"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 market 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 in time. 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 does so in a probabilistic manner.

**January 2018**

December has come to a close and continuing the trend we have seen throughout the earlier months of 2017, the S&P 500 Index rose during the month with new record highs along the way. Officially, the S&P 500 ended December at 2673.61, an increase of 0.98% from the November close of 2647.58. As I noted in my December article, the only month of 2017 that the S&P 500 was down was in February when the index declined a mere 0.04%. So as I started to look toward 2018, I wondered whether the trend would continue or whether January would bring with it the first monthly decline in almost one year. And what would the year 2018 look like? Would it be another year where the index increased almost every month? After all, 2017 gave us the largest yearly increase in the market since 2013. Or would 2018 bring the first yearly decline we have seen in a few years? To begin with, C-J looks at January 2018 in the tables below. (To interested readers, in the next couple of days I hope to publish the results of the 2018 yearly simulations).

Examining the simulation results, three things merit attention by readers. First, despite the over 19% increase in the S&P 500 Index in 2017, the median simulation continues to call for above average returns in January. In this case, the median simulation resulted in a 1.05% increase in the index, significantly above the historical average increase of 0.70%. Second, and in conjunction with the first point, C-J's simulation results suggest a 63.9% chance that the index will increase in January. Or put differently, the estimated probability of the index declining in January 2018 equals only 36.1%. That is the lowest probability of a decline since the July 2017 simulations. Taken together, these two points suggest that the year will begin in a very positive fashion.

**Negative Tail Analysis**

Third, and despite the positive simulation results reported above, an analysis of the negative tail of the January simulation results provides a reason for caution. An examination of the table below shows that the likelihood of the S&P 500 declining 5% or more in January equals 8.3%, an increase of 4.25 percentage points since the December simulations. That is more than double the probability estimated for December and is the highest estimated probability of a negative tail event since September 2017. Furthermore, the increase in the estimated probabilities is in the loss of 9% to 11.74% and 11.74% or more ranges. The estimated probability of a loss of 9% to 11.74% equals 1.5% for January. That is just slightly higher than the historical rate of occurrence and almost double the rate implied by traditional finance theory. Furthermore, the probability of a monthly decline of 11.74% or more (three standard deviations if you use the normal distribution) is estimated for January at 0.4%. That is three times the rate implied by traditional finance theory. So while C-J's January simulation results are, as noted in the preceding section, very positive, a dose of caution is clearly in order.

**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.