Endowment Awards

Date: 05/09/97 at 02:33:50
From: donna
Subject: Business calculus

An endowment of $15,000 is made. It earns 7.25 percent interest 
compounded continously, which is allowed to accumulate. How large must 
the endowment grow for it to support an annual award of $1,200.00 in 
perpetuity (that is, without dimishing the capital of the endowment)?

I started: 15,000e^.0725(k+1) - 15,000e^-.0725t = 1,200.00
            15e^1e^.0725k - 15e^.0725k = 12

Now I'm stuck!

Date: 05/09/97 at 16:32:08
From: Doctor Anthony
Subject: Re: Business calculus

If interest is compounded continuously, the expression for the amount 
after t years is:

         A = Pe^(rt/100)    where P = 15000, r = 7.25

So the endowment will grow to 15000e^(.0725t) = 15000(1.0752)^t in t 
years.

We must allow this to grow to a total which will generate $1200 
interest per year at 7.25 percent compounded continuously:

     Ae^.0725 - A = 1200       
   A(e^.0725 - 1) = 1200
   A = 1200/.0752 = $15959   

So once we have accumulated $15959, the endowment will support the 
$1200 annual award:

    15000(1.0752)^t = 15959
           1.0752^t = 1.06393    
        tln(1.0752) = ln(1.06393)
                  t = ln(1.06393)/ln(1.0752)
                  t = 0.8547 of a year
                  t = 10.25 months

We allow the $15000 to accumulate for 10 and a quarter months, and it 
will then be able to maintain the award.
 
-Doctor Anthony,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/