University of Amsterdam
MSc Economics Thesis
__________________________________________
An empirical assessment of Basel III’s IRB asset correlation
parameterization
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Thomas Rang Student n° 5726360
Supervisor: Sweder van Wijnbergen
2
Abstract
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This paper sets out to empirically assess the Basel Accords’ asset correlation assumptions. To that end, a sample of 22 EU banks over the 2004 to 2013 period is subjected to the methodology presented in Byström (2011) complemented by a panel data approach. The resulting asset correlations average 0.43 for this regionally and industrially concentrated sample over the time frame that covers two consecutive episodes of economic stress. Paradoxically, the results lend most support to the assumed stylized increasing relationship of asset correlation with firm size and decreasing in firm probability of default only after accounting for a significant rise in asset correlations during and ensuing the outbreak of the global financial crisis. This effect is not accounted for in the current capital requirements framework, and as such may provide for policy implications.3
TABLE OF CONTENTS
1. Introduction ... 4
2. Literature review... 5
2.1: The ASFR framework ... 5
2.1: Methodology of Basel III’s IRB approach to credit risk... 7
2.2: Asset correlation estimation methodologies ... 11
2.3: Previous asset correlation estimates ... 12
2.4: Asset correlation, PD and size: Evidence ... 13
3. Data and methodology ... 15
3.1: The empirical dataset ... 15
3.2: Equity correlations: the direct approach ... 16
3.3: Asset correlations: the indirect approach ... 16
3.4: Panel data regressions ... 18
4. Results ... 19
6. Conclusion ... 30
Bibliography ... 32
4 1. Introduction
Credit risk modelling stands out as one of the topics most focused on in the last decennia by the banking and finance industries, academia and accordingly, regulators. One of the most influential products of this focus of attention is, beyond any doubt, the application of Vasicek’s asymptotic single risk factor (ASRF) framework to portfolio credit risk modelling. First embraced by regulation in 2004 with the renewed Basel Capital Accord (Basel II1), it has since remained the standard, now still laying the methodological foundations of the latest Basel Capital Accord (Basel III2).
Default correlation among loan-portfolio debtors is a crucial component of the ASRF framework in that it determines the distribution of bank loan portfolio losses. These correlations measure the degree of sensitivity of the obligors’ joint probability of default (PD) to the systematic risk factors that represent the influence of the “state of the economy”. Portfolio risk will be greater the more bank exposures loans tend to vary simultaneously in reaction to the realization of these risk factors. Hence, a crucial element in the estimation of a portfolio’s loan loss distribution is a good calibration of parameters – a troublesome task given default correlations are notoriously hard to estimate directly due to the scarcity of available default data. Therefore, asset correlations, theoretically related by identity to default correlations (a default is expected when asset values drop below a certain critical value, i.e. the value of total liabilities) are often used as a stand-in. So when the second Basel Capital Accord was enacted - in which obligor-specific asset correlation parameters are prescribed as a function of obligor type, its respective PD and size - scrutiny was sparked by empirical validity concerns both for the absolute range as well as the relationships of asset correlation implied by the accord. Accordingly, the number of studies estimating asset correlation from different datasets using different methodologies has since steadily increased. All evidence taken together to date return globally ambiguous results.
1
See BCBS (2006)
2
5
This aim of this paper is to contribute to this discussion by providing for further evidence and analysis in the form an empirical study of asset correlation values and their relationship to PD, size and geographical domiciliation. The sample and estimation methodology used to that end follow and extend from Byström (2011) and include 22 EU banks during the 2004 to 2013 period. Of course, the business nature of the sample’s cross-sections (banking), is but coincidental in the context of this study as we are only interested in their role as loan portfolio exposures on (other) banks’ balance sheets.
The paper is organised as follows. Section 2 presents the topic’s context and includes an explanatory overview of the ASRF and Basel III frameworks as well as a summary of the literature. Section 3 outlines the data sample and methodologies. Section 4 presents and analyses the results. Section 5 concludes.
2. Literature review 2.1 The ASRF framework
The ASRF framework assumes the asset value of an obligor to be driven by a factor model:
, = √∅+ √1 − , (1)
where , is the asset return3 of obligor i in period t, ∅ is the systematic factor representing the “state of the economy” in period t, is the R-squared, the percentage of systematic risk, and , is the idiosyncratic factor of obligor i in period t. Two borrowers are correlated with one another because they are both exposed to the same single systematic factor. Hence, the correlation of borrower i with borrower j is given by:
(, ) = (2)
As such, credit quality correlations among obligors are driven by their correlation to the common single systematic factor.
3
Here the “asset return” can be broadly interpreted as the variable that drives the credit quality change of the borrower.
6
This framework's principal asset to regulatory bodies is its simplifying power in that it can be reconciled with portfolio-invariant capital charges. That is, that an instrument's marginal capital requirement depends not, on the properties of the portfolio in which it is held, but on its own characteristics exclusively. This is a highly convenient property for regulators and regulated alike, in that it limits the amount of data and inputs needed for its administration. For this property of portfolio-invariance to hold however, Gordy (2003) demonstrates that two conditions must hold: (a) there must only be a single systematic risk factor driving correlations across obligors, and (b) the portfolio must be infinitely fine-grained. And although Gordy finds a way to deal with violations of the granularity assumption by means of a simple and accurate portfolio-level charge for undiversified idiosyncratic risk, he finds no similarly simple way to address potential violations of the more crucial single factor assumption. Regrettably, the latest Basel Accord (Basel III) considered but failed to have Gordy's simple and accurate granularity adjustment incorporated in favour of the more flexible approach of delegating granularity considerations to national supervisors at their own discretion. However, the critical question regards the first, more crucial assumption of a single systematic risk factor: Does this framework’s assumption capture reality accurately enough to justify the validity of the capital charges it sets? Many argue it does not, as does Gordy: “The single risk factor assumption, in effect, imposes a single monolithic business cycle on all obligors. A revised Basel Accord must apply to the largest international banks, so the single risk factor should in principle represent the global business cycle. By assumption, all other credit risk is strictly idiosyncratic to the obligor. In reality, the global business cycle is a composite of a multiplicity of cycles tied to geography and to prices of production inputs. A single factor model cannot capture any clustering of firm defaults due to common sensitivity to these smaller-scale components of the global business cycle. Holding fixed the state of the global economy, local events in, for example, Spain are permitted to contribute nothing to the default rate of Spanish obligors. If there are indeed pockets of risk, then calibrating a single factor model to a broadly diversified international credit index may significantly understate the capital needed to support a regional or specialized lender.” In effect, this violation opens the door for credit portfolio risk concentrations to become a potential form of
7
regulatory arbitrage. One may therefore conclude that practical matters in the accord's methodological design have so far outweighed methodological validity and potential bias concerns in the absence of any obvious alternative.
2.2 Methodology of Basel III’s IRB approach to credit risk
The latest Basel accords' methodology to quantify regulatory capital charges to credit risk in its advanced form, the Internal Rating Based (IRB) approach, is dedicated to larger financial institutions only for it requires more complex, internal inputs. The framework and its formulae are quoted and explained below4:
Correlation (R) = (1−
−⋅)
(1−(−)) + 1 −
1−−⋅
1−(−) (3) Parameters a, b and c depend on borrower type. a and b are correlation upper and lower bounds respectively, c is the k-factor that determines the pace with which the exponential function decreases in PD.
Figure 1
Decomposition of the asset correlation function
4
8
For corporate borrowers, on which this paper focuses, a=0.12, b=0.24, and c=50. Hence the range of asset correlations prescribed for medium sized corporate exposures is [0.12, 0.24]. The k-factor of 50 is relatively high which means correlations decrease at a high pace compared to retail exposures, for example, where the dictated k-factor is 35 instead.
The probabilities of default are estimated by the financial institutions themselves and should be reflective of “normal” or average business conditions. According to the BCBS, the systematic dependency of asset correlation to PD displayed in the above formula is based on empirical evidence which could be explained by the intuition that the higher the PD, the higher the idiosyncratic risk components of a borrower will be, hence the lower the influence of the common systematic factor and as such, the lower the asset correlation5.
Moreover, two asset correlation adjustments were also introduced for borrower size, namely one for small- and medium-sized entities (SMEs) and one for large financial institutions (LFIs), the latter of which only with the drafting of Basel III. Again according to the BCBS, though in their opinion empirically not as conclusive as for PD, the dependency of asset correlation on size can be explained by the intuition that larger firm’s dependency upon the overall state of the economy will be greater than that of its smaller counterparts. The firm size adjustments follow from:
SME Correlation (R) = (1−
−⋅)
(1−(−)) + 1 −
1−−⋅
1−(−) − 0.04 ⋅"1 − #−545 % (4) Here, the adjustment applies for SMEs with a yearly sales turnover of $50 million or less. The parameter s represents that turnover.
LFI Correlation (R) = 1.25 ⋅ ' 1−
−⋅
1−(−) + 1 −
1−−⋅
1−(−) ( (5)
The LFI asset value correlation (AVC) multiplier (factor 1.25) above applies for institutions whose total assets are greater than or equal to $100 bln.
5
9
Maturity adjustment (b) = (0.11852 − 0.05478 ⋅ +,())- (6)
The next parameter being estimated is maturity. A longer maturity increases the risk of an exposure and thus requires greater capital requirements. This effect is dealt with in formula (7). However, these maturity effects are also exacerbated if a borrower has a low-PD as they have more “potential” and more room for downgrading than its high-PD counterparts. It is this effect that is corrected for in the above formula (6) and it was reverse-engineered from an empirical regression.
Capital requirement (K) = [/0 ⋅ 1 1
21−⋅ 0() + 31− ⋅ 0(0.999) − ⋅ /0] (7)
⋅
(1+(6−2.5)⋅()) (1−1.5())The asset correlation and maturity adjustment parameters enter the formula above to output the capital requirement. LGD, N and G stand for loss given default, standard normal distribution and inverse standard normal distribution respectively. There are two primary components to this equation: the unexpected loss (UL) capital requirement (first line of eq. (7)) and the (second step of the) maturity adjustment (second line of eq. (7)).
The computation of the “UL-only” capital requirement become clearer by listing the following definitions and identities, and deriving accordingly:
Unexpected Loss (UL) = Conditional Expected Loss (CEL) - Expected Loss (EL) 7/ = /0 ⋅ and 87/ = /0 ⋅ 9:;<:;=> hence ?/ = /0 ⋅ 9:;<:;=>− ⋅ /0 and UL= [/0 ⋅ 1 1 21−⋅ 0() + 31− ⋅ 0(0.999) − 7/] thus 1 @ 1 √1 − ⋅ 0() + A 1 − ⋅ 0(0.999)B = 9:;<:;=> :C <:D;EC; (FCGHF:><)
10
The PDs used in the Basel IRB formula, estimated internally by the supervised entities, are PDs that reflect expected default rates under normal business conditions. To calculate the conditional expected loss (CEL), bank-reported average PDs are transformed into conditional (or downturn) threshold PDs by means of a supervisory mapping function that assumes a more conservative value of the systematic risk factor. In contrast to PDs, banks are asked to report LGDs that already reflect economic downturn conditions directly rather than having them transformed within the supervisory framework.
Figure 2
Decomposition of the capital requirements (K) equation
The way PDs are transformed into conditional PDs follows from the Merton model. For further detail regarding the theoretical foundations and methodology behind this transformation refer to BCBS (2005, #4.2).
From the second step of the maturity adjustment (second line of eq. (7)) it is clear that the BCBS decided to adhere by a maturity standard of 2.5 years and as such penalise (compensate) exposures in excess (below) of this standard.
Risk-weighted assets (RWA) = I ⋅ 12.5 ⋅ 7J (8)
11 2.3 Asset correlation estimation methodologies
From the existing literature, one can discern two general strains of asset correlation estimation methodologies, i.e., those relying on default data and those relying on equity data. It is not obvious from the outset which method is more accurate. On the one hand default events are scarce thus require longer time intervals between observations compared to stock data that is available on a daily basis. Hence, default rate time series generally cover long time spans, which make them more susceptible to regime changes such as structural breaks in the economy or changes in the bankruptcy code. On the other hand, equity time series are vulnerable to significant noise or factors unrelated to credit risk such as a drop in demand or supply of assets.
Both asset correlation estimation methodologies have two distinct approaches, i.e., model-free estimations as opposed to model based ones. Lucas (1995) was the first to estimate them from default rates by reverting them back directly from default correlations. An approach De Servigny and Renault (2002) later refined. However, consensus since seems to have skewed in favour of the model-supported, default-generating approach, having been noted by Gordy and Heitfield (2000), for example, that they may provide more accurate estimates. In the case of correlations drawn from equity data, Duan (2003) was the first to do so in the direct method by means of pairwise equity correlations that are estimated from stock returns. Estimating asset correlations directly from equity prices is widespread in empirical studies. Indeed, using pairwise equity correlations as an estimator has nonetheless often been criticized for ignoring the leverage in the capital structure. This concern is diminished however for high-grade borrowers6 as correlation estimation from equity data competes with that extracted from default rates especially for less risky borrowers for which default events are rarest. The alternative “indirect”, model-based approach has an extra step in the estimation procedure in which asset values are estimated from equity prices and liabilities. Different models are used for this process; one example would be the MKMV7 model which uses a two-step algorithm to produce maximum
6
See Mashal et al. (2003)
7
12
likelihood (ML) asset value estimates as used in Lopez (2004), Pitts (2004) and Düllmann
et al. (2007). More recently however, Byström (2011) provides for an alternative methodology within this approach by estimating asset values through CDS-priced zero-coupon bonds in proportions consistent with the yearly reported leverage ratios.
2.4 Previous asset correlation estimates
A comprehensive overview of asset correlation estimates from the existing empirical work is provided by in Grundke (2007). Overall, the numerous empirical studies show quite different results. The lowest estimate known of is reported by Roesch (2003), in the range of 0.5% to 3.5% depending on business sectors and is based on default rates. So is the study by Düllmann and Scheule (2003) though it focuses on a sample of German firms and differentiates between buckets by size and PD. Results herein lie in the 1-14% range. In a similar study on samples of French and German firms, Dietsch and Petey (2004) obtain results that lie in between the two aforementioned studies.
In a study based on equity data, Lopez (2004) returns correlation estimates for a sample of medium and large US corporates in the 10-26% range, roughly in line with BCBS assumptions and studies by Zeng and Zhang (2001) and Düllmann et al. (2007). Certainly worthy to mention is the fact that discriminating between inter- and intra-sector asset correlations on the basis of rating classes, industries or countries is a factor of importance in the interpretation of estimation results. The majority of the literature focuses on the estimation of intra-sector correlations and these are often higher than inter-sector correlation estimations. In one of the few studies that does estimate inter-sector correlations, Demey et al. (2004) reports an average correlation of 6.1%, smaller than the smallest intra-sector correlation they find. The rationale for this latter is that intra-sectorial groups share more common risk factors than inter-sectorial groups.
In short, the aforementioned studies show significant disparity in the asset correlations they report. One noticeable trend across all studies however, is that estimates arising from methodologies based on equity prices are typically and significantly higher than those based on default rates. As a consequence, Düllmann et al. (2010) set out on a simulation
13
study to try to find out why this is. Their evidence suggests the existence of a downwards bias in estimates based on default rates. They conclude that although this methodological bias does not completely account for differences in estimates, correlation estimates from equity data are more accurate.
2.5 Asset correlation against PD and size: Evidence
Some of the aforementioned studies also address the question about which relationship exists between asset-return correlations, the credit quality of the obligors, and the firm size. As summarised earlier in section 2.2, the Basel Committee for Banking Supervision (BCBS) exercises the assumption that the asset return correlation is increasing in debtor firm size and decreasing in its respective default probability. The empirical results however, are not unequivocal and are partially controversial to the regulatory assumptions. Reported evidence of the empirical studies by Nagpal and Bahar (2001), Düllmann and Scheule (2003), Lopez (2004) and Chernih et al. (2006) do concur either completely or in the most part with both assumed relationships generally; that is, with the direction of the relationships but not necessarily with the specific quantification or qualification (rationale) of these relationships. In probably the most influential study, seen as it is published on the BIS website8 and that its proposed rationale is put forward to explain for the dependency of asset correlation on PD and firm size in their explanatory note on Basel II IRB risk weight functions9, Lopez (2004) proposes the following: asset correlations decreasing in probability of default may suggest that the drivers for rising probability of default are of idiosyncratic nature rather than tied to the general economic environment as summarized by the common risk factor. Larger firms moreover, can generally be viewed as portfolios of smaller firms, and such portfolios would be relatively more sensitive to common risks than to idiosyncratic risks. Essentially, Lopez argues that larger firms are able to diversify away idiosyncratic risk. 8 www.bis.org/bcbs/events/b2ealop.pdf 9 See BCBS (2005) p.12, #5.2
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In another paper, Düllmann and Scheule (2003) revisit, develop and critically assess the theoretical intuitions proposed as rationales for these dependencies against empirical evidence. With regards to PD, they correctly criticize the BCBS rationale for excluding the case in which a deterioration of a firm’s credit quality was initiated by the business cycle rather than by firm-specific events. Nonetheless, they include another potential explanation for this dependency, namely that firms pertaining to cyclical sectors (higher asset correlation) may choose for safer capital structures to account for this relative risk surplus, which would lead to lower PDs. On the dependency of asset correlation to firm size they further contribute to the discussion by pointing out the ambiguous nature of empirical evidence regarding the argument of size-driven diversification of idiosyncratic risk. They quote the results of the empirical work by Roll (1988) which suggest that larger firms tend to be less diversified than size-matched portfolios of small firms and, therefore, should yield lower asset correlations in general. Furthermore, they outline an alternative rationale that they call the “business cycle argument”. That is, different sectors differ in their dependency on the business cycle (cyclical vs. non-cyclical) and in the distribution of their firm size. Therefore, if sectors that are highly cyclical have a structure skewed in favour of large firms, they would expect a dependency between large firms and greater systematic risk. In other words, firm size would serve as a proxy for the business sector’s dependency to asset correlations. Their own empirical study supports the stylised dependency increasing in size but not decreasing in PD for their sample of German SMEs.
Similarly, Dietsch and Petey (2004), report ambiguous results with respect to the probability of default within their sample of SMEs. Their results in some cases even return positive values for the relationship of asset correlation and PD. Furthermore, Lee et al. (2009) also report contradictory findings with respect to this relationship for numerous asset classes, including corporate exposures. They show that if the size factor assumed by the BCBS is properly controlled for, PD returns ambiguous and weak relationships with asset correlations. Accordingly, they quite firmly maintain that: “the stylized decreasing relationship in the IRB approach of the Basel II Accord does not appear to have theoretical nor empirical support and may underestimate portfolio risk for high default probability portfolios.”
15 3. Data and Methodology
3.1 The empirical dataset
The dataset extends on the one used in Byström (2011). In the latter paper, a subsample of 26 out of the 91 banks chosen for the EU-wide stress test performed by the Committee of European Banking Supervisors (CEBS) in 2010 was drawn on the basis that they satisfy certain minimum conditions. These conditions specified that CDS data as well as stock price and balance sheet data be available for the time period starting July 1st 2004 (because CDS data improves significantly that year) ending September 30th, 2010. The dataset subject to analysis in this paper lengthens this period to December 31st, 2013 but excludes 4 banks formerly included for data availability reasons and in one occasion due to a merger. The reduced sample hence includes 22 banks from 10 EU countries - Austria, France, Germany, Greece, Ireland, Italy, Portugal, Spain, Sweden and the U.K. – enough, we believe, for it to qualify as representative for the European (larger bank) banking system. In total, 253 pairwise correlations are computed. Let it be clear that the firms’ nature as banks is trivial in context of the asset correlation estimations as we are only interested in their role as loan portfolio debtors. Nevertheless, it is a lot less trivial in context of the discussion regarding the implications these results might suggest about the degree of systemic risk caused by how interrelated the banking industry is. CDS spreads (10-year), stock prices, market capitalizations and the risk-free rate (average 10-year euro-area government bond yield over the sample period) were downloaded through Datastream. Yearly leverage ratios however, were downloaded from the website of Prof. A. Damodaran.10
10
16 3.2 Equity correlations: the direct approach
As outlined in section 2.3, the estimation of asset correlations from stock prices has two main approaches: direct and indirect. Following Duan (2003), the direct approach estimates asset correlations by the mean of the pair-wise correlations of all firms’ equity price returns:
KLMN= O(OPQ)- ∑UQOPQ∑OUVQ[7ST, 7ST] (9)
where 7ST denotes the vector that collects the equity returns of firm i over time.
3.3 Asset correlations: the indirect approach
In general, indirect approaches differ from their direct counterparts in that they include an intermediate step in which asset values are estimated from equity prices and liabilities. As aforementioned, this can be done in a number of ways. In this case, following Byström (2011), asset value estimations are backed out from equity prices using CDS and leverage ratio data.
More specifically, CDS spreads are used to price (hypothetical) zero-coupon bonds issued by the firm and added to the market capitalization in proportions consistent with the yearly-observed leverage ratios to compute the asset value proxy. That is, the debt value W, of the debt portion of the portfolio relied on as the asset value proxy is computed as: W
=
1(1+ X+#)Y
(10)
where Z is the “risk-free” rate of interest and s is the spread of a T-year CDS contract with the firm as the reference entity. The principal, N, is chosen so that the debt portion contributes to the initial asset value in proportions consistent with initial leverage ratio. In other words, initially, W equals the market capitalization, [. Consequently, the asset value proxy, \, is equal to:
17
\ = ]^+7_`a^b]^ W+ ]^+7_`a^b7_`a^b [ (11)
From the above, daily (three-month trailing window) asset correlations are computed by means of pairwise correlation estimates drawn from each cross-section’s asset value return series.
18 3.4 Panel Data Regression
A number of OLS type regressions will be run on a data panel of daily correlations generated by the asset value proxy methodology (11) for the 22 banks against their respective PD, size and a PIIGS domiciliation binary variable.
The pooled regressions, thus excluding any period or entity unobserved effects, will be specified as follows:
, = c + dQ,+ d-efg7,+ dh1i1ff0e + , (12)
The panel regressions, thus including period and/or entity unobserved effects, will be equivalently specified, namely:
, = c(,)+ dQ,+ d-efg7,+ dh1i1ff0e + , (13)
Table 1
Description of Variables (Daily frequency)
Variable Definition Unit of
measurement Data sources
Independent variables Probability of Default (PD)
Cross section CDS spread, three-month rolling average, log transformed
Basis points Datastream
SIZE
Market capitalization, three-month rolling average, log transformed
Normalized w.r.t. 09-09-2004
Datastream
Non-PIIGS (NP)
PIIGS: Portugal, Ireland, Italy, Greece and Spain domiciled banks Binary variable: 1 = Non-PIIGS 0 = PIIGS N/A Dependent variable Asset-return correlation (CORR)
Average three-month trailing window correlation of cross section (bank) i w.r.t. all others.
Percent Generated
19 4. Results
For confirmation that the alternative asset value computation method was accurately reproduced, Figure 3 portrays the extended time frame equivalent of exhibit 1 in Byström (2011), i.e., the normalized estimated asset values for the sample’s PIIGS and non-PIIGS domiciled banks from June 9th, 2004 to December 31st, 2013.
Figure 3
Normalized Asset Values for Non-PIIGS and PIIGS Domiciled Banks, June 9th 2004(=100) – December 31st 2013
The trends observable clearly corroborate with known system-wide events, i.e., the rise in asset values prior to the 2007 financial crisis, more accentuated for PIIGS domiciled banks and a hint of recovery before the European sovereign-debt crisis in which asset value divergences between the two groups are driven to extremes by the relatively severe impact on the periphery. By the end of 2013, initial asset values were recovered on average by the Non-PIIGS domiciled banks and nearly so for their PIIGS counterparts. Descriptive statistics for the asset return series of the 22 individual banks, to be found in appendix
20
Table 5, corroborate with the reference paper’s findings in that they are not normally distributed.
Table 2 not only presents average asset return correlations but also equity correlations. The latter equity return correlations have been backed out from daily market capitalization series to prevent extraordinary equity issues to interfere with true market value correlation estimates. These correlations are calculated for both the total sample of banks and for groups sorted on probability of default (PD), by means of the average CDS premium proxy over the whole time period, on market capitalization (SIZE) and on geographic domiciliation (PIIGS or non-PIIGS). The PD and SIZE groups are sorted in quartiles.
Table 2
Asset and Equity Return Correlations (Average Pairwise), September 9th 2004 - December 30th 2013
Average Pairwise Asset Correlation Average Pairwise Equity Correlation All Banks 0.434 0.390 Non-PIIGS 0.530 0.529 PIIGS 0.343 0.256 PIIGS vs. NON-PIIGS 0.399 0.339 PD Quartile 1 (Lowest PD) 0.474 0.603 PD Quartile 2 0.714 0.530 PD Quartile 3 0.525 0.473 PD Quartile 4 0.249 0.153
SIZE Quartile 1 (Largest Banks) 0.642 0.547
SIZE Quartile 2 0.426 0.602
SIZE Quartile 3 0.540 0.449
SIZE Quartile 4 0.249 0.153
As can be seen above, correlation estimates drawn from equities are in all but one case lower than those drawn from the estimated asset values. The information of banks’ capital structures evidently contributes significantly to asset correlation estimates. Disregarding such information may hence lead to a downwards bias in the asset correlation estimates from the direct equity correlation methodology.
The sample’s average correlation, consisting of 253 pairwise correlations, is 0.43. This result is slightly lower than that in Byström (2011) (0.50) for the sample ending October 2010 but still significantly larger than the upper bound of the range prescribed by the Basel
21
Accords for corporate exposures, namely 0.24. It must be said however, that these estimates are concentrated intra-sector correlations, both geographically (the EU) and sector wise (the financial industry). Should this study have encompassed a sample wider in range of corporates both geographically as sector-wise, as we would expect a bank portfolio to do, we would expect estimated correlations to have been lower. Our estimates are also high in context of previous estimates. Another explanation for this result could also be the inclusion of severe downturns, i.e., the financial crisis and the sovereign-debt crisis. Though this is a reasonable argument, referred to by Zhang et al. (2008), it may be less so in the context of banking regulation whose goal is precisely to prevent bank bankruptcies during tail events of the kind where correlations may overshoot their long-term averages. The difference in asset correlation among PIIGS and non-PIIGS domiciled banks, though of the same order as for the sample ending October 2010, namely larger for non-PIIGS, has become significantly more important. In Byström’s paper (2011), the correlations differ by a mere 0.05. Including another three years has brought this difference up to 0.19. The numbers show that a significant decrease in correlation among PIIGS domiciled banks is single-handedly responsible for this. An explanation for this might be that these three years, roughly encompassing the European sovereign-debt crisis, has seen PIIGS domiciled banks exposed to increased levels of idiosyncratic risk. Another interesting observation regarding correlations across domiciliation groups is that PIIGS show greater correlations with their non-PIIGS counterparts than among themselves. Again, this might be seen as another manifestation of a high level of idiosyncratic risk within the group of PIIGS domiciled banks, perhaps driven by national and/or regional factors, such that their asset returns correlate least among each other. In any case, it points towards the conclusion that PIIGS domiciled banks may be less homogenous of a group than is commonly believed.
The variation in domiciliation-discriminated asset correlations over the 10-year sample period is shown in Figure 4. As in Byström (2011), a clear upwards-sloping trend is observable for both groups, with slightly greater values for non-PIIGS, until mid-2010. Starting in 2011 however, a dramatic change in pattern across groups arises: reported correlations among PIIGS domiciled banks increases to a peak of 0.8 in year-end for it to consequently drop back down to its 2011 starting point over another year whereas PIIGS
22
domiciled banks follow an almost symmetrically inverse path with a through nearly as low as 0.2. Yet it seems that this happens just for correlation to revert back to their most consistent trend in the overall sample, namely roughly tracking each other with varying differentials. This noteworthy trend-reversal yet again coincides with the European sovereign-debt crisis. The dramatic drop in correlations for PIIGS domiciled banks could be explained using the same rationale as the one aforementioned. However, I can formulate no such equivalent for the non-PIIGS correlations pattern during this odd episode, and certainly not for the trend-reversal as a whole.
Figure 4
Average Pairwise Three-Month Trailing Window Asset Return Correlations for Non-PIIGS and PIIGS Domiciled Banks, September 9th 2004-December 31st 2013.
The nominal development of correlations also yields noteworthy observations. While the correlation at the start of the sample fluctuates between 0.2 and 0.3, the correlation at the end of the sample fluctuates between 0.4 and 0.6 having reached peaks as high as 0.85 and troughs as low as 0.15 throughout its rather volatile development in-between the two. Whether these trends are a result of different estimation methodologies, different firm samples or perhaps something to do with the tail events included in the sample is left for
23
future studies. Yet this is interesting because the estimates of asset correlation in the first two years are roughly in line with asset correlations assumed by the Basel III accord (0.12-0.24) and those typically reported in earlier studies. Assuming this lends support to the comparability of our results would warrant for careful reassessment of the assumed range and characteristics of asset correlations, generally presumed as static and low in magnitude relative to the correlations estimated throughout the remainder of the sample.
When we look at the evidence presented by the correlation analysis of our sample against the size (market capitalization) and probability of default (CDS spread) factors, on average such as in Table 2 and across time such as in Figure 5 and 6, the relationships returned seem to be in line with the majority of the literature11. That is that, asset correlations decrease in PD and increase in size.
Figure 5
Average Pairwise Three-Month Trailing Window Asset Return Correlations for Two of the Four Size-Sorted Groups of Banks, September 9th 2004-December 31st 2013.
11
24
Table 2 shows estimates for each quartile group. Except for one inconsistency in each scaled asset correlations across PD and size (PD: Quartile 1 < 2, SIZE: Quartile 2 < 3), results corroborate with the assumed relationships. Moreover, Figures 3 and 4 also seem to confirm this pattern’s consistence over time, albeit in a somewhat volatile way.
Figure 6
Average Pairwise Three-Month Trailing Window Asset Return Correlations for Two of the Four PD-Sorted Groups of Banks, September 9th 2004-December 31st 2013.
It may come to one’s attention however, that figures 4 to 6 share a significant resemblance across the whole length of the sample. And indeed this is the case, as it also is in Byström (2011), simply because the quartile groups involved in these figures are constituted by many banks common to each other. That is, the highest PD group, the smallest size group and the PIIGS group share many common cross-sections (banks) and vice versa (lowest PD, largest size and non-PIIGS). Whether this may be driven by specific trends or pure coincidence does not take away that this is as important as it can be misleading, as Byström (2011) does, to conclude from these figures that the sample lends support to the assumed dependencies of asset correlation with firm PD, size and
25
domiciliation. Especially, seen as Lee et al. (2009) report contradictory results for the asset correlation – PD relationship when size is properly controlled for, and all the more so when there is such a strong negative correlation between size and PD in this sample. Therefore we resort to a regression in an attempt to control for these effects simultaneously and establish what each of these factors’ true interactions with asset correlations are within this sample, if any at all.
The regressions and datasets used therein take the form specified in section (3.4). Summary descriptive statistics for the variables involved can be found in Table 6 in the appendix. Table 3 presents the panel data regression output.
Table 3
Panel Data Output: Average Asset-Return Correlations w.r.t. PD, SIZE and NP
Dependent Variable
Independent
Variables Average Asset-Return Correlation
Intercept 0.469*** (0.000) 0.356*** (0.006) 0.414*** (0.001) 0.466*** (0.008) 0.378*** (0.008) PD -1.717*** (0.072) -1.695*** (0.099) -0.345*** (0.102) SIZE 0.050*** (0.003) (0.004) 0.001 0.020*** (0.004) NP 0.071*** (0.001) 0.067*** (0.001) N 53438 53438 53438 53438 53438 - 0.01 0.05 0.04 0.01 0.17
Significance at the 10*, 5** and 1*** percent level. Probability of Default (PD) is proxied by means of the firms’ 10-year CDS spread. Size is based on the market capitalization. NP is Non-PIIGS dummy variable.
Interestingly, under the assumptions inherent to the panel data specification, all independent variables return estimations in line with the assumed relationships systematically over all specifications. Firm size however, turns out to be insignificant in the specification excluding the Non-PIIGS variable. All other estimated regressors are highly significant, partly due to the large amount of time observations, but the returned - values are rather low. This could indicate potentially omitted variable bias, though noise in the
26
dependent variable is potentially a significant factor herein as well. In an attempt to investigate this while extending the instruments used in the analysis, we resort to an alternative specification by introducing unobservable effects into the specification.
The choice for fixed effects to deal with unobservable effects is motivated by economic rationale. The observation that there are significant deviations in the average level of pairwise asset correlations across banks motivates the use of cross-section fixed effects. Clusters of domiciliation concentration within the sample, i.e., there is only one Greece-domiciled bank but four U.K.-Greece-domiciled banks, likely drive greater average pairwise correlations for the latter as they may well share more risk factors common to their country of domiciliation. Entity-fixed effects may hence adjust, at least partly, for such effects. Moreover, time-fixed effects may also correct for unobservable effects common to all entities but variable across periods, such as the business cycle and shocks therein, as was observed by Lee et al. (2011). The two downturns in the sample period, namely the global financial crisis (GFC) and the European sovereign-debt or Eurozone (EZ) crisis may have had effects common, to an uncertain degree, to all entities’ pairwise asset correlations. The relative symmetry in correlation trends across individual banks, PD, size and domiciliation groups leading up to the sovereign-debt crisis lend support to this. The period following the outbreak of the Eurozone crisis however, does less so, due to the potentially asymmetric nature of the shock, which may have been responsible, as mentioned earlier, for the dramatic correlation-trend reversal visible in Figures 4,5 and 6. The fixed effects redundancy likelihood ratio test does nonetheless lend overall support to their implementation. Table 4 presents the output of the panel data regressions with fixed unobserved effects.
The signs of the relationships of firm PD and size reverse depending on which fixed-effects are implemented along with which independent variables. This shows how important simultaneously controlling for all variables is to the outcome. Most noteworthy is the effect on firm size, which returns a negative relationship though often statistically insignificant. Firm PD and country group of domiciliation (NP) return the expected relationships to asset correlation, suggesting that PD does have an effect on asset correlation after controlling for size, country of domiciliation and the economic cycle.
27
Table 4
LSDV Fixed Effects (FE) Output
Average Asset-Return Correlations w.r.t. PD, SIZE and NP
Dependent Variable
Independent
Variables Average Asset-Return Correlation
Intercept 0.445*** (0.000) 0.512*** (0.001) 0.539*** (0.006) 0.440*** (0.009) 0.538*** (0.006) 0.522*** (0.006) PD 1.587*** (0.069) -8.653*** (0.064) -9.099*** (0.075) 1.621*** (0.100) -6.189*** (0.076) -8.728*** (0.082) SIZE -0.008* (0.003) 0.002 (0.004) -0.018*** (0.003) -0.004 (0.003) NP 0.011*** (0.001) 0.011*** (0.001) Fixed Effects ____ Bank- specific Time ____ Time ____ ____ Bank- specific Time Bank- specific Time ____ N 53438 53438 53438 53438 53438 53438 - 0.26 0.63 0.63 0.26 0.77 0.63
Significance at the 10*, 5** and 1*** percent level. Probability of Default (PD) is proxied by means of the firms’ 10-year CDS spread. Size is based on the market capitalization. NP is Non-PIIGS dummy variable.
The inclusion of fixed effects significantly increases the - across the board; since they account for a significant share of the variance, especially time-fixed effects, it may imply that the business cycle has a significant effect on asset correlations. This result would corroborate with Lee et al. (2011) in that asset correlations are asymmetric and have a procyclical impact on the economy. Taking a closer look at the development time-fixed effects over time, displayed in Figure 4 below, further reinforces this argument: time-fixed effects are positive exclusively in the period starting in September 2007, which captures both crises. The pattern is noteworthy: time-fixed effects are relatively stable around -0.25 for the period ending September 2007, then display an unstable increasing path through to September 2008, the period which coincides with the outbreak and propagation of the global financial crisis, where after fixed effects stabilize at the higher level of roughly +0.15 through to the end of 2013 but for a maximum peak of +0.4 in September 2010 coinciding with the outbreak of the Eurozone crisis.
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Figure 4
Time-Fixed Effects for Specifications 1 to 4 (Left to Right in Table 4, only including Time Effects)
As a follow up on the evidence presented by the regression output including time fixed-effect suggesting the possibility that the global financial crisis (GFC) and the Eurozone crisis (EZC) may have contributed to higher asset correlations, dummy variables for the latter two events are introduced into a new set of regressions presented in Table 5. The GFC dummy variable is set to 0 until September 2007, where it begins a linear path increase to 1 by September 2008 and stays as such until the end of the sample period. This is motivated by allowing a crescendo-type dynamic to the event, supported by the dynamics of time-fixed effects throughout that period. For the EZC binary, we set the beginning date at January 22nd, 2010 when 10 central and eastern European banks simultaneously applied for bailouts, with no ending date by 2013’s year-end.
29
Table 5
Panel Data Output: Binary Variables for the Global Financial Crisis (GFC) and the Eurozone Crisis (EZ)
Dependent Variable
Independent
Variables Average Asset-Return Correlation
Intercept 0.348*** (0.001) 0.350*** (0.001) 0.105*** (0.007) 0.105*** (0.007) 0.039*** (0.008) 0.039*** (0.009) PD -4.906*** (0.090) -4.848*** (0.088) -2.179*** (0.086) -2.241*** (0.085) SIZE 0.106*** (0.003) 0.106*** (0.003) 0.144*** (0.003) 0.145*** (0.003) NP 0.029*** (0.001) 0.029*** (0.001) GFC 0.210*** (0.002) 0.169*** (0.001) 0.240*** (0.002) 0.247*** (0.001) 0.234*** (0.002) 0.228*** (0.001) EZC -0.044*** (0.002) 0.008*** (0.002) -0.008*** (0.002)
Fixed Effects ____ ____ ____ ____ specific Bank- specific Bank-
N 53438 53438 53438 53438 53438 53438
- 0.20 0.20 0.35 0.35 0.51 0.51
Significance at the 10*, 5** and 1*** percent level. Probability of Default (PD) is proxied by means of the firms’ 10-year CDS spread. Size is based on the market capitalization. NP is Non-PIIGS dummy variable. GFC and EZC are binary variables that represent the active periods of the Global Financial Crisis and the Eurozone Crisis respectively.
As shown in Table 5 above, results suggest that the period during and ensuing the global financial was characterized by a sharp increase in asset correlations within the sample and consistently across all specifications after controlling for firm size, PD and domiciliation. This further supports an empirical dependency between the state of the economy and asset correlations. The average increase in asset correlations following the crisis’ outbreak amounted to roughly 0.23. This is equivalent to an asset correlation jump of more than 50 per cent of the sample’s average or a doubling of average correlations from its pre-crisis average.
30
No such conclusive evidence is found for the Eurozone crisis however, as returned regressors for the event are inconsistent in sign and magnitude across specification. There is one issue that likely contributes to accounting for this result however, namely that the GFC dummy variable may have absorbed some of the effect that should have been attributed to the EZC seen as they overlap. Moreover, I doubt the correlation-peak in July 2010, which coincides almost perfectly with the outbreak of the EZC is just a coincidence.
Paradoxically, controlling for these crisis events, unaccounted for by the BCBS in their prescribed asset correlations, returns the assumed decreasing relationship of asset correlations in firm PD and increasing in firm size.
6. Conclusion
The general aim of this paper was to assess the validity of the Basel Accord’s parametric assumptions regarding asset correlations, both with respect to its prescribed magnitudes as its dependency to firm probability of default and firm size. To do this, the asset value computation methodology proposed by Byström (2011) was applied to an updated and extended version of the same paper’s sample, consisting of 22 EU banks from 2004 to 2013.
In stark contrast to the static properties assumed in the Basel Accord’s credit risk capital requirements regulation, our results suggest asset correlations to be significantly dynamic, even after controlling for the dependencies to firm size and PD. Estimated asset correlations within our sample displayed a rising trend, initially in the 0.2-0.3 range as assumed by BCBS, to an average 0.43 over the whole period, with peaks as high as 0.85. It should be duly noted however that these results originate from a sample that considers firms within the same sector (banking) and region (EU) over a period subject to multiple economic shocks (financial crises and sovereign-debt crisis), all factors likely to bias correlation estimates upwards. The correlation range prescribed by the Basel Accord is meant for banks’ corporate loan portfolios, assumed to be more diversified than the sample
31
this paper subjects to analysis. Still, this paper’s results can be considered as a warning for more concentrated portfolios, especially during periods of economic stress.
Evidence regarding the relationship of asset correlation with size and PD within the sample, mapped in a two-dimensional setting, seemed to corroborate with the generally accepted view that asset correlations increase in company size but decrease in PD. However, due to concerns that simultaneously controlling for both effects may return different results, as it did in Lee et al. (2009), we turned to panel data regressions. Paradoxically, results supported these dependencies best only after simultaneously controlling for the staggering asset correlation increases during and ensuing the outbreak of the global financial crisis, an effect unaccounted for by the BCBS. This is not new, as evidence pointing to this link already exists. Lee et al. (2011) namely conclude that asset correlation increase during downturns and decrease during upturns. Our results suggest that the period after the direct aftermath of the global financial crisis was characterised by a doubling of asset correlations. However, there is no evidence suggesting asset correlations are (yet) reverting back to pre-crisis levels. This is worrying, and as such may have strong policy implications both regarding capital requirements, and potentially also, the interbank lending market.
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35 Appendix:
Table 5
Descriptive statistics by cross-section: Daily asset return series for the 22 EU banks within the sample, June 9th 2004 – December 31st 2013
Mean Std Max. Min. Skw Kur
Erste Bank 0.0059 1.11 6.74 -10.28 -0.25 13.9 BNP Paribas 0.0001 0.61 6.48 -4.36 0.25 16.9 Credit Agricole 0.0029 0.67 8.22 -4.95 0.45 19.4 Societe Generale -0.0016 0.71 8.26 -6.25 0.06 20.5 Deutsche -0.0019 0.52 4.26 -4.16 0.19 14.3 Commerzbank -0.0014 0.62 4.05 -7.13 -0.66 18.2 Alpha 0.0554 3.39 65.7 -43.82 4.85 105.0 Bank of Ireland 0.0206 2.43 40.46 -26.22 2.43 79.0 Unicredit 0.0076 1.49 20.33 -12.97 1.48 35.0
Intesa San Paolo 0.0088 1.36 20.65 -11.68 1.58 35.5
Monte dei Paschi -0.0078 1.07 12.44 -9.00 0.52 20.6
Banco Comercial -0.0054 1.40 23.26 -15.84 3.49 73.3
Banco Espirito Santo -0.0033 1.11 14.08 -7.56 1.06 24.1
BBVA 0.0049 0.93 11.36 -4.07 0.85 14.9 Banco Popular -0.0027 0.97 13.44 -10.34 0.92 37.3 Nordea 0.0103 0.89 5.85 -16.33 -1.85 51.7 Handelsbanken 0.0032 0.28 2.51 -1.94 -0.09 10.7 Swedbank 0.0091 0.51 7.00 -7.73 -0.91 49.9 RBS -0.0030 0.72 10.51 -7.74 2.14 45.5 HSBC 0.0052 0.63 4.71 -4.06 0.01 9.8 Barclays -0.0009 0.63 7.82 -6.30 0.57 26.5 Lloyds 0.0102 0.78 10.76 -5.43 1.68 26.1
Notes: Mean, Std, Min and Max are all multiplied by 100 (i.e., they are expressed in daily percentages). Skw indicates skewness, and Kur indicates excess kurtosis.
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Table 6
Regression Data Descriptive Statistics
CORR PD SIZE NP Mean 0.457 0.007 2.005 0.591 Median 0.485 0.004 2.049 1 Maximum 0.883 0.089 2.581 1 Minimum -0.179 0.0003 0.697 0 Std. Dev. 0.174 0.010 0.248 0.492 Skewness -0.445 3.493 -1.32 -0.369 Kurtosis -0.202 19.442 2.824 -1.863 Jarque-Bera 1859.8 710640.6 33404.6 8947.9 Probability 0.000 0.000 0.000 0.000 Sum 24405.6 389.8 107164.5 31577 Sum Sq. Dev. 1621.42 5.776 3306.89 12917.86 Obs. 53438 53438 53438 53438 Table 7
Data Series Correlation Matrix
PD SIZE NP CORR
PD 1.000 ____ ____ ____
SIZE -0.6269 1.000 ____ ____
NP -0.206 0.148 1.000 ____
