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Hi Tylor,

While I've been binge-reading your blog over the past couple of days and enjoying it greatly, I see an issue with this claim:
"A 50% gain followed by a 33% loss brings you a total gain of 0%, but a mean of 8.5%" - how so? The adequate mean in this case is the geometric mean and it has to be calculated not on the raw data but by converting it to growth percentages/proportions first due to the fact geometric means cannot be computed with negative numbers. So 50% becomes 150% and -33% becomes 67% The result is then a geometric mean of 0.249688278817% which exactly matches the average return after two periods wherein you gain 50% and then lose 33% ($100 x 1.00249688278817 x 1.00249688278817 = $100.4999999999998). Here is a calculator I've coded a few years ago which handles this scenario nicely and is afaik the only publicly available one which does so (e.g. Excel flat out refuses to compute it): https://www.gigacalculator.com/calculators/geometric-mean-calculator.php?numbers=50%25+-33%25

I'll have to think if this changes anything with your argument made using the example or if it is just a poor example.


P.S. Apologies for using your family name Yuval, I just got so used to typing 'Tylor' in front of the many posts I've saved for future reference :-)

Yuval Taylor

Georgi: your math is a bit off. The arithmetic mean of 50% and -33% is 8.5% and the geometric mean is 0% ((1.5 x 0.67)^0.5 - 1). Excel does caculate geometric mean with the GEOMEAN command.


Definitely not off. $100 x 50% x -33% = $150 x -33% = $100.5. The arithmetic mean does not apply to ratios, it is the geometric mean that you need to use in any such case.


Definitely not off. 50% growth followed by a 33% loss is $100 x 150% x -33% = $150 x -33% = $100.5. The arithmetic mean does not apply to ratios, it is the geometric mean that you need to use in any such case.


And here is proof that Excel (latest version part of Office 365) doesn't compute geomean with negative values, percentages or proportions: https://imgur.com/a/RRXrOCS


FYI using the geomean calculation you correctly provided (((1.5 x 0.67)^0.5 - 1)) results in 0.25%, not 0%: https://imgur.com/a/6wL1O9u . Sorry for the multiple comments, but there is no "edit" feature for these comments on my side.


Final note. I had to come back to this since initially looking at the formula you provided I thought it is the correct formula as given in multiple sources and should usually do the job. Comparing your result of 0.250% and mine of 0.249688278871% I thought the difference, though large, was just due to a rounding error. But then I used your result to check the return and with 0.250% it actually comes out to $100.500625 (($100 x 100.25%) x 100.25%) which is a definite overestimation of the true value of $100.50. I had to see why that is and checked my code and I actually use the log>exp solution I explain in the second paragraph of "How to calculate Geometric mean?" in my calculator. It seems at least in some cases log>exp leads to more accurate results. Using my outcome you get $100.4999999999998 where the difference to $100.50 is indeed minuscule and due to errors inherent in floating number calculations, however it is many times smaller than the 0.000625 discrepancy from the canonical formula you provided. Excel seems to also be using the log>exp solution as it confirms the result from my geomean calculator: https://imgur.com/a/jIveRFA

Yuval Taylor

Wow. This is a lot of commenting on the fact that I wrote "33%" instead of "33.333333333%." I assumed that most people would intuit what I meant by a 33% loss negating a 50% gain.


It is primarily about the incorrect average being used - the arithmetic one, rather than the geometric one. Kind of spiraled out of control, I agree :-)


Excellent as usual!

Did your mutual fund data have a survivor-ship bias? That might explain why the single best predictor of future performance was low one year returns.

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