一站式論文代寫,英国、美国、澳洲留学生Essay代寫—FreePass代写

R代寫 - ACTL5106 Insurance Risk Models
時間:2020-11-28
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 12 3a) 3b) 3c) 4a) 4b) 4c) 4d) 4e) 5a) 5b) 6a) 6b) 6c) 6d) 78 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 Cram′er-Lundberg surplus process C(t) = c0 + πt XN(t) i=1 Yi , t ≥ 0, where ? C(t) is the insurer’s surplus level at time t; ? c0 is the initial surplus; ? π 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) = 25e 2y (3 + 4y), y > 0. Suppose λ = 5, π = 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 π = 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 αe = α +XTt=1 xt and βe = β + 2T. PLEASE TURN OVER Page 9 of 24 (d) [5 marks] Prove that the Bayes premium is P Bayes = 2T β + 2T 1 PTt=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. Page 22 of 24 ADDITIONAL PAGE Answer any unfinished questions here, or use for rough work. Page 23 of 24 ADDITIONAL PAGE Answer any unfinished questions here, or use for rough work.

在線客服

售前咨詢
售后咨詢
微信號
Essay_Cheery
微信
专业essay代写|留学生论文,作业,网课,考试|代做功課服務-PROESSAY HKG 专业留学Essay|Assignment代写|毕业论文代写-rushmyessay,绝对靠谱负责 代写essay,代写assignment,「立减5%」网课代修-Australiaway 代写essay,代写assignment,代写PAPER,留学生论文代写网 毕业论文代写,代写paper,北美CS代写-编程代码,代写金融-第一代写网 作业代写:CS代写|代写论文|统计,数学,物理代写-天天论文网 提供高质量的essay代写,Paper代写,留学作业代写-天才代写 全优代写 - 北美Essay代写,Report代写,留学生论文代写作业代写 北美顶级代写|加拿大美国论文作业代写服务-最靠谱价格低-CoursePass 论文代写等留学生作业代做服务,北美网课代修领导者AssignmentBack