A Theoretical and Empirical Comparison of Systemic Risk Measures: MES versus CoVaR Sylvain Benoit, Gilbert Colletaz, Christophe Hurlin and Christophe Pérignon June 2012. Benoit, G.Colletaz, C. Hurlin, C. Pérignon (2012) () Systemic Risk Measures June 2012 1 / 27
Macroprudential Regulation and Systemic Risk If nancial regulation and supervision were historically focused on banks risk in isolation, with the 2008 crisis it became clear that macro-prudential rules need to be established to limit systemic risk. Basel III proposes that capital surcharges need to be imposed for systemically important nancial institutions (SIFI). Recently (12/20/2011), the FED has introduced such a surcharge for eight banks (Bank of America, Bank of New York Mellon, Citigroup, Goldman Sachs, JPMorgan, Morgan Stanley, State Street et Wells Fargo) that will be implement between 2016 and 2019. S. Benoit, G.Colletaz, C. Hurlin, C. Pérignon (2012) () Systemic Risk Measures June 2012 2 / 27
1 Market risk 2 Credit risk 3 Liquidity risk 4 Operational risk 5 Systemic risk S. Benoit, G.Colletaz, C. Hurlin, C. Pérignon (2012) () Systemic Risk Measures June 2012 3 / 27
Macroprudential Regulation and Systemic Risk S. Benoit, G.Colletaz, C. Hurlin, C. Pérignon (2012) () Systemic Risk Measures June 2012 4 / 27
Macroprudential Regulation and Systemic Risk 1 How to modi y the reglementary capital requirement for the nancial institutions that contribute the more to the systemic risk? 2 How to identify these nancial institutions? 1 Balance sheet approach: total asset, cross positions, etc. => Econometrics is useless... 2 Approach based on publicly available data ( nancial returns, leverage) => Econometrics is essential 3 Both approaches give the same results (Engle and Brownlees, 2012): BIS versus SRisK So, let do some econometrics S. Benoit, G.Colletaz, C. Hurlin, C. Pérignon (2012) () Systemic Risk Measures June 2012 5 / 27
The Econometrics of Systemic Risk How nancial econometricians measure the systemic risk? Brownless, T. C. and R. Engle (2012), Volatility, correlation and tails for systemic risk measurement, forthcoming in Review of Financial Studies. Adrian, T., and M. K. Brunnermeier (2011), CoVaR, Technical report, Federal Reserve Bank of New York. Sta report No. 348. Acharya, V. V., Pedersen L. H., Philippon T., and R. Richardson (2010), Measuring Systemic Risk, Technical report, NYU-Stern. White, H., Kim, T.-H. and S. Manganelli (2010), VAR for VaR: Measuring Systemic Risk Using Multivariate Regression Quantiles, Working Paper, ECB.. Benoit, G.Colletaz, C. Hurlin, C. Pérignon (2012) () Systemic Risk Measures June 2012 6 / 27
The Econometrics of Systemic Risk SRISK and Marginal Expected Shortfall (MES) Brownlees and Engle (2012), build a Systemic Risk index (SRISK) that captures the expected capital shortage of a rm given its degree of leverage and MES. De nition The MES is de ned as the expected equity loss per dollar invested in a particular nancial institution if the overall market declines by a certain amount.. Benoit, G.Colletaz, C. Hurlin, C. Pérignon (2012) () Systemic Risk Measures June 2012 7 / 27
The Econometrics of Systemic Risk CoVaR and CoVaR The second popular systemic risk measure is the CoVaR, introduced by Adrian and Brunnermeier (2011). De nition The CoVaR corresponds to the VaR of the market returns obtained given the e ect of a speci c event on the rm s returns. In this framework, it is possible to de ne the contribution of the institution to systemic risk, termed CoVaR, as the di erence between its CoVaR and the CoVaR calculated in the median state.. Benoit, G.Colletaz, C. Hurlin, C. Pérignon (2012) () Systemic Risk Measures June 2012 8 / 27
Objectives of the paper Objectives of the paper In this paper, we propose an uni ed and theoretical framework similar to the one used by Brownlees and Engle (2011) to compare both measures. This paper aims to determine whether they are convergent - going in the same direction - or whether they are complementary - capturing di erent components of systemic risk. This paper does not aim to determine which of the two measure is superior. S. Benoit, G.Colletaz, C. Hurlin, C. Pérignon (2012) () Systemic Risk Measures June 2012 9 / 27
If you forgot what are the VaR and ES... De nition (Value-at-Risk) VaR is an estimate of how much a certain portfolio can lose within a given time period, for a given con dence level (Engle et Manganelli, 2004). Engle, R. F., and Manganelli, S. (2004), CAViaR: Conditional Autoregressive Value-at-Risk by regression quantiles, Journal of Business and Economic Statistics, 22, pp. 367-381. Pr [r t < VaR t (α)] = α VaR t (α) = F 1 t (α) where F t (.) denotes the cdf of the returns r t at time t.. Benoit, G.Colletaz, C. Hurlin, C. Pérignon (2012) () Systemic Risk Measures June 2012 10 / 27
Distribution de P&L If you forgot what are the VaR and ES... De nition L Expected Shortfall (ES) associée à un taux de couverture de α% correspond à la moyenne des α% pires pertes attendues telle que : ES t (α) = E (r t j r t < VaR t (α) ) = 1 α Z α 0 F 1 t (p) dp 0.4 VaR et ES sous distribution Normale 0.35 0.3 0.25 0.2 0.15 P&L Distribution Normale 1% VaR = 2.3263 1% ES = 2.6652 0.1 0.05 0 5 4 3 2 1 0 1 2 3 4 5 x. Benoit, G.Colletaz, C. Hurlin, C. Pérignon (2012) () Systemic Risk Measures June 2012 11 / 27
Marginal Expected Shortfall (MES) Marginal Expected Shortfall De nition MES measures the marginal contribution of an institution i to systemic risk, measured by the ES of the system. We consider N rms and denote as r it the return of each rm i at time t. The market return, r mt, is de ned as the value-weighted average of all rms r mt = N i=1 w i r it, where w i denotes the market size weight of each rm i.. Benoit, G.Colletaz, C. Hurlin, C. Pérignon (2012) () Systemic Risk Measures June 2012 12 / 27
Marginal Expected Shortfall (MES) Strictly, the ES at the α% level is the expected return in the worst α% of the cases, but it can be extended to the general case, in which the returns are beyond a given threshold C. Formally, the conditional ES of the system is de ned as follows: ES m,t 1 (C ) = E t 1 (r mt j r mt < C ) = N w i E t 1 (r it j r mt < C ). i=1 S. Benoit, G.Colletaz, C. Hurlin, C. Pérignon (2012) () Systemic Risk Measures June 2012 13 / 27
Marginal Expected Shortfall (MES) De nition (Brownlees and Engle, 2012) The MES is then de ned as the partial derivative of the system s ES with respect to the weight of rm i in the economy. MES it (C ) = ES m,t 1 (C ) w i = E t 1 (r it j r mt < C ).. Benoit, G.Colletaz, C. Hurlin, C. Pérignon (2012) () Systemic Risk Measures June 2012 14 / 27
CoVaR CoVaR and CoVaR De nition The CoVaR corresponds to the α%-var of the market returns obtained conditionally on the nancial stress for the rm i: Pr r mt CoVaR mjc(r it ) t r it = VaR it (α) = α. The CoVaR is then de ned as the di erence between the VaR of the nancial system conditional on the distress of a particular nancial institution i and the VaR of the nancial system conditional on the median state of the institution i. CoVaR it (α) = CoVaR mjr it =VaR it (α) t CoVaR mjr it =Median(r it ) t.. Benoit, G.Colletaz, C. Hurlin, C. Pérignon (2012) () Systemic Risk Measures June 2012 15 / 27
A Theoretical Comparison of Systemic Risk Measures The Framework Thus, we consider the following bivariate process of rm and market returns: r mt = σ mt ε mt, r it = σ it ρ it ε mt + σ it q1 ρ 2 it ξ it, (ε mt, ξ it ) D, where ν t = (ε mt, ξ it ) 0 satis es E (ν t ) = 0 and E (ν t ν 0 t) = I 2, and D denotes the bivariate distribution of the standardized innovations. S. Benoit, G.Colletaz, C. Hurlin, C. Pérignon (2012) () Systemic Risk Measures June 2012 16 / 27
An Empirical Comparison of Systemic Risk Measures Marginal Expected Shortfall S. Benoit, G.Colletaz, C. Hurlin, C. Pérignon (2012) () Systemic Risk Measures June 2012 17 / 27
An Empirical Comparison of Systemic Risk Measures The conditional MES can be expressed as a function of the rm s equity price volatility, its correlation with the market return and the comovement of the tails of the distribution: MES it (C ) = σ it ρ it E t 1 ε mt j ε mt < C +σ it q1 ρ 2 it E t 1 σ mt ξ it j ε mt < C σ mt. S. Benoit, G.Colletaz, C. Hurlin, C. Pérignon (2012) () Systemic Risk Measures June 2012 18 / 27
An Empirical Comparison of Systemic Risk Measures If the standardized innovations ε mt and ζ it are i.i.d. over time, then the nonparametric estimates of the tail expectations are given by be t 1 (ε mt j ε mt < κ) = T t=1 ε mt K ( κ T K ( κ t=1 ɛ mt h ) ε mt h ), be t 1 (ξ it j ε mt < κ) = T t=1 ξ mt K ( κ T K ( κ t=1 ɛ mt h ) ε mt h ) where κ = VaR m (α) /σ mt, K (x) = R x /h k(u) du, k(u) is a kernel function and h is a positive bandwidth., S. Benoit, G.Colletaz, C. Hurlin, C. Pérignon (2012) () Systemic Risk Measures June 2012 19 / 27
An Empirical Comparison of Systemic Risk Measures S. Benoit, G.Colletaz, C. Hurlin, C. Pérignon (2012) () Systemic Risk Measures June 2012 20 / 27
An Empirical Comparison of Systemic Risk Measures De nition The SRISK is simply given by the capital shortfall which tells us how much capital does the rm need to add if an other crisis were to happen. SRISK SR it = k D it (1 k) W it MES it, where k is the prudential capital ratio of equity to asset equal to 8%, D it is the quarterly book value of total liabilities, W it is the daily market value and MES it the short term marginal expected shortfall of institution i. Benoit, G.Colletaz, C. Hurlin, C. Pérignon (2012) () Systemic Risk Measures June 2012 21 / 27
An Empirical Comparison of Systemic Risk Measures S. Benoit, G.Colletaz, C. Hurlin, C. Pérignon (2012) () Systemic Risk Measures June 2012 22 / 27
An Empirical Comparison of Systemic Risk Measures S. Benoit, G.Colletaz, C. Hurlin, C. Pérignon (2012) () Systemic Risk Measures June 2012 23 / 27
An Empirical Comparison of Systemic Risk Measures CoVaR and CoVaR S. Benoit, G.Colletaz, C. Hurlin, C. Pérignon (2012) () Systemic Risk Measures June 2012 24 / 27
An Empirical Comparison of Systemic Risk Measures This unconditional CoVaR can be estimated using a standard quantile regression (Koenker and Bassett (1978)). r mt = µ i α + γi α r it. Then, the estimated conditional CoVaR is de ned as CoVaR mjvar i (α) t = bµ i α + bγi α d VaR it (α) By de nition, the quantile-regression-based CoVaR is equal to h i CoVaR it (α) = bγ i α VaR d it (α) dvar it (0.5). S. Benoit, G.Colletaz, C. Hurlin, C. Pérignon (2012) () Systemic Risk Measures June 2012 25 / 27
An Empirical Comparison of Systemic Risk Measures According to proposition 2, the estimated DCC- CoVaR is de ned as h i CoVaR it (α) = bγ it VaR d it (α) dvar it (0.5), where bγ it = bρ it bσ mt /bσ it S. Benoit, G.Colletaz, C. Hurlin, C. Pérignon (2012) () Systemic Risk Measures June 2012 26 / 27
An Empirical Comparison of Systemic Risk Measures 0.0713 AIG 0.0555 HBAN 0.0534 0.0416 0.0356 0.0278 0.0178 0.0139 0 03/01/00 02/09/03 02/05/07 31/12/10 0 03/01/00 02/09/03 02/05/07 31/12/10 0.0604 BEN 0.0813 EV 0.0453 0.061 0.0302 0.0406 0.0151 0.0203 0 03/01/00 02/09/03 02/05/07 31/12/10 0 03/01/00 02/09/03 02/05/07 31/12/10 S. Benoit, G.Colletaz, C. Hurlin, C. Pérignon (2012) () Systemic Risk Measures June 2012 27 / 27