Figure 8-1 illustrates standard deviation. One standard deviation away from the average in both directions on the horizontal axis (the green area) accounts for approximately 68% of the annual returns in the time period. Two standard deviations away from the mean (the green and blue areas) account for approximately 95% of the annual returns. And three standard deviations (the green, blue, and red areas) account for approximately 99.7% of the outcomes, or for returns in a certain period for investments. For normal distributions, the two points of the curve which are one standard deviation from the mean are also the inflection points, which is a point on a curve where the curvature changes signs.

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 returns 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 average return and the uncertainty of future returns. Standard deviation provides a quantified estimate of the uncertainty of those future returns.

For example, let's assume an investor had to choose between two stocks.

1) The Walt Disney Company (DIS) over the last 20 years (1988-2007) had an average annualized return of 10%, with an annualized standard deviation of 26% (a return/risk ratio of 0.385) or

2) Hewlett Packard (HPQ), over the same period, had average annualized return of 12%, but a higher annualized standard deviation of 36% (a return/risk ratio of 0.333) (see the chart below ).

Figure 8-1a
Enlarge

On the basis of risk and return, an investor may decide that The Walt Disney Company is the better choice, because Hewlett Packard's additional 2% points of return is not worth the additional 10% standard deviation (greater risk or uncertainty of the expected return). Hewlett Packard has a lower risk/reward ratio (0.333 for HP versus 0.385 for Disney) and is more likely to fall short of the initial investment under the same circumstances, and is estimated to return only 2% more on average. In this example, The Walt Disney Company is expected to earn about 10%, plus or minus 26% (a range from 36% to -16%), about two-thirds of the future annual returns. When considering more extreme possible returns or outcomes in the future, an investor should expect results of up to 10% plus or minus 78% (3 x one standard deviation), or a range from 88% to -68%, which includes outcomes for three standard deviations from the average return (about 99.7% of probable returns).

Actually, neither stock is a good choice because an S&P 500 index fund over the same period earned 11.63% annualized average return with a much lower standard deviation of 13.48% (a higher return/risk ratio of 0.863), because of the diversification benefit of 500 stocks versus one stock, making it always the statistically preferred investment over individual stocks. If you diversified into other global markets and small value indexes, in addition to the S&P 500, like Index Portfolio 100, the average annualized return in this same period was 13.39%, with a standard deviation of 13.49% (a very high return/risk ratio of 0.993), making it always the statistically preferred investment over the S&P 500 and anything else that we can find.

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 repesented 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 90 is highly volatile. Figure 8-2 shows the annual returns over 35 years in the top graph and a histogram based on the average return and standard deviation on the bottom graph. Figure 8-3 shows a low standard deviation investment, Index Portfolio 5. Note the lack of volatility over the 35 years on the top graph and the narrow distribution of the bell curve on the bottom graph.

Figure 8-2

Figure 8-3

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.

8.2.2 Probability Distributions and Histograms

Figure 8-4

A probability distribution is a mathematical function that describes the probabilities of possible outcomes. For example, if two dice are rolled, the range of possible outcomes or the possible results of the dice toss are two through 12. The corresponding probability distribution for the dice toss is reflected in Figure 8-4.

The mean of a probability distribution is its average or expected value. Figure 8-5 shows a distribution of 600 monthly returns of Index Portfolio 90, which is a histogram of simulated past results over the last 50 years. The average monthly return was 1.14% and the monthly standard deviation was 4.01%. Based on the average return and standard deviation of long-term historic data, a probability distribution of future outcomes can be estimated. Figure 8-6 provides a comparison of a more narrow histogram of monthly returns. The lower risk level of Index Portfolio 30 is illustrated in the narrower range of past monthly returns.

Figure 8-5

Figure 8-6

8.2.3 Mean Reversion

Figure 8-7

Mean reversion is a tendency for certain random variables to remain at or return over time to a long-run average level. For example, interest rates and broadly diversified indexes tend to be mean reverting. Individual stock prices, mutual fund manager performances, and short-term market performance tend not to be mean reverting. Large diversified portfolios of stocks, such as the CRSP 1-10 Total Market Index or the S&P 500 Index rates of return tend to be mean reverting. They may be high or low from one year to the next. However, over 80 years, the rate of return of the S&P 500 has tended to average in the 10% range plus or minus 20% two-thirds of the time.

It is impossible to tell for certain if a variable is mean reverting by looking at its performance over any short period of time, such as a three or five-year track record of a stock, time, manager or style. This is because a tendency toward mean reversion may only reveal itself over very long horizons of 20 years or more. Figure 8-7 illustrates the difference between mean reverting and non-mean reverting outcomes.

8.2.4 Expected Return

Figure 8-8

Expected return refers to the expected or average rate of return of an investment. The term refers to a theoretical future performance, and it is definitely not a guarantee. The expected return of an index can only be derived from its very long-term historical past performance. Because returns are full of uncertainty, higher variables and the actual return for short periods of time is unpredictable, but the expected return remains the same. Only the uncertainty or standard deviation of expected returns changes with time.

An investment’s expected return is simply the middle value of the probability distribution or bell-shaped curve, which is shown as 5% in Figure 8-8. Investors are constantly surprised by short-term results, which will look nothing like the distribution below. In practice, sophisticated investors often base their expected return and volatility assumptions on historical returns of 20 years or more of an index or asset class.

8.2.5 Rolling Period Returns

One problem for investors is the high level of error when drawing conclusions from the small sample of stock market data. One way to get an improvement in the number of observed periods is to create rolling periods with monthly or annual data that overlap from one period to the next. The overlapping period returns do have some statistical problems because each period contains a substantial amount of the same data. But, since monthly returns are random and have no correlation to each other, a rolling period of 144 consecutive months out of 600 months should be similar to a random sample of 144 months out of 600 months. This random sampling is another methodology of looking at returns called bootstrapping. A third method that is used to increase sample size is Monte Carlo simulations, which simulate future outcomes of portfolios based on the historical average and standard deviation.

Figure 8-9 illustrates how rolling periods are obtained using monthly frequency. As you can see Period #1 is the 12 years from January 1957 to December 31, 1968. Period #2 starts one month later on February 1, 1957 to January 31, 1969. Imagine this occurring 457 times. This analysis helps capture the various experiences of 12-year investors who start their investments in July 1957, January 1963 or just about any month within the 50 years. You may notice that those investors who started in January 1963 experience the single lowest 12-year annualized return of 7.56%.

Table 8-1 is a rolling period analysis of Index Portfolio 100. If you look at the red highlighted row, you will see that it covers 12-year rolling periods (144 months each) and in the period from January 1955 to December 2006, there are 457 12-year rolling periods. As you read across the red highlighted row, you can see lots of interesting data about those rolling periods.

Figure 8-9

Table 8-1