An effective and common method to measure the deviation of investment returns from the average is the standard deviation of returns. Standard deviation provides a statistical measure of historical volatility and sets forth a distribution of the ranges of probable outcomes. In investing, measuring standard deviation of returns shows the extent to which returns (daily, monthly or annual) are distributed around the average return, estimating a range of probable outcomes and establishing a likely framework of risk and return trade-offs.

The normal distribution in the form of a bell-shaped curve shown in Figure 8-1 illustrates the concept of standard deviation. The curve represents a set of outcomes. In this case, let's say the outcomes are the monthly returns of an investment. The yellow area covered in one standard deviation away from the average in both directions accounts for approximately 68% of the outcomes in a period. The area within two standard deviations from the average, the yellow and green shaded areas, accounts for 95.6% of outcomes, and the area up to three standard deviations away from the mean, illustrated by the yellow, green and orange shaded areas, accounts for 99.7% of all outcomes. The higher an investment's standard deviation, the greater the chance that future returns will lie farther away from the average return.

*Figure 8-1*(permalink)

Francis Galton

Francis Galton, an English mathematician who was an expert in many scientific fields, created his "Quincunx" machine to demonstrate how a normal distribution is formed through the occurrence of multiple random events. He expressed his fascination with this phenomenon by stating, "I know of scarcely anything so apt to impress the imagination as the wonderful form of cosmic order expressed by the 'Law of Frequency of Error.' It reigns with serenity… amidst the wildest confusion. The huger the mob, and the greater the apparent anarchy, the more perfect is its sway. It is the supreme law of Unreason. Whenever a large sample of chaotic elements are taken in hand and marshalled in the order of their magnitude, an unsuspected and most beautiful form of regularity proves to have been latent all along."^{1}

I commissioned the creation of the Probability Machine to educate investors about the probability of outcomes that result from a series of random events. In the stock market, the random events are news stories about a company or about capitalism in general, and the resulting prices of securities. The random drop of the beads, starting with a central point of a fair price, simulates a series of fair prices, ultimately forming a normal distribution in the shape of a bell curve. The distribution of the beads bears a strong resemblance to the distribution of monthly returns shown in red bars behind the beads, also shown in Figure 8-2, reflecting 600 monthly returns (50 years) for an all-equity index portfolio. Like the Probability Machine's normal distribution, the portfolio carries a wide range of outcomes or a high standard deviation. It maintains an approximate normal distribution that accumulates to an average 1.14% monthly fair return over 600 months, but with a standard deviation of 4.67%, which is a high level of short-term volatility.

*Figure 8-2*(permalink)

Note the comparison to Figure 8-3, which shows a lower-risk index portfolio comprised of 90% fixed-income funds and 10% stock funds with a narrow range of outcomes. The 100% stock fund index portfolio in Figure 8-2 experienced greater price swings but had higher returns. Over the simulated 50-year period, a dollar invested in the low-risk index portfolio would have grown to $20.85, while a dollar invested in the higher risk index portfolio would have grown to $472.43. This historical data supports the presumption that investors who have the capacity to hold higher risks are expected to earn substantially higher returns.

*Figure 8-3*(permalink)

- -1 J. R. Newman (ed.)
*The World of Mathematics*, New York: Simon and Schuster, 1956. pg. 1482.

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