MATH 403A, fall 98, CHAPTER 6

Benefit Premiums not defined by the Equivalence Principle: Percentile and Exponential Premiums

Percentile Premiums: You need to know the Loss function L. A typical qquestion would be: what should the premiums be if probability of loss is at most .25.

Exponential Premiums: The premiums are determined using a utility function (exponential). This translates to solving an equation of the type

E[e^{α L}] = 1

Problems

1. An insurance company issues a fully continuous 45–year endowment insurance of 1 to (35). The annual premium is determined such that the probability of the loss random variable being positive on a particular policy is .5. You are given that mortality follows deMoivre’s law with w = 110, and δ = .06. Find the annual premium.

2. An insurance company issues a fully discrete whole life insurance of 1 to (45). The annual premium is to be as small as possible subject to the requirement that the probability of the loss random variable being greater than .3 on a particular policy not be more than .15. You are given that μ_{x} = .075 fro all x ≥ 0, and δ = .06. Find the annual premium.

3. the XYZ Insurance Company has utility function u(w) = −e^{−.005w}. It is contemplating issuing a fully discrete 1–year term insurance of 1000 to a life aged 85. You are given that q_{85} = .2 and that i = .25. Find the premium π at which the company will be indifferent as to whether to issue the policy.

4. (Continuation of problem 2) The company issues 100 of such polices to independent lives all aged 45. Find the probability that the loss on the block of policies is at least -100 (that is, the probability that the gain is no more than 100).

5. A company issues identical single–premium 10–year term policies to 5000 independent lives, each aged 35. The death benefit of 5000 is payable at the end of the year of death. Mortality follows deMoivre’s law with w = 110 and the force of interest is .07. Find the premium that must be charged for each policy in order to have a 99% probability of a positive profit on the block of policies.