# 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

## 1 Answer

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 Commented Nov 28, 2014 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? Commented Nov 28, 2014 at 7:58
• To show coherence, all 4 properties must be checked. en.wikipedia.org/wiki/Coherent_risk_measure#Properties Commented Nov 28, 2014 at 10:36