Question

The trustees of a college have accepted a gift of ​$​400,000, but are required to deposit it in an account paying 12​% per​ year, compounded semiannually. They may make equal withdrawals at the end of each​ six-month period, but the money must last 5 years.

a. Find the amount of each withdrawal.
b. Find the amount of each withdrawal if the money must last 7 years.

Answer 1

a. To find the amount of each withdrawal if the money must last for 5 years, we first need to calculate the future value of the $400,000 investment after 5 years, using the formula:

FV = PV * (1 + r/n)^(n*t)

where:
PV = present value = $400,000
r = annual interest rate = 12% = 0.12
n = number of compounding periods per year = 2 (since interest is compounded semiannually)
t = number of years = 5

Substituting the values into the formula, we get:

FV = 400000 * (1 + 0.12/2)^(2*5) = $734,449.73

The total number of withdrawals over 5 years will be 2 * 5 = 10 (since withdrawals are made every six-month period). Therefore, the amount of each withdrawal can be found by dividing the future value by the total number of withdrawals:

Withdrawal amount = FV / number of withdrawals = $734,449.73 / 10 = $73,444.97

So the amount of each withdrawal must be approximately $73,444.97.

b. To find the amount of each withdrawal if the money must last for 7 years, we can follow a similar approach as in part (a), but with t = 7 and a total of 2 * 7 = 14 withdrawals.

The future value of the $400,000 investment after 7 years is:

FV = 400000 * (1 + 0.12/2)^(2*7) = $1,007,128.23

The amount of each withdrawal is:

Withdrawal amount = FV / number of withdrawals = $1,007,128.23 / 14 = $71,937.73

So the amount of each withdrawal must be approximately $71,937.73.