After an amazing blog post by Howard Lindzon last night mentioning probability investing I decided to write a few things about it but the topic is vast and I will have to come back. All quant traders should be aware of these issues with probability.
Discretionary trading or investing based on probabilities is fraught with many difficulties. Most of the difficulties arise from the very nature of probability.
Let us start with the wrong claim that one often sees in financial blogs:
Claim: If the probability of success is high and I can make more than I can lose, then the opportunity I am considering has high expectation.
The claim is wrong. Some of those who make it did not pay enough attention to their probability and statistics teacher. Let us assume that the win ratio is w > 0.50. Actually, let us assume that it is 0.7, or that the trader is right 70% of the time. Also, let us assume that the profit potential is W and the loss potential is L, with W >> L. Actually, let us assume that W = 3 x L. Usually, this is how the claim is expressed mathematically:
Expectation = w x W – (1-w) x L = 0.7 x (3L) – (1-0.7)xL = 1.8L (1)
Therefore, the claim is that the expectation from the next investment or trade is 1.8 times the potential loss. This is not only wrong, it is flat wrong.
For those that paid attention to their professors, the expectation is also the mean of a probability distribution, usually denoted as E[X] or μ. In this case, it is the mean of the population of the values a random variable X takes with some probability p. Now this is the most important part:
Averages converge to the expectation only in the presence of sufficient samples
What does the above statement mean? Let us first look at the theoretical expectation that is given as the sum of all possible values the random variable X can take times their associated probability:
E[X] = ∑Xipi (2)
Do equations (1) and (2) have anything in common? No, but equation (1) will converge to equation (2) in the presence of sufficient samples. In statistics, they say that the sample mean targets the population mean. But this is true only for sufficient samples.
So what is a sufficient sample? In finance, nobody knows the answer but it is a large sample, possibly consisting of hundreds or even thousands of trades.
What this boils down to, is that equation (1) is not a measure of expectation of the next trade, not even close. It only says that in the longer-term, an unspecified time interval, an investor or trader that is right w% of the time and with average profit W and average loss L, where W = 3L, will realize an average trade equal to 1.8L. The total profit will be equal to the sample size times the expectation (less any friction).
Here is the danger discretionary but also systematic traders face: consecutive losers and associated clusters.
Let us go back to the original claim. The win rate of the investor is 70%. This is like a biased coin. However, the next time the coin is tossed, tails can show up instead of heads. Thus, the trade can be a loss. This can happen again for the next toss. The probability of one loss is 30%. The probability of two consecutive losers is 30% x 30% =9%. This is high probability in finance but also in statistics. According to the rare event rule, if the probability of an event is small, it will probably not occur:
Rare event rule: if p(event) is small, then the event will probably not occur
What is low probability in finance? Remember that the 1987 crash was a 6-sigma event, a highly unlikely event but it did happen. However, if we structure our lives around extreme events, or left fat tails, we may accomplish nothing or little. We have to set a “reasonable” probability of rare events. Some think that p = 0.01 is a reasonable value in finance. But note that in the case of the original claim, three consecutive losers have probability 0.027, or 2.7%. This means that a drawdown caused by three losers in a raw could occur and it is not a rare event.
There are more problems with the naive expectation formula
One may get a streak of consecutive losers, then a winner or two, and then another streak of consecutive losers. Although the method of investing or trading may be sound, the longer-term convergence to the expectation, or mean of the population, may take longer than expected and one may suffer significant losses or even face ruin. In a nutshell, trading or investing with probabilities can be tricky and even dangerous. There are ways of dealing with the uncertainties that are covered in my new book, Fooled by Technical Analysis: the perils of charting, backtesting and data-mining. But one thing is for sure as Howard wrote in his post:
Unless you are an insider, if you want to trade or invest you will have to deal with probabilities and you must understand how they work and the dangers they hide. I hope this blog post will provide an initial direction. The first step is to understand the correct notion of expectation and not get fooled by self-proclaimed experts and wannabe quants that lack understanding of it. When you average a small number of trades you are not calculating the expectation. This is only a random sample mean. You cannot make inferences about the population mean unless you have a large representative sample or you are willing to accept a large error. On top of this, non-stationarity can affect the results and render the investing method ineffective. For example, the U.S equity markets switched from momentum-driven to mean reversion-driven in the 1990s and this destroyed a large class of strategies used by traders and investors while causing massive wealth transfer. Always be prepared for a change in market conditions that will invalidate representative samples of the past. This is the name of the game and it can be played profitably but only if some misconceptions are first cleared and one stays away from hype and naive claims.
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