ANNUITIES.

173

living at 40 years of age in Dr. Halley’s table, subtract 377, thenumber ol persons living at 47 years of age, and tiie remainder,68, is the number of persons that died during these seven years ;and the probability, or chance that the person in question shalllive these seven years, is as 377 to 68, or nearly as 51 to 1. ByMr. Simpson’s table, the chance is something less than that of fourto one.

II. Jf the year, to which a person of a given age has an equalchance ol aniving before he dies, be required, it may be found j(lms: Find half the number of persons living at the given age in(lie tables, and in the column ol age you have the year required.

Thus, if the question be put with respect to a person of30 yearsof age, the number of that age in Dr. Halley’s table is 331, thehalf whereof is 265, which is found in the table between 57 and58 years; so that a person of 30 years has an equal chance of livingbetween 27 and 28 years longer.

III. By the tables, the premium of insurance upon lives may insome measure be regulated.

Thus, the chance that a person of 25 years has to live anotheryear, is, by Dr. Halley’s table,-as SO to one; but the chance thata person of 50 years has to live a year longer is only 30 to one.And, consequently, the premium for insuring the former ought tobe to the premium for insuring the latter for one year, as 30 to 80,or as three to eight.

In order to facilitate computations of annuities on lives, Mons.de Moivre assumed the age of 86 as the utmost probable extent oflife, and supposed that equal members died at all ages, so that of86 persons born at the same time, it mighfrbe expected one woulddie each year till the whole be extinct. The difference betweenany proposed age and 86 is called the complement of life.

Prob. I. To determine the value of an annuity of 11. to con-tinue during the life of a person of a given age, allowing compoundinterest at a given rate.

It has been shewn, concerning annuities certain, that if r be putfor the amount of 1/. and its interest for a year, the present valueof 1/. payable at the end of one, two, or more years is expressed11

by —, — &c. But as the right to each annual payment dependsr r' 1

on the continuance of life, we must multiply each of these terms,by the number expressing the probability, that the annuitant shalllive to the end of one, two, or more years. Now, if n representsthe number of persons alive at any age, and a the number of per-sons trat die in the iirst year thereafter, then die number of per-sons that live to the end otthat year, is n — a. But the probabilityof the happening of any event, is measured by a fraction, whosenumerator is the number of chances it has to happen, and whosedenominator is the number of chances it has both to happen andn—a

fail, therefore-will represent the chance a person of that age

n

has to live another year. In like manner putting b for the numberof persons that die in the course of the second year, the probability

n—b

that the annuitant shall live to the end of two years is-and so

n

n—a n—a

on. Therefore the value of the annuity is-1-, &c. con-

n r n r-

tinued to the extremity of life.

If we assume Mr. de Moivre ’s hypothesis, n is equal to thecomplement of life, and a, b, &c. each equal to 1, and the pre-n— 1 it —2

ceding series becomes-)-> Ac. which series is the dif-

n r nr 2

111 1x123

ference between-(-[- ■— &c. and — -)-(-]-1-,

r r 2 3 3 n \ r r' 2 r 3

«ec. ^ The former of these being a geometrical series is equal to

1 1 lx 1 n 1 ~P

-- -=[ if we put p = --- I-» and the lat-

r —l r— 1 r n V r n J r—l

vol. i.— no. 11.

1

ter is equal to — xn

r

np \

■- ) x

r-\S

1

n

n —1 n—2

Therefore-1-, &c, to n

nr n r' 2

1 —p 1

terms is equal to-— x

r—l n

• 1 — p. r

,~T\~

(n — 1 +p). r — n

the value of the annuity required.

Here p

is the present value of 1/. due at the end of the complement ofthe lite in question, which may be either computed arithmeti-cally, or taken from a table of the present value of it. But theabove theorem may be also accommodated to our II. table, forputting tire present value of 1/. annuity for the complement of1 —p

life, or-= A, we get the present value of the life annuity' of

r— 1

r

\l. equal to 1-A. This theorem may' be expressed in words

n

r—l

at length, as follows:

Rule. Find the complement of life ; and, by Table II.find the value of it. annuity for the years denoted by the saidcomplement; multiply this value by the amount of 1/. for a year,and divide the product by the complement of life; then subtractthe quot. from 1; divide the remainder by the interest of 1/. tora year; and this last quot. will be the value of the annuity sought;or, iix other words, the number of years purchase the annuity' isworth.

Examp. What i- the value of an annuity' of 1/. for an age of50 years, interest at five per cent. ?

86

50 are given.

36 complement of life.

By the tables, the value is, 16,5468Amount of 1 1. for a year, 1.05

82.7340

1654.68

Complement of life, 36)17.374140(.581615From unity, viz. 1.000000Subtract .482615

Interest of 1/. .05).517385(10.3477 valuesought.

By the preceding problem is constructed the following table .-

TABLE V.

The Value of ll. Annuity/op a single Life.

Age.

3 per c.

S^perc.'A per c.^pcrc.^5per c.

6 per c.

9= 10

8 = 117= 1213

6= 1415

19.87

19-74

19-60

19-47

19-33

19.19

18.27

18.16

18.05

17.94

17.82

17.71

16.88

16.79

16.70

16.60

16.51

16.41

15.67

15.59

15.52

15.44

15.36

15.27

14.61

14.54

14.48

14.41

14.34

14.27

12.84

12.79

12.74

12.69

12.63

12.58

16

5 = 171819

4= 20

19.05

18.90

18.76

18.61

18.46

17.59

17.47

17.34

17.22

17.09

16.31

16.21

16.10

16.00

15.89

15.19

15.10

15.01

14.92

14.83

14.20

14.12

14.05

13.97

13.89

12.53

12.48

12.42

12.36

12.30

2 Y Age