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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!