| Risk management is utilized thronghout all the process of insurance. Risk theory is one of the most important part of the actuarial theory. Based on life insurance and annuity insurance, some random models are set as a new risk management tool for institution deeling, product design, premium calculation and reserve calculation in this paper. This paper discusses risk order, enriches the conception of correlation order, investigates its properties, and discusses the relationship between correlation order, stop-loss order and exponential order.In life insurance and annuity insurance, interest rate and mortality are two very important random factors. In the traditional actuarial theory, only the mortality is ran-dom and the interest rate is certain. However, the interest rate is also random.With the development of actuarial studies, interest randomness in life insurance has received con-siderable attention . People have reallized that the risk accrued by interest randomness may be very large to insurance organizations(insurance company and social insurance development). Generally,the risk accrued by mortality randomness can spread out by issuing amounts of policies. However, the risk accrued by interest randomness can't spread out by increasing ciroulation. From this point, interest risk is more important than mortality risk(Wu chao biao (1995)). In 1970's, interest randomness began to be an assumption in actuarial science ( Pollard,J.H.(1971), Boyle(1976) ) , and study of interest randomness (i.e.the mortality and interest rate are both random ) affect-ing life insurance and social pension insurance gradually became an important field of actuarial science (Beekman and Fuelling(1990,1991.1993), De Schepper,De Vylder and Goovaerts(1993), Gary Parker(1994,1996),Andrew J.G.Cairns (1995), Griselda Deel-stra(2000),L.C.G.Rogers, Wolfgang Stummer(2000)).Based on the work of Beekman, Fuelling,De Schepper , De Vylder , Goovaerts R.kaas and Gary Parker, we establish some models about variable life insurance and annuity insurance. As one application of the models, we succeed in calculating the government debt during the transformation of social pension insurance in China.In Chapter2, we establish the model of present value of benefits, which is fitted to two situations -single life situations and multiple life situation:where B(t) is a positive function, is force of interest accumulation function, I(t) is indicator function and T is the remaining life of the insured.The benefit amountB(t) of the above product is variable(certain benefit is its special example. Generally, the benefits were assumed to be certain , the payment is instantaneous (i.e.benefit is payed as soon as the insured dies. The benefit was assumed to be paid at the end of the year which the insured dies , such as in Beekman and Fuelling (1993), De Schepper, De Vylder and Goovaerts(1992), Gary Parker(1994),A.De Schepper. M.J.Goovaerts and R.Kaas(1997)). We use a common independent increment random process to set the assumption of interest randomness. Under a more common and suitable assumption, we obtain the kth moment of the present value of benefit . To a portfolio of life policies of one kind , we study the limiting distribution and strong law of large numbers of the average lost of policies, and simulation.Generally, it assumes that the benefit is certain in daul randomness model of annuity insurance, such as Beekman and Fuelling (1990,1991), De Schepper, De Vylder and Goovaerts(1992). However, we establish a more commom truncated annuity model of present value of benefits (annuity certain and the whole life annuity are its special examples)in Chapter 3:where c(t) is a positive function and u(i) is force of interest accumulation function.Then we obtain its Arth moment. And we establish a model of present value of benefits of a portfolio of whole life annuity policies:where n is the number of policies. Then we obtain its kth moment. Further more, we study the limiting distribution of the average cost of policies, the mai...
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