Question

4. How much must you deposit each year into your retirement account starting now and continuing through year 8 if you want to be able to withdraw $50,000 per year forever, beginning 29 years from now? Assume the account earns interest at 13% per year.

Answer 1

First, we will calculate the amount requirement at the end of 28th year from now to enable the withdrawal of $50,000 forever.
Amount required = $50,000 / 0.13
= $384,615.38

Now we will calculate the present value of above calculated amount as on end of 8th year.
Present value as on 8th Year beginning = $384,615.38 / (1.13)^20
= $384,615.38 / 11.5231
= $33,377.81

Now we know the amount required at the end of Year 8, so we will calculated the annual investment required.
Future value of annuity due = P * [ ((1 + r)^n - 1) / r] * (1 + r)
Here P = annuity
r = interest rate = 0.13
n = no. of payments = 8

Future value of annuity due = P * [ ((1 + r)^n - 1) / r] * (1 + r)
$33,377.81 = P * [ ((1 + 0.13)^8 - 1) / 0.13] * (1 + 0.13)
$33,377.81 = P * 12.7573 * 1.13
$33,377.81 = P * 14.4157
P = $33,377.81 / 14.4157
P = $2315.38 i.e. annual investment.