Help with the questions in the attachment.

10 Individual claim amounts in a Poisson claims process with a frequency of 50 claims per year for the whole portfolio have mean $5,000 and standard deviation f2,500. If the annual premium rate is f300,000, calculate the tightest upper bound for the adjustment coefficient. 1 An insurer calculates the annual premiums for fire insurance of flats by increasing the risk premium by 30% and adding a f30 loading. The claim frequency is 3% and individual claim amounts can be assumed to be: E2,000 with probability 0.9 f15,000 with probability 0.1. Calculate the insurer's adjustment coefficient for these policies, to 2 significant figures. 12 A Poisson claims process has security loading =2/5 and claim size density function: He f(x)=$ex+7e-7x, x>0 (i) Derive the moment generating function (MGF) for the claim size distribution, and state the values of t for which it is valid. [3] (ii) Calculate the value of the adjustment coefficient. [4] [Total 7] 13 (i) Show that the adjustment coefficient for a compound Poisson claims process satisfies the inequality: 2 [c/ )-E(XX)] E(X2 ) and define what each of the symbols represents. [9] (ii) An insurer considers that claims of a certain type occur in accordance with a compound Poisson process. The claim frequency for the whole portfolio is 100 per annum and individual claims have an exponential distribution with a mean of $8,000. (a) Calculate the adjustment coefficient if the total premium rate for the portfolio is (1,000,000 per annum. (b) Verify that the value calculated in (ii)(a) satisfies the inequality in (i). (c) The insurer decides to take out excess of loss reinsurance for this portfolio. The reinsurer has agreed to pay the excess of any individual claim above $20,000 in return for an annual premium of $80,000. Calculate the adjustment coefficient for the direct insurer when the reinsurance is in operation. (d) Estimate the direct insurer's probability of ultimate ruin with and without the reinsurance arrangement, assuming that the initial surplus is $20,000 and that future premiums remain at the same level