Understanding the Mathematics of Personal Finance

INTRODUCTION TO THE LIFE TABLES

Table 10.1 is the 2004 Life Table for all males in the United States. The first column to the left is the age of the man at his last birthday. In the rest of the table entry defi­nitions, this single number is just referred to by the letter x.

The second column is called q(x) . This notation is referred to as functional notation. It’s saying that the variable q “is a function of x.” Don’t worry about this. Column q is the probability of dying sometime between your xth and (x + 1)st birthday. For example, the probability of dying between your twentieth and twenty - first birthday, that is, when you’re 20 years old, is 0.001266. This definition, as stated, is correct—but should be elaborated on. This probability that you ’ tl die during your twentieth year is 0.001266 assumes that you made it to your twentieth birthday. It is not the same as the probability that, when you’re born, you’ll live to be 20 years old. I’ll get to that calculation in a few pages.

Note that the probability that a man will die at age 0 (before his first birthday) is higher than the probability of his dying at any other age up to age 54. Getting born and living your first year are relatively risky activities.

The bottom entry in the leftmost column is “100 or over.” In other words, all the probabilities of living to 100, 101, 102, and so on are lumped together into one catchall entry. This is because there simply aren’t enough people living past 100 in the United States today to make probability calculations meaningful. This is analo­gous to the 100-person life insurance group example above; relatively small changes in the number of people dying in a small group swing calculations so much that you can’t draw very reliable conclusions.

The value of q associated with age = 100 or more is 1.000. A probability of 1.00 is called the “certain event.” The table is reflecting the fact that eventually, everyone will die. If you make it to age 100, you’ll die during your 100th year, your 101st year, and so on, but it’s a certainty that you will die.

The entries in the third and fourth columns of the table are calculated in an interactive manner, so they must be described together. As I’ve discussed, to do meaningful probability calculations on a relatively unlikely event (an event that has a probability much less than 1.00), we need a large group of people. It is conven­tional to use a group of 100,000 as a “standard starting point.” I don’t know if this is the best size to start with, or even how to calculate if it’s the best size to start with, but it is the standard starting point, so that’s where this table starts the third column: for x = 0 (age = 0-1), l = 100,000.

If there are 100,000 men starting out and the probability of each of them dying during their first year of life is 0.007475, then during this year (0.007475) (100,000) = 747 men will die.[27]

If I start with 100,000 men and during the year 747 of them die, then going into the next year of life (x = 1), there are 100,000 - 747 = 99,253 men alive. This is the second entry in the l column.

Now the work gets repetitive: Looking at the x = 1 line in the table, there are

99.253 men starting out the year, and there’s a probability of 0.000508 that each of them will die. The number of men most likely to die during this year is therefore (99,253)(0.000508) = 50. This is the second entry in the d column.

Going to the third entry of column 1, 99,253 men started the year and 50 of them died during the year. The column l entry for the x = 2 line is therefore

99.253 - 50 = 99,202 (again, you can’t see the rounding that actually occurred). Following this pattern all the way down to the x = 100 or more line, the column

l entry is 1,261 men reaching their one hundredth birthday. Since the probability of each of them dying during their one hundredth year or any year thereafter is 1.000, the column d entry is (1,261)(1.00) = 1,261. At this point, there’s nobody left of the original 100,000 men and we stop.

Figure 10.3 is a graph of column d, This graph peaks at about age 83. This means that the most likely age for a man to die is 83 years old. It does not mean
that the average life expectancy of all men, looking ahead from when they’re born, is 83 years. To find this average life expectancy, you first need to understand how to calculate expected, or average, values from a graph.