ANNUITIES.

History.
ANNUITIES have been treated of in the body of the work, but in such a manner as to render it necessary, that so useful a branch of knowledge should form the subject of an entirely new article in this place.

The doctrine of Compound Interest and Annuities—certain, is too simple ever to have occupied much of the attention of Mathematicians: inquiries into the values of interests dependent upon the continuance or the failure of human life, being more interesting and difficult, have occupied them more, but yet not so much as their importance would seem to demand; the discoveries, both in pure Mathematics and Physics, especially those of Newton, which distinguished the close of the seventeenth century, having provided them with ample employment of a more interesting kind, ever since the subjects of this article were submitted to calculation.

Fermat, Pascal, and Huygens, by laying the foundation of the doctrine of Probabilities about the middle of that century, first opened the way to the solution of problems of this kind. The earliest mathematical publication on Probabilities, the little tract of Huygens, De ratiociniis in ludo aleæ, appeared in 1658; and in 1671, his celebrated countryman, John De Witt, published a treatise on Life-annuities, in Dutch. (Montucla, Hist. des Math. Tom. III. p. 407.) This, however, appears to have been very little known, or read, and to have had no sensible influence on the subsequent progress of the science; the origin of which may be properly dated from the publication of Dr Halley's paper on the subject, in the Philosophical Transactions for the year 1693 (No. 196.) That celebrated Mathematician there first gave a table of mortality, which he had constructed from observations made at Breslaw, and showed how the probabilities of life and death, and the values of annuities and assurances on lives, might be determined by such tables; which, he informs us, had, till then, been only done by an imaginary valuation. Besides his algebraical reasonings, he illustrated the subject by the properties of parallelograms, and parallelopipedons: there are, perhaps, no other mathematical inquiries, in the prosecution of which, algebra is entitled to so decided a preference to the elementary geometry as in these, and this example of the application of geometry has not been followed by any of the succeeding writers.

VOL. I. PART II.

In the year 1724, Mr De Moivre published the first edition of his tract, entitled, Annuities on Lives. In order to shorten the calculation of the values of such annuities, he assumed the annual decrements of life to be equal; that is, that out of a given number of persons living at any age, an equal number dies every year until they are all extinct; and, upon that hypothesis, he gave a general theorem, by which the values of annuities on single lives might be easily determined: this approximation, when the utmost limit of life was supposed to be 86 years, agreed very well with the true values between 30 and 70 years of age, as deduced from Dr Halley's table, and the method was of great use at the time; as no tables of the true values of annuities had then been calculated, except a very contracted one inserted by Dr Halley in the paper mentioned above. But, upon the whole, this hypothesis of De Moivre has probably contributed to retard the progress of the science, by turning the attention of Mathematicians from the investigation of the true law of mortality, and the best methods of constructing tables of the real values of annuities.

The same distinguished Analyst also endeavoured to approximate the values of joint lives; but it has since been found, that the formulae he gave for that purpose are too incorrect for use. Mr Thomas Simpson published his Doctrine of Annuities and Reversions in the year 1742, in which the subject is treated in a manner much more general and perspicuous than it had been previously; his formulae are adapted to any table of mortality, and, in the 7th corollary to his first problem, he gave the theorem demonstrated in the 149th number of this article, to which we owe all the best tables of the values of life-annuities that have since been published.

In the same work, he also gave a table of mortality deduced from the London observations, and four others calculated from it, of the values of annuities on lives, each at three rates of interest; the first for single lives, the three others for two and three equal joint lives, and for the longest of two or of three lives.

These were the first tables of the values of joint lives that had been calculated; for although Dr Halley had shown, half a century before, how such tables might be computed, and had taken considerable pains to facilitate the work; the necessary calculations, by the known methods, previous to the publication of Mr Simpson's Treatise, were so very laborious.

rious, that no one had had the courage to undertake them. And, unfortunately, the mortality, according to the London table, was so much above the common average, that the values of annuities in Mr Simpson's tables were much too small for general use.

In the year 1746, M. Deparcieux published his Essai sur les probabilités de la durée de la vie humaine, in which he gave several valuable tables of mortality deduced from the mortuary registers of different religious houses, and from the lists of the Nominees in the French Tontines; also a table of the values of annuities on single lives, at three rates of interest, calculated from his table of mortality for the tontine annuitants. These tables were a great acquisition to the science, as, before their publication, there were only two extant that gave tolerably exact representations of the true law of mortality;—Dr Halley's for Breslaw, and one constructed but a short time before by M. Kerseboom, principally from registers of Dutch annuitants. Those of M. Deparcieux for the Monks and Nuns, were the first ever constructed for the two sexes separately; and by them, the greater longevity of females was made evident.

The work commences with an algebraical theory of Annuities-certain; but the principal essay, On the Probabilities of the Duration of Human Life, is perfectly intelligible to those who have not studied Mathematics; it is written with great judgment and perspicuity, but contains very little more than the explanation of the construction of his tables, some of which relate to Tontines; and he did not avail himself to the extent he might have done, of the excellent tract of Thomas Simpson.

This work, however, appears to have been more read upon the Continent, and to have contributed more to the diffusion of this kind of information there, than all the other writings on the subject. The article Rentes viagères in the Encyclopédie, is acknowledged to have been taken entirely from it, as was also the article Vie, durée de la; and these are proofs, among many others that might be produced, how little M. D'Alembert and the principal Mathematicians, his contemporaries, attended to the subject.

In the year 1752, Mr Simpson published, in his Select Exercises, a supplement to his Doctrine of Annuities; wherein he gave new tables of the values of annuities on two joint lives, and on the survivor of two lives, much more copious than those he had inserted in the principal work; but these also were calculated from his London table of mortality.

The celebrated Euler, in a paper inserted in the Memoirs of the Royal Academy of Sciences at Berlin for the year 1760, gave a formula by which the value of an annuity on a single life of any age, may be derived from that of an annuity on a life one year older; which formula was included in that given by Mr Simpson eighteen years before, for effecting the same purpose in the case of any number of joint lives; and by this compendious method, M. Euler calculated a table of the values of single lives from M. Kerseboom's table of mortality.

The first edition of Dr Price's Observations on Re-

versionary Payments was published in 1769; and its chief object was, to give information to persons desirous of forming themselves into societies for the purpose of making provision for themselves in old age, or for their widows. When tables of the values of single lives, and of two joint lives are given, the methods of determining the terms on which such provisions can be made with safety to all the parties concerned, are very simple, and were, at that time, well understood in theory, by the Mathematicians who had studied the subject; but, for want of the requisite tables, the algebraical formulæ had, till then, been of little practical utility.

In the prosecution of this laudable design, Dr Price was obliged to have recourse to approximations. He informs us, that by following M. De Moivre too implicitly in his rules for determining the value of two joint lives, he was led into difficulties which convinced him that they were not only useless, but dangerous; he therefore calculated a table of these values upon M. De Moivre's hypothesis of the decrements of life being equal, and its utmost limit 86 years, from a correct formula given by Mr Simpson in his Doctrine of Annuities (Cor. 5. Prob. 1.); by this, and a table of the values of single lives, calculated by Mr Dodson on M. De Moivre's hypothesis, he was enabled to give answers tolerably near the truth, to some of the most interesting questions of this kind, and to show that the plans of several of the societies then recently established, were quite inadequate; and instead of the benefits they promised, could only, in the end, produce disappointment and distress, unless they either dissolved or reformed themselves.

The work also contained instructive dissertations on the probabilities and expectations of life, and on the mean duration of marriage and of widowhood; besides accounts of some of the principal societies which had then been formed for the benefit of old age, and of widows; with observations on the method of forming tables of mortality for towns, and two new tables of that kind, constructed from registers kept at Norwich and Northampton. Mr Morgan's Doctrine of Annuities and Assurances was published in 1779, containing tables of the values of single lives, of two equal joint lives, and of two lives differing in age by 60 years, calculated from the Northampton table of mortality. And in the same year, M. De Saint-Cyrán published his Calcul des Rentes viagères sur une et sur plusieurs têtes, wherein the valuation of annuities on lives is treated algebraically, but in a manner much inferior in all respects to that of Mr Simpson; and six tables are given of the values of annuities,—on single lives, on the survivor of two lives, and on the last survivor of three, calculated from M. Kerseboom's table of mortality. Although the values in the cases of two, and of three lives, were only determined by approximation, these tables were, just then, a valuable acquisition to the science; but their use was entirely superseded only four years after, by the publication of others much more valuable.

The fourth edition of Dr Price's Observations on Reversionary Payments appeared in 1783. One of the best effects of the preceding editions on the pro-

History. Progress of the science, had been, to direct the public attention to these inquiries, by showing their important uses in the affairs of life; and to procure the requisite data for forming tables of mortality, that should illustrate the laws according to which human life wastes under different circumstances, by exciting the curiosity of intelligent men who had the necessary leisure and means of information. The ingenious author had, accordingly, been furnished with the necessary abstracts of mortuary registers which had been kept with these views, by Dr Haygarth at Chester, Dr Aikin at Warrington, and the Rev. Mr Gorsuch at Holy-Cross, near Shrewsbury, since the publication of the first edition; also by Mr Wargentin, with the mean numbers both of the living, and the annual deaths in all Sweden and Finland, for twenty-one successive years, in all of which the sexes were distinguished; and from these data, he constructed tables of mortality that threw great light on the subject. He also inserted in this edition, an improved table of mortality for Northampton; and, what had been so long wanted, a complete set of tables of the values of annuities on single lives, at six rates of interest, and on two joint lives at four, all calculated from the new Northampton table. The combinations of joint lives were sufficiently numerous to admit of all the values not included being easily interpolated. Besides these, he also gave tables of the values of annuities on single lives from the Swedish observations, both with and without distinction of the sexes, and on two joint lives without that distinction.

The values given in these tables are too low for the general average of lives, at all ages under 60; but in the treatise of Mr Baron Maseres on the Principles of the Doctrine of Life-Annuities, which was published in the same year (1783), others were given, calculated from the table of mortality which M. Deparcieux constructed from the lists of the Normines in the French Tontines. The tables for single lives are calculated at twelve different rates of interest from 2 to 10 per cent.; but those for joint lives, only at $3\backslash frac\{1\}\{2\}$ and $4\backslash frac\{1\}\{2\}$ per cent.; and the combinations they include are only those of ages that are equal, or that differ by 5 or 10 years, and the multiples of 10.

There is reason to believe that the values in these tables, at all ages under 75 or 80 years, are nearer the truth, for the average of this country, than any others then extant; but certainly for the average of lives on which annuities and reversions depend. After that period of life, however, they are too small; and, in most cases, it is difficult to derive the values of joint lives from them with sufficient accuracy, on account of the contracted scale they have been calculated upon.

It was not Dr Price's object to deliver the elements of the science systematically; but he treated most parts of it with great judgment, enriched it with a vast collection of valuable facts and observations, and corrected several errors into which some of the most eminent writers upon it had fallen. The mathematical demonstrations (which are given in the notes) are much inferior to the rest of the work.

History. The values of reversionary sums and annuities, which depend upon some of the lives involved failing according to assigned orders of precedence, had been approximated by Mr Simpson in his Select Exercises, and by Mr Morgan in his Doctrine of Annuities; but the latter gentleman first gave accurate solutions of problems of this kind, in the Philosophical Transactions for the years 1788, 1789, 1791, 1794, and 1800.

Except by the solution of these problems, the science had not been materially advanced, during a period of more than thirty years that had elapsed since the appearance of the fourth edition of Dr Price's work, when Mr Milne published his Treatise on the Valuation of Annuities and Assurances, on Lives and Survivorships, in the beginning of last year (1815).

The work consists of two volumes; the first is mathematical, the second entirely popular, except the notes, and a few of the tables. The algebraical part of this article is merely a short abstract of the first volume, and may serve as a specimen of the manner in which the subject has been treated there; but the construction of tables of mortality, which forms the subject of the third chapter, has not been noticed here; neither is the valuation of reversionary sums or annuities depending upon assigned orders of survivorship, treated in the present article; and these are parts of the work, which will not be found the least interesting to mathematicians.

The second volume contains upwards of fifty new tables, with a few others that had been published before, but have been reprinted either on account of their value, or scarcity, or both. Four of the new ones are tables of mortality constructed by the author, from registers kept at Carlisle and Montpellier, and in all Sweden and Finland, since the period of the observations Dr Price made use of; the sexes are distinguished in the tables for Sweden and Montpellier, but not in that for Carlisle. This last is the only table, besides those for Sweden and Finland, that has been formed from the necessary data,—enumerations of the living, as well as registers of the deaths in every interval of age.

Twenty-one of these tables, being the seventeenth to the thirty-seventh inclusive, in the collection at the end of the work, render it easy to apply the algebraical formulae to practical purposes, and numerous examples of such applications are given. They have all been calculated from the Carlisle table of mortality; those of the values of life-annuities on the same extensive scale, with those which Dr Price derived from the Northampton table. It is the author's opinion that the values of interests dependent upon the continuance or the failure of life, may be derived from them more correctly than from any others extant, and he has taken considerable pains to assist his readers in judging of this for themselves.

Besides the tables, the principal contents of the second volume, are explanations of their construction and uses; many of them relate to the progress of population,—the comparative mortality of different diseases—of different seasons,—and of the two sexes at every age—the proportion of the sexes at birth—and that of the born alive to the still-born of each sex.

It will be found that the author has been furnished with facts and observations of great value, and that he has endeavoured to present the information they afford, in the forms best calculated for the further prosecution of these inquiries.

In treating of annuities, we think that it may be useful in a work of this kind, to address ourselves as well to those readers who have not, as to those who have, an acquaintance with Algebra; and we shall, accordingly, divide what follows into two Parts, corresponding to these two views of the subject.

We shall, in this Part, demonstrate all that is most useful and important in the doctrine of Annuities and Assurances on lives, without using Algebra, or introducing the idea of probability; but the reader is, of course, supposed to understand common Arithmetic. In the first 30 numbers of this Part, Compound Interest and Annuities-certain are treated of; from the 31st to the 76th, the doctrine of Annuities on Lives is delivered; and that of Assurances on Lives, from thence to the 108th, where the popular view terminates.

What is demonstrated in this Part, will be sufficient to give the reader clear and scientific views of the subjects treated; and, with the assistance of the necessary tables, will enable him to solve the more common and simple problems respecting the values of Annuities and Assurances. He will also understand clearly the general principles on which problems of greater difficulty are resolved; but these he cannot undertake with propriety, when the object is, to make a fair valuation of any claims or interests, with a view to an equitable distribution of property, unless he has studied the subject carefully, with the assistance of Algebra; for intricate problems of this kind can hardly be solved without it; and those who are not much exercised in such inquiries, often think they have arrived at a complete solution, when they have overlooked some circumstance or event, or some possible combination of events or circumstances, which materially affect the value sought. Eminent Mathematicians have, in this way, fallen into considerable errors, and it can hardly be doubted, that those who are not mathematicians, must (ceteris paribus) be more liable to them.

No. 1. When the rate is 5 per cent., L.1 improved at simple interest during one year, will amount to L.1.05; which, improved in the same manner during the second year, will be augmented in the same ratio of 1 to 1.05; the amount then, will therefore be $1.05\; \backslash times\; 1.05$, or $(1.05)^2\; =\; 1.1025$.

In the same manner it appears, that this last amount, improved at interest during the third year, will be increased to $(1.05)^3\; =\; 1.157625$; at the end of the fourth year, it will be $(1.05)^4$; at the end of the fifth $(1.05)^5$, and so on; the amount at the end of any number of years being always determined, by raising the number which expresses the amount at the end of the first year, to the power of which the exponent is the number of years. So that when the rate of interest is 5 per cent., L.1 improved at com-

pound interest, will, in seven years, amount to $(1.05)^7$, and in 21 years, to $(1.05)^\{21\}$.

But if the rate of interest were only 3 per cent., these amounts would only be $(1.03)^7$, and $(1.03)^\{21\}$ respectively.

2. The present value of L.1 to be received certainly at the end of any assigned term, is such a less sum, as, being improved at compound interest during the term, will just amount to one pound. It must therefore be less than L.1, in the same ratio as L.1 is less than its amount in that time; but in three years, at 5 per cent., L.1 will amount to $L.(1.05)^3$.

(1). And $(1.05)^3\; :\; 1\; ::\; 1\; :\; \backslash frac\{1\}\{(1.05)^3\}$, so that $\backslash frac\{1\}\{(1.05)^3\}\; =\; \backslash frac\{1\}\{1.157625\}\; =\; 0.863838$ is the present value of L.1 to be received at the expiration of three years.

In the same manner it appears that, at 4 per cent. interest, the present value of L.1 to be received at the end of a year, is $\backslash frac\{1\}\{1.04\}\; =\; 0.961538$; and if it were not to be received until the expiration of 21 years, its present value would be $\backslash frac\{1\}\{(1.04)^\{21\}\}\; =\; (0.961538)^\{21\}\; =\; 0.438834$.

Hence it appears, that if unity be divided by the amount of L.1, improved at compound interest during any number of years, the quotient will be the present value of L.1 to be received at the expiration of the term: which may also be obtained by raising the number which expresses the present value of L.1 receivable at the expiration of a year, to the power of which the exponent is the number of years in the term.

3. When a certain sum of money is receivable annually, it is called an ANNUITY, and its quantum is expressed by saying it is an annuity of so much; thus, according as the annual payment is L.1, L.10, or L.100; it is called an annuity of L.1, of L.10, or of L.100.

4. When the annual payment does not depend upon any contingent event, but is to be made certainly, either in perpetuity or during an assigned term, it is called an Annuity-certain.

5. In calculating the value of an annuity, the first payment is always considered to be made at the end of the first year from the time of the valuation, unless the contrary be expressly stated.

6. The whole number, and part or parts of one annual payment of an annuity, which all the future payments are worth in present money, is called the number of years purchase the annuity is worth; and,

being the sum of the present values of all the future payments, is also the sum which, being put out and improved at compound interest, will just suffice for the payment of the annuity (2).

7. Hence it follows, that when the annuity is L. 1, the number of years purchase and parts of a year, is the same as the number of pounds and parts of a pound in its present value.

And throughout this article, whenever the quantum of an annuity is not mentioned, it is to be understood to be L. 1.

8. The sum of which the simple interest for one year is L. 1, is evidently that which, being put out at interest, will just suffice for the payment of L. 1 at the end of every year, without any augmentation or diminution of the principal; and, being equivalent to the title to L. 1 per annum for ever, is called the value of the perpetuity, or the number of years purchase the perpetuity is worth.

But, while the rate remains the same, the annual interests produced by any two sums, are to each other as the principals which produce them; therefore, since $5\; :\; 1\; ::\; 100\; :\; \backslash frac\{100\}\{5\}\; =\; 20$, when the rate of interest is 5 per cent., the value of the perpetuity is 20 years purchase. In the same manner it appears, that according as the rate may be 3 or 6 per cent. the value of the perpetuity will be $\backslash frac\{100\}\{3\}\; =\; 33\backslash frac\{1\}\{3\}$, or $\backslash frac\{100\}\{6\}\; =\; 16\backslash frac\{2\}\{3\}$ years purchase; and may be found in every case, by dividing any sum by its interest for a year.

9. All the most common and useful questions in the doctrines of compound interest and annuities, certain, may be easily resolved by means of the first four tables at the end of this article. Their construction may be explained by the following specimen, rate of interest 5 per cent.

10. The calculation must begin with Table III., the first number in which should evidently be 1.05, the amount of L. 1 improved at interest during one year; which, being multiplied by 1.05, the product is 1.025, the second number; this second number being multiplied by 1.05, the product is 1.157625, the amount at the end of three years. And so the calculation proceeds throughout the whole of the column;

each number after the first, being the product of the multiplication of the preceding number, by the amount of L. 1 in a year (1).

11. The number against any year in Table I. is found by dividing unity by the number against the same year in Table III. (2); thus, the number against the term of six years in Table I. is $\backslash frac\{1\}\{1.340096\}\; =\; .746215$. All the numbers in that table after the first, may also be found, by multiplying that first number continually into itself (2).

12. The number against any year in Table II. being the sum of the numbers against that and all the preceding years in Table I.; is found by adding the number against that year in Table I. to the number against the preceding year in Table II.; thus, the number against 4 years in Table II., being

the sum of 0.822702
and 2.723248
is 3.545950.

13. If each payment of an annuity of L. 1 be put out as it becomes due, and improved at compound interest during the remainder of the term, it is evident that at the expiration of the term, the payment then due will be but L. 1, having received no improvement at interest. That received one year before will be augmented to the amount of L. 1 in a year; that received two years before will be augmented to the amount of L. 1 in two years; that received three years before to the amount of L. 1 in three years, and so on until the first payment, which will be augmented to the amount of L. 1 in a term one year less than that of the annuity.

Hence, it is manifest, that the number against any year in Table IV. will be unity added to the sum of all those against the preceding years in Table III.

And, therefore, that the number against any year in Table IV. is the sum of those in Tables III. and IV. against the next preceding year.

Thus, the number against seven years in Table IV., being

the sum of 1.340096
and 6.801913
is 8.142009.

14. The method of construction is obviously the same at any other rate of interest.

15. All the amounts and values which are the objects of this inquiry, evidently depend upon the improvement of money at compound interest; it is, therefore, that the first, second, and fourth tables, all depend upon the third.

But every pound, and every part of a pound, when put out at interest, is improved in the same manner as any single pound considered separately. Whence, it is obvious, that while the term and the rate of interest remain the same, both the amount and the present value, either of any sum, or of any annuity, will be the same multiple, and part or parts of the amount or the present value found against the same term, and under the same rate of interest in these tables, as the sum or the annuity proposed is of L. 1.

So that to find the amount or the present value of

any sum or annuity for a given term and rate of interest, we have only to multiply the corresponding tabular value by the sum or the annuity proposed; the product will be the amount or the value sought, according as the case may be.

16. Example 1. To what sum will L. 100 amount, when improved at compound interest during 20 years? the rate of interest being 4 per cent. per annum.

By Table III., it appears, that L. 1 so improved, would, at the expiration of the term, amount to L. 2191123, therefore L. 100 would amount to 100 times as much, that is, to L. 2191123, or L. 219, 2s. 3d.

17. Ex. 2. What is the present value of L. 400, which is not to be received until the expiration of 14 years, when the rate of interest is 5 per cent.?

The present value of L. 1 to be received then, will be found by Table I. to be L. 0.505068: L. 400 to be received at the same time, will therefore be worth, in present money, 400 times as much, or L. 202.0272, that is, L. 202, 0s. 6$\backslash frac\{1\}\{2\}$d.

18. Ex. 3. Required the present value of an annuity of L. 50 for 21 years, when the rate of interest is 5 per cent.

Table II. shows the value of an annuity of L. 1 for the same term to be L. 12.8212; the required value must therefore be 50 times as much, or L. 641.06, that is, L. 641, 1s. 2$\backslash frac\{1\}\{2\}$d.

19. Ex. 4. What will an annuity of L. 10, 10s. or L. 10.5, for thirty years, amount to, when each payment is put out as it becomes due, and improved at compound interest until the end of the term? The rate of interest being 4 per cent.

The amount of an annuity of L. 1 so improved, would be L. 56.084938, as appears by Table IV., the amount required will therefore be 10.5 times this, or L. 588.89185, that is, L. 588, 17s. 10d.

20. When the interval between the time of the purchase of an annuity and the first payment thereof, exceeds that which is interposed between each two immediately successive payments; such annuity is said to be deferred for a time equal to that excess, and to be entered upon at the expiration of that time.

21. If two persons, A and B, purchase an annuity between them, which A is to enter upon immediately, and to enjoy during a certain part of the term, and B, or his heirs, or assigns, for the remainder of it; the present value of B's interest will evidently be, the excess of the value of the annuity for the whole of the term from this time, above the value of the interest of A.

So that when the entrance on an annuity is deferred for a certain term, its present value will be the excess of the value of the annuity for the term of delay and continuance together, above the value of an equal annuity for the term of delay only.

22. Example 1. Required the value of a perpetual annuity of L. 120, which is not to be entered upon until the expiration of 14 years from this time, reckoning interest at 3 per cent.

The perpetuity, with immediate possession, would be worth 33$\backslash frac\{1\}\{3\}$ years' purchase (8); and an annuity for the term of delay is worth 11.2961 (Table II.)

From 33.3333
subtract 11.2961, and multiply

the remainder 22.0372
by 120

the product 2644.464 = L. 2644, 9s. 8$\backslash frac\{1\}\{2\}$d.
is the required value.

23. Ex. 2. Allowing interest at 5 per cent. what sum should be paid down now for the renewal of 14 years lapsed, in a lease for 21 years of an estate producing L. 300 per annum, clear of all deductions?

This is the price of an annuity for 14 years, to be entered upon 7 years hence; the term of delay, therefore, is 7 years, and that of the delay and continuance together 21 years.

By Table II. it appears, that the present value of an annuity

The product - L. 2110.44, or L. 2110, 8s. 9$\backslash frac\{1\}\{2\}$d. is the price required.

24. Hitherto we have proceeded upon the supposition of the annuity being payable, and the interest convertible into principal, which shall reproduce interest, only once a-year.

But annuities are generally payable half-yearly, and sometimes quarterly; and the same circumstances that render it desirable for an annuitant to receive his annual sum in equal half-yearly or quarterly portions, also give occasion to the interest of money being paid in the same manner.

But whatever has been advanced above, concerning the present value or the amount of an annuity, when both that and the interest of money were only payable once a-year, will evidently be true when applied to half the annuity, and half the interest paid twice as often, on the supposition of half-yearly payments; or to a quarter of the annuity, and a quarter of the interest, paid four times as often, when the payments are made quarterly.

25. Half-yearly payments are, however, by far the most common, and these four tables will also enable us to answer the most useful questions concerning them.

For we have only to extract the present value, or the amount, from the table, against twice the number of years in the term, at half the annual rate of interest, and, in the case of an annuity, to multiply the number so extracted, by half the annuity proposed.

26. Ex. 1. To what sum will L. 100 amount in 20 years, when the interest at the rate of 4 per cent. per annum, is convertible into principal half-yearly?

This being the amount in 40 half years at 2 per cent. interest for every half year, will be the same as the amount in 40 years at 2 per cent. per annum, which, by Table III. will be found to be 220.804, or L. 220, 16s. 1d.; and is only L. 1, 13s. 10d. more than it would amount to if the interest were not convertible more than once a-year (16).

27. Ex. 2. What is the present value of an annuity of L. 50 for 21 years, receivable in equal half-yearly payments, when money yields an interest of $2\backslash frac\{1\}\{2\}$ per cent. every half year?

By Table II. it appears, that an annuity of L. 1 for 42 years, when the interest of money is $2\backslash frac\{1\}\{2\}$ per cent. per annum, will be worth L. 25,8206 (25); 25 times this sum, or L. 645, 10s. 3$\backslash frac\{1\}\{2\}$d. is therefore, the required value, and exceeds the value when the interest and the annuity are only payable once a-year by L. 4, 9s. 1d. (18).

28. The excess of an annuity-certain above the interest of the purchase-money, is the sum which, being put out at the time of each payment becoming due, and improved at compound interest until the expiration of the term, will just amount to the purchase-money originally paid.

But, while everything else remains the same, the longer the term of the annuity is, the less must its excess above the interest of the purchase-money be, because a less annuity will suffice for raising the same sum within the term. Therefore, the proportion of that excess to the annual interest of the purchase-money, continually diminishes as the term is extended; and when the annuity is a perpetuity, there is no such excess (8).

29. The reason why the value of an annuity is increased by that and the interest being both payable more than once in the year, is, that the grantor loses, and the purchaser gains, the interest produced by that part of each payment, which is in excess above the interest then due upon the purchase-money, from the time of such payment being made, until the expiration of the year.

Hence it is obvious, that the less this excess is, that is, the longer the term of the annuity is (28), the less must the increase of value be.

And when the annuity is a perpetuity, its value will be the same, whether it and the interest of money be both payable several times in the year, or once only.

30. When the annuity is not payable at the same intervals at which the interest is convertible into principal, its value will depend upon the frequencies both of payment and conversion; but its investigation without algebra, would be too long, and of too little use, to be worth prosecuting here.

II. OF ANNUITIES ON LIVES.

31. When the payment of an annuity depends upon the existence of some life or lives, it is called a Life-annuity.

32. The values of such annuities are calculated by means of tables of mortality, which show, out of a considerable number of individuals born, how many upon an average have lived to complete each year of their age; and, consequently, what proportion of those who attained to any one age, have survived any greater age.

The fifth Table at the end of this article is one of that kind, which has been taken from Mr Milne's Treatise on Annuities, and was constructed from ac-

curate observations made at Carlisle by Dr Heysham, during a period of 9 years, ending with 1787.

33. By this table it appears, that during the period in which these observations were made; out of 10,000 children born, 3203 died under 5 years of age, and the remaining 6797 completed their fifth year. Also, that out of 6797 children who attained to 5 years of age, 6460 survived their 10th year.

But the mortality under 10 years of age, has been greatly reduced since then, by the practice of vaccination.

This table also shows, that of 6460 individuals who attained to 10 years of age, 6047 survived 21. And that of 5075 who attained to 40, only 3643 survived their 60th year.

34. There is good reason to believe (as has been shown in another place), that the general law of mortality, that is, the average proportion of persons attaining to any one age, who survive any greater age, remains much the same now among the entire mass of the people throughout England, as it was found to be at Carlisle during the period of these observations; except among children under 10 years of age, as was noticed above (33).

If this be so, it will follow, that of 6460 children now 10 years of age, just 6047 will attain to 21; or rather, that if any great number be taken in several instances, this $\backslash left(\backslash frac\{6047\}\{6460\}\backslash right)$ will be the average proportion of them that will survive the period.

And if 6460 children were to be taken indiscriminately from the general mass of the population at 10 years of age, and an office or company were to engage to pay L. 1, eleven years hence, for each of them that might then be living; this engagement would be equivalent to that which should bind them to pay L. 6047 certainly, at the expiration of the term. Therefore, the office, in order that it might neither gain nor lose by the engagement, should, upon entering into it, be paid for the whole, the present value of L. 6047, to be received at the expiration of 11 years; and for each life, the $\backslash frac\{1\}\{6460\}$th part of it; that is, the $\backslash frac\{6047\}\{6460\}$th part of the present value of L. 1 to be received then.

But when the rate of interest is 5 per cent. the present value of L. 1, to be received at the expiration of 11 years, is L. 0.584679; therefore, at that rate of interest, there should be paid for each life $\backslash frac\{6047\; \backslash times\; 0.584679\}\{6460\}\; =\; \backslash text\{L.\; \}\; 0.5473$.

And the present value of L. 100, to be received upon a life now 10 years of age attaining to 21, will be L. 54.73, or L. 54, 14s. 7d.

In the same manner it will be found, that reckoning interest at 4 per cent. the value would be L. 60, 16s. 1d.

35. This is the method of calculating the present values of endowments for children of given ages; and the values of annuities on lives may be computed in the same manner.

For, from the above reasoning it is manifest, that if the present value of L. 1, to be received certainly at the expiration of a given term, be multiplied by the number in the table of mortality against the

age, greater than that of any proposed life by the number of years in the term, and the product be divided by the number in the same table, against the present age of that life; the quotient will be the present value of L. 1, to be received at the expiration of the term, provided that the life survive it.

And if, in this manner, the value be determined of L. 1, to be received upon any proposed life, surviving each of the years in its greatest possible continuance, according to the table of mortality adapted to it; that is, according to the Carlisle table, upon its surviving every age greater than its present, to that of 104 years inclusive; then, the sum of all these values will evidently be the present value of an annuity on the proposed life.

36. If 5642 lives at 30 years of age be proposed, and 5075 at the age of 40; since each of the 5642 younger lives may be combined with every one of the 5075 that are 10 years older, the number of different pairs, or different combinations of two lives differing in age by 10 years, that may be formed out of the proposed lives, is 5642 times 5075.

But at the expiration of 15 years, the survivors of the lives now 30 and 40 years of age, being then of the respective ages of 45 and 55, will be reduced to the numbers of 4727 and 4073 respectively; and the number of pairs, or combinations of two, differing in age by 10 years, that can be formed out of them, will be reduced from 5642 $\backslash times$ 5075 to 4727 $\backslash times$ 4073.

So that L. 1 to be paid at the expiration of 15 years for each of these 5642 $\backslash times$ 5075 pairs or combinations of two, now existing, which may survive the term, will be of the same value in present money, as 4727 times L. 4073, to be received certainly at the same time.

Now let A be any one of these lives of 30 years of age, and B any one of those aged 40; and from what has been advanced it will be evident, that the present value of L. 1 to be received upon the two lives in this particular combination jointly surviving the term, will be the same as that of the sum $L.\; 4727\; \backslash times\; 4073$ to be then received certainly.

But, when the rate of interest is 5 per cent. L. 1 to be received certainly at the expiration of 15 years, is equivalent to L. 0.481017 in present money (Table I).

Therefore, at that rate of interest, and according to the Carlisle table of mortality; the present value of L. 1 to be received upon A and B now aged 30 and 40 years respectively, jointly surviving the term of 15 years, will be $\backslash frac\{4727\; \backslash times\; 4073\; \backslash times\; L.\; 0.481017\}\{5642\; \backslash times\; 5075\}$.

37. Hence it is sufficiently evident, how the present value of L. 1 to be received upon the same two lives jointly surviving any other year may be found. And if that value for each year from this time until the eldest life attain to the limit of the table of mortality be calculated, the sum of all these will be the present value of an annuity of L. 1 dependent upon their joint continuance.

In this manner, it is obvious that the value of an annuity on the joint continuance of any other two lives might be determined.

38. If, besides the 5642 lives at 30 years of age,

and the 5075 at 40 (mentioned in No. 36), there be also proposed 3643 at 60 years of age; each of these 3643 at 60, may be combined with every one of the 5642 $\backslash times$ 5075 different combinations of a life of 30, with one of 40 years of age; and, therefore, out of these three classes of lives 5642 $\backslash times$ 5075 $\backslash times$ 3643 different combinations may be formed; each containing a life of 30 years of age, another of 40, and a third of 60.

But at the expiration of 15 years, the numbers of lives in these three classes will, according to the table of mortality, be reduced to 4727, 4073, and 1675 respectively; the respective ages of the survivors in the several classes being then 45, 55, and 75 years; and the number of different combinations of three lives (each of a different class from either of the other two), that can be formed out of them, will be reduced to 4727 $\backslash times$ 4073 $\backslash times$ 1675.

Hence, by reasoning as in No. 36, it will be found, that if A, B, and C be three such lives, now aged 30, 40, and 60 years, the present value of L. 1 to be received upon these three jointly surviving the term of 15 years from this time, will be

being reckoned at 5 per cent.

Thus it is shown, how the present value of an annuity dependent upon the joint continuance of these three lives might be calculated, that being the sum of the present values thus determined, of the rents for all the years which, according to the table of mortality, the eldest life can survive.

39. But it is easy to see, that the same method of reasoning may be used in the case of four, five, or six lives, and so on without limit. Whence, this inference is obvious.

The present value of L. 1, to be received at the expiration of a given term, provided that any given number of lives all survive it, may be found by multiplying the present value of L. 1 to be received certainly at the end of the term, by the continual product of the numbers in the table of mortality against the ages greater respectively by the number of years in the term, than the ages of the lives proposed; and dividing the last result of these operations, by the continual product of the numbers in the table of mortality against the present ages of the proposed lives.

And by a series of similar operations, the present value of an annuity on the joint continuance of all these lives might be determined.

But it should be observed, that, in calculating the value of a life-annuity in this way, the denominator of the fractions expressing the values of the several years rents, that is, the divisor used in each of the operations, remains always the same; the division should, therefore, be left till the sum of the numerators is determined, and one operation of that kind will suffice.

40. Enough has been said to show that these methods of constructing tables of the values of annuities on lives are practicable, though excessively laborious, and, in fact, all the early tables of this kind were constructed in that manner. We proceed now

Popular View. to show how such tables may be calculated with much greater facility.

41. By the method of No. 34, it will be found that, reckoning interest at 5 per cent., the present value of L. 1 to be received at the expiration of a year, provided that a life, now 89 years of age, survived till then, is $\backslash frac\{142\; \backslash times\; 0.952381\}\{181\}$. But the age of that life will then be 90 years, and the proprietor of an annuity of L. 1 now depending upon it, will, in that event, receive his annual payment of L. 1 then due; therefore, if the value then of all the subsequent payments, that is, the value of an annuity on a life of 90 be 2.339 years' purchase, the present value of what the title to this annuity may produce to the proprietor, at the end of the year, will be the same as that of L. 3.339, to be received then, if the life be still subsisting, or $\backslash frac\{142\; \backslash times\; 0.952381\}\{181\}\; \backslash times\; L.\; 3.339\; =\; L.\; 2.495$; which, therefore, will be the present value of an annuity of L. 1 on a life of 89 years of age. That is to say, an annuity on that life will now be worth 2.495 years' purchase (7).

42. In the same manner it appears generally, that, if unity be added to the number of years' purchase that an annuity on any life is worth, and the sum be multiplied by the present value of L. 1, to be received at the end of a year, provided that a life one year younger survive till then, the product will be the number of years' purchase an annuity on that younger life is worth in present money.

43. But according to the table of mortality, an annuity on the eldest life in it is worth nothing; therefore, the present value of L. 1 to be received at the end of a year, provided that the eldest life but one in the table survive till then, is the total present value of an annuity of L. 1 on that life. Which, being obtained, the value of an annuity on a life one year younger than it may be found by the preceding number; and so on for every younger life successively.

EXEMPLIFICATION.

Rate of Interest 5 per cent.

44. Proceeding as in No. 56, it will be found, that at 5 per cent. interest, and according to the Carlisle table of mortality, the present value of L. 1 to be received at the expiration of a year, provided that a person now 89 years of age, and another now 99, be then living, is $\backslash frac\{142\; \backslash times\; 9\; \backslash times\; L.\; 0.952381\}\{181\; \backslash times\; 11\}$; therefore, if the present value of an annuity of L. 1 on the joint continuance of two lives, now aged 90 and 100 years respectively, be L. 0.950; by reasoning as in No. 41, it will be found that the present value of an annuity on the joint continuance of two lives, of the respective ages of 89 and 99 years, will be worth $\backslash frac\{142\; \backslash times\; 9\; \backslash times\; 0.952381\}\{181\; \backslash times\; 11\}\; \backslash times\; 1.950\; =\; 1.192$ years' purchase.

45. In this manner, commencing with the two oldest lives in the table that differ in age by ten years, and proceeding according to No. 43, the values of annuities on all the other combinations of two lives of the same difference of age, may be determined.

The method is exemplified in the following specimen:

46. Hence, and by what has been advanced in the 39th number of this article, it is sufficiently evident, how a table may be computed of the values of annuities on the joint continuance of the lives in every combination of three, or any greater number; the differences between the ages of the lives in each combination remaining always the same in the same series of operations, while the calculation proceeds back from the combination in which the oldest life is the oldest in the table, to that in which the youngest is a child just born.

47. But, when there are more than two lives in each combination, the calculations are so very laborious, on account, principally, of the great number

of combinations, that no tables of that kind have yet been published, except three or four for three lives.

And, in the books that contain tables of the values of two joint lives, methods are given of approximating towards the values of such combinations of two and of three lives, as have not yet been calculated.

Therefore, assuming the values of annuities on single lives, and on the joint continuance of two or of three lives, to be given; we have next to show how the most useful problems respecting the values of any interests that depend upon the continuance or the failure of life, may be resolved by them.

48. Proposition 1. The value of an annuity on the survivor of two lives, $A$ and $B$, is equal to the excess of the sum of the values of annuities on the two single lives, above the value of an annuity on their joint continuance.

49. Demonstration. If annuities on each of the two lives were granted to $P$, during their joint continuance, he would have two annuities; but if $P$ were only to receive these upon condition that, during the joint lives of $A$ and $B$, he should pay one annuity to $Q$; then, there would only remain one to be enjoyed by him, or his heirs or assigns, until the lives both of $A$ and $B$ were extinct; whence the truth of the proposition is manifest.

50. Prop. 2. The value of an annuity on the joint continuance of the two last survivors out of three lives, $A$, $B$, and $C$, is equal to the excess of the sum of the values of annuities on the three combinations of two lives ($A$ with $B$, $A$ with $C$, and $B$ with $C$) that can be formed out of them, above twice the value of an annuity on the joint continuance of all the three lives.

51. Demon. If one annuity were granted to $P$ on the joint continuance of the two lives $A$ and $B$, another on the joint continuance of $A$ and $C$, and a third, on the joint continuance of $B$ and $C$; during the joint continuance of all the lives he would have three annuities.

But if he were only to receive these upon condition that he should pay two annuities to $Q$, during the joint continuance of all the three lives; then, there would only remain to himself one annuity during the joint existence of the last two survivors out of the three lives. And the truth of the proposition is manifest.

52. Prop. 3. The value of an annuity on the last survivor of three lives, $A$, $B$, and $C$, is equal to the excess of the sum of the values of annuities on each of the three single lives, together with the value of an annuity on the joint continuance of all the three, above the sum of the values of three other annuities; the first dependent upon the joint continuance of $A$ and $B$, the second, on that of $A$ and $C$, and the third, on $B$ and $C$.

53. Demon. If annuities on each of the three single lives were granted to $R$; during the joint continuance of all the three, he would have three annuities, and from the time of the extinction of the first life that failed, till the extinction of the second, he would have two.

So that he would have two annuities during the joint existence of the two last survivors out of the three lives; and besides these, a third annuity during the joint continuance of all the three.

Therefore, if out of these, $R$ were to pay one annuity to $P$ during the joint continuance of the last two survivors out of the three lives, and another to $Q$ during the joint continuance of all the three; he would only have left one annuity, which would be receivable during the life of the last survivor of the three.

But in the demonstration of the last proposition (51) it was shown, that the value of what he paid to $P$, would fall short of the sum of the values of annuities dependent respectively on the joint continuance of $A$ and $B$, of $A$ and $C$, and of $B$ and $C$, by two annuities on the joint continuance of all the three lives. Whence it is evident, that the value of the annuities he paid both to $P$ and $Q$, would fall short of the sum of these three values of joint lives, only by the value of one annuity on the joint continuance of all the three lives.

Wherefore, if from the sum of the values of all the three single lives, the sum of the values of the three combinations of two that can be formed out of them were taken; there would remain less than the value of an annuity on the last survivor, by that of an annuity on the joint continuance of the three lives.

But if, to the sum of the values of the three single lives $A$, $B$, and $C$, there be added that of an annuity on the joint continuance of the three, and from the sum of these four values, the sum of the values of the three combinations, $A$ with $B$, $A$ with $C$, and $B$ with $C$ be subtracted; then, the remainder will be the value of an annuity on the last survivor of the three lives. Which was to be demonstrated.

54. Prop. 4. Problem. The law of mortality and the values of single lives at all ages being given; to determine the present value of an annuity on any proposed life, deferred for any assigned term.

55. Solution. Find the present value of an annuity on a life older than the proposed, by the number of years during which the other annuity is deferred; multiply this by the present value of $L.\; 1$ to be received upon the proposed life surviving the term, and the product will be the value sought.

56. Demon. Upon the proposed life surviving the term, the annuity dependent upon it will be worth the same sum, that an annuity on a life so much older is now worth; therefore, it is evident, that the deferred annuity is of the same present value as that sum to be received at the expiration of the term, provided the life survive it.

57. Corollary. In the same manner it appears, that the present value of an annuity on the joint continuance of any number of lives, deferred for a given term, may be found by multiplying the present value of an annuity on the joint continuance of the same number of lives, older respectively than the proposed, by the number of years in the term; by the present value of $L.\; 1$ to be received, upon the proposed lives all surviving it.

58. A Temporary Annuity on any single life, or on the joint continuance of any number of lives, that is, an annuity which is to be paid during a certain term, provided that the single life or the other lives jointly subsist so long; together with an annuity on the same life or lives deferred for the same term, are evidently equivalent to an annuity on the whole duration of the same life or lives.

So that the value of an annuity on any life or on the joint continuance of any number of lives for an assigned term, is equal to the excess of the value of an annuity on their whole duration, with immediate possession, above the value of an annuity on them deferred for the term.

59. Whatever has been advanced from No. 48. to 53. inclusive, respecting the values of annuities for the whole duration of the lives whereon they depend, will apply equally to those which are either deferred or temporary; and, therefore, to determine the value of any deferred or temporary annuity, dependent upon the last survivor of two or of three lives; or, upon the joint continuance of the last two survivors out of three lives; we have only to substitute temporary or deferred annuities, as the case may require, for annuities on the whole duration of the lives; and the result will, accordingly, be the value of a temporary or deferred annuity on the life of the last survivor, or on the joint lives of the two last survivors.

60. Prop. 5. A and B being any two proposed lives now in existence, the present value of an annuity to be payable only while A survives B, is equal to the excess of the value of an annuity on the life of A, above that of an annuity on the joint existence of both the lives.

61. Demons. If an annuity on the life of A, and to be entered upon immediately, were now granted to P, upon condition that he should pay it to B during the joint lives of A and B; it is evident that there would only remain to P, an annuity on the life of A after the decease of B: whence the truth of the proposition is manifest.

62. When any thing is affirmed or demonstrated of any life or lives, it is to be understood as applying equally to any proposed single life, or to the joint continuance of the whole of any number of lives that may be proposed together, or to that of any assigned number of the last survivors of them, or to the last surviving life of the whole.

63. Prop. 6. The present value of the reversion of a perpetual annuity after the failure of any life or lives, is equal to the excess of the present value of the perpetuity with immediate possession, above the present value of an annuity on the same life or lives.

64. Demons. If a perpetual annuity with immediate possession were granted to P, upon condition that he should pay the annual produce to another individual, during the existence of the life or lives proposed; it is evident that there would only remain to P, the reversion after the failure of such life or lives; and the present value of that reversion would manifestly be as stated above.

65. The 6th, 7th, and 8th tables at the end of this article, which have been extracted from the 19th, 21st, and 22d, respectively, in Mr Milne's Treatise on Annuities, will serve to illustrate the application of these propositions to the solution of questions in numbers.

In all the following examples, we suppose the lives to be such, as the general average of those the Carlisle table of mortality was constructed from, and the rate of interest to be 5 per cent.

66. Ex. 1. What is the present value of an annuity on the joint lives, and the life of the survivor of two persons now aged 40 and 50 years respectively?

According to No. 48, the process is as follows:

Value of a single life of { 40 13.390 } by Table VI.
{ 50 11.660 }
sum 25.050
Subtract the value of their joint lives, { 9.984 (Table VIII.) }
remains 15.066 years' purchase, the required value.

And if the annuity be L. 200, its present value will be L. 3013.2, or L. 3013. 4s.

67. Ex. 2. The lives A, B, and C, being now aged 50, 55, and 60 years respectively, an annuity on the joint continuance of all the three, is worth 6.289 years purchase; What is the value of an annuity on the joint existence of the last two survivors of them?

According to No. 50, the process is thus:

Subtract $2\; \backslash times\; 6.289\; =\; 12.578$

remains 10.657 years' purchase, the required value.

Therefore, if the annuity were L. 300, it would be worth L. 3197. 2s. in present money.

68. Ex. 3. Required the value of an annuity on the last survivor of the three lives in the last example.

Proceeding according to No. 52, we have

Subtract the sum of the values of annuities on the three combinations of two lives, { 23.235 (No. 67.) }

remains 14.001 years' purchase, the required value. And if the annuity were L. 300, it would now be worth L. 4200. 6s.

69. Ex. 4. What is the present value of an annuity on a life now 45 years of age, which is not to be entered upon until the expiration of ten years; the first payment thereof being to be made at the expiration of eleven years from this time, if the life survive till then?

Solution.

The present value of an annuity on a life of 55 is 10.347 (Table VI.), and the present value of L. 1 to be received upon the proposed life attaining to the

age of 55, is $\backslash frac\{4073\}\{4727\}\; \backslash times\; 0.613913$; therefore, by No.

55, the required value is $\backslash frac\{4073\; \backslash times\; 0.613913\; \backslash times\; 10.347\}\{4727\}$

= 5.473 years purchase; so that if the annuity were L. 200, its present value would be L. 1094, 12s.

70. Ex. 5. Required the present value of an annuity to be received for the next ten years, provided that a person now 45 years of age, shall so long live.

Solution.

The present value of an annuity on a life of 45, to be entered upon immediately, is 12.648 (Table VI.)

Subtract that of an annuity on the same life deferred 10 years, } 5.473 (69).

the remainder, } 7.175
is the required number of years purchase. And, if the annuity were L. 200, its present value would be L. 1435.

71. Ex. 6. An annuity on a life of 45, deferred 10 years, was shown in No. 69, to be worth 5.473 years purchase in present money; let it be required to determine the equivalent annual payment for the same, to be made at the end of each of the next 10 years, but subject to failure upon the life failing in the term.

Solution.

The present value of L. 1 per annum on the proposed life for the next 10 years, has just been shown to be L. 7.175, and this, multiplied by the required annual payment, must produce L. 5.473; that payment must, therefore, be $\backslash frac\{5.473\}\{7.175\}\; =\; 0.76279$. And, since the annual payment for the deferred annuity of L. 1 per annum would be L. 0.76279, that for an annuity of L. 200 must be L. 152, 11s. 2d.

72. Ex. 7. Let the present value be required of an annuity on a life now 40 years of age, to be payable only while that life survives another now of the age of 50 years.

From the present value of a life of 40, } 13.390 (Table VI.)
Subtract that of the two joint lives, } 9.984 (Table VIII.)

the remainder, } 3.406
years purchase is the required value (60).

Therefore, if the annuity were L. 100, it would be worth L. 340, 12s. in present money.

73. If the annuity in the last example were to be paid for by a constant annual premium at the end of each year while both the lives survived; by reasoning as in No. 71, it will be found, that such annual premium for an annuity of L. 1 should be $\backslash frac\{3.406\}\{9.984\}\; =$ L. 0.341146; for an annuity of L. 100 it should therefore be L. 34, 2s. 3d.

74. But if one of the equal premiums for this annuity is to be paid down now, and another at the end of each year while both the lives survive; the

number of years purchase the whole of these premiums are worth, will evidently be increased by unity, on account of the payment made now, it will, therefore, be 10.984; and each premium for an annuity of L. 1 must, in this case, be $\backslash frac\{3.406\}\{10.984\}\; =$ L. 0.310087; for an annuity of L. 100 it should, therefore, be L. 31, 0s. 2d.

75. Ex. 8. Let it be required to determine the present value of the reversion of a perpetual annuity after the failure of a life now 50 years of age.

Solution.

The value of the perpetuity is 20 years purchase (8.)
Subtract that of an annuity on the life of 50, } 11.660 (Table VI.)

Remains 8.34 years purchase, the required value of the reversion (63.)

So that if the annuity were L. 300, its present value would be L. 2502.

76. In the same manner it will be found, by the 68th number and those referred to in the last example, that the reversion of a perpetuity, after the failure of the last survivor of three lives, now aged 50, 55, and 60 years respectively, is worth 5.999 years purchase in present money; therefore, if it were L. 100 per annum, its present value would be L. 599, 18s.

III. OF ASSURANCES ON LIVES.

77. An assurance upon a life, or lives, is a contract by which the Office or Underwriter, in consideration of a stipulated premium, engages to pay a certain sum upon such life or lives failing within the term for which the assurance is effected.

78. If the term of the assurance be the whole duration of the life or lives assured, the sum must necessarily be paid whenever the failure happens; and, in what follows, that payment is always supposed to be made at the end of the year in which the event assured against takes place. The anniversary of the assurance, or the day of the date of the policy, being accounted the beginning of each year.

79. At the end of the year in which any proposed life or lives may fail, the proprietor of the reversion of a perpetual annuity of L. 1 after their failure, will receive the pound then due, and will, at the same time, enter upon the perpetuity; therefore, the present value of the reversion is the same as that of L. 1 added to the money a perpetual annuity of L. 1 would cost, supposing this sum not to be receivable until the expiration of the year in which the failure of the life or lives might happen.

80. Hence we have this proportion. As the value of a perpetuity increased by unity is to L. 1, so is the present value of the reversion of a perpetual annuity of L. 1, after the failure of any life or lives, to the present value of L. 1, receivable at the end of the year in which such failure shall take place.

81. Therefore, if the value of an annuity of one pound on any life or lives, be subtracted from that of the perpetuity, and the remainder be divided by the value of the perpetuity increased by unity; the quo-

Popular View. tient will be the value, in present money, of the assurance of one pound on the same life or lives. (63)

82. Ex. 1. What is the present value of L. 1, to be received at the end of the year, in which a life now 50 years of age may fail?

The rate of interest being 5 per cent. the value of the perpetuity is 20 years purchase, and that of the life

11.66; the answer therefore is $\backslash frac\{20-11.66\}\{20+1\}\; =\; \backslash frac\{8.34\}\{21\}\; =$

L. 0.397143. And if the sum assured were L. 1000, the present value of the assurance would be L. 397, 7s. 10d.

83. When the term of a life assurance exceeds one year, its whole value is hardly ever paid down at the time that the contract is entered into, but, in the instrument (called a Policy) whereby the assurance is effected, an equivalent annual premium is stipulated for, payable at the commencement of each year during the term, but subject to failure with the life or lives assured.

84. But by reasoning as in No. 74, it will be found, that an annual premium payable at the commencement of each year in the whole duration of the life or lives assured, will be worth one year's purchase more, than an annuity on them payable at the end of each year; and, consequently, that if the value in present money of an assurance on any life or lives, be divided by the number of years purchase an annuity on the same life or lives is worth, increased by unity, the quotient will be the equivalent annual premium for the same assurance.

85. Ex. 2. Required the annual premium for the assurance of L. 1, on a life of 50 years of age.

In No. 82, the single premium for that assurance was shown to be 0.397143, and the value of an annuity on the life is 11.66, therefore, by the preceding number, the required annual premium will be $\backslash frac\{0.397143\}\{12.66\}\; =\; .0313699$ for the assurance of L. 1; and

for the assurance of L. 1000, it will be L. 31, 7s. 5d.

86. Ex. 3. Let both the single payment in present money, and the equivalent annual premium be required for the assurance of L. 1, on the joint continuance of two lives of the respective ages of 45 and 50 years.

The value of an annuity of L. 1 on the joint continuance of these two lives, appears by Table VII.

to be L. 9.737, therefore $\backslash frac\{20-9.737\}\{20+1\}\; =\; \backslash frac\{10.263\}\{21\}\; =$

L. 0.488714 is the single premium, and $\backslash frac\{0.488714\}\{10.737\}\; =$

L. 0.0455168, the equivalent annual one for the assurance of L. 1 to be paid at the end of the year, in which that life becomes extinct which may happen to fail the first of the two.

Therefore, if the sum assured were L. 500, the total present value of the assurance would be L. 244, 7s. 2d. and the equivalent annual premium L. 22, 15s. 2d.

87. Ex. 4. Let both the single and the equivalent annual premium be required for the assurance of L. 1, on the life of the survivor of two persons now aged 40 and 50 years respectively.

The value of an annuity on the survivor of these

two lives was shown in No. 66, to be 15.066, therefore, by No. 81, the single premium will be $\backslash frac\{20-15.066\}\{20+1\}\; =\; \backslash frac\{4.934\}\{21\}\; =$ L. 0.234952; and by No. 84,

the annual one will be $\backslash frac\{L.\; 0.234952\}\{15.066\}\; =$ L. 0.0146242.

That is, for the assurance of L. 1 to be received at the end of the year, in which the last surviving life of the two becomes extinct.

Therefore, for the assurance of L. 5000, the single premium will be L. 1174, 15s. 2d. the equivalent annual one L. 73, 2s. 5d.

88. Ex. 5. What should the single and equivalent annual premiums be for an assurance on the last survivor of three lives, of the respective ages of 50, 55, and 60 years.

The value of an annuity on the last survivor of them, was shown in No. 68, to be 14.001, the single

premium should therefore be $\backslash frac\{20-14.001\}\{20+1\}\; =\; \backslash frac\{5.999\}\{21\}\; =$

L. 0.285666, and the annual $\backslash frac\{L.\; 0.285666\}\{15.001\}\; =$

L. 0.0190431, for the assurance of L. 1, to be received at the end of the year, in which the last surviving life of the three may fail.

For the assurance of L. 2000, the single premium would therefore be L. 571, 6s. 8d. the annual one L. 38, 1s. 9d.

89. Lemma. If an annuity be payable at the commencement of each year, which some assigned life or lives may enter upon in a given term; the number of years purchase in its present value, will exceed by unity the number of years purchase, in the present value of an annuity on the same life or lives for one year less than the given term, but payable as annuities generally are, at the end of each year.

Demonstration. Since the proposed life or lives can only enter upon any year after the first, by surviving the year that precedes it; the receipt of each of the payments after the first, that are to be made at the commencement of the year, will take place at the same time, and upon the same conditions as that of the rent for the year then expired of the life-annuity, for a term one year less than the term proposed: this last mentioned annuity, will therefore, be worth in present money, just the same number of years' purchase as all the payments subsequent to the first, which may be made at the commencements of the several years.

And, since the first of these is to be made immediately, the present value of the whole of them, will evidently exceed the number of years purchase last mentioned, by unity, which was to be demonstrated.

90. If, while the rest remains the same, the payment of the annuity which depends upon the assigned life or lives entering upon any year, is not to be made until the end of that year; as the condition upon which every payment is to be made, will remain the same, but each of them will be one year later; the only alteration in the value of the whole, will arise from this increase in the remoteness of payment, by which it will be reduced in the ratio of L. 1, to the present value of L. 1, receivable at the end of a year (2).

91. When the value of an annuity on any proposed life or lives for an assigned term is given, it is evident that the value of an annuity on the same life or lives for one year less may be found, by subtracting from the given value, the present value, of the rent to be received upon the proposed life or lives surviving the term assigned.

92. Proposition. The present value of an assurance on any proposed life or lives for a given term, is equal to the excess of the value of an annuity to be paid at the end of each year, which the life or lives proposed may enter upon in the term, above the value of an annuity on them for the same term, but dependent, as usual, upon their surviving each year.

Demonstration. If an annuity payable at the end of each year, which the proposed life or lives may enter upon during the given term, be granted to $P$, upon condition that he shall pay over what he receives to $Q$, at the end of each year which the same life or lives may survive; it is manifest that there will only remain to $P$, the rent for the year in which the proposed life or lives may fail; that is, the assurance of that sum thereon for the given term (77). Which was to be demonstrated.

93. From the last four numbers (89—92) we derive the following

for determining the present value of an assurance on any life or lives for a given term.

Add unity to the value of an annuity on the proposed life or lives for the given term, and from the sum subtract the present value of one pound, to be received upon the same life or lives surviving the term; multiply the remainder by the present value of $L.\; 1$, to be received certainly at the end of a year, and from the product subtract the present value of an annuity on the proposed life or lives for the term.

This last remainder will be the value in present money of the assurance of $L.\; 1$ during the same term, on the life or lives proposed.

94. It has been shown above (34—39), how the present value of $L.\; 1$, receivable upon any single or joint lives surviving an assigned term, may be found. And all that was demonstrated from No. 48. to 58. inclusive, respecting annuities on the last survivor of two, or of three lives, or on the joint continuance of the two last survivors out of three lives, is equally true of any particular year's rent of those annuities. Hence it is evident, how the present value of $L.\; 1$, to be received upon the last survivor of two, or of three lives, or upon the last two survivors out of three lives, surviving any assigned term, may be found.

95. Example. Required the present value of $L.\; 1$, to be received at the end of the year, in which a life, now forty-five years of age, may fail, provided that such failure happens before the expiration of ten years.

Here the present value of $L.\; 1$, to be received upon the life surviving the term, will be found to be $L.\; 0.528976$, and the value of an annuity on the proposed life for the term, is $7.175$ (70.)

From 8.175
subtract 0.528976

the remainder 7.646024
being multiplied by 0.952381

produces 7.28193
from this subtract 7.17500

remains $L.\; 0.10693$, the required value of the assurance; and if the sum assured were $L.\; 3000$, the value of the assurance in present money would be $L.\; 320,\; 15s.\; 7d.$

96. By numbers 89, 91, and 95, it appears, that an annuity, payable at the commencement of each of the next ten years that a life of 45 enters upon, is worth 7.646 years purchase: therefore, $\backslash frac\{0.10693\}\{7.646\}\; =\; L.\; 0.013985$ will be the annual premium for the assurance of $L.\; 1$ for ten years on that life. For the assurance of $L.\; 3000$, it will therefore be $L.\; 41,\; 19s.\; 1d.$

97. When the term of the assurance is the whole duration of the life or lives assured, $L.\; 1$ to be received upon their surviving the term is worth nothing; and an annuity on the lives for the term, is also for their whole duration.

Therefore, from No. 93. we derive the following

for determining the present value of an assurance on any life or lives.

Add unity to the value of an annuity on the proposed life or lives; multiply the sum by the present value of $L.\; 1$, to be received certainly at the end of a year; and from the product subtract the value of an annuity on the same life or lives.

The remainder will be the value of the assurance in present money.

98. Example. Required the present value of $L.\; 1$, to be received at the end of the year, in which the survivor of two lives may fail, their ages now being 40 and 50 years respectively.

The value of an annuity on these lives was shown in No. 66. to be $15.066$.

Multiply $15.066$ by $0.952381$, from the product $15.3009$, subtract $15.066$, the remainder $L.\; 0.2349$ is the required value, agreeably to No. 87.

And, in all other cases, the values determined by the rule in the preceding number, will be found to agree with those obtained by the method of No. 81.

99. When an assurance on any life or lives has been effected at a constant annual premium, and kept up for some time, by the regular payment of that premium; the annual premium required for a new assurance of the same sum on the same life or lives, will, on account of the increase of age, be greater than that at which the first assurance was effected: Therefore, the present value of all these greater annual premiums, that is, the total present value of the new assurance, will exceed the present value of all the premiums that may hereafter be received under the existing policy. And the excess

Popular View. will evidently be the value of the policy, supposing the life or lives to be still insurable; that being the only advantage that can now be derived from the premiums already paid.

So that, if the present value of all the future annual premiums to be paid under an existing policy for the assurance of a certain sum upon any life or lives, be subtracted from the present value of the assurance of the same sum on the same life or lives; the remainder will be the value of the policy.

100. Example. L. 1000 has been assured some years, on a life now 50 years of age, for its whole duration, at the annual premium of L. 20, one of which has just now been paid: What is the value of the policy?

The present value of the assurance of L. 1000 on that life, has been shown in No. 82. to be L. 397, 2s. 10d.; and an annuity on the life, being worth 11.66 years purchase (Table VI.), the present value of all the premiums to be paid in future under the existing policy, is $11.66\; \backslash times\; L.\; 20\; =\; L.\; 233,\; 4s.\; 0d.$; the value of the policy, therefore, is L. 163, 18s. 10d.

Immediately before the payment of the premium, the value of the policy would evidently have been less by the premium then due.

101. In our investigations of the values of annuities on lives, we have hitherto assumed, that no part of the rent is to be received for the year in which the life wherewith the annuity may terminate fails.

But if a part of the annuity is to be received at the end of that year, proportional to the part of the year which may have elapsed at the time of such failure; as, in a great number of such cases, some of the lives wherewith the annuity terminates will fail in every part of the year, and as many, or very nearly so, in any one part of it as in any other: we may assume, that, upon an average, half a year's rent will be received at the end of the year in which such failure happens; and, therefore, that by the title to what may be received after the failure of the life or lives whereon the annuity depends, the present value of that annuity will be increased by the present value of the assurance of half a year's rent on the same life or lives.

102. Thus, for example: the present value of the assurance of L. 1 on a life of 50 years of age, having in No. 82. been shown to be L. 0.397143, the value of an annuity of L. 1 on that life, when payable, till the last moment of its existence, will exceed L. 11.66, its value, if only payable, until the expiration of the last year it survives, by $\backslash left(\; \backslash frac\{L.\; 0.397143\}\{2\}\; \backslash right)\; =\; L.\; 0.199$; it will therefore be L. 11.859.

103. If, at the end of the year in which an assigned life A may fail, Q or his heirs are to receive L. 1; and are, at the same time, to enter upon an annuity of L. 1, to be enjoyed during another life P, to be then fixed upon: the present value of Q's interest will evidently be the same as that of the assurance on the life of A, of a number of pounds, exceeding

by unity the number of years purchase in the value of an annuity on the life of P, at the time of nomination.

104. What is the present value of the next presentation to a living of the clear annual value of L. 500; A, the present incumbent, being now 50 years of age; supposing the age of the clerk presented to be 25, at the end of the year in which the present incumbent dies; also, that he takes the whole produce of the living for that year?

By Table VI. it will be found, that the value of an annuity of L. 1 on a life of 25, is L. 15.303; and in No. 82. it has been shown, that the present value of the assurance of L. 1 on a life of 50, is L. 0.397143. Hence, and by the last number, it appears, that if the annual produce of the living were but L. 1, the present value of the next presentation would be $L.\; 15.303\; \backslash times\; 0.397143\; =\; L.\; 6.07467$. The required value, therefore, is L. 3237, 6s. 9d.

105. If, to the value of the succeeding life, determined according to No. 103, the value of the present be added, the sum of these will evidently be the present value of both the lives in succession; and, in the case of the preceding number, will be $6.075\; +\; 11.66\; =\; 18.135$ years' purchase.

106. In No. 103, we proceeded upon the supposition that the annuity on the present life is only payable up to the expiration of the last year it survives; and, consequently, that the succeeding life takes the whole rent for the year in which the present fails.

But, if the succeeding life is only to take a part of that rent, in the same proportion to the whole, as the portion of the year which intervenes between the failure of the present life and the end of the year, is to the whole year, then, by reasoning as in No. 101, it will be found, that the portion of that rent which the succeeding life will receive, may be properly assumed to be one half. And, instead of increasing the number of years' purchase the annuity on the succeeding life will be worth at the end of the year in which the other fails, by unity, we must only add one half to that number, in order that the present value of the assurance of the sum on the existing life, may be the number of years' purchase, which all that may be received during the succeeding life, is now worth.

107. The value of the succeeding life, in the case of No. 104, will, upon this hypothesis, be $15.803\; \backslash times\; 0.397143\; =\; 6.27605$ years' purchase.

And this appears to be the most correct way of calculating the value of an annuity on a succeeding life; although that of No. 103. proceeds upon the principle on which life interests are generally valued.

108. But the value of two lives in succession, will be the same on both hypotheses. The rent for the year in which the first may fail, being, in the one case, given entirely to the successor; and, in the other, divided equally between the two.

This is also true of any greater number of successive lives.

109. We now proceed to treat the subject of Annuities Algebraically.

Let $r$ denote the simple interest of L. 1 for one year.

$s$, any sum put out at interest.

$n$, the number of years for which it is lent.

$m$, its amount in that time.

$a$, an annuity for the same time (3 and 4.)

$M$, the amount to which that annuity will increase, when each payment is laid up as it becomes due, and improved at compound interest until the end of the term.

$v$, the present value of the same annuity (6.)

110. Then, reasoning as in the first number of this article, it will be found that $m\; =\; s(1+r)^n$. And by No. 2. it appears, that the present value of $s$ pounds to be received certainly at the expiration of $n$ years, is $s\; \backslash frac\{1\}\{(1+r)^n\}$, or $s(1+r)^\{-n\}$.

111. The amount of L. 1 in $n$ years being $(1+r)^n$, its increase in that time will be $(1+r)^n\; -\; 1$, and when it is considered that this increase arises entirely from the simple interest ($r$) of L. 1 being laid up at the end of each year, and improved at compound interest during the remainder of the term; it must be obvious that $(1+r)^n\; -\; 1$ is the amount of an annuity of $r$ pounds in that time, but $r\; :\; a\; ::\; (1+r)^n\; -\; 1\; :\; \backslash frac\{a\}\{r\}[(1+r)^n\; -\; 1]$, which, therefore, is equal to $M$, the amount of an annuity of $a$ pounds in $n$ years.

112. Reasoning as in No. 8. it will be found, that since $r\; :\; 1\; ::\; a\; :\; \backslash frac\{a\}\{r\}$, the present value of a perpetual annuity of $a$ pounds is $\backslash frac\{a\}\{r\}$.

113. If two persons, $A$ and $B$, purchase a perpetuity of $a$ pounds between them, which $A$ and his heirs or assigns are to enjoy during the first $n$ years, and $B$, and his heirs and assigns, for ever after. Since the value of the perpetuity to be entered upon immediately, has just been shown to be $\backslash frac\{a\}\{r\}$, the present value of $B$'s share, that is, the present value of the same perpetuity when the entrance thereon is deferred until the expiration of $n$ years, will be $\backslash frac\{a\}\{r\}(1+r)^\{-n\}$, (110); and the value of the share of $A$ will be thus much less than that of the whole perpetuity (21), and therefore equal to $\backslash frac\{a\}\{r\}[1\; -\; (1+r)^\{-n\}]\; =\; v$, the present value of an annuity of $a$ pounds for the term of $n$ years certain.

114. If the annuity is not to be entered upon until the expiration of $d$ years, but is then to continue

$n$ years, its value at the time of entering upon it will be $\backslash frac\{a\}\{r\}[1\; -\; (1+r)^\{-n\}]$, as has just been shown; therefore its present value will be

115. In the same manner, it appears that, when the entrance on a perpetuity of $a$ pounds is deferred $d$ years, its present value will be $\backslash frac\{a\}\{r\}(1+r)^\{-d\}$ (110 and 112.)

116. $q$ being any number whatever, whole, fractional, or mixed, let $\backslash lambda\; q$ denote its logarithm, and $xq$ the arithmetical complement of that logarithm; so that these equations may obtain, $\backslash lambda\; \backslash frac\{1\}\{q\}\; =\; -\backslash lambda\; q\; =\; xq$. Then, for the resolution of the principal questions of this kind by logarithms, we shall have these formulae.

1. Amount of a sum improved at interest.

2. Amount of an annuity when each payment is laid up as it becomes due, and improved at interest until the expiration of the term.

3. Value of a lease or an annuity.

4. Value of a deferred annuity, or the renewal of any number of years lapsed in the term of a lease.

5. Value of a deferred perpetuity, or the reversion of an estate in fee simple, after an assigned term.

By means of each of these equations, it is manifest that any one of the quantities involved in it may be found, when the rest are given.

117. If the interest be convertible into principal $p$ times in the year, at $v$ equal intervals, since the interest of L. 1 for one of these intervals will be $\backslash frac\{r\}\{p\}$, (109), and the number of conversions of interest into principal in $n$ years $pn$; to adapt the formula in No. 110. to this case, we have only to substitute $\backslash frac\{r\}\{p\}$ for $r$, and $pn$ for $n$, in the equation $m\; =\; s(1+r)^n$ there given, whereby it will be transformed to this, $m\; =\; s\backslash left(1\; +\; \backslash frac\{r\}\{p\}\backslash right)^\{pn\}$.

118. According as $p$ is equal to 1, 2, 4, or is infinite; that is, according as the interest is convertible into principal yearly, half-yearly, quarterly, or continually, let $m$ be equal to $x$, $H$, $Q$, or $C$; so shall

and $c\; =\; s\; \backslash cdot\; N$

$N$ being the number whereof $nr$ is the hyperbolic logarithm, and $nr\; \backslash times\; 0.43429448$ its logarithm in Briggs' System, and the common tables.

119. From No. 117 and 110, it follows, that the present value of $s$ pounds to be received at the end of $n$ years, when the interest is convertible into principal at $v$ equal intervals in each year, is $s\; \backslash left(1\; +\; \backslash frac\{r\}\{v\}\backslash right)^\{-vn\}$.

120. When the present values and the amounts of annuities are desired, let the interest be convertible into principal at $v$ equal intervals in the year, while the annuity is payable at $\backslash pi$ intervals therein, the amount of each payment being $\backslash frac\{a\}\{\backslash pi\}$.

121. Case I. $\backslash mu$ being any whole number not greater than $v$, let $\backslash frac\{1\}\{\backslash pi\}\; =\; \backslash frac\{\backslash mu\}\{v\}$, so that the interest may be convertible into principal $\backslash mu$ times in each of the intervals between the payments of the annuity.

Then will the amount of L.1, at the expiration of the period $\backslash frac\{1\}\{\backslash pi\}$ be $\backslash left(1\; +\; \backslash frac\{r\}\{v\}\backslash right)^\{\backslash mu\}$ (117), and the interest of L.1 for the same time will be $\backslash left(1\; +\; \backslash frac\{r\}\{v\}\backslash right)^\{\backslash mu\}\; -\; 1$; whence the present value of the perpetuity will be $\backslash frac\{\backslash frac\{1\}\{\backslash pi\}a\}\{\backslash left(1\; +\; \backslash frac\{r\}\{v\}\backslash right)^\{\backslash mu\}\; -\; 1\}$ (8), and the value of the same defer-

red $n$ years, will be $\backslash frac\{a\}\{\backslash pi\}\; \backslash frac\{\backslash left(1\; +\; \backslash frac\{r\}\{v\}\backslash right)^\{-vn\}\}\{\backslash left(1\; +\; \backslash frac\{r\}\{v\}\backslash right)^\{\backslash mu\}\; -\; 1\}$ (119), there-

fore the present value of the annuity to be entered upon immediately, and to continue $n$ years, will be

122. Case 2. $\backslash mu$ being any whole number greater than $\backslash pi$, let $\backslash frac\{1\}\{\backslash pi\}\; =\; \backslash frac\{\backslash mu\}\{v\}$, so that the annuity may be payable $\backslash mu$ times in each of the intervals between the payments of interest, or the conversion thereof into principal.

Then, at the expiration of the $\backslash frac\{1\}\{\backslash mu\}$th of a year, when the interest on the purchase-money is first payable or convertible, the interest on all the $\backslash mu-1$ payments of the annuity previously made, will be

VOL. I. PART II.

of $\backslash frac\{a\}\{\backslash pi\}$ each (including the one only then due), the sum, $\backslash frac\{a\}\{\backslash pi\}\; \backslash left[\; \backslash mu\; +\; \backslash frac\{r\backslash mu(\backslash mu-1)\}\{2\backslash pi\}\; \backslash right]$, is the simple interest which the value of the perpetuity should yield at the expiration of each $v$th part of a year, in order to supply the deficiency (both of principal and interest) that would be occasioned during each of those periods, in any fund out of which the several payments of the annuity might be taken, as they respectively became due; and since $\backslash frac\{r\}\{v\}\; :\; \backslash frac\{a\}\{\backslash pi\}\; \backslash left[\; \backslash mu\; +\; \backslash frac\{r\backslash mu(\backslash mu-1)\}\{2\backslash pi\}\; \backslash right]\; ::\; 1\; :$

$\backslash frac\{av\}\{r\backslash pi\}\; \backslash left[\; \backslash mu\; +\; \backslash frac\{r\backslash mu(\backslash mu-1)\}\{2\backslash pi\}\; \backslash right]\; =\; a\; \backslash left(\; \backslash frac\{1\}\{r\}\; +\; \backslash frac\{\backslash mu-1\}\{2\backslash pi\}\; \backslash right)$, this last expression will be the value of such perpetuity with immediate possession (8); the value of the same deferred $n$ years, will therefore be $a\; \backslash left(\; \backslash frac\{1\}\{r\}\; +\; \backslash frac\{\backslash mu-1\}\{2\backslash pi\}\; \backslash right)\; \backslash times\; \backslash left(\; 1\; +\; \backslash frac\{r\}\{v\}\; \backslash right)^\{-vn\}$ (119). Whence it appears, that the present value of the annuity to be entered upon immediately, and to continue $n$ years, will be

123. Case 3. When, in consequence of the annuity being always payable at the same time that the interest is convertible, $v\; =\; \backslash pi$.

Since the interest of L.1 at the expiration of the period $\backslash frac\{1\}\{\backslash pi\}$ will be $\backslash frac\{r\}\{\backslash pi\}$, the value of the perpetuity will be $\backslash frac\{\backslash frac\{1\}\{\backslash pi\}a\}\{\backslash frac\{r\}\{\backslash pi\}\}\; =\; \backslash frac\{a\}\{r\}$ (8), whence, proceeding as before, we

obtain the present value of the annuity, $\backslash frac\{a\}\{r\}\; \backslash left[\; 1\; -\; \backslash left(\; 1\; +\; \backslash frac\{r\}\{\backslash pi\}\; \backslash right)^\{-vn\}\; \backslash right]\; =\; v$. When $v\; =\; \backslash pi$, and consequently $\backslash mu\; =\; 1$, the values of $v$, given in the two preceding cases, will be found to coincide with this.

124. According as $v$ and $\backslash pi$ are each equal to 1, 2, 4, or are infinite; that is, according as the interest and the annuity are each payable yearly, half-yearly, quarterly, or continually, let $v$ be equal to $y,\; h,\; q$, or $c$, then will

125. The amount of an annuity is equal to the sum to which the purchase money would amount, if it were put out and improved at interest during the whole term.

For, from the time of the purchase of the annuity, whatever part of the money that was paid for it may be in the hands of the grantor, he must improve thus to provide for the payments thereof; and if the purchaser also improve in the same manner all he receives, the original purchase money must evident-

Algebraical View. If receive the same improvement during the term, as if it had been laid up at interest at its commencement.

126. The periods of conversion of interest into principal, and of the payment of the annuity being still designated as in No. 120; since in $n$ years, the number of periods of conversion will be $rn$, in the

1st Case, Where the interest is convertible $\backslash mu$ times in each of the intervals between the payments of the

$=\; M$, (117, 121, and 125). In the 2d Case, when the annuity is payable $\backslash mu$ times, in each interval between the conversions of interest,

And, in the 3d Case, when the annuity is always payable at the same time that the interest is convertible, $\backslash left(1\; +\; \backslash frac\{r\}\{v\}\backslash right)^\{rn\}\; v\; =\; \backslash frac\{a\}\{r\}\; \backslash left[\; \backslash left(1\; +\; \backslash frac\{r\}\{v\}\backslash right)^\{rn\}\; -\; 1\; \backslash right]\; =\; M$, (117, 123, and 125).

127. According as $v$ and $\backslash pi$ are each equal to 1, 2, 4, or are infinite; that is, according as the interest and the annuity are each payable yearly, half-yearly, quarterly, or continually, let $M$ be denoted by $y\text{'}$, $h\text{'}$, $q\text{'}$, or $c\text{'}$;

128. Example 1. What will L. 320 amount to, when improved at compound interest during 40 years; the rate of interest being 4 per cent. per annum?

By the first formula in No. 116, the operation will be as follows:

And the answer is L. 1536, 6s. 6$\backslash frac\{1\}\{2\}$d.

129. Ex. 2. If the interest were convertible into principal every half-year, the operation, according to No. 117, would be thus:

So that in this case the amount would be L. 1560, 2s. 9$\backslash frac\{1\}\{2\}$d. Algebraical View.

130. Ex. 3. Required the present value of an annuity of L. 250 for 30 years, reckoning interest at 5 per cent.

By the third formula in No. 116, the operation will be thus:

And the required value is L. 3843, 2s. 3$\backslash frac\{1\}\{2\}$d.

131. Ex. 4. The rest being still the same, if the annuity in the last example be payable half-yearly, in the formula of No. 122, $v$ will be equal to 1, $\backslash pi\; =\; 2$, and $\backslash mu\; =\; 2$; that formula will therefore become $a\; \backslash left(\; \backslash frac\{1\}\{r\}\; +\; \backslash frac\{1\}\{4\}\; \backslash right)\; \backslash cdot\; \backslash left[\; 1\; -\; (1+r)^\{-n\}\; \backslash right]\; =\; v$; and the operation will be thus:

The value of the annuity will, therefore, in this case, be L. 3891, 3s.

132. Ex. 5. To what sum will an annuity of L. 120 for 20 years amount, when each payment is improved at compound interest, from the time of its becoming due until the expiration of the term; the rate of interest being 6 per cent.?

The operation by the second formula in No. 116 is thus:

And the amount required is L. 4414, 5s. 5d.

133. Ex. 6. The rest being the same as in the last example; if both the interest and the annuity be payable half-yearly, the amount will be determined by the second of the formulae given in No. 127; which, in this case, will become $\backslash frac\{120\}\{0.6\}\; \backslash left[\; (1.03)^\{40\}\; -\; 1\; \backslash right]$, and the operation will be as follows:

So that the amount in this case, would be L. 4524, 18. 7½d.

II. ON THE PROBABILITIES OF LIFE.

134. Any persons A, B, C, &c. being proposed, let the numbers which tables of mortality (32) adapted to them, represent to attain to their respective ages, be denoted by the symbols a, b, c, &c.; while lives n years older than those respectively, are denoted thus: ⁿA, ⁿB, ⁿC, &c. and the numbers that attain to their ages, by the symbols ⁿa, ⁿb, ⁿc, &c.; also let lives n years younger than A, B, C, &c. be denoted thus: A_n, B_n, C_n, &c., while the numbers which the tables show to attain to those younger ages, are designated by the symbols a_n, b_n, c_n, &c.

Then, if A be 21 years of age, and we use the Carlisle Table, we shall have a = 6047, and ⁿa = 5362, the number that attain to the age of thirty-five, or that live to be fourteen years older than A.

Hence the number that are represented by the table to die in n years from the age of A, will be a - ⁿa, that is in 14 years, a - ⁿa; and by the Carlisle Table, n 14 years from the age of 21, that is, between 21 and 35, it will be 6047 - 5362 = 685.

135. Problem. To determine the probability that a proposed life A, will survive n years.

Solution. a being the number of lives in the table of mortality, that attain to the age of that which is proposed, conceive that number of lives to be so selected, (of which A must be one,) that they may each have the same prospect with regard to longevity, as the proposed life and those in the table, or the average of those from which it was constructed; then will the number of them that survive the term be ⁿa (134).

So that the number of ways all equally probable, or the number of equal chances for the happening of the event in question is ⁿa; and the whole number for its either happening or failing is a; therefore, according to the first principles of the doctrine of probabilities, the probability of the event happening, that is, of A surviving the term, is $\backslash frac\{\backslash supset\; n\; a\}\{a\}$.

If the age of A be 14, the probability of that life surviving 7 years, or the age of 21, will, according to the Carlisle Table, be $\backslash frac\{6047\}\{6335\}$, or 0.95454.

136. Since the number that die in n years from the age of A is a - ⁿa (134), it appears, in the same manner, that the probability of that life failing in n

years will be $\backslash frac\{a\; -\; \backslash supset\; n\; a\}\{a\}\; =\; 1\; -\; \backslash frac\{\backslash supset\; n\; a\}\{a\}$ which probability, when the life, term, and table of mortality, are the same as in the last No. will be 0.04546.

137. If two lives A and B be proposed, since the probability of A surviving n years will be $\backslash frac\{\backslash supset\; n\; a\}\{a\}$, and that of B surviving the same term will be $\backslash frac\{\backslash supset\; n\; b\}\{b\}$; it appears from the doctrine of probabilities that $\backslash frac\{\backslash supset\; n\; a\}\{a\}\; \backslash cdot\; \backslash frac\{\backslash supset\; n\; b\}\{b\}$ or $\backslash frac\{\backslash supset\; n\; (ab)\}\{ab\}$ will be the measure of the probability that these lives will both survive n years.

In the same manner it may be shown, that the probability of the three lives A, B, and C all surviving n years, will be measured by $\backslash frac\{\backslash supset\; n\; a\}\{a\}\; \backslash cdot\; \backslash frac\{\backslash supset\; n\; b\}\{b\}\; \backslash cdot\; \backslash frac\{\backslash supset\; n\; c\}\{c\}$ or $\backslash frac\{\backslash supset\; n\; (abc)\}\{abc\}$. And, universally, that any number of lives A, B, C, &c. will jointly survive n years, the probability is

138. Let $\backslash frac\{\backslash supset\; n\; a\}\{a\}\; =\; \backslash supset\; n\; a$, $\backslash frac\{\backslash supset\; n\; b\}\{b\}\; =\; \backslash supset\; n\; b$, $\backslash frac\{\backslash supset\; n\; c\}\{c\}\; =\; \backslash supset\; n\; c$, &c.; also let ; so that the probabilities of A, B, C, &c. surviving n years may be denoted by ⁿa, ⁿb, ⁿc, &c. respectively; and that of all those lives jointly surviving that term by ⁿ(abc, &c.)

Then will the probability that none of those lives will survive n years, be

139. But the probability that some one or more of these lives will survive n years, will be just what the probability last mentioned is deficient of certainty; its measure therefore, being just what the measure of that probability is deficient of unity, will be

140. Corol. 1. When there is only one life A, this will be ⁿa.

141. Corol. 2. When there are two lives A and B, it becomes ⁿa + ⁿb - ⁿ(ab).

142. Corol. 3. When there are three lives A, B, and C, it becomes ⁿa + ⁿb + ⁿc - ⁿ(ab) - ⁿ(ac) - ⁿ(bc) + ⁿ(abc).

143. When three lives A, B, and C are proposed, that at the expiration of n years there will be

And the sum of these four $\backslash frac\{1\}\{n\}(ab)\; +\; \backslash frac\{1\}\{n\}(ac)\; +\; \backslash frac\{1\}\{n\}(bc)\; -\; \backslash frac\{2\}\{n^2\}(abc)$, is the measure of the probability that some two at the least, out of these three lives, will survive the term.

III. OF ANNUITIES ON LIVES.

144. Let the number of years purchase that an annuity on the life of A is worth, that is, the present value of L. 1, to be received at the end of every year during the continuance of that life, be denoted by $\backslash Lambda$; while the present value of an annuity on any number of joint lives A, B, C, &c. that is, of an annuity which is to continue during the joint existence of all the lives, but to cease with the first that fails, is denoted by $\backslash Lambda\_\{ABC\}$, &c.

Then will the value of an annuity on the joint continuance of the three lives A, B, and C, be denoted by $\backslash Lambda\_\{ABC\}$.

And on the joint continuance of the two A and B, by $\backslash Lambda\_\{AB\}$.

145. Also let $\backslash Lambda\_t$ and $\backslash Lambda\_\{t\_i\}$ denote the value of annuities on lives respectively older and younger than A, by t years: While designates the value of an annuity on the joint continuance of lives t years older than A, B, C, &c. respectively; and that of an annuity on the same number of joint lives, as many years younger than these respectively.

146. Let $\backslash frac\{1\}\{1+r\}$, the present value of L. 1 to be received certainly at the expiration of a year, be denoted by v.

Then will $v^n$ be the present value of that sum certain to be received at the expiration of n years.

But if its receipt at the end of that time, be dependent upon an assigned life A, surviving the term, its present value will, by that condition, be reduced in the ratio of certainty to the probability of A surviving the term, that is, in the ratio of unity to $\backslash frac\{1\}\{n\}a$, and will therefore be $\backslash frac\{v^n\}\{n\}a$.

In the same manner it appears, that if the receipt of the money at the expiration of the term be dependent upon any assigned lives, as A, B, C, &c. jointly surviving that period, its present value will be .

147. Let us denote the sum of any series, as $\backslash frac\{1\}\{n\}(abc)v\; +\; \backslash frac\{1\}\{n\}(abc)v^2\; +\; \backslash frac\{1\}\{n\}(abc)v^3\; +\; \backslash dots$ thus, $S\_n(abc)v^n$, by prefixing the italic capital S to the

general term thereof. Then, from what has just been advanced, it will be evident, that . Algebraical View.

When there are but three lives A, B, and C; this becomes $\backslash Lambda\_\{ABC\}\; =\; S\_n(abc)v^n$.

When there are but two, A and B, it becomes $\backslash Lambda\_\{AB\}\; =\; S\_n(ab)v^n$.

And in the same manner it appears, that for a single life A, $\backslash Lambda\; =\; S\_n(av^n)$.

148. (138), where the denominator is constant, while the numerator varies with the variable exponent n. And the most obvious method of finding the value of an annuity on any assigned single or joint lives, is to calculate the numerical value of the term for each value of n, and then to divide the sum of all these values by ; for

In calculating a table of the values of annuities on lives in that manner, for every combination of joint lives, it would be necessary to calculate the term for as many years as there might be between the age of the oldest life involved and the oldest in the table; and the same number of the terms $\backslash frac\{1\}\{n\}av^n$ for any single life of the same age.

But this labour may be greatly abridged as follows:

PROB. I.

149. Given , the value of an annuity on any number of joint lives, to determine that of an annuity on the same number of joint lives respectively one year younger than them.

Solution.

If it were certain that the lives ABC, &c. would all survive one year, the proprietor of an annuity of L. 1, dependent upon their joint continuance, would, at the expiration of a year, be in possession of L. 1, (the first year's rent,) and an annuity on the same number of lives, one year older respectively than ABC, &c. therefore, in that case, the required present value of the annuity would be , (146.)

But the probability of the lives A, B, C, &c. jointly surviving one year, is less than certainty, in the ratio of to unity; therefore .

150. Corol. 1. When there are but three lives, A, B, and C, this becomes $\backslash Lambda\_\{ABC\}\; =\; \backslash Lambda\_\{ABC\}\; v[1\; +\; \backslash Lambda\_\{ABC\}]$.

151. Corol. 2. When there are only two, A and B, $\backslash Lambda\_\{AB\}\; =\; \backslash Lambda\_\{AB\}\; v[1\; +\; \backslash Lambda\_\{AB\}]$.

152. Corol. 3. And for a single life A, it appears, in the same manner, that $\backslash Lambda\; =\; \backslash Lambda\; v(1\; +\; \backslash Lambda)$.

153. Hence, in logarithms, we have these equations,

Algebraical View. $\backslash lambda\; A\; =\; \backslash lambda\; v\; +\; \backslash lambda\; a\; +\; \backslash lambda(1\; +\; \text{'}A)$,
$\backslash lambda\; AB\; =\; \backslash lambda\; v\; +\; \backslash lambda\; a\; +\; \backslash lambda\; b\; +\; \backslash lambda[1\; +\; \text{'}(AB)]$,
$\backslash lambda\; ABC\; =\; \backslash lambda\; v\; +\; \backslash lambda\; a\; +\; \backslash lambda\; b\; +\; \backslash lambda\; c\; +\; \backslash lambda[1\; +\; \text{'}(ABC)]$,
&c. &c. &c.

PROB. II.

158. To determine the present value of an annuity on the survivor of the two lives $A$ and $B$, (155), which we designate thus, $\backslash overline\{AB\}$.

Solution.

The probability that the survivor of these two lives will outlive the term of $n$ years, was shown in No. 141, to be $\{\}\_n\; a\; +\; \{\}\_n\; b\; -\; \{\}\_n(ab)$; therefore, reasoning as in No. 146, it will be found, that the present value of the $n$th year's rent of this annuity is $[\{\}\_n\; a\; +\; \{\}\_n\; b\; -\; \{\}\_n(ab)]v^n$, and the value of all the rents thereof will be $S[\{\}\_n\; a\; +\; \{\}\_n\; b\; -\; \{\}\_n(ab)]v^n$, or $S\_a\; v^n\; +\; S\_b\; v^n\; -\; S\_\{ab\}\; v^n$; so that $\backslash overline\{AB\}\; =\; A\; +\; B\; -\; AB$ (147), agreeably to No. 48.

PROB. III.

159. To determine the present value of an annuity on the last survivor of three lives, $A,\; B$, and $C$, (155); which we denote thus, $\backslash overline\{ABC\}$.

Solution.

The present value of the $n$th year's rent is $[\{\}\_n\; a\; +\; \{\}\_n\; b\; +\; \{\}\_n\; c\; -\; \{\}\_n(ab)\; -\; \{\}\_n(ac)\; -\; \{\}\_n(bc)\; +\; \{\}\_n(abc)]v^n$ (142 and 146); whence, it appears, as in the preceding number, that $\backslash overline\{ABC\}\; =\; A\; +\; B\; +\; C\; -\; AB\; -\; AC\; -\; BC\; +\; ABC$, agreeably to No. 52.

PROB. IV.

160. To determine the present value of an annuity on the joint existence of the last two survivors out of three lives, $A,\; B,\; C$, (155); which we denote thus, $\backslash overline\{2ABC\}$.

Solution.

The present value of the $n$th year's rent is $[\{\}\_n(ab)\; +\; \{\}\_n(ac)\; +\; \{\}\_n(bc)\; -\; \{\}\_n(abc)]v^n$ (143 and 146); whence, reasoning as in the two preceding numbers, we infer, that $\backslash overline\{2ABC\}\; =\; AB\; +\; AC\; +\; BC\; -\; 2ABC$, as was demonstrated otherwise in No. 51.

161. Since the solutions of the last three problems were all obtained by showing each year's rent (as for instance the $n$th) of the annuity in question, to be of the same value with the aggregate of the rents for the same year, of all the annuities (taken with their proper signs) on the single and joint lives exhibited in the resulting formula: if any term of years be assigned, it is manifest that the value of such annuity for the term, must be the same as that of the aggregate of the annuities above mentioned, each for the same term.

PROB. V.

162. $A$ and $B$ being any two proposed lives now both existing, to determine the present value of an annuity receivable only while $A$ survives $B$.

Upon which it may be observed, that $\backslash lambda\; v\; +\; \backslash lambda\; a$, the sum of the first two logarithms that are employed in determining $\backslash lambda$ from $\text{'}\backslash lambda$, also enters the operation whereby $\backslash lambda\; B$ is determined from $\text{'}(AB)$. And that $\backslash lambda\; v\; +\; \backslash lambda\; a\; +\; \backslash lambda\; b$, the sum of the first three logarithms that serve to determine $\backslash lambda\; B$ from $\text{'}(AB)$, is also required to determine $\backslash lambda\; BC$ from $\text{'}(ABC)$; which observation may be extended in a similar manner to any greater number of joint lives.

154. By these means it is easy to complete a table of the values of annuities on single lives of all ages; beginning with the oldest in the table, and proceeding regularly age by age to the youngest.

Also a table of the values of any number of joint lives, the lives in each succeeding combination, in any one series of operations, (according to the retrograde order of the ages in which they are computed), being one year younger respectively than those in the preceding combination.

And, if a table of single lives be computed first, then of two joint lives, next of three joint lives, and so on; the calculations made for the preceding tables will be of great use for the succeeding.

155. Having shown how to compute tables of the values of annuities on single and joint lives, we shall, in what follows, always suppose those values to be given.

156. Let the value of an annuity on the joint continuance of any number of lives, $A,\; B,\; C$, &c. that is not to be entered upon until the expiration of $t$ years be denoted by

Then, if it were certain that all the lives would survive the term, since the value of the annuity at the expiration of the term would be ,

(145), its present value would be , (146).

But the measure of the probability that all the lives will survive the term is , therefore

In the same manner, it appears, that for a single life $A$, $\{\}\_t\; A\; =\; \{\}\_t\; a\; v^t\; \backslash cdot\; \{\}\_t\; A$.

157. Let an annuity for the term of $t$ years only, dependent upon the joint continuance of any number of lives, $A,\; B,\; C$, &c. be denoted by ; and, since this temporary annuity, together with an annuity on the joint continuance of the same lives deferred for the same term, will evidently be of the same value as an annuity to be entered upon immediately, and enjoyed during their whole joint continuance, we have ; whence,

And for a single life $A$, $\{\}\_t\; A\; =\; A\; -\; \{\}\_t\; A$.

A rent of this annuity will only be payable at the end of the $n$th year, provided that $B$ be then dead, and $A$ living; but the probability of $B$ being then dead is $1\; -\; \{\}\_n\; b$, and that of $A$ being then living $a$, and these two events are independent; therefore, the probability of their both happening, or that of the rent being received, is $(1\; -\; \{\}\_n\; b)\; \{\}\_n\; a\; =\; \{\}\_n\; a\; -\; \{\}\_n\; (ab)$; the present value of that rent is, therefore, $\{\}\_n\; a\; -\; \{\}\_n\; (ab)\; p^n$; whence, it follows, that the required value of the annuity on the life of $A$ after that of $B$, is $A\; -\; AB$, agreeably to No. 60.

163. If the payment for the annuity which was the subject of the last problem, is not to be made in present money, but by a constant annual premium $p$ at the end of each year, while both the lives survive; since $AB$ is the number of years purchase (6) that an annuity on the joint continuance of those lives is worth, the value of $p$ will be determined by this equation, $p\; \backslash cdot\; AB\; =\; A\; -\; AB$, whence we have $p\; =\; \backslash frac\{A\}\{AB\}\; -\; 1$.

164. But if one premium $p$ is to be paid down now, and an equal premium at the end of each year while both the lives survive, we shall have $p\; \backslash cdot\; (1\; +\; AB)\; =\; A\; -\; AB$, and $p\; =\; \backslash frac\{A\; -\; AB\}\{1\; +\; AB\}$.

165. For numerical examples illustrative of the formulae given from No. 158 to the present; see Nos. 66-74.

166. $A$ and $B$ are in possession of an annuity on the life of the survivor of them, which, if either of them die before a third person $C$, is then to be divided equally between $C$ and the survivor during their joint lives; to determine the value of $C$'s interest.

and the sum of these being $\backslash frac\{1\}\{2\}\; (ac)\; +\; \backslash frac\{1\}\{2\}\; (bc)\; -\; \{\}\_n\; (abc)$, the value of $C$'s interest is $\backslash frac\{1\}\{2\}\; AC\; +\; \backslash frac\{1\}\{2\}\; BC\; -\; ABC$.

167. An annuity after the decease of $A$, is to be equally divided between $B$ and $C$ during their joint lives, and is then to go entirely to the last survivor for his life; it is proposed to find the value of $B$'s interest therein.

168. $A$, $B$, and $C$ purchase an annuity on the life of the last survivor of them, which is to be divided equally at the end of every year among such of them as may then be living; what should $A$ contribute towards the purchase of this annuity?

169. As soon as any two of the three lives, $A$, $B$, and $C$, are extinct, $D$ or his heirs are to enter upon an annuity; which they are to enjoy during the remainder of the survivor's life; to determine the value of $D$'s interest therein.

170. The last four may be sufficient to show the method of proceeding with any similar problems.

171. Let denote the probability that the

Algebraical last $m$ survivors out of $m\; +\; \backslash mu$ lives $A,\; B,\; C,\; \backslash dots$ will jointly survive the term of $t$ years. And when $\backslash mu\; =\; 0$, the expression will become $\{\}\_t(\backslash overline\{abc,\; \backslash dots\})$ the probability that the lives will all survive the term (138).

When $m\; =\; 1$ it will become $\{\}\_t(\backslash overline\{abc,\; \backslash dots\})$, the measure of the probability that the last survivor of them will outlive the term; which it will be better to write thus, $\{\}\_t(\backslash overline\{abc,\; \backslash dots\})$, retaining the vinculum, but omitting the unit over it, as in the notation of powers.

Also let $\backslash overline\{ABC,\; \backslash dots\}$ denote the value of an annuity on the joint continuance of the same number of last survivors out of the same lives. Then, if $\backslash mu$ be equal to 0, it will be $\backslash overline\{ABC,\; \backslash dots\}$ the value of an annuity on the joint continuance of all the lives; when $m\; =\; 1$, it will be $\backslash overline\{ABC,\; \backslash dots\}$ the value of an annuity on the last survivor of them. The values of annuities on the last survivor of two and of three lives, will be denoted as in Nos. 158 and 159 respectively; and that of an annuity on the joint continuance of the last two survivors out of three lives, as in No. 160.

The value of an annuity on the last $m$ survivors out of these $m\; +\; \backslash mu$ lives, according as it is limited to the term of $t$ years, or deferred during that term, will also be denoted by $\backslash overline\{\backslash overline\{ABC,\; \backslash dots\}\}^m$ or $\backslash overline\{\backslash overline\{ABC,\; \backslash dots\}\}^m$. (156 and 157.)

172. An annuity certain for the term of $t\; +\; y$ years, is to be enjoyed by $P$ and his heirs during the joint existence of the last $m$ survivors out of $m\; +\; \backslash mu$ lives, $A,\; B,\; C,\; \backslash dots$; and if that joint existence fail before the expiration of $t$ years, the annuity is to go to $Q$ and his heirs, for the remainder of the term; to determine the value of $Q$'s interest in that annuity.

$Q$'s expectation may be distinguished into two parts:

1st, That of enjoying the annuity during the term of $t$ years.

2d, That of enjoying it after the expiration of that term.

The sum of the present values of the interests of $P$ and $Q$, together in the annuity for the term of $t$ years, is manifestly equal to the whole present value of the annuity certain for that term; that is, equal to $\backslash frac\{1-v^t\}\{r\}$ (113 and 146); and the value of $P$'s interest

for the term of $t$ years, is $\backslash overline\{\backslash overline\{ABC,\; \backslash dots\}\}^m$ (171); therefore the value of $Q$'s interest for the same term is

The present value of the annuity certain for $y$ years after $t$ years is $\backslash frac\{v^t(1-v^y)\}\{r\}$ (114 and 146); and $Q$ and his heirs will receive this annuity, if the joint conti-

nuation of the last $m$ survivors above mentioned fail before the expiration of $t$ years; but the probability of their joint continuance failing in the term, is

$1\; -\; \{\}\_t(\backslash overline\{abc,\; \backslash dots\})^m$; therefore, the value of $Q$'s interest in the annuity to be received after $t$ years, is

of $Q$'s interest, is $\backslash frac\{1\}\{r\}\; \backslash left[1\; -\; v^\{y+t\}\; -\; v^t(1-v^y)\; \{\}\_t(\backslash overline\{abc,\; \backslash dots\})^m\backslash right]$

173. Corol. 1. When the whole annuity certain is a perpetuity, $v^\{t+y\}\; =\; 0$, and the value of $Q$'s interest is $\backslash frac\{1\}\{r\}\; \backslash left[1\; -\; \{\}\_t(\backslash overline\{abc,\; \backslash dots\})^m\backslash right]\; =\; \backslash overline\{\backslash overline\{ABC,\; \backslash dots\}\}^m$.

174. Corol. 2. When the term $t$ is not less than the greatest joint continuance of any $m$ of the proposed lives, according to the tables of mortality adapted to them, $\{\}\_t(\backslash overline\{abc,\; \backslash dots\})^m\; =\; 0$, and $\backslash overline\{\backslash overline\{ABC,\; \backslash dots\}\}^m\; =\; \backslash overline\{ABC,\; \backslash dots\}$; therefore, in that case, the general formula of No. 172 becomes $\backslash frac\{1-v^\{y+t\}\}\{r\}\; =\; \backslash overline\{ABC,\; \backslash dots\}$; that

is, the excess of the value of an annuity certain for the whole term $y+t$, above that of an annuity on the whole duration of joint continuance of the last $m$ surviving lives.

175. And if, in the case proposed in the last No. the annuity certain be a perpetuity, as in No. 173, the formula will become $\backslash frac\{1\}\{r\}\; =\; \backslash overline\{ABC,\; \backslash dots\}$ the excess of the value of the perpetuity above the value of an annuity on the joint lives of the last $m$ survivors; agreeably to No. 63.

176. Example 1. Required the present value of the absolute reversion of an estate in fee simple, after the extinction of the last survivor of three lives, $A,\; B,\; C$, now aged 50, 55, and 60 years respectively; reckoning interest at 5 per cent.

The general Algebraical expression of this value has just been shown to be $\backslash frac\{1\}\{r\}\; =\; \backslash overline\{ABC\}$.

Therefore 5.999 years' purchase is the value required. And if the annual produce of the estate, clear of all deductions, were L. 100, the title to the reversion would now be worth L. 599, 18s., agreeably to No. 76.

177. Ex. 2. An annuity for the term of 70 years certain (from this time), is to revert to $Q$ and his heirs at the failure of a life $A$, now 45 years of age; what is the present value of $Q$'s interest therein; reckoning the interest of money at 5 per cent.?

In No. 174, the Algebraical expression of the required value is shown to be $\backslash frac\{1-v^\{70\}\}\{r\}\; -\; A$.

remains 6.69468 years' purchase; so that if the annuity were L. 1000, the value of the reversion would be L. 6694, 13s. 7d.

178. Ex. 3. An annuity for the term of 70 years certain from this time, is to revert to Q and his heirs at the extinction of the survivor of two lives, A and B, now aged 40 and 50 years respectively; the interest of money being 5 per cent., it is required to determine the value of Q's interest in this annuity.

The algebraical expression of the value is,

So that the required value is $\backslash overline\{4.27668\; \backslash text\{\; years\text{'}\; purchase\}\}$; and if the annuity be L. 1000, the present value of the reversion will be L. 4276, 13s. 7d.

IV. OF ASSURANCES ON LIVES.

179. Let the present value of the assurance (77 and 78) of L. 1 on the life of A be denoted by the Old English capital $\backslash mathcal\{A\}$, and that of an assurance on the joint continuance of any number of lives A, B, C, &c. by . Also, let the value of an assurance on the joint continuance of any m of them, out

of the whole number m + $\backslash mu$ be denoted by .

180. And, in every case, let us designate the annual premium (83) for an assurance, by prefixing the character $\backslash odot$ to the symbol for the single premium; so that $\backslash odot\; \backslash mathcal\{A\}$ may denote the annual premium for an assurance on the life of A; the same for an assurance on the joint continuance of all the

lives, A, B, C, &c.; and the annual premium for an assurance on the joint continuance of the last m survivors out of the whole number m + $\backslash mu$ of those lives.

181. Then will $\backslash overline\{\backslash mathcal\{A\}\}$ and $\backslash odot\; \backslash overline\{\backslash mathcal\{A\}\}$,

and , and designate the single and annual premiums for assurances on the same life or lives for the term of t years only.

182. To determine the present value of an assurance on the last m survivors out of m + $\backslash mu$ lives A, B, C, &c. for the term of t years only; that is, the present value of L. 1, to be received upon the joint continuance of these last m survivors failing in the term.

By reasoning as in No. 79, it will be found, that a perpetuity, the first payment of which is to be made at the end of the year in which the last m survivors out of these m + $\backslash mu$ lives may fail in the term, will be

of the same present value as $\backslash left(\; 1\; +\; \backslash frac\{1\}\{r\}\; \backslash right)\; \backslash frac\{1\}\{1-v\}$ pounds to be received in the same event (112 and 146); but, in No. 173, the value of the reversion of such a perpetuity in that event, was shown to be

; whence it is manifest, that

183. Since the annual premium for this assurance must be paid at the commencement of every year in the term, while the last m surviving lives all subsist (83); besides the premium paid down now, one must be paid at the expiration of every year in the term except the last, provided that these last m survivors all outlive it; but the present value of L. 1 to be received upon their surviving that last year is

, therefore all the future premiums are

now worth years' purchase, and the present value of all the premiums, or the total present value of the assurance, is

whence we have

184. Corol. 1. When (t) the term of the assurance is not less than the greatest possible joint duration of any m of the proposed lives, , and the general formulae of the

rebrical two preceding numbers become respectively

185. Corol. 2. In the same manner it appears, that, for a single life, $\backslash overline\{A\}\; =\; v\; -\; (1-v)\; \backslash overline\{A\}$,

186. Corol. 3. Also that $\backslash overline\{A\}\; =\; v(1\; -\; \{\}\_tA)\; -\; (1-v)\; \backslash overline\{A\}$

187. Corol. 4. When the assurance is on the joint continuance of all the lives, the formulae of No. 184 become respectively

And those of numbers 182 and 183,

188. Corol. 5. According as the assurance is in the last survivor of two, or of three lives, the formulae of No. 184 become respectively

And those of numbers 182 and 183 become

and $\backslash odot\; \backslash overline\{ABC\}\; =\; \backslash frac\{1\; -\; \{\}\_t(abc)\; v^t\}\{1\; -\; \{\}\_t(abc)\; v^t\; +\; \backslash overline\{ABC\}\}\; +\; v\; -\; 1$ respectively.

For the values of $\backslash overline\{AB\}$, $\backslash overline\{ABC\}$, $\backslash overline\{AB\}$, and $\backslash overline\{ABC\}$, see numbers 157-159, and 161.

189. Corol. 6. When the assurance is on the joint continuance of the two last survivors out of the three lives $A,\; B,\; C$; the formulae of No. 184 become respectively

Those of numbers 182 and 183,

For the values of $\backslash overline\{ABC\}$ and $\backslash overline\{ABC\}$ see numbers 157, 160, and 161.

190. the value of an assurance on any life or lives for the term of $t$ years, which was given in No. 182, may also be expressed thus:

And this, in words at length, is the rule given in No. 93.

191. When $t$ is not less than the greatest possible joint duration of any $m$ of the proposed lives, the last expression becomes which is also equivalent to the first in No. 184; and, in words at length, is the rule given in No. 97, for determining the value of an assurance on any life or lives for their whole duration.

192. By substituting $\backslash frac\{1\}\{1+r\}$ for $v$ (146) in the last

Algebraical View. is the proposition enunci-

ated in No. 81; $\backslash frac\{1\}\{r\}$ being the value of the perpetuity (112).

193. Examples of the determination of the single premiums for assurances, and of the derivation of the annual premiums from them, have been given in numbers 82-88, also in 95 and 96; but by the algebraical formulæ given here, the annual premiums may be determined directly, without first finding the total present values of the assurances.

194. Example 1. Required the annual premium for an assurance on the life $A$ now 50 years of age, interest 5 per cent.

According to No. 185, the operation is thus,

adding $v\; =\; .9523809$, and subtracting unity,

we have $\backslash odot\; A\; =\; .0313699$, agreeably to No. 85.

195. Ex. 2. What should the annual premium be for an assurance on the last survivor of three lives $A,\; B,\; C$, now aged 50, 55, and 60 years respectively, rate of interest 5 per cent.?

Operation by No. 188.

$v\; =\; .9523809$

$\backslash odot\; \backslash overline\{ABC\}\; =\; .0190431$, agreeably to

No. 88.

196. Ex. 3. Required the annual premium for an assurance for 10 years only, on a life now 45 years of age, interest 5 per cent.

Operation according to No. 186.

$\backslash odot\; \{\}^\{10\}A\; =\; .013985$, agreeably to No. 96.

What has been advanced from numbers 99 to 109, needs no algebraical illustration. (u.)

Showing the present Value of One Pound to be received at the End of any Number of Years not exceeding 50.

(See No. 9—12 of the preceding Article.)

Showing the present Value of an Annuity of One Pound for any Number of Years not exceeding 50.

TABLE III.

Showing the Sum to which One Pound will increase when improved at Compound Interest during any Number of Years not exceeding 50.

(No. 9—12.)

Showing the Amount to which One Pound per Annum will increase at Compound Interest in any Number of Years not exceeding 50.

ANNUITIES.

TABLE V.

Exhibiting the Law of Mortality at Carlisle. (No. 32.)

TABLE VI.

Showing the Value of an Annuity on a Single Life at every Age, according to the Carlisle Table of Mortality, when the rate of Interest is Five per Cent. (No. 65.)

Showing the Value of an Annuity on the Joint Continuance of Two Lives, according to the Carlisle Table of Mortality, when the difference of their Ages is Five Years, and the rate of Interest Five per Cent. (No. 65.)

Showing the Value of an Annuity on the Joint Continuance of Two Lives, according to the Carlisle Table of Mortality, when the difference of their Ages is Ten Years, and the rate of Interest Five per Cent. (No. 65.)