Discount Rate for Workout Recoveries: An Empirical Study*

Dscount Rate for Workout Recoveres: An Emprcal Study* Brooks Brady Amercan Express Peter Chang Standard & Poor s Peter Mu** McMaster Unversty Boge Ozdemr Standard & Poor s Davd Schwartz Federal Reserve Bank of Rchmond Abstract Banks must measure the loss arsng from counterparty default n order to acheve Advanced Internal Ratngs-Based (IRB) complance under the proposed Basel II mnmum regulatory captal framework. The dscount rate to be used on cash receved post-default s not agreed upon amongst practtoners and bankng supervsors. We revew alternatve extant proposals and develop a framework for choosng an approprate dscount rate contngent upon the rsk of the recovery cash flow. Emprcal results are presented by usng a comprehensve database of tradng prces and workout recoveres of both dstressed bonds and loans. We fnd that dscount rates vary sgnfcantly by ntal ssuer ratngs, whether or not the ndustry s n stress at the tme of default, relatve senorty to other debt and nstrument type. The conclusons are found to be robust to potentally confoundng determnants of dscount rate. Date: September 2006 * The vews and opnons expressed here are those of the authors, not necessarly the vews and opnons of the authors employers, Standard & Poor s, Federal Reserve Bank of Rchmond, the Federal Reserve Board of Governors, or the Federal Reserve System. ** Correspondence should be addressed to Peter Mu, DeGroote School of Busness, McMaster Unversty, 1280 Man Street West, Hamlton, Ontaro L8S 4M4, Canada, tel: 1-905-525-9140 ext. 23981, fax: 1-905- 521-8995, emal: mupete@mcmaster.ca. 1

I- Introducton Implementaton of the advanced nternal ratngs-based (IRB) approach under the Basel II mnmum regulatory captal framework has been a major area of effort for many fnancal nsttutons. Among the ssues debated n the ndustry s the approprate dscount rate to be used to estmate the loss gven default (LGD). Whch dscount rate to use on workout recoveres post-default, n order to arrve at the economc value of the loan at the tme of default, s a queston that s the subject of consderable dsagreement amongst practtoners and bankng supervsors. 1 Theoretcally, the dscount rate used n the estmaton of LGD should be commensurate wth the (opportunty) costs of holdng the defaulted asset over the workout perod, ncludng an approprate rsk premum requred by the asset holders. 2 Ths s also the typcal nterpretaton adopted by the practtoners and Basel. For example, the Basel document states that: when recovery streams are uncertan and nvolve rsks that cannot be dversfed away, net present value calculatons must reflect the tme value of money and a rsk premum approprate to the undversfable rsk. 3 Both practtoners and academcans approach the ssue from a mostly theoretcal standpont and several qute dfferent arguments have been made. The dscount rate used ranges from fundng rate, dstress loan rate, contract rate, current comparable market rates, opportunty cost of funds, cost of captal, an accountng conventon of the dscount rate, and the current rsk-free rate. 4 Moreover, t s rarely the case that practtoners attempt to assgn dfferentated dscount rates to dfferent nstruments accordng to the nherent degree of rskness. 5 Wthout an approprate rsk adjustment, one could have consstently over- (under-) estmated the LGD of nstruments wth low (hgh) recovery rsk. For an advanced IRB nsttuton, t can translate nto assgnng dsproportonately hgh (low) regulatory captal to nstruments wth low (hgh) recovery rsk. In ths study, we attempt to shed some lght on ths ssue by conductng an emprcal study utlzng Standard & Poor (S&P) s LossStats Database, whch s one of the largest databases commercally avalable capturng both workout recoveres and 1 LGD can be estmated ether by observng the market prce of the defaulted nstrument mmedately after default or by computng the present value of future workout recoveres. Snce the tradng of most defaulted bank loans n the secondary market s llqud, fnancal nsttutons typcally have to rely on the latter approach n assessng ther LGD. In the latter approach, LGD can be defned as the sum of the workout recoveres dscounted at the approprate dscount rate dvded by the total exposure at default. 2 In the context of ths study, the dscount rate s equvalent to the expected or requred rate of return of the asset holders (or nvestors of defaulted nstruments). These terms are used nterchangeably throughout the paper. 3 See Gudance on Paragraph 468 of the Framework Document, Basel Commttee on Bankng Supervson July 2005. 4 For example, see Maclachlan (2004) and Davs (2004). 5 In ths paper, we are focusng on LGD rsk rather than probablty of default (PD) rsk. The former measures recovery uncertanty after default has occurred. 2

defaulted loan and bond prces. We examne the relaton between the market prces and the actual workout recoveres of 1,128 defaulted loans and bonds from 1987 to 2005 ssued by 446 defaulted oblgors. We develop an estmaton methodology and estmate the dscount rates, whch are most lkely used at the tme of prcng by the market, reflectng LGD uncertanty for the defaulted loans and bonds. We solve for the dscount rate by nterpretng t as the expected return of an nvestment n the defaulted nstrument rght after default has occurred. Such an nvestor pays for the nstrument at the market prce observed mmedately after default n return for the future cash flows that could be realzed durng the workout process. In order to prce the nstrument approprately, the nvestor needs to assess the expected future recoveres and the requred rsk premum, whch s commensurate wth the LGD uncertanty. In ths study, we want to solve for the latter by estmatng the former. Expected future recoveres are, however, not observable. Nevertheless, f the market s ratonal, the expected recovery should be dentcal to the average of the actual realzed recoveres for a reasonably large and homogenous sample of default nstruments n terms of ther LGD uncertanty. Under ths assumpton, for each homogeneous segment of the dataset, we solve for the dscount rate that equates the dscounted average realzed recoveres to the average market prce. We segment the data accordng to whether the nstrument s secured or not, whether the nstrument s rated speculatve grade or not, the ndustry sector t belongs to, whether the nstrument defaulted durng the market-wde and ndustry-specfc stress perods or not, the exstence of debt above (DA) and debt cushon (DC) n the captal structure of the oblgor, and ts nstrument types. Through the emprcal analyss, we want to dentfy whch are the mportant determnants of the estmated dscount rates. We also provde the theoretcal explanatons together wth the related busness ntutons for the emprcal results. We attempt to valdate our results by also conductng the analyss at the sub-segment level. We ask ourselves queston lke: If we only consder Senor Unsecured Bonds, s the dscount rate of an nstrument defaulted durng an ndustry-specfc stress perod sgnfcantly dfferent from that defaulted outsde such perod? The sub-segment level analyss allows us to control for the potentally confoundng determnants of dscount rate. Furthermore, we confrm the robustness of our fndngs by conductng a multple regresson analyss, whch caters for the nteracton among all the sgnfcant determnants of dscount rates. For each segment, as data permt, we compute the pont estmate of the dscount rate together wth the correspondng confdence ntervals around t. These results can be used by practtoners n estmatng LGD based on ther nternal workout recovery data. For a quck reference, the most robust determnants together wth the pont estmates of dscount rates are presented n Table 1. More detaled nformaton s provded n the subsequent sectons regardng the defnton of segments, the estmaton methodology and the nterpretaton of the results reported n Table 1. 3

Table 1: Pont Estmate of Dscount Rate Pont Estmate of Dscount Rate (%) Overall 14.0 Investment Grade 22.8 Non-Investment Grade 6.4 Industry n stress 21.5 Industry not n stress 8.1 No Debt Above, Some Debt Cushon 21.2 No Debt Above and No Debt Cushon 0.9 No Debt Cushon, Some Debt Above 8.6 Some Debt Above and Some Debt Cushon 29.3 Bank Debt 13.3 Senor Secured Bonds 11.0 Senor Unsecured Bonds 27.5 Senor Subordnated Bonds 3.8 Subordnated Bonds 8.9 The followng further summarzes the results: (1) Our analyss suggests that whether the nstrument s nvestment grade (IG) or non-nvestment grade (NIG), whether t defaults durng an ndustry-specfc stress perod, the exstence of DA and/or DC, and nstrument type are mportant determnants of recovery rsk and thus dscount rate. Ther statstcal sgnfcances are confrmed n the subsequent sub-segment level analyss and the multple regresson analyss on the nternal rate of returns of defaulted nstruments. (2) Defaulted debts of an orgnally hghly-rated oblgor tend to be more rsky and command a hgher rsk premum. The mplct dscount rate of an nstrument defaulted from IG can be as much as four tmes hgher than that of an nstrument defaulted from NIG. Moreover, the fallen angles (.e. those are IG n the earlest ratng, whle NIG one year before default) appear to have the hghest uncertanty around ther expected recoveres and also the hghest expected return. (3) The estmated dscount rate of those debts defaulted durng ndustry-specfc stress perods s more than double of those defaulted outsde those perods. Moreover, our results suggest the ndustry-specfc stress condton s more mportant than the market-wde stress condton n determnng the approprate dscount rates. 4

(4) The sub-segment results and the regresson analyss do not support the Secured vs. Unsecured dfferentaton. It s however lkely to be due to data constrants. 6 The mpact of ndustry characterstcs (e.g. technology vs. non-technology) and market-wde stress condton are found to be nconclusve. (5) Durng market-wde stress perods when default rates are hgh, not only expected recoveres can be lower but also the dscount rate s larger, whch results n an even larger economc LGD. Ths s the dea that, durng the stress perods, not only the expected recovery goes down but also the recovery uncertanly ncreases. As a result, the correlaton between default rates and economc LGD would be more pronounced when ths ncrease n the dscount rate s accounted for durng market downturns. (6) Realzed recoveres have a very large dsperson. As a result, the mean LGD calculated from a small sample of workout recoveres s subject to a hgh level of error. When ths s the case, market prce appears to be a better alternatve to the dscounted workout recoveres n determnng the economc LGD. (7) In general, nvestors demand a hgher rate of return (.e. dscount rate of future cash flows) on defaulted nstruments wth a hgher LGD rsk, provdng emprcal support for an approprate rsk-return tradeoff. The sum of square errors of the realzed recoveres s found to be postvely related to the expected return on the defaulted nstruments. In the rest of ths paper, we frst ntroduce the dataset n Secton II and the proposed segmentatons together wth ther justfcatons n Secton III. In Secton IV, we explan the estmaton methodology n detal. We report and nterpret the emprcal results n Secton V where relevant busness ntutons are also dscussed. In Secton VI, we valdate our results by conductng sub-segment level analyss. We further examne the robustness of our conclusons by performng multple regresson analyss n Secton VII. Fnally, we conclude wth a few remarks n Secton VIII. 6 The data does not allow us to dstngush the level of securty, only the securty type. That s, we could have two loans lsted as secured, but the frst one may be 100% secured, whereas the second one may only be 5% secured. It s very possble that ths lmtaton weakens the sgnfcance of the Secured vs. Unsecured dfferentaton. 5

II-Data To estmate the dscount rates, we extract the market prces, workout recoveres and default rates data from the CredtPro and the LossStats Database, whch are lcensed by Standard & Poor s Rsk Solutons. LGD data LossStats Database ncorporates a comprehensve set of commercally assembled credt loss nformaton on defaulted loans and bonds. 7 Publc and prvate companes, both rated and non-rated, that have U.S.-ssued bank loans and/or bonds of more than ffty mllon dollars are analyzed and ncluded n the database. Fnancal, real estate, and nsurance companes are excluded. The companes must have fully completed ther restructurng, and all recovery nformaton must be avalable n order to be ncluded n the LossStats Database. The database contans recovery nformaton from 1987 through the second quarter of 2005. We choose the LossStats Database as a relable source of data due to ts unque feature n that t contans both the 30-day dstressed debt tradng prces and the ultmate recovery values of the defaulted nstruments, both of whch are requred n ths study. The 30-day dstressed debt tradng prces are smply the average tradng prces of the defaulted nstruments 30 days after the gven default events. In contrast, ultmate recovery s the value a pre-petton credtor would have receved had they held onto ther poston from the pont of default through the emergence date from the restructurng event. 8 A total of 1,128 defaulted nstruments wth both ultmate recovery values and 30-day dstressed tradng prces are ncluded n the analyss. 9 These nstruments are from 446 separate oblgor default events from 1987 to 2005, and from a varety of ndustres. Table 2 reports the breakdown of the dataset accordng to securty, S&P s ratng and nstrument type. 7 The October 2005 release v1.5 of the LossStats Database s utlzed for ths paper. 8 Pre-petton credtors are credtors that were n place pror to flng a petton for bankruptcy. 9 A total of 28 defaulted nstruments wth nternal rate of returns of more than 800% are removed from the orgnal dataset to ensure the results are not affected by these outlers. Moreover, among the remanng 1,128 defaulted nstruments, some of them potentally represent duplcated observatons referrng to an dentcal nstrument ssued by the same oblgor. To check for the robustness of our results, we repeat the analyss but wth all the potentally duplcated observatons excluded. The resultng dscount rates are not materally dfferent from those reported n ths paper. 6

Table 2: Composton of LGD Data Securty S&P s Ratng 10 Type Secured Unsecured 317 811 Investment Nonnvestment Others grade grade 88 398 642 Bank debt Senor Senor secured unsecured bonds bonds Senor subordnated bonds Subordnated bonds Junor subordnated bonds 222 102 395 237 161 11 Although the LossStats Database contans recovery nformaton on bankruptces, dstressed exchanges, and other reorganzaton events, we only take nto account companes, whch had gone through a formal bankruptcy. Ths helps ensure the consstency between the 30-day dstressed debt tradng prces and the ultmate recovery values. 11 Default rate data We use the commercally avalable CredtPro Database of ratngs performance statstcs to comple the aggregate default rates by ndustry. Ths nformaton s then used n segmentng the data n the subsequent analyss. CredtPro, a subscrpton database of S&P s corporate ratng hstores globally, allows us to produce the tme seres of default rates for Global Industry Classfcaton Standard (GICS) ndustry code classfcatons for S&P s-rated companes located n the U.S. 12 III-Segmentaton We consder several factors that may nfluence the approprate dscount rate to apply to post-default recoveres when estmatng economc LGD. In ths secton we descrbe the crtera we use to segment our dataset. Secured/Unsecured descrbes whether the nstrument s secured by collateral or not. 10 The classfcaton s based on the earlest S&P s ratngs of the nstruments. Instruments from oblgors that were not rated by S&P are classfed as Others. 11 Snce a dstressed exchange (exchangng an nstrument for an nstrument of lesser value), can be consdered a default event, and the completon of the exchange s consdered a recovery event, they would often occur on or close to the same date. Therefore f we were to nclude these cases and use tradng prces of the defaulted nstrument 30 days after the default event, all nformaton regardng the exchange would be known and there would be no uncertanty as to the recovery event. 12 The Global Industry Classfcaton Standard (GICS ) s an nternatonal ndustry classfcaton system. It was jontly developed by S&P's and Morgan Stanley Captal Internatonal (MSCI) n 1999, as a response to the global fnancal communty's need for one complete, consstent set of global sector and ndustry defntons. The hghest ndustry sectors of GICS are found to be broader than those of the correspondng SIC codes. 7

S&P s Ratngs: We segment our data based on whether the oblgor s rated BBB- and above (nvestment grade (IG)) or BB+ and below (non-nvestment grade (NIG)) by S&P. We consder segmentng accordng to the earlest S&P ratng and the S&P ratng one year pror to default respectvely. S&P s oblgor ratngs ndcate the amount of certanty that credtors have n ther ablty to receve all clams n full, meanng that companes wth hgher credt ratngs would be less lkely to default. Clams on ssuers wth hgher ratngs may lack provsons that would provde credtors wth hgher recovery rates; however, hghly-rated companes may also have more relable busness plans that would lead to a hgher prce on ext from default. GICS Industry Codes descrbe the ndustry groupng to whch each oblgor belongs. Industry may affect expected recoveres because nvestors expect companes n certan ndustres to have a greater ablty to delver on a post-default busness plan or to produce superor recoveres to other ndustres. GICS Groupngs are broad ndustry groupngs that combne varous 2-dgt GICS codes. We aggregate 2-dgt GICS n order to produce ndustry groups of suffcent sample sze for statstcal analyss. In the subsequent analyss, we focus our attenton on the dfference between the technology and nontechnology sector. Default n Market-Wde Stress Perod descrbes whether the oblgor defaults durng a market-wde stress perod or not. In ths study, market-wde stress perods are defned as those years (1986, 1990-1992 and 1999-2003) where the speculatve grade default rates are greater than ts 25-year average of 4.7%. Market-wde stress perods may be flled wth more uncertanty and pessmsm, whch would drve down the post-default tradng prce as well as ncrease the expected dscount rates for post-default recoveres. It s also plausble that due to the correlaton between probablty of default (PD) and LGD, oblgors asset and collateral values are depressed durng perods of hgh default rates. Default n Industry-Specfc Stress Perod descrbes whether the oblgor defaults durng those perods where the oblgor s ndustry (based on the GICS code) experences a speculatve-grade default rate of more than 5% (whch s the long-run average default rate for all speculatve-grade oblgors n the U.S.). In combnng wth the prevous consderaton of market-wde stress perod, we can therefore examne whether t s ndustry-specfc or market-wde downturn, whch drves the potentally lower expectatons of future recoveres. Debt Above (DA) and Debt Cushon (DC) descrbe whether there s debt that s superor to or subordnated to each bond and bank loan for the gven default event. DA s the sum of the amount of debt nstruments whch are contractually superor to the nstrument that s beng analyzed dvded by total amount of debt. Ths s n contrast to DC, whch s the sum of the amount of debt nstruments whch are contractually nferor to the nstrument that s beng analyzed dvded by total amount of debt. Due to the varablty of the captal structure of defaulted oblgor, by defnng accordng to ts DA and DC, nstrument can be more readly compared than by classfyng accordng to the name of the nstrument alone (e.g. Senor Secured Bonds). In ths study, we segment our 8

sample nto: (1) those wth no DA and some DC; (2) those wth no DA and no DC (3) those wth no DC and some DA; and (4) those wth some DA and some DC. Instrument Type groupngs are based on the legal descrpton of the nstrument. Instrument type s frequently used n practce n classfyng nstruments for LGD assessments. Smlar to DA and DC, nstrument type provdes nformaton about the senorty of the credtor wthn the lst of clamants. However, the same nstrument type can represent very dfferent nstruments when we compare oblgors wth very dfferent debt structure. For example, a subordnated bond ssued by an oblgor havng only a sngle class of debt may have a much lower recovery rsk than a subordnated bond ssued by another oblgor whch also ssues senor bonds. Classfyng by DA and DC s consdered to be a more approprate way to control for any dfferences n debt structure across oblgors. In ths study, we consder the nstrument types of: (1) Bank Debt (2) Senor Secured Bonds, (3) Senor Unsecured Bonds, (4) Senor Subordnated Bonds, (5) Subordnated Bonds, and (6) Junor Subordnated Bonds. IV-Estmaton of Dscount Rate We estmate the dscount rate for each segment by modelng t as the expected rate of return on the nvestments n defaulted nstruments belongng to that segment. We assume all nstruments wthn a partcular segment are dentcal n terms of ther LGD rsk and thus share the same expected rate of return. The fact that the realzed recovery turns out to be dfferent from the expected recovery s solely because of LGD uncertanty durng the recovery process. Specfcally, consder an nstrument defaults at tme t D, whle a sngle recovery cash flow R s realzed at tme t R. We observe the market prce ( P ) of ths defaulted nstrument at 30 days after t defaults. If we use d to denote the expected return (.e. the dscount rate) on nvestng n ths defaulted nstrument, the prce of the nstrument s the expected recovery dscounted at d, whch s the parameter we want to estmate n ths study. E[ R ] R t 30 ( 1+ ) t d P (1) = D Expected recovery, E[R ], s however not observable. We can only observe the realzed recovery, R. Because of LGD uncertanty, we do not antcpate the expected and realzed recovery to be dentcal unless by concdence. However, f market s ratonal, the dfference between the expected recovery and the average realzed recovery should be small for a suffcently large enough group of nstruments that are homogeneous wth respect to LGD rsk. We can formalze ths dea as below. Let ε represent the dfference between the expected and realzed recovery, beng normalzed by the prce of the nstrument, P. 9

R E [ R ] ε = or equvalently (2) P t ( 1+ d ) R D t 30 R P ε = (3) P It can be nterpreted as the unexpected return of the defaulted nstrument. It s, therefore, a unt-free measure of recovery uncertanty. However, before we can solve for the dscount rate, we also need to model for the fact that recovery uncertanty can be a functon of the tme-to-recovery. As suggested by Mu and Ozdemr (2005), recovery uncertanty s lkely to be an ncreasng functon of the tme-to-recovery snce more and more nformaton on LGD s revealed over tme (.e. less recovery uncertanty) as we approach the tme when recovery s fnally realzed. In ths study, we assume the standard devaton of ε s proportonal to the square root of tme-to-recovery (.e. R t D t 30 days). Ths assumpton s consstent wth an economy where nformaton s revealed unformly and ndependently through tme. 13 Wth ths assumpton and f the R market s ratonal, ε t t D 30 days should be small on average across a homogenous group of defaulted nstruments. The most-lkely dscount rate ( dˆ ) can R therefore be obtaned by mnmzng the sum of the square of ε D t t 30 days of all the defaulted nstruments belongng to a segment. 14 It serves as our pont estmate of the dscount rate for that partcular segment. Formally, N ˆ arg mn = 1 30 days ε d = R D t t 2 (4a) where N s the number of defaulted nstruments n that segment. A by-product of the above algorthm s the mnmum sum of the squares of ε (SSE) normalzed by the square root of the tme-to-recovery. It can be nterpreted as a standardzed measure of the recovery uncertanty (.e. LGD rsk) of that specfc segment. The dscount rate estmate obtaned by solvng equaton (4a) s n fact the maxmum lkelhood estmate. That s, 13 Gven the fact that the tme-to-recovery of the nstruments n our dataset ranges from a few days to more than a couple of years, the results can be sgnfcantly based f we gnore the relaton between LGD uncertanty and tme-to-recovery. If we had assumed no tme adjustment, the dscount rates would have been slghtly dfferent from those presented here. However, the conclusons drawn n ths study are robust whether we adjust for tme or not. 14 Ths sum of the squares of errors (SSE) of recovery returns can be used to proxy for LGD rsk for each segment. An approprate rsk-return tradeoff suggests the dscount rate (.e. expected return) should be hgh when SSE (.e. LGD uncertanty) s hgh. 10

N d ˆ = arg max L = ε = arg max log φ (4b) R D 1 t t 30 days where L s the log-lkelhood functon and φ ( ) s the probablty densty of the standard normal dstrbuton. We can also estmate the asymptotc standard devaton σ of dˆ by dˆ evaluatng the second dervatve of L wth respect to d at the pont estmate dˆ. 2 0.5 d L = ˆ 2 d dd d = d ˆ σ (5) We can therefore establsh confdence nterval around our pont estmate of dscount rate. For example, the confdence nterval at the 90% confdence level s equal to: 15 dˆ ± 1.6449 σ (6) dˆ The above methodology can be readly extended to nclude default nstruments wth multple recoveres. Suppose there are a total of m recovery cash flows ( R, 1, R, 2,, R, j,, R, m ) whch are realzed at tmes R R t, 1, t, 2,, t R, j, and t R, m respectvely for defaulted nstrument. The percentage dfference between expected and realzed recovery cash flow j s therefore equal to: 16 where m = j= 1 t ( 1+ d ), j R D, j t 30 R, j p, j ε, j = (7) p P p, j. The most-lkely dscount rate (.e. our pont estmate of the dscount rate) can therefore be solved by: dˆ = arg mn N m t t ε, j 30 days R D = 1 j= 1, j where the outer summaton s across all N defaulted nstruments of the segment. The correspondng asymptotc standard devaton and confdence nterval of the pont estmate can also be computed n a smlar fashon as outlned above. 2 (8) 15 Note that (1-2 (1-Φ(1.6449))) = 90%, where Φ s the cumulatve standard normal dstrbuton functon. 16 p, are estmated n an teratve fashon such that they are proportonal to the present values of the j realzed recovery cash flows majorty of the default nstruments have only sngle recovery. R, dscounted at the most-lkely dscount rate dˆ. In our sample, the j 11

V-Dscount Rate In ths secton, we report the dscount rates estmated for each segment usng the methodology outlned above. We also provde some nterpretatons on the results. It should be noted that the ultmate recoveres obtaned from LossStats Database are gross recoveres rather than net recoveres. That s, any drect and ndrect collecton costs such as legal fees, overheads and other collecton expenses typcally encountered by a fnancal nsttuton are not yet subtracted from the ultmate recoveres. For the purpose of measurng economc LGD, banks should apply the dscount rate on ther net recoveres rather than gross recoveres. By conductng our analyss usng gross recoveres, we are therefore overestmatng recoveres and thus the dscount rate to be used for the above purpose. As a result, f a fnancal nsttuton apples the dscount rates estmated n ths study to ther net recoveres, the resultng LGD would be somewhat conservatve. 17 Before reportng the dscount rates, we present the hstogram (Fgure 1) of the nternal rate of returns of all ndvdual nstruments wthn our sample. Fgure 1: Hstogram of Internal Rate of Returns of All Instruments 140 120 100 Frequenc 80 60 40 20 0-95% -75% -55% -35% -15% 5% 25% 45% 65% 85% 105% 125% 145% 165% 185% 205% 225% 245% 265% 285% IRR The negatve rates presented n Fgure 1 may appear to be counterntutve, ndcaton of non-economc behavour. Ths s not actually the case, as the negatve return realzed 17 As an ndcaton, usng the full dataset, when a flat $5 (.e. 5% of notonal), representng the collecton cost s subtracted from the ultmate undscounted recoveres, overall dscount rate decreases from 14.0% to 12.1%. It s more approprate to assgn a cost of collecton as a fxed % of the notonal amount (.e. $1,000) rather than a fxed % of the actual recovery n order to accommodate zero recoveres. 12

may smply be due to recovery uncertanty. In practce, we expect to observe negatve return on some nvestments. A negatve return does not necessarly suggest that the nvestor over-estmated the value of the asset, but nstead she may smply be unlucky. Investors prce the defaulted nstrument based on the expected value over a dstrbuton of possble recovery cash flows. A negatve rate of return may then be realzed f the actual cash flows turn out to be on the "loss" sde of the dstrbuton The nternal rates of returns have a large varaton as observed from Fgure 1. It s a bmodal dstrbuton wth a thck rght-hand tal. The dstrbuton s truncated at -100% and at the same tme very large postve returns wth low frequency are observed, suggestng an opton-lke behavour of the defaulted assets. If an nvestor purchases a defaulted nstrument randomly mmedately after default and holds t untl t emerges, there s about a 43% chance that the nvestment wll generate a negatve return, whle a 5% chance that she cannot recover anythng from her nvestment. The potental proft however can also be very substantal. Some banks, n ther estmaton of LGD, rely on both workout recoveres and market prces, and they consder them as substtutes to each other. 18 If the bank s prmary practce s to sell ts defaulted assets, LGD estmaton needs to be based on market prces. However, f t s the bank s polcy to work out ts defaulted assets, LGD estmaton needs to be based on dscounted workout recoveres. Fgure 1 suggests cauton needs to be pad as realzed recoveres have a very large dsperson; as a result, the mean LGD calculated from a small sample of workout recoveres s subject to a hgh level of error. Dscountng workout recovery can therefore only serve as a good substtuton of the respectve market prce when we have a suffcently large and homogeneous group of facltes. The average of dscounted workout recoveres, computed from ndvdual or a small group of defaulted nstruments s unlkely to be an accurate representaton of the economc value of LGD. Before we present the dscount rates obtaned by segmentng the dataset n dfferent ways, we report n Table 3 the Overall dscount rate estmated by usng the whole dataset. The most-lkely estmate s obtaned by usng the estmaton methodology outlned n the prevous secton. We also report the correspondng asymptotc standard devaton and 90% confdence nterval of the estmate. The results n Table 3 suggest that the dscount rate s most-lkely to be 14.0% f we assume all defaulted nstruments are homogenous n ts recovery rsk. There s a 90% chance that the actual dscount rate les between 11.1% and 16.9%. These results can therefore serve as benchmarks when we consder the dscount rates obtaned for ndvdual segments. 18 Some banks only use market prces for those defaults that are stll beng worked-out and consder them as proxes for the ultmate recoveres. 13

Table 3: Overall Dscount Rate n Percent Most Lkely Estmate Standard 90% Confdence Interval Devaton Lower Upper Overall 14.0 1.8 11.1 16.9 Secured vs. Unsecured The results n Table 4 suggest unsecured debts have hgher dscount rate than secured debts, whch s as expected. Unsecured defaulted debts are rsker than secured ones, as credtors wll only get what s left after secured are pad based upon the absolute prorty rule, therefore resultng n greater uncertanty to the recovery, and thus nvestors requre a larger rsk premum. 19 Not only are the recoveres more uncertan (total rsks) for unsecured debts, there s also lkely to be a larger component of systematc rsks. For example, the fndngs of Araten, Jacobs Jr. and Varshney (2004) suggest LGD of unsecured loans s more correlated wth PD (and thus the general economy) than LGD of secured loans. The dea of unsecured defaulted debts beng rsker than secured debts s also supported by examnng the realzed annualzed returns (.e. nternal rate of return) of ndvdual nstruments. The hstogram reported n Fgure 2 suggests t s more lkely for the unsecured defaulted debts to generate sgnfcantly negatve return (less than -25%) and sgnfcantly postve return (more than 75%) than the secured ones. Table 4: Dscount Rate n Percent: Secured vs. Unsecured Instruments Most Lkely Estmate Standard 90% Confdence Interval Devaton Lower Upper Secured 11.8 4.8 3.9 19.7 Unsecured 14.3 1.9 11.2 17.4 19 The absolute prorty rule however may not necessarly always be observed n practce. 14

Fgure 2: Hstogram of Internal Rate of Returns: Secured vs. Unsecured Dstrbuton of IRR (Secured vs. Unsecured) 30% 25% Secured Unsecured 20% 15% 10% 5% 0% -100% to -75% -75% to -50% -50% to -25% -25% to 0% 0% to 25% 25% to 50% 50% to 75% 75% to 100% IRR 100% to 125% 125% to 150% 150% to 175% 175% to 200% 200% to 225% Despte of the strong ntuton and the evdence from the earler studes, the wde confdence ntervals reported n Table 4 ndcate that the results are not as strong as expected (refer to secton VI for further dscusson). Ths s most lkely due to the fact that the data does not allow us to dstngush the level of securty, only the securty type. That s, we could have two loans lsted as secured, but the frst one may be 100% secured and where as the second one may be only 5% secured. It s lkely that ths weakens the dfferentaton between the secured and unsecured credts. The dfference between the hghly secured nstruments and unsecured nstruments mght be sgnfcantly more pronounced f we were able to account for the exact degree of securtzaton. Investment Grade (IG) vs. Non-Investment Grade (NIG) From Table 5, the estmated dscount rate of IG s much larger (almost 4 tmes) than that of NIG based on the earlest S&P ratng. Non-overlappng confdence ntervals ndcate the power of dscrmnaton. Ths fndng s also found to be robust when we consder sub-segment results (refer to secton VI for further dscusson). It s actually not surprsng that a hgher level of recovery rsk s mplct n an IG nstrument rather than a NIG one. Snce t s hghly unantcpated that an IG debt defaults, the fact that t defaults ncreases the rskness of the dstress nstruments. Moreover, credtors of an orgnally hghly-rated oblgor are much less concern about ts LGD when the oblgor s stll a gong concern snce ts default rsk s perceved to be slm. They are lkely to pay more attenton n montorng and mtgatng the LGD rsk of lowly-rated oblgors rather than hghly-rated ones. As a result, defaulted debts of an orgnally hghly-rated oblgor tend to be more rsky and command a hgher rsk premum. 15

Table 5: Dscount Rate n Percent: IG vs. NIG Most Lkely Estmate Standard 90% Confdence Interval Devaton Lower Upper IG (earlest) 22.8 5.0 14.6 31.0 NIG (earlest) 6.4 3.8 0.2 12.7 We also examne f the ratng hstory has any bearng on the dscount rate by segmentng the data jontly accordng to whether they are nvestment grade n the earlest S&P ratngs and n the S&P ratngs assgned one year before the respectve default events. The results are reported n Table 6. Table 6: Dscount Rate (n Percent) of Dfferent Ratng Hstores. The correspondng 90% confdence ntervals are presented n brackets. Most-Lkely Dscount Rate (%) IG (earlest) NIG (earlest) 14.9 N/A IG (1 yr before) (3.6-26.2) NIG (1 yr before) 34.8 (21.1 48.6) 5.3 (0.0 12.1) Note: Wthn our dataset, there are only three observatons whch are of NIG accordng to ther earlest ratngs and subsequently become IG one year before the respectve default events. Results are therefore not reported for ths case. The fallen angles (.e. those rated as IG n ther earlest ratngs whle subsequently becomng NIG one year before ther defaults) appear to have the hghest uncertanty around ther expected recoveres and thus requre the hghest dscount rate. Fallen angels havng hgher recovery rates than orgnal ssued speculatvely-rated debts may be explaned by the nature of fallen angels. That s, they are oblgors that used to have hgh credt qualty before they defaulted. Some of those advantageous qualtes may carry over nto the default work out process. Addtonally, when fallen angels default, market partcpants may be less lkely to nvest n ther defaulted debt due to the extreme change n the fortunes of the oblgor, thus demandng a larger dscount rate than an otherwse dentcal oblgor whch has always been rated as NIG. It s nterestng that fallen angels seem to have a combnaton of hgh dscount rates and hgh recovery rates. These fndngs are n lne wth Vazza, Aurora and Schneck (2005) whch note that the postdefault experence among nvestors s more favorable for fallen angels n the aggregate than for ther counterparts. Even after adjustng for securty type, fallen angels show more favorable recovery characterstcs 16

Industry Our ntuton s that, those ndustres wth exposures typcally collateralzed by tangble assets and real estate - of whch values can readly be determned would requre a lower dscount rate than those wth exposures collateralzed by assets of whch values are more uncertan. We study the ndustry effect by classfyng the oblgors n the dataset accordng to ther GICS. We frst compare technology based aganst non-technology based ndustres. 20 We expect defaulted nstruments ssued by the technology sector to have hgher recovery uncertanty snce they are collateralzed by typcally ntangble assets. An orgnally secured nstrument can become essentally unsecured when the collateral loses ts perceved value. As an example the supposedly fully secured credtors nvolved n the Wnstar Communcatons, Inc. s 2001 bankruptcy, saw ther postons practcally beng wped out. As the overcapacty n the telecom ndustry leads to a large devaluaton of the underlyng collateral, the credtors saw ther collateral values declne along wth the rest of the telecom ndustry, essentally makng the secured credtors nto unsecured credtors. The dscount rates reported n Table 7 substantate the above hypothess. The estmated dscount rate of defaulted nstruments of the technology ndustres s hgher than that of the non-technology ones. Table 7: Dscount Rate n Percent: Technology vs. Non-Technology Most Lkely Estmate Standard 90% Confdence Interval Devaton Lower Upper Technology 24.4 5.8 14.8 34.0 Non-Technology 13.0 1.9 9.8 16.2 The dea that defaulted nstruments of technology based ndustres should be rsker than those of non-technology based s also supported by examnng the nternal rate of return of ndvdual defaulted nstruments. The hstogram reported n Fgure 3 suggests t s more lkely for the former to generate sgnfcantly negatve return (less than -75%) and sgnfcantly postve return (more than 50%) than the latter. 20 In ths study, Informaton Technology (GICS 40-45) and Telecommuncaton Servces (GICS 50) are consdered technology based ndustres, whle others are non-technology based. 17

Fgure 3: Hstogram of Internal Rate of Returns: Technology vs. Non-Technology 35% Dstrbuton of IRR (Technology vs. Non-technology) 30% Technology Non-technology 25% 20% 15% 10% 5% 0% -100% to -75% -75% to -50% -50% to -25% -25% to 0% 0% to 25% 25% to 50% 50% to 75% 75% to 100% IRR 100% to 125% 125% to 150% 150% to 175% 175% to 200% 200% to 225% The overlappng confdence ntervals reported n Table 7 however ndcate that the ndustry effect s not as strong as the other factor consdered n ths study (refer to secton VI for further analyss of the related sub-segment results). We further examne the effect of collateral values by breakng down the non-technology sector nto further sub-sectors, namely Energy & Utltes, Consumer Staples/Materals, but the results (not reported) are found to be nconclusve. Default durng Market-Wde Stress Perod In Table 8, we compare the estmated dscount rate of those nstruments defaulted durng the market-wde stress perods wth those defaulted outsde the stress perods. In the last two columns of Table 8, we also decompose the estmated dscount rates nto ther respectve rsk-free nterest rates and rsk premum components. To compute the mplct rsk premum, we subtract from the respectve dscount rates the average yelds of the 1- year US Treasures, whch serve as proxes for the rsk-free rates over the respectve perods. 21 As expected, gven the lower rsk-free rate durng an economc downturn, the dfference n rsk premums between the two states of the economy s even larger than that of the estmated dscount rates. 21 Monthly US Treasury yelds from 1987 to 2005 are obtaned from the US Federal Reserve Board. 18

Table 8: Dscount Rate and Decomposton nto Rsk-Free Rate and Rsk Premum (n Percent): Market-Wde Stress Condton Estmated Dscount Rate Decomposton Most Lkely Standard 90% Confdence Interval Ave. US Treasury Implct Rsk Estmate Devaton Lower Upper Yeld Premum In Market-Wde Stress Perod 15.7 4.2 8.8 22.6 4.4 11.3 Not n Market-Wde Stress Perod 13.6 2.0 10.3 16.9 5.5 8.1 The fndng that nvestors requre a larger rsk premum durng a stress perod s consstent wth the fact that LGD rsk s hgher durng the recessonary perod. For example, the emprcal studes of Araten, Jacobs Jr. and Varshney (2004) and Acharya, Bharath and Srnvasan (2003) document larger LGD varaton durng stress perods. The hgher rsk can be due to the fact that uncertanly around the values of the collaterals and the defaulted companes assets may ncrease durng a market-wde stress perod. Ths fndng also has nterestng mplcatons wth respect to the correlaton between default rates and LGD. A postve correlaton s already documented by, for example, Araten, Jacobs Jr. and Varshney (2004) and Mu and Ozdemr (2006), whch s prmarly due to the fact that durng the perod of hgh defaults, recoveres tend to be low as assets and collateral values are depressed. These studes use a sngle dscount rate to dscount the workout recoveres for all defaulted nstruments and thus the dscount rate effect studed here s not accounted for. Unlke these studes, we can attrbute the postve relaton between default rate and LGD to the varaton of rsk premum across the dfferent states of the economy. In other words, our study shows that durng recessonary perods when default rates are hgh, not only expected recoveres may be lower but also the dscount rate s larger, whch results n an even larger economc LGD. Ths s the dea that, durng the recessonary perods, not only the expected recovery decreases but the recovery uncertanly also ncreases. As a result, the correlaton between default rate and economc LGD would be more pronounced when the ncrease n dscount rate documented n ths study s accounted for durng market downturns. The hgher dscount rate durng a stress perod s also consstent wth the theory of excess supply of defaulted debts durng a market-wde stress perod. Altman et al (2005) suggests that the ncrease n LGD durng recessonary perods s manly due to an ncrease n the supply of default nstruments. Judgng from the overlappng confdence ntervals reported n Table 8, the dfference n dscount rates between the market-wde stress and non-stress perods s, however, not strong. Specfcally, the effect s much weaker than that between the ndustry-specfc stress and non-stress perods as reported subsequently. It, therefore, suggests the ndustry-specfc effect s a more mportant determnant of the expected return than the 19

market-wde effect. Ths weaker market-wde effect s also documented n secton VI when we conduct the sub-segment analyss. Default durng Industry-Specfc Stress Perod In Table 9, we compare the estmated dscount rate of those nstruments defaulted durng the ndustry-specfc stress perods wth those defaulted outsde the stress perods. Industry-specfc stress perods are defned as those perods when the ndustry experences a speculatve-grade default rate of more than 5%. The estmated dscount rate durng the ndustry-specfc stress perod s found to be more than double of that when otherwse. Judgng from the non-overlappng confdence ntervals, the dfference s hghly statstcally sgnfcant. The result s also found to be robust when we conduct the analyss at the sub-segment level n secton VI. Table 9: Dscount Rate n Percent: Industry-Specfc Stress Condton Most Lkely Standard 90% Confdence Interval Estmate Devaton Lower Upper In Industry-Specfc Stress Perod 21.5 2.7 17.1 25.8 Not n Industry- Specfc Stress Perod 8.1 3.0 3.1 13.1 The dea that defaulted nstruments durng an ndustry stress perod are rsker than those not n a stress perod s also supported by examnng the nternal rate of return of ndvdual defaulted nstruments. The hstogram reported n Fgure 4 suggests t s more lkely for the former to generate sgnfcantly negatve return (less than -50%) and sgnfcantly postve return (more than 50%) than the latter. 20

Fgure 4: Hstogram of Internal Rate of Returns: Industry-Specfc Stress Condton 35% Dstrbuton of IRR (ndustry stress perod vs. not ndustry stress perod) 30% Industry stress perod Not n ndustry stress perod 25% 20% 15% 10% 5% 0% -100% to -75% -75% to -50% -50% to -25% -25% to 0% 0% to 25% 25% to 50% 50% to 75% 75% to 100% IRR 100% to 125% 125% to 150% 150% to 175% 175% to 200% 200% to 225% Comparng the statstcal sgnfcance of the results reported n Table 8 and 9, we can conclude ndustry-specfc stress condton s more mportant than market-wde stress condton n governng the expected return on dstress debt. Ths s consstent wth the emprcal fndngs of Acharya, Bharath and Srnvasan (2003) and the theoretcal model proposed by Shlefer and Vshny (1992). Shlefer and Vshny suggest fnancal dstress s more costly to borrowers f they default when ther compettors n the same ndustry are experencng cash flow problems. It therefore supports the concept that there are cross ndustry dversfcaton effects n play, meanng that dfferent ndustres have dfferent LGD cycles. Another plausble argument for ndustry undergong stress stuaton leadng to greater LGD uncertanty s that the uncertanly around the values of the collaterals ncreases durng an ndustry-specfc stress perod, assumng that collaterals are mostly ndustry specfc, for nstance, fber-optc cable for telecom sectors. The observaton that ndustry-specfc stress condton s more mportant than marketwde stress condton has an nterestng mplcaton on the computaton of captal requrement. Under Basel II Advanced IRB approach of regulatory captal calculaton or under an economc captal framework, banks need to estmate LGD by dscountng the workout recoveres at the approprate dscount rate. Captal s estmated at the extreme confdence nterval, correspondng to extreme events, whch s most lkely to occur n 21

stress perods. To fulfll ths objectve, the approprate dscount rate s arguably the one estmated under the market-wde stress perod documented n ths study. However, an argument can be made that the dversfcaton effects across dfferent ndustres should also be consdered, snce not all ndustres are lkely to be under stress stuaton at the same tme. Nevertheless, a captal event would be an extreme stress scenaro causng a bank's captal to be wped-out by credt losses. Durng such an extreme event, t s lkely that many, maybe most, ndustres would be experencng stress stuaton at the same tme. Snce we are extractng the dscount rate by usng the market prce of the defaulted nstrument, our dscount rate mght also ncorporate any short-term effects and rsk measures specfc to the secondary market of dstressed nstruments. Namely, as suggested by Altman et al (2005), the excess supply of defaulted nstruments durng a dstressed perod can by tself exert a downward pressure on the market prce and thus translate nto a further ncrease n the mplct dscount rate. For those banks wth a polcy to always work out ther defaulted assets rather than sellng them n the secondary market, they would not need to concern themselves wth ths short term demand/supply ssue n the secondary market and, thus, may need to adjust the dscount rates reported n the study before usng them n ther calculatons of the economc values of LGDs. A bank whch does not sell dstressed assets should not be expected to hold captal aganst the market rsk that the prces of dstressed nstruments n the secondary market are depressed by oversupply durng a market downturn. DA and DC From Table 10, the prorty of the nstruments on the balance sheet as measured by DA and DC appears to be an mportant drvng factor n determnng the requred dscount rate. Specfcally, judgng from the pont estmates and ther confdence ntervals, those nstruments wth some DC (whether wth or wthout DA) have statstcally sgnfcantly hgher requred dscount rate than those wthout DC (agan, whether wth or wthout DA). Ths effect of nstrument structure on recovery s consstent wth the fndngs of VandeCastle (2000). Ths result s also found to be robust when we conduct the subsegment analyss n Secton VI. Table 10: Dscount Rate n Percent: DA vs. DC Most Lkely Estmate Standard 90% Confdence Interval Devaton Lower Upper No DA, Some DC 21.2 3.7 15.1 27.3 No DA and No DC 0.9 7.9-12.1 13.8 No DC, Some DA 8.6 3.0 3.7 13.6 Some DA and Some DC 29.3 4.0 22.7 35.8 22

The amount of debt cushon and collateral securng a poston not only has an effect on the expected recovery rate. The results n Table 10 suggests that knowng the debt structure of the gven oblgor and the poston of the nstrument beng analyzed n ths structure may also shed some lght on the amount of uncertanty surroundng the expected recoveres. If a gven nstrument has no DA or DC, ths would mean that there are no other bonds and loans wth ether hgher or lower prorty on the balance sheet. All credtors therefore share equally n the underlyng assets securng the postons; resultng n a farly predctable recovery rate. As expected, we observe the lowest requred dscount rate for nstruments wth no DA and no DC. Followng the same argument, those nstruments wth some DA and some DC wll have the most uncertanty surroundng the expected recovery, as there are both senor and junor postons who wll be vyng for a porton of the defaulted oblgors assets. As the credtors do not know how much they need to pay the subordnated credtors n order to move the debt restructure plan forward, or how much the senor credtors agree to pay them for the same reason. It s also consstent wth the dea suggested by Acharya, Bharath and Srnvasan (2003) that more demandng the coordnaton effort among credtors n the stuaton of larger "debt dsperson" leads to hgher bankruptcy and lqudaton costs, thus translatng nto a hgher requred rate of return. As expected, ths effect s found to be even stronger f we confne ourselves to only unsecured nstruments. The estmated dscount rate of those unsecured nstruments wth some DA and some DC s found to be equal to 32.1% (refer to the sub-segment results reported n Table 12 of secton VI). For those nstruments belongng to the categores of No DA, Some DC and No DC, Some DA, the estmated dscount rates are n between those of the pror two categores dscussed. From Table 10, nstruments belongng to the category No DC, Some DA have a lower estmated dscount rate between the two. Snce these nstruments are n the bottom of the lst of absolute prorty, credtors expect not only a relatvely low level of recovery rate but also a low varaton around the expected value. 23 It therefore results n a lower recovery uncertanty and n turn a lower requred dscount rate. Fnally, those nstruments belongng to the category of No DA, Some DC should have the hghest expected recovery of all the four categores; however there may stll be a large level of uncertanty to the ultmate recovery. Ths uncertanty therefore leads to a hgh requred dscount rate. Instrument Type From Table 11 and judgng from the confdence ntervals, we can conclude the estmated dscount rate of senor unsecured nstruments are statstcally sgnfcantly hgher than those of senor secured bonds (see also secured vs. unsecured dscussed above), senor subordnated bonds and subordnated bonds. Senor subordnated bonds have the lowest 23 Recovery cash flow always has a lower bound of zero. 23

pont estmate of dscount rate among all the types. The dfferences n dscount rates should reflect the dfferent degrees of uncertanty n the recovery process for each nstrument type. For example, the relatvely hgher (lower) dscount rate for senor unsecured bonds (subordnated bonds) suggests a hgher (lower) degree of recovery uncertanty. They are found to be consstent wth a number of emprcal studes (e.g. Altman and Arman (2002) and Acharya, Bharath and Srnvasan (2003)) whch suggest standard devaton of LGD s smaller for subordnated notes than for senor unsecured bonds. These results are also confrmed n our sub-segment analyss conducted n secton VI. Table 11: Dscount Rate n Percent: Instrument Type Most Lkely Estmate Standard 90% Confdence Interval Devaton Lower Upper Bank Debt 13.3 6.7 2.3 24.3 Senor Secured Bonds 11.0 6.9-0.3 22.2 Senor Unsecured Bonds 27.5 3.1 22.4 32.7 Senor Subordnated Bonds 3.8 5.7-5.6 13.2 Subordnated Bonds 8.9 3.8 2.7 15.1 Rsk-Return Trade-Off To complete ths secton, we would lke to examne f there s an approprate rsk-return trade-off n the nvestment of defaulted nstruments. Does a defaulted nstrument of a hgher LGD rsk generate a hgher return than one of lower LGD rsk? We use the mnmum sum of the squares of errors (SSE) ε of realzed recoveres (normalzed by the square root of tme-to-recovery) to proxy for LGD rsk for each segment. If there s a certan degree of rsk-return tradeoff, we expect to fnd a postve relaton between the dscount rate and SSE. It s ndeed the case when we regress the estmated dscount rate aganst an ntercept and SSE across all segments (wth more than 10 observatons) examned n ths secton. The results are reported n equaton (9). The R-square s found to be 11% and the slope coeffcent of 0.123 s hghly statstcally sgnfcant wth a t-statstc of 12.4. dˆ = -0.01205 + 0.123*SEE (9) 24

VI Valdaton of Results: Sub-Segment Analyss In ths secton, we examne the robustness of the dfferences n dscount rates across the segments by controllng for other ways to segment the data. For example, we would lke to answer the followng questons: Is the dscount rate durng an ndustry-specfc stress perod stll statstcally sgnfcantly hgher than that durng a non-stress perod f we only consder senor unsecured bonds? Is the dscount rate of those nstruments wth some DA and some DC stll statstcally sgnfcantly hgher than those wth no DA and no DC f we only consder facltes ssued by the non-technology sector? 24 In order to answer these questons, we repeat the analyss at the sub-segment level crossng all the segments we consdered prevously. For example, usng the proposed methodology, we solve for the pont estmate and the (asymptotc) confdence nterval of the dscount rate of the sub-segment consstng of nstruments ssued by the nontechnology sector and at the same tme wth some DA and some DC. The pont estmate s found to be equal to 33.3% and the correspondng 95% confdence nterval s from 26.0% to 40.5%. We can then check f the estmated dscount rate of ths sub-segment s statstcally dfferent from the correspondng estmated dscount rates at the segmentlevel. Specfcally, we want to check f respectvely the dscount rate obtaned for all nstruments n the non-technology sector and the dscount rate obtaned for all nstruments wth some DA and some DC actually le wthn the above nterval. As reported n the prevous secton, the former s 13.0%, whle the latter s 29.3%. Snce the latter les wthn the above confdence nterval of the sub-segment dscount rate estmate, we cannot conclude that the non-technology sub-segment wthn the segment of some DA and some DC behaves dfferently from other sub-segments. However, snce the former les outsde the confdence nterval, we can conclude that the sub-segment of some DA and some DC s lkely to have hgher dscount rate wthn the non-technology sector. We conduct the above analyss for each of the sub-segments and the results are reported n Table 12. We only present the estmated dscount rates of those sub-segments whch are statstcally (at 95% confdence level) sgnfcantly dfferent from that of the correspondng segment (those bold fgures presented along the dagonal of Table 12). 25 Those dscount rates that are sgnfcantly hgher (lower) than the correspondng segment dscount rate are presented n blue (red) and n brackets (square brackets). We should nterpret the dscount rates of the sub-segments presented along each row and compare them wth the value at the dagonal (.e. the segment dscount rate) when we look for statstcally sgnfcant dfferences. Let us agan use the non-technology sector as an example. The segment dscount rate s 13.0% (the value along the non-technology row of Table 12 whch s at the dagonal). Wthn ths segment, the sub-segment of No DA and No DC has an estmated dscount rate of -7.0% whch s sgnfcantly lower than 13%. On the other hand, the sub-segments of In Industry-Specfc Stress Perod, Some DA and Some DC and Senor Unsecured Bonds have dscount rates of 21.0%, 33.3% and 26.2% respectvely whch are all sgnfcantly hgher than 13.0%. Snce 24 The reader wll fnd out later that the answers are yes for both of these questons. 25 To make sure the results wll not be dstorted by a few outlers, we only consder those sub-segments wth more than or equal to ffty LGD observatons. 25

dscount rates are not reported for all the other sub-segments, they are not found to be statstcally dfferent from 13.0%. The other rows can therefore be nterpreted n the same fashon. We can examne the robustness of the results presented n the prevous secton by consderng the results reported n Table 12. Except for wthn the segment No DA, Some DC, estmated dscount rates of secured and unsecured nstruments are not sgnfcantly dfferent from the correspondng segment dscount rate. 26 Any dfference between the secured and unsecured nstruments n terms of dscount rate s therefore not as strong as those obtaned by segmentng the nstruments n other ways (dscussed subsequently). The dfference n dscount rates between IG and NIG s found to be robust after controllng for a number of other varables. Wthn fve segments (namely Unsecured, Technology, In Market-Wde Stress Perod, In Industry-Specfc Stress Perod and Senor Unsecured Bonds ), the NIG sub-segments have sgnfcantly lower dscount rates than the correspondng segment dscount rates. At the same tme, the IG subsegment wthn the No DC, Some DA segment has estmated dscount rate whch s sgnfcantly hgher than the segment value. From Table 12, the robustness of the prevous fndngs of the dscount rate for the technology sector beng hgher than that of the non-technology sector and the dscount rate for nstruments defaulted durng the market-wde stress perods beng hgher than durng the non-stress perods are however questonable. For example, f we only consder senor unsecured bonds, the dscount rate of those defaulted durng the marketwde stress (non-stress) perods s actually sgnfcantly lower (hgher) than the segment dscount rate of 27.5%. From Table 12, the dfference between ndustry-specfc stress and non-stress perods s found to be very robust. For all the cases whch are statstcally sgnfcant, nstruments defaulted durng the ndustry-specfc stress (non-stress) perods consstently have sgnfcantly hgher (lower) dscount rate after controllng for other varables. The fndngs n the prevous secton regardng DA, DC and nstrument type are also confrmed by the results reported n Table 12. For example, No DA, Some DC has sgnfcantly hgher dscount rate even one confnes herself to consder only unsecured nstruments, the non-technology sector or durng perods other than the market-wde stress perods. On the other hand, the dscount rates estmated for the No DA and No DC sub-segments are consstently lower than the correspondng segment dscount rates. In terms of nstrument type, senor unsecured bonds always have hgher dscount rate, whle senor subordnate bonds lower dscount rate than the segment value. The results reported n Table 12 also suggest that nstrument type by tself cannot fully characterze the recovery rsk (and thus the dscount rate) of the nstrument. Controllng 26 Wthn the No DA, Some DC segment, the dscount rate of unsecured nstruments sub-segment s estmated to be equal to 31.8%, whch s sgnfcantly dfferent from the segment dscount rate of 21.2%. 26

for nstrument type, one can stll observe a large varaton of dscount rates across dfferent DA and DC. For example, wthn the segment of senor unsecured bonds, there s sgnfcantly large varaton n dscount rates across dfferent sub-segments. Specfcally, those wth some DA and some DC have sgnfcantly hgher dscount rate (70.3%), whle those wth no DA and no DC have sgnfcantly lower dscount rate (1.6%). To summarze, the analyss conducted n ths secton confrm a number of our fndngs n the prevous secton. Specfcally, (1) whether the nstrument s IG or NIG; (2) whether t defaults durng the ndustry-specfc stress perods or not; (3) the exstence of DA and/or DC; and (4) ts nstrument type are found to be mportant determnants of recovery rsks. However, the results do not strongly support the Secured vs. Unsecured dfferentaton, whch s lkely due to data constrants. 27 The mpact of ndustry characterstcs (e.g. technology vs. non-technology) and market-wde factors (e.g. stress vs. non-stress) s found to be nconclusve. 27 The data does not allow us to dstngush the level of securty, only the securty type. That s, we could have two loans lsted as secured, but the frst one may be 100% secured, whereas the second one may only be 5% secured. It s very possble that ths lmtaton weakens the sgnfcance of the Secured vs. Unsecured dfferentaton. 27

Table 12: Estmated Dscount Rates of Sub-Segments Secured Unsecured IG (earlest) NIG (earlest) Technology Non-Technology Mkt. stress perod n stress not n stress Industry stress perod n stress not n stress DA & DC No DA, some DC No DA and DC No DC, some DA Some DA and DC Instrument type Bank debt Senor secured bonds Senor unsecured bonds Senor sub bonds Sub bonds Secured 11.8 Unsecured 14.3 [4.4] (22.0) (31.8) [-6.8] (32.1) (27.5) [3.8] IG (earlest) 22.8 NIG (earlest) 6.4 Technology [-2.4] 24.4 [-28.0] (44.5) [-49.0] Non-Technology 13.0 (21.0) [-7.0] (33.3) (26.2) Mkt. stress perod n stress [-28.0] 15.7 not n stress [3.8] (37.0) 13.6 (24.7) [3.9] (24.9) [-5.1] [4.7] (30.5) (36.1) [-9.8] Industry stress perod n stress [9.6] 21.5 [11.0] (35.9) (46.6) [4.5] [5.6] not n stress (21.1) 8.1 DA & DC No DA, some DC (31.8) (44.5) [-2.2] 21.2 (33.0) No DA and DC 0.9 No DC, some DA (23.8) 8.6 [-16.4] Some DA and DC [-49.0] 29.3 (70.3) Instrument type Bank debt 13.3 Senor secured bonds 11.0 Senor unsecured bonds [2.2] [11.8] (36.1) (46.6) [8.8] [1.6] [16.6] (70.3) 27.5 Senor sub bonds (28.5) [-16.4] (19.5) 3.8 Sub bonds 8.9 Note: Those estmated sub-segment dscount rates that are statstcally sgnfcantly hgher (lower) than the correspondng segment dscount rate (at the dagonal) are presented n blue (red) and n brackets (square brackets). 28

VII- Regresson Analyss In secton VI, we check for the robustness of our results obtaned n secton V by consderng the nteractons of dfferent ways to segment the dataset n a two-dmenson analyss. For example, we nvestgate the nteracton between ndustry-specfc stress perod and nstrument type, and conclude that wthn the segment of Senor Unsecured Bonds, the estmated dscount rate durng the ndustry-specfc stress perods s stll sgnfcantly hgher than average. In ths secton, we would lke to take one step further by conductng a mult-dmensonal nvestgaton through regresson analyss. We want to study f the varaton of the returns on defaulted nstruments can be explaned by the varous characterstcs of the nstruments f we consder all the factors at the same tme. The dependent varable s the annualzed nternal rate of return on an ndvdual defaulted nstrument, assumng the nstrument s bought 30 days after default and s held untl t emerges from bankruptcy. 29 The explanatory varables consdered here are those found out to be mportant determnants of recovery rsk n secton V and VI, namely (1) whether the nstrument s secured or not; (2) whether the earlest S&P s ratng s nvestment grade or not (IG vs. NIG); (3) whether the default event occurs durng an ndustry-specfc stress perod; (4) whether there s No DA and No DC or Some DA and Some DC ; (5) whether t s Senor Unsecured Bonds or Senor Subordnated Bonds (.e. ts nstrument type). 31 In the regresson analyss, we also control for the varaton of tradng prces (30 days after default) and the weghted average tme-to-recoveres of the defaulted nstruments. 32 29 The nternal rate of return on an ndvdual defaulted nstrument s not the expected return on the nstrument, whch we dscuss n the prevous sectons. The former s the realzed return and can be substantally dfferent from the expected return due to recovery uncertanty. A negatve realzed return, therefore, does not suggest the nvestors prce the nstrument usng a negatve dscount rate. In order to ensure our results wll not be affected by any potental outlers, we gnore those realzed returns whch are ether hgher than 500% or lower than -75% n the subsequent regresson analyss. 31 We gnore the ndustry effect (.e. technology vs. non-technology) and market-wde stress stuaton, of whch ther relatons wth estmated dscount rates are found to be weak or nconclusve n secton V and VI. 32 Because there can be multple recoveres pad out at dfferent tmes durng the recovery process, we use average tme-to-recovery weghted by the nomnal recovery values. Our analyss suggests that usng the tme-to-recovery of the fnal cash flow rather than the weghted average tme-to-recovery does not affect the conclusons drawn n ths study. 29

The most general verson of the regresson equaton we consder s: 33 r = c + a P + a Sec + a IG + a IndS + a DADC 1 + a Ty 7 2 1, + a Ty 8 3 2, 4 + + a TTR + ε 9 5 1, + a 6 DADC 2, (10) where r = nternal rate of return on default nstrument ; P = tradng prce of nstrument 30 days after default (n $ per $1 nomnal value); Sec = dummy varable: equal to 1 f nstrument s secured, 0 otherwse; IG = dummy varable: equal to 1 f nstrument s earlest S&P s ratng belongs to nvestment grade, 0 otherwse; IndS = dummy varable: equal to 1 f nstrument defaults durng ndustry-specfc stress perod, 0 otherwse; DADC 1, = dummy varable: equal to 1 f there s no DA and no DC, 0 otherwse; DADC 2, = dummy varable: equal to 1 f there s some DA and some DC, 0 otherwse; Ty 1, = dummy varable: equal to 1 f nstrument s Senor Unsecured Bond, 0 otherwse; Ty 2, = dummy varable: equal to 1 f nstrument s Senor Subordnated Bond, 0 otherwse; TTR = the weghted average tme-to-recovery of nstrument (n years). The regresson results are reported n Table 13. We present the coeffcents together wth the correspondng t-statstcs (n talc). 33 In ths study, we consder varous sngle-explanatory varable versons of equaton (10) together wth the general verson of the multple regresson. 30

Table 13: Regresson Analyss: Coeffcents together wth the correspondng t-statstcs (n talc) (1) (2) (3) (4) (5) (6) (7) Constant 0.428*** 0.417*** 0.335*** 0.462*** 0.426*** 0.591*** 0.412*** 9.264 10.386 6.290 10.899 9.083 6.641 5.020 Tradng prce (per $1 nomnal) -0.484*** -4.890 Secured -0.015 0.104-0.062-0.293 1.274-0.819 IG (earlest S&P ratng) 0.187** 0.264*** 0.182** 2.210 2.956 2.052 Industry-Specfc Stress Perod 0.120** 0.085* 0.144*** 2.454 1.684 2.902 DA and DC No DA and No DC -0.249*** -0.231*** -0.265*** -3.534-3.251-3.708 Some DA and Some DC -0.056-0.033-0.022-0.837-0.475-0.312 Instrument Type Sen. Unsecured Bonds 0.033 0.033-0.020 0.620 0.437-0.261 Sen. Sub. Bonds -0.088-0.135-0.144* -1.353-1.608-1.695 Weghted ave. tme-torecovery (year) -0.103*** -0.110*** -0.093*** -0.102*** -0.103*** -0.116*** -0.103*** -4.902-5.241-4.414-4.956-4.958-5.407-4.753 R-square (adjusted) 0.025 0.030 0.031 0.036 0.027 0.071 0.048 p-value of F test 0.000 0.000 0.000 0.000 0.000 0.000 0.000 Note: - Coeffcents sgnfcant at 10%, 5% and 1% level are marked wth *, ** and *** respectvely - p-value of F test s for the hypothess test where all coeffcents are zero Consstent wth the fndngs n the prevous sectons, the sngle-varable regresson results (regresson equatons (1) to (5) n Table 13) suggest the return on a defaulted nstrument s statstcally sgnfcantly hgher f ts earlest S&P ratng s IG, f t defaults durng the ndustry-specfc stress perod and f t has no DA and DC. The sgns of coeffcents of Secured and Instrument Type are consstent wth the results documented n secton V and VI, however they are not statstcally sgnfcant. Moreover, there s a negatve relaton between realzed return and tme-to-recovery. 31

The multple regressons are reported as regresson equatons (6) and (7) n Table 13. The coeffcents of IG, Industry-Specfc Stress Perod and No DA and No DC reman statstcally sgnfcant and of the expected sgn. In addton, the negatve coeffcent for Senor Subordnated Bonds becomes statstcally sgnfcant at 10% level (n regresson equaton (7) n Table 13). It therefore confrms our prevous concluson that senor subordnated bonds have sgnfcantly lower estmated dscount rate. As expected, the coeffcent of Secured s negatve, however t s not found to be statstcally sgnfcant. Fnally, the realzed return s found to be negatvely related to the tradng prce (see regresson equaton (6) n Table 13). It s as expected gven the lmted upsde gan of fx-ncome nstruments. VIII- Concluson In ths study, we emprcally solve for the dscount rate requred to estmate economc LGD from workout recoveres. We fnd out that dscount rates vary by securty, ntal oblgor s ratng, whether or not the ndustry s n stress condton at the tme of default, relatve senorty to other debt and nstrument type. The sub-segment level analyss and the multple regresson analyss conducted on the nternal rates of returns of defaulted nstruments confrm the statstcal sgnfcance of the results and the robustness of our conclusons. Dscount rates for those nstruments whch default durng the ndustry-specfc stress perods are double those whch default durng other perods. Durng the market-wde stress perods when default rates are, n general, hgher, not only expected recoveres are lower, but also the dscount rate tends to be larger, whch results n an even larger economc LGD and more pronounced PD and LGD correlatons. We provde a framework for banks to determne the approprate dscount rate to be used n ther LGD estmaton process n order to fulfll the requrement under the advanced IRB approach of Basel II regulatory captal calculaton and ther nternal economc captal measurement. 32

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