Life Tables STA 220. Life Tables A is a table which indicates the probability of someone (or...

Life Tables

STA 220

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.

2

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 3

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

4

Life Tables

5

Life Tables• Life tables can be used to predict the following

values:– What’s the probability of being alive at a certain age?– What’s the probability of not being alive at a certain

age?– What’s the probability of dying (also known as risk)

between two ages?– Given that a certain age has been reached, what’s the

probability of living to a specific older age?– What’s the life expectancy?– Given that a certain age has been reached, what’s the

life expectancy?6

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)=

7

P(alive at a certain age)





8

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)

=

9


• 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)=

10


• 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)=

11


• 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)=

12

Risk

• What’s the risk between this age and that age?

• What’s 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)

13

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)

=

14


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)

=

15


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)

=

16

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)

17age)younger at P(alive

age)older at P(alive


• What’s 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) = •

18

85)at P(alive

90)at P(alive

%9.46496.00.262

0.130 85)at alivegiven 90at P(alive


• Suppose your best friend’s aunt is 45. What’s the probability that she will be alive to see your best friend graduate from medical school in 10 years?

• P(alive at 55 given alive at 45)

• P(alive at 55)= , P(alive at 45)=

19

45)at P(alive

55)at P(alive

%5.95955.00.942



• What’s 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)=

20


• 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)=

21

45)at P(alive

60)at P(alive

834.00.905


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.

22

Calculation of Life Expectancy• To calculate a weighted average for the age at

death:

• Risk*Age + Risk*Age + Risk*Age+…+Risk*Age– Where risk for a 0 year old is 82/1,000 and risk for a 1

year old is 19/1,000, etc

23

96*000,1

2...2*

000,1

71*

000,1

190*

000,1

82

1,000

2(96)...7(2)19(1)82(0)average weighted

24

Age P(alive

) Risk Age1

Risk * Age1

0 1.00

> 0.1 x 5 = 0.5

10 0.90

> 0.1 x 20 = 2

30 0.80

> 0.2 x 40 = 8

50 0.60

> 0.4 x 60 = 24

70 0.20

> 0.2 x 80 = 16

90 0.00

SUM 50.5 years


• 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) =

25


• Age1 column– Average of the previous age and the current age–

– Average age between 0 and 10 = (0+10)/2 = 5 years

– Average age between 30 and 50= (30+50)/2 = 40 years

26

2

agecurrent age previousAge1


• 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

27


• Life expectancy– Add up the values in the last column– 0.5+2.0+8.0+24.0+16.0 = 50.5 years

28

Procedure for Computing Life Expectancy

• 5 steps to computing the life expectancy given a life table:

1. Create a table with column headings of Age, P(alive), Risk, Age1, and Risk*Age1

2. Compute risk for each consecutive pair of ages: P(alive at previous age) – P(alive at current age)

3. Compute the average age for each consecutive pair of ages: Age1=(Previous age+Current age)/2

4. Multiply each risk and Age1 together5. Add up all the values in the last column

29

Calculation of Conditional Life Expectancy

• We already found that if you’re already 20, then 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 with one additional step– Must compute all of the conditional probabilities

to use in the life table

30


• 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)

31

32

Age P(alive

) Risk Age1

Risk * Age1

0 1

> 0.1 x 5 = 0.5

10 0.9

> 0.1 x 20 = 2

30 0.8

> 0.2 x 40 = 8

50 0.6

> 0.4 x 60 = 24

70 0.2

> 0.2 x 80 = 16

90 0

SUM 50.5 years


• P(alive at 30 given alive at 30)= P(alive at 30)/P(alive at 30)=



• P(alive at 90 given alive at 30)= P(alive at 90)/P(alive at 30)= 33


• 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

34

35

Age P(alive

) Risk Age1

Risk * Age1

30 1.00

> 0.25 x 40 = 10

50 0.75

> 0.5 x 60 = 30

70 0.25

> 0.25 x 80 = 20

90 0.00

SUM 60 years

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

36

Life Tables STA 220. Life Tables A is a table which indicates the probability of someone (or...

Documents

Transcript of Life Tables STA 220. Life Tables A is a table which indicates the probability of someone (or...