caballero on bubbles

7/28/2019 Caballero on Bubbles

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BubblesMacroeconomicsIV

RicardoJ.CaballeroMIT

Spring2011

R.J. Caballero (MIT) Bubbles Spring 2011 1 / 29


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1

2

3

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


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


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)


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)


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


The Setup


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TheSetup


(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(.)


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)


Abreu-Brunnermeier: Market Timing and Delayed


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Ifallarbitrageurstradewithadelay,thenthepricecorrectionoccursatt0+(),wherethe latter isdefined implicitlyfrom

( ) = +(())

Usingthe linearexpressionfor(.),wehave+

() = 0 +0

Abreu-Brunnermeier: MarketTimingandDelayedArbitrage


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


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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