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problemset4_b - 11(SOA Aggregate losses are modeled as...

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Unformatted text preview: 11. (SOA) Aggregate losses are modeled as follows: (i) The number of losses has a Poisson distribution with A = 3. (ii) The amount of each loss has a Burr (Burr Type XII, Singh-Maddala) distribution with a 2 3,6 = 2, and? = 1. (iii) The number of losses and the amounts of the losses are mutually independent. Calculate the variance of aggregate losses. A) 12 B) 14 C) 16 D) 18 E) 20 £2. (CA8 May 06) Prior to the application of any deductible, aggregate claim counts during 2005 followed a Poisson distribution with A = 14. Similarly, individual clairn sizes followed a Pareto distribution with a = 3 and 6 = 1000. Annual severity inflation is 10%. If all policies have a $250 ordinary deductible in 2005 and 2006, calculate the expected increase in the number of claims that will exceed the deductible in 2006. A) Fewer than 0.41 claims B) At least 0.41, but fewer than 0.45 C) At least 0.45, but fewer than 0.49 D) At least 0.49, but fewer than 0.53 E) At least 0.53 13. (CAS) An insurance portfolio produces N claims with the following distributionfi; Q P I N = n) 0 0.1 l 0.5 2 0.4 Individual claim amounts have the following distribution: .3; fxlml 0 0.7 10 0.2 20 0.1 Individual claim amounts and claim counts are independent. Calculate the probability that the ratio of aggregate claim amounts to expected aggregate claim amounts will exceed 4. A) Less than 3% B) At least 3%, but less than 7% C) At least 7%, but less than 11% D) At least 11%, but less than 15% B) At least 15% 14. (SOA) For an insured portfolio, you are given: (i) the number of claims has a geometric distribution with = % ; (ii) individual claim amounts can take on values 3, 4 or 5, with equal probability; (iii) the number of claims and claim amounts are independent; and (iv) the premium charged equals expected aggregate claims plus the variance of aggregate claims. Determine the exact probability that aggregate claims exceeds the premium. A) 0.01 B) 0.03 C) 0.05 D) 0.07 E) 0.09 15. (SOA) For a collective risk model the number of losses, X, has a Poisson distribution with A = 20. The common distribution of the individual losses has the following characteristics: (i) E[X] = 70 (ii) E[X /\ 30] = 25 (iii) Pr(X > 30) = 0.75 (iv) E{X2[X > 30] = 9000 An insurance covers aggregate losses subject to an ordinary deductible of 30 per loss. Calculate the variance of the aggregate payments of the insurance. A) 54,000 E) 67,500 C) 81,000 D) 94,500 B) 108,000 16. (SOA) You are given: (i) S has a compound Poisson distribution with /\ 2 2; and (ii) individual claim amounts 3 are distributed as follows: a: : l 2 f X(:r:) : 0.4 0.6 Determine f3(4) . A) 0.05 B) 0.07 C) 0.10 D) 0.15 E) 0.21 17. (SOA) The number of accidents follows a Poisson distribution with mean 152. Each accident 1 1 1 generates l, 2, or 3 claimants with probabilities 3-, g, 6’ respectively. Calculate the variance in the total number of claimants. A) 20 B) 25 C) 30 D) 35 E) 40 18. (SOA) Aggregate claims 5' has a compound Poisson distribution with discrete individual claim amount distribution: fx(1) = % , fX (3) = ~32 . Also, f3(4) = f3(3) +6f3(l) . Determine Var[S] . A) 76 B) 78 C) 80 D) 82 E) s4 19. (SOA) A compound Poisson distribution has A = 5 and claim amount distribution as follows: 32 13(3) 100 0.80 500 0.16 1000 0.04 Calculate the probability that aggregate claims will be exactly 600. A) 0.022 B) 0.038 C) 0.049 D) 0.060 E) 0.070 . “A“...mwmm 20. (SOA May 07) Annual aggregate losses for a dental policy follow the compound Poisson distribution with A = 3. The distribution of individual iosses is: Loss Probability 1 0.4 2 0.3 3 0.2 4 0.1 Calculate the probability that aggregate losses in one year do not exceed 3. A) Less than 0.20 B) At least 0.20, but less than 0.40 C) At least 0.40, but less than 0.60 D) At least 0.60, but less than 0.80 E) At least 0.80 ...
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