Math 101: Is profit table correct? (Why adding percents leads to nowhere)

OK, seems that we will have a cycle (tag: math101) that explains some basics.

That is another question that surfaces from time to time, usually in the context of "I found a bug" or "I don't know how profit table works" or "these numbers make no sense".

So the user asks: I look at the Profit Table and yearly profit for 2009 is +139.25%, yet when I manually add up monthly percent gains it should be +97.80%

This is yet another case that shows that percents do NOT sum up arithmetically.

Say $20 stock grows one month +10% then next month falls down -10%.
If you sum percents you will get zero, but two month gain is NOT zero.

Initial price = $20
After one month = $20 + 10% = $22
After second month = $22 - 10% = $19.80

Two month gain = (19.80-20)/20 = -1%

The same way one year percent gain is NOT arithmetic sum of monthly percent gains.

Proper way to calculate final percent gain from compounded percent gains is to multiply:

(1+10%)*(1-10%)= 1.1 * 0.9 = 0.99 (of initial value) = -1% gain

In case presented above compounded gain is smaller than the sum of percents but it can be larger than sum too. If you have two months each gaining 10%, compounded gain is not 20% but more

(1+10%)*(1+10%)= 1.1 * 1.1 = 1.21 (of initial value) = +21% gain

Bottom line: percents don't sum up