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Risk measures [26]

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Bullet points include: For (multi-variate) normally distributed random losses, VaR is coherent. Proof, recall that and verify axioms. Subadditivity holds because standard deviation of sum of two random variables is less than or equal to sum of the standard deviations. Monotonicity holds since normally distributed random variables have positive support on real line, so x1 > x2 in all states of the world if and only if x1 and x2 are perfectly correlated, when x1 = x2 + c, hence VaR(x1) = VaR(x2) + c for any alpha. Homogeneity and translation invariance are trivially satisfied. In fact, VaR coherent for a broader class of loss distributions, notably elliptic distributions. Special cases include multi-variate normal and Student’s t-distribution

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