From the December 01, 2011 issue of Futures Magazine • Subscribe!

# The law of large numbers and avoiding the Siren’s song

Ulysses made a pact with his men as they sailed near the Sirens’ rocky island. The Sirens lured sailors to their death through their wondrous singing that caused men to lose all rational thought. Ulysses wanted to hear their song, so he had his men bind him to the mast and he filled their ears with wax.

Ulysses knew well that a siren song is truly sweet and hard to resist but, if heeded, eventually will lead to one’s demise. Traders should beware the similar sweet song of unproven profits. It is all too easy even for the astute trader to succumb to empty promises of success while forgetting the shoals brought on by the law of large numbers.

A modern Odyssey

Meet modern day S&P trader Neil Ulysses. Like his namesake, Neil is on an arduous journey to find the perfect guru of trading. Things went swimmingly until he heard the siren song of Martin Gale.

Neil, like all other non-statisticians — that is, almost everyone in the world — thinks about the law of averages. He assumes if a coin is flipped 20 times, it will come out heads roughly 10 of them because of the law of averages. After all, he reasons, the probability on any one flip is 50/50, so about half of the flips should be heads.

Not so. It is perfectly possible, although not terribly likely, to get 15 heads and five tails. Or, even 20 heads. As any Bayesian would tell Neil gleefully, if he flipped 19 heads, the chances of that coin coming down heads again is a steadfast 50/50. There is no law of averages; only a law of large numbers.

Nevertheless, the typical person gladly will bet you the chance of 20 heads in a row is small and will stake large sums on it the more heads are flipped. This results from a psychological phenomenon known as a Taleb distribution and leads to the downfall of many a trader.

A Taleb distribution — a term coined by economists Wolf and Kay and named after Nassim Taleb — describes a return’s profile that appears at times deceptively low-risk with steady returns, but periodically experiences catastrophic drawdowns. It does not describe a statistical probability distribution, and does not have an associated mathematical formula. The term is meant to refer to an investment return’s profile in which there is a high probability of a small gain, and a small probability of an extremely large loss that more than outweighs the gains. In these situations, the expected value is (much) less than zero, but this is camouflaged by the appearance of low-risk and steady returns; think option writing.

The ultimate downside is a combination of kurtosis risk and skewness risk: Overall returns are dominated by extreme events (kurtosis), which are to the downside (skew). Ah, but how sweet the song sounds while it is working.

Gambling man

Although apparently new to economists, gamblers know this as the Martingale, a class of betting strategies popular in 18th century France. The simplest was designed for a game in which Neil wins his stake if our coin comes up heads and loses it if the coin comes up tails. The strategy has Neil double his bet after every loss, so that the first win would recover all previous losses plus win a profit equal to the original stake.

Because a gambler with infinite wealth will, almost surely, eventually flip heads, the Martingale betting strategy is a sure thing. Of course, Neil is not possessed of infinite wealth, and the exponential growth of the bets would eventually bankrupt those who chose to use the Martingale, and will do so to our friend Neil.

One round of the idealized Martingale without time or credit constraints can be formulated mathematically. Let the coin tosses be represented by a sequence X0, X1, … of independent random variables, each of which is equal to H with probability p, and T with probability q = 1 – p.

We’ll let N be the time of appearance of the first H. So, in other words, X0, X1, …, XN–1 = T, and XN = H. If the coin never shows H, we write N = ∞. N is itself a random variable because it depends on the random outcomes of the coin tosses.

In the first N – 1 coin tosses, the player following the Martingale strategy loses 1, 2, …, 2N–1 units, accumulating a total loss of 2N − 1. On the Nth toss, there is a win of 2N units, resulting in a net gain of one unit over the first N tosses. For example, suppose the first four coin tosses are tails, tails, tails, heads, making N = 3. The player loses one, two and four units on the first three tosses, for a total loss of seven units, then wins eight units on the fourth toss, for a net gain of one unit. As long as the coin eventually shows heads, the player realizes a gain. But, always of only one unit.

What is the probability that N = ∞, i.e., that the coin never shows heads? Clearly it can be no greater than the probability that the first k tosses are all T. This probability is qk. Unless q = 1, the only non-negative number less than or equal to qk for all values of k is zero. It follows that N is finite with probability one. Therefore, with probability of one, the coin eventually will show heads and the player will realize a net gain of one unit. How does this work if the probability of a winning trade is just slightly less than of a losing trade? What if it is even a "winning" strategy?

The law of large numbers is a theorem that describes the result of performing the same experiment a large number of times. According to the law, the average of the results obtained from a large number of trials should be close to the expected value, and will tend to become closer as more trials are performed. However, the more trials, the more likely the "unlikely" events also will occur. If you perform an infinite number of tests, then all possible events will occur, even improbable consecutive runs of losers.

Perfect system

This brings us back to Neil, who has started to follow Chicago trading guru Martin Gale. Martin Gale boasts of phenomenal results. Even though he has never heard of Taleb or the law of large numbers, Martin has it all figured out. For a price, he will tell you exactly what that is. Neil wants to know, so he pays Martin \$5,000 for a four-day training session and flies to Chicago.

Martin explains this wondrous system using the E-mini S&P index where each tick is equal to \$12.50. If the hourly, daily, weekly and monthly bars are all above their moving average and the five-minute bar trades down from a high and touches the moving average, go long one contract. It is going to bounce. Take a 10-tick profit, so Martin will be up \$125. If the market goes against Martin by 20 ticks, he buys three more contracts and will exit on a five-tick increase. Observe that Martin loses \$187.50 on his original purchase and gains \$187.50 so he comes out flat; less four commissions, of course.

If the market goes down another 20 ticks, Martin adds 12 more contracts to his existing four and will exit on a five-tick bounce. That gives him 12 x 5 or 60 ticks profit for \$750. Of the original contracts, Martin loses 3 x 15= 45 and 1 x 35, for a total of 80 ticks, so he is down \$250 plus commissions on all 16 contracts. Assume Martin holds out for six ticks. He would gain 72 and lose 76. To post a profit, the market must bounce seven ticks. But, what if it doesn’t?

If it goes down another 20 ticks, Martin now adds 48 more contracts, giving him 64 in play. Again if the market bounces five ticks, he wins 240 on the new, but loses 12 x 15 for 180 plus 3 x 35 for 105 plus 1 x 55, or a total of 340 ticks. At a six tick goal, Martin loses 36 ticks before slippage and commissions. At seven ticks, he makes \$350. If he pays only a \$12.50 round turn commission and slippage, Martin must make an "eight-tick" bounce to net \$350.

And, Martin tells Neil it always does in the S&P, Russell 2000 or the Euro FX contract. It never gets beyond 64 contracts and always has made the eight-tick bounce. Martin claims the ‘method’ will work on virtually anything.

New voyage

Neil is ecstatic. He’s heard the siren’s song and is ready to jump overboard. Along on this voyage are four other traders. Unfortunately, their ears are not filled with wax.

Tuesday, Oct. 4, 2011 arrives. Everyone is trading merrily along in either the E-mini S&P, the Russell or the Euro, and it is roughly 2:40 p.m. Chicago time. The stock market is in a downtrend on the hourly, daily, weekly and monthly. The five-minute just touched the moving average on a rally. The conditions set up for a trade and Martin yells to his devoted new acolytes: "Go!" All short one E-mini SP or Russell 2000 contract except Neil, chained to his personal mast. Neil refuses to enter a trade after 3:30 pm Eastern because another former guru told him not to. The market rises 20 ticks. They short three more. It rises another 20, they short 12 more. There does not seem to be any reason for the rise. It adds 20 more ticks, they short 48 more. Another 15 seconds later, it goes up another 20, so they add…wait for it… 192 contracts. The market goes up another 20 in less than 30 seconds. All four students are shut down by the brokerage firm.

Guru Martin, however, has a bigger account. Lucky for him. He adds — sit down before you read this — 768 contracts, giving him a total of 1,024 contracts! The market continues to rise 11 more ticks and the broker closes him down. "Road to ruin" (below) shows how it ended at 2:46 p.m. Chicago time.

Martin Gale lost \$226,050 at \$12.50 per tick in a six-minute period, plus commissions and slippage. And, it only took six failures of the hypothesis for Gale to be blown away. Even if his system has a respectable 60% success rate, it has a 40% chance of losing on any one decision. The chances of losing on six consecutive decisions are 0.41%. Although it sounds small, that is one time in every 200 series of tests. Roughly one time every six months of trading Martin will get wiped out. And, it’s good the broker stopped him out. The market continued to soar an additional 150 ticks thereafter (see "It only gets worse," below). All of Martin’s legendary profits were gone in one trade in under 10 minutes, because the "impossible" thing happened.

Bottom line protection

Could this have been avoided? One way would have been to use a stop at an earlier level. Unfortunately, that would guarantee a loss and also eliminate those trades where the market does perform as expected on the next group of contracts added. It almost assuredly would reduce the overall profitability of the method, making things even worse. Because the method is flawed because the underlying assumption is flawed, any tinkering with the method still will include the baseless underlying assumption.

A different form of protection is required. The assumptions must be changed. Recall the basis was that the market would bounce because all of the momentum was in the direction of the trade. If only someone had the wisdom to ask: What happens if the market is changing direction?

As we know, the market does that periodically and market tops and bottoms are not taken into account by Martin. His method requires a circuit breaker device to detect a fundamental direction change in the market and either eliminate the trade entirely or prevent the horrific loss from accruing.

Research has shown when popular momentum indicators fall outside of 3.25 standard deviations, the markets are ripe for sudden whipsaw reversals. Had Martin incorporated such a filter, he might avoid the losers. For example, on that particular day, the %k stochastic (a common indicator available in essentially all charting software) would have stood under 15 when Martin executed the first trade.

Or, perhaps certain times of the day are better for his system’s performance than others. The trade initiated after 2:30 p.m. Chicago time, and the market rallied from 1085.50 to 1119.50 in the next 25 minutes, eventually closing the bar at 1118.00, having started at a low of 1075.00 in the 2:15 p.m. bar. Research of prior failures could help determine a suitable filter.

It isn’t enough merely to have a money management system if the underlying concept is flawed. Nor is it enough to have a great trading method without money management and a method of confirming each trade or protecting against the impossible event. Both are essential for the successful trader, as is a price-level definition of being wrong. Every system will be wrong many times. Accept that and work to improve it, but never risk all your capital — or even more than is prudent — on any one set-up.

Neil flew home that night, \$5,000 poorer but a whole lot wiser. The moral is that the law of large numbers is always there. Beware any trading method that sings so sweetly, but does not take into account impossible events. As the White Queen said in Wonderland, "Sometimes I’ve believed as many as six impossible things before breakfast."

NOTE: This is a true story. The names involved have been changed and the price levels and markets may not be precise, but the events happened and the money lost is real.

Arthur Field has a Ph.D. from Clemson and is a former fund manager for Fidelity International. He wrote "The Magic 8: The Only 8 Indicators You Need to Make Millions in the Markets," available at www.themagic8.com. Email him at themagiceight@hotmail.com.

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