Insurance, Part I

Suppose that you own a business worth $100000. With probability 1 = 0.05, a disaster a fire, lets say occurs that reduces the value of the business to $50000. Let x denote the premium on an insurance policy that will protect you fully against that loss.

Your choices are as follows. You can take out the insurance policy, in which case your wealth will be $(100000 x) no matter what: youll have to pay the premium of x up front, but the insurance company will pay you $50000 to compensate for the loss if it occurs. Or you can forego buying insurance and take your chances, in which case your wealth will be $100000 with probability 0.95 and $50000 with probability 0.05.

Assuming that your preferences are described by a vN-M expected utility function with logarithmic Bernoulli utility function

u(Y ) = ln(Y ),

what is the maximum premium x that you will be willing to pay for the insurance policy? (Note: To solve this problem, you may need to use the fact that if x = ln(y), then y = exp(x). That is, the exponential function is the inverse function of the natural logarithm.

3. Insurance, Part II

2. Risk Premium Part II

1

Extending the example from question 2, above, suppose that in addition to the 5 percent chance of fire, there is an even smaller probability of an even bigger disaster: a flood, lets say, that occurs with probability 2 = 0.01 but reduces the value of your business to $1.

Your choices are now as follows. You can take out the insurance policy, in which case your wealth will be $(100000 x) no matter what. Or you can forgo buying insurance, in which case your wealth will be $100000 with probability 0.94, $50000 with probability 0.05, and $1 with probability 0.01.

Still assuming you have vN-M expected utility with logarithmic Bernoulli utility function, what is the maximum premium x that you will be willing to pay for insurance now?