Frog in the Pan: Identifying the Highest Quality Momentum Stocks

/Frog in the Pan: Identifying the Highest Quality Momentum Stocks

Frog in the Pan: Identifying the Highest Quality Momentum Stocks

By | 2017-08-18T17:04:46+00:00 November 23rd, 2015|Research Insights, Momentum Investing Research|35 Comments
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(Last Updated On: August 18, 2017)

Frog in the Pan: Continuous Information and Momentum


We test a frog-in-the-pan (FIP) hypothesis that predicts investors are inattentive to information arriving continuously in small amounts. Intuitively, we hypothesize that a series of frequent gradual changes attracts less attention than infrequent dramatic changes. Consistent with the FIP hypothesis, we find that continuous information induces strong persistent return continuation that does not reverse in the long run. Momentum decreases monotonically from 5.94% for stocks with continuous information during their formation period to –2.07% for stocks with discrete information but similar cumulative formation period returns. Higher media coverage coincides with discrete information and mitigates the stronger momentum following continuous information.

Alpha Highlight:

The account of the boiling frog is an anecdote describing a frog in a pan of water. If the frog is put into boiling water, it will immediately jump out. If it’s placed, however, in cold water that is slowly warming up, it won’t be aware of the gradual heat change, and it will be cooked to death.

frog in the pan pic

Investors act in a similar manner with respect to gradual stock price changes. For example, for a stock with an immediate 100% gain (boiling water) the new fair value is immediately recognized by all investors, whereas, gradual stock price changes (cold water slowly warmed up to boiling temperature) often receive less attention. In behavioral finance academic parlance, investors suffer from limited attention when it comes to gradual stock price changes.

In this paper, Da, Gurun and Warachka (2012) investigate investors’ limited attention to gradual-information diffusion and hypothesize that it has a conditional relationship with momentum.

  • Frog-in-the-pan (FIP) hypothesis: “a series of frequent gradual changes attracts less attention than infrequent dramatic changes. Investors therefore underreact to continuous information.”

The conclusions are clear: a more sophisticated momentum strategy that focuses on the path-dependency of momentum generates a much stronger momentum effect.

Key Findings:

This paper constructs a proxy for information discreteness (ID) that measures the relative frequency of small signals. A large ID means more discrete information, and a small ID denotes continuous information. For past winners with a high past return, a high percentage of positive returns (% pos> % neg) implies there are a large number of small positive returns.

proxy for information discreteness

Next, the authors double-sorted portfolios that condition first on a 12-month formation-period returns (Jegadeesh and Titman 1993), then second on ID during the 1927 to 2007 sample period. They find that over a six-month holding period, momentum decreases monotonically from 5.94% for stocks with continuous information to –2.07% for stocks with discrete information.

Frog in the pan

The results are hypothetical results and are NOT an indicator of future results and do NOT represent returns that any investor actually attained. Indexes are unmanaged, do not reflect management or trading fees, and one cannot invest directly in an index. Additional information regarding the construction of these results is available upon request.

Wow. The graph below shows the momentum alphas following continuous and discrete information from 1 to 10 months after portfolio formation. The results are consistent with the FIP hypothesis–continuous, or “quality,” momentum seems to account for much of the momentum effect. Specifically:

  • Higher profits: High-quality, or continuous, momentum stocks have higher three-factor alphas than low-quality, or discrete, momentum stocks.
  • Longer persistence: Momentum profits following continuous information persist longer (the t-stat remains significant for about 8 months); while Momentum profits following discrete information persist only 2 months (i.e., an investor can trade momentum less frequent and still win).
Frog in the pan momentum profits

The results are hypothetical results and are NOT an indicator of future results and do NOT represent returns that any investor actually attained. Indexes are unmanaged, do not reflect management or trading fees, and one cannot invest directly in an index. Additional information regarding the construction of these results is available upon request.


Momentum has been studied for many years and was officially documented by researchers in 1937 (h.t., Gary Antonacci).

Here is the chart from the original Econometrica paper:

cowles jones 1937 momentum

The results are hypothetical results and are NOT an indicator of future results and do NOT represent returns that any investor actually attained. Indexes are unmanaged, do not reflect management or trading fees, and one cannot invest directly in an index. Additional information regarding the construction of these results is available upon request.

The authors document that “inertia” is significant in stock prices. The authors also highlight that they have limited data, and therefore, limited statistical power to assess longer term inertia. Well, we now have 200+ years of data to verify their claims.

Remarkably, by simply reviewing the path by which momentum is achieved, as highlighted by the authors of the paper discussed above in this post, the momentum anomaly can be enhanced via concentration in those high momentum stocks with the highest quality, or most “continuous” momentum. (Note: An older paper from Grinblatt and Moskowitz (2004), finds a similar finding.)

To make this clear, I drew up a simple chart to highlight the concept.

barney style momentum

The results are hypothetical results and are NOT an indicator of future results and do NOT represent returns that any investor actually attained. Indexes are unmanaged, do not reflect management or trading fees, and one cannot invest directly in an index. Additional information regarding the construction of these results is available upon request.




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About the Author:

After serving as a Captain in the United States Marine Corps, Dr. Gray earned a PhD, and worked as a finance professor at Drexel University. Dr. Gray’s interest in bridging the research gap between academia and industry led him to found Alpha Architect, an asset management that delivers affordable active exposures for tax-sensitive investors. Dr. Gray has published four books and a number of academic articles. Wes is a regular contributor to multiple industry outlets, to include the following: Wall Street Journal, Forbes,, and the CFA Institute. Dr. Gray earned an MBA and a PhD in finance from the University of Chicago and graduated magna cum laude with a BS from The Wharton School of the University of Pennsylvania.
  • IlyaKipnis

    Seems to me that the final hand-drawn diagram implies that we simply want the price series with the smallest standard deviation of returns. That is, draw a straight line between the initial price and final price, and measure how far away both lines stray from that.

  • Doug

    The MTUM ETF uses this principle in its stock selection. Basically rank orders on 12-1 Sharpe Ratio instead of raw return. It has performed fairly well, although it’s a relatively new ETF.

  • similar idea, yes. Frankly, anything that quantifies that chart will work better, at the margin, than simply buying all 2-12 momentum stocks

  • Yep, a reasonable approach, in theory, but in their index cookbook and press release you’ll notice the strategy is built for “scalability.”

  • Ben

    Hi Wes,
    How is the 2-12 momentum time frame justified? It seems from other long-range and multi-asset classes momentum studies that 1-12 works for everything but for US Equities (including pretty much every other Equities in the world). I’ve never found a compelling argument for why we have to exclude the last month for US equities… I know this comes from the 1993 paper – but has it been re-tested since then? Could it just be a case of data fitting/mining then that isn’t supported by history since and/or is not robust given the wealth of data showing that 1-12 is better for every other markets?

  • you can include the first month and do 1-12–still works fine. The first month is eliminated because 1 month returns are known to be mean-reverting…so when we include it in a 12 month indicator we make it noisier than it has to be…of course, there is an element of data-mining associated with that approach…take the good with the bad. Honestly, I don’t think it matters that much…Novy-Marx has a paper showing that 1-6 and 7-12 also work. I think he finds 7-12 is the best.

  • Steve

    Just for clarification, Novy-Marx thought he found 7-12 worked best, yes. Goyal came along and showed (convincingly, in my opinion), when looking at international data, that Novy-Marx was simply observing a statistical bump in the data, that just happened to be the US data.

  • We’ve done our fair share of Novy-Marx commenting in the past so I didn’t want to elaborate…but yes, thanks for sharing…

  • Steve

    Holding period is always the consideration with momentum. Skip month is relevant to short-term trading, not so much if your trying to hold longer. That’s my understanding from the sum of my reading on momentum. Interestingly, no-skip-month seems to be better on indices / asset classes.

    It’s the same with acceleration of momentum. Holding period matters. There’s a paper (relying on memory alone, without looking for it – I think by Ibbotson?) that acceleration is a bad thing. But – the holding period is short-term (next month return). That paper uses that finding to discuss / explain the short-term reversal effect (again, just from memory).
    However, other papers (there’s an old one called, Acceleration by Marks and a more recent one, ‘Trend Salience’ which both look at a longer holding period and find Acceleration to be a very positive factor.
    The Moving Average Ratio and Momentum (Park) I believe, also inadvertently explores this topic (even though he didn’t realise it).

    That’s why, for longer holding periods (e.g. one year) you’ll often see shorter lookback periods (3 or 6 months) being favoured (e.g. O’Shaughnessy). Which kinda makes sense…a one year look back and a one year hold = a 2 year window. Much too long for momentum.

    *Please note: in all my comments that I ever make on momentum, I am always talking from the perspective of a long only investor.

    At the risk of this comment now being off-topic, I’m going to hit the post button, anyway – I just can’t help myself 🙂 Love this stuff.

  • Steve

    Heh heh, I remember the posts and sharing the sentiment.

    Having said that – he has an absolute cracker of a paper just released this year, on multi-factor investing, which really emphasises the take home point: if you’re gonna have multi-factors than they should be worthy of stand alone investing (which, when considering profitability, I find a little ironic, but let’s not mention that). Anyway, very cool paper: Kidding aside, the only way to learn (well, the only way I’ve learned) is to struggle with the tension that comes from reading these different papers, and then reconsidering them when new light emerges. So I’m grateful for all the academics that I’ve read, that made an honest effort with the data…whether I now agree or disagree with their findings.

  • Steve

    I’ve seen volatility adjusted momentum in papers being performed by stuff like dividing the price return by the beta or standard deviation, or by including standard deviation / beta in a multifactor ranking with price return.

    I imagine all of these, as Wes mentions, would have a high correlation with Frog in the pan / momentum consistency.

  • Steve

    Can I get some help, to make sure I haven’t concluded incorrectly…

    When I read this paper a couple months back, I filed it away as interesting, but another paper that only showed real benefit to the short side, in a long-short framework. I’d concluded that, long only, the double sort on continuous, didn’t really add that much (except perhaps for an argument to avoid the very most discrete stocks).

    Have I got that right?

  • umair usman

    cant wait for your next book

  • Karl

    Sometimes (always?) simplicity rules.
    I think the hand drawn diagram was a nice touch for me to “get it”.

    My question to you (or anyone basically) is – so how can you possibly quantify that?

    To be more specific – is there anyway you can possibly come close to that by doing some kind of sorting and/or calculation using simple 52w,26w,13w price changes?

  • Steve

    The artists’ drawing was illustrating what the paper is trying to quantify – via a consistency (my term) or continuous vs discrete (their terms) measurement. Did you read the paper? Consistency can be measured in various ways, and the paper mentions some of those previous papers. Volatility adjusted momentum (various ways of doing that) is another possibility.

  • Steve

    Me too, but unfortunately we’re gonna be waiting a while 🙁 Did you read DIY Fin Advisor?

  • Chris

    Do they look at drawdowns of continuous info Momentum vs discrete momentum( or have you guys) ? i don’t seem to be able to find it in the paper

  • Chris, we are talking about academic researchers here…so the answer is no 🙂 That would be entirely too practical.
    Fear not…we’ll have some stuff coming out in the next week or so that will dig into the details. To answer your question, drawdowns are similar to traditional 2-12, but expected returns are higher.

  • Steve, thanks for pointing out. The paper goes through a ton of different ways to capture the main idea. The hope was that my art enabled folks to understand the concept before they got buried in the math…but the various algos outlined in the paper capture the idea in the illustration.

  • yeah, I’m trying but we are buried in our business and I have 3 little rugrats under 6 yrs of age…slow and steady wins the race

  • We’ll post some of our own internal results on this very soon. Frog in the pan definitely adds some real value, historically

  • yes, as long as you are capturing the big muscle movement outlined in my terrible piece of art, you’ll be in the ballpark.

  • Steve

    “slow and steady wins the race” is quite apt, considering the topic of this post! 😀

  • Steve

    You guys are still rockin’ the research. Going to be a fun 2016 on the blog!

  • umair usman

    im still going through it.
    but it was Quant Value that canged my whole idea about Quant investing in the first place. it was an eye opener for me

  • Dominic Pazzula

    So basically

    abs( 0.5*cumulative return[T]*T – sum( cumulative return[t] ) )

    or perhaps

    sum( abs( cumulative return[T]*t/T – cumulative return[t] ) )

    Both seem pretty easy to implement and test.

  • TY

    The paper sounds like another illustration of the fact: momentum performs better in low volatility space than in high volatility space. It would be interesting to know how the continuous momentum perform during the 2009 market reversal (traditional momentum meltdown) period.

  • Mark

    I think calculating the least squares regression line then taking the product of the slope and the r-value (or some other function that takes the slope and r-value as variables) would be a good way to measure quality momentum.

  • TheChrisp1231

    Do you think ‘frog in the pan’ explains the low volatility anomaly or the other way around ?

  • Possibly related, but value and low-vol are more tied at the hip

  • Mark

    I attempted to integrate the FIP algorithm into the RAA
    framework using Quantopian. However as far as I can tell the FIP doesn’t add
    much value. Selecting a smaller number of winners using the simple 12-month
    lookback algorithm has the same effect as double sorting using the FIP ID
    metric. Results here,

  • Hey Mark,

    Thanks for sharing. A few questions for you on the replication process:

    1) What is your rebalance frequency?
    2) We actually cut to the top 20% mom, then the top 50% frog names…so the portfolio will be ~50 stocks. 20 is fine, but mom names are so vol and your period is so short, it is hard to differentiate signal from noise.

    Nice work. Its unfortunate that you guys have such limited data — you know when Quantopian will expand their stock data? It’s really hard to ascertain much of anything with a ~10yr dataset.

  • Mark

    Hi Wes,

    Thanks for getting back to me. I reran the strategy using a 50 stock cutoff for Frog names and posted the results on the same thread,

    I use a monthly rebalance frequency for all tests. I take your point about 10 years being a short time frame from an academic perspective. On the other hand from a practical point of view it seems like a long time to wait for a strategy to add value. In your testing do you observe a degradation in the signal in the last 10 years?

    I don’t know when or if Quantopian is planning to increase its dataset, though I would certainly welcome it.



  • janvrots

    Am I the only one struggling with the ID formula
    1) assume sign(pret)=1, %neg = 30%, %pos=70%, id = -40%
    2) assume sign(pret)=1, %neg=70%,%pos=30%, id=+40%

    So stocks with better quality momentum have a negative ID. I would have used (%pos-%neg), why do they do (%neg-%pos)

    check out the appendix to get a better grasp on theory they are going after.

    Here is a quote from the paper that is helpful:
    “The model illustrates that momentum originates from the truncation of small signals whose signs are the same as the formation-period return. Conditional on a specific formation-period return, momentum strengthens with the frequency of these small signals.”

    Basically, for past winners who have positive PRET, investors ignore small positive returns (which reflect “rational investor” activity as per their model in the appendix). In short, the small positive returns are an indication that smart money is saying something, but the rest of the market hasn’t figured it out yet.
    Also, for past winners who have a negative PRET, investors ignore small negative returns, which are an indication that smart money is ahead of the game, but the mkt won’t find out till later.

    You can flip equation if it is more intuitive: sign(PRET)*(%pos-%neg).
    So let’s say A and B are winners and both have PRET = 10%. A has 60% pos, B has 40% pos.
    ID_A =1*(20)=20 = good
    ID_B = 1*(-20)=bad

    Why? Well, if both have a PRET =10% and A has more positive than negative signals realized during the period, and the positive signals are likely to be relatively small on average

    What if A and B are winners and PRET = -10% for A and B?
    Translation: The top-performing winners have less negative returns than everyone else in the market. In this world where everyone is doing poorly on an absolute basis, lot’s of small negative are akin to lots of small positive when PRET>0

    Let’s say A and B both have PRET = -10%. A has 60% pos, B has 40% pos.
    ID_A =-1*(20)= -20 = bad
    ID_B = -1*(-20)= 20 = good

    If flipped? Now A is bad and B is good. Why? Well, if both have a PRET = -10 and B has more negative days, it must mean that a lot small negative days contributed to its relative outperformance to all the other stocks out there.

    Anyway, I agree this is confusing. Luckily, when examining the winner portfolios, PRET is almost always >0