Question

There is a town with 100 identical residents with initial wealth of $1000 and a utility function u = √ (wealth). Every person owns a car and there is a probability p = .2 that a person’s car will need a costly repair in a given year. If a person needs a repair, they must pay $500 to fix it.

1. What is the maximum amount that a person would be willing to pay to insure their self against the cost of repairing their car?

2. Suppose a single company exists to offer these insurance policies. What price do they charge? What are their expected profits?

3. What is the probability that the insurance company incurs a loss in a given year?

Answer 1

1).

So, here a person have “$1000” as an initial wealth. Now, there is a probability that “p=0.2” a car need repair and must pay “$500” to fix it, => a person will be willing to pay “$500*0.2 = $100” to insure their self against the cost of repairing their car.

2).

So, here the company will charge “$100” for the insurance in all the case. So, here the company will get “$100” in all the case and the company have to pay “$500” if a car need repair. So, the expected profit is given by.

=> 100*[$100 - 0.2*$500] = 100*0 = $0, the expected profit of the company is “$0”.

3).

Here the per person return of the company is “$100” and the cost is “$500”,=> out of 100 people if “20” car need repair, => the company will get “100*$100=$10,000” and the company have to pay “$500*20=$10,000”, => the company is getting “0” profit. So the company will incur loss if “N > 20”, where “N=numbers of cars need repair. So, the required probability is given by.

=> P(N > 20) = 1 - P(N < = 20) = 1- P(N=20) - P(N=19) - P(N=18) - ….- P(N=0).

=> P(N > 20) = 1 - P(N < = 20) = 1- ¹⁰⁰C₂₀*0.2^20*0.8^80 - ¹⁰⁰C19*0.2^19*0.8^81 - ¹⁰⁰C18*0.2^18*0.8^82 - ….- ¹⁰⁰C0*0.2^0*0.8^100.