Real Risk-Free Rate, the Central Bank, and Stock Market Bubbles

420 Jou nal o Re iews on Global Economics, 2017, 6, 420-425 E-ISSN: 1929-7092/17 © 2017 Li escience Global Real Risk-F ee Ra e, he Cen al Bank, and S ock Ma ke Bubbles Jukka Ilomäki* and Hannu Lau ila Facul y o Managemen , FI-33014 Uni e si y o Tampe e, Finland Abs ac : The cen al bank ac s as a social planne , and adjus s he eal isk- ee a e o e u n o co ec any misp icing in he s ock ma ke so ha he eme gence o posi i e o nega i e bubbles is a oided. The analysis shows ha he cen al bank mus aise he isk- ee a e in he case o a posi i e bubble, and ice e sa. Mo eo e , he cen al bank should in e ene in he s ock ma ke e en i i does no ha e pe ec in o ma ion abou he bubble. This is because he sequen ial di idend yields in he p icing equa ions a e s a iona y. Thus, e en he delayed eac ion o he cen al bank p e en s he undamen al alue and he equilib ium p ice om d i ing apa o ex ended pe iods. Keywo ds: Real In e es Ra e, Mone a y Policy, Po olio Choice. 1. INTRODUCTION Wha should be he ole o he cen al banks in he con ex o s ock ma ke bubbles? Fo example, Be nanke and Ge le (2001), G eenspan (2004), and Posen (2006) a gue ha he cen al bank should concen a e on in la ion a ge ing and s able g ow h, while Bo do & Jeanne (2002), Bean (2004), and Roubini (2006) claim ha he cen al bank should also ackle s ock ma ke bubbles. Yellen (2010) suppo s he la e iew by s a ing ha he cen al bank should in e ene in s ock ma ke p icing, i he bubble ge s dange ously la ge. Viewing he ques ion om ano he angle, Maio (2014) p o ides empi ical suppo o “don’ igh he Fed” –policy. He shows ha he policy yields economically and s a is ically signi ican sho - e m gains compa ing o buy-and-hold s a egy. Obeying he “don’ igh he Fed” p inciple means ha in es o s should eac by inc easing/dec easing isky in es men s in he s ock ma ke s when he cen al bank educes/ aises he isk- ee a e o e u n. Fu he mo e, he e a e some p ac ical examples o cen al banks’ a acks on cu ency ma ke bubbles by adjus ing he isk- ee a e. Fo example, he cen al bank o Russia eac ed o i s na ional cu ency ma ke bubble in 2014 by aising he key in e es a e (Gilenko 2017). Conce ning he means o in e en ion, Conlon (2015) sugges ha he cen al bank should use i s in o ma ional ad an age, and in e ene by wa ning he *Add ess o co espondence o his au ho a he Facul y o Managemen , FI-33014 Uni e si y o Tampe e, Finland; Tel: +358401366117; E-mail: jukka.ilomaki@u a. i JEL Classi ica ion: E43, E52, G11 in es o s abou no iceable bubbles. Thus, inac ion o he cen al bank implici ly means ha he s ock p ices can ise. Fu he mo e, he in e en ion by he cen al bank may make hings e en wo se i i does no ha e p i a e in o ma ion abou he bubble. Fische (2016) makes a s onge poin by a guing ha , in he case o economy-wide bubbles, he cen al bank should in e ene by mone a y policy. A qui e common pe cep ion abou he p ope cu e o bubbles is he so called “leaning agains he wind” ype policy, which means ha he isk- ee a e mus be li ed in o de o de-in la e a posi i e bubble. Howe e , he e a e con on ing iews, oo. Fo example Gali (2014) claims ha such a policy may ac ually make he bubble in la e e en u he . This may happen because he undamen al componen o an asse p ice co esponds o he discoun ed s eam o payo s, bu he bubble componen has no payo s o discoun . Thus, he la e componen g ows a he a e o in e es , a leas in expec a ion. Inspi ed by he abo e a gumen a ions, we p esen a simple inancial ma ke model, whe e he cen al bank is in cha ge o he adjus men o s ock ma ke bubbles. The adjus men is achie ed by con olling he ma ke eal e u n o isk- ee asse s ia mone a y policy. The maneu e abili y o bubbles is s udied bo h wi h and wi hou pe ec in o ma ion abou e icien s ock ma ke p icing and hus abou he exis ence o bubbles. The model yields a “leaning agains he wind” ype ule o moni o ing and co ec ing any misp icing, and s abilizing he inancial ma ke . The pape p oceeds as ollows. Sec ion 2 p esen s ela ed li e a u e. Sec ion 3 de ines he basic model wi h in o med and unin o med in es o s. Sec ion 4 p esen s he analysis o he equilib ium in he inancial Real Risk-F ee Ra e, he Cen al Bank, and S ock Ma ke Bubbles Jou nal o Re iews on Global Economics, 2017, Vol. 6 421 ma ke , and desc ibes he in e en ions needed om he cen al bank o s abilize he inancial ma ke . Sec ion 5 discusses he indings. 2. LITERATURE REVIEW In ou model, he key building block is he possibili y o bubbles. The seminal e iciency a gumen by Samuelson (1973) is ha , wi h pe ec ly a ional isk- neu al in es o s, he equilib ium s ock p ice (P ) is equal o he expec ed discoun ed di idends o he sha eholde , ha is he undamen al alue (V ). Ti ole (1982) shows ha P =V holds also in he a ional expec a ions equilib ium wi h long-li ed isk a e se in es o s so ha in ini e bubbles a e impossible. Howe e , Ti ole (1985) a gues ha , in an o e lapping gene a ions model wi h sho li ed in es o s and in ini ely li ed asse s bubbles a e possible so ha P !V may occasionally happen. San os & Wood o d (1997) indica es ha bubbles a e impossible in a ional ma ke s only in he long un, bu ha he e is a possibili y o P !V in he sho un. The o e lapping gene a ions model is he e in e p e ed so ha a ional in es o s ha e a sho - e m in es ing ho izon. Fo example Shlei e and Vishny (1997) a gue ha he sho -li ed-in es o assump ion can be mo i a ed so ha he majo i y o in es o s in he ma ke a e p o essional weal h manage s who handle o he people’s money wi h pe o mance moni o ing in sho in e als. Delong e al. (1990), F oo e al. (1992), Campbell and Kyle (1993), Kyle and Xiong (2001), Shille (2014), and Ilomäki (2016) a gue ha , i he e a e in o med and unin o med sho - e m ( ha is sho - li ed) in es o s in he ma ke , P !V exis s because o he asymme y and co ela ed beha io o he uni o med in es o s. Fu he mo e, Allen e al. (2006), and Bacche a & Van Wincoop (2008) show ha P !V occu s in he case o sho -li ed in es o s, because hei noisy p i a e in o ma ion inco po a es a ional highe -o de belie s in o he equilib ium. Cespa & Vi es (2015) also uses a model o sho -li ed in es o s wi h asymme ic in o ma ion, and ends o wo ex eme equilib iums: a high in o ma ion equilib ium P =V wi h low ola ili y and high liquidi y, and a low in o ma ion equilib ium P !V wi h high ola ili y and low liquidi y. The e o e, ou model builds on he assump ion o asymme ic in o ma ion among he in es o s, and on he a gumen o Loewens ein and Willa d (2006) ha he a ia ion o he isk- ee a e o e u n assu es ha P =V . O iginally, he la e poin de i es om he p ope y ha he limi ed supply o isk- ee asse s pulls oge he he equilib ium p ice and he undamen al alue, implying ha P !V is impossible ega dless o he in es o s’ beha io . In ou model, he cen al bank in e enes he ma ke by con olling he supply o isk- ee asse s hus se ing he isk- ee a e. 3. THE MODEL The model ollows Ilomäki (2016) wi h he ex ension o a ime- a ying eal isk- ee a e o e u n. The eal a e is co ec ed o in la ion so ha whe e he g oss in la ion is de e mined as 1+ ! "p p #1 , (1) whe e p is he gene al p ice le el in he economy. The economy consis s o an in ini e se o a ional cons an absolu e isk-a e se (CARA) in es o s, who ha e asymme ic in o ma ion so ha 0< µ <1 o hem a e in o med, and 1! µ o hem a e unin o med in e e y pe iod. The a omis ic in es o s li e o wo pe iods, in es ing in he i s pe iod, and consuming in he second pe iod. The e is an in ini ely li ed isky asse (sha e o i m F), and a isk- ee al e na i e wi h ime- a ying eal isk- ee a e o e u n . The in es o s alloca e hei in es men s be ween isk- ee and isky asse s. The po olio choice is simpli ied, because he assump ion o wo-pe iod li ed CARA in es o s omi s he possibili y o hedging agains changes in expec ed e u ns, and because he assump ion o an in ini ely li ed isky asse cons i u es limi s o a bi age in he o e lapping gene a ions model (Shlei e and Vishny, 1997). In addi ion, assuming no mally dis ibu ed excess consump ion, CARA in es o s assu e ha he condi ional a iance in excess e u ns is cons an . The e a e ou ypes o a ional in es o s in e e y pe iod : young in o med and unin o med in es o s who open hei posi ions ( he demand side), and old in o med and unin o med in es o s who close hei posi ions ( he supply side). Fo simplici y, excess e u ns a e no mally dis ibu ed, he ime- a ying isk p emium is common o all in es o s, and he e a e nei he ansac ion cos s no axes. The budge cons ain comes om he assump ion ha all young in es o s a ime ha e he same indi idual endowmen w y. The na u al loga i hm o he di idend D on i m F’s s ock ollows andom walk so ha ln D =ln D !1+e d , (2) 422 Jou nal o Re iews on Global Economics, 2017, Vol. 6 Ilomäki and Lau ila whe e e d~WN (0, ! d 2) . This means ha he change in he di idend a is pe manen . In pe iod , he di idends a e paid o old in es o s. In pe iod , in o ma ion common o all in o med and unin o med in es o s consis s o he his o y o equilib ium eal p ices (...P !3,P !2,P !1) , and he cu en alue o he ime a ying eal isk- ee a e ( ). Mo eo e , all in es o s obse e cu en eal di idends ( D ), bu he young in o med in es o s ha e also p i a e in o ma ion on D +1. Assume ha , in any pe iod , he cen al bank ac s as a social planne wi h he aim o s abilize he inancial ma ke s by adjus ing he eal isk- ee a e o e u n. The cen al bank may ha e pe ec o impe ec in o ma ion on he inancial ma ke . In any case, he cu en alue o he eal isk- ee a e o e u n is de e mined by he adjus men o possible misp icing so ha i is cons an wi h ze o a iance om pe iod o pe iod +1. Supposing ha he supply o he isk- ee asse is in sole con ol o he cen al bank, he ma ke clea ing condi ion o he isk- ee asse eads ay y !"bcb cb !=0 , (3) whe e ay is he o al demand by he young in es o s, and bcb is he o al supply by he cen al bank in e e y pe iod. Fu he mo e, assume exogenous noise ade s wi h he dis ibu ion e n ~N(0, ! n 2) in he s ock ma ke . Thus, he ma ke clea ing condi ion o he isky asse eads xy!so o " y "+e n =0 , (4) whe e xy e e s o o al demand o he s ock by he young in es o s, and so is he o al supply o he s ock by he old in es o s. The op imal demand decisions p oduce he equilib ium p ice in pe iod hus ul illing he ma ke clea ing condi ion. This happens because he old in es o s ha e o close hei posi ion o consume in he second pe iod. Ra ional in es o s ca e also abou he isk o he in es men . LeRoy (1973) shows ha i he isk- ee a e o e u n is ime- a ying, and i all in es o s a e isk a e se, he p ope discoun a e includes he isk- ee a e and a isk p emium. S a ing om Ma kowi z (1952) and Sha pe (1964), isk is de ined as he a iance o e u ns. A a ional young in es o maximizes indi idual u ili y by alloca ing he ini ial endowmen be ween isky and isk- ee asse s. The maximiza ion p oblem eads Max[E(!e! " c +1| # y,w y)] s. . c +1=x (1+ )+x E (R +1) w y=x +x , (5) whe e ! y is he in o ma ion se , > 0 is he coe icien o isk a e sion, c +1 is consump ion when old, w y is he endowmen , and x and x deno e he amoun o money in es ed in isk- ee and isky asse s, espec i ely. The ne excess e u n on a isky sha e is R +1!P +1"P +D +1 P " . (6) Assuming no mally dis ibu ed ex a consump ion, and plugging he consump ion cons ain in o he u ili y unc ion yields E U(c +1) [ ] =!e ! " x E (R +1)+ " 2 2 x 2 # 2 , (7) whe e ! 2 is he cons an a iance o excess e u ns. No e ha , since he in es o s obse e , i s a iance is ze o. Maximize (7) wi h espec o x , and use equa ion (6) o w i e he i s o de p icing condi ion o he isky asse , P =P +1+D +1 1+ +x !" 2 . (8) Manipula ion o (8) esul s in he op imal in es men decision, saying ha he s ock demand o a a ional young in es o eads x =(P +1+D +1) / P !(1+ ) "# 2 . (9) In equa ion (8), he equi ed ne eal a e o e u n is n= +x ! 2, whe e x ! 2=P +1+D +1 P "(1+ )= # (10) deno es he ime- a ying isk p emium. In any pe iod , he in es o s know i upon he choice o x . 4. THE FINANCIAL MARKET EQUILIBRIUM In he model, all in o med in es o s ha e he bes possible in o ma ion, and old/young in es o s Real Risk-F ee Ra e, he Cen al Bank, and S ock Ma ke Bubbles Jou nal o Re iews on Global Economics, 2017, Vol. 6 423 ecognize ha young/old in es o s obse e his. Recalling ha he p ope ies o andom walk say ha he change in he di idend a ime is pe manen , he a ional choice ep esen ed by equa ion (8) esul s o e ime in he pe pe ui y model, P =D +1 n=V . (11) whe e n= +x i ! 2, in which xi is he s ock demand o an in o med in es o . The e o e, i can be concluded ha equa ion (11) e lec s he undamen al alue o he isky asse , and ha he in o med in es o s ollow he Samuelsonian p icing pa e n, P i=V . The unin o med in es o s obse e he cu en eal di idend D . Hence, he a ional p icing ule o he unin o med in es o s is P =D n , (12) whe e n= +x u ! 2, in which xu is he s ock demand o an unin o med in es o . Using equa ions (11) and (12), and ecalling ha µ is he sha e o he in o med in es o s, and 1-µ is he sha e o he unin o med in es o s, and i deno es in o med and u deno es unin o med in es o s, he agg ega e p icing ule in he inancial ma ke eads P = µ D +1 + ! i+(1" µ )D + ! u . (13) The ma ke p ice gi en by equa ion (13) esul s om he asymme y o in o ma ion among he in es o s. P oposi ion 1: I he isk- ee a e o e u n ises/ alls, he in es o s’ a ac ion o isky in es men s diminishes/inc eases. P oo : Di e en ia ion o equa ions (11) and (12) agains x i (x u ) and yields !x i ! =!x u ! ="1 # 2<0, (14) showing ha an inc ease in he isk- ee a e educes isky in es men s o bo h in o med and unin o med in es o s. Q.E.D. Co olla y 1: I he isk- ee a e ises/ alls, he ma ke p ice alls/ ises, ce e is pa ibus. P oo : Di e en ia e Equa ion (13) agains P and , manipula e, and ge !P ! =" µ D +1 ( + # i)2"(1" µ )D ( + # u)2<0 , (15) which shows ha a ise/ all o he isk- ee a e makes he ma ke p ice o he s ock all/ ise. Q.E.D. Conside an al e na i e case, whe e he unin o med in es o s ope a e as echnical ade s saying ha hey use pas p ices o es ima e he ai p ice o oday. Manipula ing Equa ion (8), and aking one s ep backwa ds yields P u=(1+ !1 + " !1 u)P !1!D , which educes o P u=(1+ + ! u)P "1"D (16) since !1 = because he eal isk- ee a e emains unal e ed unless he cen al bank in e e es, and ! "1 u=x "1 u # 2=x u $ 2= ! u because he unin o med in es o makes he in es men decision upon pas p ices. Di e en ia ion o Equa ion (16) agains x u and yields again !x u ! ="1 # 2<0, showing ha a ise/ all in he isk- ee a e makes in es men s in he isky asse all/ ise also o echnical ade s. By Equa ions (11) and (16), he agg ega e p icing ule in he inancial ma ke eads P = µ V +(1! µ ) (1+ + " )P !1!D # $% & . (17) Co olla y 2: I he isk- ee a e ises/ alls, he ma ke p ice alls/ ises, ce e is pa ibus. P oo : Recall Equa ion (11), di e en ia e Equa ion (16) agains P and , and ge !P ! =" µ D +1 ( + # )2+(1" µ )P "1 . (18) The e ec is nega i e i µ D +1 (1! µ )P !1 > + " ( ) 2 . Tha is he equilib ium p ice alls, i he weigh ed di idend e u n is la ge han squa ed ne equi ed e u n, which is a easonable assump ion. Q.E.D. 5. THE CENTRAL BANK The ma ke p ice equa ions (13) and (17) imply a possibili y o a bubble, and he ask o he cen al bank is o a oid such bubbles by manipula ing he eal isk- 424 Jou nal o Re iews on Global Economics, 2017, Vol. 6 Ilomäki and Lau ila ee a e o e u n in o de o cause desi ed ma ke eac ions desc ibed by equa ions (15) and (18). I he cen al bank possesses pe ec in o ma ion abou he s ock ma ke and hus on bubbles, i is able o calcula e he ma ke s abilizing eal isk- ee a e om Equa ion (13) as ollows: = µ D +1 P +(1! µ )D P ! " . (19) Conside also he case ha he cen al bank does no possess pe ec in o ma ion abou he ma ke and bubbles. Assume ha he cen al bank has he same impe ec in o ma ion as he uni o med in es o s, knowing only D , and an icipa ing ha P = P -1 and D +1=D . Thus, Equa ion (19) eads != µ D P "1 +(1" µ )D P "1 " # . (20) P oposi ion 2 The cen al bank should in e ene in he s ock ma ke e en i i does no ha e p i a e in o ma ion abou i . P oo : Sub ac Equa ion (20) om Equa ion (19), and w i e ! "= µ D +1 P !D P !1 # $ %& ' ( . (21) By Equa ion (21), he di e ence o he eal isk- ee a es ha a e se wi h pe ec and impe ec in o ma ion equals he weigh ed di e ence be ween he eal di idend yield a pe iod +1 and he eal di idend yield a pe iod . The le -hand side o Equa ion (21) is s a iona y, i he igh -hand side is s a iona y. Recall Equa ion (2), ln D =ln D !1+e d . Hence, bo h componen s in he pa en hesis on he igh -hand side a e s a iona y in Equa ion (21). The s a iona i y o he di e ence be ween he wo eal isk- ee a es implies ha P oposi ion 1 and Co olla ies 1-2 apply also i he cen al bank possesses impe ec in o ma ion on he s ock ma ke . Q.E.D. 6. CONCLUSIONS The pape p esen s an o e lapping gene a ions model o a ional in es o s wi h asymme ic in o ma ion. The o e lapping gene a ions model is in e p e ed so ha he weal h manage s con ol people’s money, moni o ed in sho - e m in e als. The cen al bank, as a social planne , aims o dampen possible misp icing in he s ock ma ke by adjus ing he eal isk- ee a e o e u n. The ac i e ole o he cen al bank is jus i ied because unin o med in es o s make mis akes on p icing. By p e en ing bubbles o in la e he cen al bank a oids social losses caused by he de elopmen and e en ual bu s ing o a supe bubble. The analysis shows ha he cen al bank esponds o a ma ginal eme gence o a posi i e bubble by using i s a ailable means o manipula e he eal isk- ee a e upwa ds hus ealloca ing he in es men s o in es o s om isky o isk- ee asse s. Likewise, a nega i e bubble is dampened by making he eal isk- ee a e all hus di ec ing in es men s om isk- ee o isky asse s. Thus, he policy adjus men o he eal isk- ee a e o e u n le els ou any bubbles. Mos , impo an ly, he me hod is e ec i e e en i he cen al bank has no p i a e in o ma ion abou bubbles. The esul s o he pape a e wo h highligh ing, because he e does no exis a ull consensus nei he among academics no among p ac i ione s abou he p ope way o in e ene in ma ke bubbles. One key issue conce ns he alidi y o he “leaning agains he wind” policy. Ou s esul s clea ly jus i y he policy. To be mo e speci ic, a posi i e bubble mus be ackled by aising he isk- ee a e, and ice e sa. 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