1 UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN Actuarial Science Program DEPARTMENT OF MATHEMATICS Math 478 / 568 Prof. Rick Gorvett Actuarial Modeling Spring, 2013 Old Actuarial Exam Problems Classical (Limited Fluctuation) Credibility (1) You are given the following information about a commercial auto liability book of business: (i) Each insured’s claim count has a Poisson distribution with mean , where has a gamma distribution with = 1.5 and = 0.2. (ii) Individual claim size amounts are independent and exponentially distributed with mean 5000. (iii) The full credibility standard is for aggregate losses to be within 5% of the expected with probability 0.90. Using classical credibility, determine the expected number of claims required for full credibility. (A) 2165 (B) 2381 (C) 3514 (D) 7216 (E) 7938 (Sample Questions, Exam C, Problem # 39) (2) You are given: (i) The number of claims follows a Poisson distribution. (ii) Claim sizes follow a gamma distribution with parameters (unknown) and = 10,000. (iii) The number of claims and claim sizes are independent. (iv) The full credibility standard has been selected so that actual aggregate losses will be within 10% of expected aggregate losses 95% of the time. Using limited fluctuation (classical) credibility, determine the expected number of claims required for full credibility. (A) Less than 400 (B) At least 400, but less than 450 (C) At least 450, but less than 500 (D) At least 500 (E) The expected number of claims required for full credibility cannot be determined from the information given. (Sample Questions, Exam C, Problem # 245)

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UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN Actuarial Science Program

DEPARTMENT OF MATHEMATICS Math 478 / 568 Prof. Rick Gorvett Actuarial Modeling Spring, 2013

Old Actuarial Exam Problems

Classical (Limited Fluctuation) Credibility (1) You are given the following information about a commercial auto liability book of business:

(i) Each insured’s claim count has a Poisson distribution with mean , where has a gamma distribution with = 1.5 and = 0.2.

(ii) Individual claim size amounts are independent and exponentially distributed with mean 5000.

(iii) The full credibility standard is for aggregate losses to be within 5% of the expected with probability 0.90.

Using classical credibility, determine the expected number of claims required for full credibility. (A) 2165 (B) 2381 (C) 3514 (D) 7216 (E) 7938

(Sample Questions, Exam C, Problem # 39) (2) You are given:

(i) The number of claims follows a Poisson distribution. (ii) Claim sizes follow a gamma distribution with parameters (unknown) and =

10,000. (iii) The number of claims and claim sizes are independent. (iv) The full credibility standard has been selected so that actual aggregate losses will be

within 10% of expected aggregate losses 95% of the time. Using limited fluctuation (classical) credibility, determine the expected number of claims required for full credibility. (A) Less than 400 (B) At least 400, but less than 450 (C) At least 450, but less than 500 (D) At least 500 (E) The expected number of claims required for full credibility cannot be determined

from the information given. (Sample Questions, Exam C, Problem # 245)

(3) A company has determined that the limited fluctuation full credibility standard is 2000 claims if: (i) The total number of claims is to be within 3% of the true value with probability p. (ii) The number of claims follows a Poisson distribution. The standard is changed so that the total cost of claims is to be within 5% of the true value with probability p, where claim severity has probability density function:

.000,100 ,000,10

1)( xxf

Using limited fluctuation credibility, determine the expected number of claims necessary to obtain full credibility under the new standard. (A) 720 (B) 960 (C) 2160 (D) 2667 (E) 2880

(Sample Questions, Exam C, Problem # 273)

(Exam 4/C, Spring 2005)