
113 F.2d 833 (1940)
COMMISSIONER OF INTERNAL REVENUE
v.
KELLER'S ESTATE et al.
No. 7242.
Circuit Court of Appeals, Third Circuit.
May 28, 1940.
Samuel O. Clark, Jr., Asst. Atty. Gen., and Sewall Key, Berryman Green, and Carlton Fox, Sp. Assts. to Atty. Gen., for petitioner.
Ferdinand T. Weil and Weil, Christy & Weil, all of Pittsburgh, Pa., for respondents.
Before BIGGS, CLARK, and JONES, Circuit Judges.
CLARK, Circuit Judge.
No one knows better than insurance salesmen that only the "excess over $40,000" *834 of life insurance proceeds receivable by beneficiaries other than the insured's executor are subject to the estate tax, 26 U.S.C.A. Int.Rev.Code, § 811 (g). This $40,000 exemption, a unique characteristic of their general stock in trade, is quite naturally stressed to the customer. Sometimes, however, it is stressed to a paradoxical extreme. The paradox consists in applying a life insurance exemption to the estate of an uninsurable prospect.[1]
Mrs. Keller, the prospect (or rather decedent) at bar, had at the age of seventy-five no choice but to follow the course prescribed for uninsurables. She took out a single premium life policy in conjunction with a single premium annuity. Her estate now seeks to complete that course by securing the benefit of the $40,000 exemption. Opinions are divided as to whether it may do so. A majority of the Board of Tax Appeals and a unanimous Second Circuit Court of Appeals accept the paradox and allow the exemption.[2] On the other hand, a unanimous Eighth Circuit Court of Appeals and several students of the question have taken the opposite view.[3] We are constrained to do likewise:
One may ask at the outset: How does an uninsurable life become insured? The explanation lies in the mathematical correspondence between annual interest and annuity payments. $100 due in ten years at 2% is precisely the same thing as an annuity certain of $2 for ten years combined with a promise to pay $100 without interest at the end of ten years. The discounted sums required to meet the annuity payments plus the discounted sum required to pay the $100 when due will always total $100, the amount of the loan.[4] The same holds true if the loan is repayable at the death of the lender. The obligation of the borrower may be divided into a complete annuity of $2 combined with a whole life insurance of $100. One merely substitutes for the ten year term a series of probabilities of being alive (annuity) and of being dead (life insurance) in the future. The relation between these converse probabilities is such that the discounted sum required to provide the annuity plus that required to produce the life insurance will always be equal to the amount of the loan. We say always because the relation between the probabilities of living and dying is from the very nature of the mortality tables, on which they are calculated, constant, no matter what the age of the lender. So, an insurance company may borrow $100 at 2% from one doomed to die within ten minutes and issue in return a $2 annuity and a $100 life policy. By an actuarial tour de force (because the continuance of life is a matter of minutes, not probabilities) the amount of the loan advanced is split into the single premiums appropriate to each policy as if taken out by a normal person. In other words, the sure thing (loan) is artificially separated into doubtful bet (life insurance) and hedge (annuity). It is on this general principle that an uninsurable life becomes "insured". See Appendix to this opinion.
The transaction between Mrs. Keller and the insurance company was cast in the same mould. There are, to be sure, arithmetical *835 ramifications. Life and annuity premiums were computed on different mortality tables at different rates of interest, and included "loading charges" to meet premium taxes and agents' commissions. These, however, present no essential point of dissimilarity. The two premiums, exclusive of loading charges, add up to $20,000, the face amount of the life policy. Apart from the effects of a slight miscalculation in loading charges, Mrs. Keller's death, no matter when it occurred, could never require the company to look to funds contributed by other policy holders in order to make the life insurance payment. That being so, we can detect no substantial economic distinction between the conjoint effect of the two policies issued, and that of an engagement to repay Mrs. Keller, or her order, $20,000 upon her death, with interest at almost exactly 2% per annum ($390.84 being payable under the annuity) in the meantime. It is plain, furthermore, that the economic consequences of a loan were intended. Mrs. Keller did not undergo a physical examination, and it is conceded that the company would not have issued the life policy without the annuity.
Nevertheless, Mrs. Keller died the holder of a paid up, standard form, life insurance policy, with a cash surrender value, a right to change the beneficiary, and all the usual paraphernalia of such contracts. Its proceeds were payable to her daughter. The legal problem, therefore, is whether those proceeds constitute an "amount receivable by * * * beneficiaries as insurance under policies taken out by the decedent upon his own life" within the meaning of the taxing statute.
We think the question turns upon whether the adverbial phrase, "as insurance" refers to the legal, or to the economic, method whereby the amount is received. The legal mechanism employed at bar is unquestionably a contract of life insurance. An action could be brought on that policy to recover its proceeds, without the annuity being in any wise involved. The annuity, on the other hand, is not only a cog but also an indispensible cog in the economic machine which produced the amount received by the beneficiary. That machine, therefore, fulfilled none of the composite functions of life insurance. Being in its essence a loan transaction it neither "built up an estate", nor caused the company to assume, on loose principles of indemnity, any risk of loss occasioned by premature death. Vance, Handbook of the Law of Insurance, pp. 80, 123 et seq.; Cooley, Briefs on the Law of Insurance p. 80. What is more important, such a risk, being non-existent, could not be so assumed pursuant to a plan of spreading it among a large section of the population, i. e., other policy holders. See An Analysis of "Insurance" and "Insurance Corporation", 36 Columbia Law Review 456 (note). Mrs. Keller merely paid $20,000 to obtain $20,000 at her death. We may observe parenthetically that promissory notes payable at the death of the obligor have long been countenanced in the law. 1 Daniel on Negotiable Instruments p. 45; 2 A.L.R. 1471, note. The makers of such engagements (to say nothing of the learned framers of the Uniform Negotiable Instruments Law § 4 (3)) might be more than mildly shocked to find that they had defied the local Insurance Commissioner and written a policy of life insurance on their own lives.
It is plain in our estimation that the Congress used the word "insurance" with the economic rather than the purely contractual aspects of the term in mind. The statute prescribes an exemption prompted by a settled policy favoring the institution of life insurance.[5] The economic workings of that institution produce, upon premature death, the sudden "springing up"[6] of a fund which rescues the insured's beneficiaries, who are ordinarily within the family circle, from want. This result is both salutary and unique. The fund insofar as it springs up comprises not only the accumulated interest on premiums paid (as is more or less the case with nonexempt savings bank accounts) but also a multitude of painlessly infinitesimal contributions from others who have escaped premature death. Life insurance is, therefore, the least costly and most equitable device yet evolved to preserve families in a wage-earning society from destitution. Obviously *836 the exemption is meant to encourage life insurance as such a device. The language of encouragement, then, must be taken to refer to the economic functions, not the bare legal attributes of insurance.
We hold, accordingly, that the amounts receivable under Mrs. Keller's life policy were not receivable as insurance. They may, moreover, be properly included in her gross estate. Old Colony Trust Co. v. Commissioner, above cited. The Commissioner suggests that the transfer section of the statute is applicable, but cites no authority. The respondent executors make no argument on the point whatever. In view of this apparent concession the appropriateness of any particular section becomes of academic importance only and does not merit discussion, see 38 Michigan Law Review 526 (note), above cited; and cf. Paul, Life Insurance and the Federal Estate Tax, 52 Harvard Law Review 1037, above cited; Oppenheimer, Proceeds of Life Insurance Policies Under the Federal Estate Tax, 43 Harvard Law Review 724; 82 University of Pennsylvania Law Review 384 (note).
The decision of the Board of Tax Appeals is reversed.

APPENDIX
The actuarial symbol for the single premium on a life insurance of $1 due at the end of the year of death of a person now aged x years is Ax. The symbol for the single premium on an annuity of $1 payable at the end of each year the annuitant survives is ax. Both are calculated with appropriate discount in accordance with the probabilities of continued life (annuity) or of discontinued life (insurance) as shown by mortality tables. The relation between them may be expressed as

i being the interest rate. See King, Life Contingencies, Institute of Actuaries Textbook, Vol. 2, p. 125. That being so, a single premium on a whole life insurance of $1 (payable at the instant of death) combined with a single premium on a complete annuity (apportioned in the year of death) of the interest (i) on $1 will always equal $1. For, expressing the combination of the whole life insurance (&Amacr;x) and the complete annuity (åx) algebraically, we have:

This, of course, is merely a roundabout way of stating that the sum required to repay a loan of $1 at i per cent interest is $1 where the prevailing rate of interest is i. The mortality factors ax and Ax cancel out no matter what age is used. See generally, Wolfe, Inheritance Tax Calculations, pp. 40, 85.
The equation &Amacr;x + i åx = 1 is frequently exploited in the statutory valuation of remainders after life estates. The life estate valued as an annuity (or more accurately, as a complete annuity) is deducted from the whole estate and the difference taken as the value of the remainder, Wolfe, above cited, pp. 4, 5.
NOTES
[1]  Salesmen are instructed: "If the prospect has a reasonable income and is insurable he unquestionably should be advised to purchase annual premium life insurance to take advantage of his full insurance exemption. If he is not insurable a single premium life or endowment policy combined with an annuity provides the prospect with income comparable with that received from high grade bonds, and at the same time secures him this additional tax saving." Wright & Lowe, Selling Life Insurance Through A Tax Approach, pp. 80, 81.
[2]  Estate of Anna M. Keller, 39 BTA 1047; Commissioner v. Le Gierse, 110 F. 2d 734, decided March 24, 1940; Fisher v. Commissioner, 40 BTA 1377, June 22, 1939 (unreported memorandum opinion now on appeal to the 7th Circuit Court of Appeals; see also Bowman v. Tax Com., 135 Ohio St. 295, 20 N.E.2d 916; Matter of Ruppel, March 11, 1937 [no opinion for publication]; Clark, Inheritance and Estate Taxes on Life Insurance, pp. 184, 199, 200.
[3]  Helvering v. Tyler, 111 F.2d 422, decided April 29, 1940; Greenfield, Federal Taxation of Combined Annuity and Life Insurance Contracts, 49 Yale Law Journal 946 (note); Life Insurance and the Federal Estate Tax, 40 Columbia Law Review 86 (note); 52 Harvard Law Review 1181 (note); 38 Michigan Law Review 526 (note); 32 Illinois Law Review 223, 232 (note); see, also, State ex rel. Thornton v. Probate Court, 186 Minn. 351, 243 N.W. 389; Old Colony Trust Co. v. Commissioner, 1 Cir., 102 F.2d 380, and cases there cited.
[4]  See, Wolfe, Inheritance Tax Calculations, pp. 15, 16.
[5]  Hughes, The Federal Death Tax pp. 88-91. For criticisms as to the actual working out of this policy, see Paul, Life Insurance and the Federal Estate Tax, 52 Harvard Law Review 1037, 1076. The British death duty affords no exemption, 13 Halsbury's Laws of England pp. 241, 242.
[6]  Attorney General v. Robinson, 2 I.R. 67, 90.
