L I F
Value of an Annuity for life of I £. Intereft being Age 3 per Ct. 3 fper Ct. 4 per Ct. 5 per Ct.
L I F
56 57 58 59 60 61 62
63 64
65
66
67 68 69
70
7'
72 73
74 75 76
7I
10. 90 10. 61 10. 32 10. 03
9. 11
8. 79
7. 10 6. 75 6- 38 6- 01
10. 44 10. 18 9. 91 9- 6 4 9- 36 9. 08 8. 79 8. 49
6- 57 6- 22
87 51
77
98
57 16
74
3 1
9- 77
9. 52
9. 27
9. 01
8. 75
8. 48
8. 20
7. 92
7- 63
7- 33
7. 02
6. 75
6- 39 6- 06
5-
5-
5-
4-
4-
3-
3-
3- H
2. 70
2. 28
7^ 38 02 66 29 9 1 52
9. 24 9. 04 8. 83 8. 61
8*39 8. 16
93 68
43 18
9i 64
36
°7 77 47 >5 82
+9 4. 14
3- 78 3- 41 3- °3 2. 64 2. 23
The columns marked 3 per Ct. &c. fbew the values of annu- ities for Life in years purchafe, and decimals of a year ; thus an annuity for Life at 3 per Ct. for an age of 56, will be 10. 90. that is worth 10 T % years purchafe. Dr. Hailey alio publifhed a Table » for estimating the proba- bilities of life, grounded on the Brcllau Bills of Mortality ; and the values of annuities for life have been commonly de- termined from this Table, and from the rate of intereft. See De Moivre, DocTr. of Chances, p. 211, feq. We fhall here infert Dr. Halley's Table, divided into feveral columns, fhewing alternately the age, and the number of perfons living of that age.— [ " Philof. Tranf. N° 196. Low- thorp's Abridg. Vol. III. p. 671.]
13 H
IS 16
»7
18
19
By the help of this table we may find what the refpeflive pro- babilities are for a man of a certain age, 30, for inftance living 1, 2, 3, 4, »«:. years. Thus, look for the number 30 in one of the columns of age ; then over-againll that number in the nest adjacent column on the right hand you will find 531, under which are written 523, 515, 507 Yqa &c. each correfpopding, refpeaively, to the numbers' writ- ten in the column bf age ; the meaning of which is, that out of 531 perfons living of the age of 30, there remain but 5^3' 5*5> 5°7> 499' & c -; *« attain the refpeaive ages of 3 1 ' 3 2 > 33' 34> l&c. and who confequently do from that term of 30, live 1, 2, 3,4, &c. years refpeaively. Hence fuppofing the quantities A, B, C, D, E, &c. to re- prefent refpeaively the perfons living at a given age, and the fubfequent years, it is evident, that there being A perfons of the age given, and one year after B perfons remaining, the probability which the perfon of the given age has to con- tinue in life, 'for one year, is meafured by the fraflion 5
A and that the probability which he has to continue in life for two years, is meafured by the fraaioni: and foon. There-
A fore, if money bore no intereft, it would be fuffieient to mul- tiply thofe probabilities by the fum to be received annually, and the fum of the produas would exprefs the prefent value of the annuity. But as money bears intereft, all thole values mult be properly, discounted at compound intereft according to
Perfons
Age
Perfor
1000
22
586
855
23
579
798
24
573
760
25
567
732
26
560
710
? 7
553
692
28
546
680
29
539
670
30
53i
661
31
523
6 53
32
5 '5
646
33
507
640
34
499
634
35
490
628
36
481
622
37
472
616
3»
463
610
39
454
604
40
445
598
41:
436
592
42
427
Age
Perfons
43
417
44
407
45
- 397
46
387
47
377
48
367
49
357
5°
346
51
335
52
324
53
3'3
54
302
55
292
56
282
57
272
58
262
59
252
60
242
61
232
62
222
63
212
Age
Perfons
64
202
65
192
66
182
67
172
68
162
6 9
152
70
142
71
131
72
120
73
109
74
98
75
88
76
78
77
68
78
58
79
49
80
41
81
34
02
28
83 84
23, 20
a given rate, and the new refulting value will be the true value of an annuity for a given life at a given rate of intereft. "Mr. De Moivre obferved, that in Dr. Halley's Table the probabilities of life decreafed nearly in an arithmetic progref- iion, when confidered from a term given, and hence he found an eafy rule for the value of an annuity on a life of a given
- where P reprefents the value
age. His rule is, _
of an annuity certain of I £ for as many years as are inter- cepted between the age given, and the extremity of old aoe, fuppofed at 86, and that interval of life is exprefled by n. " r Stands for the amount of the principal and intereft of 1 £ in one year. *
The rule, therefore, in words at length, will be, Take the value of an annuity certain for fo many years as are denoted by the complement of life ; multiply this value by the rate of intereft, and divide the produft by the complement of life ; then let the quotient he fubtraaed from 1, and let the remainder be divided by the intereft of I £ ; then this laft quotient will cxprefs the value of an annuity for an age given. See Compliment of Life, infra. Thus fuppofe it were required to find the prefent value of an annuity of 1 £ for an age of 50, intereft being at 5 per cent. The complement of life being 36 ; let the value of an annuity certain, according to the given rate of intereft, be taken from the tables of fuch annuities ', and this value will be found to be 16.5468. Let this value be multiplied by the rate of intereft 1. 05 ; the produa will be 17. 3741. Let this produa be divided by the complement of life, that is, in this cafe, by 36, the quotient will be o. 04826 ; fubtradt this quotient from unity, the remainder will be 0.5174. Laftlv, divide this quotient by the intereft of I £ ; that is, in the pre- fent cafe, 0. 05, and the new quotient will be 1 0. 35, which will exprefs the value of an annuity of I £ to continue during a life ot 50, or, in other words, how many years purchafe a
life of 50 is worth » [ ■ See DoJfon's Calculator, p. 1 1 J
b De Moron's Annuit. Probl. 1. and Do&;. of Chances, p 213, feqq.
The following questions being of frequent ufe, we have here inferted them, with the rules for their folution.
I. The values of two (ingle lives being given, to find the value of an annuity granted for the time of their joint continuance; or, the value of two Single lives being given, to find the value of the joint lives.
Multiply together the values of the two lives, and referee the produa. Let that produfi be again multiplied by the in- tereft of I £ ; and let that new product be fubtraaed from the fum of the values of the lives, and referve the remainder. Divide the firft quantity referved by the Second, and the quo^ tient will exprefs the value of the two joint lives. Thus fuppofing one life of 40 years of age, the other of 50, and intereft at 5 per Cent ; the value of the firft life will be found in the tables to be 11. 83 ; the value of the fecorul 10-35 i and the produfl will be 122. 4405, which product muft be referved. Multiply this again by the intereft of 1 /; that is, by o. 05, and this new produa will be 6. 122C25. which being fubtraaed from the fum of the lives-, or 22. 18, the remainder will be 16. 057975, and this is the fecond quantity referved. Now dividing the firft quantity referved by the fecond, the quotient will be 7. 62 nearly; and this ex- prefies the values of the two joint lives.
II. The values of two Single lives being given, to find the value of an annuity upon the longcft of them ; that is, of an annuity to continue fo long as cither of them is in being, irom the fum of the values of the joint lives, fubtraa the value of the joint lives, and the remainder will be the value of the longeft.
Suppofe for inftance, two lives, one worth 13 years purchafe, the other 14, and intereft at 4 per Cent. The fum of the values of the lives is 27 ; the value of the two joint lives by the rule before given is 9. 23 ; and fubrrafling 9. 23 from 27, the remainder 17. 77 is the value of the longeft 'of the two lives.
III. The values of three Single lives being given, to find the value of an annuity upon the longeft of them :
Take the fum of the three fingle lives, from which fum fub. traa the fum of all the joint lives combined two and two ; then to the remainder add the value of the three joint lives and the refult will be the value of the longeft of he three lives! Thus fuppofing the fingle lives to be 13, 14, and 15 years purchafe, the fum of the values Will be 42 ; the values of the firft and fecond joint lives are 9. 24 ; of the firft and third 9. 65; of the fecond and third 10. 18 ; the fum of all which is 29. 06 ; which being fubtraaed from the fmu of the lives, that is, from 42, the remainder will be 12. 94 ; to which adding the value of the three joint lives 7. 41, the fum 20. 35, will be the value of the longeft of the three joint lives.
IV. To find the prefent value of a remainder in fee, after a life of a given age. That is, fuppofing A to be in pofief- fion of an annuity for his life ; and that B after the deccafe of A, is to have the annuity for him and his heirs for ever, to find the prefent value of the remainder; or, as fomc call it, the reverfion. From
