problemset2 - 1 Under a group insurance policy an insurer...

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Unformatted text preview: 1. Under a group insurance policy, an insurer agrees to pay 100% of the medical bills incurred during the year by employees of a small company, up to a maximum total of one million dollars. The total amount of bills incurred, X, has probability density function w(4«~m) f (x) = 9 ’ 0 < a: < 3 ,Where a: is measured in millions. 0, otherwise Calculate the total amount, in millions of dollars, the insurer would expect to pay under this policy. A) 0.120 B) 0.301 C) 0.935 D) 2.338 E) 3.495 2. Claim amounts follow a Weibull distribution with 7- = 2 and unknown 6. An insurer sets a policy limit of £00 and finds that 50% of the claims are beiow the policy limit. After a uniform inflation adjustment of 10% on ail claims amounts, find the percentage of the claims that will be below the policy limit of 100. A) 40% B) 42% C) 44% D) 46% E) 48% 3. A loss distribution is uniformly distributed on the interval from 0 to 100 . Two insurance policies are being considered to cover part of the loss. Insurance policy 1 insures 80% of the loss. Insurance policy 2 covers the loss up to a maximum insurance payment of L < 100. Both policies have the same expected payment by the insurer. Find the ratio Var insurer payment under policy 2 Vwr[insurer payment under policy 1] (HCarest _1)_ A) 1.5 B)«1.2 C) .9 D) .6 E) .3 4. (SOA) The unlimited severity distribution for claim amounts under an auto liability insurance policy is given by the cumulative distribution: F(a:) = 1 ... 0.863025” — 0.26—0‘00137, a: 2 0 The insurance policy pays amounts up to a limit of 1000 per claim. Caieulate the expected payment under this policy for one claim. A) 57 B) 108 C) 166 D) 205 E) 240 S. X and Y are random losses with joint density function f(3:,y) = 500mm) for 0 < a: < 100 and 0 < y < 100. An insurance policy on the tosses pays the total of the two losses to a maximum payment of 100. Find the expected payment the insurer will make on this policy (nearest 1). A) 90 B) 92 C) 94 D) 96 E) 98 t9, “Darwin "w—.-.... , ,m—«\ uwnvmw-wwwmwemww W;mm;man-”film.w_,w_tgimgmyyyruflgm awakening? mam t, m.m-i._.mm——wflmmmmmumxwmm“wflmsmPmfiMfiflmm ..... 7. (SOA) The length of time, in years, that a person will remember an actuarial statistic is modeied by an exponential distribution with mean 1/Y . In a certain population, Y has a gamma distribution with a = 6 2 2. Calculate the probability that a person drawn at random from this population will remember an actuarial statistic less than 1/2 year. A)0.125 amass C)0.500 marso E)0.875 8. (80A) You are given: (i) For Q m g, X1, X2, ..., Xm are independent, identically distributed Bernoulli random variables. with parameter q. (Ii) Sm = X1+X2+"+Xm (iii) The prior distribution of Q is beta with a 2: 1, b = 99, and 6 = 1 . Determine the smallest value of m such that the mean of the marginal distribution of Sm is greater than or equal to 50. A) 1082 B) 2164 C) 3246 D) 4950 E) 5000 9. (SOA) Bob is a carnival operator of a game in which a player receives a prize worth W = 2” if the player has N successes, N : 0, 1, 2, 3, Bob models the probability of success for a player as follows: (i) N has a Poisson distribution with mean A. (ii) A has a uniform distribution on the interval (0, 4). Calculate E[W] A) 5 B) 7 C) 9 D)ll E)13 10. The number of claims in one exposure period follows a Bernoulli distribution with mean p. The density function of p is assumed to be f(p) = % sin % , O < p < l . 2 Determine the expected number of claims. Hint: fol 3%? sin 1; dp m E . A) a in 3: Q a mi B> 23:3) ll. (SOA) An actuary for an automobile insurance company determines that the distribution of the annual number of claims for an insured chosen at tandem is modeled by the negative binomial distribution with mean 0.2 and variance 0.4. The number of claims for each individual insured has a Poisson distribution and the means of these Poisson distributions are gamma distributed over the population of insureds. Calculate the variance of this gamma distribution. A) 0.20 B) 0.25 C) 0.30 D) 0.35 E) 0.40 ‘g-(CAS) High-Roller Insurance Company insures the cost of injuries to employees of ACME Dynamite Manufacturing , Inc. ' 30% of injuries are fatal and the rest are "Pennanent Total." (PT). There are no other injury types. 1' Fatal injuries follow a log-logistic distribution with 6 2-. 400 and qr = 2. 0 PT injuries follow a log-logistic distribution with 6 x 600 and "y = 2. ' There is a $750 deductible per policy. Calculate the probability that an injury will result in a claim to High—Roller. A) Less than 30% B) At least 30%, but less than 35% C) At least 35%, but less than 40% B) At least 40%, but less than 45% B) At least 45% if, (CA8 May 2005) An insurance company offers two types of polices. Type 62 and type R. Type Q has non deductible, but a policy limit of 3,000—. Type R has no limit, but an ordinary deductible of d. Losses follow a Pareto distribution with 6 = 2, 000 and a = 3. Calculate the deductible d, such that both policies have the same expected cost per loss. A) Less than 50 B) At least 50, but less than 100 C) At least 100, but less than 150 D) At least 150, but less than 200 E) 200 or more [if (SOA) For an insurance: 0.023: 0 < a: < 10 (i) Losses have a density function fx(s;) : { 0 elsewhere (ii) The insurance has an ordinary deductible of 4 per loss. (iii) YP is the claim payment per payment random variable. Calculate E[YP] . A) 2.9 B) 3.0 C) 3.2 D) 3.3 E) 3.4 [5 (SOA) Insurance agent Hunt N. Quoturn will receive no annual bonus if the ratio of incurred csses to earned premiums for his book of business is 60% or more for the year. If the ratio is less than 60%, Hunt’s bonus will be a percentage of his earned premium equal to 15% of the difference between his ratio and 60%. Hunt‘s annual earned premium is 800,000. Incurred losses are distributed according to the Pareto distribution, with 8 = 500, 000 and o: = 2. Calculate the expected value ofI—Iunt’s bonus. A) 13,000 B) 17,000 C) 24,000 D) 29,000 E) 35,000 $5 (CA8 Nov 2005) In year 2005, claim amounts have the following Pareto distribution: 3 Fa) =1~ (.i‘tto) The annual inflation rate is 3%. A franchise deductible of 300 wiii be implemented in 2006. Calculate the loss elimination ratio of the franchise deductible. A) Less than 0.15 B) At least 0.15, but less than 0.20 C) At least 0.20, but less than 0.25 D) At least 0.25, but less than 0.30 B) At least 0.30 0? (SOA) The annual number of doctor visits for each individual in a famiiy of 4 has a geometric distribution with mean 1.5. The annual numbers of visits for the family members are mutually independent. An insurance pays 100 per doctor visit beginning with the 4-01 visit per family. Calculate the expected payments per year for this family. A) 320 B) 323 C) 326 D) 329 E) 332 fig {SOA May 07) You are given: (i) The frequency distribution for the number of losses for a policy with no deductible is negative binomial with r = 3 and )6 m 5. (ii) Loss amounts for this policy foiiow the Weibull distribution with 9 m 1000 and “if: 0.3. Determine the expected number of payments when a deductible of 200 is applied. A) Less than S B) At least 5, but less than 7 C) At least 7, but less than 9 D) At least 9, but less than i 1 B) At least 11 ...
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