actl2003-finalexam-2013

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THE UNIVERSITY OF NEW SOUTH WALES

SCHOOL OF RISK AND ACTUARIAL STUDIES

SEMESTER 2 2013

ACTL2003: STOCHASTIC MODELS FOR ACTUARIALAPPLICATIONS

Final Examination

INSTRUCTIONS:

• Time allowed: 2 hours

• Reading time: 10 minutes

• This examination paper has 24 pages

• Total number of questions: 7

• Total marks available: 100 points

• Marks allocated for each part of the questions are indicated in the examinationpaper. 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 WHOLLYUNANNOTATED.

• 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.

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Question Mark

1 [26 marks]

2 [6 marks]

3 [7 marks]

4 [23 marks]

5 [15 marks]

6 [8 marks]

7 [15 marks]

Total

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Question 1. (26 marks)Phone enquiries arriving at a big technology company wait for an average of 10minutes before being answered by a receptionist who directs the enquires eitherto junior consultants (J), senior consultants (S) or the further investigation team(I). Four enquiries in ten are directed to junior consultants, five in ten to seniorconsultants, and one in ten to the further investigation team.

An enquiry that is directed to the further investigation team spends an averageof 20 minutes there, after which 50% of cases are closed immediately (C), 25% aresent to a senior consultant and 25% to a junior consultant.

An enquiry requiring attendance from a junior consultant takes an average of20 minutes to close, and an enquiry requiring attendance from a senior consultanttakes an average of 60 minutes to close.

It is suggested that a time-homogeneous Markov process with states A, J , S, I,and C could be used to model the progress of enquires through the system. Here, anenquiry in state A means that it is waiting to be answered by a receptionist. And anenquiry in state J , S and I means that it is being handled by a junior consultant,a senior consultant and the further investigation team, respectively. An enquiry instate C means that it has been closed.

(i) Write down the matrix of transition rates, {qij : i, j = A, I, J, S, C}, of such aMarkov model. [5 marks]

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(ii) Calculate the proportion of enquires which are eventually directed to juniorconsultants. [2 marks]

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(iii) Derive expressions for the probability that an enquiry arriving at time 0 is:

(a) yet to be answered by the receptionist at time t; [4 marks]

(b) undergoing further investigation at time t. [6 marks]

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(iv) Let mi denote the expectation of the time until being closed for an enquirycurrently in state i. Calculate the expectation of the time until being closed for anewly-arrived enquiry. [9 marks]

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Question 2. (6 marks)Let {X(t) : t ≥ 0} be a continuous time-homogeneous Markov jump process withtwo states 0 and 1, and transition rates q01 = λ and q10 = µ.

Let Oi(t) denote the total amount of time spent in state i up until time t (theoccupation time in state i by time t). Suppose λ = 0.1 and µ = 0.2. Calculate theexpected occupation time in state 1 by time t = 10 given that the continuous-timeMarkov chain is starting in state 0 at time 0. (Hint: P0,0(t) = µ

λ+µ+ λ

λ+µe−(λ+µ)t

and Oi(t) =∫ t

0I{X(s) = i}ds where I{·} is the indicator function.)

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Question 3. (7 marks)Consider the time series model: Xt = 2cXt−1 − c2Xt−2 + Zt, where Zt is a whitenoise process with variance σ2.

(i) Find the values of c such that the process is stationary. [5 marks]

(ii) For c taking values such that the time series process is stationary, the model isan ARIMA(p, d, q) process. Identify the values of p, d and q. [2 marks]

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Question 4. (23 marks)Consider the time series model: Xt = a + bXt−1 + cXt−2 + Zt, where Zt is a whitenoise process with variance σ2.

(i) Now suppose that a = 0.7, b = 0.4 and c = 0.12.

(a) Calculate E[Xt]. [4 marks]

(b) Calculate the autocorrelations ρ(1), ρ(2) and ρ(3). [9 marks]

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(This page can only be used to answer Question 4(i)(b).)

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(c) Describe the behavior of the autocorrelation function ρ(k) and the kth partialautocorrelation function α(k) as k → ∞. [4 marks]

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(ii) Now suppose a = c = 0 and you have obtained the following sample autocorre-lations:

ρ(1) ρ(2) ρ(3) ρ(4) ρ(5) ρ(6) ρ(7) ρ(8) ρ(9)0.800 0.645 0.519 0.414 0.331 0.265 0.212 0.170 0.137

Suggest an appropriate value for b and justify your answer. [6 marks]

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Question 5. (15 marks)The following sequence of independent pseudo random numbers from a uniformdistribution over the interval [0, 1] has been generated by a computer:

U1, U2, · · · , Un,

where n is a known positive integer.

(i) Let X be an exponential random variable with mean 12. Outline a procedure for

generating n independent random numbers for 11+X3 by making use of the available

pseudo random numbers. [3 marks]

(ii) Write down the Monte Carlo estimator for E[ 11+X3 ] using the random numbers

obtained in (i). [2 marks]

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(iii) Provide a method to simulate E[ 11+X3 ] with reduced variance compared with

the estimator obtained in (ii) using the same sequence U1, U2, · · · , Un. You shouldprovide the algorithm and prove in detail that the variance is reduced when themethod you provide is used. [10 marks]

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Question 6. (8 marks)Let {Wt}t≥0 be a standard Brownian motion. Is {

√tWt}t≥0 a Brownian motion?

Justify your answer.

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Question 7. (15 marks)Let {Yt} be a stationary AR(3) process satisfying

Yt = ψ1Yt−1 + ψ2Yt−2 + ψ3Yt−3 + Zt,

where Zt is a white noise process with variance σ2.Define γ(h) = Cor(Yt, Yt+h) and the matrices

Γh =

γ(0) γ(1) · · · γ(h− 1)γ(1) γ(0) · · · γ(h− 2)...

.... . .

...γ(h− 1) γ(h− 2) · · · γ(0)

, γh =

γ(1)...γ(h)

.

Write xh =

xh1xh2...xhh

= Γ−1h γh. Prove that x33 = ψ3 and xhh = 0 for h > 3.

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(This page can only be used to answer Question 7.)

END OF PAPER

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ADDITIONAL PAGEAnswer unfinished questions here, or use for rough work.

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ADDITIONAL PAGEAnswer any unfinished questions here, or use for rough work.

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ADDITIONAL PAGEAnswer unfinished questions here, or use for rough work.

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actl2003-finalexam-2013

Documents

time t

state j

i j s cg

senior consultants s

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eitherto junior consultants

average of20 minutes

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