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Charley --

When doing the upside SD, modify the downside returns so that -50% is -100% (the upside equivilant to recover).

When doing the downside calculation, change the upside returns to mirror what its downside equivilant would be. That is +100% would be changed to +50%. and +200% would be change to +66%.

No problem. We accomplish this simply by adding 1 to the percentages:

-66% -> 0.34
-50% -> 0.50
+100% -> 2.00
+200% -> 3.00
You'll see that 0.50 x 2.00 = 1.00, and that 0.34 x 3.00 = 1.00.

This is why some people on these boards report returns as, for example, 1.43 rather than +43%.

If you don't introduce logs, then this step of adding 1 doesn't change the results when calculating arithmetic means and SDs. If you are going to introduce logs, though, then you must add 1 as above (since the log of a percentage change doesn't have any obvious meaning).

I know I don't understand the implications of the log converted numbers.

The point of introducing logs (I think) is that here we really care about products, not sums.

For example, at first glance the difference between returns of +10% and +20% appears to be of the same magnitude as the difference between +80% and +90%, namely "10 percentage points". Even if you add 1 to get growth rates (1.10 and 1.20 vs. 1.80 and 1.90) these still both look like a difference of "0.1 in growth rate factor". But in fact the important observation is that the first difference represents a difference in your returns of a factor of 1.09 (1.20/1.10), while the second difference represents a difference of a factor of 1.06 (1.90/1.80). In other words, if you can improve your return from +10% to +20%, this represents a bigger difference than an improvement from +80% to +90% --- a 9% improvement vs. a 6% improvement.

Converting to logs allows us to then do addition and subtraction on the logs, which is equivalent to doing multiplication and division on the original numbers. For example, the arithmetic mean of the logs is equivalent to the geometric mean of the originals. When you graph these things (on paper or, as in the preceding paragraph, in your head) our naturally linear psychology does a better job of appropriately interpreting the log-converted graphs.

Dave Goldman
Portland, OR
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