How to account for interdependence of risk in …nancial markets? A multivariate GARCH approach to CoVaR estimation Giulio Girardi A. Tolga Erguny Department of Economics Su¤olk University WORK IN PROGRESS This version: March 2, 2011 Abstract Key words: Conditional Value at Risk, Systemic Risk, Correlations, Multivariate GARCH Model. JEL codes: G10, G11, G18, G21, G28, G32, G38 1 Introduction The current economic recession has drawn the attention of many to the fragility of the …nancial system. The most common measure of risk used by …nancial institutions, the Value-at-Risk (VaR), has been seriously put under attack both by regulators as well as …nancial analysts. For a given portfolio, probability and time horizon, VaR is de…ned as the threshold value such that the probability that the loss on the portfolio exceeds this value is the given probability level. Value at Risk has become in the past …fteen years the risk management instrument most widely used by regulators to determine the appropriate capital level needed to be set aside to cover for market risk by each institution. Generally, the higher the risk, the higher is the potential ‡uctuation - and therefore VaR - of a portfolio. This translates into higher liquidity that is required to be set aside to cover potential losses. ARCH (Engle (1982)) and GARCH (Bollerslev (1986)) models have been widely used since the 1980s to estimate the mean and the time-varying second moment of …nancial time series. Once estimates of the …rst two moments are obtained, it is possible to model the time varying pdf of the series and therefore to indirectly compute VaR measures at each point in time. Since the mid1990s, di¤erent variations of these models have been employed to obtain more re…ned VaR estimates (Du¢ e and Pan (1997), Engle (2001), and Giot and Laurent (2003) among others). Although these works di¤er in the employed estimation methodologies, they …nd common ground in that they all y [email protected]¤olk.edu [email protected]¤olk.edu 1 obtain VaR measures which consider exclusively the risk that institutions face when considered in isolation. The recent economic downturn that originated in the …nancial market, however, has pointed out severe shortcoming of such VaR measures. The inability to account for interdependence of risk in …nancial market, in particular, has been identi…ed by many analysts as VaR’s main ‡aw. Several authors argued in favor of alternative risk measures. Adrian and Brunnermeier (2010) - from now on referred as AB (2010) - introduced a new risk measure: CoVaRijj . CoVaRijj - Conditional Value at Risk - is de…ned as the Value at Risk of a …nancial institution i conditional upon another institution’s j Value at Risk. The novelty of this measure, as opposed to previous VaR measures, lies in the fact that the institution’s returns are now conditioned upon another institution being in …nancial distress. By conditioning on another institution’s …nancial distress, the two authors are able to construct a measure which, going beyond Idiosyncratic Risk, is able to consider also possible risk spillovers among …nancial institutions. Using quantile regression and employing exclusively in-sample analysis, AB (2010) compute time-varying CoVaR measures. Whereas AB (2010) are the …rst to our knowledge to attempt enhancing VaR measures to explicitly account for interdependence of risk, they are not the …rst to employ quantile regression in VaR estimation. Engle and Manganelli (2004) make use of such technique to develop CAViar estimates. CAViar - Conditional Autoregressive Value at Risk by Regression Quantiles - measures are derived by the two authors via quantile regression. Rather than estimating the distribution of returns …rst and later recovering the quantiles (and therefore the VaRs), by employing quantile regression the two authors directly model the quantiles and their evolution over time. While CoVaRijj measures can be estimated for two generic institutions i and j, the empirical part of this work considers the speci…c case in which institution i is the whole …nancial system while institution j is any generic …nancial institution. In this case, because CoVaRsystemjj is the Value at Risk of the whole …nancial system conditional on institution j being in …nancial distress, this measure can be employed to determine a …nancial institution’s contribution to systemic risk. A useful de…nition of systemic risk was given in July 2009 before the Committee on Banking, Housing, and Urban A¤airs of the U.S. Senate by Federal Reserve Governor Daniel Tarullo1 : "Financial institutions are systemically important if the failure of the …rm to meet its obligations to creditors and customers would have signi…cant adverse consequences for the …nancial system and the broader economy." Beside AB (2010), several other recent works attempt explicitly to derive measures to quantify systemic risk. Among others, Billio et al. (2010) use principal components analysis and Grangercausality tests to propose several econometric measures of systemic risk to capture the interconnectedness among the monthly returns of hedge funds, banks, brokers, and insurance companies. Zhou (2010) uses the multivariate Extreme Value Theory framework to provide two measures of systemic risk - the systemic impact index SII and the vulnerability index VI - which assess respectively the risk that one institution imposes to the system and the risk that system imposes to the institution. Huang, Zhou and Zhu (2009) use data on credit default swaps (CDS) of …nancial institutions and equity return correlations to model systemic risk as the price of insurance against …nancial distress. Segoviano and Goodhart (2009) also look at CDS data and measure individual …rms’contribution 1 http://www.federalreserve.gov/newsevents/testimony/tarullo20090723a.htm 2 to the distress of the …nancial system within a multivariate setting. Finally, Acharya et al. (2009) introduce Systemic Expected Shortfall (SES) measure. SES of institution j is computed by the authors as weighted average of two components: institution’s j Marginal Expected Shortfall (MES) which is the institution average loss when the …nancial system is in its left tail, and institution’s j leverage measured as quasi-market value of assets to market value of equity. While Acharya et al. (2009) derive time invariant MES measures, Brownlees and Engle (2010) derive their time series by employing asymmetric versions of a bivariate GARCH DCC model and non-parametric tail estimators. Among these works, AB (2010) has certainly the advantage of framing the analysis using the standard regulator tool of VaR. When comparing CoVaR with the MES measure introduced by Acharya et al. (2009), moreover, there appears to be two noticeable di¤erences. First, while MES corresponds to an expected return conditional on a distress event, CoVaR corresponds to a Value at Risk measure conditional on a distress event. Given that MES reports the average state in bad times, while CoVaR reports a bad state in bad times, CoVaR represents a measure of risk which attempts to capture more extreme types of events. The second di¤erence is not necessarily intrinsic to the two measures, however it arises because of the usage that has been done of the two. While MES looks at the returns of a speci…c institution j when the whole market is in …nancial distress, CoVaR does exactly the opposite, i.e. looks at the returns of the …nancial market when the speci…c institution j is in …nancial distress. For both measure, however, is it possible to reverse the analysis. Despite the fact that AB (2010) focus on CoVaRsystemjj measures, it is straightforward in fact to obtain CoVaRjjsystem estimates. In this case CoVaR would correspond to the Value at Risk of institution j conditional on the whole …nancial system being in …nancial distress. This reverse CoVaR would be more in the spirit of MES measures. In the systemic risk de…nition reported above, however, it is the failure of a …rm that is the cause of systemic distress. A measure intended to quantify the systemic importance of a …nancial institution appears therefore to be more suitably estimated by evaluating the impact which that institution’s default may generate on the …nancial system rather than the impact that the …nancial system’s default may have on the speci…c institution. Because of this, in our analysis we consider only CoVaRsystemjj measures. Despite the fact that the literature on systemic risk has been growing fast in recent years, there have been very few attempts to test for the statistically accuracy of systemic risk measures. In this work we …rst modify the de…nition of CoVaR suggested by AB (2010) by assuming that the conditioning distress event refers to institution j being at least at its Value at Risk level as opposed to exactly its Value at Risk level. By modifying the CoVaR de…nition we are …rst able to consider more general cases of …nancial distress of institution j (i.e. those tail events below the j V aRq;t level) and we also able to test for accuracy of CoVaR measures. We carry out in sample backtesting analysis in the spirit of Kupiec (1995) and Christo¤ersen (1998) evaluating CoVaR measures both in terms of number of failures and concentration of failures. We obtain time varying CoVaR estimates by employing a three steps procedure based on a bivariate DCC GARCH model. Finally, we carry out cross section analysis among …nancial institutions to investigate both the link between institutions’systemic risk (de…ned latter as average their CoV aRqj ) and institutions’risk in isolation (measured by their average V aRqj ) as well as the link between the former and size of …nancial institutions (measured by their total assets value). The rest of the paper is set out as follows: Section 2 formally de…nes CoVaR and employs a new procedure to estimate it. Section 3 describes the estimation analysis and the backtesting procedure. 3 Section 4 presents the preliminary results. Section 5 discusses future versions of the paper. Section 6 concludes the work. 2 Methodology i of portfolio Ri This section begins by reminding that, given returns Rti of institution i, the V aRq;t t is de…ned as the q quantile, i.e.: i P r Rti V aRq;t =q i For example given q=0.05, V aR0:05;t is simply the 5th quantile of the returns distribution. AB ijj (2010) de…ne the CoVaRq;t as the VaRt of institution i conditional upon another portfolio Rtj being on …nancial distress, i.e. conditional upon the (unconditional) VaR of institution j at time t: ijj j i CoV aRq;t = V aRq;t j V aRq;t ijj That is, CoVaRq;t is implicitly de…ned by the q-quantile of the conditional probability distribution: ijj j P r Rti CoV aRq;t j Rtj = V aRq;t =q (1) ijj j To estimate time-varying CoV aRq;t and V aRq;t the two authors include a set of seven state variables Mt that capture time variation in conditional means and volatilities of asset returns. j Given these state variables Mt , the two authors …rst retrieve V aRq;t j Mt 1 measures by estimating j a q quantile regression of returns Rt on the state variables Mt 1 : Rtj = j j + Mt 1 + "t (2) j bj where R bj are From the de…nition of Value at Risk, in fact, it follows directly that V aRq;t =R t t i j Rj the q quantiles of returns Rtj estimated via regression (2). Analogously, they estimate V aRq;t t by q quantile regressing returns Rti on the state variables Mt 1 and returns Rtj : Rti = ij + ijj Mt 1 + ijj Rtj + ut (3) ijj d ijj , d Then, given the coe¢ cients d , ijj and substituting the particular return realization ijj j j Rt = V aRt they obtain time varying CoVaRq;t measures: 2.1 ijj ijj ijj + d CoV aRq;t = d Mt De…nition of CoVaR 1 ijj V aRj +d t In this work the de…nition of CoVaR presented by AB (2010) is slightly modi…ed. We assume now that the conditioning distress event refers to institution j being at least at its Value at Risk level, ijj as opposed to exactly its Value at Risk level. CoVaRq;t is de…ned therefore by the q-quantile of the conditional probability distribution: P r Rti ijj CoV aRq;t j Rtj 4 j V aRq;t =q (4) There are two reasons to advocate for this change of de…nition. The …rst is that conditioning j on Rtj V aRq;t constitutes a more general case of …nancial distress of institution j which allows j also to consider for extreme tail events (i.e. those after V aRq;t ). The second reason is to facilitate CoVaR backtesting analysis. When we express CoVaR as in equation 1) the evaluation becomes extremely complicated because institution’s j returns are a continuos variables and the exact event j Rtj = V aRq;t has near zero probability of happening. By changing the de…nition to equation 4) CoVaR evaluation becomes a straightforward extension of the standard Kupiec (1995) and j Christo¤ersen (1998) analysis in those periods in which Rtj V aRq;t . In the next section we will describe in details the in sample evaluation analysis of CoVaR measures. In the rest of this work we refer to CoVaR as described by equation (4) rather than (1). 2.2 CoV ar and Systemic Risk contribution The recent …nancial events of 2007-2009 pointed out how the stability of the …nancial system can be seriously threatened by the externalities that an institution’s risk imposes on the …nancial system. Accounting for the interdependence of risk seems an important …rst step toward the prevention of future …nancial crises. While the de…nition of CoVaR reported in equation 4) refers to i and j as two generic institutions, in the rest of this work we will focus on the case where i = System (i.e. systemjj the portfolio of all …nancial institutions) so that CoV aRq;t is the Value at Risk of the whole …nancial system conditional upon institution j being in …nancial distress. AB (2010) de…ne: systemjj CoV aRq;t systemjRj =V aRj = CoV aRq;t systemjRj =M edianj CoV aRq;t (5) as the marginal contribution of a particular institution j to …nancial system distress. In other words systemjj CoV aRq;t represents the increase between the VaR of the …nancial system conditional on the distress of a particular …nancial institution j and the VaR of the …nancial system conditional on the median state of the institution j. systemjj Given the slight change in de…nition of CoV aRq;t reported in equation 4), in this work we compute systemjj CoV aRq;t systemjj CoV aRq;t as: systemjRj V aRj = (CoV aRq;t systemjRj M edianj CoV aRq;t systemjRj M edianj )=CoV aRq;t (6) which reports the percentage increase in the Value at Risk of the whole …nancial system due to the …nancial distress of the speci…c institution j. 2.3 Three-step GARCH procedure The three steps procedure employed in this work to retrieve time varying CoVaR measures is based on a bivariate version of the Engle (2002) DCC GARCH model. We model the …nancial return series via normal distribution2 . Step1. First, the unconditional VaR of each institution j is computed by employing a GARCH (1,1) speci…cation: Rtj = jt + ajp Ap (L)Rtj 1 + j Mt 1 + bjq Bq (L)"j;t (7) 2 Future version of this work we employ also symmetric student t distribution, and Hansen (1994) skew-student t distribution. 5 where A(L) and B(L) are lag polynomials whose orders p and q are determined according to AIC and SBC information criteria, and Mt 1 are the seven state variables selected by AB (2010) to capture time variation in conditional mean and volatilities of asset returns: (i) V IX index, which captures the implied volatility in the stock market. (ii) Liquidity Spread, de…ned as the di¤erence between the 3 month repo rate and the 3 month Treasury bill rate, which captures the risk or liquidity premium of the market. (iii) The change in the three month term T reasury bill rate; which works as indicator of the current state of the economy. (iv) The change in the slope of the yield curve, measured by the yield-spread between the 10 year treasury rate and the 3 month bill rate, which measures the forward-looking expected change in interest rates. (v) The change in the credit spread between BAA rated bonds and the Treasury rate with same maturity of 10 years, which measures the risk premiums earned by holders of corporate bonds. (vi) The daily equity market return from CRSP. (vii) The one year cumulative real estate sector return. The error term "j;t = zj;t modeled by the series: j;t , 2 j;t where zj;t = j 0 + iid j 2 1 j;t 1 #(0; 1); and the variance of the error term "j;t is + j 2 2 j;t 1 + j 3 Mt 1 (8) The distribution #(0; 1) can be any continuous distribution with zero mean and unit variance. Through this GARCH(1,1) model time-varying estimates of the mean and variance of the returns Rtj are obtained. Given the distributional assumption it is possible to compute at each each point in time the density function pdft Rj and, therefore, to retrieve the estimates for unconditional j V aRq;t measures: j bj + Q(q)bj;t V aRq;t =R (9) t where Q(q) for q (0; 1) is the quantile for the estimated conditional distribution. Given our norc mality assumption, the one-step-ahead 5% VaR computed at time t 1 is equal to Rtj 1:645bj;t . Step 2. The second step consists of estimating a bivariate DCC GARCH model for returns of institution j and …nancial system. Consider the random vector Rt = (Rtsystem ; Rtj ) whose joint dynamics is given by: Rt = t + M t 1 + "t (10) 1=2 t zt "t = (11) where t denotes the (2x2) conditional covariance matrix of the error term "t and t is the 1=2 (2x1) vector of conditional means. The standardized innovation vector zt = t (Rt t ) is iid with E[zt ] = 0 and V ar[zt ] = It. We de…ne Dt the (2x2) diagonal matrix with the conditional variances 2x;t along the diagonal, so that fDxx gt = f xx gt and fDxy gt = 0; for x; y = system; j. The conditional variances 2x;t are modeled by the series: 2 x;t = x 0 + x 2 1 x;t 1 + 6 x 2 2 x;t 1 + x 3 Mt 1 (12) 2 y;t y 0 = + while the o¤-diagonal covariances y 2 1 y;t 1 y 2 2 y;t 1 + + y 3 Mt 1 xy;t : xy;t = xy;t q 2 2 x;t y;t 1=2 We de…ne Rt , the (2x2) matrix of conditional correlations of "t , as Rt = Dt Finally following Engle (2002) we specify the conditional correlation matrix by: 1=2 Rt = diag(Qt ) Qt = (1 1 2 )Q Qt + diag(Qt ) 1 (ut 1 ut 1 0) + t Dt 1=2 =f xy gt . 1=2 2 Qt 1 where Q is the unconditional covariance matrix of ut = f"x;t = x;t gx=system;j and diag(Qt ) is the (2x2) matrix with the diagonal of Qt on the diagonal and zeros o¤-diagonal. Given that in step 2 we assume normality, we carry out this assumption in step 3 by modeling the joint vector Rt = (Rtsystem ; Rtj ) via a bivariate normal distribution. The density of the (2x1) vector of standardized innovations zt therefore takes the form: f (zt ) = 1 exp (2 ) 1 zt 0zt 2 Step 3. Once that we estimate the bivariate distribution in step 2, in step 3 we proceed to systemjj obtain our CoV aRq;t measures. Given the CoVaR de…nition reported in equation 4) it follows in fact that: systemjj j =q P r Rti CoV aRq;t j Rtj V aRq;t P r Rti systemjj CoV aRq;t P r Rtj j V aRq;t j V aRq;t =q j = q so that: V aRq;t j we have that P r Rtj By de…nition of V aRq;t P r Rti ; Rtj systemjj CoV aRq;t ; Rtj j = q2 V aRq;t (13) systemjj j Given the V aRq;t estimates obtained in step 1, CoV aRq;t is easily computed by solving the joint probability reported in equation 13). Figure 1 shows a graphical representation of a joint pdf systemjj j function for the random vector Rt = (Rtsystem ; Rtj ) with depicted both V aRq;t and CoV aRq;t . 3 Backtesting Analysis systemjj By changing the de…nition of CoV aRq;t to equation 4), its evaluation becomes a straightforward extension of the standard Kupiec (1995) and Christo¤ersen (1998) analysis in those periods in j which Rtj V aRq;t . Let our sample include N observations with t = 1; :::; N . For each institution 7 j j, once that we observe the past ex-ante V aRq;t forecasts and we compare them with the past j ex-post losses, we can de…ne the "hit sequence" of V aRq;t violations as: j It+1 = 1 j j if Rt+1 < V aRq;t+1 0 j j if Rt+1 > V aRq;t+1 The hit sequence returns on a speci…c day a 1 if the loss for institution j on that day was larger j than its V aRq;t level predicted for that day, and zero otherwise. For the sub-sample periods T j j in which institution j is in …nancial distress, i.e. those days in which Rt+1 < V aRq;t+1 , we can systemjj construct a second hit sequence which compare the past ex-ante CoV aRq;t+1 compare them with the past ex-post losses of the …nancial system: systemjj It+1 = forecasts and we systemjj 1 if system Rt+1 < CoV aRq;t 0 if system Rt+1 > CoV aRq;t systemjj This second hit sequence has a number of observations T equals to the violations of the …rst hit sequence, i.e. equals those days in which institution j was in …nancial distress, and returns a 1 systemjj if the loss for the …nancial system on that day was larger than its CoV aRq;t predicted level, and zero otherwise. systemjj Unconditional coverage property. When backtesting our risk model, we say that CoV aRq;t systemjj satisfy the unconditional coverage property if Pr(It+1 = 1) = q. To test whether the uncon- systemjj aRq;t ditional probability p of a violation of CoV is signi…cantly di¤erent from the promised probability q, we can write the unconditional coverage hypothesis as: systemjj H0 : E[It ] p=q The expected value of this sequence can be estimated by the sample average, pb = systemjj It+1 1 T PT t=1 = T1 =T , where T1 is the number of 1s in the sub-sample T . Note that if the null hypothesis P systemjj = q. To test the unconditional coverage property we is true we have that E[b p] = T1 Tt=1 It+1 implement the standard likelihood ratio test. For this we write the likelihood of an i:i:d: Bernoulli(p) hit sequence T Y systemjj systemjj pIt+1 = (1 p)T0 pT1 L(p) = (1 p)1 It+1 t=1 where T0 is the number of 0s in the sub-sample T . Given the estimate of p from pb = systemjj It+1 = T1 =T we can write the likelihood as: L(b p) = (1 1 T PT T1 =T )T0 (T1 =T )T1 Under the null hypothesis p = q the likelihood will be: L(q 2 ) = (1 q)T0 q T1 so that the likelihood ratio test used to check the unconditional coverage property will be: 8 t=1 LRucp = 2 1 2 ln[L(q)=L(b p)] (14) systemjj Conditional coverage property. When backtesting our risk model, we say that CoV aRq;t systemjj satisfy the conditional coverage property if Prt (It+1 = 1) = q: The unconditional coverage property assumes particular relevance because of the time-varying volatility of …nancial time series. If systemjj CoV aRq;t measures ignore such dynamics then they would react slowly to changing market conditions and their violations will appear clustered in time. systemjj Assume that the hit sequence It is dependent over time and that it can be described as a …rst-order Markov sequence with transition probability matrix: P1 = 1 1 p01 p11 p01 p11 The transition probabilities reported in the matrix are: systemjj p01 : probability that conditional on today being a non-violation (i.e. It = 0) tomorrow systemjj is a violation (i.e. It+1 = 1) systemjj p11 : probability that conditional on today being a violation (i.e. It systemjj a violation (i.e. It+1 = 1) = 1) tomorrow is also systemjj 1 p01 : probability that conditional on today being a non-violation (i.e. It systemjj is also a non violation (i.e. It+1 = 0) = 0) tomorrow systemjj 1 p11 : probability that conditional on today being a violation (i.e. It = 1) tomorrow is systemjj a non violation (i.e. It+1 = 0) Given these probability we can express the likelihood function of the …rst-order Markov process as: L(P1 ) = (1 p01 )T00 pT0101 (1 p11 )T10 pT1111 where Tz;w with z,w = 0; 1 is the number of observations with a z following a w. Given the 01 11 maximum likelihood estimates pb01 = T00T+T , pb11 = T10T+T the likelihood function becomes: 01 11 L(Pb1 ) = 1 T01 T00 + T01 T00 T01 T01 T00 + T01 1 T11 T10 + T11 T10 T11 T10 + T11 T11 systemjj If the hits of the It sequence are independent over time, i.e. satisfy the conditional coverage property, the probability of a violation tomorrow does not depend on today being a violation or not. The null hypothesis of this test will therefore be systemjj H0 : E[It ] p = p01 = p11 so that the likelihood ratio test takes the form of: LRindp = 2 ln[L(b p)=L(Pb1 )] 2 1 (15) where L(b p) is the likelihood under the alternative hypothesis from the LRucp . systemjj Following Christo¤ersen and Pelletier (2004) for those institutions whose CoV aRq;t violations reports that T11 = 0, we calculate the …rst-order Markov likelihood as: p01 )T00 pT0101 L(P1 ) = (1 and then proceed to carry out the LRindp test as reported above. 9 4 Data and Preliminary Results 4.1 The data Following the work of Acharya et al. (2009) we consider 63 of the 102 …nancial …rms in the US …nancial sector with equity market capitalization as of end of June 2007 in excess of 5bln USD.3 Table 1 lists these …nancial institutions and their “type”based on two-digit SIC code classi…cation (Depository Institutions, Securities Dealers and Commodity Brokers, Insurance, and Others). The Dow Jones U.S. Financials Index (DJUSFN) is used as a proxy for the …nancial system. The sample of data, starting 06/26/2000 and ending 2/29/2008, includes 1930 observations for each time series. VaR and CoVaR measures are computed at the q=5% con…dence level. We decide to restrict ourselves to a sub-sample St of three of the seven state variables reported in the previous section Mt . In particular we include only state variables (i), (ii), and (vi) as they are the only ones found to be statistically signi…cant in our preliminary analysis. Table 2 reports the summary statistics relative to the selected state variables, the returns of the …nancial institutions and the …nancial system. We obtain all our data from Bloomberg Terminal. 4.2 Preliminary Results As …rst stage of our three-step GARCH methodology, we estimate time-varying Value at Risk of j each institution. Once estimates for the V aRq;t of the …nancial institutions are obtained, the second systemjj and third steps consist of estimating the CoV aRq;t institution and the relative CoV systemjj aRq;t of the …nancial system on each …nancial measures. Table 3 reports the summary statistics for systemjj aRq;t the CoV estimates aggregated at the industry group level. For each group are also reported the name of the two institutions registering respectively the highest and the lowest contribution to systemic risk. The …gures reported in the table are of easy interpretation. The number in the …rst row and …rst column, for instance, suggests that during the sample period a distress event of a Depository Institution (intended as the return of the institution being below its 5% VaR level) would have imposed an average increase to the 5% Value at Risk of the …nancial system equal to 38.33%. The second and third columns indicate that, among Depository Institutions, the highest and lowest contributors to systemic risk were Citigroup (CIT) and Hudson City Bancorp (HBAN) whose …nancial distress would have increase the 5% …nancial system Value at Risk respectively by 41.79% and 29.22%. The fourth column reports that the average standard deviation of systemjj the estimated CoV aRq;t time series was 2.93. The table reveals how Broker-Dealers are the group which contributes the most to systemic risk followed by Depositories Institutions, Others, and Insurance companies. These results are similar to the MES rankings computed in Acharya et. al (2009) and Brownlees and Engle (2010) for the same four industry groups considered in our systemjj work. The only exception is that while CoV aRq;t measures rank Depositories Institutions second and Others third, MES measures reverse this ordering. systemjj Table 4 reports the average CoV aRq;t measures at the individual institution level. Among the top 10 contributors to systemic risk, 6 are Depositories Institutions and 4 Broker-Dealers. Given the group rankings discussed before, this result is not surprising. Among the bottom 10 contributors to systemic risk instead, 5 are Insurance Companies, 3 Others and 2 Depositories 3 In later versions of the paper we will consider the entire sample of 102 …rms. 10 Institutions; among the bottom four institutions, all of them are Insurance Companies. This seems to be in line again with the group rankings results. 4.3 Backtesting Analysis systemjj CoV aRq;t measures are backtested here according to the LRucp and LRindp tests described in the previous section. For LRucp the null hypothesis H0: p = q tests whether the number of failures of systemjj the estimated CoV aRq;t series p is close to the theoretical one q. For LRindp the null hypothesis H0 : p01 = p11 tests whether the probability of a violation tomorrow depends on today being a violation or not. The …rst column of table 5 reports the average LRucp and LRindp tests for each industry group, while the second and third the percentage of institutions within each group which systemjj fail to reject the null hypothesis. As it appears clearly from the table, CoV aRq;t measures retrieved via normal distribution performs bad in terms of unconditional coverage property. For all four groups of …nancial institutions the average statistics of LRucp test fall well beyond both critical values at the 5% and 1% con…dence levels (respectively 3.84 and 6.64). Not surprisingly only a few systemjj …nancial institutions fail to reject the null hypothesis. In terms of LRindp , instead, CoV aRq;t measures seem to perform much better. The average statistics for all institutions fall within the 5% and 1% critical values and as a consequence the null hypothesis is failed to reject within each industry group for the large part of institutions. Across industry groups, Depositories Institutions systemjj seem to be ones whose CoV aRq;t measures perform the worse both in terms of unconditional as well as conditional coverage property. These results are not surprising at all and seem to be in line with the literature relative to backtesting analysis of VaR when the latter is computed assuming normality of …nancial returns. It is well known that …nancial series do not exhibit normal distributions. The distribution of prices in …nancial markets are subject to both skewness and kurtosis. Skewness translates into …nancial series not being symmetrical around the mean. Kurtosis is also known informally as "fat tails". That means that events far away from the mean are more likely to happen than a normal distribution would suggest. The literature relative to Value at Risk backtesting analysis seems to agree in …nding that, when Value at Risk is computed assuming normality, one one side the number of failures of the j is usually too high compared to the theoretical but on the other side the "hit sequence" It+1 concentration of these violations does not tend to cluster in times. As a result most of the times, the LRucp test tends to reject the null hypothesis while the LRindp tends to accept it. Assuming normality underestimates Value at Risk because ignores the fat tails of …nancial series and their skewness but it does not necessarily lead to under estimation of risk clustered in time. For exact the same reasons normality assumption tends also to underestimate CoVaR measures without however determining an underestimation of risk during any speci…c time. This results are to be expected. After all, CoVaR is nothing else than a conditional Value at Risk and, as a consequence, tends to behaves like it. 4.4 Systemic Risk, Size, Leverage and Individual Risk systemjj , is the relationship Equally as important as the magnitudes of the estimated CoV aRq between the latter and the institutions’V aRqj . As noticed by AB (2010), there does not appear to be an high correlation between an institution CoV aRqj and its V aRqj . Figure 1 shows the weak link 11 between institutions’risk in isolation (measured by average V aRqj ) and institutions’contribution to …nancial system risk (measured by average CoV aRqj ). When we regress CoV aRqj on each institution’s V aRqj , the coe¢ cient appears statistically insigni…cant at any con…dence level and the R squared of the regression extremely low. From the regulators’ perspective, because these two measures seem not to be strongly correlated, capital requirements determined exclusively by V aRqj levels would di¤er substantially from capital requirements which consider also the CoV aRqj of each institution and would fail to account for the contribution of risk that each institution imposes on the …nancial system. Moving from a regulatory framework which determines institutions capital requirements based exclusively on the individual VaRs to one which also considers the CoVaRs of each institution would translate into shifting from a system which determines capital requirements strictly based on the risk that institutions self-impose to one where requirements are determined based also on the risk that institutions impose on the …nancial system. Given the weak correlation between V aRs and CoV aRs, such change does not seem to be trivial. In order to overcome these issues, the recent proposal of reform of the …nancial system considers the size of the …nancial institutions as proxy to account for the externalities that an institution’s risk imposes on the …nancial system. The proposal suggests to regulate more strictly institutions that are considered “too big to fail”and whose possible bankruptcy is likely to impose a signi…cant threat to the …nancial system. Figure 2 illustrates the positive correlation between institutions’size and institutions’contribution to …nancial system risk. When we regress CoV aRqj on the size of the …nancial institutions, the coe¢ cient appears statistically signi…cant (at least at the 5% con…dence level). The R squared of the OLS regression however seems quite low (below 9%). Although the correlation appears to be signi…cant, in fact, the scatter plot reveals how this relationship is far away from being linear. In particular, there appears to be an higher correlation between size and systemic risk contributions when size falls within $800,000 billions, while this relationship almost disappears when size exceeds this threshold. These …ndings seems to suggest that institutions which become very large should not necessarily be considered "too big to fail". Many …nancial institutions whose size is much smaller compared to that of those few very large institutions register, in fact, similar level of contributions to systemic risk. Finally Figure 3 shows the strong postive relationship between institutions’leverage and contributions to systemic risk. The regression coe¢ cient is signi…cant at the 1% level and the relative p value extremely low. Although the R squared of the regression close to 19% seems to suggest that leverage does a much better job than size at explaning contributions to systemic risk, once again from the scatter plot it appears evident how this relationship is not linear. There seems to be, in fact, a much higher correlation between contributions to systemic risk and leverage when the latter falls below a certain threshold, while this relationship almost disappears after it. 5 5.1 Future versions of the paper Accounting for Skewness and Kurtosis In order to access the relevance of both skewness and kurtosis in CoVaR estimation, in step 2 of our estimation analysis we intend to model the …nancial return series of each institution j via symmetric t and Hansen (1994) skewed t distribution. When in step 2 we will assume symmetric t distribution, we will carry out this assumption in step 3 by modeling the joint vector Rt = (Rtsystem ; Rtj ) via bivariate symmetric t distribution. In case we model instead in step 2 the …nancial returns via 12 Hansen (1994) skewed t distribution, we will extend this assumption in step 3 by using the Bauwens and Laurent (2005) bivariate skewed t distribution. We will backtest CoVaR measures retrieved via normal, symmetric t, and skewed t distribution to determine which model performs better both in term of unconditional as well as conditional coverage property. 5.2 Explaining Systemic Risk At the end of section 4 we saw how institutions’risk in isolation does not help at all in explaining institutions’contribution to systemic risk. At the same time we saw how size and leverage seem to be positively correlated with systemic risk contributions but also how these relationships appear to be higly non linear. On these lines we intend to deeper investigate the determinants of systemic risk both within and across institutions types. 5.3 Predicting the 2007-2008 crisis Following the work of Acharya et al. (2009) we will select the 12 months prior the beginning of July 2007 as the pre-crisis sample (250 daily observations from June 2006 until June 2007). We systemjj systemjj will compute CoV aRq;t and CoV aRq;t measures for the pre-crisis sample to determine their predicting power in forecasting which institutions have contributed the most to the recent …nancial crisis. To evaluate the predictive power of these rankings, we follow the work of Billio et al. (2010) and Acharya et. all (2009) by computing the maximum percentage …nancial loss of each institution during the crisis period from July 2007 to December 2008 (the “event” period). We will rank all …nancial institutions according to their Max % Loss and we will estimate cross-section regressions j of Max % Loss rankings on institutions’ CoV aRq;t rankings. We look forward to compare our results with those of Billio et al. (2010) and Acharya et. all (2009). 6 Conclusion In this work we …rst modify the de…nition of CoVaR suggested by AB (2010) by assuming that the conditioning distress event refers now to institution j being at least at its Value at Risk level as opposed to exactly its Value at Risk level. This change in de…nition allows to consider for more j extreme distress events of institution j (i.e. those after V aRq;t ) and facilitates CoVaR backtesting analysis. We propose a three steps procedure based on a bivariate DCC GARCH model to retrieve CoVaR estimates. In this version of the paper we model …nancial return series via normal distribution. Daily equity returns of 63 major …nancial institutions are taken into consideration. VaR, CoVaR, and CoVaR estimates obtained through the proposed three-steps GARCH methodology are obtained. Our backtesting analysis reveals how CoVaR measures based on Gaussian errors perform bad in terms of conditional coverage property but well in terms of unconditional coverage property. These results are not surprising at all and seem to be in line with the literature relative to backtesting analysis of VaR when the latter is computed assuming normality of …nancial returns. Finally, cross section analysis reveals the weak link between institutions’risk in isolation (measured by average V aRqj ) and institutions’contribution to …nancial system risk (measured by average CoV aRqj ). If systemic risk does not seem to be correlated with individual risk, positive signi…cant 13 correlation is found instead between the former and both institutions’size (measured by Total Assets value) as well as institutions’leverage (measured as the ratio of Total Assets to Book Equity). These relationships however appear not to be linear. Contrary to common beliefs, there seems to be for instance higher correlation between size and systemic risk when size is relatively small. These …ndings seems to suggest that institutions which become very large should not necessarily be considered "too big to fail". References [1] Acharya, V. V., L. H. Pedersen, T. Philipphon, and M. Richardson, (2009) "Measuring Systemic Risk", In Viral V. Acharya and Matthew Richardson, op. cit. [2] Adrian, T., and M. K. Brunnermeier, (2010) “CoVaR”, Federal Reserve bank of New York, Sta¤ Reports. [3] Bauwens, L. and S. Laurent (2005) "A New Class of Multivariate Skew Densities, With Application to Generalized Autoregressive Conditional Heteroscedasticity Models" Journal of Business & Economic Statistics, 23, 346-354. [4] Bauwens, L., S. Laurent, and J.V.K. Rombouts, (2006) “Multivariate GARCH models: A survey”, Journal of Applied Econometrics, 21, 79-109. [5] Billio, M., M. Getmansky, A. W. Lo, and L. Pelizzon, (2010) "Econometric Measures of Systemic Risk in the Finance and Insurance Sectors", MIT Sloan Research Paper 4774-10. [6] Bollerslev T., (1986), "Generalized autoregressive conditional heteroscedasticity", Journal of Econometrics, 31, 307-327. [7] Brownlees, C., and R. Engle (2010) "Volatility, Correlation and Tails for Systemic Risk Measurement," Working Paper, NYU-Stern. [8] Christo¤ersen, P. (1998) "Evaluating Interval Forecasts", International Economic Review, 39, 841-862. [9] Christo¤ersen, P. and D. Pelletier (2004) "Backtesting Value-at-Risk: A Duration-Based Approach", Journal of Financial Econometrics, 2, 84-108. [10] Christo¤ersen, P. and K. Jacobs (2004) "The Importance of the Loss Function in Option Valuation", Journal of Financial Economics, 72, 291-318. [11] Du¢ e, D. and P. Jun, (1997) "An Overview of Value at Risk", The Journal of Derivatives, Spring, 7-49. 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Ng, (1990) “Correlation in Price Changes and Volatility Across International Stock Markets”, Review of Financial Studies, 3. [18] Hansen, B. (1994) “Autoregressive Conditional Density Estimation”, International Economic Review, 35, 705–730. [19] Huang, X., H. Zhou, and H. Zhu, (2009) "A Framework for Assessing the Systemic Risk of Major Financial Institutions", Journal of Banking & Finance, 33:11, 2036-2049. [20] Kupiec, P. (1995) "Techniques for Verifying the Accuracy of Risk Measurement Models", Journal of Derivatives, 3, 73-84. [21] Segoviano, M., and Charles Goodhart, (2009) "Banking Stability Measures", IMF Working Paper 09/04. [22] Zhou, C., (2010) "Are Banks Too Big to Fail? Measuring Systemic Importance of Financial Institutions", International Journal of Central Banking, 6:4, 205-250. 15 µ tsystem system j q ,t CoVaR µ tj VaRqj,t Figure 1. Joint pdf function of the random vector Rt = (Rtsystem ; Rtj ) with systemjj means ( jt ; system ): CoV aRq;t corresponds to the lower-left limits of the t j rectangle with lower-right limit given by V aRq;t and whose area equals q 2 : 16 17 DEPOSITORIES BAC Bank of America BBT BB & T C Citigroup CBH Commerce Bankcorp Inc NJ CMA Comerica Inc HBAN Huntington Bancshares Inc HCBK Hudson City Bancorp Inc JPM JP Morgan Chase KEY Keycorp New MI Marshall Isley MTB M & T Bank Corp NCC National City Corp PBCT People United Financial PNC PNC Financial Services RF Regions Financials SNV Synovus Financial Corp STI Suntrust Banks Inc STT State Street Corp UB Unionbancal Corp USB US Bancorp Del WB Wachovia WFC Wells Fargo ZION Zions Bancorp OTHERS ACAS American capital Strategies AXP American Express BEN Franklin Resources Inc BLK Blackrock Inc COF Capital One Financial EV Eaton Vance Corp FITB Fifth Third Bancorp FNM Federal National Mortgage Assn FRE Federal Home Loan Mortgage JNS Janus Cap Group Inc LM Legg Mason Inc LUK Leucadia national SEIC Sei Investments Company UNP Union Paci…c INSURANCE AFL AFLA Inc AIG American International Group ALL Allstate Corp BRKA Berkshire Hathaway Inc Del BRKB Berkshire Hathaway Inc Del CFC Countrywide Financial Corp CINF Cincinnati Financial Corp CNA Can Financial Corp HIG Hartford Financial Svcs Group HUM Humana Inc LNC Lincoln national Corp MBI MBIA Inc MET Metlife Inc MMC Marsh and Mclennan Cos Inc PGR Progressive Corp OH SAF Safeco Corp TMK Torchmark Corp TRV Travelers Companies Inc UNH United Health Group BROKER-DEALERS BSC Bear Stearns GS Goldman sachs LEH Lehman Brothers MER Merrill Lynch MS Morgan Stanley SCHW Schwab Charles Corp TROW T Rowe Price Table 1. Names and classi…cation of 63 U.S. …nancial institutions whose market cap at the end of June 2007 was in excess of 5bln USD. Although Goldman Sachs has a SIC code of 6282 thus initially belonging to the group called Others we put in the group of Brokers-Dealers. Table 2. Summary statistics for state variables, …nancial institutions and the …nancial system. Sample period June 2000 - February 2008. Depositories Mean 0.043 Median 0.052 St. Dev. 1.237 Skewness 0.176 Kurtosis 6.533 Others 0.054 0.06 1.368 0.182 5.454 Insurance 0.050 0.059 1.151 0.198 5.811 Broker-Dealers 0.047 0.072 2.029 0.447 6.688 DJUSFN 0.017 0.017 1.278 0.175 6.082 VIX 19.456 18.320 6.960 0.838 3.181 Liquid Spread 0.122 0.074 0.171 2.352 12.899 Equity Mkt 0.0143 0.058 1.081 0.113 5.276 systemjj Table 3. Summary statistics of CoV aRq;t measures for industry group (q=5%) Average Max Min Std DEPOSITORIES 38.33% 41.79% CIT 29.22% HCBK 2.93 OTHERS 33.65% 39.89% AXP 23.91% BLK 3.57 INSURANCE 31.13% 38.69% LNC 14.94% BRKB 3.11 BROKER-DEALERS 39.64% 40.80% MS 36.50% SCHW 2.33 18 Table 4. Average 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 systemjj CoV aRq;t C Citigroup JPM JP Morgan Chase BBT BB and T MS Morgan Stanley MER Merrill Lynch LEH Lehman Brothers BAC Bank of America WB Wachovia WFC Wells Fargo GS Goldman sachs STI Suntrust Banks Inc KEY Keycorp New BSC Bear Stearns AXP American Express MI Marshall Isley RF Regions Financials NCC National City Corp PNC PNC Financial Services BEN Franklin Resources Inc CMA Comerica Inc SNV Synovus Financial Corp TROW T Rowe Price HBAN Huntington Bancshares Inc LNC Lincoln national Corp USB US Bancorp Del STT State Street Corp AIG American International Group MTB M and T Bank Corp TMK Torchmark Corp HIG hartford Financial Svcs Group FITB Fifth Third Bancorp ZION Zions Bancorp 41.79% 41.66% 41.44% 40.80% 40.68% 40.50% 40.46% 40.38% 40.26% 40.15% 40.00% 39.99% 39.95% 39.89% 39.82% 39.76% 39.73% 39.58% 39.51% 39.44% 39.27% 38.93% 38.87% 38.69% 38.56% 38.45% 38.14% 38.14% 37.91% 37.71% 37.65% 36.83% measures for individual institutions (q=5%). 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 19 EV Eaton Vance Corp CINF Cincinnati Financial Corp SCHW Schwab Charles Corp MBI MBIA Inc LM Legg Mason Inc COF Capital One Financial SEIC Sei Investments Company TRV Travelers Companies Inc UB Unionbancal Corp MET Metlife Inc JNS Janus Cap Group Inc CBH Commerce Bankcorp Inc NJ FRE Federal Home Loan Mortgage ALL Allstate Corp MMC marsh and Mclennan Cos Inc FNM Federal National Mortgage Assn CFC Countrywide Financial Corp SAF Safeco Corp PGR Progressive Corp OH LUK Leucadia national AFL AFLA Inc CNA Can Financial Corp PBCT People United Financial HCBK Hudson City Bancorp Inc ACAS American capital Strategies UNP Union Paci…c BLK Blackrock Inc UNH United Health Group HUM Humana Inc BRKA A Berkshire Hathaway Inc Del BRKB A Berkshire Hathaway Inc Del 36.73% 36.64% 36.50% 36.01% 35.52% 35.32% 34.95% 34.64% 34.62% 34.09% 33.75% 33.36% 33.31% 33.25% 33.11% 32.92% 32.88% 32.87% 31.93% 30.97% 30.88% 30.75% 29.96% 29.22% 28.53% 28.14% 23.91% 21.27% 20.91% 17.43% 14.94% Table 5. Summary statistics for LRucp and LRindp test. The critical values for the 21 at the 5% and 1% levels are respectively 3.84 and 6.64. The columns %0.05 and %0.01 represent the percentage of institutions failing to reject the null hypothesis at 5% and 1% levels. Average %0.05 %0.01 LRucp 11.85 0% 0% LRindp 2.74 70% 96% LRucp 7.81 21% 57% LRindp 1.96 93% 100% LRucp 8.04 26% 47% LRindp 2.64 68% 95% LRucp 9.79 0% 43% LRindp 1.96 71% 100% DEPOSITORIES OTHERS INSURANCE BROKER-DEALERS Figure 1. Link between institutions’risk in isolation (measured by VaR) and contribution to systemic risk (measured by DeltaCovar). 20 Figure 2. Link between institutions’size (measured by Billions of Total Assets) and contribution to systemic risk (measured by DeltaCovar). Figure 3. Link between institutions’leverage (measured as the ratio of Total Assets to Book Equity) and contribution to systemic risk (measured by DeltaCovar). 21

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