Exercise 4.3

(a) Give a financial interpretation for each of the following axioms:

(i) Translation invariance : ρ(X +

a) = ρ(X) +

a, for any a ∈ R.

(ii) Monotonicity : X ≤ Y ⇒ ρ(X) ≤ ρ(Y ).

(iii) Sub-additivity : ρ(λX + (1 − λ)Y ) ≤ λρ(X) + (1 − λ)ρ(Y ).

(iv) Positive homogeneity : ρ(λX) = λρ(X) for any λ ≥ 0.

(b) Are the following measures of risk coherent with the axioms?

(i) VaR.

(ii) ES.

(iii) WCE.

(c) Give an example of a peculiar coherent measure of risk. Are there better measures of risk?

(d) Give an interpretation of the simple order of stochastic dominance and give an example of a coherent measure of risk with respect to that order.

(e) Give an interpretation of the hazard rate order and give an example of a coherent measure of risk with respect to that order.