In Modern Portfolio Theory, I often see that people seem to view Standard Deviation and Variance as equivalent. Example from Markowitz himself: "Thus far I have used the standard deviation (or equivalently, the variance) of return as a measure of the risk involved in the portfolio” (p.50, see reference below).
I have the following thoughts about this (in one case it is harmless/correct, but wrong in the second case):
If we do ordinary portfolio optimization, i.e. maximize the expected return for a given variance. Then variance and Standard Deviation is equivalent since there is a one-to-one correspondence between them. So it does not matter which measure of risk I use in this situation since we can go from one of the measures to the other in a unique way.
If we want to understand risk, I believe Variance and Standard Deviation is NOT Equivalent. This because Standard Deviation is in general sub-additive (i.e. Diversification never leaves you worse off) whereas Variance is in general NOT sub-additive (i.e. Diversification can leave you worse off).
- For a proof of the sub-additive properties of standard deviation and variance, see: https://courses.edx.org/c4x/DelftX/TW3421x/asset/coherence.pdf
So as a conclusion: Standard Deviation and Variance is NOT equivalent.
Am I overlooking something?
Source: [Markowitz, H. M. (1976). Markowitz revisited.Financial Analysts Journal,32(5):47–52]