Question

Hello, I have a question regarding perpetuties. I only need to answer Question #3, but thought having the info of #1 and #2 could be helpful.

1. You decide to give SCU an endowment that will pay out $50 K per year forever,
with a continuously compounded annual increase of 3%. Assuming that you can
lock in an interest rate of 5%, figure out how much this endowment would cost.
What is the total value of this income stream?
2. Suppose Aunt Grace wanted to give annual increases of $2,000 per year. How
would this change the computations above? Give values for the amount Aunt
Grace would have to pay to fund the income stream for 25 years, 50 years, 100
years, 200 years, and forever. (Hint: You need only integrate by parts once.)

3. You take all the information about Aunt Grace’s gift to your not-quite-so-
wealthy Aunt Margaret. In addition to the $1 M already deposited there by Aunt Grace, how much would Aunt Margaret have to add to the fund to enable it to pay out an income stream of

R(t) = 60 + t. forever?

Answer 1

Case 1: Assuming the first payout is one year from now
The stream can be broken into two streams:

First stream: Constant perpetuity of 61K every year forever
Present value=cash flow/rate=61/5%

Second stream: Arithmetic gradient of 1K forever
Present value=gradient/(rate)^2=1/(5%)^2

Total value needed=61/5%+1/(5%)^2
=1620 K

Additional funds needed=1620-1000=620K=620000

Case 2: Assuming the first payout starts immediately

Additional funds needed=620000+60000=680000