Semideviation Definition

Semideviation is a measurement of dispersion for the values of a data set falling below the observed mean or target value. Semideviation can be used instead of standard deviation or variance.

The formula for semideviation is:

Where:

• n = the total number of observations below the mean
• r_t = the observed value
• average = the mean or target value of a data set

In portfolio theory, semideviation evaluates the fluctuations in returns below the mean. It provides an effective measure of downside risk for a portfolio. It's similar to standard deviation, but it only looks at periods where the portfolio's return was less than the target or average level. This allows investors to see how much loss can be expected from a portfolio, instead of only looking at its expected fluctuations.

It was Markowitz and Roy who, in the 1950s, first applied downside risk measures to portfolio theory. They developed decision-making tools based on semideviation aimed at helping to manage risky investment portfolios.

Markowitz demonstrated how to exploit the averages, variances and covariances of the return distributions of assets of a portfolio, in order to compute an efficient frontier on which every portfolio achieves expected return for a given variance (i.e., risk level), or minimizes the variance for a given expected return. In Markowitz' explanation, a utility function, defining the investor’s sensitivity to changing wealth and risk, is used to pick an appropriate portfolio on the statistical border.

Roy, meanwhile, used semideviation to determine the optimum trade-off of risk-return. He didn't think it was feasible to practically model the sensitivity to risk of a human being with a utility function. Instead, he assumed that investors would want the investment with the smallest likelihood of coming in below a disaster level (or a target return.) Understanding the wisdom of this claim, Markowitz realized two very important principles:

• Downside risk is relevant for an investor.
• Return distributions might be skewed, or not symmetrically distributed, in practice.

As such, Markowitz recommended using a variability measure, which he called a semivariance, as it only takes into account a subset of the return distribution. In many functions like Markowitz optimization, semideviation may be used. Instead of the variance, or the full vector of returns, the covariance matrix can be constructed from semideviation or the vector of returns below the mean.

In semideviation, n is set to the full number of observations. In semivariance, n is the subset of returns below the mean. However, while this is the correct mathematical definition of semivariance, this result doesn't make any sense if you use the time series of returns below the mean or below a MAR to construct a semi-covariance matrix for portfolio optimization.