Liquidity and Short-term Debt Crises

Transcription

1 Lqudty and Short-term Debt Crses Zhguo He y We Xong z September 2009 Abstract We examne the role of deteroratng market lqudty n exacerbatng debt crses. We extend Leland s structural credt rsk model wth two realstc features: llqud secondary bond markets and a mx of short-term and long-term bonds n a rm s debt structure. As deteroratng market lqudty pushes down bond prces, t ampl es the con ct of nterest between the debt and equty holders because, to avod bankruptcy, the equty holders have to absorb all of the short-fall from rollng over maturng bonds at the reduced market values. As a result, the equty holders choose to default at a hgher fundamental threshold even f there s no frcton for rms to rase more equty. A greater fracton of short-term debt further exacerbates the debt crss by forcng the equty holders to realze the rollover loss at a hgher frequency. Our model llustrates the nancal nstablty brought by overnght repos, an extreme form of short-term nancng, to many nancal rms, and provdes a new explanaton to the wdely observed ght-to-qualty phenomenon. We also examne a tradeo between short-term debt s cheaper nancng cost and hgher future bankruptcy cost n determnng rms optmal debt maturty structure and lqudty management strategy. Keywords: Rollover rsk, Credt Rsk, Debt Maturty Structure, Lqudty Management, Flght to Qualty PRELIMINARY. We thank Xng Zhou and semnar partcpants at Rutgers Unversty for helpful comments. y Unversty of Chcago, Booth School of Busness. Emal: [email protected]. z Prnceton Unversty and NBER. Emal: [email protected].

2 Introducton The recent debt crss on Wall Street llustrates an ntertwned relatonshp between market lqudty and nancal rms credt rsk. On one hand, as nancal rms credt rsk worsens, they are reluctant/unable to provde market lqudty; on the other, as market lqudty deterorates, nancal rms debt crss also ntens es. Understandng ths ntertwned relatonshp s probably one of the most mportant research topcs confrontng the nance lterature. Ths paper ams to analyze the second part of ths relatonshp how does deteroratng market lqudty ntensfy a debt crss? The extant lterature has proposed several mechansms to ths queston. Market llqudty, together wth the d culty of credtors n coordnatng ther rollover decsons of a rm s short-term debt, could lead to runs on nancal rms, e.g., He and Xong (2009) and Morrs and Shn (2009). Deteroratng lqudty and rsng volatlty also motvate credtors to ncrease the requred margns on ther collateralzed loans to the rms, whch n turn could force the rms to lqudate ther postons n llqud markets, e.g., Brunnermeer and Pedersen (2009), Acharya, Gale, and Yorulmzer (2009), and Shlefer and Vshny (2009). These mechansms all rely on an mplct assumpton that rms are constraned from rasng more equty durng nancal dstresses. However, ths assumpton seems at odds wth the observaton that n the current crss many nancal rms pad a substantal amount of dvdends despte ther nancal dstresses and the angry credtors. 2 Ths ndcates an ntrcate nteracton between debt and equty holders, whch s mssng from the aforementoned theores. We provde a theoretcal model to explctly analyze the con ct of nterest between debt and equty holders n debt crses. We show that even n the absence of any constrant on rasng more equty, deteroratng market lqudty could exacerbate the con ct and lead equty holders to choose default at a hgher fundamental threshold. Spec cally, we buld on the structural credt model of Leland (994, 998) and Leland and Toft (996). We extend the Leland framework wth two realstc features. Frst, bond holders are subject to random lqudty shocks. Upon the arrval of a lqudty shock, they have to sell ther bond holdngs at a cost. Ths tradng cost represents llqudty of the bond markets, and can be broadly nterpreted ether as market mpact of trade, e.g., Kyle (985), See Brunnermeer (2009), Damond and Rajan (2009), Gorton (2009) and Krshnamurthy (2009) for comprehensve descrptons of the recent nancal crss. 2 See Scharfsten and Sten (2008) for a dscusson about the nancal rms dvdend payout durng the crss. 2

3 or as bd-ask spread, e.g., Amhud and Mendelson (986). Second, the rm uses a mx of short-term and long-term debt, n addton to equty, to nance ts operaton. The rm repays maturng bonds by ssung new bonds wth dentcal maturty, prncpal value, coupon rate and senorty at the market prces. When the bond prces fall, the equty holders have to absorb the short-fall from rollng over maturng bonds to avod bankruptcy. Otherwse, the rm wll be lqudated at a resale prce to pay o the debt holders. A key result of our model s that, even f there s no frcton for the rm to rase more equty, as deteroratng bond market lqudty pushes down the rm s bond prces, equty value would become zero and equty holders would choose to default when the loss from rollng over maturng bonds becomes su cently hgh. The reason les wth the standard con ct of nterest between debt and equty holders. The equty holders have to bear all the rollover loss to avod bankruptcy, whle debt holders get pad n full. Ths unequal sharng of losses makes the equty value senstve to the drop n bond prces and ultmately causes the equty holders to trgger costly bankruptcy at a hgh fundamental threshold. Ths default mechansm, smlar n sprt to the debt overhang problem suggested by Myers (977), hghlghts the ntrnsc con ct of nterest between debt and equty holders n debt crses. Ths endogenous default problem becomes even more severe when the maturty of the rm s short-term debt becomes shorter. As we observed n the current crss, nancal rms ncreasngly rely on overnght repos, an extreme form of short-term nancng wth a maturty of one day, to fund ther nvestment postons. Rght before the bankruptcy of Lehman Brothers, t had to roll over 25% of ts debt every day through overnght repos. At such a rapd rollover frequency, our model shows that the equty holders nancal burden becomes hghly senstve to bond market lqudty, and that the rm could choose to default even when the rm fundamental s stll solvent. Our model thus calls for more attenton on rms maturty structure n assessng ther default rsk, n addton to ther hgh leverage, whose role n the ongong debt crss s hghlghted by Adran and Shn (2009), Brunnermeer and Pedersen (2009), and Geanakoplos (2009). Our model provdes an nterestng mplcaton about the mpact of a market lqudty breakdown on d erent rms. It s ntutve that the lqudty breakdown, by pushng down bond prces and rasng rms endogenous default thresholds, has a greater mpact on the credt spreads and default probabltes of rms wth weaker fundamentals. Ths mplcaton provdes a new explanaton to the wdely observed ght-to-qualty phenomenon: after major 3

4 market lqudty dsruptons, the prces of low qualty bonds drop much more than those of hgh qualty bonds. The bond market uctuaton durng the ongong nancal crss provdes a nce llustraton of ths phenomenon. Accordng to a BIS report by Fender, Ho and Hordahl (2009), n a two-month perod around the bankruptcy of Lehman Brothers n September 2008, the US ve-year CDX hgh yeld ndex spread shot up from around 700 bass ponts to over 500, whle the ncrease n the correspondng nvestment grade ndex spread was more modest. D erent from the exstng explanatons of ths phenomenon based on the changes n nvestors nvestment constrants and preferences, e.g., Vayanos (2004) and Caballero and Krshnamurthy (2008), our model predcts that a surge n nvestor demand for market lqudty not only leads to a hgher lqudty premun bond prces, but also hgher bond default probabltes. Ths addtonal predcton s consstent wth the quckly rsng default rates of speculatve-grade bonds from the very low levels (around %) n early 2008 to near 5% n March Our model also provdes an nsght on rms optmal maturty structure, based on two opposng forces. On one hand, t s cheaper for rms to ssue short-term debt, because short-term debt tends to be more lqud, e.g., Bao, Pan, and Wang (2009), and thus has a lower lqudty premum. On the other hand, the hgher rollover frequency of short-term debt mposes a heaver nancal burden on the rf bond prces fall and thus makes future bankruptcy more lkely. By tradng o the short-term debt s cheaper nancng cost and hgher expected bankruptcy cost, our model suggests that rms wth lower asset volatlty, hgher bankruptcy recovery rates, and hgher secondary market debt lqudty tend to use a greater fracton of short-term debt. Our focus on market lqudty and future nancal stablty s d erent from the exstng theores of optmal debt maturty based on the dscplnary role of short-term debt n preventng managers asset substtuton, e.g., Flannery (994) and Leland (998), and the theores based on prvate nformaton of borrowers about ther future credt ratngs, e.g., Flannery (986) and Damond (99). Our analyss shows that debt maturty structure should be used as part of a rm s lqudty management strategy. Despte ts hgher cost, long-term debt gves the rm more exblty to delay realzng nancal losses n adverse states, ether when the rm s fundamental or market lqudty deterorates. Ths bene t s analogous to the role of cash reserves, the standard tool for rsk management, e.g., Holmstrom and Trole (200) and Bolton, Chen, and Wang (2009). Ths mplcaton of our model also echoes a related pont made by Brunnermeer and Yogo (2009). 4

5 Our model s related to the credt rsk lterature, e.g., Colln-Dufresne, Goldsten, and Martn (200), Huang and Huang (2003), Longsta, Mthal, and Nes (2005), Ercsson and Renault (2006) and Chen, Lesmond, and We (2007). These studes provde evdence for lqudty as an mportant factor n rms credt spreads. Our model adds to ther results by showng that a hgher tradng cost not only leads to a hgher lqudty premum, but also a hgher default probablty through the endogenous default channel. The paper s organzed as follows. Secton 2 presents the model settng. We derve the debt and equty valuaton and the rm s endogenous bankruptcy boundary n Secton 3. Secton 4 dscusses the mplcatons of the model for debt crses. We analyze rms optmal maturty structure n Secton 5. Secton 6 concludes the paper. 2 The Model We extend the structural credt rsk model of Leland (994, 998) and Leland and Toft (996) wth two realstc features. Frst, the bond markets are llqud. When a bond holder su ers a lqudty shock, he has to sell hs bond poston at a proportonal tradng cost. Second, a rm uses a mx of short-term and long-term bonds, n addton to equty, to nance ts operaton. The settng of our model s generc and apples to both nancal and non- nancal rms, although the e ects llustrated by our model are stronger for nancal rms because they tend to use hgher leverage and shorter debt maturty. 2. Frm Assets The unlevered rm asset value fv t g follows a geometrc Brownan moton n the rsk-neutral probablty measure: dv t = (r ) dt + dz t : () V t where r s the rsk-free rate n ths economy, s the rm s cash payout rate, s the asset volatlty, and fz t g s a standard Brownan moton. Throughout the paper, we refer to V t as the rm fundamental. When the rm bankrupts, we assume that credtors can only recover fracton of the rm s asset value from lqudaton. The lqudaton loss can be nterpreted n d erent ways, such as the loss from sellng the rm s real asset to second best users, loss of customers because of the bankruptcy, asset resale, legal fees, etc. An mportant ssue to keep n mnd s that the lqudaton loss represents a dead weght loss of bankruptcy ex ante to both debt and equty holders, but ex post s borne only by the debt holders. 5

6 2.2 Statonary Debt Structure The rm mantans two classes of debts wth maturtes m and m 2, respectvely. Wthout loss of generalty, we let class- debt to have a shorter maturty,.e., m < m 2 : Each class of debt s the one studed n Leland and Toft (996). At each moment n tme, the th class debt has a constant prncpal P outstandng and a constant annual coupon payment of C. The expraton of each class of debt s unformly spread out across tme. That s, durng a tme nterval (t; t + dt) ; dt fracton of class- debt matures and needs to be rolled over. Gven the shorter maturty of class- debt, t has to be rolled over at a hgher frequency =m. To focus on the rm s debt maturty structure and lqudty e ects, we take the rm s total debt prncpal P = P P and total coupon payment C = P C as gven. By takng the leverage level as gven, we gnore many nterestng ssues related to the tradeo between tax bene ts and bankruptcy costs, whch s analyzed by Leland (994) and other followng work such as Goldsten, Ju, and Leland (200), Strebulaev (2007), and He (2009). smplcty, we also assume that the prncpals and coupon payments of the two debt classes are n proporton: For C = C; P = P; (2) where represents the fracton of the th class debt, wth + 2 =. Furthermore, followng the Leland framework, we assume that the rm can commt to a statonary debt structure denoted by (C; P; m ; m 2 ; ; 2 ) : That s, the rm always mantans the ntally spec ed debt level represented by C and P and the maturty structure spec ed by (m ; m 2 ; ; 2 ) : Thus, when a bond matures, the rm replaces t by ssung a new bond wth dentcal maturty, prncpal value, coupon rate and senorty. For the th class debt, there s a contnuum of bonds wth the remanng tme-to-maturty rangng from 0 to : We measure these bonds by unts. Then each unt has a prncpal value of and an annual coupon payment of These bonds only d er n the tme-to-maturty 2 [0; ]. p = P = P; (3) c = C = C: (4) The two classes of debts have the same prorty n dvdng the rm s asset durng bankruptcy,.e., the rm s lqudaton value s dvded among all debt holders on a pro rata bass. 6

7 Ths assumpton smpl es a complcaton n realty that long-term bonds are often secured by rm assets. We beleve ths smpl caton s nnocuous to our man results. 2.3 Debt Rollover and Endogenous Bankruptcy When the rm pays o maturng bonds by ssung new bonds, the uctuaton n bond prces could generate a rollover gan/loss, whch needs to be absorbed by the equty holders. Spec cally, over a short tme nterval (t; t + dt), the net cash ow to the equty holders (omttng dt) s P NC t = V t ( c ) C + 2 [d (V t ; ) p ] : (5) The rst ters the rm s dvdend payout. = The second ters the after-tax coupon payment, where c denotes the margnal corporate tax rate. The thrd term captures the rollover gan/loss when the rm pays o maturng bonds by ssung new bonds at market prces. In ths transacton, dt unts of both class- and class-2 bonds mature. The maturng class- bonds have a prncpal value of p dt: We denote the market value of the newly ssued bonds wth dentcal prncpal value and maturty as d (V t ; ) dt, whch depends on the rm fundamental V t and bond maturty : When the bond value d (V t ; ) dt drops, the equty holders have to absorb the rollover loss P [d (V t ; ) bankruptcy. 3 p ] dt to prevent the rm from As n the Leland framework, we assume that the equty market s functonal and lqud,.e., the rm can freely rase more equty to pay for the rollover loss and the current coupon payments, as long as the equty value remans postve. In other words, equty holders have the opton to keep servcng the debt (coupons and prncpals) n order to mantan the rght to collect the future cash ows generated by the rm. Bankruptcy occurs endogenously when the rm fundamental drops to a certan threshold V B so that the equty value becomes zero. At ths pont, equty holders are no longer wllng to nject more captal to meet the coupon and prncpal payments, and the rs bankrupt. When the rs bankrupt, equty holders walk away, whle the short-term and long-term bond holders dvde the rm lqudaton value V B on a pro rata bass. Under the statonary debt structure spec ed earler, the rm s bankruptcy boundary V B s constant. We derve V B n the next secton based on a smooth pastng condton regardng 3 Followng the Leland framework, we assume that the rm cannot reduce the rollover loss by ncreasng the coupon payments of the newly ssued bonds. Such an ncrease rases the rm s leverage, and hurts the exstng debt holders. We also gnore renegotaton between debt and equty holders, whch s analyzed n Anderson and Sundaresan (996) and Mella-Barral and Perraudn (997). 7

8 the rm s equty value at the boundary. As n any trade-o theory, bankruptcy nvolves a dead-weght loss. The endogenous bankruptcy s a re ecton of the debt-overhang problem orgnated from the con ct of nterest between the debt and equty holders: when the bond prces are low, the equty holders are not wllng to bear all of the rollover loss to avod the deadweght loss n bankruptcy. suggested by Myers (977). 2.4 The Secondary Bond Markets Ths stuaton resembles the debt-overhang problem We assume that each bond holder s subject to a random lqudty shock, whch arrves accordng to a Posson process wth ntensty : Upon the arrval of the lqudty shock, the bond holder has to ext by sellng hs bond holdng n the secondary market. The lqudty shocks are ndependent across nvestors. As documented by a seres of emprcal papers, e.g., Bessembnder, Maxwell, and Venkataraman (2006), Edwards, Harrs, and Pwowar (2007), Mahant et al (2008), and Bao, Pan, and Wang (2009), the secondary markets for corporate bonds are hghly llqud. The llqudty s re ected by a large bd-ask spread that bond nvestors have to pay n tradng wth dealers, as well as a potental prce mpact of the trade. Edwards, Harrs, and Pwowar (2007) show that the average e ectve bd-ask spread on corporate bonds ranges from 8 bass ponts for large trades to 50 bass ponts for small trades. Bao, Pan, and Wang (2009) estmate that the average e ectve tradng cost, whch ncorporates bd-ask spread, prce mpact and other factors, ranges from 74 to 22 bass ponts dependng on the trade sze. There s also large varaton across d erent bonds wth the same trade sze. In partcular, Mahant et al (2008) and Bao, Pan, and Wang (2009) document an ncreasng pattern of the net tradng cost wth respect to bond maturty (the sum of bond age and tme-to-maturty). Ths result suggests that short-term debt s more lqud than long-term debt. 4 Furthermore, the bonds analyzed n ther sample have a turnover rate of about once a year. Motvated by these observatons, we assume that when an nvestor sells a class- bond n the secondary market, he only recovers a fracton ( ) of the bond value. The other fracton represents the net tradng cost. We shall broadly nterpret ths cost ether as market mpact of trade, e.g., Kyle (985), or as bd-ask spread, e.g., Amhud and Mendelson (986). Snce short-term debt s more lqud, we mpose that < 2 : 4 Intutvely, the default probabltes of long-term bonds are usually hgher than those of short-term bonds. As a result, there s a more uncertanty n valung long-term bonds, whch n turn makes long-term bonds less lqud. 8

9 The bond ssuance cost n the prmary markets tends to be much lower than the tradng cost n the secondary markets. about 5 bass ponts. 5 Issung commercal papers through dealers usually costs Whle the average cost of rasng captal through long-term debt s about 220 bass ponts, e.g., Lee, Lochhead, and Rtter (996), the e ectve cost when spread out across the debt maturty, whch s typcally 5-0 years, s stll low relatve to the secondary market tradng cost. Thus, we gnore the ssuance cost n the model. In summary, our model captures the lqudty of the bond markets usng three model parameters:, whch represents the demand of bond nvestors for market lqudty, and and 2, whch represent the llqudty of the short-term and long-term bonds. In our later analyss, we wll use an unexpected rse n the value of to proxy for deteroratng market lqudty as t causes the lqudty premum to surge. We wll focus on the mpact of ths surge on the rm s bankruptcy boundary and credt spreads. We wll also use unexpected rses n the values of and 2 to proxy for lqudty shocks to sepec c segments of the bond markets and analyze the spllover e ects to other segments. 3 Valuaton and Bankruptcy Boundary In ths secton, we derve the debt and equty valuaton and the rm s endogenous bankruptcy boundary. 3. Debt Value We rst derve the debt valuaton by takng the rm s bankruptcy boundary V B as gven. Smlar dervatons can be found n Leland and Toft (996). Denote d (V; ; V B ) as the value of one unt of class- bond wth tme-to-maturty <. We have the followng partal d erental equaton for the bond value d (V; ; V B ): + (r V 2 + c d : (6) The left-hand sde s the requred return for the bond. Ths term should be equal to the expected ncrement n the bond value, whch s the sum of the terms on the rght-hand sde. The rst three + (r ) V d capture the expected change 2 the bond value caused by the change n the tme-to-maturty, and the uctuaton n the rm s asset value V t. 5 See the Wkpeda webste at for more background nformaton on commercal paper. 9

10 The fourth term c s the coupon payment per unt of tme. The fth term gves the bond holders value change due to the lqudty shocks. Durng a tme nterval (t; t + dt), wth probablty dt; a bond holder su ers a lqudty shock and has to sell the bond n the market for a cost of d ( ) d = d. We have two boundary condtons. At the bankruptcy boundary V B, the bond holders share the rm s lqudaton value proportonally. Thus, each unt of class- bond gets d (V B ; ; V B ) = P = P V B = V B ; for all 2 [0; ] : (7) When = 0, the bond matures and the bond holder gets the prncpal value p : d (V; 0; V B ) = p, for all V > V B. (8) Indvdual Bond Value Smlar to Leland and Toft (996), we solve the ndvdual bond value d (V; ; V B ) based on equaton (6) and the boundary condtons (7) and (8). De ne the e ectve dscount rate for ths bond as r r + ; and let We have where d (V; ; V B ) = c + e r p r V v ln : V B c r ( F ()) + V B V B c r G () ; (9) 2a V F () = N (h ()) + N (h 2 ()) ; (0) G () = a+z a z V V N (q ()) + N (q 2 ()) ; V B h () = ( v a2 ) p q () = ( v z 2 ) p V B ; h 2 () = ( v + a2 ) p ; ; q 2 () = ( v + z 2 ) p ; a = r 2 =2 2 ; z = [a r 2 ] =2 2 ; and N (x) R x p 2 e y2 2 dy s the cumulatve standard normal dstrbuton. 0

11 Bond Yeld The bond yeld s typcally computed as the equvalent return on a bond condtonal on t beng held to maturty wthout default and tradng. Gven the bond prce derved n equaton (9), the bond yeld y s gven by the followng equaton: d (V t ; ) = c y e y + p e y () where the rght hand sde s the prce of a bond wth a constant cash ows c over tme t 2 [0; ] and a prncpal payment p at maturty t = ; condtonal on that there are no default and lqudty shocks. The spread between y and the rsk-free rate s often called the credt spread of the bond. Snce the bond prce n equaton (9) contans the e ects of tradng cost and bankruptcy cost, the credt yeld contans a lqudty premum and a default premum. The focus of our analyss s to uncover the ntrcate nteracton between the lqudty and default prema. It s easer to see the lqudty premum when there s no default rsk. Consder a rsk-free bond wth coupon rate c, prncpal value p, and maturty of. Antcpatng ther own future lqudty shocks and the cost n tradng the bond, nvestors would value ths rsk-free bond at 6 d rskfree ( ) = c e (r+ ) + p e (r+ ) : r Based on equaton (), the yeld of ths rsk-free bond s y rskfree = r = r +. Consstent wth Amhud and Mendelson (986), ths bond yeld contans a lqudty premum determned by the arrval rate of nvestors future lqudty shocks multpled by the tradng cost : Ths lqudty premus consstent wth the emprcal ndngs of Longsta, Mthal, and Nes (2005) and Chen, Lesmond, and We (2007) that less lqud bonds tend to have hgher credt spreads. Ths also suggests that the lqudty premun long-term debt s hgher than that n short-term debt because t s more costly to trade long-term debt ( 2 > ). The cheaper cost of short-term debt s a key factor n our analyss of the rm s optmal debt maturty structure n Secton 5. 6 Spec cally, the value of a rsk-free bond wth a tme-to-maturty sats es rd rskfree wth boundary condton d rskfree p e (r+ ). () + c d rskfree () (0) = p. Therefore (as r = r + ) d rskfree () = c r e (r+ ) +

12 Total Debt Value of all outstandng bonds n class as Wth the value of ndvdual bonds, we can calculate the total value where D (V ; V B ) = Z m 0 = C r + d (V; ; V B ) d P C r e r I ( ) + V B r C J ( ) ; r I ( ) = J ( ) = G ( ) e r F ( ) ; r " a+z a z V V z p N (q ( )) q ( ) + N (q 2 ( )) q 2 ( )# : V B V B 3.2 Equty Value and Endogenous Bankruptcy Boundary V B Leland (994, 998) and Leland and Toft (996) ndrectly derve the equty value as the d erence between the total rm value and the debt value. The total rm value s the unlevered rm value V t, plus the total tax-bene t, mnus the bankruptcy cost. Ths approach does not apply to our model because part of the total rm value s consumed by future tradng costs. Thus, we drectly compute the equty value E (V t ) through the followng d erental equaton: re = (r ) V E V V 2 E V V + V ( c ) C + X [d (V; ) p ] : (2) The left-hand sde s the requred return for the equty. Ths term should be equal to the expected ncrement n the equty value, whch s the sum of the terms on the rght-hand sde. The rst two terms (r ) V E V V 2 E V V capture the expected change n the equty value caused by the uctuaton n the rm s asset value V t : The thrd term V s the dvdend ow generated by the rm per unt of tme. The fourth term ( c ) C s the after-tax coupon payment per unt of tme. The fth term X [d (V; ) p ] gves the equty holders rollover gan/loss from payng o the maturng bonds by ssung new bonds at the market values. 2

13 Lmted lablty of equty holders provdes the followng boundary condton at V B : E (V B ) = 0: (3) Solvng the d erental equaton n (2) s challengng because t contans the complcated bond s valuaton functon d (V; ) gven n (9). We manage to solve ths equaton usng the Laplace transformaton technque detaled n the Appendx. Based on the equty value, we then derve the equty holders endogenous bankruptcy boundary V B based on the smooth pastng condton that E 0 (V B ) = 0: The equty value and the endogenous bankruptcy boundary are gven n the followng proposton. Proposton Let v ln (V=V B ). The equty value E (V ) s = V E (V ) + V B z 8 2 2X >< = >: e v ( c )C P h 2 = ( e r P ) z 2 e r m P C h r K (v;a;a;)+k (v;a; a; ) z 2 C V B h r k (v;a; z ; ) z 2 a z + K (v;a; z ;) + C r + K (v;a; a;)+k (v;a;a; ) a+z + k (v;a;z ; ) a+z + where a r 2 =2 2, z [a2 4 +2r 2 ] =2 2, a + z > 0, a + z > ; and (4) e v 9 >= >; ; + K (v;a;z ;) a z and K (v; x; w; ) n N (w p ) e 2[( x) 2 w 2 ] 2 N (( x) p o ) e v +e 2[( x) 2 w 2 ] 2 v + ( x) e v 2 N p e (x+w)v N v + w 2 p ; V B = k (v; x; w; ) e 2[( x) 2 w 2 ] 2 v + ( x) e v 2 N p The endogenous bankruptcy boundary V B s gven by P 2 h ( ( c)c+ =( e r ) P C P r + P 2 = e (x+w)v N h C r + P 2 = [B ( z ) + B (z )] : 3 v + w 2 p : a) + b (a) b ( + C r [B ( z ) + B (z )] ) (5)

14 where b (x) = e r [N (x p ) e r N ( z p )] ; h B (x) = N (x p ) e 2[z 2 x 2 ] 2 N ( z p ) : a + x + 4 Lqudty Shocks and Debt Crses We analyze the e ects of deteroratng bond market lqudty on debt crses based on the model derved n the prevous secton. We rst show that deteroratng lqudty can exacerbate the con ct of nterest between debt and equty holders and lead to a debt crss. We then dscuss whether short-term debt can further amplfy lqudty e ects. Next, we dscuss the ght-to-qualty phenomenon caused by the d erent mpacts of a market lqudty shock on d erent rms. Fnally, we analyze the spllover of lqudty shocks across d erent market segments. To facltate our dscusson, we use a set of baselne parameters: r = 0%; = 3%; = 0:5; c = 35%; = 7%; V 0 = 00; P = 90; C = 9; m = 0:25; m 2 = 5; = 0:%; 2 = 2%; = : (6) We choose the rsk-free rate r to be 0%; the dvdend payout rate to be 3%; the rm s lqudaton recovery rate to be 0:5, the tax rate c to be 35%; and the asset volatlty to be 7%: These values are farly standard, except that s on the low end relatve to the value used by Leland. We choose a small volatlty because we ntend to analyze a nancal rm whch uses hgh leverage to nance a relatvely safe asset poston. We let the ntal value of the rm asset to be 00; the total prncpal value of the rm s debts P to be 90, and the total annual coupon payment C to be 9: These values mply that the rm s ntal leverage s 73%. We choose the short-term debt maturty m to be 3 months (commercal paper) and the long-term debt maturty m 2 to be 5 years (long-term corporate bond). We set ther tradng costs and 2 to be 0:% and 2%: These values are consstent wth the emprcal estmates of Bao, Pan, and Wang (2009). Fnally, we let the arrval rate of the bond holders lqudty shocks to be ; whch s consstent wth the average turnover rate of the corporate bonds n the data sample of Bao, Pan, and Wang (2009). The rm can choose ts short-term debt fracton ; whch we wll dscuss n Secton 5. For the analyss n ths secton, we treat as exogenously gven and wth a value of 44:5%; whch s the optmal value as we wll show later. Ths value s close to the short-term debt fracton of a 4

15 typcal nancal rn the Compustat data base. 7 Under ths set of parameters, the rm s optmal default boundary V B s 88: Market Lqudty and Endogenous Defaults Deteroratng bond market lqudty can exacerbate the con ct of nterest between debt and equty holders when the rs n a nancal dstress. Ths s because as we see n Eq. (5) when deteroratng lqudty pushes the market prces of the rm s newly ssued bonds to be below ther prncpal values, equty holders have to absorb the rollover short-fall n payng o the maturng debts: 2P (d (V t ; ; ) p ) ; = where we wrte d as a functon of to emphasze the dependence of rollover losses on the bond market lqudty. As a result, when the rollover short-fall becomes su cently large, the equty holders wll choose to stop servcng the debts even f the fallng bond prces are caused by lqudty reasons. To llustrate the e ects of a lqudty shock, we conduct the followng thought experment. Suppose that the arrval rate of lqudty shocks to the bond holders has an unexpected change frots baselne value : One can broadly nterpret the unexpected ncrease n as a surge n the demand for lqudty after a major market dsrupton, such as the recent falure of Lehman Brothers or the crss of LTCM. After the shock, nvestors wll demand a hgher lqudty premum and drve down the bond prces. To analyze the e ects of the shock, we hold the rm s short-term debt fracton at the ntal value. For example, bond covenants and other operatonal restrctons prevent rms n real lfe from swftly modfyng ther debt structures n response to sudden market uctuatons. Fgure llustrates the e ects of a change n on the equty holders rollover loss and bankruptcy boundary. Panel A plots the rollover loss aganst when V = 97. The lne shows that the rollover loss ncreases wth. That s, as the arrval rate of bond holders lqudty shocks ncreases, the ncreased lqudty premun bond prces makes t more costly for the equty holders to roll over the maturng bonds. Consequently, Panel B shows that the rm s endogenous bankruptcy boundary V B also ncreases wth. In other words, after a lqudty shock, the equty holders wll choose to default at a hgher fundamental threshold. We formally prove these results n the followng proposton. 7 For nancal rms wth CDS data n ***, the average fracton of short-term debt was 44% rght before the falure of Bear Stearns n March

16 Rollover Loss Bankruptcy Boundary 0.5 Panel A 9 Panel B Arrval Rate of Lqudty Shocks ξ Arrval Rate of Lqudty Shocks ξ Fgure : Ths gure shows the e ects of a change n the arrval rate of lqudty shocks on the rm s rollover loss and endogenous bankruptcy boundary, based on the baselne parameters gven n (6) and = 44:5%. Panel A plots the rm s rollover loss aganst when V = 97. Panel B plots the rm s endogenous bankruptcy boundary V B aganst. Proposton 2 All else equal, the bond prces d s decrease wth the arrval rate of bond holders lqudty shocks. Consequently, equty holders endogenous default boundary V B ncreases wth. The rm s endogenous bankruptcy decson s rooted n the con ct of nterest between the debt and equty holders. When the rm s bond values fall (even for lqudty reasons as we llustrated here), the equty holders have to bear all of the rollover losses to avod bankruptcy, whle the maturng debt holders get pad n full. Ths unequal sharng of losses causes the equty value to drop down to zero at V B ; at whch pont the equty holders stop servcng the debts. Could the debt and equty holders share the rm s losses, they would have avoded the socal loss nduced by bankruptcy. The mplcaton of Proposton 2 s consstent wth the emprcal ndngs of Colln- Dufresne, Goldsten, and Martn (200). They nd that proxes for both changes n the probablty of future default based on standard fundamental-drven credt rsk models and for changes n the recovery rate can only explan about 25% of the observed credt spread changes. On the other hand, they nd that the resduals from these regressons are hghly cross-correlated, and that over 75% of the varaton n the resduals s due to the rst prncpal component. Whle they cannot explan ths systematc component, they attrbute t to lqudty shocks. Our model explans ther ndngs by suggestng that shocks to the ag- 6

17 Rollover Loss 0 Panel A 92 Panel B λ =0.445 λ =0.3 Bankruptcy Boundary V B λ =0.445 λ = Arrval Rate of Lqudty Shocks ξ Arrval Rate of Lqudty Shocks ξ Fgure 2: Ths gure plots the rollover loss and bankruptcy boundary aganst the arrval rate of lqudty shocks for two otherwse dentcal rms, except one wth a short-term debt fracton of 44:5% and the other wth 30%: Ths gure s based on the baselne parameters gven n (6). The rollover loss s measured when the rm fundamental s at V = 93. gregate demand for bond market lqudty can act as a common factor n ndvdual bonds credt spreads. Furthermore, our model shows that ths lqudty factor a ects not only the lqudty premum, but also ther future default probabltes. Ths ampl caton mechansm through rms endogenous defaults helps explan the large mpact of the lqudty factor observed n the data. 4.2 Further Ampl caton by Short-term Debt The ongong nancal crss reveals that many nancal rms heavly rely on short-term debt such as commercal paper and overnght repos to nance ther nvestment postons. Commercal paper typcally has a maturty of less than 270 days, whle overnght repos have an extremely short maturty of one day. What s the e ect of short-term debt on the rm s exposure to the lqudty shocks? To examne ths queston, we compare two otherwse dentcal rms, one wth a shortterm debt fracton of 44:5% and the other wth 30%: Fgure 2 plots both rms rollover loss and endogenous bankruptcy boundary aganst the arrval rate of the bond holders lqudty shocks : Panel A of the gure shows that the rollover loss of the rm wth hgher rses much faster wth the ncrease n : Ths s because short-term debt needs to be rolled over at a hgher frequency. As a result, when bond prces drop below ther prncpal values, the equty holders have to pay o the losses accumulated n the debt at a faster 7

18 speed. The heaver nancal burden could n turn cause the equty value to fall down to zero and the equty holders to qut servcng the debts at a hgher fundamental threshold. Indeed, Panel B shows that the rm wth the hgher short-term debt fracton has hgher bankruptcy boundary V B. Taken together, ths gure llustrates that short-term debt can further exacerbate the con ct of nterest between the debt and equty holders n nancal dstresses, and thus amplfy a debt crss. Why does short-term debt exacerbate the debt crss? To see the ntuton, we ntroduce d to normalze the market value of the newly ssued short-term and long-term bonds so that d (V t ; ) = d (V t ; ) (7) correspond to the value of the -th bond wth a coupon rate C and a prncpal P (recall equatons (3) and (4)). The two normalzed bonds d er only n ther maturtes. Ths normalzaton allows us to rewrte the rm net rollover gan/loss n (t; t + dt) as P 2 d (V t ; ) = P dt: (8) For each class of debt, the rollover loss s proportonal to the normalzed loss d (V t ; ) and the rollver frequency m. To understand the role of maturty n rollover losses n equaton (8), let us rst dscuss the normalzed loss d (V t ; ) P. Snce short-term debt s more lqud than long-term debt ( < 2 ), ths mples that f default s not a concern,.e., when the rm s fundamental s strong, then we have d > d 2. As a result, short-term debt has a smaller rollover loss for each unt of normalzed bond. Ths then makes the dramatc e ect of short-term debt on the rm s bankruptcy boundary more surprsng. The key to ths e ect les n the rollover frequency m,.e., a shorter maturty means a hgher rollover frequency and therefore ampl es the rollover loss. Ths e ect s at the heart of the mountng nancal burden caused by short-term debt exactly when the rm s fundamental s weak. For llustraton, Fgure 3 plots the rm s loss from rollng over ts short-term and long-term debts wth respect to the rm fundamental. P The gure shows that when the rm s fundamental s strong, short-term debt does provde a smaller rollover loss than long-term debt. 8 However, when the rm s fundamental s weak and thus close 8 Because of the low coupon rate spec ed n ths llustraton (.e., the bonds are not ssued at par when C = P=r), the rm always has a rollover short-fall, whch serves n ths stuaton as part of the nterest payment for ts debts. Ths amounts to a level shft n rollover losses n Fgure 3 and wll not a ect the relatve comparson between long-term and short-term debt. 8

19 Rollover Loss Short term Debt Long term Debt Frm Fundamental V Fgure 3: Ths gure plots the rm s rollover loss at d erent rm fundamentals for each $00 face value of short-term and short-term debt. Ths gure s based on the parameters gven n (6) and = 44:5%: to bankruptcy, short-term debt generates much larger rollover loss than long-term debt. In other words, the rollover gan/loss from short-term debt s hghly skewed on the downsde, whle that from long-term debt s relatvely at. These d erent rollover gan/loss pro les have a drect mpact on the value of the equty holders embedded opton of keepng the rm alve. Even f the current fundamental s weak, equty holders could choose to absorb the rollover losses n hope for a future fundamental comeback. The negatvely skewed payo from short-term debt makes keepng the rm alve costly and the opton less valuable, whle the at payo from long-term debt makes the opton more valuable. In ths sense, long-term debt gves the rm more exblty, and, as we wll dscuss n Secton 5, should be used as part of the rm s lqudty management strategy. We can formally prove a set of results related to the dscusson above. Frst, we can show that between two rms, one purely nanced by the short-term debt and the other purely nanced by the long-term debt, the short-term nanced rm has a hgher default boundary under some su cent condtons. Proposton 3 Suppose that = 0 for 8, and P = C. Then V r B () > V B (0),.e., the endogenous bankruptcy boundary of a 00% short-term nanced rs hgher than that of a 00% long-term nanced rm. Proposton 3 mposes two su cent condtons. Frst, ether the arrval rate of bond 9

20 Long term Debt Credt Spread (bps) Bankruptcy Boundary V B Panel A Short term Debt Rollover Frequency δ Panel B Exogenous V B Endogenous V B Short term Debt Rollover Frequency δ Fgure 4: Ths gure shows the e ect of shortenng the maturty of the short-term debt, based on the model parameters gven n (6) and xng the short-term debt fracton at 5%: Panel A plots the rm s bankruptcy boundary V B aganst the short-term debt rollover frequency ; Panel B plots the credt spread of newly ssued long-term bond aganst : holders lqudty shocks or the tradng cost s s zero. Under ths condton the bond valuaton s not a ected by market lqudty. Second, the bond s prncpal value s dentcal to the dscounted value of the perpetual stream of ts coupons,.e., the rm faces no rollover short-fall when there s no default rsk. These condtons are somewhat strong, as the complex expresson of V B n equaton (5) prevents us from dervng the result of Proposton 3 under more relaxed condtons. However, by contnuty arguments, the result must hold when the model parameters are close to the spec ed condtons (.e., ether or s small and the prncpal P s close to C=r). Furthermore, our numercal exercses show that the result holds n a wde range of parameter values. We can further prove that f the result of Proposton 3 holds, the rm s bankruptcy boundary s monotoncally ncreasng wth the fracton of ts short-term nancng. Proposton 4 Suppose that V B () > V B (0),.e., the endogenous bankruptcy boundary of a 00% short-term nanced rs hgher than that of a 00% long-term nanced rm. Then V B ( ) s monotoncally ncreasng wth,.e., the greater the fracton of the rm s short-term debt, the hgher ts endogenous bankruptcy boundary. Ths proposton provdes another key factor n our analyss of the rm s optmal debt maturty structure n Secton 5. 20

21 Repo Fnancng Rght before the bankruptcy of Lehman Brothers, t had to roll over 25% of ts debt every day through overnght repos. Repos are a type of collateralzed lendng agreement, n whch a borrower nances the purchase of a nancal securty usng the securty as collateral. Overnght repos have an extremely short maturty of one day. What s the e ect of overnght repos on the bankruptcy rsk of a rm? To llustrate the mpact of repo nancng, we consder a hypothetcal rm wth the baselne parameters gven n (6). We reduce the maturty of the short-term debt m from 3 months to day (overnght repo). We denote = m as the rollover frequency of the short-term debt, whch s 4 for commercal paper wth a 3-month maturty and 250 for overnght repos. We also x the short-term debt fracton at 5% to focus on the e ect of shortenng the maturty. 9 Fgure 4 shows that even wth a small fracton of short-term debt, shortenng ts maturty to day generates a large mpact on the rm s default probablty. Panel A shows that as the rollover frequency ncreases from 4 to 250; the endogenous bankruptcy boundary V B ncreases from slghtly above 75 to 96. As a result of the substantal ncrease n V B ; the credt spread of newly ssued long-term debt rses from 200 bass ponts to 575. Ths gures shows that smply shortenng the maturty of a small fracton of the rm s debt from 3 months to day could have a dramatc mpact on the rm s nancal stablty. There are several studes on the role played by repos n the ongong nancal crss, e.g., Brunnermeer and Pedersen (2009), Geanakoplos (2009), and Acharya, Gale, and Yorulmzer (2009). These studes all focus on the destablzng e ect of the harcut (or margn) of the repos,.e., credtors wll ncrease the harcut when the market lqudty deterorates or when the prce volatlty spkes. The ncreased margn requrement forces cash-constraned borrowers to lqudate ther postons at resale prces, resultng n a margn spral. Our model shows that even n the absence of any cash constrant on borrowers, the fast rollover frequency of overnght repos can already lead to a debt crss. Essentally, the repo rollover acts as a mark-to-market mechansm to force the borrowers to absorb the losses accumulated n ther postons every day through margn calls. The heavy nancal burden on the borrowers can n turn motvate them to default at a hgher fundamental threshold. 0 9 We reduce the short-term debt fracton from 44:5% n the prevous llustratons to 5% here because a hgher fracton, when combned wth the fast rollover frequency of overnght repos, would have caused the rm s bankruptcy boundary to be hgher than the ntal rm fundamental V 0 = 00: 0 Snce bankruptcy leads to a socal loss, t s temptng to argue that debt restructurng, such as swappng debt nto equty or lengthenng the debt maturtes, would be pareto mprovng. However, such mod catons of the debt agreements are already de ned as a credt event, whch would trgger many credt default swap contracts to pay out. As a result, even f such debt mod catons avod the socal loss, they would hurt some partes and thus run nto resstance n practce. 2

22 Short term Credt Spread (bps) Long term Credt Spread (bps) V=00 V=97 Panel A Arrval Rate of Lqudty Shocks ξ V=00 V=97 Panel B Arrval Rate of Lqudty Shocks ξ Fgure 5: Ths gure plots the credt spreads of the short-term and long-term bonds of two rms wth d erent fundamentals aganst the arrval rate of bond holders lqudty shocks, based on the baselne parameters gven n (6) and by xng the rms short-term debt fracton at = 44:5%. One of the rm has a fundamental of V = 00; whle the other rm has V = 97: 4.3 Flght to Qualty It s common to observe the so called ght-to-qualty phenomenon after major lqudty dsruptons n the nancal markets prces (credt spreads) of low qualty bonds drop (rse) much more than those of hgh qualty bonds. The recent epsodes nclude the stock market crash of 987, the events surroundng the Russan default and the crss of LTCM n 998, the events after the attacks of 9/ n 200, and the ongong nancal crss. The CGFS (999) report documents that durng the 998 LTCM crss, whch s wdely regarded as a market-wde lqudty shock, the yelds of speculatve-grade corporate bonds and emergng market bonds ncreased much more than nvestment-grade bonds. A recent BIS report by Fender, Ho, and Hordahl (2009) shows that n a two-month perod around the bankruptcy of Lehman Brothers n September 2008, the US ve-year CDX hgh yeld ndex spread shot up from around 700 bass ponts to over 500, whle the correspondng nvestment grade ndex spread wdened from 50 bass ponts to a lttle above 250. Can our model explan the ght-to-qualty phenomenon? To address ths queston, we examne two otherwse dentcal rms, except one wth a fundamental of V = 00 and the other wth V = 97: We compare the changes n the credt spreads of these two rms newly ssued short-term and long-term bonds as the arrval rate of the bond holders lqudty shocks ncreases from to :5. Fgure 5 shows that the credt spreads of the weaker rm are sgn cantly more senstve to the ncrease n than those of the stronger rm. The 22

23 ntuton s smple. As the ncrease n the arrval rate of the bond holders lqudty shocks pushes up the rms endogenous bankruptcy boundary, the weaker rs now closer to bankruptcy. As a result, ts credt spreads shoot up more than those of the stronger rm. Our explanaton of the ght-to-qualty phenomenon s d erent from the exstng ones. The CGFS (999) report attrbutes them to suddenly ncreased rsk averson of market partcpants. Vayanos (2004) provdes an explanaton based on professonal fund managers career concerns, and Caballero and Krshnamurthy (2008) argue for Knghtan uncertanty. These explanatons are all based on consderatons from the nvestor sde. Our model focuses on the nancng ssues from the rm sde and shows that deteroraton of market lqudty would ncrease rms re nancng cost of ther maturng debt, exacerbate con cts of nterest nsde the rms, and eventually cause the weaker rms to fal. Corroboratng to our theory, Fender, Ho, and Hordahl (2009) show that soon after the market lqudty breakdown caused by the falure of Lehman Brothers n September 2008, the default rates of speculatve-grade bonds ncreased sgn cantly from the very low levels (around %) observed n early 2008 to near 5% n March 2009, and were expected to rse further. The recent bankruptcy of General Growth Propertes, one of the largest mall operators n the US, n Aprl 2009 ncely llustrates how the deteroratng credt market lqudty put pressure on lower-qualty rms. Accordng to the New York Tmes (Aprl 6, 2009), "Despte barganng for months wth ts credtors, General Growth faced dwndlng optons for handlng ts more than $25 bllon n debt, largely n the form of short-term mortgages that wll come due by next year. The company has been severely wounded by the trouble n the nancal markets, whch has wreaked havoc on ts ablty to re nance that debt." 4.4 Lqudty Spllover E ect As s well known, bond markets are hghly segmented. For example, the market for commercal paper (short-term debt wth maturtes less than 9 months) operates on d erent quote conventons from the market for long-term corporate bonds. These markets also have separate sets of nsttutonal nvestors. Despte the segmentaton between these markets, our model shows that lqudty shocks to one market could stll a ect bonds n the other market through the rms endogenous default channel. To llustrate ths spllover e ect, we use unexpected changes n the tradng cost of shortterm and long-term debts, and 2 ; to proxy for lqudty shocks to these two d erent market segments. Spec cally, we use the model parameters gven n (6) and x the rm s 23

24 Short term Credt Spread (bps) Long term Credt Spread (bps) 400 Bankruptcy Boundary V B Panel B Panel A Long term Debt Tradng Cost β (bps) Panel C 300 Exogenous V B Endogenous V B Exogenous V B Endogenous V B Long term Debt Tradng Cost β (bps) Long term Debt Tradng Cost β (bps) 2 Fgure 6: Ths gure shows the e ects of the long-term debt tradng cost 2 on the rm s credt rsk, based on the baselne parameters gven n (6) and by xng the rm s short-term debt fracton at = 44:5%. Panel A plots the rm s endogenous bankruptcy boundary V B aganst 2 ; Panels B and C plot the credt spreads of the newly ssued short-term and long-term bonds aganst 2 : long-term debt fracton at = 44:5%: Fgure 6 shows the e ects of an unexpected ncrease n the long-term bond tradng cost 2 on the rm s endogenous bankruptcy boundary V B and the short-term and long-term credt spreads (credt spreads of the newly ssued short-term and long-term bonds). Panel A shows that as 2 ncreases from 200 bass ponts to 400, V B ncreases from 88.2 to Ths s because a hgher tradng cost reduces the long-term bond prces and ncreases the equty holders rollover loss, thus causng the rm to default at a hgher fundamental threshold. If the rm s bankruptcy boundary s xed at 88.2, the short-term credt spread s not a ected by the change n the long-term debt tradng cost. However, Panel B shows that the short-term credt spread ncreases from below 0 bass ponts to above 300, as 2 ncreases from 200 bass ponts to 400. Ths dramatc ncrease s exactly generated by the rm s hgher bankruptcy boundary. Ths plot thus demonstrates 24

25 Short term Credt Spread (bps) Long term Credt Spread (bps) 9 Panel A 50 Bankruptcy Boundary V B Panel B Short term Debt Tradng Cost β (bps) 280 Panel C Exogenous V B 270 Exogenous V B 00 Endogenous V B 260 Endogenous V B Short term Debt Tradng Cost β (bps) Short term Debt Tradng Cost β (bps) Fgure 7: Ths gure shows the e ects of the short-term debt tradng cost on the rm s credt rsk, based on the baselne parameters gven n (6) and by xng the short-term debt fracton at = 44:5%; the optmal level under the baselne = 0:%. Panel A plots the rm s endogenous bankruptcy boundary V B aganst ; Panels B and C plot the credt spreads of the short-term and long-term bonds aganst : the lqudty spllover e ect from the long-term debt market to the short-term debt market. Panel C also shows that as 2 ncreases from 200 bass ponts to 400, the long-term credt spread ncreases from 230 bass ponts to near 550. However, when the rm commts to x V B at the ntal level 88.2, the ncrease n the long-term credt spread s smaller, only from 230 to 430. Ths plot suggests that the rm s endogenous default also ampl es the e ect of the ncrease n the long-term debt tradng cost on the long-term credt spread. Fgure 7 con rms smlar e ects by an unexpected ncrease n the short-term debt tradng cost : Panel A shows that the bankruptcy boundary V B ncreases wth : Panel C shows that the long-term credt spread ncreases sgn cantly wth ; a clear lqudty spllover e ect from the short-term debt market to the long-term debt market. Panel B also shows that the rm s endogenous default sgn cantly ampl es the e ect of the ncrease n the short-term debt tradng cost on the short-term credt spread. 25

26 We can formally prove the followng proposton: Proposton 5 The rm s endogenous bankruptcy boundary V B ncreases wth both and 2 ; the tradng cost of the short-term and long-term debts. Ths proposton con rms that the rm s endogenous bankruptcy can serve as a channel for lqudty shocks to one segment of the bond markets to a ect credt spreads n other segments. 5 Optmal Debt Maturty Structure In our model, there are two opposng forces workng together to determne the rm s ex ante optmal debt maturty structure. On one hand, short-term debts are more lqud and therefore are cheaper for the rm. On the other, short-term debts exacerbate the con ct of nterest between the debt and equty holders and therefore ncrease the rm s future default probablty. In ths secton, we examne ths tradeo between the short-term debt s cheaper nancng cost and hgher expected bankruptcy cost n determnng the rm s optmal maturty structure. Consder the rm s optmal maturty structure choce at tme 0: The rm s objectve s to maxmze the total rm value, the sum of equty, short-term and long-term bonds: max 2[0;] E (V ) + D (V ) + D 2 (V ) : Ths objectve s also consstent wth that of the equty holders at tme 0 before the bonds are ssued. Snce the objectve s a contnuous functon of and takes values n a closed set [0; ] ; there must exst an optmum. Fgure 8 plots the rm s endogenous bankruptcy boundary V B and the total rm value aganst the rm s short-term debt fracton. Panel A shows that V B ncreases wth ; consstent wth our dscusson before. Panel B shows that the total rm value s maxmzed when = 44:5%: Ths nteror optmum re ects the tradeo between the short-term debt s cheaper nancng cost and hgher expected bankruptcy cost. Fgure 9 llustrates how rm characterstcs a ect ts optmal short-term debt fracton, based on the baselne parameters gven n (6). Panel A shows that decreases wth the rm s asset volatlty : As the volatlty ncreases, t rases the rm s future default probablty and therefore expected bankruptcy cost. As a result, t s desrable for the rm to use less short-term debt to reduce the bankruptcy cost. The gure also shows that, n the 26

27 Total Frm Value 05 Panel A 50 Panel B Bankruptcy Boundary V B Optmal λ * Short term Debt Fracton λ Short term Debt Fracton λ Fgure 8: Ths gure plots the rm s endogenous bankruptcy boundary V B and the total rm value aganst the rm s short-term debt fracton, based on the baselne parameters gven n (6). regon where the asset volatlty s low (lower than 5:5%), the cheaper cost e ect domnates and nduces the rm to use 00% short-term debt. Panel B shows that ncreases wth the rm s bankruptcy recovery rate : As ncreases, the expected bankruptcy cost becomes smaller. As a result, the rm could take advantage of the cheaper nancng cost of short-term debt more aggressvely. Panel C shows that there s a non-monotonc relatonshp between and the long-term debt tradng cost 2 : rst ncreases wth 2 when t s relatvely low and decreases wth 2 when t becomes hgh. Ths plot agan re ects the tradeo between the nancng cost and expected bankruptcy cost. As 2 ncreases, the drect e ect s that the long-term debt becomes more expensve. Ths e ect makes the short-term debt more attractve, and thus explans the ncreasng part of the plot. When 2 ncreases, an ndrect e ect s that t nduces the rm to use a hgher bankruptcy threshold, resultng n a hgher future default probablty. Ths ndrect e ect makes the short-term debt less attractve on the margn, and explans the decreasng part of the plot. If the tradng cost of both the short-term and long-term debts, and 2 ; ncrease together, then the substtuton e ect between the types of bonds s vod and only the second (ndrect) e ect through the rm s endogenous default s n operaton. Panel D of Fgure 9 shows that as the common component n and 2 ncreases, the optmal short-term debt fracton decreases. Ths s because the ncreased market llqudty makes the rm more lkely to bankrupt n the future. As a result, the bankruptcy cost e ect becomes more 27

28 Panel A 0.8 Panel B Optmal λ * Optmal λ * Asset Volatltyσ 0.46 Panel C Bankruptcy Recovery Rate α Panel D 0.46 Optmal λ * Optmal λ * Long term Bond Tradng Cost β (bps) Common Bond Market Illqudty (bps) Fgure 9: Ths gure shows how rm characterstcs a ect the rm s optmal short-term debt fracton, based on the baselne parameters gven n (6). Panel A plots aganst the rm s asset volatlty. Panel B plots aganst the bankruptcy recovery rate. Panel C plots aganst the long-term debt tradng cost 2. Fnally, Panel D plots aganst the common component n the tradng cost and 2 of the rm s short-term and long-term bonds. mportant. The extant theores on rms debt maturty choce have mostly focused on the dscplnary role of short-term debt n preventng managers asset substtuton, e.g., Flannery (994) and Leland (998), and prvate nformaton of borrowers about ther future credt ratngs, e.g., Flannery (986) and Damond (99). These theores have had some success n explanng the data, as shown by Barclay and Smth (995) and Guedes and Opler (996). Our model provdes a new hypothess, whch relates rms debt maturty structure to market llqudty consderatons. Maturty Structure as Lqudty Management In fact, our analyss shows that debt maturty structure should be used as part of a rm s lqudty management strategy. As dscussed n Secton 4.2, despte ts hgher cost, long-term debt gves the rm more exblty 28

29 to delay realzng nancal losses n adverse states, ether when the rm s fundamental or bond market lqudty deterorates. Ths bene t s analogous to the role of cash reserves, the standard tool for rsk management, e.g., Holmstrom and Trole (200) and Bolton, Chen, and Wang (2009). Keepng cash s costly for a rm, but t allows the rm to avod future nancal constrants and to take advantage of future nvestment opportuntes. Brunnermeer and Yogo (2009) argue that rms should shft to long-term debt as ther fundamentals deterorate. Ths argument s consstent wth the basc result of our model that the bankruptcy cost (or the loss of exblty) from usng short-term debt becomes hgher as the rm s fundamental or bond market lqudty deterorates. Ths e ect motvates the rm to use less short-term debt n these states. Ths argument s appealng, but t s also countered by other con cts between debt and equty holders. As ponted out by Leland (994), adjustng debt polcy n hs model by ssung or retrng debt ex post s nfeasble to the extent that t wll hurt ether equty or debt holders. Ths argument also apples to our settng. A systematc analyss of these arguments s mportant, but s a challenge beyond our current framework. We wll leave t for future research. 6 Concluson We examne the role played by deteroratng market lqudty n debt crses. We extend Leland s structural credt rsk model wth two realstc features: llqud secondary bond markets and a mx of short-term and long-term bonds n a rm s debt structure. As lqudty shocks push down bond prces, they amplfy the con ct of nterest between the debt and equty holders because, to avod bankruptcy, the equty holders have to absorb all of the short-fall from rollng over maturng bonds at the reduced market values. As a result, the equty holders choose to default at a hgher fundamental threshold even f rms can freely rase more equty. A greater fracton of short-term debt further exacerbates the debt crss It s clear that ncreasng wll lead to a hgher bankruptcy boundary V B, and therefore hurts longterm debt holders. Now suppose that the rm adjusts downward. One realstc polcy s to ssue more long-term debt to replace the maturng short-term debt, untl the desred maturty structure s acheved. In the nterm perod, the replaced short-term debt has coupon (prncpal) m C ( m P ). Therefore, to mantan the rm s net coupon and debt prncpal, the rm needs to ssue m 2 m 2 unts of long-term debt. The net nancng e ect, relatve to the base case wthout the maturty adjustment, s m d (V t ; m ) + m m m 2 d 2 (V t ; m 2 ) / d (V t ; m ) + d 2 (V t ; m 2 ) Snce the short-term debt s safer than the long-term debt, the above ters negatve. Ths heurstc argument mples that the nancal burden on the equty holders s ncreased durng the adjustment process, therefore t s not n the nterest of the equty holders to ncrease the long-term debt fracton. 29

30 by forcng the equty holders to realze the rollover loss at a hgher frequency. Our model llustrates the nancal nstablty brought by overnght repos, an extreme form of short-term nancng, to many nancal rms, and provdes a new explanaton to the wdely observed ght-to-qualty phenomenon. We also examne a tradeo between short-term debt s cheaper nancng cost and hgher future bankruptcy cost n determnng rms optmal debt maturty structure and lqudty management strategy. A Appendx A. Proof for Proposton The equty sats es the followng ODE: re = (r De ne Then, re = r )V E V V 2 E V V + d (V; m ) + d 2 (V; m 2 ) + V V v ln : V B ( c )C + P + P 2 : m m E v E vv +d (v; m )+d 2 (v; m 2 )+V B e v ( c )C + P + P 2 : m m 2 wth the boundary condton that E (0) = 0 and E v (0) = l; where the free parameter l s to be determned by the boundary condton when v!. De ne the Laplace transformaton of E (v) as F (s) = L [E (v)] = Z 0 e sv E (v) dv: Then, apply the Laplace transformaton to both sdes of the ODE, we have: rf (s) = r 2 2 L [E v ] L [E vv ] + L [d (v; m ) + d 2 (v; m 2 )] Note that + V B s ( c )C + P m + P 2 m 2 : s L [E v ] = sf (s) 30 E (0) = sf (s)

31 and L [E vv ] = s 2 F (s) se (0) E v (0) = s 2 F (s) l: Thus, r r 2 2 s 2 2 s 2 F (s) = L [d (v; m )] + L [d 2 (v; m 2 )] 2 2 l + V B s ( c )C + P m + P 2 m 2 : s Let r r where > and > 0: Then, 2 2 s 2 2 s 2 = 2 2 (s ) (s + ) = = 2 2 F (s) (s ) (s + ) + s L [d (v; m )] + L [d 2 (v; m 2 )] + V B s s + ( c )C + P + 2 P 2 s ( ) L [d (v; m )] + L [d 2 (v; m 2 )] + V B s ( c)c+ P m + P 2 m 2 s 2 2 l (9) 2 2 l Snce d (v; ) = C + e r P r C r ( F ( )) + V B C r G ( ) ; where F ( ) and G ( ) are gven n Eq. (0),.e., ( v a 2 ) ( v + a F ( ) = N p + e 2av 2 ) N p ; ( v G ( ) = e ( a+z)v z 2 ) ( v + N p + e (a+z)v z 2 ) N p ; where a = r 2 =2 2 ; z = [a r 2 ] =2 2 ; z = [a r 2 ] =2 2 : Pluggng d (v; ) n (9), we have 2 2 F (s) = s s + + s+ s+ 8 < V B : s P ( c )C + P ( e r ) e r P C r s P C r L [F ( )] + V B 9 = 2 2 l ; C r (20) L [G ( )] 3

32 Call the rst lne n (20) as F b (s), and t s easy to work out ts nverse as V B be(v) = + (ev e v ) + + e v e v ( c )C + P h ( e r P ) C r + + (ev ) e v = V l + ev e v V B + ev + + e v ( c )C + P ( e r ) l + ev e v : h P C r (ev ) e v Call the second lne n (20) as P F (s). One can show that ( + ) F = e r P C h N ( r s s a p ) e 2((s+a) 2 e r P +e r P e r P m V B m + V B m V B m V B C r s s + C r 2a + s s + 2a C r 2a s + 2a s + k 2 C r a z + s s + a z C r k 2 a + z s + a z s + C r a + z + s s + a + z C r a z s + a + z s + h N ( a p ) e 2((s+a) 2 a 2 ) 2 h N (a p ) e 2((s+a) 2 h N (a p ) e 2((s+a) 2 a 2 ) 2 a 2 ) 2 a 2 ) 2 h N ( z p ) e 2((s+a) 2 h N ( z p ) e 2((s+a) 2 h N (z p ) e 2((s+a) 2 h N (z p ) e 2((s+a) 2 z 2 ) 2 z 2 ) 2 z 2 ) 2 z 2 ) 2 : 32

33 De ne M (v; x; w; p; q) L h N (y p ) e 2((s+x) 2 w 2 ) 2 s + p s + q n = N (w p ) e 2[(p x) 2 w 2 ] 2 N ((p x) p o ) e pv +e 2[(p x) 2 w 2 ] 2 v + (p x) e pv 2 N p n N (w p ) e 2[(q x) 2 w 2 ] 2 N ((q x) p o ) e qv e 2[(q x) 2 w 2 ] 2 v + (q x) e qv 2 N p and K (x; w; p) n N (w p ) e 2[(p x) 2 w 2 ] 2 N ((p x) p o ) e pv +e 2[(p x) 2 w 2 ] 2 v + (p x) e pv 2 N p e (x+y)v N v + w 2 p Then M (v; x; w; x + w; q) = K (x; w; q) ; M (v; x; w; p; x + w) = K (x; w; p) : Therefore (note that = z 2 ) E(v) = 2 2 be(v) + P E = V V B z 2 e v + e v + ( c )C + P ( e r ) + z P m V B e r P + l 2z ev e v h P C r z 2 C r (ev ) e v (2) C K r (v; a; a; ) + K 3 (v; a; a; ) z 2 + K (v; a; a; ) + K 2a (v; a; a; ) K 7 a z + (v; a; z ; ) a+z K (v; a; z ; ) 5 K a+z + (v; a; z ; ) a z K (v; a; z ; ) There s one free parameter l = E 0 (0) to be pnned down by the boundary condton at v!. Equty value has to grow lnearly when V!. Snce e v V = V B and >, to avod exploson we need the coe cent of e v n E(v) s zero. Collectng coe cents of e v, 33

34 we need (note that a = z, = 2a +, 2 [z2 a 2 ] 2 = r ) 0 = V B P >< >: ( c )C + P e r P e r P + V B C r C r ( fn( a p ) e rn( z p )g C r l (22) fn(ap ) e rn( z p )g N( z p ) e 2[z 2 z 2 ] 2 N( z p ) ( a z + N(z p ) e 2[z 2 z 2 ] 2 N( z p ) a+z + 5 ) ) 3 >= 7 5 >; Ths condton gves l as an expresson of prmtve parameters and the bankruptcy boundary V B. More mportant, ths mples that the constant coe cent of e v should be zero. Ths smpl es the expresson of K that s nvolvng to be K (x; w; ) = e 2[(p x) 2 w 2 ] 2 v + ( x) e v 2 N p k (x; w; ) : e (x+y)v N v + w 2 p The smooth pastng condton mples that E 0 (V B ) = 0, or E 0 v (0) = l = 0. Then we can use condton (22) to obtan V B. Wth these results, we have the closed-form expresson for E(v) and V B stated n Proposton. A.2 Proof of Proposton 2 We rst x the default boundary V B. Accordng to the Feynman-Kac formula, PDE (6) mples that at tme 0; the prce of a bond wth tme-to-maturty sats es d (V 0 ; ; V B ) = E R ^ B 0 e (r+ )s c ds + e (r+ )(^ B) d ( ^ B ) ; where B = nf ft : V t = V B g s the rst httng tme of V t to V B. V t follows (), and d ( ^ B ) s de ned by the boundary condtons n (7) and (8): d ( ^ B ) = V B f ^ B = B p f ^ B = : Because enters d as rasng the dscount rate, a path-by-path argument mples that d decreases wth. Smlarly, the equty value can be wrtten as E(V 0 ; ; V B ) = E R B 0 e rs V s ( c ) C + P 2 = (d (V s ; s; ) p ) ds ; 34

35 where we wrte the dependence of d on explctly. Agan, a path-by-path argument mples that once xng V B, E decreases wth. We now consder two d erent values of : < 2. Denote the correspondng default boundares as V B; and V B;2. We need to show that V B; < V B;2. Suppose that the opposte s true,.e., V B; V B;2. Snce the equty value s zero on the default boundary, we have E (V B; ; V B; ; ) = E (V B;2 ; V B;2 ; 2 ) = 0; where we expand the notaton to let the equty value E(V t ; V B ; ) to explctly depend on the default boundary V B and the lqudty shock arrval rate. Also, the optmalty of default boundary mples that 0 = E (V B; ; V B; ; ) > E (V B; ; V B;2 ; ) Snce E decreases wth, E (V B; ; V B;2 ; ) > E (V B; ; V B;2 ; 2 ). Therefore E (V B; ; V B;2 ; 2 ) < 0. Ths contradcts to lmted lablty whch says that A.3 Proof of Proposton 3 E (V ; V B;2 ; 2 ) 0 for all V V B;2 : When the rs only nanced by one class of debt, and = 0, ths settng s exactly Leland and Toft (996). After translatng to our notaton, page 993 n Leland and Toft (996) gves the endogenous default boundary as + BLT where V B = C r A LT rm P ALT rm C r + ( ) + B LT ; A LT = 2ae rm N a p m 2zN z p m B LT = 2 p m n zp m + 2z + 2 z 2 m 2e rm p m n ap m + z a; N z p m + 2 p m n zp m z + a z 2 m : Wth potental abuse of notaton, we denote V B as a functon of maturty m. If P = C r, V B = C r BLT C r + ( ) + B LT = r C +( ) C rb LT B LT + : 35

36 Note z p m > 0) Therefore, V B (m ) > V B (m 2 has the same sgn as. Then, smple algebra shows that (note A.4 Proof of Proposton = z 2 m 2 2N z p m < 0; We have V B ( ) = ( c)c+p ( e r ) h P = V B () +V B (0) h + P h 2 + P C r + P + P + [B ( z )+B (z )] m ( h P C b ( a) + b (a) r + C r [B ( z ) + B (z )] [B ( z ) + B (z )] [B ( z ) + B (z )] + [B 2( z 2 )+B 2 (z 2 )] m 2 [B ( z ) + B (z )] : ) De ne w ( ) h + P + [B ( z )+B (z )] m then we can wrte V B ( ) = V B () w ( ) + V B (0) ( [B ( z ) + B (z )] ; w ( )) : Because w ( ) = + P h = + [B ( z ) + B (z )] + [B ( z )+B (z )] m + [B 2( z 2 )+B 2 (z 2 )] m 2 + [B ( z )+B (z )] m s decreasng n, w ( ) s ncreasng n. Therefore our clam follows. A.5 Proof of Proposton 5 The proof of ths proposton follows drectly from the proof of Proposton 2, as and 2 play the same role as n each step provded n Appendx A.2. 36

37 References Acharya, Vral, Douglas Gale, Tanju Yorulmazer (2009), Rollover rsk and market freezes, Workng paper, NYU. Adran, Tobas and Hyun Song Shn (2009), Lqudty and leverage, Journal of Fnancal Intermedaton, forthcomng. Amhud, Yakov and Ham Mendelson (986), Asset prcng and the bd-ask spread, Journal of Fnancal Economcs 7, Anderson, Ronald and Suresh Sundaresan (996), Desgn and valuaton of debt contracts, Revew of Fnancal Studes 9, Bao, Jack, Jun Pan and Jang Wang (2008), Lqudty of corporate bonds, Workng paper, MIT. Barclay, Mchael and Cl ord Smth (995), The maturty structure of corporate debt, Journal of Fnance 50, Bessembnder, Hendrk, Wllam Maxwell, and Kumar Venkataraman (2006), Market transparency, lqudty externaltes, and nsttutonal tradng costs n corporate bonds, Journal of Fnancal Economcs 82, Bolton, Patrck, Hu Chen and Neng Wang (2009), A un ed theory of Tobn s q, corporate nvestment, nancng, and rsk management, Workng paper, Columba Unversty. Brunnermeer, Markus (2009), Decpherng the lqudty and credt crunch , Journal of Economc Perspectves 23, Brunnermeer, Markus and Lasse Pedersen (2009), Market lqudty and fundng lqudty, Revew of Fnancal Studes, forthcomng. Brunnermeer, Markus and Motohro Yogo (2009), A note on lqudty rsk management, Amercan Economc Revew: Papers and Proceedngs 99, Caballero, Rcardo and Arvnd Krshnamurthy (2008), Collectve rsk management n a ght to qualty epsode, Journal of Fnance 63, Chen, Long, Davd Lesmond, and Jason We (2007), Corporate yeld spreads and bond lqudty, Journal of Fnance 62, Colln-Dufresne, Perre, Robert Goldsten, J Spencer Martn (200), The determnants of credt spread changes, Journal of Fnance 56, Damond, Douglas (99), Debt maturty structure and lqudty rsk, Quarterly Journal of Economcs 06, Damond, Douglas and Raghuram Rajan (2009), Fear of re sales and the credt freeze, NBER workng paper Edwards, Amy, Lawrence Harrs, and Mchael Pwowar (2007), Corporate bond market transparency and transacton costs, Journal of Fnance 62, Ercsson, Jan and Olver Renault (2006), Lqudty and credt rsk, Journal of Fnance 6, Fender, Ingo, Corrnne Ho, and Peter Hördahl (2009), Overvew: nvestors ponder depth and duraton of global downturn, BIS Report. 37

38 Flannery, Mark (986), Asymmetrc nformaton and rsky debt maturty choce, Journal of Fnance 4, Flannery, Mark (994), Debt maturty and the deadweght cost of leverage: Optmally nancng bankng rms, Amercan Economc Revew 84, Geanakoplos, John (2009), The leverage cycle, Workng paper, Yale Unversty. Goldsten, Robert, Nengju Ju, and Hayne Leland (200), An ebt-based model of dynamc captal structure, Journal of Busness 74, Gorton, Gary (2008), The panc of 2007, Workng paper, Yale Unversty. Guedes, Jose and Tm Opler (996), The determnants of the maturty of corporate debt ssues, Journal of Fnance 5, He, Zhguo (2009), A model of dynamc compensaton and captal structure, Workng paper, Unversty of Chcago. He, Zhguo and We Xong (2009), Dynamc Debt Runs, Workng paper, Unversty of Chcago. Holmström, Bengt and Jean Trole (200), LAPM: A lqudty-based asset prcng model, Journal of Fnance 56, Huang, Jngzh and Mng Huang (2003), How much of the corporate-treasury yeld spread s due to credt rsk? Workng paper, Penn State Unversty. Krshnamurthy, Arvnd (2009), The nancal meltdown: data and dagnoses, Workng paper, Northwestern Unversty. Kyle, Albert (985), Contnuous auctons and nsder tradng, Econometrca 53, Lee, Inmoo, Scott Lochhead, and Jay Rtter (996), The cost of rasng captal, Journal of Fnancal Research 9, Leland, Hayne (994), Corporate debt value, bond covenants, and optmal captal structure, Journal of Fnance 49, Leland, Hayne (998), Agency costs, rsk management, and captal structure, Journal of Fnance 53, Leland, Hayne and Klaus Bjerre Toft (996), Optmal captal structure, endogenous bankruptcy, and the term structure of credt spreads, Journal of Fnance 5, Longsta, Francs, Sanjay Mthal, and Erc Nes (2004), Corporate yeld spreads: Default rsk or lqudty? New evdence from the credt default swap market, Journal of Fnance 59, Mahant, Srketan, Amrut Nashkkar, Mart Subrahmanyam, George Chacko, and Gaurav Mallk (2008), Latent lqudty: A new measure of lqudty, wth an applcaton to corporate bonds, Journal of Fnancal Economcs 88, Mella-Barral, Perre and Wllam Perraudn (997), Strategc debt servce, Journal of Fnance 52, Morrs, Stephen and Hyun Song Shn (2009), Illqudty component of credt rsk, Workng paper, Prnceton Unversty. 38

39 Myers, Stewart (977), The determnants of corporate borrowng, Journal of Fnancal Economcs 5, Scharfsten, Davd and Jeremy Sten (2008), Ths balout doesn t pay dvdends, New York Tmes, October 20, Shlefer, Andre and Robert Vshny (2009), Unstable bankng, Workng paper, Harvard Unversty. Strebulaev, Ilya (2007), Do tests of captal structure theory mean what they say?, Journal of Fnance 62, Vayanos, Dmtr (2004), Flght to qualty, ght to lqudty, and the prcng of rsk, Workng paper, LSE. 39