caballero on bubbles
TRANSCRIPT
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BubblesMacroeconomicsIV
RicardoJ.CaballeroMIT
Spring2011
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2
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References
Allen,F.andD.Gale,BubblesandCrises, EconomicJournal,110:236-255,2000.Tirole,J.,AssetBubblesandOverlappingGenerations, Econometrica,53,(6),1499-1528,November1985.AbreuD.andM.Brunnermeier,BubblesandCrashes, Econometrica,71:173-204,2003.
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Introduction
Historical: DutchTulipmania,SouthSea... GreatCrashof1929SouthSeaBubble(1710-1720)
IsaacNewton: 04/20/1720soldsharesat7,000,profiting3,500.Re-enteredthemarket later endedup losing20,000Icancalculatethemotionsoftheheavenlybodies,butnotthemadnessofpeople
Japanboom-bust(a lostdecade);EMEs,Nasdaq,realestate(allaroundthedevelopedworld),commoditiesWheredotheycomefrom? Whattodoaboutthem?
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Mainpoints
Twobroad(andpolar)views:There isashortageofstoreofvalue bubbleshelpfixingthisproblemAgentsmisbehave(eitheranagencyproblemorabehavioralproblem)
Myview: Theseviewsaremore intertwinedthan itmayseemThe former isaboutmacroenvironmentswherethere isshortageofassetsThe latter isaboutthe locationofbubbles
Other: irrationalexuberance andmoreformalbehavioralstoriesMyview: More likelytoarisewhentheaboveconditionsarepresent
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Riskshifting
Allen-Gale(2000) BubblesandcrisesThere isapattern:
Phase1: financial liberalizationorsomeexpansionarypolicy fuelsabubblehase
2:
the
bubble
bursts
and
asset
prices
collapse
hase3: widespreaddefaultsby leveragedassetbuyers, leadingtoabankingnd/orexchange ratecrisis,andapersistentrecession
PPa
Mainingredient(thisisallwelldiscusshere): Uncertaintyaboutpayoffs(realorfinancialsector)can leadtobubbles inan intermediatedfinancialsystem(riskshifting/assetsubstitution)
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Model
Twodates,t=1,2andasingleconsumptiongoodTwoassets:
Safeand invariablesupplyata rateriskyand infixedsupply. Stochastic return isR perunit,withdensityh(R)ndsupport [0,R ]
R
a MAXThereturnonthesafeasset isdeterminedbymarginalproductofcapital:r =f(x)wherex areunitsoftheconsumptiongood(standardassumptionsonf)Non-pecuniaryconvexcostof investing inriskyassetsc(x)(torestrictportfoliosizesandtoensureequilibriumprofitforborrowers)
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Model
There isacontinuumofsmall,riskneutral investors; idemforbanks.InvestorshavenowealthwhilebankshaveafixedamountB (whichtheysupply inelastically). Only investorsknowhowto invest,sobanks onlychoice isto lendto investorsBanksand investorsarerestrictedtousedsimpledebtcontracts(inparticular,theydontdependonsize)Since investorscanborrowasmuchastheywantatthegoingrate, inequilibriumthecontractedrateon loansmustbeequaltotheriskless interestrateSymmetriceq. All investorsare identicalex-post. Xs andXR aretherepresentative investorsholdingsofthesafeandriskyassets
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RiskshiftingBecausebanksusedebtcontractsandcannotobserve investmentdecisionsbyborrowers,the latterdoesnotbearthefullcostof investment iftheoutcome isbad,while itgetsthebenefit iftheoutcome isgoodIfrepresentative investorbuysXs andXR, itborrowsXs +PXR (whereP isrel. priceofriskyasset)andtherepayment(ifnotbankrupt) isr(Xs +PXR)The
liquidation
value
of
the
portfolio
is
rXs +RXR,sothepayofffortheinvestor is:
max{(RrP)XR,0}
andthe
decision
problem
is:
RMAX
max XR
(RrP)h(R)dRc(XR)XR0 R=rP
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EquilibriumMarketclearingconditions:
XR = 1XS +P = B
r = f(XS),
thefocsevaluatedattheequilibriumare:R
MAX
(RrP)h(R)dR =c(1)R=rP
r =f(BP)fromwhichwecansolvefor(r,P)
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Wecanre-writethefocwrttoP toget:RMAX Rh(R)dR c(1)
1 R=rP
P =r
Pr[RR]
1 c(1)= E[R|RR]
r
Pr[R
R
]
Definethefundamentalasthepriceanagentwouldbewillingtopay intheabsenceofriskshifting,then:
1 Pf = E[R]c(1)r
It iseasytoshowthat,as longasPr[RR]>0,P>Pf
P
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Proof
rPPr[R R]=RMAX
R=rPRh(R)dRc(1)
=>
rPf R
0Rh(R)dR
rPf (rP)(1Pr[RR])
rP>rPf
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Finalremarks
Hence,duetoriskshifting,P ishigherthanfundamental(bubble)Thecounterpartofthebubble isthebank losses,andhencetherestofthestory...Inasense it isnotaGEbubble,asthepriceofbanksshouldgodown... butitmaywellbethathouseholdsarestuck... thistakesustothestandardmodelofREbubbles inmacro,whichhighlightstheshortageofassets..
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Assetshortages
Letsremoveuncertaintytohighlightthefactthatthenatureofthesebubbles isverydifferentfromtherisk-shiftingargumentReadTiroles1982"OnthepossibilityofspeculationunderRE"(EMA).... soyourealizethatrationalbubblesarenoteasytoget...ButweknowfromSamuelsons(1958)consumption-loanmodelthatbubbles (i.e. assetswithpositivepricebutno intrinsicvalue)canexist inOLGstructures(infinitenewtraders inthehorizon)andthattheycanbegood
Money inSamuelsonsmodel,butnot for itstransactionservicebuttostorevalue. Paretogain from solvingdynamic inefficiency(nocapitalwastedtostorevalue).
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AbarebonesversionofSamuelsonsmodel
OLG.Individuals livefortwoperiods,theyarebornwithanendowmentwtWhichtheysave in itsentiretyandonlyconsumewhenold(hencewecanindexthegenerationswelfarebyct,t+1) .There isnopopulationgrowth,buttheendowmentgrowsatarate.
wt+1 = (1+)wtct,t+1 = (1+rt)wt
What isthe interestrate inthiseconomy?
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Mothernature...Theanswerdependsonwhichassetsareavailabletostorevalue.Samuelsonfirstobservedthattheyoungcouldnotsaveby lendingtotheoldsincethe latterwillnotbearoundtorepaythem later(financialmarketincompleteness). Theonlyoptionoftheyoung istotradewithmothernature, i.e. to invest inphysicalcapital.Letssimplifythetechnologysideandassumethat ithasconstantreturns:. That is,oneunitofsavingsatt produce1+ att+1(wecouldhaveamorestandardf(k)... butmain insightswouldbeunchanged). Itfollowsthatthe interestrate inthiseconomymustbe:
rt =
andutility is:UMN = (1+)wtt
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B bbl
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Bubbles
Morebroadly: themarketeconomy isoveraccumulatingcapitaltofacilitatestoreofvalueDoesthismeanthatthemarketeconomy issuboptimal? Notnecessarily.Naturally, if,themarketcanreachthesameallocationasthesocialcontract,providedweenlargethesavingoptionsoftheyoungto includeoneirreproducibleanduselessobjectwithpriceBt suchthat:
Bt+1 = (1+rt)Bt
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W lf
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Welfare
LetxB/w. ThenBt+1 (1+rt)Bt 1+rt
xt+1 = = = xtwt+1 (1+)wt 1+
Ifx
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Finalremarks
TherearetwoPareto-rankablestationaryequilibria(bubblebetterthanfundamental);andacontinuumofnon-stationaryequilibriathatconvergetothefundamentalequilibriumthatprovide intermediatewelfare(note: alltheseequilibriacontainbubbles,butthesebecomesmallrelativetotheeconomy)Bubblesariseasaresultofcoordinationacrossdifferentgenerations. Butthisisjustoneofthepossibleequilibria,andhencethepossibilityofacrash islatent
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A di Ab d B i
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Appendix: AbreuandBrunnermeier
Behavioralbiases leadtobubbles(theytakethisasgiven)Assumingthatrationalarbitrageursunderstandthatthemarketwilleventuallycollapse,willtheystillridethebubble?Delayedarbitragemodel(ridingthebubbleforawhilemaybeoptimal)[connectionwithearlierdiscussionon limitedarbitrage]Amodelofmarkettiming
Dispersion inexitstrategiesmakesthebubblepossibleAtsome random timet0 pricesurpassesthe fundamentalvalue. Thereafter,rationalarbitrageursbecomesequentiallyawarethatthepricehasdepartedfrom fundamentals. Theydontknowwhethertheyareearlyor late relativetoothersBubbleburstswhenasufficientmassofarbitrageurshavesoldout(coordination)
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Th S t
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TheSetup
InBubblesandCrashes, theydiscussanirrationalexuberance episodewhereaftersamerandomdatet0 thepricecontinuestoriseatsomerateg>r,whilethefundamentalonlyrisesatrThemaineconomicforces intheirEMApaperarealsofound intheirsimpler,JFE,paper:SynchronizationRiskandDelayedArbitrage (wewilldevelopthisone,althoughtheconnectionwithabubble is lessdirect)There isasingleriskyassetwithpricept andfundamentalvt. Priortothearrivalofashockatarandomtimet ,thefundamental value isert0 andafterthat(1+)ert,with takingvaluesand withequalprob.,andF(t0) =1e t0Priortotheshockatt0,pt =vt. Aftert0 thepricedeviatesfromfundamentalsuntilfullarbitragetakesplace(thecrash if,whichweassumehenceforth)
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The Setup
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TheSetup
Therearetwotypesofagents: rationalarbitrageursandbehavioraltradersTheonlyroleofthe latteragents istosupportthemispricingandmaintainthepriceatpt =ert as longasthesellingpressurebyrationalarbitrageursliesbelowathreshold(.)Thefocusofthepaper isontheformeragents. Arbitrageursareex-anteidenticalbutreceive informationaboutthedeviationsequentially(uniformly)betweent0 andt0+An individualarbitrageurwho learnsaboutthechange infundamentalatti(denoted
by
ti)thinks
that
t0
isdistributed
between
ti
andt
i
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The Setup
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TheSetup
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(1+b)e
rt
ert
p,v
p0= 1
t0 t0 + hk0 t0 + h
1/h
t0 + tt
Random
starting
point
k0arbs areaware of
the mispricing
All arbitrageurs
are aware of
the mispricing
Pricing correction
for exogenous
reasons
+b
-b (1_b)e
rt
Image by MIT OpenCourseWare.
The Setup
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TheSetup
Arbitrageursare
risk
neutral
but
the
maximum
short
position
is
xi = 1.Thenormal/neutral position isx i =0. Departingfromthisbenchmark
generates(large)holdingcostsofcpt|xi|Thepricecorrectionoccursassoonastheaggregateorder imbalanceofallarbitrageursexceeds(tt0),with(reducedformfrombehavioralagents)
(tt0) =0[1(1/ )(tt0)][Ifthetradingorderexceeds,there isarandomizationofthepriceatwhichordersareexecuted]Motivation: The longerthemispricingpersists,thesmaller isthemassofbehavioraltradersthatremainconfidentthattheprice isright Sincetherearenopricechanges,arbitrageurscannot infert0 fromthemwhilepressure isbelow(.)
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Market Timing and Delayed Arbitrage
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MarketTimingandDelayedArbitrageArbitrageurti specifiesatradingstrategyasfunctionofi =tti. A-Bfocusontriggerstrategiessuchthatthearbitrageursetsxi =0untiladateti +i andxi =1afterthat(untilthecorrectiontakesplace)Anarbitrageurthattradesjustbeforethecorrectionachievesthehighestpayoff. Bypostponingthetradehereducesholdingcostsbutrisksmissingthearbitrageopportunity(Keynes: beatthegun terminology)Leth(t|ti)bearbitrageurtisperceivedhazardratethatthepricecorrectionoccurs inthenext instantt. Thus,hisestimateofacorrection inthenext(small)time interval ish(t|ti),whiletheholdingcost iscpt
Thusthearbitrageurwillonlytrade iftheexpectedbenefitpth(t|ti)exceedstheexpectedcostofholdinganondiversifiedportfolio(1h(tti))cpt
|Ofcoursethehazardratedependsonotherarbitrageurs tradingstrategies.A-Brestrictattentiontosymmetrictriggerstrategyequilibria(basedonEMAarticle)
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Abreu-Brunnermeier: Market Timing and Delayed
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Ifallarbitrageurstradewithadelay,thenthepricecorrectionoccursatt0+(),wherethe latter isdefined implicitlyfrom
( ) = +(())
Usingthe linearexpressionfor(.),wehave+
() = 0 +0
Abreu-Brunnermeier: MarketTimingandDelayedArbitrage
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Market Timing and Delayed Arbitrage
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Arbitrageurti knowsthat, inequilibrium,thepricecorrectionwilloccurnolaterthanti +()butafterti +()Giventhepriordistributionont0,the latterobservationyieldsasimpleposterior:
0 for t
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MarketTimingandDelayedArbitrage
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bh(ti + t|ti)
c
* j
Arbitrageur ti's Hazard Rate
Image by MIT OpenCourseWare.
Final Remarks
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FinalRemarks
Arbitrageis
delayed.
This
is
possible
because
mispricing
is
never
common
knowledge,whichpreservesthedisagreementaboutthetimingofpricecorrections
Thearbitrageurwhobecomes immediatelyawareofthemispricingatt0 knowsthatatt0+ everybodyknowsaboutthemispricing. However,thetraderwhoonlybecomesawareatt0+ thinksthathemightbethefirsttohearofitandhedoesnotknow thatalltradersalreadyknow it. Hence,even ifeverybodyknowsofthemispricingatt0+,onlythefirsttraderknowsthateverybodyknowsAtt0+2,eventhe lasttraderknowsthateverybodyknows,buthedoesnotknowthateverybodyknowsthateverybodyknowsofthemispricing,andsoon
Themaindistinctionwithnoise-tradersisthatmostoftheactioncomesfromtherationaltraders. It istheuncertaintyaboutthebehaviorofotherrationaltradersthat leadstodelayedarbitrage
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14.454 Economic CrisesSpring 2011
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