June 1, 2015

A stock price series can be viewed as a stochastic, erratic, chaotic, and random-like time function with shocks, gaps, and fat tails. Mostly unpredictable. Accepting this has a consequence: one can't predict with any significant accuracy the price of any stock, be it today, tomorrow, next week, next year, or 20 years from now, for that matter. Saying that a stock might be between \$0.00, \$10,000, or whatever with a 95% confidence level in some 20 years does not help at all.

An acceptable mathematical representation of a price series can be a Stochastic Differential Equation: dp = µdt + σdW, which not only shows its regression line (drift) but also its quasi-random nature. That one has such an equation at hand does not help in predicting prices, only in showing the quasi-random nature of the series. Detrend the price series (dp – µdt), and what's left is the quasi-random part of the equation (σdW). Personally, I'm accepting this equation with all it implies, even if it is a rough representation of what is.

When you do some signal analysis of an SDE, most of the time, you are confronted with relatively small background noise over a signal line function. The Gaussian nature of the noise can be smoothed out to uncover the underlying function. But when analyzing stock price series, most of those concepts go out the window since the function not only loses its Gaussian nature, it loses its underlying function which is being drowned in this random-like noise. So much so that more than 90% of short-term price movements could be attributed to noise.

This is illustrated in the following chart of an SDE when viewed and applied to stocks from a 20+ year perspective:

Stochastic Differential Equation - Long-Term Perspective

(click to enlarge)

The curves were smoothed out to their theoretical values. Be assured that they are much more erratic in real life, but their sum would still add up to dp: dp = µdt + σdW.

At the two extremes, start and finish, the origin of generated profits is a combination of the two right-hand side components. The chart says that at the beginning of a long-term trend, most generated profits will be from randomness, while at the finish line, the trend itself will be responsible for most of the profits. A consequence of such a chart is that short-term generated profits could be more the result of luck than of skill. And that a short-term trading strategy might more resemble a betting system, like gambling coincidentally with a randomly biased coin-tossing playscript.

On the right side of that chart, the long-term, 20-year+ trend is the major contributor to the total generated profits. So much so that it dwarfs the random-like component of the SDE. This could also be why you don't find any day traders on the Forbes 400 list of richest people. But what you will find in that list are bag holders; people that have opted to hold on to their shares holdings.

In fact, the probability of winning the stock trading/investing "game" increases with the length of the holding time. It's another sigmoid function (s-shaped as in the chart above) starting slightly above 0.50 to reach asymptotically 1.00 for time intervals in excess of 20 years. The stock market game is definitely biased to the upside for long-term players.

To me, it is evident that I have to design trading strategies with a long-term perspective. Whatever a trading script does, it must show that it can not only survive but thrive over the long term, and this is in excess of a simple Buy & Hold. That's the real challenge.

One can design short-term positive edge trading strategies, but the real question is not whether it can be done. It is will these be sustainable over the long term?