An eum customer of an insurance company has a total wealth of 100 (in some units) and is facing a random loss ? distributed as follows: p(? = 0) = 0.9, p(? = 50) = 0.05, p(? = 100) = 0.05.
(a) let the utility function of the customer be u(x) = x ? 0.005x2 for 0 ? x ? 100. graph it. is the customer a risk averter?
(b) (x)=x+0.005x2?
(c) for the case 18a, find the maximal premium the customer would be willing to pay to insure his wealth against the loss mentioned. first, set the equation clearly and explain it, then solve. is the premium you found greater or less than e{x}? might you predict it from the beginning?
(d) find the minimal premium which an insurance company would accept to cover the risk mentioned, if the company’s preferences are characterized by the utility function u1(x) = ? x, and the company takes 300 as an initial wealth. is the premium you found greater or less than e{? }. might you predict it?
(e) solveexercise18dforthecasewhenther. v.? isuniformlydistributedon[0,100].
(f) solve exercise 18c for u(x) = 200x ? x2 + 349. (advice: look at this function atten-
tively before starting calculations.)