THE UNIVERSITY OF NEW SOUTH WALES

SEMESTER 2

ACTL 3003: Insurance Risk Models

Class Test

Monday, 16 August 2010

Write the required information on the space provided:

Name: Student ID:

Circle the tutorial session which you are enrolled for:

Luke -A (Tue 3pm-4pm) Jinxia (Thu 11am-12pm)Kieran -A (Thu 12pm-1pm) Luke -B (Fri 2pm-3pm)Kieran -B (Fri 3pm-4pm)

INSTRUCTIONS:

Read through the following information carefully. DO NOT COMMENCE WRIT-

ING UNTIL YOU ARE TOLD TO DO SO.

• Time Allowed: 60 minutes

• Total Assessment credit: 10%

• Total Marks available: 100 points

• This examination paper has 11 pages

• Total number of questions: 6

• All questions are not of equal value. Marks allocated for each part of the questions areindicated.

• This is a closed-book test and no formula sheets are allowed except for the Formulae andTables for Actuarial Exams (any edition). IT MUST BE WHOLLY UNANNOTATED.

• Use your own calculator for this exam. If your calculators are not UNSW approved, thecalculators must be hand-held, internally powered and silent, and any programmablememory must be cleared prior to entering an examination room.

• Show all necessary steps in your solutions in the space provided (if necessary, you canalso use the back pages). If there is no written solution, then no marks will be

awarded.

• STUDENTS WRITING AFTER THE EXAM TIME HAS EXPIRED WILL

SCORE A MARK OF ZERO. ACADEMIC MISCONDUCT ACTION MAY

ALSO RESULT.

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Question 1. (25 marks)Consider an automobile insurance policy where the policyholder has probability q of gettinginto an accident. The distribution of the damage, given that the accident has occurred,follows an exponential distribution with parameter β.

(a) [10 marks] Find the cumulative distribution function F (x) of the loss of this policy.Clearly define your notation.

(b) [5 marks] Find the moment generating function of the loss.

Additionally, suppose the policy to have deductible amount d.

(c) [5 marks] Find the probability that the insurer is required to make a payment.

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(d) [3 marks] Find the amount the insurer expects to pay.

(e) [2 marks] Suppose the insurer purchases reinsurance and as a result retains a propor-tion, α, of the loss. Find the fair price of the reinsurance contract.

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Question 2. (15 marks)Consider the collective risk model

S = X1 + X2 + . . . + XN ,

where N ∼ Binomial(m, p) and X ∼ P (x).

(a) [5 marks] Find the moment generating function of S in terms of the moment generatingfunction of X.

(b) [5 marks] Using the cumulant generating function, find the first cumulant of S in termsof p1 = E[X]. Show your work.

(c) [5 marks] Suppose X to be a strictly positive random variable. Rewrite S as anindividual risk model with modified losses X. Find the distribution of X in terms ofthe distribution of X.

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Question 3. (10 marks)Suppose S1 ∼ Compound Poisson(λ = 5, P (x)), where the distribution of X is given byp(1) = 0.4 and p(2) = 0.6. Let Ni ∼ Poisson(λ = i), i = 1, 2, and N1, N2 and S1 bemutually independent. Find the cumulative distribution function of

S = 2N1 + 3N2 + S1.

.

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Question 4. (14 marks)Suppose that the random variable Y has the following probability mass function:

fY (y) =

(

y − 1

r − 1

)

pr(1 − p)y−r, y = r, r + 1, · · · ,

where r is a positive integer.Let N = Y − r and S =

∑N

i=1 Xi, where X1, X2, · · · are independent and identicallydistributed, and independent of N . The probability mass function of the random variableX1 is p(x) = 0.1x, x = 1, 2, 3, 4.

(a) [8 marks] Show that the random variable N belongs to the (a, b) family and identifya and b in terms of r and p.

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(b) [6 marks] Let p = 0.1 and r = 2. Suppose that you know fS(1) = 0.0018 andfS(2) = 0.003843. Find fS(0) and use Panjer’s recursion to determine fS(3).

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Question 5. (19 marks)Consider a random variable S =

∑N

i=1 Xi where X1, X2 · · · are independent and identicallydistributed, and independent of N . Suppose that X1 is a continuous random variable. LetmS(t) denote the moment generating function of S. Suppose that you have found

m′

S(0)

mS(0)= 6,

m′′

S(0)mS(0) − (m′

S(0))2

(mS(0))2= 4,

[m(3)S (0)mS(0) − m′

S(0)m′′

S(0)](mS(0))2 − 2[m′′

S(0)mS(0) − (m′

S(0))2]mS(0)m′

S(0)

(mS(0))4= 3.

(a) [12 marks] Determine E[S], V ar(S) and E[(S − E[S])3]. Show all working.

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(b) [7 marks] Let G(x; α, β) denote the Gamma distribution function with parametersα and β. Find the translated gamma approximation for Pr(S ≤ 10) in terms of G(x; α, β)with the appropriate values of x, α and β being specified.

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Question 6. (17 marks)

Consider the Cramer-Lundberg surplus process for an insurance company with positiveinitial surplus, where the premium rate is c (c > 0), the number of claims follows a Poissonprocess with parameter λ and the claim amount distribution is exponential with parameter β.

(a) [2 marks] Determine the security loading θ. In other words, express θ in terms of c,λ and β only.

(b) [5 marks] Find an expression for the adjustment coefficient R in terms of c, λ and β

only.

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(c) [10 marks] Now, instead of assuming that the claim amount random variable is ex-ponentially distributed, we assume that the claim amount random variable is uniformlydistributed over the interval [a, b] (0 ≤ a < b). Define T to be the time of ruin, i.e., the firsttime that the surplus process becomes negative. Given that T < ∞, find the range of U(T ).

—END—

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