Question

​(Complex annuity and future value​)

​Milhouse, 25​, is about to begin his career as a rocket scientist for a NASA contractor. Being a rocket​ scientist, Milhouse knows that he should begin saving for retirement immediately. Part of his inspiration came from reading an article on Social Security in Newsweek​, where he saw that the ratio of workers paying taxes to retirees collecting checks will drop dramatically in the future. In​ fact, the ratio was 8.6 workers for every retiree in​ 1955; today the ratio is​ 3.0, and it will drop to 2 workers for every retiree in 2035.​ Milhouse's retirement plan pays 14 percent interest annually and allows him to make equal yearly contributions. Upon​ retirement, Milhouse plans to buy a new​ boat, which he estimates will cost him

​$200,000 in 40 years​ (he plans to retire at age 65​). He also estimates that in order to live​ comfortably, he will require a yearly income of $70,000 for each year after he retires. Based on family​ history, Milhouse expects to live until age 85 (that is, he would like to receive 2020 payments of $70,000 at the end of each​ year). When he​ retires, Milhouse will purchase his boat in one lump sum and place the remaining balance of his savings into an​ account, which pays 88​% interest, from which he will withdraw his $70,000 per year. If​ Milhouse's first contribution is made 1 year from​ today, and his last is made the day he​ retires, how much money must he contribute each year to his retirement​ fund?

a. In order for him to withdraw 2020 payments of $7,000 at the end of each year after he​ retires, how much money will Milhouse need to have in the retirement account earning 88​% when he​ retires?

b. Upon​ retirement, Milhouse also plans to buy a new​ boat, which he estimates will cost him $200,000 in 40 years. How much money should Milhouse have in the retirement fund to achieve the goal when he​ retires?

c. Milhouse, 25, is about to begin saving for retirement immediately. His retirement plan pays 14 percent interest annually and allows him to make equal yearly contributions. If​ Milhouse's first contribution is made 1 year from​ today, and his last is made the day he retires at age 65​, how much money must he contribute each year to his retirement fund to achieve the goal when he​ retires?

Answer 1

Part (a):

Amount required in retirement account on retirement for enabling 20 withdrawals is the present value of annuity of $7,000 each. The amount required is $68,727.03 calculated as follows:

Part (b): Amount required to buy the boat, on retirement is given as $200,000

Part (c):

Amount required in 40 years in order to facilitate purchase of boat and 20 withdrawals of $7,000 each = $68,727.03 + $40,000 = $108,727.03

Yearly contribution required (interest rate 14%) = $81.02 as follows: