“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 for the problems discussed above but does so in a probabilistic manner.
January 2019
Another month, another round of volatility. When it was over, having closed November at 2760.17, the S&P 500 fell by 9.18% in December to end the year at 2506.85. Given the December 2017 close in the S&P 500 at 2673.61, the 2506.85 closing number also represented a 6.24% decline in the Index for the year. The 9.18% December decline represents the 11^{th} largest one-month loss in the Index since 1950, and the largest loss since February 2009.
While the numbers are significant, one of the facts about the current market that I find most intriguing is that while volatility is clustering intra-month, the recent pattern has not shown up as expected with regard to inter-month volatility. Recall in my November article, I discussed how large market moves in one month tend to be followed by large market moves in the following month. While history makes this point abundantly clear, that is not what has happened recently. After losing 6.9% in October, the S&P 500 gained 1.8% in November, only to lose 9.18% in December. This is truly an unusual pattern, at least on a historical basis. In fact, since 1950, there have only been three other instances where the S&P 500 declined by 5% or more in one month, had a movement of less than 2% in either direction the following month, and then declined by 5% or more again in the third month. Interestingly enough, all three of those episodes happened this century, twice in 2002 and most recently in 2008.
With that said, I was curious how C-J would view January. Would it again suggest the kind of inter-month volatility we have historically seen? Or, would it suggest a pattern similar to the recent market moves where a dramatic decrease one month is followed by a mild move the next month? The January simulations are shown below:
The results are staggering and very similar to the November scenario noted earlier. The likelihood of fat-tail event, in either direction, has again increased dramatically. The results suggest an over 50% chance that the S&P 500 Index will close January either up 5% or more or down 5% or more. The probability of a decline of 5% or more now equals 24.6%. A 5% decline in the Index, given the December close, would put the S&P 500 at 2381.51 at the end of January. In fact, C-J estimates a 12.6% chance the Index will end January below the 2018 low of 2346.58, a low established on December 26^{th}. In contrast, the probability of an increase in the Index of 5% or more for January is estimated at 28.3%. A 5% increase in the Index would put the S&P 500 at 2757.54 at the end of January. This would effectively get the Index back to where it was in late November.
On the other hand, C-J estimates a 15.9% chance that the January market move will be between +1% and -1% and a 31.2% chance the market will be between +3% and -3%. Looking at the 2,000 simulations in total, the median simulation for January equals -0.17%, with a 50.6% likelihood that the S&P 500 Index will be below 2506.85 at the end of January. That 50.6% estimated probability is below the historical rate of occurrence for a 1-month move. With that said, the results suggest an increased likelihood of a large movement in the Index - but in what direction? One also can’t ignore an almost 1 in 3 chance of a movement of plus or minus 3% - but again, in what direction?
Negative Tail Analysis
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 category in the table below, as it corresponds to three standard deviations below the average monthly percentage change. Broken out into more detail, the January negative tail results can be seen as:
If you look at the fourth column, you immediately see that the large increase in the estimated likelihood of a negative tail event is primarily concentrated in the -5% to -7% and the -7 to -9% ranges. In both cases, the estimated probability is two to four times higher than the rate of historical occurrence and the likelihood implied by traditional finance theory. In fact, when put together, C-J suggests a 22.9% likelihood that January will end with a decline in the S&P 500 Index of 5-9%. And while not an overwhelmingly high probability, it thus remains too high to be ignored.
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.