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INSURANCE

A PRACTICAL EXPOSITION FOR THE

STUDENT AND BUSINESS MAN.

T. E. YOUNG, B.A., F.R.A.S..Ex-President of the Institute of Actuaries ; ex-Chairman of the Life Offices'Association ; member of the Actuarial Societies of Belgium, France, andAmerica ; late lecturer on the Principles of Insurance at the London School

of Economics and Political Science.

' Alternate progress and impediment,' And yet a growing prospect in the main."

The Recluse Wordsworth,

London :SIR ISAAC PITMAN & SONS, Ltd., i Amen Corner, E.C.

ISAAC PITMAN & SONSfci3i UNION SQUARE,

Printing OppicbOF THE

PUUlilSHGKS.

PREFACE

This book is designed to afford a clear and succinct exposition, inan elementary form, of the principles, theoretical and practical,upon which insurance administration is based and conducted.

Art in every department of executive work naturally precedesscience or organized knowledge : actual experience must providethe materials out of which the intellect must educe into generalisa-tions and formulae the scientific principles which the practiceimplicitly involves ; and the rational prosecution of Art into itsmost manifold and widest forms is conditional upon the clearer andxacter manner in which its intellectual elements are unfolded andsystematized.

The art, or practical conduct, of insurance will thus prove to bemore readily intelligible to the apprehension of the man of businessif the principles which it embodies are explicitly presented ; andthe scope of this book accordingly comprises, with this object inview, the concurrent description of administration and of therational methods, derived from experience, on which the actuailprocesses are founded.

In pursuance of this course, an attempt has been made to explainthe mode in which the interpretation of the Board of Trade Returnsof Companies may be rendered practically serviceable in discerningthe financial position of Assurance Offices and the prospects ofprofits which they are competent to present.An exposition has also been introduced of the various modes in

which the universal and flexible principles of insurance have beenapplied to provide for popular and commercial needs which thecomplex development of social and industrial activities demands.The book, it will be observed, is not intended for the use of

experienced experts ; but for the student who aspires to thecharacter of an expert, and for the busy practical man, everyeffort has been employed throughout the volume, though in a rudi-mentary form, to produce a clear, sound, and intelligible guide tothis department of business in its multiform shapes.

In the Examples furnished in the book, different Tables ofMortality have been utilized according to convenience : and thiscourse introduces no difficulty of perception or result since each

IV. PREFACE.

illustration is of homogeneous character in itself, and the expresspurpose, moreover, in every part, has been simply the exhibition ofcomparative effects.

It need hardly be necessary to state that no particular Companyor method of administration is recommended or disapproved

where any existing or suggested method, however, required definitecomment for its complete understanding, a reasonable criticismhas naturally been submitted.The author, while drawing mainly upon his own experience and

thought as a practical manager and actuary, gratefully acknow-ledges his deep indebtedness to the numerous valuable papers anddiscussions which the Journal of the Institute of Actuaries contains;nor should he omit a special reference to Part II of the Text Bookwhich has been issued by that Institutean Association, it mayjustly be added, which has signally conferred upon the scheme oflife assurance, and incidentally upon the general principles under-lying every phase of insurance, a distinctive character as a bodyof scientific truths.

The author is also greatly indebted to Mr. C. J. Bunyon's admir-able works upon the Law of Life and Fire Insurance, and to Mr.William Gow's excellent Handbook upon Marine Insurance; hehas also helpfully drawn upon the resources of other writers.A certain amount of repetition has, unhappily, been unavoidable,

since some subjects possess aspects which relate to questionsdiscussed in more than one chapter. This fault, he trusts, will bereadily pardoned, and he has endeavoured to reduce it to the briefestextent.

ANALYTICAL TABLE OF CONTENTS.

PAGEPreface ... ... ... ... ... iii.Preliminary Remarks ... ... ... ... xvi.

Chapter I.The Numerical basis of Assurance Calculations, with

Explanations and Discussions of Cognate Subjects.i. An exposition of BernouilH's Theorem of Large Numbers as

the basis of the General Doctrine of Probabilities and ofthe system of Life Assurance ... ... ... 19

ii. Illustrations of this principle, and the analogies in LifeAssurance observations : Defect of Tables showing theprogress of a Life fund ..

.

... ......

20iii. The Probable Error, with numerical illustrations ... 23iv. The meaning of Error ... ...

...... 24

V. The detection, by observation, of data more precise than theobservation itself

...... ... ... 24

vi. Detailed illustration of the Probable Error ... ... 24vii. The scientific distinction between an Average and a Mean ... 25viii. Explanation of an Average ... ... ... 26ix. Explanation of a Mean ... ... ... ... 27X. The Weights of observations, and their use ... ... 27xi. An illustration of the formation of a Mean ... ... 29xii. Recapitulation of the distinction between an Average and a

Mean, and their respective Values ..

.

... ... 30xiii. Illustration of erroneous modes of expression in Life Assur-

ance language ... ... ... ... 32xiv. Criticism of the expressionthe Expectation of Life ... 32XV. The meaning of the Value of a Life Annuity and a Reversion 34xvi. The fallacy of Percentages ... ... ... 34xvii. The defects of many statistical deductions

... ...35

xviii. An enumeration of the accidental errors which occur in allobservations, and the consequent need of the adjustmentor graduation of the results ... ... ... 35

xix. A brief description of some methods of Graduation ... 36XX. An explanation of " portions " and " proportions " of diminu-

tion in the power to resist death ...... ...

37xxi. The conditions of Validity of a method of Graduation ... 39xxii. The assignment of weights to observations in any system of

Graduation ... ... ... ... ... 39xxiii. The necessity of Graduation in forming a basis for Assurance

Premiums ... ... ... 40xxiv. The deletion of actual physical phenomena occasionally pro-

duced by some systems of Graduation, with an illustration 40Ttxv. A suggestion for the concurrent presentation of original and

adjusted data... ... ...

... 41xxvi. The Method of Least Squares, with illustrations of its appli-

cation, .,. ... ,,.41

VI. CONTENTS

Chapter II.The Risk and its Limitation ; with Explanations and

Discussions of Cognate Subjects. pagei. The Nature of Risk general!}' in the doctrine of Probabilities 44ii. The risk in Life Assurance ... .. 44iii. The diminution ot the amount at risk in consequence of the

Reserves created... ...

... ... 44iv. The mode in which the Limit of Risk in Assurance Com-

panies is determined ... ... ... ... 45V. As the area of distribution of risk is widened, the chance of

inordinate loss and the chance of the absence of loss con-currently diminish

... ... ...... 46

vi. Illustrations and examples ... ... ... 46vii. The practical method of settling the amount of the Limit on

a single contingency ... ... ... ... 47viii. The successive increase of the limit of risk ... ... 49'ix. The adoption of the maximum limit in the acceptance of

" rated-up " lives... ... ...

49X. The assurance of special risks, and the conditions on which

this course depends, with explanations and illustrations ... 51xi. The Expectation of Loss

... ... ...... 54

xii. The most probable number of claims and losses... ...

55xiii. The number of observations on lives required to provide for

fluctuations of occurrence within specified limits ; andvarious illustrations of the mode of calculation

... 55xiv. Remarks upon the necessity of an adequate number of risks

in the Valuation of various Societies... ...

59XV. The mode of constructing the "loading" added to the pure

premium for expenses and profits, with illustrations ineach instance

......

... ...60'

xvi. Will a percentage addition to the pure premium compensatethe same percentage addition to the rate of mortality ? ... 6S

xvii. A consideration of the uniform or level, and Assessment,scales of Premium

... ... ... ... 63xviii. Tontine schemes

... ... ... ...65

xix. The mode of acceptance of lives for Assurance ... ... 66XX. The question of "rated-up" lives: the incidence of addi-

tional mortality ; the removal of extra premiums ; theschemes proposed for avoiding the imposition of an extrapremium in the ordinary form ; and the lives of teetotallers 67

xxi. The mode of assessing a general extra premium, with ex-amples of Female lives and Publicans

... ...73-

Chapter III.The Meaning and Effect of Selection.

i. The meaning of the term Selection in Life Assurance...

75>ii. The twofold process of selection concurrently operating

... 75iii. The selection exercised by the Company : its object and effect 75iv. The mode of " rating-up " inferior lives

...... 76

V. The selection exercised by the Policy holder, and its effects... 77vi. A numerical example of the probabilities of life, deduced from

a Mixed Table of Mortality and a Table of Select Mortality 77vii. Explanation of these examples

... ... ...78.

viii. The operation of Mortality in successive years of Assurance 78-ix. The rate of Mortality among Assured lives is a function of

two variables, ,

,

... ...78-

X. Explanation of the terms employed in connection withSelection ...

., 78

xi. The duration of select effects 79

CONTENTS VU.PAGE

xii. The consequent construction of the Hm (5) and Om (5) Tables,and their employment

,.

.

... ... ...79

xiii. Table of the Mortahty per cent, for certain periods ofAssurance

...... ... ... ...

79xiv. Notes on the Table

... ... ... ...80

XV. Explanation of the annual rate of Mortality... ...

80xvi. Mixed and Select Tables : the mode of construction of a

Mixed Table... ... ... ...

80xvii. Composite character of the rate of Mortality among Assured

lives... ... ... ...

81xviii. The mode of construction of a Select Table of Mortality, and

its character...... ,,, ... ... 81

xix. The advantage of the employment of Select Tables, especiallyin the formation of Premiums

... .... 82

XX. The scientific character of Select Tables... ...

82xxi. An explanation of the effect of employing Mixed Tables upon

the Premiums and Annuity-Values for young and old lives 82xxii. A rough illustration of the working of Mixed and Select Tables 85xxiii. Examples of the probabilities of life, premiums, annuities,

and reserves by a Mixed and a Select Table ... ... 86xxiv. The appropriateness of Select Tables in the settlement of

Surrender-Values... ... ...

... 86XXV. A description of the symbols used for the functions derived

from Select Tables ... ... ... ... 86xxvi. An explanation of the differences between the values of pre-

miums, annuities, and reserves produced by Mixed andSelect Tables ... ... ... ... 87

xxvii. An explanation of the comparative effect of Mixed and SelectTables at old ages ... ... ... ... 88

xxviii. Examples of reserves according to Mixed and Select Tables 89xxix. An explanation of the differences in the reserves... ... 89XXX. The addition to be made to the Annuity-Value in obtaining

the Reserve in a single instance and in a. classified mass ofPolicies ... ... ... ... ... 90

Chapter IV.Surrender Values.

i. Popular misapprehension upon the subject ... ... 91ii. The absorption of part of the contract premium in the pay-

ment of claims and of the other portion in expenditure ... 91iii. An illustration of Beserves on the basis of a Select Table in

relation to the Premiums received ... ... ... 92iv. In an Assurance Company the interests of all members must

be equally considered ... ... ... ... 92V. Lapses and surrenders increase the rate of mortality among

the remaining members, since on the whole, the with-drawals occur upon sound lives ... ... ... 93

vi. The surrender-value, therefore, will be governed by equitabletreatment of all the members ... ... ... 93

vii. A discussion of the effects produced by withdrawals ... 94viii. A numerical illustration of the proposition that the with-

drawal of members exercises the same effect upon thegeneral rate of mortality of the continuing members as theintroduction of a number of inferior lives ... ... 95

ix. A description of an attempt to measure numerically the resultof withdrawals, and the defect of that attempt : the truemethod mentioned ... ... ... ... 98

A. The adverse effect of withdrawals on the rate of expenditurein a twofold aspect ... ... ... ... 99

VUl. CONTENTSPAGE

xi. The conditions on which surrender-values may be granted inexcess of the ordinary scale ... ... ... 101

xii. The preceding considerations apply to forfeited Policies ... 102xiii. Companies endeavour to prevent lapses and surrenders by

means of advances ... ... ... ... 102xiv. The Policy may be maintained in force while the surrender-

value suffices to provide the premiums ... ... 102XV. The amount of surrender-value to be granted in the case of

" rated-up " lives ......

... 103xvi. A history of the published methods suggested for the compu-

tation of surrender-values...

... 105xvii. A summary of the principles underlying the calculation of

surrender-values ... ... ... ...110xviii. The adoption of a uniform scale of surrender-values ... Ill>ix. Extra premiums possess no surrender-value ... ... IllXX. The amount paid in surrender-values in a, year by Assurance

Companies ... ... ... ... Illxxi. Popular misconception of the amount of surrender-values

corrected by a numerical illustration of the progress of alife fund ... ... ... ... 112

xxii. The coincidence thereby exhibited of the retrospective andprospective methods of valuation

... ...114

xxiii. The percentage of the surrender-value upon the reserveshould be an increasing one

... ... ...114

xxiv. The value to be allowed for the surrender of a Bonus...

114

Chapter V.The Meaning and Process of the Periodical Valuations

OF THE Liabilities and Assets.i. A valuation ascertains the relation of past experience to that

anticipated with respect to mortality, interest, and expenses,and by means of a distribution of profits effects an equali-zation of these sets of elements ...

... ... 115ii. The three-fold meaning of a valuation explained...

... 116iii. The skill, knowledge, and prevision which a valuation involves 116iv. An illustration of the distinction between the vitality of males

and females, with an explanatory analysis, and the conse-quent difference of judgment in selecting the appropriaterates of mortality and interest

... ... ... 117V. The process of a valuation described ...

... 119vi. The identity of results obtained by a valuation of Policies

individually and in classes... ... .120

vii. The mode of classifying Policies for valuation at a commonage

... ... ... ... 121viii. The classification for valuation purposes of " rated-up " lives 122ix. A complete example at a. particular age of the method of

valuation, extracted from the Board of Trade Returns (withproportionately modified figures to avoid identification) ... 123

A. The production of the valuation balance-sheet ... ... 125xi. The surplus, equilibrium, or deficiency resulting ; and the

mode of rectification in the latter event... ...

126xii. The consideration of the proper proportion of the resulting

surplus to be distributed... ...

127xiii. The conservation of the interests of existing members, and

the regard also of the advantages to fresh entrants... 127

xiv. The question of a considerably reduced rate of interest forvaluation purposes

... ... ... ...128

XV. Definition of the pure premium. . ... ...

129

CONTENTS IX.PACB-

xvi. Definition of a pure premium valuation J 12i) "

ivii. The homogeneity of a pure premium valuation , 129xviii. A statement of a heterogeneous though sound mode of pure

premium valuation in respect of the rate of interest em-ployed

... ... ... ... ...129

xix. An illustration of an instance where different rates of mor-tality are involved

... ... ... ...130

XX. The mode of valuing Assurances upon " rated-up " lives ... 131xxi. The method of forming the adjustment of the customary

valuation factors... ... ...

... 132xxii. The adjustment of the values of the reversions employed ... 133xxiii. The adjustment for the unequal incidence of the premium

revenue ...... ...

......

134x\iv. The adjustment for provision of loading on Assurances

subject to single premiums and to a limited number ofpremiums ... ... ... ... ... 135

XXV. The reserve for Policies subject to discounted-bonus premiums 136jcxvi. The reserve for temporary Assurances

... ...137

xxvii. The reserve for extra premiums consequent upon foreignresidence or hazardous occupation ..

.

... ... 137xxviii. The question of negative values ...

... ...138

jcxix. The treatment of the loading in a valuation, with illustrations 1 39jtxx. The necessity, for an analysis of resources, of obtaining the

value of the loading, and the mode of comparing thereserved ratio of loading to premiums with the rate ofexpenditure actually incurred ...

...... 140

xxxi. Remarks upon a valuation at 2J % interest in relation to theloading

... ... ......

... 141xxxii. A qualification of the criticism upon the absorption of a por-

tion of the loading involved in a valuation at 2J % ; with anumerical illustration of the results obtained in two Com-panies, one valuing at 2J % and the other at 3 % ... 142

:xxxiii. The effect of employing a mixed table of mortality for thevaluation of Assurances upon lives recently selected ... 1 43

xxxiv. The effect upon valuation reserves of an alteration in therates of interest and mortality

... ... ...144

XXXV. The effect upon pure premiums and reserves of the absenceof interest altogether ... ... ... .. 151

xxxvi. A description of the retrospective and prospective methods ofvaluation, with a numerical illustration

... ...152

xxxvii. The re-insurance or hypothetical method of valuation ... 1 55xxxviii. A statement of the conditions under which the valuation-

reserves of the hypothetical and pure premium methodscoincide ...

......

... ... 156xxxix. A discussion of the pure premium method of valuation under

varying circumstances... ... ... ...

157jcl. Consideration of the suggestion of a special modification of

the pure premium method where the new business of aCompany is procured at inordinate expense ... ... 158

xli. The actual rate of interest in relation to income-tax anddepreciations of value ... ... ... ... 161

xlii. The mode of treatment of the investments and securities at avaluation

... ... ... ... ... 161xliii. The considerations which should govern the assessment of

the values of the investments ... ... ... 162xliv. Illustration of the effect of variations in prices ...

...163

xlv. The general practice of Companies in this respect...

163jilvi. The " quarter-average " plan ,., ... ... 163

X. CONTENTSPAGK

xlvii. Suggestion of a basis founded upon the relation between solarenergy, physical conditions, and human enterprise ... 164

xlviii. The course recommended until seme such scientific mcde isfound to be feasible ... ... ., ... 165

xlix. Appendix on the proper value and amount of a sum in respectof fractions of a year ... ... ... ... 165

Chapter VI.The Sources of Profit and its Distribution.

i. Confusion of terms : the definitions of surplus and profit ... 167ii. Considerations to be kept in view in deciding upon the

amount of surplus to be distributed as profit ... 168iii. The sources of surplus : excess interest, moderate expendi-

ture, and favourable mortality... ...

168iv. The question of surplus from lapses and surrenders ... 169V. The surplus from reversions purchased, and its treatment ... 169vi. Interest as the chief source of surplus

... ...170

vii. Capacity in Finance the supreme requirement of the Actuary 170viii. Surplus from loading

... ...... ... 171

ix. Economy of management a principal duty and test of theActuary or Administrator

... ... ...171

X. The considerations which should regulate the amount of newbusiness ...

... ... ... ...172.

xi. A tabular statement of the proportion of the new premiumsin a year which is absorbed in expenditure

...175-

xii. Consideration of the suggestion that the analysis of the totalexpenditure between the new and renewal business affordsa test of comparative economy

... ... ...175

xiii. The conditions which should control the extent of newbusiness

,. ... ... ... ... 176xiv. The case of a Company which has fallen into an enfeebled

condition ...... ... ... ...

176XV. The suggestion that the benefit of selection should be utilized

as a counterpoise to increased expenditure ...... 177

xvi. Should the first year's premium with the same object beexcluded from participation in profits ?

... ... 177xvii. Life Companies exist for family provision and profit, and not

for general social ends... ... ... ...

178xviii. The progress of Life Assurance expenditure ...

... 178xix. The demands which new premiums are required to satisfy,

and the effect on the bonuses of existing Policy holders 178XX. An illustration from the Board of Trade Returns of the

relation between rates of expenditure and profits ... 179xxi. Every aspect of Life Assurance administration possesses

actuarial implications... ... ... ...

180xxii. Practical suggestions in assessing the true burden of expendi-

ture upon the premium revenue ... ... ... 181xxiii. Mortality as a source of surplus

... ... ...182

xxiv. The case where a Company would genuinely obtain a profitfrom mortality

... ... ... ...1 82

XXV. The case where Companies adopt a general Table of mortalityas the basis of premiums and reserves correspondent withthe class of lives admitted : does the excess of the expectednumber of deaths (or claims) above the actual number con-stitute surplus in the sense of profit ?

... ...183-

xxvi. A greater number of deaths and a larger demand in claimsthan those anticipated may be more favourable than afewer number and a reduced amount, and Dj wsii

... 185

CONTENTS XI.PAGE

xxvii. Description of a mode of ascertaining the surplus or deficiencydue to the mortality experienced ... ... ... 185

xxviii. Numerical illustration of the method, showing the futility ofthe usual statements made respecting the favourable oradverse result of the mortality experience ...

...187

xxix. Description of a method for deducing the beneficial or un-favourable results ol the mortality experienced during thecourse of a quinquennium

... ... ...189

xxx. The surplus on mortality a function of the individual amountassured, or of the rate of premium per unit assured ... 190

xxxi. Analysis of a Company's actual experience of the proportionsof surplus from interest, mortality, and loading ... 191

xxxii. The principles upon which the method of distribution ofprofits should be founded ; with incidental remarks dealingwith the creation of separate series of participating Policyholders ; the attempted analysis of different scales of con-tribution to profits ; the probably main future factor inmodes of distribution, and other questions ... ... 191

xxxiii. A statement of some curious modes of division in the past ... 195xxxiv. Other methods described and criticised :

(1) In proportion to accumulated premiums ... ... 197(2) In proportion to the reserves ... ... ... 197(3) The deferment of participation until the accumulated

premiums equal the sum assured ... ... 198(4) The compound reversionary bonus scheme ... 199(5) Consideration of the conditions on which the compound

system is feasible, with a numerical illustration . . . 199(6) The mode of obtaining the uniform annual reversionary

addition... ... ... ...

201(7) A method based upon the specific appropriation of the

surplus-interest earned... ... ...

202XXXV. The difficulty introduced into the distribution of profits by

the present preponderant increase of Endowment Assurances 203xxxvi. The extension of Endowment Assurances largely aided by the

agitation relating to Old Age Pensions... ...

204xxxvii. The mode in which a change in the rate of mortality affects

the two component elements of an Endowment Assurance 204xxxviii. The prospect of profit from Endowment Assurances ... 205xxxix. The unsatisfactory effect on the genuine scheme of Life

Assurance of the increase of Endowment Assurances ... 206xl. The mode of converting a cash bonus into its equivalent

reversionary amount ... ... ... ... 208xli. The experience of a Company in respect of the rate of mor-

tality in Bonus options ... ... ... ... 208xlii. The plan to be pursued when a change in the method of dis-

tribution has become imperative ... ... ... 209xliii. The expenses of Industrial Companies ... ... 210

Chapter VII.The Investments of a Life Office.

i. Life Assurance administration a commercial trust with widepowers ... ... ... ... ... 213

ii. The paramount value of capacity in finance and economy ofmanagement in successful Assurance work ... ... 213

iii. Advice to students upon these points... ... ... 213iv. The principles of Life Assurance finance described, with

illustrations... ... ... ... 215V. A statement of securities which as a rule should be avoided... 218

xn. CONTENTS

PAGE-vi. The question of deposits in discount and banking institutions 219vii. Mortgages upon real estate in the Colonies and Foreign

countries ...... ... ... ... 219

viii. The purchase by a Company of its own shares discussed ... 220ix. Examples, with observations, of the mode in which mortgages

upon leasehold properties, and upon reversions, and advanceson personal security, should be scrutinized and conducted 221

X. The fallacy of the dictum that high interest signifies imper-fect security...

... ... ... ... 231xi. The mode of accounting when reversionary interests are

purchased... ... ... ... ... 232

xii. The retention of a portion of the surplus from the early reali-zation of reversions for the purpose of compensating thedeficiency of return in reversions whose falling-in is post-poned

... ... ... ... ... 233xiii. The elements of computation of the value of a Life Interest

and an Absolute Reversion... ... ...

234xiv. The analysis of the conditions in reversions in order to pro-

vide a formula, with an explanatory example...

... 237XV. The mode of expressing a professional opinion upon the

value of reversionary estates... ... ...

239xvi. The duties of the Administration of a Life Office in obtaining

the highest reasonable rate of interest... ...

239

Chapter VIII.The Interpretation of the Board of Trade Returns.

i. The value of the Life Assurance Companies' Act of 1870 ... 241ii. The relation of State control over the basis of valuation to

Governmental responsibility for solvency... ... 242

iii. A brief analysis of the requirements of the Act ... ... 243iv. The two primary elements to be considered are the ratio of

loading reserved compared with the rate of expenditure,and the realized return from interest in relation to thevaluation rate

... ... ... ...244

V. The particulars of a Company selected, and an analysisexhibited of its returns in the preceding respects

...244

vi. Explanation of the formula for obtaining the effective rate ofinterest

... ......

... ...246

Chapter IX.The Winding-up, Transfer, and Amalgamation of Assurance

Companies.

I. Winding-up.i. Causes of insolvency in Life Assurance enumerated

...250

ii. The provisions of the Act of 1870...

... ...250

iii. An explanation of the fact that in a reconstruction the Policiesof longer duration sustain a heavier reduction than morerecent Assurances

... .... ...251

iv. Methods of valuation as a basis for reconstruction... 253

II. Transfer,

i. Definition of the term ......

... ...254

ii. Basis of valuation precedent to a Transfer... ...

255iii. The limits of the scale of valuation

...... ...

255

CONTENTS Xiiu

III. Amalgamation.PAGE

1. Definition of the term... ...

...256

ii. Adjustment of the interests of each Company for equivalenceof future conditions

... ......

256iii. A description of methods proposed for this purpose

.,25T

iv. Powers of Transfer and Amalgamation contained in somedeeds of settlement

...... ... ...

261V. Novation

... ... ... ... ...261

vi. Observations upon Amalgamations generally ......

261

Chapter X.The uses of Life Assurance to the Business Man.

1. A Pure Endowment, or the provision of a fund at a deferredage ... ... ... ... ... 263

ii. An Endowment Assurance, or the provision of a fund at adeferred age or at earlier death ...

... ...263

iii. An Endowment Assurance, with an income receivable afterthe deferred age

... ... ... ... 263iv. Policies where the sum assured at death is paid by instalments 264V. Policies remaining at death in the Company's guardianship,

with the payment of a settled rate of interest... ...

264vi. Joint-Life Policies on husband and wife

... ...264

vii. Policies to secure a life-estate which will be forfeited to thefamily by death

... ... ... ...264

viii. The provision of a fund for children on the attainment ofadult age ...

... ... ... ...264

ix. The provision of a Whole-Life or Endowment AssurancePolicy for a child at a low rate of premium, provided thatdeath occur after adult age

... ... ...265

A. A Policy for a term of years to protect temporary businessinterests ... ... ... ... ... 265

xi. Partnership Assurances to compensate the abstraction ofcapital from the firm on the death of a partner

. ..

265xii. Policies to render absolute and marketable a contingent

reversionary interest ... ... ... 266-xiii. Policies to protect the expectancy to property which would

be defeated by the birth of issue ... ... ... 266xiv. Policies to protect the value of estates which are held on

condition of the retention of a special " Name and Arms " 266XV. Policies to protect a pecuniary interest in the event of a

present lunatic holder recovering sanity, and otherwisedisposing of it ... ... ... ... 267

xvi. Policies to provide against the loss of capital invested in thepurchase of leasehold properties, and the cancelment ofthe premium on Bonds redeemable at par ... ... 267

xvii. Policies to protect debts ..

.

... ... ...267

xviii. Policies effected by the wife or the husband for the benefit ofthe family under the Married Women's Property Acts ... 267

xix. Policies which include exemption from the payment of pre-miums during physical or mental incapacity produced byaccident ... ... ... ... ... 267

XX. Policies for the provision of estate duty under the FinanceAct of 1894 ... ... ... ... ... 268

xxi. Policies effected at minimum rates of premium in two modes 268xxii. Policies without medical examination ... ... 269xxiii. Annuities of various kinds ... ... ... 269-

JCIV. CONTENTS

Chaptek XI.A Brief Summary of some chief legal principles affecting

Life Assurance.PAGE

i. The nature of the contract of Life Assurance ... ... 270ii. The principle of Cat)ra< h//o)' inapplicable ... ... 270iii. All material facts to be disclosed, and the interpretation of

materiality of statement ... ... ... 270iv. The nature of the answer to be returned to any question if

positive evidence be not possessed ... ... ... 271V. In the employment of an agent in the submission of a pro-

posal, the obligation of complete disclosure exists...

271vi. The distinction Between'warranties and representations ... 271vii. The responsibilities of referees

... ... ...272

viii. A curious experience of the author cited respecting a referee'sambiguous replies

... ... ... ...272

ix. The proposer is not responsible for the fraud of a referee ifhe himself be innocent

... ... ... ...273

X. The nature of the requisite insurable interest, and a state-ment of its constitution ...

... ... ...273

xi. The form of Policy : its construction and conditions ; withthe mode of rectification of any errors ; the payment of thepremium, and renewal notices ...

... ...275

xii. The Assignment of Policies, and Notices of Assignment ; thepriorities of encumbrances ; a statutory form of Assignment 278

xiii. The priorities of charges upon a Policy..

279jtiv. Death, and the presumption of death ; a striking experience

of the author... ... ... ...

282

Chapter XII.Fire Insurance

i. The question of the reduction of fire insurance statistics to ascientific form ...

... ... ...284

ii. The difficulties of an attempt of this nature involved inpartial loss, moral hazard, and other elements of risk ... 284

iii. Description of two attempts to introduce a quasi-scientificmethod into the tabulation of Fire Insurance statistics .

.

.

286iv. The Contract : its nature and conditions ...

...289

V. The Proposal and its statements ; the Deposit Receipt

;

special incidents that may occur ; warranties and repre-sentations

; the survey ... ... ... ... 289"vi. The Insurable Interest : the terms " legal ' ' and " equitable '

'

defined : Trustees : Mortgagees : protection of purchasers 291-vii. The nature of the Policy ; specific : the average clause

:

Floating Policies : Valued Policies... ......

292viii. The period of protection under Fire Policies ...

...295

ix. Alterations in tlie premises ... ... ... 296X. The causes leading to the forfeiture of a Policy ... ... 297xi. Assignments and Notices ... ... ... ... 297xii. The Loss, with its incidents and conditions ... ... 298xiii. Reinstatement... ... ... ... ... 301xiv. Arbitration ... ... ... ... ... 301XV. The adjustment of the loss ... ... ... 302xvi. Successive losses ... ... ... ... 302xvii. Double Insurance ... ... ... ... 302xviii. The Insurance of rent ... ... ... ... 303xix. Insurance by a tenant ... ... ... ... 303iNt. The Tariff Association an

CONTENTS XV.

xxu. The preservation of Property from Fire a National concern ... 304xxiii. Fire inquests ...

... ...... 305

xxiv. Municipal Insurance... ... ...

...305

XXV. The change of style in the Calendar : an incident ... 306xxvi. The Reserve for unexpired risk ...

... ... 307

Chapter XIII.Marine Insurance.

i. A brief history of Lloyd's... ... ...... 309

ii. Other historical notes .. ... ......

312

Chapter XIV.Marine Insurance {continued).

i. A Contract of Indemnity... ... ... 317

ii. Consideration of the anticipated profits from the sale of thegoods insured ..

... ... ...317iii. The conduct of the business ...

... ... 317iv. Insurable Interest ... .. ... ...319V. The Policy : its forms and conditions ; the course of the

voyage : explanation of terms ... ...... 321

vi. Assignments... ... ... ... ... 325

vii. Perils of the sea : Jettison : Barratry... ...

...325

viii. Loss of the vessel and cargo : Abandonment : Constructivetotal loss ... ... ...

...... 327

ix. Average : its meaning and effects : General Average : Par-ticular Average ...

....

,

...328

x. Free from Particular Average, and from All Average. .

.

330xi. Wear and tear... ... ... ... ... 330xii. The stranding of a vessel

... ... ......

331xiii. Burning ... ... ...

...331

xiv. Time Insurance, and the presumption of loss ......

332XV. Essential elements of a valid Insurance ...

...332

xvi. The adjustment of losses... ... ...

...332

xvii. Bottomry ... ... ... ... ... 333xviii. Demurrage ... ... ... ...

... 333

Chapter XV.Accident Insurance; and other Descriptions of Insurance

against Casualities.

i. A short history of Railway Accident Insurance ......

334ii. A description, in detail, of other forms of Accident Insurance 335

PRELIMINARY REMARKS.

A few suggestions may be offered to the student who mayperuse this volume, both in relation to his mode of study and hisfuture career.

The primary conditions of success, equally in the earlier and thelater stages of his professional labour, consist of accuracy andexhaustiveness of work ; and in no sphere of intellectual and prac-tical activity is this element more imperative than in the actuarial

profession. Important financial interests are committed to theactuary's trustinterests not limited in their consequences to the

present time or to the immediate future, but ranging in influenceand effect over a prolonged series of years. And not infrequentlya result of investigation, on which judgment requires to be based,may appear to be valid which a more exhaustive analysis (accom-panied by a numerical measurement) will prove to be fallacious.The student, again, should never be satisfied with simply the

symbolical or analytical process of reasoning by which a resultmay be attained. Analytical language is merely the shorthandexpression of the natural modes of induction and deduction whichalone retain throughout the process a steady and clear perceptionand apprehension of the facts investigated ; and the student, thoughhe may employ the mathematical method as an abbreviation ofmental labour, should always, prior to his decision, translate thesymbolic language into ordinary terms, and justify or modify theanalytical course and its deductions by the usual mode of reasoningwhich deals directly, without the intervention of algebraicalmechanism, with the realities which the symbols represent. Numer-ical computation and comparison of results should invariablyaccompany the mathematical processes. The aim of accuracy notalone suggests this procedure : mental discipline and power ofgrasp are more adequately evoked into exercise and expressionwhen the method of deduction is conducted with the actual factsof the inquiry ever present to the mind without their completerelegation to conventional and representative sjonbols.As the student advances to a higher post of responsibihty, he will

discover that a primary and essential distinction in competency ofadministration lies in the circumstance that one actuary will burden

PRELIMINARY REMARKS XVll.

himself with the details of his work and thus effectually impede thecapacity and opportunity of serene judgment, while another willattain a more substantial result by delegating the minor portionsof service to carefully selected subordinates, and thus possess,while necessarily controlling and supervising their labours generally,the time and mental freedom requisite for attention to that widerand more fruitful origination and direction on which achievementdepends. This devolution of charge in less important concerns isconditional for its success upon skill and insight in the choice ofmen ; and we thus arrive at the practical conclusion that the stydent,from the outset, should cultivate, by actual personal intercoursewith his fellows, the faculty of analyzing and judging humancharacter and its gifts and deficiencies.At every stage of his career the student should sedulously develope

the power of original thought. However limited may be the degreein which it is inherently possessed, it still exists to some extent,and by judicious and assiduous toil and concentration can beexpanded. The universe, whether in the practical or the moreintellectual sphere, would prove a weary and monotonous abodewere the vast majority of voices to consist simply of iterated echoes.This force of originality can be developed by avoiding servileacquiescence, and pursuing a serious personal investigation, withindependency of judgment, into every subject on which the studentis required to decide ; and, in addition to the value conferred uponthe character by the impress of original mental and persistent effort,the student will gradually become a master, and not a mere servant,of his methods and principles, using and adapting them intelli-gently as ins'truments instead of simply performing, as a servitor,

the work they suggest. The obligation is again enforced upon thestudent, for the acquisiton and expansion of this power, of ceasingto be an occupant chiefly of the study or a dweller in the speculative

domain, and of entering into commercial intercourse with men ofbusiness, with a vigilant observation and a tenacious retention ofthe modes in which he perceives affairs to be actually conducted,and keen, though kindly, attention to the diverse characters of men.The author may here record a result of personal experience. It

is too frequently the fashion, in correspondence, to address all men

in virtually an identical form of expression or exhortation or sug-

gestion : by personally mingling with men of different dispositions,2

XVlll. PRELIMINARY REMARKS

diversely responsive to the same appeal, he gained the valuablepractical lesson that, for the purpose of securing the most effectivesupport and the most earnest and vigorous efforts of men, he mustaddress them hy letter in a mode of special overture, individuallyadapted to their several peculiarities and idiosyncrasies of character.The end of all practical life, and the potent instrument of suc-

cessful administration, consists of judgment, or the delicate andexhaustive weighing of evidence in favour of alternative lines ofaction, and a steady forecast of each probable result. This power isnaturally an acquisition of gradual growth and widening experience

;

of serene demeanour in success and teachable humility in defeat

;

incompetent of cultivation by mere study or speculation, it simplyarrives as the slow and finished product of intimate communionwith men and affairs, and the patient observance and analyzedretention in memory of the varied results which ensue upon differentcourses and under diverse conditions. The requisite traininginvolves the remembered perception and experience of the causalconnexions between actions and consequences, motives and effects,so that, when a novel problem is presented for decision, the memorymay promptly revive and supply the analogous example from itsstorehouse of past instances which is adapted to the circumstancesof the case. In every question, the views of capable and experiencedmen should be sought, but they should be utilized as data only,since, in each critical epoch of life, the individual judgment mustbe responsibly supreme.

CHAPTER I

The Numerical Basis of Assurance Calculations, withExplanations and Discussions of Cognate

Subjects.

The student is presumed to be acquainted with the elementaryprinciples of Probabilities and the modes of their calculation,the essential element being that if an event may occur in x ways,and fail to happen in y ways, the probability of its occurrence at

the next trial is , on the assumption that the whole of theseX + y

courses possess an equal likelihood of appearance.

It is accordingly intended in this chapter to describe briefly"the basis on which the Doctrine of Probabilities in general andthe system of Life Assurance, as a practical application, are founded,

with a view especially to directing attention to current errors ofspeech and the paramount necessity (particularly in the complexproblems which occur in Life Contingencies) of accuracy of expres-sion. For obscurity of statement is not merely the product ofindeterminateness of thought, but the origin again of wider mentalconfusion, with the consequent negation of valid reasoning.

This general basis is presented in James Bernouilli's Theorem

of Large Numbers, which may be annunciated in the followingpropositions: (i) That in a very extensive number of trials orobservations there exists a demonstrably greater probability thatthe events under examination will occur in numbers proportionedto their respective chances in a single trial than in any other specified"proportion. Thus, if an accurately constructed coin be spun in

"the air without any bias in either the mode or the extent of thespin, the chance of a head appearing is i, or an even chance, witha similar probability of the obverse being presented. The Theorem^accordingly shows that, in an adequately prolonged number oftossings, the probability of the total appearances of a head andof the obverse will be identical with the respective results of the

individual trial. Proposition (2) affirms that a number of trialsor observations may always be assigned of so large an extent;as to make the probabihty of the events happening in numbers"within specified limits of deviation from the proportion just

20 INSURANCE

mentioned, however narrowly those hmits may be fixed, approach tO'certainty as closely as we please by multiplication of the experi-ments.

The latter proposition is especially valuable in estimating theprobable results of a given number of trials.The following illustrations will elucidate the significance of these

propositions and explain their relationship to life contingencies.Turning to proposition (i), the births of 1,000 children are accuratelyregistered, and the ages at which the children successively die arerecorded : if, then, goo remain alive at the end of the first year, theprobability of an infant, on actual experience, surviving for one

year, under conditions corresponding with those affecting the

original observations, is expressed by the fraction-^ or -^ ; and1000 10

assuming that the number of registered children were infinite, thisresult would cease to be approximate, and the proportion of thenumbers alive on the termination of the year to the entire numberobserved at birth would be precisely and absolutely equal to the-probability of an infant under similar circumstances surviving oneyear.

And, obviously, with every increase of the number of observationswithin this limit, the proportion exhibited would approach withcloser and closer approximation to equality with the ratio ascer-tained at the outset. Let us adapt the proposition to the illustrationusually furnished in treatises on probability. If we know the odds-in favour of an event to be 3 : 2, as, for example, in that of drawinga white ball from an urn containing three white and two black balls,.we should judge that if five trials were made we should be morelikely to draw white three times and black twice than any othercombination. We should still feel, however, that this result was.uncertain in so few a number of experiments : instead of three whiteballs we might draw white o, i, 2, 4, or 5 times. But if 1,000 trialswere attempted we should feel assured that, although the numbers,of white and black balls might not occur in the ratio of 3 : 2, theywould be ascertained to be very nearly in that proportion. Andthe more the experiments were multiplied, the more preciselywould this proportion be discovered to exist.

Laplace justly stated that the theory of probabilities consistedsimply of good sense reduced to a system of calculation. In the;

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ordinary affairs of life, whether in private, social, or commercialtransactions, every mind, basing its expectations upon experiencedresults without transforming those records into a calculus, actuallyemploys the relations, whether concurrent or sequent, observedbetween past occurrences as the foundation of judgments respectingfuture events of corresponding nature and happening under con-gruent conditions. And the doctrine of probabilities simply erectsthis inherent tendency and necessity of all minds into a general andorganized scheme of numerical expression.We thus observe that probabilities are transmuted into certainties

when the series of experiments or trials is infinite, and approachto practical certainties when the observations are exceedinglynumerous ; since in the long run, every event that can occur will

occur, and the action of fortuitous causes in operation will disappearas the amplitude of survey is enlarged.*

Experiments have frequently been made upon the results of thetossing of a coin. In a well-known instanceone, indeed, whichmay be termed historicala coin was spun, and, in 4,040 trials,heads appeared 2,048 times and tails 1,992 ; while, in a secondattempt, the corresponding events were 2,048 heads and 2,044tails out of 4,092 instances, or practically each result occurred

an equal number of times. The chance of head appearing at thefirst trial is J ; and the extended observations just described expressin these particular cases the doctrine of proposition (i), that if the

experiments be multiplied the events will occur in the proportionindicated at the single trial. In a third set of experiences, it was

ascertained that, out of 12,000 tossings of a coin, the occurrence of

tails happened in 50" 16 per cent, of the total observations, while,when the experiment was extended to 24,000 trials, tails showedthe closer equality to heads of 50-05 per cent.

The more extensive, therefore, be the mass of observations uponthe" duration of human life, the more profoundly is the expectationconfirmed that a persistent regularity of results will be displayed.

In the trials already mentioned, moreover, sequences of the

* These probabilities, which become more allied to certitude as the observa-tions are extended, are frequently termed moral certainties : probabilitiesin practical life, that is to say, which assume so sure a form that, thoughfailing in final demonstration, they are adequate as a, basis for " mores " or" manners," and afford consequently a clear and sufficient guide for thepractical conduct of the affairs of life.

;J2 INSURANCE

following order were observed. In the first set of the preceding-

experiments, heads appeared at the first throw in i,o6i cases ; inthe second set, in 1,048 ; no head occurred until the second throwin 494 instances in the one set, and in 507 cases in the second ; andomitting intermediate events of the same kind, no head appeareduntil the seventh toss in twenty-five cases in the one series, and inseventeen instances in the second. That is to say, selecting onespecimen only, the seventh, in the first set there occurred a run inthe appearance of tails in twenty-five instances before a head turnedup. It will thus be observed that, since the contrary events in the

totality coincide in extent, a consecutive appearance of one event

during any duration of experiment will be followed by a diminishedsequence of occurrence of that event at another stage, and vice versa.This phenomenon is paralleled in life assurance by the occurrenceat one or more periods of numerous deaths followed by a briefersequence, or a short succession preceding a more extended one, butin such a mode as, in the mass, to render the total result accordantwith the observations from which, for example, the premiums arededuced. It follows that the basis of life assurance consists inthe condition that those who survive longest and contribute mostlargely to the common fund,in excess of the payments which willbe received by their representatives,form a necessary constituentof the scheme by balancing the experiences in which death occurs atan early date and where consequently the representatives of theassured obtain a benefit considerably superior to the premiumsdischarged.

We revert now to the second proposition, which asserts that,with a multiplied series of observations or trials, the chance of theevents occurring in numbers comprised within specified limits ofdeparture from the result shown in a single experiment, howeverbrief may be the interval between those limits, may be made tO'approach as closely to certainty as is desired, according to theextent of the series of experiments employed : that is to say, theprobability of occurrence within those boundaries can be renderedas near as we please, by enlarged trials, to certainty or unity.(Although the certainty of an event is not customarily conceived asa probability, or the numerical measure of the expectation orbelief that the event will occur, yet the continuity of mathematicallanguage requires the identification, in the doctrine of probabilities

INSURANCE 23

of the chance which possesses unity as its measure withcertainty, just as the impossibiUty of the happening of an eventis registered on the scale of probabilities as zero, or o,the termsand I forming really, in mathematical symbolism, the limits ofthe probability of occurrence.)

This proposition has been illustrated by anticipation in thepreceding examples. The subject will also be specially consideredand exemplified in Chapter II, where the question of the stabilityof results, and the range of deviation from an expected experience,are discussed.

Interesting tables, it may be anticipated, are frequently fur-nished, both for Life Assurance and Friendly Societies, where theprogress of the Assurance Fund is exhibited year by year, withincrements by premiums and interest and diminution by claims,so that, on the extinction of the lives involved, the final balance isexactly sufficient to discharge the concluding claim. This is, ofcourse, theoretically accurate ; but the student will observe thatit is practically impossible. When the numbers at risk become soreduced that uniform and regular results (whose occurrence impliesan adequate mass of cases) fail to be practicable, the Companywould then require, if it ceased to supplement its diminishingarea by fresh entrants, to transfer its liabilities and assets toanother Office which possessed the requisite basis of numbers.The meaning of the term " probable " error should here be intro-

duced. This function expresses the range of deviation from thetruth within which, taken positively and negatively, it is an evenchance (or

-J) that an observation will appear. If there exist an

even chance (A being the exact value) that the result of an observa-tion shall lie between ^aand A -[ a, a is termed the probableerror of the observation. In other words, the extent of divergence

from the precise (or true) result within which, on one side or theother, in excess or deficiency, it is an even chance (or it is as likely

as not) that the truth will be discovered, is named the probableerror of the observation to which the calculation refers. Thus, ifthe chances be i to i (or equal) that the error shall be foundbetween o and 10, and the same chance exists that it shall exceed10, 10 is named the probable error. Or again, if 5-45 be the meanof all the determinations of the density of the earth, and -2 beapproximately the probable error, the meaning implied is that the

24 INSURANCE

probability of the actual density of the earth falling between 5-25(5-45 --2) and 5-65 (5-45 + -2) is one-half.The term " error " signifies a discordance in observations of an

accidental characterthat is, one of which the cause is unknown,and which in the long run occurs as frequently in the one directionas in the opposite direction ; if the cause could be assigned, theerror would disappear, or cease to be an error, in consequence ofits dependence upon a recognized cause, and thus proving pre-dictable. An error is thus, in mathematical reasonings, thedifference between the real value in nature of an observed quantityand the result of its actual measurement. If Z represent the truevalue, and M be the product of measurement, the differenceZM is designated the Error.

It may appear impracticable by means of observation to discoverdata more precise than the observation itself. Numerical errorbeing necessarily incidental to every observation, the latitude ofthis error (or digression from the truth) is obtainable by repetitionof measurements under varied conditions : it is, therefore, thenumber of careful observations and the ascertainment of theirmean value whichwith the probability that in this proceedingthe errors of opposite kinds will at length be compensatory

bridges over the interval, and conducts with minuter nearness tothe truth.

The student must proceed further in his mathematical educationbefore he can appreciate the doctrine of the probable error, andas an illustration, he must accept at present on trust the followingexample of the mode of calculation :

1. Find the mean of all the observed results.2. Ascertain the excess or defect of each observation from this

meanthat is to say, the error of each result from the meanquantity,

3. Square each of these reputed errors.

4. Sum the squares thus obtained (which will be, of course, allpositive).

5. Divide this sum by one less than the number of observations,which furnishes the square 0/ the mean error.

6. Extract the square root of this final result, which exhibitsthe mean error of a single observation.

INSURANCE 25

7. Divide by the square root of the number of observations, andthus obtain the mean error of the mean result.

8. Multiply by the natural constant-6745 (or, say, f) (a con-

stant entering into formulae for determining the probable error),and we thus ascertain the probable error of the mean result.

In illustration of this process, let five measurements be takenof the height of a hill which produce severally the numbers of feetof 293, 301, 306, 307, and 313 ; and assume that the goodness orvalidity of the observations is equal : we desire to ascertain theprobable error of the mean or 304, that is to say, its departure fromthe true value of the altitude.

The differences between this mean and the preceding numbers,without regard to their direction in excess or deficiency, are 11, 3,2, 3, and 9 ; their squares are 121, 9, 4, 9, and 81 ; and the sum ofthese squares of errors is consequently 224. The number of ob-servations being 5, divide the preceding sum by 5 1 or 4, andthe quotient is 56. This number constitutes the square of themean error ; and extracting its square root, we obtain 7-48 (or

7J), which expresses the mean error of a single observation.Dividing by 2-236, the square root of 5 (the number of observations)we find 3-35, or the mean error of the mean result : then multi-

plpng by-6745, we arrive at the probable error of the mean

result, or 2-259, o'^ about 2J. This final number indicates that theprobability is \, or the odds are even, that the exact height of thehill lies between 30if (i.e. 304 2J) and 306^ (i.e. 304 + 2J) feet.We deal more fully with this question in Chapter II in relation

to the necessary extent of assurance risks for securing the occur-

rence of results within stated limits of deviation.An important distinction has been pointed out by physicists

and mathematicians between the significance of an average and amean ; and this accuracy of scientific expression possesses adefinite bearing upon our current modes of speech. In ordinary-language, and even in the statements of scientists themselves,accustomed though they be to verbal precision, the average isfrequently confounded with the mean. So prevalent has provedthis confusion that it has been proposed, for the purpose of adefinite separation of the terms, that the ordinary arithmetical

average should be designated the " fictitious mean." If there be

6 quantities2, 3, 4, 6, 13, and 14the arithmetical average is

26 INSURANCE

obtained by dividing the sum of the quantities by their number,when a quantity is produced intermediate at some place in the set,so that the sum total of the excesses of certain quantities over the

average is equal to the sum total of the deficiencies of the other

quantities which fall below the average. In the preceding example

the average is = 7 : the excesses are 6 and 7 or 13, and6

the deficiencies are 5, 4, 3, and i or 13. Now it has been justlypointed out that an average value involves no conception of anatural and recognizable central magnitude, all divergencies fromwhich in the observed results should be regarded as deviationsfrom a natural standard. The mere average of the altitudes ofseveral trees of the same species probably fails to furnish theheight even of any individual specimen, and exhibits no indicationof the type of altitude expressed in Nature as that to which thevariously divergent heights constantly tend.An average merely furnishes a general notion of the objects com-

pared in respect of the particular feature selected for examination :it represents no actually existent quantity, and simply forms aconvenient mode of conception and expression, enabling us toapprehend and convey in a single result a multitude of details, andto compare, with mental ease, the results of one set of observationswith those of another.

It would be quite impracticable, for example, to form a definiteconception of the duration of human life, except in the concen-trated mode of expressing the length of life (or expectation oflife) which each individual would possess if the years attained byeach were equally distributed among the whole body of personsobserved. But though this compact form of statement presentsa convenient and helpful aid in comparison, it obviously involvesno indication of an actually existing fact pertaining to the orderof human mortality in the shape of a uniformity of nature.The average thus affords, in an individual result which can be

readily pictured to the mind and retained in memory with facility,a general notion of a number of quantities, and provides us withthe means of conveying the same rtotion to other minds withoutthe necessity of presenting a lengthy list of the isolated quantities

themselves from which, in their multiplicity, no lucid and definiteconception could be constructed with ease. If, for example, we

INSURANCE 27

formed the average in the ordinary manner of the heights of lOOhouses in a street, the result would not represent any real entity,any individual house, but would merely furnish a fictitious heightindicating as nearly as possible the several altitudes of the indi-vidual buildings. An average accordingly fails to serve as a guideto the future based upon the experience of the past.A mean, on the other hand, it is rightly contended, does indicate

an approximation to a definite existing quantity ; and, applyinglanguage of a metaphorical character, points to some model orstandard inherent in Nature, some real existence expressive oflaw or uniformity or intention, to which the various observationsobtained by human measurement or induction refer, by the sym-metric nature of their deviations, as the type to which theyseverally tend.

A well-known illustration may be considered, but for its ap-preciation and a complete conception of this scientific and naturalsignificance of a mean a brief explanation of a certain mode ofmeasurement, to which the name of " Weights " is applied, re-quires attention.

An observer of some event, repeatedly examined, may considerthat some of his measurements possess a greater trustworthinessand accuracy than others : he then, by the impression producedupon his mind, at the time of observation, of their respectivemerits of value, affixes to each trial its relative weight. Supposethat ^,, ^4, . . . be the results of n observations of a statedobject or event, and that the numbers c^, c^ . . . express theirpresumed proportional validity or scientific goodness (i.e. 0,0^. . . are the " weights " measuring the respective values of the

observations) : then, instead of employing . (where 2 is then

symbol of summation) as the Average, the observer substitutes the

mean of -=

, where He A consists of A^ c^, A^c^,and 2 c

is the sum of the " weights " of c c^Expressing the proposition in a numerical form, assume that the

observations conducted cannot all be accepted with equal con-fidence : the observer must then, from inquiry and judgment,make the different observations be reckoned as different numbersof observations, allowing any one more acceptable result to count

28 INSURANCE

as more than one if it appear to be superior to the remainder inaccuracy and care of measurementtaken, for example, undermore favourable physical conditions, or with the mind and sensesmore alert, or with an instrument of observation more exact.Three observations, we will suppose, furnish the results of 26, 28,and 29 ; a close examination suggests that 28 is to be preferred tothe rest, and 29 to 26, so that, for example, 28 ranks in order ofsubstantial correctness as though it possessed the value of a mean

of 8 observations, 29 of 6, and 26 of 4. It must then be consideredthat 8 +6 +4 or 18 observations have been effected, of which8 have exhibited the result of 28, 6 that of 29, and 4 that of 26.These numbers8, 6, and 4are termed the "weights" at-tached to the several observations 28, 29 and 26 as the measuresof their values ; and in obtaining the mean result of the total ob-servations, each is multiplied by its specialized weight ; the sumis then divided by the sum of the weights. Thus we obtain,8 X 28 + 6 X 29 + 4 X 26 or 502 ; and dividing by 8 + 6 -f 4,or 18, the number 27-89 is produced as exhibiting the most probableresult, or that which most closely represents the actual truth inNature. The mere arithmetical mean or average would be26

-t- 28 -f 29or 27-6

3Without affixing weights of value dependent upon the observer's

personal judgment, and therefore liable to error through defectiverecollection, obscure impression, or imperfect surmise, the weightsthemselves may be obtained direct from Nature. Thus, thequantity of each observed phenomenon, amid a multitude ofdivergent observations, should be multiplied by the number oftimes which that individual fact is perceived in Nature. Anactual example which is frequently adduced in books on prob-ability will exhibit this genuine and trustworthy method ofascertaining the mean in the clearest light.

First recurring, however, to our previous illustration, if, insteadof taking the average measurement of the heights of 100 differenthouses, we made 100 equally sound measurements of the heightof the same house, the mean of these results would very approxi-mately express the actual altitude of the house, or the value ofthe real existent fact. Ora happier illustration still^we revertto the former reference to the height of the trees of a selected

INSURANCE 29

species. If we construct a schedule of heights of various trees,and multiply each height by the number of trees which as exactlyas possible present it, we obtain a mean value (as distinguishedfrom a mere Average resulting from a purely arithmetical process),which genuinely furnishes an index to that altitude in Naturewhich constitutes, so to speak, the ideal or typical altitude, fromwhich the isolated heights are deviations, and towards which, inthe order of Nature, and persistently, they tend.We now proceed to present the example, to which a reference has

just been made, which exhibits the mode of formation of a Meanquantity, and which, from the manner in which it is obtained,ascends beyond the mechanical conception of an Average, andjustly and distinctly expresses a phenomenon or type in Nature.The dimensions of the chests of 5,738 soldiers of different regi-

ments belonging to the same country were measured, and theresults, represented in inches, were expressed in tabular form,proceeding by differences of one inch, and treating 2-5 to 3 inchesinclusive as 3 inches, while 2-4 inches would be regarded as 2. Theappended table contains the facts

Inches

30 INSURANCE

The mean, thus scientifically obtained, it is contended, may beaccepted as furnishiiig a confident index to the future, while theaverage possesses no such inherent property of assurance ; andthe validity of all statistical records depends upon the apprehen-sion of this distinction between these two modes of measurement.The ordinary course of averaging without the introduction of theweights would have afforded a result of 40-5, or an error of aboutI.-'] per cent.

In the application of this doctrine of weights to the subject ofmortality, it is obvious that greater influence should be attachedin a Table of Mortality to a considerable number of observationsat one age or set of ages than to the smaller numbers appearingat other ages, if the natural phenomena of the rate of humanmortality are to be educed. It is evident that the more extensive

be the numbers of the living or of the deaths observed, thegreater are the " weights " assigned by Nature for detection ofthe inherent uniformity of progression which the phenomenamanifest. We refer to this question at a later stage.The student should carefully ponder the deep divergence of

significance and value in statistical inquiries between the resulttermed an average, which is the product of a mere arithmeticaloperation, and the Mean, which indicates some permanent fact inNature and in the numbers exhibited in natural processes. Hismodes of expression, consequently, in describing the results ofinvestigation, will then assume a determinate and scientificform.

A brief recapitulation accordingly will be valuable.1. The object of research, in whatever sphere of science it be

pursued, is the discovery of the facts and uniformities of co-existence or succession displayed in Nature itself, as an integralportion of its constitution ; in Physics, the elucidation of laws,as they are termed, which shall serve, by rendered obedience, asguides to the mastery of natural processes ; in the region of ob-servations upon human beings, relevant to the structure of assur-ance, the regularity exhibited, under corresponding conditions, ofthe force of mortality.

2. An average presents no feature of intrinsic value in thisaspect : it furnishes merely a convenient measure of comparisonof a purely arithmetical and convent'.onal character.

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3. The mean genuinely possesses ' a natural capacity of indi-cating the actual types and standards towards which aberrantobservations, when duly and precisely weighted in respect ofaccuracy and existence, apparently and normally tend.

This precision of measurement and the resulting approximatecloseness to the natural phenomena themselves are obtainedeither (a) by applying " weights " expressive of the degrees ofintimate accuracy, furnished by the observer's own inspection andjudgment ; or (6) by adopting the " weights " provided directlyby Nature, as exemplified in the ascertainment of the standardcircumference of soldiers' chests which has been cited. The rele-vant " weights " presented in Mortality statistics for the discoveryof any uniformity in the intensity and incidence of the force ofdeath would consist of the adequate extensiveness at the severalages of the areas of observations of the existing and the deaths,which are utilized in our search.The Mean implies an assurance that the future will be similar to

the past (which an average cannot indicate), and the recognition ofa mean, among a series of results distributed in order, dependsupon the observance of a conformity between the law of progres-sion in the magnitude of the distributed groups and the realityof some central truth.

4. It will be noticed that in the formation of an arithmeticalaverage the mere multiplication of the different quantities bythe numbers of each is not of the nature of the process of multipli-cation by natural " weights." The former numbers furnishsimply the respective areas in which these quantities occur, butwithout the genuine implication of supplying, so to speak, naturalindications of the gradation by which the various quantities maybe assumed to approximate to an existent standard inherent inthe constitution of things.

5. The average is concerned with different quantities of thesame kind, and provides a general, superficial notion of magnitudeand comparison only ; the mean deals with the one object andeduces by its process a true phenomenon embodied in Nature. Inbriefand adopting an ancient philosophic formulathe meanseeks to decipher, with exactitude and clearness, the " one in themany," while the average presents merely a superficial and generalresemblance between the " many."

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Certain current improprieties of speech in life assurance may beexposed on tlie teaching of these remarks.

It is constantly stated that a Table of Mortality must be em-ployed which furnishes " average results " ; that " on the average "

a certain consequence may be expected ; that the doctrine of" averages " applies. It is clear from the preceding expositionthat the term " average " is improperly and irrelevantly intro-duced into these expressions, and that what is intended to be con-veyed, and should be definitely affirmed, istaking the severalstatements in theif' orderthat a Table of Mortality must be em-ployed which is " sufficiently extensive in range " ; that " in thelong run," or " in the mass," or " on the whole," or " taking allevents together," a certain result may be expected ; and that thedoctrine of " large numbers of observations '' applies. It is un-fortunate that we do not possess abbreviated forms of expressionwhich would convey the significance we rightly intend ; but thisexplanation will be useful if it teach the student to measure his.terms by the exactness of his meaning.

If the new assurances of a Company comprise one for 200,another for ^^300, and a third for ;f4oo, we properly employ theterm " average " in stating that the average amount is 300,.provided, of course, that there exists no serious divergency betweenthe several sums, for the result is a mere piece of arithmetic.

The phrase of the " expectation of life " may be speciallymentioned. The term " expectation " is here erroneously em-ployed : " expectation " involves a belief in the occurrence or theorder of occurrence of events in the future ; but no such state ofmind is implied in the preceding phrase, and the adoption of theterm has frequently conduced to the fallacious notion that thefunction expresses the period which a person may be " expected "

to survive. The quantity denoting the expectation of hfe is.simply the numerical result of distributing the total durations oflife severally enjoyed by persons successively surviving after agiven age among the whole number of persons existing at that age,,so that the term " average " applies, since no mean result is dis-

played. Examine the process of calculation by the reverse mode.In the O Table it is observed that the expectation of life at ageninety-five is i'i88 years. Now the following facts appear in thattable :

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2

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statement, and through reflex action (to employ a relevant phrase),from much obscurity of thought.Two writers of the highest scientific eminence have mentioned

respectively, in standard treatises, the probability of a specified

individual surviving for a year, and the probability of an individualperson attaining a deferred age. Probabilities, it is obvious, are

merely expressive when involving numbers and not units.

The value of an annuity at age x (independent of the accompany-ing life assurance in calculating the market-value) is clearly notthe value in relation to an individual life ; and should be expressedin the phrase that the value at any age is the average value of alarge number of annuities completed at that age, taken one withanother. We have just referred to the introduction of a protectinglife policy in ascertaining the market value of an annuitj',

( i=fl): but although the isolated security is thus^d + p

'

rendered complete in itself, and we may accordingly describe thevalue in this aspect as appertaining to a single life, the element of

mass or number of transactions is still inherent, for the effectedpolicy involves the notion of large numbers in the data employedby the Company by which the assurance is granted.

The purchase of an absolute reversion as an isolated transactionmay range in profit between the limits of cent, per cent, and loss ;and hence we must speak of its value as expressive of the results cfan extensive group of investments. The market-value, again, ofan absolute reversion is not the value of an individual purchase,

but should be expressed as the value that may be granted with awell-founded expectation of the required return of interest in an

adequate number of instances.The student may also be cautioned to refuse the acceptance of

stated percentages of observations without production of the numbersout of which they have been formed. A disease which destroysone person or one hundred may equally be affirmed to be fatal infifty per cent, of the cases affected whether the number of personsattacked be two or two hundred. But one out of two will probablybe purely accidental, while one hundred in relation to two hundredwill be significant of vast importance.

No department of inquiry is so densely strewn with fallacious

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deductions, and hopeless and mischievous hypotheses constructedon their basis, than that of statistics of every description, in conse-quence not merely of comparisons between incongruent materials,in whole or part, or failure to observe whether the conditions affect-ing the several sets of observations compared are fundamentallycorrespondent, but largely also from the error of adopting theprocess of the simple arithmetical mean or average in the calcula-tions in place of the genuine and indicative mean.

All observations collected, and facts observed, are unavoidablysubjectfrom the limitations of human faculty, the lapse of atten-tion or mental clearness, the failure of precise accuracy, and thedeficiencies or want of delicacy in the instruments employedtomanifold errors. Errors of mal-observation or mis-observationoccur ; errors of cleissification of incompatible materials ; the " per-sonal equation " enters, or the individual inherent rapidity or slow-ness in sensation, perception, and record of each observer (thispeculiarity, however, can be determined in each instance andallowed for : the men of larger physical frame should be found toexhibit the least quickness in this respect, caeteris paribus, sincethe nervous current, constituting the medium for the transmissionof sensation and volition, and travelling at the rate of ninety feetper second, has a longer tract to traverse, and moreover personalinterest in the subject of investigation counts considerably) ; mis-takes of description happen ; failures in discrimination ; errors oftabulation and transcription ; errors of summation and otherarithmetical operations ; oversights in checking results ; andpaucity of data either generally or at specific stages.

Before a series of observations can be employed as the basis oflife assurance transactions, a method of adjustment or gradua-tion, tending to produce that regularity of progression which theorder and continuity of Nature in every sphere entitle us to antici-pate, must be adopted.The present book is of too elementary a character to include a

detailed or critical account of the various systems which have beenpursued to effect this object, and a brief summary alone can beattempted.

Among the essentially mathematical schemes may be mentionedthe summation of the numbers living at each of five consecutiveages, and the adoption of the one-fifth part as the adjusted central

q6 insurance

value for each sectiona process, it will be noticed, of simple

averaging.

Weights may be introduced by the following plan :

TfP P PP. P^, be obtained from the formula (modi-

fied fol each P) P^ = >

where ^'^ expresses the probability according to the observations

of mortality employed of surviving a year at age x, then theadjusted probability of living a year at age x, or p^, is

"x 2 + "it I + "a: + ^x + T + ^3r+2

,

5

where each P is formed in the mode above described. A weightis thus accorded to the unadjusted probabilities at the several agesinversely proportioned to their distances from the age considered.

This purely mathematical plan has been most ingeniouslyexpanded into a formal and rigorous system by Mr. Woolhouse.The numbers living at quinquennial intervals of age, such as lo,15, 20, etc., to the close of life or observation, are extracted ; then

the numbers existing at ages 11, 16; . . . ; those living at 12, 17,. .

.

; those at 13, 18, ... ; and those surviving to 14, 19. . . .By separate interpolation by the ordinary formula in finite differ-ences, the intermediate values for each series are obtained, and theaverage is then taken of the five completed sets of results. Wethus, as it were, draw a curve through the original data at ages10, 15, . . .; again at 11, 16, ... , and so on ; and thus producefive different curves closely approximating in outline to one another

;

and at each age the adjusted curve possesses an ordinate which isthe arithmetical average of the several ordinates of the variouscurves ; the whole of the curves are thus massed into one whichpursues a central course. (It may be noted that, in the originalexposition of this arrangement, the expressions " arithmeticalaverage " and " mean " are erroneously employed as thoue;h theywere synonymous.)

Or we may assume that the observations form the concrete andvisible expression of a law of Nature, or (to speak in more preciselyscientific language) a Uniformity of Nature. Extracting as far aspossible the character of this uniformity from a comparison ofdifferent and adequate masses of materialsthe numbers exhibited

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as surviving to successive ages throughout Hfe by various tablespossessing reasonably common features of congruency of con-ditionsit may then be possible to express the regularity of resultdisplayed in a mathematical formula of practicable compass, fromwhich the adjusted numbers may be calculated by applying theformula to the actual data themselves. Mr. Gompertz's conceptionof the uniformity in Nature in respect of the duration of life supposedthe existence of two operative causes producing deathone actingwith a constant intensity at all ages, the other of such a characterthat the power of vitality or the capacity to resist death sufferedequal proportions of diminution in equal small periods of time, sothat the intensity of mortality would assume the form of an increas-ing geometrical progression. The construction of his actual formulawas restricted to the second supposition only, but Mr. Makeham'sextended formula included boththe constant and the geometricallyaugmenting series.

Let the constant force of mortality operating with uniformintensity at all ages be represented by A , then Be'' (which should becompared with the following illustration) will express the secondconcurrent force, which increases in a geometrical ratio of intensity :hence the force of mortality furnished by Mr. Gompertz is B c^

;

while Mr. Makeham's formula shows the forces in co-operant action,or ^4 -!- B c'^, where x denotes the age.

In examining the expression of an equal proportionate reductionin the vital force by which death is resisted, the student shouldcarefully note the distinction between a loss of opposing power byequal successive portions and by equal successive proportions.

If X be the time, the function abx (representing, say, the powerto oppose death at any epoch) indicates a diminution of equalportions of power in equal small intervals : thus we have (substitutingthe periods of o, i, 2, 3, etc.), a (where at the origin of observa-tion, :;=o, andtherefore6=o),a 6, a 26, a 36, etc. : the loss in thefirst interval is a{ab)ov b; the diminution in the second instant is(ab){a2b) or b : and in the third interval (a 26) ( 36) =b:that is, the loss at each stage is constant or equal.The function of the form a 6* represents a loss or reduction of

equal proportions in corresponding minute spaces of time. Thus,at the origin of observations, the power to resist death is expressedby ab" or a ; at the beginning of the second instant by a 6 "^ ; at the

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commencement of the third, hy a '*' ', and so on. The losses,therefore (substituting numerical values), are, taking the first two

elements as examples, a ah, ah - ab^ ; the proportion of the first

loss to the power which is reduced is = i 6 ; the ratio ofa

the second loss to the value at the beginning of its epoch is

= 16, or the same proportion as beforethat is,a h

a : a ah : : ah : ah ab^, which is obvious.A concrete and familiar illustration of a proportional loss is

afforded by the present value of a sum, a, due at the expiration of aperiod, x, or a if. In the interval, h , this function loses

av' av"'^^ = a if (i w'"), which bears to the initial value of a w*the ratio av" (i w*)

;i xi_ j. ii, i.- t h u- u!^ =1 V ; that is the ratio of i

v : i whichav''

is independent of the time, x, or, av" (i f*) : av' :: x w *: iAgain, a graphical method may be adopted which has rendered

admirable service for many years in the domains of astronomyand meteorology. This process is sometimes described in scientifictreatises as the method of curves.Having marked along a horizontal line at equal intervals the

ages attained by the lives observed either from birth or from aspecified age, an ordinate (or perpendicular line) is erected at each

age, the length of which (measured on a uniform scale) representsthe number of the successive survivors at each sequent age. If acurve be then drawn through the upper extremities of these ordi-nates, a broken and irregular line will be produced, due to accidentalerrors or to paucity of observations at some stages, or to both causesin combination, though, on the whole, if the observations have beennumerous, exactly obtained, and well distributed, as in a mortalitytable of adequate compass, the abrupt and capricious deviationsfrom a continuous flow of outline will prove to be less marked :still, imperfection of progression will exist, but by a skilled observera curve of symmetric form can be drawn by cutting off all smalland irregular sinuosities, some ly