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Theories On the Square Root of Time

The square root of time is a mathematical computation. Select a period of time, such as 9 days. Using 9 as time, the square root of that time is 3. Theories on the square root of time have evolved in many fields, most notably finance. Albert Einstein was the first to propose a theory about the square root of time, using it to explain the movement of molecules in a liquid.

Brownian Motion
- The concept of Brownian motion dates back to Robert Brown, a Scottish botanist. In the early 1800s, he discovered that grains of pollen vibrate randomly when suspended in water. In the early 1900s, Albert Einstein took this idea further by explaining that the random motion was due to the collision of molecules. These collisions take place with any substance in a liquid; the molecules of the substance and liquid strike each other. Einstein also discovered how Brownian motion functioned over time: it increased with the square root of time. Einstein's work involved calculating that the root mean square displacement for a suspended particle increased with the square root of time. Displacement refers to the distance particles move over time. That displacement is squared and then the mean for all the displaced particles is calculated. You then take the square root of the mean and you have the root mean square displacement, which turns out to be the square root of time.
Random Walk Theory
- The square root of time is part of the random walk theory used in finance, specifically, to track movements of stock prices. Stocks have a standard deviation which can be determined over time. An example is the ten year movement of stocks on a daily basis in the S&P 500. The standard deviation for one day, either up or down, is 1.1 percent. That deviation does not increase in a cumulative fashion; rather, the deviation for five days is not 5.5 percent, but 2.4 percent. The random walk theory shows that the deviation for five days is 1.1 times the square root of time, or 5, which equals 2.4.
Value At Risk
- Value at risk is another concept used in work with stocks that employs the theory of the square root of time. It calculates with some reasonable certainty the worst case scenario for loss from a stock over a predetermined period of time. The calculation works by taking the amount the stock could reasonably go down in one day, for example, three percent. To run that amount out over a month, multiply three times the square root of time, which is 20 trading days. The total is your worst case or the amount of value you place at risk.
Static Filtration Theory
- Static filtration theory refers to the collection of filtrate from an activity such as drilling in an oilfield. The theory states that the amount of filtrate collected increases with the square root of time. To double the amount collected, you have to increase the time four fold.

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