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