Value at Risk: Coherent risk measure for normal distribution

Question

I know that there are cases where VaR does not satisfy the subadditivity property (coherent risk measure properties) for coherent risk measures. But I would like to show that in the case of normal distributions, VaR does satisfy this property and therefore is a coherent risk measure. However, I couldn't find a proof for this question so I wonder how this question could be approached.

Thank you!

Answer 1

For a normal return distribution, VaR can be defined as multiple of standard deviation and is subadditive.

If the returns $X$ and $Y$ are jointly normal with correlation $\rho$, then $X+Y$ is normally distributed.

Since $-1 \leqslant \rho \leqslant 1$, the standard deviation of the sum , $\sigma_{X+Y}$, is given by

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

With all return distributions normal, the same multiplier $\alpha$ is used to define VaR for a given confidence level, whence,

$$\alpha \sigma_{X+Y} = \text{VaR}(X+Y) \leqslant \text{VaR}(X) + \text{VaR}(Y)$$