SOCIETY OF ACTUARIES/CASUALTY ACTUARIAL SOCIETY EXAM C CONSTRUCTION AND EVALUATION OF ACTUARIAL MODELS EXAM C SAMPLE QUESTIONS Copyright 2008 by the Society of Actuaries and the Casualty Actuarial Society Some of the questions in this study note are taken from past SOA/CAS examinations. C-09-08 PRINTED IN U.S.A. 1. You are given: (i) Losses follow a loglogistic distribution with cumulative distribution function: b g b g x/θγ F x = b g γ 1+ x/θ (ii) The sample of losses is: 10 35 80 86 90 120 158 180 200 210 1500 Calculate the estimate of θ by percentile matching, using the 40th and 80th empirically smoothed percentile estimates. (A) Less than 77 (B) At least 77, but less than 87 (C) At least 87, but less than 97 (D) At least 97, but less than 107 (E) At least 107 2. You are given: (i) The number of claims has a Poisson distribution. (ii) Claim sizes have a Pareto distribution with parameters θ= 0.5 and α=6. (iii) The number of claims and claim sizes are independent. (iv) The observed pure premium should be within 2% of the expected pure premium 90% of the time. Determine the expected number of claims needed for full credibility. (A) Less than 7,000 (B) At least 7,000, but less than 10,000 (C) At least 10,000, but less than 13,000 (D) At least 13,000, but less than 16,000 (E) At least 16,000 C-09-08 - 1 - 3. You study five lives to estimate the time from the onset of a disease to death. The times to death are: 2 3 3 3 7 Using a triangular kernel with bandwidth 2, estimate the density function at 2.5. (A) 8/40 (B) 12/40 (C) 14/40 (D) 16/40 (E) 17/40 4. You are given: (i) Losses follow a Single-parameter Pareto distribution with density function: α f (x)= , x>1, 0 < α < ∞ (α+1) x (ii) A random sample of size five produced three losses with values 3, 6 and 14, and two losses exceeding 25. Determine the maximum likelihood estimate of α. (A) 0.25 (B) 0.30 (C) 0.34 (D) 0.38 (E) 0.42 C-09-08 - 2 - 5. You are given: (i) The annual number of claims for a policyholder has a binomial distribution with probability function: ⎛2⎞ p(x q)= qx(1−q)2−x, x = 0, 1, 2 ⎜ ⎟ ⎝x⎠ (ii) The prior distribution is: π(q)=4q3, 0<q<1 This policyholder had one claim in each of Years 1 and 2. Determine the Bayesian estimate of the number of claims in Year 3. (A) Less than 1.1 (B) At least 1.1, but less than 1.3 (C) At least 1.3, but less than 1.5 (D) At least 1.5, but less than 1.7 (E) At least 1.7 6. For a sample of dental claims x , x ,..., x , you are given: 1 2 10 (i) ∑x =3860 and ∑x2 = 4,574,802 i i (ii) Claims are assumed to follow a lognormal distribution with parameters μ and σ. (iii) μ and σ are estimated using the method of moments. Calculate E X ∧500 for the fitted distribution. (A) Less than 125 (B) At least 125, but less than 175 (C) At least 175, but less than 225 (D) At least 225, but less than 275 (E) At least 275 C-09-08 - 3 - 7. DELETED 8. You are given: (i) Claim counts follow a Poisson distribution with mean θ. (ii) Claim sizes follow an exponential distribution with mean 10θ. (iii) Claim counts and claim sizes are independent, given θ. (iv) The prior distribution has probability density function: b g 5 πθ = , θ>1 θ6 Calculate Bühlmann’s k for aggregate losses. (A) Less than 1 (B) At least 1, but less than 2 (C) At least 2, but less than 3 (D) At least 3, but less than 4 (E) At least 4 9. DELETED 10. DELETED 11. You are given: (i) Losses on a company’s insurance policies follow a Pareto distribution with probability density function: θ ( ) f xθ = , 0< x<∞ (x+θ)2 (ii) For half of the company’s policies θ=1, while for the other half θ=3. For a randomly selected policy, losses in Year 1 were 5. C-09-08 - 4 - Determine the posterior probability that losses for this policy in Year 2 will exceed 8. (A) 0.11 (B) 0.15 (C) 0.19 (D) 0.21 (E) 0.27 12. You are given total claims for two policyholders: Year Policyholder 1 2 3 4 X 730 800 650 700 Y 655 650 625 750 Using the nonparametric empirical Bayes method, determine the Bühlmann credibility premium for Policyholder Y. (A) 655 (B) 670 (C) 687 (D) 703 (E) 719 13. A particular line of business has three types of claims. The historical probability and the number of claims for each type in the current year are: Historical Number of Claims Type Probability in Current Year A 0.2744 112 B 0.3512 180 C 0.3744 138 You test the null hypothesis that the probability of each type of claim in the current year is the same as the historical probability. C-09-08 - 5 - Calculate the chi-square goodness-of-fit test statistic. (A) Less than 9 (B) At least 9, but less than 10 (C) At least 10, but less than 11 (D) At least 11, but less than 12 (E) At least 12 14. The information associated with the maximum likelihood estimator of a parameter θ is 4n, where n is the number of observations. Calculate the asymptotic variance of the maximum likelihood estimator of 2θ. 1 (A) 2n 1 (B) n 4 (C) n (D) 8n (E) 16n C-09-08 - 6 - 15. You are given: (i) The probability that an insured will have at least one loss during any year is p. (ii) The prior distribution for p is uniform on [0,0.5]. (iii) An insured is observed for 8 years and has at least one loss every year. Determine the posterior probability that the insured will have at least one loss during Year 9. (A) 0.450 (B) 0.475 (C) 0.500 (D) 0.550 (E) 0.625 16-17. Use the following information for questions 21 and 22. For a survival study with censored and truncated data, you are given: Number at Risk Time (t) at Time t Failures at Time t 1 30 5 2 27 9 3 32 6 4 25 5 5 20 4 16. The probability of failing at or before Time 4, given survival past Time 1, is q . 3 1 Calculate Greenwood’s approximation of the variance of q(cid:3) . 3 1 (A) 0.0067 (B) 0.0073 (C) 0.0080 (D) 0.0091 (E) 0.0105 C-09-08 - 7 - b g 17. Calculate the 95% log-transformed confidence interval for H 3 , based on the Nelson-Aalen estimate. (A) (0.30, 0.89) (B) (0.31, 1.54) (C) (0.39, 0.99) (D) (0.44, 1.07) (E) (0.56, 0.79) 18. You are given: (i) Two risks have the following severity distributions: Probability of Claim Probability of Claim Amount of Claim Amount for Risk 1 Amount for Risk 2 250 0.5 0.7 2,500 0.3 0.2 60,000 0.2 0.1 (ii) Risk 1 is twice as likely to be observed as Risk 2. A claim of 250 is observed. Determine the Bühlmann credibility estimate of the second claim amount from the same risk. (A) Less than 10,200 (B) At least 10,200, but less than 10,400 (C) At least 10,400, but less than 10,600 (D) At least 10,600, but less than 10,800 (E) At least 10,800 C-09-08 - 8 - 19. You are given: (i) A sample x ,x ,…,x is drawn from a distribution with probability density function: 1 2 10 1[1exp(− x)+ 1 exp(− x)], 0< x<∞ 2 θ θ σ σ (ii) θ>σ (iii) ∑x =150 and ∑x2 =5000 i i Estimate θ by matching the first two sample moments to the corresponding population quantities. (A) 9 (B) 10 (C) 15 (D) 20 (E) 21 20. You are given a sample of two values, 5 and 9. 1 You estimate Var(X) using the estimator g(X , X ) = ∑(X − X)2. 1 2 i 2 Determine the bootstrap approximation to the mean square error of g. (A) 1 (B) 2 (C) 4 (D) 8 (E) 16 C-09-08 - 9 -