William Barnett. Abstract

Transcription

1 Inrmporally non sparabl monary ass risk adjusmn aggrgaion William Barn Univrsiy o ansas Shu Wu Univrsiy o ansas Absrac Modrn aggrgaion hory indx numbr hory wr inroducd ino monary aggrgaion by Barn (980. Th widly usd Divisia monary aggrgas wr basd upon ha papr. A ky rsul upon which h rs o h hory dpndd was Barn s drivaion o h usr cos pric o monary asss. To mak ha criical par o Barn s rsuls availabl prior o publicaion o ha papr in h Journal o Economrics Barn rpad ha proo wo yars arlir in Economics rs. Boh paprs hav bcom sminal o h subsqun liraur on monary ass quaniy usr cos aggrgaion. Th xnsion o ha liraur o risk wih inrmporally non sparabl prrncs now has bcom availabl in a working papr by Barn Wu (2004 ha papr will appar in volum numbr o h nw journal Annals o Financ. W ar making availabl h ky rsuls rom ha papr blow wihou h proos which will b availabl in h longr papr. W hank paricipans a h h Global Financ Annual Conrnc Yuqing Huang or hlpul commns. Ciaion: Barn William Shu Wu (2004 "Inrmporally non sparabl monary ass risk adjusmn aggrgaion." Economics Bullin Vol. 5 No. 3 pp. 9 Submid: Jun Accpd: July UR: hp:// 04E00005A.pd

2 Inroducion Modrn aggrgaion hory indx numbr hory wr inroducd ino monary aggrgaion by Barn (980. Th widly usd Divisia monary aggrgas wr basd upon ha papr. A ky rsul upon which h rs o h hory dpndd was Barn s drivaion o h usr-cos pric o monary asss. To mak ha criical par o Barn s (980 rsuls availabl prior o publicaion o h ull papr in h Journal o Economrics Barn (978 rpad ha par o h hory wo yars arlir in Economics rs. Boh paprs hav bcom sminal o h subsqun liraur on monary ass quaniy usr cos aggrgaion. Th xnsion o ha liraur o risk wih inrmporally non-sparabl prrncs now has bcom availabl in a working papr by Barn Wu (2004 ha papr will appar in volum numbr o h nw journal Annals o Financ. Tha compl papr in working papr orm is onlin in h Economics Working Papr Archiv a hp://conwpa.wusl.du/prins/mac/paprs/0406/ abs. In his papr w ar making availabl h ky rsuls rom h working papr wihou h proos which will b availabl in h longr papr. W xnd h monary-ass usr-cos risk adjusmn o Barn iu Jnsn (997 hir risk-adjusd Divisia monary aggrgas o h cas o mulipl nonmonary asss inrmporal non-sparabiliy. Our modl can gnra ponially largr mor accura CCAPM usr-cos risk adjusmns han hos ound in Barn iu Jnsn (997. Barn ( producd h microconomic hory o monary aggrgaion undr prc crainy drivd h ormula or h usr cos o monary asss originad h Divisia monary aggrgas o rack h hory s quaniy pric aggrgaor uncions nonparamrically. Th monary aggrgaion hory was xndd o risk by Barn (995 Barn Hinich Yu (2000. In producing h Divisia indx approximaions o h hory s aggrgaor uncions undr risk Barn iu Jnsn (997 Barn iu (2000 showd ha a risk adjusmn rm should b addd o h crainy-quivaln usr cos in a consumpion-basd capial ass pricing modl (CCAPM. Th risk adjusmn dpnds upon h covarianc bwn h ras o rurn on monary asss h growh ra o consumpion. Using h componns o h usual Fdral Rsrv Sysm monary aggrgas Barn iu Jnsn (997 showd howvr ha h CCAPM risk adjusmn is sligh h gain rom rplacing h unadjusd Divisia indx wih h xndd indx is usually small. An ovrviw o h rlvan liraur is providd in Barn Srlis (2000. As in h quiy prmium puzzl liraur h small risk adjusmn is causd by h low conmporanous covarianc undr inrmporal sparabiliy bwn h ra o rurn on h ass consumpion growh. W also xnd h modl in Barn iu Jnsn (997 o includ mulipl risky non-monary asss which ar h asss ha provid no liquidiy srvics ohr han hir ras o rurn w show ha any non-monary ass can b usd as h bnchmark ass whn is ra o rurn is corrcly adjusd. 2. Consumr s opimizaion problm W assum ha h rprsnaiv consumr has an inrmporally non-sparabl gnral uiliy uncion U( m c c c dind ovr currn pas consumpion a - -n vcor o currn-priod monary asss m ( m m m. Th consumr s holdings o 2

3 non-monary asss ar k ( k k2... k. To nsur h xisnc o a monary aggrga w urhr assum ha hr xiss a linarly homognous aggrgaor uncion M ( such ha U can b wrin in h orm U( m c c c V( M( m c c c. (2. Givn iniial walh uncion n n W h consumr sks o maximiz hr xpcd liim uiliy s ( m+ s + s + s + s n s 0 E U c c c subjc o h ollowing budg consrains * + i + j i j W p c p m p k pc+ pa + i + i + j + j i j W R p m R p k +Y + (2.2 (2.3 (2.4 * whr (0 is h consumr s subjciv discoun acor is h ru cos-o-living indx A mi + k j i j is h ral valu o h ass porolio. Non-monary ass j provids gross ra o rurn. Monary ass i having quaniy has a gross ra o rurn R +. R j + p m i i Th consumr s incom rom any ohr sourcs rcivd a h bginning o priod + is Y +. Th consumr also is subjc o h ransvrsaliy condiion s * lim pa 0. (2.5 + s s Th irs ordr condiions can b obaind as λ E λ R + j + p p + (2.6 U mi λ E λ + Ri + p p + (2.7 n whr U U( c c c λ E ( U c + U c + + U c. m n + + n 3. Risk-adjusd usr cos o monary asss 3.. Th hory W din h conmporanous ral usr-cos pric o h srvics o monary ass i o b h raio o h marginal uiliy o h monary ass h marginal uiliy o consumpion so ha U U mi m i i. U U+ n U+ n E ( c c c λ (3. 2

4 W dno h vcor o monary ass usr coss by ( 2. Wih h usr coss dind abov w can show ha h soluion valu o h xac monary aggrga M ( m can b rackd accuraly in coninuous im by h gnralizd Divisia indx as provd in h prc crainy spcial cas by Barn (980. i m l m l l i i Proposiion. s b h usr-cos-valuad xpndiur shar. Undr h wak- sparabiliy assumpion (2. w hav or any linarly homognous monary aggrgaor uncion M ( ha whr M M(m. dlog M s dlog m i i i (3.2 Th xac pric aggrga dual Π Π( o h monary quaniy aggrgaor τ uncion M M(m is asily compud rom acor rvrsal Π ( M ( m im i so ha imi i Π ( M ( m. (3.3 In coninuous im h usr cos pric dual can b rackd wihou rror by h Divisia usr cos pric indx o b Π i log i i dlog s d. (3.4 To g a mor convnin xprssion or h usr cos i w din h pricing krnl Q i λ λ. ( r j + R j + p p b h ral gross ra o rurn on non-monary ass + k j i + i + l r R p p b h ral gross ra o rurn on monary ass m + i. W can prov h ollowing. Proposiion 2. Givn h ral ra o rurn r i + on a monary ass h ral ra o rurn r j + on an arbirary non-monary ass h risk-adjusd ral usr-cos pric o h srvics o h monary ass can b obaind as whr i ( + ω E r ( + ω Er Er i j + j i + j + ω i + i+ (3.6 Cov ( Q r (3.7 Cov ( Q r. (3.8 ω j + j+ 3

5 Corollary. Undr uncrainy w can choos any non-monary ass as h bnchmark ass whn compuing h risk-adjusd usr-cos prics o h srvics o monary asss. Noic ha Proposiion 2 dosn rquir xisnc o a risk-r non-monary ass (in ral-rms. To s h inuiion o Proposiion 2 assum ha on o h non-monary asss is risk-r wih gross ral inrs ra o r a im. Furhr as provd by Barn (978 h crainy-quivaln usr cos i o a monary ass is W can prov h ollowing: whr i i + m i i + i + ω i i + ωi r ω i Cov( Q+ ri +. Thror i r Er r Er. (3.9 r (3.0 could b largr or smallr han h crainy- quivaln usr cos i dpnding on h sign o h covarianc bwn i Approximaion o h hory r + Q + All o h consumpion-basd ass pricing modls rquir us o mak xplici assumpions abou invsors uiliy uncions. An alrnaiv approach which is commonly pracicd in inanc is o approxima Q + by som simpl uncion o obsrvabl macroconomic acors ha ar blivd o b closly rlad o invsors marginal uiliy growh. For xampl h wll-known CAPM [Sharp (964 innr (965] approximas Q + by a linar uncion o h ra o rurn on h mark porolio. W show ha hr also xiss a similar CAPM-yp rlaionship among usr coss o risky monary asss undr h assumpion ha Q + is a linar uncion o h ra o rurn on a wll-divrsiid walh porolio Spciically din r A + o b h shar-wighd ra o rurn on h consumr s ass porolio including boh h monary asss mi ( i h non-monary asss k j ( j. Thn h radiional CAPM approximaion o Q + mniond abov is o h orm Q a + br A + whr a b can b im dpndn. Economric mhodology or possibl xnsions o h spciicaion can b ound in Barn Binnr (2004. φ i ϕ j dno h shar o m i k j rspcivly in h porolio s sock valu so ha mi mi φ i (3. A m + k l j l j 4

6 Thn by consrucion r ϕ j m k j + k l i l i φ r + ϕ r + A + i i + j j i j i j i j k j A. (3.2 whr φ + ϕ. (3.3 ΠA φi i + ϕi j whr j is h usr cos o non-monary ass j. W i j din Π A o b h usr cos o h consumr s ass walh porolio. Bu h usr cos j o vry non-monary ass is simply 0. Hnc quivalnly Π A φi i. Th rason is ha consumrs do no pay a pric in rms o orgon inrs or h monary srvics o non-monary asss sinc hy provid no monary srvics provid only hir invsmn ra o rurn. W can show ha our diniion o Π A is consisn wih Fishr s acor rvrsal s as ollows. Rsul: Th pair ( A Π A saisis acor rvrsal dind by: Π A m + A i i j j i j k i. (3.4 Obsrv ha h walh porolio is dirn rom h monary srvics aggrga M ( m. Th porolio wighs in h ass walh sock ar h mark-valu-basd shars whil h growh ra wighs in h monary srvics low aggrga ar h usr-cos-valuad shars. Suppos on o h non-monary asss is (locally risk-r wih gross ral inrs ra r l r E r j + or all j. I ollows ha h crainy quivaln usr cos o h ass r E r A + walh porolio is Π. W can prov h ollowing proposiion. Α r Proposiion 3. I on o h non-monary asss is (locally risk-r wih gross ral inrs ra r i Q+ a br A + whr r A + is h gross ral ra o rurn on h consumr s walh porolio hn h usr cos o any monary ass i is givn by ( Π Π (3.5 i i i A A 5

7 whr i Π r Eri + i r A ar h usr coss o ass i o h ass walh porolio rspcivly r ErA + Π A r ar h crainy-quivaln usr coss o ass i h ass walh porolio rspcivly. Th ba o ass i in quaion (3.5 is givn by i Cov ( r r Var ( r A + i + A +. (3.6 Proposiion 3 is vry similar o h sard CAPM ormula or ass rurns. In CAPM hory h xpcd xcss ra o rurn Er i + r on an individual ass is drmind by is covarianc wih h xcss ra o rurn on h mark porolio Er M + r in accordanc wih Er r + ( Er + r (3.7 i i M whr Cov( ri + r rm + r i Var( rm + r. This rsul implis ha ass i s risk prmium dpnds on is mark porolio risk xposur which is masurd by h ba o his ass. 4. Concluding rmarks Simpl sum monary aggrgas ra monary asss wih dirn ras o rurn as prc subsius. Barn ( showd ha h Divisia indx wih usr cos prics is a mor appropria masur or monary srvics drivd h ormula or h usr cos o monary ass srvics in h absnc o uncrainy. Barn iu Jnsn (997 xndd h Divisia monary quaniy indx o h cas o uncrain rurns risk avrsion. For risky monary asss howvr h magniud o h risk adjusmn o h crainy quivaln usr cos is unclar. Using a sard im-sparabl powr uiliy uncion Barn iu Jnsn (997 showd ha h dirnc bwn h unadjusd Divisia indx h indx xndd or risk is usually small. Howvr his rsul could b a consqunc o h sam problm ha causs h quiy prmium puzzl in h ass pricing liraur. Th consumpion-basd ass pricing modl wih mor gnral uiliy uncions mos noably hos ha ar inrmporally non-sparabl can rproduc h larg im-varying risk prmium obsrvd in h daa [Campbll Cochran (999]. W bliv ha similarly xndd ass pricing modls will provid largr mor accura CCAPM adjusmn o h usr coss o monary asss han hos ound in Barn iu Jnsn (997. Th currn papr xnds h basic rsul in Barn iu Jnsn (997 in ha mannr. Th proos mor discussion will b availabl in h ull papr now in working papr orm o appar in vol. no.. o h Annals o Financ as Barn Wu (

8 Rrncs Barn W. A. (978 Th usr cos o mony Economic rs Rprind in W. A. Barn A. Srlis (ds. (2000 Th Thory o Monary Aggrgaion Norh-Holl Amsrdam Chapr 6-0. Barn W. A. (980 Economic monary aggrgas: An applicaion o indx numbr aggrgaion hory Journal o Economrics Rprind in W. A. Barn A. Srlis (ds. (2000 Th Thory o Monary Aggrgaion Norh-Holl Amsrdam Chapr Barn W. A. (995 Exac aggrgaion undr risk in W. A. Barn M. Salls H. Moulin N. Schoild (ds Social Choic Wlar Ehics Cambridg Univrsiy Prss Cambridg Rprind in W. A. Barn A. Srlis (ds. (2000 Th Thory o Monary Aggrgaion Norh-Holl Amsrdam Chapr Barn W. A. (997 Th microconomic hory o monary aggrgaion in W. A. Barn (ds Nw Approachs o Monary Economics Cambridg Univrsiy Prss Cambridg Rprind in W. A. Barn A. Srlis (ds. (2000 Th Thory o Monary Aggrgaion Norh-Holl Amsrdam Chapr Barn W. A. (2003 Aggrgaion-horic monary aggrgaion ovr h Euro ara whn counris ar hrognous ECB Working Papr No. 260 Europan Cnral Bank Frankur. Barn W. A. J. Binnr (ds. (2004 Funcional Srucur Approximaion in Economrics Norh-Holl Amsrdam. Barn W. A. M. Hinich P. Yu (2000 Th xac horical raional xpcaions monary aggrga Macroconomic Dynamics Barn W. A. Y. iu (2000 Byond h risk nural uiliy uncion in M. T. Blongia J. E. Binnr (ds. Divisia Monary Aggrgas: Thory Pracic ondon Palgrav -27. Barn W. A. Y. iu M. Jnsn (997 CAPM risk adjusmn or xac aggrgaion ovr inancial asss Macroconomic Dynamics Rprind in W. A. Barn A. Srlis (ds (2000 Th Thory o Monary Aggrgaion Norh-Holl Amsrdam Chapr Barn W. A. A. Srlis (ds (2000 Th Thory o Monary Aggrgaion Norh- Holl Amsrdam Scion Barn W. A. Shu Wu (2004 On usr coss o risky monary asss Annals o Financ vol no. orhcoming. 7

9 Campbll J. Y. J. H. Cochran (999 By orc o habi: A consumpion-basd xplanaion o aggrga sock mark bhavior Journal o Poliical Economy innr J. (965 Th valuaion o risky asss h slcion o risky invsmns in sock porolios capial budgs Rviw o Economics Saisics Sharp W. F. (964 Capial ass prics: A hory o mark quilibrium undr condiions o risk Th Journal o Financ