on Banking Supervision
An Explanatory Note on the I IRB Risk Weight Functions
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Table of Contents
1. Introduction……………………………………………………………………………………………………….1
2. Economic foundations of the risk weight formulas ………………………………………………….1
3. Regulatory requirements to the Basel credit risk model…………………………………………..4
4. Model specification …………………………………………………………………………………………….4
4.1. The ASRF framework…………………………………………………………………………………4
4.2. Average and conditional PDs ………………………………………………………………………5
4.3. Loss Given Default …………………………………………………………………………………….6
4.4. Expected versus Unexpected Losses …………………………………………………………..7
4.5. Asset correlations………………………………………………………………………………………8
4.6. Maturity adjustments ………………………………………………………………………………….9
4.7. Exposure at Default and risk weighted assets ……………………………………………..11
5. Calibration of the model ……………………………………………………………………………………11
5.1. Confidence level………………………………………………………………………………………11
5.2. Supervisory estimates of asset correlations for corporate, bank and sovereign exposures……………………………………………………………………………………………….12
5.3. Specification of the retail risk weight curves ………………………………………………..14
6. References ……………………………………………………………………………………………………..15
An Explanatory Note on the I IRB Risk Weight Functions 1. Introduction
In June 2004, the issued a Revised Framework on International Convergence of Capital Measurement and Capital Standards (hereinafter “Revised Framework” or I).1 This framework will serve as the basis for national rulemaking and implementation processes. The June 2004 paper takes account of new developments in the measurement and management of banking risks for those banks that move onto the “internal ratings-based” (IRB) approach. In this approach, institutions will be allowed to use their own internal measures for key drivers of credit risk as primary inputs to the capital calculation, subject to meeting certain conditions and to explicit supervisory approval. All institutions using the IRB approach will be allowed to determine the borrowers’ probabilities of default while those using the advanced IRB approach will also be permitted to rely on own estimates of loss given default and exposure at default on an exposure-by-exposure basis. These risk measures are converted into risk weights and regulatory capital requirements by means of risk weight formulas specified by the .
This paper purely focuses on explaining the I risk weight formulas in a non-technical way by describing the economic foundations as well as the underlying mathematical model and its input parameters. By its very nature this means that this document cannot describe the full depth of the ’s thinking as it developed the IRB framework. For further, more technical reading, references to background papers are provided.
2. Economic foundations of the risk weight formulas
In the credit business, losses of interest and principal occur all the time – there are always some borrowers that default on their obligations. The losses that are actually experienced in a particular year vary from year to year, depending on the number and severity of default events, even if we assume that the quality of the portfolio is consistent over time. Figure 1 illustrates how variation in realised losses over time leads to a distribution of losses for a bank:
BCBS (2004).
Expected Loss (EL)
Unexpected Loss (UL)
Time Frequency
While it is never possible to know in advance the losses a bank will suffer in a particular year, a bank can forecast the average level of credit losses it can reasonably expect to experience. These losses are referred to as Expected Losses (EL) and are shown in Figure1 by the dashed line. Financial institutions view Expected Losses as a cost component of doing business, and manage them by a number of means, including through the pricing of credit exposures and through provisioning.
One of the functions of bank capital is to provide a buffer to protect a bank’s debt holders against peak losses that exceed expected levels. Such peaks are illustrated by the spikes above the dashed line in Figure 1. Peak losses do not occur every year, but when they occur, they can potentially be very large. Losses above expected levels are usually referred to as Unexpected Losses (UL) – institutions know they will occur now and then, but they cannot know in advance their timing or severity. Interest rates, including risk premia, charged on credit exposures may absorb some components of unexpected losses, but the market will not support prices sufficient to cover all unexpected losses. Capital is needed to cover the risks of such peak losses, and therefore it has a loss-absorbing function.
The worst case one could imagine would be that banks lose their entire credit portfolio in a given year. This event, though, is highly unlikely, and holding capital against it would be economically inefficient. Banks have an incentive to minimise the capital they hold, because reducing capital frees up economic resources that can be directed to profitable investments. On the other hand, the less capital a bank holds, the greater is the likelihood that it will not be able to meet its own debt obligations, i.e. that losses in a given year will not be covered by profit plus available capital, and that the bank will become insolvent. Thus, banks and their supervisors must carefully balance the risks and rewards of holding capital.
There are a number of approaches to determining how much capital a bank should hold. The IRB approach adopted for I focuses on the frequency of bank insolvencies2 arising from credit losses that supervisors are willing to accept. By means of a stochastic credit portfolio model, it is possible to estimate the amount of loss which will be exceeded with a small, pre-defined probability. This probability can be considered the probability of bank insolvency. Capital is set to ensure that unexpected losses will exceed this level of capital
the bank failing to meet its senior obligations.
Insolvency here and in the following is understood in a broad sense. This includes, for instance, the case of
with only this very low, fixed probability. This approach to setting capital is illustrated in Figure 2.
100% minus Confidence Level
Potential Losses
Expected Loss (EL)
Value-at-Risk (VaR)
Unexpected Loss (UL)
The curve in Figure 2 describes the likelihood of losses of a certain magnitude. The area under the entire curve is equal to 100% (i.e. it is the graph of a probability density). The curve shows that small losses around or slightly below the Expected Loss occur more frequently than large losses. The likelihood that losses will exceed the sum of Expected Loss (EL) and Unexpected Loss (UL) – i.e. the likelihood that a bank will not be able to meet its own credit obligations by its profits and capital – equals the hatched area under the right hand side of the curve. 100% minus this likelihood is called the confidence level and the corresponding threshold is called Value-at-Risk (VaR) at this confidence level. If capital is set according to the gap between EL and VaR, and if EL is covered by provisions or revenues, then the likelihood that the bank will remain solvent over a one-year horizon is equal to the confidence level. Under I, capital is set to maintain a supervisory fixed confidence level.
So far the Expected Loss has been regarded from a top-down perspective, i.e. from a portfolio view. It can also be viewed bottom-up, namely from its components. The Expected Loss of a portfolio is assumed to equal the proportion of obligors that might default within a given time frame (1 year in the Basel context), multiplied by the outstanding exposure at default, and once more multiplied by the loss given default rate (i.e. the percentage of exposure that will not be recovered by sale of collateral etc.). Of course, banks will not know in advance the exact number of defaults in a given year, nor the exact amount outstanding nor the actual loss rate; these factors are random variables. But banks can estimate average or expected figures. As such, the three factors mentioned above correspond to the risk parameters upon which the I IRB approach is built:
• probability of default (PD) per rating grade, which gives the average percentage of obligors that default in this rating grade in the course of one year
• exposure at default (EAD), which gives an estimate of the amount outstanding (drawn amounts plus likely future drawdowns of yet undrawn lines) in case the borrower defaults
• loss given default (LGD), which gives the percentage of exposure the bank might lose in case the borrower defaults. These losses are usually shown as a percentage
of EAD, and depend, amongst others, on the type and amount of collateral as well as the type of borrower and the expected proceeds from the work-out of the assets.
The Expected Loss (in currency amounts) can then be written as
EL = PD * EAD * LGD or, if expressed as a percentage figure of the EAD, as
EL = PD * LGD.
3. Regulatory requirements to the Basel credit risk model
The Basel risk weight functions used for the derivation of supervisory capital charges for Unexpected Losses (UL) are based on a specific model developed by the on Banking Supervision (cf. Gordy, 2003). The model specification was subject to an important restriction in order to fit supervisory needs:
The model should be portfolio invariant, i.e. the capital required for any given loan should only depend on the risk of that loan and must not depend on the portfolio it is added to. This characteristic has been deemed vital in order to make the new IRB framework applicable to a wider range of countries and institutions. Taking into account the actual portfolio composition when determining capital for each loan – as is done in more advanced credit portfolio models – would have been a too complex task for most banks and supervisors alike. The desire for portfolio invariance, however, makes recognition of institution-specific diversification effects within the framework difficult: diversification effects would depend on how well a new loan fits into an existing portfolio. As a result the Revised Framework was calibrated to well diversified banks. Where a bank deviates from this ideal it is expected to address this under Pillar 2 of the framework. If a bank failed at this, supervisors would have to take action under the supervisory review process (pillar 2).
In the context of regulatory capital allocation, portfolio invariant allocation schemes are also called ratings-based. This notion stems from the fact that, by portfolio invariance, obligor- specific attributes like probability of default, loss given default and exposure at default suffice to determine the capital charges of credit instruments. If banks apply such a model type they use exactly the same risk parameters for EL and UL, namely PD, LGD and EAD.
4. Modelspecification 4.1. The ASRF framework
In the specification process of the I model, it turned out that portfolio invariance of the capital requirements is a property with a strong influence on the structure of the portfolio model. It can be shown that essentially only so-called Asymptotic Single Risk Factor (ASRF) models are portfolio invariant (Gordy, 2003). ASRF models are derived from “ordinary” credit portfolio models by the law of large numbers. When a portfolio consists of a large number of
relatively small exposures, idiosyncratic risks associated with individual exposures tend to cancel out one-another and only systematic risks that affect many exposures have a material effect on portfolio losses. In the ASRF model, all systematic (or system-wide) risks, that affect all borrowers to a certain degree, like industry or regional risks, are modelled with only one (the “single”) systematic risk factor.
It should be noted that the choice of the ASRF for use in the Basel risk weight functions does by no means express any preference of the towards one model over others. Rather, the choice was entirely driven by above considerations. Banks are encouraged to use whatever credit risk models fit best for their internal risk measurement and risk management needs.
Given the ASRF framework, it is possible to estimate the sum of the expected and unexpected losses associated with each credit exposure. This is accomplished by calculating the conditional expected loss for an exposure given an appropriately conservative value of the single systematic risk factor. Under the particular implementation of the ASRF model adopted for I, the conditional expected loss for an exposure is expressed as a product of a probability of default (PD), which describes the likelihood that an obligor will default, and a loss-given-default (LGD) parameter, which describes the loss rate on the exposure in the event of default.
The implementation of the ASRF model developed for I makes use of average PDs that reflect expected default rates under normal business conditions. These average PDs are estimated by banks. To calculate the conditional expected loss, bank-reported average PDs are transformed into conditional PDs using a supervisory mapping function (described below). The conditional PDs reflect default rates given an appropriately conservative value of the systematic risk factor. The same value of the systematic risk factor is used for all instruments in the portfolio. Diversification or concentration aspects of an actual portfolio are not specifically treated within an ASRF model.
In contrast to the treatment of PDs, I does not contain an explicit function that transforms average LGDs expected to occur under normal business conditions into conditional LGDs consistent with an appropriately conservative value of the systematic risk factor. Instead, banks are asked to report LGDs that reflect economic-downturn conditions in circumstances where loss severities are expected to be higher during cyclical downturns than during typical business conditions.
The conditional expected loss for an exposure is estimated as the product of the conditional PD and the “downturn” LGD for that exposure. Under the ASRF model the total economic resources (capital plus provisions and write-offs) that a bank must hold to cover the sum of UL and EL for an exposure is equal to that exposure’s conditional expected loss. Adding up these resources across all exposures yields sufficient resources to meet a portfolio-wide Value-at-Risk target.
4.2. Average and conditional PDs
The mapping function used to derive conditional PDs from average PDs is derived from an adaptation of Merton’s (1974) single asset model to credit portfolios. According to Merton’s model, borrowers default if they cannot completely meet their obligations at a fixed assessment horizon (e.g. one year) because the value of their assets is lower than the due amount. Merton modelled the value of assets of a borrower as a variable whose value can change over time. He described the change in value of the borrower’s assets with a normally distributed random variable.
Vasicek (cf. Vasicek, 2002) showed that under certain conditions, Merton’s model can naturally be extended to a specific ASRF credit portfolio model. With a view on Merton’s and Vasicek’s ground work, the decided to adopt the assumptions of a normal distribution for the systematic and idiosyncratic risk factors.
The appropriate default threshold for “average” conditions is determined by applying a reverse of the Merton model to the average PDs. Since in Merton’s model the default threshold and the borrower’s PD are connected through the normal distribution function, the default threshold can be inferred from the PD by applying the inverse normal distribution function to the average PD in order to derive the model input from the already known model output. Likewise, the required “appropriately conservative value” of the systematic risk factor can be derived by applying the inverse of the normal distribution function to the pre- determined supervisory confidence level. A correlation-weighted sum of the default threshold and the conservative value of the systematic factor yields a “conditional (or downturn) default threshold”.
In a second step, the conditional default threshold is used as an input into the original Merton model and is put forward in order to derive a PD again – but this time a conditional PD. The transformation is performed by the application of the normal distribution function of the original Merton model.
Standard normal distribution (N) applied to threshold and conservative value of systematic factor
Inverse of the standard normal distribution (G) applied to PD to derive default threshold
Inverse of the standard normal distribution (G) applied to confidence level to derive conservative value of systematic factor
Capitalrequirement(K)= [LGD*N[(1-R)^-0.5*G(PD)+(R/(1-R))^0.5*G(0.999)] – PD * LGD] * (1 – 1.5 x b(PD))^ -1 × (1 + (M – 2.5) * b (PD)
In addition, the Revised Framework requires banks to undertake credit risk stress tests to underpin these calculations. Stress testing must involve identifying possible events or future changes in economic conditions that could have unfavourable effects on a bank’s credit exposures and assessment of the bank’s ability to withstand such changes. As a result of the stress test, banks should ensure that they have sufficient capital to meet the Pillar 1 capital requirements. The results of the credit risk stress test form part of the IRB minimum standards. Since this paper is restricted to an explanation of the risk weight formulas, no more detail of the stress testing issue is presented here.
4.3. Loss Given Default
Under the implementation of the ASRF model used for I, the sum of UL and EL for an exposure (i.e. its conditional expected loss) is equal to the product of a conditional PD and a “downturn” LGD. As discussed earlier, the conditional PD is derived by means of a supervisory mapping function that depends on the exposure’s average PD. The LGD parameter used to calculate an exposure’s conditional expected loss must also reflect adverse economic scenarios. During an economic downturn losses on defaulted loans are likely to be higher than those under normal business conditions because, for example, collateral values may decline. Average loss severity figures over long periods of time can understate loss
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