CFA Institute Journal Review November 2013 Volume 43 Issue 4
What’s Wrong with Multiplying by the Square Root of Twelve (Digest Summary)
Journal of Performance Measurement
The motivation to multiply the standard deviation of monthly returns by the square root of 12 to express it in the same unit as annual return is not clear, and this approach introduces a bias. Two alternative measures of return volatility may offer a better approach.
Standard deviation, a commonly used measure of return volatility in annualized terms, is obtained by multiplying the standard deviation of monthly returns by the square root of 12. The author illustrates the bias introduced by using this approach rather than the correct method and presents two alternative measures of return volatility in which multiplying by the square root of 12 is appropriate to annualize the monthly measure.
How Is This Research Useful to Practitioners?
Portfolio managers, performance analysts, and investment consultants commonly use standard deviation in annualized terms as a measure of return volatility. Annual return is a product of monthly returns rather than a sum of monthly returns. Thus, multiplying the standard deviation of monthly returns by the square root of 12 to get annualized standard deviation cannot be correct. The bias from this approach is a function of the average monthly return as well as the standard deviation. Extreme biases at extreme average returns reflect the asymmetrical nature of return distributions.
The author derives a new formula using monthly standard deviation and monthly average return to calculate the correct value of annualized standard deviation. The result can be quite sensitive to the average monthly return because of the intrinsic asymmetrical nature of return distributions.
The author presents two alternative measures of return volatility whose monthly values can be annualized by multiplying by the square root of 12 without introducing any bias. The first alternative measure is to sum monthly logarithmic return relatives (i.e., returns plus 1) to arrive at annual logarithmic return relatives. Because an annual logarithmic return is the sum of its monthly constituents, multiplying by the square root of 12 works. The second alternative measure of return volatility involves estimating the logarithmic monthly standard deviation by using monthly average return and monthly standard deviation. Thus, the obtained monthly standard deviation can be multiplied by the square root of 12 to obtain the annualized standard deviation.
How Did the Author Conduct This Research?
To demonstrate the extent of bias in the annual measure of standard deviation obtained by multiplying the monthly measure by the square root of 12, the author uses a monthly return series with a standard deviation of 6%. A plot of monthly average return versus the difference between the correct value of annual standard deviation and the annual measure of standard deviation obtained from multiplying the monthly measure by the square root of 12 shows extreme biases at extreme returns.
The author calculates direct and estimated logarithmic standard deviations using returns for 1,824 Canadian open-end funds for the 60-month period from November 2007 to October 2012. A graph of direct versus estimated logarithmic standard deviation shows less than ±1% difference between the two values for 96% of the funds, which validates the formula that uses monthly standard deviation and monthly average return to calculate annualized standard deviation.
Despite being mathematically invalid, the most common method of annualizing the standard deviation of monthly returns is to multiply it by the square root of 12. The author suggests that it may be more appropriate to measure the volatility of annual logarithmic return rather than level returns because annual logarithmic return is the sum of its monthly constituents, thus making multiplication by the square root of 12 appropriate. In my view, it is important for asset managers to encourage the use of mathematically sound procedures for calculating the annualized volatility measure rather than to opt for an expedient but mathematically invalid procedure.