The Studies On Stochastic Reserving Basing On Individual Information Data

Stochastic loss reserving in general insurance plays a vital role in the risk management of insurance companies.Excessive reserves can lead to the low utilization of funds,while insufficient reserves make the insurance companies fail to settle the claimed debts such that they may confront huge risk.Therefore,it is an important task of actuaries to improve the accuracy of reserving.Nowadays,there are aggregated data models(ADM)and individual data models(IDM)for the reserving,where the former are based on the run-off triangle with features such as small size of data,easy computation and others,while the latter consider micro claims data mainly including occurrence times of claim events,reporting delays,settlement delays or payment developments.Some scholars have showed theoretically or numerically that IDM is better than ADM because the former more sufficiently use the available information than the latter.Obviously,the more related information utilized produces higher accuracy in reserving.Thanks to modern storage and computing capabilities of computers,more useful information should be incorporated into the reserving.However,IDM ignore policy-related features called individual information in this thesis.The information characterizes the heterogeneity between insureds,that is,the development pattern of claims is affected by some factors,which reflect the individual differences among policyholders.For example,in auto insurance,these factors include the age of the policyholder,driving age,gender and the price of the insured vehicle,fuel,brand type and geographic location,insurance contract type,etc.Hence,we investigate some methods for stochastic loss reserving based on individual information data and the main contents are as follows.(1)We study the impacts of individual information on stochastic loss reserving under a framework of discrete time and a case with settlement delay conditionally independent of the reporting delay.We first study the estimation for parameters and asymptotic behaviors of the corresponding estimators under a distribution model.Secondly,an analytic expression of individual loss reserve with individual information is derived and an intuitive algorithm is designed based on the expression.Then compare the asymptotic behaviors of loss reserving based on individual information data and individual data,by which we demonstrates the positive impacts of individual information on loss reserving.At last,we verify the conclusions by simulations and analyze the health insurance data.(2)We study the accuracy's improvement of loss reserving based on individual information data compared to that based on individual data in terms of large sample under a framework of discrete time and an over-dispersed Poisson model.Firstly,we utilize the conditional mean square error of prediction(MSEP)to measure the accuracy of the loss reserving.Then we study the ratio of the conditional MSEP of the two kinds of loss reserving and investigate the asymptotic behavior of the ration.At last,we conduct some analyses of simulations and health insurance data.(3)We study the loss reserving based on individual information data under a framework of continuous time and the case where each claim may be paid more than once.Firstly,we build a granular individual information model and study the estimation for parameters and the asymptotic behaviors of the corresponding estimators.Secondly,we investigate the analytic expressions of conditional moments of outstanding liabilities.Then we study the asymptotic behavior of loss reserving.At last,we conduct some Monte Carlo simulations for one hand to verify the analytic expressions of those moments and for other hand to show the accuracy's improvement of loss reserving brought by individual information under a continuous time framework.(4)We compare EM algorithm and Newton's method which are both used to obtain the estimates of the parameters in the individual information model with settlement delay relative with reporting delay.Firstly,the reasons for the incomplete observations are analyzed and the likelihoods of complete data and incomplete observations are derived according to the model assumptions.Then the model parameters are estimated by EM algorithm and Newton's method,respectively.At last,we compare EM algorithm and Newton's method in terms of accuracy of estimates and convergence speed by some numerical simulations.The innovations of this thesis are mainly as follows.Firstly,we incorporate individual information into stochastic loss reserving for the first time and put forward some new loss reserving methods called individual information models(IIM)in this thesis as well as show the advantages of IIM over IDM under a distribution assumption.Secondly,we study the accuracy's improvement of loss reserving by IIM with respect to IDM in terms of asymptotic theory under some weak moments assumptions,where the accuracy is measured by conditional MSEP which is approximated by semi-analytic method in a finite size of sample.Thirdly,we study a more general IIM for loss reserving under a continuous time framework and the case where each claim may be paid more than once.Fourthly,the claim data analyzed by stochastic loss reserving models is incomplete because of the truncation and censoring,we compare EM algorithm and Newton's method in terms of accuracy of estimates and convergence rates of the algorithms.