The Pure Premium and Reserve Analysis with Dynamic Mortality Rate and Stochastic Interest Rate

[[abstract]]近年來，台灣的市場利率水準呈現明顯的下降走勢，國人的平均壽命逐年遞增。面對低利率長壽時代的來臨，保險公司除了考量市場低利率水平，進而販賣低預定利率與利率變動型商品，以維持清償能力外，仍需面對低估平均餘命延長所可能造成的年金與終身醫療險給付提存不足的問題。 本文採用二種實際台灣市場利率資料以Vasicek Model及CIR Model配適隨機利率模型，並分別使用最小平方法、EM演算法 (Expectation- Maximization Algorithm)、非線性最小平方法 (Nonlinear Least Squares Method) 與複迴歸 (Multiple Regression) 來做模型的參數估計；另以HMD (Human Mortality Database ) 中台灣單齡死亡率資料，考慮世代效應 (Cohort Effect)，配適Lee-Carter動態死亡率模型，其中參數估計是採用SVD (Singular Value Decomposition) 方法。本文研究的目的，是藉由隨機利率與隨機死亡率的模型配適選擇，分別應用於終身壽險、定期壽險、生死合險與終身即期年金險，以男性30歲、50歲及65歲，繳費期間為10年、20年及30年，預定利率為2%、4%、6%及8%下，分別在四種組合情況： 1).固定利率、壽險經驗 (或年金) 生命表死亡率、2).隨機利率、壽險經驗 (或年金) 生命表死亡率、3). 固定利率、動態死亡率、4).隨機利率、動態死亡率下，模擬計算保費與責任準備金變化的差異。 研究發現，壽險經驗 (或年金) 生命表的修訂有迫切的需要，並且在本文的假設條件下試算結果顯示，對保險公司而言，不管任何險種如果同時採用隨機利率與動態死亡率，保費均比目前低，但對即期終身年金而言反而比較高，這也表示高齡人口死亡率的改善，使得保險公司在年金商品準備金的提存，恐有不足之虞。[[abstract]]In recent years, the market interest rate showed significant decreasing trend and the average life expectancy is increased progressively in every year in Taiwan. To face the coming era of the low interest rate and the longevity, insurance companies, in considering the low market interest rate, need sell low Assumed Interest rate and interest sensitive related products to keep the solvency. The insurance companies still have to face the shortage problem on the annuities and permanent medical health insurance payments caused by underestimating the extension of the average life expectancy. In this research paper, we use two actual Taiwan market interest rate data and use Vasicek Model and CIR Model as proper stochastic interest rate models and use Least Squares Method, EM Algorithm, Nonlinear Least Squares Method and Multiple Regression respectively to do the model parameter estimation. We also use the Taiwan single age mortality data from HMD (Human Mortality Database), in considering the Cohort Effect, and use Lee-Carter Mortality Model as proper Mortality Model and use SVD (Singular Value Decomposition) method to do the parameter estimation. The purpose of this research paper is to simulate the variances between the calculated insurance expenses and the changes of the reserves by proper choice on random interest rate and random mortality model to apply into the Whole life insurance, Term insurance, Endowment insurance and Immediate Annuity respectively, in using the male ages of 30, 50 and 65 and the payment periods of 10 years, 20 years and 30 years and the booking interest rates of 2%, 4%, 6% and 8% under 4 combined situation respectively: 1) Fixed interest rate, Taiwan Standard Ordinary Experience Mortality Table or Annuity table mortality, 2) Stochasitc Interest Rate, Taiwan Standard Ordinary Experience Mortality Table or Annuity table mortality, 3) Fixed interest rate, Dynamic mortality, 4) Stochasitc Interest Rate, Dynamic mortality. From this research, we found there is an urgent need to amend the Taiwan Standard Ordinary Experience Mortality table and Annuity table. Moreover, under this assumption, the result, to insurance companies, showed, no matter what type of insurance, the insurance expenses were lower than current insurance expenses if we use random interest rate and dynamic mortality. But in terms of Immediate Annuity, the insurance expenses were higher than current insurance expenses instead. This also means the improvement on high age population mortality may put the insurance companies in the risk of the shortage on annuity product reserve allowances