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If a price has first a +15% return and then second a -3% return, the arithmetic average is 6% and it is said that the geometric average is 5.617% (square root of 1.15*0.97). The term "geometric average" is to be contrasted with "geometric difference" as explained next.

If the total return is said to be 12% (summing 15% and -3%) and we want to invert the operation (take a difference) we can subtract the second term -3% to get the original 15%.

If the total return is calculated properly it is 11.55% (1.15 x 0.97 - 1) and to invert the operation we would divide 1.1155 by 0.97 to get the original 1.15 that represents 15%. Can this be said to be the "geometric difference" or would that be a misnomer or would that clash with other applications of the term?

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With a geometric mean you're supposed to multiply the two and take the square root, but seeing that one of those percentages is negative, then apparently a nontraditional definition is at play. –  J. M. May 11 '11 at 5:21
    
I think the traditional definition works with a negative and I have updated the question to show that. –  broiyan May 11 '11 at 5:27
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