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The long story short, I have developed an index based on a certain distribution. Then I aligned NYSE stocks according to this index i.e. the stocks with the best fit are first and the worst are last. The index is irrelevant with economic theory.

Afterwards, I created a group of portfolios with equal number of stocks and equal observations and then I calculated the minimum variance per portfolio.

The result suggests that the portfolios with better fit have lower variances than the ones with "bad" fit. Does this prove market inefficiency assuming that it didn't occur just by chance?

Edit: According to efficient market hypothesis, asset prices reflect all information given and "patterns" should not occur. However, according to the results, patterns occur and they are independent to the information given.

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    $\begingroup$ "According to efficient market hypothesis, asset prices reflect all information given and "patterns" should not occur." - that's not quite true. It's more accurate to say that any patterns that do occur, don't enable you to make returns that are higher than those commensurate with the risks taken. And I'm going to guess that that's the case here, because if you could make those returns, you wouldn't be telling us anything about it ... $\endgroup$ – EnergyNumbers Jul 24 '17 at 13:35
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    $\begingroup$ Are you asking if forecastable volatility violates the efficient market hypothesis? Does it violate the efficient market hypothesis if stock returns have higher volatility than government bonds? Perhaps examine the definition of the efficient market hypothesis here. $\endgroup$ – Matthew Gunn Jul 28 '17 at 1:58
  • $\begingroup$ What do you specifically mean by, "I calculated the minimum variance per portfolio." What does, "I developed an index based on a certain distribution." It's quite ambiguous what you're doing. $\endgroup$ – Matthew Gunn Jul 29 '17 at 20:39
  • $\begingroup$ I think laboratory evidence of sunspots in and the creation of bubbles in asset markets both do more to cast doubt on the EMH than does what you've written about here. $\endgroup$ – 123 Jul 31 '17 at 11:41
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The efficient market hypothesis does not imply that there are no patterns!

As Eugene Fama pointed out decades ago, any test of market efficiency is a joint test of market efficiency and an asset pricing model. The EMH on its own is not a testable theory.

If I understand your statement properly, you're claiming that forecasting variance would violate market efficiency? For that to be true, you would need to assume an asset pricing model where variance is not forecastable. That's a strange claim. It would imply that risk free government bonds must have the same volatility as small biotech firms (otherwise, I can forecast volatility based upon whether a security is a stock or a bond).

It well known that volatility is forecastable, both in the cross section and in the time series. For example, there is the widely used GARCH model and all kinds of other stochastic volatility models. Forecasting volatility is interesting, but it in no way constitutes a per se violation of market efficiency.

This answer of mine goes into greater depth on what the efficient market hypothesis actually says and how to test it.

References

Fama, Eugene F., 1991, "Efficient Capital Markets: II," Journal of Finance

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  • $\begingroup$ I cannot see how I assumed that assets should have the same variance. Nevertheless, the minimum variance of my portfolios should have no pattern. Thus, the distribution fitting of a stock should have been an irrelevant variable, since it does not reflect any information given. Also, patterns indeed may occur, but having a pattern with something irrelevant is either irrational or an outlier. $\endgroup$ – Commissar Vasili Karlovic Jul 29 '17 at 12:28
  • $\begingroup$ @CommissarVasiliKarlovic Please formulate a precise, mathematical statement about stock returns that: (1) you believe should be true and (2) you show is violated (based upon the work you have done). $\endgroup$ – Matthew Gunn Jul 29 '17 at 20:43
  • $\begingroup$ I will upload a graph as soon as I have some time to spare. $\endgroup$ – Commissar Vasili Karlovic Jul 30 '17 at 11:10
  • $\begingroup$ It amazes me that people openly discus without contest that Fama's theory cannot be tested yet still treat it as a meaningful theory. What use is there for untedtable theory in science?? What can we possibly learn? $\endgroup$ – 123 Aug 3 '17 at 5:53
  • $\begingroup$ @123 If you put zero constraints on asset pricing models, that's basically right. But rationality puts some constraints on the space of sensible asset pricing models. Eg. we know that stocks tend to decline dramatically after an earnings miss. One conclusion is that markets aren't strong-form efficient, that public markets aren't incorporating the earnings miss into the price until it's publicly announced. It would be hard to create a rational, sensible asset pricing model where markets know a firm is going to miss earnings but the price doesn't decline until after the earnings call. $\endgroup$ – Matthew Gunn Aug 7 '17 at 20:16
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By itself, no it does not, at least how I am understanding your post. While I am an opponent of the Efficient Market Hypothesis, what you would need to show is that there is a "free lunch," with your methodology. You would also have to show it persists out-of-sample and does so for decades. You would need to measure commissions as well. A $\$3$ gain that costs $\$5$ to get isn't a gain.

The second way to show it would be if your distribution implied a strong informational asymmetry by its structure.

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The latest Freakonomics podcast topic Stupidest money may shed some light.

I just quote part of the conversation from John Bogle:

The markets are highly efficient — although, importantly, not perfectly efficient. Sometimes they’re very efficient and sometimes they’re not. It’s hard for we poor souls on Earth to know which is which

It is a gamble of randomness.

(update) Extreme cases like Comcast and Enron show so call "public offering" are not immunized to information opacity and wrong doing. In addition, fake news, manipulation using various rumor mills or even human error may skew "market efficiency".

In addition, so call "market analytical tools" are anything but hindsight self-fulfillment prophecy. Currently, there is little proof that so call market-analytical tools are able to produce reliable results when compare to solid information collected (e.g. whether the business is managed by competence hand )

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  • $\begingroup$ I just upvoted your answer for your cool quote! But let's get serious. Philosophically speaking, this may answer to my dilemma-question. If there was a quantitative methodology to evaluate market efficiency and the probability of stock market not to be efficient is 0.009%, then modeling would have correctly evaluated it as perfectly efficient, although there may still be methodologies to expose that inefficiency. Burned. $\endgroup$ – Commissar Vasili Karlovic Jul 29 '17 at 18:42
  • $\begingroup$ @CommissarVasiliKarlovic The randomness occur every now and then, mostly due to irrational human behavior. Behavior economic are mean to study rationality of the irrational behavior. $\endgroup$ – mootmoot Jul 31 '17 at 7:12

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