1)[20 points] An insurer undertakes a risk X with RX = {0, 2000, 5000} and P{X = 0} = 0.5,P{X = 2000} = 0.3,P{X = 5000} = 0.2. After collecting the premium, the insurer owns a capital of w = 10000. The insurer has utility function a : (0, oo) I> R given by My) = log{y}. a) [15 points] Calculate the maximum premium the insurer is willing to pay to a reinsurer to take over the full risk X. b) [10 points] Apply Remark 1.2.3(pages 32-33) of the Notes to approximate the maximum premium the insurer is willing to pay to a reinsurer to take over the full risk X. \fHence. 1 (H - P+)u'(w - H) W 5621-!"(w - 1')- Therefore. the maximum premium for a risk P+ can be approximated by :3 E[X] $VAR[X]H