A Semi-Parametric Measure for Systemic Risk

Transcription

1 Natalia Sirotko-Sibirskaya Ladislaus von Bortkiewicz Chair of Statistics C.A.S.E. - Center for Applied Statistics and Economics Humboldt Universität zu Berlin

2 Motivation 1-1 Challenges & Objectives FHWZ 2013 High-dimensionality: Variable selection via least absolute shrinkage and selection operator (LASSO) Non-linearity: Single-Index Model (SIM) Tail behaviour: Generalized Quantile regression (GQR)

3 Outline 1. Motivation 2. Theoretical Framework 3. CoVaR SIM 4. Applications 5. Future extensions

4 Theoretical Framework 2-1 Value-at-Risk (VaR) The VaR of a financial institution i at time t at τ (0, 1) is defined as P(X i t VaR i t,τ ) def = τ, where X i t represents the log return of a financial institution i at time t.

5 Theoretical Framework 2-2 AB 2011: CoVaR and CoVaR The VaR j t,τ conditional on some events S(X i ) is denoted by CoVaR j i t,τ and defined as P { X j t CoVaR j S(X i ) t,τ S(X i ) } def = τ. where X i are insitutions i-relevant risk-drivers. The institution i s contribution to institution j s risk is defined as CoVaR j i def t,τ = CoVaR j X t i =VaRi t,τ t,τ CoVaR j X t,τ i =Mediani t t,τ.

6 Theoretical Framework 2-3 AB 2011: Linear quantile regressions Step 1. Estimate linear quantile regressions X system t X i t = α i + γ i M t 1 + ε i t, = α system i + β system i X i t + γ system i M t 1 + ε system i t. where Xt i is the log return of institution i, X system t is the log return on financial system, M t 1 are lagged macroprudential variables. Macroprudential variables Step 2. Generate predicted values under assumption F 1 (τ M ε t 1) = 0 and i t F 1 (τ M t 1, Xt i ) = 0 ε system i t ĈoVaR system i t,τ ĈoVaR system i t,τ VaR i t,τ = ˆα i τ + ˆγ i τ M t 1, = ˆα system i + ˆβ system i VaR i t,τ + ˆγ system i M t 1, = ˆβ system i ( VaR i t,τ VaR i t,50%).

7 Theoretical Framework 2-4 HSS 2013: Linear quantile regressions Step 1. Estimate linear quantile regressions X i t = W (i)t t γ i + ε i t, where W i t ((1), M t 1, E i t, C i t 1, Xt 1) i are the risk-drivers: E i t are the contemporaneous loss exceedances defined as E i t = Xt i 1(Xt i ˆQ 1 0.1), where ˆQ is the unconditional 10% sample quantile of X i ; C i t 1 are lagged company characteristics; X i t 1 are the lagged asset returns of institution i. Company characteristics Belloni/Chernozhukov 2011

8 Theoretical Framework 2-5 HSS 2013: (Realized) systemic beta Step 2. Generate predicted values under assumption F 1 (τ W (i) ε i t ) = 0 and t perform another quantile regression VaR i t,τ = W (i)t t ˆγ i τ, X system i t,τ = ˆV (i)t t γτ system i + β system i τ VaR i t,τ. where ˆV i i t = ((1), M t 1, VaR t,τ ). Note: only the risk-relevant variables for institution i identified by LASSO in Step 1 are included. Systemic beta, of VaR i t,τ on ˆβ system i τ VaR system t,τ. A counterparty to CoVaR is defined as systemic beta., of institution i denotes the effect of a marginal change ˆβ system i τ VaR i t,τ and called realized

9 CoVaR SIM 3-1 FHWZ 2013: SIM and selection of regressors Step 0. Estimate linear quantile regressions X i t = α i + γ i M t 1 + ε i t Step 1. Estimate SIM with generalized quantile regressions X j t = g(s T β j i ) + ε j t, where g is a link function; β j i is a p-dimensional vector; S j t ((1), M t 1, R j t j with R j t R j t = X j t, C j t 1 ) is the set of risk-drivers relevant for institution being the contemporaneous risk-spillovers defined as 1(ˆσ j t ˆσmarket t ). Estimation details

10 CoVaR SIM 3-2 FHWZ 2013: LASSO The relevant variables are selected by LASSO technique derived by Li and Zhu (2008): ˆβ institution τ = arg min T 2 g,g,β T T j=1 i=1 ρ τ ( Yi g(β X j ) g (β X j )X ij β ) ω ij (β) + λ i P β p, p=1 where X ij def = (X i X j ); ω ij (β) def = K h (X ij β)/ T i=1 K h (X ij β); K h is a kernel function and h is a bandwidth. Numerical procedure

11 CoVaR SIM 3-3 FHWZ 2013: More on lambda Criterion to choose the set of relevant variables: Generalized approximate cross-validation criterion (GACV) (Yuan 2006) n i=1 GACV(λ) = ρτ {y i f (x i )}, n df where df is a measure of the effective dimensionality of the fitted model.

12 CoVaR SIM 3-4 FHWZ 2013: Systemic risk measure(s) Step 2. Generate predicted values under assumption F 1 (τ M ε t 1) = 0 and i t F 1 (τ S) = 0 ε j t VaR i t,τ = ˆα i τ + ˆγ i τ M t 1, ĈoVaR j i t,τ = ĝ(ŝ T ˆβj i τ ) Step 3. Define systemic risk measure ĈoVaR system i t,τ =? β system i τ =?

13 Application 4-1 Empirical application Data Regressand: Wells Fargo & Co. Regressors: 199 financial firms and 7 macroprudential variables. Time period: January 6, 2006 to September 6, 2012, T = Frequency: daily Window size: n = 126 τ = 0.05 and 0.01

14 Application 4-2 VaR and Linear CoVaR Linear CoVaR and VaR for Wells Fargo & Co (WFC), tau = 0.05 Figure 1: VaR of WFC (blue), Linear CoVaR of WFC (red)

15 Application 4-3 Frequency of selected risk-drivers Selected variables for WFC in w=126, tau = 0.05 Number of variables Figure 2: Frequency of selected variables.

16 Application 4-4 Selected risk-drivers Selected variables for WFC in w=126, tau = 0.05 Absolute frequency V1 V21 V44 V67 V90 V115 V142 V169 V196 Figure 3: Risk-drivers selected by linear CoVaR for WFC. Selected institutions

17 Application 4-5 Estimated lambda Lambdas for Wells Fargo & Co (WFC) lambda Figure 4: λ WFC for τ = 0.05 (black) and τ = 0.01 (blue).

18 Conclusion 5-1 Outlook Comparison of performance of parametric and semi-parametric networks. Including other measures in semi-parametric modelling, e.g., expected shortfall. Using a different underlying value for network building. Overcoming atheoreticalness of systemic risk measures/network construction via state-dependent/time-varying risk parameters (high-dimensional) dynamic factor models. Integration of non-stationarity.

19 Natalia Sirotko-Sibirskaya Ladislaus von Bortkiewicz Chair of Statistics C.A.S.E. - Center for Applied Statistics and Economics Humboldt Universität zu Berlin

20 Appendix 6-1 Local linear generalized quantile regressions Define average contrast (AC): L n(β) n def = n 1 j=1 n 1 n ρ τ {Y i g(β T x) g (β T x)β T (X i x)}k h {β T (X i x)} i=1 } {{ } L n,x (β) n n } = n 2 ρ τ {Y i g(β X j ) g (β X j )β (X i X j ) j=1 i=1 K h {β (X i X j )}, where K h is kernel function and h is a bandwidth ; ρ qr τ (u) = τu1{u (0, )} (1 τ)u1{u (, 0)}; ρ er τ (u) = τu 2 1{u (0, )} (1 τ)u 2 1{u (, 0)}. Return

21 Appendix 6-2 Numerical procedure β (0) initial estimator of β (linear QR with variable selection). For t = 0, 1, 2,, given β (t), standardize β (t), β (t) (t) = 1, β 1 = 1, (t) def d l = γ λ ( β (t) ). Then compute l (â (t) (t) j, b j ) def = arg min (a j,b j ) s Given (â (t) (t) j, b j ), solve β (t+1) = arg min β n 1 n n j=1 i=1 n i=1 + p l=1 d l (t) βl. ( ρ w Yi a j b j Xij β (t)) ω ij ( β (t) ) ρ w ( Yi â (t) j (t) b j Xij β ) ω ij ( β (t) ) Return

22 Appendix 6-3 Company characteristics Company characteristics: 1. Leverage 2. Maturity mismatch 3. Market-to-book value 4. Market capitalization 5. The equity return volatility Source: AB 2011 Return

23 Appendix 6-4 The macroprudential variables The macroprudential variables: 1. VIX 2. Short term liquidity spread (liquidity) 3. Daily change in the 3-month Treasury maturities (3MT) 4. Change in the slope of the yield curve (yield) 5. Change in the credit spread (credit) 6. Daily Dow Jones U.S. Real Estate index returns (D_J) 7. S&P500 returns (S&P) Return

24 Appendix 6-5 HSS 2013: LASSO The relevant variables are selected by LASSO technique derived by Belloni and Chernozhukov (2011): where γ τ i = arg min T 1 γ i T t=1 ρ τ (X i t + W T t γ i ) + λ i T 1 τ(1 τ) P ˆσ p γp, i ρ τ (u) = τu1{u (0, )} (1 τ)u1{u (, 0)}; ˆσ p 2 = T 1 T t=1 (Wt,p)2 is componentwise variation with W t,p being the set of all potential regressors. Note: Linear quantile regressions are reestimated with the selected variables to correct for the downside bias. Return p=1

25 Appendix 6-6 Expectile-Quantile Correspondence Let v(x) represents expectile regression, I (x) represents quantile regression. Fixed x, define w(τ) such that v w(τ) (x) = I (x) then w(τ) is related to I (x) via w(τ) = τi (x) I (x) ydf (y x) 2 E(Y x) 2 I (x) ydf (y x) (1 2τ)I (x) For example, Y U( 1, 1), then w(τ) = τ 2 /(2τ 2 2τ + 1) Expectile corresponds to quantile with transformation w.

26 Appendix 6-7 The selected financial firms and macroprudential variable(s) Top 7 influential covariates Frequency No. 187 Radian Group Inc. (RDN) 711 No. 095 CNO Financial Group Inc. (CNO) 281 No. 003 Bank of America (BAC) 274 No. 088 E-Trade Financial Corporation, Inc. (ETFC) 272 No. 117 MBIA Inc. (MBI) 259 No. 197 First source corporation (SRCE) 178 No. 000 Change in the slope of the yield curve (yield) 146 Return

27 References 7-1 References Adrian, T. and Brunnermeier, M. K. CoVaR Staff Reports 348, Federal Reserve Bank of New York, 2011 Fan, Y., Härdle, W. K., Wang, W. and Zhu L. Composite Quantile Regression for the Single-Index Model Discussion Paper , CRC 649, Humboldt-Universität zu Berlin, 2013 Hautsch, N., Schaumburg, J. and Schienle, M. Financial Network System Risk Contributions Discussion Paper , CRC 649, Humboldt-Universität zu Berlin, 2011 Koenker, R. and Hallock, K. F. Quantile regression Journal of Econometric Perspectives 15(4): , 2001

28 References 7-2 References Kong, E. and Xia, Y. Variable selection for the single-index model Biometrika. 94: , 1994 Li, Y. and Zhu, J. L1-norm quantile regression J. Comput. Graph. Stat., 17: , 2008 Yuan, M. GACV for Quantile Smoothing Splines Comput. Stat. Data. An., 5, , 2006