Universiteit van AmsterdamAmsterdam School of EconomicsModern Actuarial Risk Theory Using R Rob Kaas Date: M町 1，2010Note: This file contains solutions to many exercises in Modern Actuarial Risk Theory一 Using R used in my classes at the University of Amsterdam. They are made availableto teachers using MART as is, without warranty or pretense Users are urged to shareany criticism with the author of this text, at R.KaasUvA.nl, to the benefit of all users.()1.2.1 Prove Jensen，s inequality: if v(x) is convex, then Ev(X) v(EX). Consider especially v(x) = x2. The Hint (App B) suggests to use the following characterization: a function v(x) is convex if, and only if, for every xq a line Iqx) = ax + Z?o exists such that Zo(o) = v(x0) and moreover /()(兀) Iqx) Vx, we have Ev(X) E/0(X) = EQoX + 加=QoEX +% = Z0(EX) = v(EX). The tangent line in x0 satisfies Zo(o) = v(%o) and /q(x0) = vz(x0)- If v(x) = x2, then /q(x) =%q + 2兀0(兀%o) must hold.We have v(x) /0(x) = (x-x0)2 0, so EX2 = Ev(X) EZ0(X) = g2. But also: EX2 = VarX + (EX)2 (EX)2.()1.2.2 Also prove the reverse of Jensens inequality: if Ev(X) v(EX) for every random variable X, then v(-) is convex Hint (App. B): If it holds for each random variable X, then also if especially PrX = X。+ /z = PrX = x() h = | for some arbitrary %o and /z 0.In that case, Ev(X)=知(兀o 方)+ |v(%o + 力)and EX = xq, so Ev(X) v(EX) implies |v(%o h) + 扣(羽 + 方) v(xo).Since this holds for all x0 and h, v(-) must be convex()1.2.3 Prove: if Ev(X) = v(EX) holds for every random variable X, then v(-) is linear. Because of the previous exercise, both v(-) and v(-) are convex So the derivative of v(-) is both non-decreasing and non-increasing, hence a constant. So v(-) is a linear function.(0) 1.2.4 A decision maker has utility function ux) = y/x, x 0. He is given the choice between two random amounts X and Y, in exchange for his entire present capital w. The probability distributions of X and Y arex PrX = x y Pry = y1-2 1-23-5 2-5Show that he prefers X to Y. For which values of w should he decline the offer? Are there utility functions with which he would prefer Y to X ? IfEu(X) Ew(y) or Ew(X) u(w), he prefers X toYX to w. Em(X) =|x a/400 + Ix /900 = 25; Ew(y) = 22: he prefers X to Y. 况(w) = y/w, so if w 625 it is better for him to choose random proposition X. If w EX for risk averse insurers. P- is the solution to EU(W + P- -X) =U(W).A risk averse insurer has a concave, increasing utility function, so in view of Jensens inequality we have EU(WP X) U(EW + P X).As U() is increasing, from U(W) U(EW -P X) we deduce W EX.(1.11)() 1.2.6 Including the premium for risk X, an insurer owns w = 100. What maximum premium is he willing to pay to a reinsurer to take over the complete risk, if u(w) = log(w) and PtX = 0 = PrX = 36 = |?Find not only the exact value, but also the approximation (1.18). Utility equilibrium: log(100 P+) = Elog(100 X) = |log 100 + | log 64 = log 10 + log 8 = log 80 = P+ = 20. With 1 = 1& a2 = (36)2 x | and 厂(w “)= (w (1.18) givesF+ u 卩 + |cr2r(w y) = 19.9756.()1.2.7 Assume that the reinsurer minimum premium for X as in ex. 1.2.6 is 19 and that he has the same utility function. Determine his capital W Utility equilibrium: Elog(W + 19 X) = log(W), therefore (W + 19)(W + 19 -36) = W2, so W = I x 17 x 19 = 161.5. 1.2.8 Describe the utility function of a person with the following risk behavior: after winning 1, he says yes9 when asked double or quits? after winning again, he agrees after a long huddle; the third time he says no5. Suppose, without loss of generality, that the current capital w = 0, whilefor the utility function we have w(0) = 0 and w(l) = 1. Then: *(%(0) + w(2) w(l), hence w(2) 2; |(w(0) + w(4) = w(2), hence w(4) = 2w(2);|(w(0) + w(8) w(4), hence w(8) 0 somewhere in (0,2); /(兀) 0 somewhere in (2,8).This person is neither consistently risk averse, nor consistently risk seeking.()1.2.9 Verify that P+ 2X 3/4, + 1 for 一 1 兀 3/4,3兀 + 2 for x We have w(0) 0,_ 1) _ 1, w(一3/4) = _ 1/2, w(_2) = 一4.Also, ux) = 2 for x = 4/3.Then P+X solves Ew(-X) = w(-P+X) = 1/2, so P+X = 3/4, and P+ 2X solves E讥一2X) = w(-P+2X) = -2, so P+2X = 4/3.The maximal premium for the double risk 2X for this combination of risk X and utility function w(-) is less than twice the one for the original risk.(A) 1