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Mortality Compression and Longevity Risk

Jack C. YueNational Chengchi Univ.

Sept. 26, 2009

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Summary

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

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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.

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The Life Expectancy of U.S. and Taiwan

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Incre-ment Male

Female

Increments of Life Expectancy in Taiwan (Complete Life)

Complete Life Table

National Health Insurance

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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.

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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 10.62 and 13.25 years for the male and female, increased to 17.26 and 20.18 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.

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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.

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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.

10Survival Curves of Taiwan Female

11Mortality Compression (Wilmoth and Horiuchi, 1999)

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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).

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There is a Limit vs. No Limits

14Mortality Compression (Cheung et al., 2005 )

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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).

16Mode and Standard Deviations (Kannisto, 2000 )

17Shortest Age Intervals (Kannisto, 2000 )

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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 (1976-2001) 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.

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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

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

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

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3-D Measures in Hong Kong (Male)

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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.

24Graduated Mortality Rates (Taiwan Male, 1999-2001)

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2005 Taiwan Abridged Life Table (Male)

Age Death Pob. # Survivors # Deaths Stationary PopulationsLife

Expectancy

X ~ (X+n) qx lx dx Lx Tx ex

0 0.00562 100000 562 99539 7449253 74.49

1 － 4 0.00179 99438 178 397340 7349714 73.91

5 － 9 0.00091 99260 90 496041 6952374 70.04

10 － 14 0.00110 99170 109 495641 6456333 65.10

15 － 19 0.00341 99061 337 494536 5960692 60.17

20 － 24 0.00473 98723 467 492513 5466156 55.37

25 － 29 0.00694 98256 682 489666 4973643 50.62

30 － 34 0.00936 97574 914 485701 4483977 45.95

35 － 39 0.01336 96661 1291 480261 3998276 41.36

40 － 44 0.01907 95369 1819 472530 3518015 36.89

45 － 49 0.02598 93550 2430 461932 3045485 32.55

50 － 54 0.03479 91120 3170 448040 2583553 28.35

55 － 59 0.04993 87951 4391 429420 2135513 24.28

60 － 64 0.07461 83560 6235 402977 1706093 20.42

65 － 69 0.10857 77325 8395 366663 1303117 16.85

70 － 74 0.16262 68930 11209 317783 936454 13.59

75 － 79 0.23879 57720 13783 254988 618670 10.72

80 － 84 0.34130 43937 14996 182252 363683 8.28

85+ 1.00000 28942 28942 181431 181431 6.27

26Possible Assumption of Highest Attained Age

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Male Female

Whittaker Gompertz Whittaker Gompertz

M 83 83 85 86

6.77 6.68 6.35 5.82

Life Expectancy

73.64 73.52 79.44 79.32

Two Graduation Methods in Taiwan 1999-2001

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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.

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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.

30Survival Curves of Sweden Male (Kannisto, 2000 )

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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.

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Empirical Studies We use the data from Human Mortality

Database (HMD): Japan (1947-2006), Sweden (1901-2007), and U.S. (1933-2005).

Male and Female

Raw Data

Single Age

Results computed up to age 100

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The age with the max. deaths (Female Mode)

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The prob. of death before age 50 (Female)

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

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Standard Deviation of Number of Deaths （ Female ）

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Standard Deviation (3-period Moving Average)

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

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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.

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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?

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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.

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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.

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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.

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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)

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Thank you for your attention!