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Applying CoV aR to Measure Systemic Market

Risk: the Colombian Case∗

Mauricio Arias

Juan Carlos Mendoza

David P´ erez-Reyna†

April 2010

Abstract

In Colombia, the exposition to market risk has increased significantly since 2009.

Nonetheless, the risk codependence among agents has not been analyzed yet from the

perspective of this risk. Therefore, this paper presents an approach to estimate such

relevance based on CoV aR and quantile regressions. This methodology is flexible

enough to allow the estimation of the systemic market risk contribution of banks,

pension funds, and between different types of financial institutions. Results suggest

that risk codependence among entities increases during distress periods.

JEL classification numbers: C20, G14, G21.

Keywords: Systemic Market Risk, CoV aR, Value at Risk, Quantile Regression.

∗The authors thank Dairo Estrada and the staff of the Financial Stability Department at the Banco

de la Rep´ ublica (Central Bank of Colombia) for valuable comments. The views expressed in this paper

are those of the authors and do not necessarily reflect those of the Banco de la Rep´ ublica, nor of its

Board of Directors. The authors are solely responsible for any errors or omissions.

†The authors are, respectively, Senior Professional, Professional and Specialized Professional

of the Financial Stability Department at the Banco de la Rep´ ublica.

jmendogu@banrep.gov.co.

Corresponding e-mail:

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Introduction

Negative shocks suffered by individual financial institutions can easily propagate and

affect other entities. Due to this, measuring and analyzing the phenomena derived from

systemic risk has been a common interest among policy makers. Moreover, since the

recent financial crisis, this analysis has gained even more importance.

Systemic risk may not be analyzed only by using individual risk measurements of

institutions. Herding behavior by financial entities may cause a high exposition to neg-

ative systemic events, even if individually all institutions have low risk measurements.

Additionally, the risk assumed by a systemic institution may cause negative spillovers

not internalized in risk requirements. To deal with these issues, several papers have ap-

proached systemic risk from different perspectives, according to what authors perceive

is more relevant to their analysis.

For Rochet and Tirole (1996) systemic risk is materialized when a bank’s economic

distress propagates to other economic agents linked to that bank through financial trans-

actions. This paper studies whether the flexibility offered by decentralized interbank

transactions can be maintained, while the corresponding financial authority can be pro-

tected against undesired rescue operations. If not, centralizing interbank systems would

be more efficient in terms of liquidity allocation and prudential control. In particular,

the authors analyze the “too big to fail”policy: proper authorities bail-out a bank with

short positions in the interbank market because the bank’s distress may affect solvent

lending banks.

According to Furfine (2003), there are two types of systemic risk: 1) the risk that

a financial shock causes a set of markets or institutions to simultaneously fail to func-

tion efficiently, and 2) the risk that failure of one or a small number of institutions will

be transmitted to others due to explicit financial linkages across institutions. To an-

alyze contagion, Furfine estimates it by examining federal funds exposures across U.S.

banks, which are used to simulate the impact of exogenous failure scenarios. This paper

concludes that, although the exposures are not large enough to cause a great risk of

contagion, illiquidity could pose a threat to the banking system.

For Acharya (2009) systemic risk, defined as joint failure risk, arises from the corre-

lation of banks’ assets returns. To analyze this, the author considers a model in which

banks invest in risky assets in various industries. The investment decision determines

the correlation among banks’ assets, which, in case it is high enough, results in a rising

exposition to systemic risk. The paper concludes that the effect of regulation of banks’

optimal investment decisions deserves careful scrutiny: requirements should depend both

on banks’ joint and individual risk.

On the other hand, Allen and Gale (2000) address systemic risk from a liquidity risk

perspective. They find that the resilience of the interbank market to adverse liquidity

shocks depends on the market’s structure.Similarly, Saade Ospina (2010) analyzes

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the Colombian interbank collateralized market. He develops a centrality index using

cooperative game theory and concludes that when the interbank network is disconnected,

bid ask spreads are farther apart and their volatility is higher. This implies that banks

are more exposed to liquidity market risk under this scenario.

Nonetheless, in Colombia systemic risk has not been analyzed yet from a market risk

perspective. The exposition of financial institutions to this risk has increased since 2009

as lower rates and slower credit dynamics have caused asset restructuring. Treasury

bonds (TES) holdings and volatility in yields reached levels similar to the observed by

mid 2006, when a setback in this market caused the most important losses during the

past decade. In the context of the model proposed by Acharya (2009), this behavior has

increased the correlation of the different entities’ assets, especially among commercial

banks, which could cause a higher systemic risk. Due to these reasons, it is impera-

tive to analyze market risk codependence among Colombian commercial banks, pension

funds and financial institutions to identify which institutions have a high contribution

to systemic market risk.

The objective of this paper is to analyze market risk codependence among Colombian

financial institutions in order to identify institutions with the highest contribution to

systemic market risk. We define systemic market risk as the aggregate market risk of

the financial system. We follow the definition of CoV aR introduced by Adrian and

Brunnermeier (2009), which is measured as the Value at Risk (V aR) of a financial

institution or sector conditional on the V aR of another institution or sector. In this

way, if CoV aR increases relative to V aR, so does spillover risk among institutions.

By defining the difference between these measures as ∆CoV aR, we can estimate the

contribution of each institution to systemic market risk.

Additionally, since ∆CoV aR is not necessarily symmetric (that is, the contribution

that institution i’s V aR has on institution j’s market risk does not necessarily equals

the contribution of j’s V aR on i’s V aR), this measure can be used to analyze the risk

across the Colombian financial system. We focus on the public debt portfolio of financial

entities and define the portfolio of the financial system as the aggregate public debt

holdings of these institutions. Results suggest that risk codependence among entities

increases during distress periods.

As mentioned by Adrian and Brunnermeier (2009), one advantage of CoV aR is that it

can be applied with any other tail measure to analyze other risks. For instance, Chan-Lau

(2008) follows a similar approach and assesses systemic credit risk by measuring default

risk codependence among financial institutions through an analysis of CDS spreads of

25 entities in Europe, Japan and the US.

Also, Gauthier et al (2010) compare ∆CoV aR and other four approaches to assign

systemic capital requirements to individual banks based on each bank’s contribution to

systemic risk. The authors conclude that financial stability can be enhanced substantially

by implementing a system perspective on bank regulation.

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The remainder of this paper is structured as follows: section 1 describes the spec-

ification of the model used. In section 2 we analyze the Colombian Treasury Market.

Section 3 shows the main results. Finally section 4 includes the concluding remarks.

1Methodology

To study the systemic market risk contribution of each entity it is important to analyze

the risk codependence among financial institutions in the context of a high market risk

exposure scenario. Several methodologies have been used to measure systemic risk and

risk codependence. Hartmann et al (2001) and Chan-Lau et al (2004), for instance, used

extreme value theory for this purpose. However, a common problem of this methodology

is that a large amount of data is needed because only tail observations are used.

An adequate way to measure market risk codependence is through quantile regres-

sion.1This methodology provides a more extensive analysis than ordinary least squares

in the sense that it estimates the relationship among random variables under different

quantiles. For this reason, it can be used to estimate the risk codependence among fi-

nancial institutions under different risk scenarios. Additionally, this is a methodology

that can be easily estimated with a large number of independent variables.

In general, the estimation of quantile regression consists in minimizing the sum of

residuals, weighted asymmetrically by a function that depends on the quantile τ. That

is, the τ regression quantile, 0 < τ < 1, can be represented as a solution to the following

expression:

min

β

t

where y is the dependent variable, f(xt,β) is a linear function of the parameters and

the variables used to explain the behavior of y, and ρτ is the weight assigned to each

observation, depending on the analyzed quantile τ. Specifically, Koenker and Bassett

(1978) propose the following representation of equation (1):

t∈{t:yt≥f(xt,β)}

In this paper we measure how the risk level of a financial institution j is affected by the

risk level of another financial institution i or by the whole financial sector. Following

Chan-Lau (2008), equation (2) is estimated with

?

ρτ(yt− f(xt,β)),

(1)

min

β

?

τ |yt− f(xt,β)| +

?

t∈{t:yt<f(xt,β)}

(1 − τ)|yt− f(xt,β)|

.

(2)

yt= Riskj,t

(3)

f(xt,β) = βR

ji,τ?R + βji,τRiski,t,

1This methodology was proposed by Koenker and Bassett (1978).

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where Riski,tdenotes an indicator that measures the market risk of entity i in t. For this

purpose we use the daily V aR of entity i’s TES portfolio, with a weekly frequency. βji,τ

is a vector of parameters, which indicate risk codependence between i and j for quantile

τ. These parameters were estimated for different quantiles in order to analyze if the risk

codependence between any two entities or sectors increases under higher levels of risk.

In addition, we consider a matrix with exogenous variables that can affect the market

risk level (R). R contains different aggregate risk factors that are used to explain the

evolution of TES prices and its market risk, such as inflation expectations, weekly stock

market returns and exchange rate returns, the slope of the yield curves, weekly credit

growth, EMBI+ for Colombia, VIX, five-year CDS for Colombia and the Colombian

interbank rate. To avoid multicollinearity, we estimated the principal components that

explain the 80% of the volatility of the standardized variables in R. The resulting vectors

(?R) were used in the quantile regressions. In this sense βR

market risk.

ji,τ, can be understood as the

effect of these exogenous variables over entity j’s market risk on τ quantile, given i’s

The estimation process required the calculation of 1360 regressions for banks: for

each of the 16 Commercial Banks (CB) we calculated a regression against each other

banks’ V aR, and against an aggregate V aR for the banking sector, for five different

quantiles. Similarly, we estimated 210 regressions for Pension Funds (PF), due to the

fact that we analyzed six PF and an aggregate V aR that comprised the market risk of

the PF sector. Finally, we calculated an aggregate V aR for each consolidated sector of

other Credit Institutions: Financial Corporations (FC), Financing Companies (CFC),

and Financial Cooperatives (Coop).We did the same for each sector comprised in

the other Non-Banking Financial Institutions (NBFI): Brokerage Firms (BF), Insurance

Companies (Ins) and Hedge Funds (HF), and for the whole Financial System (FS). Then,

we estimated 360 regressions among each sector of the financial system. The main results

are shown in section 3.2

Additionally, to extend the systemic risk analysis, Adrian and Brunnermeier (2009)

proposed a conditional risk codependence measure, or co-risk measure, which they de-

noted CoV aR.3CoV aRj|i

entity i. That is,

α stands for the V aRαof entity j conditional on the V aRαof

P(Xi≤ V aRi

α|Xi= V aRi

α) = α

α) = α,P(Xj≤ CoV aRj|i

where Xistands for weekly returns of the TES portfolio of entity i. A more general way

to define CoV aRαis:

CoV aRj|i

α= {V aRj

α|V aRi

α,R}.

2Regressions were estimated with 360 weekly observations for the mentioned variables, with data from

February 14th, 2003 to January 1st, 2010.

3For a detailed explanation of the definition and properties of CoV aR see Adrian and Brunnermeier

(2009).

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