A Collateral Theory of Endogenous Debt Maturity

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. A Model of Endogenous Debt Maturity with Heterogeneous Beliefs R. Matthew Darst and Ehraz Refayet 2017-057 Please cite this paper as: Darst, R. Matthew, and Ehraz Refayet (2017). “A Model of Endogenous Debt Maturity with Heterogeneous Beliefs,” Finance and Economics Discussion Series 2017-057. Washington: Board of Governors of the Federal Reserve System, https://doi.org/10.17016/FEDS.2017.057r1. 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.

A Model of Endogenous Debt Maturity with Heterogeneous Beliefs ∗ R. Matthew Darst† Ehraz Refayet‡ March 29, 2018 Abstract Thispaperstudiesoptimaldebtmaturityinaneconomywithrepaymentenforcementfrictionsandinvestorsdisagreeaboutrepaymentprobabilities. Theoptimaldebt maturitychoiceisamixoflong-andshort-termdebtsecurities. Spreadingriskydebt claims on cash flows over time allows debt to be priced by investors most willing to holdriskateachpointintime,therebyincreasinginvestmentandoutput. Bycontrast, a single maturity, either all long- or short-term, will be priced by investors less willing to hold risk, which reduces investment and output. The model provides a novel explanationforthestylizedfactthatlargeandmaturecompaniesalmostalwaysissue debt with multiple maturities rather than a single maturity, and is broadly consistent with empirical debt maturity results. Lastly, we show that non-financial covenants thatpreventdebtdilutiononlyserveassubstitutesforshort-termdebtanddonotaffectrealoutcomesastheydonotallowthefirmtocreateadditionalcollateralagainst whichtoborrow. Keywords: Debtmaturity,investment,costofcapital,covenants,debtdilution JELCodes: D92,G11,G12,G31,G32,E22 We are immensely grateful to Ana Fostel for numerous discussions and guidance that significantly ∗ improvedtheinsightsinthispaper. WewouldliketoespeciallythankYeLi(discussant),SebnemKalemli- Ozcan, Jesse Davis, Alex Vardoulakis, Greg Phalen, Dan Cao, Yulyian Mitkhov, Francesca Zucchi, and Zhiguo He for useful suggestions. We thank seminar participants at the Federal Reserve Board, George Washington University, 2017 Midwest Finance Meeting, 2016 Fall Midwest Macro Meeting, 2016 48th Money, Macro, and Finance Conference, IFABS - 2016 Barcelona, Econometric Society - 2016 European Meeting.Allerrorsareours.Theviewsexpressedinthispaperarethoseoftheauthorsanddonotnecessarily representthoseoftheFederalReserveBoardofGovernorsoranyoneintheFederalReserveSystem,U.S. DepartmentofTreasury,orOfficeoftheComptrolleroftheCurrency. †FederalReserveBoardofGovernors: matt.darst@frb.gov ‡OfficeoftheComptrolleroftheCurrency: ehraz.refayet@occ.treas.gov 1

1 Introduction Many large and mature corporations typically raise capital by issuing debt of various maturities. For example, IBM tapped the bond market 5 times in the first 6 months of 2017 with a different maturity offering each time. In July 2017, AT&T raised $22.5 billion issuing maturities ranging from 5.5 years to 41 years. In 2013, Verizon raised $49 billion issuing debt through 6 different maturities. Microsoft offered 7 different maturities when itraised$19.75billionin2016. 1 Inaddition,Kalemli-Ozcan,Laeven,andMoreno(2018) showthatbothlargefirmsandSMEsinEuropeutilizebothlong-andshort-termdebt. Theoretically explaining why the largest, most mature firms with low default and rollover risk issue a mix of debt maturity is a challenge. For example, the scant existing explanationsforwhyanyfirmwouldissuemultiplematuritiesrelyonbalancinginefficient liquidationriskfromshort-termdebtandmaintainingcontrolrentsoroptimalcontinuation policies from long-term debt (Diamond (1991), (1993), and Houston and Venkataraman (1994)). Yet, empirical and survey evidence suggests that liquidation and information asymmetries are not likely to have significant effects on large, mature corporations with abundantpublicinformationandanalystcoverage.2 This paper presents a novel theory to explain why issuing multiple debt maturities is cost minimizing and value maximizing for firms in the absence of agency conflicts or liquidityrisk. Firmsusedebtmaturitytointer-temporallycaterriskyclaimsoncash-flows to investors most willing to hold them. For any given investment, a firm has three ways to structure its debt maturity: 1) use long-term debt that matches the timing of its assets and liabilities. Long-termdebtrequirespayingpositivecreditspreadsduetodefaultrisk,butit insulatesthefirmfromchangesinthecostofissuingdebtinthefuture;2)issueshort-term 1Infact,virtuallyeveryrecentmajorpublicdebtofferinginvolvesmultiplematurities. 2GrahamandHarvey(2001)findthatlistedfirmsaretypicallynotconcernedwithinformationasymmetriesandthemis-pricingofsecuritiesduetoagencyproblems. Johnson(2003)findsthattheliquidityrisk effectsofshort-termdebtmatteralmostexclusivelyforunratedfirms,andarevirtuallynon-existentforrated firms. Billet, King, andMauer(2007)donotfindevidencethatliquidityriskdrivesshort-termdebtuseat allforratedfirms. 1

debtthatneedstoberolledover. Short-termdebtisinitiallysafewithoutliquidityrisk,but completely exposes the firm to price fluctuations in the future; or 3) issue a combination of the two maturities in which only the short-term component is subject to price changes, whichgeneratesadilutioncostforlong-termdebt.3 Debtmaturitychoiceisanalyzedasatradeoffbetweenthecostofriskydebtovertime due to changes in expected firm cash-flows and heterogeneity in the price investors are willingtopaytoholdriskydebtclaimsonthosecashflows. Foragiveninvestment,issuing morelong-termdebtrequiresofferinghighercompensationtolessoptimisticinvestors. At thesametime,morelong-termdebtalsoreducestheamountofshort-termdebtthatneeds to be rolled over, which results in more optimistic investors determining the market value of short-term debt and reduces dilution costs. Therefore, changes in the aggregate supply of debt issued across time lead to changes in the relative cost of debt financing firms face duringthosetimes. Themainresultisthatissuingamixofdebtmaturitiesallowsfirmsto issuetheoptimalamountofriskydebtineachtimeperiodandfirmscapitalizeoninvestors’ different willingness to hold risk. By contrast, issuing a single debt maturity only allows theoptimalamountofriskydebttobeissuedinasingletimeperiodandforcesthefirmto raise a larger portion of its capital from investors who seek higher compensation. Issuing a mix of debt maturities allows firms to raise additional capital for investment on better termsandincreaseoverallfirmvalue. Our model is an incomplete markets economy with binomial states and three-periods: 1) an initial state, 2) intermediate states, and 3) terminal states. The key frictions are investor heterogeneity and repayment enforcement problems.4 Creditors cannot coerce 3Recent studies of Hugonnier, Malamud, and Morellec (2014 and 2015) have used search frictions to highlight the point that capital supply frictions can generate new predictions for firm financial policy. A simple collateral constraints and investor heterogeneity in our model produces a similar environment in whichonecaneasilystudyhowdebtmaturityinteractswithinvestmentdecisions. 4OurfocusisonlimitedenforcementfrictionsasinRampiniandVishwanathan(2010)andFosteland Geanakoplos(2015,2016).Distinctfromthesepapers,weaskhowafirmshouldoptimallystructureitsdebt maturityinaneconomywithcollateralconstraintsandheterogeneousinvestors. RampiniandVishwanathan (2010) focus on how a firm should allocate scarce collateral between investment and risk management. FostelandGeanakoplos(2016)focusonhowcollateralconstraintsandfinancialinnovationaffectinvestment 2

debtors into repayment, and collateral is used as enforcement. When a debtor fails to honor its debts, the creditor has the right to seize collateral to be made whole, but no more.5 We consider risk neutral creditors with heterogeneous beliefs over the expected value of repayment. Investors are willing to pay different prices to hold risky debt, and these prices change in intermediate states as uncertainty is either resolved in good states orgrowsinbadstates. We begin with simple examples that show how investor heterogeneity permits debt maturitytoaffectfirmvalue,evenwithoutintermediateliquidityriskorbankruptcycosts. Comparedtoacommonbeliefinahomogeneousinvestoreconomy,investorheterogeneity allowsthefirmtoutilizethedebtmaturitychoiceinawaythatreducesfinancingcostsand increases firm value. Firm value is increased in the heterogeneous agent economy when the marginal investor pricing debt in equilibrium has a higher expectation of being repaid than the common belief. But this need not always be the case if the common belief is sufficiently high. However, allowing the firm to utilize both long and short-term debt in the heterogeneous agent economy spreads the debt issuance over time which actually consolidates risk to investors with higher repayment expectations. Therefore, a mix of long- and short-term debt will always be the optimal choice in the heterogeneous agent economyandwillgenerallyincreasethevalueofthefirm. Theintuitionisthefollowing: Consideralllong-termdebtfunding. Therearetwobenefitsfromsubstitutingaportionofthelong-termdebtforshort-termdebt. First,short-term debt is risk free in the initial period. Second, because less long-term debt will be issued, a more optimistic investor will price it in equilibrium, raising long-term bond prices. The costs of partially substituting into short-term debt are the following: The first is the expected cost to rollover short-term debt. The second is the dilution cost to long-term debt efficiency. 5Our choice to highlight a collateral friction is supported by recent empirical evidence suggesting that collateral plays an important role in the design of debt contracts and the provision of credit (Cerqueiro, Ongena,andRoszbach(2016)). 3

due to the expected increase in short-term debt face value needed to rollover expiring claims. In general, the substitution benefits always outweigh the costs for two reasons. Substituting risky for risk-free debt is always cheaper, and short-term refinancing costs are paid in expectation rather than with certainty as with long-term debt. Moreover, the dilutioneffectonlong-termdebtpricesismitigatedbythefactthatincreasinglyoptimistic investors finance long-term debt as long-term debt is substituted for short-term debt. This means that the marginal investor cares less about the dilution costs the more the firm substitutesawayfromlong-termdebt. By similar reasoning, multiple debt maturities generally dominate all short-term debt financing. The benefits of substituting a portion of debt financing into long-term debt are the following: Raising one dollar long-term is cheaper than raising one dollar short-term conditional on bad news because short-term debt is information sensitive. In addition, issuing less short-term debt lowers the dilution cost to long-term debt, which further increases long-term debt prices. The only cost of substituting into long-term debt is paying a positive credit spread with certainty and giving up the opportunity to finance short-term debtriskfree. Aninterestingimplicationofamultipledebtmaturityequilibriumisthatdebtdilution can actually increase the asset value of the firm. The intuition is that a more optimistic marginalbondbuyerendsuppricinglong-termdebtasthefirmsubstituteslong-forshortterm. Therefore, even though the value of long-term debt is being diluted, the marginal investorcareslessaboutthedilutedrecoveryvalueofherdebt. Inotherwords,thesubstitution effect of reducing the face value of long-term debt in leiu of some short-term debt outweighsthedilutioncost. We then ask how non-financial debt covenants that prevent debt dilution from occurring may impact our maturity results. Specifically, we allow long-term debt to be secured byspecificfirmassetsandshowthattherecoveryvalueoflong-termdebtinamixeddebt maturityequilibriumisthesameastherecoveryvalueinalong-termonlyequilibriumi.e. 4

there is no dilution effect.6 We show that utilizing a mix of long- and short-term debt remainstheoptimaldebtmaturitystrategyeveninthepresenceofprotectedlong-termdebt. The intuition is the following: Ceteris paribus, the price any investor is willing to pay for a protected long-term bond rises when their claims cannot be diluted. This causes the firmtoraisemorelong-termdebtresultinginamorepessimisticmarginallong-termbond buyercompletelyundoingtheinitialpriceincrease. Inequilibrium,thepriceoflong-term debt remainsthe sameand there isno relativecost advantage forthe firm. Theonly effect ofthecovenantisthatrelativelymorelong-termdebtisissuedinequilibrium,andissuing multipledebtmaturitiesremainstheoptimaldebtissuancestrategy. More broadly, debt maturity at issuance in emerging markets has lengthened as firms have increase their reliance on bond finance (Fuertes and Serena (2014) and Shin (2014)) A recent IMF report on emerging market corporate leverage suggests that global factors explain most of the increase in debt maturity (IMF 2015). Our model suggests that global factorsmaybeareflectionofthedifferencesamonginvestorstoholdriskyclaimsonfirm cash flows and hence firms cater the maturity of their debt issuance to these investors and increaseleverage. The organization of the paper is as follows: related literature is below. Section 2 introduces the model, agents, the different debt contracts considered, and works through 3 simple examples. Section 3 characterizes the equilibrium debt liability structure and comparative static results. Section 4 introduces the covenant and provides a numerical examplehighlightingitseffectsandthecomparativestaticresult. Section5concludes. All proofsthatarenotobviousfromthetextarecontainedintheappendix. Relatedliterature In a series of papers, Hart and Moore (1994, 1995, 1998) show that debt is an optimal contract to resolve agency problems and discipline management to payout cash flows and 6Insection4andtheappendix,wediscussthatoursecuredcovenantcanbeinterpretationasanegative pledgecovenant,oneofmostcommoncovenantsfoundinlong-termcorporatedebtindentures. 5

undertakeefficientinvestmentprojects. Thesemodelseitheronlyexaminelong-termdebt (Hart and Moore (1995)), or characterize repayment paths as either the fastest or slowest (Hart and Moore (1994, 1998)), but never a combination of the two. Zwiebel (1996) shows that multiple repayment paths will constrain empire building and prevent control takeovers, but only for the most risky firms for whom debt is a possible financing source, but is at odds with empirical observations. While agency frictions certainly explain why management may use debt in its capital structure, they do not appear to adequately describe why multiple maturities are simultaneously used to raise capital (see also Jensen andMeckling(1986),BoltonandScharfstein(1990,1998)). Privateinformationcanaffectthetypesofdebtsecuritiesfirmsissue. Whenfirmshave inside information, Flannery (1986) shows that firms will use short-term debt to signal quality. Diamond(1991,1993)andHoustonandVenkataraman(1994)showthatliquidity riskbreakstherelianceonshort-termdebtandgeneratesdifferentdebtmaturitychoicesin the cross section based on credit ratings.7 We show that multiple maturity debt is optimal withoutliquidationriskandfirmsdonotuseprivateinformationinanyway. There are many models in which debt affects firm value. For example, debt maturity can improve investment incentives due to debt overhang (Myers (1977), He and Diamond (2014)), optimal default timing (He and Milbradt (2016)), and information asymmetries (Flannery (1986), Kale and Noe (1990)). Debt maturity affects corporate financial policy in our model because the same risky debt claim will be priced differently in different periods when investors disagree about repayment probabilities.8 Another distinction in our model are non-exclusive relationships between debtors and creditors. In practice, non- 7Proposition2ofDiamond(1991)showsthatshort-termdebtistheuniquefundingoutcomeinthemodel absentliquidationriskorlossofcontrolrents. Theagencyproblemisthereforenecessaryinhismodelto obtain an equilibrium with multiple maturities. Firm borrowing is fixed in Diamond (1991) and firms can borrowuptothefixedamountatthesameinterestrate. 8Heterogeneity is at the heart of Jung and Subramanian (2014), but an agency problem gives maturity a role in their model. Specifically, heterogeneous beliefs between managers and equity holders leads to a tradeoffbetweenmanageroptimismandlong-termdebtissuance. Ourmodelalsohasaflavorofthiseffect, but heterogeneity between the firm and investors is only material for determining what portion long-term debtconstitutesoftotaldebtissuance. 6

exclusive relationships are common for large corporations. For example, Dass and Massa (2014), using Lipper eMAXX data, highlight that the average corporation has 17 institutional investors acting as creditors at any point in time (see also Detragiache, Garella, and Guiso(2000)).9 DiamondandHe(2014)highlightthesubtleeffectsofdebtmaturityondebtoverhang and investment incentives. The optimal debt maturity balances the symptoms of shortand long-term debt overhang; respectively, earlier default versus reduced investment incentives. However, they consider different debt maturities with equivalent market values and a fixed asset. We do not consider overhang effects because how debt maturities are structuredinourmodelaffectstheexantevalueoftheasset/projectthefirmundertakes. Dynamic debt maturity models in continuous time focus on refinancing policies, optimal leverage ratios, and target average debt maturity (see Leland (1994, 1998)). He and Milbradt (2016) bring debt maturity to the forefront of these models. They emphasize the joint determination of default and maturity by showing that a firm actively manages maturingdebtdependingonthefirm’sdistancetodefault. Thefirmissuesshort-orlong-term debt and commits to a constant book leverage policy, i.e. maintains a constant aggregate face value of outstanding debt. Rollover losses arise as equity holders must absorb any cash flow shortfall when maturing bonds are refinanced when credit conditions deteriorate. The rollover losses feed back to the default decision by equity holders, leading to earlier default. Our model characterizes the firm’s optimal debt financing strategy with multiple maturity issuances. Though we do not focus on the timing of default, our model hasasimilarfeedbackmechanisminwhichtheanticipatedshort-termdebtrolloverlosses induce the firm to substitute toward more initial long-term debt. We view our paper as complementary. 9Largefirmstypicallyraisecapitalfromasyndicateofcreditorsratherthanasinglecreditorevenwhen considering private loan markets. Using supervisory data on bank holding companies, Caglio, Darst, and Parolin(2016)showthatlargercorporatesborrowfrom, onaverage, 8bankscomparedtosmallfirmsthat tendtoborrowfromone. 7

Brunnermeier and Oehmke (2013) show that financial firms’ inability to commit to a maturity structure leads short-term debt to dilute long-term debt. Their model has a fixed supply of assets where the firm increases aggregate debt liabilities when new debt is issued to repay expiring claims, diluting the per-claim value of existing debt. Equity holderscannotabsorblossesintheirmodelastheycaninourmodelandHeandMilbradt (2016). He and Xiong (2012b), with a fixed maturity structure, show how short-term debt canamplifydefaultriskswhenliquidityriskispresentbecauseequityholderswilldefault atearliervaluationthresholds. Defaulttimingisfixedinourmodel,butmaturityisallowed toadjust. Geanakoplos (2009) and He and Xiong (2012a) study debt financing in incomplete asset markets with heterogeneous agents. In their models, short-term debt is the unique equilibriumbecauseasequenceofshort-termclaimsallowsagentstotakemaximumleverage. Thedifferenceintheirmodelsisthatallagentsownriskyassets,someofwhomhave higher valuations than others, and agents can borrow against the assets by issuing safe promises to obtain leverage. Issuing consecutive short-term claims allows optimists to borrow against the lowest value of the asset one period in the future, while a long-term claim only allows an agent to borrow against the final period worst case outcome. We adopt the same uncertainty structure to highlight that our result is not a special case of what one assumes about uncertainty, but that introducing an optimizing agent with productionintoaheterogeneousagentframeworkdeliversthemultiplematurityphenomenon thatwecommonlyseeinpractice. 8

2 Model 2.1 Time and uncertainty The model is a dynamic three-period production economy with incomplete asset markets. Timeisdenotedt = 0,1,2 .Uncertaintyisgivenbyatreeofstateeventss S withroot { } ∈ s , intermediate states s S that take values U,D , and a set of terminal nodes denoted 0 ∈ { } S = UU,UD,DU,UU S. Let state realizationU be up or a “good” state and D be T { }⊂ downora“bad”state. The only uncertainty in the model is an aggregate shock that affects output at t = 2. The parameter A captures the effect of the shock to production. The expected value of s T theshockisconditionalontheinformationrevealedatt=1. Weassumeforsimplicitythat good news at t =1 resolves uncertainty at t =2 and there is no shock: A =A =1. UU UD Bad news at t =1 raises uncertainty at t =2 about the ability of the firm to repay debts, akinto“scarybadnews”inGeanakoplos(2009). Specifically,thereisnoshockatterminal nodes=DU,butthereisashockatterminalnodes=DD,A <A =1. Notethatthis DD DU uncertaintystructureisthesameasthesimplificationmadeinthecontinuoustimeversion ofDiamondandHe(2014).10 Figure1depictstheeconomy’sstatetree. The assumptions about time and uncertainty are made for simplicity. We show in the appendix that equilibrium is qualitatively unaffected under different assumptions. For example, one may assume that good news is more likely to follow good news rather than badnews,whichalleviatestheconcernthatbadnewsiseffectivelynotasbad. Thiswould change only quantitatively how much long- versus short-term debt the firm would issue. Alternatively, one could allow for uncertainty conditional on s=U and that A =A . UD DU (cid:54) This alternative would also only affect the relative amounts of long- and short-term debt issued, and would not change the result that issuing both maturities is optimal. The lat- 10Inexample2ofDiamondandHe(2014), theyassumethatassetvolatilityisstate-contingent. Specifically, σ =ε >0=σ where σ is asset volatility conditional on state i. Clearly uncertainly is resolved H L i whenσ =0. L 9

Figure1: EconomyStateTree s=U A =1 s/DD s 0 s = D (1 ) t=0 (1 ) A <1 t=1 DD t=2 Tree3.pdf ter alternative would equate the model’s uncertainty structure with what He and Xiong (2012a)consider. Moreover,FostelandGeanakoplos(2010)provethatagentshavetheincentive to produce projects that become more volatile in bad times. The reason is simple: uncertainty following bad news is not very informative, which implies that price declines in bad intermediate periods are relatively small. Alternatively, if uncertainty was completely resolved after bad news, then prices in bad intermediate periods would fall much further and reflect the certain bad outcome in the final period. Lower intermediate prices wouldlimitexantehowmuchagentscouldleverageandborrowagainsttheirprojects. 2.2 Debt contracts There is a single durable consumption good available in the economy at t = 0, which is thenumeraire. Therearetwotypesofdebtcontractsthatcanbemade,eachwithdifferent maturity. Short-term debt matures after one period and long-term debt matures after two periods. All debt contracts are non-contingent and pay zero-coupons. For simplicity, we normalizetherepaymentvalueofeachcontractto1. 10

Letthequantityofdebtissuedatanystateandtimebeq . Thequantityoflong-term s/S T debtissuedatt =0isdenotedq(cid:96) andthemarketpricedenotedby p(cid:96). Short-termdebtmay ς beissuedatt =0,1. Thequantityofshort-termdebtissuedatt =0isgivenbyq andthe 0 ς quantity of short-term debt issued at t =1 is given by q , s=U,D. The prices of shorts ς ς ς term debt at t = 0,1 are respectively p ,p , and p . Following much of the literature, 0 U D we assume equal seniority between short- and long-term debt. All market prices of debt securitieswillbedeterminedthroughequilibriummarketclearingconditions. The key friction in our model is that agents cannot be coerced to repay debts. As in RampiniandVishwanathan(2010)andFostelandGeanakoplos(2016),collateralservesas the payment enforcement mechanism. Specifically, creditors have the right to confiscate debtor collateral up to the value of the promise but nothing more. “Collateral” in our economy will be the firm itself, and can be thought of as the physical assets it produces from its investment decision. The collateral value of a debt contract is given by a statecontingent delivery function, d ( ). Implicitly we are assuming there are no collateral S T · cashflowproblemsinthatallagentsanticipatethestate-contingentvalueofcollateral(see FostelandGeanakoplos(2015)). Wereturntothedebtdeliveryfunctionsinsection2.4. 2.3 Agents Wefirstdescribethefirmanditsobjective,followedbytheinvestors’problem. 2.3.1 Firm We assume a representative firm is owned and operated by a manager (equity claimant) with access to a two-period decreasing returns to scale production technology.11 The production function is denoted by f (I;α,A ) = A Iα,α < 1 where I is the amount of s s capital the manager raises and puts into production. We assume the firm has no cash 11Allagencyproblemsareresolvedwiththeassumptionthatthefirmisownedandoperatedbythesame agent. Thisisdonetocontrastourmaturityresultsfromextantagency-basedmodels. 11

endowment,doesnotgeneratecashflowatt =1,andthatnewpromisesissuedatt =1do notscaletheproject’soriginalsize.12 The firm’s objective is to maximize expected profits by choosing how much capital to raiseandthematurityofthedebtcontractsitissues.13 Letρ denotetheportionofdebtthat is raised long-term, ρ = p(cid:96)q(cid:96) , and let γ denote the probability of good news.14 Formally, I thefirmmaximizesthefollowingproblem:  (cid:26) (cid:27)      m I, a ρ x∏= max γ (cid:0) Iα − q(cid:96) − q U ς (cid:1) +(1 − γ)γ (cid:0) Iα − q(cid:96) − q ς D (cid:1) ,0      s.t. I = p(cid:96)q(cid:96)+p ς q ς 0 0 (1)      p ς s q ς s =q ς 0 ,s=U,D       0 ρ = p(cid:96)q(cid:96) 1 ≤ I ≤ Conditionalons=U,γ fullycharacterizesthefirm’sdecisionforbothterminalstates, because firm always repays. At t = 1 the firm must decide whether it is beneficial to roll-over the short-term component of its debt portfolio. The firm repays short-term debt ς ς ς holders by raising p q = q ,s = U,D . At s =U, the firm can always repay debts s s 0 { } and p ς = p ς =1.Thefirmowesq ς +q(cid:96) att =2. Ats=D,thereisuncertaintyregarding 0 U U whetherthefirmcanrepaydebtatt=2.Inthiscase, p ς <p ς =1andthefirmowesq ς +q(cid:96) D 0 D 12Alternatively, one could assume that there is an extreme form of limited commitment at the interim dateinwhichnocashflowscanbeverifiedatareasonablecostsodebtrepaymentscannotcomefromcash flow. Underthisalternative,cash-flowisindependentofhowtheprojectisfinanced. Thedebtmaturitymix will affect the investment cost that generates the cash-flow. Even if management could abscond with all intermediatecash,theywouldstillissuethetypesofdebtsecuritiesthatwouldreducecoststomakehigher profitsinthefinalperiod. 13Werestricttheanalysistodebtissuanceanddonotallowforequityfinancing. Thisallowsustofocus theanalysisentirelyontheendogenouscompositionofdebtissuanceintermsofthedebtliabilitystructure. Incorporatingequityisanaturalextensiontothemodel. 14Wewillshowthatγ doesnotdeterminethegeneralexistenceofmultiplematuritiesasanequilibrium outcome.Thefactthatγisknowntothefirmandmaynotbeequivalenttothemarginalinvestor’sexpectation of good news is not completely without loss of generality. γ will determine the relative amount of longversus short-term claims, 0<ρ <1, that makeup the optimal debt liability structure. However, one can solve the model by restricting γ to almost surely equal the marginal buyer’s expectation so that there is a “true”stateprobability. Thisapproachwillpindownauniqueρ forallA ratherthanhaveastate-space DD consistingof(A ,γ)-pairs. DD 12

at t =2. In the event of default, the firm makes no profits and all assets are distributed to creditors pro rata. The maximization problem is subject to the following constraints: the amountofcapitalthefirmcanuseforproductionhastoberaisedbyissuingbondsatt =0. Conditional on rolling over short-term debt at t =1, the firm issues new short-term debt, ς ς ς q = p q ,s = U,D . Lastly, ρ, is bound between 0 and 1. To derive the firm’s first 0 s s { } p(cid:96)q(cid:96) order conditions for a maximum, first use the definition of ρ = to write the problem I intermsofchoicevariablesI andρ : (cid:32) (cid:33) (cid:26) (cid:18) (cid:19) (cid:27) ρI (1 ρ)I ρI (1 ρ)I max∏=max γ Iα − +(1 γ)γ Iα − ,0 . I,ρ − p(cid:96) − p ς − − p(cid:96) − p ς 0 D If an interior maximum for ρ exists, the first order necessary conditions with respect to I andρ respectively,are (cid:104) (cid:105) 2 ρ 1 (1 γ) (cid:20) (cid:21) (cid:104) (cid:105) (1 ρ) γ(1 γ) αIα 1 1 (1 γ) 2 = − − + − γ+ − (2) − − − p(cid:96) p ς p ς 0 D (cid:104) (cid:105) 2 1 (1 γ) (cid:20) (cid:21) 1 γ(1 γ) − − = γ+ − . (3) p(cid:96) p ς p ς 0 D The necessary conditions for the corner solutions are easily obtained by plugging either ρ = 0or1 into the firm’s maximization problem–there will be no first order condition with respect to ρ. Equation (2) says that the marginal product of capital in states 2 where the firm makes profits–which occurs with probability 1 (1 γ) –must equal the − − maturity-weighted expected marginal cost of debt. The marginal cost of long-term debt [1 (1 γ)2] is given by − − and the marginal cost of a sequence of short-term bonds is given p(cid:96) (cid:104) (cid:105) by p 1 ς γ+ γ(1 p −ς γ) . With probability γ, the firm will issue a sequence of risk-free bonds U D ς (p = 1). With probability γ(1 γ), the firm pays a higher short-term borrowing cost U − ς ς ς p < p = p = 1 per bond to roll over existing claims. Equation (3) says that in an D U 0 interior debt maturity optimum, the marginal cost of a long-term bond must equal the 13

marginalcostofasequenceofshort-termbonds. Intuitively,ifmarginalcostofonematurity is lower than the other, the firm will find it optimal to always issue the less expensive maturity,raisemorecapital,andmakehigherprofits. Butsincethemarginalcostsmustbe equivalentinanyinterioroptimum,wecancombineequations(2)and(3)intoasimplified versionofeitherlong-orshort-termdebtrespectively: 1 αIα 1 = , (4) − p(cid:96) (cid:20) (cid:21) (cid:16) (cid:17) 1 γ(1 γ) αIα 1 1 (1 γ) 2 = γ+ − . (5) − ς ς − − p p 0 D The equilibrium debt maturity the firm chooses will be determined by the relative market prices of risky debt securities, (cid:0) p(cid:96),p ς (cid:1) . In order to find debt prices and characterize D equilibrium,wemustsolvetheinvestors’problemwiththedebtdeliveryfunctionsforthe differentmaturitystrategies. 2.3.2 Investors There exists at t = 0 a continuum of uniformly distributed investors with unit mass, h ∈ H U[0,1],eachofwhomisendowedwithaunitofthedurableconsumptiongoodinall ∼ non-terminal states, eh,eh,s = S . The uniform distribution allows one to rank investors s T (cid:54) according to the likelihood each places on the subsequent state being good, denoted by h. Investors are risk-neutral, expected utility maximizers that consume at t = 2, and do not discount the future. Without loss of generality, we assume investors have different priors (seeFostelandGeanakoplos(2015)).15 Investorsalsohaveaccesstoarisklessstoragetechnologyandformportfoliosconsisting of cash and bonds purchased from the firm. The von-Neumann-Morgenstern prefer- 15One could assume investors differ in a measure of risk aversion; have different endowments across states,whichproducesdifferentmarginalutilitiesacrossstates;orhavedifferentdegreesof“patience.” The critical assumption is the heterogeneity of marginal utilities across investors. We choose to think about beliefsbecauseitismostfamiliarinthesemodels. 14

encesaregivenby: Uh (x ,x ,x ,x ) =h2x +h(1 h)x +(1 h)hx +(1 h) 2x . (6) UU UD DU DD UU UD DU DD − − − We now characterize the investors’ budget sets. Given debt prices, (cid:0) p(cid:96),p ς ,p ς ,p ς (cid:1) , each 0 U D (cid:110) (cid:111) investor,h H,choosescashholdings, (cid:8) xh,xh,xh(cid:9) ,debtholdings, q(cid:96),h,q ς,h ,q ς,h ,q ς,h , ∈ 0 D U 0 U D and final period consumption decisions, (cid:8) xh(cid:9) ,s S , to maximize utility given by (6) s T ∈ subjecttothebudgetsetdefinedby: (cid:16) (cid:17) (cid:110)(cid:16) (cid:17) Bh p(cid:96),p ς ,p ς ,p ς = x ,x ,x ,q(cid:96),q ς ,q ς ,q ς ,x 0 U D 0 D U 0 U D s h H ∈ xh +p(cid:96)q(cid:96),h +p ς q ς,h =eh , 0 0 0 0 xh +p ς q ς =eh +q ς d (cid:0) q ς(cid:1) +xh U U U 1 U U 0 0 xh +p ς q ς =eh +q ς d (cid:0) q ς(cid:1) +xh D D D 1 D D 0 0 (cid:16) (cid:17) (cid:111) xh =xh +x h 1 +q(cid:96)d q(cid:96) +q ς d (cid:0) q ς(cid:1) ,s S . (7) s 0 U,D s s s s ∈ T Each investor may use their initial cash endowment to purchase either type of debt at t = 0. The endowment received at t = 1 plus any returns from short-term debt holdings and cash carried forward are used to either purchase short-term debt at t = 1 or held for finalconsumption. Allcashthatisnotusedtopurchasedebtiscarriedforwardtoconsume att =2. Allfinalperiodconsumptioncomesfromdebtpurchasesandcashholdings. Itisclearthateachinvestorwillchoosethedebtmaturitythatdeliversrepaymentinthe state that the investor finds most likely. And because the contracts that investors purchase are debt contracts that pay out 1 in repayment states, the most optimistic investors simply purchase the debt security that they can purchase in the largest quantity. Put differently, theoptimistspurchasethecheapestdebtsecuritieswiththehighestexpectedyield. Consider a conjectured equilibrium with both long- and short-term debt: at t = 0, optimistswillusealloftheirendowmenttopurchaselong-termbonds,whilerelativepes- 15

simists will hold a mix of short-term debt and cash. If the firm issues short-term debt that it rolls over at t =1, the optimists will use their t =1 endowment to purchase risky short-term debt at s = D as well. Alternatively, consider a conjectured short-term only debt equilibrium: all investors at t = 0 will hold a portfolio of safe short-term debt and cash,andoptimistswillusealloftheirt =1endowmenttopurchaseriskyshort-termdebt at s = D. Lastly, consider a conjectured long-term only debt equilibrium: optimistic investors at t =0 will purchase long-term debt and all other investors will remain in cash. All investors will use their t = 1 endowment for consumption because no further bonds areissued. In order to close the model and characterize which type of debt securities will trade, we must determine exactly how investors price long- and short-term debt based on the recoveryvalueofdebtgivendefault. 2.4 Debt repayment Due to the repayment enforcement friction, debt is effectively collateralized by future output as in Rampini and Vishwanathan (2010) and Fostel and Geanakoplos (2016). The implicitassumptionunderlyingthesemodelsisthattherearenocollateral“cash-flowproblems.”16 Short-termDebt (cid:0) ς(cid:1) Let d q describe short-term debt delivery at time 1. Short-term debt repayment U,D 0 isconditionalonwhetherthefirmrollsoverdebtatt =1.Specifically,short-termdebtwill be“safe”ifitisalwaysrolledover,andbothrecoveryandpricesareequalto1;otherwise, short-term debt will be risky due to potential liquidation. To highlight the novelty of investorheterogeneityratherthanliquidationriskalaDiamond(1991),weassumethereis 16Traditionalmacro/financemodelssuchasKiyotakiandMoore(1993)assumethatcreditorscanconfiscateland,butnotthefruitproducedbytheland. CorporatefinancemodelsfollowingHolmstromandTirole (1997) assume an information asymmetry between borrowers and lenders. Borrowing too much in these modelsreducescash-flowandreducesincentivestoworkhardtoproducegoodcashflows. 16

Figure2: LongTermFinancing 1 h h (1 h) 1 p` h 1 (1 h) (1 h) d (q`) DD t=0 t=1 t=2 no inefficiency from default in any state. Risky debt, regardless of its maturity, is always fairlypriced. Assuch,liquidationdoesnotprovideanyadditionalinsights,andweproceed byassumingdebtisalwaysrolledover. Welaterderivethatrollingovershort-termdebtis indeedoptimal.17 ς The firm must issue q one-period debt contracts to rollover expiring claims. Shorts termdebtdeliveryinthefinalperiodisstate-contingentwithdefaultonlyats=DD.18    1, s=DD d (qς)= (cid:54) . (8) DD    A qς D + D I q α (cid:96) , s=DD D Equation (8) says that all debts are honored as long as two periods of bad news do not occur. Firms default after two periods of bad news due to the technology shock and all 17Flannery(1986)considersonlysafeshort-termdebtatt=0aswell. Hisanalysishighlightstheimportanceonasymmetricinformationindeterminingdebtmaturity. 18ThisissimplyrestatingabsolutepriorityalaMerton(1974)viaacollateralconstraint. Equityreceives nothingwhendebtholdersarenotrepaidexpost, butcollateraldeliveryisrequiredtoobtaindebtexante. We will show that A <α is sufficient for d ()<1, and consider this parametrization throughout the DD DD · papertofocusonriskydebt. 17

Figure3: Short-TermFinancing h 1 p& h 1 U p& (1 h) 1 0 h 1 (1 h) 1 p& D dDD r ( e q p D & a ) y d m e e n n o t t f e u s n ction (1 h) dDD(q D &) t=0 t=1 t=2 firm assets are divided pro rata to debt holders. The sequence of short-term debt contract payoutsisdepictedinfigure3. Long-termDebt Long-term debt is very simple to describe as it matches the maturity of assets with (cid:0) (cid:1) liabilities. Letd q(cid:96) denotethelong-termdebtdeliveryfunction. Givenourassumptions s onshort-termdebtbeingrolledover,bothlong-andshort-termdeliveriesaregivenby(8), (cid:0) (cid:1) d q(cid:96) =d (qς), or generically d ( ),s S .19 Equation (8) gives the recovery value of s s s T · ∈ ς firmassetsforanyofthepossibledebtmaturityequilibriabysimplysettingeitherq =0 D for a long-term only equilibrium or q(cid:96) = 0 for a short-term only equilibrium. We now characterizeequilibrium. 2.5 Equilibrium Definition1 Equilibrium is a collection of the following non-negative items: debt prices, firm investment decision, investor cash holdings, debt holdings, and final consumption 19We are implicitly assuming that the collateral value of the firm is not being split as explicit collateral pieces for long- versus short-term debt. We return to this assumption in section 4 of the paper when we considerrestrictivenegativecovenants. 18

decisions that maximize firm profits given by (1), investor utility given by (6) subject to theirbudgetconstraintsin(7),andboththegoodsanddebtmarketsclearinallperiods. To highlight the importance of heterogeneity and catering debt securities to investors, we begin by solving the special case of the model with homogeneous investors. We then show how moving to heterogeneous investors delivers both similar predictions as Diamond (1991) and the new prediction that issuing both a combination of debt maturities is generallytheleastcostlyfinancingoption,evenabsentliquidityrisk. 2.5.1 Examplesofinvestorbeliefsanddebtmaturitychoice Example1 Homogeneousinvestorsanddebtmaturityirrelevance. Assume a unit mass of competitive investors all share the common prior that the up stateoccurswithprobability,γ.Theonlychangeacommonbeliefchangesishowinvestors valuedebt. Wewillusethefollowingparametersthroughouttheexamples: A =.5,α = DD 0.8,γ = 0.7. There is no risk of liquidation at t = 1 so that all short-term debt issued at t =0isriskfree. Long-term–If long-term debt is the only security traded, ρ = 1. A competitive equilibrium requires that investors make zero profits in expectation. If γ is the probability of s=U,thenlong-termdebtisvaluedbyallinvestorsaccordingto (cid:16) (cid:17) (cid:16) (cid:17) 1 (1 γ) 2 1+(1 γ) 2d q(cid:96) = p(cid:96). (9) DD − − − Oneneedstosolveforthedebtdeliveryfunction,(8),withρ =1inordertopricethedebt security. From the firm’s funding condition, 1 = p(cid:96) . Lastly, use (4) to solve for Iα 1 to q(cid:96) I − arriveatthefollowing: (cid:16) (cid:17) A d q(cid:96) = DD . (10) DD α 19

Table1: Long-termhomogeneousequilibrium (cid:0) (cid:1) MC(cid:96) p(cid:96) I V(cid:96) Π(cid:96) d q(cid:96) γ γ γ γ (α,A γ)=(.8,.5,.7) 1.034 .9663 .2760 .3570 .0650 .625 DD, Intuitively, the less production is affected by the technology shock (high A ), the more DD assets are available for investors to recover. Likewise, the more output a firm generates perunitofinvestmentcapital(lowα),themoreinvestorsrecoveronaperclaimbasis. We definethefollowingobjectsforcomparisonacrosseconomies: themarginalcostofissuing long-term debt, 1 , investment, I = (cid:0) αp(cid:96) (cid:1) 1 1 α, the value of the firm output,V(cid:96) =Iα, and p(cid:96) − γ (cid:16) (cid:17)(cid:16) (cid:17) expected profits are Π(cid:96) = 1 [1 γ] 2 V(cid:96) q(cid:96) , where the superscript (cid:96) denotes the γ γ − − − long-term debt regime and the subscript γ denotes all agents’ homogeneous belief. Table 1containsthevalueofthekeyobjects. Short-term–If short-term debt is the only security traded, ρ =0. All risky short-term debtpurchasedats=Dmustyieldzeroprofittoinvestorsinexpectation: (cid:0) ς (cid:1) ς γ1+(1 γ)d q = p . (11) DD D D − Following the same proceedure laid out in the long-term debt regime, the debt delivery functionforshort-termdebtfunding,(8),becomes (cid:32) (cid:33) ς d (cid:0) q ς (cid:1) = A DD p D γ+γ(1 − γ) . (12) DD D α 1 (1 γ) 2 − − (cid:124) (cid:123)(cid:122) (cid:125) dilutionfactor Debtdeliverywithshort-termfundingistheproductoftwocomponents. Thefirstcompo- A nent is the same fundamental recovery value as long-term debt delivery, DD. The second α component represents the dilution effect of issuing more short-term debt in expectation pς γ+γ(1 γ) at t = 1 to honor expiring claims. The dilution effect is given by, D − < 1. The 1 (1 γ)2 (cid:104) − (cid:105)− marginal cost of short-term debt is given by MC γ ς = 1 (1 1 γ)2 γ+ γ(1 p −ς γ) . Solving (11) − − D 20

Table2: Short-termhomogeneousequilibrium ς ς ς ς (cid:0) ς (cid:1) MC p I V Π d q γ D γ γ D (α,A γ)=(.8,.5,.7) 1.034 .8685 .2760 .3570 .0650 .524 DD, ς and(12)simultaneously,gives p =.8685,andarecoveryrateof.5242.Table2provides D thevaluesofthekeyobjectsfortheshort-termdebteconomy. The examples show that the economies are equivalent from the firm’s perspective. Intuitively, the firm’s efficient investment scale is given by the expected marginal cost of issuing debt. The expected marginal costs are equivalent across the two economies (MC ς =MC(cid:96) =1.034) because all agents have the same common information. The dilu- γ γ tion effect of short-term debt is correctly priced in expectation and the firm is indifferent between issuing long- or short-term debt. However, the firm is not indifferent between issuing a combination of long- and short-term debt and issuing either long- or short-term debt. Thereasonisthatassoonasthefirmissuesanycombinationoflong-andshort-term debt,thedilutioneffectofshort-termdebtreducesthepriceoflong-termdebtasinvestors will not pay the same price for a security whose value falls in expectation. Moreover, the more short-term debt the firm issues, the more long-term bond prices must fall due to the dilution effect. Put differently, risky long- and short-term debt prices positively co-move with homogeneous investors because of debt dilution. In order for the firm to issue both long- and short-term debt, the expected costs of the two debt securities must be equal.20 But the only way that the expected costs can be equivalent is if the prices are lower than what they are in either corner solution, because the prices positively co-move. Therefore, the firm cannot operate at the same efficient scale by simultaneously issuing both longandshort-termdebt. Example2 Debtmaturitywithheterogeneousinvestors Now consider the heterogeneous investor case described in section 2.3.2. All of the 20Seeequation(3). 21

Table3: Long-termheterogeneousequilibrium (cid:0) (cid:1) MC(cid:96) p(cid:96) h I V(cid:96) Π(cid:96) d q(cid:96) h 0 h h (α,A γ)=(.8,.5,.7) 1.030 .9702 .7182 .2817 .3629 .0665 .625 DD, firm’s first order conditions for an optimal are the same. The only thing that changes are the debt pricing equations. We ask the following question: Is the firm indifferent between short- and long-term debt when investors are heterogeneous, and if not, what will be the equilibriumfinancingstructure? Long-term–The marginal long-term bond buyer at t = 0 must be indifferent between buying the bond and cash. The break even condition for the marginal long-term bond buyeris: (cid:16) (cid:17) 1 (1 h ) 2 +(1 h ) 2d q(cid:96) = p(cid:96). (13) 0 0 DD − − − Inequilibrium,long-termbondpricesaredeterminedbymarketclearinginthebondmarket. Specifically,investordemandforriskybondsmustequalthesupplyofbondsthefirm issues: 1 h − 0 =q(cid:96),Long-termdebtmarketclearing (14) p(cid:96) (cid:0) (cid:1) There are four unknowns in this economy: I,p(cid:96),q(cid:96),h , and four equations: (4), (13), 0 (14),andI = p(cid:96)q(cid:96).Thevalueoftheendogenousvariableinthiseconomyarecontainedin table3. Noticethevalueoffirmoutputandprofitsarehigherundertheheterogeneousregime: Π(cid:96) = .0660 > Π(cid:96) = .0650 and V(cid:96) = .3629 >V(cid:96) = .3570. This is because the marginal h γ h γ investor’s prior is higher than the common belief h = .7182 > γ = .70, which leads to lower credit spreads and more investment. The results show that the firm benefits when issuing debt in a heterogeneous agent economy compared to the homogeneous investor economy. 22

Table4: Short-termheterogeneousequilibrium ς ς ς ς (cid:0) ς (cid:1) MC p h I V Π d q h D 1 h h D (α,A γ)=(.8,.5,.7) 1.0318 .8786 .7199 .2800 .3612 .0657 .5286 DD, Short-term–The marginal short-term debt holder at t =1 must be indifferent between riskyshort-termdebtandcash: (cid:0) ς (cid:1) ς h +(1 h )d q = p . (15) 1 1 DD D D − The short-term debt market must clear at both t = 0,1. Without liquidity risk, all short- ς term debt is initially risk free, p =1. Risk-free short-term debt implies that all investors 0 ς ς at t =0 hold a combination of short-term debt and cash. The firm must issue q =q in U 0 ς qς the good state and q = 0 in the bad state to ensure short-term debt is risk free. This D pς D reflects the fact that, conditional on s =U, all short-term debt is risk free and repaid, but conditional on s = D the face value of short-term debt must rise to clear the market. Conditionalons=D,investordemandforriskybondsmustequalsupply: 1 h 1 ς − =q ,Short-termdebtmarketclearing. (16) ς D p D Market clearing, (16) along with the pricing equation, (15), the debt delivery function, ς ς (12), and the firm’s funding condition, I = p q can be simultaneously solved for the 0 0 (cid:0) ς ς (cid:1) equilibrium endogenous variables in the short-term debt economy, I,p ,q ,h . The D D 1 solutiontothiseconomyisgivenintable4. Unlike the homogeneous investor case, the value of firm output is lower under the short-termdebtregimethanthelong-termdebtregimefortheexactsamesetofparameters. To understand why debt maturity affects the value of firm output, note that, for a given investment, I, the expected marginal product of capital across the two economies will be 23

thesame: (cid:16) (cid:17) E [MP]=αI(α 1) 1 (1 γ) 2 . 0 − − − However, in general, the expected marginal costs across the economies will not be the same. The equilibrium price of long-term debt is determined by market clearing in equation (14). Likewise, market clearing for short-term debt is given by (16). It must be the case that h = h if the firm operates at the same scale if debt maturity is irrelevant. 0 1 However, there is more uncertainty at s = D than at s = 0, which means that the same investor will never price risky short- and long-term debt equivalently. In fact, the same investor will always price risky short-term debt at s = D lower than long-term debt at s=0;otherwise,short-termdebtwillalwaysdominatelong-termdebt. Thefirm’sfinancial policyinfluencesthevalueoffirmoutputandtheefficientinvestmentscalewheninvestors areheterogeneousandthepriceofriskistimevarying. The following example asks whether or not debt liabilities can be more efficiently structured by spreading risk across time through a combination of long- and short-term debt. Example3 Optimaldebtmaturitymixwithheterogeneousinvestors In this example, the firm can choose 0 < ρ < 1 in addition to either corner solution. This economy has eight unknowns (cid:0) I,ρ,p(cid:96),q(cid:96),p ς ,q ς ,h ,h (cid:1) and eight equations, seven D D 0 1 24

ofwhichareacollectionofequationsfromthelong-andshort-termdebtfundingregimes: 1 αIα 1= ,combinedfirstorder − p(cid:96) (cid:16) (cid:17) 1 (1 h )2+(1 h )2d q(cid:96) =p(cid:96),long-termdebtpricing 0 0 DD − − − h +(1 h )d (cid:0) qς(cid:1) =pς ,short-termdebtpricing D − D DD D D 1 h − 0 =q(cid:96),long-termdebtmarketclearing p(cid:96) 1 h − D =qς ,short-termdebtmarketclearing pς D D I=p(cid:96)q(cid:96)+pςqς ,firmfundingcondition D D p(cid:96)q(cid:96) ρ = ,long-termdebtportion. I The only new equation is the definition of ρ–the long-term portion of total debt. Notice the first order conditions for an interior optimum can be collapsed into a single equation, because in expectation, the marginal costs of long- and short-term debt must be equivalent.21 The eighth equation is the debt delivery equation. Through the same procedure as the long- and short-term debt economies, the recovery value of all debt in the mixed maturityeconomyisgivenby: (cid:18) ς (cid:19) A p d ( )= DD D . (17) DD · α (1 ρ)p(cid:96)+ρp ς D − (cid:124) (cid:123)(cid:122) (cid:125) dilutionfactor Thedilutionfactorisafunctionofriskydebtpricesandthedebtmaturitymixdetermined byρ. The solution to this system is found in table 5. First, all bond pricing is significantly higher under the maturity mix regime than either of the respective corner solutions for the same parameters. The two marginal buyers in the multiple debt maturity regime are h =.8227 and h =.8690. Both marginal buyers are significantly more optimistic about 0 1 firmcashflowsthaneitheroftheircounterpartsinthecornersolutionregimes. Firmscater 21Collapsingintothelong-termfirstorderconditioneasestheexposition. 25

Table5: Maturitymixheterogeneousequilibrium pς p(cid:96) MC I ρ h h V Π d( ) D 0 1 · Maturitymix .9495 .9879 1.012 .3083 .5752 .8227 .8690 .3900 .0710 .6159 Long-term - .9702 1.030 .2817 1 .7182 - .3629 .0665 .625 Short-term .8786 - 1.032 .2800 0 - .7199 .3612 .0657 .5286 riskydebtsecuritiestoinvestorsacrosstimeandreceivebetterpricesintheheterogeneous investor economy. Second, this has real positive effects on the value of firm output and investment. Firms are able to finance more investment and production, which increases the firm’s scale. Third, the recovery rate in the maturity-mix regime is slightly lower than the fundamental value of the recovery rate in the long-term regime. Interestingly, debt dilution can actually raise the value of firm ouput. The intution for this result is that, on the one hand, short-term debt dilutes long-term debt under equal seniority, which lowerslong-termdebtprices. Ontheotherhand,short-termdebtsubstitutesforlong-term debt and concentrates the placement of long-term debt to investors with higher marginal valuations, which raises long-term debt prices. In this example, the substitution effect dominates the dilution effect and all prices rise compared to issuing only long- or shortterm debt. We show in the following section that the substitution effect always dominates the dilution effect and a maturity mix is generally the optimal funding regime for firms facingheterogeneousinvestors. 3 The general debt maturity solution with heterogeneous investors The examples in the previous section show that heterogeneous investors alone lead to equilibrium outcomes where firm debt maturity choice matters in a novel way. There is no liquidation risk in the examples making the model similar to Flannery (1996) and the special case of Diamond (1991), but the optimal debt maturity choice is very different. In 26

those models, short-term debt is the unique maturity choice. A combination of long- and short-term debt emerges in our model. In this section we show that no liquidation is an endogenousoutcomeandthatamixoflong-andshort-termdebtisingeneraltheoptimal debtmaturitychoice. Short-term debt is “safe” at t = 0 if and only if it is unconditionally rolled over at t =1. The rollover condition states that profits must be greater than or equal to zero after repayingbothlong-andshort-termdebts: Iα q(cid:96)+q ς . (18) D ≥ Wecanfocusonlyonthedown-statewithoutlossofgeneralitybecausethefirmisalways better off conditional on s=U than s=D. Note that state probabilities, γ, do not factor in this decision because the firm only retains profits when it fully repays all debts, both of which occur with probability 1 (1 γ)2. The price of short-term debt at t =0 must be − − ς p = 1 if the firm continues and produces at t = 2. Lemma 1 states the condition when 0 short-termdebtisalwayssuccessfullyrepaidandthereisnoliquidationinequilibrium. Lemma1 Short-term Debt Rollover: Suppose Q= (cid:0) q(cid:96),q ς ,q ς(cid:1) >0,s= U,D . Short- 0 s { } termdebtatt =0issafeifandonlyif ς 1 ρ p ε α − D <1. (19) ≡ (1 αρ) ≤ p(cid:96) − Safe short-term debt is possible in equilibrium as long as there is a balance between the price of risky long-term debt today versus risky short-term debt conditional on bad pς news tomorrow, D. The firm will always choose short-term debt if there is no difference p(cid:96) between long-term and risky short-term debt because short-term debt is financed risk free conditional on s=U. Therefore, in expectation, short-term debt will dominate long-term debtiftheriskycomponentofshort-termdebtisthesameaslong-termdebt. Alternatively, 27

risky short-term debt prices increase as the portion of financing through short-term debt rises. This drives a wedge between long- and risky short-term debt prices, and short-term debtwillberelativelyexpensiveinexpectationcomparedtolong-termdebt. To gain some intuition for when equation (19) holds, let ρ vary in the limit between 0 pς and1. Therelativepriceofriskydebt, D,att=0,1isdeterminedbyinvestorexpectations p(cid:96) at each point in time, h and h . For any given investment, I, ρ approaches 1 as more 0 D long-term debt is chosen. Substitution into long-term debt does two things: 1) it lowers long-term debt prices at t =0 as more pessimistic investors finance investment; and 2) it reducesshort-termdebtissuance,whichlowersthedilutioncosttolong-termdebtholders. Thus (19) holds trivially. Alternatively, let ρ approach 0, which increases short-term debt issuance. Substitution into short-term debt has the opposite effect on relative debt prices, p ς fallsand p(cid:96) rises. Aslongastheratioofriskydebtprices, pς D,isgreaterthanameasure D p(cid:96) offirmproductionandcollateral,α,equation(19)holds,whichbecomestrivialasα falls. Intuitively, α measures the return to a unit of capital input (firm marginal productivity rises as α falls, 0 <α <1). Higher marginal returns to production make it very easy to repay small amounts of debt. This reasoning implies that there is an upper boundary on α¯ (cid:47) 1 for which equation (19) holds with equality from below. While we cannot give a precise mathematical expression for α¯, numerical simulations suggest α (cid:47) 0.9 satisfies thecondition. Proposition1 Multiple debt maturity structure: When there is no liquidity risk and investorsareheterogeneous,amixoflong-andshort-termdebtisoptimal,Q (cid:0) q(cid:96),q ς ,q ς(cid:1) > ≡ 0 s 0,s= U,D . { } For any given investment, I, moving from ρ = 1 to ρ < 1 has two benefits. First, issuing a mix a debt maturities allows a more optimistic buyer to price long-term debt at t =0, which lowers the cost of long-term capital. Second, the portion of long-term debt thatissubstitutedforshort-termdebtallowsthefirmtoborrowriskfreeatt =0andgives 28

thefirmtheopportunitytoborrowriskfreeatt =1,bothofwhichloweroverallfinancing costs. The costs of substituting into short-term debt are: 1) borrowing costs rise as more short-term debt is issued, and 2) the introduction of the dilution effect on long-term debt. The dilution effect is increasing in the amount of short-term debt issued and helps temper theincreaseinlong-termbondprices. For the same investment, I, moving from ρ = 0 to ρ > 0 lowers short-term rollover costs and reduces the dilution effect, both of which represent the substitution benefit of moving into long-term debt.22 The substitution cost is that the firm must pay a positive credit spread on long-term debt rather than the risk free rate on short-term debt at t =0. However, long-term debt buyers are generally optimists meaning that long-term credit spreads are relatively low and close to the risk-free rate. Thus, on the margin, the cost of substituting into long-term debt is very small compared to the benefit of reducing both short-termcreditspreadsandthedilutioneffect. Figure4capturestheessenseofthebenefitsofsubstitutingfromasingledebtmaturity to a combination of debt maturities. Using a combination of debt maturities concentrates fewer total long-term bonds to investors most willing to hold risk att =0 than a maturity ς withnoshort-termdebt,q =0.Concurrently,thedebtneededtoensureshort-termdebtis 0 rolled over att =1 is also more concentrated to investors with higher willingness to hold ς risk than if the firm only issued long-term debt, q =0. A combination of debt maturities 0 reallocatesriskydebtawayfrominvestorstoday,whorequiremorecollateraltoborrowat a given credit spread, to investors tomorrow who require less collateral to borrow at the same rate, and vice versa. Therefore, the firm is able to optimize the amount of risky debt itissuesateachpointintime. Foragivenamountofpledgablecollateral,issuingamixof debt a maturities allows the firm to invest and produce more than if it issued a single debt maturity. 22Equation(17)collapsesto ADD asρ goesto1. α 29

semigerreyublanigraM :4erugiF 1= ℎ 1= ℎ tbed mret-trohs yksiR ,sredloh mret-gnol yksiR lan 1 ig < ra 𝜌𝜌 M <0 mret-trohs yksiR ,sredloh tbed mret-gnol yksiR ,reyub ,sredloh tbed ,sredloh tbed 𝐷𝐷ℎ 1<𝜌𝜌<0 lanigraM 1 tb<ed 𝜌𝜌mr<et-0trohs yksiR lan0igr=aM𝜌𝜌 ,reyub 1=𝜌𝜌 yksir rof detutitsbus ,tbed mret-gnol ,reyub 0ℎ lanigraM 𝐷𝐷ℎ 1<𝜌𝜌<0 ,reyub mret-gnol yksiR 1<𝜌𝜌< hs0 a C 0=𝜌𝜌 rof detutitsbus tbed 0ℎ mret-trohs efas -trohs efaS 1=𝜌𝜌 ,tbed tbed mret 1<𝜌𝜌<0 hsaC & ,hsaC hsaC 1=𝜌𝜌 0=𝜌𝜌 1= 𝑡𝑡 0= 𝑡𝑡 fdp.egamIemigeR 30

3.1 Debt maturity optimization and comparative statics Thissectionbrieflydiscussesthemodel’scomparativestaticresultsrelatedtohowthematurityprofileisoptimizedtowardlong-orshort-termdebtdependingonmodelparameters. Let 0 < ρ (γ,A ) < 1 denote the equilibrium amount of long-term debt issued for ∗ DD any given set of parameters. Specifically, γ is the likelihood that good news arrives in the followingperiod,fromthefirm’sperspective. A determinestheamountofcollateralthe DD firm can pledge at s=DD and is a measure of down risk, while α is the returns to scale parameter. Moreshort-termdebtisissuedthemorelikelygoodnewsarrivesint =1, ∂ρ <0. The ∂γ reason is that the likelihood of rolling over short-term debt at the risk-free rate increases, whichlowersexpectedrollovercostsrelativetolong-termfinancing. More short-term debt is issued the more collateral the firm can pledge at s = DD, ∂ρ <0. The reason is that risky short-term debt prices at t =1 are more responsive to ∂A DD movements in A than risky long-term debt prices at t = 0. To see why, consider any DD given investor, h. This investor puts more weight on s=DD at time t =1 than she does at t = 0, (1 h) > (1 h) 2 . The value of an investor’s claim at s = DD, irrespective of − − maturity, is the delivery rate given by (17). Investor h values the recoverable claim more att =1thanatt =0. Proposition2 With no liquidity risk and heterogeneous investors, debt maturity is optimizedmorelong-term: thelowerthelikelihoodofgoodstates,lowγ, ∂ρ <0; • ∂γ thelowerareexpectedcash-flowsorthehigherisdownrisk,lowA , ∂ρ <0; • DD ∂A DD 31

3.2 Discussion and interpretation In this section we discuss the predictions of the model relative to the existing heterogeneous agent models on which it is based, the general consistency of the model’s comparitive statics results with the broad empirical literature, and the new insights that reconcile differencesbetweenempiricaldebtmaturitystudies. Short-term debt is the unique funding outcome in the class of heterogeneous agent models of Geanakoplos (2003, 2009), Fostel and Geanakoplos (2008, 2010), and He and Xiong (2012a). In these models, all agents are endowed with both a risk-less and risky asset. Optimistswanttoholdmoreriskyassetsthanpessimists. Optimistspurchaseallthe risky assets by issuing a riskless claim backed by the maximum value of the asset in each period. Issuing a sequence of short-term claims allows optimists to borrow against the asset’s worst intermediate-state and terminal-state values. By contrast, long-term claims only allow agents to borrow against the value of the asset in the worst terminal state . Thus, for optimists who price the asset in equilibrium, short-term debt always dominates long-termdebt. Thesemodelsarebestsuitedtodescribedebtfinancingoffinancialassets forwhichtheuseofleverageisparamount. Banks,hedgefunds,andinstitutionalinvestors typicallyuseleveragetomaketheirassetpurchases. By contrast, the “firm” in our model is endowed with a risky production technology and issues debt backed by its technology–similar to Fostel and Geanakoplos (2016) and Rampini Vishwanathan (2010)–while investors have riskless assets that they use to purchase firm debt. Optimists, use their riskless asset to buy risky debt. The firm maximizes the expected value of equity value by concentrating risky claims across time because that ishowitmostefficientlyaccessesoptimisticinvestorcapital. Ourmodelisparticularlyrelevantforlargecorporationswhereliquidityriskandinformation asymmetries are likely second order concerns. Lemma 1 and proposition 1 imply that safe short-term debt should be used in conjunction with risky-long term debt because 32

it will help lower aggregate risky financing costs. This intuition rationalizes the existence of corporate commercial paper (CP) programs for large safe corporations. In our model, short-term CP is safe short-term debt issued att =0.The CP issuance must be refinanced at t = 1. This interpretation is consistent with the “bridge financing” findings of Kahl, Shivdasani, and Wang (2015). Moreover, safe-debt is sufficient for a debt maturity mix, which contrasts the liquidation risk stories underpinning Diamond (1991) and Houston andVenkataraman(1994). The predictions of our model are broadly consistent with existing empirical studies. For example, we can interpret γ as a measure of management “optimism.” Landier and Thesmar (2008) and Graham et. al (2013) find that management optimism leads to more short-term debt issuance, controlling for firm risk factors and leverage. Choi, Hackbarth, andZechner(2016)showthatcorporationstypicallyissuedebtinto,onaverage,morethan 3 distinct maturity bins, and that large and mature corporations are more likely to issue multiple debt maturities. Norden, Rooenboom, and Wang (2016) show that borrowing costs are lower and leverage is positively associated with debt granularity i.e. a mix of debtmaturitiesratherthanasingledebtmaturity. Empiricalstudiesmeasuregrowthoptionsasthemarket-to-bookvalueofassets. Inour model,themarketvalueoftheassetsistheamountthefirmproducesbecausethereisonly oneassetwhose price isnormalizedto1.Thebookvalue ofthefirm’sassetisthe amount ofcapitalitraisestoproduce,orthebookvalueofitsliabilities. Themarket-to-bookvalue ofthefirmisgivenby23 Iα 1 market-to-book= =I(α 1) = . (20) − I αp(cid:96)(α,A ,γ) DD Noticethatthegrowthoptionofthefirmisinextricablylinkedtotheexogenousparameters 23We use the first order conditions (2) and (3) to derive the market-to-book in terms of the long-term bondprice, p(cid:96). Itcanalsobeexpressedintermsofshort-termbondpricessincetheexpectedcostsacross maturitiesmustbethesameinaninteriormaturityequilibrium. 33

ofthemodelthroughthemarketpriceofdebt. Therefore,growthoptionsareendogenously determined along with the firm’s maturity choice and leverage through an asset’s fundamentalcollateralvalueviaα. This suggeststhatempiricalstudiesshouldnottreatgrowth options as exogenous to leverage and maturity choices, which is typical. The endogeneity of growth options with maturity choice may help explain why empirical debt maturity results are often mixed. Barclay and Smith (1995) and Guedes and Opler (1996) find that growth options and maturity are negatively related. Stohs and Mauer (1996) and Johnson (2003) find a positive relationship, while Billet et. al. (2007) find no relationship when controllingforcovenants. Lastly,investorheterogeneitygeneratesdifferencesincostofcapitalthroughtimeand helps reconcile puzzling survey evidence that firms try to time the market when choosing debt financing. For example, Graham and Harvey (2001) and Servaes and Tufano (2006) find that global CFO respondents largely issue debt maturity to time market interest rates and limit the amount of debt that needs to be refinanced at any point in time. Economists typically view a market timing response with circumspect. Our model suggests that debt maturity choice is not about market timing per se. Rather, debt maturity is used to smooth financing costs by limiting the amount of risky debt that firms issue at anypointintime. Lastly,bothsurveysfindverylittlesupportforinformationasymmetries (Flannery (1986)) and debt overhang (Myers (1977)) as the main factors driving maturity choice, while credit ratings are important insofar as they affect the terms of borrowing, butexpectationsaboutcreditratingchangesaresecondorderatbest(Diamond(1991)and HoustonandVenkataraman(1994)). 4 Protected debt with endogenous maturity Inthissectionweaskwhathappenswheninvestorsstructuredebtcontractswithcovenents thatpreventdebtdilution. Afterall,acombinationofdebtmaturitieswithoutanycovenents 34

does in fact lead short-term debt to dilute long-term debt. If one does not allow long-term debttobediluted,doesusingacombinationoflong-andshort-termdebthaveanybenefit relative to only long-term debt. The answer is yes. Protecting long-term debt raises the value any individual investor is willing to pay for the debt contract. The firm responds by re-optimizing its maturity more toward the relatively cheap long-term debt and away fromtherelativelymoreexpensiveshort-termdebt. Inequilibrium,themarginallong-term bond buyer will be more pessimistic but the recovery value is higher. These two effects cancel one another and the relative prices between long- and short-term debt remain the same. We begin by showing how to protect long-term debt from dilution and what the effect isontheequilibriumamountoflong-versusshort-termdebt. Wethenshowthatfirmvalue isunaffectedbyanon-financialcovenant. To see this, assume that a covenant ensures ρ portion of the firm assets are used as exclusive collateral for long-term debt, irrespective of short-term debt financing at t =1. Theseassetscannotbeusedascollateralforshort-termdebtwithoutviolatingthecovenant and opening the firm up to costly litigation. The remainder of the assets, (1 ρ), are − used as collateral to secure short-term debt. A natural interpretation of this covenant is a negative pledge covenant. Negative pledges are among the most common non-financial covenants found in public long-term debt indentures. A description of the covenant and its relevance can be found in the appendix. With the covenant, the recovery values given by(8)become    d DD (cid:0) q(cid:96) (cid:1) = ρA D q(cid:96) D Iα , long-termrecovery .   d DD (cid:0) q ς D (cid:1) = (1 − ρ q ) ς A DD Iα , short-termrecovery D Firmoutputindefaultissplitbetweenprotectedlong-termcreditorsandshort-termcreditorswhofundtheshort-termdebtrolloveratt =1.Followingthesameprocedureoutlined 35

intheexamplesection,onecanshowthatthedebtdeliveryfunctionsbecome:    d D ˆ D (cid:0) qˆ(cid:96) (cid:1) = A α DD . (21)   d D ˆ D (cid:0) qˆ ς D (cid:1) = A α DD (cid:16) p p ˆ ˆ ς D (cid:96) (cid:17) The hats represent variables in an economy with the covenant. The maturity specific recoveryvaluesin(21)behaveasifρ =1forlong-termdebtandρ =0forshort-termdebtin (17),eventhough0<ρˆ <1.Alllong-termdebtholdersareprotectedfromdebtdilution. Proposition3 Consider any dˆ (cid:0) qˆ(cid:96) (cid:1) defined by (21) with a secured covenant and cor- DD responding d ( ) defined by (17) without a covenant. In any debt financing strategy for D∗D · whichQ (cid:0) q(cid:96),q ς ,q ς(cid:1) >0,s= U,D ,thefollowinghold ≡ 0 s { } • d D ˆ D (cid:0) qˆ(cid:96) (cid:1) > d D∗D ( · ). Moreover, d D ˆ D (cid:0) qˆ(cid:96) (cid:1) = d DD (cid:0) q(cid:96) (cid:1)(cid:12) (cid:12) qς 0 =0 –long-term debt with securedcovenantsareprotectedfromshort-termdebtdilution. 1 − (1 − h 0)2+(1 − h 0)2[dˆ DD (q(cid:96))] > 1 − (1 − h 0)2+(1 − h 0)2[d DD( · )] –anygiveninvestoriswillingto • p(cid:96) p(cid:96) paymoreforlong-termdebtwithasecuredcovenantthanwithout. Animmediateimplicationofproposition3isthatdebtmaturitywillbeoptimizedmore long-term. Corollary1 Let ρˆ be the equilibrium portion of long-term debt when collateral values are determined by (21). Let ρ be the equilibrium portion of debt issued long-term debt ∗ whencollateralvaluesaredeterminedby(17). Itfollowsthatρˆ >ρ . ∗ Corollary 1 says that protective covenants act as substitutes for risky short-term debt. This prediction is consistent with the empirical findings of Billet et. al. (2007). Do these non-financial covenants affect firm value? The answer is no. The firm increases the supply of risky long-term bonds it issues but reduces the supply of risky short-term 36

bonds. In equilibrium, the relative prices of the two debt maturities must be equivalent in expectation(seeequation(2)). (cid:16) (cid:17) Proposition4 Let Q ∗ ≡ q(cid:96) ∗,q ς 0 ∗,q ς s ∗ ,s= { U,D } , ∀ q ∈ Q ∗ >0 be a set of equilibrium (cid:16) (cid:17) bond quantities with corresponding investment and price functions, I ∗ ,p(cid:96) ∗,p ς 0 ∗,p ς s ∗ , as thesolutiontoprogram(1)withdebtdeliveriesgivenby(17). Covenantsthatsecurelongterm debt and prevent dilution alter debt deliveries via (21). The resulting equilibrium withthesecuredcovenanthasthefollowingproperties: 1. The optimal debt financing strategy is given by Qˆ (cid:0) qˆ(cid:96),qˆ ς ,qˆ ς(cid:1) ,s= U,D , q ≡ 0 s { } ∀ ∈ Qˆ >0suchthatq(cid:96) ∗ >qˆ(cid:96) andq s ς ∗ <qˆ ς s ,s= 0,U,D ;and { } (cid:16) (cid:17) 2. I ∗ ,p(cid:96) ∗,p ς 0 ∗,p ς s ∗ isunchanged. The intuition is the following. Collateral is required due to the payment enforcement friction. Making debt even more “secure” by explicitly preventing dilution has no real impactbecauseitdoesnotallowthefirmtocreateadditionalvaluegivenitscollateralconstraint. Thecovenantsimplyreallocatescollateraltotheprotecteddebtinstrumentandthe firm substitutes unprotected for protected debt due to changes in the relative equilibrium prices investors require. Our model provides a simple rationalization for the substitution effectsofcovenantsandshort-termdebtempiricallydocumentedbyBilletet. al(2007). The notion that the covenant prevents dilution is equivalent to what Hart and Moore (1995) consider to be a hard claim on firm cash flows. They show that hard claims on the value of assets in place prevent managers with empire building motives from undertaking negativenetpresentvalueprojects. Theresourcesneededtofundsuchintermediateinvestment projects are encumbered by existing long-term debt claims and cannot be diluted. Secured debt improves investment incentives in models with agency concerns. Secured debt does not improve firm value with repayment enforcement frictions and multiple debt maturities. 37

Thefollowingnumericalexamplehighlightsthemajorcomparativestaticresultsfrom proposition2andcovenantresultsofcorollary1andproposition4. 4.1 Numerical example Wekeepα thesameasinthepreviousexampleandshowhowdifferentvaluesthetechnologyshock,A ,thegoodstateprobability,γ,andthecovenantchangetherelativeamount DD of long- versus short-term debt given by ρ. Figure (5) shows the equilibrium marginal investor regime for A = 0.5 and γ = 0.8. Table 6 highlights the major effects of the DD secured debt covenant for various (A γ)-pairs. The top (bottom) panel contains the en- DD, dogenous variables for the economy with (without) the covenant. The numbers in red highlight the key changes. First, note that all debt prices, investment levels and profits are unchangedacrossthetwopanels. More(Less)long-term(short-term)debtisissuedinthe economy with the covenant. The covenant simply tilts the maturity in favor of long-term debt,ρ ,andthefirmsubstitutesawayfromshort-termdebt. ↑ Thecomparativestaticresultsarecontainedbycomparingrowswithineachpanel. The firsttworowsofeitherpanelshowhowvariableschangeasA decreases,whilethethird DD rowshowschangesinγ forthesameA asthefirstrow. Moredownriskatt =2lowers DD all risky debt prices, resulting in lower investment and profits. The firm re-optimizes its debt maturity more toward long-term debt, ρ . Second, consider a decrease in γ for the ↑ same A as in the first row. The bottom row shows that the firm re-optimizes its debt DD maturity more toward long-term debt, ρ , resulting in lower long-term debt prices, but ↑ higherriskyshort-termdebtprices. Thefirmalsoinvestslessandislessprofitable. 38

Table6: EndogenousVariables Covenant p ς p ς p(cid:96) q ς q ς q(cid:96) I ρ Π 0 D 0 D (A ,γ)=(.5,.8) 1 .941 .989 .145 .154 .167 .311 .533 .075 DD (A ,γ)=(.2,.8) 1 .894 .980 .136 .153 .163 .297 .539 .072 DD (A ,γ)=(.5,.5) 1 .957 .985 .107 .112 .199 .304 .646 .057 DD NoCovenant p ς p ς p(cid:96) q ς q ς q(cid:96) I ρ Π 0 D 0 D (A ,γ)=(.5,.8) 1 .941 .989 .149 .158 .163 .311 .519 .075 DD (A ,γ)=(.2,.8) 1 .894 .980 .138 .154 .162 .297 .534 .072 DD (A ,γ)=(.5,.5) 1 .957 .985 .113 .115 .197 .304 .638 .057 DD Figure5: Regime: Portfolio-Rollover h=1 Marginal Buyers h=1 LT Bond ST Bond Buyers! h1,D = .8922 Buyers h0=.8032 Cash & ST Cash Bond Buyers! h=0 h=0 t=0 t=1 5 Conclusion This paper characterizes optimal debt maturity in an economy with payment enforcement frictions and heterogeneous lenders. A debt financing strategy with both long- and shorttermdebtisgenerallytheleastcostlywayforthefirmtoobtainfinancing. Issuingmultiple debt maturities to heterogeneous investors allows firms to cater risky debt securities to investorsmostwillingtoholdrisk,whichfacilitatesareductioninborrowingcostsandan increase in firm value. Moreover, a combination of debt maturities arises naturally when firms issue “safe” short-term debt and rationalizes why large corporates use commercial paper as bridge financing to finance long-term investment projects. Further, the model predictsthatfirmswillusemoreshort-termdebtwhenmanagersoperatinginshareholders’ bestinterestareoptimisticaboutinvestmentreturns,orwhenexpectedcashflowsarehigh, 39

or down-side risk is low. We also show how growth options and leverage are endogenous tothefirm’sdebtmaturitychoicebecausethepriceofthesecuritiesissuedareaffectedby maturity, which in turn affects investment. Finally, we show that protective non-financial debtcovenantspreventdilution,leadstomorelong-termfinancingandasubstitutionaway from short-term financing. However, protective covenants do not affect real outcomes becausetheysimplyreallocatecollateralclaimsamonglong-andshort-termdebtholders. Our model also rationalizes one of Myers’ (1993) most striking findings. Myers notes thatalmostallleverageincreasingactionsaregoodnewsandleveragedecreasingactivities arebadnews. Inourmodel,thefirmusesdebtmaturitytoreducefinancingcostsandissue more debt for any fundamental collateral value it can pledge to lenders. This raises the value of the firm because it invests and produces more. We have abstracted away from agency concerns to highlight that the mechanism operates through investor heterogeneity rather than liquidation risk or asymmetric information and signaling. In so doing, our model suggests that leverage, debt maturity and proxies for growth options are all jointly determined,whichmayhelpexplainthedifferentfindingsofvariousempiricalstudies. An important assumption of the model is that there are no collateral cash flow problems, which means that the future value of the firm that serves as collateral can be rationallyanticipated. Oneoutcomeofthisassumptionisthatinvestorswillalwaysdemandto holdriskyclaimsonfirmcashflowsevenfollowingbadnews. Alteringthemodeltoallow for investor coordination failure and self-fulfilling debt runs with collateral may produce new and interesting interactions between debt maturity, liquidity risk and the design of corporatesecuritiesthatinternalizesuchoutcomes. 40

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A Appendix Omitted Proofs Proof of Lemma 1: Combining (2) and (3) and plugging into (18) immediately gives (19). Notethatε(ρ;α) (α,0),0<ρ <1andclearlydecreasesintheargumentsthatincrease ∈ (cid:12) ρ. Proposition 2 shows that ∂ρ < 0 and ∂ρ < 0, meaning that ∂ε(ρ;A DD,γ)(cid:12) < 0 and ∂γ ∂A DD ∂A DD (cid:12) α (cid:12) ∂ε(ρ;A DD,γ)(cid:12) <0. Therefore, A 0 and γ 0 = ε lim 0. Any risky bond price ratio ∂γ (cid:12) α DD ↓ ↓ ⇒ → pς lim D >0 will satisfy (19) for small values of A and γ because ρ 1. It is less obvious p(cid:96), DD → 0 lim lim that (19) is always satisfied when ρ 0 because ε α. The reason is that moving from → → all short- to an interior solution involves reducing the safe short-term debt issued att =0 infavororriskylong-termdebtwhichisalwayscostlyatt =0.Bycontrast,movingfrom all long to an interior involves issuing less risky long-term for safe short-term at t = 0, lim for which the cost benefits are always clear. ε α as γ 1 because ρ 0. As long as → ↑ → pς pς D α asγ 1,condition(19)willholdforallγ because D asγ 0andε .Similarly, p(cid:96), ≥ ↑ p(cid:96), ↑ ↓ ↑ 0 0 pς pς if D α holds for A 0, then it will hold for all A α because D as A . p(cid:96), ≥ DD → DD → p(cid:96), ↑ DD ↓ 0 0 For the numerical example in Table 1 of appendix B, pς D (cid:117)0.95, with α =0.7, γ =0.8, p(cid:96), 0 and A = 0.5. We can show numerically that (19) does indeed hold (A ,γ) pairs. DD DD ∀ − Q.E.D. ProofofProposition1: WeshowforanyinvestmentamountI ,issuingq(cid:96) >0andq ς >0 0 0 D is cost reducing relative to either q(cid:96) = 0 or q ς = 0. First, note that (2) and (3) can be 0 0 combined to express the firm’s marginal product equal to either only the marginal cost of long-term debt or the marginal cost of short-term debt. This relationship simply reflects the fact that an interior maximum must be characterized by maturity cost equivalence at ς themargin. Supposeallshort-termdebtisrolledoversothat p =1always. Next,suppose 0 maturity is irrelevant, and the firm can obtain the same terms of financing all long-term or via interior solution. Let I be the optimal investment amount for some parameter 0∗ set Γ(α,A ,γ). If maturity is irrelevant, the firm must be indifferent to raising I by DD 0∗ issuingalllong-termdebt,Q=q˜(cid:96),atprice p˜(cid:96) ortoissuingbothlong-andshort-termdebt, 0 0 47

Q=qˆ(cid:96) +qˆ ς , at prices pˆ(cid:96) and pˆ ς =1. Clearly it must be the case that q˜(cid:96) >qˆ(cid:96), qˆ ς >0, 0 0 0 0 0 0 ∀ 0 and since the firm takes prices as given, it must be the case that p˜(cid:96) > pˆ(cid:96). Market clearing 0 0 implies the supply of financing equals the firm’s demand for financing. For only longterm debt, market clearing is given by (cid:0) 1 h˜ (cid:1) = p˜(cid:96)q˜(cid:96) =I and for both long- and short- − 0 0 0 0∗ term debt by (cid:0) 1 hˆ (cid:1) + (cid:0) 1 hˆ (cid:1) = pˆ(cid:96)qˆ(cid:96)+pˆ ς qˆ ς =I . Equating the two market clearing − 0 − D 0 0 D D 0∗ conditions for the same I gives (cid:0) 1 h˜ (cid:1) = (cid:0) 1 hˆ (cid:1) + (cid:0) 1 hˆ (cid:1) . This can only hold if 0∗ − 0 − 0 − D hˆ =1meaningthatqˆ =0–noshort-termdebtisissued–orifh˜ <hˆ –themarginallong- D D 0 0 termbondbuyerinaninteriorsolutionismoreoptimisticthanthemarginalbondbuyerin thecornersolution. But,themoreoptimistictheinvestor,thehigherthepricesheiswilling to pay = p˜(cid:96) < pˆ(cid:96), which contradicts q˜(cid:96) >qˆ(cid:96), qˆ ς >0. The same logic will also show ⇒ 0 0 0 0 ∀ 0 that the firm can never be indifferent between all short-term financing and a combination ofshort-andlong-termdebt. Q.E.D. ProofofProposition2 (cid:16) (cid:17) ∂ρ <0: From(2)andagivensetofriskydebtprices p(cid:96) ∗,p ς ∗ , γ increasesthel.h.s ∂γ 0 D ↑ more than the right. If the firm issues more long-term debt, ρ , long-term debt prices ↑ fall and short-term debt prices rise, causing further deviation from the necessary equality. ∂ρ Thus,thefirmmustissuemoreshort-termdebt, <0. ∂γ ∂ρ <0: From (17) we know that long-term debt holders and risky short-term debt ∂A DD holders expect the same delivery at s=DD. An increase in A raises expected delivery DD forbondholdersofallmaturities. However,long-termdebtholdersatt=0place(1 h ) 2 0 − (cid:0) ς (cid:1) weight on s = DD while short-term debt holders at t = 1 place 1 h weight on s = D − DD. Therefore, unless h h , short-term debt holders place more weight on recovery 0 D (cid:28) for which A has an ultimate affect. This implies that an increase in A tilts debt DD DD maturity towards short-term funding and ρ falls. Alternatively, suppose to the contrary that(1 h ) 2 >(1 h ) h h sothatlong-termmarginalbuyerplacesmoreweight 0 D 0 D − − ⇔ (cid:28) on the down state than the short-term marginal buyer. This condition can be re-written as 48

h (2 h ) > h . It then follows that 1 (1 h ) 2 > h h (2 h ) > h . The long- 0 0 D 0 D 0 0 D − − − ⇔ − termmarginalbuyerisalsomoreoptimisticabouttheup-statethantheshort-termmarginal buyer. The only way the long-term price is more responsive to changes in A is if the DD long-term buyer is simultaneously more optimistic and pessimistic than the short-term marginalbuyer. Acontradiction. Q.E.D ProofofProposition3: Anecessaryconditionforany0<ρ <1inanycollateraleconomy with or without the covenant is p(cid:96) > p ς from dΠ =0. Thus, the necessary condition also 0 D dρ ensuresdˆ (cid:0) qˆ(cid:96) (cid:1) >d ( ).Letρ =1inwhichcaseq ς =0.From(17),d ( )= A DD = DD 0 D∗D · ∗ 0 D∗D · α dˆ (cid:0) qˆ(cid:96) (cid:1) in (21). For the second item in the proof, it is clear from (13) and (15) that DD 0 any given buyer with the same marginal utilities across states must pay a higher price for securitieswithhigherdeliveries. Q.E.D. Proof of Corollary 1: From Proposition 5 and (21) we know that d (cid:0) q(cid:96) (cid:1) > d (cid:0) q ς (cid:1) DD 0 DD D (cid:0) (cid:1) when long-term indentures include the covenant for a given I ,ρ . Suppose the firm 0∗ ∗ does not alter its debt structure and ρ is unchanged. Then, long-term debt prices must ∗ rise to a new level reflecting greater marginal valuations, p(cid:96)c > p(cid:96) ∗, where the superscript 0 0 c denotes prices with the covenant. But if long-term debt is now cheaper in equilibrium, (cid:0) (cid:1) then the maturity structure for a given I ,ρ cannot be optimizing and the firm must 0∗ ∗ adjust. Thus the firm issues more long-term debt and reduces its short-term debt, leaving I unchangedandρc >ρ solowering p(cid:96)c = p(cid:96) ∗.Q.E.D. 0∗ ∗ ↓ 0 0 ProofofProposition4: FollowsimmediatelyfromtheproofofCorollary1andinvestment optimalityin(2)and(3). Q.E.D. 49

B Appendix B.1 Multiple debt maturity funding Thetenendogenousvariablesare (cid:0) p ς ,p(cid:96),p ς ,q ς ,q(cid:96),q ς ,I ,ρ,h ,h (cid:1) . Thesystemofequa- 0 0 D 0 0 D 0 0 D tions,alongwith(2)and(3)is: ς p =1 (22) 0 (cid:0) (cid:1) 1 (1 h )2+(1 h )2d q(cid:96) 1= − − 0 − 0 DD 0 (23) p(cid:96) 0 (cid:0) ς (cid:1) h +(1 h )d q D D DD D 1= − (24) ς p D I = p(cid:96)q(cid:96)+p ς q ς (25) 0 0 0 0 0 p(cid:96)q(cid:96) ρ = 0 0 (26) I 0 ς ς ς q = p q (27) 0 s s 1 h = p(cid:96)q(cid:96) (28) 0 0 0 − ς ς 1 h = p q (29) D D D − The first three equations are bond pricing equations. Equation (22) shows that shortterm bonds issued at time 0 are risk free because all short-term debt is rolled over at time 1. Equation (23) states that long-term bonds are priced based on the time 0 marginal investor’s expectations because he is indifferent between buying the bond and holding a cash equivalent asset. Similarly, equation (24) states that time 1 short-term bonds are priced based on the time 1 marginal investor’s expectations because cash is the only other alternative asset. Equation (25) says that the amount of capital the firm raises in the bond market is equal to the investment it puts into its production technology. Equations (2) and (3) are the first order conditions w.r.t. the portfolio allocation ρ and investment level I , 0 respectively. The necessary condition for the firm to issue a portfolio of both long and 50

short-term bonds in (2) says that on the margin the expected cost of issuing either type of bondmustbethesame. Thelefthandsideof(3)istheexpectedmarginalproductofcapital irrespectiveofwhetherornotitisissuedvialong-termorshort-termbonds. Therighthand side is the expected-weighted marginal cost of capital. Equation (26) sets ρ equal to the portionofthefirm’sinvestmentthatisraisedvialong-termdebt. Equation(27)showsthat the firm will issue as many short-term bonds at time 1 as it takes to fully repay its time 0 short-term creditors. Equations (28) and (29) are, respectively, the long-term and time 1 short-termbondmarketclearingconditions. C Appendix Here we show that changing the uncertainty structure of the economy does not materially alter the optimal choice to issue both long- and short-term debt. Instead of the structure given by figure 1 where γ =γ let γ =Pr(s=U)>γ =Pr(s=DU ) so that the s=D 1 2 s=D | | likelihood of receive a good state following a bad state is less that receiving an unconditional good state. Breaking the firm’s problem given by (1) into its constituent pieces, we canwriteprofitsas (cid:26) (cid:20) (cid:21) (cid:20) (cid:21)(cid:27) I I I I max∏= γ Iα ρ 0 (1 ρ) 0 +(1 γ )γ Iα ρ 0 (1 ρ) 0 . I 0,ρ 1 0 − p(cid:96) 0 − − 1 − 1 2 0 − p(cid:96) 0 − − p ς D Thisprofitexpressionsimplystatesthatconditionalongoodnewsatt =1,bothlong-and short-term debt is repaid, and conditional on bad news att =1 long- and short-term debts are repaid only if good news arrives att =2. Notice that the only difference between this problem and the one presented in the main body of the paper is that γ <γ =γ. The first 2 1 51

orderconditionsforamaximumsimplybecome (cid:20) (cid:21) [γ +γ (1 γ )] 1 γ (1 γ ) 1 2 1 2 1 − = γ + − p(cid:96) p ς 1 p ς 0 0 D (cid:20) (cid:21) ρ[γ +γ (1 γ )] (1 ρ) γ (1 γ ) αIα 1 [γ +γ (1 γ )] = 1 2 − 1 + − γ + 2 − 1 . 0− 1 2 − 1 p(cid:96) p ς 1 p ς 0 0 D Plugging into the other we obtain αIα 1 = 1 which of course arises because in equi- 0− p(cid:96) 0 librium the marginal cost of a long-term bond must equal the marginal cost of a shortterm bond for 0<ρ <1 allowing us to express the first order condition for a maximum as a function of either long- or short-term debt. Let A γ +γ(1 γ) when γ = γ s=D ≡ − | and B γ +γ (1 γ ) from the restated problem above and A > B. Then, (I ,ρ) : 1 2 1 0 ≡ − ∀ αI (α − 1) A>αI (α − 1) B. This implies that 1 (cid:12) (cid:12) > 1 (cid:12) (cid:12) p(cid:96) (cid:12) (cid:12) > p(cid:96) (cid:12) (cid:12) at the optimum. In 0 0 p(cid:96) A p(cid:96) B ⇒ 0 B 0 A 0 0 other words, for a given ρ, the firm will only raise the same amount of capital across the two economies if long-term bond prices are higher in the economy with more uncertainty at s=D, which is a contradiction because the firm is less likely to repay debt at s=DU with in the more uncertainty case. Alternatively, the firm can raise less long-term debt and more short-term debt in the economy with more uncertainty at s = D, leaving total I unchanged and tilting ρ more toward short-term debt. This results in lower short-term 0 bond prices and higher long-term bond prices. And by proposition 3, starting from a corner solution, it will always be less costly to balance long- and short-term borrowing costs against one another rather than issuing all long- or short-term debt. The only thing that willchangeistherelativematuritytilt. The same logic applies if we were to allow for uncertainty at s =U and default at s=UD.Forthis,assumethatfirmdeliverats=UDishigherthans=DD,wheregenerically d (Q)=d (Q)+ε <1. This simply reflects the fact that the ultimate shock to UD DD collateral is worse in two consecutive bad states than in an up state followed by a down 52

state. Thefirm’smaximizationproblemcanbesplitandwrittenasfollows: (cid:34) (cid:35) (cid:26) (cid:20) (cid:21)(cid:27) I I I I max∏= γ2 Iα ρ 0 (1 ρ) 0 +(1 γ)γ Iα ρ 0 (1 ρ) 0 . I 0,ρ 0 − p(cid:96) 0 − − p U ς − 0 − p(cid:96) 0 − − p ς D Onlytwothingschangeintheproblem. 1)Debtsarenolongerrepaidconditionalons=U so that now the first set of repayment states are given by γ2 rather than γ. 2) p ς = 1 as U (cid:54) it does with full repayment. Taking first order conditions for an interior maximum and plugging in, one can express the same marginal product equals marginal cost as αIα 1 = 0− 1 .Andbyproposition3,weknowthatforanygivenI andacandidatecornersolution,it p(cid:96) 0 0 is always be cheaper to fund a portion of the investment outlay by substituting into either longorshort-termdebtrathersothatbothdebtmaturitiesareutilized. QED. D Negative pledge covenant Our treatment of protected long-term debt can be thought either as an explicit collateral pledgeorearmark,ortheinclusionofanegativepledgecovenantthatexplicitlyspellsout howlong-termdebtissecuredfromshort-termdebtdilution. Thebenefitofthinkingabout negative pledge covenants, as detailed below, is two fold: 1) negative pledges are among the most common covenants found in public debt indentures, 2) given their prominence, surprisingly little is known in the academic literature of their impact. We thus attempt to fillthisvoidwiththesupportofstrongpracticalrelevance. Negative pledges are widely recognized by the law and economics profession (see Bjerre (1999), Wood (2007, 2008)). The covenant stipulates that the firm cannot issue secured debt in the future without securing the current debt issue. For example, Billet et. al. (2007) classify negative pledge covenants as “Secured Debt Restrictions” because they restrict the security of future debt issues. Table III in their paper shows that negative pledgesaretypicallythe3rdor4thmostcommoncovenant,behindcrossdefaultoraccel- 53

Table7: Negativepledgecovenant Negativepledge covenant Yes No Non-financial 14,783 11,424 Financial 3,117 4,825 <5yr 2,244 2,376 5yr-30yr 15,284 13,401 Total 17,900 16,249 eration,assetsale,andmergerclauses. Negativepledgesaremorecommonthanleverage, dividend, and share repurchase restrictions. Table 7 gives a general sense for the basic statistics on types of bonds that contain a negative pledge covenant. They are more prone inmedium-to-long-termnon-financialcorporateindentures. 54