Given are two very similar insurance companies each having 10000 customers. At t

Question

Given are two very similar insurance companies each having 10000 customers. At the beginning of the year each customer pays their insurance company a fee of $200, and during the year each customer independently files a claim for $800 of damages with probability 1/4. Both companies have a capital of $40 000 from the previous year. An insurance company goes bankrupt if it cannot pay for these claims. Should these two companies unite? Let p1 be the probability that at least one of the two companies goes bankrupt, and p2 the probability that the united company goes bankrupt. Find the numerical values of p1 and p2 and conclude whether joining the two companies is a good idea.

Explanation / Answer

Initially we consider just one company.

The total fee paid by 10,000 customers at the beginning of the year = $10000*200 = $2,000,000.
Therefore, adding the capital that the company has from the previous year, the total capital at the beginning of the year = $2,000,000 + $40,000 = $2,040,000.
     Now, each customer can file a claim for $800.
Therefore, the number of claims that the company can settle before it goes bankrupt = 2,040,000/800 = 2,550.
If the number of claims in a particular year is greater than 2550, i.e. 2551, then the company will go bankrupt.

Since the probability of filing a claim by each customer is independent, therefore, the probability that the company will go bankrupt = (0.25)^2551.

Since the bankruptcy of the two companies is independent, therefore the probability that both these companies will simultaneously go bankrupt = (0.25)^2551 * (0.25)^2551 = (0.25)^5102.

Therefore, p1 = (0.25)^2551 + (0.25)^2551 - (0.25)^5102 0.

Now, we consider the second case. If the companies combine, the new entity will have 20,000 customers, each paying $200 at the beginning of the year.

Therefore, the total capital at the beginning of the year = 20000 * $200 + $80000 = $4,080,000.
     Therefore, the number of claims that the company can settle before it goes bankrupt = 4,080,000/800 = 5,100.
If the number of claims in a particular year is greater than 5100, i.e. 5101, then the company will go bankrupt.

Therefore, p2 = (0.25)^5101 0.

Since 0.25 is a proper fraction and we are raising a proper fraction to a certain power, p2 should be much less compared to p1. Therefore, joining the companies should be a good idea.

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