Life Contingencies Review

8/18/2019 Life Contingencies Review

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Life ontingencies. y Professor . WALLACE JORDAN, F.S.A.

[Society f ctuaries’ ext ook. p. i+ 31. ublished y the Society fActuaries, hicago, llinois, 952. 8.00.1

MEMBERS of the Institute ay well ake ride n the fact hat, n the past,the ext-book n Life ontingencies hich has been recommended for readingby the ociety f ctuaries, nd by ita arents he ctuarial ociety f Americaand the merican nstitute f ctuaries, as een the fficial ext-book f theInstitute. he study f he heory f ife ontingencies s not an end in itself,but only means to an end, nd accordingly ny text-book hat is to serve itspurpose roperly ust deal ith he ractical s ell s he heoretical spectsof he ubject. he practice f ife ssurance n he nited tates nd Canadadiffers n many respects rom that n Great ritain nd it s herefore greattribute o the work of George King and E. F. Spurgeon that the Institute ofActuaries’ ext ook, art I, ife ontingencies y King, ublished n 1887,and subsequently ife Contingencies y Spurgeon, irst ublished for theInstitute n 1922, hould or o long ave served he eeds f students n thecontinent f North America.

Spurgeon’s ook is ow out of date nd the cope f the ew official ext-

book of the Institute nd Faculty, ife nd Other ontingencies, hich, at thetime hen this eview s eing ritten, s waiting ublication, as been mainlydetermined y the equirements f the resent yllabus f the Institute. heSociety f Actuaries s herefore ortunate n having secured the services ofso able n authority s Mr C. Wallace ordan, r., ssociate rofessor fMathematics t illiams ollege nd a Fellow f the Society, o write newtext-book hich ives pecial ttention o he enefits ommonly offered y thelife ssurance ompanies f he nited tates nd Canada.

The author xplains n is reface hat he has been mindful of the needs ofthose any actuarial tudents ho must acquire their knowledge of LifeContingencies ithout he assistance f an instructor. ccordingly, e hasgiven ull nd painstaking xplanations hich should make it easy for thestudent ho is-approaching he ubject or he irst ime o ollow he demon-strations f the arious ormulae. e has also ppended opious xamples toall the chapters.

The book is ivided nto hree arts. art , onsisting f seven chapters,deals ith the mortality able nd with net premiums, ffice remiums andreserves or single-life ssurances nd annuities. he American flavour sspecially vident n Chapter , which explains ome of the more importantmodified eserve ystems nd non-forfeiture rovisions. he development fthe ubject n his art f he ook is eat nd follows logical rder, lthoughlaws of mortality nd annuities ayable times year re ntroduced t anearlier tage han n Spurgeon’s ook. hose portions f Chapter I in whichthe uthor nvestigates he ffect f ssuming simple lgebraic ormula or lx area little edious nd one wonders hat benefit he tudent s xpected o derivefrom the arious xamples, t he nd of his hapter, hich require n arith-metical valuation f functions ased on such formulae. In the chapter on

JIA 79 (1953) 0233-0237


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annuities t s pity hat he uthor as ollowed he onvention hat he erm'immediate' eans 'payable n rrear', ecause his as ed im to he wkward

expression deferred mmediate ife nnuity’. art I contains ix chapters,covering oint-life nnuities nd assurances, ontingent ssurances nd re-versionary nnuities. lthough hese atters re isposed f in about alf s

many pages s Spurgeon equired, pace as een found or ome of he orecomplicated ontingent ssurances nd reversionary nnuities hich rarely, fever, ccur n practice. n Part III a chapter n stationary opulations splaced, omewhat illogically, ith wo chapters n multiple-decrement heoryand combined ables. or multiple-decrement heory he uthor ses is wnnotation: he hanter n combined ables ontains useful xplanation f thenotation sed n merica or isabilitv nsurance.

An appendix o he ook contains he Commissioners 941 tandard rdinary(CSO) Mortality able, ith onetary unctions t 2½%, nd the Life Tablefor U.S. White ales, 939-41. hen the atter able as added, he act hata slightly ifferent ersion f t ad already een included n Chapter I seemsto ave een overlooked. lso appended to the book are some finite-differenceformulae, alculus heorems, erivatives f actuarial unctions, nswers to theexamples, short ibliography nd an index to the notation.

When George King wrote he nstitute f ctuaries’ ext ook, art I, ecould ave had no inkling f the evelopments hat were to take place in thetheory f probability. e was therefore uite ontent o write, n the firstpage of his book:

Could e find OO,OOO hildren ll orn t he ame oment, nd ould e followthem throughout ife, nd enter n a column the numbers who remain at the end ofeach uccessive ear ntil ll ave assed way, e should orm the column living,headed ith he ymbol x; here lx epresents he umber ho attain he preciseage x.

This interpretation f the mortality able, o simple nd satisfactory o theunsophisticated tudent f 1887, as n air f nreality o-day. s Mr CharlesA. Spoerl as so forcibly ut it n his aper, ife nsurance nd the Theory ofProbability Proc. ent. ssembly nst. ctuaries, I, 89),

To one who is teeped n the ntricacies f the odem developments, ust thethought f eorge ing’s land ssemblage f x‘s nd dz’s nshrined t the base ofthe hole heory f ife nsurance ust be well igh ntolerable. t uggests very-thing hat s nadequate, utmoded and oversimplified.

This does ot mean that he ortality able tself s outmoded. It means,among other hings, or o it eems o he eviewer) hat mortality able itha radix f IOO,OOO t age o may be regarded s representing, ot the actualnumbers of lives urviving o successive ges ut of 1OO,OOO irths, ut thenumbers who are xpected o survive o successive ges out of 1OO,OOO livesaged o who are aken t random from an indefinitely arge number of liveswhose mortality s epresented y the able. imilarly, hen mortality ablefunctions re pplied o an individual ife r a group of lives, he ife r livesmust be regarded s having een taken t random from an indefinitely argenumber of lives ho are ssumed o experience, s a whole, he ortality nwhich the able s based.

With these houghts n mind, the reviewer urned agerly o the openingpages f rof. ordan’s ook in rder o see hether e shared hese iews, rwhether e had some other deas s o a modem presentation f the mortality


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table hich would enable he tudent o take a firm grasp of the fundamentalprinciples nderlying he application’ f the able o the solution f practicalproblems. t was extremely isappointing herefore o find the author ex-plaining mortality able ith radix f 1OO,OOO y the tatement hat

the igures n he lx olumn ndicate he umber f urvivors t ach ge and he x

figures ndicate he umber f eaths n the year of age x to x + 1.Worse still, aving ntroduced he eader o he unction po, hich e calls he‘survival unction’ nd which e denotes y the ymbol (x), e says (althoughhe has efined he unction s a probability) :

Let s se he urvival unction (x) o nswer he ollowing uestion; ow manylives,ut f group f 100,000 irths, ill urvive o age I? The answer is clearlyl00,000 (I).

No doubt rof. ordan ad some good reasons or resenting hese inadequate,outmoded nd oversimplified’ deas o he tudent, ut it s urprising hat hisreasons re not explained.

This ailure o et o grips ith undamental rinciples s, ot surprisingly,evident gain n Chapter 5 on multiple-decrement heory, here the authoromits o oint ut hat e is ssuming n everal laces hat he ecrements renon-selective, n assumption hich is ardly ver ulfilled n practice. t is,however, ildly efreshing o see he ultiple-decrement able eferred o asa mathematical odel.

In a work of his ature t s nly o e expected hat he uthor hould odoccasionally. erhaps herefore t s little ngracious o point ut that n

p. 8 e has allen nto he ommon error f uggesting hat a group of Iives anbe subject o the nfluence f a force f mortality. n p. 29 he says hat, na select able ith 3-year elect eriod, l25 epresents he urvivors t ge 25of he l[20] ives nsured t ge 0 and of he l[22] ives nsured t ge, 2 his asthe isconception hat ay t he oot f Sprague’s amous heory f amagedives. here is curious apse n pp. 118 and 119 where the rather ricky

problem f quality f olicy alues y two different ables s investigated. nhis nxiety o void purgeon’s rror f irst roving hat condition s necessaryand then assuming hat t s also uffcient, he author lindly ollowsrThomas N. E. Greville T.S.A. II, 33) n thinking hat the most convenientway to complete he roof s y induction; e apparently ails o realize hatwhen the roblem s imited o range f ges ne has nly o everse he stepsin he roof y which the ecessary ondition as een established n order toshow that he condition s sufficient. e also isleads he student in thesentence eginning t the bottom of p. 118, where he omits to point out thatthe condition s ot sufficient nless is efined s he defines t in line 3 ofp. 119.

The index hows igns f asty reparation; few may be mentioned here.It s ot lear hy the age eferences or 'Gross remiums' and 'Premiums,

gross’ re iven s 128 f. nd 128 . espectively. o entry s ncluded orTemporary annuities or Annuities, emporary), eferred nnuities orAnnuities, eferred), et premiums or Premiums payable times year. hepage reference or Reserves, ontingent nsurances’ s incorrect.

Notwithstanding hese arious hortcomings, he book, on the whole, isexcellent. t s bvious’ hat a great deal of loving labour has gone into thewriting f it nd it s ell rinted. he author as the gift f being able toexplain hings learly nd he has pared o pains o ake full se f this ift.


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Students f the ociety ill, o doubt, find the book of great value in theirpreparation or he xaminations nd they will have good cause to be gratefulto Prof. ordan or he are nd thoroughness ith which he has developedclear nd logical xplanation f the subject. tudents f the Institute ightwell rofit y reading he ook, ut hey hould e advised ot to take ChaptersI and 15 too seriously.

Logarithmetica ritannica, eing standard able f ogarithms o wenty ecimalplaces. y A. J. THOMPSON, Ph.D. (Lond.)

[Issued n 9 parts y the Department f Statistics, niversity ollege, ondon.Part I, he inth nd last art o e issued, ambridge niversity ress, 952.45s. ll ther arts, IS.]

THIRTY years fter e took p the ask, r Thompson’s major work has been

completed y the publication f the ninth and last part of LogarithmeticaBritannica. ach of the ine arts ontains he ogarithms o twenty decimalplaces f 10,000 umbers, ight arts eing ssued t fairly egular ntervalsfrom 1924 to 1937. he ninth nd last art had been completed and wouldhave een ready or ublication n 1940, but for the war. With the completionof the ull esign it can be seen as a worthy memorial of the tercentenary fthe ublication f riggs’s rithmetica ogatithmica n 1624.

The older eaders f the ournal ay remember an article n The Ter-centenary f ommon Logarithms J.I.A. VI, 2) which welcomed the publicationof he irst art o e issued, art X, containing he ogarithms f the numbers90,000 o 1OO,OOO. he article ncluded ome information bout Henry Briggs.One of the features f the ables as been the eries f prefaces o, ach partwhich have recorded ll inds f material f historical nd general nterestrelative o Briggs’s ogarithms.

What is he eed for table f ogarithms o o any decimal laces? ostpersons, uch as actuaries, hose work may entail ome computing will neverneed a large umber of significant igures. alculating achines will servetheir urposes, robably ith ore speed n alculation han ogarithmic ableswith , 5, or 7 decimal laces. he author rgues ith cogency that the day

of such ables s past.There are, owever, ccasions hen calculations ave o e made with ore

significant igures han an be conveniently andled n a calculating achine.For any computer n that redicament he ew tables ill be a boon. Prof.Pearson tates hat n tatistical nd computing aboratories he riginal riggsand original ega are n reater emand than ny more contracted ogarithmictables. he following re nstances ithin ctuarial xperience hen extendedlogarithmic ables ay be required :

(a) The calculation f high powers, .g. solated r check values in com-

pound interest ables,(b) Computations ased on recurrence elationships-because ignificant

figures ay be lost t ach tage f the computation,(c) Computations ased n differences r derivatives,(d) Any calculations nvolving perations n equence, nd(e) Standard ables or he reparation f other tables.

Readers ay be interested o know that ogatithmetica ritannica as usedfor the calculations f log, 100m nd log, 100 n the expansions f ee’ and

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the problem being the summation of n infinite eries ith terms ofalternating ign J.l.A. xxvI, 155).

The second nd fourth entral ifferences, eparately omputed, are tabu-lated eside he ogarithms f ll -figure umbers. he lay-out s clear-and iseasy to read. he Introduction urveys the methods of interpolation or‘reading etween he ines’ f the ables, amely nterpolation y the methodof factors, y Everett’s ormula nd other ethods of differences, nd byLagrange’s ethod. he author ives wo tables o facilitate he method offactors: he irst s f o g l o g a n d o g t h esecond s f antilogarithms or ogs p to .0000450. n the eries f ‘Tractsfor Computers’ here s a Table f coeficients f Everett’s entral-diffrenceinterpolation ormula which was also computed by Dr Thompson.

The Introduction escribes ully nd clearly he methods used in thecalculation f he ables. he author onstructed n integrating nd differencingmachine hich onsisted f our imple achines f he otary ype, o arranged

in steps hat number on the roduct egister f one machine could be trans-ferred echanically o the setting evers f the machine elow it nd thatnumber on the etting evers f one machine ould e similarly ransferred othe roduct egister f the one above it.

The Logarithmetica ritannica s standard ork of utstanding mportance,a magnificent onception agnificently arried ut. The author has enhancedthe alue f is ork as memorial o Henry Briggs y the istorical aterialwhich he has ublished omprising :

Part I. The will of Henry Briggs.Part II. Translation f a memoir on the life nd work of Henry Briggs

written n Latin y Thomas Smith, .D., and published n 1707. List oferrors n Briggs’s rithmetica ogarithmica f 1624.

Part II. our letters o Sam Ward, Master f Sidney ollege, nd one toThomas Lydiat n which Briggs ays I am still t my logarithmes nd cannether inishe hem to y minde nor lett hem alone’.

Part V. Title age of the ork in which Henry Briggs’s reatise n theNorth-West assage o the South Sea was published.

Part . Title age f riggs’s rithmetica ogarithmica, 624.

Part I. Letter o ohn Pell t rinity ollege, ambridge.Part II. Title age nd five ther ages f ohn Napier’s anonis escriptio

from the later dition ublished n 1619, fter is eath. he reproductionswere chosen o illustrate he espective ontributions f Briggs and Napier tothe subject.

Part X. Title age and two other ages f Briggs’s ogarithmorum hiliasPrima, 1617.

It s o be hoped that he author ill e able o round off is abours ypublishing he life of Henry Briggs for which he has been collecting he

material.M. E. 0.

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