7. (Expected shortfall for a normal distribution) = (a) Show that the expected shortfall at the 99% level if losses follow a normal distri- bution with mean 0 and standard deviation 1 is given by ES0.99 2.667 using the fact that the density function for a normal with mean 0 and standard deviation 1 is f(x) = (1/2) exp(-x2/2), and the formula: * r exp(-r/2)dr = exp(-v/2). (b) Use the result of part (a) to find the 99% expected shortfall if the losses in project A have a normal distribution with mean -$5000 and standard deviation $3000 (thus on average we make a profit). 8. (Expected shortfall for a discrete distribution) An investment opportunity is estimated to have a 95% chance of a profit of $50,000, but a 5% chance of a loss. The best estimates are that the loss has a 1% chance of being 2 $10,000, 1% chance of being $20,000, 1% chance of being $30,000, 1% chance of being $50,000, and 1% chance of being $100,000. Calculate the expected shortfall at the 97.5% level.