Claims in an insurance company that follow a Pareto distribution I'm trying to simulate the profits of an insurance company that has $n=1000$ clients. Suppose that each client pays a yearly premuium of $\$5,500$  and makes a yearly claim with probability $p=0.1$ independently of what the other clients do. Let $X$ be the size of the claim. Suppose $X$ has a Pareto distribution with parameters $a$ and $b$. 
The profists $Z(t)$ of the company in yeat $t$ are given by 
$$Z(t)=Z(t-1)+5500n-claims$$
The problem are the claims. Each client makes a claim with probability $p$ and the size of the claim follows a Pareto distribution. So 
$$claims=\sum_n\mathbb{P}(X=x|A)$$
where $A$ is the event that a client makes a claim (that we know happens with probability $p=0.1$). 
I'm not sure how to compute that condicional probability, if that is the correct reasoning. I'm not familiar with actuarial science, but this part is just to make some simulations (and are part of a larger problem of simulation).  

To simulate the claims you need to simulate Binomial and Pareto distributions.
Step 1: Simulate the number of claims in a year. 
Since each of the $n$ clients makes a claim or not (Bernoulli variable, 0: no claim, 1: claim) independently with probability $p$, the total number of claims is the sum of the $n$ Bernoulli variables, which has a Binomial distribution with parameters $n$ and $p$. Let's call this number $k$.
Step 2: Simulate each claim amount
Simulate $k$ independent Pareto variables with the given parameters $a$ and $b$.
Step 3:
Sum the claim amounts to arrive at the total claims from the portfolio.
Remark:
Your formula for $Z(t)$ is incorrect. The annual profits are just premiums minus claims. If you add $Z(t-1)$ to this, as you suggested, you get the cumulated profits up to time $t$ not the annual profits.
