You want to have $5 million when you retire in 40 years. You believe you can earn 9 percent per year on your investment. How much must you invest each year to achieve your goal when you retire? (Ignore all taxes.)
A)$10,412
B)$11,619
C)$14,798
D)$15,295
E)None of these choices are correct.

Question

Quizplus · Accepted Answer

This is a case of annuity due, where you invest the same amount every year. Using the formula for present value of an annuity due:

PV = PMT × [(1 + r) × ((1 + r)^n - 1) / r]
where PV is the present value (which we want to solve for), PMT is the payment each year, r is the annual interest rate, and n is the number of years.

Plugging in the numbers:
PV = PMT × [(1 + 0.09) × ((1 + 0.09)^40 - 1) / 0.09]
PV = PMT × 214.4622

We want PV to be $5 million, so:
$5,000,000 = PMT × 214.4622
PMT = $23,286.66

However, we need to adjust for the fact that this is an annuity due, where the payment is made at the beginning of the year. We can use the formula for the future value of an annuity due to adjust:
FVAD = FV + (PMT × ((1 + r)^n - 1) / r) × (1 + r)
where FV is the future value, FVAD is the future value of an annuity due, and the rest are the same as before.

Plugging in the numbers:
$5,000,000 = FV + (PMT × ((1 + 0.09)^40 - 1) / 0.09) × (1 + 0.09)
$5,000,000 = FV + PMT × 994.5357
FV = $2,400,000.00

So the payment each year, adjusted for annuity due, is:
PMT = ($5,000,000 - $2,400,000) / 214.4622
PMT = $14,798.26

Therefore, the answer is choice C.

You Want to Have $5 Million When You Retire in 40