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to your lordships the unchallengeable proof of my assertions, and at the same time to respond to the wish manifested by the members of this body to have such proof in writing. That proof, founded on a solid basis, is proposed to your High Mightinesses in the following manner :
Value of Life Annuities in proportion to Redeemable Annuities.
I lay down the following presuppositions in order to determine the proportion of a life annuity to a redeemable annuity. For example, in presupposing that the redeemable annuity is and will be current at 25 years' purchase, or at the rate of 4 p.c. p.a., we must find at how many years' purchase the life annuity should be sold, to be in proportion to the aforesaid redeemable annuity, in such manner that the life annuity may, if not with mathematical precision, at least in its discovered value, be more advantageous to the purchaser than an annuity redeemable with the same capital.
I presuppose that the real value of certain expectations or chances of objects of different value, must be estimated by that which we can obtain from equal expectations or chantes dependent on one of several equal contracts. Let us take, for example, a small matter, and under circumstances intelligible at first sight: A person has two different expectations or chances, which may easily lead, the one to nothing, the other to 20 stuyvers. If, by one or several equal contracts, he can obtain for 10 stuyvers two like expectations or chances, we must estimate that the two aforesaid chances are worth to him exactly 10 stuyvers, because he can really obtain for 10 stuyvers these two expectations or chances by making an agreement with another person that each of them shall take 10 stuyvers, and then gamble or draw lots, by odd or even, head or tail, blank or prize, or in some such way, to determine which of the two should have the 20 stuyvers; thus by the said contract, equal in every regard, he evidently finds himself in the position of having in reality the two expectations or chances, the one of nothing, the other of 20 stuyvers.
That in taking at pleasure some years of a man's life, limited to the time when he is in his full vigour, and neither too young nor too advanced in age (this space of time shall be here 50 years, namely, from the third or fourth year of his age up to the fifty-third or fifty-fourth year), it is not more likely that this man should die in the first half-year of a given year than in the second half: similarly, it is not more likely that he should die in the second half-year of the aforesaid year than in the first half. although it depends entirely on chance whether this man, after having lived to the given year, and dying in the course of that year, should demise in its first or second half, one finds nevertheless, in this regard, an equality of likelihood or chance similar to the case of a tossed penny, where there is an absolute equality of likelihood or chance that it will fall head or tail, although it depends entirely upon chance as to the side on which it shall turn, and this to so high a degree that the penny may fall head 10, 20, or more times following without once falling tail, or vice versa.
That a man having passed the aforesaid vigorous time of his life, namely, the fifty-third or fifty-fourth year of his age, it begins to be more likely that he should die in a given year or half-year of the second period than has previously been the case; or that it is not likely, with respect to another man of like constitution or state of body, that the latter should die in less than a year or half-year of the said vigorous time of his life; whilst this likelihood or chance of dying in a given year or half-year of the ten first following years, namely, from 53 to 63 years of his age, taken inclusively, does not exceed more than in the proportion of 3 to 2 the likelihood or chance of dying in a given year or half-year during the aforesaid vigorous period of life: so that taking, for example, two persons of equal constitution, one aged 40 years, and the other 58 years, if these two persons made such a contract that in case the person of 58 years shall happen to die in less than six months, the one aged 40 were to inherit a sum of 2000 florins from the property of the defunct; but that if on the other hand the person aged 40 years should die in less than six months, the one aged 58 were to have 3000 florins from the property of the deceased; such a contract cannot be considered disadvantageous for the person who would have the 3000 florins, if the event were favourable to him, and who in the contrary event would only lose 2000 florins.
I then presuppose that the greatest likelihood of dying in a given year or half-year of the second series of the ten following years (that is from 63 years to 73, taken one with the other, rather than in a given year or half-year of the period of the vigour of life), cannot be estimated at more than double, or as 2 is to 1; and as the triple, or as 3 is to 1, during the seven following years, that is from 73 years to 80. Finally, in supposing that life necessarily ends at the twenty-seventh year after the expiration of 50 years of age above presupposed, this time is neither assumed at too high nor too low a standard, as experience manifestly teaches us that the life of some men exceeds by a considerable period the age of 80 years, the age of 81 years and even more.
These three articles being presupposed, we have, by a demonstrative calculation, mathematically discovered and proved that the redeemable annuity being fixed at 25 years' purchase as above, the life annuity should be sold at 16 years' purchase, and even higher, to be in equality one with the other; so that in the purchase of one florin of life annuity, on a young and vigorous nominee, more than 16 florins should be paid, as is proved by the following demonstration.
We do not propose to follow the learned author through his demonstrations. We have accompanied him thus far from a belief that many even popular readers will desire to see something more than a mere outline of such an historically important document; and for the further purpose of showing how early the then young theory of mathematical prob. began to be applied to demonstrate the science of life contingencies. Nor is it necessary that we should follow De Wit's illustrations in detail, for Mr. Hendriks has prepared a summary of the result, which is at once concise, and comprehensive of the author's meaning:
Firstly. Out of 128 lives, aged say 3 years, 1 is supposed to die in every half-year of the first 100 halfyears or 2 per annum for 50 years, leaving 28 alive, aged 53 years, at the end of the term; out of whom i dies in every 9 months, being o'66 per half-year, during the next 20 half-years, or 1'33 per annum for 10 years, leaving 15'66 alive, aged 63 years, at end of second term; of whom 1 dies in every year for 10 years, being o'5 per half-year during the next 20 half-years, leaving 5'66 alive, aged 73 years, at the end of the third term; of whom 1 dies in every year and a half for 7 years, being o'33 per half-year during the next 14 half-years, leaving 1 alive, aged 80, at the end of the fourth term; which survivor does not live over another half-year. Secondly. Out of the 128 lives, those who die in the respective half-years between the ages of 3 and 80, will receive an annuity certain in half-yearly instalments, for a term equal in continuance to the number of completed half-years elapsed between age 3 and the date of their death; therefore, the sum of the present values of half-yearly annuities certain, for the corresponding terms multiplied into the numbers dying within such respective terms, gives the present worth of all the annuities which will be enjoyed by the 128 lives, 1/128 of which represents the present value of the single life annuity at age of, say, 3 years.
This system of valuation Mr. Hendriks finds to be identical with the fifth method described by Tetens.-(Assu. Mag., No. 1, pp. 9 and 18; No. 2, p. 18.)
In the third place
We must make one or two further selections from the original document for the purpose of showing how some of the practical points incident to dealings in Life Contingencies were perceived from the very beginning. Thus, on the question of selection against the office: The person who for 16 florins has purchased 1 florin p.a. on a young, vigorous, and healthy life, has made a remarkably advantageous contract. I assert it to be remarkably advantageous for the following reasons: Because, in the first place, we have not been able to rate at a certain price, by perfect calculation or correct estimation, the power which the annuitant possesses (power which is of very great value to him), of choosing a life, or person in full health, and with a manifest likelihood of prolonged existence, upon whom to constitute or purchase his annuity; and there is much less risk or danger of a select, vigorous, and healthy life dying in the first half-year than in some of the following half years at the beginning of which the aforesaid life might perhaps prove to be in a weak state of health or even in a fatal illness; and such greater likelihood of prolongation of life in the purchase of an annuity upon a select, healthy, and robust life, may further extend itself to the second, third, and some other following terms or half-years. it is, however, certain, when we examine the matter very scrupulously, that the likelihood of the decease of the nominees upon whom life annuities are usually purchased is less considerable and smaller in the first years after the purchase than in the subsequent years, seeing that the said life annuities are oftenest purchased and sunk upon lives of young and healthy children of 3, 4, 5, 6, 7, 8, 9, and 10 years or thereabout. During that time and for some years ensuing, these young lives, having become more robust, are less subject to mortality than about 50 years afterwards, and than for some years anterior to these 50 years; and so much the more as during the first aforesaid years they either are not, or are but little, exposed to external accidents, and extraordinary causes of death, such as those from war, dangerous voyages, debauch or excess of drink, of the sex, and other dangers; for females there are also confinements, and other like causes; so that the first years after the purchase or foundation of the annuity are the least dangerous, which is a considerable advantage for the annuitant, particularly if we reflect, as I have above stated, that one of the said first years may, as regards the original price of purchase, balance a great number of the subsequent years. so that the annuitant or his heirs were to receive 46 more entire half-years of annuity after the expiration of the term of the aforesaid 77 years, this could not, however, increase the price of the life annuity by more than 14 stuyvers of the same capital; and even if the annuitant could be assured that his heirs were, after the expiration of the above 100 years, to enjoy the life annuity from half-year to half-year, and that perpetually the value of the capital at the time of first purchase would not thereby be increased by ten stuyvers.
Finally, and in the fourth place,
In consequence of all these reasons, we may assume it as established and demonstrated, that the value of a life annuity in proportion to the redeemable annuity at 25 years' purchase, is really not below, but certainly above, 16 years' purchase; so that a person wishing to purchase a life annuity in such proportion, and according to its real value, ought to pay more than 16 florins for 1 florin of annuity p.a.
Regarding the rate of int. employed in the calculation, he thus sagaciously observes: Besides the consideration that this calculation has been made on the principle of a deduction of 4 p.c. p.a. at compound interest, and this with such benefit to the purchaser of the life annuity that he would realize not only the interest p.a., but also without any intermission interest upon interest at 4 p.c. p.a., as though he could always thus advantageously make use of his money in purchase of annuity; it is constant that one could not always find such opportunity of investing it, and that one is sometimes obliged to let it lie fallow for some time, and often to lend it at a materially smaller interest, to provide against a greater loss.
Even besides this, as the capital of life annuities is not subject to taxation, nor to a reduction to a lower amount of annuity or interest, it follows, that if the blessing of the Almighty continue to be vouchsafed to this country, we may consider the life annuity as much more advantageous to the annuitant than the redeemable annuity. As may manifestly be judged by the example of foregoing times-by reflecting in fact that My Lords the States of Holland and West Friesland have in the course of a few years not only increased the charge for life annuities from 11 years' purchase to 12 years' purchase, and from 12 years' purchase to 14 years' purchase, but that these annuities have been sold, even in the present century, first at 6 years' purchase, then at 7 and at 8, and that the majority of all life annuities now current, and at the country's expense, were obtained at 9 years' purchase; which annuities by reason of the successive reductions of the rate of interest from 6 to 5 per cent., and then from 5 per cent. to 4 per cent., produce to the annuitants an actual profit of nearly one-half of such half-year's payment, and of more than one-half in the case of those annuities which were obtained at 8 years' purchase or under.
The Burgomaster of Holland and West Friesland in 1671, when the preceding report was made, was Herr J. Hudde, and he added a certificate to the report, which has caused some speculation: first, as to whether he understood the question sufficiently to impart any value to his certificate; and next, if he had any reservation as to "some fault of a figure" which does not appear upon the surface: for there was a fault of a figure in the addition. Both points have more recently been decided, by competent authorities, in Hudde's favour.
Certificate: I the undersigned declare, that having attentively read and examined, at the request of my Lord the Grand Pensionary of Holland and West Friesland, the above propositions, and the conclusions thence derived, for finding the value of a life annuity compared with that of a redeemable annuity at 4 p.c. p.a. I am of opinion that the method employed for that effect is perfectly discovered, and that the conclusion made therefrom (namely that the purchaser of a life annuity still makes a gain in stipulating for 1 florin of annuity p.a. for 16 florins of capital) depends upon solid and incontestable mathematical foundations, provided that some fault of a figure has not been involuntarily made in the calculation of the Table, or in the copy, and addition of the items, which is an ordinary and well-known computation, but which I have not made.-J. Hudde.
The researches of Mr. Hendriks have estab. two facts of importance: first: that De Wit and Hudde were jointly prosecuting these inquiries and extending the calculations to two joint lives, and even three, four, five, or more lives; and next, that their attention was directed to the subject in consequence of the then recent discoveries in the doctrine of probability by Pascal and Fermat. We have, however, the authority of Mr. Milne for
stating that De Wit's treatise was very little known in this country, and had "no sensible influence on the subsequent progress of the science." The reasons for this will appear in our notice of De Wit. Leibnitz appears to have been one of the first who drew the attention of Europe to this report of De Wit's; but it was Mr. Fredk. Hendriks who discovered the document itself.
In 1674 there was issued in Lond. a scheme of Tontine annuities, of which the following is an outline: "An advertisement and demonstration concerning the improvement of moneys to the great benefit and advantage of all persons of what nation, sex, age, degree, or quality soever, willing to advance any sum or sums, according to the method hereinafter mentioned, propounded to the Right Honorable the Lord Maior, Aldermen and Commons in Common Council assembled." The sum to be advanced was not less than £20 upon one life; but several advances might be made upon the same life. The prospectus further said: "This extraordinary gain being not only lawful, but very advantageous, there can be no other way proposed whereby, in laying out so small a sum as £20, there can be produced so great an increase as by survivorship will most certainly accrue to many persons, and especially to the longest liver of this rank." We, however, hear no more of the project.
In 1677 there was pub. by Michael Dary: Interest Epitomized, both Compound and Simple: "very useful for every one that lendeth or borroweth ; and for purchasing and selling annuities or pensions, and leases in reversion. Whereunto is added, a short appendix for the solution of adfected equations in numbers by approachment: performed by logarithms."
In 1681, a paper by Adam Martindale was published in the Phil. Trans., under the following title: Twelve Problems in Compound Interest and Annuities resolved.
The following, taken from a work on church and college leases, pub. at Cambridge in 1686, Tables for renewing of leases and purchasing lives [LEASES], gives a curious indication of the conjectural methods then commonly adopted in life valuations, and also confirms what we have already seen regarding the small values put upon lives:
The way of purchasing by lives was commonly to reckon one life as a lease of seven years, two lives as a lease of fourteen years; and three lives as a lease of twenty-one years. But this way seeming unequal, there is another way, which is more agreeable to reason; and it is this, viz., for every life to decrease one year; as, if one life be reckoned as a lease for ten years, then two will be as a lease for nineteen, and three as a lease for twenty-seven years, etc. So if you reckon one life as a lease of nine years, then two will be a lease of seventeen, three as a lease for twenty-four, etc. So if one single life be reckoned as a lease of twelve years, then two will be as a lease of 23, three as a lease of 33 years, etc.
After some further exemplifications, there follows a Table which (as Mr. Farren has already pointed out) appears to constitute the first species of life annuity table offered for public use in Gt. Brit. The following is the Table, and the necessary explanation thereto : A Table for the purchasing of Lives.
The repetition of 7 p.c. (2nd division of Table) may on first perusal appear merely a misprint, but on considering the fanciful construction of the Table, the cause of the duplicature will become apparent. In the first division one life is assumed as equal in duration to 10 years, in the second, 9 years are assigned to it: consequently the two divisions, though in connexion with similar rates of int., represent different values. It seems scarcely possible to afford to the modern reader a more obvious example of the inconsistencies of the methods formerly prevailing.-Farren.
The authorship of this work was very generally and for some years attributed to Sir Isaac Newton, no doubt from the fact of his having given a certificate of his belief in the accuracy of its Tables. It is now known to have been written by one Mabbot, of King's College, Cambridge.
The most unpractised reader will at once perceive that until distinctions of age were
recognized as influencing the relative annuity values, all life estimates were too imaginary and inconsistent to be even worthy of the name of calculated results.-Farren.
In 1692 the first attempt was made by the English Government to raise money by means of life annuities; and hence there arises the first mention of life annuities in the English Statute book. This was by 4 Wm. & Mary, c. 3, known as the "Million Act." Its object was to raise one million sterling, "to carry on the war against France," by means of tontine annuities, for the int. upon which £100,000 p.a. was to be set apart until A.D. 1700, and then £70,000, but in the event of the entire million not being so subscribed by a given date, those who did sub. were to have, in lieu of the tontine advantages, an annuity of 14 in respect of every £100 subs., during the remainder of their own or their nominee's lives. There were no provisions or restrictions as to age. Even on these terms however, only £881,493 12s. 2d. could be raised.
Great efforts were made to float this loan, and there was pub., apparently by authority, the following Table predicting the course of death of the entire 10,000 nominees. A copy of the table still exists in the British Museum; and we think it sufficiently remarkable to be given here entire, with the following explanation, which, in the original, is given at the foot of the Table :
This Table designed for the encouragement of contributors to advance monies upon the funds and terms expressed in the Act of Parl. newly passed, for granting to their Majesties certain rates and duties of excise upon Beer, Ale, etc., and being applicable to the first proposal therein mentioned, upon terms of yearly payments during life, with advantage of survivorship, was calculated upon this supposition, that the whole sum of Ten Hundred Thousand Pounds may be advanced, and consequently as many nominees as there are shares or £100 advanced, viz. 10,000. The following example will show the use of the table. Look in the left-hand column for the number of years desired. Suppose 22; you will find in the columns answerable to it that at the end of 22 years, there are 5034 nominees dead, and 4966 living, and that each contributor (whose nominee is then alive) will receive that year for interest £14 1s. for each share or £100 by him advanced.
A table shewing how many out of 10,000 Lives or Nominees will be likely to die in any number of years proposed during the term of 99 years, and the yearly interest due, etc.