INSTRUCTIONS:

1. TIME ALLOWED - 3 HOURS.

2. TOTAL NUMBER OF QUESTIONS - 8. 3. TOTAL MARKS - 100. 4. THERE ARE 2 SECTIONS. EACH SECTION SHOULD BE ANSWERED IN A SEPARATE EXAMINATION BOOK. 5. SECTION I HAS 6 QUESTIONS. USE A SEPARATE EXAMINATION BOOK AND INDICATE THE SECTION NUMBER ON THE FRONT PAGE. ANSWER EACH QUESTION STARTING ON A NEW PAGE. 6. SECTION 11 HAS 2 QUESTIONS. USE A SEPARATE EXAMINATION BOOK AND INDICATE THE SECTION NUMBER ON THE FRONT PAGE. ANSWER EACH QUESTION STARTING ON A NEW PAGE. 7. QUESTIONS ARE NOT OF EQUAL VALUE. 8. CANDIDATES MAY BRING THE "FORMULAE AND TABLES FOR ACTUARIAL EXAMINATIONS" BOOKLET INTO THE EXAMINATION. 9. CANDIDATES MAY BRING THEIR OWN CALCULATORS OR HAND HELD COMPUTERS. ALL ANSWERS MUST BE WRITTEN IN INK. EXCEPT WHERE THEY ARE EXPRESSLY REQUIRED, PENCILS MAY BE USED ONLY FOR DRAWING, SKETCHING OR GRAPHICAL WORK. Answer each question starting on a new page 1

SECTIOX I [73 MARKS] START A NEW EXAMIl\ATIO~ BOOK. ANSWER ALL QUESTIONS. START EACH QUESTION ON A NEW PAGE. Question 1 (6 marks)

(a) Specify the classes of the following Markov chains, and determine whether they are transient or recurrent: 1 1 0 "2 "2 1 1 "2 0 "2 1 1 "2 "2 0 1 0 "2 0 1 1 "2 '4 0 1 0 2 0 1 0 0 0 "2 1 0 0 0 "2 1

P2

0 0 0

"2

=

0 0 0 0 0 0 1 1 "2 "2 0 0 0 1

1 1

1 1 0 0

.:i

4

1

"2 1 '4 1 "2

P4

=

~

1

0 0 0 "2 "2 0 0 0 0 0 1 0 0 1 2 0 0 3 3 0 1 0 0 0 0

4

[2 marks] (b) Let the transition probability matrix of a two-state Markov chain be given by p =

III ~

p

1;

p

11

Show that

Hint: Use mathematical induction. [4 marks]

2

Question 2 (10 marks) An insurance company is modelling its motor insurance claims. It has determined that the probability of a claim depends on the number of claims in the previous two years. If a motor insurance policyholder has had claims in both of the previous two years the probability of a claim in the current year is 0.25, if they had claims in only one of the previous two years then the probability of a claim in the current year is 0.15 and if they had no claims in the previous two years then the probability of a claim in the current year is 0.05. (a) Describe the claims process as a Markov chain taking into account the previous two years claims, specifying the states of the process and the transition matrix. [2 marks] (b) Explain the requirements for this process to have a long run stationary distribution and show that the Markov chain satisfies these requirements. [2 marks] (c) Determine the long proportion of policyholders making at least one claim in a year. [6 marks] Question 3 (15 marks) An insurance company checks policyholder claims to determine if they are fraudulent or not. Initially every claim is checked until there are i consecutive genuine claims. Once i genuine claims have been processed, the company then checks only one in r claims by random sampling. If a fraudulent claim is detected then the company returns to checking every claim until i consecutive genuine claims are processed. The probability that any claim is fraudulent is p. Consider the stochastic process {Xn : n = 1,2, ... } where X n is the state of the checking system at the nth claim. This checking system can be modelled as a Markov chain using i + 1 states. State 0 is the detection of a fraudulent claim, states k = 1, ... , i - I are the states when the checking system has processed k consecutive genuine claims, and state i is the state when random sampling for one in r claims is being used for checking. (a) Write down the probability transition matrix for the chain and justify your answer. Hint: you should have i + 1 states. [3 marks] (b) Using your transition matrix in (a), derive expressions for the long run probabilities for each of the states, showing all your working. [8 marks] (c) A student...