Looking at Compound Interest
Date: 01/08/2007 at 11:34:09 From: Ken Subject: Nature of compounding interest. I was playing with compounding interest. In the process of looking at how interest works I noticed something that seemed counter to what I initially would have expected. Then when I looked at it another way it acted yet differently. Say you have $100. You invest it and over the next four years you get 20% returns for two years and 10% returns for two years. I donít know why, but I just assumed that if you earned the higher interest first you would make more in the long run. It turns out that itís the same result either way: Low Interest then higher interest $100 x 1.1 = $110 $110 x 1.1 = $121 $121 x 1.2 = $145.20 $145.20 x 1.2 = $174.24 High interest then low interest $100 x 1.2 = $120 $120 x 1.2 = $144 $144 x 1.1 = $158.40 $158.40 x 1.1 = $174.24 Next, I tried averaging the interest to see what would happen. 20 + 20 + 10 + 10 = 60 60/4 = 15 $100 x 1.15 = $115 $115 x 1.15 = $132.25 $132.25 x 1.15 = $152.0875 $152.0875 x 1.15 = $174.90 It appears as though there is a slight gain by using a constant amount rather than a rate that fluctuates but averages the constant amount. None of this makes sense to me. I would have assumed that earning a higher percentage early would pay off, but obviously I was wrong. Then I assumed that being there is no difference if the rate fluctuates that there would be no difference if the rate was averaged, and again I was wrong. Why?
Date: 01/08/2007 at 16:29:53 From: Doctor Peterson Subject: Re: Nature of compounding interest. Hi, Ken. In your work you came very close to seeing WHY the order doesn't matter. If you write out the work for the first two cases (low first, and high first) in one line each, it looks like this: $100 * 1.1 * 1.1 * 1.2 * 1.2 = $174.24 $100 * 1.2 * 1.2 * 1.1 * 1.1 = $174.24 You're doing the same four multiplications in a different order, so you get the same result--that's called the commutative property of multiplication, and you probably never thought it would show up in such a practical way! When you averaged, you used what is called the "arithmetic mean": AM = 1/n * sum of n numbers This gives the number with which each number could be replaced if you wanted them to give the same SUM. But here, you are MULTIPLYING by the numbers, so the "average" that works is what we call the "geometric mean": GM = nth root of product of n numbers There is a theorem that says the GM is always less than the AM: Arithmetic/Geometric Mean Inequality Theorem http://mathforum.org/library/drmath/view/51596.html So when you took the arithmetic mean, you used an interest rate slightly larger than that which would give the same total interest over the four years. This shows that if you want to find the average effect of a fluctuating interest rate, you should take the geometric mean of 100% plus the rate. You're discovering some interesting practical facts based on some basic (and some not-so-basic) math! If you have any further questions, feel free to write back. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
Date: 01/08/2007 at 20:02:29 From: Ken Subject: Thank you (Nature of compounding interest.) Thank you so very much for your explanation. You answered all aspects of my question in a very thorough and professional manner. Your response should be used as an example in customer service and customer support training. I am extremely impressed and will pass word about this site to all my friends, co-workers and online programmer communities.
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