*But as we cannot predict such external influences very well, the only reliable crystal ball is a probabilistic one. - Benoit Mandelbrot*

**The C-J Monte Carlo Simulation Model**

As noted in my previous articles, C-J is a Monte Carlo simulation model used to assess risk in the S&P 500. C-J uses a series of conditional statistical distributions based on S&P 500 data going back to 1950 to correct for problems often ignored in traditional stock market models. These problems include fat tails (leptokurtosis), serial correlation, and volatility clustering. If you have read Benoit Mandelbrot's book (with Richard Hudson), *The (Mis)behavior of Markets: A Fractal View of Financial Turbulence*, then you should note that there exists a fractal nature to the C-J model.

Given data on the market, C-J runs 2,000 simulations of the S&P 500 for future periods. I then use the results to assess the probability of various percentage changes in the monthly S&P 500 Index. My purpose is not to provide a single point estimate of where the S&P 500 will be at some future point in time; history has shown us that such predictions are usually very wrong. Nor do I try to guess what factor will drive the market going forward; invariably, those lists are incomplete. As Nassim Taleb has written in *Black Swan*, "Most models, of course, attempt to be precisely predictive, and not just descriptive in nature. I find this infuriating". To that end, C-J is intended to be descriptive in nature, with a particular emphasis on describing the likelihood of negative tail events.

**February 2017**

The S&P 500 Index finished the month of January at 2278.87. That is an increase of just under 1.8% from the December close at 2238.83. It also represents the third month in a row that the S&P 500 Index has increased and the first increase in the month of January since 2013. Furthermore, the index hit a record closing high of 2298.37 on January 25. Yet despite the good results in January, there exists considerable pessimism as the post-inauguration period appears to have brought with it heightened political uncertainty. I understand that uncertainty as I worked for three different presidential administrations during my fourteen years at the U.S. Treasury Department in Washington, D.C. and I have never seen anything quite like what has transpired over the last few months. With that said, I was curious as to what the C-J simulation results for February would like. Those results are shown below.

To summarize the results, it is clear that more of the distribution of possible outcomes has shifted to the tails, both positive and negative, with less of the distribution in the middle. There are a few ways to see this. First, suppose we think of changes in the range of -1% to +1% as being small changes of no particular significance. From the table above, the probability of the S&P 500 moving within that range for the month of February is only 15.9%. Historically, that range accounts for almost 20% of monthly movements in the S&P 500. Such a historical result is not surprising given that the average and median monthly percentage changes both fall within that range. But the 15.9% estimate is the lowest estimated probability for that range since C-J's May 2016 simulations. So the distribution of likely outcomes has shifted away from small changes.

Correspondingly, there has been a dramatic increase in the probability of a tail event. In my previous article, I noted that the estimated likelihood of an increase of 5% or more for January was 6.1%. That probability has now increased to 11.8%. That is the highest estimated probability of a 5% or more increase since October 2016 and is almost equal to the historical rate of occurrence of such a 1-month move which is just over 12 percent.

**Negative Tail Analysis**

As I have done in recent articles, I break out into greater detail the negative tail results from the table above. While by some definitions not necessarily a negative fat-tail event, my particular interest is in the likelihood of losses of 5% or more. (While C-J does not use the normal distribution, I include the -11.74% or worse category as it corresponds to three standard deviations). Broken out into more detail, the February results can be seen as:

Similar to the story I told for the overall results above, the simulations suggest a dramatic increase in the likelihood of a negative fat-tail event in February. The estimated probability of the index ending February down 5% to 9% now equals 7.8%, while the estimated probabilities of declines of 9% to 11.74% and 11.74% or more are now 2.25% and 0.4%, respectively. In most cases, those probabilities are higher than the historical rate of occurrence, as well as the rate implied by traditional finance theory using the normal distribution. In total, the likelihood of the S&P 500 ending the month of February down 5% or more stands at 10.45%, well in excess of the historical rate of 8.7% and the rate based on the normal distribution of 8.5%. That is almost twice as high as the estimated probability of a negative tail event in the January simulations and is the highest that probability has been since just before the presidential election.

**To readers: I try to publish the results from C-J once or twice a month. If you would like to read more of C-J's simulation results in the future, please click on the follow button at the top of this article next to my name.**

*Disclaimer: This article contains model-based projections that are forward-looking and, as with any quantitative model, are subject to uncertainties and modeling assumptions. The C-J model is intended as a tool to assess risk in the S&P 500, and not as a forecast of the future value of the S&P 500 or any other market. The results of C-J are for informational purposes only. Nothing in this article should be construed as specific investment advice.*

**Disclosure:** I/we have no positions in any stocks mentioned, and no plans to initiate any positions within the next 72 hours.

I wrote this article myself, and it expresses my own opinions. I am not receiving compensation for it (other than from Seeking Alpha). I have no business relationship with any company whose stock is mentioned in this article.

**Additional disclosure: **I have a long position in an S&P 500 Index fund in a retirement account.