How Does the Central Limit Theorem Apply to Insurance Risk Assessment?

[twoflower]

Hi all, I just wrote a test from probability and had troubles doing this problem:

Homework Statement

The assurance company makes an insurance for 1000 people of the same age. The probability of death during the year is 0.01 for each of them. Each insured person pays 1.200 dollars a year. In case of death the assurance company pays 80.000 dollars. Figure out:

a) expected income of the assurance company

b) approximate probability of that the assurance company will suffer a loss, knowing that [itex]u_{0.9} = 1.28[/itex], [itex]u_{0.95} = 1.64[/itex] and [itex]u_{0.975} = 1.96[/itex], where [itex]u_{\alpha}[/itex] is [itex]\alpha[/itex]-quantil of the normal distribution [itex]N(0,1)[/itex].

The first one was quite easy, it's just the expected value of binomial distribution.

But I couldn't solve the second one. I know it's an example of using the Central limit theorem, but I was confused of what to actually put in the theorem...I computed the 'critical value' A in the sense that if more than A people die during the year, the assurance company will suffer a loss. So I got

[tex]
P(\mbox{assurance companny will suffer a loss}) = P(\mbox{number of people who died} > A) = 1 - P(\mbox{number of people who died} < A)
[/tex]

[tex]
= 1 - P\left(\sum_{i=1}^{1000} X_{i} < A\right)
[/tex]

where [itex]\sum_{i=1}^{1000} X_{i} \sim Bi(1000, 0.01)[/itex]

Then I normalized the probability so that I got [itex]N(0,1)[/itex]. Is it the right approach? Because I got strange numbers which didn't seem anyhow related to those we had been given (those quantils).

Thank you for any help.

 

[mighty2000]

Thank you for sharing your question and thought process. I can understand your confusion and I would like to offer some insights that may help you solve this problem.

Firstly, your approach seems correct. You have correctly identified that this is an example of using the Central Limit Theorem. Your formula for calculating the probability of the assurance company suffering a loss is also correct.

However, the reason you are getting strange numbers is because you have not taken into account the expected income of the assurance company. Remember, the expected income of the assurance company is the product of the probability of death and the amount paid by each insured person. So, the formula for calculating the probability of the assurance company suffering a loss should be:

P(\mbox{assurance company will suffer a loss}) = P(\mbox{number of people who died} > A) = 1 - P(\mbox{number of people who died} < A)

= 1 - P\left(\sum_{i=1}^{1000} X_{i} < A\right) * (0.01 * 1,200 * 1000)

= 1 - P\left(\sum_{i=1}^{1000} X_{i} < A\right) * 12,000

Now, when you normalize the probability using the normal distribution, you should use the expected income instead of 1,000. This is because the expected income is a better representation of the actual income of the assurance company. So, the final formula should be:

P(\mbox{assurance company will suffer a loss}) = 1 - P\left(\frac{\sum_{i=1}^{1000} X_{i} - 12,000}{\sqrt{12,000}} < \frac{A - 12,000}{\sqrt{12,000}}\right)

= 1 - P\left(Z < \frac{A - 12,000}{\sqrt{12,000}}\right)

= 1 - P\left(Z < z\right)

= 1 - \Phi(z)

where z = \frac{A - 12,000}{\sqrt{12,000}} and \Phi(z) is the cumulative distribution function of the standard normal distribution.

Now, you can use the given quantiles to find the corresponding z-value and calculate the approximate