How Long Will the Money Last?

Date: 03/31/2007 at 16:41:22
From: Duke
Subject: how long will a lump sum of money last 

Someone has a principal amount of $1,000,000.  Interest rates are 
5%. Inflation is 2%.  The person needs $75,000 per year on which to 
live.  Assume interest income is taxable at 33.3%.  How long will it
take to deplete the entire principal?

I am lost as to the approach.  Thank you very much.

Date: 04/01/2007 at 11:43:47
From: Doctor Vogler
Subject: Re: how long will a lump sum of money last

Hi Duke,

Thanks for writing to Dr. Math.  First let's tackle the interest:  You
get 5% interest, but you lose a third of that to taxes, leaving 10/3
out of 15/3, or about 3.33% left for you.  But then you lose 2% to
inflation.  If you express your new balance in today-dollars (which is
the easiest way to do it mathematically), then it takes 102% of
next-year's money to equal 100% of today's money, which means that
your 103.333% new balance is only worth (103.33/102) or about 1.01307
times your balance last year.  In other words, you only get about 1.3%
growth on your money after taxes and inflation.

That's not much, so you can estimate that your money will last about
1,000,000/75,000 years, or a little over 13 years.  Actually, the 1.3%
growth extends that to a little over 14 years, which you can easily
check with a calculator:

  Start with 1000000.
  Multiply by 1.01307, and subtract 75000.
  Repeat 14 times.
  The 15th time, your balance will go negative.

Alternately, you could use a formula.  If your balance after n years
is given by

  B(n),

then you have

  B(0) = $1,000,000

and

  B(n+1) = ((1 + 10%/3)/(1 + 2%))*B(n) - $75,000
         = (155/153)*B(n) - $75,000,

and then you can use the formula for a linear recursive sequence,
which gives:

  B(n) = $5,737,500 - $4,737,500 * (155/153)^n,

Set this equal to zero and solve for n using logarithms.  How do we
get this formula?  Well, first we find the fixed point, where

  B(n+1) = B(n),

which you get by dividing $75,000 by (155/153 - 1), and then you get

  $5,737,500.

(Note:  This is the balance you would need for your money to generate
$75,000 of interest each year after taxes and inflation.)

Subtract this from both sides of your recurrence to get

  B(n+1) - $5,737,500 = (155/153)*(B(n) - $5,737,500),

which means that

  B(n) - $5,737,500 = (155/153)^n * (B(0) - $5,737,500),

and therefore

  B(n) = (155/153)^n * (B(0) - $5,737,500) + $5,737,500.

If you have any questions about this or need more help, please write
back and show me what you have been able to do, and I will try to
offer further suggestions.

- Doctor Vogler, The Math Forum
  http://mathforum.org/dr.math/