Assignment 2, ACMA 315, Spring, 2005
Due Date: March 24 in class (10 marks each)
Q.1 Past data on a portfolio of group policyholders are given in the following table. Estimate the Bühlmann credibility premiums to be charged to each group in year 4.
Policyholder / Year1 / 2 / 3 / 4
Claims / 1 / - / 20000 / 24000 / ?
No. in group / - / 100 / 120 / 110
Claims / 2 / 18000 / 18000 / 15400 / ?
No. in group / 90 / 75 / 70 / 80
Claims / 3 / 28500 / 36000 / 34200 / ?
No. in group / 160 / 180 / 190 / 200
Q.2 You are given the following information on towing losses for two classes of insureds:
Exposures / Pure Premium per individualYear / Male / Female / Total / Year / Male / Female / Total
2000 / 1000 / 400 / 1400 / 2000 / 1 / 15 / 5.000
2001 / 2000 / 300 / 2300 / 2001 / 5 / 2 / 4.609
2002 / 1500 / 200 / 1700 / 2002 / 6 / 15 / 7.059
2003 / 500 / 100 / 600 / 2003 / 4 / 4 / 4.000
Total / 5000 / 1000 / 6000 / Total / 4.4 / 10 / 5.333
Determine the nonparametric credibility premiums for the male and female classes, respectively, using the method that preserves total losses.
Q. 3 You are given the following experience for two insured groups:
Group / Year 1 / Year 2 / Year 31 / Number of members / 8 / 12 / 5
Average loss per member / 80 / 100 / 32
2 / Number of members / 10 / 20 / 20
Average loss per member / 90 / 80 / 75
Determine the nonparametric credibility premium for groups 1 and 2, respectively, using the method that preserves total losses.
Q.4 The distribution of automobile insurance policyholders by number of claims is given in the following table. Assume a (conditional) Poisson distribution for the number of claims per policyholder, estimate the Bühlmann credibility premium for the number of claims next year.
No. of claims / 0 / 1 / 2 / 3 / 4 / totalNo. of insureds / 2400 / 240 / 24 / 5 / 2 / 2671
Q.5 Suppose that, given Θ, X1, …, Xn are independently geometrically distributed with parameter Θ. (a) Show that µ(θ) = θ and v(θ) = θ(1+θ).
(b) prove that a= v - µ - µ2.
(c) Rework Q.4 assuming a (conditional) geometric distribution.
Q.6 A group of 500 insureds submit 400 injury claims in a one-year period as given in the table below. Each insured is assumed to have a Poisson distribution for the number of injuries, but the mean of such a distribution may vary from one insured to another. If a particular insured experienced three claims in the observation period, determine the Bühlmann credibility estimate for the number of claims for this insured in the next period.
No. of claims / 0 / 1 / 2 / 3 / 4No. of insureds / 245 / 150 / 75 / 20 / 10
Q.7 The number of claims follows a negative binomial distribution with parameters β and r, where β is unknown and r is known. You wish to estimate β based on n observations, where is the mean of these observations. Determine the maximum likelihood estimate of β .
Q.8 You are given the following information about a credibility model:
First Observation / Unconditional Probability / Bayesian Estimate of Second Observation1 / 1/3 / 1.5
2 / 1/3 / 1.5
3 / 1/3 / 3.0
Determine the Bühlmann credibility estimates of the second observation, given that the first observation is 1, 2 and 3, respectively.
Q.9 You are given four classes of insureds, each of whom may have zero or one claim, with the
following probabilities:
class / 1 / 2 / 3 / 4No. of Claim / 0 / 0.9 / 0.8 / 0.4 / 0.6
1 / 0.1 / 0.2 / 0.6 / 0.4
Probability(class) / 1/4 / 1/8 / 1/2 / 1/8
A class is selected at random (with probability given above), and five insureds are selected at random from the class. The total number of claims is three. If six insureds are selected at random from the same class, estimate the total number of claims using Bühlmann-Straub credibility.
Q.10 You are given:
(1) Claim counts follow a Poisson distribution with mean θ.
(2) Claim sizes follow an exponential distribution with mean 12θ.
(3) Claim counts and claim sizes are independent, given θ.
(4) The prior distribution has probability density function:π(θ) = 6/θ7, θ > 1.
Calculate Bühlmann’s k for aggregate losses.
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