# Using the Month of September to Understand Probability and Expectation

September is the month when the U.S. stock market’s three most popular indexes usually perform the poorest. So say the headlines every September.

I first wrote this in September 2013 after many commentators had published information about the seasonality of the month of September. Seasonality is the historical tendency for certain calendar periods to gain or lose value. However, when commentators speak of such probabilities, they rarely provide a clear probability and almost never the full mathematical expectation.  Without the mathematical expectation, probability alone is of little value or no value. I’ll explain why.

For those of us focused on actual directional price trends it may seem a little silly to discuss the historical probability of gain or loss for a single month. However, even though I wouldn’t make decisions based on it, we can use the seasonal theme to explain the critical importance of both probability and mathematical expectation.

“From 1928-2012 the S&P 500 was up 39 months and down 46 months in September. It is down 55% of the time in September…”

“Dow Jones Industrial Average 1886-2004 (116 years) 49 years the Dow was up in September, in 67 years the Dow was down in September. It’s down 58% of the time in September…”

Those are probability statements. But they say nothing about how much it was up or down.

First, let’s define probability.

Probability is likelihood. It is a measure or estimation of how likely it is that something will happen or that a statement is true. Probabilities are given a range of value between 0% chance (it will not happen) and 100% chance (it will happen). There are few things so certain as 0% and 100%, so most probabilities fall in between. The higher the degree of probability, the more likely the event is to happen, or, in a longer series of samples, the greater the number of times such event is expected to happen.

But that says nothing about how to calculate probability and apply it. One thing to realize about probability is that it is the math for dealing with uncertainty. When we don’t know an outcome, it is uncertain. It is probabilistic, not a sure thing. Probability provides us our best estimation of the outcome.

As I see it, there are two ways to calculate probability: subjectively and objectively.

Subjective Probability: assigns a likelihood based on opinions and confidence (degree of belief) in those opinions. It may include “expert” knowledge as well as experimental data. For example, the majority of the research and news is based on “expert opinion”. They may state their belief and then assign a probability: “I believe the stock market has a X% chance of going down.” They may go on to add a good sounding story to support their hypothesis. You may see how that is subjective.

Objective Probability: assigns a likelihood based on numbers. Objective probability is data-driven. The popular method is frequentist probability: the probability of a random event means the relative frequency of occurrence of an experiment’s outcome when the experiment is repeated. This method believes probability is the relative frequency of outcomes over the long run. We can think of it as the historical tendency of the outcome. For example, if we flip a fair coin, its probability of landing on heads is 50% and tails is 50%. If we flip it 10 times, it could land on heads 7 and tails 3. That outcome implies 70%/30%. To prove the coin is “fair” (balanced on both sides), we would need to flip it more times to get a large enough sample size to realize the full probability. If we flip it 30 times or more it is likely to get closer and closer to 50%/50%. The more frequency, the closer it gets to its probability. You may see see why I say this is more objective: it’s based on actual historical data.

If you are a math person and logical thinker, you may get this. I have a hunch many people don’t like math, so they’d rather hear a good story. Rather than checking the stats on a game, they’d rather hear some guru’s opinion about who will win.

Which has more predictive power? An expert opinion or the fact that historically the month of September has been down more often than it’s up? Predictive ability needs to be quantified by math to determine if it exists and opinions are often far too subjective to do that. We can do the math based on historical data and determine if it is probable, or not.

As I said in September is statistically the worst month for the stock market the data shows it is indeed statistically significant and does indeed have predictive ability, but not necessarily enough to act on it. Instead, I suggest it be used to set expectations of what may happen: the month of September has historically been the worst performance month for the stock indexes. So, we shouldn’t be surprised if it ends in the red. It’s that simple.

Theory-driven researchers want a cause and effect story to go with their beliefs. If they can’t figure out a good reason behind the phenomenon, they may reject it even though the data is what it is. One person commented to me that he didn’t believe the September data has predictive value, even though it does, and he provided nothing to disprove it. Probabilities do need to make sense. Correlations can occur randomly, so logical reasoning behind the numbers may be useful. For example, one theory for a losing September is it is the fiscal year end of many mutual funds and fund managers typically sell losing positions before year end to realize losses to offset gains.

I previously stated a few different probabilities about September: what percentage of time the month is down. In September is statistically the worst month for the stock market I didn’t mention the percent of time the month is negative, only that on average it’s down X% since Y. It occurred to me that most people don’t seem to understand probability and more importantly, the more complete equation of expectation.

Expectation

There are many different ways to define expectation. We may initially think of it as “what we expect to happen”. In many ways, it’s best not to have expectations about the future. Our expectations may not play out as we’d hoped. If we base our investment decisions on opinion and expectations don’t pan out, we may stick with our opinion anyway and eventually lose money. The expectation I’m talking about is the kind that I apply: mathematical expectation.

So far, we have determined probability of September based on how many months it’s down or up. However, probability alone isn’t enough information to make a logical decision. First of all, going back to 1950 using the S&P 500 stock index, the month of September is down about 53% of the time and ends the month positive about 47% of the time. That alone isn’t a huge difference, but what makes it more meaningful is the expectation. When it’s down 53% of the time, it’s down -3.8% and when it’s up 47% of the time it’s up an average of 3.3%. That results in an expected value of -0.50% for the month of September. If we go back further to 1928, which includes the Great Depression, it’s about  -1.12%.

The bottom line is the math says “based on historical data, September has been the worst month for the stock market”. We could then say “it can be expected to be”. But as I said before, it may not be! And, another point I have made is the use of multiple time frames for looking at the data, which is a reminder that by intention: probability is not exact. It can’t be, it’s not supposed to be, and doesn’t need to be! Probability and expectation are the maths of uncertainty. We don’t know in advance many outcomes in life, but we can estimate them mathematically and that provides a sound logic and a mathematical basis for believing what we do.

We’ve made a whole lot of the month of September, but I think it made for a good opportunity to explain probability and expectation that are the essence of portfolio management. It doesn’t matter so much how often we are right or wrong, but instead the probability and the magnitude. Asymmetric returns are created by more profit, less loss. Mathematical expectation provides us a mathematical basis for believing a method works, or not. Not knowing the future; it’s the best we have.

Rather than seasonal tendencies, I prefer to focus on the actual direction of global price trends and directly manage the risk in individual my positions.