Probability Machine

Standard Deviation

Probability Machine

In the field of finance, standard deviation represents the risk associated with a security (stocks or bonds), or the risk of a portfolio of securities (including actively managed mutual funds, index mutual funds, or ETFs). Risk is an important factor in determining how to efficiently manage investments because it determines the variation in returns on the asset and/or portfolio and gives investors a mathematical basis for investment decisions (the basis for mean-variance optimization). The overall concept of risk is that as it increases, the expected return on the asset should increase as a result of the risk premium earned – in other words, investors should not expect a higher return on an investment without that investment having a higher degree of risk, or uncertainty of those returns.

When evaluating investments, investors should always estimate both the expected return and the uncertainty of that return. Standard deviation provides a quantified estimate of the uncertainty of those future returns, the average of which is the expected return. The chart below illustrates this point.

For example, let's assume an investor had to choose between two stocks and an S&P 500 Index Fund.

1) General Electric (GE)
2) Proctor & Gamble (PG)
3) S&P 500 Index

Neither stock is a good choice because an S&P 500 index over the period shown in Figure 8-1a earned a similar annualized return with a lower volatiliy (uncertainty or standard deviation), because of the diversification benefit of 500 stocks versus one stock, making the S&P 500 Index always the statistically preferred investment over individual stocks. You can see from the chart that even stocks that had higher returns than the S&P 500 required higher levels of risk. If you drew a diagonal line along the IFA Index Portfolios, you can see that all stocks but WalMart fell below the line.

If you diversified into other global markets and small value indexes, in addition to the S&P 500, like Index Portfolio 100, the annualized return in this same period was higher, with a similar standard deviation, making Index Portfolio 100 always the statistically preferred investment over the S&P 500 and anything else that we can find. Please note that investors must first determine their risk capacity before making investment choices. Then they can determine which investments have the highest expected return for the standard deviation that is right for them. According to IFA, the maximum risk exposure for any investor should be Index Portfolio 100.

If you want to calculate the standard deviation of a certain stock or index, start by calculating the average return (or arithmetic mean) of the security over a given number of periods, like 20 years or more. For each year, subtract the average return from the actual return for that year, which will give you the "variances". Square the variance in each period. The larger the variance in a period, the greater risk the security carries. Calculate the average of the squared variances and then calculate the square root of this average variance and you will get the standard deviation of the investment in question. The arithmetic mean is used to calculate the standard deviation, but when comparing returns of stocks or indexes, it is customary to use the annualized average return, not the arithmetic mean return. For more information on Standard Deviation, see the Wikipedia page.

Volatility or standard deviation relative to the market, also known as beta, is one way to look at risk. There are three risk factors that have explained 95% of the variability of stock market returns, dating back to 1929. (The 23 page academic study that supports this can be found here.)

According to a landmark study by Eugene Fama and Kenneth French in 1992, the three factors that explain 95% of the variability of stock market returns are Beta (Market Risk); Market Capitalization, (Size Risk); and Value (High Book to Market Ratio Risk).

Small value stocks have the highest expected return. In Step 9, you will see the historical difference in the performance of various asset classes. Evidence of the accuracy of the Three Factor Model is represented below. Since 1927, the increased risk of owning small size and high value companies has delivered higher returns. Non-US and emerging markets provide opportunities for the diversification of more small and even more value-tilted stocks in addition to additional country risks which all results in higher cost of capital for those firms and higher expected returns for investors.

An investment like Index Portfolio 100 is highly volatile. The top chart below shows the annual returns over 50 years in the top graph and a histogram based on the average return and standard deviation on the bottom graph. The bottom chart shows a low standard deviation investment, Index Portfolio 10. Note the lack of volatility over the 50 years on the top graph and the narrow distribution of the bell curve on the bottom graph. The middle chart shows a more moderate volatility portfolio.




Another important statistical concept is the standard error of the mean, which indicates the degree of uncertainty in calculating an estimate from a sample, like a series of returns data. A standard error can be calculated from the standard deviation by dividing the standard deviation by a square root of n (with n representing the number of years measured). So with only 3 years of returns data on the S&P 500, the error in the average return is 2.6 times larger than having 20 years of data. See the Khan Academy video below for further description of the Standard Error of the mean.