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.

Life Tables Age Male Female 1.000 55 0.824 0.900 1 0.974 0.980 60 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 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 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

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

Life Tables

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?

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)

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)

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) =

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) =

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) =

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) =

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)

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)

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)

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(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)

P(alive at an older age given alive at a younger age) 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) =

P(alive at an older age given alive at a younger age) 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)=

P(alive at an older age given alive at a younger age) 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) =

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) =

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 82 1 19 2 7 … 96

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

Age P(alive)   Risk Age1 Risk * Age1 1.00 > 0.1 x 5 = 0.5 10 0.90 20 2 30 0.80 0.2 40 8 50 0.60 0.4 60 24 70 0.20 80 16 90 0.00 SUM 50.5 years

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) =

Calculation of Life Expectancy 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

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

Calculation of Life Expectancy Add up the values in the last column 0.5+2.0+8.0+24.0+16.0 = 50.5 years

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 Risk*Age1 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 age+Current age)/2 Multiply each risk and Age1 together Add up all the values in the last column

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

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)

Age P(alive)   Risk Age1 Risk * Age1 1 > 0.1 x 5 = 0.5 10 0.9 20 2 30 0.8 0.2 40 8 50 0.6 0.4 60 24 70 80 16 90 SUM 50.5 years

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)

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

Age P(alive)   Risk Age1 Risk * Age1 30 1.00 > 0.25 x 40 = 10 50 0.75 0.5 60 70 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