Standard deviation,
as used by investors, is a statistical measure of the historical volatility
of a stock, mutual fund or portfolio, usually computed from a minimum
of 36 monthly returns. More specifically, it is a measure of the extent
to which numbers are spread around their average. It also quantifies
the uncertainty in a random variable, such as historical stock market
returns. To be precise, a standard deviation is the root-mean-square deviation of values from their average, or the square root of the variance.
Figure
8-1
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 ).
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.
