“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.

**June 2019**

In my May article, I asked whether trends we had seen in the S&P 500 in 2019 would continue. After large declines in the Index in October and December of 2018, the S&P 500 had increased every month of 2019, putting the Index up 17.5% after the first four months. My May article noted a contrast in results. C-J estimated only a 57.6% chance the Index would increase in May (below the historical rate of 61.3%) but only a 4% chance the decline in the Index would be 5% or more.

With that said, the trend did not continue in May as it was an ugly month. The Index, which hit a record high of 2,954.13 on May 1, ended the month at 2,752.06. That represents a decline of 6.6% in the Index for the month. And that 2,954.13 record seems much further in the past than just a little over four weeks ago.

Now when looking at the data in preparation for the June simulations, I was struck by the number of large declines in the market we have seen over a relatively short period of time. While not necessarily a large decline in a statistical sense, I define a large decline as a decrease in the Index of 5% or more in one month. Put another way, we have now seen three declines of 5% or more in the last eight months – the other two being a 6.9% decline in October 2018 and a 9.2% decline in December 2018. This is clearly outside the realm of likely outcomes under traditional theory. So with that said, I was curious what the C-J simulations would say about June. The results are reported in the table below.

Here are my key takeaways from the results. First, and of no great surprise, the tails just got much fatter. As I noted back in my November 2018 article, volatility begets volatility as volatility tends to cluster. So the question, at least based on historical trends, is not will we see a large movement in the following month (history says we probably will), but in what direction will that movement be. The results in the June table are consistent with that trend. C-J estimates a 27.7% likelihood the S&P 500 Index will decrease by 5% or more in June. That is a significant increase in comparison to the May probabilities. And the simulation results also suggest a 25.3% likelihood the S&P 500 Index will increase by 5% or more in June. Again, this is much larger than the May simulation results.

Further breaking out the simulation results suggests only a 46.2% likelihood the Index will increase in June. That is below the historical rate of 61.3%. And the median simulation calls for a change in the Index in June equal to -0.67%. Finally, while not shown in the table, C-J estimates a 32.1% likelihood that the S&P 500 will end June in correction territory (2,658.72) or worse.

**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 June negative tail results can be seen as:

This is also dramatically different than we saw in the May simulations. Of particular note is the increase in the -5% to -7% and -7% to -9% ranges. In fact, in all three ranges from -5% to -11.74%, the probabilities of such a movement in June now exceed those implied by historical outcomes as well as traditional finance theory.

*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.