*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 the problems discussed above, but does so in a probabilistic manner.

### The Year 2018

The S&P 500 Index finished the month of December at 2506.85. That is a decrease of 6.2% in the Index from the December 2017 close of 2673.61. While I don’t use C-J for “predicting” the market, if you view my January 2018 article, you will see that C-J estimated only a 14.5% probability that the S&P 500 Index would decrease by 5% or more for the year 2018. But what is perhaps most interesting to me is not the decline in the Index, but rather the way it occurred. First, in my January 2019 article I noted the unusual historical pattern of a large decline in the Index, followed not by another large movement, but rather a small movement and then another large decline. In this case, October through December brought changes in the Index of -6.9%, +1.8%, and -9.2%, respectively. Historically, such a pattern is unusual having occurred only three times since 1950.

But there is another unusual pattern present in the 2018 results. For the first nine months of 2018, the S&P 500 was up 9%. In the fourth quarter the Index fell 14%, thus taking the S&P 500 into negative territory for the year. This is the only time since 1950 that the Index was up for the first nine months, but then lost enough in the last three months to make the change negative for the year.

So with all that said, I had 3 questions I was looking to answer with regard to 2019.

1. From January 1 through the end of the period is the probability the S&P 500 will be higher?

2. How likely is the S&P 500 to end the period above the record high of 2940.41, set on September 21, 2018?

3. How likely is the S&P 500 to end the period below the bear market level of 2352.73?

The 2019 simulation results are shown below.

### First Quarter 2019

The table above shows the percentage change in the S&P 500 from the beginning of the year through March 31, 2019. With regard to the three questions the answers are as follows. First, C-J estimates a 61.3% chance the S&P 500 will end the first quarter above 2506.85. But I would caution readers that the results also suggest an 18.1% chance the first quarter will end with the S&P 500 down an additional 5% or more. Second, C-J estimates only a 1.45% chance the S&P 500 will end the first quarter above the record high of 2940.91. Finally, C-J estimates a 15.7% chance the market will end the quarter in a bear market territory (below 2352.73).

### Second Quarter 2019

The results in the table above show the simulation results for the period January 1 through the end of June 2019. In answer to the three questions. First, C-J estimates a 68.2% probability that the S&P 500 Index will be higher at the end of June 2019 than at the beginning of the year. Noteworthy here is the 50% probability of an increase in the Index of 5% or more over the first six months. Second, C-J estimates a 9.7% chance the Index will end the second quarter above the 2940.91 record. And finally, C-J estimates a 14.2% chance the second quarter will end with the Index in a bear market territory. While I generally view these results as more positive than the first quarter numbers, I can’t help but notice the 1 in 7 chance that six months from now we will still be talking about stocks being in a bear market.

### Third Quarter 2019

The results of the simulations through the third quarter of 2019 are more positive than the results for the first six months. First, C-J estimates a 72% chance the Index will end the third quarter above its 2018 close. Second, C-J estimates a 19.7% chance the Index will end the third quarter in record territory. And finally, the simulation results suggest a 14.7% chance the third quarter will end with the S&P 500 in a bear market territory. While these results are more positive than the results through the second quarter, particularly for the positive tail, it is noteworthy that the probability of the market being in a bear market territory is continuing to increase, albeit marginally.

### Fourth Quarter 2019

Finally, I present the results for the full year. To me, this table is the most interesting table as I am a long-term investor and this gives me an estimated distribution of the yearly movements in the Index. First, the simulation results suggest a 73.6% likelihood the S&P 500 will increase for the year. Particularly noteworthy here is that the median simulation suggests a 9.2% increase in the Index for the year. But more troublesome, C-J also estimates a 16.6% chance the Index will decline another 5% or more in 2019. With that said, C-J now estimates a 31.9% chance the Index will end the year above the 2940.91 record and a 24.2% chance the Index will end the year above 3000. Finally, C-J estimates a 14% chance the year will end with the market in a bear market territory. It is interesting to note that despite the upbeat simulation results, the probability of the S&P 500 ending the various time periods in a bear market territory remains stubbornly within the 14% to 16% range throughout the year. The risk in this market does not appear to be going away anytime soon.

*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.