Posts filed under “Mathematics”

Momentum vs. Mean Reversion

Salil Mehta is a statistician and risk strategist. He served for two years as Director of Analytics in the U.S. Department of the Treasury for the Administration’s $700 billion TARP program. He is the former Director of the Policy, Research, and Analysis Department in the Pension Benefit Guaranty Corporation. Salil is on the Editorial Board for the American Statistical Association, is a Chartered Financial Analyst, a fellow member of the American Statistical Association and Royal Statistical Society, as well as being a current dual candidate member of the Society of Actuaries. He is the author of the mathematics book, Statistics Topics


Which is a better trading strategy, momentum or mean-reversion?

Try this mathematical thought experiment.  Look at these four anonymized stocks below, each trading from the mid-1980s.  Each stock chart starts indexed at $100.  What do you see in each of these stocks?  Are there some hints in each of them, which indicate that one of the two trading strategies is better?  Are stocks A and C better buys, versus stock D?






Let’s look at applying a trading strategy to these companies.  Say a momentum strategy, where we focus on if the stock moves up (relative to its trend) for two years straight, then one invests in that stock until the first relative down-year of that company.  What types of returns would you make from this (“don’t fight the uptrend”) strategy?

Next let’s look at a mean-reversion strategy, where we focus on if the stock moves down (relative to its trend) two years straight,then one invests in that stock until the first relative up-year of that company.  Which of these strategies is easier to use (momentum, or the “buy the dip”)?  And what sort of returns would you get from just these simple trading rules described here?

By answering such math questions, we learn a lot about the risks and rewards, of these fundamentally exclusive trading ideas. These are the same ideas that are mathematically embedded into modern ARIMA econometric models.  Pause for a moment and try to determine by looking at the four stock returns above, which sort of trading signal and profile would be generated:

I.    <$100
II.   $100-$200
III.  >$200

What we see below is that both technical rules produce the same results!  About $75 growth (or answer I above), off of the $100 initial price.  So in other words the implementation of one successful strategy (after all we profited $75) mathematically means we would not be able to profit on the other strategy.  Aren’t we therefore -by nature- still leaving money on the table?





Inspecting the above stock chart more closely, we might also notice that both trading signals only worked about 7 or 8 of the years, per company (or a quarter of the time).  This also implies nearly half of the time neither trading style exclusively applied, even though the stocks were in an overall uptrend.  We also notice that fairly equal returns from stocks A and C, have differing results within the same strategies!  Worse yet, we notice the risks involved in any strategy, since neither strategy just moved in an up direction over time.  Both can suffer losses from mistiming still.

Putting this altogether, we see all of the mixed prospects of using one of these trading strategies.  And we only reach results of$175 after 30 years, falling short of the results of typically $400 if we instead just kept a buy-and-hold strategy the entire time (see the top-most chart.)  This is likely less than any optimistic market participant would have guessed a-priori (e.g., the IIIIII choices above).

Also notice that the interpretation of both momentum and mean-reversion are the same, in a short-term view, in that one is waiting for a a slight trend in order to same the same decision (in this case to buy stocks).  This means that both strategies are in fact the same to some degree (and offer the same returns as noted above!), differing only in the hope of the investor about where stocks should head.  We say “hope” because again at the end of waiting for the same signal, the future direction of the stock is still random.

Still not convinced buy-and-hold would be the best approach to these stocks above?  Think there is a better rule-based algorithm you could apply to better tune your approach to each of these four stocks?  Unfortunately, think again.  The stock chart above isbogus, and in fact each series is just a random number generator.  See the de-trended chart below.



Source: Statistical Ideas


While starting at $100 each variable A, through D, annually fluctuates between -$20, to +$20.  Whatever happens in any given year has no influence on what happens the following year.  And the key probability insight for this article is: ogling at a random trend to provide a signal also has no value.  We added a +$10 trend each year, on top of the random value of -$20, to +$20.  Of course the reality of economic compounding should have been an early tip-off that the top-most chart isn’t perfectly right.

One is essentially being fooled.  Not by me, but by randomness, and that can happen in more ways then we might think. Articulating ideas from stock chart patterns, which can speak to someone if they hope them to.  But signals mislead with false-positives in this article, just as they often do in real-life (herehereherehere).  In this case we also see that we could unfortunately have been fooled into thinking that the underlying trend of the “companies” A and C, were different from D.  In fact they were, and are all the same.

Also recall just a month ago that all chart-devoted technicians adamantly accepted that markets were “consolidating” into a multi-month top, with nowhere to go but down.  Instead volatility fell and we are now yet again at all-time highs, including the once-bubbled Nasdaq.  And just as quick, the same folks have gone back to update and rely on the charts to pick up new “signs” of what this all might mean.  It means nothing.  More importantly, it is just random where it will ever matter to you.
A legendary portfolio manager, who self-terminated his portfolio management career after a nice run, said of those who try timing the market:
Far more money has been lost by investors preparing for corrections, or trying to anticipate corrections, than has been lost in corrections themselves.
Peter Lynch’s wisdom in this regard works for individuals who consider the mutually-exclusive success of momentum or mean-reversion strategies, and falsely believe that either is the best way to gain an upper-hand over the market.

Category: Investing, Mathematics, Research

Dilbert: Beating the Average

Source: Dilbert

Category: Bad Math, Cognitive Foibles, Humor, Investing


As one of those folks who has spent a lot of time bashing economic and stock-market forecasters (see this, this, and this), I have no choice but to take issue with an argument made by former hedge-fund manager Jesse Felder, who asserts “that everything is a forecast.” To quote Felder: Can we please stop bashing forecasters already? There is a…Read More

Category: Asset Allocation, Bad Math, Investing, Mathematics

The Amazing Disconnect Between Hedge Fund Performance & AUM

Earlier this week, Greg Zuckerman of the Wall Street Journal pointed out one of the great mysteries of today’s investment landscape: Despite underperforming by a substantial margin, hedge funds keep attracting more investors and assets under management. It is almost as if (to borrow the headline on Zuckerman’s article), “Hedge Funds Keep Winning Despite Losing.”…Read More

Category: Asset Allocation, Bad Math, Hedge Funds, Investing, Psychology

The Unreasonable Effectiveness of Math

Raw Story:

Broadcaster and physicist Brian Cox recently explained why the discovery of the Higgs particle was so amazing.

“We sort of do know what the fuck is going on at some level with subatomic particles,” he said on the Joe Rogan Experience podcast. “If you look to the LHC — the Large Hadron Collider — which is the place where we generate the highest energy, so it is the biggest microscope in the world in that sense, we have an extremely good understanding of the laws of physics at that level, up to and including the discovery of the Higgs particle.”


Brian Cox excitedly explains the unreasonable effectiveness of math


Category: Mathematics, Science, Video

The Odds of Dying in America

Source: The Economist     Previously: You Are Worrying About the Wrong Things (October 22nd, 2014) You Are More Likely to Be Killed By Boring, Mundane Things than Terrorism (May 20th, 2014)

Category: Bad Math, Data Analysis, Digital Media, Mathematics

Don’t Play Probalities Like the Seahawks in Investing

Last week, I was in Seattle for an event sponsored by the CFA Institute. The trip was booked long before any of us knew the Seahawks were going to defend their championship title in Super Bowl XLIX. Following the Seahawks’ amazing comeback in the NFC Championship versus the Green Bay Packers on Jan. 18, the city…Read More

Category: Bad Math, Investing, Really, really bad calls, Sports

Job Recovery All Oil & Fracking ? Hardly

  “All of the job growth from 2007 to today can easily be attributed to the shale oil fracking situation and the oil Renaissance. If you take Texas and North Dakota out of the data series for job employment, what you see is that we haven’t added any jobs in the United States other than…Read More

Category: Bad Math, Data Analysis, Employment, Energy, Really, really bad calls

As goes early-January, so goes Nothing

Salil Mehta is a popular statistician and risk strategist, who has developed a unique method to teach quantitative techniques. He blogs at Statistical Ideas. ~~~ We’ve started the year with a sizable downward market pattern, which is making market participants think in ill-advised ways.  Eight of the first eleven trading days (S&P 500) were negative.  If…Read More

Category: Bad Math, Data Analysis, Markets

NFP, Inflation & Conspiracy Theories

    Unbelievable jobs numbers..these Chicago guys will do anything..can’t debate so change numbers — Jack Welch (@jack_welch) October 5, 2012     Today’s column is about stupidity. Perhaps that’s overstating it; to be more precise, it is about the conspiracy-theorist combination of bias, innumeracy and laziness, with a pinch of arrogance thrown in for…Read More

Category: Bad Math, Cognitive Foibles, Data Analysis, Economy, Really, really bad calls