The DAX and its Power Law

To continue my story of yesterday, here is the development of the DAX.

black-swan-distribution-012It shows, by the way, that the current fuzz about dropping stock rates is somewhat exaggerated. The DAX is still higher than 2003. The plot also shows that stock markets got crazy around 1998. Probably, because people got crazy.

Now, I like to study the deltas. Here is a plot of the deltas (scaled to delta per day). The function is the corresponding Gauß distribution, which does obviously not fit very well. There are too little intermediate deltas.

black-swan-distribution-0143Instead, we look at the excessive deltas (in absolute value). And indeed, we find a power law.

black-swan-distribution-016The plot shows excess(x)*x^3.6, where excess(x) is the number of deltas larger than x*var in absolute value (var is the observed variance of the deltas).

So we can explain the excessive deltas by the power law with exponent 3.6, just as claimed by Taleb.

8 Gedanken zu „The DAX and its Power Law

  1. Canadian Money

    I have a question that I hope you may be able to help me answer.

    As a statistical expert, how could one prove or disprove that an analyst can forecast the stock market? The alternative being that an investor’s successes are simple due to that person being „a lucky coin-flipper“ or one of „the survivors“?

    For some my forecasts are discussed in my blog.

    I look forward to any comments you might have on this question.


  2. Canadian Money

    Your March 7th response to my question, posted on my blog is copied here…

    „There is a simple answer: Make a verifiable forecast, and if it fails, you can forget your forecasts in the future. If it holds, that proves nothing, but accumulates credit to you.

    Mind you: A singe failure and you are out of business, at least with me.

    Now, what is a verifiable forecast? You probably do not make any precise statemenst. You might say: „This specfic rate will drop most likely“. To verify this, I will collect a few forcasts like this, and see, if you are right in 90% of the cases (which is a conservative definition of „most likely“). If not, you failed.

    Of course, the credit you get for any right case depends on the non-tiviality of your forecasts, which is not easy to judge in real life.

    In fact, my feeling is, that it is impossible to judge in the area you are working in. That is the meaning of the Black Swan: While you may seem to do well for a long time, you may still blunder terribly the moment it counts. Stock rates have this property.“

    I’m sorry… I mistook you for someone who was interested in statistics, related research, predicting the stock market, and perhaps discussing things in a scientific manner.

    „A singe failure and you are out of business, at least with me.“ Is this the statistical expert speaking? You also begin by assuming that I do not make verifiable forecasts without doing any research about what forecasts I have made and whether or not I was successful or not.


  3. mga010 Beitragsautor

    Yes, you are right. I am not interested in predicting stock prices, cause they are unprepredictable imho. That is why I simulate them as random walks.

    Sometimes, insider knowledge helps to preview what the herd will do when the here the message you already know, and often predictions are obvious.

    About my „statistical expertise“. I tried to make clear, that predictions must be precise and right. A statistical prediction can also be precise, if it contains an error estimate. But I repeat: If too few of your predictions are right (compared to your error estimate), you failed. That is statistical precision to me.

  4. Canadian Money

    Thanks for the above response.

    You appear to be interested in proving/demonstrating that stock prices are a random walk. And, you have concluded, that any sequence of correct predictions are simply a string of variations within a purely random data set. Something like flipping 5 heads in a row even though the probability of flipping a head is always 0.5 with each and every throw.

    If I understand this rationale correctly…it is based on „the null hypothesis“ or in other words… that we must assume there is no correlation with any other variable until proven otherwise.

    Here is another question I have.

    Over long periods of time, the Dow Jones Industrial Average, typical of all averages, shows an increase in the order of 5 % compound per year. For example, 41 in 1932 to ~14,000 peak in late 2007. How should we frame this observation within the context of a random walk?


  5. mga010 Beitragsautor

    Are you willing to bet that the Dow Jones is continuing to increase with 5% per year (on average)? I don’t advise you to do that. You may easily loose. Indeed, I’d rather bet that it will fall for quite some time given the present situation.

    You cannot prove the increase. You can only observe it from the past. To project that into the future means to behave like the turkey, who is surprised at thanksgiving.

    I have to thank you for this discussion. It makes me rethink my ideas. It is very valuable for me.

  6. Canadian Money

    You may find this of interest.

    My lastest short-term forecast, for a rally having started upwards at 667 on the SP500 on Friday, March 6 has so far worked out well. The index has rallied by 7 percent so far, to a high of 716 today (Tuesday, March 10th). Post and chart at

    The 5 % increase over the long term is just for academic purposes. like you I expect the bear market to go lower.



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