Systemic risk

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.

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 RCDI_l 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 10^th 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 6^th 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 9^th 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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