Kindly assist me to obtain clear solutions for the question below.

Claims occur on a portfolio of insurance policies according to a Poisson process with Poisson parameter 1. Claim amounts, X1, Xy,.., are assumed to be identically distributed with moment generating function My [t) . The insurer calculates premiums using a loading factor (2 0). The insurer's adjustment coefficient, R. is defined to be the smallest positive root of the equation: 2 + cr = AM x (r) where c is the insurer's premium income rate. (i) Using the above equation for R, or otherwise, show that, provided R is small, an approximation to R is R , where: where / = E[X; ] and of = var[X;]. [4] (ii) Describe how the adjustment coefficient can be used to assess reinsurance arrangements on the basis of security. [3] (iii) The Poisson parameter, 1, for this portfolio is 20 and all individual claims are for a fixed amount of f5,000. The insurer's premium loading factor, 0, is 0.15 and proportional reinsurance can be purchased from a reinsurer who calculates premiums using a loading factor of 0.25. Calculate the maximum proportion of cach claim that could be reinsured so that the insurer's security, measured by A, is greater than the insurer's security without reinsurance. [9] [Total 16]