Choosing expected shortfall over VaR in Basel III

International Review of Economics and Finance 60 (2019) 95–113

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International Review of Economics and Finance journal homepage: www.elsevier.com/locate/iref

Choosing expected shortfall over VaR in Basel III using stochastic dominance☆ Chia-Lin Chang a, Juan-Angel Jimenez-Martin b, *, Esfandiar Maasoumi c, Michael McAleer b, d, e, f, g, **, Teodosio Perez-Amaral b a

Department of Applied Economics, Department of Finance, National Chung Hsing University, Taichung, Taiwan Instituto Complutense de An alisis Economico (ICAE), Facultad de Ciencias Economicas y Empresariales, Universidad Complutense de Madrid, Spain c Department of Economics, Emory University, USA d Department of Finance, Asia University, Taiwan e Econometric Institute, Erasmus School of Economics, Erasmus University Rotterdam, the Netherlands f Discipline of Business Analytics, University of Sydney Business School, Australia g Institute of Advanced Sciences, Yokohama National University, Japan b

A R T I C L E I N F O

A B S T R A C T

JEL classification: G32 G11 G17 C53 C22

In this paper we use stochastic dominance to evaluate the consequences of moving from Value-atRisk (VaR) to Expected Shortfall (ES) from a policy maker's perspective. In particular, we compare VaR at the 99% level (VaR99) and ES at the 97.5% level (ES97.5). We contemplate VaR99 and ES97.5 as two alternative risk metrics according to the capital adequacy bank regulation, as suggested by Basel III. Moving from VaR99 to ES97.5 will have effects in terms of the quantity and quality of the capital required to banks. According to the Basel Committee on Banking Supervision (2013, page 18): “the Committee believes that moving to a confidence level of 97.5% (relative to the 99th percentile confidence level for the current VaR measure) is appropriate.” Stochastic dominance of the capital requirement distribution after the change in bank regulations, as suggested by Basel III, over the capital requirement under the old regulation, implies that a policy maker with any positive marginal utility of capital requirements (and a negative second derivative for risk aversion) would prefer it. SD tests examines if the rankings of outcomes are utility function specific, or uniform, over all decision makers with preferences in the class considered.

Keywords and phrases: Stochastic dominance Value-at-Risk Expected shortfall Basel III accord Daily capital charges

1. Introduction In 2016, the Basel Committee on Banking Supervision (2016) published a document which presented the Basel Committee's proposals with regard to trading book capital requirement policies. A key element of the proposal was moving the quantitative risk metrics system from VaR to Expected Shorfall, ES, and decreasing the confidence level from 99% to 97.5%. The Basel Committee (2013, p. 3) ☆ The authors wish to thank four referees for very helpful comments and suggestions, and the Ministry of Science and Technology (MOST), Taiwan, Ministerio de Economía, Industria y Competitividad, Spain (research project ECO2015-67305-P), and Banco de Espa~ na, Spain (research project PR71/15-20229), and the Australian Research Council, Australia, for research support. * Corresponding author. ** Corresponding author. Department of Finance, Asia University, Taiwan. E-mail address: [email protected] (J.-A. Jimenez-Martin).

https://doi.org/10.1016/j.iref.2018.12.016 Received 30 March 2016; Received in revised form 7 November 2018; Accepted 20 December 2018 Available online 24 December 2018 1059-0560/© 2018 Elsevier Inc. All rights reserved.

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observed that “a number of weaknesses have been identified in using Value-at-Risk (VaR) for determining regulatory capital requirements, including its inability to capture tail risk”. Artzner, Delbaen, Eber, and Heath (1997) proposed the use of ES, which considers losses beyond the VaR level and is shown to be sub-additive in order to alleviate the problems inherent in VaR, which disregards losses beyond the percentile, and is not sub-additive. In addition to previous statistical issues, policy makers are confronted with two policy alternatives which yield different distribution outcomes, namely capital requirements. There is a trade-off between limiting capital requirements, supervision and compliance costs, on the one hand, and reducing the probability of bank failure, on the other (see Van den Heuvel, 2008). There is, however, a consensus among policy makers in favor of higher capital requirements. According to Repullo and Suarez (2012), the primary microprudential role of increased requirements is clear: having more capital helps banks better absorb adverse shocks, and reduces the probability of financial distress. Greater access to capital would also reduce bank risk-taking incentives, and thereby improve investment efficiency and overall welfare. Additionally, due to transactions and dilution costs that are associated with equity issuance, variation in the capital requirements may be costly. This cost is implicit in the literature on procyclicality of capital requirements (see Gordy and Howells (2006); Repullo and Suarez (2012)). Gordy and Howells (2006) conclude that flattening the capital function in order to reduce the sensitivity of capital charges to changes in the probability of default might reduce the procyclicality of capital requirements. Consequently, from a decision-theoretic view point, ordering uncertain outcomes arising from applying VaR99 or ES97.5 is valuable for policy evaluation. In this paper, we use stochastic dominance (SD) as a criterion for choosing between two distributions of capital requirements that are the outcome of applying two alternative risk metrics, namely VaR99 or ES97.5. The interest in comparing capital requirements distributions is to know, for instance, if the distribution of the capital requirements after the change in bank regulation, as suggested by Basel III, stochastically dominates the capital requirements distribution under the old regulation. If so, a policy maker with any positive marginal utility of daily capital charges (and a negative second derivative for risk aversion) would prefer it. SD tests check if rankings of outcomes are utility function specific, or are uniform, over all decision makers with preferences in the class considered. Concavity of such preferences is a well-known aversion to uncertainty which impacts all investment and planning decisions. Such a policy maker's utility function definition would fit the suggestion of Repullo and Suarez (2012) regarding the primary microprudential role of increased requirements. In addition, the costly variance in the capital requirements (see Gordy and Howells (2006)) could explain the policy maker's risk aversion. In the context of risk management, First-order Stochastic Dominance (FSD) can be seen as a stochastic version of preferring more capital requirements to less in order to reduce the default probability of banks. Second-order Stochastic Dominance (SSD) can be seen as a characterization of risk aversion, as a greater variance in the capital requirement might prove to be costly because of greater uncertainty. Dominated outcomes and rules/decisions can be identified by stochastic dominance, and thereby eliminated, as appropriate. The novel contribution of the paper is to introduce formal SD tests to evaluate the effect on capital requirements as a consequence of moving from VaR99 to ES97.5, as the new bank regulation capital adequacy suggests. We develop a new strategy for testing the welfare cost of bank regulation changes. SD has not yet been used to choose between alternative regulatory rules by policy makers. Nevertheless, SD is a well-developed branch of “Decision Theory under Risk”, with important applications in Economics, Finance, Portfolio Theory, and Financial Risk Management, among others. For instance, SD is regarded as one of the most useful tools to rank investment prospects (see, for example, Levy (1992)) as the ranking of the assets has been shown to be equivalent to expected utility maximization for the preferences of risk averters and risk lovers (see Lean, McAleer, and Wong (2015), Quirk and Saposnik (1962), Hanoch and Levy (1969), Hammond (1974), Stoyan (1983), and Li and Wong (1999)). Davidson and Duclos (2000) use SD to order income distributions in terms of poverty, social welfare and inequality. Chang, Jimenez-Martin, Maasoumi, and Perez-Amaral (2015) use SD tests for choosing among several VaR forecasting models to analyze whether the daily capital charges produced by one model would stochastically dominate the daily capital charges produced by an alternative model. For purposes of clarity, it is worth highlighting that our paper is distinct from most research in the debate about VaR and ES. For example, Yamai and Yoshiba (2002a, 2002b, 2002c, 2002d, 2005) emphasize that portfolios minimizing ES would theoretically outperform portfolios that minimize VaR in quantifying tail risk. They also argue that VaR and ES are consistent with first- and second-order Stochastic Dominance, and hence are consistent with expected utility maximization. Yamai and Yoshiba do not run formal test of SD as ES and VaR are quantiles, and quantile functions and alternative definitions of SD in terms of quantiles are quite revealing. “Consistency” with SD orders appears to be an obvious issue. Ranking by Expected Shortfall follows from Second-order Stochastic Dominance, and ranking by VaR follows from First-order Stochastic Dominance. According to this, we deviate from Yamai and Yoshiba in several aspects: (1) we conduct statistically sound stochastic dominance tests of whether such orders exist to a degree of confidence on the capital requirements produced by using the two different risk metrics; (2) we do not test dominance between two bank portfolio distributions; (3) we do not compare VaR versus ES at the same level of confidence; and (4) we do not choose between VaR and ES as the appropriate measure of tail risk. Although it is beyond the scope of the paper, it is worth mentioning that elicitability is always present when debating the appropriate choice of a risk metric. Acerbi and Szekely (2014) and Acerbi and Tasche (2002) discuss the amenity of ES for backtesting. While backtesting of VaR is simple and intuitive, that of ES is not straightforward because it is not elicitable (Gneiting, 2011). Acerbi and Szekely (2014) argue that elicitability is relevant for model selection but not for model testing, and so is not required for evaluation and regulatory purposes. Therefore, one can find a backtest that does not exploit the property of elicitability, and there is no reason why backtesting would not work. Wong (2008), Righi and Ceretta (2013), Acerbi and Szekely (2014) and Emmer, Kratz and Tashe (2013) propose different approaches to the backtesting of ES like those based on distribution forecasts, linear approximation of ES with VaR, or directly with Monte 96

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Carlo tests. More recently, Fissler, Ziegel, and Gneiting (2015) proved that Expected Shortfall is jointly elicitable with Value-at-Risk and proposed replacing traditional backtests by comparative backtesting approaches based on strictly consistent scoring functions. Nolde and Ziegel (2017) also highlight the deficiency of the traditional backtests, and argue that the existing regulatory framework can be enhanced by including such a comparative backtesting approach. They consider the joint elicitability of VaR and ES on a simulation exercise to show that the comparative backtesting approach has the ability to differentiate among methods relying on both correctly specified and misspecified models. It is worth noting that elicitability is a relevant requirement in connection with backtesting and with the comparison of the accuracy of different forecasts of a risk measure. In this paper, we do not focus on a comparison (backtesting) of forecasting and estimation methods for forecasting either VaR or ES. Closer to our research, but using a different approach, is Danielsson (2013) who has examined the quantitative impact of shifting from VaR99 to ES97.5. Using analytical calculations, numerical simulations, and empirical results from observed data, he analyzes one of the main issues raised by this change, namely that estimating ES conditional on VaR might be such that estimation and model risk for ES will be strictly higher than for VaR. Having found that 97.5% ES and 99% VaR are exactly the same for the conditional normal distribution, with ES being slightly greater than VaR for the conditional Student-t distribution, the analysis concludes that the ES97.5 risk forecasts are generally more volatile than are their VaR99 counterparts. Consequently, while there are numerous papers on the statistical properties of VaR versus ES, less is known about the policy implications of using VaR and ES. In this paper, we analyze the SD relations induced by the sampling distribution of the daily capital charges produced by using either ES97.5 or VaR99. The Basel III Accord requires that banks and other Authorized Deposit-taking Institutions (ADIs) communicate their daily risk forecasts to the appropriate monetary authorities at the beginning of each trading day, using one of a range of alternative financial risk models to forecast risk. The risk estimates from these models are used to determine the capital requirements, depending in part on the number of previous violations, whereby realized losses exceed the estimated risk measure (for further details see, for example, Chang, Jimenez-Martin, McAleer, and Perez-Amaral (2011)). This paper examines several standard univariate conditional volatility models for forecasting VaR and ES, including GARCH, EGARCH, and GJR, paired with Gaussian and Student-t distributions. The empirical results show that the daily capital charges (DCC) produced using ES97.5, or DCC_ES, are statistically First-order Stochastic Dominant (FSD) over the daily capital charges produced using VaR99, or DCC_VaR, for the four models and the two distributions considered. This would imply that the high mean of DCC_ES is likely to be higher than the mean of DCC_VaR. Moreover, as First-order Stochastic Dominance implies Second-order Stochastic Dominance, we would conclude that the volatility inherent in the estimated DCC_VaR is greater than the volatility in DCC_ES, at least in the case of extreme events, which is contrary to Danielsson's (2013) results. Given Second-order Stochastic Dominance, a risk-averse policy maker should prefer ES97.5 to VaR99 as a risk measurement because the expected utility of the daily capital charges produced by ES is greater than the expected utility of the daily capital charges produced by VaR, whatever the specific preference function that might be chosen, as long as it is increasing and concave. Three tests proposing different resampling procedures for estimating the critical values of an extended Kolmogorov-Smirnov (KS) test for stochastic dominance will be used, as follows: (i) Barrett and Donald (2003), who propose a standard bootstrap simulation method to mimic the asymptotic null distribution of the least favorable case (LFC); (ii) Linton, Maasoumi, and Whang (2005), LMW, who estimate the critical values using the sub-sampling method proposed in Politis and Romano (1992), which allow for general dependence among the prospects, and for observations not to be iid, so that the critical values for this test do not rely on the (LFC); and (iii) Donald and Hsu (2013), whose test extends Hansen's (2005) re-centering method to obtain critical values for the KS test: this increases the power properties compared with the unadjusted test mounted at the composite boundary of the null and alternative spaces, namely the so-called Least Favorable Case (LFC). The remainder of the paper is organized as follows. Section 2 describes VaR and ES risk measures, and how to derive daily capital charges. In Section 3 the definition, notation and properties of stochastic dominance are presented, together with an outline of the SD tests. Section 4 introduces the data, describes the block bootstrapping method to simulate the time series, and presents alternative conditional volatility models of VaR and ES in order to obtain the daily capital charges. Section 5 presents the main empirical results, and Section 6 gives some concluding comments. 2. Forecasting value-at-risk, expected shortfall and daily capital charges eIn this section we introduce the definitions and explain the forecasting of Value-at-Risk (VaR) and Expected Shortfall (ES). In addition, we describe how to compute Daily Capital Charges (DCC) under VaR and ES as a basic criterion for choosing between risk measures. The Basel II Accord stipulates that Daily Capital Charges must be set at the higher of the previous day's VaR or the average VaR over the last 60 business days (still valid under Basel III), multiplied by a factor (3 þ k) for a violation penalty, where a violation occurs when the actual negative returns exceed the VaR forecast negative returns for a given day. Basel III modifies the computation of DCC charges by using Expected Shortfall instead (see Basel Committee on Banking Supervision, 2016).

2.1. Value-at-Risk The VaR for a given confidence level q 2 ð0; 1Þ and time t is given by the smallest number yq such that the loss Ytþ1 at time tþ1 will fall below yq with probability q: 97

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VaRqt ¼ inf yq 2 < : P Ytþ1 yq q ¼ inf yq 2 < : P Ytþ1 > yq 1 q :

(1)

Thus, VaR is a quantile of the distribution of the loss function, and q is usually taken to be in the range [0.9, 1). For example, the Basel II accord refers to the “VaR99”. Sometimes the level of significance or coverage rate, α ¼ 1 q; is used instead. If the random variable, Ytþ1 , is normally distributed with mean μtþ1 and standard deviation σ tþ1 , for q 2 ð0; 1Þ the VaR of Yt is given by: VaRqt ðYtþ1 Þ ¼ μtþ1 þ σ tþ1 Φ1 ðqÞ;

(2)

where Φ is the cumulative distribution function of a standard normal variable. ~ t ¼ ðYt μt Þ=σ t has a standardized t-distribution with ν > 2 degrees of freedom, that is, the If the normalized random variable Z ~ t would be given as: Student t distribution with mean 0 and variance 1, the VaR of Yt ¼ μt þ σ t Z VaRqt ðYtþ1 Þ ¼ μtþ1 þ σ tþ1

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ν1 ðν 2Þ tν ðqÞ;

(3)

where t 1 ν ðqÞ is the q quantile of the standard Student-t distribution. As quantiles translate under monotonic transformations, the q pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 quantile of the standardized Student-t distribution (mean 0 and variance 1) with ν degrees of freedom is given as t 1 ν ðqÞ ν ðν 2Þ:. In addition to these parametric VaR calculations, we include the analysis based on the non-parametric historical VaR, which does not have to make an assumption about the parametric form of the distribution of the returns. The q% historical VaR is the q quantile of the random variable, Ytþ1 . 2.2. Expected shortfall The expected shortfall at level q is the expected value at time t of the loss in the next period, Ytþ1 , conditional on the loss exceeding VaRqtþ1 : ESqtþ1 ðYtþ1 Þ ¼ Et ½Ytþ1 jYtþ1 > VaRqtþ1 :

(4)

For q 2 ð0; 1Þ; the expected shortfall for a normally distributed random variable, Yt Nðμt ; σ ESqt ðYtþ1 Þ ¼ μtþ1 þ σ tþ1

ϕðΦ1 ðqÞÞ ; 1q

2 t Þ;

is given as: (5)

where ϕ is the density of a standard normal variable. ~ t ¼ ðYt μt Þ=σ t has a standardized t-distribution with ν > 2 degrees of freedom, then the exIf the normalized random variable Z pected shortfall of εt is given by: ESqt ðYtþ1 Þ

¼ μtþ1 þ σ tþ1

f *v t*1 v ðqÞ 1q

2 v 2 þ t*1 v ðqÞ ; q

(6)

where t*1 v ðqÞ denotes the q quantile of the standardized Student-t distribution (that is, with zero mean and unit variance) having ν degrees of freedom, and f *v ðt*1 v ðqÞÞ is the value of its density function at that point. The standardized Student-t density is given as: f *v ðxÞ ¼ ððv 2Þπ Þ1=2 Γ

v v þ 1 ð1þvÞ=2 Γ 1 þ ðv 2Þ1 x2 ; 2 2

(7)

where the gamma function, Γ; is an extension of the factorial function to non-integer values (see Alexander, 2009, p. 130). Analogous to the VaR calculations, for the calculation of ES it is possible for σ t to be replaced by alternative estimates of the conditional standard deviation in order to obtain an appropriate VaR (for useful reviews of theoretical results for conditional volatility models, see Li, Ling, and McAleer (2002) and McAleer (2005), where several univariate and multivariate, conditional, stochastic and realized volatility models are discussed). In the historical VaR model, the ES can be estimated directly, simply by taking the average of all the losses in the tail above the VaR. 2.3. Forecasting daily capital charges In this section, which follows McAleer, Jimenez-Martin, and Perez-Amaral (2013a, 2013b, 2013c) closely, we introduce the calculation of DCC. The Basel II Accord stipulates that DCC must be set at the higher of the previous day's VaR or the average VaR over the last 60 business days, multiplied by a factor (3 þ k) for a violation penalty, where a violation occurs when the actual negative returns exceed the VaR forecast negative returns for a given day. Assuming that the new risk measure might be ES, we can generalize the DCC expression changing VaR to RiskM, which is the risk measure that can be either VaR or ES for day t: DCCt ¼ sup ð3 þ kÞRiskM60 ; RiskMt1

(8)

98

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where DCCt ¼ daily capital charges, RiskM

60 ¼ mean RiskM over the previous 60 working days, 0 k 1 is the Basel II violation penalty (see Table 1).

It is well known that the formula given in equation (8) is contained in the 1995 amendment to Basel I, while Table 1 appears for the first time in the Basel II Accord in 2004. The multiplication factor (or penalty), k, depends on the central authority's assessment of the ADI's risk management practices and the results of a simple backtest. It is determined by the number of times actual losses exceed a particular day's VaR forecast (see Basel Committee on Banking Supervision, 1996, 2006). As stated in a number of previous papers (see, for example, McAleer et al. (2013a; 2013b; 2013c)), the minimum multiplication factor of 3 is intended to compensate for various errors that can arise in model implementation, such as simplifying assumptions, analytical approximations, small sample biases, and numerical errors that tend to reduce the true risk coverage of the model (see Stahl (1997)). Increases in the multiplication factor are designed to increase the confidence level that is implied by the observed number of violations at the 99% confidence level, as required by regulators. For a detailed discussion of VaR, as well as exogenous and endogenous violations, see McAleer (2009) and McAleer, Jimenez-Martin, and Perez-Amaral (2010). In calculating the number of violations, it is well known that ADIs are required to compare the forecasts of VaR with realized profit and loss figures for the previous 250 trading days. In 1995, the 1988 Basel Accord (Basel Committee on Banking Supervision (1988)) was amended to allow ADIs to use internal models to determine their VaR thresholds (Basel Committee on Banking Supervision (1995)). However, ADIs that propose using internal models are required to demonstrate that their models are sound. Movement from the green zone to the red zone (see Table 1) arises through an excessive number of violations. Although this will lead to a higher value of k, and hence a higher penalty, violations will also tend to be associated with lower daily capital charges. It should be noted that the number of violations in a given period is an important, though not the only, guide for regulators to approve a given VaR model. 3. Stochastic dominance and risk measures The primary purpose of the paper is to evaluate the optimality of ES97.5 and VaR99 with respect to the stochastic dominance relations induced by the sampling distribution of the Daily Capital Charges produced by using both risk measures. It is worth noting that each measure will yield different values of DCC. The stochastic dominance concept is applied to determine which risk measure should be used to maximize the expected utility of a risk averse regulator. The objective is to rank capital requirements distributions according to the expected value of the utility-of-DCC function introduced below in this section. In addition, we define first- and second-order stochastic dominance and its relation to the decision making process of risk averse regulators. Finally, we briefly describe the SD tests that are used in this paper.

3.1. Daily capital charges and evaluation framework: stochastic dominance SD rules have been shown to offer superior and more efficient criteria on which to base investment decisions than those derived from traditional strategies based on first and second moments. In this paper we use SD and utility function theory to compare DCC produced by VaR or ES. We focus on the behaviour of the policy decision maker when the potential DCC is used as the variable of interest. An attractive aspect is that SD criteria do not require a parameterized utility function, but rather rely on a general preference assumption. The regulator's utility-of-DCC function may be represented by U (DCC), that denotes the class of all utility functions, u, such that assuming u' 0 (increasing) and u'' 0 (strict concavity), where u' is the first derivative and u'' is the second derivative, reflects the diminishing marginal utility of DCC associated with risk-averse decision making. Consequently, the definition of first- and second-order stochastic dominance applied to risk averse-decision regulator is defined as follows: Table 1 Basel accord penalty zones. Zone

Number of Violations

k

Green Yellow

0 to 4 5 6 7 8 9 10þ

0.00 0.40 0.50 0.65 0.75 0.85 1.00

Red

Note: The number of violations is given for 250 business days. The penalty structure under the Basel II Accord is specified for the number of violations and not their magnitude, either individually or cumulatively. 99

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Definition 1. Consider two distributions of DCC, DCC1 and DCC2, characterized by the cumulative distribution functions, FX and FY, respectively. It is defined that DCC2 First-order stochastically dominates DCC1, denoted DCC2 FSD DCC1, if and only if either: a) E½uðDCC2 Þ E½uðDCC1 Þ for all u with strict inequality for some u; or b) FY ðzÞ FX ðzÞ for all z with strict inequality for some z. Definition 2. It is defined that DCC2 Second-order stochastically dominates DCC2, denoted DCC2 SSD DCC1, if and only if either: a) b)

E½uðDCC2 Þ E½uðDCC1 Þ for all u with strict inequality for some u; or Rz Rz ∞ FY ðtÞ dt ∞ FX ðtÞ for all z with strict inequality for some z. ðsÞ

Davidson and Duclos (2000) offer a very useful characterization of any SD order. They define Dk ðzÞ ¼ ð1Þ Dk ðzÞ

ðsÞ DY ðzÞ

ðsÞ DX ðzÞ

Rz ∞

ðs1Þ

Dk

ðtÞdt; k ¼ Y; X;

where ¼ Fk ðzÞ: Then Y stochastically dominates X at order s, if for all z with strict inequality for some z. In addition, for a better understanding of stochastic dominance in this context, it can be shown that for any integer s 2, is easy to check inductively that: ðsÞ

Dk ¼

1 ðs 1Þ!

Z

z ∞

ðz tÞðs1Þ dFk ðtÞ

ð2Þ

In particular, Dk ðzÞ corresponds to the Domar and Musgrave (1944) measure of risk, expected down side risk, or the well-known poverty gap of Atkinson (1987). Therefore, FSD would imply that the probability of higher capital requirements is always greater in FY than in FX. Additionally, if Y stochastically dominates X at the second order, capital requirements under the FY distribution would imply a reduction in costs derived from a lower variance in capital requirements. In the next section, we describe the statistics to be used for testing SD. 3.2. Test statistics and critical values1 We examine three different resampling procedures for estimating the critical values of an extended Kolmogorov-Smirnov (KS) test for dominance, namely: (i) Barret and Donald (BD) (2003), who propose a standard bootstrap simulation method to mimic the asymptotic null distribution in the least favourable case (LFC); (ii) Linton et al. (2005) (LMW), who proposed subsampling methods proposed in Politis and Romano (1992), and allow for general dependence amongst the prospects, and for non i.i.d. data, with critical values that do not rely on the (LFC); and (iii) the Donald and Hsu (2013) test that extends Hansen's (2005) recentering method, as an attempt to increase power compared with the uncentered tests mounted at the composite boundary of the null and alternative spaces (LFC). We are interested in knowing if DCC2 first order dominates DCC1, requiring FY ðzÞ FX ðzÞ for all z 2 R. The technical assumptions required for the underlying statistical theory include the following (see Linton et al. (2005) (hereafter LMW), Linton, Song, and Whang (2010) and Donald and Hsu (2013) for further details): Assumption 3.1: 1. Z ¼ ½0; z; where z < ∞: 2. FX and FY are continuous functions on Z such that FX (z) ¼ FY(z) ¼ 0 iff z ¼ 0, and FX(z) ¼ FY (z) ¼ 1 iffz ¼ z: The second part of this assumption rules out other cases where FX (z) ¼ FY(z) ¼ 0. This is adhered to in implementation of the tests. Assumption 3.2: 1. fXi gNi¼1 and fYi gM i¼1 are samples from distributions FX and FY, respectively. Linton et al. (2005) allow dependent time series, and possibly dependent X and Y. 2. The relation between M and N: M → ∞ and N/(N þM) → λ 2 ð0; 1Þ as N → ∞. Assumption 3.2 requires that N and M grow at the same rate. The hypothesis, Y FSD X, is stated as follows: H0 : FY ðzÞ FX ðzÞ for all z 2 Z;

(9)

H1 : FY ðzÞ > FX ðzÞ for some z 2 Z;

(10)

Under the null of dominance Y would provide a higher probability of larger DCCs. The empirical CDFs that are used in the tests are:

1

This section is based on Donald and Hsu (2013), and Linton et al. (2005). 100

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N N X X b Y;M ðzÞ ¼ 1 b X;N ðzÞ ¼ 1 1ðXi zÞ; F 1ðYi zÞ F N i¼1 M i¼1

where 1(⋅) denotes the indicator function. The Kolmogorov-Smirnov test statistic is given by: b SN ¼

rffiffiffiffiffiffiffiffiffiffiffiffiffiffi NM b X;M ðzÞ b Y;N ðzÞ F sup F N þ M z2Z

In order to test if Y stochastically dominates X at any order, we can formulate the null and alternative hypotheses by “incomplete moments”, as in Davidson and Duclos’ characterization: ðsÞ

ðsÞ

ðsÞ

ðsÞ

H0 : DY ðzÞ DX ðzÞ for all z 2 Z;

(11)

H1 : DY ðzÞ > DX ðzÞ for some z 2 Z;

(12)

Under the null hypothesis of Second Order Dominance (s ¼ 2), over the entire range of DCC outcomes, risk adverse regulators would prefer alternative Y as it provides a greater expected utility of DCCs. The test statistic for second order dominance is: b S 2;N ¼

rffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 NM b 2X;M ðzÞ b Y;N ðzÞ D sup D N þ M z2Z ð2Þ

where a natural estimator of Dk ðzÞ is based on empirical CDFs: b ð2Þ D k ðzÞ ¼

Z

z 0

b k ðtÞ ¼ ðz tÞd F

N N 1 X 1 X ðz ti Þ Iðti zÞ ¼ ðz ti Þþ N i¼1 N i¼1

Under the previously stated assumptions these tests are known to be asymptotically Gaussian. The asymptotic approximation is also known to be poor for reliable empirical guidance. In response, many resampling techniques and approaches have been proposed in the literature. We implement three of these suggested resampling techniques in this paper. 3.2.1. and Donald (BD) (2003) Barret and Donald apply three different simulation techniques to mimic the asymptotic null distribution over the LFC: the multiplier BD method and bootstrap, with both separate and combined samples. In this paper, the critical values b q N are computed using the bootstrap with separate samples. Consider drawing a random sample of size N from f X1 ; : : : ; XN g and sample size M from f Y1 ; : : : ; YM g to form: b ðsÞBD D k;N ðzÞ ¼

Z

z ∞

ðs1ÞBD b ð1ÞBD b DB b kN ðtÞdt; where D ðzÞ ¼ F D kN kN ðzÞ; k ¼ X; Y:

s b ðsÞBD ðzÞ D b s ðzÞÞ ð D b ðsÞBD ðzÞ D b s ðzÞÞ, and let Pu denote the conditional probability measure given the Define b S YX ðzÞ ¼ ð D Y;N Y;N X;M X;M BD observed sample. Let α be the significance level, then b q N can be computed as:

( ! ) rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s NM u b b sup S ¼ sup q P ðzÞ q 1 α q BD : z2Z YX N N þM

(13)

BD b q N is bounded away from zero in probability.

3.2.2. Linton et al. (2005) BD simulation methods do not work well enough when the data are weakly dependent, as in time series samples. In these cases, one has to appeal to either the subsampling technique of Linton et al. (2005), or a variant of the block bootstrap. Donald and Hsu (2016) provide a comparative examination of these alternatives Linton et al. (2005), estimate the critical value by the subsampling method (proposed by Politis and Romano (1992)), but also allow Y and X to be mutually dependent. Let fðXi ; Yi ÞgNi¼1 be a strictly stationary time series with joint distribution function FXY on Z2 and marginal CDF's, FX and FY, respectively. Suppose that Assumption 1 of LMW (2005) holds. Then under the null hypothesis that H0: ðsÞ

ðsÞ

DY ðzÞ DX ðzÞ for all z 2 Z, the SD tests defined earlier are asymptotically Gaussian. Donald and Hsu (2016) further modify LMW's (2005) test to allow for different sample sizes, j. For j 1, let Xj denote the collection of all of the subsets of size j from {X1, ..., XN}:

2 LMW (2005) extend their test to residuals of models that control for some explanatory factors (styles). If the estimators of these regression models permit certain common expansions, the limiting distribution theory will be preserved with re-centering.

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Xj


Xr1 ; : : : ; Xrj r1 ; : : : ; rj f1; : : : ; Ng :

A random draw denoted by fX b1 ; :::; X bj g from Xj would be a random sample of size j without replacement from the original data. Let b

b be the empirical CDF based on the random draw, fX b ; : : : ;X b g. Define Fb b similarly, let jN and jM denote the subsampling sizes for F 1 j X;j Y;j Rz ðsÞb ðsÞb ðsÞb ðsÞb b ðs1Þb ðtÞdt; and D b b b b ð1Þb ðzÞ ¼ Fb b ðzÞ; k ¼ Y; b the X and Y, respectively, and define D X;Y ðzÞ ¼ D Y;jM ðzÞ D X;jN ðzÞ; where D k;jN ðzÞ ¼ ∞ D k;jN k;jN k;jN ðsÞLMW

X: Then the subsampling critical value bc N bc NðsÞLMW

for any dominance order s is given by:

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ) ( ! jN jM u b ðsÞb supz2Z D ¼ sup cP 1α : YX ðzÞ c jN þ jM

Assume that: 1. jN→ ∞, jM→ ∞, jN/N → 0 and jM/M → 0 as N → ∞. 2. jN/(jNþ jM) → λ, where λ is defined in Assumption 4.2. The subsampling SD tests are known to be asymptotically Gaussian under these assumptions, and also provide consistent tests. The limiting distribution theory in LMW covers weakly stationary, dependent samples, with certain mixing conditions, such as in our applications2 3.2.3. Donald and Hsu (2013) and Re-centering functions Donald and Hsu (2013) and Linton et al. (2005) propose re-centering methods introduced by Hansen (2005) to construct critical values for Kolmogorov-Smirnov type tests. This approach provides a test with improved size and power properties compared with the unadjusted test mounted at the composite boundary of the null and alternative spaces, the so-called Least Favorable Case (LFC). b bb ðzÞ and Fb bb ðzÞ be the empirical cumulative distribution function of the bootstrap sample of X and Y, respectively, where the Let F X;N Y;N b bb ðzÞ be: superscript bb denotes the blockwise bootstrap method that uses a sample of size b. Let D N bb bb b b b b b bb D YX ðzÞ ¼ F Y;N F Y;N ðzÞ F X;N F X;N ðzÞ P P where Fb X;N ðzÞ ¼ N1 Ni¼1 1ðXi zÞ; Fb Y;N ðzÞ ¼ N1 Ni¼1 1ðYi zÞ: For a negative number aN, Donald and Hsu (2013) define the re-centering function b μ N ðzÞ as: pffiffiffiffi b Y;N ðzÞ F b X;N ðzÞ ⋅ 1 N F b Y;N ðzÞ F b X;N ðzÞ < aN ; b μ N ðzÞ ¼ F (see Donald and Hsu (2013) for further details of re-centering functions). The blockwise bootstrap is used to compute the critical values based on the sum of the simulated processes and the re-centering function. For α ½: bb bc bb ~c ; η ; η;N ¼ max n N o pffiffiffiffi bb u bb b YX ðzÞ þ b ~cN ¼ sup cP supz2Z N D μ NðzÞ c 1 α ;

μ NðzÞ is the recentering function defined above. If the decision rule is to reject the where η is an arbitrarily small positive number and b bb null hypothesis, H0: FY ðzÞ FX ðzÞ for all z 2 Z when b S N > bc , then the corresponding test has the same size properties as in the inη;N

dependent random samples case, and analogously for tests of stochastic dominance of a higher order. 4. Data and implementation of tests In this section we describe the data used together with the block bootstrapping procedure for simulating the time series of the stock prices. In addition, for computing ES97.5 and VaR99, the conditional variances must be estimated. We use three standard univariate conditional volatility models that are also described in this section. 4.1. Data description The data used for estimation and forecasting are the closing daily prices for Standard and Poor's Composite 500 Index (S&P500), which are obtained from the Thomson Reuters-Datastream database for the period 1 January 1999 to 26 June 2014, giving 4040 observations. The returns ðRt Þ at time t are defined as: Rt ¼ logðPt =Pt1 Þ;

(14)

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where Pt is the market price. Fig. 1 shows the S&P500 returns. The extremely high positive and negative returns are evident from September 2008 onward, after the Lehman Brothers bankruptcy, and have continued well into 2009. In spring, 2010, the European debt crisis began, with the European Union together with the International Monetary Fund providing 110 million Euros to Greece that became unable to borrow from the market. Greece required a second bailout in mid-2011. Thereafter, Ireland and Portugal also received bailouts in November 2010 and May 2011, respectively. Higher volatility in the S&P500 returns is observed during these periods. Regarding the descriptive statistics, the median (0.022) is above the mean (0.012), and the range is between 11% and 9.5%, with a standard deviation of 1.27. S&P500 returns show a negative skewness (0.17) and high kurtosis (10.99), which would seem to indicate the existence of extreme observations and non-Gaussianity. The Jarque-Bera test takes the value of 10,771.95, which strongly rejects the null hypothesis of Gaussianity. Fig. 2 shows several graphs that provide valuable information for identifying the returns probability distribution. Panel A displays the empirical histogram, together with the density function of the Gaussian distribution and a kernel density estimate of the distribution that show fatter tails than under the normal distribution, and slight asymmetry. Panels B and C exhibit two theoretical quantile-quantile plots (QQ-plot) comparing the quantiles of the S&P500 returns with the quantiles of both a fitted normal, Panel B, and Student-t, Panel C, distributions. For the Gaussian case, the QQ-plot does not lie on a straight line, overall on the tails, supporting the non-normality of returns. According to the QQ-plots, the Student-t distribution seems to fit the observed data better than does the Gaussian distribution. Finally, Panel D displays a boxplot that summarizes the returns distributions showing the extreme observation mentioned above. Figs. 1 and 2 show that stock markets have been working under stress during the last seven years. Traditional risk measurement, specifically VaR, might not work properly under these extreme price fluctuations. The BIS Committee on the Global Financial System (2000) discussed the shortcomings of VaR for measuring and monitoring market risk when many such events are taking place in the tails of the distributions. VaR that suffers tail risk only measures the distribution quantile, and disregards the extreme loss beyond the VaR level, ignoring important information regarding the tails of the distribution. Expected shortfall might be a more appropriate tool for risk monitoring under stress circumstances. As Yamai and Yoshiba (2002c) state, expected shortfall has no tail risk under more lenient conditions than VaR. Fat tails might be explained by clusters of volatility that seem to appear in Fig. 1. A closer examination of the volatility of returns, using a measure proposed in Franses and van Dijk (1999), is given as: Vt ¼ ðRt EðRt jFt1 ÞÞ2 ;

(15)

where Ft1 is the information set at time t-1, highlights the volatility clustering (see Fig. 3). The above conditional expectation, in our case, can be approximated using an AR(1) model.

4.2. Block bootstrapping In order to increase the power of the three tests used in the empirical analysis, we will use block bootstrapping for simulating 500 time series observations of the S&P500 returns for the 3000-observation rolling window that are used for producing a total of 500 onestep-ahead ES97.5 and VaR99 forecasts. We implement the Circular Block Bootstrapping (CBB) method developed in Politis and Romano (1992) for resampling the S&P500 through the MFE toolbox of Sheppard (2013). The block bootstrap is widely used for implementing the bootstrap with time series data, which consists of dividing the data into blocks of observations and sampling the blocks randomly, with replacement. In the CBB, let the data consist of observations fXi : i ¼ 1; :::; ng; and let l 2 f1; :::; ng and b 1 denote the length and the number of

Fig. 1. S&P500 returns 1 January 1999–24 June 2014. 103

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Fig. 2. SP500 Returns distribution analysis 1 January 1999–24 June 2014.

Fig. 3. Volatility of S&P500 returns 1 January 1999–24 June 2014.

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blocks, respectively, such that l b n. Let n and m be the initial data size and the bootstrap sample size, m n and k the number of blocks chosen. Circular Block Bootstrapping consists of dividing the time series into b blocks of consecutive observations, denoted by: Bi ¼ Xði1Þlþ1 ; :::; Xil ;

i ¼ 1; :::; n:

A random sample of k blocks, k 1, B*1 ; :::; B*k is selected with replacement from B*1 ; :::; B*k . Joining the k blocks with m ¼ k l observations, the bootstrap sample is given as: X *1 ; :::; X *l ; :::; X *ðk1Þlþ1 :::; X *l : The Circular Block Bootstrapping procedure is based on wrapping the data around a circle and forming additional blocks using the “circularly defined” observations. For i n, define X1 ¼ Xin , where in ¼ i mod n (where mod refers to the remainder after division of one number by another), and X0 ¼ Xn . The Circular Block Bootstrapping method resamples overlapping and periodically extended blocks of length l. Notice that each Xi appears exactly l times in the collection of blocks and, as the Circular Block Bootstrapping resamples the blocks from this collection with equal probability, each of the original observations X1, ..., Xn receives equal weights under the Circular Block Bootstrapping. This property distinguishes the Circular Block Bootstrapping from previous methods, such as the non-overlapping block bootstrap of Carlstein (1992). Note that ES97.5 and VaR99 are estimated for each sample that is drawn, thereby generating the bootstrap (subsample) distribution of the test statistics. The next section describes several volatility models that are widely used to forecast the 1-day ahead conditional variances and VaR thresholds for the parametric cases.

4.3. Models for forecasting VaR ADIs can use internal models to determine their VaR thresholds. There are alternative univariate time series models for estimating conditional volatility. In what follows, we present several well-known univariate conditional volatility models that can be used to evaluate strategic market risk disclosure, namely GARCH, GJR and EGARCH, with Gaussian and Student-t distributions. These univariate models are chosen because they are widely used in the literature. For an extensive discussion of the theoretical properties of several of these models see, for example, Ling and McAleer (2002a, b, 2003a), McAleer (2005), Caporin and McAleer (2012), McAleer and Hafner (2014), and McAleer (2014). 4.3.1. GARCH For a wide range of financial data series, time-varying conditional variances can be explained empirically through the autoregressive conditional heteroskedasticity (ARCH) model, which was proposed by Engle (1982). When the time-varying conditional variance has both autoregressive and moving average components, this leads to the generalized ARCH(p,q), or GARCH(p,q), model of Bollerslev (1986). It is very common in practice to impose the widely estimated GARCH(1,1) specification in advance, which seems to work well in practice. Consider the stationary AR(1)-GARCH(1,1) model for daily returns, yt : yt ¼ ϕ1 þ ϕ2 yt1 þ εt ;

jϕ2 j < 1

(16)

For t ¼ 1,…, n, where the shocks to returns are given by: pffiffiffiffi

εt ¼ ηt ht ; ηt iidð0; 1Þ ht ¼ ω þ αε2t1 þ βht1 ;

(17)

and ω > 0; α 0; β 0 are necessary and sufficient conditions to ensure that the conditional variance ht > 0: For further details, see McAleer (2014).The stationary AR(1)-GARCH(1,1) model can be modified to incorporate a non-stationary ARMA(p,q) conditional mean and a stationary GARCH(r,s) conditional variance, as in Ling and McAleer (2003b). Tsay (1987) shows that α > 0 in the derivation of the GARCH model. 4.3.2. Glosten, Jagannathan and Runkle (GJR) In the symmetric GARCH model, the effects of positive shocks (or upward movements in daily returns) on the conditional variance, ht , are assumed to be the same as the effects of negative shocks (or downward movements in daily returns) of equal magnitude. In order to accommodate asymmetric behaviour, Glosten, Jagannathan, and Runkle (1992) proposed a model (GJR), for which GJR(1,1) is defined as follows: ht ¼ ω þ ðα þ γ Iðηt1 ÞÞ ε2t1 þ βht1 ; where by:

(18)

ω > 0; α 0; α þ γ 0; β 0 are necessary and sufficient conditions for ht > 0; and Iðηt Þ is an indicator variable defined

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Iðηt Þ ¼

1; 0;

εt < 0 εt 0

(19)

as ηt has the same sign as εt . The indicator variable differentiates between positive and negative shocks, so that asymmetric effects in the data are captured by the coefficient γ. For financial data, it is expected that γ 0 empirically as negative shocks have a greater impact on risk than do positive shocks of similar magnitude. The asymmetric effect, γ; measures the contribution of shocks to both short run persistence, α þ γ=2, and to long run persistence, α þ β þ γ=2. Although GJR permits asymmetric effects of positive and negative shocks of equal magnitude on conditional volatility, the special case of leverage, whereby negative shocks increase volatility while positive shocks decrease volatility (see Black (1976) for an argument using the debt/equity ratio), cannot be accommodated, in practice (for further details on asymmetry versus leverage in the GJR model, see Caporin and McAleer (2012)). McAleer (2014) showed that α > 0 and γ > 0 in the derivation of the GJR model. 4.3.3. EGARCH An alternative model to capture asymmetric behaviour in the conditional variance is the Exponential GARCH, or EGARCH(1,1), model of Nelson (1991), namely: εt1 þ γ εt1 þ β log ht1 ; loght ¼ ω þ α ht1 ht1

β < 1;

(20)

where the parameters α; β and γ have different interpretations from those in the GARCH(1,1) and GJR(1,1) models presented above. EGARCH captures asymmetries differently from GJR. The parameters α and γ in EGARCH(1,1) represent the magnitude (or size) and sign effects of the standardized residuals, respectively, on the conditional variance, whereas α and α þ γ represent the effects of positive and negative shocks, respectively, on the conditional variance in GJR(1,1). As in the case of GJR, EGARCH cannot accommodate leverage (for further details, see McAleer, Chan, and Marinova (2007) and McAleer (2014)). McAleer and Hafner (2014) show that α > 0 and γ > 0 in the derivation of the EGARCH model, which prevents a consideration of leverage in the EGARCH model. In the empirical analysis, the three conditional volatility models given above are estimated under the following distributional assumptions on the conditional shocks: (1) Gaussian; and (2) Student-t, with estimated degrees of freedom. As the models that incorporate the t distributed errors are estimated by QMLE, the resulting estimators are consistent and asymptotically normal, so they can be used for estimation, inference and forecasting. 5. Empirical results Let DCC_ES and DCC_ VaR be the Daily Capital Charges produced using ES97.5 and VaR99, respectively. Based on Definitions 1 and 2, if DCC_ES first-order stochastically dominates DCC_VaR, then the DCC_ES will involve a higher probability of larger Daily Capital Charges than the latter. Similarly, if DCC_ES distribution dominates the DCC_VaR distribution stochastically at the second order, it would imply that a risk averse manager would prefer the DCC_ES distribution as it will provide greater expected utility. In essence, a stochastic dominance preference ordering rule divides the two alternative risk measurements to compute DCC into the efficient set of undominated alternatives and the inefficient set of dominated alternatives. The expected utility of those alternatives in the efficient set is larger than the expected utility of those in the inefficient set. According to Definitions 1 and 2, graphically, DCC_ES would dominate DCC_VaR when the cumulative distribution function of DCC_VAR is above the CDF of DCC_ES. SSD implies that the difference between the integrated cumulative distribution functions (ICDF) (area under CDFs) DCC_VaR and ICDF DCC_ES is always positive for every level of probability. As neither ES nor VaR is observed, they have to be estimated, so we proceed as follows: (1) We use a 3000 observations rolling window (from 1 January 1999 to 1 the July 2010, around 75% of the total number of observations available) for smoothing out spikes of volatility during the Global Financial Crisis, while estimating the conditional volatility models for producing one-step-ahead ES97.5 and VaR99 forecasts.

Table 2 Rejection rates for first-order stochastic dominance tests. Design

BB BD LMW

Gaussian

Student-t

HS

GARCH

EGARCH

GJR

GARCH

EGARCH

GJR

0.0001 0.0000 0.0061

0.0112 0.0092 0.0112

0.0010 0.0000 0.0031

0.0000 0.0000 0.0000

0.0000 0.0000 0.0000

0.0000 0.0000 0.0000

0.2002 0.1848 0.1848

Note: Rejection rates are from three different tests, namely Donald and Hsu (2013) (BB), Barrett and Donald (2003) (BD), and Linton et al. (2005) (LMW), for the null hypothesis: H0: DCC_ES FSD DCC_VaR, where DCC_ES and DCC_VaR denote the DCC produced using ES and VaR risk measurements, respectively. Forecast risk measures are produced using two probability distributions, Gaussian (left panel of the table) and Student t, and three conditional volatility models, stated in the first row of each table. In the last column, VaR and ES are computed using the Historical Simulation (HS) procedure. 106

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(2) In order to obtain the empirical distribution of DCC using both risk measures, block bootstrapping is used for simulating 500 time series of the S&P500 returns for the 3000-observations rolling window chosen in step 1 that will be used for producing a total of 500 one-step-ahead ES97.5 and VaR99 forecasts. (3) Steps 1 and 2 are then repeated for the 1040 days remaining in the total sample (from 2 July 2010 until 26 June 2014), each time leaving out the first observation and adding a new observation at the end. This procedure yields a 500 1040 matrix for computing the cumulative distribution functions needed for testing stochastic dominance. Table 2 presents the rejection rates for three different tests, namely Donald and Hsu (2013) (BB), Barrett and Donald (2003) (BD), and Linton et al. (2005) (LMW), for the null hypothesis: H0: DCC_ES First-order Stochastically Dominates DCC_VaR. Forecast risk measures, needed for computing Daily Capital Charges, are produced using two probability distributions, Gaussian (left panel of the table) and Student-t, and three conditional volatility models given in the first row of each table. In addition, the historical simulation (HS) results are shown in the analysis. Table 3 shows the rejection rates for the Second-order Stochastic Dominance tests of BB, BD and LMW for the same distributions and volatility models. The p-values for the blockwise bootstrap method are approximated based on 200 replications, and the p-values for the subsampling method are approximated based on the 176 possible subsamples. The significance level is set to 5%. For example, in Table 2, under the Gaussian distribution and using GARCH to produce DCC_ES97.5 and DCC_VaR99 forecasts, the BB test obtained a 0.0001% rejection rate for the null hypothesis that DCC_ES First order Stochatically Dominates DCC_VaR. Following Donald and Hsu (2013), when implementing the blockwise bootstrap, the block sizes are set to 12 and the subsample size is set to 25. In summary, the main results are as follows: (1) The BB, BD and LMW tests in Table 2 show that DCC_ES, assuming both Gaussian and Student-t distributions and for every conditional volatility model, first-order stochastically dominates DCC_VaR. This is equivalent to saying that using ES to produce daily capital charges provides a higher probability of obtaining larger DCCs than using VaR. Therefore, the expected utility of the policy maker/regulator would be higher using ES rather than VaR. Nonetheless, DCC_ES does not first-order stochastically dominate DCC-VaR when using Historical Simulation, where the rejection rates are greater than 0.05. Fig. 4 shows the cumulative density functions CDF and integrated cumulative density functions ICDFs for observation 3558 (21 August, 2012) under historical simulation, HS. (2) As the CDFs shown in Fig. 4 cross, first-order stochastic dominance cannot be shown. The DCC_VaR model has a much smaller risk of higher costs at levels of DCC lower than 12% and greater than 13.5%. Nevertheless, the risk of higher costs for DCC levels between 12.0 and 13.5% is higher. Thus, the policy maker should test for second-order stochastic dominance by plotting the difference in the areas under the CDFs at all DCC levels. This is shown in the right panel in Fig. 4. In this example, the area between the cumulative density functions of the DCC_VaR and the DCC_ES is always greater than or equal to zero for all possible outcomes. Therefore, DCC_ES second-order dominates DCC_VaR, as the uncertainty associated with DCC_VaR is greater than that associated with DCC_ES. (3) As First-order Stochastic Dominance implies Second-order Stochastic Dominance, then DCC_ES second-order stochastically dominates DCC_VaR for all the conditional volatility models and distribution errors considered. Therefore, using ES to produce DCC would be chosen by regulators who prefer larger capital requirements to less, and have risk aversion. Contradictory objectives might exist between regulators and risk bank managers. Regulators would prefer ES97.5 to the non-coherent VaR99 of a risk measure as it only provides the amount that is at risk for a particular probability. (4) VaR does not suggest how much is at risk at twice that probability, or at half that probability, as it only tells part of the risk condition. In addition, as stated above, ES would provide capital requirements that will lower the probability of default of banks. On the other hand, VaR might be the preferred risk measure of risk managers, provided that the probability of larger capital requirements will be lower. In addition, it seems that, contrary to the analysis of Danielsson (2013), which was based only on a comparison of standard deviations, VaR99 turns out to be a stochastically different measure of tail-risk when compared with 97.5-ES. (5) The fact that the Student-t distribution has heavier tails than its Gaussian counterpart explains why CDFs and ICDFs are very close under Gaussianity, while this is not the case when the Student-t is used. This can be seen in Fig. 5, assuming the Gaussian

Table 3 Rejection rates for second-order stochastic dominance tests. Design

BB BD LMW

Gaussian

Student-t

HS

GARCH

EGARCH

GJR

GARCH

EGARCH

GJR

0.0000 0.0000 0.0000

0.0000 0.0000 0.0000

0.0000 0.0000 0.0000

0.0000 0.0000 0.0000

0.0000 0.0000 0.0000

0.0000 0.0000 0.0000

0.0000 0.0000 0.0000

Note: Rejection rates are from three different tests, namely Donald and Hsu (2013) (BB), Barrett and Donald (2003) (BD), and Linton et al. (2005) (LMW), for the null hypothesis: H0: H0: DCC_ES SSD DCC_VaR, where DCC_ES and DCC_VaR denote the DCC produced using ES and VaR risk measurements, respectively. Forecast risk measures are produced using two probability distributions, Gaussian (left panel of the table) and Student t, and three conditional volatility models, stated in the first row of each table. In the last column, VaR and ES are computed using the Historical Simulation (HS) procedure. 107

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108 International Review of Economics and Finance 60 (2019) 95–113

Fig. 4. CDF and ICDF for DCC_ES97.5 and DCC_VaR99. Note: In the left panel, the solid line is the CDF of DCC_ES97.5 and the dashed line is the CDF of DCC_VaR99 produced by historical simulation. In the right panel, the solid and dashed lines depict the integrated cumulative distribution functions (ICDF) of the CDFs shown in the left panel. These are the empirical distributions of DCC_ES97.5 and DCC_VaR99 of observation 3559 (21 August, 2012).

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Fig. 5. CDF and ICDF for DCC_ES97.5 and DCC_VaR99. Note: In the left panel, the solid line is the CDF of DCC_ES97.5 and the dashed line is the CDF of DCC_VaR99 produced by an EGARCH model assuming a Gaussian distribution of S&P500 returns. In the right panel, the solid and dashed lines depict the integrated cumulative distribution functions (ICDF) of the CDFs shown in the left panel. These are the empirical distributions of DCC_ES97.5 and DCC_VaR99 of observation 3559 (21 August, 2012).

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Fig. 6. CDF and ICDF for DCC_ES97.5 and DCC_VaR99. Note: In the left panel, the solid line is the CDF of DCC_ES97.5 and the dashed line is the CDF of DCC_VaR99 produced by an EGARCH model assuming a Student-t distribution of S&P500 returns. In the right panel, the solid and dashed lines depict the integrated cumulative distribution functions (ICDF) of the CDFs shown in the left panel. These are the empirical distributions of DCC_ES97.5and DCC_VaR99 of observation 3559 (21 August, 2012).

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distribution, and Fig. 6, under Student-t with estimated degrees of freedom. Cumulative Distribution Functions (CDF) in the left panel and the integrated CDFs (ICDF) in the right panel are for observation 3558 (21 August, 2012). (6) These figures represent DCC_VaR and DCC_ES produced using VaR99 and ES97.5 for GARCH. The outcomes of the SD tests for the Gaussian distribution, showing that DCC_ES FSD DCC_VaR, are illustrative in the light of previous results (see Danielsson, 2013), in which ES97.5 and VaR99 had similar statistical properties (namely mean and standard deviation), thereby making it difficult to uncover any inherent empirical differences. Expected Shortfall has the advantage that it reveals information about the magnitude of losses in the case of extreme events, besides being a coherent risk measure which encourages diversification to reduce overall risk for a portfolio. On the one hand, First-order Stochastic Dominance implies that using ES97.5 would result in a higher probability of larger estimates of the daily risk forecast. On the other hand, Second-order Stochastic Dominance has revealed that ES provides more accurate estimates of the magnitude of losses under extreme events than does VaR. This difference is due to the fact that Expected Shortfall is the average of the daily losses that may occur in the tails, whereas VaR is simply the α percentile of a returns distribution. Moreover, VaR does not factor in the magnitude of losses above such a percentile, whereas ES does. 6. Conclusions In this paper we introduced the stochastic dominance approach as a tool to make comparisons of the implication of banking regulation changes, and used it to compare VaR and ES as risk measurements. The SD approach is consistent with the spirit of Basel III concerning tail outcomes, and considers a more general comparison that is robust to particular loss functions (for example, mean and variance/quadratic), and underlying (unknown) exact distributions of the estimated daily capital charges. While VaR is still useful, Expected Shortfall, ES, may be a preferable risk measure (Dowd, 2005). Specifically, ES considers losses beyond the VaR level, and is sub-additive, while VaR disregards losses beyond the percentile, and is not sub-additive. Using S&P500 data, the paper evaluated the optimality of ES97.5 and VaR99 with respect to the stochastic dominance relations induced by the sampling distribution of the daily capital charges produced by using both risk measures. Stochastic dominance provides the pairwise comparison of the two risk measures, such that regulators whose utility functions belong to some set, will prefer one to another. Stochastic dominance ordering is theoretically superior to statistical moment rules (for example, mean-variance analysis). The SD approach uses as much information as possible from the DCC probability distribution. We showed some empirical results that should shed light on the Basel Committee dilemma of moving from VaR99 to ES97.5. First, the null hypothesis of the dominance of the Daily Capital Charges distribution, produced using the VaR99, by the distribution of the DCC, produced using ES97.5, cannot be rejected. Therefore, using Expected Shortfall, for producing DCC would be chosen by regulators who prefer larger DCCs to lower, and have risk aversion. SD is even more easily perceived for fat-tailed conditional distributions. Second, the ES97.5 not only accounts for the tail-risk, but also provides a more stable measurement of risk, being less sensitive to extreme observations, and hence should be the preferred option by regulators. However, there is not a unique winner in terms of risk management. Welfare comparisons between different regulations involve nontrivial trade-offs. Although most regulators and researchers would agree on the benefits of increasing capital requirements to reduce the probability of bank defaults, few would deny that larger capital requirements can be socially costly as they reduce a banks’ ability to create liquidity. In this paper, we focused on the perspective of a regulator who is interested in analyzing whether using ES as a risk measure reduces the social cost of bank failures, thereby making the capital requirements faced by banks larger and less sensitive to the cycle than using VaR. Although providing worthwhile information about the statistical properties of ES and VaR, this paper also suggests that regulators should weigh the advantages of moving from VaR to ES in terms of providing a more stable risk measure, but bearing in mind that this change can jeopardize bank profits. An important caveat of the paper is that stochastic dominance between ES and VaR can only be obtained for the class of utility function defined that might fit the regulator's preferences. We do not determine the best risk measure with respect to other classes of utility functions. Stochastic dominance has not been used to identify the risk measure that is preferred under a bank risk manager's utility function. The risk manager's choice of risk measure using the SD approach would require the specification of a particular utility function. In certain situations, there may be no dominant risk measure. Finally, while these empirical results are obtained using a single asset and a limited range of models and distributions, they are likely to hold in a more general setting. The next step in this research agenda is to use a variety of assets, both from the USA and other countries, and alternative univariate and multivariate risk models to ascertain the validity of the empirical results. 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