Suppose it is desired to minimize expected shortfall and the current market price for a futures contract

(CFD) is 114.5 per MWh for peak electricity (i.e. in the period 8 am to 8 pm). This is a purely financial

contract. Assume the same pattern of outages as in Part 3 (In Part 3 we just calculated Value at Risk and Expected Shortfall when 4% of days in a year are outage - generation is halved).

(A) What quantity of electricity futures would you recommend buying (or selling) in order to

minimize the 95% expected shortfall for daily profit? (20%)

(B) Now assume that there are also gas futures available at a price of 44.5 per MWh. We need to

determine both a quantity of electricity futures , and a quantity of gas futures. What

quantities would you recommend buying (or selling) in order to minimize the 95% expected

shortfall for daily profit? (15%)

(C) Can you make any comments on the values you end up with for and in parts (A) and (B)

above, and why these values are optimal? (10%)

This part involves a minimization over three parameters, the quantity of futures contracts for gas ,

the quantity of futures contracts for electricity , and a parameter . Here will end up being the

Value at Risk. We use the minimization form of the expression for expected shortfall. Thus we calculate

for each simulated day , the value max ( , 0), where the loss depends both on the operation

and prices as in part 3, and also on the money gained or lost on financial contracts, and hence on the

values of and . Then we minimize the expression

+ ( 1/0.05) (1/) max ( , 0)

over choices of , and for part (A) and add in part (B)

We need to pay attention to the sign of the quantities which will determine whether the generator is

buying or selling the contracts.

Please explain how to do the optimisation in R, if possible.