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1. Calculate the arithmetic mean return for the asset over some time period.
2. Subtract each individual return from the mean return to get the deviations.
3. Square each deviation, sum the deviations, and divide this sum by the number of
observations minus 1, producing the variance.
4. Take the square root of the variance, which produces the standard deviation of returns for
the asset.
The formula for standard deviation of returns is:
where σ is the standard deviation of returns, R i is each individual return, R is the mean return, and n
is the number of observed returns.
With a series of returns for any asset, the standard deviation of returns for the asset can be
calculated with this formula. The Center for Research in Security Prices (CRSP) at the University of
Chicago captures and reports returns for many investment securities, and CRSP data is frequently used
by academics and practitioners when researching investment returns.
We will illustrate the calculation and interpretation of standard deviation of returns using common
stock as the asset, specifically the common stock of the Walt Disney Company and IBM. The annual
returns earned on the common stock of Disney and IBM, as captured and reported by CRSP, are
shown in Exhibit 5.4.
Based on these 10 years of data, the mean annual return and standard deviation of annual returns for
the common stock of Disney are 3.12% and 25.17%, respectively. For the common stock of IBM, the
mean annual return is 12.54% and the standard deviation of annual returns is 32.43%. These results
show the common stock of IBM has a higher standard deviation of annual returns than the common
stock of Disney. Since standard deviation of returns is the appropriate measure of stand-alone risk,
IBM‟s common stock is riskier than Disney‟s common stock for investors who choose to hold all of
their wealth in a single asset. A stand-alone investment in IBM common stock is more variable and
volatile. In up years, IBM‟s stock performs better than Disney‟s stock, but in down years IBM‟s stock
performs worse than Disney‟s stock. IBM‟s stock does have a higher mean annual return than
Disney‟s stock, but IBM‟s stock is also riskier, reflecting the positive relationship between risk and
return.
Exhibit 5.4 Annual returns of Disney and IBM, 1998-2007.
Source: Center for Research in Security Prices, Graduate School of Business of the University of
Chicago.