Moments Of The Risk Models For The Natural Disaster And Optimal Pricing Strategies Problems

With the improvement and development of modern financial markets, actuarial sci-ence is attracting more and more attention. In actuarial research field, it is becoming a hot topic that is establishing appropriate insurance risk model, pricing insurance product and researching bankruptcy for their broad application potential and important theoreti-cal value. Establishing a reasonable risk model is the scientific basis on studying insurance risk and providing insurance policies for insurance companies. Modeling a rational port-folio is a effective method for insurance companies and customers to avoid the risk. Thus, pricing for the heterogeneous risk portfolio is an important topic of insurance researches. In the real world, natural disasters are significant risks which must be confronted by human beings. Taking into account its character of low-frequency and high-loss, it is inevitable to implement reinsurance. Corresponding to these problems, the main work has been arranged as follows:For the natural disaster problem, we study the moments of risk models. Firstly, the natural disaster aggregation claims risk model with zero interest of continuous is proposed, where the variables are dependent. Then, we introduce circumstance process to deal with the relationship between variables in risk models, and use Laplace transform to establish expected expression of the discounted aggregate claims in small period of time. By the moment properties of moment generating function, we derive the explicit expressions of the first and second-order moments of the aggregation of claims. Secondly, the constant interest rate is introduced to risk model. Then we get similar moments. At last, the moments of multi-dependent insurance claims are considered. Noting the feature of low frequency of natural disasters and the fact that the general data obtained is discrete data, it has a great practical significance to study discrete state risk model. We first establish discrete risk model, then consider the moments of discounted aggregate claims by integral-differential equations. Similar to the previous continuous models, the first two moments' explicit expressions are obtained. As the extent of this part, the stochastic interest rate risk model is studied.For pricing insurance product, firstly, we consider optimal pricing problem of a class of heterogeneous risk portfolio in convex distance measure. The purpose is to obtain the premium's explicit expression of every risk. This part study four constrained optimization problems and the unique optimal solutions obtained under priority order theory and con-vex distance measure. Secondly, using Lagrange multipliers method, we study an optimal pricing problem of heterogeneous risk portfolio which has been given a risk factor and convex distance measure. At the beginning, we study the primal problem, given a small probability of the bankruptcy, and find suitable premiums for_all risks which make the dis-tance function achieve the minimum. And then we study its dual problem. These results provide theoretical approaches and broaden the ideas for practice of insurance pricing, and provide a theoretical basis for pricing portfolio risk for the actuaries. Considering the tremendous catastrophe losses, it is necessary to carry out reinsurance strategies. We study the reinsurance issues on a number of insurance companies and describe some optimal methods of pricing theory.At last, we proposes a generalized linear model with measurement error and estimates the unknown parameters using robust M-estimator. The proposed M-estimators are proved to be consistent and asymptotically normal. We also evaluate the finite sample performance of our estimator through a Monte Carlo method.