How to Calculate Downside Deviation

In finance, deviation calculations are a great way to look at the volatility of an investment, meaning the extent to which that investment moves around in value over a period of time. Investments with a high amount of deviation are riskier, which can lead to a greater reward or greater loss. While standard deviation may seem like a straightforward way to calculate an investment's volatility, downside deviation can add a new chapter to that story to further explain the risk of losses to the investor.

The concept of standard deviation originated in statistics, where it's used to produce a value measuring the difference between individual data points and a calculated mean. If the data points are centered around the mean, deviance will be small; if data is more spread out from the mean, the standard deviation will be higher.

For example, a volatile stock will show a higher standard deviation whereas stable stocks should show a very low standard deviation. While standard deviation does tell you something about the volatility of an investment over a certain time period, the calculation considers both positive and negative changes with equal impact. In the investment realm, positive changes are actually good for the investor, whereas stability is not.

Downside deviation, also called downside volatility, corrects this by only looking at negative changes. The downside risk is related to the probability that an asset falls in price; this represents a potential loss to an investor based on past performance. Investors are usually only concerned with changes that reflect loss, and downside deviation allows them to compare values with other deviation calculations with a focus on volatility that negatively affects the stock.

Downside deviation doesn't always need to be calculated against a mean; since investments are expected to return a certain percentage, downside deviations can be calculated against that investor's expected return, or against the current risk-free rate, represented by short-term Treasury security investments. The calculation helps investors quantify worst-case scenarios and know what to expect.

To calculate downside volatility, you'll need to identify a minimum acceptable return (MAR) on your investment. In some circumstances, your MAR might be zero; in other cases, it may correspond to an industry-expected return like the risk-free rate, or it may be a value you define based on your expectations as the investor. In this example, we'll be assuming a MAR of ​5 percent return​ as a baseline and look at a period of ​six months​.

To measure the downside deviation (calculated monthly) over the last ​six months​, you'll need to know the investment's monthly returns for each month. These can be calculated using the downside deviation formula (also known as the downside volatility formula) (E - B + D)/B, where B is the price at the beginning of the month and E is the price at the end. D represents any dividends or interest that were paid out from the investment, such as stock dividends.

​Consider also:​ How to Calculate Stock Prices With Standard Deviations

For this example, assume a stock started the month at ​$10​ and finished it at ​$15​ with ​$1​ paid in dividends: the formula then gives ​($15 - $10 + $1)/$10​, which amounts to ​0.6​, or ​6 percent​. Complete this calculation for each month. For this example, we'll assume the monthly returns were ​6, -2, 2, 8, 0 and -3 percent​.

Identify any months that were less than the MAR. In this example, with the MAR at ​5 percent​, we would consider ​2, -2, 0 and -3 percent​. Then subtract the low returns from the MAR and square each result. For this example:

• ​0.05 - 0.02 = 0.03; squared is 0.0009​
• ​0.05 - (-0.02) = 0.07; squared is 0.0049​
• ​0.05 - 0.00 = 0.05; squared is 0.0025​
• ​0.05 - (-0.03) = 0.08; squared is 0.0064​

Sum these squares and divide it by the number of returns considered earlier. Since this example looks at ​six months​, take ​(0.0009+0.0049+0.0025+0.0064) = 0.0147​ and divide by ​6​ to get ​0.00245​. Then calculate the square root of this result; for this example, the square root of ​0.002033​ is ​0.049497​. Converted into a percentage, this example results in a downside deviation of ​4.95 percent​. This is very close to the MAR, which means the investor will need to consider risk, but may mean this is a favorable investment.

Standard deviation can be found by calculating the sum of ​(6, -2, 2, 8, 0, -3 percent) = 0.11 (11 percent)​ divided by ​6​, yielding ​0.01833 (1.833 percent​). Variance then subtracts the mean from each data point, which is then squared. In this example, we start with ​(0.06 - 0.01833) = 0.041​ squared to give ​0.001681 (0.1681 percent).​ Each squared value is added together and then divided by ​(6-1) = 5​ to give variance. Using the numbers in this example we get:

• ​1st month with 6 percent: 0.001681​
• ​2nd month with -2 percent: 0.001469​
• ​3rd month with 2 percent: 0.00000279​
• ​4th month with 8 percent: 0.0038​
• ​5th month with 0 percent: 0.00036​
• ​6th month with 3 percent: 0.0000195​

The sum is then ​0.0073​. This sum divided by ​(N-1) = 5 gives us 0.0014​; the square root of ​0.0014 is 0.0374, or 3.74 percent.​ This overall standard deviation can be compared to the downside deviation of ​4.95​ and the minimum acceptable return of ​5 percent​ to help an investor decide whether they want to put their money here or elsewhere.