Banks | The Case of Australia’s Big
Four Banks
Xincen Xie
Master’s Thesis to obtain the degree in Actuarial Science and Mathematical Finance University of Amsterdam
Faculty of Economics and Business Amsterdam School of Economics
Author: Xincen Xie Student nr: 11384476
Email: xincen.xiie@gmail.com Date: August 15, 2017
Supervisor: dhr. Z. (Merrick) Li MPhil Second reader: Prof. dr. Roger J. A. Laeven
Statement of Originality
This document is written by Student Xincen Xie who declares to take full responsibility for the contents of this document. I declare that the text and the work presented in this document is original and that no sources other than those mentioned in the text and its references have been used in creating it. The Faculty of Economics and Business is responsible solely for the supervision of completion of the work, not for the contents.
iv Xincen Xie — Systemic Capital Requirement for Banks
Abstract
In the aftermath of the global financial crisis, it is of great interests in resetting the capital requirements regarding the systemic risks. This thesis uses the structural credit risk model to simulate the system loss distributions incorporating two systemic risk factors, the contagion ef-fect and correlation efef-fect. Having set up the systemic risk constraints, two approaches are proposed to obtain the optimal capital requirement for banks in aggregate and individual level.
Keywords Banking, Systemic risk, Capital requirement, Interbank liability network, Bayesian methodology, Gibbs sampling, Credit risk model.
1 Introduction 1
2 Literature Review 4
3 Methodology 6
3.1 Interbank Liability Estimation . . . 7
3.1.1 Bayesian Basic Model . . . 8
3.1.2 Gibbs Sampling. . . 9
3.1.3 Components Update . . . 10
3.2 Credit Risk Modeling . . . 11
3.2.1 Contagion Effect . . . 13
3.2.2 Correlation Effect . . . 14
3.2.3 System Loss Simulation . . . 14
3.3 Optimal Capital Requirement . . . 16
3.3.1 First Approach . . . 16
3.3.2 Second Approach . . . 17
4 Empirical Results 19 4.1 Data Analysis . . . 19
4.1.1 Big Four Banks in Australia. . . 19
4.1.2 Interbank Deposits and Loans. . . 20
4.1.3 Merton Model Calibration . . . 22
4.1.4 Optimization Procedure . . . 23
4.2 Result Reporting . . . 24
4.2.1 Interbank Liability Network . . . 24
4.2.2 Estimated Asset Value . . . 25
4.2.3 Simulated System Loss Distribution . . . 26
4.2.4 Capital Surcharges . . . 28
5 Conclusion 31
Appendix A: MCMC Sampling 33
Appendix B: Interbank Liability Network 35 v
vi Xincen Xie — Systemic Capital Requirement for Banks
Appendix C: Correlation Matrices Between Asset Values 39
Introduction
Systemic risk, referred to the contagious default events across the financial system, has been emphasized again in the aftermath of the global financial crisis in 2007-2009. This global crisis first started with the collapse in the housing market and soon spread into the banking sector and other financial institutions through their mortgages and other securities. The intensification of interlinkages in the banking sector expanded the scope of initial shocks and amplified the magnitude of losses. At the end, a full-blown world-wide financial crisis was triggered by a string of banks’ failures, e.g., Lehman Brothers, Merrill Lynch, AIG, Freddie Mac, etc. Undoubtedly, the financial crisis of 2007-2009 has pointed out the fact that the regulators have underestimated the systemic risk and its potential to destabilize the banking system. In particular, two important channels of contagions are left unaccounted for by the capital requirements for banks. The intercon-nectedness between banks, stemming from interbank activities such as interbank loans, acts as a channel for fund collecting as well as for risk transmission. The second channel is the commonality of assets held by banks. The commonality of asset holdings, which arises from syndicated loans as well as fire sales externalities, indicates the potential dependencies between banks in distress and therefore contributes to systemic risk.
Apparently, regulating the banking system from microprudential perspective is not nearly enough. To ensure the financial stability, systemic risk should be considered in capital requirement settings. Various studies have been conducted to quantify the systemic risk into capital requirements. Saunders and Allen (2010) point out that the capital requirements for banks should be tailored individually, depending on their contri-bution to the systemic risk. However, the implications of theoretic models and practical goals of regulators are sometimes mismatched. Regulators are usually more interested in maximizing the efficiency in the economic system, whereas scholars are more concerned about the financial stability of the system and ignoring the difficulties of the imple-mentations. Some researchers suggest that the capital requirement should be increased extensively since the mix of debt and equity is costless to the companies (Modigliani and Miller,1958) and by doing so, the default risk will be reduced sharply (Admati and Hell-wig,2014). However, in fact, higher capital requirements could also lead to an increase
2 Xincen Xie — Systemic Capital Requirement for Banks
in bank debt and further raise the cost of capital. As a result, the economic activities will be dampened and the economic growth will be slowed down (Cline, 2015). Hence, higher capital requirements are not always beneficial to society. The trade-off between the efficiency and risk taking has challenged banking regulators. From a mathematical perspective, this can indeed be interpreted as a constraint optimization problem. The banking regulators are seeking for a level of capital requirements to minimize the prob-ability of default, whilst leaving some rooms to banks to operate their normal economic activities in an efficient way.
The primary focus of this paper is to determine the optimal capital requirements for the big four banks in Australia: National Australia Bank (NAB), Commonwealth Bank (CBA), Australia and New Zealand Banking Group (ANZ) and Westpac (WBC). Based on the public data, together with the structural credit risk model, this thesis attempts to incorporate systemic risk by quantifying two major systemic risk drivers: (i) the correlation between banks’ assets which generates the co-movements of banks in distress and (ii) the interconnectedness between banks that creates the potential con-tagion risk. The Australian banks are chosen for two reasons. First, according to the Financial Stability Review of Australia 2006, the banking sector plays a fundamental role in the Australian financial system by holding the majority financial assets includ-ing but not limit to business bankinclud-ing, stockbrokinclud-ing, insurance and tradinclud-ing in financial markets. Second, the banking sector in Australia is highly concentrated with four major banks taking the leading role in many aspects. Such concentration creates risks to its financial system and make it a good example to be discussed as the Australian financial system can be effectively capitalized focusing only on these four banks.
To determine the optimal systemic capital requirement, first of all, this thesis em-ploys two quantitative techniques to quantify the aforementioned two risk drivers. The first technique is used to estimate the interbank linkage. Although in principle, the ac-tual bilateral interbank lending data is the most desirable to assess the systemic risk, it cannot be accessed in public for security reasons. Fortunately, the aggregate interbank liabilities and assets can be extracted from the public data. To fill in the huge gaps between the aggregate data and the bilateral data, this thesis employs the Bayesian methodology developed byGandy and Veraart(2016) to simulate the interbank lending market for systemic risk assessment. Having the information on bilateral interbank lia-bilities, the interbank exposures will be cleared to justify the contagion risk. To do so, the clearing algorithm proposed by Eisenberg and Noe (2001) is applied to markdown the assets of defaulting banks. Next to it, to compute the correlation between holding assets, this paper closely follows the algorithm proposed by Elsinger et al. (2006) to calibrate the asset value regarding the correlation effect. After these two risk drivers are weighted in the credit risk model, the systemic loss distribution will be simulated and the percentile of the system loss distribution will be identified to achieve the given
systemic risk constraints. This thesis uses two different approaches to determine the optimal capital requirement in aggregate and individual level. The first approach, in-troduced byWebber and Willison(2011), aims at the optimal capital level subject to a given risk level. The second approach, adopting from Gauthier et al.(2010), focuses on the optimal distribution of capital across banks such that the minimum level of systemic risk can be achieved given the aggregate capital level.
Relying on a simplified assumption, this thesis proves that these two approaches in-duce the same capital allocation. After the optimal capital level is obtained, this thesis compares it with the holding capital of individual banks. It is found that CBA is the only under capitalized bank in Australia Big Four and at least 10 billion AUD short in capital to achieve the desired system risk level. WBC is the most over capitalized bank. ANZ and NAB are the most efficient banks in the sense that the capital they hold are well just above the requirement.
This thesis makes three main contributions. First, it incorporates two systemic risk drivers into credit risk modeling. The original Merton’s credit risk model does not take into account of systemic risk. As the financial crisis in 2007-2009 has clearly demon-strated the importance of systemic risk in stabilizing financial systems, this thesis models the correlation effect and simulates the bilateral interbank liability so that the systemic risk can be mapped and integrated into Merton model. Second, it illustrates two differ-ent optimization strategies that can be used to estimate the optimal capital requiremdiffer-ent from different perspectives. In this case, the banking regulators are assumed to be inter-ested in meeting the systemic risk requirements with the minimum capital. Alternatively, they target at minimizing the systemic risk with given level of capital. It is proven that these two approaches lead to the same results though their rationale are quite distant. Third, the resulting systemic capital requirement for Australian Big Four can serve as a supporting indicator for Australian banking regulators as well as Australian banks by giving them the idea of incorporating the systemic risk into the capital buffers settings for banks.
The structure of this thesis is as follows. Chapter 2 provides a brief literature review on systemic risk assessment and credit risk modeling. Chapter 3 presents the details of methodologies used in this paper including the interbank liability estimation and credit risk model calibration and implementation. Chapter 4 consists of two parts, data analysis and result reporting. Conclusion follows in Chapter 5.
Chapter 2
Literature Review
Systemic risk, as defined in De Bandt and Hartmann (2000), is a systemic event that could severely impair a vast number of financial institutions and hence, affects the finan-cial system in a strong sense. Under this definition, a series of recent papers attempts to measure the systemic risk of banking system. The majority of these approaches are based on Value-at-Risk(VaR) methodology which measures the banks’ contribution to systemic risk individually. For instance, as one of the important extensions of VaR, Brunnermeier et al.(2009) propose using Conditional Value-at-Risk(CVaR) to quantify the extent of banks’ tail risks moving together. More specifically, the α% CVaR is the expected loss that the bank makes, conditional on making losses equal to or greater than the α% VaR. Brunnermeier et al. (2009) aim to protect market participants in the fi-nancial system from bankruptcy by holding enough capitals.Acharya et al.(2017) share the similar idea. Instead of holding enough capital, they suggest that banks can pay a certain amount of premiums which are based on banks’ systemic expected shortfalls. However, they all fail to address how the systemic risks should be mapped into capital requirement except for Tarashev et al.(2010), who use Shapley values to measure the banks’ contribution to the systemic risk and then seek to allocate the the systemic risk fairly across banks.
Another research direction to assess systemic risk is to discuss the amplification mechanisms exist in the financial system. Allen and Gale (1994) show that the asset prices depend on the cash-in-the-market pricing during the crisis and later in2000, they propose a contagion model by introducing the network of interbank exposures. They model the financial contagion as an equilibrium phenomenon, which is financially frag-ile in reality. Next to it, Krishnamurthy(2010) describes two amplification mechanisms involving asset price, balance sheet and investors’ Knightian uncertainty. In account of the amplification caused by interbank linkages, one of the popular approaches is to derive the individual interbank liability by Kullback Leibler(KL) divergence. The KL divergence is used to estimate the interbank exposures of the banks in Germany (Upper and Worms, 2004), Belgium (Degryse et al., 2007) and the United Kingdom (Elsinger et al., 2006). It turns out that the resulting interbank liabilities using KL
method are all complete networks which is very sparse in practice. Moreover,Mistrulli (2011) points out that the KL method underestimates the systemic risk. This thesis employs a Bayesian methodology (Gandy and Veraart,2016) to generate the interbank liabilities. The Bayesian methodology is superior in the sense that it allows the entries in liability matrix to be non-positive, meaning that the interbank lending relationship can be nonexistent between two banks, which is consistent with the empirical study of Craig and Von Peter (2014).
To assess the systemic risk associating to the amplification mechanisms, this the-sis chooses a structural credit risk model, Merton (1974). The Merton model provides tractability by relying on the many simplified assumptions. For instance, in Merton’s framework, the companies are assumed to only default at the maturity date, which is quite unrealistic. The First-passage-time model proposed byBlack and Cox(1976), is a modified version of Merton model with a distant feature that default can occur prior to the maturity data and the recovery payoff resulting from default can be discussed with great flexibility. Moreover, the empirical evidence (Ong, 1999) shows that companies are more likely to default when their asset values reach a certain point between the total liabilities and the short-term debt, which contrasts the definition of default in the Merton model. Kealhofer Merton Vasicek model (KMV) (Kealhofer,2003) improves it by adjusting the face value of the bonds. Besides, researchers have always been skeptical of a constant interest rate that is assumed in Merton model, claiming that it is better to incorporate the stochastic interest rate model and consider the correlation between stochastic components. Though there are more sophisticated credit risk models, this thesis only sheds light on the original structure credit risk model, Merton model, hop-ing that could be helpful to incorporate the amplification mechanism in other extended models as well.
Chapter 3
Methodology
To quantify two channels of systemic credit risks, this thesis combines several quan-titative techniques to generate the interbank exposure and calibrate the asset value. In essence, the optimal capital requirements and the optimal capital allocation depend on the loss distribution of the internal capital. This capital, the difference between the internal assets and liabilities, is influenced in two ways. The first is general asset value changes. The other one is related to interbank liability networks, in which changes in the asset value at another bank, which lead to default, can work through the liability network of banks and affect the asset value of other banks. Given the asset value and interbank liability network, one can simulate the system loss distribution. The optimal capital requirements are set in individual and aggregate level such that the systemic risk constraints are satisfied. In principle, the goal of this paper is to derive the capital requirements for banks that can protect them from bankruptcy whereas maximizing the efficiency of the system. The methodology can be structured in three sections.
In Section 3.1, to capture the interconnectedness existing in the banking system, the interbank lending relationships will be estimated. One can see the interbank lend-ing market as a double-edged sword. On one hand, interbank markets play a crucial role in the financial systems, enabling banks to redistribute the liquidity by borrowing and lending money within the banking system. On the other hand, interbank lending markets are channels through which financial shocks can be transmitted from one bank to another and soon spreading into the whole system. The pattern of interbank lend-ing relationships has been a vividly discussed topic among economists as it determines the potential of contagion through the whole banking system. However, in most of the cases, the the bilateral interbank network is intransparent to the public for the sake of confidentiality, which poses difficulties in modeling the systemic risk. To overcome this issue, the Bayesian methodology proposed byGandy and Veraart (2016) is used in this thesis to construct the interbank liability matrix.
The focus of Section 3.2 is on credit risk modeling. Except for the interconnected-ness, the correlation between banks’ assets has always been another critical driver in
modeling systemic risk as it generates a tendency for creditworthiness of banks to dete-riorate together in the time of recession. The correlation between banks’ assets mainly comes from the common risk exposure, i.e., overlap in the asset portfolio, information contagion, etc. To incorporate the correlation effect in credit risk modeling, this thesis applies the clearing algorithm, introduced byEisenberg and Noe (2001), to capture the contagion risk caused by interconnectedness. Then the system loss distribution will be simulated regarding these two systemic risk drivers.
The optimization strategies are demonstrated in Section 3.3. Two approaches are proposed in the spirit of Webber and Willison (2011) and Gauthier et al. (2010) to determine the optimal capital requirement for the whole banking system as well as the individual banks.
3.1
Interbank Liability Estimation
Gandy and Veraart (2016) describe a network model that generates thev distribution of the liability matrix. Along these lines, their model can be applied to the Australian bank case. Given that there are n banks with indices N = (1, ..., n) in the system, an interbank liability matrix is defined as the n × n matrix L with nonnegative entries Lij,
where i, j ∈ N , indicating that bank i is obligated to pay Lij to bank j. (Elsinger and
Summer, 2013) transforms the problem into a network model, in which nodes can be viewed as banks and the directed edges from node i to node j with weight Lij can be
seen as the liability that bank i has to bank j. By definition, the interbank liability matrix L satisfies
r(L) = l, (3.1)
c(L) = a, (3.2)
where l is the vector of interbank liabilities with li equal to the ith row sum (the total
liabilities of bank i), a is the vector of interbank assets with aj equal to the jth column
sum (the total assets of bank j). Due to the unavailability of bilateral banks’ exposures, the liability matrix L can not be fully observed. So the admissible liability matrix L∗ is introduced, where L∗ij = 0 if i = j ∗ if i 6= j
The diagonal elements are always zero, as one bank does not have liabilities to itself. L∗ij = ∗ means the interbank liability between bank i and bank j is unknown. The total interbank liabilities and assets of bank i = 1, ..., n are regarded as the row sums and the column sums. By definition, the matrix L is called the interbank liability matrix
8 Xincen Xie — Systemic Capital Requirement for Banks
respecting L∗ if ∀i, j ∈ N : L∗ij 6= ∗ implies Lij = L∗ij. Then we set L ≡ L∗.
Depending on the size of the matrix, a great number of degrees of freedom is left when estimating the interbank liability matrix L. For instance, there are 2n linear con-straints are given in the minimal observation setting where only the row sums and columns sums are known. Nonetheless, one of these constraints is redundant to satisfy the condition that the total liabilities has to be equal to the total assets within the system. Diagonals are set to zero for the aforementioned reason, giving another n re-strictions. Hence, there are n2− (2n − 1) − n = n2− 3n + 1 degrees of freedom left in
n × n matrix. This brings up another interesting question: in what circumstances does the admissible liability matrix L∗ exist?
Theorem 2.3 in Gandy and Veraart(2016) provides a necessary condition. Consider two vectors a ∈ [0, ∞)n, l ∈ [0, ∞)n and the admissible matrix L∗ satisfies Pn
i=1ai =
Pn
i=1li, r(L∗) ≤ l and c(L∗) ≤ a. The necessary condition is:
For all I ⊂ N, J ⊂ N, with L∗ij 6= ∗ ∀i, j ∈ N, we have
X i∈I ˜ li+ X j∈J ˜ ai ≤ S (3.3) where ˜l = l − r(L∗), ˜a = a − c(L∗), S =Pn i=1˜li.
This can be transformed into a maximum flow problem. Again the proof is given in the paper by Gandy and Veraart(2016). Therefore under the minimum observation setting, this is equivalent to
ai ≤
X
j6=i
lj ∀i ∈ N (3.4)
When the equal sign holds, it is assumed that the interbank network is as dispersed as possible. This assumption is in line with the idea of Maximum Entropy theorem (Mistrulli,2011) that banks are trying to distribute their lending evenly to the rest.
3.1.1 Bayesian Basic Model
The Bayesian model by Gandy and Veraart (2016) is rather easy to comprehend. Sup-pose that the network model with n vertices is generated in such a way that each possible edge has the same probability of existing. In other words, it is assumed that bank i is indifferent in building interbank linkage with any other banks. Note that it is possible that two banks have no liability to each other, indicating that there is no edge exists between two banks. Therefore, in this case, an adjacency matrix A is introduced as below.
Aij = 1 if Lij > 0 0 otherwise According to Erdos and Renyi(1959),
P(Aij = 1) = p, (3.5)
Lij|(Aij = 1) ∼ Exp(λ), (3.6)
where the directed edges are generated by an Bernoulli distribution with the probability p of success and the attached weight follows an exponential distribution with parameter λ.
As a matter of fact, this basic model can be adjusted into a more generalized version where p and λ are the random matrices. A hierarchical model incorporates the random variables p and λ. A random value p reflects the fact that banks have preferences making interconnections with the rest. Iori et al. (2008) point out that less connected banks have a tendency making contacts with larger banks. Boss et al. (2004) find that the distribution of the degree of interbank networks follow the power laws. While making λ random, it allows us to model the distribution of interbank liabilities, i.e., Poisson distribution. The conjugate distribution model assumes that all components of p and λ are equal but random. The Fitness model (Servedio et al., 2004) is used to model the undirected networks that are unweighted.
As Gandy and Veraart note, the interbank liability matrix L gives the informa-tion about the liability structure and interconnectedness of banks. From this, not only can the (conditional) default probabilities be derived, but also many other informative statistics, for example, the loss given default. Therefore, one can define a function h(L) which transforms the liability matrix into the statistics on the topic that we want to investigate (like default probabilities as a vector outcome of h(L)). The function h(L) is therefore conditional on the row sums, column sums and admissible liability matrix. As L is usually for a great deal unknown and therefore has to be simulated (to create a distribution of feasible liability structures), the function h(L) also becomes stochastic. Unfortunately, the distribution of h(L)|l, a, L∗ does not exist in a closed form. Thus, the Markov Chain Monte Carlo (MCMC) algorithm will be carried out next to make the approximation of the h(L) function. The key idea of MCMC sampling is to estimate the desired expectations by ergodic averages. Plummer et al. (2006) have shown the convergence of MCMC samplers in the software program R.
3.1.2 Gibbs Sampling
Gibbs algorithm is chosen to sample from the conditional distribution for two reasons. Firstly, all the entries in the interbank liability matrix are continuous random variables
10 Xincen Xie — Systemic Capital Requirement for Banks
with a point mass at zero. Secondly, Gibbs sampling allows a certain value to be fixed. In this case, the diagonal values are set to be zero.
The underlying logic of Gibbs sampling is to update one or more independent vari-able(s) iteratively by sampling them from their joint distribution while keeping the remaining variables fixed. To estimate the liability matrix L, this thesis chooses to up-date the random size submatrices such that the new matrix L is still feasible. This can be seen as constructing a Markov chain of which the distribution will converge to the target distribution as the iteration number goes to infinity.
In practice, the first few samples in MCMC runs are usually discarded as the ini-tial transient has no value for determining the equilibrium distribution. Moreover, to remove the autocorrelation, the so called thinning process is applied to make sure that each iteration is an independent sample. Consequently, the point estimates of h(L) can be computed as follows: E[h(L)|l, a] ≈ 1 N N X i=1 h(Liδ+b) (3.7)
where N is the number of samples, b is the burn-in period which represents the dis-carded length of the sample period, δ can be interpreted as the amount of thinning. Thinning reduces the autocorrelation of the samples by only taking every δth sample into account. Plummer et al. (2006) propose to use pilot runs of chains together with the estimates of an effective sample size to determine an appropriate amount of thinning.
3.1.3 Components Update
As mentioned in the previous section, Gibbs sampling is used to update the components of liability matrix L individually or jointly leaving the rest fixed. However, updating individual components is not pragmatic anymore given that it is updated conditional on the column sums and row sums. That is to say, conditioning on the column sums and row sums and other components are equivalent to conditioning on the row sums and column sums of any submatrices keeping the rest fixed. Figure 1 illustrates the minimum submatrix that can be updated. Note that there are three conditions needed to be satisfied in 2 × 2 submatrix estimation. They are row sums, column sums and these two sums have to be equal. That means that given L11 = x, it satisfies that
L22= l2−a1+x = a2−l1+x. Thus, these conditions leave only 1 degree of freedom in 2×2
submatrix. Without the loss of generosity, this thesis decides to update subcomponents jointly by using such k × k submatrices where k ∈ 2, ...n.
Robert(2004) proves that by a finite number of iterations, Gibbs sampler will even-tually converge to the correct invariance distribution. So the Gibbs sampler is used in
Figure 1: The left figure represents any 2 × 2 submatrix in n × n interbank liability matrix L. Suppose that the row sums and column sums are known vectors l and a, where the row sum and column sum are equal (a1+ a2= l1+ l2). Letting L11be the free parameter such that L11= x, one can easily see that the rest entries
are fixed as shown in the right figure, where L12= l1− x, L21= a1− x, L22= l2− a1+ x = a2− l1+ x.
this thesis to construct the interbank liability matrix works as follows:
• Sample a random matrix L following Erdo and Renyi model such that the degree existence probability equals to average degree distribution of unconditional model.
• Use maximum flow algorithm (Connen et al., 1990) to construct the matrix L which satisfies equation 3.1 and 3.2.
• Derive a new matrix Lnew again satisfying equation 3.1 and 3.2 by selecting a
submatrix and replacing the corresponding values of L with conditionally sampled new values and by leaving the remaining values untouched.
• Repeat step 3 until L converges. Return L.
Theoretically, there are numerous submatrices can be selected which leaves a great flexibility. The submatrix selection can have a great influence on the efficiency of MCMC sampler. To improve the efficiency, as Gandy and Veraart (2016) suggest in their pa-per, a submatrix of size k is chosen to update with the probability 2n−12n−k−1 for k ∈ 2, ..., n.
Appendix A provides the Gibbs updates in details conditioning on row and column sums.
3.2
Credit Risk Modeling
To model the credit risk, this thesis chooses the Merton structural credit risk model (Merton,1974) for two reasons. Firstly, this model can incorporate two systemic risks, which are created by the correlation and interconnectedness between banks respectively. Secondly, this model uses public available data only such as balance sheet data and market data, which is quite handy. The rationale of the Merton model is simple. The company’s equity can be viewed as a call option on its assets. In this way, Black-Scholes option pricing model can be applied to estimate the value of its equity. In other words, the equity holders are offered a call option on the company’s assets with the strike price K. If the asset value A is larger than the strike price K, equity holders will receive the difference between these two. Otherwise, equity holders get nothing. Analogously,
12 Xincen Xie — Systemic Capital Requirement for Banks
bondholders can be regarded as who short a put option. The Bondholders can gain value of K when asset value A is larger than K and receive the asset value at the maturity date Va otherwise.
The Merton model (1974) is as follows:
E = A · N (d1) − B · e−rTN (d2) (3.8) d1= logAB + (r +12σA2)T σA √ T (3.9) d2 = d1− σA √ T (3.10)
where the equity E, debt B and risk-free rate r can be obtained from the market data. The function N denotes the cumulative distribution function of the standard normal random variable. σA is the asset volatility. This equation can be easily rewritten as a
function of asset value A with the independent variable σA. It is assumed that the asset
value at time t follows geometric Brownian motion:
dAi,t= Ai,tµidt + Ai,tσidWi,tP for i = 1, 2, ..., n (3.11)
Recall the Itˆo’s lemma:
df = ∂f ∂tdt + ∂f ∂xdx + 1 2 ∂2f ∂x2dx 2 (3.12)
So the log asset diffusion is
d(log Ai,t) = ∂(lnA∂ti,t)dt +∂(lnA∂Ai,ti,t)dAi,t+12∂
2(lnA i,t) ∂A2 i,t dA2i,t = (µi− 1 2σ 2 i)dt + σidWi,tP, (3.13)
which leads to the log-normal distribution of asset returns:
xi,t = log( Ai,t+∆t Ai,t ) ∼ N (µi− 1 2σ 2 i)∆t, σi2∆t . (3.14)
Hence, the mean of such return can be estimated as:
ˆ µi=
E[log Ai,t+∆t] − E[log Ai,t]
∆t +
1 2σ
2
i. (3.15)
The likelihood function L = QT
t=1f (xi,t) can be derived from here as in Elsinger
et al. (2006), where f (xi,t) is the density function of the Normal distribution. So the
likelihood function is:
Li= (2π)− T 2(σ2∆t)− T 2exp − 1 2σ2∆t T X t=1 xi,t− (µ − 1 2σ 2)∆t2 . (3.16)
The log likelihood function is:
li= log Li = − 1 2σ2 i∆t T X t=1 xi,t− (µ − 1 2σ 2)∆t2−T 2 log(σ 2 i∆t) − T 2 log(2π). (3.17)
Given that the asset volatility is unobserved and unpredicted, this thesis calibrates the asset value as below, strictly following the paper by Webber and Willison(2011).
• Choose an initial value of asset volatility σi exogenously.
• Use the Merton model in equation 3.8-3.10, calculate the asset value Ai,t for bank
i at time t.
• Obtain the estimate of asset drift rate ˜µi from equation 3.15.
• Compute the log likelihood function in equation 3.17 based on σi, Ai,t and ˜µi
• Search for the best ˜σisuch that log likelihood function l( ˜σi) reaches the maximum
value.
• If optimal asset volatility ˜σiis equal to the estimated σicalculated from estimated
Ai,t, algorithm stops.
Else, go back to step 1.
3.2.1 Contagion Effect
To track down the contagion effect generated by interbank lending relationships, this thesis employs the clearing algorithm which is first developed by Eisenberg and Noe (2001) and further adjusted by Rogers and Veraart (2013) by introducing the default costs. Recall the interbank liability matrix L. Suppose that there are n banks in the system. Each bank is characterized by the given nominal liability Lij and the net asset
ei. By definition, default happens when ei ≤ di where di = Pj∈NLij. Define a new
matrix π such that
πij = Lij di if di> 0 0 otherwise
The clearing vector R∗ is defined in the paper byRogers and Veraart (2013) such that
R∗ = Φ(R∗), (3.18)
where the function Φ is :
Φ(Li) = di ifPNj=1πijLj+ ei ≥ di αei+ β PN j=1πijLj if di > PN j=1πijLj+ ei ≥ 0 0 ifPN j=1πijLj+ ei < 0
where, α and β are the constants between 0 and 1, representing the default costs, ei is
the exogenous operating cash flow, which is defined in this thesis as total asset minus the total liability of bank i.
The clearing vector R∗ can be viewed as the total money bank i has available to pay out the debts to other banks. The asset value of bank i is the sum of interbank liabilities
14 Xincen Xie — Systemic Capital Requirement for Banks
PN
j=1πij plus the net asset ei. If the value of asset is larger or equal to the value of
liability, bank i pays out exactly the same amount as the liability to meet its obligation. If the value of the liability outweighs the value of asset, the bank defaults automatically. In this case, the asset must be called in and unfortunately only a fraction of the net asset α × ei and a fraction of the interbank asset β ×PNj=1πijLj can be recovered.
Usually, α is observed to be a very small number in the sense that banks are reluctant to sell their investment portfolios. β, on the contrary, is expected to be close to 1 in the sense that the default bank would try to take back as much interbank loans as possible to repay the full amount of debts (if applicable).
Based on the idea of Rogers and Veraart (2013), an iterative algorithm is used to determine the clearing vector R∗. The algorithm works as follows:
• Set R∗ = L, where L is the interbank liability matrix constructed in Section 3.1), calculate the net worth Vi of each bank i where Vi = ei+
PN
j=1πijLj− L∗i.
• If Vi≤ 0, then replace Lij with θ ∗ Lij, where θ =
αei+βPNj=1πijLj
Li .
return to step 1. Else, return Lij
3.2.2 Correlation Effect
The correlation between banks rising from the common risk exposure in the banking sector increases the likelihood of systemic banking crises in the event of simultaneous defaults. To introduce the asset correlation into Merton model, Webber and Willison (2011) provide the iteration procedure. Details are as follows:
For each bank i = 1, 2, ..., n
• Calculate the variance-covariance matrix P between banks’ asset returns from Ai,t over period t ∈ [0, T ], where Ai,t is computed in section 3.2.
• Recompute the asset values for each bank using new variances σi = pdiagi(P)
(by equation 3.8-3.10).
3.2.3 System Loss Simulation
To better understand the systemic effects (contagion and correlation effects) on system loss, Figure 2 depicts an example of the balance sheet model of three banks. In the model, the balance sheet comprises assets, debt and (share holder) capital, denoted by A, D and C respectively. The correlated assets are marked in gray, meaning that these assets are moving towards the same direction. The colored areas represent the interbank assets and liabilities among three banks. This model demonstrates the contagion risk stemming from the interbank linkage in the banking system. If bank 1 defaults and
unable to repay the full amount of its debt back to bank 2, a portion of assets of bank 2 hence vanishes and there is a chance that bank 2 also defaults. The default of bank 2 could result in the loss in asset of bank 3 and therefore bank 3 is in danger as well. That is how the losses get transferred from one to another and eventually the default cascades is triggered.
Figure 2 illustrates an example of the balance sheet model within three banks. Assume that the balance sheet consists of asset (A), debt (D) and capital (C). Part of the assets held by three banks are correlated (in gray) due to the common risk exposures. The interbank linkage arising from interbank lending relationship are marked in corresponding colors at bilateral level. For instance, bank 1 is in debt to bank 2 (in orange) while there exists a payment that is due from bank 2 to bank 3 (in blue). Bank 3, as the creditor to bank 2, is also the debtor to bank 1 (in purple). Therefore, a default of bank 1 could be the cause of the failure of bank 2, which leads to the possible bankruptcy of bank 3. That is how the loss can be amplified through their interconnections.
The system loss is defined as the aggregate asset shortfalls below the aggregate debt liabilities. Taking into account of systemic risks, the system loss distribution is shifted to the left and has fatter tails compare to the original as in Figure 3.
Not surprisingly, the system (aggregate) loss is smaller than the total loss (sum of losses of each bank) due to the existence of interbank linkage. Hence, as Figure 3 suggests, the optimal capital has to be adjusted such that the tail of the system loss distribution is equal to the chosen target, which will be discussed later in Section 3.3. Webber and Willison (2011) provide the idea of system loss simulation as follows:
• Simulate forward the correlated asset value distribution for each bank by using the Cholesky decomposition of the correlation matrix. The asset value is computed in
16 Xincen Xie — Systemic Capital Requirement for Banks
Figure 3: The aggregate (system) loss is defined as the aggregate asset shortfalls below the aggregate debt liabilities. The blue line represents the total losses excluding the systemic risk, which is the sum of the losses of individual banks in the system. The red line displays the aggregate loss with regards to the systemic risk. In this case, the capital, as the difference between the aggregate asset and liability, has to be calibrated to meet the chosen target.
Section 3.2.2.
• For bank i = 1, 2, ..., n, calculate the distribution of asset shortfalls below the promised debt liabilities (without contagion).
• Use network clearing presented in Section 3.2.1 until there is no further rounds of contagious defaults. Mark down the assets of any default banks from the value reached under the diffusion process from equation 3.11.
• Identify the zth percentile of system loss distribution.
3.3
Optimal Capital Requirement
The previous sections have shown that system-wide losses can far exceed the initial shocks due to the inherent vulnerability of the banking system. The primary focus of this thesis is to derive the systemic capital requirements for banks taking into account of the correlation of banks’ assets and the interlinkages between banks. This section will describe two system-wide risk management approaches. The first approach can be viewed as a constraint optimization problem where the policymakers are interested in the minimum capital satisfying the given systemic risk constraints. The second approach seeks for the optimal distribution of capital across banks such that the systemic risk is minimized. In principle, the policymakers aim to maximize the efficiency at the minimum cots of capital.
3.3.1 First Approach
This approach relies on the optimization strategy mentioned in the paper by Webber and Willison(2011). The idea of this approach is to derive the optimal capital for banks in aggregate and individual level. The optimization strategy can be split up into two steps. First, the optimal aggregate capital level is derived for the banking system as a whole subject to the given systemic risk level. Second, the system-wide risk is minimized
by distributing capitals to individual banks depending on their balance sheet structure. This approach is developed based on the assumption that the efficiency of the whole banking system is a decreasing function of total capital, which is the sum of the capital held by every individual bank. Recall that the goal of optimization is to mitigate the risk subject to the minimum costs. So the problem can be formulated as:
min n X i Ci s.t. P r( X i Ai(Ci) < X i Di) = 1 − z (3.19)
where Aiis the asset value of bank i, Ci is the capital of bank i, Diis the liability of bank
i. The systemic risk constraint can also be expressed by V arsystemz (Ci
i=1,...,n) = 0,
meaning that z% chance of the capital Ci is less than zero. Note that the constrained
optimization problem is non-linear and hence makes the computations cumbersome. This thesis uses an iterative algorithm which is originally from Webber and Willison (2011) and works as follows:
• Determine the approximate level of aggregate capital such that the systemic risk measure V arsystemz (Ci
i=1,...,n) = 0.
• Reallocate the capital held by each bank such that V arsystemz ( ˜Ci) < 0 (i.e. the
worst case capital position of all banks together becomes better), where ˜Ci is new
capital allocation subject to P
iC˜i =
P
iCi.
• Reduce the level of systemic capital by a small amount such that V arzsystem(
˜ Ci
i=1,...,n) = 0. Go back to step 2.
• Repeat step 2 and 3 until the systemic capital cannot be reduced anymore without increasing the systemic risk level. It yields the minimum level of aggregate capital that can be allocated across banks while satisfying the systemic risk requirement.
3.3.2 Second Approach
The second approach is close to the idea ofGauthier et al.(2010), aiming at achieving the minimum level of systemic risk given the aggregate capital. Gauthier et al.(2010) propose to reallocate the bank capital such that the allocated capital for bank i is equal to the risk contribution of itself. This thesis uses Value-at-risk from the risk management literature by Jorion et al. (2007) to compute the risk distribution of each bank to systemic risk, so the problem can be reduced to:
Ci0 = βi n
X
i=1
Ci (3.20)
where Ci0 is the reallocated capital for bank i, βi = cov(lσ2(lip,l)p), li is the loss of bank i, lp is the loss of the whole banking system, Ci is the pre-existing capital held by each
bank. This thesis completes the optimization strategy by incorporating the aggregate capital optimization. Note that it is assumed that the capital is costly for banks and the society. The algorithm goes as follows:
18 Xincen Xie — Systemic Capital Requirement for Banks
• Determine the approximate level of aggregate capital such that V arzsystem(Ci
i=1,...,n) = 0.
• Update C0
i by equation 3.20, recalculate the individual capital level Ci0 for each
bank.
• Repeat step 1 and 2 until Ci0 = Ci
In this way, the optimal aggregate capital level is determined as well as the capital reallocation.
Empirical Results
4.1
Data Analysis
4.1.1 Big Four Banks in Australia
The Australian economy, like all other advanced economies, has been hit hard by the global financial crisis. However, while experiencing difficult times, the Australian econ-omy turned out to be more resilient and performed better in terms of the GDP growth. Figure 4 compares the Australian GDP growth with three other countries and the Eu-ropean Union (28 countries). The GDP growth of Australia went down briefly in 2008 and rebounded in the early times of 2009, whilst other countries which suffered more severe declines had an extremely slow pace of recovery. Except for the timely policy re-sponses of the Australian government and the Reserve Bank of Australia, Australia was supported by the booming trade with China. Next to that, notably, the Australian fi-nancial system has remained functioning throughout the crisis. In particular, Australian banking sector has managed to stay profitable, which served as a great boost for the economy. Shortly after the recovery, in 2010, the European debt crisis occurred and it has brought down the real GDP growth of Australia, Japan, United States and surely, the European Union. Fortunately, the spillover effects from this crisis did not last long for Australia. Soon in 2011, the real GDP growth went back to positive level for all countries/regions except European Union.
According to the Australian Bureau of Statistics (2015) , Australian financial sys-tem can be divided into three overlapping components: the financial institutions (like banks), the financial markets (like the bond market) and the payment system (cash, cheque, etc). Banks play a significant role in all three components by offering the finan-cial services, enabling funds for investments and creating payment access.
Besides, the Australian banking system is highly concentrated. There are four sys-temic important banks: Australia and New Zealand Grouping Bank (ANZ), Common-wealth Bank (CBA), National Australia Bank (NAB), Westpac (CBA). Table 1 illus-trates the top 8 banks in Australia by book asset value. National Australia Bank is the
20 Xincen Xie — Systemic Capital Requirement for Banks
Source: OECD Statistics
Figure 4: This figure compares the real GDP growth of Australia and four other countries/regions: Japan, United Kingdom, United States and European Union in 2006-2015. In the times of crisis in 2008 and 2010, Australia economy turns out to be more resilient than others, supported by its stable financial system and the booming trade with China.
biggest bank with the book asset value of 955,052 million AUD. The Australia and New Zealand Banking Group ranks second with 889,900 million AUD, closely followed by Commonwealth Bank of Australia with 873,446 million AUD. Westpac Banking Corpo-ration takes the forth place with slightly less book asset value of 812,156 million AUD. Together, the top four banks possess over 85% of the Australian banking system assets. The Macquarie Bank is at the fifth place with 187,976 million AUD of book asset value, which is almost one fifth of the asset value of National Australia Bank. Suncorp-Metway, Bendigo and Adelaide Bank and Bank of Queensland are the last ones on the list, hold-ing the book asset less than a billion AUD each. In summary, out of 33 Australia owned banks, the Big Four turn out to be ’Too Big To Fail’. That means that any default of the Big Four would have severe repercussions for the banking sector, in turn, the whole financial system. Such concentration in the banking sector creates the risk in the financial stability that cannot be neglected.
4.1.2 Interbank Deposits and Loans
The information about the interbank assets and the interbank liabilities used in this thesis can be found on the balance sheet of each bank individually. More specifically, the deposits/loans to other financial institutions are extracted to construct the interbank network. Note that the data of ANZ in 2014 and 2015 are missing due to the structural change of its balance sheet. Thus, linear regression is used to forecast the interbank deposits and loans given the observed linear trend in the history. Figure 5 depicts the percentage of the Big Four’s interbank network in Australia banking system. Not surprisingly, the Big Four dominate the domestic interbank lending market. They are
Table 1: The summary of top 8 Australian banks in terms of book asset value in Australia from 2011 to 2015. All values are in million AUD. One can see that the top 4 banks are ’Too Big To Fail’. Any default of these four banks would have severe repercussions for Australian banking sector, in turn, the whole financial system. Such concentration creates the systemic risk in the financial stability in Australian banking sector that cannot be ignored.
together holding over 90% of interbank deposit and the percentage has been increasing since 2005. Up till 2015, over 96% of interbank deposits are under their control. As for interbank loan, the shares of the Big Four have been decreasing over the last decade. However, still 86% of interbank loans are claimed by the Big Four. To estimate the interbank network, the fifth bank is added to make a closed economy, representing the rest of the banks in the system based on the interbank loans percentage. Table 2 gives an overview of the interbank loans of five banks.
Source: Australian Prudential Regulation Authority.
Figure 5: The Percentage of Big Four’s Interbank Network in Domestic Banking System. Among 33 domestic banks in Australia, the Big Four are dominated in the interbank lending market. Together 90% of interbank deposit can held by them and over 85% of interbank loan are under their control.
The interbank matrix of bilateral exposures is estimated yearly for 10 year horizon from 2006-2015. Given the concentration in banking sector, it is reasonable to assume that five banks are all connected. That means the interbank network is complete and there exists an direct edge between every two banks. So the probability of Aij = 1
22 Xincen Xie — Systemic Capital Requirement for Banks
Table 2: Interbank Loans. Note that the data of Australia and New Zealand Banking group in 2014 and 2015 are missing due to the structural change of its balance sheets. Thus, the interbank loan of ANZ in 2014 and 2015 are estimated values instead of observed values.
The fifth bank ”OTHER” is added to construct a closed economy. It can be viewed as the rest domestic banks in the banking system. So the interbank loans of ”OTHER” is computed as (1/percentage of interbank loan of Big Four - 1) * sum of Big Four. All values are in million AUD.
is one. Out of simplicity, this thesis only considers a homogeneous setup in which λ is fixed and set to be 0.75. This thesis implements 1000 iterations in the step of com-ponent updating to ensure that the distribution of interbank liability matrix converges1.
4.1.3 Merton Model Calibration
The estimation of credit risk requires both market data and balance sheet data. To obtain the equity value, this thesis uses the stock prices (close price) and outstanding shares provided in Yahoo Finance. The equity value is computed as the product of the two on weekly basis. The stock prices of Big Four are presented in Figure 6. Given that the balance sheet data are settled in Australian dollar, this thesis has transformed all stock prices into the same measurement, the Australian dollar. Figure 6 demonstrates a remarkable decline for all stocks starting from 2007 indicating that the Australia bank-ing system was not immuned from the global financial crisis. Consistent with Australian real GDP growth, the stock prices of Big Four came back on track at the beginning of 2009. Besides, the stock prices of the ANZ, NAB and WBC are found to be highly correlated during the given time horizon. The CBA performs a definite growth starting from 2012 and affected massively by the Chinese stock market turbulence in 2015.
Total liabilities, regarding as the debt value, can be found in the balance sheet of each bank. Assume the independence of the debt, the weekly debt is annual debt divided by the number of weeks in a year, which is 52 approximately. Zero coupon rate is perceived as the annual risk free interest rate. With regards to the weekly data, the formula rweek = (ryear− 1)
1
52 − 1 is applied. The maturity time T is set to be one. To simulate the asset value, 1000 unobserved asset volatilities are picked iteratively until the optimal asset volatility is close (± 10) to the estimated asset volatility. Besides, this thesis assumes that the net assets of defaulting bank can only be recovered by 10% and only 90% interbank liabilities can be repaid, which means α and β in the clearing mechanism are 0.1 and 0.9 respectively. Furthermore, to simulate the loss distribution, the default
1
To make sure the distribution of interbank liability matrix converges, the author compares the result from 1000 iterations and 10000 iterations. It turns out that 1000 iterations are sufficient in the sense that the result is similar to the one after 10000 iterations.
Source: YahooFinance
Figure 6: this figure provides the stock price of Big Four from 2006-2015 on weekly basis. Consistent with the real GDP growth, a prolonged slump can be observed in the stock prices of Big Four in late 2007 till the beginning of 2009. Shortly after the recovery, in 2010, the European debt crisis has hit the Australia stock market again and slowed down the growth of Big Four’s stocks. Another downturn can be captured in 2015 when the Chinese stock market collapses. Moreover, it is interesting to see that the stock prices of ANZ, NAB and WBC are found to be very similar in their movements throughout the last five years. Note that all prices are settled in AUD.
threshold is defined as aggregate assets minus the aggregate interbank liabilities.
4.1.4 Optimization Procedure
In the balance sheet model, the capital Ci is set to be equal to asset value Ai minus
the debt Di. The optimal capital is defined as the level such that the probability of
aggregate capital being negative is 1%. To simulate the capital distribution, this thesis makes a strong assumption that the capital is normally distributed (i.i.d.) as it provides a valuable insight into the nature of these two approaches. The results obtained from these two approaches are proven to be the same although the ideas are quite distant. This thesis implements 1000 simulations to determine the 99% value at risk level.
The first step of the two approaches is exactly the same. After the approximate level of the aggregate capital is set, the first approach tries to find a superior capital allocation such that the systemic risk can be reduced, V arsystemz (
˜ Ci
i=1,...,n) < 0, where ˜Ci,i=1,...,n
is new capital allocation. Given the assumption that capital is normally distributed, the quantile ofP Ci is additive. Hence,
V arsystemz (Ci i=1,...,n) = n X i=1 V aRz(Ci) = n X i=1 Ci+ δz n X i=1 σi (4.1)
where δz is the inverse normal cumulative distribution function. As the σi is fixed, the
systemic risk cannot be reduced anymore. So the one and only optimal solution is
n X i=1 Ci+ δz n X i=1 σi = 0 (4.2)
which boils down to
24 Xincen Xie — Systemic Capital Requirement for Banks
The second approach suggests that the capitalP Ci should be reallocated such that
Ci0 = β n X i=1 Ci (4.4) where β is βi = cov(lσ2(li,lp)
p) . li is the loss of bank i, by definition, equals to the asset of bank i minus its liability. The asset value is calibrated for its correlation in Section 3.2.2. According to the i.i.d. assumption, β can be reduced to:
β = σ 2 i Pn i σ2 . (4.5)
By equation 4.3, the reallocated capital is
Ci0 = σ 2 i Pn i σ2 (−δz) n X i σi, = −δzσi (4.6)
which equals to the new capital allocation derived by the first approach.
4.2
Result Reporting
4.2.1 Interbank Liability Network
The interbank liability is estimated on a yearly basis from 2006 to 2015. Table 3 lists the interbank liability network between the Big Four in 2006. Note that originally the inter-bank liability network is generated between the Big Four and the fifth inter-bank, a pseudo bank. The fifth bank is left out as the focus of this thesis is on the Big Four only. From Table 3, it is observed that the National Australia bank is the most ”well-connected” bank in the sense that it not only has the largest volumes of liability but also acts as the biggest creditor of all other three banks. The numbers with * represent the liabilities that National Australia Bank is obligated to pay back to the other three banks. The sum of National Australia Bank’s liabilities turns out to be the greatest, which makes it the biggest debtor in the system. At the same time, the numbers with ** implies that National Australia bank has been acting as the most important funding source for the other three banks in the interbank lending market. This confirms the idea of the Maximum Entropy Theorem (Mistrulli, 2011) that the estimated interbank links are proportional to the total values of banks. Appendix 2 provides the estimated interbank liability matrices on yearly basis throughout the entire sample period.
In the given time period, National Australia Bank has dominated the interbank lend-ing market as the largest debtor and creditor. It makes sure that any financial changes in National Australia Bank can have great influences on others. Alternatively, National Australia Bank can easily transfer the idiosyncratic shocks into other banks through the interbank liability network, which indirectly increases the systemic risk.
Table 3: Interbank Liability Network between Australia Big Four in 2006. Recall that Lijrepresents the nominal
liability of bank i to bank j. It is observed that the National Australia Bank has the largest volumes of liability to other banks (see the numbers with *). Meanwhile it also serves as the biggest creditor in the system (see the numbers with **). Not surprisingly, the National Australia Bank, the biggest bank in terms of book asset value, is the most ”well-connected” bank in the system.
4.2.2 Estimated Asset Value
The asset values of the Big Four are estimated by applying the Merton model as dis-cussed in Section 3.2. As Figure 7 shows, the pattern of four banks are found to be very similar in the given period. They all suffered huge losses in the global financial crisis in 2008, almost half of their asset values vanished in the year of 2008. After that, CBA and WBC have successfully recovered its asset back to the level before the crisis within a year. However, ANZ and NAB were not as lucky as the other two, especially NAB. It possesses the largest asset in 2006 and dropped to the third place after the crisis and even down to the forth place in 2010 when the European Debt crisis hap-pened. Moreover, Figure 7 also captures another severe crisis in 2015, the slump of the Chinese stock market. Due to the close economic relationship with China, this crisis has shed at least 20% asset values and more than 135,648 million AUD vanished in 2015.
Figure 7: Asset values in 2006-2015 on weekly basis. The patterns of Big Four are found to be very similar. Severe losses are observed in the period of the global financial crisis in 2008. After a short recovery, in 2010, the negative growths of Big Four are recorded. Starting from 2012, the estimated asset values of Big Four show an increasing trend until 2015, massive declines in asset values are captured when Chinese stock market crashes.
26 Xincen Xie — Systemic Capital Requirement for Banks
Table 4 gives an overview of the asset correlation in 2008. As predicted, the cor-relations between four banks are proven to be positive. In particular, the Pearson’s correlation coefficients between ANZ, CBA and NAB are above 0.8, reflecting that they move in a linear relation to each other in more than 80% of the time in 2008. More importantly, it is found that the asset are more likely to be correlated in the time of financial distress. For instance, as Table 5 and 6 shows, in 2010 and 2015, when Eu-ropean debt crisis hits and Chinese stock market crashes, the correlation coefficients are positive which can be interpreted as a negative sign for the banking system in the sense that it increases the probability of simultaneous defaults. In this case, throughout the interbank liability network, any negative shock will be amplified and lead to greater losses. More correlation matrices can be found in Appendix C for the given time horizon.
Table 4, 5 and 6 demonstrate the asset correlation between Big Four in 2008, 2010 and 2015, respectively. As predicted, the estimated asset value of Big Four are proven to be positively correlated in times of the crises, which poses huge threats to the systemic financial stability.
4.2.3 Simulated System Loss Distribution
Before revealing the optimal capital level and its allocation, it is worth looking at the system loss distribution. This section will compare the system loss distribution
gener-ated from different credit risk model settings. As discussed in Chapter 3, the system loss in the original Merton model is described as the number that the total liabilities exceed the total assets, where the total liabilities/assets are the sum of liabilities/assets for each bank individually as the original model ignores the correlation between banks’ asset values and the interbank exposures. Having estimated the interbank liability net-work, the aggregate liabilities after the calibration is set to be the sum of the liabilities minus the interbank liabilities for the four banks in the system (there are five banks in total). Moreover, the asset values are calibrated regarding the asset value correlation as described in Section 3.2.2. Figure 8 shows the evolution of aggregate losses in the banking system. Loss is expressed in the percentage of aggregate liability. The original Merton model is presented in blue line whereas the Merton model with asset correlation is shown in black. The Merton model, incorporating two risk factors (interconnectedness and correlation), is marked in red.
Figure 8: 99th Percentile of System Loss Simulations from 2006-2015. The system loss distribution is estimated every 18 weeks. Note that the loss here is expressed in percentage of the aggregate liability. The loss at time t is defined as the asset shortfalls below the promised debt liabilities. Figure 8 compares the system loss under different model settings. As predicted, incorporating the systemic risk factors widens the distribution of losses at systemic level as there exist positive correlations between assets and interbank network which could amplify the losses in distress.
From Figure 8, it is observed that the system losses are always below zero, meaning that the aggregate asset is always above the aggregate liability throughout the sample period. The major peaks are observed in the system loss distribution associating with the global financial crisis that happened in 2007-2009. Not surprisingly, the original Merton model has underestimated the system risk to the great extent. As there exist positive correlations between the asset value of the Big Four, the system loss distri-bution with correlation is shifted upwards, indicating a greater loss occurs considering asset correlations compared to original system loss distribution. With regards to the in-terbank exposure, as expected, the tail of system loss distribution becomes even fatter due to the increasing likelihood of contagious defaults.
28 Xincen Xie — Systemic Capital Requirement for Banks
4.2.4 Capital Surcharges
Following the optimization procedure described in Chapter 3, the first step is to deter-mine the optimal capital requirement for the whole banking system. The systemic risk constraint is set to ensure that there is only 1% probability that capital is less than zero. Figure 9 depicts the aggregate capital from 2006 till 2015 where the aggregate capital is defined as P
iCi =
P
iAi−
P
iDi. The sample period has witnessed a series
of costly financial crises. Remarkable declines have been recorded in 2008 and 2015. The European debt crisis in 2010 also slows down the growth of the aggregate capital. In general, the aggregate capital soars in sample period and reaches 333 billion AUD in 2015, which is more than triple of the capital 10 years ago.
Figure 9: Aggregate Capital. According to the balance sheet model setting, the capital is defined as the difference between the asset and the promised liabilities. So the aggregate capital is defined asP
iCi=PiAi−PiDi, i =
1, 2, 3, 4. Not surprisingly, three major declines are observed in the time of crises associating with the global financial crisis in 2008, the European debt crisis in 2010 and the Chinese stock market turbulence in 2015.
Based on the assumption that the capital is normally distributed, this thesis runs 1000 simulations and each simulation has 100,000 draws to assure that the VaR con-verges. Figure 10 shows the first 5000 draws of one simulation. There exist several draws that fall below zero, meaning that the whole banking system could have collapsed in those cases. After 1000 simulations, 99% VaR is computed to be -2618. To satisfy the systemic risk constraint that 99% VaR has to be non-negative, indicating that another 2618 million AUD of capital is required. This yields to the optimal capital level for the Big Four.
It is proven that the result generated by the two approaches are the same under the assumption that capital follows normal distribution. Hence, by equation 4.7, the optimal capital requirement is shown in Figure 12 while the capital allocation before the optimization is provided in Figure 11. The betas computed by the second approach are 0.21, 0.40, 0.17 and 0.22 for ANZ, CBA, NAB, WBC respectively. Note that the
Figure 10: Capital Simulation. Assume that the capital is normally distributed, the first 5000 draws are shown in this figure. Several draws fall below zero indicating that the whole banking system could have collapsed at some points. This thesis runs 1000 simulations and each simulation has 100.000 draws. The 99% VaR is computed to be -2618, which implies that another 2618 million AUD capital is required to meet the systemic constraints.
capital requirements for the Big Four are set to be proportional to the optimal aggregate capital. Hence, all patterns are found to be the same.
Figure 11 shows the capital allocation before the optimization. The optimization strategy indicates that another 2618 million AUD dollar should be added to the aggregate capital. Besides, to minimize the systemic risk, the capital should be reallocated as in Figure 12. It is obvious that CBA, which has the most volatile assets, has always been under capitalized. The short in capital of CBA not only poses serious threats to its own solvency position but also the financial stability of the system.
By comparison, one can see that CBA has been under capitalized since the capital requirement for CBA (see blue line in Figure 12) has always exceeded the capital it actually holds (see blue line in Figure 11) throughout the given period. For instance, in 2006, the optimal capital requirement for CBA is to possess 39,326 million capital, while the capital it actually holds is only 24,446 million. The 14,880 million short in
30 Xincen Xie — Systemic Capital Requirement for Banks
capital poses a serious threat to the solvency position of CBA. As the bank with the most volatile asset value, more capital is assigned to CBA to reduce its default risk as well as the risk in systemic level. In principle, the systemic capital requirement depends on the default probability of each bank and such resulting loss for the system.
The systemic capital surcharges of the Big Four are given in Figure 13. The cap-ital surcharges are defined as the difference between the capcap-ital requirement and the capital it has in practice. On the contrary to CBA, WBC is the most over capitalized bank. WBC holds the second largest amount of capital before the optimization while the liability it has is almost at the same level of ANZ and NAB. Next to it, ANZ claims the minimum capital surcharges, implying that ANZ is the most efficient bank of the Big Four. However, the financial crisis in 2007-2009 has increased its debt significantly and disrupted its capital. Thus, ANZ fails to satisfy the capital requirement during the crisis. As for NAB, it holds supernumerary capital before 2008 and after that, it starts to calibrate its capital level to the optimal capital requirement. Starting from 2013, NAB decides to increase the capital buffer again.
Figure 12: Capital Surcharges, which are defined as the difference between the capital before and after the optimization. In other words, it indicates the capital that needs to be added/reduced to achieve the systemic risk constraints. Clearly, Commonwealth bank has been under capitalized throughout the whole period whilst Westpac has been over capitalized. Australia and New Zealand Banking group is reported as the most efficient bank out of Big Four in the sense that it has the minimum capital surcharges. However, Australia and New Zealand Banking group sometimes fails to meet the capital requirements during the crises. As for National Australia Bank, it holds supernumerary capital before 2008 and after that, it starts to calibrate its capital to the optimal capital requirement. Starting from 2013, it decides to increase the capital buffer again.
Conclusion
The goal of this thesis is to set up the banks’ capital requirements, in part, reflecting the systemic risks. The motivation comes from the large-scale breakdown of the banking system occurred during the Global Financial Crisis in 2007-2009. The failure of Lehman Brothers sparked off a series of bank defaults and soon triggered a worldwide economic crisis. This crisis has drawn the public attention to the systemic risk factors that can lead to simultaneous insolvencies. In particular, the contagion effect and correlation ef-fect are of the essence. This thesis has outlined a framework to set up the individual capital requirement for banks from macroprudential perspective regarding these two systemic factors.
To find the optimal capital level for the four biggest Australian banks (ANZ, CBA, NAB, WBC), the Merton model is used and mapped into a constrained optimization problem. To capture the contagion effect resulting from interbank exposures, this thesis implemented a Bayesian methodology proposed by Gandy and Veraart (2016) to esti-mate the interbank liability network and use the network clearing algorithm (Eisenberg and Noe,2001) to clear the interbank exposures and calibrate the liability. Meanwhile, closely following the paper byWebber and Willison(2011), the asset value is calibrated with regards to its correlation, which stems from the common risk exposure. Taking into account of two systemic risks, this thesis simulates the system loss distribution and uses two different approaches to obtain the optimal capital level in individual and aggregate level. The systemic risk constraint is determined such that 99% of time the capital for each bank is non-negative. In other words, the default probability for each bank is less than 1%. Under the assumption that capital is normally distributed, the results of two approaches are proven to be the same.
It turns out that the financial stability in Australia banking system can still be improved by adding another 2618 million AUD of capital. CBA is found to be under capitalized while WBC is over capitalized throughout the period of 2006-2015. There-fore, CBA is confirmed to be vulnerable to the risk of systemic nature while WBC slows down its own business at the level of development by holding too much capital. ANZ
32 Xincen Xie — Systemic Capital Requirement for Banks
and NAB are reported to be the most ”efficient” banks of the Big Four.
The author hopes that the results in this thesis will be useful for assessing the sys-temic risk for the banking systems. The input data are all available in public, therefore, it is easier to be applied when investigating the systemic financial stability. Moreover, this thesis explored the dynamics of systemic capital requirements for the Australian Big Four banks, hoping that it could serve as an indicator in the future research.
