Question

You are saving for retirement. To live​ comfortably, you decide you will need to save $3,000,000 by the time you are age 65. Today is your 27th ​birthday, and you​ decide, starting today and continuing on every birthday up to and including your 65th ​birthday, that you will put the same amount into a savings account. If the interest rate is 10 %​, you set aside $ 7473 each year to make sure that you will have $3,000,000 in the account on your 65th birthday. You realize that your plan has a flaw. Because your income will increase over your​ lifetime, it would be more realistic to save less now and more later. Instead of putting the same amount aside each​ year, you decide to let the amount that you set aside grow by 12 % per year. Under this​ plan, how much will you put into the account​ today? (Recall that you are planning to make the first contribution to the account​ today.)

Answer 1

If we have $3,000,000 at the age of 65 then,

when interest rate is 10%, which are compounded annually, then installment for the same is -

Future value = Present value ( 1 + r )ⁿ

$3,000,000 = Present value ( 1 + 10/100 )^{(65 - 27)}

Present value = $3,000,000 / ( 1 + 0.1)³⁸

Present Value = $ 3,000,000 / 37.4043434

Present value = $ 80,204.589

If he contribute $ 80,204.589 at the interest rate of 10% then, at the age of 65, he has $3,000,000.

If the interest rate is 12%, then amount to be contributed per year to have $3,000,000 at the age of 65.

$3,000,000 = Present value ( 1 + 12/100)³⁸

Present value = $3,000,000 / ( 1 + 0.12)³⁸

Present value = $3,000,000 / 74.1796639

Present value = $ 40,442.351

If he invest $ 40,442.351 at the interest rate of 12% per year he can get $3,000,000 at the age of 65.