Systemic risk and contagion: the concepts

In this way one can obtain a sufficient amount of constraints to replicate a more or less close to reality banking system. The remaining procedure is purely a computational one - the program is written in a way to generate missing values in the Exposure matrix such that the values satisfy all the listed above constraints. The solution is likely to be not the only one but the important thing is for it to satisfy all the constraints.

As a final note on the data reconstruction procedure, one must realize that there were too many fairly restrictive assumptions for the reconstruction to be feasible. Thus, the results obtained from such simulation could not be directly transferred to the real Russian banking system, that is the absolute results will almost certainly be different. Nevertheless, due to the emphasized importance of the reconstructed graph having the same properties as in real world, one should expect relative results to be more applicable. For example, if superiority of one type of delicensing policy is proven for these reconstructed numbers, then the type of delicensing policy should be expected to be superior in reality as well. Furthermore, all the complications with data vanish once the analysis is performed by the Central Bank (because it actually has all the missing information). Then the theoretical model developed in the main body of this paper may become a handy tool that will not only simplify the delicensing policy but also allow it to be more cautious from the point of view of systemic risks.

4.2 Empirical comparison of the delicensing policies

Unfortunately, the described data reconstruction procedure cannot be realized due to the insufficient power of the available computational resources The procedure uses more than 200,000 variables (while, for example, Excel allows only 200) that are subject to many relatively complicated constraints (as described in Subsection 5.1) and no obvious method of optimizing the variables (thus, some iterative procedure is to be used). All together the complications generate a problem that simply cannot be solved by the available computational powers.. The procedure itself is to minimize the value of the sum of squared deviations between the theoretical values (from constraints) and the empirical ones that are obtained through changing the variables in the varying (non-zero) range of Exposure matrix The procedure can be found in the attached Excel file “Interbank”.. If there were no computational restrictions one would obtain a filled Exposure matrix with all the parameters satisfying the listed constraints.

Since it is impossible to reconstruct the Exposure matrix, the further numerical analysis of delicensing policies and default contagion becomes impossible as well. However, generally the next step would have been modelling of the default contagion process under “strict” delicensing policy for a specified initially delicensed bank and comparing it to the alternative suggested policy (based on a policy rule that uses RCDI and RCCI measures) The procedure can also be found in the attached Excel file “Interbank” written as a VBA program that could be found in the Macros menu.. The comparison of the two policies could then be done by comparing the values of Default Impact, Contagion Index, the number of delicensed banks and the amount of lost individuals' deposits. Even though one is not able to actually perform the comparison (outside the Central Bank that has access to the real data in the Exposure matrix) and show numerically whether the results of the two policies are significantly different or not, one can be absolutely sure that the alternative model is at least not worse than the “strict” one. This follows directly from the definitions of the two policies - the “strict” one allows all the banks-violators of CAR to default while the alternative policy may bail out some of the banks when this reduces the CB's losses. Thus, if some banks are actually saved, then the losses of the CB are automatically lower in case of the alternative delicensing policy.

Conclusion

In this work the interconnection of two extremely important problems was discussed. Those problems are systemic risk and delicensing policy of the Central Bank. Using the presented model, it was thoroughly explained how systemic risk and, particularly, effects of financial default contagion could be quantified. The quantification allows regulators to see not only the overall amount of capital losses after a default of a specific financial institution but also the relative systemic significance of each financial institution. Then this model has been modified in such a way that it enables one to quantify the consequences of the delicensing policy based on capital adequacy requirements. It has been explained why the delicensing policy leads to many problems including drop in depositors' confidence in the banking system and a possible decrease in competition between commercial banks. Hence, it was suggested that such policy may not always be the best one to conduct. Several alternative approaches have been proposed and explained. One of the alternatives is based on a further extension of the described model. It suggests calculating two additional measures of a bank's relative importance in the default contagion process and then using them as a policy rule by the Central Bank when deciding whether to revoke a bank's license. A particularly important aspect of such an alternative delicensing policy is that it is more flexible and takes into account the Central Bank's possibility to intervene at any step of the default contagion. Thus, the suggested policy is in general better than the existing “strict” one. The point could have been proven by a numerical example using either real world data or a developed in this paper algorithm of data reconstruction. Unfortunately, the real world data is available only to the Central Bank while the data reconstruction procedure appears to be too computationally extensive to be performed. Nevertheless, the theoretical superiority of the alternative delicensing policy was proven (though the extent of this superiority is still unknown).

The paper in general and the presented approach in particular provide a ground for a potential further research in the area. First of all, it is obvious that an analysis using the actual data should be done in order to show the extent to which the described alternative policy is better than the existing “strict” one. Then, it may be useful to explore what other factors should be included in the Central Bank's loss function and in which functional way do all the factors enter this loss function. Another interesting issue to study could be obtaining the best performing (the most meaningful) dependence of the threshold on the values of k, A and DI. Finally, as the very main point of this paper was to demonstrate the importance of accounting for systemic risk when conducting delicensing policy (something, the existing policy fails to account for), a more general direction of further research could be searching for other more complex, more broad and more appropriate alternatives to the existing delicensing policy.

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Appendix 1

Alternative Definition 4

In the Definition 4 an alternative way to express capital at step (k+1) could be written using Heaviside function

:

where and could be interpreted as capital of financial institution j after losses of step k.

Appendix 2

Illustration of complex node interconnectedness)

In what follows I show a part of an imaginary directed weighted graph of an interbank system. The part is the default cluster after the default of bank A. I show only defaulted banks and only their links with other defaulted banks.

Steps:

Step 0

Step 1

Step 2

Step 3

Step 4

When one calculates RCDI or RCCI he/she should be careful about banks like E, G and H. For example, banks E and G can default after the default of bank C but can survive if bank C is saved (while A, B and D all default), that is these banks might be vulnerable only to the default of bank C while defaults of their other debtors (D for G and both A and B for E) would only insignificantly reduce their amounts of capital. On the other hand, it may be the case that, for example, G is vulnerable only to default of D; or both defaults of C and D may be needed in order for bank G not to default et cetera. That is why one has to be cautious when dealing with complex networks.

Representation of RCDIl through DI(l, c, E) is extremely complicated because one will have to account for the interconnectedness. In order to express the additional capital losses to banking system due to the default of bank l through DI(l, c, E) one has to not only add the immediate capital losses of l's creditors but also account for those banks' losses that default due to a collective default of l and some other previously defaulted bank(s). And then one should also consider the losses of creditors of those newly defaulted banks and so on. In the picture above this would be the case for l=C and the conditions that E and G do not default after defaults of only A, B and D but do default when all four banks (A-D) default => when C is added to the default cluster of A, B and D - E and G default => one adds their additional losses but, in addition to that, bank I defaults because of the default of G => one adds losses of I as well. The process is not over yet because bank K defaults as both C and I default => K's capital losses should also be added. Moreover, one should be interested not only in defaulted banks' losses but in overall losses => one should add all losses of other creditors of C, G, E, I and K (that are not depicted above). Obviously, the contagion process may be even more tangled when there are more nodes and more links between the nodes. The situation is worsened by the fact that cautiousness should be taken when adding losses so that not to add losses to already defaulted banks (having zero capital). Thus, such expression (through DI(l, c, E)) will not only be complicated analytically (as a formula) but it will be much more time consuming for an automated program to calculate the values of RCDI and RCCI, making the approach totally impractical. That is why I have chosen a different representation of the two relative measures.

Appendix 3

Delicensing vs Default conditions example

As it was noted, the losses under the delicensing policy rule based on the capital adequacy requirements are almost always greater (and are never less) than those under the initial default conditions from the original model. The following simple example will demonstrate this idea. Also note that the presented banking system is fully an imaginary one and all the featured numbers do not have a formal or maybe even logical explanation. The example's sole role is to show the difference in the values of DI (which is an estimator of losses) for two different default rules used.

Consider a hypothetical banking system with 10 banks with the following matrix of bilateral exposures E and the following initial amounts of capital c(-1):

The values A(i) and L(i) are total interbank assets and total interbank liabilities of bank i respectively and are calculated in the way defined by (2). For now the colours do not matter.

Now the default process can be initiated. Suppose bank #10 defaults, its capital becomes 0 and all of its interbank liabilities become zero as well. So one models Loss cascade according to (3) and obtains the following capital amounts after each step:

At step 0 the 10th bank defaults (red) which nullifies all its liabilities and particularly the loan to bank #8 of 201. This amount happens to be bigger than the capital of bank #8 and therefore this bank defaults as well (orange). In the same way the 6th bank goes bankrupt (yellow) and the process ends. The total loss expressed as Default Impact (DI) is:

However, if the default rule is the one that uses Capital Adequacy Ratio, the losses will be larger. Consider the following risk-weight table of interbank loans taken from each of the ten banks:

According to this table one can calculate the value of Risk Weighted Assets (RWA) of the banks (assuming that all other assets held by the banks are riskless, that is л=0):

Then, supposing that the value of Market risk and Operational risk for all banks equals 20, one calculates the CAR for all the banks as explained in (11):

As one may see all values are comfortably >8% => the system is stable.

Again, consider that bank #10 defaults. In this case the process is essentially the same as the previous one. The difference appears after the default of bank #6. Even though the capital level of bank #9 was 15 (the value of c(3) for 9th bank above), this bank does not meet the CAR rule anymore (it is <8%) and becomes delicensed, that is defaults (blue). Here are those values:

The values of capital levels were calculated as described in Definition 4.

Note that for the direst comparison of the two DIs, the capital loss of the first bank is also not included in the DI under the delicensing policy (even though in general in the model, according to Definition 5, the initial loss is included in the DI calculation)

Hence, in this case the amount of total losses is:

So, .

Thus, as it was stated before, the result is expected. Losses from the default cascade are bigger in the case when the default rule is the CAR one (and not the original one).

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