Copyright
Australia. Commonwealth Bureau of Census and Stati.

Australian joint life tables; online

. (page 1 of 18)
Online LibraryAustralia. Commonwealth Bureau of Census and StatiAustralian joint life tables; → online text (page 1 of 18)
Font size
QR-code for this ebook


UC-NRLF




13784



B 3 131 753



CO

O



)



'..■■g;,%*v: ■■.fr^ .■■ ij. .■*■>,' '\




W-



^:;::r<»



3




AUSTRALIAN



JOINT LIFE TABLES



1901-1910.







Q. H. KNiBBS. C.M.G., F-S.S., ETC.
Commonwealth Statistician.



* » • • • • •
• •••«♦ •



* •



c _ • t • «







uC<,-C'tZn_rfuO.



Il



Commonwealth Bureau of Census and Statistics,

MELBOURNE.



Australian Joint Life Tables.

COMPILED AND ISSUED UNDER THE AUTHORITY

OF THE
MINISTER OF STATE FOR HOME AND TERRITORIES,



BY



G. H. KNIBBS, C.M.G.,

Fellow of the Royal Statistical Society, Membre de I'lnstitut International de Statistique,
Honorary Member American Statistical Association, and of the Societe de Statistique

de Paris, etc., etc.

COMMONWEALTH STATISTICIAN.



By Authority :
McCARRON. BIRD & CO.. PRINTERS, 479 COLLINS STREET, MELBOURNE.



PREFACE. /^/y



A& A



1. In the preface to the Australian Life Table, 1901-10, published
on 30th September, 1914, it was stated that the compilation of joint life
annuity tables was under consideration. The disorganisation resultant
upon the! outbreak oi t;he;war has delayed the preparation of these tables,
but opportunity has now b6en found to effect their publication.

2. "The' tables ha^6 been based on the Commonwealth male and
female experience for the decennium 1901-10, and comprise four distinct
sets, viz. :■ —

(i.) Annuities on 2 Male Lives ;

(ii.) Annuities on 2 Female Lives ;

(iii.) Annuities on 1 Male and 1 Female Life, the Male the Elder ;
(iv.) Aiuiuities on 1 Male and 1 Female Life, the Female the Elder.

For the sake of completeness the elementary values and single life
amiuity values for the same rates of interest have also been included.
In all the joint life tables the values are given m single years of age for
the older life combmed with ages of the younger life at quinquennial
intervals. The rates of interest for which the annuities have been
tabulated are 2|, 3, 3|, 4, 4^, 5, 5|, and 6 per cent.

3. The arrangement of the jomt life tables differs somewhat from
that which is usually adopted. For each set of tables the whole of the
results for all the tabulated combinations of ages for any rate of uiterest
are given at one opening, thus facilitating the work of mteri)olation, which
is necessary in most cases to determme values for the given ages.

The usual method of presentmg jouit life values is that of givmg several
rates of iiiterest at the one opening for a given difference in the ages of the
two joint lives. This method of presentation has an advantage in cases
where it is desired to uiterpolate for rates of interest other than those
tabulated. With rates tabulated for every ^ per cent, of uiterval, how-
ever, such interpolations are rare in practice, \\hile interpolations for
intermediate ages are of constant occurrence. It a\ ill thus be seen that
the balance of advantage lies with the arrangement adopted m the present
tables.

A further imiovation consists in commencing the tables with the
oldest ages of each life, and working downwards and outwards to the
youngest ages. The reasons for this arrangement are : —



Preface.

(i.) that the jihiciug on the same line of all the values in which the
older age occurs facilitates interpolation iii respect of the
younger age ;

(ii.) that by placing the oldest age first in the age column, the table
can be conveniently and clearly set out without any space
being wasted, if the device of folding back the end of the
table be adopted.



4. In the computation of the joint Ufe tables contained in the
present volume, the " millionaire" calculating machine has been used
throughout, the formula on which the calculations were based being the
following, viz. : —

O'xy = I^Vxu (1 + «cr + l : y+l)-

The initial computation consisted of the calculation of values of vp^;
for all ages for male and female lives, and according to each of the rates of
interest decided upon. Values of vp^y for age differences 0, 5, 10, 15,
etc., were then computed by multiplying the values of vjij. by the
appropriate values of jjy. Finally, the values of a^^ were determined
from the values of vj^xy by means of the formula quoted above.

The whole of the Mork was carefully checked at each stage by differ-
ent computers and different machines, and an elaborate and exhaustive
■check by differences was applied to the whole of the tables.

In addition, numerous sample checks by means of summation for-
mulae were applied.

As a consequence of this care it is believed that a liigh degree of
accuracy has been attained.



5. Where the ages involved in any question differ by a multiple of
5, the accompanying tables will give the corresponding annuity innnedi-
ately, the age of the older life being found in the margin on the right or left
of the page, and the difference between the ages being indicated at the
head of the appropriate column. For example, a joint life amiuity of 1
on two male lives aged respectively 62 and 42, interest being at 4| per
cent., is found on p. 26 to be 7.988. The value is found in line with age
62, which appears in the left margin, and in the colunm relating to age
differences of 20 {u: = x — 20).,

Where the ages differ by a number which is not a multiple of 5, inter-
polation will be necessary, and for many purposes interpolation by first
differences wiU be sufficient. This will be effected as follows : the ages
being 62 and 40, the lives male, and the interest as before, 4^ per cent.
For ages 62 and 42 the value is, as above, 7.988, while for ages 62 and 37
it is 8.162. Hence, as a decrease of 5 years in the age of the younger life
increases the anmiity value by .174 (viz., 8.102—7.988), a decrease of 2
years (from 42 to 40) Asould increase the annuity value by a])]n'oxi)nately
2-oths of .174, that is, by .070, giving a value of (7.988 -F.070), or 8.058.'

3



904^235



^ Preface.

If greater accuracy is required, interpolation by second differences
should be used. This will require three values to be taken from the table,
all on the same line as the older age. Taking the data as above, the work
is as follows : —



Joint Ages.


dxy


A


A^


62 : 42


7.988


.174


.044


62 : 37


8.162


.130




62 : 32


8.292







The values in the a^y column are taken direct from the table on
p. 26. The values headed /\ are obtained by subtracting the first from
the second and the second from the third values in the preceding column.
The value A^ is obtamed by subtracting the first from the second of the
A values.

The values of /\ and /\^ on the upper line must now be multiplied
by the appropriate coefficient in the attached table, and the product
must be added to the a^y value on the upper Ime : —



t


Coefficient of A


Coefficient of A^


1

2
3
4


.2
.4
.6
.8


.08
.12
.12

.08



In this table t denotes the difference between the age of the younger
Hfe in the problem and the age of the younger life in the upper line of
the above working process. In this case the difference is 2, that is, the
difference between 42 and 40. Hence the amount to be added is
(.174 X .4) + ](— .044)x(-.12);- = .070 + .005 = .075. The correct

value of the annuity is thus 7.988 + .075 = 8.063.

In the calculations involvuig interpolation care must be taken to
employ the correct signs, and it must be remembered that when a number
to be added is of the minus sign, the process is one of subtraction.



6. The notation used is that devised by the Institute of Actuaries,
London, and adopted by the International Actuarial Congress at its London
session in 1898 as the international actuarial notation. The following
explanations of the various sjanbols used herein are furnished for
convenience of reference.

ly. denotes the number of persons M'ho reach the exact age x out of
an arbitrary number (say 100,000) who are assumed to come under
observation at a specified age. In a life table relative to the general
po]iulation, the lives aio usually assumed to come under observation at
age 0, that is, at the juoment of birth.

d;^ denotes the number of persons who die after reacMng age x, but.
before reaching age x + 1. Hence d^ —- Ix — K+i-



Preface.

7)^ doiiotos the probability that a person aged x Avill survive a year,
or, in other \\ords, denotes the proportion of the persons who reach age x
that will live to reach age x +1. Hence p^ = l^+i/lx-

q^, usually known as the " rate of mortality at age .r," denotes the
probability that a person aged x will die within a year, or, in other
\\ords, denotes the proportion of the persons who reach age x that will die
before reaching age .r + 1-

Hence q^=l — j^^ = dx / h-

jx-g, usually known as the "' force of mortality at age x," denotes the
rate per unit per annum at w hich deaths are occurring at the moment of
attaining the age x. In other words, it represents the proportion of
persons of that age who would die in a year, if the intensity of mortality
remained constant for a year, and if the number of persons under ob-
servation also remained constant, the places of those who die being
constantly occupied by fresh lives.

Hence ^.= ^ ^^^ ^^'^'^^



Ix dx dx

rriy., usually known as the '" central death rate at age x," denotes
the ratio of the number of deaths between the ages x and x^ 1 to the mean
population between these ages. This mean population is usually denoted
by L^.

TT ^X 2(1— Px) , X 20j. ,

Hence m^. = - = — ^; — (approx.) — ~ — - — (approx.)

^x ^ -^-Px 2 - g^

= 5'z (1 ~r ^) (approx.), w^hen (7^ is small.

o

C.J., usually known as " the complete expectation of life at age x,"
denotes the average future lifetime of persons who reached age x.



Hence e ^ =



La; + La; + i + La; + 2 +



I



X



i denotes the effective rate of interest per unit accruing in one year.

1 -j- i denotes the sum to which a capital of 1 will amount in 1 year
at the effective rate i.

(1 + ij"' denotes the sum to which a capital of 1 wdll amount in n
years at compound interest at the effective rate i, where n may be any
number, integral or fractional.

V denotes the present value of 1 due 1 year hence at the effective
rate i.

Hence z; = 1 / (1 -j- {)

v" denotes the present value of 1 due n years hence at compound
interest at the effective rate i, w^here n may be any number, integral or
fractional.

Hence ?;" = 1/(1 4- i)«



Preface.

d denotes the discount on 1 due 1 year hence at the effective rate of
interest i.

Hence d = \ — v = i / (1 -\- i) = iv.

j(m) denotes the nominal rate of interest per unit per annum \\ hich,
onvertible m times a year, is equivalent to an effective rate i.

Hence J(w) = m I (1 + i)m — If

8, usually known as " the force of interest," denotes the nominal
rate of interest per unit per aiuium which, convertible momently, is
equivalent to an effective rate i.

Hence h = j^ = loQe (1 + i).

a^ denotes the present value of " a curtate annuity" of I payable
at the end of each year which a life aged x survives, but providmg no
payment for the fraction of the year in «hich (x) dies. [Note. — The
expression (x) is used as an abbreviation for " a person whose exact
age is X years."]

a^ denotes the present value of " a complete annuity" on (.r), and
differs from " a curtate amiuity," i.e., a^, in making provision for a
proportionate payment in respect of the fraction of the year elapsing
between the last payment of the curtate annuity and the date of death of

{X).

a*"*> denotes the value of a curtate annuity of 1 per annum payable
in instalments m times a year, the last payment benig made at the end
of the last completed - th part of a year prior to the death of x.

dj. denotes the value of " a continuous annuity. "" that is, an annuity
of 1 per annum payable in momently instalments.

dxy^ f^xyz, ciwxyz, etc., denote the values of joint life curtate annuities
of 1, payable j^early, the last payment being made at the end of the last
year completed prior to the failure of the joint lives by the first death
amongst them. Jomt life annuities may be "' com})lete," '" payable
fractionally" or " continuous," the notation bemg modified for such cases
in the same maimer as for single life annuities.

O'xyy cixyz, (hjoxyz, etc., denote the values of curtate annuities of 1
per annum, payable on the last survivor of the lives concerned

ay\ X known as "a reversionary annuity," denotes a curtate annuity
of 1 per annum to (.r) after the death of (//), the first payment being made
to {x) at the end of the contract year m which {y) dies, and the last being
made at the end of the contract year immediately preceding the death
of [x).

A

a^7*r denotes a complete reversionary annuity of 1 per annum to (.r)
after the death of {y), ])ayable m times a year, the first payment being
made l^ th of a year after the deatli of (?/), and the last payment bemg
in respect of the fractional period to the death of [x).

Kx denotes the value of 1, payable at the end of the contract year
in which (.r) dies. Similarly Aa;^, A.,:,/z, etc., denote the value of 1,
payable at the end of the contract year in which the joint lives fail by
the first deatli amongst them.



Preface.

7. For convenience of reference the following working formulae for
the valuation of benefits involving the values of annuities on smgle and
joint lives are given without demonstration : —

A^ = 1 - fZ (1 + aj
% =■■ a^ 4- lA^ (1 + i)i

(frequently taken in practice as a^ H ^ )

''a = «x + i — hkl^x + S)

(frequently taken in practice as a^. + |)



Ctxi/2 = '^x 'T ^y \ ^z ^xy ^'.C2 ^i/2 I ^xijz
^yz 1 a; = ^'a; *:j;!/2

^Vz\x^ ^X "" ^X'-yZ = ^X ^Xi/ ^.1-2 I Ct_jj/2

'^zl xy = ^«(/ ~~ ^u;2/ : 2 = ^a; r ^(/ ^xi/ ^J2 ^1/2 ~r ^xyz

8. By means of Milne's modification of Simpson's Rule for joint
life annuities, the values of annuities on two joint lives may be employed
to give a fair approximation to the values of annuities on three joint
lives. Using modern notation, Mihies modification may be stated as
follows : —

" Let {x) be the youngest and (z) the oldest of the three proposed
lives (;r), (//), and (2). Fmd the value of the two joint lives {if) and (2),
and let {w) be the equivalent single life. Then if (z), the oldest life pro-
posed, be under 45 years of age, let the age of the substituted life be the
whole number next greater than that which expresses the age of {%o).

" But if the age of (2) be not under 45 years, let the age of the sub-
stituted life be the next greater than that of {w), w hich does not require
more than one decimal figure to express it."

In other words, when 2 < 45 and xo is the next higher integer in the
equation Uy^ = «m;, or when 2 = or > 45 and w is the next greater
number which can be represented with one decimal place in the eqiiation

O'yz = C^w, then Uj^yz = Uxw

9. For annuities on four or more joint lives the best method of
valuation is by means of a summation formula.



Peeface.

When (x> is the Hmiting age of the hfe table, and 71 is so taken that
Urn falls just within or just beyond the table, the following convenient
fonnula, devised by the late Sir G. F. Hardy, gives satisfactory results : —

u
{ujx = %|-28w„ + l-62w«+2-2%n+l-62%n+-o6w6« + l-62w7„-

For joint life annuity calculations Un = '^^nPxy The value so

obtained is that of a continuous annuity on the joint lives involved.

10. On pages 9 are given the values of certain interest functions
which are frequently required in the solution of problems involving
annuity values.

In connection Mith these tables I desire to acknowledge the pro-
fessional services of the Supervisor for Census in this Bureau, Mr. C. H.
Wickens, A. I. A.

G. H. KNIBBS,

Commonwealth Statistician.



Commonwealth Bureau of Census and Statistics,
30th September, 1917.



CONTENTS.



Interest Functions, p. 9.

Elementary Values, Ix, dx, Jpx, Qx, fJ-x, rux, h- —

Male Lives, pp. 10, 11. Female Lives, pp. 12, 13.

Values of ax- — Male Lives, pp. 14, 15. Female Lives, pp. 16, 17.



Joint Life Annuities.







Rate of Interest.




Class of Lives.














2r/o


3<


Yo


3^%


4%


4i%


5%


H%


6%




pp.


pp.


pp.


pp.


pp.


pp.


pp.


pp.


Two ^lale Lives


18, 19


20,


21


22, 23


24, 25 26, 27


28, 29 1 30, 31 1 32,


33


Two Female Lives


34, 35 36,


37


38, 39


40, 41 42, 43


44, 45 46, 47 48,


49


Male and Female —














1






Male the Elder


50, 51


52,


53


54, 55


56, 57


58, 59


60, 61 : 62, 63


64,


65


Female the Elder


66, 67


68,


69


70,71


72, 73


74, 75


76, 77


78, 79


80,


81



Interest Functions.









Rate Per


Cent.










2i


3


3+


4


4*


5


5*


6


i


.025


.030


.035


.040


.045


.050


.055


.060


l+i


1.025


1.030


1.035


1.040


1.045


1.050


1.055


1.060


{l + i)i


1.01242


1.01489


1.01735


1.01980


1.02225


1.02470


1.02713


1.02956


(l + i)i


1.00619


1.00742


1.00864


1.00985


1.01107


1.01227


1.01348


1.01467


V


.97561


.97087


.9()618


.96154


.95694


.952.38


.94787


.94340


v\


.98773


.98533


.98295


.980.38


.97823


.97.590


.97358


.97129


vl


.99385


.99264


.99144


.99024


.98906


.98788


.98670


.98554


d


.02439


.02913


.03382


.03846


.04306


.04762


.05213


.05660


Jii)


.02485


.02978


.03470


.03961 !


.04450


.04939


.05426


.05913


Jii)


.02477


.02967


.03455


.03941


.04426


.04909


.05390


.05870


5


.02469


.02956


.03440


.03922


.04402


.04879


.05354


.0.5827



AUSTRALIAN MALE LIVES, 1901-1910.



ELEMENTARY VALUES.



X


Ix

\


^.


Vx


<lx


V-^


m^


o
^X





100 000


9 510


.90490


.09510


.2279


.10112


55.200


1


00 490


1 611


.98220


.01780


.0344


.01804


.59.962


2


88 879


599


.99325


.00675


.0093


.00677


60.044


3


88 280


388


.99561


.00439


.0052


.00441


.-)9.449


4


87 892


307


.99651


.00349


.0040


.00350


58.709


5


87 585


246


.99719


.00281


.0031


.00281


57.913


6


87 339


205


.9976.1


.00235


.0025


.00235


.57.075


i


87 134


182


.99791


.00209


.0022


.00209


.56.208


8


86 952


170


.99804


.00196


.0020


.00196


55.325


9


86 782


160


.99816


.00184


.0019


.00185


.54.432


10


86 622


155


.99821


.00179


.0018


.00179


53.532


11


86 467


155


.99821


.00179


.0018


.00179


52.627


12


86 312


159


.99816


.00184


.0018


.00184


51.720


13


86 153


171


.99802


.00198


.0019


.00199


50.815


14


85 982


193


.99775


.00225


.0021


, .00225


49.915


15


85 789


219


.99745


.00255


.0024


.00256


49.026


16


85 570


240


.99719


.00281


.0027


.00281


48.150


17


85 330


259


.99697


.00303


.0029


.00304


47.284


18


85 071


282


.99669


.00331


.0032


.00332


46.427


19


84 789


296


.99651


.00349


.0034


.00350


45.579


20


84 493


313


.99630


.00370


.0036


.00371


44.7.37


21


84 180


329


.99609


.00391


.0038


.00.392


43.902


22


83 851


339


.99596


.00404


.0040


' .00405


43.072


23


83 512


349


.99582


.00418


.0041


.00419


42.245


24


83 163


361


.99566


.00434


.0043


.0043.")


41.420


25


82 802


371


.99552


.00448


.0044


.00449


40.599


26


82 431


383


.99536


.00464


.0046


.00466


39.779


27


82 048


392


.99522


.00478


.0047


.00479


.38.962


28


81 656


403


.99506


.00494


.0049


.00495


38.147


29


8 1 253


409


.99497


.00503


.0050


.00505


37.333


30


80 844


419


.99481


.00519


.00.') 1


.00520


36.520


31


80 425


434


.99460


.00540


.0053


.00.541


35.707


32


79 991


447


.99442


.005.-)8


.0055


.00560


34.898


33


79 544


462


.99419


.0058 1


.0057


.00582


34.092


34


79 082


475


.99399


.((0601


.0059


.00602


33.288


35


78 607


498


.99367


.00633


.0062


.00636


32.486


36


78 109


518


.99337


.00663


.006.')


.((0665


31.690


37


77 591


541


.99302


.00698


.0()6S


.((0700


30.898


38


77 050


568


.99264


.00736


.0072


.00740


30.112


89


76 482


595


.99222


.00778


.0076


.0078 1


29.331


40


75 887


619


.99184


.00816


.0080


.00819


28.557


41


75 268


647


.99140


.00860


.0084


.00863


27.788


42


74 621


679


.99090


.00910


.0089


.00914


27.025


43


73 942


714


.99035


.00965


.0094


.00970


26.268


44


73 228


749


.98976


.01024


.0100


.01028


25.520


45


72 479


785


.98917


.01083


.0106


.01089


24.778


46


71 694


819


.98858


.01142


.0112


.01149


24.044


47


70 875


854


.98796


.01204


.0118


.01212


23.316


48


70 021


882


.98739


.01261


.0124


.((1268


22.594


49


69 139


918


.98673


.01327


.0130


.01337


21.876


50


68 221


951


.98605


.01395


.0137


.01404


21.163


51


67 270


984


.98537


.01463


.0144


.01473


20.456


52

1


66 286


1 020


.98462


.01538


.0151


.01551


19.752



AUSTRALIAN MALE LIVES, 1901-1910.



ELEMENTARY VALUES.



X


h


''


Px


<lx


^'x


Wa,


o


63


65 266


1 058


.98378


.01622


.0159


.01634


19.0.53


54


64 208


1 101


.98286


.01714


.0168


.01729


18.358


55


63 107


1 146


.98184


.01816


.0178


.01832


17.670


56


61 961


1 198


.98066


.01934


.0189


.01952


16.987


57


60 763


1 258


.97929


.02071


.0202


.02092


16.312


58 ■'>!> 50.1


1 327


.97771


.02229


.0217


.02255


15.646


59 ^S 178


1396


.97600


.02400


.0234


.02428


14.992


60


56 782


1 467


.97416


.02.584


.0252


.02617


14.348


61


.55 315


1 543


.97212


.02788


.0272


.02829


13.715


62


53 772


1 619


.96988


.03012


.0294


.03057


13.094


63


52 153


1 698


.96743


.03257


.0318


.03309


12.485


64


50 455


1785


.96463


.03537


.0345


.03601


11.888


65


48 670


1878


.96141


.038.59


.0376


.03934


11.306


66


46 792


1979


.95770


.04230


.0412


.04320


10.7.39


67


44813


2 081


.95356


.04644


.0453


.04753


10.191


68


42 732


2 182


.94894


.05106


.0499


.05239


9.663


69


40 550


2 275


.94389


.05611


.0550


.0.5771


9.156


70


38 275


2 359


.93838


.06162


.0606


.06358


8.670


71


35 916


2 428


.93240


.06760


.0667


.06996


8.207


72


33 488


2 483


.92.585


.07415


.0734


.07699


7.765


73


31 005


2 518


.91878


.08122


.0808


.08464


7.347


74


28 487


2 525


.91138


.08862


.0887


.09275


6.952


75 25 962


2 495


.90390


.09610


.0969


.10097


6.580


76 23 467


2 433


.89631


.10369


.1052


.10938


6.226


77 2 1 034


2 347


.88842


.11158


.1138


.11822


5.889


78


18 687


2 240


.88012


.11988


.1229


.12758


5.566


79


16 447


2 117


.87132


.12868


.1326


.13766


5.257


80


14 330


1976


.86205


.13795


.1430


.14826


4.960


81


12 354


1826


.85226


.14774


.1540


.15977


4.675


82


10 528


1 671.1


.84124


.1.5876


.1660


.17264


4.400


83


8 856.9


1 513.8


.82909


.17091


.1800


.18720


4.137


84


7 343.1


1 348.6


.81634


.18366


.1950


.20265


3.889


85


5 994.5


1 181.0


.80299


.19701


.2110


.21911


3.654


86 4 813.5


1 015.3


.78908


.21092


.2280


.23653


3.431


87 3 798.2


857.4


.77427


.22573


.2460


.25542


3.218


88 i 2 940.8


711.1


.75818


.24182


.2660


.27631


3.014


89 i 2 229.7


577.7


.74093


.25907


.2880


.29928


2.821


90


1 652.0


4.58.2


.72264


.27736


.3120


.32414


2.639


91


1 193.8


354.07


.70340


1 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

Online LibraryAustralia. Commonwealth Bureau of Census and StatiAustralian joint life tables; → online text (page 1 of 18)