Examining the Validity of Credit Ratings Assigned to Credit Derivatives

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1 Examnng the Valdty of redt atngs Assgned to redt Dervatves hh-we Lee Department of Fnance, Natonal Tape ollege of Busness No. 321, Sec. 1, h-nan d., Tape 100, Tawan heng-kun Kuo Department of Internatonal Busness, Natonal Tawan Unversty No. 50, Lane 144, Sec. 4, Keelung d., Tape, 106, Tawan ABSTAT Ths paper examnes the valdty of ratngs assgned by ratng agences on structured products: ABS and ABS DO. The ratng agences have been crtczed for assgnng AAA ratngs to the structured products created from mortgages. The ratngs mght gve a false sense of confdence to nvestors wth whch a vast market was created and then collapsed. Usng a new loss functon to compute the attachment ponts, we show that the prevously consdered reasonable ratngs may need more scrutny. Keywords: atng agency, Asset-backed securty (ABS), ollateral debt oblgaton (DO), Loss functon. 1. INTODUTION The ratng agences have been crtczed for assgnng AAA ratngs to the structured products created from mortgages. The ratngs mght gve a false sense of confdence to nvestors wth whch a vast market was created and then collapsed. There have been varous attempts to check whether the assgned ratngs to structured products are reasonable. For example, Fender, et al. (2008) concludes that nvestors who narrowly focus on ratngs can serously msjudge the value-at-rsk of collateral debt oblgatons (DOs). Benmelech and Dlugosz (2009) examne the ratng practces of ratng companes. Hull and Whte (2010) examnes the rsk n the tranches of assetbacked securtes (ABSs) and ABS DOs usng the crtera of the ratng agences. The ratng agences test the should-be attachment pont for an AAA-rated tranche f t has the same probablty of losses as a AAA-rated corporate bond. The test nvolves: 1. The expected default rate (ED) for the mortgages portfolo. 2. A correlaton model that s needed to lnk the ED to a probablty dstrbuton for the actual default rate. 3. A specfed rato of the expected loss gven default (ELGD) to the ntal mortgage prncpal. Hull and Whte emprcally show that some BBB tranches of ABS cannot be consdered BBB bonds for the purposes of subsequent ABS DO securtzatons. Therefore, nvestors may be lulled nto a false sense of confdence. Ths conforms to the market observaton that nvestors suffer tremendous losses after the outbreak of subprme crss n July However, Hull and Whte also conclude that the AAA ratngs assgned by ratng agences are not totally unreasonable. Ths concluson appears contrary to the extreme default rates observed n the fnancal crss perod, namely, the wdespread default of AAA rated securtes. In ths paper, we use the loss functons proposed by Lee, et al. (2004), and Kuo and Lee (2007), for computng the mnmum attachment pont n order to better assess the valdty of credt ratngs assgned to credt dervatves. The man dfference between Hull and Whte model and our approach s the assumpton of default correlaton. In Hull and Whte model, the probablty of default s assumed to follow a cumulatve normal dstrbuton functon. Therefore the realzed default rate, condtonal on a common macro factor, s a specfc factor copula. The dsadvantage or ths model s that the specfc copula does not ft the real stuaton precsely. In our approach, default loss s descrbed by a Posson dstrbuton, whch has the potental to descrbe the mortgage default loss more accurately wthout usng an ad-hoc copula functon. The next secton descrbes the structure desgn of ABS DO tranches. The thrd secton derves the loss functon for calculatng the attachment ponts of ABS DO tranches. Fnally, a smple concluson follows. 2. METHODOLOGY ABS DO s created from a portfolo of ABS tranches. Hence, there are two levels n ABS DO structure. In the frst level, ABS conssts of several mortgages. The underlyng mortgage collateral

2 was allocated to senor tranches rated AAA, mezzanne tranches, and subordnated tranches that were ether unrated or rated BB. The rules for allocatng cash flows from mortgages to tranches were defned by what was known as a waterfall. In the second level of securtzaton, ABS DOs were formed by creatng tranches from tranches. Two types were common: a Hgh Grade ABS DO, created from AAA, AA, and A tranches of ABSs, and a Mezz ABS DO, created from the BBB tranches of ABSs. The structure s llustrated n Fgure 1 and Fgure 2. the cumulatve losses of the reference entty exceed the AP. The tranche experences total loss when the cumulatve losses reach the DP. The loss of ABS tranches wll pass through to the ABS DO. Smlarly, tranches of ABS DO suffer loss when the cumulatve losses of underlyng ABS tranches exceed the AP. The ABS DO tranche experences total loss when the cumulatve losses reach the DP. In other words, the tranche s defned by establshng the AP and DP. 3. THE LOSS FUNTION To calculate the mnmum attachment pont for AAA tranche of ABS DO formed from varous ABS tranches, we use mult-pool correlaton loss functon. To justfy the AAA ratng assgned to tranches of ABS DO, we have to compare attachment ponts for AAA tranche of ABS DO wth that for ABS tranches of the same ratng. The latter attachment pont can be calculated usng a sngle pool correlaton loss functon. Fgure 1: The structure of Hgh Grade ABS DO, created from senor tranches of ABSs. Frst, we derve a sngle pool correlaton loss functon. Assume mortgages n the pool to have equal prncpals and to have same probablty of default. Also denote ABS DO loss functon as L = ρl + ( 1 ρ) L (1) where L follows a Posson dstrbuton wth hazard rate parameter, and L, a factor specfc to mortgage, also follows a Posson dstrbuton wth hazard rate arameter. The parameter ρ s the correlaton between the losses of any two mortgages. Fgure 2: The structure of Mezz ABS DO, created from mezzanne tranches of ABSs. Each tranche n an ABS or an ABS DO has ts own attachment pont (AP) and detachment pont (DP). The AP s the lower bound of the loss covered by a tranche. The upper bound s called the DP. The range between the two ponts s called the thckness of the tranche. The tranche of an ABS begns to suffer loss when In ths way, L s related to ratngs of mortgages. The purpose of L s to decde the attachment pont for a specfc ratng ABS tranche. The loss functon s the sum of two Posson dstrbutons, and would be a random varable ndependent of the Posson dstrbuton. By usng moment generatng functon (MGF), we can derve that expected value E(L) of the loss functon L as E(L) = E ρl ) + E((1 ρ) L ) = ρ + ( 1 ρ) (2) Hence, ( = ( E ( L) ρ ) /(1 ρ) (3)

3 It should be noted that E(L) dvded by the tranche prncpal s equvalent to the expected default rate (ED). Ths s the expected proporton of the mortgages n the portfolo that wll default. Secondly, we calculate the attachment ponts of ABS tranches gven a specfc (AAA or BBB) ratng. In the prevous loss functon framework, the two-taled 10% confdence level for DO loss, condtonal on L, s dversfcaton beneft. The parameter α measures ths beneft. If α s low, ths extra dversfcaton s valuable to nvestors, but f α s hgh, t has lttle value. Fnally, we calculate attachment ponts of ABS DO tranches by usng the expected loss crteron. Denotng L ( L ) the loss on the mortgage portfolo for a partcular value of L, the expected loss on the ABS DO when the attachment pont X for the senor tranche (wth one hundred mortgages) s + z0. 1 (4) 100 * [ L( L ) X ] θ ( ) (7) L L Assume the realzed loss follow a normal dstrbuton wth mean and varance, where z 0. 1 = s the 10th quantle of the Normal dstrbuton. After the realzed loss s calculated, we then know what the attachment pont should be f t s to have the same expected loss of prncpal as a specfcally rated corporate bond. In Hull and Whte (2010), they use onefactor Gaussan copula for calculatng the attachment pont. Ther copula model has both a factor common to all mortgages, whch s denoted by M, and a factor specfc to mortgages, whch s denoted by Z. The factors M and Z are assumed to have ndependent standard normal dstrbutons. Therefore, the attachment pont s 1 1 N ( ED) ρ N ( PD) (1 ) N (5) 1 ρ where s the recovery rate, and PD s the probablty of default for a specfc ratng bond. To calculate the attachment pont of ABS DO tranche, multpool correlaton model has to be used. To consder several pools smultaneously, we defne a between-pool factor L, and wthnpool factor L W. The factor L affects probabltes of default for all mortgages, where as factor L W affects expected losses of ABS DO. Thus, ABS DO loss functon n a mult-pool case can be wrtten as L αρl + ( 1 α ) ρl + (1 ρ) L = (6) The parameter ρ s the total wthn-pool correlaton. The parameter α s the proporton of the default correlaton that comes from a factor common to all pools. A mult-pool correlaton model s useful when consderng ABS DOs. One of the potental advantages of ABS DOs over ABSs s that nvestors beneft from both between-pool and wthn-pool dversfcaton. Suppose that half the underlyng pools of an ABS DO consst of mortgages on homes n Pennsylvana and the other half consst of mortgages n alforna. If the mortgage default n alforna s not so correlated wth that n Pennsylvana, nvestors receve a W where L * s the value of L. By adjustng factor L, we let the expected loss on the ABS DO equals to a gven expected loss, whch s often calculated by ratng agences. After L s decded, the attachment pont X s thus obtaned. By examnng whether the attachment pont of ABS tranches can wthstand the total losses of the underlyng mortgage pool, we can determne whether the assgned ratngs are reasonable. 1 Gven the probablty of default for a specfc ratng bond, the attachment ponts of ABS are then calculated. In addton to the probablty of default, expected loss of a specfcally rated bond s needed when calculatng attachment ponts of ABS DO tranches. The reason s rsks of tranches n an ABS DO are crtcally dependent on correlaton between dfferent asset pools. Therefore, valung ABS DO needs an addtonal parameter α, a between-pool correlaton whch descrbes correlaton between underlyng ABS tranches pools. It s for calculatng ths addtonal parameter that the expected loss s needed. For ratng, S&P/Ftch crteron gnores between-pool correlaton and depends only on probablty of default. Moody s crteron depends on both probablty of default and expected loss for a specfc ratng bond. Varous researches make dfferent assumptons about the correlaton structure. For example, oval, et al. (2008) assumed that the asset pools underlyng ABS DOs have zero default correlaton wth each other. By comparng attachment ponts of ABS DO tranches wth those of same ratng bonds, one can determne whether the ratngs assgned to ABS DO tranches are reasonable. In the next secton, we use a test to determne whether means of two attachment ponts are equal. If the null hypothess s accepted, we can conclude that the ratngs assgned to ABS DO tranches are reasonable. 1 For example, on page 62 of John and Hull (2010), wth attachment ponts result from ther Table 2, they calculate the correspondng losses and conclude that the AAA ratngs were not totally unreasonable.

4 4. TESTING ESULTS In ths secton, we use statstcs from Moody s for concernng the cumulatve fve-year probablty of default for AAA and BBB bonds. Probabltes of loss for AAA and BBB bonds are 0.1% and 1.8%, respectvely. We assume the underlyng ABS tranches are responsble for losses of 4-9 percent. onstant recovery rate 40% s gven. Table 1 shows results usng Hull and Whte (2010) model for senor ABS DO formed from AAA ABS rated tranche. When, wth probablty of loss crtera, the DO attachment wll be 28.2% on average. Wth expected loss crtera used n mult-pool correlaton model, the attachment s 35.4% on average. The two-taled t test for equalty of means s 1.224, whch s not sgnfcant at the 5% level. It mples that the AAA ratngs assgned to the senor tranche of ABS DO s reasonable. Table 1: Mnmum Attachment Ponts Usng Hull and Whte Model for Hgh Grade ABS DO senor Tranche ρ= % 30.15% 35.75% 30.92% 16.34% ρ= ρ= ρ= Average= ED = 5% ρ= % 23.01% 28.55% 24.14% 9.83% ρ= ρ= ρ= Average= Notes: 1. Assumng constant recovery rate 40%, and usng Gaussan copula. 2. olumn Mult denotes average of attachment pont usng mult-pool correlaton model gven a specfcρvalue. 3. olumn Sngle denotes attachment pont usng sngle pool correlaton model. 4. The two-taled t test wth df = 3 for equalty of means for and 5% are and 1.549, respectvely. Both t values are not sgnfcant at the 5% level. Table 2 shows results usng loss functon model for senor ABS DO formed from AAA ABS tranche. When, wth probablty of loss crtera, the DO attachment wll be 33.9% on average. Wth expected loss crtera used n mult-pool correlaton model, the attachment s 23.9% on average. The two-taled t test for equalty of means s , whch s sgnfcant at the 5% level. The attachment pont 23.9% for ABS tranche less than 33.9% for corporate bond mples that that the AAA ratngs assgned to the senor tranche of ABS DO s over-valued. Table 2: Mnmum Attachment Ponts Usng Loss Functon Model for Hgh Grade ABS DO senor Tranche ρ= % 21.07% 45.12% 21.84% 31.25% ρ= ρ= ρ= ED = 5% Average= ρ= % 12.87% 28.85% 13.61% 20.79% ρ= ρ= ρ= Average= Notes: The two-taled t test wth df = 3 for equalty of means for ED = 10% and 5% are and , respectvely. Both t values are sgnfcant at the 5% level. Table 3 shows results usng Hull and Whte (2010) model for mezz ABS DO formed from BBB rated corporate bond. When, wth probablty of loss crtera, the DO attachment wll be 18.7% on average. Wth expected loss crtera used n mult-pool correlaton model, the attachment s 31.3% on average. The two-taled t test for equalty of means s 3.842, whch s sgnfcant at the 5% level. The attachment pont 31.3% for ABS tranche more than 18.7% for corporate bond mples that that the BBB ratngs assgned to the mezz tranche of ABS DO s undervalued. Table 3: Mnmum Attachment Ponts Usng Hull and Whte Model for Mezz ABS DO senor Tranche ρ= % 28.80% 34.89% 28.78% 12.13% ρ= ρ= ρ= Average=

5 ED = 5% ρ= % 22.28% 27.79% 21.60% 6.83% ρ= ρ= 0.2 ρ= Average= Notes: The two-taled t test wth df = 3 for equalty of means for and 5% are and 5.371, respectvely. Both t values are sgnfcant at the 5% level. Table 4 shows results usng losng functon model for mezz ABS DO formed from BBB rated corporate bond. When, wth probablty of loss crtera, the DO attachment wll be 33.3% on average. Wth expected loss crtera used n mult-pool correlaton model, the attachment s 23.3% on average. The twotaled t test for equalty of means s , whch s sgnfcant at the 5% level. The attachment pont 23.3% for ABS tranche less than 33.3% for corporate bond mples that that the BBB ratngs assgned to the mezz tranche of ABS DO s over-valued. Table 4: Mnmum Attachment Ponts Usng Loss Functon Model for Mezz ABS DO senor Tranche ρ= % 20.94% 44.87% 21.72% 31.09% ρ= ρ= ρ= ED = 5% Average= ρ= % 12.70% 28.51% 13.43% 31.09% ρ= ρ= ρ= Average = Notes: The two-taled t test wth df = 3 for equalty of means for and 5% are and , respectvely. Both t values are sgnfcant at the 1% level. 5. ONLUSION atng agences have come under great scrutny after the subprme crss started n Investment banks created securtes from underlyng mortgages. redt ratng agences assgned ratngs on these nstruments by calculatng probablty of loss. We evaluated whether ratngs assgned to structured products by ratng agences were reasonable. We looked at both hgh grade and mezzanne ABS DO. Usng Hull and Whte (2010) model, we confrm ther fndng that the AAA ratng assgned to the senor tranches of ABS s not unreasonable, and ths concluson cannot be extended to the ratng assgned to tranches created from mezzanne ABS. or ABS DO. However, usng our loss functon model, we show that not only the ratng assgned to tranches created from mezzanne ABS s unreasonable, the AAA ratng assgned the senor tranches of ABS s also not reasonable. The reason s that the probablty dstrbuton of loss from a BBB tranche s qute dfferent from the probablty dstrbuton of loss from a BBB bond. The same reasonng can also be appled to the AAA ratng case. 6. EFEENES [1] Benmelech E. and J. Dlugosz (2009), The Alchemy of DO redt atngs, Journal of Monetary Economcs, Vol. 56, Issue 5, pp [2] oval, J.D., J. Jurek, and E. Stafford (2008), The Economcs of Structured Fnance, workng paper #09-060, Harvard Busness School. [3] Hull, J. and A. Whte (2010), The sk of Tranches reated from Mortgages, Fnancal Analysts Journal, Vol. 66, No. 5, pp [4] Fender, I., N. Tarashev and H. Zhu (2008), redt Fundamentals, atngs and Value-at-sk: DOs versus orporate Exposures, BIS Quarterly evew, March, pp [5] Kuo,.-K. and.-w. Lee, (2007), Integratng Market and redt sk Usng a Smplfed Fralty Default orrelaton Structure, The Journal of Fxed Income, Vol.17, No.1, pp [6] Lee,.-W,.-K. Kuo and J. Urruta, (2004), A Posson Model wth ommon Shocks for DO Valuaton, The Journal of Fxed Income, Vol.14, No.3, pp Our results also show the mpact of ED on attachment ponts. As mght be expected, attachment ponts become lower because of lower probablty of default.