and ^ X 500 , so that the entire value of the bequest is 300 X p 25 + 400 X p 35 + 500 X p 45 p20 From this example we perceive the convenience of having a table of the values of pay a as well as of their logarithms. ANNUITIES. 28. When a fixed payment is to be repeated annually during the life of a specified nominee it is called a life annuity; such an annuity is the aggregate of a number of endowments, and its value is to be computed as such. If the annual payment be to begin to-day, as in the case of an annual premium, on account of a nominee aged a years, I call it an annuity for age a; but if it be to begin twelve months hence, as in the case of an annuity purchased from an office, I call it an annuity deferred one year. In this respect I differ from some writers on the subject who place what we may call the sero point of time one year hence, and say of an annuity beginning two years hence that it is an annuity deferred one year. The propriety of placing the zero point at the piesent moment does not stand in need of any argument. I shall therefore hold an annuity of which the first payment is to be made n years hence as an annuity deferred n years. 29. An annuity of £1 payable during the life of a nominee A may be regarded as the aggregate of separate endowments of £1 to be paid now, and 1, 2, 3, etc., years hence till the end of life; its present value, there- fore, is , »(« + ]) p (a + 2) p (a + 3) l + £-i - + i--^ i+^^--^ ^ + ete., or pa pa pa pa+p(ffl + l)+p (a + 2) +p (a + 3) + etc. _ pa ' this formula may be written more concisely by help of the usual symbol ANNUITIES. 19 for summation ; adopting the obvious notation ann a for the value of an annual payment during the life of a person aged a we thus have 2»a ann a = —^— . pa 30. In order to use this formula conveniently we must prepare a table of the values of 2 p a. Now we observe that 2 p 100 means the sum of all the p a's from 100 to the end of the table, that is 2 p 100 =p 100 +p 101 + p 102 + etc., while 2p 99 =p 99 +p 100 + etc., that is 2p 99 =p 99 + 2p 100 : therefore the summation has to be made from the bottom of the table up- wards. The writing in this way is somewhat troublesome. The best pro- cess is to take the sum of each group of ten or of five lines and then the total of these sums, this total is 2 p , and has to be written in the first line of the column titled ipaya. The column pay a is then placed beside it and the values of 2 pi, 2p2, etc., found by subtraction, care being taken to check the work at each tenth or fifth line by help of the partial sums previously found. In this way a careful computer may avoid the necessity of having the work done in duplicate. 31. The values of 2pa are very frequently used as factors or divisors, and on that account we next proceed to make a table of their logarithms ; the title of this table is hog ipaya. 32. Placing now the columns Log pay a and Log 2 pay a side by side we take their difference and so obtain a table containing the values oiLog anna. For a reason to be explained afterwards, it is not requisite to perform this subtraction in duplicate. 33. Having now obtained the logarithms of the immediate annuities we take out the corresponding natural numbers and form the table Annuity a. 34. The value of a deferred annuity is obtained exactly in the same way. Thus the present value of an annuity to begin n years hence, and to continue during the life of a person now aged a years, is, evidently, , „ 2p(a + m) ann a aei". n years = —^-^ ' •' pa 35. Hence in order to make a table of the values of annuities first pay- ment one year hence we place the column Log pa alongside of the column Log 'S.pa, arranging the latter one hne higher than the former, so that p 5 may be opposite 2p6, and subtract as before; the remainders form the table Log annuity deferred 1 year. 20 ANNUITIES. 36. We now take out the natural numbers for these logarithms, and so obtain the table of annuity deferred 1 year. 37. Since the present value of an annuity of £1 deferred one year is just £1 less than that of an immediate annuity, a comparison of the two will serve to detect any error in the computation of either, and this check is much more severe than a duplicate calculation would have been. 38. If we wish to construct a table of the values of annuities deferred 5 or any other number of years, we have only to place the column Log 'Spa five (or the corresponding number of) hues higher than Log pa, and to perform the subtraction as before. It is in order to facihtate this displacement that the division of the ruled spaces has to be made with care. 39. It is easy to obtain the values of short-period annuities by subtract- ing the deferred from the whole life annuity. Thus the present value of an annual payment of £l beginning to-day and to continue for n pay- ments is ann a - ann a deferred n years . The same value may be obtained directly from the commutation tables thus. Short ann . of n pay' = -^ ^—^ ' ; ' ■' pa in the course of this calculation we get the logarithm of the short annuity, and the one process serves as a complete check upon the other. 40. The value of an intercepted annuity is the difference between two deferred annuities ; thus an annuity of £1 to begin n years hence and to consist of t payments is the excess of an annuity deferred n years, over an annuity deferred u + t years. It is also given by the formula Sp(a + n)-'Sp (a + n + t) pa and we have thus two independent computations. 41. It often happens that an annuity is payable half-yearly, sometimes quarterly, monthly, or even weekly; and we now proceed to consider these cases. Taking first the case of half-yearly payments, let us endeavour to com- HALF-YEAELY PAYMENTS. 21 pute the present value of £l payable at each half year. That value is, evidently, expressed by the formula pa+p{a + ^)+p{a + l)+p{a + ^) +etc. pa and the only subject for consideration is the interpolation of the values P(« + i)'/'(« + f). etc. If we adopt the rudest scheme of interpolation and suppose that p{a + ^) is the arithmetical mean between pa and p{a + 1) , the calculation becomes easy, since, in that case, we may write for the numerator of the above fraction pa + i{pa +p{a + 1)} +p{a + 1) + i{p{a + 1) +p{a + 2)} + etc. which, since pa decreases to be zero at the end of the table, is ^pa + 2 p{a + 1) + 2 p{a + 2) + etc. = 2i.pa-\pa wherefore the value of the half-yearly payment is 2^pa-\pa ^^.^^^^ pa and consequently the value of an annuity of £1 per annum payable half- yearly, that is of £0,5 payable every six months, is arm a-i This is the formula commonly given ; it is, in the present state of our knowledge of the law of the decrement of human life, sufficiently exact for business purposes ; but the intelligent student will readily perceive that there is no more reason for supposing that p{a + -g) is the arithmetical mean between pa and p{a + 1) than there is for assuming that p{a + 1) is the mean between p{a + ^) and p{a + %); he will then scarcely rest satisfied with the preceding investigation. 42. The Values of p(a + ^), p(a + f), etc., will be more satisfactorily obtained if we take into consideration the higher orders of differences. These give us p{a + ^)=pa +^bpa -^^i^pa +iif^^F« -etc. p(a + f) = p(a + l) + i 3p(a + l) - i i h^p{a + V) + \ i f b^p{a + l) - etc. etc. etc. etc. etc. etc. wherefore the sum of these is "2 pa + ^'S8pa-^'2d^pa + -^2d'^pa- ^^-g 2 d'^pa + etc. Now 2 8pa= -pa, 2 d^pa - - dpa and so on, wherefore the entire numerator becomes 2 'Spa-^pa + ^ dpa--^-g S^pa + y|^ 6^ pa - etc.. 22 HALF-YEARLY PAYMENTS. and consequently the present value of an annuity of £1 payable half- yearly, first payment now, is half-yearly anna= ^pa-ipa + ^Spa-^&^pa + et,. * 16 pa the fractional part showing the correction which ought to be applied to the usual formula. For the purpose of obtaining some idea of the actual amount of this cor- rection we may propose the case of an annuity payable half-yearly during the life of a party now aged 15 . Here we have a pay a Ipa Ppa Ppa 15 4 043,73 - 142,08 + 3,03 + 1.17 16 3 901,65 - 139,05 + 4,20 17 3 762,60 - 134,85 18 3 627,75 whence the correction is ' — ' ' which amounts to - ,002 21 , so that, ultimately, the value of the half-yearly annuity is awn 15 = 23,581 99 -i =-,25 correct" = - ,002 21 half yearly ann 15 = 23,329 78 43. This result may be obtained from tables of deferred annuities ; thus, using my Life Tables, vol. i., we find Age defd. Annuity. 1st diff. 2ddiiF. 3d diflf. 15 1 2 3 23,582 22,582 21,617 20,687 - 1,000 - ,965 - ,930 + ,035 + ,035 ,000 whence ann 15 deferred | year = 23,582 - ,500 - ,004 . = 23,078 HALF-YEARLY PAYMENTS. 23 and consequently the value of the half yearly annuity is i{23,582 + 23,078} =23,330 as before. It is thus apparent that while the ordinary method of interpolation induces, at this part of the table, no very serious inaccuracy, it still affects the results in the third decimal place. 44. When the payment of a half yearly annuity is only to begin at next term, that is when it is deferred half a year, one payment must be sub- tracted from the above value, the result being half yearly ann a, defer* ^ year = anna - 1 . 45. It may be proper here to introduce a formula for ordinary inter- polation, which is very convenient, and, at the same time, is not given in every treatise on algebra. If we wish to divide the interval between any two quantities A and B into n equal parts, that is if we wish to insert w - 1 arithmetical means between A and B, the simple process is to take the nth part of the differ- ence B- A and add it successively thus A; A + ^{B-A); A + ^{B-A); etc. by changing these into the form ^{nA + OB]; ^{{n-l)A + lB]; ^{{n-2)A + 2B]; etc. we render them more convenient for our present purpose. 46. When we have to ascertain the value of an annuity payable quar- terly, we proceed just in the same way ; the formula being „ , , pa +p(a + i)+p(a + h)+p(a + i)+p(a + l)+ etc. Quarterly ann a = — ' — — r — as is apparent from the nature of the question. Using the formula of article 45 , the numerator of this fraction may be put under the form \{ipa + Op(a + l)}+l{Spa + I p{a + V)] + \{2 pa +2p{a + l)} + l{\pa + 3p{a + V)}+ i{4p(a + 1) + Op(a + 2)} + \{Bp{a + 1) + lp{a + 2)} + i{2p(a+l)+2p(a + 2)}+i{lK« + l) + 3K« + 2)}+ete. on the supposition that the values of p{a + \), p{a + i), p{a + 1) are three arithmetical means inserted between pa and p{a + 1) . The term pa occurs in this expression 4 + 3 + 2 + 1, that is 10 times, 24 QUARTERLY PAYMENTS. but every other term as p{a + 1) occurs 16 times, viz., + 1+2 + 3 + 4 + 3 + 2 + 1 times ; hence the result is Iffg +y(a + 1) +p(a + 2) +p{a + 3) + etc. pa , . 'S.pa-%pa that IS — ^-^- = ann a-% . pa " 47. If the quarterly annuity be to begin one term hence we have Quarterly ann a deP 3 months = ann a - f . 48. The above investigation has been founded on the supposition that the value of pa decreases uniformly during the year. If we wish to take into account the variation of the decrease we must use the higher orders of differences. These give pa =pa p{a + \) = fa + \ipa + ^ =^b^pa + ^^=^h^pa + etc. p{a + -^) =pa + -^hpa + -^ ~ b'^pa + \ ^ ^ b^pa + etc. p{a + \) =pa + -^bpa + ^ ~- ^^/>« + t ^ t|- ^^ pa + etc. for the first year ; the sum of these is 4pa + f 3pa--j^ b^ pa + -^-^ &^ pa +etc. and there must be a similar sum for each succeeding year, wherefore the whole numerator is 4 2pa + f 2 ipa-^^ 2 h^pa + ^'^ 2 S^pa-eic. so that Quarterly anna^ ^^^-^y^-^^^^^-^^^^^^^*^- : ann a - § + bbpa-^h'^pa + etc. 64 ^a The first and second terms of this expression agree with the usual for- mula, the remaining terms show the correction on account of the higher orders of differences ; this correction is about ;^th part more than in the case of half-yearly payments. 49. Instead of discussing next the subject of monthly payments, I shall WEEKLY PAYMENTS. 25 take the general case of t payments during the year. The general expres- sion for the value of such an annuity is •pa +p{a + y) + p{a + y) + p{a + 7-) + ^^^- +p{'' + 1) + ^*c. tpa Assuming that the values oi p{a + Y), p(a + 7-), etc., are arithmetical means between pa and p{a + 1) we have pa =t{* P*^ +0^(ffl + l)} f(«+I) =!{(«- 1)/"* +iK«+i)} P(.<^ + t) =|{(«-2)p« +2p{a + l)} p{a + l ) = \{tpia + -i) +0K« + 2)} p{a + l|) = !{(*- 1) p{a + 1) + 1 p(« + 2)} etc. etc. the sum of which is \{{t + {t-l) + (t-2)...l) pa + {1 + 2 + 3. ...... 3 + 2 + 1) 2p(a + 1)} , or *-Y- pa + ip(a + l) + tp(a + 2) + etc. = fspa ^pa wherefore the value of an annuity payable t times during the year is t^pa-*-^pa (_i = ann a--^ tpa " and if the first payment be deferred for one term the value becomes t+i ann a — -^ . 50. By making « = 12 we obtain the value of an annuity of £1 payable in monthly instalments to be Monthly ann a = ann a-\^ and Monthly ann deferred one month = ann a - |-f . 51. So also by putting «=52, or i = 365, we shall obtain the values of a weekly or daily payment. 52. An annuity payable so frequently as once a day occurs only in alimentary arrangements by which the payments, though made weekly. 26 ALIMENTS. monthly, or even quarterly, yet are made for the fraction of the term during which the nominee has lived, although he may not have lived to complete the term. In this case we may suppose t to be indefinitely large, so that the frac- tion ^ comes to be I , and thus Aliment a - ann a - i . 53. In the preceding investigations we have neglected the higher orders of differences, and the result will consequently deviate from the true value. If we divide the year into t parts, and wish to compute the value of pa at the end of the s'* of these terms, we have the formula f'(« + t) = />« + f ^Z'" + T ^ «> + T ^ '-ir «'P« + etc- and, in order to find the value of all the payments to be made in the first year, we must assign to s each value, 0, 1, 2, ... up to t-\, and then sum the whole. For the purpose of this summation we may write the above expression thus K'' + t)=P^+T ^ pa{s)+ ~ b^pa{^-st) + -6P- ^^pai^-Zs^t + lsf) + 2^ h^pa{s^-G^t+lls^f-Qsf) or, arranging those terms which contain the same power of s in one group + «' lir ^^P<^ - ^ «*P« + w «'P« - etc.} + s*f2ii5^V-w.«'F«-etc.} + «^{T^^^P«-etc.| p{a + \)=pa^^{hfa-\h^pa + \h^pa- \ h^pa^ \ 3«j9a-etc.} + fI +ia2pa-f S3/>a + HS>a-T¥i7^^pa + etc.} + %{ +\h^pa-^h''pa + -^h^pa-Qi^\ + |^{ +i>3r3*;ja-T35^«*pa + etc.} + f{ +TiT7^^i'a-etc.} etc. etc ALIMENTS. 27 Now we observe that the sums of the powers of the natural numbers from up to « - 1 are (-1 t 2{0 +1 +2 +...{t-l)}- 1 2 ■ I t_ it- 1 2 3 2{0 + 12 + 22 +...(«- 1)2} =1: 2{0 +P + 23+...(«-l)3}='- 2{0* + l* + 2*+...(«-l)*}=-^-! 1?^^' ?il:^' 2{0* + 15 + 2«+...(if-l)5}: wherefore the total value of a payment of \th part of £1 payable at intervals of -^th part of a year, beginning at the date a and ending at date 1 t_ t-_i t^ 2 nr 1 t-i • —r- IS pa + ^f {Spa-^ d^pa + etc] *j{i&^ pa -etc.] + t- 1 t-i t 2t 2* + ^' ^ 24 ^^^i^ ItJtt S^;'* - etc.| + etc. When the number t becomes indefinitely great, the above expression merges into this one pa + ^{dpa-id^pa + id^pa- i d*pa+ i a^^^a-etc.} + 3 { +^S^pa + id^pa + ii S*pa - ■fy')^ d^pa + etc.} + i{ + i d^ pa - -^ d^ pa + ^^^d^ pa -etc.} + i{ +-iT^^pa-^T^h^pa + eic.} + i{ +xk^^ pa- etc.} which on being summed gives pa + i &pa -^ d^pa + -^8^ pa- ^V^ d* pa + -^hr d^ pa - etc., as the value of the payments to be made in the year a to a + 1 . If we now take the sum of this and of the similar expressions for the succeeding years, and observe that 2dpa= —pa, :sd'^pa= —bpa, and so on, we obtain as the true value of an aliment of £1 per annum payable during the life of a person aged a years, the expression AT i. 'S.pa-\pa + ^hpa-^-^i^pa + ^\b^pa--^b^pa + etc. Aliment a = pa , dpa-^d^pa + i^d^pa-:^8^pa + etc. = ann a-i + -^ — — — — — — ^ ^^ pa 54. The numerator of the above fraction expresses the sum of an 28 INCREASING ANNUITIES. infinite number of infinitely small payments made during the life of the nominee, and is thus what, in the language of the higher calculus, is called an integral ; it may therefore be appropriately represented by the usual symboiy for integration, and our formula would then be fpa.?)a Aliment a ="^ • pa 55. Ai-THOUGH SHORT period or intercepted ahments very rarely occur in business, it may be proper to indicate the process by which their values may be computed. If an aliment be to commence when a person now aged a shall attain the age of 6 years, and thereafter continue during his life, its present value is obviously deferred aliment = ^^^ pa If the aliment be to be paid between the ages a and h, its value is fpa .9a- fpb . 3 a short period aliment = ^ pa And, lastly, if the aliment be payable to a person now aged a, so long as his age is between 6 and c years, the present value is fpb .3a- fpc . 3 a intercepted ahment = ^^ ■ • ■^ pa INCREASING ANNUITIES. 56. When the payment is to increase at each successive term, by some constant quantity, it is said to be a uniformly increasing annuity, or more shortly an increasing annuity. If, for example, beginning with £40 to- day, the annuity be to be £42 next year, £44 the year after, and so on, increasing £2 every year during the life of a nominee now aged a years, the general formula for the payment n years hence will be £40 + 2 n, and the present value of the whole is the sum of the progression 40pa + 42p(a + 1) + 44p(a + 2) + 46p(a + 3) +etc. pa INCREASING ANNUITY. 29 and in general if we put s for the sum payable at the beginning, and i for the annual increment, the present value of the annuity is spa + (s + i) p(a + !) + (« + 2i) p(a + 2) + etc. incr. annuity = —i- ^^ '-^^ — ^^ '-^-^ •' pa. The numerator of this fraction may be divided into two parts thus, s 2^a + i{lp(a+ l) + 2p(a + 2) + 3p(a + 3) + etc.}, so that the value of the increasing annuity becomes . 1 jo( a + ] ) + 2ff(g + 2) + 3ff(a + 3) + etc. pa and thus we have only to seek for a convenient method of obtaining the numerator of the latter fraction. Now if we had the sum of p80 + 2^81+3p82 + etc. to the end of the table, we could obtain p 79 + 2 j9 80 + 3 ;? 81 + 4p 82 + etc. by adding p 79 + p 80 + j9 81 + etc. to it, so that jo 79 + 2 p 80 + 3^9 81 + etc. = j9 80 + 2 p 81 + etc. + 2 p 79 and in the same way p 78 + 2p 79 + 3p 80 + etc. =p 79 + 2^ 80 + etc. + 2 p 78 = p80 + 2p81 + etc. + 2p79 + 2p78 and thus we see that lp(a + 1) + 2 p{a + 2) + etc. = l.p{a + 1) + 2 p{a + 2) + 2^(a + 3) + etc. The numerator of which we are in search is thus the sum of the sums of p{a + 1), p(a + 2) , and so on to the end of the table. The sum of the sums of any series of quantities is called their second sum, and its symbol is 22 or 2^ . The use of this mode of notation puts our formula thus ; — , .2V(a + l) Incr. ann = s. ann a + i — ^-^ ' . pa We have therefore to make a table of the values of 2^ pa by adding up the column 2 pa from the bottom upwards, exactly as we formed the column 2 pa from pa; and, because these second sums occur very fre- quently in multiplication and division, it is convenient also to make a table of their logarithms. The multiple of the value of the simple increasing annuity, viz., of £l one year hence, £2 two years hence, and so on, added to that of the con- stant annuity, gives us the present value of any proposed increasing annuity ; its formula is 2^p(a + l) Incr. ann a = — *-^ ■ pa 30 INCREASING ANNUITY. and the logarithms of its values are easily obtained by placing the column Log ^pa beside Log fa, but one step higher, and subtracting. The beginner will do well to observe that if we place the two columns at the same height we obtain the logarithms of which expresses the value of £1 to-day, £2 one year hence, £3 two years hence, and so on ; which would need to be added to the present value of the annuity of £ (s - 1) per annum in order to give the value of £ s down, £ (s + 1) one year hence, and so on. 57. To compute the value of an annuity of £ s beginning n years hence with an annual increase of £ i thereafter, we use the formula s 2 f{a + m) + « J?"p{a + m + 1) pa or the equivalent one (s - i) 'S.pia + n) + i li' p{a + n) pa as is quite evident from the preceding investigation. 58. If an annuity of £ s to increase by £ i for n years and thereafter to remain constant were proposed, we have s'S,pa + i{'2'' p{a + V) -"2^ p{a + n+V)} pa the subtraction of '%^p{a + n + l) stopping the further increase, and so keeping the latter part of the annuity constant. 59. But if an annuity of £ s to increase annually for n years by £ i, and thereafter to cease altogether, were proposed, we must not only stop the farther increase, but must stop the annuity altogether, wherefore we must subtract {s + ni)'S,p{a + n) as well as i2^p(a + w + l); the formula thus becomes s'S,pa-{s + ni) •2,p{a + n)+ i{'S?p{a + 1) - 2'p(a + w + 1)} pa 60. Again, we may have to compute the value of an annuity beginning with £s and decreasing by £i every year; but in this case we cannot apply the converse formula sl.pa-iy?p{a + l) pa ASSUEANCES. 31 without circumspection, because if the multiple of i, ni, were to become equal to s, there would be no payment at the w'* year, while at the {n + iy* year the formula would imply a repayment by the annuitant. Wherefore in computing the value of a decreasing annuity we must con- sider that it naturally stops at the -j = n** term. However, if the quotient 4- give a value of n which would carry us be- yond the limits of the life table, the above formula is applicable. Other- wise the problem must be held as belonging to the following proposition. 61. To compute the value of an annuity beginning with £ 5 and decreas- ing annually by £ i for n years, and then ceasing entirely ; it being understood that n is less than 4 . In this case the formula is merely a transcript of that of article 59, changing the sign of i, thus s'2pa-{s-ni)'2 p(a + n)- i{2' p{a + 1)-!,' r(a + n + l)} pa ASSTJRAXCES. 62. When a sum of money is to be paid at the death of a specified individual it is said to be assured, that is assured to the heirs or family of the deceased ; sometimes also such a payment takes the name of a succes- sion or a reversion ; but, throughout this part of the treatise I shall use the word assurance to denote a sum of money payable at the death of the nominee, whether it be payable to or by the heirs. The leading problem of the present section is to compute the value of a given sum of money payable at the death of the nominee. 63. In order to examine thoroughly the general problem, let us restrict the period during which the sum is payable, and seek to compute the pre- sent value of £1 payable at the death of a nominee aged a years, provided that death happen between the ages a + n and a + n + 1 years, or putting Aiov a + n, between the ages A and A + 1 years. If each one of those entered in the life table as alive at age a were nominated, we should have la transactions, and it is obvious that, on account of the whole of these transactions, I A - 1{A + 1) payments would have to be made, namely, one for each person that dies during the year. Now these payments are to be made during the specified year, some of 32 ASSUEANCES. them near its beginning and some near its end, so that, one with another, we may hold them as payable w + 1 years hence. In this way the present value of the expected payments may be taken as {lA-liA + l)}r—i , and this being the present value of the payments on account of la trans- actions, that of one of them must be {lA-l{A + l)}r-''-i la 64. Here as in former cases we have assumed that the middle of the year gives the true average ; but this assumption is quite gratuitous, for neither are the deaths uniformly distributed through the year, nor is the discount for the middle of the year an arithmetical mean between those for the beginning and for the end of it. In order that we may see the separate effects of those two inequalities I shall first suppose that the deaths are uniformly distributed, and take the true discount, and afterwards take note of the unequable distribution. 65. In the first place, then, if the deaths happen uniformly during the year, and if the year be divided into t equal parts, the number of deaths happening in each of those parts must be ^{lA-l{A + l)}=-\dlA, and if we regard the payments as to be made at the end of each period the total value becomes _i- -i. -£ _JL -J 8lAr-''^r'+r*+r*+ ,,. r'l this total being evidently too small, while -J. __£ «-i --^SlAr-^^r" + r* +r* +...r * | , formed on the supposition that they are made at the beginning of each period, is too great. By supposing the number of periods to be indefinitely increased these two values may be brought nearer and nearer to each other and to the true result, so that in seeking the ultimate limit we may use either of them. Our business then is to discover the amount of the geometrical progression when t is indefinitely great. Taking the sum in the usual way we find A JLil tr r ' — 1 ASSURANCES. 33 On substituting 1 + p for r this expression becomes, when the denomina- tor is developed by the binomial theorem, , l-« „ 1-41- 2«q, , , '•{P + ^p' + ^-sr? +etc.} which, when t is made indefinitely great, is P P »'{p- ^ p^ + ^ />^- etc.} r.neplogr and therefore the value of the expectation is -blA .r--"-' 4 • nep Log r The above fraction -, — may be put m the form r nep log r ^ ^ or 0- + p){p-ip^ + i p'-etc) 1 1 + rb P - 2^ /"^ + rh ''' - 475 f-* + etc. which enables us to contrast the true result for deaths equably distributed through the year, with the discount to the middle of the year; for then instead of the above fraction we have (1 +p)"i or 1 1.1., 1 ■ 1 2 I 1 ■ I ■ 3 3 1 . 1 . 3 . 5 . , ; '■ + 2 P 2.4:P ^2.4.6" 2.4.6.8^ + *^l^' the true result being thus somewhat greater than that obtained by the common average. 66. Let us now take into account the varying rate of mortality during the year. It has been shown in article 9 that the number alive at any fractional age is given by the formula lw{A + }) = livA+^dlA + ^^ SUA + ^^ '-^bHA + eto. and therefore the number of those that die in each successive part of the year will be found by attributing to s the values 1 , 2 , , . . to f , and taking the differences between the results. Making, then, one step backwards we have liv {A + -V') = IA + ^HA + ^ i^' &^ la + etc. so that the number of deaths occurring between the dates A + ^ and A + -j is found by subtracting the former from the latter of these two expressions. 34 A8SUEANCES. If we develop the former expression according to the powers of j we obtain i,A Sv ,A Sf,T. IIHA 21HA 68< lA ^ 248° lA Z(.l + -) = Z^ + -^|aZ^-^ +J7273- im+TTTS-^t^- «2 C 18' '-1 3^8 Zyl 118* M 50S»L1 «s ( 1S'?4 68* M 358'' M . + IT 1 + 1-7271 - TTTT + TTTS - e*°- 18* M 108» lA 5 185 Zyl ( IS* lA \W i + TTTi-TT + etc, + etc. which, for the sake of shortness, we shall write and similarly l{A + '-^) = lA + '-^ B + ^^ C+^' i> + etc., whence the number of deaths in the interval before ^ + ^ is -5+^^C+ z:3£i+3£^ 2) + -^^' + ?:-"'+' J^ + etc.. and this represents the number of payments that have to be made during this interval. The values of these payments, estimated as at the begin- ning of the year, may be obtained by multiplying the above expression by s - 1 s some factor between r ' and r * , the one giving a result slightly too great, the other a result too small. As we shall afterwards suppose t to be indefinitely great, these two values will coalesce, so that it becomes a matter of indifference which of them we take ; I shall take the latter. Again the sum of any powers of the natural numbers 1, 2, 3 ... up to t contains the next higher power of ^, so that the sum of such fractions as \f will have t^ both in the numerator and denominator, and will thus have a finite value when t is supposed infinite : whereas such fractions as \, will give, when summed, i* in the numerator, and so their sum becomes zero when { is infinite. Hence we may neglect all but the first part of each of the above numerators, and write {|5 + ?^C-H^ D + ^E^ etc.| ; as the value of the payments in one fraction of the year discounted to its commencement. In order to find the corresponding value for the whole year we must ASSUEANCE8. 35 assign to s every value from 1 to t and sum up the whole of the results ; this sum becomes _^ ll" r'^ + 2' r'^ + i' ,--T+ ... t' r~^ _i_ _2_ 3^ _t^ - etc. + etc. and thus the solution of the problem which we have proposed requires the summation of series having the general form 1 -2 _-i -1. -Izl J. and the subsequent determination of the value to which this sum approaches when t is made indefinitely great. Taking these series in succession, if we put 1-1 r ' + J* ' + ... r multiply by r ' and subtract, we obtain as above r-1 «« = *•" r' -1 and the limit to which -^ approaches when t becomes infinite is r . nep log r " putting Sq to denote the limit. Again, if we make i_ i_ -ini * 1 r-~*'+ 2 r" *+...(«- l)r * +tr~^ =Sj^, multiply by r * and subtract, we find 1 -1 -— -1 - (l + 7-"'+r * + ...r *)-tr =(r*-l)Sj^, in which the series inclosed in the parenthesis is «„ r- ' , so that 36 ASSURANCES. By inserting in this the previously found value of Sg we may express s^ in terms of r and t ; but as our present object is to discover the limit of 1^ , it may be better to put this fraction in the form i_ t^~ 1 t{r' -1) and to observe that while the limit of r* is unit, and of f(r* -1) is nep log r, that of j is as above ; so that ^ ^ ^-1 1 ''■ r{nep log rf r . nep log r I Similarly if we put s^ for the third series, multiply by r * and subtract, we obtain (2-l) + (4-l)r"'+(6-l)>-"«+ ,..(2«-l)*-" ' -t^ r =(/•« -l)s^ or 1 2 - 1 i r ' (2«j -s^-t r = (r « - 1) ^2 , whence 12£l--£5 «2 *•* t' -r-^ t(r' -1) Now here we observe that while the limit of |1 is finite, that of ^ must be zero when t becomes indefinitely great, wherefore s - 2(^)-^'' -3 ^-1 2 L_ ^ nep . log r r{nep log rf r{nep log rf r , nep log r or putting H = nep log r Further, if we put S3 for the sum of the fourth series, and proceed as before, we have - i - ? -Izk L (P-0) + (23-P)r" « +(35-23)r « +...(<3-(f_l)3)r ' -tV-" =(r«-l>, or, since p^-{p- 1)^ = 3p^ -3p + l, ± i^ r* (Ssj -3fii +sj-<3r-' = (»-'-l)s3, that is ASSURANCES. 37 Sg = j , of which the limit is t{r* - 1) r a , of which we seek out the logarithms. In order now to form a table of the present values of Assurances we place the columns Log pa and Log i ma side by side, and subtract the former from the latter, the difference is the Log assurance, from which the values of assurances can at once be found. 70. According to this notation the symbol die a or da is equivalent to -ila, and therefore dda = -&^la ; S^da = '- d^ la, and so on ; wherefore the true value of m^r a becomes mora = r-'^{da{S^)-^ {1S„-2S^) + ^^{2S^ -GS^+3SJ ^' '^^ (65'o - 22S^ + 18S^ - 4S^) + etc,} 1 ...4 71. Hence the sum of the mortuary payments for the whole range of the life table takes the very involved form 5„ 2da.r-»- (l/S^o - 25,) 2 \^ r-« + (25'„ -6S^+BS,p^^r-<^-etc. the first term of which differs, as we have already seen, very slightly from 2£ia.r-°i. 72. The above sum of the mortuary payments may be regarded as a common integral of which the usual symbol is yda .r-'^da or -f'bla.r-'^. 73. The value of a deferred assurance is very easily obtained : thus if 40 AS8UEANCES, £1 be to be paid at the death of a nominee aged a, provided that death happen after n years, we have only to consider the mortuary payments after age a + n, and thus the value is , „ , 2 mor (a + n) deferred assurance a„= -■ • pay a 74. Similarly in computing the value of a short period assurance for n years we take into account the mortuary payments from age a to age a + n, and obtain , 2 mor a - 2 mor (a + n) short assurance a„ = ■ pay a 75. The value of an assurance given in article 68 may be deduced from the annuity table thus ; mora = la ,7-"-* - l(a + l) .r-^-i mor{a + l) = l{a + l) .»--»-H_ Z(a + 2) . r-^-'i mor{a + 2) = l{a + 2) . r-'^-'^i- Z(a + 3) . r-»-2i etc., or mora = pa . »— J ~p{a + Vj .r^ mor{a + \) = p{a + \) .r-i -p{a + 2).r^ mor {a + 2) = p{a + 2) . r- * -pia + 3) . r* etc., wherefore taking the sums we have "Smor.a = r-i 2pa-r^ 2p(a+ 1) whence 2 mor a 1 2 »a ,2 pla + 1) p{a + 1) ass a = = r-i -^ r* —^7 i-r- pa pa p[a + 1) pa a-r^ ^-^ arm {a + \) = r'^ ann pa and if we observe here that p(a + l) = Z(a + 1)»"-"-', pa = la . r'" we obtain finally , ( l{a + l) , 1 , ■) ass a = r-i |anii a - -^ — ■ ann (a + 1) 5 the annuities being understood as beginning immediately ; that is as including to-day's payment. Those students who have perused the works of Baily and others on the subject will perceive that there is a difference of six month's discount between the above formula and those usually given ; the discrepancy arises from these authors having regarded the payment as not due xmtil the end of the year in which the death may happen. ASSUEANCES. 41 76. This conversion of annuity into assurance may also be effected thus, ri mora - la .r-" -l{a+l) .v" oy ri.mora = pa -—rp^a + l) ri . mor (a + 1) = p{a +!)-»• p(a + 2) and so on, whence ri 2 mor a = 'S.pa-r ip{a + 1) = 'S.pa-r '2 pa +rpa and dividing by pa , and thus r^ ass a = r - (*• - 1) arm a OSS a = r* 3— ann a r -r^ ass a ann a = = r —\ which formulse of mutual conversion enable us to test the accuracy of our results. 77. It is evident that these formulse for computing the assurance from the annuity, or the annuity from the assurance, are not apphcable when the assurance is calculated strictly. 78. When the assurance is to increase or to decrease periodically we proceed exactly as in the case of annuities. Thus if an assurance be granted for £500, with an augmentation of £50 every seventh year, the present value must be 500 2 ma + 50 2 m{a + 7) + 50 2 m{a + 14) + etc. pa = 500 assur a + 50 deferred assur a^ + 50 dteferred assur a^ ^ + 50 deferred assur a^^+ etc. In this case the agreement is to pay £500 at the death if within 7 years, £550 if between 7 and 14 years hence, and so on. 79. If the assurance be to increase annually by a given sum, we shall have to take the second sum 2 2 mora or '2^ mor a, and then we have for the value of an assurance of £s to begin with, together with an increase of £i for each year, the value is given by the formula s . 2 mora + i . 2^ mor {a + 1) pa As such cases occur in business the student would do well to make up the 42 ANNUAL PKEMIUMS. columns 2^ mora, Log 'Z'^ mora, and Log incr . assurance as well as iricreas . assurance ; the value of the increasing assurance being obtained from the formula 2^ mor {a + 1) pa in order that the addition may begin one year hence. By help of this table the above value becomes s . ass a + i . incr ass a . 80. In computing the value of a decreasing assurance, beginning with £s and diminishing by £2" each year, we can only apply the formula s . ass a-i, incr ass a if the fraction 4- added to the age a carry us beyond the limits of the life table ; if otherwise we must proceed according to the principles laid down in article 61. 81. The values of assurances increasing or decreasing for a term of years, and then remaining constant or ceasing entirely, are computed exactly as analogous annuities, mortuary payments mor a merely taking the place of life payments pay a. It is unnecessary here to repeat the investigations. 82. If the assurance, instead of increasing suddenly at each year, were made to increase gradually from day to day, so that the additional sum may be proportional to the duration of the life from the present date, the computation would have to be made in the same manner as that for an alimentary payment. Such assurances do not occur in business, and there- fore I leave the investigation as an exercise for advanced students, who, after the example already given, should find no difficulty in conducting the inquiry. ANNUAL PREMIUMS. 83. When, instead of paying down the present value or single premium as it is called, the purchaser of an assurance agrees to make an equivalent annual payment, the payment takes the name of Annual Premium, or simply premium; and this premium may, by the terms of the agreement. PEEMIUMS. 43 be made payable during the life of the nominee or only during so many years of that life ; it may also be made to increase or decrease. 84. If a person now aged 20 wish to secure for himself an annuity after having reached the age of 60, by paying annually a sum of money until he reach that age, the calculation of the relative sums of money must be made in this way. Let us put s for the sum to be paid on and after 60 , y for the annual premium ; then if all the persons 1 20 were to purchase such annuities, the value of the payments to be made to them would be s.sp60 while that of the payments to be made by them would be y{'Sp 20 - sp 60} . On equating these two values we obtain y{'2 p20 - '2 p GO} = s.^p 60 whence 2p60 _ 2p20-2p60 ^'^ *2p20-2p60 °'' ^-^ 2p60 Dividing each side of the equation by p 20 we have y . ann from 20 till 60 = s . ann deferred till 60. 85. The principle of the preceding operation is too simple to need any farther explanation : it is that which guides us in converting a life-contin- gency of one kind into an equivalent contingency of another kind. Thus if we desire to pay an annual premium during the life of a nominee aged a , for an assurance of £s payable at his death, we have, putting y for the annual premium, y . ann a = s . ass a whence y -- s • Or otherwise ■^ anna , 2 mor a V .'Spa = 5.2 ma, whence y = s — • " t- ''2 pay a In this way we see that the annual premium for an assurance of £1 at death is 2 ma ass a prem a = — — = , ^ 2 pa ann a and thus that the table of Log Premium may be formed by placing the columns Log 2 pa and Log 2 ma side by side and subtracting the numbers in the former from those in the latter, or by using the columns Log Annuity and Log Assurance in the same way. The comparison of the results of 44 ACCUMULATING AS8UEANCE. the two processes will serve to check the previous subtractions. The table of premiums is then formed by taking the corresponding- natural numbers. 86. When the premium for an assurance is restricted to a certain number of years, say n years, its value, obtained from the same considera- tions, is , ass a 2 mor a short. prem an = -^ — r = r r — -, ; • ^ short aim a„ 2 pa - 2 p{a + n) 87. And again, when a short-period assurance is to be purchased by an annual premium continued during its term, the premium for £1 is short assurance _ 2 ma - 2 m{a v w) short annuity 2pa-'Sp{a + n) After these examples it is needless to insist farther on this department of our subject : the student may find plenty of examples in the collection of Exercises given at the end of the present Treatise. 88. If a person aged a years agree to pay £1 each year to an assurance office for the purpose of purchasing an increasing assurance, each premium being regarded as a single payment for the corresponding increase in the sum assured, the sum to be paid at his death will be dependent on the number of premiums which he has paid, and may be called the accumu- lated assurance. The assurance to be purchased by a payment of £1 at the age a is {ass a)- , 2 ma that secured by the next payment of £1 is {ass (a + 1) ) -•, and so on, so that the assurance accumulated in n years is {assays + (ass(a + l) )-i + (cess (a + 2) )"'+ ... {ass{a + n- 1) )"' or if we sum up the values of (ass a)-', which may be called accumulating assurances, to the end of the life table, and denote the sum as usual by the mark 2 (ass a)-', the assurance accumulated in n years, that is the sum payable at the death if after n payments of premium, is accum . ass = 2 (ass a) " ' - 2 (ass (a + n) ) - ' . And this sum would be payable at the death of the nominee even although the payment of the premium were to cease. In making a table of such accumulating assurances we should first take the arithmetical complement of Log assurances, then seek out the natural numbers, and sum them in the usual way from the bottom. 45 POLICY. 89. The agreement between two parties as to a Life Contingency receives the general name of Policy. When a policy or agreement has subsisted for some time, it may happen that one or both parties may wish its dis- continuance, and then it becomes necessary to compute the payment which the one should make to the other party in order to purchase the surrender or cancelment of the deed of contract ; this is generally called the surren- der-value of the policy. In actual business the computation of the ralues of policies is complicated by surcharges or loadings which are put on the strictly computed premiums in order to give a margin of security ; but these considerations are foreign to the present work, in which it is proposed only to treat of nett values. The general principle which guides us in the valuation of a policy, is that which guides us in any other business-transaction. We value the obligations on the one hand, and set that against the value of the obliga- tions on the other hand ; the difference of the two is the value of the policy. Thus if a person A have paid for n years a premium for an assurance of £1 granted by an assurer O, and if it be wished now to close the transac- tion, it is clear that on the one hand A is bound to continue the payment of the premium, while on the other hand O is bound to pay £1 at A's death whenever that may happen. Each of these obligations may have its money-value computed, and the difference is the value of the policy. If the transaction had been entered into at the age a, the premium must have been ass a 2 ma prem a = = , •^ ann a ^pa and therefore the present value of all the payments to be made by A, this day's premium included, is , . 2 ma 2 p(a + n) prem a . ann (a + n) = — -h ^ , ^ ^ ' ^fa p[a + n) while the value of the obligation on O's part is , . 2 m(a + n) ass (a + n) = — t r- ^ ' p{a + n) and therefore the balance as in favour of A, the policy-holder, is ass (a + n) -prem a . ann {a + n) = , , 2 p(a + n) ^ ■S.m(a + n)- ^;^„ ^ 2 ma -. f = vol an p{a + n) ^ 46 POLICY. The calculation by the first formula is sufficiently easy : we compute the 'pvodnct prem a . ann {a + n) by placing the columns log premium and log annuity side by side, but shpping the latter up by n lines : the addi- tion then gives us the logarithms of the products for all poUcies of n years' standing. Taking out the natural numbers of these sums we obtain the products themselves, and placing this column beside column of assurance, the latter placed n lines higher, the differences are the values of all policies for £1 of n years standing, estimated as just before the payment of the premium. The second formula becomes , _ 2 pa . 2 m{a + n)- 'S.p{a + n) . 2 ma ^ p{a + 9i) . 2 pa now it has been shown in article 76 that 'S.ma = r-i{rpa ~{r-l)^pa ) whence 2?n(a + n) = r-i{rp{a + n)-(r-l)'2p{a + n)}, therefore substituting these values we have pola =ri h P"" Sp(a + n) ) ■^ " I ^pa p{a + n) S - r^ \l ann{a + n) j c ann a > This same formula might also have been obtained by observing that 2 ma x{ r , ,, i 'S.pa ianna ^ ' ) Spa \/r r -\ while ann a vr , . 2m(a + m) it,,. , ,i This last form for the value of a poUcy may be otherwise put \ann a - ann (a + n)i = pol an anna which affords a ready means of computation. The value of a policy may also be expressed in terms of assurances in this way : multiplying by »• - 1 or p we have (r - 1) pol a„ = (^-l)spa.2m(a + «)-(r-l)2p(a + «).2>wa " p{a + n) . 2 pa POLICY. 47 now (r- 1)2 pa = r pa -ri'Sma (r - 1) 2p(a + n) = rp{a + n) - ri 2 m{a + n) wherefore by substitution , ** pa . 2 m^a + n) - p{a + w) . 2 ma pot aji — 1 7 r ■* r-l p[a + n) 2 pa r I ass (a + n) ) = 1 1 - prem a t r - 1 1 ann a '^ > r ass (a + n)- ass a ~ r-l ann a Hence ass (a+n)- ass a : ann a - ann (a+n) : : r - 1 : ri . 90. The preceding investigation gives us the value of an ordinary policy of assurance just before the premium is paid. The moment this payment is made the value of the policy rises by as much, seeing that, by this pay- ment, the obligation of the pohcy-holder is lessened, and thus the value of a pohcy at the middle of a year is not the mean between its two tabulated values. For example, the value of a policy opened at age 20, is, just before the payment of the 16th premium, £,14585, and just before that of the 17th premium, £,15704, so that, apparently, the policy has increased in value £,01119 between the ages 35 and 36 . But when we come to look more closely at the matter we find that at age 35 a premium of £,01516 had been paid, making the value of the policy at the beginning of the year £,16101 , and thus during the twelve months the policy has actually fallen in value by £,00397 • Supposing that this decrease is uniform during the year, the value of the policy at age 35^ is found by adding the half of this to the value at 36 , making for the middle of the year £,15902 . The difference between the values of the policy at ages a + n, and a + n + 1, each time just before the payment of the premium, is , , , ann (a + n)- ann (a + n + 1) pot a„ + 1 - pel an = ri ^^ ^ ■' ^ T X- anna _ r assur (a + n + 1) - ass (a + n) r-l ann a which represents the apparent increase in the value of the policy, and deducting this from the premium we have , , (r-l)assa + rass(a + n)-rass(a + n + l) prema+polan-pola^ + , = {r-l) ann a ^ for the decrease in the value of the pohcy between the payment at age a + n, 48 POLICY. and the falling due of the next premium at age a + n + 1. This is what, in the First Volume of my Life Tables, I have entered as risk for difference of ages n + 1 . This fall in value of the policy during the currency of the year arises from the circumstance that while the value of the assurance due by the office increases, the expectation of receiving the future pre- miums increases also ; the latter being, in general, more than the former, 91. These principles guide us in estimating the values of policies of any other kind. It would be mere tedious repetition to exemplify the various applications ; suffice it to observe that when, as in premiums payable for a limited number of years, the payments by the policy-holder are exhausted, the account has only one side; and that if a contract for accumulated assurance had been entered into, the value of the policy would be simply that of the accumulated assurance, seeing that the value of the prospective additions to the sum assured is just that of the future premiums. 92. Before taking leave of calculations involving the contingency of one life, it may be well to consider what would be the effect of the dis- covery of an analytic expression for the value of human life : in other words, what would be the effect of a knowledge of the nature of the func- tion la . Since annuities are paid at definite intervals to the parties then alive, without any reference to the manner in which the deaths may have hap- pened during the interval, the value of an annuity would still be obtained by a common summation, and its symbol would still be . 2 (/«./•-«) Spa Anna = — ^ = -^-- : la.r-'^ pa but the value of an alimentary payment would then be obtained by a pro- per integration. If we regard a, the age or time, as the primary variable, and if we sup- pose the ahment to be at the rate of £1 per annum, the payment due for a minute portion of time da would be da, and the value of this estimated as at the day of birth would be r- » . da; but this payment has to be made for each one alive at age a, and thus the element or differential of the sum of all the payments becomes la . r-" . da so that the total of all the elementary payments would be expressed by the integral C-yia.r-".da C representing the sum corresponding to age 0, and /"la.r-" .da standing for the sum of all the payments from age till age a . The CONCLUSION. 49 mode of performing this integration would, of course, depend on the analy- tic form of the function la ; and adopting the common notation of inter- cepted integrals, we should have f "" la- »•-». 'da alim . a = s ;; • The value of an assurance would also be obtained by integration ; for in the minute portion of time 3a, the number of deaths would be represented by "dla, which would also stand for the number of payments of £1 each ; and thus the value of all the payments would be yi ""»■-" 3 fa wherefore the value of one assurance would be given by the formula / '^r-'^'dla J a ass a la. ?■-"' The relation between the values of annuities and of assurances which we have found when examining the value of a policy, is sufficiently remark- able to deserve some farther notice. The two values of the policy are , , ann a - ann (a + n) polan = n '^ ann a r ass {a + n)- ass a ~ r—1 ann a ' from which we obtain the proportion given at the end of article 89 ; which proportion generalised becomes, for any two ages a and a , ass a — ass a vi ann a — ■ann a, »■ - 1 Now if this proposition be true for payments made at intervals of one year, it must also be true for payments made at any other interval, as six months, or one month, and even for continuous or ahmentary payments. However, in extending the formula to such cases it is clear that we must suppose the value of the assurance to have been strictly computed. When the payments are made at intervals of half a year, the rate of improvement for that interval is r^, while each payment is reduced to ^ . Supposing, then, that the values of the assurance is computed as from a half-yearly life table, we shall have ass a - ass a _ ri half-yearly ann a - half-yearly ann a 2{ri - 1) 50 CONCLUSION. And in general, under a similar provision, if the number of payments per annum be t, each payment being reduced to -ji ass a, - ass a r^' fi-eq anna-Jreq anna t(r*-l) And therefore on making t indefinitely large we arrive at the conclusion ass a - ass a _ 1 all a —alia. nep log r In our investigation of the value of an alimentary payment, the result was deduced from the previously obtained values oi pa, whereas the assur- ance was deduced from the fundamental table la combined with the rate of interest. This was done in order to exhibit both methods ; the student may exercise himself by obtaining a value for alimentary payments analo- gous to that given in article 66, and thence demonstrating the truth of the above theorem. This relation between the values of assurances and alimentary payments may be shown to be true for every law of the duration of life, in the fol- lowing manner. The integral yr-^.dla treated by the method of partial integration becomes, la . r-'^-J'la . ?— " . nep log r , wherefore the value of an assurance at age a is fla.r-" assa = 1 - nep logr -, ^-^ , that is to say ass a = \— nep log r .alia , and sinailarly ass « = 1 - nep log r .alia , whence , ass a, — ass a , as above — r. tt— = nep . loq r . ah a — alia ^ " Thus the discovery of the law of mortahty as an analytical function would make the computation of Life Contingencies a part of the Differen- tial and Integral Calculus. TWO LIVES. JOINT ENDOWMENT. 93. When a sum of money is made payable at a given date, in the event of both of two nominees being then alive, we may, for the sake of uniformity of notation, give to it the name of Joint JEndowment . Supposing the nomi- nees to be A and B, their ages a and 6, and the number of years to elapse before payment n, the symbol of such an endowment may be end {a, 6)„ . In this symbol it is not necessary to place the word joint, because its being a joint endowment is sufficiently indicated by inclosing the two present ages in a parenthesis ; and this symbol may be read endowment to A and B jointly, n years hence. 94. When there is any difference between the ages we shall always understand that A is the elder ; and afterwards in calculations involving three or more lives we shall give to the nominees the names A, B, C, D, etc., in the order of their births. 95. In order to compute the present value of £1 payable n years hence, if A and B be then both alive, we suppose that each one of those entered in the table as ahve at age A is named against every one alive at age B ; in this way we obtain transactions equal in number to the product of the two numbers liv a and liv b . The number of payments which will have to be made n years hence on account of all of these transactions is the product of the numbers liv (a + n) and liv (b + n); wherefore the present value of one endowment is ,, ,. Uv(a + n) X liv(b + n) end(a, b)n = — p ,. , ^ r-" . ^ ' hva X livb 96. The above is the simplest expression for the value of a joint endow- 52 JOINT ENDOWMENT. II. ment ; it may be rendered more convenient for our investigations by a transformation analogous to that which was used in one-hfe calculations. We may discount the payment back to any fixed epoch, as to the birth of A or to the birth of B . There is no matter of principle involved in the choice ; yet, as in single life calculations, we have discounted to the birth of the nominee, and as the later birth is truly that of the couple A, B; we shall be consistent in preferring the birth of B as the epoch. Multiplying, then, each member of the above fraction by r " '' we obtain ,, .N _ liv{a + n) X liv{b + n) x ,•- (!■+«) liva X livb x r"' liv (a + n) . pay (b + n) liv a . pay b 97. The product la . lb expresses the number of couples existing at ages a, b; while l{a + m) x l(b + n) is the number existing at ages a + n and b + n. Thus the denominator of the above fraction. may be regarded as bear- ing to the couple the same relation that pay a bears to the single nominee : it indicates the value of £1 payable on account of each couple alive, dis- counted to the birth of that couple ; and may thus appropriately be denoted by the symbol pay {a, b), while the numerator takes the corresponding form pay {a + n, b + n). In this way the value of the endowment becomes p{a + n,b + n) end {a, bu = ^-^ — -, — jt — - , p{a, b) which is quite analogous to that of a single-life endowment. 98. If it were proposed to compute the value of £ao, n years hence, £y, q years hence, £z, s years hence, each sum payable if both A and B be then alive, we shall have the formula x.p{a + n, b + n) + y.p(a + q, b + q) + z .p{a + s, b + a ) p{a, b) exactly as in calculations for one life. 99. From this we see that the values ofp{a, b) are needed for the solu- tion of many problems connected with two lives. In beginning to prepare tables of these values we must first observe that for each difference of age there must be a complete set of tables, because of any couple A, B, the difference of age remains constant during life. The columns therefore must be marked conspicuously with the difference, in order to prevent any confusion. I have found it convenient to mark the difference in large figures at the top of each column ; thus all columns belonging to Two Lives, II. JOINT ENDOWMENT. 53 Carlisle Bills, 3 per cent, difference of ages 5 years are marked as in the subjoined example, article 100 . 100. The logarithm of p(a, b) is evidently the sum of the logarithms of la and oipb,. Carlisle Log liv a 5 C 3 Log pay a GC 3 Logp(a, F) 4.000 0000 3.927 4217 3.890 9238 3.861 7733 3.844 9739 3.8 3 2 3173 4.000 0000 7.832 3173 3.824 5163 3.914 5845 7.739 1008 3.819 1489 3.865 2494 7.684 3983 3.815 3120 3.823 2616 7.638 5736 3.812 4454 3.793 6250 7.606 0704 3.810 2325 3.768 1312 7.5 7 8 3637 3.808 2785 3.747 4930 7.555 7715 3.806 1800 3.729 2883 7.535 4683 3.804 0031 3.712 6142 7.516 6173 3.801 7466 3.696 9104 7.498 6570 wherefore having prepared and titled a blank column we place the log liv a along side of log pay a, but sliding it up as many lines as mark the differ- ence between the ages ; and then placing the blank column beside Log pay a and on the same level with it, we inscribe the sums of each pair of logarithms. In the adjoining illustration the arrangement is shown for difference of age 5 years, from which it will be apparent that the column log pay a of one life becomes by its position log pay h of our present calculation. It is 54 JOINT ANNUITIES. H. also to be noticed that the age of the couple is the age of its younger member B . A complete two-life table must contain the values of logp{a, b) for every diiFerence of age from to the limit of the life table. Now as the difference of the ages is augmented the column is shortened, and thus me lower parts of the slips of paper would be left blank. In order to avoid this waste we use the under parts for higher differences ; thus in the Carlisle Tables the extreme length of the column is 104; so, to allow room for the sub- title, we carry on the formation of the columns as above until we reach difference of age 55, and then place the numbers for difference 56 in the lower part of the slip, taking care to make the work terminate on the same line as for single life calculations. By this arrangement differences 55, 56; 5'4, 57; 53,58; and in general every pair of differences which make up 111 are found on one piece of paper ; while, for the lower part, the age of the elder is in its usual place in the column. The next operation is to form the tables of p{a , 6) , by taking out the natural numbers of the preceding logarithms. JOINT ANNUITIES. 101. An annuity payable during the joint life of two nominees is a series of endowments, and its present value is therefore 2p(a, 6) annia, b) - —^ — ^v^ , p{a, b) when the first payment is to be now. 102. And when the payment is to begin n years hence the value is , _e . . ,, 2p(a + n, 6 + n) defeir ann (a , 6)„ = — -, rr • •^ p{a, b) Wherefore, in order to be ready to make computations in annuities we shall have to sum up the column p{a, b) and take the logarithms of the sums. From these we are able to construct the columns of log annuity {a, b) thence annuity {a, b) ; or the logarithms of annuities deferred any number of years. II. SURVIVOR'S ANNUITY. 55 103. The value of a short period annuity is obtained in the same way, thus shortannia, b\ = ^Pi<^,b)-^p{a^n, b + n) p{a, b) gives the value of an annuity of n payments beginning to-day and contin- gent on the joint life of A and B , 104. In the second volume of my Life Tables the values of annuities for every difference of age are given ; and also a table of Log joint Endow- ments as valued at the birth of the younger. By help of these we can readily compute the value of a two-life deferred annuity for we have , , . , ,s ^p(a + n,b + n) 'Sp(a + n, b + n) p(a + n,b + n) •^ \ ' ." p{a,b) p(a + n, b + n) p{a,b) = ann {a + n, b + n)x end (a , b)n . end{a-b, 0)6+„ = ann {a + n, 6 + w) x — j^ =— j-^ — ^ ^ end {a-o, O)^ which agrees with the formula given in the explanatory part of the work. The value of the short-period annuity, or of any intercepted annuity, may thus be deduced from these tables, though not so readily as from the tables of pay (a, b) . 105. An increasing joint-annuity, that is a payment of £1 one year hence, £2 two years hence, and so on, has its value , , , 22 »(« + 1,6 + 1) mcr ann (a.b) = •* . — ~ — p{a, b) while its applications, seldom if ever occurring in practice, are identic, with those of an annuity depending on one life, 106. When an annuity is only to be paid to the one after the death of the other, it is called a survivorship annuity. We very easily see that the value of an annuity payable to B after A's death is just that of one payable during the whole of B's hfe, less that of another payable during the joint-life ; and I do not think that any techni- cal demonstration can make the matter clearer, yet we may look into the technical proof for the sake of the hght which it may throw upon other parts of the general subject. The total number of transactions entered into being la. lb, the number of payments to be made n years hence must be the product of the number of B's then alive, viz., l{b + n) by the number of deaths among the A's, 56 SUKVIVOE'S ANNUITY. II. which is la - l{a + n), wherefore the value of these payments, as at the pre- sent moment, must be {la-l{a + n)}l{b + n) . r-" and the share of that belonging to each combination A B must be la .l{b + n) .r-'^ -l{a + n) .l{b + n) r-" la • lb l{b+n)r-'^ l{a + n) . l{b + n).r-" ' lb la , lb Now the first of these quantities, taken for every value of n within the limits of the table, make up an annuity payable during the life of B, while the second indicates one payable during the joint-life. 107. In order to denote shortly an annuity payable to B after the death of ^, I shall use the character ann-^, while, conversely, ann-^ will denote an annuity payable to A after B's death. In this way we have the formulae ann —^ = arm b - awn {a , b) ann -|- = ann a — ann {a, b) . 108. Since the expression {la- l{a + n)]l{b + w)r-" la . lb is the value of an endowment payable to B n years hence, if ^ be then dead, it follows that similar formulas hold good for deferred, for short period, and for intercepted survivorship annuities, thus Defer'' Ann-—- = def^ ann b - def* ann (a, 6) and so on. 109. When an annuity is made payable to either after the death of the other it is called simply the survivorship annuity. It is clearly the sum of the two individual survivorships, and therefore ann (-^ or ■^) = ann a + ann 6-2 ann (a, b) . 110. And, lastly, when an annuity is payable so long as one of the couple may bo ahve, it is called a longest life annuity, its value being an7i {a or b) = ann a + ann b - ann (a, b) , 111. The values of joint annuities payable half-yearly or quarterly are II. QUAETERLY PAYMENTS. 57 computed exactly as those of single life annuities, there being no difference either in the principles or in the resulting formulae ; so that if the payment be made t times during the year, the value is ann{a, h)- -^ as shown in article 49 . 112. But it is to be noticed that the present value of a survivorship annuity is the same whether the payments be once a year or t times during the year, for the annuity payable to B after A's death would be {ann h - *-^) - {ann {a, 6) - ^) from which the fraction -^ disappears. 113. The value of an interpolated payment may be computed strictly from the table of the values of p(a, h), just as has been done for annuities depending on a single life. This process does not exhibit the separate parts which the unequable decrease of life and the rate of interest perform in the result ; and therefore it may be expedient to examine the subject from the beginning. In order to find the present value of a payment to be made at a frac- tional part of a year we must take account of the number alive at that time. Now, as shown in article 5, the fraction -f- being put for n, liv{a + {) = la + {bla + \'-^bHa + ^'-^'-^bHa + etc. and the number alive at the age 6 + y is liv{b + \) = lb + \Hb + \-'-i^hHb + \'-^'^bHb + ^t ^-Tmo" r-'T^^o -r . :{!%"' +2 r'' +3"»-"* + ... +t\'*} + -|-{l%" + 2%-"'^+3%"+ ... +ir'^ +etc. an expression analogous to that which has been given in article 66 . 115. If, as in article &Q , we represent the above series by the symbols Sq, s^, s^, etc., and the limits to which these series approach by the cor- responding symbols S^, 8.^, S^, etc., the value of the alimentary pay- ments to be made during the year a to a+1, 6 to 6 + 1, becomes, restoring the factor »•-' la. lb. r-" Sa+{A.S^+B.S^ + C. S^+etc.) /•-» where A, B, 0, are, for conciseness, written for the co-efficients given in article 113 . 116. Lastly, to find the present value of an aUmentary payment to be continued during the joint hfe of A and B, we must compute the values of the multipliers A, B, C, etc., for each succeeding year to the end of II. FRACTIONAL AGES. 59 the life-table, take the sum of the whole, and divide by pay (a , 6) . In this way we obtain the formula pay {a, 6) x joint alt {a, b) = So.Sp{a,b) + S^r-^-2{ la.i Ib + lb.h la -\{la.h^lb + lb.b^la) + eiG.} + ^2 r-» 2{ 1 (Za.32 lb + lb. d^ la) - i {la.d^ lb + lb.8« la.) + etc.} + Sg r-» 2{ i {la. 8^ lb + lb.d^ la) - i [la. 8^ lb + lb.d^ la.) + etc.} + S^r-'' ^{^\{la. d* lb + lb.d* la) + etc.} + etc. 117. It can very seldom happen that the difference between the ages of two parties A and B is an exact number of years, and still seldomer that the date of payment should happen on the birth-day; and therefore we must examine the process for interpolating when the ages and their dif- ference are all fractional. In the second volume of my Life Tables, the values of joint annuities are given for every age and for every difference of age in entire years, and the problem before us may be regarded as this : to compute the value of a joint annuity for the ages a + — , b + — , those for integer ages being known. This interpolation may be effected by three common interpolations, thus : From the values of ann {a , b) and ann (a + 1 , 5) we deduce ann (a + -^, b) = — Vf - «) ann {a, b) + a, ann (o + 1 , b)> and again between ann (a , fe + 1) and ann (a + 1 , 6 + 1) we interpolate ann{a + -^, 6 + 1) = — |(^-a)anw(fl!, b + 1) + a ann{a + 1) {b + 1)1 Lastly, between these two we interpolate ajm(a + -f ,6 + ^) = -^\{u-^)annia + -^ , b) + l3ann{a + -^,b + l)^ = -— ](*-«)(«- |S) an7i {a,b) + a{u-l3) ann (a + l,b) + (t-a) ^ ann {a, b + l) + aj3 ann (a + 1 , b + l)i This formula, which serves for common interpolation with all tables of double entry, is suscep- tible of a very neat geo- metrical illustration. If we place the argu- ments age of A in a. vertical column, those of jB in a horizontal one, and write the values of the joint annuities in the intersections of these col- umns, we obtain a rectangular arrangement such as is shown in the 34 35 36 20 21 22 23 17,192 17,128 17,056 16,979 17,031 16,969 16,900 16,825 16,861 16,801 16,735 16,663 37 16,686 16,628 16,564 16,495 60 FRACTIONAL AGES, IT. R W tJ adjoining abstract. Let us suppose that the annuity for ages 35^, 21f , is wanted. Having drawn a rectangle of any size, let us write the four values of ann (35, 21) , ann {35, 22), anM(36, 21), arm {36; 22), which, however, for the sake of conve- nience, I shall represent by the letters P, Q, B, S, at the four corners; then having made P T one-third of PB, and P V three-quarters of P Q , let us draw the parallels VW, TCT intersecting each other at X, X may be imagined to be the place at which, in an extended table, the value of ann {35^, 21|) should be inscribed. Or, in general, if PBhQ divided into t equal parts, and P T be made a of these, TB being t - a of them ; and if P Q be divided into u equal parts, P V being made /3 of these, and VQ, therefore being u- (3 of the same, then X may repre- sent the value of the annuity corresponding to the ages « + -7- and 6 + — . The area of the rectangle P Q SB is proportional to the product tu, that of P VX T to a, 13, while, similarly, we may put VQ UX = «(« - /3), BTXW = («-a)/3, and WXUS = {t-a.) {u-j3). It may now be seen that the above formula takes the form Px WXUS+QxBTXW+BxXVQU+SxPVXT PQSB ~^ that is to say, the sum of the products of each value in the table by the area of the opposite rectangle, is equal to the product of the interpolated value by the area of the whole rectangle. In the actual performance of the calcu- lation we may leave off any part that is common to all the four tabulated quan- tities, and treat only the remainders ; thus in our example we deduct 16,700 from all the four, and then our diagram takes the form shown in the margin; and the formula becomes ,269 X 1 . 2 + ,200 X 2 . 3 + ,101 x 1 . 1 + ,035 x 1 . 3 tseff t2Da tVS.S 3x4 = ,162 and so adding to this the 16,700 left off we have ann (34^, 21f) = 16,862. These diagrams show, very clearly, the nature of the double interpola- tion ; thus if we put p = ann{a, 6), q = ann{a, b + l), r = ann {a + l, b), s = ann {a + 1, b + l); II. FRACTIONAL AGES. 61 and suppose these to be written at the points P, Q, JR, S, then we shall have by the interpolations above explained t = ann{a + -^, b) u = ann(a + -^, 6 + 1) x = ann(a + -^, b + —) and it is evident that we should have obtained the same ultimate result had we made the interpolations for Fand W, and then between these interpolated for X. 118. If at the points P, Q, R, S, we were to erect perpendiculars or normals to the plane of the paper, and were to make the lengths of these normals proportional to the values of the joint annuities, continuing an analogous construction for other pairs of ages, the upper ends of these normals would indicate a curved surface, the configuration of which would exhibit the variations in the values of such annuities. Confining our attention for the present to the preceding interpolation, the adjoining figure may represent such a structure seen obliquely. P QSR is the rectangle, and the normals Pp, Qq> Rr, 8s, may stand for the four tabulated values. In order to interpolate be- tween Pp and iZr- we join pr and raise at T the normal Tt to meet it, the length of this normal corresponds to arm (a + -^ , 6) . Similarly joining qs, and raising the normal Uu, we obtain arm (a + -|- , 6 + 1) . Lastly, Ob & joining tu, and raising the normal Xx, we have ann {a + —j-,b + -£-) . If we had joined pq, rs, raised the normals Vv, Ww; joined VW and raised a normal at X, we should have obtained the same value Xx . 119. In this investigation we have supposed that the value of the joint annuity changes uniformly with a change in one of the ages ; and this supposition is sufficiently exact for business purposes in the actual state of our information ; yet it is proper that we should ascertain how far the result so obtained may possibly differ from the truth. Let, in the adjoining figure, the rectangle P QSR belong to the ages 62 FRACTIONAL AGES. II within which the interpolation is to be made, the line P Q belonging to age a, B S to age« + l, while PR belongs to age 6, and Q-StoS + l . For the sake of symmetry let the lines for ages a-1 and a + 2, and for 6-1 and 6 + 2, be drawn, thus inclosing nine rectangles of which P QSR occupies the middle. Then at the sixteen intersections let normals be raised to represent the annuities ann {a- I, 6-1) ann (a ,6-1) ann (a + 1 , 6-1) ann (a + 2, 6-1) ann{a-l,b) ann{a-l,b + l) anw (a -1,6 + 2) ann{a , b) ann (a ,6 + 1) ann {a ,6 + 2) awn (a + 1,6) ann (a +1,6 + 1) ann(a+ 1,6 + 2) ann(a + 2, b) ann{a + 2, b + l) ann{a + 2,b + 2) and let the tops of these be connected by lines which, in general, will be curved, these will form a net-work of quadrangular meshes, corresponding to the rectangles on the flat surface below. Having divided the year represented by P J? in the proper ratio at T and drawn through T, extended both ways, a line parallel to PQ, draw normals Tt, Uu, etc., to meet the curves, these, as in the former case, represent the values of the annuities ann (a + -|- , 6) and ann (a + -^ , 6 + 1) , only this time exactly instead of approximately. The plane erected on LT UM cuts the curved surface along a line hum, of which the ordinates LI, Tt, Uu, Mm, etc., represent the values of all annuities in which the age a + — 11. FRACTIONAL AGES. 63 is combined with ages b±n; and, of course, that plane contains the ordi- nate Xx, which represents the value of ann{a + ~ , b + —), the annuity sought. The calculation of this value is as follows, going only to diflferences of the third order ; — From the values of ann {a-1, b-1), arm (a, 6-1), ann (a + 1 , b-1), and ann{a + 2, 6-1), we interpolate the value oi LI, which represents ann (a + —r- , 6 - 1) . Similarly we find Tt, Uu,Mm, which stand for ann{a + — , 6), ann (a + -|- , 6 + 1) , and ann (a + -^ , 6 + 2), respectively Lastly, from these we compute Xm the value of ann (a + -^ , 6 + — ) . 120. Such being the outline of the operation, we may observe that the five interpolations are all similar ; and we may therefore shorten our work by help of a general investigation. Let, then, (x-\) 0{x-l)i (p X (p(a;+l) (, and therefore according to what has been shown in article 9, -T^M'« + 2)-2^(^ + l) + H + T V TT k* + 2) - 3?(« + 1) + 3^;. - x ^^ {x + r) p(a; + 2) j ^^ t) Qfi \ fj^g g t_g 2t-s } which is remarkable for its symmetry. 121. Applying the above formula to the business in hand, we obtain, -|- taking the place of— , , a , IS _ {t + a) a{t - a) {2t - a) I ann{a-\, 6-1) ann( a + 2,h-\) \ 2t-a, ) -ann(a,b-l) _anM(a+ 1, &- 1) ann{i + o ■ 1- o a t — a, and similarly for ann (a + -|- , 6), ann {a + ~,b + l), and ann {a + -|-, 6 + 2) And again putting — for j we have ^ (m + /3)/3(m-/3)(2m-)8) f ann(a+-^,b-l) an«(a + -f,6 + — )= ^^^3 \ ^;^ ann (a + — , b) ann (a + — , 6 + 1) ann (a + -|-, 6 + 2)) + 3 ^ +3 ^^-g ^—^ \ so that, finally, II. ASSUEANCE. 65 annia + — ,b + —) ^^3 x ^^3 "^ ) arm(a-l, 6-1) _ n onw (a-l, 6 ) ggnn (a-1, 6 + 1) an?i(a-l. ft + 2) (<+«)(« + /3) (<+«)/3 (< + «)(u-/3) "^ it+a.)(2u-fl) „ ann(a ■,b-l) q aww (ra ,6 ) „ ann (a ,b + l) „ anw (a , 6 +2) «(a + /3) «;3 "^^ «(m-/3) "^ «(2m-/3) ann (g+1, &- 1) -annfa + l.ft ) ^ ann (a + 1, 6+ 1) „ ann (a + 1, 6 + 2) ~ (<-«)(k + /3) ■'"'^ (,«-«) /3 "^ (<-«)(«-/3) - {t-a.)(2u-fi) arm (a + 2, ft - 1) „ ann (a + 2, ft ) „ ann (a + 2, ft + 1) ann (a + 2, ft + 2) I """ (2<-»)(m + ^)~ (tt-«.)fi (2«-«)(M-/3) "*■ (2«-«)(2m-/3) ) This formula is universal ; if the student translate it into geometrical language he may observe that the divisor of each tabulated quantity is the area of the rectangle having X in the one corner and the position of the ordinate in the opposite corner. ASSURANCES. 122. In order to investigate the value of a payment to be made at the death of one of two nominees, I shall, as in the case of one-life transactions, at first restrict the inquiry to deaths happening within a specified year, and shall propose the problem : To find the present value of £1 to be paid at the death of A, provided that death happen in the year n to n + 1 years hence, and provided B be then alive. The whole number of transactions being taken as la .lb, the number of deaths which may be expected to happen among the A's within the speci- fied year is l{a + n)-l{a + n + \) or d{a + n) ; and these have to be combined with the number of B's alive at the deaths but, as these deaths are dis- persed through the year, and as the number of B's alive is decreasing during the year, the true number of payments can not be found by a simple multiplication. The product d{a + n) x l{b + n) must be too great, because that would be to suppose that all the B's alive at the beginning of the year live till the end of it ; while the product d{a + n) x l[b + n + 1) must be too small, for an analogous reason. The true number of payments, then, lies between these two, and the product of d{a + n), the number of A's who die during the year, by l{J> + n + ^), the number of B's alive at its middle can- not be far wrong. 66 ASSUKANCE. II. Taking the discount also from the middle of the year, the value of all the payments to be made may be assumed as d{a + n). Z(6 + w + -^) »•-"-*, and therefore the value for one of the transactions is d{a + n). l{h + n + \)r-''-i la. lb 123. In this rough way of averaging it is possible that an error may have been made. We may, on the assumption that the deaths are uni- formly distributed during the year, determine the number of payments thus. Let the year be divided into t equal parts, then, putting A and B for a + n and h + n respectively, the number of deaths in each part is — dA ; while the number of B's alive at s + ^ parts of the year is W-^^-^ dB, and thus the number of payments to be made during that portion of the year must be nearly \dA{lB-^dB}; and so the number of payments to be made during the whole year must be the sum of all such expressions obtained by giving to s every integer value from to f - 1 ; the first part IB remains the same throughout, so that for it the sum is dA . IB, and from this falls to be deducted dA if Hi + 3 + 5 +{2t-l)}dB now the sum of the odd numbers up to 2< - 1 is just t^, wherefore the num- ber of payments, on the supposition of a uniform decrement of life, is dA{lb-\dB} or dAxl{B + \), an expression which does not contain t, and which, therefore, remains the same, into however many parts the year may be subdivided. The average of the preceding article, then, is in accordance with the supposition of a uniform distribution of the deaths through the year. 124. If the proposition had been to compute the value of £1 payable at B's death, A being alive, with the same restriction as to time, the formula for the number of deaths would have been d{b + n) -x. l{a + n + ^) and the value of a single expectation l{a + n + \). d{b + n). r-"-J la . Ih II. ASSURANCE. 67 125. The value of £1 payable in the year n to n + 1 years hence, at the first death of the two, must clearly be the sum of the two preceding values, and the total number of payments on such conditions must be d{a + 'n)y.l{b + n + \) + l{a + n + \)y. d{b + n) . But the number of first deaths is just the number of couples which dis- appear ; now the number of couples existing at the beginning of the year is l{a + n) . l{b + n) , and the number at the end is l(a + n + l) ,l(b + n + l), wherefore the number of first deaths during the year is l{a + n) . l{b + n) - l{a + n + l) , l(b + n + l) , since a couple disappears at each first death. This expression ought to be equal to the preceding one; the equality becomes evident if we write l{a + n)- d{a + n) for l{a + n + \), and l{a + n)- \d{a + n) for l{a + n + ^) . 126. An agreement to pay a certain sum of money at the death of A, provided B be then alive, may be called an assurance at A!s death if first, or, when B is the heir of ^, it may be regarded as B's right of succession to A ; for the sake of uniformity of language I shall use the expression assurance at A's death if first, or assurance to B at A's death, and shall mark by the symbol B b ass —r- , ass — the present value of £1 payable to B at A's death. The value of such an assurance is the sum of the temporary assurances treated of in article 122, taken for every value of n from to the end of the life table. Resuming the expression for that assurance, and multiplying both nume- rator and denominator by r~*, that is, discounting back to the birth of B, we obtain d{a + n).l(b + n + ^) . r-(!'+"+i) la.lb .r-^ a fraction of which the denominator is identic with that used in the compu- tation of annuities. This fraction may be written, according to the notation for one-life calculations, d{a + n) . p{b + n + ^) ^ la .pb ' the denominator is p{a, b), and the numerator expresses the value of all payments made to the B's alive on account of the deaths of the A's which happen in that year ; it may, therefore, appropriately be written h + n 68 ASSUEANCE. II. and thus the value of an assurance payable at A's death, if first, takes the form ,, 2 mor ~~~ a ass~ — pay {a, b) 127. Conversely an assurance at jB's death if first, may be written ass -|~; its value is the sum of the contingencies treated of in article 124 for every value of n till the end of life. The fraction given in article 124 may be written l{a + n + \) .d{b + n)r-^-'^-i _ l(a + n + ^) . mor {b + n ) la.lb.r-^ ~ la.puy{b + n) ' of which the denominator is, as before, paj/ (a, b); while the numerator may be expressed by the symbol mor-^ and thus the value of A!s succession to B becomes 2 mor 4- ass ~s- = pay(a,b) 128. In order to apply these formulae to the actual business of calcula- tion we must begin by forming a table (single-life) of the number alive at the middle of each year, the title of which is liv{a f ^), and also a column of the corresponding logarithms ; and here it may be observed that we do not title such columns liv (6 + 5), because they are made to refer to A or B , or, when more lives are concerned, to C or D by their position. By adding together the numbers in the columns Loff l{a + \) and logr-'^-i we obtain Logp{a + ^), and then by another addition, slipping the column for a to the proper distance upwards, we obtain log da + logp{b + |) = log mor -^ of these the natural numbers must be extracted, the values summed in the usual way, to give 2 mor-^ , and the logarithms of these again tabulated. By subtracting log pay {a, b) from log 2 mor -^ we obtain Log ass~^, and thence the value of the assurance itself. Again, to compute the values of assurances at Bs death, if first, we use the formula log mor -^ = log l{a + \) + log mor b and the computer will observe that this change is made in order that the discounting may be to the latest birth in both branches of the calculation. II. ASSURANCE. 69 129. When the sum assured is payable at the first death, as at the dis- solution of a partnership by the death of either of the partners, the value of the contingency is the sum of those of the two separate contingencies; that is to say ass {-J- or •-") = ass-™ +ass-^ = ass {a, b) and thus we obtain a table of the values of assurances at first death, by adding the numbers in the column ass-^ to those in the column ass-^. 130. Regarding the couple AB as an individual, the product la , lb is the number of such couples alive at the ages a, b, and this product we may write l{a,b). Thus a column containing the products la. lb, l{a + 1) . l{b + 1) , etc., may be regarded as a life table for such couples. If such a column were prepared, and if the differences of the numbers in it were taken, these would indicate the number of couples which disappear in each year ; they would constitute, indeed, the table d{a, b) ; that is, the deaths of the couples. Here we may notice that b is properly the age of the couple. Treating this life table, with its accompanying column of deaths, just as we did those for single-lives, we obtain j)ay[a, b) = l{a, b)r~^; mor{a, b) = d{a, b)r-''-i ■2p{a,b) ■S.m{a,h) annia, b) = — ^ — rr ; ass (a, b) = — r-^ — rr , ' p(a, b) ' "• ' ^ p{a, b) and the coincidence of the values of the assurance at first death obtained in this way, with the sum of the separate assurances, affords a complete check upon the whole work. The multiplications la .lb, when a complete set of tables is to be con- structed, can be performed by successive additions ; or, when the tables for only one difference of age are to be made out, by help of Crelle's Rechentafeln or Multiplication Table, a work which should be on every computer's desk. 131. The present value of a payment which is to be made at B's death, if A have died previously, is, obviously, the excess oi an assurance at B's death absolutely over an assurance at B's death if first. Thus if an office grant a bond to pay a certain sum at B's death whenever that may happen, and if the policy-holder grant a back-bond to pay to the office the same sum at B's death if A be then ahve, he, the policy-holder, will be able to make any claim against the office at B's death only in case A have died previously ; and thus ass at B's death if 2'' = ass -^ = assb-ass'^ ■ 70 ASSURANCE. II. 132. Similarly the value of an assurance at the death of A, if B have previously died, is ass at A's death if 2* = ass -r- = ass a- ass b 133. When the sum assured is to be paid at the later of the two deaths, the value of the expectation is the sum of the two previous values, hence ass at 2^ death = ass a, b = ass a + ass b — ass ~ — ass -^ = ass a + ass b - ass (a, b) . The correctness of this formula may be seen by supposing that an office grants two bonds each for the same sum, the one payable at the death of A, the other at the death of B, taking at the same time from the policy- holder a back-bond for a like sum payable to the office at the first death of the two. 134. We may give to these investigations the appearance of profundity, without adding anything to their clearness, in the following manner. Let it be proposed to compute the present value of £1 payable at the death of B, provided A be previously dead; this being the problem of article 131 . The number of transactions being la, lb, that of the payments to be made in the time between n and n + 1 years hence must be the product of d{b + n), the number of deaths which happen among the B's, by the num- ber of A's previously dead, which number may be taken as to the middle of the year, viz., la~l{a + n + ^); hence the present value of all the pay- ments to be made during that year is {la-l{a + n + \)}d(b + n)r-'^-i and consequently that share of this sum which belongs to a single couple is {la-l{a + n + \)}d(li> + n)r-'^'^ ^ d{b + n)r-'-i l{a + n + -l) d(b + n) r-^- i la . lb lb la . lb mor (b + n) mor -^ pay b pay {a , b) The value of the assurance for the whole of life is the sum of all such terms taken for w = 0, w == 1, n = 2 to the end of the life table, and thus ass at B's death if 2^ = 2m(6 + w) 2m -~~ pb p{a, b) or ass -—■ = ass b-ass^ b II. ASSURANCE. 71 It is easy to see how the same investigation may be applied to the prob- lem of article 132, or extended to that of article 133 . 135. If, not contented with the crude assumption that the deaths occur uniformly during each year, we seek to allow for the inequality, as shown by the table la, the investigation becomes complex, and still more so if we make the discount strictly. I shall, in the first place, put the discount out of view, and show how to compute the true number of pay- ments to be made in each year, and shall afterwards indicate the allowance for the discount. We shall take the problem, " to compute the present value of £1 pay- able at A's death, if B be then alive," and first confine our attention to this part of it, " how many payments shall have to be made in the year n to n + 1 on account of the total number la . lb of transactions ? " Putting, as before, A, B, for a + n, b + n, the number of payments must be between the limits dA . IB and dA . 1{B + 1) . By dividing the year into t equal parts, and treating each of those parts in succession, we shall bring the limits nearer to each other, and so by making t indefinitely great, shall obtain the true result. We have seen, article 66, that the number of A's who die in the interval between ^ and -j is ] <3 \ +1.2.3 ~ 1.2.3.4 +^i<'-; 4s»-6s' + 4s-l C l^dA , > , 4- ^ I + j-2-3-^ - etcf + etc. in which dA is written for its equivalent -hlA. Now the number of B's alive at the end of that fraction of the year is -^ + etc.| IB-^^ {d5- ■*■¥ - + 4 s* 1.2 ' 1.2.3 UdB SI' dB 1.2 1.2.3 ■ 4 XV dB 1.2.3 1.2.3 118^ dB "1.2.3.4 el^dB 1.2.3.4 - etcf- + etcj- { + 1.2.?.4 -^W ^^t« and consequently the number of payments to be made in that portion of time is the product of these two. Since the sum of the n"^ powers of the 72 ASSUKANCE. II. integer numbers up to t, involves the (« + 1)" power of <, it follows that all fractions of the form ^ in which x is less than y-l may be neglected, since after the summation, when we come to make t indefinitely great, the fraction would disappear. The product, after these rejections, becomes ; c l.dA l^'dA I'dA^ , ) (,_, idB I'dB VdB , ) IB t- 2s' I IdA 3VdA UP dA ) s» (,- l.dA l^dA , ) (IdB SP dB 11¥ dB , ) + :^\dA--^+^—eto. j j_____ + .____etc.| 2s'' (odA SydA , m^dA , ) (,_ IdB PdB l^ dB . ) -^lT:2-r:^ + 1:^:3:4 -^t°-} r^-r72+-3 ^+^''^-} Ss' ( l^dA 6l^dA . ) ,n ^ +-*^{r:2r3-r72-:3:i+^*''- p^ +^*<'- By assigning to s in this expression every value from 1 to i, taking the sum of all the results, and in that sum supposing t to become indefinitely great, we obtain the true expression for the number of payments to be made during the year on account of the first deaths among the B's . This operation is shortened by observing that the fraction l'' + 2" + 3''+ t" approaches the nearer to ;^qri the greater the number t is taken. Again, multiplying the above expression by r ' , and proceeding in the manner already explained, we can obtain the true value of these payments, each one discounted from the instant at which it is to be made. These formulje are far too complex to be of any use in calculation, the more so that the greater degree of accuracy that would be obtained from them is far beyond that of any life table which we as yet possess. 136. The values of these assurances may also be obtained from the tables of j oint-annuities. We have (article 126) moi — ~ = da. ?(6 + |-)r-'-i = {la -l{a + l)}i{lb + l{b + l)}r-'>-i = ^{la . lb - l{a + l)lb + la. l{b + 1) - i(a + 1) l{b + 1)} r-»-i = ir-i{p{a, b)-p{a + l, b)}+rHp{a, b + l)-p(a+l, 4 + 1)} II. ASSUKANCE. 73 wherefore, taking the sum to the end of the life tahle, 2 . ass -— = r-i y ' r-i — ^, ^-^^ " p{a, b) la.pb + ^ 2p(a, 6 + 1) ^., 2p{a + l,b + l) la pb p{a, b) The first of these four terms contains the value of the joint annuity a, b, and by proper changes in the denominators the others can be brought into the form of annuities, giving the result 2 . ass -~ = r-i arm {a,b)-r-i —, — - arm (a + 1 , 6) + *'"*"^i — ann{a,b + \) - r -i ^ — - —^,-^a}in(a + 1,6 + 1). Similarly we obtain o o ir / TN ^(08 + 1) / ■• 7s ^(6 + 1) , , ,v ^ .ass^-g- = r-i{ann{a, b) + ^— arm{a + l, b)-^j—^ ann{a, 6 + 1) l{a + V) l{b + l) , , , ,,, + -^^"2 — - • J, ^ ann{a + l, 6 + 1)} and adding these together /IN Xr / 7N ^(a + 1) • ^(6 + 1) , 1 . ,M ass {a, b) = r-i{ann {a, 6) - -^ . ' arm (a + 1, o + 1)} which last result is in accordance with the analogous proposition concern- ing assurance on a single life. The computation of assurances by help of these formulae is very trouble- some ; much more so than that which has been previously given. In the same way we may represent annuities as derived from assurances ; but, as the formulae are of little use, the investigation of them is left as an exercise for the student. 137. If the law of life were so well known as to be represented by an analytical formula, we should have, in the language of the higher calculus, -/( la.lb.r-^db) _ "^*("'^^ = la.lb.r-o ' J - /(dla.lb.r-^m) '***~S' = la.lb.r-" ' -/( la.dlb.r-^ ) ^^ '5~ " la. Ib.r-" ' assia b) - -A^Ka.b)r-^) ass{a,b)- ia,lb.r-^ and 74 DEFERKED ANNUITIES. 11. By applying the process of partial integration to the last of these we can obtain a relation between the joint aliment and the assurance at first death analogous to that which is given at the end of article 92 ; but when we apply the same process to the second, or to the third, we only repro- duce the truism ass -J- + ass -j- = ass {a, b) . DEFERMENT. 138. The value of a deferred joint annuity is computed exactly in the same way as that of a deferred single-life annuity ; thus we have , ,x J /. a 2 »(a + n, b + n) ann[a, b) dej&r n years = —^-^ — -, — jr , as is evident from what has been already shown ; this expression may be put under the form 2p(a + w, b + n) p(a + n, b + n) p{a + n, b + n) p{a, b) which is obviously the product of the value of ann {a + n,b + n)hj that of a joint endowment n years hence. In my tables for two hves I have given what are there called the logarithms of joint endowments, the values of these endowments being estimated as at the birth of the couple. These are inserted for the pur- pose of computing deferred annuities. Inserting in the numerator and denominator of the above fraction the arbitrary factor p{a-b, 0) we have ann (a, b) def n years = ann(a + n,b + n) P(« + »^^^ + ") K«-ft>0) p{a-b,0) p{a,b) , , . endia + n, b + n) = ami (a + n, b + n) ^ ,, , , — ' . end {a, b) 139. But the value of a deferred survivorship annuity cannot be calcu- lated in this way, because it is the difference between a one-hfe and a two- hfe annuity, the values of which vary in different ratios. Thus, adopting II. SHOKT PERIODS. 75 the obvious notation cum™ to signify an annuity payable to B after A!s death, we have ann ■—- = ann b - ann (a, b) and similarly for deferred annuities, that is ann -^ def^ n years = ann b def^ n years — ann (a, b) def^ n years p(b + n) ,, . )-yiaj-n, b + n) short ass ~rr = - • p{a, b) 2,1ft ^^ — ^ ffl ^ " p{a, b) shortass" ^^^'^^'^-^-^ short ass (a b) ^"'C^' ^)-^M« + w, & + «) ^ ^ P(a, b) But when survivorships or second deaths are involved, the short period 76 LEGACY-ANNUITIES. H. annuities and assurances for one life must be used : of these a complete set of values is given in the volume of one-life tables. 143. There is yet another class of expectations which may be called Legacy-Annuities; they arise in this way : A testator may direct that an annuity, or alimentary annuity, be payable to a nominee after the testator's death. In this manner such a problem as the following may arise : — " To compute the present value of an annuity of £1 payable to B at A's death, and annually thereafter." The elementary consideration here is, " what is the present value of £l to be paid to B q years after the death of A ? " In the first year the number of deaths among the A's is da, and for each of these £1 will have to be paid to the individuals B alive q years after the death ; that is between q and q + 1 years hence. In this way the value of all these payments becomes da + l{b + q + ^).r-^-i this has to be divided by la. lb, and the quotient is da.l{b + q + i)r-<-'> + ^ + i') _ mort±S la . lb. r"* l>ay{a, b) ' wherefore the value of the whole annuity is Smoj*™-' b + q p{a,b + n) , — ,c = ass X — - — __ pay{a,b) a p{a,b) b + q p(b + n) = ass — * X ^^ — J — ' • a pb 144. This being the general expression for the value of £1 to be paid to B q years after the death of A, that of £1 payable at the death, and at each anniversary thereafter, may be found by giving to q every integer value from 0, 1, 2, to the end of the life table, and thus we have J ass^ .pb + ass ™~1 . p{b + 1) + ass ™? . p{b + 2) + etc. Legacy ann ™= -^ This very inconvenient formula may be simplified in the following manner : — In the first year of the currency of the transaction the number of pay- ments on account of all the cases la . lb must be da . l{b + ^) , and the value of these discounted back to the usual epoch da . l{b + ^)r-''-i. In the second year each B alive will claim against every one of the A's previously II. PREMIUMS. 77 dead, and therefore taking the average for the year the value of the pay- ments will be {da + d{a + 1)} l{b + 1^) r-<-'' + '« and so on ; or in 1^' year {la - l{a + 1)} p(b + 1) in 2« year {la - l{a + 2)} p{h + H) in 3"^ year {la - l{a + 3)} p{h + 2\) etc., vrhence the value for one transaction is p{b + ^)+p{b + \\)+ etc. l{a + V) p{b + 1) + l{a + 2) . pjb + 1^) + etc. pb la .pb The first of these fractions is the value of an aliment payable during the life of B ; the second may be changed in its form by putting l{a + \)-^da for l{a + 1) , and becomes l(a + ^) jo(6 + i) + Jja + H) p(6 + H) + etc. ^ da. p{h + ^) + d(a + 1) .p(b + Ij) + etc. la.pb la.pb whence altogether Log . ann ™ = alib — all {a, b) + \ ass -^ = aU~~ + iass-~~ a ^ a a result at which we might easily have arrived without any profound algebraic investigation. 145. It may afford a good exercise to the advanced student to supply a strict proof for the above formula on the supposition of equable decre- ments during the year, the minutice of discount being neglected. PREMIUMS. 146. When the price of an assurance or other deferred benefit is to be paid in annual instalments, the premium is computed in the same way as for analogous one-life transactions ; the only difference being that, in two- life business, there is a greater variety of cases. If an assurance at the death of A be to be purchased by an annual pay- ment during the joint-life of A and B, the premium is ass a ann (a , 6) 78 CONCLUSION. II. The premium for an assurance at a first death is, naturally, made pay- able during the joint-life, its amount being ass-^ ass~^ ass (a, b) arm {a, b)' ann {a, b)' ann (a, 6) ' according to the transaction. The premium for assurance at a second death, on the other hand, may be made payable during the joint-life, during the life of A, during that of B, or during the longest life. Premiums for survivorship annuities have for their natural limit the duration of the joint-life. CONCLUSION. 147. Throughout these calculations concerning two lives we have sup- posed the use of a single-life table. Now it is a notable fact that the law of decrement of life among females differs considerably from that among males ; so that if A were a husband, and B his wife, it would be proper to take this difference into account. For this purpose it is enough to take all the numbers la, da from the table for males; all those lb, db from that for females ; and so of all their derivatives pa, ma, pb, mb, etc. In this way the value of a joint annuity becomes ann {am, 6/) = — p — ' A and so on of all other contingencies. im a . p^ o) In addition to this difference between the expectations of male and female life, we have variations from one class to another. When the character of these variations shall have been well ascertained, it may become proper to use, for each of the individual nominees, the appropriate life table. It is hardly necessary to observe that the introduction of the distinction between male and female life would quadruple the extent of a complete set of two-life tables. THREE LIVES. JOINT ENDOWMENTS. 148. In order to estimate the value of a payment depending upon three lives we suppose that la . lb . Ic transactions are entered upon, compute the number of these on account of which the payments may be expected, arid thence deduce the value of each share. If, then, it be proposed to compute the value of £1 payable n years hence, provided three nominees A, B, C, be then all alive, we observe that, on account of the total number la .lb . Ic of transactions l{a + n) , l{b + n) . l{o + n) payments are to be expected, and thence find ,, , . l{a + n).l{b + n).l{c + n) .r-'" end{a, b, c\ = j^^-j^^ l{a + n) . l(b + n) . p{c + n) ~ la . Ib.po The product la .lb . po is the value of £l payable to each triplet A, B, Q, in existence at the ages a, b, c, estimated as at the birth of c, that is, c being the youngest, at the birth of the triplet : it may therefore be represented by the symbol />(a, b, c), while the corresponding symbol l{a, b, c) indicates the number of triplets existing. The value of a joint endowment takes the form ,, , . p{a + n,b + n, c + n) end{a, b, c)„ = ^(^7^,-5 . and is quite analogous to those already obtained for one and for two-life endowments. 149. From this we obtain at once the value of an annuity payable so long as the three nominees are alive : it is , , ^ ^p{a,b,c) 80 SUEVIVOESHIPS, III. 150. In every three-life transaction we have to consider the two differ- ences between the ages, viz., the difference a-b, and the difference b-c, and must classify the columns according to these differences : thus a ques- tion in which the ages are 70, 52, 21, belongs to that set of combinations of which the differences are 18, 31 ; and all the columns referring to these combinations should have the differences conspicuously marked upon them. The product la.lb.pc may also be written la.p{b, c), so that the column logp{a, b, c) may be obtained by adding log la to logp{a, b) of the two- life calculations, p{a, h) being converted into p{b, c) by the proper change in its position in regard to la. The same product may be written lb.p{a, c), so that we have another way of obtaining log p[a, b, c) to be a check upon the former. From Logpia, b, c) we obtain p(a, b, c), thence by summation 'Sp{a, b, c)^ from that again Log 'Sp{a, b, c); and with these columns we are ready to compute joint annuities, immediate, deferred, or for short periods, exactly as for single-life annuities ; the resemblance of the processes being so com- plete that it would be mere repetition to go over them. 151. The value of an annuity payable to B and C jointly after the death of A is obviously the excess of the annuity to B and C jointly over an annuity payable so long as all the three are alive ; hence ann -—- = ann (6, c) - ann {a, b, c), ann -2™- = ann {a, c) - ann {a, b, c), ann ~~~ = ann {a,b)- ann (a, b, c) . 152. And hence the value of an annuity payable so long as two, and only two, of the three may be alive is ann{b, c) + ann(a, c) + ann{a, fe)-3a?m(a, b, c) . 153. Hence also the value of an annuity payable so long as two of the three may be alive is ann {b, c) + ann (a, c) + ann {a, b)-2 ann {a, b, c) . 154. An annuity payable to C after the death of one or other of A and B is the excess of an annuity to C simply over a joint annuity to A, B, and O; that is awM™-g = ann c~ ann {a, b, c) ; anwj-jj-^ = ann b- ann {a, b, c) ; aniij--— = anna~ann(a, b, c) . III. SUEVIV0ESHIF8. 81 155. Hence the value of annuities of £1 to be paid to each one of the survivors after the first death among the three is ann a + ann h + ann c - 3 ann {a, b, c) . 156. In order to compute the value of an annuity payable to C after the deaths of both A and B, we observe that the value of an annuity payable to C after A's death is ann -~ = ann c - ann {a, c) . The purchase of such a provision would secure to C an annuity after A's death ; this, however, is more than is wished ; so long as B is alive there should be no payment; let us then subtract ann{b, c) and we obtain ann c — ann {a, c) - ann (b , c) but neither is this what is wanted, for in virtue of the three contracts here represented, £1 annually would be claimed by the oiRce so long as all the three may be alive, wherefore, in order to cancel this repayment, we must add an annuity to ^, J3, C, jointly ; thus, finally, ann^-^^ = annc-ann{a, c)-ann{b, c) + ann(a, b, c) ; ann ——g-; = ann b - ann {a,, b) - ann (b, c) + ann (a, b, c) ; anng-^g-^ = anna-ann{a, b)-ann{a,e)+ann{a, b, c) • 157. The value of an annuity payable so long as one, and only one, of the three may be alive is, evidently, the sum of the three preceding sur- vivorships ; that is r ann a +ann b +ann c J Single survivorship = J.- 2 {ann {b, c) + ann (a, c) + ann {a, b)}> ( +3ann{a, b, c) * 158. When an annuity is payable so long as any one of the three nomi- nees may be alive, its value is t ann a +ann b +ann o '\ Ann (a ov b ov c) = <- ann (b, c) - ann (a, c) - ann (a, 6) >■ ( +ann{a, b, c) } 159. An annuity payable after A's death, so long as either B or may be alive, is obtained by omitting ann a from the above expression ; that is to say ann '-™"° = ann {a or 5 or c) - ann a ann b r ann b + ann c J -^ - ann {b, c) - ann {a, c) — ann {a, b)V i +ann{a, b, c) ) and similarly for the other alternative survivorships. 82 ASSURANCES. III. 160. There remains yet another class of survivorship annuities, viz., those which are payable to the last survivor only on condition that the previous deaths have happened in a specified order. Thus an annuity may be payable during the life of C after B's death, provided A have died before B . The symbol for such an annuity is ann ( The investigation of its value involves considerations analogous to those which enter into the valuation of assurances, and I therefore postpone it to article 192. m ASSURANCES. 161. The first problem in assurances connected with three lives is " to compute the value of a payment to be made at the death of one nominee, as ^, if two other nominees B and Cbe then both alive." This may be called an assurance at A's death if first, or it may be called B and C's joint succession to ^. I shall continue to use the word assurance, and shall use the symbol a In order to compute this value we suppose la .lb . Ic transactions, and seek to discover how many payments will fall to be made in each year on account of all of these transactions. Putting A, B, {or the ages at the beginning of a year, lA, IB, IC are the numbers then alive, and lA - dA, IB-dB, IC-dC the numbers alive at the end of the year; from these we have to extract our result. For each one of the deaths dA claims will be made ; the question is, " how many ? " Since these deaths are distributed over the year, some will happen when almost all the IB are alive, and some near the end of the year, when only IB - dB are alive to prefer a claim ; hence, one with another, we may sup- pose that the mean IB -^ dB may be taken ; and similarly concerning the C's, wherefore, adopting this rude method of proceeding, we may say that the total number of claims during the year on account of A's death first is dA{lB-^dB){lO-^dO). III. ASSUKANCES. 83 Following the same line of argument we find that the claims for similar assurances at B's death if first should be dB{lA-\dA){lC-idC) in number ; and again that the number of claims for assurances at C's death if first is dC{lA-^dA){lB-^dB). Now the number of triplets existing at the beginning of the year is lA .IB . IC, while the number existing at the end is (lA - dA) (IB - dB) x {IQ- dC) . The total number that disappear during the year is the differ- ence between these two, viz., lA.lB. dC+lA . dB. IC + dA . IB . IC-IA .dB.dC -dA.lB. dC- dA.dB.lC + dA.dB. dC and this is, certainly, the number of first deaths during the year. It ought, therefore, to be the sum of the three preceding quantities. How- ever, on expanding these, and collecting the results, we find lA.lB.dC+lA.dB.lC+dA.lB.lC-lA.dB. dC -dA.lB.dC-dA.dB.lC+ldA.dB.dC This does not agree with what is known to be right, the defect being \ dA . dB . dC, and therefore we conclude that our mode of averaging has been erroneous. Nor need this surprise us, for we might as well have taken the geometric or any other mean between the numbers alive at the beginning and at the end of the year. It becomes, therefore, incumbent upon us to examine critically the method of taking the average. This we shall do by dividing the year into a number of equal parts, computing approximately the number of first deaths in each of those parts, and taking the sum : by this means we obtain a result which will be nearer to the truth as the number of parts is augmented ; so that the limit to which this sum approaches as the number of parts is made very great must be the true number of first deaths in the whole year. It is obvious, from this outline of the process, that the sum- mation of series and the doctrine of limits are needed ; and therefore I must here take leave of those students who have not so far prosecuted the study of algebra, recommending them to take up the subject from the beginning, without omitting any of the articles that are marked as belong- ing to the higher branch of the study. 162. Supposing the deaths to happen uniformly during the year, and dividing that portion of time into t equal parts, the number of A!s who die in each part is \ dA . At the end of s of those parts the number of B'a 84 ASSUKANCES. HI- and C's alive are lB-\dB and ZC--f-dC respectively, wherefore the number of triplets which disappear by deaths among the A's during the interval from y till ^^ must be less than \ dA {IB - -f dB) {IC- \ dO) and greater than \dA{}.B-'-^dB){ia--^dC). It is enough for us to consider one of these limits ; expanding the former of them we have \dA.lB.lC--j,{dA.dB. iC+dA .IB .dC) + ^ dA .dB . dC. By giving to s every value 1, 2, 3 ... to ^, and taking the sum of the results, we obtain ^dA.lB.lC-*-^i^(dA.dB.lC+dA . IB . dC) + '^' + ^H!' '^^^ dA.dB.dC . for an approximation to the true number for the whole year. Lastly, by supposing ^ to be a very large number, the ultimate result becomes dA{lB . IG- \{dB . 10+ IB .dC) + ^dA.dB. dC) . This result exceeds that which was obtained by taking the numbers of B's and C's alive at the middle of the year by the quantity -^ dA . dB . dO, which again is just one-third part of the total deficiency from the number of first deaths. The above expression may also be written dA{{lB-^dB){lQ-^dO) + -^^dB.dC} or dA{l{B + \) . l{C+i) + -r\dB. dC} and again, since dB = IB- l{B + \), in the form ,, ^ IB.IC+1{B+\).1{C+V) l B.l{■ = fa la .da. 171. Similarly, as has been shown in article 54, the value of an ali- mentary payment is indicated by the symbol ,. /^"'la-r-O'-da fj^pa.-da all a = j = • la . r-' pa And again, as in article 92, -/"^dla.r-" ass a = ; 172. So ALSO IN two-life transactions we have f"" la. lb. da. r-^ all {a, b) = ass b ass~~ la. -lb .r-" -r la. dlb.r- ■ b la .lb .r" -r lb. "dla .r -b la .lb .r-" III. ASSURANCES. 91 173. And in three-life transactions f^ la.lb .Ic .r-' ."da uce^u, (/, W - la.lb.lc.r-" nsa '■" - fj^?)la.lb.lc.r-'' ass g la. lb Ac. r-" a,c = - f^ la.'blb.lc.r-" ass J la.lb.lc.r-" a,b = - J^ la .lb .'die .r-" ass „ la.lb .Ic .r-" On attentively examining any one of these expressions the student may perceive that it recals the whole of the operations which conducted us to the actual results; thus, to take the second of the last group for an example, in computing the value of an assurance payable to B and C jointly at A's death, we selected some small portion of time, as, in article 163, one year, and sought for the number of payments to be made in that year. Now the number of deaths was -SIA, and therefore on the sup- position that all of these happen at the beginning of the year, before any of the IB and IC have died, the number of payments would be — IC.IB.81A. This supposition, however, necessarily gives a result too great, and we sought to get a nearer approximation by taking the num- bers of B's and G's alive at the middle of the year. Afterwards, in article 164, we subdivided the year into t parts, and multiplied the number of deaths among the A's in one of those parts by the numbers of B's and C's alive at the beginning of it, and summed up all these products; then, lastly, taking note of the discount, we supposed the intervals of time to become indefinitely small, and ascertained the limit to which the sum reached. Now the symbol /^"^{-dla.lb.lc.r-") indicates all of these operations ; for first the character — 3 Za represents the number of deaths which occur among the A's in a minute portion of time da ; —dla.lb. Ic the number of payments to be made on account of those deaths ; r-" the discount, and J a denotes the summing of these from the age a to infinity ; that is, to the end of the life table. 174. The import of the above symbols may be more clearly seen by observing that the expression -dla .lb. Ic 92 SECOND DEATH. III. is necessarily too great, because it supposes that lb and lo are alive all through the interval of time da, while the expression -dla{lb-dlb){lc-dlc) is, for a like reason, necessarily too small. The difference between these two is d la . lb . d Ic + 'd la .d lb . Ic -d la .d lb .d le . Now if we render the subdivisions of the time twice as small, we shall, roughly speaking, halve the differences d la, d lb, 3 Ic, and thus make each of the terms ^i la . lb ."die; "dla ,c)lb .Ic four times less ; and so, as there are now twice as many of them, their total will be half as much as before: wherefore, if we take the element of time da excessively minute, those terms which contain the product of two differentials become so minute that even the sum of the whole of them may be neglected. Much more insignificant, then, must be the terms which involve the product of three or more dif- ferentials ; and therefore only those terms which contain a single differen- tial need be kept. 175. In the above expressions the ages a, b, c which are not governed by the sign of integration are the given ages of the problem, and do not change, whereas those ages which are under that sign are variable ; they change from a, b, c to every higher age within the limits of the table. On this account it may be proper to make a distinction between them, and I shall write, as in some of the preceding articles. A, B, Cfor the variable ages ; this change does not alter the import of the formulae. 176. Let it be proposed to compute the value of £1 to be paid to C at the death of B provided A have died previously. The present ages of the three nominees being a, 6, c, we may suppose that la .lb . Ic transactions of this kind are gone into ; and we may inquire how many payments are to be made during the interval of time da, when the three ages have advanced to become A, B, O. The number of C's alive at that time is IC, and claims will be made on account of each of them. In the minute interval of time there will occur -dlB deaths among the B's, and thus we have -dlB.lC grounds for claim : but those claims can only be sustained in regard to the A's who are dead, the number of whom is la - lA, wherefore the total number of claims that can be made good is -{la-lA).-dlB.lC and the value of these claims, estimated as at the present moment, {-la.dlB .lO+lA.dlB .10) >•- (c- «) in. SECOND DEATH. 93 and this divided among the la .lb. Ic transactions gives for each -'dlB.W.r-'^ lA.dlB.l0.r-<' ■ + ■ lb. lev" la . lb . lo . r-" and thus the total value of this assurance is /, "^ i-dlB.lC.r-o) /"= {-lA.dlB.lC.r-<^ ass (±) = Ib.lc.r-" la.lb.lc.r-" now the first of these expressions denotes the value of £1 payable to C at B's death, and the second that of £1 payable to A and O jointly at B's death, v^herefore ii)- ass I B 1 = ass -g- - ass ~s- as, indeed, we might have discovered by an inspection of the nature of the contingency ; for if C were to obtain a bond for £1 payable to him at B's death, and, at the same time, were to grant a back obligation to pay the like sum at B's death if both he and A be then alive, he could only, in virtue of these two bonds, claim the payment at B's death if A be pre- viously dead. In the very same way we obtain the values of the other five assurances of this class, viz., ass 1-11 = ass ~r - < A ass I _ c j = ass~;g-- ass"-^ (?) = /-4-\ A A, 1 <^s [JLj - ass~^ - ass-~^ ass C^j = ass-g- ass 14 1 = ass - A A^B^ ~^ — ass -~-«~ 177. The value of £1 payable to C at the last death of the others, without regard to the order in which they may have died, is evidently the sum of the first and second of the above expectations ; that is, c c c A,C B,C ass ~g~A' ~ ^^^ "jB" "*" '^^ ~A '^^^ ~B '^^^ ~-^~» ; and similarly ass "^c- = ass-~^ + ass -^ — ass — g ass -^^ ; ass ~^^ = ass-^ + ass—^ - ass~g ass-^^ . 94 THIRD DEATH. lU. 178. Again the value of an assurance of £1 payable at A's death, if it be the second, without regard to which of the others may have died before him, is the sum of the second and fourth of the same expectations ; that is, ass {A second) = ass —^ + ass ~2 — 2 ass "-^^ ass {B second) — ass ^g- + ass ^-2 ass —^ ass ( C second) = ass --^ + ass ~^ — 2 ass ~~g- . 179. Lastly, the value of an assurance of £1 payable at the second death is the sum of the assurances in any of these groups ; or ass at 2^ death — ass -~^ + ass —^ — •■ ciss ~-g — i- ass -^ + ass -g- B n B, C n A, C n A,B + ass -^-2, ass -^ 2 ass ~g 2 ass —g— • which completes the varieties of assurance at the second death. THIRD DEATH. 180. The deaths of A, B, and C may occur in six diiferent orders : it is enough for us to examine one of these. Let it be proposed to com- pute the value of £1 payable at the death of C, if C have survived B, and B have survived A . Proceeding in the same way as before, the number of deaths of Cs in a minute interval of time 3a is, at the age C, - 3 10, and for each one of these claims will be made as against each one of the couples A, B, which has been totally extinguished by the death of B. Now the number of couples which have entirely disappeared between the dates a, b and A, Bis (la -lA) {lb -IB) and therefore we have to ascertain how many of these have been extin- guished in the order A, B. In article 123 it has been shown that if the deaths be uniformly distri- buted during the year, one-half would happen in the order A B, the other half in the order B A, and the same reasoning could be applied to any longer period ; so that if the yearly number of deaths were uniform we should have to take one-half of the above product. But actually the in. THIKD DEATH. 95 deaths are not at uniform rates, and, therefore, in order to obtain the true number of claims we must consider the matter in detail, 181. In the minute interval of time da, the number of deaths among the B's is -d IB, and these combined with the A's previously dead give -?)lB{la-lA) for the number of couples which are completely extin- guished by the deaths of B's during that interval, wherefore the number of which we are in search is to be obtained by performing the integration represented by the symbol f^^{-?)lB{la-lA)), or by the two separate integrations /j'{--d IB . la) - /f{-dlB . lA) . In the former of these it is to be observed that la is constant, and that y^^( -dlB) = lb- IB wherefore the number sought for is expressed by la. lb-la. lB-/f{-dlB . lA) ; and here again the remaining integral represents the number of cases in which B dies before A during the whole time between the ages a and A. It is evidently the difference between the two complete integrals f^"^ {--dlb.la) and /^""{-dlB.lA) that is, between the total number of cases in which B dies before A from the ages a , 6 to the end of the life table, and the number of them which happen after the ages A, B ; hence our expression becomes la.lh- fj" {-■dlb.la)-la.lB + f^"" (-dlB.lA). Here we observe that the number of couples which disappear by the occurrence of A's death first, together with the number of those which dis- appear by B's death first, necessarily make up the whole number of couples la. lb, and that, consequently, la.lb- /J" {--dlb.la) = fj^ {--dla.lb) , whence the farther simplification fj^ {-dla.lb)-la.lB+ f^"" {--dlB.lA). The expression y {-dla. lb) denotes the number of cases in which 96 THIED DEATH. I". the death of A occurs first, and is analogous to the formula 2 mor — . dif- fering from it only in this, that no interest is allowed; it is therefore equivalent to 2 mor--^ at per cent ; and similarly J^ (-3 IB.IA) may be regarded as 2 mor -g- at per cent. 182. The number of deaths which occur among the C's during the minute interval of time 3^ is -dlC, and hence the number of cases in which claims can be preferred during this time is -■blC.f^ °° {-'dla.lb) + la.lB.?>lC-?ilC.f^'"{--dlB.lA); this being the number of third deaths among the O's, the A's having died first : we shall have to make use of this number in computations concerning four lives. 183. The value of these claims for £1 each estimated as at the birth of Cis -■dlC.r-(^f^ °" {-Zla.lh) + la.lB.'dlC.r-<^--blG.r-<'f^ "^ {-ZIB .lA) and therefore the total value of similar claims for the whole duration of life is y^°° {-'blC.r-^.fJ' l^-?)la.lh))-la.fj° {-IB .dlC .r-c) ^/^"^ {-dlC.r-o/^^ (.^IB.IA)} whence the value of £l payable at O's death, O having survived B, and B having survived A, is I /"(-3Za.;6) e f,"^ {-dlC.r-cf "" {-^IB.IA)} ass S. = — = — =r ass C - ass •^+ 5 — ^ A la .lb " la. lb .pc 184. Being unable, from our ignorance of the nature of the function la, to perform the integrations indicated in the above formulae, we are forced to have recourse to the summation of the annual steps, in order to ascertain in how many cases out of the combinations la . lb the death of A will occur first. Now in the first year we have da deaths among the A's , and these hap- pen, supposing the deaths to be uniformly distributed through the year, before those of l{b + ^) of the B's, wherefore the number in the first year is da . lib + \) , and, similarly, in the next year it is d{a + 1) .l{b + ll), and so on ; wherefore the total number of first deaths of A's becomes 2 da . l{b + ^) III. THIKD DEATH. 97 and this is the representative of the integral on the imperfect supposition that the deaths are uniformly distributed. Having constructed a table of the products da . l{b + \), and also of the complementary ones db . l{a. + \), and taken their sums in the usual way, we can at once determine in how many cases the B is alive at A's death within two given limits, as, for example, between the ages a and A ; this number being evidently ^da.l{b + l)-sdA.l{B + i]. 185. But the question before us is to discover in how many cases the ffs who hiive died in this interval of time were alive at the death of A. In the first year we have db deaths among the B's and da among the A's, so that \da .db is the number for that year. In the next year there are d{b + 1) deaths among the B's, and in the two years la - l{a + 1|) previous deaths among the A's : and in general in the n + Isi year we have d{b + n) deaths among the B's, and la - l{a + n + \) previous deaths among the A!s ; so that in n years the number of couples in which B dies after having sur- vived A is {la-l{a + ^) }db + {la-l(a + l\) }d{b + \) + {la-l{a + 2^) }d{b + 2) + {la -l{a + n- ^)} d{b + n - 1) the sum of which is evidently la{lb-l{b + n)}--S.db. l{a + ^) + -2d{b + n) .l{a + n + J) = la .lb-'S,db .l{a + \)-la . l{b + n) + '2d{b + n) .l{a + n + ^) = ^da.l{b + ^)-la. l{b + n) + 'S d{b + n) . l{a + n + ^) ; which, if we replace a + n and b + nhj A and B , takes the form •s. da .lib + \) -la .IB + 1. dB . l[A + 1) an exact counterpart of that given in article 181. 186. Or THE DEATHS dQ which occur among the C's during the next year, every one must happen after the above number of cases in which both A and B are dead, A having predeceased B; these give dC.^da.l{b + ^)-la.lB.dC+dC^dB.l{A + \) a 98 THIRD DEATH. IH. claims ; but to these must be added the number of claims which arise from the deaths of A's and B's during the year. In the first place, of the dB . dC couples which entirely disappear, one- half occur with B's death first, and each of these has to be combined with the la - lA previously dead of the A's, so that to the above expression we must add ^{la-lA)dB.dG. And in the second place, of the dA , dB . dC triplets completely extinct, it can be shown that one-sixth part have the deaths in the order A , B, O; so that altogether the number of claims arising in the course of the year is dC.-s.da. l{b + \) - la . lB.dC+\la . dB . dC-^lA .dB.dC + idA.dB.dG+dC-2dB.l{A + i), which may be written more concisely dC.sda.l{b + ^)-la.l(B + ^).dC-^l{A + ^).dB.dC+dC'2dB.l{A + ^). I leave as an exercise the proof of the assertion that, of the triplets dA . dB . dC which entirely disappear, one-sixth part occur in each of the six orders A,B, C; A, C, B; B, A, C; B, C, A; C, A, B; and C, B, A. This assertion is only true when the deaths are uniformly dis- tributed through the year ; and the demonstration is obtained by supposing the year divided into a great number of equal parts in the manner already sufficiently exemplified. It may be more instructive to obtain the last cor- rections by help of integrals. Let us imagine the year from ^ to ^ + 1 to be divided into an infinitely great number of minute parts ?>t, and let t be the interval of time after the date A, B, C; then, on the supposition that the deaths are uniform, we have l{A + t) = lA-tdA l{B + t) = IB-tdB l{C + t) = IC-tdC. Now, in the minute portion of time dt, the number of deaths among the B's is dB . "dt , and these have to be combined with all the A's previously dead, viz., with la-lA + t dA, giving the product {la - lA) .dB .dt + dA.dB.tdt so that the integral of this, which is {la -lA).dB.t + \dA.dB .t^ expresses the number of cases in which, during the time t, B dies having survived A . Multiplying this by dO.dt, the product {la - lA) . dB . dO. tdt + ^dA.dB. dC. i^dt is the number of cases occurring during the time dt in which C dies having III. THIED DEATH. 99 survived B, B having died in tbe time t and having survived A ; and its integral \{la-lA) . dB . dC. t^ + idA.dB. dO. fi is the number of such cases happening during the currency of the time t. Making < = 1 , we have, for the whole year, i {la - lA) dB . dC+idA .dB.dC as already found. 187. If it were desired to make the computation strictly, that is allow- ing for the varying rate of death during the year, we should have to revert to the formula given in article 5 for the number alive at any intermediate part of the year : assuming t to represent the time or fractional part of the year after the age A, the expression takes the form l{A + t) = lA-tdA-t*-^&dA-t'-^'-^h^dA-Qic.; and if we confine our attention to differences of the first and second orders it becomes l{A + t) = lA-{dA-^ddA)t-i6dA.t^ Writing B ior A, and taking the differential of this, we find that the number of B's who die in the minute portion of time dt is {dB - i SdB)dt + SdB.tdt; but the number of A's previously dead is Ia-IA + {dA - i8dA)t + iSdA . t^ and consequently the number of couples completely extinguished by the deaths of C's during that time is {la-lA)(dB-i^dB)dt+{{la-lA)ldB + {dA-iidA){dB-^^dB)}.tdt + {{dA-ildA)ldB + ildA{dB-iidB)} .fdf + ildA.idB.fdt. The integral of this, viz., {la.dB-^la.ldB-lA.dB + ^lA.ldBj.t + {la.ldB-lA.ldB + dA.dB-^dA.ldB-^ldA.dB + ildA .IdB}^ f + {dA.ldB + ^ldA.dB-lldA.8dB}^f + ^ldA.ldB.t^ gives the number of cases in which, during the time t, B dies having sur- vived A . Taking this for the whole year by making ^ = 1, we obtain la . dB-l{A + i) . dB + ^{dA .hdB-hdA.dB) for the number of couples which are extinguished by the deaths of B's during the year from age A to age A + 1, 100 THIRD DEATH. HI. In order to find the corresponding number for the whole time between age a and age A, we must give to A and B all integer values from a, h to ^ - 1 , B-1, and take the sum of the results ; this sum may be put in the form •s.da.l{b + i)-la.lB+-s.l{A + \).dB + ^ri{-s.da.bdb--^hda. db -sdA.ddB + ^ddA.dB} . The first three terms of this formula agree with those given in article 185 ; the latter part shows the correction to be made on account of second difi^erences. Thus it appears that the total number of couples A B which have been completely extinguished by the deaths of B, between the dates a, b and A+t, B + t,is •2da .l{h + \) -la .IB + 1. 1{A + ^) .dB + ^{^da .ddb-Sdda.db -:sda.8dB + -2ddA. dB] + {la.dB-\la.&dB-lA.dB + ^lA.bdB}.t + {^la.ddB-^lA.ddB + idA.dB-idA.8dB-^SdA.dB + iddA.ddB}.f + {^dA.ddB+^ddA.dB-iddA.ddB}.f + ^ddA.SdB.t*. This, multiplied by the number of deaths which occur among the C's during the minute time dt, viz., by {dC-^ddC)dt+ddC.tdt gives the number of triplets A, B, (7 which, during the same time, are completely extinguished by the deaths of C, B having died after A ; this number is {1, da .l{b + \) -la .IB + -s.l{A + \) .dB + -^-s.da Adb - ^ -zhda .db ~^-s.dA.bdB + i^-s.idA.dB}{dC-\&dC}.^t + [hdC.^da.l{b + \)~la.lB.bdC+hdC.l.l{A + \).dB + -^bd0^da.idb -^8dO.-2Sda.db-^8d0.sdA.SdB + ^^ddC.sddA.dB + la.dB.dC-\la.bdB.dC-lA.dB.dC+llA.hdB.dC -^la.dB.SdO+ila.SdB.ddC+llA.dB.SdO -^lA.SdB .8dC} .tdt + {la.dB.8dC-ila.ddB.d dC- lA.dB.d dC+i lA.ddB.ddO + ila.ddB.dO-llA.ddB.dC+idA.dB.dC-idA.ddB.dC -iddA.dB.dC+^ddA.ddB.dC-ila.SdB.ddC + ilA.&dB.ddC-idA.dB.ddO+^dA.ddB.8dO + iddA.dB. bdO-^bdA . bdB . bdC} . t^dt III. THIRD DEATH. 101 + {ila.SdB.8dC-^lA.&dB.SdC+idA.dB.SdO-idA.ddB.ddC -^SdA.dB.ddC+^ddA.ddB.ddC+^dA.SdB.dC + iBdA.dB.dO-^ddA.SdB.dC-idA.ddB.8dC -^BdA.dB.8dC + i8dA.ddB.ddC}.^dt + {^dA.ddB.6dC+^8dA.dB.SdC-iddA.SdB.ddC+i8dA.ddB.dC -■^ddA.ddB.ddO}.t^dt + ^SdA.SdB.8dC.t^dt. The number of triplets ■which are completely extinguished in the order A, B, C, between the beginning of the year and the end of the time t, is the integral of this, viz., {dC .■sda.l(b + ^)-la.lB.dC+dC.sl{A + ^).dB + ^dC.-2da.ddb -■^d0.s8da.db--^dC. ^ dA . 8dB + ^dO .S SdA. dB -^hdC.^da.l{b + \) + \la.lB .bdC~\bda.-^l{A + \).dB -^8dO.-S,da.&db + ^8dC.-2 8da.db + ^-^ddC.'2dA.8dB fyddC.-^SdA.dB} .t + {bdC.-S.da.l{b + \)-la.lB.&dC+8dQ.-S.l{A + ^).dB + ^bda.^daJdb -■^8d0.^8da.db-^^8dO.-2dA.SdB + ^8dC.^8dA.dB + la.dB.dC-\la.bdB.dC-lA.dB.dC+\lA.hdB.dC -^la.dB.bdO+^la.&dB.bdC+llA.dB.bdG -ilA.ddB .ddCj^t^ + {la.dB.hdC-%la.bdB.b dC- lA.dB.h dC+ ^lA.SdB.ddO + ila.ddB.dO-ilA.ddB.dC + i^dA.dB.dC-idA.ddB.dC -IhdA.dB.dC + ^bdA.&dB.dC-ldA.dB.bdC +^dA.8dB.ddC+i8dA.dB.ddO -■^BdA.SdB.ddCj^f + {ila.ddB.ddO-ilA.ddB.ddO+^dA.dB.ddC+idA.BdB.dC + ^ddA.dB .dC-^dA.8dB.ddC-^8dA.dB.ddC -^8dA.ddB.dC + iSdA.8dB.8dO}^t^ + {^dA.8dB.8dC + ^SdA.dB.SdC+iSdA.8dB.dC -^8dA.8dB.8dO}i^ + :^8dA.SdB.SdC.t^ . And by making, in this expression, t = 1 , we obtain the total number 102 THIRD DEATH. HI of triplets extinguished in the course of the year, the deaths being in the order A, B, C, viz., {^da.l(b + ^)+-^^da.ddb--^^8da.db} .dC + la{-l{B + i).dC + ^dB.SdO--^8dB.dC} + dC{sl{A + ^).dB-^sdA.ddB + ^-2ddA.dB} + dC{{-^lA + idA-^SdA).dB + {-^lA + -^ddA).8 dB} + ddO{-^l{A + ^).dB + ^^dA.ddB-Thx^dA.dB} In this example the student may perceive the advantage of employing the fluxional process ; for though, even with its help, the investigation be laborious, it would have been much more so had we followed the method of dividing the year into parts, and supposing those parts to become inde- finitely numerous. The resulting formula is so complex as to put out of the question the determination of value of a third death by the use of second differences; rejecting from it all those terms which contain the symbol 8 we are brought back to the value given in article 186. 188. The number of claims which will be made in the year between the ages C and 0+ 1 is thus dO{^da.lib + ^)-la.liB + ^)-il{A + i).dB + ^l{A + i).dB} and the value of the payments, estimated as at the birth of O, becomes ^da.l{b + i).mC-la.l{B + i).mC-}I{A + -k).dB.mO+mC.sl(A+i).dB. If we take the sum of the values of such expressions for each integer value of C from C to the end of the life table, and divide the amount by la .lb .pc, we shall obtain the value of a deferred assurance payable at the death of C if O have survived B and B have survived A, the assurance only to take effect if C die after C-c years from the present moment : this value is ^da.ljb + i) _ smC i.l{B + l).mC ^ s. 1{A + ^) . dB . mC La . lb pc lb .pc ^ la. lb . pc s(m0.sl(A + i).dB) la .lb . pc and thus the value of such an assurance from the present time is ags-^ ^ ^da.l{b + i) smc ^l{b + i).mo ~a" la . lb pc lb , pc 2l{a + ^).db.mc "^ (mc . 2 l{a + i) . db) la .lb .pc la .lb . pc m. THIRD DEATH. 103 Here it may be observed that 2 da . l{h + \) is the number of cases in which A dies before B among the couples la . lb ; that is an assurance at C's death ; and that — //, ' '"'^ is the expression for an assurance pay- able to B at C's death. The numerators of the remaining fractions involve three lives, and require independent calculations ; the last one involves a double summation. The expression may, more conveniently, be written ass^ = ^da l{b^^)^^^^_^^^^^{Uia^i).db-^l{a-.^).db}mc ~A~ la .10 " la .lb .pc 189. By transposing the letters of the above formula we can obtain the value of any other assurance at third death, subject to a restriction as to the order of the deaths. 190. The value of an assurance payable at the death of C if J5 and A be previously dead, but without regard to the order of these previous deaths, is the sum of the two assurances for the orders A, B, C and B, A, C; hence ass at C's death if last = ass jb^ + ass ^ A B 2 da . l{b + 1) b ^{^l{a + i) . db - ^ l(a + ^) , db}mc — ; Ti OjSS C — €tSS _ > ^ 7j ~ — la. lb " la.lb.pc , '2l{a + \).db __ a H^da.l{b + \)-\ l{b + ^) ■ da}mc "T 5 Yi abb O — ilbb "-Z r J J- La .lb " la . lb . pc Now we may observe that 2 da . l{b + ^) is the number of couples A B which are completely extinguished by the death of A, while 2 l{a + ^) . db is the number of those extinguished by the death of B , and that these two numbers necessarily make up the complete number la . lb of couples exist- ing at first ; wherefore the above sum may be written t a "Sila .lb-il(a + -k).db-^da .l(b + i)}mc assc-ass^-ass-^+^ ^^7167^3 ^ and here again it is to be observed that the numerator of the fraction is the sum of the payments to be made at C's deaths if first, so that ulti- mately ""c /^ B A A, B ass -^^ = ass C - ass-^ - ass-^ + ass -^^ , a theorem which might readily have been obtained from ordinary business considerations. 104 THIRD DEATH. HI. 191. The assurance at G's death if last may also be obtained directly in the following manner, Taking, as before, t to represent a part of the year, and confining our attention to differences of the first order, the numbers of A's and B's who have died between the ages a , b and A+t, B + t, are la-lA + tdA and Ib-lB + tdB wherefore the number of couples AB oi which both members are dead is {la - lA) {Ih - IB) + {{la - lA) dB + dA {lb - IB)} t + dA.dB.t^ now the number of O's who die during the minute interval of time ?)t is dC ."dt, wherefore the number of triplets completely extinguished by the deaths of O's in that time is {la-lA) {lb-IB) dC.dt+{{la^lA)dB + dA{lb-lB)}dC.tdt + dA.dB.dC.t'df, the integral of which, {la-lA) {lb-IB) dC.f + ^{{la-U)dB + dA{lb-lB)}dC.t' + }dA.dB.dC.f is the number of such cases which happen during the currency of the time f. Making t = 1, we obtain, for the number of third deaths among the C's during the year, {la.lb-la.l{B + ^)-lb.l{A + i)+lA.lB-^lA.dB-^dA.lB + ^dA.dB}dC discounting this to the birth of C, taking the sum to the end of the life table, and dividing by la ,lb .pc, we obtain the value of a deferred assur- ance payable at O's death if it be the last •s.mC •s,l{B + \).mC ^ •2l{A + \).mC ^coef{A, B).mC pc lb .po la . pc la , lb . pc and thence that for an assurance without deferment ass "2^ = «ss C - ass -g- ^ ass ~yf + ass -^^ as before. 192. When the sum assured is payable at the last death of the three, without regard to the order in which the deaths may happen, its present value is the sum of the three assurances at O's death if last, at B's death if last, and at A's death if last, and therefore assA,B,C=assA +assB +assC - ass ™- - ass -~- - OSS ~~<~ Cob - ass --3-^- - OSS -~g~ - ass — g— + assS^/--,ass^+ass^. III. SUKVIVORSHIP ANNUITY, 105 193. It is now proper to return to a subject which was left incomplete in article 162 ; and to seek the value of an annuity payable to C after A and B are both dead, subject to the condition that A have died before B . We have shown, in article 185, that the number of cases in which B dies after having survived A, between the ages a, b and A, B, is ■s.da.l{b + ^)-la.lB + -2.dB. 1{A + +) ^and therefore each one of the C's then alive, in number IC , will prefer the above number of claims, so that the value of all these claims estimated as at the birth of C is 2 da . l{b + i) . pC- la. IB. pO+pC. 2 1{A + i).dB and consequently the value of this annuity from the ages A, B, C till the end of the life table is ■sda.l{b + ^) :spO ^IB.pO ■s{pC.-2l(A + i).dB) la . lb pc lb .pc la. lb. pc Hence, ultimately, the value of this contingent annuity from the present moment onwards is '(¥) = ^da.l(b + i) ,^ , 2(pc.2Z(a + i).c?i) ; — j7 — - ann c - arm {o, c) + , — j, - la. lb ^ ' la.lb.pc 194. By transposing the letters A, B, in the above expression, we obtain the value of an annuity payable to C after the death of ^, ^ hav- ing survived B, viz., , /J^\ l.l(a + \).dh , , •S.{pc.-S.da.l(b + h)) Ann { _5_ ) = — H — ri «ww c - ann (a, c) + ; r, \ I / la. lb ^ ' la .lb .pc and it is obvious that the sum of these two agrees with the value of an annuity payable to C after both A and B are dead. 195. In these cases we have supposed that the payments are to be made at a fixed date, viz., at the anniversary of the present moment ; but it may happen that the payments are to fall due on the day of B's death, and thereafter annually. In order to compute the value of such an obli- gation we must regard the first payment as an assurance at B's death, the second as an assurance payable to C one year after B's death, and so on ; wherefore the general question is, to compute the value of £1 payable to C n years after the death of B, provided B have survived A; the sum of such assurances taken forK=0, w=l, w = 2, etc., is the value of the 106 FEACTIONAL AGES. III. proposed annuity. The nature of the investigation has been already explained in regard to two-life transactions. 196. Let it be proposed to compute the present value of £1 payable to C w years after the death of B, subject to the condition that B shall have survived A . Between the ages a, b and A, B, the number of cases in which B has died after A is l.da .l(b + \)-la .IB + -s.dB .l{A + y) and against each one of these there will be preferred d{C+n) claims dur- ing the year between n and n+ 1 after the ages A, B ; and also for a portion of the cases of complete extinction of the couples A, B during the course of the year from A, B to A+\, B + 1; and thus we have only to substitute C+n for C in the investigation of article 186, in order to obtain the entire number of claims ; this number is {•2da.l{b + \)-la.l{B + ^)-\l{A + ^)dB+^dB.l{A + \)}d{C+n) and the value of these payments estimated as at the birth of C is got by changing the factor d{C+n) into m{C+n). Making this change, and taking the sum from the ages A, B, C+n, onwards, dividing also this sum by la .Ib.pc, we have for the value of the postponed assurance :2da.l{b+l) 2 m(c + n) _ 2 l(b + j) ■ m{c + n) ^ 2 l{a + ^) .db . m(c + n) la . lb pc lb .pc ^ la.lb .po 2 (m(c + n).'sl{a + i) . db) la ,lb . po £+5 n(c + h) which is evidently ass b x— ^ ~r pc PRACTIONAL AGES. 197. Before leaving the subject of transactions involving three lives, it is proper that I should indicate the manner of interpolating for fractional ages. Thus, supposing that we have a complete set of tables of annuities or of assurances for every combination of integer ages a, 6, c, it may be III. FKACTIONAL AGES. 107 required thence to compute the value for ages a + r, b + s, c + t, in which r, s, tare parts of a year. Such a problem belongs, not to Ufe calculations in particular, but to all trinomial functions : it belongs to the general sub- ject of interpolation. Using the character p to denote any function, the symbol p(a, b, c) may denote some quantity depending on the three ages a, b, c; and so p(a + 1 , 6 + 1 , c) would denote the corresponding quantity for the ages a + b,b + l, c. If we restrict ourselves to differences of the first order we obtain, from the four values, {a , 6 + 1 , c + 1) + r . s . p(a + 1 , 6 + 1 , c + 1) ; lastly, by interpolating between these two, we have q){a + r,h + s,c + t) = {\-r) (1 - ») (1 - f) . atE ntF at G at JET , after the occurrence of the first death. In this case the number of payments n years hence, on account of the total number la.lb .Ic . Id of quartets, must be {la - lA) IB ACID + [lb - IB) lA . 10. ID + {Ic - W) lA . IB. ID + {ld-lD)lA.lB.lG because these four products exhibit all the possible combinations on account of which claims may be made ; hence the total value of all the subsequent payments is la.^lB.lC.pD + lb.^lA.lC.pD + lC.^lA.lB.pD + ld.^lA.lB.l0.r-^-4tilA.lB.lC.pD which gives to each one of the combinations the share 120 ANNUITIES. IV. ilB.lC.yD ilAACpD s.lA.lB.pD ^lA.lB.pC Ib.lc.pd la.lc.pd la.lb.pd la.lb.pc ■2lA.lB.lC.pD la. lb, Ic .pd Each of these fractions expresses the value of a deferred joint annuity : making A = a, etc., we obtain as the value of the specified annuity, ann(b, c, d) + ann(a, c, d) + ann{a, b, d) + ann{a, b, c) -4:ann{a, b,c, d) 214. Again, let the annuity be payable so long as two, and two only, of the four are alive. The number of payments to be made n years hence in this case is {la - lA) {lb -IB)IC.ID + {la - lA) {Ic -IC)IB .ID + {la - lA) {Id -ID)IB.IC + {lb - IB) {Ic - IC) lA . ID + {lb - IB) {Id - ID) IA.IG + {Ic - 10) {Id - ID) lA . IB from which we deduce the value ann {a, b) + ami {a, c) + ann {a, d) + ann {b, c) + ann (6, d)+ ann {c, d) - 3 ann {a, b, c) - 3 ann {a, b, d) — 3 ann {a, c, d) - 3 ann {b, c, d) + 6 ann {a, b, c, d) . 215. Or if the annuity be payable so long as one, and only one, of the four is alive, we have for the number of payments n years hence {la - I A) {lb - IB) {Ic - IC) ID + {la - lA) {lb - IB) {Id -ID)IC + {la - lA) {Ic - IC) {Id - ID) IB + {lb - IB) {Ic - IC) {Id - ID) lA ; from which the value of the single survivorship comes out ann a + ann b + ann c + ann d — 2 ann {a, 6) - 2 an7i {a, c) - 2 ann {a, d)-2 ann {b, c) - 2 ann {b, d)-2 ann (c, d) + 3 ann {a, b, c) + 3 ann (a, b, d) + 3 ann {a, c, d) + 3 ann {b, c, d) -4: ann {a, b, c, d) . 216. As a last example, I may take that of an annuity payable so long as any one of the four may be alive. This is, clearly, the sum of, 1st, the joint annuity ; 2d, the survivorship of three ; 3d, the survivorship of two ; and, lastly, the survivorship of one. Hence the value of the longest-life annuity is ann a + ann b + ann c + ann d - ann {a, b) - a7in {a, c) - ann {a, d) - ann (6, c) - ann {b, d)- ann {c, d) + ann {a, b, c) + ann{a, b, d) + ann{a, c, d) + ann{b, c, d) -ann {a, b, c, d) . 121 ASSURANCES. 217. The values of assurances are more easily represented by integrals than are those of annuities ; but the facility is, as has been repeatedly explained, of no practical value for want of a knowledge of the nature of the function la . If a sum of money be payable at the death of A to the joint survivors B, C, D, the present value of the expectation is obtained by considering that, at the time A + t, B + t, etc., the number of triplets B, C, D in existence is (IB -tdB){lC-t dC) {ID - 1 dD) while the number of A's who die in the minute time "dt is Zt . dA ; and that, therefore, the number of claims to be made during that time is {IB AC .W-{IB AC .dD + lB .dO .ID + dB .IQ .lD)t + {lB .dC .dD + dB AC .dD + dB .dO AD)t''-dB .dC .dD.f}dA.c>t, that is on the supposition that the deaths are uniformly distributed through the year. If it were desired to take into account the gradually varying rate of mortality we should have to use the values IB-t dB + ^{t^-t)8dB — etc. for the number alive at the fraction t of the year ; and to make the corres- ponding change on the value of - 3 {lA), which would then become dt . dA — ^(2t-l)'dt.S dA + etc. The introduction of these niceties would not change the nature of the investigation ; it would, however, add greatly to the complexity of the formulae. Taking the integral of the above product we find that the number of claims arising between the beginning of the year A and the time t is {lBACAD.t-i{lBAC.dD + lB.dCAD + dBACAD).t^ + ^{lB.dC.dD + dBAC.dD + dB.dOAD).t^-\dB.dC.dD.^}dA. Putting, in this, t = 1 , we have, for the total number of claims during the year from Aio A+\, {IB AC. ID-\{IB .IC.dD + lB. dC.lD + dB.lC. ID) + ^{lB.dC.dD + dB.lC.dD + dB.dOAD)-idB.dC.dD}dA. 218. The multiplier of dA in the above formula may be called the co- efficient of joint succession of the three parties B, C, D to any fourth nominee A ; and we may denote it by the symbol coef{B, C, D) . 122 ASSURANCES. ^^■ The numbers of B's, C's, and D's alive at the middle of the year are lB-\dB, IQ-^dC and ID-^dD respectively; now the product of these is lB.lC.lD-\{lB.lC.dD + lB.dC.lD + dB.lC.lD) + \{lB.dC.dD + dB.lC.dD + dB.dC.lD)-^dB.dC.dD wherefore the co-efficient of joint succession may otherwise be put l{B + \).l{C+^).l{D + ^) + ^{lB.dC.dD + dB.lC.dD + dB.dC.lD) -idB.dC.dD under which form only half as many multiplications are required as in the former. 219. The number of claims arising within the year being, according to this notation, coef{B, C, D) . dA their value, estimated as if, one with another, they should be discounted from the middle of the year to the birth oi A, is coef{B, C, D) . morA and therefore the value of the assurance is expressed by the equation t,c.d 2 coefib , c, d) . mor a ^** » pa. lb. Ic. Id or if we discount, as usual, to the latest birth, the formula becomes hj^ _' 2da.coef(b, c, d)r-^-i " la.lb .Ic .pd 220. If, in the expression for the co-efficient of joint succession, we per- mute the letters A, B, C, D, and take the sum of the four products coef{B, C, D).dA + coef(A , 0,D).dB + coef{A, B,D).dC + coef{A, B,C).dD we obtain the entire number of first deaths occurring during the year, which number ought evidently to be the difference between the number lA .IB .IC .ID of groups existing at the beginning, and the number 1{A + 1) . 1{B + 1) . l{0+ 1) . 1{D + 1) of these groups existing at the end of the year. 221. The value of an assurance to be paid on the occurrence of the first death among the four is the sum of the four assurances at A's death if IV. ASSUEANCES. 123 first, of B's, of C's, and of D's death if first; it may also be obtained by treating the product lA .IB .10 . ID as if it were the number of quartets alive; that is, as if it were liv{A, B, C, D), and by considering the dif- ference in this number for one year, as if it were die {A, B, C, D), the calculation would then be exactly as for a single-life assurance. 222. We may now proceed to investigate the value of a payment to be made to two survivors, Cand D, jointly at the death of B, on the condi- tion that B shall have survived A . At the date A + t, B + t, etc., the number of couples CD existing is (IC-tdC) (ID-tdD), and each of these has a claim as against every death among the B's; now the number of these deaths during the minute time dt is dt . dB, while the number of A's previously dead is la - lA + 1 dA ; therefore the total number of claims arising during this minute time is {la-lA + tdA)(lO-tdC)(lD-tdD)dB.dt or {{la -IA)IC.ID+ (IC .ID.dA- {la - lA) ID.dQ- {la - lA) lC.dD)t + {{la-lA)d0.dD-l0.dA.dD-lD.dA.dO)t^ + dA.dC.dDt^}dt.dB. Integrating this quantity we find, for the number of such claims arising during the time t, {{la-lA).lG.lD.t + \{dA.lC.lD-{la-lA)lD.dG-{la-lA)lC.dDy + ^{{la-lA) dC. dD -IC. dA.dD -ID.dA. dCy + \dA. dC. dDt^}dB By making « = 1 we obtain the total number of claims during the year, viz., {la{lC. ID-^{IC. dD + dC. lD) + ^dC. dD) - lA . 10. ID + \{lA.lC.dD + lA.dC.lD + dA. 10. ID)-^{IA . dC . dD + dA.lO. dD + dA. dO . lD) + ^dA . dC.dD} dB which may be more concisely written {la . coef{C, D) - coef{A, C, D)} dB . This has to be discounted to a fixed epoch as the birth of D ; the results then have to be summed for the whole of life, and the amount divided by la.lh .Ic . pd in order to give the value of the conditional assurance. The result is -^ — c, rt a.c.d ass _J_^ - ass ~^ - ass ~™- . a The student may advantageously compare this demonstration with the 124 ASSURANCES. IV. line of argument followed in article 176 ; the superiority of the fluxional process in exhibiting general principles is apparent by the contrast. 223. When the payment is to be made to C and D jointly after the deaths of A and B , but without regard to the order of these deaths, its value is the sum of the two conditional assurances, that is ' a 6 — ass -^ + ass ~^ — ass -^ ass -^— • 224. We may now examine the value of an assurance payable to one of the nominees as soon as the three others are dead, it being stipulated that these shall have died in a particular order. We shall suppose the sum to be payable to D on the death of C, C hav- ing survived B, and B having survived A . The number of cases in which B dies having succeeded to ^, in the time between age a and age A + t, is ^cla.l(b + i)-la.lB + ^l{A + ^).dB + {la-lA).dB.t + ^dA.dBe; the number of deaths among the Cs in the minute interval of time 'dt is dC ."dt, while the number of D's alive is ID -tdD , wherefore the number of cases in which payment may be claimed during the differential of the time is the product of these three factors, viz., {ID .i.da.lib + \)-la.lB .ID + ID.^ 1{A + i).dB + {{la - 1 A) dB.lD-dD.-2da.l{b + i) + la.lB.dD-dD.-2 1{A + J) . dB)t + {^dA.dB.lD-{la-lA)dB.dn)t^-^dA.dB.dnf}dC.dt The integral of this, viz., {dC.W.^da.l{b + ^)-la.lB.W.dC+dC.W.sl{A + ^).dB}t + ^{{la-lA)dB.dC.lD-dC.dD.-2da.l(b + i) + la.lB.dC.dD -dQ.dD^l{A + \)dB}t^ + ^{\dA.dB.d0.lD-{la-lA)dB.dC.dD}fi -^dA.dB.dO.dD.t' shows the number of such cases occurring during the fractional part t of the year; and making t - 1, the number for the whole year is found to be dC.l{D + ^).^da.l{b + i)-la.coef{B,D).dC+dC.l{D + ^).^l{A + ^).dB -^l{A + ^).dB .dC.lD + iilA-^ dA) dB.dC.dD . TV. ASSUEANCES. 125 Discounting these, taking the sum for the whole of life, and dividing by la. lb. Ic .pd, we obtain "f _ ^da.l(b + i) a „„ b,d ^dc.pid + i).^l(a + i).db ""'X - la. lb «««"r -«*«~S" + la.lb.lc.pd J 2 Z(a + ^) . d6 . mc . M 2 (^ ?a - ^ da) .db.dc. md ^ la. lb .pc. Id la.lb.lc. pd 225. By exchanging the positions of the letters a and b in the above formula, we obtain the value of a sum payable to D on the death of C, provided that Chave survived A, and that A have survived B ; and by adding this to the preceding we obtain the value of an assurance payable to D at C's death, it being stipulated that C shall have outlived both A and B. On performing the addition and simplifying we obtain --^-. d T,,d a.,d ^coef(a,b,d).mc ass _s„ = ass — — ass "r — ass -V -i , ,, t3 — a,h la. lb . po. Id — ass ~ — ass ~ — ass --r + ass -^-r— • c c c c 226. This result might have been arrived at by considering that the number of couplets A B entirely extinct at the date A + tis the product of la-lA + tdA by Ib-lB + tdB, and that, therefore, the number of pay- ments to be made during the minute time 3i is {la -lA + t dA) {lb -lB + tdB)dC{lD-t dD) dt . When the expansion of this product is integrated, and when in the integral so obtained t is made equal to 1, we have, for the number of pay- ments to be made during the year {la . lb . 1{D + i)-la{lB .W-ilB.dD-^dB.W + ^dB . dD) -lb{lA.lD-^lA.dD-\dA.lD + ^dA.dD) + lA.lB.lD-^lA.lB.dD + ^lA.dB.dD -\lA.dB.lD + ^dA.lB .dD -^dA.lB.W + ^dA.dB.lD-idA.dB.dDjdC. Tliis expression may be put in the more concise form la.lb.dC. l{D + \) - la . coef{B, D) . dO- lb . coef{A, D) . dC + coef{A,B,D).dC which at once pictures the result above given. d 227. By exchanging the positions of the letters in the value of ass _£_ a, b we can obtain those of ass _5_ and ass ^ ; the sum of these three is the a, c bfC 126 ASSURANCES. IV. value of an assurance payable to D on the complete extinction of the trip- let ^, 5, O, without regard to the order of the deaths. This value is d d h,d \ years when he shall have attained the age of 60 and thereafter. 13. Required the half-yearly premium for the purchase of £26 to be paid to A now aged 23^ years, when he shall have attained the age of 60, and thereafter half-yearly during his life. 14. Required the half-yearly premium for the purchase of £26 payable 37 years hence to A now aged 23J years, and to be repeated half-yearly thereafter during his life. Article 42. Revise the exercises on the preceding article, using second differences. Article 46. 1- Required the present value of £13 payable every three months during the life of A aged 50 years ; first payment now. 2. Required the value of a quarterly payment of £13 to begin three months hence, and to continue during the life of A aged 50 . 3. Required the value of a quarterly payment of £13, to begin when A now aged 23 shall have reached the age of 60 years. 4. Required the value of £1 payable quarterly for 148 payments during the life of A now aged 23 years. 5. Required the quarterly premium for the purchase of £13 to be paid quarterly to A now aged 23 years, when he shall have attained the age of 60 and thereafter. 6. Required the quarterly premium for an endowment of £800 to A aged 12 when he shall have reached the age of 25 years. Article 49. 1. Required the present value of £1 payable monthly dur- ing the life of A aged 50 years ; the first payment to be now. 2. Required the present value of £l , payable weekly, beginning one week hence, and to continue during the hfe of A aged 30 . 3. Required the present value of £1 to be paid weekly to A now aged 23 when he shall have reached the age of 60 and thereafter. 4. Required the value of an aliment of £52 per annum, to begin when A aged 23 shall have reached the age of 60 . EXERCISES. 135 5. What is the value of an aliment of one shilling per day payable to a person now aged 30 years ? Article 53. Make some of the preceding calculations, using differences of the third order. Article 56. 1. Construct a table of "2^ pa . 2. Make also a table of log ^pa . Article 57. 1. Required the present value of an annuity payable to a party now aged 23 years, beginning with £30 one year hence, and increas- ing by £4 every year thereafter. 2. What is the present value of a present payment of £40, with an addition of £3 annually, during the life of a person aged 37 years ? 3. What is the present value of an increasing annuity, beginning with £70 payable 5 years hence, with an augmentation of £10 each succeeding year, to a person now aged 43 years ? Article 58. 1. Required the present value of £43 to be paid seven years hence, £45 eight years hence, and so on until the payment reach £51, beyond which it is not to increase, to a nominee now aged 19 years. 2. Required the present value of an increasing annuity of £50 seven years hence, with an annual augmentation of £5 until it become £100, to be paid to a person now aged 23 years. Article 59. 1. Required the present value of thirteen payments, the first of £87 to be made 5 years hence, and the others of £90, £93, etc., annually thereafter during the life of a nominee now aged 15 years. Article 60. 1. Required the present value of a decreasing annuity beginning one year hence with £300, and decreasing each year by £5, during the life of a nominee now aged 50 years. 2. An annuity beginning with £200 one year hence, and diminishing every year thereafter by £5, is to be paid to a person now aged 30 years. Required its value. 3. A person now aged 40 wishes to lay out £4000 in the purchase of an increasing annuity ; the annual increase to be £10. What must the first payment be ? 4. £5000 is to be expended in purchasing an annuity for a person now aged 30 years, the annuity is to increase annually by one-twentieth part 136 EXERCISES. of the first payment, to be made one year hence. How much must that payment be ? Article 63. 1. Required the value at January 1, 1864, of £1300, to be paid at the death of a person then aged 23, if that death happen during the year 1894. 2. Required the present value of £1000, to be paid at the death of a person now aged 20 years, provided that death happen in the 28th year of his age. 3. Required the value as at January 1, 1864, of £1300, to be paid at the death of a person then aged 23 , if that death happen during any of the years 1894 to 1903 inclusive. Article 65. 1. Compute the ratio of the result obtained by the com- mon method of average to that deduced strictly from the hypothesis that the deaths are equally distributed during the year, for the rates 3 , 4 , 5 , and 6 per cent. 2. Thence calculate the values of the expectations proposed in the pre- ceding article. Article 66. 1. Compute the values of the quantities represented by S^, 8^, 8^, 8^, at the rates 3, 4, 5, and 6 per cent. 2. Required the value of £1000 payable at the death of a person now aged 40 years, provided that death happen between 20 and 21 years hence ; according to the common method of averaging ; according to the hypothesis of uniform distribution of deaths; and strictly using fourth diiferences. Article 67. Compute the values of mor 20 , mor 40 , mor 60 . Article 69. 1. Construct a table of Zo^ 1,03-"-*, or for any other rate of interest. 2. Make out the columns da and log da . 3. Compute Zo^ mor a, and mora. 4. Thence form the columns 2 mor a and log 2 mor a . 5. Compute log assurance and assurance. 6. Required the present value of an assurance of £1200 to be paid at the death of A aged 23 years. EXEECISES. 137 7. A person aged 37 wishes to inyest £3000 in the purchase of an assur- ance at his death. What sum may he assure for ? 8. Required the present value of £1200 to be paid at the death of A aged 43| years, — 1st, Using common interpolation. 2d, Using differences of the second and third orders. Article 71. 1. Make a table of the logarithms of assurance, on the supposition of a uniform distribution of deaths during the year. 2. Examine the correction due to second differences ; for this purpose it will be necessary to form a column 2 8da .r-". Article 73. 1. Required the present value of £1000 to be paid on the death of A aged 30 years, provided that death happen after 10 years. 2. Required the present value of £1200 to be paid at the death of B aged 35 , if it happen after 20 years. 3. Construct a table of assurances deferred fifteen years. Article 74. 1. What is the present value of an assurance for £1200 at the death of B aged 35 , if that death happen within 20 years. 2. What is the present value of £1000 payable at the death of C aged 43, if that death happen within 10 years ? 3. Construct a table of short assurance, say for 15 years. Article 75. 1. Compute the value of ass 45 by the formula given in this article. Article 76. 1. Compute ass 45 from ann 45 by the formula of article 76. Article 78. 1. Required the present value of an assurance of £1000, with a guaranteed quinquennial bonus of 5 per cent, on the life of a per- son aged 30 . 2. Required the present value of an assurance of £1200, with a septen- nial bonus of £50, on the life of a person aged 23 . Article 79. 1. Form tables of 2^ mora and log l.^ mor a . 2. Thence deduce loff incr . ass , and incr . ass . 3. Required the present value of £1000, with £10 additional for each year, to be paid at the death of a person now aged 30 . 138 EXEECISES. Contrast this with the first exercise in article 78 . Article 80. 1. What is the value of £1000, less £10 for every year, to be paid at the death of a person now aged 30 ? 2. Required the value of an assurance of £1000, with an annual deduc- tion of £20, to be paid at the death of a person now aged 30 . Article 84. 1. Required the premium to be paid annually by a person now aged 20 years in order to secure a sum of £1000, to be paid to him on his reaching the age of 60 . 2. Required the premium to be paid annually by a person now aged 20 in order to secure an annuity of £80 , to be paid to him at age 60 and thereafter. 3. Required the premium to be paid annually for 15 years by a person now aged 20 in order to purchase an annuity of £80 , to begin at age 60 . Article 85. 1. Required the annual premium for an assurance of £1200 to be paid at the death of A aged 23 years. 2. What sum must be paid half-yearly to procure an assurance of £1200 to be paid at the death of A aged 23 years ? 3. What is the monthly premium for an assurance of £1200 at the death of A aged 30 ? 4. Construct a table of log premiuvn, by the two processes, and compare the results. 5. Required the annual premium for an assurance of £1000, with an addition of £100 every 5 years, on the life of A aged 23 . 6. An assurance of £1000 on the life of a person aged 23 , with a quin- quennial bonus of £100, is to be paid for by a premium decreasing quin- quennially by 5 per cent of the first payment. Required that first payment. 7. A person aged 25 assigns out of the rent of his estate £200 annually to purchase an assurance at his death. What should the assurance be ? 8. A person aged 27 wishes to purchase an assurance for £1000 by a premium decreasing 10 per cent each year. Required the first and sub- sequent premiums. 9. What must be the annual premium for an assurance of £1000 at the death of a person aged 27 , with a return of one-half of the premiums paid (without interest) ? 10. What must be the annual premium for an assurance of £1000 at EXEECISES. 139 the death of a person aged 27 , with a return of one-half of the premiums paid, and interest thereon. 11. Required the annual premium for an assurance of £1000 at the death of a person aged 27, with a return of two-thirds of the premiums, and interest thereon. Article 86. 1. Required the annual premium during 10 years for an assurance of £1000 payable at the death of A aged 27 . 2. Required the five years premium for the above assurance. Article 87. 1. Required the annual premium for a short period assurance of £1000 to be paid at the death of A aged 27, if that death happen within ten years. 2. Required the premium to be paid for £1000 payable at the death of A aged 27 , if that death happen within one year. 3. What is the annual premium for £1000 payable to A aged 27 , on his attaining the age of 60, or to his heirs at his death, if it happen previously ? 4. What is the annual premium for assuring £1000 payable at the death of A aged 27 , if that death happen after 33 years ? 5. A life-tenant who owes £5000 has assigned £500 per annum of the rents in liquidation of the debt, and proposes to secure his creditor by a further assignation of rent to purchase an assurance of the balance as at his death. Required the annual premium for that assurance. 6. A person aged 23 wishes to have £1000 at his command 27 years hence, and proposes to purchase the endowment by five annual payments. What must each payment be ? Article 88. 1. Ten years ago A , then aged 25 , paid £20 in purchase of an assurance at his death, and has continued to pay a like sum annually. What is the amount of the assurance standing at his credit ? 2. What is the present value of that assurance ? 3. To what sum would the premiums, with interest, have amounted ? 4. A now proposes to double his annual payment ; why would a medical certificate be required ? 5. Construct a table of accumulated assurance. Article 89. 1. A policy for an assurance of £4700, payable at the death 140 EXERCISES. of a person then aged 25, was opened 23 years ago. Required its pre- sent value, the next premium being just due. 2. A pohcy of assurance for £1200 was opened 20 years ago on the life of a person then aged 23 years. What is its present value, the 21st premium having been just paid ? 3. A policy of assurance for £1200 was opened 20|^ years ago on the life of a person then aged 23 years. What is its present value ? 4. A policy of assurance for £1200 was opened 20| years ago on the life of a person then aged 23 years. What is its present value ? 5. A policy of assurance for £1200 was opened 20^ years ago on the life of a person then aged 23 years : he wishes to discontinue the annual payments, and to commute the obligation into an assurance at his death. For what sum should the new policy be granted ? Article 91. 1. Required the present value of a policy for an annuity of £80 to begin at age 60, opened 10 years ago by a person then aged 25, and paid for by annual premiums. 2. Required the present value of a policy for £1000 payable at age 60, opened 10 years ago by a person then aged 25, and paid for by annual premiums. 3. A policy for short-period assurance of £1000, at the death of a party then aged 23 , was opened six years ago ; four years of the term are still to run. Required the value of the policy. 4. At the age a a policy was opened for an endowment of £1 payable at age A ; the annual premiums have been paid up to age a ; show that the value of the poUcy, the premium being just due, is pA "ipa, - "SpA pa 'S.pa- ^pA ' 5. At the age a a policy was opened for an annuity of £1 payable at age ^ and thereafter ; the annual premiums have been paid up to age a; show that the value of the policy, just before payment of the premium, is " SpA ^pa-'SpA pa ^pa-'SpA 6. Find the general expression for the value of a policy of assurance payable by a limited number of premiums. 7. At what ages is it improper for an Office to grant a whole life pohcy of assurance, without security for the payment of the future premiums ? II. EXERCISES. 141 TWO LIYES, Article 95. 1. Required the present value of £1000 payable 17 years hence, provided A aged 40 and B aged 27 be then both aUve. 2. What is the present value of £500 payable 43 years hence, provided two persons, each aged 21 years, be then both alive ? 3. What is the present value of £500 payable 30 years hence, provided A aged 31 and his wife B aged 26 be then both alive ? 4. What is the present value of £500 payable 30 years hence, provided A aged 31 be then dead, and his wife B aged 26 be then alive ? 6. What is the present value of £500 payable 30 years hence, provided A aged 31 be alive, and his wife B aged 26 be then dead ? 6. What is the present value of £500 payable 30 years hence, provided, of the above couple, one be alive and the other dead 1 7. What is the present value of £500 payable 30 years hence, provided both A aged 31 and his wife B aged 26 be then dead ? 8. Required the present value of £1000 payable 20 years hence if both A aged 43 and B aged 23 be then alive, 9. Required the present value of £1000 payable 20 years hence to A aged 43 , provided B aged 23 be then dead. 10. Required the present value of £1000 payable 20 years hence to B aged 23, provided A aged 43 be then dead. 11. Required the present value of £1000 payable 20 years hence, pro- vided both A aged 43 and B aged 23 be then dead. 12. Show that the sum of the four expectations 8 , 9, 10 , 11 , make up the present value of £1000 certain, payable 20 years hence. 13. What is the present value of £1000 payable 20 years hence, pro- vided, of the two, A aged 43 and B aged 23, one be alive and one dead? 14. What is the present value of £1000 payable 20 years hence, pro- vided two persons, A and B, each aged 33 years, be then both alive ? 15. What is the present value of £1000 payable 20 years hence to A aged 33, provided B, also aged 33, be then dead ? 142 EXERCISES. H. 16. What is the present value of £1000 payable 20 years hence, pro- vided A and B, each aged 33, be then both dead ? 17. Required the present value of £1000 payable 20 years hence to the husband A, aged 38, and the wife B, aged 28, jointly. 18. Required the present value of £1000 payable 20 years hence to the wife B aged 28, provided her husband A aged 38 be dead. 19. Required the present value of payments of £1300 twenty-seven years hence, £1000 thirty years hence, and £700 thirty-three years hence, on condition that A aged 43 and B aged 23 be then both alive. Article 100. 1. Construct a table of Logp{a, b) for a difference of 20 years in the ages. 2. Form also a table of p(a, b) . 3. Thence form the column 2p(<2, b) . 4. And thereafter Log 2p (a, 6) . 5. By help of these columns solve the Exercises 95, 1, 8, 19. Article 101. 1. Make a table of log Joint annuity (beginning now). 2. Thence find the values of joint annuity. 3. What is the present value of an annuity of £700 payable so long as A aged 43 and B aged 23 are both alive ; first payment now ? 4. Required the present value of an annuity of £600 payable to A aged 50 and B aged 30 jointly. 5. A present payment of £2000 is to be commuted into an annual pay- ment during the joint hves of ^'aged 35 and B aged 15. Required the annual payment. 6. An assurance for £1200 payable at the death of A aged 50 is to be purchased by an annual premium payable during the joint-hves of the above A, and B aged 30. Required the premium. 7. An assurance for £1200 at the death of A aged 50 is to be purchased by an annual premium during the joint lives of B aged 40 and C aged 20. What is the premium ? and what precautions must be taken in drawing up the agreement ? 8. A, aged 43, wishes to have an assurance of £2000 payable at his death ; for which annual premiums are to be paid during the joint lives of himself and his wife B aged 23. Required the premium. 9. The above arrangement having subsisted for 10 years, required the value of the poUcy. II. EXERCISES. 143 10. Exhibit the general formula for the value of such a policy opened at the ages a,b, and having subsisted for n years. Article 102. 1. Similarly make a table of log joint annuity (beginning one year hence). 2. Find thence the values of joint annuity (first payment one year hence). 3. Compare the two columns of annuity as a check upon the work. 4. What is the present value of an annuity of £52 to begin one year hence, and to continue so long as both A aged 40 and B aged 20 may be alive. 5. Make a table of log joint annuity deferred jive years. 6. And thence one oi joint annuity deferred jive years. 7. Required the present value of an annuity of £500, beginning 25 years hence, and payable to A aged 35 and B aged 15 jointly. 8. What is the present value of 23 payments of £500 each to begin 6 years hence, and to be made at intervals of one year, provided both A aged 39 and B aged 19 be both alive ? 9. What is the present value of £500 payable 3 years hence, £700 pay- able 5 years hence, and £900 payable 9 years hence, subject to the condi- tion that A aged 41 and B aged 21 be both alive ? 10. Required the present value of an annual payment of £80, begin- ning to-day and to continue for 20 payments, so long as A aged 37 and B aged 17 may be both alive. 11. A aged 37 and B aged 17 wish to purchase a joint annuity of £200, to begin 28 years hence, for which they offer to pay an annual premium. Required the premium. 12. A aged 47 desires to secure an annuity of £200 payable to his wife B aged 27, when and after she shall have reached the age of 60, by the payment of an annual premium. Required the premium. Article 106. 1. Form a table of A's survivorship annuity. 2. Similarly make a table of ami -^ • 3. A aged 37 wishes to secure £200 a-year to his wife B, aged 27, after his death, and will pay for it by annual instalments. Required the premium. 4. A policy of the above nature having subsisted for ten years, required its value. 144 EXEECI8ES. II. 5. Exhibit the general formula for the value of a policy of this kind, a and h being the ages at the commencement, and n the number of years during which the policy has lasted. 6. Required the present value of an annuity of £52, to be paid at inter- vals of one year from the present date to B aged 20 after the death of A aged 40 . 7. Required the present value of an annuity of £52, to be paid at inter- vals of one year from the present date to A aged 40 after the death of B aged 20 . Article 109. 1. Construct a table of survivorship annuities ; that is, payable to one after the death of the other. 2. Required the present value of an annuity of £52, to be paid at inter- vals of one year from the present date to the survivor after the death of either A aged 40 or B aged 20 . Article 110. 1. Construct a table of longest-life annuity. 2. What is the present value of an annuity of £52 , to be paid so long as one of the two, A aged 40 and B aged 20 may be alive ? 3. What is the present value of £70 payable annually so long as one of the couple A aged 43, B aged 23, may be alive ? 4. A yearly payment of £500 is to be made to A aged 41 , and B aged 21 , jointly, each receiving one-half so long as they are both alive, and the survivor to receive the whole after the death of the other. Required the present value of each expectation. 5. An allowance of £240 annually is directed to be paid to A aged 45 and B aged 25, so long as both may be ahve; the allowance to be reduced to £192 at A's death, and to £144 at B's death if first. Required the present value of the legacy. 6. What is the present value of a perpetuity of £52 payable at inter- vals of one year from the present date after both A aged 40 and B aged 20 are dead ? 7. Required the present value of an annuity of £52, to begin 15 years hence, and to continue during the joint lives of A aged 40 and B aged 20 years. 8. Required the present value of £52, payable annually for 15 years during the joint lives of A aged 40 and B aged 20 . 9. Required the present value of £52 , payable annually for 15 years from this date, to B aged 20, A aged 40 being dead. II. EXEECISES. 145 10. Required the present value of £52 payable annually for 15 years from this date to A aged 40, B aged 20 being dead. 11. Required the present value of an annuity of £52, beginning 15 years hence, and payable to B aged 20 after the death of A aged 40 . 12. Required the present value of an annuity of £52, beginning 15 years hence, and payable to A aged 40 after the death of B aged 20 . 13. Required the present value of an annuity of £52, beginning 15 years hence, and payable to either A aged 40 or B aged 20, the other being dead. 14. Required the present value of an annuity of £52, beginning 15 years hence, and payable to either A aged 40 or to B aged 20 ; but pay- able once only at each date. 15. A aged 30 wishes to procure an endowment of £1000 in favour of B aged 10, on his reaching the age of 25 years, by a premium payable during their joint life. What is the premium ? > 16. Required the annual premium to be paid during the joint lives of A aged 50 and B aged 30 in order to purchase an annuity of £52 to be paid to B after A's death. 17. Seven years after the agreement in the preceding exercise, A and B are both alive. Required the value of the policy. 18. Investigate the general formula for the valuation of such policies. Article 111. 1. Required the present value of an annuity of £52, pay- able quarterly, during the joint lives of A aged 47 and B aged 27. 2. What premium must be paid quarterly during the joint lives of A aged 47 and B aged 27 in order to secure an annuity of £520, to be paid quarterly to B after the death of A ? 3. An allowance of £20 per month is directed to be paid to A aged 45 and B aged 25 jointly, so long as both may be alive ; this allowance to be reduced to £16 at A's death, and to £12 at B's deatii if first. Required the present value of the legacy. Article 113. 1. Compute the value of p(43^, 23^) by common inter- polation, and also by the method given in this article, and contrast the results. 2. In computing strictly the value oip{43>^, 23 J) show what corrections are due to differences of the second and third orders. Article 116. Compute the value of an alimentary payment to A aged 146 EXEECISES. H. 43 and B aged 23 jointly ; and contrast the true value with that obtained in the ordinary way. Article 117. 1. Required the present value of an annuity beginning to-day, and payable during the joint lives of A aged 35 years 7 months and B aged 21 years exactly. 2. Required the present value of an annuity beginning to-day, and pay- able during the joint lives of A aged 35 years 7 months and B aged 21 years 9 months. 3. Required the present value of an annuity payable to B aged 27 after the death of A aged 39]^. 4. Required the present value of an annuity payable to B aged 27^ after the death of ^ aged 39 1^. 5. Required the present value of an annuity payable to the survivor of A aged 53 j^ and B aged 38xV' Article 119. 1. Required the present value of an annuity beginning to-day, and payable during the joint lives of A aged 35xV ^nd B aged 21 1^. Contrast the result with that of the second example under article 117. 2. Perform the interpolation for the third exercise in article 117, and show the difference between the two results. Article 122. 1. The sum of £1000 is to be paid to B, now aged 29 years, at the death of A aged 49 years, provided that death happen between 10 and 11 years hence. Required the limits of the value. 2. Required the value, taking the number of B's alive at the middle of the year. 3. What is the value on January 1, 1865, of £1300 to be paid in the year 1895 to A aged 50 on the death of B aged 30 years ? 4. What is the value on January 1, 1865, of £1300 to be paid to B aged 30 on the death of A aged 50, provided that death happen in the year 1895 ? Article 125. 1. What is the value on January 1, 1865, of £1300 to be paid in the year 1895 to A on the death of jB, or to 5 on the death of ^, the ages being 50 and 30 ? 2. What is the value of £1000 payable to A at Es death, or to B at A's death, if that death happen between 20 and 21 years hence, the pre- sent ages being 40 and 20 ? II. EXERCISES. 147 3. What is the value at January 1, 1865, of £1300 to be paid to B aged 30 at the death of A aged 50 , if that death happen in any of the years from 1895 to 1905 inclusive ? Article 128. 1. Form a table of the logarithms of the numbers alive at the middle of each year, that is log l{a + \) . 2. Make a table of logp{b + \) . 3. Thence compute log mor -^ and log mar -|~ . 4. Extract the values of mor -j- and of mor -~ . 5. Compute 2 mor™ and 2mor-^. 6. Take out the logarithms, log^m-^ and log 2 m -j- . 7. Thence find log ass ™ and log ass -^ . 8. Extract the values of ass -^ and of ass -^~ . 9. Thence by addition find ass {a, b) . Article 130. 1. Make by multiplication the table l{a, b) . 2. By taking the difi'erences find d{a, b) . 3. Take out the logarithms logd{a, b) . 4. Make the table log mor {a, b). 5. Extract the values of mor (a, b) . 6. Thence find 2 mor {a, b) and log 2 mor {a, b) . 7. Form the table log ass {a, b). 8. Take out the values of ass {a , 6) and compare them with the result of No. 9 in article 128. 9. Required the present value of £1300 to be paid to B aged 30 on the death of A aged 50 . 10. Required the present value of £1300 to be paid to A aged 50 on the death of B aged 30 . 11. Required the present value of £1300 to be paid at the first death of A aged 50 or B aged 30 . 12. Required the present value of £1300 to be paid to B aged 30xV at the death of A aged 50^^ . 13. Required the present value of £1300 to be paid to A aged SO^V at the death of B aged 30xV ■ 148 EXEECISES. H. 14. Eequired the present value of £1300 to be paid at the first death of A aged 50^^ or B aged 30t^ . Article 131. 1. Form a table of the values of assurance at A's death if second. 2. Form a table of the values of assurance at B's death if second. Article 133. 1. Form a table of the values of assurance at the second death. 2. Required the present value of £1300 payable atihe death of B aged 30, provided A aged 50 be previously dead. 3. Required the present value of £1300 payable at the death of A aged 50 , provided B aged 30 be previously dead. 4. What is the present value of the reversion of £1300 of which the use is held jointly and severally by A aged 50 and B aged 30 1 5. A aged 40 and B aged 20 draw jointly in equal shares the interest of £4000 in 3 per cent stock, and the principal is to fall to the longest liver. Required the present value of each one's expectation. 6. A aged 40 and B aged 20 draw jointly the dividend on £4000 of 3 per cent stock, A receiving two-thirds and B one-third. The principal is to fall to the longest liver. Required the value of each expectation. 7. A sum of £2000 is to be expended in the purchase of an assurance to B aged 30 at the death of A aged 50. What sum should be assured ? 8. £2000 is to be laid out in the purchase of an assurance to A aged 50 at the death of B aged 30. Required the amount of the assurance. 9. What assurance at the last death of A aged 50 and B aged 30 can be purchased for a present payment of £2000 ? Article 136. 1. Compute the value of ass~^ from the annuity tables. 2. Compute the value of ass -^ from the annuity tables. 3. Compute the value of ass (a, b) from the annuity tables. 4. Show how annuity {a , 6) may be deduced from assurances. 5. Show that the difference between the values of ass {a, b) and ass {A , B) bears a constant ratio to the difference between the correspond- ing joint annuities. Article 138. 1. Required the present value of an annuity of £200, to II. EXEKCISES. 149 begin 20 years hence, and to continue during the joint lives of A aged 40 and B aged 20 . 2. A aged 40 and B aged 20 wish to purchase a joint annuity of £200 , to begin 30 years hence, by an annual premium. Required the premium, 3. A policy, arranged as in the above exercise, having subsisted for 15 years, required its value. 4. Exhibit the general formula for the value of a policy of the above nature. 5. A aged 40 and B aged 20 wish to purchase for B an annuity of £200, to begin 30 years hence, by an annual premium payable during their joint lives. Required the premium. 6. A policy as in No. 5 having subsisted for 15 years, required its value. 7. Exhibit the general formula for the value of such a policy. Article 139. 1. Required the present value of an annuity of £200, to begin 30 years hence, payable to B aged 20, provided A aged 40 be dead. 2. Required the present value of an annuity of £200, to begin 30 years hence, payable to A aged 40 after the death of B aged 20 . 3. Required the present value of an annuity of £200, to begin 30 years hence, payable to either survivor of A aged 40 and B aged 20 . 4. Required the annual premium, payable during the joint life, for No. 1. 5. Required the annual premium, payable during the joint life, for No. 2. 6. Exhibit the general formulse for the values of policies to suit the transactions Nos. 4 and 5 . Article 140. 1. What is the value of an assurance of £1300 payable to B aged 30 on the death of A aged 50 , provided that death happen after 30 years. 2. Required the premium, payable annually during the joint lives, for the above deferred assurance. 3. Exhibit the general formula for the value of a pohcy as above. 4. What is the value of an assurance of £1300 payable at the first death of A aged 50 and B aged 30, provided that death happen after 30 years ? 150 EXERCISES. 11. 5. Required the annual premium for the above deferred assurance. 6. Give the general formula for the value of a policy of the above nature. Article 141. 1. Give the general formulae for the values of deferred assurances payable at As death if second, at B's death if second, and at the second death. 2. What is the present value of an assurance of £1300 payable at the death of B aged 30, provided he have survived A aged 50, and provided B do not die within 30 years ? 3. What is the present value of £1300 payable at the death of B aged 30, provided he have survived A aged 50, and provided that neither of them die within 30 years ? 4. Show the distinction between the formulae for these two cases. 5. What is the present value of £1300 payable at the death of the sur- vivor of the two, A aged 50 and B aged 30, provided the survivor do not die within 30 years ? 6. What is the present value of £1300 payable at the last death of A aged 50 and B aged 40 , provided neither of them die within 30 years ? 7. Give the formula for both cases. 8. Required the premium to be paid annually, during the joint lives, for the assurance in No. 2. 9. Required the premium to be paid annually, during the life of B, for the assurance in No. 2. 10. Exhibit the formula for the value of a policy arranged as in No. 8. 11. Exhibit the formula for the value of a policy arranged as in No. 9. 12. Required the premium to be paid annually, during the joint lives, for the assurance as in No. 5. 13. Required the premium to be paid annually, during the joint Uves, for the assurance as in No. 6. 14. Exhibit the formulae for the values of policies arranged as in Nos. 12 and 13. Article 142. 1. Required the value of an annual payment of £200, to be continued for 30 payments, during the joint lives of A aged 50 and B aged 30. II. EXEECISES. 151 2. Required the value of a payment of £200, to be made 10 years hence, and repeated annually for 10 payments, subject to the condition that A aged 50 and B aged 20 be both alive. 3. What is the present value of £1300, to be paid to B aged 30 at the death of A aged 50, if that death happen within 30 years ? 4. What is the premium, payable annually during the joint lives, for the assurance as in No. 3 ? 5. What is the present value of £1300 payable at the first death of A aged 50 and B aged 30 , if within 30 years ? 6. Required the annual premium for the assurance as in No. 5. 7. What is the present value of an assurance of £1300, to be paid at the death of B aged 30, if that death happen within 30 years, and if B have survived A aged 50 ? 8. Required the annual premium, payable during the joint lives, for the assurance in No. 7. 9. Required the payment to be made annually during B's life for the assurance in No. 7. 10. Give the formulae for the values of policies arranged as in Nos. 8 and 9. 11. What is the value of £1300 payable at the last death of A aged 50 and B aged 30 , if within 30 years ? Article 143. 1. Required the present value of £1300 , to be paid to B aged 30, six years after the death of A aged 50. 2. Required the value of seven payments of £100 each to be made to B, one at the death of A, and the rest annually thereafter; the present ages being 50 and 30 years. Article 144. What is the value of an annuity of £100 payable to B aged 50 at the death of A aged 30 , and annually thereafter ? Article 146. 1. An assurance of £1300 , payable at the death of A aged 50, is to be purchased by a premium during the joint lives of A and of B aged 30 . Required the annual premium. 2. What is the value of such a policy after 10 years, supposing both to be alive ? 3. What is the value of such a policy after 10 years, supposing B to be dead? 152 EXEECISES. II. 4. Required the premium for an assurance of £1300 payable to B aged 30 at the death of J. aged 50. 5. What is the value of the policy after 10 years ? 6. Give the general formula for the value of such a policy. 7. Required the premium for an assurance of £1300 payable at the first death of A aged 50 and B aged 30. 8. What is the value of the policy after 10 years ? 9. Give the general formula, and show the analogy between it and the formula for an ordinary single-life policy. 10. Required the premium, payable during the joint lives, for an assur- ance of £1300 at the death of B aged 30 , if he have survived A aged 50 . 11. What is the value of the policy after 10 years ? 12. Give the general formula for the policy. 13. Required the premium, payable during the life of B, for an assur- ance of £1300 at the death of B, if he have survived A. 14. What is the value of the policy after 10 years ? 15. Give the general formula. 16. Required the premium, payable during the joint life, for £1300, to be paid at the last death of A and B. 17. Required the value of the policy after 10 years. 18. Give the general formula. 19. Required the premium, payable during A's life, for £1300, to be paid at the last death of A and B. 20. Give the general formula for the value of such a policy. 21. Required the premium, payable so long as either may be alive, for £1300, to be paid at the last death of A aged 50 and B aged 30. 22. What is the value of the policy after 10 years? 23. Exhibit the general formula for the value of such a policy. 24. Required the annual premium for an annuity of £200 , to be paid to B aged 30 after the death of A aged 50. 25. Compute the value of the policy after 10 years. 26. Exhibit the formula for the value of such a policy. A testator has directed that an annual payment of £600 out of the rents of his estate shall be paid so long as ^ or £ may be alive, to be divided between them in proportion to their ages at each division. Required the value of each legacy, the ages of the legatees being 40 and 20 at the tes- tator's death. III. EXERCISES. 153 THKEE LIYES. Article 148. 1. Required the value of £1000 payable 25 years hence, if A aged 40, B aged 20, and C aged 10 years be then all alive. 2. Required the value of £1000 payable 25 years hence, if A aged 40 be then dead, and B aged 20, C aged 10, be then both aUve. 3. Required the value of £1000 payable 25 years hence, if A and C be both alive, and B dead. 4. Required the value of £1000 payable 25 years hence, if A and B be both alive, and C be dead. 5. Required the value of £1000 payable 25 years hence, if A and B be both dead, and if C be then alive. 6. Required the present value of £1000 payable 25 years hence, if A and C be both dead, and B alive. 7. Required the value of £1000 payable 25 years hence, if B and C be both dead, and A alive. 8. Required the value of £1000 payable 25 years hence, if all the three A, B, C be then dead. 9. Required the value of £1000 payable 25 years hence, if, of the three A, B, C, one be dead, and the others alive. 10. Required the value of £1000 payable 25 years hence, if, of the three A, B, Q, two be dead, and one alive. 11. Required the value of £1000 payable 25 years hence, if one of the three A, B, C be then dead. 12. Required the value of £1000 payable 25 years hence, if one of the three A, B, Cbe then aUve. 13. Required the value of £1000 payable 25 years hence, if two of the three A, B, Cbe then dead. 14. Required the present value of £1000 payable 25 years hence, if two of the three A, B, C be then alive. 15. Write out the formulae for the above values. M 154 EXEECISES. III. Article 150. 1. Make a table of logp{a, b, c). 2. Form the column p{a, b, c) . 3. Take the sums of these, viz,, 2p(a, b, c) , 4. Seek out log^p{a, b, c). 5. Thence form the tables log annuity (immediate) ; log annuity deferred one year. 6. Extract the values of annuity (a, b, c) and of ann{a, b, c) deferred one year, and verify the work by comparison. 7. Construct a table of ann{a, b, c) deferred 10 years. Article 151. 1. Required the value of an annuity of £200, payable to B and C jointly after the death of ^, 2. Required the value of an annuity of £200, payable to A and Q jointly after the death of B. 3. Required the value of an annuity of £200, payable to A and B jointly after the death of C. 4. Give a detailed proof of the three equations in this article. Article 152. Required the value of an annuity of £200, payable jointly to the two survivors of the three A, B, C. Article 153. Required the value of an annuity of £200 , payable so long as two of the three A, B, Cmay be alive. Article 154. 1. Required the value of an annuity of £200 , payable to C after the death of either J. or ^. 2. Required the value of an annuity of £200 , payable to B after the death of either A or O. 3. Required the value of an annuity of £200 , payable to A after the death of either B or O. Article 155. Annuities of £200 each are to be paid to the survivors after the first death among the three A, B, O. Required the present value. Article 156. 1. Required the value of an annuity of £200, payable to C after the deaths of both A and B, 2. Required the value of an annuity of £200, payable to B after both J. and are dead. in. EXEECISES. 155 3. Required the value of an annuity of £200 , payable to A after both B and C are dead. 4. Give a detailed demonstration of the equations in this article. Article 157. Required the value of an annuity of £200 , payable to the single survivor of the three A, B, C after the others are dead. Article 158. Required the value of an annuity of £200, payable so long as one of the three A, B, Cmay be alive. Article 159. 1. An annuity of £200 is to be paid after A's death, so long as either B or C may be alive. Required its value. 2. An annuity of £200 is to be paid after the death of -S, so long as either ^ or C may be alive. Required its value. 3. An annuity of £200 is to be paid after the death of C, so long as either A or B may be aUve. Required its value. 4. A testator has directed that the sum of £600 annually from the rents of his estate shall be paid to A, B, and C, so long as any of them may be alive, to be equally divided. Required the value of each legacy, the ages of the legatees being 40, 20, and 10 years at the time of the testator's death. 5. A testator has directed that a sum of £F annually shall be equally divided among A, B, and C, so long as they are all alive, a sum of £Q annually so long as two of them may be alive , and a third sum of £R to the single survivor. Give the formula for the value of each legacy. 6. Show that when the numbers P, Q, R are as 3, 2, 1, the value of each legacy is that of a single annuity of £B . 7. A testator has directed that £600 annually shall be divided amongst A , B, and C in proportion to their ages at each division. Required the values of the several legacies. 8. Required the annual premium, payable during the joint lives of A, B, C, for the purchase of an annuity of £200, payable to B and C jointly after A's death. 9. Investigate the formula for the value of a policy of the above nature, 10. Required the annual premium, during the joint lives of A, B, C, for the purchase of an annuity of £200 to after the deaths of both A and B. 11. Investigate the formula for the value of the policy as in No. 10. 156 EXEKCISE8. III. 12. Eequired the annual premium during the joint lives oi A, B, Cfor the purchase of an annuity of £200 to B after the deaths of A and C. 13. Required the annual premium for the purchase of an annuity of £200, payable after the death of J., so long as either B or Cmay be alive. 14. Required the annual premium for the purchase of annuities of £400 during the joint survivorship of B and C, of £300 during B's survivorship of A and C, and of £200 during C's survivorship of A and B. 15. Exhibit the general formula for the value of a policy as in No. 14. Article 161. 1. Compute the value, as at Jan. 1, 1865, of £1000, pay- able in the year 1895 to B and C jointly on the death of A, using the averages for the middle of the year. 2. Compute, for the same dates, the value of £1000, payable to A and C jointly at the death oi B. 3. Compute also the value of £1000, payable to A and B jointly at the death of C. 4. Compute the value, as at Jan. 1, 1865, of £1000, payable in the year 1895, if the first death among the three A, B, C happen in that year, 5. Contrast the result in No. 4 with the sum of the three values in 1 , 2, and 3. Article 162. 1. Compute the value of £1000, payable as in No. 1 of article 161, strictly. 2. Compute strictly the value of £1000, payable as in No. 2 of the pre- ceding article. 3. Compute strictly the value of £1000, payable as in No. 3 of the pre- ceding article. 4. Take the sum of the three values and compare it with No. 4 of article 161. Article 163. 1. Compute a table of the co-efficients of joint succession for difference of ages 10 years. 2. Compute a table of co-efficients of joint succession for difference of age 20 years. 3. Compute a table of co-efficients of joint succession for difference of age 30 years. III. EXEECISES. 157 4. Take out the logarithms of the above co-efficients. 5. Make the columns logmor^, logmor^, and logmor~~. 6. Extract the values of mor ™, mor ^, and of mor ^ . Article 164. 1. Take the sums 'S.mor^, '2mor^, and "Zmor^ . 2. Form columns of log^s mor -™ , log's mor ~^, and log 2 mor •— - . 3. Thence deduce logass--—, log ass ^, log ass ^. 4. Make tables of the values of the three assurances. 5. Required the present value of £1300, to be paid to B and G jointly on the death of A . 6. What is the annual premium for an assurance of £1300, to be paid to B and C jointly on the death of A ? 7. Required the present value of £1300, to be paid to A and C jointly on the death of B . 8. Required the annual premium for the same. 9. Required the present value of £1300, to be paid to A and jB jointly on the death of 0. 10. What is the annual premium for the same ? 11. Make a table l{a, b, c) of the number of triplets existing at each successive year. 12. Make a table d{a, b, c) of the number of triplets which disappear from year to year. 13. Take out the logarithms of dia, b, c). 14. Thence form the table log mor {a, b, c). 15. Afterwards extract the values mor (a, 6, c), take their sums 2 mor (a, b, c); and the logarithms of these sums. 16. Thence compute log ass (a, b, c), and ass {a, b, c). 17. Contrast the values of ass (a, b, c) thus obtained with the sum of the three previously found assurances. Article 176. 1. Required the present value of £1300, to be paid to C at the death of B, if A have been previously dead. 2. What premium, payable annually during the joint lives of A, B, C, will purchase an assurance of £1300, to be paid to O at B's death, B having survived A ? 158 EXERCISES. HI. 3. Give the formula for the value of a policy arranged as in No. 2. 4. What premium , payable annually during the joint lives of B and C, will purchase an assurance of £1300 to C at B's death, B having sur- vived A ? 5. Give the formulae for the values of a policy arranged as in No. 3, after n years, on the two suppositions that A is then alive, and that A is then dead. 6. Required the present value of £1300, to be paid to Cat the death of u4, if ^ be previously dead. 7. What premium must be paid annually during the joint lives oi A, B, C for the assurance in No. 6 ? 8. What premium must be paid annually during the joint lives of A and C for the assurance as in No. 6 ? 9. What objection is there to making the premium payable during the joint lives of B and G? Article 177. 1. Required the present value of £1300, payable to C as soon as A and^^ are both dead. 2. What premium must be paid during the joint lives of A, B, Cfor the above assurance ? 3. What premium must be paid during the joint lives of A and C for the assurance as in No. 1 ? 4. What premium must be paid during the joint lives of B and O for the same assurance ? 5. And what premium must be paid annually so long as 0, and either or both of the two B and A may be alive ? 6. Give the values of the policies as in the above arrangements, after the lapse of n years, and on the three suppositions, A and B both alive, A dead, and B dead (12 varieties) . Article 179. 1. Required the value of £1300, payable at the second death of the three A, B, 0. 2. Required the annual premium, payable during the joint lives of A, B, C, for £1300 at the second death, 3. What is the value of a poUcy, as in No. 2, after n years, on the four suppositions J , B, C, all alive; A and B aUve, C dead ; ^ and C alive, B dead ; and B and G alive, A dead ? III. EXEECISE8. 159 4. Required the premium, payable annually during the joint lives of A and B, for the same assurance. Article 184. 1. Show that, when the ages a and b are ahke, the sum 2 da . l{b + i) becomes just the half of la.lb . 2. Construct the tables da . l{h + ^),db.l(a + \),'2da.l(f> + \),'2db.l{a + \), for the difference 20 years. Article 186. 1. Compute the value, as at Jan. 1, 1865, of £1300, pay- able in the year 1895 at the death of C, provided B have died previously, after having survived A . 2. Compute the value, as at Jan. 1, 1865, of £1300, payable in the year 1895 at the death of C, provided A have died previously, after hav- ing survived B , 3. Compute the value, as at Jan. 1, 1865, of £1300, payable in the year 1895 at the death of G, provided A and B be both previously dead. Article 188. 1. Form a table of the products mc . 2 ?(a + 1) . db ; and by summation one of 2[mc . 2 Z(a + ■^) . db} 2. Form a table of liv (a + f), and thence the column -J l{a + ^) .db. 3. From that deduce J l{a + ^) .db .mc. 4. Thence take the sums '2^l{a + ^) .db .mc. 5. By adding 1 and 4 together construct the table of 2{2i(a + J) . db - -^ Z(a + ^) . db}mc ; and take out the logarithms. Article 189. By exchanging the positions of the letters a and b in the above, we obtain the formula for ass JL . The computation may be varied as under. 1. Make a table of ^l{b + ^).da; subtract these from the values of :sda.l{b+^), and so obtain 2da.l{b + i)-^da.l{b + ^) with the loga- rithms. 2. Compute the product {'2da.l{b + l)-^da . l{b + ^)} mc. 3. Sum these and take their logarithms. 4. Required the value of £1300, payable at the death of C, if C have survived B, and if B have survived A . 5. Required the annual premium, payable during the joint Uves of A, B, C, for the assurance as in No. 4. 160 EXEECISES. III. 6. Give the formulae for the values of a policy arranged as in No. 5, after the lapse of n years, on the various suppositions A, B, C , all alive ; B and C alive, A dead ; O alive, A and B dead. 7. Required the value of £1300, payable at the death of C, if C have survived A, and if A have survived B . 8. Required the annual premium, payable during the joint lives of A and C, for the assurance as in No. 7. 9. Give the formulae for the values of the policies arranged as in No. 8, after n years, on the suppositions A, B, C all alive; A and C alive, B dead ; C alive, A and B dead. 10. Required the annual premium, payable during the life of C, for the assurance as in No. 7. 11. Give the formulae for the values of the policy as in No. 10, on the various suppositions. Article 190. By adding together the two columns of assj^ and ass j;^ a & form the table of assurance --^ ; that is of assurance at C's death, if it be the last of the three. Article 191, 1. Verify the preceding calculation by the formula of this article. 2. The verification may be made by examining whether the sum of 'S,l{a + \) .db-\ l{a + ^) db and its converse agree with the value of the co-efficient of joint succession. 3. Required the present value of £1300, payable at the death of C, if it be the last of the three. 4. Required the annual premium, during the joint lives oi A, B, C, for the assurance as in No. 3. 5. Give the formulae for the values of the policy as in No. 4, after n years, on the suppositions A, B, Call alive; B, G alive, A dead; A, alive, B dead ; and C only alive. 6. Required the annual premium for the assurance as in No. 3, to be paid so long as A and C may be both alive. 7. Give the formulae for the values of a policy as in No. 6. 8. Required the annual premium, payable so long as C and both or either of the others may be alive, for £1300, to be paid at C's death if last. 9. Give the formulae for the values of the above pohcy. III. EXEECISES. 161 10. Eequired the annual premium, during C's life, for the assurance as in No. 3. 11. Give the formulae for the values of the policy as in No. 10. Article 192. 1. Required the present value of £1300, payable at the last death of the three A, B, C. 2. Required the premium payable during the joint lives A, B, C. 3. Give the values of the policy No. 2 on the various suppositions, A, B, C alive; B, alive, A dead; A, C alive, B dead; A, B alive, C dead ; C only alive ; B only alive ; A only alive. 4. Required the premium, payable during the joint lives of A and B , for the assurance of £1300 at the last death of the three A, B, C. 5. Give the formulae for the values of the policy as in No. 4. 6. Required the premium, payable annually so long as two of the three may be alive, for an assurance of £1300 at the last death of J., B, G. 7. Give the formulae for the values of the policy as in No. 6. Article 193. 1. Compute the values of pa . 2Z(a + ^) . db, and of their sums ; take also the logarithms of the sums. 2. Required the present value of an annuity of £200, payable by C after the death oi B, \i B have succeeded to A . 3. Required the present value of an annuity of £200, payable by C after the death oi A, provided A have survived B. 4. C being the heir of B, and B the heir of A, he wishes to raise £2000. What annual sum must he assign out of the rents of the estate in order to obtain the advance ? N 162 EXERCISES. ^^^ FOUE LIYES. Article 199. 1. Required the present value of £1000, payable 20 years hence, provided the nominees A aged 45, B aged 25, C aged 15, and D aged 10 , be then all alive. 2. Required the present value of £1000, payable 20 years hence, pro- vided A be then dead, and B, G, D alive. 3. Required the present value of £1000, payable 20 years hence, pro- vided B be dead, and A, C, D alive. 4. Required the present value of £1000, payable 20 years hence, pro- vided Q be dead, and A, B, D alive. 5. Required the present value of £1000, payable 20 years hence, if D be dead, and A, B, C alive. 6. Required the value of £1000, payable 20 years hence, if A and B be dead, C and D alive. 7. Required the value of £1000, payable 20 years hence, if A and Cbe dead, B and D alive. 8. Required the value of £1000, payable 20 years hence, if A and D be dead, B and C alive. 9. Required the value of £1000 , payable 20 years hence, if B and C be dead, A and D alive. 10. Required the value of £1000, payable 20 years hence, if B and D ■ be dead, A and G alive. 11. Required the value of £1000, payable 20 years hence, if Cand D be dead, A and B ahve. 12. Required the value of £1000, payable 20 years hence, if D alone be alive. 13. Required the value of £1000, payable 20 years hence, if C alone be aUve. 14. Required the value of £1000, payable 20 years hence, if B alone be alive. IV. EXEECISES. 163 15. Required the value of £1000, payable 20 years hence, if A alone be alive. 16. Required the value of £1000, payable 20 years hence, if all the four be dead. 17. Show that the sum of the above sixteen values is the present value of £1000 due 20 years hence. 18. What is the value of £1000,- payable 20 years hence, if one and only one of the four be dead ? 19. What is the value of £1000, payable 20 years hence, if two be ahve and two dead ? 20. What is the value of £1000, payable 20 years hence, if only one of the four be alive ? 21. Show that the values in Nos. 1, 2, 3, 4, 6, 7, 9, and 12 make up the value of an endowment to D, payable 20 years hence. 22. Which of the above values make up an endowment to C simply ? 23. Which of the above values make up an endowment to C and D jointly ? 24. Which of the above values make up an endowment to C and D jointly or severally ? 25. Construct a table of logfia, b, c, d). 26. Thence compute p(a, b, c, d) . 27. Next take the sums 2/i(a, b, c, d). 28. And then the logarithms log 2p(a, b, c, d). Article 200. 1. What is the value of an annuity of £200, beginning to-day and payable so long as all the four A, B, C, D may be alive ? 2. Construct the tables logann(a, b, c, d) and arm {a, b, c, d) . 3. What is the value of an annuity of £200, beginning one year hence, and payable durmg the joint lives oi A, B, C, D? 4. Construct the tables logann{a,b, c,d) deferred one year, and ann{a, b, c, d) deferred one year. 5. Verify the work by comparing the values of the annuities. 6. What is the value of an annuity of £200, beginning 20 years hence, and payable so long as the four A, B, O, D may be all aUve ? 7. Required the value of twenty payments of £200 each, beginning to-day, and to be made annually so long a,s A, B, C, D may be all alive. 164 EXEECISES. IV. 8. Required the value of annual payments of £200 each, to begin 15 years hence, and end 25 years hence, ii A, B, C, D be all alive. Article 201. 1. Required the value of an annuity of £200, to be paid after the death of A to B, C, and D jointly. 2. Required the value of an annuity of £200, to be paid to ^, O, and D jointly after the death of B . 3. Required the value of an annuity of £200, to be paid io A, B, and J) jointly after the death of C 4. Required the value of an annuity of £200, to be paid io A, B, and C jointly after the death of D , 5. Required the value of an annuity of £200, to be paid after the death of one of the four, so long as there may be three survivors. 6. Required the premium, payable annually during the joint lives of A, B, C, D, for the purchase of an annuity of £200 to the three survi- vors jointly after A's death. 7. Give the formula for the value of a policy as in No. 6. 8. Required the premium, payable annually during the joint lives of A, B, G, D, for the purchase of an annuity of £200 to the three survi- vors jointly after the death of one. 9. Give the formula for the value of a policy as in No. 8 . Article 203. 1 . Construct a table of the logarithms of Ic.pd.^db. l(a + ^). 2. Thence take out the values oilc.pd .l.dh , l{a + 1) 3. Fill up the column l.ilc . pd . '2 db . l{a + ^)} . 4. Required the value of an annuity of £200, payable to O and D jointly after the death of B, provided B have survived A. Article 204. 1. Construct a table of the logarithms of Zc.jod. 2 da. Z(6 + ^). 2. Thence take out the values oilc . pd . l. da . l{b + 1) . 3. Take the sums 2{Zc .pd .'S,da.l{b + 1)} . 4. Required the value of an annuity of £200, payable to C and D jointly after the death of A, provided A have survived B. 5. Give the formula for the value of an annuity, payable to D and B jointly after the death of C, C having survived A . 6. Give the formula for the value of an annuity, payable to D and B jointly after the death oi A, A having survived C. IV. EXERCISES. 165 7. Give the formula for the vahie of an annuity, payable to D and A jointly after the death of C, having survived B . 8. Give the formula for the value of an annuity, payable to D and A jointly after the death oi B, B having survived C. 9. Arrange the formula for the value of an annuity, payable to C and 5 jointly after the death oi D, D having survived A, in such a way as to have the divisor in the form la.lb.lc .pd. 10. Give the formula for the value of an annuity, payable to C and B jointly after the death oi A, A having survived D . 11. Give the formula for the value of an annuity payable to C and A jointly after the death oi D, D having survived B . 12. Give the formula for the value of an annuity, payable to C and A jointly after the death oi B, B having survived D , 13. Give the formula for the value of an annuity, payable to B and A jointly after the death oi D, D having survived C . 14. Lastly, give the formula for the value of an annuity, payable to B and A jointly after the death of 0, having survived D . 15. What is the formula for the value of an annuity, payable so long as one of the two D and C may be alive after the death oi B, B having sur- B or C vived A; which annuity may be indicated by the symbol ann ,b^ 1 Article 205. 1. Find the value of an annuity of £200, payable to C and D jointly after A and B are both dead ; and compare the result with the sum of those in the preceding articles. 2. Give the formula for an annuity, payable to D and B jointly after the deaths of A and C. 3. Give the formula for an annuity, payable to D and A jointly after the deaths of B and C. 4. Give the formula for an annuity, payable to C and B jointly after the deaths of A and D . 5. Give the formula for an annuity, payable to Cand A jointly after the deaths of B and D . 6. Give the formula for an annuity, payable to B and A jointly after the deaths of C and D . 7. Required the value of an annuity, payable to C and D jointly after the death of either of the two A and B . 166 EXERCISES. IV. 8. "Write out the expressions for the five other survivorships of this class. 9. Required the value of an annuity, payable to two survivors jointly after the deaths of the others. 10. Required the value of an annuity, payable to two survivors after the second death, the first death being that of A . 11. Required the value of an annuity, payable to two survivors after the second death, one of those dead being A . 12. Required the value of an annuity, payable to two survivors, one of whom must be D, after the second death. 13. Required the value of an annuity, payable to two survivors, one of whom must be D, after the second death, the first death having been that of J. 14. Required the value of an annuity, payable to two survivors, one of whom must be B, after the second death, one of the two deceased being A . 15. Required the value of an annuity, payable to two survivors after the death oi A, if one of the others have died before A . 16. Required the value of an annuity, payable to two survivors, of whom Z) must be one, after the death of A, provided either B or C have prede- ceased A . 17. Give a few of the varieties of the above alternative survivorship annuities. Article 206. 1. Write out the formula for the value of an annuity pay- able to i>, if the deaths of the others have occurred in the order A, B, C. 2. Show that when the ages of A, B, and C are all alike, the fraction by which annd is multiplied in the above formula has the value ^, while the co-efficient of ann {c, d) is ^ . Exhibit also the changes which the other terms of the expression undergo on this supposition of equality in the ages. 3. Give the value of the above annuity when the four ages are all alike. 4. Write out the formulae for the remaining five cases in which D is the solitary survivor. 5. Give the formulas for the annuity to the solitary survivor C, in such a way that the discount may be to the birth of D . Article 208. 1. Write out the formula for an annuity, payable to D after the death of 0, provided A and B have both died before C. IV. EXEECISES. 167 2. Give the formula for an annuity, payable to D after the death of B, provided both A and C have died before B. Article 209, 1. Write out the formula for an annuity to D after the deaths oi A, B, 0, provided A's death have been the first. 2. Required the value of an annuity, payable to D after the deaths of the other three, provided B's death have been the second. 3. How many varieties are there of the above survivorship annuity ? Article 210. Write out in full the demonstration of this theorem in the case of four nominees. Article 211. 1. Give, at length, the investigation in the manner here explained. 2. Required the value of an annuity, payable during Cs solitary survi- vorship of the three A, B, D. Article 212. 1. Investigate the value of an annuity, payable to E, D, C, B jointly after the death oi A. 2. Investigate the value of an annuity, payable to E, D,C jointly after the deaths of A and B . 3. Required the value of an annuity, payable to E and D jointly after A, B, C are all dead. 4. Required the value of an annuity, payable to E after A, B, C, D are all dead. 5. Required the value of an annuity, payable while four and only four of the five A, B, O, D, E may be living. 6. Required the value of an annuity, payable while three and only three oi A, B, O, D, E may be living. 7. Required the value of an annuity, payable while two and only two of A, B, C, D, E may be living. 8. Required the value of an annuity, payable while one and only one of A, B, 0, D, E may be living. 9. Required the value of an annuity, payable while four of the five may be living. 10. Required the value of an annuity, payable while three of the five may be living. 168 EXEECI8ES. IV. 11. Eequired the value of an annuity, payable while two of the five may be living. 12. Required the value of an annuity, payable so long as one of the five may be living. Article 216. 1. The sum of £1200 from the rent of an estate is secured jointly and severally to the four nominees A, B, G, D, to be divided equally among such of them as may be alive. Required the value of each person's expectation. 2. Required the annual premium, payable during the joint Hves of A, B, C, D,to secure an annuity payable to D after the deaths of all the others. 3. Give the formulae for the values of the policy on the various possible suppositions. 4. Required the annual premium, payable so long as D and any of the others may be alive, to secure an annuity to D after A, B, C are all dead. 5. Give the formulae for the values of the policy on the various possible suppositions. 6. The sum of £1200 from the rent of an estate is secured, jointly and severally, to the four nominees A, B, C, D,tohe divided amongst them in proportion to their ages at each division. Required the value of each person's expectation. Article 217. 1. The ages oi A, B, C, D being as before, required, as at Jan. 1, 1865, the value of £1000, payable in 1885 at the death of A, provided B , C, and D be all alive at the time of the death. 2. Give the formula on the supposition that B, C, and D are all of one age. Article 218. 1. Construct a table of the co-efficient of joint succession to suit the ages b, c, d, 2. Form the column log coef{b, c, d) , Article 219. 1. Thence make the table log{coef(b, c, d)r -<*-*}. 2. Afterwards compute log{da . coef{b, c, d) . r-^-^. 3. Thence the values of wior^j^, and 2mor^. IV. EXERCISES. 169 4. Required the present value of £1300, payable to B, C, D jointly at the death of A . 5. Required the premium, payable annually during the joint lives of A, B, C, D, to purchase an assurance of £1300 at the death of A if first. 6. Required the value of £1300, payable to D and C jointly at the first death of A and B . 7. Required the value of £1300, payable to D at the first death among the three A, B, C. Article 220. 1. Requiredthe value of £1300, payable at the first death among the four. 2. Required the annual premium for the above assurance. 3. Show that the value of a policy as in No. 2 may be put in a form analogous to that of single-Ufe policies. Article 222. 1. Required the value of £1300, payable to C and D jointly on the death of B, provided A have died before B . 2. In what ways may the annual premium for the above assurance be arranged ? 3. Give the formulae for the values of the policies according to the various arrangements, and in all possible combinations of survivors. Article 223. 1. Required the present value of £1300, payable to C and D jointly as soon as A and B are both dead. 2. In what ways may the annual premium for the above assurance be arranged ? 3. How are the values of current policies, according to the different arrangements, to be computed ? Article 224. 1. Design the scheme for computing the value of an assur- ance, payable to D at the death of C, provided that have survived B, and that B have survived A . 2. What change would need to be made in the formula for assurance to at D's death, provided that D have survived B, and that B have survived A, in order to bring all the discounts to the birth of D ? Article 225. 1. Give a detailed proof of the formula in this article. h 2. Exhibit the formula for ass ^a_ . 170 EXEECISES. IV. Article 227. Give a detailed investigation for the value of ass^^^ . Article 228. Write out at length the formula for an assurance at the third death among the four. Article 229. 1. Design the calculation of the value of an assurance payable at the death of Z), if Z) have survived G, if O have survived B, and if JB have survived A . 2. Arrange the formula for an assurance at £'s death, if the deaths take place in the order A, D, C, B, in such a way as to have the discount to the birth of D . _™ Article 230. 1. Give the formula for the assurance ass^ , arrang- ing it so as to have the discounts to the birth of D . 2. Design the scheme for computing the above assurance. Article 231. 1.' Give the detailed demonstration of this formula. 2. Give the formula for ass j-f-g • Article 232. 1. Write out the different orders in which the four deaths may happen, and the symbols for the corresponding assurances. 2. Reckoning from the very beginning of the calculation, how many columns are needed for the computation of an assurance at the last death of the four ? 3. In how many different ways may the annual premium be made pay- able? 4. In computing the value of a current policy for assurance at the last death of the four, what are the different cases that may arise ? 5. Give the formulae for the values of policies typical of the various classes of cases. V. EXEKCI8E8. 171 FIYE LIYES. 1. Investigate the co-efficient of joint succession of four nominees to a fifth, and give it in the most compact form. 2. Give the formula for an assurance at the first death among five nominees. 3. Required the value of an assurance, payable to E, D, C jointly at the death oi B, B having survived A . 4. Required the value of an assurance, payable to E, D, C jointly as soon as B and A are both dead. 5. What is the formula for an assurance payable at the second death among five nominees ? 6. Required the value of an assurance, payable to E and D jointly at the death of C, if have survived B, and if B have survived A . 7. Required the value of an assurance, payable to E and D jointly so soon as O, B, A are all dead. 8. What is the value of an assurance, payable to E at the death of D, if D have survived C, if C have survived B, and if B have survived A ? 9. What is the value of an assurance, payable to E at the death of D, if D have survived C, B, and A ? 10. What is the value of an assurance, payable to E so soon as all the four A, B, O, D are dead? 11. What is the value of an assurance, payable at the fourth death among the five ? 12. Required the value of an assurance, payable at the death of E, if E have survived D, if D have survived 0, if Chave survived B, and if B have survived A , 172 EXEKCISES. V. 13. Required the value of an assurance, payable at the death of E, if E have survived all the four A, B, C, and I) . 14. What is the value of an assurance at the last death of the five ? 15. How many classes of cases are there, and how many cases in each class of assurances among five nominees ? 16. How are these cases modified when all the ages happen to be alike ? 17. Investigate the co-efficient of joint succession of five nominees to a sixth. APPENDIX. Separcieuz. Tontine N'ominees. 1689 to 1696. Age. Living. Die. Age. Living. Die. 50 581 10 1 51 571 11 2 52 560 11 3 1000 30 63 549 11 4 970 22 54 538 12 5 948 18 55 526 12 6 930 15 56 514 12 7 915 13 57 502 13 8 902 12 58 489 13 9 890 10 59 476 13 10 880 8 60 463 13 11 872 6 61 450 ]3 12 866 6 62 437 14 13 860 6 63 423 14 14 854 6 64 409 14 15 848 6 65 395 15 16 842 7 66 380 16 17 835 7 67 364 17 18 828 7 68 347 18 19 821 7 69 329 19 20 814 8 70 310 19 21 806 8 71 291 20 22 798 8 72 271 20 23 790 8 73 251 20 24 782 8 74 231 20 25 774 8 75 211 19 26 766 8 76 192 19 27 758 8 77 173 19 28 750 8 78 154 18 29 742 8 79 136 18 30 734 8 80 118 17 31 726 8 81 101 16 32 718 8 82 85 14 33 710 8 83 71 12. 34 702 8 84 59 11 35 694 8 85 48 10 36 686 8 86 38 9 37 678 7 87 29 7 38 671 7 88 22 6 39 664 7 89 16 5 40 657 7 90 11 4 41 650 7 91 7 3 42 643 7 92 4 2 43 636 7 93 2 1 44 629 7 H 1 1 45 622 7 95 46 615 8 47 607 8 48 599 9 49 590 9 Price's Northampton Table. 1735 to 1780. Age. Living. Die. Age. Living. Die. 11650 3 000 50 2 857 81 1 8 650 1367 51 2 776 82 2 7 283 502 52 2 694 82 3 6 781 335 53 2 612 82 4 6 446 197 54 2 530 82 5 6 249 184 55 2 448 82 6 6 065 140 56 2 366 82 7 5 925 110 57 2 284 82 8 5 815 80 58 2 202 82 9 5 735 60 59 2120 82 10 5 675 52 60 2 038 82 11 5 623 50 61 1956 82 12 5 573 50 62 1874 81 13 5 523 50 63 1793 81 14 5 473 50 64 1712 80 15 5 423 50 65 1632 80 16 5 373 53 66 1552 80 17 5 320 58 67 ] 472 80 18 5 262 63 68 1392 80 19 5199 67 69 1312 80 20 5132 72 70 1232 80 21 5 060 75 71 1152 80 22 4 985 75 72 1072 80 23 4 910 75 73 992 80 24 4 835 75 74 912 80 25 4 760 75 75 832 80 26 4 685 75 76 752 77 27 4 610 75 77 675 73 28 4 535 75 78 602 68 29 4 460 75 79 534 65 30 4 385 75 80 469 63 31 4 310 75 81 406 60 32 4 235 75 82 346 57 33 4160 75 83 289 55 34 4 085 75 84 234 48 35 4 010 75 85 186 41 36 3 935 75 86 145 34 37 3 860 75 87 111 28 38 3 785 75 88 83 21 39 3 710 75 89 62 16 40 3 635 76 90 46 12 41 3 559 77 91 34 10 42 3 482 7S 92 24 8 43 3 404 78 93 16 7 44 3 326 78 94 9 5 45 3 248 78 95 4 3 46 3170 78 96 1 1 47 3 092 78 97 48 3 014 78 49 2 936 79 =:^==^^= Price's Swedish Table. 1755 to 1776. Males. Females. Males. Females. AoE. AGE. Living. Die. Living. Die. Living. Die. Living. Die. 10 000 2 300 10 000 2 090 50 3 666 95 4 027 75 1 7 700 500 7 910 518 51 3 571 95 3 952 80 2 7 200 337 7 392 350 52 3 476 95 3 872 85 3 6 863 240 7 042 250 53 3 381 95 3 787 85 4 6 623 150 6 792 135 54 3286 95 3 702 85 5 6 473 125 6 657 120 55 3191 95 3 617 85 6 6 348 105 6 537 105 56 3 096 95 3 532 85 7 6 243 90 6 432 85 57 3 001 100 3 447 90 8 6153 75 6 347 70 58 2 901 100 3 357 90 9 6 078 65 6 277 60 59 2 801 100 3 267 100 10 6 013 55 6 217 52 60 2 701 105 3 167 110 11 5 958 45 6165 46 61 2 596 110 3 057 118 12 5 913 45 6119 40 62 2 486 115 2 939 120 13 5 868 40 6 079 35 63 2 371 115 2 819 120 14 5 828 40 6 044 35 64 2 256 115 2 699 120 15 5 788 39 6 009 35 65 2141 115 2 579 120 16 5 749 39 5 974 40 66 2 026 115 2 459 120 17 5 710 39 5 934 40 67 1911 120 2 339 120 18 5 671 44 5 894 42 68 1791 125 2 219 120 19 5 627 44 5 852 43 69 1666 125 2 099 120 20 5 583 50 5 809 43 70 1541 125 1979 130 21 5 533 50 5 766 43 71 1416 125 1849 140 22 5 483 50 5 723 43 72 1291 120 1709 150 23 5 433 55 5 680 44 73 1171 120 1559 160 24 5 378 55 5 636 45 74 1051 110 1399 150 23 5 323 55 5 591 45 75 941 105 1249 140 26 5 268 55 5 546 50 76 836- 100 1109 130 27 5 213 55 5 496 52 77 736 90 979 120 28 5158 55 5 444 55 78 646 85 859 110 29 5103 56 5 389 55 79 561 80 749 100 30 5 047 59 5 334 60 80 481 75 649 95 31 4 988 60 5 274 60 81 406 70 554 90 32 4 928 60 5 214 65 82 336 65 464 85 33 4 868 60 5 149 65 83 271 60 379 80 34 4 808 60 5 084 65 84 211 50 299 75 35 4 748 60 5 019 60 85 161 40 224 55 36 4 688 60 4 959 56 86 121 30 169 40 37 4 628 60 4 903 56 87 91 22 129 30 38 4 568 60 4 847 56 88 69 17 99 23 39 4 508 60 4 791 58 89 52 14 76 18 40 4 448 65 4 733 65 90 38 12 58 15 41 4 383 72 4 668 75 91 26 9 43 12 42 4 311 80 4 593 76 92 17 7 31 10 43 4 231 80 4 517 76 93 10 6 21 8 44 4151 80 4 441 75 94 4 3 13 6 45 4 071 80 4 366 72 95 1 1 7 4 46 3 991 80 4 294 67 96 3 2 47 3 911 80 4 227 65 97 1 1 48 3 831 80 4162 65 98 49 3 751 85 4 097 70 Duvillard. Loi de Mortalite en France. 1806. Age. Living. Die. Age. Living. Die. Age. Living. Die. 1000 0000 232 4753 40 369 4042 6 9857 80 34 7048 5 8186 1 767 5247 95 6906 41 362 4185 7 0186 81 28 8862 5 2062 2 671 8341 47 1657 42 355 3999 7 0584 82 23 6800 4 5736 3 624 6684 25 9550 43 348 3415 71064 83 19 1064 3 9311 4 598 7134 15 5625 44 341 2351 71629 84 15 1753 3 2897 5 583 1509 10 1259 45 334 0722 7 2289 85 11 8856 2 6613 6 573 0250 71871 46 326 8433 7 3046 86 9 2243 2 0590 7 565 8379 5 5933 47 319 5387 7 3904 87 71653 14963 8 660 2446 4 7682 48 312 1483 7 4864 88 5 6700 9843 9 555 4864 4 3648 49 304 6619 7 6924 89 4 6857 8557 . 10 551 1216 4 2334 50 297 0695 7 7084 90 3 8300 7365 11 546 8882 4 2581 51 289 3611 7 8337 91 3 0936 6272 12 542 6301 4 3751 62 281 5274 7 9677 92 2 4663 5281 13 538 2550 4 5445 63 273 5597 81093 93 19382 4388 14 533 7105 4 7412 54 265 4504 8 2575 94 14994 3594 15 528 9693 4 9490 55 257 1929 8 4108 95 11400 2898 16 524 0203 51576 56 248 7821 8 5677 96 8602 2295 17 518 8627 5 3605 57 240 2144 8 7260 97 6207 1783 18 513 5022 5 5532 58 231 4884 8 8836 98 4424 1353 19 507 9490 5 7331 59 222 6048 9 0380 99 3071 1003 20 502 2159 5 8989 60 213 5668 91866 100 2068 722 21 496 .3170 6 0496 61 204 3802 9 3262 101 1346 503 22 490 2674 61849 62 195 0540 9 4535 102 843 338 23 484 0825 6 3055 63 185 6005 9 5653 103 505 217 24 477 7770 6 4110 64 176 0352 9 6577 104 288 133 25 471 3660 6 5026 65 166 3775 9 7268 105 155 77 26 464 8634 6 5812 66 156 6607 9 7688 106 78 42 27 458 2822 6 6476 67 146 8819 9 7795 107 36 21 28 451 6346 6 7030 68 137 1024 9 7551 108 15 9 29 444 9316 6 7484 69 127 3473 9 6917 109 6 4 30 438 1832 6 7854 70 117 6556 9 5855 110 2 2 31 431 3978 6 8150 71 108 0701 9 4334 111 32 424 5828 6 8388 72 98 6367 9 2328 33 417 7440 6 8581 73 89 4039 8 9811 34 410 8859 6 8743 74 80 4228 8 6776 35 4040116 6 8890 75 71 7453 8 3211 36 397 1226 6 9035 76 63 4242 7 9129 37 390 2191 6 9190 77 55 6113 7 4547 38 383 3001 6 9370 78 48 0566 6 9496 39 376 3631 6 9589 79 41 1070 6 4022 MUne s CarliRle Table. 1779 to 1787. Age. Living. Die. Age. Living. Die. Age. 80 Living. Die. 10 000 1539 40 5 075 66 953 116 1 8 461 682 41 5 009 69 81 837 112 2 7 779 505 42 4 940 71 82 725 102 3 7 274 276 43 4 869 71 83 623 94 4 6 998 201 44 4 798 71 84 529 84 5 6 797 121 45 4 727 70 85 445 78 6 6 676 82 46 4 657 69 86 367 71 7 6 594 58 47 4 588 67 87 296 64 8 6 536 43 48 4 521 63 88 232 51 9 6 493 33 49 4 458 61 89 181 39 10 6 460 29 50 4 397 59 90 142 37 11 6 431 31 51 4 338 62 91 105 30 12 6 400 32 52 4 276 65 92 75 21 13 6 368 33 53 4 211 68 93 54 14 14 6 335 35 54 4143 70 94 40 10 15 6 300 39 55 4 073 73 95 30 7 16 6 261 42 56 4 000 76 96 23 5 17 6 219 43 57 3 924 82 97 18 4 18 6176 43 58 3 842 93 98 14 3 19 6133 43 59 3 749 106 99 11 2 20 6 090 43 60 3 643 122 100 9 2 21 6 047 42 61 3 521 126 101 7 2 22 6 005 42 62 3 395 127 102 5 2 23 5 963 42 63 3 268 125 103 3 2 24 5 921 42 64 3143 125 104 1 1 25 5 879 43 65 3 018 124 105 26 6 836 43 66 2 894 123 27 5 793 45 67 2 771 123 28 5 748 50 68 2 648 123 29 5 698 56 69 2 525 124 30 5 642 57 70 2 401 124 31 5 585 57 71 2 277 134 32 5 528 56 72 2143 146 33 5 472 55 73 1997 156 34 5 417 55 74 1841 166 35 5 362 55 75 1675 160 36 5 307 56 76 1515 156 37 5 251 57 77 1359 146 38 5194 58 78 1213 132 39 5136 61 79 1081 128 1 p= Milne's Swedish Table. 1776 to 1795. Age. Males. Females. AaE. Males. Females. Living. Die. Living. Die. Living. Die. Living. Die. 10 210 2169 9 790 1861 50 3 967 86 4 206 73 1 8 041 554 7 929 534 51 3 881 89 4133 75 2 7 487 317 7 395 297 52 3 792 90 4 058 76 3 7170 228 7 098 219 53 3 702 92 3 982 78 4 6 942 169 6 879 159 54 3 610 93 3 904 79 5 6 773 154 6 720 141 55 3 517 94 3 825 81 6 6 619 114 6 579 105 56 3 423 97 3 744 86 7 6 505 83 6 474 73 57 3 320 99 3 658 89 8 6 422 61 6 401 56 58 3 227 100 3 569 92 9 6 361 51 6 345 49 59 3127 101 3 477 95 10 6 310 47 6 296 45 60 3 026 108 3 382 105 11 6 263 44 6 251 41 61 2 918 116 3 277 113 12 6 219 42 6 210 39 62 2 802 124 3164 119 13 6177 40 6171 37 63 2 678 126 3 045 125 14 6137 39 6134 36 64 2 552 127 2 920 129 15 6 098 39 6 098 36 65 2 425 132 2 791 134 16 6 059 39 6 062 36. 66 2 293 138 2 657 138 17 6 020 40 6 026 37 67 2155 141 2 519 144 18 5 980 42 5 989 38 68 2 014 141 2 375 148 19 5 938 44 5 951 39 69 1873 139 2 227 151 20 5 894 48 5 912 40 70 1734 136 2 076 151 21 5 846 50 5 872 41 71 1598 133 1925 154 22 5 796 52 5 831 43 72 1465 131 1771 156 23 5 744 54 5 788 44 73 1334 127 1615 152 24 5 690 55 5 744 45 74 1207 124 1463 148 25 5 635 56 5 699 48 75 1083 119 1315 140 26 5 579 57 5 651 49 76 964 113 1175 133 27 5 522 58 5 602 50 77 851 106 1042 125 28 5 464 59 5 552 51 78 745 99 917 115 29 5 405 60 5 501 53 79 646 96 802 105 30 5 345 61 5 448 56 80 550 86 697 95 31 5 284 61 5 392 57 81 464 76 602 92 32 5 223 61 5 335 58 82 388 66 510 88 33 5162 61 5 277 57 83 322 58 422 79 34 5101 61 5 220 56 84 264 53 343 69 35 5 040 61 5164 56 85 211 45 274 59 36 4 979 61 5 108 56 86 166 37 215 44 37 4 918 61 5 052 56 87 129 29 171 34 38 4 857 61 4 996 57 88 100 21 137 28 39 4 796 64 4 939 61 89 79 19 109 24 40 4 732 71 4 878 67 90 60 14 85 19 41 4 661 73 4 811 68 91 46 12 66 16 42 4 588 73 4 743 68 92 34 9 50 13 43 4 515 73 4 675 67 93 25 8 37 10 44 4 442 74 4 608 67 94 17 6 27 8 45 4 368 78 4 541 67 95 11 4 19 7 46 4 290 80 4 474 66 96 7 3 12 6 47 4 210 80 4 408 66 97 4 2 6 3 48 4130 80 4 342 67 98 2 1 3 2 49 4 050 83 4 275 69 99 1 1 1 1 Milne's Table for llontpellier. 1772 to 1792. AOE. Males. Females. Aqe. Males. Females. Living. Die. Living. Die. Living. Die. Living. Die. 12 239 3 574 12145 2 831 50 3 017 75 3 521 64 1 8 665 1264 9 314 1177 51 2 942 76 3 45r 67 2 7 401 754 8137 825 52 2 866 77 3 390 70 3 6 647 549 7 312 633 53 2 789 78 3 320 73 4 6 098 450 6 679 458 54 2 711 80 3 247 75 5 5 648 261 6 221 260 55 2 631 83 3172 77 6 5 387 165 5 961 147 56 2 548 87 3 095 79 7 5 222 102 5 814 93 -57 2 461 89 3 016 81 8 5120 51 5 721 66 58 2 372 90 2 935 82 9 5 069 36 5 655 44 59 2 282 91 2 853 84 10 5 033 24 5 611 34 60 2191 93 2 769 86' 11 5 009 22 5 577 31 61 2 098 93 2 683 87 12 4 987 21 5 546 30 62 2 005 93 2 596 88 13 4 966 21 5 516 29 63 1912 93 2 508 89 14 4 945 22 5 487 30 64 1819 93 2 419 90 15 4 923 25 5 457 31 65 1 726 92 2 329 90 16 4 898 29 5 426 33 66 1634 91 2 239 90 17 4 869 38 5 393 37 67 1543 90 2 149 90 18 4 831 46 5 356 41 68 1453 89 2 059 90 19 4 785 49 5 315 44 69 1364 88 1969 90 20 4 736 51 5 271 47 70 1276 85 1879 91 21 4 685 52 5 224 49 71 1191 83 1788 91 22 4 633 52 5175 51 72 1108 81 1697 91 23 4 581 52 5124 53 73 1027 79 1606 91 24 4 529 52 5 071 54 74 948 77 1515 91 25 4 477 52 5 017 55 75 871 75 1424 91 26 4 425 53 4 962 56 76 796 72 1333 90 27 4 372 53 4 906 57 77 724 70 1243 89 28 4 319 53 4 849 57 78 654 68 1154 89 29 4 266 53 4 792 58 79 586 64 1065 89 30 4 213 53 4 734 59 80 522 62 976 88 31 4160 53 4 675 60 81 460 59 888 87 32 4107 53 4 615 60 82 401 55 801 86 33 4 054 53 4 555 60 83 346 52 715 85 34 4 001 52 4 495 60 84 294 49 630 84 35 3 949 52 4 435 60 85 245 45 546 82 36 3 897 52 4 375 60 86 200 41 464 79 37 3 845 52 4 315 60 87 159 39 385 77 38 3 793 52 4 255 61 88 120 34 308 72 39 3 741 53 4194 61 89 86 30 236 62 40 3 688 53 4133 61 90 56 23 174 44 41 3 635 68 4 072 61 91 33 14 130 36 42 3 577 62 4 011 61 92 19 8 94 29 43 3 515 66 3 950 61 93 11 5 65 22 44 3 449 68 3 889 61 94 6 3 43 16 45 3 381 70 3 828 61 95 3 2 27 11 46 3 311 72 3 767 61 96 1 1 16 7 47 3 239 73 3 706 61 97 9 5 48 3166 74 3 645 62 98 4 3 49 3 092 75 3 583 62 99 1 1 - APFENDIX. 181 Griffith Davies. EquitaMe Society. 1768 to 1825. Age. 1 2 3 4 5 6 7 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 Living. Die. 2 844 2 833 2 822 2 810 2 798 2 785 2 771 2 756 2 740 2 723 2 705 2 687 2 669 2 650 2 631 2 611 2 591 2 570 2 548 2 525 2 501 2 477 2 452 2 426 2 400 2 374 2 347 2 320 2 292 2 264 2 236 2 208 2180 2152 2123 2 093 2 063 2 033 2 002 1970 11 11 12 12 13 14 15 16 17 18 18 18 19 19 20 20 21 22 23 24 24 25 26 26 26 27 27 28 28 28 28 28 28 29 30 30 30 31 32 33 Age. 60 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 Living. 1937 1902 1865 1826 1785 1744 1702 1659 1615 1570 1524 1478 1432 1385 1337 1288 1238 1 187 1135 1082 1028 974 919 864 808 752 697 642 588 534 480 426 373 321 271 224 181 143 111 85 65 49 36 25 16 9 4 1 Die. 35 37 39 41 41 42 43 44 45 46 46 46 47 48 49 50 51 52 53 54 54 55 55 56 56 55 55 54 54 54 54 53 52 50 47 43 38 32 26 20 16 13 11 9 7 5 3 1 Combined Experience of 17 Offices. 1843. Age. Living. Die. Age. Living. Die. 50 69 517 1108 1 51 68 409 1156, 2 52 67 253 1207 3 53 66 046 1261- 4 54 64 785 1316 5 55 63 469 1375 6 56 62 094 1436 7 57 60 658 1497 8 58 59161 1561 9 59 57 600 1627- 10 100 000 676 60 55 973 1698. 11 99 324 674 61 54 275 1770 12 98 650 672 62 52 505 1844 13 97 978 671 63 50 661 1917. 14 97 307 671 64 48 744 1990 15 96 636 671 65 46 754 2 061- 16 95 965 672 66 44 693 2128 17 95 293 673 67 42 565 2191 18 94 620 675 68 40 374 2 246 19 93 945 677 69 38128 2 291- 20 93 268 680 70 35 837 2 327. 21 92 588 683 71 33 510 2 351- 22 91905 686 72 31159 2 362 23 91219 690 73 28 797 , 2 358 • 24 90 529 694 74 26 439 2 339- 25 89 835 698. 75 24100 2 303. 26 89137 703 76 21797 2 249- 27 88 434 708 77 19 548 2179- 28 87 726 714 78 17 369 2 092 29 87 012 720 79 15 277 1987 30 86 292 727 80 13 290 1866- 31 85 565 734 81 11424 1730- 32 84 831 742- 82 9 694 1582 33 84 089 750 83 8112 1427 34 83 339 758 84 6 685 1268 35 82 581 767 85 5 417 1111 36 81814 776- 86 4 306 958 37 81038 785 87 3 348 811 38 80 253 795 88 2 537 673- 39 79 458 805- 89 1864 545- 40 78 653 815 90 1319 427- 41 77 838 826 91 892 322 42 77 012 839 92 570 231 43 76173 857- 93 339 155 44 75 316 881 94 184 95 45 74 435 909 95 89 52 46 73 526 944- 96 37 24 47 72 582 981- 97 13 9 48 71601 1021- 98 4 3 49 70 580 1063 99 1 1 ; 'eeooQ APPENDIX. 183 Parr's Northampton Table. 1838 to 1844. Age. Living. Die. Age. Living. Die. 10 000 1705 50 4 388 82 1 8 295 832 51 4 306 85 2 7 463 373 52 4 221 86 3 7 090 185 53 4135 89 4 6 905 140 54 4 046 91 5 6 765 107 55 3 955 90 6 6 658 86 56 3 865 90 7 6 572 72 57 3 775 93 8 6 500 62 58 3 682 95 9 6 438 30 59 3 587 96 10 6 408 29 60 3 491 100 11 6 379 30 61 3 391 148 12 6 349 31 62 3 243 158 13 6 318 32 63 3 085 165 14 6 286 34 64 2 920 169 15 6 252 35 65 2 751 173 16 6 217 35 66 2 578 174 17 6182 37 67 2 404 173 18 6145 37 68 2 231 170 19 6108 39 69 2 061 165 20 6 069 39 70 1896 161 21 6 030 40 71 1735 154 22 5 990 41 72 1581 146 23 5 949 42 73 1435 138 24 5 907 42 74 1297 130 25 5 865 44 75 1167 122 26 5 821 44 76 1045 112 27 5 777 45 77 933 104 28 5 732 46 78 829 95 29 5 686 48 79 734 86 30 5 638 48 80 648 78 31 5 590 50 81 570 71 32 5 540 50 82 499 64 33 5 490 52 83 435 56 34 5 438 53 84 379 56 35 5 385 54 85 323 56 36 5 331 56 86 267 56 37 5 275 58 87 211 56 38 5 217 59 88 155 47 39 5158 60 89 108 36 40 5 098 62 90 72 27 41 5 036 64 91 45 19 42 4 972 66 92 26 12 43 4 906 68 93 14 7 44 4 838 70 94 7 4 45 4 768 72 95 3 2 46 4 696 73 96 1 1 47 4 623 76 97 48 4 547 78 49 4 469 81 184 APPENDIX. Fair's English life Table, Xo. 2. Males . 1838 to 1844. Age. Living. Die. Age. Living. Die. Age. 80 Living. Die. 5 126 235 817 331 40 2 748 678 34 678 435 034 60 540 1 4 308 904 281 493 41 2 714 000 35 334 81 874 494 55 914 2 4 027 411 145 352 42 2 678 666 36 024 82 318 580 50 979 3 3 882 059 95 786 43 2 642 642 36 743 83 267 601 45 839 4 3 786 273 69 451 44 2 605 899 37 495 84 221 762 40 614 5 3 716 822 50 073 45 2 568 404 38 272 85 181 148 35 425 6 3 666 749 36 653 46 2 530 132 39 077 86 145 723 30 387 7 3 630 096 31331 47 2 491 055 39 908 87 115 336 25 611 8 3 598 765 26 047 48 2 451 147 40 759 88 89 725 21186 9 3 572 718 22 976 49 2 410 388 41629 89 68 539 17184 10 3 549 742 19 260 50 2 368 759 42 514 90 51355 13 652 11 3 530 482 16 926 51 2 326 245 43 412 91 37 703 10 611 12 3 513 556 16 668 52 2 282 833 44 315 92 27 092 8 060 13 3 496 888 16 496 53 2 238 518 45 219 93 19 032 5 977 14 3 480 392 19 061 54 2 193 299 46119 94 13 055 4 321 15 3 461 331 17 203 55 2 147 180 47 003 95 8 734 3 043 16 3 444 128 19 532 56 2 100 177 48 530 96 5 691 2 083 17 3 424 596 22 674 57 2 051 647 51921 97 3 608 1386 18 3 401 922 25 802 58 1 999 726 55 033 98 2 222 894 19 3 376 120 26 861 59 1 944 693 57 914 99 1328 559 20 3 349 259 27125 60 1 886 779 60 599 100 769 338 21 3 322 134 27 380 61 1 826 180 63119 101 431 198 22 3 294 754 27 629 62 1 763 061 65 497 102 233 111 23 3 267 125 27 879 63 1 697 564 67 744 103 122 61 24 3 239 246 28128 64 1 629 820 69 861 104 61 31 25 3 211118 28 383 65 1 559 959 71841 105 30 16 26 3 182 735 28 647 66 1 488 118 73 663 106 14 8 27 3 154 088 28 924 67 1 414 455 75 302 107 6 3 28 3 125 164 29 215 68 1 339 153 76 718 108 3 2 29 3 095 949 29 525 69 1 262 435 77 871 109 1 1 30 3 066 424 29 856 70 1 184 564 78 709 110 31 3 036 568 30 208 71 1 105 855 79182 32 3 006 360 30 585 72 1 026 673 79 234 33 2 975 775 30 990 73 947 439 78 817 34 2 944 785 31420 74 868 622 77 884 35 2 913 365 31886 75 790 738 76 400 36 2 881 479 32 379 76 714 338 74 342 37 2 849 100 32 905 77 639 996 71704 38 2 816 195 33 464 78 568 292 68 499 39 2 782 731 34 053 79 499 793 64 759 WORKS BY THE SAME AUTHOR. ASSURANCE AND ANNUITY TABLES. Vol. I. ONE LIFE, CARIISLE BIILS, 3 Per Cent, ... £4, 4s. This work contains Short-period and Defen-ed Annuities and Assurances for every year of age and of duration, as also the Values of Policies for every year. It renders the Periodical Investigations quite easy. A few copies only remain unsold. Vol. II. TWO LIVES, CARLISLE BILLS, 3 Per Cent, . . £4, 4s. In this volume the Values of every combination of Joint and Survivor- ship Annuities and of Eeversions are given from year to year. TREATISE ON THE VALUATION OF LIFE CONTINGENCIES, arranged for the nse of Students, ... ... £1, Is. A NEW GENERAL THEORY OF THE TEETH OF WHEELS, lOs. 6d. The above sent free on receipt of Post-office Order addressed to the Author at 35 Cockhwn Street, Edinburgh. ELEMENTARY ARITHMETIC, • 5s. HIGHER ARITHMETIC, 5s- In this Treatise an easy method of extracting Roots of all orders is ex- plained, and also a process for computing directly the Logarithm of a proposed number to any basis. WILLIAM BLACKWOOD & SONS, Edinburgh and London. The Travelers Insurance Company Actuarial Department LIBRARY No. This book is to be returned within two months (or renewal or for the use of others. It is to be held in any case only as long as needed. Books are subject to recall by the librarian at etny time. DATE 1 LIB. 1 DATE LIB. Return book to librarian's desk — not to shelf. 8630. 4-24-'14. * j:i £ -f ^