*"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. Fat tails, or leptokurtosis as it is called in statistics, recognizes that rare events, and hence large movements in the market, occur far more frequently than implied by traditional finance theory. As a result, there often exists a very significant underestimation of the amount of risk in financial markets. On the other hand, serial correlation implies that what happens in the market over one time period, say a month, influences what happens in future periods. If you think of it in the context of the stock market, it seems kind of obvious as when one month ends and another begins investors don't suddenly forget everything that happened last month. But many traditional stock market models assume that serial correlation doesn't exist.

With regard to C-J, it uses data on valuation, earnings, and short-term historical patterns in the stock market going back to 1950 to correct for the problems mentioned above. If you have read Mandelbrot's book (with Richard Hudson), *The (Mis)behavior of Markets: A Fractal View of Financial Turbulence*, then you should note that there exists a fractal nature by design to the C-J model. In the model, Monte Carlo simulation is used to assess the uncertainty inherent in the market going forward. In fact, C-J runs 2,000 simulations of the S&P 500 for future periods as short as one month to as far in the future as thirteen months. My purpose is not to provide a single point estimate of where the S&P 500 will be at some future point in time; history has shown us that such predictions are usually very 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, with a particular emphasis on describing the likelihood of negative tail events.

**February 2017**

The S&P 500 Index finished the month of February at 2363.64. That is an increase of just over 3.7% in the index from its January close at 2278.87. And the record closing high was established just a few days ago, on February 27 at 2379.63. To put it in perspective, the S&P 500 Index has now advanced over 200 points since the presidential election on November 8. Yet despite the good results in February, there exists considerable uncertainty over forthcoming economic policy. In fact, the Baker, Bloom, and Davis Economic Policy Uncertainty Index continues to measure policy uncertainty at heightened levels, as it has since the election. With that said, I was curious as to what the C-J simulation results for February would like. Those results are shown below.

Given the results there are a number of points worth noting. In my article last month, I noted that the distribution of possible outcomes had shifted outward toward the tails, both positive and negative. Such a result would suggest an increase in risk. In fact, last month's simulations estimated a 29.4% chance the market would increase by 3% or more and the market ended up increasing by 3.7%. If you look at the percentage point change from previous month column in the chart above it is obvious that the distribution for March has moved back to the center and away from the tails. That brings with it some good news in that the probability of a decline in the S&P 500 Index of 5% or more by the end of March has now decreased to only 4.2%. Unfortunately, the positive tail also exhibits a lower probability of occurrence as the likelihood of the market rising by 5% or more by the end of March is now forecast at 5.8%. In total, the likelihood of a tail event has declined by a whopping 12.3 percentage points.

Consistent with the movement of the distribution to the center, the median forecast result is also lower than that in February. In my February article, I noted that the median simulation forecast was 0.91%, a number which is higher than the historical average change of 0.69% since 1950. The March simulations result in a median forecast change in the S&P 500 of 0.72%. Furthermore, the likelihood of the S&P 500 being higher at the end of March now stands at 56.9%, effectively identical to the February result of 57.0%.

**Negative Tail Analysis**

Finally, as I have done in recent articles, I break out the negative tail results in more detail. While by some definitions not necessarily a negative fat-tail event, I define the analysis to include losses of 5% or more. While a 5% decline is not technically a negative fat-tail event, it certainly marks a level of losses investors will 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. Broken out into more detail, the March results can be seen as:

To put it in perspective, the risk of a negative tail event has been dramatically reduced from the February simulation results. This can be seen by noting the reduction by 6.25 percentage points shown in the percentage point change from previous month column. (A special thank you to Seeking Alpha reader JeromeB who inspired this addition to my charts.) The decline is most pronounced in the -9% to -11.74% range, but is present in all four of the ranges I analyze. It is also noteworthy that the likelihood of losses of 5% or more from the March simulation results are now less than half of those implied by traditional finance theory and those associated with the historical rate of occurrence.

**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 mutual fund in a retirement account.