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AUSTRALIAN

JOINT LIFE TABLES

1901-1910.

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

Commonwealth Statistician.

* Â» â€¢ â€¢ â€¢ â€¢ â€¢

â€¢ â€¢â€¢â€¢Â«â™¦ â€¢

* â€¢

c _ â€¢ t â€¢ Â«

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. :â– â€”

(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 : â€”

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. : â€”

O'xy = I^Vxu (1 + Â«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

â– 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 â€” 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., 8.102â€”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 : â€”

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 : â€”

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) + ](â€” .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^ â€”- Ix â€” 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 â€” 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(1â€” Px) , X 20j. ,

Hence m^. = - = â€” ^; â€” (approx.) â€” ~ â€” - â€” (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)Â«

Preface.

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

interest i.

Hence d = \ â€” 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 â€” 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 Â«hich (x) dies. [Note. â€” 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 : â€”

A^ = 1 - fZ (1 + aj

% =â– â– a^ 4- lA^ (1 + i)i

(frequently taken in practice as a^ H ^ )

''a = Â«x + i â€” 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 = ^Â«(/ ~~ ^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 : â€”

" 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^ = Â«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 : â€”

u

{ujx = %|-28wâ€ž + l-62wÂ«+2-2%n+l-62%n+-o6w6Â« + l-62w7â€ž-

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- â€”

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

Values of ax- â€” 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 â€”

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 â– '>!> 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

13784

B 3 131 753

CO

O

)

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

W-

^:;::r<Â»

3

AUSTRALIAN

JOINT LIFE TABLES

1901-1910.

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

Commonwealth Statistician.

* Â» â€¢ â€¢ â€¢ â€¢ â€¢

â€¢ â€¢â€¢â€¢Â«â™¦ â€¢

* â€¢

c _ â€¢ t â€¢ Â«

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. :â– â€”

(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 : â€”

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. : â€”

O'xy = I^Vxu (1 + Â«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

â– 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 â€” 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., 8.102â€”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 : â€”

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 : â€”

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) + ](â€” .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^ â€”- Ix â€” 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 â€” 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(1â€” Px) , X 20j. ,

Hence m^. = - = â€” ^; â€” (approx.) â€” ~ â€” - â€” (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)Â«

Preface.

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

interest i.

Hence d = \ â€” 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 â€” 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 Â«hich (x) dies. [Note. â€” 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 : â€”

A^ = 1 - fZ (1 + aj

% =â– â– a^ 4- lA^ (1 + i)i

(frequently taken in practice as a^ H ^ )

''a = Â«x + i â€” 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 = ^Â«(/ ~~ ^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 : â€”

" 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^ = Â«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 : â€”

u

{ujx = %|-28wâ€ž + l-62wÂ«+2-2%n+l-62%n+-o6w6Â« + l-62w7â€ž-

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- â€”

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

Values of ax- â€” 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 â€”

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 â– '>!> 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

Online Library → Australia. Commonwealth Bureau of Census and Stati → Australian joint life tables; → online text (page 1 of 18)