Debt Financing in Asset Markets† 88 Debt Financing in Asset Markets†

American Economic Review: Papers & Proceedings 2012, 102(3): 88–94 Contents http://dx.doi.org/10.1257/aer.102.3.88

I. The Model 89 By Zhiguo He and Wei Xiong*

II. The Static Setting 90 Short-term debt, such as overnight repurchase creditors, causing a leverage cycle to emerge. agreements and commercial paper, was heav- The presence of such rollover risk prompts a III. The Dynamic Setting 90 ily used by financial institutions to fund their fundamental question regarding the optimality investment positions during the asset market of short-term debt financing. boom preceding the financial crisis of 2007– Moreover, one might argue that as credi- A. Risk-Neutral Representation of Prices 90 2008, and later played a major role in leading to tors are less optimistic and undervalue risky the distress of these institutions during the cri- debt issued by optimistic asset buyers, it is not sis. What explains the popularity of short-term desirable for the buyers to use risky financing. B. An Optimist’s Problem 91 debt in financing asset market investments? Following Geanakoplos’ framework, Simsek Geanakoplos 2010 presents a dynamic model 2011 examines a setting in which asset buy- of the joint equilibrium( ) of asset markets and ers( have) to hold their positions until the asset C. The Equilibrium 92 credit markets, in which optimistic buyers of a matures. He shows that optimistic asset buyers risky asset use the asset as collateral to raise debt may use risky debt financing despite their debt financing from less optimistic creditors. The being undervalued by the creditors. The contrast References 93 asset’s collateral value depends on the marginal between Simsek’s and Geanakoplos’ analysis creditor’s asset valuation, and determines the suggests that the asset’s tradability in the interim buyers’ purchasing capacity to bid up the asset periods may affect the debt financing of asset price. Geanakoplos posits that short-term debt buyers. allows the asset buyers to maximize riskless This paper examines the optimal use of debt leverage.1 Short-term debt, however, exposes financing by extending the dynamic framework the asset buyers to rollover risk e.g., Acharya, of Geanakoplos along two dimensions. First, we Gale, and Yorulmaker 2011; and (He and Xiong allow the asset’s fundamental value to follow 2012, forthcoming , which is reflected in the a generic binomial tree, which takes multiple fact that the asset buyers) in Geanakoplos’ model values after multiple periods, as opposed to the may not be able to obtain refinancing after a two values in Geanakoplos’ model. Second, we negative fundamental shock. As a result, they allow two groups of agents to have time-vary- are forced to transfer their asset holdings to the ing beliefs about the asset’s fundamental value, and, in particular, to have more dispersed beliefs after a negative fundamental shock. The more * He: Booth School of Business, University of Chicago e-mail: [email protected] ; Xiong: Department dispersed beliefs make debt refinancing more of( Economics, Princeton University) e-mail: wxiong@ expensive to the optimistic asset buyers, and thus princeton.edu . We thank Dan Cao, (Christian Hellwig, expose them to greater rollover risk. The roll- ) Zhenyu Gao, Adriano Rampini, Alp Simsek, and seminar over risk motivates long-term debt financing. To participants at 3rd Theory Workshop on Corporate Finance uncover the role played by the asset’s tradability, and Financial Markets, 2011 AFA Meetings, 2012 AEA Meetings, Northwestern University, Stockholm School of we also contrast two settings: the main setting, Economics, and the University of Minnesota for helpful whereby agents can trade the asset freely on any comments. date, and a benchmark setting whereby agents † To view additional materials, visit the article page at cannot trade after the initial date. The nontrad- http://dx.doi.org/10.1257/aer.102.3.88. 1 The existing literature also offers several other advan- ability is due to the presence of informational tages of short-term debt. First, short-term debt can act as and trading frictions that prevent agents from a disciplinary device by giving the creditors the option to promptly responding to fundamental fluctua- pull out if they discover that firm managers are pursuing tions, and makes the benchmark setting essen- value-destroying projects e.g., Calomiris and Kahn 1991 . Second, the short maturity( also makes short-term debt less) tially static and analogous to that of Simsek. information sensitive and thus less exposed to adverse-selec- Our analysis delivers several results. First, to tion problems (e.g., Gorton and Pennacchi 1990 . our surprise, the maximum riskless short-term ) 88 VOL. 102 NO. 3 Debt Financing in Asset Markets 89

1 leverage is optimal despite the presence of roll- i πu uu over risk. While long-term debt financing can u dominate short-term debt financing when the i π0 rollover risk is sufficiently high i.e., belief i ud ( 1 u dispersion becomes sufficiently high after −π i θ a negative shock , the improved investment πd du ) 1 t opportunity after interim negative fundamental −π0 shocks makes saving cash for that state more d i dd 1 d desirable than acquiring the position. Thus, −π 2 long-term debt financing is dominated either θ t 0 t 1 t 2 by short-term debt financing or by saving cash. = = = This result explains the pervasive use of short- term debt in practice. It also verifies the state- Figure 1. ment put forward by Geanakoplos, albeit due to optimists’ cash-saving incentives rather than his argument that short-term debt allows optimists the end of ud and du is , and at the end of dd to maximize riskless leverage. is 2, where 0, 1 . θThe probability of the Second, we highlight the role played by the treeθ going upθ in∈ (each) period is unobservable. asset’s tradability. By making it possible to buy Two groups of risk-neutral agents differ in the liquidated collateral on an interim date, the their beliefs about these probabilities. We col- asset’s tradability motivates some optimists to lect an agent’s beliefs on the three intermediate save cash, which, in turn, boosts the creditors’ states, one on date 0 and two on date 1 u and i i i ( valuation of the collateral on the initial date, as d , by 0 , u , d where i h, l indicates the well as the optimists’ purchasing capacity to bid agent’s) { type.π π We π } assume that∈ {the }h-type agents up the asset price. are always more optimistic than the l-type Finally, despite the market incompleteness agents the superscript “h” and “l ” stand for ( h l due to the borrowing constraints faced by the high and low : n n for any n 0, u, d . optimists, we derive a risk-neutral representa- We emphasize) thatπ ≥each π agent’s belief∈ { changes} tion of the equilibrium prices of the asset and over time, driven by either his learning process debt contracts collateralized by the asset. This or sentimental fluctuation. As a result, the belief is because the payoff of a collateralized debt dispersion between the optimists and pessimists contract is monotonic with respect to the asset’s varies over time. Our analysis highlights the value, and thus shares the same marginal inves- case where the optimists face rollover risk in the tor as the asset. This representation facilitates intermediate state d when the belief dispersion our analysis and could prove useful in analyzing between the optimists and pessimists diverges h l h l 2 issues related to collateralized debt financing in i.e., d d 0 0 0 . more complex settings. ( Theπ pessimists − π ≥ π l -type− π agents≥ ) who have an The paper is organized as follows. Section I infinite amount of( cash deep pockets) are ini- describes the model. We derive the static bench- tially endowed with all (of the asset,) normal- mark setting in Section II, and analyze the ized to be one unit. On date 0, each optimist dynamic setting in Section III. We provide an an h-type agent is endowed with c dollars of online Appendix to cover more details and all cash.( We assume) that both the risk-free interest the technical derivations. The online Appendix rate and the agents’ discount rate are zero and also extends the two-period model described in short sales of the asset are not allowed. We focus the paper to N periods. on noncontingent debt contracts, which specify constant debt payments at maturity unless the I. The Model borrowers default.

Consider a model with three dates t 0, 1, 2 and a long-term risky asset. The asset’s = final 2 We simplify the continuum of constant beliefs consid- payoff on date 2 is determined by a publicly ered by Geanakoplos 2010 to two types. By focusing on observable binomial tree Figure 1 . The asset’s ( ) ( ) homogeneous optimists, we can isolate an optimist’s incen- final payoff ˜ at the end of the path uu is 1, at tive to save cash from a pure belief effect. θ 90 AEA PAPERS AND PROCEEDINGS MAY 2012

II. The Static Setting 2011 . To facilitate our interpretation, suppose ( h ) l u u i.e., there is no belief divergence at We first consider a benchmark setting in πstate = uπ after ( the interim good news . Then, 1 which agents cannot trade the asset on date 1 can be rewritten as ) ( ) and are restricted from using debt contracts h l h 1 d ( − π 0 ) π 0 that mature on date 1. This setting is effectively _π l _h l . ≤ 1 static with three possible final states on date 2. π d ( − π 0 ) π 0 Like Geanakoplos 2010 , we allow the opti- As the optimists with short-term debt financ- mists to use their asset( holdings) as collateral to ing have to roll over their debt after the interim obtain debt financing. bad news in state d of date 1 and as the belief h l Suppose that an optimist uses a debt contract, divergence in this state d d captures their collateralized by one unit of the asset with a prom- refinancing cost, 1 requires(π /π the ) rollover risk ( ) ised payment F due on date 2. The debt’s eventual faced by the optimists be modest relative to their payment is ˜ F min F, ˜ . A pessimistic initial speculative incentives on date 0. As we θ ∧ ≡ ( θ ) l ˜ creditor will grant a credit of D 0 0 F , are interested mainly in scenarios with signifi- l = 피 [ θ ∧ ] where 0 is the expectation operator under cant rollover risk, our later analysis will focus the pessimist’s피 belief. To establish the asset on the case whereby 1 fails. l ˜ ( ) position, the optimist has to use p 0 0 F Under the monotone belief dispersion speci- of his own cash. This amount is the− 피 so-called [ θ ∧ ] fied in 1 , the optimists will use risky debt haircut. With a cash endowment of c, the opti- with a (promised) payment to finance their l ˜ θ mist can purchase c p0 0 F units of asset acquisition when their cash endowment the asset. Each unit /(gives −an 피 expected [θ ∧ ]) payoff of is sufficiently low Proposition 2 of the online h ˜ ˜ ( 0 F . Maximizing the expected profit Appendix . Despite their debt being underval- leads피 [θ −to θ the ∧ optimist’s] optimal choice of lever- ued by the) creditors, the optimists believe that age F *. the asset price is sufficiently attractive to offset In the joint equilibrium of the asset and credit the high financing cost. This result is consistent markets, each optimist maximizes his expected with that of Simsek 2011 , and builds on the profit. As an optimist could also choose to three-point distribution( of the) asset’s fundamen- acquire the debt issued by other optimists tal on the final date, as opposed to the two-point instead of the asset, another equilibrium condi- distribution in Geanakoplos 2010 . On the tion requires that the marginal value of investing other hand, if 1 is not satisfied,( the) optimists one dollar in the asset is as good as investing in use only riskless( )debt with a promised payment debt issued by other optimists. Furthermore, if 2 to finance their asset acquisition Proposition the optimists are the marginal investors of the 3θ of the online Appendix . This is the( case that asset, the market clearing condition requires we focus on. ) that the asset price is determined by the optimists’ aggregate purchasing power the sum of III. The Dynamic Setting their cash endowment c and the borrowed( credit l ˜ * 0 F . In the online Appendix, we We now allow agents to trade the risky asset describe피 [θ ∧ these ]) conditions in detail and derive the on the interim date—date 1—after the two pos- equilibrium. Here, we highlight the key charac- sible states, u and d, are revealed to the public. teristics of the equilibrium. As a result, each optimist can also make endoge- The equilibrium depends crucially on whether nous investment and leverage decisions in these the belief dispersion between the optimists and two interim states. Moreover, each optimist has pessimists is concentrated in the highest state uu the choice to use either short-term debt maturing i.e., the belief dispersion about uu being higher on date 1 or long-term debt maturing on date 2 than( about the three upper states uu, ud, du : to finance his asset position on date 0. { }) h h h h h h 0 u 0 d 0 d A. Risk-Neutral Representation of Prices 1 _π l π l __π l + π l − π l π l . ( ) 0 u ≥ 0 d 0 d π π π + π − π π Despite the financing constraints faced by the This condition is consistent with the mono- optimists, we can still establish a risk-neutral tonic belief ordering imposed by Simsek representation of the prices of the asset and debt VOL. 102 NO. 3 Debt Financing in Asset Markets 91 contracts collateralized by the asset. It is realis- long-term debt financing; or he can simply save tic to allow the collateralized debt contracts to cash for establishing a position later on date 1. be tradable on dates 0 and 1. The payoff of a Suppose that he uses a debt contract F ˜ either collateralized debt contract is monotonic with long-term or short-term collateralized by( each respect to the asset’s value, and thus must share unit of asset to finance) a position. By selling the same marginal investor, whose identity can the debt to the marginal investor in the mar- ˜ ˜ be either optimist or pessimist and can be differ- ket, he obtains a credit of D0F 0 DuF ˜ ( ) = ϕ ( ) + ent across states. We can represent the price P s 1 0 DdF , where we denote Ds the debt of the asset or any debt contract collateralized by (value − ϕin )state ( s). Thus, each dollar of cash allows the asset in any intermediate state s u, d, 0 him to establish a position of x 1 p0 D0 by a risk-neutral form: ∈ { } units of the asset. This position gives = /(an expected − ) profit of P P 1 P , u = ϕu uu + ( − ϕu) ud ˜ 2 V0F ( ) ( ) = P P 1 P , h h h ˜ h ˜ d d ud d dd 0 vu pu 1 0 vd pd 0 vu DuF 1 0 vdDdF = ϕ + ( − ϕ) _____(π + ( − π ) ) − (π ( ) + ( − π ) ( )) . p D F˜ 0 − 0( ) P0 Pu 1 Pd , = ϕ0 + ( − ϕ0) This equation reflects two effects. One is a leverage effect: a more aggressive debt contract where s is the marginal investor’s effective allows the optimist to establish a greater posi- belief inϕ state s. tion, as shown by the denominator of 2 . The These effective beliefs are directly linked with other is a debt-cost effect: a more aggressive( ) the optimists’ marginal value of cash. Consider debt contract also implies a higher expected debt h the intermediate state d. The asset price pd must be payment in the future, as shown by the term 0 h π between the optimists’ and pessimists’ asset val- vuDu 1 0 vdDd in the numerator of 2 . l ˜ h ˜ l h + ( − π ) ( ) uations: pd d , d and d d , d . One dollar of∈ [cash피 ( θallows) 피 ( θ an) ] optimistϕ ∈ to[ π acquireπ ] Debt Maturity Choice.—In light of 2 , when the asset at a discount to his valuation. Each dol- the optimist compares a pair of long-term( ) and lar, levered up by using a debt contract collater- short-term debt contracts with the same initial alized by one unit of the asset and with a promise credit, he prefers the one with a lower expected of 2 the maximum riskless promise , allows debt cost. For these two contracts to give the himθ to( acquire a position of 1 p ) 2 units same initial credit, we must have /( d − θ) of the asset and thus gives a marginal value of vd h 2 2 h h l S S L L d pd d d 1, d d . 0 Du 1 0 Dd 0Du 1 0 Dd, Similarly,= π (θ − hisθ)/( marginal − θ )value = π of /ϕ cash ∈ [ inπ state/π ]u ϕ + ( − ϕ) = ϕ + ( − ϕ) h h is vu u1 pu u u. On date where, with potential abuse of notation, = π ( − θ)/( − θ) = π /ϕ S L 0, the optimists can either use the cash, levered Ds Ds denotes the value of the long-term up by collateralized debt, to acquire the asset short-term ( ) debt in state s s, d . We define or save it for date 1. Thus, his marginal value the( optimist’s) cash value–adjusted∈ { } probabil- h h h h of cash at date 0 is the maximum of the two ity as ˆ 0 0 vu 0 vu 1 0 vd . The choices: short-termπ ≡contract π / (πgives +the ( lower− π ) expected) debt cost to the optimist, and thus is preferable h h h to the long-term debt, if and only if ˆ 0 0 v0 max u0, 0 vu 1 0 vd , π ≥ ϕ = π + ( − π ) Proposition 7 of the online Appendix . ( ) ( ) h h This result establishes that the optimist’s debt 0 vu pu pd 0 vu where u0 π ( − ) π . maturity choice depends on his cash value– = __p0 pd = _ − ϕ 0 adjusted belief relative to the creditor’s belief. The basic intuition works as follows: refinanc- B. An Optimist’s Problem ing the short-term debt allows the borrower to trade a higher payment in state d for a lower pay- An individual optimist faces three alterna- ment in state u. Whether this trade is preferable tives on date 0: he can establish an asset position depends on whether the borrower’s belief about by using short-term debt financing; he can use the probability of the state u, after adjusting for 92 AEA PAPERS AND PROCEEDINGS MAY 2012 his marginal value of cash in the future states, is equilibrium, each optimist and each pessimist higher than the creditor’s belief. solve his optimal decisions based on our earlier We can further show in the online Appendix discussion, and the markets for the asset and the h that ˆ 0 0 holds when 1 holds. Thus, when debt used by optimists must clear. In particu- rolloverπ ≥ risk ϕ is relatively (high,) long-term debt lar, if the optimists are the marginal investor in could dominate short-term debt in financing the the markets, their aggregate purchasing power optimists’ asset positions. This result contrasts needs to be sufficient to support the asset price. the standard intuition that optimists always pre- We present the specific equilibrium conditions fer short-term debt financing if they have to bor- for each intermediate state s 0, s, d in the row from pessimistic creditors. online Appendix, which also characterizes∈ { } vari- ous cases that arise in the equilibrium. Cash Saving.—The optimist can also choose Now we provide a numerical illustration to save cash for the next date. Interestingly, sav- to contrast the dynamic setting with the static h ing cash dominates over using long-term debt to benchmark setting. We set 0.4, 0 0.6, h l h l θ = h π = finance an asset position if ˆ 0 0 Proposition 0 0.55, u u 0.5, d 0.75, and π < ϕ ( π l = π = π = π = 8 of the online Appendix . This is because the d 0.4. The specified belief structure does not asset appears to be overvalued) to the optimist satisfyπ = the monotone belief dispersion given in after adjusting for his marginal value of cash on equation 1 , and thus captures the rollover risk the following date. This result shows that when discussed( after) equation 1 in Section II. long-term debt dominates short-term debt for Figure 2 illustrates the( ) effects of varying financing the optimist’s asset position, saving optimists’ cash endowment c from 0.42 to 0.07. cash dominates for establishing the position. Panel A depicts the date-0 asset price p0. The Thus, the optimist will never strictly prefer long- two horizontal lines at the levels of 0.5 and 0.556 term debt in the equilibrium. He could be indif- represent the pessimists’ and the optimists’ ferent between using long-term and short-term expectation of the asset’s fundamental value; h l ˜ h ˜ debt in financing his position only if ˆ 0 0. i.e., 0 and 0 . The equilibrium price has π = ϕ to fall피 between (θ ) 피these (θ ) two benchmark levels. The Optimal Debt Financing.—In summary, dotted-and-dashed line is the asset price under h if ˆ 0 0, the optimist will establish an asset the static setting discussed in Section II. When positionπ > ϕby using short-term debt with the c is above 0.4, the equilibrium price is equal to maximum riskless promise to pay p d on date 1; the optimists’ valuation. Once c falls below 0.4, h h if ˆ 0 0, he will save cash; if ˆ 0 0, he is the asset price starts to fall with c and is deter- indifferentπ < ϕ between saving cash andπ establishing= ϕ mined by the optimists’ purchasing power rather the position using either long-term or short- than their valuation. As c falls below 0.336, the term debt . ( marginal investor shifts to the pessimists and the ) asset price is equal to the pessimists’ valuation. C. The Equilibrium The solid line in panel A gives the asset price in the dynamic setting discussed in this sec- The pessimists are initially endowed with all tion. Like the dotted-and-dashed line, when c is of the asset. Due to their risk neutrality and deep above 0.4, the equilibrium price is equal to the pockets, their asset valuations put lower bounds optimists’ valuation; and the asset price starts to on the prices of the asset and any collateralized fall with c once c falls below 0.4. Interestingly, debt contract. On date 0, the pessimists are the as c drops below 0.357 but above 0.14, the price l marginal investor if and only if 0 0 . They levels off at 0.523. In this range, the asset price are indifferent between selling andϕ retaining= π the is substantially higher than that in the static l asset in this case, and prefer selling if 0 0 . setting. Panel C reveals the key source of this The joint equilibrium of the asset andϕ > credit π difference—as c falls, a greater fraction of the markets can be characterized by the prices of optimists 1 choose to save cash on date the asset on date 0 and in states u and d of date 0 and more( pessimists− α) hold their asset endow- 1: p0, pu, pd; the fraction of optimists who ments . As further shown by panels B and establish{ asset } positions on date 0: 0, 1 ; D, in this(λ) range the optimists’ cash saving sus- α ∈ [ ] and the fraction of pessimists who sell their tains the asset price pd and the implied belief of asset endowments on date 0: 0, 1 . In this the marginal investor in state d of date 1 at λ ∈ [ ] ϕd VOL. 102 NO. 3 Debt Financing in Asset Markets 93

Panel A. Date−0 price marginal investor on date 0 is the pessimists l 0.56 0.55 . Despite this, the date-0 asset (ϕ0 = π 0 = ) 0.55 price p is higher than that in the static setting p0 (dynamic) 0 p (static) 0.54 0 because the pessimists anticipate that p will be Optimist valuation d 0.53 Pessimist valuation determined by the optimists. As c gets lower than 0.14 but higher than 0.09, 0.52 all optimists save cash on date 0 0 and all 0.51 pessimists hold their asset endowments(α = ) 0 . 0.5 In this case, the optimists remain the marginal(λ = ) 0.49 investor of the asset in state d. As the optimists 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 c have saved all of their cash endowments to state Panel B. State−d price d, the asset price in this state now falls with c, 0.35 which in turn causes the asset price on date 0 to fall with c as well. Nevertheless, the date-0 price p0 is higher than that in the static setting because the pessimists, the marginal investor on date 0, 0.3 anticipate selling their asset holdings to optimists in state d. Only when c falls below 0.09 do pd Optimist valuation Pessimist valuation the pessimists become the marginal investor of the asset both on date 0 and in state d of date 1.

0.25 As a result, the prices in the dynamic and static 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 settings coincide. c Taken together, Figure 2 demonstrates a sig- Panel C. Date−0 agent distribution nificant impact of the asset’s tradability when 1 Fraction of optimists buying the optimists’ cash endowment c is in the inter- Fraction of pessimists selling 0.8 mediate range between 0.09 and 0.357. In this range, the tradability induces at least some 0.6 optimists to preserve cash from date 0 to state

0.4 d, where the marginal value of cash is highest. These optimists’ cash saving supports the asset 0.2 price in state d, which in turn motivates the pessimists to assign a higher collateral value to the 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 asset on date 0. c Panel D. Marginal investor belief References 0.75

0.7 ϕ0 Acharya, Viral V., Douglas Gale, and Tanju Yor- ϕd 0.65 ulmazer. 2011. “Rollover Risk and Market Freezes.” Journal of Finance 66 4 : 1177– 0.6 ( ) 0.55 209. Calomiris, Charles W., and Charles M. Kahn. 0.5 1991. “The Role of Demandable Debt in Struc- 0.45 turing Optimal Banking Arrangements.” Amer- 0.4 ican Economic Review 81 3 : 497–513. 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 ( ) c Geanakoplos, John. 2010. “The Leverage Cycle.” In National Bureau of Economic Research Figure 2. The Effects of Optimists’ Cash Endowments Based on the Following Parameters: Macroeconomics Annual 2009, edited by h l h l Daron Acemoglu, Kenneth Rogoff, and 0.4, 0 0.6, 0 0.55, u u 0.5, θ = π h= π = l π = π = 0.75, and 0.4. Michael Woodford, 1–65. Chicago: University π d = π d = of Chicago Press. constant levels, and, in particular, ensures the Gorton, Gary, and George Pennacchi. 1990. optimists as the marginal investor in this state “Financial Intermediaries and Liquidity Cre- l i.e., . Panel D also shows that the ation.” Journal of Finance 45 1 : 49–71. ( ϕd > π d ) ( ) 94 AEA PAPERS AND PROCEEDINGS MAY 2012

He, Zhiguo, and Wei Xiong. Forthcoming. and Credit Risk.” Journal of Finance 67 2 : “Dynamic Debt Runs.” Review of Financial 391–429. ( ) Studies. Simsek, Alp. 2011. “Belief Disagreements and He, Zhiguo, and Wei Xiong. 2012. “Rollover Risk Collateral Constraints.” Unpublished.