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The asymptotic Basel II Tier One Capital formula

The Basel II Capital Accord seeks to improve on the existing rules by aligning regulatory capital requirements more closely to the underlying risks that banks face. One of the risk types described in the Capital Accord is credit risk. Banks need to hold capital to cover the credit risk on their credit portfolio. On the basis of the work of Merton and Vasicek, the Basel Committee on Banking Supervision (BCBS) decided to adopt the assumptions of a normal distribution for the systematic and idiosyncratic risk factors of a credit portfolio. The model behind the capital requirements function is called an Asymptotic Single Risk Factor (ASRF) model. The Basel Committee had as important requirement that the capital requirements function should be portfolio invariant. Gordy has shown that essentially only ASRF models are portfolio invariant. Therefore the Basel Committee has chosen for an ASRF model.

Under Basel II computation of the Tier One Capital ratio is implemented by following a simple arithmetic rule in both the banking book and the trading book. In the banking book the arithmetic rule is an oversimplified implementation of a sophisticated statistical approximation technique called ASRF (Asymptotic Single Risk Factor) generally associated with Professor Michael Gordy of the Federal Reserve. In the trading book regulatory capital is predicated on the VaR technique an essentially backward looking and oversimplified modelling approach, which Basel II seems to recognize in adding a fudge factor of 3. Recently the Basel Committee has issued proposals to update and revise the Basel II market risk framework for computation of tier one regulatory capital[1]. We will look at the recent proposals to strengthen Basel II in detail below but the parameters of the calculation of Tier One regulatory capital in the banking book do not look as if they are going to change, the emphasis in strengthening Basel II post-CC appears to be in the Pillar II arena, in regard to economic capital and the Supervisory Review Process (SRP).

J P Morgan describes an overview of the operation of the Basel II regulatory capital calculation in a research paper from 2006[2];

 “The output of each Basel 2 formula (which varies by asset type) is the regulatory capital requirement, and is sized to cover only the unexpected losses, since banks are expected to provision for expected losses. …Unexpected losses, under Basel 2, are modeled as a function of a systematic risk factor (a macroeconomic variable). The capital requirement, consequently, also depends on a correlation input (R) which captures the relationship between unexpected loss and the systematic risk factor. Basel 2 formulas assume that idiosyncratic risks are eliminated by diversification. However, individual regulators may adjust the total capital charge for lumpy (less diversified) or highly correlated pools. Adjustments made by regulators to capital requirements fall under “Pillar 2” of the Accord, which relates to (supervisory) oversight.

Economic Concepts behind the IRB Capital Requirements Function

The IRB risk weight equation seems arbitrary and mechanical at first sight. However, there are some important concepts embedded in this equation

Expected Loss (EL) is the mean of the loss distribution. The EL is defined as the average level of credit losses a financial institution can reasonably expect to experience. Note that, in contrast to a normal distribution, the mean is not at the centre of the distribution but rather is right of the peak. That occurs because the typical loss distribution of a credit portfolio is asymmetric. It has a long right-hand tail.  Financial institutions view EL as a cost component of doing business. EL is managed by financial institutions in a number of ways, including through the pricing of credit exposures and through provisioning.

Unexpected Loss (UL) is the standard deviation of the loss distribution. In contrast to EL, UL is a risk associated with being in the business, rather than a cost of doing business. One of the functions of bank capital is to provide a buffer to protect a bank’s debt holders against losses that exceed expected levels. Banks often have an incentive to minimise the capital they hold. Reducing capital frees up economic resources that can be directed to profitable investments. However, if a bank holds less capital, its chance that is will not be able to meet its own debt obligations will increase.

Economic Capital (EC) is a risk measure that can be viewed, as the amount of capital the bank needs to retain to cope with unexpected loan losses.  The more risky the assets, the more capital will be required to support them. The main difference between EC & Regulatory Capital (RC) is that RC is the minimal capital required by the regulator (Basel II), whereas economic capital is the capital level bank shareholders would choose in absence of capital regulation. The target insolvency rate of a financial institution determines the amount of EC a bank needs to retain. Many large commercial banks are using a target insolvency rate of 0,03%. The target insolvency rate is directly linked with the credit rating of the financial institution. Looking at historical one-year default rates, the probability of default for a financial institution rated AA is 0,03%.   

The Capital Treatment of Expected & Unexpected Losses by BIS

Below is presented twelve references from BIS documents on the subject of Unexpected versus Expected Loss. Clearly in the definition of the Basel II P1 RWA calculation for Regulatory Capital, the BIS unilaterally defined the new Regulatory Capital calculation approach as being capital for unexpected loss whereas prior to that regulatory capital had been capital for expected loss. Further BIS publications clearly do not necessarily agree with this dichotomy particularly since it makes it unclear as to exactly what Economic Capital is for. In the introduction to the reference document for P1 (reference zero below) BIS states clearly that further work is necessary to clarify the new conceptual distinction. For the moment we have to work with the BIS definitions in P1 and P2. In the last pages of this document I have made some more conceptual comments about Economic Capital in particular from an economic theory standpoint.

The important point I suppose is that BIS sets the rules but the conceptual views of the distinctions of UL and EL and r.cap and e.cap will differ amongst the users of those rules (the banks being reulated) as they do amongst the BIS committee members and their economists.











One area where the Committee intends to undertake additional work of a longer term nature is in relation to the definition of eligible capital. One motivation for this is the fact that the changes in the treatment of expected and unexpected losses and related changes in the treatment of provisions in the Framework set out here generally tend to reduce Tier 1 capital requirements relative to total capital requirements. Moreover, converging on a uniform international capital standard under this Framework will ultimately require the identification of an agreed set of capital instruments that are available to absorb unanticipated losses on a going-concern basis. The Committee announced its intention to review the definition of capital as a follow-up to the revised approach to Tier 1 eligibility as announced in its October 1998 press release, “Instruments eligible for inclusion in Tier 1 capital”. It will explore further issues surrounding the definition of regulatory capital, but does not intend to propose changes as a result of this longer-term review prior to the implementation of the revised Framework set out in this document. In the meantime, the Committee will continue its efforts to ensure the consistent application of its 1998 decisions regarding the composition of regulatory capital across jurisdictions.


International Convergence

of Capital Measurement

and Capital Standards

A Revised Framework

June 2004





In its 11 October 2003 press release, the Basel Committee on Banking Supervision (the Committee) announced its intention to move to a UL-only risk weighting construct. The Committee requested comments on this revision, and received 52 comment letters. Respondents generally welcomed the Committee’s solution and agreed that it will align regulatory capital more closely with the concepts underpinning banks’ economic capital modeling processes. Many commenters, however, requested the Committee to provide more detailed information on the new framework. Responding to such requests, the Committee, at its meeting on 14 and 15 January 2004, took decisions on a number of questions arising from the move to the UL-only construct. The purpose of this paper is to provide information on the concrete modifications that have been decided. In summary, for the IRB approach, expected losses will be removed from the risk weight functions. However, banks will be required to compare their actual provisions with expected losses. Any shortfall should be deducted equally from Tier 1 and Tier 2 capital and any excess will be eligible for inclusion in Tier 2 capital subject to a cap. Therefore, the current treatment of general provisions will be withdrawn from the IRB approach. The Committee is not intending to make any related changes to the standardized approach. Where banks are partly on the standardized approach and partly on the IRB approach, an element of general provisions may be retained in Tier 2 capital.


Modifications to the capital treatment for expected and unexpected credit losses in the New Basel Accord

30 January 2004







The IRB measurement of expected losses (EAD x PD x LGD)




[Default] intensity is calibrated using the average .five-year-ahead default rate of corporate issuers which is an estimate of the unconditional five-year default probability based on data from Moody.s. Expected loss is computed as the product of this default probability and a constant rate of loss in the event of default. Loss given default is set equal to one minus the average recovery rate on senior unsecured debt based on Moody.s recovery rate data.

BIS Working Papers

No 190

The Pricing of Unexpected Credit Losses

by Jeffery D Amato and Eli M Remolona

Monetary and Economic Department

November 2005





“Risk” is a multifaceted concept. In order to understand better the themes developed in what follows, it is necessary to clarify various definitions and dimensions of risk. Of particular relevance are the distinctions between expected and unexpected losses (in the statistical sense), between relative and absolute risk, between idiosyncratic and systematic risk and between the risk of individual portfolios and that of the financial system as a whole.


One popular way of characterising risk is to describe it in terms of a probability distribution over future outcomes. In the case of credit risk – the focus of this paper – the term “risk” is normally used to refer to at least two quite distinct concepts, namely expected and unexpected losses, depending on which features of the distribution one focuses on. “Expected losses” refer to the average or mean losses anticipated over a particular period, while “unexpected losses” refer to a measure of the dispersion, or degree of uncertainty that surrounds that outcome. This second notion of risk is closer in spirit to classical definitions of risk.


Procyclicality of the financial system and financial stability:

issues and policy options

Claudio Borio, Craig Furfine and Philip Lowe





As has now become widely accepted, it is best to think of the role of capital as that of providing protection against unexpected losses, and of the role of provisions as that of providing cover against expected losses.


This distinction between capital and provisions is sometimes seen as artificial, since both provide the bank with similar protection against losses. It could be argued that all that matters is that the sum of capital and provisions is sufficient to cover expected and unexpected losses; if provisions are “too low”, then the bank can simply hold additional capital to achieve its acceptable probability of failure


Procyclicality of the financial system and financial stability:

issues and policy options

Claudio Borio, Craig Furfine and Philip Lowe





There are many different modeling approaches to compute expected and unexpected losses on a portfolio and several alternative risk measures that can be drawn from a given loss distribution. A number of studies have applied to macro stress-testing value-at-risk (VaR) measures, so far mainly used for the risk management of individual financial institutions.


BIS Working Papers

No 165

Stress-testing financial systems: an overview of current methodologies

by Marco Sorge





Regarding the estimation of LGD, the BCBS published a number of principles that banks are expected to adhere to in order to become eligible to use own estimates of LGDs within the internal ratings-based approach of Basel II. The Committee recognised that potentially higher than average realised losses during times of high default rates might prove a material source of unexpected credit losses for some exposures. The BCBS considered that a principles-based approach, which provides a significant degree of flexibility, was most appropriate at this time


BIS Quarterly Review, September 2005





Existing studies of procyclicality in the proposed New Accord have focused on Pillar 1 IRB treatment of whole commercial loans.4 Regulatory capital charges are set at the individual loan level and are given by a formula with five inputs: the borrower’s one-year PD, the instrument’s expected loss given default (LGD) and remaining maturity (M), the asset-value correlation which parameterizes dependence across borrowers, and a target one-year solvency probability (q) for the bank.


Procyclicality in Basel II: Can We Treat the Disease Without Killing the Patient?

Michael B. Gordy and Bradley Howells

Board of Governors of the Federal Reserve System





The capital rule is derived from the large-portfolio asymptotic behavior of a Merton model with a single common risk-factor. As shown by Gordy (2003), an instrument’s marginal contribution to portfolio VaR converges in the asymptotic limit to its expected loss conditional on the common factor suffering a qth percentile “stress event.” In the IRB implementation, this conditional expected loss is approximated as a separable equation of three terms: LGD; the one-year actuarial conditional expected loss, V (PD; _; q); and a maturity adjustment h(PD;M) that approximates the ratio of mark-to-market capital charges to actuarial loss capital charges


In their internal capital systems, large banks typically set target solvency probabilities to at least 99.95%. If the IRB capital formula is reasonably consistent with the internal VaR model, then the bank’s internal capital requirement would exceed the regulatory minimum (calculated setting q to 99.9%) by 15% to 20%. It is often argued that a regulatory solvency target of 99.9% implies a rather higher solvency target in practice, as the bank must hold a buffer sufficient to avoid falling below the regulatory minimum in the event of a downturn (Furfine 2000). Jokivuolle and Peura (2001) model this buffer as satisfying a meta-VaR requirement: the bank sets a target probability of being unconstrained by the regulatory minimum at the horizon.13 Adding this constraint significantly increases the regulation’s effective one-year solvency standard. While this argument has merit, it ought to apply equally to the bank’s internal economic capital calculations. At least for large international institutions, franchise value at the horizon is dependent on retaining a favorable agency rating. So, for example, an institution that seeks to have capital sufficient for an AA rating today (say, a one-year target solvency probability of 99.95%) might also want to have a 95% probability of remaining investment grade at the horizon. This meta-VaR constraint creates a buffer for economic capital.


Procyclicality in Basel II: Can We Treat the Disease Without Killing the Patient?

Michael B. Gordy and Bradley Howells_

Board of Governors of the Federal Reserve System





Some firms split risk factors into groups according to liquidity profile. An appropriate liquidity horizon is then attributed to each group. This is a particularly important distinction for those institutions involved in emerging markets. For those institutions covering very illiquid assets, such as property or unlisted equities, expected losses need to be covered by capital in the form of provisions (thereby obviating the need to sell the asset); as a result, a much longer period, such as half a year, might be assumed to verify that there is enough capital to cover the risk.



Stress testing at major

financial institutions: survey results and practice

Report by a working group established by the Committee on the

Global Financial System





Every loan intrinsically has an expected (or potential) loss that should be recognised as a cost by means of an early provision. Otherwise, the picture of the true profitability and solvency of the bank over time could be distorted. More dangerously, the overvaluation of profits might lead to an increase in dividends that could undermine the solvency of the bank



Credit growth, problem loans and credit risk provisioning in Spain

Santiago Fernández de Lis, Jorge Martínez Pagés and Jesús Saurina1





The concepts of Expected and Unexpected Loss (EL and UL) have a long history of refinement and debate in Economics. At one level it is a meaningless distinction since it is impossible to definitively answer the question about at which point one clearly defines an expectation as expected or unexpected; that would depend upon a likelihood definition or an arbitrarily selected confidence interval. It will be different for different people. We should remember that the BIS is not preparing its prescriptions of RWA calculations for professional economists or statisticians but that it is likely that interpreters of the regulations in the banks and the their consultants will be one or other or both and will be well aware of confusions in the accords.

It was these confusions which allowed the Banks to delay the implementation of B2 for so long. It does have to be said however that it beggars belief that a mechanistic formula can be viewed as a good approach to uncertainty. I think this flaw in the BIS logic strengthens the argument for taking an holistic approach to e.cap which is not predicated upon the formulaically defined r.cap, doing that allows the institution and the supervisor to double check that both numbers are sane. The path to Basel 3 is clearer in the treatments of the Trading Book and Securitized Instruments. If the BIS wishes the Banking system to be moved to a capital regime which treats unexpected loss (properly) then maybe an A-IRB approach based exclusively upon internally modeled capital in the holistic e.cap manner is the end-game envisaged.

[2] Basel 2 and Securitisation A Paradigm Shift, J P Morgan, European Securitized Products Research, 2006



The Asymptotic Single-Risk-Factor (ASRF) model Specification and Calibration Errors


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