Life Contingencies

Citation preview

:

.

,

(QnrneU Hniuetaitg Blihrarg ithara. JJmh

fork

BOUGHT WITH THE INCOME OF THE

SAGE ENDOWMENT FUND THE GIFT OF

HENRY W. SAGE 1891

The

original of this

book

is in

the Cornell University Library.

There are no known copyright

restrictions in

the United States on the use of the

text.

http://www.archive.org/details/cu31924001080633

Cornell University Library

HG8781.S77 Life contingents.

3 1924 001 080 633

LIFE CONTINGENCIES

LIFE CONTINGENCIES BY E. F.

SPURGEON,

F.I.A.

PUBLISHED "BY THE AUTHORITY AND ON BEHALF OF

THE INSTITUTE OF ACTUARIES BY Charles

&

Edwin Layton, Farringdon

LONDON 1922 All Rights Reser-ved

Street, E.C.4

PRINTED IN GREAT BRITAIN

LIFE CONTINGENCIES—ERRATA In the following table the second that of the § (paragraph). is

indicated

The copies,

13

line,

Where the

by the

letter

table includes a

first

column gives the number of the page and the

except where otherwise indicated by line is

/.

(formula) or

counted from the bottom of the page, this

b.

number

of items

but incorrectly in others.

which are correctly printed

in

some

160

INTRODUCTION Part II of the Text Book of the Institute of Actuaries, dealing with the Theory of Life Contingencies, was first issued in 1887. This work, for which the Institute must ever be indebted to its distinguished author

Mr

George King, did more than simplify the

progress of the actuarial student to his desired goal;

it

syste-

matised and co-ordinated the presentation of the complex theory with which it dealt, thus elevating to the status of a definite

branch of

knowledge a subject which, though fully ripe for such recognition, had up to that time suffered from the disadvantage of comparative inaccessibility. During the long period which has elapsed since the Text Book was first published considerable changes have been made in the scientific

educational course prescribed by the Institute

;

in particular the

value to the actuarial student of a competent knowledge of the

elements of the Differential and Integral Calculus has been more fully realised

and these subjects are now included in the mathe-

matical course with which the training of the student begins. secure a consistent educational scheme

it

To

has therefore been found

necessary to re-arrange the volume hitherto entitled Part II of the

Text Book and

to bring into greater

prominence those mathematical

demonstrations which were formerly treated as subjects of optional study.

The

theoretical basis of these demonstrations

in the mathematical text book which

is

is

now included

issued as a separate work,

and certain chapters of the old Text Book, Part II, are omitted from the present Volume which thus deals exclusively with the theory of actuarial science so far as relates to Life Contingencies.

These are the principal changes introduced, but the oppormake certain alterations which

tunity has also been taken to

bring the work more fully into conformity with modern require-

ments.

new

At the request

treatise

of the Council the compilation of the

was undertaken by

Mr

E. F.

Spurgeon who brought

to his task the indispensable qualifications of long experience

and a conspicuous gift of exposition. The resulting by the Council in the belief that it will fully meet the needs of students and promote the attainment of those high professional qualifications which are connoted by the Fellowas a Tutor

Volume

is

issued

ship of the Institute of Actuaries.

A.

May

1922.

W. W.

AUTHOR'S PREFACE In the preparation of

this

Volume and

in determining the order

in which the various subjects should be dealt with, special attention has (i)

been directed to the following considerations, namely,

That the student should,

at

an early stage of his work,

acquire a sound knowledge of the principles of the construction of Mortality Tables,

and of the evaluation

Annuities and Assurances, on the simplest possible (ii)

That in applying these principles

to the

more

of

basis.

intricate

parts of his work, the student should automatically revise

knowledge of the earlier portions, on a sound grasp of which so much dependshis

(iii)

That such information on

practical points as will

be of

more advanced studies be given, even though at times

assistance to the student in his

should, where

these

may

possible,

not be strictly within the limits of Part II of

the Institute Examinations.

The Book has been divided

into three parts:

—Part

I deals

with Mortality Tables and Single-Life Functions; Part II with Functions involving two or more lives; and Part III with those subjects which could not conveniently be fitted into either of the earlier parts.

In Part I

it

was

essential that the first subject dealt with

should be the Mortality Table, and in the second chapter

deemed advisable

it

was

to proceed directly to Mortality Tables con-

structed from Life Assurance Statistics, with particular reference

This enables us, in Chapter III, to deal at once with Single-Life Annuities payable annually, Single-Life Assurances payable at the end of the year of death and Annual Premiums to Select Tables.

In Chapter IV, Premium Conversion and in Chapter V, varying Annuities and Assurances and Premiums for special classes of assurances with reference to many practical points. Chapter VI deals with the

based on Select Tables. Tables are dealt with

;

values of Single-Life Policies subject to annual Premiums.

AUTHOE S PREFACE It will

Vll

be seen that by taking the subjects in

this order the

by the end of Chapter VI, have acquired a knowledge of all the principles involved in the calculation of probabilities and in the evaluation of Annuities and Assurances, on the basis of Single Life Annual Functions, all the later work being an exten-

student

will,

sion or modification of the matters dealt with in these Chapters.

As a

result of this arrangement, we are able in a single chapter (Chapter VII) to dispose of Annuities and Premiums payable more

frequently than once a year, together with the values of policies subject to such premiums.

payable at the

moment

In Chapters VIII and

of death

IX

Assurances

and Complete Single-Life An-

nuities are discussed.

Chapter X, dealing with Life

Office Valuations, has

been

in-

cluded in order to avoid the difficulty which Part II students have

met with

in the past, namely, that they have acquired a

of Policy Values only to find that in their later

knowledge

work Policy Values,

as such, are seldom used.

In order to prepare

now becomes

for subjects included in

Part II of the Book

Makeham's and in Chapter XII to deal with statistical applications of the Mortality Table and the Expectation of Life. In Part II, when considering Joint-Life and Contingent Probabilities, Joint-Life and Last-Survivor Annuities and Assurances, and Contingent Assurances, the student will be merely extending it

Law

necessary, in Chapter XI, to consider

of Mortality,

the principles studied in the

first

few chapters.

Part III consists of two chapters only.

In Chapter

XX (Con-

struction of Tables) the advantage of the Arithmometer has been

emphasised and

it

has been considered necessary again to include

a plate of a machine showing a particular calculation, the description of the

machine being taken word

Book. In Chapter

XXI

for

word from the

old

Text

the question of Tables involving two or

more causes of decrement has been introduced in the simplest possible manner.

At the end of the Volume it was firstly,

considered advisable to include,

a Mortality Table based on population data (The English Life [NM]

) and thirdly, M of its value for Graduation) because (Makeham Table the H on the Functions instructional purposes. The Tables of Monetary basis of the English Life Table, No. 8, which have not previously

Table, No.

8),

secondly, a Select Life Table (The

V1I1

AUTHOR S PREFACE

been published, have been supplied entirely by the Prudential [NM1 experience Assurance Company; the Tables based on the have been taken from those published by the Institute of Actuaries and Faculty of Actuaries jointly and those based on the H M experience from the old Text Book Part II. It has been a distinct advantage to have a great part of the work already mapped out by Mr George King in the old Text Book, and to have been able to make use of his collection of symbols and expressions in such chapters as that relating to Compound Survivorship Annuities and Assurances. It would have added very materially to the work of preparation of the present volume had these not been available. ;

In conclusion I wish to express very valuable

assistance

rendered

my to

high appreciation of the

me by

Messrs

W. W.

Williamson, F.I.A., and C. C. Barrett, F.I.A., both in criticising

my

manuscript and in reading the proofs.

If this book prove of

the value to students and to the profession which I hope, their exceedingly helpful criticisms and suggestions will have contributed thereto in no small degree. E. F. S.

5

TABLE OF CONTENTS PART CHAPTERS

I

TO XII

I

MORTALITY TABLES SINGLE-LIFE PROBABILITIES OF LIFE

AND DEATH

AND ASSURANCES STATISTICAL APPLICATION OF THE MORTALITY TABLE

SINGLE-LIFE ANNUITIES*

CHAPTER

I (Pages

1—18)

THE MORTALITY TABLE— MORTALITY TABLES CONSTRUCTED FROM POPULATION STATISTICS PAGE Definition of Mortality Table

The number

living

Deaths dx Rate of Mortality

lx

:

and

1

.

.

Lx

1, 2,

2

:

Method

2

...

qx

:

of Construction of Mortality Table

.

3

—

Radix of Mortality Table Limiting Age a Central Death Rate mx

3 3

:

4

:

Probability of surviving one year

5

px

:

Fallacy of constructing Mortality Table from deaths alone

.

6,

.

Graduation English Life Table No. 8 Probabilities of Life

Force of Mortality

and Death

:

npx n \qx ,

,

\

n Qx>

.

may

n\mQx

•

.

:

Force of Mortality

7 7

px Relation between px and q x Formulae for value of /ix Relation between /ix and colog e px

When

3

8

—

10,

12

.

11

7—8 & 13

— 15 12

.

.

exceed unity

.

....

13,

14

15,

16

15

Differential Coefficients of various functions

16,

17

Examples

17,

18

X

CONTENTS

CHAPTER

II (Pages

19—30)

MOETALITY TABLES CONSTRUCTED FROM LIFE ASSURANCE STATISTICS— SELECT LIFE TABLES PAGE Assumptions made Exposed to Risk

for purposes of explanation

Ex

... .... :

Ultimate Tables

20

.20

.

Aggregate Tables Select Tables

19,

20, 21

...

.

21—25

.

24

.

British Offices Experience

M 0™, (0

M(6) ),

,

Institute of Actuaries Life Tables

(H

Brief reference to

Brief reference to

),

.

.

.

28

27

Value of inx]

and Death by Select Table

Examples

.

CHAPTER

26 27

Practical Application of Select Tables

Probabilities of Life

.

.

III (Pages

.

.

.

.

.28 .

29—30

31—59)

AND ASSURANCES. ANNUAL PREMIUMS

SINGLE-LIFE ANNUITIES

Assumption made in calculation of monetary values Pure Endowment

Ax

:

Whole-Life Annuity

:

.

31 31

ji

ax

31—32

.

.

Annuity-due a. x Deferred Annuity n \ax Temporary Annuity ax .^\ or nax Deferred Temporary Annuity n m ax Commutation Functions D x and N^ Formulae for Annuity values in terms of Commutation Functions Temporary and Deferred Annuities- due &x-.nl, n a^ Older form of Commutation Function N,. :

.

:

:

|

:

.

\

:

.

:

Commutation Functions based on

.

|

.

:

Select Tables

Annuities deferred a fraction of payment period

Dy, Ny,

:

1

:

:

Ax: ^],

etc

Premiums based on

Select Tables

Annual Premiums

P(J]: SI, Pr^,

:

PM: S1

\

n

Ax A ,

.

,

.... :

Endowment Assurances

34 34 35 35

36

37—38 39 40

Temporary and Whole-Life Assurances A^.^I or Commutation Functions for Assurances G x M x Deferred Whole-Life and Temporary Assurances Assurances based on Select Tables

33 33

Tc

Table of Expressions for Annuity Values Approximation to isolated Annuity Value

:

etc.

i|aj;, i| as-.nl

:

32

32—33

n \A x , n

41,42 43 43

44

44—45 45

46—49

.

CONTENTS

XI PAGE

Annual Premiums

for

Limited Payment Policies

tP[xh tP[aO:n]j

:

48,

tP[x]-.n\

Table of Expressions for Single Premiums for Single-Life Assurances Table of Expressions for Annual Premiums for Single-Life Assurances

51

Numerical Example of Annuity Fund Numerical Example of Assurance Fund Examples

CHAPTER IV

49 50

.

52 53,

55

(Pages

54

— 59

60—67)

RELATION BETWEEN ASSURANCES AND ANNUITIES. PREMIUM CONVERSION TABLES for Single and Annual Premiums in terms of Annuities Premium Conversion Tables Annual Premium Conversion Tables Comparison of Annual Premiums by Select and Aggregate Tables

—62 — 63 63 — 66

Formulae

60

Single

62

...

.

Whole-Life and Temporary Annuity Values in terms of

CHAPTER V

(Pages

P and d

.

66

.

67

68—94)

VARYING SINGLE-LIFE ANNUITIES AND ASSURANCES. OFFICE PREMIUMS AND SPECIAL CLASSES OF ASSURANCES Commutation Function Bx Increasing Temporary and Whole-Life Annuity-due (lai) X :n\, (la^ Increasing Temporary and Whole-Life Annuities (Ia) x: JJ, (Ia) x Old form of Commutation Function S^ Examples of Varying Annuities Commutation Function ~RX

68

:

:

:

:

.

:

(IA)i

:

^I, (IA)j.

....

69

69,

70

70, 71

.71 71,72

Increasing Assurance in terms of ordinary Assurances

.

.

.72, 73 73 77

—

Participating or With-Profit Policies

.78

Guaranteed Bonus Policies Policy with varying Sum Assured and Varying Premiums .

.

Premiums Premiums for Instalment Policies Premiums for Debenture Policies Premiums for Double Endowment Assurances Assurances subject to Increasing Premiums Assurances with Return of Premiums Pure Endowment with Return of Premiums

79,

...

Deferred Assurances with Return of Premiums

80 81

....

Options

78

.

Office

.81 82 .82 83 .86

....

Discounted Bonus Policies Mortality Experience of Different Classes of Assurances

Examples

69

.69

Examples of Varying Assurances

.

68,

.

.

:

Increasing Assurances

.

87

88 90

...

91

92

—94

CONTENTS

Xll

CHAPTER VI

(Pages

95—127)

VALUES OF SINGLE-LIFE POLICIES SUBJECT TO ANNUAL PREMIUMS PAGE 95

Definition of Policy Value

Whole-Life Assurance policy value Endowment Assurance policy value

:

(

:

V[ X]

95

tV[x]:n\

96

Prospective and Retrospective Methods of obtaining Policy Values

96,

.

97

Values of limited payment whole-life and endowment assurance policies 97, 98 99 Policy Values when net premiums are not valued ,V 99 101 Fractional Durations Values at _, Policy iV , L ' :

t+ - [xY t+j

.... — .

.

fo;]:»|

Values of Participating Policies Pure Endowment Policy Values

101

Deferred Assurance Policy Values

103

Endowment Assurances with

102

return of premiums, policy values

.

103

— 104

Instalment and Debenture policy values

104

Double Endowment Assurance policy values Temporary Assurance policy values Tables showing progress of various assurance funds Table Whole-Life Assurances Table B Temporary Assurances Table C— Pure Endowment Table D Endowment Assurances Alternative formulae for Whole-Life policy value Alternative formulae for Endowment Assurance policy value Comparison of Policy Values by different Mortality Tables Condition for equal Whole-Life policy values by two different

105

.

.

.

A—

105

106

107—108

—

—

.... .

.

.

— 110 109

109

110

—111 Ill — 112 110

112

— 115

114—115

Tables

Surrender Values

116

Free or Paid-up Policies

116—119

Alterations of Policies

119

.

Whole-Life to Endowment Assurance

Endowment Assurance

to

mature at

119 earlier age

than under

original contract

Limitation of

number

121 of future

premiums

Application of cash value of bonus to alter policies

Examples

— 123 — 121 121

.

122 123

— 123

— 127

:

CONTENTS

CHAPTER

Xlll

VII (Pages 128—150)

SINGLE-LIFE ANNUITIES AND PREMIUMS PAYABLE MORE FREQUENTLY THAN ONCE A YEAR. VALUES OF SINGLE-LIFE POLICIES SUBJECT TO PREMIUMS PAYABLE MORE FREQUENTLY THAN ONCE A YEAR PAGE Whole-Life Annuity payable

m times a year

:

a£

B) .

m)

Practical Value for c4

.

....

Whole-Life annuity-due payable m times a year a?' Temporary annuity and annuity-due payable m times :

(m)

m

times a year,

yearhence: i|air\i|a (mL x

year:

for

i

payment

first

— th

than

less

of a 131

|

t

t

Form

a

129 129

(m)

Annuities payable

General

128—129

.

|

a^m) for all values of

m

132

....

2m

Annuity values

in terms of Commutation Functions Continuous Annuities dx a x -.n\ :

132 133

,

134

Differential coefficients of various functions

Constant addition to rate of mortality or rate of interest

Premiums payable m times a year True premiums for Whole-Life Assurances True premiums True premiums

P|3

:

P

.

for

Endowment Assurances

for

Limited Payment policies: jP^

:

|a

.].^

Instalment premiums for Whole-Life Assurances

.

.

.134 135

— 139

135

— 136

.

,

137

•

(P[a;]:n]

.

137

— 138

1

:

Instalment premiums for Limited Payment policies

.

Pjj,].^]

:

jPf^j

1 ,

.

.

138

.

•

139

*P[™]1^

139

(Industrial Life Assurances)

p«,p.:5i Values of Policies subject to true premiums payable Whole-Life Assurances, Integral duration

Endowment

.

1

Pf^j

:

Instalment premiums for Endowment Assurances

Premiums payable weekly or monthly

.

:

Assurances, Integral duration

t

:

Whole-Life Assurances, Fractional duration

139

m times a year

Vf™|

tV^.^] :

140

....

L Vf™]

m

.

,

.

,

.

140

.141

_i_Vf™]

+ to + sm

142—144 Whole-Life Assurances with practical formulae and numerical 144

example

Endowment

145

Assurances, fractional duration

Values of Policies subject to Instalment Premiums Whole-Life Assurances t V$, etc :

Endowment Assurances (V^.^j, :

146—147 147

etc

148

Values of Limited Payment Policies

Examples s.

149—150 b

XIV

CONTENTS

CHAPTER

VIII (Pages 151—160)

ANY OTHER MOMENT THAN AT THE END OF THE YEAR OF DEATH

SINGLE-LIFE ASSURANCES PAYABLE AT

page Assurances payable at

moment

Temporary Assurances Whole-Life Assurances

of death (Continuous Assurances)

:

A., .^j

:

A[x ]

Endowment Assurances

152 152

-A[x] n]

:

:

Alternative formulae by Aggregate Tables Single

151

....

Premium Conversion Table

:

A

152- -153

.•jr

153

.

Practical approximation

154

Deferred Assurances: „|A b ], n

Annual Premiums:

'"'Pfr],

l

|

m A fc

155

]

155

"'P[a;]:n]

Premiums payable m times a year °°'P[™]', etc. Continuous Premiums: ("'P^], etc. Continuous annual premium conversion table :

155

(

155 155- -156

Policy Values

156

Differential Coefficient of

M^

Differential Coefficient of

A,

156

157

.

Assurances payable at end of

— th m

interval in

— 158 158 — 159

which death occurs

Increasing Assurances payable at moment of death Continuously Increasing Assurances in terms of a x and (la) x .

.

157 .

.

.

Examples

CHAPTER IX

159 160

(Pages 161—171)

COMPLETE SINGLE-LIFE ANNUITIES Definition of Complete Annuity

161 m)

Complete Whole- Life Annuity

:

«L

Complete Temporary Annuity

:

dj"^|

...

Expression for value of A^ in terms of a x Why there cannot be a complete annuity-due

161 .

—164 164

v

164

....

Table of values of « 30 and d 60 by various formulae in any event until payments amount to purchase

165 165

Annuity to continue price

Annuity with return of balance of purchase price Examples

166—167 167

168—171

CONTENTS

CHAPTER X

XV

(Pages 172—190)

LIFE OFFICE VALUATIONS PAGE Policies valued in

Groups

172

Whole-Life Assurances, Valuation of Method of Grouping

By

172

— 176 172

Aggregate Tables

Assumptions as to age and net premium Sums Assured and Bonuses Annual Premiums True Half- Yearly and Quarterly Premiums

.... ....

172 172

— 173 173 173

174

— 175

Instalment Premiums

175

Year of Birth

Office

Endowment Method

.

176

.

Assurances, Valuation of

176

176

Meaa Valuation Age for Group Sums Assured and Bonuses

(see also Chap. XI, pp.

200—202)

176

.

176

Annual Premiums True Half- Yearly and Quarterly Premiums Instalment Premiums

....

Gross Premium Valuation Limited Payment Policies, Valuation of Profit or

— 179

of Grouping

176 176

— 178

178

— 179

181

— 182

180

Loss of a Life Office

.

182—184

Death Strain at Risk 184 Expected Death Strain .184 Equation of Equilibrium 185 185 190 Effect on Whole-Life policy values of variation in Valuation Bases Effect on Whole-Life policy values of variations in the Rate of Interest 188 Effect on Whole-Life policy values of variations in the Rate of

—

188—190

Mortality

CHAPTER XI

(Pages 191—202)

MATHEMATICAL REPRESENTATION OF THE LAW OF MORTALITY; GOMPERTZ'S AND MAKEHAM'S LAWS Gompertz's

Law

of Mortality

191

Formulae for fix and lx Formulae for t px and log px Makeham's Law of Mortality Formulae for /%, lx and p x Value of A x

— 192 191

192

t

192 192

t

193

How to obtain the values of the constants k, c, g How to construct a table of lx when the values of the constants s,

are

known

Curve of Deaths

.

.

193

— 194

194

— 195

196—198

&2

CONTENTS

XVI

PAGE

Makeham's Law of Mortality (cont.) Age when dx has a maximum value Age when dx has a minimum value

....

Effect of constant increase in px Effect of increase in constant B in the formula for

197 198 198

px

198

.

198—199

.... —

Applied to Select Tables

Makeham's Second Development of Gompertz's Law Mean Age for Valuation of Endowment Assurances in Groups

CHAPTER

.

200

199 202

XII (Pages 203—220)

STATISTICAL APPLICATIONS OF THE MORTALITY TABLE Population aged x and upwards

:

Tx

203 204

Differential Coefficient of T^.

Ratio of number of deaths to population Expectation of Life ex and ex

204

205

:

—205

—206, 208

Average age at death of persons who attain age x Temporary complete expectation of Life nSx Average age at death of persons who die between ages x and x + n 207 Comparison of average ages at death amongst different populations Effect of immigration on average age at death Average age at death of present population aged x and upwards 210 Value of p x in terms of complete expectation of life

207 207

:

\

—208 209

.

.... .

Proof that a^\

>ax

210

— 211 212

212—213

—220

Examples

214

PAET CHAPTERS

II

XIII

TO XIX

FUNCTIONS INVOLVING TWO OB MORE LIVES PROBABILITIES OF LIFE AND DEATH JOINT-LIFE AND LAST SURVIVOR ANNUITIES AND ASSURANCES CONTINGENT ASSURANCES REVERSIONARY ANNUITIES

CHAPTER JOINT-LIFE

XIII (Pages 223—237)

AND SURVIVORSHIP PROBABILITIES

Expressions for probability that (x)

and

(y) will

both survive n years

m lives will all survive n years

:

n p xy

223

223

:

CONTENTS

XV11 PAGE

Expressions for probability that (#)

and

(y) will

both die within n years

: \

n Qiy

223, 226

.

m lives will all die within n years One only

of (x)

and

223

survive n years

(y) will

n jo—

:

.

At least one of (x) and (y) will survive n years n p^j The joint existence of (x) and (y) will fail within n years Value of dxy .

:

Probabilities relating to the

the joint

both

(x)

life- time

and

(