# Insurance claim with normal approximation

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1. Dec 2, 2015

### bonfire09

1. The problem statement, all variables and given/known data
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

2. Relevant equations

3. 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 2. Dec 2, 2015 ### 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.

3. Dec 2, 2015