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### Demonstrating that VaR (for worse enough outcomes) is a coherent risk measure for a Gaussian, i.e. multi-variate normal, distribution

For a risk measure to be coherent it must satisfy:

(a)    Subadditivity: for any pair of loss variables,  and

(b)   Monotonicity: if, for all states of the world,  then

(c)    Homogeneity: for any constant  and random loss variable

(d)   Translational invariance: for any constant  and random loss variable x

For a normal distribution the VaR at the  confidence level is as follows, if the distribution (for the given variable of interest) is distributed :

Homogeneity and translational invariance therefore immediately apply.

Subadditivity holds (as long as , i.e. )  because the standard deviation of the sum of two random variables is less than or equal to the sum of their standard deviations

Monotonicity holds because Normally distributed random variables have positive support on the real line, so  in all states of the world only if  and  are perfectly correlated, in which case , where  is constant, hence  for any .