Does Tail Dependence Make A Difference In the ... - Boston College

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2.2.2 ∆CoV aR Estimation With the stress condition being X V aR x (τ), the modified version of CoV aR proposed by Girardi et al. (2011) can also be analyzed under the framework of the copula model. Starting from the above equation ∫ CoV aRτ ∫ V aRq −∞ −∞ f(x, y)dxdy = τq The joint density function f(x, y) can be substituted by the cross product of marginal density and copula density f(x, y) = ∂2 C(u,v) ∂u∂v f(x)f(y). By a simple manipulation, we have: ∫ CoV aRτ ∫ V aRq −∞ −∞ ∂ 2 C(u, v) f(x)f(y)dxdy = τq ∂u∂v ∫ Fy(CoV aR τ ) ∫ Fx(V aR q) 0 0 ∂ 2 C(u, v) dudv = τq ∂u∂v (2) Let q = τ, we can solve CoV aR y|xV aRx(τ) y|xV aRx(0.5) numerically from Equation (2). Analogously, let q = 0.5, CoV aR can be obtained as well. Therefore, ∆CoV aR can be estimated as y|xV aRτ ∆CoV aR τ = CoV aRτ y|xV aR0.5 − CoV aRτ Girardi et al. (2011) compared the bivariate skewed t distribution (Bauwens and Laurent 2005) with bivariate Gaussian distribution to model the density function f(x, y), and they found that the former joint density outperformed the latter in the estimation of CoVaR. But they fail to explain whether this performance improvement comes from the difference in the specification of marginal distribution or the dependence structure in the tail. In our paper, we only focus on the discrepancy of dependence structure and eliminate the effect of marginal distribution as we believe that what really matters in the estimation of systemic risk is the dependence structure (especially tail dependence) rather than marginal characteristics. Figure 3 illustrates the CoV aR and ∆CoV aR under two copula models with different dependence structures in the tail of the distribution. The left panel present the Clayton copula joint distribution with lower tail dependence, while the right panel displays the joint distribution of the Gumbel copula with upper tail dependence. Suppose that bank i financial condition moves from its normal state (at Median) to its financial distress (at 1% quantile). We want to investigate what is the change of 1% quantile for the financial system. 8000 random draws are simulated from two bivariate joint distributions. The parameters of each copula model are chosen to provide a linear correlation of random variates equal to 0.8. It is not surprising that the systemic risk measure ∆CoV aR under the Clayton copula is larger than that under the Gumbel copula because the former joint distribution is characterized by the larger lower tail dependence, which results in stronger co-movement in the tail risk of financial institutions and the market. 12
Figure 3: This figure displays 8000 random draws from two bivariate joint distributions: Clayton-copula (left) and Gumbel-copula (right). Both margins are set to be standard Gaussian N(0, 1). In each panel, the parameter is chosen such that the linear correlation ρ of random variates is equal to 0.8. CoVaR is estimated at 1% quantile of financial market return. Visually, ∆CoV aR in the left panel is obviously greater than that in the right panel 2.2.3 MES Estimation Analogous to the calculation of CoVaR, the value of MES is associated with the joint distribution of downside risk between the returns of the market and financial firms. Proposition 2.2 Suppose X and Y are the returns of two assets with the marginal distributions being X ∼ F x and Y ∼ F y . The joint distribution of X and Y are defined by a parametric copula C(u, v; θ). The closed form solution of marginal expected shortfall MES = −E(X|Y < V aR y (τ)) can be expressed as (see the Proof in Appendix 2): MES(τ) = − 1 τ ∫ 1 0 Fx −1 ∂C(u, τ; θ) (u) du ∂u where τ is the quantile percentage for the distribution of Y which defines the tail event Y < V aR y (τ). Figure 4 compares the copula-based estimation of MES with the nonparametric kernel estimation following Brownless-Engle (2011). 12 As we can see, Brownless–Engle’s approach provides a good approximation to the estimation of MES even when the data are generated by student t copula (symmetrical tail dependence) or Rotated Gumbel Copula characterized by lower tail dependence. This simulation exercise seems to justify the nonparametric kernel estimation in capturing nonlinear tail dependence. Following the models proposed by Brownlees and Engle (2011), the return of market and financial institution i is specified as r m,t = σ m,t ɛ m,t r i,t = σ i,t ɛ i,t 12 For the details of nonparametric kernel estimation of MES, see Brownless–Engle (2011) 13
Page 1 and 2: Does Tail Dependence Make A Differe
Page 3 and 4: Figure 1: The upper panel graphs di
Page 5 and 6: more related to the “too big to f
Page 7 and 8: CoV aR (hereafter called CoV aR )
Page 9 and 10: Theorem 1 (Sklar)(1959) Let G be a
Page 11: Therefore the systemic risk measure
Page 15 and 16: (lower dependence with the market).
Page 17 and 18: higher value of systemic risk. Figu
Page 19 and 20: oth estimated based on the joint di
Page 21 and 22: Figure 8: This figure displays the
Page 23 and 24: the Goodness of Fit (GoF) of copula
Page 25 and 26: Depositories and Broker-Dealer comp
Page 27 and 28: 5.2 Empirical Results 5.2.1 Estimat
Page 29 and 30: [ Insert Table 6 Here ] Table 6 pre
Page 31 and 32: isk measure (last panel of Table 9)
Page 33 and 34: References Acharya, V. V., Pedersen
Page 35 and 36: Appendix Proof on the relationship
Page 37 and 38: Therefore v = F Y (y) = Φ( y −
Page 39 and 40: In the case of continuous random va
Page 41 and 42: Table 2: Bivariate Copula Functions
Page 43 and 44: Table 5: AIC for the Different Copu
Page 45: Table 8: Systemic Risk Rankings:

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