounts. FVIF for the 1st year = 1.05^(30-1) = 4.1161, for the 2nd year = 1.05^(30-2) = 3.9201 and so on till it becomes 1 for the 30th year.

FV = amount put in retirement account*FVIF and total of all FV's is the amount available in your retirement account when you retire.

Year	Amount put in retirement account	1+r	FVIF	FV
1	8,400.00	1.05	4.1161	34,575.54
2	8,652.00		3.9201	33,916.96
3	8,911.56		3.7335	33,270.92
4	9,178.91		3.5557	32,637.19
5	9,454.27		3.3864	32,015.53
6	9,737.90		3.2251	31,405.71
7	10,030.04		3.0715	30,807.50
8	10,330.94		2.9253	30,220.69
9	10,640.87		2.7860	29,645.06
10	10,960.09		2.6533	29,080.39
11	11,288.90		2.5270	28,526.48
12	11,627.56		2.4066	27,983.12
13	11,976.39		2.2920	27,450.11
14	12,335.68		2.1829	26,927.25
15	12,705.75		2.0789	26,414.35
16	13,086.93		1.9799	25,911.22
17	13,479.53		1.8856	25,417.67
18	13,883.92		1.7959	24,933.53
19	14,300.44		1.7103	24,458.60
20	14,729.45		1.6289	23,992.72
21	15,171.33		1.5513	23,535.72
22	15,626.47		1.4775	23,087.42
23	16,095.27		1.4071	22,647.66
24	16,578.13		1.3401	22,216.28
25	17,075.47		1.2763	21,793.11
26	17,587.73		1.2155	21,378.00
27	18,115.37		1.1576	20,970.80
28	18,658.83		1.1025	20,571.36
29	19,218.59		1.0500	20,179.52
30	19,795.15		1.0000	19,795.15
			Total	795,765.56

Thus  the amount available in your retirement account when you retire = $795,765.56