10 Cumulative claims incurred on a motor insurance account are as follows: Cumulative claims Development year incurred (f'000) 0 1 2 2010 1,417 1,923 2,101 Policy year 2011 1,701 2,140 2012 1,582 The data have already been adjusted for inflation. Annual premiums written in 2012 were f3,073,000 and the ultimate loss ratio has been estimated as 92%. Claims paid to date for policy year 2012 are $441,000, and claims are assumed to be fully run-off by the end of Development year 2. Estimate the outstanding claims to be paid arising from policies written in 2012 only, using the Bornhuetter-Ferguson technique. [6] 11 The following table gives the cumulative incurred claims data, by years of accident and reporting development for a portfolio of motor insurance policies: Cumulative incurred Development year claims (f'000) 0 1 2 2010 252 375 438 Accident 2011 230 343 year 2012 208Number of reported Development year claims 0 1 2 2010 56 74 87 Accident 2011 49 65 year 2012 44 (1) Given that the total claims paid to date are 1950,000 for Accident years 2010 to 2012 calculate the outstanding claims reserve for this cohort using the average cost per claim method with grossing-up factors. [7] (ii) State the assumptions that underlie your result. [2] [Total 9] 12 Aggregate claims on a general insurance company's portfolio form a compound Poisson process with parameter 1. Individual claims have an exponential distribution with mean 100. The company applies a 20% premium loading. The insurer effects proportional reinsurance with a retained proportion of a . The reinsurer applies a 30% premium loading. (1) Calculate the minimum value of a such that the insurer's net income is greater than the expected net claims. [2] (ii) Hence, show that the direct insurer's adjustment coefficient, R , satisfies: 1-3a R = [4] 1000 -1,300q2 (iii) By differentiating the result from (ii), show that a =0.6257 maximises the adjustment coefficient and calculate the corresponding optimal value of R. You may assume that the turning point is a maximum. [4] [Total 10]