# Popular Science Monthly/Volume 5/June 1874/Correspondence

*To the Editor of the Popular Science Monthly:*

IT is one of the most difficult things to popularize mathematical science, either abstract or applied. On this account, and not for want of good intention, I have failed, in my "Politics and Mysteries of Life Insurance," to make myself understood on some points, even by my able and candid reviewer in The Popular Science Monthly for May. The reviewer says: "If the holder of a 'single-premium policy,' having twenty or more years to run, becomes desirous to surrender his policy at the end of five years, he should get back from the company, not only the 'reserve,' but also that portion of the 'insurance value' that has been set apart by the company to compensate for the risk attached to the remaining fifteen or more years of the policy's term."

This, I regret to be obliged to say, is not quite correct. But I must confess it is not an altogether unnatural inference from the definition of "insurance value" given on page 12 of "Politics and Mysteries." "Insurance value" is there spoken of *as if* it were a sum "paid in advance." It is really only a *function* of the sum so paid, not a part of it. The "single premium," apart from the margin added to defray office expenses, that is, the *net* single premium, is itself the reserve. The cost of the first year's insurance, so far as it is done by the company, comes from the *interest* of that premium. It is the present value of so much of the interest as is not needed to make up the "self-insurance" or reserve at the end of the year. And this reserve is the net single premium at the party's present age. And just so comes the cost of the next year's insurance, so far as it is done by the company. The "insurance value" of the policy at the start is the present value of all these costs, or partial interests, discounted both by the interest and mortality rates. Consequently it is nothing to be returned in addition to the reserve. On the contrary, if the life is a good one, which may be expected to live long enough to pay more than the average toward death-claims, something must be deducted from the reserve, which will bear some proportion to the "insurance value" of the policy at the time of surrender, to compensate the company for the loss of the *future wherewith* to make up the deficiencies of lives that are not good. This is on the principle that, other things being equal, the profitableness of policies will be as their "insurance values."

If we speak of "insurance value" as being actually contained in the single premium (net), then the balance thereof is less than the "self-insurance" or reserve. We have no technical name for that balance. Prof. Bartlett calls it (*see* "Politics and Mysteries," page 73), the fund which "works at compound interest till it amounts to the sum assured." But it is more, as will appear presently.

The two new technical terms "self-insurance" and "insurance value," which I have felt obliged to introduce into the discussion of this subject, cannot be well understood without noting their relation to each other.

*Self-insurance* is the amount in the hands of the company at the end of a policy year, which the insured party has paid beyond the normal cost of the past insurance. In the fact of paying so much beyond the normal or assumed cost, he insured himself to that amount. And the law has stepped in and made it emphatically a self-insurance, by virtually forbidding the company ever to apply it to the payment of a claim on any other policy.

*Insurance value*—and I should have done better by so defining it—is the present value, discounting by the assumed rates both of interest and mortality, of which the policy *may be expected* to contribute toward the payment of death-claims, including its own, so far as that, when it occurs, shall not be self-insured.

This "insurance value" has of course nothing to do with the margin arbitrarily 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 ^{468.40}⁄_{989.64} $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 ^{473.31}⁄_{1,000} $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.

If there were any practical utility in it, we might analyze the premium into three functions: the insurance value of the insurance by the company, the insurance value of the self-insured, and the endowment-value function, making, in the present case, $511.15 = $50.42 + $135.97 + $324.76.

At the end of five years, in the case of this policy, the reserve or self-insurance is $595.82, and the "insurance value" is reduced to $37.90. According to the absurd rule, imported from England, no regard is had to "insurance value" withdrawn, but only to reserve, and the "surrender charge" is from one-third to two-thirds of the latter. Of course, no one would think of sacrificing a "paid-up" policy at such a rate. Prof. Bartlett recommends that in this case the company should deduct the entire "insurance value" and pay $557.92 as the "surrender value." My own opinion is, that eight per cent. of the "insurance value," or $4.03, is a sufficient charge to keep the company whole. This charge of eight per cent. is based on two assumptions, either of which seems to me reasonable: First, that the members who will select themselves out of a mutual company will not be collectively as much as eight per cent, better than the average. Secondly, that eight per cent. of the "insurance value" deducted from the reserve will be more than sufficient to replace that withdrawn with others as good.

Elizur Weight. |

Boston, April 22, 1874. |