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

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2 Methodology In this section, we introduce a formal definition of systemic risk measures proposed by Adrian and Brunnermeier (2011) and Acharya et al. (2010). Let us assume a financial system composed of a large number of institutions. Denote by r mt and r it the daily return for the market, i.e. the financial system and firm i on time t, respectively. The market return can be considered as the portfolio of all firms’ returns r mt = N∑ ω it r it where ω it denotes the weight (market capitalization) of firm i in the portfolio at time t. i=1 2.1 Definitions 2.1.1 ∆ CoVaR Paralleling the definition of Value at Risk (VaR), the conditional Value at Risk (CoVaR) is defined as the expected maximal loss of a certain portfolio at some confidence interval (τ quantile) given another portfolio at the same time experiences expected maximal loss. Formally, the CoVaR of financial institution i corresponds to the Value at Risk of the financial system m conditioning on the occurrence of tail event r it = V aRτ i for the institution ∗ i. ) P r (r mt ≤ CoV aR m|i (τ, τ ∗ )|r it = V aRτ i = τ ∗ When τ ∗ = τ, we simply denote CoV aR m|i (τ, τ) = CoV aR m|rit=V aRi τ τ . The Value at Risk of the financial institution i: V aRτ i is defined as the τ ∗ quantile of its loss probability distribution ∗ P r ( r it ≤ V aR i τ ∗ ) = τ ∗ Further, AB (2011) 6 proposed ∆CoVaR as the difference between CoV aR m|rit=V aRi τ τ i is in distress and CoV aR m|rit=Mediani τ when the financial institution i is in the normal state. ∆CoV aRτ m|i = CoV aR m|rit=V aRi τ τ − CoV aR m|rit=Median τ = CoV aR m|i (τ, τ) − CoV aR m|i (τ, 0.5) when the financial institution Therefore ∆CoVaR measures the increment of the VaR (difference between two conditional VaR), which aims to capture the contribution of a particular institution i to the tail risk of financial system as a whole. 2.1.2 ∆CoV aR Defining the financial distress being exactly at its VaR is arguably too restrictive. Girardi et al. (2011), among others, extended the definition of stress event to be below (r it V aR it (τ)) rather than exactly being at its VaR (r it = V aR it (τ)), which allows for more severe distress event being further in the tail. Formally, the alternative 6 Adrian and Brunnermeier (2011) further introduce “Exposure CoVaR”, which reverse the conditioning set to be the tail event for market return: r mt = V aR m τ ∗ . 6
CoV aR (hereafter called CoV aR ) can be defined as: ) P r (Y CoV aR τ |X V aR q = τ ) P r (Y CoV aR τ , X V aR q P r ( ) = τ X V aR q ) P r (Y CoV aR τ , X V aR q = τq where q(τ) is the quantile defining the tail event of the financial institution (market). Therefore the alternative CoVaR at confident interval τ can be solved from the following equation: ∫ CoV aRτ ∫ V aRq −∞ −∞ f(x, y)dxdy = τq Where f(x, y) is the joint density function of X and Y . By analogy, the modified version of Delta CoVaR: ∆CoV aR can be specified as ∆CoV aR τ = CoV aR m|rit V aR it(τ) τ − CoV aR i|rit Median τ Such a slight change of tail event from r it = V aR it (τ) to r it V aR it (τ) seems to be trivial, but will prove to make a significant difference in both the estimation strategy and its statistical properties. 2.1.3 MES Marginal Expected Shortfall (MES) measures the marginal contribution of firm i loss to systemic loss, or the average return of firm i on the α% worse days when the market as a whole is in the tail of its distribution. Formally, the coherent risk of system can be measured by its conditional expected shortfall (ES): ES m,t−1 = E t−1 (r mt |r mt < C) = N∑ ω i E t−1 (r it |r mt < C) Where C is the threshold value which defines the tail event when the market return exceeds it r mt < C. The MES of firm i is defined as the partial derivative of the system’s aggregate risk (ES) with respect to the weight of firm i in the market portfolio: ω i i=1 MES it = ∂ES m,t−1(C) ∂ω i = E t−1 (r it |r mt < C) This measures the marginal contribution of firm i to the systemic risk. The higher the value of MES, the higher the individual contribution of the firm i to the risk of the financial system as a whole. Brownlees and Engle (2011) recently show that the estimation of MES can be reduced into the estimations of three components such as volatility, correlation and tail expectation. The linear market model of Brownlees and Engle (2011) can be summarized as: r mt = σ mt ɛ mt r it = σ it ρ it ɛ mt + σ it √1 − ρ 2 it ξ it (ɛ mt , ξ it ) ∼ F 7
Page 1 and 2: Does Tail Dependence Make A Differe
Page 3 and 4: Figure 1: The upper panel graphs di
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Page 9 and 10: Theorem 1 (Sklar)(1959) Let G be a
Page 11 and 12: Therefore the systemic risk measure
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Page 23 and 24: the Goodness of Fit (GoF) of copula
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Page 27 and 28: 5.2 Empirical Results 5.2.1 Estimat
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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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