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

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

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.