*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 using such an approach is that traditional portfolio theory underestimates (and in some cases significantly underestimates) the downside risk in the market.

C-J uses data on valuation, earnings, and short-term historical patterns in the stock 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 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.

**2018: The First Six Months**

After finishing 2017 at 2673.61, the S&P 500 advanced 44.76 points, or 1.67%, over the first six months of the year to end the second quarter at 2718.37. Back in early April, I published the results of C-J's Monte Carlo simulations for the last 9 months of 2018. The median result from the 2,000 simulations called for the S&P 500 Index to end the second quarter at 2715.25. And while I take some comfort in that result, I am struck that the overriding characteristic of the market in the first six months has been the return of volatility, a volatility that has reminded me of the need to assess the probability of likely outcomes going forward. Now with that said, there are plenty of market experts, both here on Seeking Alpha and on other sources, forecasting either new record highs, corrections, or bear markets before year end. And there are so many diverse opinions about what will cause the market to move through the remainder of the year including Federal Reserve policy, trade wars, geopolitical risk, market technicals, or even the upcoming mid-term elections. With all that said, however, the S&P 500, which hit a record closing high of 2872.87 on January 26 and a year-to-date closing low of 2581.00 shortly thereafter on February 8, has remained within that 291-point range for the last 4½ months.

So, as I began writing this article, I decided to follow my previous approach and frame the C-J results as the answer to three of the questions I find most interesting.

1. What are the likely changes in the S&P 500 in the second half of 2018?

2. How likely are we to remain within the 2581 to 2872 range over the next six months?

3. What about 3000? At the beginning of 2018, so many analysts were calling for the S&P 500 to end 2018 at 3000 or above. How likely is that result?

**Third Quarter 2018**

The results of the simulations show the associated probabilities of various outcomes from C-J's third quarter simulations. The median simulation calls for an increase in the S&P 500 Index of 2.97%. That would result in the S&P 500 ending September at 2799.11. Also of note, there is a 31.7% likelihood that September will end with the S&P 500 above 2872.87 and an 11.6% likelihood that September will end with the Index below 2581. Or stated differently, there is a 56.7% chance that the Index will still be within the 2581 to 2872 range come the end of the third quarter.

And if you are curious about the low end of the range (I was), C-J estimates a 24% chance the S&P 500 will end September below the 2017 year-end close of 2673.61. Finally, C-J estimates a 10.5% likelihood that the third quarter will end above 3000.

**Fourth Quarter 2018**

To me, this table is interesting when combined with the results from some of my previous articles. As you will note, the median simulation calls for the Index to increase by 5.35% in the second half of the year. This would put the S&P 500 at 2863.79 at year end. At the beginning of the year, I published a 2018 preview article. The median simulation, in that case, called for the Index to end 2018 at 2911.29. And finally, my April article looking out at the remaining 9 months had a median simulation calling for a year-end value of 2834.59. So, in all three simulations, the median result has the S&P 500 at or near record levels by year end.

In response to the second of my questions, C-J estimates a 48.4% likelihood the S&P 500 will end December above 2872.87 and a 13.6% chance it will end the year below 2581. Or put differently, C-J estimates a 38% chance the S&P 500 will end 2018 within the 2581 to 2872 range we have been in for the last 4½ months. Also noteworthy is that C-J estimates a 21.8% probability the Index will end 2018 below the 2673.61 closing value for 2017.

Finally, in regards to the Index ending the year above 3000, C-J now estimates that probability at 28.1%. Now this result sets up an interesting trichotomy that helps me summarize C-J's year-end simulations. I think of it this way: The is a 28% chance the S&P 500 ends the year above 3000 (analysts forecasts), a 22% chance the Index ends the year below 2673 (year-end 2017 level), and a 50% chance the Index ends the year somewhere in between.

**To readers: I usually publish the results from C-J once a month. At the beginning of January, April, July, and October, I write an additional article examining the simulations through year end. 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 own a long position in an S&P 500 Index fund in a retirement account.