I’ve seen some rank charts that look like power laws (e.g., ranking on fundamentals only). I would argue that graphs that look more like logistic functions are better. When your model is very specific about placing winners and losers in the extreme quantiles, then it explains a greater amount of the variation in returns.
Sort of stretch to apply Pareto and power law distributions to rank/returns graphs. I think that the distribution of returns among designer models may be a better use case for Pareto.
According to Wikipedia, things that have a Pareto distribution include:
“The standardized price returns on individual stocks”
A separate source about the Zipf plot: “…constructed on binned observations.”
Obvious conclusion: the better your ranking system is at ordering stock returns (which is a Pareto distribution) the better your (binned) rank performance will be at approximating a Pareto distribution (or a binned Zipf plot) [Reference] .
Practical use: trying to make your rank performance test results linear (or assuming they are linear) MAY not be helping you.
Some of you may have observed that I have essentially rediscovered the main point of Nassim Taleb’s book “The Black Swan”
Yep, nothing new on my part. Since Taleb claims power-laws apply to pretty much everything in the stock market, my noticing the pattern in rank performance is not a great intellectual leap (which I am sure surprises no one).
I do not think that it is an exaggeration to say that the whole book was about stocks returns (and indexes) following a power law.
Mandelbrot (closely associate with Taleb’s ideas) wrote a lot about “scale invariance.” Pareto models are scale invariant as are fractals. Mandelbrot is most know for developing fractal theory but he has written much about power laws.
For those of you who have not liked my use of classical statistics, this book and these ideas could provide much ammunition. But of course they are both (all) correct………. Well, if you are sure about whether you can use the central limit theorem assumptions: See recent post about ergodicity and its relation to the central limit theorem.
I am not sure we can prove that our models were ergodic (which is necessary for the central limit theorem) until we are done with our models in the future. Looking at our models in that past we will be able to tell if they WERE ergodic. Looking forward, it is just a convent assumption.
Uhhhh….So you have a point. There are risks—Black Swans and more—that are not adequately covered in Statistics 101.
Sadly, the risk are still there: whether you do the math or not. The example of an Ostrich (burying their head in the sand) would be more appropriate than that of a Black Swan in this instance.
Taleb’s and Mandlebrot’s main points were regarding the behaviors of man made phenomena in the extremes. Mandelbrot modeled these things as fractal Brownian Motions in which extreme and unlikely outcomes happen much more frequently than predicted under smooth Brownian Motions. Taleb just marketed this idea by giving context and a nice title. The Black Swan was about more than just stock market returns.
Anyhow, no credible economist (to me) believes that the market is ergodic. But most have explicitly or implicitly treated it as one whenever modeling market variables as having known and time-invariant distributions.
Market outcomes don’t behave as physical constants. They evolve as participation grows and becomes increasingly knowledgeable. Gravity behaves as a constant no matter how many people observe it and increase their understandings of it.
Markets today are not the same as they were a decades ago, a century ago, and even multiple centuries. Yet, they were there fulfilling some purpose, which itself has also evolved.
But while this is a wrong modeling assumption (as are most), it can be useful. For example, I am assuming in my port that the underlying ranking systems’ rank/return distribution is a constant. It’s not true, but it’s a useful assumption if I manage to leverage the assumption to earn excess returns.
But [b]if you think much of Mandlebrot’s work (and The Black Swan) was not about the stock market I will have to give a big:
Okey-Dokey to that one.[/b]
Uh, so I guess maybe you mean the most EXTREME bucket in a rank performance test (with 200 buckets say) “being a man made phenomenon”…….Uh, would NOT be expected to ever be extreme? Yea, I am having trouble following.
I am sure you have an excellent point but I am clearly missing something.
Also, this is NOT JUST ABOUT MAN MADE PHENOMENON. Sizes of asteroids, size of bomb fragments, word frequency (okay man made), bacterial colonies, Clusters of Bose–Einstein condensate near absolute zero, Sizes of sand particles, maximum one-day rainfalls……
Mandlebrot is actually best know for pointing out the scale-invariance of things in nature (fractals).
Maybe you think markets are different today:
The Black Swan is pretty recent and there is no indication that—Taleb at least— has come to a new opinion about the markets.
Copyright 2007
MAYBE WE AGREE ON THE FOLLOWING.
First, the power laws allow for stocks that DO PARTICULARLY WELL. So maybe (probably) I do some of the things you do to find those.
While Taleb has some good points (all good perhaps). Keeping a negative view and pretending that he predicted the 2008 recession might have sold some books.
My point would only be that there are reasons, often little understood, for why what we do may work. And reasons why it (sometimes) doesn’t.
Anyway, there are a lot of rank performance test that are not so linear. If you have a better explanation for this I would love to hear it. Even then, I would have to be convinced that Taleb is 100% completely wrong.
Maybe you are sure it SHOULD be linear? I would really like to hear why you are so sure (if that is the case). I cannot find a reference for that anywhere (I already gave a specific quote from Wikipedia).
We seem to agree that the ergodic (and mixing) assumption does not always workout over the period of the investment: