Rankings and Cycles, from Victor Niederhoffer
One way to see if cycles occur is to rank closing prices and see if the distribution of ranks is consistent with independence.
take 5 prices 800, 801, 799, 804, 803,
let's rank them 2 3 1 5 4 from lowest to highest.
The distribution of actual ranks for the last 10years, 1, 1 99 to present in S&P adjusted futures prices is as follows:
rank number of times
1 637
2 418
3 338
4 404
5 733
Queries: Is this consistent with independence? What does this show about the shape of the distribution? How can this be generalized? How does it compare to other markets? Is there any predictive value to such studies?
Victor Niederhoffer adds:
mkt rank S&P Bonds Bund Yen Gold Crude
1 637 612 624 672 395 430
2 418 328 354 409 277 294
3 338 382 352 405 265 257
4 404 406 404 431 318 295
5 733 781 789 709 596 529
drift -03 01 02 00 03 00 per day
Sd 154 69 37 27 81 142
The starting point is 1/1/1999 to present daily for all except gold and crude. Note unusal number of rank 2 for bonds. The surprises are the number of rank 5 for stocks which exceed the rank 1 even though there was a drift of -03 a day. This is consistent with larger big declines than big rises.
Charles Pennington comments:
You got 733 "fives" and 637 "ones", a difference of 96.
I looked at ten simulations of a 2500 step random walk, using numbers drawn from a normal distribution. In only one of the ten was there a difference between the number of "fives" and "ones" that exceeded 96. In that one instance, the difference is 137. The standard deviation of the 10 differences was 42.
So an effect this big should show up randomness alone on the order of 1 in 10 times. It's probably a non-random effect.
Victor Niederhoffer asks:
Excuse me, Professor, but did you use the drift of -03 a day and given standard deviation of 137?
Charles Pennington replies:
That seems like false precision to me, with the market's own real-time estimate of volatility having varied by a factor of ten over the period, and with the drift flattish until the last year of the ten year period. If the point is that "the market has gone down rapidly over the past six months", then count me in. Even over that period, though, four of the top ten 1-day magnitude moves have been in the positive direction, including the biggest and the second biggest. And even from 1929-1939, the biggest 1-year magnitude move was the UP year of 1933 with the market up 78%. I think if you or I had lived through that, then..*, **
My friend, you would not tell with such high zest; To children ardent for short selling glory, "Teres quod tardus est, ut venalicium oriri".
* mangled version of poem "Dulce et decorum est",
** Online English-to-Latin translation of "Smooth and slow it is, when markets rise". I have no idea if the translation is right, but it looks nice.
Victor Niederhoffer requests:
I'd appreciate it if someone would test whether the distribution of ranks for S&P for the 5 day periods is consistent with independence with actual changes since 1/1/1999.
An Artful Simulator writes in:
Using 1000 randomly resampled (w/replacement) data series, I get
where
obs = observed distribution of ranks
exp = expected number of ranks from simulations
exp = 95% empirical confidence interval from the simulations
rk obs exp 95% conf
1 658 702 [644,767]
2 434 417 [382,454]
3 349 366 [332,397]
4 419 396 [360,432]
5 741 720 [656,782]
(i added some random noise to break ties)
doesn't look that non random to me. Also, the chi squared statistic
for observed as a function of expected is 6.28 with p val of 18%
Victor Niederhoffer comments:
The Professor was right.