(b) Consider a time period from 0 to T > 0 and suppose there is a risk free asset with return R0 over that period. A portfolio has known value V₀ at time 0 and random value V_T at time T The future net worth is X = V₁ V₀R₀. For the portfolio you must

(i) Define the risk measure Value-at-Risk in terms of V₀, V_T , and R₀,

(ii) Define the risk measure Expected Shortfall in terms of the Value-at-Risk,

(iii) Express the Expected Shortfall in terms of the quantile function for the dis-counted loss L = V₀ V_T /R₀.