Research On Interest Rate Modeling And Fractional Age Assumptions In Life Contingencies

The interest rate and the mortality rate are the key elements in life contingencies.Plenty of works,such as Policy pricing,evaluation of life insurance reserve,financial management and investment in life insurance companies etc.,are based on interest rate and mortality rate.Especially to the long-term policies,the impact of interest rates on the actuarial data is usually crucial.Random variations of interest rate give the huge uncertainty to life insurance companies,and it is one of the main ways to solve this problem to choose the appropriate stochastic interest rate model.Life table is the standard reference to characterize the mortality rate,but the tabular only specifies survival and death probability for integer ages.However,there are large numbers of policies which are signed at non-integer ages,and all life insurances need to assess the reserves according to the calendar year.In addition,there are also many life insurance and annuity products with multiple payments or continuous payment.To solve all these actuarial problems,we must utilize fractional age assumptions,and the rationality of these assumptions determines the calculation precision of relevant actuarial data.In view of the key roles of interest rate model and fractional age assumptions in actuarial research and actuarial practices,these two aspects are systematically studied in this paper.On the one hand,we put forward some kinds of stochastic interest rate models with random jumps,and these models' probability properties,expressions of expected discounted functions and application problems in life contingencies are studied with detailed numerical analyses and simulations.On the other hand,we introduce two new kinds of fractional age assumptions based on the theory of spline interpolations.In our research on each assumption,we discuss the probabilistic properties of time-until-death random variables,the geometric properties of survival functions and forces of mortality etc.Especially,we give a new optimality criterion to compare different fractional age assumptions with detailed numerical analyses.Besides,above theoretical results are also used to research the pricing of life insurance products and the evaluation of life insurance reserve.In this paper,the main innovations in the research of interest rate modeling are as follows:(1)we investigate the random jumps of interest rates,and then construct stochastic interest rate models by combining this jumping features with other continuous interest rate models;(2)considering the adjustment mechanism of interest rate jumps driven by multiple economic factors,we propose a new idea of interest rate modeling by means of Erlang distributions;(3)as for research methods,the conversions of integral direction on the stochastic process integral are used to solve the technical problems in the study as well as solving sumfunctions by means of constructing and solving a kind of ordinary differential equations.In addition,the major innovations in the research of fractional age assumptions are:(1)the survival(death)information in three adjacent years is involved in the estimation of fractional age mortality based on spline interpolation theory which makes the death information be utilized more efficiently,hence we can estimate the fractional age mortality more reasonably and more accurately;(2)A new evaluation criterion for different hypotheses and a selection algorithm for the optimal parameters are put forward in the research of fractional age assumption based on rational spline.According to main contents of this paper,there are seven chapters which can be divided into four parts.The major contents and conclusions of each chapter are introduced briefly as follows.Chapter one is the exordium which introduces the research background,research significance and the guidance function to the operation of the life insurance company,such as development,pricing and reserving of insurance products etc.Chapter two mainly summarizes the current status of research issues in this paper and illustrates the necessity and innovations of this research content.At the same time,we also briefly introduce the relevant theories of stochastic processes and geometric interpolations needed in this paper.In Chapter three,we study three kinds of stochastic interest models with Poisson jumps which indicate three cases of stochastic interest force functions respectively: only Poisson jumps,combining Brown motion with Poisson jumps and combining Ornstein-Uhlenbeck process with Poisson jumps.Through the conversions of integral direction on the stochastic process integral,the mathematical expressions of the expected discounted functions under different interest models are obtained.We further study the corresponding probabilistic properties,the validity of models,and discuss the applications of interest models in life contingencies.Meanwhile,the detailed numerical analyses are carried out in above research.Considering the market mechanism that there are multiple economic factors which jointly drive the interest rates to jump,Chapter four describes the time intervals of interest rate jumping by means of Erlang distributions.Based on this idea of modeling,we study two types of stochastic interest models which both combine Erlang distribution with OrnsteinUhlenbeck process.In the two models,Erlang distribution is used to describe the accumulated interest force process and the force of interest respectively.These two types of models have different application environments and application areas in life contingencies.In the analysis of the models,we first obtain the series formula of the expected discounted function under each model,and then we further deduce their sum-functions by means of constructing and solving a kind of 9)order ordinary differential equations.At the same time,we discuss the probabilistic properties of these two interest models,carry out the corresponding numerical analyses,and study the applications of these models in life contingencies.In terms of the continuity of the force function of mortality and the correlation among the mortality rates at adjacent integer ages,Chapter five and Chapter six introduce two types of fractional age assumption methods based on cubic polynomial interpolation and cubic rational spline respectively.Meanwhile,both the validities of the two kinds of assumption methods and the properties of the corresponding death probability distributions under these assumptions are discussed.In the research of fractional age assumption under cubic rational spline,the stochastic order relations between the time-until-death random variables under parameter conditions are analyzed and a new optimality criterion to compare different fractional age assumptions is introduced.In order to show the advantages of these two kinds of assumptions,we compare them with other fractional age assumptions by means of different numerical methods.Lastly,applications of these two assumptions in life contingencies are discussed with numerical analyses.Chapter seven summarizes the full text,further extracts the main conclusions and innovations of this paper,and emphasizes the research significance and application value of this paper.All in all,the research in two aspects can provide new theoretical support and more precise data protection for China's life insurance industry which can made some contributions to the development of China's life insurance industry from two aspects of academic and practical aspects.