Proof That Properly Discounted Present Values of Assets Vibrate Randomly

Abstract

Even the best investors seem to find it hard to do better than the comprehensive common-stock averages, or better on the average than random selection among stocks of comparable variability. Examination of historical samples of percentage changes in a stock's price show that, when these relative price changes are properly adjusted for expected dividends paid out, they are more or less indistinguishable from white noise, or, at the least, their expected percentage movements constitute a driftless random walk (or random walk with mean drift specifiable in terms of an interest factor appropriate to the stock's variability or riskiness). The present contribution shows that such observable patterns can be deduced rigorously from a model which hypothesizes that a stock's present price is set at the expected discounted value of its future dividends, where the future dividends are supposed to be random variables generated according to any general (but known) stochastic process. This fundamental theorem follows by an easy superposition applied to the 1965 Samuelson theorem that properly anticipated futures prices fluctuate randomly -- i.e., constitute a martingale sequence, or a generalized martingale with specifiable mean drift. Examples demonstrate that even when the economy is not free to wander randomly, intelligent speculation is able to whiten the spectrum of observed stock-price changes. A subset of investors might have better information or modes of analysis and get above average gains in the random-walk model; and the model's underlying probabilities could be shaped by fundamentalists' economic forces.