Question

(pleqase explain througougly with sentences and correct numbers)(need to walk me step by step how got the answer)

3.Mr. D. plans to retire exactly twenty years from now (t=0), and he would like to have accumulated, by retirement, enough money to enjoy a $100,000 per year retirement income beginning in year 21 and continuing in perpetuity thereafter. So far he has saved up $50,000, all in stocks (that is, at t=0 his pension account contains $50,000).

1. What must his annual contributions be if he is to achieve his goal (assume he makes 20 payments)? On average he expects to earn 10% on his money.
2. The stock market collapses. By the end of the day (it is still t=0)his accumulated wealth has fallen to $30,000. Assuming he still expects on average to earn 10%, how much must he now contribute (assume 20 equal payments)?

Answer 1

a. Mr. D needs to have enough accumulation at end yr.20 to draw $ 100000 every year , beginning Yr, 21 end, perpetually ,at 10% interest rate

so,we need to find the Present value of year-end perpetuity of $ 100000 at 10% opportunity cost of interest lost on savings ,that he would have other-wise earned--if not for setting aside for retirement.

which can be found by the formula,

PV of perpetuity=Annual retirement Income needed/Interest Rate%

ie. 100000/10%=

1000000

So, he needs to have a total of

$1,000,000

at end of his 20 yrs.,

The $ 50000 he has saved now, will be

using the Future Value of a single sum at the end of n years at a specified interest rate , formula

FV=PV*(1+r)^n

ie. FV at end Yr. 20=50000*(1+10%)^20=

336375

so, the balance he needs to accumulate by end of yr. 20=

1000000-336375=

663625

Now, the equal contributions   at 20 years-end ,whose FV at end -yr. 20 will be $ 663625 ,at 10% interest rate

can be found by using the Future value of year-end,ordinary annuity formula,

FVOA=Pmt.*((1+r)^n-1)/r

Plugging in all the available values, we can find the pmt.(annual contribution)

663625=Pmt.*((1+0.10)^20-1)/0.10

Solving for pmt. In an online equation solver,

we get the pmt.,ie. Annual contribution as

11586.64

or

$11,587

ANSWER: $ 11587

b.If his accumulated wealth at t=0, is only $ 30000

The $ 30000 he has saved now, will be

using the Future Value of a single sum at the end of n years at a specified imterest rate , formula

FV=PV*(1+r)^n

ie. FV at end Yr. 20=30000*(1+10%)^20=

201825

so,now the balance he needs to accumulate by end of yr. 20=

1000000-201825=

798175

Now, the equal contributions   at 20 years-end ,whose FV at end-yr. 20 will be $ 798175 ,at 10% interest rate

can be found by using the Future value of year-end,ordinary annuity formula,

FVOA=Pmt.*((1+r)^n-1)/r

Plugging in all the available values, we can find the pmt.(annual contribution)

798175=Pmt.*((1+0.10)^20-1)/0.10

Solving for pmt. In an online equation solver,

we get the pmt.,ie. Annual contribution as

13935.84

or

$13,936

ANSWER: $ 13936