Advanced Approaches to Calculation of Capital Requirements under Basel II

The new capital accord, Basel II, will influence the credit institutions' risk management and capital requirements and will thus have a direct impact on financial stability.

As a new element, Basel II allows credit institutions to apply internal models to calculation of the capital requirement for credit risk. This will contribute to greater consistency between the credit institutions' capital requirements and risk. In Denmark, the relatively few large credit institutions that will apply the new internal models will account for a large market share.

The framework for determining the capital requirement under the advanced approaches has been designed to accommodate the credit institutions' own systems and methods for risk management and assessment. Consequently, the capital requirements for a given credit portfolio may vary substantially. In order to perform an accurate assessment of a credit institution's robustness it is therefore necessary to have access to detailed information about the structure of the credit institution's systems for risk management and assessment.

This chapter illustrates a few of the methods and calculations for determining capital requirements under the advanced approaches, as well as the impacts of various choices. The analyses are partial and limited by the data available to Danmarks Nationalbank, e.g. on the estimated failure rates of companies. The chapter does not provide an overview of all options.

INTRODUCTION OF NEW CAPITAL-ADEQUACY RULES IN DENMARK

On 1 January 2007, the new capital-adequacy rules, Basel II,^[1] will be introduced in Danish legislation. The existing, uniform capital-adequacy rules will be replaced by rules that ensure greater consistency between the credit institutions' capital requirements and the individual risks to which they are exposed.

Under the new rules, credit institutions have a choice of three different approaches to calculation of the capital requirement for credit risk. The approaches, which vary in complexity, are a standardised approach, whereby the credit institutions' various assets are allocated fixed, standardised risk weights, and two internal ratings-based approaches (IRB), under which the credit institutions' own knowledge of their credit risks is used to estimate risk weights.

The vast majority of small and medium-sized Danish banking institutions will opt for the standardised approach, which is the most simple and is based on the current compilation method, but divides exposures into additional risk classes with fixed weights. 5-10 large Danish credit institutions are expected to apply the IRB approaches. These credit institutions account for a substantial market share in Denmark.^[2]

The framework for determining the capital requirement under the IRB approaches is relatively flexible, in order to accommodate e.g. the credit institutions' own systems and methods for risk management and assessment, but also because the use of credit models is a new area without a well-established best practice. Credit institutions applying the IRB approaches must therefore make a number of choices in relation to estimation of credit risks. These choices may have a considerable impact on the capital requirements that are calculated.

For Danish credit institutions, the new rules are expected to entail an overall reduction of the capital requirement. One reason is that Danish credit institutions have a relatively large number of loan exposures with households and small and medium sized enterprises, for which the capital requirements are lower under the new rules.^[3] The differences between the present and future capital requirements will not be discussed further in this chapter.

THEORETICAL BACKGROUND TO THE CREDIT INSTITUTIONS' CAPITAL REQUIREMENTS

A credit institution will frequently have borrowers that do not meet their payment obligations. To be able to operate, a banking institution must have the special skills required for assessing, managing and pricing such credit risks.

Credit losses typically vary over time, as illustrated in Chart 41. A credit institution can never know for certain which losses will be realised in a given year, but the expected loss can be estimated, cf. the dashed line in Chart 41. The expected loss on a lending portfolio can be taken into account via loan pricing, e.g. by charging a sufficiently high rate of interest. However, in some cases the actual loss will exceed the expected loss.

The actual loss less the expected loss is known as the unexpected loss. To some extent, credit institutions hedge this loss via their capital reserves.

Under the new capital-adequacy rules, the formula for calculating the capital requirement entails that the capital covers the difference between the expected loss and the maximum loss to the credit institution within a confidence interval of 99.9 per cent.^[4] Chart 42 illustrates the distribution function for a credit institution's credit losses. The distribution of the credit loss is right-skewed with a long tail since the probability of small losses is substantially greater than the probability of large losses. Consequently, the average (the expected loss) does not correspond to the maximum of the distribution function. In Chart 42, the unexpected loss is shown as the area below the curve from the expected loss to the 99.9th percentile. In the chapter on the corporate sector and the households, the development in the Danish banking sector's credit losses on corporate lending is illustrated using an equivalent distribution function.

CALCULATION OF THE CAPITAL REQUIREMENT USING IRB APPROACHES

Credit institutions must apply to the supervisory authorities for permission to use either the foundation or the advanced IRB approach. The latter gives greater scope for applying the credit institution's own estimates.

When calculating the capital requirement for a loan according to the IRB approaches, four parameters are used:

• the probability of default on a loan within the coming year (PD)
• the loss given default (LGD)^[5]
• the expected exposure at the time of default (EAD)
• the maturity of the loan (M).^[6]

Credit institutions applying the advanced IRB approach must estimate all parameters themselves, while those opting for the foundation IRB approach must only estimate PD.^[7] The risk weights for the individual loans are calculated by inserting PD, LGD and if required M into fixed formulae and multiplying by the exposure, EAD, cf. Chart 43. The capital requirement can then be calculated as 8 per cent of the risk-weighted lending (0.08 x risk weight x EAD).

The Basel Committee's formulae for calculation of capital requirements are described in Box 12.

There is considerable flexibility with regard to the estimation methods the credit institutions can use to determine the value of the parameters. The analysis below focuses primarily on the methods for estimating PD, supplemented with a simple sensitivity analysis of LGD. The other parameters, EAD and M, are not included.

Estimation of risk parameters under the IRB approaches
In order to obtain the Danish Financial Supervisory Authority's approval to use the IRB approaches, a credit institution must, among other things, establish one or more rating systems for meaningful classification of borrowers on the basis of their credit quality. The system(s) may be developed by the credit institution itself or purchased from a third party. A rating system is typically based on a credit-score system whereby the credit institutions rank their customers on the basis of both quantitative data (e.g. financial accounts) and qualitative data (e.g. knowledge of the customer).

For lending to the corporate sector, governments and other credit institutions, the credit institutions' rating systems must include at least seven classes for non-defaulted loans and a separate class for defaulted loans. The credit institutions must also establish rating systems for their retail portfolios^[8]. Lending to small and medium-sized enterprises (SME) is in principle included in the corporate portfolio, but may under certain conditions be treated as part of the retail portfolio, cf. Box 12.

IRB FORMULA	Box 12
This Box describes the formula for calculation of the capital requirement for corporate lending, including lending to small and medium-sized enterprises (SME). The capital requirement (C) is calculated as 8 per cent of the risk weight (RW) multiplied by the exposure at default (EAD): C = 0.08 X RW X EAD As the Chart shows, the risk weight (RW) for a corporate loan is to a large extent determined by whether the loan is considered to be part of the corporate portfolio, including the SME subcategory, or transferred to the retail portfolio. The risk weight for lending to SMEs is thus considerably lower than for corporate lending, and the gap widens as PD increases. Furthermore, the risk weight can be approximately halved if the loan meets the requirements for inclusion in the retail portfolio. This requires that the loan to the individual enterprise does not exceed 1 million euro. In addition, the credit institution must have a significant number of SME loans that are handled in a similar manner.
RISK WEIGHTS FOR LENDING TO THE CORPORATE SECTOR, SME S AND RETAIL CUSTOMERS, IRB APPROACH
Note: The SME curve is calculated for a company with an annual turnover of 25 million euro. An LGD value of 45 per cent and an M value of 2.5 years are applied. Source: Danmarks Nationalbank.
For non-defaulted exposures 1, the IRB formula for calculation of the risk weight (RW) is:   PD (Probability of Default) is the probability that a borrower defaults within the next year, LGD (Loss Given Default) is the expected loss ratio in the event of default, and R expresses the correlation between the value of the credit institutions' loans. N(x) is the cumulative distribution function for a standardised, normal-distribution stochastic variable. The confidence level, N -1(0.999), has been set at 99.9 per cent. M is the maturity of the loan in years. If the maturity, M, is set at 2.5 years, the expression in the last fraction of the formula can be reduced to a function of the maturity factor b, which is determined by: R is inversely dependent on the borrowers' credit quality (PD). For corporate lending of the highest credit quality (i.e. with a low PD) the correlation approaches 24 per cent, while for corporate lending of the lowest credit quality (i.e. with a high PD) it approaches 12 per cent, cf. the R formula. 2 The R formula also includes a correction factor wSME for SMEs with turnover not exceeding 50 million euro. The correction factor is expressed by: where S is the total annual turnover in million euro for enterprises with turnover in the range of 5 to 50 million euro. For enterprises with annual turnover of less than 5 million euro, S will be set at 5 million euro. For enterprises with turnover of 50 million euro or more, wSME is zero, so that the correlation band remains unchanged at 12-24 per cent. For enterprises with turnover not exceeding 5 million euro, wSME is 0.04, whereby the correlation band is reduced to 8-20 per cent. The Basel Committee has determined that PD for corporate lending, etc. must be at least 0.03 per cent. Under the foundation IRB approach, LGD for uncollateralised lending must be set at 45 per cent if the loan is not subordinate to other creditors. If, however, the uncollateralised loan is subordinate, LGD must be set at 75 per cent. For loans for which recognised collateral has been pledged, LGD can be reduced in accordance with special rules for credit risk mitigation. LGD can thus be reduced to less than 45 per cent for credit institutions applying the foundation IRB approach. For credit institutions opting for the advanced IRB approach, which must estimate all parameters themselves, the starting point for LGD, i.e. not taking pledged collateral into account, may be higher than 45 per cent, since LGD must as a minimum, among other things, reflect the conditions in an economic downturn. On the other hand, more types of collateral are recognised according to the rules for credit risk mitigation, and consequently LGD may be reduced further. However, the Basel Committee has set a provisional lower limit of 10 per cent for loans secured by real estate. Under the foundation IRB approach, M has been set at 2.5 years.
1 For defaulted loans, PD=1 and RW thus 0. 2 For retail lending, the correlation range has been set at 3-16 per cent with the exception that the correlation for housing loans has been set at 15 per cent and for overdraft facilities at 4 per cent.

The rules give the credit institutions two options when it comes to estimating PD values:

• Estimation on the basis of the rating system: a representative PD is estimated for all companies within the same rating class.
• Direct estimation: the credit institution's internal statistical credit-risk models are applied, after which an individual PD is estimated for each loan.

Credit institutions applying the advanced IRB approach must also estimate LGD and EAD and calculate M. As a minimum, the LGD and EAD estimates must reflect conditions in an economic downturn, unlike the 1-year term of the PD estimate.

The rules enable estimation of LGD either directly or on the basis of a rating system corresponding to the rules for PD estimation. Pledged collateral may be included in the LGD estimate, in which case provision must be made for the fact that it is very difficult for the credit institutions to gain control of their collateral and liquidate it at market value within a short timeframe.

When estimating EAD, the credit institutions must take account of the uncertainty related to the size of the exposure at the time of default. For instance, an overdraft facility may not have been fully utilised.

When the risk parameters for the individual loans have been estimated, they are entered into the Basel Committee's formula for calculation of the capital requirement, cf. Box 12. The capital requirements for the individual loans are added to give the credit institution's total capital requirement for credit risk. Consequently, the estimations of the risk parameters, including the use and structure of rating systems, influence the capital requirement.

Further requirements of rating systems
The credit institutions must test the accuracy and robustness of their rating systems, e.g. by back-testing and stress-testing against a mild recession scenario.

The rating systems must be established on a sound and meaningful basis, which requires that the credit institutions have access to extensive data on the individual assets, including whether or not the borrowers have previously defaulted on their payment obligations. Consequently, it may be necessary for a credit institution to supplement its own data with external data, i.e. data pooling, with a view to obtaining a sufficient number of observations to distinguish between risks in the various rating classes. If a credit institution needs to pool data, it must be able to demonstrate that the overall data set is representative of its own loan portfolio, including that a uniform definition is applied to determining when a loan is classified as defaulted.

Besides forming the basis for calculation of the capital requirement, the system must e.g. be used in day-to-day credit management. This also applies if a credit institution uses direct PD estimation. Credit institutions applying the IRB approaches thus have a very direct incentive to develop and maintain systems and methods that provide the best possible overview of the assets' risk characteristics.

Developing, maintaining, testing and documenting the systems and methods required in order to apply the IRB approaches is a considerable task. These approaches are therefore only relevant for large credit institutions.

QUANTIFICATION

This section quantifies differences in the capital requirements for a portfolio of lending to Danish non-financial enterprises. The point of departure is the failure rates for Danish companies estimated using Danmarks Nationalbank's failure-rate model, cf. Box 13. The estimated failure rate for each company in the model is used to express PD.

DATA BASIS	Box 13
Danmarks Nationalbank's failure-rate model primarily applies historical accounting data for Danish companies to estimate the failure rate for each company. The same basic model is used to analyse the credit risk of the Danish banking sector in this publication's chapters on, respectively, the financial sector and the corporate sector and the households. In order to calculate banks' capital requirements under the Basel II Accord, it is necessary to have an expression of the probability of default on a loan (PD). The estimated failure rate for each company in Danmarks Nationalbank's failure-rate model is used to express PD. Establishment of a rating system for calculation of PD A rating system can be structured in many ways. First, the companies must be rated by credit quality. In this analysis, primarily accounting data has been applied, and the companies are ranked by the value of the estimated PD. Then, the number of rating classes must be determined. This analysis operates with rating systems with, respectively, 8 and 14 rating classes. Under the capital-adequacy rules, a rating system must have at least 8 rating classes. 14 rating classes have been selected for the simple reason that this is the highest number stated by a Danish banking institution. The individual rating classes are determined by an upper and a lower band for the companies' estimated failure rates. Companies with high estimated failure rates are in the low rating classes, while companies with low estimated failure rates are in the high rating classes. Both active and failed companies are allocated to rating classes. When all companies have been placed in a rating class, it is possible to calculate the number of companies in each rating class that have actually failed. Typically, rating classes are designed so that the actual failure rate across rating classes approximates to an exponential function, cf. the Chart. The same approach is used to establish the various rating systems in this analysis. The PD of the individual rating class is calculated as the simple average of estimated failure rates for companies in that rating class. Direct PD estimation In direct PD estimation, the estimated failure rate for each company is used as an expression of PD. EAD, exposure at default In order to calculate the capital requirement for each individual company, it is necessary to know the bank's exposure with the company in question (EAD). For this purpose, the bank debt stated in the company's annual accounts is used. The data basis for the failure-rate model does not include data on bank debt for all companies; where no data is available, the company is included in the calculations with a capital requirement of zero.
EXAMPLE OF FAILURE RATES IN A RATING SYSTEM
Note: The Chart shows Moody's historical 1-year failure rates for various rating classes. Source: Default and Recovery Rates of Corporate Bond Issuers, 1920-2004, Hamilton et al., 2005.
Basic scenario For comparing results, a basic scenario has been defined, which has been set at 100. In this basic scenario, the rating system is based on all companies in the failure-rate model and has 8 rating classes. The development in the actual failure rates is relatively flat across rating classes. On calculation of the capital requirement, LGD has been set at 45 per cent and M at 2.5 years.

Danmarks Nationalbank's data basis cannot be used to derive parameter estimates of LGD and EAD, but like PD estimates they will be subject to uncertainty, and the size of the parameters will have a significant impact on the capital requirements calculated.^[9] This is illustrated by a simple sensitivity analysis of various parameter values for LGD.

The analysis is not a systematic account of the sensitivity of the capital requirement, but an illustration on the basis of available data.

Both the number of rating classes and the distribution of companies within the rating classes may affect the capital requirement.

Chart 44 shows two different rating systems with eight rating classes each. The difference between the two is that the rating system with the "steep" slope has a higher ratio of failed companies in the low rating classes (classes 6-8) than the "flat" rating system, and a lower ratio of failed companies in the high rating classes. Chart 45 presents two rating systems with, respectively, 8 and 14 rating classes.

The capital requirements calculated on the basis of the various rating systems and via direct PD estimation are shown in Table 9. For the rating system giving the highest capital requirement in the analysed portfolio of Danish companies, the capital requirement has been set at 100. For the lending portfolio applied, the capital requirement depends on the "steepness" of the rating system.

The "steep" system entails lower capital requirements than the "flat" system. At the same time, the analyses show that in this case the capital requirement is reduced when the number of rating classes in the rating system is increased, the highest effect being achieved for the "flat" system.

The lowest capital requirement is obtained using direct PD estimation, whereby the number of rating classes in principle matches the number of companies. In general, the distribution of lending within each rating class will determine whether direct PD estimation results in higher or lower capital requirements. Especially the distribution of loans has an impact on rating systems based on a relatively small lending portfolio, since some rating classes may include only few loans.

The capital requirement for credit risk is also affected by the estimation of LGD. The higher LGD, the higher the capital requirement. This is evident from the Basel Committee's formula for calculation of capital requirements, cf. Box 12. If it is assumed that all loans in the portfolio have the same LGD, the capital requirement changes by the same factor as LGD. This sensitivity is illustrated in Table 10, which is based on the range of parameter values for LGD that the rules provide for^[10].

The new capital-adequacy rules allow credit institutions to treat part of their lending to SMEs as retail lending. Since capital requirements are generally lower for retail lending, the overall capital requirement decreases as more SME lending is transferred to the retail portfolio, cf. the Chart in Box 12. SMEs may be included in the retail portfolio under certain conditions, e.g. that lending to the individual SME does not exceed 1 million euro, and that the credit institution has a significant number of SME loans that are handled in a similar manner. In a Danish context, the 1-million-euro exposure limit is relatively high, i.e. many loans could potentially be comprised.

To illustrate the impact, rating systems are established and the capital requirement for credit risk is calculated for different lending portfolios from which small enterprises are increasingly eliminated. Small enterprises typically have a high PD, i.e. there are relatively many small enterprises in the lowest rating classes. When the small enterprises are transferred from the corporate portfolio to the retail portfolio, the greatest impact is therefore seen in the lowest rating classes, where the average PD declines. The result is a lower capital requirement, cf. Table 11.

[1] For a description of the overall framework for the new capital-adequacy rules, reference is made to Lisbeth Borup and Dorte Kurek, Proposal for a Directive on New Capital-Adequacy Rules (Basel II), Danmarks Nationalbank, Monetary Review, 1st Quarter 2005.

[2] According to the financial statements for 2005, Danske Bank, Nordea, Jyske Bank, Sydbank, Alm. Brand Bank, FIH Erhvervsbank, Nykredit, BRFkredit and DLR, among others, intend to apply the IRB approaches.

[3] For an assessment of the expected changes in the capital requirements as a result of the new rules, reference is made to the report Study on the financial and macroeconomic consequences of the draft proposed new capital requirements for banks and investment firms in the EU, PriceWaterhouseCoopers, 8 April 2004. The report can be downloaded from the European Commission's website, http://europa.eu.int/comm/internal_market/bank/regcapital/impact_en.htm#consequences.

[4] This means that in 1,000 different simulated instances, the loss within the following year will exceed the capital requirement in one case.

[5] For credit institutions applying the advanced IRB approach, the LGD and EAD estimates must as a minimum reflect the conditions in an economic downturn.

[6] M must be calculated for lending to governments, credit institutions and corporates.

[7] For retail lending, only the advanced approach can be applied.

[8] For the retail lending portfolio, the credit institutions shall base the PD estimation on a rating system based on either obligor grades or pools, which – in addition to counterparty risk – also includes facility-specific risk, i.e. whether the loan is a guarantee, an overdraft facility, etc.

[9] In principle, determining the impact on the capital requirement of various parameter values for e.g. LGD and EAD requires access to confidential data on e.g. collateral pledged for the individual loans, customers' utilisation of overdraft facilities, etc.

[10] See Box 12.