The First Mathematically Correct Life Annuity Valuation Formula *


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1 James E. Ciecka he Firs Mahemaicay Correc Life Annuiy. Journa of Lega Economics 5(): pp he Firs Mahemaicay Correc Life Annuiy Vauaion Formua * he sory of he firs acuariay correc specificaion of a ife annuiy revoves around Jan (Johan) de Wi (6567), Jan Hudde (68704), and Chrisiaan Huygens (69695). A were Duch and mahemaics sudens of Frans van Schooen (65660). De Wi was born ino a poiicay prominen famiy, was a nava officer who fough agains he Briish on he high seas, and was he grand pensionary (prime miniser) of Hoand a age 8. Hudde was head of he Duch admiray, a mayor and ongerm burgomaser (a high profie poiica pos) of Amserdam, and a poiica opponen of de Wi. Huygens was a prominen mahemaician and conribuor o probabiiy heory and considered by some o be he eading naura phiosopher immediaey prior o he age of Isaac Newon. In 656 Huygens wroe a shor reaise in Duch on probabiiy heory. Van Schooen convinced Huygens o pubish his reaise as an appendix a he end of a book ha van Schooen was preparing. Van Schooen himsef prepared a Lain ransaion of Huygens work and pubished i, under Huygens name, in his book in 657. he foowing proposiion, pubished in he van Schooen appendix, was among Huygens conribuions o probabiiy heory (Had, 003): If he number of chances of geing a is p, and he number of chances of geing b is q, assuming aways ha any chance occurs equay easiy, hen his is worh ( pa qb)/( p q). We migh rewrie Huygens as expression as p q a b p q p q ; * James E. Ciecka, Professor, Deparmen of Economics, DePau Universiy, Eas Jackson Bouevard, Chicago IL, Phone: , Emai: I wish o hank Gary R. Skoog for reading his noe, suggesing improvemens, and many peasan hours discussing mahemaics, acuaria science, and forensic economics. Ciecka: he Firs Mahemaicay Correc Life Annuiy Vauaion Formua 59
2 and, in more modern erminoogy, e X be a random variabe ha akes on he vaue a wih probabiiy p /(p q) and he vaue b wih probabiiy q/( p q). hen he expeced vaue of he random variabe X is () p q E[ X ] = a b p q p q. hus, he concep of mahemaica expecaion or expeced vaue was born, or a eas pu in prin for he firs ime, approximaey 350 years ago. In 669 Huygens and his broher Ludwig engaged in correspondence wih each oher ha ceary appied he idea of expeced vaue o he cacuaion of ife expecancy. hey focused on he disribuion of deahs; or, ooked a from anoher poin of view, wha migh be ermed he disribuion of remaining ife or he disribuion of addiiona years of ife given a person age x. Le x denoe he number of survivors a age x, and dx = x x denoes he number who die beween ages x and x. hen he probabiiy of iving years (no ess and no more) beyond age x is given by d x /xbecause o ive years (no more and no ess) requires ha one survives years bu no years from age x. ha is, deah mus occur beween ages x and x. Le be a random variabe for remaining years of ife, hen ife expecancy, from he Huygens brohers poin of view, is he expeced vaue of and can be wrien as () ω = x x x = 0 E( ) ( d / ) whereω is he younges age a which here are no survivors (ω = 0). 3 o Huygens, he mahemaica expecaion was he fair price of a gambe. Hacking (975) indicaes ha Huygens was concerned wih deermining a fair vaue or price of a gambe, and he answer o ha quesion was he expeced vaue of he gambe. Siger (999) has he foowing ransaion of Huygens: If he number of chances eading o a is p, and he number of chances eading o b is q, and a chances are equay ikey, hen he expecaion is vaued a ( pa qb)/( p q ). Ludwig Huygens may have been he firs o make a ife expecancy cacuaion. his correspondence aso reveas ha he Huygens brohers ceary undersood he difference beween he expeced vaue and he median vaue of a random variabe. hey disinguished beween he average number of addiiona years of ife and he number of years uni haf of a popuaion had died and haf si ived. 3 oday, we woud ca his he curae ife expecancy. ha is, a person mus ive a fu year for ha year o coun in ife expecancy. More ypicay we assume ha deahs occur uniformy hroughou a year; and herefore peope who die wihin a year wi ive, on average, onehaf of he year in which hey die. he compee ife expecancy becomes he cu Journa of Lega Economics 60 Voume 5, Number, Augus 008, pp.5963
3 In he midseveneenh cenury, saes of ife annuiies were a common source of oca and naiona finance in Hoand and some oher European counries. Sandard pracice of he day dicaed seing annuiies a one price regardess of he age of he nominee (i.e., he person on whose coninued ife annuiy paymens depend). Mos nominees, as woud be prediced when he price is unreaed o age, were very young. Hudde s daa showed ha 80% of a nominees were under age 0 and haf were ess han 0 years od (Poiras, 000). As prime miniser, de Wi dea wih he financia consequences of AngoDuch wars and in 67 was preparing for war wih France. In shor, he Duch governmen needed money and de Wi proposed seing more ife annuiies o generae funds; bu he correc mahemaica formua for pricing annuiies was hereofore unknown. In 67 de Wi issued a repor which showed he correc acuaria vauaion of a ife annuiy, a repor which demonsraed for he firs ime how o correcy inegrae compound ineres and moraiy probabiiies ino he vauaion of a ife annuiy. Hudde read de Wi s repor and aesed o he mahemaica appropriaeness of his mehods. Here is de Wi s mehod. Le a denoe an ordinary annuiy cerain (i.e., an annuiy immediae) defined in he usua way as (3) j= a = ( i) j where i denoes he ineres rae. Now, hink of his annuiy cerain as a random variabe by aowing he erm o vary from one o an arbirariy arge number. Define o be a random variabe measuring remaining ife ime as in formua () above, hen he random variabe a akes on he vaues a, a, a, wih probabiiies d / 3 x x, d / x x, d / x 3 x,. ha is, a nominee ges he annuiy cerain a if he survives one year bu no wo years (which occurs wih probabiiy d x / x ), a nominee ges he annuiy cerain a if he survives wo years bu no hree years (which occurs wih probabiiy d / x x), and so on. hus, de Wi s formua (de Wi, 67 and Hendricks, 85, 853) for a ife annuiy becomes 4 (4) ω = x x x = E( a ) a ( d / ). rae expecancy pus.5. Using he ypica noaion, compee ife expecancy for a ife age x is e x =.5 E( ), where E ( ) is defined in formua (). 4 We can imagine a ead erm in his sum o be a 0 ( d / x ); bu we define a = 0, meaning 0 ha nohing is paid if a nominee dies beween age x and x. A nominee mus survive one year o ge he firs annuiy paymen. Ciecka: he Firs Mahemaicay Correc Life Annuiy Vauaion Formua 6
4 A his poin, we noice he simiariy beween de Wi s formua (4) and he Huygens brohers formua (). One simpy repaces he ime variabe in formua () wih he annuiy cerain variabe a and formua (4) emerges (Had, 003). In modern erminoogy, we woud say ha de Wi defined an annuiy cerain wih a random erm as a random variabe and specified is probabiiy disribuion. Formua (4) is insighfu because i says ha he vaue of a ife annuiy is he expeced presen vaue of he annuiy cerain random variabe. In modern imes, we ofen speak of he presen vaue of a ife annuiy; bu de Wi s formuaion was ceary an expeced presen vaue. His approach aso enabes us o compue higher order momens and any oher characerisic of he a random variabe (Haberman and Sibbe, 995). De Wi showed ha he correc price of a ife annuiy for a hreeyear od nominee o be6 forins (forins being he sandard of vaue of he day) when Hoand was seing such annuiies for 4 forins, using he ineres rae of 4% per annum. Afer discovering formua (4), de Wi s bigges pracica probem was obaining accurae moraiy probabiiies. He assumed eve moraiy beween ages hree and 53 and decining moraiy hereafer for age groups 5363, 6373, and (Had, 003 and Poiras, 000). Subsequen correspondence beween de Wi and Hudde discussed probems ike annuiies on join ives, wha moraiy probabiiies shoud be used in ife annuiy cacuaions, and sef seecion of nominees imporan probems o his day. In 67 Hoand sared seing ife annuiies a prices ha were inversey reaed o age. However, prices were beow hose cacuaed by de Wi and Hudde. As an epiog, we noe ha afer 67 Hudde and Huygens coninued heir careers and evenuay died of naura causes. De Wi was no so forune. France invaded Hoand in 67, and de Wi resigned as prime miniser. A mob, supporing he Prince of Orange, murdered him and his broher and hen muiaed heir bodies. he gruesome deed is capured in a paining (exhibied a he Rijksmuseum in Amserdam) enied he Bodies of he Brohers de Wi, Maryrs of he Repubic by Jan de Baen showing heir evisceraed bodies hanging upside down. A saue of de Wi was ereced in 98 in he Hague on he spo of his murder; he inscripion honors him for his conribuions o poiics, he navy, sae finance, and mahemaics. he saue was dedicaed by Queen Wihemina who, somewha ironicay, was from he House of OrangeNassau. James E. Ciecka Journa of Lega Economics 6 Voume 5, Number, Augus 008, pp.5963
5 References De Wi, Jan, Vaue of Life Annuiies in Proporion o Redeemabe Annuiies, 67, pubished in Duch wih an Engish ransaion in Hendricks (85, 853). Haberman, Seven and revor A. Sibbe, Hisory of Acuaria Science, Voume I, London, UK: Wiiam Pickering, 995. Hacking, Ian, he Emergence of Probabiiy, Cambridge, UK: Cambridge Universiy Press, 975. Had, Anders, Hisory of Probabiiy and Saisics and heir Appicaions before 750, Hoboken, NJ : John Wiey and Sons, Inc, 003. Hendricks, F., Conribuions of he Hisory of Insurance and he heory of Life Coningencies, he Assurance Magazine, 85; he Assurance Magazine 3, 853. Poiras, Geoffrey, he Eary Hisory of Financia Economics, , Cheenham, UK: Edward Egar, 000. Siger, Sephen M., Saisics on he abe, Cambridge, MA: Harvard Universiy Press, 999. Ciecka: he Firs Mahemaicay Correc Life Annuiy Vauaion Formua 63
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