How to reduce capital requirement? The case of retail portfolios with low probability of default

How to reduce capital requirement?

The case of retail portfolios with low probability of

default

Marie-Paule Laurent

1 Research Fellow FNRS-Bernheim Centre E. Bernheim – Solvay Business School

Université Libre de Bruxelles

Preliminary Draft– Do not quote February 2004

Abstract

This paper focuses on the internal rating-based advanced approach (IRBA) of the Basel Committee proposal for calculating adequate capital requirements to cover credit risks. For “other retail” portfolios, the capital required is a function of loss given default (LGD) and probability of default (PD). For a given LGD, this function is concave with respect to the probability of default, especially for low PD. This implies that for a given credit portfolio, a segmentation that pool contracts on the basis of their PDs allow significant reduction of the total capital requirement. We claim that the use of an asset return correlation adjusted for the volatility of PD, as it is the case in a one factor model, eliminate this regulatory arbitrage. These statements are tested on a portfolio of over 35,000 retail lease contracts. Results show that even for simple segmentation techniques (i.e. based on univariate ex ante characteristics), the total capital requirement may be reduced by over 10%. However, when using the adjusted asset return correlation, no segmentations allow for a reduction in capital requirement. This enlightens the necessity for the Basel Committee to verify the accuracy of its asset return correlation estimation.

1 _{50 av. Roosevelt – ULB CP 145/1 – 1050 Bruxelles – Belgium –}

1.

Introduction

The Basel Committee, a working group of the BIS2, has released the third consultative document (CP3) since June 1999 with a view to establishing a revised capital adequacy Accord. The aim is to provide a number of new approaches that are both more comprehensive and more sensitive to risks than the 1988 Accord, while maintaining the overall level of regulatory capital. The New Accord on regulatory capital is expected to be implemented in the European Union through a directive by 2005, so that all EU financial institutions will be subject to the new provisions.

The current proposal provides three approaches for calculating adequate capital requirements to cover credit risks: the standardised approach (SA), the internal rating-based foundation approach (IRBF) and the internal rating-based advanced approach (IRBA). Under the SA, risk weights for capital requirement are evaluated according to the credit ratings given by external institutions or agencies in the case of corporate exposures, or set at a given fixed regulatory level (75%) in the case of retail exposures. For both other approaches, financial institutions have to use their own rating system. Under the IRBF approach, only the probability of default (PD) of borrowers has to be reliably estimated, the other parameters are set by regulators. Under the IRBA approach, loss given default (LGD), exposure at default (EAD) and maturity (M) also have to be estimated.

The present paper focuses on the general specification of the model defined by the Basel Committee for the calculation of regulatory capital requirements under the IRB approaches. More specifically, we determine how it is theoretically possible to significantly reduce the required capital by choosing a specific segmentation of the total portfolio. We claim that this possibility to optimize capital requirement of a given portfolio is due to a bad estimation of a risk characteristic: asset return correlation.

The empirical testing of these statements is realized on a large retail lease portfolio characterised by low PD. Results shows the method identified theoretically yields significant capital requirement reduction. Moreover, this peculiarity disappears when the volatility of PD is taken into account for measuring asset return correlation Next section presents the Basel framework for retail portfolios and its implications for portfolio segmentation. The database is described in section 3 and results of the Basel approach are shown in section 4. Section 5 presents the one factor model used to measure asset return correlation. The empirical results of this model are analysed in section 6. Section 7 concludes

2 The Basel Committee on Banking Supervision is composed of central banks’ and supervisory authorities’ representatives from Belgium, Canada, France, Germany, Italy, Japan, Luxembourg, the Netherlands, Sweden, Switzerland, the United Kingdom and the United States.

2.

Basel Proposals for Retail Portfolio

Presentation of the three approaches

The current proposal provides three approaches for calculating adequate capital requirements to cover credit risks: the standardised approach, the internal rating-based foundation approach (IRBF) and the internal rating-rating-based advanced approach (IRBA).

The capital allocation (KA) is calculated as the product of the regulatory capital ratio (K), i.e. 8% of the risk-weighting ratio (RW), and the exposure at default (EAD).

EAD RW EAD K KA × × = × = % 8 [1] STANDARDIZED APPROACH

Under the standardised approach, risk weights for capital requirement are set at a given fixed regulatory level (75%) in the case of retail exposures.

EAD

KA=8%×0.75× [2]

For both other approaches, financial institutions have to use their own rating system.

INTERNAL RATING-BASED FOUNDATION APPROACH

Under the IRBF approach, only the probability of default (PD) of borrowers has to be reliably estimated, the other parameters are set by regulators. Loss given default (LGD) is set at respectively 45% and 75% for secured and subordinated claims without specifically recognised collaterals. It may be adjusted in order to take into account the risk-mitigation effect of recognised collaterals, subject to operational requirements and regulatory floors.

However, retail exposures are excluded from IRBF Approach. The corporate exposure case of IRBF must be used instead.

In this case, capital requirement is formulated as:

[

R PD R R

]

Madj

LGD

K = ×_φ(1− ₎−0.5×_φ−1_{( )}+_{( 1}− ₎0.5×_φ−1_(0.999) ×

[3] Where

φ

(.)

is the normal standard cumulative distribution function,

_φ

−1

(.)

is the inverse of the normal standard cumulative distribution function, R is the asset return correlation and Madj is the adjustment for maturity.

]

45

5

1

[

04

.

0

]

1

1

1

[

%

24

1

1

%

12

50₅₀ 50₅₀

−

×

−

−

−

−

−

×

+

−

−

×

=

−₋× −₋×

S

e

e

e

e

R

PD PD [4]

It corresponds to an average of two extreme values (12% and 24%), minus an adjustment for the firm size with S being the total annual sales (in millions €).

The maturity adjustment is taken into account to reflect the possibility of long-term loans experiencing a decrease in their fair value because the obligor has been downgraded. It is expressed as:

)] ( ) 5 . 2 ( 1 [ )] ( 5 . 1 1 [ _{b PD} 1 _{M b PD} M_adj = − × − × + − × [5]

with M the effective maturity of exposure and

2

)]

ln(

05898

.

0

08451

.

0

[

)

(

PD

PD

b

=

−

×

.

INTERNAL RATING-BASED ADVANCED APPROACH

Under the IRBA approach, probability of default (PD) as well as loss given default (LGD), exposure at default (EAD) and maturity (M) have to be estimated.

According to CP3, capital requirement for retail exposure is formulated as:

[

(1₋ )−0.5_× −1( )₊( 1₋ )0.5_× −1(0.999)

]

×

=LGD φ R φ PD R R φ

K [6]

The asset return correlation is again calculated as a function of PD. and is defined as an average of two extreme values: 2% and 17%.

]

1

1

1

[

%

17

1

1

%

2

₃₅ 35 35 35 − × − − × −

−

−

−

×

+

−

−

×

=

e

e

e

e

R

PD PD [7]

Study of the IRBA capital requirement function

Under IRBA, the capital requirement is a function of two variables: LGD and PD but we will focus here on the influence of PD on the general level of K. This is done through the asset return correlation measure (eq. [7]) and through the general definition of capital requirement (eq. [6]).

First, asset return correlation is defined as a decreasing function of PD. This implies that the asset return correlation assigned to a low-quality borrower converges to the minimum value. It expresses the assumption that riskier firms (e.g. small, low-quality firms) are driven mostly by idiosyncratic risk and are therefore less sensitive to systematic risk than larger companies. Moreover, the correlation converges rapidly to its minimum. The influence of the correlation definition is thus most important for small PD.

Secondly capital requirement is an increasing function of PD. However, although the function is nearly linear for most of the PD, it function presents a strong concavity for low PD (less than 5%). The mathematical study is presented in the appendix.

Figure 1: Asset return correlation as a function of PD under the IRBA approach for retail credit 0,00 0,04 0,08 0,12 0,16 0,20 0% 5% 10% 15% 20% 25% 30% PD C o rre la ti o n

Figure 2: Capital requirement as a function of PD under the IRBA approach for retail credit (LGD=50%) 0 5 10 15 20 25 0% 5% 10% 15% 20% 25% 30% PD K

Implication of the definition of the IRBA function

THEORETICALLY

For probabilities of default inferior 5%, capital requirement is a concave function of PD, given the LGD. Therefore, the following equation holds:

(

x PD1 (1 x) PD2

)

x K

(

PD1

)

(1 x) K

(

PD2

)

K × + − × ≥ × + − × [8]

Where

x

∈

[

0

;

1

]

.

Equation 9 is the decomposition propriety of concave function. It can be interpreted as the fact that a decomposition of a portfolio into sub-portfolios allows the reduction

of the total capital requirement. The size of the reduction depends on the level of concavity. Moreover, the reduction is the largest for PD1 and PD2 being “extreme”

(i.e. largely different from one another).

EXAMPLE OF SEGMENTATION

In order to determine whether the concavity of the capital requirement function is sufficient to allow significant reduction of capital requirement by segmentation of the portfolio, we construct a simple example.

Let’s suppose we have a total portfolio of 1000 retail credit loans with maturity of 1 year of whose 30 have defaulted. All loans are similar: EAD =1 and LGD =100%. The global PD of this portfolio is 3%. Thus, under the Basel framework, the asset return correlation is calculated as R= 0.072 and the required capital is K =0.1381. Suppose now it is possible to find a criteria in order to split the total portfolio into two sub-portfolios such that the first (portfolio A) is composed of the 30 defaulted loans and the other (portfolio B) is composed of the 970 non-defaulted loans. In this extreme case, the capital required for portfolio A is K=1 and for portfolio B is K=03_. This induce a capital requirement for the total portfolio of K =30/1000 x 1 + 970/1000 x 0 = 0.03.

Figure 3 presents the level of the capital requirement on the basis of the size of portfolio B. The criteria of selection of the portfolio is such that portfolio A is composed of 100% of defaulted loans.

We observe that the capital requirement decreases rapidly until the extreme case of a perfect segmentation of the total portfolio. After that, the segmentation is not perfect anymore and the level of capital requirement increases. However, this increase is quite slow compared to the decrease at the beginning of the process. Obviously, at the end of the process, all loans are transferred to Portfolio A and the capital requirement equals 0.1381 again.

Figure 3: Evolution of the capital requirement with respect to the size of portfolio A.

3 _{The Basel Proposal imposes a minimum on estimated PD of 0.003%. This restriction does not change}

0,00 0,02 0,04 0,06 0,08 0,10 0,12 0,14 0,16 1000 995 990 985 980 975 970 965 960 955 950 945 Size of Portfolio B K o f th e to tal p o rtfo li o

If another criterion is used, the composition of the sub-portfolio will be different. Let’s characterise the criterion by the proportion of defaulted contract in portfolio A. Figure 4 presents several evolution of the total capital requirement depending on the percentage of defaulted loans in portfolio A.

Figure 4: Evolution of the capital requirement with respect to the size of portfolio B depending on the percentage of defaulted loans in portfolio A.

0,00 0,02 0,04 0,06 0,08 0,10 0,12 0,14 0,16 1000 975 950 925 900 875 850 825 800 Size of Portfolio B K o f th e to ta l p o rtfo li o 100% 80% 60% 40% 20%

We observe that the maximum reduction of the total capital requirement is larger for good segmentation (i.e. for which the proportion of defaulted loans in portfolio A is the largest). Moreover, a good segmentation implies more rapid reduction of capital requirement.

REGULATORY IMPLICATION

The concavity characteristic of the capital requirement function may lead to arbitrage especially in the case of portfolio with small probabilities of default. Indeed, as the Basel Proposal allows segmentation of the loan portfolio based on past data. It is

possible to find ex post criteria in order to determine a portfolio decomposition that clearly identify defaulted loans and thus allows for a lower level of capital requirement.

3.

Data

Characteristics of lease financing

Lease is defined ‘as an agreement whereby the lessor conveys to the lessee, in return for a payment or series of payments, the right to use an asset for an agreed period of time’ (IAS17). This definition covers various types of contracts. Our empirical analysis is based on automotive lease contracts. The main characteristics are that the contracts are mainly non-cancellable and that lessees are responsible for the selection, acquisition, maintenance and payment of associated charges (taxes and insurance premiums) of the asset. At maturity, the residual value of the leased asset returns to the lessor but the lessee has usually the right to buy it. A lease contract is defined as defaulted when the lessor has unilaterally cancelled the agreement because the lessee did not pay the scheduled rentals (interest and/or capital). In that case, the lessor can repossess the asset, declare the remaining payment due and payable and claim any losses incurred.

Lease is not a marginal mean of financing. Indeed, according to Leaseurope’s4 estimates, lease financing in the EU represents more than €199 billions in 2002 including €82 billions in automotive leasing (i.e. motocars and road transport vehicles). This corresponds to a penetration rate of equipment lease in comparison with total equipment investments of 12.5%. However, although lease financing lies within the scope of the Basel Accord, most of the empirical studies focussing on non-traded financial products are conducted on different financing means like private debts [Carey (1998)], SMEs [Dietsch and Petey (2002)] or mortgages [Calem and Lacour-Little (forthcoming)].

Nevertheless, the few recent empirical studies conducted with a focus on leasing peculiarities conclude that leasing is a relatively low-risk activity with low asset return correlation and that physical collaterals play a major role in reducing the credit risk. De Laurentis and Geranio (2001) and Schmit and Stuyck (2002), analysing the severity of loss when a lease defaults, show that recovery rates are relatively high as compared with other means of financing (especially in the automotive segment). Two empirical studies using parametric [Schmit (2003)] and non-parametric

4 Leaseurope is the acronym of the Brussels-based “European Federation of Leasing Company Associations”, founded in 1973 to represent the leasing industry. Leaseurope comprises 30 member and correspondent national associations which in turn represent more than 1,300 leasing companies.

estimation [Schmit (forthcoming)] conclude that the Basel Proposal as in CP3 imposes excessively conservative capital requirements on leasing businesses. The reason invoked is the too little recognition of physical collaterals. Pirotte, Schmit and Vaessen (2004) show the effective mitigation provided by the physical collateral for leasing and how it substitute a market risk to the original credit risk exposure. Finally, Duchemin, Laurent and Schmit (2003) show that the estimated asset return correlations in leasing portfolio is significantly lower than those assumed under the Basel Proposal implying that the Basel Proposal inadequately reflects the risk profile.

Descriptive statistics of the database

The database consists of a portfolio of 35,787 individual completed automotive lease contracts issued between 1990 and 2000 by a major European leasing companies. The database contains all relevant information concerning the leases throughout their life divided into ex ante and ex post variables. Ex ante variables are origination date of the contract, cost and type of the leased asset, maturity of the lease, periodicity of forecasted payments, amounts of any up-front payments, amount of any broker commissions, estimated residual value, estimated funding rate, internal rates of return (purchase option included or excluded), due dates and the amounts to be paid. The ex post variables are effective payments (reimbursement), amount of any prepayments with the payment dates, final status of the contract (re-rented, terminated or defaulted) and date of the declaration of the status.

Descriptive statistics and frequency distribution are presented in Table 1. The sample is divided into seven segments (panels A to G) based on respectively issuance date of the lease contract, term to maturity, cost of the leased asset, distribution network of the lease contract, region of origin of the lessor, interest premium of the contract and final status of the lease.

Table 1: Descriptive statistics characterising of the portfolio Panel A: Frequency distribution by issuance date of the lease

Date of issuance Number of contracts Percent of total Cumulative percent

1990 3108 8,7% 8,7% 1991 3593 10,0% 18,7% 1992 4328 12,1% 30,8% 1993 4414 12,3% 43,2% 1994 3943 11,0% 54,2% 1995 4518 12,6% 66,8% 1996 4421 12,4% 79,1% 1997 3631 10,1% 89,3% 1998 1793 5,0% 94,3% 1999 1367 3,8% 98,1% 2000 671 1,9% 100,0% Total 35787 100,0%

Panel B: Frequency distribution by the term-to-maturity of the lease

Term-to-maturity in months Number of contracts Percent of total Cumulative percent

0 to 11 5961 16,7% 16,7% 12 to 23 1086 3,0% 19,7% 24 to 35 1682 4,7% 24,4% 36 to 47 7724 21,6% 46,0% 48 to 59 11515 32,2% 78,2% 60 to 71 7738 21,6% 99,8% over 71 81 0,2% 100,0% Total 35787 100,0%

Minimum Maximum Mean Median

0 120 39 48

Panel C: Frequency distribution by cost of the leased asset

Cost of the asset in € Number of contracts Percent of total Cumulative percent

7,400 to 25,000 26303 73,5% 73,5% 25,001 to 50,000 6238 17,4% 90,9% 50,001 to 100,000 2955 8,3% 99,2% 100,001 to 200,000 264 0,7% 99,9% 200,001 to 300,000 17 0,0% 100,0% 300,001 to 400,000 7 0,0% 100,0% 400,001 to 500,000 3 0,0% 100,0% Total 35787 100,0%

Minimum Maximum Mean Median

7437 495787 23302 17291

Panel D: Frequency distribution by the distribution network

Distribution network Number of contracts Percent of total Cumulative percent

DN 1 756 2,1% 2,1% DN 2 10688 29,9% 32,0% DN 3 10688 29,9% 61,8% DN 4 8936 25,0% 86,8% DN 5 4719 13,2% 100,0% Total 35787 100,0%

Panel E: Frequency distribution by the region of origin of the lessor

Region of origin Number of contracts Percent of total Cumulative percent

A 8589 24,0% 24,0% B 17204 48,1% 72,1% C 9931 27,8% 99,8% D 51 0,1% 100,0% E 12 0,0% 100,0% Total 35787 100,0%

Panel F1: Frequency distribution by the interest premium

Interest premium Number of contracts Percent of total Cumulative percent

Less o% 565 1,6% 1,6% 0% to 0,99% 251 0,7% 2,3% 1% to 1,99% 5165 14,4% 16,7% 2% to 2,99% 6906 19,3% 36,0% 3% to 3,99% 9653 27,0% 63,0% 4% to 4,99% 9382 26,2% 89,2% 5% to 5,99% 2821 7,9% 97,1% 6% to 6,99% 459 1,3% 98,4% 7% to 7,99% 164 0,5% 98,8% 8% to 8,99% 107 0,3% 99,1%

9% to 9,99% 79 0,2% 99,3%

Over 10% 235 0,7% 100,0%

Total 35787 100,0%

Minimum Maximum Mean Median

-103,49% 110,00% 3,09% 3,06%

Panel F2: Frequency distribution by the interest premium (decile)

Interest premium Number of contracts Percent of total Cumulative percent

-103,49% to 1,29% 3572 10,0% 10,0% 1,29% to 1,59% 3585 10,0% 20,0% 1,59% to 2,21% 3594 10,0% 30,0% 2,25% to 2,65% 3570 10,0% 40,0% 2,65% to 3,05% 3591 10,0% 50,1% 3,07% to 3,40% 3585 10,0% 60,1% 3,40% to 3,70% 3548 9,9% 70,0% 3,71% to 4,05% 3561 10,0% 79,9% 4,05% to 4,56% 3601 10,1% 90,0% 4,56% to 110,00% 3580 10,0% 100,0% Total 35787 100,0%

Minimum Maximum Mean Median

-103,49% 110,00% 3,09% 3,06%

Panel G: Frequency distribution by the state of the contract

State of the contract Number of contracts Percent of total Cumulative percent

Re-rented 511 1,4% 1,4%

Completed 32021 89,5% 90,9%

Defaulted 3255 9,1% 100,0%

Total 35787 100,0%

First, it should be pointed out that fewer data on leases are available for the most recent years, since the database only consists of completed contracts. Nevertheless, the oldest leases were issued in 1990 and the most recent in 2000. The median contractual term-to-maturity of lease contracts is 48 months with a minimum term of 0 month5 _{and a maximum term of 120. The average cost of the leased asset is €23,302.} This lease portfolio falls into the definition of retail exposure as 99% of the sample has an original value of less than €100,000 and no lease value represents more than 0.2% of the total portfolio value. The interest premium of the lease, defined as the difference between the ex ante internal rate of return (option included) and the cost of funding, is on average 3%. Moreover, lease contracts may be sorted on the basis of 5 distribution networks, and 5 regions of origins of the lessor. Overall, 9.1% of the contracts defaulted.

Estimation of the portfolio risk measures

The risk measures necessary to evaluate capital requirement of a segment k (Kk)

under the Basel Proposal are probability of default (PDk), loss given default (LGDk)

and earnings at default (EADk). The probability of default is estimated following

Altman’s (1989) life-table methodology. For each period t (from 1990 to 2000), the one year probability of default (PDt) is measured as the default proportion observed

during that period. PDk is calculated as the average PDt weighted by the number of

existing contracts. This procedure takes into account that the risk associated with lease contracts can vary through time until maturity. The earning at default of a defaulted contract i (EADi) is the total amount due at time of default. EADk is thus

calculated as the sum of EADi of all contracts belonging to segment k. The loss given

default of a defaulted contract i (LGDi) is computed as one minus the discounted

value of amount recovered in comparison with EADi. The recovered cash flows arise

from asset net liquidation, other guaranties and collaterals and late payments. The discount rate applied to each cash flow is the ex ante yield to maturity for the lease contract in defaults. LGDi may be positive or negative expressing net losses or net

gains to the lessor. LGDk is measured as the average of observed LGDi weighted by

EADi.

The Basel formula for capital requirement is applied to the risk measures for each segment k. Subsequently, the total capital required for the global portfolio is computed as the average of Kk weighted by the size of the segment (Nk).

4.

Results (Basel approach)

Total capital requirement of the lease portfolio is computed for 7 different segmentations based on ex ante variables. The interest premium variable is used twice: first on the basis of absolute level clusters and second via the deciles. A control segmentation is also tested. In this case, the total portfolio is randomly divided into 10 segments. Table 2 provides a summary of the results6_.

Table 2: Summary of the capital requirement by segmentation (Basel only approach)

Capital required % of reduction Capital required % of reduction Mean Mean Asset

LGD included LGD not included LGD Correlation

No segmentation 4,00% 12,83% 3,21 8,71%

Segmentation by:

A - Issuance date 3,94% 1,5% 12,74% 0,8% 3,24 8,77%

B – Term-to-maturity 3,55% 11,3% 11,29% 12,1% 3,18 9,89%

C - Cost of the leased asset 3,88% 2,9% 12,85% -0,1% 3,31 8,68%

D - Distribution network 3,94% 1,3% 12,69% 1,1% 3,22 8,89%

E - Region of origin of the lessor 4,01% -0,3% 12,79% 0,4% 3,19 8,77%

F1 - Interest premium 3,70% 7,4% 12,15% 5,4% 3,28 9,36%

F2 - Interest premium (decile) 3,69% 7,7% 11,97% 6,7% 3,25 9,48%

H - Control 3,99% 0,1% 12,83% 0,1% 3,21 8,72%

LGD not included indicates that the capital requirement was calculated using LGD=100%.

The capital required for the total lease portfolio without segmentation is 4.00%. In general, the use of segment of the portfolio yields to a decrease of the total capital required. This is especially true for the segmentation based on the term-to-maturity (B) and on the interest premium (F1 and F2). For both segmentations average PD varies widely across segments. When LGD is fixed, the total capital requirement is 12.8%. We observe similar capital requirement pattern through segmentation type but with slightly lower reduction. LGD seems not to be the key variable in determining optimal segmentation. The control panel presents virtually no capital reduction either with or without taking LGD into account. Indeed, we observe close PD and LGD for all deciles. Thus, the number of segment is not the leading variable in optimising total capital requirement.

These results tend to follow the theoretical statement that capital requirement can be reduced through a segmentation of the portfolio that yields the most “extreme” PDs. This is confirmed by the good performance of the interest premium segmentation. Indeed, the interest premium reflects the lender (lessee) confidence on the payments of all due amounts by the borrower (lessor). This premium is most probably derived from a scoring system. Thus, it is no surprise that the segments defined on the basis of this variable present different PDs and allow nearly 7.5% reduction of the capital requirement.

Nevertheless, reducing the risk (and thus the capital requirement) by grouping similar assets (i.e. lease contracts with close risk premium) is somewhat striking from a portfolio management point of view and unsatisfactory from a regulatory one. Without departing from the one factor model used in the Basel framework, we concentrate in the rest of this paper on asset return correlation. Under IRBA, this risk characteristic is not estimated but directly computed from PD. However, as it represents the specific contribution of every asset to the systematic risk of the overall portfolio, it significantly affects the estimation of a portfolio’s credit risk.

5.

Estimation of the asset return correlation

The asset correlation debate

The Basel Proposal introduced asset return correlation as a parameter for calculating regulatory capital requirements. However, as it quickly becomes unrealistic to consider the unique asset return correlation for each obligator of a large credit portfolio, the Basel Committee proposed using an average correlation for every obligor. In the Basel Committee’s paper of January 2001, this average asset return correlation was set at 20%. Empirical studies testing this assumption pointed out that the risk-weighting ratio was too high (e.g. Sironi and Zazzara (2001) who analysed corporate portfolios of Italian banks). In its November 2001 release, the Basel Committee proposed an alternative formula defining the asset return correlation as a

decreasing function of PD. Resti (2002) supported theoretically this modification arguing that the use of a lower asset return correlation coefficient for riskier borrowers is a reasonable way of making the weighting function less steep. However, empirically results were less supportive. On the one hand, Lopez (2002) using the KMV methodology model on 14000 US, EU and Japanese firms indicate that asset return correlation is a decreasing function of PD and an increasing one of the firm’s asset size. On the other hand, Dietsch and Petey (2003), using an ordered probit model on French and German SMEs, argue for a different risk-weight function for SMEs. Indeed, their estimated correlations are significantly lower than the correlation levels assumed by the Basel Committee and their results show no negative relationship between asset return correlations and PDs. Based on a leasing portfolio, Duchemin, Laurent and Schmit (2003) conclude similarly. Moreover, they challenge the underlying Basel assumption of an increasing and concave relationship between standard deviation of conditional probabilities of default and unconditional probabilities of default.

The one factor model

We use a one systematic factor probit ordered model for deriving asset return correlation. It is a restricted version of CreditMetricsTM _{(Gordy (2000)). This one} factor model has been used by the Basel Committee for the determination of appropriate capital requirements.

The asset value return of an obligator i (Zi) in a given portfolio is defined as a linear

function of a single systematic factor (x), which represents the state of the economy, and an idiosyncratic factor (εi).

Z

i =

wx

+ (1-

w

²)

0.5

_ε

_{i [9]}

The loading factor w indicates the extent to which any obligor of the given portfolio is exposed to systematic risk. If both factors (x and εi) are assumed to be independent

standard normal variables, Zi is also a standard normal variable.

Default is stated when the asset value return falls below a certain threshold (

τ

). The probability of default for each obligator in the portfolio (PD) is thus directly identified as:

PD = Proba (Zi <

τ

) =

φ

(

τ

)

[10]

where

φ

(.)

is the cumulative distribution function of a standard normal. In other words, obligator i defaults when:

i.e. when wx + (1-w²)0.5_{εi <}

φ

-1_(PD)

_[12] or when εi < [

φ

-1(PD) – wx ] / (1-w²)0.5 [13]

where

_φ

−1

(.)

_{is the inverse cumulative distribution function of a standard normal.} Hence, the default probability for obligor i, conditional on the realization of x is

PD(x) =

φ

[ (

φ

-1_{(PD) - wx)/(1-w²)}0.5_]

_[14] In this framework, two distinct formulations of the probability of default are given: on the one hand, the conditional probability of default (PD(x)) which is the one that is observed given the state of the economy and on the other hand, the unconditional probability of default (PD).

Moreover, the correlation between the asset value returns of two obligors i and j from the same portfolio can be easily derived:

ρ

(Zi,Zj) = w²

[15]

As asset value return is not observable, a new observable variable is introduced in order to derive the average asset return correlation. The dummy variable (Di) reflects

the emergence of defaults. It is defined as follows:

   − = ∞ < < − − = − − ≤ < ∞ − = ₋ − ) ( 1 ) 1 /( ] ) ( [ 0 ) ( ) 1 /( ] ) ( [ 1 5 . 0 2 1 5 . 0 2 1 x PD proba w wx PD if x PD proba w wx PD if D i i i

_φ

_ε

φ

ε

[16] The joint probability of default of two obligors can be expressed as:

E[DiDj] = E[ Proba (Zi <

φ

-1(PD) & Zj <

φ

-1(PD)|x) ] [17]

An thus,

E[DiDj] =

φ

2(

φ

-1(PD),

φ

-1(PD), w²) [18] where

φ

2(.) is the cumulative distribution function of a bivariate standard normal with correlation w².

Finally, one can show that:

Var[PD(x)] = E[PD(x)²] - E[PD(x)]² = E[DiDj] – PD²

[19]

Therefore, the calibration of the asset value return correlation involves solving Equations 18 and 19 simultaneously. In the remainder of the paper, STD will refer to

the unconditional standard deviation7 _{and R to asset return correlation calibrated as} described previously.

Influence of STD on capital requirement

We assess the influence of STD on the capital requirement through its relation with asset return correlation. For each couple (PD, STD), we can derive asset return correlation using the one factor model. The measured R is then introduced with PD in the Basel Committee formula to calculate capital requirement.

Figure 5 presents the 3D-graph of the asset return correlation as a function of the probability of default and unconditional standard deviation. We first observe a negative relationship between R and PD as is defined in the Basel Proposal. Moreover, STD has a large impact on R which is reinforced for low PDs. Overall, Basel correlation is higher than estimated correlation except for high STD and low PD (see Figure 6).

Figure 5: Asset return correlation as a function of PD and STD

0,01 0,06 0,11 0,16 0,5% 1,0% 1,5%2,0% 0% 2% 4% 6% 8% 10% 12% 14% 16% 18% 20% 22% 24% 26% 28% R PD _S

Figure 6: Comparison of asset return correlation depending on STD

0% 5% 10% 15% 20% 25% 30% 0,01 0,03 0,05 0,07 0,09 0,11 0,13 0,15 0,17 0,19 PD R S=0.5% S=1% S=1.5% S=2% Basel

The influence of STD and PD on the capital requirement in presented in figure 7. We observe the influence of STD through asset return correlation on the capital requirement. Indeed, K is relatively high for high STD and low PD. In general, K increases with STD and with PD. Finally, the capital required using only the Basel formula is significantly higher than the one calculated using STD, except for high STD and low PD (see figure 8).

Figure 7: Capital requirement as a function of PD and STD (LGD=100%)

0,01 _0,06 0,11 _0,16 0,5% 1,0% 1,5% 2,0% 0% 2% 4% 6% 8% 10% 12% 14% 16% 18% 20% 22% 24% 26% 28% K PD S

0,00 0,05 0,10 0,15 0,20 0,25 0,30 0,35 0,40 0,01 0,03 0,05 0,07 0,09 0,11 0,13 0,15 0,17 0,19 PD K S=0,5% S=1% S=1,5% S=2% Basel

6.

Results (Model approach)

In order to apply the one factor model to calculate asset return correlation, we need to measure first STD from the data. We used the method presented by Gordy (2000) to estimate unconditional variance of the default rate for each of the N segments (Var [pk(x)]=STD²). The assumptions for equation 20 to hold are that the realised values of

the systematic factor (x) are serially independent, that in segment k the measured defaults (dk) and the initial number of contracts (nk) are independent of the state of

nature.

[

]

    − − × ×     − = k k k k k k n E p p n E p Var x p Var 1 1 ) 1 ( 1 ) ( ) ( , k= 1, … , N _[20]

Table 3 presents the level of capital requirement for each the segmentation of the portfolio.

Table 3: Summary of the capital requirement by segmentation (Model approach)

required Capital reduction % of required Capital reduction % of Mean Mean Mean Asset

LGD included LGD not included LGD STD Correlation

No segmentation 1,35% 4,32% 3,21 0,513% 0,87%

Segmentation by:

A - Issuance date 3,09% -129,8% 9,61% -122,5% 3,11 1,346% 5,32%

B – Term-to-maturity 1,81% -34,5% 5,19% -20,2% 2,87 0,620% 4,94%

C – Cost of the leased asset 1,34% 0,6% 4,41% -2,2% 3,30 0,518% 0,99%

D - Distribution network 1,48% -9,9% 4,77% -10,5% 3,23 0,598% 1,34%

E - Region of origin of the lessor 1,45% -7,9% 4,65% -7,6% 3,20 0,581% 1,14%

F1 - Interest premium 3,68% -173,6% 12,12% -180,7% 3,29 0,883% 11,07%

F2 - Interest premium (decile) 2,12% -57,4% 6,80% -57,4% 3,21 0,847% 5,35%

A first observation is that capital requirement using the model approach is on average 50% lower than the capital requirement calculated on the Basel framework. For the total portfolio, K reaches 4.00% in the Basel model but only 1.35% in the one factor model. This is explained by the large difference between regulatory and estimated asset return correlation. Secondly, when using estimated R, the segmentation of the total portfolio does not reduce the capital requirement. Instead, K estimated rises. The increase in absolute term is relatively small for most of the segmentation but is significant for segmentation A and F1. For both clustering, STD is quite large in some segment implying high K. The influence of LGD estimate is again not substantial. Similar capital requirement patterns through segmentation type are observed. Finally, the control segmentation again yields no sizable8 modification of the capital requirement.

In this framework, the segmentation of the total portfolio leads to higher total capital requirement if this procedure defines sub-portfolios whose PD are more volatile than the average. This feature reconciles the capital requirement calculation with portfolio management and regulatory objectives.

7.

Conclusion

This paper focuses on the current Basel Committee proposal for calculating adequate capital requirements to cover credit risks. More specifically, the internal rating-based advanced approach (IRBA) for the retail segment is analysed. The capital requirement function with respect of probability of default is strongly concave for small PDs. Thus, theoretically a sensitive the segmentation of a loan portfolio into sub-portfolios with largely different PDs will reduce the overall capital requirement. Several segmentations were tested on a large retail lease portfolio (over 35,000 contracts) which is characterised by low PDs. Results show that even for simple segmentation techniques, the total capital requirement may be reduced by over 10%. The analysis also confirms the theoretical statement that capital requirement can be reduced through a segmentation of the portfolio that yields the most “extreme” PDs. Pooling the portfolio on the basis of ex ante interest premium, as a proxy for credit scores, allows for 30bp capital reduction.

It enlightens the opportunity for regulatory arbitrage. Indeed, as the segmentation criterion is not set by the regulator, each financial institution may use historical data (including the ex post variables) in order to define pools of credits that optimize capital requirement.

We claim that this inefficiency can be ruled out by estimating asset return correlation on the basis of PD and its volatility (STD). This is done through a one factor model.

The empirical results using estimated R show no reduction in capital requirement is achieved through segmentation of the portfolio. This enlightens the necessity for the Basel Committee to verify the accuracy of its asset return correlation estimation and argue for taking another characteristic of the portfolio risk profile into account (like volatility of PD).

8.

Bibliography

Altman, E., (1989), ‘Measuring Corporate Bond Mortality and Performance’, Journal of Finance, vol. 44. pp. 909-922.

Basel Committee on Banking Supervision (2001a), ‘The Internal rating-based Approach:

Supporting Document to the New Basel Capital Accord’, Consultative Document, BIS

January, 108 pages.

Basel Committee on Banking Supervision (2001b), ‘Potential Modifications of the Committee’s Proposals’, Press release dated November 5, 6 pages.

Basel Committee on Banking Supervision (2002), ‘Quantitative Impact Study 3 –

Technical Guidance’, BIS, Basel Switzerland, 164 pages.

Basel Committee on Banking Supervision (2003), ‘The New Basel Capital Accord’ Consultative Document, BIS, Basel, Switzerland.

Calem, P. and M. Lacour-Little (forthcoming), ‘Risk-based capital requirements for

mortgage loans’,

Journal of Banking and Finance

.

Carey M. (1998), ‘Credit risk in private debt portfolios’,

Journal of Finance

, Vol. 53, No.4,

pp.1363-1387.

De Laurentis G. and Geriano M. (2001), ‘Leasing recovery rates’, Leaseurope – Bocconi

University Business School Research, 21 pages.

Dietsch M. and Petey J. (2002), ‘The credit risk in SME loans portfolios: Modelling issues, pricing and capital requirements’, Journal of Banking and Finance, 26, pp. 303-322.

Duchemin S., M-P. Laurent and M. Schmit (2003), “Asset return correlation and Basel II: The case of automotive lease portfolios”, Working Paper CEB, n°03/007.

Gordy M. (2000), ‘A comparative anatomy of credit risk models’,

Journal of Banking and

Finance

, 24, pp. 119-149.

International Accounting Standards Board, (2002), International Accounting Standards, IAS 17 (revised 1997), p. 17-8.

Leaseurope (2002), ‘Leasing Activity in Europe. Key Facts and Figures’, available from <http://www.leaseurope.org/pages/ Statistic/Stat.asp >

Pirotte H., M. Schmit and C. Vaessen (2004), “Credit Risk Mitigation Evidence in Auto Leases: LGD and Residual Value Risk”, Working Paper.

Schmit M. (2003), ‘Is automotive leasing a risky business?’, Working Paper CEB, n°03/009.

Schmit M. (forthcoming), “Credit Risk in the Leasing Industry”,

Journal of Banking and

Finance

.

Schmit M. and Stuyck J. (2002), ‘Recovery rates in the leasing industry’. Working Paper,

Leaseurope, 39 pages.

Sironi A. and Zazzara C. (2001), ‘The New Basel Accord: Possible Implications for Italians Banks’, September, available on <http://www.defaultrisk.com>, 36 pages.

9.

Appendix

Derivation of the capital requirement formula with respect to the

probability of default

The correlation and capital requirement, R(p) and K(p), are defined as follows:













−

−

−

−

−

×

+

−

−

−

−

×

=

)

35

exp(

1

)

35

exp(

1

1

17

.

0

)

35

exp(

1

)

35

exp(

1

02

.

0

)

(

p

p

p

R

[A1]

















−

+

−

=

(

0

.

999

)

)

(

1

)

(

)

(

)

(

1

1

)

(

G

p

R

p

R

p

G

p

R

N

p

K

[A2]

with N(.) the cumulative standard normal distribution function and G(.) the inverse of N(.).

Remember that the cumulative distribution function is:

∫

∞ −

−

=

x

z

dz

x

N

)

2

exp(

2

1

)

(

2

π

[A3]

This function is twice derivable on IR. Moreover, N(.) is bijective implying that G(.) exists. As it is not possible to define the inverse explicitly, it is computed through an approximation method.

Therefore, the differentiation of K(p) is not straightforward and we use the following result :

( )

(

1

)

1 1

)

(

− − −









=

f

x

dx

df

x

x

d

f

d

[A4]

In our case, it can be written as:

( )

(

(

)

)

1 −













=

G

p

dp

dN

p

dp

dG

[A5]

In order to compute the first and second derivatives, we first define K(.) as a function of p and G(p), which must be seen as an unknown function of p. The first derivative

with respect to p

( )

p

dp

dK

is a function of p, G(p) and

( )

p

dp

dG

. The latter term is replaced by its equivalent (equation [A5]). The second derivative is computed in the same way. Thus

( )

p

dp

dK

and

( )

p dp K d 2 2

are both functions of p and G(p).

As G(p) can be calculated for p∈

[ ]

0,1 , we can derive the graphs of K(p) and its first and second derivatives.

Moreover, the study of K(p) shows that it is: increasing and concave over [0 ; 0.04903[ increasing and convex over ]0.04903 ; 0.15184[ increasing and concave over ]0.15184 ; 1]

Graph A2: First derivative of K(p)

Capital requirement by segmentation (Basel approach)

Table A1: Capital requirement by segmentation (Basel only approach) Panel A: Segmentation by issuance date of the lease

Date of issuance Number of contracts Average PD Average LGD Asset Correlation Capital requirement K/LGD

1990 3108 3,5% 30,2% 6,4% 4,3% 14,3% 1991 3593 2,3% 26,7% 8,6% 3,4% 12,9% 1992 4328 2,3% 30,1% 8,7% 3,9% 12,8% 1993 4414 2,4% 33,3% 8,5% 4,3% 13,0% 1994 3943 2,0% 31,9% 9,4% 3,9% 12,3% 1995 4518 1,7% 28,5% 10,4% 3,3% 11,5% 1996 4421 1,5% 23,4% 10,8% 2,6% 11,2% 1997 3631 2,0% 29,4% 9,4% 3,6% 12,3% 1998-2000 3831 4,1% 42,4% 5,6% 6,3% 14,9%

Total Capital Required 3,9%

Panel B: Segmentation by the term-to-maturity of the lease

Term-to-maturity in months Number of contracts Average PD Average LGD Asset Correlation Capital requirement K/LGD

0 to 11 5961 0,2% 52,6% 16,0% 1,9% 3,5% 12 to 23 1086 1,7% 42,8% 10,4% 5,0% 11,6% 24 to 35 1682 2,1% 36,8% 9,3% 4,6% 12,4% 36 to 47 7724 1,9% 23,9% 9,7% 2,9% 12,1% 48 to 59 11515 2,3% 30,8% 8,8% 3,9% 12,8% over 60 7819 3,1% 32,2% 7,0% 4,5% 14,0%

Total Capital Required 3,5%

Panel C: Segmentation by cost of the leased asset

Cost of the asset in € Number of contracts Average PD Average LGD Asset Correlation Capital requirement K/LGD

7,400 to 25,000 26303 2,4% 29,0% 8,4% 3,8% 13,1%

25,001 to 50,000 6238 2,0% 34,4% 9,5% 4,2% 12,3%

50,001 to 100,000 2955 2,0% 32,7% 9,4% 4,0% 12,3%

100,001 to 500,000 291 1,6% 27,1% 10,6% 3,1% 11,4%

Total Capital Required 3,9%

Panel D: Segmentation by the distribution network

Distribution network Number of contracts Average PD Average LGD Asset Correlation Capital requirement K/LGD

DN 1 756 4,6% 26,7% 5,0% 4,1% 15,4%

DN 2 10688 2,7% 28,5% 7,9% 3,8% 13,4%

DN 3 10688 2,0% 30,3% 9,3% 3,8% 12,4%

DN 4 8936 1,9% 34,1% 9,8% 4,1% 12,1%

DN 5 4719 2,1% 34,3% 9,1% 4,3% 12,5%

Total Capital Required 3,9%

Panel E: Segmentation by the region of origin of the lessor

Region of origin Number of contracts Average PD Average LGD Asset Correlation Capital requirement K/LGD

A 8589 1,9% 32,9% 9,6% 4,0% 12,2%

B 17204 2,4% 28,3% 8,5% 3,7% 12,9%

C 9931 2,4% 35,4% 8,4% 4,6% 13,0%

D and E 63 3,7% 41,7% 6,0% 6,1% 14,6%

Total Capital Required 4,0%

Interest premium Number of contracts Average PD Average LGD Asset Correlation Capital requirement K/LGD Less o,99% 816 1,5% 23,2% 10,8% 2,6% 11,2% 1% to 1,99% 5165 0,8% 33,6% 13,3% 2,9% 8,5% 2% to 2,99% 6906 1,4% 25,7% 11,1% 2,8% 10,9% 3% to 3,99% 9653 2,2% 29,8% 9,0% 3,8% 12,6% 4% to 4,99% 9382 2,9% 29,9% 7,4% 4,1% 13,7% 5% to 5,99% 2821 3,2% 32,8% 6,8% 4,6% 14,1% Over 6% 1044 4,6% 52,1% 5,0% 8,0% 15,4%

Total Capital Required 3,7%

Panel F2: Segmentation by the interest premium (decile)

Interest premium Number of contracts Average PD Average LGD Asset Correlation Capital requirement K/LGD

-103,49% to 1,29% 3572 1,1% 35,6% 12,3% 3,4% 9,7% 1,29% to 1,59% 3585 0,5% 20,7% 14,5% 1,4% 6,7% 1,59% to 2,21% 3594 1,3% 23,5% 11,4% 2,5% 10,7% 2,25% to 2,65% 3570 1,8% 29,5% 9,9% 3,5% 11,9% 2,65% to 3,05% 3591 2,0% 25,0% 9,4% 3,1% 12,3% 3,07% to 3,40% 3585 2,3% 32,8% 8,6% 4,2% 12,9% 3,40% to 3,70% 3548 2,6% 31,8% 8,1% 4,2% 13,3% 3,71% to 4,05% 3561 2,9% 28,3% 7,5% 3,9% 13,6% 4,05% to 4,56% 3601 3,3% 31,3% 6,8% 4,4% 14,1% 4,56% to 110,00% 3580 3,6% 43,0% 6,2% 6,2% 14,5%

Total Capital Required 3,7%

Panel G: No segmentation

Number of contracts Average PD Average LGD Asset Correlation Capital requirement K/LGD

Total portfolio 35787 2,3% 31,1% 8,7% 4,0% 12,8%

Panel H: Control segmentation (decile)

Number of contracts Average PD Average LGD Asset Correlation Capital requirement K/LGD

Decile 1 3537 2,4% 32,1% 8,6% 4,1% 12,9% Decile 2 3596 2,3% 32,9% 8,6% 4,2% 12,9% Decile 3 3541 2,5% 32,3% 8,2% 4,3% 13,2% Decile 4 3660 2,2% 29,7% 8,9% 3,8% 12,7% Decile 5 3559 2,3% 29,6% 8,7% 3,8% 12,9% Decile 6 3635 2,3% 31,7% 8,6% 4,1% 12,9% Decile 7 3592 2,4% 30,2% 8,4% 3,9% 13,0% Decile 8 3647 2,2% 33,4% 8,9% 4,2% 12,7% Decile 9 3629 2,2% 32,7% 8,9% 4,2% 12,7% Decile 10 3391 2,1% 26,4% 9,3% 3,3% 12,4%

Capital requirement by segmentation (One factor model approach)

Table A2: Capital requirement by segmentation (One factor model approach) Panel A: Segmentation by issuance date of the lease

Date of issuance Number of contracts Average PD Average LGD STD Asset correlation Capital requirement K/LGD

1990 3108 3,5% 30,2% 1,4% 3,4% 3,1% 10,2% 1991 3593 2,3% 26,7% 1,2% 4,1% 2,2% 8,2% 1992 4328 2,3% 30,1% 1,9% 9,7% 4,1% 13,8% 1993 4414 2,4% 33,3% 1,2% 4,4% 2,9% 8,7% 1994 3943 2,0% 31,9% 1,1% 4,3% 2,4% 7,5% 1995 4518 1,7% 28,5% 0,7% 2,4% 1,3% 4,7% 1996 4421 1,5% 23,4% 0,8% 3,8% 1,3% 5,6% 1997 3631 2,0% 29,4% 1,5% 7,8% 3,2% 10,8% 1998-2000 3831 4,1% 42,4% 2,6% 8,1% 7,8% 18,3%

Total Capital Required 3,1%

Panel B: Segmentation by the term-to-maturity of the lease

Term-to-maturity in months Number of contracts Average PD Average LGD STD Asset correlation Capital requirement K/LGD

0 to 11 5961 0,2% 52,6% 0,4% 21,6% 2,6% 5,0% 12 to 23 1086 1,7% 42,8% 1,6% 12,1% 5,6% 13,1% 24 to 35 1682 2,1% 36,8% 0,9% 3,4% 2,5% 6,7% 36 to 47 7724 1,9% 23,9% 0,5% 1,1% 0,9% 3,9% 48 to 59 11515 2,3% 30,8% 0,5% 0,8% 1,3% 4,2% over 60 7819 3,1% 32,2% 0,9% 1,4% 2,1% 6,6%

Total Capital Required 1,8%

Panel C: Segmentation by cost of the leased asset

Cost of the asset in € Number of contracts Average PD Average LGD STD Asset correlation Capital requirement K/LGD

7,400 to 25,000 26303 2,4% 29,0% 0,5% 0,7% 1,2% 4,3%

25,001 to 50,000 6238 2,0% 34,4% 0,5% 0,9% 1,3% 3,8%

50,001 to 100,000 2955 2,0% 32,7% 0,9% 3,5% 2,2% 6,7%

100,001 to 500,000 291 1,6% 27,1% 0,9% 4,5% 1,7% 6,3%

Total Capital Required 1,3%

Panel D: Segmentation by the distribution network

Distribution network Number of contracts Average PD Average LGD STD Asset correlation Capital requirement K/LGD

DN 1 756 4,6% 26,7% 1,6% 2,8% 3,1% 11,8%

DN 2 10688 2,7% 28,5% 0,4% 0,5% 1,2% 4,3%

DN 3 10688 2,0% 30,3% 0,7% 2,0% 1,6% 5,3%

DN 4 8936 1,9% 34,1% 0,5% 1,2% 1,4% 4,0%

DN 5 4719 2,1% 34,3% 0,7% 1,7% 1,7% 5,1%

Total Capital Required 1,5%

Panel E: Segmentation by the region of origin of the lessor

Region of origin Number of contracts Average PD Average LGD STD Asset correlation Capital requirement K/LGD

A 8589 1,9% 32,9% 0,5% 1,3% 1,4% 4,2%

B 17204 2,4% 28,3% 0,6% 1,2% 1,4% 4,9%

C 9931 2,4% 35,4% 0,6% 0,9% 1,6% 4,6%

D and E 63 3,7% 41,7% 0,0% 0,0% 1,6% 3,7%

Total Capital Required 1,5%

Interest premium Number of contracts Average PD Average LGD STD Asset correlation Capital requirement K/LGD Less o,99% 816 1,5% 23,2% 2,1% 19,0% 4,2% 18,2% 1% to 1,99% 5165 0,8% 33,6% 1,5% 29,6% 6,6% 19,5% 2% to 2,99% 6906 1,4% 25,7% 0,8% 23,6% 5,5% 21,4% 3% to 3,99% 9653 2,2% 29,8% 0,5% 4,5% 2,4% 8,1% 4% to 4,99% 9382 2,9% 29,9% 0,6% 1,1% 1,7% 5,7% 5% to 5,99% 2821 3,2% 32,8% 0,8% 0,9% 1,9% 5,9% Over 6% 1044 4,6% 52,1% 3,1% 9,2% 11,3% 21,7%

Total Capital Required 3,7%

Panel F2: Segmentation by the interest premium (decile)

Interest premium Number of contracts Average PD Average LGD STD Asset correlation Capital requirement K/LGD

-103,49% to 1,29% 3572 1,1% 35,6% 1,7% 21,5% 5,8% 16,3% 1,29% to 1,59% 3585 0,5% 20,7% 0,8% 16,0% 1,5% 7,5% 1,59% to 2,21% 3594 1,3% 23,5% 0,8% 4,9% 1,4% 5,8% 2,25% to 2,65% 3570 1,8% 29,5% 0,9% 4,0% 2,0% 6,6% 2,65% to 3,05% 3591 2,0% 25,0% 0,8% 2,5% 1,4% 5,7% 3,07% to 3,40% 3585 2,3% 32,8% 0,6% 1,0% 1,5% 4,6% 3,40% to 3,70% 3548 2,6% 31,8% 0,6% 0,8% 1,5% 4,7% 3,71% to 4,05% 3561 2,9% 28,3% 0,6% 0,8% 1,5% 5,2% 4,05% to 4,56% 3601 3,3% 31,3% 0,6% 0,0% 1,0% 3,3% 4,56% to 110,00% 3580 3,6% 43,0% 1,1% 2,0% 3,6% 8,4%

Total Capital Required 2,1%

Panel G: No segmentation

Number of contracts Average PD Average LGD STD Asset correlation Capital requirement K/LGD

Total portfolio 35787 2,3% 31,1% 0,5% 0,9% 1,3% 4,3%

Panel H: Control segmentation (decile)

Number of contracts Average PD Average LGD STD Asset correlation Capital requirement K/LGD

Decile 1 3537 2,4% 32,1% 0,6% 1,0% 1,5% 4,6% Decile 2 3596 2,3% 32,9% 0,2% 0,2% 1,0% 3,2% Decile 3 3541 2,5% 32,3% 0,6% 0,9% 1,5% 4,7% Decile 4 3660 2,2% 29,7% 0,5% 0,7% 1,2% 4,0% Decile 5 3559 2,3% 29,6% 0,6% 1,1% 1,4% 4,7% Decile 6 3635 2,3% 31,7% 0,4% 0,5% 1,2% 3,9% Decile 7 3592 2,4% 30,2% 0,5% 0,8% 1,3% 4,4% Decile 8 3647 2,2% 33,4% 0,5% 0,9% 1,4% 4,2% Decile 9 3629 2,2% 32,7% 0,5% 0,7% 1,3% 4,0% Decile 10 3391 2,1% 26,4% 0,5% 0,9% 1,0% 3,9%