Understanding the Mathematics of Personal Finance
LISTS AND SUBSCRIPTED VARIABLES
Throughout this book, I make frequent use of tables. Tables are lists of numbers that relate variables in different situations. This isn’t as bad as it first sounds. I’m sure you’ve all seen this many times—everything from income tax tables that the Internal Revenue Service provides to automobile value depreciation tables.
Table 1.1 is a hypothetical automobile value depreciation table. Don’t worry about what kind of car it is—I just made up the numbers for the sake of this example.
Looking from left to right, you see two columns: the age of the car and the car’s wholesale price. Looking from top to bottom you see six rows. The top row contains the headings, or descriptions, of what the numbers beneath mean. Then there are
|
Table 1.1 Hypothetical Automobile Value Depreciation Table
|
|
Table 1.2 Hypothetical Automobile Depreciation Table with Air-Conditioning Option
|
five rows of numbers. The numbers on each row “belong together.” For example, when the car is 2 years old, the wholesale price is $21,300.
An important point about the headings is that whenever appropriate, the units should be listed. In Table 1.1, the age of the car is expressed in years. If I didn’t say so, how would you know I didn’t mean months, or decades? The value of the car is expressed in dollars. To be very precise, maybe I should have said U. S. dollars (if that’s what I meant). Someone in Great Britain could easily assume that the prices are in pounds if I didn’t clearly state otherwise.
Very often a table will have many columns. Table 1.2 is a repeat of Table 1.1, but with a third column added: How much more the car is worth if it has airconditioning. Notice that I was a little sloppy here. I didn’t say that the extra amount was in dollars. In this case, however, a little sloppiness is harmless. Once you know that we’re dealing in dollars, you can be pretty sure that things will be consistent.
Again, the numbers in a given row belong together: A 3-year-old car is worth $18,000, and it is worth $650 more if it has air-conditioning.
Tables 1.1 and 1.2 tell you some dollar amounts based on the age of the car. It’s therefore typical for the age of the car to appear in the leftmost column. I could have put the car’s age in the middle column (of Table 1.2) or in the right column. Even though doing this wouldn’t introduce any real errors, it makes things harder to read.
Whenever convenient, columns are organized from left to right in order of decreasing importance. That is, I could have made the air-conditioning increment the second column and the car value the third column (always count columns from the left), but again it’s clearer if I put the more important number to the left of the less important number.
Some tables have many, many rows. The Life Tables presented in Chapter 10, the chapter about life insurance, have 102 rows—representing ages from 0 to 100, plus the heading row. The second column in the Life Tables is a number represented by the variable q, the third by the variable -, and so on. Don’- worry about what these letters mean now; this is a topic in Chapter 10.
In Table 1.3, I’ve extracted a piece of the Life Table shown in Table 10.1, As you can see, for every age there are six associated pieces of information. Suppose I wanted to compare the values of q for two different ages, or to make some
|
Table 1.3 Part of the 2004 U. S. Life Table for All Men
|
generalizations of some sort. As I go through my discussion, I find that it’s very cumbersome repeating terms like “the value of q for age 10” over and over again.
I can develop a much more concise, easy to read, notation by taking advantage of the fact that the left-hand column is a list of nonrepeating numbers that increase monotonically. By this I mean that 1 is below 0, 2 is below 1, 3 is below 2, and so on, so that it’s easy to understand what row I’m looking at just by referring to the age (the left-hand column). Then I use a subscript (a little number placed low down on the right) tied to any variable that I want to discuss to tell you what I’m looking at. This is hard to describe but easy to show with examples:
q3 refers to the value of q for age 3: q3 = 0.000250.
q12 refers to the value of q for age 12: q12 = 0.000162.
d15 refers to the value of d for age 15: d15 = 60.
Now I can easily discuss the table using this subscript notation. In Table 1.3, qio is the smallest of all the values of q, l22 is about 2% smaller than l0, and so on. Asking why I’d want to be saying these things depends on the topic and the table
under discussion. It’s like asking why I’d ever want to multiply two numbers together.
