ACTL2102 Exam 2018 PDF

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Paper ID: 00424

FAMILY NAME: OTHER NAME(S): STUDENT ID: SIGNATURE:

SCHOOL OF RISK AND ACTUARIAL STUDIES SESSION 2, 2018

ACTL 2102: FOUNDATIONS OF ACTUARIAL MODELS MIDTERM EXAM INSTRUCTIONS: 1. TIME ALLOWED—1 HOURS 2. READING TIME—5 MINUTES 3. THIS EXAMINATION PAPER HAS 14 PAGES. 4. TOTAL NUMBER OF QUESTIONS—3 5. TOTAL MARKS AVAILABLE—100 6. MARKS AVAILABLE FOR EACH QUESTION ARE SHOWN IN THE EXAMINATION PAPER (AND OVERLEAF). ALL QUESTIONS ARE NOT OF EQUAL VALUE. 7. ANSWER ALL QUESTIONS IN THE SPACE ALLOCATED TO THEM. IF MORE SPACE IS REQUIRED, USE THE ADDITIONAL PAGES AT THE END. 8. CANDIDATES MAY BRING a. THE TEXT “FORMULÆ AND TABLES FOR ACTUARIAL EXAMINATIONS” (ANY EDITION) INTO THE EXAMINATION. IT MUST BE WHOLLY UNANNOTATED. b. THEIR OWN CALCULATORS. ALL CALCULATORS MUST BE UNSW APPROVED. 9. ALL ANSWERS MUST BE WRITTEN IN INK. EXCEPT WHERE THEY ARE EXPRESSLY REQUIRED, PENCILS MAY BE USED ONLY FOR DRAWING, SKETCHING OR GRAPHICAL WORKS. 10. THIS PAPER MAY NOT BE RETAINED BY THE CANDIDATE. CANDIDATES MUST CEASE WRITING IMMEDIATELY WHEN INSTRUCTED TO DO SO BY THE SUPERVISOR AT THE END OF THE EXAMINATION. FAILURE TO DO SO WILL RESULT IN A MARK OF ZERO.

Question

Mark

1(a)

[5 marks]

1(b)

[5 marks]

1(c)

[15 marks]

1(d)

[10 marks]

2(a)

[5 marks]

2(b)

[5 marks]

2(c)

[5 marks]

2(d)(i)

[10 marks]

2(d)(ii)

[10 marks]

3(a)

[10 marks]

3(b)

[5 marks]

3(c)

[5 marks]

3(d)

[5 marks]

3(e)

[5 marks]

Total [100 marks]

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Question 1 [35 marks] An insurance company allows the following No-Claim-Discount (“NCD”) levels for its domestic motor insurance policies: NCD Level 0 1 2

Discount % 0 10 20

The rules for the transitions between NCD levels for a policyholder are given in the following. - If a policyholder is at level 0 and has no claim in a year, then he/she moves to level 1. - If a policyholder has been at level 1 for two years without making a claim, then he/she moves to level 2. - If a policyholder is at level 1 or 2 and has at least one claim in a year, then he/she moves to the next lower level of discount. - In all other circumstances, a policyholder stays at the current level. The probability for a policyholder making one or more claim in a year is 0.1. (a) [5 marks] A discrete-time stochastic process {Yn ; n ≥ 0} on the states {0, 1, 2} is used to represent the NCD level of a policyholder at year n. Show that {Yn ; n ≥ 0} is not a Markov chain.

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(b) [5 marks] An alternative way of representing the NCD level of a policyholder is to use a Markov chain {Xn ; n ≥ 0} with the following 4 states - state 0: if a policyholder is at NCD level 0 in year n - state 1: if a policyholder is at NCD level 1 in year n but not in year n − 1 - state 2: if a policyholder is at NCD level 1 in both years n and n − 1 - state 3: if a policyholder is at NCD level 2 in year n Show that the one-step transition matrix, denoted by P is 

 0.1 0.9 0 0  0.1 0 0.9 0     0.1 0 0 0.9  0 0.1 0 0.9

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(c) [15 marks] Calculate the proportion of policyholders in each of the four NCD levels in the long run. You can assume that the limiting distribution exists for the Markov chain {Xn ; n ≥ 0}.

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(d) [10 marks] If a policyholder is at NCD level 0 in year n, what is the probability that the policyholder is at NCD level 1 in year n + 2?

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Question 2 [35 marks] A factory has 2 identical engines, namely Engine A and Engine B. Engine A is currently used to provide electricity to the factory; if Engine A breaks down, then Engine B is utilised immediately while Engine A is under repair. The length of period an engine can function without repairing follows an exponential distribution with an average of 1 year. If an engine is broken, then the amount of time to fix it follows an exponential distribution with an average of 1 month. Here we assume that repair times and operating times are independent. (a) [5 marks] Calculate the probability that Engine A can function for at least 2 years without repairing.

(b) [5 marks] Calculate the probability that Engine A will break down in less than 2 years (from the time it starts working), given that it has been working for 6 months.

(c) [5 marks] Suppose that Engine A breaks down. Calculate the probability that Engine A can be repaired before Engine B breaks down.

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(d) Suppose that the factory has decided not to use the current engines any more. They are planning to purchase n engines from another supplier, each of which has a life time that follows an exponential distribution with an average life time of 1 year. Furthermore, at any point of time, only one engine is utilised and another engine will be used immediately if the current engines break down. There is no repairing any more, hence the system of n engines will break down once all n engines are broken. (i) [10 marks] Calculate the expected number of engines that will break down in 5 years time. You should assume that n is large enough that the factory will never have 0 engines.

(ii) [10 marks] Calculate the minimum value for n such that the system will not break down in 2 years time with a probability of at least 50%.

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Question 3 [30 marks] Assume that the weather conditions of a city are modelled by a continuous-time Markov Chain with three states: Sunny (S), Cloudy (C ), and Rainy (R). The transition rate matrix is given by S "−0.25 Q = 0.5 0.2

C 0.15 −1 0.3

R 0.1 # S 0.5 C −0.5 R

The unit of time is day. (a) [10 marks] Calculate the limiting probabilities for each of the three states of the continuous-time Markov chain.

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(b) [5 marks] If the weather has been sunny for 5 days, what is the probability that it will be sunny for another 10 days? You should show the steps of your calculations.

(c) [5 marks] Calculate the transition probabilities for the corresponding embedded Markov chain.

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∂ (d) [5 marks] Write down the Kolmogorov’s forward equation for ∂t PSC (t). You should express all the parameters with the numerical values given in the question. You should also state the initial condition.

(e) [5 marks] The Kolmogorov equations can be used the evaluate the transition probabilities of a continuous-time Markov chain over a given period of time. This involves evaluating the powers of a matrix. State one issue when computing the exponential of a matrix in this context.

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ADDITIONAL PAGE Answer any unfinished questions here.

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ADDITIONAL PAGE Answer any unfinished questions here.

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ADDITIONAL PAGE Answer any unfinished questions here.

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