Insurance claim with normal approximation

[bonfire09]

Homework Statement

There are $N = 10000$ clients of an insurance company. One-half of them will file claims with probability $p_1 = .05$, another half of them will file claims with probability $p_2 = .03$. Each claim is worth $1000$. Find the Value-at-Risk at the level α = 0.99, that is, the amount of money the company should accumulate to be able to pay its customers with probability greater than or equal to α

Homework Equations

The Attempt at a Solution

What I did was let $\sum_{i=1}^{5000} X_i$ where $X_i \sim$ Bernoulli($p_1$) be the proportion of 5000 customers with probability of filing a claim with $p_1=.05$ probability and $\sum_{i=1}^{5000} Y_i$ be the proportion of the other 5000 customers with probability $p_2=.03$ of filing a claim. Again $Y_i\sim$Bernoulli($p_2$).Then we know $\sum_{i=1}^{5000} X_i$ and $\sum_{i=1}^{5000} Y_i$ both follow a binomial distribution. Since $p_1$ and $p_2$ are large enough with each sample size being large enough we can use a normal approximation. From here

$\sum_{i=1}^{5000} X_i\sim\text{N(5000*.05,5000*.05*.95)=N(250,237.5)}$
$\sum_{i=1}^{5000} Y_i\sim \text{ N(5000*.03,5000*.03*.97)=N(150,145.5)}$
Then we let $Z=\sum_{i=1}^{5000} X_i+\sum_{i=1}^{5000} Y_i\sim\text{N(400,383)}$
From here since each claim is $1000$ dollars then $1000Z$ is total amount in claims and 10000p is the amount the company should collect where p is the amount of money we collect from each individual. which is a constant value to make this problem work. We want to find
$P(1000Z \leq 10000p)=.99 \implies P(Z\leq 10p)=.99 \implies P(\dfrac{z-400}{\sqrt{383}}\leq\dfrac{10p-400}{\sqrt{383}})=.99\implies \dfrac{10p-400}{\sqrt{383}}=2.33\implies p=44.559$. So we should collect about $44.56 from each person to keep the value at risk at 99%. But I am not sure if this is correct or not. Any help would be greatly appreciated. Thanks

 

[RUber]

It looks like all your steps are correct.
I am not 100% clear on why you are using 10000p, is that some per-customer cost? The question asks how much money should be accumulated, so I would expect the answer to be given in total dollars i.e. 10000p or about $45K.
If you just let Z be number of claims, and then multiply that by 1000 later, you get the same result result.
In all, the company should expect no more than 446 claims at $1000 each, which as you pointed out would give a shared cost per customer of about $44.6.

 

[bonfire09]

I assumed that like each customer pays a certain amount $p$ into the insurance company so $10000p$ is just the total value the customers pay into the company. I take that back I think your right. I don't need $10000p$ rather just $p$ itself## which should be the amount of money the company has.

 

[RUber]

That makes sense. However, the question asked for how much money the company should accumulate...this could be from sources other than insurance principals. Also, it would make sense to charge the higher risk customers more than the lower risk customers. I am simply saying from my perspective, it seems like the question is looking for the answer of 10000p or 1000z.

 

[Ray Vickson]

I assumed that like each customer pays a certain amount $p$ into the insurance company so $10000p$ is just the total value the customers pay into the company. I take that back I think your right. I don't need $10000p$ rather just $p$ itself## which should be the amount of money the company has.

Probably each customer pays more than $p$, because the company wants to be profitable. All you can really say here is that the company must put aside at least $446,000 from somewhere. That is all the question asked you.