Question

Aloma, a university graduate who started a successful business, wants to start an endowment in her name that will provide scholarships to ME students. She wants the scholarship to provide $11,000 per year and expects the first one to be awarded on the day she fulfills the endowment obligation. If Aloma plans to donate $180,000, what rate of return must the university realize in order to award the annual scholarship forever? The rate of return that the university must realize in order to award the annual scholarship forever is--------%.

Answer 1

In order to determine rate of return we must have to find rate of return at which Present Value of all scholarships must be equal to what Aloma Plans to donate.

If Aloma plans to donate $180,000, thus Present Value of all scholarships must equal 180,000

Present Value of amount A given after n years is given by;

PV = A/(1 + r)ⁿ

where r = rate of return and n = time in years

Here Annual amount of scholarships = 11000

Thus Present Value(PV) of all the scholarships is given by:

PV = 11000/(1 + r)⁰ + 11000/(1 + r)¹ + 11000/(1 + r)² + 11000/(1 + r)³ + 11000/(1 + r)⁴ + --------------infinity

=> PV = 11000 + 11000/(1 + r)¹ + 11000/(1 + r)² + 11000/(1 + r)³ + 11000/(1 + r)⁴ + --------------infinity

This series is of the form S = a + ax + ax² + ax³ - ------------------infinity

For such a series If -1 < x < 1 then S = a/(1 - x)

Here a= 11,000 , x = 1/(1 + r) and as r > 0 => -1 <1/(1+ r) < 1

=> NV = 11000/(1 - 1/(1 + r)) = 11000(1 + r)/r

Here NV must equals 180,000

=> 180,000 = 11000(1 + r)/r

=> r = 11000/(180000 - 11000) = 0.065

Hence r = 6.5%

Thus, The rate of return that the university must realize in order to award the annual scholarship forever is 6.5%