Designing a Main Street Lending Facility

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. Designing a Main Street Lending Facility Alexandros P. Vardoulakis 2020-052 Please cite this paper as: Vardoulakis, Alexandros P. (2020). “Designing a Main Street Lending Facility,” Finance and Economics Discussion Series 2020-052. Washington: Board of Governors of the Federal Reserve System, https://doi.org/10.17016/FEDS.2020.052. NOTE: Staff working papers in the Finance and Economics Discussion Series (FEDS) are preliminary materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. References in publications to the Finance and Economics Discussion Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers.

∗ Designing a Main Street Lending Facility AlexandrosP.Vardoulakis† June25,2020 Abstract Banksaddvaluebymonitoringborrowers. Highfundingcostsmakebanksreluctanttolend. A centralbankcaneasefundingbypurchasingloans,butcannotdistinguishwhichloansrequire more or less monitoring, exposing it to adverse selection. A multi-tier loan pricing facility arises as the optimal institutional design setting both the purchase price and banks’ risk retention for given loan characteristics. This design dominates uniform (flat) structure for loan purchases,providestherightincentivestobanksandachievesmaximumlendingatlowerrates tobusinesses. Boththemulti-tierandflatstructuresdeliverwelfaregainscomparedtonointervention,buttherelativegainbetweenthetwodependsonthreesufficientstatistics: theshareof loansrequiringmonitoring,therisk-retentionratio,andtheliquiditypremium. Keywords: MainStreet, centralbanklendingfacilities, monitoring, smallbusiness, sufficient statistics,COVID-19 JELClassification: E58,G01,G28 ∗IamthankfultoseminarparticipantsattheFederalReserveBoard,andtoLeventAltinoglu,DavidArseneau,Garth Baughman,FrancescaCarapella,Jin-wookChang,MattDarst,SebastianInfante,ElizabethKlee,DavidRappoport,and SkanderVandenHeuvelforhelpfulcommentsandsuggestions. Allerrorshereinaremine. Theviewsexpressedinthis paperarethoseoftheauthoranddonotnecessarilyrepresentthoseofFederalReserveBoardofGovernors,oranyonein theFederalReserveSystem. †BoardofGovernorsoftheFederalReserveSystem,UnitedStates;email:alexandros.vardoulakis@frb.gov

1 Introduction A distinguishing feature of the economic consequences of the COVID-19 pandemic is the total shutdown of large parts of the economy. Small businesses are expected to be disproportionally affected given the inability to market their products and services remotely, but also due to a likely lack of established credit lines. Moreover, small businesses usually do not have good—tangible and fairly liquid—collateral that would allow them to borrow more easily, which also raises the importanceofbankmonitoring. Inturn,theeffectivebankfundingcostsmaybeincreasingrapidly and banks may be unwilling to extend credit even to good businesses with low probabilities of default. The collapse in revenue and the inability to raise funds to cover working capital would likely forcesmallbusinessestoclosuresandlayoffswithadversesecond-roundeffectsfortheeconomyas whole. Policymakersandcommentatorshaverecognizedtheseissuesandtheneedtosupportsmall businesses(see,forexample,Bigio,2020,DrechselandKalemli-Özcan,2020). Onewaytoprovide liquidity support is for central banks to set up a lending facility that would buy from banks loans extended to small businesses. For example, the Federal Reserve has announced it is establishing a MainStreetLendingProgram.1 TheProgramwilloperatethroughthreefacilities: theMainStreet NewLoanFacility(MSNLF),theMainStreetPriorityLoanFacility(MSPLF),andtheMainStreet Expanded Loan Facility (MSELF). The first two are designed to facilitate/stimulate new loans to eligiblebusinesses,whilethethirdcoverstheupsizingofexistingloans. Thefacilitieswillpurchase at par value a uniform/flat participation for each category of Eligible loans, which also have a uniform/flatinterestrate(LIBOR+300basispoints). 2 Anotheralternativewouldbetohavepublicly owneddevelopmentbanksextendthenecessarylinesofcredit. Forexample,Germany’sstateowned development bank KfW provides unlimited access to loans.3 A central bank lending facility may 1The lending facilities established under section 13(3) of the Federal Reserve Act are meant to support liquidity fundingandtheflowofcreditinresponsetotheCOVID-19.Allofthesefacilitieshavebeenundertakenwiththeapproval oftheTreasurySecretary,andmanyofthemaresupportedbyfundingfromtheCARESAct.Assuch,theFederalReserve isprimarilyprovidingthenecessaryliquiditysupportwithoutintentionallyundertakingcreditrisk. Thesefeaturesare presentinthemodelherein. 2One difference between the MSNLF and the MSPLF is that the latter will purchase loans that have priority over other debt and that Eligible firms may have higher debt to EBITDA ratios (earnings before interest, taxes, depreciation, and amortization) at origination, which is an observable firm characteristic driving its riskiness. Both facilities will purchase a 95 percent (uniform/flat) participation in all Eligible Loans at par value. See https://www.federalreserve.gov/monetarypolicy/mainstreetlending.htmfordetails. 3See https://www.bundesfinanzministerium.de/Content/DE/Pressemitteilungen/Finanzpolitik/2020/03/2020-03-13download-en.pdffordetails. 2

provide more flexibility and can be quickly deployed (and rolled back when not needed), but its design should be such that banks maintain the incentives to continue monitoring borrowers after sellingtheirloanstothecentralbank. Asmentioned,thisisparticularlyimportantforsmallbusiness loans.4 Thispaperstudiesthedesignofalendingfacilitywithaspecialfocusonunobservableheterogeneityacrossfirmsratherthansimplydifferentriskprofilesorotherobservablefirmcharacteristics thatcanbeeasilyaccountedforinthepricingschemeofthefacility.5 Tostudytheoptimalfacility design, consider a stylized model where banks choose either to extend loans to entrepreneurs with no own capital and no collateral, or to invest in a storage technology. 6 Entrepreneurs have access to projects that deliver a positive net present value, but only if banks engage in monitoring for the whole term of the loan akin to Diamond (1984) and Holmström and Tirole (1997). Sufi (2007) showsthatmonitoringisimportantforsyndicatedloans. Gustafson,IvanovandMeisenzahl(2020) providefurtherempiricalevidenceaboutbankmonitoringforsyndicatedloansandshowthatabout 20percentofloansinvolveactivemonitoring,whichissoftinformationsimilartowhatthismodel assumes. They also show that monitoring takes place throughout the life of the loan with 55 percent of loans being monitored monthly or daily, and 30 percent annually. Arguably, the need for monitoringmaybeashighorevenhigherforthesmallerbusinessescoveredinthelendingfacility. Monitoringisvaluable,butitiscostlyforthebankandcannotbeobservedbyoutsiders. Banks value liquidity and, thus, may be unwilling to extend loans unless they anticipate to sell them in a secondary market at a price that covers the premium for liquidity. After origination, loan sales are possible if banks maintain a portion of the loan on their balance sheet such that they can credibly 4Thepaperfocusesoncentralbankfacilitiesthatprovidefundingforprivatelenderstoinducethemtolendtosmall businesses, butabstractsfor othertypesofpoliciesadvocated, such asfiscaltransfers(SaezandZucman, 2020). See Dreyer,McNamara,Nye,NygaardandSankar(2020)forasummaryofthevarioustypesofpoliciesemployedworldwide tosupportsmallbusinesses. 5Themodelhereinalsoderivesthepricingofloansandrisk-retentionrequirementsgivenobservable,orothereasy toinfer,firmcharacteristics. Butdoesnotnotuseestimateddistributionsofdefaultorotherdatatofullycomputethe loantermsthatlendingfacilitiesshouldset. Thefocusisoncomputingthewelfaregainsfromoptimallydesigninga facility that tackles the adverse selection arising from unobserved monitoring intensity. As it will be clear, just three sufficientstatisticsareadequateforthis.Nevertheless,themodel,anditsextensionintheappendix,couldalsobeuseful forcalculatingallloantermsusingasinputsdataaboutprobabilitiesofdefault,lossgivendefault,andotherobservable firmandbankcharacteristics.SeeEnglishandLiang(2020)foranattempttoderiveloanpricingtermsusingobservable firmcharactersticsandinferredprobabilitiesofdefault,butwithnomentionoftheroleofmonitoring,whichiscentral herein. 6Thebaselinemodelcoversthecaseoflendingagainstfirms’cash-flowswithsameobservablecharacteristicsand probabilitiesofdefault. SeetheAppendixforanextensionofthemodeltofirmswithdifferentdefaultprobabilitiesand loanssecuredbycollateralwithimplicationsforasset-basedlendingfacilities. 3

committocontinuemonitoringborrowers,asshowninGortonandPennacchi(1995)andPennacchi (1988). In a systemic crisis, all banks may suffer from the same shock, which pushes up their cost of funding and deters them from lending. At the same time, selling loans to outside investor without the ability to monitor may not be feasible even if banks retain a big portion of the loan to signal their commitment to continue monitoring. A central bank can step in and provide the required funding/liquidityatalowcost,butalsoneedstomakesurethatbankscontinuetomonitortoavoid bad behavior rendering loans non-performing down the road. In turn, this requires banks to retain a portion of loans and be adequately compensated for monitoring. If all small businesses require equalmonitoring,thiswoulddictateaflatstructurespecifyingthesamerisk-retentionrequirements and interest rates for all loans to firms with the same observable characteristics (for example, the MSNLFandtheMSPLFapplytofirmswithdifferentdebttoEBITDA,buttheydonotdistinguish amongfirmswiththesamedebttoEBITDA). The novelty of this paper is that entrepreneurs are heterogeneous in the monitoring intensity required. Contrary to banks, the central bank cannot distinguish the entrepreneurs that require monitoringfromthosethatdonot. Importantly,therequiredmonitoringintensitydoesnotcorrelate inasystematicwaywithobservablefirms’characteristicsand, thus, itcannotbeinferred.7 Hence, thecentralbankfacesadverseselectionifitnegotiatesthetermsofloanpurchaseswitheachbank separately,orifitcommitstouniform/flattermsforallloans. Ifitoffersalowerpricethanwhatis requiredtocompensatebanksformonitoring,bankswillbetemptedtooffloadmonitoring-intensive loansandstopmonitoringthereafter. Ifitoffersapricecompensatingformonitoring,bankswillbe temptedtoalsosellloansthatdonotrequiremonitoring,extractingextrapayment. Theoptimaldesignconsistsofamulti-tierpricingschemethatthecentralbankfacilitycommits to. The scheme sets differential purchase prices, loan rates and risk-retention ratios restoring the incentivesofbankstodifferentiatebetweendifferenttypesofentrepreneursandcontinuemonitoring. Inotherwords,loanstofirmswiththesameobservablecharacteristics,suchastheprobability ofdefault,couldrequiredifferentialrisk-retentionratios. Thisdesignachievesthehighestliquidity supporttobusinessesatthelowestfeasibleratesmaximizingthesurplustotherealeconomy. 7Gustafson, IvanovandMeisenzahl(2020)arguethatthereislittleevidenceofasignificanttradeoffbetweentheir monitoringfrequencymeasureandeitherloanamountorloanspreadsdespitethefactthat,intheirdata,loanswithhigher creditspreadstendtobemonitoredlessoften. 4

Alternatively,thecentralbankmaynotattempttoaccountfortheunobservableborrowerheterogeneityintermsofmonitoringintensityand,instead,offerauniform/flatrisk-retentionrequirement forallloans. Thewelfaregainoftheoptimaldesigncomparedtoauniform/flatschemeforallloans depends on three sufficient statistics: the liquidity premium, the (appropriately chosen) flat riskretentionratio,andtheshareofloansthatrequiremonitoring. Therisk-retentionratioisaproxyfor themonitoringcost, whiletheliquiditypremiumproxiesfortheincrementalcostofbankfunding. Intuitively, the gain is increasing in the liquidity premium and the monitoring cost, and decreasing in the share of loans requiring monitoring. The reason is that the optimal design avoids compensating banks for loans that do not require monitoring, while at the same time all of these loans are sold to the central bank eliminating the liquidity premium banks would demand to hold a portion ofthemontheirbalancesheets. Forempiricallyplausiblevalueforthesethreesufficientstatistics, thewelfarelossfromusingauniform/flatschemewithaflatrisk-retentionrequirementrangesfrom 0.02%to0.31%ofthetotalfacevalueofloanpurchasesbythecentralbank. Nevertheless,theoperationalcomplexitiesofimplementingtheoptimaldesignmaybedaunting. In that case offering a uniform/flat scheme may not provide all the right incentives to banks and mayberelativelymorecostlyforbusinesses,butitisstillpreferredthannotprovidinganyliquidity support. Moreover,forvaluesforthesufficientstatisticsobservedintheoutbreakofthetheCOVID- 19crisis,thewelfarelossisrelativelysmall. Thus,uniform/flatpricingcouldappearasanattractive practicalsolution. Therestofthepaperproceedsasfollows. Section2presentsthemodelandtheprivateequilibriumintheabsenceofcentralbankintervention. Section4derivestheoptimalinstitutionaldesign, while section 5 estimates the welfare cost from using uniform/flat structure using a minimal set of sufficientstatistics. Section6discussessomepracticalimplementationissuesandconcludes. 2 Model Theeconomyhasthreedates,t =0,1,and2,andispopulatedbyacentralbankandacontinuumof threetypesofagents: entrepreneursoftype1withmassm ,entrepreneursoftype2withmassm , 1 2 andbankswithmass1. Allagentsareriskneutralandhaveatimediscountfactorof1. Eachentrepreneurofeithertype1ortype2isendowedwithaprojectthatrequiresaninvestment 5

of1att =0andgeneratesatt =2apayoffR>0inthecaseofsuccessand0inthecaseoffailure. Projectsyieldnothingifthereareliquidatedearlyatt =1. Entrepreneursdonothavefundsoftheir own and need to borrow from banks. For simplicity, banks have a fixed weighted average cost of capital,normalizedto1,andtotalresourcese≥m +m att=0. Thedifferencebetweentype1and 1 2 type2entrepreneursaccruesfromamoralhazardproblem. Type1entrepreneurshaveaprobability ofsuccessθirrespectiveofbeingmonitoredornot,withθR>1. Type2entrepreneursmaychoose to shirk, which destroys the firms’ value and reduces the probability of success from θ to θ˜, with θ˜R < 1, i.e., resulting in a negative net present value.8 Shirking can be avoided by continuous (active)bankmonitoring,whichintroducesanadditionalcostX ∈(0,θR−1). Moreover,apooling equilibriumwithlendingtoallentrepreneursandnomonitoringfortype2onesisnotfeasible,i.e., (m θ+m θ˜)R<1.9 For simplicity, banks know what entrepreneurs are of type 1 and type 2, but 1 2 thecentralbankdoesnot. Nooneknowswhetheraprojectpaysoffuntilperiod2.10 Apart from lending to entrepreneurs att =0, banks may choose to invest some resources in a riskless one-period storage technology with zero net yield. At t =1 all banks receive with probability q, and at the same time, an investment opportunity that yields a riskless payoff K ≥1 per unit of funds invested. Importantly, all banks receive this opportunity at the same time, so interbanklendingisnothelpful,andtheyareeitherunabletoobtainoutsidefundingoritscostishigher thanK. Thus,thisstateresemblesasystemicfunding/liquidityshortageandtheonlywaytoinvest in the new technology is to carry over funds from t =0 using the storage technology. Instead of receivinganinvestmentopportunity,bankscouldsufferliquidityshocksorfacebindingregulatory constraints,withKbeingtheshadowcostofliquidityortheshadowvalueofrelaxingtheconstraint. The exact nature of the demand for liquidity is not important; the systemic nature of the liquidity shortageis. Thedecisiontolendtoentrepreneursorhoardliquiditytoinvestinthenewtechnology willdependontheirrespectiveexpectedreturns. Hence,theopportunitycostoflendingadollarat 8For simplicity, entrepreneurs are assumed to be identical aside from the moral hazard problem. In practice, entrepreneursmaydifferinotherdimensionsaswell.Forexample,theymayhavedifferentprobabilitiesofsuccessθ.Such additionalheterogeneitydoesnotimpacttheresultsinthepaperasshowninthemodelextensionintheAppendix. 9Alltheresultsinthepapergothroughevenif(m 1θ+m 2θ˜)R>1,but(θ−θ˜)R>X,i.e.,monitoringaddsvalue. 10Themodelabstractsfromotherfrictionsthatmayimpedetheabilitytosellloans. Forexample,ParlourandPlantin (2008)assumethatmonitoringloansbeforetheyaresoldrevealsinformationabouttheirperformance,whichintroduces anadditionaladverseselectionproblem. Arguably,thisfrictionwouldbemorerelevantforexistingloanrelationships with banks (Main Street Expanded Loan Facility) compared to newly issued loans (Main Street New Loan Facility). Extendingthemodelinthisdirectionwouldbeinteresting. Note,however,thatthecentralbankalreadyfacesadverse selectionbecauseoftheheterogeneityintherequiredmonitoringintensityevenifbanksdonothavesuperiorinformation abouttheprobabilitiesofsuccessofdifferentborrowers. 6

t=0is1+q(K−1)irrespectiveofthenatureoftheliquidityshock,whereq(K−1)istheliquidity premium.11 Moregenerally,1+q(K−1)couldbethoughtastheshadowvalueofadollaratt =0,withthe realizedshadowvaluebeing1withprobability(1−q)andK withprobabilityq. Underthisgeneral interpretation the model can be applied to a situation where the cost of capital increases to K with probabilityqinducingbankstorequirehigherreturnstoextendloans. Thus,banksmaybeunwilling toextendcredittopositivenetpresentvalueprojects,becauseofbindingfinancialfrictions,tighter balance sheet constraints, or elevated aversion to aggregate risk. Although the micro-foundations forthesealternativefrictionsdiffer,thebasicprincipleandqualitativeresultsinthepapercontinue to hold, as long as the central bank can alleviate these constraints and/or has higher tolerance for aggregate risk. However, the quantitative results may be markedly different and the welfare gain fromimplementingtheoptimaldesignmaybemuchhigher,asdiscussedinsection5. Finally,notethattherearenorestrictionsonqandforq→1theaggregateshockhasmaterializedandtheneedforliquidity/centralbanksupportisimminent,whichmanymaybelievehasbeen thesituationaftertheoutbreakoftheCOVID-19pandemic. Thevalueofqisnotimportantforthe qualitative and quantitative results (given the use of sufficient statistics). However, it is important that banks know at t =0, when setting the terms for the new loans, that the central bank will intervene with probability q. Also, time periods do not need to have the same length. For example, the time between t =0 and t =1 can be several months, while the time between t =1 and t =2 can be several years. For example, the Main Street Lending Facilities of the Federal Reserve will purchaseloansthatareoriginatedbetweenApril24,2020andSeptember30,2020,andhaveamaturityoffouryears. Despitebeingstylized,thisframeworkhasthenecessaryingredientstojustifya centralbanklendingfacilitythatbuysbankloans. Amoreelaborateframeworkwouldendogenize thereasonswhybankscannotraiseenoughfunds, fromexample, similartoHolmströmandTirole (1998). Figure 1 presents the timeline of the model where the role of the central bank is suppressed. Optimal central bank intervention is discussed in section 4 after the private equilibrium in section 11Notethat,forsimplicity,theoccurrenceofthesystemicshockdoesnotaffecttheprobabilitythatprojectssucceedat t=2. Onemightexpectthatthesystemicshockwouldreducethisprobabilityfromθtosomeθ(cid:48). Aslongastheproject continuestohavepositivenetpresentvalue,i.e.,θ(cid:48)R>1,alltheresultsinthepapergothrough.Naturally,ifθ(cid:48)R<1,the centralbankwouldnotinterveneasitwouldviolateitsnolossconstraintbybuyingloansofinsolventfirms. 7

3. Notethatthemonitoringtakesplacebetweent=1andt=2tocapturethefindingofcontinuous activemonitoringinGustafsonetal. (2020). t=0 t=1 t=2 Banksdecidewhethertolendtotype Liquidityshockmaterializesornot Projectssucceedorfail 1and/ortype2entrepreneurs Banksinvestinnewtechnology Entrepreneursrepaytheirloans Bankssettype1and2loanrates ifshockhits Allagentsconsume Banksstoreremainingfunds Banksusestorageifshockdoesn’thit Banksmonitortype2loans Figure1: ModelTimelinewithoutCentralBankFacility 3 Private Equilibrium Suppose that θR≥1+q(K−1)+X, i.e., the payoff from lending to either type 1 or type 2 entrepreneurs is higher than the payoff from hoarding liquidity and investing in the new technology if it arrives at t =1, accounting for any monitoring costs. Given perfect competition, banks will lend to both types at competitive rates Rˆ = (1+q(K−1))/θ and Rˆ = (1+q(K−1)+X)/θ, 1 2 respectively. Moreover, they will invest the remaining resources e−m −m in the storage tech- 1 2 nology. Equivalently, for θR∈[1+q(K−1),1+q(K−1)+X), banks will voluntarily lend only to type 1 entrepreneurs and invest the remaining resources, e−m , in the storage technology. For 1 θR<1+q(K−1) banks are unwilling to lend to either type 1 or type 2 entrepreneurs. This case represent a systemic liquidity crisis in the sense that there are positive net present value projects available at t = 0, but banks choose to hoard their existing funds. The next section derives the optimalinstitutionaldesigntodealwiththisextremesituation. 4 Central Bank Lending Facility Thecentralbankcansetupafacilitytoprovidebankswithliquidityinordertosupportlendingto businesses. In particular, the central bank sets a Special Purpose Vehicle (SPV) that buys business loansfrombanksatt =1ifandwhentheliquidityshockhits,whichiscommonknowledge. Theobjectiveofthecentralbankistomaximizethesurplusaccruingtoentrepreneurs,without incurring any losses in its loan portfolio, making sure that banks are willing to intermediate, and 8

maintaining bank incentives for monitoring.12 The pricing of the loans by the SPV is important to provide the right incentives to banks such that the SPV, first, achieves its purpose of supporting business lending for positive net present value projects, second, it does not overpay on aggregate, which would constitute a direct transfer of surplus from businesses to banks, and, third, that monitoring takes place. The main difficulty that the SPV faces is that it does not know whether it will bebuyingtype1ortype2loans,sincethisisprivateinformationtobanks. Hence,thecentralbank facesadverseselection. Figure2presentstheaugmentedtimelinewiththecentralbank. t=0 t=1 t=2 SPVsetstermsforpurchasingloans Liquidityshockmaterializesornot Projectssucceedorfail Banksdecidetolendtotype1and BankssellsloanstoSPVandinvest Entrepreneursrepaytheirloans type2entrepreneurs innewtechnologyifshockhits Allagentsconsume Banksstoreremainingfunds Banksusestorageifshockdoesn’thit Banksmonitortype2loansgiven adequateincentives Figure2: ModelTimelinewithCentralBankFacility There are three problems that the SPV needs to resolve. First, consider that the SPV does not condition pricingterms toloan interestrates thatcan help distinguishtype 1and type2 loans—or, equivalently, commits to a uniform/flat structure for all loans. If banks are not compensated for monitoring, they would be tempted to sell to the SPV type 2 loans pretending that they are type 1, andnotengageinmonitoringthereafter. Thisisundesirablebecausetheaggregateloanpoolhasa negativenetpresentvalueiftype2loansarenotmonitored. If,instead,banksarecompensatedfor monitoringtype2loans,theywillbetemptedtoselltype1loansatthosetermsaswell,extracting a higher surplus. Second, there is no guarantee that banks will actually monitor type 2 loans even if they are offered a payment that compensates them for it. Third, the central bank needs to price loanssuchthatitdoesnotlosemoneyonitsaggregateportfolio. An SPV design featuring two pricing schemes for loan purchases can resolve the three aforementioned problems and achieve maximum lending support to businesses. The pricing schemes definethepricepaidforloanswithcertainloanratesaswellastherisk-retentionratio. Importantly, 12Recallthattheprojectsofbothentrepreneurialtypeshavepositivenetpresentvalue(ifmonitored), solendingto bothtypesissociallyoptimal. 9

theSPVpre-commitstothispricingschemeatt =0. ThefirstpricingschemesetsapriceP thatthebankiswillingtopayfortype1loanswithrate 1 R and zero risk-retention. The price offered to banks should make them willing to extend credit 1 insteadofhoardingliquidityatt =0,i.e.,thefollowingparticipationconstraintshouldhold: (1−q)θR +qP K ≥1+q(K−1). (1) 1 1 As shown later, the loan rate R will be chosen such that the SPV does not lose money on its 1 aggregateloanportfolio. ThesecondpricingschemesetsapriceP andrisk-retentionratiosγ fortype2loanswithrate 2 2 R . TheSPVchoosesthetermssuchthatbanks(i)usethisschemetosellonlytype2loans,(ii)have 2 anincentivetocontinuemonitoringaftersellingsomeloanstotheSPV,and(iii)haveanincentive to extend lending to type 2 entrepreneurs. We show how these conditions determine the terms in reverseorder. Banks will lend to good type 2 entrepreneurs att =0, anticipating that they can sell a portion 1−γ totheSPVatpriceP att =1withprobabilityq, ifthefollowingparticipationconstraintis 2 2 satisfied:13 (1−q)θR +q(γ θR +(1−γ )P K)≥1+q(K−1)+X. (2) 2 2 2 2 2 Bankswillhavetheincentivetocontinuemonitoringaftersellingaportionoftheirtype2loan portfoliototheSPVatt =1ifthefollowingincentivecompatibilityconstraintissatisfied: γ θR ≥X, (3) 2 2 Banks should not have an incentive to misrepresent type 2 loans for type 1 loans (and vice versa). RecallthattheSPVpaysP andsetzeroriskretentionforloanswithobservablerateR . So, 1 1 banks could restructure type 2 loans before selling them to the SPV to change the rate from R to 2 R , which entrepreneurs would be happy to accept, and sell the whole loan to the SPV at price P . 1 1 13Multiplying(1)bym and(2)bym ,andaddingthem,onecanobtaintheaggregateparticipationconstraint. 1 2 10

Bankswilltruthfullyrevealtype2loansifthefollowingconstraintholds:14 γ θR +(1−γ )P K−X ≥P K. (4) 2 2 2 2 1 Banksmayalsobetemptedtomisrepresenttype1loanstoreceiveahigherpaymentaccountingfor a monitoring cost that they do not need to incur. But, they are unable to do so because this would requirerestructuringtype1loanstochangetheratefromR toR , whichentrepreneurswouldnot 1 2 accept(recalltheSPVoffersP forloanswithobservablerateR ). Equivalently,bankswillnotbe 2 2 able to issue type 1 loans at rate R att =0, because there is a profitable deviation of offering R 2 1 andattractingalltype1entrepreneurs. Finally,theSPVneedstoguaranteethatitsaggregateportfoliodoesnotlosemoney. Underthe twopricingschemesdescribedabove,bankswillextendloanstoalltype1andtype2entrepreneurs, i.e.,therequirementbecomes m (θR −P )+m (1−γ )(θR −P )≥0. (5) 1 1 1 2 2 2 2 Formally,theSPVchoosesterms(R ,P )and(R ,P ,γ )tomaximizeanobjectivefunction 1 1 2 2 2 W =m θ(R−R )+m θ(R−R )−L, (6) 1 1 2 2 subject to constraints (1)-(5). L >0 represents an unmodeled loss to the central bank and could potentially be associated to reputational concerns from intervening in credit markets. Thus, the centralbankwillinterveneonlyifthesurplustotherealeconomyishighenoughtojustifyincurring thelossandwoulddefinitelynotintervenewhentheliquiditypremiumislowandbanksvoluntarily lend to entrepreneurs. Note that L is not a pecuniary loss from the loan portfolio given constraint (5).15 Because of perfect competition in the banking sector, the SPV chooses P and P such that (1) 1 2 14Notethatbankswillnotwanttoissuetype2loansatrateR giventhattheycanrestructureatt=1,becausefor 1 q<1,thereisachancethattheydon’tsellthemtotheSPV.But,evenforq=1,theywouldnotissuetype2loansatrate R ,andtheywilltruthfullyrevealtheirtype,if(4)holds 1 15NoneoftheresultsdependonthelossL,whichcanbeeneasilysettozero. Thoughinthatcasethemodelwould predictthatthecentralbankshouldalwaysintervene,evenwhentheliquiditypremiumisverylow,because,contraryto banks,thecentralbankdoesnotrequirecompensationforitandcanofferbetterloanrates(seecorollary1below). 11

and (2) hold with equality, i.e., it pays the lowest possible price to banks. Moreover, (3) and (4) hold with equality because higher risk retention makes banks require a higher payment without improvingfurthermonitoringincentives. Finally,(5)alsobindbecausetheSPVwouldratheroffer entrepreneursbetterratestomaximizetheirsurplusthanmakingapositiveprofit. Altogetherthese yield a closed-form solution for the the optimal pricing schemes characterized in the following proposition. Proposition 1. For θR<1+q(K−1) and (m +m )(θR−1)−m X >L the central bank inter- 1 2 2 venes. The optimal SPV design involves two distinct pricing schemes: A price P = 1 for loans 1 withrateR =1/θandzerorisk-retentionrequirements,andapriceP =1+X forloanswithrate 1 2 R =(1+X)/θ and risk-retention requirement γ =X/(1+X). Banks will choose to sell type 1 2 2 andtype2loanstothefirstandsecondschemes,respectively. Thisdesignprovidestherightincentivetobankstotruthfullyreportthetypeofloanandachieves maximumlendingtoentrepreneurs. Banksselltype2loansatthesecondscheme,retainaportion, and maintain the incentive to monitor, while they fully sell type 1 loan at the first scheme. But, as the following corollary shows, the central bank facility is also allowing businesses to borrow at thelowestratespossiblemaximizingtheirsurplus,becausebanksarecompensatedfortheliquidity premium. Corollary1. Loanratestobothtype1andtype2entrepreneursarelowerundertheoptimalSPV design compared to the competitive rates in normal times by an amount equal to the liquidity premiumq(K−1). Now, suppose that the SPV does not condition pricing terms to loan interest rates. Instead, it setsatt=0auniform/flatpricingscheme,P∗,forallloanswiththesameloanrate,R∗,alongwitha uniform/flatrisk-retentionratio,γ∗,tomaintaintheincentivesformonitoring. Giventhatmonitoring is necessary only for type 2 entrepreneurs, banks will be willing to lend att =0 to a pool of type 1 and type 2 entrepreneurs at the same rate if the following participation (aggregate) constraint is satisfied: (1−q)θR∗+q[γ∗θR∗+(1−γ∗)P∗K]≥1+q(K−1)+m /(m +m )X. (7) 2 1 2 12

Moreover, banks will maintain incentives to monitor type 2 loans after selling a portion att =1 if thefollowingincentivecompatibilityconstraintissatisfied: γ∗θR∗≥X. (8) Finally,theSPVpaysbanksapricethatresultsinnolossesontheloanportfolio,i.e., (m +m )(1−γ∗)(θR∗−P∗)≥0 (9) 1 2 TheSPVchooses(R∗,P∗,γ∗)tomaximize W∗=(m +m )θ(R−R∗)−L, (10) 1 2 subjecttoconstraints(7)-(9). Asabove,allconstraintsbindinequilibriumyieldingtheuniform/flatpricingschemedescribed inthefollowingproposition. Proposition2. Auniform/flatpricingschemesetspriceP∗=1+X[m /(m +m )+q(K−1)]/[1+ 2 1 2 q(K−1)]forloanswithrateR∗=P∗/θ,andrequiresbankstoretainγ∗=X/P∗. Thefollowingpropositioncomparestheoptimaltwo-tieranduniform/flatpricingschemes. Proposition 3. Compared to the optimal multi-tier pricing scheme, 0 < (m +m )θ(R−R∗) < 1 2 m θ(R−R )+m θ(R−R ), i.e., the surplus to entrepreneurs is smaller under the uniform/flat 1 1 2 2 pricingscheme. Thesurplusgainisequaltom q(K−1)X/(1+q(K−1)). 1 Proposition3showsthattheoptimaltwo-tierpricingsystemispreferredtouniform/flatpricing. The gain in surplus/welfare is increasing in the number of type 1 entrepreneurs, the cost of monitoring,andtheliquiditypremium. Thereasonisthattheoptimaldesignavoidscompensatingbanks for loans that do not require monitoring, while at the same time all of these loans are sold to the central bank eliminating the liquidity premium banks would demand to hold a portion of them on theirbalancesheets. Uniform pricing also delivers economic benefits and should be preferred to taking no action. Note, however, that uniform pricing also needs to be accompanied by a positive risk-retention re- 13

quirement; otherwise, monitoring for type 2 loans will not take place and the overall pool of loans purchasedbytheSPVwillhavenegativenetpresentvalueas(m θ+m θ˜)R<1.16 1 B 5 Sufficient Statistics for the Welfare Gain In practice, central banks may not know what is the cost of monitoring for all the different banks usingthefacility, whichis akeyparameterinthemodel. However, itcouldbepotentiallyinferred byarelationshipofthetypedescribedincondition(3)or(8). Inotherwords,policymakersmayuse thevoluntaryrisk-retentionratiosonpastloanstogaugetheshadowcostofmonitoringaccounting for all other loan characteristics. For example, Sufi (2007) shows that the lead share—share of a loanheldbytheleadbankinsyndicate—isaproxyformonitoring, andisonaverage28.5percent (23.5 percent median) in his database. Alternatively, the 5 percent risk-retention requirement for issuersofAssetBackedSecuritiesinDodd-Frankcouldbeusedtocalibrateγinthemodel.17 Other parameters could also be calibrated using existing empirical studies. The share of type 2 entrepreneurs could be calibrated to 20 percent, i.e., equal the percentage of syndicated loans actively monitored in Gustafson et al. (2020). Moreover, the liquidity premium q(K−1) captures the additional return from investing the available fund in the risk-free, short-term, storage technology and having them available for use att =1. In other words, the price of a risk-free, long-term (two period), discount bond would be 1/(1+q(K−1)). As such, the liquidity premium can be approximatedbythespreadbetweentheone-monthovernightindexswap(OIS)andthefour-week Treasury bill (T-bill)—a spread known as the convenience yield (see, Infante, 2020; Cashin et al. 2017). Given that T-bills are publicly produced short-term safe assets, trading with very narrow bid-ask spreads, and the OIS is merely a contractual agreement promising a risk-free payoff, the spreadmeasurestheconvenienceofholdingsafeassets. Alternatively,theliquiditypremiumcould be approximated by the spread between the General Collateral (GC) repo rate and the (4-week) T-bill (Nagel, 2016) or the spread between the 1-week Treasury repo rate and the (4-week) T-bill, becausethesereporatesarefreeofriskastheyarebackedbysafecollateral,butinvestinginthem isilliquidduringthetermoftheloan.18 Figure3plotsthemonthlyaveragesforthesethreeproxies 16Naturally,ifmonitoringisnotessentialbutuseful,i.e.,(m 1θ+mBθ˜)R>1and(θ−θ˜)R>X,thenuniformpricing withnorisk-retentionrequirementsisviable,yetstillnotoptimal. 17https://www.govinfo.gov/content/pkg/FR-2014-12-24/pdf/2014-29256.pdf. 18Nagel(2016)discussedadditionalproxiessuchasthespreadbetweentheCDsrateandtheT-bill,thespreadbetween 14

Figure3: EstimatesfortheLiquidityPremium fortheliquiditypremium. Thefollowingcorollaryrecaststhesurplus/welfare(percentage)gainintermsoftheaforementionedsufficientstatistics: therisk-retentionrequirement,γ∗,theshareoffirmsrequiringmonitoring m¯ ≡m /(m +m ),andtheliquiditypremium,r ≡q(K−1),usingtheequilibriumconditions. 2 1 2 (cid:96) Corollary 2. The percentage welfare gain of using the optimal design compared to a uniform/flat pricingschemeisgivenby (1−m¯)γ∗r (cid:96) %WelfareGain= . 1+(1−γ∗)r −γ∗m¯ (cid:96) Note that the flat risk-retention requirement, γ∗, is a sufficient statistic, which means that the corollarycanbeusedtoevaluatethewelfarelossfromtheuniform/flatpricingschemeobservedin actuallendingfacilities. theoff-the-runandon-the-runTreasurenotes,andthespreadbetweentheoff-the-runTreasurynotesandtheT-bill.These measuresadditionallycaptureeithersomecreditrisk(CDs)orapremiumforimperfectmarketliquidity(Treasurynotes). 15

Table 1 reports the welfare gain for different levels of the liquidity premium and risk-retention requirement when the share of loans that require monitoring is set to 20 percent. For a 5 percent flatrisk-retentionrequirementandaliquiditypremiumclosetotheonesuggestedbytheCOVID-19 spike in the three liquidity measures, the gain is minuscule. But even for a higher liquidity premiumclosetowhatwasobservedduringtheFinancialCrisisof2007-2008andaflatrisk-retention requirementcalibratedclosetotheleadshareinSufi(2007),thewelfaregainsarerelativelysmall. Hence,uniform/flatpricingmaynotbeoptimal,butitdoesnotimplyabigwelfareloss. γ r 0.05 0.15 0.25 (cid:96) 0.50% 0.02% 0.06% 0.10% 1.00% 0.04% 0.12% 0.21% 1.50% 0.06% 0.18% 0.31% Table 1: Welfare gain of optimal design relative to uniform/flat pricing for different levels of riskretentionrequirementsandliquiditypremium. Theshareofloansthatrequiredmonitoringissetto 20percent. The model interprets the rise in banks’ funding costs as an increase in the liquidity premium. Alternatively,onecouldconsiderthatfundingcostsincreaseforadditionalreasonsduringthecrisis. As mentioned, r could more generally be interpreted as the incremental increase in the shadow (cid:96) cost of bank funding, which may be higher than the increase in the liquidity premium due to, for example,bindingbalancesheetconstraintsandahighercostofbankequity. Inotherwords,banks wouldneedtocommitequitycapitaltofundloanstobusinesses,thepriceofwhichmayincreasea lotduringacrisistranslatingintoahighr inthemodel. Naturally,thismayresultinhigherwelfare (cid:96) lossesthantheonesreportedinTable1,whichcanbeseenasareasonablelowerbound,rendering the implementation of the optimal design more important. Given estimates for the increase in the shadowcostofbankequitycapitalduringcrises,onecouldderiveadifferentestimateforr anduse (cid:96) corollary2tocomputethelossesfromimplementingauniform/flatstructure.19 Notethattheserelativewelfare-gaincalculationspresumethattheSPVhassettheriskretention at the appropriate level to resolve the moral hazard problem, or in other words, the risk-retention ratioisasufficientstatisticforthemonitoringcost. Boththeoptimalandflat-pricingdesignswould 19Ofnote,regulatoryactionsallowingtheexclusionofnewloanstosmallbusinessesfromthecalculationofregulatory capitalrequirementscouldcomplementcentralbankpurchasesinordertoalleviatethepressurefrombindingbalance sheetconstraints.Thiscombinedpolicyresponseisinteresting,butbeyondthescopeofthispaper. 16

resultinnegativenetpresentvalueiftherisk-retentionrequirementisnothighenoughtosatisfythe incentivecompatibilityconstraintformonitoring. 6 Conclusion The model captures a key difficulty central banks may face when purchasing loans extended to small businesses; namely that bank monitoring may be necessary, but not observable/contractible. Establishing a lending facility with tiered pricing schemes, for given observable borrower characteristics,isoptimaltoprovidetherightincentivestobanks,whilemaximizinglendingtobusinesses at the lowest possible rates. The model focuses on small business loans because they are likely candidatesforrequiringcontinuousmonitoring. But, themechanismandconclusionsinthispaper couldapplytoothercaseswhereactivemonitoringisimportant,suchasthesyndicatedloanmarket (Gustafson et al., 2020). However, being stylized, the model leaves a number of implementation issuesunanswered. Forinstance,loansmayhavedifferentcharacteristicsandcovenantsmandatingwhenownership shouldbetransferredorotheractionsaretobetaken. Notallthesecovenantsarepresenttoaddress moral hazard issues by borrowers, but some of them may act as an ex ante screening mechanism. The baseline model is extended in the appendix to incorporate not only ex post monitoring, but also ex ante screening to separate entrepreneurs with good projects from entrepreneurs with bad projects. Totheextendthatthevariouscovenantscannottackleperfectlythemoralhazardproblem andmonitoringisstilluseful,thetwo-tierpricingschemealongwithdifferentialrisk-retentionratios isstilltheoptimalinstitutionaldesignforthecentralbanklendingfacility. Inturn,thisisreasonable for small business loans given that they do not have a long track record and are typically unrated. Similarly, small businesses may lack tangible and liquid collateral, and they usually collateralize cashflowsorhavecovenantsthatmandatethetransferofcontrolrights,whichmaybeonlyvaluable ifentrepreneursdonotengageindestructivebehavior. Moreover, the operational complexity of implementing a tiered pricing facility may be high. Taking such important concerns into consideration, the SPV may decide to offer a uniform/flat pricing scheme similar to the one described in proposition 2. That would be still preferable to not providing the necessary liquidity to the economy, while the welfare cost is not expected to be that 17

high. Overall, practical implementation can be daunting. Nevertheless, the main principle of having pricingschemesthatdependnotonlyonobservableborrowercharacteristics,butalsoonmonitoring intensityandmonitoringcostshouldcontinuetoholdinmoreelaborateenvironments. Itisalsotrue that the central bank lending facility in this model mainly delivers benefits, because central banks do not need to be compensated for the shadow value of liquidity. Central banks will weigh the benefits to the real economy from intervening to the relevant costs, which may be reputational or operational, given that the terms are chosen such that the loan portfolio does not lose money in expectation. Only when the shadow value of liquidity is high enough, and in accordance to their mandate, would lending facilities being employed. Such considerations do not affect the message of the paper, as the shadow value of liquidity can be made arbitrarily high to represent systemic liquidityshortagesandjustifycentralbankinterventiononlyincrisisperiods. References Cashin, David, Erin Syron Ferris, Beth Klee and Cailey Stevens (2017), ‘Take it to the limit: The debtceilingandtreasuryyields’,FinanceandEconomicsDiscussionSeries2017-052. Diamond,DouglasW.(1984),‘Financialintermediationanddelegatedmonitoring’,ReviewofEconomicStudies51(3),393–414. Drechsel,ThomasandSebnemKalemli-Özcan(2020),‘Arestandardmacropoliciesenoughtodeal withtheeconomicfalloutfromaglobalpandemic?’. URL: https://econfip.org/policy-brief/are-standard-macro-policies-enough-to-deal-with-theeconomic-fallout-from-a-global-pandemic/ Dreyer, Mallory, Christian McNamara, Alexander Nye, Kaleb Nygaard and Priya Sankar (2020), ‘Large-scaleassistanceprogramsforsmallbusinesses’. URL:https://som.yale.edu/blog/large-scale-assistance-programs-for-small-businesses English, WilliamB.andNellieLiang(2020), ‘DesigningtheMainStreetLendingProgram: ChallengesandOptions’,HutchinsCenterWorkingPaper#64. 18

Gorton,GaryB.andGeorgeG.Pennacchi(1995),‘Banksandloansalesmarketingnonmarketable assets’,JournalofMonetaryEconomics35(3),389–411. Gustafson,MatthewT.,IvanT.IvanovandRalfR.Meisenzahl(2020),‘Bankmonitoring: Evidence fromsyndicatedloans’,JournalofFinancialEconomics,forthcoming. Holmström, Bengt and Jean Tirole (1997), ‘Financial intermediation, loanable funds, and the real sector’,TheQuarterlyJournalofEconomics112(3),663–691. Holmström, Bengt and Jean Tirole (1998), ‘Private and public supply of liquidity’, The Journal of PoliticalEconomy106(1),1–40. Infante, Sebastian (2020), ‘Private money creation with safe assets and term premia’, Journal of FinancialEconomics136(3),828–856. Nagel, Stefan (2016), ‘The Liquidity Premium of Near-Money Assets’, The Quarterly Journal of Economics131(4),1927–1971. Parlour, Christine A. and Guillaume Plantin (2008), ‘Loan sales and relationship banking’, The JournalofFinance63(3),1291–1314. Pennacchi, George G. (1988), ‘Loan sales and the cost of bank capital’, The Journal of Finance 43(2),375–396. Saki,Bigio(2020),‘AconcretepolicyresponsetotheCovid-19crisis’. URL:http://saki-bigio.squarespace.com/policy-recommendation Sufi,Amir(2007),‘Informationasymmetryandfinancingarrangements: Evidencefromsyndicated loans’,TheJournalofFinance62(2),629–668. 19

Appendix The appendix extends the baseline model to account for adverse selection. Within each type of entrepreneurs, there are two sub-types: good and bad. Good entrepreneurs of either type 1 or type 2 have a probability of success θ , while bad entrepreneurs of either type have a probability of G success θ satisfying θ R > 1+X > 1 > θ R. In other words, only good entrepreneurs have a B G B positive net present value project, even if monitoring is not required. Denote by α the relative percentage of goodentrepreneurs, which, forsimplicity and without loss ofgenerality, is the same forbothtypes1and2. Then,theaveragepayoffforapooloftype1or2entrepreneursisR¯=R¯ = 1 R¯ =(αθ +(1−α)θ )R, and the average payoff for the whole population is (m +m )R¯, where 2 G B 1 2 R¯>1+X. Hence,theaverageprojectissimilartotheprojectinthebaselinemodel. Before extending credit, banks can choose to screen entrepreneurs by setting non-price terms along with the loan rate. For simplicity, we assume that the non-price loan terms take the form of a payment to banks in the case of failure. Thus, banks can offer a two-dimensional loan contract (R ,C ), where R is the loan rate and C are the non-price loan terms. This characterization is s s s s meant to proxy for collateral posted or for other covenants mandating, for example, the transfer of ownership to banks or other restrictions decreasing the value to entrepreneurs and increasing the valuetobanks. EntrepreneursloseC intheeventoffailure,whilebanksgainκC ,where1−κ≥0 s s is the per unit bank cost of enforcing/servicing the non-price terms. For conciseness, C will be s referred to as collateral.20 Instead of offering a two-dimensional contract, banks may choose to poolallborrowerstogetherbyofferingaone-dimensionalloancontractdefiningonlytheloanrate R ,asisthecaseinthebaselinemodel. p With respect to the moral hazard friction, monitoring type 2 entrepreneurs—either good or bad—guarantees both that their project will not fail with certainty and that the collateral will not be useless. The latter is meant to capture the fact that the value of the various non-price terms dependsonentrepreneurialeffort. Thereasoncouldbethatsmallbusinessescanonlypostintangible collateralintheformofearlypayments,accountsreceivable(tradecredit),ortransferofownership rights, which may be worthless if the entrepreneur engages in destructive behavior (for example 20Inreality,smallbusinessloansaremultidimensionalcharacterizedbymanycovenants. Thesimplifiedstructureof themodelextensionismeanttocapturetheideathattheappropriatesetofcovenantscanbeusedtoseparategoodfrom badborrowers. 20

damaging organizational structures, or using them to establish other businesses). In sum, this assumption implies that non-price loan terms are not sufficient to discourage bad behavior, and that bankmonitoringcaneffectivelyputastoptothisagencyproblem.21 Consider type 1 entrepreneurs. The separating contract, (Rˆ1,Cˆ1), should separate good from s s bad entrepreneurs and be profitable for each bank. The former condition requires that p (R− B Rˆs)−(1−p )Cˆs ≤ 0, while the second requires p Rˆs +κ(1−p )Cˆs ≥ 1+q(K−1). Because 1 B 1 G 1 G 1 collateral enforcement is costly and because of perfect competition in the banking sector, these two constraints hold with equality Rˆs =((1+q(K−1))(1−p )−κp (1−p )R)(p (1−p )− 1 B B G G B κp (1−p ))andCˆ1=p /(1−p )·((p R−(1+q(K−1)))(1−p ))(p (1−p )−κp (1−p )). B G s B B G B G B B G Similarly, theseparatingcontracttermsfortype2entrepreneursareRˆs =((1+q(K−1)+X)(1− 2 p )−κp (1−p )R)(p (1−p )−κp (1−p ))andCˆ2= p /(1−p )·((p R−(1+q(K−1)+ B B G G B B G s B B G X))(1−p ))(p (1−p )−κp (1−p )). Instead,thepoolingcontractsetsonlytheloanrateequal B G B B G toRˆp =(1+q(K−1))/(αp +(1−α)p )andRˆp =(1+q(K−1))/(αp +(1−α)p )fortype1 1 G B 1 G B andtype2entrepreneurs,respectively. Thefollowingpropositionderiveswhichcontractwillbechoseninequilibrium. Proposition A.1. Consider θ R≥1+q(K−1). For α>α¯ ∈(((1+q(K−1))/R−p )/(p − G 1 B G p ),1),thereexistsκ¯ ∈(0,1)suchthatbankschoosethepoolingcontractfortype1entrepreneurs B 1 for κ<κ¯ (α) and they choose the separating contract with terms otherwise. For α∈((1+q(K− 1 1))/R−p )/(p −p )onlytheseparatingcontractischosen. B G B Proof. The separating contract constitutes an equilibrium if no individual bank has an incentive to deviate and offer the pooling contract. In turn, this is true if the effective payment for good entrepreneurs, i.e., Rˆ1+(1−p )/p Cˆ1 is lower that the pooling rate, i.e., Rˆp . First, note that s G G s 1 for α ≤((1+q(K−1))/R−p )/(p −p ), Rˆ1 is higher that R and, thus, no entrepreneur will B G B p choosetoborrow. Hence,onlytheseparatingcontractwouldbeviable. Next,observethat: d(Rˆ1+ s (1−p )/p Cˆ1)/dκ < 0; that Rˆ1+(1−p )/p Cˆ1 < Rˆ1 for κ = 1 irrespective of the value of G G s s G G s p α; and that for κ=0 Rˆ1+(1−p )/p Cˆ1 >Rˆ1 for α→1, while Rˆ1+(1−p )/p Cˆ1 <Rˆ1 for s G G s p s G G s p α→((1+q(K−1))/R−p )/(p −p ). All these together imply, by continuity, that there is an B G B 21Thisisnotanunreasonableassumption. Gustafson,IvanovandMeisenzahl(2020)showthatbankmonitoringfor syndicatedloanscaneithercomplementorsubstituteforcovenant-basedmonitoring,dependingonwhetherthemonitoringinformscovenantcompliance. 21

α¯ such that Rˆ1+(1−p )/p Cˆ1 >Rˆ1 for α>α¯ and κ. In turn, this implies that there exists a 1 s G G s p 1 κ¯ (α)asafunctionαsuchthatthepoolingcontractistheequilibriumcontractforκ<κ¯ (α)(and 1 1 theseparatingotherwise). PropositionA.2. Considerthatθ R≥1+q(K−1)+X. Forα>α¯ ∈(((1+q(K−1)+X)/R− G 2 p )/(p −p ),1), there exists κ¯ ∈(0,1) such that banks choose the pooling contract for type 2 B G B 2 entrepreneurs for κ < κ¯ (α) and they choose the separating contract with terms otherwise. For 2 α≤(((1+q(K−1)+X)/R−p )/(p −p )onlytheseparatingcontractischosen. B G B Proof. The proof is the same as for Proposition (A.1) above where 1+q(K−1) is replaced by 1+q(K−1)+X. Corollary A.1. The separating contract for type 2 entrepreneurs carries a higher loan rate and lowercollateralthanfortype1entrepreneurs,whilethesameistruefortheloanrateintherespectivepoolingcontracts. Moreover,separatingcontractsareeasiertoobtainfortype2entrepreneurs κ¯ <κ¯ . 2 1 Proof. The first part is trivial given that X >0. To prove the second part, observe that, for κ=1, Rˆ2+(1−p )/p Cˆs−Rˆp <Rˆ1+(1−p )/p Cˆs−Rˆp <0andd(Rˆ1+(1−p )/p Cˆs)/dκ<d(Rˆ2+ s G G 2 2 s G G 1 1 s G G 1 s (1−p )/p Cˆs)/dκ<0. G G 2 For 1+q(K−1)>θ R, banks do not engage in lending and, instead, hoard liquidity. As in G the baseline, model the central bank can set an SPV to provide the necessary liquidity and improve outcomes. In the case of pooling contracts, the optimal design is the same one described in proposition 1 for θ=αθ +(1−α)θ . The two-tier pricing system is also optimal for the case of G B separating contracts given that collateral cannot address the moral hazard problem. The solution is obtained by the set of conditions (1)-(5) by replacing θR and θR with effective expected loan 1 2 paymentsθ Rs+(1−θ )Cs andθ Rs+(1−θ )Cs,respectively,withCs = p /(1−p )(R−Rs) G 1 G 1 G 2 G 2 1 B B 1 andCs = p /(1−p )(R−Rs). 2 B B 2 PropositionA.3. Inthepresenceofbothexantescreeningandexpostmonitoring,theoptimalSPV designinvolvestwodistinctpricingschemesconsistingoftwo-dimensionaland/orone-dimensional contractsdependingonthelevelofκ. 22

- For κ<κ(cid:48), the optimal design sets a price P =1 for loans with rate R =1/(αθ +(1− 1 1 G α)θ ),zerocollateral,andzerorisk-retentionrequirements,andapriceP =1+X forloans B 2 with rate R = (1+X)/(αθ +(1−α)θ ), zero collateral and risk-retention requirement 2 G B γ =X/(1+X). 2 - For κ∈[κ(cid:48),κ(cid:48)(cid:48)), the optimal design sets a price P =1 for loans with rate R =((1−p )− 1 1 B κp (1−p )R)(p (1−p )−κp (1−p )), collateral C = p /(1−p )·((p R−1)(1− B G G B B G 1 B B G p ))(p (1−p )−κp (1−p )), and zero risk-retention requirements, and a price P = B G B B G 2 1+X forloanswithrateR =(1+X)/(αθ +(1−α)θ ),zerocollateral,andrisk-retention 2 G B requirementγ =X/(1+X). 2 - For κ∈[κ(cid:48)(cid:48),1), the optimal design sets a price P =1 for loans with rate R =((1−p )− 1 1 B κp (1−p )R)(p (1−p )−κp (1−p )), collateral C = p /(1−p )·((p R−1)(1− B G G B B G 1 B B G p ))(p (1−p )−κp (1−p ))andzerorisk-retentionrequirements,andapriceP =1+X B G B B G 2 for loans with rate R = ((1+X)(1−p )−κp (1−p )R)(p (1−p )−κp (1−p )), 2 B B G G B B G collateral C = p /(1−p )·((p R−(1+X))(1−p ))(p (1−p )−κp (1−p )) and 2 B B G B G B B G risk-retentionrequirementγ =X/(1+X). 2 Proof. Theprices,contractterms,andrisk-retentionratioscanbederivedusingthesamestepsasin proposition1andrealizingthatforκ<κ(cid:48) onlythepoolingcontractswillbechosenbybanks(and only the separating ones for κ≥κ(cid:48)(cid:48)). There thresholds are derived as the thresholds κ¯ and κ¯ in 1 2 propositionsA.1andA.2. 23