"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**

C-J is a Monte Carlo simulation model used to assess risk in the S&P 500. Traditional stock market models suffer from a number of problems including fat tails, serial correlation, and the failure to account for volatility clustering. The fat-tail problem arises because traditional finance theory uses the normal distribution. For investors, the practical implication of such an approach to explain movements in the stock market is that traditional portfolio theory underestimates (and in some cases significantly underestimates) risk in the market.

C-J uses data on valuation, earnings, and short-term historical patterns in the stock to correct for the problems noted above. C-J does this by using a series of non-normal conditional distributions. If you have read former Yale mathematician Benoit Mandelbrot's book (with Richard Hudson), *The (Mis)behavior of Markets: A Fractal View of Financial Turbulence*, then you should note that C-J is fractal by design. And while the model maintains a fractal nature, because of its design, it also maintains statistical properties similar to the behavior of the S&P 500 over the last 60+ years.

The purpose of C-J is not to provide a single point estimate of where the S&P 500 will be at some future point. As investors we don't see the process generating movements in the market, we only see the outcomes, thus explaining why "expert" predictions are often wrong. 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 by providing not only a model that corrects for the problems discussed above but also does so in a probabilistic manner.

**July 2018**

June ended with the S&P 500 at 2,718.37. Given the May close at 2,705.27, the index recorded a gain of 0.48% for the month. For those of you who read my June article, you know that the median simulation result called for an increase in the S&P 500 of 0.59%. Furthermore, given the March close of 2,640.87, my beginning-of-the-second quarter article (published in early April) had a median simulation resulting in the second quarter ending at 2,715.87.

Despite those results, the daily fluctuations in the S&P 500 still leave me with a rather unsettling feeling. When the market has been rising, I find myself wondering if a new closing high is at hand. And when the S&P is in decline, I can't help wondering if a correction (or worse) is at hand. Plenty of market experts, both here on Seeking Alpha and on other sources, have given forecasts of either a new record high, a correction, or even a bear market in the near future. But alas, the S&P 500, which hit a record closing high of 2,872.87 on January 26, and a year-to-date closing low of 2,581.00 shortly thereafter on February 8, remains within that range despite all the market happenings. With so many reasons for the market to move, and so many diverse expert forecasts about the direction and magnitude of such a movement, I'm not sure if I'm confused or simply exhausted. Either way, with the exception of a few intraday movements, the market has now been fluctuating within the range noted above for over 4½ months.

So with all that in mind, I had C-J take a look at July. The results are shown below:

A few points are worth noting. First, as can be seen from the column on the right, there has been a noticeable increase in the likelihood of a tail event either positive or negative. In the case of a decrease of 5% or more, that likelihood is now estimated at 8%, a move which would take the S&P 500 to 2,582.45 or lower, a level near the lower end of the range I noted above. In contrast, C-J estimates a 10.6% chance the S&P 500 will increase by 5% or more, a move that would take the index to at least 2,854.29, a level just shy of the all-time closing high. Looked at with an eye toward the trading range, C-J estimates a 16.1% chance July will end outside of the range it has been in for the last 4½ months. Specifically, C-J estimates an 8.5% chance July will end with the S&P 500 Index above 2,872.87 and a 7.6% chance it will end below 2,581.

Second, the median simulation for July suggests an increase in the S&P 500 of 1.11%. Such a move would result in the index ending July at 2,748.54. Furthermore, the simulation results suggest a 64.4% chance the S&P 500 will increase in July. That is above the historical average going back to 1950.

**Negative Tail Analysis**

Given the underestimation of negative tail risk in traditional financial theory, I break out the negative tail results in more detail. And while C-J does not use the normal distribution, I include the -11.74% or worse category in the table below as it corresponds to three standard deviations below the average monthly percentage change. Broken out into more detail, the July negative tail results can be seen below:

This table simply provides a more detailed breakout of the reduction in negative tail risk noted earlier in this article. In particular, we see an increase in each of the negative tail ranges, particularly in the -9% to -11.74% range. You will note that while the estimated likelihood of a decline of 5% or more is below both the historical average and the likelihood associated with traditional finance theory, the likelihood of a decline of 9% or more is estimated to be greater than that witnessed historically or implied by finance theory.

**To readers: I try to publish the results from C-J once or twice a month. In the next few days I hope to publish the simulation results for the remaining six months of 2018. 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 own a long position in an S&P 500 Index fund in a retirement account.