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 up-front 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.

10.2 PROBABILITY

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 – of-the-pants 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 well-shuffled 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

 Age q(x) l(x) d(x) L(x) T(x) e(x) 42 0.002853 95,043 271 94,907 3,404,448 35.8 43 0.003113 94,772 295 94,624 3,309,541 34.9 44 0.003412 94,477 322 94,316 3,214,917 34.0 45 0.003735 94,154 352 93,979 3,120,601 33.1 46 0.004071 93,803 382 93,612 3,026,622 32.3 47 0.004428 93,421 414 93,214 2,933,010 31.4 48 0.004806 93,007 447 92,784 2,839,796 30.5 49 0.005206 92,560 482 92,319 2,747,012 29.7 50 0.005648 92,078 520 91,818 2,654,693 28.8 51 0.006121 91,558 560 91,278 2,562,875 28.0 52 0.006594 90,998 600 90,698 2,471,597 27.2 53 0.007045 90,398 637 90,079 2,380,899 26.3 54 0.007488 89,761 672 89,425 2,290,819 25.5 55 0.007946 89,089 708 88,735 2,201,394 24.7 56 0.008459 88,381 748 88,007 2,112,659 23.9 57 0.009064 87,633 794 87,236 2,024,652 23.1 58 0.009810 86,839 852 86,413 1,937,416 22.3 59 0.010706 85,987 921 85,527 1,851,002 21.5 60 0.011763 85,067 1,001 84,566 1,765,476 20.8 61 0.012934 84,066 1,087 83,522 1,680,909 20.0 62 0.014159 82,979 1,175 82,391 1,597,387 19.3 63 0.015362 81,804 1,257 81,175 1,514,996 18.5 64 0.016558 80,547 1,334 79,880 1,433,820 17.8 65 0.017847 79,213 1,414 78,507 1,353,940 17.1 66 0.019331 77,800 1,504 77,048 1,275,433 16.4 67 0.020992 76,296 1,602 75,495 1,198,386 15.7 68 0.022858 74,694 1,707 73,840 1,122,891 15.0 69 0.024921 72,987 1,819 72,077 1,049,050 14.4 70 0.027065 71,168 1,926 70,205 976,973 13.7 71 0.029363 69,242 2,033 68,225 906,768 13.1 72 0.032031 67,209 2,153 66,132 838,543 12.5 73 0.035178 65,056 2,289 63,912 772,411 11.9 74 0.038734 62,767 2,431 61,552 708,499 11.3 75 0.042414 60,336 2,559 59,057 646,947 10.7 76 0.046171 57,777 2,668 56,443 587,891 10.2 77 0.050325 55,109 2,773 53,723 531,448 9.6 78 0.055085 52,336 2,883 50,894 477,725 9.1 79 0.060498 49,453 2,992 47,957 426,831 8.6 80 0.066557 46,461 3,092 44,915 378,873 8.2 81 0.072986 43,369 3,165 41,786 333,958 7.7 82 0.079682 40,204 3,204 38,602 292,172 7.3 83 0.086593 37,000 3,204 35,398 253,570 6.9

Table 10.1 Continued

 Age q(x) l(x) d(x) L(x) T(x) e(x) 84 0.094013 33,796 3,177 32,207 218,172 6.5 85 0.102498 30,619 3,138 29,050 185,965 6.1 86 0.111640 27,481 3,068 25,947 156,915 5.7 87 0.121472 24,413 2,965 22,930 130,968 5.4 88 0.132023 21,447 2,832 20,031 108,039 5.0 89 0.143319 18,616 2,668 17,282 88,007 4.7 90 0.155383 15,948 2,478 14,709 70,726 4.4 91 0.168232 13,470 2,266 12,337 56,017 4.2 92 0.181880 11,204 2,038 10,185 43,680 3.9 93 0.196334 9,166 1,800 8,266 33,496 3.7 94 0.211592 7,366 1,559 6,587 25,229 3.4 95 0.227645 5,808 1,322 5,147 18,642 3.2 96 0.244476 4,486 1,097 3,937 13,496 3.0 97 0.262057 3,389 888 2,945 9,559 2.8 98 0.280351 2,501 701 2,150 6,614 2.6 99 0.299312 1,800 539 1,530 4,463 2.5 100 or over 1.00000 1,261 1,261 2,933 2,933 2.3

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 1-year, \$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., 50-year-old men) requires more funding than was predicted for this year, there will be another group (e. g., 60-year-old women) that will require less funding than was predicted this year. When all else fails, insurance companies can cross-insure 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 100-person 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 one-third as likely for four people to die as it is for two people to die and about one-tenth 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 100,000-person 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 one-third 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.