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## Life Tables

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### Life Tables STA 220 Calculation of Conditional Life Expectancy These values can be substituted into the table for P(alive) Need to recalculate column using these ... – PowerPoint PPT presentation

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Title: Life Tables

1
Life Tables
• STA 220

2
Life Tables
• A is a table which indicates the probability
of someone (or something) being alive at a
certain age
• Used extensively in the insurance business
• Life Insurance
• Based on gender, smoking status, current age, etc.

3
Life Tables
Age Male Female Age Male Female
0 1.000 1.000 55 0.824 0.900
1 0.974 0.980 60 0.755 0.863
5 0.970 0.977 65 0.658 0.807
10 0.967 0.975 70 0.538 0.725
15 0.965 0.974 75 0.402 0.607
20 0.959 0.971 80 0.261 0.448
25 0.951 0.968 85 0.131 0.262
30 0.944 0.965 90 0.075 0.130
35 0.936 0.960 95 0.035 0.089
40 0.924 0.953 100 0.020 0.060
45 0.905 0.942 105 0.010 0.040
50 0.874 0.925 Finally 0.000 0.000
4
Life Tables
• Refer to previous slide
• P(male alive at 60)
• P(Female alive at 60)
• Males are not expected to live as long as females

5
Life Tables
6
Life Tables
• Life tables can be used to predict the following
values
• Whats the probability of being alive at a
certain age?
• Whats the probability of not being alive at a
certain age?
• Whats the probability of dying (also known as
risk) between two ages?
• Given that a certain age has been reached, whats
the probability of living to a specific older
age?
• Whats the life expectancy?
• Given that a certain age has been reached, whats
the life expectancy?

7
P(alive at a certain age)
• To determine the probability of being alive at a
certain age, go to the life table and read off
the probability of being alive
• P(male alive at 25)
• P(female alive at 50)

8
P(alive at a certain age)
• P(male alive at 80)
• P(female alive at 95)
• P(male alive at 105)
• P(female alive at 106)

9
P(not alive at a certain age)
• People or things have 2 basic states alive (or
functioning) and not alive (or not functioning)
• Sum of these 2 states must equal 1
• P(alive at some age) P(not alive at some age)
1
• P(not alive at some age)

10
P(not alive at a certain age)
• Remember the life table contains the value for
P(alive at some age)
• Suppose we want to find the probability that a
male is not alive at age 20
• P(male alive at 20) 0.959
• P(male not alive at 20)
• 1 P(male alive at 20)

11
P(not alive at a certain age)
• Probability a female is not alive at 35
• P(female alive at 35)
• 0.960
• P(female not alive at 35)
• 1- P(female alive at 35)

12
P(not alive at a certain age)
• Probability a female is not alive at 65
• P(female alive at 65)
• 0.807
• P(female not alive at 65)
• 1- P(female alive at 65)

13
Risk
• Whats the risk between this age and that age?
• Whats the probability of dying between this age
and that age?
• P(dying between a younger age and an older age)
P(alive at a younger age) P(alive at older age)

14
Risk
• Suppose you want to find the probability of a
male dying between the ages of 20 and 30
• P(male dying between 20 and 30)
• P(alive at 20) P(alive at 30)
• P(male alive at 20)
• P(male alive at 30)
• P(male dying between 20 and 30)

15
Risk
• Suppose you want to find the probability of a
female dying between the ages of 20 and 40
• P(female dying between 20 and 40)
• P(alive at 20) P(alive at 40)
• P(female alive at 20)
• P(female alive at 40)
• P(female dying between 20 and 40)

16
Risk
• Suppose you want to find the probability of a
female dying between the ages of 85 and 90
• P(female dying between 85 and 90)
• P(alive at 85) P(alive at 90)
• P(female alive at 85)
• P(female alive at 90)
• P(female dying between 85 and 90)

17
P(alive at an older age given alive at a younger
age)
• Life tables reflect the values for a newborn
child or a brand new thing
• Once a person ages, their probability of reaching
an older age as the size of the population
• P(alive at an older age given alive at a younger
age)

18
P(alive at an older age given alive at a younger
age)
• Whats the probability that a female will be
alive at 90 given that she lived to 85?
• P(alive at 90 given alive 85)
• P(alive at 90)
• P(alive at 85)

19
P(alive at an older age given alive at a younger
age)
• Suppose your best friends aunt is 45. Whats
the probability that she will be alive to see
10 years?
• P(alive at 55 given alive at 45)
• P(alive at 55) , P(alive at 45)

20
P(alive at an older age given alive at a younger
age)
• Whats the probability that a male will not be
alive at 60 given that he is alive at 45?
• Recall P(not alive at some age)
• 1 P(alive at some age)
• So, P(not alive at older age given alive at
younger age)

21
P(alive at an older age given alive at a younger
age)
• P(alive at 60 given alive at 45)
• P(alive at 60) , P(alive at 45)
• P(not alive at 60 given alive at 45)
• 1- P(alive at 60 given alive at 45)

22
Calculation of Life Expectancy
• Suppose you wanted to determine the life
expectancy of New Englanders in 1900. To do
this, you went to various town halls and obtained
death records for 1,000 individuals.

Age at Death How Many
0 82
1 19
2 7

96 2
23
Calculation of Life Expectancy
• To calculate a weighted average for the age at
death
• RiskAge RiskAge RiskAgeRiskAge
• Where risk for a 0 year old is 82/1,000 and risk
for a 1 year old is 19/1,000, etc

24
Age P(alive)   Risk   Age1   Risk Age1
0 1.00
0 1.00 gt 0.1 x 5 0.5
10 0.90 gt 0.1 x 5 0.5
10 0.90 gt 0.1 x 20 2
30 0.80 gt 0.1 x 20 2
30 0.80 gt 0.2 x 40 8
50 0.60 gt 0.2 x 40 8
50 0.60 gt 0.4 x 60 24
70 0.20 gt 0.4 x 60 24
70 0.20 gt 0.2 x 80 16
90 0.00 gt 0.2 x 80 16
90 0.00
SUM 50.5 years
25
Calculation of Life Expectancy
• Risk column
• Subtract probability of being alive at the
current age from the previous age
• P(dying between a younger age and an older age)
P(alive at younger age)-P(alive at older age)
• Risk between 0 and 10
• P(dying between 0 and 10)
• Risk between 50 and 70
• P(dying between 50 and 70)

26
Calculation of Life Expectancy
• Age1 column
• Average of the previous age and the current age
• Average age between 0 and 10
• (010)/2 5 years
• Average age between 30 and 50
• (3050)/2 40 years

27
Calculation of Life Expectancy
• Risk Age1 column
• Product of the Risk and Age1
• 0.10 x 5 0.5 years
• 0.10 x 20 2.0 years
• 0.20 x 40 8.0 years
• 0.40 x 60 24.0 years
• 0.20 x 80 16.0 years

28
Calculation of Life Expectancy
• Life expectancy
• Add up the values in the last column
• 0.52.08.024.016.0 50.5 years

29
Procedure for Computing Life Expectancy
• 5 steps to computing the life expectancy given a
life table
• Create a table with column headings of Age,
P(alive), Risk, Age1, and RiskAge1
• Compute risk for each consecutive pair of ages
P(alive at previous age) P(alive at current
age)
• Compute the average age for each consecutive pair
of ages Age1(Previous ageCurrent age)/2
• Multiply each risk and Age1 together
• Add up all the values in the last column

30
Calculation of Conditional Life Expectancy
the probability of being alive at 30 is higher
than the value in the life table
• The same holds true for
• Procedure is same as computing life expectancy
• Must compute all of the conditional probabilities
to use in the life table

31
Calculation of Conditional Life Expectancy
• Suppose we had the same life tables as in the
previous example and we wanted to determine the
life expectancy for a person that was 30 years
old.
• Compute following conditional probabilities first
• P(alive at 30 given alive at 30)
• P(alive at 50 given alive at 30)
• P(alive at 70 given alive at 30)
• P(alive at 90 given alive at 30)

32
Age P(alive)   Risk   Age1   Risk Age1
0 1
0 1 gt 0.1 x 5 0.5
10 0.9 gt 0.1 x 5 0.5
10 0.9 gt 0.1 x 20 2
30 0.8 gt 0.1 x 20 2
30 0.8 gt 0.2 x 40 8
50 0.6 gt 0.2 x 40 8
50 0.6 gt 0.4 x 60 24
70 0.2 gt 0.4 x 60 24
70 0.2 gt 0.2 x 80 16
90 0 gt 0.2 x 80 16
90 0
SUM 50.5 years
33
Calculation of Conditional Life Expectancy
• P(alive at 30 given alive at 30)
• P(alive at 30)/P(alive at 30)
• P(alive at 50 given alive at 30)
• P(alive at 50)/P(alive at 30)
• P(alive at 70 given alive at 30)
• P(alive at 70)/P(alive at 30)
• P(alive at 90 given alive at 30)
• P(alive at 90)/P(alive at 30)

34
Calculation of Conditional Life Expectancy
• These values can be substituted into the table
for P(alive)
• Need to recalculate column using these
conditional probabilities
• The column stays the same because the ages have
not changed
• Last column multiplies the new column with the
column
• Finally, add up all the values in the last column
to arrive at the conditional life expectancy

35
Age P(alive)   Risk   Age1   Risk Age1
30 1.00
30 1.00 gt 0.25 x 40 10
50 0.75 gt 0.25 x 40 10
50 0.75 gt 0.5 x 60 30
70 0.25 gt 0.5 x 60 30
70 0.25 gt 0.25 x 80 20
90 0.00 gt 0.25 x 80 20
90 0.00
SUM 60 years
36
Life Expectancy
• Useful in determining payouts of Social Security
to retirees
• Should women get the same amount per month as
men?
• Average life expectancy can be used as a measure
of quality or reliability