# Does the existence of anomalies disprove the efficient markets hypotheses?

Is the existence of anomalies to the EMH sufficient to disprove the EMH, or do we need a persistent anomaly which the financial markets cannot correct to disprove the EMH?

This question is for the weak, semi-strong, and strong versions of EMH.

• The award of the Nobel Prize in Economic Sciences to Robert Shiller confirms that the EMH has been rejected. However the fact that Eugene Fama also got the prize shows that the Swedish Academy cannot make up its mind on this issue. – Alex C Jul 6 '17 at 9:16
• @AlexC I'm not sure the nobel prize award can be used as a criteria, substituting academic research... thanks for your comment anyway ;) – An old man in the sea. Jul 6 '17 at 9:24
• Famous Chicago saying "it takes a theory to beat a theory" (not just a bunch of anomalies). – Alex C Jul 6 '17 at 9:24
• Market efficiency cannot be disproven, as showed below by @MatthewGunn. On the other hand, Market inefficiency CAN be disproven: it is enough to find the "right" model for excess returns. Hence, until we find the right model we should accept inefficiency...At least if we believe in Popperian Science – fni Jul 6 '17 at 19:35

If one finds an anomaly relative to some asset pricing model, there are three possibilities:

1. The anomaly you "found" actually isn't there: you're overfitting the data, found a spurious result.
2. The efficient market hypothesis is wrong.
3. The asset pricing model you're using is wrong.

Any test of market efficiency is a joint test of market efficiency and an asset pricing model. The efficient market hypothesis on its own doesn't generate testable predictions.

# Some definitions:

The efficient market hypothesis (EMH) of Prof. Eugene Fama is that market prices reflect all available information.

• Strong form: prices reflect literally all information, public or private.
• Semi-strong form: prices reflect all publicly available information.
• Weak form: prices reflect all past price movements.

Let's also review some things the EMH on its own does NOT say:

• The EMH does NOT say that expected returns are constant.
• The EMH does NOT say that expected returns are the same for all securities.
• The EMH does NOT say that returns are entirely unforecastable.
• The EMH does NOT say that stock prices follow geometric Brownian motion etc...

The efficient market hypothesis, on its own, says FAR less than a lot of people think.

# Testing market efficiency and the joint-hypothesis problem

Early in the history of academic finance, Fama pointed out that any test of market efficiency is a joint test of market efficiency and an asset pricing model. To say prices are wrong, you need to say something about what they should be!

### Example: Joint test of efficient markets and the capital asset pricing model (CAPM)

For example, the CAPM asset pricing model is that expected returns are linear in market betas: $$\operatorname{E}[R_{it} - R^f_{t}] = \beta_i \operatorname{E}[R^m_t - R^f_t]$$

Using various proxies for the market portfolio, joint tests of this model and market efficiency are strongly rejected: since the 1980s, average returns actually decline in market beta rather than increase.

You also had the anomalies of size and value. Does the existence of these anomalies relative to the CAPM imply the efficient market hypothesis is violated? No. The CAPM could simply be the wrong asset pricing model.

That begs the question, what's an allowable asset pricing model? Could one test market efficiency more directly by making fewer assumptions on the asset pricing model?

### Toward cleaner tests of market efficiency

A basic commonality of any risk based asset pricing model is the law of one price: a linear pricing function. If you can show two securities have the same payoffs and trade at different prices, that's hard to reconcile with market efficiency and any risk based asset pricing model.

Along that line, you have Richard Thaler's paper "Can the market add and subtract?" on tech stock carveouts. Thaler argues that in the case of Palm and 3-Com, linearity of the pricing function was violated.

More broadly, one can argue against market efficiency and risk based asset pricing models by showing:

• Some strategy earns significant expected returns
• A risk based story doesn't make sense, and that it's far more plausible that positive returns reflect the market not incorporating information.

For example, the abnormal return to earnings surprises is fairly compelling evidence evidence against the strong form efficient market hypothesis. It suggests that the market was not already aware of the earnings info until it was announced.

Fama and French have been unwilling to add a momentum factor to their asset pricing models. They're not willing to call it a risk factor. Is momentum an anomaly that falsifies the joint hypothesis of efficient markets and risk based asset pricing models? Perhaps?

### Unexploitable anomalies?

A line of research in the broader market efficiency literature is that you can have unexploitable anomalies due to transactions costs and various other frictions. Due to short sale constraints, transaction costs, stale prices, etc... a strategy may not actually earn the returns that a naive backtest says it would have.

I have a paper that argues that the distribution of returns cannot have a mean. I argue that prices are data and that returns are not data. Rather, returns are transformations of data. Therefore, it is the statistical distribution of prices that must determine the distribution of returns. Since returns are $$\frac{p_{t+1}}{p_t}-1,$$ it follows that returns are a ratio distribution. In the simplest case, the Markowitz case, with many buyers and sellers and infinite liquidity, it can be shown through the central limit theorem that prices would be normally distributed. Depending slightly on how you construct the process, this implies that the distribution will have no mean or variance. Because negative prices don't exist, the distribution of returns for going concerns must be: $$\left[\frac{\pi}{2}+\tan^{-1}\left(\frac{\mu}{\gamma}\right)\right]^{-1}\frac{\gamma}{\gamma^2+(r-\mu)^2},$$ if you accept the Markowitzian assumption. This also assumes the distribution is conditioned on the equilibrium prices so that: $$\mu=\frac{p^*_{t+1}}{p_t^*}.$$

If you do not exist in a Markowitzian world, then the distribution will vary, but it will have the Cauchy distribution buried in it. You can see this by transforming the process into polar coordinates. In doing so, you will find that the limits of integration of the angles that the cumulative density of the Cauchy distribution must be present, regardless of any assumptions you may otherwise make. You will end up with some type of integral similar to: $$\int_{-\frac{\pi}{2}}^{\tan^{-1}(r)}.$$ It will depend a little bit on how you orient your problem. Nonetheless, the arctangent will be present because the arctangent is opposite over adjacent and if the current price is the length of the adjacent and the future price is the length of the opposite side, then you can see the linkage.

I have also replaced the rules of mathematics in other papers to accommodate this. The efficient market hypothesis is invalid because it depends upon a mathematics that cannot be correct. This does not mean that markets are inefficient, though, it just means we have been discussing what efficiency means the wrong way. Having looked at the data in quite some depth, I would argue that prices, conditioned upon dividends, are quite efficient, but in a sense much weaker than any version of the efficient market hypothesis.

This weakening comes from the nature of the allowable cost functions and the rules of regression. The implied cost function for the distribution to find $\mu$ is the "all-or-nothing" cost function. It is $$C(\hat{\mu},\mu)=\begin{matrix}c_1&\hat{\mu}<\mu\\0&\hat{\mu}=\mu\\ c_2&\hat{\mu}>\mu\end{matrix},c_1,c_2>0$$

A gamble in this form means that any mistake is a bad mistake, though it may be the case that a mistake to the left is a different cost than a mistake to the right. Overvaluing $\mu$ causes you to buy too much at too high a price while undervaluing $\mu$ causes you to buy too little and you suffer an opportunity loss of profits. A stronger gamble can be placed by regressing price over dividends. That leads to the absolute linear loss function, still weaker than an expectation, but about fifty percent of the time you should make too much money and about fifty percent of the time you should make too little. Since it is linear, the median loss should be zero, though the mean loss could be any value in the real number line.

The anomalies are being created by the difference in the math. People are measuring behavior under an assumption of some form of normality or lognormality and seeing behavior under distributions without a mean, variance or covariance. There are tons of persisting anomalies. Until people quit using the tools of mean-variance finance, the ideas behind the EMH will persist.

People are receiving compensation based upon the EMH. For example, the Uniform Prudent Investor Act presumes the validity of the EMH and declares alternative behavior imprudent and hence tortious unless you can document why you behaved in a manner different from how the hypothesis would recommend. Everyone knows it is not empirically supported. Indeed, it has been a bonanza under "publish or perish," because you don't have to discover anything to discover something because you can always find a new way to finagle a prediction into an anomaly.

So many careers implicitly depend upon the EMH that while paying lip service to it being empirically incorrect, people are not stupid. They don't want lawsuits. I found 3800 articles on just one anomaly in an EconLit search. Empirical disproof has existed since 1963 with Mandelbrot's article "On the Variation of Certain Speculative Prices." Four hundred and eighty-two trillion dollars in options are priced on the EMH or a variant according to the Bank of International Settlements. That is a lot of money to say "oops" on. The EMH will persist until something forces a sea change.

If you are interested in math that will work. Buy a copy of Parmigiani's book called "Decision Theory." Its ISBN-10: 047149657X is a way to search for it. If you have no background at all in Bayesian methods, you should first pick up "Introduction to Bayesian Statistics," by William Bolstad. It is ISBN-10: 1118091566.

Edit Just a note, this probably should have been asked and answered under the Economics forum instead and should probably be migrated to there.

• You may like this article papers.ssrn.com/sol3/papers.cfm?abstract_id=2365294 by R. Rebonato – zer0hedge Jul 7 '17 at 19:11
• How would prices be normally distributed?? Prices are A.S. positive, and so are certainly not normally distributed. – user9403 Jul 9 '17 at 13:28
• @user9403 sorry, poor choice of words. I will go back later and edit it. In the Markowitzian world, where all assumptions are true, with many buyers and sellers, in equilibrium, a winner's curse will not obtain because of the double auction. The rational activity is to bid your expectation and so the static limit book would be normally distributed. This is functionally the same as saying the appraisal errors are normally distributed. Only one transaction will happen, so prices across time will be skewed as prices tend to rise. – Dave Harris Jul 9 '17 at 16:39
• @user9403 if you disaggregate the market into individual trades, remove shell companies and closed-end funds, and remove mergers and bankruptcies then the empirical distribution is very close to the truncated Cauchy distribution. If you account for the intertemporal budget constraint it is almost the truncated Cauchy as that removes the skew. – Dave Harris Jul 9 '17 at 16:44

I agree with most answers posted. But the research on this subject has been highly contradictory, and the answer to your question is highly dependant on the hypothesis you are trying to test. If your hypothesis is looking for 6-sigma events within an "efficient" market, then anomalies are what you need to come by, and will disprove your hypothesis if not found.

On the other hand, if you are trying to test for arbitrage opportunities on the basis that markets are not as frictionless as they have been theorized to be, then anomalies aren't that normal and should not be that evident in your findings, unless the stock/company/government has underwent a surprising event which has led to that anomaly.

If you follow the Martingale logic for equity prices, then the price of a stock should be the sum of all past events and should reflect such "known" events. On the flip side, if you are a random walk advocate, you can fairly certainly assume that prices move randomly while following a log normal distribution, which in turn forces your returns to have a normal distribution. Stock predictability in that sense has been very insignificant, but you can run millions of Monte Carlo or Geometric Brownian motions and claim that you have predicted a future price, when in fact that would just be an assumption dependant on the set of inputs you have specified. In that sense, if you believe the prices to be following a random walk, then the price change itself follows a random walk and not prices themselves. This leads to the assumption that in this case, markets are of the weak form in which prices reflect all past information, but this also leads to the fact that todays price is a martingale of all previously known event data points of that asset. Then again, you should also test for autocorrelation within your variables because that may as well be the reason why anomalies are evident. This definitely doesn't work for assets with high storage costs such as commodities, or illiquid asset classes.

Long story short, your interpretation should be highly dependent on what assumptions you are inputting to begin with. As you fiddle around with your inputs, your outputs might vary highly, so knowing the dataset you are working with would provide you with better insights as to whether the anomalies are true anomalies, or just a result of auto-correlation or multi-collinearity between the variables you are trying to test.

I find the argument binary. As long as there is self interest in human behavior, EMH is a fairy tail.

• Hmm, I thought the self-interest is what eliminates inefficiencies from the market, making the markets tend toward efficiency. – Alex C Jul 8 '17 at 19:59