RATE-MAKING ONE-YEAR TERM OR NAT URAL PREMIUM. Because of its simplicity the method of insuring lives which naturally first suggests itself is insurance from the date of the payment of one premium to the date when another premium becomes due, collecting just sufficient to cover the current cost of the protection. Thus, if pre miums are payable annually, this will be an insurance for the term of one year, renewable, if at all, by the payment of a premium varying with the risk as the age increases, the premium for each year just taking care of the insur ance for that year.
Of course, the annual probability of death does not remain the same throughout life, but must be conceived of as constantly varying in a more or less regular man ner. From birth to age io, for instance, it is usually found that it constantly diminishes, though at a diminishing ratio. Then it usually begins to increase, very slowly at first, but at a constantly increasing ratio.
The custom in life insurance, though it is not without exceptions, is to deal with the deaths as occurring uni formly throughout the year, and with the death-rate as changing at the end of the year, so that the deaths then occur uniformly at the new rate throughout another year. This, as applied to renewable term or current cost insur ance, involves no material error when premiums are paid annually; and at age 10 and over the error is also not material from a financial standpoint when premiums are paid at intervals of less than one year because, as the risk is actually increasing, to charge a pro rata part of an annual one-year term premium, representing the risk of the entire year, yields in the aggregate of premiums paid more than current cost for all parts of the year until the whole year's insurance is paid for.
This assumption that the risk does not vary throughout the year is called the assumption of uniform deaths throughout the year. On the basis of this assumption, insurances at current cost, with premiums more frequent than annual, may be treated as one-year term insurances, with premiums adjusted to the varying cost at the begin ning of each year and the remainder of the year's pre mium as a deferred premium.
Another assumption, also common in British and Amer ican life insurance companies, is that death-claims are payable on the average at the end of the policy year in which the deaths occur. This was probably very nearly true when claims were not payable until three months after approval of proofs of loss; for deaths occur on the average in the middle of the year, and if we allow three months within which to make claim and complete proofs and then three months for payment, the claims would be paid on the average at the end of the year. But in these days, when proofs are made promptly and claims paid almost at once, it would doubtless be more nearly correct to assume that death-claims are payable at the middle of the year. Such is the assumption in France and in some other countries; and doubtless the change would be made also in Great Britain and America if it were not, first, that there are so many valuable working tables based on the former assumption, and, second, that the large margins in the standard mortality tables make the distinction of minor importance.
Let us then compute the annual premium, payable at the beginning of the year, for one year's insurance from age 1o. And let us assume that deaths will be according to the Actuaries' Table, and that money in hand will earn 4 per cent. interest per annum until required.
For convenience, assume, also, that 'woo° such insur ances start at the same moment, each for $1,000 and each on a life aged 1o. Then there will be altogether $Too,000,: 000 of insurance running for one year. According to this table 676 persons out of the mo,000 will die during that year, calling for $676,00o to be paid at the end of the year. The sum which must be on hand at the beginning of the year to meet payments of $676,000 at the end of the year, 4 per cent. interest being realized, is $676,000 1.04 = $65o,00o. If each of the 100,00 persons insured were to pay his proportionate part of this sum in advance, he would pay a premium of $6.50 for an insurance of $1,000 for one year.