Assume that there are N banks in the market and x i j is the ratio of bank i’s deposits in bank j to total interbank lending. x i j is unobservable, but the ratio of the total value of assets deposited by all other banks in bank i to total interbank

lending ( b i = ∑ j = 1 N x i j ) is known, as is the ratio of the total value of liabilities of bank j to all other banks to total interbank lending ( l j = ∑ i = 1 N x i j ) . b i and l j are approximately the marginal distributions f ( b ) and f ( 1 ) , respectively. ∑ i = 1 N b i = 1 , ∑ l = 1 N l j = 1 .

X = [ x 11 ⋯ x 1 j ⋯ x 1 N ⋮ ⋱ ⋮ ⋱ ⋮ x i 1 ⋯ x i j ⋯ x i N ⋮ ⋱ ⋮ ⋱ ⋮ x N 1 ⋯ x N j ⋯ x N N ] (1)

Under market equilibrium, given certain constraints, the most reasonable solution is that which satisfies the principle of maximum entropy. The maximum entropy principle in relation to this problem is expressed mathematically as follows:

Min ∑ i = 1 n ∑ j = 1 n x i j ln x i j s . t . ∑ j = 1 n x i j = b i ; ∑ i = 1 n x i j = l j , x i j ≥ 0 (2)

The equation x i j = a i ⋅ l j is provable.

A bank would not lend money to itself, so each diagonal is zero, namely, x i i = 0 .

X ∗ = [ 0 ⋯ x 1 j ∗ ⋯ x 1 N ∗ ⋮ ⋱ ⋮ ⋱ ⋮ x i 1 ∗ ⋯ 0 ⋯ x i N ∗ ⋮ ⋱ ⋮ ⋱ ⋮ x N 1 ∗ ⋯ x N j ∗ ⋯ 0 ] (3)

The adjusted matrix should be consistent with the original matrix wherever possible. In general, the minimum cross-entropy principle should be followed:

Min ∑ i = 1 n ∑ j = 1 n x i j ∗ log x i j ∗ x i j s . t . ∑ j = 1 n x i j ∗ = b i ; ∑ i = 1 n x i j ∗ = l j , x i j ≥ 0 ; x i i ∗ = 0 (4)

This problem may be solved using the RAS algorithm [14] :

step 0 k = 0 , X 0 = X step 1 for i = 1 , 2 , ⋯ N η i k = b i ∑ j x i j k , x i j k ← η i k x i j k step 2 for j = 1 , 2 , ⋯ N λ j k = l i ∑ i x i j k , x i j k ← λ i k x i j k step 3 Re place k ← k + 1 , Re turntostep 1

Assuming that a bank goes out of business, it transfers its losses to other banks through the interbank lending market so as to cause losses and even bankruptcies among other banks. The default loss ratio is θ and the repayment capital is c i . In the event that the first round of contagion starts when θ x i j > c i bank j triggers the bankruptcy of bank i. The combination of the failure of both bank j and bank i will cause the bankruptcy of other banks that survived the first round, in other words when θ ( x k j + x k i ) > c k , bank k is bankrupted and causes the bankruptcies of other banks in the second round. In this way, the wave of bankruptcies continues until there are no more banks left to be bankrupted. As a result of this high level of contagion, the number of bankrupted banks increases and the process leading to bankruptcy accelerates so that even those banks that were initially in a sound position are eventually affected. The process of risk contagion mainly depends on the level of repayment capital, the scale of lending, and the default loss ratio. Various industries differ in relation to the repayment ratio, which is lower in the financial industry, averaging 36% [15] . A study of the Pudong Development Bank’s loan defaults from 1993 to 2006 showed that the default ratio was between 50% and 85%. To simplify their studies, researchers generally assume that the default ratio is the same for all banks in all rounds of contagion, irrespective of factors such as invisible government guarantees and bankruptcy costs.

The HITS algorithm was developed by Jon Kleinberg to rank Web pages in order of importance. The authority score (a) and hub score (h) of a Web page are the core components of this algorithm. The former focuses on the quality of the Web page [16] , while the latter reflects the quality of the linkage [17] . The core principle of the HITS algorithm is that Web pages with a higher a value will be linked to more Web pages with a higher h value, while Web pages with a higher h value will be linked to more Web pages with a higher a value. The pseudo code of the HITS algorithm is as follows:

a 0 = 1 , h 0 = 1 t = 1 do foreach v in V do a t ( v ) = ∑ ( w , v ) ∈ E h t − 1 ( w ) h t ( v ) = ∑ ( v , w ) ∈ E h t − 1 ( w ) a t = a t / ‖ a t ‖ h t = h t / ‖ h t ‖ t = t + 1 While ‖ a t − a t − 1 ‖ + ‖ h t − h t − 1 ‖ < ε Re turn ( a t , h t )

This paper proposes two indicators, “contagion” and “vulnerability”, guided by the HITS algorithm principle, to reflect the degrees of contagion and vulnerability, respectively. “Contagion” refers to the fact that, for an individual bank, assuming it is closed at a certain default loss ratio, the more influenced (bankrupted) banks there are, the greater the contagion that this bank experiences. This concept corresponds to the h values in the HITS algorithm. “Vulnerability” refers to the fact that, for an individual bank, the more influenced (bankrupted) banks there are, the more vulnerable this bank is. This concept corresponds to the a values in the HITS algorithm.

This study is based on the lending data and core capital data for 21 banks during the period from 2008 to 2015. The banks are listed in Appendix 1. The total assets of these banks account for more than 70% of total interbank assets, so these banks are a representative sample. There are also nonbanking institutions involved in the lending market, so the net amount of interbank lending is not zero, the balance reflecting the activities of the nonbanking institutions. In accordance with the rules of the financial industry, a bank will give priority to internal lending in the case of illiquidity. The maximum interbank lending scale is accordingly the sum of interbank lending.

Yu’E Bao is a service for individuals’ fund payment with balance. The transfer of funds to Yu’E Bao enables the purchase of relevant financial products from both the Yu’E Bao fund and other institutions. Fund management companies place most of the funds under their control into the interbank lending market and mainly trade with 29 banks including policy-oriented banks, the four major state-owned banks, and joint-stock banks, i.e. basically the same banks that comprise the sample used in this study. Yu’E Bao lends money but never borrows, so it cannot be deemed a bank, and thus it cannot be included in the matrix of the 21 banks. Considering that, in the absence of other financing channels, the banks must depend on more frequent interbank lending to make up any shortfall, and thus the scale of lending will increase if Yu’E Bao is excluded, the following steps are taken to process the data (see Table 1).

1) Calculate the interbank lending relationships among the 21 banks during the period from 2008 to 2015; calculate the systemic risk according to the actual scale of lending.

¹The amount of capital provided by Yu’E Bao in the period from 2012 to 2013 was relatively small, and thus can be ignored.

2) Correct the scale of lending¹ during the period from 2014 to 2015 to include funds provided by Yu’E Bao; calculate the interbank lending relationships among the 21 banks using the corrected scale, and identify the systemic risk without Yu’E Bao during the same period. During the period from 2014 to 2015, the change in the scale of bank lending was able to be ignored, but the scale of bank borrowing rose rapidly, mainly as a result of the development of innovative online financial products. Taking the 2013 data as a baseline, it is clear that the increase in total bank borrowings was influenced by nonbank-

Source: Almanac of China’s Finance and Banking, Tianhong Fund Annual Report, and annual reports of banks in 2015.

ing institutions, and thus the scale of interbank lending during 2014 and 2015 can be estimated.

The a value represents the degree of contagion of banks, while the h value represents the number of sources of contagion and contagiousness. The stronger contagiousness is, the more important the bank in the market. Table 2 shows each bank’s a and h values when the default loss ratio θ = 1. During the period from 2008 to 2015, a values greater than zero were limited to EXIMBC, ADBC, PSBC, and CZB. There were only four banks closed down in China, even in the year that witnessed the greatest number of bankruptcies in extreme cases. Therefore, interbank systemic risk is low in China.

As to individual banks, EXIMBC, PSBC, and CZB were vulnerable with greater bankruptcies probability. Among them, EXIMBC was closed in every tide of bankruptcies with higher Value a, showing that it was always vulnerable during the period from 2008 to 2014 and was affected by many other banks. PSBC had experienced six bankruptcies contagion with low Value a, showing that it was vulnerable during the period in question, but was only affected by a limited number of banks. CZB had been bankrupted for three times with higher Value a, showing that it was more vulnerable than any other bank to the performance of the other banks, even after a reduction in vulnerability during the period in question.

Regarding the contagiousness, 11 banks had become sources of contagion four or more times during the period from 2008 to 2015. Among them, ICBC, ABC,

BOC, CCB, and BCM were sources of contagion every time with higher Value h. BOC and ICBC led the way, followed by CCB, ABC, and BCM, CDB, CIB, CMBC, CCB, CMB, and SPDB, and CZB, PSBC, and CBHB.

The number of sources of contagion increased at first, then decreased, then increased again before finally decreasing again during the period from 2008 to 2015, as shown in Figure 1.

As shown in Figure 2, the number of closed banks initially increased and then decreased during the period from 2008 to 2015. The ratio of core capital losses to total losses slowly increased and then slowly decreased during the same period. Comprehensive risk in the three indicators rose from 2008 to 2010, fell from 2010 to 2013, and then falling to the bottom in fluctuation from 2013 to 2015. In 2015, the remarkable increase in core capital of the most vulnerable banks, namely, EXIMBC and PSBC, reduced the systemic risk.

Under corrected condition, or estimated condition, the number of sources of contagion, number of bankrupted banks, and ratio of core capital losses to total losses were all higher than under actual condition in the period from 2014 to 2015, indicating that the risk under the estimated condition was higher than that under both the corrected condition and the actual condition. This means that the entrance of Yu’E Bao and similar innovative online financial products into the interbank lending market reduced the level of interbank systemic risk. Their involvement in the interbank lending market helped to reduce collaboration among banks and avoided a domino effect in times of crisis.

²Critical value of bankruptcy means the loss ratio when the first bank is bankrupted. The lower the critical value, the easier it is for the bank to enter bankruptcy, and thus the greater the risk.

As shown in Figure 3, there were negative relationships between the critical value of bankruptcy² and both the number of sources of contagion and the number of bankrupted banks during the period from 2008 to 2015. In other words, the initial decrease, followed by an increase, and then another decrease, and finally an increase shows that the risk was rising from 2008 to 2010, falling from 2010 to 2013, and then fluctuating before falling to an even lower level

from 2013 to 2015.

The critical value of bankruptcy under both the corrected condition and the estimated condition was lower than under the actual condition in the period from 2014 to 2015, i.e. the bank is at greater risk of bankruptcy in comparison with actual condition. This suggests that the entrance of Yu’E Bao and similar innovative online financial products into the interbank lending market helped to reduce interbank systemic risk.