Impact of CDO Tranches on Economc Captal of Credt Portfolos Ym T. Lee Market & Investment Bankng UnCredt Group Moor House, 120 London Wall London, EC2Y 5ET KEYWORDS: Credt rsk, Collateralzaton Debt Oblgaton, Economc captal, Captal allocaton, Expected shortfall, Monte Carlo smulaton. ABSTRACT: Collateralzaton Debt Oblgatons (CDOs) are fnancal securtes backed by a pool of assets. Clams on the underlyng assets of a CDO are dvded nto several tranches from rskest to least rsky. Many fnancal nsttutons ncludng banks are holdng CDO tranches of dfferent transactons n ther credt portfolos as a result of ther recent actvtes n CDO nvestng and structurng. Modelng the mpact of CDO tranches on the economc captal of credt portfolos s thus becomng ncreasngly mportant. Ths paper presents the methodology to measure the mpact of CDO tranches on the economc captal and captal allocaton of credt portfolos. We llustrate the methodology by presentng an example of calculatng captal allocaton for a synthetc CDO tranche n a credt portfolo comprsed of a tranche and other loans. We demonstrate that addng a CDO tranche to a credt portfolo ncreases the systematc rsk of the portfolo and consequently, a CDO tranche could requre much larger economc captal compared to a loan of smlar credt qualty and same maturty. An mportant mplcaton of our fndng s that t s a very poor approxmaton n calculatng economc captal to treat a CDO tranche as a loan equvalent. 1
1. BACKGROUND Collateralzaton Debt Oblgatons (CDOs) are fnancal securtes backed by a pool of assets. CDOs are credt market products smlar to collateralzed mortgage oblgatons (CMOs) whch have been actvely traded n secondary mortgage markets for many years. In a CDO, clams on a pool of assets are dvded nto several tranches from rskest to least rsky. The rskest pays the hghest return, whle the least rsky pays the lowest. Each tranche s then sold at the market to dfferent nvestors. CDOs are typcally based on assets of one asset class such as leveraged loans, small and medum enterprse loans, or credt default swaps. However, n some complex CDOs such as CDOs of Asset Backed Securtes (ABS) and CDO squareds, asset composton n ther pools s a mx of multple asset classes ncludng resdental and commercal mortgage backed securtes and tranches of other CDOs. Snce CDO tranches are fxed ncome securtes, nterest potentally receved by a tranche nvestor s specfed at the tme of orgnaton. However, the actual amount of nterest a tranche nvestor would receve over the lfe of the transacton would depend on the performance of the assets n the pool. Cash flows generated from the assets n the pool are used to servce the CDO s notes whch nclude payng ther nterests, repayng ther prncpal balances to mantan the qualty of the structure, and renvestng n new assets followng the waterfall structure as specfed. How well assets n the pool perform, n terms of default and credt mgraton, wll contrbute to whether an nvestor of a CDO tranche wll fully recover ts ntal nvestment at maturty. However, especally for cash CDOs, any loss on a tranche does not entrely depend on the losses of ts underlyng assets. At a gven level of losses n the pool, the loss of a CDO tranche can vary sgnfcantly 2
dependng on the tranche s subordnaton and waterfall structure. For example, Interest and Par Value Coverage tests of the waterfall structure could trgger early redemptons of senor tranches at the expense of reducng cash flows avalable to pay the junor and equty tranches. Renvestment n new assets for the pool could result n a pool of assets wth hgher default rsk, thereby ncreasng potental losses on senor tranches. To capture all of these complextes n the evaluaton of a CDO tranche, a Monte Carlo smulaton coupled wth a cash flow analyss based on the waterfall structure of the CDO s employed. The Monte Carlo smulaton generates the cash flows from the assets n the pool through modelng ther performance. The cash flow analyss uses these cash flows to pay the tranches of the CDO transacton and renvest n new assets followng the specfcatons of the waterfall structure. Wth the recent rapd growth n the CDO market snce ts ncepton n the md- 1990s, many fnancal nsttutons n partcular banks are holdng CDO tranches of many dfferent transactons n ther credt portfolos. Ths s one of the consequences of warehousng CDO tranches to support CDO structurng. It s also a consequence of CDO nvestment by these nsttutons over the last decade. Pror to the recent growth n the CDO market, a bank s credt portfolo would be comprsed of manly loans to ther clents and to a lesser degree, credt default swaps, whch are purchased for the purpose of hedgng credt rsk n ther portfolos. However, wth the recent change of many banks busness models from orgnaton and holdng of assets to maturty to orgnaton and dstrbuton of assets, the asset composton of ther credt portfolos have altered to nclude structured credt products such as CDO tranches. It s becomng ncreasngly common to fnd CDO tranches n the credt portfolos of many fnancal nsttutons. Thus, evaluaton of the mpact of 3
CDO tranches on the economc captal of a credt portfolo s becomng ncreasngly mportant. 2. PURPOSE OF THE PAPER Ths paper dscusses how to measure the mpact of CDO tranches on economc captal and captal allocaton n credt portfolos. The economc captal of a credt portfolo s the amount of captal reserved to pay for any unexpected losses up to a confdence level as requred by the fnancal nsttuton. The purpose of economc captal s to buffer the effect of large losses n the portfolo. Captal allocaton measures the ncremental economc captal requrement of a portfolo as a result of addng an asset, such as a CDO tranche. Economc captal and captal allocaton are calculated from the probablty dstrbuton of portfolo losses. Monte Carlo smulaton has been wdely appled to calculate probablty dstrbuton of portfolo loss n credt portfolos comprsed of bonds and loans. However, applyng Monte Carlo smulatons to credt portfolos comprsed of CDO tranches n addton to bonds and loans s relatvely new. The methodology presented here s our frst attempt to address ths mportant problem n a consstent framework. Ths paper llustrates the Monte Carlo smulaton methodology to calculate portfolo loss dstrbuton of a credt portfolo comprsed of CDO tranches n addton to bonds and loans. The man advantage of usng Monte Carlo smulaton s ts easy mplementaton and t correctly calculates the correlated defaults of CDO tranches wth the bonds and loans of the portfolo by capturng the correlated defaults of these assets wth the CDOs underlyng assets. Ths s mportant snce the calculaton of economc captal s very senstve to such default clusters n a portfolo. 4
The paper s organzed as follows. In Secton 3, we gve a bref ntroducton to the methodology for calculatng portfolo loss dstrbuton of credt portfolos comprsed of bonds, loans and CDO tranches. Secton 4 dscusses the calculaton of economc captal and captal allocaton from the portfolo loss dstrbuton. In Secton 5, we present an example of calculatng probablty dstrbuton of portfolo loss for a portfolo comprsed of loans and one synthetc CDO tranche. From the loss dstrbuton, we calculate economc allocaton to the tranche and compare t to the captal allocaton of a loan of the same maturty and smlar credt qualty. 5
3. MONTE CARLO SIMULATION OF CREDIT PORTFOLIOS COMPRISED OF BONDS, LOANS AND CDO TRANCHES In credt portfolo management Monte Carlo methods are the ndustry standard approach for calculatng rsk fgures from a portfolo loss dstrbuton. Even though the synthetc ndex-based market has seen strong development of analytc and sem-analytc approaches, these methods do not fully apply to credt portfolo management. Ths s manly a consequence of the heterogenety of underlyng assets and the need for default ndcators at ndvdual asset levels. Followng standard ndustry conventon, we use a bottom-up approach to model correlated defaults n a pool of names by assumng ther asset values depend on certan rsk factors. These types of dependences are commonly captured wth an asset value factor model frst ntroduced by Merton [1] and Vascek [2]. The underlyng dea s very straghtforward. If there s a severe downturn n the overall economy, the lkelhood of multple defaults wll ncrease because many assets n the portfolo wll declne together n qualty. Ths s because of ther common dependence on the overall economy. Applyng ths dea to model correlated defaults n a portfolo holdng CDO tranches n addton to bonds and loans, we assume a tranche s dependence on rsk factors s fully captured through the dependence of ts underlyng assets. Any severe downturn n the overall economy wll result n a declne n credt qualty of the portfolo s bonds and loans together wth the underlyng assets of the CDO tranches. Thus, n ths bottom-up approach, the calculaton conssts of two basc steps. The frst step s determnng the default or credt mgraton of bonds and loans n the portfolo together wth the underlyng assets of 6
CDO tranches over a horzon. The second step s calculatng the values of CDO tranches at horzon. For a CDO tranche, ts value would be a sum of the cash flows receved over the horzon plus a forward value calculated based on the future credt state of ts underlyng assets at horzon. Cash flows receved by a synthetc tranche over the horzon would consst manly of nterest payments, whle those receved by a cash CDO tranche would consst of nterest payments and prncpal repayments. The loss of a tranche over a horzon s then calculated as the dfference between the tranche value at horzon and ts current value. Ths yelds a loss dstrbuton at horzon for a credt portfolo comprsed of bonds, loans, and CDO tranches, from whch ts economc captal and captal allocaton are calculated. Although the methodology dscussed here for the calculaton of economc captal n a credt portfolo s conceptually smple, t s hghly complex n practce because t requres nested Monte Carlo smulatons. A nested Monte Carlo smulaton conssts of an outer smulaton and an nner smulaton at every scenaro of the outer smulaton. In the outer smulaton, systematc and dosyncratc rsk factors are drawn at each scenaro over a horzon to calculate the defaults and new credt states of non-defaulted assets. The purpose of the nner smulaton s to value a CDO tranche condtonal on the credt states of ts underlyng assets at horzon. Valuaton of the bonds and loans based on the future credt states of ther oblgors s also requred at a horzon, but ther valuaton generally does not requre a smulaton. Even though developng of effcent technques to perform nested Monte Carlo smulatons s an nterestng and mportant area of research, a more thorough dscusson would be well beyond the scope of the present paper. 7
For the purpose of llustratng the Monte Carlo smulaton methodology, we dscuss an example of calculatng the probablty dstrbuton of portfolo loss for a credt portfolo comprsed of N loans and one CDO tranche, T, over a horzon of one year. Furthermore, we assume the CDO s backed by these loans of the portfolo. r To determne the defaults and credt mgratons of the loans at horzon, we calculate ther asset value correlaton wth a sngle factor model. The asset value and credt state of a loan n ths paper refer to those of ts oblgor. In the sngle factor model framework, the senstvty of the asset value of a loan, X, to the systematc rsk factor s gven n terms of the correlaton parameter ρ as follows: X = ρ Y + 1 ρ Z (1) 2 where Y and Z are the systematc rsk factor and the dosyncratc rsk factor of the loan, respectvely. Both Y and Z are ndependent and have standard normal dstrbuton. Furthermore, we assume the calculaton of the default and credt mgraton of a loan at horzon s based on the change of ts asset value as gven by Equaton (1). Specfcally, a loan s defaulted f ts asset value falls below ts default threshold whch s defned by ts default probablty to horzon. Analogously, a loan ntally n the state j s mgrated to a state r f ts asset value at horzon falls wthn the nterval defned by ts lower edge, rlower, k j k j, and upper edge, r, upper, as follows: r 1 r rlower, rupper, 1 1 kj, k j = Φ PDj + Tj, k, Φ PDj + Tj, k k k, (2) 8
Φ s the nverse of the standard normal dstrbuton and Tj, k s the probablty 1 where [ ] for the transton from state j to state k. The calculaton of the defaults and credt mgratons of the portfolo s loans from ths frst step would be used to determne the loss of portfolo value over the horzon. The contrbuton to the loss of portfolo value conssts of the loss of the loans defaulted over the horzon and the loss n value of the loans whch are not defaulted and the CDO tranche. The loss from loans defaulted over the horzon s calculated as follows: L = ω I LGD, (3) D D N where ω s the weght of the notonal value of the loan relatve to the total notonal value of the portfolo. LGD s the fxed percentage loss of the notonal value of the loan at default. D I s the default ndcator of the loan whch s defned as I = 1, f D X < k and D D I = 0, f X k D. The loss from those loans whch are not defaulted and the CDO tranche at horzon H relatve to ther values as of today t 0 s calculated as follows: D ( 1 ) (,, ) L I PV PV DF, (4) H M = ω H t 0 t0 > NT, r where and PV are the values of asset at H and, respectvely, and s the PV H, t, 0 t0 DFt > H dscount factor from t0 to H. For a CDO tranche, the default ndcator assumes a value of zero. Valuaton of a loan at the horzon assumes that the future credt state of the loan suffcently determnes ts value. Valuaton of a CDO tranche at horzon assumes that the future credt states of ts underlyng assets at horzon suffcently determnes ts value although ths could requre another smulaton. By mergng the two loss random varables, 9
defned n Equatons (3) and (4), we obtan a loss dstrbuton ( L l) P < for the portfolo. The tal of the loss dstrbuton gves the probablty that the portfolo wll suffer large losses exceedng a specfc loss value. Although the calculaton of loss dstrbuton dscussed here s based on a sngle factor asset value model, extenson of the calculaton to a mult-factor asset model s farly straghtforward. Thus, employng a mult-factor asset model n the calculaton of loss dstrbuton allows one to apply the methodology dscussed n ths paper to portfolos wth heterogeneous asset compostons. 4. ECONOMIC CAPITAL AND CAPITAL ALLOCATION Economc captal of a credt portfolo s defned as the loss exceedng the portfolo expected loss EL for the quantles of the loss dstrbuton to a gven confdence level α,.e. EC α = q α EL wth q ( α ) = mn { x : P( L < x) α}. Ths confdence level s p ( ) ( ) nterpreted as the probablty that the credt portfolo of a fnancal nsttuton would suffer a loss whch wll use up the nsttuton s captal. Snce captal exhauston mples nsttutonal falure, the confdence level α s equvalent to the default rsk of an nsttuton. In the calculaton of captal allocaton for the CDO tranche of the credt portfolo dscussed n the prevous secton, we apply the methodology of expected shortfall. The 10
expected shortfall 1 of an asset, ether of the loans or the CDO tranche, s defned n terms of the portfolo loss dstrbuton as follows: ( α ) ES =Ε L, H Lportfolo, H > q Ε L, H, (5) where LH, and Lportfolo, H are the losses of asset and the portfolo at H. Then, the captal allocaton of the CDO tranche n terms of ts expected shortfall shortfalls of the loans n the portfolo s calculated as follows: ES tranche and the expected EC tranche ( α) = EC ( α) p ES NT, r tranche ES. (6) 5. EXAMPLE OF CALCULATING PORTFOLIO LOSS DISTRIBUTION We dscuss an example of calculatng the portfolo loss dstrbuton over a horzon of one year for a credt portfolo of loans and a CDO tranche. 5.1 Data and modelng assumptons A credt portfolo s assumed to be comprsed of 125 loans and one synthetc CDO tranche. Each loan s a unque entty whch s taken from the enttes of the Traxx Europe S8 ndex. The Traxx Europe ndex s a credt default swap (CDS) ndex composed of the most lqud 125 CDSs referencng European nvestment grade credts. Each loan assumes a notonal amount of one mllon Euros, a maturty of one year, and a fxed recovery rate of 25%. The synthetc CDO s based on a portfolo of 125 CDSs. Each CDS assumes a fveyear maturty, the same maturty as the CDO. Each CDS references the entty of one of the 1 The other approach of calculatng captal allocaton s based on margnal Value-at-Rsk contrbuton. For a recent revew of the two approaches of calculatng captal allocaton, please refer to the followng paper: Glasserman, P (2006) Measurng Margnal Rsk Contrbuton n Credt Portfolos, Journal of Computatonal Fnance 9, 2. 11
loans of the credt portfolo. Therefore, ths example consders a case where there s a strong overlap between the credt portfolo s assets and the underlyng assets of the CDO. The nested Monte Carlo smulaton as dscussed n Secton 2 s appled to calculate the contrbuton of the synthetc CDO tranche to the portfolo loss dstrbuton. At each scenaro of the outer smulaton, we determne whch loan s defaulted and a future credt state for the one whch s not defaulted. Then, snce the loans mature at horzon, we calculate the loss n the notonal value of the portfolo from the defaults of the loans over the horzon. To calculate the loss of the synthetc CDO tranche, we value the tranche at horzon based on the future credt states of ts underlyng assets and compare the value at horzon to ts current value. In ths example, the future credt states of the underlyng assets are exactly those of the loans of the credt portfolo. The standard ndustry practce s to use the rsk neutral default probabltes of the underlyng assets based on ther future credt states to value the synthetc CDO tranche at horzon. One would calculate a Mark to Market (MTM) value of the tranche by usng the rsk neutral default probabltes of the underlyng assets calbrated to ther market spreads and the correlaton parameter calbrated to the prcng of tranches of smlar structures 2. The calculaton s farly straghtforward once the parameters are determned. However, calculatng a MTM value of the tranche at horzon s much more challengng because one must determne the future credt states of ts underlyng assets and applyng them to derve ther forward rsk neutral default probabltes. 2 See Burtschell, X., Gregory, J., and Laurent, L.-P. (2005) A Comparatve Analyss of CDO Prcng Models, Workng paper, BNP-Parbas. 12
Instead of calculatng the loss of the CDO tranche at horzon from ts MTM values, we approxmate ts loss at each scenaro of the outer smulaton by the condtonal expected loss of the tranche. The condtonal expected loss of the tranche at horzon n each scenaro of the outer smulaton s calculated as the average loss over the scenaros of the nner smulaton. We employ the followng assumptons n the calculaton of condtonal expected loss at horzon. Frst, the future credt state of a loan s represented by an S&P credt ratng. The S&P credt ratng of a loan at horzon s calculated from ts current S&P credt ratng wth the emprcal S&P transton matrx [3]. Second, the best estmaton of the forward default probablty of a loan at horzon s the forward default probablty derved from the default term structure based on ts S&P ratng. We are assumng that default s a Markov process, but properly nhomogeneous n tme. Lastly, we nclude only the loss of prncpal n the calculaton of condtonal expected loss, thereby neglectng any loss from nterest payments to the tranche. In addton, we assume a zero nterest rate. Under these assumptons, we use a sem-analytcal approach [4] to calculate the condtonal expected loss of the CDO tranche at horzon nstead of performng an nner smulaton. We calculate economc captal for the credt portfolo as the loss of ts value exceedng the portfolo expected loss at the confdence level of 99% and calculate captal allocaton based on expected shortfall methodology at the confdence level of 95%. We assume a correlaton parameter of 31% for the underlyng assets. We consder a synthetc CDO tranche wth an attachment pont of 0.0, 3% or 6% and a correspondng detachment pont of 3%, 6% or 9%. Table 1 shows the expected loss of a tranche as of today and at horzon and Table 2 shows the economc captal allocaton of the tranche at dfferent 13
notonal amounts. Expected loss of the tranche as of today s calculated wth a fve-year cumulatve default probablty for the underlyng assets based on ther current S&P ratngs. Expected loss of the tranche at horzon s calculated by averagng the condtonal expected loss of the tranche at horzon over the scenaros of the outer smulaton of the nested Monte Carlo smulaton. 5.2 Analyss of results Now, we would lke to dscuss the results for the EL of the tranche and ts captal allocaton as shown n Tables 1 and 2. Table 1 clearly shows that the EL values of the tranche at horzon are larger than the values as of today. In partcular, for the 6-9% tranche, the dfference s as much as 50%. However, the dfference decreases wth decreasng tranche subordnaton and becomes much smaller for the 0-3% tranche. Ths should suggest that calculatng the correlated defaults of the underlyng assets plays an mportant role n determnng these results. The EL of the 6-9% tranche s more senstve to default clusters n the portfolo than the EL of the 0-3% tranche. Calculatng the EL of a tranche as of today and at horzon requres modelng the assets defaults n the pool over the lfe of a transacton. The calculaton of the EL of a tranche as of today s based on the approach of Merton [1] whch calculates the defaults of the underlyng assets over a horzon of 5 years. The tmng of the defaults s not requred because the calculaton of the expected loss assumes a zero nterest rate. It s a one-step calculaton of default snce both the systematc and dosyncratc rsk factors are drawn once at each scenaro of the Monte Carlo smulaton. However, the calculaton of the EL of the tranche at horzon assumes two steps although the calculaton of the defaults n each of 14
these steps s also based on the approach of Merton. It s a two-step calculaton of default because both the systematc and dosyncratc rsk factors are drawn ndependently twce n the outer and nner smulatons of the nested Monte Carlo smulaton. As a result of calculatng the default n a sngle step n the calculaton of EL as of today and n two steps n the calculaton of EL at horzon, one should expect the results to be dfferent because of the dfference n capturng the correlated defaults n the portfolo 3 Snce the calculaton of portfolo loss dstrbuton requres usng ths two-step approach n calculatng defaults, we should adopt ths approach n the calculaton of EL of the tranche as of today and at horzon. Consequently, both ELs would be calculated to have the same value. The results n Table 3 also show that the EL of a fve-year loan at horzon s larger than ts EL as of today for dfferent S&P ratngs although the dfference s less than 10%. Ths s smply the consequence of usng emprcally measured S&P transton matrx and default probablty term structures. However, snce n comparng a CDO tranche to a loan of smlar credt rsk, we have selected to measure credt rsk by an asset s EL at horzon, ths knd of dscrepancy from usng emprcally measured transton matrx and default probablty term structures should not affect the comparson of the results of ths paper. In Table 2, we present the tranche s captal allocatons at dfferent notonal values. Captal allocaton s reported as per unt exposure of the tranche. Snce the maxmum loss of a tranche depends on ts thckness defned as the dfference between ts detachment pont and attachment pont, t s more meanngful to report captal per unt of exposure rather 3 In valung a synthetc CDO tranche, one can uses a sngle-step default model because the valuaton s based on the calbraton of asset correlaton parameters to the prcng of smlar tranches n the market. 15
than per unt of notonal amount. These results clearly show that captal allocaton per unt of exposure ncreases wth ncreasng exposure. However, the ncrease s much larger for the 0-3% tranche than for the 6-9% tranche. Ths s because the 6-9% tranche s mostly senstve to systematc rsk, whle the 0-3% tranche s senstve to both systematc and dosyncratc rsks. An asset whch s mostly senstve to systematc rsk and an asset n a very granular portfolo should have captal requrements scalng approxmately the same wth the sze of exposures. Captal allocaton per unt of exposure of an asset n a very granular portfolo s approxmately constant ndependent of the asset s exposure sze. Thus, as shown n Table 2, the captal allocaton of a senor tranche ncreases slowly wth exposure sze as compared to a junor tranche. Now, we would lke to compare the captal allocaton of a tranche to that of a loan of smlar credt qualty and the same maturty. In a separate calculaton, we substtute the CDO tranche n the credt portfolo wth a loan of 5 years maturty and re-run the Monte Carlo smulaton to calculate the captal allocaton of the substtuted loan. Table 4 presents the captal allocatons for the substtuted loans wth dfferent S&P ratngs. To compare the captal allocaton of a CDO tranche to that of ts loan equvalent, we defne a loan equvalent of a CDO tranche as a loan whch has the same value of EL at horzon as the CDO tranche. We consder a CDO tranche and ts loan equvalent have smlar credt rsks as of today. For example, the loan wth AA- ratng of Table 4 s a loan equvalent of the 6-9% tranche. However, for the 3-6% tranche, we can not fne a loan from Table 4 wth ts EL at horzon matchng that of the CDO tranche. For the purpose of comparng the captal allocaton of the 3-6% tranche to ts loan equvalent, we scale the 16
captal allocatons of the loan wth BBB- ratng by a rato of the tranche s EL at horzon over the loan s EL at horzon. We assume that the scaled captal allocatons are for the loan equvalent of the 3-6% tranche. In Table 5 we compare the captal allocatons of the 3-6% and 6-9% tranches to that of ther loan equvalents. The results clearly show that the captal allocaton of a CDO tranche can be larger than that of ts loan equvalent by as much as 80%. We can understand these results n terms of the ncrease n the systematc rsk of the credt portfolo as a result of addng a CDO tranche or a loan. In general, a credt portfolo wth larger systematc rsk would requre hgher economc captal. A CDO transacton backed by a pool of assets s mostly senstve to systematc rsks snce ts exposures to dosyncratc rsk factors are already sgnfcantly reduced through the dversfcaton of the assets n the pool. Therefore, addng CDO tranches to a credt portfolo could sgnfcantly ncrease the portfolo s systematc rsk than addng a loan equvalent. As a result, one should expect holdng a CDO tranche n a credt portfolo would requre extra economc captal compared to holdng a loan of smlar credt ratng and the same maturty. 6. CONCLUSION In ths paper, we present the methodology based on nested Monte Carlo smulaton to measure the mpact of CDO tranches on economc captal and captal allocaton of credt portfolos. One of the advantages of usng the methodology presented here s the calculaton of the correlated defaults of the assets n a portfolo. Calculaton of correlated defaults s mportant n calculatng economc captal for a portfolo especally for those 17
holdng CDO tranches because CDO tranches are mostly senstve to systematc rsk. As an example, we apply the methodology of calculatng captal allocaton to a synthetc CDO tranche n a credt portfolo. Ths portfolo also comprses of the loans to whch the underlyng assets of the CDO are referencng. The results of our calculaton shows that the captal allocaton for addng a CDO tranche to a credt portfolo can be much larger than that of addng a loan of smlar credt qualty and the same maturty. In some cases, the ncrease n captal allocaton s as much as 80%. Our fndng also clearly suggests that t s a poor approxmaton n calculatng economc captal to treat a CDO tranche as a loan equvalent. Ths explans why usng the methodology presented n ths paper s requred to measure the mpact of CDO tranches n credt portfolos. 7. ACKNOWLEDGMENT I would lke to thank Davd Cao, Mchele Freed, Sayd Islam, Bll Morokoff, and Lmng Yang for many useful dscussons. I especally want to thank Dr. Chrstoff Goessl for careful readng of the manuscrpt and provdng me wth many useful comments. The vews expressed n ths paper are the author s own and do not necessarly represent those of Market & Investment Bankng of the UnCredt Group, Lloyds TSB Group or Moody s KMV. All errors reman my responsblty. 8. REFERENCES [1] Merton, R.C. (1974) On the Prcng of Corporate Debt: The Rsk Structure of Interest Rates, Journal of Fnance 29, 449-470. [2] Vascek, O. (2002) Loan Portfolo Value, Rsk 15, 160-162. [3] Okunev, P. (2005) A Fast Algorthm for Computng Expected Loan Portfolo Tranche Loss n 18
the Gaussan Factor Model, LBNL-57676. [4] Standard & Poor s (1996) Credt Week-Aprl. 19
9. TABLES AND CAPTIONS Table 1: Tranche Statstcs 0-3% 3-6% 6-9% EL today 34.71% 2.61% 0.23% EL at horzon 38.81% 3.43% 0.33% Table 2: Economc Captal Allocatons to CDO Tranches Notonal (MM) Exposure(MM) 0-3% 3-6% 6-9% 10 0.3 28.13% 10.46% 1.36% 20 0.6 32.04% 10.48% 1.37% 30 0.9 35.49% 11.12% 1.38% 40 1.2 38.44% 11.41% 1.39% 50 1.5 40.94% 11.69% 1.39% Table 3: Loan Statstcs AA- BBB+ BBB- BB EL today 0.30% 1.45% 3.71% 7.49% EL at horzon 0.33% 1.58% 3.69% 7.49% Table 4: Economc Captal Allocaton to a Loan Substtutng the CDO Tranche Exposure(MM) AA- BBB+ BBB- BB 0.3 0.87% 3.88% 5.94% 11.03% 0.6 1.11% 4.49% 6.95% 12.55% 0.9 1.18% 4.85% 7.53% 13.35% 1.2 1.24% 5.03% 7.95% 13.66% 1.5 1.24% 5.10% 8.24% 14.49% 20
Table 5: Captal Allocatons: CDO Tranche Versus Loan Equvalent 3-6% Exposure(MM) AA- 6-9% Scaled of BBB- * 0.3 0.87% 1.36% 5.52% 10.46% 0.6 1.11% 1.37% 6.46% 10.48% 0.9 1.18% 1.38% 7.00% 11.12% 1.2 1.24% 1.39% 7.39% 11.41% 1.5 1.24% 1.39% 7.66% 11.69% *Captal allocatons calculated by multplyng a constant factor of 0.93 wth the captal allocatons of the loan wth a BBB- ratng. 21