*"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.

## May 2019

In my April article, I asked the question of whether the recent trends in the S&P 500 would continue into April. You will recall that the fourth quarter of 2018 brought two significant drops in the S&P 500 Index - a 6.9% decline in October followed by a 9.2% decline in December. Since dipping below 2350 on December 26, the S&P 500 Index has been on an uphill climb with April marking not only the fourth consecutive month of an increase, but also a new record high. More specifically, the Index, which closed March at 2834.40, ended the month of April at 2945.83. The previous record of 2940.91, set on September 21, was eclipsed on April 29 with an intraday high of 2949.52. So after a first quarter of 2019 that saw the Index up 13.1%, one of the best quarters in years, the Index added another 3.93% in April. Year-to-date, the S&P 500 is now up 17.5%.

So with that said I began to look toward May. More specifically, after four consecutive months with gains in the S&P 500, and the Index showing a double-digit gain after four months, I was wondering if this trend would continue. Or given that it is the start of the baseball season, and I am a big fan of baseball history, I asked myself, in the words of Yogi Berra, will May be like "déjà vu all over again"? To that question, here is what C-J had to say:

Here are my key takeaways from the results. First, the simulation results for May look very similar to those of April. One need only look at the far right column to notice how small the changes were in the distribution of the simulation results. One noticeable difference however, is that the median simulation for May calls for the S&P 500 to increase 0.71%, a number which in the grand scheme of stock market history is very average. But that is higher than the median simulation of 0.48% for the April simulations. You will also note that the simulation results suggest over a 41% chance that the Index increases between 1% and 5% in May. A second noteworthy result is that C-J estimates a 57.6% likelihood the market will end May higher than the April close. While that is below the historical rate of occurrence for a market increasing in a given month, it does suggest this market is headed higher. Finally, with a new all-time high at 2949.52, I was curious what C-J would say is the likelihood that May ends with the month higher than the current record. C-J's simulations suggest that answer is 55.7%. Taken from the perspective of this table, May looks very similar to March and April and suggests the current market uptrend will continue through the end of May.

## 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 May negative tail results can be seen as:

Again, this result looks very similar to the negative tail analysis for April. The likelihood of the month of May ending with the market down 5% or more is less than half that associated with both historical outcomes and traditional financial theory.

**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.

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.