Dynamic expected shortfall: A spectral decomposition of tail risk across time horizons

Journal of Economic Dynamics & Control 108 (2019) 103753

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Journal of Economic Dynamics & Control journal homepage: www.elsevier.com/locate/jedc

Dynamic expected shortfall: A spectral decomposition of tail risk across time horizons Di Bu b, Yin Liao b, Jing Shi a,b, Hongfeng Peng a,∗ a b

School of Finance, Shandong University of Finance and Economics, China Macquarie University, Australia

a r t i c l e

i n f o

Article history: Received 22 August 2018 Revised 23 August 2019 Accepted 10 September 2019 Available online 16 September 2019 JEL classification: G13 C22 Keywords: Financial institution Tail risk Expected shortfall Wavelet analysis Time horizon

a b s t r a c t The tail risk of financial institutions is traditionally measured by Expected Shortfall (ES) that does not characterize risk changes over investment horizons. Using wavelet analysis, we propose a new method to capture the dynamics of ES across time horizons. The new method decomposes the stock return of financial institutions into different frequency (e.g., short-, mid-, and long-run) components, and then, models the dynamics of these components separately to produce an aggregated ES forecast. We provide numerical and empirical examples to illustrate the new method. We also study the relevance of each frequency component to out-of-sample ES forecasts over different predictive horizons. Our empirical results confirm that the different frequency components of stock returns exhibit different persistence. Explicitly considering this distinction when modeling ES significantly improves the out-of-sample forecasting performance. In addition, excluding the long-run (e.g., yearly) return component can largely reduce short-run (e.g., weekly or monthly) ES forecasts without impacting the regulatory quality of the risk assessment. © 2019 Elsevier B.V. All rights reserved.

1. Introduction The financial asset returns exhibit a multi-scale nature that reflects a mixture of investor trading activities (e.g., Constantinides et al., 2002; Jappelli and Modigliani, 1998; Modigliani, 1986; Poterba, 2001). In relation to risk management, the multi-scale asset returns typically cause financial risk to vary across investment horizons, but we find that there are few risk measures capturing this type of dynamics. In this paper, we propose a novel approach to model and forecast financial institution expected shortfall (ES) across investment horizons, in which the frequency components of the financial institution stock returns are separately considered. Using this new method, we also evaluate the out-of-sample forecasting performance of each frequency component over a variety of predictive horizons. Expected shortfall, defined as the average return on a risky asset conditional on the return being below some quantile of its distribution, has gained increasing popularity among both practitioners and academics. Compared with VaR, the ES possesses several additional merits. First, ES measures the magnitude of losses beyond the level of VaR, whereas the VaR only reflects the probability of the loss. Second, VaR is not sub-additive that causes the VaR of a combined portfolio can be larger than the sum of the VaRs of its components. ES successfully overcomes this shortcoming. But unlike VaR, there are ∗

Corresponding author. E-mail address: [email protected] (H. Peng).

https://doi.org/10.1016/j.jedc.2019.103753 0165-1889/© 2019 Elsevier B.V. All rights reserved.

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D. Bu, Y. Liao and J. Shi et al. / Journal of Economic Dynamics & Control 108 (2019) 103753

few studies that model ES. This dearth is perhaps in part because regulatory interest in this risk measure is only recent, and may also be due to the fact that this measure is not elicitable (Patton et al., 2018).1 Drawing on recent work by Fissler and Ziegel (2016), which shows the ES and VaR are jointly elicitable, two recent works (Patton et al., 2018; Taylor, 2017) model the dynamics of VaR and ES. Although these studies open up new directions for ES modeling and prediction, they still largely ignore the fact that the ES (and VaR) varies over time horizons. Specifically, shorter time horizons typically imply lower risk for various probability level (e.g., daily V aR99% <= one-year VaR99% ). In this paper, we develop a new approach to model and forecast ES over various investment horizons. We first use the wavelet analysis of Percival and Walden (20 0 0) to decompose the financial asset returns into frequency based (e.g. short-, mid- and long-run) information components. And then, we apply the joint model of ES and VaR separately to these frequency components to study that whether the dynamics of ES (and VaR) differ significantly in the frequencies with which the return series are generated. We find that the ES (and VaR) exhibits more persistence at higher frequencies, and less persistence at lower frequencies. Our new modeling framework is also useful for identifying the relevance of return frequency components to ES (and VaR) forecasts over particular predictive horizons. Our results show that for short-run (e.g., weekly) ES forecasts, excluding the long-run (e.g., yearly) return frequency component can lower forecasts, which is preferred by institutions, without impacting the regulatory quality of the risk assessment. Our paper contributes to the finance literature from several perspectives. First, our modeling framework for ES relates to a growing body of literature on the modeling dynamics of the downside risk of financial institution. The downside risk consists of two components: exposure (the amount of the highest possible loss, measured by VaR) and uncertainty (a certain probability, measured by ES), both of which have to be addressed in order to design a valid and powerful risk barometer (Holton, 2014). While modeling risk uncertainty has been thoroughly studied, risk exposure has received much less attention. Taylor (2017) and Patton et al. (2018) model ES jointly with VaR, and our work extends these models by utilizing them in conjunction with wavelet decomposition. Our new framework is able to disentangle the time-scale components of the underlying asset returns, and model the dynamics of tail risk across time horizons. It is worthwhile to note that the option (particularly put option) implied risk measures can also achieve this target by relying on options with different time maturities. Hence, we compute the option implied ES to validate our proposed method in empirical analysis. The results are qualitatively similar. However, the option implied risk measures have some limitations that our method can mitigate. First, option prices are readily available for stock indices and some exchange-traded funds (ETFs), but not for generic portfolios. Second, option time to maturity does not always equal the desired risk horizon. In this sense, our method offers a more general way of capturing the dynamics of tail risk across investment horizons. Our second contribution to the literature is to improve the ES out-of-sample forecasting for ES, which is a “rising star” used to regulate the capital requirements of financial institutions. Although traditional risk models include a time dimension, the time horizon has rarely been the main focus of attention, except in the recent work of Berger and Gençay (2018). These authors show that conditional volatility exhibits multi-scale property and removing long-run information can lower shortrun VaR forecasts without influencing the forecasting quality. We extend this work to propose a multi-scale approach for ES modeling and forecasting. Our approach exploits the relevance of the time scale based return information for ES forecasts over a specific predictive horizon. Our empirical results show that for weekly ES forecasts, the short-run return component can provide sufficient information to achieve adequately lower out-of-sample forecasts, although the long-run information components of the underlying return series are quantitatively irrelevant. However, when we increase the predictive horizon to one-year ahead, using short-run returns alone underestimates the ES, while the long-run components play an important role in improving the forecasts. Third, we add to the literature on the use of spectral analysis in finance. The spectral analysis, (e.g., Fourier or wavelet transformations), is a powerful tool that can be used to disentangle the variability of financial asset returns resulting from fluctuations at a specific frequency (or time horizon). Our frequency-specific measures allow us to distinguish between short- and long-run return information components, and provide new insights on asset pricing, portfolio allocation, and risk management. In response to advances in machine learning and signal processing, there has been a rebirth of interest in using spectral analysis for asset pricing models2 However, thus far the use of spectral analysis in risk management is largely unexplored. Our work links spectral analysis with ES modeling and forecasting, providing a new way to incorporate time-frequency analysis into financial risk management. Finally, research into ES forecasts enhances our understanding of risk management, not only for corporate managers interested in hedging risk, and institutional and retail investors interested in stock return behaviours, but also for regulators as the issue is related to regulatory features and may impinge the efficiency and operation of capital markets. In particular, this research is important when the capital requirements of financial institutions differ across time horizons or when investors wish to target specific horizons. The remainder of this paper is structured as follows. In Section 2, we present the methodology we use to analyze the multi-scale dynamics of ES as well as the evaluation criteria we use to assess the out-of-sample forecasting performance

1

A risk measure is said to be elicitable if there exists a loss function such that the risk measure is the solution to minimizing the expected loss. For example, Ortu et al. (2013) disaggregate consumption growth into cyclical components classified by their level of persistence and develop an asset pricing model in which consumption responds to shocks due to heterogeneous preference choices. Bandi and Tamoni (2016) argues that consumption growth should be separated into a variety of cyclical components because investors may not focus on high-frequency components of consumption representing short-term noise. 2

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of ES. In Section 3, we discuss the Monte Carlo simulation methods we use to evaluate the finite-sample property of our model and examine the relevance of the frequency components with respect to ES forecasts over multiple time horizons. An empirical application to nine banks is provided in Section 4. We conclude in Section 5. 2. Methodology 2.1. Expected shortfall ES is the expected return on an asset conditional on the return being below a given quantile (or VaR) of its distribution. The asset return at time t is denoted as yt and has conditional distribution Ft . The α -level ES can be expressed as:

E St = E [yt |yt <= V aRt ],

(1)

where V aRt = Ft−1 (α ) for 0 < α < 1. To develop a model to describe the dynamics of ES, we start with a simple scenario in which an asset return yt follows a Gaussian distribution with mean μt and standard deviation σ t . φ and  respectively denote the density and distribution functions of the standardized return. The VaR, denoted by VaRt , for a given probability τ can be expressed as:

V aRt = μt + σt [−1] (τ ; 0, 1 )

(2)

and the corresponding ES can be computed as:

ESt = E (yt |yt < V aRt )  VaRt yt h(yt )dyt = −∞ VaRt −∞ h (yt )dyt

  φ [−1] (τ ) = μt − σt · , τ

where h (yt ) is the density function of asset returns, yt . Thus far, both VaRt and ESt are clearly proportional to the standard deviation σ t , and hence we can assume the same functional form for the dynamics of ES as the one used in the CaViaR model of Engle and Manganelli (2004) for VaR. 2.2. A multi-scale approach to modeling and forecasting ES In this subsection, we first introduce two recently developed dynamic models of ES, and then, describe the wavelet analysis of the decomposition of the return series into their frequency components. Last, we describe how to incorporate these frequency components into the two ES models to model and forecast the multi-scale dynamics of ES, and improve the ES forecasts over different predictive horizons. 2.2.1. The dynamic model of ES There are two challenges associated with modeling the dynamics of ES. First, the ES is not elicitable on its own. This simply means that there is no loss function available to build up the dynamic models for ES. Second, ES relates to the tail portion of the asset return distribution. Building a parametric dynamic model for the tail part, one often needs to specify a conditional distribution for the asset returns. Two recent models successfully solve or avoid the two issues through jointly modeling VaR and ES; one model is developed by Taylor (2017) and the other by Patton et al. (2018). Taylor’s model includes conditional VaR and ES, which can be estimated by maximizing an asymmetric Laplace (AL) loglikelihood. We label this model as the “AL model”. We assume that yt is the value of the asset return series at time t, and VaRt and ESt are the conditional quantile and ES respectively. Taylor (2017) shows that the density of yt in terms of VaRt and ESt follows the AL density, which can be written as:

 (y − V aRt ) ∗ (α − I (yt ≤ V aRt ) ) f (yt ) = exp , ESt α ESt α−1



(3)

where α is the probability level to define the VaRt and ESt . The model parameters associated with the dynamics of VaRt and ESt , therefore, can be estimated by maximizing the AL density from Eq. (3). In terms of modeling the dynamics of VaRt , Taylor (2017) simply follows the CaViaR model of Engle and Manganelli (2004),3 and defines the dynamics of ESt based on 3 There are three examples of CaViaR model specifications we use as described below: Symmetric absolute value:

gt (γ (τ )) = γ1 + γ2 gt−1 (γ (τ )) + γ3 |yt−1 | Asymmetric slope:

gt (γ (τ )) = γ1 + γ2 gt−1 (γ (τ )) + γ3 (yt−1 )+ + γ4 (yt−1 )−

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the principle that ES and VaR are, to some extent, likely to vary together as follows:

ESt = (1 + exp(γ0 ))V aRt ,

(4)

where γ 0 is another parameter that needs to be estimated. We next consider the score driven joint dynamic model of VaR and ES proposed by Patton et al. (2018). Drawing on the “generalized autoregressive score” (GAS) work of Creal et al. (2013) and Harvey (2013), the model starts from an assumption that the target variable has some parametric conditional distribution where the parameters of that distribution follow a GARCH-like equation. The forcing variable in the equation is the lagged score of the log-likelihood. Specifically, in the model, we assume that both ES and VaR are driven by a common factor κ t :

V aRt = a ∗ exp(κt ),

(5)

ESt = b ∗ exp(κt ),

(6) −1 βκt−1 + γ Ht−1 st−1 .

where the common factor follows a dynamic process, that is, κt = ω + obtained from a joint loss function of VaRt and ESt (Fissler and Ziegel, 2016) as follows:

LF Z (yt , V aRt , ESt ; α ) = − where:

st =

1

α ESt

1yt ≤VaRt (V aRt − yt ) +

V aRt + log(−ESt ) − 1, ESt

 1 1 ∂ LF Z =− 1y ≤VaRt (yt − ESt ), ∂κt ESt α t

The forcing variable

−1 Ht−1 st−1

is

(7)

(8)

and the Hessian matrix Ht turns out to be a constant, which we set to be one. Fissler and Ziegel (2016) prove that minimizing the loss function LFZ returns the true VaR and ES, therefore, the above model can be estimated by minimizing the loss function, LFZ . We label this the “GAS one factor model”. 2.2.2. Wavelet decomposition of return series Next, to distinguish the information components of the return series over different time horizons, we employ the wavelet analysis of Percival and Walden (20 0 0) to decompose the underlying return series into different scales, in which each scale is comprised of frequency information (e.g., short-run, mid-run and long-run) for the original return series. We follow Berger and Gençay (2018) to apply the maximal overlap discrete wavelet transform (MODWT) to the return series, which is a modified version of the discrete wavelet transform (DWT). The MODWT transforms the return series into coefficients that are related to variations of the original return series over a set of time scales. We refer to Percival and Walden (20 0 0) for a detailed technical introduction of the wavelet analysis. We assume that there is a daily return series of y1:T = {yt , t = 1, 2, . . . , T }. The MODWT filter produces a set of timedependent wavelet and scaling coefficients, based on vectors associated with a time location t and a unitless scale τk = 2k−1 for each frequency level k = 1, . . . , K. For example, at each time point t, the MODWT filter at the kth frequency level consists of applying a wavelet (high-pass) filter hˆ k,l to yield a set of wavelet coefficients: Lk −1

ˆ k,t = W



hˆ k,l yt−l ,

(9)

l=0

and a scaling (low-pass) filter gˆk,l to yield a set of scaling coefficients: Lk −1

Vˆk,t =



gˆk,l yt−l ,

(10)

l=0

where the wavelet and scaling coefficients have a width of Lk = (2k − 1 )(L − 1 ) + 1, and L is the width of the k = 1 base filter. In fact, the wavelet and scaling filters for the kth level (that is, gˆk,l and hˆ k,l ) are a set of scale dependent localized differencing and averaging operators, extracting the short-term and long-term components of the return series, respectively. Note that the MODWT filter needs a time series over an infinite time interval, but our real sample of the return series are often over a finite time interval sampled at discrete time points. Therefore, for the finite sample of time series yt , t = 1, . . . , T , the MODWT treats them as if they were periodic, whereby the unobserved samples y−T , . . . , y−1 are assigned the observed values at y1 , . . . , yT . The MODWT coefficients are thus given by: Lk −1

ˆ k,t = W



hˆ k,l yt−l mod T ,

l=0

Indirect GARCH(1,1): 2 gt (γ (τ )) = (γ1 + γ2 gt−1 (γ (τ )) + γ3 yt−1 )1/2

(11)

D. Bu, Y. Liao and J. Shi et al. / Journal of Economic Dynamics & Control 108 (2019) 103753

5

and Lk −1

Vˆk,t =



gˆk,l yt−l mod T ,

(12)

l=0

for all time points t = 0, . . . , T . Based on the MODWT coefficients, the original return series can be described by the sum of all detailed coefficients and the smoothed version of the decomposed components as follows:

y1:T =

K 

ωkT Wˆ k,t + νKT VˆK,t =

k=1

K 

Dˆ j + SˆJ ,

(13)

k=1

where ωk and ν k are K × K matrices, and each row of ωk and ν k contains the circularly shifted hˆ k,l and gˆk,l . The Dˆ j = K T ˆ T ˆ k=1 ωk Wk,t describes the detailed coefficients at frequency k, and νK VK,t is the smoothed version of the return series at frequency K. For low levels of k, Dˆ j captures the short-term dynamics of the original return series, and the long-term fluctuations are extracted when k increases. Hence, using the MODWT filter, we can differentiate the original return series into frequency based information components. For example, we can use y1: k to describe the reconstructed return series that comprise the information components from the 1st to kth frequencies, where:

y1:k =

k 

Dˆ j .

(14)

j=1

The y1: k does not include the information of the return signals from the (k + 1 )th to Kth frequencies. 2.2.3. Modeling the multi-scale dynamics of ES Without considering the changes of ES over different time horizons, the standard way of modeling and forecasting the dynamics of ES is to simply apply the above two models to the original return series, y1:T = {yt , t = 1, 2, . . . , T }. To further distinguish the multi-scale dynamics of ES over different time horizons, we apply the two models to different frequency components of the return series (e.g., y1: k ), and generate the corresponding ES (and VaR) forecasts to capture the tail risk variation across time horizons. Specifically, we use the MODWT filter to decompose the different frequency components of the original return series from the in-sample periods into eight scales4 ; each scale contains different frequency based information regarding the underlying return series. Linking the original return series with the eight scales of the decomposed component, we have:

yt =

7 

Dˆ i,t + Sˆ8,t ,

(15)

i=1

where Dˆ i,t denotes the ith frequency components, and Sˆ8,t represents the long-run information of the original series not captured by Dˆ 1,t to Dˆ 7,t . To attach economic meaning to these frequency components, we iteratively reconstruct them, leading to three reconstructed returns:

y1:2,t =

2 

Dˆ i,t ,

(16)

Dˆ i,t ,

(17)

Dˆ i,t ,

(18)

i=1

y1:4,t =

4  i=1

y1:7,t =

7  i=1

where y1: 2,t , y1: 4,t , and y1: 7,t comprise the first two, four, and seven frequency components, representing the short-, mid-, and long-run information components of the original return series and corresponding to 8 days (one week), 32 days (one month), and 256 days (one year) time horizons. We refer to Berger and Gençay (2018) for the detailed economic interpretation regarding each frequency component. With the three reconstructed return series (y1: 2,t , y1: 4,t , and y1: 7,t ), we can use the above two models to differentiate the short-, mid-, and long-run dynamics of ES, capturing by the β coefficients in the GAS one factor model and γ 3 in the AL model for each series. Furthermore, using these models, we can generate and evaluate ES forecasts over different predictive horizons (e.g., one-week, one-month, and one-year ahead) by using the forecasts based on the original return series as a benchmark (or competing approach). In addition, for a particular predictive horizon (e.g., one-week ahead), we can study the relevance of the frequency components to the ES forecasts by examining whether excluding the long-run frequency component can reduce short-run ES forecasts without impacting the regulatory quality of the risk assessment. 4 Berger and Gençay (2018) show that in most cases, the first seven scales would be able to describe more than 95% of the variations in the original return series, while the eighth scale would describe less than 1% of the variation. We therefore stop the decomposition at the eighth scale.

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2.3. Back-testing We rely on two sets of back-testing to evaluate our new ES (and VaR) forecasts. First, we use three score (or loss) functions to judge the statistical property of the forecasts. Fissler and Ziegel (2016) show that ES and VaR are jointly elicitable, and provide a class of strictly consistent scoring functions (or loss functions) for jointly evaluating ES and VaR forecasts as follows:

S(V aRt , ESt , yt ) = (I (yt <= V aRt ) − α )G1 (V aRt ) − I (y <= V aRt )G1 (yt ) + G2 (ESt )(ESt − V aRt + I (yt <= V aRt )(V aRt − yt )/α ) − ϕ2 (ESt ) + a(yt ), where G1 , G2 , ϕ 2 , and a are functions satisfying a number of conditions. These conditions allow a variety of alternative functions to be chosen. We consider the three most commonly used function forms. The first scoring function we consider here is produced by setting G1 (x ) = x, G2 (x ) = exp(x )/(1 + exp(x )) and a = ln(2 ) to ensure positive values for the scoring function.5 We refer this score function as the FZG score:

SF ZG (V aRt , ESt , yt ) = (I (yt <= V aRt ) − α ) − I (yt <= V aRt )yt ESt + (ES − V aRt + I (yt <= V aRt )(V aRt − yt )/α ) (1 + exp(ESt )) t 

2 + ln . 1 + exp(ESt ) A second scoring function we consider is one proposed by Acerbi and Szkeley (2014). We refer to this function as the AS score. The function is defined as:

SAS (V aRt , ESt , yt ) =

  α ESt2 /2 + W V aRt2 /2 − V aRt ESt + I (yt <= V aRt )(−ESt (yt − V aRt )  + W (yt2 − V aRt2 )/2 ,

where we implement the AS score with W = 4. The third scoring function we consider is produced by setting G1 = 0, G2 (x ) = −1/x, ϕ2 (x ) = −ln(−x ), and a = 1 − ln(1 − α ). The scoring function takes the following form:

SAL (V aRt , ESt , yt ) = −ln

α − 1

ESt

−

(yt − V aRt )(α − I (yt <= V aRt )) yt + . α ESt ESt

(19)

As this function is basically the same as the negative of the asymmetric Laplace log-likelihood, we refer to it as the AL score. To further evaluate the quality of our ES estimates from a regulatory risk assessment perspective, we employ the backtesting procedure introduced by Acerbi and Szkeley (2014). This procedure is analogous to the standard VaR back-testing used by the Basel committee. Given the observed loss realization yt , where yt <= V aRt , the Acerbi–Szekely procedure tests the ESt (and VaRt ) estimates from a model against the loss realization under the following null and alternative hypotheses:

H0 : Pt = Ft H1 : Ft

is riskier than Pt , that is,

E StF > E StP ,

where Ft is the empirical distribution of the asset return data and Pt is the model implied distribution. The procedure basically tests whether the model accurately depicts the tail distribution (that is, H0 ) or underestimates the tail risk (that is, H1 ), involving both unconditional and conditional tests. The unconditional test (also known as the second Acerbi–Szekely test) is formed based on the unconditional expectation:

E Sα ,t = −E

y I

t t

α

,

(20)

where It = (yt + V aRα ,t < 0 ), that is the VaR failure indicator of period t with a value of 1 if yt < −V aRα ,t , and 0 otherwise. The corresponding test statistic is constructed as follows:

zuncon,t =

T  t=1

yt It + 1, T α ESα ,t

(21)

where the losses yt It are scaled by the corresponding ES values. Under the null hypothesis that the return distributional assumption is correct, the expected value of the test statistic zuncon,t is 0, whereas the negative value of the test statistic indicates that the tail risk has been underestimated. The critical values of the zuncon,t test statistic are very stable across a range of distributional assumptions on returns. We run the test statistics under the normality distributional assumption. The conditional test is inspired by the conditional expectation (and also known as the first Acerbi–Szekely test):

E Sα ,t = −E (yt |yt + V aRα ,t < 0 ), 5

This score function is also used in Fissler and Ziegel (2016).

(22)

D. Bu, Y. Liao and J. Shi et al. / Journal of Economic Dynamics & Control 108 (2019) 103753

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from which the test statistics can be defined as:

T

zcon,t =

yt It t=1 ESα ,t

N f ailure

+ 1,

(23)

T where N f ailure = t=1 It > 0. Under the null hypothesis that the distributional assumptions are correct, the conditional test has two parts. A VaR backtest is run first for the accuracy of the number of failures, and a stand-alone conditional test is performed for the conditional test statistic zcon,t . The conditional test accepts the model only when both the VaR test and the stand-alone conditional ES test accept the model. Negative values of the test statistic indicate an underestimation of risk. The conditional test is a one-sided test that rejects when there is sufficient evidence that the model underestimates risk (for technical details on the null and alternative hypotheses, see Acerbi and Szkeley (2014). The conditional test rejects the model when the p-value is less than one minus the confidence level of the test. 3. Numerical example To develop additional intuition for our new method, we provide the following numerical examples to illustrate how the wavelet decomposition can help us to model the dynamics of ES (and VaR) over different time horizons, and how this new modeling framework helps improve ES (and VaR) forecasts over different predictive horizons. Following Berger and Gençay (2018), we simulate return series using the following process:

yt = σt zt ,

zt N (0, 1 ),

(24)

where the return is described by zero mean, and the residual term zt follows a standard normal distribution. To mimic the relevance of asset-specific information regarding the memory of volatility components, we simulate conditional variance processes using a FIGARCH(1,d,1) model with different parameter values of d, using d = 0.05 (short memory), d = 0.25 (mid memory), and d = 0.45 (long memory), to describe different memory schemes. Next, we decompose the return series yt , t = 1, . . . , T , into eight scales and re-constructed the decomposed components into three return series, y1: 2,t , y1: 4,t , and y1: 7,t , as described in Section 2.2.3. We then apply the two dynamic models of ES (and VaR) to each of the three re-constructed return series to produce one-week, one-month, and one-year ahead ES (and VaR) forecasts. For each scenario (short memory (d = 0.05), mid memory (d = 0.25), and long memory (d = 0.45)), we simulate 20 0 0 return observations, and use the first 10 0 0 observations for model estimation and the last 10 0 0 observations for out-of-sample forecasting evaluation. We repeat the simulation by 10 0 0 times. Table 1 reports the estimation results for the two models using the original return series and the three re-constructed series, y1: 2,t , y1: 4,t , and y1: 7,t . The numbers in the table are the average value of each coefficient across the 10 0 0 simulations.6 We can see that the coefficients showing dynamics of ES (and VaR) (that is, the β s in GAS one factor model and the γ 3 s in AL model) vary across the return series. The y1: 2,t (which only contains short-run frequency components) exhibits the strongest persistence, followed by y1: 4,t (which contains both the short- and mid-run frequency components) and y1: 7,t (which contains short-, mid-, and long-run frequency components). The results confirm the dynamics of ES changes over time horizons, and the ES is more persistent over shorter time horizons. We further evaluate the out-of-sample forecasting performance of our new modeling framework using the two sets of back-testing procedures, and the forecasts from the models based on the original returns yt , t = 1, . . . , T , as a benchmark. Tables 2 and 3 present the out-of-sample 95% ES (and VaR) forecasting results based on the GAS one factor model.7 Panel A, B, and C present the results of one-week-, one-month-, and one-year-ahead forecasts, respectively. The results reveal several worthy findings. First, it is not surprising that the ES (and VaR) forecasts based on the original return series are the most accurate as they have the smallest score (or loss) function values, regardless of the length of the predictive horizons and the memory property of the original return volatilities. Second, focusing on the relatively shorter predictive horizon (e.g., one-week and one-month-ahead), the forecasts from the models using the short- and mid-run return series components (that is, y1: 2,t and y1: 4,t ) are always lower than those from the models using the original returns. More important, the Diebold-Mariano test shows that the score (or loss) function values of the former forecasts are not statistically different from those based on the latter counterparts. In addition, both the unconditional and the conditional Acerbi–Szekely test accept the model forecasts using only the short- and mid-run return information components, implying that excluding the long-run return components leads to lower ES forecasts without impacting the regulatory quality of the risk assessment, which are preferred by financial institutions to minimize capital requirements. Third, when we move towards the longer predictive horizon (e.g., one-year-ahead), most of the forecasts that use only the short-run return information components are significantly worse than those based on the original return series. These forecasts offer higher score (or loss) function values and are rejected by Acerbi–Szekely unconditional and conditional tests in most cases. This finding suggests that including the long-run return information component can largely improve the forecasts over longer time horizons. Overall, our results confirm the benefit of exploring the relevance of the return frequency components of 6 To save space, for VaR specification in AL model, the table only reports results of symmetric absolute value specifications. The results associated with the other two specifications are available upon request. 7 We also implement the AL model, which generates similar patterns. These results are not reported but available upon request.

8

D. Bu, Y. Liao and J. Shi et al. / Journal of Economic Dynamics & Control 108 (2019) 103753 Table 1 ES (and VaR) dynamic model estimation results. This table reports the estimation results of the GAS one factor dynamic model of Patton et al. (2018) and the AL model of Taylor (2017). Panels A to C report the results of different memory series for the volatility of the return series with d = 0.05, d = 0.25, and d = 0.45. “Full” denotes the original return series; y1: 2,t , y1: 4,t , and y1: 7,t comprise the first two, five, and seven frequency components, representing the short-, mid-, and long-run information components of the original return series. Panel A: d = 0.05 GAS one factor model Full

ω β γ a b

0.0152 0.8626 0.0765 −1.5231 −2.0345

y1: 2,t 0.0081 0.9321 0.0651 −1.6231 −2.0221

AL model y1: 4,t 0.0115 0.9034 0.0545 −1.5322 −1.9982

y1: 7,t 0.0143 0.8821 0.0702 −1.5141 −2.0432

Full

γ1 γ2 γ3 γ0

−0.0004 0.0135 0.8976 −1.1325

y1: 2,t −0.0003 0.0149 0.9124 −1.1221

y1: 4,t −0.0003 0.0153 0.9089 −1.0897

y1: 7,t −0.0002 0.0151 0.9021 −1.1334

Panel B: d = 0.25 GAS one factor model Full

ω β γ a b

0.0164 0.9021 0.0823 −1.4032 −1.9987

y1: 2,t 0.0099 0.9533 0.0667 −1.5654 −2.1345

AL model y1: 4,t 0.0123 0.9218 0.0598 −1.5021 −1.8734

y1: 7,t 0.0152 0.9113 0.0734 −1.4948 −1.9897

Full

γ1 γ2 γ3 γ0

−0.0005 0.0133 0.9037 −1.1145

y1: 2,t −0.0004 0.0146 0.9325 −1.1232

y1: 4,t −0.0002 0.0143 0.9231 −1.1206

y1: 7,t −0.0003 0.0131 0.9105 −1.1133

Panel C: d = 0.45 GAS one factor model Full

ω β γ a b

0.0178 0.9345 0.0902 −1.6089 −1.9223

y1: 2,t 0.0107 0.9769 0.0898 −1.5958 −2.0345

AL model y1: 4,t 0.0134 0.9554 0.0912 −1.6023 −1.9453

y1: 7,t 0.0165 0.9421 0.0909 −1.5923 −2.0231

Full

γ1 γ2 γ3 γ0

−0.0003 0.0124 0.9472 −1.1145

y1: 2,t −0.0004 0.0138 0.9678 −1.1232

y1: 4,t −0.0003 0.0135 0.9523 −1.1206

y1: 7,t −0.0002 0.0129 0.9501 −1.1133

the ES (and VaR) forecasts over different time horizons. In particular, using only the short- and mid-run return information components can reduce the one-week and one-month-ahead ES forecasts without impacting the regulatory quality of the risk assessment. 4. Empirical example 4.1. Data To provide a real example of the new method, we apply the multi-scale ES modeling approach to the stock data of eight banks: Bank of America(BOA), China Citic, Credit Suisse, Deutsche Bank (Deutsche), Mitsubishi Bank (Mitsubishi), National Australia Bank (NAB), Royal Bank of Canada (RBC), Societe Generale (Societe). We obtain the daily stock prices of the eight banks from Datastream for 2001–2014. We use the first 1800 observations from 2001 to 2008 as a training period to fit the in-sample models, and leave the 1350 observations from 2009 to 2014 to conduct our out-of-sample forecasting evaluation. The daily returns are calculated as the difference in the logarithms of the prices, expressed as a percentage. Table 4 provides the descriptive statistics of the daily log returns of the eight banks. While the Jarque–Bera test rejects the assumption of normality for all banks returns, the Ljung-Box test results indicate that the return series are characterized by strong autocorrelation. As described in Section 2.2.3, we implement the MODWT wavelet filter to decompose these return series into eight scales. Table 5 gives the results of the decomposed return components, including the economic interpretations and the average contribution to the variance of the original return series. We find that the contribution of each information component decreases as the frequency decreases. The first scale of information component describes approximately (50%) of the original variance, whereas the seventh scale component describes less than 5% of the banks. 4.2. Empirical results We re-construct the decomposed return components by successively including the first two, four, and seven scales to form the short-, mid-, and long-run return information components. We then apply the GAS one factor dynamic model of Patton et al. (2018) and the AL model of Taylor (2017) to these reconstructed return series to generate one-week-, onemonth-, and one-year-ahead ES forecasts. We estimate the GAS one factor model by minimizing the loss function (7), and

Panel A: One-week-ahead forecast Short-memory (d = 0.05)

Full (benchmark) (yt ) Short run (y1: 2,t ) Mid run (y1: 4,t ) Long run (y1: 7,t )

Mid-memory (d = 0.25)

Long-memory (d = 0.45)

Size(ES)

FZG

AS

AL

Size(ES)

FZG

AS

AL

Size(ES)

FZG

AS

AL

−0.83 −0.71 −0.75 −0.81

3321.992 3527.892∗ ∗ ∗ 3508.125∗ ∗ ∗ 3425.786∗ ∗ ∗

1243.157 1255.236∗ ∗ ∗ 1251.723∗ ∗ ∗ 1245.238∗ ∗ ∗

652.678 663.353∗ ∗ ∗ 659.232∗ ∗ ∗ 655.353∗ ∗ ∗

−1.67 −1.56 −1.61 −1.65

3980.126 4032.108∗ ∗ ∗ 4013.246∗ ∗ ∗ 3989.567∗ ∗ ∗

1515.237 1523.189∗ ∗ ∗ 1519.331∗ ∗ ∗ 1503.238∗ ∗ ∗

690.341 701.221∗ ∗ ∗ 698.002∗ ∗ ∗ 695.126∗ ∗ ∗

−1.94 −1.85 −1.89 −1.91

4081.104 4101.034∗ ∗ ∗ 4096.115∗ ∗ ∗ 4089.223∗ ∗ ∗

1523.147 1531.018∗ ∗ ∗ 1528.126∗ ∗ ∗ 1525.132∗ ∗ ∗

702.323 711.326∗ ∗ ∗ 709.231∗ ∗ ∗ 705.115∗ ∗ ∗

Panel B: One-month-ahead forecast

Full (benchmark) (yt ) Short run (y1: 2,t ) Mid run (y1: 4,t ) Long run (y1: 7,t )

Size(ES)

FZG

AS

AL

Size(ES)

FZG

AS

AL

Size(ES)

FZG

AS

AL

−0.91 −0.79 −0.84 −0.89

3567.823 3785.125∗ ∗ 3631.298∗ ∗ ∗ 4590.981∗ ∗ ∗

1456.723 1602.554∗ ∗ 1578.723∗ ∗ ∗ 1503.567∗ ∗ ∗

664.198 681.231∗ 678.567∗ ∗ ∗ 671.223∗ ∗ ∗

−1.88 −1.75 −1.80 −1.83

4012.455 4213.189∗ ∗ 4189.343∗ ∗ ∗ 4659.083∗ ∗ ∗

1525.205 1548.143∗ ∗ 1539.452∗ ∗ ∗ 1531.154∗ ∗ ∗

702.154 721.673∗ ∗ 714.451∗ ∗ ∗ 709.223∗ ∗ ∗

−2.09 −1.94 −1.99 −2.03

4102.335 4254.223∗ ∗ 4198.202∗ ∗ ∗ 4153.184∗ ∗ ∗

1538.321 1551.225∗ 1547.302∗ ∗ ∗ 1540.214∗ ∗ ∗

718.204 735.567∗ ∗ 728.152∗ ∗ ∗ 723.187∗ ∗ ∗

Size(ES)

FZG

AS

AL

Size(ES)

FZG

AS

AL

Size(ES)

FZG

AS

AL

−1.23 −1.11 −1.19 −1.21

3983.021 4171.563 4086.125∗ 3998.786∗ ∗ ∗

1647.218 1698.012∗ 1672.723∗ 1653.238∗ ∗ ∗

683.215 704.431∗ ∗ 697.121∗ ∗ ∗ 688.894∗ ∗ ∗

−2.01 −1.87 −1.93 −1.98

4236.676 4278.123∗ ∗ 4254.525∗ ∗ ∗ 4243.187∗ ∗ ∗

1783.654 1806.245∗ 1798.437∗ 1787.665∗ ∗ ∗

721.218 745.554 732.105∗ ∗ ∗ 729.114∗ ∗ ∗

−2.15 −1.98 −2.09 −2.11

4115.678 4389.104 4286.878∗ 4193.215∗ ∗ ∗

1553.098 1578.012∗ 1564.141∗ ∗ 1559.326∗ ∗ ∗

729.153 752.645∗ 748.098∗ ∗ 735.231∗ ∗ ∗

Panel C: One-year-ahead forecast

Full(benchmark) (yt ) Short run (y1: 2,t ) Mid run (y1: 4,t ) Long run (y1: 7,t )

D. Bu, Y. Liao and J. Shi et al. / Journal of Economic Dynamics & Control 108 (2019) 103753

Table 2 Out-of-sample Forecasting Performance of the Simulated Return Series (loss function value). This table reports the three score function values to jointly evaluate the 95% VaR and ES out-of-sample performance using different frequency components of the return series. Short-memory is described by d = 0.05; mid-memory, by d = 0.25; and long-memory, by d = 0.45. “Size” indicates the magnitude of the average ES forecasts. “Full” indicates the original return series, denoted by yt ; “Long Run” indicates the reconstructed return series which reflects the long-run return information, denoted by y1: 7,t ; “Mid Run” indicates the reconstructed return series which reflects the mid-run return information, denoted by y1: 4,t ; and “Short Run” indicates the reconstructed return series which reflects the short-run information, denoted by y1: 2,t . “FZG”, “AS”, and “AL” denote the FZG, AS, and AL negative score function values, respectively. ∗ , ∗ ∗ , and ∗ ∗ ∗ indicates the cases where the score function values of the forecasts from the frequency based return components are not significantly worse than the ones generated by the original return series at the 1%, 5%, and 10% levels based on the Diebold-Mariano test.

9

10

D. Bu, Y. Liao and J. Shi et al. / Journal of Economic Dynamics & Control 108 (2019) 103753 Table 3 Out-of-sample Forecasting Performance of the Simulated Return Series (statistical test). This table reports the results of two statistical tests on VaR and ES forecasts. zuncon and zcon represent the unconditional and conditional tests of Acerbi and Szkeley (2014). “Accept” (or ‘Reject”) indicates the cases where the model provides accurate risk estimates (or underestimates the risk) in more than 95% out of 10 0 0 replications. Panel A: One-week-ahead forecast Short-memory (d = 0.05)

Mid-memory (d = 0.25)

Long-memory (d = 0.45)

Return series component

zuncon

zcon

zuncon

zcon

zuncon

zcon

Full (benchmark) (yt ) Short run (y1: 2,t ) Mid run (y1: 4,t ) Long run (y1: 7,t )

accept accept accept accept

accept accept accept accept

accept accept accept accept

accept accept accept accept

accept accept accept accept

accept accept accept accept

Panel B: One-month-ahead forecast Return series component

zuncon

zcon

zuncon

zcon

zuncon

zcon

Full (benchmark) (yt ) Short run (y1: 2,t ) Mid run (y1: 4,t ) Long run (y1: 7,t )

accept accept accept accept

accept accept accept accept

accept accept accept accept

accept accept accept accept

reject accept accept accept

reject accept accept accept

Panel C: One-year-ahead forecast Return series component

zuncon

zcon

zuncon

zcon

zuncon

zcon

Full(benchmark) (yt ) Short run (y1: 2,t ) Mid run (y1: 4,t ) Long run (y1: 7,t )

accept accept accept accept

accept accept accept accept

accept reject accept accept

accept reject accept accept

accept reject reject accept

accept reject reject accept

Table 4 Summary Statistics of Real Data. The table presents descriptive statistics for the return series of eight banks for 2001–2014. The Jarque–Bera test is a goodness-of-fit test of whether the sample data have the skewness and kurtosis matching a normal distribution, with the test statistic equals to 5.44 for a normal distribution. Q-Stat gives the Ljung-Box test statistic for serial correlation with a lag size equal to ten.

Mean Std Dev Max Min Skew Kurt JB-Stat Q-Stat

BOA

China Citic

Credit Suisse

Deutsche

Mitsubishi

NAB

RBC

Societe

0.00 0.03 0.05 −0.26 1.059 10.218 3218.985 32.67

0.00 0.04 0.07 −0.34 −0.601 11.011 2390.887 51.43

0.00 0.04 0.08 −0.53 0.273 8.005 3219.874 45.66

0.00 0.03 0.05 −0.65 −0.162 14.621 4342.178 32.88

0.00 0.01 0.08 −0.74 0.291 9.837 2426.890 51.23

0.00 0.04 0.06 −0.82 1.056 12.795 1908.763 43.78

0.00 0.02 0.06 −0.73 −14.692 53.409 3290.878 34.89

0.00 0.03 0.08 −0.39 −1.994 24.006 4321.870 45.67

Table 5 The Decomposed Information Components of the eight banks’ returns. This table presents the daily return series and the average contribution to the variance of the original return series of each time horizon. The numbers are percentages. Information component

Horizon days

Detail

BoA

ChinaCitic

CreditSuisse

Deutsche

Mitsubishi

NAB

RBC

Societe

Scale Scale Scale Scale Scale Scale Scale Scale

2–4 4–8 8–16 16–32 32–64 64–128 128–256 256–512

D1 D2 D3 D4 D5 D6 D7 D8

50.78 20.104 9.235 5.342 4.897 3.953 2.623 3.066

56.707 20.804 11.382 5.445 2.567 1.441 1.043 0.611

47.036 22.532 11.949 6.737 4.325 3.959 2.554 0.908

61.238 12.133 7.238 4.171 3.875 3.943 3.685 3.717

55.915 20.921 10.658 6.278 3.038 1.649 0.879 0.662

58.753 18.007 8.627 4.115 2.46 1.869 2.724 3.445

57.194 19.519 10.615 6.009 2.741 1.446 0.949 1.527

47.792 22.765 10.196 6.383 3.692 3.318 2.962 2.892

1 2 3 4 5 6 7 8

estimate the AL model by maximizing the AL density function (3). For the VaR component of the AL model, we simply use the three specifications of the CaViaR model as described in footnote 3.8

Bank Names

Original return (yt ) FZG

AS

Short run (y1: 2,t )

Mid run (y1: 4,t )

Long run (y1: 7,t )

AL

FZG

AS

AL

FZG

AS

AL

FZG

AS

AL

Panel A: One-week-ahead forecast BoA 4521.334 1787.249 ChinaCitic 3984.897 1219.763 CreditSuisse 3897.914 1198.372 Deutsche 4219.768 1689.268 Mitsubishi 3908.134 1206.234 NAB 4902.187 1893.267 RBC 5012.334 1956.238 Societe 4512.387 1743.889

872.135 623.908 598.092 812.896 601.673 989.223 1012.334 815.673

4654.234∗ ∗ 4219.345∗ 4198.129∗ ∗ 4431.034∗ ∗ 4190.506∗ 5025.321∗ 5290.945∗ ∗ ∗ 4735.342

1913.684∗ 1389.239∗ 1301.569∗ ∗ 1878.893∗ 1367.129∗ ∗ 2078.345∗ ∗ 2197.673∗ ∗ 1912.34

987.109∗ 754.372∗ ∗ 654.031∗ 897.464∗ 654.078∗ 1109.021∗ ∗ 1187.456∗ ∗ ∗ 868.121

4622.167∗ ∗ 4135.113∗ 4120.445∗ ∗ ∗ 4390.121∗ ∗ ∗ 4135.223∗ ∗ 5009.434∗ ∗ ∗ 5128.908∗ ∗ ∗ 4713.129

1890.334∗ ∗ 1301.435∗ 1289.343∗ ∗ 1804.235∗ ∗ 1308.543∗ ∗ ∗ 2018.438∗ ∗ 2187.503∗ ∗ ∗ 1898.676

903.232∗ ∗ 706.897∗ ∗ 629.675∗ ∗ 878.323∗ ∗ 622.659∗ ∗ 1086.043∗ ∗ ∗ 1121.543∗ ∗ ∗ 835.879

4578.908∗ ∗ ∗ 4089.231∗ ∗ 4018.543∗ ∗ 4329.682∗ ∗ ∗ 4089.29∗ ∗ 4990.545∗ ∗ ∗ 5098.045∗ ∗ ∗ 4678.343

1809.154∗ ∗ ∗ 1287.657∗ ∗ 1216.454∗ ∗ 1786.043∗ ∗ 1295.141∗ ∗ 1977.167∗ ∗ ∗ 2098.443∗ ∗ ∗ 1807.523

890.576∗ ∗ ∗ 698.772∗ ∗ 613.876∗ ∗ 856.114∗ ∗ 613.567∗ ∗ ∗ 998.332∗ ∗ ∗ 1090.897∗ ∗ ∗ 823.787

Panel B: One-month-ahead forecast BoA 4768.203 1892.189 ChinaCitic 4015.665 1309.234 CreditSuisse 3993.276 1214.653 Deutsche 4321.089 1764.128 Mitsubishi 4018.453 1324.675 NAB 4987.376 1908.464 RBC 5234.675 2087.143 Societe 4674.892 1805.212

925.167 705.342 612.467 822.673 623.565 994.976 1067.294 823.765

4908.359∗ ∗ 4251.923∗ 4300.836∗ 4537.428 4308.796∗ 5112.651∗ 5525.645∗ 4905.876

2026.048∗ ∗ 1491.141∗ 1319.252 1962.156∗ ∗ 1501.368∗ 2095.028∗ 2344.734 1979.586∗

1047.132∗ ∗ 852.8345∗ 669.7505∗ 908.2581∗ 677.8768∗ 1115.471∗ 1251.923∗ ∗ 876.7333∗ ∗

4874.542∗ ∗ ∗ 4167.041∗ ∗ 4221.251∗ 4495.532∗ ∗ 4251.952∗ ∗ 5096.487∗ ∗ 5356.42∗ ∗ 4882.863∗

2001.327∗ ∗ ∗ 1396.897∗ ∗ 1306.86∗ 1884.19∗ ∗ 1437.03∗ ∗ 2034.64∗ ∗ 2333.883∗ ∗ 1965.442∗ ∗ ∗

958.1549∗ ∗ ∗ 799.1629∗ ∗ 644.8091∗ 888.8869∗ ∗ 645.3146∗ ∗ ∗ 1092.359∗ ∗ 1182.432∗ ∗ 844.1715∗ ∗ ∗

4828.921∗ ∗ ∗ 4120.805∗ ∗ 4116.856∗ ∗ 4433.642∗ ∗ 4204.723∗ ∗ ∗ 5077.269∗ ∗ ∗ 5324.188∗ ∗ 4846.825∗ ∗ ∗

1915.38∗ ∗ ∗ 1382.108∗ ∗ 1232.981∗ ∗ 1865.192∗ ∗ ∗ 1422.312∗ ∗ ∗ 1993.037∗ ∗ ∗ 2238.864∗ ∗ 1871.084∗ ∗ ∗

944.7293∗ ∗ ∗ 789.9774∗ ∗ 628.6304∗ ∗ 866.4108∗ ∗ 635.8918∗ ∗ ∗ 1004.138∗ ∗ ∗ 1150.122∗ ∗ 831.9595∗ ∗ ∗

Panel C: One-year-ahead forecast BoA 4923.154 1998.054 ChinaCitic 4214.784 1543.289 CreditSuisse 4021.332 1318.334 Deutsche 4567.767 1876.356 Mitsubishi 4218.254 1413.908 NAB 5021.675 2013.125 RBC 5334.908 2218.544 Societe 4895.224 1936.367

989.023 773.281 665.189 887.554 698.434 1009.238 1123.675 921.467

5067.865∗ 4462.757 4331.053∗ 4796.456∗ 4523.033∗ 5147.81∗ 5631.449 5137.095∗

2139.402∗ 1757.716∗ 1431.861∗ 2086.982 1602.504∗ 2209.92∗ 2492.352∗ 2123.41

1119.406∗ 934.9801∗ ∗ 727.4035 979.8889∗ 759.2668 1131.46∗ 1318.058∗ 980.7176∗

5032.948∗ ∗ 4373.666∗ ∗ 4250.909∗ ∗ ∗ 4752.169∗ ∗ ∗ 4463.363∗ ∗ 5131.536∗ ∗ 5458.984∗ 5112.997∗ ∗

2113.298∗ ∗ 1646.623∗ ∗ 1418.412∗ ∗ 2004.056∗ ∗ 1533.831∗ 2146.22∗ ∗ 2480.819∗ 2108.238∗

1024.288∗ 876.1388∗ ∗ 700.3151∗ ∗ 958.9899∗ ∗ 722.795∗ 1108.017∗ ∗ 1244.895∗ 944.2937∗

4985.845∗ ∗ ∗ 4325.137∗ ∗ ∗ 4145.78∗ ∗ ∗ 4686.745∗ ∗ ∗ 4413.785∗ ∗ 5112.187∗ ∗ 5426.135∗ ∗ 5075.26∗ ∗

2022.543∗ ∗ 1629.191∗ ∗ 1338.226∗ ∗ ∗ 1983.849∗ ∗ ∗ 1518.122∗ ∗ ∗ 2102.336∗ ∗ 2379.817∗ ∗ 2007.024∗ ∗

1009.936∗ ∗ 866.0686∗ ∗ ∗ 682.7437∗ ∗ ∗ 934.7412∗ ∗ ∗ 712.2408∗ ∗ ∗ 1018.531∗ ∗ ∗ 1210.879∗ ∗ 930.6334∗ ∗

D. Bu, Y. Liao and J. Shi et al. / Journal of Economic Dynamics & Control 108 (2019) 103753

Table 6 Out-of-Sample Evaluation for 95% VaR and ES forecasts (loss function value). This table reports the three score function values to jointly evaluate the 95% VaR and ES out-of-sample forecasts using the original returns and the three reconstructed return series. “Long Run” comprises the first seven scales of the decomposed return components, denoted by y1: 7,t ; “Mid Run” comprises the first four scales of the decomposed return components, denoted by y1: 4,t ; and “Short Run” comprises the first two scales of the decomposed return components, denoted by y1: 2,t . “FZG”, “AS”, and “AL” denote the FZG, AS, and AL negative score functions, respectively. ∗ , ∗ ∗ , ∗ ∗ ∗ indicate the cases where the score function values of the forecasts from the frequency based return components are not significantly worse than the ones generated by the original return series at the 1%, 5%, and 10% levels, respectively, based on the Diebold-Mariano test.

11

12

D. Bu, Y. Liao and J. Shi et al. / Journal of Economic Dynamics & Control 108 (2019) 103753 Table 7 Out-of-Sample Evaluation for 95% VaR and ES forecasts (statistical test). This table reports the results of two statistical tests on VaR and ES forecasts. zuncon and zcon represent the unconditional and conditional tests of Acerbi and Szkeley (2014). “Accept” (or ‘Reject”) indicates the cases where the model provides accurate risk estimates (or underestimate the risk). Bank names

Original return (yt )

Short run (y1: 2,t )

Mid run (y1: 4,t )

Long run (y1: 7,t )

zuncon

zuncon

zcon

zuncon

zcon

zuncon

zcon

Panel A: One-week-ahead forecast BoA accept accept ChinaCitic accept accept CreditSuisse accept accept Deutsche accept accept Mitsubishi accept accept accept NAB accept RBC accept accept Societe accept accept

accept accept accept accept accept accept accept accept

accept accept accept accept accept accept accept accept

accept accept accept accept accept accept accept accept

accept accept accept accept accept accept accept accept

accept accept accept accept accept accept accept accept

accept accept accept accept accept accept accept accept

Panel B: One-month-ahead forecast BoA accept accept ChinaCitic accept accept CreditSuisse accept accept Deutsche accept accept Mitsubishi accept accept NAB accept accept RBC accept accept Societe accept accept

accept reject reject accept reject reject accept accept

accept reject reject accept reject reject accept accept

accept accept accept accept accept accept accept accept

accept accept accept accept accept accept accept accept

accept accept accept accept accept accept accept accept

accept accept accept accept accept accept accept accept

Panel C: One-year-ahead BoA accept ChinaCitic accept CreditSuisse accept Deutsche accept Mitsubishi accept accept NAB RBC accept Societe accept

reject reject reject reject reject reject reject reject

reject reject reject reject reject reject reject reject

reject accept accept accept reject accept reject reject

reject accept accept accept reject accept reject reject

accept accept accept accept accept accept accept accept

accept accept accept accept accept accept accept accept

zcon

forecast accept accept accept accept accept accept accept accept

Tables 6 and 7 present the out-of-sample 95% ES (and VaR) forecasting results based on the GAS one factor model.9 The table reports the three scoring (or loss) function values as well as the Acerbi–Szekely unconditional and conditional test results. Again, the forecasts using full original return series achieve the smallest loss function values, suggesting that the full sample of the original return series offers the most accurate forecasts. Moreover, a successive exclusion of irrelevant information components from the underlying return series reduces the ES (and VaR) forecasts without lowering the quality of the forecasts. For example, when we focus on one-week and one-month-ahead forecasts, the Diebold-Mariano test results show that the forecasts based on the short- and mid-run return components are not significantly different from those based on the full original return series. Also, the Acerbi–Szekely tests accept the forecasts using only the short- and mid-run return components in most cases. These results indicate that excluding long-run information components leads to lower ES (and VaR) forecasts without impacting the forecasting performance. The short- and mid-run return components are able to provide the crucial information to determine the ES forecasts over one-week- and one-month-ahead horizons. For the one-year-ahead forecasts, those using only the short-run return components are always worse than those based on the full return series, as the former reaches significantly higher loss function values than the latter in all cases. Moreover, the former forecasts are rejected by Acerbi–Szekely unconditional and conditional tests in all the cases. These findings confirm that the short-run components are not able to provide sufficient information for ES forecasts over a relatively longer time horizon. The ES forecasts using only the short-run return components underestimate the risk over the one-year predictive horizon. Tables 8 and 9 report the results of the 99% ES (and VaR) forecasts. We observe a similar pattern in the 95% and 99% ES (and VaR) forecasts. Collectively, those results emanate a clear message: for one-week-ahead ES (and VaR) forecasts, the long-run information components of the underlying return series are quantitatively irrelevant as the short-run information component alone can

8 Note that the two baseline ES modeling approaches we employ do not require the choice of return distribution, which is different from the traditional VaR models. In the GAS one factor model of Patton et al. (2018), both the ES and VaR are proportional to the return volatility, and the return volatility follows a score driven dynamic process, as described in Section 2.2.1. Similarly, in the AL model of Taylor (2017), the VaR dynamics are described by the CaViaR model of Engle and Manganelli (2004), where the return distribution function is also not required. 9 Again, we also implement the AL model but the results are not reported. The results show a similar pattern and are available upon request.

Bank Names

Original return (yt ) FZG

AS

Short run (y1: 2,t )

Mid run (y1: 4,t )

Long run (y1: 7,t )

AL

FZG

AS

AL

FZG

AS

AL

FZG

AS

AL

Panel A: One-week-ahead forecast BoA 4619.232 1809.121 ChinaCitic 4098.454 1298.434 CreditSuisse 3912.667 1212.656 Deutsche 4320.545 1734.563 Mitsubishi 3997.026 1287.128 NAB 5098.237 1924.456 RBC 5012.334 1990.565 Societe 4598.188 1812.543

894.576 656.923 612.114 867.342 656.897 1089.025 1098.675 889.432

4755.01∗ ∗ 4339.583∗ ∗ ∗ 4214.018∗ ∗ ∗ 4536.856 4285.821 5226.295∗ ∗ 5290.945 4825.382∗ ∗

1937.103∗ ∗ 1478.841∗ ∗ 1317.083∗ ∗ ∗ 1929.272∗ ∗ 1458.813∗ 2112.583∗ ∗ 2236.237 1987.626∗ ∗ ∗

1012.508∗ ∗ 794.2907∗ ∗ ∗ 669.3645∗ ∗ 957.5742∗ 714.112∗ 1220.909∗ ∗ 1288.733 946.6227∗ ∗ ∗

4722.248∗ ∗ ∗ 4252.951∗ ∗ ∗ 4136.04∗ ∗ ∗ 4494.966∗ ∗ ∗ 4229.283∗ ∗ ∗ 5209.773∗ ∗ 5128.908∗ ∗ ∗ 4802.747∗ ∗ ∗

1913.468∗ ∗ 1385.374∗ ∗ ∗ 1304.711∗ ∗ ∗ 1852.613∗ ∗ 1396.298∗ ∗ ∗ 2051.689∗ ∗ 2225.888∗ ∗ 1973.424∗ ∗ ∗

926.4732∗ ∗ ∗ 744.3035∗ ∗ ∗ 644.4374∗ ∗ ∗ 937.1512∗ ∗ 679.8092∗ ∗ ∗ 1195.613∗ ∗ ∗ 1217.198∗ ∗ 911.466∗ ∗ ∗

4678.053∗ ∗ ∗ 4205.761∗ ∗ ∗ 4033.753∗ ∗ ∗ 4433.084∗ ∗ ∗ 4182.302∗ ∗ ∗ 5190.129∗ ∗ ∗ 5098.045∗ ∗ ∗ 4767.321∗ ∗ ∗

1831.294∗ ∗ ∗ 1370.707∗ ∗ ∗ 1230.954∗ ∗ ∗ 1833.933∗ ∗ 1381.997∗ ∗ ∗ 2009.738∗ ∗ ∗ 2135.265∗ ∗ ∗ 1878.682∗ ∗ ∗

913.4915∗ ∗ ∗ 735.7485∗ ∗ ∗ 628.268∗ ∗ ∗ 913.4546∗ ∗ ∗ 669.883∗ ∗ ∗ 1099.053∗ ∗ ∗ 1183.939∗ ∗ ∗ 898.280∗ ∗ ∗

Panel B: One-month-ahead forecast BoA 4521.334 1787.249 ChinaCitic 3984.897 1219.763 CreditSuisse 3897.914 1198.372 Deutsche 4219.768 1689.268 Mitsubishi 3908.134 1206.234 NAB 4902.187 1893.267 RBC 5012.334 1956.238 Societe 4512.387 1743.889

872.135 623.908 598.092 812.896 601.673 989.223 1012.334 815.673

5014.638∗ ∗ 4373.091∗ ∗ ∗ 4317.114∗ ∗ 4645.791∗ ∗ 4406.801∗ 5317.117 5525.645 4999.159

2050.842∗ ∗ 1587.315∗ ∗ 1334.977 2014.768∗ 1602.055 2129.543∗ 2385.878 2057.521∗

1074.076∗ ∗ 897.964∗ ∗ 685.4525∗ 969.091 740.095∗ 1228.019∗ 1358.699∗ 956.014

4980.087∗ ∗ ∗ 4285.788∗ ∗ ∗ 4237.228∗ ∗ ∗ 4602.895∗ ∗ ∗ 4348.665∗ ∗ 5300.307∗ ∗ 5356.423∗ 4975.709∗ ∗

2025.819∗ ∗ ∗ 1486.992∗ ∗ ∗ 1322.437∗ 1934.711∗ ∗ 1533.402∗ 2068.158∗ ∗ 2374.837∗ ∗ 2042.818∗ ∗

982.810∗ ∗ ∗ 841.452∗ ∗ ∗ 659.927∗ 948.423∗ 704.544∗ ∗ 1202.566∗ ∗ 1283.283∗ ∗ ∗ 920.508∗ ∗

4933.479∗ ∗ ∗ 4238.235∗ ∗ ∗ 4132.438∗ ∗ ∗ 4539.527∗ ∗ ∗ 4300.361∗ ∗ ∗ 5280.321∗ ∗ ∗ 5324.188∗ ∗ ∗ 4938.985∗ ∗ ∗

1938.823∗ ∗ ∗ 1471.253∗ ∗ ∗ 1247.677∗ ∗ ∗ 1915.204∗ ∗ 1517.697∗ ∗ ∗ 2025.873∗ ∗ ∗ 2278.153∗ ∗ ∗ 1944.745∗ ∗ ∗

969.039∗ ∗ ∗ 831.781∗ ∗ ∗ 643.368∗ ∗ ∗ 924.4411∗ ∗ ∗ 694.257∗ ∗ ∗ 1105.445∗ ∗ ∗ 1248.215∗ ∗ ∗ 907.191∗ ∗ ∗

Panel C: One-year-ahead forecast BoA 5029.752 2022.506 ChinaCitic 4334.892 1642.826 CreditSuisse 4036.552 1334.048 Deutsche 4676.855 1926.667 Mitsubishi 4314.2 1508.729 NAB 5222.504 2046.288 RBC 5334.908 2257.474 Societe 4988.304 2012.599

1014.472 814.200 680.784 947.000 762.539 1111.059 1219.512 1004.793

5177.597 4589.932 4347.445∗ 4911.005 4625.911∗ 5353.683 5631.449∗ 5234.774∗

2165.584∗ 1871.084 1448.929 2142.941∗ 1709.973∗ 2246.325∗ 2536.087∗ ∗ 2207.006

1148.21∗ 984.4559∗ 744.4572∗ 1045.523∗ 828.955∗ 1245.612∗ 1430.474∗ 1069.401

5141.924∗ ∗ 4498.302∗ 4266.998∗ ∗ 4865.661∗ ∗ 4564.884∗ ∗ 5336.758∗ 5458.984∗ ∗ ∗ 5210.219∗ ∗

2139.163∗ ∗ 1752.826∗ 1435.318∗ ∗ 2057.791∗ ∗ 1636.695∗ 2181.576∗ ∗ 2524.351∗ ∗ ∗ 2191.236∗ ∗

1050.644∗ ∗ ∗ 922.501∗ ∗ 716.734∗ ∗ 1023.221∗ ∗ 789.136∗ ∗ 1219.804∗ ∗ 1351.071∗ ∗ ∗ 1029.684∗ ∗

5093.801∗ ∗ ∗ 4448.396∗ ∗ 4161.472∗ ∗ ∗ 4798.675∗ ∗ 4514.178∗ ∗ 5316.635∗ ∗ 5426.135∗ ∗ ∗ 5171.764∗ ∗ ∗

2047.294∗ ∗ ∗ 1734.269∗ ∗ 1354.177∗ ∗ ∗ 2037.043∗ ∗ ∗ 1619.932∗ ∗ ∗ 2136.97∗ ∗ 2421.577∗ ∗ ∗ 2086.038∗ ∗

1035.922∗ ∗ ∗ 911.898∗ ∗ ∗ 698.750∗ ∗ ∗ 997.348∗ ∗ ∗ 777.613∗ ∗ ∗ 1121.296∗ ∗ ∗ 1314.153∗ ∗ ∗ 1014.788∗ ∗ ∗

D. Bu, Y. Liao and J. Shi et al. / Journal of Economic Dynamics & Control 108 (2019) 103753

Table 8 Out-of-Sample Evaluation for 99% VaR and ES forecasts (loss function value). This table reports the three score function values to jointly evaluate the 99% VaR and ES out-of-sample forecasts using the original returns and the three reconstructed return series. “Long Run” comprises the first seven scales of the decomposed return components, denoted by y1: 7,t ; “Mid Run” comprises the first four scales of the decomposed return components, denoted by y1: 4,t ; and “Short Run” comprises the first two scales of the decomposed return components, denoted by y1: 2,t . “FZG”, “AS”, and “AL” denote the FZG, AS, and AL negative score functions, respectively. ∗ , ∗ ∗ , ∗ ∗ ∗ indicate the cases where the score function values of the forecasts from the frequency based return components are not significantly worse than the ones generated by the original return series at the 1%, 5%, and 10% levels based on the Diebold-Mariano test.

13

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D. Bu, Y. Liao and J. Shi et al. / Journal of Economic Dynamics & Control 108 (2019) 103753 Table 9 Out-of-Sample Evaluation for 99% VaR and ES forecasts (statistical test). This table reports the results of two statistical tests on VaR and ES forecasts. zuncon and zcon represent the unconditional and conditional tests of Acerbi and Szkeley (2014). “Accept” (or “Reject”) indicates the cases where the model provides accurate risk estimates (or underestimate the risk) in more than 95% out of 10 0 0 replications. Bank names

Original return (yt )

Short run (y1: 2,t )

Mid run (y1: 4,t )

Long run (y1: 7,t )

zuncon

zuncon

zcon

zuncon

zcon

zuncon

zcon

Panel A: One-week-ahead forecast BoA accept accept ChinaCitic accept accept CreditSuisse accept accept Deutsche accept accept Mitsubishi accept accept accept NAB accept RBC accept accept Societe accept accept

accept accept accept accept accept accept accept accept

accept accept accept accept accept accept accept accept

accept accept accept accept accept accept accept accept

accept accept accept accept accept accept accept accept

accept accept accept accept accept accept accept accept

accept accept accept accept accept accept accept accept

Panel B: One-month-ahead forecast BoA accept accept ChinaCitic accept accept CreditSuisse accept accept Deutsche accept accept Mitsubishi accept accept NAB accept accept RBC accept accept Societe accept accept

accept accept reject reject reject reject reject reject

accept accept reject reject reject reject reject reject

accept accept accept accept accept accept accept accept

accept accept accept accept accept accept accept accept

accept accept accept accept accept accept accept accept

accept accept accept accept accept accept accept accept

Panel C: One-year-ahead BoA accept ChinaCitic accept CreditSuisse accept Deutsche accept Mitsubishi accept accept NAB RBC accept Societe accept

reject reject reject reject reject reject reject reject

reject reject reject reject reject reject reject reject

accept accept accept accept accept accept accept accept

accept accept accept accept accept accept accept accept

accept accept accept accept accept accept accept accept

accept accept accept accept accept accept accept accept

zcon

forecast accept accept accept accept accept accept accept accept

provide sufficient information to achieve adequate out-of-sample ES forecasts. However, when we increase the predictive horizon, the long-run information components offer additional information to generate accurate forecasts.10

5. Conclusion In this paper, we propose a multi-scale approach for ES modeling and forecasting, with which the capital requirements of financial institutions can be more accurately regulated over multiple time horizons. Using this approach, we decompose the original stock return series into different frequency based components, and then model and forecast the ES (and VaR) over different predictive horizons. In this method, we can combine the benefits of two recently developed ES models, which jointly deal with the ES and VaR, with the wavelet analysis, which explores the different frequency based information from the return series. The results of our Monte Carlo simulation study show that using only short- and mid-run return components can strike a balance between lower ES forecasts (to minimize the capital requirements of financial institutions) and the forecast accuracy over short-run predictive horizons (e.g., one-week and one-month forecasts). However, the long run return component plays an important role in achieving accurate forecasts over a longer horizon (e.g., one-year-ahead forecasts). Furthermore, we apply the new method to real data of eight banks. We find that the wavelet based multi-scale approach delivers noteworthy forecasting gains over a wide range of forecasting horizons. In the current study, we solely focus on univariate ES forecast for a single financial institution, ignoring the crossdependence or spill-over effects of the risk levels of different institutions. Clearly, the wavelet based techniques can be easily extended to analyze multivariate time series, which should improve the multivariate tail risk forecasts. We leave this for future research.

10 To validate our new method, we compared our model based one-year-ahead ES forecasts with a put option (with one-year maturity) implied ES for two banks from our sample: Bank of America(BOA) and National Australian Bank (NAB). Our results confirm that our model forecasts with original stock returns are comparatively as good as the option implied ES, and our model forecasts using the ”long run” return component are slightly better than the option implied ES. We do not report these results here but they are available upon request.

D. Bu, Y. Liao and J. Shi et al. / Journal of Economic Dynamics & Control 108 (2019) 103753

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Acknowledgments The author, Hongfeng Peng, acknowledge financial supports from Major Program of National Social Science of China (Grant No. 16ZDA032), and Taishan Scholars Program (Grant No. TS201712059). Supplementary material Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.jedc.2019.103753. References Acerbi, C., Szkeley, B., 2014. Backtesting expected shortfall. Risk 76–81. Bandi, F.M., Tamoni, A., 2016. Business-Cycle Consumption Risk and Asset Prices. Technical Report. Johns Hopkins University working paper. Berger, T., Gençay, R., 2018. Improving daily value-at-risk forecasts: the relevance of short-run volatility for regulatory quality assessment. J. Econ. Dyn. Control 92, 30–46. Constantinides, G.M., Donaldson, J.B., Mehra, R., 2002. Junior can’t borrow: a new perspective on the equity premium puzzle. Q. J. Econ. 117 (1), 269–296. Creal, D., Koopman, S.J., Lucas, A., 2013. Generalized autoregressive score models with applications. J. Appl. Econ. 28 (5), 777–795. Engle, R., Manganelli, S., 2004. Caviar: conditional autoregressive value at risk by regression quantiles. J. Bus. Econ. Stat. 22, 367–381. Fissler, T., Ziegel, J.F., 2016. Higher order elicitability and osbanddel?’s principle. Ann. Stat. 44, 1680–1707. Harvey, A., 2013. Dynamic Models for Volatility and Heavy Tails. Cambridge University Press. Holton, G., 2014. Value-at-Risk: Theory and Practice, second ed. e-book. Jappelli, T., Modigliani, F., 1998. The Age-Saving Profile and the Life-Cycle Hypothesis. CSEF Working Papers. Centre for Studies in Economics and Finance (CSEF), University of Naples, Italy. Modigliani, F., 1986. Life cycle, individual thrift, and the wealth of nations. Science 234 (4777), 704–712. Ortu, F., Tamoni, A., Tebaldi, C., 2013. Long-run risk and the persistence of consumption shocks. Rev. Financ. Stud. 26, 2876??2915. Patton, A.J., Ziegel, J.F., Chen, R., 2019. Dynamic semiparametric models for expected shortfall (and Value-at-Risk). J. Econom. 211 (2), 388–413. Percival, D., Walden, A., 20 0 0. Wavelet Methods for Time Series Analysis. Cambridge University Press. Poterba, J.M., 2001. Demographic structure and asset returns. Rev. Econ. Stat. 83 (4), 565–584. Taylor, J.W., 2017. Forecasting value at risk and expected shortfall using a semiparametric approach based on the asymmetric laplace distribution. J. Bus. Econ. Stat. 0 (0), 1–13. doi:10.1080/07350015.2017.1281815.