International Styles

Term Insurance

Term Insurance. Term insurance is the simplest type of contract issued insuring against premature death. Term policies usually run for five, ten, fifteen, or twenty years, and promise to pay the sum insured if the policyholder should die within this period, nothing being paid if death does not occur during the designated term. Term policies are therefore a distinct type of temporary insurance. Attempts have been made to popularize a one-year term policy which is renewable from year to year at the option of the insured, thereby granting current cost insurance which is paid for at the beginning of each year, the premium furnishing protection for that year only, and a different rate being chargeable for the following year's insurance. This type of policy offers an excellent opportunity to explain the simple elements of rate-making. Suppose, therefore, that the net single premium is to be ascertained on a renewable one-year term insurance of $1,000 on a life aged 45. Immediate use will now be found for two of the assumptions used in rate-making which were mentioned in the preceding chapter, viz, that premiums are paid in advance and. that matured claims are paid at the close of the policy year. Accordingly, it is required to find the amount of money which must be paid in at the beginning of the year by a policyholder in order to enable the company to return $1,000 at the close of the year in case the policy has matured. The question must now be asked: What is the risk insured against? It follows from the definition of .term insurance that it is the chance of dying during the year. This will be determined by means of the mortality table. This shows that, of 74,173 persons living at age 45, 828 die during the year. Suppose now that an insurance company should issue 74,173 one-year term policies to persons aged 45. If the mortality experienced among this group coincides with the experience indicated in the mortality table there will be 828 deaths during the year. Since each of these deaths represents a liability of $1,000 to the company, and since the claims are payable at the close of the year, the company must have on hand at that time $828,000 to pay claims. But this entire amount need not have been collected from the policyholders since they were required to pay their premiums at the beginning of the year and the company was able to invest the money at interest for one year and earn 3 percent, thereon. For every $1 collected, therefore, the company will have on hand $1.03 when the claims mature. Eight hundred and twenty-eight thousand dollars, therefore, bears the same ratio to the amount of money which must be collected from the group of 74,173 persons as $1.03 bears to $1. Put in the form of a proportion this may be stated as follows:

X may here be defined as the present value of $828,000 discounted for one year at 3 percent. This amount of money ($803,883.50) therefore must be paid at the beginning of the year by the group of 74,173 persons in order that there may be on hand at the end of the year sufficient funds to pay $1,000 for each of the 828 deaths. To obtain the premium which each individual should pay, it is only necessary to divide the total fund by the number contributing, viz: 803,883.50 ÷ 74,173 = $10.84

The "net single premium" for a one-year term insurance at age 45, or the amount of money that must be paid at the beginning of the year to supply each individual's contributions to the death losses of the group for the year is, therefore, $10.84.

This same problem may be approached in a different way and a formula stated for determining costs. The original assumption required the insurance of a group of 74,173 persons of identical age. But this is impossible to obtain in practice. Suppose now that it is desired to insure a single individual aged 45 against death during the year and that the net single premium for this insurance is to be ascertained. Clearly, if the event occurs against which protection is desired, it will cost the insurance company $1,000. But what is the probability of death occurring during the year? It has been shown that 828 persons aged 45 die out of a group of 74,173. Reference to the discussion of the theory of probabilities in Chapter XI, Measurement of Risk in Life Insurance, will show that this is equivalent to saying that the probability of death during the forty-fifth year is 828/74173. The cost to a single person, therefore, will be 828/74173 of $1,000. But since this value needs to be on hand at the end of the year and money earns 3 percent, interest, the amount to be paid in by the insured will be the value of the above amount discounted for one year at 3 percent. This result is found as follows:

828/74173 X 1000 ÷ 1.03 = $10.84

It must not be assumed from this that an insurance company can insure a single person; instead, it must always deal with a group sufficiently large to guarantee a close approximation of its actual mortality experience with the table mortality. It must, as was explained earlier, be sure of the operation of the law of average. But it does not need to insure this entire, group with the same kind of policy or at the same age. The law will operate if only the entire group of policyholders including all ages and all kinds of policies be sufficiently large.

If the method here used in determining the cost of this insurance is carefully studied it will be found to embody the following process: Multiply the probability insured against by the amount of the policy and divide by the amount of $1 at the assumed rate of interest for one year; and from this formula it is possible to construct a general formula to apply in computing all net single premiums, viz, the probability insured against multiplied by the amount of the policy multiplied by the value of $1 discounted for the period the money is held. One dollar discounted for one year at 3 percent, equals 1.00/1.03 = .970874. Multiplying by this factor gives the same result as dividing by 1.03. This formula will be used hereafter in computing net single premiums. It would be possible now to compute the net single premium paid at the beginning of the second year for the second year's insurance under our renewable one-year term policy issued at age 45. The probability of death during this year is the yearly probability of death at age 46, or 828/74173 and the cost of the year's insurance would be:

828/74173 X 1000 X .970874

In like manner, the yearly cost of insurance can be computed for any age from 10 to 95, inclusive, the years covered by the American Experience table.

While much emphasis has here been placed upon the one-year term policy because of its appropriateness in developing the elementary principles of rate computation, the fact must not be lost sight of that one-year term policies are rarely sold. The usual term policies extend for five years or longer, and .this fact brings complications into the matter of rate-making. Suppose it is desired to compute the net single premium for a five-year-term insurance issued at age 45, i.e. the amount of money which, paid in a single sum at age 45, will purchase insurance against death at any time within the next five years. Two facts are apparent upon a moment's reflection : (1) the premium is paid only once, in a single sum at the inception of the risk; (2) death claims will be paid at the end of the year in which they occur, and not at the end of the five-year period. This latter fact has an important bearing on the interest which will be earned and therefore on the method of computing the five years' cost. Manifestly, the cost cannot be correctly determined by multiplying the total probability of dying during the five years by the face value of the policy and discounting this amount in one operation since some of the money collected will draw interest for only one year while another part will be earning interest for five years. It is necessary to compute the cost of each year's mortality separately. The probabilities insured against in this case are the chances that a person aged 45 will die during the first year following, during the second year, the third year, etc. These probabilities are respectively 828/74173, 848/74173, 870/74173, 896/74173 and 927/74173. Each of these figures must be multiplied by the amount insured and by the present value of $1.00 discounted in each instance, by the length of time the money is held. The money available for the first year's death claims will be held one year; for the second year's claims, two years, etc., the funds for the last year's claims being held five years. The discounted values of one dollar for one, two, three, four and five years at 3 percent, interest are respectively $.970874, $.942596, $.915142, $.888487 and $.862609. The cost of the five years' insurance, therefore, can be shown as follows :

This computation shows that $53.86 deposited with the company and placed at 3 percent, interest will furnish enough money to pay all the death claims on this five-year term policy.

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