Thumbs up to Woodshedder and his post On Fractals and Market Crashes.
This topic has been covered a gazillion times by folks with far denser cranial matter than my own, but it bears repeating (again and again and again ad infinitum or at least until the financial community finally “gets it”).
The short version…most of the theories that form the backbone of “modern” finance are based on the assumption that markets follow a normal (AKA Gaussian) distribution. Prices are on a “random walk down Wall Street” so to speak.
This assumption allows financial practitioners to do all sorts of nifty stuff like tell you the exact probability of portfolio X losing X%, or bundle a bunch of crappy mortgages together and magically create a AAA-rated security.
But here’s the rub. The normal distribution (as applied to stock market returns) is an utterly and completely debunked sham. A simple example:
If the market followed a normal distribution, we would expect to see days greater or less than 5 standard deviations (or roughly +/- 4.7% by my calculations) about once every 6,922 years…we’ve seen FIVE so far this month.
This archaic assumption does not accurately model financial markets of any kind in any part of the world – period – because real markets in the real world have fat-tails (or a greater propensity for large portfolio-busting returns). Enough said.
My Take on Modeling the Distribution of Returns (a Little Bit of Geekery)
I break from some financial minds on this topic.
Because the distribution of stock market returns is impossible to accurately model, yes, it’s impossible to make statements like portfolio X has an X% chance of an –X% loss or justify rating this pile of junk (even if it appears to be disparate junk) as anything but a pile of junk.
But modeling the distribution of returns is useful for comparing two things to each other to give some perspective on returns. For example, in this table from my recent post Asia vs US Stock Market Performance I used a volatility-adjusted return in column 3 (similar to a Sharpe Ratio) to show whether the average return from the second set of observations (shaded green) was large or small relative to the first set.
The point was not to model the future, or model the observation within a portfolio, or model how the observation will respond to fat-tails. The point was to determine whether +/- 0.3% really has been “3-times better” than +/- 0.1%, or just a function of the volatility of the underlying instrument.
Subtle difference I know, but an important one.
Happy Trading,
ms
P.S. The fact that this is still a topic of discussion 10 years after the fall of LTCM (which was largely influenced both directly and in spirit by the grandfathers of this nonsense) is a sad reminder of the idol-worship that persists in academia and Wall Street, but also a happy reminder of the reason little guys like me (coupled with a lot of sweat equity) can continue to outperform the high-brows.
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Filed under: Random Stuff | 11 Comments
Hey Michael, thanks!
Good point. These tools can still be useful.
One thing I’m struggling with, that maybe you’ve thought about before: I really like using a runs test over the trade by trades of backtested systems. The resulting Z-score I believe helps to determine if the system has winners that begat winners, and losers that begat losers. Of does it tend to flip indiscriminantly between winners and losers (many short streaks).
However, since the runs test assumes a standard deviation, is it flawed for the purpose for which I’m using it?
I believe that it may still be worthwhile since the only data used is whether the trade was a win or loss. In other words, does the system trade like a fair sided coin being tossed? Or, do the trades show some dependence?
RE to Wood: I’m not an expert on the subject, but my personal opinion would be…
If you’re using the runs test to compare two different strategies to say this one has produced more/less streaks than the other so that you can decide which one better suits you, that’s okay.
If you’re using the runs test to say I expect this probability of streaks in the future to be X, that’s NOT okay. As we saw this month, the number of consecutive up/down days in the market (which would implicitly impact consecutive up/down days/streaks of a strategy) suffer from fat-tails as well.
Just my personal opinion.
Haven’t had a chance to read your part 2 yet, but looking fwd to it.
Hope all is well,
michael
Nice analysis. It’s been awhile since my Stats courses but i seem to remeber that data is increasingly descriptive as we move up the data chain from: nominal to ordinal to interval to ratio data.
You are on to something by REINTERPRETING THE NOMINAL or interval market DATA INTO RATIO DATA. This should provide more powerful information.
“As we saw this month, the number of consecutive up/down days in the market (which would implicitly impact consecutive up/down days/streaks of a strategy) suffer from fat-tails as well.”
I agree, sort of. This is the crux of my issue: the runs test is not really being applied to the changes of an index, or an equity. It is simply measuring whether a particular mechanical system tends to win and lose in streaks.
To be more clear, in terms of a system, if it uses stops, or has like your YK some criteria that limits long streaks of losers, or is contrarian, the recent action would have caused an exit of the system, or possibly a long string of winning trades.
I guess maybe I’m thinking that a system could be engineered to be streaky, even when the market is and isn’t doing the same thing.
RE to Shed: I had that same thought when I was writing that part (about the system returns not necessarily being tied to underlying market returns).
Similar conclusion though – the number of actual consecutive winners/losers in the future will exceed what run stats predict for the future (so have to be used with caution).
Is the idea that you want to determine the “normal” streakiness of the strategy so that you can no where to place a filter/stop that limits streaks of losers?
michael
Well, the test would determine whether the trades exhibit dependence. The question to me is is the dependence determined by the system, or the market, or a combination of both?
Your questions though do make we wonder about running standard deviations of the length of losing streaks, and having a filter kick in when they exceed the acceptable limit above the standard.
Of course, there is that damned standard distribution again. If I am already assuming that the system’s win / loss results exhibit dependence, will not that flaw any standard statistical analysis of the length of the losing streaks?
markets are social phenoms not physical. a heard of sheep is not to be confused with a gaseous dispersion. VaR, MPT etc. are all flawed a bad case of paradigmarhea. http://www.gogerty.com
I couldn’t have said it better myself nickgogerty. Paradigmarhea? Well done sir.
These I-banks weren’t using normal distributions to model these derivatives, but they were mathematical models to socioeconomic behavior (and only a sample of the populations at that). In any case, I don’t believe this is the root of our problems. Any statistician can tell you the evils of using models without discretion or any mind to the assumptions being violated and so could the quants building these models. People made rational decisions in a flawed system because of perverse incentives. Our fail-safes were cracked and everybody was left standing with there pants down. Thanks for a great website!
RE to Jeff: I agree that (at least I hope) their risk models were more sophisticated than the very simple discussed in this post. And I agree that Wall Street lost its collective common sense because of “perverse incenties” (good adjective).
But I do think that the root of the entire financial crises (outside of fundamental economic stressses) was failed risk models. Again, you name the crises and I’d be willing to bet that the root of it was thinking that we could lever up our gains because our risk models showed that we were safe – ignoring contagion, fat-tails, non-representative samples, and all that jazz.
I guess in a nuthsell, you’re right, it’s not as simple as what I covered in this post, but fundamentally, I think it’s still the same issue.
Thanks for the comment and the kind word!
michael