Target Semideviation: Calculation and Use
Sometimes you want to just focus on the downside. Not to be pessimistic, but just to have a measure of dispersion on the low end of the distribution.
If you are familiar with the variance calculation, then there is only one change you need to make to compute __target semideviation__: rather than using all observations, you focus on observations below some target value.
Target semideviation works like this: start by considering all sample returns below some chosen target, _B_. The equation for target semideviation is then
$$ \displaystyle \sqrt{ \sum_{for \, all \, X_i\leq{B}}^{}\frac{(X_i-B)^2}{n-1}} $$
See if you can put all of that into context. The mean annual returns for TarSem in the past 10 years are shown in the table.
| Year | Annual Return |
|----|---------------|
| 1 | 26% |
| 2 | 25% |
| 3 | 24% |
| 4 | -30% |
| 5 | 5% |
| 6 | 42% |
| 7 | -18% |
| 8 | 32% |
| 9 | 36% |
| 10 | -22% |
Not exactly.
That would be a focus on downside risk, but since it incorporates observations below the mean, it does not restrict itself to purely negative returns; it would use all returns below 12% rather than just returns below 0%, which is the true target here.
Correct!
Target semideviation requires a target, and if the choice is to focus on just negative returns, then that target is 0%. So it will focus on only returns below zero.
For example, suppose throughout its history, TarSem Corp. has generally produced positive returns for its shareholders, and it has enjoyed a respectable mean annual return of 12% over the past 10 years. Occasionally, however, TarSem produces large negative annual returns.
If a TarSem analyst wanted to focus only on these negative returns, what would that analyst compute?
What intuitive comparison can you make between standard deviation and target semideviation in most cases?
That's right!
Never.
Note that both of these measures are the square root of summed deviations divided by _n_-1. That _n_ means the whole sample, not just the observations chosen. So any chosen target that limits the observations used will have to reduce the sum of squared deviations, and any target below the mean will also make those lowest values represent smaller deviations, making target semideviation smaller than standard deviation.
Given the analyst's focus, how many observations should enter the calculation for target semideviation?
Not exactly. The target semideviation uses only those observations below the target value and then compares those values with the target. A mean is not used, so many observations are ignored.
No, recall that the target value is 0%, so it's a matter of counting all negative values.
Exactly!
The three negative returns are those in focus. The target semivariance is then calculated as
$$\displaystyle \frac{(-0.30 - 0)^2 + (-0.18 - 0)^2 + (-0.22)^2}{10-1} = 0.018978 $$.
And what is the value of the target semideviation for TarSem?
No, this value is likely the result of using 2 in the denominator. But note that there are 10 observations, so the divisor should be 9 instead.
To summarize:
[[summary]]
No, that's not right. Don't forget to take the square root of the fractional value at the end.
Correct!
The target semideviation is calculated as:
$$\displaystyle \sqrt{\frac{(-0.30-0)^2+(-0.18-0)^2+(-0.22-0)^2}{10-1}} \approx 0.13776 \approx 13.8 \% $$
While this is an interesting metric for skewed distributions, it's a little more obscure and mathematically challenging than the traditional measures of variance and standard deviation, so you might not see it used in practice quite as often.
Measures focusing on all returns below the mean
Measures focusing only on returns below zero
Target semideviation should be smaller than the standard deviation
Target semideviation should be greater than the standard deviation
3
4
10
1.9%
13.8%
29.2%
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