Why some Distressed Firms Have Low Expected Returns

Transcription

1 CIRJE-F-504 Why some Disressed Firms Have Low Expeced Reurns Ryoichi Ikeda Graduae School of Economics, Universiy of Tokyo Takao Kobayashi Universiy of Tokyo July 2007: Revised in Sepember 2007 CIRJE Discussion Papers can be downloaded wihou charge from: hp:// Discussion Papers are a series of manuscrips in heir draf form. They are no inended for circulaion or disribuion excep as indicaed by he auhor. For ha reason Discussion Papers may no be reproduced or disribued wihou he wrien consen of he auhor.

2 Why some Disressed Firms Have Low Expeced Reurns Ryoichi Ikeda Universiy of Tokyo JSPS Research Fellow Takao Kobayashi Universiy of Tokyo Sepember 3, 2007.

3 Absrac In recen years, empirical researchers show ha firms wih higher credi risk have much smaller average sock reurns. This finding is opposie o he risk-reward principle and is ofen aribued o mispricing and marke anomalies. We invesigae how credi risk and expeced sock reurn are deermined in a model wih producion, capial srucure and aggregae uncerainy. We show ha, conrary o he convenional wisdom, a firm wih higher credi risk can have less risky sock han he one wih lower credi risk. 2

4 Inroducion Mos of he analyses on credi risk have focused on defaul probabiliy or he pricing of defaulable securiies such as bonds or credi derivaives. The relaionship beween credi risk and sock reurn is rarely horoughly invesigaed. In recen years, however, researchers have invesigaed how he credi risk is refleced in he cross-secional sock reurn, and mos of he papers show resuls opposie o our inuiion: firms wih higher credi risk have much smaller average sock reurns (e.g. Dichev(998), Griffin and Lemmon(2002) and Campbell, Hilscher, and Szilagyi(2005)). Furhermore, his phenomenon appears even afer he marke, value and size effec are adjused. We invesigae how credi risk and expeced sock reurn are deermined by economic primiives, such as ases and echnology, in he neoclassical framework wih raional expecaions. We show ha, conrary o he convenional wisdom, a firm wih higher credi risk can have less risky sock han he one wih lower credi risk. In he model we consider a firm wih a zero-coupon bond. When he firm faces high defaul risk, he value of he cash flow afer he mauriy for sockholders is small. However, he firm pays dividend before he mauriy o he sockholders, which is he main deerminan of he sock value. Since he value of he cash flow before he mauriy is more sable han he value of he cash flow afer he mauriy, he sock has low risk and low reurn, which explains he anomaly of he relaionship beween credi risk and sock reurn. This paper is relaed o wo srands of he financial economics lieraure. On one hand i shares wih a series of recen papers such as Berk, Green and Naik (999), Gomes, Kogan and Zhang (2003), Carlson, Fisher and Giammarino (2004), Zhang (2005) and Obreja (2006), wih he objecive of explaining boh he predicable variaions in equiy reurns hrough ime and he cross-secional relaion beween equiy reurns and characerisics. However, hese models focus on all-equiy firms only excep for Obreja (2006), while ours focuses on financially leveraged firms. 3

5 Our heoreical model also shares wih he credi risk lieraure on defaul probabiliy and pricing of bonds. Imporan conribuion o his lieraure include: Meron (974), Black and Cox (976), Geske (977), Leland (994), Leland and Tof(996). From his perspecive, he novely in our model sems from he fac ha he defaul probabiliy is relaed o ime-varying price of risk. This paper is organized as follows. In secion 2, we presen he empirical research on he relaionship beween credi risk and sock reurn. In secion 3, we presen he model, oulining he echnology, he ownership srucure, he objecives, and he decisions of he firms. Then we discuss he valuaion of he corporae bond and socks. In secion 4, we presen he resuls of he paper, and finally we conclude in secion 5. The proofs of he proposiions are in Appendix. 2. Empirical research Empirical research on he relaionship beween credi risk and sock reurn can be classified ino hree groups according o he mehod of performing he measuremen of he likelihood of bankrupcy; () using accouning informaion, such as O score and Z score, (2) saisical compuaion using hazard models, and (3) using srucural models such as Meron (974). Dichev (998) shows ha here is a negaive correlaion beween sock reurn and he likelihood of bankrupcy wih mehod of (), using Z-score. This is confirmed using he same mehod by Griffin and Lemmon (2002), using O-score. Griffin and Lemmon (2002) also claim ha he phenomenon is caused by ha he bad performance of firms wih low Book-o-Marke raios and high disress risks. They claim ha such socks are ofen mispriced in he marke. Campbell, Hilscher, and Szilagyi (2005) analyze he same issues using (2). In ha paper, a firm wih high likelihood of bankrupcy has also a low sock average reurn and hey asser ha he sock marke is mispricing he disress risk. Vassalou and Xing (2004) examine he problem wih mehod (3). In ha paper, conrary o he oher research, i is repored ha a firm wih high likelihood of bankrupcy has a high average sock reurn. And in such firms, hose wih lower capial size and higher book-o-marke raio have even higher average reurns. Da and Gao (2006) dispue he conclusion of Vassalou and Xing (2004). They propose ha he shor-erm reurn reversal and he liquidiy risk can explain mos of he resuls in Vassalou and Xing (2004), and conclude ha here is lile evidence ha he See Alman (968) for Z score and Ohlson (980) for O score. 4

6 likelihood of bankrupcy and he subsequen average sock reurn have posiive correlaion. Garlappi, Shu, and Yan (2006) also analyze he relaionship beween he likelihood of bankrupcy and he sock expeced reurn, using EDF (Expeced Defaul Frequency). They show ha when likelihood of bankrupcy is high, he average sock reurn does no necessarily become high. In he paper, hey sugges ha when a firm has fallen ino serious financial crisis here will be a negoiaion beween a sockholder and a bondholder. And hey claim ha likelihood of bankrupcy and a sock expeced reurn are decided depending on bargaining abiliy using he model of Fan and Sundaresan (2000). 3. The model We consruc a neoclassical producion model (e.g., Lucas (967), Lucas and Presco (97)) augmened wih aggregae uncerainy and capial srucure. Secion 3.A describes he economic environmen. Secion 3.B pus in he capial srucure and defaul. Then secion 3.C discusses valuaion of socks and debs. Appendix conains he proof. A. Environmen A.. Technology Firms own echnologies ha use one inpu, capial, and exhibi decreasing-reurns-o-scale. The oupu in dollar uni, y, is uncerain, depending on he realizaion of boh an aggregae shock, x, and an idiosyncraic shock z. In his model, differen firms have differen idiosyncraic shocks, leading o heerogeneiy in he producion economy. Le k denoe he sock of capial a ime and he producion funcion be given by α y = f( k ; x, z ) = k exp( x + z ), (3.A.) where 0< α <, which denoes he capial share. We assume ha he produciviy shocks have saionary and monoone Markov ransiion funcions. Specifically, x and z follow saionary auonomous dynamics of he following form: x = x + ( ρ )( x x ) + σε (3.A.2) x + x x + 5

7 z = z + ( ρ )( z ) + σε, (3.A.3) z + z z + x z where ε + and ε + are I.I.D. sandard normal shocks. The cenral endency of he aggregae shock is capured by x, while he speed of he reversion and he condiional volailiy are capured by ρ x and σ x, respecively. The parameers ρ z and σ z for idiosyncraic process, carry similar inerpreaions. A.2. Sochasic Discoun Facor We parameerize direcly he pricing kernel wihou explicily modeling he consumers problem. We assume he process of he pricing kernel M follows he specificaions in Zhang (2005), ha is log M + = log β γ ( x+ x) (3.A.3) γ ( ) = γ0 + γ x x, (3.A.4) where β, γ 0 > 0 and γ < 0 are consan parameers. γ is ime-varying and decreasing wih he demeaned aggregae produciviy x x. γ can be inerpreed as he marke price of aggregae risk, while he consrain on α < 0 ensures ha his price of risk is counercyclical. The cenral endency of he marke price of aggregae risk is capured byγ 0. A.3. Capial Accumulaion, Adjusmen Coss, and Tax The law of moion for capial is driven by depreciaion, expressed as a fracion,δ, of he level of capial, and he rae of invesmen, as a fracion of capial, i. The rae of capial growh is given by he following equaion: k = k + ( i + δ ) k. (3.A.5) We assume ha invesmen is reversible, bu o adjus he level of capial, firms incur adjusmen coss. Following he lieraure on invesmen wih adjusmen coss, we assume ha he cos funcion is given by he quadraic: θ 2 hi (, k) = i k. (3.A.6) 2 Noice ha firms are subjec o adjusmen coss no only when hey inves bu also when hey disinves. Furher, A firm pays axes on realized profis, τ ( y δ k). Here, τ denoes he fla ax rae on corporae profis. 6

8 B. Capial Srucure and Defaul The enire corporae deb of a firm is modeled as a single zero-coupon bond of face value F mauring a ime T. If he afer-ax firm value a ime T does no exceed he acual principal, hen defaul occurs and he shareholders give he ownership o he bondholders. Oherwise we assume ha he firm redeems he bond and coninues wih no liabiliy. The bankrupcy process is assumed o be insananeous bu cosly. We assume ha he bankrupcy coss are expressed as a fracion,ξ, of he asse value 2. C. Valuaion This secion deals wih he valuaion of he ousanding socks and deb. Upon observing he shocks a he beginning of period, a firm makes opimal invesmen decisions o maximize expeced presen cash flows for sockholders. In our model, since ime-homogeneiy holds only afer he redempion of a zero-coupon bond, sock value before he mauriy canno be represened as he soluion of a Bellman s equaion. In order o solve he problem, firs we need o obain he firm value a he mauriy by solving a Bellman s equaion, and hen solve he dynamic program from T recursively. C. Sock We sar wih he valuaion of he ousanding socks. The value of he firm a ime afer a firm redeems he bond is priced as he soluion o he following dynamic program: { } W( k, x, z ) = ( τ)( y δk ) + δ k + max ( ik + h( i, k )) + E ( M + W( k +, x +, z + )), i subjec o k = k + ( i + δ ) k. In his case, we obain he following resuls: (3.C.) 2 There is no incenive for a firm o decide o defaul before mauriy because we assume ha he producion funcion is always posiive. Therefore, defaul can occur only a mauriy. 7

9 Proposiion There exiss a unique value funcion W( k, x, z ) ha saisfies (3.C.) and is coninuous, increasing and differeniable in k, x and z, and concave in k. Proof: See appendix. A mauriy, he value of he socks is, VT( kt, xt, zt; F) = max( W( kt, xt, zt) ( τ ) F,0), (3.C.2) Then, we obain sock value a < T recursively: V( k, x, z; F) = ( τ )( y δk) + δk (3.C.3) + max ( + (, )) + ( (,, ; )), subjec o k = k + ( i + δ ) k. i { ik h i k E M V k x z F } Proposiion2 There exiss i * ha maximizes ( ( ik + h( i, k)) + E( M+ V+ ( k+, x+, z+ ; F))), and V( k, x, z; F ) a < T is coninuous, increasing and differeniable in k, x and z, and concave in k. Proof: I is almos obvious from Proposiion. Noice ha, V depends on ime since he model is no ime-homogeneiy because of he exisence of a zero-coupon bond, C.2 Deb We price a zero-coupon bond. A mauriy, he bond price is as follows: F if W( kt, xt, zt) > ( τ ) F BT( kt, xt, zt; F) =, (3.C.4) ξ W( k, x, z ) else T T T We obain he bond value a < T recursively: B ( k, x, z ; F) = E[ M B ( k, x, z ; F)]. (3.C.5)

10 4. Resuls This chaper presens he main resuls of our model. Firs, we describe he calibraed values of he parameers used in solving he model. Second, we analyze he relaionship beween he credi risk and expeced sock reurns and opimal invesmen raes. 4. Calibraion The parameers of he model and heir calibraed values are summarized in Table. Parameer Value Produciviy Shocks Monhly persisence of aggregae produciviy Monhly condiional volailiy of aggregae produciviy Long-run average of he aggregae produciviy Monhly persisence of idiosyncraic produciviy 0.97 Monhly condiional volailiy of idiosyncraic produciviy 0. Capial Dynamics Capial share 0.3 Monhly depreciaion rae 0.0 Adjusmen cos of invesmen 5 Pricing Kernel β γ 0 50 γ -000 Tax Tax rae on corporae income 0.35 Bankrupcy Process Bankrupcy coss 0.03 Table : Calibraion values for he parameers of he model The monhly persisence, ρ x, and condiional volailiy, σ x, of he aggregae produciviy shock, are consisen wih he quarerly esimaes of Cooley and Presco(995). The long run average, is calibraed so ha he rae of reurn on a uni of invesed capial, for an average firm, is, abou 30 percen per year, consisen wih he choice of Berk, Green, and Naik (999). The monhly persisence, ρ z, and volailiy, σ z, 9

11 of he idiosyncraic produciviy shock correspond o hose in Zhang (2005), and hey are consisen wih he empirical evidence of Pasor and Veronesi (2003). The parameers driving he capial flow are calibraed as follows: he capial share α is similar o he choice in Kydland and Presco (982), he monhly depreciaion rae,δ, is consisen wih he esimaes in Cooper and Haliwanger (2000), and he adjusmen cos parameer,θ, corresponds o he choice in Zhang (2005) and is consisen wih he empirical esimaes of Whied (992). The parameers driving he dynamics of pricing kernel, γ 0, γ and β are similar o hose in Zhang (2005), and hey are calibraed o mach he average Sharp raio, he, average real ineres rae and he volailiy of ineres raes. The parameers governing he ax sysem,τ, is consisen wih he annual calibraed values employed by Moyen (2004). Finally, he bankrupcy parameer,ξ, in line wih he empirical esimaes of Alman (99). 4.2 Analysis We analyze he relaionship beween expeced sock reurn and defaul probabiliy 3. Expeced sock reurn is defined as follows: EV [ + ] E[ R+ ] =, (4.2.) V d where d denoes dividend a ime : d = d( k, x, z) = τ δ + δ + V d is ex-dividend marke value of equiy. Defaul probabiliy a is defined as: * * ( )( y k) k ( ik h( i, k)). (4.2.2) DP = E ( > 0) (4.2.3) V T A. Afer he mauriy The plos of Figure (a) capure expeced sock reurns afer he mauriy T in good 3 In following sudy, we se k = 0.5, F = 2 and T = 20, and following expeced reurn and defaul probabiliy are all condiional. 0

12 and bad economy 4. The expeced reurns are higher in bad economy. This paern is driven by he counercyclical price of risk: In bad economy, since invesors become more risk averse, he price of risk is higher and he fuure cash flow is priced lower. However, wih respec o idiosyncraic shocks, he expeced reurn is almos consan in boh good and bad economy. This implies ha in his model, wih no liabiliy, invesors price aggregae shocks, while invesors hardly care abou idiosyncraic shocks 5. The plos of Figure (b) capure opimal invesmen rae afer he mauriy T in good and bad economy Wih respec o opimal invesmen rae, firms wih lower idiosyncraic or aggregae shocks choose lower invesmen rae. We can inerpre his as follows. When he idiosyncraic shock is lower, because of is persisence, he fuure cash flow is also lower, which means ha he value of he capial is smaller. Wha is more, when he aggregae shock is lower, since no only he produciviy is lower, bu also he price of risk is higher, he value of he fuure cash flow, or capial, becomes even lower. Then, in such case, since marginal value of invesmen is also lower, a firm choose lower invesmen rae. If he produciviy is exremely low, a firm may even disinves capials if i is no resriced by he bond covenans. B. Before he mauriy B. Relaionship beween expeced sock reurn and credi risk The plos of Figure 2 capure defaul probabiliy o idiosyncraic shocks before he mauriy T in bad economy a differen imes. As Figure2 shows, wih hese parameers, he defaul probabiliy of each firm is very high, especially when he ime o mauriy is shor. In he following sudy, we invesigae he relaionship beween expeced sock reurn and defaul probabiliy. The plos of Figure 3 capure expeced sock reurns o idiosyncraic shocks before he mauriy T in bad economy a differen imes. Figure 3 shows an ineresing resul. Firs when he ime o mauriy is long ( = 0 ), as idiosyncraic shock is higher, he expeced sock reurn becomes lower. However, when he ime o mauriy is shorer ( = 0,5,8 ), as idiosyncraic shock is higher he expeced reurn is no always lower: 4 In following sudy, we se x = in good economy, and x = in bad economy. Noice ha, since a firm loads no debs afer he mauriy, expeced sock reurns and opimal invesmen raes of Figure does no depend on he ime. 5 Zhang (2005) shows ha inroducing asymmeric capial adjusmen cos, he expeced reurns is significanly high when he idiosyncraic shock is low, which generaes value premium in cross-secion.

13 Raher, in lower idiosyncraic shocks, expeced reurn is increasing. From Figure 2 and 3, when a firm has very high defaul probabiliy, he expeced sock reurn of he firm can be lower han he one of he firm wih lower defaul probabiliy. We analyze his puzzling resul as follows. We can decompose he ex-dividend marke value of equiy, V d, ino he cash flows before he mauriy and he ones afer he mauriy: where V d = E ( M V ) + + = E ( M ( d + M V )) = T = E ( M d ) + E ( M V ) i=+ T i=+ +, i i +, T T = E ( M d ) + E ( M max( W ( τ ) F,0)) = SW + CO, +, i i +, T T s j= (4.2.4) M s, M j (4.2.5) T SW E ( M d ) (4.2.6) +, i i i=+ CO E ( M max( W ( τ ) F,0)) (4.2.7) +, T T In his decomposiion, we can inerpre he firs erm as he value of a swap receiving floaing ineres, SW, and he second erm as he value of a call opion CO on he firm value. We also decompose he sock reurn ino reurns of swap and call opion: V + R + = V d d+ + SW+ + CO+ = (4.2.8) SW + CO SW SW CO CO = w R + w R where w w R SW CO, + + SW SW + CO (4.2.9) CO SW + CO (4.2.0) d + + SW + SW (4.2.) SW + 2

14 CO R + (4.2.2) CO +. CO In his way, we can inerpre ha he sock is a porfolio of he swap and he call opion, and ha he expeced reurn and variaion of he sock are dependen upon he proporions of he swap and he call opion as conribuions o he oal sock value. To undersand he relaionship of expeced sock reurns and defaul probabiliy, his decomposiion is useful. The poin is as follows. When he defaul probabiliy of a firm is high, he value of he call opion is small, while ha of he swap is no as small as ha of he call opion. The reason is ha, even if here is lile possibiliy of survival a he mauriy, dividends are sill paid o he sockholders before he mauriy. This cash flow is less risky han he cash flow afer he mauriy. Therefore, when he defaul probabiliy is high, he expeced reurn of he swap is also much lower han ha of call opion. This explains he puzzling resul of Figure 3. When he defaul probabiliy is low, as he plos = 0 in Figure3, we can inerpre ha he porfolio ha represens he oal sock value has high weigh on he call opion. If idiosyncraic shock is higher, he weigh on he call opion of he porfolio becomes higher and, a he same ime, he risk of he call opion becomes lower. As a resul, he risk of porfolio becomes lower and he expeced reurn of i becomes lower. This is he reason why he plo of = 0 in Figure 3 is decreasing. On he oher hand, ake a look a he case when he defaul probabiliy is high as he plos = 0,5,8 in Figure3. When he defaul probabiliy is high, he porfolio has high weigh on he swap. If idiosyncraic shock is higher, since he value of swap does no goes up as much as ha of call opion, he weigh on he call opion becomes higher. As idiosyncraic shock is higher, he call opion becomes less risky. Sill, when he defaul probabiliy is very high, he risk of call opion is much higher han ha of swap. Therefore, by raising he weigh on he riskier call opion, he porfolio becomes riskier and is expeced reurn becomes higher as idiosyncraic shock is higher. This is why he plos of = 5,8 in Figure3 are increasing. In case idiosyncraic shock is much higher, since he expeced reurn of call opion is no much higher compared o ha of he swap, he porfolio becomes less risky and is expeced reurn becomes lower. This is he reason ha he plo of = 0 sars o decrease if he idiosyncraic shock is high. B.2 Opimal invesmen rae The plos of Figure 4 capure opimal invesmen rae o idiosyncraic shocks before 3

15 he mauriy T in bad economy a differen imes. This figure implies ha he higher defaul probabiliy is, he lower he opimal invesmen rae is. By invesmen, a firm exchanges oday s cash for fuure cash flow. If i expecs good fuure cash flow by invesmen, a firm invess a lo. However, if i doesn, a firm invess lile, or even disinvess and increases dividend if allowed by he bond covenans. In case a firm has liabiliy and defaul probabiliy is high, wih low expeced cash flow afer he mauriy, a firm will make lile invesmen. How does invesmen change he expeced sock reurn, compared wih he case in which a firm does no inves a any ime? If a firm has no debs, expeced sock reurn will be low because by invesmen sockholders can increase he sock value and decrease is risk. Then consider a firm wih liabiliy. In such case, invesmen will also increase sock value. However, if he firm has high defaul probabiliy, since he weigh of call opion will increase, he porfolio will be riskier. Therefore, in a high defaul probabiliy, invesmen can increase he value of call opion more han, he one of swap, and make he expeced sock reurn higher. 4.3 Discussion The resul ha he cross-secional expeced sock reurn is lower when he cross-secional credi risk is higher is consisen wih he empirical research of Dichev(998), Griffin and Lemmon(2002) and Campbell, Hilscher, and Szilagyi (2005). However, some of he resuls from he model need furher discussion. Firs, he resuls implies ha if defaul probabiliy is higher, since he weigh of swap value is higher, dividend yield, d / V, becomes higher. Though his seems a odds, here is no empirical research on he relaionship beween credi risk and dividend yield, furher research is needed. Second, in he model, even if a firm defauls a he mauriy, he firm sill has always posiive profi. One of he soluions is he inroducion of nonnegaive fixed cos f o profi funcion (3.A.), α y = k exp( x + z ) f. (4.3.) Wih his inroducion, here will be he possibiliy of defaul wih negaive profi and herefore here is possibiliy of defaul before he mauriy of he deb because of he fixed cos. Due o he poenial negaive profi, here will be he case of a negaive sock value before he mauriy if a firm coninues. Even if we inroduce he fixed cos, since he weigh of swap value is larger han ha of call opion in higher defaul probabiliy, 4

16 he lower expeced sock reurn sill holds. In his paper for simpliciy, he profi funcion has no fixed cos. Finally, he relaionship beween he invesmen and credi risk has no been confirmed in empirical research. I can provide fresh direcions for fuure research. 5. Conclusion In his paper, we propose a neoclassical producion augmened wih capial srucure and aggregae uncerainy. In our model, we show ha, alhough he credi risk is high, under invesors expecaion for sable dividend paymens before he mauriy, expeced sock reurn can be lower han when credi risk is low. The puzzling phenomenon, inerpreed by Griffin and Lemmon (2002) and Campbell, Hilscher, and Szilagyi (2005) as mispricing, is herefore consisen wih raional expecaions. Sill, o be persuasive, he model needs fuure research in his area. Appendix Proof of Proposiion I is a special case of Zhang (2005). We prove he former half of he proposiion, he uniqueness of he funcion of he asse value only. Le T be he funcional operaor associaed wih he dynamic program in (3.5.). Tha is: { } TW ( )( k, x, z) = y+ max ( ik+ hi (, k)) + E( M Wk (, x, z )) (A.) i To see wheher he operaor T has a unique soluion, i is sufficien o check he Blackwell s sufficien condiions for a conracion mapping. The monooniciy condiion follows immediaely from he lineariy of he T and V. The discouning condiion follows from he fac ha { } TW ( + a)( k, x, z) = y+ max ( ik+ hi (, k)) + E( M ( Wk (, x, z ) + a) i { } = y + max ( ik + h( i, k )) + E ( M W( k, x, z )) + E ( M ) a i (A.2) for any real a > 0. Therefore, as long as E( M + ) <, he discouning propery is saisfied and T is a conracion mapping. This ensures ha he operaor T has a unique fixed poin in he space of coninuous and bounded funcion. 5

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19 Figure (a): Condiional Expeced sock reurn o idiosyncraic shocks afer he mauriy Figure (b): Opimal invesmen rae o idiosyncraic shocks afer he mauriy 8

20 Figure 2: Condiional defaul probabiliy o idiosyncraic shocks in bad economy Figure 3: Condiional expeced sock reurns o idiosyncraic shocks in bad economy 9

21 Figure 4: Opimal invesmen rae o idiosyncraic shocks in bad economy 20