Compound Poisson Approximation For Individual Risk Model

The Insurance Company usually have to consider the claim amount of a portfolio over a given period in practice. In the individual risk model consisting of n insurance policies, the risk X_i corresponding to the ith contract is usually expressed in the form:where I_i is a Bernoulli random variable taking the value 1 when at least one claim is filed while the contract is valid, and V_i is a strictly positive random variable. The (V'_i)s are typically assumed to be independent of each other and of all the (I'_i)s. Of partically interest is the cumulative distribution function F_s of the aggregate claim amount S =Σ_i=1ⁿX_i and functions thereof such as the stop-loss premium Ï€s(d)= E{max(S -d, 0)}, d ≥ 0, the Value-at-risk VaR_α = F_s^-1(α), α∈ (0, 1). But it is generally difficult to compute these quantities for large n. Usual way is to approach S by T =Σ_i=1^N Y_i .where N follows a Poisson distribution with mean A and Y₁, Y₂,... is a sequence of independent and inentically distributed random variables and also independent of N. Thus, we can simplify the calculation by using the character of Compound Poisson Distribution. So far, people do a lot of effort to find the best Compound Poisson risk model which approach the individual risk model.This paper discuss the different ways to approach the individual risk model by Compound Poisson risk model, and measure the quality of the approximation of S by T. Do a farther work when the approximation result depends on the distribution of cliam amount. Give some examples in the insurance field and take some calculation in the special situation. Finally, this paper introduces the Compound Poisson approximations for individual models with dependent risks.