In one of the Minister’s original studies, the monthly Dow moves are sorted by month, and the average % move for each month is calculated, e.g., the average of all January moves is +1.44%, February +0.36%, and so on.

The mean and standard deviation are calculated for the resulting 12 percentages: mean = +0.62%, sd = 0.88%. Then the Minister ran a few sims, re-sorting the monthly % moves amongst the months to see what kind of standard deviations resulted. All the sim runs produced standard deviations below the actual one. This implies some non-random structure in the pattern of returns over the calendar.

Taking essentially the same data (with an extra year added, hence a slightly different sd), and running 1000 re-sorts, produced the following:

mean of 1000 re-sort sd’s: +0.59%
sd of the series of sd’s: +0.12%
max re-sort sd: +1.02%
min re-sort sd: +0.25%
>> actual sd: +0.86%
z of actual: 2.25

So, the standard deviation of the actual was not outside the range of the sd’s of the sim run, but did have a significant z.

Because one is dealing with percentages, one can’t help but wonder about the effect of changing volatility regimes. For example, in a high-volatility year perhaps September is the worst month with a move of -2% and December the best with +2%; then in a low volatility year, September is the best with +1% and December the worst with -1%. The “average” move for September is now -0.5%, and for December +0.5%. But it may be that September was simply “unlucky” in having its big up-move in a low volatility year and its big down-move in a high-volatility year.

To attempt to factor in the volatility-regime effect, all monthly % moves were converted to z-scores relative to the mean and sd of the preceding 12-month period. Then the same analysis was conducted, with 1000 sim runs resorted the z-scores amongst the months.

mean sd of the 1000 re-sorts: 0.164
sd of the sd’s: 0.035
max sd: 0.290
min sd: 0.080
>>actual sd: 0.219
z of actual: 1.58

So, one proposes that the “volatility regime factor” is the difference between the z = 2.25 of the study done with raw % moves, and the z = 1.58 of the same study done with moves normalized against the preceding 12 months.


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