| This thesis uses collective risk model in actuarial models as example,combined with generalized linear model,to study the application of Copula functions in insurance industry.Collective risk model contains two parts,one is the number of claims,and the other is the claim size every time.Previous study indicates that the number of claims in a risk group often obeys to Poisson distribution,while the claim size every time often obeys to Gamma distribution.Actuaries are always interested in the total loss of a risk group.So obtaining the composite distribution of the number of claims and claim size every time is the most important point in actuarial process.In general situation,due to the complexity of the composite distribution,the actuaries often assume that the distribution of the number of claims and the distribution of claim size every time are independent.Based on this assumption,the expectation of the total loss is the product of the expectation of the number of claims and the expectation of claim size every time.However,in many situations,the assumption of independence will not hold.Therefore,the result calculated by this method will have a large deviation from the real total loss.To get a better estimate for the total loss,this paper takes the relation of the number of claims and claim size every time into consideration,using Gaussian Copula to combine the number of claims and claim size every time.Then the better joint distribution and the more precise estimate will be obtained.This thesis is made up with four sections.Section 1 derives the PoissonGamma joint distribution based on the Gaussian Copula theory.Section 2 derives the maximum likelihood estimates for the generalized linear model parameters using MBP algorithm.Section 3 applies the theoretical result into actual data to get the detailed model and obtain the estimate of the total loss.Section 4 compares the estimate of the total loss with the real total loss and draws the conclusion.The result indicates that the generalized linear model based on Poisson-Gamma joint distribution is more precise than the generalized linear model based on the independence assumption.Moreover,the result is more reasonable. |
