# The Assessment of Systemic Risk in the Kenyan Banking Sector

Authors: Hong Fan; Allan Alvin Lee Lukaya Amalia; Qian Qian Gao
Journal: Complexity
 (2018)

---

## Abstract

The present paper aims to assess the systemic risk of the Kenyan banking system. We propose a theoretical framework to reveal the time evolution of the systemic risk using sequences of financial data and use the framework to assess the systemic risk of the Kenyan banking system that is regarded as the largest in the East and Central African region. Firstly, we estimate the bilateral exposures matrix using aggregate financial data on loans and deposits from annual reports and analyze the interconnectedness in the market using network centrality measures. Next, we extend the Eisenberg–Noe method to a multiperiod setting to the systemic risk of the Kenyan banking system, in which the multiperiod includes the dynamic evolutions of the Kenyan banking system of every bank and the structure of the interbank network system. We apply this framework to assess dynamically the systemic risk of the Kenyan banking system between 2009 and 2015. The main findings are the following. The theoretical network analysis using network centrality measures showed several banks displaying characteristics of systematically important banks (SIBs). The theoretical default analysis showed that a bank suffering a basic default will trigger a contagious default that caused several other banks in the sector to go bankrupt. Further stress test proved that the KCB bank theoretically caused a few contagious defaults due to an unusually high interconnectedness. This methodology can contribute by being part of monitoring system of the Central Bank of Kenya (regulatory body) as well as the implementation of policies (such as bank-internal stress tests) that assist in preventing default contagion.


---

## Body

## 1. Introduction

Events beginning in the United States in 2008, including the collapse of some major financial institutions and the rescue of others, ideally depict the effect that a systemic crisis can have on an economy. Many financial institutions around the world felt the impact of the default of some of these institutions in the United States. Thus, systemic risk emerged as one of the most challenging aspects. Previous to this occurrence, there was a limited knowledge of how systemic risk affected institutions and how to assess the systemic risk. Therefore, it is essential to have an effective assessment of systemic risk exposure to an institution.A few measures of systemic risk have been proposed in recent empirical studies. Among them, De Jonghe [1] uses the extreme-value analysis to measure the contribution of each single financial institution to systemic risk. Using CDS (Credit Default Swap) of financial firms and correlations between their stock returns, Huang et al. [2, 3] and Segoviano Basurto [4] propose portfolio credit risk measurement methods, such as CoPoD and CIMDO methods. The conditional Value at Risk (CoVaR) is proposed as measure of systemic importance of financial institutions proposed by Adrian and Brunnermeier [5]. The CoVaR can capture how much the distress of one institution can affect another institution and it provides a clear direction on the relation between the risks involved between two financial institutions. Another measure that was proposed and used as an indicator to measure the systemic risk is the System Expected Shortfall (SES). It was introduced to measure listed financial institutions’ contributions to systemic risk. SES is defined as an institution’s tendency to be undercapitalized when the financial system as a whole is undercapitalized. Marginal Expected Shortfall (MES) was also introduced as a measure of an institution’s loss in the tail of the system’s loss distribution. MES is a systemic fragility metric that can also be used to determine an optimal taxation policy based on systemic risk [6]. Both the SES and the MES methods were proposed by Acharya et al. [7].However, a limitation of those approaches is that they measure a financial institution’s loss only if the system is in normal time and only indirectly take into account the size, the probability of default, and the correlation of each financial institution. Furthermore, the correlation does not capture the interconnectedness adequately because it does not consider the various interactions (such as contagious defaults) or the relationship between interconnectedness and systemic importance in a financial system.In normal times, the interbank market ensures efficient liquidity redistribution from banks with surplus liquidity to banks with a shortage of liquidity and thus serves as an absorber of idiosyncratic liquidity shocks. In turbulent times, however, interbank markets can become a channel for liquidity contagion due to liquidity hoarding by banks and/or credit risk contagion due to credit losses on interbank exposures. Interbank market contagion is more likely to occur in banking sectors that are highly dependent on wholesale financing [8]. In an extensive study of the US financial system, Hautsch et al. [9] show that it is mainly the interconnectedness within the financial sector that increases the risk of failure of the entire system, denoted as systemic risk [10].The intricate structure of linkages can be naturally captured via a network representation of the financial system. Such a network models the interlinking exposures between financial institutions and can thus assist in detecting important shock transmission mechanisms. The use of network theories can enrich our understanding of financial systems, helping to answer questions related to how resilient financial networks are to contagion and how financial institutions form connections when exposed to risk of contagion [11]. Two types of sources for the risk of contagion have been studied in the literature. One is the network of banks investing in similar types of assets, in which one bank failure can lead to a fall in the price of its assets and then affect the solvency of other banks that hold the same assets [12, 13]. The other is the risk of contagion in the interbank market, which concerns the liquidity risk of contagion at a form of interlocking exposure; such exposure is very short term, mainly overnight. The focus of the present paper is on the risk of contagion in the interbank market. The empirical studies of the risk of contagion in the interbank market have been done in those references [14–17]. In the theoretical studies of the risk of contagion in the interbank market, Allen and Gale [18] studied the effect of the static network structure on the risk of contagion. Their results found that an interbank network system with a complete network structure is more stable than that with an incomplete network structure. Besides Allen and Gale, many researchers theoretically study the effect of the network structure on the risk of contagion [19–24]. Although studies on the risk of contagion in the interbank network systems have made the great progress, there are some limitations in existing research. Most of the studies are based on static network structures (fixed bank lending matrixes) and static bank systems (fixed the bank balance sheets). However, the reality of an interbank network system is with high complex dynamics.This is where the present paper is useful in assessing systemic risk in relation to dynamic financial networks. In theory, the Eisenberg–Noe framework [25] describes simultaneous defaults for one period and not for a dynamic multiperiod scenario which applies to our case. We have to mention that we extend the framework to a multiperiod setting, which borrows from the framework of Kanno [26] and theoretically analyze the Kenyan interbank market using aggregate data on loans and deposits from the portal-African Markets. Kanno [26] and Lehar [27] considered the multiperiod scenario on the systemic risk; however, they did not consider the dynamic change of the interbank network. Besides, Kanno only uses the maximum entropy algorithms to estimate the topology of the interbank system. Here, in the present paper, we use two methods (the maximum entropy algorithms and the minimum density approach) to estimate the network structure and make the structure of the network system change with time. Therefore, the present paper proposes a theoretical framework to reveal the time evolution of the systemic risk using sequences of financial data and uses the framework to assess the systemic risk of the Kenyan banking system which is the largest in East and Central Africa. This study will help us understand the impact of systemic risk in the Kenyan banking sector. The theoretical frame work we proposed merges existing algorithms, maximum entropy estimation method [14], the minimum density approach [28], the asset value estimation algorithm [26, 27], and obligation clearing algorithm [25], seamlessly to calculate the time evolution of the systemic risk.The key results of the present paper are summarized as follows. First, we explain the network structure of the Kenyan interbank market and theoretically examine its structure using the estimated bilateral exposures matrix. Based on the measures on in- and out-strength, we show that most of the banks designated as systematically important banks (SIBs) are quite significant in the role they play in the Kenyan interbank market. Second the contagious defaults are then modeled in the Kenyan interbank market to then analyze the mechanism behind contagious defaults. Thirdly stress tests are conducted to analyze the possibility of contagious defaults conditional on a banks basic default at an evaluation time point.The present paper is organized as follows. Section2 proposes a theoretic framework of the systemic risk. In the theoretic framework, Section 2.1 first proposes the method of measuring systemic risk that includes basic defaults and contagious defaults, then Section 2.2 deliberates the methodology of the bilateral exposures matrix, and Section 2.3 explains the estimation methodology for the market values of assets. Section 3 describes the data used in this study. Section 4 presents the results of the risk analysis and finally Section 5 concludes the paper.


## 2. The Dynamic Theory Framework of Assessing Systemic Risk

We first construct a dynamic theory framework of assessing systemic risk, which is described as Figure1. Figures 1(a) and 1(b) describe the network structure of the interbank system in Kenyan, which can be estimated by the methods described in Section 2.1. Figure 1(a) shows the complete network, which can be estimated by the Maximum Entropy Method, while Figure 1(b) represents the sparse network structure that can be estimated by the minimum density approach, which is first proposed by Anand et al. [28]. Figure 1(c) is the dynamic estimation method of banks’ assets, which is described in Section 2.2. When we get the parameters ui and σi, then, according to (7), we get the evolution of banks’ assets like Figure 1(d). In Figure 1(d), in each time step, we first calculate the basic default of banks; then, due to the connection of banks like Figures 1(a) and 1(b), the basic default of banks, namely, total assets smaller than the total liabilities, cause the default of other banks that are connected to the basic default banks. The method of computing losses from basic default and contagion default is described in Section 2.3. At time t=0, the structure of the interbank network system is estimated by the balance sheet of end of each year. However, due to the time evolution, banks may be defaulted by basic default or by contagion, then banks that are default will be removed from the network, and, therefore, the structure of network will change with time, which is showed in Figure 1(e), the evolution equations of which will be described in Section 2.4.Figure 1
The dynamic theory framework of assessing systemic risk.


### 2.1. Estimation of Bilateral Exposures Matrix

The lending relationship in the Kenyan interbank market is represented by the following nominal interbank matrixX: (1)where xij denotes the amount of money that bank i borrows from bank j. aj=∑ixij denotes the total value of bank j’s interbank assets and li=∑jxij denotes bank i’s total liabilities. It has to hold that(2)∑jaj=∑ili=XΣ,where XΣ is size of the interbank market. Next we adopt two methods to estimate the matrix X. One is the method of the maximum entropy estimation [14] in Section 2.1.1, and the other is the minimum density approach [28] in Section 2.1.2.


#### 2.1.1. Method of the Maximum Entropy Estimation

We know that the diagonal elements ofX have to be zero. Therefore, we set the prior matrix of X0 as follows:(3)X0=0foranyi=jailjotherwise.Matrix X0 violates the summing constraints expressed in (2). Consequently, a new matrix X must be found to satisfy the constraints. The solution is provided by solving the optimization problem as follows: (4)min∑i=1N∑j=1Nxijln⁡xijxij0Subjectto∑j=1Nxij=li,∑i=1Nxij=aj,xij≥0.


#### 2.1.2. The Minimum Density Approach

The minimum density approach minimizes the network’s density, the share of actual to potential bilateral links. It minimizes the total number of linkages necessary for allocating interbank positions, consistent with total lending and borrowing observed for each bank. LetC represent the fixed cost of establishing a link. Then the minimum density approach can be formulated as a constrained optimization problem as follows:(5)minxC∑i=1N∑j=1N1xij>0Subjectto∑j=1Nxij=li,∑i=1Nxij=aj,xij≥0,where the integer function 1 equals one only if bank i lends bank j.


### 2.2. Estimation Methodology of Market Values of Assets

Asset valueVi(t) is not daily observable. However, we can get the asset value in the bank balance sheet at the end of each year, while the equity market price of banks can be observed by stock price on each day. The time t is measured in units of day in the present paper. Next we will give a method to estimate asset values of each day (time evolution of asset value) according to the equity market data of banks. Assume that the asset value Vi(t) of bank i follows a geometric Brownian motion with drift ui and volatility σi:(6)dViVi=uidt+σidWt,where W(t) is the standard Brownian motion. The solution to this equation is obtained as(7)Vit=Vi0eui-σi2/2t+σitz,where z is a standard normal random variable. If we know the parameters ui and σi, then, according to (7), we can get the evolution of Vi(t). Next we can observe a time series of equity price Ei(t) from the stock market; then we can use the Black-Scholes model to estimate the parameters ui and σi as follows: (8)Eit=VitNdt-DitNdt-σiT,where T=365 days, t represents the evolution of days, and (9)dt=ln⁡Vit/Dit+σi2/2TσiT.In the stock market, one can observe a time series of Ei(t) and read the face value of bank debt Di(0) from the balance sheet. We assume that all bank debt is insured and will therefore grow at the risk-free rate r (the interest rates used have been obtained from the Central Bank of Kenya website, for the relevant years (2009–2015) (https://www.centralbank.go.ke/statistics/interest-rates/)). Then, Di(t)=Di(0)ert. Given the initial data of Vi(0), time series data of Ei(0),Ei(1),…,Ei(T),Di(0),Di(1),…,Di(T), and the arbitrary initial value of ui(0),σi(0), we can get the estimation of V^i(1),V^i(2),…,V^i(T) according to (8) and (9). Then, we use the following maximization likelihood function to estimate the parameters ui and σi, which is proposed by Duan et al. [29–31]:(10)Lui,σi;V^i1,V^i2,…,V^iT=-T2ln⁡2πσi2h-T2∑k=1TRik-ui-σi2/2h2σi2h-∑k=1Tln⁡V^ik,where Ri(k)=ln⁡(V^i(t)/V^i(t-1)) and h=1/365. Here, h represents business days instead of calendar days. According to the stock market data source, h may equal about 250, which will be different at each year.The estimated parameters are denoted asu⌢i,σ⌢i respectively. Then, we compare the estimated parameters u⌢i,σ⌢i with the initial values ui(0),σi(0); if u⌢i,σ⌢i do not equal ui(0),σi(0), we replace the initial values ui0,σi0,andVi(0) with u⌢i,σ⌢i,andV⌢i(0), and then we repeat the estimation method from (8), (9), and (10) once again until the estimated parameters u⌢i,σ⌢i equal ui(0),σi(0). Accordingly, we get the estimated parameters ui and σi. Then, according to (7), we get the evolution of Vi(t). The estimation method of Vi(t) is the same as [27].


### 2.3. Method of Measuring Systemic Risk

Here, we study how to assess the systemic risk in the financial network. We extend a fundamental framework proposed by Eisenberg and Noe [25] to a multiperiod setting. There is a clearing payment system that deals with the interbank payment amounts of all the banks in the system daily.Let us look at a set of banksN=1,…,N at time t. The interbank structure is represented as (X(t),e(t)), where X is a (N×N) nominal interbank liabilities matrix and e is the noninterbank net claims which is the difference between the market value of assets and the value of liabilities, namely, Vi(t)-Di(t).If the total value of a bank becomes negative for a pair (X(t),e(t)), then the bank becomes bankrupt. Let xi(t)=∑j=1Nxij(t) represent the total interbank liability of bank i to all banks j of the system. Furthermore, we consider a matrix Π, which is brought about by normalizing the entries to the total claims: (11)Πijt=xijtxit,ifxit>00,otherwise.A banking system is designated as a tuple (Π(t),e(t),andX(t)), for which we describe a clearing payment vector p∗(t). The clearing payment vector represents the limited liabilities of the banks and the proportional distribution in the event of a collapse.A payment vectorpi∗(t) is a clearing payment vector subject to the following happening:(12)pi∗t=xit∑j=1NΠjitpj∗t+eit≥xit,∑j=1NΠjitpj∗t+eit0≤∑j=1NΠjitpj∗t+eit<xit,0∑j=1NΠjitpj∗t+eit<0.Therefore, bank j is solvent in case  1 and insolvent in cases  2 and 3.We identify an insolvent bankj under the condition (pi∗(t)<xi(t)), which holds for cases  2 and 3.We implement the default algorithm established by Eisenberg and Noe [25] to find a clearing payment vector. They demonstrate that, under mild regulatory conditions, a unique clearing payment vector always exists for (Π(t),e(t),andX(t)). These regularity conditions refer to properties that the network structure must have in order that there is a unique clearing vector.These results apply to our multiperiod setting. The number of defaulted banks is computed by comparing the clearing payment vector to the nominal liability vector. A theoretical default algorithm is applied to compute the clearing payment vector and is summarized as follows.Type 1: Basic Default
(13)
V
i
t
-
D
i
t
≤
0
,where Vi(t) is the market value of the total assets of bank i at time t (days) and Di(t) is the total face value of the interest-bearing debt of bank i at time t. The basic default is an idiosyncratic default, caused by the condition of the defaulting node itself. According to Section 2.3, Vi(t) is a random walk variable causing the idiosyncratic default at time step t.Type 2: Contagious Default
(14)
∑
j
=
1
N
Π
j
i
t
x
j
t
-
x
i
t
+
e
i
t
>
0
,
(15)
∑
j
=
1
N
Π
j
i
t
p
j
∗
t
-
x
i
t
+
e
i
t
≤
0
.If the claims of bank i are positive, but its obligations banks pay less liability to bank i, which results in the fact that the net claim of bank i is negative, then a contagious default occurs on bank i.


### 2.4. The Evolution of the Bilateral Exposuresx(t+1), ai(t+1), and li(t+1)

After calculating a clearing payment vector at time stept, we can calculate the new matrix of X at the time step t+1. We should note that when bank i defaults, bank i can pay only part of their liabilities to other banks. χi, the ratio, is defined as follows:(16)χi=∑j=1NΠjitpj∗t+eitlit.The total assets and liabilities of bank j from time steps t+1 to T will be updated as follows:(17)Vjt+1:T=Vjt-xji,Djt+1:T=Djt-xji,Vjt+1:T=Vjt-1-χixij.When bank i defaults, we set xi,j(t)=0andxj,i(t)=0 and clear out bank i from the network bank system. In real world, the interbank exposures may change from day to day. And, in the present paper, due to the default banks, the number of banks decreases, which causes the interbank exposures to change by time step. Note that if there is no default bank in t, the network estimated does not change, because the interbank totals remain the same in t and t+1.Then, we need to recalculate the bilateral exposures matrixXt+1 according to the algorithm in Section 2.1. Thus, the evolution of aj(t+1) and li(t+1) is described as follows:(18)ajt+1=∑i=1Nxi,jt+1,lit+1=∑j=1Nxi,jt+1.


## 2.1. Estimation of Bilateral Exposures Matrix

The lending relationship in the Kenyan interbank market is represented by the following nominal interbank matrixX: (1)where xij denotes the amount of money that bank i borrows from bank j. aj=∑ixij denotes the total value of bank j’s interbank assets and li=∑jxij denotes bank i’s total liabilities. It has to hold that(2)∑jaj=∑ili=XΣ,where XΣ is size of the interbank market. Next we adopt two methods to estimate the matrix X. One is the method of the maximum entropy estimation [14] in Section 2.1.1, and the other is the minimum density approach [28] in Section 2.1.2.


### 2.1.1. Method of the Maximum Entropy Estimation

We know that the diagonal elements ofX have to be zero. Therefore, we set the prior matrix of X0 as follows:(3)X0=0foranyi=jailjotherwise.Matrix X0 violates the summing constraints expressed in (2). Consequently, a new matrix X must be found to satisfy the constraints. The solution is provided by solving the optimization problem as follows: (4)min∑i=1N∑j=1Nxijln⁡xijxij0Subjectto∑j=1Nxij=li,∑i=1Nxij=aj,xij≥0.


### 2.1.2. The Minimum Density Approach

The minimum density approach minimizes the network’s density, the share of actual to potential bilateral links. It minimizes the total number of linkages necessary for allocating interbank positions, consistent with total lending and borrowing observed for each bank. LetC represent the fixed cost of establishing a link. Then the minimum density approach can be formulated as a constrained optimization problem as follows:(5)minxC∑i=1N∑j=1N1xij>0Subjectto∑j=1Nxij=li,∑i=1Nxij=aj,xij≥0,where the integer function 1 equals one only if bank i lends bank j.


## 2.1.1. Method of the Maximum Entropy Estimation

We know that the diagonal elements ofX have to be zero. Therefore, we set the prior matrix of X0 as follows:(3)X0=0foranyi=jailjotherwise.Matrix X0 violates the summing constraints expressed in (2). Consequently, a new matrix X must be found to satisfy the constraints. The solution is provided by solving the optimization problem as follows: (4)min∑i=1N∑j=1Nxijln⁡xijxij0Subjectto∑j=1Nxij=li,∑i=1Nxij=aj,xij≥0.


## 2.1.2. The Minimum Density Approach

The minimum density approach minimizes the network’s density, the share of actual to potential bilateral links. It minimizes the total number of linkages necessary for allocating interbank positions, consistent with total lending and borrowing observed for each bank. LetC represent the fixed cost of establishing a link. Then the minimum density approach can be formulated as a constrained optimization problem as follows:(5)minxC∑i=1N∑j=1N1xij>0Subjectto∑j=1Nxij=li,∑i=1Nxij=aj,xij≥0,where the integer function 1 equals one only if bank i lends bank j.


## 2.2. Estimation Methodology of Market Values of Assets

Asset valueVi(t) is not daily observable. However, we can get the asset value in the bank balance sheet at the end of each year, while the equity market price of banks can be observed by stock price on each day. The time t is measured in units of day in the present paper. Next we will give a method to estimate asset values of each day (time evolution of asset value) according to the equity market data of banks. Assume that the asset value Vi(t) of bank i follows a geometric Brownian motion with drift ui and volatility σi:(6)dViVi=uidt+σidWt,where W(t) is the standard Brownian motion. The solution to this equation is obtained as(7)Vit=Vi0eui-σi2/2t+σitz,where z is a standard normal random variable. If we know the parameters ui and σi, then, according to (7), we can get the evolution of Vi(t). Next we can observe a time series of equity price Ei(t) from the stock market; then we can use the Black-Scholes model to estimate the parameters ui and σi as follows: (8)Eit=VitNdt-DitNdt-σiT,where T=365 days, t represents the evolution of days, and (9)dt=ln⁡Vit/Dit+σi2/2TσiT.In the stock market, one can observe a time series of Ei(t) and read the face value of bank debt Di(0) from the balance sheet. We assume that all bank debt is insured and will therefore grow at the risk-free rate r (the interest rates used have been obtained from the Central Bank of Kenya website, for the relevant years (2009–2015) (https://www.centralbank.go.ke/statistics/interest-rates/)). Then, Di(t)=Di(0)ert. Given the initial data of Vi(0), time series data of Ei(0),Ei(1),…,Ei(T),Di(0),Di(1),…,Di(T), and the arbitrary initial value of ui(0),σi(0), we can get the estimation of V^i(1),V^i(2),…,V^i(T) according to (8) and (9). Then, we use the following maximization likelihood function to estimate the parameters ui and σi, which is proposed by Duan et al. [29–31]:(10)Lui,σi;V^i1,V^i2,…,V^iT=-T2ln⁡2πσi2h-T2∑k=1TRik-ui-σi2/2h2σi2h-∑k=1Tln⁡V^ik,where Ri(k)=ln⁡(V^i(t)/V^i(t-1)) and h=1/365. Here, h represents business days instead of calendar days. According to the stock market data source, h may equal about 250, which will be different at each year.The estimated parameters are denoted asu⌢i,σ⌢i respectively. Then, we compare the estimated parameters u⌢i,σ⌢i with the initial values ui(0),σi(0); if u⌢i,σ⌢i do not equal ui(0),σi(0), we replace the initial values ui0,σi0,andVi(0) with u⌢i,σ⌢i,andV⌢i(0), and then we repeat the estimation method from (8), (9), and (10) once again until the estimated parameters u⌢i,σ⌢i equal ui(0),σi(0). Accordingly, we get the estimated parameters ui and σi. Then, according to (7), we get the evolution of Vi(t). The estimation method of Vi(t) is the same as [27].


## 2.3. Method of Measuring Systemic Risk

Here, we study how to assess the systemic risk in the financial network. We extend a fundamental framework proposed by Eisenberg and Noe [25] to a multiperiod setting. There is a clearing payment system that deals with the interbank payment amounts of all the banks in the system daily.Let us look at a set of banksN=1,…,N at time t. The interbank structure is represented as (X(t),e(t)), where X is a (N×N) nominal interbank liabilities matrix and e is the noninterbank net claims which is the difference between the market value of assets and the value of liabilities, namely, Vi(t)-Di(t).If the total value of a bank becomes negative for a pair (X(t),e(t)), then the bank becomes bankrupt. Let xi(t)=∑j=1Nxij(t) represent the total interbank liability of bank i to all banks j of the system. Furthermore, we consider a matrix Π, which is brought about by normalizing the entries to the total claims: (11)Πijt=xijtxit,ifxit>00,otherwise.A banking system is designated as a tuple (Π(t),e(t),andX(t)), for which we describe a clearing payment vector p∗(t). The clearing payment vector represents the limited liabilities of the banks and the proportional distribution in the event of a collapse.A payment vectorpi∗(t) is a clearing payment vector subject to the following happening:(12)pi∗t=xit∑j=1NΠjitpj∗t+eit≥xit,∑j=1NΠjitpj∗t+eit0≤∑j=1NΠjitpj∗t+eit<xit,0∑j=1NΠjitpj∗t+eit<0.Therefore, bank j is solvent in case  1 and insolvent in cases  2 and 3.We identify an insolvent bankj under the condition (pi∗(t)<xi(t)), which holds for cases  2 and 3.We implement the default algorithm established by Eisenberg and Noe [25] to find a clearing payment vector. They demonstrate that, under mild regulatory conditions, a unique clearing payment vector always exists for (Π(t),e(t),andX(t)). These regularity conditions refer to properties that the network structure must have in order that there is a unique clearing vector.These results apply to our multiperiod setting. The number of defaulted banks is computed by comparing the clearing payment vector to the nominal liability vector. A theoretical default algorithm is applied to compute the clearing payment vector and is summarized as follows.Type 1: Basic Default
(13)
V
i
t
-
D
i
t
≤
0
,where Vi(t) is the market value of the total assets of bank i at time t (days) and Di(t) is the total face value of the interest-bearing debt of bank i at time t. The basic default is an idiosyncratic default, caused by the condition of the defaulting node itself. According to Section 2.3, Vi(t) is a random walk variable causing the idiosyncratic default at time step t.Type 2: Contagious Default
(14)
∑
j
=
1
N
Π
j
i
t
x
j
t
-
x
i
t
+
e
i
t
>
0
,
(15)
∑
j
=
1
N
Π
j
i
t
p
j
∗
t
-
x
i
t
+
e
i
t
≤
0
.If the claims of bank i are positive, but its obligations banks pay less liability to bank i, which results in the fact that the net claim of bank i is negative, then a contagious default occurs on bank i.


## 2.4. The Evolution of the Bilateral Exposuresx(t+1), ai(t+1), and li(t+1)

After calculating a clearing payment vector at time stept, we can calculate the new matrix of X at the time step t+1. We should note that when bank i defaults, bank i can pay only part of their liabilities to other banks. χi, the ratio, is defined as follows:(16)χi=∑j=1NΠjitpj∗t+eitlit.The total assets and liabilities of bank j from time steps t+1 to T will be updated as follows:(17)Vjt+1:T=Vjt-xji,Djt+1:T=Djt-xji,Vjt+1:T=Vjt-1-χixij.When bank i defaults, we set xi,j(t)=0andxj,i(t)=0 and clear out bank i from the network bank system. In real world, the interbank exposures may change from day to day. And, in the present paper, due to the default banks, the number of banks decreases, which causes the interbank exposures to change by time step. Note that if there is no default bank in t, the network estimated does not change, because the interbank totals remain the same in t and t+1.Then, we need to recalculate the bilateral exposures matrixXt+1 according to the algorithm in Section 2.1. Thus, the evolution of aj(t+1) and li(t+1) is described as follows:(18)ajt+1=∑i=1Nxi,jt+1,lit+1=∑j=1Nxi,jt+1.


## 3. Data

In the present paper, we use data from the portal of African Markets. The portal has historical share price data and has annual reports of the listed companies of the biggest economies on the African continent. The data we have used from this site was obtained from annual reports from the years 2008 to 2015 from eight listed banks, as well as market data (share prices) for the same. We also obtained monthly interest rates from the Central Bank of Kenya for the same banks. The banks selected for this research are big banks in the Kenyan banking sector and they essentially have big market share of banking clients in the market. Financial sector in the Kenyan banking system includes the Central Bank of Kenya (CBK), the primary regulator of the banking industry; 28 domestic and 14 foreign commercial banks with branches, agencies, and other outlets throughout the country; one mortgage finance company; eight representative offices of foreign banks; eleven licensed deposit taking microfinance institutions. However, the banking sector is essentially dominated by four major commercial banks, namely, Equity Bank, Kenya Commercial Bank, Barclays Bank of Kenya, and Standard Chartered. In addition, smaller banks have emerged and experienced tremendous growth in recent years. According to the Central Bank of Kenya, 66.7 percent of the adult population in 2013 had formal access to financial services through commercial banks and the government-owned Post Bank. With the advent of mobile money and its recent linkages to the formal banking system, however, the number of Kenyans with access to electronic financial services has grown rapidly. Kenya has now become a leader in financial inclusion and its example is being replicated in countries around the world.To be able to achieve the objectives of our research, we need to identify the interbank exposures and noninterbank exposures (net claims cash flowe for each bank). The interbank exposures considered in the analysis are interbank loans and advances to banks. These items are yearly interbank transactions in the interbank markets. We do not consider interbank transfers between parent firms and subsidiaries. There are plenty of interbank transactions in this market; therefore, we can estimate the bilateral exposures matrix from the data available in the annual reports. Because we also need to estimate the market value of the assets from equity data, we only consider publicly traded banks. We therefore acquired daily market share price data of the said eight banks. The data of end-of-year period for 2009–2015 in the Kenya banking system is listed in Table 1.Table 1
Data of the end-of-year period for the years 2009–2015 in the Kenya banking system. We use the letters s1, s2, s3, s4, s5, s6, s7, and s8 to stand for Barclays Bank, Coop, Diamond Trust, Equity Bank, HFCK, KCB, NBK, and NIC respectively. The unit of currency is Shs.
Year 2009
Bank name
Total assets
Total interbank lending
Total liabilities
Total interbank borrowing
S1
Barclays Bank
155151000
1061000
140941000
262000
S2
Coop
110531373
4642338
94428114
1010216
S3
Diamond Trust
47146767
5638340
40883446
1310761
S4
Equity Bank
96512000
2022000
73174000
6114000
S5
HFCK
18280761
2106419
14196530
1700000
S6
KCB
172384128
5936128
149986213
9309706
S7
NBK
51404408
1154271
43496716
50290
S8
NIC
47558241
4936616
40765987
56767
Year 2010
Bank name
Total assets
Total interbank lending
Total liabilities
Total interbank borrowing
S1
Barclays Bank
171151000
935000
140818000
92000
S2
Coop
153983533
6671257
134396691
5348291
S3
Diamond Trust
58605823
4404753
50548446
1584787
S4
Equity Bank
133890000
1238000
105583000
7464
S5
HFCK
29325842
7866266
25056311
1573369
S6
KCB
223024556
3959963
182148111
10807301
S7
NBK
60026694
1408699
50097083
780203
S8
NIC
54776432
4611935
46880578
101479
Year 2011
Bank name
Total assets
Total interbank lending
Total liabilities
Total interbank borrowing
S1
Barclays Bank
166269000
913000
139276000
123000
S2
Coop
167772390
7437716
147395935
1881284
S3
Diamond Trust
77453024
9452751
67086550
1658627
S4
Equity Bank
176911000
1094000
141864000
13769000
S5
HFCK
31972113
4724183
27189895
847507
S6
KCB
282493553
17648880
237330895
11502571
S7
NBK
5564998
3388191
58208042
155613
S8
NIC
73581321
4486475
63681461
206149
Year 2012
Bank name
Total assets
Total interbank lending
Total liabilities
Total interbank borrowing
S1
Barclays Bank
185100000
1282000
155514000
1724000
S2
Coop
199662956
8886167
171270727
1065302
S3
Diamond Trust
94511818
4404676
79633326
1002394
S4
Equity Bank
215829000
983000
173158000
25755000
S5
HFCK
40685928
6395958
35540111
10212633
S6
KCB
304751807
4491511
250870082
8261878
S7
NBK
67154805
1356954
56704829
254694
S8
NIC
101771705
6569964
86707098
3044959
Year 2013
Bank name
Total assets
Total interbank lending
Total liabilities
Total interbank borrowing
S1
Barclays Bank
207011000
1386000
174639000
4738000
S2
Coop
228874484
10040915
193776707
5462337
S3
Diamond Trust
114136429
4120113
95568152
3734712
S4
Equity Bank
238194000
1063000
187508000
25612000
S5
HFCK
46755111
6878601
41073258
10212469
S6
KCB
322684854
5222915
260921815
5516617
S7
NBK
92493034
8283461
80644743
824858
S8
NIC
112916814
4189408
95285885
4910171
Year 2014
Bank name
Total assets
Total interbank lending
Total liabilities
Total interbank borrowing
S1
Barclays Bank
226116000
1207000
187931000
121000
S2
Coop
282689098
12814861
240337979
3241726
S3
Diamond Trust
141175794
7413484
115391380
2176219
S4
Equity Bank
276115727
16904704
236382382
199136
S5
HFCK
60490883
8583509
54214800
0
S6
KCB
376969401
2577662
304802062
8733510
S7
NBK
122864886
2033662
110750974
5077625
S8
NIC
137087464
7843903
113829447
4625406
Year 2015
Bank name
Total assets
Total interbank lending
Total liabilities
Total interbank borrowing
S1
Barclays Bank
241152698
252867
201436326
187385
S2
Coop
339549808
13869273
290239033
2905405
S3
Diamond Trust
190947903
4973737
160951702
8149404
S4
Equity Bank
341329318
16554308
293889704
565208
S5
HFCK
68808654
5517670
59718700
0
S6
KCB
467741173
9254721
386855215
14759625
S7
NBK
117789712
1850368
104345832
6806967
S8
NIC
156762225
5464120
130307839
7734613


## 4. Results

This section describes our analysis results. In Section4.1, we discuss the significance of the bilateral exposures matrix estimation and the network analysis. Section 4.2 deals with the default analysis. Finally, in Section 4.3 we report on the results of the stress test.


### 4.1. Network Analysis and Estimation of Bilateral Exposures Matrix

We estimate the bilateral exposures matrixX stated in (1) and use the matrix to examine the network structure of the Kenyan interbank market. We investigate the global interbank network using network centrality measures. Usually the degree of a node is considered as a proxy variable for interconnectedness and explains the number of edges connected to a node. In the present paper, we define the in-strength that shows the ratio of the money lent to all the other banks to the total money. For simplicity, the in-strength of bank i is ai/∑ai. Similarly, the out-strength of bank i is li/∑li. The total strength of a bank is the summation of its in-strength and out-strength. These measures, hence, give a sense of investment and funding diversifications. Figure 2 highlights the time variations of the in-strength (red line) and out-strength (blue line) for eight listed banks. The bigger the increase in the in-strength, the more the debtors the bank would have. In contrast, the bigger the increase in the out-strength, the more the creditors the bank would have. Therefore, in terms of contagious default, the out-strength is more important than the in-strength. Figure 2 depicts a group of banks that borrow more than they lend and others that lend more than they borrow. Banks like NIC Bank, Diamond Trust Bank, National Bank of Kenya, Cooperative Bank, and Barclays Bank lend more than they borrow to other banks by averages of 55.3%, 53.3%, 46.2%, 38.9%, and 35.8%, respectively, from 2008 to 2015.Figure 2
Time variations of in-strength and out-strength for 8 banks. The total strength is the sum of the in-strength (red line) and the out-strength (blue line). The vertical axis (the strength) shows the percentage of the amount of money borrowed or lent by a bank while horizontal axis represents time in years.Meanwhile, banks like Kenya Commercial Bank and Equity Bank lend more than they borrow by an average of 24.3% and 19.2%, respectively, from 2008 to 2015. Therefore, we can examine which banks borrow (lend) more than they lend (borrow) in the Kenyan network in terms of percentage. Following is a breakdown of the percentage figures that these banks borrow in order of magnitude and they include Kenya Commercial Bank (24.3%) and Equity Bank (19.2%). The list of banks that lends includes NIC Bank (55.3%), Diamond Trust Bank (53.3%), HFCK Bank (49.5%), National Bank of Kenya (46.2%), Cooperative Bank (38.9%), and Barclays Bank (35.8%).


### 4.2. Bank Default Analysis

We estimate the theoretical number of defaulting banks during the estimation period of 2009–2015, which is presented in Figure3. Figure 3 indicates the time variations of the number of defaulting banks in the Kenyan banking industry with the complete network that is estimated by the Maximum Entropy Method. It basically shows that in 2015 the Kenyan banking system is more unstable than other years. The years in 2013 and 2010 are more stable than other years, because no banks defaulted.Figure 3
Time variations in number of defaulting banks in Kenyan banking system with the network estimated by maximum entropy estimation method.There has also been a banking crisis in Kenya since 2015 mostly due to weak supervision and outright fraud by bank directors. An example of this is the nonlisted Chase Bank which was put under receivership in the same year. Several other banks including National bank also showed signs of collapsing due to the same. Analysts have been warning banks since 2012 to stop understating loan provisions and to increase their capitalization. A lot of work is still needed especially with the regulators including the CMA (Capital Market Authority) and Kenyan banking sector poses a challenge of lack of trust in the banking industry as most clients move to other rudimentary means of saving their money.Since the Kenyan banking system in the year 2015 is most unstable, we compare the effect of network structure on the time variations in the number of defaulting banks in 2015, which is presented in Figure4. From Figure 4, we can see that the network topology estimation methods cause not much effect on the time variations in the number of defaulting banks. Next, we compare the evolution of the network topology, which is showed in Figures 5 and 6, respectively. In Figures 5 and 6, the time step is showed in each bank defaults; for example, in Figure 5, when the time step is 158, then HFCK defaults. After HFCK is removed from the bank system, the topology of the network system is changed after 158 time steps. Figures 5 and 6 show that although the estimation methods are not similar, the results are similar, perhaps due to the low number of banks in the network. However, we note in Figure 4 that the number of defaults increases earlier in the case of the minimum density approach. Considering that the maximum entropy estimation produces a complete network, which lowers systemic risk measures, and that the minimum density approach produces a network that increases systemic risk measures, we can claim that the time variations in the number of defaulting banks in the Kenyan banking system, for the real (nonobserved) network, is around the range provided by the two methods.Figure 4
Time variations in number of defaulting banks in Kenyan banking system with the network estimated by maximum entropy estimation method and the minimum density approach in the year 2015 with parameterc=10.Figure 5
The evolution of the network topology in 2015 estimated by maximum entropy; the size of each node represents the total percentage of in-strength and out-strength of each bank and the number of size is marked beside each node.Figure 6
The evolution of the network topology in 2015 estimated by the minimum density approach with parameterc=10.


### 4.3. Stress Testing

Since the effect of the topology of the bank network system estimated by two methods is not relevant, we conduct a stress test to confirm the strength of the Kenyan banking system with the minimum density approach in 2009, 2011, and 2015, which would provide higher systemic risk measures (more conservative). Our test is somewhat different from typical macro stress tests, which first remove a bank from the Kenyan banking system and then find how many banks defaults the removed bank can cause, namely, contagious defaults. The results of stress test are listed in Tables2, 3, and 4 as follows. Table 2 shows that in the Kenyan banking system the KCB bank defaulting can result in four defaults banks (namely, contagious default), because the KCB bank is the highest interconnectedness and its total assets and interbank assets are the largest.Table 2
Results of stress test in 2009; CDs number: number of contagious defaults; IC: interconnectedness: the total strength, namely, sum of in-strength and out-strength; TA: total assets; IA: interbank assets (loans and advances to banks). Total assets are measured at market value, whereas interbank assets are measured at book value. We use the letters s1, s2, s3, s4, s5, s6, s7, and s8 to stand for Barclays Bank, Coop, Diamond Trust, Equity Bank, HFCK, KCB, NBK, and NIC, respectively. The unit of currency is Shs.
Number
Bank name
Collapsed banks
CDs number
IC
TA
IA
S1
Barclays Bank
S1, S6, S7, S8
3
0.0518
155,151,000
1061000
S2
Coop
S1, S2, S6, S7, S8
4
0.2198
110531373
4642338
S3
Diamond Trust
S1, S3, S6, S7, S8
4
0.2713
47146767
5638340
S4
Equity Bank
S1, S4, S6, S7, S8
4
0.3821
96512000
2022000
S5
HFCK
S1, S5, S6, S7, S8
4
0.1624
18280761
2106419
S6
KCB
S1, S2, S3, S6, S7, S8
5
0.6858
172384128
5936128
S7
NBK
S1, S6, S7, S8
3
0.0445
51404408
1154271
S8
NIC
S1, S6, S7, S8
3
0.1824
47558241
4936616Table 3
Results of stress test in 2011.
Number
Bank name
Collapsed banks
CDs Number
IC
TA
IA
S1
Barclays Bank
S1, S2, S5, S7
3
0.0227
166269000
913000
S2
Coop
S2, S5, S7
2
0.2137
167772390
7437716
S3
Diamond Trust
S2, S3, S5, S7
3
0.2473
77453024
9452751
S4
Equity Bank
S2, S3, S4, S5, S7
4
0.4791
176911000
1094000
S5
HFCK
S2, S5, S7
2
0.1242
31972113
4724183
S6
KCB
S2, S3, S5, S6, S7, S8
5
0.7407
282493553
17648880
S7
NBK
S2, S5, S7
2
0.0741
5564998
3388191
S8
NIC
S2, S5, S7, S8
3
0.0981
73581321
4486475Table 4
Results of stress test in 2015.
Number
Bank name
Collapsed banks
CDs number
IC
TA
IA
S1
Barclays Bank
S1, S3, S5, S6, S7, S8
5
0.009
241152698
252867
S2
Coop
S2, S3, S5, S6, S7, S8
5
0.3109
339549808
13869273
S3
Diamond Trust
S3, S5, S6, S7, S8
4
0.2843
190947903
4973737
S4
Equity Bank
S3, S4, S5, S6, S7, S8
5
0.3004
341329318
16554308
S5
HFCK
S3, S5, S6, S7, S8
4
0.0956
68808654
5517670
S6
KCB
S3, S4, S5, S6, S7, S8
5
0.5193
467741173
9254721
S7
NBK
S3, S5, S6, S7, S8
4
0.1976
117789712
1850368
S8
NIC
S1, S3, S5, S6, S7, S8
5
0.2828
156762225
5464120In 2011, Table3 shows the relationship between the number of contagious defaults and interconnectedness, the total assets, and the interbank assets. With high interconnectedness, the total assets, and the interbank assets, the default banks cause more contagious defaults banks, for example, Equity Bank and KCB bank, which cause the number of contagious defaults to be 4 and 5, respectively. In 2015, the Kenyan banking system is more unstable seen from Table 4, because any bank defaults can cause more than 4 banks to have contagious defaults.The stress tests results indicate the number of contagious defaults caused by a SIB’s (systematically important bank) default. In general, the banks that trigger over four contagious defaults have significantly more interbank exposures as well as greater interconnectedness measured in terms of strength than the other banks do. In contrast, the banks that trigger less contagious defaults do not necessarily have more interbank exposures or greater interconnectedness compared to SIB’s banks. As per the above results, we do not have any bank that has triggered five contagious results, only KCB. Therefore, we found the KCB is the systematically important bank in the Kenyan banking system.


## 4.1. Network Analysis and Estimation of Bilateral Exposures Matrix

We estimate the bilateral exposures matrixX stated in (1) and use the matrix to examine the network structure of the Kenyan interbank market. We investigate the global interbank network using network centrality measures. Usually the degree of a node is considered as a proxy variable for interconnectedness and explains the number of edges connected to a node. In the present paper, we define the in-strength that shows the ratio of the money lent to all the other banks to the total money. For simplicity, the in-strength of bank i is ai/∑ai. Similarly, the out-strength of bank i is li/∑li. The total strength of a bank is the summation of its in-strength and out-strength. These measures, hence, give a sense of investment and funding diversifications. Figure 2 highlights the time variations of the in-strength (red line) and out-strength (blue line) for eight listed banks. The bigger the increase in the in-strength, the more the debtors the bank would have. In contrast, the bigger the increase in the out-strength, the more the creditors the bank would have. Therefore, in terms of contagious default, the out-strength is more important than the in-strength. Figure 2 depicts a group of banks that borrow more than they lend and others that lend more than they borrow. Banks like NIC Bank, Diamond Trust Bank, National Bank of Kenya, Cooperative Bank, and Barclays Bank lend more than they borrow to other banks by averages of 55.3%, 53.3%, 46.2%, 38.9%, and 35.8%, respectively, from 2008 to 2015.Figure 2
Time variations of in-strength and out-strength for 8 banks. The total strength is the sum of the in-strength (red line) and the out-strength (blue line). The vertical axis (the strength) shows the percentage of the amount of money borrowed or lent by a bank while horizontal axis represents time in years.Meanwhile, banks like Kenya Commercial Bank and Equity Bank lend more than they borrow by an average of 24.3% and 19.2%, respectively, from 2008 to 2015. Therefore, we can examine which banks borrow (lend) more than they lend (borrow) in the Kenyan network in terms of percentage. Following is a breakdown of the percentage figures that these banks borrow in order of magnitude and they include Kenya Commercial Bank (24.3%) and Equity Bank (19.2%). The list of banks that lends includes NIC Bank (55.3%), Diamond Trust Bank (53.3%), HFCK Bank (49.5%), National Bank of Kenya (46.2%), Cooperative Bank (38.9%), and Barclays Bank (35.8%).


## 4.2. Bank Default Analysis

We estimate the theoretical number of defaulting banks during the estimation period of 2009–2015, which is presented in Figure3. Figure 3 indicates the time variations of the number of defaulting banks in the Kenyan banking industry with the complete network that is estimated by the Maximum Entropy Method. It basically shows that in 2015 the Kenyan banking system is more unstable than other years. The years in 2013 and 2010 are more stable than other years, because no banks defaulted.Figure 3
Time variations in number of defaulting banks in Kenyan banking system with the network estimated by maximum entropy estimation method.There has also been a banking crisis in Kenya since 2015 mostly due to weak supervision and outright fraud by bank directors. An example of this is the nonlisted Chase Bank which was put under receivership in the same year. Several other banks including National bank also showed signs of collapsing due to the same. Analysts have been warning banks since 2012 to stop understating loan provisions and to increase their capitalization. A lot of work is still needed especially with the regulators including the CMA (Capital Market Authority) and Kenyan banking sector poses a challenge of lack of trust in the banking industry as most clients move to other rudimentary means of saving their money.Since the Kenyan banking system in the year 2015 is most unstable, we compare the effect of network structure on the time variations in the number of defaulting banks in 2015, which is presented in Figure4. From Figure 4, we can see that the network topology estimation methods cause not much effect on the time variations in the number of defaulting banks. Next, we compare the evolution of the network topology, which is showed in Figures 5 and 6, respectively. In Figures 5 and 6, the time step is showed in each bank defaults; for example, in Figure 5, when the time step is 158, then HFCK defaults. After HFCK is removed from the bank system, the topology of the network system is changed after 158 time steps. Figures 5 and 6 show that although the estimation methods are not similar, the results are similar, perhaps due to the low number of banks in the network. However, we note in Figure 4 that the number of defaults increases earlier in the case of the minimum density approach. Considering that the maximum entropy estimation produces a complete network, which lowers systemic risk measures, and that the minimum density approach produces a network that increases systemic risk measures, we can claim that the time variations in the number of defaulting banks in the Kenyan banking system, for the real (nonobserved) network, is around the range provided by the two methods.Figure 4
Time variations in number of defaulting banks in Kenyan banking system with the network estimated by maximum entropy estimation method and the minimum density approach in the year 2015 with parameterc=10.Figure 5
The evolution of the network topology in 2015 estimated by maximum entropy; the size of each node represents the total percentage of in-strength and out-strength of each bank and the number of size is marked beside each node.Figure 6
The evolution of the network topology in 2015 estimated by the minimum density approach with parameterc=10.


## 4.3. Stress Testing

Since the effect of the topology of the bank network system estimated by two methods is not relevant, we conduct a stress test to confirm the strength of the Kenyan banking system with the minimum density approach in 2009, 2011, and 2015, which would provide higher systemic risk measures (more conservative). Our test is somewhat different from typical macro stress tests, which first remove a bank from the Kenyan banking system and then find how many banks defaults the removed bank can cause, namely, contagious defaults. The results of stress test are listed in Tables2, 3, and 4 as follows. Table 2 shows that in the Kenyan banking system the KCB bank defaulting can result in four defaults banks (namely, contagious default), because the KCB bank is the highest interconnectedness and its total assets and interbank assets are the largest.Table 2
Results of stress test in 2009; CDs number: number of contagious defaults; IC: interconnectedness: the total strength, namely, sum of in-strength and out-strength; TA: total assets; IA: interbank assets (loans and advances to banks). Total assets are measured at market value, whereas interbank assets are measured at book value. We use the letters s1, s2, s3, s4, s5, s6, s7, and s8 to stand for Barclays Bank, Coop, Diamond Trust, Equity Bank, HFCK, KCB, NBK, and NIC, respectively. The unit of currency is Shs.
Number
Bank name
Collapsed banks
CDs number
IC
TA
IA
S1
Barclays Bank
S1, S6, S7, S8
3
0.0518
155,151,000
1061000
S2
Coop
S1, S2, S6, S7, S8
4
0.2198
110531373
4642338
S3
Diamond Trust
S1, S3, S6, S7, S8
4
0.2713
47146767
5638340
S4
Equity Bank
S1, S4, S6, S7, S8
4
0.3821
96512000
2022000
S5
HFCK
S1, S5, S6, S7, S8
4
0.1624
18280761
2106419
S6
KCB
S1, S2, S3, S6, S7, S8
5
0.6858
172384128
5936128
S7
NBK
S1, S6, S7, S8
3
0.0445
51404408
1154271
S8
NIC
S1, S6, S7, S8
3
0.1824
47558241
4936616Table 3
Results of stress test in 2011.
Number
Bank name
Collapsed banks
CDs Number
IC
TA
IA
S1
Barclays Bank
S1, S2, S5, S7
3
0.0227
166269000
913000
S2
Coop
S2, S5, S7
2
0.2137
167772390
7437716
S3
Diamond Trust
S2, S3, S5, S7
3
0.2473
77453024
9452751
S4
Equity Bank
S2, S3, S4, S5, S7
4
0.4791
176911000
1094000
S5
HFCK
S2, S5, S7
2
0.1242
31972113
4724183
S6
KCB
S2, S3, S5, S6, S7, S8
5
0.7407
282493553
17648880
S7
NBK
S2, S5, S7
2
0.0741
5564998
3388191
S8
NIC
S2, S5, S7, S8
3
0.0981
73581321
4486475Table 4
Results of stress test in 2015.
Number
Bank name
Collapsed banks
CDs number
IC
TA
IA
S1
Barclays Bank
S1, S3, S5, S6, S7, S8
5
0.009
241152698
252867
S2
Coop
S2, S3, S5, S6, S7, S8
5
0.3109
339549808
13869273
S3
Diamond Trust
S3, S5, S6, S7, S8
4
0.2843
190947903
4973737
S4
Equity Bank
S3, S4, S5, S6, S7, S8
5
0.3004
341329318
16554308
S5
HFCK
S3, S5, S6, S7, S8
4
0.0956
68808654
5517670
S6
KCB
S3, S4, S5, S6, S7, S8
5
0.5193
467741173
9254721
S7
NBK
S3, S5, S6, S7, S8
4
0.1976
117789712
1850368
S8
NIC
S1, S3, S5, S6, S7, S8
5
0.2828
156762225
5464120In 2011, Table3 shows the relationship between the number of contagious defaults and interconnectedness, the total assets, and the interbank assets. With high interconnectedness, the total assets, and the interbank assets, the default banks cause more contagious defaults banks, for example, Equity Bank and KCB bank, which cause the number of contagious defaults to be 4 and 5, respectively. In 2015, the Kenyan banking system is more unstable seen from Table 4, because any bank defaults can cause more than 4 banks to have contagious defaults.The stress tests results indicate the number of contagious defaults caused by a SIB’s (systematically important bank) default. In general, the banks that trigger over four contagious defaults have significantly more interbank exposures as well as greater interconnectedness measured in terms of strength than the other banks do. In contrast, the banks that trigger less contagious defaults do not necessarily have more interbank exposures or greater interconnectedness compared to SIB’s banks. As per the above results, we do not have any bank that has triggered five contagious results, only KCB. Therefore, we found the KCB is the systematically important bank in the Kenyan banking system.


## 5. Conclusion

The present paper proposed a theoretical framework to find the time evolution of the systemic risk by calculating the number of defaults of banks using sequences of daily financial data. The framework combines the asset value estimation algorithm [26, 27], maximum entropy estimation method [14], the minimum density approach [28], and obligation clearing algorithm [25], effortlessly to deal with the dynamic problem—the time evolution of the systemic risk. The asset value estimation algorithm is used to approximate the asset values of the banks at each day which are required to calculate the time evolution of systemic risk. The obligation clearing algorithm is used to calculate the systemic risk given the tuples of data on a daily basis.In the present paper, we evaluated the systemic risk of the Kenyan banking system using the theoretical framework proposed. The Kenyan interbank market involves various domestic contracts and transactions. First, we clarified the network structure of the Kenyan interbank market and theoretically analyzed its network structure using the estimated bilateral exposures matrix. We also analyzed the interconnectedness of each bank in the Kenyan interbank market using the in- and out-strength measure. Significantly, we found that the banks designated as systematically important banks (SIBs) play a central role in the Kenyan interbank market and these are Kenya Commercial Bank (KCB) and Equity Bank.We modeled contagious defaults in the Kenyan interbank network using real aggregate banking data from the portal of African Markets and theoretically analyzed the mechanism of contagious defaults conditional on a basic default during a seven-year period (2009–2015).Further analysis theoretically showed the occurrence of some contagious defaults in 2009, 2011, and 2015, and these years are very unstable than other years. We also conducted a stress test and analyzed the likelihood of contagious defaults conditional on a bank’s basic default at an evaluation time point in the future. Some banks designated as SIBs were confirmed to have the potential to trigger the contagious defaults of other banks. In general, the banks that trigger over more contagious defaults have significantly more interbank exposures as well as greater interconnectedness measured in terms of strength than the other banks do. We found that the KCB is the most systematically important bank in the Kenyan banking system.To finalize, we are convinced that, in order to uphold the stability of the Kenyan banking system, there is a need to apply systemic risk assessment practices. These could also be useful in the execution of bank-internal systemic stress tests of default contagion.



---

*Source: 8767836-2018-01-23.xml*