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I'm sure that someone with some knowledge of statistics will find this laughable, but here's one suggestion for separating upside and downside volatility.

Let's take the returns posted by Kevin Louche in Message #31867, for PEG4 Dual Semi, 1987-1998. In the table below, these appear under “Actual distribution”, where I have sorted them by CAGR rather than by year.

The traditional calculation of Standard Deviation assumes that these returns are distributed normally about their mean. We are having this discussion, though, because we are instead assuming that the distribution is skewed.

As an approximation, we'll follow Kevin's lead and separate the below-mean returns from the above-mean returns. Now, let's pretend that the below-mean returns represent the lower half of a normal distribution around the original mean. And that the above-mean returns represent the upper half of a normal distribution around the original mean.

In the second column of the following table I have reflected the below-mean returns across the original mean. For example, 6.00% is 40.84 percentage points below the mean, so we'll create a corresponding pretend return that is 40.84 percentage points above the mean: 46.84% + 40.84% = 87.68%.

So, with these reflected pretend returns shown in italics, we can calculate a S.D. for each of the two pretend distributions:

Actual Downside Upside
distribution distribution distribution
-------- -------- --------
6.00% 6.00% 9.18%
16.70% 16.70% 14.78%
28.60% 28.60% 25.88%
33.40% 33.40% 28.28%
33.60% 33.60% 32.88%
39.20% 39.20% 46.48%
47.20% 54.48% 47.20%
60.80% 60.08% 60.80%
65.40% 60.28% 65.40%
67.80% 65.08% 67.80%
78.90% 76.98% 78.90%
84.50% 87.68% 84.50%

Mean 46.84% 46.84% 46.84%
StDev 24.77% 24.58% 24.95%
Thus in this case, rather than reporting the arithmetic mean annual return as 46.8% (+/-24.8%), we would report it as 46.8% (+25.0%/-24.6%).

I guess if you instead do all this in log-space you'd proceed in the same way, except that the result would represent the geometric mean of annual returns. Hmm... let me fire up Excel again...

Okay. I've taken each percentage value, added 1 to it, and taken the natural log. Then I've reflected across the mean of these logs, as I did above, and calculated means and S.D.s. Finally, I've taken the anti-log of the results:

Actual Downside Upside
distribution distribution distribution
-------- -------- --------
Mean 44.89% 44.89% 44.89%
StDev 18.81% 19.45% 18.16%
Thus, the geometric mean annual return is 44.9% (+18.2%/-19.5%).

Dave Goldman
Portland, OR
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