Thursday, 6 March, 2008

The Square Root Theory

The Square Root Theory

References to the Square Root Theory as a predictor of stock prices pops up every now and then in financial writings. Norman Fosback used the theory in a 1976 publication called Stock Market Logic to make the case that the normal trading range of low price stocks provides greater profit opportunities than the normal trading range of high price stocks. In 1983, a book entitled The Templeton Touch, by William Proctor, disclosed that one of Templeton's 22 principles for stock market investing was that stock price fluctuations are proportional to the square root of the price.

In the 1950s William Dunnigan developed two stock trading systems called the Thrust Method and the One Way Formula. Both methods had several advantageous entry techniques but each had an absence of exit techniques. Dunnigan was above all a portfolio manager and not happy with the risk-reward aspects of his own trading methods, Dunnigan supported and publicized the Square Root Theory. He went so far as to call this theory the "golden key" and claimed recognition from some economics and statistical trade journals of the era.

The theory holds that stock prices move over the long and short term in a square root relationship to prior highs and lows. For example, IBM made a monthly closing low of 4.52 in June, 1962 and monthly closing high of 125.69 in July, 1999. This is within a few percentage points of the square of the sum of the square root of the low price + 9 or (2.12+9)^2. GM made a low of 15 in November, 1974 and a high of 95 in May, 1999. Again, a few percentage points from the square of the sum of the square root of the low + 6 or (3.87+6)^2. There are hundreds of these examples across the stock, financial and commodity markets. Even a few minutes with a pile of stock charts and a calculator will build confidence that major highs and lows are related to each other by additions and subtractions to their square roots.

Let’s go through a recent example and see how it works. The chart is Eurodollars continuous futures.

Eurodollars made a high of 1.37 on December 30, 2004. First step is to convert the actual price into a useable number so that we will not be dealing with tiny decimals. In this case multiply the actual Eurodollar price by 1,000. That makes the December high 1370. The square root of 1370= 37.01. Subtract 1 from the square root 1370 (37.01) = 36.01. Square 36.01 to get 1297. The low on February 9, 2005 was 1.28. Not bad. Now that you know the drill let’s look at the remaining swings on the Eurodollar chart.

Feb 9 low = 1.28 = 1280 = Square Root 35.77
35.77 + 1 = 36.77
36.77 ^2 = 1352. Bingo!

March 14 high = 1.35 = 1350 = Square Root 36.74
36.74 – 2 = 34.74
34.74 ^2 = 1207 = June 13, 2005 low!

Before Dunnigan and Templeton, probably starting in the early 1900s, W.D. Gann was using square roots to forecast stock and commodities prices. His method was more complex and appears to have been based on some ideas he picked up on during his trips to India or Egypt. Gann used an ennegram, a diagram of numbers constructed in such a way to show square and square root relationships. This ennegram is what’s come to be known as the Square of Nine from the Greek root “enneas” which is the word for “nine.”

Although Gann never revealed exactly how he used the ennegram we can gather from his words that it was probably very important to him: "We use the square of odd and even numbers to get not only the proof of market movements, but the cause." W. D. Gann, "The Basis of My Forecasting Method" (the Geometrical Angles course), p. 1