Understanding the Mathematics of Personal Finance
Life Insurance
10.1 WHAT IS AN INSURANCE POLICY?
An insurance policy is a contract, usually valid over a specified period of time or term, between you and an insurance company that basically reads the following:
1. You will give the insurance company some money. This money may be an upfront payment or it may be made up of periodic payments. The details will be spelled out in the insurance policy.
2. The insurance policy will describe some relatively unlikely occurrence during the term of the contract such as you breaking an elbow or having a fender bender or having your house robbed or dying and so on.
3. If the specified event occurs, the insurance company will pay you some money. The amount may be a specified fixed amount such as in a “$25,000 life insurance policy,” or it might be an amount determined by the circumstances, such as the medical costs for setting your elbow or the cost of getting a new left front fender for your car.
The event that triggers the insurance company to pay you is an event that could not have been predicted, except in statistical generalities. This means that understanding insurance requires a bit of the understanding of mathematical probabilities.
Another name for buying an insurance policy is “placing a bet.” Insurance companies don’t dwell on this aspect, but buying an auto collision policy is essentially you saying, “I’ll bet I’m going to have an auto accident next year,” and the insurance company replying, “We’ll take that bet. We don’l think you’re going to have an auto accident next year.” Since having an auto accident is a relatively unlikely event, there are odds involved in the bet. For example, you might pay $1,000 for a policy that will cover up to $25,000 in auto repairs. What differentiates this kind of bet from a gambling casino bet is that when you buy an insurance policy, you don’t want to win your bet. That is, you don’t want to have an auto accident or to die.
Understanding the Mathematics of Personal Finance: An Introduction to Financial Literacy, by
Lawrence N. Dworsky
Copyright © 2009 John Wiley & Sons, Inc.
An insurance policy that pays out if you die is called a life insurance policy. The amount you pay for it is called the premium.
Life insurance premiums are based on life insurance tables, which are issued annually by the U. S. federal government.[25] The basic idea is that nobody knows when they are going to die, but there’s a lot of reliable information about the average life span of a large group of people like them. There are tables for the entire population, for all men, for all women, for men by race, for women by race, and so on. Table 10.1 shows the 2004 Life Table for all men. There’s a lot of information in this table, much of it not useful to us right now. On the other hand, there’s a lot of information that is useful to us. Before going through the table and explaining what the entries tell us, it’ s necessary to talk briefly about two topics: probability and expected, or average, value.
The mathematical field of probability (and its closely related field of statistical inference, or more simply, statistics) is fascinating and subtle. Defining some of the basic terms often takes chapters in textbooks. For our purposes, I’ll offer some seat  ofthepants definitions that should get us through what we need without detailed study but would not compel a mathematician to send this book through the garbage disposal.
Some events that we see every day are determined by so many subtle causes, most of which are always changing, that the event itself seems somewhat random. When you shuffle a deck of cards for example, the motion of each card is determined by where it started in the deck, how clean the surfaces of the cards are, the humidity that day, how stiffly you are holding the deck, and so on. All of these factors, and a host more that I’ve overlooked, contribute in such a complicated manner to the motion and ultimate position of each card in the deck that even the most cynical gambling houses from Las Vegas to Monte Carlo consider the locations of the individual cards in a wellshuffled deck to be random. When you turn over the top card of a shuffled deck, you don’t know what to expect.
Of the 52 choices of cards you might turn over, there are 13 clubs, 13 spades, 13 hearts, and 13 diamonds (adding up to the 52 cards in a deck that has no jokers). If you repeat the basic experiment of shuffling the deck and turning over the top card many, many times, you would expect to see a diamond about one time out of four. That is, we say that the probability of turning over a diamond is equal to the number of opportunities of getting a diamond (13) divided by the total number of different cards that could appear (52). Doing the math, 13/52 = 1/4 = 0.25.
The definition above is a simple but perfectly usable definition of probability: For a random event with D possible outcomes, if we’re interested in N of them, then the probability of N occurring is N/D.
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 
10.2 Probability
The 2004 U. S. Life Table for All Men
q(x) 
l(x) 
d(x) 
L(x) 
T(x) 
0.007475 
100,000 
747 
99,344 
7,517,501 
0.000508 
99,253 
50 
99,227 
7,418,157 
0.000326 
99,202 
32 
99,186 
7,318,929 
0.000250 
99,170 
25 
99,157 
7,219,744 
0.000208 
99,145 
21 
99,135 
7,120,586 
0.000191 
99,124 
19 
99,115 
7,021,451 
0.000182 
99,105 
18 
99,096 
6,922,336 
0.000171 
99,087 
17 
99,079 
6,823,240 
0.000152 
99,070 
15 
99,063 
6,724,161 
0.000125 
99,055 
12 
99,049 
6,625,098 
0.000105 
99,043 
10 
99,038 
6,526,049 
0.000111 
99,033 
11 
99,027 
6,427,011 
0.000162 
99,022 
16 
99,014 
6,327,984 
0.000274 
99,006 
27 
98,992 
6,228,970 
0.000431 
98,978 
43 
98,957 
6,129,978 
0.000608 
98,936 
60 
98,906 
6,031,021 
0.000777 
98,876 
77 
98,837 
5,932,116 
0.000935 
98,799 
92 
98,753 
5,833,278 
0.001064 
98,706 
105 
98,654 
5,734,526 
0.001166 
98,601 
115 
98,544 
5,635,872 
0.001266 
98,486 
125 
98,424 
5,537,328 
0.001360 
98,362 
134 
98,295 
5,438,904 
0.001419 
98,228 
139 
98,158 
5,340,609 
0.001435 
98,089 
141 
98,018 
5,242,451 
0.001419 
97,948 
139 
97,878 
5,144,433 
0.001390 
97,809 
136 
97,741 
5,046,554 
0.001365 
97,673 
133 
97,606 
4,948,813 
0.001344 
97,540 
131 
97,474 
4,851,207 
0.001336 
97,408 
130 
97,343 
4,753,733 
0.001341 
97,278 
130 
97,213 
4,656,390 
0.001352 
97,148 
131 
97,082 
4,559,177 
0.001371 
97,017 
133 
96,950 
4,462,094 
0.001408 
96,884 
136 
96,815 
4,365,144 
0.001469 
96,747 
142 
96,676 
4,268,329 
0.001553 
96,605 
150 
96,530 
4,171,653 
0.001653 
96,455 
159 
96,375 
4,075,123 
0.001770 
96,296 
170 
96,210 
3,978,747 
0.001911 
96,125 
184 
96,033 
3,882,537 
0.002075 
95,942 
199 
95,842 
3,786,504 
0.002254 
95,742 
216 
95,635 
3,690,662 
0.002438 
95,527 
233 
95,410 
3,595,027 
0.002632 
95,294 
251 
95,168 
3,499,617 
Table 10.1 Continued

Table 10.1 Continued

If we flip a coin, there are two possible outcomes: heads or tails. The probability of heads is therefore 1/2 = 0.5, as is the probability of tails. The total of all the probabilities (a long way of saying that one of the possible choices must occur) is always 1.0.
Extending this a bit, the probability of getting 5 heads when I flip 10 coins (or equivalently, flip one coin 10 times) is 5/10 = 0.5. If you try this, however, don’t be surprised if you don’t get exactly 5 heads out of 10 flips. What happens in practice is that the larger the number of times you try, the more likely it is that the percentage of error (from exactly 50% heads) gets smaller.
Let’s take this basic idea and see how this works with a hypothetical life insurance company. Suppose that you and your friends get together and decide to all put some money into a bank for life insurance for all of you. I’m setting the situation up this way so that I can avoid issues of company profits, operating expenses, and so on. This is an idealized group—there are no profits, no taxes, no cost of doing business, and so on.
Imagine that the life insurance tables tell you (and I’ll show you soon how they tell you this) that the probability of each of the people in your group dying within the next year is 0.02 (= 1/50). This means that if there are 100 of you, it’s most probable that two of you will die during the next year. If you each put $1,000 into the bank account, that’s a total of $100,000. If two of you die, there’s funding to give the families of these two people $50,000 each. Note that in this ideal situation, there is no money left in the bank account at the end of the year and you have to start over again next year. In the jargon of life insurance, you’ve each paid a $1,000 premium for a 1year, $50,000 term insurance policy.
This scenario works fine unless more than two people die during the year. If, say, four people die, then there’s only enough money to pay $25,000 to each of the decedents’ families. This is an example of where betting on the probabilities can get you into trouble. When you have a low probability of something happening and a small enough group for this thing to happen to, the actual number of occurrences of your event of interest can wander all over the place.
We don ’ t want to resolve this dilemma by somehow making it more likely for people to die during the next year. Instead, we avoid 100 people insurance “groups” and try to deal with very large numbers of people. For example, if we had 100,000 people, each contributing $1,000, we would collect $10,000,000 dollars into our insurance pool. Using the same probability of dying, 2% of 100,000 is 2,000 and we’d still be able to give $50,000 to the family of each person who died.
With our small group, if we wanted to collect enough money to pay for more people than the ideal probability tells us we’l l have to pay for, we have to collect another $500, or a total of $1,500 from each person. With our large group, we’d only have to collect another $2 from each person to pay for each additional person. We could cover a whole bunch of extra people.
What if we still haven’t put aside enough money for the actual number of people who die? Insurance companies handle this problem by noting that averages tend to be averages. That is, if one of the policy groups (e. g., 50yearold men) requires more funding than was predicted for this year, there will be another group (e. g., 60yearold women) that will require less funding than was predicted this year. When all else fails, insurance companies can crossinsure each other.
Calculating how much extra money should be available “just in case” involves discussions of confidence intervals that are, in a sense, probabilities of probabilities. Such discussion is beyond the scope of this book. It isn’t really that hard to understand; it’s just a lengthy discussion that heads off in a direction that’s not central to this book. Figures 10.1 and 10.2 show the results of one of these calculations for the examples presented above.
In Figure 10.1, the horizontal axis is the number of people who will die during the year in our 100person group, and the vertical axis is the relative probability of this number of people dying.[26] As can be seen, the maximum probability is for two people to die. Interestingly, it is almost as likely (i. e., as probable) for only one person do die. Of more importance, however, is that it is about onethird as likely for four people to die as it is for two people to die and about onetenth as likely for five people to die. In other words, it’s not unlikely that twice or even more than twice the number of people will die than our basic funding model provides for.
Now look at Figure 10.2. In this case, the horizontal axis is the number of people who will die during the year in our 100,000person group. The most likely case is for 2,000 people to die (2% of 100,000). It’s almost as likely that 2,001 or 2,002 or 1,999 or 1,998 people will die as it is that exactly 2,000 people to die. This is another
way of saying that there’s “nothing special” about exactly 2,000 people dying. On the other hand, looking at this graph for the number of people greater than 2,000 that have about onethird the probability of dying as in the most likely case, we see that this probability is about 2,070 people. This is only a 3.5% deviation from the most likely case. It doesn’t take much extra contribution from each person to cover this contingency. This comparison, in a nutshell, shows why insurance companies work.