Perpetual Annuity

Question

I'm working on the question:

A benefactor wishes to endow in perpetuity, five annual scholarships of $5,000 at a university. What is the minimum amount she must invest at an annual interest rate of 2.32%, in order to fund the scholarship?

This question seems easy, but I don't think I'm going about it the right way. "In perpetuity" means with no defined end. So presumably she continues on for a long time. I assume we want to find the amount she will invest at the beginning of the year in order to get the $25000 by the end of the year. So, if we use the future value equation, we'd get 25000 = PV * (1.0232)^1, where we solve for PV.

Is this question truly this easy, or am I making a mistake here? Any help is appreciated!

Answer 1

Assuming that the scholarships are awarded at the end of the year (this is a reasonable assumption - the language of the problem doesn't really tell you whether the scholarship starts immediately or later), then you need to put enough money in the investment to have an extra \$25,000 ($= 5 \times \$5,000$) at the end of the year.

So you need the interest: $I=Prt = P(0.0232)(1) = 25000$. This means you need $P = 25000/0.0232 = \$1,077,586.21$ endowed.

Perpetuities just pull off the interest (they don't touch the principal).

By the way, if you start giving out the scholarship immediately, just up the endowment by that much: $ \$1,077,586.21 + \$25,000$ :)

Answer 2

Hint: Your computation only seems to take into account a single withdrawal.

The present value of all future withdrawals (assuming the first occurs immediately), writing $v=\frac{1}{1.0232}$, is $$25000\cdot(1 + v + v^2 + v^3+\cdots)$$

If the first occurs in a year, you will have another factor of $v$ on every term. You should be able to write a closed form for the sum of this geometric series.