Coin Tossing - Risk & Money Management

Is the Trend Really Your Friend? A Professional Trader Flips a Coin 10,000 Times to Find Out
By David Silverman SFO Mag- May, Vol 4 No 5

What can coin flipping tell us about basic chart patterns that traders believe are their friend?

Often when I am instructing a novice in the esoterica of the active trader’s thought process, I will begin by presenting a coin-flipping scenario: You have a dollar and can double or lose it on the flip of a coin.

Alternatively, you can choose to win two cents or lose a penny with every flip. Which is a better bet? I have yet to encounter anyone who wants to argue that risking their entire stake on a single bet is preferable. Yet once many begin trading they completely ignore what they have told me, apparently believing that making big bets, irrespective of the risk involved, is the quickest way to riches. In fact, it is the quickest way to a margin call.

Some extremely complicated risk management models rely on this basic principle, namely that a large potential reward does not justify the assumption of excessive risk; that the law of averages guarantees that the more flips (trades) you are able to make, the better the chances for a successful outcome. And we learn this from a simple coin toss.

With the theme of this month’s issue revolving around trends, I thought it would be interesting to consider what we can learn from coin flipping about the basic chart pattern that traders believe is their friend.

It occurred to me that, while I have studied probability in school, read many academic papers on the subject and deal with the underlying concepts every time I make a trade, I have never actually conducted my own research. So late one night, I decided to write a stream-of-consciousness piece focusing on important principles of probability relating to trends, while actually flipping a coin 10,000 times. It seemed like a clever idea at the moment I conceived it, but at one point during my college days, I also thought it was a good idea to try to catch a major league baseball thrown off of the top of a tall building; what seems clever after 1 a.m. is not always easily explained to the attending physician in the emergency room. So I commenced my experiment, but did not appreciate that the human thumb is perfectly constructed to hitch a ride, indicate approval or disapproval, wipe away a stray tear from the corner of an eye, but not to flip a coin 10,000 times.

After the first 100 tosses, I was ready to call the pharmacist and beg for some needed Vicodin, but then my young son, Joshua, walked into the room. Luckily, his only goal in life is to please me. I am not proud of myself, but thanks to Joshua, I was able to complete this article. Let me tell you, it’s not easy to watch your children suffer, but there is a silver lining to every cloud, and I am happy to report that my son won’t be sucking his thumb anymore.

Heads: 5,209; Tails: 4,721; Buy Heads, Sell Tails?
First, some statistics (Table 1- see attachment below). Heads beat tails by 418 flips. Throughout the experiment there were numerous series of at least five consecutive flips in favor of heads or tails. There were five series of 10 consecutive flips in favor of one or the other; four of those series were in favor of heads. The longest consecutive series was 12, near the end of the contest, in favor of tails. There were only four occasions when the tally stood even: on the first, third, ninth and seventeenth flips. After flip #17, heads took the lead with a run of seven consecutive flips, a lead it never relinquished.

Even if one doesn’t know a standard deviation from a regression to the mean, results such as these might be surprising. The casual observer might ask, if each flip of the coin, by definition, has an equal chance of coming up heads or tails, how can heads hold such a commanding lead after 10,000 flips? And why did heads spend so much time in the lead? These are good questions, but what our casual observer really wants to understand is how a random series of events can appear to indicate a trend.

John Allen Paulos, in his wonderful book, A Mathematician Plays the Stock Market, explains this phenomenon in the following way: “One odd and little-known fact about coin flips concerns the proportion of time that the number of heads [in our case] exceeds the number of tails. It’s seldom close to 50 percent!... it’s considerably more probable [when there are a large number of coin flips] that heads has been ahead more than, say, 96 percent of the time than that either has been ahead between 48 and 52 percent of the time.” The shocking principle here is that there is no “probabilistic rubber band,” as Paulos calls it, snapping the tally back into its rightful 50/50 proportion. Yet at the same time, there remains an equal chance that each flip will be either heads or tails.

Therefore, it is only natural that “patterns” favoring the winning side develop. When these patterns are graphed, they look remarkably like the charts traders use to make their trading decisions, with head and shoulders formations, double tops and bottoms and trend lines that practically beg you to place an order to buy or sell. And yet, because these chart patterns were generated by completely random events, no rational person would assume that the past performance as expressed on the chart could predict the outcome of the next toss. Or would they?

Consider the following: If we took 10,000 traders rather than coin flips, we can theorize that some percentage of them – let’s say 50 percent – might be profitable from year to year simply by chance. After year one, there would be 5,000 winners, year two, 2,500, year three, 1,250, and so on, until by year 10 there would remain three traders that were profitable for ten consecutive years, solely by chance. Nonetheless, it is likely that these individuals would be very well regarded in the investment community; they probably would be inundated with offers to manage other people’s money.

Another way to think about this scenario is suggested by Nassim Nicholas Taleb, the author of Fooled By Randomness: The Hidden Role of Chance in the Markets and in Life, a brilliant book that every trader should read before even thinking about making his next trade. The author poses the following scenario: An eccentric billionaire offers $10 million to anyone who wins at a game of Russian roulette. One has to agree that the odds, strictly speaking, are favorable. A participant has a 5/6 chance of walking away rich, and only a 1/6 chance of, let’s just say, suffering the consequences of an extremely bad risk/reward ratio. The successful risk-takers in this “market” likely would be revered by the public, whose view tends to be that the attainment of riches is worthwhile irrespective of how the riches are attained. Were one to start playing this game at age 25 and commit to playing it every year, it is more likely than not that the individual would win before losing, yet it also is probable that he or she would not die of old age.

Now imagine that we have hundreds of thousands of 25-year-old players in the game. A few of these individuals will survive to old age, and in the process become very wealthy simply by chance (remember, even tails, which trailed badly in our experiment, had a winning streak of 12 tosses). Perhaps you think the analogy flippant: the market is the market, not Russian roulette (although it would explain the origin of the ubiquitous trader’s phrase, “pulling the trigger”). Maybe so. But how then are we to explain the relatively small number of successful traders out of all of the hundreds of thousands of individuals who test their skills in the market? Perhaps some traders know how to pull the trigger better than others.

It’s a Beautiful Day in the Market. Would You Like to Take a Random Walk?
One cannot discuss this subject without acknowledging that the market may be a random walk, that is, completely unpredictable. And if the market is unpredictable, what does that mean for those of us who treat trend lines as if they were drawn by the finger of God? Burton Malkiel, author of one of the most famous and important books about the market,
A Random Walk Down Wall Street, would say we are delusional and that past market information reveals no more about the future than the “wallpaper behind the mirror” can predict the pattern of wallpaper above the mirror. He scoffs at the notion that prices moving in a particular direction are like a “fullback, who once having gained some momentum, is expected to carry on for a long gain.” He states, unequivocally, that the patterns suggesting trends are nothing more than a “statistical illusion.”

If reading Malkiel doesn’t make you want to cancel your subscription to your charting service, consider the words of Warren Buffett. While Buffett, the great value investor, does not buy into the random walk theory – after all, there can be no such thing as a “value” investment in a truly efficient market – he has no patience for technical analysis and nothing but contempt for those who use it to make investment decisions.

In a 1984 speech delivered at Columbia University in honor of the 50th anniversary of the publication of The Intelligent Investor (one of whose co-authors, Benjamin Graham, was Buffett’s mentor), he said, “I find it extraordinary that so many studies are made of price and volume behavior, the stuff of chartists. Can you imagine buying… simply because the price… had been marked up substantially last week and the week before?... It isn’t necessarily because such studies have any utility; it’s simply that the data are there and the academicians have worked hard to learn the mathematical skills needed to manipulate them. Once these skills are acquired, it seems sinful not to use them, even if the usage has no utility. As a friend said, to a man with a hammer, ‘everything looks like a nail.’”

If anyone has the right to scoff at the way I make my living it is Buffett. His strategy is to “buy a dollar for 40 cents,” and no one has ever done it better than he does. Yet to be dismissed so summarily hurts a bit. My only consolation is that with his haughty attitude Buffett probably got beat up a lot when he was a youth, whereas I had a very nice childhood, even if I did grow up to be a hopeless trend follower.

A Comforting Message to All Hopeless Trend Followers
Probability can explain a lot, but it can’t explain everything. If an operation with a one-percent mortality rate has been performed 99 times successfully, and I am about to be rolled into the operating room for the 100th operation, it would hardly concern me that I had a 100-percent chance of dying, although statistically speaking this could be said. I would rely on my belief in the skill of the doctors and nurses to get me through the operation notwithstanding the odds against survival. In other words, my chances of survival could not be said to be random.

Similarly with chart patterns, and specifically trends, I expect that there is more to rely on than simply an arbitrary line connecting some points on a graph. Something significant is happening when a trend is forming. For whatever reason, discernible or not, people are deciding to play follow the leader to a destination unknown. That it often is impossible to figure out who the leader is or why they are leading the way is largely immaterial; the fact is that something palpable is occurring, and it is not in your best interest to get in the way of the crowd. Usually, if you are paying close enough attention, you can join the group as it makes its way toward its objective. You can befriend the trend. Sometimes the trend will betray you, but this happens in life. If we are smart, we learn from our mistakes, and if not, we suffer the same disappointments again and again.

Among other things, George Soros is famous for admonishing his traders that they – and he – are a bunch of “mistake-prone idiots, who know nothing, but are endowed with the rare privilege of knowing it.” This is one of those statements that is so simple, it must be profound, and most traders could use an occasional profundity amidst the nonsense that normally clutters their brains.

But how can we know if a market move is real? Nassim Taleb offers some excellent perspective on this. He asks you to consider you are in a bicycle race across Siberia. If a month later you have beaten your opponent by one second, it isn’t particularly meaningful to say you are a faster rider. Perhaps #2 went over a pothole at some crucial point, or some other random event influenced the outcome. But if you beat him by three days, it would remove any doubt that random causes influenced the outcome of the race. Applying this approach to the markets, Taleb suggests that while it can be very complex to perceive what he calls “causality,” it can be found if you know how and where to look for it.

In his case, he knows “if something is real” by the magnitude of the move. As he watches the markets, he reacts only when the percentage move is large enough to ensure that the move could not have taken place as a result of random events. How does he measure the “realness” of a move? He does not divulge his proprietary models, except to say something extraordinarily important: “The interpretation is not linear; a two-percent move is not twice as significant an event as one percent; it is rather like four times. A move of 1.3 points in the Dow has less than one-millionth of the significance of the serious seven-percent drop of October 1997.

People might ask, “Why do I want everyone to learn statistics?” The answer is that we cannot instinctively understand the non-linear aspect of probability.

For me, as well, the significance of the move is tied to its scale, and I learned this as a young trader in one of the currency pits of the Chicago Mercantile Exchange. I stood next to an order-filling group that each day entrusted one of its members with its stack of customer sell orders and another with the stack of buys (this was in the not-too-distant past when orders were transcribed to paper). On days when the market was moving up, the stack of buy orders would grow so large that clerks would have to hold some of it in their pockets lest it spill onto the trading floor, and the order filler with the deck of sales stood mostly idle, having few orders to execute in a rising market. The mirror opposite occurred on days when the market was plummeting. If the market is truly random, this should not have been the case; one would expect that at least on some of those days the order filler with the smaller deck should have been doing some business. Yet in 17 years on the trading floor, this is the reality I knew – a market that did not seem random, and I used that knowledge to my advantage. In today’s electronic world, the cues are different but no less palpable. Like Taleb, when a move of significance is underway, I usually can spot it.

I have been thinking and talking about coin flips since the beginning of my career 24 years ago. To this day, I struggle with the subject of randomness, which is far too complex to deal with adequately in an article such as this. This is not, after all, some obscure area of market analysis. The vast number of studies that have been performed is only exceeded by the number of traders who utterly ignore the findings. The random walk debate preceded me and likely will continue long after I have pulled the trigger on my final trade.

So what do I really believe? My intellect tells me that the academics are right, that the numbers don’t lie. But at the gut level, where for better or worse most of my trading decisions are made, I continue to think that there is a chance that not all is governed by chance. So I draw my trend lines and make my trades. While I am not as successful as Warren Buffett, there’s food on the table every night, a roof overhead, and I can afford to pay for the physical therapy that Joshua so desperately needs to regain the use of his thumbs.

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