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FinalexamACTL2003-Yr2009

# FinalexamACTL2003-Yr2009 - THE UNIVERSITY OF NEW SOUTH...

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THE UNIVERSITY OF NEW SOUTH WALES SEMESTER 2 ACTL 2003: Stochastic Models for Actuarial Application Final Examination 2009 INSTRUCTIONS: Time Allowed: 3 hours Reading time: 10 minutes Total Assessment credit: 75% Total Marks available: 100 points This examination paper has 13 pages Total number of questions: NINE All questions are not of equal value. Marks allocated for each part of the questions are indicated. This is a closed-book test and no formula sheets are allowed except for the Formulae and Tables for Actuarial Exams (any edition). IT MUST BE WHOLLY UNANNOTATED. Use your own calculator for this exam. Calculators are permitted, but must be UNSW and/or Actuarial Studies approved. Show all necessary steps in your solutions. If there is no written solu- tion, then no marks will be awarded . Answer each question starting on a new page 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 BE RETAINED BY THE CANDIDATE 1

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Question 1 (12 marks) Consider a Markov chain { X n , n = 0 , 1 , · · · } with state space { 1 , 2 , · · · , 5 } . Suppose its probability transition matrix is given as below: 1 5 0 0 0 4 5 0 1 3 0 2 3 0 0 0 1 0 0 0 1 2 0 1 2 0 1 2 0 1 2 0 0 (a) Classify the states, and determine whether they are recurrent or transient. [2 marks] . (b) Find the expected number of visits in each transient state starting in any transient state. [2 marks] (c) Given that the probability of the initial state X 0 being i is i 15 , i = 1 , 2 , · · · , 5, calculate the probability that two steps later, the process is in state 3. [2 marks] (d) Is the Markov chain ergodic? Why? [2 marks] (e) Does the limiting probability lim n →∞ p 24 exist? If yes, calculate the lim- iting probability. [4 marks] 2
Question 2 (12 marks) (Questions (a) and (b) are unrelated.) (a) In some automobile insurance, annual premiums are determined by use of a Bonus Malus system. Each policyholder is given a state (1, 2, or 3) and the annual premium depends on this state. For states 1, 2, and 3, the premium is 100, 200 and 300, respectively. A policyholder’s state changes from year to year in response to the number of claims made by that policyholder. For a policyholder at state k this year, if the policyholder makes no claim the state for next year will become max( k - 1 , 1); if 1 claim, the state for next

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FinalexamACTL2003-Yr2009 - THE UNIVERSITY OF NEW SOUTH...

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