Downside deviation, similar to semi deviation, eliminates
positive returns when calculating risk. Instead of using
the mean return or zero, it uses the Minimum Acceptable
Return as proposed by Sharpe (which may be the mean
historical return or zero). It measures the variability
of underperformance below a minimum targer rate. The
downside variance is the square of the downside
potential. To calculate it, we take the subset of returns that are
less than the target (or Minimum Acceptable Returns
(MAR)) returns and take the differences of those to the
target. We sum the squares and divide by the total
number of returns to get a below-target semi-variance.
$$DownsideDeviation(R , MAR) = \delta_{MAR} =
\sqrt{\sum^{n}_{t=1}\frac{min[(R_{t} - MAR),
0]^2}{n}}$$
$$DownsideVariance(R, MAR) =
\sum^{n}_{t=1}\frac{min[(R_{t} - MAR),
0]^2}{n}$$
$$DownsidePotential(R, MAR) =
\sum^{n}_{t=1}\frac{min[(R_{t} - MAR), 0]}
{n}$$
where $n$ is either the number of observations of the
entire series or the number of observations in the subset
of the series falling below the MAR.
SemiDeviation or SemiVariance is a popular alternative
downside risk measure that may be used in place of
standard deviation or variance. SemiDeviation and
SemiVariance are implemented as a wrapper of
DownsideDeviation with MAR=mean(R).
In many functions like Markowitz optimization,
semideviation may be substituted directly, and the
covariance matrix may be constructed from semideviation
or the vector of returns below the mean rather than from
variance or the full vector of returns.
In semideviation, by convention, the value of $n$ is
set to the full number of observations. In semivariance
the the value of $n$ is set to the subset of returns
below the mean. It should be noted that while this is
the correct mathematical definition of semivariance, this
result doesn't make any sense if you are also going to be
using the time series of returns below the mean or below
a MAR to construct a semi-covariance matrix for portfolio
optimization.
Sortino recommends calculating downside deviation
utilizing a continuous fitted distribution rather than
the discrete distribution of observations. This would
have significant utility, especially in cases of a small
number of observations. He recommends using a lognormal
distribution, or a fitted distribution based on a
relevant style index, to construct the returns below the
MAR to increase the confidence in the final result.
Hopefully, in the future, we'll add a fitted option to
this function, and would be happy to accept a
contribution of this nature.