How informative are variance risk premium and implied volatility for Value-at-Risk prediction? International evidence

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How informative are variance risk premium and implied volatility for Value-at-Risk prediction? International evidence Skander Slim a,b,∗ , Meriam Dahmene a , Adel Boughrara a a b

University of Sousse, IHEC, LaREMFiQ, BP: 40, 4054 Sousse, Tunisia Mena College of Management, Al Safa 1, Street 8B, Building 43, Jumeirah (P.O. Box 430470), Dubai, UAE

a r t i c l e

i n f o

Article history: Received 28 November 2017 Received in revised form 11 July 2019 Accepted 20 August 2019 Available online xxx Keywords: Variance risk premium Implied volatility Value-at-Risk Risk management GARCH

a b s t r a c t The aim of this paper is to examine the information embedded in the implied volatility index and the variance risk premium in terms of quantifying market risk for developed and emerging stock markets. The backtesting results indicate that incorporating the relative variance risk premium into the GARCH model, greatly enhances the forecasts of one-day-ahead Value-at-Risk (VaR) for a long trading position in developed markets, while the standard GARCH is the most relevant specification in capturing risk in emerging markets. Results are found to be robust against distressed financial markets and alternative measures of the variance risk premium. Moreover, the empirical evidence shows that the superior performance of these models cannot completely reduce the scope of implied volatility as a risk management tool. Including implied volatility into the GARCH model incurs substantial savings in terms of efficient regulatory capital provisions. © 2019 Board of Trustees of the University of Illinois. Published by Elsevier Inc. All rights reserved.

1. Introduction Erecting as one of the most influential variable among academicians, market practitioners, policymakers and media, market volatility is at the centre of numerous financial applications such as portfolio selection/optimization, risk management, derivatives pricing and hedging strategies. It all depends on how the volatility is characterized and measured. The rapid growth of financial markets over the recent decades and the recurrence of financial crises along with the development of new and more sophisticated financial instruments have reinforced the need for accurate and efficient volatility forecasting tools. Furthermore, the latent character of the actual stock market volatility makes the forecasting exercise more challenging. Unlike the underlying security market, the option market is often considered as a market for trading volatility which entails that the implied volatility backed out from option prices could be a good predictor of subsequent volatility “quotes” provided that the option market is efficient and the option pricing model is well specified. The VIX, publicly disseminated by the Chicago Board Option Exchange (CBOE) and commonly referred to as the “world’s barometer of market volatility”, was the first successful attempt at

∗ Corresponding author. E-mail address: slim [email protected] (S. Slim).

creating and implementing a volatility index. Its success has paved the way for other marketplaces to build their own implied volatility indexes. The VIX relevance arises not only from its forward-looking nature, since it embeds market sentiment and/or investors’ expectations about future states, but also because it forms the backbone of a host of volatility derivatives, particularly variance swaps. The expected premium from selling a stock market variance in a swap contract reflects the variance risk premium as the difference between the expected realized variance and its risk-neutral counterpart (i.e., variance swap rate). In this regard and using a stylized self-contained general equilibrium model, Bollerslev, Tauchen, and Zhou (2009) suggest that the equity risk premium is explained by at least two factors related to the volatility dynamics of consumption growth. The first one represents time-varying economic uncertainty in consumption growth while the second is associated to the volatility-of-volatility (vol-of-vol) of consumption growth. More recently, Conrad and Loch (2015) provide compelling empirical evidence that fundamental uncertainty (vol-of-vol) is the main driving factor for the variance risk premium. Hence, the variance risk premium may be regarded as a second source of (fundamental) uncertainty when investing in stock markets. A body of financial literature has focused on examining the ability of implied volatility to forecast market volatility and to a lesser extent to estimate the VaR (see the literature review in Section 2), whereas the related studies on the variance risk premium have remained very limited and mostly fall within the vast literature

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on the predictability of asset returns. In any event, the empirical evidence strongly suggests that the variance risk premium predicts medium-term aggregate stock market returns (see Bekaert & Hoerova, 2014; Bollerslev et al., 2009; Bollerslev, Marrone, Xu, & Zhou, 2014, among others). Furthermore, despite the financial literature-based consensus on the information content of optionimplied measures, the answer to the question whether they supply additional information beyond historical volatility-based measures is far from being without controversy. In fact, little has been known so far about the practical usefulness of the variance risk premium as a risk management tool. The current paper attempts to fill this gap by investigating the information content of both implied volatility and variance risk premium in terms of forecasting the VaR for international stock market indexes. More specifically, this paper aims to analyse and appraise the practical usefulness of volatility estimates from various volatility forecasting methodologies including RiskMetrics, implied volatility and GARCH models with and without implied volatility under the VaR framework. This paper contributes to the literature in a number of ways. The first contribution is to further account for the variance risk premium as a potential predictor, and to fairly compare its performance to both implied volatility and time series models. Nevertheless, computing the variance premium is not uncontested for at least two reasons. From the outset, the empirical evidence (Carr & Wu, 2009) shows that the variance premium is increasing in variance -the so-called level dependency- in a stylized manner (Prokopczuk & Siemen, 2014). Then, whilst “model-free implied volatility” can be constructed from option prices, the expected realized variance has to be empirically estimated. To this end, the most common approach is to assume that the realized variance follows a martingale difference, and next to estimate the ex post realized variance from high-frequency intraday returns. In this paper, we additionally employ the relative variance risk premium to circumvent the level dependency issue and we use the heterogeneous autoregressive model for the realized variance (Corsi, 2009) to estimate the expected variance premium as a robustness check while making full use of overlapping daily data rather than the standard sparse end-of-month data. The second contribution consists in expanding upon the financial literature regarding the market risk modelling by performing a direct comparison of the forecasting performance of implied volatility and variance risk premium measures across two groups of countries with different development levels of derivatives markets. Of course, the VaR forecasting is not limited to the volatility specification, and it is commonly recognized that distributional assumptions play a key role in obtaining accurate models. The empirical evidence has demonstrated that the introduction of flexible distributions, allowing for skewness and heavy tails, greatly enhances the predictive performance of GARCH-type models (Bao, Lee, & Saltoglu, 2007; Slim, Koubaa, & BenSaïda, 2017). Therefore, we select the skewed-t distribution in an attempt to jointly account for both long and short positions risk in equity markets. Besides, since the focus is on models of higher accuracy and seeing that the statistical analysis made during periods of stability does not provide much guidance in times of market meltdowns, the forecasting performance of these heavy-tailed models is assessed over long sample periods covering the recent global financial crisis. We empirically evaluate the accuracy of the VaR estimates for twelve international stock market indexes by means of a new set of improved backtests recently introduced by Ziggel, Berens, Weiß, and Wied (2014) and which was proven to outperform traditional and state-of-the-art backtests. Furthermore, we evaluate the economic importance of our results by computing daily capital requirements under the Basel III Accord (BCBS, 2009).

Finally, the third contribution consists in conducting an empirical analysis while carefully selecting seven developed and five emerging markets to provide insights into each financial market’s unique trait. The results show that incorporating the variance risk premium into the GARCH model improves the empirical performance of long position VaR for most of developed markets. However, the non-augmented GARCH is the most relevant specification in capturing risk in emerging markets followed by the RiskMetrics. Unlike previous studies, implied volatility with and without GARCH effect is outperformed by the competing models. More importantly, the implied volatility-augmented GARCH incurs substantial capital savings for both long and short positions, thereby indicating that the volatility index is particularly an efficient input for regulatory capital allocation. The article unfolds as follows. Section 2 provides a brief overview of extant studies on the usefulness of both implied volatility and variance risk premium for volatility forecasting and market risk modelling. Section 3 presents the research design. Section 4 describes the dataset and discusses the empirical results. Finally, Section 5 concludes the paper and suggests some policy implications along with future research lines.

2. Literature review In order to identify how our study is related to volatility forecasting literature and its risk management applications, we start by reviewing this already growing strand of research. Although numerous studies have used several models to forecast future volatility, no consensus has been reached regarding the best predictor of future volatility. At present, two different streams of empirical financial research can be identified. The first stream highlights the superiority of historical models over implied volatility in predicting future volatility. Many previous studies report that implied volatility may not contain information on realized volatility, or it may be neither efficient nor unbiased predictor of market volatility. For example, Kumar and Shastri (1990) argue that implied volatility is a biased and inefficient estimator whilst historical volatility models significantly improve volatility forecasts. These findings are corroborated by Canina and Figlewski (1993) who report that historical volatility forecasts are superior to option-implied forecasts. In the same vein, Martens and Zein (2004) compare the forecasting performance of implied volatility and time series models based on historical intraday returns. Their results indicate that time series models have incremental information over that contained in implied volatility. In addition, Bentes (2015) assesses the information content, the bias and the efficiency of the implied volatility and the GARCH forecasts using stock market data on three Asian countries along with the US. The author provides strong evidence in favour of GARCH models in terms of forecasting accuracy. Agnolucci (2009) comes to the same conclusion for the crude oil market. Taken literally, such findings cast doubt on the widespread belief in informational efficiency of options markets. Nevertheless, a second body of research points to the relevance of implied volatility in forecasting future volatility. Lamoureux and Lastrapes (1993) find that implied volatility extracted from options on individual stocks contains predictive information about future volatility beyond that contained in GARCH. Using data on the S&P100 index, Christersen and Prabhala (1998) report that implied volatility is both unbiased and efficient, subsuming all information contained in the historical data after the 1987 stock market crash. Similar results are reached by Chiras and Manaster (1978), Day and Lewis (1992), Jorion (1995), Fleming (1998), Blair, Poon, and Taylor (2002) and Giot (2003) who conclude that the implied volatility contains relevant information regarding future volatility, although the implied

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volatility seems to be a biased volatility forecast. Charoenwong, Jenwittayaroje, and Low (2009) assess the predictive power of implied volatility extracted from currency options traded on different venues, namely the Philadelphia Stock Exchange, the Chicago Mercantile Exchange, and the over-the-counter (OTC) market. The authors conclude that implied volatility conveys more information about future volatility than time series-based volatility, regardless of the trading venue. Likewise, Covrig and Low (2003) report that quoted implied volatility in OTC currency market subsumes the information content of historically-based forecasts at short horizons, whereas both forecasts perform equally well at long horizons. In a comprehensive study covering a wide variety of asset classes and exchanges, Szakmary, Ors, Kim, and Davidson (2003) prove the superiority of the option-implied forecasts compared with the GARCH and moving average models. More recently, Benavides and Capistràn (2012), Pati, Barai, and Rajib (2018) and Chun, Cho, and Ryu (2019) report that implied volatility is highly informative for forecasting future volatility dynamics when combined with time series models. Collectively, these studies provide evidence that implied volatility is a biased, albeit a more efficient forecast for future volatility than historical volatility. The implied volatility bias raises questions as to whether the assumptions imposed by traditional option pricing models are appropriate, particularly those regarding a zero price of volatility risk and frictionless markets (see for instance Doran, 2005; Lamoureux & Lastrapes, 1993; Neely, 2009). A study pioneered by Jiang and Tian (2005) has provided guidance for subsequent research to improve upon the forecast quality of implied volatility, and more generally, to improve the performance of option valuation models. The authors introduce the model-free implied volatility (MFIV) estimator that does not rely on any specific option pricing model or a particular strike price and they assess its information content for the S&P500 index. They find that the model-free measure subsumes all relevant information contained in both historical volatility and Black-Scholes (BS) implied volatility. Efforts in this direction include Taylor, Yadav, and Zhang (2010) and Cheng and Fung (2012) who point to the superiority of MFIV over popular time series models for long prediction horizons. The aforementioned studies have focused on the forecasting accuracy of implied volatility but have overlooked the fact that the volatility risk is priced in option markets. One of the main reasons for the implied volatility bias is the existence of an economically significant variance risk premium (see Bates, 2003; Bollerslev & Zhou, 2006; Chernov, 2007) giving rise to a wedge between physical and risk-neutral variances. More recently, Rompolis and Tzavalis (2010) put forth similar findings and provide strong evidence that the risk premium-generated bias can be sufficiently explained by the negative skewness of the risk-neutral density. This reflects the compensation for investors for bearing the risk of extreme negative shifts in stock markets due to crashes or other extreme events. In an attempt to reconcile the relationship between expected and realized volatility, DeMiguel, Plyakha, Uppal, and Vilkov (2013) and Prokopczuk and Siemen (2014) adjust the MFIV for the volatility risk premium and employ this adjustment in the context of portfolio allocation and volatility forecasting, respectively. They show that adjusting for the volatility risk premium leads to superior volatility forecasts compared to time series models, BS implied volatility and non-adjusted MFIV. Similarly, Kourtis, Markellos, and Symeonidis (2016) evaluate the importance of this adjustment by comparing the predictive ability of implied, realized, and GARCH volatility models for international equity indexes and report that an implied volatility model that corrects the volatility risk premium affords the most accurate predictions at a monthly horizon. While these findings suggest that accounting for the variance risk premium mitigates the implied volatility bias and, hence, improves the

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forecasting performance of volatility models, Bams, Blanchard, and Lehnert (2017) show that such an adjustment remains insufficient for quantile-based risk measures. Another promising line of research, straightforwardly related to the current paper, focuses on the usefulness of implied volatility as a risk management tool. In this respect, Chong (2004) compares daily VaR estimates derived from univariate and multivariate time series models with those implied by currency option prices. The author reports that time series-induced VaR models perform better than those based on implied volatility and, consequently, concludes that market expectation as determined from option market may not be a useful tool to assess the market risk. Similarly, Christoffersen and Mazzotta (2005) assess the quality of the information embedded in OTC currency options along three dimensions of exchange rate forecasting, namely volatility, density and interval forecasting. They find that the implied volatility outperforms historical and GARCH volatility forecasts for all considered currency exchange rates and forecast horizons. Nonetheless, their findings show that, somewhat disappointingly, implied volatility is not particularly a useful input in density and interval forecasts since it fails to capture the tail behaviour of the distribution. Giot (2005) predicts one-dayahead VaR using the VIX with a skewed-t distributional assumption and concludes that the VaR model relying exclusively on lagged implied volatility performs as well as VaR estimates based on the volatility forecasts of GARCH-type models. Besides, Jeon and Taylor (2013) enrich the CAViaR model of Engle and Manganelli (2004) with implied volatility, not only directly using a forecast combination strategy, but also by using implied volatility as an additional regressor. They find that implied volatility has more explanatory power than that of standard VaR models as one moves further into the left tail of the conditional distribution of the S&P500 daily returns. Kambouroudis, McMillan, and Tsakou (2016) compare the forecasting ability of a wide range of GARCH, implied volatility, and realized volatility models for major European and US stock markets. Their findings indicate that as far as implied volatility forecasts are considered individually, they generally perform poorly. Again, a linear combination of implied volatility, realized volatility and asymmetric GARCH is preferred for forecasting both volatility and VaR. Kim and Ryu (2015) show that incorporating implied volatility into volatility specifications greatly enhances the performance of VaR models in the Korean stock exchange. They also consider the distinct characteristics of implied volatilities across option moneyness and argue that their information content can contribute differently to forecasting a certain quantile of the return distribution. In sum, their results point to the superiority of the BS at-the-money implied volatility during and after the 2007–2008 financial crisis whereas VaR models incorporating the BS out-ofthe-money implied volatility outperform the competing models before the crisis. For the S&P500 and five other Asia-Pacific market indexes, Siu (2018) finds that their respective volatility indexes contain fairly useful information about the maximum cumulative loss in the context of VaR and conditional VaR. Moreover, Leiss and Nax (2018) underscore the importance of using option-implied information to predict large market downturns. Probably closest to our study is Bams et al. (2017) who propose implied volatility adjustments in order to account for the variance risk premium in the same spirit as Prokopczuk and Siemen (2014), and implement the procedure to forecast the VaR for three US equity indexes. Their findings show that both adjusted and non-adjusted implied volatilities do not outperform the GJR-GARCH. More precisely, the implied volatility-based VaR model excessively overstates market risk incurring an excess capital charge for financial institutions. Hence, the authors conclude that the well-documented superiority of implied volatility over historical volatility models, according to the volatility forecasting literature, cannot be translated easily into the risk management area of the VaR. Unlike Bams et al. (2017),

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who viewed the variance risk premium as a simple adjustment of implied volatility, our study goes further by considering the forecasting ability of the variance risk premium itself. The current paper shows that there is room for improvement on how to make use of the variance risk premium to handle both long and short position risks, and find that implied volatility indexes offer scope for financial institutions to devise economically efficient VaR models. 3. Methodology 3.1. Variance risk premium measures The ultimate aim of this study is to investigate the role of variance risk premium for the VaR forecasts. We assume a nonparametric form of variance risk premium inspired by previous studies. Following Bollerslev et al. (2009) and Bekaert and Hoerova (2014), among others, the market price of variance risk is simply defined as the difference between the risk-neutral and physical expectations of variance:

 Q

VRP t,T = Et where

2 Vt,T



 P

2 2 Vt,T − Et Vt,T



(1)

and VRPt,T refer to the return variation and the variance





2 risk premium between t and T, respectively. EQ Vt,T is the ex ante t forecast of the variance under the risk-neutralprobability, which  2 is measured by the implied volatility index. EPt Vt,T is the ex ante forecast of the variance under the physical measure, proxied by the ex post realized variance RVt . For a given trading day t, the realized nt variance is measured in a “model-free” fashion by RV t = r2 , j=1 j,t 2 is the jth intraday log-return and n is the number of where rj,t t intraday returns observed on that day. Then, the volatility index observed at the end of day t is scaled such  that it forecasts the 1/365 × volatilityindex. next one-day implied volatility: IV t = The daily expected variance risk premium is thus given by:

VRP t ≡ IV 2t − RV t

(2)

It is worth noting that Eqs. (1) and (2) reflect the negative of the return to buying variance in a variance swap contract. Hence, VRP refers to the negative of the variance risk premium. By doing so, we obtain a variance risk premium measure which values are mostly positive and hence can be easily used as a key input in the VaR framework. Economically, the negative sign of the volatility risk premium suggests that investors are willing to pay a premium to hedge away upward movements in future volatility. It stems from the fact that investors regard increases in the market volatility as unfavourable shocks to the investment opportunity. Therefore, it is hardly surprising that the VRP is increasing in variance, in absolute terms. This level dependency has been empirically documented by Carr and Wu (2009) and Prokopczuk and Siemen (2012), and has important implications for our VaR analysis that echo the Prokopczuk and Siemen (2014)’s arguments regarding a two-period setting with different volatility states. The first period is characterized by a high volatility level, while the second is characterized by a low volatility. This implies that the VRP has a larger magnitude in the first period than in the second one. If we use the higher market price of volatility risk of the first period to forecast the realized variance in the second period, this may lead to biased forecasts. Building on these insights, we use the ratio of the expected implied variance to the expected realized variance as a second measure of the variance premium. That is: RVRP t ≡

IV 2t RV t

(3)

where RVRPt is the relative variance risk premium on day t. We also discuss in Section 4.3.2 complementary empirical results in which we employ an estimate for the expected premium instead of its

observed counterparts in Eqs. (2) and (3). Specifically, we propose to base both the expected VRPt and RVRPt on the one-day-ahead forecasts for the RVt from a simple reduced form time series model for the realized variance. 3.2. VaR and volatility specifications The VaR has emerged as a statistical risk metric that expresses the maximum expected loss of a trading position at a given confidence level over a target forecast horizon. Thus, assuming that the VaR model is correct, the realized loss will exceed the VaR threshold with only a small target probability ˛, typically chosen between 1% and 5%. More specifically, conditional on the information set available until time t, the VaR (for long position) on time t + h of one unit of investment is the ˛-quantile of the conditional return distribution, that is: VaRt+h = q˛ (rt+h |Ft ) = inf





x ∈ R|P (rt+h ≤ x|Ft ) ≥ ˛ ,

0 < ˛ < 1,

(4)

where q˛ denotes the quantile function, rt+h is the index return in period t + h, and Ft designates the information available at date t. For simplicity and without loss of generality, let us assume the following scaled martingale difference:

t = t zt ,

(5)

where t denotes the demeaned index return on day t,  t is a scale parameter and zt is an independent and identically distributed (i.i.d.) random variable that follows the standardized skewedt distribution of Hansen (1994) and implemented into the VaR framework in Giot (2005).1 In our empirical analysis, we calculate the VaR for portfolio managers who have taken either a long position or a short position. In the former case, the risk arises from a drop in the price of the asset, while in the latter case the trader will incur a loss when the asset price increases. To fully characterize the risk exposure arising from both long and short trading positions, one needs an adequate modelling of the left and the right tails of the expected return distribution as well as the conditional scale parameter. The one-day-ahead VaR forecast is then reduced to calculating q˛  t+1 and q1−˛  t+1 for the long and short trading positions, respectively. q˛ and q1−˛ denote the left and right quantile, implied by the skewed-t distribution, at the level ˛ and 1 − ˛, respectively.  t+1 is the forecasted volatility of the market index return. To forecast the one-day-ahead volatility, numerous econometric models have been suggested to capture its time-varying nature as well as some prominent stylized facts of stock returns. Postulating the dependence of current volatility on past volatilities and returns, conditional volatility models appear as attractive candidates for seizing the unconditional non-normality of the data and mimicking the volatility clustering effect. Pursuing strictly practical motivations, this research will unfold within the boundaries of three simple representations. First, we use the RiskMetrics (RM) approach that was originally developed by JP Morgan. The RM approach is known as an exponential smoother in that it assigns more weight to the recent observations and less weight to the old ones. Second, we employ the GARCH model introduced by Bollerslev (1986), and since then arguably standing as one of

1 The conditional mean is set to zero in Eq. (5). This assumption arises from the fact that the location parameter is of little relevance in estimating the VaR on a daily basis, and it is largely dominated by the conditional scale parameter (see for instance Figlewski, 1997; Kim, Malz, & Mina, 1999). Alternatively, one can apply an AR filter to account for the serial autocorrelation, even though the autocorrelation might be a statistical artefact of stale quotes (see the discussion in Kuester, Mittnik, & Paolella, 2006).

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the most popular among financial community. It is worth noting that the current RM model, also considered as a fixed-parameter integrated GARCH, is an enhanced version of the typical normal specification as it makes use of the heavy-tailed skewed-t distribution. Finally, as we focus on the contribution of both implied volatility and variance premium in terms of quantifying market risk, they are separately included as additional input variables into the GARCH to exploit the information from both stock and options markets. In addition, following Giot (2005) and Kim and Ryu (2015), we specify a simple implied volatility model in which we use solely the volatility index series to forecast future volatility. Specifically, the volatility forecasting models are described as follows: • RiskMetrics (RM) 2 t+1 = 0.042t + 0.96t2

(6)

• Implied volatility (IVM) 2 t+1 = ω + ˇIV 2t

(7)

• GARCH 2 t+1 = ω + ˛2t + ˇt2

(8)

• GARCH-X (9)

where GARCH-X in Eq. (9) denotes augmented GARCH and it will be referred to as either G-IV, G-VRP or G-RVRP, depending on the selected exogenous regressor Xt which could be either IVt , VRPt or RVRPt . 3.3. VaR backtesting The backtesting methodology is a formal statistical framework that consists in verifying whether or not actual trading losses are in line with model-generated VaR forecasts or not and relies on testing over VaR violations (also called the hit). A violation is said to occur when the realized trading loss exceeds the VaR forecast. A variety of backtesting methods have been proposed to gauge the accuracy of VaR estimates (see Campbell, 2007; Ziggel et al., 2014, for a review) with tests focusing on the frequency of violations (unconditional coverage), the independence of violations sequence and both properties of independence and correct number of violations (conditional coverage). Ziggel et al. (2014) have recently introduced a set of tests that significantly improve upon the existing backtesting procedures mainly because of their finitesample size and power properties. In this section we briefly present this new set of improved tests used in our empirical assessment of VaR models. 3.3.1. Unconditional coverage test The unconditional coverage (UC) test is an industry standard mostly due to the fact that it is implicitly incorporated in the “traffic light” system proposed by BCBS (2006, 2009) which remains the reference backtest methodology for the banking regulators. The test consists of checking whether or not the realized coverage rate (˛) ˆ equals the theoretical coverage rate (˛) of the VaR for a backtesting sample of T non-overlapping observations. Under the assumption of a possibly non-stationary VaR violations sequence, Ziggel et al. (2014) define the UC property as E



1 It (˛) T T

t=1

where It (˛) is the hit variable on day t, which takes 1 if the loss exceeds the reported VaR measure and 0 otherwise. Under the null hypothesis of correct UC, the test statistic is given by: MCS uc =

T

It (˛) + 

(11)

t=1

where  is a normally distributed random variable. 3.3.2. Test of i.i.d. VaR violations In the financial econometrics literature, an enhancement of the unconditional backtesting framework is achieved by additionally testing for the independence property of violations yielding a combined test of conditional coverage (CC). Examples include the test of Christoffersen (1998) against an explicit first-order Markov alternative, the regression-based test for higher-order dependence in the violation series (Engle & Manganelli, 2004) and the durationbased tests of Christoffersen and Pelletier (2004) and Candelon, Colletaz, Hurlin, and Tokpavi (2011). While these standard backtests embed the null hypothesis of i.i.d. violations, they focus solely on testing the independence of the VaR violations. Extending the current state-of-the-art, Ziggel et al. (2014) suggest a new test for both the property of independent as well as the property of identically distributed VaR violations. Specifically, the i.i.d. property is i.i.d.

2 t+1 = ω + ˛2t + ˇt2 + Xt



5

=˛

simply formulated as It (˛) ∼ Bern(˛), ∀t, and the test statistic is given by: MCS iid,m = t12 + (T − tm )2 +





m

(ti − ti−1 )2 + 

(12)

i=2

where V = i|Ii = 1 = {t1 , . . ., tm } denotes the set of points in time associated to the sequence of VaR violations. Based on squared durations between consecutive violations, the test explicitly allows to detect clusters in the violation process. This feature is economically meaningful as the test would reject the models that yield inaccurate forecasts of large losses that occur in rapid succession during financial turmoil. 3.3.3. Conditional coverage test The CC hypothesis can be tested by the following weighted test statistic:





MCS cc,m = a · f (MCS uc ) + (1 − a) · g MCS iid,m ,

0 ≤ a ≤ 1,

(13)

   where a is the weight of the UC test, f (MCS uc ) =  MCS uc /T˛ − 1,      and rˆ is an estig MCS iid,m = MCS iid,m /ˆr − 1 · 1 MCS iid,m ≥ˆr

mator of the expected value of the test statistic MCSiid,m under the null hypothesis of i.i.d. VaR violations. In the empirical application, we set a = 0.5. For all the three tests (i.e., UC, IID and CC), we use Monte Carlo simulations to approximate the distribution of the test statistics under the corresponding null hypothesis and obtain the associated p-values of the tests.2 4. Empirical results 4.1. Data and preliminary analysis For the empirical analysis, data from seven developed markets (France, Germany, Japan, Netherlands, Switzerland, UK and US) and five emerging markets (Hong Kong, India, Mexico, South Africa and South Korea) are employed, covering the geographical regions of Africa, Asia, America and Europe. For each market, the dataset is

(10) 2

For further details on these tests see Ziggel et al. (2014).

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6 Table 1 Sample indexes. Country

Market index

Volatility index

Starting date

Developed markets (Sample period: January 3, 2000 – August 28, 2015) CAC40 VCAC France Germany DAX30 VDAX Japan NIKKEI225 JNIV AEX VAEX Netherlands Switzerland SMI VSMI FTSE100 VFTSE UK S&P500 VIX US Emerging markets Hong Kong India Mexico South Africa South Korea

HSI S&P CNX Nifty IPC TOP40 KOSPI200

VHSI INDIA VIX VIMEX SAVI VKOSPI

January 2, 2001 March 2, 2009 March 26, 2004 February 1, 2007 April 10, 2009

This table gives information on the list of countries, stock market indexes, implied volatility indexes and the sample period for each market.

daily and it consists of implied volatility index and closing price of the underlying stock market index, obtained from Thomson Reuters Eikon. The continuously compounded daily returns are calculated as the logarithmic difference of daily closing prices. The sample period starts on January 3, 2000 for developed countries, while it varies for emerging markets, as reflected in Table 1, depending on the availability of implied volatility data. All datasets end on August 28, 2015. Since recent studies have demonstrated the usefulness of the MFIV to provide accurate volatility forecasts, the CBOE employs this approach to compute the volatility index of the S&P500 (the so-called VIX) based on the highly liquid index options for a wide range of strike prices. Most of the volatility indexes used in our empirical analysis are computed utilizing the VIX methodology explicitly tailored to replicate the risk-neutral variance of a fixed 30-day maturity, except for the VDAX, the VIMEX and the SAVI (see Siriopoulos & Fassas, 2013, for a detailed exposition on the construction methodology of all publicly available volatility indexes). The VDAX approximates the implied volatility using the Black (1976) model for at-the-money options with a constant maturity of 45 days. The VIMEX and the SAVI are calculated utilizing the Black and Scholes (1973) model to represent a three-month index of market volatility, instead of the typical one-month period. In order to estimate the variance risk premium, we additionally employ daily realized variance constructed using five-minute intraday returns readily available at the Oxford-Man Institute (realized.oxford-man.ox.ac.uk).3 Table 2 provides summary statistics for daily index returns, implied volatility and the estimated variance risk premium over the full sample period. The return series display similar statistical properties with respect to skewness and kurtosis. Most of them exhibit negative skewness. The deviations of the median from the mean and the values of the excess kurtosis justify the use of the skewed and leptokurtic distribution-based VaR. As expected, the VRP is positive on average for most of the investigated markets. Fig. 1 depicts the VRP for the S&P500 and HSI indexes over the full sample period. Looking at Fig. 1, two things stand out immediately. First, VRP is almost always positive and displays pronounced spikes in uncertain episodes. For both market indexes, the largest spikes broadly coincide and they are observed in the third quarter of 2002 following the Enron debacle, during the Lehman aftermath in 2008 and also following the euro area debt crisis at the end of 2011.

3 High-frequency based realized variance estimator is not available for the KOSPI200 and TOP40 indexes. For these markets, realized variance is measured as the sum of squared daily returns for a particular month.

Second, during these periods of massive uncertainty, the realized variance exhibits extreme peaks, and consequently, a few pockets where the VRP goes negative are observed. To the extent that positive VRP represents investors being risk-averse, it is unlikely that there was a sudden increase in risk appetite. One possible explanation for the sign reversal is that the option market might have underpriced the actual volatility level during financial distress. Another view owing to Fan, Imerman, and Dai (2016) relates the VRP, as a systematically priced bias, to supply and demand forces in option markets. On the one hand, the magnitude of the VRP can, therefore, be interpreted as the compensation that market makers receive for providing liquidity to investors who seek to hedge against downside tail risk. As such, the deviation between implied and realized volatility is expected to widen when the hedging demand increases. Similarly, when dealers switch from being liquidity suppliers to being liquidity demanders, the deviation widens as well. Thus, it is possible that the significant sign reversals, in Fig. 1, represent liquidity supply shocks to the market for hedging market-wide crashes. On the other hand, Fan et al. (2016) advocate an alternative hypothesis to the practitioners’ belief that the direction of VRP reflects expectations about future levels of volatility. They provide evidence that the sign of VRP is indicative of gains or losses on market makers’ delta-hedged positions. More specifically, a negative (positive) VRP indicates a delta-hedged gain (loss) for the market maker. 4.2. VaR performance Based on the volatility forecasts estimated from the six volatility forecasting models, the predicted one-day-ahead VaR for both long and short positions is compared to the actual index return. All the models are updated on a weekly basis, and the forecasting performance is assessed based on the out-of-sample framework. For developed markets, the out-of-sample period spans from January 30, 2004 to August 28, 2015. Due to the dual requirements of reliable implied volatility data and a sufficient rolling window length to calibrate models parameters, the following starting dates are selected for the forecasting exercise in emerging markets: April 18, 2006 for Mexico; February 24, 2009 for Hong Kong; March 5, 2009 for South Africa; April 15, 2011 for India and May 11, 2011 for South Korea. First, we compute the empirical violation rate and the significance based on the UC test. Second, we use the IID and the CC tests to evaluate the accuracy of the VaR estimates. Tables 3 and 4 report the backtesting results over the entire outof-sample period for the 1% and 5%-VaR, respectively, depending on the investor’s position. Regarding the empirical violation rate and the UC test, all the models broadly exhibit satisfactory performance. Yet, the RM model exhibits the worst performance for long trading positions. As shown in Table 3, except for few cases of emerging markets including India, Mexico and South Korea, all the empirical violation rates for the RM model are significantly higher than the theoretical value. Such findings suggest that the RM model underestimates the true VaR for most of the developed markets. Regarding the CC test, the highest p-values are attributed to the standard GARCH and its augmented specification with the relative VRP in three and five out of twelve cases, respectively. Moreover, for these two models, the hypothesis of correct conditional coverage cannot be rejected at standard significance levels for most of the investigated markets. Hence, for the long position VaR, the best performing model is the G-RVRP followed by the GARCH, while both the RM and implied volatility models (i.e., IVM and G-IV) exhibit the worst performance. The results emerging from the estimated right quantile highlight the predictability of the VaR for short trading position, where most of the models perform accurately well. Interestingly, the RM exhibits the best performance and the implied volatility models improve consid-

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Table 2 Descriptive statistics. Mean

Std

Kurtosis

Skewness

Median

Max

Min

R IV VRP RVRP

−0.005 0.017 0.244 1.410

1.511 0.016 0.878 0.488

7.780 17.698 44.186 4.548

−0.009 3.175 3.136 0.849

0.034 0.013 0.264 1.363

10.595 0.167 6.425 4.823

−9.472 0.002 −11.615 0.284

Germany

R IV VRP RVRP

0.010 0.016 −0.276 1.093

1.560 0.015 1.239 0.382

7.667 12.633 30.621 4.041

−0.099 2.740 −4.368 0.674

0.079 0.011 0.0525 1.062

10.797 0.150 5.116 3.027

−10.163 0.003 −14.339 0.246

Japan

R IV VRP RVRP

−0.001 0.021 0.949 2.183

1.555 0.019 1.102 0.994

9.231 35.229 29.862 21.923

−0.440 4.912 4.093 2.555

0.032 0.016 0.716 1.985

13.235 0.229 13.350 16.281

−12.111 0.004 −2.068 0.398

Netherlands

R IV VRP RVRP

−0.008 0.018 0.542 1.674

1.486 0.020 0.883 0.582

9.396 14.346 18.089 4.562

−0.096 3.028 2.037 0.847

0.044 0.011 0.367 1.617

10.028 0.181 8.418 5.370

−9.590 0.001 −6.545 0.211

Switzerland

R IV VRP RVRP

0.005 0.012 0.298 1.585

1.223 0.014 0.642 0.499

10.547 31.374 56.121 6.278

−0.251 4.275 2.992 0.766

0.054 0.008 0.242 1.556

10.788 0.197 12.339 6.534

−9.385 0.002 −4.398 0.265

UK

R IV VRP RVRP

0.000 0.013 0.437 1.852

1.236 0.014 0.691 0.692

9.494 24.967 41.577 4.931

−0.203 3.814 3.593 0.964

0.033 0.009 0.317 1.763

9.384 0.156 10.364 6.509

−9.266 0.002 −4.710 0.431

US

R IV VRP RVRP

0.009 0.014 0.148 1.452

1.275 0.015 0.901 0.596

11.124 30.913 80.521 13.184

−0.182 4.394 −6.327 1.885

0.056 0.009 0.195 1.358

10.957 0.179 5.713 9.008

−9.470 0.003 −14.770 0.297

R IV VRP RVRP

0.008 0.018 0.487 1.764

1.510 0.021 1.041 0.650

11.830 31.312 28.234 4.834

−0.028 4.328 0.141 0.949

0.038 0.012 0.410 1.664

13.407 0.298 14.660 5.099

−13.582 0.003 −7.805 0.389

India

R IV VRP RVRP

0.056 0.014 0.461 1.615

1.238 0.011 0.556 0.830

22.301 12.591 11.823 37.245

1.297 2.747 2.386 4.920

0.046 0.011 0.333 1.484

16.226 0.086 4.837 10.159

−6.097 0.004 −0.585 0.443

Mexico

R IV VRP RVRP

0.051 0.015 0.006 1.971

1.288 0.016 0.010 0.998

9.236 19.083 22.390 5.562

0.082 3.686 3.238 1.430

0.087 0.010 0.004 1.739

10.441 0.127 0.085 6.831

−7.266 0.003 −0.044 0.298

South Africa

R IV VRP RVRP

0.032 0.016 −0.396 1.200

1.428 0.012 1.653 0.613

6.455 12.779 17.608 5.810

−0.080 2.777 −3.529 1.323

0.083 0.013 0.090 1.091

7.707 0.092 2.429 4.650

−7.959 0.003 −10.436 0.329

South Korea

R IV VRP RVRP

0.016 0.010 −0.002 1.002

1.111 0.008 0.008 0.361

6.256 13.041 20.294 4.868

−0.315 2.751 −3.517 0.998

0.019 0.008 0.000 0.952

5.057 0.069 0.031 2.988

−6.649 0.003 −0.059 0.267

Developed markets France

Emerging markets Hong Kong

This table reports the summary statistics for daily returns (R), implied variance (IV), variance risk premium (VRP) and relative VRP (RVRP). All variables are reported in percentage form over the full sample period for each market index as described in Table 1.

erably. More importantly, comparing the GARCH to the G-RVRP sheds light on the information content of the volatility risk premium. The results indicate that accounting for the market price of volatility risk significantly improves the forecasting performance of the GARCH in most cases (i.e., seven and eight out of twelve markets for long and short positions, respectively). In the relatively less extreme case of 5%-VaR, we can see, in Table 4, that both the RM and the implied volatility models are outperformed by the GARCH for either long or short positions. Although the RM does a fairly good job in modelling the VaR for most of the markets, its performance deteriorates significantly compared to that incurred for the 1%-VaR. Again, the inclusion of

the variance risk premium improves the forecasting accuracy of the GARCH. The superiority of the variance risk premium in combination with the GARCH seems to be more pronounced on developed markets, whereas the non-augmented GARCH is the most relevant specification in capturing risk in emerging markets. Taken together, the results of the G-RVRP and the G-VRP are very close in terms of modelling positive and negative returns for different ˛-VaR levels, but the G-RVRP performs slightly better due to the level dependency of the VRP as explained in Section 3.1. The picture arising from the implied volatility models shows that they do not significantly perform better than the remaining VaR specifications. This result may be explained by either inappro-

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Fig. 1. Daily time series of VRP for the US and Hong Kong as the difference between implied and realized variance.

priate options pricing models or the existence of a volatility risk premium in index options, even though the former is not likely to hold since most of the implied volatility indexes are computed by the established model-free approach, and hence they are not contaminated by model bias. In line with Bams et al. (2017), the limited utility of the implied volatility for predicting the VaR may be due to the non-linear and regime switching dynamics of the volatility risk premium. However, our findings are in stark contrast to those of Giot (2005) who documents the superiority of the VIX for estimating VaR forecasts. This verdict is surprising because most of the volatility indexes under consideration are constructed using the VIX methodology. Consequently, at least a priori, the volatility index is likely to have advantages over GARCH volatility, which contains only historical information. The more mundane explanation for our results is as follows. While they provide accurate empirical violation rate, the inferior performance of implied volatility models merely arises from violation clustering. As it is common in the VaR literature, Giot (2005) judges the efficiency of the VaR estimates solely on the basis of the traditional likelihood ratio test of independence that centres around first order autocorrelation in the violation series, leaving the property of identically distributed violations unexamined. Clearly, the IID test we employ is more prone to explicitly test for the presence of clusters in the VaR violation process and it allows detecting misspecified VaR models that react too slowly to changes in market conditions, thus inducing solvency issues. The robustness of these results is investigated in the following Sections.

4.3. Robustness check In the first robustness check, we investigate the backtesting performance of the VaR models during crises. Then, we analyse the robustness of our main results against an alternative approach to estimate the variance risk premium.

4.3.1. Performance of VaR models during crises The role of sophisticated risk measurement tools in igniting the recent crisis has been at the centre of the debate between the financial industry and regulators. Since the VaR has been singled out as the major culprit (see Triana, 2011), it is worthy to investigate the validity of the proposed VaR models during periods of financial crises. Therefore, a forecasting period of significant financial turmoil between August 1, 2007 and May 14, 2012 is examined separately. This sample period covers the 2007–2008 US subprime crisis as well as the subsequent global financial crisis. However, given that our data sample does not cover this period for South Africa and South Korea, distinct periods of challenging trading environment are selected according to the level of the volatility index in each market. The selected sample periods overlay extreme events and they span from August 13, 2008 to July 1, 2010 and from June 22, 2011 to October 2, 2012 for South Africa and South Korea, respectively. Besides, the Indian market is discarded from the analysis because the sole period of high volatility pertains to the early part of the sample.4 From the results in Table 5, we find that the overall number of the 1%-VaR violations during the crisis period is consistent with that during the whole sample period. The empirical failure rate is equal to the promised probability, and the null hypothesis of correct UC for the various VaR models and large quantiles cannot be rejected. The excellent performance of all the VaR models during this period of acute economic and financial strain highlights the usefulness of modelling large negative and positive returns with skewed and fat-tailed distributions. Yet, for the less extreme 5%-quantile, the results in Table 6 show that the forecasting performance of the VaR models deteriorates during the crisis period, predominantly in the case of developed markets. The significant increase in VaR

4 Spanning from April 6, 2009 to April 19, 2010, this sample period is employed to calculate stressed VaR in compliance with the Basel III capital accord (see Section 4.4).

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Table 3 1%-VaR results during whole sample period. Long trading position

Developed markets ˛ ˆ France IID CC

Short trading position

RM

IVM

GARCH

G-IV

G-VRP

G-RVRP

RM

IVM

GARCH

G-IV

G-VRP

G-RVRP

1.36%* 0.394 0.149

1.12% 0.283 0.508

1.15% 0.492 0.560

1.15% 0.287 0.404

1.08% 0.319 0.608

1.08% 0.448 0.721

0.98% 0.805 0.979

1.05% 0.440 0.820

0.95% 0.640 0.859

1.08% 0.438 0.728

1.15% 0.944 0.570

0.95% 0.718 0.838

Germany

˛ ˆ IID CC

1.23% 0.227 0.212

0.88% 0.033 0.078

0.99% 0.544 0.959

0.92% 0.042 0.074

0.99% 0.253 0.758

0.99% 0.561 0.969

1.09% 0.629 0.693

0.92% 0.688 0.793

1.06% 0.953 0.860

0.99% 0.779 0.959

1.06% 0.886 0.822

0.99% 0.410 0.949

Japan

˛ ˆ IID CC

1.44%** 0.125 0.031

1.26% 0.147 0.113

1.15% 0.400 0.548

1.41%** 0.220 0.053

1.23% 0.252 0.232

1.23% 0.354 0.306

0.79% 0.440 0.432

0.69%* 0.267 0.119

0.58%*** 0.910 0.108

0.72% 0.178 0.115

0.65%** 0.759 0.169

0.61%** 0.958 0.128

Netherlands

˛ ˆ IID CC

1.49%** 0.258 0.044

1.22% 0.060 0.051

1.18% 0.786 0.462

1.28% 0.046 0.046

1.35%* 0.410 0.158

1.18% 0.833 0.479

0.95% 0.386 0.811

0.81% 0.339 0.386

0.84% 0.502 0.510

1.05% 0.251 0.626

1.22% 0.915 0.424

0.95% 0.868 0.832

Switzerland

˛ ˆ IID CC

1.49%*** 0.102 0.016

1.28% 0.009 0.008

1.14% 0.818 0.577

1.21% 0.023 0.043

1.21% 0.864 0.390

1.14% 0.831 0.615

1.07% 0.706 0.795

0.93% 0.231 0.543

1.04% 0.434 0.884

0.97% 0.106 0.376

1.32% 0.493 0.200

1.04% 0.265 0.653

UK

˛ ˆ IID CC

1.60%*** 0.101 0.011

1.25% 0.045 0.049

1.22% 0.076 0.079

1.29% 0.061 0.055

1.22% 0.031 0.046

1.22% 0.033 0.028

0.91% 0.245 0.513

0.94% 0.136 0.390

0.87% 0.072 0.136

0.98% 0.115 0.346

1.15% 0.025 0.040

0.77% 0.090 0.082

US

˛ ˆ IID CC

1.59%*** 0.099 0.007

1.25% 0.000 0.004

1.04% 0.080 0.335

1.25% 0.000 0.001

1.18% 0.064 0.116

1.04% 0.033 0.191

1.07% 0.630 0.782

0.69% 0.036 0.012

0.73% 0.330 0.227

0.73% 0.034 0.006

0.80% 0.344 0.370

0.69%* 0.315 0.147

1.50%** 0.388 0.068

1.42%* 0.257 0.067

1.31% 0.087 0.052

1.46%** 0.137 0.037

1.46%** 0.189 0.036

1.23% 0.137 0.145

1.04% 0.755 0.891

1.04% 0.258 0.672

0.73% 0.493 0.310

0.96% 0.135 0.436

0.89% 0.266 0.425

0.77% 0.386 0.428

Emerging markets ˛ ˆ Hong Kong IID CC India

˛ ˆ IID CC

1.31% 0.044 0.071

0.84% 0.252 0.531

1.22% 0.073 0.154

0.84% 0.233 0.481

1.31% 0.044 0.061

1.22% 0.050 0.143

1.03% 0.504 0.948

0.94% 0.160 0.500

0.56% 0.704 0.266

1.03% 0.548 0.980

0.65% 0.671 0.388

0.56% 0.733 0.278

Mexico

˛ ˆ IID CC

1.28% 0.198 0.166

1.23% 0.001 0.008

1.36%* 0.186 0.092

1.53%** 0.078 0.008

1.36%* 0.188 0.115

1.36%* 0.195 0.110

1.19% 0.706 0.554

1.15% 0.003 0.019

1.28% 0.053 0.048

1.32% 0.138 0.094

1.28% 0.057 0.059

1.28% 0.053 0.053

South Africa

˛ ˆ IID CC

1.39%* 0.069 0.051

1.45%* 0.008 0.009

0.99% 0.423 0.968

1.22% 0.131 0.212

1.10% 0.638 0.780

1.10% 0.627 0.777

1.34% 0.101 0.082

1.16% 0.288 0.471

1.34% 0.339 0.253

1.28% 0.374 0.327

1.34% 0.340 0.239

1.22% 0.371 0.510

South Korea

˛ ˆ IID CC

1.13% 0.296 0.645

1.22% 0.020 0.058

0.84% 0.051 0.079

1.22% 0.031 0.090

1.03% 0.085 0.429

0.84% 0.050 0.091

1.22% 0.382 0.571

1.31% 0.895 0.452

1.31% 0.136 0.208

1.31% 0.887 0.444

1.22% 0.215 0.358

1.22% 0.197 0.335

This table reports the 1%-VaR backtesting results for long and short trading positions during the whole forecasting period. ˛ ˆ denotes the empirical violation rate, and the significance of the empirical violation rate is determined based on the unconditional coverage test. IID is the p-value for the i.i.d. test, and CC is the p-value for the conditional coverage test. Entries in boldface denote the best outcomes. * Indicates that the empirical violation rate is significantly different from the theoretical value at 10% level. ** Indicates that the empirical violation rate is significantly different from the theoretical value at 5% level. *** Indicates that the empirical violation rate is significantly different from the theoretical value at 1% level.

violations demonstrates that the VaR estimates during the global financial crisis are underestimated. This is mainly induced by an overreaction to bad news as most of the VaR violations are found in the left quantile. The overreaction of investors is not surprising during this turbulent period and makes the estimation of long position risk more challenging. Nevertheless, emerging markets seem to be affected to a much lesser extent by the crisis, as evidenced by the high statistical accuracy of most of the investigated models for both long and short positions. In line with our previous findings based on the entire sample period, the G-RVRP is the best performing model, in terms of modelling large positive and negative returns, followed by the GARCH and the G-VRP. Implied volatility models still have a difficult job in generating non-clustered violations, albeit they are highly recommended on the ground of violation frequency. The notable accuracy of the implied volatility regarding the UC test echoes the Giot

(2005)’s suggestion that the performance of VaR models including lagged implied volatility does not deteriorate under challenging market conditions and it remains stable through time.

4.3.2. Expected variance premium The variance risk premium measures defined in Eqs. (2) and (3) are based on the difference between the index (risk-neutral) expected variation and the current realized variation. Both measures are directly observable and, in turn, assume that RVt follows a martingale difference sequence, an assumption which may be inappropriate (Bekaert & Hoerova, 2014), leading to potentially biased variance premiums. As alternative measures, we suggest to base the expected variance premium on the Heterogeneous Autoregressive model of Realized Volatility (HAR-RV) advocated by Corsi (2009). The HAR-RV model accommodates long memory in the realized

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10 Table 4 5%-VaR results during whole sample period.

Long trading position

Short trading position

RM

IVM

GARCH

G-IV

G-VRP

G-RVRP

RM

IVM

GARCH

G-IV

G-VRP

G-RVRP

Developed Markets ˛ ˆ France IID CC

5.63% 0.010 0.005

5.59% 0.038 0.022

5.59% 0.062 0.045

5.59% 0.038 0.031

5.63% 0.133 0.076

5.56% 0.065 0.067

5.83%** 1.000 0.140

5.12% 0.968 0.851

5.15% 0.995 0.833

5.29% 0.964 0.609

5.32% 1.000 0.581

5.15% 0.990 0.829

Germany

˛ ˆ IID CC

5.68* 0.006 0.005

5.55% 0.025 0.017

5.41% 0.110 0.152

5.68%* 0.010 0.006

5.58% 0.085 0.061

5.34% 0.178 0.261

6.54%*** 0.970 0.009

5.75%* 0.834 0.193

5.65% 0.881 0.247

5.72%* 0.482 0.224

5.75%* 0.887 0.177

5.58 0.910 0.292

Japan

˛ ˆ IID CC

5.20% 0.009 0.020

4.84% 0.016 0.039

5.09% 0.049 0.136

5.05% 0.024 0.075

5.02% 0.068 0.249

4.87% 0.013 0.031

5.85%** 0.147 0.040

5.27% 0.000 0.000

4.94% 0.193 0.588

5.30% 0.003 0.011

4.91% 0.060 0.218

4.76 0.149 0.265

Netherlands

˛ ˆ IID CC

5.88%** 0.005 0.001

5.51% 0.001 0.000

5.57% 0.067 0.063

5.71%* 0.022 0.017

5.64% 0.068 0.053

5.37% 0.255 0.299

6.18%*** 0.997 0.042

5.17% 0.575 0.782

5.10% 0.999 0.858

5.34% 0.897 0.578

5.47% 0.993 0.385

5.30% 0.988 0.600

Switzerland

˛ ˆ IID CC

5.43% 0.032 0.033

4.98% 0.044 0.223

5.23% 0.217 0.397

5.16% 0.032 0.099

5.47% 0.292 0.288

5.12% 0.121 0.389

6.37%*** 0.292 0.011

5.61% 0.392 0.260

5.12% 0.574 0.842

5.57% 0.346 0.244

5.64% 0.685 0.274

5.19% 0.602 0.752

UK

˛ ˆ IID CC

5.65% 0.093 0.042

5.51% 0.006 0.012

5.30% 0.483 0.606

5.68% 0.117 0.070

5.68% 0.033 0.022

5.40% 0.600 0.497

6.13%*** 0.915 0.049

5.47% 0.604 0.403

5.51% 0.999 0.393

5.58% 0.864 0.321

5.61% 0.949 0.280

5.23% 0.996 0.709

US

˛ ˆ IID CC

5.85%** 0.066 0.012

5.68% 0.033 0.017

5.47% 0.219 0.196

5.68% 0.063 0.032

5.68% 0.597 0.231

5.57% 0.323 0.230

6.30%*** 0.789 0.031

5.54% 0.350 0.255

5.16% 0.644 0.791

5.61% 0.468 0.311

5.50% 0.949 0.429

5.30% 0.977 0.627

5.85%* 0.022 0.012

6.01%** 0.021 0.004

5.31% 0.054 0.085

6.01%** 0.026 0.006

5.28% 0.074 0.119

5.12% 0.083 0.234

5.97%** 0.878 0.116

5.28% 0.705 0.664

5.51% 0.715 0.387

5.35% 0.633 0.573

5.54 0.457 0.386

5.39 0.934 0.582

Emerging markets ˛ ˆ Hong Kong IID CC India

˛ ˆ IID CC

5.52% 0.281 0.423

5.61% 0.004 0.010

5.24% 0.383 0.744

5.52% 0.005 0.012

5.33% 0.149 0.299

5.33% 0.157 0.348

5.43% 0.896 0.670

5.89% 0.987 0.354

5.05% 0.925 0.965

5.33% 0.978 0.749

5.52 0.989 0.597

5.14 0.952 0.892

Mexico

f IID CC

5.70% 0.065 0.038

5.49% 0.000 0.000

5.19% 0.257 0.585

5.32% 0.209 0.378

5.32% 0.299 0.437

5.19% 0.222 0.465

6.21%** 0.997 0.059

5.78%* 0.001 0.001

5.95%** 0.981 0.135

6.08%** 0.964 0.085

5.95%** 0.971 0.128

5.95%** 0.975 0.121

South Africa

˛ ˆ IID CC

5.23% 0.007 0.010

4.82% 0.012 0.023

5.29% 0.170 0.365

5.23% 0.083 0.212

5.35% 0.110 0.196

5.40% 0.080 0.155

5.17% 0.624 0.849

5.40% 0.000 0.000

5.00% 0.925 0.982

5.17% 0.418 0.838

5.17% 0.817 0.856

4.88% 0.974 0.908

South Korea

˛ ˆ IID CC

6.75%*** 0.085 0.015

6.29%** 0.018 0.015

5.72% 0.040 0.067

6.38%** 0.019 0.010

5.91% 0.037 0.041

5.91% 0.034 0.033

5.25% 0.374 0.717

5.25% 0.388 0.723

4.60% 0.765 0.702

5.07% 0.471 0.937

5.16% 0.618 0.864

4.88% 0.747 0.909

This table reports the 5%-VaR backtesting results for long and short trading positions during the whole forecasting period. ˛ ˆ denotes the empirical violation rate, and the significance of the empirical violation rate is determined based on the unconditional coverage test. IID is the p-value for the i.i.d. test, and CC is the p-value for the conditional coverage test. Entries in boldface denote the best outcomes. * Indicates that the empirical violation rate is significantly different from the theoretical value at 10% level. ** Indicates that the empirical violation rate is significantly different from the theoretical value at 5% level. *** Indicates that the empirical violation rate is significantly different from the theoretical value at 1% level.

variance, and it is found to be empirically very accurate, which is impressive given its relatively simple time series representation: (d)

RV t+1 = ω + ˇd RV t where 1 22

21

(d)

RV t

(w)

+ ˇw RV t

= RV t ,

(w)

RV t

(m)

+ ˇm RV t =

4 1 5

h=0

+ t+1

RV t−h

(14) and

(m)

RV t

=

RV t−h are respectively the daily, weekly, and monthly (ex h=0 post) observed realized variances. Table 7 provides evidence on how robust are the results against using the expected variance premium EVRP t = IV 2t − Et (RV t+1 ) and the expected relative variance premium ERVRP t = IV 2t /Et (RV t+1 ). The results are qualitatively (and almost quantitatively) similar to those reported in Tables 3 and 4 and the observed variance premium measures are not significantly outperformed by their expected counterparts, yet the G-EVRP exhibits fairly higher pvalues from the CC test in nine and ten out of twelve cases for long and short position 5%-VaR, respectively. Whereas the highest p-values are attributed to the G-VRP in eight out of twelve

markets for long position 1%-VaR. Compared to the VaR estimates of the non-augmented GARCH, the VaR estimates of the G-EVRP are dominant, particularly in developed markets, and the performance ranking does not seem to be affected by the forward-looking variance premium. 4.4. Daily capital charges The regulatory framework has not only established the VaR as the official measure of market risk but also allowed financial institutions to employ the Internal Model Approach to determine their market risk minimum capital requirements from their own VaR estimates.5 Formally, the daily risk-capital charge is defined as

5 In the modified Basel III Accord (BCBS, 2019), a number of amendments are intended to establish new standards for the MCR and the market risk including a

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11

Table 5 1%-VaR results during crisis periods. Long trading position

Short trading position

RM

IVM

GARCH

G-IV

G-VRP

G-RVRP

RM

IVM

GARCH

G-IV

G-VRP

G-RVRP

Developed markets ˛ ˆ France IID CC

1.22% 0.226 0.390

1.14% 0.106 0.278

0.98% 0.394 0.933

1.14% 0.106 0.285

0.82% 0.451 0.625

0.90% 0.504 0.844

0.65% 0.676 0.358

1.14% 0.109 0.284

0.90% 0.637 0.819

1.14% 0.114 0.292

1.14% 0.848 0.714

0.98% 0.652 0.993

Germany

˛ ˆ IID CC

1.07% 0.048 0.207

0.74% 0.019 0.023

0.74% 0.801 0.517

0.83% 0.015 0.024

0.74% 0.289 0.353

0.83% 0.693 0.640

0.91% 0.496 0.829

0.91% 0.476 0.826

1.32% 0.927 0.399

0.99% 0.522 0.940

1.24% 0.973 0.554

1.24% 0.952 0.554

Japan

˛ ˆ IID CC

1.30% 0.084 0.115

1.13% 0.046 0.135

0.96% 0.265 0.688

1.22% 0.151 0.281

0.87% 0.222 0.511

0.96% 0.285 0.698

0.70% 0.434 0.410

0.35%** 0.948 0.091

0.78% 0.581 0.561

0.43%** 0.290 0.099

0.52%* 0.825 0.204

0.78% 0.585 0.547

Netherlands

˛ ˆ IID CC

1.39% 0.075 0.065

0.73% 0.130 0.182

1.22% 0.579 0.574

0.73% 0.140 0.183

1.22% 0.304 0.456

1.22% 0.682 0.583

0.49% 0.565 0.176

0.73% 0.134 0.181

0.98% 0.121 0.531

0.82% 0.100 0.218

1.22% 0.562 0.584

1.14% 0.684 0.732

Switzerland

˛ ˆ IID CC

1.33% 0.356 0.328

1.00% 0.015 0.074

1.17% 0.830 0.651

1.08% 0.028 0.099

1.33% 0.951 0.375

1.17% 0.858 0.663

1.08% 0.756 0.818

0.75% 0.166 0.237

1.42% 0.472 0.244

0.75% 0.060 0.083

1.50% 0.073 0.042

1.50%* 0.111 0.059

UK

˛ ˆ IID CC

1.42% 0.225 0.152

1.26% 0.085 0.144

1.34% 0.702 0.373

1.26% 0.080 0.144

1.34% 0.270 0.236

1.42% 0.592 0.249

0.59% 0.169 0.103

1.09% 0.021 0.086

0.92% 0.046 0.153

1.17% 0.013 0.035

1.34% 0.003 0.004

0.92% 0.043 0.171

US

˛ ˆ IID CC

1.50% 0.019 0.012

1.33% 0.000 0.000

1.17% 0.027 0.073

1.33% 0.000 0.000

1.42% 0.007 0.010

1.25% 0.010 0.018

1.08% 0.682 0.849

0.92% 0.001 0.002

1.00% 0.376 0.926

0.92% 0.001 0.002

1.00% 0.270 0.912

1.00% 0.343 0.927

Emerging markets ˛ ˆ Hong Kong IID CC

1.27% 0.130 0.211

1.27% 0.047 0.091

1.10% 0.026 0.076

1.27% 0.055 0.092

1.44% 0.039 0.031

1.02% 0.040 0.205

1.02% 0.539 0.966

1.02% 0.102 0.451

0.93% 0.183 0.588

1.02% 0.103 0.444

1.02% 0.116 0.510

0.93% 0.183 0.525

Mexico

˛ ˆ IID CC

1.25% 0.299 0.436

1.00% 0.044 0.227

1.50%* 0.269 0.132

1.59%* 0.267 0.083

1.49% 0.267 0.140

1.49% 0.271 0.125

1.42% 0.485 0.246

1.17% 0.006 0.016

1.34% 0.100 0.108

1.34% 0.107 0.109

1.34% 0.102 0.112

1.34% 0.100 0.109

South Africa

˛ ˆ IID CC

1.49% 0.000 0.002

1.06% 0.001 0.006

0.85% 0.046 0.137

1.06% 0.014 0.068

1.06% 0.312 0.848

1.06% 0.308 0.852

0.85% 0.192 0.553

0.85% 0.157 0.463

1.06% 0.134 0.429

0.85% 0.207 0.558

0.85% 0.187 0.545

0.85% 0.186 0.587

South Korea

˛ ˆ IID CC

1.56% 0.375 0.210

1.87% 0.260 0.088

1.25% 0.627 0.806

1.87% 0.251 0.092

1.87% 0.257 0.065

1.56% 0.376 0.208

1.25% 0.025 0.063

1.56% 0.183 0.141

1.87% 0.119 0.069

1.56% 0.213 0.149

1.87% 0.117 0.051

1.87% 0.118 0.058

This table reports the 1%-VaR backtesting results for long and short trading positions during crisis periods. ˛ ˆ denotes the empirical violation rate, and the significance of the empirical violation rate is determined based on the unconditional coverage test. IID is the p-value for the i.i.d. test, and CC is the p-value for the conditional coverage test. Entries in boldface denote the best outcomes. * Indicates that the empirical violation rate is significantly different from the theoretical value at 10% level. ** Indicates that the empirical violation rate is significantly different from the theoretical value at 5% level.

follows6 :



MCRt = max

 + max

3 + k VaRt−i ; VaRt−1 60 60

i=1

ms sVaRt−i ; sVaRt−1 60 60



 (15)

i=1

where VaRt−1 denotes the 1%-VaR for day t − 1 and k is a penalty factor depending on the number of VaR violations in the previous 250 trading days such that 0 ≤ k ≤ 1 (see BCBS, 2009); sVaR stands

shift from the VaR to an Expected Shortfall measure of risk under stress. The intended scheme, coming into effect in 2022, plans to ensure a more prudent capture of fat-tail risks and capital adequacy during periods of significant financial market stress. 6 BCBS demands the use of a 10-day holding period through the square-root-oftime rule. However, the present research omits this specification and works with a 1-day holding period instead. This is mainly due to the inconsistencies and the excessive conservatism attained using the scaling rule based on strong distribution assumption such as normally distributed returns and time-independent volatility (Danielsson, 2002; Danielsson & Zigrand, 2006).

for the stressed 1%-VaR for day t − 1; and ms is the multiplier for the sVaR, with ms = 3 (1 + k) and k arises from backtesting results for the VaR. The introduction of the sVaR component in the MCR formula is intended to strengthen the protection against heavy losses, particularly in view of serious problems of shortfall provisions in capital levels of most financial institutions following the unfolding of the US subprime crisis in 2007 and subsequently the 2008–2012 global financial crisis (see Rossignolo, Fethi, & Shaban, 2012). In compliance with the Basel III guidelines, the sVaR is estimated using a dataset that belongs to a continuous twelve-month period of significant financial stress as follows. We calibrate models parameters using an estimation window from July 1, 2008 to July 31, 2009 for developed markets as well as for Kong Kong and Mexico. For South Korea and South Africa, the estimation window pertains to the periods of high volatility as explained in Section 4.3.1, and it spans from April 6, 2009 to April 19, 2010 for India. After that, we use the calibrated models to predict the one-day-ahead sVaR over the forecasting sample. Table 8 reports the Average Minimum Capital Requirements (AMCR) over the forecasting period, using both VaR and sVaR fore-

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12 Table 6 5%-VaR results during crisis period.

Long trading position

Short trading position

RM

IVM

GARCH

G-IV

G-VRP

G-RVRP

RM

IVM

GARCH

G-IV

G-VRP

G-RVRP

Developed markets ˛ ˆ France IID CC

6.45%** 0.008 0.003

6.77%*** 0.026 0.004

7.02%*** 0.130 0.011

6.77%*** 0.026 0.003

6.85%*** 0.307 0.028

6.94%*** 0.098 0.010

5.06% 0.912 0.945

4.73% 0.889 0.803

5.47% 0.922 0.618

4.73% 0.889 0.798

5.30% 0.913 0.760

5.39% 0.933 0.670

Germany

˛ ˆ IID CC

6.28%** 0.025 0.012

6.12%* 0.030 0.016

6.36%** 0.255 0.070

6.28%** 0.028 0.011

6.69%*** 0.278 0.037

6.44%** 0.367 0.079

6.03% 0.998 0.229

5.21% 0.973 0.821

5.86% 0.976 0.323

5.20% 0.970 0.850

5.86% 0.995 0.324

5.86% 0.984 0.341

Japan

˛ ˆ IID CC

5.64% 0.001 0.001

4.51% 0.021 0.032

5.73% 0.008 0.011

4.95% 0.017 0.074

4.95% 0.113 0.398

5.21% 0.043 0.125

5.21% 0.096 0.283

4.34% 0.001 0.001

4.86% 0.752 0.901

4.34% 0.026 0.027

3.91%* 0.602 0.216

4.170 0.728 0.352

Netherlands

˛ ˆ IID CC

6.53%** 0.022 0.005

6.69%** 0.011 0.002

7.26%*** 0.071 0.004

6.77%*** 0.037 0.007

6.77%*** 0.034 0.005

6.77%*** 0.162 0.019

5.38% 0.997 0.686

5.38% 0.963 0.681

5.38% 0.999 0.684

5.22% 0.989 0.829

5.46% 0.999 0.620

5.55% 0.998 0.546

Switzerland

˛ ˆ IID CC

5.42% 0.029 0.055

5.42% 0.030 0.061

5.75% 0.188 0.171

5.50% 0.026 0.051

6.08%* 0.189 0.097

5.83% 0.184 0.139

5.50% 0.588 0.563

4.42% 0.334 0.422

5.17% 0.918 0.871

4.42% 0.489 0.543

5.33% 0.822 0.726

5.17% 0.935 0.858

UK

˛ ˆ IID CC

5.86% 0.029 0.026

6.03%* 0.046 0.026

5.78% 0.264 0.232

5.94% 0.053 0.041

6.45%** 0.061 0.016

5.86% 0.310 0.234

6.28%** 0.387 0.134

5.78% 0.338 0.296

6.45%** 0.941 0.108

5.69% 0.479 0.438

6.19%* 0.715 0.174

6.36%** 0.936 0.126

US

˛ ˆ IID CC

6.16%* 0.078 0.034

6.41%** 0.003 0.001

6.25%** 0.310 0.107

6.33%** 0.011 0.007

6.25%** 0.275 0.095

6.50%** 0.329 0.071

6.00% 0.244 0.144

5.41% 0.120 0.239

6.16%* 0.998 0.180

5.50% 0.088 0.137

5.91% 0.991 0.306

6.25%* 1.000 0.155

Emerging markets ˛ ˆ Hong Kong IID CC

5.58% 0.042 0.067

5.67% 0.044 0.058

5.25% 0.170 0.410

5.67% 0.044 0.059

5.67% 0.191 0.201

5.25% 0.192 0.415

5.33% 0.647 0.748

4.57% 0.593 0.652

5.42% 0.670 0.685

4.65% 0.533 0.715

5.92% 0.659 0.297

5.42% 0.871 0.658

Mexico

˛ ˆ IID CC

5.93% 0.209 0.139

5.59% 0.002 0.003

5.09% 0.414 0.902

5.17% 0.353 0.771

5.34% 0.468 0.700

5.09% 0.284 0.764

6.01%* 0.963 0.245

5.17% 0.002 0.004

6.09%* 0.924 0.209

6.01% 0.935 0.240

6.09%* 0.926 0.218

6.01% 0.932 0.251

South Africa

˛ ˆ IID CC

5.10% 0.016 0.119

4.67% 0.014 0.040

5.52% 0.563 0.721

5.73% 0.160 0.272

5.52% 0.238 0.571

5.73% 0.482 0.637

4.25% 0.507 0.570

4.03% 0.001 0.000

4.03% 0.688 0.442

3.82% 0.164 0.131

4.46% 0.300 0.567

3.82 0.782 0.376

South Korea

˛ ˆ IID CC

6.23% 0.018 0.034

5.92% 0.020 0.033

5.61% 0.032 0.147

6.23% 0.035 0.040

5.61% 0.045 0.124

5.92% 0.019 0.073

5.61% 0.027 0.069

7.17% 0.016 0.014

5.61% 0.096 0.289

6.54% 0.037 0.039

6.23% 0.063 0.108

5.92 0.073 0.180

This table reports the 5%-VaR backtesting results for long and short trading positions during crisis periods. ˛ ˆ denotes the empirical violation rate, and the significance of the empirical violation rate is determined based on the unconditional coverage test. IID is the p-value for the i.i.d. test, and CC is the p-value for the conditional coverage test. Entries in boldface denote the best outcomes. * Indicates that the empirical violation rate is significantly different from the theoretical value at 10% level. ** Indicates that the empirical violation rate is significantly different from the theoretical value at 5% level. *** Indicates that the empirical violation rate is significantly different from the theoretical value at 1% level.

casts generated from the different models. The G-IV generates significantly lower AMCR values than the competing models in developed markets, except for long position risk in Switzerland, albeit they are very close to those induced by the IVM, and we cannot reject the hypothesis of mean equality in most cases. The results show that the implied volatility models incur substantial capital savings for both long and short positions. The best performing models regarding the backtesting results in Section 4.2 (i.e., GARCH and G-RVRP) are significantly less demanding than the G-IV, in terms of capital requirements, only when it comes to emerging markets. This is the case for Mexico and South Africa for both long and short positions, and for South Korea when it comes to provision against long position risk. Interestingly, even in these cases, the G-IV cannot be significantly outperformed. The benefit of using the MCR is twofold. First, the MCR can be used as an additional measure for model selection since it is straightforwardly related to the cost of risk management. Hence,

it can help internal model builders to decide on the trade-off between highly accurate VaR models and those that would likely to immobilize unnecessary funds. Fig. 2 illustrates the cost of risk management in terms of additional MCR that should be provisioned by financial institutions if they implement the GARCH (the most successful model) instead of the G-IV (the most efficient model in terms of AMCR) for long position VaR in the US market. Even if this cost exhibits a somewhat unstable dynamics before 2007, its relatively high level during and after the full unfolding of the global financial crisis shows that augmenting the GARCH model with implied volatility affords a substantial improvement in the efficient allocation of regulatory capital. Second, the MCR can be viewed as an early warning indicator for a forthcoming sharp market downturn. For instance, this indicator plotted in Fig. 3, as the daily MCR implied by the G-RVRP for long position VaR in the US, starts to increase in July 2007 (one year prior to the market crash of 2008), reaching its maximum value in the last months of 2008.

Please cite this article in press as: Slim, S., et al. How informative are variance risk premium and implied volatility for Value-at-Risk prediction? International evidence. The Quarterly Review of Economics and Finance (2019), https://doi.org/10.1016/j.qref.2019.08.006

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13

Table 7 VaR results for the GARCH augmented with expected variance risk premiums. ˛ = 1%

˛ = 5%

Long position

Short position

Long position

Short position

G-EVRP

G-ERVRP

G-EVRP

G-ERVRP

G-EVRP

G-ERVRP

G-EVRP

G-ERVRP

˛ ˆ IID CC

1.22% 0.052 0.051

1.15% 0.109 0.181

1.05% 0.292 0.695

1.08% 0.130 0.316

5.42% 0.328 0.352

5.63% 0.172 0.094

5.29% 0.985 0.646

5.46% 0.995 0.438

Germany

˛ ˆ IID CC

1.02% 0.221 0.654

1.02% 0.554 0.931

0.92% 0.243 0.506

0.99% 0.227 0.704

5.72%** 0.084 0.035

5.72%** 0.063 0.029

6.23%*** 0.795 0.038

6.26%*** 0.942 0.036

Japan

˛ ˆ IID CC

1.12% 0.358 0.581

1.15% 0.306 0.415

0.69% 0.355 0.182

0.61% 0.346 0.105

5.09% 0.037 0.129

4.94% 0.093 0.307

4.98% 0.021 0.090

4.83% 0.576 0.794

Netherlands

˛ ˆ IID CC

1.49%*** 0.147 0.023

1.32%** 0.460 0.203

1.11% 0.259 0.437

1.05% 0.538 0.897

5.37% 0.085 0.107

5.47% 0.106 0.101

5.61%* 0.944 0.291

5.61%* 0.951 0.296

Switzerland

˛ ˆ IID CC

1.07% 0.580 0.788

1.11% 0.907 0.687

1.11% 0.391 0.659

1.04% 0.237 0.647

5.43% 0.132 0.153

5.36% 0.248 0.330

5.78%** 0.342 0.133

5.43% 0.489 0.468

UK

˛ ˆ IID CC

1.36%** 0.048 0.021

1.29%* 0.009 0.007

1.05% 0.046 0.168

1.08% 0.162 0.380

5.23% 0.010 0.024

5.37% 0.306 0.384

5.23% 0.303 0.530

5.47% 0.362 0.337

US

˛ ˆ IID CC

1.14% 0.230 0.375

1.00% 0.293 0.860

0.87% 0.260 0.411

0.865% 0.266 0.422

5.571%* 0.149 0.101

5.502% 0.099 0.099

5.502% 0.988 0.393

5.329% 0.999 0.610

˛ ˆ IID CC

1.35%** 0.097 0.050

1.15% 0.098 0.173

0.89% 0.376 0.637

0.96% 0.455 0.930

5.66%* 0.028 0.020

5.16% 0.075 0.202

5.23% 0.007 0.021

5.31% 0.005 0.011

India

˛ ˆ IID CC

1.24% 0.009 0.022

1.44%* 0.023 0.022

0.62% 0.904 0.380

0.41% 0.808 0.163

5.15% 0.102 0.299

5.26% 0.046 0.135

4.95% 0.877 0.939

5.36% 0.779 0.717

Mexico

˛ ˆ IID CC

1.51%** 0.151 0.031

1.47%** 0.154 0.041

1.15% 0.402 0.573

1.11% 0.413 0.723

5.42% 0.218 0.272

5.24% 0.162 0.309

5.86%** 0.757 0.194

5.86%** 0.973 0.184

South Africa

˛ ˆ IID CC

1.16% 0.602 0.703

1.29% 0.389 0.395

1.63%*** 0.046 0.014

1.36% 0.135 0.110

5.30% 0.111 0.270

5.23% 0.058 0.146

5.37% 0.003 0.008

4.89% 0.495 0.907

South Korea

˛ ˆ IID CC

0.93% 0.055 0.222

0.83% 0.097 0.265

1.14% 0.041 0.140

1.14% 0.194 0.468

5.59% 0.222 0.323

5.38% 0.156 0.339

4.66% 0.937 0.759

4.35% 0.967 0.511

Developed markets France

Emerging markets Hong Kong

This table reports VaR backtesting results for the GARCH augmented with either the expected variance risk premium (G-EVRP) or the expected relative variance risk ˆ denotes the empirical violation rate, and the significance of the empirical violation rate is determined based on premium (G-ERVRP) during the whole forecasting period. ˛ the unconditional coverage test. IID is the p-value for the i.i.d. test, and CC is the p-value for the conditional coverage test. Entries in boldface denote the best outcomes. * Indicates that the empirical violation rate is significantly different from the theoretical value at the 10% level. ** Indicates that the empirical violation rate is significantly different from the theoretical value at the 5% level. *** Indicates that the empirical violation rate is significantly different from the theoretical value at the 1% level.

5. Conclusion When investing in security markets, financial institutions are exposed to at least two sources of uncertainty, namely the uncertainty about the return as captured by the return variance and the uncertainty about the return variance itself. Besides, with the growing need for proper modelling and forecasting asset volatility, the accuracy of implied volatility versus volatility forecasts based on historical returns is an important question in modern finance. It is within this context that this paper provides the first comparison of the information content of implied volatility, variance risk premium and state-of-the-art time-series models in terms of quantifying market risk for international stock market indexes. On the methodological side, we apply the RM, the GARCH and the standalone implied volatility model under the skewed-t distribution. We separately include the implied volatility, the VRP and

its relative form as additional regressors into the GARCH model. To assess the performance of the daily VaR estimates, we use newlydeveloped Monte Carlo-based backtests with a key emphasis on the property of conditional efficiency for the violation process featuring both unconditional efficiency and absence of clustering. The results support earlier evidence that GARCH models including asymmetric and heavy-tailed distributions are quite useful in quantifying and predicting market risk seeing that the statistical sufficiency is achieved effortlessly for most of the investigated markets. More importantly, we find that the accuracy of VaR forecasts can be significantly enhanced by accounting for the variance risk effect, especially for long trading positions and most markedly by including the relative VRP into the GARCH model rather than its level. The performance ranking appears remarkably stable across challenging trading environments and alternative measures of the variance risk premium.

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Table 8 Average Minimum Capital Requirements (AMCR) under the Basel III Accord. Long trading position RM

Short trading position

IVM

GARCH

G-IV

G-VRP

G-RVRP

RM

IVM

GARCH

G-IV

G-VRP

G-RVRP

Developed markets 18.845 France (0.000)

17.422 (0.066)

18.127 (0.000)

17.279 (0.500)

17.929 (0.000)

18.045 (0.000)

16.130 (0.000)

15.557 (0.075)

15.995 (0.000)

15.428 (0.500)

15.978 (0.000)

16.030 (0.000)

Germany

18.881 (0.000)

17.998 (0.016)

18.292 (0.000)

17.837 (0.500)

18.370 (0.000)

18.365 (0.000)

16.143 (0.000)

15.881 (0.048)

16.624 (0.000)

15.759 (0.500)

16.437 (0.000)

16.496 (0.000)

Japan

21.466 (0.000)

20.878 (0.500)

21.220 (0.001)

20.999 (0.160)

21.299 (0.000)

21.554 (0.000)

18.821 (0.000)

18.466 (0.310)

18.545 (0.063)

18.418 (0.500)

18.894 (0.000)

18.839 (0.000)

Netherlands

18.254 (0.000)

17.977 (0.000)

17.638 (0.060)

17.488 (0.500)

17.796 (0.002)

17.746 (0.006)

15.527 (0.000)

15.118 (0.237)

15.389 (0.000)

15.061 (0.500)

15.799 (0.000)

15.459 (0.000)

Switzerland

16.179 (0.000)

15.750 (0.000)

15.596 (0.019)

15.708 (0.001)

15.429 (0.500)

15.572 (0.037)

13.127 (0.039)

13.093 (0.135)

13.483 (0.000)

13.016 (0.500)

13.682 (0.000)

13.664 (0.000)

UK

16.652 (0.000)

15.368 (0.270)

16.039 (0.000)

15.309 (0.500)

15.636 (0.000)

16.040 (0.000)

13.337 (0.000)

12.906 (0.500)

13.526 (0.000)

12.971 (0.240)

13.766 (0.000)

13.329 (0.000)

US

16.827 (0.000)

16.141 (0.445)

16.752 (0.000)

16.126 (0.500)

16.314 (0.035)

16.723 (0.000)

13.895 (0.000)

13.365 (0.500)

14.156 (0.000)

13.371 (0.471)

14.006 (0.000)

14.081 (0.000)

Emerging markets 23.713 Hong Kong (0.000)

21.954 (0.500)

23.805 (0.000)

22.409 (0.002)

23.613 (0.000)

23.711 (0.000)

21.220 (0.000)

20.057 (0.442)

20.674 (0.000)

20.036 (0.500)

20.523 (0.000)

20.859 (0.000)

India

17.847 (0.000)

15.500 (0.072)

17.715 (0.000)

15.435 (0.500)

18.203 (0.000)

18.015 (0.000)

17.760 (0.000)

16.058 (0.122)

16.346 (0.000)

15.992 (0.500)

16.288 (0.000)

16.639 (0.000)

Mexico

24.021 (0.324)

26.371 (0.000)

24.026 (0.333)

24.961 (0.000)

23.963 (0.500)

24.002 (0.394)

20.349 (0.500)

35.150 (0.000)

20.542 (0.060)

20.591 (0.027)

20.518 (0.088)

20.535 (0.068)

South Africa

17.958 (0.000)

20.787 (0.000)

17.467 (0.500)

17.789 (0.000)

17.540 (0.026)

17.555 (0.011)

16.307 (0.500)

18.806 (0.000)

16.687 (0.000)

17.263 (0.000)

17.419 (0.000)

17.333 (0.000)

South Korea

20.111 (0.000)

19.280 (0.113)

19.058 (0.500)

19.128 (0.350)

19.536 (0.005)

19.439 (0.011)

18.219 (0.000)

16.135 (0.500)

17.771 (0.000)

16.221 (0.215)

17.565 (0.000)

17.774 (0.000)

This table reports the AMCR induced by the competing VaR models during the whole forecasting period. Bold numbers indicate the most appropriate models in terms of the minimum AMCR. Entries in parentheses denote p-values for the one-sided t-test that the AMCR associated to model i is greater than the minimum AMCR.

Fig. 2. Cost of risk management as daily MCR difference between GARCH and G-IV for long position risk in US.

Better, the results are in line with the Basel Accord’s tolerance for financial institutions to build their own internal approaches to forecast the VaR as none of the investigated models outperforms the others in terms of both statistical accuracy and capital requirements. While the VaR models incorporating implied volatility are outperformed, regarding the conditional efficiency test, they

cannot be rejected on the ground of unconditional efficiency. Furthermore, as far as regulatory capital allocation for equity exposure is concerned, implied volatility models are strongly recommended, predominately in developed markets, thereby highlighting that options markets are relatively more informationally efficient in developed countries.

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Fig. 3. Time series of daily MCR induced by G-RVRP for long position risk in US.

In light of the promising results regarding the usefulness of option-implied VRP, a primary direction for future research is to explore the information content of higher-order moment swaps.7 For example, both skewness and kurtosis risk premiums would enrich the information set available to risk managers (Ruan, Zhu, Huang, & Zhang, 2016). Alternatively, an in-depth understanding of the VRP dynamics would complement the findings of the present paper. Based on the asset pricing model proposed by Li and Zinna (2018), one may consider different VRP components to improve the forecasting ability of both VaR and Expected Shortfall models, especially during financial crises which are notoriously challenging for market players. The short-term jump component of the VRP would be useful reflecting investors’ fear of market crashes, according to Li and Zinna (2018).

Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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