# Is volatility really a coherent risk measure?

Why people say that volatility is a coherent risk measure?

I don't see it clearly because what happen if the two assets are correlated positively? subadditivity would not be preserved.

That affirmation is in some papers online or even in this question

Well, if you assume $X$ has volatility $\sigma_X$ and $Y$ has volatility $\sigma_Y$, then

$$\sigma_{X+Y} = \sqrt{ Var( X + Y) } = \sqrt{ \sigma_X^2+\sigma_Y^2 + 2 \sigma_X \sigma_Y \rho }$$

Then, you want to show

$$\sigma_{X+Y} = \sqrt{ \sigma_X^2+\sigma_Y^2 + 2 \sigma_X \sigma_Y \rho } \leq \sigma_X + \sigma_Y$$

Squaring both sides:

$$\sigma_X^2+\sigma_Y^2 + 2 \sigma_X \sigma_Y \rho \leq \sigma_X^2 + \sigma_Y^2 + 2 \sigma_X \sigma_Y$$

Given the fact that, by definition, $\sigma_X \geq 0$, $\sigma_Y \geq 0$ and $\rho \in [ -1, 1]$, it looks to me that the property holds.

• Was a confusion from my part. If one would like to define a measure of risk as the variance, then that not be a coherent risk measure. Thanks – rlartiga Nov 28 '14 at 4:14
• This answer on subadditivity is great. But doesn't volatility lack the property of translation invariance and therefore it is not coherent? – Ric Nov 28 '14 at 7:58
• To show coherence, all 4 properties must be checked. en.wikipedia.org/wiki/Coherent_risk_measure#Properties – emcor Nov 28 '14 at 10:36