Solving an Actuarial Problem: Investing $10,000 for Annual Scholarships

[ToxicBug]

This is a problem on an assignment for my actuarial class.

A sum of 10,000 was invested on September 1, 1970 at an effective annual interest rate of 5% in order to provide an annual scholarship of 2000 every September 1 forever, starting as soon as possible. In what year will the first payment of 2000 be made? What smaller payment could be made one year earlier while still permitting the annual scholarships of 2000 thereafter? Assume that interest is credited every August 31.

First of all I found how much time it would take for the investment to reach the present value of the perpetuity:

10000(1 + i)^n = 2000/i
10000(1 + 0.05)^n = 2000/0.05

n = ln(4)/ln(1.05)
n = 28.41339817

Then for the second part I did this:

X + 10000(1.05)^(28.41339817 - 1) = 2000/0.05
X = 1904.7618

But the answer in the back of the book is 1161.36

Anyone know what is my mistake?

 

[courtrigrad]

do you mean $$10000(1+i)^{t}$$?

 

[shmoe]

Then for the second part I did this:

X + 10000(1.05)^(28.41339817 - 1) = 2000/0.05
X = 1904.7618

But the answer in the back of the book is 1161.36

Anyone know what is my mistake?

The scholarship payments aren't starting at year 28.41339... Year 29 is the first year you have enough to sustain the perpetuity, the first payment is at year 30 though. The excess payment would be at year 29.

 

[ToxicBug]

Brilliant, thanks!

 

[ToxicBug]

Another question if you don't mind, I would like to get a hint on what I'm supposed to do:

There is $40,000 in a fund which is accumulating at 4% per annum convertible continuously. If money is withdrawn continuously at a rate of $2300 per year, how long will the fund last?

 

[courtrigrad]

$$P = P_{0}e^{rt}$$

 

[ToxicBug]

Tried that, didn't work.

 

[ToxicBug]

Any ideas?

 

[physics girl phd]

there are two things involved each year -- a 4% growth, but a subtraction of $2300... seems like your first try only took into account one of those. Any ideas on how to stick in the other?