PSTAT 173 FINAL WINTER 2020 CONTINUE TO NEXT PAGE FINAL EXAM PSTAT 173 – Winter 2020 By writing my name and signing below I swear that I have not engaged in any form of academic dishonesty or cheating. Name:__________________________________________________________________________ Signature:_______________________________________________________________________ Section meeting date & time:_______________________________________________________ • Partial credit will be awarded for all significant work. Full credit will not be awarded if the grader cannot understand how you arrived at your final answer, even if the final answer is correct. BOX or clearly MARK your final answers. • Write legibly, unreadable scans will result in a 0 point score. • Scan this entire completed exam, including this page, create a single pdf file, and e-mail that pdf file to BOTH [email protected] and [email protected]. • Do not write anything in the boxes shown below. Good luck! Problem 1 (20 Points) 2 (10 Points) 3 (10 Points) Score Problem 4 (10 Points) 5 (20 Points) 6 (10 Points) Score Problem 7 (10 Points) 8 (10 Points) TOTAL SCORE Score PSTAT 173 FINAL WINTER 2020 CONTINUE TO NEXT PAGE 1. (20 points) You are given: (i) First dollar claim sizes follow an exponential distribution with mean 10θ. (ii) The number of first dollar claims follow a Poisson distribution with mean 1.25θ. (iii) There is an ordinary deductible of 10θ⋅(ln 1.25). (iv) Claim counts and claim sizes are independent, given θ. (v) The prior distribution has probability density function: ( ) 6 5 , 1π θ θ θ = > Calculate Bühlmann’s k for aggregate losses. ` PSTAT 173 FINAL WINTER 2020 CONTINUE TO NEXT PAGE 2. (10 points) Suppose X has a Pareto distribution with α = 2 and θ = 1000. Calculate VaR0.99 and TVaR0.99 for X PSTAT 173 FINAL WINTER 2020 CONTINUE TO NEXT PAGE 3. (10 points) Suppose X has a Pareto distribution with α = 2 and θ = 1000 and Y has a loglogistic distribution with parameters γ = r and θ = 1000. Using the ratio of distribution functions, for what values of r is (a) Y heavier tailed than X? (b) Y lighter tailed than X? (c) X and Y have similar tails? PSTAT 173 FINAL WINTER 2020 CONTINUE TO NEXT PAGE 4. (10 points) You are given the following regarding coverage for automobile crash losses: (i) Claim counts are expected to follow a Poisson distribution with λ = 0.10. (ii) Claim sizes are expected to follow a Pareto distribution with α = 2.5 and θ = 4,500 (iii) We have observed three years of experience from a particular insured: Year 1 Year 2 Year 3 Vehicles 1,000 2,000 1,500 Claims 100 220 175 Total Losses 325,000 640,000 575,000 Using limited fluctuation credibility, with the full credibility standard to be within 5% of the true frequency 90% of the time and the normal approximation, find the credibility frequency estimate for this insured for the fourth year. PSTAT 173 FINAL WINTER 2020 CONTINUE TO NEXT PAGE 5. (20 points) You are given: (i) The annual number of claims for an individual risk follows a Poisson distribution with mean λ. (ii) For 75% of the risks, λ = 1. (iii) For 25% of the risks, λ = 3. A randomly selected risk had r claims in Year 1. The Bayesian estimate of this risk’s expected number of claims in Year 2 is 2.98. Determine the Bühlmann credibility estimate of the expected number of claims for this risk in Year 2. PSTAT 173 FINAL WINTER 2020 CONTINUE TO NEXT PAGE 6. (10 points) You are given total claims for two policyholders: Policyholder Year 1 Year 2 Year 3 Year 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. PSTAT 173 FINAL WINTER 2020 CONTINUE TO NEXT PAGE 7. (10 points) Losses in 2013 follow a two-parameter Pareto distribution with α = 2 and θ = 5. Losses in 2014 are uniformly 20% higher than in 2013. An insurance covers each loss subject to an ordinary deductible of 10 in each of the two years. Calculate the Loss Elimination Ratio in 2014. PSTAT 173 FINAL WINTER 2020 END OF EXAM 8. (10 points) A company insures a fleet of vehicles. Aggregate losses have a compound Poisson distribution. The expected number of losses is 20. Loss amounts, regardless of vehicle type, have exponential distribution with an expected value of 200. In order to reduce the cost of the insurance, two modifications are to be made: (i) a certain type of vehicle will not be insured. It is estimated that this will reduce loss frequency by 20%. (ii) a deductible of 100 per loss will be imposed. Calculate the expected aggregate amount paid by the insurer after the modifications.
