The same result may be reached thus : The contribution of each towards the amount required to pay the total losses of $676,000, is $6.76. But this would not need to be paid until the end of the year in which case the premiums of those who had died could be deducted from the amounts of their insurances and since it is to be paid at the be ginning of the year, it must be discounted at 4 per cent. and the net amount becomes $6.76 ± 1.04 = $6.5o, as before.
Or that result may be reached by a more general pro cess, thus : for each the risk of dying during the year is 676 This, then, would be the premium, payable at the end of the year, for an insurance of 1; therefore, the premium payable at the beginning of the year for an insur ance of I isioegoo 1.04 = 0000o, and the premium, payable at the beginning of the year for an insurance of $i,000, is to 000 X $1,000 = $6.5o, as before.
This last method is most instructive. It shows that, on the assumption of losses payable at the end of the year, the value, at the beginning of the year, of the year's insur ance is the product of the probability of death, the dis counted value of $1 due at the end of the year and the sum insured. From that fact many rules are derived.
In the foregoing we have dealt with "net premiums" only, i. e., premiums sufficient merely to provide for the mortality expected according to the table employed with out addition for expenses and contingencies.
Historically, one-year term or natural premium insur ance was not the earliest, though, because of its simplicity, it will here be considered first. Current cost insurance came earliest, to be sure, but no care was taken to appor tion the cost correctly. Insurances for short terms, such as less than a year, a single year or a short terms of years, were granted at an early period, but without developing the method of computing the premiums for such insur ances. Even whole life insurance, paid for by level pre miums or by a single premium, preceded whole life insur ance divided up into renewable one-year term insurances, each paid for by its own appropriate premium.
The late Sheppard Homans, an eminent American actu ary, devoted a large part of his life to an attempt to in troduce and popularize renewable one-year term life in surance, which he named the "natural premium" system. The demand for low-priced insurance was then, as always, very great; and numerous societies and orders, operating under inequitable plans of apportioning the costs, had sprung up to supply this demand.
The history of Mr. Homans' experiment, which was undertaken by the Provident Savings Life Assurance So ciety, which he founded and which survived until this year, 1911, when it was reinsured and merged with another company, though the natural premium plan which it at first employed was long ago recognized to be a fail ure, is as follows : The gross premiums were computed by adding one-third to the net premium as a contingency and reserve provision, and then $4 per each $1,000 in sured, for expenses. This was on renewal premiums; the first year the entire margin over the net premium was available for expenses. Savings in mortuary cost were to be apportioned at the end of each year in reduction of the next year's premiums. The premium was to advance each year to the premium for the attained age. At the end of ten-year periods the unused reserve was to be applied to reduce subsequent premiums. The dividends derived from mortality savings were at first large and served to disguise the increase in rates of premiums; and the reserve provision, together with this fact, enabled uninformed and unscrupulous agents to represent that premiums would not increase. When, owing to the wear ing off of fresh selection and to the advancing ages, the cash payments had to be rather sharply increased, the pro tests of agents and the defections of policyholders caused the company to abandon the original plan so far as to employ the reserve portion to offset the increase in the one-year term premiums and to return the remainder in increased dividends in the form of reductions of current premiums.
Thus, it will be seen, the pure natural premium, or one year renewal plan, sold for what it was and then rigidly adhered to, did not get a fair trial, in spite of the re nowned actuary's good intentions. The same company offered a non-participating, pure natural premium plan at low rates and in two forms, viz.: one with the amount of insurance fixed and the premiums increasing and one with the amount of the premium fixed and the amounts of in surance decreasing. Neither of these found a ready sale, but it is not clear whether this was due to the plan itself or to these facts : A larger commission was paid for selling insurance on the other plan; the latter seemed, also, in view of the dividends, to be even cheaper, while the policy holders did not expect the premiums to increase; and cur rent cost protection was furnished much cheaper by assessment societies and orders, though on unscientific and ultimately unsafe plans.
The practicability of the plan, then, was not put directly to the test; but for certain reasons, now to be considered, it is believed that the result of such a test, if attempted, would have been and must be unfavorable. First, the participating plan actually employed by the Provident Savings becomes pure natural premium at about age 6o and upward, the dividends not availing to hold the rate level; second, the discontinuances of good lives were per haps not greater before those ages under the plan actually used than they would have been under a pure natural premittm plan; and, third, statistics as to mortality at advanced ages on current cost plans, though unscientific and calling for much less than the true one-year term premiums at these ages, strongly confirm the experience of the Provident Savings that under this plan excessive mortality must be expected at the older ages.
Consideration of the nature of the plan shows that ad verse selection at advanced ages is to be expected, because the continually increasing premiums more and more dis pose all who do not feel the need for the insurance, to with draw. Persons who expect to die very soon always consider that they need the insurance.
Of course, at age 95 under the American Experience Table and age 99 under the Actuaries' Table, where the one-year term premium for $1,000 insurance is $1,000 1.04 = $961.36 on the basis of 4 per cent. interest, no person would pay the premium at all, and the premiums for many ages before that are prohibitive, also, for all lives except such as are consciously moribund.
Yet such are one-year term premiums, if based on the usual tables, for average lives at these ages. If, say from age 6o, higher mortality were assumed to cover the ad verse selection caused by the increase of premiums, would they not drive out the best of the lives which might have remained at rates computed on the lower mortality as sumptions? In other words, is there not a force at work in natural premium insurance, when continued in force at the older ages, which renders it impracticable, and perhaps even impossible, to have a mortality table making a sufficient provision, because the larger the rates, the greater the adverse selection? On a priori grounds, this is now believed to be the case, and actual experience, so far as available, bears out that conclusion.
It will be shown, hereinafter, and may now be stated, that on sound plans under all forms of life insurance the insured really pays for all the protection he receives, over and above the reserve or "self-insurance" fund, at the natural premium or one-year term rate for the attained age. If this were fully understood by everyone, perhaps the natural premium plan especially if with decreasing benefits instead of increasing premiums might be more popular and also might not exhibit so marked adverse selection at the higher ages.