university alumni group wants to provide a semiannual scholarship to current students. If the scholarship amount of $2,800 is awarded semiannually beginning 6 months from now, and the fund earns interest at a rate of 7% per year compounded continuously, find the amount of the endowment the alumni are required to make now.
To determine the amount of the endowment required by the alumni group, we need to calculate the present value of the semiannual scholarship payments. The present value represents the current value of future cash flows.
Here's how you can calculate it:
Step 1: Convert the annual interest rate to the continuously compounded interest rate (r).
r = ln(1 + i)
Where i is the annual interest rate divided by 100.
In this case, i = 7, so r = ln(1 + 0.07) = 0.0677
Step 2: Determine the number of periods (n) and the periodic interest rate (m).
Semiannual payments mean there are two payment periods per year.
The total number of periods in this case would be twice the number of years of scholarship payments.
Since the scholarships start 6 months (0.5 years) from now, and they are awarded semiannually, each scholarship payment would be made for a period of 0.5 years.
Therefore, n = 0.5 * 2 = 1 and m = r/2 = 0.0677/2 = 0.03385
Step 3: Calculate the present value (PV) using the formula:
PV = FV / (1 + m)^n
Where FV is the future value of the scholarship payment.
In this case, FV = $2,800
Substitute the values into the formula:
PV = $2,800 / (1 + 0.03385)^1 = $2,800 / (1.03385) = $2,700.51 (rounded to two decimal places)
Thus, the amount of the endowment the alumni are required to make now is approximately $2,700.51.