I'm currently looking into the mean-variance approach to portfolio theory and I wonder, why the standard deviation $\sigma$ is graphed on the x-axis and not the variance $\sigma^2$ as a measure of volatility (as the name would indicate). Does anybody know the reason for that?
Suppose you have a risk-free security R and a risky security B. A portfolio with a 0.50, 0.50 combination will have a standard deviation of $0.5 \sigma_B$, but a variance of $0.25 \sigma_B^2$. So if you draw it in Standard Deviation space it will be half way between R and B, in Variance space it won't be. This linearity is the reason it is more convenient to draw (std dev, return) space rather than (variance, return) space.
The locus of all combinations of R and B will be a straight line in the (std dev, ret) diagram, it will be a curve in the other diagram. Hence the former is usually preferred (although the other is also sometimes used).