The investment home bias with peer effect

Le y, Haim A icle The in es men home bias wi h pee e ec Jou nal o Risk and Financial Managemen P o ided in Coope a ion wi h: MDPI – Mul idisciplina y Digi al Publishing Ins i u e, Basel Sugges ed Ci a ion: Le y, Haim (2020) : The in es men home bias wi h pee e ec , Jou nal o Risk and Financial Managemen , ISSN 1911-8074, MDPI, Basel, Vol. 13, Iss. 5, pp. 1-19, h ps://doi.o g/10.3390/j m13050094 This Ve sion is a ailable a : h ps://hdl.handle.ne /10419/239182 S anda d-Nu zungsbedingungen: Die Dokumen e au EconS o dü en zu eigenen wissenscha lichen Zwecken und zum P i a geb auch gespeiche und kopie we den. Sie dü en die Dokumen e nich ü ö en liche ode komme zielle Zwecke e iel äl igen, ö en lich auss ellen, ö en lich zugänglich machen, e eiben ode ande wei ig nu zen. So e n die Ve asse die Dokumen e un e Open-Con en -Lizenzen (insbesonde e CC-Lizenzen) zu Ve ügung ges ell haben soll en, gel en abweichend on diesen Nu zungsbedingungen die in de do genann en Lizenz gewäh en Nu zungs ech e. Te ms o use: Documen s in EconS o may be sa ed and copied o you pe sonal and schola ly pu poses. You a e no o copy documen s o public o comme cial pu poses, o exhibi he documen s publicly, o make hem publicly a ailable on he in e ne , o o dis ibu e o o he wise use he documen s in public. I he documen s ha e been made a ailable unde an Open Con en Licence (especially C ea i e Commons Licences), you may exe cise u he usage igh s as speci ied in he indica ed licence. h ps://c ea i ecommons.o g/licenses/by/4.0/ Jou nal o Risk and Financial Managemen A icle The In es men Home Bias wi h Pee E ec Haim Le y Depa men o Finance, Heb ew Uni e si y, Je usalem 9190401, Is ael; [email p o ec ed] Recei ed: 26 Feb ua y 2020; Accep ed: 7 May 2020; Published: 11 May 2020   Abs ac : Obse ed in e na ional di e si ica ion implies an in es men home bias (IHB). Can bi a ia e p e e ences wi h a local domes ic pee g oup a ionalize he IHB? Fo example, i is a gued ha wishing o ha e a la ge co ela ion wi h he S anda d and Poo ’s 500 s ock index (S&P 500 s ock index) may induce an inc ease in he domes ic in es men weigh by Ame ican in es o s and, hence, a ionalize he IHB. While his a gumen is alid in he mean- a iance amewo k, employing bi a ia e i s -deg ee s ochas ic dominance (BFSD), we p o e ha his in ui ion is gene ally in alid. Coun e in ui i ely, employing “keeping up wi h he Joneses” (KUJ) p e e ence wi h ac ual in e na ional da a e en enhances he IHB phenomenon. Keywo ds: in es men home bias (IHB); bi a ia e i s -deg ee s ochas ic dominance (BFSD); keeping up wi h he Joneses (KUJ); co ela ion lo ing (CL) JEL Classi ica ion: D81; C91 1. In oduc ion The in es men home bias (IHB) is well documen ed. Fo example, he US equi y ma ke accoun s o abou 35% o he wo ld equi y ma ke , ye abou 75% o Ame icans’ equi y in es men is alloca ed o he US ma ke . Hence, he US equi y IHB is in he magni ude o 40%. Fo mos commonly employed u ili y unc ions, he uni a ia e expec ed u ili y maximiza ion also does no suppo he ela i ely la ge domes ic in es men weigh ; hence, he IHB puzzle eme ges. I is ad oca ed ha he bi a ia e expec ed u ili y maximiza ion a ionalizes pa ially o ully he IHB. In his s udy, we employ he “keeping up wi h he Joneses” (KUJ) p e e ence, whe e he in es o ’s weal h and he pee g oup’s weal h a e he wo a ibu es o his u ili y unc ion, analyzing he pee e ec on he empi ically obse ed IHB phenomenon. The basic idea and he in ui ion o he KUJ a gumen o a ionalizing he IHB phenomenon is as ollows: suppose ha you know you uni a ia e u ili y unc ion and ha , o a gi en join dis ibu ion o e u ns co esponding o he a ious in e na ional ma ke s, you de i e wi h his u ili y unc ion he op imal in es men weigh s in he domes ic ma ke as well as in he o eign ma ke s unde conside a ion. Fu he mo e, suppose ha he op imal domes ic in es men weigh is, say, p%. Now, suppose ha you decide o conside , in addi ion o he join dis ibu ion o e u ns, one mo e ac o : you also wan he pe o mance o you po olio o be as close as possible o he pe o mance o a ce ain local s ock index. Fo example, he Ame ican in es o wan s he e u n on he po olio o be as close as possible o he e u n on he S&P 500 index, which, o simplici y o he discussion, is assumed o be he pee ’s po olio ( he same analysis applies o any o he local s ock index). Thus, i he in es o bene i s om ha ing a ela i ely la ge co ela ion wi h he S&P index, she may ha e an incen i e o inc ease he domes ic in es men weigh (which gene ally inc eases he co ela ion wi h he S&P s ock index, and i he domes ic in es men weigh is 100%, his co ela ion is +1) beyond J. Risk Financial Manag. 2020,13, 94; doi:10.3390/j m13050094 www.mdpi.com/jou nal/j m J. Risk Financial Manag. 2020,13, 94 2 o 19 wha is ob ained by a maximiza ion o a uni a ia e expec ed u ili y unc ion. The e o e, employing KUJ p e e ences may a ionalize he IHB.1 Indeed, Lau e bach and Reisman (2004) use he KUJ p e e ence p o e ha he IHB is a ionalized by inco po a ing he pee e ec . Howe e , hey use he mean- a iance model wi h some app oxima ions o achie e his esul . We analyze in his pape he impac o inco po a ing he pee e ec on he IHB in he mos gene al bi a ia e expec ed u ili y case, no elying on he mean- a iance amewo k and whe e no app oxima ions o he a ious ma hema ical o mulas a e employed. We de ine he p ecise condi ions, which gua an ee he IHB a ionaliza ion, by adding he pee e ec . We ind ha , in his un es ic ed analysis, he appealing in ui i e explana ion o he IHB a ionaliza ion by he pee e ec is gene ally w ong. Thus, we conclude ha one should seek o he economic explana ions o he obse ed IHB phenomenon. Fo some in e es ing economic sugges ions, see Coeu dacie and Rey (2013) and Be iel and Bha a ai (2013)2o o he beha io al explana ions. We employ in his s udy dis ibu ion- ee bi a ia e i s -deg ee s ochas ic dominance (BFSD), wi h no assump ions on he shape o he bi a ia e p e e ences and no app oxima ions. We p o e ha , despi e he abo e appealing in ui ion o he pee e ec on he op imal domes ic in es men weigh , using he bi a ia e p e e ences, he IHB may inc ease o dec ease ela i e o he uni a ia e op imal domes ic in es men weigh . Mo eo e , we demons a e wi h ac ual in e na ional da a ha adding he pee e ec , coun e in ui i ely, e en in ensi ies he IHB om he Ame ican in es o ’s poin o iew. Hence, he IHB s ill exis s. The s uc u e o he es o his pape is as ollows. Sec ion 2p o ides a b ie li e a u e e iew. Sec ion 3p esen s bi a ia e i s -deg ee s ochas ic dominance (BFSD) ule and he implied heo e ical esul s. We analyze he a ious ac o s a ec ing he IHB and show ha bi a ia e p e e ences a ionalize he IHB phenomenon only in a limi ed and un ealis ic case. Sec ion 4is de o ed o he commonly employed KUJ p e e ences, which is a speci ic se o all he bi a ia e p e e ences. We show empi ically ha he pee e ec wi h KUJ p e e ences e en enhances he IHB puzzle. Sec ion 5concludes. 2. Li e a u e Re iew Vanp é e and DeMoo e (2012) show ha he IHB exis s in i ually all coun ies. The magni ude o he IHB phenomenon is ela i ely la ge, cha ac e izing a ious pe iods, asse s, and coun ies. While abou h ee decades ago he Ame ican in es men in he local ma ke was mo e han 90%, implying a e y la ge IHB, in ecen yea s he IHB phenomenon has been mi iga ed, ye i is s ill abou 40%. When i comes o ixed-income asse s, he home bias is e en la ge . This phenomenon is no unique o he US and cha ac e izes many capi al ma ke s ( o a epo on he IHB in a ious coun ies, ega ding equi y and ixed-income asse s, see (Philips e al. 2012)). Ac ually, he e is e idence ha he home bias is e en wo se han epo ed (see Bax e and Je mann 1997). Resea che s ha e analyzed a ious possible key explana ions o he IHB. I is ag eed ha some po ion o he domes ic o e in es men may be induced by in e na ional ade ba ie s, o eign exchange isk, and egula ion, as well as by a domes ic pee g oup e ec . Howe e , wi h he inc ease in he apid low o in o ma ion and ma ke e iciency obse ed o e he las ew decades, he ade ba ie s, including possible asymme ical in o ma ion, ha e d as ically declined. This may accoun o he obse ed sligh dec ease in he domes ic o e in es men phenomenon. Howe e , since 1998, he equi y IHB o Ame ican in es o s has s abilized a abou 40% (see Le y and Le y 2014). 1 No e, we analyze whe he he pee e ec inc eases he op imal domes ic weigh , which pa ially o ully a ionalizes he IHB. The eason is ha i is possible ha he pee e ec inc eases he op imal domes ic weigh by, say, 1%, bu he IHB is, say, 40%, a case whe e o he ac o s a e needed o explain he obse ed IHB. In ou s udy, we ind empi ically ha he pee e ec e en enhanced he IHB; hence, he dis inc ion be ween pa ial and ull IHB a ionaliza ion is i ele an . 2 They conside po olio di e si ica ion when mac oeconomic ac o s a e inco po a ed in o a wo-coun y gene al equilib ium model, called he “Open Economy Financial Mac oeconomics” model. They conclude ha , wi h his equilib ium model, he home bias is less o a puzzle. Be iel and Bha a ai (2013) also sugges a mac oeconomic model ( ela ed o he posi i e associa ion be ween go e nmen spending and e u n on local s ocks) o explain he home bias. J. Risk Financial Manag. 2020,13, 94 3 o 19 While mos empi ical s udies analyze he IHB a he coun y le el (see F ench and Po e ba 1991; Tesa and We ne 1995), Kang and S ul z (1997), who s udy he IHB puzzle in Japan, analyze i a he indi idual i m le el, showing ha o eign in es o s hold disp opo ionally mo e Japanese sha es o i ms in he manu ac u ing indus ies, la ge i ms, and i ms wi h good accoun ing pe o mance. Simila ly, Dahlquis and Robe sson (2001) iden i y he cha ac e is ics o Swedish i ms ha a ac o eign in es o s. Lewis (1999), who analyzes he e ec o each economic ac o ha is conside ed as a ba ie o e icien in e na ional di e si ica ion on he IHB, concludes ha he ade ba ie s canno explain he magni ude o he exis ing IHB. The e o e, he IHB puzzle is s ill an in e es ing esea ch opic.3 Ob iously, i he IHB does no incu economic loss, i does no cons i u e an economic puzzle. Indeed, he in ensi y o he IHB economic cos changes o e ime. Le y (2016) analyzes he end in he IHB phenomenon o e ime. Mo eo e , he dis inguishes be ween he economic home bias (EHB), which measu es he economic loss in e ms o he di e ences in he ce ain y equi alen o wo al e na i e in e na ional di e si ica ion s a egies (wi h and wi hou a home bias) and he IHB, which simply measu es he de ia ions be ween he op imal in e na ional in es men weigh s and he ac ual in es men weigh s. He epo s ha , while he EHB was e y la ge in he pas , in he las 15 yea s, he EHB om he Ame ican in es men poin o iew has become negligible, despi e he exis ence o abou 40% IHB. This educ ion in he EHB is induced by he inc easing end in he in e na ional co ela ions. Thus, i seems ha o he Ame ican in es o s he IHB is no a majo economic puzzle. Howe e , he also epo s ha o o he coun ies, e.g., F ance, he EHB is s ill e y la ge, and he economic puzzle exis s. Mo eo e , in ecen yea s, we ha e end e e sal in co ela ions, and a dec ease in he a e age co ela ion be ween a ious ma ke s has been eco ded. As a esul o his end e e sal, he EHB has ecen ly inc eased, e en o Ame ican in es o s. Thus, o mos coun ies and wi h he ecen end e e sal in co ela ion also o he US, he IHB s ill cons i u es an economic puzzle ha needs an explana ion. The employmen o he KUJ p e e ence, namely inco po a ion o he pee e ec , is conside ed as one o he p omising pa hs in explaining he IHB puzzle. We employ in his pape a bi a ia e p e e ence. Gene ally, wi h bi a ia e p e e ence, he wo a iables can ake many o ms, e.g., weal h and heal h, clima e and income, e c. Ou s udy deals wi h in es men choices. Hence, he wo a iables a e he indi idual’s weal h and he pee g oup’s weal h. The pee g oup’s weal h can be he e u n on a ce ain domes ic po olio, and in ou case, as men ioned abo e, we assume, o he simplici y o he discussion and wi hou loss o gene ali y, ha i is he e u n on S&P 500 s ock index. The common iew is ha he ele an bi a ia e u ili y unc ion has a posi i e c oss de i a i e (we will elabo a e on his issue below) and ha in es o s wan , among o he hings, he pe o mance o hei po olio o be as close as possible o he pe o mance o he pee ’s po olio, i.e., a la ge co ela ion wi h he S&P s ock index is desi ed. 4 The e o e, we ocus ou analysis on he posi i e c oss-de i a i e case. Ob iously, despi e he desi e o ha ing a ela i ely la ge co ela ion wi h he S&P index, he in es o will shi om a po olio wi h a small co ela ion o a po olio wi h a la ge co ela ion, only i he bi a ia e expec ed u ili y inc eases by such a shi . We u n o analyze he condi ions unde which indeed such shi akes place, namely ha he IHB can be a ionalized wi h he pee e ec . 3 I is in e es ing o no e ha , e en in a case in which he e a e no anspa en ade ba ie s, he e is a endency o in es in i ms ha a e geog aphically loca ed close o he in es o ’s loca ion. This phenomenon is well documen ed wi hin he US (see Co al and Moskowi z 1999,2001;Hube man 2001). This indica es ha he home bias is a complex phenomenon ha is no easy o explain wi h con en ional economic ac o s. 4 Tse lin and Winkle (2009) ad oca e ha co ela ion a e sion p e ails. Howe e , in hei model, he wo a ibu es o he bi a ia e p e e ence di ec ly a ec he u ili y o he decision make , o example, income and quali y o li e. In ou model, he wo a ibu es a e di e en : he indi idual’s weal h and he pee g oup’s weal h. As ela i e weal h may a ec he indi idual’s u ili y, i is ad oca ed in he li e a u e ha , when some condi ions hold, co ela ion lo ing p e ails. J. Risk Financial Manag. 2020,13, 94 4 o 19 3. Bi a ia e Fi s -Deg ee S ochas ic Dominance (BFSD) and he IHB We would like o s ess a he ou se ha mos o he ma hema ical o mulas gi en in he i s pa o his sec ion a e no new and exis in he li e a u e, albei in di e en o ms and in di e en conno a ions. Howe e , we use hese ma hema ical esul s, o he bes o ou knowledge o he i s ime o analyze he pee e ec wi h KUJ p e e ences on he IHB phenomenon. 3.1. The Su icien Condi ions o BFSD Implying he IHB Ra ionaliza ion Conside an indi idual wi h a bi a ia e p e e ence U(w , wP) , whe e w deno es he e u n on he selec ed in e na ional po olio by he in es o unde conside a ion, and wp deno es he e u n on he pee ’s po olio. We compa e wo bi a ia e in es men po olios, Fand G, whe e he domes ic in es men weigh in po olio Fis la ge han he domes ic in es men weigh in po olio G(we will elabo a e la e on he selec ed po olios, Fand G). Ou aim is o examine he condi ions unde which F domina es Gby BFSD wi h he abo e bi a ia e u ili y unc ion, whe e we i s assume wo assump ions on he p e e ences: ∂Uw,wp/∂w≡U1≥ 0 (mono onici y) and ∂2Uw,wp/∂w∂wp≡U12 ≥ 0 (la e on we conside also U12 ≤ 0, a case usually no conside ed in KUJ economic esea ch bu eme ges as impo an o ou analysis). The e is no cons ain on he de i a i e ∂Uw,wp/∂wp≡U2 , which can be nega i e, ze o, o posi i e. 5 I such dominance exis s, hen all in es o s, ega dless o he p ecise shape o he bi a ia e p e e ence, will swi ch om G o F. Hence, he op imal domes ic in es men weigh inc eases, and he e o e he pee e ec a ionalizes he IHB phenomenon. No e ha he main ing edien o he KUJ p e e ence is ha he c oss de i a i e (U12) is posi i e, implying ha he indi idual’s ma ginal u ili y inc eases wi h an inc ease in he pee g oup weal h (see Ljungq is and Uhlig 2000). 6 The e o e, as explained be o e, i seems ha he in es o wi h a posi i e c oss de i a i e would incline o o e in es domes ically, as she p e e s he weal h o be posi i ely co ela ed wi h he pee ’s weal h. While he abo e in ui i e explana ion is appealing, in he ollowing p oposi ion, i is o mally shown ha gene ally only unde some speci ic condi ions, indeed a posi i e c oss de i a i e is an amoun o co ela ion lo ing, whe e co ela ion lo ing implies ha , by inc easing he domes ic in es men weigh , he bi a ia e expec ed u ili y inc eases. Namely, i he condi ions equi ed in he p oposi ion a e in ac , he in es o inc eases he bi a ia e expec ed u ili y by o e in es ing domes ically ( ela i e o he op imal uni a ia e expec ed u ili y maximiza ion op imal domes ic in es men weigh ), and by doing so, he co ela ion inc eases. Thus, i he p oposi ion equi ed condi ions hold in p ac ice, we ha e by he KUJ p e e ences a a ionaliza ion o he IHB, and he IHB puzzle may anish. As we explain below, in p ac ice, he equi ed condi ions o IHB a ionaliza ion a e no in ac . Be o e s a ing he p oposi ion, we need he ollowing de ini ion: De ini ion 1. De ini ion o co ela ion lo ing (CL): The in es o is CL i and only i , by inc easing he co ela ion be ween he po olio and he pee ’s po olio, he expec ed bi a ia e u ili y inc eases. Hence, CR in es o s who maximize he bi a ia e expec ed u ili y would inc ease he domes ic in es men weigh ela i e o he op imal uni a ia e expec ed u ili y weigh . 5No e ha a nega i e sign implies jealousy, and a posi i e sign implies al uism (see Dupo and Liu 2003). 6 Nume ous s udies sugges eplacing he uni a ia e expec ed u ili y analysis wi h he expec ed bi a ia e u ili y analysis wi h a ious de ini ions o he wo a iables: pas and p esen consump ion, consump ion o he indi idual, and consump ion o he pee g oup, he weal h ob ained by he indi idual and he opponen in an ul ima um game, and so o h. Fo s udies ha assume ha he u ili y is de i ed no om he absolu e weal h (o consump ion) o he indi idual bu om he ela i e weal h (o consump ion), in which he weal h’s posi ion ela i e o he pee g oup plays an impo an ole, as well as o o he ac o s ha do no a ec he classic uni a ia e expec ed u ili y bu a ec he bi a ia e expec ed u ili y, see, o example, Abel (1990), Cons an inides (1990), Bol on (1991), Rabin (1993,1998), Gal í (1994), Campbell and Coch ane (1999), Bol on and Ocken els (2000), Dupo and Liu (2003), Zizzo (2003), and Dema zo e al. (2008). J. Risk Financial Manag. 2020,13, 94 5 o 19 P oposi ion 1. Suppose ha he in es o aces wo al e na e bi a ia e p ospec s, F(w , wp) and G(w , wp) , whe e w as well as wp can ake only wo di e en ou comes. As he e a e only wo ou comes, hey can be ea anged o ha e ei he a co ela ion o +1 o a co ela ion o − 1. Di e si ica ion be ween w and wp is no allowed, implying ha he ma ginal dis ibu ions a e iden ical (namely, Fw=Gw and FwP=GwP , ega dless o he ou comes a angemen ; see, o example, Table 1). Unde hese speci ic condi ions, he in es o wi h a bi a ia e p e e ence is CL i and only i he c oss de i a i e is posi i e, namely U12 ≥ 0. Speci ically, unde he condi ions o he p oposi ion, wi h CL, he p ospec wi h a co ela ion o +1 yields a highe bi a ia e expec ed u ili y han any o he possible p ospec . (Fo p oo , wi h some o he no a ion, see (Eeckhoud e al. 2007)). Thus, i he condi ions o he p oposi ion we e in ac , he Ame ican in es o who likes he in es men pe o mance o be as close as possible o he S&P index would ha e a highe expec ed u ili y by inc easing he domes ic in es men weigh . Ac ually, unde he condi ions o he p oposi ion, ha ing a co ela ion o +1 wi h he S&P index is op imal, implying ha in es ing 100% domes ically is op imal, which c ea es a nega i e IHB puzzle (because in p ac ice less han 100% is in es ed domes ically). In sho , i he condi ions o P oposi ion 1 a e in ac , we ha e: CL ⇔U12 >0 (1) No e ha in es ing mo e in ensi ely domes ically, hence inc easing he co ela ion be ween he in es o ’s po olio and he pee ’s po olio, gene ally does no imply CR as de ined abo e. The eason is ha , wi h in es men in p ac ice, by inc easing he domes ic in es men weigh , al hough he co ela ion inc eases, gene ally, o he pa ame e s o he po olio may also change, he ma ginal dis ibu ions may change (hence, he condi ions o he p oposi ion a e iola ed), and he bi a ia e expec ed u ili y may dec ease. The e o e, he Ame ican in es o may decide no o dec ease he domes ic in es men weigh , despi e he desi e o ha e la ge co ela ion wi h he S&P index. Howe e , by he abo e de ini ion, he in es o is CL only i , a e conside ing all e ec s, he bi a ia e expec ed u ili y inc eases. Howe e , no e ha , by P oposi ion 1, he ma ginal dis ibu ions a e kep unchanged, and he co ela ion can ake only he ex eme alues o ei he +1 o − 1. This is because in Eeckhoud e al. (2007) o iginal p oposi ion, each a iable can ge only wo possible alues. Hence, by eo de ing hese alues, he ma ginal dis ibu ions a e kep unchanged. Also, di e si ica ion be ween wand wp is no allowed, because i i is allowed, he ma ginal dis ibu ion o he indi idual’s weal h, w, gene ally will no be kep cons an . Thus, he s a emen gi en in P oposi ion 1 is sui able o some choices, whe e he a iables a e, o example, weal h and heal h, wi h only wo ou comes (say, bad and good heal h, high and low income, e c.). As we shall see below, wi h in e na ional di e si ica ion, we ha e mo e han wo ou comes co esponding o each p ospec , and di e si ica ion is allowed. Hence, he ma ginal dis ibu ions gene ally change when he selec ed di e si ica ion changes. The e o e, a posi i e c oss de i a i e in ou analysis does no necessa ily imply CL. As a esul , we may e en ob ain an IHB phenomenon enhanced wi h bi a ia e p e e ences ela i e o he uni a ia e IHB, despi e he ac ha a posi i e c oss de i a i e is assumed. Le us u n now o he condi ions o BFSD o he dis ibu ion o e u ns o he po olio wi h he IHB o e he dis ibu ion o e u ns wi h no IHB. The wo po olios ha we compa e, Fand G, ha e bi a ia e densi y unc ions, deno ed by (w , wP) and g(w , wP) , espec i ely. As we ocus on he possible IHB a ionaliza ion, i is assumed, as explained be o e, ha Fs ands o a po olio wi h an IHB, ha is, he domes ic weigh in his po olio is la ge han he co esponding weigh in G. Thus, i he domes ic in es men weigh in Gis equal o he op imal heo e ical uni a ia e expec ed u ili y maximiza ion domes ic weigh (say, he in e na ional ma ke po olio), he BFSD o Fo e G implies ha he pee e ec a ionalizes he IHB phenomenon, as all in es o s would p e e Fo e J. Risk Financial Manag. 2020,13, 94 6 o 19 G. 7 Assuming ha ∂2w, wp/∂w∂wp)≡U12 > 0 wi h KUJ p e e ence 8 o explain a ious obse ed economic phenomena is e y common. As seen in P oposi ion 1, his assump ion is an impo an ing edien also needed o a ionalize he IHB phenomenon, so long as he condi ions o P oposi ion 1 hold. The e o e, we examine he ole o he c oss de i a i e on he BFSD ela ion. To examine possible a ionaliza ion o he obse ed IHB wi h KUJ p e e ences, we ex end he expec ed u ili y uni a ia e analysis o he bi a ia e expec ed u ili y analysis by adding he pee e ec . The expec ed bi a ia e u ili y o po olios Fand Gis gi en by: EFU=Rw wRwp wpU(w,wp) w,wpdwdwp EGU=Rw wRwp wpU(w,wp)gw,wpdwdwp (2) whe e w and w deno e he minimal and maximal alues o w (which can be −∞ and ∞ ); simila ly, wP and wPdeno e he minimal and maximal alues o wP. Thus, ∆i≡EFU−EGU=Zw wZwp wp U(w,wp)h w,wp−gw,wpidwdwp In eg a ing by pa s he abo e equa ion wi h espec o bo h a iables yields: ∆i≡EFU−EGU=Rwp wpRw wU12[F(w,wp)−G(w,wp)]dwdwp+Rw wU1[G(w)− F(w)]dw +Rwp wpU2[G(wp)−F(wp)]dwp ≡A+B+C (3) whe e ∆i deno es he expec ed u ili y di e ence co esponding o he i h in es o , F(w , wP) and G(w , wP) a e he wo bi a ia e cumula i e dis ibu ions, F(w)=Fw,wp is he ma ginal cumula i e dis ibu ion unc ion o w , Fwp=Fw,wp is he ma ginal cumula i e dis ibu ion unc ion o wP , and U1 , U2 , and U12 deno e he pa ial de i a i es: U1≡∂U/∂w , U2≡∂U/∂wP , and U12 ≡∂2U/∂w∂wP , espec i ely. Fo he de i a ion o Equa ion (3) wi h sligh ly di e en no a ions, see Le y and Pa oush (1974, p. 131) and A kinson and Bou guignon (1982, pp. 185–86). 9 No e ha , as he ma ginal u ili y o he pee ’s po olio is iden ical unde he a ious in es men s a egies (wi h and wi hou in ensi e domes ic in es men ). Namely, we ha e Gwp=Fwp , he e o e e m Cin Equa ion (3) is equal o ze o. Thus, he es o he pape ela e only o e ms Aand B. Le us i s analyze he ela ion be ween Equa ion (3) (wi h C=0) and he condi ions gi en in P oposi ion 1. I each o he wo andom a iables, w and wp , has only wo possible di e en ou comes, he co ela ion is ei he +1 o − 1. Also, when di e si ica ion be ween wand wp is no allowed, he ma ginal dis ibu ions a e equal (namely, G(w)=F(w) , see also he example gi en in 7 Ob iously, we ha e a di e en op imum po olio o each u ili y unc ion, bu , as we shall see below, he analysis is in ac , independen o he assumed p e e ence. 8 The KUJ and CUJ li e a u e is e y ex ensi e; hence, we men ion he e only a ew o hese s udies. Abel (1990) and Gal í (1994) use his bi a ia e amewo k o explain op imal choices. Ljungq is and Uhlig (2000) examine he ole o ax policies in economics wi h CUJ u ili y unc ions. Campbell and Coch ane (1999) assume ha he p e e ence is a unc ion o he ela i e consump ion, when he indi idual’s consump ion is measu ed ela i e o he weigh ed a e age o he pas consump ion o all indi iduals. In hese models, when he pee g oup’s a iable (e.g., consump ion) is a lagged a iable, he model is commonly called he CUJ model, and when he indi idual’s a iable and he pee g oup a iable ela e o he same ime pe iod (e.g., e u n on in es men ), i is commonly called he KUJ model. In his pape , we analyze he op imal po olio in es men decision in he KUJ se -up. 9 No e ha Equa ion (2) is educed o he well-known uni a ia e o mula employed o de i e he FSD ule, whe e U12 =U2= 0. Fo mo e de ails, see Hada and Russell (1969) and Hanoch and Le y (1969). Al hough we ocus in his pape on FSD, one can assume isk a e sion and employ s onge in es men ules; o example, see Ro hschild and S igli z (1970) and Le y (2015). J. Risk Financial Manag. 2020,13, 94 7 o 19 Table 1. Hence, in his speci ic case also e m Bis equal o ze o, and we a e le wi h e m A. I F ep esen s he +1 co ela ion and G he − 1 co ela ion, we mus ha e wi h he wo ou comes case ha Fw,wp≥Gw,wp (see example 1 in Table 1. Hence, in his case by Equa ion (3), wi h B=C=0, he condi ion U12 ≥ 0 is a su icien condi ion o dominance o he join dis ibu ion wi h he +1 co ela ion o e he join dis ibu ion wi h he − 1 co ela ion (see Equa ion (3)). 10 I is easy o e i y ha , in his speci ic case, U12 ≥ 0 is a necessa y and su icien condi ion o dominance. 11 Thus, Equa ion (3) is pe ec ly consis en wi h P oposi ion 1, so long as he condi ions gi en in he p oposi ion a e in ac . Howe e , Equa ion (3) co esponds o he gene al case, as i co e s he mo e ealis ic scena ios whe e mo e han wo ou comes a e possible. Di e si ica ion is allowed, and he ma ginal dis ibu ions a e no necessa ily equal, hence e m Bis no necessa ily equal o ze o. As we shall see in his gene al and ealis ic case, U12 ≥ 0 is nei he a necessa y no a su icien condi ion o dominance. We u n now o analyze he possible dominance o he po olio wi h he IHB o e a po olio wi h no IHB in he mos gene al case. To examine possible a ionaliza ion o he IHB phenomenon wi h bi a ia e p e e ences, le us i s ake a deepe look a he ma ginal dis ibu ions co esponding o he in e na ional di e si ica ion issue analyzed in his pape . Re u ning o Equa ion (3), no e ha , as men ioned abo e, i is easonable o assume ha he hi d e m on he igh -hand side o Equa ion (3), e m C, is equal o ze o, as he in es o in he capi al ma ke gene ally canno a ec he pee g oup in es men decision; hence, he pee ’s g oup ma ginal dis ibu ion is iden ical unde Fand G. This condi ion con o ms o he equi emen in P oposi ion 1, e en in he case whe e mo e han wo ou comes exis . This is a easonable assump ion wi h he in es men choices ha we analyze in his s udy bu no wi h ul ima um games in which he indi idual decision a ec s he opponen ’s ou come. Mo eo e , in he po olio in es men case, his e m is equal o ze o, ega dless o whe he he pee g oup po olio is domes ic o in e na ional. Thus, ega ding he issue ha we in es iga e in his pape (in es men wi h a pa icula s ock index as he pee g oup’s po olio), as ad oca ed abo e, he sign o he de i a i e U2 is i ele an . Namely, e m C=0 and he e is no need o assume jealousy (U2<0) o al uism (U2> 0 ) o ob ain ou esul s co esponding o he po olio in es men case. Thus, as o he analysis o he IHB, e m Cis equal o ze o, and Equa ion (3) is educed o: ∆i=A+B. (4) Howe e , no e ha gene ally we canno assume ha also e m Bis equal o ze o, as by changing he di e si ica ion s a egy, we change he ma ginal dis ibu ion o he indi idual’s weal h. Thus, wi h in e na ional po olio di e si ica ion, he condi ion o equal ma ginal dis ibu ions (see e m Bo Equa ion (3)) equi ed by P oposi ion 1 does no hold. To be able o de e mine whe he he pee e ec induces an inc ease in he op imal domes ic in es men ela i e o he uni a ia e expec ed u ili y op imal domes ic in es men weigh , we need o be mo e speci ic ega ding he de ini ions o po olios Fand Gunde conside a ion. We examine he e he possible exis ence o BFSD by conside ing he wo speci ic po olios wi h di ec implica ion o he IHB issue analyzed in his pape . These wo po olios a e deno ed by FAand GM, as de ined below. De ini ion 2. GM is he po olio wi h he in e na ional ma ke weigh s. I he Ame ican in es o holds his ma ke po olio, she would in es 35% (which is he weigh o he Ame ican ma ke in he wo ld ma ke ) domes ically; hence, he IHB does no exis . Dis ibu ion FA s ands o he ac ual agg ega e po olio held by he Ame ican in es o s. Namely, he ac ual domes ic weigh held by he Ame ican in es o is 75%; hence, holding his po olio implies an IHB o 40%. 10 Ac ually, i is equi ed o ha e a leas one s ic inequali y wi h he dis ibu ion unc ions as well as wi h he c oss de i a i e o a oid he i ial case o ha ing ∆i= 0. In he es o he pape , when we w i e such inequali ies, we always mean ha he e is a leas one s ic inequali y, bu o a oid a complex w i ing, we will no w i e i down e e ywhe e. 11 I U12 < 0, in some ange, one can always ind a bi a ia e p e e ence, such ha ou side his ange he c oss de i a i e is close o ze o; hence, ∆iis nega i e. The e o e, o gua an ee ha ∆iis non-nega i e, he c oss de i a i e canno be nega i e. J. Risk Financial Manag. 2020,13, 94 8 o 19 Assuming ha e m Cis equal o ze o, and ew i ing Equa ion (4) in e ms o he abo e wo po olios, we ob ain: ∆i≡EFAU−EGMU=Rwp wpRw wU12hFAw,wp−GMw,wpidwdwp +Rw wU1[GM(w)−FA(w)]dw ≡A+B (5) Suppose ha wi hou he pee e ec he ma ke po olio is op imal. I ∆i> 0, he i h in es o unde conside a ion who conside s also he pee e ec p e e s he ac ual po olio o he ma ke po olio; hence, he in es o inc eases he bi a ia e expec ed u ili y by inc easing he domes ic in es men weigh . Howe e , o ha e BFSD and IHB a ionaliza ion, we need o ha e ha ∆i> 0 o all in es o s i=1, 2, . . . n, ega dless o he p ecise shape o hei p e e ences. A ew conclusions, some o hem in con adic ion o he common iew ega ding he ole o he c oss de i a i e, can be d awn om Equa ion (5). Fi s , i U12 ≥ 0 is assumed (as needed in P oposi ion 1 o jus i y he a ionaliza ion o he IHB), we ind ha he e is no BFSD; hence, in his se ing he e is no IHB a ionaliza ion. The eason is ha i U12 ≥ 0, e m Ao Equa ion (5) is posi i e only i FAw,wp≥GMw,wp , bu his implies ha FA(w,∞)=FA(w)≥G(w,∞)=G(w) , and he e o e, e m Bis nega i e. The sum A+Bmay be nega i e, implying ha he e is no BFSD. Su p isingly, in con as o P oposi ion 1, he condi ion U12 ≤ 0 may allow BFSD; hence, i may allow IHB a ionaliza ion. We ha e BFSD and IHB a ionaliza ion wi h U12 ≤ 0 i he ollowing wo condi ions hold: FAw,wp≤GMw,wp(6a) FA(w)≤GM(w)(6b) Bu as condi ion (a) implies condi ion (b), unlike he posi i e c oss-de i a i e case, hese wo condi ions can simul aneously hold. The e o e, i condi ion (a) on he join dis ibu ion holds, bo h e ms Aand Ba e posi i e (wi h U12 <0)and he e o e ∆i≥0 o i=1, 2, . . . n. Example 1. The ma ginal dis ibu ions and he BFSD. In his example, we demons a e he ela ion be ween he BFSD and he posi i e c oss de i a i e in he case whe e he condi ions o P oposi ion 1 a e in ac , and hen we demons a e he mo e ealis ic case, whe e he ma ginal dis ibu ions a e no kep cons an ; hence, BFSD does no exis , despi e he posi i e c oss-de i a i e assump ion. Suppose ha he S&P index e u n wp is equal o 3 o 4, each ou come wi h an equal p obabili y o 0.5. We conside in es ing in ei he po olio Fo po olio G, bo h yielding e u n w o ei he 2 o 5 wi h equal p obabili y o 0.5. Howe e , Fhas a co ela ion o +1 wi h he S&P index (wi h join e u ns o (2, 3) wi h a p obabili y o 0.5 and join e u ns o (5, 4) wi h a p obabili y o 0.5). Ghas a nega i e co ela ion o −1 wi h he S&P index wi h join e u ns o (2, 4) wi h a p obabili y o 0.5 and join e u ns (5, 3) wi h a p obabili y o 0.5 (see Table 1). All o he join p obabili ies a e equal o ze o. Deno ing he join dis ibu ion co esponding o he co ela ion +1 by FA and he join dis ibu ion co esponding o co ela ion − 1 by GM , we ha e wi h he abo e example wi h he join p obabili ies he ollowing ela ionship: FAw,wp≥GMw,wp o all alues w,wp(7) wi h a leas one s ic inequali y (see lowe pa o Table 1Pa a), e.g., FA(2, 3)=0.5 >GM(2, 3)=0, J. Risk Financial Manag. 2020,13, 94 15 o 19 We employ he bi a ia e i s -deg ee s ochas ic dominance (BFSD) ule and p o e heo e ically ha bi a ia e p e e ences wi h a posi i e c oss de i a i e a ionalizes he obse ed IHB, only in he un ealis ic case in which he ma ginal dis ibu ions o all possible po olio unde conside a ion a e iden ical. O cou se, his does no hold in p ac ice, as no all in e na ional ma ke s a e iden ical, and he e o e also he ma ginal dis ibu ions o a ious selec ed di e si ied po olios a e no iden ical. Thus, e en wi h pee e ec , o e in es ing domes ically may be an in e io in es men s a egy, hence he IHB canno be explained by he pee e ec . Wi h ac ual empi ical in e na ional s ock ma ke da a (ob iously, wi h unequal empi ical ma ginal dis ibu ions), we ind ha he commonly employed KUJ p e e ence wi h a posi i e c oss de i a i e, which in ui i ely implies a desi e o inc ease he co ela ion by o e in es ing domes ically, dec eases a he han inc eases he domes ic in es men weigh , hence he pee e ec e en enhances he IHB puzzle. Mo eo e , once again coun e in ui i ely, we ind ha , wi h a bi a ia e p e e ence wi h a nega i e c oss de i a i e, he op imal domes ic in es men inc eases. Thus, a posi i e c oss de i a i e is nei he necessa y no su icien o IHB a ionaliza ion. In sum, employing a gene al bi a ia e u ili y unc ion wi h pee e ec , wi h no cons ain s on he p e e ence employed, gene ally canno a ionalize he empi ically obse ed IHB. Employing he commonly employed speci ic KUJ bi a ia e p e e ences also does no a ionalize he IHB. As he IHB is an empi ical ac , o a ionalize his phenomenon, one needs o seek o he explana ions and o he esea ch s ands, as he in ui i e explana ion o he pee e ec o a ionalizing he IHB phenomenon is misleading. Funding: This esea ch ecei ed no ex e nal unding. Acknowledgmen s: I would like o hank h ee anonymous e e ees o his Jou nal o hei help ul commen s which g ea ly imp o ed he pape . Con lic s o In e es : The au ho s decla e no con lic o in e es . J. Risk Financial Manag. 2020,13, 94 16 o 19 Appendix A Table A1. The annual a es o e u n 1988–2012. Yea USA Canada Ge many F ance The Ne he lands No way Sweden UK Aus alia Japan Eme ging Ma ke s 1988 0.16 0.18 0.21 0.39 0.16 0.43 0.49 0.06 0.38 0.36 0.40 1989 0.31 0.25 0.47 0.37 0.37 0.46 0.33 0.22 0.11 0.02 0.65 1990 −0.02 −0.12 −0.09 −0.13 −0.02 0.01 −0.20 0.10 −0.16 −0.36 −0.11 1991 0.31 0.12 0.09 0.19 0.19 −0.15 0.15 0.16 0.36 0.09 0.60 1992 0.07 −0.11 −0.10 0.03 0.03 −0.22 −0.14 −0.04 −0.10 −0.21 0.11 1993 0.10 0.18 0.36 0.22 0.37 0.43 0.38 0.24 0.37 0.26 0.75 1994 0.02 −0.02 0.05 −0.05 0.13 0.24 0.19 −0.02 0.06 0.22 −0.07 1995 0.38 0.19 0.17 0.15 0.29 0.07 0.34 0.21 0.12 0.01 −0.05 1996 0.24 0.29 0.14 0.22 0.29 0.29 0.38 0.27 0.18 −0.15 0.06 1997 0.34 0.13 0.25 0.12 0.25 0.07 0.13 0.23 −0.10 −0.24 −0.12 1998 0.31 −0.06 0.30 0.42 0.24 −0.30 0.15 0.18 0.07 0.05 −0.25 1999 0.22 0.54 0.21 0.30 0.07 0.32 0.81 0.12 0.19 0.62 0.66 2000 −0.13 0.06 −0.15 −0.04 −0.04 0.00 −0.21 −0.12 −0.09 −0.28 −0.31 2001 −0.12 −0.20 −0.22 −0.22 −0.22 −0.12 −0.27 −0.14 0.03 −0.29 −0.02 2002 −0.23 −0.13 −0.33 −0.21 −0.20 −0.07 −0.30 −0.15 0.00 −0.10 −0.06 2003 0.29 0.55 0.65 0.41 0.29 0.50 0.66 0.32 0.51 0.36 0.56 2004 0.11 0.23 0.17 0.19 0.13 0.54 0.37 0.20 0.32 0.16 0.26 2005 0.06 0.29 0.11 0.11 0.15 0.26 0.11 0.07 0.18 0.26 0.35 2006 0.15 0.18 0.37 0.35 0.32 0.46 0.45 0.31 0.33 0.06 0.33 2007 0.06 0.30 0.36 0.14 0.21 0.32 0.01 0.08 0.30 −0.04 0.40 2008 −0.37 −0.45 −0.45 −0.43 −0.48 −0.64 −0.49 −0.48 −0.50 −0.29 −0.53 2009 0.27 0.57 0.27 0.33 0.43 0.89 0.66 0.43 0.77 0.06 0.79 2010 0.15 0.21 0.09 −0.03 0.02 0.12 0.35 0.09 0.15 0.16 0.19 2011 0.02 −0.12 −0.17 −0.16 −0.12 −0.09 −0.15 −0.03 −0.11 −0.14 −0.18 2012 0.16 0.10 0.32 0.23 0.21 0.20 0.23 0.15 0.22 0.08 0.19 J. Risk Financial Manag. 2020,13, 94 17 o 19 Appendix B Table A2. The co ela ion ma ix o he pe iod 1988–2012. USA Canada Ge many F ance The Ne he lands No way Sweden UK Aus alia Japan Eme ging Ma ke s USA 1 0.68 0.79 0.83 0.85 0.48 0.76 0.86 0.58 0.44 0.52 Canada 0.68 1 0.77 0.76 0.75 0.84 0.88 0.79 0.81 0.67 0.78 Ge many 0.79 0.77 1 0.90 0.89 0.70 0.80 0.84 0.72 0.59 0.68 F ance 0.83 0.76 0.90 1 0.88 0.66 0.84 0.83 0.75 0.62 0.67 The Ne he lands 0.85 0.75 0.89 0.88 1 0.73 0.77 0.93 0.75 0.45 0.66 No way 0.48 0.84 0.70 0.66 0.73 1 0.79 0.75 0.83 0.55 0.76 Sweden 0.76 0.88 0.80 0.84 0.77 0.79 1 0.80 0.79 0.80 0.74 UK 0.86 0.79 0.84 0.83 0.93 0.75 0.80 1 0.78 0.42 0.65 Aus alia 0.58 0.81 0.72 0.75 0.75 0.83 0.79 0.78 1 0.63 0.83 Japan 0.44 0.67 0.59 0.62 0.45 0.55 0.80 0.42 0.63 1 0.67 Eme ging Ma ke s 0.52 0.78 0.68 0.67 0.66 0.76 0.74 0.65 0.83 0.67 1 J. Risk Financial Manag. 2020,13, 94 18 o 19 Re e ences Abel, And ew B. 1990. Asse p ices unde habi o ma ion and ca ching up wi h he Joneses. Ame ican Economic Re iew 80: 38–42. A kinson, An hony B., and F ançois Bou guignon. 1982. The compa ison o mul i-dimensioned dis ibu ions o economic s a us. Re iew o Economic S udies 49: 183–201. [C ossRe ] Bax e , Ma ianne, and U ban J. Je mann. 1997. The in e na ional di e si ica ion puzzle is wo se han wha you hink. Ame ican Economic Re iew 87: 170–80. Be iel, Tiago C., and Sa oj Bha a ai. 2013. Hedging agains he go e nmen : A solu ion o he home asse bias puzzle. Ame ican Economic Jou nal: Mac oeconomics 5: 102–34. [C ossRe ] Bol on, Ge y E. 1991. A compa a i e model o ba gaining: Theo y and e idence. Ame ican Economic Re iew 81: 1096–136. Bol on, Ge y E., and Axel Ocken els. 2000. ERC: A heo y o equi y, ecip oci y, and compe i ion. Ame ican Economic Re iew 90: 166–93. [C ossRe ] B i en-Jones, Ma k. 1999. The sampling e o in es ima es o mean- a iance e icien po olio weigh s. Jou nal o Finance 54: 655–71. [C ossRe ] Campbell, Y. John, and John H. Coch ane. 1999. By o ce o habi : A consump ion based explana ion o agg ega e s ock ma ke beha io . Jou nal o Poli ical Economy 107: 205–51. [C ossRe ] Coeu dacie , Nicolas, and Helene Rey. 2013. Home bias in open economy inancial mac oeconomics. Jou nal o Economic Li e a u e 51: 63–115. [C ossRe ] Cons an inides, Geo ge M. 1990. Habi o ma ion: A esolu ion o he equi y p emium puzzle. Jou nal o Poli ical Economy 98: 519–43. [C ossRe ] Co al, Joshua D., and Tobias J. Moskowi z. 1999. Home bias a home: Local equi y p e e ence in domes ic po olios. Jou nal o Finance 54: 2045–73. [C ossRe ] Co al, Joshua D., and Tobias J. Moskowi z. 2001. The geog aphy o in es men : In o med ading and asse p ices. Jou nal o Poli ical Economy 109: 811–41. [C ossRe ] Dahlquis , Magnus, and Go an Robe sson. 2001. Di ec o eign owne ship, indi idual in es o s, and i m cha ac e is ics. Jou nal o Financial Economics 59: 413–40. [C ossRe ] Dema zo, Pe e M., Ron Kaniel, and Ilan K eme . 2008. Rela i e weal h conce ns and inancial bubbles. Re iew o Financial S udies 21: 19–50. [C ossRe ] Dupo , Bill, and Wen-Fang Liu. 2003. Jealousy and equilib ium o e consump ion. Ame ican Economic Re iew 93: 423–28. [C ossRe ] Eeckhoud , Louis, B é a ice Rey, and Ha is Schlesinge . 2007. A good sign o mul i a ia e isk aking. Managemen Science 53: 117–24. [C ossRe ] F ench, Kenne h R., and James M. Po e ba. 1991. In e na ional di e si ica ion and in e na ional equi y ma ke s. Ame ican Economic Re iew 81: 222–26. Gal í , J ó di. 1994. Keeping up wi h he Joneses: Consump ion ex e nali ies, po olio choice, and asse p ices. Jou nal o Money, C edi and Banking 26: 1–8. [C ossRe ] Hada , Jose , and William R. Russell. 1969. Rules o o de ing unce ain p ospec s. Ame ican Economic Re iew 59: 25–34. Hanoch, Gio a, and Haim Le y. 1969. The e iciency analysis o choices in ol ing isk. Re iew o Economic S udies 36: 335–46. [C ossRe ] Hube man, Gu . 2001. Familia i y b eeds in es men . Re iew o Financial S udies 14: 659–80. [C ossRe ] Kang, Jun-Koo, and Rene M. S ul z. 1997. Why is he e a home bias? An analysis o o eign po olio equi y owne ship in Japan. Jou nal o Financial Economics 46: 3–28. [C ossRe ] Lau e bach, Beni, and Haim Reisman. 2004. Keeping up wi h he Joneses and he home bias. Eu opean Financial Managemen 10: 225–34. [C ossRe ] Le y, Haim. 2015. S ochas ic Dominance: In es men Decision-Making unde Unce ain y, 3 d ed. New Yo k: Sp inge . Le y, Haim. 2016. Economic Home Bias. Jou nal o Money, C edi and Banking 26. o hcoming. Le y, Moshe, and Haim Le y. 2014. The home bias is he e o s ay. Jou nal o Banking and Finance 47: 27–40. [C ossRe ] Le y, Haim, and Jacob Pa oush. 1974. Towa d mul i a ia e e iciency c i e ia. Jou nal o Economic Theo y 7: 129–42. [C ossRe ] J. Risk Financial Manag. 2020,13, 94 19 o 19 Le y, Moshe, and Richa d Roll. 2010. The ma ke po olio may be mean- a iance e icien a e all. Re iew o Financial S udies 23: 2464–491. [C ossRe ] Lewis, Ka en K. 1999. T ying o explain home bias in equi ies and consump ion. Jou nal o Economic Li e a u e 37: 571–608. [C ossRe ] Ljungq is , La s, and Ha ald Uhlig. 2000. Tax policy and agg ega e demand managemen unde ca ching up wi h he Joneses. Ame ican Economic Re iew 90: 356–66. [C ossRe ] Me on, Robe C., and Paul Samuelson. 1974. Fallacy o logno mal app oxima ion o op imal po olio decision-making o e many pe iods. Jou nal o Financial Economics 1: 67–94. [C ossRe ] Philips, Ch is ophe B., F ancis M. Kinni y, and Sco J. Donaldson. 2012. The Role o Home Bias in Global Asse Alloca ion Decisions. Vangua d: Vangua d Resea ch. Rabin, Ma hew. 1993. Inco po a ing ai ness in o game heo y and economics. Ame ican Economic Re iew 83: 1281–302. Rabin, Ma hew. 1998. Psychology and economics. Jou nal o Economic Li e a u e 36: 11–46. Ro hschild, Michael, and Joseph E. S igli z. 1970. Inc easing isk: I. A de ini ion. Jou nal o Economic Theo y 2: 225–43. [C ossRe ] Tesa , Linda, and Ing id We ne . 1995. Home bias and home u no e . Jou nal o In e na ional Money and Finance 14: 467–92. [C ossRe ] Tse lin, Ilia, and Robe L. Winkle . 2009. Mul i a ia e u ili y sa is ying a p e e ence o combining good wi h bad. Managemen Science 55: 1942–952. [C ossRe ] Vanpée, Rosanne, and Lie en DeMoo e. 2012. Bond and Equi y Home Bias and Fo eign Bias: An In e na ional S udy. Wo king Pape . B uxelles: Hogesschool Uni e si ei B ussel. Zizzo, Daniel John. 2003. Money bu ning and ank egali a ianism wi h andom dic a o s. Economics Le e s 81: 263–66. [C ossRe ] © 2020 by he au ho . Licensee MDPI, Basel, Swi ze land. This a icle is an open access a icle dis ibu ed unde he e ms and condi ions o he C ea i e Commons A ibu ion (CC BY) license (h p://c ea i ecommons.o g/licenses/by/4.0/).