"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 is that traditional portfolio theory underestimates (and in some cases significantly underestimates) the downside 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.

**June 2018**

In my May article, I noted that negative tail risk was gradually rising, but continuing to remain below the historical average probability. In fact, May ended with the S&P 500 at 2,705.27, an increase of 2.16% from the April close at 2,648.05. And while the month was decidedly volatile, the increase in the index was the second largest this year, as February and March brought back-to-back monthly declines in the S&P 500 for the first time since October 2016. Nevertheless, if you read some of my earlier articles, particularly my February 2018 piece, you note that the likelihood of tail events, as estimated by the C-J simulations, has shown considerable fluctuation over the first five months of the year. So with that in mind, I was curious whether the gradual increase in negative tail risk probabilities would continue or whether this would be another month of large and abrupt changes in the estimated likelihood of a tail event. The simulation results for June are shown below:

A few points are worth noting. First, the simulation results suggest a dramatic reduction for June in the probability of a tail event in either direction. To put numbers to that comment, there is an 89.3% probability the S&P 500 Index will end June with a change between -4.9% and +4.9%. That implies the lowest estimated probability of a tail event so far this year. The estimated likelihood of June ending with a decline of 5% or worse is estimated at only 4.8% (more on that below), while the estimated likelihood of a gain of 5% or more is only 5.9%. As shown by the far right column in the table, there has been a dramatic move away from the tails of the distribution, with very large increases in the -1 to -2.9% range and the +1 to 2.9% range.

Those estimates piqued my interest with regard to the following questions. In late January of this year, the S&P 500 Index set a record closing high of 2,872.87. Given the results of the June simulations, how likely is it that the S&P 500 will end June at a new record high? C-J's simulation results place that probability at 3.4%. Second, how likely is it that the S&P 500 Index will end June in correction territory or worse? Given the all-time high of 2,872.87, a correction would take the index down to 2,585.58 or lower. Here, C-J's simulation results estimate the probability at 5.4%.

Finally, as a general overview of the S&P 500 for June, I note that the median simulation calls for an increase of 0.59%, a result that is below the historical monthly average change in the index of 0.705%. Furthermore, C-J's simulation results suggest a 45.1% probability the S&P 500 will decline in June. That is higher than the historical rate of 40%.

**Negative Tail Analysis**

Given the underestimation of negative tail risk in traditional financial theory, I break out the negative tail results in more detail. 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 June negative tail results can be seen as:

This table provides a more detailed breakout of the reduction in a tail risk event, albeit in this case I am only examining the negative tail. Particularly noteworthy is that for all of the loss ranges, the simulation results suggest estimated probabilities that are lower than the historical rate of occurrence (column 5) and the likelihood according to traditional finance theory (column 6).

**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 own a long position in an S&P 500 Index fund in a retirement account.