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

 Age of car (years) Wholesale price (\$) 0 32,000 1 26,500 2 21,300 3 18,000 4 15,500 5 13,250

Table 1.2 Hypothetical Automobile Depreciation Table with Air-Conditioning Option

 Age of car (years) Wholesale price (\$) Extra for air- conditioning 0 32,000 1,200 1 26,500 1,050 2 21,300 850 3 18,000 650 4 15,500 550 5 13,250 450

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

 Age q l d L T e 0 0.007475 100,000 747 99,344 7,517,501 75.2 1 0.000508 99,253 50 99,227 7,418,157 74.7 2 0.000326 99,202 32 99,186 7,318,929 73.8 3 0.000250 99,170 25 99,157 7,219,744 72.8 4 0.000208 99,145 21 99,135 7,120,586 71.8 5 0.000191 99,124 19 99,115 7,021,451 70.8 6 0.000182 99,105 18 99,096 6,922,336 69.8 7 0.000171 99,087 17 99,079 6,823,240 68.9 8 0.000152 99,070 15 99,063 6,724,161 67.9 9 0.000125 99,055 12 99,049 6,625,098 66.9 10 0.000105 99,043 10 99,038 6,526,049 65.9 11 0.000111 99,033 и 99,027 6,427,011 64.9 12 0.000162 99,022 16 99,014 6,327,984 63.9 13 0.000274 99,006 27 98,992 6,228,970 62.9 14 0.000431 98,978 43 98,957 6,129,978 61.9 15 0.000608 98,936 60 98,906 6,031,021 61.0 16 0.000777 98,876 77 98,837 5,932,116 60.0 17 0.000935 98,799 92 98,753 5,833,278 59.0 18 0.001064 98,706 105 98,654 5,734,526 58.1 19 0.001166 98,601 115 98,544 5,635,872 57.2 20 0.001266 98,486 125 98,424 5,537,328 56.2 21 0.001360 98,362 134 98,295 5,438,904 55.3 22 0.001419 98,228 139 98,158 5,340,609 54.4

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.

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