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.

image081
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.

Leave a reply

You may use these HTML tags and attributes: <a href="" title=""> <abbr title=""> <acronym title=""> <b> <blockquote cite=""> <cite> <code> <del datetime=""> <em> <i> <q cite=""> <s> <strike> <strong>