**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*, or Nassim Taleb's *Black Swan*, 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 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, but rather to assess market risk, in general, with a particular emphasis on negative fat-tail risk.

**August 2016**

The S&P 500 Index finished the month of July at 2173.60. That is an increase of almost 3.6% in the index from the June close at 2098.86. With the first seven months of 2016 behind us, the S&P 500 now stands almost 130 points, or 6.3%, above its year-end 2015 close at 2043.94. July also marks the fifth consecutive month the S&P 500 Index has increased, the last decline being a 0.4% decrease in February. And on July 22, the S&P 500 established a record closing high of 2175.03. So as the month comes to a close, the S&P 500 stands near its all-time high. With that said, C-J looks ahead to August in the table below:

A number of interesting things stand out from the August simulations. First, the results are slightly more positive than the simulation results for July. The simulation results for August suggest a 59.3% chance that the index will increase in August. That is slightly more positive than the 56.7% chance estimated for July and is in line with the historical rate of 59.5%. Furthermore, C-J's median forecast for the S&P 500 at the end of August is 2189.27, a gain of 0.72%, slightly higher than the historical average monthly percentage change in the market since 1950.

Second, while the forecast probability of an increase has risen for August, the probability of a positive fat-tail event has fallen from 10.2% in July to 8.4% in August. This probably reflects the fact that the index has risen for 5 consecutive months and that the increase in August put the index above its previous record high established in May 2015. Also noteworthy in the results is that none of the simulations calls for an increase of 10% or more. Historically, when the S&P 500 has increased on the order of 2% to 4% in a given month, dating back to 1950, it has never had a double-digit increase (or decrease for that matter) the following month. This is not to say that a double-digit increase in August can't happen. Rather, none of C-J's 2,000 simulations resulted in such an outcome. But it is interesting to note that the simulations now suggest a much lower probability of a tail event, either positive or negative, when compared to the July simulations.

**Negative Tail Analysis**

Finally, similar to my July article, I break out into greater detail the negative fat-tail results from the table above. The simulation results suggest a 40.7% chance of a decline of some magnitude in the S&P 500 by the end of August. So despite the record high established during July, there is still an estimated 2 in 5 chance that the market will be down at the end of August. But my interest is primarily in the likelihood of larger losses (-5% or worse). The August simulations above suggest that probability equals only 5.7%. Broken out further the results can be seen as:

It is interesting to note that the simulation results suggest a much lower probability of a negative fat-tail event in August than existed in July. As noted in my previous article, the estimated probability of such an event for July was 10.4%. The likelihood of a 5% to 7% decline has remained constant relative to July, and that probability is just slightly below the historical rate. But the results for losses of 9% or more in August are not only dramatically lower than the simulated probabilities for July, they are also lower than those associated with historical outcomes and the rate implied by the normal distribution (traditional finance theory). In fact, of C-J's 2,000 August simulations, the single worst outcome was a drop in the S&P 500 to 1856.69 by the close of August. That is well in excess of a 3 standard deviation decrease (-11.74% is the 3 standard deviation mark) and such a move would return the S&P 500 to within its 300-point trading range that existed for much of the last two years. But that would still be above its February 2016 low of 1829.08. So C-J's simulation results suggest that while the probability of a decline in the market in August is in line with historical data, the likelihood of a negative fat-tail event occurring in the monthly percentage change is considerably below average. In part, these results probably derive from historical trends as well as the fact that the market has shown considerable upward momentum recently.

**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 mutual fund.