Estimating Expected Shortfall Using a Conditional Autoregressive Model: CARES

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1 Estimating Expected Shortfall Using a Conditional Autoregressive Model: CARES Yin Liao and Daniel Smith Queensland University of echnology Brisbane, QLD, 4001 April 21, 2015 Abstract Expected shortfall (ES) has recently become an increasingly popular measure of downside risk because of its conceptual appeal and desirable properties. his article proposes a new conditional autoregressive model for estimating ES (CARES) by specifying the evolution of ES over time using an autoregressive process. We develop a tail-based least-squares method of estimating the model parameters and establish the consistency and asymptotic normality of the resultant estimator. Our simulation results show that the CARES model demonstrates superior finitesample forecasting performance compared with other existing methods. o use examples with real data, we implement the model to evaluate the ESs of one stock index and two individual stocks. 1 Introduction he recent financial crisis has spurred renewed interest in developing accurate downside risk measures for financial markets and institutions. Value at risk (VaR), which measures the possible maximum loss of an asset or a certain portfolio within a given time period We thank conference participants at the Princeton-QU-SMU Financial Frontiers Workshop at the Queensland University of echnology, the Current Research in Empirical Finance and Macroeconomics Workshop at the University of Queensland and the 2014 Australasian Econometric Society Meeting at the University of asmania. 1

2 at a given confidence level, has been the most popular measure of downside risk used in practice over the past decade. Despite its great success, VaR has two major drawbacks. First, this measure lacks subadditivity and is not a coherent measure. Hence, the VaR of a portfolio can be larger than the sum of its individual VaRs, which contradicts the conventional wisdom that diversification reduces risk. Second, the VaR measure ignores the magnitude of the loss because it accounts for only the probabilities of losses but not their sizes. In response to these shortcomings, a number of alternative risk measures have been gaining traction, particularly expected shortfall (ES) (see Artzner, Delbaen, Eber, and Heath (1999)). ES is defined as the conditional expectation of a loss given that the loss is larger than the VaR. In contrast to VaR, ES provides information regarding the magnitude of the loss beyond the VaR level and is subadditive. However, the calculation of ES can be an intricate computational exercise given the lack of a closed-form formula (Yuan and Wong (2010)), and little research has examined the methods of ES estimation. Our work here contributes to filling this gap. Let X t, t = 1,..., n denote the price of an asset or a portfolio over n periods, which is a stationary process with the marginal distribution function F, and let y t = log(x t /X t 1 ) represent the negative logarithmic return over the t th period. he VaR satisfies V ar τ = inf{u : F (u) τ}, which is essentially the τ th quantile of the return distribution. Consequently, the ES is defined as ES τ = E(y t y t > V ar τ ). Although the ES is conceptually superior to the VaR, ES modeling and estimation remain a challenging problem whose solution lacks consensus. First, the ES measures the tail magnitude of the return distribution beyond a certain point (VaR) such that its modeling and estimation cannot be independent of the modeling and estimation of the VaR. Second, because the distribution of returns typically changes over time, estimating the ES requires a suitable method of modeling the time-varying tail magnitude beyond a certain quantile. A variety of techniques for VaR estimation have been proposed in the literature. Of these methods, most focus on first modeling and estimating the entire distribution of returns and then extracting a certain quantile of the estimated distribution. Although parametric methods for this purpose involve the parameterization of the time-varying financial asset return distribution, nonparametric methods simply rely on historical data for the estimation of the return distribution without imposing any assumptions. However, each of these types of methods can be easily criticized. First, the parametric approach requires an assumption regarding the return distribution, but unfortunately, these assumptions are typically inconsistent with the real data. Meanwhile, the nonparametric approach is notoriously difficult to apply when the data sample is limited, and it assumes that returns are independent and identically distributed (i.i.d.) and hence does not allow for time-varying volatility. 2

3 In a recent exception to traditional approaches, the quantile is modeled directly rather than the entire distribution. Engle and Manganelli (2004) proposes a conditional autoregressive value-at-risk (CAViaR) model, which specifies the evolution of the VaR over time using a specialized type of autoregressive process. Unknown parameters are estimated using quantile regression. he CAViaR model relies on the empirical finding that stock market return volatilities cluster over time. Consequently, the VaR, which is tightly linked to the standard deviation of the distribution, must exhibit similar behavior. his approach has strong appeal in that it provides a modeling framework but does not rely on distributional assumptions. However, the focus of this model is solely on VaR estimation, with no consideration of how to estimate the corresponding ES. We pursue a similar concept in proposing a conditional autoregressive model for the calculation of ES (henceforth referred to as the CARES model). he CARES model specifies the evolution of the ES itself over time using an autoregressive process, and the model parameters are estimated by minimizing the squared loss function in the region of losses in excess of the VaR. he resultant parameter estimates can be regarded as tail-based least-squares estimators. More specifically, to solve the problem of minimizing the squared loss in excess of the VaR, we jointly address two other minimization 1 problems. he first involves minimizing the check loss function min β t=1 (τ I(y t < f t (β)))(y t f t (β)) to obtain an estimate for f t, where f t (β) is a dynamic specification for V ar t that is known up to β, and the second involves minimizing the squared loss 1 function min γ t=1 I(y t < f t (β))(y t g t (γ)) 2, where g t (γ) is a dynamic specification for ES t that is known up to γ. he first-order conditions of these two minimization problems provide two moment conditions that are useful in generalized method of moments (GMM) estimation, and we therefore derive the asymptotic theory of these model parameter estimators under a GMM framework to study their theoretical properties. Meanwhile, we conduct a series of Monte Carlo studies to investigate the finite-sample properties of the model parameter estimators and the finite-sample forecasting performance of the CARES model. Finally, a recent similar work by aylor (2008) should be considered. he work of aylor (2008) also develops an autoregressive model for conditional ES (henceforth referred to as the CARE model), but this model differs from our CARES model in several aspects. First, given the one-to-one mapping between quantiles and expectiles for a given distribution (Efron (1991); Yuan and Wong (2010); Yao and ong (1996)), the CARE model employs the expectile to obtain both VaR and ES estimates. herefore, the success of the CARE model largely depends on how the coverage probability of the expectile (α) is formulated as a function of the coverage probability of the quantile (τ). Because the return distribution varies over time, the value of α changes over time even 3

4 for a fixed τ. he need to estimate α at each time point causes the CARE model not only to be computationally demanding but also to incur greater estimation errors. his effect will degrade the ES forecast, as shown in Section 3. Second, rather than directly estimating the autoregressive model for the conditional ES, aylor (2008) relies on an algebraic link (which is a function of α and τ) between the conditional expectile and quantile to infer the CARE model parameters based on an autoregressive model for the estimated conditional expectile. his link holds for many widely used distributions, but it is violated when the underlying return distribution becomes overly complicated. In this sense, our model allows for a greater degree of flexibility in the model parameters, and the CARE model is therefore nested in our CARES model as a special case when the set of restrictions on the model parameters implied by the link between the conditional expectile and quantile holds. We formally test these restrictions on simulated and real data in Section 3 and Section 4. he remainder of this paper is structured as follows. Section 2 introduces the CARES model and establishes the consistency and asymptotic normality of the model parameter estimators. Section 3 presents a range of Monte Carlo simulations performed to study the finite-sample properties of the model. Several empirical applications are presented in Section 4. Section 5 concludes the paper. 2 CARES model In this section, we provide a description of the CARES model, introduce the model estimation procedure, and discuss the asymptotic theory of the model parameter estimators. 2.1 Model Description o motivate the functional form, we consider a simple scenario in which an asset return r t follows a Gaussian distribution with mean µ t and standard deviation σ t. φ(r t ; µ t, σ 2 t ) and Φ(r t ; µ t, σ 2 t ) respectively denote the density and distribution functions of the return. V ar t for a given probability τ can be simply expressed as V ar t = µ t + σ t Φ [ 1] (τ) (1) 4

5 and the corresponding ES can be computed as ES t = E(r t r t V ar t ) = V ar t φ(r t ; µ t, σt 2 ) r t Φ(V ar t ; µ t, σt 2 ) dr = µ t + = µ t σ 2 t [ σ 2 t φ(r t ; 0, σt 2 ) Φ(V ar t ; 0, σt 2 ) φ(v ar t ; 0, σ 2 t ) Φ(V ar t ; 0, σ 2 t ). Because (V ar t µ t )/σ t = Φ [ 1] (τ), we can rearrange the above equation to obtain V ar t ES t = E(r t r t V ar t ) = µ t σ t φ(φ[ 1] (τ)). (2) τ hus far, both V ar t and ES t are clearly proportional to the standard deviation σ t, and we can therefore reasonably assume the same functional form for the ES as used for the VaR. Note that we use the Gaussian distribution as an example here to illustrate the link between volatility and ES (and VaR). his relationship should hold for any other distribution with a different functional form. Consequently, we draw inspiration from the CAViaR model of Engle and Manganelli (2004)) to propose a CARES model to formalize the dynamic characteristics. Recall the CAViaR model, in which the conditional quantile is specified as an autoregressive function f t (β) that depends on the parameter vector β as follows: f t (β) = β 0 + q β i f t i (β) + i=1 r β q+i l(x t i ), (3) i=1 where the β i f t i (β), i = 1,..., q are the autoregressive terms, which ensure that the quantile changes smoothly over time, and the role of l(x t j ) is to link f t (β) to observable variables that belong to the information set. We introduce a similar functional form for the conditional ES to specify its dynamics as g t (γ) = γ 0 + q γ i g t i (γ) + i=1 r γ q+i m(x t i ), (4) i=1 where γ i g t i (γ), i = 1,..., q are the autoregressive terms and m(x t j ) contains the exogenous variables that belong to the information set. Some examples of the CARES model 5

6 can be readily obtained as follows: Symmetric absolute value: Asymmetric slope: Indirect GARCH(1,1): f t (γ) = γ 1 + γ 2 f t 1 (γ) + γ 3 r t 1 f t (γ) = γ 1 + γ 2 f t 1 (γ) + γ 3 (r t 1 ) + + γ 4 (r t 1 ) f t (γ) = (γ 1 + γ 2 f 2 t 1(γ) + γ 3 r 2 t 1) 1/2 Including the lagged returns as exogenous variables is a natural choice, as we would indeed expect the VaR and ES to increase during the next period when the return becomes either very negative or very positive. he first and third models allow the ES to respond symmetrically to past returns, whereas the second model allows for variation in the responses to positive and negative returns. As discussed in Engle and Manganelli (2004)), the indirect GARCH model will correctly specify the ES when the underlying return data truly follow a GARCH(1,1) process with an i.i.d. error distribution. he symmetric absolute value or asymmetric slope quantile specification will be correct when the actual return follows a GARCH process in which the standard deviation rather than the variance is modeled symmetrically or asymmetrically with i.i.d. errors. In the following analysis, we primarily focus on these three example models. 2.2 Model Estimation Next, we estimate the model parameters using a tail-based least-squares method. his method minimizes the mean of the squared deviation between the return observations beyond a certain quantile (VaR) and the estimated ES from the model. he resulting parameter estimates can be regarded as GMM-type estimators, and we derive the asymptotic theory for these estimators under a GMM framework. In practice, the tail-based least-squares method can be implemented using a two-stage process. Assuming that the τ th quantile of a sample of return observations y 1,..., y follows the CAViaR model, that is, V ar t (τ) = f t (β 0 (τ)), (5) where f is assumed to be known up to the vector of parameters β 0 and the corresponding 6

7 ES depends on another vector of parameters γ 0 such that ES t (τ) = g t (γ 0 (τ)), (6) then both V ar t and ES t are determined by the θ 0 = (θ 01, θ 02 ) = (β 0 (τ), γ 0 (τ) ) that minimizes the loss function E[I(y t < V ar t (τ)) (y t ES t (τ)) 2 ]. (7) A standard method of obtaining the estimator for θ 0, denoted as ˆθ, is to minimize the sample counterpart 1 I(y t < V ar t (τ)) (y t ES t (τ)) 2. (8) t=1 We solve this minimization problem using a two-stage procedure. In the first stage, we estimate equation (5) by solving 1 min β (τ I(y t < V ar t (τ))) (y t V ar t (τ)) (9) t=1 to obtain ˆβ and V arˆ t (τ). his standard quantile regression estimation is used in Engle and Manganelli (2004)) to estimate the CAViaR model. In the second stage, the estimated V ar t (τ) is used as an observation in estimating the parameters of (6) by solving 1 min γ I(y t < t=1 ˆ V ar t (τ)) (y t ES t (τ)) 2. (10) his method can be regarded as a variation of the least-squares approach that focuses only on the tail observations to minimize the sum of the squared estimation errors. Alternatively, the parameters of equations (5) and (6) can be jointly estimated by solving the two problems defined in (9) and (10) simultaneously. he two first-order conditions involved, as specified below, yield moments that are useful in GMM estimation: ( ) 1 t=1 βf t (β(τ)) (τ I(y t < f t (β(τ)))) = 0 1 t=1, (11) γg t (γ(τ)) (y t g t (γ(τ))) I(y t < f t (β(τ)) = 0 where f t (β) = d dβ f t(β) (12) 7

8 and g t (γ) = d dγ g t(γ). (13) herefore, ˆθ is actually the resulting GMM estimator given the two moment conditions presented above. he asymptotic distribution of ˆθ can then be established within the GMM framework as follows. heorem 2.1 and heorem 2.2 show that the GMM estimator ˆθ is consistent and asymptotically normal. he relevant assumptions and a detailed proof are provided in Appendix A. heorem 2.1. (Consistency) Under assumptions 5.1 and 5.2, we have as. ˆθ(τ) P θ 0 (τ) Proof. See Appendix A. heorem 2.2. (Asymptotic normality) Given assumptions , we have as, (ˆθ θ0 ) D N(0, Σ(θ 0 )) (14) where Σ(θ 0 ) = D(θ 0 ) 1 S(θ 0 )(D(θ 0 ) 1 ) with [ ] D11 D D(θ 0 ) = 12 D 21 D [ 22 ] E( = β f t (β 0 (τ)) β f t (β 0 (τ)) h(0)) 0 E( γ g t (γ 0 (τ)) β f t (β 0 (τ)) (f t (β 0 (τ)) g t (γ 0 (τ)))h(0)) E( γ g t (γ 0 (τ)) γ g t (γ 0 (τ)) τ) (15) and = [ ] S11 S S(θ 0 ) = 12 S 21 S [ 22 ] τ(1 τ)e( β f t (β 0 (τ)) β f t (β 0 (τ)) ) 0, 0 E( g t (γ(τ)) γ g t (γ 0 (τ)) ) V (16) 8

9 where h(.) is the density function and V = E((y t g t (γ 0 (τ)) 2 I(y t f t (β 0 (τ)) < 0)). Proof. See Appendix A. he basic concept is that we approximate the (discontinuous) gradient of the objective function as its continuously differentiable expectation and then relate this approximation to the asymptotic first-order condition to set the approximation of the gradient asymptotically equal to zero. his approach enables the use of the standard aylor expansion to derive the asymptotic theory for the model parameter estimators. he method used to obtain such an approximation is provided by the theorem of Huber (1967). his technique is widely used in quantile and expectile regression; see Engle and Manganelli (2004) and Kuan, Yeh, and Hsu (2009) for recent applications. Under the assumptions and conditions of heorem 2.1 and heorem 2.2, the asymptotic variance-covariance matrix Σ(θ) can be consistently estimated as ˆΣ(θ) = ˆD(θ) 1 Ŝ(θ) ˆD(θ) 1, where [ ] D ˆ ˆD(θ) = 11 (θ) Dˆ 12 (θ) Dˆ 21 (θ) Dˆ 22 (θ) Dˆ 11 (θ) = 1 2 c ˆ D 12 (θ) = 0 ˆ D 21 (θ) = ˆ D 22 (θ) = 1 β f t ( ˆβ(τ)) β f t ( ˆβ(τ)) I( y t f t ( ˆβ(τ)) < c ) t=1 1 γ g t (ˆγ(τ)) β f t ( 2 c ˆβ(τ)) (f t ( ˆβ(τ)) g t (ˆγ(τ)))I( y t f t ( ˆβ(τ)) < c ) t=1 γ g t (ˆγ(τ)) γ g t (ˆγ(τ)) τ t=1 [ S11 ˆ (θ) Ŝ(θ) = Sˆ 21 (θ) ˆ ˆ S12 (θ) S22 (θ) ] ˆ S 11 (θ) = 1 Sˆ 12 (θ) = 0 Sˆ 21 (θ) = 0 ˆ S 22 (θ) = 1 τ(1 τ) β f t ( ˆβ(τ)) β f t ( ˆβ(τ)) t=1 g t (ˆγ(τ)) γ g t (ˆγ(τ)) )(y t g t (ˆγ(τ)) 2 I(y t f t ( ˆβ(τ)) < 0), t=1 where c is a bandwidth. We can compute the bandwidth c in several ways. First, we 9

10 follow Engle and Manganelli (2004) in estimating c using a k-nearest neighbor estimator with k = 40 for 1% coverage probability and k = 60 for 5% coverage probability. Alternatively, we follow Koenker (2005) to estimate c as c = ŝ(φ 1 (τ + h ) Φ 1 (τ h )), (17) where ŝ = min(sd(y t f t ( ˆβ)), IQR(y t f t ( ˆβ)))/1.34, and h = 1/5 [ 4.5φ4 (Φ 1 (t)) ] 1/5 (2Φ 1 (t) 2 +1) 2 (see Bofinger (1975)) or h = 1/3 Φ 1 ( ) 2/3 [ 1.5φ2 (Φ 1 (τ)) ] 1/3 (see Hall and (2Φ 1 (τ) 2 +1) 2 Sheather (1988)). Next, we perform a small simulation study to examine the finite-sample performance of the model parameter estimators. In doing so, we assume that an asset or portfolio s return follows a GARCH(1,1) model: r t = σ t z t, σ 2 t = a 0 + a 1 r 2 t 1 + a 2 σ 2 t 1, where the parameters are set to a 0 = 0.025, b 0 = , and c 0 = and the disturbance z t follows a standard Gaussian distribution. Based on the relationship between the conditional VaR/ES and the standard deviation of the return, as shown in Section 2, the true values of the parameters of the CARES model with the indirect GARCH(1,1) specification are implied to be β 0 = a 0 (Φ 1 (τ)) 2, β 1 = a 2, β 2 = a 1 (Φ 1 (τ)) 2, γ 0 = a 0 ( φ(φ 1 (τ))/τ) 2, γ 1 = a 2, and γ 2 = a 1 ( φ(φ 1 (τ))/τ) 2, where Φ and φ are the cumulative density function and probability density function, respectively, of the standard Gaussian distribution and τ is the coverage probability. See Appendix B for more details on the derivation. We then generate 10,000 samples from the GARCH(1,1) model with sample sizes of 1,000, 2,000, 5,000 and 10,000. he initial return and volatility values are drawn from their unconditional distributions. For each sample, we estimate the parameters of the indirect GARCH(1,1) CARES model when the coverage probability is 5% or 1%. he means and standard deviations of these estimators computed based on 10,000 simulation iterations are reported in able 1. 10

11 able 1: Finite-Sample Properties of Each Parameter Estimator in the CARES Model 11 Panel A: τ = 0.05 Sample Size = 1000 = 2000 = 5000 = rue Parameter Mean Estimated Parameter (Standard Deviation) β 0 = a 0 (Φ ( 1) (τ)) 2 = (0.0470) (0.0329) (0.0194) (0.0140) [0.0356] [0.0314] [0.0192] [0.0123] β 1 = a 2 = (0.0248) (0.0173) (0.0108) (0.0086) [0.0211] [0.0111] [0.0102] [0.0082] β 2 = a 1 (Φ ( 1) (τ)) 2 = (0.0561) (0.0388) (0.0248) (0.0171) [0.0402] [0.0285] [0.0214] [0.0141] γ 0 = a 0 ( φ(φ ( 1) (τ))/τ) 2 = (0.1881) (0.0914) (0.0432) (0.0283) [0.1054] [0.0825] [0.0394] [0.0223] γ 1 = a 2 = (0.0496) (0.0264) (0.0147) (0.0103) [0.0309] [0.0213] [0.0134] [0.0101] γ 2 = a 1 ( φ(φ ( 1) (τ))/τ) 2 = (0.0938) (0.0633) (0.0401) (0.0294) [0.0828] [0.0529] [0.0392] [0.0261] Note: his table reports the means and standard deviations of the CARES model parameter estimators (when the coverage probability is 5%) computed from 10,000 simulation iterations with sample sizes of = 1, 000, = 2, 000, = 5, 000 and = 10, 000 when the underlying return follows a GARCH(1,1) process. he standard deviations of the parameter estimators over all 10,000 simulations and the average theoretical standard deviations implied by the asymptotic theory of these parameters are presented in round and square brackets, respectively.

12 12 Panel B: τ = 0.01 Sample Size = 1000 = 2000 = 5000 = rue Parameter Mean Estimated Parameter (Standard Deviation) β 0 = a 0 (Φ ( 1) (τ)) 2 = (0.0989) (0.0815) (0.0523) (0.0366) [0.2360] [0.1218] [0.0685] [0.0281] β 1 = a 2 = (0.0279) (0.0217) (0.0144) (0.0099) [0.0489] [0.0250] [0.0141] [0.0099] β 2 = a 1 (Φ ( 1) (τ)) 2 = (0.1395) (0.0963) (0.0611) (0.0312) [0.1826] [0.1175] [0.0780] [0.0302] γ 0 = a 0 ( φ(φ ( 1) (τ))/τ) 2 = (0.2089) (0.1382) (0.0762) (0.0406) [0.3049] [0.2156] [0.0950] [0.0404] γ 1 = a 2 = (0.0388) (0.0144) (0.0139) (0.0096) [0.0711] [0.0308] [0.0217] [0.0095] γ 2 = a 1 ( φ(φ ( 1) (τ))/τ) 2 = (0.2177) (0.1416) (0.0860) (0.0498) [0.4288] [0.2618] [0.1073] [0.0470] Note: his table reports the means and standard deviations of the CARES model parameter estimators (when the coverage probability is 1%) computed from 10,000 simulation iterations with sample sizes of = 1, 000, = 2, 000, = 5, 000 and = 10, 000 when the underlying return follows a GARCH(1,1) process. he standard deviations of the parameter estimators over all 10,000 simulations and the average theoretical standard deviations implied by the asymptotic theory of these parameters are presented in round and square brackets, respectively.

13 hese results reveal several noteworthy observations. First, these estimators perform well even when the sample size is moderate ( = 1, 000). heir bias and standard deviations decline, as expected, with increasing sample size. he observation that each parameter estimator converges to the true value of the parameter as increases confirms the consistency of these estimators. Meanwhile, we calculate the average theoretical standard error of each parameter estimator (the values reported in square brackets in able 1) using the estimated values of the parameters from each simulation along with the asymptotic theory described above and compare these theoretical values with the standard deviation of each parameter estimator over all simulation iterations (the values reported in parentheses in able 1). he small deviation between the asymptotic and finite-sample standard errors confirms the validity of the asymptotic theory derived above. Finally, to investigate the degree of efficiency loss in the CARES model estimation, we alternatively compute the theoretical standard errors of the parameter estimators in the CARES model based on the asymptotic standard errors of the above GARCH(1,1) model parameters (a 0, a 1, and a 2.), with an appropriate scaling based on the relationship between the GARCH(1,1) model parameters and the CARES model parameters. Using 10,000 samples with a sample size of = 10, 000, the implied standard errors of the parameter estimators in the CARES model with respect to the GARCH(1,1) model are computed to be for β 0, for β 1, for β 2, for γ 0, for γ 1 and for γ 2 when τ = 0.05 and are computed to be for β 0, for β 1, for β 2, for γ 0, for γ 1 and for γ 2 when τ = he closeness between these implied standard errors and those computed based on the simulation suggests that our estimation procedure yields efficient estimates of the model parameters. 3 Simulation Study In this section, we present a series of simulation studies to illustrate the finite-sample properties of the CARES model. In all cases, we examine the model performance in terms of the one-step-ahead ES forecast, and performance is measured in terms of the root mean squared error (RMSE). he RMSE of an ES forecast ÊS from an arbitrary model has the standard definition E((ES ES) ˆ 2 ), where ES is the true value of the ES. Our one-step-ahead ES forecast procedure involves the following: Step 1. Return data are simulated with a sample size of , where = 500, = 1, 250, = 2, 500 and = 5, 000. he first 500 observations are discarded to allow for a sufficiently long burn-in period. 13

14 Step 2. With the remainder of the samples, the first observations are used to fit the three CARES models described in Section 2, and the ( + 1)th observation is reserved for one-step-ahead out-of-sample forecast evaluation. Step 3. he above steps are repeated 10 4 times, and the RMSE is approximated as the square root of the average of all 10 4 simulated values of (ES ÊS)2. We consider the following three experimental designs. Design 1 A simple GARCH(1,1) model, as described in Section 2.2. Design 2 A GARCH(1,1) model with time-varying skewness and kurtosis: r t = σ t z t, σ 2 t = a 0 + b + 0 (r + t 1) 2 + b 0 (r t 1) 2 + c 0 σ 2 t 1. he disturbance z t follows a generalized Student s t-distribution 1 with a time-varying asymmetry parameter λ t and a time-varying tail-fatness parameter η t, namely, z t G (z t η t, λ t ), where η t = a 1 + b + 1 y t b 1 yt 1 + c 1 η t 1, λ t = a 2 + b 2 yt c 2 λt 1, (18) η t = g [2,+30] η t, λ t = g [ 1,1] λt, with g representing the logistic map. Following the S&P 500 stock index return analysis of Jondeau and Rockinger (2003), we set the model parameters to a 0 = , b + 0 = , b 0 = , c 0 = , a 1 = , b + 1 = , b 1 = , c 1 = , a 2 = , b 2 = , and c 2 = his model not only accommodates a time-varying volatility but also allows for dynamics in higher-order moments: skewness and kurtosis. Design 3 A Markov switching stochastic volatility (MS-SV) model: y t = µ st + σ t u t (19) σ 2 t = ω st + α st ε 2 t 1 + β st σ 2 t 1 + ɛ st, (20) 1 he density of the generalized t distribution (G) is defined as gt(z η, λ) = { bc(1 + 1 η 2 ( bz+a 1 λ )2 ) (η+1)/2 if z < a/b bc(1 + 1 η 2 ( bz+a 1+λ )2 ) (η+1)/2 if z a/b, where a 4λc η 2 η 1, b λ 2 a 2, and c Γ((η+1)/2). π(η 2)Γ(η/2) 14

15 CAViaR IGARCH VaR rue VaR CARES IGARCH ES rue ES Figure 1: he ES forecasts of the CARES model vs. the true ES for a GARCH(1,1)- GAUSSIAN model where s t is an ergodic homogeneous Markov chain on a finite set S = 1,..., n, with a transition matrix P defined by the probabilities η ij = P (s t = i s t 1 = j). For simplicity, we set n = 2 to reflect a two-regime switching SV model. Following the S&P 500 index analysis of Bauwens, Preminger, and Jeroen (2010), the parameters are set to η 11 = 0.979, η 22 = 0.986, µ 1 = 0.069, µ 2 = 0.012, ω 1 = 0.313, ω 2 = 0.049, α 1 = 0, α 2 = 0.055, β 1 = 0, and β 2 = o provide an initial visual impression of the performance of the CARES model, Figure 1 shows the 5% ES forecasts obtained using the CARES model (with the indirect GARCH(1,1) specification) against the true values of the 5% ES when the sample size is 2,000. We observe that the true value of the ES exhibits strong dynamic clustering and that the 5% ES forecasts are able to properly capture this pattern. For the sake of comparison, we also compute the RMSEs of the ES forecasts obtained using three alternative methods: historical simulation (HS), a kernel density estimator 15

16 (KDE), and the CARE model of aylor (2008). Assuming i.i.d. asset returns, the HS and KDE approaches estimate empirical distributions of the return using past observations to obtain the VaR and ES. When using HS, we vary the length of recent past observations from 250 to 500 to construct the empirical distribution. he ES determined using the KDE approach takes the form ES KDE = (np) 1 n t=1 r tg h (V ar KDE r t ), where V ar KDE is the kernel-based VaR estimator, G h (t) = G(t/h), G(t) = t K(u)du, and K and h denote the standard Gaussian kernel and the optimal bandwidth, respectively. Because the standard KDE estimator is known to be biased 2, we use the jackknife technique to correct this bias. In addition, to better describe the time-varying features of the return distribution, we also apply the exponentially weighted HS (EWHS) and KDE (EWKDE) approaches, in which the empirical distribution is constructed from exponentially weighted past observations. Following aylor (2008), we set the exponential decay parameter λ to its optimal value to minimize the RMSE of the ES forecast. In the CARE model, the function that describes the relationship between the coverage probability of the expectile (α) and the coverage probability of the quantile (τ) has a closed form for the GARCH(1,1) design; thus, the true value of α is known for a given value of τ. herefore, in the GARCH(1,1) data-generating process (DGP), we obtain ES forecasts using the CARE model for two scenarios to study the effect of estimating α on the ES forecast: in one scenario, α is estimated (the estimator is denoted as ˆα) using a grid search method 3, and the other scenario uses the true value of α (denoted by α 0 ). For the other two simulation designs, because the true value of α is unknown, we obtain ES forecasts using only the CARE model with estimated α. he results of the 5% and 1% VaR and ES forecasts for the three simulation designs are shown in able 2, able 3 and able 4 respectively 4. he far left-hand column contains the model names, and the remaining columns present the bias and RMSEs of the one-step-ahead ES forecasts for sample sizes of = 500, = 1, 250, = 2, 500 and = 5, 000. Regarding the outcome, the ranking of these methods in terms of the RMSE is invariant with respect to the sample size. he CARES model with the indirect GARCH specification yields the lowest RMSE, followed by the CARES model with the other specifications, the CARE model with the true value of α, the CARE model with estimated α, the exponentially weighted HS and KDE approaches, and standard HS and KDE approaches, in that order. 2 See heorem 2 of Chen (2008) for more details. 3 Following aylor (2008), the optimal value of α is determined by estimating models with different values on a grid with a step size of Consistent with aylor (2008), we find that the asymmetric-slope CARE model and CARES model are outperformed by the other versions of the two models. herefore, we do not report the results for the asymmetric-slope CARE model and CARES model throughout the remainder of this analysis. 16

17 Although comparing the RMSEs provides an indication of the relative forecast accuracy, it provides no information regarding whether any of the observed differences in performance are significant. o this end, we employ the test of equal predictive accuracy of Diebold and Mariano (1995) (DM), and we use asterisks to indicate the results that are found to be significant when the CARES model with the indirect GARCH specification is used as the benchmark. he DMW test results confirm that the CARES model, particularly that with the indirect GARCH specification, demonstrates significantly superior performance compared with the other alternatives, except for the CARE model with the true value of α. he significant reduction in RMSE observed for the CARES model compared with the HS and KDE approaches reflects the benefit of directly modeling the time-varying tail of the return distribution. he significant reduction in RMSE observed for the CARES model compared with the CARE model with estimated α provides evidence that the error that arises as a result of estimating α significantly diminishes the model s forecasting performance. Unsurprisingly, the CARES model with the indirect GARCH specification performs the best, as it correctly specifies the ES when the underlying return data truly follow a GARCH(1,1) process with an i.i.d. error distribution. o further understand the differences between our CARES model and the CARE model of aylor (2008), we apply the GMM Wald test to the demeaned simulated data to empirically examine whether the relationship between the conditional quantile and ES used in the CARE model holds. he test details are provided in Appendix C. We employ the 5% significance level for the test; therefore, if the relationship holds, the ideal rate of rejection of the null hypothesis should be approximately 5%. he test results are reported in able 5 based on 10 4 times of simulations. While the rejection rates are and respectively in the GARCH(1,1)-GAUSSIAN model and the GARCH(1,1) model with time-varying skewness and kurtosis, the NM-SV model yields a higher rejection rate of hese test results imply that the relationship used in the CARE model holds when the underlying return follows a GARCH(1,1) process or a GARCH(1,1) process with time-varying high-order moments but that the relationship is violated when the return follows a more complicated NM-SV model. 4 Empirical Analysis o evaluate our CARES model using real data, we perform a simple empirical study to assess the ES of a stock index and two individual stocks. We apply several different CARES model specifications and then evaluate both the in-sample and out-of-sample forecasting performance of these specifications. 17

18 able 2: Model VaR and ES Forecasts Obtained for Data from the GARCH(1,1)-GAUSSIAN Panel A: 5% VaR =500 =1250 =2500 =5000 Bias RMSE Bias RMSE Bias RMSE Bias RMSE HS(250) * * * * HS(500) * * * * KDE * * * * KDE-JK * * * * EWHS(500) * * * * EWKDE * * * * EWKDE-JK * * * * CARE-SAV(α 0 ) * * * * CARE-IG(α 0 ) * CARE-SAV(ˆα) * * * * CARE-IG(ˆα) * * * * CARES-SAV * * * * CARES-IG Panel B: 1% VaR HS(250) * * * * HS(500) * * * * KDE * * * * KDE-JK * * * * EWHS(500) * * * * EWKDE * * * * EWKDE-JK * * * * CARE-SAV(α 0 ) * * * * CARE-IG(α 0 ) * * * * CARE-SAV(ˆα) * * * * CARE-IG(ˆα) * * * * CARES-SAV * * * * CARES-IG

19 Panel A: 5% ES =500 =1250 =2500 =5000 Bias RMSE Bias RMSE Bias RMSE Bias RMSE HS(250) * * * * HS(500) * * * * KDE * * * * KDE-JK * * * * EWHS(500) * * * * EWKDE * * * * EWKDE-JK * * * * CARE-SAV(α 0 ) * * * * CARE-IG(α 0 ) * CARE-SAV(ˆα) * * * * CARE-IG(ˆα) * * * * CARES-SAV * * * * CARES-IG Panel B: 1% ES HS(250) * * * * HS(500) * * * * KDE * * * * KDE-JK * * * * EWHS(500) * * * * EWKDE * * * * EWKDE-JK * * * * CARE-SAV(α 0 ) * * * * CARE-IG(α 0 ) * * * * CARE-SAV(ˆα) * * * * CARE-IG(ˆα) * * * * CARES-SAV * * * * CARES-IG Note: his table reports the bias and RMSEs of one-step-ahead ES forecasts obtained using CARES models and other competing models when the underlying return follows a GARCH(1,1) process. Panels A and B report the results obtained based on 5% and 1% coverage probabilities, respectively. HS(250) and HS(500) denote the historical simulation method based on the most recent 250 and 500 observations. KDE and KDE-JK denote the kernel density estimator before and after jackknife bias correction. CARE-SAV(α 0 ) (CARE-SAV(ˆα)) and CARE-IG(α 0 ) (CARE-IG(ˆα)) denote the CARE model with the symmetric absolute value specification and the indirect GARCH(1,1) specification, respectively, when α takes its true value (and when α is estimated). CARES-SAV and CARES-IG denote the CARES model with the symmetric absolute value specification and the indirect GARCH(1,1) specification, respectively. We use the CARES-IG model as a benchmark (because it always yields the best performance with the smallest RMSE). Based on the test of Diebold and Mariano (1995) at the 5% level of significance, an asterisk indicates that the RMSE of the ES forecast obtained using the corresponding model is significantly larger than that of the benchmark for a sample size of 2,

20 able 3: ES Forecasts Obtained for Data from the GARCH(1,1) Model with imevarying Skewness and Kurtosis Panel A: 5% VaR =500 =1250 =2500 =5000 Bias RMSE Bias RMSE Bias RMSE Bias RMSE HS(250) * * * * HS(500) * * * * KDE * * * * KDE-JK * * * * EWHS(500) * * * * EWKDE * * * * EWKDE-JK * * * * CARE-SAV(α 0 ) * * * * CARE-IG(α 0 ) * * * * CARE-SAV(ˆα) * * * * CARE-IG(ˆα) * * * * CARES-SAV * * * * CARES-IG Panel B: 1% VaR HS(250) * * * * HS(500) * * * * KDE * * * * KDE-JK * * * * EWHS(500) * * * * EWKDE * * * * EWKDE-JK * * * * CARE-SAV(α 0 ) * * * * CARE-IG(α 0 ) * * * * CARE-SAV(ˆα) * * * * CARE-IG(ˆα) * * * * CARES-SAV * * * * CARES-IG

21 Panel A: 5% ES =500 =1250 =2500 =5000 Bias RMSE Bias RMSE Bias RMSE Bias RMSE HS(250) * * * * HS(500) * * * * KDE * * * * KDE-JK * * * * EWHS(500) * * * * EWKDE * * * * EWKDE-JK * * * * CARE-SAV(α 0 ) * * * * CARE-IG(α 0 ) * * * * CARE-SAV(ˆα) * * * * CARE-IG(ˆα) * * * * CARES-SAV * CARES-IG * Panel B: 1% ES HS(250) * * * * HS(500) * * * * KDE * * * * KDE-JK * * * * EWHS(500) * * * * EWKDE * * * * EWKDE-JK * * * * CARE-SAV(α 0 ) * * * * CARE-IG(α 0 ) * * * * CARE-SAV(ˆα) * * * * CARE-IG(ˆα) * * * * CARES-SAV CARES-IG * * * * Note: his table reports the bias and RMSEs of one-step-ahead ES forecasts obtained using CARES models and other competing models when the underlying return follows a GARCH(1,1) process with time-varying skewness and kurtosis. Panels A and B report the results obtained based on 5% and 1% coverage probabilities, respectively. HS(250) and HS(500) denote the historical simulation method based on the most recent 250 and 500 observations. KDE and KDE-JK denote the kernel density estimator before and after jackknife bias correction. CARE-SAV(α 0 ) (CARE-SAV(ˆα)) and CARE-IG(α 0 ) (CARE-IG(ˆα)) denote the CARE model with the symmetric absolute value specification and the indirect GARCH(1,1) specification, respectively, when α takes its true value (and when α is estimated). CARES-SAV and CARES-IG denote the CARES model with the symmetric absolute value specification and the indirect GARCH(1,1) specification, respectively. We use the CARES-IG model as a benchmark (because it always yields the best performance with the smallest RMSE). Based on the test of Diebold and Mariano (1995) at the 5% level of significance, an asterisk indicates that the RMSE of the ES forecast obtained using the corresponding model is significantly larger than that of the benchmark 21 for a sample size of 2,000.

22 able 4: ES Forecasts Obtained for Data from the MS(2)-SV Model Panel A: 5% VaR =500 =1250 =2500 =5000 Bias RMSE Bias RMSE Bias RMSE Bias RMSE HS(250) * * * * HS(500) * * * * KDE * * * * KDE-JK * * * * EWHS(500) * * * * EWKDE-JK * * * * EWKDE * * * * CARE-SAV(α 0 ) * * * * CARE-IG(α 0 ) * * * * CARE-SAV(ˆα) * * * * CARE-IG(ˆα) * * * * CARES-SAV * * * CARES-IG Panel B: 1% VaR HS(250) * * * * HS(500) * * * * KDE * * * * KDE-JK * * * * EWHS(500) * * * * EWKDE * * * * EWKDE-JK * * * * CARE-SAV(α 0 ) * * * * CARE-IG(α 0 ) * * * * CARE-SAV(ˆα) * * * * CARE-IG(ˆα) * * * * CARES-SAV * * * CARES-IG

23 Panel A: 5% ES =500 =1250 =2500 =5000 Bias RMSE Bias RMSE Bias RMSE Bias RMSE HS(250) * * * * HS(500) * * * * KDE * * * * KDE-JK * * * * EWHS(500) * * * * EWKDE * * * * EWKDE-JK * * * * CARE-SAV(α 0 ) * * * * CARE-IG(α 0 ) * * * * CARE-SAV(ˆα) * * * * CARE-IG(ˆα) * * * * CARES-SAV * * * * CARES-IG Panel B: 1% ES HS(250) * * * * HS(500) * * * * KDE * * * * KDE-JK * * * * EWHS(500) * * * * EWKDE * * * * EWKDE-JK * * * * CARE-SAV(α 0 ) * * * * CARE-IG(α 0 ) * * * * CARE-SAV(ˆα) * * * * CARE-IG(ˆα) * * * * CARES-SAV * * * * CARES-IG Note: his table reports the bias and RMSEs of one-step-ahead ES forecasts obtained using CARES models and other competing models when the underlying return follows the MS(2)-SV model. Panels A and B report the results obtained based on 5% and 1% coverage probabilities, respectively. HS(250) and HS(500) denote the historical simulation method based on the most recent 250 and 500 observations. KDE and KDE-JK denote the kernel density estimator before and after jackknife bias correction. CARE-SAV(α 0 ) (CARE-SAV(ˆα)) and CARE-IG(α 0 ) (CARE-IG(ˆα)) denote the CARE model with the symmetric absolute value specification and the indirect GARCH(1,1) specification, respectively, when α takes its true value (and when α is estimated). CARES-SAV and CARES-IG denote the CARES model with the symmetric absolute value specification and the indirect GARCH(1,1) specification, respectively. We use the CARES-IG model as a benchmark (because it always yields the best performance with the smallest RMSE). Based on the test of Diebold and Mariano (1995) at the 5% level of significance, an asterisk indicates that the RMSE of the ES forecast obtained using the corresponding model is significantly larger than that of the benchmark for a sample size of 2,

24 able 5: CARE Model Specification est Rejection Rate GARCH(1,1)-GAUSSIAN model GARCH(1,1) model with time-varying skewness and kurtosis MS(2) GARCH model Note: his table reports the specification test (GMM Wald test) results for the CARE model obtained from simulations of 1,000 samples based on the GARCH(1,1)- GAUSSIAN model, the GARCH(1,1) model with time-varying skewness and kurtosis, and the MS(2) GARCH model with a sample size of 2,500. We perform the test at the 5% significance level; therefore, the ideal rejection rate should be 5%. 4.1 Data We consider two individual stocks, General Motors (GM) and IBM, and one stock index, S&P 500, to conduct this empirical study. Following Engle and Manganelli (2004), we first obtain a sample of 3,392 daily prices from Datastream for each study target, spanning the period from April 7, 1986, to April 7, his sample is useful for testing whether the ES estimates produced by our CARES model can provide the same risk indication provided by the VaR estimates produced by the CAViaR model of Engle and Manganelli (2004). Second, we obtain a recent sample of daily prices from Wharton Research Data Services (WRDS) for the above two stocks and one index, ranging from January 1, 2005, to December 31, his sample period overlaps with the recent global financial crisis to allow us to study the ability of our model to adapt to new risk environments. he daily returns are calculated as 100 times the difference in the logarithms of the prices. 4.2 Empirical Results For the first sample, we use the first 2,892 observations to establish the CARES models and reserve the last 500 observations for one-step-ahead out-of-sample forecasting. he 5% VaR and ES forecasts for GM are plotted in Figure 2, and the model estimation results are reported in able 6 and able 7. As expected, the VaR results are similar to those reported in Engle and Manganelli (2004); consistent with Engle and Manganelli (2004), the always significant coefficient of the autoregressive term (β 2 ) in both the VaR and ES models confirms that the phenomenon of volatility clustering is also relevant in the tails. 24

25 able 6: Estimation Results of the CARES Models for Real Data (Part A) Symmetric Absolute Value Indirect GARCH GM IBM S&P 500 GM IBM S&P 500 VaR: 1% Coverage Probability β * (Std1) (0.2691) (0.1533) (0.0532) (0.1346) (0.2107) (0.0248) (Std2) (0.3218) (0.1422) (0.0573) (0.1600) (0.3519) (0.0290) (Std3) (0.3100) (0.1394) (0.0543) (0.1430) (0.4966) (0.0260) β *** *** *** *** *** *** (Std1) (0.0929) (0.0950) (0.0390) (0.0094) (0.0136) (0.0042) (Std2) (0.1127) (0.0933) (0.0430) (0.0117) (0.0230) (0.0050) (Std3) (0.1059) (0.0961) (0.0411) (0.0105) (0.0317) (0.0043) β *** * (Std1) (0.1650) (0.2374) (0.1307) (1.4698) (0.3609) (1.2146) (Std2) (0.1915) (0.2390) (0.1497) (1.7949) (0.3387) (1.3364) (Std3) (0.1774) (0.2587) (0.1398) (1.6078) (0.9364) (1.2704) VaR: 5% Coverage Probability β *** * *** ** ** (Std1) (0.0580) (0.0217) (0.0055) (0.1037) (0.1816) (0.0099) (Std2) (0.0741) (0.0237) (0.0083) (0.1277) (0.0826) (0.0149) (Std3) (0.0870) (0.0251) (0.0073) (0.1521) (0.1103) (0.0117) β *** *** *** *** *** *** (Std1) (0.0248) (0.0161) (0.0211) (0.0133) (0.0394) (0.0061) (Std2) (0.0344) (0.0193) (0.0251) (0.0185) (0.0165) (0.0108) (Std3) (0.0376) (0.0192) (0.0239) (0.0223) (0.0217) (0.0088) β *** *** * *** (Std1) (0.0419) (0.0284) (0.0427) (0.1151) (0.0481) (0.6198) (Std2) (0.0329) (0.0341) (0.0512) (0.0860) (0.0699) (0.3561) (Std3) (0.0388) (0.0313) (0.0482) (0.1375) (0.0553) (0.4446) Note: his table reports the CARES model estimation results for two individual stocks, General Motors (GM) and IBM, and for one stock index, S&P 500. he sample contains 2,892 daily prices for each study target, spanning the period from April 7, 1986, to April 7, We report the standard errors (where c is estimated using a k-nearest neighbor estimator as in Engle and Manganelli (2004)) as Std1 and two bias-corrected standard errors as Std2 (where c is estimated using Koenker s bandwidth with Bofinger s h ) and Std3 (where c is estimated using Koenker s bandwidth with Hall and Sheather s h ). *, ** and *** indicate that the coefficients are significant at the 5% significance level based on Std1, both Std1 and Std2, and all three standard errors, respectively. 25

26 able 7: Estimation Results of the CARES Models for Real Data (Part B) Symmetric Absolute Value Indirect GARCH GM IBM S&P 500 GM IBM S&P 500 ES: 1% Coverage Probability ˆγ *** (Std1) (1.5962) (1.2069) (0.5396) (1.0177) (5.4666) (0.0703) (Std2) (1.4586) (1.2582) (0.6173) (1.0245) (9.1299) (0.0710) (Std3) (1.6313) (1.1975) (0.5791) (1.0161) ( ) (0.0705) ˆγ *** *** *** *** (Std1) (0.2300) (0.9532) (0.4026) (0.0878) (0.1034) (0.0214) (Std2) (0.2058) (1.0297) (0.4128) (0.0883) (0.1744) (0.0216) (Std3) (0.2391) (1.0202) (0.4085) (0.0876) (0.2327) (0.0215) ˆγ *** * *** *** *** *** (Std1) (0.8261) (0.7204) (0.4376) (0.5860) (0.3782) (0.1030) (Std2) (0.8147) (0.1286) (0.4309) (0.5899) (0.4315) (0.0990) (Std3) (0.8416) (0.1226) (0.4376) (0.5832) (0.3252) (0.1010) ES: 5% Coverage Probability ˆγ (Std1) (0.5707) (0.3806) (0.3854) (1.8022) (2.2157) (3.3992) (Std2) (0.9244) (0.3916) (0.4094) (2.1404) (1.0939) (3.1835) (Std3) (0.9737) (0.3896) (0.3812) (2.5239) (1.2997) (3.1485) ˆγ *** *** *** *** *** (Std1) (0.2449) (0.2156) (0.2775) (0.0993) (0.1320) (0.3156) (Std2) (0.3578) (0.2084) (0.2769) (0.1141) (0.0628) (0.2621) (Std3) (0.3710) (0.2095) (0.2588) (0.1336) (0.0756) (0.2628) ˆγ *** *** *** (Std1) (0.2205) (0.0890) (0.8126) (0.5039) (0.5361) (2.6250) (Std2) (0.2077) (0.0584) (0.8020) (0.5336) (0.4246) (2.6027) (Std3) (0.2125) (0.0629) (0.7828) (0.5024) (0.4433) (2.6138) Note: his table reports the CARES model estimation results for two individual stocks, General Motors (GM) and IBM, and for one stock index, S&P 500. he sample contains 2,892 daily prices for each study target, spanning the dates from April 7, 1986, to April 7, We report the standard errors (where c is estimated using a k-nearest neighbor estimator as in Engle and Manganelli (2004)) as Std1 and two bias-corrected standard errors as Std2 (where c is estimated using Koenker s bandwidth with Bofinger s h ) and Std3 (where c is estimated using Koenker s bandwidth with Hall and Sheather s h ). *, ** and *** indicate that the coefficients are significant at the 5% significance level based on Std1, both Std1 and Std2, and all three standard errors, respectively. 26

27 Figure 2: 5% VaR and ES Estimates Obtained Using the CARES Models for GM he top panel of Figure 2 presents a plot of the 5% VaR and ES estimates obtained using the CARES model with the symmetric absolute value specification for GM 5, and the bottom panel of Figure 2 presents a plot of the 5% VaR and ES estimates obtained using the CARES model with the indirect GARCH specification for GM. he ES plot exhibits a pattern similar to that of the VaR plot, with a spike near the beginning of the sample corresponding to the 1987 crash and an increase toward the end of the sample that reflects the increase in volatility following the Russian and Asian crises. hese findings indicate that the ES estimates produced by the CARES model are able to provide the same risk indication as the VaR estimates produced by the CAViaR model of Engle and Manganelli (2004) and can therefore be regarded as an alternative risk measure. With respect to the model estimation results, most coefficients related to both VaR and ES are statistically significant at the 5% significance level, strongly supporting the time-varying nature of the tail of the distribution. For the second sample, we use the first 1,262 observations to establish the CARES models and again reserve the last 500 observations for out-of-sample forecasting. We estimate the 1% and 5% one-day-ahead ESs using the two CARES specifications discussed in 5 he plot exhibits the same trend shown in Figure 1 of Engle and Manganelli (2004); the only difference is that the VaR is reported as a negative rather than positive value. 27

28 Figure 3: 5% VaR and ES Estimates Obtained Using the CARES Models for IBM Section 2.1. he 5% VaR and ES estimates for IBM and S&P 500 are plotted in Figure 3 and Figure 4, respectively. he VaR and ES estimates are reported as negative numbers in these plots. he common spike in the middle of the sample (between the end of 2008 and 2009) reflects the recent global financial crisis, and the increased risk toward the end of the sample reflects the recent euro zone crisis. he estimation results are reported in able 8 and able 9. Again, the coefficients of the autoregressive terms in the CARES models are always significant. his finding confirms that the phenomenon of the clustering of volatilities and higher-order moments is also relevant in the tails. Finally, we empirically test whether the relationship between the parameters of the autoregressive VaR model (β 1,β 2 and β 3 ) and the parameters of the autoregressive ES model implied by the expectile-based models (CARE models) hold in our CARES model. We also apply the GMM Wald test to the estimated parameters, with the results indicating that the linear proportionality between the conditional quantile and ES used in aylor (2008) does not hold in these two empirical data sets. 28