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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.


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 

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 

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.
Associated Topics:
High School Interest

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