The Analysts Journal
Volume 1, Number 2, April 1945
Published by The New York Society of Security Analysts11
Low Priced Versus High Priced Stocks
The Square Root Rule and Its Significance
By Harry D. Comer
The habits of security price movements have long been a fruitful field for research. Of the many "habit-theories" evolved from such studies, one appears of unusual interest. This is the "square-root rule", For the lessons it teaches cast doubt upon many other commonly accepted market-habit theories. Further, this "rule" makes questionable the validity of certain appraisal-techniques practiced throughout the financial community.
In the following paragraphs the square root rule is explained. The data developed in a test of the rule are given. And the more obvious weaknesses of conventional financial appraisals, in the light of the square-root rule, are indicated.
Pygmy versus Colossus
Use of an extreme example will serve to emphasize what is involved in the problems dealt with. The buyer of Bush Terminal at 21/2 on April 28, 1942, had a gross profit of 170% at closing price 63;" on July 10, 1944. In the same period American Telephone and Telegraph rose only 59\"0, from 1023/. to 1633,4. Obviously, per dollar of commitment, the gross gain on "B8H" was almost 3 times as large as on "T"". There can be no question as to the correctness of these percentage comparisons in reflecting the relative financial results to the holder. But are we to conclude that they measure the true effectiveness of security selection?
It is my contention that the Colossus-American Telephone-really was the better performer; the Pygmy-Bush Terminal-was the laggard. Considering quality (price-class, risk), "T" outdistanced the general market; "BSH" failed to parallel the general rise. The basis for these and other perhaps startling conclusions is demonstrated in this study.
Big Profit from Little Stocks
Data in Table I (p. 16) represent the essence of a monumental compilation of statistics. They show, by price-groups, the average percentage gains achieved by all common stocks listed on the New York Stock Exchange, in all bull markets from 1897 to 1929.
In general, the lower the price the larger was the ensuing percentage gain. For example, stocks under $5 rose 24170, whereas those in the $30-40 class rose only 790/0 or roughly one-third as much on a percentage basis.
The phenomenon of big profits from little stocks is not peculiar to bull markets prior to 1929. The same tendency prevailed in subsequent years, even under new regulations and war conditions. Obviously, equities do not move in equal percentage amount. It is perfectly normal for low-priced speculative stocks to show larger profits per dollar of commitment than high-priced investment-type stocks. But how much larger and how consistently?
Mathematical analysis results in a simple formula expressing the inverse curvilinear relationship between prices of stocks and ensuing percentage advances. The formula, known as the "Square-Root Rule", follows: Bull swings tend to add equal increments to the square roots of stock prices. Conversely, bear swings tend to subtract equal amounts from the square roots of stock prices.
This rule provides a surprisingly accurate description of what actually happens in the stock market. Table II serves to illustrate what the rule means in a bull market. An upswing which lifts $100 stocks-as a group--to $144 (up 440/0) tends to lift $25 stocks by 960/0. $4 stocks by 300%. and so on for other price groups. In each case in the table the number 2 has been added to the square foot of the starting price to produce the figure in the third column. (For simplicity's sake only perfect squares are shown in the first column of-the table. It should be understood, however, that the curve representing the relationship described is a smooth one; in actuality, there are no gaps in the table between price-groups.)
Similarly, a bull market which added 1 or 1.5 or 2.6 or any other amount to the square root of, say $15 stocks, would tend to add approximately the same amount to the square roots of stocks starting from other price-levels.
An Acid Test of the Rule
An "acid test" of the Square Root Rule is exhibited in Table III. Ten common stocks in each price-class were taken at random (in alphabetical order) from the page of a newspaper showing New York Stock Exchange prices for April 28, 1942, the day the general averages bottomed. For each price class, average closing prices were computed for that day, and also for July 10, 1944.
The actual percentage gains are shown in the fifth column of Table III. The sixth column shows "estimated price". This is the average level which would have been reached by each price-class if each had added the same amount to its square root as was added by the $30-32 stocks. (This group was selected as a "base group". It added 1.40 to its square foot in rising to an average price of $48.43. The figure 1.40 also represents the median in the series, lending support to its choice as a measure of the force of the bull market in the period covered.)
Observe the smallness of errors of estimate as shown in column seven. This must be considered a remarkable showing; only 10 stocks had been selected at random in each group--a small statistical sampling indeed.
A Test of the Security Analyst
If we were to stop here, it could be concluded that security analysis is a pleasant pastime, but that to get results in investment management all one need do is to acquire low-priced stocks at the time that market analysis suggested a rise was ahead. Let ·us proceed, however.
In Table IV we see each of our random-selected group of 10 stocks in the $10 class. As a class, these stocks have risen over 100% in a market which increased the value of $30-32 stocks by only 56%.
Inasmuch as it was normal for $10-$11 stocks to rise somewhat more than 100% in the period covered, Baldwin Locomotive's move from 103,4 to 22 V4 was just about "right" considering its price-class and the extent of rise in the general list. According to my computations, only 4 issues in the table truly outdistanced the market. The outstanding performer was Consolidated Cigar, followed in order by Crane, Flintkote and Addressograph-M ultigraph.
The addition of 1.40 to the square root of the price of Consolidated Cigar on April 28, 1942 would have equated its price to the rise in the general market. At 21, it would have shown a gain of 107%, about the same percentage as the average for the group. Actually the stock reached 29, the added 8 points being the measure of its "super-performance". Thus it may be reasoned that out of Consolidated Cigar's total rise of 18 Vs points, all but 8 points were attributable to the upswing in the general list (after adjusting to allow for the fact that it started from the $10 class in the period under scrutiny). These 8 points constitute a measure of the field in which the security analyst could have made a contribution. Do not the other 10 Vs points lie in the market technician's field?
Similarly, the area in which the security analyst could have made a contribution in the case of Crane was only 5 points, and only 3 or 4 points in Flintkote and Addressograph-Multigraph. Moreover, the perfectly successful analyst should have avoided such "laggards" as Allied Kid, American Machine & Foundry, American Stores, Beneficial Industrial Loan and Crown Zellerbach, despite the fact that these 5 issues were destined to rise an average of 679'0 in the period covered.
If the square-root rule is to be taken as a fundamental "truth" in the behavior pattern of the market, then these conclusions logically follow:
The measure of performance of a security selection is not its percentage change as against that of a general market index, during an ensuing period. Nor is it the percentage difference in price as against that of an issue of another company in the same industry. Rather, it is the percentage change as against that of other securities in its price class at the beginning of the period.
In the management of a general securities portfolio, the timing of purchases and sales is more important than the selection of holdings. "When" is more important that "what".
Assuming their purchase is properly timed, low priced stocks generally will prove more profitable to own than high priced stocks. That low priced issues usually embody greater risks goes without saying.
These and other derivative conclusions are each a proper subject for separate study and test-and for reporting in a Journal such as this.
He who tampers with the currency robs labor of its bread. He panders indeed to greedy capital, which is keen sighted and may shift for itself; … The prosperity of the working classes lives, moves, and has its being in established credit, and a steady medium of payment. All sudden changes destroy it. Honest industry never comes in for any part of the spoils in that scramble which takes place when the currency of a country is disordered. Did wild schemes and projects ever benefit the industrious? Did irredeemable bank paper ever enrich the laborious? Did violent fluctuations ever do good to him who depends on his daily labor for his daily bread: Certainly never.
—Daniel Webster (1857).