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“But as we cannot predict such external influences very well, the only reliable crystal ball is a probabilistic one.” - Benoit MandelbrotThe C-J Monte Carlo Simulation ModelC-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. Fat tails, or leptokurtosis as it is called in statistics, 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 movements in the stock market, traditional portfolio theory underestimates (and in some cases significantly underestimates) the downside risk 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 maintains statistical properties similar to the behavior of the S&P 500 over the last 60+ years. In the model, C-J runs 2,000 simulations of the S&P 500 for future periods. My purpose is not to provide a single point estimate of where the S&P 500 will be at some future point or to what factor will drive the market. As investor 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. July 2017In my June article, that the bulk of the simulations had shifted to the 1% to 5% ranges on both the positive and negative sides, with a median simulation for June of 0.38%. While my intention in writing the articles is not to predict the level of the S&P 500, now that June is over we can see that the actual change in the S&P 500 was an increase of 0.48% as the S&P closed out the month of June at 2423.41. June also marked the second month in a row where the median C-J simulation was for a positive, but significantly smaller than the historical average percentage change in the index. And once again we had a record closing high in the S&P 500, this time established at 2453.46 on June 19. With that as a backdrop, I was curious as to what the C-J simulation results for July would look like. The results are shown below. What immediately jumps out at you from the simulation results is the movement of the distribution back toward the positive tail. This can be seen in three ways. First, the median simulation result calls for a 1.14% increase in the S&P 500 by the end of July. This is in comparison to the median result from the June simulations that called for a 0.38% increase. This 1.14% increase is also considerably larger than the historical average monthly change in the S&P 500 of 0.70%. Second, the simulation results suggest a 65.4% chance that the S&P 500 will increase in July. This compares to a 54.6% probability of an increase in the June simulations. Going back to 1950, the historical likelihood of the S&P 500 increasing in a given month equals 59.5%. Finally, C-J estimates the likelihood of a positive tail (5% increase or more) result at 11.3%. That is the highest probability of a positive tail event since the February 2017. In this case a 5% increase in the S&P 500 would put the index at 2544.58 by the end of July. Furthermore, the S&P 500 ended the month of June 30.05 points (1.2%) below its all-time closing high of 2453.46. With regard to that record high, C-J simulations estimate a 47.9% chance that July will end with the S&P 500 Index at a new record level. Negative Tail AnalysisFinally, as I have done in recent articles, I break out the negative tail results in more detail. For purposes of these articles, I define the negative fat tail to include losses of 5% or more. While a 5% decline is not technically a negative fat-tail event, it certainly marks a level of losses investors recognize in their portfolios. Furthermore, while C-J does not use the normal distribution, I include the -11.74% or worse category as it corresponds to three standard deviations. Broken out into more detail, the July results can be seen as: As noted in the fourth column, the risk of a negative tail event has increased by 4 percentage points since the June simulation results. The estimated probability of a loss of 5% or more in the S&P 500 for July now equals 8.3%, an estimated probability slightly below the historical rate of occurrence and the likelihood from traditional financial theory. While readers may take that as good news, a word of caution is in order. While the overall probability of a negative tail event remains low in comparison, the estimated probability for particularly large losses 0f 9% or more are above both the historical rate of occurrence and the rate implied by financial theory. 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.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. ................
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