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

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Figure 4: This figure compares the copula-based estimation of MES with the nonparametric kernel estimation following Brownless–Engle (2011). In the left panel, data are generated by the student t copula with degree of freedom df = 5. In the right panel, data are generated by the Rotated Gumbel Copula. Both margins are set to be student t with df = 5. The parameters of both copula models are selected such that the Kendall correlation of generated data is 0.5. The X-axis denotes the unconditional cut-off value τ which determines the tail event of the market return when r mt V aR rm (τ). The simulations are implemented for 5000 times. Proposition 2.2 implies that the marginal expected shortfall of financial institution i can be written as: MSE i,t−1 (K) = −E t−1 (r i,t |r m,t K) ( = −σ i,t E t−1 ɛ i,t |ɛ m,t K ) σ m,t = − σ ∫ 1 i,t τ t 0 ɛ i,t (u) ∂C(u, τ t) du ∂u F −1 where the threshold value K = r m (τ) is the unconditional quantile at τ% of the market return r mt , which defines the tail event when the market return exceeds K (namely, r m,t K). τ t = F ɛm,t ( K σ m,t ) is the CDF of the market return evaluated at the scaled threshold value K σ m,t . C(u, τ) is the copula function between the innovation process ɛ i,t and ɛ m,t . It is noteworthy that the value of MES relies not only on the dependence structure determined by copula function C(u, v; θ), but also on the marginal characteristics such as volatility σ it and F −1 ɛ i,t . In other words, two firms with the same dependence structure (C(u, v; θ)) with market but with different volatility (σ it ) would otherwise be considered as similarly systemically important, but will be treated differently by the measures of MES. This has been highlighted by Archaya et al. (2012) in the case of the joint Gaussian distribution where the linear correlation ρ captures the full dependence structure. Therefore, unlike ∆CoV aR or ∆CoV aR , which solely depend on dependence structure determined by the copula model, MES is a firm-specific value, which is determined by the dependence structure between firms and market as well as the marginal characteristics of financial firms such as volatility σ it . Thus, it is possible to observe a situation where a financial institution is more systemically risky according to the MES measure (higher volatility), but could be less risky based on the ranking of ∆CoV aR 14
(lower dependence with the market). 13 Although MES is associated with firms’ specific volatility, it still fails to account for some important firm level characteristics such as size and leverage which related directly to the distress condition of financial institutions 2.2.4 SRISK Estimation The discussion so far disregards the question of causality. Tail dependence only describes the extent of interconnectedness between the tail risks of financial firms and the market, but fails to indicate whether the crisis happens because the firms fail, or conversely, firms fail because of the crisis 14 . Therefore, tail dependence alone cannot accurately measure systemic risk. It is likely that a small, unlevered firm (with higher tail dependence on the market because of the fragility of a small firm per se) can appear more “dangerous” for the financial system than a big, levered one (with lower tail dependence on the market because of robustness of the big firm per se). Brownlees & Engle (2011), Acbarya, Engle and Richardson (2012) extended MES and proposed the systemic risk measures (SRISK) to account for the levels of firms’ characteristics (size, leverage, etc.). We have shown in Section 2.1.4 that SRISK is a linear function of the long run MES, liability and market capitalization ) SRISK it = max (0; kDebt it − (1 − k)(1 − LRMES it )Equity it In the followings, we elaborate on the simulation procedure and illustrate how to obtain LRMES it in the framework of the copula model. The following five step procedures are implemented for each “market-institution” pair. Step 1. Draw S sequences of length h of pairs of marginal distribution (u it , v mt ) from the parametric copula model C(u, v; θ it ) for the innovation series ɛ i,t and ɛ m,t ( ) T +h u s it, vmt s , for s = 1, 2, · · · , S t=T +1 Step 2. Obtain the sequence (ɛ s it , ɛs mt) T +h t=T +1 by setting ɛs it = F −1 i (u s it ) and ɛs mt = Fm −1 (vmt), s where F i (F m ) is the empirical marginal distribution for asset i (market). Step 3. Obtain the sequence of firm and market returns by setting rit s = σ itɛ s it and rs mt = σ mt ɛ s mt, where σ it (σ mt ) is the forecast standard deviation by GJR-GARCH model for asset i (market). Step 4. Calculate the simulated cumulative return of firm i: RiT s +1:T +h RmT s +1:T +h ) relying on the properties of logarithm returns (analogously for the market return ( h∑ ) RiT s +1:T +h = exp ri,T s +k − 1 k=1 13 Sylvain et al(2012) discussed the condition when systemic risk measure ∆CoV aR and MES are consistent or converge under the assumption of joint normal distribution,where ∆CoV aR and MES have a clean closed form solution. 14 Acbarya, Engle and Richardson(2012) argued that both of them are jointly endogenous variables. 15
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 and 12: Therefore the systemic risk measure
Page 13: Figure 3: This figure displays 8000
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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