Question

You have 35 years left until retirement and want to retire with $1.3 million. Your salary is paid annually, and you will receive $63,000 at the end of the current year. Your salary will increase at 2 percent per year, and you can earn a return of 8 percent on the money you invest. If you save a constant percentage of your salary, what percentage of your salary must you save each year?

Answer 1

Amount required at the time of retirement = $1.3 million (this is future value i.e. value at the end of 35th year)

Let the % saved of salary be x% each year. Thus amount saved at the end of current year = 63,000*x/100. Future value of this amount = 63,000*x/100*1.08^(35-1)

Salary in year 2 = 63,000*1.02 = 64,260. Future value of this amount = 64,260*x/100*1.08^(35-2)

Thus 63,000*x/100*1.08^(35-1)+63,000*1.02^1*x/100*1.08^(35-2).......+63,000*1.02^34*x/100*1.08^(35-35) = $1.3 million

Solving the above equation we get x = 9.6836% (or 9.68% after rounding off to 2 decimal place)

Year	Salary	% saved	Savings	Future value
1	63,000.00	9.68	6,100.68	83,519.16
2	64,260.00		6,222.70	78,879.21
3	65,545.20		6,347.15	74,497.03
4	66,856.10		6,474.09	70,358.30
5	68,193.23		6,603.57	66,449.51
6	69,557.09		6,735.65	62,757.87
7	70,948.23		6,870.36	59,271.32
8	72,367.20		7,007.77	55,978.47
9	73,814.54		7,147.92	52,868.55
10	75,290.83		7,290.88	49,931.41
11	76,796.65		7,436.70	47,157.45
12	78,332.58		7,585.43	44,537.59
13	79,899.23		7,737.14	42,063.28
14	81,497.22		7,891.88	39,726.43
15	83,127.16		8,049.72	37,519.40
16	84,789.71		8,210.72	35,434.99
17	86,485.50		8,374.93	33,466.38
18	88,215.21		8,542.43	31,607.14
19	89,979.51		8,713.28	29,851.19
20	91,779.10		8,887.54	28,192.79
21	93,614.69		9,065.29	26,626.52
22	95,486.98		9,246.60	25,147.27
23	97,396.72		9,431.53	23,750.20
24	99,344.65		9,620.16	22,430.74
25	101,331.55		9,812.56	21,184.59
26	103,358.18		10,008.82	20,007.67
27	105,425.34		10,208.99	18,896.13
28	107,533.85		10,413.17	17,846.35
29	109,684.52		10,621.44	16,854.88
30	111,878.22		10,833.86	15,918.50
31	114,115.78		11,050.54	15,034.14
32	116,398.10		11,271.55	14,198.91
33	118,726.06		11,496.98	13,410.08
34	121,100.58		11,726.92	12,665.08
35	123,522.59		11,961.46	11,961.46
			Total	1,300,000.00