A rank-dependent utility model of uncertain lifetime, time consistency and life insurance

Transcription

1 A rank-dependen uiliy model of uncerain lifeime, ime consisency and life insurance Nicolas Drouhin To cie his version: Nicolas Drouhin. A rank-dependen uiliy model of uncerain lifeime, ime consisency and life insurance <halshs v3> HAL Id: halshs hps://halshs.archives-ouveres.fr/halshs v3 Submied on 5 Dec 2012 HAL is a muli-disciplinary open access archive for he deposi and disseminaion of scienific research documens, wheher hey are published or no. The documens may come from eaching and research insiuions in France or abroad, or from public or privae research ceners. L archive ouvere pluridisciplinaire HAL, es desinée au dépô e à la diffusion de documens scienifiques de niveau recherche, publiés ou non, émanan des éablissemens d enseignemen e de recherche français ou érangers, des laboraoires publics ou privés.

2 A rank-dependen uiliy model of uncerain lifeime, ime consisency and life insurance Nicolas Drouhin December 5, 2012 Absrac In a coninuous ime life cycle model of consumpion wih uncerain lifeime and no pure ime preference, we use a non-parameric specificaion of rank dependen uiliy heory o characerize he preferences of he agens. From a normaive poin of view, he paper discusses he implicaion of adding an axiom of ime consisency o he former model. We prove ha ime consisency holds for a much wider class of probabiliy weighing funcions han he ideniy one characerizing he expeced uiliy model. This special class of probabiliy weighing funcions provides foundaions for a consan subjecive rae of discoun which inerac muliplicaively wih he insananeous condiional probabiliy of dying. We show ha even if agen are ime consisen, life annuiies no more provide perfec insurance agains he risk o live. Code JEL: D81 D91 Key words : ineremporal choice; life cycle heory of consumpion and saving; uncerain lifeime; life insurance; ime consisency; rank dependen uiliy. Thanks: As par of he behavioral revoluion, he radiional exponenial discouning model has been a sack in he las hiry years. Following he pioneering work of Ainslie (1975) and Thaler (1981), many anomalies in ineremporal choice (Loewensein and Prelec, 1992) have been documened (see Frederick e al., 2002, for a survey). Ecole normale supérieure de Cachan and Cenre d économie de la Sorbonne (UMR CNRS 8174) 1

3 There is a long radiion in economics o disinguish wo kinds of primiive for explaining discouning. Firs, discouning can be explained by purely psychological facors, such as impaience, capured by he discoun funcion. If he discoun funcion is exponenial as in he seminal model proposed by Samuelson (1937), hen ime preference is characerized by a pure rae of ime preference (i.e. he log derivaive of he discoun funcion) ha is invarian wih ime and he level of consumpion. Even if some auhors have considered early he possibiliy for he discoun facor o be non-exponeial (for example Yaari, 1964; Harvey, 1986), i is only wih he behavioral revoluion ha alernaive ad hoc paramerical discoun funcions have been proposed and used sysemaically in he applied economics lieraure. Among hem he quasi hyperbolic discoun funcion (Phelps and Pollak, 1968; Laibson, 1997) is probably he mos popular. The second explanaion for discouning is jus o consider ha fuure prospec are uncerain. In his case, i is reasonable o consider ha he uiliy of fuure prospecs will be weighed according o heir probabiliy o be effecively consumed a he given dae (see Sozou, 1998; Dasgupa and Maskin, 2005, for a general discussion of ha opic). Among his lieraure, models of ineremporal choice wih uncerain lifeime, pioneered by Yaari (1965), are good ools o invesigae he heory of discouning. Yaari (1965) s seminal paper was considering expeced uiliy maximizers wih known probabiliy disribuions of he age of deah, and sandard exponeial discouning life cycle-uiliy. More recen models have considered various kind of more sophisicaed uiliy framework o deal wih lifeime uncerainy. For example, Moresi (1999) is considering an applicaion of Selden (1978) s ordinal cerainy equivalen hypohesis. Bommier (2006) considers a concave ransformaion of he lifecycle uiliy o explain risk aversion wih respec o lengh of live. Halevy (2008) uses Yaari (1987) s Dual heory of choice. Ludwig and Zimper (2012) and Groneck e al. (2012) consider a non-bayesian reamens of ambiguous survival probabiliy. The imbricaion of risk and ime preference is also of firs imporance for analyzing life insurance, which was Yaari (1965) s iniial purpose. In his paper, we build a model of ineremporal choice of consumpion and saving wih uncerain lifeime in which he agen psychologically ransforms her survival probabiliy disribuion, like in Quiggin (1982) s rank-dependen uiliy model, or in Tversky and Kahneman (1992), cumulaive prospec heory. The idea of inroducing rank dependen uiliy in his seing has been already explored by Drouhin (2001) and Bleichrod and Eeckhoud (2006). The originaliy of his paper is ha we use coninuous ime modeling 1 and 1 Like in Yaari (1965). 2

4 opimal conrol o solve he model. Wih ha mehodology, we are able o discuss he imporan opic of ime consisency, he main crieria of raionaliy over ime. Following Sroz (1956), here exiss a convenional wisdom in considering ha any deparure from exponenially discouned expeced uiliy will imply ime inconsisency. This paper will give a counerexample. In paricular, i shows ha any agen who ransforms he probabiliy disribuions of he age of deah wih a power funcion are ime consisen. Tha provides foundaions for a coefficien of ime preference ha inerac muliplicaively wih he probabiliy of dying, insead of addiively in Yaari s approach. However, i also demonsraes ha life annuiies do no necessarily provide perfec insurance, even in he case in which agens are ime consisen 2. The plan of he paper is as follows. Secion 1 presens he uiliy funcional used. Secion 2 solves he model in absence of life annuiies. Secion 3 discusses ime consisency. Secion 4 solves he model when agen has access o life annuiies. Finally secion 5 concludes. 1 A rank dependen uiliy model of consumpion and savings wih uncerain lifeime We consider an agen s choice of her consumpion profile. A consumpion profile is a funcion of ime defined on he inerval [0,T], wih 0 he age of birh and T an arbirary consan, inerpreed as he maximum possible life duraion for he agen. Because we are ineresed in undersanding he way he iming of decision influences he choice of he consumpion profile, we will denoe by [0,T), he dae of he decision. Le us consider, in a firs sep, he case in which he agen, alive a dae, knows for sure her age of deah s. H1 If an agen knows wih cerainy her dae of deah s, her ineremporal preferences can be represened by an ineremporal uiliy funcional assumed o be addiive, and saionary, wih no pure ime preference : V (c,s) = s u(c(τ))dτ (1) wih u (c(τ)) > 0 and u (c(τ)) < 0. H2 (monooniciy according o lifespan). c : s > s V (c,s ) > V (c,s) H1 and H2 implies ha u is posiive. H2 means ha for a given consumpion profile, oucomes will be always ranked according o lifespan. When 2 The only excepion being he case of expeced uiliy. 3

5 inroducing uncerainy our model will be a naural candidae for using rank dependen uiliy. The fac ha we do no posulae any kind of pure ime preference implies ha, in our model, lifeime uncerainy is he only primiive of ime preference. The agen acually does no know wih cerainy her age of deah. We assume ha for a living agen, a each age, he age of deah s is an absoluely coninuous random variable defined on he inerval [, T]. We denoe by π (s) > 0 he probabiliy densiy funcion of he random variable, assumed o be differeniable a leas once and Π (s) he cumulaive disribuion funcion. We hus have: Π (s) = s π (τ)dτ wih Π (T) = 1. Π (s) can be inerpreed as he probabiliy of being dead a age s, knowing you are alive a dae, and, (1 Π (s)) he probabiliy of being alive a dae s, knowing you are alive a dae. We can derive from Bayes formula ha for s : and (1 Π (s))(1 Π ( )) = (1 Π (s)) (2) π (s) = π (s) 1 Π ( ) In he special case where s =, we ge he hazard rae a dae s: π s (s) = π (s) 1 Π (s) We are hus facing a special problem of choice in uncerainy. If we assume ha he agen is an expeced uiliy maximizer as in Yaari (1965), we have: EV (c) = T s (3) (4) u(c(τ)dτdπ (s) (5) If we now reain a more general model of choice under uncerainy, in which agens ransform oucomes and probabiliy disribuions as in Quiggin (1982, 1993) or Tversky and Kahneman (1992), we have : RDU (c) = T s u(c(τ)dτdh(π (s)) (6) wih (h) a probabiliy weighing funcion assumed o be coninuous an wice differeniable, and such ha: h(0) = 0, h(1) = 1 and h (Π (s)) 0. 4

6 Le us noice ha (5) is a special case of (6) when h(π (s)) = Π (s). Inegraing (6) by pars, we obain: RDU (c) = T (1 h(π (s)))u(c(s))ds (7) For he agen, he expeced presen value a dae of he uiliy sream beweenandt isheinegraloverhisinervalofheproducofheuiliyof consumpionaeachdaesofheinervalwihhesubjeciveweighgivenby he agen o he even being alive a dae his dae s. Equaion (11) makes explici our iniial inuiion wihin he coninuous ime framework. The facor f (s) = (1 h(π (s))) is he discoun facor applied o uiliy of he consumpion a dae s viewed from dae. I depends only on he probabiliy disribuion of he ages of deah and he subjecive ransformaion of his probabiliy disribuion. I is coninuous, derivable and sricly decreasing from one o zero on he inerval [,T]. Taking he log-derivaive of his discoun facor, we can also define he rae of discoun of uiliy a dae s viewed from dae : θ (s) def = h (Π (s)))π (s) 1 h(π (s)) (8) Thus we can rewrie he ineremporal rank-dependen uiliy funcional (7): RDU (c) = T ( ) exp θ (τ)dτ u(c(s))ds (9) s A his sage, we jus wan o noice ha he mahemaical srucures of hose discoun facor and rae provides a very ineresing case. On he one hand, his mahemaical srucure is much more general han he one ha prevails in radiional exponenial or hyperbolic model of ineremporal choice. On he oher hand, he mahemaical srucure is also much more precise han he mos general form sudied by Yaari(1964) where he discoun facor is only assumed o be posiive and differeniable. As in Drouhin (2001), he rae of discoun can be decomposed in wo facors. Thefirsone, π (s),heprobabiliydensiyassociaedwihheeven dyingadaes, knowinghayouraliveadae, canbeinerpreedashe objecive par of ime preference. The second one, h (Π (s))/(1 h(π (s))), depends on he way he agen ransforms probabiliy disribuions. I can be inerpreed as he subjecive par of ime preference. The objecive par is exacly he same as in Yaari (1965). The subjecive par sems from he rankdependen formulaion. I will be of firs imporance when we will discuss 5

7 he consequences of his formulaion for sandard resuls on ime consisency and life insurance. Wearenowgoingoinvesigaeheproperiesofhechoiceofheopimal consumpion pah made by an agen a dae. 2 The opimal consumpion pah wih no life annuiies To express he opimal consumpion pah, we have firs o define he feasible se of consumpion profiles. We assume ha a each dae s he living agen receives a flow of non-financial income w(s) assumed o be coninuous and differeniable and a flow of financial income proporional o her asse (a(s)). Those incomes are eiher used for curren consumpion or saved for fuure consumpion. In his par we assume ha here are no life annuiies. The only asse available for savings is sandard bond earning a consan rae of ineres r. Thus, a each ime s, he sandard ineremporal budgeary consrain holds: s [0,T], ȧ(s) = w(s)+ra(s) c(s) (10) We also assume ha here is no beques moive implying ha an agen living is maximum possible life-duraion will choose o leave no beques a(t) = 0. Thus, if beween and T, we sum he differenial consrains (10) a each dae weighed by he economical discoun facor exp( rs), we obain afer some simple manipulaions ha, for an agen living he maximum possible life-duraion : a()+ T w(s)e r(s ) ds = T c(s)e r(s ) ds (11) This is he very sandard life cycle budgeary consrains, he presen value of all incomes over he life cycle is equal o he presen value of he consumpion sream. Thus we can express he oal sock of asses a dae : a() = s 0 (w(s) c(s))e r(s ) ds (12) We now consider an agen a dae who has o decide her opimal consumpion pah beween and T. We denoe c (s) he opimal consumpion pah 6

8 decided a dae for he ime inerval [,T]. Thus c (s) is he soluion of he following program: P T maxrdu(c) = (1 h(π (s)))u(c(s))ds c u.c. ȧ(s) = w(s)+r a(s) c(s)) a() = cs a(t) = 0 Because of he coninuiy of (w) (h) and (Π ) and he coninuiy and sric concaviy of (u) his program can be shown o admi a unique soluion ha will be coninuous and derivable 3. Applying Ponryagin s maximum principle, he resoluion of such problem implies o solve a sysem of differenial equaions. If, for no losing generaliy of he resuls, we refuse o specify special easy o use funcional form for uiliy, earnings and probabiliy disribuion of he age of deah, he only hing we can do is o derive he rae of growh of he opimal consumpion pah planned a dae. Proposiion 1. Wihou life annuiies, a each dae s, he rae of growh of he opimal consumpion pah planned a dae is: ċ (s) c (s) = r θ (s) γ (s) (13) wih γ (s) = def u (c (s)) u (c (s)) c (s) (14) he coefficien of relaive resisance oward ineremporal subsiuion Proof: The Hamilonian of agen s program is: H = (1 h(π (s))) u(c(s))+λ(s)(w(s)+ra(s) c (s)) Firs order condiions gives: H c = 0 λ(s) = (1 h(π (s)))u (c (s)) (15) H a = dλ ds (s) = λ s λ s λ s = r (16) 3 For a purpose of simpliciy, we have no aken ino accoun he borrowing consrains ha is usually associaed wih he model wih no life annuiies/insurance. See Leung (1994) for an exensive discussion of his opic. 7

9 Taking he logarihm of (15) and differeniaing according o s e we ge: λ s = h (Π (s))π (s) λ s 1 h(π (s)) + u (c (s)) dc (s) (17) u (c (s)) ds Comparing (16) and (17), and using definiion (8), we deduce proposiion 1. Proposiion 1 is he mos general predicion one can make wihin he life cycle heory of consumpion and saving. The rae of growh of he consumpion pah is he difference beween he rae of ineres (economic discoun rae) and he rae of ime preference, boh divided by an index of he curvaure of he uiliy funcion usually referred as he coefficien of relaive risk aversion or more properly in his conex, according o Gollier (2001), as he resisance o ineremporal subsiuion. The imporan poin is ha, as in Yaari (1965) he rae of ime discouning is no more consan and can give a wide variey of possible dynamic for consumpion. Bu conrarily o Yaari (1965) i is no only he properies of he probabiliy disribuion of he ages of deah ha maers. The way agens ransform subjecively his probabiliy disribuion will also maer. If we wan o go furher, we have o specify some more resricions o he model. 3 Time consisency The ineremporal choice model wih uncerain lifeime combine boh risk and ime. We can specify he model for being consisen wih some crieria of raionaliy. Because i uses rank- dependen uiliy, our model fulfils necessarily, and by consrucion, he main axiom of raionaliy oward risk, firs order sochasic dominance (Quiggin, 1993). According o decision in ime, he main crieria of raionaliy is ime consisency proposed by Sroz (1956). Wha resricion do we have o impose o he probabiliy ransformaion funcion o fulfil ime consisency? To answer his quesion we have o define properly he noion of ime consisency. Using dynamic opimizaion, we will use he same definiion as Sroz (1956); Capuo (2005); Drouhin (2009, 2012). In he absence of new informaion, an agen is said o be ime consisen if she behave in he fuure as she has planned in he pas. Definiion 1 (Time consisency). If we denoe (c )and (a ) he soluion of he program P. If we denoe (c )and (a ), he opimal soluion of he 8

10 program P, wih [,T] and: T maxrdu(c) = (1 h(π (s)))u(c(s))ds c u.c. ȧ(s) = w(s)+r a(s) c(s)) P a( ) = a ( ) a(t) = 0 Then an agen is ime consisen if and only if: [0,T], [,T], s [,T] : c (s) = c (s) (18) A well known corollary of his definiion is ha for being ime consisen he rae of discoun a each dae s has o be independen from he decision dae. 4 In he special case of expeced uiliy, θ (s) = π s (s), whaever he form of he probabiliy disribuion, i does no depend on he planning decision dae, i is hus ime disance independen, ime consisency holds. Bu for oher cases he disribuion probabiliy of deah and he rankdependen uiliy give a special mahemaical srucure o he discoun rae and facor. We can noice ha in he mos general case θ (s) is ime-disance dependen because i depends on Π (s). I is a srong presumpion for ime inconsisency. Is here neverheless some oher cases where ime consisency holds? Proposiion 2. Agen is ime consisen if and only if her probabiliy disribuion ransformaion funcion is of he form h(x) = 1 (1 x) α wih (α > 0). Proof: (sufficiency) h(x) = 1 (1 x) α θ (s) = απ s (s) The rae of discoun is independen of he planning dae so he choice of consumpion is ime consisen. (necessiy) Ifheagenisimeconsisen, shefulfillsequaion(18). Asc (s)issricly posiive and differeniable, i implies ha: [0,T], [,T], s [,T] : ċ (s) c (s) = ċ (s) c (s) Taking ino accoun(13), (8), and (3) i implies ha: 4 See, for example, Drouhin (2012) for a rigorous proof. 9

11 [0,T], [,T], s [,T] : h (Π (s)) 1 h(π (s)) = h (Π (s)) (1 h(π (s)))(1 Π ( )) (19) Thisequaionshouldholdinheparicularcasewheres =. Considering his case and remarking ha Π () = 0, equaion (19) also implies : [0,T], [,T] : h (Π ( )) 1 h(π ( )) = h (0) (1 Π ( )) (20) This is a firs order differenial equaion wih a se of soluions fully described by h(x) = 1 (1 x) α, wih α = h (0). The disribuion of he probabiliy of dying and is reamen wihin Rank-Dependen Uiliy Theory of choice added wih an axiom of ime consisency gives behavioral foundaions o a model of ineremporal choice ha is raher simple and no less inuiive han he sandard discouned uiliy model. Some poins should be noiced. 1. The expeced uiliy model is no he only one compaible wih ime consisency. The rank-dependen uiliy model wih a power funcion for ransforming he probabiliy disribuion 5 implies also ime consisency. This gives behavioral foundaions for a model of ineremporal choice ha is differen from he original discouned expeced uiliy model. 2. When agen is ime consisen we have: The uiliy funcional can be rewrien : RDU (c) = θ (s) = απ s (s) (21) T e α s πτ(τ)dτ u(c(s))ds (22) 3. The parameer α can be inerpreed as muliplicaive facor of ime preference. If α > 1, his means ha he agen gives a psychological weigh o presen consumpion more imporan han he insananeous probabiliy of dying. In his case he agen will demonsrae preference for presen consumpion. In a RDU/ Cumulaive prospec heory, he behavior is inerpreed as pessimisic in he sense ha she ends o 5 Diecidue e al. (2009) provide axiomaic foundaions for such probabiliy ransformaion funcion. 10

12 overweigh her probabiliy of dying 6. In he opposie, if α < 1, he agen will demonsrae a kind of preference for fuure consumpion, she underweigh her probabiliy of dying ( opimisic behaviour). 4. InYaari (1965) he rae of discoun was: θ (s) = θ+π s (s) (23) I means ha our model has he same level of complexiy as Yaari s model. I depends only on one parameer and on he condiional probabiliy densiy of he even dying a dae s knowing you are alive a dae s. In Yaari (1965), he parameer θ inerac addiively wih he probabiliy of dying. In our s he parameer α inerac muliplicaively. Boh parameers can be inerpreed as measuring ime preference. Bu a huge difference is ha in Yaari (1965) discouned expeced uiliy model, as in sandard economic heory, he parameer θ is posulaed as a coefficien of pure ime preference, in a raher ad hoc manner. In he conrary, in our ime consisen rank dependen uiliy model, we have no posulaed any ad-hoc form of ime preference. The coefficien α jus sems from he axiom of ime consisency, as we have demonsraed 7. To commen furher, we have o consider special disribuions of he age of deah. The mos simple case (and no so unrealisic) is he one of a consan hazard rae (sandard Poisson process). For ha o be possible we have o allow he maximum duraion of life, T, o go o infiniy. Le us assume ha, for all s R +, π s (s) = π = cs. In his case he ineremporal uiliy funcional is equivalen o he exponenial discouning model (wih α π = θ): RDU (c) = + e απ(s ) u(c(s))ds (24) The sandard model of exponenial discouning is jus a simplified version of our general rank-dependen uiliy model of ineremporal choice wih uncerain lifeime in he special case of ime consisency and consan hazard rae. If we adop a more realisic model of uncerain lifeime, like for example he Gomperz law, hen he hazard rae is increasing wih age. This has an imporan consequence : 6 See, for example, Wakker (2010), , for an exensive discussion of probabiliy ransformaion as pesimism/opimism 7 Same degree of complexiy, less ad-hociy, according o Occam s razor principle, our model is a beer candidae for modeling uncerain lifeime and ime insurance! 11

13 Proposiion 3. If he consumpion sream is bounded and if here exiss an π age ˆ such ha for all > ˆ, () > 0, hen for all T R + +{+ } hen he ineremporal rank dependen uiliy funcional (22) is always definie. Proof: Obvious. This proposiion means ha our concep of muliplicaive ime preference is more robus han he usual rae of pure ime preference of he exponenial discouning model. In paricular preference for presen consumpion in our model (α > 1) is no a prerequisie for our uiliy funcional o be definie even when he horizon is infinie. From a behavioral perspecive i means ha our model is a ool o explore possibiliies han canno be addressed wih he sandard discouned expeced uiliy model. 4 The opimal consumpion pah wih life annuiies As in Yaari (1965), we will now assume ha agens have access o acuarial noes issued by insurance companies or pension funds. Those noes are coningen asse ha pay R() as long as he agen is alive, and 0 afer her deah. If insurance companies refund hemselves on he bond marke a he rae r, and if hose noes are acuarially fair, hen i is well known ha: R() = r+π () (25) When here is no beques moives, sandard bonds are sricly dominaed by life annuiies. Thus differenial consrain of he program can be rewrien: s [0,T], ȧ(s) = w(s)+r(s)a(s) c(s) (26) We can now deduce he propery of he opimal ineremporal consumpion profile when he agen have access o life annuiies. Proposiion 4. When he agen has access o life annuiies, hen a each dae s, he rae of growh of he opimal consumpion pah planned a dae is: ċ (s) c (s) = R(s) θ (s) γ (s) = r+π s(s) h (Π (s)))π (s) 1 h(π (s)) γ (s) (27) 12

14 Proof: We proceed exacly he same way as in Proposiion 1. The main resul of Yaari s model was ha when he agen has access o life annuiies he rae of growh of he ineremporal consumpion profile was no more deermined by he condiional probabiliy of dying, and hus was he same as he one of he model wih a cerain life duraion. This is why life annuiies are considered as offering perfec insurance wih he meaning ha uncerainy has no more influence on he rae of growh of opimal consumpion. Obviously, as Proposiion 4 shows, ha is generally no more he case in our rank-dependen uiliy model of ineremporal choice. Corollary 4-a. If he agen is ime consisen and has access o life annuiies, he rae of growh of he opimal consumpion pah planned a dae is: ċ (s) c (s) = r+(1 α)π s(s) (28) γ (s) This resul is imporan. In he case α = 1 (expeced uiliy), we rerieve Yaari s resul. Bu if α 1, hen we have no perfec insurance, i.e. he condiional probabiliy of dying sill deermine he rae of growh of he consumpion profile. Wha is is imporan o noice is ha, in his las case, he agen is fully raional, she fulfils simulaneously firs order sochasic dominance and ime consancy. Neverheless, here is no more perfec insurance in his case 8. This resul is imporan for analyzing social securiy. Wihin Yaari s model fully funded social securiy is considered as equivalen wih life annuiies, and hus provides perfec insurance. In our model his is no more necessary rue, even for fully raional agens. 5 Conclusion Convenional wisdom generally considers exponenial discouning and expeced uiliy as being he only models of choice compaible wih full raionaliy. If hose models do no fi he acual behavior of agens, hen i is legiimae o consider alernaive descripive/behavioral models. If you believe in he convenional wisdom, you will deduce ha agens are no more raional. Obviously, from a normaive poin of view, ha has very imporan implicaions for policy design. However, his paper proves, in he case of life 8 Using Selden (1978) s ordinal cerainy equivalen insead of rank dependen uiliy, Moresi (1999) arrives o a very similar conclusion, in he special case of an iso-elasic per period uiliy funcion. RDU is more general because i implies no resricion on he per-period uiliy funcion. 13

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