Michel Dietsch
email: michel.dietsch@unistra.fr

Henri Fraisse
email: henri.fraisse@acpr.banque-france.fr

Mathias Lé
email: mathias.le@banque-france.fr

Sandrine Lecarpentier
email: sandrine.lecarpentier@acpr.banque-france.fr

Lower bank capital requirements as a policy tool to support credit to SMEs: evidence from a policy experiment *

Keywords: SME finance, Credit supply, Basel III, Credit risk modelling, SME Supporting Factor JEL-Classification : C13, G21, G33

Starting in 2014 with the implementation of the European Commission Capital Requirement Directive, banks operating in the Euro area were benefiting from a 25% reduction (the Supporting Factor or "SF" hereafter) in their own funds requirements against Small and Medium-sized enterprises ("SMEs" hereafter) loans. We investigate empirically whether this reduction has supported SME financing and to which extent it is consistent with SME credit risk. Economic capital computations based on multifactor models do confirm that capital requirements should be lower for SMEs. Taking into account the uncertainty surrounding their estimates and adopting a conservative approach, we show that the SF is consistent with the difference in economic capital between SMEs and large corporates. As for the impact on credit distribution, our difference-in-differences specification enables us to find a positive and significant impact of the SF on the credit supply.

Introduction

Small and Medium-sized Enterprises ("SMEs" hereafter) finance is a growing concern in Europe. While SMEs are a crucial engine for growth in Europe, they are largely dependent on bank credit regarding their external financing and are much more likely to report issues with bank financing than large corporates. As an illustration, in 2014, bank lending to SMEs was still below its pre-crisis level, in contrast with large corporates (EBA, 2016). To improve SMEs' access to credit in time of crisis, policymakers and central bankers traditionally rely on monetary and targeted fiscal policies.1 However, in a context of changing bank regulation and rising bank capital requirements ("CRs" hereafter), bank lending decision has also become increasingly more sensitive to the regulatory framework, as illustrated by recent contributions to the empirical literature.2 Against this background, this paper investigates the effectiveness and the consistency of a new regulatory tool implemented specifically to promote SMEs' access to bank credit: a targeted reduction in bank CRs associated with SMEs loans.

In 2014, the transposition of the Basel III standards into EU law introduced a 24% reduction in CRs for exposures to SMEs -labelled Supporting Factor ("SF" hereafter)-with the aim of fostering the provision of credit to SMEs. The idea behind this proposal is that any reduction of regulatory capital requirements (CRs) should boost credit availability for businesses. The European legislators have required credit institutions to use this CRs relief for the "exclusive purpose of providing an adequate flow of credit to SMEs established in the Union". 3 But, a necessary condition for the Supporting Factor to be effective is that regulatory capital requirements should be consistent with the underlying SME credit risk. Hence, as required by the legislators themselves, this targeted reduction in CRs needs to be regularly evaluated according to two criteria: (i) an easier access to bank credit for SMEs and (ii) the consistency of capital requirements with SME credit risk.

In this paper, we assess this policy experiment along with these two dimensions. For this purpose, we exploit the French credit register. This dataset is maintained by the Banque de France and offers an almost comprehensive sample of loans granted to corporate businesses operating in France over the last two decades. Importantly, in this dataset, we also observe the credit rating granted by the Banque de France to a large sub-sample of firms, which enables the computation of historical time series of default rates. Based on this dataset, the consistency of the reform regarding the intrinsic riskiness of SMEs is gauged through the computation of banks' economic capital requirements using the structural credit risk framework underlying the computation of the regulatory capital requirements. Then, the impact of the reform on the credit supply to targeted SMEs is estimated through the difference-in-differences methodology.

To assess the consistency issue (we refer to this part as the risk analysis hereafter), we use the credit risk structural approach, as devoted by Merton (1974). This approach underlies the Basel II/III regulatory capital requirements formula (Gordy, 2003). However, the regulatory capital requirements do not necessarily reflect SMEs credit risk in a bank portfolio. The reason is that, in the regulatory framework, the asset correlation, which measures the sensitivity of exposures to the single risk factor, is invariant with the characteristics of the real portfolios and is assumed to depend mainly on the probability of default. Consequently, the value of this risk parameter appears to be overestimated, as shown by numerous academic papers using real portfolios data. 4 The overestimation of SMEs capital charges comes largely from the fact that SMEs show higher probabilities of default than large corporates.

To compute in a consistent way the capital requirements, we expand the asymptotic single risk factor model (ASRF) to a multifactor framework to take into account differences in risk associated with firms' size. We consider several risk factors, depending on the firm's size, and we compute the marginal contribution of specific groups of firms to the total potential losses on bank loans portfolio, depending on their size. This marginal contribution measures the amount of "economic capital" required to cover the losses associated with each size class, economic capital being defined as the estimate of the worst possible decline in the bank's amount of capital at a specified level of confidence (99.9% in Basel formulas) within a chosen time horizon (one year). Thus, in this paper, we adopt a lender's perspective and address the consistency issue by assessing firms' size as a driver of systematic credit risk in loans portfolios. Therefore, our multifactor approach with firms' size as a risk factor departs from the Basel II/III framework, which considers only a single general systematic risk factor, and in which the sensitivity of exposures to risk does not vary with the size of the firms.

In fact, firms of different sizes could not be equally sensitive to the general common single risk factor. Moreover, they could be sensitive to risk factors which are specific to their own size class. Ignoring firms' heterogeneity could generate an overestimation of SMEs credit risk if the regulatory correlation is significantly higher than the empirical correlation estimated by using data of banks' real loans portfolios. On the contrary, by comparing estimated economic capital requirements for specific size classes with the Basel II/III regulatory capital requirements for the same size classes, our approach takes into account borrowers' heterogeneity and allows us to verify the consistency of regulatory CRs with the contribution of specific groups of firms to portfolio credit risk. Moreover, the multifactor approach also takes into account the potential correlations of exposures not only within a group of borrowers of the same size but also between groups of borrowers of different sizes to measure economic capital requirements. To summarize, estimating risk parameters by using real data and adopting a multifactor approach also allows us to assess whether the difference in capital requirements covering eligible loans and non-eligible loans is consistent with the capital relief induced by the Supporting Factor.

For being eligible to the Supporting Factor, SMEs must have (i) an annual turnover lower than e50 million and (ii) a total outstanding amount of credit with a given banking group lower than e1.5 million. We take advantage of this setting to estimate the effect of the SF on the provision of credit to targeted SMEs (we refer to this part as the credit analysis hereafter) using a difference-in-differences approach. We first restrain the sample to SMEs, identified as the firms with a turnover lower than e50 million. Then, we define a treatment group made of eligible exposures from pairs of bank-firm with a total outstanding amount below e1.5 million and a control group made of (ineligible) exposures from the remaining pairs of bank-firm. We then compare the evolution of the outstanding amount of credit of eligible and ineligible exposures after the reform (vs. before the reform). We deal carefully with possible identification issues by running several robustness checks.

After having estimated the average effect of the SF on the credit supply to eligible firms, we then investigate the dynamics of this effect over time. Did the banks respond immediately to the reform or, on the contrary, has the SF become increasingly effective quarter after quarter ? To that end, we estimate the effect of the SF within each quarter, both before and after the reform. In doing so, we not only gather information about the evolution of the effectiveness of SF over time but we also test a fundamental assumption of the difference-in-differences estimator: the parallel trend assumption. In a third time, we investigate possible sources of heterogeneity in the effect of the SF. We focus on two crucial firms' characteristics: the size and the riskiness of firms. We classified exposures according to the turnover and the Banque de France rating of firms in the pre-reform period (to avoid any endogeneous feedback loops) and we test whether the effect of the SF is the same for the various groups of firms.

Finally, we explore possible non-linearities by estimating the effect of the SF conditional on the ex ante size of the exposures. For this purpose, we classify exposures into three buckets based on their average outstanding amount computed over the pre-reform period and we estimate an effect of the SF separetely for each of these buckets. As we will show, there are some reasons to think that all exposures have not benefited in the same way from the SF, owing to the design of the SF.

Our main results can be summarized as follows. Regarding the risk analysis, the computation of economic CRs based on a multifactor framework and its comparison with Basel III regulatory CRs do confirm that the CRs should be lower for SMEs than for large corporates. We first find that the largest firms are the most exposed to systematic risk, i.e. they are the most exposed to general economic conditions even if their default rates are low. Second, the results of the estimation confirm the potential for diversification provided by the presence of exposures on SMEs in the total bank loans portfolio: while the classes of medium-sized and large firms are highly correlated with each other, we find negative or very small correlations across small firms and medium-sized firms, on the one hand, and large firms, on the other hand.

The higher values of the ratio of regulatory CRs to economic CRs for small size classes reflect an overestimation of SMEs risk relative to large corporates in the regulatory frameworks, even after taking into account the SF. Overall, after considering the uncertainty surrounding these estimates and adopting a conservative approach, we find strong evidence that the SF is consistent with the difference in economic CRs between SMEs and large corporates.

Regarding the credit analysis, we find evidence showing that the SF has been effective in supporting bank lending to targeted SMEs. First, we show that eligible exposures have increased by 5% to 10% on average as compared to ineligible exposures after the implementation of the SF (vs. before the reform) depending on the specification. In the most conservative estimation including group specific trends, we still find that the SF has boosted eligible exposures by 2%. This average effect is corroborated to various robustness checks. Then, we find that the magnitude of the effect of the SF has increased over time: the effect is almost zero in the first year after the entry into force of the SF but it has then intensified to reach a magnitude of 8% to 10% two years after the entry into force. At the same time, we do confirm that the trends of eligible and ineligible exposures did not diverge in a significant way before the reform. This test is an important validation of our empirical strategy based on the difference-in-differences estimator.

Concerning the possible sources of heterogeneity, we first find that the effect of the SF seems much stronger on eligible exposures of small and, most notably micro enterprises (i.e. firms with a pre-reform turnover lower than e7.5 million) than on eligible exposures of medium-sized SMEs.5 Then, we find convincing evidence showing that exposures of SMEs with a no Banque de France credit rating tend to be more affected by the implementation of the SF than exposures of SMEs considered as safe or risky based on their credit rating. This last result tends to indicate that safe firms are only rarely facing credit constraints (and as a result, they do not really benefit from the SF) while risky firms remain too risky from the point of view of banks, even after taking into account the capital relief provided by the SF. Hence, banks tend to target the firms with no credit ratings that have default rates much lower than firms considered as risky but that are more likely to face credit constraints than safe firms. These two results provide interesting insights regarding the effectiveness of the SF.

Finally, our analysis reveals that the effect of the SF is non-linear. Indeed, one might suspect that the impact of the reform could be limited by the loss of the capital relief as soon as the exposure breaches the e1.5 million threshold. Since the SF applies to the total outstanding amount (i.e. the existing stock of credit) and not just to the new loans, not only banks can benefit from the SF without extending any additional loans but also, as increasing the outstanding amount of loans makes them closer to the threshold, the risk to pass above the threshold and, as a result, to lose the CRs discount on the total outstanding amount increases. Given that, banks may have incentives to limit the growth of exposures that are "too large", meaning those exposures that are originally in the vinicity of the threshold of eligibility.

We test this hypothesis by estimating the effect of the SF separately on small, medium and large exposures. We find that eligible exposures classified as small (i.e. exposures with an average outstanding amount of credit over the pre-reform period lower than e500,000, namely those for which an increase is unlikely to lead to overcome the threshold) have strongly benefited from the SF. In contrast, medium and large eligible exposures have decreased (as compared to the ineligible exposures) in the post-reform period. Overall our results suggest that this disincentive feature of the reform is at play in our data.

Our paper contributes to three strands of the literature. First, our work relates to the literature exploring empirically the relationship between CRs and lending. This relation has been recently reassessed exploiting the strong capital shortfall induced by the financial crisis, the recent changes in regulation and an easier access to granular data that allows us to control for demand and supply shocks. Recent contributions tend to support a negative impact of higher CRs on credit distribution (Aiyar et al. (2014), Behn et al. (2016), Jimenez et al. (2017) or Fraisse, Lé, and Thesmar (forthcoming)). In contrast to these studies that generally consider the impact of tighter CRs, our paper exploits a policy experiment explicitly designed to support credit growth through a targeted decrease in CRs. So far, to the best of our knowledge, this is the first study that analyses the consequences of reduced regulatory CRs, which can provide a more comprehensive understanding of this relationship because there are no reasons to think that the effect is symmetric. For instance our results might provide guidance to macroprudential authorities, when relaxing macroprudential buffers targeting SME lending.

Our paper is also related to the literature about the risk assessment of banks credit portfolio. From the seminal work of Merton (1974) having shaped the credit risk structural approach, a lot of progress has been made regarding credit risk assessment. In particular, this approach underlies the Basel II/III regulatory capital requirements formula (Gordy, 2003). However, the latter does not take into account borrowers' heterogeneity and possible concentration effects coming from potentially correlated defaults across borrowers whose financial situation is driven by "sectoral" systematic risk factors. We contribute to the existing literature by explicitly accounting for concentration/diversification effect using a multifactor framework where firm size acts as risk factors, a choice that is motivated by the SF issue. 6 Lastly, despite its consequences on the regulatory capital of the European banks, the evaluations of the effectiveness of the SF are scarce. To our knowledge, Mayordomo and Rodríguez-Moreno (2018) is the first and unique academic contribution to assess the reform. Our work complements their analysis by considering the French banking sector over a long time period covering the implementation of the reform, exploiting a very granular data set and analysing the effects of the reform both on the credit distribution and on the risk taken by the banks. 7 Overall, our results support largely their findings.

Our paper contributes to these different strands of the literature, providing a new perspective, with both an analytic examination of the consistency of the CRs associated with SME risks and an empirical evaluation of the effective impact of the SF on the credit supply. The remainder of the paper is organized as follows: Section 2 presents the institutional background and the detailed definition of the SF. Section 3 describes the data and provides descriptive statistics. Section 4 details the methodology used for our empirical analysis. Section 5 is devoted to the presentation of our findings, some alternative specifications and related comments. Section 6 provides concluding remarks.

Institutional Framework

SMEs are generally considered as a key driving force for job creation and economic growth in Europe. Furthermore, bank credit is a crucial source of finance for SMEs (Beck and Demirguc-Kunt (2006)) and this is why bank lending to SMEs is a salient political issue in Europe. 8 In 2013, the European Commission assessed that the transition from Basel II to Basel III would lead to an increase in CRs from 8% to 10.5% for the average European bank. Against this background, the European legislator decided to introduce a deduction in the CRs for exposures to SMEs when transposing the Basel III standards into EU law. 9

6 With respect to this strand of literature, our paper has also the benefit to exploit longer time series of default rates of SMEs.

7 Mayordomo and Rodríguez-Moreno (2018) use data from the Survey on the Access to finance and Enterprise conducted by the ECB and the European Commission to assess the effects of the SF on the European banking sector. The authors use as key indicator to gauge the reform efficiency the question in the survey asking SMEs' managers whether they applied for a loan and whether their application was fully or partially rejected. The authors also run an additional experiment focusing on the exposures around the e1.5 million threshold (e.g. between e1 million and e2 million) and around the introduction date of the SF in Spain (from August to December 2013). In addition to differences in methods and data between their approach and ours, it is worth noting that the French and Spanish banking sectors differ significantly both on their structure -the French one being much more concentrated-and on how they fared through the financial crisis -the Spanish one being confronted with the burst of a housing bubble. Those differences also make additional analysis on the SF impact to our opinion worthwile.

8 "SMEs are the backbone of the European economy, providing a potential source for jobs and economic growth", European Commission, Regulation of the European Parliament and of the Council, 2015.

9 See Article 501 of the Capital Requirements Regulation (CRR).

This deduction aims at offsetting the expected increase in CRs due to the transition from Basel II to Basel III for SMEs. It reflects the policy willingness and the general concern that SMEs should not suffer from the consequences of financial crises they are not responsible for. Therefore, under certain conditions, CRs associated with SMEs loans will be reduced by 23.81% 10 or subject to a so called Supporting Factor of 0.7619.

The regulation comes into force the 1st of January, 2014. The new regulation defines precisely the SMEs targeted by this CRs relief. Banks can alleviate their CRs for credit risk associated with a given exposure by multiplying these CRs by 0.7619 provided that 11 :

-the exposure is included either in the retail, corporate or secured by mortgages on immovable property regulatory portfolio, -the borrower/debtor is a firm with a turnover below e50 million. (See Appendix 10.1

for more details about the definition of SMEs), -the total amount owed to the institution and parent undertakings and its subsidiaries, including any exposure in default but excluding claims or contingent claims secured on residential property collateral, does not exceed e1.5 million (See Appendix 10.1 for more details about the total amount owed to the institution).

Note also that :

-exposures in default shall be included for the purpose of determining the eligibility, but excluded from the application of the SF, -these precited criteria should be met on an ongoing basis.

Overall, it is expected that the implementation of the SF leads banks to provide relatively more credit to eligible exposures/SMEs than to ineligible exposures/SMEs. 12 Indeed, for each additional euro of credit granted, the relative cost in terms of CRs is 24% lower than before when this additional euro of credit is granted to eligible SMEs as compared to ineligible SMEs or to non-SMEs. At the margin, banks have thus incentives to increase their lending to eligible SMEs as compared to any other firms.

Nevertheless, as soon as 2014-Q1, the alleviation in CRs applies to the total outstanding amount of credit, and not just at the margin i.e. on newly granted loans. That means that any banks whose exposures are eligible will immediately benefit from a 24% discount in CRs on their actual stock of eligible exposures, whatever be their response to the reform. Consequently, we cannot exclude that banks also use the relaxation of capital constraints (resulting from the application of the SF to the (actual) outstanding amount of eligible 10 The magnitude of this discount was calibrated from the anticipated increase in CRs for the average European bank : 1-8%/10.5%

11 https://www.eba.europa.eu/regulation-and-policy/single-rulebook/interactivesingle-rulebook/-/interactive-single-rulebook/toc/504/article-id/4902;jsessionid= 1BB645BAF83F701B15ABF1B26949F02A

12 Formally, a firm per se is not considered as eligible or ineligible. Only the exposures are deemed as eligible/ineligible. For instance, a given SMEs can be eligible to the SF with a given bank and, at the same time, ineligible with the other banks. For the sake of simplicity, in the present paper, we will nonetheless use indiscriminately the terms eligible firms and eligible exposures. credit) to provide more credit to non-eligible SMEs, to large companies or even to invest in other classes of assets that are far from the objective of improving credit supply to SMEs.

In this case, the ineligible exposures will also increase following the implementation of the Supporting Factor. However, if this is true, this mecanism acts as a downward bias in our estimation of the effect of the Supporting Factor. Said differently, it will make more difficult to find a positive effect of the SF in our setting as it will become clear in the section 3.

Data

The French credit register

We use the French national credit register maintained by the Banque de France ("Centrale des risques"). This register reports all the credits granted by any resident credit institution as well as some specific institutions like the Sociétés de financement (which are entitled to make credit but not to receive deposits on demand) or the investment firms providing credit. The population of borrowers/debtors includes any resident and nonresident legal entity (firms, local governments and administrations) as well as any natural person having a professional activity operating nationwide. Firms are defined here as legal units (they are not consolidated under their holding company when they are affiliated with a corporate group) and identified by a unique national identification number (called a "SIREN" number). They include single businesses, corporations, and sole proprietorships engaged in professional activities. A bank has to report its credit exposure to a given firm as soon as the total outstanding exposure on this firm is larger than e25,000. The credit register provides quarterly information regarding the type of credit granted among 12 distinct types of loans belonging to 6 broad categories (see Appendix 10.2).

The credit register also provides detailed information regarding the size and the creditworthiness of borrowing firms when they have a turnover above e0.75 million or a total outstanding amount higher than e380K. Indeed, the Banque de France estimates internally its own credit ratings for a large population of resident firms (about 300,000) and in particular for small firms that are generally not under the scope of the private rating agencies. The Banque de France has been recognized as an external credit assessment institution (ECAI) for its company rating activity. This enables credit institutions to rely on this Banque de France rating to calculate their regulatory capital requirements. The Banque de France has also been recognized as an ICAS -In-house Credit Assessment System -under the General Documentation governing the Eurosystem's monetary policy operations. Therefore, ratings are also used for refinancing bank loans in the Eurosystem Credit Assessment Framework (ECAF).

Both the credit and the risk analyses are run using this common dataset but they are subject to slightly different restrictions that we explain in the following subsections.

Risk analysis

Regarding the risk analysis, we restrict the sample to firms that have at least one exposure reported in the French credit register (see Appendix 10.3 for more details about the reporting requirement of this register). This population constitutes more than 3 million of observations over the period. We restrict the dataset to the years going from 2004 to 2015, i.e. 66 quarters, and to firms having a Banque de France rating. The sample is representative of the French businesses population. For instance, the Banque de France indicates that the database containing all accounting information used to assess the creditworthiness of firms (Centrale de Bilans, Fiben) represents at least 75% of the turnover of the population of French firms.13

Credit analysis

Regarding the credit analysis, we are no longer limited by the availability of the credit risk rating. We run the analysis over the period 2010-2016, i.e. 4 years before the entry into force of the SF (the pre period) and 3 years after (the post period). For the purpose of identifying eligible exposures, we aggregate them at the firm-banking group-quarter level using an auxiliary dataset allowing us to identify banking groups and their affiliates. As we suspect that the credit effect of the SF might be observed on very small firms, we do not restrict the sample to firms with available information regarding the Banque de France credit rating.

We limit our dataset to small and medium-sized enterprises (SMEs) and exclude large companies with a turnover higher than e50 million. The aim there is to contrast the most comparable firms, using the eligibility threshold as the main component of our differencein-differences specification (see 4.2). We could have included non-SME firms such that the eligibility criterium would have been two-folded: we could have compared SMEs and non-SMEs with an outstanding amount of loan lower than e1.5 million. We do not proceed in this way because firms with a turnover higher than e50 million are significantly different from SMEs, especially regarding their relation with bank lending (in particular, they have an easier access to a range of substitutes to bank credit). This is why we limit the sample to SMEs and we discriminate across them using the threshold for eligibility to the SF. We provide additional information about the restrictions made to the dataset in Appendix 10.4.

We also limit the sample of firms to independent firms. Anecdotical evidence indicates that banks have sometimes many difficulties to identify precisely the scope of consolidation of companies. By limiting the sample to independent firms, we overcome this uncertainty.

As a result, we end up with an extremely large, unbalanced, dataset of more than 18.5 million of observations corresponding to 1,093,817 unique firms over 28 quarters. Overall, this dataset has several advantages. First, it can be considered as quasi-comprehensive given the low reporting threshold. We only miss few loans to very small firms that are economically insignificant at the aggregate level. Second, we have a long time series at a quarterly frequency. It thus covers a sufficient period before and after the implementation of the SF, which enables us to explore the effects of the reform while allowing banks to take time to react and adjust their lending. Third, we have very granular information, even if we do not have a flow information (new credit issuance), but a stock information (outstanding credit amount). This information at the firm-bank-quarter level enables us to distinguish a given firm-bank pair through many dimensions (e.g. time series, cross-section, banking product) that we exploit in this paper.

Empirical Strategy: methodology

Assessing the risk consistency of the Supporting Factor

To assess the consistency of regulatory CRs -including the SF-with SMEs intrinsic credit risk, we compare regulatory CRs with CRs computed by using a more comprehensive economic approach provided by a multifactor portfolio credit risk model. 14 This model grounds on the structural credit risk approach, as devoted by Merton (1974). In our implementation of this model, firm size acts as a systematic risk factor. Even if the regulatory formulas for corporate exposures introduce some adjustments related to firm size, the regulatory models do only consider a single general risk factor and they do not account for the impact of other systematic risk factors (Gordy, 2003).

It is important to emphasize why the multifactor model can be used as a benchmark to check the consistency of the capital deduction induced by the Supporting Factor on SMEs loans with the SMEs risk. Therefore, in this section, we first explain why the CRs measures derived from a multifactor framework can be used as benchmarks. Then, we provide a short description of the multifactor framework and describe how we apply it for our risk analysis.

The economic capital as benchmark for CR measurement

There is a relationship between regulatory CRs and economic CRs derived from a multifactor model. The Basel II/III risk weight regulatory formulas were calibrated using the standard Asymptotic Single Risk Factor (ASRF) model (Gordy, 2003). In this framework, bank's total CRs are computed by using two parameters which refer to firm's individual risk, the probability of default (P D) and the loss given default (LGD), and a third parameterthe asset correlation R -which measures the sensitivity of borrowers to a common single systematic risk factor. So, regulatory Risk Weighted Assets (RWAs) are consistent measures 14 See Appendix 10.5 for more details about this methodology.

of credit risk. However, two calibration choices determine potential differences between regulatory and economic CRs, which justifies to compare the two types of measures. Firstly, in the regulatory formulas, asset correlation R is entirely determined by the P D. But, as measures of the sensitivity of loans to a risk factor, they should in fact vary from one portfolio to another one, depending on its composition. In practice, Basel II/III regulation provides banks with the formulas to compute R, instead of leaving them computing this risk parameter using internal information. Consequently, a main difference between regulatory and economic CRs comes from the value of assets' correlations. Secondly, in the ASRF model, there is only one single risk. However, borrowers' financial health is linked to multiple sources of credit risk which are more or less specific to the risk segment to which they belong. Consequently, risk measurement should account for borrowers' heterogeneity.

Taking account for borrowers' heterogeneity obliges to expand the standard single risk factor model and to adopt a multifactor framework. Moreover, a multifactor model allows the detection of potential concentration (diversification) effects coming from the strong (weak) dependence of borrowers to risk factors which are specific to their own risk segment. In case of realization of unfavorable value of one systematic risk factor, the number of defaults will increase and losses will climb to higher levels. In such a case, the contribution to the portfolio's segment which is exposed to this risk factor will raise, inducing an increase in total losses. More generally, if the sensibility of exposures to the systematic risk factor which is specific to their segment is high, the relative contribution of this segment to the portfolio's total losses will be high, which corresponds to a situation of credit risk concentration in that segment. So, in a portfolio composed of several segments, using a multifactor model allows us to compute the marginal contribution of each segment to total losses and to observe either the impact of this segment on the concentration of losses or, on the contrary, the role the segment plays in the diversification of the portfolio credit risk.

In practice, this marginal contribution can be expressed under the form of a capital ratio by relating economic CRs needed to cover potential unexpected losses produced to this segment (computed at a given percentile -for instance 99.9 percent -of the probability distribution function of losses) to total exposures of the segment. In this way, we can assess portfolio's concentration and diversification in terms of capital ratio as a common metrics, showing how size factors could contribute to increase or decrease the level of CRs relative to the level given by a single risk factor model.

A short view of the multifactor model

The multifactor model we use in this paper belongs to the class of structural credit risk models. 15 It is an extended version of the ASRF model. The extension of the ASRF framework consists in introducing risk factors which can be linked to observable characteristics of borrowers and vary across groups of borrowers. As mentioned before, such an extension improves substantially the computation of the dependency structure across exposures in a loans portfolio, by allowing us to account for potential credit risk concentration which is linked to borrowers' heterogeneity. Here, we assume that firm size reflects borrowers' heterogeneity and we expand the ASRF model by considering a latent risk factor for each size class.

To compute economic capital in this framework, we proceed in two steps. First, we compute portfolios' main risk parameters and in particular the dependence structure among exposures measured by the matrix of variance-covariance within each size class and between classes. Then, we use Monte-Carlo simulations to build the probability distribution function of losses, determine the total portfolio potential losses and compute the marginal contribution of each size class to potential losses, which measures the buffer of economic capital required to cover the losses in each size class.

To estimate risk parameters, we use an econometric model that belongs to the class of generalized linear mixed models (GLMMs) that combines fixed and random effects for observable and (latent) unobservable factors. Indeed, as shown by Frey and McNeil (2003) and McNeil and Wendin (2007a), the GLMM model implements in a coherent way the Merton latent factor default modeling approach, in which the default occurs when the value of the firm's assets becomes smaller than the value of its debt, that is, because firm's assets values are difficult to observe, when the value of a latent variable describing the financial situation of the firm -which depends on the realization of a set of risk factors -crosses an unobservable threshold which determines the default. In this framework, the default threshold is considered as the fixed effect of the GLMM. The systematic risk factors are supposed to be latent factors and then correspond to the random effects of the GLMM. Here, random effects are linked to the firm size segmentation of the portfolio. Thus, in this framework, the default rate is modeled as :

P (Y ti = 1|γ t ) = Φ(x ti µ r + z ti γ t ) (1)
in which Y ti the default indicator variable of borrower i at time t depends on (i) a fixed effect measured by the borrower's internal rating µ r , and (ii) random effects γ t which are related to a set of factors corresponding to the size segmentation of the portfolio. x ti is a (1 × R) vector of dummies that defines the rating of borrower i during time period t, while z ti is the design matrix of the random effects. Taking firm's credit rating histories to build time series of rates of default by portfolio segments, we get estimates of portfolio's credit risk parameters in a multi-factor context. The GLMM model provides estimates of default thresholds considered as fixed effects and covariance matrixes of a set of latent random effects corresponding to the set of systematic size factors. The estimation of such parameters allows the computation of economic capital as buffer of losses.

Once the credit risk parameters are estimated, the distribution of losses at the portfolio level is computed by Monte Carlo simulations, with each simulated realization of the systematic risk factors being converted into a conditional default probability at the rating/size segment level and, finally, into conditional expected losses at the portfolio level. Various quantiles based on risk measures such as Value-at-Risk (VaR) can then be retrieved from the simulated distribution of portfolio-wide losses. The computation of the portfolio's valueat-risk (VaR) and marginal risk contributions are made by using a methodology proposed by Tasche (2009), which grounds on an importance sampling based simulation of expected conditional losses. This methodology has the advantage to take into account the impact of borrowers' heterogeneity on economic capital charges and capital allocation.

Identifying the effect of the Supporting Factor on credit supply 4.2.1 The difference-in-differences framework

To assess the effectiveness of the Supporting Factor regarding the provision of credit to SMEs, we rely on the difference-in-differences framework. In this setting, we compare a treated group composed of all individuals affected by the reform to a control group made of comparable individuals non-affected by the reform. In our case, the sample is made of French SMEs, i.e. firms with a turnover lower than e50 million (see Section 3 and Appendix 10.3 for more details about the sample selection). The treatment group will then refer to exposures/SMEs eligible to the SF and the control group will refer to exposures/SMEs non-eligible to the SF.

As described in Section 2, a pair of bank-firm {b, f } (or more precisely an exposure {b, f }) is considered as being eligible to the SF when the total eligible outstanding amount of credit from bank b toward firm f is lower than e1.5 million. 16 More precisely, we carefully dissociate the exposure used to assess the eligibility to the SF, denoted Lf,b,t , from the exposure that will benefit from the CRs deduction, denoted L f,b,t . 17

Starting from the first quarter of 2014, all exposures L f,b,t eligible to the SF (i.e. exposures where Lf,b,t <e1.5 million) have immediately benefited from the 23.8% discount in CRs. We denote by EL f,b,t the variable that indicates the eligibility status of bank b when lending to firm f at quarter t:

EL f,b,t =    1 if Lf,b,t ≤ e1.5 million 0 if Lf,b,t > e1.5 million
(2) 16 We could have included non-SMEs firms and the eligibility criteria would have been two-folded : we could have compared SMEs and non-SMEs with an outstanding amount lower than e1.5 million. We do not proceed in this way because we think that firms with a turnover higher than e50 million are very different from SMEs, especially regarding their relation with bank lending (in particular, they have access to a lot of substitutes to bank credit). This is why we limit the sample to SMEs and we classify them as treated/non-treated according to the eligibility threshold.

17 See Appendix 10.1 for more details about the differences between the two quantities.

However, under this definition, a pair of bank-firm {b, f } may switch from the treated group to the control group (and vice versa) from one quarter to another as the amount Lf,b,t used to assess the eligibility to the SF fluctuates over time. Hence, we are facing an important composition issue that could affect the stability of our treatment/control groups. To overcome this issue, we decide to keep only exposures {b, f } whose eligibility status is stable over the whole period, i.e. we keep all exposures from pairs of bank-firm {b, f } that are continuously eligible or ineligible to the SF over the entire period. We thus define ĒL f,b as follows :

ĒL f,b =    1 if ẼL f,b,t = 1 ∀ t 0 if ẼL f,b,t = 0 ∀ t (3)
This restriction is rather conservative and does not threaten our identification strategy. Indeed, the restriction that we impose leads to exclude (i) eligible exposures that would become at some point ineligible and (ii) ineligible exposures that would become eligible at some point. In the first case, these are fast-growing "treated" exposures that pass above the e1.5 million threshold at some point. Ignoring them tends to reduce the intensity of the response of the treated exposures to the treatment. In the second case, these are likely to be exposures classified in the control group that tends to decrease significantly over time until they pass below the threshold. By ignoring them, the control group as a whole has a dynamics more favorable than it would be otherwise. In both case, this restriction creates a downward bias in our identification strategy, i.e. it makes more difficult for us to detect an effect of the SF. 18 We finally denote by P ost t the variable indicating the period where the SF has entered into force:

P ost t =    1 if t ≥ 2014Q1 0 if t < 2014Q1 (4)

The baseline specification

The goal of the credit analysis is to test whether the entry into force of the SF in 2014-Q1 has fostered credit supply of banks to eligible SMEs (as compared to ineligible SMEs). For this purpose, we estimate the following classical difference-in-differences specification :

ln(L f,b,t ) = α + β • ĒL f,b • P ost t + γ • ĒL f,b + θ • P ost t + µ b,t + ω b + ρ f + b,f,t (5)
where:

18 Alternatively, we could be tempted to classify exposures based on their status in the pre-reform period. However, in doing so, we would have created an upward bias. Indeed, in this case, we could misclassify an exposure as "treated" in post (because it is truly a "treated" exposure in pre) while it is not. Such an exposure would have grown significantly between the two periods. As a result, we would have overestimated the dynamics of the group of treated exposures. The opposite is true for the exposures classified in the control group.

-L f,b,t refers to the total outstanding amount of loans granted to the firm f by the bank b at the quarter t µ b,t , ω b , and ρ f denote respectively bank-time fixed effects, bank fixed effects and firm fixed effects (FEs)

In these regressions, the coefficient of interest is β. It indicates to which extent the credit supply evolves differently for eligible pairs of bank-firm {b, f } relative to ineligible pairs of bank-firm {b, f } after the implementation of the SF compared with the pre-implementation period. We gradually saturate the regressions with firm, bank and time fixed effects to control for possible confounding factors. In some specifications, we even include bank-time fixed effects to control for bank funding shocks among other things (think to the TLTRO for instance). In all regressions, we systematically control for the Banque de France rating and the size of the firms as well as their industrial sector and their geographic location. We cluster our standard errors at the firm level, i.e. we allow for some dependency in the standard errors within firms but we consider that these standard errors are i.i.d across firms (Abadie et al., 2017).

Dynamics over time and firm characteristics

Testing the parallel trends assumption We identify the effect of the SF on the credit supply using a difference-in-differences framework. An important identifying assumption of the difference-in-differences setting is the parallel trends assumption. We could test this identifying assumption by running a dynamic version of the baseline specification (5). Rather than identifying the effect of the SF on the entire post-reform period (as compared to the pre-reform period), we now estimate the differences in the (log of) outstanding amount of credit between eligible and ineligible pairs of bank-firm {b, f } within each quarter. The specification writes as follows :

ln(L f,b,t ) = α + t β t • ĒL f,b • 1 t + t γ t • ĒL f,b + t θ t • 1 t + µ b,t + ω b + ρ f + b,f,t (6)
where 1 t denotes a dummy taking the value of 1 in quarter t and zero otherwise. The set of coefficients {β t } estimates the effect of the SF on the provision of credit within each quarter t. We thus expect to find the {β t } to be indistinguishable from zero for any quarter t in the pre-implementation period. In this case, we can conclude that the conditional dynamics of the credit received by eligible and ineligible exposures {b, f } are not significantly different in the pre-reform period, i.e. our setting satisfies the parallel trends assumption.

By constrast, we expect to find the {β t } to be significantly positive for any quarter t in the post-implementation period. Above all, this specification is also very informative regarding the dynamics of the reform : does the magnitude of the effect tend to increase over time or, in constrast does this effect overshoot and then fade out after several quarters?

Heterogeneity and firm characteristics

We then investigate to which extent the magnitude of the effect of the SF varies along with firm characteristics. We examine three dimensions: (i) the size of the firm, (ii) the riskiness of the firm and (iii) the size of the exposure. For this purpose, we run slightly modified versions of the specification (5). First, firms are sorted according to their size and their riskiness. 19 We define a firm as :

-Risky f when its Bank of France rating is below to the notch 4 (this range of ratings corresponds to the speculative grade category) -Saf e f when its Bank of France rating is above or equal to the notch 4 (this range of ratings corresponds to the investment grade category) -U nknown f when its Bank of France rating is equal to the notch 0 ("No unfavourable information gathered")

In Table 8, we present quasi default rates computed for each Banque de France ratings. For this purpose, we use two definitions of default : (i) when an SME receives the Banque de France rating for which at least one trade bill payment incident has been reported and (ii) a more restrictive definition corresponding to SME subject to insolvency proceedings (recovery or judicial liquidation proceedings). The default rates are computed at a one year horizon. Interestingly, we observe that the default rates increase monotonically and continuously as we downgrade in the rating scale. The rating corresponding to the situation where the Banque de France has "no unfavourable information" displays a default rate that stands between the default rates of the investment grade category and the default rates of the speculative grade category. As a result, we could consider this category as less risky than speculative grade but more risky than investment grade.

Then, regarding the size of firms, we define a firm as :

-M edium f when its turnover is higher than e7.5 million but lower than e50 million -Small f when its turnover is higher than e1.5 million but lower than e7.5 million -M icro f when its turnover is lower than e1.5 million Our size variable allows us to consider several thresholds (e0.75 million, e1.5 million, e7.5 million, e15 million . . . ). We have built our size categories to follow the most closely the traditional decomposition used by both the OECD and the European Commission20 of the population of SMEs into micro (Turnover< e2 million), small (turnover in [ e2 million; e10 million]) and medium-sized enterprises (turnover in [ e10 million; e50 million]).

We define these variables based on the characteristics of firms in the pre-reform period because both size and riskiness of firms can be affected by their indebteness. However, the size and the creditworthiness of firms are likely to vary from one quarter to another. Rather than using the value at a specific quarter in the pre-reform (2013-Q4 for instance), we prefer to take the mode of the size and the riskiness, i.e. we classify the firm as small or risky based on the size bucket and the Banque de France rating that are the most frequent in the pre-reform period. By doing so, we nonetheless lose few observations.21

Finally, we also classify pairs of bank-firm according to the size of their initial exposure. Indeed, we suspect that the incentives provided by the reform become increasingly mixed and ambiguous as the firm's outstanding amount of credit approaches the threshold of eligibility: to avoid loosing the entire capital relief when breaching the e1.5 million threshold, banks may become more reluctant to provide additional credit to firms with an initial outstanding amount of credit too close from the threshold. To test this hypothesis, we classify pairs of bank-firm {b, f } as :

-Small b,f when the average pre-reform total "eligible" outstanding amount of credit is lower than e500,000 -M edium b,f when the average pre-reform total "eligible" outstanding amount of credit is comprised between e500,000 and e1 million -Large b,f when the average pre-reform total "eligible" outstanding amount of credit is comprised between e1 million and e1.5 million

We favor this ad hoc classification over a breakdown by quartiles because the very unbalanced nature of the distribution of initial outstanding amount of credit (i.e. there is a disproportionate share of very small exposures, those with an outstanding amount of credit lower than e500,000) would prevent us to analyse accurately the effect of the SF around the threshold. For instance, the cut-off point for the fourth quartiles is just at e155M, far below the threshold.

After that, to test each of these sources of heterogeneity, we run a generic specification where we interact all the terms of the baseline equation ( 5) with the various dummy variables built just before22 :

ln(L f,b,t ) = α + k β k • ĒL f,b • P ost t • 1 T ype=k + k γ k • ĒL f,b • 1 T ype=k + k θ k • P ost t • 1 T ype=k + µ b,t + ω b + ρ f + b,f,t (7)
Overall, these specifications allow us to test whether the magnitude of the effect of the SF is stronger or weaker depending on (i) the size of the firm f , (ii) the riskiness of the firm f , as measured by the Banque de France rating, and (iii) the size of the initial exposure between bank b and firm f .

Results

The effect of the Supporting Factor on bank risk portfolio

To assess the consistency of regulatory CRs for SMEs, and by the way the effectiveness of the SF, we compare, for each size class, the ratio of CRs measured when using the multifactor model parameters with the ratio of CRs given by the regulatory Basel II/III formulas. The economic CRs are computed by using the multifactor risk model presented above. They measure the marginal contributions of the different firm size classes to the total potential losses on a comprehensive bank business loans portfolio. The multifactor model provides a more comprehensive measure of portfolio credit risk, taking into account borrowers' heterogeneity and possible concentration or diversification effects coming from the interactions between systematic risk factors associated to firm size classes. Thus the comparison between the two types of CRs provides an information about the possible over(under)estimation of effective credit risk by the regulatory formula and the possible compensation provided by the SF.

Recall that, in order to estimate the model, we built a portfolio containing the sum of the business loans held by French banks on each firm registered in the French Credit Register. To compute rate of defaults and other portfolio risk parameters by size classes, we used each firm's history of quarterly ratings (including default) in the ratings system of the Banque de France. To compute CRs, we assume a 45% Loss Given Default (LGD) and a 99.9% quantile of the probability distribution function of losses. These parameter values are those we find in the Basel II/III regulatory framework. 23 All models are estimated using annual default rates. Since we are ultimately concerned with the calibration of CRs, we consider not only the credit risk parameters estimates but also CRs dependent on these estimates. More precisely, we compare in each size class the ratio of CRs measured when using the multifactor model parameters with the ratio of CRs given by the regulatory Basel III formulas.

In this section, we first assess the level of sensitivity of firms to systematic risk, depending of their size, and the potential for diversification across size segments in the portfolio. Table 1 displays the random effect variances -which measure the exposure to systematic riskand the correlations of the random effects -the correlations between size systematic risk factors -provided by the GLMM model. 24 More precisely, it shows that the largest firms are the most exposed to systematic risk, i.e. are the most exposed to general economic conditions, even if their default rates are low. Additionally, a joint equality test across random effects variances rejects the null hypothesis. Moreover, the random effects across the classes of medium-sized and large firms are highly correlated, with correlations ranging between 95% and 100%. However, correlations across small firms and medium-sized, on the one hand, and large firms, on the other hand, are negative or very small, showing a potential for diversification effects between these size classes. Finally, what appears clearly from the results of the estimation is the potential for diversification provided by the presence of exposures on SMEs in the total bank loans portfolio.

The computation of economic CRs at the size level allows taking into account all different estimated dimensions of credit risk in a consistent way. Table 2 allows to compare the value of the economic CRs with the level of the regulatory CRs, under two regulatory regimes: the standard Basel II/III IRB regime and the CRD IV/CRR regime including the SF impact. 25 Table 2 shows an increasing relationship between size and the three distinct CRs, reflecting the growing sensitivity to systematic risk factor (a general factor in the regulatory models, the size risk factor specific to each size class in the economic model) with firm size. Moreover, the level of the two regulatory CRs is far superior to the level of the economic CRs, showing a potential overestimation of CRs by the Basel II/III regulatory formulas or the CRD IV/CRR regulatory formulas with a Supporting Factor. Here, we consider the CRs on large corporates (i.e. corporates with a turnover of more than e50 million) as a benchmark, which could be motivated by the fact that the SF introduces a deduction of CRs for SME loans with respect to the lack of deduction of CRs for larger corporates. We compute the ratios of the two regulatory CRs relative to the economic CRs (last two columns of Table 2).

The comparison of the values of these ratios between size classes allows us to determine whether the size dependence of the regulatory CRs is consistent with that of the estimated economic CRs. The results confirm that the higher values of the ratios for small size classes reflect an overestimation of SMEs risk relative to large corporates in the two regulatory frameworks. In addition, the results also show the CRs reduction provided by the imple-25 Under the Basel II/ III regime the regulatory CRs (for exposures on corporate) are computed accordingly to the following formula :

RW = LGD•N (1-R) -0.5 •G(P D)+ R (1 -R) 0.5 •G(0.99) -P D•LGD •(1-1.5•b) -1 •(1+(M -2.5)•b)•12.5•1.06 (8)
where: RW denotes the risk-weight or the capital requirements, R the correlation, b an adjustment factor, S the total annual sales in millions, P D the probability of default, LGD, the loss given default, M the maturity, N (x) is the cdf of a normal distribution N (0, 1) and G(z) is the reciprocal of this cdf. Under the CRD IV/CRR regime, the RW is multiplied by the Supporting Factor for the eligible firms. For a conservative approach, every firm of a size class is given the upper bound of the turnover sales. For instance, firms belonging to the [e7.5M-e15M] class are given a e15 million annual total sale. A similar IRB formula is provided for exposure on "other retail", i.e. on firms with exposures lower than e1.5 million.

R = 0.12 • (1 -e (-50•P D) ) (1 -e (-50) ) + 0.24 • 1 - (1 -e (-50•P D) ) (1 -e (- 50
mentation of the Supporting Factor. The ratio of the regulatory CRs to the economic CRs is lower for the CRD IV/CRR model than for the Basel II/III model. But, despite this reduction of the CRs, the ratio of regulatory CRs to the economic CRs still remains largely higher for SMEs than for large corporates. Notice that the last row of Table 2 (called "all") shows the diversification benefits provided by the existence of exposures on different size classes within the same portfolio. The row corresponds to the weighted average of the different size classes CRs. It shows that the weighted average value of the economic CRs (3.2%) is far below the value of economic CRs for the large corporate (6.3 %), thanks to the presence of SMEs exposures less demanding in CRs in the loans portfolio. This saving in CRs is smaller for the regulatory CRs, the regulatory model failing to account for diversification benefits.

There is obviously some model uncertainty in economic CRs measurement. To deal with this issue, we use the values of random effect variance displayed in Table 1 and we inflate the estimates of the random effect variance of the SME by two standard deviations26 and we reduce the estimate for the large corporate by two standard deviations (0.225-2*0.07615=0.0727). With this new set of random effects, we compute both the economic CRs and the Basel II/III regulatory CRs. We find a ratio regulatory CRs to economic CRs of 9.10% for SMEs and of 6.47% for large corporates. In order to have the same economic CRs for SMEs and for large corporates as for the regulatory ratio, SMEs should benefit from a 71% discount which is very close to the SF calibration (76%).

In sum, economic CRs computations do confirm that the CRs should be lower for SMEs. According to a multiple factor economic capital framework, the SF should be much higher than 25% in order to be consistent with the difference in economic capital between large and small firms. Nevertheless, taking into account uncertainty surrounding the estimates of the multifactor models and adopting a conservative approach, the SF is consistent with the difference in economic capital between SMEs and large corporates.

The effect of the Supporting Factor on the credit supply

As explained in the econometric framework described in Section 4.2, we assess the average effect of the SF on the distribution of credit by banks by relying on the difference-in-differences specification (5). With this specification, we compare the (log of) total outstanding amount of credit of eligible exposures vs. ineligible exposures in the post-reform period (as compared to the pre-reform period). After analyzing the impact of the SF on the average exposure/firm, we implement the other tests detailed in Section 4.2.3 in order to provide a comprehensive overview of the impact of the SF.

The average impact of the Supporting Factor on the credit distribution

Table 5 presents the results associated with the baseline specification (5). The dependent variable is the logarithm of the total outstanding amount of credit between bank b and firm f at time t. As a result, we could interpret the estimated coefficient as a semi-elasticity, i.e. the estimated coefficient of interest indicates the change in the initial exposure in percent resulting from being eligible to the SF. Importantly, we only consider the effect of the SF at the intensive margin, i.e. the effect of the SF on existing and positive bank-firm relationships.

We gradually include a set of fixed effects (FEs) in the regressions. We start with a set of quarter, location, industry and size FEs. The result can be found in column (1). Then, we introduce bank, bank-quarter and firm FEs. The role of the bank-quarter FEs is to control for bank funding shocks among other things. For instance, in the case where (i) some banks face a positive funding shock in the post-reform period and (ii) these banks have credit portfolios biased toward small eligible exposures, we could observe a positive coefficient β but for reasons unrelated to the SF. The results are shown in columns ( 2) to (4). In column (5), we even include size-quarter FEs to control for all the shocks specific to a given firm size class as it is a crucial dimension of the SF.

The point estimate ranges from 0.043 to 0.095 indicating that, when an exposure is considered as eligible to the SF, it receives on average between 4.3% and 9.95%27 more credit than ineligible exposures in the post-reform period as compared to the pre-reform period.

In the last column, we include linear group-specific time trends. This allows treatment and control groups to follow different trends in a limited but potentially revealing way. With this specification, the effect of the SF is now identified through a relative deviation from the group-specific trends. Not suprisingly, the magnitude of the point estimate decreases but remains nonetheless sizable and significant (+1.8%). Importantly, we have enough quarters in the pre-reform period to identify accurately the trends from the period preceeding the reform (Wolfers, 2006). Overall, we found that the average exposure between bank b and firm f increases by 2% to 10% more after the introduction of the SF when the exposure is eligible to the SF than when it is not.

Robustness tests

In order to ensure the robustness of this finding, we perform additional tests that are presented in the Table 6. In the first column, we remove exposures with an average outstanding amount comprised in [e1M-e2M]. Indeed, as the exposure gets closer to the threshold, the incentives for banks are increasingly mixed. At the margin, banks still have incentives to provide more credit to eligible exposures as compared to ineligible exposures. However, the bank has to make sure that the exposure will never pass above the threshold because it will then lose the 24% discount on CRs associated with the total outstanding amount of credit. This will provide strong incentives to limit the growth of exposures as they approach the threshold. This is why we remove exposures higher than e1M and lower than e2M where the incentives for banks to extend credit are mixed. 28 The effect is now identified by comparing exposures below e1M (eligible) with those above e2M (ineligible). We continue to observe a sizable effect of the SF (+6%).

Then, in column (2), we remove exposures just around the threshold, i.e. exposures with an average outstanding amount comprised in [e1.4M-e1.6M]. Indeed, we suspect that the outstanding amount of credit we observe in the credit register and the regulatory definition of exposures may not be perfectly aligned. Alternatively, banks may experience difficulties in identifying the total outstanding amount of a given counterpart at the group level on an ongoing basis. In either case, this could give rise to some misclassification. To avoid this issue, we estimate the effect of the SF after removing exposures around the threshold: the main finding remains unchanged.

In column (3), we adress the classical serial correlation issue (Bertrand et al., 2004) by collapsing the dataset into two periods (pre and post). After that, we rerun the baseline specification and still find a positive and significant effect of the SF. In column (4) we drop firms whose size (as reported by the Banque de France) is unknown. These are generally very small firms and ignoring them does not affect the initial findings. In column (5), we remove the two quarters surrounding the entry into force of the SF. This is particularly important in the case where banks tend to anticipate a bit the reform. We continue to observe a significant effect of the SF. Finally, in our last robustness check in column (6), we estimate the effect of the SF on a perfectly balanced sample. Said differently, all pairs of bank-firm {b, f } have now a positive exposure all along the period studied (2010Q1-2016Q4). We lose a lot of observations but we now have perfectly stable groups over time. This is a way to avoid any composition effect. We observe that the estimated effect of the SF remains unchanged.

Dynamics of the effect over time and the parallel trend assumption

The difference-in-differences estimator hinges on a crucial assumption: the fact that the treated group and the control group have similar trends in the outcome variable throughout the period preceeding the reform. We can test this assumption by estimating a dynamic version of the baseline specification in which we estimate the effect of the SF within each quarter. Above all, this regression is informative regarding the dynamics of the effect over time : have banks responded immediately to the reform or, on the contrary, has the SF become increasingly effective quarter after quarter ? Has the effect of the SF on the credit supply persisted over time or has the initial impulse faded out after few quarters (for in-stance, as a result of the uncertainty surrounding the nature of the reform : temporary or permanent) ?

To answer these questions, we run the specification (6). Rather than presenting the numerous coefficients in an extended table, we plot the results in the Figure 1. In this figure, we represent the coefficients estimated within each quarter as well as the corresponding 95% confidence bands. We define the reference period as 2014Q1 and materialize it with the vertical black line. All the coefficients must be interpreted with respect to this reference period. The underlying regression includes time, location, industry, size, rating, bank, firm and bank-quarter FEs.

First, we do not observe significant differences between the eligible and the ineligible exposures in the period before the reform. Except for few quarters where the difference between the control and the treated group is marginally significant at the 5% level (albeit negative), the figure reveals that the two groups have similar dynamics in the pre-reform period. This is a validation of the common trends assumption. Second, when we look at the post-reform period, we observe that the effect of the SF on the credit supply is not distinguishable from zero for the first four quarters following the entry into force of the SF. However, starting from 2015-Q1, the dynamics of the outstanding amount of credit received by eligible and ineligible groups tends to diverge increasingly and significantly. Said differently, the effect of the SF tends to be stronger over time. In 2016, the magnitude of the effect lies between 5% and 10%, at a much higher level than the baseline estimate.

Heterogeneity of the effect of the SF

We now investigate in more details to which extent the effect of the Supporting Factor varies along with firm/exposures characteristics. We focus on three dimensions : the size of firms, their riskiness (as measured by the Banque de France rating) and the size of the exposure. We run the generic specification (7) using these three dimensions one after another. The results of these tests are presented in the Tables 7 and9.

Firm characteristics

In the first column of the Table 7, we test whether the SF has the same effect on eligible exposures from medium-sized, small and micro firms. For that purpose, we classify SMEs as medium-sized, small or micro according to their size classes reported in the credit register and based on the turnover of firms. Then, in the second column of the Table 7, we test the magnitude of the effect of the SF on the provision of credit by banks depending on the riskiness of SMEs. We classify the SMEs as safe or risky according to their Banque de France rating.

As explained in Section 4.2.3, we classify an SME as large or risky according to the most frequent value observed in the pre-reform period. By doing so, we want to avoid endogeneous feedback loops where the response to the SF affects the size or the riskiness of the firm. Indeed, a firm receiving relatively more credit is likely to grow faster or to have an increasing leverage. Both of these mechanisms would affect the size or the riskiness of firms.

The result of this first test in column (1) is unambiguous. While eligible exposures from medium-sized enterprises do not grow at all (as compared to ineligible exposures), the eligible exposures from micro enterprises grow by 3% (as compared to ineligible exposures) and those from small enterprises grow by 13% (as compared to ineligible exposures). The effect of the Supporting Factor is the strongest for enterprises with a turnover higher than e1.5M but lower than e7.5M. In contrast, no effect can be found on the largest SMEs. This finding is not constistent with those of Mayordomo and Rodríguez-Moreno (2018). Using the Survey on the Access to Finance of Enterprises, they found that "the SF alleviates credit rationing for medium-sized firms that are eligible for the application of the SF but not for micro/small firms". But we have to keep in mind that these findings come from two differents sources and use two distinct methodologies. Furthermore, in their analysis the authors take firm size as a proxy for firm riskiness and conclude that "this finding is in line with the fact that micro/small firms are riskier than medium firms, and hence, they are not treated equally to medium-sized firms by banks.". Interestingly, we are able to analyse separately the size and the riskiness of firms thanks to the Banque de France credit rating. As a result, we can push the analysis further.

The result of the second test in column ( 2) is less clear-cut. We observe that the effect of the SF on exposures from SMEs classified as risky (speculative grade) does not diverge from the effect of the SF on exposures from firms deemed as safe (investment grade) : in both cases, the effect is largely insignificant.29 However, we find that exposures from SMEs for which the Banque de France has "no unfavourable information gathered" tend to benefit from the SF.30 Interestingly, the SMEs with "no unfavourable information" -i.e. firms with unknown ratings-have a quasi default rate that stands between the quasi default rates of risky and safe SMEs as shown in Table 8. Hence, the results of this test indicate that neither the most risky firms nor the safest ones have been targeted by banks when responding to the SF. One possibility to explain this result is that the safest SMEs are largely insensitive to the SF because their are rarely facing credit constraints while riskier SMEs remain too risky from the banks' point of view, even after accounting for the capital relief coming from the SF.

How do these two results interact with each other ? Indeed, the vast majority of firms with no Banque de France credit ratings are firms classified as micro enterprises. To understand in more depth the relation between size and riskiness, and to understand whether the former can be considered as a good proxy for the latter, we replicate the results from column (2) in terms of riskiness over the various size categories. In columns (3), ( 4) and

(5), we examine how the effect of the SF varies along with Banque de France ratings for the subsamples of respectively medium-size, small enterprises and micro enterprises. The results confirm that these two dimensions do not convey the same information, albeit they are interelated. We observe that for medium-sized firms, no specific pattern can be found: the magnitude of the coefficients tends to support the hypothesis that only firms with unknown credit ratings benefit from the SF, but this result lacks statistical significance to be conclusive. Regarding the sample of small firms, our result indicates that the effect is the strongest for firms considered as safe (investment grade) and, to a lesser extent for firms with no credit ratings information. Finally, among the population of micro firms, only those with no information regarding the Banque de France credit ratings seem to benefit from the SF. These results are consistent given that the micro firms with no credit ratings have default rates that are similar to those of small firms considered as safe (see Table 8).

Overall, our results show that the SF primarily benefits to small and micro enterprises and firms with no credit rating. More precisely, the effect is found to be the strongest for small enterprises categorized as safe and micro enterprises with no credit rating. Thus, the SF gave banks an extra incentive to grant credit to firms which are not closely monitored and which suffer more from asymmetry of information (i.e. firms with no Banque de France credit rating). It clearly shows that the credit distribution to those firms was somewhat constrained by the regulatory weights before the implementation of the SF.

Non linearities

Our third test is slightly different. Rather than contrasting the effect of the SF with respect to firm characteristics, we now explore how the effect of the SF differs depending on the size of the exposure. Say differently, we test for non linearities in the effect of the SF on the credit supply. The rationale behind such a test comes from the fact that, as explained before in Section 5.2.1, as the outstanding amount of credit associated with the pair of bank-firm {b, f } approaches the threshold of eligibility, the incentives given to banks by the SF becomes increasingly ambiguous. On the one hand, it is still relatively less costly in terms of CRs to lend to eligible pair of bank-firm at the margin but on the other hand, the risk to pass above the threshold (and therefore to lose the CRs discount on the total outstanding amount of credit) increases at the same time. Consequently, as the exposure gets closer to the threshold, banks may become increasingly reluctant to extend additional credit to the relevant firms.

As a result, we expect that the effect of the SF should become proportionally weaker as the size of the exposure increases. Given that the coefficient of interest β indicates an effect of the SF in relative terms, the coefficient associated with the largest eligible exposures could even become negative, indicating that these exposures have decreased (as compared to ineligible exposures) after the implementation of the SF.

To implement this test, we classify exposures according to their average outstanding amount computed over the pre-reform period as explained in Section 4.2.3. We define three buckets : small ([0-e500,000]), medium ([e500,000-e1M]) and large ([e1M-e1.5M]). Then, we estimate simultaneously a coefficient β for each of these three buckets using the generic specification (7). This test is in line with the identification strategy implemented by Mayordomo and Rodríguez-Moreno (2018). However, we favor a discrete functional form while they use a continuous functional form.

The results of this test are displayed in Table 9. We report several specifications including various fixed effects. We observe that the coefficient associated with exposures categorized as small is systematically positive and significant while the coefficient associated with the two other buckets of exposures (medium and large) are significantly negative: as compared to the ineligible exposures, the eligible exposures considered as small tend to grow more after the entry into force of the reform (between +9% and +15%) while this is not the case for medium and large eligible exposures. We even observe that the medium and large exposures tend to decrease in the post-reform period.

These results confirm that banks have primarily supported exposures of small size as a result of the implementation of the Supporting Factor. This finding can be rationalized if we consider that the design of the SF provides ambiguous incentives. This ambiguity comes from the fact that the SF does not target flows of new credit but rather provides a CRs relief on the existing stock of credit. As a result banks may benefit from the SF without any action being required on their part. Around the threshold of eligibility, banks may even have incentives to curb credit growth in order to avoid passing the threshold and losing the CRs relief on the total outstanding amount of credit. This last hypothesis is consistent with our results: the exposures classified as medium or large tend to decrease (in relative terms) after the entry into force of the SF. To the best of our knowledge, we are the first to highlight this drawback in the design of the Supporting Factor and to provide evidence showing it. Note that the current Commission's proposal to maintain the Supporting Factor and to extend its scope with no upper limit31 is a way to resolve this drawback of the current scheme.

Overall, we find that the effect of the SF is highly heterogeneous and that not all SMEs with eligible exposures have benefited from its implementation. Our findings show that the SF has mainly benefited to large and safe SMEs with rather small exposures. While the first result indicates that the SF has not been very effective for the smallest SMEs, the firms that are presumably the most credit constrained as a result of asymmetry of information and those for which a marginal euro of additional credit has the highest value, the second finding provides a more optimistic view regarding the effectiveness of the SF : banks have directed their lending to the healthiest borrowers.

Overall, we find that the effect of the SF is highly heterogeneous and that not all SMEs with eligible exposures have benefited from its implementation. Our findings show that the SF has mainly benefited to small or micro enterprises, to those deemed as relatively safe and to firms with rather small exposures. Interestingly, small and micro firms and those with no Banque de France credit rating are firms which are likely to suffer the most from asymmetry of information in gaining access to external finance. As a result, we see our findings as optimistic : the SF has boosted the credit supply toward those firms which are presumably facing the most severe credit constaints.

Conclusion

This paper investigates the effectiveness and the consistency of a new regulatory tool implemented specifically to promote SMEs' access to bank credit: a targeted reduction in bank CRs associated with SMEs loans. In particular, the objectives of this reform are to provide an easier access to bank credit for SMEs and to ensure adequate capital requirements for SME credit risk. That is why we examine this policy experiment along with these two dimensions.

First, the consistency of the reform regarding the intrinsic riskiness of SMEs is gauged through the computation of banks' economic capital requirements using the structural credit risk framework underlying the computation of the regulatory capital requirements. This method allows us to compute the contribution of each size class to the total risk of the portfolio, taking into account the potential diversification or concentration effects within the portfolio. We finally compare the "economic" CRs resulting from our multifactor model with the regulatory ones, with and without considering the reduction associated to the SF. We find that for each size class, the level of the regulatory CRs is far superior to the level of the economic CRs, even after the application of the SF. Overall, after considering the uncertainty surrounding these estimates and adopting a conservative approach, we find strong evidence that the SF is consistent with the difference in economic CRs between SMEs and large corporates.

Then, the impact of the reform on the credit supply to targeted SMEs is estimated through the difference-in-differences methodology. We thus compare the evolution of the outstanding amount of credit of eligible and ineligible exposures after the reform (vs. before the reform). We find evidence showing that the SF has been effective in supporting bank lending to targeted SMEs. Specifically, we find that the magnitude of the effect of the SF has increased over time: the effect was almost zero in the first year after the entry into force of the SF but it has then intensified to reach a magnitude of 8% to 10% two years after the entry into force. Concerning the possible sources of heterogeneity, results indicate that the effect of the SF is much stronger on eligible exposures of small and micro enterprises than those on medium-sized enterprises. Then, we find convincing evidence showing that exposures of SMEs with no Banque de France credit rating tend to be more affected by the implementation of the SF than exposures of SMEs considered as safe or risky according to their credit rating, the former rarely facing credit constraints while the latter remaining too risky even after considering the capital relief provided by the SF. Finally, we find that the smallest eligible exposures benefited the most from the SF (as compared to larger eligible exposures). This result indicates that the threshold at e1.5 million can provide perverse incentives to banks regarding exposures slightly below the threshold. Overall, this supports the relaxation of the threshold as it is planned in the new version of the SF to come. 9 Tables This table provides descriptive statistics on our dependent variable, the total outstanding amount of loans. The samples described are before and after the implementation of the SF. Source: Banque de France, French national Credit Register and authors' calculations. This table reports the estimates associated with the difference-in-differences specification (5) on various subsamples. The dependent variable is the logarithm of the total outstanding amount of credit. The dummy Eligible denotes SMEs whose total "eligible" outstanding amount of credit is lower than e1.5M. The dummy Post denotes the period after the implementation of the SF in 2014. The variable of interest Eligible • Post is the product of these two latter dummies. In column (1), we run the difference-in-differences estimation on a subsample excluding pairs of bank-firm with an average outstanding amount of credit between e1M and e2M. Column (2) estimates the effect of the SF on a subsample excluding pairs of bank-firm with average outstanding amount of credit between e1.4M and e1.6M. Column (3) reports the coefficient of interest after collapsing the data into 2 time periods (pre and post) to overcome serial correlation issues. Column (4) reports our estimations on a subsample excluding firms whose size (turnover) is unknown. Column (5) reports estimations after dropping the period surrounding the implementation of the SF, i.e. from 2013Q3 to 2014Q2. Finally, column (6) shows estimates based on a perfectly balanced sample, i.e. we keep all pairs of bank-firm b, f that have a positive exposure over the entire period considered (2010Q1-2016Q4 This table reports the estimates associated with the difference-in-differences specification (5). The dependent variable is the logarithm of the total outstanding amount of credit. The dummy Eligible denotes SMEs whose total eligible outstanding amount of credit is lower than e1.5M. The dummy Post denotes the period after the implementation of the SF in 2014. The variable Eligible • Post is the product of these two latter dummies. We interact this last variable with 2 characteristics of the firms: the size as measured by the turnover and the riskiness as assessed by the rating provided by the Banque de France. Firms are classified according to the most frequent value observed in the pre-reform period. We distinguish the riskiness of the firms according to their rating in 3 classes: risky (speculative grade), non-rated and safe (investment grade). Likewise, we distinguish the size of the SME in 3 classes: medium-sized, small and micro enterprises. All regressions control for size, rating, geographic location, industry, bank, bank-time, firm, as well as year-quarter FEs. This table reports quasi default rates broken down by Banque de France ratings. We build these quasi default rates at a one year horizon. The first definition of default (Default 1 ) is a dummy variable taking the value of 1 as soon as a firm received the ratings Banque de France for which at least one trade bill payment incident has been reported. The second definition of default (Default 2 ) is more restrective and is a dummy taking the value of 1 as soon as a firm is the subject of insolvency proceedings (recovery or judicial liquidation proceedings). The default rates are computed over the entire period. The table reads as follow : 0.10% (resp. 0.05%) of the firms having a rating Banque de France 3++ over the pre reform period will be in defaults one year later.

Figures

Source: Banque de France -ACPR, French national credit register and authors' calculations. Sample: Independent SMEs that borrow from one of the 7 main French banking groups over the period 2010-2016.

Risk analysis: data restrictions

This appendix reports more detailed information about the databases used in the risk analysis. To do that, we need to restrict the sample of firms to the following conditions:

-Firms must have exposures in the French credit register. In France, every bank should declare business loans provided that the loan amount is over e25,000 starting from 2006. However, before 2006, this threshold was e75,000. To avoid creating artificial entries of firms in 2006, we apply the e75,000 threshold over the entire sample period considered, i.e. 2004-2015. -The Banque de France rating directorate gives to these firms a rating (including a default grade). This means that firms' annual turnover is above e0.75 million and firms obtain credit from at least one large banking group operating in the French loans to businesses market. -We also exclude exposures toward the financial sector. Indeed, we neutralize a break related to the end of the reporting of interbank exposures with non-resident counterparts in 2006. -We exclude exposures toward individual entrepreneurs. They stopped reporting their exposures within the credit register in 2012, so we drop them from the total sample to avoid artificial exits of the sample.

This framework is known as the asymptotic multi-factor framework of credit risk (see Lucas et al. (2001)) and is an extension of the asymptotic latent single risk factor (ASRF) model underlying the Basel II CRs for credit risk. Equation (4) assumes that each obligor can be characterized by his individual default threshold and factor sensitivities. However, in retail loan portfolios, default rates are generally computed based on rating grades, and sensitivities to risk factors cannot be computed on an individual basis. Thus, assumptions are required to reduce the number of parameters of the loss variable. A common assumption is that obligors who belong to the same rating j will share the same default threshold. Moreover, one could assume that the vector of risk factor sensitivities is the same for obligors sharing a set of common characteristics. Assuming that the portfolio is portioned in K segments (here the firm size), that credit exposures are rated using a scale with J grades, and denoting n kj the number of exposures with rating j in segment k, losses can be rewritten as:

L(s) ≈ K k=1 J j=1 n kj i=1 u i Φ   Φ -1 ( pj ) -w k s 1 -w k Rw k   (14)
The calibration of this credit risk model requires the estimation of J default thresholds Φ -1 ( pj ) of the rating scale, the factor loadings w k , and the correlation matrix R. A first order choice is the specification of the systematic risk factors. However, we are interested in capturing the risk heterogeneity for firms of different sizes. Thus, we expand the latent factor approach underlying the ASRF model by considering a latent risk factor for each size class. These factors are possibly correlated, with correlation matrix R. In other words, we consider that credit risk within each portfolio size segment can be described by a single risk factor model, taking into account correlations of risk exposures across segments. The different parameters are estimated using a binomial probit generalized linear mixed model (see McNeil and Wendin (2007b)). The generalized linear mixed model (GLMM) provides a straightforward econometric framework to estimate the parameters of our multifactor credit risk model. Indeed, the choice of this specification leads to consider the default thresholds as fixed effects and the factor loadings and factor correlations as described by a multivariate vector of Gaussian random effects.

Within the framework of GLMM models, the default probability in Equation ( 5) is defined as follows. Let Y t be an (N × 1) vector of observed default data at time t and γ t be the (K × 1) vector of random effects. The conditional expected default probability of obligor i at time t is then:

P (Y ti = 1|γ t ) = Φ(x ti β + z i γ t ) (15)
where Φ(.) is the standard normal cdf 32 , β denotes the vector of parameters associated 32 We focus on the probit link function because the normal distribution is the underlying link function that is assumed by the Basel II/III framework of credit risk.

with the fixed effects (the borrower's rating class) and z i is the design matrix of the random effects, here an identity matrix with size the number of random effects. If the rating scale is properly built, we expect the β parameters that correspond to the default thresholds to be associated with the ratings to be ordered and to increase as credit quality decreases.

x ti = [0, ..., 1, ..., 0] is a (1 × J) vector of dummies defining the rating of borrower i at time t.

Once the credit risk parameters are estimated, the distribution of losses at the portfolio level is computed by Monte Carlo simulations, with each simulated realization of the systematic risk factors being converted into a conditional default probability at the rating/size segment level as defined by Equation ( 5) and, finally, into conditional expected losses at the portfolio level. Various quantiles based on risk measures such as Value-at-Risk (VaR) or Expected Shortfall can then be retrieved from the simulated distribution of portfolio-wide losses.

Our multifactor model provides the economic capital necessary to cover losses of a portfolio of loans by firm size buckets. We use this model as a benchmark to check whether the capital deduction induced by the supporting factor on SME loans (about 24%) is consistent with the difference in economic capital between the SME loans and the rest of the corporate loans portfolio (the "large" corporate businesses).

Figure 1 :

 1 Figure 1: The effect of the SF on the credit supply: dynamics over time

10. 2

 2 French national credit register: breakdown by loan type -Short-term loans (i.e. with an initial maturity shorter than 1 year) -Overdrafts on ordinary account (including short term credit line drawdown) -Accounts receivable financing -Factoring -Other short-term loans -Medium and Long-term loans (i.e. with an initial maturity longer than 1 year)

Table 1 :

 1 Random effects variances and correlationsThis table shows the estimated variances of the random effects and their correlation matrix. All parameters in Table1are significantly different from 0 with p-values lower than 1%.

		Panel A: Random effects variances (%)
		Retail		Corporate
	Size Class:	[0.75 -1.5] [1.5 -5]	[5 -15] [15 -50] [> 50]
	Estimates	0.0094	0.0034	0.0163	0.0723	0.225
	Standard Errors	0.0101	0.0012	0.0144	0.0360	0.0762
	Panel B: Correlation matrix of random effects
	Size Class:	[0.75 -1.5] [1.5 -7.5] [7.5 -15] [15 -50] [> 50]
	[0.75 -1.5]	1
	[1.5 -7.5]	0.6454	1
	[7.5 -15]	-0.5802	0.2520	1
	[15 -50]	-0.7361	0.04326	0.9721	1	1
	[> 50]	-0.7698	-0.04406	0.9519	1	1

Source: Banque de France, authors' calculations.

Table 3 :

 3 Descriptive Statistics : Distribution of the outstanding amount of loans

	N	Mean	SD	P10 P25 Median P75	P90
	Sample including firms whose eligibility status changes over time
	18 599 438 145.40 247.37 0.00 25.00 36.00 70.00 154.00
	Sample of firms whose eligibility status is constant over time
	18 369 085 130.97 195.50 0.00 25.00 36.00 69.00 149.00
		Before the implementation of the SF
	10 391 672 129.19 192.84 0.00 25.00 36.00 68.00 147.00
		After the implementation of the SF
	7 977 413 133.30 198.88 0.00 25.00 35.00 69.00 151.00

Table 4 :

 4 Descriptive StatisticsThis table shows the distribution of eligible and non eligible exposures, for the two periods before and after the implementation of the SF.

	Eligibility Status	Frequency	Percent
		Before the implementation of the SF
	Non eligible (Exposures > e1,5M)	17,437	0.17
	Eligible (Exposures < e1,5M)	10,374,235	99.83
	Total	10,391,672	100.00
		After the implementation of the SF
	Non eligible (Exposures > e1,5M)	17,907	0.22
	Eligible (Exposures < e1,5M)	7,959,506	99.78
	Total	7,977,413	100.00

Source: Banque de France, French national Credit Register and authors' calculations.

Table 6 :

 6 The effect of the SF on the credit supply: robustness checks

			Logarithm of the total outstanding amount of credit
		(1)	(2)	(3)	(4)	(5)	(6)
	Eligible • Post	0.059***	0.044***	0.029**	0.036***	0.063***	0.051***
		(0.010)	(0.010)	(0.014)	(0.010)	(0.012)	(0.011)
	Observations	16,214,490	16,274,136	1,665,354	8,930,159	13,808,816	5,144,383
	Adjusted R-squared	0.728	0.733	0.583	0.697	0.727	0.787
	Bank FE	Yes	Yes	Yes	Yes	Yes	Yes
	Bank*Time FE	Yes	Yes	Yes	Yes	Yes	Yes
	Firm FE	Yes	Yes	Yes	Yes	Yes	Yes
	Sample	Avg exp. in [0;1000[ Avg exp. in [0;1400[	Collapse	Drop firms with Drop quarters in Balanced
		& [2000-5000[	& [1600-5000[	into 2 periods	unknown size	[2013Q3-2014Q2]	sample

Table 7 :

 7 The effect of the SF on the credit supply: breakdown by firm's characteristics

		Logarithm of the total outstanding amount of credit
		(1)	(2)	(3)	(4)	(5)
	Eligible • Post • Medium-sized	-0.018
		(0.056)
	Eligible • Post • Small	0.128***
		(0.016)
	Eligible • Post • Micro	0.029**
		(0.012)
	Eligible • Post • Risky		-0.001	-0.085	0.026	-0.019
			(0.015)	(0.075)	(0.017)	(0.027)
	Eligible • Post • Unknown risk		0.049***	0.234	0.118*	0.051***
			(0.012)	(0.150)	(0.067)	(0.012)
	Eligible • Post • Safe		0.027	0.009	0.129***	-0.107
			(0.041)	(0.093)	(0.046)	(0.089)
	Observations	15,050,896 15,132,625	150,195	1,318,019 13,327,670
	Adjusted R-squared	0.723	0.724	0.472	0.541	0.747
	Sample	All firms	All firms Medium-sized	Small	Micro
	Bank FE	Yes	Yes	Yes	Yes	Yes
	Bank*Time FE	Yes	Yes	Yes	Yes	Yes
	Firm FE	Yes	Yes	Yes	Yes	Yes
	Test medium vs small	.01
	Test small vs micro	0
	Test risky vs unknown		.01	.06	.18	.02
	Test safe vs unknown		.61	.2	.89	.08

Table 8 :

 8 Quasi default rates across Banque de France ratings

	Classification	Banque de France Rating	Default 1 Default 2 All firms	Default 1 Medium-sized Small	Micro
		3++	0.10%	0.05%	0.02%	0.15%	.
		3+	0.31%	0.19%	0.03%	0.37% 0.23%
	Safe (Investment Grade)	3	0.44%	0.28%	0.42%	0.41% 0.54%
		4+	1.48%	0.97%	1.25%	1.55% 1.39%
		4	3.76%	2.16%	2.41%	3.73% 4.04%
	No unfavorable information No rating	4.54%	2.73%	6.37%	7.10% 4.52%
		5+	6.98%	3.26%	5.66%	7.39% 6.67%
		5	12.71%	8.00%	8.60%	12.49% 12.99%
		6	14.02%	10.45%	15.98%	16.19% 13.88%
	Risky (Speculative Grade)	7	58.85%	12.59%	66.67%	60.51% 58.65%
		8	72.01%	16.23%	87.69%	68.01% 72.53%
		9	79.60%	19.59%	79.59%	83.25% 79.16%
		P	96.77%	95.84%	60.61%	78.82% 97.09%
	Total	11.37%	9.11%	2.99%	5.59% 12.15%

During the financial crisis, targeted monetary policy instruments such as the TLTRO or the ACC have been implemented in Europe. In France, SMEs benefit from specific fiscal deductions related to their investment plan.

See for instance[START_REF] Behn | Procyclical Capital Regulation and Lending[END_REF],[START_REF] Bonner | The Impact of Liquidity Regulation on Bank Intermediation[END_REF],[START_REF] Jimenez | Macroprudential Policy, Countercyclical Bank Capital Buffers, and Credit Supply: Evidence from the Spanish Dynamic Provisioning Experiments[END_REF], or Fraisse, Lé, and Thesmar (forthcoming) respectively in the case of Germany, the Netherlands, Spain and France.

See paragraph

of the Capital Requirement Directive (CRDIV) published in the Official Journal of the European Union the 27 June 2013.

See Dietsch et al. (2016) for a comprehensive overview of the existing empirical studies on the relationship between asset correlations and firm size.

This decomposition aims to follow the most accurately the traditional decomposition used by both the OECD and the European Commission.

See Banque de France, 2016, Rapport de l'Observatoire des délais de paiement.

In the Appendix 10.5, we offer a complete and technical presentation of the model.

More details about the Bank of France rating scale we use can be found here: https://www.banquefrance.fr/sites/default/files/media/2016/12/29/the-banque-de-france-rating-reference-guide.pdf.

 20 See https://stats.oecd.org/glossary/detail.asp?ID=3123 and http://ec.europa.eu/growth/ smes/business-friendly-environment/sme-definition_en.

In the case where the firm has two values for the mode, we ignore the firm.

For instance, k ∈ {Small; Large}.

The 45% are the LGD that were used in the so-called Basel II fondation approach that the banks could use in absence of a validated LGD model but with a validated PD model.

The estimation also yields

(5x5) default thresholds, not shown here for the sake of simplicity. As expected, these thresholds are ordered, reflecting the increasing likelihood of default for lower ratings, and statistically different from 0.

For illustration : 0.0723+2*0.03602=0.1443 for the [e15M-e50M] size class, 0.0163+2*0.0144=0.0451 for the [e5M-e15M] size class and so on...

See[START_REF] Kennedy | Estimation with correctly interpreted dummy variables in semilogarithmic equations [the interpretation of dummy variables in semilogarithmic equations[END_REF] on how to interpret accurately semi-logarithmic elasticity with a dummy variable.

At the same time, regarding exposures slightly above the threshold, banks may have incentives to let the exposures diminish as the loan is amortized in order to benefit from the 24% discount on CRs once the exposure falls below the threshold of eligibility.

We still find no significant differences between risky and safe SMEs when we restrict the sample to firms with a known Banque de France rating.

Importantly, as soon as one default on trade bills is reported, the firms will be immediately classified as risky. As a result, this category indicates at least that the firm is not performing too poorly.

More specifically, above the e1.5 million limit, a 15% reduction for the remaining part of the exposure would apply.

This table shows the value of capital ratios when using the multifactor model (economic capital) and the regulatory Basel III model or the regulatory CRD IV/CRR model taking into account the supporting factor (SF). For the regulatory models, we used the IRB other retail formula for the computation of assets correlation in the smallest size class [0.75-1.5], and the IRB corporate formula (with the corresponding size-turnover-adjustment) for the four last classes of medium and large enterprises. Source: Banque de France, French national Credit Register and authors' calculations. The identification of SMEs is precisely defined by the 2003 European Commission Recommendation as follows: "The category of micro, small and medium-sized enterprises (SMEs) is made up of enterprises which employ fewer than 250 persons and which have an annual turnover not exceeding e50 million or an annual balance sheet total not exceeding e43 million." Among these criteria, the CRR indicates that only the annual turnover must be considered to qualify a company as an SME allowed to benefit from the SF.

Conditions of eligibility

Regarding the amount "owed" to the institution, the CRR also brings precisions about exposures eligible to the SME-SF. In the case of a credit line, only the drawn amount must be considered as regard to the e1.5 million compliance limit. However, provided that all conditions are met, the exposure as a whole, including its undrawn part will benefit from the capital relief. Thus, there exists a discrepancy between (i) the exposure amount considered for the eligibility to the SF and (ii) the exposure amount that will benefit from the CRs deduction (SF enforcement). Practically, off-balance sheet exposures and claims or contingent claims secured on residential property collateral must not be considered when assessing the amount owed and eligible to the SF. Though, the SF, as deduction in CRs, applies to the entire bank's exposure.

Credit analysis: data restrictions

This section reports more detailed information about the databases used in the credit analysis. To do that, we need to restrict the sample to the following conditions:

-We restrict the sample to the 7 largest banking groups operating in France. The other credit institutions of the sample are very specific credit suppliers that do not reflect the bank lending in France. We thus keep the following banking groups: BNP-Paribas, Société Générale, BPCE, Crédit Agricole, Crédit Mutuel, HSBC, la Banque Postale, which represents 90% of the corporate lending market. -Firms are restricted to SMEs according to the sole turnover criterium, i.e. we only keep firms with an annual turnover lower than e50 million. As mentioned above, the identification of SMEs can be tricky. For instance, 80% of legal entities constituted by 50 employees belong to a larger group. Therefore, to avoid any misclassification of companies, we restrict the sample to independent firms only, i.e. firms that are not affiliated with a corporate group. -We exclude exposures toward the financial sector, the real estate sector, the public sector and the non-profit sector. We also drop holding companies. -We exclude exposures toward individual entrepreneurs. They stopped reporting their exposures within the credit register in 2012, so we drop them from the total sample. -In order to run clean specifications, we drop firms whose SME status or eligiblity status vary over the period. There, the aim is to keep firms for which the status remains constant over time (additional information on this important point can be found in Section 4.2).

Risk analysis: the detailed methodology

The computation of banks portfolio risk under the Basel regulatory framework derives from the structural credit risk approach proposed by Merton (1974). We use an extended version of this approach in order to assess the risk increase induced by the SF. In this section we provide a short description of this framework and describe how we apply it for our risk analysis.

In the Merton (1974) framework, losses at the sub-portfolio level are defined as the sum of losses on defaulting loans. Thus, if u i is defined as the loss given default (LGD) of obligor i and Y i is the default indicator variable of obligor i ( Y i takes the value of 1 if there is a default and 0 otherwise), total portfolio losses L are given by

where n denotes the number of obligors.

In structural credit-risk models, default occurs if the financial health of borrower i crosses a default threshold. Here, financial health is represented by a latent (unobservable) variable U i , which is determined by the realizations s of a set of S multivariate Gaussian systematic risk factors with loadings w i and correlation matrix R, and the realization of a standard normal specific factor ε i . Denoting Φ the standard normal cdf, default occurs when U i crosses downwards a threshold. This threshold is calibrated from the borrower's historical default probability pi :

Specific risk factors ε i are assumed to be uncorrelated among obligors and also independent from the systematic factors S. The factor loading can be interpreted as the sensitivity of the obligor i to systematic factors or more commonly expressed as the general macroeconomic state of the economy. Thus, given a realization s of the systematic risk factors, Equation (2) can be rewritten such as a default occurs when:

As the borrower's specific risk factor is normally distributed, the default probability conditional to s is also standard normal. Moreover, assuming that specific risk can be entirely diversified away, losses can be approximated by their expected value conditional to s (see Gordy (2003)). Conditional portfolio losses are then defined by: 13)