Net Result of Consecutive Percentage Increases

Date: 12/11/2003 at 12:59:32
From: Perplexed Percenter
Subject: Percentage increase of a percentage

Dr Math,

I haven't worked with percentages for a good while, so I am trying to
get my head around the theory of increasing a percentage by a
percentage.  What I'm trying to figure out is if I have an investment
that has returned a 20% increase in performance from the start of the
year until October, and then increased a further 3% in November, what
the year to date increase has been.

I was given the formula (1 + 20%)*(1 + 3%) - 1 which appears to work,
but I am trying to understand how the underlying math works.  What
role do the 1's play in that calculation? 

Date: 12/12/2003 at 10:52:31
From: Doctor Peterson
Subject: Re: Percentage increase of a percentage

Hi, Perplexed.

Suppose you start with X dollars, and add a fraction A to it (for 
example, A = 0.20 represents 20%); then the amount you have now is

  X + AX = (1 + A)X

In our example, that is 120% of what we started with.

Now increase this by another fraction B (your 3%); you now have

  (1 + B)(1 + A)X

Do you follow that?  We've done the same thing with a different 
percentage, applying it to the result of the first increase.  So we 
have 103% of 120% of the original amount.

Now, we have increased this much from the original amount:

  (1 + B)(1 + A)X - X

And the percentage increase is that divided by the original amount:

  [(1 + B)(1 + A)X - X]/X = (1 + B)(1 + A) - 1

That's your formula.

Now let's break this down a bit by expanding it:

  (1 + A + B + AB) - 1 = A + B + AB

This says that, in your example, we increased the original by 20%, and
we also increased it by 3%, and we ALSO increased the 20% increase by
3%!  That is, our overall percentage increase is

  20% + 3% + 20%*3% = 0.20 + 0.03 + 0.20*0.03 = 0.20 + 0.03 + 0.006
                    = 0.236 = 23.6%

We can picture it like this:

  +------------+----+
  |            |    |
  |            |    |
  |     X      | BX |
  |            |    |
  |            |    |
  |            |    |
  +------------+----+
  |     AX     |ABX |
  +------------+----+

The change is AX + BX + ABX, and the fractional increase is A + B + AB.

Now you should see where each part of the formula comes from, and also
how to extend it to any number of increases.

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

Date: 12/15/2003 at 12:31:15
From: Perplexed Percenter
Subject: Thank you (Percentage increase of a percentage)

Dr Peterson--I've got it!  Many thanks for your time and trouble
explaining this.  This website is terrific, keep up the good work!