1 Mortality Compression and Longevity Risk Jack C. Yue National Chengchi Univ. Sept. 26, 2009

2 Summary Motivation What is Mortality Compression Measuring the Mortality Compression New Measurements and Results Discussions

3 Prolonging Life Expectancy The average life-span has being experienced a significant increase since turning to the 20th century.  For example, the life expectancies of U.S. male and female were at the upper 40’s in 1900’s and reached upper 70’s in 2000’s.  The life expectancies of Taiwan male and female have similar increasing trend.

4 The Life Expectancy of U.S. and Taiwan

5 Incre- ment Male Female Increments of Life Expectancy in Taiwan (Complete Life) Complete Life Table National Health Insurance

6 Increments in Life Expectancy The life expectancy in U.S. has an increment of 0.3 year annually during the 20th century. The trend in Taiwan is similar but it seems that the slope is steeper.  According to U.N., the world has an annual increment of 0.25 year, during the second half of 20th century. The trend is likely to continue, at least for a while.

7 Impacts of Prolonging Life We are experiencing the longest life ever in the history, and this has changed our lives.  In Taiwan, the life expectancies at age 65 were and years for the male and female, increased to and years, respectively. It increases about 50% more financial burden for retirement preparation.  Taiwan started national pension in 2008, in addition to other social insurance programs.

8 The Potential Problem It is not appropriate to use the current information (e.g., period life table) to plan the future.  Stochastic mortality models are one of the popular choices to deal with the problem. Still, there is no guarantee if the future mortality will follow the historical trend. We will use the idea of mortality compression to explore the life expectancy.

9 Mortality Compression According to Fries (1980), Mortality Compression is  Rectangularization of the survival curve  A state in which mortality from exogenous causes (e.g., infectious diseases) is eliminated and the remaining variability in the age at death is caused by genetic factors. Mortality compression is linked with morbidity compression.

10 Survival Curves of Taiwan Female

11 Mortality Compression (Wilmoth and Horiuchi, 1999)

12 Rectangularization and Lifespan Regarding the theory of lifespan, there are two opinions: life with or without a limit. In either case, the rectangularization seems to be a consensus.  Premature deaths (including infants) will gradually decrease and some postulates that the distribution of death number will behave like a normal curve (at least for the part with age higher than the mode).

13 There is a Limit vs. No Limits

14 Mortality Compression (Cheung et al., 2005 )

15 Measuring Compression Wilmoth and Horiuchi (1999) proposed 10 measurements and they recommended the Interquartile (IQR). Kannisto (2000, 2001) calculated IQR and percentiles on numbers of deaths and other life table values, and based on data from 22 countries, he found signs of mortality compression (e.g., C50, the shortest age interval covering 50% of deaths).

16 Mode and Standard Deviations (Kannisto, 2000 )

17 Shortest Age Intervals (Kannisto, 2000 )

18 3D Measures (Hong Kong data) Cheung et al.(2005) proposed 3-D measures, Horizontalization, Verticalization, and Longevity Extension.  They applied the idea to complete life tables in Hong Kong ( ) and found “the increase in human longevity is meeting some resistance.” Note: The mortality rates of ages 85 to 120 were graduated using logistic curve.

19 3 Dimensions of Survival Curve Cheung et al. (2005) proposed 3-dimension measurements to describe the survival curve.  Horizontalization: measure the descending speed at the mode  Verticalization: measure the descending speed at the mode  Longevity Extension: measure the expansion of lifespan

20 3 Dimensions of Survival Curve (Cheung et al., 2005 )

21 Hong Kong Survival Curves (Cheung et al., 2005 )

22 3-D Measures in Hong Kong (Male)

23 Data and Results The past studies rely on values from the life tables, which are being graduated, and these values (numbers of deaths) can be changed. The graduation methods and the highest age assumption usually have the largest impacts.  The elderly mortality rates are influenced the most.  The life expectancies, especially for the elderly, also are affected.

24 Graduated Mortality Rates (Taiwan Male, )

Taiwan Abridged Life Table (Male) AgeDeath Pob.# Survivors# DeathsStationary Populations Life Expectancy X ~ (X+n)qxlxdxLxTxex － － － － － － － － － － － － － － － － －

26 Possible Assumption of Highest Attained Age

27 MaleFemale WhittakerGompertzWhittakerGompertz M  Life Expectancy Two Graduation Methods in Taiwan

28 Modified the Measurements We suggest use the raw data to verify the mortality compression, but use the mortality rates instead (not the number of deaths).  It is equivalent to using the mortality rates to explore the mortality compression.

29 Measuring with Mortality Rates Based on the mortality rates, we suggest 5 measurements for the mortality compression.  Mode age (M) with the maximum # of deaths  The probability of pre-mature deaths  Shortest age interval covering certain prob.  Standard deviation  2 of normal distribution  Surviving beyond a high age, P(X > M + k  ) Note: The results would be highly discrete, since the raw data are used.

30 Survival Curves of Sweden Male (Kannisto, 2000 )

31 Normal Curve and Death Numbers Some believes the death numbers beyond mode M behave like a normal curve.  We only have the values at integer and can use the  2 -test (less powerful).  We can also apply the ratio and this can be used to verify the normality.

32 Empirical Studies We use the data from Human Mortality Database (HMD): Japan ( ), Sweden ( ), and U.S. ( ).  Male and Female  Raw Data  Single Age  Results computed up to age 100

33 The age with the max. deaths (Female Mode)

34 The prob. of death before age 50 (Female)

35 Shortest age Interval for 25% 、 50% 、 75% (Female)

36 Standard Deviation of Number of Deaths （ Female ）

37 Standard Deviation (3-period Moving Average)

38 Female High age Survival Prob. P(X > M + k  ), k = 1 & 2

39 Findings The graduation methods have impacts on the measurements of mortality compression.  We found that the mode age (& the life expectancy) continues to increase. The prob. of premature death and the shortest age interval of covering certain death prob. decrease.  But the standard deviation of death number and the prob. of survival beyond a high age do not show obvious signs of decrease.

40 Possible Implications It seems the life expectancy will continue to extend in the future.  The future life expectancy of most stochastic models will also increase. But there are not enough data for the old-aged group (e.g., oldest-old, 85+) and their mortality rates rely heavily on model assumption (such as extrapolation). Question: Is Gompertz-type assumption feasible?

41 Possible Implications (conti.) The values of st.d.  and P(X > M + k  ) suggest that we should pay more attention to the higher age groups (such as oldest-old).  Mortality compression is not fully confirmed and the increase in human longevity is not meeting apparent resistance.  We are still not sure about the survival probability beyond very high ages.

42 Possible Implications (conti.) Maybe we can use the result P(X > M + k  ) is close to a constant.  For example, set the highest attained age to M + 2  (it is close to 100 years old currently). This would be plenty for most people and reduce the mortality risk for the insurance company.  Still, the need of people who outlive the age M + 2  is not satisfied.

43 Conclusion There are signs of mortality compression and the possibility of premature death continue to decrease. The life expectancy is likely to increase and does not show signs of slowing down.  The probability of survival beyond very high age (increasing as well) looks like a constant.

44 Suggestions There are not enough data for the elderly, in order to obtain reliable mortality models.  The highest attained age? Countries and areas with small population  How do we obtain reliable estimates? Mortality compression and the mortality models  The mortality ratios? (Discount Sequence)

45 Thank you for your attention!