a framework for loss given default validation of retail ...€¦ · the journal of risk model...

The Journal of Risk Model Validation (23–48) Volume 4/Number 1, Spring 2010

A framework for loss given default validationof retail portfolios

Stefan HlawatschDepartment of Banking and Finance, Otto-von-Guericke University Magdeburg,Postfach 4120, 39106 Magdeburg, Germany; email: [email protected]

Peter ReichlingDepartment of Banking and Finance, Otto-von-Guericke University Magdeburg,Postfach 4120, 39106 Magdeburg, Germany; email: [email protected]

Modeling and estimating loss given default (LGD) is necessary for banksthat apply for the internal ratings based approach for retail portfolios.To validate LGD estimations, there are only a few approaches discussedin the literature. In this paper, two models for validating relative LGDand absolute losses are developed. The validation of relative LGD isimportant for risk-adjusted credit pricing and interest rate calculations. Thevalidation of absolute losses is important to meet the capital requirementsof Basel II. Both models are tested with real data from a bank. Estimationsare tested for robustness with in-sample and out-of-sample tests.

1 INTRODUCTION

According to Basel II, banks can choose between two approaches to measure theircredit risk: the standardized approach and the internal ratings based approach. So-called internal ratings based approach banks have to underlay credits with equitydepending on the unexpected loss (denoted as UL in equations) according toEquation (1):1

UL =

relative unexpected loss︷︸︸︷[�

(�−1(PD) +√

R · �−1(0.999)√1 − R

)· LGD︸︷︷︸

maximum loss

− PD · LGD︸︷︷︸expected loss

]· EAD (1)

The unexpected loss equals the difference between the so-called maximum loss,which is computed as the value-at-risk of the loss, and the expected loss, which iscomputed as the product of probability of default (PD) and loss given default (LGD).Here, R denotes the correlation coefficient of the PD with the systematic risk factor.

We thank Hendrik Ritter and two anonymous referees for helpful comments.1� indicates the standard normal distribution function and �−1 denotes the inverse of �.

23© 2010 Incisive Media. Copying or distributing in print or electronic forms without written permission of Incisive Media is prohibited.

24 S. Hlawatsch and P. Reichling

FIGURE 1 PD and LGD elasticity of risk-weighted assets for the retail internal ratingsbased approach.

1

00.1 0.2 0.3 0.4

PD

PD elasticity (secured by residential properties)

0.5 0.6 0.7 0.8

–1

–2

–3

–4

PD elasticity (other)

LGD elasticityPD elasticity (qualifying revolving exposures)

The product of the thus computed relative unexpected loss and the exposure at default(EAD) results in the unexpected loss in monetary units.

According to Article 87 No. 6 and 7 of Europe’s Capital Requirement Directive,2

banks have to estimate PD and LGD for retail claims or contingent retail claimson their own. Furthermore, the estimation procedures have to be validated forrobustness and accuracy of the models. This validation should transcend the simplecomparison of historical data with estimated parameters, as it is mentioned in AnnexVII, Part 2 No. 112, Capital Requirement Directive. While validation techniques forPD estimations are discussed extensively in the literature, research on quantitativevalidation instruments for LGD estimation models is rare.

Validating LGD estimations is crucial because the required capital reacts moresensitively to changes in LGD than to changes in PD. By way of illustration, theLGD and the PD elasticities of unexpected loss are shown in Figure 1.3

The LGD elasticity is constant and amounts to one. However, the PD elasticitiesfor the shown subcategories for retail claims (up to a PD of 50%) are absolutelysmaller than the LGD elasticity. Therefore, the risk-weighted assets react moresensitively to changes in LGD.4 Thus, the high sensitivity of the risk-weight functionwith respect to the LGD (in the relevant PD range up to 50%) implies the necessity

2Directive 2006/48/EC of the European Parliament and of the Council.3The right hand side of Equation (1) is multiplied by factor 12.5 in Figure 1, so that unexpectedloss equals the risk-weighted assets.4The partly negative PD elasticity shown in Figure 1 results from the fact that, with an increasingPD, the unexpected loss becomes smaller and the expected loss, which reduces the capitalrequirement, rises. Expected loss’s are considered by depreciations, provisions or write-offs and,therefore, no capital underlay is required.

The Journal of Risk Model Validation Volume 4/Number 1, Spring 2010

© 2010 Incisive Media. Copying or distributing in print or electronic forms without written permission of Incisive Media is prohibited.

A framework for LGD validation of retail portfolios 25

for precise estimations of LGD.5 An evaluation of the accuracy of LGD estimationscan be done by the technique of validation.

Our paper is organized as follows. After a brief review of the literature of differentLGD estimation models, two validation models are developed in Section 2. For anempirical analysis, real data from a bank is used. Section 3.1 describes the data andempirical analysis and Section 3.2 reports the results. Section 4 concludes.

2 LOSS GIVEN DEFAULT VALIDATION

2.1 Literature review

There are four different approaches to compute the LGD: workout LGD, marketLGD, implied market LGD, and implied historical LGD.6

The workout LGD belongs to the group of so-called explicit methods of LGDestimation; “explicit” here refers to the data used. Explicit methods use historicalLGDs of defaulted credits in order to derive prognoses for future LGDs. The workoutLGD is cashflow-oriented. To compute the workout LGD, all recoveries, as well asall costs, are considered in the period from the day of the credit event up to finalrecovery. In order to consider different points in time where costs and recoveriesemerge, payments have to be discounted to the day of the credit event. Therefore, theworkout LGD is computed as follows:7

LGDi =EADi −

∑nj=1 Ei,j (r) +

∑mk=1 Ki,k(r)

EADi

= 1 −∑n

j=1 Ei,j (r) −∑m

k=1 Ki,k(r)

EADi

(2)

where:

Ei,j (r) = discounted recoveries j of credit i

Ki,k(r) = discounted costs or losses k of credit i

r = discount rate

Typical recoveries are collaterals or securities.8 Examples of costs and losses are aloss on interest payments, opportunity costs for equity, handling costs, and workoutcosts such as overhead costs of the recovery department.

5Despite the higher LGD sensitivity within the relevant range, this does not imply a less precise orless robust estimation of PD.6See Basel Committee on Banking Supervision (2005b, pp. 61–62).7See Basel Committee on Banking Supervision (2005b, p. 66).8Especially for tangible fixed assets, for example real-estate or machinery, market prices canchange until the collateral utilization. Therefore, haircuts should be computed to consider this lossin value. See also Basel Committee on Banking Supervision (2005b, p. 67).

Research Paper www.thejournalofriskmodelvalidation.com



The discount rate to determine the economic loss has to be risk-adjusted. Inparticular, for parameters such as collaterals or workout costs, no market exists.Therefore, the determination of the discount rate is difficult. If historical interestrates are used, the risk-free interest rate plus a loss impact or the initially agreedinterest rate can be used.9

After computing the workout LGD, the estimation model can be developedusing, for example, regressions.10 The independent variables that should be usedhere depend on the institution and branch. Commonly accepted variables includeprovisions of security, repayment priority, industry affiliation, macro-economicfactors such as economic growth or ratings.11

A further explicit method to determine the LGD is the so-called market LGDmethod. In this approach, market prices of publicly traded defaulted loans orsecuritized credits are used. After a default, the recovery rate can be determinedby the market value of the loan because investors anticipate possible proceeds fromrealizations as well as possible costs. Thus, loss results in the difference betweenthe par offering price and the market price after default. For publicly traded loans,this data is collected by rating agencies. The appeal of this concept comes from thefact that only the recovery rate is needed for the LGD computation. However, thisrecovery rate corresponds to the market price after default for initially priced at parloans.

In this approach, it is critical that several parameters are based on subjectiveestimates. It remains doubtful which time horizon should have been taken after thepoint in time of default of the loan, such that all investors anticipate possible earningsand costs.12 In addition, internal workout costs of the bank are not reflected by themarket price. Moreover, market prices are influenced by both supply and demand.Therefore, on illiquid markets, the use of market prices can lead to false estimates ofLGD.

After computing the market LGD, the development of the estimation model issimilar to the workout LGD. A well-known model for LGD estimation based onmarket LGD is LossCalc 2.0 from Moody’s KMV.13 The market LGD is onlysuitable for securitized loans or credits due to the need for market data. Unincludedworkout costs have to be integrated by an adjustment of the LGD. Therefore, internalbank data is required. Alternatively, the workout LGD can be used.

A further possibility to determine LGD is the implied market LGD, which belongsto the group of implicit methods. Non-defaulted securitized loans or credits form the

9Brady et al (2006) empirically show that discount rates significantly differ for different branchesof industry and ratings. The determined discount rate ranges from 0.9% to 29.3%.10Hamerle et al (2006) use a two stage regression. Siddiqi and Zhang (2004) assume that theLGD is beta distributed and transform the LGD into a normally distributed random variable beforeestimation.11For more examples and analyses of influencing factors see Schuermann (2005).12Moody’s, for example, uses a time horizon of one month after default. See also Gupton (2005).13Gupton (2005) gives an overview of the model and procedure of LossCalc 2.0.




database for this approach. Here, it is assumed that the spread between a loan-specificinterest rate and the risk-free interest rate equals the expected loss as a percentage. Ifthe spread is known, the LGD comes from the ratio of spread and default probability.This concept is based on the model of Jarrow et al (1997). The value of a loan V

equals the value of a loan without risk Vrf multiplied by the probability of non-default plus the value of a loan without risk multiplied by the recovery rate (RR)and the PD. Then, the following valuation equation holds for risky bonds (see Jarrowet al (1997)):

V = Vrf · (1 − PD) + Vrf · PD · RR (3)

Implied market LGD models differ only in the statistic modeling of the parametersof Equation (3) and different interpretations of the recovery rate. In general, thereare three possible interpretations. The recovery rate is defined as a portion of theissue price, a portion of the current present value or a portion of the value of the loanshortly before default (see Bakshi et al (2006)).

Madan and Unal (2000) and Bakshi et al (2006) use a hazard process in orderto model the default probability. The recovery rate, together with the process of therisk-free interest rate term structure, is modeled by stochastic processes.14 Also, theuse of alternative interest rate spreads is discussed.15 For the implied market LGD,the decomposition of the interest rate spread into its components is crucial. Since theinterest rate spread may contain a liquidity premium as well as a risk premium forthe unexpected loss, the applied asset pricing models must be able to determine thesingle components separately.

The implied historical LGD is also a concept of implicit LGD determination.According to the Basel Committee on Banking Supervision, the use of the impliedhistorical LGD is only allowed for retail portfolios.16 According to this approach,banks are allowed to determine their LGD on the basis of PD estimations. Thedatabase consists of historical loss data of retail portfolios. Here, the LGD of a retailcredit is computed similarly to that of the implied market LGD: EL = PD · LGD.

All the mentioned LGD models possess the same shortcoming, namely thatLGD and PD are estimated independently. Therefore, the Basel Committee onBanking Supervision has tried to overcome this weakness by introducing a so-calleddownturn LGD. Downturn LGD is defined as “an LGD for each facility that aimsto reflect economic downturn conditions where necessary to capture the relevantrisks.” Furthermore, banks “must consider the extent of any dependence betweenthe risk of the borrower and that of the collateral or collateral provider.”17 Banksare free to choose an appropriate downturn LGD model. However, the bank has to

14For a detailed overview of the modeling of the parameters see Ong (1999, pp. 61–92).15For the use of credit default swap spreads to determine implicit recovery rates see Pan andSingleton (2008).16See Basel Committee on Banking Supervision (2004, paragraph 465).17Basel Committee on Banking Supervision (2004, paragraphs 468 and 469).




identify the relevant risk drivers for each debt type to verify possible dependenciesbetween PD and LGD and to integrate these dependencies into the downturn LGDmodel.18

Hartmann-Wendels and Honal (2006) use a linear regression model includinga downturn dummy variable. As a result, the downturn LGD exceeds the default-weighted average LGD by up to eight percentage points. Miu and Ozdemir (2006)analyze the correlation between PD and LGD with respect to a single risk driver andestimate the increase in the mean LGD needed to achieve the adequate economiccapital. Barco (2007) and Giese (2005) integrate the correlation between PD andLGD directly into Merton’s framework under the assumption that PD and LGD bothdepend on the same systematic risk factor. Rösch and Scheule (2009) extend theseapproaches assuming that PD and LGD depend on different systematic risk factors,which are correlated.

2.2 Proportional decomposition of the credits

As seen in the previous section, the discussed concepts of LGD computation aredeveloped on defaulted credits, especially for retail claims. If realized LGDs areknown, a validation model should use those realized LGDs as a benchmark forLGD estimation models.19 This idea is also used in PD validation.20 Therefore, thebasic idea of our validation models is to use well known PD validation techniquesfor LGD validation purposes. Firstly, transformed ratios of the PD validation, forexample area under curve (AUC) or accuracy ratio, are computed for realized LGDs.Since the interpretation of these ratios is different for PD validation, it is necessaryto compare the ratios generated on realized LGDs with those ratios generated onestimated LGDs. As a result, the quality of the LGD estimation model does notdepend on the ratio itself but on the comparison of the ratios of realized andestimated LGDs. The better the estimation model fits, the more equal the comparedratios are.

In order to simplify the interpretation of the following model, a sample retailportfolio of a bank is analyzed. At first, the technique of proportional decompositionis developed on this sample retail portfolio. Subsequently, the model is applied to realdata. Assume the portfolio consists of 100 credits with individual EADs and losses.The number of defaulted credits amounts to 54 with a total loss of e600,000. Thetotal EAD adds up to e3 million, which results in an average LGD of 20% accordingto Appendix E.

18See Basel Committee on Banking Supervision (2005a) for further details.19Li et al (2009) provide an overview of different validation techniques for LGD models, wherethey focus on descriptive metrics of LGD models using discretized LGD rating scales.20See Engelmann et al (2003) and Sobehart and Keenan (2001) for an overview of PD validationtechniques.




Each EAD is divided into n portions of equal size.21 The number of portionsshould realign to the EAD amount.22 For every portion i = 1, . . . , n of creditk = 1, . . . , K , a binary variable di,k is defined as follows:

di,k ={

1 if portion i of the EADk is defaulted

0 if portion i of the EADk is not defaulted(4)

Furthermore, a second binary variable ndi,k is defined by ndi,k ≡ 1 − di,k . Credit 1of our sample retail portfolio is not defaulted. Therefore, all di possess a valueof zero and all ndi possess a value of one. For credit 12, the variables d1,12 tod462,12 possess a value of one and the variables d463,12 to d1,000,12 possess a valueof zero.23 After computing the variables di and ndi for all credits K , the variablesare added up for each portion. Therefore, Di ≡

∑Kk=1 di,k represents the number of

credits where portion i is defaulted. Similarly, NDi is defined as NDi ≡∑K

k=1 ndi,k

and corresponds to the number of credits where portion i is not defaulted. As aconsequence, the sum of Di and NDi for each portion must be equal to the numberof credits. The results for our sample retail portfolio are shown in extracts in Table 1(see page 30).

The decomposition in Table 1 is based on the assumption that all credits exhibitan LGD smaller or equal to one. However, this is not always ensured. For creditswith high workout costs and small EADs, LGDs above one are possible. In order toprevent this, the variables di and ndi can be redefined as follows:

di,k ={

1 if portion i of the double EADk is defaulted

0 if portion i of the double EADk is not defaulted(5)

With this redefinition, credits with LGDs smaller or equal to 200% are possible.24

The further computations are carried out similarly to the model with simple EADs.Analogously to measures of the accuracy of rating functions, hit rates and false

alarm rates can be computed for each portion. The interpretation of these rates is,however, not equal to those of rating functions. The hit rate hri and the false alarm

21If n = 100 is chosen the decomposition corresponds to a percental decomposition, for n = 1,000it corresponds to an one-tenth of a percent decomposition.22If the portfolio consists of credits with EADs below e1,000, a more precise decomposition thann = 1,000 makes little sense, since every portion then corresponds to an amount of less than oneeuro. For our sample retail portfolio, n = 1,000 was selected. Here, the resulting portion amountsto e51.60 in maximum.23The figure 462 was rounded. For a more precise decomposition, the deviation converges to zero.In our case, the deviation of the loss due to rounding amounts to e2.80 and therefore lies in anegligible range.24To reduce rounding errors, the number of portions should be increased when rising the permittedLGD.




TABLE 1 Proportional decomposition of our sample retail portfolio.

Portion i in % Di NDi

1 54 462 54 463 54 464 54 465 54 46...

......

450 12 88451 11 89452 11 89453 9 91454 8 92455 8 92456 8 92457 8 92458 8 92459 8 92

......

...

996 3 97997 3 97998 3 97999 3 97

1,000 3 97

The second column indicatesthe number of defaults withinportion i and the third col-umn shows the number of non-defaults within portion i.

rate fari are computed as follows:

hri = Di∑ni=1 Di

and fari = NDi∑ni=1 NDi

(6)

The hit rate hri is the fraction of portion i of all defaulted portions. The false alarmrate fari is the fraction of portion i of all portions that are not defaulted. If the hitrates and the false alarm rates are summed up, the cumulated hit rate HRj and thecumulated false alarm rate FARj for portion j , respectively, result in:25

HRj =j∑

i=1

hri = j

n· LGDj

LGDwhere LGDj =

∑j

i=1 Di

j · K LGD =∑n

i=1 Di

n · K

25For the derivation see Appendix A.




FARj =j∑

i=1

fari = j

n· RRj

RRwhere RRj =

∑j

i=1 NDi

j · K RR =∑n

i=1 NDi

n · K (7)

The cumulated hit rate corresponds to the ratio of the average LGD of the firstj portions (denoted by LGDj ) to the average LGD over all portions (denoted byLGD), multiplied by a weighting factor. The cumulated false alarm rate correspondsto the ratio of the average recovery rate of the first j portions (denoted by RRj )to the average recovery rate over all portions (denoted by RR), multiplied by thesame weighting factor. The average LGDs LGDj and average recovery rates RRj

are unweighted. This implies the advantage that LGDs can be validated without anyinfluence of the size of EADs. Thus, it can be ruled out that banks arrange theirmodels so that LGDs of credits with high EADs are estimated more precisely whileestimations for credits with smaller EADs are imprecise. The LGD for our sampleretail portfolio equals 18.42%.

The receiver operating characteristic (ROC) curve evolves by plotting the cumu-lated hit rates against the cumulated false alarm rates. This curve shows thehomogeneity of the credit portfolio with respect to the LGDs. The more steeply itruns, the more homogeneous the LGDs of single credits are. If all credits exhibitthe same LGD, the portfolio is perfectly homogeneous concerning the LGDs. Thus,the ROC curve runs vertically along the y-axis and then horizontally at the level ofone. If the credit portfolio consists of two disjunct quantities, one with credits thatpossess an LGD of 100% and the other with only non-defaulted credits, the portfoliois perfectly heterogeneous and the ROC curve shows a slope of one. Since the latterportion can only default if the first one also defaults, Di cannot be larger than Di−1.Therefore, the ROC curve is linear in sections but concave overall. For our example,a ROC curve arises according to Figure 2 (see page 32).

The AUC measures the steepness of the ROC curve. It corresponds to theprobability that the rank of a defaulted portion (Rankd

Por) is higher than the rankof a non-defaulted portion (Ranknd

Por) plus half of the probability that the rank of adefaulted portion is identical to the rank of a non-defaulted portion:26

AUC =n∑

i=1

((FARi − FARi−1) · HRi + HRi−1

2

)

=n∑

i=1

(fari · HRi + HRi−1

2

)(8)

In probabilistic terms, AUC reads as:

AUC =n∑

i=1

(Prob(Ranknd

Por = i) · Prob(RankdPor ≤ i) + Prob(Rankd

Por ≤ i − 1)

2

)

= Prob(RankdPor < Ranknd

Por) + 0.5 · Prob(RankdPor = Ranknd

Por) (9)

26See Bamber (1975) for a derivation in the context of the accuracy of discriminative power.




FIGURE 2 ROC curve of the sample retail portfolio.

–0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0FAR

HR

ROC curve of the sample retail portfolio

ROC curve of heterogeneous portfolioROC curve of homogeneous portfolio

0

0.2

0.4

0.6

0.8

1.0

The figure shows the ROC curves of a perfect heterogeneous, perfect homogeneous and of the sampleportfolio.

AUC equals one for a perfectly homogeneous portfolio. For a perfectly heteroge-neous portfolio AUC becomes 0.5. The AUC of our sample retail portfolio shows avalue of 0.8342.

Another well-known validation measure is the cumulative accuracy profile (CAP).The CAP curve measures, analogously but not equal to the Lorenz curve, the degreeof inequality, ie, how the hit rates are distributed over all portions. Therefore, thecumulated hit rates are plotted against the cumulated portions. The CAP curve, likethe ROC curve, is linear in sections but concave overall. If the hit rates are equallydistributed over all portions, the CAP curve possesses a slope of one and the portfoliois perfectly heterogeneous. If all credits possess the same LGD, the portfolio isperfectly homogeneous and the CAP curve runs linearly rising up to the unweightedaverage LGD and then horizontally. For our sample retail portfolio, a CAP curveresults in accordance with Figure 3 (see page 33).27

Additionally, an inequality coefficient can be formed analogously to the Ginicoefficient. In order to retain the notation of accuracy measures for rating functions,the coefficient is called the accuracy ratio (denoted as AR in equations) and iscomputed as follows:28

AR =∑n

i=1((1/n) · ((HRi + HRi−1)/2)) − 0.5

1 − 0.5 − LGD/2

= 2 · ∑ni=1 HRi − 1 − n

n · (1 − LGD)(10)

27The CAP curve of the perfectly homogeneous portfolio corresponds to a portfolio with anunweighted average LGD (LGD) of 18.42%28See Appendix B for the derivation.




FIGURE 3 CAP curve of the sample retail portfolio:

0

0.2

0.4

0.6

0.8

1.0

–0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0Cumulated percentiles

HR

CAP curve of the sample retail portfolio

CAP curve of heterogeneous portfolioCAP curve of homogeneous portfolio

The figure shows the CAP curves of a perfect heterogeneous, perfect homogeneous and of the sampleportfolio.

The accuracy ratio increases the more unequally the hit rates are distributed over allportions, ie, the more homogeneous the portfolio is with respect to its LGDs. Fora perfectly heterogeneous portfolio the accuracy ratio becomes zero, for a perfectlyhomogeneous portfolio the accuracy ratio equals one. Furthermore, the relationshipbetween AUC and the accuracy ratio is AR = 2 · AUC − 1.29 Our sample retailportfolio possesses an accuracy ratio of 0.6683.

The ROC curve, the CAP curve, AUC and accuracy ratio of the realized LGDsprovide a benchmark for the parameters and curves of the dataset of the LGDestimation model, which have to be computed similarly. Afterwards, the results ofboth datasets can be compared with each other. Nevertheless, it has to be consideredthat the interpretations of ROC, CAP, AUC and accuracy ratio in our LGD frameworkdiffer from the interpretation of these measures in the framework of rating accuracy.There are no perfect curves. Rather, the parameters and curves of the LGD estimationmodel should deviate as little as possible from those of the historical dataset. If AUCand accuracy ratio of the realized and estimated LGDs are almost equal, then theLGD estimation model can be seen as a good forecasting tool for future LGDs.

ROC curves can intersect each other. In this case, area differences might becompensated. It can happen that the datasets of the historical losses and of the LGDestimation model possess the same AUC and accuracy ratio but the results of theLGD estimation model are not related with the historical loss dataset. Therefore, amodified AUC, that we call MAUC, should be computed additionally, where MAUC

29This relation is well-known for the accuracy measure of rating functions. For a proof with respectto our LGD framework see Appendix C.




equals the sum of the absolute values of the area differences for each portion i:

MAUC =n∑

i=1

∣∣∣∣(

(FARhisti − FARhist

i−1) ·HRhist

i + HRhisti−1

2

)

−(

(FAResti − FARest

i−1) ·HRest

i + HResti−1

2

)∣∣∣∣ (11)

where:

FARhisti = false alarm rate of portion i of the historical loss distribution

FAResti = false alarm rate of portion i of the estimation model

HRhisti = hit rate of portion i of the historical loss distribution

HResti = hit rate of portion i of the estimation model.

A goal within validation should be to develop LGD estimations that minimize thevalue of MAUC.

In order to create a figure that measures AUC differences of the two ROC curvesof the historical and estimation dataset, we compute the following single AUCi :

AUCi = (FARi − FARi−1) · HRi + HRi−1

2

= fari · HRi + HRi−1

2(12)

Subsequently, we suggest the following linear regression:

AUCesti = α + β · AUChist

i (13)

where:

AUChisti = AUCi of the historical loss distribution

AUCesti = AUCi of the estimation model.

If the LGD estimation model perfectly forecasts future LGDs, α should be zero andβ should be one. If α is significantly different from zero, there is a bias in the LGDestimation. A further linear regression with suppression of the location parametershows whether the AUC of the historical dataset is, on average, systematicallyunderestimated (β < 1) or overestimated (β > 1). A measure of the quality of theestimation model is the coefficient of determination R2(45◦).30 It is computed

30A correlation analysis of the AUC of the historical and estimated datasets is not meaningfulsince a similar movement does not imply a high accuracy of the estimation model. Therefore, acorrelation factor like R2(45◦) should be chosen.




as follows:

R2(45◦) = 1 −∑n

i=1(AUCesti − AUChist

i )2∑ni=1(AUCest

i − AUCest)2(14)

As mentioned before, the proportional decomposition of credits has the appealthat it validates LGD estimations without size considerations of the EADs. This isreasonable if LGDs are estimated as exactly as possible to price credits and calculateinterest rates. Imprecise estimations of LGDs can lead to the wrong credit interestrates for new contracts.

If the LGDs are used to compute losses in euros, additional validation instrumentsshould be implemented to guarantee that losses of large credits are estimatedprecisely.31 For this purpose, a marginal decomposition of the credit should be used.This is the subject of the following subsection.

2.3 Marginal decomposition of the credit

In the framework of a marginal decomposition of credits, each credit is divided intosingle euros.32 Here, the number of euros differs for each credit because each creditamount varies. For each single euro i = 1, . . . , EADk of credit k = 1, . . . , K , now,a binary variable de

i,k is defined as follows:33

dei,k =

{1 if single euro i of EADk is defaulted

0 if single euro i of EADk is not defaulted(15)

Corresponding to the proportional decomposition, again a second binary variablende

i,k ≡ 1 − dei,k is defined.34 Subsequently, the sum of de

i,k and ndei,k over all credits

for each single euro, denoted by Dei and NDe

i , respectively, is computed. In contrastto the proportional decomposition, the sum of De

i and NDei is not equal for every

single euro. For the marginal decomposition, the same sample retail portfolio as forthe proportional decomposition is used in order to simplify the interpretation. Theresults of the marginal decomposition for our sample retail portfolio are shown inTable 2 (see page 36).

The hit rate hrei and the false alarm rate farei are again computed according toEquation (6). The hit rate hrei equals the fraction of the single euro i over all defaulted

31Clearly, for a credit with an EAD of e100, a stronger deviation of the relative LGD estimationis less problematic than for a credit with an EAD of e10,000.32It is also possible to define a different decomposition. For example a decomposition into singlehundred euros is more reasonable for large credit amounts.33The superscript e denotes equations and parameters for the marginal decomposition. For adecomposition in single hundred euros, i lies between one and arg maxk(EADk/100).34For losses larger than the EAD, a modification of the variables de

i,kand nde

i,k, such as that for

the proportional decomposition, is also possible.




TABLE 2 Marginal decomposition of our sample retail portfolio.

Single euro i Dei NDe

i

1st 54 462nd 54 463rd 54 464th 54 465th 54 46...

......

20,595th 6 5820,596th 6 5820,597th 6 5820,598th 6 5820,599th 6 5820,600th 6 5820,601st 6 5720,602nd 6 5720,603rd 6 5720,604th 6 57

......

...

51,596th 0 151,597th 0 151,598th 0 151,599th 0 151,600th 0 1

The second column indicatesthe number of defaults of thesingle euro i and the third col-umn shows the number of non-defaults of the the single euro i.

euros. The false alarm rate farei equals the fraction of the single euro i over all eurosthat are not defaulted. The function of the hit rates is monotonic, decreasing in i

because the second single euro can only default if the first single euro also defaults.The function of the false alarm rates can increase or decrease in i because of differentamounts of EADs of every credit. The sum of hit rate and false alarm rate is again adecreasing function in i. The cumulated hit rate HRe

j and the cumulated false alarm

rate FARej are computed as follows:35

HRej =

j∑i=1

hrei =∑j

i=1 EADi∑maxi=1 EADi

·LGDe

j

LGDe

35For the derivation see Appendix D.




where:

LGDej =

∑j

i=1 Dei∑j

i=1 EADi

LGDe =∑max

i=1 Dei∑max

i=1 EADi

FARej =

j∑i=1

farei =∑j

i=1 EADi∑maxi=1 EADi

·RRe

j

RRe

where:

RRej =

∑j

i=1 NDei∑j

i=1 EADi

RRe =∑max

i=1 NDei∑max

i=1 EADi

(16)

The variable max is computed as arg maxk(EADk) and equals the largest EADamount of all credits of the portfolio when assuming a decomposition into singleeuros. Here, LGDe

j , LGDe, RRej and RRe are credit weighted averages. The credit

weighted LGD of our sample retail portfolio is 0.2.The ROCe curve, the CAPe curve, AUCe and ARe are computed similarly to the

proportional decomposition. AUCe can be interpreted as the probability that the rankof a defaulted single euro is higher than the rank of a non-defaulted single euro plushalf the probability that a defaulted single euro ranks on the same position of a non-defaulted single euro. The formula for AUCe reads as follows:

AUCe =max∑i=1

(FARe

i − FARei−1 ·

HRei + HRe

i−1

2

)

=max∑i=1

(farei ·

HRei + HRe

i−1

2

)(17)

In probabilistic terms, AUCe reads as:

AUCe =max∑i=1

(Prob(Ranknd

Eur = i) · Prob(RankdEur ≤ i) + Prob(Rankd

Eur ≤ i − 1)

2

)

= Prob(RankdEur < Ranknd

Eur) + 0.5 · Prob(RankdEur = Ranknd

Eur) (18)

If all credits show the same amount of loss, AUCe assumes a value of one. If the ratioof defaulted to non-defaulted euros is equal for every single euro i, AUCe assumesa value of 0.5. In contrast to the proportional decomposition, AUCe and ARe canassume values below 0.5.

The validation procedure is similar to the procedure for the proportional decom-position. After computing the validation ratios for realized and estimated losses, theyare compared with each other. If the ratios are almost equal, the estimation model canbe seen as a good forecast instrument for future losses. In order to avoid mistakes inthe interpretation caused by intersections of ROC curves, a modified AUCe, denoted




as MAUCe, should again be computed corresponding to Equation (11). For statisticalvalidation of the results, R2(45◦)e can be computed analogously to Equation (14).

Both approaches, the proportional and marginal decomposition, are developed tocompare the estimated LGD with the realized LGD. To validate the accuracy of thedownturn LGD, we suggest the comparison of the estimated downturn LGDs withrealized LGDs of an out-of-sample dataset. This dataset should contain an economicdownturn to verify that the estimated downturn premium is high enough to cover therelative and absolute increase in LGDs. If the downturn premium is not a constantbut a result of an estimation model as mentioned in Section 2.1, the accuracy of thedownturn model with an out-of-sample dataset containing an economic boom canalso be tested. Such data enables us to consider whether or not the downturn modeloverestimates the downturn LGD in a healthy economic environment.

3 EMPIRICAL ANALYSIS

3.1 Data

For our empirical analysis, real loss data from a retail portfolio of a commercial bankis used. The LGD estimation model is based on workout LGDs. It consists of twoparts. Initially, a logistic regression for estimating the probability of a recovery or awrite-off is carried out and, subsequently, a linear regression for estimating LGDsfor each case is run. Afterwards, the sum of both LGDs, weighted by the rates ofa recovery and a write-off, is computed and is used as the LGD for the credit inquestion.36 The retail portfolio is split up into four subportfolios, distinct as to privateand commercial clients and collateralized and uncollateralized credits at default date.

The four subportfolios are analyzed with the proportional and marginal decompo-sition models of Section 2. Both in-sample and out-of-sample tests for the robustnessof the estimation model are implemented. For the in-sample test, the completemodeling database is used for validation. Therefore, the AUCs of the realized LGDs,estimated LGDs, realized losses and estimated losses are computed. Afterwards, theMAUC and the R2(45◦) are calculated.

For the out-of-sample test, a rejection level is computed for the MAUC at a90%, 95% and 99% confidence levels and for the R2(45◦) at a 10%, 5% and 1%confidence level using the bootstrapping method. For bootstrapping, a random subsetof the modeling database, reduced to each single subportfolio, is drawn 100 times.Afterwards, a validation database is used, which was not in the modeling set.37 Thevalidation data time period follows the modeling data time period. Therefore, theout-of-sample test also represents an out-of-universe test. The computed MAUC andR2(45◦) of the validation database are compared with the confidence levels of themodeling database. If MAUC and/or R2(45◦) of the validation database exceed the

36Also Hamerle et al (2006) divide the LGD estimation into two parts.37The size of the random subset for bootstrapping equals the portion of the validation database onthe modeling database, computed for each subportfolio.




TABLE 3 Results for the proportional decomposition of the modeling database.

Subportfolio Portions MAUC R2(45◦)Private clients, collateralized 100 0.2751 0.1658Private clients, uncollateralized 100 0.2900 0.5721Commercial clients, collateralized 100 0.2314 0.4879Commercial clients, uncollateralized 100 0.2325 0.7785

Private clients, collateralized 500 0.2749 0.1666Private clients, uncollateralized 500 0.2898 0.5723Commercial clients, collateralized 500 0.2313 0.4883Commercial clients, uncollateralized 500 0.2331 0.7785

Private clients, collateralized 1,000 0.2749 0.1668Private clients, uncollateralized 1,000 0.2898 0.5723Commercial clients, collateralized 1,000 0.2313 0.4883Commercial clients, uncollateralized 1,000 0.2331 0.7782

The first column indicates the analyzed subportfolio and the second columndenotes the chosen proportional decomposition. MAUC and R2(45◦) are pre-sented in the third and fourth column, respectively.

rejection levels, the analyzed estimation model is not robust with respect to time orsample changes.

3.2 Results

Firstly, relative LGDs are validated using the proportional decomposition. Decom-positions into 100, 500 and 1,000 portions are chosen. Because some contracts of thesubportfolios possess LGDs larger than one, a modification according to Equation (5)is used. For each subportfolio, an EAD multiple is chosen, so that at least 95% of thedatabase does not have to be modified.38 After computing the validation ratios for therealized and estimated LGDs for all portfolios, MAUC and R2(45◦) are calculatedfor further analysis and are shown in Table 3.

The LGD estimation model analyzed in our paper is mainly designed to meet theBasel II capital requirements. Therefore, we are interested in precise estimations ofabsolute losses. However, for risk-adjusted credit pricing, a different approach will beused. Hence, the validation for relative LGDs is only of secondary importance in thelatter case. Nevertheless, three of the subportfolios possess an R2(45◦) larger than0.48. Only the LGD estimation of the subportfolio “private clients, collateralized”exhibits an inadequate accuracy.

The LGD estimation model is robust with respect to changes in the decomposition.Therefore, for the rejection levels of MAUC and R2(45◦), a proportional decompo-sition of 100 for bootstrapping is used. The rejection levels for the out-of-universe

38The 95% level was chosen to avoid data shortening due to outliers.




TABLE 4 Confidence levels of MAUC and R2(45◦).

MAUC subportfolio 90% c.l. 95% c.l. 99% c.l.

Private clients, collateralized 0.2777 0.2781 0.2793Private clients, uncollateralized 0.3005 0.3038 0.3099Commercial clients, collateralized 0.2354 0.2365 0.2403Commercial clients, uncollateralized 0.2795 0.2905 0.3112

R2(45◦) subportfolio 10% c.l. 5% c.l. 1% c.l.


The abbreviation c.l. means confidence level.

TABLE 5 Results for the proportional decomposition of the validation database.

Subportfolio MAUC R2(45◦) Rejection

Private clients, collateralized 0.3160 −0.2955 YesPrivate clients, uncollateralized 0.5152 −1.4947 YesCommercial clients, collateralized 0.2299 0.3581 No/yesCommercial clients, uncollateralized 0.7479 −3.9974 Yes

test are presented in Table 4. For MAUC, the 90%, 95% and 99% confidence levelare computed. However, a lower MAUC in the validation database in contrast to themodeling database is not a problem, since the estimation model then works evenbetter for the validation database than for the modeling database. For R2(45◦), the10%, 5% and 1% confidence levels are computed. In the case of a higher R2(45◦)in the validation database in contrast to the modeling database, again the estimationmodel works even better for the validation database.

For the out-of-universe test, MAUC and R2(45◦) for the validation database arecomputed using a proportional decomposition with, again, 100 portions. Subse-quently, these ratios are compared with the corresponding rejection levels. Table 5presents the results for the validation database. For every subportfolio, the R2(45◦)figures of the validation database are smaller than the confidence level. The resultsare similar for MAUC. Only for the subportfolio “commercial clients, collateral-ized”, the MAUC figure would not lead to rejection. Here, it can be seen that dataquality is important. Since the validation data time period follows the modelingdata time period, an increase in data computation and data collection quality can beassumed. Also changes in the portfolio structure or in debtor-specific characteristicsmay lead to this result.




TABLE 6 Results for the marginal decomposition of the modeling database.

Subportfolio Marginal MAUC R2(45◦)Private clients, collateralized 100 0.2549 0.6677Private clients, uncollateralized 100 0.1029 0.9322Commercial clients, collateralized 100 0.2164 0.7788Commercial clients, uncollateralized 100 0.0955 0.9178



The first column indicates the analyzed subportfolio and the second columndenotes the chosen marginal decomposition. MAUC and R2(45◦) are shownin the third and fourth column, respectively.

The next step is the validation of losses in euros using the marginal decomposition.Therefore, decompositions into single e100, e50 and e10 are chosen. The EADmultiples remain the same as for the proportional decompositions. Again, AUCsof realized and estimated losses are calculated at the beginning. Subsequently, thecomparison of both realized and estimated losses is done by using MAUC andR2(45◦). The results of the in-sample test for the marginal decompositions arepresented in Table 6.

The estimations of losses in euros are more precise for every subportfolio than theestimations of relative losses. Every subportfolio possesses an R2(45◦) above 66%,two subportfolios even show an R2(45◦) above 90%. Therefore, the estimation modelrepresents an appropriate model for estimating absolute losses, which are needed todetermine the capital requirements according to Basel II. Thus, the aim of the bank,ie, the development of an estimation model for absolute losses, is achieved.

The results are robust with respect to changes in the marginal size. Therefore,for computing the rejection levels via bootstrapping, a marginal decomposition intosingle e100 is chosen. The rejection levels for the out-of-universe test are presentedin Table 7 (see page 42).

To compare the modeling database with the validation database, firstly, a marginaldecomposition into single e100 of the validation database is done. Afterwards,MAUC and R2(45◦) of the validation database are compared with the correspondingrejection levels. If the validation ratios exceed the rejection levels, the estimationmodel is not robust with respect to additional new data. Table 8 (see page 42) showsthe corresponding results.




TABLE 7 Confidence levels of MAUC and R2(45◦).

MAUC subportfolio 90% c.l. 95% c.l. 99% c.l.


R2(45◦) subportfolio 10% c.l. 5% c.l. 1% c.l.


The abbreviation c.l. means confidence level.

TABLE 8 Results for the marginal decomposition of the validation database.

Subportfolio MAUC R2(45◦) Rejection

Private clients, collateralized 0.3048 0.5460 YesPrivate clients, uncollateralized 0.1110 0.9578 NoCommercial clients, collateralized 0.2614 0.5281 YesCommercial clients, uncollateralized 0.2682 0.7833 Yes/no

The reasons for rejections are the same as for the proportional decomposition.Again, an increase in data quality and possible changes in the portfolio structure maylead to differences. However, the magnitude of misspecification of absolute losses issmaller than the misspecification of relative LGDs. Though two subportfolios arerejected by both measures, the R2(45◦) figures of all subportfolios are above 50%and for two subportfolios even above 75%. Thus, the estimation model for absolutelosses is still applicable to forecast future losses. Therefore, the estimation modelcan be implemented to determine Basel II capital requirements.

4 CONCLUSION

The idea of this paper was to develop an LGD validation method for retail portfolios.This topic is important because banks have to estimate LGDs if they want to applyfor the internal ratings based approach for retail portfolios. The Basel II regulationspostulate that retail portfolios have to be homogeneous concerning, at least, riskdrivers such as borrower and transaction risk characteristics and delinquency ofexposure.39 This makes it difficult to develop LGD rating or scoring models for retailportfolios because of their similar characteristics.

39See Basel Committee on Banking Supervision (2004, paragraph 402).




In contrast, our suggested proportional and marginal decomposition methods areapplicable without LGD ratings. Furthermore, there are no specific requirements forthe LGD estimation model. Even for arithmetic mean estimations of the LGD forretail portfolios, both methods can be used. Our methods use validation instruments,that are well-known from PD validation, where the interpretation is different.The used instruments, eg, AUC and accuracy ratio, are accepted by supervisoryauthorities. Because of the different interpretation, the ratio itself contains noinformation about the accuracy of the estimation model. In fact, the ratio describesthe composition and structure of the portfolio. To validate the estimation model,it is necessary to compare the ratios calculated on realized LGDs or losses withthose based on estimated LGDs or losses. Therefore, AUC and accuracy ratio of thehistorical database provide a benchmark for the ratios of the estimated values. If themeasures are similar, the estimation model can be seen as a good forecasting tool forfuture losses.

It also turns out that the proportional decomposition is credit size independent.Thus, the method is also an instrument to prove the functionality of the estimationmodel over all single credits in the portfolio. The proportional decomposition canreveal possible weaknesses in the estimation model. Therefore, banks can avoidarranging their models such that LGDs for credits with high EADs are estimatedmore precisely, while estimations for credits with smaller EADs are imprecise. Thisfact is important if the estimation model is to be used for credit pricing, wherea precise estimation of the LGD is important for the calculation of the contract’sinterest rate. Furthermore, our approaches can also be used to validate downturnpremiums by using out-of-sample datasets containing either an economic downturnor an economic boom. We also showed that the models work on real data and thatout-of-sample and out-of-time tests can easily be implemented.

APPENDIX A DERIVATION OF HIT RATES AND FALSE ALARM RATES

In the context of LGD validation, the single and cumulative hit rates and false alarmrates, respectively, can be transformed as follows:

hri = Di∑ni=1 Di

= K − NDi∑ni=1 Di

= K∑ni=1 Di

− NDi∑ni=1 Di

= 1

LGD · n − NDi

LGD · n · K

HRj =j∑

i=1

hri = j

LGD · n −∑j

i=1 NDi

LGD · n · K = j

LGD · n − (1 − LGDj ) · jLGD · n

= j − (1 − LGDj ) · jLGD · n = j

n· LGDj

LGD




fari = NDi∑ni=1 NDi

= K − Di∑ni=1 NDi

= K∑ni=1 NDi

− Di∑ni=1 NDi

= 1

RR · n − Di

RR · n · K

FARj =j∑

i=1

fari = j

RR · n −∑j

i=1 Di

RR · n · K = j

RR · n − (LGDj ) · jRR · n

= j − (1 − RRj ) · jRR · n = j

n· RRj

RR

APPENDIX B DERIVATION OF ACCURACY RATIO

The accuracy ratio, associated with the LGD-based CAP curve, can be rearranged asfollows:

AR =∑n

i=1((1/n) · ((HRi + HRi−1)/2)) − 0.5

1 − 0.5 − LGD/2

=∑n

i=1((1/n)·(i·LGDi/(n·LGD)+(i − 1)·LGDi−1/(n·LGD))/2) − 0.5

0.5 − LGD/2

= (∑n

i=1(i · LGDi ) − 0.5 · n · LGD)/(n2 · LGD) − 0.5

0.5 − 0.5 · LGD

= 2 · ∑ni=1(i · LGDi ) − n · LGD − n2 · LGD

n2 · LGD · (1 − LGD)

= 2 · ∑ni=1 HRi − 1 − n

n · (1 − LGD)

APPENDIX C LINEAR RELATIONSHIP BETWEEN ACCURACY RATIOAND AREA UNDER CURVE

We prove that the well-known linear relationship of AUC and accuracy ratio inthe context of measuring PD-based accuracy also holds in the framework of LGDvalidation:

AUC =n∑

i=1

(FARi − FARi−1 · HRi + HRi−1

2

)

=n∑

i=1

(i · RRi

n · RR− (i − 1) · RRi−1

n · RR

)

· i · LGDi/(n · LGD) + (i − 1) · LGDi−1/(n · LGD)

2




=∑n

i=1(i · LGDi ) − 0.5 · n · LGD − 0.5 · n2 · LGD2

n2 · RR · LGD

=∑n

i=1 HRi − 0.5 − 0.5 · n · LGD

n · (1 − LGD)

2 · AUC − 1 = 2 · ∑ni=1 HRi − 1 − n · LGD

n · (1 − LGD)− 1

= 2 · ∑ni=1 HRi − 1 − n · LGD − n · (1 − LGD)

n · (1 − LGD)

= 2 · ∑ni=1 HRi − 1 − n

n · (1 − LGD)= AR �

APPENDIX D DERIVATION OF HRe AND FARe

For the marginal decomposition of the analyzed credit portfolio, the hit rates andfalse alarm rates, respectively, can be determined as follows:

hrei =De

i∑maxi=1 De

i

HRej =

j∑i=1

hrei =j∑

l=1

Del∑max

i=1 Dei

=∑j

l=1 Del∑max

i=1 Dei

·∑max

i=1 EADi∑j

l=1 EADl

·∑j

l=1 EADl∑maxi=1 EADi

=∑j


·LGDe

j

LGDe

farei =NDe

i∑maxi=1 NDe

i

FARej =

j∑i=1

farei =j∑

l=1

NDel∑max

i=1 NDei

=∑j

l=1 NDel∑max

i=1 NDei

·∑max

i=1 EADi∑j

l=1 EADl

·∑j


=∑j


·RRe

j

RRe

APPENDIX E DATA OF THE SAMPLE RETAIL PORTFOLIO

Credit EAD in e Loss in e Credit EAD in e Loss in e

1 7,100 0 51 29,900 02 7,400 0 52 30,900 03 7,500 0 53 31,300 04 8,300 0 54 32,200 6,500




APPENDIX E Continued

Credit EAD in e Loss in e Credit EAD in e Loss in e

5 8,300 0 55 32,700 06 8,500 0 56 33,500 11,7007 9,000 0 57 34,000 08 10,100 0 58 34,800 09 10,500 0 59 35,400 11,500

10 10,500 0 60 36,100 10,90011 10,500 0 61 36,700 7,30012 10,600 4,900 62 37,400 013 11,000 2,600 63 38,000 13,70014 11,400 0 64 38,600 015 11,900 4,400 65 39,300 016 12,200 2,800 66 39,800 18,60017 12,400 0 67 40,500 1,70018 12,500 4,000 68 41,000 019 12,900 0 69 41,700 16,50020 13,400 13,400 70 42,200 10,40021 13,800 0 71 42,800 10,80022 13,800 13,800 72 43,200 4,00023 14,500 0 73 43,900 024 14,700 3,600 74 44,300 2,20025 14,900 0 75 44,900 10,30026 15,000 0 76 45,300 027 15,600 4,300 77 45,900 7,90028 16,000 0 78 46,200 17,60029 16,800 0 79 46,800 030 16,900 8,300 80 47,100 031 17,000 7,700 81 47,600 032 18,000 5,700 82 47,900 23,40033 18,200 0 83 48,400 11,10034 19,200 0 84 48,600 13,50035 19,400 0 85 49,100 17,60036 20,500 3,500 86 49,200 037 20,600 0 87 49,700 7,60038 21,800 4,700 88 49,800 22,50039 21,800 0 89 50,200 19,10040 23,000 2,100 90 50,300 11,50041 23,100 0 91 50,600 16,00042 24,300 10,300 92 50,700 25,10043 24,400 24,400 93 51,000 5,10044 25,600 9,100 94 51,100 20,00045 25,700 6,400 95 51,200 18,70046 26,900 10,900 96 51,300 047 27,200 0 97 51,400 048 28,200 8,800 98 51,400 17,90049 28,500 7,100 99 51,500 23,20050 29,600 0 100 51,600 23,300

Sum 3,000,000 600,000




REFERENCES

Bakshi, G., Madan, D., and Zhang, F. (2006). Understanding the role of recovery in default riskmodels: empirical comparisons and implied recovery rates. Working Paper, Federal DepositInsurance Corporation, Center for Financial Research.

Bamber, D. (1975). The area above the ordinal dominance graph and the area belowthe receiver operating characteristic graph. Journal of Mathematical Psychology 12(4),387–415.

Barco, M. (2007). Going downturn. Risk 20(9), 39–44.

Basel Committee on Banking Supervision. (2004). International Convergence of CapitalMeasurements and Capital Standards – A Revised Framework. Bank for InternationalSettlements.

Basel Committee on Banking Supervision. (2005a). Guidance on Paragraph 468 of theFramework Document. Bank for International Settlements.

Basel Committee on Banking Supervision. (2005b). Studies on the validation of internal ratingsystems. Working Paper No. 14, Bank for International Settlements.

Brady, B., Chang, P., Miu, P., Ozdemir, B., and Schwartz, D. (2006). Discount rate for workoutrecoveries: an empirical study. Working Paper.

Engelmann, B., Hayden, E., and Tasche, D. (2003). Testing rating accuracy. Risk 16(1), 82–86.

Giese, G. (2005). The impact of PD/LGD correlations on credit risk capital. Risk 18(4), 79–84.

Gupton, G. M. (2005). Advancing loss given default prediction models: how the quiet havequickened. Economic Notes 34(2), 185–230.

Hamerle, A., Knapp, M., and Wildenauer, N. (2006). Modelling loss given default: a “pointin time”-approach. The Basel II Risk Parameters, Engelmann, B., and Rauhmeier, R. (eds).Berlin, pp. 127–142.

Hartmann-Wendels, T., and Honal, M. (2006). Do economic downturns have an impact onthe loss given default of mobile lease contracts? an empirical study for the German leasingmarket. Working Paper, University of Cologne.

Jarrow, R. A., Lando, D., and Turnbull, S. (1997). A Markov model for the term structure ofcredit risk spreads. Review of Financial Studies 10(2), 481–523.

Li, D., Bhariok, R., Keenan, S., and Santilli, S. (2009). Validation techniques and performancemetrics for loss given default models. The Journal of Risk Model Validation 3(3), 3–26.

Madan, D., and Unal, H. (2000). A two-factor hazard rate model for risky debt and the termstructure of credit spreads. Journal of Financial and Quantitative Analysis 35(1), 43–65.

Miu, P., and Ozdemir, B. (2006). Basel requirements of downturn loss given default: modelingand estimating probability of default and loss given default correlations. The Journal ofCredit Risk 2(2), 43–68.

Ong, M. K. (1999). Internal Credit Risk Models, Capital Allocation and Performance Measure-ment. Risk Books, London.

Pan, J., and Singleton, K. J. (2008). Default and recovery implicit in the term structure ofSovereign CDS spreads. Journal of Finance 63(5), 2345–2384.




Rösch, D., and Scheule, H. (2009). Credit portfolio loss forecasts for economic downturns.Financial Markets, Institutions & Instruments 18(1), 1–26.

Schuermann, T. (2005). What do we know about loss given default? Recovery Risk, TheNext Challenge in Credit Risk Management, Altman, E., Resti, A., and Sironi, A. (eds). RiskBooks, London, pp. 3–24.

Siddiqi, N. A., and Zhang, M. (2004). A general methodology for modeling loss given default.Risk Management Association Journal 86(8), 92–95.

Sobehart, J. R., and Keenan, S. C. (2001). Measuring default accurately. Risk 14(3), 31–33.



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