“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 finance 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 market 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 underlying 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 does so in a probabilistic manner.

**November 2018**

In my column last month, I raised the question of whether or not there would be an October surprise. While I’m not the most creative person, at least in terms of titles for articles, that was my way of saying to readers that at least from the perspective of C-J’s simulation results, October presented a significant contrast.

Overall, the probabilities suggested a significant likelihood of an above average monthly return. But the article also noted, “C-J’s results also show a significantly greater likelihood of a negative tail event in October. In fact, the estimated likelihood of a negative tail event is the highest it has been since C-J’s February simulations. And we all remember what happened last February.” (See the February article here).

So after ending the month of September at 2913.98, in October, the S&P 500 recorded its worst month since the 7.18% decline in the Index in September 2011. In fact, October 2018 ended with the S&P 500 at 2711.74, a decline of 6.94%. It now seems like a long time ago when the S&P 500 hit a record high of 2940.91 – in fact, that was only back on September 21.

With that said, I was curious to see what C-J had to say about the market in November. If you study financial markets long enough, you realize that volatility has a tendency to cluster. That is to say that highly volatile periods (say a given month) are often followed by other volatile periods (the next month), and quiet periods tend to be followed by quiet periods. The question is often not one of whether there will be heightened volatility the next month; rather the question tends to be in what direction will volatility manifest itself and how large will the movement be.

Or to give it a historical context, there have been 35 instances since 1950 when the S&P 500 fell by more than 6.5% in one month. In 21 of those cases, the S&P 500 moved by 5% or more the next month – 9 of those months had increases of 5% or more and the remaining 12 months had losses of 5% or more. Like I said, volatility begets volatility. With that in mind, the table below summarizes C-J’s simulations for November.

Some of the results for November are quite striking. In particular, C-J’s simulation results show a considerable contrast in the November outcomes. On the negative tail, C-J estimates a 25.7% likelihood that the S&P 500 will end November down an additional 5% or more from the October close. A 5% decline from here would put the S&P 500 at 2576.15. That puts the S&P 500 well into correction territory and well below the December 2017 close of 2673.61.

On the positive tail, C-J estimates a 28.3% likelihood that the S&P 500 will end November up by 5% or more. A 5% increase would put the S&P 500 at 2847.33, well below the record high set back in September. In fact, C-J currently estimates a 4.2% chance the Index will end November at a new record high.

In total, C-J estimates a 54% chance the S&P 500 will move by 5% or more in November. Furthermore, the median simulation calls for a decline in the Index in November by 0.41%. The negative tail probabilities above are C-J’s largest estimated probabilities of such an event since I began writing for Seeking Alpha in March 2016. But in contrast, the positive tail probabilities are also the largest estimated probabilities of such an event since I began writing this column. Like I said, the November simulations show considerable contrast.

**Negative Tail Analysis**

Given the underestimation of negative tail risk in traditional financial theory, I break out the negative tail estimates 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 November negative tail results can be seen as:

If you look at the fourth column, you will immediately see the very large increase in the estimated likelihood of a negative tail event from the October simulations. This is particularly concentrated in the -5% to -7% and the -7 to -9% ranges. Furthermore, you will note that the estimated probabilities are well in excess of the rate of occurrence implied by both historical outcomes and traditional finance theory. In total, the likelihood of a negative tail event is up 17.1 percentage points for November as compared to October.

**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.