1. Random future lifetime
In a life insurance product the payment of a benefit depends on whether the insured is alive or not. Such products require a model for human mortality. Since the future lifetime of an insured is unknown, you will use a random variable to express this person's future lifetime.
2. The random future lifetime
An actuary writes (x) in brackets for a life aged x, typically a policyholder who buys a policy at age x. The random variable Tx then expresses the remaining future lifetime of (x). It is a positive and continuous random variable. x plus Tx then gives you the random age at death of the insured. The goal of this chapter is to study the distribution of Tx, the future lifetime of the insured. This distribution captures the possible outcomes of Tx and also how likely these outcomes are.
3. The life table in R
From the Human Mortality Database you can download mortality data for a wide range of countries. You see a preloaded excerpt from the life table for Belgian males, as registered in the year 2013. The data are stored in life_table. In this video you'll focus on the variables age, qx and ex.
4. Mortality rates and survival probabilities
Now, what is qx? Actuaries use the notation qx for the one-year probability of dying of an individual who is x years old. Put otherwise: qx is the probability that your remaining lifetime is less than or equal to 1 year. qx is also called the mortality rate at age x.
Actuaries also use a short notation for survival probabilities, and write px for the one year survival probability. Thus, px equals 1 minus qx.
5. Mortality rates of Belgian sportsmen in R
Ready to meet some famous Belgian sportsmen? Eden Hazard is a Belgian footballer who plays for Chelsea and is currently 27 years old. What is his mortality rate? Using life_table you extract the qx that corresponds with age 27. The code shows you two ways to do this. You either explicitly use the vectors qx and age and look for the entry where age equals 27, or you realize that the vector qx runs over all ages, starting from 0. Hence, age 27 corresponds with entry 28 in this vector.
You can repeat this operation for Eddy Merckx, who is currently 72 years old. Do note that the mortality rate for Eddy is much higher than the rate for Eden!
6. Picturing Belgian mortality rates $q_x$ in R
You will now plot the evolution of qx as a function of age x. Age x is on the x-axis, and the log mortality rate is on the y-axis. The R code demonstrates how to display the label on the y-axis in a nice way.
7. Picturing Belgian mortality rates $q_x$ in R
Looking more carefully into this graph, you'll see three important features of the evolution of mortality rates over ages. The rates for infants (in blue) are locally high for newborns. Rates then decrease but show an upward hiccup (in red) around the age of 18 and then straighten again. This is the accident hump reflecting testosterone fueled males and their increased mortality rates. Rates then increase for higher ages (in green), reflecting the human aging process.
8. The life expectancy
Another interesting feature of the random variable Tx is its expected value. This is the life expectancy at age x, and it is stored in the variable ex in the life_table object. The R code tells you that Eden's life expectancy is 51-point-74 years, whereas Eddy Merckx is expected to live another 12-point-67 years.
9. Picturing the life expectancy in R
This plot shows how life expectancy decreases with age. Age x is on the x-axis and the corresponding life expectancy for an x-year-old is then on the y-axis.
10. Let's practice!
Now it's your turn to start digging into mortality data!