added to the net premium to defray working expenses, etc., nor with the excess of interest that may be realized on investments over the assumed rate. It would be a very extraordinary emergency indeed in any well-established company that would call these into requisition for the payment of death-claims, and, when not required for expenses, they are returned annually as surplus in mutual companies. It is strenuously argued that, because they may, if necessary, be used for the payment of claims, they should, along with insurance-value as above defined, be taken into account in fixing the maximum "surrender charge." But, as their discounted values, reduced by the improbability of their requisition for company use, would not materially differ in proportion from the insurance values, the argument for taking them into account is a good deal more nice than wise.
As the surrender value which can be fairly and wisely stipulated to be paid, if not paid when not stipulated, is a matter of great practical importance, let me explain particularly the case put by my reviewer of a single premium endowment insurance for twenty years, to be surrendered at the end of five years. If entered at forty, by the actuary's table, at four per cent., its net single premium would be $511.15 for $1,000. At the age of forty, the natural net premium to insure $1,000 for one year is $9.96. This, at four per cent., will amount at the end of the year to $10.36, so that the company, exclusive of the party himself, by insuring $1,000, runs the risk of losing $989.64 if he dies, and of gaining $10.36, if he does not. There is in this case no reserve at the end of the year, and no occasion to use the term "self-insurance." But if $511.15, which at the end of the year will amount to $531.60, it runs the risk of losing only $468.40. And, if in the former case the company may properly be said to have insured $1,000, in this it insures only 468.40989.64 $1,000=$473.31. For this insurance the party paid in advance (by part of his interest discounted) at the same rate as for the $1,000 in the other case, that is 473.311,000 $9.96=$4.72 nearly. But if he had died during the year, his heirs would have received $1,000, the same as in the other case. Therefore, since the company insured only $473.31, the party himself must have insured the remaining $526.69; and this is precisely the reserve which, in Massachusetts, the law requires the company to show on hand at the end of the year, if the party is then alive, as the sum necessary under the assumptions to enable it to carry through the remaining nineteen years of the contract. Hence, the whole of the insurance operation of the first year, under this contract, is, that the company, exclusive of the party himself, will lose $473.31 if he dies, and gain $4.90 if he does not. The amount of the $511.15, after deducting the $4.90, which belongs to the company in case of survival, to pay other death-claims with, is a mere deposit held in trust, subject to the terms of the contract and the provisions of law.
If, having paid $4.72, the normal cost of the first year's insurance, we find those of all the succeeding years and discount them back to the start by both the assumed interest and mortality, their sum will be what I call the "insurance value" of the policy at its inception—"the sum which, paid in advance, under the assumptions, would exactly pay for all the insurance which the company is to do under the policy, as distinguished from what the party is to do himself."
Now, if we conceive that this sum, which in this case is $50.42, is a part of the $511.15, the balance, $460.73, is less than what will make the self-insurance or reserve required at the end of the year, and it is more than a sum which "works at compound interest till it amounts to the sum assured" at the end of the term. That would be only $456.39. The truth is, that the $50.42 is not a part of the sum in hand, but one of its functions. The complementary function, exhausting the power of the sum, is the $460.73—the "insurance value" of the self-insurance, so to speak, including in the self-insurance the endowment. The endowment, as will easily be perceived, becomes self-insurance, by assuming, what is the same to the company, that the party, if he lives to enter it, will be sure to die in the last year of the term.
