Fire Sales, Indirect Contagion and Systemic Stress Testing

Cont, Rama; Schaanning, Eric Working Paper Fire Sales, Indirect Contagion and Systemic Stress Testing Working Paper, No. 2/2017 Provided in Cooperation with: Norges Bank, Oslo Suggested Citation: Cont, Rama; Schaanning, Eric (2017) : Fire Sales, Indirect Contagion and Systemic Stress Testing, Working Paper, No. 2/2017, ISBN 978-82-7553-963-0, Norges Bank, Oslo, https://hdl.handle.net/11250/2495596 This Version is available at: https://hdl.handle.net/10419/210112 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc-nd/4.0/deed.no Fire sales, indirect contagion and systemic stress testing Norges BaNk research 2 | 2017 Rama Cont and ERiC SChaanning WorkiNg PaPer Norges BaNk Working PaPer xx | 2014 rapportNavN 2 Working papers fra Norges Bank, fra 1992/1 til 2009/2 kan bestilles over e-post: [email protected] Fra 1999 og senere er publikasjonene tilgjengelige på www.norges-bank.no Working papers inneholder forskningsarbeider og utredninger som vanligvis ikke har fått sin endelige form. hensikten er blant annet at forfatteren kan motta kommentarer fra kolleger og andre interesserte. Synspunkter og konklusjoner i arbeidene står for forfatternes regning. Working papers from Norges Bank, from 1992/1 to 2009/2 can be ordered by e-mail: [email protected] Working papers from 1999 onwards are available on www.norges-bank.no norges Bank’s working papers present research projects and reports (not usually in their final form) and are intended inter alia to enable the author to benefit from the comments of colleagues and other interested parties. Views and conclusions expressed in working papers are the responsibility of the authors alone. ISSN 1502-819-0 (online) ISBN 978-82-7553-963-0 (online) Fire sales, indirect contagion and systemic stress testing Rama Cont Eric Schaanning∗ Imperial College London Norges Bank Research March 17, 2017 Abstract We present a framework for quantifying the impact of fire sales in a network of financial institutions with common asset holdings, subject to leverage or capital constraints. Asset losses triggered by macro-shocks may interact with one-sided portfolio constraints, such as leverage or capital constraints, resulting in liquidation of assets, which in turn affects market prices, leading to contagion of losses and possibly new rounds of fire sales when portfolios are marked to market. Price-mediated contagion occurs through common asset holdings, which we quantify through liquidity-weighted overlaps across portfolios. Exposure to pricemediated contagion leads to the concept of indirect exposure to an asset class, as a consequence of which the risk of a portfolio depends on the matrix of asset holdings of other large and leveraged portfolios with similar assets. Our model provides an operational stress testing method for quantifying the systemic risk arising from these effects. Using data from the European Banking Authority, we examine the exposure of the EU banking system to price-mediated contagion. Our results indicate that, even with optimistic estimates of market depth, moderately large macro-shocks may trigger fire sales which may then lead to substantial losses across bank portfolios, modifying the outcome of bank stress tests. Price-mediated contagion leads to a heterogeneous cross-sectional loss distribution across banks, which cannot be replicated simply by applying a macro-shock to bank portfolios in absence of fire sales. Unlike models based on ‘leverage targeting’, which assume symmetric reactions to gains or losses, our approach is based on the asymmetric interaction of portfolio losses with one-sided constraints, distinguishes between insolvency and illiquidity and leads to substantially different loss estimates in stress scenarios. ∗This working paper was prepared as part of the authors’ work at Norges Bank Research. The views expressed are those of the authors and do not necessarily reflect those of Norges Bank and at the Isaac Newton Institute for Mathematical Sciences, Cambridge, during the programme on Systemic Risk: Mathematical modeling and Interdisciplinary Approaches, supported by EPSRC grant no EP/K032208/1. We thank Farooq Akram, Tobias Adrian, Laurent Clerc, Fernando Duarte, Darrell Duffie, Thomas Eisenbach, Martin Hellwig, Anil Kashyap, Andrei Kirilenko, David Levermore, Barbara Meller, David Murphy, Sergio Nicoletti-Altimari, Martin Summer and Lakshithe Wagalath for stimulating discussions on this topic. Eric Schaanning is grateful for funding from the Fonds National de la Recherche Luxembourg (FNR) for his PhD studies under the AFR Grant scheme. We are grateful for the helpful and constructive suggestions of an anonymous referee that improved the presentation of our results. 1 Contents 1 Introduction 3 1.1 The need for macroprudential stress tests . . . . . . . . . . . . . . . . . . . 3 1.2 Summary and main findings . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 Outline...................................... 6 2 Modeling spillover effects from fire sales 7 2.1 Balance sheets and portfolio constraints . . . . . . . . . . . . . . . . . . . . 7 2.2 Stressscenarios ................................. 8 2.3 Deleveraging................................... 9 2.4 Market impact and price-mediated contagion . . . . . . . . . . . . . . . . . 11 2.5 Feedback loops, insolvency and illiquidity . . . . . . . . . . . . . . . . . . . 14 2.6 Comparison with “leverage targeting” . . . . . . . . . . . . . . . . . . . . . 15 3 A systemic stress test of the European banking system 16 3.1 Data....................................... 16 3.2 Market impact and market depth . . . . . . . . . . . . . . . . . . . . . . . 18 3.3 Portfoliooverlaps................................ 21 3.4 Stressscenarios ................................. 24 3.5 Systemic stress test of European banks: results . . . . . . . . . . . . . . . 26 4 Indirect exposures 33 4.1 Notional vs effective exposures . . . . . . . . . . . . . . . . . . . . . . . . . 33 4.2 Indirect exposures: empirical evidence . . . . . . . . . . . . . . . . . . . . 35 4.3 Relevance of indirect contagion for bank stress tests . . . . . . . . . . . . . 38 4.4 Why indirect exposures cannot be reproduced in single-bank stress tests . . 41 5 Implications for macroprudential stress testing and regulation 43 A Appendix 48 A.1 Sourcesformarketdata ............................ 48 A.2 EBA: data identifiers and residual exposures . . . . . . . . . . . . . . . . . 49 2 1 Introduction 1.1 The need for macroprudential stress tests In October 2008, the IMF’s estimate of losses in the US subprime mortgage sector was on the order of 500 bn USD, a large loss but still lower than, say, the loss from the Dotcom bubble burst Hellwig (2009). However, as the US subprime crisis developed into a full-blown global financial crisis, losses spilled over into other asset classes, sectors and countries and ballooned into trillions of dollars Hellwig (2009), exceeding the losses of the Dot-com bubble by an order of magnitude. Supervisory stress tests for banks, which have become a cornerstone of financial regulation, have focused on examining the resilience of bank balance sheets to severe stress scenarios. However, as the above example illustrates, loss amplification mechanisms which may compound initial losses may be just as important for understanding the nature of systemic risk. Indeed, as pointed out in the Basel Committee on Banking Supervision’s recent report, stress tests conducted by bank supervisors still lack a genuine macroprudential component Basel Committee on Banking Supervision (2015). The report identifies the key missing ingredients as “endogenous reactions and feedback effects to initial stress”. As noted by ECB Vice-President Vitor Constˆancio Anderson (2016), in the current approach to bank stress tests “no bank reaction is considered. It would be far more realistic to assume that market participants could react to adverse conditions, rather than assuming passive bank behaviour throughout the entire stress test period. Bank behaviour or reaction could take the form of deleveraging, straight capital increases or working out of non-performing loans.” There is ample empirical evidence that such deleveraging occurred on a large scale in 2008, leading to cross-asset contagion and amplification of losses in the financial system Kashyap et al. (2008); Brunnermeier and Pedersen (2009); Khandani and Lo (2011); Manconi et al. (2012); Cont and Wagalath (2016). The well-documented occurrence of fire sales during market downturns Ellul et al. (2011); Coval and Stafford (2007); Shleifer and Vishny (2011); Jotikasthira et al. (2012) is not a coincidence: portfolio constraints -capital, leverage or liquidity constraintsthat financial institutions are subject to forces them to deleverage when these constraints are breached as a result of losses, leading to fire sales of assets Kyle and Xiong (2001); Cont and Wagalath (2013). Similar largescale deleveraging is also foreseeable in future stress scenarios, and foreseen by financial institutions themselves: “If we are unable to raise needed funds in the capital markets (including through offerings of equity and regulatory capital securities), we may need to liquidate unencumbered assets to meet our liabilities. In a time of reduced liquidity, we may be unable to sell some of our assets, or we may need to sell assets at depressed prices, which in either case could adversely affect our results of operations and financial condition” Credit Suisse (2015). Fire sales generate endogenous risk Shin (2010) and can act as channel of loss contagion across asset classes and across financial institutions holding these assets Cont and Wagalath (2016); Caccioli et al. (2014). Unlike direct contagion through counterparty exposures Cont et al. (2013), fire-sales spillovers are mediated by prices and thus defy limits on counterparty exposures and institutional ring-fencing measures. As noted by Glasserman and Young (2014), following the introduction of large exposure limits and collateral requirements, the likelihood of direct contagion through counterparty exposures has diminished in the banking system. 3 It is therefore important for supervisors to include in macro-stress testing frameworks used for assessing bank capital adequacy the impact of fire sales and the deleveraging of portfolios in stress scenarios. This is especially relevant given that, post-crisis, supervisory stress tests have set a binding constraint for bank capital adequacy.1 Fire sales and the resulting destabilizing feedback effects have been extensively studied in the literature Kyle and Xiong (2001); Shleifer and Vishny (2011) from a conceptual viewpoint. The challenge is to develop a quantitative framework versatile enough to be taken to empirical data and used in an operational macro-stress testing framework to quantify the endogenous risk and spillover effects arising from fire sales. The goal of the present work is to address this challenge, by proposing a modeling framework for quantifying the exposure of the financial system to the endogenous losses and feedback effects resulting from fire sales in a macro-stress scenario. We provide a detailed discussion of the model, its use for the design of systemic stress tests, and the results obtained by applying the methodology to EU bank portfolios. Previous attempts by regulators to account indirectly for the impact of fire sales in bank stress tests include the use of (exogenously specified) liquidation costs or increasing the severity of shocks in single-bank stress tests to account for possible loss amplification due to feedback from fire sales. These adjustments may mimick the severity of potential losses which may result from fire sales but fail to capture key cross-sectional features of fire-sales spillovers, such as the contagion across asset classes and the heterogeneous distribution of fire-sales losses across financial institutions and across asset classes of varying liquidity. More recently, Greenwood et al. (2015); Duarte and Eisenbach (2013) have proposed a stress testing approach incorporating the impact on asset prices of deleveraging in bank portfolios, based on the assumption of leverage-targeting Adrian and Shin (2010) i.e. that, in response to a shock, financial institutions rebalance their portfolios to maintain a constant leverage, which leads to a linear deleveraging rule in reaction to market shocks. This approach was used to analyze fire-sales spillovers in the EU banking system by Greenwood et al. (2015) and in the US banking system by Duarte and Eisenbach (2013). Both studies find evidence of potentially large exposures of the banking system to contagion via fire sales. This approach has been explored by supervisors as a possible method for incorporating fire sales in macro stress tests Henry et al. (2013); Cappiello et al. (2015). There is some empirical evidence that in the medium term large financial institutions maintain fairly stable levels of leverage Adrian and Shin (2010) but it is not clear why the same institutions would enforce such leverage targets in the short term, especially in stress scenarios where this could entail high liquidation costs. We propose a different approach for modeling fire sales, based on the premise that deleveraging by financial institutions occurs in reaction to losses in their portfolios Kyle and Xiong (2001); Cont and Wagalath (2013). This deleveraging may be the result of investor redemptions for funds, as evidenced in Ellul et al. (2011); Coval and Stafford (2007); Shleifer and Vishny (2011); Jotikasthira et al. (2012); but for regulated financial institutions such as banks, large scale deleveraging is mainly driven by portfolio constraints – capital, leverage or liquidity constraints – which may be breached when large losses occur. We have focused for simplicity on leverage constraints, but the model is easily extendable to multiple constraints on portfolios. Given the one-sided nature of these constraints, such institutions react asymmetrically to large losses and large gains Ang 1See the Section “Process and Requirements after CCAR 2016: https://www.federalreserve.gov/ newsevents/press/bcreg/bcreg20160629a1.pdf. 4 et al. (2006). The asymmetry and the threshold nature of deleveraging differentiates our approach from models based on leverage targeting, leading to quite different outcomes. Deleveraging by financial institutions impacts market prices and, when portfolios are marked to market, leads to further losses which may in turn trigger further deleveraging. We quantify this impact and the resulting endogenous risk, paying attention to the estimation of market impact parameters; the magnitude of these parameters, and their heterogeneity across asset classes, is shown to greatly influence the results of the stress tests. We use these ingredients to design a systemic stress testing framework for banking systems. Application of the method to the EU banking system shows that this approach may lead to outcomes which are substantially different from single-bank stress tests. We emphasize in particular the concept of indirect exposure, which we show to be relevant for bank stress testing. 1.2 Summary and main findings We present a framework for quantitative modeling of fire sales in a network of financial institutions with common asset holdings, subject to leverage or capital constraints. Asset sales may be triggered in reaction to external shocks to asset values when portfolios are subject to capital or leverage constraints; the market impact of these asset sales then leads to contagion of mark-to-market losses to other portfolios, which may in turn be led to deleverage if their constraints are breached. In contrast to balance sheet contagion which arises through direct bilateral exposures, this price-mediated contagion occurs through common asset holdings, even in absence of direct linkages between financial institutions. The resulting feedback loop may lead to loss amplification, systemic risk and large-scale instability of the financial system. Our model provides a method for quantifying the exposure of the financial system to these effects, which we apply to data on European banks. This leads to several interesting findings. •Existence of a tipping point: we show the existence of a critical macro-shock level beyond which fire sales trigger considerable contagion. The level of this critical shock depends on the institutions’ leverage, as well as the concentration, commonality and liquidity of their asset holdings. In the European banking system, this critical shock size is found to correspond to large, but not extreme, losses in asset values. •Magnitude and heterogeneity of losses due to fire sales: we find that fire sales contribute significantly to system-wide losses in stress scenarios, accounting for more than 35% of the total losses and between 20 to 40% of system bank equity. These results are significant enough to modify the outcome of bank stress tests. Moreover, while the total system-wide loss can always be replicated in a stress test without fire sales by applying a larger shock to assets, the heterogeneous crosssectional distribution of losses across banks cannot be reproduced in absence of fire sales by simply applying a larger macro-shock. •Importance of gain-loss asymmetry and the threshold nature of fire sales: We argue that fire sales arise when portfolio constraints such as leverage, liquidity or capital ratios are breached as a result of large portfolio losses. The one-sided nature of these constraints leads to an asymmetric reaction of banks to gains and losses, which differentiates our model from the ‘leverage targeting’ model used in some 5 previous studies of fire sales Adrian and Shin (2010); Duarte and Eisenbach (2013); Greenwood et al. (2015). Comparison with stress tests based on leverage targeting shows substantial differences in outcomes: leverage targeting models overestimate the magnitude of fire sales, especially at smaller shock levels, but underestimate the acceleration (convexity) of fire sales with increasing shock size present in the threshold model. •Distinction between insolvency and illiquidity: unlike previous models of contagion, which have mainly focused on modeling insolvency, our model distinguishes between failure due to insolvency and failure due to illiquidity. We observe that, while insolvency is the dominant mode of failure of banks in scenarios associated with extremely large initial shocks, illiquidity is the dominant mode of failure in scenarios associated with moderate shocks which are nevertheless large enough to trigger fire sales. •Indirect exposures: As a result of fire-sales spillovers, a portfolio’s exposure to an asset class in a stress scenario may be larger than its notional exposure. This naturally leads to the notion of indirect exposure, which is scenario-dependent, and can be quantified using our model. One striking finding is that many EU banks have significant (indirect) exposures to asset classes they do not hold, such as commercial and residential mortgages in EU countries where they do not issue loans. •Sensitivity to market depth across asset classes: Calibration to data on market prices, trading volumes and turnover reveals a significant dispersion in market depth across asset classes. We show that ignoring this heterogeneity of market depth across asset classes leads to a considerable bias in the estimation of fire-sales losses in stress tests. This highlights the importance of conducting a rigorous sensitivity analysis on liquidity estimates. •Second-round effects are significant: fire sales may lead to a feedback loop which generates market losses and further fire sales across other financial institutions. Ignoring these feedback effects and the corresponding secondand higherround deleveraging may lead to a significant underestimation of system-wide losses. These findings have many implications for risk management in financial institutions and for the monitoring systemic risk in the financial sector. In particular, they underline the need for a systemic approach to stress testing and the necessity of macroprudential tools for tackling the risks resulting from price-mediated contagion. We believe the model presented here provides a useful tool for monitoring of system-wide and bank-level exposure to price-mediated contagion. 1.3 Outline The paper is structured as follows. Section 2 introduces our modeling framework and outlines a method for systemic stress testing in presence of fire sales. Section 3 describes the application of this stress testing approach to the European banking system using data from the European Banking Authority (EBA) and details our empirical findings. In Section 4, we introduce the concept of indirect exposure resulting from fire-sales spillovers and illustrate the magnitude of indirect exposures in the European banking system. Section 5 discusses implications of our findings for systemic stress testing, risk management in institutions and macroprudential policy. 6 prices. In Cont and Schaanning (2017) it is shown that α≥1 2is a plausible assumption, also from a modeling perspective. Summing (15) with (14) yields the total loss of portfolio iat the k-th round of develeraging: Li(Πk−1, Ck−1) = Mi(Πk−1, Ck−1) + Ri(Πk−1, Ck−1) (16) =1−(1 −α)Γi(Πk−1, Ck−1)M X µ=1 Πi,µ k−1Ψµ N X j=1 Γj(Πk−1, Ck−1)Πj,µ k−1!. This loss reduces the equity of institution iby the same amount: Ci k=Ci k−1−Li(Πk−1, Ck−1)+(17) Linearizing the market impact function Ψµyields Li(Πk−1, Ck−1)≈(1 −(1 −α)Γi) M X j=1 N X µ=1 Πi,µ k−1Πj,µ k−1 Dµ | {z } Ωij (Πk−1) Γj= (1 −(1 −α)Γi) M X j=1 Ωij(Πk−1)Γj, (18) which shows that the magnitude of fire-sales spillovers from institution ito institution j is proportional to the liquidity-weighted overlap Ωij between portfolios iand jCont and Wagalath (2013): Ωij(Π) := M X µ=1 Πi,µΠj,µ Dµ .(19) The matrix of portfolio overlaps Ω(Π) = ΠD−1Π>,(20) where Dis the diagonal matrix of market depths Dµ, can be viewed as a weighted adjacency matrix of the underlying network, linking portfolios through their common exposures. We will further analyze the properties of this matrix in Section 3. In summary, an initial loss in asset values may trigger a feedback loop, schematically represented in Figure 3, in which, at each iteration, portfolio deleveraging leads to fire sales, leading to price declines and mark-to-market losses which may in turn trigger further fire sales. The state variables representing the matrix Π of portfolio holdings in marketable assets and the equity levels C= (Ci, i = 1..N) are initialized as described in equations (3)-(4) (for k= 0) and updated at each round of deleveraging (Πk, Ck) = f(Πk−1, Ck−1),(21) where Πkis defined in (13) and Ci k=Ci k−1−Li(Πk−1, Ck−1)+(22) where the loss Liis defined in (16). 13 Initial shock Deleveraging Mark to market losses Market impact to assets Figure 3: An initial loss in asset values may generate a feedback loop which may lead to multiple rounds of deleveraging and further declines in asset values. 2.5 Feedback loops, insolvency and illiquidity The iteration described above continues in principle as long as at least one institution is in breach of its leverage/ capital constraint after losses due to deleveraging are accounted for. However, in the (realistic) situation where we assume that institutions build a nonzero buffer beyond the minimal capital requirements (i.e. λb< λmax in the notation of Section 2.3 ), this fire-sales cascade terminates after a finite number Tof iterations Cont and Schaanning (2017). As we will discuss below, this is not the case in leverage targeting models, which lead to infinite fire-sales cascades. Along the way, some institutions may become insolvent: this occurs if at any point in the iterations the loss Li(Πk, Ck) exceeds the capital Ck. Then instution ibecomes insolvent and does not play any further role in subsequent rounds. Another type of failure which may occur along the cascade is failure due to illiquidity: this occurs when an institution has sold all of its marketable assets and is left with no further liquid assets. This may occur even though the institution is still solvent. This distinction between failure due to insolvency and failure due to illiquidity is highly relevant in practice. In fact, one can note that this was precisely the scenario that occurred in the failure of Bear Stearns and Lehman Brothers.4In contrast to most default risk models and previous studies on fire sales, our model distinguishes between these two causes of failure and highlights the fact that institutions can fail even when they have positive equity. In contrast to models of default contagion, contagion of losses across institutions occurs not just at default but actually before the default of an institution, and its scope is not limited to counterparties. Deleveraging by distressed institutions, which is precisely aimed at preventing their default, is in fact what triggers this contagion. Denoting by Tthe length of the cascade, the fire-sales loss for bank itriggered by the stress scenario is given by FLoss(i, ) = T X k=1 Li(Πk−1, Ck−1).(23) and the total system-wide fire-sales loss in this scenario is SLoss() = N X i=1 FLoss(i, ).(24) 4See letter by the then SEC chairman Christopher Cox to the Basel Committee on Banking Supervision https://www.sec.gov/news/press/2008/2008-48.htm. 14 Note that the fire-sales loss (23) does not include the initial loss which triggers the deleveraging: in absence of deleveraging and price-mediated contagion FLoss(i) = SLoss = 0. Table 1 summarizes the notations of the model and references the equations where they are defined. Variable Notation Defined in Financial institutions i, j = 1..N - Asset class: illiquid assets κ= 1..K Section 2.1 Asset class: marketable assets µ= 1..M Section 2.1 Number of iterations (rounds) kState variables (in EUR) Marketable assets Πi,µ Section 2.1 Capital CiSection 2.1 Parameters Illiquid asset (in EUR) Θi,κ Section 2.1 Initial shock (in %) κ(4) Market depth for asset class µ Dµ(13) Key quantities Deleveraging proportion at round kΓi k(9) Leverage of institution i λi(1) Fire-sales loss (k-th round) Li(Πk−1, Ck−1) (16) Fire-sales loss for bank i(all rounds) FLoss(i, ) (23) System-wide fire-sales loss SLoss() (24) Table 1: Overview of model notations. 2.6 Comparison with “leverage targeting” Recent empirical studies on fire-sales spillovers Duarte and Eisenbach (2013); Greenwood et al. (2015) have explored a different mechanism for fire sales, based on the idea that banks maintain a ‘leverage target’, which leads them to rebalance their portfolios in a procyclical manner following changes in asset values. An oft-cited argument to support this model is the empirical correlations between quarterly changes in asset size and debt size for banks Adrian and Shin (2010, 2014). While both models incorporate the idea of market impact of deleveraging and the resulting endogenous portfolio losses and contagion effects, they differ in some important ways: 1. The threshold nature of fire sales: in the leverage targeting model, bank (de)leveraging in response to arbitrarily small changes in asset values, regardless of their capital or liquidity buffers, generates fire-sales losses even for low market stress levels. In our model, deleveraging only occurs when losses are large enough to trigger portfolio constraints: for shocks below this critical level, there is no deleveraging. By assuming that all institutions constantly respond to arbitrarily small changes in asset values, the leverage targeting model overestimates the magnitude of deleveraging in response to small shocks. This is illustrated in Figure 2, which compares the overall deleveraging across EU banks in response to losses on exposures to the Spanish housing market. 15 2. Dependence of deleveraging on magnitude of losses: Leverage targeting implies a volume of deleveraging linear in the size of the portfolio loss; since the volume of deleveraging is capped at 100% of assets, this leads to a concave dependence of the volume of asset sales on the shock size. By contrast, for small to moderate shocks, the volume of deleveraging has a convex dependence on the loss size in our model, as shown in Section 2.3 and illustrated in the example of Figure 2: deleveraging accelerates as we increase the shock size to more extreme levels, leading to a ‘multiplier effect’, absent in the leverage targeting model. 3. Finite length of fire-sales cascades: The assumption of leverage targeting leads to an infinite sequence of iterations which never cease since at each round further mark-to-market losses are generated endogenously, which leads to a deviation from the target leverage and in turn generates new asset sales or purchases. In stress tests, one then needs to choose an ad-hoc number of iterations to compute the loss. Although losses converge as we iterate this cascade, in general estimates of fire-sales losses depend on the actual number of iterations that chosen in a simulation. By contrast, as shown in Cont and Schaanning (2017), in a threshold model with a capital buffer λb< λmax, the fire-sales cascade always terminates after a finite number of iterations, typically 5 to 10 rounds in most empirical examples, as shown in the next section. The consequences of these differences are explored in more detail in the next section, where we compare the results of stress tests performed using the two approaches, and in the companion paper Cont and Schaanning (2017). To implement the leverage targeting model in our stress test, we simply replace the deleveraging function by Γi(Π, C0()) = PM µ=1 Πi,µ +Ii()−λbCi 0() PM µ=1 Πi,µ ∧1!. Only marketable assets are assumed to be available for deleveraging. This assumption is different from Duarte and Eisenbach (2013); Greenwood et al. (2015), where deleveraging is applied to the entire portfolio. 3 A systemic stress test of the European banking system We now describe how the model may be used to perform a systemic stress test, in order to quantify the exposure of the banking system to fire-sales spillovers, and apply the framework to data on the European banking system. 3.1 Data Our empirical study is based on data from the European Banking Authority (EBA), which provides information, collected in 2011 and 2016, on notional exposures of 90 European banks across 148 asset classes.5Holdings are given by asset class and geographical region. 5This dataset was also used in the study by Greenwood et al. (2015), and facilitates comparison with the literature. 16 C Capital Assets Deleveraging zone Leverage constraint No deleveraging zone max b leverage target I(✏) ⇧+I(✏) Illiquidity Insolvency Figure 4: Evolution of asset values (vertical axis) and capital (horizontal axis) in a firesales cascade. The solid red line corresponds to the leverage constraint, the dashed green line denotes a target leverage corresponding to an excess capital buffer, and the dotted blue line is the path of a sample portfolio. Losses in asset values erode the equity, moving the portfolio closer to the origin; when the portfolio crosses the red line corresponding to the leverage constraint, deleveraging occurs: the institution tries to reconstitute a buffer by returning to the target leverage (dotted line). The market impact of these asset sales leads to further losses and displaces the state to the left; if it crosses the red line again, a new round of deleveraging follows etc. An institution becomes insolvent when it reaches the boundary C= 0 (vertical axis) and illiquid when it reaches the boundary Π = 0 (horizontal dashed line). Asset classes are specified in Table 2. Greenwood et al. (2015) assumed all assets to be available for liquidation; we only consider a subset to be marketable, i.e. available for liquidation at short notice. We identify four classes of marketable assets (“securities”), which may be liquidated in a stress scenario; the other asset classes are classified as illiquid assets.6Assets are further labelled by 37 geographical regions, which correspond to the 27 countries of the EU (i.e. without Croatia at the time) plus the United States (US), Norway (NO), Iceland (IS), Liechtenstein (LI), Japan (JP), Asia (A1), Other non-EEA non-emerging countries (E3), Eastern Europe non-EEA (E5), Middle and South America (M1) and Rest of the world (R5). Hence, with the four marketable asset classes and 37 geographical regions, the matrix of marketable assets Π is given by a 90 ×148 matrix. 6Table 10 in the Appendix, we provide the data identifiers that allow correspondence with the EBA dataset. 17 The illiquid asset holdings are given by a 90 ×75 matrix Θ. This corresponds to 74 asset classes for commercial and residential mortgage exposures respectively in the 37 regions and a 75th entry consisting of all remaining illiquid asset holdings.7 Illiquid assets Residential mortgage exposures Commercial real estate exposure Retail exposures: Revolving credits, SME, other Indirect sovereign exposures in the trading book Defaulted exposures Residual exposure (cf. Appendix Table 10) Securities / marketable assets Corporate bonds Sovereign debt Direct sovereign exposures in derivatives Institutional client exposures: interbank, CCPs,... Table 2: Asset classes used for the stress test. 3.2 Market impact and market depth A key assumption in models of fire-sales spillovers concerns the impact of asset liquidations on market prices. This may be summarized in the choice of a market impact function Ψµ, which defines the correspondence between the liquidation size q(in monetary units) and the relative price change for each asset class µ:∆Sµ Sµ=−Ψµ(q).An adequate choice for Ψµshould be increasing, concave, satisfy Ψµ(0) = 0 and lead to non-negative prices. Common specifications are the linear model Kyle (1985); Bertsimas and Lo (1998), Almgren and Chriss (2000); Obizhaeva (2012); Cont et al. (2014) Ψµ(q) = q Dµ with Dµ=cADVµ σµ (25) where ADVµis the average daily trading volume (in EUR), σµthe daily volatility (in %) of the asset, ca coefficient close to 0.5, estimated from transactions data, and the square root model Bouchaud (2010) Ψµ(q) = c σµrq ADVµ (26) We note that both trading volume and volatility are associated with a liquidation horizon τ, taken in most studies to be daily by default. If we assume the liquidation horizon τ to be longer than a day, then the market depth parameter needs to be adjusted. In the linear impact model, the adjustment is: Dµ(τ) = cADVµτ σµ√τ=cADVµ σµ×√τ. (27) 7Our representation differs slightly from Greenwood et al. (2015), who considered 42 asset classes consisting of the 37 sovereign exposures by geographical region and five further classes, aggregated across all geographical regions: “commercial real estate”, “mortages”, “corporate loans”, “small and medium enterprise loans” and “retail revolving credit lines”.This leads to a less granular model compared to ours as we distinguish assets both by type and country. 18 This adjustment is important, and corresponds to the intuitive observation that liquidating the same portfolio over a longer horizon reduces impact. The liquidation horizon τ may be interpreted as the time window the banks dispose of to comply with portfolio constraints. In the case study below we will use τ= 20 days. By contrast, in the square root model, the impact of a transaction is invariant to a change in the liquidation horizon since the denominator and the numerator in (26) scale in the same way. This leads to the counterintuitive (and, we believe, incorrect) conclusion that impact is insensitive to the rate of liquidation. For this reason, we refrain from using the square-root model in the sequel. In a linear impact model, asset classes are differentiated according to their market depth Dµ. Greenwood et al. (2015) assume a uniform depth Dµ= 1013 (EUR) for all asset classes. This homogeneity assumption is not supported by empirical studies on market impact Bouchaud (2010); Obizhaeva (2012); Cont et al. (2014), which indicate that market impact varies widely across assets. Ignoring this heterogeneity may lead to biased results, overestimating losses in more liquid asset classes while underestimating losses in less liquid asset classes. Duarte and Eisenbach (2013) use haircuts and repo rates for determining the liquidity of different asset classes; this does introduce some heterogeneity across asset classes but the relation between these quantities and market impact is not clear. For instance, haircuts may simply reflect the volatility of an asset, rather than its liquidity or market depth. We use a direct, data-driven approach to the modeling of market impact. To estimate the market depth parameters for each asset class using (27), we •estimate volatility parameters σµusing daily returns of S&P sector indices8. •obtain average daily volume estimates ADVµfrom annual volume data provided by the US Treasury and various central banks (Appendix A.1). As noted above, we use τ= 20, which corresponds to a liquidation horizon of 4 weeks, a fairly lenient assumption. Cont and Wagalath (2016) and Obizhaeva (2012) find c≈0.33. Ellul et al. (2011) find c≈0.2−0.3 for US corporate bonds under fire-sales pressure by insurance companies. An important difference is that in Ellul et al. (2011) the bonds are being liquidated due to the bond issuer’s credit rating being downgraded, while in our analysis, we assume that the fire sales are exogenous and not linked to the security issuers. We have used c= 0.4 here. ADV for US corporate bonds was obtained from SIFMA (see Appendix A.1). For European bonds, we simply use the same sovereign-to-corporate ratio of ADV as observed in the US (567.81bn/269.8bn ≈0.48) to estimate the ADV of the corporate and “institutional” asset classes for European corporate bonds. For some asset classes, data on trading volume are unavailable (or difficult to obtain). We work around this issue by estimating, based on OECD data, the following regression model for the relationship between average daily volume (ADV) and outstanding notional log ADVµ:= c1log (Nµ) + c0+εµ(28) where Nµdenotes outstanding notional, and using it to estimate volume for the remaining asset classes. Table 3 shows the results of this regression analysis. 8http://us.spindices.com/indices/fixed-income/sp-eurozone-sovereign-bond-index http://us.spindices.com/index-family/us-treasury-and-us-agency/all http://us.spindices.com/indices/fixed-income/sp-500-investment-grade-corporate-bond-index 19 Coefficient US treas US corp UK DE ES c1: 0.53*** 0.64*** 0.56*** 0.48 0.94*** std. dev. 0.13 0.15 0.05 0.30 0.14 c0: 0.63 -1.1** -0.35** 0.4 -1.35*** std. dev. 0.55 0.55 0.14 0.99 0.37 adj. R20.39 0.60 0.93 0.15 0.76 n19 19 11 10 15 Table 3: Logarithmic regression of bond trading volume on outstanding notional (Equation 28). Figure 5 shows the distribution of market depth estimates for all asset classes in the EBA dataset, on a logarithmic scale. The histogram reveals considerable heterogeneity in the cross-sectional distribution of market depth, with 4 orders of magnitude separating the most liquid from the least liquid assets. Market depth (EUR) Percent 0.0 0.2 0.4 0.6 1081091010 1011 1012 1013 1014 1015 Holdings (EUR) Market depth (EUR) 1061071081091010 1011 1012 1091010 1011 1012 1013 1014 1015 Figure 5: Left: Histogram of estimated market depths for the 148 asset classes in the EBA dataset. The dashed vertical line indicates the holdings-weighted average depth given by (31).Right: Scatter plot of depth vs. holdings. Table 4 provides values for average daily volumes (ADV) and market depths for the asset classes representing the largest holdings. These values are used as base values; we will later perform a sensitivity analysis of our results with respect to changes in these values. Extrapolation to large volumes When applied to large transaction volumes, linear or square-root impact models may lead to negative prices. In Greenwood et al. (2015) this was addressed by capping the loss at 100%: Ψµ(q) = min n1,q Dµo. Cifuentes et al. (2005) use an exponential specification Ψµ(q) = 1 −exp −q Dµ(29) which also ensures that prices remain non-negative but gives a concave impact. Nevertheless, prices can get arbitrarily close to zero in both of these models. However, it is 20 Asset class ADV Market depth DµImpact of 10 bn (sovereign unless specified) (bn EUR) (1012 EUR) US 567.8 428.8 0.2332 US (corp.) 269.8 70.2 1.424 IT 28.82 21.8 4.585 ES 28.0 21.1 4.737 DE 24.7 18.7 5.345 GB 24 18.1 5.522 FR 10 7.58 13.18 GR 8.17 6.18 16.16 SE 3.92 2.96 33.67 PT 2.27 1.71 58.14 Table 4: Average daily trading volume and estimated market depth over τ= 20 days, for the largest holdings in the EBA dataset. The impact in basis points are also given for a liquidation of 10 bn EUR over τ= 20 days. realistic to assume that long before the price level reaches zero, arbitrageurs will step in to purchase assets subject to fire sales at a discount Shleifer and Vishny (1992). To capture this effect we introduce a price floor Bµ>0 and consider a two-parameter level-dependent price impact function: Ψµ(q, S) := 1−Bµ S1−exp(−q δµ ).(30) Bµdetermines how far the price can fall in a fire-sales scenario. Note that this is a lower bound for the price and in a given stress test the price may not actually fall to this level. In the empirical examples below, we set Bµat 50 % of the market price levels. Choosing δµ=1−Bµ Sµ 0Dµ makes the specification (30) compatible with the linear specification (25) for small volumes. 3.3 Portfolio overlaps As shown in Eq. (18), the transmission of fire-sales losses from portfolio ito jdepends on the liquidity-weighted overlap between portfolio iand j: Ωij(Π) := M X µ=1 Πi,µΠj,µ Dµ . The matrix Ω of liquidity-weighted overlaps thus plays an important role in the transmission of fire-sales losses, which may be viewed as a contagion process on a network of financial institutions in which the link from ito jis weighted according to the liquidityweighted overlap Ωij . We call this network the indirect contagion network. Similar network structures were explored by Braverman and Minca (2016); Guo et al. (2015) for 21 mutual funds. Figure 6 displays the indirect contagion network for European banks as implied by EBA data collected in 2011. The nodes correspond to different banks, with node size proportional to balance sheet size, and edges correspond to non-zero portfolio overlaps, with edge widths proportional to the logarithm of the liquidity-weighted overlap Ωij. Figure 7 (left) shows the distribution of Ωij in this network. The peak at zero reflects the fact that many pairs of banks have no common asset holdings (so, zero overlap), i.e. the network is sparse. On the other hand, the values are dispersed over five orders of magnitude, which illustrates the heterogeneity of the network. Figure 6: The core of the European indirect contagion network: Node sizes are proportional to balance sheet size. Edge widths are proportional to the liquidity-weighted overlap. Red nodes correspond to the banks with highest loading in the first principal component of the portfolio overlap matrix Ω. The overlap matrix Ω also gives a glimpse of the nature of ‘second-round’ contagion effects in this network. The element (i, j) of the matrix Ω2may be interpreted as the 22 Figure 13: The fire-sales losses as a function of the initial shock and the market depth, scaled via the liquidation horizon, for the leverage targeting model. The leverage targeting model predicts large-scale losses for all combinations of shock sizes and market depths. Figure 14: Bank-level fire-sales losses, compared with losses under the leverage targeting model, using the same market depth estimates (logarithmic scale). Loss estimates are higher in the leverage targeting model by several orders of magnitude. a model with uniform market depth Dcomputed as a holdings-weighted average: X i,µ Πi,µ D=X i,µ Πi,µ Dµ .(31) The result, displayed in Figure 15, clearly shows a huge impact of heterogeneity in market depth when estimating fire-sales losses: clearly, institutions are differentiated according 29 Figure 15: Bank-level fire-sales losses computed using heterogeneous market depth parameters vs uniform market depth. The use of a uniform market depth for all assets increases fire-sales loss estimates considerably. to the liquidity of their holdings, sometimes by three orders of magnitude. This clearly shows that bank stress tests with fire sales require a careful estimation of market impact/ market depth parameters for each asset class. Figure 16 shows that the same experiment in the leverage targeting model leads to quite different results: heterogeneity of market depth does not seem to have much impact in this case. The reason is that the leverage targeting model overestimates fire-sales losses, which are so large that they trigger the insolvency of many large leveraged institutions in the first rounds of deleveraging. These defaulted institutions then cease to further contribute to the loss estimates, which then do not have any further dependence on market depth parameters. For completeness, we also compare in Figure 17 our results with the ones obtained in the leverage targeting model using a uniform market impact parameter Dfor all assets, as in Greenwood et al. (2015). The models agree in the ‘meltdown’ region where all banks default, but for intermediate stress levels the difference between loss estimates in the two models is substantial, up to several orders of magnitude (note the logarithmic scales in Figure 17). As discussed above, the main difference in the loss estimates arises from the deleveraging rule (leverage targeting vs one-sided leverage constraint) but the difference in market depth parameters accentuates the difference. Bank failures: illiquidity and insolvency. Most theoretical models of financial contagion have focused exclusively either on insolvency or on illiquidity as the cause of bank failure, while bank stress tests have traditionally focused exclusively bank solvency. As discussed in Section 2.5, our model allows for both possibilities: a bank may become insolvent due to asset losses, or become illiquid when all marketable assets have been sold. The model thus allows to examine which is the principal mode of failure in the stress test. Note that this distinction is not available in Greenwood et al. (2015); Duarte and 30 Figure 16: Impact of heterogeneity in market depth in the leverage targeting model. Eisenbach (2013): their assumption that all assets are available for liquidation entails that the only situation where an institution becomes illiquid is when there are no more assets available for liquidation, which implies that it is also insolvent. Our implementation of the leverage targeting model distinguishes between marketable and illiquid assets, to allow a meaningful comparison with our model. We now proceed to analyse the number of failures respectively due to illiquidity and insolvency as a function of the initial stress level. The left (resp. right) panel in Figure 18 displays the number of banks which become insolvent (resp. illiquid) after a given number of rounds of deleveraging, as a function of the initial stress level. We observe that insolvency is far from being the only mode of failure: depending on stress levels, 10 to 20 banks are expected to default due to illiquidity, yet remain solvent. Figure 19 shows respectively the number of failures due to insolvency (left) and those due to illiquidity (right) as a function of the liquidation horizon (market depth scaling factor) and the initial shock size. We can see that while the number of insolvent banks increases with the stress level, the number of failures due to illiquidity is non-monotone: it is maximal in the range of relevant shocks (between 5% and 10%) but then decreases as the shock level increases and insolvency becomes the main mode of default. Second and higher round effects. We end this section by discussing the impact of further rounds of deleveraging. Although in theory deleveraging may continue for many rounds (and, in the leverage targeting model, for an infinite number of rounds), previous empirical studies have mostly focused on a single round of deleveraging. (Greenwood et al., 2015, Appendix B) find that, with their parameter choices, iterating the cascade leads to all banks defaulting; we confirm this feature of the leverage targeting model in Figure 13 for more general parameter combinations. Duarte and Eisenbach (2013) do not find evidence for significant higher round effects for US banks. A possible explanation for this may be their choice of parameters (large market depth, and/or large initial shock). We attempt to clarify these findings by exploring systematically the impact of second or 31 Figure 17: Bank-level fire-sales losses in the leverage targeting model with uniform market depth differ by several orders of magnitude from those in the threshold model using heterogeneous market depths. Figure 18: Left: Number of insolvent banks as a function of number of rounds of deleveraging and initial stress level. Right: Number of illiquid banks as a function of number of rounds of deleveraging and initial stress level. 32 Figure 19: Number of bank failures due to insolvency (left) and illiquidity (right), as a function of the initial stress level and liquidation horizon (market depth). higher rounds of deleveraging. First, as noted in Section 3.3, even though the matrix Ω of portfolio overlaps is sparse, the matrix Ω2of second-order overlaps is dense, which implies that second round spillovers can potentially lead to contagion from any institution to any other, quite unlike what happens at the first round. By comparing the distribution of liquidity-weighted overlaps in Figure 7 to second-round overlaps in Figure 8, we see that some shocks need at least two rounds to propagate from one bank to another. Second, as observed in Figure 18, many bank failures occur only at higher rounds, especially failures due to illiquidity. Only examining a single round of the feedback loops thus leads to an underestimation of the number and severity of bank failures. The left panel in Figure 20 shows the fire-sales loss at each round kon a log scale for the estimated market depth with τ= 20. The right panel in Figure 20 shows the ratio of total loss (20 rounds) to the first round loss, as a function of the initial stress level. The non-monotone feature of this dependence shows that fire-sales contagion is most important for moderate stress levels, which trigger deleveraging but are not extreme enough to generate insolvency at the first round. In this case, second and higher rounds considerably change the outcome in terms of total loss level and number of bank failures. The results are even starker for the leverage targeting model: subsequent rounds lead to an increase by a factor 10 to 40 of estimated fire-sales losses, especially when the initial shock is small. This makes the “number of rounds” an important implicit parameter in the leverage targeting model. 4 Indirect exposures 4.1 Notional vs effective exposures The starting point in portfolio risk analysis is the notion of exposure to an asset class, usually quantified by the notional volume of holdings, in monetary units, in that asset class. In absence of contagion, losses in a stress scenario for this asset class will be a linear function of the percentage shock to the asset value, the proportionality coefficient being this notional exposure. 33 Figure 20: Left: fire-sales losses as a function of the iteration step at the estimated market depth (τ= 20) for the leverage targeting model (dashed), and the threshold models with λb=λmax (full) and λb< λmax (circles). Right: Ratio of total loss to first round loss, as a function of initial stress level. However, the endogenous risk arising from fire-sales may amplify this initial loss in a non-linear way, leading to a total loss that is higher than the one given by the notional exposure, leading to an effective exposure higher than the notional one. Furthermore, spillover effects from fire-sales may further increase this loss through indirect contagion. The end result is that the (marginal) ratio of a portfolio’s loss in an asset class to shocks affecting this asset class may be higher than its notional exposure. We capture this effect by defining the notion of indirect exposure and quantifying it using our model. Denoting as above by Θi,κ the holdings of institution iin the (illiquid) asset class κ, consider a stress scenario in which the asset class κdepreciates by κ. Then institution i has a direct loss κΘi,κ and, as long as the shock size is small, no deleveraging occurs and the latter also represents the total loss, which increases linearly with κ. However, as stress levels increase, feedback effects and price-mediated contagion may amplify the initial losses and result in a loss Loss(i, κ) = κΘi,κ +FLoss(i, )> κΘi,κ. The effective exposure to the asset class κaccounts for these additional losses and is defined as Ei,κ(κ) := Loss(i, κ) κ = Θi,κ |{z} Notional exposure +FLoss(i, κ) κ | {z } Indirect exposure .(32) Unlike the notional exposure, the indirect exposure depends on the shock size κand, more importantly, on the configuration – size, leverage – of other financial institutions with common asset holdings. Clearly, unlike the first term in (32), the second term cannot be computed by examining the portfolio of ialone and depends on the entire network of overlapping portfolios which contribute to fire-sales losses. Given the discussion in the previous section, this should not come as a surprise, but it clearly departs from the common assumption that notional exposures of a portfolio are sufficient to quantify its risk. Here, the risk of a portfolio cannot be quantified in isolation, but depends on the network of overlapping portfolios and the constraints that these portfolios face. Interestingly, a financial institution may even have a non-zero (indirect) exposure to an asset class which it does not hold in its portfolio, i.e. the second term in (32) may be non-zero even if the first term is zero. Consider for example a bank A which does not hold any subprime asset-backed securities, but has common holdings with a bank B with large 34 holdings in subprime ABS. A loss in subprime ABS entails no direct loss to A but, if large enough, may force B to deleverage by selling its marketable assets, also held by A, leading to mark-to-market losses for A. This means that A will have an indirect exposure to large losses in subprime ABS, an asset it does not hold! In this example, deleveraging by B only occurs if subprime losses are significant, so A has no exposure to a small depreciation in subprime ABS: the indirect exposure of A is clearly scenario-dependent. However, the threshold beyond which spillover from B to A occurs depends on the leverage, size and composition of B’s portfolio, which is unknown to A. This example, far from being a curiosity, is in fact illustrative of the mechanism which led to amplification and contagion of losses from the subprime asset class to the entire global financial system in 2007-2008 Longstaff (2010). 4.2 Indirect exposures: empirical evidence To assess how large such indirect exposures may be in the European banking system, we use the results of the previous section to examine the magnitude of indirect exposures of European banks to residential and commercial mortgages. A first observation is that most European banks tend to have very few mortgage loans outside of their own country, so direct exposure to foreign residential and commercial mortgage is zero in most cases, except for a few multinational banks. Given the large volume of mortgages on bank portfolios, it is generally assumed that domestic house prices are the main risk factor for commercial bank portfolios. However, as we shall see now, European banks also have substantial indirect exposures to residential and commercial real estate asset classes in other European countries where they do not issue mortgages. Figure 21 displays the (total) loss for two banks in our sample, HSBC and Santander, in the scenario where stress is applied to the Spanish real estate sector (Scenario 1). Santander, a major Spanish bank, holds a lot of Spanish mortgages on its portfolio, representing a notional exposure of EUR 82.7 bn, which corresponds to the slope of the loss for shocks less than 3%, where no deleveraging occurs. HSBC, on the other hand, holds very few Spanish mortgages (less than EUR 0.5 bn) so its notional exposure to this asset class is much lower. As losses on Spanish real estate exposures exceed 3.5%, some Spanish banks start deleveraging and indirect losses lead to a sharp increase in the loss level. For a 5% shock level, the total loss for Santander sharply increases to around EUR 25 bn, of which around EUR 22 bn are indirect fire-sales losses and 3bn are direct losses. This corresponds to an indirect exposure exceeding EUR 440 bn, which is significantly larger than its actual direct exposure! In this regime, the indirect exposure is an important source of losses. This deleveraging by Spanish banks then affects HSBC through price-mediated contagion: as the shock level exceeds 5%, the fire-sales loss for HSBC starts to become important. For a 5.5% shock level, HSBC’s losses through price-mediated contagion are equal to 2.73 bn EUR, which corresponds to an indirect exposure to Spanish real estate close to EUR 44 bn. Note that in the case of HSBC, the indirect loss corresponds to 100 times its notional exposure to this asset class! This example suggests that in stress scenarios which are sufficiently severe to trigger fire sales, actual losses can be much higher than what is suggested by notional exposures. The difference corresponds to indirect exposures. The right-hand graph in Figure 21 displays the magnitude of indirect exposures corresponding to these losses. We see that the indirect exposure of Santander, which mainly stems from its own deleveraging and deleveraging by other Spanish banks with similar as35 sets, sharply increases when the shock size reaches the threshold of 3%, then peaks around 4%: as the stress level increases beyond a certain level, some banks become insolvent and cease to contribute to fire-sales spillovers. As illustrated in the right panel in Figure 21, these indirect exposures clearly depend on the severity of the stress scenario considered, the bank’s initial exposures, and the exposure and constraints of other portfolios holding similar marketable assets. An implication is that one cannot mimic the impact of indirect contagion by simply applying a higher stress level for all banks, as is implicitly done in current supervisory stress tests. Table 7 reports indirect exposures to Spanish real estate for several other European banks. Comparing with their notional exposures, we observe that non-Spanish banks have a substantial indirect exposure to Spanish real estate, which is not revealed by inspecting their notional exposures to this asset class, which are between ten and a hundred times smaller. This indirect exposure is due to the overlap in marketable assets with banks that are directly exposed to the Spanish housing market. These observations are not particular to the Spanish real estate sector. Other examples, which we do not report here for the sake of brevity, show that in the European banking system a stress scenario affecting the commercial or residential real estate sector of a given country may generate significant losses arising from indirect exposures for foreign banks. Figure 21: Left: Losses arising from a depreciation in Spanish real estate, for HSBC and Santander, as a function of the depreciation level (horizontal axis). Right: The indirect exposures corresponding to these losses. The concept of indirect exposures may also be defined at the country level. Figure 22 shows the indirect exposures of European banks (aggregated by country) to the Spanish housing market. Shocks below 1% do not appear to trigger fire sales, and do not generate indirect exposures. As the stress level is increased, fire sales are triggered and spillover losses materialise; when the initial shock size exceeds 3%, contagion increases fire-sales losses by two to three orders of magnitude. What is striking is that these thresholds – 1%, 3% – are not large, suggesting that indirect exposures and losses arising from price-mediated contagion cannot be ignored in stress tests. The left panel of Figure 22 reveals that, after Spanish banks, Portuguese and British banks have the largest indirect exposure to the Spanish housing market. The right panel of 22 shows these indirect exposures. Table 8 reports fire sales losses and estimated indirect exposures for different stress levels for the countries highlighted in Figure 22. 36 Bank Bank equity (EUR bn) Fire-sales loss (EUR bn) Fire-sales loss (EUR bn) Fire-sales loss (EUR bn) Stress level 1% Stress level 5.5% Stress level 10% Santander 42.0 0.826 22.0 22.2 BBVA 24.9 0.771 18.8 20.0 Deutsche Bank 37.6 0.045 3.00 3.07 HSBC 86.90 0.024 2.73 2.76 RBS 59.0 0.069 4.36 4.42 Nordea 19.1 0 0.17 0.19 Bank Direct exposure (EUR bn) Indirect exposure (EUR bn) Indirect exposure (EUR bn) Indirect exposure (EUR bn) Stress level 1% Stress level 5.5% Stress level 10% Santander 82.7 30.0 354 202 BBVA 82.5 28.0 302 177 Deutsche Bank 8.88 1.63 48.4 27.8 HSBC 0.452 0.86 43.9 25.0 RBS 2.68 2.50 70.2 40.1 Nordea 0.003 0.05 2.76 1.69 Table 7: Capital, direct exposures and fire-sales losses and indirect exposures for three selected shock sizes. When price-mediated contagion occurs, fire-sales losses erode a considerable fraction of the equity. This corresponds to indirect exposures that often exceed the direct exposures. Overall, the direct exposure of European banks to the Spanish residential and commercial real estate sector is EUR 740 bn. When averaged both across all banks and all shock sizes (between 0 and 20%), the indirect exposures amount in this example to 160% of the direct exposures, which is non-negligible. To be clear, in the event of a meltdown of the Spanish housing sector, non-Spanish European banks will, on average, be affected for up to 160% of their actual holdings in Spanish mortgages. This clearly means that housing prices in one European country can strongly affect banks of other European countries, an issue which can only be tackled by macroprudential policies applied at a Europe-wide level.12 These indirect exposure figures, as all other results obtained in the stress tests, are sensitive to assumptions on market depth, or, alternatively, on the liquidation horizon. Figure 23 shows the fire sales losses of the UK banking system to the Spanish housing market. Reading the graph from front to back, we see that a contraction of market liquidity can greatly exacerbate fire-sales losses: there is a tipping point at which losses increase from EUR 10 bn to EUR 100 bn. Figure 24 shows the estimated indirect exposure of the UK banking system to the Spanish housing market as a function of the stress level and the liquidation horizon (or market depth scaling factor). In the case of the UK, the main asset classes contributing to this indirect exposure to Spanish residential and commercial mortgages are via portfolio overlaps in Spanish, UK, US, and Asian corporate bonds as well as Spanish, Asian, non-emerging EEA, and US government bonds. Indirect exposures and the losses arising from them provide a more palpable explanation of how an initial loss of about USD 500 bn in the US subprime sector was able to 12On this point, we note that Brexit does not act as a barrier for price-mediated contagion. 37 balloon into trillions of dollars of losses across multiple asset classes during the financial crisis. Figure 22: Indirect exposures of European banks, aggregated by country, to the Spanish housing market. Country Total bank equity (EUR bn) Fire-sales loss (EUR bn) Fire-sales loss (EUR bn) Fire-sales loss (EUR bn) Stress level: 1% Stress level: 2% Stress level: 3.5% ES 22.8 0 2.14 71.6 GB 7.9 0 0.08 12.6 FR 15.9 0 0.09 9.88 DE 2.4 0 0.07 8.23 PT 697.0 0 0.03 14.5 Country Direct exposure (EUR bn) Indirect exposure (EUR bn) Indirect exposure (EUR bn) Indirect exposure (EUR bn) Stress level: 1% Stress level: 2% Stress level: 3.5% ES 22.8 0 0.6 7 GB 7.9 0 0.02 0.51 FR 15.9 0 0.03 0.43 DE 2.4 0 0.02 0.37 PT 697.0 0 0.01 0.13 Table 8: Fire-sales losses and indirect exposures at country level for the most heavily exposed countries. 4.3 Relevance of indirect contagion for bank stress tests Given the magnitude of indirect contagion, it is not surprising that accounting for indirect losses can modify the outcome of bank stress tests. 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A Appendix A.1 Sources for market data US treasuries: http://www.sifma.org/uploadedFiles/Research/Statistics/StatisticsFiles/ SF-US-SF-Trading-Volume-SIFMA.xls US corporate bonds: http://www.sifma.org/uploadedFiles/Research/Statistics/ StatisticsFiles/Corporate-US-Corporate-Trading-Volume-SIFMA.xls DE treasuries: http://www.deutsche-finanzagentur.de/en/institutional-investors/ secondary-market/ UK treasuries: http://www.dmo.gov.uk/index.aspx?page=Gilts/Turnover ES treasuries: http://www.tesoro.es/sites/default/files/estadisticas/18.pdf 48 FR treasuries: http://www.aft.gouv.fr/rubriques/trading-volume_109.html IT treasuries: http://www.dt.tesoro.it/export/sites/sitodt/modules/documenti_ en/debito_pubblico/presentazioni_studi_relazioni/3_3_2000_13_43_The-Italian-Treasury-. pdf BE treasuries: http://www.debtagency.be/fr_products_olo_volume.htm SE treasuries: http://www.riksbank.se/Documents/Rapporter/Finansmarknaden/2014/ rap_finansm_140829_eng.pdf PT treasuries: http://www.igcp.pt/fotos/editor2/2015/Estatisticas/12_Transacies_ medias_diarias_OT_e_BT_Dez15_1.pdf GR treasuries: http://www.bankofgreece.gr/Pages/en/Markets/HDAT/statistics. aspx A.2 EBA: data identifiers and residual exposures Model variable EBA dataset identifier Assets Illiquid assets Θ Residential mortgage exposures (κ≥0) 33013 Commercial real estate exposures (κ≥0) 33018 Retail: Revolving exposures (κ≡0) 33015 Retail: SME exposures (κ≡0) 33016 Retail: other exposures (κ≡0) 33017 Indirect sovereign exp. in the trading book (κ≡0) 34017 Defaulted exposures (κ≡0) 33020 Remaining exposures* (κ≡0) - Securities Π Institutional client exposures 33010 Corporate exposures 33011 Sovereign exposures 34013 , 34014 , 34015 Direct sovereign exposures in derivatives 34016 Liabilities Tier 1 capital 30014 Debt - Table 10: Mapping of EBA data to model variables. At the time the data was collected, banks were following Basel II guidelines, which corresponds to 8% ratio of capital to risk-weighted assets (RWA), and the Basel 3 leverage constraint was not yet in place. Some banks in the sample have leverage higher than the Basel III limit of 33. In order to avoid fire sales in absence of a shock, we scale these banks’ capital levels to bring their leverage within the interval of [29.7,31.35] = 33 ×[90%,95%] as shown in Figure 28.13 Correcting for data inconsistencies. The EBA data provides information on notional exposures of each bank to 148 asset classes.14 Bank BE005 records a zero value 13 Similarly, Greenwood et al. (2015) cap the leverage at 30 in their analysis. 14For more details on the regulatory definition of “exposure” cf. the EBA methodological note: https://www.eba.europa.eu/documents/10180/15932/EBA-ST-2011-004-Detailed-Methodological-Note_1.pdf as well 49 Figure 28: Leverage of banks in the EBA dataset and adjusted leverage used as model input: By adding capital to the banks with leverage above 33, the leverage of these banks is brought slightly below 33. for total exposures despite positive exposures in the individual asset classes. As a proxy for this bank, we use information from the “Total assets after the effects of mandatory restructuring plans”15, which is usually quite close to the values of “total exposures”across the dataset. Another consistency issue is that the individual exposures in the dataset do not always sum up to the total exposure figure. Indeed, EBA explains in a footnote to the “total exposure” data that: “Total exposures is the total EAD according to the CRD definition based on which the bank computes RWA for credit risk. Total exposures, in addition to the exposures broken down by regulatory portfolios in this table [corresponding to the four asset classes in the portfolio Π above] include EAD for securitisation transactions, counterparty credit risk, sovereigns, guaranteed by sovereigns, public sector entities and central banks”. Due to this, the sum of balance sheet items deviates from the “total exposures” information recorded in the dataset. In order to correct for this deviation we add a “remaining exposures” item to the illiquid assets category. The average size of the negative correction terms is 5.9% of the corresponding balance sheet sizes. Double counting is thus a minor issue. The average size of the positive correction terms is 13.4%. This average is inflated by five small banks that are outliers and have correction terms above 50%. Excluding these outliers, the average of the correction terms reduces to 9%. The average size of the correction term over the entire data set is 8.9%. On average, we thus underestimate the size of the banks’ balance sheets. This may only bias results in the sense of underestimating the impact of fire sales. We adjust for this with the “other illiquid assets” category, as they play no role in the fire sale cascade. as the identifier 33021 in the dataset. 15This information is recorded under identifier 30029 in the EBA dataset. 50