Further Note on the Scottish Bankers' Mortality and Marriage Experience, 1903-1923, with special reference to the use of "Equivalent Ages"

Institute and Faculty of Actuaries

Further Note on the Scottish Bankers' Mortality and Marriage Experience, 1903-1923, withspecial reference to the use of "Equivalent Ages"Author(s): Alexander FraserSource: Transactions of the Faculty of Actuaries, Vol. 10, No. 98 (1924-1925), pp. 351-365Published by: Cambridge University Press on behalf of the Institute and Faculty of ActuariesStable URL: http://www.jstor.org/stable/41218054 .

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No. 98.] 351

TRANSACTIONS OF THE

FACULTY OF ACTUARIES

.Further Note on the Scottish Bankers' Mortality and Marriage Experience, 1903-1923, with special reference to the use of "Equivalent Ages.1'

By Alexander Fraser, F.F.A., F.I.A., F.R.S.E.

IN my recent paper on the "Scottish Bankers' Mortality and Marriage Experience, 1903-1 923," when referring to the use of equivalent ages in obtaining the values of reversionary annuities as being a disadvantage of the Reversionary Annuity method of valuing Widows' Funds (see page 243), I stated that the particular table used influenced the results appreciably, and that there was room for an inquiry on the subject. In the discussion Mr. A. E. King suggested that the approximation probably gave an error of from 3 to 5 per cent., this estimate having been based on a series of values at representative ages, and in my reply I stated that I might have an opportunity of making a more extensive test, and also of examining the question from a general point of view. I have accordingly looked into the point, and as the results seem to be interesting, and the deviations in some cases surprising in extent, a note of my investigation is now given.

The first step was to ascertain whether any general principle could be found underlying the use of equivalent ages, and a very simple demonstration may be given, which although perhaps not of universal application, yet clearly shows that in certain circumstances the approximation is bound to give reserves consistently in excess, or in defect, of the true figures.

In the expression {ay-axy) assume that the mortality of the life y follows a standard table, for which joint annuities are tabulated, but that the mortality of the life x does not follow that table, and that an equivalent age is to be used, say z. Such an

vol. x. 2 F



352 Further Note on the Scottish Bankers'

equivalent age can always be found so that a'z=aXi i.e., so that ^vttpřz = ̂ tPx- But it does not follow that ̂tftp' zy^^tPxy ', as a rule the equation will not hold, and the same applies if equivalent ages are used for both lives. The two graphs which have been drawn out enable one to follow easily the effect of the inequality in the examples selected. The actual mortality assumed is that of the Om table at ages 30 and 50, and equivalent ages are taken both from the Carlisle table and from the Scottish Bankers' table for Wives, to the nearest integral age for convenience. There are two sets of curves on each graph ; the values of qx+n are plotted for each year of the annuity (the vertical scale being enlarged ten times for the earlier years), and the corresponding values of vnnpx at 3 per cent, interest.

For age 30 (actual) the values of qx by the Carlisle table (equivalent age 27) are too high for the first 19 years, and thereafter too low ; the values of vnnpx are correspondingly too low for about 30 years, and thereafter too high, the later excess balancing the initial defect, as by hypothesis the areas under the two curves are (approximately) equal. But when a second life is brought in this equality no longer holds ; greater weight is given to the errors in the earlier years when the values are multiplied by the corresponding values of npy, as these are larger than in the later years, the value of the joint life annuity is too small, and of the reversionary annuity too great. (The effect is particularly obvious if in this case the age of the second life is within 30 years of the limiting age of the table.) Taking the second life as also age 30, and subject also to Om mortality, the true joint annuity value is 16 933, and the reversionary annuity 3 331. Taking the exact equivalent Carlisle age of 26*77, the joint annuity is 16*493, and the reversionary annuity 3*771, or 13*2 per cent, too high.

For age 50 (actual) the values of qx by the Carlisle table (equivalent age 51) cross and recross the true Om values several times in the earlier years, and the net effect is that the two eurves of vnnpx are very close, so that one would expect the approximation to be good. Again taking the two ages to be equal, the true joint annuity by the Om table is 10*700, and the reversionary annuity 3*354. Taking the equivalent Carlisle age of 50*67, the joint annuity is 10*699, and the reversionary annuity 3*355, so that the result is practically exact.

Dealing now with ages by the Scottish Bankers' table equivalent to the Om actual ages, in the graph for age 30 the values of qx by the Bankers' table for Wives (equivalent age 40) are considerably too low until after 22 years, and thereafter too high;



Mortality and Marriage Experience, 1903-1923 353 the values of vnnpx are too high for 30 years, and thereafter too low. Taking the exact equivalent age (4026) from the table for Wives for the annuitant, and that from the Married Mens* tablé (35*28) for the counter-life, the joint annuity is 17*491, and the reversionary annuity 2*773, or 16*75 per cent, too low.

Taking actual age 50, the values of qx by the Bankers' table for Wives (equivalent age 57) are much too low until the tenth year, and thereafter too high, except for a few ages round the thirtieth year ; while those of v'px are too high until the seven- teenth year, and thereafter too low. The exact equivalent age from the table for Wives is 57*03, and that from the Married Mens' table 52*15, the joint annuity is 10*908, and the reversionary annuity 3*146, or 6*20 per cent, too low.

Aseries of representative values is now given, using the true annuity values by the Scottish Bankers' tables, and comparing them with the values found by using the exact Carlisle equivalent ages.

Values of (ay - axy) Scottish Bankers' 3 per cent.

age x-y = 0. x-y = 6. x-y='2. OF ._ p

Hus- band. TV,,« Carlisle Excess m „_ Carlisle Excess т,,.,- Carlisle Excess TV,,« mie* Bquivt. yQ

m lnie* „_ Equivt. °/

т,,.,- llue* Equivt. °/

36 3*500 4*569 30 5 4*424 5*423 ! 22 6 44 3*982 4*693 179 5*213 6050 16*1 6*441 7261 12*7 63 4*185 4*505 7*6 5*639 6 122 8*6 7 207 7*817 8*5 62 3*981 4188 5*2 5*711 5*949 42 7*553 7*912 4*8 71 3*483 3*489 *2 5146 5306 3*1 7*241 7*405 2*3

Except at the oldest ages, the differences are very large. The high percentage errors at the young ages are of course due to the joint annuity values being large compared with the reversionary annuity values ; for equal ages 35, the approximate joint annuity is only 5*8 per cent, less than the true value of 18384, while the corresponding reversionary annuity is 30*5 per cent, too great.

The position may perhaps be put in general, that if the equivalent ages are throughout very close to the actual, or if the curve of mortality is throughout of the same shape although the points on it correspond to different ages in the two tables (that is, if the rating up or down is by approximately a constant number of years), there is probably no great error introduced by using equivalent ages, although it may be safer to make a few tests by approximate summations. But when the rate of mortality is




entirely different, especially where as in most modern tables it is much lighter at the younger ages than by the older standard tables, the results may be far out, and in the latter case the use of equivalent ages must very considerably overstate the liability at the young ages.

In all the circumstances I have thought it desirable to calculate the true values of joint life annuities by the Scottish Bankers' table for Husbands and Wives, the values being tabulated for triennial combinations of ages as is done in the new Office Annui- tants' Experience. The values were calculated accurately for differences between ages of husband and wife for multiples of 6, ranging from -18 to +48, and tabulated for each third pair of ages, and thereafter the values for the odd multiples of 3 were interpolated by second differences. These values are given for the three rates of interest, 3, 3J, and 4 per cent. Tables XII. to XVII. of the original paper contain values arrived at by the use of equivalent ages ; I have recalculated the figures for Table XIL, the annuities to the widows by the first marriage of bachelors, as these are important, and for Table XVII., the annuities to widows from the remarriage of widowers, as these required comparatively little extra work. In both cases the new values are considerably smaller than the original, especially at the younger ages. I have not recalculated the values of annuities to the widows by remarriage of men at present married, or by the second marriages of bachelors; the calculations are very lengthy, and the liabilities in any valuation form a very small proportion of the whole.

To show the effect of the changes, I have recalculated the items of the specimen valuation affected by them, and the following table gives the amended liabilities, arranged as before by decen- nial groups of ages. The figures give the total liabilities except for the annuities to existing widows.

"s* Ж' trence. ***> %•

-19 15,913 19,186 3,273 120*6 20-29 40,630 49,177 8,547 12Г0 30-39 50,056 60,724 10,668 121*3 40-49 38,449 43,795 5,346 113 9 60-69 56,313 60,966 4,653 108*3 60-69 38,160 39,801 1,641 104*3 70-79 14,151 14,371 220 10Г6 80- 901 910 9 101*0

Total 254,573 288,930 34,357 П3*5 !



Mortality and Marriage Experience, 1903-1923 355 The difference is accordingly far greater than permissible in a

valuation, and this is a case where the use of equivalent ages, at least by the Carlisle or a similar table, is not suitable.

A comparison is now made with the valuation by the Collective method, the figures being given in the following statement :-

AGES- A Reversionary Collective Diffprpupp Diflerence. Ratio/. Ratio °/ AGES- A Annuity Method. Method.

Diffprpupp Diflerence. Ratio/. Ratio °/

-19 15,913 17,087 1,174 107'4 20-29 40,630 43,069 2,439 1064 30-39 50,056 52,533 2,477 1049 40-49 38,449 40,923 2,474 106 4 60-69 56,313 60,991 4,678 108*3 60-69 38,160 40,730 2,570 1067 70-79 14,151 14,452 301 102*1 80- 901 1,612 711 1789

Total . 254,573 271,397 16,824 1066

In this revaluation the reserves by the Collective method come out in all groups in excess of the other, the average excess being 6*6 per cent. The tendency to a smaller excess is shown at the early middle ages, though not to such a marked extent as in the original valuation (see page 242).

The Collective method figures at the early ages are now greater, instead of less, than those by the Reversionary Annuity method, the difference being 7*4 per cent, for ages up to 20. In the original paper the difference was attributed to want of satisfactory data at the younger ages, and to the various graduations at these ages being somewhat empirical. On reconsideration I am inclined to doubt whether that was the true explanation, particularly as a substantial modification in the graduation of the earlier values of fx had very little effect on the corresponding values of Wx.

It seems more probable that the difference is partly due to the marked decrease in the differences between the ages of husbands and wives, to which attention was drawn at page 229 ; the Collec- tive method (based on deaths) took into account to a considerable extent marriages dating from a period when the difference was greater, and where the liability on the death of a husband would tend also to be greater. At the same time an absolute corre- spondence between the two results could perhaps hardly be expected, especially where the experience of several Banks com- bined has been applied to value the Fund of only one of them.

In order to test further the effect of using equivalent ages with



356 Scottish Bankers9 Expérience, 1903-1923 other tables, I have used these in valuing another fund. The valuation was made by the Om table for males, and by the Oaf ultimate for females, at 4 per cent, interest. There were over 500 married members, whose average age was 52, and the annuity was £50. Joint annuities were calculated for triennial differences of age, as with the Scottish Bankers' Experience. Using the true annuity values, the liability for the reversionary annuities was «£108,126, and using equivalent ages by the Carlisle table the liability was «£109,144, or only £1018 too much. The percentage excess is under 1 per cent, on the liability for the reversionary annuities, or about half of that on the total reserve, which may almost be regarded as negligible. . The Carlisle equivalent to the Oaf age runs steadily at from 3 to 4 years lower up to age 70, which probably accounts for the small difference in this case. Owing to special .circumstances there were comparatively few young married members in this particular fund, or the difference would have been somewhat greater.



Шс 1 л Diagram I *W

Ж

Ç*+» "NÛ уП n/>oc *r

%.. 0м ос -30 ''' Carlisle ас = 27

ûn V'' SB. W/VES X = 4O 'OU

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1 - n - i 1 1 1 'ioo ! n i I 1 1 I I 1 . ! ! Diagram II.

U . <frX + n*»° Vnn/>XAT3% 9Q

| 0M JC= 50 i Carl/sle jc= 5/ ! SBIV/VES ОС- 40

¡I f l/èrí/ca/ sea /e en/arged/O ¿//nes ' I /br q,jc+nafeQr//era<;esj

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SO SS 60 65 70 75



TABLES



358 Further Note on the Scottish Bankers' Table ХШ.

Value of Annuity to Widow by First Marriage of Bachelor.

Ao«. 3% Ц% 4% AoE. 3% Ц% 4%

15 1702 1-316 1023 60 526 469 419 16 1-768 Г365 1066 61 470 420 376 17 1*815 1-417 1112 62 -416 373 336 18 1*874 1*470 1159 68 *365 '328 '296 19 1*935 Г525 1*209 64 '316 285 257

ao 1*998 1*583 1*260 66 *269 *243 '220 21 2*063 1*642 1*313 66 *22б *204 186 22 2128 1*702 1*368 67 185 168 153 28 2*195 1*763 1*423 68 147 134 '123 24 2*262 1-825 1*480 69 114 104 '095

26 2-328 1-886 1*536 70 *084 077 070 26 2*391 Г945 1*591 71 057 053 049 27 2*448 1*999 1*640 72 035 033 030 28 ¿'493 2*043 1'682 73 018 '017 '015 29 2-532 2081 1*719 74 006 006 005

80 2*559 2110 1*748 31 2*573 2127 1*767 82 2*571 2131 1*774' 88 2*550 2118 1*767 M 2-510 2090 1*747

86 2*463 2055 1*722 86 2*409 2*014 1*692 . 87 2*348 1*968 Г657 88 2*283 1*918 1*619 89 2*207 1*859 1*572

40 2124 Г794 1*521 41 2038 1*725 1*466 42 1*950 1*655 1*410 48 1*861 1*584 1*353 M 1*769 1*509 1*292

i6 1*677 1*434 1*231 46 1*584 1*359 1*170 47 1-490 1*281 1*106 «8 1-394 1*203 1*041 49 1*299 1124 '975

60 1*208 1048 912 61 1123 *976 -851 62 1040 '907 -793 68 -963 *842 739 M *892 782 *688

66 '827 *727 -641 66 763 -673 595 67 701 '620 -550 68 *642 '569 '506 69 '583 '519 '462



Mortality and Marriage Experience, 1903-1923 359 Table XVIIa.

Value of Annuity to Widow of Widower.

Age. 3 % 3J % 4 % Age. 3 % 3* % 4 %

60 1-882 Г674 1-493 61 1*736 Г548 1-384 62 Г591 1*422 Г275 63 1-450 Г300 1-168 64 Г311 1-178 Г061

66 Г 175 1058 -955 66 Г044 -942 -853 67 -918 -831 -754 68 800 *726 -660 69 -691 -628 '572

25 ... ... ... 70 '587 -535 *489 26 ... ... ... 71 493 -451 -413 27 3138 2-580 2133 72 408 *374 *343 28 3*236 2*667 2-210 73 332 '305 *280 29 3*346 2 764 2*295 74 265 244 225

30 3457 2-863 2-383 75 207 190 '176 31 3*568 2*962 2-470 76 *157 *145 '134 32 3-680 3063 2-560 77 115 '106 . 4)98 33 3-791 3-162 2*649 78 *081 *075 *069 34 3898 3-259 2*736 79 4)54 '050 '046

35 4 001 3-353 2 822 80 031 '029 '027 36 4*097 3*441 2*903 81 *016 '015 4)14 37 4-185 3*523 2*979 82 4)05 4)04 004 38 4-267 3*601 3*051 39 4*325 3-658 3*107

40 4*363 3*699 3*150 41 4-384 3-725 3*179 42 4-386 3736 3-195 43 4*367 3*729 3*190 44 4*313 3*691 3170

46 4-234 3-632 3127 46 4-132 3-553 3-066 47 4 007 3 453 2*987 48 3*859 3*335 2*891 49 3*695 3*200 2*781

50 3*527 3 063 2*668 61 3362 2926 2*555 | I 62 3190 2*784 2-436 I 63 3015 2*637 2*313 64 2845 2494 2'193

66 2-681 2*356 2077 66 2514 2-215 Г957 67 2-346 2072 Г835 68 2*185 Г935 Г717 69 2030 1-801 1-603



Збо Further Note on the Scottish Bankers*

Table XX. Value of Joint Annuities - Husband and Wife - 3 %.

Aoe of Age of Wife Hus- ■ . 1 . . BAKD 20 23 26 29 32 | 35 38 41

20 22*260 22010 21708 21349 20933 20*381 19719 23 21-825 2Г609 21*368 2Г045 20*641 20156 19*521 18757 26 2Г326 2Г145 20*918 20*638 20*305 19*836 19 258 18*552 29 20720 20*571 20*407 20166 19 850 19*459 18*915 18*246 32 20*040 19*923 19*767 19*569 19*329 18*959 18*486 17*883 36 19*247 19*154 19*055 18-895 18*672 18*384 17*951 17*400 38 18*380 18*311 18*215 18091 17*939 17*670 17*316 16*835 41 17*426 17*373 17*325 17*230 17*088 16*900 16*582 16*161 44 16*449 16*412 16*358 16*290 16*208 16033 15*789 15*433 47 15*419 15*390 15*372 15*324 15*245 15135 14 918 14*622 60 14*366 14*344 14*319 14*289 14*254 14*149 13*997 13*753 63 13*267 13*249 13*248 13*228 13191 13136 13001 12*806 66 12166 12152 12*143 12*135 12128 12*072 11*984 11*828 69 11*029 11018 11*023 11*019 11*007 10 987 10*908 10787 62 9*857 9*849 9*845 9*846 9*854 9*828 9782 9*689 66 ' 8*680 8*672 8*677 8*680 8*681 8*679 8*637 8*567 68 7*581 7573 7*572 7*577 7*589 7*581 7*561 7*508 71 6*528 6*531 6*535 6*540 6*547 6*527 6*490 74 5*531 5*540 5*545 5*545 5*541 5*514 77 4*564 4*576 4*577 4*570 4*554 80 3*634 3*641 3*639 3*628 83 2*674 2*675 2*669 86 1795 1792 89 Г084 92

Age of Age of Wife Hus- j . . i_ BAND 62 66

j I 68 71 74

. 77 80 83

44 10*638 '

47 10*363 9*373 60 9*990 9109 8*164 63 9*594 8752 7*936 7*102 66 9*105 8*380 7*581 6*881 6 221 69 8*570 7*907 7*243 6*546 5*970 5*465 62 7*912 7*366 6743 6*189 5*654 5176 4*742 66 7*203 6733 6*242 5709 5*271 4*881 4*452 4013 68 6'474 6*102 5*659 5*259 4*863 4*512 4194 3757 71 5*737 5*440 5*108 4732 4*428 4156 3*853 3*540 74 4*984 4760 4*481 4*218 3*950 3*722 3*520 3*219 77 4-200 4*044 3*851 3*620 3*435 3*267 3088 2*908 80 3*407 3*301 3162 3021 2*868 2745 2*638 2*479 83 2545 2*487 2*405 2*302 2*217 2136- 2058 1*984 86 1730 Г697 Г655 1*606 1*548 1*503 1*464 1*412 89 1*056 1*044 Г023 *996 #972 *948 *926 *907 92 -473 *469 -463 '456 '446 *439 *432 '425



Mortality and Marriage Experience, 1903-1923 361

Table XX.- (continued). Value of Joint Annuities - Husband and Wife - 3 %.

Age of Age of Wife Hus- - - ; : . - .- band 41 ^ 47 ' 50 53 66 69 62

23 18 757 i 26 18-552 17*724 ! 29 .18-246 17*499 16*659 32 17*883 17-159 16*399 15*574 36 17*400 16*766 16038 15-265 14433 38 16-835 16*240 15*601 14897 14*098 13*214 41 16*161 15-658 15*068 14-426 13*720 12'883 1Г936 44 15*433 14*979 14480 13'916 13*254 12-506 1Г629 10*638 47 14*622 14*252 13*809 13*311 12*754 12*061 1Г257 10*363 60 13*753 13*431 13071 12*658 12150 11*560 10*834 9*990 63 12-806 12*556 12*249 11*897 11*498 10'967 10*332 9*594 66 1Г828 11*616 1Г379 1Г106 10-751 10*329- 9773 9*105 69 10*787 10*630 10*435 10212 9*958 9*588 9*133 8*570 62 9*689 9558 9*418 9*257 9038 8*773 8*390 7*912 66 8'567 8-475 8*360 8*235 8 '097 7*874 7*588 7*203 68 7*508 7*430 7*353 7*267 7*150 7 008 6*775 6*474 71 6*490 6-439 6372 6*308 6*242 6124 5*971 5*737 74 5*514 5-470 5*430 5*385 5*330 5*267 5*147 4*984 77 4*554 4*528 4*492 4*462 4*435 4*384 4*317 4*200 80 3*628 3*608 3*589 3*568 3p46 3*524 3*474 3*407 83 2*669 2*659 2*644 2*632 2*623 2*607 2*586 2*545 86 Г792 1787 Г781 1*773 1*768 Г762 1*749 Г730 89 Г084 1080 Г078 1075 1073 1069 1*065 1*056 92 -480 '479 *478 478 477 475 *473

Age of Age of Wife Hus- BAND 83 86 ' 89 92 96 98

66 4 013 68 3 757 3-252 71 3*540 3 035 2 397 74 3*219 2*862 2306 1*599 77 2*908 2*561 2101 Г571 *956 80 2*479 2*292 1*912 Г394 911 *407 83 Г984 1*817 Г588 Г247 *811 *380 86 Г412 1*355 1*191 *952 '668 *330 89 *907 *863 *802 *672 *491 *258 92 *425 -417 *387 *344 *267 164



362 Further Note on the Scottish Bankers

Table XXI. Value of Joint Annuities - Husband and Wife - 3J %.

Age of Age of Wife Hus- BAND 20 23 26 29 | 32 35 | 38 41

20 20-455 20-254 204)07 19713 19*372 18'908 18'346 23 20*095 19*921 19727 19'462 19*126 18720 18*176 17*516 26 19*686 19*540 19*351 19'120 18*846 18*448 17955 17*342 29 19*177 19*055 18*921 18722 18*457 18*127 17*659 17*076 32 18*603 18*506 18*375 18*210 18*010 17'694 17*287 16759 36 17*923 17*845 17764 17*629 17*439 17*195 16*818 16*335 38 17*174 17*115 17*033 16*929 16'801 16569 16*261 15*836 41 16*341 16 294 16*254 16*173 16*052 15*890 15*609 15*236 44 15*477 15*443 15*398 15*341 15*272 15*117 14903 14*584 47 14*563 14*536 14*521 14*480 14*411 14*316 14*123 13*856 60 13*619 13*599 13*576 13*550 13*523 13*429 13*292 13070 63 12*623 12-607 12*606 12*589 12*557 12*509 12*387 12*209 66 1Г621 Ll'609 1Г599 1Г593 1Г588 1Г538 11*457 11*314 69 10*575 10565 10-570 10*567 10*557 10*538 10*465 10*353 62 9*487 9 479 9 476 9 '478 9*486 9*463 9*421 9*334 66 8*385 8*377 8*382 8*385 8*386 8*385 8*345 8*280 68 7*351 7*343 7342 7*347 7*358 7*351 7*333 7*283 71 6*352 6-355 6*358 6*363 6*371 6*351 6*316 74 5-400 5-409 5414 5414 5410 5*384 77 4*472 4*482 4*484 4*477 4*462 80 3*573 3*579 3*577 3*566 83 2*637 2*637 2*632 86 1775 1772 89 Г075 92

Age of Age of Wife Hus- BAND

62 | 66 68 71 74 77 80 83

44 10212 ' 47 9*957 ; 9*031 60 9*610 I 8784 7*893 63 9*240 | 8*450 7*678 | 6 887 66 8786 8*102 7344 | 6*679 6*051 69 8*285 7657 7026 I 6361 5'812 5*330 62 7*665 7*147 6*552 6022 5510 5053 4638 66 6*993 6*545 6075 5*563 5*143 4769 4*358 3937 68 6-300 5*944 5*519 5133 4*752 4*415 4*109 3*688 71 5*597 5-311 4992 4628 4 334 4*071 3778 3*478 74 4874 4658 4388 4134 3*874 3*653 3*457 3166 77 4*119 3968 3780 3*556 3*376 3 211 3*037 2*863 80 3352 3*249 3*113 2*975 2*825 2704 2*600 2*445 83 2-510 2*455 2*375 2*273 2189 2*110 2*034 Г961 86 1712 Г680 1*638 1*589 1*532 1488 1*449 1398 89 Г048 Г035 1*015 I *988 964 941 *920 900 92 -470 *466 -461 -453 *444 *437 *430 *422



Mortality and Marriage Experience, 1903-1923 363 Table XXI. - {continued)

Value of Joint Annuities - Husband and Wife - 3£ %.

Age of Aoe of Wipe Hus- = band 41 M 47 60 j

58 66 69 62

23 17*516 26 17*342 16*616 29 17076 16*420 15*679 32 16*759 16*121 15*448 14*715 36 16*335 .15*774 15127 14*438 13*692 38 15*836 15*307 14*738 14107 13*387 12*587 41 15*236 14*787 14*259 13*682 13045 12*282 11*415 44 14*584 14176 13*727 13*219 ¡ 12*619 1Г938 1Г131 10*212 47 13*856 13*522 13*120 12*670 12164 11*528 10*788 9*957 60 13070 12*777 12*449 12074 11*609 11*068 10*396 9*610 63 12*209 11*979 11*698 11*375 1Г009 10*518 9*930 9*240 66 11*314 11*118 10*899 10*648 10*320 9*930 9*411 8*786 69 10*353 10207 10026 9*819 9*584 9*241 8*815 8*285 62 9*334 9*210 9078 8*928 8*724 8*477 8*116 7*665 66 8*280 8*193 8084 7*967 7*837 7*626 7*358 6*993 68 7*283 7*208 7*134 7053 6*941 6*808 6*587 6*300 71 6*316 6*266 6*203 6*143 6*080 5*968 5*821 5*597 74 5*384 5*342 5*303 5*260 5*207 5*146 5030 4*874 77 4*462 4*437 4*402 4*373 4*347 4*296 4*232 4*119 80 3*566 3*547 3*529 3*508 3*488 3*465 3*418 3*352 83 2*632 2*622 2*608 2*597 2*588 2*572 2*551 2610 86 1772 1767 1761 1754 1748 1743 1730 1712 89 1*075 1*071 Г069 Г067 Г064 1*060 1*056 Г048 92 *478 *477 *476 *475 *474 *472 *470

Age of Age of Wife Hus- BAND 83 86 89 92 96 98

¡ 66 3*937 ! 68 3*688 ! 3*201 71 3*478 i 2*987 2*366 74 3*166 I 2*820 2*276 Г582 77 2*863 2*524 2075 Г554 *948 80 2*445 2*262 1*889 1*381 *905 *405 83 Г961 1797 Г572 Г236 *805 *377 86 Г398 1*342 Г179 *944 *663 *328 89 *900 *856 796 *667 *488 *257 92 *422 -414 *385 *342 *266 *164




Table XXII. Value of Joint Annuities - Husband and Wife - 4 %.

Age of Age of Wife Hus- - BAND 20 23 26 29 32 36 38 41

20 18*878 187H 18*012 18271 ! 17*991 17*600 17121 23 18*581 18*439 18*281 18063 17*783 17*444 16*977 16*403 26 18*243 18-121 17*965 17*774 17*548 17*211 16*787 16*252 29 17*813 17-712 17*603 17*437 17*215 16937 16*532 16*023 32 17*327 17*246 17*136 16*998 16*832 16*561 16*209 15*745 36 16-741 16*675 16*609 16*496 16-336 16*127 15*798 15*372 38 16*091 16040 15*970 15*882 15*776 15*576 15*305 14*928 41 15-358 15*318 15*285 15*217 15113 14*973 14*725 14*393 44 14*596 14*566 14*527 14*478 14*419 14*284 14*095 13*809 47 13-780 13*756 13*744 13*709 13*650 13'567 13*394 13*153 60 12-933 12*915 12*895 12*874 12*850 12768 12*645 12*443 63 12*030 12016 12015 12001 11*973 11*931 1Г819 1Г656 66 11-114 1Г103 11*095 11*090 1Г086 1Г041 10*968 10*836 69 10151 10-142 10*147 10*145 10*135 10*120 10*052 9*949 62 9-140 9132 9*129 9*131 9*139 , 9*118 9*079 8*997 66 8*108 8*100 8*105 8*108 8*109 8*109 8*073 8*010 68 7*132 7*124 7*123 7*128 7*139 7*132 7*116 7*068 71 6184 6*186 6*190 6*195 6*202 6*184 6*150 74 5*275 5*284 5*288 5*289 5*285 5*260 77 4*382 4*392 4*394 4*388 4*374 80 3*513 3*519 3*517 3*506 83 2*601 2*602 2-597 86 1756 1751 89 1*066 92 I

AGE OF AGE 0F WlFE Hus- ■ BANI) 62 66 68 71 74 77 80 83

44 9*814 47 9-576 8709 60 9*255 8*479 7*637 63 8-910 8-166 7*437 6*684 66 8*484 7*839 7*120 6*487 5*888

* 69 8*014 7*421 6*821 6*186 5*663 5*202 62 7*430 6*937 6-370 5*863 5*372 4*934 4*538 66 6793 6*366 5*917 5*425 5*022 4*663 4*268 3*863 68 6134 5792 5*385 5014 4*646 4*322 4027 - 3*621 71 5*462 5*188 4*879 4*528 4*244 3*990 3707 3417 74 4769 4*561 4-300 44)54 3*800 3*586 3*396 3*115 77 4041 3*895 3712 3*494 3*318 3*158 2*989 2*819 80 3*298 3*197 3065 2*930 2783 2*665 2*563 2*412 83 2*477 2*423 2*344 2*244 2*162 2*084 2009 Г938 86 Г694 1*662 | 1*621 1*573 1*517 Г473 1*435 1*385 89 1*040 1*027 1*007 *980 *957 *933 *912 *893 92 '468 *464 *458 *450 *441 *434 *427 *420



Mortality and Marriage Experience, 1903-1923 365 Table XXII. - (continued).

Value of Joint Annuities - Husband and Wife - 4 %.

Aok ok A(se of Wife Huh- BAND 41 44 47 50 I 63 66 69 62

23 16-403 26 16*252 15616 29 16023 15446 14*788 32 15*745 15*182 14584 13*931 36 15*372 14875 14*298 13*682 13011 38 14*928 14*458 13*949 13*383 12*733 12007 41 14*393 13*992 13*516 12*996 12*422 11*725 10*929 44 13*809 13*441 13035 12*578 12032 11*411 10667 9*814 47 13153 12*851 12*485 12076 11*617 1Г034 10350 9*576 60 12*443 12174 1Г875 1Г533 1Г107 10*611 9*987 9*255 63 1Г656 11*445 11*186 10*891 10*556 10102 9*556 8*910 66 10*836 10*653 10*451 10*219 9*916 9*555 9071 8*484 69 9*949 9*813 9*643 . 9*451 9*233 8*912 8*514 8*014 62 8*997 8*881 8*757 8*617 8*426 8*195 7*856 7*430 66 8010 7*928 7*824 7*714 7*592 7*394 7*140 6*793 68 7068 6*997 6*926 6*849 6*743 6*618 6*408 6*134 71 6*150 6103 6041 5*983 5*923 5*816 5*677 5462 74 5*260 5*219 5*181 5140 5090 5031 4*919 4*769 77 4*374 4*350 4*315 4*287 4*261 4*213 4*151 4041 80 3*506 3*488 3*470 3*450 3*429 3*408 3*361 3*298 83 2*597 2*588 j 2*573 2*562 2*553 2*537 2*517 2*477 86 Г751 1*748 1*742 1*736 1*730 1*724 1*711 Г694 89 1066 1063 1060 1058 1055 Г051 Г048 1040 92 *475 *474 -473 *472 *471 '470 *468

A«E OF AOE OF WlFE Hus- i ^-i BAND 83 86 89 92 96 98

66 3*863 68 3-621 3150 71 3*417 2*940 2*336 74 3115 2*779 2*247 Г566 77 2*819 2*488 2*050 Г538 *940 80 2*412 2*234 1*868 Г368 *897 *403 83 1*938 1*777 Г556 Г224 *799 *376 86 Г385 1*329 1-169 *937 *659 *327 89 -893 -850 *790 #663 *485 '256 92 '420 -412 -383 *340 *264 *163



Documents

Further Note on the Scottish Bankers' Mortality and Marriage Experience, 1903-1923, with special reference to the use of "Equivalent Ages"