Interchangeability of the median operator with the present value operator

Transcription

1 Appled Economcs Letters, 20, 5, Frst Interchangeablty of the medan operator wth the present value operator Gary R. Skoog and James E. Cecka* Department of Economcs, DePaul Unversty, East Jackson Boulevard, Chcago, IL 60604, USA Downloaded By: [Cecka, James] At: 9:56 7 June 20 It s well known that the expected present value of a lfe annuty s smaller than the present value of an annuty certan wth term equal to lfe expectancy. Ths result can be vewed as a consequence of the lack of nterchangeablty between the present value operator and the mathematcal expectaton operaton. However, we prove that the medan and present value operators can be nterchanged; that s, the medan of the present value of a lfe annuty equals the present value of an annuty certan whose term s the medan addtonal years of lfe. At young ages and through late mddle age, medan addtonal years of lfe exceed lfe expectancy. Therefore, the medan value of a lfe annuty exceeds an annuty certan pad to lfe expectancy whch exceeds the expected present value of a lfe annuty - the far prce of a lfe annuty. Keywords: annuty; medan present value; expected value; lfe annuty JEL Classfcaton: G00; G9 I. Introducton Modern understandng of lfe expectancy and medan years of addtonal lfe dates to the work done by Chrstaan Huygens (669) and hs brother Lodewjk. They drew a sharp dstncton between these concepts but recognzed the usefulness of both. Before the end of the seventeenth century, Jan de Wtt (67) and Edmond Halley (693) provded the correct understandng of the expected present value of a lfe annuty although they approached lfe annutes dfferently. Expected present value s undoubtedly the correct concept upon whch to prce lfe annutes. However, a purchaser of a lfe annuty also mght be nterested n the medan present value of the lfe annuty just as a person mght be nterested n medan addtonal years of lfe. It s well known that a lfe annuty s worth less than an annuty certan pad to lfe expectancy. In ths artcle, we consder the relatonshp between the medan present value of a lfe annuty and the present value of an annuty certan pad to the medan addtonal years of lfe. We show that what s not true for expectatons s ndeed true for medans. That s, the medan operator and present value operator can be nterchanged even though the expectaton and present value operators cannot. II. Notaton Let l x denote the number of survvors n a populaton at exact age x, andyal x denote the years-of-addtonallfe random varable for that populaton. YAL x takes on the value t 0.5 wth probablty ðl xþt l xþt Þ=l x f a person survves for t years beyond age x and des before the completon of t years beyond age x, where t ¼ ; 2;...; x and s taken to be the *Correspondng author. E-mal: jcecka@depaul.edu Appled Economcs Letters ISSN prnt/issn onlne # 20 Taylor & Francs DOI: 0.080/

2 2 G. R. Skoog and J. E. Cecka Downloaded By: [Cecka, James] At: 9:56 7 June 20 youngest age at whch there are no survvors. Complete lfe expectancy at age x s defned as x _e x ; EðYAL x Þ¼ X ðt 0:5Þ l xþt l xþt l x ðþ t¼ for x ¼ 0; ;...; 0 The present-value-of-an-annuty-certan random varable payng one at md-year for each addtonal year of lfe and wth postve dscount rate r s defned as PVðYAL x Þ ; a YALxj ; ¼YAL X x ¼0:5 ð þ rþ ¼ h r ð þ rþ0:5 ð þ rþ YALx for r>0 ð2þ When YAL x ¼ t 0:5, PVðYAL x Þ¼a t 0:5j by constructon from Equaton 2; and both YAL x ¼ t 0:5 and PVðYAL x Þ¼a t 0:5j occur wth the same probablty of ðl xþt l xþt Þ=l x. Takng expectatons n Equaton 2 results n the followng lfe annuty expresson : ( X ) ¼YAL x a x ; E½PVðYAL x ÞŠ ¼ E ¼0:5 ð þ rþ ¼ h n o r ð þ rþ0:5 E ð þ rþ YALx ð3þ whle nsertng _e x ¼ EðYAL x Þ nto the annuty-certan defnton results n a _exj ; ¼EðYAL X xþ ¼0:5 ð þ rþ ; h r ð þ rþ0:5 ð þ rþ EðYALxÞ ð4þ The well-known actuaral nequalty a x <a _exj arses when comparng Equaton 3 wth Equaton 4. 2 III. Medan Annuty Theorem The two most mportant descrptve measures of central tendency for addtonal years of lfe are _e x ; EðYAL x Þ average years of addtonal lfe, or lfe expectancy; and med(yal x ) the medan years of lfe, when 50% of a populaton has ded. An ndvdual may contemplate purchasng an annuty, a random varable, payng the person whle alve. Wth the purchase of such an annuty, the present value of the uncertan stream has two related measures of central tendency n whch a purchaser may be nterested: a x ; E½PVðYAL x ÞŠ the expected (or average) present value of the annuty; and med½pvðyal x ÞŠ the medan present value, the result acheved by 50% of purchasers. There also are two annutes certan, assocated wth each of the YAL x characterstcs: a exj the annuty certan to lfe expectancy; and a medðyalx Þj the annuty certan to the medan years of lfe remanng. We may now state the motvaton for ths artcle. Beyond the nequalty a x <a _exj, what s true theoretcally and emprcally about relatonshps among the foregong concepts? Equaton 3 dffers from Equaton 4 because the present value PV operaton and the mathematcal expectaton operaton E cannot The fundamental dea for ths representaton of a lfe annuty dates to Jan de Wtt (67) (Hendrcks, 852/853; Haberman and Sbbett, 995 for de Wtt s report on lfe annutes, and Potras, 2000). De Wtt (67) used a weghted average of annutes certan where the weghts were mortalty probabltes, thereby producng the expected value of the present value of the lfe annuty. Edmond Halley s (693) representaton of the lfe annuty dates to 693 when he expressed the lfe annuty as /( + r) multpled by the probablty of survvng at least years and summed over. 2 George Kng (878, 887/902) dscussed an analogy between an annuty certan and a lfe annuty n 878 and gave a rgorous proof n 887 as dd E. F. Spurgeon (929). Kng s (878, 887/902) proof uses the dea that survval probabltes occurrng n the more dstant future (and thereby most affected by dscountng n a lfe annuty) are less heavly dscounted n an annuty certan to lfe expectancy where all survval probabltes are lumped nto lfe expectancy. Spurgeon s (929) proof reles on the result that the arthmetc mean of postve quanttes exceeds the geometrc mean. Nether Kng (878, 887/902) nor Spurgeon (929) cted a prevous proof. It s unclear whether Kng (878, 887/902) produced the frst rgorous proof, or whether a proof was gven earler (Kopf, 925). Useful references nclude Steffensen (99), Sarason (960) and Olson (975). Ths nequalty has had a checkered and stored past, havng been asserted as an equalty n the early actuaral scence lterature. Indeed, some contnue to hold ths ncorrect vew n very modern tmes. For example, Hackng (999) asserted The far prce for 00 for lfe must be the same as that for a termnal annuty for n years, where n s the expectaton of lfe.

3 Interchangeablty of the medan operator wth the present value operator 3 Downloaded By: [Cecka, James] At: 9:56 7 June 20 be nterchanged. It s natural to nqure about whether there s a correspondng nequalty between the present value operaton, whch s the essence of an annuty, and the med operaton. The chef result of ths artcle s that the ntuton about lack of nterchangeablty emboded n a x <a _exj s wrong when t comes to the medan and present value operatons, that s, med and PV do nterchange. It wll be shown that a medðyalx Þj ¼ med½pvðyal xþš; an annuty certan evaluated at the medan years of addtonal lfe equals the medan of the present value random varable of addtonal years of lfe. We need to establsh a result for medans before provng the Medan Annuty Theorem below. We start wth the general defnton of the medan. Defnton : x med s the medan of the random varable X, also denoted med(x), f and only f t satsfes P½X x med Š0:5 andp½x x med Š0:5. Lemma 2: Let g(x) be monotonc ether nondecreasng or nonncreasng. Then for any random varable X, med½gðxþš ¼ g½medðxþš. Proof 2: Assume g(x) s ncreasng. The events ½X x med Š and ½gðXÞ gðx med ÞŠ are the same events, and so have the same probablty, whch by defnton equals or exceeds 0.5. The same s true of ½X x med Š and ½gðXÞ gðx med ÞŠ. Ifg(x) s decreasng, ½X x med Š and ½gðXÞ gðx med ÞŠ are the same events, and each wth probablty greater than, or equal to, 0.5; and the same statement holds for the opposte nequaltes n the second part of the medan defnton. The lemma now follows by defnton. Medan Annuty Theorem: a medðyalxþj ¼ med½pvðyal xþš. Proof: In Equaton 2, PV(YAL x ) s an ncreasng functon of YAL x. Applyng Lemma 2 wth PV() replacng g() and YAL x replacng X gves the result med½pvðyal x ÞŠ ¼ PV½medðYAL x ÞŠ ; a medðyalx Þj The result s llustrated wth the closed form expresson for PV() as follows: ( X ) ¼YAL x med½pvðyal x ÞŠ ; med ¼0:5 ð þ rþ ¼ h r ð þ rþ0:5 ð þ rþ medðyalxþ snce medans may be nterchanged wth constants, and ð þ rþ YAL x s monotonc. However, a medðyalx Þj ; ¼medðYAL X x Þ ¼0:5 ð þ rþ ¼ h r ð þ rþ0:5 ð þ rþ medðyal xþ The rght-hand sdes are equal, llustratng the general result and provng t here n partcular. 3 & Consder Table. For young to late mddleaged men (approxmately age x, 70), 4 we have Table. Lfe expectancy, medan years of lfe, lfe annuty, annuty certan to lfe expectancy and annuty certan to medan years of lfe for men usng 0.02 dscount rate Age x _e x med(yal x ) a x ($) a _exj ($) a medðyal xþj ¼ med½pvðyal xþš ($) Source: _e x from Aras (200, table 2). Notes: _e x reproduced wth Equaton. med(yal x ) computed wth the defnton P½X x med Š0:5 and P½X x med Š0:5. a x computed wth Equaton 3. a _exj computed wth Equaton 4. a medðyal xþj computed usng Equaton 2 evaluated at med(yal x). 3 A related result holds for the mode. That s, the annuty certan evaluated at the modal years of addtonal lfe equals the mode of the present value functon of addtonal years of lfe; a modðyalxþj ¼ mod½pvðyal xþš. 4 Survval data are from Unted States Lfe Tables 2006, table 2, Aras (200).

4 4 G. R. Skoog and J. E. Cecka Cumulatve probablty YAL and PV (YAL) YAL DF of YAL DF of PV PV of YAL Fg.. Illustraton of the Medan Annuty Theorem for men age 20 and r = 0.02 Downloaded By: [Cecka, James] At: 9:56 7 June 20 _e x <medðyal x Þ, but _e x >medðyal x Þ when the age nequalty changes. Combnng these emprcal results wth the above theorem, we have (for men age x, 70) a x ¼ E½PVðYAL x ÞŠ<a _exj <med½pvðyal xþš ¼ a medðyalx Þj : For older men (x 70) a _exj >med½pvðyal xþš ¼ a medðyalx Þj Fgure llustrates the Medan Annuty Theorem for men aged 20 wth r = The cumulatve Dstrbuton Functon (DF), the ntegral or ndefnte sum of the probablty mass functon, s the natural representaton wth whch to llustrate the determnaton of medans, snce the medan occurs where the horzontal lne at 0.5 crosses the DF. The DF of YAL 20 (the rght-most functon) and the DF of PV(YAL 20 ), the mddle functon, show medðyal 20 Þ¼59:5 and med½pvðyal 20 ÞŠ ¼ $35:, respectvely. Startng wth med(yal 20 ) = 59.5 on the rght-hand scale and movng to the present value of an annutycertan functon (left-most functon), we see that PV½medðYAL 20 ÞŠ ¼ a medðyal20 Þj ¼ a 59:5j ¼ $35: as well. That s, the medan of the present value equals the present value functon evaluated at the medan addtonal years of lfe the Medum Annuty Theorem. IV. Concluson The medan operator can be nterchanged wth the annuty-certan operator, unlke the relatonshp between the expectaton and annuty-certan operators. Therefore, the medan of the present value of a lfe annuty has a value equal to an annuty certan wth term equal to medan addtonal years of lfe. For young to late mddle-aged people, the medan value of a lfe annuty exceeds the value of an annuty certan wth lfe expectancy as ts term, whch always exceeds the expected value of a lfe annuty. The results for the medan easly generalze for any cumulatve probablty characterstc lke the nterquartle range. The results for the medan also hold for jont-lfe annutes and last-survvor annutes. The far prce of a lfe annuty s ts expected present value, and annuty contracts are correctly based on expected present value. However, a purchaser of a lfe annuty also mght be nterested n knowng ts medan present value. Chrstaan Huygens (669) dentfed the medan addtonal years of lfe wth the dea of wagerng. He sad There are therefore two dfferent choces for the expectaton or the value of the future age of a person, and the age to whch there s equal lkelhood that he wll survve or not survve. The frst s n order to set [a] lfe penson, and the other for wagers. In the same way, a person mght be nterested n both the expected present value and the medan present value of a lfe annuty. The Medan Annuty Theorem tells us that half of a populaton at rsk wll receve an annuty whose value s at least equal to an annuty certan whose term s fxed at medan addtonal years of lfe. Sad dfferently, more than half of the populaton wll receve an annuty that exceeds a x for x, 70. Consder a 20-year-old male n Table. Lfe expectancy s _e 20 ¼ 56: years and med(yal 20 ) = 59.5

5 Interchangeablty of the medan operator wth the present value operator 5 Downloaded By: [Cecka, James] At: 9:56 7 June 20 years. The far prce of a lfe annuty s a 20 = $33.2, but half of the populaton at rsk wll receve at least a medðyalx Þj ¼ $35:. That s, a Huygens bettng man mght be attracted to an annuty whose prce s $33.2 snce the probablty of recevng an annuty exceedng that far purchase prce s greater than 50%. References Aras, E. (200) Natonal Vtal Statstcs Reports, Unted States Lfe Tables, 2006, US Department of Health and Human Servces, Centers for Dsease Control and Preventon, Hyattsvlle, MD. De Wtt, J. (67) Value of Lfe Annutes n Proporton to Redeemable Annutes, State Archves, The Hague, publshed n Dutch wth an Englsh translaton n Hendrcks (852, 853). Haberman, S. and Sbbett, T. A. (995) Hstory of Actuaral Scence, Vol. I, Wllam Pckerng, London. Hackng, I. (999) The Emergence of Probablty, Cambrdge Unversty Press, Cambrdge. Halley, E. (693) An estmate of the degrees of mortalty of manknd, drawn from the curous tables of the brths and funerals at the cty of Breslaw; wth an attempt to ascertan the prce of annutes upon lves, Phlosophcal Transactons, 7, , Hendrcks, F. (852/853) Contrbutons of the hstory of nsurance and the theory of lfe contngences, The Assurance Magazne 2, 2 50, ; The Assurance Magazne 3, Huygens, C. (669) The correspondence of Huygens concernng the blls of mortalty of John Graunt, extracted from Volume V of the Oeuvres Completes of Chrstaan Huygens. Avalable at: Sources/Huygens/sources/cor_mort.pdf (accessed 8 May 20). Kng, G. (878) On the analogy between an annuty certan and a lfe annuty, Journal of the Insttute of Actuares, 20, Kng, G. (887/902) Lfe Contngences, Insttute of Actuares: Charles & Edwn Layton, London. Kopf, E. W. (crca 925) The early hstory of the annuty, Proceedngs, Casualty Actuary Socety, 3, Olson, G. E. (975) On some actuaral nequaltes actuaral note, Transactons of Socety of Actuares, 27, 22. Potras, G. (2000) The Early Hstory of Fnancal Economcs, , Edward Elgar, Cheltenham. Sarason, H. M. (960) A layman s nterpretaton of the expectancy annuty, Transactons of Socety of Actuares, 2, Spurgeon, E. F. (929) Lfe Contngences, 2nd edn, Unversty Press for the Insttute of Actuares, Cambrdge. Steffensen, J. F. (99) On certan nequaltes and methods of approxmaton, Journal of the Insttute of Actuares, 5,