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/
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