An essay on probabilities and their applications to life contingencies and insurance offices

You are reading a page from An Essay on Probabilities and their Application to Life Contingencies and Insurance Offices, Augustus de Morgan (1838)
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Term Life Insurance

182    eSSAY ON PROBABILITIES.
    Y.    2    2C    I    3    3i    4    I    41    Y.
    1    980    976    971    966    962    957    '    1
    2    961    952    943    934    925    916    2
    3    942    929    915    902    889    876    3
    4    924    906    888    871    8.55    839    4
    5    906    884    863    842    822    802    5
    6    888    862    837    814    790    768    6
    7    871    841    813    786    760    735    7
    8    853    821    789    759    731    703    8
    9    837    801    766    1734    703    673    9
    10    820    781    744    709    676    644    10
    11    804    762    722    685    650    616    11
    12    788    744    701    662    625    .590    12
    13    773    725    681    639    601    564    13
    14    758    708    661    618    577    540    14
    15    743    690    642    597    555    517    15
    16    728    674    623    577    534    494    16
    17    714    657    605    557    513    473    17
    18    700    641    587    538    494    453    18
    19    686    626    570    520    475    433    19
    20    673    610    554    503    456    415    20
    21    660    595    538    486    439    397    21
    22    647    581    522    469    422    380    22
    23    634    567    507    453    406    363    23
    24    622    553    492    438    390    348    24
    25    610    539    478    423    375    333    25
    26    598    526    464    409    361    318    26
    27    586    513    450    395    347    305    27
    28    574    501    437    382    1333    292    28
    29    563    489    424    369    321    279    29
    30    552    477    412    356    808    267    30
    35    500    421    355    300    253    214    35
    40    453    372    307    253    208    172    40
    45    410    329    264    213    171    138    45
    50    372    291    228    179    141    111    50
    55    337    257    197    151    116    089    55
    60    305    227    170    127    095    071    60
    65    276    201    146    107    078    057    65
    70    250    178    126    090    064    046    70
    75    226    157    109    076    053    037    75
    80 1205    139    094    064    043    030    80
    85    186    123    081    054    036    024    85
    90    168    108    070    045    029    019    90
    95    152    096    060    038    024    015    ;    95
    100    138    085    052    032    020    012    1100
ON ANNUITIES.    183
1    T.    5    6    7    ^1    89    10    Y.
    1    952    943    935    926    917    909    1
    2    907    890    873    857    842    826    2
    3    864    840    816    794    772    751    3
    4    823    792    763    735    708    683    4
    5    784    747    713    681    650    621    5
    6    746    705    666    630    596    564    6
    7    ^'711    665    623    583    547    513    7
    8    677    627    582    540    502    467    8
    9    645    592    544    500    460    424    9
    10    614    558    508    463    422    386    10
    11    1585    527    475    429    388    350    11
    12    557    497    444    397    356    319    12
    13    ¹⁵³⁰469    415    368    326    290    13
    14    505    442    388    340    299    263    14
    15    481    417    362    315    275    239    15
    16    458    394    339    292    252    218    16
    17    436    371    317    270    231    198    17
    18    416    350    296    250    212    180    18
    19    396    331    277    232    194    164    19
    20    377    312    258    215    178    149    20
    21    359    294    242    199    164    135    21
    22    342    278    226    184    150    123    22
    23    326    262    211    170    138    112    23
    24    310    247    197    158    126    102    24
    25    295    233    184    146    116    092    25
    26    281    220    172    135    106    084    26
    27    268    207    161    125    098    076    27
    28    255    196    150    116    090    069    28
    29    243    185    141    107    082    063    29
    30    231    174    131    099    075    057    30
    35    181    130    094    068    049    036    35
    40    142    097    067    046    032    022    40
    45    111    073    048    031    021    014    45
    50    087    054    034    021    013    009    50
    55    068    041    024    015    009    005    55
    60    054    030    017    010    006    003    60
    65    042    023    012    007    004    002    65
    ^'70033    017    009    005    002    001    70
    75    026    013    006    003    002    001    ^'7580    020    009    004    002    001    000    80
    85    016    007    003    001    001    000    85
    90    012    005    002    001    000    000    90
    95    010    004    002    001    000    000    95
    100    008    003    001    000    000    1 000    100
184    ESSAY ON PROBABILITIES.

When 000 appears in the table, the sum requisite is less than ten shillings. Thus, less than ten shillings improved for 85 years at 10 per cent., will produce a thousand pounds.
There are six primary results which will be neces

The present value of 11., meaning that fraction of a pound which will, when improved at interest, produce 11. in a certain number of years. Since the present value of 10001. is tabulated, that of 11. is virtually so, and may be found by placing the decimal point before any result of the preceding table. Thus, 4581. is the fraction of a pound which will, when improved at 5 per cent. compound interest for sixteen years, produce 11.; or, in technical language, the present value of 11. due sixteen years hence, at 5 per cent. To find the present value for any year not in the table, multiply the present values for any years in the table the sum of which is the number of years in question. Thus, to find the present value of 11. due fifty-seven years hence, at 3 per cent., multiply together 337 and '961, the present values for fifty-five and two years, the result of which is 324, the fraction of 11. required.

The present value of a perpetuity of 11., or the sum which will, when improved at interest, pay ll. at the end of every 365 days from the present time, for ever. The number of pounds required is found by dividing 1001. by the rate of interest ; being, in fact, nothing but the sum which will at interest produce 11. per annum. The results are in the following table : —

Rate P. V. of Perp.Rate P. V. of Perp.Rate P. V. of Perp.2£504£257£14P21£4041£ 22981 12123£33'a5£20~9£11y3142896£1610£10Thus 201. improved at 5 per cent. yields 11. a year for ever, provided the first payment be due at the interval of a year from the time of putting the money out atON ANNUITIES.185interest. This is also said to be 20 years' purchase, meaning the sum which would buy at once the annual payments of twenty years.

The present value of an annuity certain of 11. (so called to distinguish it from a life annuity.) By an annuity of 11. is meant, a right to receive ll. at the expiration of every complete year after the creation of the annuity, which is said to commence a year before any payment is made; (or a term before payment is made, whether the term be yearly, half-yearly, &c.) Thus an annuity certain of five years, commencing January 1, 1838, is paid on the first days of 1839, 1840, 1841, 1842, and 1843.

When an annuity is to be designated which makes one payment immediately, I shall call it an annuity due; and a perpetuity of which one payment is to be made immediately, a perpetuity due. Thus on January 1, 1839, the preceding annuity becomes an annuity due of four years, comprising an immediate payment of 11. and a commencing annuity of four years. To find the present value of an annuity of 11., use the followingRULe. Multiply the perpetuity (present value of the perpetuity) by the excess of one pound over the present value of the last payment. Thus, for an annuity of five years, at 4 per cent, multiply 25 by the excess of 1 over 822, or by 178 ; the result is 4^{.451., nearly
The amount of 11. at compound interest, meaning the sum to which 11. with its interest will amount in any number of years. It is found by dividing 1 by the present value of 11. due that number of years hence. Thus Il. in 25 years, at 6 per cent., will amount to (11.=233), or 4^.291.

The amount of an annuity at compound interest, meaning the sum of which the annuitant would be possessed immediately after receiving the last annual payment, if he had made compound interest upon every preceding payment. This amount is found by multi..
186ESSAY ON PROBABILITIES.plying the number of years^{' purchase in the perpetuity by the interest (not the amount) of one pound imduring the existence of the annuity. Thus, at 6 per cent., and in 50 years, 11, will become (11.=054, or 18^{.51., so that the interest is 17.51. This multiplied by 16°, or AV—), gives 2921. The more correct answer is 290^{.341., but the error is not ten shillings in a hunpounds.6. The present value of an annuity for a term of years, to begin after the expiration of another term ; techcalled a deferred annuity. To find this, multiply the present value of such an annuity created immediately by the present value of 11. due at the end of the term of deferment. Thus an annuity of 101. for five years, at four per cent., to commence at the end of twenty years, is thus found : were it to commence immediately, it would (see last page) be worth 4^{.45 x 101. or 44.51., and the present value of 11. to be received twenty years hence is 4561. Multiply 44^{.6 by 456, which gives 20.31., the present value required.Extensive tables of the results of the preceding proare given in a^{ll works on interest or annuities; the present treatise is not meant to contain more than enough to enable the student to exercise himself in first principles, or the proficient to obtain approximate rewhen no more extensive work is at hand. I now pass to the consideration of annuities in general.Though the term annuity be generally understood as meaning a sum of money paid yearly or half-yearly to an individual, yet it is important that the reader should consider the word as implying any sum of money paid at fixed intervals, for any term, definite or indefinite provided only that the payment can never be suspended and resumed again. Thus 11. to be paid every year for ten years provided A live as long, with an additional condition that payment is to cease if C in the term should die, B being alive at the time of C^{'s death, is an annuity within the meaning of the word : nothing .can make the payment cease at all, without making it cease altogether. But 11. to be paid as long as A, B,ON ANNUITIES.187and C are alive, to cease when one dies, and to re-commence when a second dies, ceasing finally with the death of the third, is not one annuity, but two annuities. And by whatever name periodical payments may be called, whether rent, salary, rent-charge, interest, &c. &c., they are considered in this subject as annuities.An annuity may be in possession or in reversion. In the first case, a payment will become due at the end of a term after the creation of the annuity, unless, in the mean time, one of the conditions on which payment depends cease to exist. In the second, the annuity is not to begin to be paid until some circumstances happen, or cease to happen, which are named in the agreement. Sometimes the term reversionary annuity implies that there is a contingency in the time of its beginning ; and an annuity not to commence till after a fixed time has elapsed, would be called a deferred annuity. There is, however, a little variation in the use of these words : reversion sometimes implies a remainder of something existing, while a deferred interest is one which does not commence till a future time. Thus if B be to take an annuity on his own life as soon as A is dead, A enjoying it during his own life, B^{'s reversion is a certain part of an annuity on his own life, reckoned from the present time. But if no annuity at all were paid until A's death, then B's interest might be called a deferred annuity. No difference arises from these distinctions in formulae: or calculations ; and they are useful in de-scribing the circumstances of complicated problems.A reversion, and also a deferred annuity, may he certain or contingent : and the same of a reversionary or deferred fixed sum. Thus A may have B^{'s estate after his death, provided he survive B; or else A and his heirs or assigns may have the same : in the first case the reversion is contingent, in the second certain. A life insurance may be of one or the other kind : thus A may covenant for his executors to receive 1001. at his death, in any case, or else if B should be then surAnd if, in return for such insurance, A should engage to pay a yearly premium, making the first pay-188ESSAY ON PROBABILITIES.ment (as is usually done) at the time of contracting, then the premiums altogether constitute an annuity due upon A^{'s life.All problems relating to this subject might be solved from first principles, as will be shown ; but, in such case, even the most simple of them would be attended with laborious calculation. Tables are therefore constructed of such results as will most facilitate the solutions of all problems : and these tables give the values of annuities on single lives, and on two joint lives. The annuity presumed is always 11. per annum, from which the value of any other annuity is found by simple multiBy the value of an annuity is meant the sum which must be paid down in order to enable the grantor of the annuity to pay it as it becomes due during the term of its continuance. If money made no interest, the average duration of such sets of circumstances as are conditions for the payment of the annuity would immediately point out its value. For instance, according to the Carlisle tables, persons aged 40 live, one with another, 27^{.6 years. Now, remembering that half a year was added (p. 163.), as being the part of their last year which, one with another, they pass through, but which will not (in our meaning of the term annuity) entitle them to any payment, we see that 27'1 is the average number of payments which such annuitants, aged 40, will receive; that is, one hundred with another, 100 annuitants will receive 2710 payConsequently, money making no interest, each of them must pay 27^{.101. for a life annuity of 11., or 27^{.10 years^{' purchase. But if each had been entitled to his fraction of the sum which would have become due had he lived to the end of the year, then 2760 years^{' purchase would have been necessary.Let us now suppose money to make interest. It has before now seemed, to more than one writer, that the value of a life annuity must be the same as that of an annuity certain during the average duration of the life. Many of my readers will not see the fallacy ofON ANNUITIES.189supposing that, money making 4 per cent., 16^{.71., the value of an annuity certain for 28 years, must be but very little more than the value of an annuity on a life aged 40. Such a rule, if it be absolutely true, must be true in any extreme case, however physically impossible. Suppose, then, that of 101 persons aged 40, one lives for ever, and the rest die between the ages of 41 and 42 : whence 100 of them will receive only one payment, and the remaining one will receive a perpetuity of 11. The present value of an annuity of 11. for one year, at 4 per cent., is 961., and 961. increased by the value of 11. for ever, is 1211., the present value of all payIf, then, it be not known which is to have the perpetuity, is what each should pay, or 1^{.211. very nearly. This result is undoubtedly correct: but the average life of the 101 persons is infinitely great, since there is no number of years which is not exceeded by them all put together. Each, then, would have to pay for a perpetuity, if the preceding fallacy were admitted ; or all together would pay 25 x 1011., that is 25251., more than 200 times the truth. It is true that, applied to any actually existing law of life, the incorrect notion cannot produce results so grossly false as the preceding: but it is quite sufficient for us to know that a rule is incorrect in its principle, to make it necessary to apply correct reasoning before we can attempt to say how far it is wrong.The rule for calculating the real value of an annuity is made up of a collection of individual cases, not more complicated than the following. A is to receive one pound ten years hence, if a halfpenny which is to be thrown up give a head; what is the present worth of his chance? If he were to receive the money immediately, in case of winning, its value, on the prinin chapter V., would be -51.: but the present value of 11. deferred for ten years at 4 per cent. is 7091.: whence that of 51. similarly deferred is 3551. This simple process contains the method of valuing the sum which must be now paid down to secure the tenth payment of an annuity to a life belonging to a190ESSAY ON PROBaBILITIES.class of which just one half die in ten years. A repetition of a similar process for every year in which the annuity may become payable gives the present values of the other payments. The sum of all the present values of the different payments is the present value of the whole annuity. Thus if we take a very old life, and suppose that of 10 alive at the present time there will be left at the end of successive years, 7, 5, 3, 0, there are three possible payments of the annuity, and the chances of having to make them are qi,,, and °,,. But the first, if made at all, is not made for a year, and the second and third are not made for two and three years. If then a, b, and c be the present values of ll. to be received at the end of one, two, and three years, the values of the several payments are _i7,,of a, i' of b, and T°_{u of c, the sum of which is the value of an annuity of 11. on any one life of the kind in question.By the status of an annuity, I mean the state or condition of things during the continuance of which the annuity is to be paid. This status may be simple or complicated : in the latter case the method of findthe chances of its continuance or termination will also be complicated ; but this does not affect the simple rule by which those chances, when found, are made, in conjunction with the rules of compound interest, to give the value of the annuity. Thus the status in question may be of a compound character, allowing of several distinct changes. For instance, A is to enjoy an annuity to the end of his life, unless B should die before C, in which case it is to cease. This annuity will be enjoyed as long as either of the following status exist.A, B, and C all living.A and B living, and C dead.A living, and B and C dead, C having died first.Tables of the values of annuities are constructed in the case of single lives and two joint lives; that is to say, the mere inspection of the tables will enable us toON ANNUITIES.191answer the questions — what is the value of an annuity to be paid so long as A shall live, and what is the value of an annuity to be paid so long as A and B shall both be alive, but to cease when either of them dies. The authorities upon the subject have been already mentioned ; namely, Mr. Milne * for the Car-lisle tables, and Dr. Price t and Mr. Morgan + for the Northampton tables. Besides these we have the work of Mr. Baily, now completely out of print; in conwith which must be mentioned the treatiseson interest and on leases by the same author, which' are also out of print and scarce. The work of Mr. Davies q( contains, in connexion with the Northampton tables, a very considerable amount of results, not elsewhere published. The work of Mr. Baily en insurances has lately been translated into French.** The results of the Carlisle and Northampton tables have been lately published on one large sheet, by Mr. M`Kean.t I' All the preceding are valuable works, and very much surpass, in useful application, all that has been done in any other country, or in all countries put together.I now give such a brief abstract of the results of the Carlisle and Northampton tables as is consistent with the plan of this work, in regard to annuities upon single lives, and upon two joint lives.^{a A Treatise on the Valuation of Annuities and Assurances, &c. &c., by Joshua Milne. London. Longman and Co. 1815.} Observations on Reversionary Payments, &c. &e., by Richard Pries^{., D.D. F.R.S.; seventh edition, edited by William Morgan, F.R.S. London. Cadell and Davies, 1812.The Principles and Doctrines of Assurances, Annuities, &c., by WilMorgan, F.R.S. London. Longman and to. 1821.The Doctrine of Annuities and Insurances, &c., by Francis Baily. London. Richardson, 1816.The Doctrine of Interest and Annuities, &c., by Francis Baily. Lon-don. Richardson, 1808.Tables for the purchasing and renewing of Leases, &c.,by Francis Bally. Landon. Richardson, 1807, (second edition).Y Tables of Life Contingencies, &c., by Griffith Davies. London. Long. man and Ca, and Richardson, 1825.* Thcorie des Annnitc> \1 aebres, &c., traduit de 1'Anglais par Alfred de Courcy. Paris. Bachelier, 1836.}} Practical Life Tables, by Alex. M'Kean. London. Richardson, Butterworth, &c.,1837. The annuities on two lives are given for every difference or age ; whereas, in the works of Messrs. Morgan and Milne, they are only given for multiples of five years, and the rest most be supplied by interThis sheet is a beautiful specimen of typography.192ESSAY ON PROBABILITIES.Values of an Annuity of 1 10, or (with the last figure
made a decimal) of '1, upon a single Life.Northampton.Carlisle.1Age.3 p. c.;4 p. c.5 p. c.I6p. c.3 p. c.4 p. c.5 p. c.i6 p. c.! Age.0123103897617314312110405205172148130237196166143510207175151133235]9616714410151971681461292261901621411520186160140124217184158138202617815413612120717615313525301 69148131117196169147130303515914012511218416014112635401481321181071711511341204045137123111101159141126114455012411310394143129117106505511210294871241131039555609890847810597898360165837873GS89837873651 70676460.57716763607075525047455552504875803836353444424039808526252524323130298590181817I1725242323909522222S27262595100000017171616100 ,According to this first table it appears, for instance, that at the Northampton rate of mortality, the value of an annuity upon a life aged 45, at 3 per cent., is 13^{.7 years purchase, or 131.14s. Od. for an annuity of 1 1., 1371. for an annuity of 101., and 13701. for one of 1001. per annum. Or rather, since the table is correct within one twentieth of a year's purchase, we should say that the value of such an annuity is between 13^{.65 and 13^{.75 years' purchase. The same remarks apply to the table of annuities on joint lives.}}}ON ANNUITIeS.193Values of an annuity of ,_{22.^{1210, or (with the last figure
made a decimal) of ;^{'1, upon two joint lives.}}}}}}}}}}}}}}}}}}}}}}}Northampton.CarlisleAges.--Ages.31415634156551561361201071981681451275510101631431271132₀017014813010j101515152134120108-1891631421,561515202014112511210218013411371222020252513411910898169148131117252530301213113103941581391241123030353511710697891471311181073535404010898908313512111010040404545989083771241121039445455050878175701091019386505055557772676391857974555560606662595673696561606065655552504760575552656570704341~39384644424))707075753130292832313073758080212120202524232380808585131313121716161585859090999911111010909051016013912311019916914612951101015158138123110194166145128101515201471301161051841591401241520202513712211010017415213412l)202525301301161059616314312711425303035121109100911521351211093035354011210293861401261141033540404510294868012911610697404545509285797411610697904550505582767167999285805055556071676359817671675560206560575451666259566065Fu704846444252504745657070753635335238375534707575802524242328272625758080851616161520191919808585901111101013131313. 85905151341351201071931651431265151020152134119107189163142126"10152514212611310217815513612115252030133119107971671471301162020253512.5112102531571581_⁰4111233530401161059688144129116103304f)35451069789213312010910045405096888216120109100921 4050I45558679746910496i898245555060757066628781⁷67150605565636057547268646155656070514947455653504460501657540383735434139³865737080282827263231302970807585191918_i182222212075 185801013i131312161))1.;158090O194ESSAY ON PROBABILITIES.Northampton.Carlisle.Ages.Ages,1314151631456}

5 20 148    130    116    104    187    161    140    124    5 20
10 25 147    130    116    105    182    158    139    123 10 25
15 30 137    122    110    100    171    149    132    118 15 _{130
20 35 127    114    104    95    160    141    126    113 20 35
25 40 119    107    98    90    148    132    119    107 25 _{i 40
30 45 109    100    91    84    137    123    111    101 30 45
35 50    99    91    84    78    123 112    102    94 35 50
40 55    89    82    77    71    107    98    90    84 40 55
45 60    78    73    68    64    91    84    78    73 4_.5 60
50 65    66,    62    59    56    77    72    68    (4 50 65
55 70    54    51    49    47    60    57    54    52 55 70
60 75    42    40    39    37    45    43    41    40 60 75
65 80    ^{31    30    29    28    35    34    ^{33    ^{32 65 80
70 85    21    20    20    19    25    24    24    23 70 85
75 90    14    14    14    14    18    17    17    16 75 90

5 25 143    126    118    102    180    156    137    121    5 25
10 30 142    126    113    102    174    152    134    120 10 30
15 35 132    118    107    97    163    143    128    115 15 35
20 40 121    109    99    91    151    134    121    109 20 40
25 45 112    102    93    86    140    125    119    103 25 45
30 50 102    93    86    80    126    114    104    96 30^{.50
35 55    91    84    78    73    109    100    92    86 35 55
40 60    80    75    70    66    92    86    80    74 40 60
45 65 1 69    65    61    58    79    74    70    66 45 65
50 70    56    53    51    48    63    60    57    54 50 70
55 75    44    42    40    39    48    46    44    42 55 75
60 80    32    31    30    29    37    36    34    33 60 F0
65 85    22    22    21    21    27    26    26    25 65 85
70 90    15    15    15    15    20    19    19    18 70 90

5 30 138    122    110    99    172    150    132    118    5 50
10 ^{35    135    121    109    99    166    146    130    116    10 135
15 40    125    112    102    93    153    1345    122    110 15 ^{140
20 45    114    103    94    87    142    127    115    105 20 45
25 50 104    95    87    81    128    116    106    97}}}}}}25 50 1
30 55    93    86    80    75    111    102    94    87 30 55
35 60    82    77    72    67    94    87    81    76 55 60
40 65    70    66    62    59    80    75    70    66 40 65
45 70    57    55    52    50    65    61    58    55 45 70
50 75    45    43    41    40    50    48    46    44 50 75
55 80    33    32    31    30    39    38    36    35 55 80
60 85    23    22    22    21    28    27    26    26 60 85
. 65 90    16    16    15    15    21    21    20    20 65 ! 90

5 ^{135 131    117    106    96    164    144    128    114    5 35
10 44) [ 128    115    104    95    156    1138    124    112    10 40
15 45 117    106    97    89    144    129    116    106 15 45
20 50 1 105    96    89    82    130    118    107    98 20 50
25 55    95    88    81    76    113    103    95    88 25 55
30 60    84    78    73    68    95    88    82    76 30 6()
35 65    72    67    64    60    81    76    71    67 35 65
40 70    59    56    53    50    63    62    58    5.5 40 70
45 75    46    44    42    40    51 I 49    46    44 45 75
50 80    34    32    31    30    41    39    37    36 50 80
55 85    23    23    02    22    30    29    28    27 5.5 85
60 90    16    16    16    15    22    21    21    20 1601 90
ON ANNUITIES.}195
    Northampton.    Carlisle.
    Ages.            Ages.

    3    4    5    [    6    3    4    5    6
    5    40    124    112    101    92    154    136    122    110    5    40
    10    45    120    109    99    91    146    131    118    107    10    45
    15    50    1 108    99    91    84    131    119    108    99    15    50
    20    55    96    89    82    76    114    105    96    89    20    55
    2.5    60    85    79    74    69    97    89    83    77    25    60
    30    65    73    68    64    61    82    77    72    68    130 ' 65
    35    70    60    57    54    51    66    62    59    56    35    70
    40    75    47    45    43    41    51    49    47    44    40 ^{1 75}45    80    34    33    32    31    41    39    38    36    45    80
    50    85    24    23    23    22    30    29    28    28    50    85
    55    90    17    16    16    16    2,    22    22    21    55    90
    5    45    116    105    96    88    144    129    116    105    5    45
    10    ⁵⁰110    101    93    85    133    120    110    100    10    50
    15    55    99    91    84    78    115    105    97    90    15    55
    20    60    86    80    75    70    98    90    84    78    20    60
    25    65    ⁷⁴    69    65    62    83    78    73    69    25    65
        60
    30    70        57    54    52    67    63    60    56    30    70
    35    75    47    45    43    42    52    49    47    45    35 1 75
    40    80    35    33    32    31    41    39    38    36    40    80
    45    85    24    24    23    22    31    30    29    28    45    85
    50    90    17    17    16    16    21    1    22    22    50    90
    5    50    107    97    89    82    131    118    108    99    5    50
    10    55    101    93    86    80    117    107    98    90    10    55
    15    60    88    82    76    71    99    91    84    79    15    60
    _20    65    74    70    66    62    84    79    74    69    20    65
    25    70    61    58    55    52    67    64    60    57    25    70
    30    ⁷⁵48    46    44    42    52    50    47    45    30    75
    80
    35        35    34    13    32    41    40    38    37    35    80
    40    85    24    24    23    23    31    30    29    28    40    85 I
    45    90    17    17    16    16    24    23    22    22    45    90
    ,
5    55    97    89    83    77    115    105    96    89    5    55
    10¹60
    90    83    78    73    100    92    85    79    101 60
15    65    76    71    67    63    85    79    74    ⁷⁰15    65
    20 , 70    61    58    55    53    68    64    61    57    20    70
25    75    48    46    44    42    53    50    48    46    25    75
30    80    35    34    33    32    42    40    38    37    130    1 80
35    85    25    24    23    23    31    30    29    ²⁸55    85
    40 , 90    17    17    16    16    24    J    23    22    22    j    40    90
5    60    86    80    75    70    98    90    84    79    5    60
10    65    77    72    68    64    85    80    75    ⁷⁰10    65
15    ⁷⁰63    59    56    54    68    64    61    58    15    70
    i
    20 . 75    48    46    44    42    53    50    46    46    20 i 75
25    80    36    14    33    32    42    40    39    37    25    80
30    85    25    24    23    23    31    30    29    28    30    85
35    90    17    17    17    16    24    23    23    22    35    90
5    65    74    70    &i i    62    84    78    73    69    5    G5
10    70    fw3    60    57    54    69    65    61    58    10 1 70
    ⁷5    49    47    45    43    53    51    48    46    15    75
15
20    80    ^{36    34    33    32}42    41    39    j    37    20    80
    95 ¹⁸³    25    24    24    23        30    1    29    j    28    f    25    85
30    90    17    17    17    16    24    /    23    23    22    30    90
o 2
196    eSSAY ON PROBABILITIES.
    Northampton.    Carlisle.
    Ages.            6    Ages.
    ³4    !    5 1    ⁶67    ⁴6
                52        64
    5    70                            50    57    5    70
        61    58    55
    10    75    50    47    45    44    54    51    49    46    10    75
    15    80    36    35    34    33    42    41    39    38    15    80
    20    85    25    24    24    23    31    30    29    28    20    85
    25    90    17    17    17    16    24    23    23    22    25    90
    5    75    48    46    44    42    52    50    48    45    5    75
    10    80    36    35    84    33    43    41    39    38    10    80
    15    85    25    25    24    23    ³¹30    29    28    15    85
    20    90    17    17    17    16    24    24    23    22    20    90
    5    tO                    42    40    38    37    5    80
    10    85                    32    31    30    29    10    85
    15    90                    24    24    23    22    15    90
    5    85    ~~~            31    30    I    29    28    5    85
    10    90                25    24    23    22    10    90

The preceding table contains the values of annuities upon two lives, for all ages which are multiples of 5. Thus, for the ages 25 and 40, look to that part of the table in which the ages differ by 1,5 years, and there, opposite to 25 40, will be found, under 4 per cent., 132 in the Carlisle table and 107 in the Northampton. That is, money making 4 per cent., an annuity of 101., which is to continue as long as lives of 25 and 40 are both in being, is worth something between 1311. 10s. Od. and 1321. 10s. Od. according to the Carlisle tables, and something between 1061. 10s. Od. and 1071. 10s. Od. according to the Northampton tables.
When the two required lives have ages which do not end with the figures 0 or 5, proceed as follows: —Let the value of annuity be required on joint lives of 38 and 47, (Carlisle tables at 3 per cent.). First take 35 and 45, and 40 and 45, and between the corresponding annuities insert such a mean as would represent 38 and 45 upon the supposition uniformly diminishing values. Then be35 and 50 and 40 and 50 insert such a mean as answers to 38 and 50. Having then 38 and 45 and 38 and 50, find such a_{,,mean as answers to 38 and 47. This process, which will be intelligible to a reader who has practised similar ones before, will only}
ON ANNUITIES.    197
be comprehended (if at all) by others from the example.
35 and 45    133    35 and 50    123
40 and 45    1 29    40 and 50    110
4    3
3    3
5) 12    5)    9
2    2
38 and 45    131    38 and 50    121
38 and 50    121
10
2    38 and 47    127 Ans
5) 20 4

Before proceeding further, I shall describe the notation of which I intend to make use. It was not the practice of the earlier writers to invent any disnotation of different contingencies, the first attempt at which is found in the work of Mr. Bai]y. Here, however, it was not carried to the full extent, and Mr. Milne endeavoured to organise a system which should take in every case, in which he succeeded perfectly as far as distinct representation of all the cases which occur. His symbols, however, are complicated and strange, though I am clearly of opinion that they are much preferable to the attempt to dispense with notation altogether. The new principle which the notation I now propose involves, lies in the treatment of terms of years certain as lives not subject to con_ tingencies. Thus, if AB represent an annuity on the joint lives of A and B, meaning that it is to cease when either A or B dies, then tB may represent an annuity on the joint term of t years and B^{'s life, to cease with the first which expires ; or what would be}

o 3
198    eSSAY ON PROBABILITIES.

called a temporary annuity on the life of B to last t years, provided B should live t years.

Any simple status on the existence or termination of which a benefit depends, is denoted by juxtaposition of large and small letters, the large letters denoting separately the values of annuities on given lives or status, the small letters (or rather certain small letters, m, n, t, for the most part) denoting given terms of years. Thus ABC t is in existence as long as t years last, provided A, B, and C (or the persons on whom annuities now granted have these values), remain alive all the time.

A compound status, or one which exists as long as either of two or more status remain, is denoted by colons placed between the symbols of the simple status ; thus A : B : t is in existence as long as A, or B, or t years, any or all, are in existence. The symbol is unbetween two certainties : thus, n : n -- t is n + t.

A bar placed over two status indicates that the one is to succeed the other and that the compound symbol denotes a status in being as long as the one, or the
other after it shall exist : thus A P denotes a status which remains in being during the life of A, and also (luring a life to be named at the end of the year in which A dies, having then the value P. If there be occasion, a thicker bar, or one with an accent, or a double bar, may be used where there are two successions involved, between which it is necessary to distinguish.

The symbol in all cases gives the whole symbol the meaning of the present value of a benefit to be reand a figure attached, as in I}_{, denotes the rate per cent. at which the value is to be calculated. This benefit is always 11. at each one or more payments. The description of the status which must end before the benefit begins will be found on the left of ; and the description of the status during which the annual payments of the said benefit are to last will be found on the right. It is further to be understood that the first payment made}
ON ANNUITIES.199under A!B will take place at the end of the year in which A drops, provided B be then in existence : thusA 1 denotes " the present value t of an annuity of 11., the first (and only) payment of which is to be made at the end of the year in which A drops ; while All denotes the present value of 11., to be paid in a year from this time, if A be then dead.
The last moment of a term certain is a part of that term, unless the contrary be expressed by symbols:
thus 6T refers to a pound payable at the end of seven years, and is 617. But when the last moment of a term is considered as having followed the end of the term, a small hyphen (considered as an abridged negasign) is placed after the term in question : thus,t-In denotes an annuity of n payments, the first of which is to be made at the end of t years; and is the

same with t—l1 n.

The absence of symbols on the left of I indicates that the first payment takes place in a year from the present time; but -I indicates an annuity now due. The absence of symbols on the right of I indicates a perpetuity in reversion ; and I itself indicates a perpethe first payment of which is to be made in a year ; while -I indicates a perpetuity now due.

Dots between two symbols of status indicate that the joint status shall be held to exist throughout the year in which the first is determined, provided the second remain at the end of the year ; and dots placed under several status denote that the succeeding benefit is not due unless all those status shall drop in the same year : thus 1A...B denotes an annuity on the joint lives of A and B, payable also at the end of the year in which A dies, if B be then

alive ; and A : Brl is the value of 11. at the end of the life of the longest survivor of A and B, provided they

* One bar may be omitted in very simple cases.
Remember particularly that in A II, 1 means one year, not one pound.

o 4

200    ESSAY ON PROBaBILITIES.

both die in the same year ; also IA B... denotes the value of an annuity on the joint lives to be paid in addition at the end of the year in which the joint existence fails.

When the condition is that a given status shall be in existence at the moment in which another status drops (whether the first last to the end of the year or
not), single dots are placed over the two : thus A 1 B means the present value of ll., to be received at the end of the year in which A dies, provided B be alive at thatmoment ; while A Ii B means the same, provided B be alive when the payment is to be made.
When it is a condition that deaths are to happen in a specified order, it must be represented by writing small figures under the status. Thus, A : BIC means
I,.an annuity on the life of C, to begin at the end of the year in which B dies, provided A have diedbefore B ; and A : (B C) 1 denotes the present value ofll. payable at the end of the year in which the longest of the two status A and BC drops, provided that the status BC is determined by the death of B.
When the joint existence of one number of lives, out of a larger number, is a condition, a figure may be annexed as follows : (A B C D)„ indicates a status which exists as long as any two out of the four are alive.
1]. The double sign 11 indicates the premium which is to be paid during the continuance of the status on the left, in consideration of the deferred benefit described on the right; premium being always interpreted as an annuity due. And where a specific event, as distinfrom the duration of a status, is a condition, the premium is to be held payable as long as any status exists out of which that event may happen. Thus A : BCIdenotes the premium which should be paid as long as A lives with B, or B after A, to secure an annuity to C when both are dead, provided A die first.ON ANNUITIES.201I now proceed to some further instances :In means the present value of an annuity of 11. to last n years.m-I n the present value of an annuity of 11., which is to commence payment at the end of m years, and then to last n—1 years, or n—1 more payments.nl the present value of a perpetuity of 11., comcumencing at the end of n years, or first paid at the end of n + 1 years.n-I the present value of a perpetuity, first paid at the end of n years.A the present value of an annuity on the life of A. IA B the present value of an annuity on the joint lives of A and B, to cease with either.t IA the present value of an annuity on the life of A, to begin in t years ; that is, the first payment to be made at the end of t-]- 1 years, if A should then be alive.At the present value of an annuity on A^{'s life, or t years, whichever drops first.Al the present value of ll. for ever, to he first received at the end of the year in which A dies.ABI the present value of 11. for ever, to be first reat the end of the first year in which A or B dies.A't the value of an annuity for t years, payment to begin at the end of the year in which A dies.ABIC the present value of an annuity which begins payment at the end of the year in which either A or B dies (the first), provided C be then alive, and which continues during the life of C.A : B signifies the present value of an annuity which is to be paid as long as either A or B is alive.A B : CID . E the present value of an annuity which is not to be paid as long as A and B are both alive, nor as long as C is alive, but which begins when the joint existence of A and B, and that of C, are both terminated; and continues as long as either D or E are alive.202ESSAY ON PROBABILITIeS.Brackets, as distinguished from colons, will serve the same purpose as in algebra, namely, to give compound terms the meaning of single ones. Thus,(AB) CID: E denotes the last-mentioned annuity, on the supposition that payment is to begin when either of two events happens, the failure of the joint existence of A and B, or that of C.A BIA : B the present value of an annuity to begin when one of the two, A and B, dies, and to continue during the life of the survivor. There is in this parcase the expression of an event which cannot happen ; for if B die first, it is only A who can receive the annuity. Thus BIB is an expression for nothing ; for the present value of an annuity on the life of B, to begin at the death of B, is nothing. In the expression ABA: B, part is nothing, and the rest has a value. ,A IT is the present value of 11. to be received at the end of the year in which A dies ; t-I 1 is the presentvalue of 11. due t years hence; ABI1C is the pre-sent value of an annuity which, commencing with the first death out of the two, A and B, lasts till either one payment, or the life of C drops: that is to say, the value of ll. to be received at the end of the year in which the joint existence of A and B fails, provided C be then alive.Let a colon placed after the final letter denote that a perpetuity is one of the status during which the annuity is to last. Thus,AI(C:) denotes the present value of an annuity to last for ever, after the death of A. The symbol denotes that C being alive at the time of the first payment is a necessary condition. This being satisfied, the longest of the two, C, or a perpetuity, is of course a perpetuity.The presence of the colon always indicates the longest of the two status, and when the colon is a final symbol, one of the status is an infinite number of years, or a perpetuity.A _{1 : _{B_{: C denotes the present value of an annuityON ANNUITIES.}}}203which is to be paid during the life of C, after the deaths of A and B, if A die before B. A : B : CjE is the present value of 11. to be paid23at the end of the year in which the last of the lives, A, B, C, drops, on condition that B shall have died second or third, and that E shall be alive.AIP denotes the present value of an annuity which is to begin payment at the end of the year in which A dies, and to last during the life which shall then have the value P. if there be several conditions, put a symbol over the status which ends and before the one which begins. Thus,IAB : P denotes the value of an annuity which is to be paid during the joint existence of A, B, and C, and further during that of a life which is to be nominated (then having the value P) at the end of the year in which either of the three drops : whileAB C[P denotes that part of the annuity which is paid during the life of P. An author^{'s interest in his works, which is now denoted by IA : 28 was proposed, in a bill lately before the House of Commons, to bechanged into A : 60.The preceding notation will admit of almost any degree of extension, and will be found perfectly capable of expressing any case which now occurs in practice. It must receive some generalisation before it can be applied to an indefinite number of lives, in the manner of Mr. Milne. But since it very rarely happens that more than three lives occur in a practical question, I shall leave farther extension to those who may find the want of it. I shall merely now add, that any case must admit of expression by means of a notation which provides for the conditions under which the benefit begins, the number of years which it is to last, and the conditions of discontinuance : and also that in problems of any degree of complexity, the invention of the notation will be a useful preliminary to the actual solu-204ESSAY ON PROBABILITIES.}tion of the problem. Thus, suppose it required to represent the value of an annuity which is to continue as long as any two of the three, A, B, and C are alive. This must be done thus :IAB:AC: BCThis might be abbreviated into (ABC)_{,,, or any such symbol; which, however, I should recommend to no one who is not very familiar with the developed form.A method of making the notation of chances analogous to that of annuities was devised by Mr. Milne, which is much too ingenious and efficient to allow of its being dispensed with in any future system. To adapt the principle of this connexion to the system which I have proposed, let nta express the chance that A shall be alive in n years, in which the small letters answering to the capitals which denote lives refer to the chances of those lives, and certain letters m, n, t (or more if necessary), are reserved to signify terms of years. Then all which precedes the sign t (or any other which may be preferred), refers to lives or terms expired, and those which follow the sign, to lives or terms which are to be then in existence. Where no given term of years is included, the chance must be understood to refer to the whole continuance of all the status mentioned. Thus,a'l'b is the chance that A shall die before B.a ni'b is the chance that one of the two, A or n years, shall fail before B.a : ntb is the chance that both A and n years shall fail before B ; that is, that B shall outlive A, and also live more than n years.atn the chance that A shall not survive n years.a : ntb the chance that A shall die in n years, and1that B shall outlive that term.ntab the chance that the joint existence of A and B (or both A and B) shall outlast n years.n : a t (n r 1) the chance that A shall die in the(n t 1)th year from this time.ON ANNUITIES.205The tables, of which an abstract has been given, will enable us to find IA and IAB, the values of an annuity on one or two lives, by simple inspection or easy interpolation, but not IA BC, the value of an annuity on three joint lives. A sufficient approximation to IABC is found by finding the single life Z, whose annuity is equal, or as nearly equal as the tables will give, to the annuity on the joint lives of the two elder of A, B, and C. This new life must be placed instead of those of B and C, which reduces the three lives to two. That is IZ being equal to IBC (A being the youngest of the three) IAB C is IAZ very nearly. Thus, the ages of A, B, and C being 45, 50, and 65, we have IBC (Carlisle table, 5 per cent.) =6^{.8, and in the table of single lives 7^{.8 and 6^{.3 are the values of a single life at 65 and 70 years, giving a diminution of P5 in five years or 3 per year of age. Hence 7^{.8 will become 6^{.8 in about three years, or 68 is the age of Z. Again, writing the ages at the feet of the letters signifying lives, ^{1A45 Z55 is 7^{.0 and IA_{,45 Z _{o is 5^{.8, giving a diminution of 24 for every year of Z^{'s age, so that 1 A45 _{Zoo will be 7^{.0—7 or 6^{.3, which is near the value ofIA_{4S B_{5„ C_{55._{The values of annuities upon single or joint lives being thus found, it is easy to solve many other proof annuities. Such annuities may either be recuquired for a temporary purpose, or as a provision for future years, or for some parties after the death of others. I write the result required in symbols at the beginning of each question, and leave the reader to refresh his memory of them by observing the demonstration.PROBLEM. An annuity of 11. on the life of A, aged n, is to be bought, to begin payment at the end of t ycars, if A should live so long : required its value.The question is proposed in the most usual form, but a little change will facilitate its solution. The right of A is evidently an annuity to begin in t— 1 years, should he live so long. And the rule is expressed thus,206ESSAY ON PROBABILITIES.}}}}}}}}}}}}}}}}}}t—11 An=(t—lta,,)X(t—21i)x ^Antt—IAt the end of t—1 years, if A be then alive (of which find the chance), he enters upon an annuity of 11., being then n + t— 1 years of age. The value of an annuity at that age, multiplied by the chance of attainthat age, and by the present value of 11., to be received t—1 years hence, is, on the principles exin p. 189., the present value of the annuity.Example n=45, t=11 (Carlisle tables 5 per cent.)¹⁰¹^-a45⁼1717^='862911 = 614I A5,= 10^.3862 x 614 x 10'3 = 5'45X5^.45 Answer.PROBLEM. An annuity of 11. is granted to A, aged n, for t years, provided he live so long : what is its value ?1Ant= ^An— ^tl^AnThe last equation is obvious ; for the whole annuity on A^{'s life is made up of a contingent annuity for t years, and a contingent annuity to commence payment in}t + 1 years. Consequently, from the whole value of an annuity for A^{'s life subtract that of an annuity to commence payment in}t+1 years, if he should be then alive, and the remainder is the value of an annuity for t years, if he should live so long.Example n=45, t=10By last problem ^{101^A,,=5'5,}I^A,,^{=12.6. Therefore I(A 10)=}I A,,— 1019_{45=?'1 1^{17'1 Answer.PROBLEM. To find A,,, 1 B,,_{, the value of an annuity on the life of B, aged}}}n, the first payment of which is to be made at the end of the year in which the life of A, aged m, fails. This is called a survivo^{rship annuity, since it can never be paid unless B survive}A. To give this annuity is evidently to give a complete annuity to B, on condition that he shall restore it as long as A is alive; that is,AFB= IB—1AB;or, from the value of an annuity on B^{'s life subtract that of an annuity on the joint lives. Thus (Carlisle tables 4 per cent.), the value of an annuity on the lifeON ANNUITIES.}207of B, aged 30, to commence after the death of A aged 35, is 16'9— 13^{.5, or 3^{.4 years^{' purchase.PROBLEM. To find AB 1 A : B, the value of an annuity on the life of the survivor, whichever it may be, after the death of the other.ABTA: B=IA+IB—2(1AB)This is the same thing as giving an annuity to both, on condition that both restore it as long as both are alive. From the sum of the annuities, therefore, on the lives of A and of B, subtract twice the value of an annuity on the joint lives.PROBLEM. To find ABCIAB : BC: CA, or ABCI(ABC)„ the present value of an annuity to begin payment at the end of the year in which one of the three dies, and to continue as long as both of the other two are alive. Give each pair an annuity on their joint existence, and withdraw all three annuities as long as all three are alive.ABCI(ABC)_=IAB+ABC+ACA—3(IABC)If the annuity be to commence immediately, without waiting for the death of one, this is an additional grant of A^{BC orCAB: BC: CA=CAB+IBC+ICA—2 (ABC)PROBLEM. To determine IA :.B, the present value of an annuity to be continued as long as either A or B shall be alive. Give both an annuity, but withdraw it from one as long as both shall be alive.IA: B=;A+jB—CABPROBLEM. To determine !A : B : C, the present value of all annuity to be continued so long as any one of the three shall be alive. Give each an annuity, and withdraw one of the annuities from any pair as long as that pair shall be both alive : but as this would take away the annuity during the joint continuance of the208ESSaY ON PROBABILITIeS.three lives, grant an additional annuity of 11. on that joint continuance. Thus,IA: B: C=IA+IB+IC—IAB—IBC—ICA+IABC;
In the process of finding which it becomes evident that
ABCIA: B: C=IA FIB+HC—IAB—IBC—ICAIt may be worth while to point out how, in every poscase, the preceding grants and withd^{rawals produce the required effect ; namely, one annuity to be paid as long as any one life lasts. The following are the payand withdrawals : —Living.I_I^IDead.Pay.}Withdraw.Balance.ABC1+1+1+11+1+1ABC1+1BCA1+1CAB1+1ABC0BAC0CBA01PROBLEM. Required CRAB, the value of an annuity on the joint lives of A and B as long as they shall surC. Grant a complete annuity on the joint lives, and withdraw it while all three are alive.CIAB=IAB—IABCPROBLEM. Required ABIC, the value of an annuity on the life of C, to commence after the failure of the joint existence of A and B. Grant an annuity on the life of C, and withdraw it as long as all three are alive. Thus,ABIC=IC—IABCPROBLEM. Required the value of an annuity B : CIA, to be paid to A after B and C are both dead. GrantON ANNUITIES.209to A an annuity on his own life, withdrawing it as long as either B or C survives with A ; that is, withdrawingIAB: AC or IAB+IAC— ABC. Hence, B: CIA=IA—IAB—IAC+IABCPROBLEM. Required CIA : B, the value of an annuity to be paid as long as either A or B shall survive C. Grant one annuity to be paid as long as A or B is alive (IA : B), and withdraw it as long as A or B lives with C ; that is, withdraw !AC: BC. Hence,CIA: B=IA: B—IAC: BC=IA+IB—IAB—(IAC+IBC—IABC) =1A+1B—(IAB+IAC+IBC)+IABCThe same case also amounts to granting IA : B : C and withdrawing IC.PROBLEM. An annuity CIA: B, payable as long as either A or B shall survive C, is to be divided equally between them, while they both live, and is then to go to the survivor. 1Vhat is the value of the interest of each?The interest of A is an annuity of half a pound while both A and B survive C, and of a whole pound as long as he shall survive both C and B : or(CIAB)+C: BIA. But,}}}}}}

i (^{CI^{AB)^{=i ^{(I^{AB)^{—i ^{(I^ABC),}}}}}}}
^{C: BIA=IA-IAB—IAC+IABC the sum of which is

!A— i (IAB)—IAC+i (IABC) and similarly, the value of B^{'s interest is IB—I (IAB)—IBC+i (IABC)}
the sum of which makes up, as it should do, the value}of CIA : B, as given above.
210    ESSAY ON PROBABILITIeS.

PnoBLxw. An annuity on the longest survivor of
A and B, or IA : B, is to be equally divided between them during their joint lives, and afterwards to go to the survivor. What is the value of the interest cf each ? That of A is evidently (CAB)+B1A, which is
1 (j AB)+jA — IAB or IA—I (I AB) Similarly that of B is I B— _{i (j A B)}which results are obtainable by a yet more evident process, since the interest of each is evidently an annuity on his own life, with half of it withdrawn as long as both are alive.
PROBLEM. To determine (BC)IA, the value of an
annuity on the life of A, to commence with the failure of the joint existence of B and C, provided it be B who dies first.
There are no tables for the accurate solution of this problem ; but the following reasoning leads to a result which cannot be far wrong, unless some of the lives be very old, and which will always be near enough for the species of application contemplated in this work. The interest of A may be divided into two annuities, one of which is NA C, and the other a portion of B: CIA. For A is certain of an annuity during C^{'s life after the death of B, and of another after both are dead, provided
B die first. Suppose A^{'s interest in the latter annuity to be worth one half of it, which is strictly true if B and C be of the same age, and not much beside the truth for considerable differences of age, particularly when A is the oldest of the three. On this supposition A^{'s interest is
BjAC+1 (B : CIA)}}}
or !AC—(ACB+1 (IA— jAB— 'AC+IABC) or _{t (jA—IAB+jAC—IABC) =(BC)IA which is obtained by supposing

1(A -1AB—jAC+!ABC)=B: C;A}ON ANNUITIeS.    211

The facility with which the preceding rules may be applied, in every part of the process except that of finding the annuities on three Iives, makes it unnecessary to present examples. In order that examples may be readily obtained in cases involving three lives, some partial tables are given both by Mr. Morgan and Mr. Milne, from which the following selection is made : —

Valucs of an Annuity of x^{'10 (or, with the last figure made a decimal, of ,1,) on the joint continuance of three lives of equal ages, from the Northampton and Carlisle tables; in the former at 4 per cent., and in the latter at 5 per cent.}
112    129    45    71    88
    122    134    50    63    79
    113    127    55    56    65
    103    122    60    48    51
    98    115    65    39    42
    92    108    70    30    32
    86    102    75    21    21
    79    94    80    14    16

The following is from the Northampton table, at 4 per cent., the annuity being 1'10, and the ages as specified.

Ages.        Annuity.    Ages.        Annuity.
    5,    15,    25    107    45,    55,    65    51
    10,    20,    30    104    50,    60,    70    42
    15,    25,    35    97    55,    65,    75    33
    20,    30,    40    90    60,    7 0,    80    24
    25,    35,    45    83    65,    75,    85    16
    30,    40,    50    76    70,    80,    90    11
    35,    45,    55    68    75,    85,    95    2
    40,    50,    60    60

5 10 15 20 25 30 35 40

P 2
212    ESSAY ON PROBaBILITIES.
The following is from the Carlisle table, at 5 per cent: —
    Ages.        Annuity.        Ages.    Annuity.
            ^{11

    5,    30,    35    111    45,    70,    75    33
    10,    35,    40    106    50,    75,    80    25
    15,    40,    45    99    55,    80,    85    18
    20,    45,    50    91    60,    85,    90    12
    25,    50,    55    80    65,    90,    95    11
    30,    55,    60    66    70,    95,    100    9
    35,    60,    65    55
    40,    65,    70    44

I shall proceed in the next chapter to consider the methods of finding the value of reversionary interests, including life insurances.

CHAPTER X.

ON THe VALUE OF REVERSIONS AND INSURANCES.

THE distinction between the problems of this chapter and the last, lies more in names and in the circumstances under which they occur for solution, than in difference of methods, principles, or (according to the scheme which I have suggested), even of notation. Every interest, the symbol of which has any thing preceding the I, is properly a reversion, being something of which the benefit is not to begin until the happening of some event, or the determination of some existing status.

It may be proper here again to remark, that all the rules in the preceding chapter, though the status menare technically called lives, are equally true for any species of circumstances, temporary or permanent, certain or contingent. Thus an annuity for t years, to}}}