Actually this is a tricky question: Even after defining a bear market, could a given decline have occurred by chance — given a random arrangement of returns? One aspect of a bear market could be down weeks clustering more than would be expected by chance, giving rise either to more frequent or deeper declines.

4194 DJIA weekly closes were partitioned into non-overlapping 40 week periods. At the end of every such period, calculated the maximum decline as:

min(this 40) / max (last 40)

Done this way the maximal decline could have been as long as 80 weeks or as short as 2 weeks; the idea was to capture large drops over various periods of interest to investors.

A simulation was used for comparison: The same 4194 DJIA weekly returns were resampled 100,000 times, and multiplied ("compounded", without dividends) out to produce a 100,000 week series. Like the actual market history, the series was partitioned into non-overlapping 40 week periods, and every 40 weeks min/max was calculated for the current and prior period.

One definition of a bear market is "a decline more than 20%". In the actual series, such declines occurred in (a surprisingly high) 26% of 40 week intervals (27 out of 104 40 week pairs). If this were more often than random, it would have occurred more often than in the simulated series. However in the simulation declines more than 20% actually occurred 32% of the time.

So if anything, declines of 20% or more occurred less often historically than by chance along.

But what if 20% is too arbitrary to capture a bear? In the actual series, here are the 40 week pair declines above the 95th percentile (ie, declines worse than 94.2% of the rest):

Date    40 min/max
06/20/32    -0.751
04/03/33    -0.642
09/14/31    -0.564
12/08/30    -0.542
03/02/09    -0.499
08/15/38    -0.469

The mean of these 6 40 week pairs is -58% (all but 5 from the depression). In the 100,000 week simulation, the 95th percentile is -34%. The actual 95th percentile and above mean of -58% is lower than even the worst simlated 40 week pair decline of -56%, which was the bottom of 2496 pairs (99.96 percentile, like the Obama cabinet SATs).

The worst 5% of actual 40 week pair declines dropped much more than would be expected by chance arrangement of down weeks. This is consistent with "fat tails" (at least on the downside), but you have to go out further than -20% to see it.

Alston Mabry comments:

Great study, Dr. Z. One thing I would want to explore would be whether in the simulation process, one intermixed different volatility regimes. That is, in the actual 4194 weeks, you may have periods of high volatility and periods of low. High volatility periods would have larger moves in absolute terms than would low volatility periods, and if the simulation mixed them together, the simulation might tend to produce lower volatility overall - this might account both for more 20% moves but fewer +50% moves. If this were a problem, one solution might be to normalize all the weeks against some preceding period, say 52 weeks.

Kim Zussman replies:

 Knowing that volatility clusters, if one is resampling a long data series this gets shuffled up. So you'll get 4% days near a bunch of 0.2% days (though the stdev of the whole series should be the same -shuffled or not). But if the question is whether the market has structure which is not random, does it make sense to stipulate whether you are in a volatile regime or not? Relatedly, maybe sticky volatile regimes translate to down markets, which is kind of the point.

Alston Mabry responds:

Exactly. To be precise, what I'm saying is that the fact that the simulated distribution produces more +20% moves but fewer +50% moves is simply an artifact of the shuffling process, especially when you shuffle individual weeks and then use 40-week stretches for calculating results. I'm thinking that the shuffling takes the actual distribution of % moves and increases the kurtosis and pulls in the tails.

This is not arguing against the hypothesis, just questioning that meaningfulness of the % comparisons.

Charles Pennington adds:

Prof POne uncontroversial hypothesis that might unify and explain many of these studies is that "markets get more volatile after they've gone down".

If you compute the skewness of the weekly or monthly returns of the Dow since 1929, it's quite negative. However if you take those same returns and divide them by some measure of the volatility over the following week(s)(*), then you'll find that both the skew and the kurtosis are close to zero, i.e. it's similar to a normal distribution of returns. That means that someone trading backwards in time, i.e. he has next week's newspaper but not last week's, would experience safe, non-Black Swannish returns if he just adjusted his position size for the volatility that he had experienced in his recent future.

* for example, one might use the following week's high/low range, 100*(h/l-1), or the average of that quantity over the following N weeks, where N is "a few".

To illustrate, here is a model.

First, create a series of random normal numbers with standard deviation 1, with one number for each trading day.

Now, use the following rule: "If the average of the last three days' numbers is negative, then today's return is 2 times today's number. Otherwise today's return is 1 times today's number."

I ran 2500 simulated trading days using that rule, and it gave 715 5-day maxes and 622 5-day mins. That's similar to what the Chair reported for the market.

More generally, I suggest that whenever you see one of these apparent anomalies of "market falls faster than it rises", try to see if it can be distinguished from the uncontroversial hypothesis that "volatility rises following down moves".

By the way, over the past 10 years, the standard deviations of daily returns of SPY under two scenarios:

all days 1.39% after up three-day move only: 1.17% after down three-day move only: 1.61%

Kim Zussman replies:

The simulation made the skew and kurtosis go away.  Here for the 40 day min/max both from actual series and simulation:
Descriptive Statistics: min/max, sim

Variable   Mean     StDev      Min    Median   Max       Skew
Kurtosis         N
min/max  -0.1435  0.1463  -0.7506  -0.1134  0.0313    -1.70      3.66
sim         -0.1604  0.1008  -0.5576  -0.1504  0.0654     -0.53
-0.04         2496

Even accepting there could be non-randomly down markets, this is a different question than whether they can be predicted.  So a small decline results in higher volatility, and trading smaller long positions can be on average profitable.  But some of the small declines go on to become big ones, and its hard to tell one from another.  Using stops (physical or otherwise) is tuchass saving, but it's hard to know whether "cutting your losses and let profits run" is worse in theory or execution. Which doesn't preclude that others can discriminate good from bad dips, or that they found work-arounds using opportunities independent of short term decline-reversal.

Phil McDonnell writes:

It may be helpful to look at the underlying hypothesis a little more closely. When we randomize by individual time periods we are deliberately randomizing any period to period dependencies. I presume that this was Dr. Zussman's point. Thus we are implicitly testing a null and alternate hypothesis something like:

Null: The original distribution or returns is similar to the distribution of a randomly ordered sequence of returns.

Alternate: The original distribution is not similar to a randomly reordered sequence of returns.

One good test of the difference between distributions is the non-parametric Kolmogorov-Smirnov test. Also one can use the more powerful D'Agostino test.

Another way to preserve the known autocorrelation in variance is to perform block resampling. From memory I believe the autocorrelation fades after about 35 days or so. Block resampling of 40 days should keep something like 97% of the variance autocorrelation and even other unknown dependencies even non-linear effects in that range. Comparing the distribution of the original returns to the 40 day resequence might tell us if there is something non-random even beyond the 40 day block level.

Dr. McDonnell is the author of Optimal Portfolio Modeling, Wiley, 2008


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