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