David will retire to Florida in ten years. After he retires, he wants to take out $28,000 at the end of each year for 15 years. If he can invest the money at 8% annually, what amount must he invest today?
To find out how much David must invest today, we can use the present value of an annuity formula. The present value (PV) of an annuity is the current value of a series of future cash flows, discounted back to today's dollars.
The formula to calculate the present value of an annuity is:
PV = C * (1 - (1 + r)^(-n)) / r
Where PV is the present value, C is the cash flow per period, r is the interest rate per period, and n is the number of periods.
Now, let's plug in the given information into the formula:
C (cash flow per period) = $28,000
r (interest rate per period) = 8% = 0.08 (as a decimal)
n (number of periods) = 15
First, let's calculate (1 + r)^(-n):
(1 + 0.08)^(-15) = 0.315972
Now, let's plug in the values into the formula:
PV = $28,000 * (1 - 0.315972) / 0.08
Calculating this, we get:
PV = $28,000 * 0.684028 / 0.08
PV = $238,281.50
Therefore, David must invest approximately $238,281.50 today in order to have enough money to withdraw $28,000 at the end of each year for 15 years, assuming an 8% annual interest rate.