INSTRUCTIONS: Time Allowed: 2 hours Reading time: 10 minutes This examination paper has 24 pages Total number of questions: 8 Total Marks available: 100 points Marks allocated for each part of the questions are indicated in the examination paper. All questions are not of equal value. This is a closed-book test and no formula sheets are allowed except for the For- mulae and Tables for Actuarial Exams (any edition). IT MUST BE WHOLLY UNANNOTATED. Use your own calculator for this exam. All calculators must be UNSW ap- proved. Answer all questions in the space allocated to them. If more space is required, use the additional pages at the end. Show all necessary steps in your solutions. If there is no written solution, then no marks will be awarded. All answers must be written in ink. Except where they are expressly required, pencils may be used only for drawing, sketching or graphical work. THE PAPER MAY NOT BE RETAINED BY THE CANDIDATE. Page 1 of 24 Question Marks 1 2 3a) 3b) 3c) 4a) 4b) 4c) 4d) 4e) 5a) 5b) 6a) 6b) 6c) 6d) 7 8 Total Page 2 of 24 Question 1. (2 marks) Let X be a loss random variable of a risk. Write down a formula for the premium of this risk using the expected value principle. Question 2. (2 marks) Which of the following statements are true (A) de Pril’s algorithm is for calculating convolutions of discrete non-negative integer valued random variables with positive probability mass at 0. (B) de Pril’s algorithm is for calculating the distribution of non-negative integer valued compound random variables with positive probability mass at 0. PLEASE TURN OVER Page 3 of 24 Question 3. (15 marks) Consider the Crame′r-Lundberg surplus process C(t) = c0 + pit N(t)∑ i=1 Yi, t ≥ 0, where C(t) is the insurer’s surplus level at time t; c0 is the initial surplus; pi is the constant premium rate; N(t) is a Poisson process with rate λ; and Yi’s are claim amounts that are independent and identically distributed and are independent of the above Poisson process. (a) [4 marks] Assume that each claim amount follows the probability density function fY (y) = 2 5 e 2y(3 + 4y), y > 0. Suppose λ = 5, pi = 6 and c0 = 1.5. Calculate the relative security loading θ. PLEASE TURN OVER Page 4 of 24 (b) [3 marks] Give two reasons why the condition θ > 0 is important from the in- surer’s point of view. PLEASE TURN OVER Page 5 of 24 (c) [8 marks] Suppose Yi ≡ 1000 for i = 1, 2, 3, . . ., and pi = 1300λ; The values of λ and c0 are unknown; The probability that ruin occurs at the first claim is 1%. Determine the numerical value of the initial surplus, c0. PLEASE TURN OVER Page 6 of 24 Question 4. (20 marks) The annual claims X1, X2, . . . for a given policyholder in an insurance portfolio are known to be (conditional on the policyholder’s risk parameter Θ = θ) independent and identically distributed with probability mass function fX|Θ(x|θ) = (x+ 1)(1 θ)2θx, x = 0, 1, 2, . . . , where 0 < θ < 1. The (unobservable) risk parameter Θ is assumed to follow a Beta distribution with parameters α, β, where α > 0 and β > 2. (a) [2 marks] State the name of the distribution that has probability mass function fX|Θ(x|θ) and identify its parameter(s). Hence, deduce that E[Xi|Θ = θ] = 2θ 1 θ for i = 1, 2, 3, . . .. PLEASE TURN OVER Page 7 of 24 (b) [6 marks] Define μ(θ) = E[Xi|Θ = θ]. Show that E[μ(Θ)] = 2α β 1 . PLEASE TURN OVER Page 8 of 24 In the parts (c)-(e) below, suppose that we have observed T years of claim amounts X = (X1, X2, . . . , XT ) to be x = (x1, x2, . . . , xT ). (c) [4 marks] Show that the posterior distribution of Θ|X = x is a Beta distribution with parameters α = α + T∑ t=1 xt and β = β + 2T. PLEASE TURN OVER Page 9 of 24 (d) [5 marks] Prove that the Bayes premium is PBayes = 2T β + 2T 1 ∑T t=1 xt T + β 1 β + 2T 1 2α β 1 . PLEASE TURN OVER Page 10 of 24 (e) [3 marks] Without performing any calculation, determine whether the Buhlmann’s credibility premium is greater than, smaller than, or equal to the Bayes premium with justification. PLEASE TURN OVER Page 11 of 24 Question 5. (12 marks) Consider two random variables X and Y , where both follow exponential distribution but with parameters α > 0 and β > 0 respectively. They are linked through the Farlie-Gumbel-Morgenstern copula defined by C(u, v) = uv + θuv(1 u)(1 v), u, v ∈ [0, 1], where θ ∈ [0, 1] is the parameter of the copula. (a) [4 marks] Explain whether the copula allows for possibility of independence between X and Y . PLEASE TURN OVER Page 12 of 24 (b) [8 marks] Show that the joint density of X and Y can be represented as fX,Y (x, y) = A(αe αx)(βe βy) +B(2αe 2αx)(βe βy) + C(αe αx)(2βe 2βy) +D(2αe 2αx)(2βe 2βy), x, y > 0, and determine the constants A, B, C and D. PLEASE TURN OVER Page 13 of 24 Question 6. (23 marks) Recall that a distribution is from an exponential dispersion family if its density has the form fY (y) = exp [ yθ b (θ) ψ + c (y;ψ) ] , θ ∈ Θ, ψ ∈ Π. (a) [4 marks] Describe the two main components of a generalized linear model and explain how the two components are linked. PLEASE TURN OVER Page 14 of 24 (b) [7 marks] Show that the distribution corresponding to the following probability density function belongs to the exponential family of distributions: g(y) = yα 1e y/β βαΓ(α) , y > 0. PLEASE TURN OVER Page 15 of 24 (c) [6 marks] Consider a distribution from the exponential dispersion family with b(θ) = 10 log(1 + eθ). Derive the expressions for the natural link function and the variance function. PLEASE TURN OVER Page 16 of 24 (d) [6 marks] Assume that you know that the following three Poisson general linear models (GLM) with the same link function, g(·), all fit the data well: Model 1: g(μi) = β1xi1 Model 2: g(μi) = β1xi1 + β2xi2 Model 3: g(μi) = β1xi1 + β2xi2 + β3xi3 + β4xi4 The scaled deviances are given as below Model Deviance Model 1 72.23 Model 2 70.64 Model 3 67.13 Which model is the best based on the available information and the likelihood ratio test at 5% significance level Explain why. PLEASE TURN OVER Page 17 of 24 Question 7. (13 marks) The cumulative paid claims on a portfolios of insurance policies are given in the following table: Accident year Development year 1 2 3 2014 5,496 x 7,982 2015 5,162 8,028 2016 6,434 where x is a positive number. Suppose the claims will completely run off in 3 years and the development factor from development year 2 to development 3 is 1.04504. By assuming that the ultimate loss ratio is 0.85, you have found that the Bornhuetter-Ferguson estimate of outstanding claims at the end of year 2016 for accident year 2016 is 6,464. Determine the numerical value of the earned premium for year 2016. PLEASE TURN OVER Page 18 of 24 (This page can only be used to answer Question 7.) PLEASE TURN OVER Page 19 of 24 Question 8. (13 marks) Consider the following payoff matrix of a zero-sum game with two players, A and B. The payoff matrix lists the gains for A and losses for the player B. B A Strategy 1 2 3 a 10 34 7 b 22 14 8 c X 30 26 where X is an exponential random variable with mean 1/λ. Determine the numerical value of λ so that the probability that there is an optimal solution is 10%. END OF PAPER Page 20 of 24 ADDITIONAL PAGE Answer any unfinished questions here, or use for rough work. Page 21 of 24 ADDITIONAL PAGE Answer any unfinished questions here, or use for rough work. 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