Assume that a random loss L has a continuous distribution with the density p(1) > 0, for 1 (, + Suppose that a confidence level ,0 < < 1, is given. Consider the following function 0)) (a) = E | max(L a, 0) (a) (3 marks) Show that f(a) is continuously differentiable with ' (a) = - (for p(l)al) Hint: assuming a(x), g(x, y) are continuously differentiable, Leibniz's rule states that + 1 ( g(x, y)dy) - d dx 1(x) d g(x, y)dy) = g(x, a(x)) = a(x) + [to (b) (3 marks) What is f" (a)? Is f(a) convex? Explain. . dx -g(x, y)dy (c) (4 marks) Let 0 < < 1 be given. Is the following optimization problem convex? Explain. mina + 1 _ 1 B Emax(L -, 0) (0)) Assume that a* is the solution to (11). Show that a* is the VaR with confidence B. (1: