# Why do some anomalies persist while others fade away?

In their 1990 book, A Non-Random Walk Down Wall Street, Andrew Lo and Craig MacKinlay document a number of persistent predictable patterns in stock prices. One of these "anomalies" is variously known as lead-lag or serial cross-correlation, and it says that the returns of larger, more liquid stocks tend to lead the returns of less liquid small-capitalization stocks. Lo and MacKinlay showed that the degree of lag is greater than what could be explained by the lower trading frequency of small-cap stocks (nonsynchronous trading). A 2005 working paper by Toth and Kertesz claims to show that the lead-lag effect has "vanished" over the past 20 years. Meanwhile, other anomalies documented at the time, such as long-horizon overreaction (first documented by DeBondt and Thaler (1985)), appears to be alive and well (see McLean (2010)).

Why do some anomalies persist even decades after they are discovered while others have seemingly been arbitraged away to nothingness? What is it about those anomalies that are still around so many years later that prevents them from being arbitraged away? Conversely, what is it about the short-lived anomalies that made them so fragile?

Note: This post was inspired by a a blog post, Of Hurricanes and Economic Equilibrium, although I do not agree with the author's conclusions.

Bounty update: As promised, I created a new bounty for RYogi's answer, which is "exemplary and worthy of an additional bounty". It will be awarded shortly, as the system requires some lag time until the bounty is awarded. Feel free to add your own up-votes to his answer in the mean time.

-

A very conservative stand is to distinguish between anomalies and arbitrage opportunities. Roughly speaking, while an arbitrage opportunity is risk-free by definition, an anomaly allows for unaccounted risk factors. It is the magnitude of these unidentified risk factors that might determine the long term persistance of certain anomalies. A good starting point is the "limits to arbitrage" entry in Wikipedia. This literature has developed to cover several aspects. I can provide more references and examples if needed.

EDIT: following Tal's comment, here are some more details.

As a working definition of "Anomaly" I use: something which is not explained within a model. That something is usually expected returns. Typical examples:

1. short run momentum,
2. long run reversal,
3. cross-industry momentum,
4. value effect,
5. post-earnings drift, and
6. many other instances of unexplained predictability of returns

The first comment is the model matters. Short run momentum (1) is an anomaly for the CAPM, but maybe not so much for Kyle's Model (Econometrica 85) sequential trading model. Resolving (1) within CAPM requires explaning why recent upward performance renders and asset riskier and more correlated with consumption.

The second comment is that unexplained is a keyword in (6). There is nothing anomalous with outsized returns here, it is the risk-adjusted returns that should be inline with the risk free rate.

The third comment is that anomalies are not the same as arbitrage opportunity. To classify as an arbitrage, a portfolio has to be costless and riskfree. While anomalies might look like arbitrage opportunities, they are not: arbitrage opportunities are a particular kind of anomalies. Therefore using "arbitraged away" when referring to anomalies is a misnomer and can create confusion.

Back to the question: Why do some anomalies persist while others fade away?

1. An anomaly that persists might have unexplained risk that distinguishes it from an arbitrage opportunity. For example: fleeting instances of mispricing across different trading venues might persist because of latency risk.
2. An anomaly might persist because there are limits to arbitrage: arbitrageuers face borrowing constraints, computational constraints, attention constraints, informational constraints etc. Often the word constraints above can be replaced with costs, and the categories I listed overlap.
-
Hi RYogi, welcome to quant.SE and thanks for posting your answer. If I understand what you've said, your thesis seems to be that anomalies are, in essence, potential arbitrages that cannot be arbitraged away because of limits to arbitrage. The logical conclusion, then, is that the anomaly only persists so long as the limit preventing it from being arbitraged away persists. I like that answer (though you could have said it a bit more clearly). – Tal Fishman Oct 16 '11 at 4:52
@TalFishman, thanks. Your interpretation is generally correct. But more specifically I argue that anomalies and arbitrage opportunities (AO) are different beasts. AO are anomalies, but anomalies are not necessarily AO. As such anomalies cannot generally be "arbitraged away". Why? (1) Unexplained risk and (2) limits to arbitrage. I edited my answer to make it a bit more complete. – Ryogi Oct 16 '11 at 22:17

Very good question! I think part of the answer lies in the structure of the financial industry.

Some anomalies have a certain kind of structure which cannot be exploited by the players that are big enough to let the anomaly disappear. I would put e.g. the Turn-of-the-month effect (TOTM) into this category since big funds just can't turn their whole portfolio every end of the month.

The following page concisely lists some of the differences between big players and smaller, independent asset managers: http://quantivity.wordpress.com/2011/08/28/fund-structure-arbitrage/

It would be interesting to see if there is a certain correlation between the persistence of anomalies and structural issues in the industry.

Additionally many interesting ideas and research concerning anomalies can be found here: http://www.cxoadvisory.com/?s=anomalies

-
Thanks for the answer. I generally agree that the structure of the financial industry can play a role in which anomalies persist. Quantivity's efforts at identifying specific anomalies based on industry structure is an interesting way of turning the question on its head. I would also be interested to see some sort of study of how structural issues affect anomalies. – Tal Fishman Sep 12 '11 at 18:47
Interesting example of this fund structure arbitrage phenomenon in Russell Futures: pimco.com/EN/Insights/Pages/… – Ram Ahluwalia Oct 19 '11 at 23:49
@QuantGuy: Thank you for the link. What I don't understand concerning their strategy is how the use of derivatives should resolve the problem they address. These have to be hedged with the underlying and in case of illiquidity of the underlying the spread will be accordingly large so the same problems will reappear eventually. Or am I missing sth here? – vonjd Oct 20 '11 at 7:43

Joel Greenblatt's "magic formula" is similar in spirit to classic value styles. He has a discussion of why he thinks it will continue to work (despite it's simplicity and public knowledge) around p. 73 in his Little Book that Beats the Market (see http://books.google.com/books?id=M5HxYZaNQEQC&lpg=PA68&dq=continue%20to%20work%20magic%20formula&pg=PA73#v=onepage&q=continue%20to%20work%20magic%20formula&f=false).

The basic argument seems to be that it doesn't always work so institutions with annual performance mandates who would be most likely to make the anomaly go away would be less likely to follow the approach. It's a kind of patience arbitrage argument, i.e. there are excess returns available to those who are patient.

This argument may apply more generally.

-
I think this is not a very good answer since the so called "magic formula" doesn't deliver on its promises: "In summary, evidence [...] does not support a belief that the “magic formula” outperforms reasonable exchange-traded fund benchmarks in real use." Source: cxoadvisory.com/15343/fundamental-valuation/… – vonjd Sep 30 '11 at 5:42
This is almost like saying the anomaly persists because it is hardly there to begin with. But what about the many persistent anomalies identified in the literature, such as momentum and post-earnings announcement drift? – Tal Fishman Oct 3 '11 at 1:03
For the Magic Formula "anomaly", there is some debate about whether it is as string as Greenblatt claims. User vonjd mentioned one example of that and here is another: blog.empiricalfinancellc.com/2011/05/…. If you accept Greenblatt's research, there is definitely a very powerful anomaly and his argument for why it would persist relies on how the returns from the anomaly are distributed through time. – user915 Oct 3 '11 at 17:14

Investing in Stock Market Anomalies by Bali, Brown, and Demirtas (2011) addresses some of my questions. I will read this paper and report back on my findings.

-

In the stock market, successful companies are the most innovative ones (esp in biotech, tech) so their individual market is new and their individual market has not been arbitraged away by competitors and governments. Therefore upcoming competitors are going to be as optimistic and correlated to the market leaders. Just from trading shares and financial derivatives, you and the other traders will not be affecting the underlying book value of that market's assets.

-
What does is this have to do with anomalies? How does this relate to my question? – Tal Fishman Oct 3 '11 at 1:05
I can elaborate..... lead-lag has to do with the optimism in an industry. This is not arbitraged away by share/options traders because the industry itself is growing. The industry itself is an efficient market but the participants will be the actual companies and regulators. The fact that the industry exists is because it is innovative and there simply hasnt been time for regulators and competitors to "arbitrage" it away. – CQM Oct 3 '11 at 1:59

anomalies like arbitrage tend to disappear once exploitation increases, once an anomaly is published if the underlying process is stationary it will get exploited, if it is non-stationary it will cease to be observed.

What about anomalies that are not published, I guess we could use the analogy of the malingering employee who skips work to play golf, lands a hole in one and has no one to tell it to.

If you have an exploitable anomaly that has remained that way for a long time, this is your forum.

-
Are you saying that the existence of this forum is an anomaly that one can exploit? Heh, funny thought. BTW, my question refers solely to published or well-known anomalies, some of which (long-horizon overreaction) have persisted for a very long time despite being no less stationary than similar anomalies which have disappeared. – Tal Fishman Oct 24 '11 at 14:16
That's a good angle too, although I was meaning to say it is not in the employee's best interest to tell his co-workers that he invoked his sick leave to play golf, just as it is not in the best interest of an prop trader to tell us which anomalies are most profitable. Will read up on long-horizon overreaction thanks. – icequations Oct 24 '11 at 16:06
at a glance, the difficulty of exploiting long horizon over-reaction or under-reaction includes the quantifying of entry, exit price levels and holding periods, especially in an environment where news is continuous and may compound the previous under-over reaction. correct me if I am wrong, wouldn't the classification of over and under reaction be post fact. – icequations Oct 24 '11 at 16:15