Inferring Default Correlation from Equity Return Correlation

Transcription

1 Inferrng Default Correlaton from Equty Return Correlaton Howard Q, a Yan Alce Xe, b heen Lu, c and Chunch Wu d October 3, 008 a chool of usness and Economcs, Mchgan ech Unversty, Emal: howardq@mtu.edu b Department of Accountng and Fnance, chool of Management, Unversty of Mchgan-Dearborn, Emal: yanxe@umd.umch.edu c Department of Fnance, Insurance, and Real Estate, Washngton tate Unversty Vancouver, Emal: lus@vancouver.wsu.edu d Department of Fnance, Robert J. rulaske, r. College of usness, Unversty of Mssour-Columba, Emal: wuchu@mssour.edu

2 Inferrng Default Correlaton from Equty Return Correlaton Abstract hs paper proposes a new approach to estmate default correlaton. It overcomes an emprcal dffculty encountered n the structural model when estmatng default correlaton from the unobservable asset process. he unque feature of ths approach s that t lnks default correlaton to equty return correlaton whle preservng the fundamental relaton between default correlaton and asset return correlaton embedded n the structural model. Emprcal results show that our model consderably outperforms Zhou s (00) model n predctng default correlaton, especally for bonds wth long maturty horzon and low credt ratng. Results ndcate that a lttle more careful specfcaton of the underlyng mechansms n the structural model can sgnfcantly mprove ts performance. Our fndng strongly suggests that structural models are a very useful tool for estmatng default correlaton. Keywords: Default correlaton, equty return correlaton, defaultable bonds, and structural model.

3 Inferrng Default Correlaton from Equty Return Correlaton. Introducton Default correlaton s an mportant pece of nformaton for rsk management of credt portfolos because portfolo managers must accurately estmate portfolo losses that depend on jont default events between oblgors n a portfolo. Das, Fong and Geng (00) fnd that default rates of debts n credt portfolos are sgnfcantly correlated and estmates of credt losses are substantally dfferent f default correlaton s gnored. he recent subprme mortgage crss s an acute example that underscores the mportance of understandng default correlaton. Cowan and Cowan (004) show that default correlatons between subprme loans are substantally hgher than those between commercal bonds and loans, and that default correlaton ncreases as the ratng of the lender declnes. Hgh default correlatons among loans compound the current problem n the subprme mortgage market. hough default correlaton s mportant for rsk management and credt analyss, ths nformaton s often unavalable because default correlaton cannot be measured drectly. In partcular, t s dffcult to uncover default correlaton based on observed default data for hgh-grade bonds because default s a rare event. A number of models have been developed to estmate default rsk and to explore the structure of default correlaton. Majorty of these models adopt ether the structural or reduced-form approach. he reduced-form approach models default as an ntensty process that s determned by exogenously state varables (see, among others, Jarrow and urnbull, 995; Madam and Unal, 998; Duffe and ngleton, 999; and Das, Duffe, Kapada,

4 and ata, 007). hs approach allows the default ntensty process to be drectly estmated from the credt rsk premum wthout relyng on parameters related to the frm s underlyng unobserved asset value. ecause of ths advantage, the reduced-form approach has been wdely used to explan credt spreads (see Duffe and ngleton, 999). Notwthstandng ths advantage, formulaton of default ntensty as an exogenous factor lmts the applcaton of the reduced-form model to predcton of default correlaton between frms. he structural approach offers an excellent alternatve to model default rsk and default correlaton. A dstnct advantage of ths approach s that t can be used to determne default probablty, debt and equty values smultaneously n a unfed framework. he structural model s poneered by Merton (974) and further developed by others (see, for example, Ingersoll, 977; mth and Warner, 979). Merton assumes that the evoluton of frm asset value follows a geometrc rownan moton and the default boundary s the face value of debt. Equty represents a European call opton on the frm s asset wth the strke prce equal to the debt face value. In Merton s model, default can only happen at debt maturty. o allow default before debt maturty, lack and Cox (976) ntroduce the frst-passage-tme model that specfes default as an event of the frst tme that the frm s asset value hts the default boundary. he default boundary can be exogenously specfed as a covenant to protect bondholders nterests (see lack and Cox, 976; Longstaff and chwartz, 995), or can be determned endogenously as a threshold at whch stockholders maxmze the equty value at default (see Leland, 994; Leland and oft, 996).

5 Gven nterdependence of two frms, the structural models can be used to derve default probablty for each frm and to nfer the default correlaton between them (Hull and Whte, 00; Löffler, 003; Overbeck and chmdt, 005). nce major components n structural models are asset, debt, equty, and default boundary, any dependence of one component on another can generate default correlaton. We can thus dfferentate varous default correlaton models by ther correlaton channels. Earler studes focus on the correlaton between two frms assets (Zhou, 00; Frey, McNel, and Nyfeler, 00). Asset values are treated as a functon of common factors and frm-specfc factors (Fnger, 999; Frey, McNel, and Nyfeler, 00) where the common factors dctate the asset return correlaton between frms. ubsequent studes (see Gesecke, 003, 006) model default correlaton by ntroducng the correlaton between frms default boundares n addton to that between frms assets to account for the contagous effects of credt rsk. Gesecke (003, 006) assumes that each tme a frm defaults, the true level of ts default boundary s revealed, and nvestors use ths new nformaton to update ther belefs about the default boundares of other frms. o model the dependence of frms default boundares, Gesecke uses copulas to lnk the probablty dstrbutons of ndvdual frm default boundares to a jont dstrbuton functon of default boundares. Whle the structural approach provdes a cohesve framework to relate asset return correlaton to default correlaton, mplementaton of ths type of models s lmted n practce because the asset value process and default boundary are unobservable. A number of studes propose dfferent ways to overcome ths problem. For nstance, Zhou (00) assumes that asset return correlaton s equal to equty return correlaton, and CredtMetrcs, an ndustral credt rsk model, approxmates asset return correlaton by 3

6 equty return correlaton. Intutvely, replacng asset return correlaton ρ by equty return correlaton ρ s more sutable for frms wth a low level of debt over a short horzon as ndcated by Zhou (00). However, a hgh-rated frm can have hgh leverage f ts assets are consdered safe whereas a frm wth rsky assets may stll has a low ratng even though ts leverage s lowered. hs suggests that the equty-asset relatonshp can be complcated not only by leverage but also by the rsk n ts asset (.e., asset volatlty). Havng a low leverage s not necessarly a vald reason for usng the approxmaton that ρ = ρ. hus, to better capture default correlaton, the naïve approxmaton of ρ = ρ should be replaced wth a more subtle relatonshp based on the structure of the frm. For nstance, as frms debt level and tme horzon ncrease, asset and equty return correlatons often dverge. herefore, addtonal mechansms should be ntroduced to capture ths effect. Zeng and Zhang (00) show that equty return correlaton s not a perfect proxy for asset return correlaton because the covarance between the assets of two frms s composed of the covarance between ther equtes and the covarance between ther rsk-free components. Deervgny and Renault (00) examne whether default correlaton can be effcently extracted from equty return correlaton. her emprcal results show that default correlaton mpled by equty return correlaton s generally not a good proxy for emprcal default correlaton. In ths paper we propose a new method to nfer default correlaton from equty return correlaton usng a structural approach wthout approxmatng the asset return correlaton wth the equty return correlaton or ntroducng addtonal correlated processes, such as dependent default boundares (e.g., Gesecke, 006), and cross holdngs between two frms (e.g., Elsnger, 007). A ratonale of workng wth the 4

7 structural model s that f the underlyng lnkage among key fnancal decson varables s more accurately modeled, default correlaton could be more relably nferred from equty data, whch n turn would result n a better understandng of the channels of default correlaton n the structural model. hs also allows us to better assess how much of default correlaton s due to the correlated asset process and whether addtonal factors are needed. In our method, we frst establsh the lnks between equty return correlaton and asset return correlaton, and between asset return correlaton and default correlaton, respectvely. We then put the two lnks together to elmnate the requrement for the nformaton of asset return correlaton. In ths way, we are able to nfer default correlaton from observed equty return correlaton more accurately based on a theoretcally sound contngent clams framework. nce frms stocks are actvely traded, we can estmate default correlaton easly usng stock returns. We develop a hybrd structural model to relate equty return correlaton to default correlaton and to provde estmates of default correlaton. he model combnes an extended Leland-oft (996, hereafter L) model and Zhou s (00) model (see Fgure ). he extended L model lnks equty return correlaton to asset return correlaton whereas Zhou s model lnks asset return correlaton to default correlaton. Combnng these two structural models establshes a lnk between equty return and default correlatons. It s straghtforward to ntegrate Zhou s (00) asset-default correlaton model to the extended L model because the former s also based on the frst-passagetme framework. We use ths ntegrated hybrd model to estmate default correlaton from emprcal data and to compare ts performance wth the well-known Merton model and Zhou s model. 5

8 Equty correlaton Extended Leland -oft model Asset correlaton Zhou s model Default correlaton Fgure. Hybrd model radtonally, the Merton-type model s used to estmate default probablty and yeld spread for each ndvdual frm. It can be extended to the two-frm case to explore the default correlaton between them. he extended Merton model has a closed-form soluton and s easy to mplement. Emprcal results show that the model can capture the fact that as bond ratng decreases, default correlaton ncreases. However, t cannot capture the tme horzon effect as shown by Lucas (995). he reason s that n the Merton-type model default can only happen at debt maturty. A frm havng negatve assets s permtted to contnue ts operaton untl debt maturty. As a result, default probablty s reduced and so s the default correlaton. In addton, the smple Mertontype model mposes restrctve assumptons, such as the equty value s a European call opton on the frm s assets, no tax beneft from the use of debts, and default boundary s exogenously determned. Zhou (00) attempts to mprove Merton s model by developng a frst-passagetme model to estmate a jont default probablty dstrbuton and default correlaton. hs model avods the restrctve assumpton n Merton s model that default can only occur at the bond maturty date. he strength of ths model s that t provdes a theoretcal framework that makes use of frm-specfc nformaton to determne default correlaton among frms. Moreover, the model provdes an analytcal formula for calculatng default Lucas estmates default correlatons usng the frm bankruptcy data n the perod. hus, hs default correlatons are based drectly on the default events observed n the market. 6

9 correlatons that can be easly mplemented for a varety of applcatons. However, a potental drawback of ths model s that t mposes a restrctve assumpton that asset return correlaton equals equty return correlaton. In addton, Zhou (00) uses an arbtrary value for the asset (equty) return correlaton n hs calculaton for default correlaton. nce one can always boost the equty return correlaton coeffcent to ncrease the sze of default correlaton, the performance of the model s called n queston. Our hybrd model overcomes the shortcomngs of the Merton-type model and Zhou s model. We adopt a frst-passage-tme model (e.g., the L model) and extend t to a two-frm settng n whch equty and default of these frms are ntrnscally lnked. We frst establsh the relaton between asset return correlaton and equty return correlaton (see Fgure ) and then examne sutablty of usng the equty return correlaton as a proxy for the asset return correlaton. mulatons show that equty return correlaton s generally not a good drect proxy for asset return correlaton. In partcular, as tme horzon ncreases and bond ratngs decrease, the gap between equty return correlaton and asset return correlaton wdens ncreasngly. o overcome ths problem, we employ the hybrd model that utlzes the fundamental relaton between asset return correlaton and default correlaton mpled by the structural model. Usng ths model, we abstract the asset return correlaton from the observed equty return correlaton, estmated from hstorcal stock returns, usng the calbrated L model and then nput t nto Zhou s model to obtan default correlaton. We compare the performance of our model wth that of the smple Merton-type model and Zhou s model usng emprcal data. Our results confrm the prevous fndng 7

10 that equty return correlaton s generally not a good proxy for asset return correlaton (see Deervgny and Renault, 00; Zeng and Zhang, 00). Our paper contrbutes to the lterature by proposng a new approach to resolve the problem of unobserved asset return correlaton whch s a key nput n a default correlaton model. Results show that our hybrd model based on ths new approach performs better than other structural models n estmatng default correlaton of multple frms. he greater estmaton accuracy s acheved wthout havng to mpose more complcated structures such as correlated default boundary, networkng, and cross holdngs. Our results strongly suggest that the structural model s a useful tool for estmatng default correlaton. he remander of the paper s organzed as follows. ecton II develops a smple Merton-type model and a generalzed frst-tme-passage (FP) model to relate equty return correlaton to default correlaton. ecton III dscusses the FP model mplementaton and Monte Carlo smulatons. ecton IV reports smulaton results and presents emprcal fndngs. Fnally, ecton V concludes the paper.. he Model In ths secton, we frst develop a smple Merton-type model to relate equty return correlaton to default correlaton. he smple model has a closed-form formula and s relatvely easy to mplement. However, the model mposes restrctve assumptons. o relax these restrctons, we extend the L model (996) to a two-frm settng to permt default before maturty. We then propose a hybrd model that ntegrates the extended L model and Zhou s model to lnk default correlaton drectly to equty return correlaton. 8

11 hs model provdes a convenent framework to estmate default correlaton, whch does not requre the nformaton for the unobserved asset return correlaton.. he mple Equty Return Correlaton Model.. Dervaton of the smple model Followng Merton (974), we assume a perfect and arbtrage-free captal market. he default-free nterest rate, r, s constant and the money market account has value rt () t e = at tme t. Denote V and V as total assets of frms and, respectvely. he dynamcs of V and V are gven by the followng stochastc process: d ln V = µ dt + Σdw () µ are column vectors. where lnv ( lnv,lnv ), = ( µ, µ ), and w = ( w w ) =, µ and µ are nstantaneous expected rates of return of frms per unt of tme, and w and w are ndependent standard rownan motons wth volatltes σ and σ, respectvely. he returns can be determned as the change n lnv over a unt of tme, denoted as lnv and lnv for frms and, respectvely. Asset return correlaton s therefore gven by and the varance-covarance matrx s ( lnv, lnv ) ( lnv ) var( lnv ) cov ρ = () var Σ σ Σ' = ρσ σ σ ρσ σ he equty values and are the call optons on frm assets wth maturty and strke prce,. hat s,. For example, f the unt of tme s month, then lnv represents monthly return. For rownan motons, return correlaton s ndependent of the frequency of the return data snce the tme factor cancels out n (). 9

12 , + ( V ), =, (3) =,, where, s the promsed payment (debt face value) of frm to ts debtholders at maturty. If the frm s asset value (V ) at maturty s greater than the face value of debt ( ), the frm does not default and shareholders receve V,,. On the other hand, f V,, <, the frm defaults on ts debts, and debtholders take control of the frm. We can derve the correlaton coeffcent of equty returns gven the correlaton coeffcent of two frms assets n (). he correlaton coeffcent of equty returns at tme s ( R, R, ) ( R, ) var ( R, ) where the covarance of equty returns s gven by cov cov, ρ =. (4) var ( ) + A ( { }),,,, E V, I V, >,, V, >, (, I{ V, >,, V, > + A, 4EI { V, >,, V, >, } (, I{ V }), >,, V, + A, 6EI{ V, >,, V,, } r ( I ) + A EI{ } + e I{ }, ( R, R ) = A E V, V, I{ V >, V > } + A E V 3 + A E V 5 + A E V 7, { V, V > },,,, 8 V, and the varance of equty returns for frm s gven by ( R ) D, E V, I{ V },, V, >, V,,, V, r ( > ) + D, E( V, I{ V > }) + D3, I{ V > } + e I{ V }. var, = (6),,,,,,,, he defnton of each term n (5) and (6) s shown n Appendx A, and the dervaton of Eq. (5) and (6) s presented n Appendx. Next, default correlaton s defned as E( I { }) E( I { }) E I V, >,, V, >, V, >, { V, >, } ρd =, (7) Var I { V,, } Var I > V, >, ( ) ( ) ( { }), (5) 0

13 where I {u} s an ndcator functon wth a value equal to one f u s true, and zero otherwse and Var( I { }) = E( I { }) E I,,,, {,, }. V> V> V> ( ) (8) he defnton of each term n (7) s gven n Appendx A. Equatons (4) and (7) ndcate that both equty return correlaton ( ρ ) and default correlaton ( ρ ) are functons of asset return correlaton. Usng these two equatons, we D can determne the relatonshp between ρ and ρ D drectly wthout the knowledge of asset return correlaton ρ. Asset return correlaton can be determned from equty return correlaton va (4) and default correlaton can be obtaned usng asset return correlaton as the nput to (7). hus, default correlaton can be computed from equty return correlaton wthout a pror knowledge of asset return correlaton. In the followng, we mplement ths procedure and present the smulaton results. For ease of notatons, we would often refer to asset (equty) return correlaton as asset (equty) correlaton henceforth... Predctons of the smple model o test the smple (Merton-type) model, we use the observed equty data to predct default correlaton over the same perod as Lucas (995). Equty correlaton and volatlty are estmated usng monthly stock return data from 970 to It s mportant to note that the Merton-type model underestmates default probablty because t allows a bankrupt frm to contnue to operate untl debt maturty. o remedy ths 3 ecton 4. gves the detaled descrpton of estmaton of equty correlaton. o estmate equty volatlty, we frst calculate equal-weghted stock returns n a partcular ratng group for each month over the perod of 970 to 993. Next, we calculate the standard devatons of the equal-weghted stock returns.

14 problem, we ntroduce a volatlty multpler (the only fudge factor n the model) n order for the model to generate default probabltes commensurate wth hstorcal default rates. able reports the predctons of default correlatons by the smple model usng stock data. We fnd that the smple model generates a pattern of default correlaton smlar to Lucas (995). For example, when the tme horzon s fve years, 4 the smple model predcts that default correlaton ncreases as ratngs of both frms declne. he correlaton between two Aa frms s.%, whle that between two frms ncreases to 3%. In addton, gven the ratng for one frm, default correlaton generally ncreases as the ratng for another frm declnes. For example, default correlaton between two A frms s 3.7%, and 5.% between A and frms. hese results are consstent wth the fndngs of Lucas (995). Overall, default correlatons estmated by the smple model tend to understate the default correlatons of lower grade bonds estmated from hstorcal data (see Lucas, 995). Moreover, the default correlaton estmates are nsenstve to tme horzon. 5 A potental caveat n the above analyss s that stock data are collected from frms that are gong concerns. hus, the analyss s subject to a survval bas when usng the data of these frms. o address ths ssue, we match the survved frm s equty value process to the observed data and then back out the asset value process for the frm to predct the default correlaton. he results are reported n the rght panel of able. As expected, the dfference s hardly notceable for nvestment grades. For junk bonds, the survval bas has a larger effect but ths effect s not statstcally sgnfcant except for the default correlaton between two -rated frms where correcton of the survval bas leads 4 We only report the results for the tme horzon of fve years for brevty. he results for the other tme horzons are avalable upon request. 5 Estmates of default correlatons at dfferent horzons are avalable upon request.

15 to an 8% default correlaton compared to 3% wthout correcton. However, compared to Lucas s result (9%), both numbers seem low, suggestng the survval bas s not a major ssue here. A possble cause for the underestmaton of default correlaton s that the Mertontype smple model allows a bankrupt frm s equty process to contnue to evolve wth a fnte chance of becomng solvent agan over the horzon before debt maturty. hs setup could lead to an ncreasng dvergence between equty correlaton and asset correlaton. he above results show a substantal room for mprovement of the model. For nstance, f we could model default n a more realstc way,.e., to allow frm default before debt maturty, t should ncrease the predctve ablty of the model. In the followng, we propose an alternatve approach to estmate default correlaton usng the frst-passage-tme method to permt default before maturty.. he Hybrd Model he hybrd model, as depcted n Fgure, conssts of two components. It frst ncorporates a model to back out the unobserved asset correlaton ρ from the observed equty data. It then nputs the resultant asset correlaton ρ nto a frst-passage-tme (FP) model to obtan default correlaton ρ D. In the former, we extend the L model to a multple-frm settng and use the equty correlaton nformaton to nfer the asset correlaton, whle n the latter we employ the model of the frst passage tme suggested by Zhou (00) to establsh the lnk between asset correlaton ρ and default correlaton ρ D. hs procedure allows us to use a more accurate relatonshp between default and equty correlatons rather than the restrctve substtuton of ρ = ρ n Zhou (00). We 3

16 summarze the jont default probabltes by Zhou (00) n Appendx C. hese jont default probabltes can be used along wth () and (7) to calculate asset correlaton ρ and default correlaton ρ... he Extended Leland-oft model D In ths secton, we extend the L model to the case wth two frms. hs extended model wll be used later to combne Zhou s (00) to yeld the hybrd model. We relax the assumptons n the smple Merton-type model to ncorporate bankruptcy costs and taxes, and allow the frm to go bankrupt as ts asset value falls below a threshold (default boundary) for the frst tme. Assume that the asset value of an unlevered frm, V, has the followng contnuous dffuson process: 6 dv V [ µ ( V t) δ ] dt + σ dz =,, (9) where µ ( V,t) s the total expected rate of return on the frm s assets, δ s the total payout rato, whch s the proporton of the frm value pad to all securty holders, Z s a standard Wener process, and σ s the constant volatlty of asset returns. he asset value V ncludes the net cash flows generated by the frm s actvty. uppose there s an dentcal but levered frm ssung a rsky debt d per unt tme wth t perods to maturty, and contnuous constant coupon flow c(t) and prncpal p (t). he frm remans solvent untl the asset value V hts a default boundary V, leadng to bankruptcy. Upon bankruptcy, bondholders receve a fracton χ = ( β ) of the asset value V, where β s the bankruptcy cost rato and β V s loss due to bankruptcy. Further we assume that r represents the contnuous nterest rate pad by a default-free 6 A smlar specfcaton s used by Leland and oft (996), Merton (974), lack and Cox (976), and rennan and chwartz (978). 4

17 asset and nvestors follow a buy-and-hold nvestment strategy. Under the rsk-neutral valuaton, t can be shown that the value of the debt, d, s gven by () () ct rt ct ct d( V, V, t) = + e p() t ( F() t ) + χ V G() t, r r r () (0) where F(t) and G(t) are gven n Leland and oft (996). hus, the total outstandng debt D s the ntegraton of the debt flow d( V V, t), over (maturty of newly ssued debt): ( V, ) = d( V, V, t) D V, dt () t= 0 he ntegral can be carred out numercally. he tradeoff between the beneft of tax shelds and bankruptcy cost suggests that there exsts an endogenously determned bankruptcy threshold V that maxmzes frm value. he equty value, as a functon of V and asset value V, s gven by, a z a z C V + V + EVV V C V DVV r V V (,, ) = + τ β (,, ), () where C s the annual coupon payment, τ C s the corporate tax rate. Parameters a and z are functons of asset volatlty σ and nterest rate r. 7 Equaton () establshes a lnk between asset process V and equty process E. If asset value V(t) follows a geometrc rownan moton, equty value E(V) as a functon of V wll also exhbt a smlar random process. Consder two frms wth asset values V and V. he dynamcs of V and V are specfed by ()-(4), where asset returns lnv and lnv are correlated wth a coeffcent ρ, and volatltes σ and σ. mlar to (), we defne equty return correlaton as 7 Detaled dervatons of (0) and () are gven by Leland and oft (996). 5

18 ( ln E( V ), ln E( V )) ( ln E( V )) var( ln E( V )) cov ρ =. (3) var he correlaton between two asset processes, V and V, wll undoubtedly result n a correlaton between two correspondng equty processes, E(V ) and E(V ). However, these two correlatons can dverge sgnfcantly because as tme evolves, both leverage rato l and asset volatlty σ may change. 3. Model Implementatons and Monte Carlo mulaton We choose nterest rate r = 8% and payout rato δ = 6%. hese fgures are n lne wth Huang and Huang (003). Corporate tax rate τ C s set at 35%. ankruptcy rato β s set at 0% based on estmates n Andrade and Kaplan (995). 8 o mplement the structural model properly, a calbraton s necessary. he objectve of calbraton s to choose equty premum and asset volatlty σ such that the model generates a default probablty consstent wth the observed default rates for each ratng gven n able A. he equty premums for ratngs Aa to are gven n able. When dealng wth heterogeneous tme horzons, from one to ten years n the present case, we carry out the model calbraton by choosng an asset return volatlty σ that mnmzes the aggregate squared dfference between the mpled and observed default probabltes, σ = arg mn ς 0 0 [ P ( σ ) P ] =, (4) 8 Personal tax rate s set to zero n ths exercse. We have also tested varous personal tax rates. For the ssues dscussed n ths study, a non-zero personal tax rate does not qualtatvely change our results. For smplcty and clarty, we abstract away from personal tax nfluence. Nevertheless, personal taxes exhbt nterestng effects whch wll be a subject n a sequel to ths study. 6

19 where P s the model-mpled default probablty by year, and P s the correspondng observed default rate. nce we nput equty premums nto the model, these probabltes are physcal probabltes. able C reports the model-mpled asset return volatlty σ for bonds wth ratng categores from Aa to. o value debt and equty wth the model, we return to the rsk-neutral measure by retanng asset return volatlty σ and forcng equty premum to zero. In the Monte Carlo smulaton, for each teraton we generate a tme seres sample path accordng to (9) wth the startng asset value V (0) normalzed to 00, where =, and denotng the two frms. For each random movement n V (0) at tme t, we apply () and () to obtan debt D (t) and equty E (t). For the next random movement n V (t + t) at tme t + t, we agan apply the model to obtan D (t + t) and E (t + t) whle keepng the coupon, prncpal and default boundary unchanged. hs s to recognze the fact that the statonary captal structure of the L model rules out any debt restructurng after the optmzaton s done. he procedure s repeated untl we reach the horzon at t = H. hs allows us to map out one sample path. For a second teraton, the same procedure s repeated to generate another sample path of V (t) (and thereby D (t) and E (t) as well) for each frm. o permt correlaton ρ between the returns of the two asset processes, we employ the followng return dynamcs: µ σ V = V + Z n n, (5) µ ρσ ρσ V = V + Z + Z n n n 7

20 where n denotes the number of tme ntervals parttoned for each year and µ s the net drft rate. he random varables Z and Z follow two ndependent standard normal dstrbutons. he volatlty for each perod s σ / n where σ s the annualzed return volatlty for frm. For example, when the tme nterval s month, n s set to and σ / s the monthly return volatlty. In each smulaton, t s represented by the number of tme steps wthn the perod [0, t]. he convergence of Monte Carlo smulaton can be acheved by a large number of teratons but at the expense of computaton tme. For each ratng pars (e.g., aa and a frms), we generate 0,000 sample paths. Return correlatons of assets, equtes and debts are calculated for each sample path and ther averages are reported. 4. mulaton Results, Emprcal Analyss and Dscussons In ths secton, we frst demonstrate that equty correlaton ρ and asset correlaton ρ can be qute dfferent and the dfference grows as the horzon lengthens and the ratng declnes. hs justfes our efforts to seek a more accurate relatonshp between ρ and ρ. Next, we establsh the lnk between ρ and ρ through the extended L model for dfferent pars of ratngs. Equty return correlaton ρ s estmated from stock return data. Combned wth the lnk between asset correlaton ρ and default correlaton ρ D gven n Zhou (00), we mplement the hybrd model to estmate default correlaton. 4.. Devaton of equty from asset correlatons horzon and ratng effects 8

21 able 3 reports the smulaton results for the relaton between asset correlaton and equty correlaton. he condton that ρ = mposed by prevous studes of default correlaton (e.g., Zhou, 00) s often volated as shown n able 3 where we fx the correlaton between the two asset processes at ρ = 40%. he model-predcted equty correlatons ncreasngly devate from asset correlaton ρ = 40% as horzon lengthens and ratngs declne. hs s due to the dffuson nature of the two asset value processes ( lnv and lnv ) and the nonlnearty n the model-predcted asset-equty relatonshp. akng the par of Aa- ratngs for example, when tme horzon = year, equty correlaton ρ s about 38%, whch s farly close to the gven asset correlaton ρ = 40%. ut for ρ = 0 years, ρ drops to %, whch s only about a half of ρ. In the case of - bonds, the dscrepancy between ρ and ρ s even larger. hus, the restrcton of the equalty of asset correlaton and equty correlaton s a very strong assumpton n the prevous default correlaton studes. 4.. Estmaton of equty and asset correlatons In emprcal nvestgaton, we estmate equty correlaton ρ from hstorcal equty return data. We calbrate the model such that the mpled equty correlaton from the model matches the emprcal equty correlaton. We calculate equty correlaton ρ for frms across ratngs from stock returns. 9 he data of stock returns are retreved from CRP and the data of frm ratngs are obtaned from Compustat. 0 We use monthly returns to calculate the correlaton for each year. hus, f we have n frms wth the same 9 Lucas (995) calculated hstorcal default correlaton based on default nformaton provded by Moody s Investors ervce from 970 through 993. o compare wth hs results, we use the equty nformaton over the same perod to calculate equty correlaton. 0 Compustat provdes both short-term and long-term ssuer credt ratngs. In our analyss, we lmt attenton to those frms wth assgned tandard & Poors Long erm Domestc Issuer Credt Ratng. 9

22 ratng, we wll have ( n ) / n pars of correlatons between two frms wthn the ratng n a gven year. If we have n frms wth one ratng and m frms wth another ratng, we wll have n m pars of correlatons across the two ratngs. We calculate parwse equty correlatons wthn and across ratngs each year over the sample perod of 970 to 993. We obtan an equally-weghted average of correlatons for each par of ratngs and report them n able 4. he summary statstcs of the sample are gven n Panel and the mean and standard error are gven n Panel. he relatonshp between asset correlaton ρ and equty correlaton ρ can be determned by smulatng the model to match emprcal estmates of ρ n able 4 (Panel ). he estmates of the asset return correlaton are reported n Panel A of able Performance of the hybrd model he results from the hybrd model are shown n Panel D of able 5 for horzons = 4, 6, 8 and 0 years, respectvely. We estmated the results for ten horzons ( to 0 years) but n the nterest of brevty we only report the results for these four horzons here. As shown, gven the calbrated asset correlaton ρ n Panel A, the hybrd model (Panel D) and Zhou s model (Panel C) generate qute dfferent default correlatons, whch ndcates that ρ = ρ n Zhou (00) s a poor assumpton. hs evdence s consstent wth prevous fndngs that equty correlaton s not a good proxy for asset correlaton (e.g., see Zeng and Zhang, 00; de ervgny and Renault, 00). In addton, n accordance wth de ervgny and Renault (00), we fnd that asset correlaton ρ he results for other horzons are consstent and avalable upon request. o compare wth Zhou s results, we use the same values of Z used by Zhou (00). pecfcally, Z for AA, A,,, and bonds are 9.3, 8.06, 6.46, 3.73, and., respectvely. 0

23 nferred from equty correlaton ρ exhbts an ncreasng trend as tme horzon lengthens (e.g., comparng dfferent horzons n Panel A of able 5). Frst, for bonds wth hgh ratng, the hybrd model predcts hgher default correlaton than Zhou s model does as shown n Panels C and D. ut the dfference between the two models s not as bg as that for bonds wth low ratng. When both ratngs are nvestment-grade, Zhou s model and the hybrd model tend to overpredct default correlaton for horzons greater than 6 years and underpredct for shorter horzons f we use Lucas s results n Panel as a benchmark. However, the over- and underpredctons are small. For example, for A-A par wth = 8 years, Lucas s estmate s % whle the hybrd model and Zhou s model predcts.86% and.%, respectvely. It should be noted that Lucas s results may not be a relable benchmark for hgh ratng bonds. Lucas s results are essentally unchanged from 4 to 0 years consderng possble roundng errors. hs mples that no (addtonal) default s recorded between 4 and 0 years n hs sample. Furthermore, estmatng default correlaton for safe frms can be extremely challengng n emprcal analyss usng hstorcal data. hs s because jont default s an even rarer event, whch may result n very few or even no observatons. For example, default correlaton beng 0 for aa-aa could mean no observed jont default n ths category as explaned n Lucas (995). Also, default correlaton s a rato of two small numbers (.e., functons of default probablty; see (7) and (8)). hs can make estmaton of default correlaton qute unrelable wth even a small error n the default probablty estmaton. econd, the hybrd model predcts much hgher default correlatons ρ D than Zhou s approach does for bonds wth lower ratngs. hese correlatons are closer to Lucas s estmates. For example, for the - bond par wth 4-year horzon, the hybrd

24 model predcts 8.% for default correlaton ρ, whch s much hgher than the default correlaton obtaned from Zhou s approach (.96%). he mprovement s even more sgnfcant for 0-year horzon the hybrd model predcts 3.3% for - par whle Zhou s approach can only generate 3.68%. From the perspectve of credt rsk management, a more accurate default correlaton estmate for lower-grade bonds s more mportant than that between hgh-grade bonds because when credt rsk s hgh, jont default becomes a major concern. y contrast, for hgh-rated bonds, the low credt rsk makes jont default a less lkely event even wth a hgh default correlaton. o the extent that there are more default and jont default events n low ratng classes, Lucas s estmates should be more relable. Indeed, hs estmates show larger and more sensble varatons as horzon changes for bonds wth lower ratngs. herefore, Lucas s estmates may serve as a reasonable benchmark n ths regme. 3 As shown, the hybrd model clearly outperforms Zhou s model by predctng much hgher default correlatons for bonds wth lower ratngs. D 5. Concluson Default correlaton nformaton s mportant for credt analyss and rsk management, but the scarcty of bankruptcy data makes t dffcult to obtan ths measure. Exstng structural models rely on the asset return correlaton to predct default correlaton. ecause asset value and asset return correlaton are unobservable, equty value and equty return correlaton are used as substtutes nstead. Prevous studes have 3 We note that Zhou (00) use Lucas s results as a benchmark across all ratng classes and horzons.

25 ndcated that these substtutons can result n poor estmates of default correlaton. Our fndngs confrm ths vew. In ths paper, we establsh a more accurate lnk between equty correlaton and default correlaton through ther relatons wth asset correlaton n a structural framework, whch allows us to use the equty return data to estmate default correlaton more relably. We frst develop a smple Merton-type model to drectly lnk default correlaton to equty correlaton. We then develop a hybrd model to overcome the problem n Zhou s model where unobservable asset return correlaton s approxmated by equty return correlaton. hs hybrd model s an ntegraton of the extended L model n a two-frm settng and Zhou s model. Emprcal results show that the hybrd model performs substantally better than other structural models. he smple Merton-type model can capture the trends of hstorcal default correlatons but t cannot capture the tme horzon effect exhbted n hstorcal default correlatons. y contrast, the hybrd model can predct the tme horzon effect as well as the ratng effect. Zhou s model predcts lower default correlatons than the hybrd model partcularly for low-grade bonds. Results show that the hybrd model performs much better than the Merton-type model and Zhou s model. Our results show that wth only one basc correlated stochastc drvng force (asset process), a hybrd structural model can yeld reasonable default correlaton estmates wthout mposng more complex factors such as correlated default threshold, ncomplete nformaton, and cross holdngs among frms. Results show that structural models are qute useful for estmatng default correlatons. he hybrd structural model developed n ths paper can be used to enhance credt rsk management. In partcular, t makes dynamc 3

26 management of credt rsk possble because the requred nputs to the model are equty data, whch are readly avalable. 4

27 APPENDIX A Relatonshp between Equty Correlaton and Asset Correlaton In ths appendx, we present the result needed to calculate equty correlaton from asset correlaton and we show the dervaton of ths result. Result Let Ψ(x,y) denote the bvarate normal dstrbuton functon, that s ( x y) = P( X x Y y) Ψ,, where X and Y are standard rownan motons wth means 0 and varances, and correlaton coeffcent ρ (we suppress the parameter ρ for smplcty). Τhen EV ( { }), V, I V, >,, V, >, (A) ς + ς m ln m ln = exp m+ m + ρςς + Ψ ς+ ρς +, ρς+ ς +, ς ς EV ( { }), I V, >,, V, >, m ln m ln exp m ς ς, ρς ς (A) + + = + Ψ, ς ς EV ( { }), I V, >,, V, >, (A3) m ln + ρςς m ln + ς = exp ς + m Ψ,, ς ς m ln m ln E( I { }),, V, >,, V, > =Ψ (A4), ς ς EV ( { }), I V, >,, V,, m ln m ln m ln exp m ς ς ς, ρς ς (A5) = + Φ Ψ, ς ς ς 5

28 EV ( { }), I V,,, V, >, m ln + ς m ln + ρςς m ln + ς = exp ς + m Φ Ψ,, ς ς ς m ln m ln m ln E( I { }) =Φ,, V, >,, V, Ψ, ς ς ς m ln m ln m ln E( I { }) =Φ,, V,,, V, > Ψ, ς ς ς (A6) (A7) (A8) where ς = σ and m ( =, ) s the expected value of lnv for frm, m = lnv,0 ς + µ s the debt value of frm at tme, and Φ(x) s the cumulatve normal dstrbuton functon, that s, Φ ( x) = P( X x) varances. lnv has a mean m and standard devaton ς :, and X s standard rownan motons wth means 0 and ln V, = Φ( m, ς ), where Φ(x,y) s the cumulatve normal dstrbuton functon wth mean x and volatlty y. Τhe expectatons n (A9)-(A) can be expressed n terms of Φ(x), that s, E E m + ( { }) ( ) ln ς V = + Φ I V > m ς ς exp (A9) ς + ( { }) m ln ς V = Φ I V > m + ς exp (A0) E m ln ( { }) I = Φ V > ς (A) Other terms n (6) are defned as below: 6

29 A A A 4 6 =,0 = =,,0,,0,0 e,,0 r, A + + e = r,,0, A + 7,0,,0,,0 e = e r,0 + e r, A r + e 8, A r = e 3, A r = 5 e =,,0,0 r,0 + e, r,,0 + e r, and ( ) E I{ V V }, m = Ψ,,,, ln ς m, ln ς. D r,,,, = D = +,,0,0,0 e D 3, =,,0 + e r and I { V } m = Φ,,, ς ln, Dervaton of Result he left-hand-sde of (A) can be expressed as E { > > } V, V lnv ψ d lnvd lnv (A) ( V V I ) = exp ( lnv) uppose that σ σ 0 and ρ <. hen where ( ) ( ) = - ψ lnv exp ( lnv - m) Σ ( lnv - m) ' (A3) π det Σ m = ( ) m,m 7

30 8 = ' Σ Σ ς ς ρς ς ρς ς y an affne transformaton ν F lnv U + = (A4) where ( ) ( ) mf, ν 0 F, U = = = = ν ν ρ ς ρ ς ρ ς U U U and U are ndependent and dentcally dstrbuted standard normal random varables and m F U lnv - + = he Jacoban of the transformaton s ( ) ln ρ ς = ς = U V U J he regon under the ntegraton n (A3) s bounded n (lnv, lnv ) plane by, V V = = he mage under the transformaton n (U,U ) plane s bounded by b a U U b U + = = ln,, ln ρ ς ρ ρ ς = = = m b a m b he change of varable theorem gves

31 = ( I ) E V V b a U + b { V >, V > } exp - ( U F + m) φ ( U ) φ( U ) J ( U) du du (A5) where φ s the standard normal probablty densty functon. (A5) can be rewrtten as where ( I ) E VV = γ b φ { V >, V > } ( U ( ς + ρς )) du φ( U ς ρ ) a U + b ς + ς γ = exp ρς ς + + m + m Integratng (A6) wth respectve to U yelds ( I ) E VV = γ b φ { V >, V > } ( U ( ς + ρς )) Φ( au b + ς ρ ) du du (A6) he ntegral can be expressed n terms of the bvarate normal dstrbuton functon, Ψ, (Chuang, 996) E ( V V I ) { V >, V > } m ln = γψ ς + ρς + ς m, ρς + ς + ln ς (A7) Usng a smlar transformaton, we have (A) to (A4). For (A5), t can be related to (A) by ( V, I { V >, V }) = E( V, I { V > }) E( V, I { V > V > }) (A8) E,,,,,,,,,,, where 9

32 ( ) exp( ς U + m ) φ( U ) E V, ς = exp I { V > },, = b m ln + m Φ ς + ς du (A9) Usng (A9) and (A), we obtan the result n (A5): E ( V ), = exp m I { V >, V },,,, ς m ln + Φ ς m ln + ς Ψ ς m + ς, ln ς + ρς (A0) y a smlar transformaton, we have (A6), (A7) and (A8). Under the equvalent martngale measure, we have r = E( V, I { V > }) e E I { V > } ( ) o,,,, ς whch s the lack-choles opton formula. nce V 0 = exp m +, we have ς m = lnv,0 + r under the equvalent martngale measure. 30

33 Appendx Dervaton of Equty Correlaton from Asset Correlaton Gven the results n Appendx A, we can derve equty correlaton from asset correlaton. he equty value of frm s the call value on the underlyng frm asset process V,t. At tme the frm value s gven by Eq. (5). he equty return for the perod from 0 to can be expressed as the call opton wth the strke equal to the frm s face value of debt,,0,0 = ( V ),,0 +, () We use the smple return rather than a contnuous compoundng n order to obtan explct solutons. 4 For a constant nterest rate and under the equvalent martngale measure, we have ( V ) +,,0,, r E = E = e (),0,0 nce the volatlty of a rownan moton s nvarant under dfferent equvalent martngale measures, we can calculate the equty correlaton between two frms under ether the physcal measure or the equvalent martngale measure. hus, by defnton the covarance between the equty returns of frms and j, we have 4 Gven the jont probablty dstrbuton of the two assets, we may alternatvely assume contnuous return and compute the equty correlaton drectly by usng the defnton,, Cov ln ln,0,0 However, ths alternatve does not provde much smplfcaton. 3

34 cov ( R, R ),,0 = cov,,0 V = E Ee, V E r, V,0,,,0,,0 e,, r e r V e e,0 r, r,0,0 I { V >, V },,, e, r, I { V, >,, V, >, } r I { V V } Ee I,,,, > +, { V,,, V,, } (3) he frst term on the rght hand sde of (3) can be expanded and rewrtten as V, E,0 = A E V + A E V 3, V r,, r e e I { V V }, >,,, >,,0 I { V >, V > } + A E V, I V >, V > (, V, ) ( { }),,,,,,,, (, I { V >, V > }) + A4 I { V >, V > },,,,,,,, (4) where A A, r, r =, A = e A3 e +,,,0,0,0 =,0 +,0,0 4 =,,0,,0 +,,0 +,,0 e r + e r (5) he second and the thrd terms on the rght hand sde of (3) can be expanded and rewrtten as V E, C = A E V 5,0, e I { V >, V } I { V >, V } r (, ) A6e I { V >, V }, r, e r,,,,,, V r,, r Ee e I { V V } C,,,, >,,0 = A7 E V, I { V, V > } A8 I V,,, ( ) {, V > },,,,,,,,, (6) (7) 3

35 A e r r, r r r, r 5 =, A6 = e + e, A7 =, A8 = e + e (8),0,0,0,0 e Fnally, the covarance of the equty returns of frms and can be expressed as cov ( ) + A E( V { }) I,,,,, V,>,, V, >, (, I { V V A EI,>,,, > +, 4 { V,>,, V, >, } (, I { V }) V A EI, >,,, +, 6 { V, >,, V,, } r (, I V, V > ) + A8 EI { V, V > } + e I { V, V } ( R R ) = A E V, V, I { V >, V > }, + A E V 3 + A E V 5 + A E V 7 { },,,,,,,,,,,, (9) where the expectatons are gven n Lemma. y defnton, the varance of the equty returns s var ( R ) = var V = E,,,0,0,,0 = E e r, r I { V > } + e I { V },,0,,0 E,,,0,,0 (0) he frst term on the rght hand sde of (0) can be expanded nto V,, r E e I { V }, >,,0 = D, E( V, I { V > }) + D, E V, I { V > },, ( ) + D3, I { V > },,,, () where r,,,,, e r D, = D = + D 3, = + e (),0,0,0,0 Combnng (0) and () yelds var ( ) + D, E( V, I{ V > }) ( R ) = D, E V, I{ V > } + D 3, r I{ V > } + e EI{ V },,,,,,,, (3) 33

36 Appendx C Jont Default Probabltes hs appendx summarzes the jont default probabltes (see Zhou, 00). When debt and equty have equal expected growth rates,.e., leverage ratos l and l are constant, the expected default probablty s wth propertes Z Z µ ς Z µ E[ ω( ) = ] = P[ ω( ) = ] =Φ + e Φ σ σ,0 [ ω ] = P[ ω = ]{ P[ ω = ]} Z = var ( ) ( ) ( ), and µ lnv where ζ s the volatlty of frm value. he jont default probablty s [ ( ) ω ( )] = E[ ω ( )] + E[ ω ( )] P[ ω ( ) = or ω ( ) ] E ω, = where the probablty when at least one frm has defaulted s lnv r α β β β nπθ nπθ n α n= α α 0 o x+ x+ t o t ( ) = ( ) = = ( ) P ω or ω e sn e sn g θ dθ, and the relevant parameters are gven as follows, σ ρ tan f ρ < 0 ρ α =, ρ π + tan otherwse, ρ Z ρ tan f ρ < 0 Z ρz θo = Z ρ π + tan otherwse, Z ρz ( θ ) 0 r t r h h rro sn( θ α) cos( θ α) nπ α t gn = re e I dr, r o Z =, sn ( θ ) o β µ ρσ µ σ =, ( ρ ) σ σ β µ ρσ µ σ =, β t = + ρaaσ σ + + βµ + β µ, ( ρ ) σ σ β σ β σ h, = β σ + ρσ σ h = β σ ρ. 34

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