考量死亡、利率、脫退與流動性風險下生死合險契約之盈餘分析 - 政大學術集成

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(1)國立政治大學商學院金融學系碩士班碩士學位論文. 考量死亡、利率、脫退與流動性風險下生死合險契約之盈餘分析政治. 大. 立. ‧ 國. 學. Surplus Analysis for Endowment Contracts Considering Mortality, Interest Rate, Surrender and Liquidity Risks. ‧. n. er. io. sit. y. Nat. al. Ch. engchi. i Un. v. 指導教授：林士貴博士研究生：林偉翔撰. 中華民國一 ○ 四年六月.

(2) Surplus Analysis for Endowment Contracts Considering Mortality, Interest Rate, Surrender and Liquidity Risks Student: Wei-Hsiang Lin. Advisor: Dr. Shih-Kuei Lin. Submitted to Department of Money and Banking College of Commerce National Chengchi University. Abstract Once insurance contracts are issued, the insurers should be capable to deal with the unknown conditions in the future as possible. In this paper, we analyze the impact. 政治大. of mortality, interest rate, surrender and liquidity risks on the surplus of endowment. 立. contract. We model the interest rate risk by Vasicek model, the surrender rate based on. ‧ 國. 學. the rational behavior of policyholders and introduce the discounted price of zero coupon bonds as the liquidity risk. Under such assumptions, we compute the premium and. ‧. reserve, demonstrate the simulated insurance surplus, and finally exhibit the statistics. y. Nat. io. sit. of the surplus from different sources. The simulated results show the sensitivity of the. n. al. er. surplus to the parameters of the risks. At the same time, we also show the probabilities. Ch. i Un. v. of insolvency and illiquidity of the insurer before the maturity date of the contract due. engchi. to the fluctuating surrender rate and liquidity risk resulting from the stochastic interest rate.. Keywords：Surrender; Stochastic Interest Rate Process; Endowment Contract; Liquidity Risk.. ii.

(3) 考量死亡、利率、脫退與流動性風險下生死合險契約之盈餘分析學生：林偉翔. 指導教授：林士貴. 博士. 國立政治大學商學院金融學系. 摘要當保險契約被發行時，保險公司必須被要求盡可能的具備承擔未來不可知的風險的能力。本文將死亡風險、利率風險、脫退風險以及流動性風險引入，並針對生死合險契約進行盈餘分析。在此以 Vasicek (1977) 所提出之隨機利率模型、. 政治大與 Nies (2005)流動性風險債券價格來描述各種風險。根據上述模型假設下計算保立根據被保險人理性行為作為基礎之脫退模型以及引入簡化後的 Longstaff、Mithal. ‧ 國. 學. 費及準備金，遂以蒙地卡羅模擬法量化源於各種風險之盈餘。最後，本文計算保險公司之盈餘對各風險參數之敏感度分析，並計算各期破產與發生流動性問題之. ‧ y. Nat. sit. 可能性。. io. n. al. er. 關鍵詞：脫退、隨機利率、生死合險、流動性風險. Ch. engchi. iii. i Un. v.

(4) Acknowledgements. I would like to thank the Department of Money and Banking for providing such an inspiring environment for academic growth. I am grateful to my examining committee, Dr. Ming-Chin Hung, Dr. Linus Chan, and Dr. Li-Hsien Sun, for their careful reading of the work and valuable comments that helped to the better version. I am deeply indebted to my supervisor Dr. Shih-Kuei Lin, Dr. Chih-Kai Chang and Dr. Cary Tsai for their guidance during my studies, for their encouragement and for giving me the opportunities to discover my interests. At the same time, many thanks to. 治政 Dr. Szu-Lang Liao, Dr. Lung-Chi Chen and Dr. Hsiu-Hau 大 Lin for passing on the skills 立 needed and their positive attitude. ‧ 國. 學. The way here would have been harder and less enjoyable without the constant. ‧. support from my close friends. I am especially grateful to Victor Huang, Саша Huang,. sit. y. Nat. Premium Ho, Benton Ku and the smart Dummy Peng for their help in the mathematical. io. er. difficulties and for sharing so many happy moments with me.. al. Last but not least, I totally owe a great debt of gratitude to my parents and my. n. iv n C lovely soulmate, Kelly Chang, forhtheir love, support, e n g c h i U understanding and patience. Without their care and strong belief in me, I do not think I would be able to go this far.. iv.

(5) Contents. Abstract .................................................................................................................................... ii Acknowledgements ................................................................................................................. iv Contents .................................................................................................................................... v List of Figures ........................................................................................................................ vii 1.. Introduction ..................................................................................................................... 1. 2.. Endowment Contract ...................................................................................................... 4. 3.. Assumptions for Models.................................................................................................. 6. 3.3.. 治政大 Interest Rate Risk ...................................................................................................... 6 立 Surrender Risk ........................................................................................................... 8. 3.4.. Liquidity Risk ............................................................................................................ 9. 3.2.. ‧. 4.1.. Reserve Based on Assumed Interest Rate ............................................................... 11. 4.2.. Reserve and Premium .............................................................................................. 12. y. sit. Case 1 : nˆ  1 .................................................................................................. 12. 4.2.2.. Case 2: nˆ  n  1 ............................................................................................ 13. 4.2.3.. Case 3: nˆ 1, n  1 ..................................................................................... 14. n. al. er. 4.2.1.. io. 6.. Premium and Reserve ................................................................................................... 11. Nat. 5.. 學. 4.. Mortality Risk............................................................................................................ 6. ‧ 國. 3.1.. Ch. engchi. i Un. v. Surplus Analysis ............................................................................................................ 16 5.1.. Surplus from Mortality Risk .................................................................................... 18. 5.2.. Surplus from Interest Rate Risk............................................................................... 18. 5.3.. Surplus from Surrender Risk ................................................................................... 19. 5.4.. Surplus from Liquidity Risk .................................................................................... 19. 5.5.. Probability of Insolvency and Illiquidity ................................................................. 19. Numerical Illustrations ................................................................................................. 20 6.1.. Sensitivity to Long-Term Interest Rate  ............................................................. 20. 6.2.. Sensitivity to Volatility Parameter of Interest Rate v.  ........................................... 23.

(6) . 6.3.. Sensitivity to Force of Reversion. 6.4.. Sensitivity to Parameter for Surrender Cash Value  ........................................... 28. 6.5.. Sensitivity to Parameter for Liquidity Risk  ....................................................... 30. 6.6.. Sensitivity to Long-term Interest rate at Different Initial Interest rate .................... 32. 7.. ...................................................................... 25. Conclusions .................................................................................................................... 34. Appendices ............................................................................................................................. 35 A.. Reserve Based on Assumed Interest Rate ................................................................... 35. B.. Mortality Table ............................................................................................................ 37. References .............................................................................................................................. 38. 立. 政治大. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. vi. i Un. v.

(7) List of Figures Figure 1. Sensitivity of Expectation of Surplus per Policy to Long-Term Interest Rate. ...................................................................................................................................... 22 Figure 2. Sensitivity of Probability of Insolvency to Long-Term Interest Rate.......... 22 Figure 3. Sensitivity to Probability of Illiquidity to Long-Term Interest Rate. .......... 23 Figure 4. Sensitivity of Expectation of Surplus per Policy to Volatility Parameter of Interest Rate. ................................................................................................................ 24 Figure 5. Sensitivity of Probability of Insolvency to Volatility Parameter of Interest Rate. ............................................................................................................................. 25 Figure 6. Sensitivity of Probability of Illiquidity to Volatility Parameter of Interest. 政治大. Rate. ............................................................................................................................. 25 Figure 7. Sensitivity of Expectation of Surplus per Policy to Force of Reversion. .... 26. 立. Figure 8. Sensitivity of Probability of Insolvency to Force of Reversion. ................. 27. ‧ 國. 學. Figure 9. Sensitivity of Probability of Illiquidity to Force of Reversion. ................... 27 Figure 10. Sensitivity of Expectation of Surplus per Policy to Parameter for Surrender. ‧. Cash Value.................................................................................................................... 28 Figure 11. Sensitivity of Probability of Insolvency to Parameter for Surrender Cash. sit. y. Nat. Value. ........................................................................................................................... 29 Figure 12. Sensitivity of Probability of Illiquidity to Parameter for Surrender Cash. io. er. Value. ........................................................................................................................... 29. al. n. iv n C hengchi U Risk. ............................................................................................................................. 30 Figure 13. Sensitivity of Expectation of Surplus per Policy to Parameter for Liquidity. Figure 14. Sensitivity of Probability of Insolvency to Parameter for Liquidity Risk. 31 Figure 15. Sensitivity of Probability of Illiquidity to Parameter for Liquidity Risk... 31 Figure 16. Sensitivity of Expectation of Surplus per Policy to Long-term Interest Rate at Different Initial Interest Rate. .................................................................................. 32 Figure 17. Sensitivity of Probability of Insolvency to Long-term Interest Rate at Different Initial Interest Rate. ...................................................................................... 33 Figure 18. Sensitivity of Probability of Illiquidity to Long-term Interest Rate at Different Initial Interest Rate. ...................................................................................... 33. vii.

(8) 1.. Introduction The analysis of surplus is an essential indicator for asset and liability management. in the life insurance industry. Nolde and Parker (2014) consider the distribution of surplus with stochastic rate of return and mortality-related cash flows for homogeneous policies. Our work extend the valuation model by taking surrender risk into consideration, which will contribute to interest rate-sensitive cash flows since policyholders may tend to surrender their insurance contracts and find externally higher yield alternatives. We focus on the distribution of surplus and point out the probability of generating negative surplus in order to seek out how profitable insurance companies. 治政 can be when issuing policies to policyholders. Also, 大the results can be used by 立 supervisors on checking whether the recent regulations are over-tightened or ‧ 國. 學. accidentally loosen.. ‧. Risks can be generated from every individual factor insurance companies may meet. sit. y. Nat. during the period since date of issue until the maturity date of the contract. Nolde and. io. er. Parker (2014) measure the surplus—including the calculation of premium and. al. reserve—for the portfolios of endowment and temporary insurance contract. n. iv n C stochastically on reinvestment rateh of return, which causes e n g c h i U interest rate risk, by using conditional AR(1) process at each period after the issuance. They concentrating on the market risk, and provide the derivation of the theoretical difference between stochastic and accounting surplus. Addition to market risk, surrender risk is also an important and common element that the insurance industry needs to deal with. As life insurance companies own a production cycle far distinct from other financial institutions, insurers usually hold assets with less liquidity than those held by banks. Since such insurance contracts are usually designed to provide financial protection for policyholders or their family members from their death as compensation , insurers are not expect to prepare cash in 1.

(9) exception of benefits. Also, policyholders do not expect the insurers give a shallow assumed interest rate lead by the high-liquidity asset bought. However, policyholder may tend to surrender their policies for cash due to the needs of personal consumption or the discovery of external financial products with higher yields while macroeconomic environment is changing. Surrender number may increase largely above the insurers’ expectation during certain periods, such as financial crises, insurance companies are required to prepare extra cash for those surrendered policyholders. While still many factors lead to surrender exists, Smink (1991), Albizzati and Geman (1994), Zenios (1999), Tsai et al.. 治政 (2009), De Giovanni (2010), Loisel and Milhaud (2011)大 and Le Courtois and Nakagawa 立 (2011) gave evidences that the difference between realized and expected market interest ‧ 國. 學. rate, namely, Δr, can be the dynamic surrender driver especially for policyholders of. ‧. traditional products. At the time while both the emergency fund hypothesis and the. sit. y. Nat. interest rate hypothesis can theoretically account for surrender behavior, Kuo at el.. io. al. er. (2003) gave their opinion that the overall impact caused by interest rate overwhelms. n. that of unemployment does. Thus, we would like to model the surrender rate according to this driver.. Ch. engchi. i Un. v. As we mentioned above, insurance company reasonably hold assets with lower liquidity than banks do. However, if the assets chosen by insurers at the time they receive premium are not being managed appropriately, during the period surrender rate rises unexpectedly those asset may occasionally be sold at an unsatisfying price. Thus, under such circumstance that herd effect of surrender occurs, the liquidity risk should also take into consideration of risk management. In this paper, the endowment contract will be mentioned in the next section. After that, the sources of risks which may be faced by insurance company and the modeling of these risks are introduced. Under the given assumptions, the calculation of the level 2.

(10) premium and reserves are derived in section 4. The methodology of the investment made by the insurance company and the quantity of surplus form different sources is computed in section 5. We will illustrate the numerical results including the sensitivity of surplus to risk parameters and the probability of insolvency and illiquidity in section 6. Finally, the conclusions are given in section 7.. 立. 政治大. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. 3. i Un. v.

(11) 2.. Endowment Contract In this paper, we consider homogenous portfolios, 1 which is the portfolios of. identical endowment policies are issued to a group of m individuals aged x with the identical risk characteristics. Under such assumption, the future life time and the behavior of surrender of the policyholders in this portfolio are independent and identically distributed. We define the time equals to 0 when the endowment contracts are issued. Each policy pays death benefit bt at the end of the year of death if death occurs during [t  1, t ) , a pure endowment benefit c if the policyholder survives to the end of policy year. 治政 n , and a surrender cash value 大 sv if the survival policyholder 立 t. ‧ 國. 學. decides to surrender at policy year t . The annual level premium  is payable at the. beginning of the initial k years since the issuance and an indicator  t corresponding. ‧. to the payment can be defined as. y. (1). er. io. sit. Nat. t  1tk .. If policyholders are required to pay premium to the insurance company at the beginning. n. al. Ch. of policy year t , then  t equals to 1, otherwise zero.. engchi. i Un. v. Since during any policy year the policyholders must be survival, died, or surrender option exercised, we define the following indicator variables: The indicator of survival I ti equal to 1 if the policyholder i (i  1,2,3,..., m) dies during [t  1, t ) (t  0,1,2,..., n). and the indicator of surrender J ti equals to 1 if the policyholder i (i  1,2,3,..., m) surrenders at time t (t  0,1,2,3,..., n) . According to the assumption of independent and identically distributed characteristic of m policyholders, we let. 1. This is the concept introduced in Nolde and Parker (2014). 4.

(12) Lt 1. Lt :  1  I ti 1  J ti  i 1. Lt 1. Lt :  1  I ti  J ti. (2). i 1. Lt 1. Dt :  I ti i 1. where L0  m and. t  1,2,3,..., n , then we can say the number of the survival. policyholders survive and not surrendered at time t (in-forced polices at time t) is Lt , the number of survival policyholders surrendered at time t is Lt and the number of policyholders dies during policy year t is Dt .. 立. 政治大. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. 5. i Un. v.

(13) 3.. Assumptions for Models Insurance companies provide financial protections to policyholders through. insurance contracts. In other word, they take the risks the policyholders might have had to face before the contract are issued. In order to make numerical illustration of the distribution of the endowment contract surplus, we model the risks the insurer will meet after the contracts are issued to policyholders.. 3.1.. Mortality Risk. 政治大. We begin by the introduction of mortality table. It is a table containing the probabilities. 立. of death at each age of an individual. The symbol. t 1|. qx stands for probability that an. ‧ 國. 學. x -year old survives to  x  t  1 -year old, and dies within that year2. Accordingly, we. ‧. use the fact that the indicator of mortality I ti which we introduced above is Bernoulli. sit. n. al. i Un. I ti ~ Bernoulli  qx t 1  .. Ch. engchi. er. io. mortality is. y. Nat. distributed, thus for the survival policyholder i at time t  1 , the indicator of his/her. v. (3). Since the probability of death is independent to interest rate, thus. E  I ti | t   qx t 1 .. 3.2.. (4). Interest Rate Risk In this paper, we choose to model the interest rate under the equivalent assumption. made by Nolde and Parker (2014) through a conditional autoregressive process of order one, AR(1), given the current force of interest. The process describe the stochastic. 2. We denote. 0|. qx as qx in convenience. 6.

(14) interest rate as a vibration around a trend toward a certain level of interest rate, which acts the same as Vasicek (1977). Let rt be the instantaneous interest rate at time t . Then under the assumption of Vasicek model, the process of the interest rate satisfies: drt    rt    dt   dwt , t  0 ,. where. . (5). is the force of reversion,  denotes the long-term mean of interest rate. wt t 0 is Wiener’s process and.   0 is a constant. Assuming that   1 is able to. ensure the stationarity of the process. We can gather the information about interest rate at time t into an information set t  rt ,  , ,   in convenience.. 立. 政治大. When it comes to interest rate, the discount factor is always discussed. Define. ‧ 國. 學. D(t, s) be the present value at time t of one dollar at future time T , T  t , given by T. rs ds. .. ‧. D  t, T   e t . (6). sit. y. Nat. The Vasicek model has been already studied and many of its propertied are.   T  t   T t re   1  e        rT | t   a t t. er. io. provided. For example, in Brigo and Mercurio (2006),.   t u . ev i l C n U hengchi Accordingly, it implies that it is normally distributed:. n. T. E  rT | t   re t.   T  t . .  1 e.   T  t . dWu .. (7).  (8). 2  2 T t Var  rT | t   1 e   .   2. Thus, the expectation of discount factor D(t, T ) given the information at time t is. E  D  t, T  | t   C  t, T  e where. 7.  F  t ,T rt. ,. (9).

(15)  2  2 2 C  t , T   exp    2   F  t , T   T  t   F t, T   2  4   1  T  t F  t , T   1  e    .   . (10). This is exactly the price of a zero coupon bond matured at time T valued at time t given the information at time t , which is denoted as B  t, T  , that is,. B  t, T  | t  C  t, T  e. 3.3..  F  t ,T rt. (11). Surrender Risk. 政治大 r , 0  t  n , is usually provided by insurance company to policyholders as reference. 立. When an endowment contract is issued, the assumed interest rate, denoted as. t. ‧ 國. 學. As time passes, if the interest rate becomes higher than the assumed interest rate given by insurers, policyholders may be yearning to give up the insurance for cash in order to. ‧. invest in the more profitable investment in the market. The more the difference between. y. Nat. er. io. sit. the interest rate and the assumed interest rate is, the policyholders are theoretically more craving to surrender the contracts. We learn from Kuo, Tsai and Chen (2003) that the. n. al. Ch. i Un. v. behavior of surrender can be accounted by emergency fund hypothesis and interest rate. engchi. hypothesis. They concluded that the interest rate overwhelms on the overall impact on the dynamics of the surrender rate. As a result, we choose the interest rate as the main factor of surrender risk. Moreover, Albizzati and Geman (1994) modeled the surrender risk from the aspect of the rational behavior of the policyholders according to the interest rate. An decision criterion t is introduced in this paper for the indicator related to whether surrender at time t is profitable for policyholders or not: T. t   rs ds  ln  t t. 8. (12).

(16) where rs represent the assumed interest rate. Surrender options will only rational be exercised by policyholders only when the accumulated value of one dollar from time t to time T , which is  rs ds , is larger than t . T. t. Given the interest rate at time t , we know from the properties under the assumption of Vasicek model that. . T. rs ds is normally distributed with mean. t. E.  r ds r      r  F t,T    T  t  , T. s. t. t. t. (13). and variance Var. . T. t.  政治大. rs ds rt . 2 2.   2 2 T  t  F t , T  F  t , T  .    2  . (14). 立. Rt  Pr    t. T.   T    E  t rs ds rt   t  rs ds  t rt     .   Var  T r ds r     t s t  . 學. ‧ 國. Thus, the surrender rate at time t is. (15). ‧. sit. y. Nat. Similarly, we define an indicator J ti for the survival policyholder i at time t  1 , the. n. al. 3.4.. er. io. indicator of his/her mortality is Bernoulli-distributed with parameter Rt , i.e.,. CJ h~ Bernoulli  R  .U n i engchi i t. t. v. (16). Liquidity Risk Berkowitz (2000) divided the liquidity risk into two aspects: 1) Asset Liquidity Risk: This is the risk that the large bid-ask spread of an asset if the investor is necessarily involving in large transaction, then he or she will not be able to fair treated. 2) Funding Liquidity Risk: This kind of risk occurs whenever the financial institution is not capable to own sufficient cash for its current payment, then. 9.

(17) selling its immature assets will be required. Usually, these assets cannot be sold at its fair price under the urgent situation. These two liquidity risks will be discusses in this paper. We simplify the liquidity risk model introduced in Longstaff, Mithal and Nies (2005) by setting  t as a constant  . Since we only consider the investment the insurance company will get involve in is default-free zero coupon bond, the default intensity indicator t equals to 0 for all t  0 . As we form the interest rate under Vasicek model, we can derive the price. considering liquidity risk at time t of a zero coupon bond which will mature at time. T , denoted as LB  t , T  :. 立.  rs  ds.     T t  | t   E e  e t  . T. rs ds.   T t  B t, T  . e . ‧. ‧ 國. T. 學.   LB  t , T   E e t . 政治大. n. er. io. sit. y. Nat. al. Ch. engchi. 10. i Un. v. (17).

(18) 4.. Premium and Reserve. 4.1.. Reserve Based on Assumed Interest Rate The assumed interest rate is the guaranteed rate of return made by the insurer to. the policyholders. Therefore, when the policyholders would like to surrender their policies at time t , the surrender cash value svt is calculated as the assumed reserve V , which is computed based on the assumed condition, multiplied by a proportion. t.  t  0,1 , that is. svt   t  tV. (18). 政治大. where  t is introduced in Albizzati and Geman (1994) by a parameter   1 :. 立.  t  e 1r .. (19). s. ‧ 國. 學. Traditionally, it is calculated under the assumption that there is no surrender, the. ‧. number of death is exactly followed by the given mortality table and the rate of return. Nat. sit. y. is the assumed interest rate announced to the policyholders. By the properties of reserve,. n. al. er. io. it can be seen as a function of the premiums paid to the insurance company followed by. Ch. i. e. i Un. v.   t n g c h t  t  V   Y X    Z    j   k    t j   Xk  j 1  j 1  k  j 1    k  j 1   t. (20). where.  1 rt  p e  x Xt    t 1 px e rt  t p x Yt  t 1  X t. ,t  1 , t  2,3, 4,..., n (21). bt  t 1| qx    t px Zt    bt  t 1| qx  c  n p x  n px 11. , t  1, 2,3,..., n  1 ,t  n.

(19) We can see the relation between the premium reserves of each period under the assumed interest rate.. 4.2.. Reserve and Premium Now that we have obtained the surrender cash value of each policy year, we can. derive the reserve of our target contract tV by the retrospective method under a given path of interest rate and the realized number of death, survival and surrender. According to the properties of reserve, t V can be recursively calculated from. V : the total. t 1. reserve at time t , Lt  tV , is equal to the sum of. 政治大. a) The accumulated value of the premium receipt and total asset from the. 立 beginning of the previous policy year,. t. r ds  Lt 1  t 1    Lt 1  t 1V   et 1 s ,. ‧ 國. 學. b) the death benefit paid to died policyholders at period [t  1, t ) , Dt  bt ,. . ‧. c) and the surrender payment to the policyholders who exercises the surrender. . y. sit. Nat.  option at the end of policy year t , Lt   t  tV .. a. t. n.  Lt 1  t 1    Lt l1  t 1V   et 1. Ch. rs ds. er. io. Thus, for t  1,2,3,..., n ,.  Dt  bt  Lt  i vt  tV   Lt  tV. n engchi U. (22). In order to consider the situation that there is no effective policy existing anymore during the n years, we define an indicator nˆ : , Lt  0 n  1 nˆ    min t  n : Lt  0 , o.w.. 4.2.1. Case 1 :. (23). nˆ  1. In the case that nˆ  1 , since policy is issued, there is no effective policy existing after that because of the surrendering and dying of the policyholders during the period. [0,1) . Since the reserve can be represent as the liability of the insurance company to the 12.

(20) policyholder, the reserve should be zero since there is neither premium receipt, death benefit nor surrender payment in the future. Accordingly, the reserve should be zero for. t  1,2,3,..., n . Since there is no liability after time 1, 1. rs ds m    e0  D1  b1  L1  1  1V   L1  1V  0. (24). Therefore, the premium under such situation can be obtained by. . (25). 1. rs ds m  L  1  U1  e 0. 政治大 .  1. 立 nˆ  n  1. ‧ 國. 學. 4.2.2. Case 2:. 1. r ds  D1  b1  L  1 W1  e 0 s .  1. By the recursive formula of reserve mentioned above, for t  1,2,3,..., n ,. ‧. y. sit. io. a lL. n. Xt. (26). t 1. t. rs ds e t 1 , L0  m. LtC. h eLn g c h i. Yt  t 1  X t  Zt .  t. Lt.  t Ut. er. where. Nat. t  t   t   t  V  Y X    Z  j   k   j   X k  ,   t j 1  j 1    k  j 1    k  j 1 . i Un. v. bt  Dt  c  Ln  1t n  Lt   t  Wt Lt. (27). .. Now that we know the reserve of the given premium, interest rate and numbers of death, survival and surrender, we can use the property that the reserve when is zero the contract matures to derive the level premium paid by policyholders. Namely, V  0.. n. 13. (28).

(21) Thus, the fair premium is obtain by n n 1   n 1  r ds  n 1  bn  Dn  c  Ln  L   n Wn  e s  Ln1    Z j   X k  j 0    k  j 1    n  n 1   n 1      rs ds n 1 Ln 1    Y j   X k    n 1   Ln   n  U n  e  j 0   k  j 1       . .  n. 4.2.3. Case 3:. (29). nˆ 1, n  1. Similarly to case 1, if there is no remaining policy before the contract matures, then the insurance company are not required to prepare for the future payment, thus the. 政治大 V  0, for t  nˆ .. reserve of the left years should be zero, that is. 立. ‧ 國. 學. For 1  t  nˆ ,. (30). t. (31). t. n. al. rs ds L X t  t 1 e t 1 , L0  m Lt. C Y h t. Zt . i Un. Lt  t Ut t 1  X t  Lt. engchi. er. io. sit. y. Nat. where. ‧. t  t   t   t  V  Y X    Z X         t j  k j  k  j 1  k  j  1 j  1 k  j  1         . v. (32). bt  Dt  Lt   t  Wt . Lt. Accordingly, the reserve at time nˆ is zero, therefore nˆ.  Lnˆ 1  nˆ1    Lnˆ1  nˆ1V   enˆ 1. rs ds.  Dnˆ  bnˆ  Lnˆ   nˆ  nˆV   Lnˆ  nˆV  0 .. (33). Thus, the level premium paid in each policy year is nˆ nˆ 1   nˆ 1  rs ds  nˆ 1 b  D  L    W e  L  Z  nˆ nˆ nˆ nˆ  nˆ 1   j   X k   j 0    k  j 1    nˆ  nˆ 1   nˆ 1      rs ds nˆ 1 Lnˆ 1    Y j   X k    nˆ 1   Lnˆ   nˆ  U nˆ  e  j 0   k  j 1       .  nˆ. . 14. (34).

(22) The above shows the calculation of premium under a certain path of interest rate. Since the surrender option is quasi-American, we use Monte Carlo method to simulate the path of interest rate, and number of death, survival and surrender. Furthermore, the distribution of the level premium and reserve can be finally obtained. In the next section, the expectation of premium ˆ will be charged and the expectation of the reserves tVˆ will be used as the level of liability per policy when analyzing the surplus gained, that is,. and. 立. ˆ  E  . (35). Vˆ  E  tV  .. (36). 政治大 t. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. 15. i Un. v.

(23) 5.. Surplus Analysis In this paper, we consider the insurance company holds zero coupon bonds to meet. the future needs of cash at the end of each policy year. Assuming that only default-free zero coupon bonds maturing during time 0, n  , are available for the insurance company to invest in. Let wti be the portfolio held by the insurance company at time t of the zero coupon bond maturing at time i . In order to establish the principle of investment made by the insurance company, we can record the expectation of cash needed of each period during our simulations for premium and reserve in the previous section.. 治政 CN  E  D  b  L  c  1  L   大 V  L    ˆ    立 t. t. t. n. t n.  t. t. t. t. t. (37). ‧ 國. 學. We then use these expectations to adjust the portfolio held by the insurance company. At the end of policy year 𝑡 the insurance company will receive an amount of invest income. ‧. 𝑖 from the matured bond of that year, whose value equals to 𝑤𝑡−1 and the payment of. Nat. sit. y. premium paid. At the same time, the insurer is ought to pay death benefits and surrender. al. n. and inflows at time 𝑡 to be Gt :. er. io. payment to those dead and surrendered. Thus, we define the gap between the cash outflows. i Un. Ch. v. n  hi ˆ Gt  Dt  bt  Ln  c  1e t n gLtc  t  tV  Lt  t    wt. (38). If Gt is negative, which means that there is extra cash held, the insurer will invest in zero coupon bonds with different maturity such that the expected cash flow needed in the future can be satisfied. If Gt is positive, which means that the insurer is running out of cash at that time, the assets held should be sold in order to generate enough cash. In the sake of decreasing the liquidity loss, the asset with lower maturity would be sold first, and so on. After the portfolio is adjusted, we then move on to the calculation of surplus. Surplus is the difference between the value of assets and liabilities. Since the insurer 16.

(24) holds only the zero coupon bonds as the assets, we can added up the value of portfolio and the value of total asset At is obtained by. At . n.  w  B  t, i  .. i t 1. i t. (39). For an insurance company after an insurance contract is issued, the total liability at the end of policy year t immediately before the receipt of the payment of the premium is Lt  tV . However, the payment of premium, at the beginning of the next period, Lt  t   , is included into the assets, this additional part is necessary to be. 政治大. eliminated, thus the surplus at the end of the policy year t is:. 立S  A  L   Vˆ    ˆ  . t. t. t. t. (40). t. ‧ 國. 學. For any policy year t before the assets are adjusted ( thus the liquidity risk is not. ‧. included yet), we can tear the current total surplus into three parts according to the risks. sit. al  iv  n C  L    V     w h e w    B  j, t i LB n g c h U  j, t    1 er. n.  t. t. rs ds ˆ ˆ  t 1  bt  Dt  c  Ln  1t n t 1V  St 1  Lt 1  t 1     e. io. Lt  St   Lt 1  . . y. Nat. we involve in:. n 1. t.   Lt 1  . t. .   Lt 1  .  j t 1 Vˆ  S. t 1. . t 1.   L  Vˆ  t t   Lt 1  t 1  ˆ  bt  Dt  c  Ln  1t n  Lt  tVˆ   j t 1. . j. wt  wtj1. t. j. (41).  t 1 rs ds  ˆ ˆ  1 t 1V  St 1  Lt 1   t 1      e  . . t.  n 1     Lt   t  tV       wtj1  wtj    B  j, t   LB  j , t    1 w j  w j   t t 1  j t 1  where the first term is referred as the surplus according to mortality risk, the following one is the surplus corresponding to interest rate risk and the last one is the surplus related to surrender risk. As the surplus from different sources has been separated, we can compare them to the expected surplus from each risk, and find out the impact of 17.

(25) then respectively. During the simulation in order to obtain the premium and reserves, we can also obtain the surpluses from these risks. However, under every simulated path, the fair premium and reserves are calculated such that the surplus of each policy year is zero, the sum of these surpluses are zero, that is, rs ds   E  Lt 1  t 1V  Lt 1  t 1     e t 1  bt  Dt  c  Ln  1t n  Lt   t  tV  Lt  tV    t.  E  Lt 1  t 1V  Lt 1  t 1    bt  Dt  c  Ln  1t n  Lt  tV   . (42).  rs ds    E  Lt 1  t 1V  Lt 1  t 1      e t 1  1   E   Lt   t  tV     0 t. 政治大 Here we also consider the expectation of these three parts as the expected surplus 立 results from these risks. Therefore, we define S , S and S respectively as M. t. I. S. t. t. ‧ 國. 學. StM  E  Lt 1  t 1V  Lt 1  t 1    bt  Dt  c  Ln  1t n  Lt  tV   . ‧. t   t 1 rs ds   St  E  Lt 1  t 1V  Lt 1  t 1      e  1    . I. io. sit. y. Nat. StS  E   Lt   t  tV  .. n. er. 5.1. Surplus from Mortality Risk. al. Ch. (43). i Un. v. Since the simulated numbers of death and survival is not necessarily the same as. engchi. the expected ones, the surplus will be generated from the difference between the realized and the expected path. Thus, the surplus form mortality risk is. StM   Lt 1  . . . Vˆ  St 1  Lt 1  t 1  ˆ  bt  Dt  c  Ln  1t n  Lt  tVˆ   StM . t 1. (44). 5.2. Surplus from Interest Rate Risk Since the simulated paths of interest are mostly deviated from the expected one, the reinvestment earned by the insurance company is accordingly different from the expected one, thus the surplus from interest rate risk is referred as. 18.

(26) S   Lt 1   I t. .  t 1 rs ds  ˆ ˆ  1   St I t 1V  St 1  Lt 1  t 1      e  . . t. (45). 5.3. Surplus from Surrender Risk Similarly, the number of surrender may not be exactly the same as the expected, thus the surplus from surrender risk is. StS    Lt   t  tV   StS. (46). 5.4. Surplus from Liquidity Risk Since the liquidity risk is not considered when the assumed reserves is calculated,. 政治大. the surplus from liquidity risk occurs only when the insurance company is required to. 立. 學. liquidity risk StL is given by. StL . n 1.  w. j t 1. j t 1.  wtj    LB  j, t   B  j, t    1 w j w j. . Nat. t. t 1. (47). io. sit. y. 5.5. Probability of Insolvency and Illiquidity. . ‧. ‧ 國. sell the immature assets held in order to satisfy its cash needs. Thus, the surplus from. n. al. er. Insolvency occurs when the asset held by the insurance company is less than the. Ch. i Un. v. liability. We are interested in the probability of insolvency of the insurance company. engchi. during the end of each policy year before the premium payment of the next period. We can count the times that the total surplus of each policy year is less than zero, and then divide the value by the total simulation time. The proportion we obtain can be the reference of how possible the insurance bankrupts at each policy year. Similarly, we can also count the number of events that the insurance company is need to sell those immature assets held for cash and then divide the value by the total simulation time. The result can be an indicator of how possible will an insurance company face liquidity loss at each period. 19.

(27) 6.. Numerical Illustrations In this paper, endowment policies maturing at 10 years after the issuance are issued. to 10,000 policyholders under the following conditions respectively: the initial interest rate r0 is 6% when the policy is issued; the force of reversion. . is 0.1; the long-. term mean interest rate  is 8%; the volatility parameter of interest rate  is 1%; the leveled assumed interest rate rt announced to the policyholders is 6%; the death benefit is 1000 dollar paid to each dead policyholders who die before the insurance matures; the endowment benefit is 1000 paid to those policyholders survival to the maturity date; the parameter corresponding to the surrender cash value  is 0.9; the. 政治大. parameter for the liquidity risk  is 1%.. 立. The above situation is considered to be the parameters used by the insurance. ‧ 國. 學. company when computing the premium and reserves by Monte Carlo method.. ‧. Sometimes things do not act like what we feel like. We would like to see what will. sit. y. Nat. happen is these situation changes as followed right after the insurance company issue. io. er. the contracts to 30-year-old policyholders and 80-year-old policyholders: 1) Long-term interest rate  : 8%;. n. al. 2). C h rate  : 1%; U n i volatility parameter of interest engchi. 3) force of reversion. :. v. 0.1;. 4) parameter for surrender cash value  : 0.8; 5) parameter for liquidity risk  : 1%.. 6.1. Sensitivity to Long-Term Interest Rate. . Figure 1 shows that the surplus will be enormously effected result from the deviation of the long-term interest rate. As the long-term interest rate is set to be 8% when the endowment contract is issued, the premium and the reserve is computed under 20.

(28) this long-term mean. If the long-term mean level goes to 10% right after the issue date, the interest rate will grow higher than it was expected, which will lead to the increased number of surrender. Since the surrender cash value is usually set to be less than the reserve at the time policyholders surrender, greater number of surrender will make the insurance company more profitable. Thus, the simulated surplus under 𝜃=10% will be greater than that of 𝜃=8%. On the other hand, if the long-term mean decrease to 6%, then the interest rate will tend to decrease more than expected. Therefore the surplus also decreases as time passes. These phenomena appear no matter when the endowment policies are issue to 30-year-old policyholders or 80-year-old ones.. 立. As the long-term interest. 政治大 rate increases with other. parameters fixed, the. ‧ 國. 學. reinvestment income of the insurance company also increases. Therefore, under such circumstances will the value of the total asset held by the insurer more possibly be. ‧. greater than its liabilities. This will lead to the decrease of probability of insolvency. io. sit. y. Nat. shown in Figure 2.. n. al. er. We consider the situation that the long-term mean level of the interest rate increase. Ch. i Un. v. to 10%. The increased growth of interest rate will lead to the unexpected increasing. engchi. number of surrender, since exercising the surrender options is a rational decision if the market interest rate grows above the assumed interest rate. As a result, more surrender paid will be announced and the insurance company will be necessary to sell the immature assets for these unexpected call for cash.. 21.

(29) % % 0%. % % 0%. 政治大. 立. ‧ 國. 學. Figure 1. Sensitivity of Expectation of Surplus per Policy to Long-Term Interest Rate.. ‧. n. Ch. % % 0%. er. io. sit. y. Nat. al. % % 0%. engchi. i Un. v. Figure 2. Sensitivity of Probability of Insolvency to Long-Term Interest Rate.. 22.

(30) % % 0%. 立. % % 0%. 政治大. Figure 3. Sensitivity to Probability of Illiquidity to Long-Term Interest Rate.. ‧ 國. 學. 6.2. Sensitivity to Volatility Parameter of Interest Rate . ‧. As the parameter of volatility increases, the interest rate will be more volatile than. y. Nat. io. sit. expected. For the analysis of surplus from the four risks, we find the evidence that the. n. al. er. surplus form interest risk and surrender risk decreases largely when the volatility of. Ch. i Un. v. interest rate grows high. This is because the policyholder may surrender more severely. engchi. than expected in the first few years. This will lead to the increasing cash outflow for surrender in these years and thus result to the increase in loss of liquidity risk for the insurance company. This is shown in Figure 4. In the case that the endowment policies are issued to 30-year-old policyholders, as the volatility of interest rate rises, the path of interest rate is more unlikely act as it is expect when the policies were issued. Thus, the investing decision that the insurance company made become unsuitable, which will make the probability of insolvency increase. On the other hand, if the volatility is not so high as it was expected, the path 23.

(31) of interest rate is more likely to stay in the smaller interval. Therefore, although there is a little decrease in investment income, the investing decision still works, leading to the decrease in the probability of insolvency in Figure 5. The surrender rate cannot be determined directly from the volatility of the interest rate, however, its asymmetric property lead the increasing surrender paid to policyholders when the interest rate is more volatile. Thus, the overall impact on the probability of illiquidity is higher when the volatility is higher for both cases.. 立. .5% % %. 政治大. .5% % %. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. i Un. v. Figure 4. Sensitivity of Expectation of Surplus per Policy to Volatility Parameter of Interest Rate.. 24.

(32) .5% % %. .5% % %. 政治大. 立. Figure 5. Sensitivity of Probability of Insolvency to Volatility Parameter of Interest Rate.. ‧. ‧ 國. 學 y. .5% % %. n. al. er. io. sit. Nat. .5% % %. Ch. engchi. i Un. v. Figure 6. Sensitivity of Probability of Illiquidity to Volatility Parameter of Interest Rate.. 6.3. Sensitivity to Force of Reversion  If the force of reversion rises, it cost shorter time for the interest rate reach the 25.

(33) long-term mean level. Therefore, the interest rate will rise faster when the force of reversion is higher. This lead to the increase in number of surrender in the first few years. Thus, the expectation of surplus increases in both cases that the endowment policies are issued to 30-year-old policyholders and 80-year-old-policyholders shown in Figure 7. Figure 8 shows that the probability of insolvency rises when the force of reversion of interest rate increases in both cases. On the other hand, if the force of reversion is lower than it is expected, the number of surrender is also lower. Thus, the surrender. 政治大 issued to 30-year-old policyholders 立 and 80-year-old-policyholders.. payment which is paid to policyholders decreases in both cases that the policies are. ‧ 國. 學. Similar to the reason above, when the force of reversion of interest rate increases, the possibility that the insurance company is short in cash rises. This is illustrated in. ‧. io. sit. y. Nat. n. al. er. Figure 9.. .05 .1 .2. Ch. engchi. i Un. v. .05 .1 .2. Figure 7. Sensitivity of Expectation of Surplus per Policy to Force of Reversion.. 26.

(34) .05 .1 .2. .05 .1 .2. 政治大. 立. ‧ 國. 學. Figure 8. Sensitivity of Probability of Insolvency to Force of Reversion.. ‧. n. .05 .1 .2. Ch. er. io. sit. y. Nat. al. n engchi U. iv. .05 .1 .2. Figure 9. Sensitivity of Probability of Illiquidity to Force of Reversion.. 27.

(35) 6.4. Sensitivity to Parameter for Surrender Cash Value. . The parameter is associated with how much proportion of reserve can a surrendering policyholder get back when he or she decides to surrender at the end of a certain policy year. The more proportion the insurance company needs to pay back, the less profitable is. In Figure 10, we can easily find out that the expectation of surplus decreases when the parameter increases in both cases that the policies are issued to 30year-old policyholders and 80-year-old-policyholders.. 治政大phenomenom happens in both paid back to the surrendered policyholders increase. This 立 Clearly, the probability of insolvency increases when the proportion of reserve. cases and is illustrated in Figure 11.. ‧ 國. 學. We can find out from Figure 12 that the parameter of surrender cash value effect. ‧. mainly at the beginning few policy years after the issue date. This is because that the. Nat. n. al. er. io. sit. y. surrender options are exercised mostly during this period.. Ch. engchi. i Un. v. Figure 10. Sensitivity of Expectation of Surplus per Policy to Parameter for Surrender Cash Value.. 28.

(36) 立. 政治大. ‧ 國. 學. Figure 11. Sensitivity of Probability of Insolvency to Parameter for Surrender Cash Value.. ‧. n. er. io. sit. y. Nat. al. Ch. engchi. i Un. v. Figure 12. Sensitivity of Probability of Illiquidity to Parameter for Surrender Cash Value.. 29.

(37) 6.5.. Sensitivity to Parameter for Liquidity Risk  The parameter decides the discount level of the zero coupon bond price when the. asset is necessarily be sold in a short period. Clearly, the insurance company may loss much more value under larger gamma. Thus, the decreasing expectation of surplus when the parameter rises in both cases are shown in Figure 13. As the loss from liquidity risk rises, which is resulted from the increased parameter for liquidity risk, the increasing probability of insolvency of the insurance company for. 政治大. both cases is illustrated in Figure 14.. 立. Since the parameter effect mainly on the level of discount of the assets sold, the. ‧ 國. 學. probability of illiquidity do not change largely in Figure 15.. ‧. n. er. io. sit. y. Nat. al. Ch. engchi. i Un. v. Figure 13. Sensitivity of Expectation of Surplus per Policy to Parameter for Liquidity Risk.. 30.

(38) 立. 政治大. ‧ 國. 學. Figure 14. Sensitivity of Probability of Insolvency to Parameter for Liquidity Risk.. ‧. n. er. io. sit. y. Nat. al. Ch. engchi. i Un. v. Figure 15. Sensitivity of Probability of Illiquidity to Parameter for Liquidity Risk.. 31.

(39) 6.6.. Sensitivity to Long-term Interest rate at Different Initial Interest rate Additional to the interest rate starting from 8%, we would also like to know how. the sensitivity of the surplus distribution of the endowment contracts to the long-term interest rate at lower initial interest rate: 3%. When the interest rate increases more than it is expected, the rate of surrender will grow higher, which may cause the endowment contract gain from the surrender penalty given to surrendered policyholders. From Figure 16 we can clearly find out that the expectation of surplus rises as the long-term interest rate increases, which acts likely as when the initial interest rate is 6%. The expectation of surplus is similar when the. 治政 endowment contracts are issued to 80-year-old policyholders. 大立 ‧ 國. 學. If the long-term interest rate increases, the probability that occurring negative surplus for the endowment contracts decreases. This is shown in Figure 17. According. ‧. to Figure 18, the probability that occurring liquidity problem rises in the second policy. Nat. n. al. er. io. sit. y. year, and then decreases in the following policy years.. Ch. engchi. i Un. v. Figure 16. Sensitivity of Expectation of Surplus per Policy to Long-term Interest Rate at Different Initial Interest Rate.. 32.

(40) 立. 政治大. Figure 17. Sensitivity of Probability of Insolvency to Long-term Interest Rate at Different Initial Interest Rate.. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. i Un. v. Figure 18. Sensitivity of Probability of Illiquidity to Long-term Interest Rate at Different Initial Interest Rate.. 33.

(41) 7.. Conclusions This paper takes stochastic interest rate process, surrender risk and liquidity risk. into consideration and shows the simulated endowment surplus at the end of each policy year. Also, the probability of insolvency and probability of illiquidity can be referred to insurance company, policyholders and even the regulators as a tool of realizing how risky the insurance company is in the future years after the endowment contracts are issued. One further extension can be done by allowing the insurance company to invest in equities rather than zero coupon bonds. Another one is to have commission fee and. 治政 other expenses taken into consideration. 大立. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. 34. i Un. v.

(42) Appendices. A.. Reserve Based on Assumed Interest Rate Under the assumed interest rate and the expected number of death and survival. given by the mortality table, we can derive the assumed reserve as a function of the premium. Since the assumed reserve can be seen retrospectively as the expected accumulated value of the premium of the policyholders paid, the reserve at time is zero when the policy is issued. Therefore, for t  1 , the reserve of an individual policy can be described by. 治政大   e  b  q  p  V . 立 1. 0. rs ds. 1. ‧ 國. x. (48). 1. 學. Thus,. x.  e r1   b1  qx  V        . 1  px   px . (49). ‧. Nat. t. n. al.  bt  t 1| qx  t px  tV. ni C h U e n g cp h ei  .  t 1 rsds  t 1 p x  e   V    tV    t 1  t 1 t px    t. rs ds. er. io. t 1 t 1 px  t 1    t 1V   e Thus. sit. y. Similarly, for t  2,3,4,..., n  1 ,. (50). v. t. t 1. t 1. x. t. px. rs ds.  b  q      t t 1| x  .   t px  . (51). And for t  n , n. n 1 n 1 px  n 1    n 1V   e. rs ds.  bn  n1| qx  c  n px  n px  nV. (52). Thus,.   n 1 rs ds  n 1 p x  e    n 1V   n 1  nV     p  n x     b  q  c  n p x    n n 1| x  n px   n. 35. n. n 1 n 1 p x  e n px. rs ds.     . (53).

(43) To sum up, we obtain. V  Y1    Z1. (54). V  X t  t 1V  Yt    Zt. (55). 1. and for t  2,3,4,..., n t. where  1 rt  p e  x Xt    t 1 p x e rt  t p x Yt  t 1  X t. ,t  1 , t  2,3, 4,..., n. (56). bt  t 1| qx    t px Zt    bt  t 1| qx  c  n p x  n px. , t  n.. 學. ‧ 國. 立. n 1 政治, t  1, 2,3,..., 大. By the recursive formula obtained from (54) and (55), we can clearly find out that.  X t  Y2 X 3 .   Z1 X 2 X 3 .  X t  Z2 X 3 .  X t  Y3   X t  Z3 . n. engchi. 36. y. sit er. io. Ch.  Z t 1 X t  Z t .  Xt . Nat. t  t  t   t    Y j   X k       Z j   X k  . j 1  j 1  k  j 1    k  j 1  . al.  Yt 1 X t  Yt   .  Xt . ‧. V  Y1 X 2 X 3 . t. i Un. v. (57).

(44) B.. x. Mortality Table. qx. x. x. qx. qx. x. x. qx. qx. 0 0.000522. 22 0.000668. 44 0.003139. 66 0.017892. 88 0.116732. 1 0.000384. 23 0.000710. 45 0.003418. 67 0.019497. 89 0.127197. 2 0.000277. 24 0.000762. 46 0.003714. 68 0.021322. 90 0.139237. 3 0.000215. 25 0.000821. 47 0.004033. 69 0.023359. 91 0.153157. 4 0.000181. 26 0.000885. 48 0.004381. 70 0.025556. 92 0.166960. 5 0.000166. 27 0.000926. 93 0.182008. 6 0.000149. 28 0.000965. 71 0.027961 治政大0.030517 50 0.005136 72. 7 0.000139. 29 0.001008. 立. 49 0.004766. 94 0.198411. 30 0.001061. 52 0.005939. 74 0.036264. 96 0.235786. 31 0.001127. 53 0.006351. 75 0.039482. 97 0.257036. 10 0.000129. 32 0.001209. 54 0.006754. 76 0.042913. 98 0.280199. 11 0.000131. 33 0.001305. 55 0.007189. 77 0.046627. 12 0.000153. 34 0.001413. 56 0.007689. 78 0.050663 100 0.332986. 13 0.000196. 35. 14 0.000255. 36 0.001661. 58 0.009084. 80 0.059942 102 0.395712. 15 0.000344. 37 0.001804. 59 0.010040. 81 0.065252 103 0.431358. 16 0.000455. 38 0.001949. 60 0.010943. 82 0.070972 104 0.470260. 17 0.000540. 39 0.002089. 61 0.011680. 83 0.077204 105 0.512597. 18 0.000584. 40 0.002254. 62 0.012592. 84 0.083852 106 0.558864. 19 0.000607. 41 0.002429. 63 0.013699. 85 0.091053 107 0.609304. 20 0.000624. 42 0.002636. 64 0.014981. 86 0.098875 108 0.664279. 21 0.000641. 43 0.002875. 65 0.016404. 87 0.107353 109 0.722667. Nat. io. al. sit. y. ‧. 9 0.000133. 95 0.216293. n. iv n C 0.001532 h57 79 0.055090 e n0.008320 gchi U. 37. 99 0.305452. er. 8 0.000134. ‧ 國. 73 0.033290. 學. 51 0.005524. 101 0.362989.

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