Refer to the attachment.

1. A premium principle is said to be sub-additive if for two risks X, and X2 (which may be dependent), Ix,+x2 - Ilx, + Ix, . Under what conditions is the variance principle sub-additive? 2. The mean value principle states that the premium, IIx, for a risk X is given by Ix = v-(E[v(X)1) where v is a function such that v'(x) > 0 and v"(x) > 0 for x > 0. (a) Calculate DIx when v(x) = x2 and X ~ y (2, 2). (b) Construct a counter example to show that this principle is not consistent. 3. Let X have probability function Pr(X] = 80) = 0.5 = 1 - Pr(X] = 120) and let X2 have probability function Pr(X2 = 90) = 0.6 = 1 - Pr(X2 = 120). An insurer has wealth 300 and calculates premiums using the principle ofzero utility with utility function u(x) = x - 0.001x2 for x