# An investment of $1000 in the stock market earns 20% during the first year, but there is a loss of 10% in year 2 and a loss of 10% in year 3. Why is the net value not equal to $1000 at the end of...

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The initial investment is $1000. In the first year, there is a gain of 20% followed by 10% loss in the subsequent 2 years.

The arithmetic mean of 20 and -10 + -10 = -20 is equal to 0, but this is not applicable here. The reason behind this is that the initial gain of 20% is of $1000 which is equal to 0.2*1000 = 200. The following losses made are 0.1*1200 = 120 and 0.1*(1200 - 120) = 108. The total loss made is 120 + 108 = 228 while the total gain is only 200, giving a net loss of $28. The percentage loss is 2.8%

To determine the percentage loss as a mean, the correct way is to find the geometric mean of the growth(or loss) factor, i.e. 1 + profit. Doing this gives: `root(3)(1.2*0.9*0.9) ~~` 0.990578 . The percentage loss every year is (1 - 0.990578)*100 = 0.9421%. In three years, the loss is `1 - (1 - 0.9421)^3 ~~` 2.8%

In problems where the percentage change is not of the same number, it is not possible to use arithmetic mean to determine the net change. A more complex method is required, as shown here.