Randomness in Momentum Everywhere

Randomness explains much of the “Momentum Effect” in stocks.  Yes, you read that correctly, much of the evidence for momentum can actually be explained through randomness. In part 2 of my evaluation of momentum, I’m going to show you how.

Randomness (and Rebalancing) Hiding In Plain Sight

In my post on when you eat matters more than what you eat, I discussed how randomness and rebalancing fools factor investors.  I believe at least some of the factors discovered in the investing world are products of the rebalancing strategy used in the studies, not the factors themselves.  Momentum is not different in this case.

Momentum investors believe stocks that rise in price are more likely to continue rising. They believe the same about stock that are falling in price. The concept mirrors the same term in physics: objects in motion stay in motion. Winning stocks are more likely to keep winning, and losing stocks are more likely to keep losing. Many academic studies have examined this effect. I’m going to focus on the first.

In the earliest paper on momentum by Jegadeesh and Titman (1993), the authors produced data showing momentum influencing stock’s returns over 12 months.  The authors found the strongest effect of momentum when they screened stocks over the prior 12 months. They selected the top 10% stocks over the prior year to test momentum.¹  They then created equal weighted portfolios of these stocks and held them for 3, 6, 9, or 12 months before repeating the process and creating a new portfolio from scratch again. Here are their results showing the average portfolio returns, converted to annual returns:

Now let’s think about what they did with this test from a rebalancing perspective.  I’ve shown here, that rebalancing increases a portfolio’s returns. The more often you rebalance the greater the benefit.  With the 30 components of the Dow, increasing the rebalancing frequency increases both the portfolio’s arithmetic and geometric returns. 

With this in mind, look at Jegadeesh and Titman’s study again.  Their methodology is actually an equal-weight rebalancing scheme, with the 3 month “holding period”, serving as a 3 month rebalancing period, and a 6 month rebalancing period, a 9 month rebalancing period, and finally a 12 month rebalancing period.  The finding that “momentum” is strongest over the shorter period and fades as the holding period grows is not a finding about momentum.  It’s exactly what you would expect from random behavior when adjusting portfolio rebalancing frequency.  Yes the slope of the momentum curve is much higher, but momentum stocks are also much, much more volatile than dow components.

If momentum fades away, but that fading is actually explained by randomness and rebalancing, was the momentum ever there to begin with?

The Nature of Momentum Filter

Momentum invests in the top recent winners (and if you’re ok with shorting, it also sells the biggest losers).  But what kind of stocks become the biggest winners?

First off, they should favor stocks that have the highest expected return.  But they will also favor the stocks with the highest volatility.  If stock returns are random, then by pure chance many of the recent top performers will have a lower expected return while displaying high volatility.²

Imagine two groups of 50 stocks.  The first has an average return of 5% but volatility of 25%.  The second has an average return of 10%, but a volatility of 15%.   If you let the stocks randomly produce returns for a short period, and then select the 10 best stocks, is your sample more likely to come from the first group or the second?

Chart for demonstration purposes only

It’s the first group, by far

Because the first group is more volatile, it is more likely to have extreme losers and winners.  And in this instance, you are most likely to end up with 7 stocks from the first group, and 3 stocks from the second group. 

The exact same effect is true for picking the “short” side of momentum.  You are far more likely to pick strong stocks that are volatile than you are weak stocks that are not volatile.   Momentum is a gigantic volatility screen, more so than a “momentum” screen. 

Jegadeesh and Titman’s data shows this (page 73).  They calculated the beta (which is related to volatility) of each decile in their sample, and found the two extremes (highest and lowest return) exhibited the most volatility, with the middle being the calmest.  Exactly the result you would expect to find from randomness in stock returns.

The momentum screen will lean toward picking stocks with higher expected returns. But importantly it will also be filled with high volatility stocks even if they have average or poor returns. 

Momentum Fades Over Time

Later on, Jegadeesh and Titman evaluate the performance of their portfolios through “event time”.  This simply means they would look at how the momentum portfolios did each month after creation over the following 3 years. Here, rebalancing is not an issue as the portfolio is left untouched over the 36 month period.  But once again, the results show extremely strong evidence of randomness.

The authors created a table of their findings, which I recreated graphically below.  In it they show that after the poor first month (the skip month), returns each month start strong, but then slowly fade, until in the 13^th month, when they actually become negative.  They stay negative for a very long time too.

Below is the cumulative return from these monthly returns. Here you can see the portfolio shows strong gains until the 12th month, when it rolls over and starts falling. This is why the authors say momentum starts strong and then fades away and disappears. The charts certainty don’t look random. So what is happening here?

What if I told you nothing except time and compounded returns in a volatile portfolio? A random process produces a very similar return stream.

Randomness Also Gains and Then Fades Away 

Let’s think back to my post on investment games. In game #7, I brought up the idea of a very simple “coin flip portfolio” in which you get returns from two coins flipping simultaneously.³ I’m going to slightly tweak the payout of the game for this example to the following:

Bet $100.  Flip a coin, heads you win 50% of the bet (receive $150 back, for a gain of $50), tails you lose 36.7% of your money (get $63.3 back, for a loss of $36.7).

You play two games simultaneously. Each game is then played with the full payout from the previous game rolled into the next round.  The game repeats, with no rebalancing.

What are the expected results from this game over time?

Remember, since we know the compound growth rate is all that matters (not an arithmetic average), we will determine the expected future wealth from the average compound growth rate of the coin flips. Our very simple portfolio of two “coin flip assets” produces the following results over time:

Whoa. That chart looks an awful lot like the chart the authors found in “momentum”. It rises sharply to start, peaks and drifts back downward. Its as if the coin’s returns have “momentum” that fades. But how can that be, these are coin flips?

Compound Randomness Starts with the Arithmetic Return, and Fades to the Geometric

All random compounded returns start out producing returns equivalent to the asset’s arithmetic returns. But with every repetition, the returns will converge toward geometric return. A portfolio of stocks slows down this degradation of returns toward the geometric return, but it still happens.

The shape of returns found in momentum studies is the shape of every random process with a positive arithmetic return and a negative geometric return. Early on, the returns track near the positive arithmetic return. But over time, the repetitions ensure the expected return moves closer and closer to the geometric return. Therefore, the original positive gains roll over and start heading downward. The slope of the gain, location of the peak, and slope of the losses are all dependent on the number of assets and the random properties of those assets. But they will all show this same overall shape.⁴

We already know momentum screens select high volatility stocks. High volatility stocks will inherently have a large spread between their arithmetic and geometric returns.⁵ Therefore, the shape of the momentum return stream over time isn’t really an anomaly at all, but is expected.

You don’t need to say the momentum effect works in the short run and then fades.  It’s all explainable with randomness and geometric compounded returns.  You don’t need stock “momentum” to explain the results of the study.  The rules of the strategy alone create the illusion of momentum, even with random coin flips.

Some Have Gotten Near This Point

Other investment research is getting close to this point.  Jesse Livermore (the anonymous blogger not the historical investor) studied momentum in stocks and concluded that:

The strategy substantially under performed buy and hold in every stock except $GE.  The pattern is not limited… – it extends out to the vast majority of stocks in the S&P 500.  The strategy [momentum] performs poorly in almost all of them despite performing very well in the index.

The fact that the strategy performs poorly in individual securities is a significant problem as it represents a failed out-of-sample test that should not occur if the popular explanations for the strategy’s [momentum] efficacy are correct.

Here Mr. Livermore notices that momentum doesn’t work on individual stocks, but it does work on portfolios of stocks (index).  Yet papers on momentum study portfolios of stocks.  Maybe the benefit isn’t in the stocks, but in the portfolio construction strategy?

Secondly, a recent article by Cory Hoffstein looked into momentum inside “sectors” of the market.  He found that the equal weight sector strategy explained the “momentum effect”, not momentum in the sectors themselves.

Rather it would appear that a top N momentum strategy was merely able to back its way into harvesting the return benefits of the equal-weight portfolio…

It would appear the real hero here in our top N momentum strategy is not momentum, but rather the equal-weight sector tilt implied in the strategy.

Pretty similar to what I said above about momentum being a product of developing equal weight portfolios and the rebalancing of them.  It’s the nature of the strategy creating results, not “stocks that go up continue to go up”.

(Please do not assume either Mr. Livermore or Mr. Hoffstein agree with my overall thesis. I am only pointing out two others who have found results in markets which align with my overall view).

This Also is Why Momentum is “Found Everywhere”

Researchers have “found” momentum everywhere.  In equity markets all over the globe, deep into history, in all different kinds of currencies, in commodities, and in bonds. There’s nowhere researchers don’t find momentum.  Now what’s the common denominator in all these studies?  The methods of the momentum studies themselves.

If you find that every object you measure over time increases in temperature, you could create the theory of universal temperature momentum.  Or you could question whether the identical tools and methods used in each test influenced your results.  Your choice. 

Investment Research Needs to Compare to Randomness

The best research compares its finding to a control group.  This is clearly very difficult to do with investment research.  Most people use a market index, like the S&P 500 as their control group.  I’ve already shown a few times why this is problematic.  A market index does not represent the market over time. What researchers should be doing is comparing their results with randomness.

Technically, I’m not saying that randomness explains ALL of the momentum effect.  It may.  I’m saying randomness and rebalancing undoubtedly explain SOME of the findings of these papers. The process of selecting high volatility stocks and rebalancing them frequently produces most of “momentum’s” performance. If researchers compared their results to a random data set, they would see this.

But now, you can see how randomness hides in unexpected places, even if the study isn’t looking for it. Understanding the powers of geometric compounding and rebalancing reveals the randomness concealed inside momentum.

Now, take this knowledge and explore other realms where randomness unknowingly lurks. It might be everywhere.

1-They also did the shame with the worse decile, shorting that group.

2- Some may point out that the standard version of the efficient market theory says stocks with lower average returns and higher volatility shouldn’t exist. This is true. But proponents of momentum don’t usually believe in the efficient market theory. And as I’ve shown, I’ve got a different view of the theory.

3- I would have loved to use more than two coin flips. The math on this gets exponentially complicated the more assets get involved. And two coin flip still shows the effect I was looking for.

4-I’m sure some professionals are thinking that I’m comparing my coin flips, which only represent the long side of momentum, against data from Jegadeesh and Titman which is the long minus the short portfolio. Your right, I am. It’s much easier to grasp that randomness can behave like the long side of momentum before trying to incorporate the short side. But once you do, it is an easy step to see that a long portfolio minus a short portfolio will produce the exact same return shape. Taking the the difference of two random return streams produces a standard deviation much higher than the standard deviation of the long and short values (longshort variance = variance #1 + variance #2), nearly ensuring the long-short portfolio has a positive arithmetic return and a negative geometric return.

5- Estimated Geometric Return = Arithmetic Return – 1/2 Volatility ²