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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 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

• ccurrence 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)

• f 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(d1) − DBe−rTN(d2) d1 = ln ⑩ A DB ❿ + ✒ r + σ2

A

2 ✓ T

σA √ T

d2 = ln ⑩ A DB ❿ + ✒ 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 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) .

Asset value

T

Actual Probability

• f Default

Risk-Neutral Probability 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 DB exp [rT] N(−d1) N(−d2). The Expected Loss Given Default considering the costs for recovering (ϕ) is defined as; LGD0 = 1 − (1 − ϕ) A0 DB exp [rT] N(−d1) N(−d2),

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

Minp(x,y)C[p, q] =

❩ ❩

p(x, y) ln[p(x, y) q(x, y ]dxdy, restrict to

❩ ❩

p(x, y)X(DBx,∞)dxdy = PDx

t

❩ ❩

p(x, y)X(DBy,∞)dydx = PDy

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

IndPD =

N

❳

j=1j=k

wjPD(Bj), where wj is the assets share of bank Bj. 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

IndPDCond =

N

❳

k=1 N

❳

j=1j=k

wjP(Bj|Bk), where wj is the assets share of bank Bj. 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 =

❳

i=j

wijPDConj(Bi ∩ Bj), where wij is the assets share of banks Bi and Bj. 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

ELmaxt = Maxi,j(LGDi.EADi + LGDj.EADj)P(Bi ∩ Bj). 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)

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% 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 moments

• f 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 the individual PD increase, but there is also an increase in stress dependency.

Final Remarks

The empirical results show that the systemic risk measures proposed present characteristics of early warning indicators. The proposed indicators are useful tools for stress tests for policy makers. The cluster analysis can be used for scenarios design or risk analysis of specific group of banks that are of interest to policy makers. Further research could focus on the use of other dependence measures to establish the clusters, such as copula dependence measures, and forecast clusters composition.

Thank you!