may exacerbate systemic risk concerns and concern about systemic risk may lead to regulatory attempts to influence rating agencies, thereby under- mining their independence and credibility.26
Although an important advantage of external ratings is their validation by the market, the credit-rating industry is not very competitive. There are only a handful of well-regarded rating agencies. This leads to the risk of rat- ing shopping.27The obligors are free to choose a rating agency, so moral hazard may lead rating agencies to shade their ratings upward in a bid to obtain business. Moreover, because there is no single, universally accepted standard for credit ratings, they may not be comparable across rating agen- cies and across countries. [See discussions in White (2001), Cantor (2001), Griep and De Stefano (2001).] This is likely to distort capital requirements more in less developed countries (LDCs), because of greater volatility in LDC sovereign ratings, less transparent financial reporting in those coun- tries, and the greater impact of the sovereign rating as a de facto ceiling for the private sector in LDCs.28
Finally, banks are also considered “delegated monitors” [see Diamond (1984)] that have a comparative advantage in assessing and monitoring the
individual borrowers is between 10 percent and 20 percent with the correla- tion a decreasing function of PD; see BIS (November 5, 2001b).32
Expected losses upon default can be calculated as follows:
Expected losses=PD×LGD
where PDis the probability of default and LGDis the loss given default.33 However, this considers only one possible credit event—default—and ig- nores the possibility of losses resulting from credit-rating downgrades. That is, deterioration in credit quality caused by increases in PD or LGD will cause the value of the loan to be written down—in a mark-to-market sense—even prior to default, thereby resulting in portfolio losses (if the loan’s value is marked to market). Thus, credit risk measurement models can be differentiated on the basis of whether the definition of a “credit event” includes only default (the default mode or DM models) or whether it also includes nondefault credit quality deterioration (the mark-to-market or MTM models). The mark-to-market approach considers the impact of credit downgrades and upgrades on market value, whereas the default mode is only concerned about the economic value of an obligation in the event of default. There are five elements to any IRB approach:
1. A classification of the obligation by credit risk exposure—the internal ratings model.
2. Risk components—PD and EAD for the foundation model and PD, EAD, LGD,andMfor the advanced model.
3. A risk weight function that uses the risk components to calculate the risk weights.
4. A set of minimum requirements of eligibility to apply the IRB approach (i.e., demonstration that the bank maintains the necessary information systems to accurately implement the IRB approach).
5. Supervisory review of compliance with the minimum requirements.
The Foundation IRB Approach
The bank is allowed to use its own estimate of probability of default (PD) over a one-year time horizon, as well as each loan’s exposure at default (EAD). However, there is a lower bound on PDthat is equal to three basis points, so as to create a nonzero floor on the credit risk weights (and hence capital required to be held against any individual loan). The average PDfor each internal grade is used to calculate the risk weight for each internal rat- ing. The PDmay be based on historical experience or even potentially on a credit scoring model. (See Chapter 2 for traditional credit scoring models
and Chapters 4 through 8 for newer models.) The EAD for on-balance- sheet transactions is equal to the nominal (book value) amount of the exposure outstanding. Credit mitigation factors (e.g., collateral, credit de- rivatives or guarantees, on-balance-sheet netting) are incorporated follow- ing the rules of the standard IRB approach by adjusting the EAD for the collateral amount, less a haircut determined by supervisory advice under Pillar II. The EADfor off-balance-sheet activities is computed using the BIS I approach of translating off-balance-sheet items into on-balance-sheet equivalents mostly using the BIS I conversion factors [see Saunders (1997), Chapter 20].34The foundation IRB approach sets a benchmark for M,ma- turity [or weighted average life (WAL) of the loan] at three years. Moreover, the foundation approach assumes that loss given default for each unsecured loan is set at LGD=50 percent for senior claims and LGD=75 percent for subordinated claims on corporate obligations.35 However, in November 2001, the Basel Committee on Banking Supervision presented potential modifications that would reduce the LGDon secured loans to 45 percent if fully secured by physical, nonreal estate collateral and 40 percent if fully se- cured by receivables.
Under the January 2001 proposal, the foundation approach formula for the risk weight (RW) on corporate obligations (loans) is:36
or
12.50×LGD
whichever is smaller; where the benchmark risk weight (BRW) is calculated for each risk classification using the following formula:
The term N(y) denotes the cumulative distribution function for a standard normal random variable (i.e., the probability that a normal random variable with mean zero and variance of one is less than or equal to y) and the term G(z) denotes the inverse cumulative distribution function for a standard normal random variable (i.e., the value y such that N(y)=z). The BRW formula is calibrated so that a three year corporate loan with a PDequal to 0.7 percent and a LGDequal to 50 percent will have a capital requirement (3.2)
BRW N G PD PD
=976 5. ×
(
1 118. ×( )
+1 288.)
× +1 .0470×(
1PD−0 44.)
(3.1) RW LGD
= BRW
× 50
of 8 percent, calibrated to an assumed loss coverage target of 99.5 percent (i.e., losses can exceed the capital allocation that occur only 0.5 percent of the time, or five years in 1,000).37Appendix 3.2 shows the calibration of equation (3.2) for retail loans, demonstrating that the BRWfor retail loans is set lower than the BRWfor corporate loans for all levels of PD.Figure 3.1 shows the continuous relationship between the BRWand the PD.Note that this continuous function allows the bank to choose the number of risk cate- gories in the internal risk rating system, as long as there is a minimum of six to nine grades for performing borrowers and two grades for nonperforming borrowers.38
Consultation between the Basel Committee on Banking Supervision and the public fueled concerns about the calibration of the foundation ap- proach as presented in equations (3.1) and (3.2). This concern was galva- nized by the results of a Quantitative Impact Study (QIS2) that examined the impact of the BIS II proposals on the capital requirements of 138 large and small banks from 25 countries. Banks that would have adopted the IRB foundation approach would have seen an unintended 14 percent in- crease in their capital requirements. Potential modifications were released on November 5, 2001 to lower the risk weights and make the risk weight- ing function less steep for the IRB foundation approach only. Moreover, the potential modifications (if incorporated into the BIS II proposals) would make the correlation coefficient a function of the PD, such that the correlation coefficient between assets decreases as the PD increases.
Finally, the confidence level built into the risk weighting function would be increased from 99.5 percent to 99.9 percent.
FIGURE 3.1 Proposed IRB risk weights for hypothetical corporate exposure having LGDequal to 50 percent. Source:“The Internal Ratings-Based Approach,” The New Basel Capital Accord, Bank for International Settlements (2001).
0
Risk Weight (Percent)
700 600 500 400 300 200 100 0
5 10
PD (Percent)
15 20
The potential modifications to equations (3.1) and (3.2) corporate loan risk weight curves are as follows:
where
and
where X=75 for a subordinated loan, X=50 for an unsecured loan,
X=45 for a loan fully secured by physical, nonreal estate collateral, and
X=40 for a loan fully secured by receivables.
In equations (3.3) through (3.6), exp stands for the natural exponential function, N(.) stands for the standard normal cumulative distribution func- tion, and G(.) stands for the inverse standard normal cumulative distribu- tion function.
Equation (3.4) denotes the maturity factor M. This is reportedly un- changed from the BIS II proposals shown in equation (3.2) in that it is still benchmarked to a fixed three year weighted average life of the loan.39The correlation coefficient R is computed in equation (3.5). The correlation ranges from 0.20 for the lowest PDvalue to 0.10 for the highest PDvalue.
(3.6)
RW X
= BRW
× 50
(3.5)
R exp
exp
exp exp
PD PD
= ×
(
−)
(
−)
+ × −
(
−)
−
−
−
−
0 10 1 −
1 0 20 1 1
1
50 50
50
. . 50
(3.4)
M PD
= + ×
(
PD−)
1 0 047 1
. 0 44.
(3.3)
BRW LGD M
N R G PD R
R G
= × ×
×
(
−)
×( )
+(
−)
×
( )
−
12 5
1 0 5 1 0 999
05
.
. .
.
This inverse relationship appears to be somewhat counterintuitive in that empirically asset correlations increase during systemic crises when PDs also tend to increase, thereby implying a direct positive (rather than inverse) re- lationship between correlation and PD; see Carey (1998) and Erlenmaier and Gersbach (2001).
Using the potential modifications of November 2001, the benchmark risk weight (BRW) is calculated from equations (3.3) through (3.5). The actual risk weight (RW) is then calculated in equation (3.6) where RW=(X/50)×BRW andX=the stipulated fixedLGD for each type of loan. For example, under the potential modifications of November 2001, the LGD takes on a value of either 40 percent (if the loan is fully secured by receivables), 45 percent (if fully secured by physical, non-real estate collateral), 50 percent (if unsecured but senior), or 75 percent (if subordinated). Risk-weighted assets (RWA) are then computed by multiplying the risk weight (RW) times the exposure at de- faultEAD. Finally, the minimum capital requirement is computed by multi- plying the risk-weighted assets times 8 percent; that is, the minimum capital requirement on the individual loan=RW×EAD×8 percent.
Table 3.6 shows the impact of the November 2001 modified risk weighting function on the capital requirements under the IRB foundation approach. For TABLE 3.6 Comparison of BIS II Proposals and Potential Modifications: Capital Requirements under the IRB Foundation Approach
January 2001 November 2001
Probability of Default BIS II Proposal BIS Modified (Basis Points) Capital Requirements (%) Capital Requirements (%)
3 1.1 1.4
10 2.3 2.7
25 4.2 4.3
50 6.4 5.9
75 8.3 7.1
100 10.0 8.0
125 11.5 8.7
150 12.9 9.3
200 15.4 10.3
250 17.6 11.1
300 19.7 11.9
400 23.3 13.4
500 26.5 14.8
1,000 38.6 21.0
2,000 50.0 30.0
Notes: The minimum capital requirements shown are a percent of EAD(exposure at default) assuming LGD=50 percent. Source: BIS (November 5, 2001b).
example, an unsecured $100 million corporate loan with aPDof 10 percent would have a 2.62 percent benchmark risk weight under the November 2001 modifications, computed using equations (3.3) through (3.5). Because the loan in our example is unsecured, using equation (3.1) the RW=(50/50)×BRW= 2.62. Thus, the loan’s minimum capital requirement would be $100m×.08× 2.62=$21 million shown in Table 3.6 column (3). In contrast, Table 3.6 shows that the same loan’s minimum capital requirement under the January 2001 pro- posals would have been $38.6 million shown in column (2). Moreover, under BIS I the capital requirement would have been $100 million×8 percent=$8 million. Table 3.6 also shows that the capital requirement for the highest qual- ity (lowest PD) exposures increases slightly in the modified proposals, whereas the capital requirement for the lowest quality (highestPD) exposures decreases significantly as compared to the January 2001 BIS II proposals.
This example is for a single loan. In practice, the BIS makes an additional adjustment for loan portfolio concentration. In the foundation model, the RWin equation (3.1) is multiplied by the EADfor each internal rating classi- fication (on a transaction by transaction basis) in order to obtain a measure of risk-weighted assets for each loan; that is, RWA=RW×EAD.The risk- weighted assets are summed across all ratings classes to obtain the baseline level of credit risk-weighted assets. Then an adjustment for granularity (i.e., the degree of single-borrower risk concentration) is applied.40The adjustment may be positive or negative and reflects the undiversified idiosyncratic risk of the portfolio. Although the granularity adjustment incorporates correlations (such that the adjustment increases as asset correlations increase), it differs from the Rfactor in equation (3.5) because it measures overall portfolio con- centration rather than pairwise asset correlation. Thus, the effect of the gran- ularity adjustment is to increase (decrease) the total risk-weighted assets of portfolios with relatively large (small) borrower risk concentration.
TheBRW in equation (3.2) is calibrated using CreditMetrics (see Chap- ter 6) to an assumed PD=0.7025 percent, LGD = 50 percent, maturity of three years, and a granularity scaling factor of 4 percent. That is, about 4 percent of baseline capital is allocated to cover the expected and unex- pected losses associated with undiversified idiosyncratic risk resulting from the fact that the portfolio does not contain an infinite number of equal- sized loans. Thus, the portion of risk-weighted assets that is levied as a granularity charge is .04×RWA. To determine the effect of actual portfolio granulation, the portfolio’s granularity must be compared to this baseline level, such that if the actual portfolio’s granularity is higher (lower), the portfolio’s minimum capital requirement is higher (lower).
Calculation of the portfolio’s granularity capital charge is based on the property that the VARof a granular portfolio consisting of n homogenous loans is equal to the VARfor an infinitely fine-grained portfolio (assumed
in calculating the baseline risk weights) plus an adjustment factor that is inversely proportional to n.The constant of proportionality is a function of PD, LGD, andF(the systematic risk sensitivity of the exposures in the port- folio). Thus, the additional capital charge (as a fraction of portfolio size) that is required to cover the undiversified idiosyncratic risk of a granular portfolio is GSF/n, where GSF is the constant factor of proportionality (shown in equation (3.8) to be a function of PD, LGD,andF)andnis the number of exposures in the portfolio. This granularity capital charge must be compared to the baseline 4 percent granularity charge. The form of the granularity adjustment is then as follows:41
where Portfolio TEAD=the portfolio’s total non-retail exposure,42 GSF=the granularity scaling factor; see equation
(3.8),
n*=effective number of loans, taking into account their size distribution,43 RWA=risk-weighted assets under the baseline
assumptions of equation (3.1).
Credit Risk Plus (see Chapter 8) is used to calibrate the granularity scaling factor (GSF); see also Gordy (2000). The form is:
where LGD=the weighted average of the portfolio’s loss given default, PD=the weighted average of the portfolio’s default probability,
F=the measure of systematic risk sensitivity, is defined as follows:
where as in equation (3.2), N(y) denotes the cumulative distribution func- tion for a standard normal random variable (i.e., the probability that a nor- mal random variable with mean zero and variance of one is less than or equal to y) and the term G(z) denotes the inverse cumulative distribution (3.9) F=N
(
α1×G PD( )
+α0)
−PD(3.8)
GSF LGD PD
=
(
+ ×)
× + × F
0 6 1 8. . 9 5 13 57. .
(3.7) Portfolio TEAD GSF
n RWA
×
−
(
×)
* .04
function for a standard normal random variable (i.e., the value ysuch that N(y)=z).α0 andα1are the same terms as in equation (3.2) (e.g., α0=1.288 andα1=1.118 for corporate loans). The granularity adjustment is applied to the entire portfolio as a whole (excluding the retail portfolio, which is generally assumed to be infinitely granular) after the sum of all baseline risk-weighted assets for all portfolio exposures is computed.
The Advanced IRB Approach
Sophisticated banks are encouraged to move from the foundation approach to the advanced approach. A primary source for this incentive results from the use of the bank’s actual LGDexperience in place of the fixed assump- tion of a 40, 45, 50, or 75 percent LGD.Evidence suggests that historical LGDfor bank loans is significantly lower than 50 percent44and, therefore, the shift to the advanced approach is expected to reduce bank capital re- quirements by 2 to 3 percent. However, the quid pro quo for permission to use actual LGDis compliance with an additional set of minimum require- ments attesting to the efficacy of the bank’s information systems in main- taining data on LGD.
Another adjustment to the foundation approach’s benchmark risk weight (BRW) is the incorporation of a maturity adjustment that reflects the transaction’s effective maturity, defined as the greater of either one year or nominal maturity, which is the weighted average life (= ΣttPt/ΣtPtwherePtis the minimum amount of principal contractually payable at time t) for all in- struments with a predetermined minimum amortization schedule. The ma- turity is capped at seven years to avoid overstating the impact of maturity on credit risk exposure.
The advanced IRB approach allows the bank to use its own credit risk mitigation estimates to adjust PD, LGD,andEADfor collateral, credit de- rivatives, guarantees, and on-balance-sheet netting. The risk weights for the mark-to-market Advanced IRB approach are calculated as follows:
where
(3.11) b PD
PD
PD PD
( )
=[
× −( ) ]
+ × −
( )
[ ]
.
. .
0235 1 0470 1
0 44
(3.10) RW LGD
BRW b PD M
=
× × +
[ ( )
×(
−) ]
50 1 3
andBRWis as defined in the foundation IRB approach. The effect of the [1 +b(PD)×M(−3)] term in equation (3.10) is to adjust the risk of loans for its maturity.45For longer maturity instruments, the maturity adjustments in- crease for low PD-rated borrowers (i.e., higher rated borrowers). The intu- ition is that maturity matters most for low PDborrowers since they can move only in one direction (downward) and the longer the maturity of the loan, the more this is likely to occur. For high PD(low-quality) borrowers who are near default, the maturity adjustment will not matter as much be- cause they may be close to default regardless of the length of the maturity of the loan.46
The advanced IRB approach entails the estimation of parameters re- quiring long histories of data that are unavailable to most banks [see the Basel Committee on Banking Supervision (April 1999) for a survey of cur- rent credit risk modeling practices at 20 large international banks located in 10 countries]. Given the costs of developing these models and databases, there is the possibility of dichotomizing the banking industry into “haves and have-nots.” For example, some anecdotal estimates suggest that no more than 15 U.S. banks will choose to use either IRB approach. Moreover, capital requirements are highly sensitive to the accuracy of certain parame- ter values; in particular, estimates of LGD and the granularity in PDare im- portant [see Gordy (2000) and Carey (2000)]. Because credit losses are affected by economic conditions, the model parameters should also be ad- justed to reflect expected levels of economic activity. Thus, the data require- ments are so substantial that full implementation of the advanced IRB approach lies far in the future, even for the most sophisticated banks. When that date comes, regulators will have commensurate challenges in obtaining the necessary data to validate the banks’ models.