Understanding the Mathematics of Personal Finance

# TERM INSURANCE

The simplest kind of life insurance policy you can buy is term insurance. As the name implies, a term insurance policy insures you for a specific term. You pay a premium up front for the policy. If you don’t die during the term, you and the insur­ance company no longer have any contractual relationship. Whatever happens, the insurance company keeps the premium. Actually, term insurance is the only kind of life insurance there is. All the other life insurance products that you can buy are built on term insurance policies, sometimes combined with savings/investment accounts.

Term insurance policies themselves can get very complicated. They might be written for many years, and payments made might be spread out over these years rather than up front. There are so many possible variations that it’s impossible to cover them all. But let’s look at a few simple cases. I will be losing accuracy by considering only average values and by not considering the business costs, the return on investment of all the money that the insurance company is holding, the need for the insurance company to make a profit for its owners (investors), and so on. Also, life insurance companies have many sets of life tables that get far more specific than the few sets I’m using for my examples. In other words, I will show you what goes on in putting together a life insurance policy. I can’t actually price your policy.

Suppose that a 50-year-old woman wants a 1-year, \$100,000 term life insurance policy for the following year. Table 10.2 shows the 2004 Life Table for all women. This table just shows the age (x) and the probability of dying at that age (q) because that ’s all that ’s needed.

Table 10.3 shows a part of Table 10.2 from age 50 to age 100. In addition, I have added column l and column d, with column l starting at 100,000 people. The numbers for column l and column d were generated following the procedure shown above.

Looking at the top line (age 50), the table shows that if 100,000 people sign up for this policy, on the average, 320 will die during the year. The insurance company will have to pay out 320(\$100,000). Since 100,000 people bought these policies, the cost per person is just 320(\$100,000)/100,000 = \$320.

Let’s make this a little more complicated. If the insurance company sold a lot of these policies, it would expect to see a few women die every day of the year, since some women die early in the year, and some die very late in the year. The average date of death is the middle of the year. This means that the insurance company gets to hold everybody’s money for an average time of half a year. If an average woman has to pay an amount today that will be worth \$320 half a year from now, then she should only pay the present value of that amount. At 4% interest, therefore, she should only have to pay (assuming annual compounding)

What if this same woman, who’s just turning 50 years old, wants to buy a 2- or more year term policy? At first blush, you might ask why she would do this. If she

Table 10.2 The 2004 U. S. Life Table for All Women (Age and q Columns Only)

 Age q Age q Age q 0-1 0.006091 34-35 0.000825 68 - 69 0.014966 1-2 0.000457 35-36 0.000892 69-70 0.016407 2-3 0.000267 36-37 0.000971 70-71 0.017945 3-4 0.000197 37-38 0.001071 71-72 0.019617 4-5 0.000168 38-39 0.001190 72-73 0.021503 5-6 0.000151 39-40 0.001321 73-74 0.023635 6-7 0.000138 40-41 0.001453 74-75 0.025987 7-8 0.000129 41-42 0.001586 75-76 0.028358 8-9 0.000120 42-43 0.001727 76-77 0.030849 9-10 0.000112 43-44 0.001883 77-78 0.033818 10-11 0.000107 44-45 0.002055 78-79 0.037481 11-12 0.000113 45-46 0.002243 79-80 0.041792 12-13 0.000135 46-47 0.002439 80-81 0.046463 13 - 14 0.000178 47-48 0.002633 81-82 0.051306 14-15 0.000237 48-49 0.002819 82 - 83 0.056613 15 - 16 0.000306 49-50 0.003005 83 - 84 0.062608 16-17 0.000371 50-51 0.003204 84-85 0.069533 17-18 0.000421 51-52 0.003432 85 - 86 0.076645 18-19 0.000446 52-53 0.003695 86 - 87 0.084411 19-20 0.000453 53-54 0.004000 87 - 88 0.092876 20-21 0.000456 54-55 0.004346 88 - 89 0.102085 21-22 0.000464 55-56 0.004725 89-90 0.112081 22-23 0.000471 56-57 0.005137 90-91 0.122907 23-24 0.000481 57-58 0.005594 91-92 0.134602 24-25 0.000492 58-59 0.006110 92-93 0.147201 25-26 0.000506 59-60 0.006697 93-94 0.160735 26-27 0.000522 60-61 0.007389 94-95 0.175225 27-28 0.000541 61-62 0.008167 95-96 0.190689 28-29 0.000565 62-63 0.008977 96-97 0.207132 29-30 0.000593 63 - 64 0.009776 97-98 0.224550 30-31 0.000627 64-65 0.010581 98-99 0.242924 31-32 0.000667 65 - 66 0.011466 99-100 0.262224 32-33 0.000712 66-67 0.012498 100 or over 1.00000 33-34 0.000764 67-68 0.013661

just planned to buy term policies year by year and then died, say, during the first year, her premiums for the subsequent years would still be part of her estate rather than in the hands of the insurance company. The need to buy such a policy can arise, for example, if there is a business loan with the repayment due as a lump sum, say, 5 years from today. The creditor might want to guarantee his or her repayment in case the borrower dies before the repayment is due, without having to get involved

Table 10.3 Life Table for Women Aged 50 and Up

 X q l d X q l d 50-51 0.003204 100,000 320 76-77 0.030849 74,058 2,285 51-52 0.003432 99,680 342 77-78 0.033818 71,774 2,427 52-53 0.003695 99,337 367 78-79 0.037481 69,347 2,599 53-54 0.004000 98,970 396 79 - 80 0.041792 66,747 2,790 54-55 0.004346 98,574 428 80-81 0.046463 63,958 2,972 55-56 0.004725 98,146 464 81-82 0.051306 60,986 3,129 56-57 0.005137 97,682 502 82-83 0.056613 57,857 3,275 57-58 0.005594 97,181 544 83 - 84 0.062608 54,582 3,417 58-59 0.006110 96,637 590 84-85 0.069533 51,165 3,558 59- 60 0.006697 96,046 643 85 - 86 0.076645 47,607 3,649 60- 61 0.007389 95,403 705 86 - 87 0.084411 43,958 3,711 61-62 0.008167 94,698 773 87 - 88 0.092876 40,247 3,738 62- 63 0.008977 93,925 843 88-89 0.102085 36,509 3,727 63- 64 0.009776 93,082 910 89-90 0.112081 32,782 3,674 64- 65 0.010581 92,172 975 90-91 0.122907 29,108 3,578 65- 66 0.011466 91,197 1,046 91-92 0.134602 25,530 3,436 66- 67 0.012498 90,151 1,127 92-93 0.147201 22,094 3,252 67- 68 0.013661 89,024 1,216 93-94 0.160735 18,842 3,029 68- 69 0.014966 87,808 1,314 94-95 0.175225 15,813 2,771 69-70 0.016407 86,494 1,419 95-96 0.190689 13,042 2,487 70-71 0.017945 85,075 1,527 96-97 0.207132 10,555 2,186 71-72 0.019617 83,548 1,639 97-98 0.224550 8,369 1,879 72-73 0.021503 81,909 1,761 98-99 0.242924 6,490 1,577 73-74 0.023635 80,148 1,894 99-100 0.262224 4,913 1,288 74-75 0.025987 78,254 2,034 100 or over 1.00000 3,625 3,625 75-76 0.028358 76,220 2,161

in chasing her estate for his or her money. If there is a life insurance policy with the creditor as beneficiary, then he or she doesn’t have to worry about getting his or her money back if she dies before the loan is due. The creditor would demand a paid-up term life insurance policy at the start of the loan period (when the woman gets the money), and the lender would have to consider the premium for this policy as part of her cost of getting the loan (i. e., increasing the effective interest rate).

Returning to Table 10.3, look at the fifty-first year. There are 100,000 - 320 = 99,680 women starting their fifty-first year, so we should expect 0.003432(99,680) = 342 deaths during this year.

During the first year of the 2-year policy, the insurance company has to pay out 320 times for every 100,000 people that signed up, as discussed above. In the second year, it has to pay out 342 times for every 100,000 people that originally signed up.

I just showed how much the 1-year term policy would be for the 50-year-old woman. This is identical to the cost of the first year of a 2-year term policy.

For the second year, the cost to the insurance company for a \$100,000 policy, following the same procedure, is \$342 per person in the original (100,000-person) group.

This cost is incurred by the insurance company, on the average, 1.5 years after it collected the premiums on the policies, and the present value of this cost at the date of collecting the premiums is

The premium for the 2-year policy is the sum of the two individual premiums, \$313.80 + \$332.46 = \$646.26.

This procedure may be continued for as many years as you want the policy.

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