*“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 and serial correlation. The fat-tail problem arises because traditional finance theory uses the normal distribution. For investors, the practical implication is that by using the normal distribution to explain movements in the stock market, traditional financial theory underestimates often significantly underestimates the downside risk in the market.

C-J uses data on valuation, earnings, and short-term historical patterns in the stock market going back to 1950 to correct for the problems of fat tails and serial correlation. 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 the fractal nature suggested by Mandelbrot, because of its design it also maintains statistical properties similar to the behavior of the S&P 500 over the last 60+ years.

In the model, C-J runs 2,000 simulations of the S&P 500 for future periods. The purpose 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 helping to explain 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.

### October 2017

September, historically the worst month of the year for the S&P 500 in terms of average monthly percentage change, has come to an end. And with it, the third quarter of the year also concluded. And rather than a loss, September registered more record highs with the S&P 500 ending the month at 2519.36. While I did not build C-J for forecasting purposes, in my September article I asked whether September would follow the historical pattern, particularly noting that the simulation results suggested an increase in the probability of large losses. But I also noted that the C-J simulations suggested a 70.8% probability of the market increasing in September with a median percentage change from the simulations of 1.05%.

As it turned out, the 2519.36 September close for the index represents an increase of 1.93% from the August close of 2471.65. Furthermore, back on July 6, I wrote a quarterly article examining simulation results for the third and fourth quarters of 2017. Those end-of-June simulation results called for a median increase in the S&P 500 of 2.87% in the third quarter to 2492.97. Obviously, September was a good month for investors and the median C-J simulations underestimated how good September, and the third quarter, would be. But despite the good news from the third quarter, many questions remain about the market as we head into October and the beginning of the fourth quarter. With that in mind, here is a look at C-J’s October simulations (readers interested in C-J simulations through December 2017 should note that I hope to publish that article in the next few days):

What is immediately noticeable from the October simulations is a return to more modest, albeit still positive, results. The September simulations had a median return of 1.05%; the October median simulation is significantly lower at 0.38%. That result is also lower than the median monthly return historically of 0.90%. If the median is correct, that would put the S&P 500 Index at 2528.93 at the end of October. Furthermore, the simulations suggest a 55% chance that the S&P 500 will be higher at the end of October. That is also lower than the historical rate of 59.9%.

The more modest results also exist in the tails of the simulation results. The estimated likelihood of a decline of 5% or more drops from 9.0% in September to 4.7% in October (more on that below). With regard to the positive tail, C-J’s October simulation results suggest a 6.3% chance of an increase of 5% or more. That result is down 4 percentage points from the September simulations. As a result, the simulations again cluster toward the center with the likelihood of a movement in the -2.9% to 2.9% range equaling 65.1%, an increase of 17.5 percentage points from the previous month.

### Negative Tail Analysis

Finally, given the underestimation of negative tail risk in traditional financial theory, and my experience with tail risk events at the U.S. Treasury Department, I break out the negative tail results in more detail. For purposes of these articles, I define the negative fat tail to include losses of 5% or more in one month. While a 5% decline is technically not a negative fat-tail event, it certainly marks a level of losses investors recognize in their portfolios. Furthermore, while C-J does not use the normal distribution, I include the -11.74% or worse category as it corresponds to 3 standard deviations below the average monthly return. Broken out into more detail, the October negative tail results can be seen as:

As noted in the fourth column, the risk of a negative tail event for October has decreased by 4.2 percentage points from C-J’s September simulations. The estimated probability of a loss of 5% or more in the S&P 500 for August now equals only 4.7%, an estimated probability that is almost half the historical rate of occurrence (8.7%) and the likelihood derived from traditional financial theory (8.53%). All told, whether it is the lower median simulation result or the lower probabilities of a tail event, the C-J’s October simulations suggest a more subdued month ahead.

*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.