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## Abstract

Within statistics, semistandard deviation is a well-known measure used to analyze the dispersion of probability distributions. In finance, semistandard deviation of returns is sometimes defined consistently with its statistical definition, but it is sometimes defined differently. The ambiguity emanates from whether the number of observations in its calculation is specified as *T*, the total number of observations in a sample, or *T**, the number of negative deviations. The authors show that the use of *T* is consistent with the statistical definition but generates a measure that cannot be directly compared to standard deviation. Practitioners should be aware of the implications of using either *T* or *T** both as a stand-alone risk measure and as the denominator of the Sortino ratio. The authors derive an alternative measure of downside risk based on *T** that provides several advantages over semistandard deviation. They term that measure *semivolatility* and demonstrate its usefulness.

Semivariance and semistandard deviation are measures of dispersion that are used both in statistics and in finance. Semistandard deviation, or *downside risk*, is used both as a stand-alone risk measure and as the denominator of the *Sortino ratio*. The Sortino ratio, a widely used risk-adjusted performance measure similar to the Sharpe ratio, is designed to provide better performance measurement through its focus on downside risk.

Statisticians use semivariance for purposes that differ markedly from those of finance practitioners who analyze return distributions. Finance practitioners desire a measure of downside risk, based on a sample of returns, that can be (1) intuitively understood, (2) used directly to compare downside risk from different assets, and (3) easily used with common statistical significance testing. Semistandard deviation as defined by statisticians fails in all three regards.

Semivariance and semistandard deviation as used in statistics are general measures that are not used simply to measure dispersion below a distribution’s mean or some threshold. Rather, statisticians use semivariance and semistandard deviation as measures of dispersion within generalized ranges and view semivariances as components of total variance. Finance practitioners use semivariance and semistandard deviation to measure the dispersion of the *downside *of a distribution below the distribution’s mean or below a specified threshold.

Using a sample of *T* returns as an illustration, semivariance for a sample relative to its mean is generally defined by statisticians as shown in Equation (1):

where *T* is the number of total observations in the sample and
is the sample mean. In many investment applications,
can be replaced with a threshold value such as a benchmark return.

Semistandard deviation is defined as the square root of semivariance:

2Statisticians define *T* in Equations (1) and (2) as the total number of observations in the sample, not simply the number of negative deviations (i.e., not the number of observations less than the sample mean or less than a specified threshold). As we will discuss, practitioners of finance often define *T* as being equal to only the number of deviations below the mean or below a specified threshold.

The advantage of the statistical definition of semivariance is that the variance is equal to the sum of the downside’s semivariance and the upside’s semivariance; more generally, statisticians view a distribution as having regions or ranges with two or more semivariances that sum to the total variance. The primary problem with the definition of semistandard deviation used by statisticians is that its value cannot be directly and easily compared to the magnitude of the overall standard deviation. For example, in the case of a symmetrical distribution, the semivariance will tend to be half of the total variance, and the semistandard deviation will tend to be times the total standard deviation. Thus, a symmetrical distribution of returns with a standard deviation of 40% will have a semistandard deviation of approximately 28%. A value of 28% for semistandard deviation in the case of an asset with a symmetric return distribution and a total standard deviation of 40% provides little information to a practitioner. Worse yet, and as we will demonstrate later in this article, the magnitude of the difference between the semistandard deviation and total standard deviation in a sample is driven by the percentage of the total observations that lie below the sample’s mean (or a user-supplied threshold).

In finance, most authors cite the statistical formulas in Equations (1) and (2) (which are based on the total number of observations) in computing downside risk (e.g., Lhabitant [2004]). However, practitioners often define *T* as being only the number of negative deviations. This alternative risk measure is also referred to as semistandard deviation. In this interpretation of the formula for semistandard deviation, the total number of observations, *T*, is replaced with the number of downside deviations, *T**, as shown in Equation (3):

where *T** is the number of observations in the sample for which
.

We do not label the measure in Equation (3) as semistandard deviation (even though practitioners often do) because it is not consistent with the more common usage of the term in statistics. The primary advantage of using the risk measure in Equation (3) as a measure of downside risk is that it offers finance practitioners a measure of dispersion that has a scale that is roughly comparable to the scale of the overall standard deviation. The main idea is that a symmetrical distribution with a standard deviation of 40% would generate a downside risk measure, using the formula in Equation (3), approximately equal to 40%. A distribution skewed to the left (downside) with a total standard deviation of 40% would tend to have a measure, again using Equation (3), that exceeds 40%. The same cannot be said of semistandard deviation as defined in Equation (2).

**DERIVATION OF SEMIVOLATILITY**

The risk measure depicted in Equation (3) has four problems. First, it causes confusion when referred to as semistandard deviation because it conflicts with the term as used in statistics. Second, it is a biased and unintuitive measure of what most financial practitioners wish to know. Third, it will vastly understate significance if applied in common statistical tests as a standard deviation. Finally, it is not defined for a sample with only one negative deviation. We propose an improved measure that provides greater clarity, intuition, and comparability. We do so with the following crucial premise: Most financial practitioners seek an indication of the downside risk of an actual sample of returns that would be the same as the standard deviation of a symmetrically distributed hypothetical sample with deviations on each side of the mean that match the downside deviations of the actual sample being analyzed.

To illustrate clearly, consider an “actual” sample of *T* returns for which a practitioner desires to compute a measure of downside risk based on the aforementioned premise. For simplicity, let us assume that the sample mean is zero and there are only 10 observations, four of which are downside deviations with values of -1%, -2%, -5%, and -10%. Our premise is that financial practitioners can benefit from a downside risk measure that matches the standard deviation of a symmetrical sample with the following eight observations: +10%, +5%, +2%, +1%, -1%, -2%, -5%, and -10%. We derive a new downside risk measure called *semivolatility* for that purpose.

To state the procedure more generally, consider a sample that has *T** observations below its mean or threshold return. The hypothetical distribution is created as follows:

1. Identify the

*T** negative deviations of the actual sample below its sample mean (or a user-supplied threshold).2. Eliminate all positive deviations from the sample.

3. Replace the positive deviations with the absolute values of the negative deviations.

4. Compute the standard deviation of the hypothesized (and symmetrical) sample as a measure of the downside risk of the actual sample.

The resulting (“hypothetical”) sample is symmetrical, with the upside forming a mirror image of the actual distribution’s downside. This approach assumes that the practitioner seeks to know the standard deviation of the returns of this hypothetical sample, in which both sides of the distribution are identical to the downside of the “actual” sample.

The term we entitle semivolatility is distinguished from semistandard deviation to clearly signal that it needs to be interpreted differently from semistandard deviation and to avoid criticism that the risk measure is inconsistent with statistics. Its formula can be derived as follows.

The standard deviation of the hypothetical sample is given by Equation (4):

4where *T* is the total number of observations in the hypothesized sample.

Separating the summation of Equation (4) into positive and negative deviations generates Equation (5):

5Noting that both summations are equal (because the sample is symmetrical), the result can be written as

6The value inside in Equation (6) can be factored as shown in Equation (7):

7where *T** = *T*/2 = the number of negative deviations in this symmetrical hypothetical sample.

Equations (6) and (7) then can be used to compute our final risk measure, semivolatility:

8where *T** is the number of observations less than the mean (or threshold).

Semivolatility addresses all of the problems with semistandard deviation discussed in previous sections: It is the same scale as total standard deviation, and it is defined for *T** = 1.

**ANALYTICAL ANALYSIS**

The relationship between the semistandard deviation in Equation (2) and the semivolatility in Equation (8) is driven entirely by the number of observations below the mean (or threshold) and is given in Equation (9):

9For a symmetrical sample or any other sample with *T** = *T*/2, Equation (9) reduces to the semistandard deviation being equal to the semivolatility divided by
Note that the relationship between semistandard deviation and semivolatility depends only on the relationship between *T* and *T**. Skew and kurtosis do not change the ratio between the semistandard deviation and the semivolatility.

Exhibit 1 demonstrates the relationship between the ratio of semistandard deviation to semivolatility and the ratio *T** to *T*. The Exhibit reflects a sample with 200 observations, but analogous diagrams based on any sample of 30 or more observations generates highly similar results. Note from Exhibit 1 that, when half of the observations are below the mean (i.e., 0.50 on the horizontal axis), the ratio of the semistandard deviation to the semivolatility is 0.707 (i.e.,1/
). However, when the downside sample size differs substantially from equaling half of the total sample size, the semistandard deviation changes when viewed as a proportion of the semivolatility.

Although downside samples based on the sample mean will tend to be approximately 50% of the size of the full sample, the semistandard deviation likely will be substantially affected when the semistandard deviation is calculated relative to a threshold value that differs substantially from the sample mean. Thus, interpretation of a particular value of semistandard deviation relative to total standard deviation requires complex adjustment for the percentages of observations in the downside sample. The effects of downside sample size on the value of semistandard deviation therefore can be problematic when the semistandard deviation is used to assess downside risk relative to standard deviation. The effects can also be problematic when semistandard deviations from different assets are compared in cases in which the assets have downside sample sizes that differ as a proportion of total sample size. Thus, semistandard deviation can fail to provide clear indications of downside risk in both the analysis of one asset and in the comparison of multiple assets.

**DOWNSIDE RISK AND THE SORTINO RATIO**

The Sortino ratio is similar to the Sharpe ratio; however, the denominator in the Sortino ratio is downside risk, whereas the denominator in the Sharpe ratio is total risk (i.e., the standard deviation of returns). As the sample size approaches infinity and in the case of symmetrical distributions, the sample semivolatility approaches the population standard deviation, and the Sortino ratio (of a symmetrical distribution) using semivolatility as the measure of downside risk will tend to be equal to the Sharpe ratio. However, the sample semistandard deviation approaches the population standard deviation divided by in the case of a symmetrical distribution. Therefore, the Sortino ratio of a symmetrical distribution will tend to be equal to the Sharpe ratio times if semistandard deviation is used in the denominator; the Sortino ratio of a symmetrical distribution will tend to be equal to the Sharpe ratio if semivolatility is used in the denominator. The Sharpe ratio will be greater than, equal to, or less than the Sortino ratio using semivolatility to the extent that the return distribution has downside risk that is greater than, equal to, or less than its upside risk, respectively.

These relationships highlight a major benefit of using semivolatility as the measure of downside risk in the Sortino ratio rather than semistandard deviation: By using semivolatility in the denominator of the Sortino ratio, an analyst can expect the ratio to be comparable to a Sharpe ratio in magnitude when the sample is approximately symmetric and to be substantially less than the Sharpe ratio when the sample is negatively skewed.

**STATISTICAL TESTING**

The improved intuition of semivolatility is its ability to be more directly compared and contrasted with total volatility, as mentioned earlier. Analysts tend to have an intuitive understanding of total volatility, such as an understanding that the annualized volatility of the returns of the US stock market tend to range between 10% and 20% and that volatility as a measure of return dispersion very roughly approximates the mean absolute deviation. Semivolatility is much better than semistandard deviation at preserving these interpretations.

This section discusses statistical testing of semivolatility and semistandard deviation. It is common to interpret total standard deviation in the context of the normal probability distribution. For example, analysts speak of the likelihood of *two sigma events*, and they understand that roughly two-thirds of the outcomes will lie within one standard deviation of the mean if the sample is roughly normally distributed. Semivolatility is more conducive to traditional statistical interpretations than is semistandard deviation.

Downside risk measures are used in place of or in addition to total standard deviation when a distribution might have a substantial skew. In the case of a distribution with a downside skew, the use of total standard deviation (and the assumption of normality) will tend to cause underestimation of the likelihood of various degrees of negative outcomes. Using semivolatility in place of volatility will tend to generate better estimates of the likelihood of negative outcomes.

**EMPIRICAL ANALYSIS**

Exhibit 2 reports a variety of statistics for the reported returns of a number of securities over the interval of April 11, 2006 to December 31, 2015. The starting date was chosen based on availability of the exchange-traded fund GLD and the desire to span the time period of the financial crisis of 2007–2009. The underlying data series reflects daily total returns based on closing prices of exchange-traded funds (ETFs). The ETFs were selected based on data availability and to represent diverse asset categories.

As discussed in the previous section, the relationship between semistandard deviation and semivolatility is driven entirely by the relationship between *T** and *T*. The primary purpose of Exhibit 2 is to highlight the relationship between semistandard deviation and total standard deviation (in the second to last column) and the relationship between semivolatility and total standard deviation (in the last column) based on actual asset returns with various levels of skew and kurtosis and with different percentages of observations below the sample mean.

Note that some skews are positive and some negative for the ETFs in Exhibit 2. The positive skews are generally attributable to the very large positive returns during the high volatility in autumn 2008 (i.e., the worst of the financial crisis). The last two columns denote the ratio of the two downside risk measures (semistandard deviation and semivolatility) to the standard deviation of the entire samples. Those columns demonstrate that the semistandard deviation is consistently and substantially much lower than the standard deviation for the entire sample, regardless of the sample’s skew. This result demonstrates the inadequacy of semistandard deviation (using the total number of observations) as an intuitive indication of downside risk relative to total standard deviation. In other words, the magnitude of the semistandard deviation for a particular asset relative to total standard deviation does not provide a clear signal regarding the asset’s downside risk. In contrast, the relationship between semivolatility and total standard deviation is strongly related to the skew and/or kurtosis of the asset.

Exhibit 3 depicts the same analysis on the same daily asset returns for the interval from January 4, 2010 to December 31, 2015 to avoid covering the period of the financial crisis of 2007–2009. Almost all skews are negative, and the semistandard deviation is still consistently and substantially lower than the standard deviation for the entire sample. The ratio of the semivolatility to the total standard deviation continues to provide an indication of an asset’s downside risk that is consistent with the indications provided by analysis of the skew and kurtosis.

The semivolatility values given in Exhibits 2 and 3 provide a much more intuitive measure of downside risk because they can be directly compared to a total standard deviation. An analyst who notes that the semivolatility exceeds the asset’s total standard deviation (i.e., volatility) receives a clear signal that the asset has higher downside risk than a symmetrical distribution with the same total standard deviation of returns. Note also that semivolatility lends itself to other applications, such as the estimation of value at risk (VaR) and tail risk probabilities based on the normal distribution. In both cases, an analyst who attempts to measure the size of an N-sigma event would vastly underestimate tail risk if using semistandard deviation. More generally, the use of semivolatility to measure N-sigma events should be superior to the use of total standard deviation.

Exhibits 2 and 3 indicate that a financial analyst who observes the semivolatility alongside the total volatility has an intuitive feel for the downside risk exhibited by the sample. Skew and kurtosis also provide indications of downside risk, but they do so with deviations raised to the third and fourth powers, respectively, which are more subject to large outliers than semivolatility, which is based on deviations raised only to the second power. Furthermore, although a measure of skew or kurtosis can be used to indicate downside risk, the advantage to using semivolatility is that it provides the analyst with an intuitive measure of the magnitude of the downside deviations. Specifically, the semivolatility of a sample, like the total standard deviation of a sample, somewhat approximates the sample or subsample’s mean absolute deviation. In contrast, the semistandard deviation is rather useless in this context because its value is driven by the extent to which *T** is less than *T*.

**SUMMARY AND CONCLUSION**

We propose the use of semivolatility as a measure of downside risk because it offers several benefits. Semivolatility provides (1) a measure of downside risk with no ambiguity with regard to the number of observations to be included in its computation, (2) a measure of downside risk that can be intuitively interpreted relative to total volatility, (3) a better comparison of downside risk across assets with different proportions of downside sample sizes relative to total sample sizes, (4) a better estimate of downside risk for use in VaR computations or N-sigma event estimation, and (5) an unbiased measure of dispersion that is defined in the case of a sample with only one negative deviation.

Although this article has emphasized the analysis of downside risk measures in the context of deviations relative to the mean of a sample, the results are even more valid for measures of downside risk relative to a threshold other than the sample mean. The use of semivolatility in place of semistandard deviation could also enhance the value of ratios such as the Sortino ratio and the usefulness of portfolio management models that rely on semistandard deviation as a measure of risk.

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