Question

Please solve the following time value of money problem: John's Retirement Contribution John, 22 is about to begin his career as a rocket scientist for a NASA contractor. Being a rocket scientist, John knows that he should begin saving for retirement immediately Part of his inspiration came from reading an article on Social Security in Newsweek. The article indicated that the ratio of workers paying taxes to retirees collecting checks will drop dramatically in the future. In fact, the number will drop to 2 workers for every retiree in 2040 John's retirement plan allows him to make equal yearly contributions, and it pays 8% interest annually. Upon retirement, John plans to buy a new boat, which he estimates will cost him $300,000 in 43 years, which is when he plans to retire (at age 65). He also estimates that in order to live comfortably he will require a yearly income of $80,000 for each year after he retires. Based on his family history, John expects to live until age 80 (that is, he would like to receive 15 payments of $80,000 at the end of each year). When he retires, John will purchase his boat in one lump sum and place the remaining balance into an account that pays 6% interest, from which he will withdraw his $80,000 per year If John's first contribution is made one year from today and his last is made the day he retires, how much money must he contribute each year to his retirement fund?

Answer 1

John's present age - 22 years

Retirement age - 65 years

Years to make annual contribution - 43

Now as this requires to estimate the amount to be invested as of today, we need to consider one important factor to understand time value of money, that is Inflation. Inflation in USA is estimated to be 2% currently. For the value we arrive at the end of age 65 needs to be equivalent to the money required by John after factoring in the time value of money.

We go by the following way:

Factors Known:

One time expense for the boat - $300,000

Money needed annually for 15 years after retirement - $80,000*15 = $1,200,000

Annual interest earned on contributions made till retirement = 8%

Annual interest earned after 65 = 6%

Inflation rate = 2%

To calculate: Amount of annual contribution for next 43 years

Contribution at the end of Age:	Years to retirement	Minimum Annual Contribution($)	Value of Contribution at the age 65 ($)
22	43	$                                      10,000	$                          273,666
23	42	$                                      10,000	$                          253,395
24	41	$                                      10,000	$                          234,625
25	40	$                                      10,000	$                          217,245
26	39	$                                      10,000	$                          201,153
27	38	$                                      10,000	$                          186,253
28	37	$                                      10,000	$                          172,456
29	36	$                                      10,000	$                          159,682
30	35	$                                      10,000	$                          147,853
31	34	$                                      10,000	$                          136,901
32	33	$                                      10,000	$                          126,760
33	32	$                                      10,000	$                          117,371
34	31	$                                      10,000	$                          108,677
35	30	$                                      10,000	$                          100,627
36	29	$                                      10,000	$                            93,173
37	28	$                                      10,000	$                            86,271
38	27	$                                      10,000	$                            79,881
39	26	$                                      10,000	$                            73,964
40	25	$                                      10,000	$                            68,485
41	24	$                                      10,000	$                            63,412
42	23	$                                      10,000	$                            58,715
43	22	$                                      10,000	$                            54,365
44	21	$                                      10,000	$                            50,338
45	20	$                                      10,000	$                            46,610
46	19	$                                      10,000	$                            43,157
47	18	$                                      10,000	$                            39,960
48	17	$                                      10,000	$                            37,000
49	16	$                                      10,000	$                            34,259
50	15	$                                      10,000	$                            31,722
51	14	$                                      10,000	$                            29,372
52	13	$                                      10,000	$                            27,196
53	12	$                                      10,000	$                            25,182
54	11	$                                      10,000	$                            23,316
55	10	$                                      10,000	$                            21,589
56	9	$                                      10,000	$                            19,990
57	8	$                                      10,000	$                            18,509
58	7	$                                      10,000	$                            17,138
59	6	$                                      10,000	$                            15,869
60	5	$                                      10,000	$                            14,693
61	4	$                                      10,000	$                            13,605
62	3	$                                      10,000	$                            12,597
63	2	$                                      10,000	$                            11,664
64	1	$                                      10,000	$                            10,800
Total		$                                   430,000	$                      3,559,496

How I have calculated Value of contribution - Annual Amount*(1+ rate of Interest %)^N {N = number of years}

Here an important factor to understand is when John will make annual payments, he won't withdraw the amount till his retirement age. Hence, he will earn the compounded interest on his lump sum every year i.e 8% of annual contribution + 8% of contribution made in earlier years

Now that we have arrived at the lump sum amount that John will get at his retirement age, we need to discount it by rate of inflation to understand whether his one time expense and annual expenses are taken care of in Present Value terms of money. The calculation is as follows:

Future Value of money at the end of 65 years - $3,559,496

Inflation Rate - 2%

Present Value of money - FV/(1+ inflation rate %)^N

Therefore, Present Value - $ 1,519,082

The above amount matches for the amount required by John at the end of retirement. Hence, we can conclude that John will need to make an annual payment of $10,000 or a monthly payment of $834 at the end of each year. The amount will be enough for John to buy a boat for $300,000 while he can keep withdrawing $80,000 annually and keep earning interest on the remaining lump sum year after year.