Page:Cyclopaedia, Chambers - Supplement, Volume 2.djvu/880

 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