“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. It 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 by doing so in a probabilistic manner.
In my August article, I noted that a significant increase (12.3 percentage points) in the likelihood of a small decline in the S&P 500 Index for August. As I stated, “C-J also estimates a 39.7% likelihood the Index will end August down between 0 and 4.9%”. As it turned out, the S&P 500, which began August at 2980.38, declined by 53.92 points, or 1.8%, to end August at 2926.46. But to me, and perhaps of more interest to many readers, is that for a period of time during the month, it looked as if the decline would be significantly larger, with the possibility once again of the Index crossing over into correction territory. And while the C-J simulation results suggested a significantly greater likelihood of a small decline in the Index, it also noted a reduced likelihood of either a positive or negative tail event.
So, with those thoughts in mind, I began to look to the September simulation. Specifically, I was curious what C-J estimated for the coming month with regard to another decline. Would it be another month of a small decline? A larger decline leading us into correction territory? Or did the simulation results suggest an increasing likelihood of a gain in the Index? And what about the 3000 level for the Index, a level that has been at the focus of many analysts? Finally, I was curious about the likelihood of the Index testing the record high of 3027.98 established in July. To me, July seems like a long time ago given the volatility in the market in August.
With those questions as a backdrop, here are C-J’s simulation results for September:
Here are my key takeaways from the results. First, the simulation results are decidedly more positive than the results for August. The median simulation result calls for a gain of 1.23% in the Index, while overall, 70.2% of the results suggest the Index will increase in September. Furthermore, an examination of the far right-hand side column shows a total increase of over 10 percentage points in the 0-0.9%, 1-2.9%, and 3-4.9% ranges. In fact, C-J estimates a 46.5% likelihood the Index will increase by 1-4.9% for the month.
Second, I was curious with regard to certain outcomes for the market. While not shown in the table above, C-J estimates a 37.6% likelihood the Index will end September above 3000 and a 22.3% likelihood the Index will end the month above the record-high level of 3027.98 established on July 26 of this year. Finally, C-J estimates only a 2.6% likelihood the Index will end September below 2725.18, a level that would put the Index in correction territory.
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 range in the table below, as it corresponds to three standard deviations below the average monthly percentage change. Broken out into more detail, the September negative tail results can be seen as under:
The results in this table simply confirm the earlier results. In particular, the likelihood of a decline of 5% or more for the month is lower than the rate that has occurred historically, as well as the rate implied by traditional finance 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 have 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.