Why are you dividing by 1.5? If you lose 50%, you are left with 1/2 of what you had. Look at it in a different way. (100% + 20%) X (100% + 20%) X (100% + 20%) X (100% + 20%) X (100% - 50%) = (120% ^ 4) X (50%) = (120% ^ 4)/2 = (1.2 ^ 4)/2
That is how to compound. The first is: 1.2*1.2*1.2*1.2/1.5 division is for loss. The second is: 0.5*1.2*1.2*1.2*1.2 and if that's not obvious it is the point of the question to determine if somebody understands percentage compounding or not. Which you don't. Because the 20% gainer in the first four years would have way more profit if it had a 50% loss in the third year, the 20% gainer after a 50% loss in the first year would have much less to compound, and if the mathematical result is not exactly what I said the right answer is, you'll need to convince yourself visually without the confusing "^" sign. Fine, let's say you need another numerical reason why. We know 1.2^4 is 2.0736, and that's only a 107.36% return. Dividing by two means it is a 207.36% return because you haven't substracted the real return correctly. However, the dummy answer of 107.36*0.5=53.68% which is still greater than 1.0368 you'll get. Really, if you have less to compound to start with, that factor will always be less than the later losing one.
Anyone with a basic understanding of math would know they are testing for the communitive property: A*B*C = A*C*B
It is a compounding question. And the answer is 1.2^4/1.5-1>0.5*1.2^4-1. If that's not obvious, nobody knows what they're talking about and has no training. Put another way, if you do better after losing half, what the hell makes you think you if you did better then lost half would be the same thing? They aren't. This is a foolish example because it is a question of what your percentage return will be if you compound first, then lose, than if you lose first, then compound with less. Just think about it, guys!
Beau all you need to see is this. A X A X A X A X B = B X A X A X AX A It's a trick question. They're trying to see if you can see beyond just the math of investment returns or losses.
You don't understand the path dependency issue. They are not the same returns and the communitive property has nothing to do with it.
Yes, it does! 1.2^4 is the first four years=2.076-1=107.6% return, followed by a 50% loss, divide by 1.5=1.3824. If 50% is the first year, 0.5*(2.076) is not greater than 1.2^4/1.5. If the loss is first, there will be a lot less to compound later, whereas if the gain is first, there's a lot more left after the later loss. You guys are idiots if you don't understand this! Losing half the first year means there's a lot less money to compound with. Losing half in the fifth year, after you've made a 107% gain, is a lot more even if you do loss half in that year. This is not the communitive property. This is simple compounding and obviously you guys don't understand what you're talking about, because think about it. I lose half first, then make 20% per year for 4 years after. If it isn't obvious that if I make 20% per year in the first four years, then lose half, and that number is not greater than losing half first, then you guys are morons.