CREDIBILITY - PROBLEM SET 2 Bayesian Analysis - Discrete Prior

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1 CREDIBILITY - PROBLEM SET 2 Bayesian Analysis - Discrete Prior Questions 1 and 2 relate to the following situation Two bowls each contain 10 similarly shaped balls Bowl 1 contains 5 red and 5 white balls equally likely to be chosen) Bowl 2 contains 2 red and 8 white balls equally likely to be chosen) A bowl is chosen at random with each bowl having the chance of being chosen A ball is chosen from that bowl 1 Find the probability that the ball chosen is red A)! B) C)! D) E) 2 Suppose that the ball chosen is red Find the probability that bowl 1 was the chosen bowl ' A) B) C) D) E) Questions 3 and 4 relate to the following situation A portfolio of insurance policies consists of two types of policies Policies of type 1 each have a Poisson claim number per month with mean 2 per period and policies of type 2 each have a Poisson claim number with mean 4 per period of the policies are of type 1 and are of type 2 A policy is chosen at random from the portfolio and the number of claims generated by that policy in the following is the random variable \ 3 Find TÒ\! A) 0321 B) 0482 C) 0642 D) 0803 E) Suppose that a policy is chosen at random and the number of claims is observed to be 1 for that period Find the probability that the policy is of type 1 A) 85 B) 88 C) 91 D) 94 E) 97 5 A risk class is made up of three equally sized groups of individuals Groups are classified as Type A, Type B and Type C Any individual of any type has probability of 5 of having no claim in the coming year and has a probability of 5 of having exactly 1 claim in the coming year Each claim is for amount 1 or 2 when a claim occurs Suppose that the claim distributions given that a claim occurs, for the three types of individuals are Î B TÐclaim of amount BlType A and a claim occurs š ß Î B Î B TÐclaim of amount BlType B and a claim occurs š ß Î B Î' B TÐclaim of amount BlType C and a claim occurs š Þ Î' B An insured is chosen at random from the risk class and is found to have a claim of amount 2 Find the probability that the insured is Type A A) ' B) C) D) E) '

2 6 You are given the following: - A portfolio consists of 75 liability risks and 25 property risks - The risks have identical claim count distributions - Loss sizes for liability risks follow a Pareto distribution with parameters ) 300 and α 4 - Loss sizes for property risks follow a Pareto distribution ) 1,000 and α a) Determine the variance of the claim size distribution for this portfolio for a single claim b) A risk is randomly selected from the portfolio and a claim of size 5 is observed Determine the limit of the posterior probability that this risk is a liability risk as 5 goes to zero 7 A portfolio consists of 100 independent risks 25 of the risks have a policy with a 5,000 per claim policy limit, 25 of the risks have a policy with a 10,000 per claim policy limit, and 50 of the risks have a policy with a 20,000 per claim policy limit The risks have identical claim count distributions Prior to censoring by policy limits, claim sizes for each risk follow a Pareto distribution with parameters ) 5,000 and α 2 A claims report is available which shows the number of claims in various claim size ranges for each policy after censoring by policy limits, but does not identify the policy limit associated with each policy The claims report shows exactly one claim for a policy selected at random This claim falls in the claim size range of 9,000-11,000 Determine the probability that this policy has a 10,000 policy limit 8 CAS May 05) An insurer selects risks from a population that consists of three independent groups The claims generation process for each group is Poisson The first group consists of 50 of the population These individuals are expected to generate one claim per year The second group consists of 35 of the population These individuals are expected to generate two claims per year Individuals in the third group are expected to generate three claims per year A certain insured has two claims in year 1 What is the probability that this insured has more than two claims in year 2? A) Less than 21 B) At least 21, but less than 25 C) At least 25, but less than 29 D) At least 29, but less than 33 E) 33 or more 9 CAS May 06) Claim counts for each policyholder are independent and follow a common Negative Binomial distribution A priori, the parameters for this distribution are Ð< ß Ðß or Ð< ß Ðß Each parameter set is considered equally likely Policy files are sampled at random The first two files samples do not contain any claims The third policy file contains a single claim Based on this information, calculate the probability that Ð<ß Ðß A) Less than 03 B) At least 03, but less than 045 C) At least 045, but less than 06 D) At least 06, but less than 075 E) At least 075

3 10 A portfolio of insurance policies consists of three types of policies The loss distribution for each type of policy is summarized as follows: Policy Type Type I Type II Type III Loss Distribution Exponential Exponential Exponential With Mean 2 With Mean 4 With Mean 8 Half of the policies are of Type I, one-quarter of the policies are of Type II and one-quarter are Type III A policy is chosen at random, and the loss amount is \ a) Find IÒ\ b) Find Z+<Ò\each of the following two ways: i) Z +<Ò\ IÒ\ ÐIÒ\ ii) Z +<Ò\ Z +<ÒIÒ\lX IÒZ +<Ò\lX, where X ÖMßMMßMMM is the random variable describing the Type of policy chosen 11 SOA) You are given: i) A portfolio of independent risks is divided into two classes, Class A and Class B ii) There are twice as many risks in Class A as in Class B iii) The number of claims for each insured during a single year follows a Bernoulli distribution iv) Classes A and B have claim size distributions as follows: Claim Size Class A Class B 50, , v) The expected number of claims per year is 022 for Class A and 011 for Class B One insured is chosen at random The insured s loss for two years combined is 100,000 Calculate the probability that the selected insured belongs to Class A A) 055 B) 057 C) 067 D) 071 E) Prior to tossing a coin, it is believed that the chance of tossing a head is equally likely to be or The coin is tossed twice, and both tosses result in a head Determine the posterior probability of tossing a head A) B) C) D) E) 13 A portfolio of insureds consists of two types of insureds Losses from the two types are: Type 1 insured loss: exponential with a mean of 1, Type 2 insured loss: exponential with a mean of 2, In the portfolio, of the insureds are of Type 1 and of the insureds are of Type 2 Two insureds are chosen at random and one loss is observed from each insured The first insured is observed to have a loss of 1 and the second insured is observed to have a loss of 2 Find the probability that the two insured are of the same Type It is assumed that the losses of the two insureds are independent of one another

4 CREDIBILITY - PROBLEM SET 2 SOLUTIONS 1 TÒredTÒredlbowl 1 TÒbowl 1 TÒredl bowl TÒbowl 2 Ð! Ð Ð! Ð! Answer: C TÒbowl 1 red TÒredlbowl 1 TÒbowl 1 Ð! Ð 2 TÒbowl 1lred T Òred Î! Î! Answer: D 3 T Ò\ 5 0\ Ð5 / / 0\l] Ð5ltype 3 0] Ð3 Ð 5x Ð Ð 5x Ð This is a mixture of two Poisson distributions 0\ Ð5 0 Ð5 + 0 Ð5 +!! / / For 5! this is T Ò\! Ð!x Ð Ð!x Ð Þ!*' Answer: E 4 TÒType 1l\ T Ò\lType 1 T ÒType 1 TÒ\ T Ò\lType 1 T ÒType 1 T Ò\lType 1 T ÒType 1 T Ò\l Type T Ò Type Ð/ x Ð Þ)) Ð/ Ð Ð/ Answer: B Ð x x T Ò\lA T ÒE T Ò\lA T ÒE 5 TÒType Al\ T Ò\ T Ò\lE T ÒE T Ò\lF T ÒF T Ò\lG T ÒG ÒÐÞÐ Ð ÒÐÞÐ Ð ÒÐÞÐ Ð ÒÐÞÐ Ð Answer: B ' 6 a) The unconditional claim model is a mixture of two Paretos The moments will be the mixture of the corresponding Pareto moments, with weights 75 and 25 ) ) We use the Pareto moments first moment) and α Ðα Ðα second moment)!!!!! IÒ] ÐÞÐ ÐÞÐ!! Ð!! Ð!!! IÒ] ÐÞ Ð Ð ÐÞ Ð Ð ß!!!ß!!! ß!! Z +<Ò] ß!! Ð!! ß!! Ð!! 0Ð5lP0ÐP ÐÞ Ð5!! b) 0ÐPl5 0Ð5lP0ÐP 0Ð5lT 0ÐT Þ Ð!! Ð!!! ÐÞ ÐÞ!! ÐÞ As 5p! the limit is Þ*! ÐÞ ÐÞ!!!!! Ð5!! Ð5!!!

5 7 P, P! and P! will denote the event that the risk has limit 5, 10 and 20 thousand, respectively The prior probabilities for policy limits for a randomly selected policy are T ÒP T ÒP! Þ and T ÒP! Þ T Ò* \ lp! T ÒP! We wish to find the posterior probability T ÒP! l* \ T Ò* \ The distribution function of the Pareto distribution with ) 5,000 and α 2 ß!!! is JÐB Ð B ß!!! The table of calculations is as follows Prior Prob TÒP Þ TÒP Þ TÒP Þ!! Model Prob T Ò* \ lp T Ò* \ lp! T Ò* \ lp!! J Ð*ß!!! J Ðß!!! J Ð*ß!!! Ð Þ' Ð Ð ' Þ!** Note that if a claim payment is between 9,000 and 11,000, it is not possible that the policy had a limit of 5,000 That is the reasoning behind T Ò* \ lp! If a policy has a limit of 10,000, then no payment will be over 10,000 so to say that the payment is between 9,000 and 11,000 is the same as saying that the payment is over 9,000 Therefore, T Ò* \ lp J Ð*ß!!!! Joint Prob TÒ* \ P TÒ* \ P! TÒ* \ P!! ÐÞ'ÐÞ ÐÞ!**ÐÞ Þ!* Þ!* Marginal Prob TÒ* \ TÒ* \ P T Ò* \ P! T Ò* \ P!! Þ!* Þ!* Þ!') T Ò* \ P! Þ!* Posterior Prob T ÒP! l* \ T Ò* \ Þ!') Þ') We have used the following conditional probability rules to solve this problem We define the following events: J 5,000 limit, J 10,000 limit, J 20,000 limit and I is the event * \ Then the expression above for TÒ* \ is TÒITÒIlJ TÒJ TÒIlJ TÒJ TÒIlJ TÒJ, and T ÒIlJ T ÒJ TÒJ li T ÒIlJ T ÒJ T ÒIlJ T ÒJ T ÒIlJ T ÒJ, which is T Ò* \ lp! T ÒP! Ð * ÐÞ T ÒP! l* \ T Ò* \ Þ')! Ð ÐÞ ÒÐ Ð ÐÞ * *

6 8 Let us label the groups as Group 1 50), Group 2 35) and Group 3 15), and let \ denote the annual number of claims of an individual chosen at random for the three groups combined The conditional distribution of \ given that the individual is chosen from Group 1 is Poisson with a mean of 1, given that the individual is chosen from Group 2 \ is Poisson with a mean of 2, and for Group 3 \ is Poisson with mean 3 We wish to find TÒ\ l\, where \ and \ are the numbers of claims years 1 and 2, respectively By the definition of conditional probability, this probability is TÒ\ \ TÒ\ l\ TÒ\ We use the rule that if F ß F ß ÞÞÞß F5 is a partition of a probability space, then TÒETÒElF TÒF TÒElF TÒF â TÒElF TÒF 5 5 With event E as \ and FßFßF as the events individual is from Group 1, 2, 3, the denominator is TÒ\ TÒ\ lgroup 1 TÒGroup 1 TÒ\ lgroup 2 TÒGroup 2 TÒ\ lgroup 3 TÒGroup 3 / / / Ð ÐÞ Ð ÐÞ Ð ÐÞ Þ! The numerator is TÒ\ \ TÒ\ \ lgroup 1 TÒGroup 1 TÒ\ \ lgroup 2 TÒGroup 2 TÒ\ \ lgroup 3 TÒGroup 3 For each particular group, the numbers of events in successive years are independent, so that TÒ\ \ lgroup 1 TÒ\ lgroup 1 TÒ\ lgroup 1 / / Ð Ò/ / Ð ÐÞ!)!!ÐÞ)*! Þ!, and TÒ\ \ lgroup 2 TÒ\ lgroup 2 TÒ\ lgroup 2 / / Ð Ò/ / Ð ÐÞÐÞ!' Þ!), and TÒ\ \ lgroup 3 TÒ\ lgroup 3 TÒ\ lgroup 3 / / Ð Ò/ / Ð ÐÞ')!ÐÞ! Þ*! Then T Ò\ \ ÐÞ!ÐÞ ÐÞ!)ÐÞ ÐÞ*!ÐÞ Þ!!!, Þ!!! and T Ò\ l\ Þ! Þ'! Answer: C 9 We are asked to find TÒÐßl\!ß\!ß\ TÒÐß \!ß\!ß\ This is TÒ\!ß\!ß\ T ÒÐß \!ß \!ß \ T Ò\!ß \!ß \ lðß T Ðß Since the conditional distribution of \ given < ß is negative binomial, we get < ÐÐ T Ò\!lÐß Ð < *, and T Ò\ lðß Ð <, so that T Ò\!ß \!ß \ lðß Ð * Ð * Ð Then, T ÒÐß \!ß \!ß \ Ð Ð, since T Ðß In a similar way, we get T ÒÐß \!ß \!ß \ T Ò\!ß \!ß \ lðß T Ðß, where TÒ\!ß\!ß\ lðß Ð Ð Ð, so that TÒÐß \!ß\!ß\ Ð Ð

7 9 continued Then TÒ\!ß\!ß\ T ÒÐß \!ß \!ß \ T ÒÐß \!ß \!ß \ 2 TÒÐß \!ß\!ß\ TÒ\!ß\!ß\ Þ)) and Answer: E, 10a) IÒ\ IÒ\lM TÐM IÒ\lMM TÐMM IÒ\lMMM TÐMMM ÐÐ ÐÐ Ð)Ð b)i) IÒ\ IÒ\ lm TÐM IÒ\ lmm TÐMM IÒ\ lmmm TÐMMM Ð Ð Ð Ð Ð ) Ð Z+<Ò\ ) X M ß prob X M ß prob ii) IÒ\lX X MMß prob and Z +<Ò\lX X MMßprob ) X MMM ß prob ) X MMM ß prob Z+<ÒIÒ\lXÐ ) Ð ) ' IÒZ +<Ò\lX Ð ) Z +<Ò\ Z +<Ò IÒ\lX IÒ Z +<Ò\lX ' ) Þ 11 This is a fairly standard application of Bayesian analysis, using conditional probability rules Let Eand F denote the events that the chosen risk is from Class A or Class B, respectively, and let \ and \ denote the claim amounts in years 1 and 2, respectively T ÒEl\ \!!ß!!! T ÒE Ð\ \!!ß!!! T Ò\ \!!ß!!! T ÒE Ð\ \!!ß!!! T ÒE Ð\ \!!ß!!! T ÒF Ð\ \!!ß!!! T Ò\ \!!ß!!!lE T ÐE T Ò\ \!!ß!!!lE T ÐE T Ò\ \!!ß!!!lF T ÐF We are given TÐE ß TÐF since there are twice as many risks in Class A as in Class B In Class A there is a 22 chance of a claim in each year, and in Class B there is a 11 chance of a claim in each year In order for the total claim in two years to be 100,000, one of three possibilities must occur: \!!ß!!! and \!, or \!ß!!! and \!ß!!!, or \! and \!!ß!!! Therefore, T Ò\ \!!ß!!!lE T ÒÐ\!!ß!!! Ð\!lE T ÒÐ\!ß!!! Ð\!ß!!!lE T ÒÐ\! Ð\!!ß!!!lE

8 11 continued In order for the annual claim to be 100,000 in Class A, there must be a claim, and it must be for 100,000; this has probability ÐÞÐÞ In order for the annual claim in Class A to be 0, there must be no claim; this has probability Þ) In order for the annual claim to be 50,000 in Class A, there must be a claim, and it must be for 50,000; this has probability ÐÞÐÞ' Then, T Ò\ \!!ß!!!lE ÐÞÐÞÐÞ) ÐÞÐÞ'ÐÞÐÞ' ÐÞ)ÐÞÐÞ Þ In a similar way, we get T Ò\ \!!ß!!!lF ÐÞÐÞ'ÐÞ)* ÐÞÐÞ'ÐÞÐÞ' ÐÞ)*ÐÞÐÞ' Þ')) ÐÞÐ Then, T ÒEl\ \!!ß!!! Þ!* ÐÞÐ ÐÞ'))Ð The order of calculations can be summarized as follows TÒE, given TÒF, given T Ò\ \!!ß!!!lE Þ T Ò\ \!!ß!!!lF Þ')) T ÒE Ð\ \!!ß!!! T ÒF Ð\ \!!ß!!! T Ò\ \!!ß!!!lE T ÒE T Ò\ \!!ß!!!lF T ÒF ÐÞÐ ÐÞÐ T Ò\ \!!ß!!! T ÒE Ð\ \!!ß!!! T ÒF Ð\ \!!ß!!! ÐÞÐ ÐÞÐ ÐÞÐ T ÒEl\ \!!ß!!! Þ!* ß ÐÞÐ ÐÞ'))Ð ÐÞ'))Ð T ÒFl\ \!!ß!!! Þ* Answer: D ÐÞÐ ÐÞ'))Ð 12 Posterior probability of H is TÒ3Hl1H and 2H l TÒ 3H,1H,2H TÒ1H,2H We will denote by H the event that the probability of tossing a head with the coin is, with a similar definition for H TÒ1H,2HTÒ1H,2Hl H TÒ H TÒ1H,2Hl H TÒ H Ð Ð Ð Ð ) TÒ3H,1H,2HTÒ3H,1H,2Hl H TÒ H TÒ3H,1H,2Hl H TÒ H Ð Ð Ð Ð ' ) Î) Then TÒ3Hl1H and 2H l Î) Answer: C

9 13 We will define \ and \ to be the loss amounts from the first and second insured, respectively We are given that \ and \ We will define A to be the Öß random variable that identifies Type, and A and A are the types of insured 1 and insured 2 We wish to find TÐA A l\ ß\ Using the definition of conditional probability, this is T ÒÐA A Ð\ ß\ TÐ\ß\ Þ Since A must be 1 or 2, the numerator can be formulated as TÒÐA A Ð\ ß\ TÒÐA A Ð\ ß\ TÒÐA A Ð\ ß\ TÐ\ ß\ la A TÐA A TÐ\ ß\ la A TÐA A Since both policies are chosen at random, TÐA A and TÐA A Also, since the insureds are independent of one another, T Ð\ ß \ la A TÐ\ la A TÐ\ la A / / / the notation T really means density of \, not probability, in this situation) Similarly, T Ð\ ß \ la A Î Î Î TÐ\ la A TÐ\ la A / / / Then, T ÒÐA A Ð\ ß \ / / Þ!'* Î Also, TÐ\ ß\ TÐ\ TÐ\ because of independence of \ and \ B BÎ Each of the \ 3's is a mixture of two exponenetials with pdf Ð/ /, so Î Î TÐ\ Ð/ /, and TÐ\ Ð/ /, and Î Î T Ð\ ß \ Ð/ / Ð/ / Þ!! Þ!'* Finally, T ÐA A l\ ß \ Þ!! Þ*