Semivariance of a normal distribution

Question

Semi variance is defined as : Link, (see Examples).As per the formula semivariance of N(0,sigma^2) distribution comes out to be sigma^2/2

However Based on what I have been reading in my finance books semi variance for standard normals is same as variance. So what is semivariance of a normal distribution?

Answer 1

Let's just deal with the standard normal distribution. Then, as I understand it, you are disregarding negative data values and taking the variance of only positive values.

Intuitively, I anticipate the mean and variance of such values to be the same as for the 'half-normal' distribution: $\sqrt{2/\pi}$ and $1 - 2/\pi,$ respectively. (Given what we know about the related integrals of ordinary standard normal distributions, I believe these values are not difficult to prove.)

A simulation in R with a million standard normal samples, each of size 100, gives the following results, verifying my guess (within 2 or 3 place accuracy):

m = 10^6;  sv = sm = numeric(m)
for (i in 1:m) {
  x = rnorm(100); x.pos = x[x >= 0] # truncate to non-neg values only
  sv[i] = var(x.pos) 
  sm[i] = mean(x.pos) }
mean(sm);  mean(sv)
## 0.7977938  # aprx E(semivar)
## 0.3633463  # aprx V(semivar)
sqrt(2/pi); 1 - 2/pi
## 0.7978846
## 0.3633802

Here is a histogram of the simulated distribution of semivariances, with a vertical red line at its mean.