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As opiniões expressas neste trabalho são exclusivamente do(s) autor(es) e não
refletem, necessariamente, a visão do Banco Central do Brasil ou de seus membros.
The views expressed in this work are those of the author(s) and do not necessarily reflect those of the Banco Central do Brasil or its
members.
Systemic Risk MeasuresSolange M. Guerra
Banco Central do Brasil
Summary of the Presentation
Introduction and Motivation
Contribution
Methodology
Probability of Default and Loss Given DefaultMultivariate DensityClusters AnalysisSystemic 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 ofoccurrence of a chain reaction of bankruptcies.
ECB (2004) describes systemic risk as the probability that thedefault of one institution will make other institutions alsodefault. This interdependency would harm liquidity, credit andthe stability and confidence of the markets.
Acharya et al (2010) claim that systemic risk may be seen asgeneralized bankruptcies or capital markets freezing, whichmay cause a substantial reduction in financial intermediationactivities.
No unique definition.
Introduction and Motivation
We will define systemic risk as a consequence of an event thatmake financial markets stop functioning properly, increasingasymmetric information. In this outlook, prices no longerprovide useful information for decision taking.
Systemic risk stems from different risk sources.
In general, a specific market suffers a shock, which isamplified through different channels to other markets(including real sector).
Credit risk is a very important risk source as well as banksconnectivity is an important amplifier.
We will focus on systemic risk that comes from banking creditrisk 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 marketvalue of a bank’s assets and the probability of the financialinstitution deplete its capital [Lehar (2005), Gray, Merton eBodie (2008)].
Some papers focus on the Expected Shortfall to measures thecontribution 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 onesingle bank of the system [Adrian e Brunnermeier (2011)].
Banking Stability Measures are derived from a BankingSystem’s Multivariate Density [Segoviano e Goodhart (2009)].
Besides the definition, the data scarcity is another challengeto 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’sMethodology (2009).
The expected Loss Given Default is included in theconstruction of Systemic Risk Indicators.
These indicators are used to analyze the effects of the recentglobal crises on the Brazilian Banking System.
Methodology
We follow five steps:
Step 1 We obtain empirical individual probability of defaultfor each bank of the system, and estimate the implied marketLoss Given Default.
Step 2 Each pair of bank is considered as a portfolio.
Step 3 For each portfolio, we estimate a Multivariate Densitytaking as input the probability of default calculated in Step 1.
Methodology
Step 4 Clusters of banks are defined using the correlationbetween 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 theprobability 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(d1)− DBe−rTN(d2)
d1 =ln
�A
DB
�+
�r +
σ2A
2
�T
σA
√T
d2 =ln
�A
DB
�+
�r − σ2
A
2
�T
σA
√T
Methodology - PD and LGD
The Risk-Neutral Probability of Default is defined as N(−d2).
The Distance to Distress (D2D) defined as
D2D = d2
gives, in terms of standard deviation, how distant the marketvalue of bank assets is from the Distress Barrier (DB).
The Distress Barrier is usually defined as
DB = (short − term debt) + α(long − term debt)
.
Asset value
T
ActualProbability
of Default
Risk-NeutralProbability of Default
A0
Time
Asset Return(µA)
Risk-Free Rate(r) Distress Barrier
Distributions of asset value at T(continuous line - actual distribution)
(dashed line - risk-neutral distribution)
Methodology - PD and LGD
The Recovery Rate is defined as
RR = E (AT
DB| AT < DB)
RR =A0
DBexp [rT ]
N(−d1)
N(−d2).
The Expected Loss Given Default considering the costs forrecovering (ϕ) is defined as;
LGD0 = 1− (1− ϕ)A0
DBexp [rT ]
N(−d1)
N(−d2),
Methodology - CIMDO
The Consistent Information Multivariate Density Optimizing(CIMDO) methodology is based on the minimumcross-entropy approach.
Under this approach, a posterior multivariate distribution p isrecovered using a optimization procedure by which a priordensity q is updated with empirical information by means of aset of constraint.
In order to formalize this idea, consider a portfolio of 2 banksX e Y , whose logarithmic returns are the random variables xand y .
Methodology - CIMDO
Choose the prior density q(x , y), taking into accounttheoretical 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 isconsistent with the empirically estimated PD.
Methodology - CIMDO
Minp(x ,y)C [p, q] =Z Z
p(x , y) ln[p(x , y)
q(x , y]dxdy ,
restrict toZ Zp(x , y)X(DBx ,∞)dxdy = PDx
tZ Zp(x , y)X(DBy ,∞)dydx = PDy
tZ Zp(x , y)dxdy = 1
p(x , y) ≥ 0.
Methodology - Clusters
The clusters were established considering banks that arestrongly related.
The relationship of banks is defined by means of the distance:
d(i , j) =È
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 thesedistances. The MST is a tree that minimizes the distancebetween the knots of a Graph.
Methodology - Risk Level Indicator
IndPD =NX
j=1j 6=k
wjPD(Bj),
where wj is the assets share of bank Bj .
This indicator is an upper bound to the PD of one or morebanks of the system. As it does not consider the dependencystructure among banks, this bound is overestimated.
An increase in this indicator suggest that the banking systemas a whole is more exposed to systemic risk.
Methodology - First round effects Indicator
IndPDCond =NX
k=1
NXj=1j 6=k
wjP(Bj |Bk),
where wj is the assets share of bank Bj .
This indicator tries to capture the first round effects of thedefault of one bank over the probability of default of othersbanks.
The higher the indicator is, the higher is the propagationpossibility of shocks to the system.
Methodology - Joint PD Indicator
IndPDConj =Xi 6=j
wijPDConj(Bi ∩ Bj),
where wij is the assets share of banks Bi and Bj .
This indicator aims to capture the macroprudential riskeffects.
An increase in this indicator means that the banking system ismore exposed to macroprudential risk.
Methodology - Expect Loss Indicator
ELmaxt = Maxi ,j(LGDi .EADi + LGDj .EADj)P(Bi ∩ Bj).
where EAD is the amount of bank assets that are exposed atdefault.
This indicator allows us to evaluate the evolution of expectedlosses in the worst case scenario, when both banks default andthe losses are maximum. We have then an upper bound toexpected losses.
The literature supports that LGD is higher in periods offinancial market distress. Thus, an increase in this indicatorsuggest the existence of vulnerabilities in the banking system.
Data
We used monthly accounting data from January 2002 to June2012.
The sample includes banks operating in Brazil with aminimum of 20 observations. We have approximately 70% oftotal assets of financial institution operating in Brazil. Wehave 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 Studentdistribution with 5 degrees of freedom (prior distributionq(x , y)).
Empirical Results
Clusters Analysis
Cluster 5Cluster 5
Cluster 1 Cluster 4
Cluster 3
Cluster 2
Empirical Results
Risk Level in the Brazilian Banking System (IndPD)
15%
20%
25%
30%
35%
0%
5%
10%
I Q
2002
III Q
2002
I Q
2003
III Q
2003
I Q
2004
III Q
2004
I Q
2005
III Q
2005
I Q
2006
III Q
2006
I Q
2007
III Q
2007
I Q
2008
III Q
2008
I Q
2009
III Q
2009
I Q
2010
III Q
2010
I Q
2011
III Q
2011
All banks Cluster 1 Cluster 2 Cluster 3 Cluster 4 Cluster 5
Empirical Results
First round effects of a bank’s default (IndPDCond)
18%
24%
30%
36%
42%
0%
6%
12%
I Q
2002
III Q
2002
I Q
2003
III Q
2003
I Q
2004
III Q
2004
I Q
2005
III Q
2005
I Q
2006
III Q
2006
I Q
2007
III Q
2007
I Q
2008
III Q
2008
I Q
2009
III Q
2009
I Q
2010
III Q
2010
I Q
2011
III Q
2011
All banks Cluster 1 Cluster 2 Cluster 3 Cluster 4 Cluster 5
Empirical Results
Joint Probability of Default of two banks (IndPDConj)
4%
6%
8%
10%
0%
2%
I Q
2002
III Q
2002
I Q
2003
III Q
2003
I Q
2004
III Q
2004
I Q
2005
III Q
2005
I Q
2006
III Q
2006
I Q
2007
III Q
2007
I Q
2008
III Q
2008
I Q
2009
III Q
2009
I Q
2010
III Q
2010
I Q
2011
III Q
2011
All banks Cluster 1 Cluster 2 Cluster 3 Cluster 4 Cluster 5
Empirical Results
Indicators of Expect Loss and Loss Given Default
0.6%
0.8%
1.0%
1.2%
1.4%
1.6%
1.8%
3,000
4,000
5,000
6,000
7,000
BRL mio
0.0%
0.2%
0.4%
0.6%
0
1,000
2,000
Jan
2002
Jul
2002
Jan
2003
Jul
2003
Jan
2004
Jul
2004
Jan
2005
Jul
2005
Jan
2006
Jul
2006
Jan
2007
Jul
2007
Jan
2008
Jul
2008
Jan
2009
Jul
2009
Jan
2010
Jul
2010
Jan
2011
Jul
2011
Jan
2012
ELmax (Left axis - BRL billion) LGD (Q .99) (Right axis) LGD (Max) (Right axis)
Final Remarks
The indicators we implement are able to capture the momentsof increasing systemic risk in the Brazilian banking system,specially within the recent global crises.
They also incorporate dependency structures between banks.Thus, the results show that in stressful moments, not only theindividual PD increase, but there is also an increase in stressdependency.
Final Remarks
The empirical results show that the systemic risk measuresproposed present characteristics of early warning indicators.
The proposed indicators are useful tools for stress tests forpolicy makers.
The cluster analysis can be used for scenarios design or riskanalysis of specific group of banks that are of interest topolicy makers.
Further research could focus on the use of other dependencemeasures to establish the clusters, such as copula dependencemeasures, and forecast clusters composition.
Thank you!