Edmond Halley s Life Table and Its Uses *

Transcription

1 James E. Ciecka Edmond Halley s Life Table and Is Uses. Journal of Legal Economics 5(): pp Edmond Halley s Life Table and Is Uses * Edmond Halley ( ) was a remarkable man of science who made imporan conribuions in asronomy, mahemaics, physics, financial economics, and acuarial science. Halley was forunae o have been born ino a wealhy family and o have had a faher who provided for a firs-rae educaion for his son. Halley enrolled in Oford Universiy a age 7, sayed for hree years and, wihou a degree in hand, se sail for S. Helena in he souh Alanic o observe and caalogue sars unobservable from Europe. The voyage ook wo years and, upon his reurn o London, he was eleced o he Royal Sociey a age 22 for his S. Helena work. Halley became he edior of Philosophical Transacions (he journal of he Royal Sociey), an Oford professor from , and Asronomer Royal a Greenwich from 720 o his deah. Isaac Newon and Halley were friends, and he urged Newon o wrie wha became he Principia Mahemaica and assised financially and ediorially in is publicaion. Halley ploed he orbis of several comes. In paricular, he conjecured ha objecs ha appeared in 53, 607, and 682 were one and he same come ha would reappear approimaely every 75 years. He correcly prediced ha he come would reurn in 758, and i was poshumously named in his honor afer is reappearance a he prediced ime. Halley made wo forays ino financial economics, demography, and acuarial science. The second work (705, 77) was on compound ineres. He derived formulae for approimaing he annual percenage rae of ineres implici in financial ransacions and annuiies. His firs conribuion (693) was seminal and is he opic of his noe. In his work, Halley developed he firs life able based on sound demographic daa; and he discussed several applicaions of his life able, including calculaions of life coningencies. Halley obained demographic daa for Breslau, a ciy in Silesia which is now he Polish ciy Wroclaw. Breslau kep deailed records of birhs, deahs, and he ages of people when hey died. In comparison, when John Graun ( ) published his famous demographic work (662), ages of deceased people were no recorded in London and would no be re- * James E. Ciecka, Professor, Deparmen of Economics, DePaul Universiy, Eas Jackson Boulevard, Chicago IL, Phone: , [email protected]. I wish o hank Gary R. Skoog for reading his noe, suggesing improvemens, and many pleasan hours discussing mahemaics, acuarial science, and forensic economics. Ciecka: Edmond Halley s Life Table and Is Uses 65

2 corded unil he 8 h cenury. Caspar Neumann, an imporan German miniser in Breslau, sen some demographic records o Gofried Leibniz who in urn sen hem o he Royal Sociey in London. Halley analyzed Newmann s daa which covered he years and published he analysis in he Philosophical Transacions. Alhough Halley had broad ineress, demography and acuarial science were quie far afield from his main areas of sudy. Hald (2003) has speculaed ha Halley himself analyzed hese daa because, as he edior of he Philosophical Transacions, he was concerned abou he Transacions publishing an adequae number of qualiy papers. 2 Apparenly, by doing he work himself, he ensured ha one more high qualiy paper would be published. The Breslau daa had he propery ha annual birhs were approimaely equal o deahs, 3 here was lile migraion in or ou of he ciy, and age specific deah raes were approimaely consan; ha is, Breslau had an approimaely saionary populaion. Afer some adjusmens and smoohing of he daa, Halley produced a combined able of male and female survivors; here reproduced as Table. He deermined he populaion was approimaely 34,000 people. To eplain his able, le l denoe he size of a populaion a eac age = 0,,2,,ω, where ω is he younges age a which everyone in he populaion has died, hen L =.5( l l ) capures he average number alive beween ages and ; or, alernaively, he number of years lived by members of he populaion beween ages and. Halley s life able gives L ; so, for eample, he very firs enry (for age = ) is L = L = L0 =.5( l0 l ) = 000, he average number of people alive beween ages zero and one. 4 Figure is he graph of Halley s able; and, for purposes of comparison, we also show he life able for he US in 2004 (CDCP, 2007). Halley made seven observaions and used his life able o eemplify hose observaions. John Graun developed a life able in 662 based on London s bills of moraliy, bu he engaged in a grea deal of guess work because age a deah was unrecorded and because London s populaion was growing in an un-quanified manner due o migraion. 2 Wihou arguing in suppor or agains Hald in his regard, we noe ha he same issue of Philosophical Transacions conained papers by he grea chemis/physicis Rober Boyle and he noed mahemaician John Wallis. 3 There was a small increase in populaion. As Halley pu i an increase of he people may be argued of 64 per annum. Here, Halley menions ha ecess birhs may perhaps be balanced by he levies of he emperor s service in his wars. 4 Table has a radi of 000. The Breslau daa had l 0 = 238 and l = 890, implying L 0 = 064. Halley seems o have rounded o 000 for convenience. Journal of Legal Economics 66 Volume 5, Number, Augus 2008, pp

3 Table. Halley s Life Table Age L Age L Age L Age L Toal Figure. Halley's 693 Breslau and 2004 US Populaion Survivor Funcions L Age Ciecka: Edmond Halley s Life Table and Is Uses 67

4 Firs, Halley looked a his able from a miliary poin of view (perhaps because Graun did eacly he same hing in 662) and calculaed he proporion of men able o bear arms. He compued he number of people beween he ages of 8 and 56, divided by wo o esimae he number of men, and epressed he laer number as a fracion of he enire populaion of 34,000 people. Halley s approimae answer was 9/34 or abou.26 of he populaion (see Table ). If one were o make a similar calculaion using he curren US life able illusraed in Figure, he corresponding fracion is.24. Lile has changed since Halley s ime in his regard even hough Figure illusraes wo very differen life ables. Second, Halley compued survival odds beween ages using L /( L L ). He gave an eample of 377 o 68 or 5.5 o for a man age 40 living o age 47 (see Table ). Third, Halley compued he age, o which i is an even wager ha a person of he age proposed shall arrive before he die. Tha is, Halley calculaed he median addiional years of life. He gave an eample for a 30 year old. There are 53 survivors a ha age and half ha many beween ages 57 and 58 (see Table ). Therefore, Halley s median was beween 27 and 28 years. Halley made no life epecancy calculaions. Fourh, in one raher long senence, Halley menioned ha he price of erm insurance ough o be regulaed, and is price relaed o he odds of survival. He poined ou ha he odds of one year survival were 00 o ha a man of 20 dies no in a year, and bu 38 o for a man of 50 years of age. Halley s poin is clear, bu here is a ypographical error in he paper because he odds of survival for a 50 year old are approimaely 30 o (see Table ). Fifh, Halley did no give an eplici mahemaical formula for a life annuiy, bu he provided e and eample calculaions ha clearly showed ha he used he following formula: 5 () ω = = a ( i) (L / L ). Halley calculaed life annuiies wih a 6% discoun rae and provided he epeced presen values shown in Table 2. The Years Purchase Columns are he epeced presen values of life annuiies of one pound. Halley noed ha he Briish governmen sold annuiies for seven years purchase regard- 5 Afer some re-wriing, Halley s life annuiy formula is similar o Jan De Wi s (67) formula as shown in he Appendi. Journal of Legal Economics 68 Volume 5, Number, Augus 2008, pp

5 less of ages of nominees. Table 2 shows his was abou half he value of an annuiy on 5, 0, or 5-year-old nominees and poor governmenal policy for all nominees under age 60, bu he Briish governmen did no change is single-price policy afer Halley s work. Table 2. Halley s Life Annuiy Table Age Years Years Years Age Age Purchase Purchase Purchase Sih, Halley urned his aenion o a join life annuiy on wo lives. He used a recangle wih lengh L and heigh Ly o represen lives age and y. In conemporary noaion, le L L D and L y L y D y, where Dand D y denoe deahs from L and Ly wihin years. The produc of L and L is y LL = L L L D L D D D. (2a) y y y y y The lef side of (2a) represens he area of Halley s recangle which he calls he oal number of chances. Halley gave he eample from Table for = 8 and y = 35 and said []here are in all or 298,900 chances. Halley coninued he eample for = 8 and said ha he number of chances was or 3650 ha hey are boh dead, which is he las erm of he righ hand side of (2a). This gives us LL D D = L L L D L D (2b) y y y y y (2c) ( D D / LL) = (/ LL)( L L L D L D) y y y y y y where (2c) is he probabiliy of a leas one life surviving. The life annuiy ha pays when a leas one of wo nominees survives becomes Ciecka: Edmond Halley s Life Table and Is Uses 69

6 (3) ω y = y y = a ( i) ( D D / L L ). Halley did no provide any numerical eamples of annuiy calculaions in his par of his paper. Sevenh, Halley considered he problem of annuiies on hree lives. He drew a complicaed looking hree dimensional figure which is he eension of he recangle he previously considered. The dimensions of his new figure, in modern noaion, are L L D, Ly Ly Dy, and Lz Lz Dz. The produc LLL y z = ( L D)( Ly Dy)( Lz Dz) has eigh erms ha correspond o various living and deah saes for hree lives. A his poin, Halley compued he value of a life annuiy ha pays whenever a leas one of he hree nominees is alive wih a formula like (4) ω yz = y z y z = a ( i) ( D D D / L L L ). He gave an eample where = 0, y =30, and z = 40 and concluded such an annuiy was worh years purchase. Finally, Halley alked abou a reversionary annuiy on he younges life age afer he older lives ages y and z. Tha is, he annuiy pays he younges nominee afer he older nominees die. The value of his annuiy is (5) ω a = ( i) ( L Dy Dz/ LLyLz) yz. = A his poin Halley seemed o ire of he laborious calculaions involved in formula (5) and he concluded his paper. To summarize, here is wha we can say abou Halley s paper: () we sill use life ables similar o he one he developed and (2) we sill make calculaions of life coningencies as he did. The main difference beween Halley and modern work lies in Halley s use of he average survivors beween ages (i.e., L, L,..., L ω ) raher han survivors a eac ages ( l, l,..., l ω ), alhough some (e.g., Poiras, 2000) inerpre Halley as using survivors a eac ages. In eiher case, Halley wroe a remarkable paper 300 years ago; formulae (), (3), (4), and (5) are especially insighful. Halley refleced on his paper in a posscrip. Four addiional paragraphs appear like a coda which he eniled Some Furher Consideraions on he Breslau Bills of Moraliy. Halley mused abou how unjusly we Journal of Legal Economics 70 Volume 5, Number, Augus 2008, pp

7 repine a he shorness of our lives and hink ourselves wronged if we aain no old age. Afer observing ha only abou half of Breslau s,238 newly born children survive 7 years, Halley added ha we should no fre abou unimely deah bu raher submi o ha dissoluion which is he necessary condiion of our perishable maerials. He concluded his rain of hough by observing he blessing we have received if we have lived more han he median years of life a birh. Halley s second, and las, commen deal wih human feriliy. He calculaed approimaely 5,000 persons beween ages 6 and 45 (see Table ) and esimaed ha a leas 7,000 were women capable o bear children. He reckoned ha,238 birhs relaive o 7,000 ferile women were bu lile more han a sih par. If all women in his age group were married, Halley hough four of si should bring a child every year. Celibacy was o be discouraged and large families encouraged because he srengh and glory of a king was in direc proporion o he magniude of his subjecs. Halley concluded wih a carro and sick policy prescripion: he sick par was ha celibacy should be discouraged hrough eraordinary aing and miliary service, and he carro was ha large families should be encouraged hrough sociey finding employmen for poor people and hrough laws such as he jus rium liberorum among he Romans. 6 James E. Ciecka 6 Augusus Caesar graned cerain privileges o fahers of hree or more children. These privileges were known by he erm jus rium liberorum. Thomas Malhus menioned hese laws in An Essay on he Principle of Populaion (798) as being ineffecive among poorer classes and of some minor influence on higher classes of Roman ciizens. Ciecka: Edmond Halley s Life Table and Is Uses 7

8 References Cener for Disease Conrol and Prevenion, U.S. Deparmen of Healh and Human Services, Naional Vial Saisics Repors, Unied Saes Life Tables, 2004, Volume 56, Number 9, Table, December, 2007, Hyasville, MD. De Wi, Jan, Value of Life Annuiies in Proporion o Redeemable Annuiies, 67, published in Duch wih an English ranslaion in Hendricks (852, 853). Graun, John, Naural and Poliical Observaions Made upon he Bills of Moraliy, Firs Ediion, 662; Fifh Ediion 676. Hald, Anders, Hisory of Probabiliy and Saisics and Their Applicaions before 750, Hoboken, NJ : John Wiley and Sons, Inc, Halley, Edmund, An Esimae of he Degrees of Moraliy of Mankind, Drawn from he Curious Tables of he Birhs and Funerals a he Ciy of Breslaw, wih an Aemp o Ascerain he Price of Annuiies upon Lives, Philosophical Transacions, Volume 7, 693, pp , Mahemaical Tables Conrived afer a Mos Comprehensive Mehod, by Henry Sherwin, 77. Google Digiized Images. Hendricks, F., Conribuions of he Hisory of Insurance and he Theory of Life Coningencies, The Assurance Magazine 2, 852, pp and ; The Assurance Magazine 3, 853, pp Malhus, Thomas Rober, An Essay on he Principle of Populaion, UK: John Murray, 6 h Ediion, 826. Poiras, Geoffrey, The Early Hisory of Financial Economics, , Chelenham, UK: Edward Elgar, Journal of Legal Economics 72 Volume 5, Number, Augus 2008, pp

9 Appendi We can ge close o Halley s life annuiy formula () from Jan de Wi s formula as shown in (A)-(A5). De Wi (67; Hendricks, 852 and 853) used he disribuion of deahs d = l l in his formula for he epeced presen value of a life annuiy; here wrien as he lef hand side of formula (A). (A) ω Ea ( ) = a( d / l) = T = ω = j= j ( i) ( d / l ) = [( i) ]( d / l) 2 (A2) [( i) ( i) ]( d 2 / l) 2 3 [( i) ( i) ( i) ]( d 3 / l) 2 ( ω ) [( i) ( i) ( i) ]( d / l ) = [( i) ]( d d 2 d 3 dω ) / l 2 (A3) [( i) ]( d 2 d 3 dω ) / l 3 [( i) ]( d d ) / l ω 3 ω ( ω ) [( i) ]( dω ) / l = [( i) ]( l ) / l 2 (A4) [( i) ]( l 2) / l 3 [( i) ]( l ) / l 3 ( ω ) [( i) ]( lω ) / l (A5) = w = ( i) ( l / l ) Ciecka: Edmond Halley s Life Table and Is Uses 73

10 In going from he lef o he righ hand side of (A), we simply use he definiion j= ( ) j a i. Summaions are epanded in (A2) and hen re-grouped in (A3). (A4) uses he propery ha he coefficien of ( i) is ( d d... d ) / l = l / l. 7 2 ω 2 Similarly, he coefficien of ( i i) sums o ( d 2... dω ) / l = l 2 / l, 3 he coefficien of ( i i) is ( d 3... dω ) / l = l 3 / l, and so on, unil we ( ) ge o he las erm ( i i) w wih coefficien ( dω )/ l = lω / l. Formula (A5) becomes Halley s life annuiy (formula ()) when we subsiue he average number of survivors beween ages for he number alive a eac ages. Tha is, Halley used L, L,..., L ω and de Wi usedl, l,..., l ω. Halley published in 693, some 22 years afer de Wi; bu here is no informaion ha Halley was aware of de Wi s work. De Wi s formulaion emphasizes he epeced presen value naure of a life annuiy and uses he disribuion of deahs d / l,, d / l ω. Halley uses he survivor disribuion L / L,, L / L ω. De Wi s formulaion allows one o compue higher order momens bu Halley s does no. However, Halley s formulaion, wih he subsiuion of l, l,..., l ω for L, L,..., L ω, has become he much more widely used mehod. 7 The number of people alive a age is l. Since all mus evenually die, we have ( d d 2... dω ) = l. A similar idea holds for all ages; all people alive a a cerain age will evenually die and he sum of hose deahs equals he number alive a ha age. Journal of Legal Economics 74 Volume 5, Number, Augus 2008, pp