Momentum investing might be a mirage. I’m positive pieces of it are. Momentum studies actually display evidence of a random market, evidence which some very very smart people use to disprove the efficient market theory. Sadly, they didn’t truly understand randomness. Open up your mind for this one, it’s going to get deep and very weird.
Momentum in Stocks
Momentum investors believe stocks that are going up in price are more likely to continue gaining in value than others. They believe the same about stock prices going down, but for now, lets just focus on the up. The concept mirrors the same term in physics: objects in motion stay in motion.
Momentum studies usually “skip” the first month after selecting high momentum stocks. As an example, if they find a stock goes up strongly from January to June, showing strong “momentum”, they will wait until August to invest in the stock, skipping July. Some people even believe you should skip two months. While strange, skipping a months makes sense on a very basic level.
Multiple studies found a “reversal” effect in stocks for the first month after identifying high momentum stocks. 1 The first month after strong momentum, the stock underperforms and “reverses”. The first study on momentum in 1993 by Jegadeesh and Tilman offered a skip month as the best way to capture the gains from “momentum” .2 Studies ever since have followed suit.
Now doesn’t this seem a little strange? If stocks show momentum why are they likely to stop the first month after you identify this momentum? Why would stocks reverse backwards anyway? Doesn’t the belief that they would reverse backwards, kind of counter act the belief that stocks that are going up, will continue to go up?
The academic explanations for this reversion vary. Some say it’s “related to liquidity or microstructure issues”. Most look to behavior finance and say there is an investor overreaction that required correction. Some people think it has to do with reversions within industry, whatever that means.
Well, what if I were to tell you that randomness caueses the momentum “skip month reversion”. Its presence should be a red flag about possible poor statistical inferences of momentum, and all backtested investment strategies in general. But before that, lets talk basketball.
The Hot Hand Fallacy
Many athletes believe a participant can get “hot”. Something takes over within the player and their prior excellent play itself leads to continued excellence. Nothing exemplifies this feeling more than a basketball player who can’t miss. Shot after shot goes in, in a surprisingly long run devoid of misses. The player clearly gets “hot” and ultimately becomes a better player during this time. Watching Klay Thomspon of the Golden State Warriors during the NBA playoffs reminded me of this phenomenon.
Mathematicians studied this phenomenon in the 80’s and concluded the hot hand was an illusion. Three famous academics 3 produced a study showing players’ shooting percentage did not improve after getting “hot”. They concluded, great shooting streaks are nothing more than pure randomness — there is no evidence a basketball player’s prior hot shooting has any effects on their future shots. They found the shooting percentage for players after a long run of made baskets was statistically identical to their long run average. There is no hot hand.
It turns out this was the wrong conclusion.
Joshua Miller and Adam Sanjurjo recently came to earth shattering conclusions which I’m convinced will, when applied to investing, change finance forever. They realized that for a historic sample, the next shot after a long string of makes mathematically should show a reduction in a player’s typical shooting percentage if there is no hot hand. The fact that the shooters in the 1985 study didn’t show any reversion to a lower shooting percentage was actually proof the hot hand existed. With the correct math, they identified Klay Thompson as one of, if not the, “hottest” shooter in the game.
Yeah I know, this sounds like nonsense at first.6 Bear with me, and keep an open mind.
Why Randomness Reverts
Miller and Sanjurjo’s breakthrough lies in comparing the historical results in basketball shooting to a purely random coin flip. Lets say I flip a fair coin 100 times (likely 50 head and 50 tails). There will be many runs of multiple tails in a row. Lets say we find a run of 5 tails in a row in our sample of flips. What are the odds the next coin is also tails (aka, the momentum trend continues)?
Most people believe its 50%.
And they would be wrong. Its 36%. 4
The authors claim the downward revision from 50% is “driven by two sources of selection bias. One is related to sampling-without-replacement, and the other to the overlapping nature of streaks.” The later is pretty complicated, but the former is very easy to understand.
When you select a streak for evaluation, you inherently remove it from the sample. A simple evaluation shows the proper odds of the next coin are the number of tails in the sample left (45 = 50 tails from original sample minus 5 from the run we already found) divided by the number of flips (95= 100 of the original sample minus 5 from the run already found).
45 / 95 = 47%.
In historical studies, what you are studying gets taken out of the sample, changing the actual statistics for what is likely for the next selection. The “overlapping nature of streaks” then exaggerates the effect even further.
There is a big difference between the results taken from a historical sample and a live event. Obviously, the chances for a live coin flip are 50/50 heads tails. But a historical sample isn’t a live event. It’s a catalog of prior events. If you take the best results out of a historical sample, the remaining results will necessarily be less than average. An infinitely long historical sample won’t show this effect, but we don’t live in a world of infinitely long samples. I can’t repeat this point enough:
Statistical percentages for historical data and live events are not the same and can’t be evaluated as if they are identical.
And this is what Miller and Sanjurjo realized. True randomness in a historical sample of events will show reversals in data after long winning streaks. The absence of those reversals in the basketball data proves shooters do get hot. They actually are influenced by their prior made shots.
Continued Reading
I know this is very hard to grasp. Don’t worry if you don’t get it yet, the writers of the original study were world class thinkers. Most people don’t fully comprehend the concept on the first pass. So I encourage you to read more on this topic as much has been written about these revolutionary findings over the past couple years. For more on this concept, read this, and this and this.
This article outlines a discussion with a hedge fund manager on the ideas. He didn’t get it at first.
If you would rather listen, here is a recent podcast with Joshua Miller on Barry Ritholtz’s Masters In Business Podcast. It’s Excellent.
The phenomenon at its roots is very similar to the infamous Monte Hall problem (I love this problem and will probably do a full post on it later).
Back to the Stock Market
If you’ve fully grasped the concept, you’ve probably already realized the issue with the “skip month” in momentum. Even though the randomness in the stock market is more complicated than a coin flip, the math still works in the same direction.
A truly random market should show a “reversal” in the first month after a long stretch of outperformance in every backtest. This is true even if randomness actually drives stock returns. True momentum would not show a reversal, just as a hot basketball player does not show a reversal in his shooting percentage.
I think I’m the first to notice this flaw in financial research. Please share this concept with others because investors need to know.
What Does This Mean for Investing?
The skip month is probably not necessary
First off, if you still want to invest in momentum strategies, you can likely ignore the skip month. The reversion isn’t what people think it is, and there might not be any reason to offset your momentum samples.
Furthermore, any trading strategy which relies on mean reversion in a backtest should immediately be suspect.
Notice how this explains many failed strategies
Secondly, the hot hand fallacy helps explain why some strategies exist in the historical data, but never show up in actual trading profits. You often hear about trading strategies that never worked properly in the real world, even though they were clearly there in past data.5 Apologists usually explain their poor returns by claiming the market quickly arbitraged the “edge” away. But it’s all just a story to make the investor feel better. The “edge” someone thought they found didn’t actually exist. It was a statistical error.
This is very powerful to understand. A coin flip is totally random. You would be nuts to make aggressive bets on a live flip of a coin because the last 5 flips were heads. Yet, if your looking at an old sample of flips, you actually should make aggressive bets on a reversion. Just as random compound interest confuses people, investors often mix these two up by foolishly taking historical investment strategies and applying them to live trading. Statistics is hard.
Is Momentum Real?
I’m not actually disproving momentum here. The momentum part people invest in comes after the first month. But the “skip month” is a foundational concept in momentum studies, which should now give people pause since I have shown momentum studies incorrectly evaluated historical data without fully capturing the true statistical message they send.
Interestingly, there is a chance what I’ve shown will actually prove momentum is real. Using the coin flip example above, if you see a 40% tails finding in a backtest when randomness says 36%, you may have just found proof of momentum. Maybe someone will look back at the studies and find something similar in stocks. Maybe. But right now, it is best to say we don’t really grasp the statistics behind momentum. Therefore, I’m not investing my hard earned money in something so poorly understood.
Question the finding of other studies.
At minimum, the fallacy should make you question the findings of many financial studies using historical data. This is very hard to wrap you head around I know. But there is a major difference between statistics used to review prior historical events, and statistics used to predict future events. Very few people fully grasp this idea, and as far as I know, no financial studies consider it in their findings.
But a least you’re now one of the rare people who do understand randomness and historical data.
In Part II, we will look into the actual returns from momentum after the first month, to uncover the randomness hiding among those bushes as well.
1-See , Narasimjan Jegadeesh “Evidence of Predictable Behavior of Security Returns” (1990) and Bruce Lehmann, “Fads, Martingales, and Market” (1990) among others.
2- Jegadeesh and Titman, “Returns to Buying Winners and Selling Losers: Implications for Stock Market Efficiency” (1993)
3-Thomas Gilovich, Robert Vallone, and Amos Tversky “The hot hand in basketball: On the misperception of random sequences” (1985)
4-I think this is correct. The math is not simple.
5-Remind anyone of factor investing or dare I say it, Momentum.
6- Klay Thompson being one of the best shooters in the game isn’t strange. Everyone knows he’s great. The strange part is that continuing to shoot “at the average” during a streak in historical data means the player was hot.
Very interesting findings, indeed. I have never thought about this effect before, but it seems to make sense. I still have to read the resources you linked. But one question that I have is this: Shouldn’t the effect be weaker than “expected”? Because if you have a finite data set, also your statistical moments will not be the ones of the distribution you sample from. So a streak of “positive events” can simply offset the average of your distribution and the remaining data set (disregarding the streak) could still look like the “ideal distribution”, no? This may make things even more complicated. But I guess I will find some answers in the linked material.
Furthermore, assuming your argumentation is indeed correct, wouldn’t that make momentum even stronger? It’s basically exactly the opposite of the hot hand fallacy. If you find momentum without considering the described effect even AFTER taking out the initial streak, then momentum should be so strong to even survive that treatment, no? Maybe I’m just missing something after an initial read.
Their argument is rubbish. Take a sequence of N i.i.d coin flips, where we will examine the next flip after any sequence of k heads. Say this flip occurs at position p (obviously, p>=k+1). Then the proportion of possible prefixes that could have led up to flip p is precisely 2^(-k) of all possible sequences of length (p-1) – the ones that ended with k or more heads in a row just before p – and because the flips are i.i.d, the set of possible suffixes, starting with p, is precisely all possible sequences (i.e. 50% H/T, or different for a biased coin, for the remaining N-p flips). So the fact that we are considering sequences does not change the ‘obvious’ intuition about the next flip (which is correct – 50%). And we can calculate the total expected number of “k+1-streaks” (or longer) across all p in all possible sequences simply as sum_{k+1}^N 2^-(k+1), where the +1 is for the 50% chance of p being another head. Which is just (N-k)/2^(k+1). There are no mean reversions or “interrupted streaks” involved.
The only thing that would affect this math is if the coin flips were *not* i.i.d. (i.e. they are conditional on recent history). Which can only be true if the “hot hand” (or “momentum”) effect actually exists. Which can only be proved (or disproved) by showing that the sequences are distinguishable from (or indistinguishable from) a sequence of i.i.d random variables. Which should only be attempted by proper statisticians. I’d rather believe the basketballers – who, at least, have a good chance of knowing what they are talking about (having spent more time doing it).
Back to the paper. Their first mistake is that they posit an independent sequence, then invoke a construction on conditional probabilities to extract the trials (duh… they’re *independent*, which makes the math pretty simple). Based on that they construct a fallacy: in A1 (the proof), page 23, we have the purported inequality between (2) and (3), which they argue is an inequality because P(\tau=t) would be smaller if X_t = 1. But everything there is already conditioned on F_t, which FIXES the value of X_t – so the probability of either (2) or (3) is zero (or, really, is zero divided by zero, since it’s in the conditional). And in any case, F_t also fixes |I_k|, which is the reciprocal of P(t=\tau). Any attempt to exempt X_t from being a member of F_t means that t isn’t in \tau. So the failure is irrecoverable.
Finally, since I’m obviously on a hot streak here, here are the corrections to the paper:
Sec 2 (proof outline) and A.1 (proof): “P(…|X_t=1…) < P(…|X_t=0…) …for t < n, which guarantees that" should be "P(…|X_t=1…) = P(…|X_t=0…)… for k < t 1″… umm…
BTW, I really enjoyed some of your articles.
For some reason this got stuck in spam and I missed it. I’m not yet fully gathering your point, so I’m going to have to think about it.
I was probably drunk when I wrote it 🙂
It was a delightful article, nonetheless. Thanks 🙂
This is the best explanation of the biases identified by Miller and Sanjurjo’s 2018 paper I’ve read. I’ve struggled to comprehend them since reading it in 2018, but now they seem pretty obvious. Thank you!
Also, I thought you (or I) might be nuts when I first read the “36% probability of tails”, but then a lightbulb came on with this – “Statistical percentages for historical data and live events are not the same and can’t be evaluated as if they are identical.”
Thanks. It is very eye opening realization that I think has application in lots and lots of studies, even outside finance.
> A simple evaluation shows the proper odds of the next coin are the number of tails in the sample left (45 = 50 tails from original sample minus 5 from the run we already found) divided by the number of flips (95= 100 of the original sample minus 5 from the run already found).
In this scenario, do we *know* there were exactly 50 tails in the history? If not, then I don’t follow. If all we know is that a coin was flipped 100 times, and the first 5 were tails, then my expected value for total tails is no longer 50.
With most historical data, we don’t know the true odds like we do with a coin. So a 50% heads/tails rate will be found first in the historical data and then used as the average. No matter what the rate is for the entire sample, it will always be lower when you take a long string out of the equation.
With a coin, we don’t “know” it will be 50/50, but that’s where it’s likely to be. Of course it can be different because of randomness, and of course it can be different after a string of tails because of randomness, but its not what’s expected.
> With a coin, we don’t “know” it will be 50/50, but that’s where it’s likely to be.
For a random sequence, yes, the expectation is 50 heads, 50 tails. But we don’t have a random sequence – we have a random sequence that happened to have a streak of 5 tails! Conditioning on that, by my calculations we should expect ~50.9 tails in the sequence, not 50.
it doesn’t take many flips to expect to find a streak of 5 within that sequence.
Well yes, but still ~19% of length 100 sequences won’t have a streak of 5. The remaining 81% are biased to be 51.9% tails, 48.1% heads.