P aper No. 907 Banks, shadow banks, and business c

2ASimpleModelofSecuritization

Theeconomylastsfortwodates,t=1,2,andispopulatedbythreetypesofagents, banks,shadowbanks,andinvestors.Wedescribetheminturn.

4ProminentexamplesincludeSmetsandWouters(2003,2007),Justiniano,Primiceri,andTambalotti (2010),IacovielloandNeri(2010),JermannandQuadrini(2012),Schmitt-GrohéandUribe(2012),Liu, Wang,andZha(2013),Christiano,Motto,andRostagno(2014),Ajello(2016),andBloometal.(2018).

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2.1Banks

frominvestorsinperiod=1.Banksuse

AmeasureoneofbankscollectdepositsD t
thedepositstoextendmanyloansBtotherealsectoroftheeconomy,whichincludes householdsandfirms.LoansarebackedbycollateralKcorrespondingtorealestate orphysicalcapital.ThemarketpriceofcollateralisQ.WeassumeKisincreasingin B,butconcave;inparticular,K(B)=B^γ,withγ∈[0,1].Thiscapturestheideathat

asbankslendmore,theyareentitledtomorecollateral,butthereisafiniteamountof goodcollateralintheeconomy,hencethedecreasingreturns.

Depositsaretotallysafeandpaytherisk-freerateRinperiodt=2.Bycontrast, loansarerisky.Int=2,theyrepayR^bwithprobability1−panddefaultwithprobability

p.Incaseofdefault,banksseizethecollateralwhosevaluedependsonabank-level idiosyncraticshockω.ThisshockisundiversifiablebythebankandtransformsQK unitsofcollateralintoωQKeffectiveunits.Letω_minbetheminimumpossiblevalue

ofω,knownbythebankint=1.

NoSecuritization.—Considerfirstthecasewherebanksdonotsecuritize.Lending tomanyborrowersturnstheexanteindividualdefaultriskintoaknownfractionof defaultingborrowers.Thatis,abank’sexpectedprofitinperiodt=2is

E[Π^ns]=(1−p)R^bB+pE[ω]QK(B)−RD.

Toensureitalwaysrepaysitsdepositors,thebankmustnottakeonmoredeposit liabilitiesthanitsassetsareworthundertheworst-casescenario

(1−p)R^bB+pω_minQK(B)≥RD.

SettingthissolvabilityconstrainttoequalityandusingthebalancesheetB=D,we obtainthemaximumamountoflendingundertheno-securitizationscenario

ns=□_{R−(1−p)Rb}□_γ−¹

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B .

pω_minQ

andobtainoptimallending,B_ns_⋆=□_{R−(1−p)Rb}□_γ−¹

Toseewhetherthesolvabilityconstraintbinds,maximizeprofitsubjectonlytoB=D

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_pγE[ω]Q .Assumingω_min<γE[ω],wehave

B^ns⋆>B^ns.Therefore,thebankalwayspreferslendingmorethanitisallowedto,ie

theconstraintbinds.

WithSecuritization.—Supposethatinperiodt=1eachbankhasthepossibilityto sellpartofitsloanbooktoashadowbankatpriceQ^˜>ω_minQ.Thisenablesthebank
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tooffloaditsidiosyncraticriskandguaranteeinsteadasaferevenuestream.Underthis arrangement,thebank’sexpectedprofitint=2writes

E[Π^s]=(1−p)R^bB+pQ^˜K(B)−RD. (1)

UsingB=Dandmaximizingprofit,weobtain

Bs=□_{R−(1−p)Rb}□_γ−¹

1

. (2)
pγQ^˜

TheconditionR−(1−p)R^b>0ensuresthatprofitandlendingarebounded.Next,as longasE[ω]≤Q^˜/Q,wefindthatforany{ω_min,γ}∈[0,1),expectedprofitislarger

whenbankssecuritize.5Therefore,itisalwaysintheinterestofthebanktoengagein securitization.Thisleadsustoourfirstresult.