5. An insurer charges a loading of 28% on its policies with limit $1,000,000, and a...

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5. An insurer charges a loading of 28% on its policies with limit $1,000,000, and a loading of 26% on its policies with limit $500,000. It purchases stop- loss reinsurance of $500,000 over $500,000 for a loading of 45%. What percentage of the insurer's premiums for a policy with limit $1,000,000 are paid to the reinsurer? Let x be the expected losses limited to $500,000 and let y be the expected losses limited to $1,000,000. The insurer's premium for a policy with limit $500,000 is 1.26x, and the insurer's premium for a policy with limit $1,000,000 is 1.28y. The reinsurer's premium is 1.45(y - x). Since this is the difference between the insurers two premiums, we have 1.26x1.45(y - x) = 1.28y 0.17y = 0.19x y = 1.11764705882x The insurer collects 1.28y = 1.43058823529x in premiums, and pays 1.45 (y- x) = 0.170588235289x to the reinsurer. The percentage paid to the rein- surer is therefore 0.170588235289 = 11.92%. 1.43058823529 6. Policyholders are assumed to have a utility function u(x) = =e 15000 e-30000. Policyholder wealth is assumed to follow an exponential distri- bution with mean $10,000. An insurance company is considering selling an insurance policy which covers a risk which causes a loss of $3,000 with probability 0.02. The expenses for this policy are $2 million plus $2 per policy sold. If there are 2 million policyholders who might buy the policy, what will the expected profit on this policy be for the insurance company if they set the premium for each policy at $65? For a policyholder with wealth x, the expected utility without insurance is E(u(x)) = == (0.98e-15000 000 + 30000 0.98e 2-3000 +0.02e 15000 +0.02e 30000 2-3000 3000 3000 = =e 15000 0.98 0.02e 15000 30000 - e 0.98 0.02e 30000 = -1.00442805516 15000 1.00210341836e30000 -66.2742025389 15000 x-63.0362781846 30000 If the policy has premium p < 66.2742025389, then a policyholder will buy it if x-P 15000 x-P 30000 e 15000 (e , 15000 66.2742025389 15000 e x-66.2742025389 15000 In particular for p = 65, the policyholder will buy it if 65 e 30000 -e 63.0362781846 30000 e 30000 e 66.2742025389 15000 65 -e15000 = 0.768843437175 x-30000 log(0.768843437175) = 7886.03768865 Since X follows an exponential distribution with mean 10000, the proba- bility e P 30000 > a a) = P(X < 30000 log (a)) = 1-e 30000 log(a) 10000 = 1-elog (a) = 1-a3 In particular P e - 30000 > 0.768843437175 = 10.7688434371753 = 0.545521089901 so the expected number of buyers is 20000000.545521089901 = 1091042.1798. The expected profit is therefore 1091042.1798 (65 - 62) - 2000000 $1,273, 126.54. = Standard Questions 5. An insurer charges a loading of 30% on its policies with limit $500,000, and a loading of 33% on its policies with limit $1,000,000. It purchases stop- loss reinsurance of $500,000 over $500,000. The cost of this reinsurance is 22% of total premiums. What is the reinsurer's loading on the reinsurance policy? = 6. Policyholders are assumed to have a utility function u(x) -e- where > 0 varies between policyholders following an exponential distribution with unknown mean. An insurance company sells an insurance policy which covers a risk which causes a loss of $6,000 with probability 0.4. There are 3,000,000 potential customers for this policy. The insurer finds that when the premium for the policy is set to $3000, they are able to sell 952,000 policies. How many policies would they sell if they increased the premium to $4,000? 5. An insurer charges a loading of 28% on its policies with limit $1,000,000, and a loading of 26% on its policies with limit $500,000. It purchases stop- loss reinsurance of $500,000 over $500,000 for a loading of 45%. What percentage of the insurer's premiums for a policy with limit $1,000,000 are paid to the reinsurer? Let x be the expected losses limited to $500,000 and let y be the expected losses limited to $1,000,000. The insurer's premium for a policy with limit $500,000 is 1.26x, and the insurer's premium for a policy with limit $1,000,000 is 1.28y. The reinsurer's premium is 1.45(y - x). Since this is the difference between the insurers two premiums, we have 1.26x1.45(y - x) = 1.28y 0.17y = 0.19x y = 1.11764705882x The insurer collects 1.28y = 1.43058823529x in premiums, and pays 1.45 (y- x) = 0.170588235289x to the reinsurer. The percentage paid to the rein- surer is therefore 0.170588235289 = 11.92%. 1.43058823529 6. Policyholders are assumed to have a utility function u(x) = =e 15000 e-30000. Policyholder wealth is assumed to follow an exponential distri- bution with mean $10,000. An insurance company is considering selling an insurance policy which covers a risk which causes a loss of $3,000 with probability 0.02. The expenses for this policy are $2 million plus $2 per policy sold. If there are 2 million policyholders who might buy the policy, what will the expected profit on this policy be for the insurance company if they set the premium for each policy at $65? For a policyholder with wealth x, the expected utility without insurance is E(u(x)) = == (0.98e-15000 000 + 30000 0.98e 2-3000 +0.02e 15000 +0.02e 30000 2-3000 3000 3000 = =e 15000 0.98 0.02e 15000 30000 - e 0.98 0.02e 30000 = -1.00442805516 15000 1.00210341836e30000 -66.2742025389 15000 x-63.0362781846 30000 If the policy has premium p < 66.2742025389, then a policyholder will buy it if x-P 15000 x-P 30000 e 15000 (e , 15000 66.2742025389 15000 e x-66.2742025389 15000 In particular for p = 65, the policyholder will buy it if 65 e 30000 -e 63.0362781846 30000 e 30000 e 66.2742025389 15000 65 -e15000 = 0.768843437175 x-30000 log(0.768843437175) = 7886.03768865 Since X follows an exponential distribution with mean 10000, the proba- bility e P 30000 > a a) = P(X < 30000 log (a)) = 1-e 30000 log(a) 10000 = 1-elog (a) = 1-a3 In particular P e - 30000 > 0.768843437175 = 10.7688434371753 = 0.545521089901 so the expected number of buyers is 20000000.545521089901 = 1091042.1798. The expected profit is therefore 1091042.1798 (65 - 62) - 2000000 $1,273, 126.54. = Standard Questions 5. An insurer charges a loading of 30% on its policies with limit $500,000, and a loading of 33% on its policies with limit $1,000,000. It purchases stop- loss reinsurance of $500,000 over $500,000. The cost of this reinsurance is 22% of total premiums. What is the reinsurer's loading on the reinsurance policy? = 6. Policyholders are assumed to have a utility function u(x) -e- where > 0 varies between policyholders following an exponential distribution with unknown mean. An insurance company sells an insurance policy which covers a risk which causes a loss of $6,000 with probability 0.4. There are 3,000,000 potential customers for this policy. The insurer finds that when the premium for the policy is set to $3000, they are able to sell 952,000 policies. How many policies would they sell if they increased the premium to $4,000?