¹ The expressions “predict the future” and “forecast” are used in this paper because they have come to be accepted terminology in the random walk literature. However, they are actually somewhat misleading; after long observation, a bystander might be able to accurately “forecast” the probability of the event: “black on the next ball” occurring on a roulette wheel. This would be true even though successive spins of the wheel are independent.

What is required as proof of dependence, for either a roulette wheel and a change in security price, is to demonstrate that the conditional probability of a future event, given information about past events, is not the same as the unconditional probability of the same future event, given no previous knowledge.

² This fact is not strictly true; the investor who sells at price X_o and repurchases the same security later at price X_o + e or X_o − e, with equal probability of either event, will improve his position. However, the “loss” of dividends and the drift of prices upward offset this tendency (reduce the probability of repurchasing at X_o − e) to the extent that it is doubtful that the investor will do better than buying and holding.

³ I should like to express my appreciation to Dean Karger and to the faculty of the Rensselaer School of Management for their outstanding support throughout this project.

⁴ Lorie, James and Fisher, Lawrence, “Rates of Return on Common Stock,” The Journal of Business, Vol. XXXVIII, No. 1 (January 1964), p. 10Google Scholar.

⁵ Ibid.

⁶ Ibid.

⁷ Tomlinson, Lucille, Practical Formulas for Successful Investing (New York: William Funk, 1955), p. 243Google Scholar.

⁸ Ibid, p. 246.

⁹ Ibid., p. 248.

¹⁰ Merritt, Robert E., Financial Independence through Common Stock (New York: Simon and Schuster, Inc., 1957), p. 264Google Scholar.

¹¹ Dahl, Curtiss, Consistent Profits in the Stock Market (Cincinnati, Ohio: Tri-State Offset Co., 1960), p. 155Google Scholar.

¹² Granville, Joseph E., A Strategy of daily Stock Market Timing for Maximum Profit (Englewood Cliffs, N. J.: Prentice-Hall, 1960), p. 155Google Scholar.

¹³ Merritt, Robert D., Financial Independence through Common Stocks (New York: Simon and Schuster, 1957), p. 156Google Scholar.

¹⁴ Barnes, Leo, Your Investments (Englewood Cliffs, N. J.: American Research Council, 1967 Edition), p. 107Google Scholar.

¹⁵ It is instructive to note that the simple moving average is nothing more than an intercept fitted by the method of least square to successive time periods. See Yamane, Taro, Statistics, An Introductory Analysis (New York, N. Y.: Harper and Row, 1967), p. 860Google Scholar.

¹⁶ Some authorities who use or present information relative to 200-day Moving Averages are Granville, op. cit., p. 237, and Barnes, op. cit., p. 182.

¹⁷ Barnes, op. cit., p. 108. Note that although the average age of the two methods is identical, they may yield somewhat different answers.

¹⁸ James, F. E. Jr., The Implications of Trend Persistency in Portfolio Management, a thesis submitted to the Faculty of the School of Management at the Rensselaer Polytechnic Institute, Troy, New York in June 1967Google Scholar.

¹⁹ Where N(0,1) represents the standardized Normal variate. The size of the sample required to make the approximation adequate for tests of significance depends on the nature of the underlying population. As Miller and Freund state: “In practice, the normal distribution provides an excellent approximation to the sampling distribution of x for n as small as 25 or 30, with hardly any restrictions on the shape of the population.” Miller, I. and Freund, J., Probability and Statistics for Engineers (Englewood Cliffs, New Jersey: Prentice-Hall, Inc., 1965), p. 134Google Scholar. In our examples n is never less than 200.

²⁰ The substitution is not direct, of course. The random variable formed by the ratio of a normalized random variable over the quantity obtained by taking the square root of an independent chi-square random variable divided by its degrees of freedom follows the “t” distribution with the degrees of freedom associated with the chi-square variate. Some algebraic manipulation of this quantity will yield a statistic in the form presented above. See Bowker, Albert and Lieberman, Gerald, Engineering Statistics (Englewood Cliffs, New Jersey: Prentice-Hall, Inc., 1964), p. 79Google Scholar.