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 by design 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 stress test 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 or to guess what factor will drive the market going forward. 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 fat-tail events.

**December 2016**

The S&P 500 Index finished the month of November at 2198.81. That is an increase of just over 3.4% in the index from the October close of 2126.15. In my article last month, I noted that the distribution of simulated outcomes had become decidedly more positive for November than it had been in the preceding months. You may recall that the probability of a tail event, either positive or negative, was 25% in September, and 27.4% in October. (For the purposes of making the simulations reader friendly, I define a tail event as an increase or decrease of 5% or more in a given month). For November, that estimate fell to 16.7% with an estimated probability of a decrease of 5% or more falling to a mere 5.3%. In that article I stated, "these results suggest that the simulations for November are more positive with a smaller likelihood of a tail event occurringâ€¦". To be honest I was quite skeptical of C-J's results - having worked at the U.S. Treasury Dept. in Washington, D.C. on issues such as systemic risk, I tend to emphasize the negative. So I was surprised by the record highs in the S&P 500 and the 3.4% increase in the index for November. Perhaps the increase is even more surprising when you consider that the S&P 500 fell for 9 straight days from the end of October through early November. With all that as a backdrop, I was really interested to see what C-J's simulations would look like for December. Those results are contained in the table below:

Here are my takeaways from the December simulations. First, the probability of a tail event, either positive or negative has decreased again. On the negative end of the tail, the probability of a loss of 5% or more is 5.5%, just slightly above the 5.3% reported in the November simulations. On the positive end of the tail, the estimated likelihood of an increase of 5% or more has shrunk from 11.4% in the November simulations to the 6.8% in the table above.

Second, the estimated probability of an increase in December has fallen. In the November simulations, the estimated probability of an increase in the S&P 500 was an unusually high 64.4%. For the December simulations, that probability is now estimated at 59.5%, much closer to the historical rate of occurrence for a one-month move in the market. Furthermore, for the 2,000 November simulations the median result was a 1.3% increase in the market. For December, the median estimate remains positive at 0.9%, but lower by 40 basis points.

Taken together these results continue to suggest a significant probability of a positive month with a lower probability of a tail event. This clearly stands in contrast to the simulation distributions I reported just a few months ago.

**Negative Tail Analysis**

As I have done in recent articles, I break out into greater detail the negative fat-tail results from the table above. But my particular interest is in the likelihood of larger losses (-5% or worse). The December simulations above suggest that probability has increased ever so slightly to 5.4%. Broken out further the results can be seen as:

Compared to the November simulations, the estimated probability of a 5% to 7% loss has increased from 2.4% to 3.7%. For losses in the 7% to 11.74% (11.74% represents 3 standard deviations for those who use the concept) range, the estimates remain very similar to what C-J suggested in the November simulations. Finally, and in sharp contrast to the November simulations, the estimated probability of a loss of 11.74% or more has shrunk to only 0.05%. In my November article, I noted that while the distribution was decidedly more positive, the likelihood of 3+ standard deviation decline had increased dramatically. The December simulations suggest that is no longer the case as only 1 of the 2,000 simulations suggested such a result. That simulation suggested the S&P 500 would decline by 12.7% to approximately 1919 by the end of December. Furthermore, as seen in the table above, the 0.05% probability is markedly lower than both the historical rate and the rate implied by traditional finance theory.

**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.