Default Risk in Equity Returns

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1 Defaul Risk in Equiy Reurns MRI VSSLOU and YUHNG XING * BSTRCT This is he firs sudy ha uses Meron s (1974) opion pricing model o compue defaul measures for individual firms and assess he effec of defaul risk on equiy reurns. The size effec is a defaul effec, and his is also largely rue for he book-o-marke (BM) effec. Boh exis only in segmens of he marke wih high defaul risk. Defaul risk is sysemaic risk. The Fama-French (FF) facors SMB and HML conain some defaul-relaed informaion, bu his is no he main reason ha he FF model can explain he cross-secion of equiy reurns. * Vassalou is a Columbia Universiy and Xing is a Ph.D candidae a Columbia Universiy. This paper was presened a he 2002 Wesern Finance ssociaion Meeings in Park Ciy, Uah; a London School of Economics; Norwegian School of Managemen; Copenhagen Business School; Ohio Sae Universiy; Darmouh College; Harvard Universiy (Economics Deparmen); he 2003 NBER sse Pricing Meeing in Chicago; and he Federal Reserve Bank of New York. We would like o hank John Campbell, John Cochrane, Long Chen (WF discussan), Ken French, David Hirshleifer, Ravi Jagannahan (NBER discussan), David Lando, Lars Tyge Nielsen, Lubos Pasor, Jay Rier, Jay Shanken, and Jeremy Sein for useful commens. Special hanks are due o Rick Green and an anonymous referee for insighful commens and suggesions ha grealy improved he qualiy and presenaion of our paper. We are responsible for any errors.

2 firm defauls when i fails o service is deb obligaions. Therefore, defaul risk induces lenders o require from borrowers a spread over he risk-free rae of ineres. This spread is an increasing funcion of he probabiliy of defaul of he individual firm. lhough considerable research effor has been pu oward modeling defaul risk for he purpose of valuing corporae deb and derivaive producs wrien on i, lile aenion has been paid o he effecs of defaul risk on equiy reurns. 1 The effec ha defaul risk may have on equiy reurns is no obvious, since equiy holders are he residual claimans on a firm s cash flows and here is no promised nominal reurn in equiies. Previous sudies ha examine he effec of defaul risk on equiies focus on he abiliy of he defaul spread o explain or predic reurns. The defaul spread is usually defined as he yield or reurn differenial beween long-erm B corporae bonds and long-erm or U.S. Treasury bonds. 2 However, as Elon e al. (2001) show, much of he informaion in he defaul spread is unrelaed o defaul risk. In fac, as much as 85 percen of he spread can be explained as reward for bearing sysemaic risk, unrelaed o defaul. Furhermore, differenial axes seem o have a more imporan influence on spreads han expeced loss from defaul. These resuls lead us o conclude ha, independenly of wheher he defaul spread can explain, predic, or oherwise relae o equiy reurns, such a relaion canno be aribued o he effecs ha defaul risk may have on equiies. In oher words, we sill know very lile abou how defaul risk affecs equiy reurns. The purpose of his paper is o address precisely his quesion. Insead of relying on informaion abou defaul obained from he bonds marke, we esimae defaul likelihood indicaors for individual firms using equiy daa. These defaul likelihood indicaors are nonlinear funcions of he defaul probabiliies of he individual firms. They are calculaed using 1

3 he coningen claims mehodology of Black and Scholes (BS) (1973) and Meron (1974). Consisen wih he Elon e al. (2001) sudy, we find ha our measure of defaul risk conains very differen informaion from he commonly used aggregae defaul spreads. This occurs despie he fac ha our defaul likelihood indicaors can indeed predic acual defauls. We find ha defaul risk is inimaely relaed o he size and book-o-marke (BM) characerisics of a firm. Our resuls poin o he conclusion ha boh he size and BM effecs can be viewed as defaul effecs. This is paricularly he case for he size effec. The size effec exiss only wihin he quinile wih he highes defaul risk. In ha segmen of he marke, he reurn difference beween small and big firms is of he order of 45 percen per annum (p.a.). The small socks in he high defaul risk quinile are ypically among he smalles of he small firms and have he highes BM raios. Furhermore, even wihin he high defaul risk quinile, small firms have much higher defaul risk han big firms, and defaul risk decreases monoonically as size increases. similar resul is obained for he BM effec. The BM effec exiss only in he wo quiniles wih he highes defaul risk. Wihin he highes defaul risk quinile, he reurn difference beween value (high BM) and growh (low BM) socks is around 30 percen p.a., and goes down o 12.7 percen for he socks in he second highes defaul risk quinile. There is no BM effec in he remaining socks of he marke. gain, he value socks in hese caegories have he highes BMs of all socks in he marke, and he smalles size. Value socks have much higher defaul risk han growh socks, and here is a monoonic relaion beween BM and defaul risk. We also find ha high defaul risk firms earn higher reurns han low defaul risk firms, only o he exen ha hey are small in size and high BM. If hese firm characerisics are no 2

4 me, hey do no earn higher reurns han low defaul risk firms, even if heir risk of defaul is acually very high. We finally examine wheher defaul risk is sysemaic. We find ha i is indeed sysemaic and herefore priced in he cross-secion of equiy reurns. Fama and French (1996) argue ha he SMB and HML facors of he Fama-French (1993) (FF) model proxy for financial disress. Our asse pricing resuls show ha, alhough SMB and HML conain defaul-relaed informaion, his is no he reason ha he FF model can explain he cross-secion. SMB and HML appear o conain imporan priced informaion, unrelaed o defaul risk. Several sudies in he corporae finance lieraure examine wheher defaul risk is sysemaic, bu heir resuls are ofen conflicing. Denis and Denis (1995), for example, show ha defaul risk is relaed o macroeconomic facors and ha i varies wih he business cycle. This resul is consisen wih ours since our measure of defaul risk also varies wih he business cycle. Opler and Timan (1994) and squih, Gerner, and Sharfsein (1994), on he oher hand, find ha bankrupcy is relaed o idiosyncraic facors and herefore does no represen sysemaic risk. The asse pricing resuls of he curren sudy show ha defaul risk is sysemaic. Conrary o he curren sudy, previous research has used eiher accouning models or bond marke informaion o esimae a firm s defaul risk and in some cases has produced differen resuls from ours. Examples of papers ha use accouning models include hose of Dichev (1998) and Griffin and Lemmon (2002). Dichev examines he relaion beween bankrupcy risk and sysemaic risk. Using lman s (1968) Z-score model and Ohlson s (1980) condiional logi model, he compues measures of financial disress and finds ha bankrupcy risk is no rewarded 3

5 by higher reurns. He concludes ha he size and BM effecs are unlikely o proxy for a disress facor relaed o bankrupcy. similar conclusion is reached in he case of he BM effec by Griffin and Lemmon (2002), who use Olson s model and conclude ha he BM effec mus be due o mispricing. There are several concerns abou he use of accouning models in esimaing he defaul risk of equiies. ccouning models use informaion derived from financial saemens. Such informaion is inherenly backward-looking, since financial saemens aim o repor a firm s pas performance, raher han is fuure prospecs. In conras, Meron s (1974) model uses he marke value of a firm s equiy in calculaing is defaul risk. I also esimaes is marke value of deb, raher han using he book value of deb, as accouning models do. Marke prices reflec invesors expecaions abou a firm s fuure performance. s a resul, hey conain forwardlooking informaion, which is beer suied for calculaing he likelihood ha a firm may defaul in he fuure. In addiion, and mos imporanly, accouning models do no ake ino accoun he volailiy of a firm s asses in esimaing is risk of defaul. ccouning models imply ha firms wih similar financial raios will have similar likelihoods of defaul. This is no he case in Meron s model, where firms may have similar levels of equiy and deb, bu very differen likelihoods o defaul, if he volailiies of heir asses differ. Clearly, he volailiy of a firm s asses provides crucial informaion abou he firm s probabiliy o defaul. Campbell e al. (2001) demonsrae ha firm level volailiy has rended upwards since he mid-1970s. Furhermore, using daa from 1995 o 1999, Campbell and Taksler (2003) show ha firm level volailiy and credi raings can explain equally well he cross-secional variaion in corporae bond yields. Clearly, a firm s volailiy is a key inpu in he Black-Scholes opion-pricing formula. 4

6 s menioned, an alernaive source of informaion for calculaing defaul risk measures is he bonds marke. One may use bond raings or individual spreads beween a firm s deb issues and an aggregae yield measure o deduce he firm s risk of defaul. When a sudy uses bond downgrades and upgrades as a measure of defaul risk, i usually relies implicily on he following assumpions: Tha all asses wihin a raing caegory share he same defaul risk and ha his defaul risk is equal o he hisorical average defaul risk. Furhermore, i assumes ha i is impossible for a firm o experience a change in is defaul probabiliy, wihou also experiencing a raing change. 3 Typically, however, a firm experiences a subsanial change in is defaul risk prior o is raing change. This change in is probabiliy of defaul is observed only wih a lag, and measured crudely hrough he raing change. Bond raings may also represen a relaively noisy esimae of a firm s likelihood o defaul because equiy and bond markes may no be perfecly inegraed, and because he corporae bond marke is much less liquid han he equiy marke. 4 Meron s model does no require any assumpions abou he inegraion of bond and equiy markes or heir efficiencies, since all informaion needed o calculae he defaul risk measures is obained from equiies. Examples of sudies ha use bond raings o examine he effec of upgrades and downgrades on equiy reurns include hose of Holhausen and Lefwich (1986), Hand, Holhausen, and Lefwich (1992), and Dichev and Pioroski (2001), among ohers. The general finding is ha bond downgrades are followed by negaive equiy reurns. The effec of an increase in defaul risk on he subsequen equiy reurns is no examined in he curren sudy. The remainder of he paper is organized as follows. Secion I discusses he mehodology used o compue defaul likelihood indicaors for individual firms. Secion II describes he daa 5

7 and provides summary saisics. Secion III examines he abiliy of he defaul likelihood indicaors o predic acual defauls. In Secion IV we repor resuls on he performance of porfolios consruced on he basis of defaul-risk informaion. In Secion V, we provide asse pricing ess ha examine wheher defaul risk is priced. We conclude in Secion VI wih a summary of our resuls. I. Measuring Defaul Risk. Theoreical Model In Meron s (1974) model, he equiy of a firm is viewed as a call opion on he firm s asses. The reason is ha equiy-holders are residual claimans on he firm s asses afer all oher obligaions have been me. The srike price of he call opion is he book value of he firm s liabiliies. When he value of he firm s asses is less han he srike price, he value of equiy is zero. Our approach in calculaing defaul risk measures using Meron s model is very similar o he one used by KMV and oulined in Crosbie (1999). 5 We assume ha he capial srucure of he firm includes boh equiy and deb. The marke value of a firm s underlying asses follows a Geomeric Brownian Moion (GBM) of he form: dv = µ V d + σ V dw, (1) where V is he firm s asses value, wih an insananeous drif µ, and an insananeous volailiy σ. sandard Wiener process is W. We denoe by X he book value of he deb a ime, ha has mauriy equal o T. s noed earlier, X plays he role of he srike price of he call, since he marke value of equiy can 6

8 be hough of as a call opion on V wih ime o expiraion equal o T. The marke value of equiy, V, will hen be given by he Black and Scholes (1973) formula for call opions: E V E = V rt N( d1) Xe N( d 2 ), (2) where d 1 ln( = V 1 / X ) + r + σ 2 σ T 2 T, d 2 = d 1 σ T, (3) and r is he risk-free rae and disribuion. N is he cumulaive densiy funcion of he sandard normal To calculae σ we adop an ieraive procedure. We use daily daa from he pas 12 monhs o obain an esimae of he volailiy of equiy, σ, which is hen used as an iniial value for he esimaion of σ. Using he Black-Scholes formula, and for each rading day of he pas 12 monhs, we compue V using as V he marke value of equiy of ha day. In his manner, we obain daily values for V. We hen compue he sandard deviaion of hose V s, which is used as he value of E σ, for he nex ieraion. This procedure is repeaed unil he values of from wo consecuive ieraions converge. Our olerance level for convergence is 10E-4. For mos firms, i akes only a few ieraions for σ o converge. Once he converged value of σ is obained, we use i o back ou V hrough equaion (2). The above process is repeaed every end of he monh, resuling in he esimaion of monhly values of σ. The esimaion window is always kep equal o 12 monhs. The risk-free rae used for each monhly ieraive process is he one-year T-bill rae observed a he end of he monh. E σ 7

9 Once daily values of V are esimaed, we can compue he drif, µ, by calculaing he mean of he change in lnv. The defaul probabiliy is he probabiliy ha he firm s asses will be less han he book value of he firm s liabiliies. In oher words, ( X V ) = Pr ob( ln( V ) ln( X ) V ) P def, = Pr ob V, T,, + T, +. (4) Since he value of he asses follows he GBM of equaion (1), he value of he asses a any ime is given by: 2 σ ln ( V T ) ( V ), + = ln, + µ T + σ Tε + T. (5) 2 ε ε + T + T W = ~ N ( + T ) W( ) ( 0,1 ). T, (6) Therefore, we can rewrie he defaul probabiliy as follows: P P def, def, = Pr ob ln V ln X = Pr ob ( V ) ln( X ),, σ + µ 2 σ + µ 2 σ T 2 T 2 T + σ ε + T. Tε + T 0 (7) We can hen define he disance o defaul (DD) as follows: DD 1 2 ln( V, / X ) + ( µ σ ) T = 2. (8) σ T 8

10 Defaul occurs when he raio of he value of asses o deb is less han one, or is log is negaive. The DD ells us by how many sandard deviaions he log of his raio needs o deviae from is mean in order for defaul o occur. Noice ha alhough he value of he call opion in (2) does no depend on µ, DD does. This is because DD depends on he fuure value of asses which is given in equaion (3). We use he heoreical disribuion implied by Meron s model, which is he normal disribuion. In ha case, he heoreical probabiliy of defaul will be given by: P def ln( V = N( DD) = N, 1 2 / X ) + ( µ σ ) T 2. (9) σ T Sricly speaking, P def is no a defaul probabiliy, because i does no correspond o he rue probabiliy of defaul in large samples. In conras, he defaul probabiliies calculaed by KMV are indeed defaul probabiliies because hey are calculaed using he empirical disribuion of defauls. For insance, in he KMV daabase, he number of companies imes he years of daa is over 100,000, and includes more han 2,000 incidens of defaul. We have a much more limied daabase. For ha reason, we do no call our measure defaul probabiliy, bu raher defaul likelihood indicaor (DLI). 6 I is imporan o noe ha he difference beween our measure of defaul risk and ha produced by KMV is no maerial for he purpose of our sudy. The defaul likelihood indicaor of a firm is a posiive nonlinear funcion of is defaul probabiliy. Since we use our measure of defaul risk o examine he relaion beween defaul risk and equiy reurns raher han price deb or credi risk derivaives, his nonlinear ransformaion canno affec he subsance of our resuls. 9

11 II. Daa and Summary Saisics We use he COMPUSTT annual files o ge he firm s Deb in One Year and Long- Term Deb series for all companies. COMPUSTT sars reporing annual financial daa in However, prior o 1971, only a few hundred firms have deb daa available. Therefore, we sar our analysis in s book value of deb we use he Deb in One Year plus half he Long-Term Deb. I is imporan o include long-erm deb in our calculaions for wo reasons. Firs, firms need o service heir long-erm deb, and hese ineres paymens are par of heir shor-erm liabiliies. Second, he size of he long-erm deb affecs he abiliy of a firm o roll over is shor-erm deb, and herefore reduce is risk of defaul. How much of he long-erm deb should ener our calculaions is arbirary, since we do no observe he coupon paymens of he individual firms. KMV uses 50 percen and argues ha his choice is sensible, and capures adequaely he financing consrains of firms. 7 We do he same. We use annual daa for he book value of deb. To avoid problems relaed o reporing delays, we do no use he book value of deb of he new fiscal year, unil four monhs have elapsed from he end of he previous fiscal year. 8 This is done in order o ensure ha all informaion used o calculae our defaul measures was available o he invesors a he ime of he calculaion. We ge he daily marke values for firms from he CRSP daily files. The book value of equiy informaion is exraced from COMPUSTT. Each monh, he book-o-marke (BM) raio of a firm is he six-monh prior book value of equiy divided by he curren monh s marke value of equiy. Firms wih negaive BM raios are excluded from our sample. 10

12 s risk-free rae for he compuaion of DLI, we use monhly observaions of he oneyear Treasury Bill rae obained from he Federal Reserve Board Saisics. Table I repors he number of firms per year for which DLI could be calculaed, as well as he number of firms ha filed for bankrupcy (Chaper 11) or were liquidaed. [Inser Table I approximaely here ] The aggregae defaul likelihood measure P(D) is defined as a simple average of he defaul likelihood indicaors of all firms. graph of he P(D) is provided in Figure 1 for he whole sample period (1971:1 o 1999:12). The shaded areas represen recession periods as defined by he NBER. The graph shows ha defaul probabiliies vary grealy wih he business cycle and increase subsanially during recessions. [Inser Figure 1 approximaely here] We define he aggregae survival rae, SV as one minus P(D). The change in aggregae survival rae (SV) a ime is given by SV SV 1. Summary saisics for SV and (SV) are presened in Panel of Table II. [Inser Table II approximaely here] The defaul reurn spread is from Ibboson ssociaes, and i is defined as he reurn difference beween B Moody s-raed bonds and Moody s raed bonds. Similarly, he defaul yield spread is defined as he yield difference beween Moody s B bonds and Moody s bonds. The series is obained from he Federal Reserve Bank of S. Louis. The change in spread (spread) is obained from Hahn and Lee (2001). The spread in Hahn and Lee is defined as he difference in he yields beween Moody s B bonds and 10-year governmen bonds. (spread) is he change in ha spread. 11

13 Panel B of Table II provides he correlaion coefficiens beween he above-defined defaul spreads and (SV ). The correlaions are very low and reveal ha he informaion capured by our measure is markedly differen from ha capured by he commonly used defaul spreads. This is consisen wih he findings in Elon e al. (2001). The Fama-French facors HML and SMB, and he marke facor EMKT are obained from Kenneh French s Web page. 9 From he same Web page we also obain daa for he one-monh T-Bill rae used in our asse pricing ess. Panel C of Table II repors he correlaion coefficiens beween (SV ) and he Fama-French facors. The correlaions of (SV ) wih EMKT and SMB are posiive and of he order of 0.5 whereas ha wih HML is negaive and equal o This suggess ha EMKT and SMB conain poenially significan defaul-relaed informaion. The regressions of Panel D in Table II show ha (SV ) can explain a subsanial porion of he imevariaion in EMKT and SMB. This does no mean, however, ha he priced informaion in EMKT and SMB is relaed o defaul risk. The defaul-relaed conen of he priced informaion in SMB and HML will be examined in Secion V. Finally, given ha he need o compue defaul likelihood indicaors for each sock consrains us o use only a subse of he U.S. equiy marke as presened in Table I, i is imporan o verify ha our resuls are represenaive of he U.S. marke as a whole. To his end, we consruc he Fama-French facors HML and SMB wihin our sample, and compare hem wih hose consruced by Fama and French using a much larger cross-secion of U.S. equiies. The resuls are repored in Panel E of Table II. The disribuional characerisics of he HML and SMB facors consruced wihin our sample are similar o hose of he HML and SMB facors provided by Fama and French. Furhermore, heir correlaions are quie large and of he order of 0.95 for 12

14 SMB and 0.86 for HML. The above comparisons reveal ha he subsample we use in our sudy is largely represenaive of he U.S. equiy samples used in oher sudies of equiy reurns. III. Measuring Model ccuracy In his secion, we evaluae he abiliy of our defaul measure o capure defaul risk. To do ha, we employ Moody s ccuracy Raio. In addiion, we compare he defaul likelihood indicaors of acually defauled firms wih hose of a conrol group ha did no defaul.. ccuracy Raio The accuracy raio (R) proposed by Moody s reveals he abiliy of a model o predic acual defauls over a five-year horizon. 10 Le us suppose a model ranks he firms according o some measure of defaul risk. Suppose here are N firms in oal in our sample and M of hose acually defaul in he nex five M years. Le θ = be he percenage of firms ha defaul. For every ineger λ beween zero and N 100, we look a how many firms acually defauled wihin he λ % of firms wih he highes defaul risk. Of course, his number of defauls canno be more han M. We divide he number of firms ha acually defauled wihin he firs λ percen of firms by M and denoe he resul by f (λ). Then f (λ) akes values beween zero and one, and is an increasing funcion of λ. Moreover, f ( 0) = 0 and f ( 100) = 1. Suppose we had he perfec measure of fuure defaul likelihood, and we were ranking socks according o ha. We would hen have been able o capure all defauls for each ineger λ, and f (λ) would be given by 13

15 λ f ( λ ) = for λ < θ and f ( λ) = 1 for λ θ. (10) θ Suppose we also calculae he average f (λ) for all monhs covered by he sample. The graph of his funcion of average f (λ) is shown as he kinked line in Figure 2, graph B. [Inser Figure 2 approximaely here] he oher exreme, suppose we had zero informaion abou fuure defaul likelihoods, and we were ranking he socks randomly. If we did ha a large number of imes, f (λ) would be equal o λ. Graphically, he average f (λ) would correspond o he 45 degree line in he graphs of Figure 2. We measure he amoun of informaion in a model by how far he graph of he average f (λ) funcion lies above he 45 degree line. Specifically, we measure i by he area beween he 45-degree line and he graph of average f (λ). The accuracy raio of a model is hen defined as he raio beween he area associaed wih ha model s average f (λ) funcion and he one associaed wih he perfec model s average f (λ) funcion. Under his definiion, he perfec model has accuracy raio of one, and he zero-informaion model has an accuracy raio of zero. The measure implied by Meron s model is he disance-o-defaul (DD). Therefore, if we rank socks according o DD, he accuracy raio we obain is equal o This means ha our measure conains subsanial informaion abou fuure defauls. By consrucion, our measure of defaul risk is relaed o size. I is herefore emping o conclude ha i conains virually he same informaion as he marke value of equiy. This is no he case, however. If we rank socks on he basis of heir marke value of equiy and compue he corresponding accuracy raio, his will be equal o only Therefore, DD conains much more informaion han ha conveyed by he size of he firms. This is an imporan poin, since 14

16 par of our analysis in Secion IV provides an inerpreaion of he size effec, based on he informaion conained in DLI. Finally, an imporan parameer in he DD measure is he volailiy of asses. Therefore, one may conjecure ha wha we capure wih our defaul measure is simply he volailiy of asses. This is again no he case. If we rank socks on he basis of heir volailiy of asses, he accuracy raio we obain is 0.290, which is much lower han ha based on DD (0.592). In oher words, our measure of defaul risk capures imporan defaul informaion beyond wha is conveyed by he marke value of equiy or he volailiy of he firm s asses alone. B. Comparison Beween Defauled Firms and Non-Defauled Firms s a furher es of he abiliy of our measure o capure defaul risk, we compare he defaul likelihood indicaors of firms ha acually defauled wih hose of a conrol group of firms ha did no defaul. Similar comparisons have been performed in he pas in lman (1968) and harony, Jones, and Swary (1980). To make he comparison meaningful, we choose firms in he conrol group ha have similar size and indusry characerisics as hose in he experimenal group. In paricular, for every firm ha defauls, we selec a firm wih a marke capializaion similar o ha of he firm in he experimenal group before i defauled. In addiion, he firm in he conrol group shares he same wo-digi indusry code as he one in he experimenal group. We compue he average defaul likelihood indicaor for each group. Figure 3 presens he resuls. We find ha he average defaul likelihood indicaor of he experimenal group goes up sharply in he five years prior o defaul. In conras, he average defaul likelihood indicaor of he conrol group says a he same level hroughou he five-year period. Noe ha in he graph, =0 corresponds o abou wo o hree years prior o defaul, since he daabase does no provide 15

17 daa up o he dae of defaul. Therefore, an average defaul likelihood indicaor of 0.57 for he experimenal group can be considered high. The resuls of his es provide furher assurance ha our defaul likelihood indicaors do indeed capure defaul risk. [Inser Figure 3 approximaely here] IV. Defaul Risk and Variaion in Equiy Reurns We sar our analysis of he relaion beween defaul risk and equiy reurns by examining wheher porfolios wih differen defaul risk characerisics provide significanly differen reurns. significan difference in he reurns would indicae ha defaul risk may be imporan for he pricing of equiies. Table III repors simple sors of socks based on heir defaul likelihood indicaors. he end of each monh from 1970:12 o 1999:11, we use he mos recen monhly defaul probabiliy for each firm o sor all socks ino porfolios. We firs sor socks ino five porfolios. We examine heir reurns when he porfolios are equally weighed or value-weighed and repor he average defaul likelihood indicaor for each one of hem. Evidenly, he lower he average defaul likelihood indicaor, he lower he risk of defaul. [Inser Table III approximaely here] Noe ha in calculaing he reurns of porfolios in Secion IV, we use he following procedure. Every ime a sock ges delised due o defaul, we se he reurn of he porion of he porfolio invesed in ha sock equal o -100 percen. In oher words, we assume ha he recovery rae for equiy-holders is zero. In his way, we fully ake ino accoun he cos of defaul in our calculaions of average porfolio reurns. In fac, he reurns we repor may be considered as he lower bounds of reurns (before ransacion coss) earned by equiy-holders. The reason is ha ofen, he recovery rae is no zero. 16

18 The -values of all ess in Secion IV are compued from Newey-Wes (1987) sandard errors. In paricular, hey are correced for Whie (1980) heeroskedasiciy and serial correlaion up o he number of lags ha are saisically significan a he five percen level. The reurn difference beween he equally weighed high-defaul-risk porfolio and lowdefaul-risk porfolio is 53 basis poins (bps) per monh or 6.36 percen per annum (p.a.). The difference is saisically significan a he five percen level. This is no he case for he valueweighed porfolios whose difference in reurns is only 14 bps per monh. When we sor socks ino 10 porfolios, he resuls we obain are similar. The difference in reurns beween he high-defaul risk porfolio and he low-defaul risk porfolio is saisically significan for he equally weighed porfolios bu no for he value-weighed porfolios. The reurn differenial for he equally weighed porfolios is 98 bps per monh or percen p.a. Noice hough ha he aggregae defaul measure for he equally weighed porfolios assumes bigger values han i does for he value-weighed porfolios. I appears ha smallcapializaion socks have on average higher defaul risk, and as a resul, hey earn higher reurns han big-capializaion socks do. In addiion, boh in he case of defaul quiniles and deciles, he average marke capializaion of a porfolio (size) and is book-o-marke (BM) raio vary monoonically wih he average defaul risk of he porfolio. In paricular, he average size increases as he defaul risk of he porfolio decreases, whereas he opposie is rue for BM. These resuls sugges ha he size and BM effecs may be linked o he defaul risk of socks. Recall ha boh effecs are considered sock marke anomalies according o he lieraure of he Capial sse Pricing Model (CPM). The reason for heir exisence remains unknown. The remainder of he paper invesigaes furher he possible link beween defaul risk and hose effecs. Our analysis will focus on equally weighed porfolios, since his is he weighing scheme ypically 17

19 employed in sudies ha consider he size and BM effecs. 11 However, all he resuls of he paper remain qualiaively he same when porfolios are value-weighed.. Size, BM, and Defaul Risk To examine he exen o which he size and BM effecs can be inerpreed as defaul effecs, we perform wo-way sors and examine each of he wo effecs wihin differen defaul risk porfolios..1. The Size Effec Table IV presens resuls from sequenial sors. Socks are firs sored ino five quiniles according o heir defaul risk. Subsequenly, he socks wihin each defaul quinile are sored ino five size porfolios. This procedure produces 25 porfolios in oal. In wha follows, we examine wheher he size effec exiss in all defaul risk quiniles, as well as in he whole sample. [Inser Table IV approximaely here] The resuls of Panel show ha he size effec is presen only wihin he quinile ha conains he socks wih he highes defaul risk (DLI 1). The effec is very srong wih an average reurn difference beween small and big firms of 3.82 percen per monh or a saggering percen p.a. Noice ha he difference in reurns drops o close o zero for he remaining defaul-sored porfolios. There is a saisically significan size effec in he whole sample, bu he reurn difference beween small and big firms is more han four imes smaller han in DLI 1. The resuls of Panel sugges ha he size effec exiss only wihin he segmen of he marke ha conains he socks wih he highes defaul risk. To wha exen, however, are we ruly capuring he size effec? Is here really subsanial variaion in he marke capializaions of 18

20 socks wihin he DLI 1 porfolio? Panel B addresses his quesion. We see ha here is indeed large variaion in he marke caps of socks wihin he highes defaul risk porfolio. Bu in erms of he average marke caps for he size quiniles formed using he whole sample, he bigges firms in DLI 1 are raher medium o large firms. On he oher hand, he DLI 1 Small porfolio conains he smalles of he small firms compared o he small size quinile formed on he basis of he whole sample. These resuls imply ha he size effec is concenraed in he smalles of he small firms, which also happen o be among hose wih he highes defaul risk. How much riskier are he socks in DLI 1 compared o he oher defaul risk quiniles? Panel C of Table IV shows ha hey are a lo riskier. The small firms in DLI 1 are almos 14 imes riskier in erms of likelihood of defaul han he small firms in DLI 2. They are also on average more han wice as risky in erms of defaul han he socks in he small size quinile consruced using he whole sample. Therefore, he large average reurns ha small high defaul socks earn compared o he res of he marke can be considered o be compensaion for he large defaul risk hey have. To see ha, noice also ha in he high DLI quinile, DLI decreases monoonically as size increases. In oher words, he large difference in reurns beween small and big socks in he DLI 1 quinile can be explained by he large difference in he defaul risk of hose porfolios. In he remaining defaul quiniles where here is no evidence of a size effec, he difference in defaul risk beween small and big socks is also very small. Panel D repors he average BM of he defaul- and size-sored porfolios. These resuls are useful in order o undersand he exen o which size, defaul risk, and BM are inerrelaed. Panel D shows ha he average BMs in he size-sored porfolios of DLI 1 are he highes in he 19

21 sample. The BM decreases monoonically wih DLI, which suggess ha he BM effec may also be relaed o defaul risk. The conclusion ha emerges from Table IV is ha he size effec is in fac a defaul effec. There is a size effec only in he segmen of he marke wih he highes defaul risk. Wihin ha segmen, he difference in reurns beween small and big socks can be explained by he difference in heir defaul risk. In he remaining socks in he marke, where here is no significan size effec, he difference in defaul risk beween small and big socks is minimal. Book-o-marke seems also o be relaed o defaul risk and size, and we will examine hese relaions in he following secion..2. The BM Effec Table V presens resuls from porfolio sorings in he same spiri as hose of Table IV. Socks are firs sored ino five defaul risk quiniles, and hen each of he five defaul quiniles is sored ino five BM porfolios. In wha follows, we will examine he BM effec wihin each defaul quinile, as well as for he marke as a whole. [Inser Table V approximaely here] Panel shows ha he BM effec is prominen only in he wo quiniles wih he highes defaul risk, wih he reurn differenial beween value (high BM) and growh (low BM) socks being almos wo and a half imes bigger in DLI 1 han in DLI 2. There is a BM effec in he whole sample, bu he reurn differenial is abou half as big as ha found in DLI 1. Noice ha wihin DLI 1, he average DLI is much higher for value socks han i is for growh socks. In DLI 2, where he BM effec is weaker, he difference in defaul risk beween value and growh socks is also small. These resuls imply ha, similarly o he size effec, he 20

22 BM effec seems o be due o defaul risk. The only difference is ha he BM effec is significan wihin he wo-fifhs of he socks wih he highes defaul risk, whereas he size effec is presen only in he one-fifh of socks wih he highes defaul risk. In oher words, he inerrelaion beween size and defaul risk seems o be a bi igher. This is confirmed in Secion IV. C. using regression analysis. There is a lo of dispersion in he average BM raios wihin he DLI porfolios. This is paricularly rue for DLI 1 and 2, which means ha indeed he reurn differenial we examine capures a BM effec. In fac, he average BM raio varies more across porfolios in DLI 1 han i does across BM porfolios formed using he whole sample. In DLI 1 and 2 where defaul risk is higher han in he oher quiniles and he marke as a whole, he average BM raios of he BM sored porfolios are also higher. This resul underlines again he inerrelaion beween BM and defaul risk discussed above. Furhermore, he average DLIs in Panel C exhibi a monoonic relaion wih BM only in he DLI 1 and 2 quiniles, ha is, he wo quiniles wih he highes defaul risk, where he BM effec is significan. For he res of he sample, he relaion beween defaul risk and BM raios does no appear o be linear. similar resul emerges from Table IV, Panel C. Defaul risk varies monoonically wih size only wihin he wo highes defaul risk quiniles. I seems ha here are linear relaions beween defaul risk and size, and defaul risk and BM, only o he exen ha ha defaul risk is sizeable. When he risk of defaul of a company is very small, he lineariy in he relaion beween defaul and size and defaul and BM disappears, probably because defauls are very unlikely o occur in hose cases. Panel D shows again ha DLI 1 conains mainly small firms. However, size does no vary monoonically wih BM, excep wihin he wo highes defaul risk quiniles. The same 21

23 conclusion can be reached from Panel D of Table IV. The average BM raios vary monoonically wih size only wihin he wo highes defaul risk quiniles. In boh cases he variaion is small. I seems ha size and BM proxy o some exen for each oher only wihin he segmen of he marke wih he highes defaul risk. This implies ha hey are no idenical phenomena. Furhermore, he reurn premium of small firms over big firms is more han one percen larger han ha of high BM socks over low BM socks. In addiion, he size effec is presen in a subse of he segmen of he marke in which he BM effec exiss. Boh are linked, however, o a common risk measure, which is defaul risk. B. The Defaul Effec Tables IV and V show ha size and BM are inimaely relaed o defaul risk. Bu does his mean ha here is also a defaul risk in he daa? nd if here is, is i confined only wihin cerain size and BM quiniles? In oher words, is defaul risk rewarded differenly depending on he size and BM characerisics of he sock? These are he quesions we address in his secion. We define he defaul effec as a posiive average reurn differenial beween high and low defaul risk firms. B.1. The Defaul Effec in Size-Sored Porfolios Table VI examines wheher here is a defaul effec in size-sored porfolios by reversing he soring procedure of Table IV. In paricular, we firs sor socks ino five size quiniles, and hen sor each size quinile ino five defaul porfolios. s we will see below, his exercise also allows us o obain a beer undersanding of small firms as an asse class. [Inser Table VI approximaely here] 22

24 Panel shows ha here is a saisically significan defaul effec only wihin he small size quinile. The average monhly reurn is 2.2 percen or 26.4 percen p.a. In mos of he remaining size quiniles, he difference in reurns beween high and low defaul risk porfolios is in fac negaive. This means ha high defaul risk firms earn a higher reurn han low defaul risk firms only if hey are also small in size. To verify his poin, see Panel B of Table VI. ll high-dli porfolios have subsanial defaul risk, independenly of he marke capializaion of he socks. Similarly, all low-dli porfolios have virually no defaul risk. However, only small high defaul risk socks earn higher reurns han low defaul risk socks. This resul may indicae ha firms differ in heir abiliy o re-emerge from Chaper 11, depending on heir size. If small firms, for insance, are less likely o emerge from he resrucuring process as public firms, invesors may require a bigger risk premium o hold hem, compared o wha hey require for bigger size-high defaul risk firms. This will induce he average reurns of small high-dli firms o be higher han hose of bigger high-dli firms. 12 Empirical evidence from he corporae bankrupcy lieraure shows ha indeed large firms are more likely o survive Chaper 11 han small firms. 13 Panel B of Table VI provides also insighs ino he profile of small firms as an asse class. Noice ha wihin he small size quinile, DLI varies beween percen and 0.09 percen. This implies ha small firms can differ a lo wih respec o heir (defaul) risk characerisics. They can also differ significanly wih respec o heir reurns, as Panel reveals. These resuls sugges ha small firms do no consiue a homogenous asse class, as is commonly believed. Finally, Panel B shows ha defaul risk decreases monoonically as size increases, confirming he close relaion beween size and defaul risk observed in Table IV. Panels C and D 23

25 show ha he small - high DLI porfolio conains he smalles of he small socks and hose wih he highes BM raio. Two imporan conclusions emerge from his able. Firs, defaul risk is rewarded only in small, value socks. Firms ha have high defaul risk, bu are no caegorized as small and high BM, will no earn higher reurns han firms wih low defaul risk and similar size and BM characerisics. This resul furher underlines he close link among size, defaul risk, and BM. Second, small firms are no made equal. They differ subsanially in erms of boh heir reurn and (defaul) risk characerisics. This resul reveals ha small firms do no consiue a homogeneous asse class. B.2. The Defaul Effec in BM-Sored Porfolios To furher examine he link beween defaul risk and BM, Table VII examines he presence of a defaul effec in BM-sored porfolios. sses are firs sored in five BM quiniles, and subsequenly, each BM-sored quinile is subdivided ino five defaul-sored porfolios. [Inser Table VII approximaely here] Panel reveals ha he defaul effec is again presen only wihin he high BM quinile. This resul is consisen wih ha of Table VI. Since he smalles high-dli firms are also ypically he highes BM firms, he same inerpreaion applies here. Specifically, defaul risk is rewarded only for small, value socks, and no for any oher socks in he marke, independenly of heir risk of defaul. This is confirmed in Panels C and D. Once again, Panel B shows ha value socks can differ a lo wih respec o heir defaul risk characerisics. Given ha hey also differ significanly in erms of heir reurns, Panels 24

26 and B sugges ha, similarly o small firms, value socks do no consiue a homogeneous asse class eiher. The resuls of Table VII are consisen and analogous o hose of Table VI. High defaul risk socks earn a higher reurn han low defaul risk socks, only o he exen ha hey are small and high BM. If he size and BM crieria are no fulfilled, hey will no earn higher reurns han low defaul risk socks, even if heir defaul risk is very high. Furhermore, our analysis implies ha small firms and value socks do no consiue homogeneous asse classes. 14 C. Examining he Ineracion of Size and Defaul, and BM and Defaul Using Regression nalysis In his secion, we summarize and quanify he degree of ineracion beween size and defaul, and BM and defaul using regression analysis. Two differen mehodologies are employed. The firs one is a porfolio-based regression approach developed in Nijman, Swinkels, and Verbeek (2002). The second one uses he Fama-MacBeh (1973) mehodology on individual sock reurns. C.1. The Porfolio-based Regression pproach The regression mehodology in Nijman, Swinkels, and Verbeek (2002) is an exension of he mehodology in Heson and Rouwenhors (1994) which allows for he presence of ineracion erms beween he variables of ineres. In he curren applicaion, we analyze average reurns of porfolios grouped on he basis of DLI, size, and BM, and examine he relaive magniudes of he individual effecs, as well as heir ineracions. 25

27 Similarly o Daniel and Timan (1997), Nijman, Swinkels, and Verbeek (2002) assume ha he condiional expeced reurn of a sock can be decomposed ino several effecs. In oher words, E Na Nb ( Ri, + 1 ) = α a, b X i, ( a, b) a= 1 b= 1 (11) where E (.) denoes he expecaion condiional on he informaion available a ime, R i, + 1 is he reurn of he sock a ime +1, ( a b) X i,, is a dummy variable ha indicaes he membership of he sock in a paricular porfolio, and α he expeced reurn of a sock wih characerisics a a,b and b. In our applicaion, a and b are eiher size and defaul risk, or BM and defaul risk. Therefore, equaion (7) simply saes he condiional expeced reurn of a sock, given is size/bm and defaul risk characerisics ha gran i membership o a paricular porfolio. 15 be wrien as: The condiional expeced reurn on a porfolio p of N socks wih weighs E N a Nb p ( R 1) X p i, ( a, b) a= 1 b= 1 a, b p w i,, can hen + = α (12) where X p i, i,, ( a b) = w p i, X ( a, b). Since he porfolios we will use for our ess are all equallyweighed and sored on he basis of he characerisics a and b, we can simplify he above equaion as follows: E N a Nb p ( R 1) X ( a, b + = α ). (13) a= 1 b= 1 The regression equaion implied can be wrien as: a, b N a Nb p R + 1 = α a, b X, + + a= 1 b= 1 ( a b) ε 1 (14) 26

28 where ε P p + 1 R+ 1 E + ( R 1 ), which is by consrucion orhogonal o he regressors. The only assumpion made is ha he cross-auocorrelaion srucure is zero, i.e., p p E( ε + hε ) = 0. However, equaion (10) can be wrien in a more parsimonious way by imposing an addiive srucure similar o ha in Roll (1992) and Heson and Rouwenhors (1994). In ha case, he condiional expeced reurn of porfolio p will be given by: E Na Nb Na Nb p ( R + 1) = + φ X ( a, ) + φ X (, b) + α X ( a, b α ) 1,1 a a= 2 b b= 2 a= 2 b= 2 a, b (15) where X (., b ), for insance, denoes ha only he argumen b is considered. In ha case, all socks in group b are considered, irrespecively of heir a characerisic. The consan denoes he reurn on he reference porfolio. The reference porfolio is arbirarily chosen and is used o avoid he dummy rap. When we examine he ineracion of size and defaul effecs, he reference porfolio we use is he porfolio ha conains big cap and low-dli socks. In he ess of he ineracion of BM and defaul effecs, he reference porfolio is he one ha conains socks wih low BM and low DLI. The esimaed coefficiens φ can be inerpreed as denoing he difference in reurn beween porfolio p and he reference porfolio aribued o a paricular effec. Similarly, he coefficiens α denoe he addiional expeced reurn for porfolio p due o he ineracion of wo effecs. The oal expeced reurn on porfolio p is given by he sum of he reurns of he reference porfolio, he individual effecs, and he ineracion effecs. Each se of esimaions uses 15 lef-hand-side porfolios. In he case of he size-defaul effecs es, hey are comprised of hree size porfolios, hree defaul-sored (DLI) porfolios, and nine porfolios creaed from he inersecion of wo independen sors on hree size and hree DLI porfolios. In he case of he BM-defaul effecs es, he porfolios include hree BM porfolios, α 1,1 27

29 hree defaul-sored porfolios, and nine porfolios from he inersecion of wo independen sors on hree BM porfolios and hree DLI porfolios. In boh ses of ess, here are eigh parameers o be esimaed. The resuls are repored in Table VIII. The firs panel refers o he ess of he size-defaul effecs, whereas he second panel conains he resuls for he BM-defaul effecs. [Inser Table VIII approximaely here] Panel shows ha he economically and saisically mos imporan coefficiens for he individual effecs are for small size and high DLI. In addiion, he sronges ineracion effec refers o he ineracion of small size and high DLI. In oher words, a porfolio will earn higher reurn, he smaller is marke cap, and he higher is defaul risk. I will also earn an addiional reurn from he ineracion of high defaul risk and small size. This addiional reurn is zero if he small firms have medium defaul risk. These resuls are consisen wih our earlier finding ha he size effec exiss only among high defaul risk socks. Noe also ha he coefficien on he ineracion erm beween high DLI and medium size is negaive and saisically insignifican. This is again in line wih our previous resul ha he defaul effec exiss only wihin small firms. The reurn on he reference porfolio (big firms, low DLI) is percen per monh (p.m.) This means ha a porfolio of small firms wih high DLI will earn ( ) 2.37 percen per monh, compared o 1.79 percen p.m ha a porfolio of small firms of medium DLI will earn. Similarly, a porfolio of medium firms of high DLI will earn 1.62 percen per monh, whereas a porfolio of medium size firms of medium DLI will earn only 1.37 percen per monh. Noice ha he reurns above are smaller han hose in Tables IV and VI. The reason is ha socks here are classified ino eriles of size and DLI porfolios raher han quiniles as in 28

30 Table IV o VII. The paern of reurns and he conclusions remain he same: The highes reurns are earned by socks wih he highes DLI and smalles size. Similar conclusions emerge for he BM-DLI porfolios in Panel B. The socks ha earn he highes reurns are socks ha are boh high BM and high DLI. The reurn of he reference porfolio here (low BM, low DLI) is 1.05 percen p.m. Therefore, he high BM, high DLI porfolio will earn a oal reurn of 2.38 percen p.m. as opposed o he 1.84 percen earned by he high BM, medium DLI porfolio. Medium defaul risk firms earn an exra reurn for defaul risk, bu i is smaller han ha earned by high defaul risk firms. In addiion, he only posiive and saisically significan ineracion coefficien is he one referring o high BM and high DLI socks. By he same oken, a porfolio of medium BM and high DLI socks will earn 1.58 percen per monh compared o 1.45 percen per monh earned by a porfolio of medium BM and medium DLI firms. In boh cases, he ineracion erm is economically and saisically equal o zero. C.2. The Fama-MacBeh Regression pproach on Individual Sock Reurns Table IX presens resuls from Fama-MacBeh regressions of individual socks on heir pas monh s size, BM, and DLI characerisics. The regressions consider boh a linear relaion beween sock reurns and characerisics, as well as a nonlinear relaion by including he characerisics squared (size2, BM2, DLI2). In addiion, here are ineracion erms proxied by he produc of size wih DLI (sizedli) and BM wih DLI (BMDLI). We render size and BM orhogonal o DLI before performing he ess, in order o avoid possible problems in he inerpreaion of he resuls. [Inser Table IX approximaely here] 29

31 The resuls show ha wha explains nex monh s equiy reurns is he curren defaul risk of securiies, heir BM, and he ineracion of defaul risk and size. Size per se does no appear o play any role. This is confirmed in ess where only he DLI and size variables are considered. Indeed, only DLI, DLI2, and sizedli are imporan for explaining he nex period s equiy reurns. In conras, BM seems o conain incremenal informaion abou nex period s reurns, over and above ha conained in DLI. The regressions ha consider only DLI and BM variables show ha he BM variables and DLI2, in addiion o he ineracion erm, are imporan for explaining nex period s equiy reurns. The regression resuls in Table IX also highligh he imporance of he squared erms, and herefore, he nonlineariy in he relaions beween equiy reurns, DLI, and BM. The boom line from hese ess is he following. The observed relaion in he lieraure beween size and equiy reurns is compleely due o defaul risk. Size proxies for defaul risk and his is why small caps earn higher reurns han big caps. They do so, because small caps have higher defaul risk han big caps. BM also proxies parially for defaul risk. Defaul risk is no however all he informaion included in BM. D. Conclusions bou he size, BM, and Defaul Effecs The resuls in Secion IV poin o he following conclusions. The size effec is a defaul effec as i exiss only wihin he quinile of firms wih he highes defaul risk. The BM effec is also relaed o defaul risk, bu i exiss among firms wih boh high and medium defaul risk. Defaul risk is rewarded only o he exen ha high defaul risk firms are also small and high BM and in no oher case. In oher words, defaul risk and size share a nonlinear relaion, and he same 30

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