# Your house is worth 500k. Every year, there is a 0.02% chance that a fire will occur and cause a complete damage. In addition, there is a 0.05% chance that a fire will occur and cause a partial...

Your house is worth 500k. Every year, there is a 0.02% chance that a fire will occur and cause a complete damage. In addition, there is a 0.05% chance that a fire will occur and cause a partial damage worth 200k.

What is the fire insurance premium per year for your house assuming that insurance companies need a 20% profit?

Suppose that the amount of your liquid wealth is 100k and that your risk aversion parameter is y=1. Will you buy the insurance?

What is your reservation price for the insurance?

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The insurance premium will be equal to the expected loss, plus the profit margin for the insurance company.

The expected loss is the probability of the loss times the magnitude of the loss, added up for all possible outcomes:

0.02*500,000 + 0.05*200,000 = 10,000 + 10,000 = 20,000

Adding a 20% profit margin for the insurance company means we multiply this by 1.20:

20,000*1.20 = 24,000

The annual insurance premium is therefore $24,000.

By "risk aversion parameter", I'm assuming you mean *relative *risk aversion parameter, which for y = 1 means that your utility function is logarithmic:

U = ln(C)

(As a reminder, normally it would be U = 1/(1-y) * (C^(1-y) - 1), but for y = 1 there is a special limit case which is U = ln(C).)

Ignoring all other spending, your consumption will be:

$76,000 if you buy insurance

$100,000 if you don't buy insurance and nothing happens (93% probability)

-$100,000 if you don't buy insurance and you get a small fire (5% probability)

-$400,000 if you don't buy insurance and you get a large fire (2% probability)

We can immediately see that you must buy the insurance, because your consumption goes negative and therefore your utility goes to *negative infinity* if you don't buy the insurance and you have a fire. You simply cannot afford to have a fire.

Under these conditions, your reservation price is actually $100,000. You are willing to spend all of your money to insure your house, because you cannot afford what would happen if you didn't.