Susan would like to set up an endowment fund which would award a student with a scholarship of $500 at the end of each year. The scholarship will continue in perpetuity. The interest rate is 6% compounded semi-annually, and the first award is made one year from today. How much should Susan invest today?
annual interest is
P((1+.06/2)^2-1)
so, you need
P(1.03^2-1) = 500
P = $8210.18
To calculate how much Susan should invest today, we can use the present value formula for perpetuity:
PV = A / r
Where:
PV = Present Value (Amount to invest)
A = Annual scholarship amount = $500
r = Interest rate per compounding period
First, we need to calculate the interest rate per compounding period:
Since the interest is compounded semi-annually, the interest rate per compounding period is half of the annual interest rate: r = 6% / 2 = 3% = 0.03 in decimal form.
Now we can calculate the present value (PV):
PV = A / r = $500 / 0.03 = $16,666.67
Therefore, Susan should invest $16,666.67 today to fund the perpetuity scholarship.
To calculate the amount Susan should invest today, we need to determine the present value of an infinite series of $500 payments occurring at the end of each year.
The present value of an infinite series can be calculated using the formula:
PV = PMT / (r / (1 + r))
Where:
PV = Present value
PMT = Payment amount
r = Interest rate per period
In this case, PMT is $500 and the interest rate is 6% compounded semi-annually. We need to convert the annual interest rate to a semi-annual rate by dividing it by 2, and then convert it to a decimal by dividing it by 100. So the semi-annual interest rate is (6/2)/100 = 0.03.
Using the formula, we can calculate the present value:
PV = $500 / (0.03 / (1 + 0.03))
Simplifying further:
PV = $500 / (0.03 / 1.03)
PV = $500 / 0.0291262
PV ≈ $17,182.36
Therefore, Susan should invest approximately $17,182.36 today to set up the endowment fund.