Standard Deviation

Standard Deviation

Investors use standard deviation to measure the risk of a portfolio. It is an investment’s average variation from the average return. Knowing the standard deviation tells investors the amount of swing in performance that an investment can be expected to have from year to year.

It is also a measure of volatility: the more a portfolio’s returns vary from the average return, the more volatile it is. The higher the standard deviation, the greater the volatility, and therefore the greater the risk.

If a set of numbers is close to the average of those values, then you may expect to see a low standard deviation. In contrast, if the set of numbers is spread across a greater range, it may present a high standard deviation. Higher standard deviation is often interpreted as higher volatility. In comparison, lower standard deviation would likely be an indicator of stability. The most consistent values will usually be the set of numbers with the lowest standard deviation.

For example, Fund ABC has earned an average of 8% per year over the past 10 years, with a standard deviation of 6. Statistically, there is a 67% probability that returns will fall within 1 standard deviation, and a 95% probability that returns will fall within 2 standard deviations. Thus, 67% of the time, Fund ABC earned between 2% and 14%, and 95% of the time, the fund earned between -4% and 20%.

If two funds have the same average return, investors should prefer the one with the lower standard deviation. To calculate it, take the square root of the variance. The variance is a measure of how spread out a distribution is. It is computed as the average squared deviation of each number from its mean. For example, for the numbers 1, 2, and 3, the mean is 2 (1+2+3/3) and the variance is:

= (1-2) ² + (2-2) ² + (3-2) ²

3

= .667

The square root of .667 (the variance) is .8167; therefore, the standard deviation is .8167.

As another example, consider the following two portfolios and their respective returns over the last six months. Both portfolios increase in value from $1,000 to $1,058. However, they differ in volatility. Portfolio A's monthly returns range from -1.5% to 3% whereas Portfolio B's range from -9% to 12%. The standard deviation of the returns is a better measure of volatility than the range because it takes all the values into account. The standard deviation of the six returns for Portfolio A is 1.52; for Portfolio B it is 7.24.

PORTFOLIO A:

Month	Value	Return %	Final Value
January	$1,000	0.75	$1,008
February	$1,008	1.00	$1,018
March	$1,018	3.00	$1,048
April	$1,048	-1.50	$1,032
May	$1,032	0.50	$1,038
June	$1,038	2.00	$1,058

PORTFOLIO B:

Month	Value	Return %	Final Value
January	$1,000	1.50	$1,015
February	$1,015	5.00	$1,066
March	$1,066	12.00	$1,194
April	$1,194	-9.00	$1,086
May	$1,086	-4.00	$1,043
June	$1,043	1.50	$1,058

Another way to look at standard deviation is to think of it as a band of probabilities. The lower the standard deviation, the narrower the band and the lower the volatility of the numbers in the series.

Think of a series of numbers, the average of which is 30. If the series has a standard deviation of 10, that means two-thirds of the numbers in the series will fall within 10 of the average, in this case between 20 and 40.

If the standard deviation were 18 (and the average were still 30), then you would know that two-thirds of the numbers would fall between 12 (30 minus 18) and 48 (30 plus 18).

The second series of numbers would be much more volatile than the first, even though their averages were identical. Therefore, the lower the standard deviation, the lower the risk. The biggest problem with standard deviation is that it makes no distinction between upward volatility (the type investors want) and downward volatility (the type investors don't want).

In reality, the best way to measure the risk of an investment is by studying what happened in the worst periods that investors had to endure.