Elimination of systemic risk in financial networks by means of a systemic risk transaction tax
Abstract
Financial markets are exposed to systemic risk (SR), the risk that a major fraction of the system ceases to function, and collapses. It has recently become possible to quantify SR in terms of underlying financial networks where nodes represent financial institutions, and links capture the size and maturity of assets (loans), liabilities, and other obligations, such as derivatives. We demonstrate that it is possible to quantify the share of SR that individual liabilities within a financial network contribute to the overall SR. We use empirical data of nationwide interbank liabilities to show that the marginal contribution to overall SR of liabilities for a given size varies by a factor of a thousand. We propose a tax on individual transactions that is proportional to their marginal contribution to overall SR. If a transaction does not increase SR it is taxfree. With an agentbased model (CRISIS macrofinancial model) we demonstrate that the proposed “Systemic Risk Tax” (SRT) leads to a selforganised restructuring of financial networks that are practically free of SR. The SRT can be seen as an insurance for the public against costs arising from cascading failure. ABM predictions are shown to be in remarkable agreement with the empirical data and can be used to understand the relation of credit risk and SR.
I Introduction
Failure to manage systemic risk (SR) has turned out to be extremely costly for society. The financial crisis of 20072008 and its consequences demonstrated the importance of reducing it. The threat of collapse of large parts of the financial system has forced national governments to bailout hundreds of banks (The Economist, 2013). As a result, one observed falling global stock and real estate markets (The Economist, 2007a), a severe and global credit crunch (The Economist, 2007b), skyrocketing and prolonged unemployment rates, and several Western governments at the verge of bankruptcy. Bank bailouts have caused dangerously high levels of sovereign debt around the world, and it has become necessary to find alternatives to finance bailouts (Klimek et al., 2015). The International Monetary Fund has proposed a tax on banks, called the “financial stability contribution” (FSC), i.e. a contribution of the financial sector to the public costs of the financial crisis, which is used to create reserves for future crises. Bank taxes have been proposed in many countries around the world, e.g. the “Financial Crisis Responsibility Fee” in the US. In several European countries, including Germany and Austria, bank taxes are currently in force. The European Commission has proposed an EUwide bank tax under the “Single Resolution Mechanism”. In addition to bank taxes, a financial transaction tax (FTT) is being considered by many countries. A FTT is not a tax on financial institutions per se, but a levy placed on specific types of financial transactions. Its main purpose, besides generating revenue for governments, is to curb the volatility of financial markets (Tobin, 1978; Summers and Summers, 1989). Related empirical studies are generally inconclusive, and a causal relation between volatility and FTTs remains ambiguous (McCulloch and Pacillo, 2011; Matheson, 2012). In response to the financial crisis of 20072008, a consensus on the need for new financial regulation is emerging (Aikman et al., 2013). New financial regulation must be designed to mitigate the risk of the financial system as a whole. This approach to financial regulation is known as “macroprudential regulation”, and is currently being put in place around the globe (Aikman et al., 2013; Bank of England, 2011, 2013). The Basel III framework recognises systemically important financial institutions (SIFI) and recommends increased capital requirements for them – the so called “SIFI surcharges” (Bank for International Settlements, 2010; Georg, 2011). Basel III further introduces the idea of “countercyclical buffers” that allow regulators to increase capital requirements during periods of high credit growth. No matter how wellintended these developments might be, they miss the central point about the nature of SR, and therefore may not be suitable to improve the stability of the financial system in a sustainable way. SR is closely related to the network structure of financial assets and liabilities in a financial system. Management of SR is essentially a matter of restructuring financial networks in such a way that the probability of cascading failure is reduced, or ideally eliminated.
Credit risk is the risk that a borrower will default on a specific debt by failing to make the full prespecified repayments. It is usually seen as a risk that emerges between two counterparties once they have engaged in a financial transaction. The lender is the sole bearer of credit risk and accounts for the likelihood of failed repayments by demanding a risk premium. Lenders usually charge higher interest rates to borrowers that are more likely to default (riskbased pricing). Credit risk is relatively wellunderstood, and can be mitigated through a number of methods and techniques (Duffie and Singleton, 2012). The Basel Accords provide an extensive framework, dealing foremost with the mitigation of credit risk (Bank for International Settlements, 1988, 2006; Balin, 2008). When two counterparties are part of a financial system, for example as nodes in a financial network, the situation changes, and their transaction may affect the financial system as a whole. The lender is no more the sole bearer of credit risk, nor does credit risk depend on the financial conditions of the borrower alone. The impact of a default of the borrower is no longer limited to the lender, but may affect other creditors of the lender, which in turn may affect their creditors. Similarly, the lender is not only vulnerable to a default of the borrower but also to defaults of all debtors of that borrower, as well as their debtors. In other words, in financial networks credit risk is no longer limited to two counterparties, but becomes systemic.
SR is the risk that the financial system as a whole, or a large fraction of it, can no longer perform its function as a credit provider and collapses. In a narrow sense, SR is the notion of contagion or impact from the failure of a financial institution, or group of institutions, on the financial system and the wider economy (De Bandt and Hartmann, 2000; Bank for International Settlements, 2010). It is a result of the interconnected nature of financial transactions, and claims or liabilities in the financial system. It unfolds as secondary cascades of credit defaults, triggered by credit defaults between individual counterparties (Eisenberg and Noe, 2001). These cascades can potentially wipe out the financial system by a deleveraging cascade (Minsky, 1992; Fostel and Geanakoplos, 2008; Geanakoplos, 2010; Adrian and Shin, 2008; Brunnermeier and Pedersen, 2009; Thurner et al., 2012; Caccioli et al., 2012; Poledna et al., 2014; Aymanns and Farmer, 2015; Caccioli et al., 2015). It is obvious that lenders have a strong incentive to mitigate credit risk. In the case of SR the situation is less clear as SR involves externalities, i.e. financial institutions manage their own risks but do not consider their impact on the system as a whole (Acharya et al., 2009). In fact, funding costs for large financial institutions are lowered due to a market expectation that the state will bailout banks that are deemed to be systemically important (Davies and Tracey, 2014). Unless financial institutions are required to internalise costs of SR, institutions will have little incentive to minimise risks that are borne by the general public (Acharya et al., 2012). Management of SR is, therefore, foremost in the public interest.
SR is a network externality resulting from contagion effects (Acemoglu et al., 2013). To cope with this externality, governments can use two main policy instruments: taxation or regulation (Masciandaro and Passarelli, 2013). Taxation is aimed at reducing the gap between public and private costs of SR, while financial regulations impose direct restrictions and requirements on financial institutions. In general, taxation is superior to regulation because a taxation scheme can be designed to produce any desired progressive impact (Masciandaro and Passarelli, 2013). In principle, marginal tax rates can be set so that they reflect the marginal cost of reducing SR. Several authors have recently advocated for a taxation of SR (Cooley et al., 2009; Acharya et al., 2012; Adrian and Brunnermeier, 2011; Markose et al., 2012; Acharya et al., 2013; Zlatić et al., 2014), while in the real world regulation policies are being put in place due to the inherent difficulties of measuring SR (Bank for International Settlements, 2010). In this context several measures for SR have recently been proposed that focus (mainly) on statistics of losses, accompanied by a potential shortfall during periods of synchronised behaviour where many institutions are simultaneously distressed (Adrian and Brunnermeier, 2011; Acharya et al., 2012; Brownlees and Engle, 2012; Huang et al., 2012). None of these measures, however, take cascading failure directly into account.
SR is predominantly a network property of liability networks (Battiston et al., 2012a; Thurner and Poledna, 2013). Recent econometric studies indicate that network measures could potentially serve as early warning indicators for crises (Caballero, 2012; Billio et al., 2012; Minoiu et al., 2013). Different financial network topologies will have different probabilities for contagion and systemic collapse, given the link density and the financial conditions of nodes are the same (Roukny et al., 2013). In this sense the management of SR becomes a technical problem of reshaping the topology of financial networks (Haldane and May, 2011). The goal is to do this in a way that neither reduces the credit provision capacity, nor the transaction volume of the financial system. Data on the topology of liability networks is available to many central banks. Several studies on historical data show typical scalefree connectivity patterns in liability networks (Upper and Worms, 2002; Boss et al., 2004, 2005; Soramäki et al., 2007; Cajueiro et al., 2009; Bech and Atalay, 2010; MartínezJaramillo et al., 2014), including overnight markets (Iori et al., 2008), financial flows (Kyriakopoulos et al., 2009) and mutual cross holdings (Huang et al., 2013). As a network property, SR can be (precisely) quantified by using network metrics (Battiston et al., 2012a; Thurner and Poledna, 2013). In particular, a relative network measure (DebtRank) can be assigned to all nodes in a financial network that specifies the fraction of SR that they contribute to the system (institution or nodespecific SR) (Thurner and Poledna, 2013). As shown later, it is natural to extend the notion of nodespecific SR to individual liabilities between two counterparties (liabilityspecific SR) and to individual transactions (transactionspecific SR).
In this paper we introduce a novel approach for the management of SR in financial networks. First, we develop a risk measure to quantify the marginal contribution of individual liabilities in financial networks to the overall SR. Second, we use this risk measure to design an incentive scheme where banks pay a Pigovian tax – the “Systemic Risk Tax” (SRT) – on each transaction, which is proportional to the increase in overall SR that it would cause. Following this approach, financial institutions would internalise their externality, as they are “taxed” according to their marginal contribution to overall SR. This incentive scheme leads to a selforganised reduction of SR in the following way: Market participants looking for credit will try to avoid this tax by looking for credit opportunities that do not increase SR and are thus taxfree. As a result, the network rearranges toward a topology that, in combination with the financial conditions of individual institutions, will lead to a de facto elimination of SR. This is due to the fact that with the new topology cascading failures can no longer occur. With the help of an agentbased model (ABM), we show that financial institutions react to the SRT by rearranging the financial network over time such that overall SR is indeed drastically reduced. A number of ABMs have been used recently to study interactions between the financial system and the real economy, focusing on destabilising feedback loops between the two sectors (Delli Gatti et al., 2009; Battiston et al., 2012b; Tedeschi et al., 2012; Gabbi et al., 2015). We test the proposed SRT within the framework of the CRISIS macrofinancial model
Ii The systemic risk tax
The SRT is a levy placed on a financial transaction to offset the SR increase associated with that transaction. We show that SR associated with a transaction can be quantified by the DebtRank methodology, which was originally suggested as a recursive method to determine the systemic importance of nodes within financial networks (Battiston et al., 2012a). It is a quantity that measures the fraction of the total economic value (\crefecovalue) in the network that is potentially affected by the default and distress of a node or a set of nodes, see \crefdebtrank_section. For simplicity’s sake let us think of the nodes in financial networks as banks. By we denote the liability (exposure
DebtRank has the precise meaning of economic loss (in Euros) that is caused by the distress or default of a node (Battiston et al., 2012a). This precise meaning of the DebtRank allows us to define the “expected systemic loss” for the entire economy. Assuming that we have banks in the system, the expected systemic loss can be approximated by
(1) 
with the probability of default of node , and the combined economic value of all nodes at time . That this is an excellent approximation has been demonstrated in (Poledna et al., 2015). For the derivation, see \crefel_approx.
measures the fraction of the total economic value (\crefecovalue) that is potentially affected by node . In general, is not known and can, in principle, also depend on the particular topology of various financial networks. Since denotes the risk of financial contagion from the liability network , the probability of default should not explicitly depend on . However, can, in principle, depend on other networks, like the network of overlapping portfolios. Besides overlapping portfolios there are a number of reasons why default correlation exists, e.g. external events can trigger joint defaults of firms in the same geographic region or sector (Hull, 2012). Note that we assume in \crefEL that denotes the risk of financial contagion and all other factors that lead to default correlations are comparably small (second order). Thus we calculate the total expected loss by summing the expected losses across banks. However, summing the expected losses across banks in general does not have the meaning of total expected loss because it ignores the joint probability of default. If the default correlation is known, additional terms containing the joint probability of default and the impact of a group can be added to \crefEL, see \crefdiscussion.
To calculate the marginal contributions to the expected systemic loss, we start by defining the net liability network . After we add a specific liability , we denote the liability network by
(2) 
where is the Kronecker symbol. The marginal contribution of the specific liability on the expected systemic loss is
(3) 
where is the DebtRank of the liability network and the total economic value with the added liability . Clearly, a positive means that increases the total SR.
Finally, the marginal contribution of a single loan (or a transaction leading to that loan) can be calculated. We denote a loan of bank to bank by . The liability network changes to
(4) 
Since and can have a number of loans at a given time , the index numbers a specific loan between and . The marginal contribution of a single loan (transaction) is obtained by substituting by in \crefmarginal_effect. In this way every existing loan in the financial system, as well as every hypothetical one, can be evaluated with respect to its marginal contribution to overall SR.
The central idea of the SRT is to tax every transaction between any two counterparties that increases SR in the system. The size of the tax is proportional to the increase of the expected systemic loss that this transaction adds to the system as seen at time . The SRT for a transaction between two banks and is given by
(5) 
Note that we assume in \crefsrteq_simple that defaults occur only on the maturity date of the loan. For simplicity’s sake we do not discount. To allow defaults at any time, valuation can be done similarly to credit risk models as for example for credit default swaps (Hull and White, 2000, 2001; Hull, 2012)
is a proportionality constant that specifies how much of the generated expected systemic loss is taxed. means that 100% of the expected systemic loss will be charged. means that only a fraction of the true SR increase is added on to the tax due from the institution responsible. can be chosen such that the efficiency (total transaction volume) of the financial system is kept the same as it would be in the untaxed world. We show below that this is indeed the case.
Iii The model to test the ability of the systemic risk tax to reduce systemic risk
To test the economic and financial implications of the SRT we use the CRISIS macrofinancial model. This is an economic simulator that combines a wellstudied macroeconomic ABM (Gaffeo et al., 2008; Delli Gatti et al., 2008, 2011) with an ABM of financial markets. We use a modified version of the macroeconomic model of Delli Gatti et al. (2011), which additionally has an interbank market and is a closed, stockflow consistent economic system that allows no in or outflows of cash. Here we give a short description of the model, for a comprehensive description, see Delli Gatti et al. (2011) or Gualdi et al. (2015) and for the modifications, see \crefmodel_details.
In the model, there are three types of agents: households, banks, and firms, as depicted in \crefmodel_fig. The agents interact on four different markets:

Firms and banks interact on the credit market.

Banks interact with banks on the interbank market.

Households and firms interact on the job market.

Households and firms interact on the consumption goods market.
Banks hold all firms’ and households’ cash as deposits. Households are randomly assigned as owners of firms and banks (shareholders). Agents repeat the following sequence of decisions at each time step:

firms define labour and capital demand,

banks raise liquidity for loans,

firms allocate capital for production (labour),

households receive wages, decide on consumption and saving,

firms and banks pay dividends, firms with negative liquidity go bankrupt,

banks and firms repay loans,

banks raise additional liquidity to manage unanticipated cash needs.
Households owning firms or banks receive dividends as income. All other households earn salaries for work done for firms. Banks and firms pay of their profits as dividends.
iii.1 The agents
We give a short description of the agents; for more details on the agents and their interactions, see Delli Gatti et al. (2011), Gualdi et al. (2015), and \crefmodel_details.
Households
There are households of which there exist two types: firm owners, and workers. Each of them has a personal account at one of the banks. indexes the worker, the bank. Household accounts are randomly assigned to banks. Workers apply for jobs at the different firms. If hired, they receive a fixed income per time step, and supply a fixed labour productivity . Firm owners receive their income through dividends from their firm’s profits. At each time step every household spends a fixed percentage of its current deposit account on the consumption market. Households compare prices of consumption goods from randomly chosen firms and buy the cheapest.
Firms
There are firms producing perfectly substitutable consumption goods. At every time step firms compute an expected demand for the next time step , and an estimated price (subscripts label the firm), based on a rule that takes into account both excess demand/supply and the deviation of the price from the average price at the present time step. Each firm computes the number of required workers to supply the expected demand. If the wages for the respective workforce exceed the firm’s current liquidity, it applies for a loan. Firms approach randomly chosen banks and choose the loan with the most favourable rate. If this rate exceeds a threshold rate , the firm only asks for percent of whatever loan was originally requested. Based on the outcome of this loan request, firms reevaluate the required workforce, and hire or fire the necessary number of workers. Firms sell the goods on the consumption goods market. Firms go bankrupt if they have negative liquidity after the goods market has closed. Each of the bankrupted firm’s debtors (banks) incurs a capital loss in proportion to their investment in the company. Firm owners of bankrupted firms are personally liable, and their personal account is divided by the debtors pro rata. They immediately (next time step) start a new company. Their initial estimates for and equals the respective current averages in the population.
Banks
There are banks that offer firm loans at rates that take into account the individual specificity of banks (modelled by a uniformly distributed random variable), and the firms’ creditworthiness quantified by their leverage ratio (see \crefmodel_details). Firms pay a credit risk premium according to their creditworthiness, which is modelled by a monotonically increasing function of their financial fragility. Banks try to provide requested loans and grant them if they have enough liquid resources. If they do not have enough cash, they approach other banks in the interbank market to obtain the necessary amount. If a bank does not have enough cash and cannot raise the full amount for the requested firm loan on the interbank market it does not pay out the loan. Interbank and firm loans have the same duration. Additional refinancing costs of banks remain with the firms. Each time step firms and banks repay percent of their outstanding debt (principal plus interest). If banks have excess liquidity they offer it on the interbank market for a nominal interest rate. The interbank market is modelled after an electronic marketplace where, in principle, all participants can enter into business relationships. In the model, banks choose the interbank offer with the most favourable rate. This does not mean that the emerging interbank network is fully connected. Emerging interbank networks are shown in \crefnetwork and (weighted) degree distributions can be found in \crefdegree_dist. Interbank rates offered by bank to bank take into account the specificity of bank , and the creditworthiness (leverage ratio) of bank . If a firm goes bankrupt the respective creditor bank writes off the respective outstanding loans as defaulted credits. If the bank does not have enough equity capital to cover these losses, it defaults. Following a bank default an iterative defaultevent unfolds for all interbank creditors. This may trigger a cascade of bank defaults. For simplicity’s sake, we assume no recovery for interbank loans. This assumption is reasonable in practice for short term liquidity (Cont et al., 2013). A cascade of bankruptcies happens within one time step. After the last bankruptcy is taken care of the simulation is stopped.
iii.2 Implementation of the systemic risk tax and the Tobin tax
A systemic risk premium, in form of the SRT, is imposed on all interbank transactions. Before entering a desired loan contract , the credit seeking banks can get quotes for the rates from the central bank, for various offering banks . They choose the interbank offer from bank with the smallest total rate, which is composed of . All other transactions are exempted from the SRT. In contrast to current market practice, the effective interest rate reflects both the creditworthiness of the borrowing counterparty and the SR increase associated with each transaction. The SRT is collected in a bailout fund. The SRT is calculated according to \crefsrteq_simple. We approximate by the financial fragility, defined by the borrower’s leverage at time . For more details, see \crefmodel_details_ib.
For comparison, we implement a Tobinlike (Tobin, 1978) FTT for interbank loans. We impose a constant tax rate of 0.2% of the transaction (this is about 5% of the interbank interest rates) on all interbank rates on offer. Other transactions are not taxed. The FTT makes lending less attractive for firms that borrow from banks requiring liquidity from the interbank market, as refinancing costs remain with the firms.
Iv Results
We implement the above model in MATLAB for banks, firms, and households. The model is run in three modes, without any tax, with the SRT, and with a Tobinlike FTT. Results are averages over independent, identical simulations across time steps. We set (see \crefsrt), and for the Tobinlike FTT we impose a constant tax rate of 0.2% of the transaction. For different tax rates for the Tobinlike FTT and an alternative mode in which the SRT is set to the true increase in SR associated with each transaction (), see \creftobin. Additionally in \creftobin there is a short discussion of the effect of the SRT on the network properties.
We compare model results to historical, anonymised, and linearly transformed interbank liability data provided by the Austrian Central Bank (OeNB), see \crefdata. In \crefnetwork(a) we show a snapshot of the Austrian interbank network at the end of the first quarter of 2006. Nodes symbolise the banks of the Austrian banking system and links represent their lending relations (weighted by the liabilities). Nodes are coloured according to their systemic importance , from systemically important banks (red) to unimportant ones (green). The nodesize represents the capitalisation of banks and the width of the links symbolises the liabilities. In \crefnetwork(b) we show the largest banks of the Austrian interbank network. Clearly, the largest banks contribute most to overall SR (red and orange dots). \Crefnetwork(c) shows results from the ABM without a tax, (d) with the FTT, and (e) with the SRT. The SRT effectively reduces the spreading of SR by preventing systemically important nodes from lending. This can be seen from the fact that there are only green links in \crefnetwork(e). In the snapshot of the Austrian interbank network and in the model without the SRT numerous red links are clearly visible. In \crefdata_vs_model(a) we show SR as measured by DebtRank . In particular, we show for the largest banks (according to total assets) of the Austrian banking sector at the end of the first quarter of 2006. Here we calculate from (see \crefdata), in \crefdata_vs_model(b) we use the net liability network . Banks are ordered by their DebtRank, the systemically most important one is to the very left, the least important one to the very right. The ABM results for are presented in \crefdata_vs_model(b): without a tax (red), with the FTT (blue) and with the SRT (green). The shown distributions are averages over independent simulations. Clearly, the SRT drastically reduces the SR contributions of individual banks. The situation without tax resembles the empirical distribution remarkably well. In \crefdata_vs_model(c) the marginal contributions on expected systemic loss from \crefmarginal_effect are presented for all individual interbank liabilities , as a function of the relative size of interbank loans. Every data point represents a single interbank liability from bank to . Interbank loans are themselves powerlaw distributed (not shown), which is known empirically (Boss et al., 2005). The loan size captures the credit risk for lenders, whereas is the SR contribution of the liability. \Crefdata_vs_model(d) shows the corresponding marginal contributions for the ABM simulations for the three modes. It is clearly visible that the SRT reduces the SR contribution of liabilities by approximately an order of magnitude (note the log scale), but leaves contractsizes practically unchanged.
The effects of the SRT and the FTT on total losses to banks (see \crefrisk_measures) that occur as a consequence of bank defaults are shown in \crefhist_losses(a). Clearly, the mode without tax (red) produces fat tails in the loss distributions of the banking sector. The Tobin tax slightly reduces losses. The SRT gets rid of big losses in the system completely (green). The remaining losses reflect those from firm defaults, which represents the economic risk in the system. Note that economic risk can hardly be managed. This elimination of losses on the interbank market is due to the fact that under the SRT the possibility for cascading defaults is drastically reduced. This is seen in \crefhist_losses(b), where the distributions of cascade sizes (see \crefrisk_measures) for the three modes are compared. While the untaxed mode produces considerable cascade sizes of up to banks, the maximum cascade sizes under the SRT is less than half. The Tobin tax more or less follows the untaxed case. As mentioned above, the interbank loan sizes are practically unchanged under the SRT. This is also true for the total transaction volume (see \crefrisk_measures) in the interbank market, as can be seen in \crefhist_losses(c), where the distributions of transaction volumes at time step 100 are shown. Obviously, the situation for the SRT (green) is very similar to the untaxed case (red), whereas the transaction volume is drastically reduced in the FTT scenario (blue), as expected.
V Discussion
We extend the notion of SR to individual liabilities within a financial network and show with empirical data of nationwide interbank liabilities that this is indeed feasible. The notion of liabilityspecific SR allows us to quantify the marginal contribution of every financial transaction to overall SR. We propose a Pigovian tax (SRT) on every SRincreasing transaction, proportional to the marginal contribution to overall SR. By trying to avoid the SRT, financial institutions effectively rearrange the financial network over time, such that cascading failures can no longer occur. This process leads to a sustainable, selforganised and selfstabilising reduction of SR.
The notion of liabilityspecific SR is based on the probability of default and the impact of a failure of a financial institution, as measured by DebtRank. A central idea of this paper is to separate default risk from contagion risk. Contagion risk is the risk that a default by one institution leads to defaults of other institutions. DebtRank denotes the risk of financial contagion from the interbank liability network. The default risk of a financial institution depends on its financial condition and on the economic situation in general. In principle, it can also depend on financial networks, for example, the network of overlapping portfolios or the network of firms and regions or industries. However, if network contributions to the probability of default are small (second order), it becomes possible to separate default risk from contagion risk, as given by DebtRank, in a meaningful way. Otherwise, it is necessary to replace or generalise DebtRank with a methodology that includes the network of overlapping portfolios and other relevant networks.
In Poledna et al. (2015) we show that DebtRank and the ideas presented here can be generalised for multilayer networks to quantify and reduce SR originating from different financial markets, such as derivatives, foreign exchange and securities. In future work we will conduct an empirical study on the network of overlapping portfolios and its implications for SR. Joint defaults can also be taken into account for example by copula methods. Once the default correlation has been estimated, it is possible to include the joint probability of default straightforwardly in the present framework. Joint defaults can be included by considering the joint probability of default of a group of financial institutions and by calculating the DebtRank for this group. Thus, additional terms containing the joint probability of default and the impact of a group can be added to \crefEL,srteq_simple.
We test the SRT within the framework of the CRISIS macrofinancial model. The model produces SRprofiles of banks that are practically identical to those of actual interbank liability data. Even on the level of individual transactions the model is fully compatible with the empirical data (see also (Poledna et al., 2015)). The SRT drastically reduces the probability of a financial collapse due to restructured liability networks that minimise the size of cascading failure. The tax is implemented in a simple way: an agent would like to make a transaction (with a given counterparty) and expresses this interest by announcing it (and the envisioned counterparty) to the central bank. The latter computes the SR increase associated with the transaction based on the knowledge of the present state of the entire liability network and the capitalisation of its agents. The SRincrease is then presented to the agent as a tax (SRT) for that particular transaction. If the SRincrease is zero, then it is taxfree. The agent can now look for other counterparties to carry out exactly the same transaction. The agent will therefore typically screen several possible counterparties and then decide on the one with the lowest tax. Once the agent decides to carry out the transaction, it is executed and the tax is paid to the central bank or the government.
We show explicitly that SR is, to a large extent, a network property. We show that the SRT is able to restructure financial liability networks without loss of transaction volume in the financial market. For an explicit comparison we implement and test a Tobinlike tax that taxes all transactions regardless of their SR contributions. The Tobinlike tax does not restructure networks and only reduces SR because it also drastically reduces transaction volume in the system. This is damaging as it makes the system less efficient; the loss of efficiency materialises as expensive credit for the real economy. We tested an alternative mode in which the SRT is set to the true increase in SR associated with the transaction, and not only a fraction (). This alternative leads to much more homogeneous SRspreading across all agents, and makes the system even safer, see \crefdata_vs_model(b) and \crefhist_srt(d), however, it is done at the cost of much reduced transaction volume.
An obvious alternative to the SRT are bank taxes for SIFIs as recently suggested by several authors (Cooley et al., 2009; Acharya et al., 2012; Adrian and Brunnermeier, 2011; Markose et al., 2012; Acharya et al., 2013; Zlatić et al., 2014), or alternatively, to increase capital requirements for them by imposing SIFI surcharges as proposed in Basel III (Bank for International Settlements, 2010). An immediate problem of a bank tax or capital surcharges, compared to the SRT, is that financial institutions sometimes have no control over their systemic importance. For instance, in case of publicly traded securities, such as bonds, financial institutions have no authority to decide who holds them and thus no influence on their systemic importance. In recent work the effects of the Basel III regulation framework on SR has been studied in the same ABM environment (Poledna et al., 2016). Results indicate that capital surcharges for SIFIs can reduce SR, but must be larger than specified in Basel III to have a measurable impact, and thus cause a loss of efficiency.
We close with a remark on the policy relevance of the SRT. We think that the concept that networkrelated SR can be managed most efficiently by restructuring the underlying network topologies, is generally true. We further believe that the concept of the SRT as presented here is directly policy relevant. In particular, the incentive scheme introduced to internalise the externalities that lead to SR, can be directly transferred to banking regulation. Technically, the requirements for its implementation would include an electronic market run by a central bank or another central authority. This electronic market would work in the same way as an airline reservations system, i.e. by holding a quote for a limited amount of time. The computational requirements for central banks to compute the SRT for several thousand banks on a minute by minute basis is by todayâs standards a technical triviality. Central banks would have to record transactions in realtime; this is done as a matter of course for several financial markets routinely, for instance at the Banco de México (SolorzanoMargain et al., 2013; Poledna et al., 2015). There are no privacy issues; obtaining information about the portfolios of other banks from the SRT quotes is impossible, in the same way as it is impossible to infer the entries of a real valued matrix from the eigenvector associated to the largest real eigenvalue. The problem that generally applies to any FTT – that it should be implemented globally to avoid free riding – also applies in the context of the SRT.
We finally stress that the current market practice of not pricing SR into transactioncosts effectively amounts to an implicit subsidy for those with the highest contribution to overall SR, and an effective tax on those with the lowest. In that sense, the SRT can be seen as an insurance premium that keeps the public neutral with respect to the SR that is introduced by the nonoptimal liability networks.
Acknowledgements
We thank P. Klimek for many stimulating conversations. We acknowledge financial support from EC FP7 projects CRISIS, agreement no. 288501 (65%), LASAGNE, agreement no. 318132 (15%) and MULTIPLEX, agreement no. 317532 (20%).
Appendix A Details of the model
In this section we describe the extensions and modifications of the macroeconomic model of Delli Gatti et al. [2011]. The modifications include the implementation of an interbank market and a closed, stockflow consistent economic system that allows no in or outflow of cash. The closed economic system is also discussed in Gualdi et al. [2015].
a.1 The credit market
There are banks that offer firm loans at rates that take the individual specificity of banks (modelled by a uniformly distributed random variable) and the firms’ creditworthiness into account. Firms pay a credit risk premium according to their creditworthiness that is modelled by a monotonically increasing function of their financial fragility. A firm’s financial fragility is defined as the ratio between the outstanding debt and the liquid financial resources of the firm [Delli Gatti et al., 2011]. Specifically the interest rate for firm , borrowing from bank is given by
(6) 
where is a benchmark interest rate, is the specificity of bank – modelled as random variations in its operating costs, strategy, etc. and captured by a uniformly distributed random variable on the interval . is a proxy for the financial fragility of the borrower – modelled by a monotonically increasing function of the borrower’s debt to liquidity ratio . The hyperbolic tangent is chosen for .
a.2 The interbank market
Banks try to provide firm loans and grant them if they have enough liquidity. If they do not have enough cash, they approach other banks in the interbank market to obtain the required amount. If a bank does not have enough cash, and cannot raise the full amount for the requested firm loan on the interbank market, it does not pay out the loan. Interbank and firm loans have the same duration. Additional refinancing costs of banks remain with the firms. Each time step firms and banks repay percent of their outstanding debt. If banks have excess liquidity they offer it on the interbank market. The interbank market is modelled after an electronic marketplace where, in principle, all participants can enter into business relationships. In the model, banks choose the interbank offer with the most favourable rate. This does not mean that the emerging interbank network is fully connected. Emerging interbank networks are shown in \crefnetwork and (weighted) degree distributions can be found in \crefdegree_dist. Interbank rates offered by bank to bank take into account the specificity of bank and the creditworthiness of bank . Specifically the interest rate on the interbank market for bank borrowing from bank is given by
(7) 
where is a benchmark interest rate, is the specificity of bank , modelled as random variations in its operating costs, strategy, etc. and captured by a uniformly distributed random variable on the interval . is a proxy for the financial fragility of the borrower, modelled by a monotonically increasing function of the borrower’s leverage . As the monotonically increasing function again the hyperbolic tangent is chosen. Banks add the additional refinancing costs to the offered interest rate for firms. Therefore the interest rate for firm , borrowing from bank , which requires additional liquidity from bank is given by
(8) 
where is the firm loan and is the ratio between the interbank and the firm loan.
a.3 Implementation of the systemic risk tax and the Tobin tax
A systemic risk premium, in form of the SRT, is imposed on all interbank transactions. Before entering a desired loan , the credit seeking banks can get quotes for the rates from the central bank, for various banks . They choose the interbank offer from bank with the smallest total rate, which is composed of . All other transactions are exempted from the SRT. In contrast to current market practice, the effective interest rate reflects both the creditworthiness of the borrowing counterparty and the SR increase associated with each transaction. The SRT is collected in a bailout fund. The SRT from the main text is given by
(9) 
For we use a proxy for the financial fragility of the borrower, modelled by a monotonically increasing function of the borrower’s leverage at time .
For comparison we implement a FTT (Tobin tax [Tobin, 1978]) for interbank loans. We impose a constant tax rate of 0.2% of the transaction (this is about 5% of the interbank interest rates) on all interbank rates on offer. Other transactions are not taxed. The FTT makes lending less attractive for firms that borrow from banks requiring liquidity from the interbank market, as refinancing costs remain with the firms.
Interbank rates offered by bank to bank including the FFT or the SRT are composed of
(10) 
In case of the FFT the tax term is simply a constant tax rate of 0.2%
(11) 
To obtain a tax rate, the SRT must be expressed as a ratio with respect to the interbank loan (). The total rate is then given by
(12) 
where is from \crefbank_ir. Banks add the additional refinancing costs, including taxes, to the offered interest rate for firms. Therefore \creffirm_ir becomes for the Tobinlike tax
(13) 
and in case of the SRT,
(14) 
a.4 Model parameters
All parameters of the model are collected in \crefparameters.
number of banks  
number of firms/capitalists  
number of workers (households)  
share of dividends  
general refinancing rate  
labour productivity  
credit demand contraction  
rate of debt reimbursement  
wage rate  
number of applications in consumption goods market  
propensity to consume  
number of applications in credit market 
Appendix B Comparison of different tax rates for the Tobinlike financial transaction tax
In \crefhist_tobin we show the distribution functions of the three measures for (a) losses , (b) cascade sizes , (c) transaction volume in the interbank market and (d) the distribution of DebtRank , for the simulations performed with different tax rates for the Tobinlike FTT, 0.1% (red), 0.2% (blue) and 0.5% (green). Clearly, the shape of the distribution of losses and cascade sizes are similar. The tail of the distributions is only reduced due to a decrease in efficiency (transaction volume), as can be seen in \crefhist_tobin(c). Evidently, average losses are reduced at the cost of a loss of efficiency by roughly the same factor.
For the comparison of different levels of the SRT we choose (red) and (blue), as shown in \crefhist_srt. Again, we compare the three measures for (a) losses , (b) cascade sizes , (c) transaction volume in the interbank market and (d) the distribution of DebtRank . Clearly, for both the SRT gets completely rid of big losses in the system. reduces average losses by a factor of compared to the case of , at the cost of a loss of efficiency by roughly the same factor, as can be seen in \crefhist_srt(c). The SRT with () leads to homogeneous SR spreading across all agents, as shown in \crefhist_srt(d).
Mode  

Normal  9.43(5)  38.4(35)  0.136(5)  0.119(3)  10.3(7)  40.2(53) 
Tobin tax  9.39(9)  32.4(32)  0.135(5)  0.117(3)  10.4(7)  40.0(55) 
SRT ()  9.15(13)  25.4(35)  0.138(6)  0.112(5)  11.9(20)  44.1(76) 
SRT ()  8.73(20)  7.4(14)  0.122(6)  0.100(4)  11.3(12)  45.9(68) 
In \crefdegree_dist we show the effect of the bank selection process induced by the SRT on the interbank liability network topology. The distributions of weighted indegrees of the interbank liability network , without FTT (red), 0.2% Tobin tax (blue), the SRT () (green) and, the SRT () (yellow) are shown in \crefdegree_dist(a). Without FTT, the emerging liability network shows Poisson distributed indegrees. The interbank network topology without FTT coincides nicely with the expected result from random linking. In the SRT modes, market participants looking for credit will try to avoid the tax by looking for credit opportunities that do not increase SR and are thus taxfree. This leads to fewer banks lending on the interbank market and is reflected in \crefdegree_dist(a) by the high number of nodes with a low weighted indegree.
The total demand for interbank loans (which is approximately the same for the SRT with as without FTT) is now serviced by fewer banks. As a result, the indegree distribution of the SRT mode broadens and has a fat tail. The outdegree distribution is mainly influenced by the cash needs of a bank. Therefore, the weighted outdegree distribution of the SRT modes is less clearly affected, which is shown in \crefdegree_dist(b).
In \crefdegree_dist(c) we show the average weighted clustering coefficient of the interbank liability network , without FTT (red), 0.2% Tobin tax (blue), the SRT () (green) and the SRT () (yellow). Average weighted clustering coefficients are calculated according to Barrat et al. [2004]. The average clustering is roughly the same without a FTT on interbank loans and for the 0.2% Tobin tax. The SRT reduces average clustering, as can be seen in \crefdegree_dist(c).
Mean values of various centrality measures, averaged over simulations runs, can be found in \crefnwanalysis. and shows the mean degree and the weighted mean degree for the different modes. is approximately the same for all modes. shows the largest value for the normal mode and lower values in all other modes. Clearly, with the SRT () is substantially reduced. With and we show values for the average clustering coefficients and the average weighted clustering coefficients from \crefdegree_dist(c). Additionally, we provide values for the average betweenness centrality () and the average weighted betweenness centrality () for the different modes. The SRT increases both the and the .
Appendix C Empirical data
Data provided by the Austrian Central Bank (OeNB) contains fully anonymised, and linearly transformed interbank liabilities/exposures from the entire Austrian banking system, comprised of about 800 banks over 12 consecutive quarters from 20062008. The data set additionally includes total assets, total liabilities, assets due from banks, liabilities due to banks, and liquid assets (without interbank assets/liabilities) for all banks again in anonymised form. The data does not contain credit ratings of banks. Therefore we assume for all banks in the data set. This corresponds approximately to Standard & Poor’s OneYear Global Corporate Default Rates for Rating Categories A+, A, and BBB+ in 2008 [Standard & Poor’s, 2009]. Representative Austrian banks are in the Rating Categories A+, A and A.
Appendix D DebtRank
DebtRank is a recursive method suggested in Battiston et al. [2012a] to determine the systemic importance of nodes in financial networks. It is a number measuring the fraction of the total economic value in the network that is potentially affected by a node or a set of nodes. denotes the interbank liability network at any given moment (loans of bank to bank ), and is the capital of bank . If bank defaults and cannot repay its loans, bank loses the loans . If does not have enough capital available to cover the loss, also defaults. The impact of bank on bank (in case of a default of ) is therefore defined as
(15) 
The value of the impact of bank on its neighbours is . The impact is measured by the economic value of bank . For the economic value we use two different proxies. Given the total outstanding interbank liabilities of bank , , its economic value is defined as
(16) 
To take into account the impact of nodes at distance two and higher, this has to be computed recursively. If the network contains cycles, the impact can exceed one. To avoid this problem an alternative was suggested in Battiston et al. [2012a], where two state variables, and , are assigned to each node. is a continuous variable between zero and one; is a discrete state variable for three possible states, undistressed, distressed, and inactive, . The initial conditions are , and (parameter quantifies the initial level of distress: , with meaning default). The dynamics of is then specified by
(17) 
The sum extends over these , for which ,
(18) 
The DebtRank of the set (set of nodes in distress at time ), is , and measures the distress in the system, excluding the initial distress. If is a single node, the DebtRank measures its systemic importance on the network. The DebtRank of containing only the single node is
(19) 
The DebtRank, as defined in \crefdebtrank, excludes the loss generated directly by the default of the node itself and measures only the impact on the rest of the system through default contagion. For some purposes, however, it is useful to include the direct loss of a default of as well. The total loss caused by the set of nodes in distress at time , including the initial distress is
(20) 
Appendix E Derivation of the expected systemic loss
To compute the expected systemic loss, we first consider the simple case where only one bank can default and all other banks survive. In this case the expected loss is given by , where is the probability of default of bank , and the survival probability of . The general case occurs when we also consider possible joint defaults, meaning that a set of banks go into distress. Taking into account all possible combinations of defaulting and surviving banks, we arrive at a combinatorial expression of the expected loss for an economy of banks
(21) 
where is the power set of the set of banks , and is the DebtRank of the set of nodes initially in distress. , the DebtRank of the empty set is defined as zero. The reason is that, by definition of DebtRank, , the value obtained in \creftotEL_normalized cannot exceed the total economic value.
totEL_normalized is only practical for situations with less than about banks. Computing the power set and calculating DebtRanks for all possible combinations of more than banks in a large financial networks is practically unfeasible. If the default probabilities are low () or the interconnectedness is low (), can be approximated by
(22) 
In an unconnected or unleveraged financial system (), is exactly equal to . If , the first terms of \creftotEL_normalized (with only one node initially in distress) contribute more to the final result. Thus the approximation \crefdebtrank_approx has only a minor impact on the final result. Typically, or holds in real word financial networks. With the approximation \crefdebtrank_approx, \crefEL can be derived from \creftotEL_normalized by
(23)  
(24)  
(25) 
The term in brackets in \crefbrace sums to 1 (proof by induction). This approximation is practical for large financial networks, for details see [Poledna et al., 2015].
Appendix F Measures for losses, default cascades and transaction volume
We use the following three observables: (1) the size of the cascade, as the number of defaulting banks triggered by an initial bank default (), (2) the total losses to banks following a default or cascade of defaults, , where is the set of defaulting banks, and (3) the average transaction volume in the interbank market in simulation runs longer than time steps,
(26) 
where represents new interbank loans at time step .
Footnotes
 http://www.crisiseconomics.eu
 Note that the entries in are the liabilities bank has towards bank . We use the convention to write liabilities in the rows (second index) of . If the matrix is read columnwise (transpose of ) we get the assets or loans banks hold with each other.
 Here is the default probability density of node at time , and the present value (at time ) of 1 Euro received at time . The default probability density is defined as , where is the hazard rate. The duration of the loan is from until and is computed at time .
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