Myopic Loss Aversion and the Equity Premium Puzzle

Transcription

1 Cand.mer. Applied Eonomis and Finane Deparmen of Eonomis Maser Thesis Myopi Loss Aversion and he Equiy Premium Puzzle An Alernaive Explanaion Applied o he Danish Sok Marke Suden: Line Isager-Nielsen Handed in: November s 2006 Insruor: Moren Bennedsen Copenhagen Business Shool 2006

2 Exeuive Summary This hesis presens an aemp o resolve he well-known equiy premium puzzle using insigh from behavioural finane namely prospe heory and he onep known as myopi loss aversion. The noion is ha he reason why eonomis have had suh a hard ime reoniling he prediions of sandard expeed uiliy heory o real world observaions is ha deision makers do no behave as suggesed by he sandard normaive model. Raher a new desripive heory is warraned sine deision makers in heir behaviour are observed o violae vial assumpions underlying uiliy maximisaion. The hesis firsly reviews he original equiy premium puzzle from 985, presening he finding ha observed sok reurns in he US by far exeed he prediions of expeed uiliy heory. Only an assumpion of implausibly high risk aversion of agens is able o reonile empiris and heory. The hesis hen proeeds o argue he failure of expeed uiliy heory as a useful desripive model of deision making behaviour. As he alernaive, prospe heory is presened. A purely desripive model ha enompasses loss aversion, i.e. he noion ha a loss o an agen is more hurful han an equivalen win is pleasurable. Prospe heory is ombined wih myopia o build he alernaive desripion of deision making behaviour. Myopia is relaed o menal aouning and apures he asserion ha individual are onsanly monioring he suess or failure of heir finanial disposiions. Thus he alernaive desripion of behaviour is myopi loss aversion agens are aggravaed exraordinarily by losses and are frequenly evaluaing resuls. Soks are volaile and will drop in value from ime o ime bu over he long run he average reurn is high. Myopi loss averse invesors will evaluae ofen and be hur by volailiy by observing losses and so hey will view soks as a less araive invesmen han if hey were raional. In urn hey will demand a higher premium o inves in soks. This is he raionale for why desribing invesors as myopi loss averse should resul in a higher observed risk premium o sok invesmen. In he empirial par of he hesis, he heoreial argumen is pu o work on Danish sok marke daa, where firsly i is onfirmed ha here exiss an equiy premium puzzle similar o he one doumened in he US weny years ago. Seondly a model based on prospe heory and myopi loss aversion is fied o he Danish daa o arrive a he implied faors of invesor behaviour. The resuls show ha he approah an reonile he puzzle in Denmark if i is assumed ha myopi invesors evaluae resuls every year and are hur approximaely wie as hard by a loss han pleased by a similar gain. These assumpion lead o he observed equiy premium bu also opimal asse alloaions very similar o observed behaviour in he Danish marke.

3 Conen.0 Inroduion Problem Saemen Problem Definiion Mehodology and Delimiaions Daa Desripion of Daa The Equiy Premium Puzzle The Represenaive Agen Model Consumpion CAPM Deduion of he Model Empirial Resuls of Mehra and Preso Furher Validaion of he Empirial Resuls Hisorial Aemps o Explain he Equiy Premium Puzzle Complee Markes Assumpion No Transaion Coss Assumpion Alernaive Preferene Sruure Habi Formaion Keeping Up Wih he Joneses Generalized Expeed Uiliy Myopi Loss Aversion Chaper Summary Expeed Uiliy Theory and Beyond The Axioms of Expeed Uiliy Theory Violaions of he Axioms of Expeed Uiliy Theory Chaper Summary Prospe Theory The Framing Phase The Evaluaion Phase The Value Funion The Weighing Funion Behavior of he Weighing Funion Tehnial Formulaions of Prospe Theory Chaper Summary Behavioral Finane Loss Aversion The Endowmen Effe The Saus Quo Bias Seion Summary Menal Aouning Menal Aouning and he Framing Phase of Prospe Theory Choie Brakeing Seion Summary The Combinaion of Loss Aversion and Choie Brakeing

4 7.0 Empirial Analysis of Myopi Loss Aversion in he US Inroduion Empirial Hypohesis Resuls of Benarzi and Thaler Evaluaion Period Opimal Asse Alloaion Implied Equiy Premium Furher Evidene of Myopi Loss Aversion Chaper Summary Empirial Analysis of he Equiy Premium Puzzle in Denmark Daa Empirial Resuls Chaper Summary Empirial Analysis of Myopi Loss Aversion in Denmark Inroduion Mehodology Evaluaion Period Opimal Asse Alloaion Implied Equiy Premium Empirial Resuls Evaluaion Period Opimal Asse Alloaion Implied Equiy Premium Seion Summary Sensiiviy Analyses Real Reurns vs. Nominal Reurns Money Marke Reurns vs. 5-year Governmen Bond Reurns Loss Aversion Parameer Value Relaed Researh Seion Summary Disussion of Poenial Limiaions Chaper Summary Conlusion Referenes Appendies

5 .0 Inroduion The equiy premium puzzle is a paradox originally presened by Mehra and Preso in 985. They found ha he risk adjused reurn o US sok invesmen over a risk free invesmen had been muh oo high over he ourse o he wenieh enury relaive o wha radiional heory would sugges. This puzzle has been horoughly sruinized over he years and numerous researhers have ried o reonile he empirial evidene o eonomi heory. Many aemps have proven fuile and he puzzle has held up remarkably well o advanes in modern finane heory. The original analysis of he equiy premium puzzle is based on expeed uiliy heory, whih has radiionally dominaed he analysis of deision making under risk and has generally been aeped as a normaive model of raional hoie, and in urn, been applied as a desripive model of eonomi behaviour. The equiy premium puzzle is hus an illusraion of a disrepany beween wha he heory predis and wha is aually observed empirially. Behavioural finane as an area of researh has gained muh suppor in he lae wenieh enury and has been posiioned as an ineresing alernaive o many radiional ways of looking a finane. In his area, psyhologiss Daniel Kahneman and Amos Tversky have been quie influenial and in 979 hey developed a onep of behavioural finane alled prospe heory as an alernaive formulaion of invesor behaviour and risk aiudes. In heir researh Kahneman and Tversky found ha he prediions of expeed uiliy heory someimes fail beause agens diverge from raionaliy and uiliy maximisaion in heir deision-making. Prospe heory is a desripive model ha apures how agens are aually observed o behave raher han predi how agens are supposed o behave. In 995, Benarzi and Thaler uilized prospe heory o presen an approah alled myopi loss aversion whih onsiss of wo behavioural oneps, namely loss aversion and menal aouning. They hypohesized ha a ombinaion of a srong aversion o losses (raher han risk per se) and frequen evaluaion of porfolio reurns is a more inuiive desripion of invesor behaviour and in doing so presened a plausible explanaion of he equiy premium puzzle using empirial US daa. This hesis will explore he usefulness of his approah as a desripive model of he Danish finanial markes. Benarzi and Thaler have reahed plausible resuls for he US finanial markes bu hus far none have horoughly invesigaed wheher he Danish finanial marke serves o be a similarly aommodaing esing ground where myopi loss aversion an gain furher suppor. 4

6 2.0 Problem Saemen The inroduion above ses he sage for wha I se ou o do in his hesis. This haper defines he aual objeive of he hesis and how I inend o reah ha objeive. The nex subseion more onreely defines he overall mission of he hesis and he issues ha I seek o resolve in order o omplee his mission. Based on his formulaion of my objeives, I will presen a moivaed desripion of he sruure of he hesis, i.e. review he differen pars, why hey are in he hesis, how he omposiion is moivaed and he limis of my subje area. 2. Problem definiion As menioned, he overall objeive of his hesis is o invesigae he usefulness of myopi loss aversion as a plausible explanaion for he equiy premium puzzle using Denmark as he empirial esing ground. Following his line of argumenaion, I an pose he main quesion ha I seek o answer in he hesis as he following; Does myopi loss aversion as a desripion of invesor behaviour onsiue an improvemen in explaining he equiy premium in Denmark? Evidenly, several subordinae asks need o be addressed in order o shed ligh on his overall objeive. Firsly, we mus review he mehanis of he equiy premium puzzle in order o answer he quesion of wha auses he puzzle and why he behavioural finane field migh heoreially resolve he paradox. Moreover, as a supporive quesion i is illusraive o address wheher oher alernaive approahes have already suessfully resolved he paradox or if he puzzle sill holds. Par of answering his quesion is obviously o sipulae wheher or no he puzzle is a produ of resriive sylised heoreial assumpions. I should be eviden by now ha I will argue ha he assumpions regarding invesor behaviour underlying he model is of vial imporane. Seondly, i is of vial imporane o argue he usefulness of behavioural finane and onreely he oneps of prospe heory and myopi loss aversion as desripive models of he behaviour of individual agens. To his end, I will show how sandard expeed uiliy heory fails o desribe he observed behaviour ha is apured by prospe heory. In order o reah hese onlusions I mus invesigae evidene on he aual behaviour of individuals and how his behaviour onradis expeed uiliy heory and I mus dediae effors o reviewing prospe heory and argue how i onsiues an aommodaing alernaive. Furher I will dediae some effor o explaining he oneps ha ombined onsiue myopi loss aversion, namely loss aversion and menal aouning. Finally, invesigaing he magniude of he equiy premium in Denmark is a enral par of he overall objeive 5

7 beause he resul of suh an invesigaion will serve as a benhmark for evaluaing he effeiveness of my approah. Thus I will submi he Danish finanial marke o a es similar o he original es for he equiy premium puzzle in he US. So in order o answer he main quesion, he following sub quesions mus be analysed: a. Wha is he equiy premium puzzle? b. Have oher alernaive approahes suessfully resolved he paradox or does he puzzle sill hold?. Is expeed uiliy heory a useful desripive model? d. Wha is prospe heory and how does i differ from radiional expeed uiliy heory? e. Wha is loss aversion and menal aouning and how an hese oneps heoreially resolve he puzzle? f. Wha is he equiy premium in Denmark and does i onsiue a puzzle? Based on his ground work I will be able o address he overall mission saemen. The nex seion pus ino onree language he overall sruure and moivaion for he individual pars of he analyses o be made. 2.2 Mehodology and Delimiaions This seion provides an overview of and argumenaion for he heoreial mehodology applied hroughou his hesis as well as he sruure of he hesis and he delimiaions ha will be made. The main objeive of he hesis is o invesigae wheher or no myopi loss aversion onsiues a reasonable explanaion o he equiy premium puzzle. So a he very basis, I mus aoun for, wha he equiy premium puzzle aually is. For his purpose I urn o Mehra and Preso who were he firs o doumen he puzzle, so naurally heir model deserves some aenion. In order o provide a more nuaned piure of he puzzle, I also hose o inlude a more saisial approah ondued by Koherlakoa (996) beause his provides me wih a more dire ool o es if an equiy premium puzzle exiss in Denmark. Throughou he las 2 years, sine Mehra and Preso firs doumened he puzzle, several aemps have been made rying o aoun for heir findings. Some of hese onerned poenial adjusmens o he empirial side of he puzzle, some onerned an exploraion of differen heoreial frameworks and ye ohers have foused on relaxing he key assumpions underlying he model. Myopi loss aversion belongs o he laer of hese groups of aemps. So I delimi he disussion of he differen aemps o hose ha have as heir objeive o relax he underlying assumpions. 6

8 A he har of myopi loss aversion lies prospe heory whih is a desripive alernaive o radiional raional deision heory, speifially expeed uiliy heory. In order o see why we mus abandon his radiional approah, I dediae some effor o aouning for some empirial and experimenal work showing how agens seem o violae he key axioms of raional deision heory. Obviously, his is done in order o give jusifiaion o prospe heory as a relevan alernaive. Prospe heory as suh has been applied in many differen onexs; however he main objeive here is o aoun for he pars of prospe heory ha are relevan for myopi loss aversion. Therefore I will no fous on he many differen appliaions of prospe heory. As will beome eviden laer on in he hesis, prospe heory is able o ome up wih some losed form funions, whih I apply when invesigaing he myopi loss aversion model. These funions onain parameers esimaed hrough experimenal researh ondued in US by he originaors of prospe heory. Sine no similar experimens have been ondued in Denmark, I assume ha hese esimaes apply o he represenaive invesor and herefore also o a Danish invesor. I should be noed ha he examinaion of prospe heory will be horough. This is done due o he belief ha sine prospe heory is he hallenger o radiional eonomi heory i is imporan o dediae a signifian fous o he deduion of he heory. Myopi loss aversion (and prospe heory) belongs o he behavioural finane field. As suh, his is a huge heoreial field and he purpose of his hesis is no o grasp he enire field of behavioural finane. Raher I limi myself o aoun for and disuss he pars relevan o myopi loss aversion, namely loss aversion and menal aouning. These wo oneps an be applied in very differen onexs, bu he examinaion in he hesis will have as is purpose o larify wha he oneps are and how hey are applied in he onex of myopi loss aversion. The empirial par of he hesis onsiss of an analysis of wheher an equiy premium puzzle exiss in Denmark in whih I perform a saisial es as desribed by Koherlakoa and wheher myopi loss aversion an aoun for he magniude of his premium in whih I apply he model by Benarzi and Thaler. So, having provided he heoreial bakground I urn o he empirial analyses of his. The daa maerial used in he empirial analyses will be desribed in he nex seion. 2.3 Daa In his seion, I will presen and disuss he daa maeriel ha forms he basis for he empirial analyses in his hesis. The analyses are arried ou on he Danish marke and will be on monhly 7

9 and annual sok, money marke and bond reurns. In he analysis onerning he equiy premium puzzle I apply he annual reurns from he period whereas in he analysis onerning myopi loss aversion I apply monhly daa from January 97- July This small inonsiseny resuls from he desire o have as many monhly observaions as possible in he laer analysis and he inonsiseny of periods is of insignifian onern wih regards o he resuls. The analysis period spreads over 35 years from January 97 o July 2006, whih resuls in 427 monhly observaions and 35 annual observaions Desripion of Daa For a desripion of he Danish sok marke I use he MSCI Denmark Toal Reurn Index 3 (in Danish Kroner). The oal reurn index measures he marke performane, inluding prie performane and inome from dividend paymens. The dividends are reinvesed he day he seuriy is quoed exdividend (ex-dae). The index overs 85% of free floa 4 on he Danish sok marke and is marke ap.-weighed. Noe ha his index is no an all-share index. This means ha here an be a risk of size- and seleion bias. However, he index has he longes hisory relaive o oher Danish sok indies and so I find ha he benefis of his longer index series as well as he fa ha reinvesed dividends are inluded ouweigh he poenial biases. Finally, in appendix A. I use saer plos and Durbin Wason ess o show ha he series exhibi no sign of auoorrelaion or heeroskedasiiy. As he risk free asse, I use a daa series onsrued from wo differen soures. From January 97 ill Deember 99, I use he disoun rae from he Naional Bank (Naionalbankens diskono) and from 992 onwards, I have used a 3-monh CIBOR rae. A ommon disussion onerns how o define he risk free asse and no formal onsensus has been developed. Some hoose a shor erm money marke ineres rae 5, whereas ohers have applied governmen bonds wih various duraions 6. The raionale for hoosing a 3-monh ineres rae as a risk free asse in his hesis is wofold. Firsly, I find i hard o jusify ha for insane a 0-year governmen bond an be onsidered risk free. Though defaul risk seems raher unlikely (a leas in my empirial field), reinvesmen risk and ineres rae risk, for insane, are sill presen. Seondly, wih regards o my empirial invesigaion on wheher or no an equiy premium puzzle an be observed in Danish daa, I wish o be ompleely faihful o he heoreial and empirial approah on Using losing pries 2 End period being Deember A horough desripion of he index an be found on msi.om 4 MSCI defines he free floa of a seuriy as he proporion of shares ousanding ha are deemed o be available for purhase in he publi equiy markes by inernaional invesors. 5 E.g. The Federaion of Danish Invesmen Assoiaions (IFR), Koherlakoa (996), Mehra and Preso (985/2003) 6 The Danish Naional Bank uses he reurn of a 0 year Governmen Bond, Quarerly Repor

10 whih I build my analysis and here 3-monh T-bills are applied as a risk free asse. Therefore, I find ha he mos appropriae proxy of a risk free asse is a shor money marke ineres rae. An exensive invesigaion revealed ha available daa for reurns on Danish Governmen Bonds bak o 97 does no seem o exis. So I onsru an approximaed daa series from 97 unil Deember 985 from whereon available daa exiss. Hereafer I use he JP Morgan Danish Governmen Bond Index. I find his o be he mos appropriae due o he fa ha i onains he mos issues of available alernaives and has a duraion of 4.5 years. From January 97 o Deember 985 I have used he monhly bond yields for 5-year Danish Governmen Bonds of whih available daa exiss and alulaed reurns using he following expression. R r v 2 r (r r n = ), where R n is he nominal bond reurn and v is an esimae of he duraion of he bond his variable has been se o 4.5. This expression approximaes he monhly reurns o 5-year bonds by one welfh of he annualised bond yield for he previous monh less an esimaed prie hange. The prie hange is esimaed as he modified duraion imes he yield hange over he previous monh, e.g. a one peren yield inrease resuls in a prie derease orresponding o he duraion. The approximaion approah and he size of he duraion faor have been horoughly disussed wih he head of Fixed Inome a Gudme Raashou Asse Managemen, Henrik Qvisgaard. Sine he approximaion onerns governmen bonds and shor erm monh-o-monhs yield hanges, no faor for he onvexiy has been implemened. If I had been esimaing reurns using a selfonsrued Danish allable morgage bond, onvexiy had been an issue due o he high level of negaive onvexiy in espeially he high oupon Danish morgage bond marke. This high degree of negaive onvexiy appears when he prie is lose o par and he opion sars o represen value. I is reasonable o ask how well his approximaion works. Obviously, I anno examine his prior o 985, bu I applied he mehodology above o he period in whih daa from he JP Morgan index is available. This invesigaion revealed ha he average (arihmei) annual reurn is only 5 bp higher for he JP Morgan index han for he onsrued series (8.86% vs. 8.8%). However, i mus be noed ha he annual sandard deviaion is somewha higher for he onsrued series (6.84% ompared o 4.26%). Bu wih hese onsideraions in mind, I find ha he approximaion 9

11 works raher well. So from , I use yields from 5-year governmen bonds and alulae an approximaed index. From 986 onwards, I use J.P Morgan Danish Governmen Bond Index. Per apia onsumpion has been alulaed as he annual privae onsumpion (all inluded) divided by he size of he annual Danish populaion. In order o arrive a real reurns and onsumpion growh, all series will be deflaed wih he onsumer prie index. I should be poined ou ha he annual inflaion rae is alulaed by annualizing he monhly raes hereby giving me he exa annual inflaion rae. Danish Saisis repor he average annual inflaion rae bu I find his o be an inappropriae measure due o he fa ha I wish o examine he aual monhly reurns based on he aual monhly inflaion. Refer o appendix A.2 for an overview of he daa series and heir soures. 3.0 The Equiy Premium Puzzle This seion presens he equiy premium puzzle as originally posed by Mehra and Preso in 985 and laer revisied in They showed how sandard heory fails o produe he large equiy premium ha has been observed empirially in he US. Inuiively, sine soks are riskier han bonds hey should ommand a higher reurn beause invesors will demand a premium for bearing his risk. This is suppored by Mehra and Preso qualiaively bu a he quaniaive level soks are shown no o be suffiienly riskier han bonds o jusify he observed equiy premium. I desribe he updaed model uilized by Mehra and Preso (2003), he inuiion behind i, and presen heir resuls for he US sok marke for he period The formal deduion of he enral equaions of he model is arried ou in appendix A.3. In his seion, I will also presen he onlusions of Koherlakoa (996) who suppors he findings of Mehra and Preso by esing he saisial signifiane of he equiy premium puzzle on he US sok marke (same period and daa). He finds ha he observed premium is signifianly higher han wha he model predis for all reasonable levels of risk aversion. This approah will be he basis for he empirial invesigaion of Danish daa in haper 8. Following his, I will give a brief overview of some of he mos predominan alernaive heoreial aemps ha have been made o explain he magniude of he equiy premium. The ommon denominaor for hese aemps is ha hey deviae from he main assumpions underlying Mehra and Preso s model in differen ways. 0

12 3. The Represenaive Agen Model Consumpion CAPM As opposed o he Capial Asse Priing Model, in whih i is assumed ha he ypial invesor s onsumpion sream is perfely orrelaed wih he reurn o he sok marke 7, Luas (978) desribed a so-alled represenaive agen model of asse reurns in whih per apia onsumpion is perfely orrelaed wih he onsumpion sream of he ypial invesor. In his ype of model, he risk of an asse an be measured using he ovariane of is reurn wih per apia onsumpion. The key idea in his ype of model is ha onsumpion oday and onsumpion in some fuure period are reaed as differen goods. The relaive pries of hese differen goods are equal o people s willingness o subsiue beween hem and businesses abiliy o ransform hese goods ino eah oher. In heir paper from 985, Mehra and Preso desribed an empirial problem for he represenaive agen paradigm. They find, ha in he period , he average annual real reurn o soks has been abou 7% whereas he average annual real reurn o T-bills has been only abou %, They show ha he differene in he ovariane of hese reurns wih onsumpion growh is only large enough o explain he differene in he average reurns if invesors are implausibly risk averse. And his is wha hey dubbed he equiy premium puzzle; in a quaniaive sense, soks are no suffiienly riskier han T-bills o jusify he spread in heir reurns. 3.. Deduion of he Model In he framework of Mehra and Preso he following basi assumpions apply. ) Invesors are raional and have preferenes assoiaed wih he sandard uiliy funion, and herefore are able o maximise expeed fuure disouned uiliy. 2) Markes are omplee, i.e. i is possible o insure agains any possible siuaion. 3) There are no ransaion oss assoiaed wih invesing. The model is of he represenaive agen ype, so he resuls for he represenaive agen are assumed o hold a he aggregae level as well. 8 Agens maximise he following uiliy funion desribing he presen expeaion of all fuure onsumpion sreams, (3.) E0 β U ( ), where 0<β< = 0 Uiliy is derived from onsumpion. Sine he model holds a he aggregae level is onsumpion per apia. The invesors ime preferene is apured by he disoun faor, β, ha people apply o 7 Whih in urn implies ha a finanial invesor an measure an asse s risk by is ovariane wih he reurn o he sok marke 8 Aording o Consandinides (982) his resriive assumpion ha all agens have homogenous preferenes is no vial sine even models wih heorogenous agens produe similar resuls a he aggregae level.

13 he uiliy derived from fuure onsumpion. β is small if people are highly impaien and hus prefer onsumpion oday raher han omorrow. Mehra and Preso resri he uiliy funion o be of he form, α (3.2) U (, α) =, where α>0. α U ''( ) This funion is of he onsan relaive risk aversion (CRRA) ype sine U '( ) = α. This preferene funion links risk preferenes wih ime preferenes. Agens wih CRRA preferenes like o smooh onsumpion over various saes of naure and hey also prefer o smooh onsumpion over ime, ha is hey dislike growh. This is beause he oeffiien of relaive risk aversion is he reiproal of he elasiiy of ineremporal subsiuion 9. The maximizaion behaviour of invesors is wha pries asses in equilibrium. The invesor purhases finanial asses if she an obain a higher marginal uiliy from invesing han from onsuming oday. So in equilibrium he marginal uiliy of he amoun paid for he sok p U ( ) mus be equal o he presen value of he expeed uiliy βe ((p y )U ( )) in he nex period, where p is he prie of he seuriy and y is he dividend i pays. I show in appendix A.3 ha equaing hese wo yields he following equilibrium expression for he expeed reurn on soks U '( ), Re, (3.3) E ( R = e, ) R f, COV, E ( U '( )) where he reurn on soks is R p y e, =. p This resul shows ha he expeed reurn on soks is equal o he risk free rae plus a risk premium ha depends on he ovariane of marginal uiliy wih sok reurns. If sok reurns are posiively orrelaed wih onsumpion his premium is high and vie versa. The inuiion behind why he equiy premium is derived from he ovariane of onsumpion and equiy reurns is ha invesors obain differen levels of uiliy from he same amoun of onsumpion a differen imes. This follows from dereasing marginal uiliy, i.e. an asse ha pays off when imes are good and onsumpion is high will be onsidered less desirable han an asse ha pays off a similar amoun when imes are bad and where addiional onsumpion herefore is more highly valued. Similar, agens are assumed o seek smooh onsumpion pahs over ime and hus like as- 9 Mehra and Preso (2003) page 20 2

14 ses ha pay off when onsumpion is low o fill he onsumpion gap. Conrarily, asses ha pay off when onsumpion is already high, ruins he sabiliy of he onsumpion pah and are hus less valuable o invesors, who in urn will demand a higher reurn o hold hem. The quesion hen is wheher he ovariane beween equiy reurns and onsumpion growh is large enough o jusify he empirially observed equiy premium. Now I proeed o derive he version of his relaionship for he equiy premium esed by Mehra and Preso. I show in appendix A.3 ha he opimum ondiion p U ( )=βe ((p y )U ( )) yields he following expression for he expeed reurn on soks and bonds respeively (3.4) (3.5) E ( x ) E ( Re, ) = and α βe ( x ) R f,, α βe ( x ) = where x is he growh rae of onsumpion, i.e. x =. The proess of onsumpion growh is assumed by Mehra and Preso o be log normally disribued. This means ha we have explii expressions for he expeed reurns for soks and risk free invesmens: 0 μ x ½σ x e (3.6) E ( Re, ) = 2 2 ( α ) μ x ½( α ) σ x βe (3.7) R = f, 2 2 αμx ½α σ x βe 2 ln E ln R 2 2 ( R e, ) = ln β αμ x ½α σ x f, = ln β αμ x ½ 2 α σ 2 x ασ 2 x 2 In hese expressions μ x =E(lnx), σ x = VAR(lnx), and lnx is he oninuously ompounded growh rae of onsumpion. From his we ge he models prediion of he equiy premium: (3.8) ln E ( R 2 e, ) ln R f, = ασ x. Thus he risk premium ommanded by sok invesmen is he produ of he oeffiien of he invesors risk aversion and he variane of onsumpion growh. Mehra and Preso assume ha in equilibrium he onsumpion growh pah is perfely orrelaed wih equiy reurns, whih means he equiy premium is also equal o he oeffiien of risk aversion 0 I derive hese expressions in appendix A.2. 3

15 imes he ovariane of sok reurns and onsumpion growh, σ x,r e. Tha is, he resul obained from he model assuming log normally disribued onsumpion growh is equivalen o he general represenaion in equaion (3.3) Mehra and Preso s Empirial Resuls In heir original analysis, Mehra and Preso used he following daa series; he real reurn o he S&P 500, he real reurn o shor erm nominally risk free bonds, and he growh rae of per apia onsumpion. The sample saisis, Mehra and Preso arrived a is shown below in able 3.: Table 3. Sample saisis Mean Risk free rae, R f.008 Mean reurn on equiy, E(R e ).0698 Mean growh rae of onsumpion E(x).08 2 Variane of he growh rae of onsumpion, σ x Mean equiy premium E(R e )-R f From Mehra and Preso (2003) As an be seen from he able, he variane of he growh rae of onsumpion is And as we observe from equaion (3.8), his will have o imply a very large oeffiien of risk aversion, α, oherwise a high equiy premium simply is no possible. Wha is furher illusraed in he able is ha he equiy premium is alulaed o be 6.8 peren p.a. Several sudies 2 have argued ha he oeffiien of risk aversion is a small number in he range of -2. Mehra and Preso use his insigh o argue ha i should a leas be less han 0. So, if we for insane se α equal o 0 and β equal o 0.99, applying equaion (3.7) yields: 2 ln R = ln ln(.08) ½ ha is a risk free rae of 2.7%. f, = Now applying equaion (3.8) we have ln E ( ) = = , whih yields E(R e ) = R e.4, ha is a reurn on equiy of 4.%. This indiaes an equiy premium of.4% and even wih a very high oeffiien of risk aversion, his is far lower han he observed premium of 6.8%. This irumsane is wha Mehra and Preso dubbed he equiy premium puzzle. I is puzzling ha even for parameer values, α and β, pushed o heir very limis, here is a huge differene beween wha he model predis and wha is aually observed empirially. As saed before, a risk aversion of 0 is onsidered oo large by several sudies. Furhermore, he value of bea is se as liberally as 90-day T-bills from , T-erifiaes from 920-3, and day Commerial Paper prior o Following Mehra and Preso (985) page 54 4

16 possible beause applying a bea value larger han one would indiae ha people s subjeive ime preferene (θ) is negaive (sine β = ). Obviously, his is ounerinuiive sine i would imply θ ha invesors are willing o pay o ransfer onsumpion from oday o omorrow Furher Validaion of he Empirial Resuls Koherlakoa (996) uilises equaion (3.4) and (3.5) o perform a saisial signifiane es of he findings of Mehra and Preso using he same daa. In appendix A.4, I illusrae ha ombining equaion (3.4) and (3.5) yields he following expression: (3.9) E ( R R ) = 0 α e, f, Tha is, when aking onsumpion risk ino aoun, he equiy premium should no be signifianly differen from zero. Koherlakoa proeeds by esimaing he expeaions on he lef hand side of he equaion by using he sample means of: (3.0) e ( R R ) α = e, f, And his relaionship is hen esed as he null hypohesis. Tha is, he sample means should no be signifianly differen from zero. Koherlakoa alulaes he sample mean for differen values of α, ranging from 0.0 o 0.0. As an be seen from he able below, for all values of alpha 8.5, he sample mean of e is signifianly posiive and herefore he null hypohesis is rejeed for all values of alpha smaller han 8.5. Table 3.2 The Equiy Premium Puzzle α e -sa Exra from Koherlakoa 996 5

17 Tha is, in order for invesors o be indifferen beween invesing in soks and bonds, invesors mus be highly risk averse. So, Koherlakoa suppors Mehra and Preso s findings showing ha only wih an unrealisially high level of risk aversion he observed equiy premium an be jusified. Tha is, he premium for bearing aggregae risk aouns for lile of he hisori equiy premium. 3 So, even hough sandard heory is onsisen wih he noion of risk, ha soks, on average, should earn a higher reurn han bonds, he quaniaive prediions of he heory are an order of magniude differen from wha have been doumened in he empirial daa above. 3.2 Hisorial Aemps o Explain he Equiy Premium Puzzle Over he las 20 years, several aemps o resolve he puzzle have been made. Generally he onsensus is ha he heoreial framework of Mehra and Preso is robus and represens an inegral par of modern maroeonomis and inernaional finane. Thus any aemps o reonile he apparen empirial defes of he represenaive agen model of asse reurns mus be based on he abandonmen of a leas one of he hree key assumpions on whih i is based (Koherlakoa 996 page 43). These were ) asse rading is osless, 2) asse markes are omplee, and 3) agens have preferenes assoiaed wih he sandard uiliy funion. In his seion, I disuss some of he aemps made o explain he equiy premium puzzle by relaxing hese assumpions. The haper loses wih a more horough inroduion o he alernaive explanaion ha is he main opi of his hesis, namely myopi loss aversion Complee Markes Assumpion A key presumpion underlying Mehra and Preso s model is ha he behaviour of per apia onsumpion growh is an appropriae proxy for he behaviour of individual onsumpion growh. This is rue if i is assumed (as Mehra and Preso do) ha markes are omplee. The assumpion ha markes are omplee implies ha agens an insure agains any oningeny, e.g. fluuaions in labour inome inome shoks. In he framework of he onsumpion based represenaive agen model of Mehra and Preso (985 and 2003), his means ha agens an insure agains fluuaions in heir onsumpion sream. This assumpion is vial in using he per apia onsumpion as a measure of onsumpion for he represenaive agen. Agens will use he finanial markes o diversify away any idiosynrai differenes beween heir own onsumpion growh and aggregae onsumpion growh making he wo series idenial. The raionale for why he abolishmen of his assumpion ould explain he equiy premium puzzle is as follows: If he realiy is ha markes are no omplee and invesors hen are no able o om- 3 Mehra and Preso (2003), page 33 6

18 pleely hedge all possible fluuaion in heir onsumpion sream, hen hey fae a more volaile onsumpion sream han wha is indiaed by per apia onsumpion. (And as have previously been noed, invesors wan o smooh onsumpion over ime and saes.) Sine he Mehra-Preso model shows ha he equiy premium equals risk aversion imes he variane of he onsumpion sream, he premium demanded by invesors wih higher onsumpion volailiy would be higher. Indeed he main empirial finding of Mehra and Preso was ha he variane of onsumpion was oo low o explain he premium. Weil (992) sudies a wo-period model in whih markes are no omplee. This means, ha variabiliy in inome mus be fully refleed in he onsumpion paern. He shows ha he exra variabiliy in individual onsumpion growh indued by he absene of markes helps explain he equiy premium puzzle. Koherlakoa (996), however, argues ha wo-period models are inomplee in he sense ha hey do no apure he use of dynami self-insurane; an inuiive proess by whih individuals (if assumed ha hey live for more han wo periods) offse fluuaions in inome and hus onsumpion by inreasing (when inome is high) or dereasing (when inome is low) savings. Tha is, individuals need no absorb he inome risk oally ino urren onsumpion. In his framework, invesors are able o smooh onsumpion quie suessfully if only inome shoks are no highly persisen. If he inome shok is permanen, dynami self-insurane anno play a role; inome shoks mus be fully absorbed ino onsumpion. Heaon and D. Luas (995a) find ha inome shoks are in fa no persisen; raher he auoorrelaion of idiosynrai inome shoks is around 0.5, whih means ha he inome shok dies ou afer some ime (an auoorrelaion of implying a permanen inome shok). Numerous empirial appliaions of dynami inomplee markes models 4 onfirm ha individuals an losely approximae he alloaions in he omplee markes environmen by dynamially selfinsuring, i.e. equilibrium asse pries are very similar. So in onlusion, even hough he omplee markes assumpion may seem unrealisi, he evidene shows ha he equiy premium need no be largely affeed by marke inompleeness No Transaion Coss Assumpion The model developed by Mehra and Preso assumes ha asse rading is osless, whih means ha here are no onsrains on or oss assoiaed wih rading finanial seuriies. This is no he ase in he real world where he ypial invesor will fae onsrains on boh borrowing and shor sales. Thus, he relaxaion of his assumpion has been pu forward as a possible resoluion of he 4 Following Koherlakoa, hese are Telmer (993), Luas (994), Heaon and Luas (995a), Mae and Singleon (99). 7

19 equiy premium puzzle. Koherlakoa (996) argues ha if invesors are onsrained on borrowing his leads o a lower demand for loans and so a lower ineres rae. All else equal his implies a higher equiy premium by simply inreasing he heoreially predied differene beween he mean reurn o equiies and he (now lower) ineres rae. Heaon and D. Luas (995b), however, find ha onsrains on he rading aiviy of invesors have lile effe on he size of he equiy premium. This is beause he ypial invesors will fae onsrains boh in he bond marke and in he sok marke. If no, invesors ould shif resoures from one marke o he oher and hereby loosen he onsrain. Wih parallel onsrains on invesmen in bonds and soks he expeed reurn in boh markes will be similarly lower hus preserving he equiy premium. The absene of rading oss in he Mehra and Preso model is also possible o quesion sine he real world feaures several levels of expenses assoiaed wih asse rading. If invesors have long horizons he magniude of rading oss will diminish over he life of he invesmen onsequenly reduing he imporane of hese oss. If invesors, however, are fored o sell invesmens premaurely, e.g. following a drop in labour inome, he invesmen horizon is oo shor o fully amorize he oss. Thus, he equiy premium should be higher in order o offse hese oss. Researh by Aiyagari and Gerler (99) and Heaon and D. Luas (995a), however, shows ha only a very large differene in he os of equiy rading relaive o bond rading an explain he equiy premium. Koherlakoa (996) finds ha his subsanial differene in oss is no suppored by empirial evidene and as suh he relaxaion of he assumpion of osless rading anno help resolve he equiy premium puzzle Alernaive Preferene Sruure Thus far, we have seen ha relaxing he assumpions regarding omplee and friionless markes have no helped refue he resuls of Mehra and Preso. So, now we urn o he hird key assumpion underlying he model, whih onerns he preferenes of he represenaive agen. I briefly review hree differen modifiaions o (3.) and hereafer urn he aenion o he main fous of his hesis, namely myopi loss aversion, whih also onsiues an alernaive in he preferene modifiaion lass Habi Formaion The sandard preferenes in (3.) assume ha he level of onsumpion in period - does no affe he marginal uiliy of onsumpion in period. I ould be argued ha i is more naural o hink ha an individual who onsumes a lo in period - will ge used o his high level of onsumpion and herefore more srongly desire onsumpion in period. A habi-formaion uiliy funion as pre- 8

20 sened by Consaninides (990) apures his inuiion: one an individual ges used o a erain sandard of living, her onsumpion level forms a habi. This level will hen beome he benhmark o whih she evaluaes fuure onsumpion. Thus, i is he deviaions from his benhmark ha maers for he individual raher han he absolue level of onsumpion and he uiliy of urren onsumpion will be a dereasing funion of onsumpion yeserday. The impliaion of his approah is ha demand for savings will be higher han in Mehra and Preso s model. This is beause individuals for any given level of urren onsumpion knows ha he desire for fuure onsumpion is ever inreasing. So a fair amoun of savings is neessary. The onsequene for he implied equiy premium is no enouraging, however. The high demand for savings drives down ineres raes and hus predis a low empirial risk free rae, bu unforunaely i is sill neessary for individuals o be highly averse o onsumpion risk o explain he magniude of he equiy premium. Mehra and Preso (2003) argue ha he puzzle is no explained by emphasizing ha wih a moderae level of risk aversion (Consaninides presens α=2.8 as a soluion) he sensiiviy o onsumpion risk as measured by he oeffiien of relaive risk aversion is five imes α. The reason for his is ha alhough Consaninides finds ha he model an generae a high equiy premium a a relaively low level of risk aversion, i is neessary o assume ha agens are exremely persisen in requiring urren onsumpion o exeed previous onsumpion Koherlakoa poins ou ha he agens in Consaninides framework requires a large amoun of onsumpion jus o survive and hus will pay a lo o avoid small onsumpion gambles. Thus alhough aversion o wealh risk an be low, onsumpion risk aversion mus sill be implausibly high o explain he puzzle Keeping Up Wih he Joneses Duesenberry (949) assumes ha agens uiliy no only depends on heir own onsumpion as in (3.) bu also on he aggregae level of onsumpion in he eonomy. This ype of preferenes has been dubbed keeping up wih he Joneses. Abel (990) applies his ype of preferene in an aemp o explain he equiy premium puzzle. In his seing he invesmen deision of an individual will depend on boh he aiude owards own onsumpion risk and he variabiliy of he general onsumpion growh in he soiey. Speifially, he uiliy funion of agens inludes individual onsumpion relaive o per apia onsumpion a ime as well as ime -. I is hen possible o esimae he risk aversion parameers assoiaed wih individual as well as per apia onsumpion. The model offers an explanaion of he high equiy premium, namely ha invesors need no be exessively averse o individual onsumpion risk as long as he sensiiviy of marginal uiliy owards he variabiliy in per apia onsumpion is suffiienly high. So, invesors do no find soks unaraive beause hey are highly averse o individual onsumpion risk bu 9

21 raher beause hey are very averse o per apia onsumpion risk. Koherlakoa noes ha he insigh gained from relaxing he preferene sruure of Mehra and Preso in his direion, is limied. In he original se-up he only looming explanaion of he puzzle was ha invesors were exremely risk averse. The relaive onsumpion approah implies ha his need no be he ase. Bu insead invesors are required o be exremely averse o any marginal variaion in per apia onsumpion in order o explain he equiy premium Generalised Expeed Uiliy A enral assumpion uilized in he sandard preferenes in (3.) is ha he oeffiien of relaive risk aversion is resried o be equal o he reiproal of he elasiiy of ineremporal subsiuion. Consequenly individuals who are sensiive o variaion in onsumpion aross differen saes are also averse o variabiliy in onsumpion over ime, i.e. will desire a smooh onsumpion pah. Several sudies sugges ha his speifiaion of preferenes is oo rigid and is he resriion ha auses he equiy premium puzzle. 5 Epsein and Zin (989) develop he onep of generalised expeed uiliy (GEU) preferenes, whih is a preferene sruure ha allows he disenanglemen of risk aversion from he elasiiy of ineremporal subsiuion. In his model agens uiliy depend parly on oal wealh and he reurn on he agens oal porfolio of asses (inluding real esae, human apial, e.). This reurn is prinipally unobservable bu Epsein and Zin use he marke reurn as a proxy (speifially, he value-weighed reurn o he NYSE). In equilibrium, he equiy premium depends on he ovariane of asse reurns wih boh onsumpion growh and he reurn on oal asses or he marke porfolio. (The model hen has as is wo limi ases he onsumpion CAPM and he sandard CAPM). The key o he speifiaion is ha agens an be risk averse wihou waning o smooh onsumpion over ime and in heir 99 paper, Epsein and Zin laim o resolve he equiy premium puzzle empirially. Mehra and Preso (2003) ouner his evidene by noing ha hey oversae he orrelaion beween he reurn on oal asses and he reurn on he marke porfolio. Koherlakoa (996) suppors his noion by poining ou ha he marke porfolio underesimaes he level of diversifiaion of agens oal asse porfolios and so overesimaes he orrelaion beween he marginal rae of subsiuion and sok reurns. This high ovariane is he reason why Epsein and Zin an explain he puzzle wih moderae risk aversion. Moreover, Koherlakoa furher develops he framework of Epsein and Zin o a model where he assumpion regarding oal asse reurn is no required. He shows ha, equivalen o sandard uiliy, he preferene sruure of Epsein and Zin requires an implausibly high level of risk aversion o explain he puzzle. 5 Epsein and Zin (990) referene Hall (985), Zin (987) and Aanasio and Weber (989). Mehra and Preso (2003) noe ha here is no a priori reason why he parameers should be linked. 20

22 Myopi Loss Aversion As I have shown above, here have been several heories rying o explain his large equiy premium, bu none, as i seems, has been able o fully aoun for he magniude of he premium. In 995, B&T (B&T heneforh) se ou o ry o give an alernaive explanaion for he size of he equiy premium hey alled heir aemp myopi loss aversion. Their explanaion finds is foundaion in he behavioural finane lieraure. The overall fous for behavioural finane is he inegraion of human psyhology and eonomi heory. The human per se is in fous and his means ha behavioural finane deviaes from he more sandard eonomi heory. The onep of myopi loss aversion ress on wo priniples from behavioural finane. These are loss aversion and menal aouning. Loss aversion means ha invesors end o be more sensiive o dereases in heir wealh han inreases. The onep of loss aversion originaes from prospe heory, whih is an alernaive o expeed uiliy heory. Prospe heory differs from expeed uiliy heory in several aspes. Firs of all, i is a purely desripive heory ha makes no normaive laims regarding how people ough o a. Raher i merely invesigaes how people aually do a. Moreover, as I will reurn o in haper 5, in prospe heory ouomes are no evaluaed in erms of final wealh; raher ouomes are evaluaed as eiher a gain or a loss relaive o a referene poin. The oher behavioural onep is menal aouning. Menal aouning is a erm ha apures he ogniive and unonsious operaions people use o organize, evaluae and keep rak of finanial aiviies. The noion is ha people end o make and evaluae deisions one a a ime and plae hem in separae menal aouns raher han evaluae hem in a broader onex. In a finanial perspeive, his refers o how ransaions are grouped boh ross-seionally (are seuriies evaluaed one a a ime or as porfolios) and ineremporally (how ofen are porfolios evaluaed). When his narrow evaluaion of he deisions and ouomes ake plae, finanial invesors will end o make shor-erm deisions raher han adop long-erm poliies regarding heir invesmens and evaluae heir gains and losses frequenly (Thaler, Tversky, Kahneman and Shwarz 997). The ombinaion of loss aversion and menal aouning onsiues he onep of myopi loss aversion. For finanial invesors his implies ha hey are averse o losses and evaluae heir porfolios a very shor horizons. And aording o B&T, his ombinaion an aoun for he magniude of he equiy premium. 2

23 They illusrae he onep of myopi loss aversion wih a problem ha Samuelson (963) posed; He asked a olleague of his wheher he would be willing o ake a be ha would eiher pay $200 or -$00 wih 50% hane. The olleague urned he be down bu said ha he would be willing o ake 00 of suh bes. Samuelson showed ha if a single be is rejeed, so mus a whole sequene of suh bes. Oherwise i would be an inonsiseny of expeed uiliy maximizaion. Several hings an be noed abou his example, and I will reurn o i laer for furher disussion. Firs, Samuelson quoes he following reason for why his olleague will no ake he single be: I will no be beause I would feel he $00 loss more han he $200 gain. The behavioural finane ranslaion of his would be I am loss averse. Moreover, why is i ha he likes a series of bes? Tha is, wha menal aouning operaions does he apply sine a series of bes seem araive when one single play is no? Assume ha Samuelson s olleague is haraerized by loss aversion and have a uiliy funion in whih U(x) = x, if x 0, and 2.5x if x is < 0 (and x being he hange in wealh due o he be). The 2.5 indiaes a loss aversion faor of 2.5, i.e. losses are weighed 2.5 imes as hard as gains. Then he expeed uiliy of one single be is negaive: ½(200) ½(2.5)(-00) = -25, and he will obviously urn down his be. Hene, Samuelson will reje one be, and even wo or more, if hey are evaluaed separaely. So, if eah play of he be is reaed as a separae even, hen wo plays of he be is wie as bad as one. Bu if wo bes are ombined ino a porfolio, he expeed uiliy of he bes are posiive: ¼(400) ½(00) ¼(-500) = 25. And as he number of repeiions inreases he porfolio of bes beome even more araive! So Samuelson s olleague should aep any number of plays of his be (>) as long as he does no have o wah hem being arried ou, i.e. evaluae afer eah be. This means, ha loss averse people (invesors) are more willing o ake on risk if hey ombine many bes (invesmens) ogeher han if hey onsider hem one a a ime. Reurning o he equiy premium puzzle, we see ha his inuiion an also be applied here by onsidering he problem faing an invesor wih he same uiliy funion as desribed above. Imagine ha an invesor mus hoose beween a risky asse offering a 7 peren expeed reurn (like soks) and a risk free asse offering peren. By he same inuiion as applied in he example above, he araiveness of he risky asse will depend on he horizon of he invesor. The longer he invesor inends o hold on o he risky asse, he more araive i will seem, as long as she do no evaluae he invesmen frequenly. 22