Question

a college has a scholarship fund that pays a sum of money twice a year. the scholarship will pay out 500 at the end of six months and another 500 at the end of one year. every year thereafter, the two semi-annual payments will be increased by 10. for example, in year two, both payments will be 510 and in year three both payments will be 520. the scholarship fund earns interest at an annual effective interest rate of 7.5%. calculate the fund balance needed today to provide this scholarship indefinitely.

Answer 1

The college needs to calculate the present value of a perpetuity with increasing payments to determine the fund balance required today. Using the given semi-annual payment, growth rate, and interest rate, we apply the formula for a perpetuity with arithmetic growth to find the amount. The present value calculated will provide the scholarship indefinitely when considering the compound interest earned by the fund.

Step-by-step explanation:

To calculate the fund balance needed today for the described scholarship fund that pays out semi-annual increasing payments indefinitely, we need to apply the concept of a perpetuity with increasing payments. The payments increase by $10 each year, which gives us a perpetuity with arithmetic growth. The formula for the present value of such a perpetuity is P = (C/r) + (G/r2), where P is the present value we are trying to find, C is the initial payment, r is the interest rate, and G is the growth in payments.

Here, the initial semi-annual payment is $500, the annual effective interest rate is 7.5%, and the growth is $10 per year, which is $5 per payment. The effective semi-annual rate is (1 + 0.075)1/2 - 1. Using this, we can calculate the present value needed to fund this scholarship in perpetuity.

The calculation will yield the required amount that must be in the fund today to sustain the scholarship payments indefinitely, considering the interest rate and the compound interest that accumulates on the fund.