What a Logistic Curve of the S&P 500 Tells Us

The S&P 500 is a broad-based stock market index. Because markets are noisy, sometimes it goes up, and sometimes it goes down. (See Mr. Market.) The long-term trend is up, as over the last century or so the US economy is generally growing.

Often people treat this growth as an exponential process. Instead I fit a logistic growth curve (S-Curve) to the data. While at some scales, and particularly in the early period, exponential and logistic curves look similar, eventually a logistic curve implies slowing growth. There is active debate about whether growth is in fact slowing, and whether this is temporary or more permanent.

If you buy this curve, it provides some insights, most notably that sometimes the value of the S&P 500 is below the long-term trend and other times it is above. When it is below, historically it has been a good time to buy stocks. When it is above it is a good time to sell stocks. Obviously the best time to sell is at the peak, and the best time to buy is at the trough. (By “at”, I mean moments before, while it is assume buyers are price-takers, there is always some aspect of “price-maker” as well).

Finding the peaks/troughs is tricky, but finding whether or not the market is high or low is straight-forward. We can speculate that dollar cost averaging into and out of the market may be a good way to avoid the traps of timing imperfections. When it is above the line, dollar cost average selling, when it is below, dollar cost average buying. History waits to prove whether this is better than the simpler dollar cost average into the market that so many personal investment advisors advise.

Currently we are above the trendline. Not as much above as in 2000, but pretty close to where we were in 2008. In my mind this is a “sell signal”: it is a time to own cash, not stocks.•

Since I tend to think this is a reliable indicator, I am mostly in cash right now. It’s not clear where the top or bottom will be, and anyone who tells you otherwise is either very rich or lying (or both).


A logistic curve has to have a maximum value (which is asymptotically but never actually reached), which we call “K“. The estimation follows the procedure outlined here, which I am my classes have used to understand technology deployment, particularly in transport. The data come from Quandl.

In this case, K=3035 gives the best fit (highest R-square), the model implies the S&P 500 will approach but never exceed 3035. This model is estimated for data from 1980 forward, and then applied for data from 1900 forward. Since tnought is 2010, the indication is that growth has slowed (we are in late growth).

Arguably there was a phase shift in the stock market around 1980, with the gold standard and oil embargo in 1973 through deregulation in the late 1970s and early 1980s changing how the economy operated compared with the post-World War II era.

It is certainly possible technology will great improve the economy and future profits, or policy will make or break the economy. But the economy is a big thing, it is hard to move much. When we worry about doubling of unemployment from 5% to 10% that is a personal tragedy for many people, but the economy as a whole sees employment drop from 95% to 90%, which is a 5% difference, not a 100% difference.

Some individual stocks may be buys at this point, even as the market as a whole is not. On the other hand, a falling market sinks all boats, to mix a metaphor.



The model is given below:

1980 start INTERCEPT -208.9
b 0.1039
RSQ 0.9129
tnought=intercept/-b tnought 2010
K 3035



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