Question

Your friend is celebrating her 25^th birthday today and wants to start saving for her anticipated retirement at age 65. She wants to be able to withdraw $200,000 from her saving account on each birthday for 15 years following her retirement; the first withdrawal will be on her 66^th birthday. Your friend intends to invest her money in a retirement account, which earns 8 percent return per year. She wants to make an equal annual deposit on each birthday into the account for her retirement fund. Assume that the annual return on the retirement account is 8 percent before and afer her retirement. Ignore taxes and transaction costs for the problem.

a) If she starts making these deposits on her 26^th birthday and continue to make deposits until she is 65 (the last deposit will be on her 65^th birthday and the total number of annual deposits is 40), what amount must she deposit annually to be able to make the desired withdrawals at retirement? (Hint: One way to solve for this problem is to first find the value on your friend’s 65^th birthday of the $200,000 withdrawal per year for 15 years after her retirement and then find the equal annual deposit that she needs to make from her 26^th birthday to 65^th birthday.)

b) Suppose that whenever your friend makes the annual deposit to the retirement account, her employer will contribute $2,000 to the same account every year as part of the company’s profit sharing plan. In addition, your friend will receive a $30,000 distribution from a family trust fund on her 25^th birthday, which she will also put into the retirement account. What amount must she deposit annually now to be able to make the desired withdrawals at retirement?

Answer 1

Given information follows below:

Friend is celebrating her birthday =  25^th

Start saving for her anticipated retirement at age = 65 year

She wants to be able to withdraw= $200,000.

First withdrawal will be on her = 66^th birthday.

Returns percentage  in year = 8%

a) Calculating amount must she deposit annually to be able to make the desired withdrawals at retirement.

Given time period = 15 years

Amount = $200,000

Calculating Present value annuity factor :

$200,000 * PVAF(8%, 15)

= $200,000 x 8.5595

= $1,711,900

Annual deposit

X * FVF(8%, 40) = $1,711,900

X * 259.06 =  $1,711,900

X = $1,711,900 / 259.06

X = $6,608.12167066

or

X = $6,608.21

b) Calculating amount must she deposit annually now to be able to make the desired withdrawals at retirement

Employer will contribute = $2,000.

Friend will receive a = $30,000

((X +Employer will contribute) * FVF(8%, 40)) + Friend will receive =  $1,711,900

subtituting values

((X + $2,000) * FVF(8%, 40)) + $30,000 =  $1,711,900

((X + $2,000) * FVF(8%, 40)) =  $1,711,900 - $30,000

((X + $2,000) * FVF(8%, 40)) = $16,81,900

(X + $2,000) * 259.06 = $16,81,900

X = $4,492.32

$4,492.32 amount must she deposit annually now to be able to make the desired withdrawals at retirement.

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