Suppose you own a house valued at $20,000. In any given year, there is a 10% chance that a fire will completely destroy your house. To protect yourself against this risk, you purchase an insurance policy that fully compensates you in the event of a fire. The annual premium for this insurance coverage is $1,000.

What is your expected wealth after purchasing the insurance?

Question

Suppose you own a house valued at $20,000. In any given year, there is a 10% chance that a fire will completely destroy your house. To protect yourself against this risk, you purchase an insurance policy that fully compensates you in the event of a fire. The annual premium for this insurance coverage is $1,000.

What is your expected wealth after purchasing the insurance?

BlackTom AI · Accepted Answer

The problem presents a scenario with a base wealth of 20,000 and a 10% risk of total loss from fire in a given year. An insurance policy is purchased that costs 1,000 annually and pays out 20,000 if a fire occurs, otherwise paying nothing.
First, consider what happens if no fire occurs. After purchasing the insurance, you have spent the premium, so your wealth would be 20,000 − 1,000 = 19,000 in that branch of the world.
Second, consider what happens if a fire does occur. You still paid the premium, so you start from 19,000, but you receive a payout of 20,000 from the insurer. Your wealth in that fire scenario becomes 19,000 + 20,000 = 39,000.
Now weight these outcomes by their probabilities. The no-fire scenario has probability 0.9, and the fire scenario has probability 0.1. Therefore, the expected wealth is:
0.9 × 19,000 + 0.1 × 39,000 = 17,100 + 3,900 = 21,000.
In summary, the insurance policy changes the distribution of wealth across states: you forego the risk of losing the house with a fire, but you pay the premium upfront, and the expected wealth calculation combines both possible outcomes with their respective probabilities. The arithmetic leads to an expected wealth of 21,000.

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