Bell Curves and Stock Prices

 

Last week our discussion took us as far as to a description of the normal or Bell curve.

See a depiction above. The numbers at the bottom of the graph refer to “standard deviations” from the norm.

We were discussing specifically mileage errors on maps. One standard deviation from the mean (the area between -1 and +1 on our graph above) accounts for roughly 68 percent of all the maps in our hypothetical database. Two standard deviations from the mean (the area between -2 and +2) account for roughly 95 percent of the maps. Three standard deviations account for 99 percent.

A normal curve for a phenomenon is taken as evidence of randomness. If there were some reason why the mapmakers of 17^th century England were inclined to make a particular error that reason would show up as some non-normality in this chart. Perhaps the roads between these two places were especially good by the standard of the day, and the ease of travel created a general impression that the cities were closer than was/is the case. That might lead to a lop-sided (or as statisticians say a skewed) curve.
 

Or perhaps some famous troubadour of the day had sung of the “191 cursed miles” between his own home and that of his beloved. The song became quite popular and the right mileage number stuck in people’s heads.  This would make mistake less likely, and it would give the resulting curve a pointier look or (again to use the right jargon) a higher kurtosis than the above.
 

Now: it isn’t difficult to apply the same general idea of probability to the movement of the price of a stock. Consider every day in which XYZ’s stock has been listed on an exchange as a distinct datum. On any given day, the price has either risen, or fallen, or held its ground. As a first approximation, we might take a price that stays at the end of the day where it was at the start at the mid-point of our curve. As a first hypothesis, we might expect small upward moves and small downward moves to be about equally likely with each other, and slightly less likely than a flat day. Huge moves in either direction would be quite unlikely, some number of standard deviations away from the center.
 

If we create a chart for XYZ stock on these principles and it turns out not to look like a Bell curve, then could we fairly conclude that there is some unexpected joker in the deck? Is that how we might make our case that the Morgan Stanley downgrade had an impact? Because such reports create non-random distortions? Maybe the downgrade is acting something like our imaginary troubadour or like those atypically good roads that might once have distorted cartography.
 

Maybe. But we have to take account of other complications.
 

For example, there is a long-run upward trend in stock market prices, so that if we really tried to use no change as a mid-point for a particular stock’s prices, we’d risk a much skewed curve. The long-run trend is in some periods the result of inflation, in some periods the result of real underlying economic growth, and in some periods includes elements of both. Still, it exists, and some modification of our method is in order.
 

Suppose we compared a particular stock’s performance to that of a broader market (thus controlling for factors like price inflation and over-all economic growth). As a second approximation, then, we might say that a stock price that at the end of the day moves just as far, up or down, in percentage terms, as did the broader index on the same day (the Dow Jones, the S&P, the Nikkei, whatever) is at the midpoint of the curve. If we think that stock prices move randomly, we would expect days in which the stock of XYZ did a little better than the broader market, or a little worse, to be about equally likely with one another, and slightly less likely than a day on which XYZ does just what the broader market does. Huge departures from the broad market in either direction would be quite unlikely, some number of standard deviations from the center.