How to reduce capital requirement?
The case of retail portfolios with low probability of
default
Marie-Paule Laurent1
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Preliminary Draft– Do not quote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.
50 av. Roosevelt – ULB CP 145/1 – 1050 Bruxelles – Belgium – [email protected]
1
How to reduce capital requirement? The case of retail portfolios with low probabilities of default
1
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.
How to reduce capital requirement? The case of retail portfolios with low probabilities of default
2
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 ratingbased foundation approach (IRBF) and the internal 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).
KA = K × EAD
[1]
= 8% × RW × EAD
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.
KA = 8% × 0.75 × EAD
[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:
[
]
K = LGD × φ (1 − R) −0.5 × φ −1 ( PD) + ( R 1 − R) 0.5 × φ −1 (0.999) × M adj
[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.
The asset return correlation is defined as a function of PD.
How to reduce capital requirement? The case of retail portfolios with low probabilities of default
3
R = 12% ×
1 − e −50× PD
1 − e −50× PD
S −5
+
24
%
×
[
1
−
] − 0.04 × [1 −
]
−50
−50
45
1− e
1− e
[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:
M adj = [1 − 1.5 × b( PD)]−1 × [1 + ( M − 2.5) × b( PD)]
with
M
the
effective
maturity
[5]
of
exposure
and
b( PD) = [0.08451 − 0.05898 × ln( PD)] .
2
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:
[
K = LGD × φ (1 − R) −0.5 × φ −1 ( PD) + ( R 1 − R) 0.5 × φ −1 (0.999)
]
[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 − e −35× PD
1 − e −35× PD
R = 2% ×
+ 17% × [1 −
]
1 − e −35
1 − e −35
[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, lowquality 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.
How to reduce capital requirement? The case of retail portfolios with low probabilities of default
4
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
0,20
Correlation
0,16
0,12
0,08
0,04
0,00
0%
5%
10%
15%
20%
25%
30%
PD
Figure 2: Capital requirement as a function of PD under the IRBA approach for retail
credit (LGD=50%)
25
20
K
15
10
5
0
0%
5%
10%
15%
20%
25%
30%
PD
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:
K ( x × PD1 + (1 − x) × PD2 ) ≥ x × K (PD1 ) + (1 − x) × K (PD2 )
Where x ∈ [0;1] .
[8]
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
How to reduce capital requirement? The case of retail portfolios with low probabilities of default
5
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
the general conclusion of this section.
How to reduce capital requirement? The case of retail portfolios with low probabilities of default
6
K of the total portfolio
0,16
0,14
0,12
0,10
0,08
0,06
0,04
0,02
0,00
1000 995 990 985 980 975 970 965 960 955 950 945
Size of Portfolio B
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.
K of the total portfolio
0,16
0,14
0,12
100%
0,10
80%
0,08
60%
0,06
40%
0,04
20%
0,02
0,00
1000
975
950
925
900
875
850
825
800
Size of Portfolio B
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
How to reduce capital requirement? The case of retail portfolios with low probabilities of default
7
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 nontraded 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.
How to reduce capital requirement? The case of retail portfolios with low probabilities of default
8
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
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
Total
Number of contracts
3108
3593
4328
4414
3943
4518
4421
3631
1793
1367
671
35787
Percent of total Cumulative percent
8,7%
8,7%
10,0%
18,7%
12,1%
30,8%
12,3%
43,2%
11,0%
54,2%
12,6%
66,8%
12,4%
79,1%
10,1%
89,3%
5,0%
94,3%
3,8%
98,1%
1,9%
100,0%
100,0%
How to reduce capital requirement? The case of retail portfolios with low probabilities of default
9
Panel B: Frequency distribution by the term-to-maturity of the lease
Term-to-maturity in months
0 to 11
12 to 23
24 to 35
36 to 47
48 to 59
60 to 71
over 71
Total
Number of contracts
5961
1086
1682
7724
11515
7738
81
35787
Minimum
0
Maximum
120
Percent of total Cumulative percent
16,7%
16,7%
3,0%
19,7%
4,7%
24,4%
21,6%
46,0%
32,2%
78,2%
21,6%
99,8%
0,2%
100,0%
100,0%
Mean
39
Median
48
Panel C: Frequency distribution by cost of the leased asset
Cost of the asset in €
7,400 to 25,000
25,001 to 50,000
50,001 to 100,000
100,001 to 200,000
200,001 to 300,000
300,001 to 400,000
400,001 to 500,000
Total
Number of contracts
26303
6238
2955
264
17
7
3
35787
Percent of total Cumulative percent
73,5%
73,5%
17,4%
90,9%
8,3%
99,2%
0,7%
99,9%
0,0%
100,0%
0,0%
100,0%
0,0%
100,0%
100,0%
Minimum
Maximum
Mean
Median
7437
495787
23302
17291
Panel D: Frequency distribution by the distribution network
Distribution network
DN 1
DN 2
DN 3
DN 4
DN 5
Number of contracts
756
10688
10688
8936
4719
Total
35787
Percent of total Cumulative percent
2,1%
2,1%
29,9%
32,0%
29,9%
61,8%
25,0%
86,8%
13,2%
100,0%
100,0%
Panel E: Frequency distribution by the region of origin of the lessor
Region of origin
A
B
C
D
E
Number of contracts
8589
17204
9931
51
12
Total
35787
Percent of total Cumulative percent
24,0%
24,0%
48,1%
72,1%
27,8%
99,8%
0,1%
100,0%
0,0%
100,0%
100,0%
Panel F1: Frequency distribution by the interest premium
Interest premium
Less o%
0% to 0,99%
1% to 1,99%
2% to 2,99%
3% to 3,99%
4% to 4,99%
5% to 5,99%
6% to 6,99%
7% to 7,99%
8% to 8,99%
Number of contracts
565
251
5165
6906
9653
9382
2821
459
164
107
Percent of total Cumulative percent
1,6%
1,6%
0,7%
2,3%
14,4%
16,7%
19,3%
36,0%
27,0%
63,0%
26,2%
89,2%
7,9%
97,1%
1,3%
98,4%
0,5%
98,8%
0,3%
99,1%
How to reduce capital requirement? The case of retail portfolios with low probabilities of default
10
9% to 9,99%
Over 10%
Total
79
235
35787
0,2%
0,7%
100,0%
99,3%
100,0%
Minimum
-103,49%
Maximum
110,00%
Mean
3,09%
Median
3,06%
Panel F2: Frequency distribution by the interest premium (decile)
Interest premium
-103,49% to 1,29%
1,29% to 1,59%
1,59% to 2,21%
2,25% to 2,65%
2,65% to 3,05%
3,07% to 3,40%
3,40% to 3,70%
3,71% to 4,05%
4,05% to 4,56%
4,56% to 110,00%
Total
Number of contracts
3572
3585
3594
3570
3591
3585
3548
3561
3601
3580
35787
Percent of total Cumulative percent
10,0%
10,0%
10,0%
20,0%
10,0%
30,0%
10,0%
40,0%
10,0%
50,1%
10,0%
60,1%
9,9%
70,0%
10,0%
79,9%
10,1%
90,0%
10,0%
100,0%
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
Re-rented
Completed
Defaulted
Number of contracts
511
32021
3255
Total
35787
Percent of total Cumulative percent
1,4%
1,4%
89,5%
90,9%
9,1%
100,0%
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)
5
A lease contract with a 0-month term-to-maturity is originated for stock financing purposes.
How to reduce capital requirement? The case of retail portfolios with low probabilities of default
11
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)
No segmentation
Segmentation by:
A - Issuance date
B – Term-to-maturity
C - Cost of the leased asset
D - Distribution network
E - Region of origin of the lessor
F1 - Interest premium
F2 - Interest premium (decile)
H - Control
Capital required % of reduction
LGD included
4,00%
3,94%
3,55%
3,88%
3,94%
4,01%
3,70%
3,69%
3,99%
1,5%
11,3%
2,9%
1,3%
-0,3%
7,4%
7,7%
0,1%
Capital required % of reduction
LGD not included
12,83%
12,74%
11,29%
12,85%
12,69%
12,79%
12,15%
11,97%
12,83%
0,8%
12,1%
-0,1%
1,1%
0,4%
5,4%
6,7%
0,1%
Mean
LGD
3,21
Mean Asset
Correlation
8,71%
3,24
3,18
3,31
3,22
3,19
3,28
3,25
3,21
8,77%
9,89%
8,68%
8,89%
8,77%
9,36%
9,48%
8,72%
LGD not included indicates that the capital requirement was calculated using LGD=100%.
6
Detailed results can be found are in the appendix.
How to reduce capital requirement? The case of retail portfolios with low probabilities of default
12
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
How to reduce capital requirement? The case of retail portfolios with low probabilities of default
13
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).
Zi = wx + (1-w²)0.5εi
[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:
Zi < φ -1(PD)
How to reduce capital requirement? The case of retail portfolios with low probabilities of default
[11]
14
wx + (1-w²)0.5εi < φ -1(PD)
εi < [ φ -1(PD) – wx ] / (1-w²)0.5
i.e. when
or when
[12]
[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 if
Di = 
0 if
− ∞ < ε i ≤ [φ −1 ( PD) − wx ] /(1 − w 2 ) 0.5
[φ −1 ( PD) − wx ] /(1 − w 2 ) 0.5 < ε i < ∞
proba = PD( x)
proba = 1 − PD( x)
[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
How to reduce capital requirement? The case of retail portfolios with low probabilities of default
15
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
28%
26%
24%
22%
20%
18%
16%
R 14%
12%
10%
8%
6%
4%
2%
0%
0,01
0,06
PD
0,11
2,0%
1,5%
0,16
1,0%
0,5%
S
Figure 6: Comparison of asset return correlation depending on STD
7
STD = (Var [ p(x)])
0.5
How to reduce capital requirement? The case of retail portfolios with low probabilities of default
16
30%
R
25%
20%
15%
10%
5%
0%
0,01
S=0.5%
0,03
0,05
S=1%
0,07
0,09
0,11
S=1.5%
0,13
S=2%
0,15
0,17
0,19
PD
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%)
28%
26%
24%
22%
20%
18%
16%
K 14%
12%
10%
8%
6%
4%
2%
0%
0,01
2,0%
1,5%
1,0%
0,06
0,11
0,16
S
0,5%
PD
Figure 8: Comparison of capital requirement depending on STD (LGD=100%)
How to reduce capital requirement? The case of retail portfolios with low probabilities of default
17
0,40
0,35
K
0,30
0,25
0,20
0,15
0,10
0,05
0,00
0,01
0,03
S=0,5%
0,05
S=1%
0,07
0,09
S=1,5%
0,11
0,13
S=2%
0,15
0,17
Basel
0,19
PD
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.
Var ( pk ) − E  1  × p k × (1 − p k )
 nk 
, k= 1, … , N
Var[ pk ( x)] =


1
1− E
 nk 
[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)
No segmentation
Segmentation by:
A - Issuance date
B – Term-to-maturity
C – Cost of the leased asset
D - Distribution network
E - Region of origin of the lessor
F1 - Interest premium
F2 - Interest premium (decile)
H – Control
Capital
% of
required
reduction
LGD included
1,35%
3,09%
1,81%
1,34%
1,48%
1,45%
3,68%
2,12%
1,29%
-129,8%
-34,5%
0,6%
-9,9%
-7,9%
-173,6%
-57,4%
4,1%
Capital
% of
required
reduction
LGD not included
4,32%
9,61%
5,19%
4,41%
4,77%
4,65%
12,12%
6,80%
4,15%
-122,5%
-20,2%
-2,2%
-10,5%
-7,6%
-180,7%
-57,4%
4,0%
How to reduce capital requirement? The case of retail portfolios with low probabilities of default
Mean
LGD
3,21
Mean
STD
0,513%
Mean Asset
Correlation
0,87%
3,11
2,87
3,30
3,23
3,20
3,29
3,21
3,22
1,346%
0,620%
0,518%
0,598%
0,581%
0,883%
0,847%
0,474%
5,32%
4,94%
0,99%
1,34%
1,14%
11,07%
5,35%
0,77%
18
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.
8
Only 6bp in absolute terms.
How to reduce capital requirement? The case of retail portfolios with low probabilities of default
19
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. 303322.
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 >
How to reduce capital requirement? The case of retail portfolios with low probabilities of default
20
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.
How to reduce capital requirement? The case of retail portfolios with low probabilities of default
21
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:
R ( p ) = 0.02 ×
 1 − exp(−35 p ) 
1 − exp(−35 p)
+ 0.17 × 1 −
1 − exp(−35)
1 − exp(−35) 



R( p)
1
K ( p) = N 
G ( p) +
G (0.999) 

 1 − R( p)
1 − R( p)


[A1]
[A2]
with N(.) the cumulative standard normal distribution function and G(.) the inverse
of N(.).
Remember that the cumulative distribution function is:
x
N ( x) =
1
− z2
exp(
)dz
∫ 2π
2
−∞
[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 :
d f −1
(x ) =  df f −1 ( x) 
dx
 dx

(
)
−1
[A4]
In our case, it can be written as:
dG
( p ) =  dN (G ( p) )
dp
 dp

−1
[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
How to reduce capital requirement? The case of retail portfolios with low probabilities of default
22
dK
dG
( p)
( p)
dp
dp
with respect to p
is a function of p, G(p) and
. The latter term is
replaced by its equivalent (equation [A5]). The second derivative is computed in the
dK
d 2K
( p)
( p)
2
same way. Thus dp
and dp
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 A1: Capital requirement with respect to the probability of default: K(p)
How to reduce capital requirement? The case of retail portfolios with low probabilities of default
23
Graph A2: First derivative of K(p)
Graph A3: Second derivative of K(p)
How to reduce capital requirement? The case of retail portfolios with low probabilities of default
24
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
1990
1991
1992
1993
1994
1995
1996
1997
1998-2000
Number of contracts
3108
3593
4328
4414
3943
4518
4421
3631
3831
Total Capital Required
Average PD
3,5%
2,3%
2,3%
2,4%
2,0%
1,7%
1,5%
2,0%
4,1%
Average LGD
30,2%
26,7%
30,1%
33,3%
31,9%
28,5%
23,4%
29,4%
42,4%
Asset Correlation
6,4%
8,6%
8,7%
8,5%
9,4%
10,4%
10,8%
9,4%
5,6%
Capital requirement
4,3%
3,4%
3,9%
4,3%
3,9%
3,3%
2,6%
3,6%
6,3%
K/LGD
14,3%
12,9%
12,8%
13,0%
12,3%
11,5%
11,2%
12,3%
14,9%
Average PD
0,2%
1,7%
2,1%
1,9%
2,3%
3,1%
Average LGD
52,6%
42,8%
36,8%
23,9%
30,8%
32,2%
Asset Correlation
16,0%
10,4%
9,3%
9,7%
8,8%
7,0%
Capital requirement
1,9%
5,0%
4,6%
2,9%
3,9%
4,5%
K/LGD
3,5%
11,6%
12,4%
12,1%
12,8%
14,0%
Average PD
2,4%
2,0%
2,0%
1,6%
Average LGD
29,0%
34,4%
32,7%
27,1%
Asset Correlation
8,4%
9,5%
9,4%
10,6%
Capital requirement
3,8%
4,2%
4,0%
3,1%
K/LGD
13,1%
12,3%
12,3%
11,4%
Average PD
4,6%
2,7%
2,0%
1,9%
2,1%
Average LGD
26,7%
28,5%
30,3%
34,1%
34,3%
Asset Correlation
5,0%
7,9%
9,3%
9,8%
9,1%
Capital requirement
4,1%
3,8%
3,8%
4,1%
4,3%
K/LGD
15,4%
13,4%
12,4%
12,1%
12,5%
Average PD
1,9%
2,4%
2,4%
3,7%
Average LGD
32,9%
28,3%
35,4%
41,7%
Asset Correlation
9,6%
8,5%
8,4%
6,0%
Capital requirement
4,0%
3,7%
4,6%
6,1%
K/LGD
12,2%
12,9%
13,0%
14,6%
3,9%
Panel B: Segmentation by the term-to-maturity of the lease
Term-to-maturity in months
0 to 11
12 to 23
24 to 35
36 to 47
48 to 59
over 60
Number of contracts
5961
1086
1682
7724
11515
7819
Total Capital Required
3,5%
Panel C: Segmentation by cost of the leased asset
Cost of the asset in €
7,400 to 25,000
25,001 to 50,000
50,001 to 100,000
100,001 to 500,000
Number of contracts
26303
6238
2955
291
Total Capital Required
3,9%
Panel D: Segmentation by the distribution network
Distribution network
DN 1
DN 2
DN 3
DN 4
DN 5
Number of contracts
756
10688
10688
8936
4719
Total Capital Required
3,9%
Panel E: Segmentation by the region of origin of the lessor
Region of origin
A
B
C
D and E
Number of contracts
8589
17204
9931
63
Total Capital Required
4,0%
Panel F1: Segmentation by the interest premium
How to reduce capital requirement? The case of retail portfolios with low probabilities of default
25
Interest premium
Less o,99%
1% to 1,99%
2% to 2,99%
3% to 3,99%
4% to 4,99%
5% to 5,99%
Over 6%
Total Capital Required
Number of contracts
816
5165
6906
9653
9382
2821
1044
Average PD
1,5%
0,8%
1,4%
2,2%
2,9%
3,2%
4,6%
Average LGD
23,2%
33,6%
25,7%
29,8%
29,9%
32,8%
52,1%
Asset Correlation
10,8%
13,3%
11,1%
9,0%
7,4%
6,8%
5,0%
Capital requirement
2,6%
2,9%
2,8%
3,8%
4,1%
4,6%
8,0%
K/LGD
11,2%
8,5%
10,9%
12,6%
13,7%
14,1%
15,4%
Average PD
1,1%
0,5%
1,3%
1,8%
2,0%
2,3%
2,6%
2,9%
3,3%
3,6%
Average LGD
35,6%
20,7%
23,5%
29,5%
25,0%
32,8%
31,8%
28,3%
31,3%
43,0%
Asset Correlation
12,3%
14,5%
11,4%
9,9%
9,4%
8,6%
8,1%
7,5%
6,8%
6,2%
Capital requirement
3,4%
1,4%
2,5%
3,5%
3,1%
4,2%
4,2%
3,9%
4,4%
6,2%
K/LGD
9,7%
6,7%
10,7%
11,9%
12,3%
12,9%
13,3%
13,6%
14,1%
14,5%
Number of contracts
35787
Average PD
2,3%
Average LGD
31,1%
Asset Correlation
8,7%
Capital requirement
4,0%
K/LGD
12,8%
Number of contracts
3537
3596
3541
3660
3559
3635
3592
3647
3629
3391
Average PD
2,4%
2,3%
2,5%
2,2%
2,3%
2,3%
2,4%
2,2%
2,2%
2,1%
Average LGD
32,1%
32,9%
32,3%
29,7%
29,6%
31,7%
30,2%
33,4%
32,7%
26,4%
Asset Correlation
8,6%
8,6%
8,2%
8,9%
8,7%
8,6%
8,4%
8,9%
8,9%
9,3%
Capital requirement
4,1%
4,2%
4,3%
3,8%
3,8%
4,1%
3,9%
4,2%
4,2%
3,3%
K/LGD
12,9%
12,9%
13,2%
12,7%
12,9%
12,9%
13,0%
12,7%
12,7%
12,4%
3,7%
Panel F2: Segmentation by the interest premium (decile)
Interest premium
-103,49% to 1,29%
1,29% to 1,59%
1,59% to 2,21%
2,25% to 2,65%
2,65% to 3,05%
3,07% to 3,40%
3,40% to 3,70%
3,71% to 4,05%
4,05% to 4,56%
4,56% to 110,00%
Total Capital Required
Number of contracts
3572
3585
3594
3570
3591
3585
3548
3561
3601
3580
3,7%
Panel G: No segmentation
Total portfolio
Panel H: Control segmentation (decile)
Decile 1
Decile 2
Decile 3
Decile 4
Decile 5
Decile 6
Decile 7
Decile 8
Decile 9
Decile 10
Total Capital Required
4,0%
How to reduce capital requirement? The case of retail portfolios with low probabilities of default
26
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
1990
1991
1992
1993
1994
1995
1996
1997
1998-2000
Number of contracts Average PD Average LGD
3108
3,5%
30,2%
3593
2,3%
26,7%
4328
2,3%
30,1%
4414
2,4%
33,3%
3943
2,0%
31,9%
4518
1,7%
28,5%
4421
1,5%
23,4%
3631
2,0%
29,4%
3831
4,1%
42,4%
Total Capital Required
STD Asset correlation Capital requirement K/LGD
1,4%
3,4%
3,1%
10,2%
1,2%
4,1%
2,2%
8,2%
1,9%
9,7%
4,1%
13,8%
1,2%
4,4%
2,9%
8,7%
1,1%
4,3%
2,4%
7,5%
0,7%
2,4%
1,3%
4,7%
0,8%
3,8%
1,3%
5,6%
1,5%
7,8%
3,2%
10,8%
2,6%
8,1%
7,8%
18,3%
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
0 to 11
5961
0,2%
52,6%
12 to 23
1086
1,7%
42,8%
24 to 35
1682
2,1%
36,8%
36 to 47
7724
1,9%
23,9%
48 to 59
11515
2,3%
30,8%
over 60
7819
3,1%
32,2%
Total Capital Required
STD Asset correlation Capital requirement K/LGD
0,4%
21,6%
2,6%
5,0%
1,6%
12,1%
5,6%
13,1%
0,9%
3,4%
2,5%
6,7%
0,5%
1,1%
0,9%
3,9%
0,5%
0,8%
1,3%
4,2%
0,9%
1,4%
2,1%
6,6%
1,8%
Panel C: Segmentation by cost of the leased asset
Cost of the asset in €
7,400 to 25,000
25,001 to 50,000
50,001 to 100,000
100,001 to 500,000
Total Capital Required
Number of contracts Average PD Average LGD
26303
2,4%
29,0%
6238
2,0%
34,4%
2955
2,0%
32,7%
291
1,6%
27,1%
STD Asset correlation Capital requirement K/LGD
0,5%
0,7%
1,2%
4,3%
0,5%
0,9%
1,3%
3,8%
0,9%
3,5%
2,2%
6,7%
0,9%
4,5%
1,7%
6,3%
1,3%
Panel D: Segmentation by the distribution network
Distribution network
DN 1
DN 2
DN 3
DN 4
DN 5
Total Capital Required
Number of contracts Average PD Average LGD
756
4,6%
26,7%
10688
2,7%
28,5%
10688
2,0%
30,3%
8936
1,9%
34,1%
4719
2,1%
34,3%
STD Asset correlation Capital requirement K/LGD
1,6%
2,8%
3,1%
11,8%
0,4%
0,5%
1,2%
4,3%
0,7%
2,0%
1,6%
5,3%
0,5%
1,2%
1,4%
4,0%
0,7%
1,7%
1,7%
5,1%
1,5%
Panel E: Segmentation by the region of origin of the lessor
Region of origin
A
B
C
D and E
Total Capital Required
Number of contracts Average PD Average LGD
8589
1,9%
32,9%
17204
2,4%
28,3%
9931
2,4%
35,4%
63
3,7%
41,7%
STD Asset correlation Capital requirement K/LGD
0,5%
1,3%
1,4%
4,2%
0,6%
1,2%
1,4%
4,9%
0,6%
0,9%
1,6%
4,6%
0,0%
0,0%
1,6%
3,7%
1,5%
Panel F1: Segmentation by the interest premium
How to reduce capital requirement? The case of retail portfolios with low probabilities of default
27
Interest premium
Less o,99%
1% to 1,99%
2% to 2,99%
3% to 3,99%
4% to 4,99%
5% to 5,99%
Over 6%
Number of contracts Average PD Average LGD
816
1,5%
23,2%
5165
0,8%
33,6%
6906
1,4%
25,7%
9653
2,2%
29,8%
9382
2,9%
29,9%
2821
3,2%
32,8%
1044
4,6%
52,1%
Total Capital Required
STD Asset correlation Capital requirement K/LGD
2,1%
19,0%
4,2%
18,2%
1,5%
29,6%
6,6%
19,5%
0,8%
23,6%
5,5%
21,4%
0,5%
4,5%
2,4%
8,1%
0,6%
1,1%
1,7%
5,7%
0,8%
0,9%
1,9%
5,9%
3,1%
9,2%
11,3%
21,7%
3,7%
Panel F2: Segmentation by the interest premium (decile)
Interest premium
-103,49% to 1,29%
1,29% to 1,59%
1,59% to 2,21%
2,25% to 2,65%
2,65% to 3,05%
3,07% to 3,40%
3,40% to 3,70%
3,71% to 4,05%
4,05% to 4,56%
4,56% to 110,00%
Number of contracts Average PD Average LGD
3572
1,1%
35,6%
3585
0,5%
20,7%
3594
1,3%
23,5%
3570
1,8%
29,5%
3591
2,0%
25,0%
3585
2,3%
32,8%
3548
2,6%
31,8%
3561
2,9%
28,3%
3601
3,3%
31,3%
3580
3,6%
43,0%
Total Capital Required
STD Asset correlation Capital requirement K/LGD
1,7%
21,5%
5,8%
16,3%
0,8%
16,0%
1,5%
7,5%
0,8%
4,9%
1,4%
5,8%
0,9%
4,0%
2,0%
6,6%
0,8%
2,5%
1,4%
5,7%
0,6%
1,0%
1,5%
4,6%
0,6%
0,8%
1,5%
4,7%
0,6%
0,8%
1,5%
5,2%
0,6%
0,0%
1,0%
3,3%
1,1%
2,0%
3,6%
8,4%
2,1%
Panel G: No segmentation
Total portfolio
Number of contracts Average PD Average LGD STD Asset correlation Capital requirement K/LGD
35787
2,3%
31,1%
0,5%
0,9%
1,3%
4,3%
Panel H: Control segmentation (decile)
Decile 1
Decile 2
Decile 3
Decile 4
Decile 5
Decile 6
Decile 7
Decile 8
Decile 9
Decile 10
Total Capital Required
Number of contracts Average PD Average LGD
3537
2,4%
32,1%
3596
2,3%
32,9%
3541
2,5%
32,3%
3660
2,2%
29,7%
3559
2,3%
29,6%
3635
2,3%
31,7%
3592
2,4%
30,2%
3647
2,2%
33,4%
3629
2,2%
32,7%
3391
2,1%
26,4%
STD Asset correlation Capital requirement K/LGD
0,6%
1,0%
1,5%
4,6%
0,2%
0,2%
1,0%
3,2%
0,6%
0,9%
1,5%
4,7%
0,5%
0,7%
1,2%
4,0%
0,6%
1,1%
1,4%
4,7%
0,4%
0,5%
1,2%
3,9%
0,5%
0,8%
1,3%
4,4%
0,5%
0,9%
1,4%
4,2%
0,5%
0,7%
1,3%
4,0%
0,5%
0,9%
1,0%
3,9%
1,3%
How to reduce capital requirement? The case of retail portfolios with low probabilities of default
28