SECT. I. ANNUITIES CERTAIN. THE PRESENT VALUE of an annuity, is that fum, which, being improved at intereft, will be fufficient to pay the annuity. The prefent value of an annuity certain, payable yearly, is calculated in the following manner. Let the annuity be 1, and let r denote the amount of 11. for a year, or 11. increased by its intereft for one year. Then, 1 being the prefent value of the fum r, and having to find the present value of the fum 1, it will be, by proportion thus, ♬ : 1 :: 1: the prefent value of tl. due a year hence. In like manner, will be the present value, of 1 I. due two years I I + ++ THEOR. 4. MP-PX ( r X I Ι r I I I I or = 1 I r I I Hence also the present worth of EXAMPLE. 2 I m r + m Let the rate of intereft be 4 per cent, and the term 5 years; and confequently = 104, n = 5, P 25; and alfo let m = 12, or the interest payable monthly in theorem 4: then the prefent value of fuch annuity of 11. a-year, for 5 years, according as it is supposed payable 1 1. yearly, or 1 half of a pound every half year, or 1 fourth of a pound every quarter, or 1 twelfth of a pound, every month, or 1 twelfth part of a year, will be as follows: =4'5251 ANNUITIES may be confidered as in arrears, cafe each payment is to be confidered as a fum or as forborn for any number of years; in which put out to intereft for the remainder of the term after the time it becomes due. And as 1. due at the end of 1 year amounts to r at the end of ano ther year, and to r at the end of the third year, and to 3 at the end of the 4th year, and fo on; therefore, by adding always the laft year's annuity, or 1, to the amounts of all the former years, the fum of all the annuities and their interefts will be the fum of the following geometrical feries till the laft term ber" -, or till the number of 1 + r + r2 + μ3 + r1 - - - - - - to r" - ', continued terms be n, the number of years the annuity is forborn. But the fum of this geometrical progreffion is which therefore is the amount I r - I of 11. annuity forborn for n years. And this quantity being multiplied by any other annuity perpetuity in reverfion, of 1 1. to commence af-, inftead of 1, will produce the amount of that VOL. II. PART I. 36 63 27594 37 66.17422 70.00760 77.59831 86-16397 95.83632 119:12087 7345787 8170225 9104134 10162814 127-26812 38 6915945 7702889 8597034 96-13820 107.70955 135 90421 39 72 23423 80 72491 40 75401268455028 | 904091510146442 114 09502 145 05846 95 02552 10703032 120.79977 154*76197 41 78-66330 88.50954 99-82654 112 84669 127 83976 165-04768 42 82 c2320 92 60737 104·81960 118.92479 135 23175 175 95054 4385-48389 96 84863 11001238 125-276404299334 187 50758 44 89 04841 101 238331541288 13191384 15114301 199 75803 45 9271986 105 78167 12102939 138.84997 159 70016 212-74351 Yrs.jat 3 per cent. 3 per cent. 4 per cent. 4 per cent. 5 per cent. 6 per cent. 30 19.60044 31 20.00043 18.73628 17.87683 16.89035 18.32703 17:28536 28 18.76411 17.66702 29 19.18845 18:03577 16:98371 18.39205 17 29203 17.58849 16.54439 17.87355 16:78889 15.80268 14:08484 33 20-76579 19.39021 18·14765 17'02266 35 21.13184 1970068 13'41120 32 20 38877 19:06887 16·00255 14/23023 17'24676 A 32 1 14'37519 13.00317 14.64303 13.21053 1574287 14-89813 13:40616 16:02189 16.28889 15'37245 13:76483 Yrs 3 percni per cent. 4 per cent. per cent 5 per cent. 6 per cent. To find the AMOUNT of an ANNUITY FOR PORN any number of years. Take out the amount from the 1ft table, for the propofed years and rate of intereft; then multiply it by the annuity in queftion; and the product will be its amount for the fame number of years, and rate of intereft. And the converfe to find the rate of time. EXAMP. I. To find how much an annuity of gol. will amount to in 20 years at 3 per cent. compound intereft.-On the line of 20 years, and in the column of 3 per cent. ftands 28.27968, which is the amount of an annuity of 11. for the 20, years; and therefore 28°27968 multiplied by 50, gives 14139841. or 14131.-198. 8d. for the answer. EXAMP. 2. In what time will an annuity of 20 1. amount to a 1000l. at 4 per cent. compound intereft?-Here the amount of 100c 1. divided by 201. the annuity, gives 50, the amount of 11. annuity for the fame time and rate. Then, the neareft tabular number in the column of 4 per cent. is 49'96758, which standing on the line of 28, fhews that 28 years is the answer. EXAMP. P. 3. If it be required to find at what rate of intereft an annuity of 20 l. will amount to Jocol. forborn for 28 years. Here 1000 divided by 20 gives 50 as before. Then looking along the line of 28 years, for the neareft to this number I find 49'96758 in the column of 4 per cent. which is therefore the rate of intereft required. 50, EXAMP. I. To find the prefent value of an an, nuity of gol. which is to continue 20 years, at 3 per cent. By the table, the prefent value of 1, for the fame 'rate and time, is 14'21240; there fore 14:21240 X 50710621. or 710l. 128. 4d. is the prefent value fought. EXAMP. 2. To find the prefent value of an annuity of 201. to commence 20 years hence, and then to continue for 40 years, or to terminate so years hence, at 4 per cent. intereft.-In fuch cafes as this, it is plain we have to find the difference between the prefent values of two equal annuities, for the two given times; which therefore will be effected by fubtracting the tabular value of the one term from that of the other, and multiplying by the annuity. Thus, tabular value or 50 years 21°48218 the difference 13°37128 gives pr ་ 20 267°4256 2671.8s. 6d. the answer. SECT. II. CONTINGENT, or LIFE ANNUITIES. THE value of annuities for life is determined from obfervations made of the bills of mortality. Dr Halley, Mr Simpfon, and Monf. de Moivre, are gentlemen of diftinguished merit in calculations of this kind. Dr HALLEY had recourfe to the bills of mortality at Breflaw, the capital of Silefia, as a proper ftandard for the other parts of Europe, being a place pretty central, at a diftance from the fea and not much crowded with traffickers or fo eigners. He pitches upon rooo perfons all born in one year, and obferves how many of these were alive every year, from their birth to the extinction of the laft, and confequently how many died each year, as in the first of the following tables; which is well adapted to Europe in general. But in the city of London, there is obferved to be a greater difparity in the births and burials than in any other place, owing probably to the vast re*fort of people thither, in the way of commerce, from all parts of the known world. Mr Simplon, therefore, in order to have a table particularly fuited fuited to this populous city, pitches upon 1280 perfons all born in the fame year, and records the number remaining alive each year till none were in life. TABLE III. From the preceding tables the probability of the continuance or extinction of human life is eftimamated as follows. I. The probability that a perfon of a given age fhall live a certain number of years, is measured by the proportion, which the number of perfons living at the propofed age has to the difference DR HALLEY'S TABLE on the BILLS of MORTA- between the faid number, and the number of perLITY at BRESLAW. fons living at the given age. Thus, if it be demanded, what chance a perfon of 40 years has to live feven years longer? From 445, the number of perfons living at 40 years of age in Dr Halley's table, fubtract 377, the number of perfons living at 47 years of age, and the remainder, 68, is the number of perfons that died during these 7 years; and the probability, or chance that the perfon in the queftion fhall live thefe 7 years, is as 377 to 68, or nearly as 5 to 1. By Mr Simpfon's table, the chance is fomething lefs than that of 4 to 1. II. If the year, to which a perfon of a given age has an equal chance of arriving before he dies, be required, it may be found thus: Find half the number of perfons living at the given age in the tables, and in the column of age you have the year required. Thus, if the question be put with respect to a perfon of 30 years of age, the number of that age in Dr Halley's table is 531, the half whereof is 265, which is found in the table between 57 and 58 years; fo that a perfon of 30 years has an equal chance of living between 27 and 28 years longer. III. By the tables, the premium of insurance upon lives may in fome measure be regulated. Thus, the chance that a perfon of 25 years has to live another year, is, by Dr Halley's table, as 80 to 1; but the chance that a perfon of 50 years MR SIMPSON'S TABLE on the BILLS of MORTA- has to live a year longer is only 30 to 1. And, LITY at LONDON, Perf 59 45 789 45678 34 confequently, the premium for infuring the former ought to be to the premium for infuring the latter for one year, as 30 to 80, or as 3 to, 8. In order to facilitate computations of annuities on lives, Monf. de Moivre affumed the age of 86 as the utmost probable extent of life, and fuppo54 fed that equal numbers died at all ages, fo that of 86 perfons born at the fame time, it might be expected one would die each year till the whole be extinct. The difference between any propofed age and 86 is called the complement of life. 49 53 180 77 38 564 30 385 555 22 32 367 56 9 532 33 583 57 ISI 81 26 58 23 11 517 35 340 59 12 510 36 331 60 504 37 322 61 38 313 62 117 86 12 63 III 87 ΙΟ 498 492 39 304 16 486 40 297 17 480 41 284 65 18 474 42 274 19 468 43 264 66 67 4 nual payment depends on the continuance of life, we muft multiply each of thefe terms, by the number expreffing the probability, that the annuitant thall live to the end of 1, 2, or more years, Now, if n represents the number of perfons alive at any age, and a the number of perfons that die in the first year thereafter, then the number of perfons |