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Systemic Risk Measures Solange M. Guerra Banco Central do Brasil
Summary of the Presentation Introduction and Motivation Contribution Methodology Probability of Default and Loss Given Default Multivariate Density Clusters Analysis Systemic Risk Indicators Data Empirical Results Final Remarks
Introduction and Motivation What is systemic risk?
Introduction and Motivation Kaufman (1995) defines systemic risk as the risk of occurrence of a chain reaction of bankruptcies. ECB (2004) describes systemic risk as the probability that the default of one institution will make other institutions also default. This interdependency would harm liquidity, credit and the stability and confidence of the markets. Acharya et al (2010) claim that systemic risk may be seen as generalized bankruptcies or capital markets freezing, which may cause a substantial reduction in financial intermediation activities. No unique definition.
Introduction and Motivation We will define systemic risk as a consequence of an event that make financial markets stop functioning properly, increasing asymmetric information. In this outlook, prices no longer provide useful information for decision taking. Systemic risk stems from different risk sources. In general, a specific market suffers a shock, which is amplified through different channels to other markets (including real sector). Credit risk is a very important risk source as well as banks connectivity is an important amplifier. We will focus on systemic risk that comes from banking credit risk and the connectivity of the banks.
Introduction and Motivation Literature presents several measures of systemic risk. The Contingent Claims Analysis is used to estimate the market value of a bank’s assets and the probability of the financial institution deplete its capital [Lehar (2005), Gray, Merton e Bodie (2008)]. Some papers focus on the Expected Shortfall to measures the contribution of each single financial institution to systemic risk [ Acharaya, Pedersen, Philippon e Richardson (2010), Brownlees e Engle (2010)].
Introduction and Motivation Conditional VaR (CoVar) estimates the Value at Risk (VaR) of the financial system conditioned by the VaR loss in one single bank of the system [Adrian e Brunnermeier (2011)]. Banking Stability Measures are derived from a Banking System’s Multivariate Density [Segoviano e Goodhart (2009)]. Besides the definition, the data scarcity is another challenge to measure systemic risk.
Contribution The paper contributes with the literature in several ways: We propose feasible systemic risk measures jointly using PD, multivariate density of pairs of banks and clusters analysis. This is an improvement of the Segoviano and Goodhart’s Methodology (2009). The expected Loss Given Default is included in the construction of Systemic Risk Indicators. These indicators are used to analyze the effects of the recent global crises on the Brazilian Banking System.
Methodology We follow five steps: Step 1 We obtain empirical individual probability of default for each bank of the system, and estimate the implied market Loss Given Default. Step 2 Each pair of bank is considered as a portfolio. Step 3 For each portfolio, we estimate a Multivariate Density taking as input the probability of default calculated in Step 1.
Methodology Step 4 Clusters of banks are defined using the correlation between the probability of default calculated in Step 1. Step 5 We estimate the proposed systemic risk indicators.
Methodology -PD and LGD We use the Merton’s Structural Model to calculate the probability of default. The idea is modeling bank capital as an European call option, with strike price equal to the promised payment for the debts (DB) and maturity T. Payoff of this option is max (0 , A − DB ) • This option is valued using the Black-Scholes pricing equation.
Methodology - PD and LGD The Black-Scholes pricing equation: E = AN ( d 1 ) − DBe − rT N ( d 2 ) ⑩ A r + σ 2 ✒ ✓ ❿ A ln + T 2 DB d 1 = √ T ⑩ A σ A r − σ 2 ✒ ✓ ❿ A ln + T 2 DB d 2 = √ T σ A
Methodology - PD and LGD The Risk-Neutral Probability of Default is defined as N ( − d 2 ). The Distance to Distress (D2D) defined as D 2 D = d 2 gives, in terms of standard deviation, how distant the market value of bank assets is from the Distress Barrier (DB). The Distress Barrier is usually defined as DB = ( short − term debt ) + α ( long − term debt ) .
Distributions of asset value at T (continuous line - actual distribution) Asset value (dashed line - risk-neutral distribution) Asset Return ( µ A ) A 0 Risk-Free Rate ( r ) Distress Barrier Actual Risk-Neutral Probability Probability of Default of Default T Time
Methodology - PD and LGD The Recovery Rate is defined as RR = E ( A T DB | A T < DB ) DB exp [ rT ] N ( − d 1 ) RR = A 0 N ( − d 2 ) . The Expected Loss Given Default considering the costs for recovering ( ϕ ) is defined as; LGD 0 = 1 − (1 − ϕ ) A 0 DB exp [ rT ] N ( − d 1 ) N ( − d 2 ) ,
Methodology - CIMDO The Consistent Information Multivariate Density Optimizing (CIMDO) methodology is based on the minimum cross-entropy approach. Under this approach, a posterior multivariate distribution p is recovered using a optimization procedure by which a prior density q is updated with empirical information by means of a set of constraint. In order to formalize this idea, consider a portfolio of 2 banks X e Y , whose logarithmic returns are the random variables x and y .
Methodology - CIMDO Choose the prior density q ( x , y ), taking into account theoretical models and economic hypothesis. From this approach, we obtain the posterior density q ( x , y ) that is closest to the prior distribution p ( x , y ) and that is consistent with the empirically estimated PD.
Methodology - CIMDO p ( x , y ) ln[ p ( x , y ) ❩ ❩ Min p ( x , y ) C [ p , q ] = q ( x , y ] dxdy , restrict to ❩ ❩ p ( x , y ) X ( DB x , ∞ ) dxdy = PD x t ❩ ❩ p ( x , y ) X ( DB y , ∞ ) dydx = PD y t ❩ ❩ p ( x , y ) dxdy = 1 p ( x , y ) ≥ 0 .
Methodology - Clusters The clusters were established considering banks that are strongly related. The relationship of banks is defined by means of the distance: ➮ d ( i , j ) = 2(1 − ρ ( i , j )) where ρ ( i , j ) is the correlation between PDs of banks i e j . • A Minimum Spanning Tree (MST) is drawn from these distances. The MST is a tree that minimizes the distance between the knots of a Graph.
Methodology - Risk Level Indicator N ❳ IndPD = w j PD ( B j ) , j =1 j � = k where w j is the assets share of bank B j . This indicator is an upper bound to the PD of one or more banks of the system. As it does not consider the dependency structure among banks, this bound is overestimated. An increase in this indicator suggest that the banking system as a whole is more exposed to systemic risk.
Methodology - First round effects Indicator N N ❳ ❳ IndPDCond = w j P ( B j | B k ) , k =1 j =1 j � = k where w j is the assets share of bank B j . This indicator tries to capture the first round effects of the default of one bank over the probability of default of others banks. The higher the indicator is, the higher is the propagation possibility of shocks to the system.
Methodology - Joint PD Indicator ❳ IndPDConj = w ij PDConj ( B i ∩ B j ) , i � = j where w ij is the assets share of banks B i and B j . This indicator aims to capture the macroprudential risk effects. An increase in this indicator means that the banking system is more exposed to macroprudential risk.
Methodology - Expect Loss Indicator ELmax t = Max i , j ( LGD i . EAD i + LGD j . EAD j ) P ( B i ∩ B j ) . where EAD is the amount of bank assets that are exposed at default. This indicator allows us to evaluate the evolution of expected losses in the worst case scenario, when both banks default and the losses are maximum. We have then an upper bound to expected losses. The literature supports that LGD is higher in periods of financial market distress. Thus, an increase in this indicator suggest the existence of vulnerabilities in the banking system.
Data We used monthly accounting data from January 2002 to June 2012. The sample includes banks operating in Brazil with a minimum of 20 observations. We have approximately 70% of total assets of financial institution operating in Brazil. We have all the major banks operating in the Brazilian economy. The costs for asset recovery were set to 15%. We considered that bank returns follow the Student distribution with 5 degrees of freedom (prior distribution q ( x , y )).
Empirical Results Clusters Analysis Cluster 5 Cluster 5 Cluster 1 Cluster 4 Cluster 3 Cluster 2
Empirical Results Risk Level in the Brazilian Banking System (IndPD) 35% 30% 25% 20% 15% 10% 5% 0% I Q III Q I Q III Q I Q III Q I Q III Q I Q III Q I Q III Q I Q III Q I Q III Q I Q III Q I Q III Q 2002 2002 2003 2003 2004 2004 2005 2005 2006 2006 2007 2007 2008 2008 2009 2009 2010 2010 2011 2011 All banks Cluster 1 Cluster 2 Cluster 3 Cluster 4 Cluster 5
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