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:

Prof PYou 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

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.


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