*But as we cannot predict such external influences very well, the only reliable crystal ball is a probabilistic one. -*Benoit Mandelbrot

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**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 and serial correlation. The fat-tail problem arises because traditional finance theory uses the normal distribution. Leptokurtosis, or fat tails as it is more commonly known, recognizes that large market movements occur far more frequently than implied by the normal distribution. The practical implication for investors is that by using the normal distribution to explain stock movements, traditional finance theory underestimates the risk of large, significant declines in the market.

With regard to C-J, it uses data on valuation, earnings, and short-term historical patterns in the stock market going back to 1950 to correct for the problems of fat tails and serial correlation. 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 the fractal nature suggested by Mandelbrot, because of its design it also possesses statistical properties similar to the S&P 500 over the last 60+ years.

In the model, C-J runs 2,000 simulations of the S&P 500 for periods as far as 12 months into the future. My purpose is not to provide a single point estimate of where the S&P 500 will be at some future point in time or to guess what factor will drive the market. 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.

### The First 9 Months of 2017

After ending 2016 at 2238.83, the S&P 500 closed the third quarter at 2519.36, a gain of just over 12.5%. That is the largest increase in the index in the first 9 months of the year since 2013 when the S&P 500 gained 17.9% to close the third quarter at 1681.55. In case you’re curious, in that year the index gained another 9.9% in the fourth quarter to end 2013 at 1848.36.

Back to the current period, in early July I ran a second half of the year preview simulation looking at the market through the end of the year. Those results suggested a 72.4% chance the S&P 500 would be positive for the last half of 2017, with the median simulation calling for the index to increase 5.3% in the second half to 2551.75. In addition, my most recent article highlighted C-J’s October simulations. Those results suggest a 55% chance the index will increase in October with a median simulation equal to a 0.38% gain. A 0.38% gain would take the index to 2528.93 at the end of October.

### The Fourth Quarter

With that in mind, I was curious as to what C-J’s results would look like through the end of the year. Those results are shown below:

A number of interesting results come out of the simulations for the fourth quarter. First, the median simulation calls for a 2.3% increase in the S&P 500 for the last 3 months of 2017. Such an increase would take the index to 2577.18. That is about 25.4 points (or almost 1%) higher than the end of year forecast published in early July. So given the recent trends, C-J has become more optimistic about the market through year end. If you now couple that with the median October forecast which formed the first month of the quarter, the median C-J year-end result suggests the S&P 500 will advance an additional 57.82 points in November and December.

While I generally take the above results as positive, a few caveats are in order. First, the simulations suggest a 66.5% chance that the S&P 500 will end December higher than it ended September. Obviously, that still leaves a very significant likelihood of a decline in the index in the fourth quarter. And looking at the chart above you can see an 11.4% chance that decline will be 5% or worse. In fact, of the 2,000 simulations C-J ran, 2 of them called for the market to decline by 20% or more while another 68 of them called for a decline between 10 and 20%. Furthermore, I would also note that in 50 of the 2,000 simulations, valuations became high enough so as to significantly increase the likelihood of a significant market decline in the final quarter. So while C-J's simulation results look promising, there still exists considerable risk in the market as we move toward the end of the year.

*To readers: I try to publish the results from C-J once or twice a month. 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.*

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*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 have a long position in an S&P 500 index fund in a retirement account.