(. (
at
Jt T
f
rf^ a
rOQfiifc c:
c c
LIBRARY
UNIVERSITY OF CALIFORNIA
GIFT OF
Class
LIFE INSURANCE
PREMIUMS AND RESERVES.
OF THE
UNIVERSITY
BY
SHEPPARD HOMANS,
ft
CONSULTING ACTUARY.
[COPYRIGHT BY THE SPECTATOR COMPANY, i388.J
1888.
"THE SPECTATOR COMPANY,
16 DEY STREET, NEW YORK.
LIFE INSURANCE PREMIUMS AND RESERVES,
BY SHEPPARD ROMANS, CONSULTING ACTUARY.
The basis of every sound system of life insurance is the MORTALITY TABLE.
While nothing is more uncertain than the duration of an individual life, the rates of
mortality,. or, in other words, the probabilities of living and dying in any one year at
each age among a large number of persons similiarly situated as regards family history,
climatic influences, etc., can be predicted with almost mathematical precision. The
rates of mortality among insured lives at the several ages have been carefully ascer-
tained by observations among a vast number of persons insured in British and Ameri-
can companies. These results are embodied in three mortality tables of standard
authority, viz :
The ACTUARIES, or COMBINED EXPERIENCE TABLE, deduced from the mortuary
statistics of seventeen British companies, and published in 1837.
The NEW ACTUARIES OR HM. TABLE, deduced from the later experience of
twenty British companies, and published in 1869.
The AMERICAN EXPERIENCE TABLE, deduced chiefly from the mortuary statistics
of the Mutual Life Insurance Company of New York.
Of these the last named table, confirmed, as it has been in a remarkable degree,
by the experience of other American companies, is by far the best index of the rates
of mortality which may be expected to prevail among insured lives in the United
States. This table has been adopted by nearly all American companies as a basis
for premiums and reserves, and by many States as a standard of valuation for contin-
gent insurance liabilities.
These tables do not differ materially from each other, and either would be a safe
basis for the transactions of American life insurance companies. Their teachings have
all the force of natural laws, and these teachings cannot be disregarded or violated
with impunity.
Columns (i) and (2) of the following Table No. i, show respectively the numbers
living and dying at each successive age out of 100,000 persons starting at the age of
ten years. Column (3) shows for each age the rate of mortality, or probability of
dying within one year. This is also the cost, without interest, to insure one dollar, or
unity, payable in case of death within the year, and is found, for any age, by
dividing the number of deaths by the number living. For instance, at age 40,
dividing 765, the number dying, by 78,106, the number living, we have .009794 as the
112686
TABLE No. i.
Probability of Dying
at Each Age, Which
Probability of
Living
Cost to Insure
in case of Deatl
$1,000 Payable
i. Am, Exp. 456.
T *
. .
is Also the Cost to
Through the
AGE.
X
Number Living at
Each Age.
JN umber Dying
at Each Age.
Insure $1.00 lor One
Year, at Each Age.
Year at Each
Age.
For One Year
Equal Yearly
Premiums Dur-
dx
x dx
Only, at Age
ing Remain-
/x
dx
'*
i-T
X
der of Life.
(I)
(2)
(3)
(4
(5)
(6)
10
100,000
749
.007490
.992510
7 20
10.53
II
99.250
746
.007516
.992484
7.23
10.70
12
98,505
743
.007543
992457
7.25
10.88
13
97,762
740
.007569
.992431
7.28
ii. 06
14
97,022
737
007596
.992404
7-30
11.26
15
96,285
735
.007634
.992366
7.34
11.47
16
95.550
732
.007661
992339
7-37
11.69
17
94,818
729
.007688
.992312
7-39
11.91
18
94,089
727
.007727
.992273
7-43
12.15
19
93,362
725
.007765
992235
7-47
12.40
20
92,637
723
.007805
.992195
7-51
12.67
21
91,914
722
.007855
.992145
12 95
22
91,192
721
.007906
.992094
7.60
13.24
2 3
90471
720
007958
.992042
7-65
13-55
2 4
89,751
719
.008011
.991989
770
13-87
89,032
718
.008065
991935
7-75
14.21
26
88,314
718
.008130
.991870
7.82
14-57
28
87.596
86,878
718
718
.008197
.008264
.991803
.991736
7.88
7-95
14-95
15-35
29
86,160
719
.008345
991655
8.02
15-77
3
85,441
720
.008427
991573
8.10
16.21
3 1
84,721
721
.008510
.991490
8.18
16.68
3 2
84,000
.008607
99 I 393
8.28
17.18
33
83,277
726
.008718
.991282
838
17.70
34
82,^51
729
.008831
.991169
8.49
18.26
81,822
73 2
.008946
.991054
8.60
18 84
36
81,090
737
.009089
.990911
874
19 46
37
80,353
742
.009234
.990766
888
2O. 12
38
79,611
.009408
.990592
905
20.82
39
78,862
756
.009586
.990414
9.22
21-57
40
78,106
765
.009794
.990206
9.42
22.35
41
42
76,567
785
.010008
.010252
989748
9 62
9.86
23.19
24.08
43
75,782
797
.010517
989483
IO.II
25-03
44
74,985
812
.010829
.989171
10.41
26.04
74,173
828
.011163
.988837
10.73
27.12
46
73-345
848
.011562
.988438
II. 12
28.27
47
72,497
870
.012000
.988000
"54
29.50
48
71.627
896
.012509
.987491
12 03
30.81
49
70,731
927
.013106
.986894
12. 60
32.21
50
Si
52
69,804
68.842
67,841
962
001
,044
.013781
.014541
.015389
.986219
985459
.984611
13.25
13.98
14 80
33.70
36.98
53
66,797
,091
016333
.983667
1571
38.79
54
65,706
.143
.017396
.982604
40.73
64-563
,199
.018571
.981429
17.86
42 79
56
63364
.260
.019885
.980115
19.12
45 oo
57
62,104
.325
021335
978665
20.52
47 35
58
60,779
,394
.022936
.977064
22.00
49 87
59
59,385
,468
.02472^
.975280
2377
52.57
60
57.917
,546
.026693
973307
2567
55-45
61
56,371
,628
.C28880
.971120
27.77
58-54
62
54-743
,713
.031292
.968708
30.09
61.84
63
53.030
,800
033943
.966057
31.90
65 39
64
51.230
,889
.036873
.963127
3545
69.18
49.341
,980
.040129
.959871
3859
73-25
66
47.36i
2,070
043707
956293
42.03
77.61
67
45,291
2,158
.047647
952353
45-82
82.28
68
43,133
2,243
.052002
.947998
50.00
87-29
69
40,890
2,321
.056762
943238
54-58
92.65
70
38,569
2.391
061993
.938007
5961
98 39
36,178
7,448
.067665
932335
65.06
104.54
TABLE No. i Continued.
Probability of Dying
Probability of
Cost to Insure $i,oco Payable
at Each Age. Which
Living
in case of Death. Am. Exp. 4%.
AGE.
X
Number Living at Number Dying
Each Age. | at Each Age.
is Also the Ccst to
Insure $1.00 for One
Year, at Each Age.
Through the
Year at Each
Age.
For One Year
Equal Yearly
Premiums Dur-
dx
i-^L.
Only, at Age
ing Remain-
4 dx
T~
l-L
X
der of Life.
(I) (2)
(3)
(4)
(5)
(6)
72
33,730 2,487
073733
.926267
70.90
111.13
73
31.243 2,505
.080178
.919822
77.09
118.21
74
28,738 2,501
.087028
.912972
83.68
125.85
75
26,237 2,476
.094371
.905629
90.74
134.14
76 23,761 2,431
.102311
.897689
9838
I43-I9
77 21,330 2,369
78 18,961 2,291
.111064
.120827
^879173
106.79
116.18
I53.I4
164.12
79 16,670 2,196
I3 1 734
.868266
126.67
176.30
80 14,474 2,091
.144466
855534
138.91
189.87
81 12,383 ,964
.158605
.841395
152-50
204.95
82 10,419 ,816
.174297
.825703
!67-59
221 . 82
83 8,603 .648
.191561
808439
184.19
240.90
84 6,955 ,470
211359
.788641
203 23
262.89
85 5485 .292
235552
764448
226.49
288.62
86 4,193 ,114
.265681
734319
255-46
318.82
87 3-079 933
.303020
.696980
291.37
354-03
88
2,140 744
.346692
.653308
334-13
394-5 2
89 1.402 555
395863
.604137
380.64
441.22
90
847 385
454545
545455
43706
497.08
91 462 246
.532466
467534
5" 99
566.28
92
216 137
634259
365741
609.87
649.34
93 79 58
94 21 18
734177
857143
265823
.142857
705.94
824.18
736.31
840.77
95
3 3
I.OOOOOO
o.oooooo
96i54
961.54
rate of mortality or probability of dying within one year, at that age. Column (4)
gives for each age the probability of surviving through one year. This is also the
cost, without interest, to provide one dollar, or unity, at the end of one year, payable
in case of surviving to the end of the year. This is found by dividing the number
living at the next higher age, or one year older, by the number living at the age indi-
cated. Thus for age 40, the probability of surviving through one year is found by divid-
ing 77>34i, the number living at age 41, by 78,106, the number living at age 40, and is
represented by the fraction .990206. This also is the value, without interest, of one
dollar, or unity, payable in case a person now aged 40 is alive at the end of one year.
As it is certain that every individual will be either alive or dead at the end of the
year, the probabilities of dying and of living in one year at age 40 may be represented
as follows :
Probability of dying in one year 009794
Probability of living through one year 990206
Certainty of living or dying in one year i.oooooo
Column (5) gives the cost, in advance, for each age to secure $1000 payable at
the end of the year in case of death within the year, assuming interest at four per cent
per annum. Thus, for age 40, the sum of $9.42 paid in advance is the net cost to
secure $1000 payable at the end of the year provided death should occur within the
year. Similiarly at age 50, the cost to insure $1000 for one year is $13.25. At age
60, $25.67 ; at age 70, $59.61, etc. This cost of insurance for one year is, of course,
independent of the form of policy contract, or of the age at which the policy was
issued, and in general increases each year as a man grows older. These yearly in-
creasing costs of insurance are called natural premiums.
Z. It may be laid down as a fundamental principle that every life insurance company
must collect each year, in some way, either by direct payments, or partly from an
accumulated fund and partly by direct payments, the cost, according to these natural
premiums, to cover the insurance for the year of the net amount at risk on each and
every policy in force, based upon the actual age attained, regardless of the age at
entry, the form of policy contract, or the scale of premium payments, y.
These natural premiums, or cost of insurance for each separate year, constitute
the basis of all sound life insurance. Theoretically, the receipt each year of the
natural premium, or yearly cost of insuring the net amount at risk, based always upon
the actual age attained, will enable any company to meet all its insurance obligations at
maturity, on each and every policy in force. Practically, it is necessary to add, under
any form of policy contract, a margin for necessary expenses, and a further margin
to guard against adverse contingencies, such as epidemics, undue withdrawal of
sound lives, etc. But it cannot be too clearly stated that natural premium payments,
properly loaded, are not only sufficient, but are all-sufficient to meet all the insurance
obligations of any company, no matter what may be the forms of its policy contracts
or the methods of its premium adjustments. In fact, any payment in excess of the
natural premium applied to the net amount at risk and to the actual age attained is
outside of, and independent of, insurance, and should go to expenses, contingent fund,
investment or surplus. The natural premium in any year pays for the entire insurance
during that year, under any and every form of policy contract in any and every com-
pany.
Column (6) gives for each age the level or uniform premiums, to continue un-
changed through the remainder of life, as the consideration for securing $1000 payable
at the end of the year when death occurs. For instance, at age 40 the payment
of $22.35 annually in advance is the net premium at that age to secure $1000, pay-
able at the end of the year when death occurs. These level premiums are the com-
muted equivalents of the natural, or increasing premiums, as shown in column (5).
We will now examine the principles upon which these level premiums are deter-
mined.
^The first step is to ascertain the net single premium or amount to be paid down
in one sum to secure $1000 payable at death, whenever that event shall happen. It
is manifest that this single premium is the sum total of the separate costs of insuring
one dollar, or unity, in each successive year, discounted at the rate of interest assumed
to the present date or age. As we have seen, the net cost without interest at age 40
TABLE No. 2.
e. S
'
U
08
^
ir*
0-5
S *""
a !
s
Sojf
rt
Sf *& M
firs c
-*^ ^3
;!! i
SQ^
3*1
rt EM .
-i
!|S
AGE.
I 3 + o
-> n'-T
J|| B +
!!:'
ft"P~"
llr
"
ji^.g
c
s|,^ * ~
|-=|>
S ^|
s'S'sJ
yQ Q ? Q
S5 03 C
US Cl o ^"' D
c3 &B^ ^
w rt rs o
P C Q ^r
S PH f
i> fLi c ^
Q *S ^ O
V h-i <*H
v P-) > -^.
PH
h
PH
p.
P.
(I)
(2)
(3)
(4)
(5)
(6)
40
.009794
.961538
.0094177
1. 000000 x
1. 000000
I.OOOOO
O
41
.O099IO
924556
.0091620
.990206
.961538
.95212
I
42
.010050
.888996
.0089348
.980296
.924556
.90634
2
43
.010204
854804
.0087225
.970246
. 888996
.86254
3
44
.010396
.821927
.0085449
.960042
.854804
.82065
4
45
.OI060I
790315
.0083781
.949645
.821927
.78054
46
.OIO857
.759918
.0082505
939045
.790315
.74214
6
48
.011139
.011471
.730690
.702587
.0081389
.0080598
.928187
.917049
.759918
.730690
.67008
I
49
.011869
675564
.0080179
905577
.702587
.63625
9
50
.012317
.649581
.0080006
.893709
675564
.60376
10
51
.OI28l6
.624597
.0080048
.881392
.649581
57254
II
52
.013366
.600574
.0080275
.868576
.624597
.54201
12
53
.013968
577475
.0080663
.855212
600574
.51362
13
54
.014634
55526=;
.0081257
.841241
577475
.48580
14
.015351
53398
.0081960
.826606
555265
.45899
15
56
.016132
513373
.0082817
.811257
533908
.43314
16
57
.016964
.493628
.0083740
795125
513373
.40820
17
58
.017848
.474642
.0084712
.778160
.493628
.38412
18
59
.018795
456387
.0085778
.760313
.474642
.36088
19
60
.019794
438834
.0086861
.741518
456387
33842
20
61
.020843
421955
.0087950
.721724
438834
.31672
21
62
.O2I932
.405726
.0088983
.700881
421955
29574
22
63
.023046
.390121
.0089906
.678949
.405726
27547
2 3
64
.024185
375"7
.0090722
655904
.390121
.25588
24
65
66
.025350
.026502
.360689
.346817
.0091435
.0091915
631718
.606367
375II7
.360689
.23697
.21871
3
67
.027629
333477
.0092131
.579866
.346817
.20111
2 7
69
.028762
.O297I6
.320651
.308319
.0092083
.0091620
552237
523519
333477
320651
.18416
.16787
2 9
70
.030612
.296460
.0090753
493803
.308319
15225
30
.031342
285058
0089343
.463191
.296460
-I373 2
31
72
.031841
.274094
.0087275
.431849
.285058
.12310
32
73
.032072
.263552
.0084526
.400008
.274094
.10964
33
74
.032021
.253415
.0081145
.367936
.263552
.09697
34
75
.O3I7OI
.243669
.0077244
335915
253415
.08513
35
76
.O3II24
234297
.0072923
303515
.243669
.07413
36
77
.030331
225285
.0068330
.273090
.234297
.06398
37
78
.02 9 332
.216621
.0063539
.242760
.225285
.05469
38
79
.028116
.208289
.0058562
.213428
.216621
.04623
39
80
.026771
.200278
.0053617
.185312
.208289
.03860
40
81
025X45 |
.192575
.0048424
158541
.200278
.03175
82
.023250
.185168
.0043052
133396
.172575
.02569
42
83
.021100
.178046
.0037567
.110145
.185168
.02040
43
84
.018821
.171198
.0032221
.089046
.178046
.01585
44
85
.016542
.164614
.0027230
.070225
.171198
.OI2O2
45
8b -
.014263
.158283
.0022575
.053684
.164614
.00884
46
87
.011946 i
.152295
.0018180
.039421
.158283
.00624
88
.009526 i
.146341
.0013940
.027476
152295
.00418
48
89
.007106 ;
.140713
.0009999
.017950
.146341
.00263
49
90
.004929
I3530I
.0006669
.010844
.140713
00153
50
QI
.003150
.130097
.0004097
.005915
I3530I
.OOOSO
51
92
.001754
.125093
.0002194
.002765
.130097
.00036
52
93
.000743
.120282
.0000893
.001011
.125093
.00013
53
94
.000230
.115656
.0000267
.000269
.120282
.00003
54
95
.000038
.112207
.0000043
.000038
.115656
II22O7
.00000
55
. j. w^sy
Totals
"367^747
1644311
J <J /O/4/
i
'
to secure $i, payable at the end of one year in case of death during the first year,
is .009794. To find its net present value, paid down, we must discount this cost for
one year at the rate of interest assumed. The present value of one dollar, payable
certain at the end of one year, at four per cent interest, is .961538. The net present
value of one dollar, or unity, payable at the end of one year in case of death, on the
basis of the American Table four per cent interest is for age 40 years .009794 X
.961538 =.0094177. [See columns (i), (2), and (3), Table No. 2.] In the same way
the net present value of one dollar, or unity, payable at the end of two years, provided
a person now aged 40 should die in the second year, or between ages 41 and 42, is
found by dividing 774, the number dying, by 78,106, the number living at age 40,
and discounting the quotient for two years. Thus ^loe =.009910; this multiplied by
924 556 =.0091620, and this is the cost at age 40 to secure one dollar, or unity, payable
at the end of two years in case of death during the second year. Again, the net pres-
ent value of one dollar, payable in case a man now aged 40 years should he die in the
eleventh year, or between ages 50 and 51, is .0080006. These separate values are
shown in column No. 3 in Table No. 2. Their sum total is .3675747, and this is the
net single premium paid down to secure one dollar, or unity, payable at the end of the
year, when a person now aged 40 years dies, whenever that event shall happen.
By a similar course of reasoning the net present value of one dollar, or unity,
payable annually in advance during the remainder of life at any age, is the sum total
of the present values of the separate chances of surviving during each successive year,
discounted to the present date or age. Thus for age 40 the present value of one dollar
in advance is unity or one dollar. The present value, without interest, of one dollar,
payable in one year, or at age 41, is, as we have seen, .990206. This multiplied by
.961538, the discount, gives .95212 as the present value of one dollar, payable at the
end of one year, or at age 41, provided a person now aged 40 be then alive. The
present value of one dollar, payable in ten years, or at age 50, provided a person now
aged 40 be then alive, is $J?{jJ =.893 7 09 multiplied by .675564 = .60376. These suc-
cessive net present values are found in column (6). Their sum total is 16.44311, and
this is the present value of one dollar per annum in advance during the lifetime of a
person now aged 40 years upon the basis adopted.
As already shown, the net single premium at age 40 to secure one dollar, or
unity, payable at the end of the year when death occurs, is .3675747. Proportionally,
a net single premium of $16.43311 would secure $44.7341 payable at death. But
$16.44311 is also the net present value at age 40 of an annual premium of one dollar.
Therefore, a net level or uniform premium of $22.3543 would, at age 40, secure
$1000 payable at death. [See column (6), Table No. i.]
Let us now suppose a company to consist of 78,106 persons, each aged 40 years,
each insured for $1000, or $78,106,000 in all, and each paying the net annual pre-
mium of $22.3543. The following table No. 3 has been prepared to show the progress
of the fund each year until the last death claim has been paid at the age of 96 years,
on the basis of the American Experience Table and four per cent interest. Column
6
TABLE No. 3.
78,106 PERSONS. AGED 40 YEARS, INSURED FOR $1,000 EACH.
Share of
Each Per-
AGE.
X
Premiums.
Fund at
Beginning of
Year.
Interest 4%.
Death Claims.
Fund at End of
Year.
son in the
Fund at
End of
Year or Net
Reserve.
(l)
(2)
<3)
. (4)
(5)
(6)
4
$l, 746,030
$1,746,030
$69,840
$765,000
$1,050,870
13-59
41
7,728,930
2,779,800
111,190
774.ooo
2,116,990
27.65
42
1.711,630
3,828,620
153-140
785,000
3,196,760
42 18
43
1,694,080
4,890,840
195,630
797.000
4,289,470
57-20
41
1,676,260
5,965.732
238,630
812,000
5,392,360
7270
4^
1,658,110
7,050,470
282,020
828,000
6,504,490
88.68
46
1,639,600
8,144,090
325,760
848,000
7,621,850
105 13
47
I 620,640
9,242,490
369,700
870,000
8,742,190
122.05
48
1,601,190
10,343,380
413,740
896,000
9.861.120
139.42
49
1,581,170
11,442,290
457.690
927,000
10 972,980
157.19
5
I 560,440
12,533,420
501,340
962,000
12,072,760
175-37
5i
1,538,940
13,611,700
544.470
i 001,000
I3.i55.i70
I93-9I
52
1,516,560
14,671,730
586,870
i 044 ooo
. 14,214,600
212.80
53
1,493,220
15,707,820
628,310
1,091,000
15,245,130
232.02
54
1,468,830
16,713,960
668560
1,143.000
16,239,520
25I-53
55
I 443,290
17,682,810
707,310
1,199000
17,191,120
271.30
5^
. 1,416,480
18,607,600
744,300
I 260,000
18,091,900
291.31
57
1,388 310
19,480,210
779 210
1,325,000
18.934,420
3II-52
58
1,358,680
20,293,100
811 720
1,394,000
19.710,820
33I-9I
59
1,327,520
21,038,340
841 530
1,408,000
20 411,870
352-43
60
1,204,710
21,706,580
868,260
i 546,000
21,028 840
373-04
61
1,260,150
22,288,990
891,560
1,628,000
21,552,550
393-70
62
1,223,750
22,776,300
911.050
1.713,000
21.974,350
4 J 4-37
63
1,185,450
23,159,800
926,390
1,800,000
22,286,190
435-01
64
I 145,210
23,431,400
937,260
1,889
22,479,660
455-59
6^
1,102,480
23,582,140
943,280
1,980,000
22,545,420
476.03
66
1,058.720
23,604.140
944.i6o
2,070.000
22,478,300
496.31
67
1,012,450
23-490-750
939.630
2,158,000
22,272,380
516.36
68
964,210
23,236,590
929,460
2,243,000
21,923.050
53 6 i5
69
914,070
22,837,120
913,490
2 321,000
21,429,610
555-62
70
862,180
22,291,790
891 670
2,391,000
20,792,460
574-73
7i
808,740
2I,6oi,22O
864,050
2,448,000
20,017,270
593-45
72
754-01
20,771,280
830,850
2,487,000
19.115,130
611 82
73
698,420
I 9.8i3,550
792,540
2,505,000
18,101,090
629.86
74
642,420
18,743,510
749-740
2,501,000
16,992.250
64764
75
586,510
17,578,760
703,150
2,476.000
15,805,910
665.^0
76
53i,i70
16,337,100
653-480
2,431,000
14.559-580
68258
77
476,830
15,036,410
601,460
2,369,000
I3,2b8,870
699.79
78
423,870
13,692,740
547.710
2,291,000
11,949,450
716.82
79
372,650
12,322,110
492,880
2,196,000
10,618,980
733-65
80
3 2 356o
10,942,540
437-700
2,091,000
9,289,240
75097
81
276,820
9,566.060
382,640
i 964 ooo
7,984,700
766.36
82
232 910
8,217,610
328,700
I 816,000
6 730,310
78232
83
192,320
6,922,630
276,900
1,648,000
5 55L530
798.20
84
I55,48o
5,707,010
228.280
1.470,000
4,465,290
814 10
8=5
122,620
4,587,910
183520
1,292,000
3,479.430
829 82
86
93.74
3.573.170
142,930
1.114,000
2,602,100
84479
87
68,630
2,670,730
106,830
933-000
1,844 560
859-54
88
47-980
1,892,540
75-900
744,000
1,224 240
87321
89
3L340
1,255,580
50,220
555.ooo
750,800
886.42
90
18,940
769,740
30,790
385,000
415.530
899.42
9 1
10,330
425,860
17830
246,000
196,890
9"-53
92
4,830
201,720
8.070
137000
72,790
921.39
93
1,770
74.56o
2.980
58,000
19.540
93049
9^
470
26,010
800
18,000
2,810
936.67
QC
7
2,880
1 20
o OOO
IOOO.OO
yj
J, V -* JV
(i) shows the total premiums paid by those alive at the beginning of each successive
year. Column (2) shows the fund at the beginning of each year just after the pre-
miums have been paid. Column (3) shows the interest on the fund each year. Column
(4) shows the death claims in each year. Column (5) shows the fund at the end of each
successive year. Column (6) shows the share held for account of each survivor in each
successive year (found by dividing the total fund by the number of persons surviving),
and this is also the net investment reserve upon each policy.
The functions of the investment reserve will be made clearly apparent by a study
of Table No. 4, which has been prepared to illustrate the appropriation each year of the
component parts of an ordinary whole life level premium of $313, paid annually in ad-
vance, to secure $10,000 at the death of a man now aged 40 years (or, rather, at the
end of the year when death occurs). Column (i) shows the net reserve at the end of
each successive year. Column (2) shows the corresponding net amount at risk borne
by the company during each successive year. This is always the difference between
the face of the policy and the net reserve, which last, being in hand, is not subject to
any insurance risks. Column (3) shows the net cost to insure $10,000 during each
separate year by the scale of natural premiums, as indicated in column (5), Table i.
Column (4) shows the cost to insure the net amounts at risk at the successive ages in-
dicated in the margins. Column (5) shows the deposit portion of the annual premium
in each year, which, until the age of 68 is attained in the example given, goes to swell
the investment reserve or accumulated deposit. After the age of 68 the yearly costs to
insure the net amount at risk exceed the entire net premiums, and hence the deficien-
cies (as indicated by the minus sign) must be supplied by drawing from the reserve
fund.
From the foregoing it will be apparent :
(i.) Every level premium policy is in reality a contract for a yearly decreasing
amount of insurance, and a yearly increasing amount of investment. It is a combina-
tion of insurance, which is one thing, with investment, which is quite another thing.
There is no necessary connection between the two. Insurance or indemnity may be
purchased without investment, as investment may be purchased without insurance.
The investment element does not add to the security of the insurance, the yearly cost
of which depends, under any and every form of policy, upon the net amount at risk
borne by the company, and the actual, present, attained age of the person whose life
is exposed to mortality. For instance, in the example given (Table No. 4) of a whole
life insurance policy of $10,000, issued at the age of 40, the reserve or invested de-
posits, at the end of twenty years, or at age 60, is $3,730.35. Now, this sum is in
hand, and is not subject to any insurance hazard, hence the net amount at risk for that
year is $6,269.65 only. The cost to insure $10,000 for one year at age 60, as shown
in column (3), is $256.67. Proportionately the cost to insure $6,269.65, the net
amount at risk, is $160.92, and this is all the insurance done by the company with re-
spect to that policy during that year. At age 70 the net amount at risk is only
$4,254.74, the cost of which for that year, $253.50, is $29.96 more than the net an-
8
TABLE No. 4.
WHOLE LIFE INSURANCE BY LEVEL OR UNIFORM PREMIUMS. AGE AT ISSUE 40 YEARS
AMOUNT INSURED $10,000. ANNUAL PREMIUM DURING LIFE, $313.
AGE.
Net Reserve or Accu-
mulated De p o s i ts,
being ^//"-Insurance
at End of Year.
Net Amount of Insur-
ance Carried by the
Company During the
Year.
Tabular Cost to Insure
$10,000 During Each
Year. Am. Exp.
Table 4 per cent.
Ditt >, to Insure the Net
Amount at risk Each
Year, being also the
Full Insurance Re-
serve each Y-ar.
Deposit Portion of j
each Premium which
is merely for Accu
mulation.
o v
,2
|l
|
llf
3 l-ti
4O
$I3S 83
(2)
$9 864 12
$q4 1 8
, (4)
$Q2 QO
(6)
S8o 46
(7)
41
276 40
723 CT
q6 23
nq cq
80 46
313 OO
42
421 83
q 578 17
q8.C.8
q4 42
8q 46
313 OO
43
572 04
q,427 q6
101.13
oc, qi
128 20
8q 46
qiq oo
44
726.98
q,273 02
104.12
q6 SS
126 qq
8q 46
qiq OO
4S ....
88682
9,113 08
107 34
q7 82
I2C 72
80 46
qiq oo
46
1,051.31
8,948 69
III.I7
00 48
124 06
80 46
qiq OO
47
1,220.50
8,779.50
115 39
101.31
122 23
80 46
313 oo
48. .
1,394.15
8,605.85
120.28
103.51
1 20 03
8q 46
313 oo
40
1,571.94
8,428.06
126.02
106 21
117 33
8q 46
313 oo
I 7Sq 66
8,246 34
132 51
ICQ 27
114 27
8 Q 46
313 OO
CJ
I q3Q o3
8,060 92
I3q 8l
H2 70
1 IO 84
8O 46
313 OO
2,127 OQ
7 872 01
147.07
116 48
107 06
80 46
qiq oo
CO
2,320 16
157 05
120 61
IO2 q3
80 46
qiq oo
CA
2,51s 2S
7,484 75
167 27
125 20
O8 34
80 46
qiq oo
ecr
2,713 O2
7,286.98
178 57
130 12
93 42
80 46
qiq oo
si. '.'..
2,913.10
7,086.90
191.20
I35-5O
89 46
qtq OO
cy . . ...
3 115.22
6 884.78
205.15
141.24
8 30
8946
3I3.OO
58
3 3IQ OQ
6 680 91
220 03
147 OO
80 46
qiq oo
Co
3,524.25
6,475.75
237-69
IS3-93
6q6i
89.46
3I3.OO
60...
3,730.35
6,269.65
256.67
' 160 92
6262
89.46
313 oo
61
3,036 os
6,063.05
277.69
160 3S
S4 iq
8q 46
qiq oo
62
4,143 66
5,856.34
300 88
176 20
47 34
8q 46
qiq oo
63
4,350.12
5,649 88
318.95
1 80 20
43 34
8q 46
qiq oo
64"
A 555 86
354.54
103. OI
3O S3
8q 46
qiq oo
4,760 33
5,239.67
385.85
202 1 8
21 36
89 46
qiq CO
66
4,963 07
5,036.93
420,26
2ii 68
11.86
8q46
qiq oo
67
5,163 64
4.836.36
458.15
221.^8
I. q6
8q 46
313 oo
68
5,361.46
4,638.54
500.02
231.04
8.4O
89.46
313 oo
60...
5,556.16
4,443.84
545.79
242 53
iS.QO
80.46
313 oo
7O
5,747 26
4,2^2.74
596.08
2S3 SO
2Q 06
80 46
313 OO
71
4,065 46
650 63
264 61
41 O7
8q 46
qiq oo
72 .... .....
6,118 19
3.881.81
708 97
27S 23
CT 60
8q 46
qiq OO
70
6,298.64
3,701.36
770 94
285.35
61 81
8q 46
qiq oo
74. . ....
6,476.42
3,523 58
836.80
2Q4 85
71 31
8q 46
313 oo
7cr
6,652.02
3,347.98
907.41
3O3.8O
80 26
8q 46
313 oo
76 :.::;
6.825.83
3,174 17
983 76
312.26
88 72
89.46
3I3.OO
77
6 QO7 Q3
3,002.07
1,067 03
32O 60
Q7 06
8q 46
313 OO
78
7,168.17
2,831.83
1,161.80
329.OO
y/ >v ~
105.46
89.46
313.00
70
7,336.51
2,663.49
1,266.67
337 22
113 68
80 46
qiq oo
80
7,509.70
2,490 30
1,389 10
34 C, qo
122 36
8q 46
qiq OO
7,663 60
2.336.40
1,525.04
356 31
132 77
8q 46
313 oo
82
7,823.20
2,176.80
1,675.93
364 83
14! 29
89 46
313 oo
7,982 oo
2,018.00
1,841.93
371 70
140 l6
89.46
3I3.OO
84 .
8,141 oo
1,859
2,032.30
377 8 1
154.27
89.46
313.00
85
8 298,20
1,701.80
2,264 9 2
385.44
161 90
89.46
313.00
86
8,447.90
1,552.40
2,554,62
396.57
173.03
89.46
313,00
8,595.40
1,404.60
2,913.66
4Oq 26
i8s 72
80 46
qiq oo
88
8,732.10
1,267.90
3,335.57
422 61
IQQ O7
8q 46
3I3.OO
80...
8,864.20
1,135.80
3,806.38
432 32
208 78
8q 46
313.00
QO . .
8,994.20
1,005 80
4,370 63
430 60
216.06
8946
313.00
91
9,115.30
884.70
5,11988
452.96
229.42
89.46
313.00
Q2
0,213.00
786.10
6,098.68
2SS 88
8q 46
qiq oo
03
0,304.00
695,10
7,059.40
4QO 6q
267 15
8q 46
qiq oo
04 . ....
9,366,70
633.30
8,241.76
208.42
89 46
313 oo
QC
10,000.00
9,615.40
3
8q.46
313.00
nual premium ($223.54). The deficiency for that year, as well as the deficiencies for
each subsequent year, as shown in column (5), must be met by drawing on the in-
vestment reserve, or accumulated fund, the express functions of which is to provide for
the excessive cost of insurance in old age when the level premium is insufficient for
that purpose.
(2). The investment reserve is occasioned solely by the artificial condition in the
level premium contract, which provides that the premiums shall not increase as the
insured grows older, and to enable the company to pay the sum insured as an endow-
ment.
(3). Whether the combination of insurance and investment is desirable or advan-
tageous, depends upon the manner in which each is administered. If either the in-
surance or the investment can be obtained on better terms separately, the combina-
tion of the two is certainly undesirable and disadvantageous to the policyholder.
Instead of contracting with a life insurance company for both insurance and in-
vestment, which together make up the sum insured, two separate contracts might be
made the one with a life company for the yearly decreasing amounts of insurance
only, see column (2) table 4, the other with a savings bank or trust company for
accumulating the deposit, or investment portions of the yearly premium, see column (5)
of the same table. In case of death in such case the insurance company would pay
the net amount insured only, column (2), while the savings bank would pay the ac-
cumulated deposits, column (i), the two together making up the full amount guar-
anteed.
To show even more clearly how the insurance and investment elements may be
completely separated the following tables have been prepared.
Table No. 5 illustrates the case of an endowment assurance issued at age of forty
years for $10,000 payable in ten years or at death if prior. The net premium only
($853.62) is considered the margin for expenses and adverse contingencies being dis-
regarded.
Tables 6 and 7 are intended to show how the same result can be secured by
purchasing a ten-year term insurance with the insurance company, annual premium
$106.03, an d a pure endowment (payable only in case of survival) by depositing the
residue ($747.59) of the endowment assurance premium for accumulation. In case of
death at any time during the ten years, the insurance company would pay the full
amount insured, and the endowment fund would be lost. In case of surviving, the
$10,000 would be paid as an endowment, and the insurance would cease.
The same principles apply to any other term of years, as a whole life policy is in
reality an endowment assurance payable on attaining the age of ninety-six years, or
at death if prior.
10
Comparison of an endowment assurance contract, a ten year term level premium
contract, and a pure endowment contract. Amount $10,000, and age at issue 40
years, in each case :
TABLE No, 5.
ENDOWMENT ASSURANCE, ANNUAL PREMIUM $853.62.
YEAR.
Net Reserve
or Accumulated
Deposits Being
Self-Insurance.
Net Amount of
Insurance at
Risk or Carried
by the Company.
Tabular Cost
Each Year
to Insure
$10,000 for
the Year.
Tabular Cost to
Insure Net
Amount at Risk
which is also
the Full Legal
and Mathemat-
ical Insurance
Reserve.
Deposit
Portion of
Annual
Premium
Which is
Merely for
Accumu-
lation.
I ,
$797.63
$9,202 37
$94 1 8
$86.67
$766 95
2
1,633 57
8,36643
96 23
80.51
773-H
3..
2,509 89
7,490.11
98.58
73-84
779.78
o 428.0^
6,571.05
101.13
6645
787 17
4 . QQQ.I6
5,606.84
104.12
58.38
795-24
< t 4<X.q6
4,494 64
107.34
48.24
805.38
6,468.51
3,531.49
111.17
39.26
814.36
g
7,586.05
2,413.95
115.39
27.85
825.77
8,761.76
1,238.24
120.28
14.89
838.73
10,000.00
Nil.
126.02
Nil!
853.62
TABLE No. 6.
TEN- YEAR TERM INSURANCE, NET ANNUAL PREMIUM $106.03.
YEAR.
Net Reserve
or Accumulated
Deposits Being
Self- Insurance.
Net Amount of
Insurance at
Risk or Carried
by the Company.
Tabular Cost
Each Year
to Insure
$10,000 for
the Year.
Tabular Cost to
Insure Net
Amount at Risk
which is also
the Full Legal
and Mathemat-
tical Insurance
Reserve.
Deposit
Portion of
Annual
Premium
Which is
Merely for
Accumu-
lation.
$12 45
$9,987.55
$94.18
$94.06
$tl 97
2
9,976.63
96.23
96.00
10.03
32.37
9,967.63
9858
98.26
7-77
H
9,960.82
IOI.I3
100.73
5.30
43.2O
9,956.80
104.12
103. 57
2.37
6
44.05
9,955.95
107.34
106.87
0.84
40.95
9,959.05
III.I7
110.72
g
33.24
9,906.76
115.39
115.01
3 gg
19.99
9,980 oi
120.28
120.04
I4.OI
10,000.00
I26.O2
126 02
iq.qq
TABLE No. 7.
PURE ENDOWMENT AGE 40 AT ISS T TE $10,000 PAYABLE ONLY IN CASE OF BEING ALIVE
AT THE END OF 10 YEARS, OR AT AGE 50.
YEAR.
Yearly
Payments.
Value (With-
out Interest) of
$t.oo Pay able
Only in Case
of Surviving to
End of Year.
Fund at
Beginning of
Year.
Vame of Ditto
Payable
Only in Case
of Survivirg.
Interest 4%.
Fund at End
of Year.
I.....
$74.7 5Q
$1 OOQ 8q
$747 50
$754 OQ
$30 2O
$78 c; in
2 .
747 5Q
I OIO II
I 532.78
I 548 28
6l Q3
I 6lO 21
74.7 ZQ
I OIO 36
4
747. 5Q
1,010.63
Q.225.IO
q 250 38
iqp Q7
q q8q 75
747 t;o
I OIO Q^
4182 61
167 .I
6
747 C;Q
i on 29
C. 007 C;Q
,iU-S. Uj
5jre 08
206 20
c q6i 28
7
747 5Q
i on 70
6 108 87
6 180 34
247 21
^'J
6 427 zz
8
747 5Q
I OI2 15
7 175 14
7 262 32
2QO 4Q
7 552 8l
747 5Q
I,OI2.67
8 300 40
8 405 c;6
336 22
8 741 78
jO
747 ^Q
I 013 28
Q Qjir OQ
11
Insurance and investment therefore have no necessary connection either one
may be obtained without the other.
(4). Pure insurance, unmixed with banking or investment, involves the payment
of natural premiums, which inevitably and inexorably increase with age. The only
way to avoid these increasing rates is to pay largely in excess of the requirements for
current death claims in the earlier years, and thus provide a fund upon which to draw
in the later years that is to say, by combining investment with insurance. The first
is known as the natural premium plan, the second as the level premium plan. Prop-
erly administered, the one is as safe and as sound as the other, as both depend upon
the application of the same laws of nature which govern the rates of mortality, or the
probability of living and dying in each successive year of life. In fact, as before
stated, level premiums are simply the commuted equivalents of the increasing or
natural premiums. In both systems, the company must alike be furnished with the
cost of insuring the net amount at risk at the actual age attained on each and every
policy in force. This cost is independent of the form of policy contract, the age at
issue, or the scale of premium charged. This cost, as previously stated, may be fur-
nished either by direct, present payments, as by natural premiums, or partly by direct
present payments, and partly by drawing upon the investment reserve or accumulated
deposits, a fund contributed by the policyholder for this express purpose.
There are only two sound systems of life insurance; the one by natural premiums,
increasing each year as a man grows older ; the other, by level premiums, which
necessitate investments or accumulated payments largely in excess during the
earlier years to meet the deficiencies of the uniform, unchanging premiums in later
years. The attempts by so many co-operative or assessment companies to furnish in-
surance by assessments based upon the age at entry, and which rates do not increase
with age must inevitably result in disappointment and disaster, Natural laws may not
be violated with impunity.
SHEPPARD ROMANS.
NEW YORK, May 10, 1888.
12
THIS BOOK IS DUE ON THE LAST DATE
STAMPED BELOW
2
MOV 6
"EP 2 9 i960
30m-6,'14
05532
<o
-Mi
LIBRARY
i UNITE:
CALIFORNIA.
w^c.^ ^^r
E3H
^
! ^m