THE RELATIONSHIP BETWEEN AVERAGE ASSET CORRELATION AND DEFAULT PROBABILITY

JULY 22, 2009 THE RELATIONSHIP BETWEEN AVERAGE ASSET CORRELATION AND DEFAULT PROBABILITY AUTHORS Joseph Lee Joy Wang Jng Zhang ABSTRACT Asset correlaton and default probablty are crtcal drvers n modelng portfolo credt rsk. It s generally assumed, as n the Basel II Accord, that average asset correlaton decreases wth default probablty. We examne the emprcal valdty of ths assumpton n ths paper. Overall, we fnd lttle emprcal support for ths decreasng relatonshp n the data for corporate, commercal real estate (CRE), and retal exposures. For corporate exposures, there s no strong decreasng relatonshp between average asset correlaton and default probablty when frm sze s properly accounted for. For CRE and retal exposures, the emprcal evdence suggests that the relatonshp s more lkely to be an ncreasng one. We also provde economc arguments aganst the assumpton that average asset correlaton decreases wth default probablty. For corporate exposures, defaultng frms do not necessarly experence ncreases n ther frm-specfc rsk f there s a systematc negatve shock that causes wdespread defaults. For retal exposures, sub-prme borrowers are more senstve to general economc condtons and thus experence hgher asset correlatons than prme borrowers. Our analyses suggest that t s mprudent to assume a decreasng relatonshp between average asset correlaton and default probablty n measurng portfolo credt rsk. In lght of these economc arguments and emprcal evdence, we encourage the Basel Commttee to revst the use of ths relatonshp n bank captal requrement.

Copyrght 2009, Moody s Analytcs, Inc. All rghts reserved. Credt Montor, CredtEdge, CredtEdge Plus, CredtMark, DealAnalyzer, EDFCalc, Prvate Frm Model, Portfolo Preprocessor, GCorr, the Moody s logo, the Moody s KMV logo, Moody s Fnancal Analyst, Moody s KMV LossCalc, Moody s KMV Portfolo Manager, Moody s Rsk Advsor, Moody s KMV RskCalc, RskAnalyst, RskFronter, Expected Default Frequency, and EDF are trademarks or regstered trademarks owned by MIS Qualty Management Corp. and used under lcense by Moody s Analytcs, Inc. ACKNOWLEDGEMENTS We are grateful to our colleagues Amnon Levy and Fanln Zhu, as well as Jose Lopez from the Federal Reserve Bank of San Francsco for helpful comments and suggestons. All errors are, of course, our own. Publshed by: Moody s KMV Company To contact Moody s KMV, vst us onlne at www.moodyskmv.com. You can also contact Moody s KMV through e-mal at nfo@mkmv.com, or call us by usng the followng phone numbers: NORTH AND SOUTH AMERICA, NEW ZEALAND, AND AUSTRALIA: 1 866 321 MKMV (6568) or 415 874 6000 EUROPE, THE MIDDLE EAST, AFRICA, AND INDIA: 44 20 7280 8300 ASIA-PACIFIC: 852 3551 3000 JAPAN: 81 3 5408 4250

TABLE OF CONTENTS 1 OVERVIEW... 5 2 BASEL II AVERAGE ASSET CORRELATION... 6 3 CORPORATE... 7 3.1 Evdence from Asset Return Data... 7 3.2 The Change n Asset Correlaton as a Frm Approaches Default... 11 3.3 Evdence from the Default Data... 13 4 COMMERCIAL REAL ESTATE... 16 5 RETAIL... 18 6 IMPACT OF CORRELATION MISSPECIFICATION ON REGULATORY CAPITAL... 20 7 CONCLUSION... 21 THE DYNAMIC RELATIONSHIP BETWEEN AVERAGE ASSET CORRELATION AND DEFAULT PROBABILITY 3

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1 OVERVIEW Asset correlaton and probablty of default (PD) are crtcal drvers n modelng portfolo credt rsk. The most common approach to modelng default correlaton n portfolo credt rsk calculaton s to combne default probabltes wth asset correlatons. Bascally, two borrowers wll default n the same perod f both of ther asset values are nsuffcent to pay ther oblgatons. Asset correlaton helps defne the jont behavor of the asset values of the two borrowers. Ths dea has become the bass for many portfolo credt rsk models, for example, the mult-factor model framework n the Moody s KMV (MKMV) Portfolo Manager and RskFronter, and the so-called Asymptotc Sngle-Rsk Factor (ASRF) model supportng the Basel II IRB credt rsk captal charge. In the context of the ASRF model, the sngle systematc rsk factor may be nterpreted as reflectng the state of the economy, and all borrowers are lnked to one other by ths sngle rsk factor. The asset correlatons determne how the asset value of one borrower depends on the asset value of another borrower. Wth ths sngle factor set-up, the asset correlatons can be consdered as the dependence of the asset value of a borrower on the general state of the economy. It s generally assumed, as n the Basel II Accord, that average asset correlaton decreases wth an ncrease n default probablty. Ths relatonshp suggests that, on average, rsky borrowers would have smaller asset correlatons. Wthn the ASRF, the decreasng relatonshp has the effect of dampenng the captal charge curve as a functon of default probablty. Ths decreasng relatonshp was frst ntroduced by Basel n 2001. 1 It s generally recognzed that ths decreasng relatonshp s justfed by the desre by regulators to reduce pro-cyclcal effects of the new accord. Snce then, some nsttutons have also utlzed the Basel II average asset correlaton functon for other portfolo credt rsk calculatons, such as measurng requred economc captal. Despte ts mportance, there have been few studes on the emprcal relatonshp between asset correlaton and default probablty. The most cted study s Lopez (2004), who studed the emprcal relatonshp between average asset correlaton, frm probablty of default, and frm sze. The study found that average asset correlaton s a decreasng functon of PD and an ncreasng functon of frm sze. Furthermore, the study argued that varables other than PD, for example, frm sze, need to be consdered n determnng average asset correlaton n the ASRF regulatory framework. The ncreasng relatonshp between average asset correlaton and PD s also found n German corporates by Düllmann and Scheule (2003), but they dd not fnd unambguous relatonshp between asset correlaton and PD. Usng SME data, Detsch and Petey (2004) found that the relatonshp s actually postve n the German populaton and U-shaped n France. Unlke the prevous studes that focus only on corporate data, ths paper examnes the valdty of ths assumpton for corporate, commercal real estate (CRE), and retal portfolos. Usng varous sources of data for these asset classes, we nvestgate the emprcal relatonshp between average asset correlaton and default probablty. Overall, we fnd lttle emprcal support for ths decreasng relatonshp n the data for corporate, CRE, and retal exposures. For corporate exposures, there s no strong decreasng relatonshp between average asset correlaton and default probablty when frm sze s properly accounted. For CRE and retal exposures, the emprcal evdence suggests that the relatonshp s more lkely to be an ncreasng one. We also provde economc arguments and emprcal evdence aganst the assumpton that average asset correlaton decreases wth default probablty. For corporate exposures, defaultng frms do not necessarly experence ncreases n ther frm-specfc rsk f there s a systematc negatve shock that causes wdespread defaults. For retal exposures, sub-prme borrowers are more senstve to the general economc condtons and thus, have hgher asset correlatons than prme borrowers. Our analyses suggest that t s mprudent to assume a decreasng relatonshp between average asset correlaton and default probablty n measurng portfolo credt rsk. In lght of these economc arguments and emprcal evdence, we encourage the Basel Commttee to revst the use of ths relatonshp n determnng bank captal requrement. The rest of ths paper proceeds as follows: Basel II Average Asset Correlaton on page 6 descrbes the Basel II average correlaton functons. Corporate on page 7 reports results for corporate exposures. Commercal Real Estate on page 16 reports results for CRE exposures. Retal on page 18 reports results for retal exposures. 1 Basel Commttee on Bankng Supervson, 2001c. Potental Modfcaton to the Commttee s Proposals, Press release dated November 5, 2001. THE DYNAMIC RELATIONSHIP BETWEEN AVERAGE ASSET CORRELATION AND DEFAULT PROBABILITY 5

Impact of Correlaton Msspecfcaton on Regulatory Captal on page 20 shows the mpact of underestmaton of asset correlaton on captal charge. Concluson on page 21 concludes our fndngs. 2 BASEL II AVERAGE ASSET CORRELATION In the ASRF model of Basel II, the asset value of a borrower s drven by a factor model: r = R φ + 1 R ε (1) where r s the asset return 2 of borrower, φ s the systematc factor representng the state of economy, s the, the percentage of systematc rsk, and ε s the dosyncratc factor of borrower. Two borrowers are correlated wth one another because they are both exposed to the systematc factor (wth potentally varyng degree). Mathematcally, the correlaton of borrower wth borrower j s gven by: R corr ( r, r ) = R R (2) j j Accordng to the Advanced Internal Ratngs Based Approach (A-IRB) of the Basel II, the asset correlaton parameter R s a decreasng functon of PD: 1 exp( c PD) 1 exp( c PD) R = a + b 1 1 exp( c) 1 exp( c) (3) Parameters a, b, and c depend on borrower type. For corporate borrowers and the so-called low asset correlaton CRE exposures, a=0.12, b=0.24, and c=50. 3 For the hgh asset correlaton CRE exposures, a=0.12, b=0.30, and c=50. For retal borrowers, Basel recommends asset correlaton of for resdental mortgage and asset correlaton of 4% for revolvng retal lnes of credt. Otherwse, a=0.03, b=0.16, and c=35 for other retal exposures. Fgure 1 plots the Basel II recommended asset correlaton functons for dfferent types of borrowers. Asset correlaton ranges from 12% to 24% for corporate borrowers 4 and t decreases as PD ncreases. A smlar pattern s assumed for CRE and retal exposures. Asset correlaton ranges from 12% to 24% for low asset correlaton CRE portfolos and 12% to 3 for hgh asset correlaton CRE portfolos. Asset correlaton for retal exposures ranges from 3% to 16%. All asset correlaton functons mply a decreasng relatonshp between asset correlaton and PD. 2 Here the asset return can be broadly nterpreted as the varable that drves the credt qualty change of the borrower. 3 There s a downward adjustment appled to European mddle market frms wth annual sales between 5 mllon and 50 mllon. 4 Ths does not nclude the small frm adjustment. For small frms the asset correlaton ranges between 8% and 2. 6

3 3 Corporate & CRE (Low) CRE (Hgh) Retal 2 Correlaton 2 1 1% 2% 3% 4% 6% 7% 8% 9% 1 PD FIGURE 1 Asset Correlaton Parameters n Basel II IRB Framework Although the decreasng relatonshp n (3) was frst specfed for regulatory captal calculaton, t has snce gathered general acceptance n the bankng ndustry for other credt rsk calculaton, for example, determnng requred economcal captal. Gven the crtcal role that asset correlaton plays n determnng portfolo credt rsk, t s mportant to understand the dynamc relatonshp between average asset correlaton and default probablty. Our paper now focuses on ths relatonshp. 3 CORPORATE Ths secton presents our results on the relatonshp between asset correlaton and PD for corporate exposures. We dscuss results from both the asset return data and the realzed default experence. We also examne the behavor of frms as they approach default. 3.1 Evdence from Asset Return Data We frst nvestgate the emprcal relatonshp between asset correlaton and PD usng Moody's KMV s asset return and Expected Default Frequency (EDF) data. Moody's KMV uses an opton theoretc approach to calculate EDF credt measures. Ths approach measures probablty of default for more than 30,000 publc traded companes worldwde. Durng EDF calculaton, asset returns can be obtaned from equty returns and fnancal statements. The sample data of asset returns of these publcly traded companes serves as the development dataset of Moody's KMV s Global Correlaton Model (GCorr). The values from the GCorr possess a smlar economc nterpretaton as the R s from Equaton (1), as both represent the percentage of systematc rsk and determne the asset correlaton. In the subsequent analyss of ths secton, we use the values from the GCorr as a measure of asset correlatons. Fgures 2 and 3 show the relatonshp between PD and for U.S. ndustral and fnancal frms, respectvely. We dvde all frms nto fve dfferent PD groups based on EDF quntles. Medan EDF for each PD group s shown along the horzontal axs. THE DYNAMIC RELATIONSHIP BETWEEN AVERAGE ASSET CORRELATION AND DEFAULT PROBABILITY 7

2 R- squared 1 PD1 [ 0.06%] PD2 [ 0.] PD3 [0.42%] PD4 [1.5] PD5 [ 12.] FIGURE 2 Medan values for dfferent PD groups Industral Frms 2 2 R- squared 1 PD1 [ 0.07%] PD2 [ 0.19%] PD3 [ 0.44%] PD4 [ 1.0] PD5 [ 6.02%] FIGURE 3 Medan values for dfferent PD groups Fnancal Frms Fgures 2 and 3 suggest asset correlaton decreases wth PD, as recommended by Basel II. However, our fndngs show that ths relatonshp s manly drven by frm sze. Large frms tend correlate more wth the economy and, hence, have hgher values. They also tend to have smaller PD values because they typcally experence lower asset volatlty. The prmary reason we observe a decreasng relatonshp between PD and s that larger frms tend to have hgher values and lower PD values, whle smaller frms have lower values and hgher PD values. To test ths hypothess, we need to nvestgate the relatonshp condtonal on frm sze. To do ths, we dvde the total sample nto fve dfferent groups based on sze quntles. Sze s measured by sales for ndustral frms and by book value of assets for fnancal frms. We further dvde each group nto fve dfferent PD groups based on EDF quntles. Fgures 4 and 5 show the medan for each sze-pd group for ndustral frms and fnancal frms, respectvely. The results ndcate that medan generally ncreases wth the sze for both ndustrals and fnancals. However, does not decrease unformly wth PD. Vsually, PD and do not show a strong decreasng relatonshp, whle PD and sze show a strong ncreasng relatonshp. 8

3 2 2 1 S4 S5 S3 PD1 PD2 PD3 PD4 PD5 S1 S2 FIGURE 4 Medan for each sze-pd Group Industral Frms 3 3 2 2 1 S4 S5 S3 PD1 PD2 PD3 PD4 PD5 S1 S2 FIGURE 5 Medan for each sze-pd Group Fnancal Frms To further llustrate the pont that sze plays an mportant role n explanng the varaton n whle PD does relatvely lttle, we run followng regresson models: Rsquared = α + β log( Sze) + ε (4) THE DYNAMIC RELATIONSHIP BETWEEN AVERAGE ASSET CORRELATION AND DEFAULT PROBABILITY 9

ε = α + β (5) PD where ε s the resdual obtaned from the frst regresson n equaton (4). We frst regress the frm s values on the log of ther szes. Table 1 summarzes the regresson result. The coeffcent for sze s postve and sgnfcant, whch ndcates that frms values ncrease wth sze. from the regresson s 45.03%, whch ndcates sze tself explans about 4 of the total varaton n. Next, we regress resduals from the frst regresson on PD. By constructon, these resduals are ndependent of sze. Thus, we can explore the relatonshp between PD and R- squared whle controllng for frm sze. Regresson result s summarzed n Table 2. The coeffcent for PD s postve and sgnfcant, whch ndcates ncreases as PD ncreases. The sgn of the coeffcent does not mply a negatve relatonshp between PD and. In addton, from the second regresson s 0.0032, mplyng the addtonal porton n total varaton n explaned by PD s very mnmal. Taken together, these regresson results suggest that sze s the man drver n explanng the varaton n. Table 1 Regresson Result - values on log(sze) Coeffcent Standard Error t Statstc Intercept 0.0749 0.0019 38.5918 0.4503 log(sze) 0.0193 0.0003 57.9019 Table 2 Regresson Result Resduals on PD Coeffcent Standard Error t Statstc Intercept -0.0014 0.0010-1.3188 0.0032 PD 0.0505 0.0139 3.6273 To further test the relatonshp n (3) whle controllng for frm sze, we also estmate the followng regresson: Rsquared = α + β log( Sze) + γ ω + ε (6) where 1 exp( 50 PD) 1 exp( 50 PD ω = 0.12 + 0.24 1 1 exp( 50) 1 exp( 50) Note that ω s the recommended, and t s a decreasng functon of PD. If there s a sgnfcant relatonshp between and PD after controllng for frm sze, we expect γ to be sgnfcant. However, as summarzed n table 3, γ turns out to be nsgnfcant at the 9 level. Furthermore, addtonal obtaned by ncludng n the model s 0.0004 (0.4507 0.4503), whch s mnmal. The result confrms that, whle sze provdes dstnct nformaton explanng the varaton n, PD provdes lttle addtonal nformaton. In other words, after controllng for the frm s sze, the asset return data does not support the decreasng relatonshp n (3). ω Table 3 Regresson Result values on log(sze) and ω Coeffcent Standard Error t Statstc Intercept 0.0670 0.0051 13.2242 0.4507 Log(Sze) 0.0190 0.0004 47.7381 ω 0.0467 0.0279 1.6773 10

3.2 The Change n Asset Correlaton as a Frm Approaches Default A major argument for the decreasng relatonshp between PD and s that, as a frm approaches default, ts frm-specfc rsk ncreases, and ts systematc rsk (hence ) decreases. On the other hand, one could argue ths relatonshp s not always true. When a sgnfcant negatve shock trggers wdespread defaults n a specfc sector, the percentages of systematc rsk of those frms could ncrease as the result of the large negatve shock. Ths secton examnes these arguments emprcally. To study the change n asset correlaton as frms approach default, we estmate the tme seres of of defaulted frms from 24 months pror to default to mmedately before default. For ths analyss, our default data spans from 1994 to 2006, ncludng defaults for U.S. non-fnancal frms. Fgure 6 shows the total number of defaults by year from 1994 to 2006. 2001 s of specfc nterest, as ths year experenced the largest number of defaults n the sample perod, a result of the large, negatve shock to the overall economy. Fgure 7 shows the tme seres dynamcs of the 25 th, medan, and 75 th percentle of values of all defaulted frms as they approach default. The medan decreased slghtly as frms approached default, but the decrease was not very steep. Fgure 8 presents the results for all frms that defaulted n 2001, durng a sgnfcant negatve economc shock. In ths case, the medan ncreased slghtly at the tme of default for frms that defaulted n 2001. Interestngly, when we focus on a specfc sector, for example, telecom, we observe ncreases n as defaulted frms n the sector approach default. The telecom ndustry experenced wdespread defaults n 2001 due to the large negatve shock to the entre sector. In ths case, actually ncreases as PD ncreases (Fgure 9), whch contradcts the decreasng relatonshp between PD and. To summarze the results from Fgures 7 through 9, we do not fnd strong, emprcal support for the argument that frms systematc rsk decreases as they approach default. On the contrary, they could actually ncrease durng a sectorwde negatve shock. 300 250 200 Defaults 150 100 50 0 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 Year FIGURE 6 Annual Number of Defaults THE DYNAMIC RELATIONSHIP BETWEEN AVERAGE ASSET CORRELATION AND DEFAULT PROBABILITY 11

18% Medan P25 P75 12% 9% 6% 3% 24m 18m 12m 6m 0m Month Pror to Default FIGURE 7 Dynamcs All Frms Defaulted 1994 to 2006 (1,261 frms) 18% Medan P25 P75 12% 9% 6% 3% 24m 18m 12m 6m 0m Month Pror to Default FIGURE 8 Dynamcs All Frms Defaulted n 2001 (263 frms) 12

2 Medan P25 P75 2 1 24m 18m 12m 6m 0m Month Pror to Default FIGURE 9 Dynamcs All Frms Defaulted n Telecom Industry 2001 (26 frms) 3.3 Evdence from the Default Data In the prevous sectons, we nvestgated the relatonshp between asset correlaton and PD usng asset returns data. In ths secton, we explore the relatonshp usng realzed defaults data. We estmate the quarterly realzed default rate for each sze PD group (as defned n secton 3.1) from Aprl 1980 through September 2008. We then estmate the defaultmpled asset correlaton from the realzed default rate seres. Default-mpled asset correlaton can be estmated usng followng equatons: 5 JDP = N N PD N PD R (7) 1 1 2 ( j ( ), ( j ), ) j j Dt Dt JDP j = w (8) t j t N N PD and PD j are the mean realzed default rates for groups and j, respectvely. JDP j s the jont default probablty t between groups and j, and t s estmated usng the realzed default rates as n equaton (8). s the weght appled to each quarter. We use equal weghts n our study. Default mpled asset correlaton s then backed out from the equaton (7). The default-mpled asset correlaton for each sze-pd group for ndustral frms 6 s presented n Fgure 10. Even after controllng for sze, we do observe a negatve relatonshp between PD and asset correlaton. However, as we shall see subsequently, ths decreasng relatonshp s, to a certan degree, ntroduced by the bases assocated wth estmatng default-mpled asset correlaton for groups wth a low number of defaults, ether as the result of low PD or a low number of frms wthn the group. t w t 5 For more detals on the estmaton methodology, please see Asset Correlaton, Realzed Default Correlaton, and Portfolo Credt Rsk. 6 For fnancal frms, there were no defaulted frms for many sze-pd groups, whch made nvestgatng the relatonshp between PD and asset correlaton dffcult. THE DYNAMIC RELATIONSHIP BETWEEN AVERAGE ASSET CORRELATION AND DEFAULT PROBABILITY 13

4 3 3 2 Asset Correlaton 2 1 S5 S4 PD1 PD2 PD3 PD4 PD5 S3 S2 S1 FIGURE 10 Default-Impled Asset Correlaton for Each Sze PD Group Industral Frms To understand the magntude of the bases assocated wth estmatng default-mpled asset correlaton, we conducted the followng smulaton exercse: 7 1. Generate correlated asset returns usng the sngle-factor model framework n equaton (1). An asset correlaton parameter s assumed to mpose a correlaton structure on the smulated asset returns. For each pont of tme (total number of tme perods = T), generate N asset returns (the number of frms = N). Thus, we generate T x N asset returns. 2. Count defaults Whenever an asset return falls below the default threshold, count defaults. PD parameter s used to determne default threshold. 3. Get default rate seres Usng the number of defaults counted for each perod and the number of frms (N), obtan default rate seres. 4. Estmate default-mpled asset correlaton Usng the default rate seres, estmate default-mpled asset correlaton. 5. Repeat steps 1 through 4 10,000 tmes to form a dstrbuton of default-mpled asset correlatons. The smulaton exercse suggests that when PD s low, asset correlaton s overestmated. Fgures 11 through 13 show the medan default-mpled asset correlaton estmates along the dfferent levels of PD. The total number of perods s assumed to be 114, the actual number of quarters between Aprl 1980 and September 2008, (28.5 years). The total number of frms s assumed to be 165, the average number of frms n each group. As shown n Fgures 11 through 13, the smulated default-mpled asset correlaton shows a decreasng relatonshp wth PD, even though the true stays constant. Default-mpled asset correlaton s overestmated n the low PD range, and the sze of bas decreases as the level of PD ncreases. The absolute dfference between true and estmated value can be as large as 33% n the case where PD s 1bp and the true s 1%. The bas assocated wth the estmates s partly, f not entrely, responsble for the observed decreasng relatonshp between PD and default-mpled asset correlatons. 7 For the n-depth analyss of how each parameter affects default correlaton estmates, please see Asset Correlaton, Realzed Default Correlaton, and Portfolo Credt Rsk. 14

4 3 3 Estmates True 2 2 1 0.01% 0.1% 1% 2% 3% PD FIGURE 11 Smulated Default-Impled Asset Correlaton (True = 1%) 4 3 3 Estmates True 2 2 1 0.01% 0.1% 1% 2% 3% PD FIGURE 12 Smulated Default-Impled Asset Correlaton (True = 1) THE DYNAMIC RELATIONSHIP BETWEEN AVERAGE ASSET CORRELATION AND DEFAULT PROBABILITY 15

4 3 3 Estmates True 2 2 1 0.01% 0.1% 1% 2% 3% PD FIGURE 13 Smulated Default-Impled Asset Correlaton (True =2) 4 COMMERCIAL REAL ESTATE Ths secton descrbes the relatonshp between asset correlaton and PD among CRE borrowers. Recall that Basel s recommendaton for asset correlaton for CRE exposures s a decreasng functon of PD and ranges from 12% to 24% for low asset correlaton CRE portfolos and from 12% to 3 for hgh asset correlaton CRE portfolos. In fact, the functon form for low asset correlaton CRE exposures s the same for corporate exposures. To test the valdty of Basel s recommendaton for CRE exposures, we borrow the results from a Moody s KMV study on the asset correlatons of US CRE asset correlatons. 8 Usng Amercan Councl of Lfe Insurers (ACLI) delnquency rates of CRE loans n dfferent regons, we can examne the relatonshp of delnquency rates mpled wth PD values for dfferent property types. Results are plotted n Fgures 14 through 18. 5 4 3 2 1 0. 0. 1. 1. 2. 2. PD FIGURE 14 GCorr CRE R-Squared Estmates: Mult-Famly Housng 8 Please see Modelng Asset Correlatons for Commercal Real Estate Exposures n Credt Portfolos. 16

4 3 2 1 0. 0. 1. 1. 2. PD FIGURE 15 GCorr CRE R-Squared Estmates: Retal 5 4 3 2 1 0. 0. 1. 1. 2. PD FIGURE 16 GCorr CRE R-Squared Estmates: Offce 5 4 3 2 1 0. 0. 1. 1. 2. PD FIGURE 17 GCorr CRE R-Squared Estmates: Industral THE DYNAMIC RELATIONSHIP BETWEEN AVERAGE ASSET CORRELATION AND DEFAULT PROBABILITY 17

7 6 5 4 3 2 1 0. 1. 2. 3. 4. PD FIGURE 18 GCorr CRE R-Squared Estmates: Hotels Clearly, the evdence n Fgures 14 through 18 does not support a decreasng relatonshp between PD and asset correlaton for CRE exposures. In fact, these estmates show a postve relatonshp between PD and, although t s not very strong. Our results suggest that we need to be more careful when explorng the relatonshp between PD and asset correlaton for dfferent asset classes. Assumng a smlar pattern obtaned from publc frm data can potentally underestmate the rsk for CRE portfolos n hgh PD range. 5 RETAIL In ths secton, we study the relatonshp between average asset correlaton and PD for retal exposures. Recall, the Basel average asset correlaton for retal exposures ranges from 3% to 16%, as a decreasng functon of PD. Ths functon suggests that a retal borrower wth a lower credt score has lower asset correlaton than a borrower wth a hgher credt score. We are not aware of any justfcatons for ths relatonshp for retal exposures. There are a number of economc arguments supportng sub-prme borrowers as more senstve to general economc condtons than prme borrowers and, hence, havng a hgher percentage of systematc rsk (.e., hgher ). Frst, the rato of fnancal oblgatons over dsposable ncome tends to be hgher for sub-prme borrowers. As we see n Fgure 19, between 2005 and 2007, fnancal oblgatons are less than of dsposable ncome for Amercan prme borrowers and more than 3 for sub-prme borrowers. Ths data mples that sub-prme borrowers are more senstve to economc changes. 40 Prme Subprme Renter 35 30 25 20 15 FIGURE 19 10 00 01 02 03 04 05 06 07 Fnancal Oblgaton Ratos, Percentage of Dsposable Income, Source: Moody s Economy.com 18

Second, sub-prme borrowers spend a hgher percentage of ther ncome on basc necesstes such as utltes and commutng. Fgure 20 shows that for the lowest 2 of consumers, most lkely sub-prme borrowers, the expense for fuel ol, electrcty, natural gas, and gas s more than 1 of ther spendng. These borrowers have fewer cushons aganst an economc downturn. 12 10 8 Fuel ol Electrcty Natural gas Gas 6 4 2 Lowest 2 Second 2 Mddle 2 Fourth 2 Top 2 FIGURE 20 Expense Percentage of Consumer Spendng, Source: Moody s Economy.com The above economc arguments suggest that retal borrowers wth hgher PD values (e.g., sub-prme borrowers) tend to have hgher asset correlatons. To test ths thess emprcally, we estmated values of varous retal loans for 250 Metropoltan Statstcal Areas (MSAs) n the U.S. usng delnquency rates for varous retal loans provded by Credtforecast.com. 9 Fgure 21 shows the medan of these values and suggests that medan asset correlatons are hgher for sub-prme borrowers than for prme borrowers. Medan Retal Asset Correlatons of 250 MSA Retal Markets Prme Sub Prme 14.0 12.0 10.0 Medan R-Squared 8.0 6.0 4.0 2.0 0.0 AUTO BANKCARD CONSUMER FIRSTMORTGAGE HOMEEQUITY STUDENTLOAN FIGURE 21 GCorr Retal Medan Estmates In addton to our emprcal evdence, Cowan (2004) showed that default correlaton ncreases as the nternal credt ratng declnes, usng a large portfolo of sub-prme loans from an anonymous sub-prme lender. In ther paper, they present the frst formal study of default correlaton wthn a sub-prme mortgage loan portfolo. They analyze sx-month default correlaton usng both actual defaults (foreclosures) and a more broad defnton of delnquency. As antcpated, 9 For more detals, please see Modelng Retal Correlatons n Credt Portfolos. THE DYNAMIC RELATIONSHIP BETWEEN AVERAGE ASSET CORRELATION AND DEFAULT PROBABILITY 19

the magntude of default correlaton ncreases as the nternally assgned rsk grade declnes. To facltate a lke-to-lke comparson, we calculated mpled asset correlaton from ther reported default correlatons and present the results n Fgure 22. Agan, the results do not support a decreasng relatonshp between average asset correlaton and PD. Default Correlaton and Asset Correlaton Impled by Default Rate and 90 Days Delnquency Rate Impled Asset Corr (Default) Default Corr (Default) Impled Asset Corr (90 Delnquent) Default Corr (90 Delnquent) Impled Asset Corr 2 18% 16% 14% 12% 1 8% 6% 4% 2% 7% 6% 4% 3% 2% 1% Default Corr AA A B C CC FIGURE 22 Default Correlatons and Impled Asset Correlatons 6 IMPACT OF CORRELATION MISSPECIFICATION ON REGULATORY CAPITAL In ths secton, we demonstrate the mpact of potental msspecfcaton of average asset correlaton on the calculaton of regulatory captal. Conceptually, the (unjustfed) decreasng relatonshp specfed n (3) would yeld (unjustfed) lower regulatory captal for borrowers wth hgh PD. To llustrate ths, we use the data from our CRE secton. Recall, under Basel IRB, the captal charge s gven by: R 1 1 1 captal = LGD N N PD N ( ) + (0.999) PD LGD (9) R R 1 1 where R s the average asset correlaton or n a sngle-factor framework. R s obtaned from the Basel asset correlaton functon and actual estmates for CRE. For each property type, we ftted a lnear lne based on the actual PD and R-square estmates. All fve property types mply a postve relatonshp between and PD. We then use these lnear functons to attan R for each PD level, and calculate captal charges usng equaton (9). Fgure 23 presents the captal charges along wth PD based on dfferent correlaton estmates, together wth the Basel II functon. 10 Fgure 23 llustrates that the decreasng relatonshp between PD and assumed n Basel s asset correlaton functon can lead to an underestmaton of captal charge. Ths underestmaton can be more severe n the hgh PD range. A smlar result s expected for retal exposures as well, as our estmates suggest that there s a postve relatonshp between PD and for retal nstruments. Ths smple analyss demonstrates that captal can vary largely wth the assumptons made to the asset correlaton parameter. Usng the parameters from the exstng Basel II requrement could lead to sgnfcant under-estmaton of captal requrements for rsky borrowers. 10 LGD s assumed to be 5 20

3 BASEL CRE - Retal CRE - Offce CRE - Mult-Famly Housng CRE - Industral CRE - Hotel 2 2 Captal 1 0.1% 0.3% 0. 0.7% 0.9% 1.1% 1.3% 1. 1.7% 1.9% PD FIGURE 23 Captal Charge Based on Dfferent Correlaton Estmates 7 CONCLUSION Our analyses fnd lttle emprcal support for the decreasng relatonshp between asset correlaton and default probablty for CRE and retal. In addton, we also fnd that, after controllng for frm sze, there s no strong negatve relatonshp between asset correlaton and default probablty for corporate. The exact relatonshp between asset correlaton and default probablty s complcated and may not be consstent across asset classes. For CRE and retal exposures, the emprcal evdence suggests that the relatonshp s more lkely to be an ncreasng one. The stylzed decreasng relatonshp n the IRB approach of the Basel II Accord does not appear to have theoretcal nor emprcal support and may underestmate portfolo rsk for hgh default probablty portfolos. In lght of our research, n addton to the prevous results by Düllmann and Scheule (2003) and Detsch and Petey (2004), we encourage the Basel Commttee to revst the use of ths relatonshp n bank captal requrement. THE DYNAMIC RELATIONSHIP BETWEEN AVERAGE ASSET CORRELATION AND DEFAULT PROBABILITY 21

REFERENCES Adran M. Cowan and Charles D. Cowan, "Default Correlaton: An Emprcal Investgaton of a Subprme Lender, workng paper, 2004. Basel Commttee on Bankng Supervson, Internatonal Convergence on Captal Measurement and Captal Standards, June 2006. Mchael Detsch, Joёl Petey, Should SME exposures be treated as retal or corporate exposures? A comparatve analyss of default probabltes and asset correlatons n French and German SMEs, 2004, Journal of Bankng and Fnance,, 773-788. Klaus Düllman, Harald Scheule, Determnants of the Asset Correlatons of German Corporatons and Implcatons for Regulatory Captal, workng paper, 2003. Jose A. Lopez, The Emprcal Relatonshp between Average Asset Correlaton, Frm Probablty of Default and Asset Sze, 2004. Journal of Fnancal Intermedaton, 13, 265 283. Jose A. Lopez, Emprcal Analyss of the Average Asset Correlaton for Real Estate Investment Trusts, 2008, Quanttatve Fnance, 9, 217 229. Joy Wang, Jng Zhang, and Amnon Levy, Modelng Retal Correlatons n Credt Portfolos, Moody s KMV whte paper, 2009. Jng Zhang, Fanln Zhu, and Joseph Lee, Asset Correlaton, Realzed Default Correlaton, and Portfolo Credt Rsk, Moody s KMV whte paper, 2008. 22