ACTL5106 INSURANCE RISK MODELSQuiz 2, Year 2004

Please READ instructions carefully. Please write your name and student numberat the spaces provided:

Name: Student ID:

There are four (4) questions here and you are to answer all four. Provide detailsof your workings. You will be marked according to the quality of the presentationand accuracy of your solutions. Please return this questionnaire. We will NOTmark your paper if you do not return it.

Question No. 1: [25 points]

An insurer has a surplus process with a compound Poisson claims process. Individual claimamounts have probability density function:

fX(x) = ex, for x > 0.

The probability of ruin is given by

(u) =4

5eu, for u 0

where u is the initial surplus.

Determine the value of the parameter in the individual claims distribution.

Question No. 2: [25 points]

Aggregate claims S = X1 + +XN has a compound Poisson distribution with = 12and

individual claim amounts have the following probability distribution:

x p(x)1 0.42 0.33 0.24 0.1

Show that you can actually write the probability distribution of S as

fS(x) =1

2

4Xh=1

hxp(h) fS(x h) .

Use this formula to compute P (S 2).

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Question No. 3: [25 points]

The amount of a claim, X, is Uniformly distributed over the interval [0, ]. The prior densityof is

() = 200 3 for > 10.

Two claims, x1 = 15 and x2 = 5 have been observed.

Calculate the Bayesian premium, E(X3 |x1, x2 ).

Question No. 4: [25 points]

You are given the following past claims data on a portfolio of three (3) classes of policyholders:

YearClass 1 2 31 70 80 602 80 50 67.53 65 85 75

From these observed claims, the sum of squares can be computed asX3j=1

X3t=1

Xjt Xj

2= 854.1667

and X3j=1

Xj X

2= 42.1296.

Estimate the Buhlmann credibility premium to be charged in year 4 for each class of poli-cyholder.

- end of Quiz 2 questions -

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