172
ANNUITIES.
The Use of -Table I.
To find the Amount of an Annuity forhorn any Number of Years.
Take out the amount from the 1st table, for the proposed yearsand rate of the interest; then multiply it by the annuity in ques-tion; and the product will be-its amount for the same number ofyears, and rate of interest.
And the converse to find the rate or time.
Examp. i. To find how much an annuity of 507. will amountto in 20 years at per cent, compound interest.—On the line of20 years, and in the column of 3| per cent, stands 2S-27968,which is the amount of an annuity of 17. for the 20 years; andtherefore 28-2796* multiplied by 50, gives 1413-9847. or 14137.19s. 8 d. for the answer.
Examp. 2. In what time will an annuity of 207. amount to10007. at 4 per cent, compound interest?—Here the amount of10007. divided by 207, the annuity, gives 50, the amount of 1/.annuity for the same time and rate. Then, the nearest tabularijumber in the column of 4 per cent, is 49-96758, which standingon the line of 28, shews that 28 years is the answer.
Examp. 3- If it be required to find at w hat rate of interest anannuity of 207. will amount to 10007. forborn for 28 years. Here1000 divided by 20 gives 50 as before. Then looking along theline of 28 years, for the nearest to this number 50,1 find 49-96758in the columu of 4 per cent, which is therefore the rate of interestrequired.
The Use of Table II.
Examp. 1. To find the present value of an annuity of 507.which is to continue 20 years, at 3J per cent.—By the table, thepresent value of 17. for the same rate and time, is 14-21240;therefore 14-2124 x 50 = 710-627. or 7107. 12r. Ad. is the pre-sent value sought.
Examp. 2. To find the present value of an annuity of 207. tocommence 10 years hence, and then to continue for 40 years, orto terminate 50 years hence, at 4 per cent, interest.—In suchcases as this, it is plain we have to find the difference between thepresent values of two equal annuities, for the two given times;which therefore will be effected by subtracting the tabular valueof the one term from that of the other, and multiplying by theannuity. Thus,
Tabular value for 50 years...21-48218Tabular value for 10 years... 8-11090
The difference.13-37128
Multiplied by. 20
Gives .267.4256
Or..2677. 8s. 6 d. the answer.
Contingent, or Life Annuities.
t
The value of annuities for life is determined from observationsmade on the bills of mortality. Dr. Ilalley, Mr. Simpson, andMons. de Moivre , are gentlemen of distinguished merit in calcu-lations of this kind.
Dr. Hailey had recourse to the bills of mortality at Breslaw,the capital of Silesia, as a proper standard for the other parts ofEurope , being a place pretty central, at a distance from the sea,and not much crowded with traffickers or foreigners. He pitchesupon 1000 persons all born in one year, and observes how manyof these were alive every year, from their birth to the extinctionof the last, and consequently how many died each year, as in thefirst of the following tables; which is well adapted to Europe ingeneral. But in the city of London , there is observed to be agreater disparity in the births and burials than in any other place,owing probably to the vast resort of people thither, in the way ofcommerce, from all parts of the known world. Mr. Simpson,therefore, in order to have a table particularly suited to this po-pulous citv, pitches upon 1280 persons all born in the same year,and records the number remaining alive each year till none werein life.
TABLE III.
Dr. IIallev’s Table on the Bills of Mor-tality at Breslaw.
Age
Pers.
Hr.
, )Tm.A - | Hr.
A.
Pers .
liv.
A.
Pers,liv .
1
1000
24
573
47
377
70
142
2
855
25
567
48
367
71
131
3
798
26
560
49
357
72
120
4
760
27
553
50
346
73
109
5
732
28
546
51
335
74
98
6
710
29
539
52
324
75
88
7
6q2
30
531
53
313
70
78
8
680
31
523
54
302
77
68
9
670
32
515
292
78
58
10
661
33
507
56
282
79
49
11
653
34
.499
57
272
80
41
12
646
35
490
58
262
81
34
13
640
36
481
59
y52
82
28
14
634
37
472
60
242
S3
23
15
628
38
463
61
232
84
20
16
622
39
454
62
222
85
15
17
616
40
445
63
212
86
11
18
610
41
436
64
202
87
8
19
604
42
427
65
192
88
5
20
598
43
417
66
182
89
3
21
592
44
407
67
172
90
1
22
586
45
397
68
162
91
0
23
579
46
387
69
152
t
Age.
Pers.
lie.
A.
Pers.
liv.
A.
Pers.
liv.
A.
Pers.
liv.
0
1280
24
434
48
220
72
59
1
870
25
426
49
212
73
54
2
700
26
418
50
204
74
49
3
635
27
410
51
196
75
45
4
600
28
402
52
188
76
41
5
580
29
394
53
180
77
38
6
564
30
385
54
172
78
35
7
551
31
376
55
165
79
32
8
541
32
367
56
158
80
29
9
532
33
358
57
151
81
26
10
524
34
349
58
144
82
23
11
517
35
340
59
137
83
20
12
510
36
331
60
130
84
17
13
504
37
322
61
123
85
14
14
498
38
313
62
117
86
12
15
492
39
304
63
111
87
10
16
486
40
294 '
64
105
88
8
17
480
41
284 :
65
99
89
6
18
474
42
274 !
66
93
90
5
19
468
43
264 '
67
87
91
’ 4
20
462
44
255 :
68
81
92
3
21
455
45
246 -
69
75
93
2
22
448
46
237
70
69
94
1
23
441
47
228
71
64
95
0
From the preceding tables the probability of the continuanceor extinction of human life is estimated as follows:
l. The probability that a person of a given age shall live acertainnumber of years, is measured by the proportion, which the numberof persons living at the proposed age has-to the difference betweenthe said number, and the number ofpersons living at the given age.
Thus if it be demanded, what chance a person of 40 years hasto live seven years longer? From 445, the number ofpersons
living