Systemic risk and contagion: the concepts

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Introduction

financial default risk delicensing

Introduction and outline

The problem of systemic risk has been discussed in economic literature for quite a long time already. The works (Sharpe, 1964) and (Lintner, 1965) could be supposed to be the first ones when the problem, in its simplest form, was referred to. However, it was not until recently that systemic risk has been recognized as one of the main problems in today's financial systems. The severity and the fast spread of the financial crisis of 2008-2010 have emphasized the great importance of systemic risk which may very generally and vaguely be described as a macro-level risk that can pose a threat to the stability of the entire financial sector. It turned out that monetary authorities and other regulators were not able to predict the consequences of defaults (particularly of large financial institutions) as well as to react to such defaults fast and accordingly. The main reasons for that are non-transparent structure of financial system and the lack of adequate indicators for measuring and monitoring systemic risk. This suggests that the emphasis should be transferred from the traditional “too big to fail” perspective to the one which focuses on the contribution to stability of financial system as a network, that is “too interconnected to fail” (Griffin, 2008, p. 2).

The highlighted importance is the reason why this work focuses on the contagion in the banking system and its effects. More specifically, the issue is analysed in the context of bank delicensing policy in Russia, the policy which is a point of interest itself. The main objectives of this paper are: 1) to stress the fact that it is vital to consider systemic risks when conducting delicensing policy; 2) to present and develop a theoretical model for measuring/quantifying systemic risks; 3) to suggest a more prudent and flexible delicensing policy based on the developed model.

The paper starts with a brief literature review where I point out and cite some of the most important research works concerned with the problem. Then, in Section 2, the concepts of systemic risk and financial default contagion are defined; their main causes and features are discussed and explained in details. In Section 3, I present the model for quantification of systemic risk and particularly of cascade default losses, the model that was firstly developed in (Cont, Moussa and Santos, 2012). This model allows one to calculate two measures (Default Impact and Contagion Index) that show the amount of capital losses in a banking system after a default contagion triggered by one or several initial defaults. In Section 4, I move on to discussing the issue of the Central Bank's policy of delicensing of commercial banks in Russia. First of all, I describe the existing situation in Russian banking system and provide the reasons behind the revocation of banking licenses. Then the focus is transferred to the delicensing that is based on satisfying (or not satisfying) capital adequacy ratios. The model from Section 3 is revisited and modified for that matter. Next, I proceed to the main topic of this paper, namely, to the analysis of the existing delicensing policy in the context of default contagion. I suggest that the current policy may be disregarding several important aspects of systemic risk. I then propose several more prudent alternative policies including the one that uses a further extension of the developed model in the form of two additional measures - Relative Contribution to Default Impact and Relative Contribution to Contagion Index. Finally, in Section 5, I look for the empirical justification of the improved results of the alternative delicensing policy. I propose the algorithm that allows one to reconstruct the missing data more or less accurately and then the algorithm that applies the theoretical model to a set of real/reconstructed data These algorithms can be found in the attached Excel file “Interbank” (both in macros, VBA, part and in the main Excel sheets).. Even though I was not able to run the algorithms and, therefore, get a numerical illustration (due to computational constraints), I was still able to make a qualitative conclusion that the suggested alternative delicensing policy is usually superior to (and is never worse than) the existing one.

Literature review

As it was already said above, the very first time the problem of systemic risk was tackled was in 1960s and particularly in the work of Sharpe. Although the work itself was quite far from dealing with the systemic risk directly, the author has for the first time underlined the importance not only of the idiosyncratic (or firm-specific) risk but also of the so-called systematic risk, that is the risk that the whole market is in the downfall. This point illustrated the crucial characteristic of interconnectedness of different agents in the financial markets.

The next big step in the direction of realizing the importance of systemic risk was made in 1980s when a lot of economic works were devoted to the problem of bank runs. Before the times it was commonly supposed that financial crises are purely random events which could not be related to changes in real economy (Kindleberger and Aliber, 1978). However, in the well-known work (Diamond and Dybvig, 1983) a different (and, as it seems now, a more reasonable) view on the subject was proposed. The authors suggested that bank run is a kind of self-fulfilling prophecy: each depositor's decision to either withdraw or not to withdraw their funds depends on their expectations about other depositors' decisions. In case the large enough part of depositors expects others to withdraw their funds (even if in reality it is not so) all of the depositors have an incentive to rush to the bank to be “the first served”. This conclusion emphasises the common for systemic risk idea that some small scale adverse events may easily lead to a large scale disaster (for instance, bank runs or crisis et cetera) in a financial sector.

Closer to the end of the millennium systemic risk has gained the popularity among the economists, regulators and practitioners. Since then the problem of systemic risk has become recognized and many working papers were directly devoted to the all-round study of the phenomenon. The literature could be logically classified in several ways. Firstly, the understanding of what systemic risk actually means was constantly changing with the concept becoming more and more general. More specifically, before the recent financial crisis the main focus was on contagion and the large scale of effects of systemic risk, while after the outbreak of the crisis attention was drawn to the negative impact on the real economy (Smaga, 2014). Naturally, the difference was reflected in the research papers: before the crisis the majority of them were aimed at analysing contagion within financial sector as a channel of transmission of initial shock, whereas after the crisis “pro-cyclical connection between the financial sector and the real economy” (Bijlsma et al., 2010, p.9) was studied as an additional channel. Secondly, the whole literature about systemic risk could be divided according to the type of research performed. It could be a purely mathematical work which suggests a new measurement tool for the systemic risk and then presents some simulation; or it could be a practical work with the analysis of a particular banking (or other financial) network; or it could be a purely descriptive work which summarizes the earlier obtained information and gives more “food for thought” for future researchers. Finally, another possible differentiation could be based on the type of author, that is whether a paper is written by a regulator or by a researcher-economist. Now let us carefully look at the papers, which are the most relevant for this work, according to these three classifications.

The vast majority of all scholar papers are concerned with the contagion effect in financial systems. One example is the work (Santos and Cont, 2010) where the Brazilian interbank network was explored in the context of its exposure to systemic risk. The authors have shown that any banking network “can be adequately modelled as a directed scale-free (weighted) graph with heavy-tailed degree and weight distributions” (Santos and Cont, 2010, p.3). They have defined default and introduced several measurements of systemic risk and applied the model to the unique set of data on interbank deals over some period in Brazil. In this way they were able to come up with the quantitative measures of the extent of systemic risk generated by each individual financial institution. This work was expanded in their subsequent work in collaboration with Amal Moussa (Cont, Moussa and Santos, 2012) where the researchers investigated characteristics of the Brazilian interbank system. Those characteristics are degree of connectivity between the nods of a graph (banks in the interbank lending market), relationship between exposure size and connectivity, clustering and assortativity property. Additionally they have analysed what particular features of financial institutions make the institutions systematically important. It was observed that the size of such institutions does matter (“too big to fail” principle holds), however, size as an explanatory variable only explained about a quarter of the variation in systematic importance index. Other two significant explanatory variables appeared to be counterparty susceptibility coefficient and local network frailty (Cont, Moussa and Santos, 2012, pp.32-36). Another closely related to the two research papers is (Leonidov and Rumyantsev, 2013). Similarly to Cont et al. the Russian authors considered the interbank lending market structure using oriented graph. Using the aggregated data for Russian interbank lending market they have verified the importance of clustering, assortativity and the other abovementioned attributes. A bit more complicated analysis of a random graph representation of financial systems was done in (Hurd and Gleeson, 2011). A probabilistic framework is introduced in such a way that the possibility of disassortative edge probabilities is explicitly incorporated. The disassortativity principle, being defined by the authors as “an above average tendency for small banks to link to large banks” (Hurd and Gleeson, 2011, p.1), could be viewed as the main focus of their work. It was observed that the so-called “graph-assortativity”, rather than node-assortativity, can greatly affect the cascading default process.

Although the random graph theory is the most common way of studying the contagion effect, it is by no means the only one. The working paper (Allen and Gale, 2000) serves as an example. The main aim of that paper is to provide microeconomic foundations for the phenomenon of financial contagion. The emphasis was made on liquidity issues: different simplified interbank lending market structures were considered (with different connections and their types) and possibilities of liquidity shortages were analysed. An alternative method was proposed in (Acharya et al., 2009). The theory developed by these researchers considers “a number of financial institutions (“banks”) that must decide on how much capital to raise and which risk profile to choose in order to maximize their risk-adjusted return” (Acharya et al., 2009, p.2). Additionally there is a regulator, with a specific expected utility function, who maximizes its own utility by choosing optimal tax on financial institutions. The tax is paid for the additional systemic risk created by the financial institution and, therefore, such risk is considered to be an externality (negative one). The tools for measurement of systemic risk are widely known ones - Value-at-Risk and Expected-Shortfall. The latter is subsequently substituted by a newly defined Systemic Expected Shortfall (SES) - “expected loss in a systemic crisis, adjusted for its leverage” (Acharya et al., 2009, p.33). Moreover, authors proposed a particular way of calculating SES from the real data. Finally, the model was applied to the data during subprime mortgage crisis. Important theoretical result was the observed dependence of SES on institution's leverage, institution's volatility, system's volatility and several other indexes. There were also several significant empirical results: (1) ex-ante SES index of financial institutions accurately predicts their losses during the subprime crisis; (2) SES is higher during periods of macroeconomic instability; (3) SES is positively correlated with capitalization of a financial institution measured as assets divided by equity (again the sign of “too big to fail” principle).

Moving on to the works concerned with pro-cyclicality, one should highlight the review paper (Bijlsma et al., 2010) (and cited literature). It is underlined that although contagion is a crucial part of systemic risk analysis, question of “how fluctuations in the financial system and the real sector may reinforce each other, i.e., how credit growth and banks' risk taking can amplify the business cycle and vice versa” (Bijlsma et al., 2010, p.39) is just equally important. The researchers distinguish several potential causes of pro-cyclicality including regulation, asymmetric information and financial markets “rational herding”. And then each of the causes is comprehensively discussed. As a result of their study, Bijlsma et al. suggested some regulatory policies to diminish the exposure to systemic risk.

Some works on the subject were purely mathematical. Two most interesting ones are (Amini et al., 2013) and (Ichiba and Fouque, 2011). The former one deals with the financial networks in a stochastic framework which allows analysing potential equilibria in the case of central counterparty clearing (CCP) - a special financial entity that has all liabilities cleared through it. Introduction of such financial entity transforms the whole interbank system in a way that each and every other financial institution in the network has either a payment to the CCP or a receivable from the CCP or both. The authors showed that the introduction of the CCP in a network does not necessarily decrease systemic risk but in general it could do so (the corresponding necessary and sufficient conditions are derived in the work). The latter of the two works above presents how contagion effect is analysed via random graph processes in continuous time. The main obtained result is quantification of the proposed model.

On the other side there are more theoretical (without specific mathematical models, simulations and any empirical analysis using real world data) papers which are primarily aimed at descripting some aspects of systemic risks. One sapid article was written by the head of financial stability department of Bank of England A. Haldane and a former Chief Scientific Adviser to the UK Government R. May (Haldane and May, 2011). They have provided a very interesting perspective on the subject by comparing systems of financial institutions to natural ecosystems. Their principal goal was to “explore the interplay between complexity and stability in deliberately simplified models of financial networks” (Haldane and May, 2011, p.351). The key finding was that, similarly to ecosystems, with the increasing complexity of a system, the stability of the system could be maintained only up to some point. And when the system becomes too complex it becomes highly unstable and difficultly predictable. This leads to the conclusion that the existence of a big number of complexly structured relations between various financial institutions not only allows diversification of individual risks but also facilitates the fast propagation of potential shocks in the system, that is it accounts for the increase in systemic risk. The authors have also suggested some regulatory measures for public policies such as setting regulatory/liquidity ratios and other regulatory requirements. An absolutely different treatment of systemic risk may be found in the recent work (Smaga, 2014). The paper attempts to summarize all existent definitions of systemic risk and proposes its own one; it also explores the essence of the concept of systemic risk and looks for the factors which contribute to its accumulation; finally it assesses impacts of contagion. It is pointed out that “systemic risk is characterized by its evolving and multidimensional nature and can both be endo- and exogenous” (Smaga, 2014, p. 21). Also the author notes that systemic risk is transmitted not only within the financial system but it can also spill-over to the real economy.

The problem of systemic risk could not have been overlooked by international as well as domestic supervisory and regulatory authorities. The abovementioned work of Haldane and May could be referred to in this way. Quite a big number of international conferences specifically devoted to discussions devoted to systemic risks have been held since 1990s. For example, there have been a series of Joint Central Bank Research Conferences in 1995, 1998, 2002 and 2005. The high frequency of such conferences indicates the increased importance of systemic risk. The presented reports are usually aimed at providing some regulatory responses to the problem of systemic risk.

Finally, outside the classification of scholar works, articles and papers suggested initially, there are several very broad themed works which try to cover a substantial part of all research directions in the field of systemic risk. Those works are usually called surveys. An all-round survey of theoretical works could be found in (De Bandt and Hartmann, 2000) while the exhaustive analysis of systemic risk analytics was done in (D. Bisias et al., 2012). The latter work is especially useful because the researchers provided an extensive description of how each step in analytical paper can be done in specialized computer program (Matlab) and have developed an open-source Matlab code for most of the analytics.

1. Systemic risk and contagion: the concepts

It is obvious that one cannot study any phenomenon without a clear understanding of what the phenomenon is and what potential methods for its measurement exist. Thus, before introducing the model for analysis of default contagion process it is necessary, firstly, to define the two notions (systemic risk and financial default contagion) and, secondly, to explain why the phenomena are of special concern in the financial system.

1.1 Defining and explaining systemic risk

As it has already been noticed, there does not exist a unique and complete definition of systemic risk. It is common to come across various definitions of the term in almost every paper dedicated to the problem. It is of great importance to stress that understanding of systemic risk comes along with the development of financial markets, with the emergence of new regulatory responses and especially with the occurrence of macroeconomic disturbances which provides data for research and shows some weak points of the system. The recent financial crisis, for example, has demonstrated that systemic risk is actually much more than simply a combination of individual types of risk. While before the crisis those individual types of risk (such as liquidity risk, credit risk, operational risk et cetera) were considered separately, the consequences of their potential correlation are much more devastating. After the crisis it has also become widely accepted that systemic risk conceals possible problems not only for the financial sector itself but also for real economy.

To sum up, one could define systemic risk as a risk that a small initial shock of one financial institution (or a small group of financial institutions) will have an immensely adverse effect, via transmission of disturbances to other financial institutions through a highly interconnected financial structure, not only on the financial system itself but also on the real sector of the economy. Later in this work, however, the concept of systemic risk will be redefined in a narrower way such that it will only deal with contagion process and will solely consider the effects on financial sector.

Having defined the notion of systemic risk, let us move on to understanding its features and causes. The first thing to note is that in general it is rather hard to determine whether an event is likely to have a systemic influence. It is even more so during periods of macroeconomic instability when several negative shocks may happen at a time. This in turn leads to overlapping effects and therefore eliminates the possibility of figuring out the effect of each single shock. Another problem created by economic fluctuations is that “assessing the extent to which it [an event or shock] affects other parts of the system may be subject to dynamic changes and the assessment might be prone to an underestimation bias” (Smaga, 2014, p. 6).

The most frequent causes of systemic risk are fully described in the work (Allen and Carletti, 2011). The list includes: (i) contagion; (ii) liquidity provision and mispricing of assets; (iii) multiple equilibria and panics (the cause of bank runs in Diamond and Dybvig model); (iv) sovereign default; (v) common exposure to asset bubbles and (vi) currency mismatches in banking system.

1.2 Financial default contagion and its peculiarities

Another crucial notion to understand is contagion. Since the word is quite widely used in non-financial contexts it could be easily inferred that the phenomenon is not limited solely to economics or financial systems. For example, people have faced contagion in the sphere of healthcare in the form of various epidemic diseases and in the sphere of information technology in the form of computer viruses. Although the three spheres (including financial one) are very different from each other, contagion in all of them appears to have many common features. At first, an initial shock happens which then leads to adverse effects on one small part of a system (liquidity shock leads to a default of a bank, infection of a person leads to his/her illness or an appearance of a computer virus in one of computers leads to its dysfunction). Afterwards, due to high interconnectedness of parts of a system, the shock is transmitted to the “neighbours” of the initially affected part of a system and then further into the system (the defaulted bank can no more serve its liabilities and other banks, which are creditors of the bank, lose part of their assets which may lead to their own default; an infected person by contacting with other people may allow the infection to be transmitted to them; a computer virus may easily spread to other computers on the same local, or even global, network as the “infected” one). A third thing in common is that once the process has started it is extremely difficult to stop it artificially and it is usually easier just to wait until it ends by itself. A final similarity is the absolutely devastating result of such events (Great Depression; Black Death plague; CIH virus also known as Chernobyl).

It could come as a surprise, though, that contagion, or contamination effects, can occur not merely in financial sector of an economy but also in other sectors. Nonetheless, it is not usually considered as a problem, or at least as a big problem, in other sectors mainly because the likelihood and severity of contagion is significantly lower than in the financial sector. There are three main reasons for the so-called “financial fragility hypothesis” that explains the vulnerability of financial systems to contagion.

The first reason is the special structure and function of banks as financial intermediaries. According to Asset Transformation Theory one of such functions is maturity transformation, that is banks borrow for a short time (the majority of deposits are short term ones) while they lend for a much longer period (credit programs are usually designed for a term longer than several years, even if not taking mortgages into account). Thus, there is in principle a possibility of a mismatch between funds available for repayment and funds demanded by lenders (depositors). Such illiquidity may be quite a serious issue which can lead to a default even of a sound and stable in the long-run bank. The potential problem is exacerbated by the common practice of holding only partial reserves and the availability (for depositors) of partial withdrawal of funds before the actual end of the depository term. The problem is partly resolved when the Law of Large Numbers applies - the mismatch is unlikely to happen and only a small fraction of assets in liquid form needs to be held in order to meet all the withdrawals. Thus, for maintaining its stability and health, a bank has to select wisely profitable investment projects (for lending funds) as well as to have an image of a reliable bank, that is to assure confidence of its depositors in the value of the loan book and, even more importantly, to assure confidence in that other depositors will not run the bank. Such confidence is partly provided by a deposit insurance system adopted in the vast majority of developed and in many developing countries (for example, in Russia there exists a specialized agency for the purpose - the Deposit Insurance Agency of Russia). This issue is extremely important and the practical consequences of a decreased confidence will be discussed further in the coursework.

The second reason is the high interconnectedness of financial institutions through direct exposures. Banks play a pivotal role in payment and settlement systems. Sometimes during a single business day banks may be so overexposed that a fail of one bank to fulfil its payment obligations could lead to an immediate similar failure of other banks. This in turn could trigger a large scale crisis via the “domino effect”.

The third reason is informational intensity of financial contracts and other credibility problems. In general it is supposed (and justified by microeconomic approach) that individuals make their optimal decisions by maximizing their utilities from their specific intertemporal consumption functions. Since intertemporal functions always depend on the future values of some variables and at a given point in time there is only partial (or even no at all) information about the future (events, state of the economy, returns et cetera), the decisions made are based on expectations. In particular, those expectations might be concerned with whether the cash flows promised in the contract are actually going to be observed in the future. Therefore, questionable credibility of financial commitment may easily change the expectations. And even a slight shift in the expectations may considerably change the individually rational decisions, which in turn could lead to the abovementioned liquidity problems and even bank runs. For example, if depositors observe that there is a revision of their bank, they, not having any further information, may think that there is something wrong with the bank and so will rush to withdraw their deposits.

These three reasons fully explain the extensive vulnerability of financial system in general and as compared to other sectors of the economy in particular.

2. Systemic risk and contagion: the model

Having defined and described the two concepts of systemic risk and contagion, let us proceed to the modelling part of the paper. The model presented below is partly adopted and modified from the work (Cont, Moussa and Santos, 2012).

2.1 Counterparty network

First of all, it is necessary to represent the interbank relations as a weighted directed graph, or a network, with nodes of the graph representing banks and edges of the graph representing exposures of banks (including directions). Such graph is defined by a triplet, where:

V is a set of financial institutions whose number is n.

E is a matrix of bilateral exposures. The notation means the exposure of node i to node j measured as a market value of all liabilities of bank (or other financial intermediary) j to bank i. In this way, therefore, it is the maximal possible loss of bank i in case of the default of bank j.

is the vector of capital of institutions which is used to determine solvency of a bank (and ).

One can also define some technical characteristics of the graph such as in-degree of a node as the number of bank's debtors and out-degree as the number of bank's creditors:

, (1)

Finally, one can define A(i) as total interbank assets of financial institution i and L(i) as its total interbank liabilities:

, (2)

2.2 Default contagion

Having defined the interbank network as a graph, it is time to describe how systemic risk is related to it. One is interested not only in identifying relative systemic importance of a financial institution in the financial system but also in observing the quantified consequences of the failure of a particular bank (node in the network). In such way one could determine the stability of which banks is vital for the stability of the system as a whole. This, in turn, shows monetary and other regulatory authorities which banks should be bailed out in the first place in case of liquidity or other problems. There are two indicators of default contagion and systemic impact described below - Default Impact and Contagion Index. The former indicator represents the total loss in capital (in the whole system) in the cascading default process started with the default of one particular bank. The later indicator is a bit more complex. By trying to incorporate the macro situation, it shows the expected value of Default Impact in a market stress scenario.

Before formally defining the two indexes one should carefully explain the default mechanism itself. “<…> default is a failure to meet the legal obligations (or conditions) of a loan <…>” Definition is taken from www.wikipedia.org.. The “legal obligations” include not only the repayment of a loan itself (its principal) but also the payment of interest and the payment for servicing of the loan. The most usual reason for this to happen is the described shortage of liquidity. In general default should be distinguished from other similar terms like insolvency, bankruptcy and liquidation. Insolvency is the situation when a debtor is unable to repay his/her debts. In the banks' case it is, in particular, a situation when an institution's net worth (or capital) is reduced to zero. Bankruptcy is slightly different because it is a legal status of an entity that is unable to repay its debts. Finally, liquidation is the process of bringing a bankrupt entity to an end. Thus, the four notions are not the same. They also differ in several juridical aspects and in the amounts of time for each of the status to be assigned to an entity and in some other ways but all the differences are not important for the paper. So, the notions of default, bankruptcy and liquidation are combined and referred to as “default”. There is an important distinction between default and insolvency, however. “Default essentially means a debtor has not paid a debt which he or she is required to have paid” Definition is taken from www.wikipedia.org. while insolvency is only the situation when a debtor is unable to pay, that is the debtor has not yet failed to repay the debt but will do so when the repayment date arrives. Thus, insolvency does not necessarily lead to default as long as the financial institution is able to refinance its debts. Still, in reality it is very hard to find examples of insolvent banks that succeeded in raising funds in such situation (insolvency) and then continued their business. An explanation for that is clear - it is usually not worth lending funds to insolvent institutions because there is an exceedingly high risk of an eventual failure of such institutions. Hence, it can be assumed that insolvency always leads to default by the following chain: insolvency > illiquidity > default. Exactly such view is adopted for the model of cascade default below.

Another key point should be made. It is quite obvious that illiquidity, being a major reason for banks' defaults, is still only one of the reasons. For example, other reasons include bank runs, overall loss of confidence and uncertainty about the future solvency of a bank, macroeconomic instability et cetera. This means that all quantified results obtained from the model are by no means the perfect ones and they can only show the lower bounds for the actual extent of default contagion in the absence of government intervention.

2.2.1 Loss cascade and Default Impact

The process is modelled as follows. The immediate consequence of a financial institution i default (either after initial shock or in the subsequent stage of contagion) is that all its liabilities to all its creditors are written off. While it is important to note that all debtors of the institution still have to fulfil their obligations to the defaulted institution. As a result the losses after the write-down are subtracted from the capital of the creditors, which is the loss of for each creditor j. However, it is not always the case that creditors get absolutely nothing after a default of their debtor(s). After quite a long process of defaulted bank liquidation (which could take sometimes up to a calendar year) all the assets of the bank are repaid to its creditors in the judicially prescribed manner. Therefore, there is a so-called recovery rate R that shows which part of the initial exposure could be eventually returned. If it happens so that the amount of final loss (after the partial recovery) exceeds the amount of capital (), the bank j becomes insolvent which in turn, as it was stated above, leads to its default. Then there is a possibility of second round effects when all the liabilities of institution j are written off causing losses for its creditors and so on. This cascading domino effect is modelled by a step-by-step procedure (loss cascade) of these write-downs and checking the stability of suffered creditors (condition of capital being not less than zero):

Definition 1. Loss cascade.

Consider the initial vector of capital reserves. The sequence of is defined as

, (3)

where Ri is the recovery rate of institution i. For simplicity it is (implicitly) assumed that the recovery rate of institution i is the same for all of its creditors. The assumption could be easily dropped and different rates may be used (if known).

In general such process ends after the step T when the condition is satisfied. Still in order not to bother oneself with checking the condition every time, a strategy of finding maximum possible k can be adopted. In total there are n institutions and the “worst scenario” for the number of steps (k) is that after each round one and only one bank defaults. Such process comes to an end when all n banks have defaulted which happens at the step (n-1). Hence, in any possible cascading process the final amount of capital of institution j (after all losses have been accounted for) can be described as a vector. And the set of insolvent institutions is represented by

(4)

It is also useful to note the difference between fundamental defaults and defaults by contagion.

The set can be decomposed into two subsets

,

where Fundamental defaults are the ones which have initiated the Loss cascade, while Defaults by contagion are all the subsequent defaulted institutions.

The default contagion process defined above transmits the initial default of one institution to several others. Thus, its effect can be measured as the total amount of losses incurred throughout the whole system:

Definition 2. Default Impact.

The Default Impact of a financial institution is the total loss in capital after the end of the Loss cascade process triggered by the default of i:

, (5)

where is defined by the relation (3) with the initial condition

The final remark of Definition 2 points out the fact that the losses of initially defaulted institutions are not included in calculation of Default Impact as they are out of interest for studying the effects of contagion and systematic importance of the institution.

It is also possible to model a situation with the initial default of several institutions (instead of just one). In this case the only change is that the Default Impact is calculated excluding losses of all Fundamental defaulted institutions. In addition to that, the process's length is reduced and so the model (programmed calculation part) can be optimized. When m institutions default at the very first step the final amount of capital of institution (after all losses have been accounted for) is described as a vector .

Moreover, the provided formula for calculating Default Impact is not the only reasonable one. It was expressed in terms of total losses of capital whereas it could have been calculated in terms of the amount of deposits lost. This approach may be particularly important for monetary authorities or other specialized agencies which are responsible for deposit insurance. In this way Default Impact is:

(6)

Also note that in such form Default Impact does include the depository loss of the fundamentally defaulted institution(s). This is so because the purpose of the indicator in such form is to show the amount of funds the responsible authorities will have to spend for compensation of lost deposits. Understandably, there exist very different systems of deposit insurance in various countries and especially there are maximum sums (let's denote them max) that are covered by the insurance. That is why the simple sum of deposits may not be a telling figure. Still it shows the upper bound of the required payments; and in the case of a need of a more accurate figure Central Bank may ask for such information (for example, the monetary amounts of insured deposits that are ? max and the total sum of all other insured deposits) to be provided by each bank.

A few other points should be clarified. The model can only be used to examine short-run effects of default contagion. That is it takes into consideration only immediate changes in balance sheets of banks because of defaults of their debtors while it totally omits possible long-run effects of a loss which have not led to default. For example, if bank's loss is huge but it is able to survive confidence of its depositors will likely be undermined which could finally lead to a default. There is another implication of the fact that the model is a short-run one. As it has already been stated, the whole process of the defaulted bank liquidation could last for quite a long time. This in turn means that in the short-run no single creditor would be able to recover any amount of the residual value of the defaulted institution. So it may be right to set the short-run recovery rate R=0 in (3). In any case the value of R must be chosen in the most reasonable and suitable way.

2.2.2 Contagion Index

It can be noticed that the magnitude of contagion default and therefore the value of the Default Index depends on the amount of capital reserves that is held by different financial institutions as a safety cushion. That is if the capital reserves of some bank are small, even a minor exposure will pose a great threat to the stability of the bank (direct implication of insolvency condition - the smaller is the capital buffer the greater is the probability of a default of the institution due to default of its debtor(s)). However, the amount of such capital buffers is not always under direct control of the financial institution itself. The amount is also dependent on the state of the economy as a whole. In case of an adverse macroeconomic shock (unfavourable stress scenario) some of banks' assets might deteriorate due to their dropped value. This in turn makes banks to cover the decreased value of their assets by capital. In this way macroeconomic shocks not only increase market risks by generating correlated losses across bank portfolios but also contribute to amplifying the severity of contagion and to destabilization of the whole interbank network. This indicates that both contagion effects and macroeconomic scenarios should be incorporated in the model so that to properly measure systemic risk in interbank networks.

In order to add macroeconomic shocks to the model it is suggested by Cont et al. to introduce a negative random variable Z which will represent the magnitude of such shocks. The variable is then scaled to generate a loss (%) in the capital of ith institution in such a way that the value depends on several characteristics of the institution which influences its probability of default in response to an adverse macroeconomic shock (for example the amount of capital, confidence of the bank's depositors, overall creditworthiness et cetera).

As it has already been stated, banks' similar exposures result in the fact that macroeconomic shocks affect bank portfolios in a highly correlated way. The correlation has been found to be significantly positive in many banking systems over the world. Even more so, many stress tests have shown that the correlation is not just strong but sometimes it occurs to be perfect. So, combining all the facts authors suggest to use a co-monotonic model for macroeconomic shocks.

In principle other specifications are possible (for example, static or dynamic, copula-based or factor-based) but authors underline that representing macro shocks in a co-monotonic way allows obtaining some desirable for further analyses monotonicity properties.

Hence, using the co-monotonic form the capital loss of a financial institution i can be represented as follows:

, (7)

where the are strictly increasing functions with values .

For every macroeconomic stress scenario, defined by a vector of capital losses , it is possible to compute the value of Default Impact of a financial institution i in the same way as it is shown in Definition 2 but now with the decreased, due to macroeconomic shock, capital buffers . The macroeconomic stress scenario corresponds to a very negative value of random variable Z. For example, it could be described by a low quantile of Z:

,

where is the percentage expression for the probability of a macroeconomic shock to be more severe than (the lower is Z the bigger is the shock). This means that if is, say, the 1% quantile of the common factor Z, the corresponding macroeconomic stress scenario has the probability of 1% to occur. Such value implies quite a severe market stress while the value of 5% would imply a mild market stress and the value of 0.1% - extremely severe market stress scenario.

Now it is possible to define an indicator which shows the consequences of default contagion and systemic importance of each particular financial institution under the condition of macroeconomic shocks.

Definition 3. Contagion Index.

The Contagion Index (at confidence level q) of institution is defined as its expected Default Impact in a market stress scenario:

, (8)

where the vector of capital losses is defined by (7) and is the q-quantile of the macro risk factor Z: .

Thus, the Contagion Index , representing the expected loss of capital in the entire banking system conditional on the severity of macroeconomic stress, takes into account both default cascade process and possible macro shocks. The Contagion Index is measured in terms of capital while it could be easily expressed in terms of lost deposits by defining Default Impact as in (6).

It is also worth noting that from the specification (7) of the capital loss caused by macroeconomic instability: which means that defaults are not caused exclusively by the macroeconomic shocks. Still, though, since which means that capital buffers of all banks are in general lowered in case of macroeconomic stress (the case of is actually the case of absence of any macroeconomic shocks). So:

Cont et al. suggest modelling the random variable Z as a negative random variable with a heavy-tailed distribution F and an exponential function :

, (9)

where is a scale factor depending on the creditworthiness (or probability of default ) of the financial institution i . A suggested way to specify is to choose it such that corresponds to the probability of losing 90% of capital in a market stress scenario:

(10)

The values for default probabilities can be obtained from historical default rates given by credit rating agencies for the financial institutions at the date of simulation.

3. Russian bank delicensing policy

Bank delicensing policy conducted by Russian Central Bank has recently become one of the most discussed topics concerning macroeconomic policy. With the appointment of Elvira Nabiullina as a new Chairman on 24th June 2013, the Bank of Russia has chosen the way of tightened policy concerning banks and other credit financial institutions. The seriousness of the policy was quickly realized as during the first month after the appointment 4 banks had their banking licenses withdrawn. Since that time an astonishing number of approximately 140 financial institutions have lost their official bank status. This becomes even more remarkable when the figure is compared to the one which was before Nabiullina's “rule” - only 37 banks and other financial institutions were subject to revocation of their banking licenses by the Central Bank (over a comparable period of time, that is slightly less than 2 years). The amount of total assets of the “ex-banks” is also of a particular interest. The total sum of all assets of those 37 banks did not exceed 110 billion roubles mark. In contrast, the value of assets of banks which had their banking licenses deprived over the last two years totalled about a trillion roubles, that is almost ten times more. Although the sum constitutes just about 1% of the total assets in the Russian banking system, it is bigger than the amount of assets of any individual bank out of top ten (by the amount of assets held) Information taken from www.banki.ru and corresponding approximate calculations can be found in the “Delicensed banks” Excel file..

Another problem for monetary authorities (apart from the partial loss of assets and capital in the banking system) is that the majority of the delicensed banks were a part of deposit insurance program. This means that substantial amounts of money should have been spent by the Deposit Insurance Agency of Russia (DIA) in order to cover the lost insured deposits. In addition to that, the increasing pace of banking license withdrawals and, therefore, the increasing amount of deposits to be repaid in the situation of a limited amount of funds available to the DIA (of about 84.7 billion roubles as at 30.03.2015 Information taken from www.asv.org.) gave rise to apprehension that DIA would not be able to cover all the insured deposits. This anxiety reached its peak when “Bank Pushkino” was deprived of its banking license on 30th September 2013. At the beginning of July 2013 the bank had around of 30 billion roubles of total assets and its delicensing had become the greatest insured event in the history of the Russian Federation Information taken from www.ria.ru..

The third issue is closely related to the above one and it deals with the decreased confidence of public in general and depositors in particular. Moreover, this is not just a decreased confidence in one specific bank (which may possibly lead to the bank run as described in previous sections) but it may be a decreased confidence in the entire banking system. The issue is even more important in the context of the level of development of banking system in Russia. It had long been supposed that public in Russia was sceptical and suspicious of banks and was not willing to lend money to the banks. Furthermore, several economists suggested that Russia did not have banking system at all and that it only had separate financial institutions. So now, when the public has gained confidence and the banking system has become indisputably recognized as such, it seems reasonable to preserve the situation and to prevent public from losing its confidence yet again. Admittedly, the delicensing policy has mainly affected medium and small banks while large and especially very large banks did not experience any negative shocks. This could be easily explained by the fact that large banks are simply “too big to be delicensed” as well as “too interconnected to be delicensed”. While on the other hand small and medium banks do not have such security which is reflected by a decreased credit rating by Standard & Poor's and is exacerbated by panics in some Russian regions Information taken from http://www.kreditnews.ru.. So, such tendency may eventually lead to the situation when all depositors become clients of top-10 or, say, top-30 banks whereas all other credit financial institutions would be redundant and finally disappear. Such scenario means little competition and does not correspond to a healthy banking system which is probably the goal of Russian Central Bank.

These problems raise several logical questions. Is it always necessary for the Central Bank to revoke a bank's license in the situations when it usually does? What are the immediate consequences of delicensing and how can they be quantified ex-ante? What could be the possible approach(es) alternative to the delicensing if any?

The questions will be tackled in the following subsections.

3.1 The reasons for banking license withdrawal

In order to answer the first questions it is first necessary to state the conditions when the Central Bank must deprive a bank of its banking license and when it may do so.

According to one of the Federal acts Federal law “on banks and banking activities” #395-1, 02.12.1990., the Central Bank is obliged to withdraw a banking license in the cases:

when the financial institution fails to meet liquidity requests of its depositors