VaR and CVaR are examples of βrisk measures.β We will use the notation π(π)
to denote a risk measure for random variable X. A risk measure is said to be
coherent if it satisfies the following properties for random variables X and Y:
(i) Monotonicity: if πΉβ1
π (πΌ) β€ πΉβ1
π (πΌ) for any value of πΌ, then π(π) β€ π(π)
(ii) Sub-additivity: |π(π + π)| β€ |π(π)| + |π(π)|
(iii) Positive homogeneity: if c is a constant, then π(ππ) = ππ(π)
(iv) Translation invariance: if b is a constant, then π(π + π) = π(π) βπ
Consider two identical but independent investments X and Y. Both investments have a
payoff of 0 with probability 0.96 and a payoff of -70 with probability 0.04. Suppose that
you put r% of your money in X and (1-r)% of your money in Y. The joint return of the
portfolio X+Y would then be r%(payoff of X) + (1-r)%(payoff of Y).