Standard Deviation

The statistical measurement of dispersion about an average, which depicts how widely a stock or portfolio's returns varied over a certain period of time. Investors use the standard deviation of historical performance to try to predict the range of returns that is most likely for a given investment. When a stock or portfolio has a high standard deviation, the predicted range of performance is wide, implying greater volatility.

If the returns for a stock or portfolio follow a normal distribution, then approximately 68 percent of the time they will fall within one standard deviation of the mean return, and 95 percent of the time within two standard deviations. For example, if the mean annual return is 10 percent and the standard deviation is 2 percent, you would expect the return to be between 8 and 12 percent about 68 percent of the time, and between 6 and 14 percent about 95 percent of the time.

Morningstar calculates standard deviation for stocks and portfolios using the trailing monthly total returns for the appropriate time period. All of the monthly standard deviations are then annualized.

Two equations make up our current standard deviation calculation. The first part calculates the monthly standard deviation. The second part annualizes the monthly number to put it in a one-year context.

Monthly Standard Deviation

where

X1 = return for first month

Xi = return for the ith month (each month between the first and last month)

Xn = return for the last month

n = total number of returns being used, in this case 36.

= average monthly total return during the 36-month period, also called the arithmetic mean. This number is arrived at by adding together all 36 monthly returns for the fund and dividing by 36.

Annualized Standard Deviation

where

sM = Monthly standard deviation

Ri = Return of the portfolio in month i

n = Number of periods

= Average monthly total return for the portfolio

is also called the arithmetic mean, and it is calculated by adding together all the monthly returns for the portfolio and dividing by the number of months.

Morningstar annualizes the monthly standard deviation by multiplying it by the square root of 12.