Mortality Improvement in China and the Influence on Life Insurance Reserving

Transcription

1 Mortality Improvement in China and the Influence on Life Insurance Reserving Manqi Xie September 4, 22 Abstract Based on the mortality sample data in each age range separately for both male and female from 98 to 29, we analyze the mortality improvement and then discuss its influence on reserves of both life insurance and annuity. Keyword: Life table; Rate; Mortality improvement; Reserve Supervisor: Xiaojun Wang Renmin University of China

2 Introduction A life table, also called a mortality table or actuarial table, is a table which shows, for each age, what the probability is that a person of that age will die before his or her next birthday, which is exactly probability of death. The life table is an important tool to calculate life insurance rate. At present Chinese insurers refer to the life table summarized by Chinese People Insurance Corporation Limited in the late 2th century with death data before the time. However, according to the result in the 5th National Census, Chinese average life expectancy improved by 2.85years comparing with it in 99. So from theoretical view, the price of life insurance product like term life insurance should decrease while it would go up with respect to high-risk group. On the contrary, the price of annuities like life annuities should increase because insurers have to give more average indemnity to insured due to longer lives and guarantee period. However, most life insurers fail to follow this law, instead, their products prices are insensitive to mortality improvement and almost are keeping stable. Mortality improvement also has influence on reserving. This paper tries to explore mortality improvement laws for each age and gender and its impact on life reserving. To simplify the problem, I take the basic insurance products, term life insurance and life annuities, as examples, and calculate five-year reserves separately based on practical mortality and improved mortality. By comparing the results, I analyzed laws and trends. 2

3 Introduction A life table, also called a mortality table or actuarial table, is a table which shows, for each age, what the probability is that a person of that age will die before his or her next birthday, which is exactly probability of death. The life table is an important tool to calculate life insurance rate. At present Chinese insurers refer to the life table summarized by Chinese People Insurance Corporation Limited in the late 2 th century with death data before the time. However, according to the result in the 5 th National Census, Chinese average life expectancy improved by 2.85years comparing with it in 99. So from theoretical view, the price of life insurance product like term life insurance should decrease while it would go up with respect to high-risk group. On the contrary, the price of annuities like life annuities should increase because insurers have to give more average indemnity to insured due to longer lives and guarantee period. However, most life insurers fail to follow this law, instead, their products prices are insensitive to mortality improvement and almost are keeping stable. Mortality improvement also has influence on reserving. This paper tries to explore mortality improvement laws for each age and gender and its impact on life reserving. To simplify the problem, I take the basic insurance products, term life insurance and life annuities, as examples, and calculate five-year reserves separately based on practical mortality and improved mortality. By comparing the results, I analyzed laws and trends. The mortality improvement is defined by: Avg Ann Improv x,t to t+k = ( q x.t+k q x,t ) k () where Avg Ann Improv x,t to t+k is mortality improvement of x-year-old insured from t th year to (t+k) th year; q x.t+k is x-year-old insured s mortality in (t+k) th year; q x,t is x-year-old insured s mortality in t th year. The modified term-life-insurance reserve is defined by: A x: (2) 2

4 Pa A x h x a h x Paxh : a : : h x (3) V x:n = Pa x:n (4) where α is one-year premium of x-year-old insured; β is correction coefficient; h a x is present value of one-year-deferred (h-)-year-paid annuities of x-year-old insured; a x:h is h-year-paid annuities present value of x-year-old insured; P is term premium; A x:n is n-year term life insurance premium of x-year-old insured; V x:n is life insurance reserves. The term annuity reserve is defined by a = xn : n - i+ v px+ i i= (5) where a x:n is present value of n-year term annuities of x-year-old insured; p x+i is probability of survival of (x+i)-year-old insured; v is factor. One problem is that insured of some ages have zero mortality due to the year sample data from Bureau of Statistics, thus I use the adjacent age average instead of zero. 2 Mortality Improvement Analysis in China 2. Both Male s and Female s Mortality Improvement From year 98 to year 29, the overall mortality improvement shows in Chart. Based on Formula (), we let k=9, thus, we got mortality improvement for every age bracket. From Chart, we can see that female s mortality improvement is higher than male s, and the difference becomes the largest from age 2 to age 5. However, the difference narrows from age 7. With age increasing, female s mortality improvement decreases. Infants with age -4 have the highest mortality improvement.7, while the mortality improvement of females whose ages are above 7 is only about.25. As with male s mortality improvement, those whose ages are between 3 and 34 have 3

5 the lowest improvement, but after this age bracket their mortality improved quickly until equaled to female s Male Female Chart Mortality Improvement from 98 to 29 Then we will discuss mortality improvement trends of each gender Chart 2 Male s Mortality Comparison in Every Age Bracket From Chart 2, there is little change of male s mortality before age 54 in every year while it increases rapidly after age 54. From year 98 to year 29, the mortality apparently improved, and in the age bracket decreases even from 2 out of one thousand to out of one thousand which is almost a half. 4

6 Chart 3 Female s Mortality Comparison in Every Age Bracket From Chart 3, there is little change of male s mortality before age 6 in every year and they are all very low, however, it increases rapidly after age 6. From year 98 to year 29, female s mortality improvement changes a lot from 8 out of thousand to 8 out of one thousand. 2.2 Mortality Improvement in Every Age Bracket Let k= in Formula (), we can get mortality improvement between every year. According to the difference between mortality improvement of male s and female s, improvement in every age bracket can be classified into three groups: Group includes people in age 5-9, age -4 and age 5-9. Take mortality improvement in age 5-9 as an example. In this age range, mortality change of male s and female s are almost the same, which means not only the trends but also every year s change are similar. Mortality decreases with large fluctuation: for male, mortality decreased sharply between year 997 and 999, and fluctuated intensely between year 27 and 29; for female, mortality also decreased sharply between year 27 and 29, while increased largely between year 22 and 25. Except for year 22 to 24, male s mortality is higher than female s in all other year. Their mortality improvement always remains between -.6 and.4. 5

7 male femal e Chart 4 Mortality Improvement in Age 男 女 Chart 5 Mortality Change in Age 5-9 Group 2 includes people in age 2-24, age 25-29, age 3-34 and age Take mortality improvement in age 2-24 as an example. Female s mortality is lower than male s and decreases more significantly. As a result, the difference between male s mortality and female s became larger and larger from year 98 to 29. From Chart 6 and Chart 7, female s mortality increased significantly between year 997 and 998, while male s mortality improvement stays stable all the way. 6

8 male female Chart 6 Mortality Improvement in Age 男 女 Chart 7 Mortality Change in Age 2-24 Group 3 includes people in age 4-44 and all age brackets after it until age 94. Take mortality improvement in age 4-44 as an example. In this age range, mortality gap between genders is much larger than that of former age brackets, and mortality decreases smoothly. Male s mortality decreased by out of one thousand in 28 years while female s decreased by 2 out of thousand in the same years. Despite the descent of mortality, mortality of individuals in age 4-44 is out one thousand higher than that of individuals in age Similar as group 2, male s mortality improvement is more stable than male s. However, female s mortality improvement in this group is more stable than that in group 2. 7

9 .4.2 male female Chart 8 Mortality Improvement in Age 男 女 Chart 9 Mortality Change in Age The Influence on Life Insurance Reserving Theoretically, the choice of mortality table would affect the amount of insurer s reserves. However, in the practice it is difficult for us to judge which mortality table would enable a policy in a policy year to reserve more or less just by life table analysis, because mortality improvement not only affects death number in an age bracket but also survive number in the future as well as premium change. So I take two basic insurance products as examples and discuss what influence on reserving by comparing improved mortality with mortality on life table. 8

10 3. Ten-year premium ten-year policy term life insurance from 995 Assuming interest is 2.5% and policy amount is RMB. I only study people in age for the older have larger risk and they are always rejected. The calculation results of male s term life insurance reserves under life table s mortality and improved mortality are showed in Table. The difference of reserves between males in age and those in other age brackets is small, but improved mortality s reserves are still a little less than life table s mortality s. Reserves of improved mortality are a little more than those of life table s mortality regarding to males in age 28-3, and the difference between that of age 28 and 29 is larger than age 3. Reserves of improved mortality are less than those of life table s mortality regarding to males in age Considering mortality trends analysis in last part, which is that the change of mortality of males in age becomes smaller with years go by, so we could conclude that with policy age increases, difference between reserves of improved mortality and those of life table s mortality becomes larger and larger, and former is much smaller than later one. Table Comparison of Term Life Insurance Reserves (Male) (9 thousand RMB)

11 UM stands for Unimproved Mortality; IM stands for Improved Mortality. Data Source: Mortality Sample Data in Beijing Female s results are showed in Table 2. Except for female in age 29 and 3, the difference between other ages is small, and in most circumstances reserves of improved mortality are lower than those of life table s mortality. This indicates that reserves of improved mortality become smaller in female. Future payments become less for improved mortality is smaller than life table s mortality, and term premium becomes less as well. Table 2 Comparison of Term Life Insurance Reserves (Female) (9 thousand RMB)

12 UM stands for Unimproved Mortality; IM stands for Improved Mortality. Data Source: Mortality Sample Data in Beijing 3.2 Five-year premium five-year policy annuities from 995 Assume that payment is RMB every year, and interest rate is 2.5%. Males reserves of improved mortality and life table s mortality are showed in Table 3. It indicates that there is little difference between these two methods, and reserves calculated by improved mortality are always larger. This is because the decrease of mortality gives rise to the increase of survive probability and amount that annuity receivers becomes more given certain number of policy insured. Table 3 Comparison of Annuity Reserves (Male) (9 thousand RMB)

13 UM stands for Unimproved Mortality; IM stands for Improved Mortality. Data Source: Mortality Sample Data in Beijing Table 4 Comparison of Annuity Reserves (Female) (9 thousand RMB) UM stands for Unimproved Mortality; IM stands for Improved Mortality. Data Source: Mortality Sample Data in Beijing 3.3 Summary From the above two examples, as with term life insurance, the situation of both genders is similar in which reserves of improved mortality are less than reserves of life table s mortality, especially for people in high ages; as for annuities, amount of reserves based on these two methods is close, and this may indicate that mortality improvement has little influence on annuities or the age bracket and policy term I have chosen are responsible for the conclusion. 4 Conclusion and Discussion Based on mortality trends analysis and its impacts on reserving, we could arrive at these conclusions: As for mortality improvement, mortality in all age brackets decrease from year 2

14 98 to 29, and the fall became larger and larger. Male s mortality improvement is more stable and smooth than female s in that female s mortality improvement fluctuates twice or third times generally while male s fluctuation is little or even no. The difference between reserves based on two methods, the result is the same as we expected for ten-year premium ten-year policy which is reserves of improved mortality are less than those of life table s mortality, and the difference between them becomes larger as age grows. However, as with five-year premium five-year policy, what we expected is incorrect for the difference is small. What we must pay attention to is our conclusion is not limited to these two examples and the conclusion may be inaccurate for it only takes two general insurance products into account. For example, as for annuities, the small difference may because the premium year and policy year is both too short. Currently deferred insurance is the most common one, so we could continue to discuss this topic by studying deferred insurance and deferred annuities. Reference [] Towers Watson. Global Mortality Improvement Experience and Projection Techniques. Society of Actuaries. 2(6): 56~97 [2] Zhongquan Li. Life Table Revision s Influence on Life Insurers Based on Actuarial Analysis[J]. South Finance, 26(4): 2~4 [3] Yan Wang. Actuarial Science on Life Insurance[M], Second Edition. Renmin University of China Press, 28: 3~78 [4] Hongxiao Hu, Jia Xie, Bing Han. Methods to Deal with Missing Data. Market Modern Analysis, 27(5): 2~3 3