Unsw Actl2003 November 2007 Final Examination

THE UNIVERSITY OF NEW SOUTH WALES MONTH OF EXAMINATION - NOVEI\1BER 2007 Final Examination ACTL2003 STOCHASTIC MODELS FOR ACTUARIAL APPLICATIONS

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