Credi spreads and Capial Srucure policy Dr. Howard Qi, Michigan echnological Universiy, USA ASRAC Wha is he effec of capial srucure policy on credi spreads? In a widely cied paper by Huang and Huang (003), a few represenaive srucural models of credi spreads have been calibraed and compared. Among hem are he model by Collin-Dufresne and Goldsein (001) and he one by Leland and of (1996). he former assumes exogenously given saionary mean-revering leverage, he laer opimizes is capial srucure endogenously. heir sudy shows ha all he models hey calibraed perform similarly in ha hey all explain a very small porion of he observed spreads. We make hree improvemens based on ha sudy. Firs, we correc a misake in heir calibraion for he CG model. Second, he Leland (1994) model of perpeual bond was used for simpliciy. We sricly calibrae he L model wih finie mauriies. hird, we use he same se of arges, so he model comparison is more meaningful. We argue ha leverage raio canno be an appropriae arge for calibraing he Leland-of model. his differeniaes our calibraion approaches. All hese hree issues have no been recognized hus far and we address hem in his paper. INRODUCION he effec of a firm s capial srucure policy on he credi spreads on he firm s deb is conroversial. In a widely cied paper by Huang and Huang (003), hereafer HH (003), a few represenaive srucural models of credi spreads have been calibraed and compared. heir sudy shows ha all he models hey calibraed perform similarly in ha hey all explain a very small porion of he observed spreads, especially for invesmen-grade bonds. heir resuls seem o be robus despie drasically differen capial srucure assumpions underlying hose models. Indeed, srucural models have long been recognized in heir inabiliy of generaing high enough spreads beween corporae and reasury bond yields. his has promped sudies o include oher missing facors. For example, Liu e. al. (006) incorporae personal axes, de Jong and Driessen (005) and Driessen (005) invesigae liquidiy facor, Johnson and Qi (006) consider personal axes, raing ransiion risk and liquidiy facors. However, before we search for oher possible sources ha conribue o he credi spreads, i is imporan o have more accurae knowledge of how much spread is due o defaul risk, and o wha exen would capial srucure policies affec he credi spreads. he well known sudy by HH (003) provides such benchmarks and hereafer has been cied by many academic papers. A quick Inerne search shows ha 186 academic papers have cied heir resuls as of he ime when his paper is being wrien. hus, i is imporan ha heir approach, resuls and claims are clear and accurae. While applauding heir ambiious work and agreeing wih heir sudy in general, we have idenified hree flaws ha are subsanial enough o warran careful re-invesigaion. o our bes knowledge, his is he firs ime and paper o poin ou and aemp o address hese hree flaws. Firs, heir calibraion of he Collin-Dufresne and Goldsein (001) model, hereafer he CG model, is based on inconsisen leverage arges by misake. We will calibrae he CG model using he same se of arge leverage raios consisen wih all oher models in heir sudy. his is a minor ye necessary correcion o heir sudy. Second, he rue Leland-of model, hereafer he L model, is for bonds wih finie mauriy. u in HH (003), he Leland (1994) model of perpeual bond (and inaccuraely called he L model) was used o replace he rue L model for is simpliciy, and hey believe ha if he L model (of finie mauriy) were used, he spreads would have been even lower. hus, all srucural models perform similarly (poorly). We will show ha his is compleely rue. o address his flaw, we calibrae he rue L model. Our resuls show ha heir claims have o be undersood in conex. In paricular, consisen wih heir sudy, he majoriy of he observed spreads canno be explained by neiher of he wo models. Neverheless, he calibraed L model appears o have significanly higher explanaory power han he CG model. In addiion, we also calibraed he L model for 4-year bonds which is missing from HH (003) because of heir use of perpeual bond model. hird, i appears ha he calibraion arges are he same for boh he L and he CG 66 Journal of Inernaional Managemen Sudies * February 008
models (and ohers oo). We argue ha he inclusion of leverage raio as a arge for he L model is problemaic because of is underlying radeoff naure. In his paper, we will explain why leverage is an appropriae a arge for calibraing he CG model bu no for L model. his differenial reamen of he CG and L model makes our calibraion approach differen from heirs. We believe ha our hree modes conribuions provide necessary correcions, clarificaions and exensions o he curren lieraure regarding credi spread and is relaion wih he capial srucure policy. he paper proceeds as follows. Secion reviews he models o be sudies. Secion 3 explains model implemenaion, calibraion and he choice of calibraion arges. Secion 4 provides he resuls and analysis. Secion 5 concludes he paper. RIEF MODEL REVIEW oh he CG (001) model and he L (1996) model are consruced on he firs-passage ime approach which models bankrupcy as a sochasic process hiing a hreshold for he firs ime. his bankrupcy hreshold can be hough of as he firm's marke deb value, hence he higher he leverage raio, he more probable his hreshold being hi. Ceeris paribus, here is a monoonically increasing relaionship beween defaul probabiliy and leverage raio. his is rue o boh models. However, wha differeniae hem is how o rea he leverage raio. he CG model assumes an exogenously deermined arge leverage raio which follows a mean-revering sochasic process. he L model is based on he radeoff beween corporae ax benefis and bankrupcy coss, which endogenously ses an opimal leverage raio. his difference urns ou o be vial for correc model calibraion. he Collin-Dufresne and Goldsein Model In he CG model, he leverage is exogenously given. he firm value V follows a geomeric rownian moion. he firm value V is specified under he risk-neural measure by dv = ( r δ ) d + σdz1( ) (1) V z is a rownian random variable, δ is he payou raio and σ is he volailiy. he spo rae r follows he where 1 Vasicek (1977) process, dr = ( r ) d dz () β θ + η where β, θ and η are consans wih he coupling bankrupcy hreshold K, or l log ( K / V ) = 0 dk = = [ y ν φ( r θ ) k ] where y = logv, ( ν ) dz dz = ς d 1. Defaul occurs when he firm value his he. he log-defaul hreshold k is mean-revering: λ d (3) y ses he arge hreshold wih ν and λ adjusing he mean-revering speed, and 0 φ. Upon defaul, bondholders can recover a fracion of he face value. Denoing he loss rae by L, hen he price of a risky zero-coupon bond is given by: P ( r0, l0 ) = D( r0, )[1 LQ ( r0, l0, )] (4) where D ( r 0, ) is he price of he risk-free zero-coupon bond and Q r, l, ) is he cumulaive defaul ( 0 0 coupon probabiliy before under he -forward measure, and is specific form is given in CG (001). For defaulable coupon bonds, we assume he coupon loss rae is 100 percen, i.e., L = 1. Coupon bond is reaed as a porfolio of zerocoupon bond. he yield o mauriy Y saisfies he following equaion: P = e Y + i= 1 ce Y i where c is he coupon. he yield spread, YS( ) = Y Y, is defined as he difference beween Y, he yield on a corporae bond, and ha on a reasury, Y. oh have he same mauriy. (5) Journal of Inernaional Managemen Sudies * February 008 67
he Leland and of Model he L model (1996) endogenously decides a bankrupcy boundary V which balances he radeoff beween he benefi of corporae ax shields and he bankrupcy coss. Firm value of an unlevered firm, V, is assumed o follow he diffusion process: dv = [ µ ( V, ) δ ] d + σdz (6) V µ is he expeced rae of reurn on he firm s asse, Z is a sandard Wiener process. he firm issues where ( V,) deb coninuously o replace he deb ha is expiring, hence mainaining saionary leverage. Wihin a uni of ime, he firm issues deb d wih a coninuous consan coupon flow c(), principal p ( ), and mauriy. Upon defaul, bondholders receive a fixed porion ρ of he asse value V. Given his deb policy, he firm would have deb d V, V, over he ousanding wih ime o mauriy from 0 o. he oal deb value D is hen given by inegraing ( ) period of : ( V, ) = d( V, V, ) D V, d (7) = 0 he levered firm value W ( V, V, ) equals he unlevered firm value V plus leverage benefis ( V V ) bankrupcy coss ( V, V ), W ( V, V, ) = V + h( V, V ) ( V, V ) (8) h, less a+ z V, V. Oher parameers are he corporae a+z where τ CC V h( V, V ) = 1 and ( V V ) = ( 1 ρ) r V V δ ( σ / ) income ax raeτ C, = r 1/ V [( aσ ) + rσ ] a, b = ln σ, z =. Applying he smooh-pasing V σ E( V, V, ) condiion = 0 o maximize he equiy value E = W D and using he addiional par-bond V V = V consrain, he closed-form soluion is obained by L (1996) for he price of he firm s newly issued deb (per uni of ime), c r c c e [ 1 F( )] + ( ) V G( ) r r ρ r p( ) =, (9) r 1 e 1 F [ ( )] V, = 1 and V where he cumulaive defaul probabiliy is F( V, V ) N[ h ( ) ] + N[ h ( ) ] a z a+ z V V (, V, V ) = N[ q ( ) ] + N[ q ( ) ], wih ( ) G N denoing he cumulaive sandard normal disribuion. he 1 V V q are funcions of volailiy σ, ineres rae r, and bankrupcy boundary V. hey can be found in L (1996). he yield of his par bond is simply Y = c / p and he yield spread is defined as he parameers h 1 ( ), h ( ), q 1 ( ) and ( ) difference beween Y and he riskfree rae r. a MODEL IMPLEMENAION AND CALIRAION Numerical Implemenaion o implemen he CG model, I choose similar parameer values as in HH (003). For example, he meanrevering coefficien λ = 0.; he long-erm average leverage raioν = 38%; he coupling coefficien ς = 0. 5 for dz1 dz = ς d ; coupon rae c = 8.13% and payou rae δ = 6%. For he Vasicek ineres, β = 0. 6, same as HH (003). We choose ineres rae volailiy η = 1.5% as in CG (001). Similarly, for he L model, I also choose 68 Journal of Inernaional Managemen Sudies * February 008
consan ineres rae r = 8%; bankrupcy cos ( 1 ρ ) = 15% of he firm value a defaul V ; he reasons for making τ 35 percen. he iniial hese choices are given in deail by HH (003). he corporae ax rae is chosen o be = C value of he unlevered firm s V = 100. Model Calibraion and he arge Choices he essence of calibraion is o une some unobserved variables such ha he model generaes he defaul probabiliy o mach he observed hisorical daa for he same sample ime period. his is in agreemen wih he general spiri of HH (003). However, wha differs from heirs is ha we rea he CG and L model differenly. In he CG case, leverage is exogenously given, hus he average leverage for each bond raing can be used as an imporan calibraion arge. While in he L case, leverage is endogenously opimized based on he radeoff beween corporae ax shields and bankrupcy coss. I is well known ha radeoff heory canno explain he observed relaionship beween bond raing and he average leverage raio. I predics ha profiable firms (e.g., AAA-raed firms) should use more deb han less profiable firms (e.g., -raed firms). herefore, based on he model s srucure, we make he following choices (also shown in able 1 and ). For he CG model s calibraion arges, we choose defaul probabiliy, equiy risk premium, leverage raio, and recovery raio (of he face value). For he L model, we choose defaul probabiliy, equiy risk premium, and recovery raio (of he firm value a defaul V ). Such differenial reamen is essenial for a meaningful calibraion. For he CG model, leverage raio is inpued, and he model will generae a credi spread for his iniial leverage inpu. However, he L model will endogenously generae boh he opimal leverage raio as well as he defaul probabiliy simulaneously. If we include leverage raio in he arges o calibrae he L model, hen i is impossible o simulaneously have he model agree o he arge leverage raio as well as he defaul probabiliy. In his case, he model is said o be over-specified. hus, he quesion boils down o which one is an appropriae arge for he L model, he defaul probabiliy or he average leverage raio? Cerainly, undersanding ha he L model is a radeoff model, we should drop leverage as a arge. his is in line wih Hile e. al. (199) ha only 11 percen of he surveyed 500 large OC firms use opimal arge capial srucure. 1 herefore, we do no expec he leverage raio of he majoriy of he firms o be explained by he radeoff heory. hen why do we sill calibrae he L model? his is because here are sill firms (say, 11 percen by Hile e. al., 199) ha follow he radeoff heory. hese firms shall have similar defaul probabiliy o ha of oher firms in he same credi raing class. RESULS AND ANALYSIS able 1 repors he credi spreads prediced by he CG model afer calibraion. Our resuls show ha he credi spreads generaed by he CG model given he correc leverage arges are generally significanly lower han hose from HH (003). he only excepion is Aaa bond, for which our credi spread is 0.6 bps while heirs is 0 bps. We believe his excepion may come from rounding errors in he numerical calculaion. he overall resuls are no surprising because hey misakenly used much lower leverage arges as follows. Leverage arges chosen for CG calibraion in HH (003) Leverage arges for CG calibraion in his sudy Aaa Aa A aa a 7.8 1.7 19. 6.0 3.1 39.4 13.1 1. 3.0 43.3 53.5 65.7 1 We also noe ha Pinegar and Wilbrich (1989) surveyed he Forune 500 firms and find 31 percen of hem use arge (i.e., opimal) capial srucure. For example, see Column of able 6 in HH (003), and compare leverage raios hey used o calibrae oher models. Journal of Inernaional Managemen Sudies * February 008 69
able shows he arge parameers and our calibraion resuls for he L model. HH (003) only calibraed for 10-year bonds because hey used a perpeual bond model o approximae he finie mauriy L model. We calibraed for boh since we implemened he rue finie mauriy L model. 3 For Aaa-raed bonds, here is a considerable difference beween spreads from he finie mauriy L model (6 bps) and HH (003) s perpeual version (37 bps). For oher invesmen grade bonds, his difference is surprisingly negligible. his finding is in conras o he saemen in HH (003) - a firs glance, he L model seems o generae higher credi spreads for invesmen-grade bonds han he LS model did in our base case. his, however, is due o he fac ha he L model considered here has a perpeual bond, which, for invesmen-grades, should have a higher credi spread han 10-year bond. he above quoed saemen is based on heir inuiion wihou quaniaive esimaes. As shown by our resuls in Column (7) and (8) of able, compared o heir perpeual bond approximaion, he L model (of finie mauriy) generaes fairly similar spreads and even higher spreads for aa-raed bonds. However, his credi spread difference on aa-raed bonds is only 4 bps. he cenral heme in HH (003) is ha prey much all srucural models perform similarly poorly in generaing credi spreads afer calibraion. Our resuls confirm his belief only o cerain exen. For invesmen-grades, boh he CG and he L model can only explain a small porion of he observed spreads. However, he calibraed L model has significanly greaer explaining power. If we compare able 1 and, i is easy o see ha he calibraed L model (of finie mauriy) can explain 8 imes as much spreads for Aaa-raed bonds as can he CG model (i.e., 6/3. = 8.1), and almos 3 imes as much for aa-raed bonds (i.e., 64/5 =.6). hus, he calibraed L model clearly can generae significanly higher spreads for invesmen-grades. I is inaccurae o claim ha all srucural models of erm srucure of credi spreads perform similarly poorly afer calibraion. here is a significan difference beween he wo models we sudied. However, he overall explanaory power is unsurprisingly unsaisfacory because here are oher facors lef ou of he picure. his indeed moivaes sudies of oher facors affecing he yield spreads. For example, Liu e. al. (006) incorporaed personal axes, de Jong and Driessen (005) and Driessen (005) invesigae liquidiy facor, Johnson and Qi (006) consider personal axes, raing ransiion risk and liquidiy facors. CONCLUSIONS In his paper, we calibraed wo srucural models of credi spreads wih differen capial srucure policies he CG and L models. his work is moivaed by a few flaws we idenified in a widely cied paper by Huang and Huang (003). While our approach agrees wih heir general calibraion spiri, we disagree wih heir reamen of he L model on he leverage arge. We argue ha leverage raios can only be used for calibraing he CG model bu no he L model because he laer is a radeoff heory model. I is impossible o calibrae he L model by having i simulaneously generae he observed leverage raios and he defaul probabiliies. I is well documened ha radeoff heory canno explain he observed rend in average leverage across bond raings. here are mainly hree conribuions from our resuls. Firs, we applied he consisen leverage arges for calibraing he CG model for boh 4- and 10-year bonds. his may serve as a minor correcion o HH (003) since heir calibraion of he CG model is based on inconsisen leverage arges by misake. Second, he rue L model (of finie mauriy) was no calibraed in HH (003). Insead, he Leland (1994) model of perpeual bond was used for is simpliciy, and i is believed here ha all srucural models perform similarly (poorly) since hey argue ha if he L model of finie mauriy were used, he generaed spreads would be even lower. However, our resuls do no fully suppor hese beliefs. oh models can only explain a small porion of he observed spreads for invesmen-grades, bu he L model has significanly more explanaory power (e.g., 8 imes as much for he Aaa-raed bonds). hird, we explain why leverage should no be chosen as a arge for calibraing he L model. his discernmen beween how we rea he CG and L model makes our calibraion approach differen from heirs. We provide a heoreical argumen for why leverage canno be chosen as one of he calibraion arges. Our sudy conribues some clarificaions o he lieraure. 3 Rigorously speaking, he perpeual bond model is no he L model. I is he Leland (1994) model. 70 Journal of Inernaional Managemen Sudies * February 008
REFERENCES Andrade, G., and S. Kaplan, 1998, How cosly is financial (no economic) disress? Evidence from highly levered ransacions ha became disressed, Journal of Finance 53, 1443 1493. Collin-Dufresne, P., and R. Goldsein, 001, Do credi spreads reflec saionary leverage raios?, Journal of Finance 56, 177 08. de Jong, F., and J. Driessen, 005, Liquidiy risk premia in corporae bond and equiy markes, working paper, Universiy of Amserdam. Driessen, J., 005, Is defaul even risk priced in corporae bonds?, Review of Financial Sudies 18, 165-195. Hile, L. C., K. Haddad, and L. J. Giman, 199, Over-he-couner firms, asymmeric informaion and financing preferences, Review of Financial Economics, 81-9. Huang, J.-Z., and M. Huang, 003, How much of he corporae-reasury yield spread is due o credi risk?, working paper, Penn Sae and Sanford Universiies. Johnson, D., and H. Qi, 006, Effecs of raing ransiion risk on bonds wih axes, working paper, Michigan ech Universiy. Leland, H., 1994, Corporae deb value, bond covenans, and opional capial srucure, Journal of Finance 49, 113 15. Leland, H., and K. of, 1996, Opimal capial srucure, endogenous bankrupcy, and he erm srucure of credi spreads, Journal of Finance 51, 987 1019. Liu, S., H. Qi, and C. Wu, 006, Personal axes, endogenous defaul, and corporae bond yield spreads, Managemen Science 5, 939-954. Pinegar, J. M., and L. Wilbrich, 1989, Wha managers hink of capial srucure heory: A Survey, Financial Managemen, 8-91 able 1. he CG Model and Credi Spreads his able liss he arge parameers ha we calibrae he model agains. he sixh column shows he observed average yield spread. he daa presened here are from HH (003) for he period of 1973 1993. he original sources are Lehman bond index and Moody's. Columns (10) and (11) are HH s resuls for comparison. 4-Year ond (1) () (3) (4) (5) (6) (7) (8) (9) (10) Raing Parameers used in he model ( rivial arge parameers)* Recovery rae (% of face value) Leverage raio Equiy premium arge parameer Cumulaive defaul probabiliy Observed spread (bps) his invesigaion From HH (003) Due o defaul (bps) (ased on differen arge leverage raios by misake) Due o defaul (bps) Aaa 51.31 13.1 5.38 0.04 55 0.6 1.0 0.0 0.1 Aa 51.3 1. 5.60 0.3 65 3.1 4.9 6.3 9.7 A 51.31 3.0 5.99 0.35 96 4.8 5.1 9.9 10.3 aa 51.3 43.3 6.55 1.4 158 17.1 11.0 31.1 19.7 a 51.31 53.5 7.30 8.51 30 114.5 37.5 168.0 5.5 51.3 65.7 8.76 3.3 470 340.1 7.4 435.3 9.6 10-Year ond Aaa 51.31 13.1 5.38 0.77 63 3. 5.1 11.4 18. Aa 51.3 1. 5.60 0.99 91 5.1 5.6 14.9 16.4 A 51.31 3.0 5.99 1.55 13 8.4 6.8.5 18.3 aa 51.3 43.3 6.55 4.39 194 5.1 13.0 5.3 6.9 a 51.31 53.5 7.30 0.63 30 17.0 39.7 18.7 57.1 51.3 65.7 8.76 43.91 470 304. 64.7 371.6 79.1 *rivial arge parameers are observed values ha can be direcly inpued ino he model. he implied asse volailiies were chosen such ha he model generaes exacly he same defaul probabiliies maching hose in column (5). Journal of Inernaional Managemen Sudies * February 008 71
able. he L Model and Credi Spreads his able liss he arge parameers and calibraion resuls. he sixh column shows he observed average yield spread. he daa presened here are from HH (003) for he period of 1973 1993. he original sources are Lehman bond index and Moody's. Columns (10) and (11) are HH s resuls for comparison (4-year bond resuls are no available). 4-Year ond (1) () (3) (4) (5) (6) (7) (8) (10) (11) From HH Parameers used in he model arge his invesigaion (003) ( rivial arge parameers) parameer Raing **Recovery rae (% of defaul firm value V ) Equiy premium Cumulaive defaul probabiliy Observed spread (bps) Due o defaul (bps) Aaa 85 5.38 0.04 55 3 5.3 Aa 85 5.60 0.3 65 1 18.3 A 85 5.99 0.35 96 17 17.4 aa 85 6.55 1.4 158 48 30.6 a 85 7.30 8.51 30 106 33.1 85 8.76 3.3 470 3 49.4 Perpeual bond approximaion Due o defaul (bps) 10-Year ond Aaa 85 5.38 0.77 63 6 41 36.9 58.6 Aa 85 5.60 0.99 91 30 33 34.5 37.9 A 85 5.99 1.55 13 38 31 38.5 31.3 aa 85 6.55 4.39 194 64 3 59.5 30.6 a 85 7.30 0.63 30 160 50 165.7 51.8 85 8.76 43.91 470 315 67 408.4 86.9 ** As in HH (003), I also choose he lower of 85% of V or 51.31% of he face value as he recovered amoun when defaul happens. N/A 7 Journal of Inernaional Managemen Sudies * February 008