Martingales and Markets

“You cannot beat a roulette table unless you steal money from it.” – Albert Einstein

Roulette is not a “fair” game, but it’s about as close as you can get in a casino. Assuming a standard wheel with 18 black slots, 18 red slots, and two green slots (0 and 00), your odds of hitting on black or red are 47.37% (18/38). The “house edge” is 5.26% as they will win 52.63% of the time. You’re expectation, if you stay at the table long enough, would be to lose money. I don’t like losing money so I tend avoid casinos and games such as roulette like the plague.

Now let’s pretend the wheel had no green slots. In this case, roulette would become a “fair” game, as your odds of winning move up to 50% (18 black slots, 18 red slots). Without the green slots, roulette would be no different than a coin flip, and your expectation betting on black/red over time would be to breakeven. I don’t like to breakeven, but if I had infinite wealth and infinite time I might consider playing such a game, just to prove a point.

And what point is that?

That I could employ a strategy to never leave the casino a loser. How is that possible? It’s just math. By using a Martingale betting strategy, whereby you double your bet until you win, I would eventually walk away a winner.

Let’s run through a few scenarios to make this point clearer. For simplicity, we’ll assume that I always start with a bet of $5 and I always bet on red…

Scenario 1: I place a $5 bet on red. It hits. I win $5.

Scenario 2: I place a $5 bet on red. It lands on black. I lose $5 and bet $10 on red during the next spin. It lands on red and I win $10 for a net gain of $5.

Scenario 3: $5 bet, lands on black, loss. $10 bet, lands on black, loss. $20 bet, lands on black, loss. $40 bet, lands on black, loss. $80 bet, lands on black, loss. $160 bet, lands on black, loss. $320 bet, lands on black, loss. $640 bet, lands on black, loss. $1,080 bet, lands on black, loss. $2,160 bet, lands on red, win.

In the last scenario, I am risking $2,160 to recoup my $2,155 in losses and win $5, not necessarily the greatest risk/reward but a positive outcome nonetheless. I walk away a winner. Theoretically, I could have lost 50 times in a row or more (given an infinite amount of time, this will happen) but eventually, the roulette wheel will land on red. And if you have infinite time/money, you’ll at a minimum walk away with $5 in your pocket when it does.

Why doesn’t everyone use the Martingale strategy? Many have (dates back to 18th-century France) and over short periods it will certainly increase your odds leaving the casino a winner. But if you play long enough, you will most certainly lose because: a) no one has infinite wealth or time, b) the casino places limits on how much you can bet, and c) roulette is not a 50/50 “fair” game.

Because no one has infinite wealth, you cannot simply double your bet forever, and all it takes is one catastrophic loss to wipe you out. And because the casino is not in the business of creating “fair” games, they place limits on bets so that someone with deep pockets can’t continue to double their bet until they win.

Martingales and Markets

“Martingale” is more than a betting strategy. It is also mathematical term used to describe a stochastic process (a collection of random variables indexed by time) where past occurrences are not predictive of future occurrences. Without getting into complex equations (which tend to scare people off), this essentially means that the next variable in a sequence of variables is totally random, as the variables preceding it have no influence on its outcome. A gambler’s profit in a “fair” game like a coin toss (heads you win, tails you lose) is a martingale because each successive observation is independent of the previous observations. Whether you won or lost in the prior toss of the coin has no effect on the outcome of the next toss.

A random walk (a succession of random steps) is said to be a martingale and is observed in chemistry (path of a molecule as it travels in a liquid), nature (path of an animal foraging for food), and as many have argued (most notably, Burton Malkiel), stocks prices.

If stock prices follow a random walk then the stock market is at the very least weak-form efficient and looking past prices will tell you nothing about the future. Over the years, many studies have poked holes in the random walk and EMH hypotheses, and our own research adds to this literature. The question of whether the stock market is entirely random/efficient is at the heart of the active/passive debate that rages on today.

But what unfortunately gets lost in this heated debate is a concept more important to investors than their decision to pursue an active/passive strategy. And that is: can they stick with it? For the easy part of buy and hold is the buying; it’s the holding that tends to be problematic for most.

Regardless of whether the stock market is a martingale or a non-random process, one truth remains: if you can’t stick with a plan/strategy for a time period long enough to put the odds of success in your favor, you are not investing but gambling. The odds of the S&P 500 being up on any given day is only 53% (see chart below). That means in a typical year with 252 trading days you will see red on your statement in 118 of those days. Some of those days will be big declines and you will invariably be tempted to sell. You can have an entirely passive approach and this will still be the case. And if you panic and sell every time your index fund drops 5-10%, are you really an investor, passive or otherwise?

No, you are a gambler. You are reducing your odds of success by shortening the game to a time frame where luck becomes paramount. On a 1 minute time-frame I suspect the odds a positive/negative S&P 500 return are close to 50/50. If you like those odds then by all means become a day trader, but over time and after commissions/slippage you’ll learn that only the house wins at that game.

We can extend this argument to even the best active strategies, where the odds of success over short time periods are effectively random and only by extending the time frame do you increase the odds sufficiently to call yourself an investor.

I suspect there are many market participants who consider themselves to be investors but whose actions are more akin to gambling at the roulette wheel. And if we studied their return streams, they would closely resemble that of a martingale. For these people, I would suggest forgetting the markets altogether and taking a trip to Las Vegas instead. If nothing else, the entertainment value will be much greater there.

For the rest of us, the investor path is the only proven road to success, and to increase your odds requires a) building a diversified portfolio and b) having the patience and time to stick with it through the ups and the downs. Whether you are an active or passive investor, this is no easy task, which is why the risk premium exists and the reward for that critical “holding” part is so great.


This writing is for informational purposes only and does not constitute an offer to sell, a solicitation to buy, or a recommendation regarding any securities transaction, or as an offer to provide advisory or other services by Pension Partners, LLC in any jurisdiction in which such offer, solicitation, purchase or sale would be unlawful under the securities laws of such jurisdiction. The information contained in this writing should not be construed as financial or investment advice on any subject matter. Pension Partners, LLC expressly disclaims all liability in respect to actions taken based on any or all of the information on this writing.



Charlie Bilello is the Director of Research at Pension Partners, LLC, an investment advisor that manages mutual funds and separate accounts.  He is the co-author of four award-winning research papers on market anomalies and investing. Mr. Bilello is responsible for strategy development, investment research and communicating the firm’s investment themes and portfolio positioning to clients. Prior to joining Pension Partners, he was the Managing Member of Momentum Global Advisors and previously held positions as a Credit, Equity and Hedge Fund Analyst at billion dollar alternative investment firms.

Mr. Bilello holds a J.D. and M.B.A. in Finance and Accounting from Fordham University and a B.A. in Economics from Binghamton University. He is a Chartered Market Technician (CMT) and a Member of the Market Technicians Association. Mr. Bilello also holds the Certified Public Accountant (CPA) certificate.

You can follow Charlie on twitter here.


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