The curriculum states that semivariance and variance are equivalent if distributions are symmetric. But here’s a problem: if we have the same denominator for semivariance and variance (and we do), any additional data in the numerator (and they are squared, i.e. only positive) would make variance increasingly different from semivariance.
A small experiment with a perfectly symmetric series of 5 returns: 2, 4, 6, 8, and 10. Arithmetic mean is 6; sample variance is 10, and sample semivariance is 5.
99.999999% the curriculum is right. But can anybody explain what’s going on in this case?
To calculate (lower) semivariance, some authors divide by the number of observations below the mean, while others divide by the total number of observations, both above and below the mean.
It would be nice if there were agreement on the correct divisor; there isn’t.
Thanks. Yet, as I understand, the purpose of semivariance is to describe downside risk, and this works both for symmetrical and skewed distributions. If semivariance described distribution in the same way as variance it would not be needed as such, right?
UPD: If a distribution is approximately symmetrical, then to find variance we just need to approximately double semivariance, or the other way around. Do you mean this kind of equivalence?