Edward I. Altman 2 NYU Stern School of Business 44 west Street New York, NY 10012, USA Ph.: altman@stern.nyu.

Transcription

1 An Integrated Pricing Model for Defaultable Loan and Bond Mario Onorato Ca Buine School, City Univerity 06, Bunhill Row London, EC2Y 8Z, U.K. Phone: M.F.Onorato@city.ac.uk Edward I. Altman 2 NYU Stern School of Buine 44 wet Street New York, NY 002, USA Ph.: altman@tern.nyu.edu M. Onorato i Reearch Fellow at Ca Buine School, London City Univerity and Viiting at Eramu Univerity Rotterdam 2 E. Altman i the Max L. Heine Profeor of Finance, NYU Stern School of Buine and Vice Director of the NYU Salomon Center

2 An Integrated Pricing Model for Defaultable Loan and Bond 3 JEL claification: C5, C69, G2, H63 Keyword: tatitical imulation method, financial r management, credit r meaurement model, aet pricing, debt & debt management. In recent year, credit r ha played a key role in r management iue. Practitioner, academic and regulator have been fully involved in the proce of developing, tudying and analying credit r model in order to find the element which characterize a ound r management ytem. In thi paper we preent an integrated model, baed on a reduced pricing approach, for market and credit r. It main feature are thoe of being mark to market and that the pread term tructure by rating cla i contingent on the eniority of debt within an arbitrage-free framework. We introduce iue uch a, the integration of market and credit r, the ue of tochatic recovery rate and recovery by eniority. Moreover, we will characterie default r by etimating migration r through a mortality rate, actuarial baed, approach. he reultant probabilitie will be the bae for determining multi-period rneutral tranition probability that allow pricing of ry debt in the trading and banking book. We thank Giovanni Barone Adei, Winfried Hallerbach, Anthony Saunder, Jaap Spronk, Rangaraian Sundaram, Contantine hanaoula, three anonymou referee, and eminar participant at Italian Banking Aociation for helpful dicuion of thi tudy. Earlier verion of thi paper were preented at the Annual Meeting of both Financial Management Aociation (Pari, 200) and European Financial Management Aociation (London, 2002) and at the Euro Working Group on Financial Modelling (Capri, 2002). 2

3 . Introduction he 988 Bale Capital Accord, which created a minimum r-baed capital adequacy requirement for bank, marked a major tep forward in introducing r differentiation into the regulatory framework but did not repreent the optimum olution. he 996 amendment to the Capital Accord aimed to correct ome of the iue concerning the original accord, but it did not change the ection regarding credit r. A a conequence, regulatory capital for credit aet wa till not an appropriate bai for the capital allocation proce. Financial intitution tarted developing their own internal credit r ytem uing economic capital. In April 999, regulator propoed a document, Credit R modelling: current practice and application [6], which aimed at aeing the potential application and limit of credit r model 2 for uperviory and regulatory purpoe, in ight of the foreeeable amendment to the Capital Accord. he Bale Committee propoed the amendment in June 999: A new capital adequacy framework [8] (which i expected to be finalied in 2003). hi event marked a breakthrough a regard the credit r concept for capital adequacy. he committee propoal dramatically modified the tandard approach for capital requirement impoed by the 988 Accord. A more realitic approach wa introduced baed on internal rating ytem. Moreover, it preented the reclaification of ecuritie taking into account credit r in all it apect: default, migration, recovery rate, credit pread, aggregation and concentration r. Over the lat decade r manager, regulator, academic and oftware vendor are devoted to define a ound credit (and market) r meaurement and management ytem. he main objective of thi tudy i to define a general framework to price ry debt. he pricing of any ry ecurity mut reflect the return on a r-free aet plu a r margin. he r margin mut compenate the invetor for the r aumed which i repreented by: both the tranition and recovery (by eniority of debt) r, liquidity r, credit expoure r, default correlation r, collateral r and concentration r. Moreover, in an integrated framework, another ource of r ha to be conidered, namely: market and credit correlation r 3. here are everal approache, which may be ued to jointly model interet rate and credit pread. In thi paper reduced 4 form model and in particular the one directly modelling credit pread component (tranition and recovery r) are conidered. Our approach i a generalization of the Da & ufano [22] (D) (996) model, which i an extenion of the Jarrow-Lando-urnbull (JL) (997) [35] model, and ue credit rating to characterize the tranition r. Unle the JL model, the D model make the recovery rate in the event of default tochatic, and provide a two-factor decompoition of credit pread. In thi paper we generalie thi approach by conidering different et-up for different eniority clae of debt. herefore, it main feature are thoe of being mark to market (MM) 5 and that the pread term tructure by rating cla (SSRC) i adjuted for a pread term tructure which i contingent on the eniority of debt (SSD) within an arbitrage free framework. Summariing, in our integrated pricing model we take into account the r coming from both interet rate variation and credit event verification 6. he remainder of thi paper i organied a follow. Section 2 define default and tranition r and 2 he document iued by the Bale Committee [6] analye in particular four widepread credit r model: Creditmetric M (JP Morgan 997) [5] ; KMW Portfolio Manager M (Kealhofer 998) [7], [38], [39], CreditR+ M (Credit Suie Firt Boton 997) [6], Credit Portfolio View M (Wilon 997 and 998) [53], [54]. A the recent literature how Koyluoglu and Hickman (998) [5], [40], thee model fit within a ingle generalied framework.. 3 Conequently, all the correlation among the market factor (which drive the price of ecuritie) and credit r factor-alo called background factor- which affect the creditworthine of obligor in the portfolio, need to be identified and modelled. For an illutrative decription of the background factor within the credit r model framework refer to Koyluoglu and Hickman (998) [40] and Wilon(997 and 998) [53], [54]. 4 Reduced form model are o called to contrat them to tructural model (Merton 974 [44] ). he difference between tructural and reduced form model i outlined in ection 2 and 4. For a complete review of the inherent literature refer to the work of Acharya, Da and Sundaram (2002) []. 5 In contrat to the default mode paradigm, within the mark to market (or, to be more accurate, mark to model) paradigm a credit lo can arie in repone to deterioration in aet credit quality hort of default. Foe a detailed dicuion on the topic pleae refer to the work of Bale Committee, pag.2 [6]. 6 When a credit event occur the credit quality of the iuer change. Example of uch event are given in ection 2. 3

4 it i aimed at illutrating both the mot common method to etimate them and the inherent empirical evidence. Section 3 analye the ame iue for the recovery rate by eniority of debt r. Section 4 i the core of the paper. In thi ection the theoretical integrated pricing model i illutrated. he bulk of the uggeted approach relie on the modelling of both the tochatic pread term tructure by rating cla and the pread contingent to the eniority of debt within a unified arbitrage-free framework. Section 5 illutrate poible application of the integrated pricing model. Section 6 conclude. 2. Default and ranition R Credit pricing and r model attempt to meaure credit loe. Loe are the conequence of the firm financial poition and aet quality deterioration, which then lead to the degradation of it creditworthine (credit migration). he determination of the creditworthine of the iuer i difficult a it i driven by many factor uch a general economic condition, indutry trend and pecific iuer factor le the iuer financial wealth, leverage, market value, equity value, aet value, capital tructure and le tangible thing uch a reputation and management kill. he probability of a cutomer migrating from it current r-rating category to any other category, within a pre-defined time horizon, i frequently expreed in term of a rating tranition matrix 7. Rating migration probabilitie are therefore collected in the tranition (migration) matrice and decribe the probability of migrating from any given credit rating to another one. Moreover, etimate of tranition probabilitie often uffer from mall ample, either in the number of rated firm or in the number of event; in particular thi happen when conidering tranition toward the mot ditant rating clae. hi often reult in biaed etimate of thee type of tranition probabilitie that have led the Bale Committee on Banking Superviion to impoe a lower minimum probability of for rare event. In our analyi we will cluter obligor into obligor rating clae. In thi matrix the wort internal grade correpond to the wort tate, that i the default tate (lat column of the matrix) 8. Let u aume to deal with K rating clae (the Kth being the default tate), then, the tranition matrix i the collection of one-tep tranition probabilitie of migrating from any cla-i to any cla-j at the given time-m, including the probabilitie of remaining in the ame cla (correponding to the off-diagonal value). he tatitical, or actual, probabilitie matrix can be repreented a: Q = { q }.... qk q.. q.... qk q.. q q q q2 22 2K q2 22 2K ij = (2.) qk qk 2.. qkk Once the default tate i reached, no other rating clae are poible to exit at the next time tep, therefore all the tranition probabilitie are defined to be null, with the exception of the probability of taying in the K-tate i.e. default. A a conequence of the definition of tranition probability, another relevant property of the migration matrix i that the um over the element K of the ame row mut equal one: q ; i. In the integrated model, the tatitical ij j= tranition probabilitie are derived from empirical data by uing the mortality approach briefly decribed in paragraph 2.. he proce determining cutomer default or rating migration can be modelled through two approache: actuarial baed method and equity baed method. 7 See Altman, Caouette and Narayanan (998) [2] for a dicuion on thi topic. 8 In a two tate default proce, within the conidered time horizon, there are only two poible event: no default and default. 4

5 2. Actuarial baed method he baic actuarial approach ue hitorical data on the default rate of borrower to predict the expected default rate for imilar cutomer. he actuarial (alo called empirical) model i baed on the etimation of tatitical (or actual) tranition probabilitie 9. hi method ue hitorical data to evaluate the migration probabilitie. In thi model the input are repreented by empirical data and the output will be the tatitical etimation concerning thee data. One of the mot important criticim to the empirical approach i the apparently tatic nature of the reulting average hitorical probabilitie. In reality, actual tranition and default probabilitie are very dynamic and can vary quite ubtantially over one year, depending on general economic condition and buine cycle 0. In our integrated pricing model we will etimate the actual tranition probability by uing the Altman (998) mortality rate approach [2]. hi method determine the (expected) default rate, uing an empirical method. An important element that need to be oberved i the aging effect, that i, the time between intrument iue up to valuation time. hi approach implie lower default probabilitie in the firt year than in the next year. he quetion i not whether a bond or another credit derivative i going to default or not but when it i going to default and what will be the lely recovery, given it original rating and it original eniority. Credit intrument are claified for iue time and rating clae. After thi claification it i poible to calculate the default probability. In order to do thi, it i neceary to calculate the marginal mortality rate and the cumulative mortality rate. he marginal mortality rate (MMR) i the probability that a credit intrument default over the firt year, over the econd and o on. he MMR can be expreed both in term of number and value. In the lat cae MMR i equal to the ratio between the total (nominal) value of the corporate bond included in a pecific rating cla defaulting over the planning horizon and the total (nominal) value of corporate bond included in the ame rating cla, at the beginning of the time horizon. MMR i = otal value of defaulted bond in the i year otal value of bond iued at the beginning of the i where i=.nn= number of year 2 th th year Conequently the urvival rate (SR i ) i equal to SR i =-MMR i. One can meaure the cumulative mortality rate (CMR ) during a pecific time period, ubtracting the product of the urviving population over the previou time, that i CMR =- SR i.altman [2] derive the migration probabilitie for each rating cla from CMR, which i= repreent the default probability. 9 Here we refer to actual (tatitical or empirical) tranition probability in contrat to r-neutral probability 0 A real dilemma concern the private companie that are neither rated by the agencie nor publicly traded. In fact a ubtantial proportion of thee portfolio do not have very clear benchmark for etimating default and tranition probabilitie Altman method (998) i different from other method determining the aging effect. In fact Moody and S&P ue tatic pool (including all credit intrument), while Altman make a ditinction among intrument according to the iue date. he (actual) tranition probability matrix in the integrated pricing model can be eaily inferred from the migration matrix, which i etimated uing the Altman mortality rate approach. For a more detailed illutration of the mortality (default) rate by rating and by age approach pleae refer to Altman (998) [2] 2 If, for example, the par vale outtanding of high-yeld debt in 997 wa ($ million) and the par value default wa ($ million), the MMR (or alternatively the default rate) wa.252% 5

6 2.2 he equity baed method he equity-baed approach, often aociated with the Merton model [44], i mainly ued for etimating the Expected Default Frequencie (EDF) 3 of large and middle-market buine cutomer, and i often ued to crocheck etimate generated by actuarial-baed method. hi technique ue publicly available information on a firm liabilitie, the hitorical and current market value of it equity and the hitorical volatility of it equity to etimate the level, rate of change and volatility (at an annual rate) of the economic value of the firm aet. here are at leat three practical limitation to implement the option (Merton model) approach:. It i neceary to know the market value of firm aet. hi i rarely poible a the typical firm ha numerou complex outtanding debt contract traded on an infrequent bai. 2. It i neceary to etimate the return volatility of the firm aet. Since the market price cannot be oberved for the firm aet, the rate of return cannot be meaured and volatilitie cannot be computed. 3. It i neceary to imultaneouly price all the different type of liabilitie enior to the corporate debt under conideration. Mot corporation have complex liabilitie tructure 4. Summarizing, the key ingredient of credit aet pricing and r modelling i the default (or, in a multi-tate framework, tranition) r, which i the uncertainty underlying a firm ability to ervice it debt and obligation. Prior to default, there i no way to dicriminate unambiguouly between firm that will default and thoe that will not. At bet we can only make probabilitic aement about the default poibility. In practice, we ue tranition probabilitie baically for two main reaon:. In the trading book, to price credit enitive intrument adjuting it through the r neutral tranition probability 2. In the banking book, to meaure the credit r of portfolio loe of loan 3. Recovery by eniority of debt r he default i one of the main type of credit event that determine lo amount occurring once a credit ha defaulted. hi credit lo, alo called lo given default (LGD) i defined a the difference between the bank credit expoure and the preent value of the future net recoverie (cah payment from the borrower le workout expene). herefore the recovery rate (RR) i equal to the ratio between -LGD and the initial expoure 5. LGD depend on a limited et of variable characteriing the tructure of a particular credit facility. hee variable may include the type of product (e.g. buine loan or credit card loan), it eniority, collateral and country 3 Expected default probabilitie can be inferred from the option model under the aumption that default occur when the value of a firm aet fall below it liabilitie. See Crobie (998) [7] for a detailed decription of how the EDF are etimated within the KMV model. 4 For an intereting and detailed analyi of the limitation of the equity approach ee Jarrow and urnbull (2000) [36] 5 he etimation of LGD depend on the availability of hitorical lo data that may be retrieved by the following poible ource: bank own hitorical LGD record, ample by r egment; trade aociation and publicly available regulatory report; conultant proprietary data on client LGD, and publihed rating agency data on the hitorical LGD of corporate bond. 6

7 of origination. In the Credit R+ M [6] model 6 the LGD i treated a a determinitic variable while in the other tructural model i treated a a random variable 7. Reduced form model aume either a contant (Litterman and Iben [4], JL [35], for example) or a tochatic recovery rate (Duffie and Singleton [28] and D [22] for example). hee model aume zero correlation among the LGD of different borrower, and hence no ytematic r due to LGD volatility 8. Moreover, the empirical evidence how that the recovery rate are both tate 9 and tructure 20 dependent. In particular, our analyi baed on the lat three decade default and recovery data on US corporate bond how that the recovery rate change depend on the eniority of debt. In fact, comparing enior ecured and unecured bond one can ee that the recovery ditribution for the latter i more pread out and ha a longer lower tail (ee table ) 2. Recovery Rate by Seniority and Original Bond Rating, Recovery Rate Number of Average Weighted Median Standard Seniority Oberv ation Price Price Price Deviation Senior Secured Invetment Grade 35 $62.00 $66.00 $56.88 $9.70 Non-Invetment Grade Senior Unecured Invetment Grade 59 $53.4 $55.88 $50.00 $26.4 Non-Invetment Grade Senior Subordinate Invetment Grade 0 $39.54 $42.04 $27.3 $24.23 Non-Invetment Grade Subordinated Invetment Grade 0 $35.64 $23.55 $35.69 $32.05 Non-Invetment Grade able. Recovery rate by eniority and original rating 6 For portfolio characteried by ditribution of expoure ize that are highly kewed, the aumption that LGD are known with certainty may tend to bia downward the etimated tail of the PDF of credit loe 7 In thee model the LGD probability ditribution i aumed to take the form of a beta ditribution becaue thi reult in a type of ditribution whoe hape i tipically kewed to the right a hown in the empirical work of Altman and Kihore (996) [2], Carty and Lieberman (996) [0], [], Duffie and Singleton (996) [28], Catle, Keiman and Yang (2000) [3] 8 Furthermore, they aume independence among LGD aociated with the ame borrower. he aumption that LGD between borrower are mutually independent may repreent a eriou hortcoming when the bank ha ignificant indutry concentration of credit. Furthermore, the independence aumption i clearly fale with repect to LGD aociated with imilar (or equally ranked) facilitie to the ame borrower. he aumption of default intenitie independence may contribute to an undertatement of loe to the extent that LGD aociated with borrower in a particular indutry may increae when the indutry a a whole i under tre. 9 Some evidence conitent with the tate-dependence of recovery rate i preented in the analyi, baed on recovery rate, compiled by Moody for the period 974 through 996 (Carty and Lieberman, 996 [0], [] ). However, even for enior ecured bond, there wa ubtantial variation in the actual recovery rate. Although thee data are alo conitent with cro-ectional variation in recovery that i not aociated with tochatic variation in time of expected recovery, Moody recovery data alo exhibit a pronounced cyclical component. here i equally trong evidence that of corporate bond vary with the buine cycle (a i een, for example, in Moody data) Speculative-grade default rate tend to be higher during receion, when interet rate and recovery rate are typically below their long-run mean. 20 See Catle, Keiman and Yang (2000) [3] 2 Source: Altman and Pompeii (2002) [3].Alo Duffie and Singleton (998) [28] found imilar reult. 7

8 he analyi alo rank the reult in invetment grade and non-invetment grade. In fact, when evaluating an intrument at the firt tep of it life the type of guarantee rate on the underlying ecurity i an extremely relevant characteritic, which according to hitorical data - implie that the credit i ubject to a global lower r. hi i hown in able, where the rating cla, at the time of iuance, non-invetment grade in particular, i not influencing the average value a much a the eniority cla doe (ee the enior ecured and enior unecured invetment grade cae). he mot relevant iue i that the dominant factor influencing the evaluation of the ecurity i the compoition of both the recovery rate and default probability. Actually, low default rate do not aure that in cae of default the recovery rate i low a well; on the other hand high recovery rate i not a credit low-r index alone, ince the ecurity might by highly defaultable, implying the elevated invetment r. In the integrated model we will etimate pread term tructure by rating cla, to explicitly conider the credit rating (r) tranition and will correct them by mean of pread term tructure of recovery rate by eniority of debt, both in a arbitrage free framework, in order to get a r-neutral price of the financial intrument. 4. he theoretical integrated pricing model here are two main approache to pricing credit ry intrument: the tructural 22 and the reduced form approach. It i argued that tructural approache are of limited value when applied to price interet rate and credit enitive intrument and, conequently, in meauring and managing market and credit r in an integrated fahion. Roen (2002) [47] how that ince the main focu of the tructural model i the meaure of the counterparty expoure r, they aume determinitic market r factor, uch a interet rate r. In contrat to tructural model, which aume a pecific microeconomic proce generating cutomer default and rating migration, reduced-form model attempt to directly decribe the arbitrage free evolution of ry debt value without reference to an underlying firm-value proce. Acharya, Da and Sundaram (2002) []23 how how thi cla of model ha reulted in ucceful conjoint implementation of term-tructure model with default model. he objective purued within the uggeted integrated pricing model i that of deriving a general framework for pricing ry debt, both plain vanilla (a for example corporate bond) and (credit) derivative. Preent value of all cah flow are calculated by uing both tochatic interet rate term tructure (market r) and tochatic credit pread term tructure (credit r). hi lat term can be decompoed in the following r ource: ) tochatic recovery rate by eniority, 2) correlation between interet rate term tructure and tochatic recovery rate (correlation of market and credit r) and 3) multi tate tranition probability at the m- th time tep for the M-period proce. In thi et up the propoed integrated pricing model may be conidered a multi-period mark to model framework. A for all reduced-form model, alo in our integrated model we tart modelling the r free term tructure by conidering an underlying proce for the evolution of r-le rate. he objective i to build a lattice of ry rate on top of the r-le rate proce in an arbitrage-free manner by directly modelling credit pread component (tranition and recovery r). We generalie the Da & ufano [22] (D) model, where the pread term tructure by rating cla i modelled through three main component: r neutral probability matrix, tochatic 22 In fact, the tructural approach aume ome explicit microeconomic model for the proce that determine default or rating migration of any ingle cutomer. A cutomer might be aumed to default if the underlying value of it aet fall below ome pecified threhold, uch a the level of the cutomer liabilitie. he change in the value of a cutomer aet in relation to variou threhold i often aumed to determine the change in it r rating over the planning horizon. Structural approach model are Merton type model. 23 Reduced-form model may differ depending on the procedure that i ued, the input information required, the ue of rating-matrix and the recovery aumption. A pointed out by Da and Sundaram (2000) [2] here are three commonly ued aumption concerning recovery rate in the event of default: recovery of par, where the recovery amount i pecified a a fraction of par value due at maturity; recovery of treaury, where recovery amount i pecified a a fraction of value of a default-free bond with the ame maturity; recovery of market value, where the recovery amount i pecified a a fraction of the immediately-preceding market value. 8

9 recovery rate and it correlation with interet rate. In the D model the firt component i aimed at etimating the tranition r; the econd one, the recovery r; the lat one, the correlation between market and credit r. In the integrated model different et-up for different eniority clae are introduced within an arbitrage free framework. A the empirical evidence how (ee ection 2 and 3) the mean recovery i mainly contingent on the eniority of debt rather than on the rating cla alone (invetment grade v. non-invetment grade in our analyi) a it i almot invariably aumed in all reduced model. he major contribution of thi work it i to correct each SSRC through a pread contingent on the eniority of debt (SSD) within a unique arbitrage free framework. A a reult, thi model allow more variability in the pread of ry debt. Moreover, by chooing different recovery rate procee for intrument within the ame credit rating cla, it allow variability of pread to be intrument pecific rather than rating cla pecific. 4. he tochatic interet rate term tructure model In the integrated model an interet term tructure i aigned to each rating cla i (where 0 i K, if we conider K rating clae). he i-th interet rate f i at which cah flow are to be dicounted i compoed of forward r-free interet rate plu a (forward) pread aociated to the ame rating cla a hown below: f i (t) = forward curve for rating cla-i = forward r free(t) + pread-i (4..) In thi context, the r-free forward interet rate (tochatic) proce can be modelled by uing any interet rate term tructure model le, for example, the Heath-Jarrow-Morton [992] [32] or the Black-Derman-oy [990] [9] model. It i not the purpoe of thi paper to detail the r neutral et up model formulation for the evolution of the interet rate free term tructure, for which pecialied literature may be addreed. More relevant to the preent paper purpoe i to illutrate how the pread i modelled for which the following paragraph are devoted to. 4.2 he tochatic pread term tructure model Recovery rate, r of default and the eniority type are relevant parameter for aeing credit r. herefore, in the integrated model, the pread i decompoed in it two main determinant a) recovery rate and b) default 24 (tranition) r. hu, in order to price the credit pread component of the interet rate term tructure, both recovery rate and default variable need to be modelled. Let q be the (r neutral) default rate 25 (i.e., the rate at which default occur). hi rate may be either contant, or function of time-to-maturity of the ecurity or of any other factor in the economy. he recovery rate will be denoted by φ and repreenting the fraction of the face value of the ecurity that i recovered in cae of default (by definition 0 φ ). Conidering the influence of recovery and default rate on credit intrument, it i poible to conider a firt imple relationhip between thee parameter and interet rate pread. Let r be the one period r-le rate of interet, then the r-neutral value B of a credit ry bond maturing in a ingle period from now mut be equal to the dicounted value of expected cah flow in the future: q q φ + B = + r ( q ) (4.2.) where the parameter and φ have been et to their r-neutral value. On the other hand the price of the ry bond B off the pread curve i given by: 24 he default r bearing alo information on the type of eniority type 25 he default rate being the rate at which default occur 9

10 B = + r + (4.2.2) By equating the right hand ide of Eq. (4.2.) and Eq. (4.2.2) the required relationhip between the pread, which i the oberved market pread for the generic ecurity I, and the determinant of the pread may be derived. Solving by we obtain: ( φ )( + r) q ( φ ) q = (4.2.3) In general, the actuarial etimation of the default rate i different from it r neutral value, becaue the way through which the actuarial value i etimated i independent from the market price of that ecurity. If, recalling eq. 2., the actuarial default rate i, q we have: ( φ )( + r) q ( φ ) q act = (4.2.4) where act differ from. o calibrate the tatitical value of the default rate one can ue pread market data. Given, it i poible to render q r neutral by umming to q the adjutment factor π. ( φ )( + r) + π )( φ ) ( q + π ) = (4.2.5) ( q Conequently q = q +π. hi approach allow coupling the model of the tochatic proce for the interet rate term-tructure to the market data, that i to ay theoretical and empirical data. From Eq. (4.2.3) it i poible to ee that: - the pread increae proportionally to the default rate increae; thi ha a financial q implication: a the default rate increae the poibility of getting value far apart form the expected average value i higher. On the contrary, in the limiting cae of default r approaching zero ( q 0; for =0 the recovery rate looe it meaning) the pread tend to zero ( 0), allowing the certain value equal to the average q - the pread decreae proportionally to the recovery rate φ increae, which mean that - in cae of default - the higher i the chance of getting back the inveted amount, the more limited fluctuation from the average price are got; in other word high recovery rate aure low credit r. In the limiting cae, approaching total recovery (φ ) the pread till tend to zero ( 0), in the ideal limiting cae =0 repreenting the evolution of a r-le proce q - when the default rate tend to one ( ) and the recovery rate tend to zero (φ 0) the pread tend aymptotically to become infinite. Of coure limiting cae are never reached but their tudy help viualiing the trend of the functional dependence of the pread from the default r and recovery rate. In fact, a pointed out by Da [20], Eq. (4.2.3) expree the pread a a function of the compoite variable q (-φ), for thi reaon the above formulation doe not allow expreing the pread a a function of default r and recovery rate independently. herefore a more elaborated interet rate pread modelling i needed. Conidering, for example, the HJM model, it i poible to oberve that it tructure allow for the required effective two-factor decompoition of credit pread. Under q 0

11 the r-neutral meaure, the expected ry cah flow dicounted at r-le rate mut be equal to the value of expected r-le cah flow dicounted at ry dicount rate: t m= + m ( ) = ( ( ) + ( ) ) E exp f i t, j * Cd exp fi t, j i j * (4.2.6) t t j = j = where C d (m) i the expected cah flow of the ry bond in cae of default at the time tep-m before maturity and i the cah flow in cae of non-default. In order to render C d (m) in explicit form it i neceary to define the cumulative and one-period default probabilitie aociated to the rating cla of the intrument at any given time, and to conider the recovery rate at the correponding default time. Including the default r, the recovery rate and the credit eniority information in C d (m), by mean of Eq. (4.2.3) it i poible to etimate the determinant of the interet rate term tructure pread aociated to the rating cla I contingent to the eniority of debt. In order to develop a conitent framework - ince for the interet rate term tructure model a r-neutral world i aumed, the actual tranition probabilitie (etimated by uing the mortality approach 26 decribed in ection 2) have to be r-neutral adjuted. After having obtained the r-neutral et up for the evolution of the term tructure of interet rate, the integrated model derive the r-neutral probabilitie of the tranition proce to default. Summariing, we will firt correlate the interet rate term with one recovery rate tructure, then, we will generalie the reult by conidering eniority type thu including the pread correction due to the recovery dependence on the eniority. Following thi et up a new tochatic framework for the arbitrage-free pricing of ry debt i depicted. hi framework i illutrated through the following three tep:. firt contruct a one period r neutral probability matrix for each eniority type 2. then extend to a multi-period framework through the definition of a cumulative r-neutral tranition matrix which allow the obligor to default at any point in time 3. third etimate the SSRC contingent to the eniority of debt 4.2. R-neutral probability tranition matrix One of the key point in which the integrated model depart from other model i in the pread dependence aumption of both the recovery rate on eniority and of the rating cla. In general, in the integrated model, the recovery rate i aumed to follow any reaonable ditribution. We ugget calibrating the model by uing a beta ditribution in according to the empirical evidence decribed in the econd ection. In practice, any value for the recovery rate i poible with a non zero probability. he probability denity function of the beta ditribution i given by: g m ( φ m, α m, β m Γ( α ) = Γ( α 0 m m + β ) Γ( β m m ) φ ) αm m ( φ β ) m m for 0 < φ for φ m m < < 0 andφ m with =,...,5 (4.2..) > φ m where repreent the fraction of recovery at time m aociated to the eniority type, () g m. repreent the probability aociated to that recovery rate (belonging to the eniority cla); the eniority type range i between and 5 becaue the conidered eniority clae are 26 he tatitical migration matrix i an input in thi pricing model. One can alo ue other approach, le the S&P or Moody method of etimating the migration matrix

12 5: enior ecured, enior unecured, enior ubordinated, ubordinated, junior ubordinated. Moreover Γ tand for the gamma ditribution and α m and βm are two generic parameter which depend from both the eniority and the time. Equation ha the required property that 0 and bound the recovery 27. If µ m = 3,73% and σ m =22,06%, a for the ubordinated non-invetment grade bond (ee table ), the pdf of the beta ditribution i depicted in figure. pdf (BEA) ubordinated non invet ment grade 0,02 probability (beta) 0,05 0,0 0, recovery % φ Figure Beta Ditribution he figure illutrate the high degree of randomne preent in recovery rate. he model objective i to develop a r neutral lattice for pricing ry debt. In order to render the forward interet and recovery rate proce tractable numerically, the correponding tate pace 28 could be dicretied conitently through both the (dicretied) time tructure, m, and through the hock vector v 29, for the r free term tructure proce, and z, for the recovery rate proce. o implement the model, we make the tandard dicrete time aumption that v and z are binomial random variable. In particular, we aume that both v and z take on the value ± with probability 0,5: + + v = ; + z = + (4.2..2) Conequently, dicretiing, the recovery rate vector, function of each eniority type at the given time m, become: 27 he parameter can be computed ince the mean µ m and variance σ 2 m of a beta ditribution are given by: µ α m m = and α m + β m αmβm σ = where µ m and σ m are the mean and tandard deviation of 2 m 2 ( α + β ) ( α + β + ) m m m m the actual empirical ditribution of credit recovery belonging to eniority debt type-. 28 he tate-pace i defined a the enemble of all poible tate related to the tochatic proce. 29 If, for example, the HJM model i ued to build the r neutral et up for the etimation of the r free term tructure, v repreent the random variable of the underlying tochatic proce, i.e.: f ( t + m, ) = f ( t, ) + a( t, ) m + σ ( t, ) v m, where α and σ repreent repectively both the drift and the volatility of the proce. In a dicrete time et up, period are taken to be of length m>0, thu a typical time point, t, ha the form lm for integer l. 2

13 φ m + φ m φm ( z) = ; =,...,5 + φm φm (4.2..3) From now on, to reduce the notational burden, we uppre the dependence from z and in the remainder we will conider φ (z) = φ m (m). We will remark the time dependence becaue we will allow in our model to chooe different beta ditribution parameteriation in different time, le for example, in different economic cycle. In a dicrete time et up, in order to conider a conitent and integrated r-neutral framework, it i neceary to correlate the tate pace recovery rate tructure with the forward rate term tructure at any given time. Let u define ρ a the (empirical) correlation between the term interet rate tructure and the recovery rate 30, the ( +, + ), ( +, ), with prob. aumed joint ditribution i: 4 f ( v, z) = (4.2..4) ρ (, + ), with prob. (, ), + ρ with prob. 4 ρ 4 + ρ with prob. 4 Moreover, let u define ρ a the r neutral probability vector 3 + ρ 4 ρ probabilitie of each branch of the lattice: ρ ' = 4. ρ + 4 ρ 4 collecting the tate For computational need and for notation eae, it i ueful to introduce another concept before getting the final explicit functional form for the forward pread: the tate-price 32. he tate price (denoted by the variable w(m)) at time m+ evaluated at time-m, i defined a the price at time-m time the r-neutral probability ρ of being in that tate at the time-m dicounted at the r-free interet rate, i.e.: w = ρ w[ ( m ) ] (4.2..5) + f m, m where the tate price are conidered a four-dimenional vector (correponding to the four poible tate defined by the double tochatic tructure) for each eniority and i the f m, m forward rate between time t=(m-) and time t=m. Both the interet rate term tructure and 30 he definition of the parameter ρ allow having one more degree of freedom, which enable to perform the proper recovery rate and interet rate correlation choice according to the overall economy time-cale conidered in the model. 3 he vector i r neutral by contruction having aumed that v and z take on the value ± with probability 0,5. 32 A pointed out by Da and Sundaram (2000) [29] State price are the current value of a ecurity that pay off a dollar in a ingle pecific tate in the future and zero in all other tate. For example, if there are only two poible tate ( up and down ) at the ame time in the future, then the tate price of the up tate would be the value of a dollar received in that tate time the r neutral probability of that tate, dicounted to the preent, uing a r-le dicount rate. State price are ueful ince they allow to compute the price of any ecurity by multiplying the payoff of the ecurity by tate price in each node (tate), and then add thee value up. Of coure at time 0 the tate price i imply unity. i.e. w(0)= 3

14 the recovery rate tructure are implied in the definition of the tate price, track of them can be found in the dicount and probability factor, repectively. he cah-flow at time tep-m i a function of recovery rate a well a tranition rate. While recovery rate are correlated to the r-neutral interet rate tructure, the tranition probabilitie have to be rendered r-neutral in order to preerve the overall framework conitency. For thi purpoe, a generally decribed in paragraph 4.2, it i neceary to introduce rating cla i and eniority pecific adjuting factor to the empirical tranition probabilitie defined for any time tep-m π i. Let u conider the one-period tranition from a generic time-m to time (m+); thi i performed by defining the unknown quantitieπ i referred to the i-th rating cla and to the -th eniority type 33 of the credit intrument at time tep-m. ( ) ( ) q m.... q m K { ( )} q q.. q Q = q m = K ij ( π ) q.. π q2 π q K q2 π ( π ) q22.. q2 π 2 2 K (4.2..6) where Q (m) i the r neutral repreentation of Q (m), when incorporating the eniority type effect in the tranition matrix by rating cla, a hown in the generic element i (m) q ij,which, by contruction, explicitly conider the adjutment factor π i m. Invoking the definition of tate price, for the credit intrument of eniority type- being in cla-i at time-m, the following condition, in a r neutral world, mut be atified: [ ] act w E C = act (4.2..7); where f i the actual forward interet in the period + fmm + mm between time-m and maturity (time-m), i the market pread and the expected cah flow at time-m for the bond of rating cla-i and eniority type- i determined by: q E C m = q m, q m,..., m,,..., φ (4.2..8) [ ( )] ( ) ( ) ( ) i [ ] [ ] i i2 Eq. (4.2..7) and Eq. (4.2..8) provide the olution for the unknown π m aociated to the rating cla-i and eniority type- by calibrating thoe equation with the (average) market pread of the conidered ry debt. In fact, making ue of imple algebra, it i poible to how that Eq. (4.2..7) i the generaliation of Eq. (4.2.5) when conidering the aumption of the uggeted integrated pricing model. Applying the above-mentioned market price calibration it i poible to find all the adjutment factor to get the r neutral tranition matrix for eniority type- at time t. Five tranition matrice for each rating cla correpondent to the five eniority type are generated. herefore the model can be plit into five parallel model yielding pecific information on the eniority for any rating cla, at any time tep. hi information i then embedded in the final expreion of the pread related to the eniority type. At thi tage it i important to oberve that, according to the data in table, default rate are not affected by the credit intrument dependence on eniority, while the recovery rate doe. Within thi unified r ( ) ( ) 33 One reaon behind the choice of K rating cla and eniority type i that there are well documented table of default frequencie for tandard rating but there i not enough data in all cae to ditinguih between different eniority type. Another reaon i that while rating are ubject to random change the eniority cla remain unchanged during the life of an aet 4

15 neutral framework it i poible to meaure the contribution of the eniority of a credit iue to the r neutral pread curve Multiple time horizon Up to now the attention wa focued on thoe variable, aumption and parameter that have a direct impact on credit r, without explicitly conidering the time at which thoe quantitie have been evaluated or defined. Another baic managerial apect of credit r i the time horizon of the r meaure. hi meaure of r of a financial intrument i a critical iue. In fact, in thi cae the problem of extrapolation, or interpolation, ha to be faced in order to achieve the correct etimation of migration probabilitie in the multiple time-horizon. Let u conider the one-period probabilitie q 0 ij a the probability of migrating from rating cla-i to rating cla-j in the time interval between time tep-(m-) and tep-m with repect to a generic recovery rate tructure. Actually, the probabilitie previouly conidered in the tranition matrice element at the generic time tep-m are regarded a cumulative probabilitie; the actual cumulative probabilitie are obtained by the one tep probabilitie by a recurive procedure. Under the aumption that the one period migration probabilitie at ubequent time tep are independent, it i poible to obtain the actual rating cla tranition probabilitie, from any cla-i to any cla-j, at the ubequent time tep by multiplying the actual migration matrix at time-m with the one at time m+. hi procedure may be applied recurively yielding for the actual tranition matrix at time. Applying the r-neutral adjutment procedure at any time tep a outlined in previou paragraph, the r-neutral tranition matrix at time period m+ i directly derived. It i important to point out that thi tructure allow embedding in any tranition probability at the given time-m all the information on tranition probabilitie at previou time tep (maintaining probabilitie independence), therefore the ingle one-period tranition probability q ij keep the information on q kl ( n) for all tate k,l and for all time tep-n (n<m). In particular, the default probabilitie q (m) contain the information on the previou time tep tranition probabilitie. hi feature ditinguihe the integrated model form the other reduced model outlined in ection,2 and 4. he tranition probability can change ignificantly over time. An invetment grade ha a higher chance of downgrade than of upgrade and vice vera (mean reverion in credit rating). hi mean that in the high rated firm tranition r (and default probability in particular) increae over time and, by contrat, high yield ry debt that do not default, are more lely to improve than deteriorate in credit quality, thu howing a decreaing default probability over time One-period and cumulative tranition probabilitie Before deriving the formula for the pread curve it i intereting to focu on tranition probabilitie. Provided that K rating clae are conidered, q0 i defined a the oneperiod default probability over the period from [(m-), m] aociated to the tate I and generic eniority, i.e. the probability of migration from the rating cla i to cla K (correponding to the default tate). he cumulative probability of default at the time period-m (t=m) i defined a q 34, and it i a function of the previou-time cumulative probability and the one-period default probability 35 a follow: q 0 = q m ) ) + [ q (( m ) ) ] q ( m (( ) (4.2.3.) 34 he previou time cumulative probability i the probability of having got default until the previou time tep. 35 he one-period default probability i the probability of getting to default between (m-) and m 5

16 Converely, the one period probability of default in the period indexed by m may be expreed q q (( m ) ) 0 a: q = ( ). q (( m ) ) hee definition are ueful to compute expected cah flow over time for a zero coupon ry bond. Since q > 0, and the cumulative probability of default mut be increaing: q (( m ), ) 0 q > ( ) then, default probabilitie lie in the range [0,] a required. In thi formaliation it i important to point out that by mean of the procedure outlined above, the r-neutral adjuted tranition probabilitie to default tranmit the information of all the actual tranition probabilitie. At thi tage all the information required for deriving the pread tructure a a function of it determinant ha been derived and may be embedded into the cah flow evaluation. With reference to Eq. (4.2.6) and Eq. (4.2..8), the expected cah flow at the m-th time period for the given eniority cla- in it explicit form i: E 0 [ C ] [ q (( m ) ) ] q ( m n) = φ ( ) d, which alo generalie Eq. (4.2.). A pointed out in the multiple time horizon approach, in the integrated model the one-period probabilitie are given by the firt tranition probability, and the cumulative probabilitie are derived recurively. he philoophy of the integrated model appear evident alo at thi tage ince the trict correlation between the underlying model tructure and the empirical data i aured at each tep of the formulation: theory and actual data are interwoven in order to aure adherence between the theoretical proce and the market dynamic. Recalling Eq. ( ) it i poible to rewrite Eq. (4.2.6) in the following way: t m = + m E exp t j = f i ( ) ; i ( t, j) C = exp f ( t, j) + p ( j) t j = i ( ) Making ue of both the definition of tate price and cah-flow in cae of default (ee Eq ) at any time-tep-m Eq. ( ) become: t m= + m w( m, n) n= [ q m 0 ] q m m n ( ) ) ( ) (, ) = exp ( f ( t, j) + p ( j) ) ; i φ ( ) i t j= 4.3 Spread term tructure by rating cla contingent to the eniority of debt he term tructure of forward credit pread etimation i the problem to be olved in lat tep of the proce. For any rating cla and eniority type the following pread, p, et i given { p } i ( t) i =,..., K ; =,...,5; t < (4.3.) In order to give the pread curve in it explicit form it i neceary to conider it integral formulation. he pread curve evaluated at time t for a given rating cla-i i defined by all pread computed at conequent time tep within the bond life-pan, pecifically in the time interval [t,]. Let u conider Eq. ( ) and define the integral pread curve S(µ,M) between the µ -th and the M-th period (correponding to any given time τ [t,] and maturity t=, repectively) 6

17 τ SPi, SPi Ln / m= τ / + = ( µ, M ) p ( j) t j = m w( m, n) n i 0 [ q ( m ) )] q ( m ) φ ( m, n) f ( τ, j) τ j = (4.3.2) In order to derive the pread curve at time tep-µ the following differential relation i ued τ pi pi ( µ ) SPi ( µ, M ) SPi ( µ, M ) (4.3.3) Finally, referring to Eq. (4.3.2) the forward interet rate pread i determined a: p i ( µ ) / m (, w m n m= Ln = / m w( m, n) m= τ / n= i =,..., K; =,...,5; t τ ) [ q ( m ) )] [ q ( m ) )] 0 q ( m ) φ ( m, n) 0 q ( m ) φ ( m, n) τ / + n f ( τ, j) f ( τ, j ) (4.3.4) τ τ j= j= he lat-period forward pread between time and time + relative to the i-th rating cla and adjuted for the eniority i denoted by p i ( ) = pi ( M ), by computing node- on the tree of the interet forward rate tructure, the pread i derived conidering the lat period expected cah-flow in cae of default without conidering previou cah-flow event. Referring to Eq. (4.3.4) it i traightforward to derive the lat-period pread a follow q ( + ) q ( ) exp [ pi ( )] = E ( φ ( + )), i (4.3.5) q ( ) 5. Application Our model require eaily available information a input, namely: the r-free yield curve, the term tructure of credit pread for each rating cla, the tatitical tranition matrix and both mean and tandard deviation of the recovery rate by eniority of debt. he mot important and ueful reultant model information i: r neutral tranition matrix and r neutral pread term tructure are both contingent on the eniority of debt. Moreover, the bivariate lattice, thorough which the SSRC and SSD ha been etimated, wa built by correlating rle interet rate and recovery rate thu conidering the integration between market and credit r. Uing thi information, the following product, among other, are priced by generating the neceary cah flow at each node on the lattice and dicounting the cah flow back by multiplication of the tate price to obtain preent value on plain-vanilla ry debt of any rating cla and any eniority of debt. Our model perform quite well to price rating-enitive debt ince the rating tranition matrix provide r-neutral information on rating change (adjuted to the eniority) which can be directly ued to generate cah flow at each node on the tree. For pread-adjuted note, the coupon may alo be indexed to the pread at each node, thi i achieved by computing the forward pread at each node on the lattice and, ince the price of the ry debt i known at each node, and o i the rle rate, it i quite imple to compute the credit pread at each node a well. It i alo poible to price pread option ince cah flow may be generated at each node by comparing the pread at the node with the tre rate. For total return wap, ince the price of any underlying ry bond i computable at each node on the tree, the total return on the bond may alo be eaily calculated. Although the model i rich and flexible enough to price many credit aet, both plain vanilla and derivative, we think i particularly appropriate to price defaultable loan and corporate bond. 7

18 6. Concluion he tochatic pread tructure model conidered within the integrated model allow taking into account effect due to rating tranition (including default event) and recovery rate depending on eniority. he overall procedure allow dicriminating the effect of the credit intrument belonging to a pecified rating cla at any given time; actually fixing the time tep in the forward interet rate term tructure, k- pread correponding to the defined rating clae are derived. More pecifically, thi model i aimed at computing the pread for credit intrument belonging to a defined rating cla and having a pecified eniority, o that to dicriminate the information relative on the given eniority. hi framework allow depicting the effect on pread curve due to the rating cla, and -for any given rating cla, the effect due to the different eniority type uing the r neutral arbitrage et-up. Further reearch on thi area will be devoted both on conidering the influence of the economic cycle and the upply/demand for defaulted aet on the etimation of recovery contingent to eniority and analye the tructural (firm related) interdependencie between recovery rate and default probability. Moreover the iue of default correlation and it impact on pricing ry debt hould alo be invetigated. Finally, from a practical point of view, there are at leat two other relevant iue that need to be carefully taken in conideration in future work, namely liquidity r and parameter calibration. Our intuition i that we need an integrated pricing and r model to exploit in a coherent framework the r and capital management banking problem. Reference [] Acharya V.V., Da S. R., Sundaram R. K., Pricing credit derivative with rating tranition, Financial Analyt Journal, (3)(2002) [2] Altman E.I., Caouette J.B., Narayanan P., Managing credit r: the next financial challenge, Ed. John Wiley & Son, Inc., 998. [3] Altman E.I., Pompeii J., Market ize and invetment performanceof defaulted bond and bank loan, Salomon Center NYU, [4] Altman E.I., Kihore V.M., Almot everything you wanted to know about recoverie on defaulted bond, Financial Analyt Journal, (6)(996) [5] Altman E.I., Onorato M., Patorello A., A general framework for credit r model, Working Paper, rondaim [6] Bale Committee on Banking Superviion, Credit r modelling: current practice and application, 999. [7] Bale Committee on Banking Superviion, Principle for credit r management, 999. [8] Bale Committee on Banking Superviion, A new capital adequacy framework, 999. [9] Black F., Derman E., and oy W., A one-factor model of interet rate and it application to treaury bond option, Financial Analyt Journal, ()(990) [0] Carty L.V., Lieberman D., Corporate bond default and default rate , Moody Invetor Service, Global Credit Reearch, Special Report, 996. [] Carty L.V., Lieberman D., Defaulted bank loan recoverie, Moody Invetor Service, Global Credit Reearch, Special Report, 996. [2] Carverhill A., A implified expoition of the Heath, Jarrow, and Morton model, Stochatic, 995, pp [3] Catle K., Keiman D. and Yang R. Suddenly tructure mattered: inight into recoverie from defaulted debt Standard & Poor, Credit Weekly, [4] Cooper I. A. and Mello A. S., Default r and derivative product, Applied Mathematical Finance (3)(996) [5] CreditMetric M, echnical document JP Morgan,

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