Thus it appears, that the whole People of Breslaw does consist of 34000 Souls, being the Sum Total of the Persons of all Ages in the Table: The first use hereof is to shew the Proportion of Men able to bear Arms in any Multitude, which are those between 18 and 56, rather than 16 and 60; the one being generally too weak to bear the Fatigues of War, and the Weight of Arms; and the other too crasie and infirm from Age, notwithstanding particular Instances to the contrary. Under 18 from the Table, are found in this City 11997 Persons, 3950 above 56, which together make 15947, so that the Residue to 34000 being 18053, are Persons between those Ages. At least one half thereof are Males, or 9027: So that the whole Force this City can raise of Fencible Men, as the Scotch call them, is about 9000, or 9⁄34, or somewhat more than a quarter of the Number of Souls; which may parhaps pass for a Rule for all other places.
The Second Use of this Table, is, to shew the differing degrees of Mortality, or rather Vitality, in all Ages; for if the Number of Persons of any Age remaining after one Year, be divided by the difference between that and the number of the Age proposed, it shews the Odds that there is, that a Person of that Age does not die in a Year. As for Instance, a Person of 25 Years of Age has the Odds of 560 to 7, or 80 to 1, that he does not die in a Year: Because that of 567, living of 25 Years of Age, there do die no more than 7 in a Year, leaving 560 of 26 Years old.
So likewise for the Odds, that any Person does not die before he attain any proposed Age: Take the number of the remaining Persons of the Age proposed, and divide it by the difference between it and the number of those of the Age of the Party proposed; and that shews the Odds there is between the Chances of the Party's living or dying. As for Instance; What is the Odds that a Man of 40 lives 7 Years: Take the number of Persons of 47 Years, which in the Table is 377, and subtract it from the number of Persons of 40 Years, which is 445, and the difference is 68: Which shews that the Persons dying in that 7 Years, are 68, and that it is 377 to 68, or 5½ to 1, that a Man of 40 does live 7 Years. And the like for any other number of Years.
Use III. But if it be enquired at what number of Years, it is an even Lay that a Person of any Age shall die, this Table readily performs it; For if the number of Persons living of the Age proposed, be halfed, it will be found by the Table at what Year the said Number is reduced to half by Mortality; and that is the Age, to which it is an even Wager, that a Person of the Age proposed shall arrive before he die. As for Instance; A Person of 30 Years of Age is proposed, the number of that Age is 531, the half thereof is 265, which number I find to be between 57 and 58 Years; so that a Man of 30 may reasonably expect to live between 27 and 28 Years.
Use IV. By what has been said, the Price of Insurance upon Lives ought to be regulated, and the difference is discovered between the Price of insuring the Life of a Man of 20 and 50. For Example; It being 100 to 1, that a Man of 20 dies not in a Year, and but 38 to 1, for a Man of 50 Years of Age.
Use V. On this depends the Valuation of Annuities upon Lives; for it is plain, that the Purchaser ought to pay for only such a part of the Value of the Annuity, as he has Chances that he is living; and this ought to be computed yearly, and the Sum of all those yearly Values being added together, will amount to the Value of the Annuity for the Life of the Person proposed. Now the present Value of Money payable after a Term of Years, at any given Rate of Interest, either may be had from Tables already computed; or almost as compendiously, by the Table of Logarithms: For the Arithmetical Complement of the Logarithm of Unity, and its yearly Interest, (that is, of 1,06 for Six per Cent. being 9,974694.) being multiplied by the number of Years proposed, gives the present Value of One Pound payable after the end of so many Years. Then by the foregoing Proposition, it will be as the number of Persons living after that Term of Years, to the number dead; so are the Odds that any one Person is alive or dead. And by consequence, as the Sum of both, or the number of Persons living of the Age first proposed, to the number remaining after so many Years, (both given by the Table) so the present Value of the yearly Sum payable after the Term proposed, to the Sum which ought to be paid for the Chance the Person has to enjoy such an Annuity after so many Years. And this being repeated for every Year of the Person's Life, the Sum of all the present Values of those Chances is the true Value of the Annuity. This will without doubt appear to be a most laborious Calculation; but it being one of the principal Uses of this Speculation, and having found some Compendia for the Work, I took the pains to compute the following Table, being the short Result of a not ordinary number of Arithmetical Operations: It shews the Value of Annuities for every Fifth Year of Age, to the Seventieth, as follows.
| Age. | Years Pur. | Age. | Years Pur. | Age. | Years Pur. |
|---|---|---|---|---|---|
| 1 | 10,28 | 25 | 12,27 | 50 | 9,21 |
| 5 | 13,40 | 30 | 11,72 | 55 | 8,51 |
| 10 | 13,44 | 35 | 11,12 | 60 | 7,60 |
| 15 | 13,33 | 40 | 10,57 | 65 | 6,54 |
| 20 | 12,78 | 45 | 9,91 | 70 | 5,32 |
This shews the great Advantage of putting Money into the present Fund lately granted to Their Majesties, giving 14 per Cent. per Annum, or at the Rate of 7 Years Purchase for a Life; when young Lives, at the usual Rate of Interest, are worth above 13 Years Purchase. It shews likewise the Advantage of young Lives over those in Years; a Life of Ten Years being almost worth 13½ Years Purchase, whereas one of 36 is worth but 11.
Use VI. Two Lives are likewise valuable by the same Rule; for the number of Chances of each single Life, found in the Table, being multiplied together, become the Chances of the Two Lives. And after any certain Term of Years, the Product of the two remaining Sums is the Chances that both the Persons are living. The Product of the two Differences, being the numbers of the Dead of both Ages, are the Chances that both the Persons are dead. And the two Products of the remaining Sums of the one Age multiplied by those dead of the other, shew the Chances that there are, that each Party survives the other: Whence is derived the Rule to estimate the Value of the Remainder of one Life after another. Now as the Product of the Two Numbers in the Table for the Two Ages proposed, is to the difference between that Product, and the Product of the two numbers of Persons deceased in any space of time; so is the Value of a Sum of Money to be paid after so much time, to the Value thereof under the Contingency of Mortality. And as the aforesaid Product of the two Numbers answering to the Ages proposed, to the Product of the Deceased of one Age multiplied by those remaining alive of the other; so the Value of a Sum of Money to be paid after any time proposed, to the Value of the Chances, that the one Party has that he survives the other, whose number of Deceased you made use of, in the second Term of the Proportion. This perhaps may be better understood, by putting N for the number of the younger Age, and n for that of the Elder; Y, y the Deceased of both Ages respectively, and R, r for the Remainders; and R + Y = N, and r + y = n. Then shall Nn be the whole Number of Chances; Nn - Yy be the Chances that one of the two Persons is living, Yy the Chances that they are both dead; Ry the Chances that the elder Person is dead, and the younger living; and rY the Chances, that the elder is living, and the younger dead. Thus two Persons of 18 and 35 are proposed, and after 8 Years these Chances are required. The Numbers for 18 and 35, are 610 and 490; and there are 50 of the First Age dead in 8 Years, and 73 of the Elder Age. There are in all 610 × 490, or 298900 Chances; of these there are 50 × 73, or 3650, that they are both dead. And as 298900, to 298900 - 3650, or 295250: So is the present Value of a Sum of Money to be paid after 8 Years, to the present Value of a Sum to be paid, if either of the two live. And as 560 × 73, so are the Chances that the Elder is dead, leaving the Younger; and as 417 × 50, so are the Chances that the Younger is dead, leaving the Elder. Wherefore as 610 × 490 to 560 × 73, so is the present Value of a Sum to be paid at 8 Years end, to the Sum to be paid for the Chance of the Younger's Survivance; and as 610 × 490 to 417 × 50, so is the same present Value to the Sum to be paid for the Chance of the Elder's Survivance.
This possibly may be yet better explained, by expounding these Products by Rectangular Parallelograms, as in Fig. 7. wherein AB or CD represents the number of Persons of the younger Age, and DE, BH those remaining alive after a certain Term of Years; whence CE will answer the number of those dead in that time: So AC, BD may represent the number of the elder Age; AF, BI the Survivors after the same Term; and CF, DI, those of that Age that are dead at that time. Then shall the whole Parallelogram ABCD be Nn, or the Product of the two Numbers of Persons, representing such a number of Persons of the two Ages given; and by what was said before, after the Term proposed, the Rectangle HD shall be as the number of Persons of the younger Age that survive, and the Rectangle AE as the number of those that die. So likewise the Rectangles AI, FD shall be as the Numbers, living and dead, of the other Age. Hence the Rectangle HI shall be as an equal number of both Ages surviving. The Rectangle FE being the Product of the Deceased, or Yy, an equal number of both dead. The Rectangle GD or Ry, a number living of the younger Age, and dead of the elder: And the Rectangle AG or rY a number living of the elder Age, but dead of the younger. This being understood, it is obvious, that as the whole Rectangle AD or Nn is to the Gnomon FABDEG or Nn - Yy, so is the whole number of Persons or Chances, to the number of Chances that one of the two Persons is living: And as AD or Nn is to FE or Yy, so are all the Chances, to the Chances that both are dead; whereby may be computed the Value of the Reversion after both Lives. And as AD to GD or Ry, so the whole number of Chances, to the Chances that the younger is living, and the other dead; whereby may be cast up what Value ought to be paid for the Reversion of one Life after another, as in the Case of providing for Clergy-men's Widows, and others, by such Reversions. And as AD to AG, or rY, so are all the Chances, to those that the elder survives the younger. I have been the more particular, and perhaps tedious, in this Matter, because it is the Key to the Case of Three Lives, which of it self would not have been so easie to comprehend.