Question

Every year, an insurance company sells a $500 policy to 100,000customers. In case of death, the company must pay $200,000 to a relative. Assumingthat the each person dies every year with probabilityp= 0.001 independently of othersanswer the following questions:

a. What is the probability that the company makes a profit on any givenyear? Use the relevant approximation.

b. The company goes bankrupt if the first year with losses or no profit isfollowed by another year of losses or no profit. What is the probability that the company goes bankrupt within the first 10 years of existence? (Denote the probability in part (a) bypand express the answer in terms of p)

Answer 1

Answer :

(a).

Comapny will make a loss when the total revenue generated, is less than the insurance claim paid, in case of death

Let n be the number of death, to happen from 10000 insurances sold, for the company not to make profit

So total revenue generated will be = 10000*500 = 5000000

And in case of n death the payment wil be made = 200000n

When this 200000n > = 5000000

Then company will not make profit

So n > = 5000000 / 200000

n > = 25

So if in a year atleast 25 death occur than company will not make profit

So probability for 1 death = 0.001

Since they are independent

So probability for 25 death = 0.025

This will be probability when company will not make any profit

Probability company make a profit = 1 - probability company will not make profit

p = 1 - 0.025 = 0.975

(b).

Company goes bankrupt with no profit for two consecutive years

So out the ten years we need two consecutive years, to choose two consecutive year, consider the two co nsicutive year as 1, then there will be nine ways to choose now for any way of the nine, there should be no profit, p is probability of no profit

= p*p + p*p+......................+p*p ( 9 times)

So probability that company goes bankrupt in 10 years

= 9p^2