The Research On Optimal Bonus-Malus System Of Automobile Insurance

Actuarial science applies mathematical methods in insurance and is a science between applied mathematics and insurance. Bonus-malus system (BMS in short) of automobile insurance is an important topic in actuarial science. Markov chain, Bayes theory, the theory of delay differential equation and dynamic programming are used to study BMS, especially optimal BMS, of automobile third-party liability insurance in this dissertation, which is organized as follows:First, we study the properties of Chinese present BMS. Based on Markov theory, some measurements, such as stationary distribution, elasticity and transparency, are used to describe Chinese BMS under the assumption of a closed portfolio. At the same time, to reflect the heterogeneous of portfolio, corresponding unconditional measurements are defined in this dissertation. Numerical analyses illustrate that Chinese present BMS is facing some problems as in most BMSs of other countries. These problems are elasticity deficient and allowance among policyholders. But all this conclusions are limited to a closed portfolio without leaving and entering of policyholders.Second, we study the limited distribution of a bonus-malus system in an open market. A model of the number and proportion of policyholder in bonus-malus levels is constructed by delay differential equation. When the leaving and entering rate of each BM level and delay factors satisfy certain conditions, sufficient conditions for the existence and global exponential stability of a unique equilibrium point are established and proved by using the fixed point theorem and differential inequality techniques.Thirdly, we study bonus-hunger of BMS. We generalize Lemaire's algorithm of calculating optimal retention of full insurance to the situation with proportional deductible and insurance coverage limit. Based on evaluation of any given retention, optimal retention is obtained through dynamic programming. Then, under the assumption of the existence of a fixed hunger proportion, the parameters of true loss are estimated for fixed deduction proportion with a coverage limit, in which maximum likelihood estimation is used.Fourthly, we study optimal uni-factor bonus-malus system in which claim frequency is used as posterior factor. First, two optimal four level BMSs with present Chinese BMS's transfer rule are obtained under two objects, minimize insurer's expected loss and maximum insurer's expected utility, respectively. We also study path-dependent BMS...