*“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.

## April 2019

In my March article, I discussed how the S&P 500 Index had increased 11.1% in the first two months of the year. While such an event is rare (only 4 times since 1951), when the S&P 500 increased by 10% or more in the first two months of the year the Index historically increased between 2% and 3.2% in March. Thus, I devoted C-J’s March simulations to looking at whether that trend would continue. The results suggested a 55.2% likelihood the Index would increase in March with an estimated 24.3% probability the increase would be within the 1 to 2.9% range.

With that said, having ended February at 2784.49, the S&P 500 ended March at 2834.40, an increase of 1.8%. So as the first quarter of the year comes to a close, the S&P 500 is up 13.1%, making it one of the best quarters in years.

But a casual reading of the investment news suggests a whole host of things that could go wrong going forward, from a global recession to earnings slowdowns to slowing U.S. growth. And never discount the possibility of animal spirits. With that said, I was curious what C-J had to say about April. The simulation results are shown in the table below.

Here are my key takeaways from the results. First, the simulation results for April look very similar to the March simulations. The median simulation for March called for an increase of 0.59% while the median April simulation calls for an increase of 0.48%. Both of those results are below the historical rate of change in the Index over a one-month period. The March simulations suggested a 55.2% likelihood the Index would increase in March; the April simulations suggest a 54.9% likelihood of the Index increasing in April. Again, both of those results are below the historical rate of occurrence. Second, both the March and April simulations suggest the likelihood of the Index increasing 5% or more is between 6.4 and 6.6%. Finally, while the market increased significantly in January and February, the March simulations estimated the likelihood of the Index decreasing by 5% or more was well below the historical rate as well as the rate implied by traditional finance theory. As discussed in the negative tail section below, that trend also appears to be continuing into April.

## 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 April negative tail results can be seen as:

As discussed earlier in the article, the negative tail results for April look very similar to March. In total, C-J estimates a 3.9% likelihood the S&P 500 Index will be down 5% or more at the end of April. That is slightly below the estimate for March (4.2%), well below the historical rate of occurrence (8.7%), and well below the rate implied by traditional finance theory (8.53%). In summary, C-J’s simulation results for April look very similar to March.

**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: **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.

I own a long position in an S&P 500 Index fund in a retirement account.