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!
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