"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 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 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.

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 investor 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 does so in a probabilistic manner.

**December 2017**

November has come to a close and much like the preceding months, it has been another period of monthly gains with a new record level for the S&P 500 along the way. For the record, the S&P 500 hit a closing high of 2647.58 on November 30. It seems like a long time ago but the S&P 500 closed out 2016 at 2238.83. That represents a gain so far this year of 18.3%. If you go back and look at C-J's simulation results for the year 2017 (made on January 1, 2017), those simulation results suggested a 45.6% chance that the S&P 500 would be up 10% or more for the year.

Now at this point you may be wandering if I've gotten ahead of myself as it is only December 1. And you are right. So as I started to think about December. With eleven months of the year now behind us, the S&P 500 has recorded gains in ten of those months. I talked about this in my article last month. The only month this year the index has been down was in February when the index fell by 0.04%. Prior to that, the last negative month for the S&P 500 was September 2016 when the index lost 1.94%. And an almost two percent decline is by no means, at least on a historical and probabilistic basis, an unusual outcome. In fact, the last time the S&P 500 had anything resembling a negative tail event was a 5.07% decline in January 2016.

So as the year comes to a close, and the S&P 500 sits at near record highs, I wondered if December would continue the trend. Or would investors see something different? To examine that question, C-J's December simulations are shown below:

When compared to the November simulations, what is immediately noticeable is that the December simulations exhibit a small shift to the negative side. In total, the probability of the market declining in December, which stood at 43.8% in November, now stands at 45.3%. That is below the historical rate of the index declining in a single month which equals just over 40%. In particular, the probability of a decline in the S&P 500 of 2.9% or less increased by 1.6 percentage points. Also noteworthy is that the median simulation result calls for a below average gain in the market of only 0.48%. So while the results still suggest a continuation of the trend of the S&P increasing, the results are clearly more muted than the November simulations.

**Negative Tail Analysis**

Given the underestimation of negative tail risk in traditional financial theory, 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. 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 three standard deviations below the average monthly return. Broken out into more detail, the December negative tail results can be seen as:

The simulation results continue to suggest an environment of reduced risk as it applies to negative tail events. In total, the estimated probability of a decline of 5% or more in the S&P 500 Index by the end of December equals 4.1%. That rate is down slightly from the November simulation results. But I would note that for the third month in a row, the estimated probability of a negative tail event is less than the historical rate of occurrence and the rate implied by traditional financial theory. The last time the simulation results were above the historical rate and the rate implied by the normal distribution was in the September 2017 simulations.

**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.