Systemic risk and contagion: the concepts

when the financial institution fails to meet requirements of the Central Bank concerning the amount of capital

when the amount of capital of the financial institution is reduced to the level below the charter capital

when the capital adequacy ratio becomes lower than some specifically stated benchmark (either the one stated in Basel Accords or the one stated by the Central Bank)

While the Bank of Russia also may (but does not have to) cancel a banking licence in the situations:

when there is a revealed existence of unreliably reported data

when the financial institution conducts an operation which is not permitted by the banking license

when there is a violation of the requirements of the Federal act concerning illegal incomes and sponsoring terrorism Federal law “on counteraction to legalization (laundering) of proceeds from crime and terrorist financing” #115-ФЗ, 07.08.2011.

when some other minor requirements of the Central Bank are not met at all or are not met on time

It could also be noted that the most widely “used” reasons for delicensing are those about liquidity problems, unreliably reported information, violation of Federal act concerning illegal incomes and sponsoring terrorism (this particular reason has recently become very popular; for example, “LINK-bank” and “Radian” bank lost their licences due to this reason), and capital inadequacy Information taken from www.banki.ru..

And while some causes from the list are perfectly grounded, some of the other ones seem to be questionable. It is definite that in the case of illiquidity situation which, as it was explained previously, is very likely to cause default, or in the case when suspicious information is reported which is likely to be a sign of illegal actions, or in the case of a bank being suspected in some illegal income transactions which poses a criminal threat to the country, in all the cases it is reasonable to deprive banks at fault of their banking licenses. It is not that crystal clear when one is talking about capital adequacy ratios though.

Let us now consider how delicensing based on the capital adequacy ratios works and analyze the effects of such policy in the context of systemic risk and financial default contagion.

3.2 Effects of the delicensing policy on the systemic risk

Suppose the Central Bank is about to revoke a banking license of financial institution i. What are the likely consequences of this action in terms of systemic risk?

The most immediate and obvious effect is the write-down by the bank's i creditors of the loans given to the bank i. This decreases the creditors' respective amounts of capital which could lead to their possible defaults Note that from now on the words “default” and “delicensing” will be used interchangeably (unless specifically said otherwise) in the sense that the bank under consideration does not operate anymore and has all borrowed sums written down by its creditors. and a subsequent default contagion process. Even if there are no follow-up defaults, the stability of the creditors of bank i is worsened and a smaller initial shock is needed to cause their defaults. Hence, from this point of view it is safe to say that the deprivation of a banking license always increases systemic risk. Since the inference was obvious it is not the only interesting thing to explore. A more intriguing issue is quantification of losses.

In order to study this question deeper it is useful to refer to the model described in section 3. It appears that the model fits perfectly in the context of delicensing policy even though in its initial form it describes cascade default process. Furthermore, it becomes more meaningful because in the original model the first step was to artificially default one of the banks while now the bank to be defaulted is a specific one (the bank i). It is also justified to consider the delicensed bank as a defaulted bank because the main consequences are the same, that is such bank's creditors lose their assets from loans to the bank (and after the bank is liquidated some of the lost assets could be recovered) whilst the bank's debtors still owe it the borrowed sums. Besides that, the incorporated (in the form of random variable Z) macroeconomic stability factor acquires an additional meaning. It now also includes the probability that banks with similar portfolios and capital structures (to those of delicensed bank i) might also be delicensed.

The main thing to be modified in the original model is the insolvency (or default) condition. Initially, the condition was the reduction of bank's capital below zero or, in terms of loss (exposure), . Now, however, the default condition is a weaker one than that of delicensing condition Another implication here is worth noting. As the default condition is weaker, it is almost certainly that the losses will be bigger under the delicensing policy than under the initial default conditions. This is illustrated by a simple example which may be found in Appendix 3.. This condition is derived from the capital adequacy ratio. In general Russian banks do not have to meet the Basel II requirements for capital adequacy and the benchmark value is simply set by the Central Bank but as Russia is to become the first country to adopt the Basel III requirements, it is reasonable to consider the capital adequacy ratio as the one determined by the accord Even though there are various requirements for different Tiers of capital, only the most general one (for the total amount of capital) will be used here. In principle, if there is a need to use a distinct definition of capital adequacy, including several additional Basel III ratios or some specific ratios set by the Bank of Russia, it could be easily incorporated as additional conditions to be met.:

, (11)

where RWA stands for risk-weighted assets, Mr is Market risk and Or is Operational risk. All of the values are calculated in a particularly prescribed way.

From (11) one can express delicensing condition in terms of loss (exposure):

, (12)

where is the risk-weight of a loan of bank j to bank i. It is worth noting that in the case when there are several different kinds of loans from j to i which fall into different risk categories, is the row vector of risk-weights while is a column vector of the corresponding loans of different riskiness from j to i. The value of risk-weighted assets has to be decreased when financial institution i fails because there is no more risk of the loan to provide for There is a minor possibility that the delicensing of bank i may also lead to a change in market risks of bank j, that is change the value . Still since the change may only be an adverse one it could have only worsen the stability of bank j. So, as it has already been said, the model will provide the lower bound for the actual extent of default contagion.. It is also vital to see that since the recovery rate is usually quite low (or even zero) the loss is always bigger than the decrease in the provision for the lost assets, that is . Therefore, the “adequate” capital after the loss must be bigger in order to keep banking license:

, (13)

where the first and the second inequalities of the system satisfy CAR in the situations before and after default of bank j respectively.

Hence, the condition (12) means that if capital of bank j is not big enough, that is (where and are the minimum amounts of and respectively that satisfy conditions from (13)), the license of the bank will be withdrawn and the cascading process may continue.

One should also remember that the Default Impact in the original definition was comprised only of capital losses of those banks which defaulted by contagion (that is excluding fundamentally defaulted banks). Such approach was reasonable there because the start of contagion was determined by artificial default of a random financial institution. In the situation of delicensing, however, the institution i under consideration is a particular one and with a positive amount of capital. Therefore, its losses should also be included in the Default Impact calculation. That is why it is necessary to slightly modify the definition of Default Impact (and Contagion Index).

For that matter it is also necessary to redefine the Loss cascade.

Definition 4. Loss cascade in the context of delicensing.

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

(14)

where An alternative definition through a more conventional Heaviside function could be found in Appendix 1..

With such definition the set of insolvent institutions is still represented by (4):

The “-1” index is of no use in a static situation while it will be very important for the modification of definitions of DI and CI. This time index will represent the time before the delicensing took place. Thus, it will allow having the amounts of capital “saved” (in an algorithm) for banks i after the default triggering condition.

Definition 5. Default Impact and Contagion Index in the context of delicensing.

(15)

(16)

Contagion Index is defined in the same manner as before but is now different because of the modified Default Impact.

Finally, Default Impact and Contagion Index when expressed in terms of lost deposits are of a special importance for the policy of revocation of banking licenses. The reason for that is the already mentioned system of depository insurance. An explanation of the ways the Central Bank can obtain the required information (for calculation of total insured repayments) may be found in Section 3 right after expression (6).

There is yet another possible consequence of the delicensing policy. It is connected with the decreased confidence of depositors and their increased cautiousness. The two behavioural changes should be recognized to be different. The former one concerns the depositors who are currently holding some of their funds on deposit in various banks and who, after analysing the current market conditions, may decide to withdraw (partially or fully) their deposited funds. Whereas the latter behavioural pattern change results in the future depositors' being very picky about the bank to put money in and about the amount of money to be deposited (if at all).

Accounting for these two changes is more difficult as there is no direct or obvious method for their quantification. One possible way to quantify potentially decreased confidence is to include it into the macroeconomic instability variable (variable Z). In this way if, for example, a q-quantile of 3% is chosen for (from ), it will correspond not to a market stress scenario with probability of 3% (as in the original model), but to a market stress scenario with probability of, say, 5%. That is a less severe market stress (of just 5%) would now lead to more severe consequences (as those of a more harsh market stress with probability of 3%). Still it is to be decided what the real severity of the decreased confidence is. For example, according to Sukhov M. (who is the vice-chairman of the Central Bank) there is no drop in confidence at all and, conversely, the growth rate of the number of depositors has reached the predicted value Information taken from http://www.kreditnews.ru.. So, according to the information, there is no need of accounting for this change at all, even though the information does not seem to be realistic.

Yet, it cannot be denied that people have become more cautious. An ever-increasing number of delicensed banks serve as a sign that no non-top bank is safe. Therefore, being afraid of losing their deposits, some depositors may decide to put their money in a safe bank, that is a bank from top-30. As a result, smaller banks, which would have otherwise obtained the deposits, are left with fewer deposits. Alternatively, the cautious depositors may now deposit only the sum which is covered by the depository insurance system but the outcome is still the same. Moreover, it will be much harder for small and medium-sized institutions to attract deposits (and other types of funds for operation) because they will not be able to offer competitive deposit rates (just by the virtue of their smaller scale activity and the threat of losing their banking license due to inappropriate provision for risk). The likely long-run outcome is, therefore, deterioration of smaller banks' assets which should be covered by capital. And the reduced capital in turn poses a threat of falling into the category of banks that do not/did not satisfy the delicensing condition. Although, as it has been already stated, the model does not take into account long-run effects this implication is still useful.

Thus, following the steps described in Section 3 and modified as it is shown above the Central Bank is able to determine the quantified consequences of its delicensing of financial institution i. It might be also a good idea to describe all the types of information needed to do the calculations. There should be a set of particular financial institutions (banks), the amount of capital these banks possess, the matrix of bilateral exposures (that is the information about the amounts of loaned and borrowed funds to and from every other financial institution), the amount of insured deposits (for calculation of insurance payments), the information concerning capital adequacy ratio (such as RWA, Mr, Or et cetera), an assumed distribution F of the random variable Z and probabilities of default of each bank (for calculation of and a subsequent calculation of capital losses caused by macroeconomic shocks, ). Additionally, the value of recovery rates is required but there is no such information and so it should be chosen in the most justifiable way. From all of the pieces of information every single one can be easily obtained by the Bank of Russia (either as reported information by banks themselves or as the information provided by credit rating agencies about probabilities of default) except for that of bilateral exposures. This is not usually the information reported by financial institutions but still the Central Bank may get it on demand (as in the case of Russian Central Bank). Hence, the calculations in the model are feasible to do in reality.

Since the delicensing policy leads to so many problems it might be questioned whether it is the best policy to be conducted.

3.3 Possible alternatives to the delicensing policy

It seems that the immediate delicensing of a financial institution in case when it fails to meet capital requirements in the form of capital adequacy ratio(s) is a too harsh policy. After all, such institution still has enough capital and enough liquidity to cover its liabilities (at least for some time and if the problem with CAR is the only one). It is true that the capital requirements are important in order for financial institutions not to take excessive risks and to have a safety cushion in the form of capital reserves, but it does not mean that a one-time negative deviation from the value must be instantly punished with the revocation of banking license. A possible alternative response could be some reorganizational measures aimed at improving the situation. Exactly the same thing has been proposed by the chairman of Russian banks Association - Tosunian G. He has underlined that it is necessary to change the philosophy of financial supervision and regulation. He also suggested that the usual revisions of banks by the Central Bank should be followed by reorganizations rather than delicensing Information taken from http://www.regnum.ru.. This is especially important considering the fact that usually, when there is a revision of a bank, depositors expect the bank to be delicensed (even if it would not be) and rush to withdraw their money. Consequently, the stability of the bank is worsened and could eventually lead to its banking license being revoked (a kind of self-fulfilling prophecy).

Even if the policy of reorganization is adopted in the place of immediate delicensing, there is still the described problem of trust. That is the depositors who realize that the bank they have money in is under Central Bank's revision are very likely to run the bank. Hence, the Central Bank either will have to carry out a “secret” revision or will have to assure all the depositors that it will take control of the bank and that the bank will not be defaulted. The first proposed solution is always subject to informational leakages while the second one may simply not be believed by depositors due to credibility issues.

So a different solution should be found. Such solution could be the one based on forecasting of financial performance indicators and on minimization of the Central Bank's intertemporal expected loss function. The idea is explained below.

Consider a financial institution i that has violated the requirements of capital adequacy ratios. The Central Bank decides whether it should revoke banking license of the institution or not (without an official revision of the bank's stability and other reorganizational methods). First of all, it is necessary to calculate the values of the Default Impact and Contagion Index, the decrease in capital in the banking system, the decrease in the number of solvent banks and possibly some other values which could be included in the Central Bank's intertemporal expected loss function One type of such loss function can be found in (Acharya et al., 2009, p.9). For the purposes of the graduate qualification paper a specific type of the function is not needed, though it will be required in case the approach is to be implemented in future.. The next step is to predict other institutions' future financial indicators while assuming the future default of the bank under consideration (as it is the worst thing which could happen to the bank in the future if it is not delicensed now). This could be done by applying neural networks, for example, in the way it was done in the scientific paper (Rud'ko-Silivanov et al., 2013) by the employees of the Bank of Russia. The paper is also important because the authors, using retrospective analysis, were able to accurately predict those financial indicators that are needed for this analysis (in particular depository amounts). After the predicted values are obtained they can be used to calculate the predicted values of Default Impact and Contagion Index, the predicted amount of lost capital et cetera. Afterwards, these values could also be substituted into the intertemporal loss function and the value of the loss (for the Central Bank) can be calculated. Finally, the two loss values of the Central Bank should be compared. In such a way the appropriate decision could be made - either to delicense the bank now or to leave it as it is (implicitly assuming the future default). Of course, the approach is very far from being easy to implement. Despite a relatively simple logic behind the procedure, that is an ordinary comparison of two values representing losses, there are several shortcomings that should be overcome before the approach could be adopted. Firstly, it is not clear what the exact form of the intertemporal loss function of the Central Bank is. Secondly, and more importantly, the result of the comparison will be highly dependent on the accuracy of the forecasted values. And even though the forecasts obtained in the abovementioned paper were quite close to the actually observed values, there is no guarantee that the precision of the technique will always be that high.

As it has been shown above, the two alternatives are quite hard to implement properly due to various reasons. So, as it is usually done, one may try to find the optimal solution somewhere in between the two. The Central Bank might create a policy rule (and let it be widely known for credibility) based on a particular loss function of current period variables (eliminating the problem of forecasting). One possible kind of such approach is presented in the following subsection.

3.4 Relative Contribution Measures

Until this point I have only discussed two quantification measures of a default contagion process (Default Impact and Contagion Index). However, those measures show only particular numbers that are a kind of total loss to a banking system induced by one or several initial defaults. Hence, neither of the two indexes tells the regulator of a banking system whether it is time to intervene and, if it is time, what exactly should be done. The “time to intervene” phrase in this case is synonymous to “the loss is big enough” and it is quite obvious that the regulator decides on the threshold of the loss being big enough on its own (and the value might even be different each time the regulator faces the problem of default contagion). Due to this fact, there is not much the model can do about determining the threshold. Nevertheless, the model can be quite telling about relevant regulatory policies.

Assume that some bank A has breached the capital adequacy ratio requirement and the Central Bank considers delicensing of bank A. For that matter, the CB analysts calculate Default Impact and Contagion Index (in the manner presented above, in the Subsection 4.3) and define the maximum amount of losses to the banking system that is acceptable. Then, if the values of Default Impact and/or Contagion Index are larger than the determined threshold, the reasonable action of the Central Bank would be bailing out bank A (either through simply ignoring the existing breach and warning the board of bank A if the mismatch is small or through sanation of the bank if the mismatch is large). At first sight the situation above seems to be perfectly reasonable but this logic may happen to miss a very crucial point. The point is that the Central Bank can actually intervene at any point of the triggered contagion (especially if the contagion is spread by the CB itself - by delicensing - and not through actual defaults). This is important because it may happen that the bank A has actually a relatively small amount of capital (so its default is not of great importance to the banking system as a whole) and at the same time has a creditor, bank B (with a relatively big amount of capital), that would breach capital adequacy ratio requirements as a result of the delicensing of bank A Note that it is not an uncommon situation for real complex networks when a node with a sole Out link is connected to a node with many Out links implicitly meaning a larger amount of capital and assets (therefore, increasing the possibility and severity of default contagion). This characteristic of networks is called disassortativity and is explained in more details in Section 5.. And so, it may be more reasonable to bail out bank B instead of bank A as bank A is poorly managed (excessively risky) and its capital loss is almost not notable while bank B is more stable and has a relatively big amount of capital. In absolutely the same fashion the argument would hold for any step of a default contagion with the only difference being the need to account for possible multiple defaults and their interrelations at each step (that is one should be careful with the banks that are not vulnerable to default of one particular bank in the chain of defaults but is vulnerable to defaults of several banks An illustration of this idea can be found in Appendix 2.). In order to take the described idea into account I introduce a measure of the relative contribution to the overall loss (DI or CI).

Definition 6. Relative Contribution to Default Impact and to Contagion Index.

Let us define Relative Contribution to Default Impact (RCDI) for each financial institution that was delicensed, l, and for each step of contagion default process, m (l defaults at step m), as the ratio of the capital of bank l at step m and the sum of additional capital losses suffered in banking system due to default of bank l, that is additional amounts of capital lost by banks that would have lost less capital without the default of bank l. Note that this sum is not necessarily a simple DI(l, c, E) because of the abovementioned interrelation of earlier defaults.

These interconnectedness of banks can be so complex that the definition expressed through the DI(l, c, E) becomes very complicated and impractical Refer to Appendix 2.. Instead one can revisit the definition and express the “additional capital losses of banking system due to default of bank l at step m” as the difference between the original Default Index and the Default Index in case when bank l does not default. The latter DI could be calculated in the same way as the former one but after increasing capital of bank l at step m sufficiently (so that it does not default). There is yet another important thing to note - when one increases capital of bank l any further defaults of other banks may continue decreasing capital of bank l which is not lost in reality (after capital is zero it doesn't go below anymore). Hence, one would need to take the losses of bank l after step m away from DI calculations.

Let be the set of banks subject to initial default (license deprivation) - the same as it was defined in the main model above.

Let be the set of banks subject to default (license deprivation) at step j of the contagion triggered by the initial default of banks from set i. It follows then that and is the set of all ultimately defaulted banks after initial default of i. Remember that and that . Also remember that one can use T and (n-1) interchangeably as .

Let the sequence be defined as it was in (14) and be defined as:

L(i) is defined as in (2). By setting capital of l equal to the constant L one takes it out from further default process triggered by the initial default of i (just as the CB would do if it was to save the bank l).

Finally, let the combined sequence of and be

Then RCDIl is defined as:

 (17)

Note that the correction terms in denominator are needed (as explained before) to account for the real amount of capital lost by bank l. The part of denominator dependent on bank l is:

,

which is exactly the “additional capital loss of bank l due to default of bank l at step m”.

Also note that RCDI for an initially delicensed bank, that is , will use which is naturally the amount of capital of the bank l before it is delicensed (at step 0). Moreover, if it happens that there is only one delicensed bank at first and it is the one under consideration, that is l=i, the formula of RCDI will convert to a simple ratio .

In the same manner one can define Relative Contribution to Contagion Index (RCCI):

(18)

Note that the formula stays the same and is not influenced by the random variable Z (it is eliminated by taking expectation).

Finally, it is obvious that both RCDI and RCCI lie between 0 and 1 because denominator is not less then and can be only bigger than the value.

These two measures (RCDI and RCCI) can play a role of a simple, yet reasonable, policy rule. They both take into account an important possibility of CB intervening at any time during contagion process and represent a (relative) quantified measure of the importance of each particular bank in the contagion. The only thing left for the CB is to define the thresholds it would consider acceptable.

The procedure would include defining the threshold, A, for the initial overall acceptable loss of capital. Then, if the DI and/or CI are in excess of A, the values of RCDI and RCCI are calculated and compared with additional percentage threshold(s), , that classify banks either as “important” (if ) or as “unimportant” (). If the bank is “important”, it is saved by the CB (not delicensed and helped in some way, for instance, sanation/new temporary management et cetera). Otherwise, the bank is delicensed. There are three things to be aware of in such policy rule.

Firstly, one may spot that RCDI and RCCI are usually increasing with an increase of the step of contagion process (as there are less banks to be delicensed after each step the relative importance of a considered bank increases). This fact may seem to suggest that it is possible to have a situation when a poorly capitalized bank is not delicensed. For example, when there is only one bank left in the contagion process and it has very little capital but, since it is the last one, RCDI=1. Thus, the bank is saved. However, such situation is not possible because of the decisions of the CB at previous steps - if it decided to delicense a bank before, then it observed a much higher capital loss at future steps, that is there has to be at least one relatively well capitalized bank that would have to be saved. This implies that the process will come to the very last bank in the contagion, allowing default of every other one at previous steps, only if the bank is exactly that well capitalized bank. Otherwise, the contagion via this branch would have been prevented earlier (by saving the well capitalized bank at some previous step).

Secondly, there is no need to define additional thresholds in absolute terms for further steps because the relative ones will work quite well to not allow default of banks with relatively big amounts of capital. It is also necessary to note that the initial threshold A is used several times and determines when the process of assessing whether a bank is “important” or “unimportant” ends. That is, when the first bank is found to be “important” and is saved, the new value of capital losses becomes equal to . This value (DI') is compared with A and, if , the CB stops intervening and lets all other banks be delicensed. While if the inequality is reversed, the CB continues comparisons of RCDI/RCCI and .

Thirdly, a static threshold may not always be perfect in capturing well capitalized banks in case of a huge DI that is much higher than A. For example, suppose that A=10 and DI=10000 and the only initially delicensed bank has capital=11 => RCDI=0.0011 which will lead to not saving the bank at the threshold as low as 0.12%. Obviously, the CB should not delicense the bank (as 11>10) but at the same time it would be irrational to set only for the RCDI approach to work (as almost any bank would then be saved). So, it seems very reasonable to set a varying (for each step k). Additionally, it might be a good idea to set it as some function of the values DI and A because, as shown above, the reasonable value for the threshold is influenced by the two parameters.

After the policy rule is worked out there is a need to deal with the second problem of any delicensing policy - the drop of confidence of depositors in the banks that are being bailed out by the CB (so that a bank run does not ruin the CB's attempts to save the banks). The confidence is, obviously, not a thing that can be improved overnight. However, a clear statement and public accessibility of the delicensing policy rule should be enough because the policy clearly states that the banks that are being saved or helped are the ones that have to be saved by the CB for the stability of the banking system (that is the CB will not have any incentives in the near future to stop helping such banks and to delicense them).

Admittedly, the proposed policy is a relatively simple one and does not include other important for the Central Bank factors. Such factors may also include the amounts of deposits to be returned to depositors through the Deposit Insurance Agency of Russia, deposits, the number of delicensed banks (as a determinant of public confidence in banking system as a whole), n, possibly loans from the CB itself (that would not be returned in case of delicensing), loans, et cetera. The given set of variables only increases the loss of the CB when more banks default, which implies that all banks should be saved. Obviously, this is not the proper way. Hence, the loss function must also include some outweighing variables that decrease the loss when the number of defaulted banks rises. Such variables could be the amount of money spent on bailing out procedures, bail-out, or the number of “bad” banks that are not delicensed, bad, et cetera. So, as a more general approach, the CB can construct a particular form of its loss function and calculate the loss values for all possible combinations of defaults This may seem quite computationally extensive due to the big amount of all possible combinations of defaults. However, there are not literally all such combinations to be analyzed since some banks can default only after a default of other bank(s). In addition to that, the default cluster is usually relatively small.. Then the scenario generating the lowest loss is chosen (while the problem of trust should be dealt with in the same way as described above).

4. Empirical considerations and example

Having defined both the model for quantification of losses of a banking system because of the delicensing policy and an alternative modification of such a policy, it is only natural to try to apply the concepts to the real world data. The idea is, firstly, to see to what extent Russian banking system is vulnerable to the existing delicensing policy of the Central Bank, that is whether there are any banks whose defaults would pose a huge threat to the stability of the banking system as a whole; what the size of an average default cluster (average number of defaulted banks after an initial default of each bank in the banking system) is; what the average capital and deposit losses in the default clusters are (for which Default Impact and Contagion Index measures are used) et cetera. Secondly, one can consider the application of the modified delicensing policy to the data and compare the outcome with that obtained at the first step (that is under an existing “strict” delicensing policy The “strict” policy is assumed to be the one when the CB withdraws banking license of an institution immediately upon violation of (any) Capital Adequacy Requirement. Realistically, this is probably not always the case in Russia and there are several exclusions but at least, according to Russian law, the violation must be followed by license withdrawal. Thus, I will consider the “strict” policy to be an actual one.).

However, such analysis turns out to be impossible not only because there is simply no publically available information about the interbank borrowed and lent amounts Not the aggregate figures but the ones showing how much each bank owes to each of the other banks, that is the data to complete Exposure matrix. but also because such information would have been kept in secret by the CB (so that no one could have obtained important information concerning the stability of the banking system).

Even though there is no possibility of carrying out the analysis using the more accurate real data, one can try to reproduce the missing pieces of data from the analysis done by the Central Bank itself - in the paper (Leonidov and Rumyantsev, 2013) - and then proceed as described above but with less accurate and relevant data. The loss in accuracy, though important for determining “problematic” banks in the system, is not so crucial in the current analysis because the main objective is to demonstrate that the modified delicensing policy is superior to the “strict” one.

The idea and procedure of reconstruction of missing data are explained in the following subsection.

4.1 Data reconstruction procedure

The main idea behind the process of data reconstruction is that the overall structure, that is general characteristics of the weighted directed graph, of interbank lending market in Russia can be supposed to be more or less the same over the last 5 years. Such characteristics were studied and presented in the abovementioned paper of Leonidov and Rumyantsev for the data corresponding to the period from 01.08.2011 to 03.11.2011 Since one of the authors is an employee of the CB, he was granted access to the type of data required to carry out the research. However, as already said, the data is not available for outsiders.. These characteristics will be also presented and explained in this work below and then they will be used as additional constraints for the reconstruction of Exposure matrix. The following step would then be to get any other relevant information for even further constraints (these would include overall numbers of interbank assets and liabilities for each bank, that is the summed values by row and by column of the Exposure matrix; as well as other more specific information), which will be explained in deeper details later.

Let us start with the characteristics of the weighted directed graph of Russian interbank lending market.

4.1.1 Size and structure

In their work, Leonidov and Rumyantsev analyzed 659 banks that had at least one transaction in the interbank lending market over the period from 01.08.2011 to 03.11.2011. The average number of banks with an open position in the market was 505 per day. The following so-called bow-tie diagrams summarize the structure of the interbank weighted graph:

Figure 1. Bow-tie structure in the Russian Interbank Lending Market

The first diagram shows the average number of banks with an open position in the interbank market for each component Remember that banks in the In component are pure creditors, those in Out component are pure debtors, while the ones that are creditors and debtors at the same time compose the In-Out Component. (standard deviation in brackets), while the second diagram demonstrates the percentage share of loans (per each component) from the total amount of interbank assets. In this way loans of banks from Out component to banks from In-Out component constitute 23% of the total interbank assets, loans of banks from Out component to banks from In component make up 2% of the total interbank assets et cetera.

The question is whether the observed structure of interbank relations is actually similar to the one that can be seen now. In this work I take all active (as at 01.01.2015) banks in Russia which is 820 banks. Additionally, I will be using the information about the total interbank amounts of assets and liabilities for each bank Taken from www.banki.ru.. Also, one should note that the data is the stock one (as at a particular date), while the data from the analysis of Leonidov and Rumyantsev is periodic. Thus, in my work I will not be able to capture the majority of over-night and other short term deals. In addition to that, 2011 year, being a post-crisis year, may have had an adverse effect on the trust between banks which, in turn, would suggest a smaller amount of banks willing to be exposed in the interbank market. This is likely to result in some difference in the quantity of banks in the components (in particular one should expect a larger number of banks in the In-Out component in my analysis and a smaller amount in the In and Out components). However, this should not significantly change the overall characteristics of the interbank lending market.

The expectations are actually proven by the data. One finds that from the 820 banks 294 are from In component, 40 are from Out, 393 from In-Out and 93 didn't have any interbank assets or liabilities at all. Since the total number of banks differs, one should compare percentage composition of the overall amount. In the work of Leonidov and Rumyantsev In component corresponds to 58.42% (=295/505), Out - 18.61%, In-Out - 22.97%, while the current data suggests In - 40.44% (=294/(820-93)), Out - 5.5%, In-Out - 54.05%. That is a much larger relative amount of banks in the In-Out component.

As for the distribution of interbank assets among the components, based on the data (that is not having a complete Exposure matrix), one can only compare the overall figures of loans from In-Out component and from In component (without subdivision). The data from the abovementioned paper suggests 87% of loans from In-Out banks and the rest 13% - from In banks. The corresponding values from the current data are 97.15% and 2.85% respectively. Again this was anticipated because of the increased relative number of banks in the In-Out component and this should not be having any drastic effect on other characteristics of the interbank lending market. Moreover, the difference in these values does not seem to be that big because of a relatively high standard deviation of the values.

Bearing in mind the outlined differences in the structure of the weighted directed graph for interbank lending market, one can now analyze other important properties of the graph. That is for each of the properties one should decide whether those differences are likely to result in the properties being different as well.

4.1.2 Probability distribution of the number of In and Out links

From this section on only descriptions of the characteristics and the empirical values from the paper of Leonidov and Rumyantsev will be presented, that is there will be no comparison to the currently observed numbers simply because they cannot be calculated without the fulfilled Exposure matrix. Conversely, these numbers will be assumed to be the same as in the mentioned work so that to reconstruct the missing values in the matrix.

One of the most important such characteristic is the marginal probability distribution of the number of In and Out links of the graph's nodes. In particular, the usually common distribution for real complex networks is the power distribution of the form

where the parameter в can be obtained from the following simple linear regression (the same type for both In and Out):

with the empirical marginal probability being calculated as

and from the domain of logarithmic function it follows that only the positive probabilities are used in the regression (zero probabilities are excluded).

The following results were obtained by Leonidov and Rumyantsev:

Table 1. Fitting parameters for In degrees

Table 2. Fitting parameters for Out degrees

Are the values likely to be different for the current interbank market (with a bigger amount of banks in total and in the In-Out component in particular)? It seems not since there is no direct and strict relationship between the number of banks, which is an absolute characteristic, and the marginal probability distribution, which is more of a relative one.

Thus, these eight values in total (four numbers from each regression - ) will be used as additional constraints.

4.1.3 Disassortativity

Another important feature of any complex network is assortativity or disassortativity. Disassortativity is the tendency of nodes with a large amount of links with other nodes to be connected to the nodes that have a small amount of links with other nodes and vice versa.

It was found that the Russian interbank lending market is characterized by an apparent disassortativity, which is shown in the following graphs.

Figure 2. Disassortativity

As one can spot, this characteristic quite obviously depends on the amount of In and Out links, which in turn depends on the overall number of nodes in the graph (naturally, the more nodes there are the more links should exist among the nodes). Thus, it seems reasonable to rescale the maximum number of both In and Out links to get 89 (?62*727/505) for In and 116 (?81*727/505) for Out. To predict the corresponding y-values for these new additional x-values, one could fit a hyperbola. Alternatively, one may just realize that for banks with a big number of In or Out links the average number of Out links of counteragents fluctuates around a constant. Hence, the prediction might be the constant corrected by a random variable taking values between -1 and 1 (for In) and -0.5 and 0.5 (for Out) Admittedly, these values are quite arbitrary but they seem to be compatible with the graphs..

Additionally, it is probably too harsh to restrict the values in Exposure matrix by the exact values from the graph (it may even happen that no solutions could be found satisfying the restrictions). Therefore, it may be reasonable to have some interval of y-values for each of the x-values, that is to have a band within which fluctuations of y-values would be considered acceptable. In optimization problem this possibility of deviation can be accounted for by simply allowing some variation in the constraints (less precision option).

4.1.4 Clustering coefficient

The next highly important but usually overlooked feature of any complex network is the so-called clustering. It is measured by a clustering coefficient which is in general “a measure of the degree to which nodes tend to cluster together” The citation and the subsequent formal definition of Local clustering coefficient are taken from www.wikipedia.org.. In other words, it is a measure that shows the extent of interconnectedness of nodes in the graph. From this definition the crucial importance of the clustering property in the context of default contagion becomes clear. When high degree of clustering is present, the shock is spread to other nodes not just via a single “infected” (defaulted) node but rather via multiple such nodes. Such process poses a much bigger threat to the stability of the system as it makes any shock more severe (compared to the effects of the same shock in the less clustered network).

Let us now formally define the Local clustering coefficient for a directed graph.

Definition 7. Local Clustering Coefficient for a directed graph.

First of all remember that a graph consists of a set of vertices (or nodes) V and a set of edges (or links) E between the nodes. An edge connects vertex with vertex .

Then, the neighbourhood for a vertex is defined as the set of vertices directly connected with it (direction does not matter here):

and the number of nodes-neighbours in the neighbourhood is defined as (note that this k is different from the ones in (1)).

Finally, clustering coefficient for each vertex is defined as the ratio of the number of actual links in its neighbourhood to the maximum possible amount of such links, that is :

Such coefficients are calculated for each node and then the averages are taken among the banks from In, Out and In-Out components. The following values were obtained by Leonidov and Rumyantsev (and the value for a random graph case is there as well for comparison):

Table 3. Clustering coefficients

The big difference between the value of the Clustering coefficient for a random graph and for the real data one suggests the already mentioned real complex network characteristic of high interconnectedness.

One can note that the values in the table in general do depend on the relative amount of the banks from In-Out component. More banks from the component mean that more banks are connected with each other, the fact that means that the Clustering coefficient values should be higher for all components (as each node's neighbourhood is now more likely to contain more In-Out banks). Hence, the values in the table should only be treated as the lower bounds for the Clustering coefficients in the present analysis. The problem is that it is not possible to obtain true values of the coefficients. One way round would be sticking to these exact values as the true ones but at the same time recognising that the lower than the actual values imply a lower degree of interconnectedness. As a consequence, the contagion process becomes less harmful, the fact that is perfectly compatible with the initial statement that the modelled quantified loss measures (Default Impact and Contagion Index) represent only the lower bound of the actual losses to be incurred.

So, the 6 values (for the average coefficients themselves and their standard errors in brackets) generate yet additional constraints.

4.1.5 Interbank assets and liabilities and the number of In- and Out-links

The final feature of the interbank lending graph that is considered is the dependence of the average amounts lent and borrowed in the interbank market from the number of counteragents (that is the number of In and Out links of a node respectively).

The dependence from the paper of Leonidov and Rumyantsev is plotted below:

Figure 3. The dependence of the borrowed/lent amounts on the number of Out/In links

This dependence is an important one but, nevertheless, there could be several problems with applying the same dependence to the current data (using the information in reconstruction of data from the Exposure matrix). Firstly, as the number of In and Out links is encountered again, it is again necessary to rescale the maximum number of links as before. Now, however, the dependence is not as clear as in the disassortativity analysis which means that predicting values out of the given sample is likely to be very inaccurate. Moreover, the amounts of lent and borrowed sums should be expected to vary quite significantly as the number of banks in the system increases and as time passes. These points suggest not using the dependence in the reconstruction procedure in the exact form. However, what can be used is the part of the relationship for small number of In and Out links, where the relation is much more clear. For example, in the first graph one may observe almost a linear dependence for the first banks with less than 20 In links. Such relation is explained by the existing limits on banks' counteragents and by regulatory restrictions imposed by the Central Bank.

In such a way the final restrictions are obtained.

4.1.6 Additional specific features

Now one can proceed with several additional assumptions about the distribution of interbank assets and liabilities inside the banking system.

In the Russian banking system, there exists a conventional division by the so-called echelons according to the rating based on the amounts of capital. All banks are usually divided into three echelons. The first one consists of about 30 banks; the second echelon includes banks placed from 31st to 200-250th position in the rating; while all the rest banks constitute the third echelon.

In practice there is a kind of a rule that makes all banks to have interbank connections mainly with the banks from the same echelon. What is more important, banks (almost) never lend to the banks from lower echelons while, in principle, do borrow from such banks sometimes Information taken from www.banki.ru.. From this fact, it is reasonable to assume that top-30 banks do not lend to or borrow from the banks that belong to the lower echelons (from 200th place in the rating). Another similarly justified assumption is that banks from the first two echelons (rated 1-200) neither lend to nor borrow from the banks in the lower half (placed 411-820).