Verify that VaR and CVaR are both risk measures. (b) Verify that CVaR is coherent risk...

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Verify that VaR and CVaR are both risk measures. (b) Verify that CVaR is coherent risk measure (proving the sub-additivity is tricky, use Lemma 1 in L2) but VaR is not coherent (sub-additivity is violated, try to construct a counterexample using some discrete distributions). (c) Suppose XN(μ, o2), show that VaRa = μ-σ-¹(a), CVaRa = = −μ + − y($−¹(a)), where (), y() is the cumulative distribution function and density function of a standard normal distribution respectively. Verify that VaR and CVaR are both risk measures. (b) Verify that CVaR is coherent risk measure (proving the sub-additivity is tricky, use Lemma 1 in L2) but VaR is not coherent (sub-additivity is violated, try to construct a counterexample using some discrete distributions). (c) Suppose XN(μ, o2), show that VaRa = μ-σ-¹(a), CVaRa = = −μ + − y($−¹(a)), where (), y() is the cumulative distribution function and density function of a standard normal distribution respectively.