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Licensed Unlicensed Requires Authentication Published by De Gruyter October 13, 2018

Risk Analysis of Cumulative Intraday Return Curves

  • Piotr Kokoszka EMAIL logo , Hong Miao , Stilian Stoev and Ben Zheng

Abstract

Motivated by the risk inherent in intraday investing, we propose several ways of quantifying extremal behavior of a time series of curves. A curve can be extreme if it has shape and/or magnitude much different than the bulk of observed curves. Our approach is at the nexus of functional data analysis and extreme value theory. The risk measures we propose allow us to assess probabilities of observing extreme curves not seen in a historical record. These measures complement risk measures based on point-to-point returns, but have different interpretation and information content. Using our approach, we study how the financial crisis of 2008 impacted the extreme behavior of intraday cumulative return curves. We discover different impacts on shares in important sectors of the US economy. The information our analysis provides is in some cases different from the conclusions based on the extreme value analysis of daily closing price returns.

Appendices

A Variances of VaRˆα and ESˆα

A Denote Su=P(Y>u) and Sˆu=Nu/N. Observe that Sˆu is the maximum likelihood estimate of Su, as the number of exceedances of u follows the binomial distribution Bin(n,Su). Since both VaRˆα and ESˆα can be viewed as a function of σˆ, γˆ, and Sˆu, i.e.

VaRˆα:=g1(Sˆu,σˆ,γˆ)=u+σˆγˆ[(Sˆuα)γˆ1]

and

ESˆα:=g2(Sˆu,σˆ,γˆ)=u+σˆ1γˆ+σˆγˆ(1γˆ)[(Sˆuα)γˆ1],

large sample standard errors or confidence intervals for VaRα and ESα can be derived from the delta method. From standard properties of the binomial distribution, VaR(Sˆu)Sˆu(1Sˆu)/n and denote the (i,j) term of the variance-covariance matrix of σˆ and γˆ by vi,j, then the complete covariance-variance matrix for (σˆ,γˆ,Sˆu) is approximately

V=Sˆu(1Sˆu)/n000v1,1v1,20v2,1v2,2.

Hence, by the delta method,

Var(VaRˆα)g1TVg1

and

Var(ESˆα)g2TVg2,

where

g1T=g1Su,g1σ,g1γ=σSuγ1αγ,1γSuαγ1,σγ2Suαγ1+σγSuαγlogSuα

and

g2T=g2Su,g2σ,g2γ=σSuγ1(1γ)αγ,1/γ(1γ)γ+Suαγ1,σ(1γ)2+σ(2γ1)γ2(1γ)2Suαγ1+σγ(1γ)SuαγlogSuσ

evaluated at (Sˆu,σˆ,γˆ).

B Estimation of the angular density

This section explains how the density of Y(u) in eq. (13) can be estimated nonparametrically. Since we consider two scores, this is a density on (π,π). The idea is that if θi,i=1,,n, are i.i.d. points from a density f, then its kernel estimator is given by

(14)fˆh(θ):=2ni=1nϕh(tan(θθi)/2))/cos2((θθi)/2),

where ϕh(x)=ϕ(x/h)/h is probability density on the real line. The idea behind this estimator is to lift a distribution on the real line to the unit circle via the transformation θtan(θ/2). By default, we will use the standard normal density for φ (see also (Hall, Watson, and Cabrera 1987)). To apply this idea to the procedure of resampling angles, suppose (ξ1,i,ξ2,i),i=1,,Nu are the points with norm greater than the threshold u and define θi:=arctan(ξ2,i/ξ1,i) for i=1,,Nu. Choosing the bandwidth h=2π/Nu2/3, we compute eq. (14) using the θis, and so obtain an estimated density function fˆh(θ) of the angles θ on (π,π). From this density, we sample angles that can be used to generate new extreme points. Figure 4 illustrates this method using the asset XLF in the “before” period.

Figure 4: Left panel: Histogram of the observed θi′s$\theta_i's$ with the estimate fˆh(θ)${\hat f}_h (\theta)$ superposed. Right panel: histogram of the same number of new θi′s$\theta_i's$ generated from fˆh(θ)${\hat f}_h (\theta)$.
Figure 4:

Left panel: Histogram of the observed θis with the estimate fˆh(θ) superposed. Right panel: histogram of the same number of new θis generated from fˆh(θ).

C Elaboration on the measures χ and χ

This section provides details of inference based on the extreme dependence measures χ and χ. Essentially, χ provides a measure with which to describe the strength of dependence within the class of asymptotically dependent variables, while χ provides a corresponding measure within the class of asymptotically independent variables. Taken together, the pair (χ,χˉ) provides more complete information characterizing the form and the degree of extremal dependence of any two random variables. For asymptotically dependent variables, χˉ=1 and the value of χ > 0 measures the strength of dependence. For asymptotically independent variables, χ = 0, and the value of χ[1,1) might be used to quantify the strength of extremal independence, which may be interpreted also as a secondary or hidden tail dependence in terms of the notion of hidden regular variation (Resnick 2007).

We begin by providing two examples where χ > 0 (asymptotic dependence) and χ = 0 (asymptotic independence).

Example C.1.

In this example, X and Y have a common risk factor. Let Z, εx and εy be independent GPD(γ,γ,0) random variables with the same index γ > 0, such that

P(Z>u)=P(εx>u)=P(εy>u)=(1+u)1/γ,

for u > 0. Consider the simple factor model

X=σzZ+aεxandY=σzZ+bεy,

where σz, a and b are positive. By applying Lemma C.1 below with a1=b1=σz, a2=a, b2=0, and a3=0, b3=b, we obtain

σX1/γ=σz1/γ+a1/γ,σY1/γ=σz1/γ+b1/γ,

and

χ(X,Y)=σz1/γσz1/γ+a1/γσz1/γσz1/γ+b1/γ=11+max(a1/γ,b1/γ)/σz1/γ.

Thus, the larger the ratio max(a1/γ,b1/γ)/σz1/γ, the smaller the extremal dependence. This is natural since in this case, σz is relatively small and the common factor σzZ has relatively smaller contribution to the simultaneous extremes of X and Y. On the other hand, if σz dominates a and b, the common risk factor leads to more frequent simultaneous extremes, which is reflected by relatively larger values of the extremal dependence measure χ.

Example C.2.

Suppose now that (X,Y) are jointly Gaussian with correlation coefficient ρ(1,1). It can be shown that χ = 0 in this case, i.e. X and Y are always asymptotically independent (unless ρ = 1). The measure χˉ is more appropriate in this case and χ=ρ. See Heffernan (2000) for the details.

Lemma C.1.

Let Z1,,Zp be independent heavy-tailed GPD(γ,γ,0) random variables, i.e. P(Zi>u)=(1+u)1/γ,u>0, for some γ > 0. Define

X=i=1paiZiandY=i=1pbiZi,

where ai>0 and bj>0, for some i,j{1,,p}. Then,

(15)P(X>u,Y>u)u1/γi=1p(aibi)+1/γ,as u,

where xnyn means xn/yn1, xy: = min{x,y} and (x)+:=max{x,0}.

Consequently, the tail-dependence coefficient between X and Y equals

(16)χ(X,Y)=i=1paiσXbiσY+1/γ,

where σX1/γ=i=1p(ai)+1/γ and σY1/γ=i=1p(bi)+1/γ.

Proof.

We shall first obtain Relation eq. (16) as a consequence of eq. (15). For simplicity, let α:=1/γ>0. By formally applying eq. (15) to ai:=bi, we obtain

P(Y>u)σYαuα,where σXα=i=1p(bi)+α,

and similarly P(X>u)σXαuα, as u. Observe that σX>0 and σY>0 since maxi=1,,pai>0 and maxi=1,,pbi>0.

Thus, P(X/σX>u)P(Y/σY>u)uα, as u, and hence

χ(X,Y)=limuP(X/σX>u|Y/σY>u)=limuuαP(X>σXu,Y>σYu).

Now, by applying eq. (15) with ai replaced by ai/σX and bi replaced by bi/σY, the formula eq. (16) follows.

Relation eq. (15) is well-known but we give a proof for completeness. This result can be understood in terms of the so-called one big jump heuristic. Namely, since the Zi’s are independent and heavy-tailed with the same tail exponent α=1/γ>0, asymptotically, a linear combination of the Zi’s is large if one and only one of its terms is extreme. That is, the probabilities that two or more components aiZi contribute to an extreme value of X is asymptotically negligible. Using this principle, one can intuitively see that

P(X>u,Y>u)i=1pP(aiZi>u,biZi>u)=uαi=1p(aibi)+α,

as u.

We will make the above heuristic precise using the notion of multivariate regular variation. We start with some terminology. A set ARp{0} is said to be bounded away from 0 if there is a ball B(0,ϵ) centered at the origin with radius ε > 0 such that AB(0,ϵ)=. A random vector Z=(Z1,,Zp) is said to be regularly varying in Rp{0} if there exists a Borel measure ν on Rp{0}, such that

(17)c(u)P(ZuA)ν(A),

for all measurable bounded away from 0 and ν-continuity sets A, i.e. such that ν(A)=0, where A=AA is the boundary of A.

In the simple case above, it is easy to see that the vector Z with independent GPD(γ,γ,0) components is regularly varying, where ν is supported on the positive orthant [0,)p. One can identify the measure ν by taking A=[0,x]c={yRp:yi>xiforsomei=1,,p}, for x0. Indeed, by using the independence of the Zi’s and the inclusion–exclusion formula, we get as u,

P(ZuA)Pi=1p{Zi>uxi}uαi=1pxiα=:uαν(A).

Thus eq. (17) holds with c(u):=uα, where the measure ν is concentrated on the positive axes i:={λei:λ>0},i=1,,p, with e1=(1,0,,0),,ep=(0,,0,1) denoting the standard basis of Rp. In general, for a measurable set A, we have the formula

(18)ν(A)=i=1pνα(πi(Ai)),

where να(x,)=xα,x>0 is a measure on the positive half-line (0,), and πi:RpR is the projection on the i-th coordinate axis.

With this general tool, we can establish the tail behavior of various functionals of Z by relating them to suitable sets A. In particular, observe that

{X>u}={ZuA},and{Y>u}={ZuB}.

where A:={z=(zi)i=1p:i=1paizi>1} and B:={z:i=1pbizi>1}. Observe that both A and B are bounded away from 0 and it can be shown, using the scaling properties of the measure ν, that A and B are both ν-continuity sets. Thus, by eq. (17) applied to the set AB, we obtain

uαP(X>u,Y>u)=uαP(Zu(AB))ν(AB),

as u. By elementary geometric considerations, however, (AB)i=, unless both ai and bi are positive. In this case, if aibi>0, we have

πi(ABi)=(1/ai,)(1/bi,)=(1/(aibi),).

This, in view of eq. (18), yields the formula eq. (15) and completes the proof.

Equivalent definition of χ

Before we discuss any estimation methods of χ and χ, it is useful to introduce an alternative way of defining the two measures. The joint distribution of a set of variables can be separated into their respective marginal distributions and dependence structure among them. This idea is also well known in the study of copulas. In order to focus on the dependence structure of two variables, it is helpful to remove the influence of marginal aspects first by transforming the raw data to a common marginal distribution. After such a transformation, differences in distributions are purely due to dependence structures. Now we transform the bivariate variables (X,Y) to unit Fréchet marginals S and T as follows:

S=1/logFX(X)andT=1/logFY(Y),

where FX and FY are the marginal distribution functions of X and Y, respectively. Notice that there are two typical ways to estimate FX and FY practically. Each of the two marginals can either be approximated from a GPD family using univariate EVT, or be simply approximated by its empirical cumulative distribution function (ECDF). It follows that S and T have the common distribution function F(s)=e1/s. Thus, we can show that χ can be equivalently defined as

χ=limq1P(F(T)>q|F(S)>q)=limsP(T>s|S>s)=limsP(T>s,S>s)P(S>s).

Estimation of χ and χ

Ledford and Tawn (1996 and 1998) established that under weak conditions

P(S>s,T>s)L(s)s1/ηass,

where 0 < η ≤ 1 is a constant and L is a slowly varying function. From this representation, we have

χ=2η1,

and if χ=1, corresponding to η = 1, then χ=limsL(s). So the estimation of η and limsL(s) provide the basis for estimating χ and χ. Here, the key point is to estimate the joint distribution P(S > s,T > s) hence η and L(s). Let Z = min(S,T), so P(S > s,T > s) = P(min(S,T) > s) = P(Z > s). Inference follows using univariate extreme value techniques to fit a GPD to the data points in Z that exceed a large fixed threshold u, then the estimated shape parameter of the fitted distribution provides an estimate of η.

Before we estimate χ, it is important to decide if there exists an asymptotic dependence. We thus first test the null hypothesis χ=1. Only if there is no significant evidence to reject it, we estimate χ.

Testing χ=1

There are basically two major different ways to test if χ=1. First, we can simply create a confidence interval for an estimate of χ. From the discussion above, the estimate and standard error of the shape parameter η can be obtained in the usual way from standard likelihood theory. So by the Delta method, one can obtain the corresponding estimate and standard error of χ as

χˆ=2ηˆ1andSE(χˆ)=2SE(ηˆ),

where SE(ηˆ) can be obtained from the standard maximum likelihood method. Hence, a 95% confidence interval of the true χ is approximated by

(χˆ1.96SE(χˆ),χˆ+1.96SE(χˆ)),

as the sample size in the study is big enough. Thus, the hypothesis χ=1 is rejected (at level 0.05) when the confidence interval does not capture 1. For the second way, given that χ=1 corresponds to η = 1, one can use anova in R to perform a likelihood ratio test for asymptotic dependence, with the null hypothesis η = 1 versus the alternative η < 1. We refer to (Stephenson 2012) for the details of R implementation.

Estimation of χ

Once the above test shows no significant evidence to reject χ=1, we can estimate χ under the assumption that χ=η=1. (Poon, Rockinger, and Tawn 2004) provided a maximum-likelihood estimator of χ, i.e. the natural non-parametric estimator under the constraint χˆ=1

(19)χˆ=unun,
SE(χˆ)=u2nu(nnu)n3,

where n is the sample size and nu is the number of observations of variable Z that exceeds the threshold u.

D Large tables and displays

Figure 5: VaR and ES estimates (with standard errors) of norms of CIDRs for the nine sector EFTs.
Figure 5:

VaR and ES estimates (with standard errors) of norms of CIDRs for the nine sector EFTs.

Figure 6: VaR and ES estimates (with standard errors) of magnitude of point-to-point returns for the seven sector EFTs.
Figure 6:

VaR and ES estimates (with standard errors) of magnitude of point-to-point returns for the seven sector EFTs.

Figure 7: VaR estimates (bootstrap): The scatter plot of XLF scores (financials) with VaR estimates. On each plot, the big black thick circle corresponds to the norm at VaR0.004${\rm VaR}_{0.004}$ (or the 1-year return level), while the bigger grey circle corresponds to the norm at VaR0.0001${\rm VaR}_{0.0001}$. The stars represent the real extreme points with norm greater than VaR0.004${\rm VaR}_{0.004}$. The little gray circles represent the simulated extreme points with norm greater than VaR0.004${\rm VaR}_{0.004}$ based on the bootstrap method.
Figure 7:

VaR estimates (bootstrap): The scatter plot of XLF scores (financials) with VaR estimates. On each plot, the big black thick circle corresponds to the norm at VaR0.004 (or the 1-year return level), while the bigger grey circle corresponds to the norm at VaR0.0001. The stars represent the real extreme points with norm greater than VaR0.004. The little gray circles represent the simulated extreme points with norm greater than VaR0.004 based on the bootstrap method.

Figure 8: VaR extremal regions (bootstrap): The extremal regions of CIDRs corresponding to extreme regions in different quadrants on the (ξ1,ξ2$\xi_1, \xi_2$) plane in Figure 7. On each panel, the two curves in the same type, respectively, represents the lower and upper bound of the extremal region of CIDRs regarding one of the four periods under comparison. The thick solid curves represent the real extremal CIDRs with norm greater than VaR0.004${\rm VaR}_{0.004}$.
Figure 8:

VaR extremal regions (bootstrap): The extremal regions of CIDRs corresponding to extreme regions in different quadrants on the (ξ1,ξ2) plane in Figure 7. On each panel, the two curves in the same type, respectively, represents the lower and upper bound of the extremal region of CIDRs regarding one of the four periods under comparison. The thick solid curves represent the real extremal CIDRs with norm greater than VaR0.004.

Figure 9: ES estimates (bootstrap): The scatter plot of XLF scores (financials) with ES estimates. On each plot, the big black thick circle corresponds to the norm at ES0.004${\rm ES}_{0.004}$ (or the 1-year expected shortfall). The stars represent the real extreme points with norms greater than VaR0.004${\rm VaR}_{0.004}$. The little gray circles represent the simulated extreme points with norm at ES0.004${\rm ES}_{0.004}$ based on the bootstrap method.
Figure 9:

ES estimates (bootstrap): The scatter plot of XLF scores (financials) with ES estimates. On each plot, the big black thick circle corresponds to the norm at ES0.004 (or the 1-year expected shortfall). The stars represent the real extreme points with norms greater than VaR0.004. The little gray circles represent the simulated extreme points with norm at ES0.004 based on the bootstrap method.

Figure 10: ES extremal regions (bootstrap): The extremal regions of CIDRs corresponding to extreme regions in different quadrants on the (ξ1,ξ2$\xi_1, \xi_2$) plane in Figure 9. On each panel, the two curves in the same type, respectively, represent the lower and upper bound of the extremal region of CIDRs regarding one of the four periods under comparison. The thick solid curves represent the real extremal CIDRs with norm greater than VaR0.004${\rm VaR}_{0.004}$.
Figure 10:

ES extremal regions (bootstrap): The extremal regions of CIDRs corresponding to extreme regions in different quadrants on the (ξ1,ξ2) plane in Figure 9. On each panel, the two curves in the same type, respectively, represent the lower and upper bound of the extremal region of CIDRs regarding one of the four periods under comparison. The thick solid curves represent the real extremal CIDRs with norm greater than VaR0.004.

Figure 11: VaR estimates (KDE): The scatter plot of XLF scores (financials) with VaR estimates. On each plot, the big black thick circle corresponds to the norm at VaR0.004${\rm VaR}_{0.004}$ (or the 1-year return level), while the bigger gray circle corresponds to the norm at VaR0.0001${\rm VaR}_{0.0001}$. The stars represent the real extreme points with norm greater than VaR0.004${\rm VaR}_{0.004}$. The little gray circles represent the simulated extreme points with norm greater than VaR0.004${\rm VaR}_{0.004}$ based on KDE.
Figure 11:

VaR estimates (KDE): The scatter plot of XLF scores (financials) with VaR estimates. On each plot, the big black thick circle corresponds to the norm at VaR0.004 (or the 1-year return level), while the bigger gray circle corresponds to the norm at VaR0.0001. The stars represent the real extreme points with norm greater than VaR0.004. The little gray circles represent the simulated extreme points with norm greater than VaR0.004 based on KDE.

Figure 12: VaR extremal regions (KDE): The extremal regions of CIDRs corresponding to extreme regions in different quadrants on the (ξ1,ξ2$\xi_1, \xi_2$) plane in Figure 11. On each panel, the two curves in the same type, respectively, represent the lower and upper bound of the extremal region of CIDRs regarding one of the four periods under comparison. The thick solid curves represent the real extremal CIDRs with norm greater than VaR0.004${\rm VaR}_{0.004}$.
Figure 12:

VaR extremal regions (KDE): The extremal regions of CIDRs corresponding to extreme regions in different quadrants on the (ξ1,ξ2) plane in Figure 11. On each panel, the two curves in the same type, respectively, represent the lower and upper bound of the extremal region of CIDRs regarding one of the four periods under comparison. The thick solid curves represent the real extremal CIDRs with norm greater than VaR0.004.

Figure 13: ES estimates (KDE): The scatter plot of XLF scores (financials) with ES estimates. On each plot, the big black thick circle corresponds to the norm at ES0.004${\rm ES}_{0.004}$ (or the 1-year expected shortfall). The stars represent the real extreme points with norms greater than VaR0.004${\rm VaR}_{0.004}$. The little gray circles represent the simulated extreme points with norm at ES0.004${\rm ES}_{0.004}$ based on KDE.
Figure 13:

ES estimates (KDE): The scatter plot of XLF scores (financials) with ES estimates. On each plot, the big black thick circle corresponds to the norm at ES0.004 (or the 1-year expected shortfall). The stars represent the real extreme points with norms greater than VaR0.004. The little gray circles represent the simulated extreme points with norm at ES0.004 based on KDE.

Figure 14: ES extremal regions (KDE): The extremal regions of CIDRs corresponding to extreme regions in different quadrants on the (ξ1,ξ2$\xi_1, \xi_2$) plane in Figure 13. On each panel, the two curves in the same type, respectively, represent the lower and upper bound of the extremal region of CIDRs regarding one of the four periods under comparison. The thick solid curves represent the real extremal CIDRs with norm greater than VaR0.004${\rm VaR}_{0.004}$.
Figure 14:

ES extremal regions (KDE): The extremal regions of CIDRs corresponding to extreme regions in different quadrants on the (ξ1,ξ2) plane in Figure 13. On each panel, the two curves in the same type, respectively, represent the lower and upper bound of the extremal region of CIDRs regarding one of the four periods under comparison. The thick solid curves represent the real extremal CIDRs with norm greater than VaR0.004.

Figure 15: Estimates of χ(q) and χ‾(q)$\overline \chi (q)$ for the squared scores of the first two CIDR FPCs for XLF. Dotted lines correspond to approximate 95% confidence intervals. From top to bottom, the panels show the results for the periods “before,” “during,” “after1,” and “after2”.
Figure 15:

Estimates of χ(q) and χ(q) for the squared scores of the first two CIDR FPCs for XLF. Dotted lines correspond to approximate 95% confidence intervals. From top to bottom, the panels show the results for the periods “before,” “during,” “after1,” and “after2”.

Table 3:

Estimation results for the norm R for nine sector ETFs. The value-at-risk estimates VaR.004 and VaR.0008, respectively, represent the 1-year and 5-year return levels. Standard errors in parentheses. Recall that u is the threshold and Nu is the sample size of values over the threshold.

ETFPrduNuParameter γˆVaR.004ES.004VaR.0008ES.0008
Bf0.8540.12 (0.16)2.46 (0.16)3.08 (0.13)3.43 (0.94)4.18 (0.34)
XLFDr2.7710.27 (0.09)7.83 (0.20)8.40 (0.70)8.78 (0.43)9.15 (0.97)
(Financials)A12.5330.24 (0.20)6.81 (0.37)7.54 (1.57)8.02 (1.23)8.52 (2.56)
A21.3510.02 (0.16)3.11 (0.13)3.65 (0.19)3.98 (0.69)4.54 (0.35)
Bf0.8670.08 (0.11)2.13 (0.04)2.39 (0.09)2.56 (0.13)2.79 (0.17)
XLKDr1.6690.08 (0.11)4.96 (0.23)5.64 (0.43)6.06 (0.79)6.66 (0.74)
(Technology)A11.1730.16 (0.12)3.08 (0.19)3.90 (0.19)4.33 (1.09)5.39 (0.65)
A20.8680.10 (0.15)1.99 (0.03)2.23 (0.11)2.37 (0.14)2.57 (0.21)
Bf0.6780.09 (0.11)1.91 (0.06)2.33 (0.05)2.58 (0.31)3.06 (0.12)
XLYDr2.2580.02 (0.13)7.06 (0.69)8.29 (1.11)9.04 (2.92)10.22 (2.03)
(Cons. Dis.)A11.4630.10 (0.17)4.20 (0.41)5.18 (0.33)5.74 (2.49)6.88 (0.64)
A20.9690.06 (0.15)2.56 (0.08)2.92 (0.16)3.14 (0.35)3.47 (0.31)
Bf0.6500.15 (0.15)1.34 (0.01)1.47 (0.08)1.55 (0.04)1.65 (0.15)
XLPDr0.61850.20 (0.08)5.38 (0.82)7.27 (0.43)8.25 (4.41)10.84 (1.37)
(Cons. Stap.)A10.8720.10 (0.12)1.92 (0.02)2.13 (0.07)2.27 (0.09)2.45 (0.14)
A20.6660.12 (0.14)1.51 (0.04)1.84 (0.06)2.03 (0.21)2.43 (0.21)
Bf1.3500.08 (0.19)2.46 (0.05)2.71 (0.16)2.87 (0.23)3.09 (0.32)
XLEDr2.5510.23 (0.14)5.79 (0.16)6.27 (0.60)6.58 (0.49)6.91 (0.94)
(Energy)A11.7510.23 (0.15)3.45 (0.05)3.71 (0.23)3.88 (0.14)4.06 (0.39)
A21.3570.12 (0.13)2.93 (0.05)3.24 (0.16)3.44 (0.19)3.70 (0.30)
Bf0.5660.11 (0.14)1.74 (0.09)2.24 (0.18)2.51 (0.54)3.16 (0.77)
XLVDr1.5530.06 (0.17)4.46 (0.29)5.15 (0.58)5.59 (1.31)6.22 (1.08)
(Health)A10.8950.12 (0.11)3.13 (0.20)3.93 (0.13)4.38 (1.01)5.35 (0.30)
A20.7710.11 (0.11)2.20 (0.08)2.73 (0.08)3.03 (0.41)3.66 (0.19)
Bf0.8480.12 (0.16)1.69 (0.02)1.87 (0.10)1.98 (0.08)2.12 (0.20)
XLIDr1.5880.14 (0.13)5.28 (0.68)6.67 (0.41)7.44 (3.92)9.17 (0.88)
(Industrials)A11.5650.10 (0.11)3.83 (0.10)4.28 (0.22)4.57 (0.31)4.95 (0.39)
A21.2480.10 (0.17)2.42 (0.04)2.69 (0.14)2.85 (0.17)3.08 (0.28)
Bf0.8850.09 (0.13)2.75 (0.16)3.35 (0.09)3.70 (0.81)4.39 (0.18)
XLBDr2.0740.02 (0.14)5.83 (0.51)6.87 (0.57)7.49 (2.40)8.57 (1.07)
(Materials)A11.5700.05 (0.14)3.99 (0.17)4.54 (0.30)4.88 (0.71)5.38 (0.54)
A21.2640.23 (0.10)2.84 (0.03)3.06 (0.13)3.21 (0.07)3.36 (0.21)
Bf0.8570.01 (0.15)2.17 (0.07)2.52 (0.10)2.74 (0.32)3.08 (0.20)
XLUDr0.91630.13 (0.09)5.78 (0.75)7.35 (0.34)8.22 (3.53)10.13 (0.67)
(Utilities)A11.1560.06 (0.14)2.83 (0.08)3.24 (0.17)3.49 (0.32)3.86 (0.32)
A20.7740.16 (0.14)2.17 (0.12)2.77 (0.17)3.10 (0.74)3.88 (0.73)
Table 4:

Estimation results for the magnitude of the first score |ξ1| for nine sector ETFs; quantities reported are the same as in Table 3.

ETFPrduNuParameter γˆVaR.004ES.004VaR.0008ES.0008
Bf0.7600.10 (0.14)2.30 (0.12)2.86 (0.10)3.17 (0.64)3.82 (0.21)
XLFDr2.5690.30 (0.10)7.57 (0.18)8.11 (0.69)8.46 (0.38)8.79 (0.95)
(Financials)A12.3340.23 (0.19)6.72 (0.40)7.49 (1.58)8.00 (1.35)8.54 (2.60)
A21.2570.02 (0.15)2.98 (0.12)3.48 (0.16)3.78 (0.58)4.30 (0.30)
Bf0.9490.07 (0.15)2.07 (0.04)2.34 (0.10)2.50 (0.16)2.74 (0.20)
XLKDr1.6600.06 (0.13)4.91 (0.29)5.66 (0.52)6.13 (1.14)6.82 (0.94)
(Technology)A11.0760.14 (0.12)3.05 (0.18)3.84 (0.16)4.27 (0.99)5.25 (0.43)
A20.9450.03 (0.23)1.97 (0.06)2.26 (0.13)2.44 (0.33)2.72 (0.27)
Bf0.6680.12 (0.13)1.87 (0.08)2.32 (0.07)2.58 (0.40)3.12 (0.22)
XLYDr2.2510.04 (0.14)6.97 (0.64)8.14 (1.17)8.86 (2.56)9.95 (2.13)
(Cons. Dis.)A11.01080.06 (0.11)4.04 (0.28)4.88 (0.21)5.37 (1.23)6.30 (0.37)
A21.0480.14 (0.17)2.45 (0.05)2.74 (0.20)2.92 (0.21)3.14 (0.39)
Bf0.6440.19 (0.14)1.31 (0.01)1.43 (0.07)1.50 (0.03)1.59 (0.15)
XLPDr0.51880.19 (0.08)5.22 (0.77)7.06 (0.40)8.03 (4.06)10.54 (1.24)
(Cons. Stap.)A10.9480.07 (0.15)1.89 (0.03)2.13 (0.08)2.28 (0.12)2.49 (0.17)
A20.5900.13 (0.12)1.47 (0.04)1.81 (0.07)2.01 (0.21)2.43 (0.29)
Bf1.2520.21 (0.15)2.39 (0.02)2.57 (0.15)2.68 (0.08)2.80 (0.26)
XLEDr2.5450.14 (0.17)5.62 (0.24)6.24 (0.71)6.64 (0.96)7.14 (1.26)
(Energy)A11.7460.23 (0.16)3.40 (0.05)3.66 (0.24)3.83 (0.14)4.01 (0.42)
A21.2570.17 (0.12)2.85 (0.04)3.13 (0.15)3.31 (0.13)3.52 (0.27)
Bf0.5580.15 (0.16)1.69 (0.09)2.17 (0.16)2.43 (0.53)3.05 (0.66)
XLVDr1.7350.18 (0.22)4.31 (0.18)4.81 (0.80)5.14 (0.79)5.51 (1.45)
(Health)A10.71040.10 (0.10)3.07 (0.18)3.83 (0.12)4.26 (0.84)5.16 (0.24)
A20.7590.08 (0.11)2.16 (0.08)2.66 (0.08)2.94 (0.35)3.51 (0.16)
Bf0.8390.25 (0.15)1.64 (0.01)1.76 (0.10)1.84 (0.03)1.93 (0.19)
XLIDr1.3880.11 (0.12)5.14 (0.60)6.43 (0.42)7.16 (3.15)8.70 (0.82)
(Industrials)A11.6530.07 (0.13)3.80 (0.12)4.30 (0.26)4.62 (0.45)5.07 (0.48)
A21.2410.09 (0.21)2.37 (0.05)2.65 (0.18)2.82 (0.24)3.06 (0.38)
Bf0.8760.13 (0.15)2.74 (0.21)3.44 (0.13)3.83 (1.20)4.68 (0.38)
XLBDr1.5940.04 (0.11)5.65 (0.39)6.52 (0.52)7.06 (1.51)7.86 (0.91)
(Materials)A11.6480.22 (0.11)3.94 (0.07)4.31 (0.27)4.55 (0.17)4.81 (0.43)
A21.3500.20 (0.12)2.79 (0.03)3.03 (0.15)3.19 (0.09)3.36 (0.26)
Bf0.8500.06 (0.19)2.16 (0.11)2.60 (0.09)2.86 (0.62)3.34 (0.16)
XLUDr0.91360.06 (0.08)5.35 (0.48)6.56 (0.34)7.26 (1.97)8.60 (0.61)
(Utilities)A11.1500.01 (0.16)2.80 (0.10)3.27 (0.18)3.55 (0.48)4.00 (0.35)
A20.6930.11 (0.12)2.04 (0.08)2.53 (0.06)2.81 (0.41)3.39 (0.17)
Table 5:

Estimation results for the magnitude of the point-to-point returns for nine sector ETFs; quantities reported are the same as in Table 3.

ETFPrduNuParameter γˆVaR.004ES.004VaR.0008ES.0008
Bf1.1590.06 (0.18)4.58 (0.65)5.72 (0.47)6.28 (3.61)7.53 (0.97)
XLFDr5.3530.23 (0.13)15.53 (1.37)16.00 (2.77)17.95 (3.82)17.96 (4.27)
(Financials)A14.9260.21 (0.23)13.52 (2.03)14.46 (4.77)16.47 (7.46)16.89 (8.80)
A22.4530.02 (0.14)7.77 (0.90)9.15 (1.08)10.06 (3.94)11.40 (2.16)
Bf1.4400.03 (0.14)2.83 (0.07)3.16 (0.08)3.44 (0.25)3.76 (0.17)
XLKDr2.9580.10 (0.14)9.20 (1.77)11.75 (1.59)12.72 (9.59)15.66 (3.46)
(Technology)A12.1460.10 (0.12)5.05 (0.19)5.51 (0.30)6.11 (0.61)6.48(0.57)
A21.8480.02 (0.18)4.89 (0.41)5.84 (0.40)6.37 (2.17)7.34 (0.85)
Bf1.1570.04 (0.15)3.26 (0.15)3.71 (0.17)4.06 (0.61)4.48 (0.35)
XLYDr3.4500.04 (0.14)9.50 (1.08)10.90 (1.33)11.98 (4.38)13.29 (2.61)
(Cons. Dis.)A12.5430.15 (0.12)5.95 (0.23)6.31 (0.42)7.07 (0.69)7.28 (0.76)
A22.0420.05 (0.16)5.35 (0.35)6.11 (0.50)6.75 (1.49)7.43 (1.02)
Bf0.8510.04 (0.14)2.27 (0.09)2.76 (0.08)2.99 (0.41)3.50 (0.16)
XLPDr1.8560.11 (0.15)6.65 (1.11)8.64 (1.01)9.43 (6.23)11.75 (2.19)
(Cons. Stap.)A11.2600.10 (0.11)3.24 (0.08)3.54 (0.13)3.91 (0.24)4.16 (0.24)
A21.2420.00 (0.16)3.25 (0.16)3.87 (0.19)4.24 (0.78)4.86 (0.41)
Bf1.7670.17 (0.17)3.82 (0.11)3.89 (0.17)4.34 (0.41)4.33 (0.35)
XLEDr3.5640.04 (0.14)15.68 (5.60)19.38 (5.57)21.36 (27.57)25.29 (11.57)
(Energy)A12.7510.12 (0.10)6.16 (0.21)6.57 (0.34)7.30 (0.58)7.59 (0.60)
A22.3570.03 (0.12)6.90 (0.57)8.00 (0.68)8.77 (2.24)9.82 (1.32)
Bf0.7640.12 (0.15)2.53 (0.08)2.78 (0.17)3.06 (0.31)3.25 (0.32)
XLVDr2.2470.06 (0.16)7.84 (1.47)9.91 (1.56)10.88 (7.99)13.16 (3.41)
(Health)A11.6380.16 (0.14)3.98 (0.12)4.25 (0.26)4.77 (0.37)4.93 (0.50)
A21.4570.02 (0.14)4.44 (0.29)5.17 (0.33)5.68 (1.26)6.38 (0.67)
Bf1.2460.10 (0.18)3.04 (0.11)3.33 (0.17)3.68 (0.45)3.92 (0.36)
XLIDr2.9550.17 (0.12)8.21 (0.45)8.67 (0.81)9.67 (1.35)9.92 (1.36)
(Industrials)A12.3580.10 (0.13)6.42 (0.35)7.03 (0.51)7.77 (1.25)8.25 (0.94)
A22.2450.12 (0.14)5.87 (0.31)6.41 (0.52)7.13 (1.09)7.55 (0.98)
Bf1.3590.19 (0.13)3.72 (0.09)3.87 (0.21)4.31 (0.28)4.36 (0.37)
XLBDr4.1400.22 (0.14)10.59 (0.70)10.93 (1.48)12.37 (1.98)12.39 (2.46)
(Materials)A12.6580.32 (0.12)5.51 (0.07)5.19 (0.20)6.03 (0.16)5.58 (0.32)
A22.6400.09 (0.19)6.66 (0.53)7.40 (0.77)8.22 (2.40)8.83 (1.60)
Bf1.3430.13 (0.21)3.80 (0.21)4.15 (0.39)4.62 (0.95)4.88 (0.82)
XLUDr2.4510.11 (0.17)9.60 (2.84)12.66 (2.66)13.91 (17.54)17.51 (6.12)
(Utilities)A11.7510.25 (0.15)3.59 (0.05)3.52 (0.14)4.03 (0.16)3.86 (0.27)
A21.1700.19 (0.14)3.90 (0.44)5.40 (0.58)5.81 (2.87)7.75 (1.78)
Table 6:

Fisher’s exact test on independence of the four regions with frequencies of points falling in each region given their norms above the threshold.

ETFPrduNu(+, +)(+, )(, )(, +)P value
Bf0.85215815140.27
XLFDr2.768132118160.62
(Financials)A12.529676100.47
A21.350141012140.78
Bf0.861141215201.00
XLKDr1.664181515161.00
(Technology)A11.169161422170.47
A20.866161320170.62
Bf0.674191618211.00
XLYDr2.253121612130.59
(Cons. Dis.)A11.461181216150.44
A20.963181510200.44
Bf0.64713813130.56
XLPDr0.6171454837410.65
(Cons. Stap.)A10.867141716200.47
A20.660151214191.00
Bf1.34613128130.56
XLEDr2.54812118170.38
(Energy)A11.750121513101.00
A21.35416815150.27
Bf0.563151716151.00
XLVDr1.55116813140.39
(Health)A10.892251929190.14
A20.765171019190.32
Bf0.844101112111.00
XLIDr1.577211817211.00
(Industrials)A11.563161517150.80
A21.246101114111.00
Bf0.881211920211.00
XLBDr2.067191211250.61
(Materials)A11.560131517151.00
A21.261201113170.61
Bf0.855151215130.59
XLUDr0.9147393834361.00
(Utilities)A11.153111314150.78
A20.769201813180.81
Table 7:

Fisher’s exact test on independence of the four regions with probabilities of points falling in each region given their norms above the threshold.

ETFPrduNuP(+, +)P(+, )P(, )P(, +)P value
Bf0.8520.290.150.290.270.27
XLFDr2.7680.190.310.260.240.62
(Financials)A12.5290.210.240.210.340.47
A21.3500.280.200.240.280.78
Bf0.8610.230.200.250.331.00
XLKDr1.6640.280.230.230.251.00
(Technology)A11.1690.230.200.320.250.47
A20.8660.240.200.300.260.62
Bf0.6740.260.220.240.281.00
XLYDr2.2530.230.300.230.250.59
(Cons. Dis.)A11.4610.300.200.260.250.44
A20.9630.290.240.160.320.44
Bf0.6470.280.170.280.280.56
XLPDr0.61710.260.280.220.240.65
(Cons. Stap.)A10.8670.210.250.240.300.47
A20.6600.250.200.230.321.00
Bf1.3460.280.260.170.280.56
XLEDr2.5480.250.230.170.350.38
(Energy)A11.7500.240.300.260.201.00
A21.3540.300.150.280.280.27
Bf0.5630.240.270.250.241.00
XLVDr1.5510.310.160.250.270.39
(Health)A10.8920.270.210.320.210.14
A20.7650.260.150.290.290.32
Bf0.8440.230.250.270.251.00
XLIDr1.5770.270.230.220.271.00
(Industrials)A11.5630.250.240.270.240.80
A21.2460.220.240.300.241.00
Bf0.8810.260.230.250.261.00
XLBDr2.0670.280.180.160.370.61
(Materials)A11.5600.220.250.280.251.00
A21.2610.330.180.210.280.61
Bf0.8550.270.220.270.240.59
XLUDr0.91470.270.260.230.241.00
(Utilities)A11.1530.210.250.260.280.78
A20.7690.290.260.190.260.81
Table 8:

Estimation results of extreme dependence between ξ12 and ξ22 for the nine ETF sectors. Standard errors in parentheses. The value of q is the quantile for the threshold. P value is for the likelihood ratio test of χ=1, and “*” indicates cases where χ is not significantly different from 1.

ETFPrdqnuχˆP valueχˆ
Bf0.80630.59 (0.43)0.21*0.40 (0.04)
XLFDr0.701060.25 (0.32)0.000
(Financials)A10.80710.53 (0.42)0.15*0.43 (0.05)
A20.80720.73 (0.41)0.39*0.35 (0.04)
Bf0.80630.67 (0.43)0.32*0.38 (0.04)
XLKDr0.701050.01 (0.33)0.000
(Technology)A10.80710.20 (0.35)0.000
A20.75890.21 (0.33)0.000
Bf0.70940.62 (0.34)0.15*0.42 (0.04)
XLYDr0.75870.28 (0.39)0.020
(Cons. Dis.)A10.651250.31 (0.28)0.000
A20.75900.37 (0.37)0.040
Bf0.80630.29 (0.41)0.040
XLPDr0.601400.62 (0.33)0.13*0.42 (0.03)
(Cons. Stap.)A10.85540.03 (0.43)0.010
A20.75900.42 (0.38)0.050
Bf0.80630.59 (0.32)0.000
XLEDr0.80700.15 (0.40)0.010
(Energy)A10.80710.21 (0.37)0.010
A20.701080.61 (0.35)0.14*0.39 (0.03)
Bf0.80630.97 (0.50)0.93*0.38 (0.04)
XLVDr0.80700.10 (0.33)0.000
(Health)A10.701060.25 (0.30)0.000
A20.85540.19 (0.42)0.020
Bf0.85470.54 (0.47)0.22*0.37 (0.05)
XLIDr0.80700.58 (0.28)0.000
(Industrials)A10.80700.16 (0.35)0.000
A20.80720.23 (0.39)0.020
Bf0.85470.44 (0.50)0.16*0.35 (0.05)
XLBDr0.80700.37 (0.34)0.000
(Materials)A10.80710.36 (0.30)0.000
A20.80720.13 (0.36)0.010
Bf0.80620.07 (0.37)0.000
XLUDr0.80700.09 (0.38)0.010
(Utilities)A10.80710.31 (0.40)0.040
A20.80720.87 (0.45)0.70*0.36 (0.04)

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Published Online: 2018-10-13

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