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Life table 2003 US mortality table, Table 1, Page 1 In actuarial science and demography,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 ("probability of death"). There are two types of life tables used in actuarial science. The period life table represents mortality rates during a specific time period of a certain population. A cohort life table, often referred to as a generation life table, is used to represent the overall mortality rates of a certain population’s entire lifetime. They must have had to been born during the same specific time interval. Both of these types of life tables are created based on an actual popu- lation from the present, as well as an educated prediction of the experience of a population in the near future. [1] Other life tables in historical demography may be based on historical records, although these often undercount infants and understate infant mortality, on comparison with other regions with better records, and on mathemat- ical adjustments for varying mortality levels and life ex- pectencies at birth. [2] From this starting point, a number of inferences can be derived. the probability of surviving any particular year of age remaining life expectancy for people at different ages Life tables are also used extensively in biology and epidemiology. An area that uses this tool is Social Secu- rity. It examines the mortality rates of all the people who have Social Security to decide which actions to take. [1] The concept is also of importance in product life cycle management. 1 Background There are two types of life tables: Period or static life tables show the current prob- ability of death (for people of different ages, in the current year) Cohort life tables show the probability of death of people from a given cohort (especially birth year) over the course of their lifetime. Static life tables sample individuals assuming a station- ary population with overlapping generations. “Static Life tables” and “cohort life tables” will be identical if popula- tion is in equilibrium and environment does not change. If a population were to have a constant number of peo- ple each year it would mean that the probabilities of death from the life table were completely accurate. Also, an ex- act number of 100,000 people were born each year with no immigration or emigration involved. [1] “Life table” primarily refers to period life tables, as cohort life tables can only be constructed using data up to the current point, and distant projections for future mortality. Life tables can be constructed using projections of future mortality rates, but more often they are a snapshot of age- specific mortality rates in the recent past, and do not nec- essarily purport to be projections. For these reasons, the older ages represented in a life table may have a greater chance of not being representative of what lives at these ages may experience in future, as it is predicated on cur- rent advances in medicine, public health, and safety stan- dards that did not exist in the early years of this cohort. A life table is created by mortality rates and census figures 1

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In actuarial science and demography, a life table (alsocalled a mortality table or actuarial table) is a tablewhich shows, for each age, what the probability is that aperson of that age will die before his or her next birthday("probability of death").There are two types of life tables used in actuarial science.The period life table represents mortality rates during aspecific time period of a certain population. A cohortlife table, often referred to as a generation life table, isused to represent the overall mortality rates of a certainpopulation’s entire lifetime. They must have had to beenborn during the same specific time interval. Both of thesetypes of life tables are created based on an actual popu-lation from the present, as well as an educated predictionof the experience of a population in the near future.[1]

Other life tables in historical demography may be basedon historical records, although these often undercountinfants and understate infant mortality, on comparisonwith other regions with better records, and on mathemat-ical adjustments for varying mortality levels and life ex-pectencies at birth.[2]

From this starting point, a number of inferences can bederived.

• the probability of surviving any particular year ofage

• remaining life expectancy for people at differentages

Life tables are also used extensively in biology andepidemiology. An area that uses this tool is Social Secu-rity. It examines the mortality rates of all the people whohave Social Security to decide which actions to take.[1]

The concept is also of importance in product life cyclemanagement.

1 Background

There are two types of life tables:

• Period or static life tables show the current prob-ability of death (for people of different ages, in thecurrent year)

• Cohort life tables show the probability of death ofpeople from a given cohort (especially birth year)over the course of their lifetime.

Static life tables sample individuals assuming a station-ary population with overlapping generations. “Static Lifetables” and “cohort life tables” will be identical if popula-tion is in equilibrium and environment does not change.If a population were to have a constant number of peo-ple each year it would mean that the probabilities of deathfrom the life table were completely accurate. Also, an ex-act number of 100,000 people were born each year withno immigration or emigration involved. [1] “Life table”primarily refers to period life tables, as cohort life tablescan only be constructed using data up to the current point,and distant projections for future mortality.Life tables can be constructed using projections of futuremortality rates, but more often they are a snapshot of age-specific mortality rates in the recent past, and do not nec-essarily purport to be projections. For these reasons, theolder ages represented in a life table may have a greaterchance of not being representative of what lives at theseages may experience in future, as it is predicated on cur-rent advances in medicine, public health, and safety stan-dards that did not exist in the early years of this cohort. Alife table is created by mortality rates and census figures

1

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2 3 THE MATHEMATICS

from a certain population, ideally under a closed demo-graphic system. This means that immigration and emi-gration do not exist when analyzing a cohort. A closed de-mographic system assumes that migration flows are ran-dom and not significant, and that immigrants from otherpopulations have the same risk of death as an individualfrom the new population. Another benefit from mortal-ity tables is that they can be used to make predictions ondemographics or different populations.[3]

The life table observes the mortality experience of a sin-gle generation, consisting of 100,000 births, at every agenumber they can live through.[1]

Life tables are usually constructed separately for menand for women because of their substantially differentmortality rates. Other characteristics can also be used todistinguish different risks, such as smoking status, occu-pation, and socioeconomic class.Life tables can be extended to include other informationin addition to mortality, for instance health informationto calculate health expectancy. Health expectancies suchas disability-adjusted life year and Healthy Life Years arethe remaining number of years a person can expect to livein a specific health state, such as free of disability. Twotypes of life tables are used to divide the life expectancyinto life spent in various states:

• Multi-state life tables (also known as increment-decrements life tables) are based on transition ratesin and out of the different states and to death

• Prevalence-based life tables (also known as theSullivan method) are based on external informationon the proportion in each state. Life tables can alsobe extended to show life expectancies in differentlabor force states or marital status states.

2 Insurance applications

In order to price insurance products, and ensure thesolvency of insurance companies through adequate re-serves, actuaries must develop projections of future in-sured events (such as death, sickness, and disability). Todo this, actuaries develop mathematical models of therates and timing of the events. They do this by studyingthe incidence of these events in the recent past, and some-times developing expectations of how these past eventswill change over time (for example, whether the progres-sive reductions in mortality rates in the past will continue)and deriving expected rates of such events in the future,usually based on the age or other relevant characteristicsof the population. When analyzing a population, one ofthe main sources used to gather the required informationis insurance by obtaining individual records that belongto a specific population.[3] These are called mortality ta-bles if they show death rates, and morbidity tables if theyshow various types of sickness or disability rates.

The availability of computers and the proliferation ofdata gathering about individuals has made possible calcu-lations that are more voluminous and intensive than thoseused in the past (i.e. they crunch more numbers) and it ismore common to attempt to provide different tables fordifferent uses, and to factor in a range of non-traditionalbehaviors (e.g. gambling, debt load) into specialized cal-culations utilized by some institutions for evaluating risk.This is particularly the case in non-life insurance (e.g. thepricing of motor insurance can allow for a large numberof risk factors, which requires a correspondingly complextable of expected claim rates). However the expression“life table” normally refers to human survival rates and isnot relevant to non-life insurance.

3 The mathematics

pₓ chart from Table 1. Life table for the total population: UnitedStates, 2003, Page 8

The basic algebra used in life tables is as follows.

• qx : the probability that someone aged exactly xwill die before reaching age (x+ 1) .

• px : the probability that someone aged exactly xwill survive to age (x+ 1) .

px = 1− qx

• lx : the number of people who survive to age x

note that this is based on a radix.,[4]or starting point, of l0 lives, typi-cally taken as 100,000lx+1 = lx · (1− qx) = lx · pxlx+1

lx= px

• dx : the number of people who die aged x lastbirthday

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3

dx = lx − lx+1 = lx · (1− px) = lx · qx

• tpx : the probability that someone aged exactly xwill survive for t more years, i.e. live up to at leastage x+ t years

tpx =lx+t

lx

• t|kqx : the probability that someone aged exactlyx will survive for t more years, then die within thefollowing k years

t|kqx = tpx · kqx+t =lx+t − lx+t+k

lx

• μx : the force of mortality, i.e. the instantaneousmortality rate at age x, i.e. the number of peopledying in a short interval starting at age x, divided bylₓ and also divided by the length of the interval.

Another common variable is

• mx

This symbol refers to Central rate of mortality. It is ap-proximately equal to the average force of mortality, aver-aged over the year of age.

4 Ending a Mortality Table

In practice, it is useful to have an ultimate age associatedwith a mortality table. Once the ultimate age is reached,the mortality rate is assumed to be 1.000. This age maybe the point at which life insurance benefits are paid to asurvivor or annuity payments cease.Four methods can be used to end mortality tables:[5]

• The Forced Method: Select an ultimate age and setthe mortality rate at that age equal to 1.000 withoutany changes to other mortality rates. This createsa discontinuity at the ultimate age compared to thepenultimate and prior ages.

• The Blended Method: Select an ultimate age andblend the rates from some earlier age to dovetailsmoothly into 1.000 at the ultimate age.

• The Pattern Method: Let the pattern of mortalitycontinue until the rate approaches or hits 1.000 andset that as the ultimate age.

• The Less-Than-One Method: This is a variation onthe Forced Method. The ultimate mortality rate isset equal to the expected mortality at a selected ul-timate age, rather 1.000 as in the Forced Method.This rate will be less than 1.000.

5 Epidemiology

In epidemiology and public health, both standard life ta-bles to calculate life expectancy and Sullivan and multi-state life tables to calculate health expectancy are com-monly used. The latter include information on health inaddition to mortality.

6 See also• Age-adjusted life expectancy• Actuarial present value• Decrement table• Gompertz-Makeham law of mortality• Strategies for Engineered Negligible Senescence• Survival analysis• Service life

7 Notes[1] Bell, Felicitie. “LIFE TABLES FOR THE UNITED

STATES SOCIAL SECURITY AREA 1900-2100”. So-cial Security. Retrieved 9 February 2015.

[2] Saskia Hin, The Demography of Roman Italy, CambridgeUniversity Press, 2013, pp. 104-118.

[3] Pavía, Jose. “Introducing Migratory Flows in Life Ta-ble Construction”. Upcommons. Retrieved 10 February2015.

[4] “Period Abridged Life Tables”. avon.nhs.uk.

[5] “Ending the Mortality Table”. soa.org.

8 References• Shepard, Jon; Robert W. Greene (2003). Sociologyand You. Ohio: Glencoe McGraw-Hill. pp. A–22.ISBN 0-07-828576-3.

• “Life Expectancies”. Office of the State Actuary.2008-09-22. Retrieved 2008-01-16.

• Preston, Samuel H.; Patrick Heuveline; MichelGuillot (2001). Demography: measuring and mod-eling population processes. Blackwell Publishers.ISBN 1-55786-214-1.

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4 9 EXTERNAL LINKS

9 External links• Canadian Human Mortality Database

• UKGovernment Actuary Department’s Interim LifeTables

• Actuarial Life Table from the U.S. Social Securitydepartment

• US CDC Vital Statistics Reports

• Ehemu Database

• Human Mortality Database

• World Health Organisation Life Tables

• Rates in the Cloud Life Tables Generator

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5

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