A critical analysis of the finalization of Basel III (Note)

Usefulness of the article: This article provides a critical analysis of the revisions to the Basel III reform of December 2017. This critical analysis focuses in particular on the lack of change to the formula that allows banks to calculate the minimum level of capital required by the regulator in approaches based on internal models (Internal Rating Based – IRB).

Summary:

· Since the Basel I Accords, regulators have required banks to maintain a minimum level of capital. This capital requirement, also known as « regulatory capital, » enables banks to cope with unexpected losses that are not covered by provisions. These would be losses arising from a sudden deterioration in economic conditions. The Basel II and Basel III agreements have allowed banks to estimate themselves (using their internal models) the input parameters of the function used to calculate this regulatory capital.

· The revisions to the Basel III reform regulate and limit the use by banks of approaches based on internal models to assess their regulatory capital (known as « IRB approaches »). However, these revisions do not call into question the mathematical formula used to calculate this regulatory capital based on the credit risk parameters that banks estimate using their internal models.

· However , the assumptions of the Gordy-Vasicek model on which this formula is based (a perfectly granular credit portfolio and the presence of a single source of common risk) are unrealistic, which may result in the underestimation of regulatory capital.

· Furthermore , there seems to be some confusion in practice between the concept of simple probability of default (PD) —a probability that is not assessed for a specific economic situation—and that of conditional probability of default —a probability assessed for a specific economic situation—yet the distinction between the two probabilities is fundamental to the Gordy-Vasicek model.

· The following solutions could be considered to address these weaknesses in the Basel II and Basel III reforms : a specific stress test (with a PD assessment based on individual counterparty data) for non-diversified portfolios (i.e., exposed to one or a few counterparties), a multifactorial generalization of the regulatory capital assessment formula, and a review and additional monitoring by the regulator of the PD estimation models used by banks to ensure that these models are compatible with the definition of this risk parameter in Gordy-Vasicek.

IRB approaches allow banks to calculate the minimum level of capital required by the regulator based on credit risk parameters estimated using their internal models. These approaches, along with the associated mathematical formula for calculating this minimum capital requirement (hereinafter referred to as the « regulatory capital assessment function »), were introduced by the Basel II reform and have been retained in the Basel III reform despite the various changes introduced by the latter. The December 2017 revisions to the Basel III reform aimed to reduce disparities in the assessment of risk-weighted assets (RWA)[1] between banks, in particular by regulating and limiting the use of IRB approaches[2]. However, these revisions do not call into question the regulatory capital assessment function relating to these approaches[3]. However, this function, which is derived from the Gordy-Vasicek model, is based on assumptions that are difficult to sustain in reality (namely, a perfectly granular credit portfolio and the existence of a single source of common risk: the economic environment).

Firstly, this function is not suitable for non-diversified credit portfolios as it may underestimate the credit loss of these portfolios. As the specific risk[6] of these portfolios is not dissipated, specific stress tests are a good way of assessing it, as they focus on the counterparty or counterparties to which a portfolio is heavily exposed.

Secondly, the regulatory capital assessment function in IRB approaches does not take into account the multiple sources of common risk that can impact different sectors. A multi-factor extension of this function would be necessary, as it would not only allow for the consideration of several common risk factors, but also enable the macroeconomic shocks of the EBA/ECB macroprudential stress tests to be applied correctly (within the meaning of the Gordy-Vasicek definitions) and in a more precise and transparent manner to sectoral portfolios.

Finally, the various concepts of default probability in the Gordy-Vasicek model do not always seem to be well understood and taken into account in practice. An understanding of the fundamentals of this function would be necessary as it would allow for a better understanding of the various concepts/definitions associated with it and would provide credit risk parameter estimation models compatible with the Gordy-Vasicek base model.

1)Failure to comply with the assumptions of the regulatory capital assessment function

1.1. Non-compliance with the assumption of a perfectly granular portfolio

If the credit portfolio is perfectly granular (i.e., perfectly diversified), the actual credit loss of this portfolio will converge toward its expected level. For non-diversified (or concentrated) portfolios, the credit loss does not converge toward its expected level; it will be higher if the default event occurs. This difference translates into the presence of an undissipated specific risk that is not captured by the expected (conditional) loss. The expected loss (EL) and the conditional expected loss (corresponding to the « unexpected » loss (UL) when the economic environment is assumed to be deteriorating) then become irrelevant risk measures. Consequently, the regulatory capital assessment function, which corresponds to the UL function, is not suitable for capturing the credit risk of these portfolios[7].

Although there are regulatory measures to limit excessive concentration in banks’ portfolios (in particular, the treatment and reporting of « large exposures »), banks must conduct internal stress tests to assess the specific credit risk of their undiversified portfolios. These stress tests would make it possible to stress the credit risk parameters of the counterparty or counterparties to which the bank is heavily exposed, in relation to their specific risk.

The specific risk must stem from factors related to the counterparty’s financial fundamentals and not from common risk factors such as macroeconomic variables. Indeed, the more concentrated the portfolios are, the more likely it is that they do not depend solely on common risk. Taking into account and stressing only common risk factors, as in the macroprudential stress tests conducted by theEuropean Banking Authority (EBA) and the European Central Bank (ECB), is therefore insufficient, if not irrelevant, for portfolios concentrated on counterparties that are not very cyclical, or even countercyclical.

Specific shocks to a counterparty’s financial fundamentals can be reflected in its simple probability of default (PD) using a structural approach derived from Merton (1974): this would involve significantly downgrading the total asset value of the counterparty in question and assessing the resulting probability of default[8]. A relevant PD estimation model must take into account not only common risk factors but also counterparty-specific factors, which structural models (notably KMV[9]) are able to do.

In the Gordy-Vasicek model, the portfolio is assumed to be perfectly granular, so the average PD in the portfolio (or the expected unconditional default rate of the portfolio) reflects only common risk. If we assume that the portfolio is not granular, the average PD must also reflect specific, undiversified risk. By calculating the expected loss of the portfolio based on an average default probability reflecting specific risk[10], a modified version of the Gordy-Vasicek model is obtained.

1.2. Non-compliance with the single common risk factor assumption

The single-factor model currently used to calculate regulatory capital in banks using IRB approaches takes into account a single source of common risk: the macroeconomic environment. In this model, the maximum credit loss is expected when the common risk factor is at its minimum level, i.e., when the economic environment is severely deteriorated. As a single-factor model, it does not allow different macroeconomic shocks to be applied directly to common risk factors, which prevents these shocks from having their own impact on the conditional probability of default of counterparties.

A multi-factor generalization of the regulatory capital assessment function would allow the EBA/ECB macro stress test shocks to be correctly applied to each sector portfolio, stressing the common risk factors identified as having an impact in a precise and transparent manner.

By applying macroeconomic shocks to the various common risk factors impacting sector portfolios[12], we pass these shocks directly on to the conditional probability of default of the counterparties in these portfolios, rather than passing them on to the simple probability of default (PD) of these counterparties, as is currently done in banks through their internal models (which is theoretically incorrect).

In this multifactor extension, it is necessary to determine, for each sector, the systematic risk factors that have an impact, their weights, and the presumed monotonicity between the bank’s credit loss and each risk factor taken separately (i.e., determine a priori for which extreme of the common risk factor we are supposed to obtain a maximum credit loss).

The main advantage of the multifactor model for assessing capital requirements is that it includes any common risk factor estimated to have a medium-term impact on counterparties in a sector. In this multifactor model, it is assumed that counterparties in the same sector should, a priori, have equal or similar sensitivity to the same systematic risk factors. However, the sensitivity coefficients for each sector must be estimated at regular intervals corresponding to the frequency of publication of banks’ RWA (in this case, quarterly). Ideally, this estimation should be carried out by the regulator (in particular the EBA or the ECB at European level), as it has a more comprehensive view of the various sectors within a geographical area, and this would avoid very different estimates between banks.

With a view to sustainable economics and a resilient banking system in the long term, the regulator could even go further when choosing common risk factors and consider factors that have a long-term impact, such as environmental and climate factors, which have so far been largely overlooked in the literature and in banks’ internal models. This would lead to an assessment of a banking institution’s long-term capital charge (thus exceeding the current annual horizon) and would thus provide a better picture of the long-term resilience of the banking system.

2)Confusion between simple default probability and conditional default probability

In the Gordy-Vasicek model, the conditional probability of default is a function of the simple probability of default and the common risk factor alone. This probability is therefore obtained for a specific value of the common risk factor: it is assumed that the maximum credit loss is obtained when the common risk factor is at its minimum level. On the other hand, the simple default probability is unconditional at a specific value of the common risk factor.

Qualitative variable models (notably Logit and Probit) are among the models used by banks to estimate simple default probabilities. In the use of these models, and particularly in the translation of the macroeconomic shock scenarios of the EBA/ECB stress tests into changes in the PD and LGD credit risk parameters, there appears to be some confusion between simple default probability and conditional default probability. For example, reflecting a macroeconomic shock on the simple default probability is theoretically incorrect because this shock consists of assigning a precise value to the common risk factor(s), which is incompatible with the unconditional nature of the simple default probability in Gordy-Vasicek. This methodology would only be relevant if the stress scenario is considered sufficiently likely to influence the normal estimate of the unconditional probability of default. Indeed, an unconditional probability of default must correspond to the average of the conditional probabilities of default assessed across the various possible realizations of the common risk factor(s).

Although, under the IRB approach, banks can estimate the credit risk parameters (PD, LGD, CCF, M) used to assess regulatory capital themselves, additional monitoring should be carried out by the regulator on estimates of simple default probabilities (PD) to ensure that these estimates comply with the definition of these probabilities in the Gordy-Vasicek model. The regulator should also review the EBA/ECB macroprudential stress test methodology in the context of assessing stressed capital requirements, so that the proposed macroeconomic shocks do not result in a change in the simple probability of default (which, as a reminder, is unconditional on the common risk factor in Gordy-Vasicek); the multifactor generalization proposed in section 1.2 could be a way to remedy this.

Conclusion

Although the finalization of Basel III regulates and limits the use of IRB approaches, it does not call into question the regulatory capital assessment function in these approaches. However, the two fundamental assumptions of this function are unrealistic. Furthermore, the definitions of the different types of default probability (simple and conditional) involved in this function seem to be poorly understood by banks and not taken into account in the EBA-ECB macroprudential stress test methodology for assessing stressed capital requirements. Specific stress tests (with a PD assessment based on individual counterparty data) would be an appropriate solution to compensate for the lack of granularity in undiversified credit portfolios and thus enhance the effectiveness of the regulation. In addition, a multi-factor generalization of the regulatory capital assessment function would make it possible to take into account several common risk factors and to correctly reflect macroeconomic shocks on the capital charge for each sectoral portfolio. Finally, additional oversight of the modeling of simple default probabilities estimated by banks would be necessary to ensure that the definition of these probabilities corresponds to that of the Gordy-Vasicek model, as would a review of the EBA/ECB macroprudential stress test methodology in the context of stress capital charge assessment.

Appendix

Formula for calculating regulatory capital in IRB approaches

The detailed formula for calculating regulatory capital for credit exposures (i.e., for debt exposures) and for derivatives, imposed by the Basel II Committee and maintained by the Basel III Committee for IRB approaches, is as follows:

where:

LGD : Loss given default estimated by the bank for a downturn scenario (economic slowdown);

PD : Simple or unconditional probability of default estimated by the bank. This probability is supposed to reflect the average of the conditional probabilities of default assessed across the various scenarios of the common risk factor (in this case, the economic situation). It corresponds, in a way, to the probability of default assessed for an « average » economic context.

EAD : Exposure at default, where the conversion factor for off-balance sheet items into balance sheet equivalents (CCF) is estimated by the bank.

_MAdj : Adjusted maturity.

N[…] : Probability of default conditional on a deteriorated economic environment (known as the « unexpected » default rate). This probability is a distribution function of the centered-reduced normal distribution N(0;1), where N-1(0.001)  value of X (here the economic situation) such that there is a 99.9% chance of exceeding it: P (X ≤ N-1(0.001)) = 0.001; – N-1(0.001) = N-1(0.999), where the « – » sign is in the Gordy-Vasicek model formula (see formulas (6.3) to (7.2) in Dhima (2019)).

R : Coefficient measuring the degree of dependence of a borrower on the overall state of the economy (i.e., the economic situation). It is referred to as the « correlation coefficient » by the regulator and in practice. This terminology is not entirely correct. According to Vasicek (2002), R should correspond to the correlation of each pair of exposures in the same credit portfolio, and √R to the exposure of any debt security in the portfolio to the common risk factor.

The formula for calculating regulatory capital is derived from the Gordy-Vasicek model. The Basel II Committee made a number of extensions to this model: in particular, it introduced a fixed formula for calculating the R coefficient, the adjusted maturity _{MAdj and} its calculation formula, and the multiplier factor. These extensions have been retained by the Basel III Committee.

a) ^First extension related to the calculation of the R coefficient (R function)

The expression of N[…] itself uses a sub-formula for calculating the R coefficient . This coefficient is calculated according to the following different scenarios:

– For large corporations, financial institutions, and sovereigns (annual revenue S ≥ €50 million):

– For large corporations, financial institutions, and sovereigns (annual revenue S ≥ €50 million):

– For small and medium-sized enterprises (annual turnover S < €50 million):

–For retail customers :

– The degree of dependence of borrowers on the global economic environment is therefore a function o

The degree to which borrowers are dependent on the global economic context therefore depends on:

–the type of counterparty (sovereign, financial institution, large enterprise, small or medium-sized enterprise, retail customers): for example, an enterprise is more sensitive to the overall state of the economy than an individual;

–the size of the company as measured by its annual revenue S: a large company (S ≥ €50 million) is more sensitive to the overall state of the economy than an SME (S < €50 million);

– the type of product: for example, there is historically a much higher correlation between borrowers who have taken out residential mortgages than between credit card holders;

– the credit quality of the counterparty, measured by its probability of default (PD): healthy customers are more sensitive to the overall state of the economy than customers already on the brink of bankruptcy, which explains why the R coefficient determined by the formula is highest (24% if we consider the first formula) for a minimum PD (0%) and lowest (12% if we consider the first formula) for a maximum PD (100%).

The higher R is, the higher the calculated capital charge. Indeed, the more borrowers are linked to each other by a common dependence on the overall state of the economy, the higher the peaks of unexpected losses that may occur in an unfavorable economic environment. For identical expected loss (EL) values, two different populations of borrowers may have very different levels of unexpected losses.

b) ^Second extension related to adjusted maturity ( _MAdj)

The unexpected loss (UL) is subject to a maturity adjustment (_MAdj). _MAdj=1 for retail customers, while it is a function of simple maturity (M) and simple probability of default (PD) for wholesale customers :

_{MAdj =} (1 – 1.5*b)^-1(1 + (M – 2.5)*b), where b = (0.11852 – 0.05478*ln(PD)) ²

For a constant M , an increase in PD causes _MAdj to decrease . This corresponds to the fact that a healthy customer has more potential for deterioration than a customer already on the verge of bankruptcy.

c) ^Third extension related to the multiplier factor

Based on quantitative impact studies (QIS) conducted with banks around the world, the Basel Committee estimated that the capital charge as determined by the K function was insufficient to achieve the objective of maintaining the overall level of capital in the banking sector. It therefore decided to introduce a scaling factor, set at 1.06 when the revised framework was published in 2004. This means that the revised capital charge is: K’ = 1.06*K

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