Empirical analysis of long memory,leverage, and distribution effects for stock market risk estimates

In this study, eight generalized autoregressive conditional heteroskedasticity (GARCH) types of variance specifications and two return distribution settings, the normal and skewed generalized Student's t (SGT) of Theodossiou (1998), totaling nine GARCH-based models, are utilized to forecast the volatility of six stock indices, and then both the out-of-sample-period value-at-risk (VaR) and the expected shortfall (ES) are estimated following the rolling window approach. Moreover, the in-sample VaR is estimated for both the global financial crisis (GFC) period and the non-GFC period. Subsequently, through several accuracy measures, nine models are evaluated in order to explore the influence of long memory, leverage, and distribution effects on the performance of VaR and ES forecasts. As shown by the empirical results of the nine models, the long memory, leverage, and distribution effects subsist in the stock markets. Moreover, regarding the out-of-sample VaR forecasts, long memory is the most important effect, followed by the leverage effect for the low level, whereas the distribution effect is crucial for the high level. As for the three VaR approaches, weighted historical simulation achieves the best VaR forecasting performance, followed by filtered historical simulation, whereas the parametric approach has the worst VaR forecasting performance for all the levels. Furthermore, VaR models underestimate the true risk, whereas ES models overestimate the true risk, indicating that the ES risk measure is more conservative than the VaR risk measure. Additionally, based on back-testing, the VaR provides a better risk forecast than the ES since the ES highly overestimates the true risk. Notably, long memory is important for the ES estimate, whereas both the long memory and the leverage effect are crucial for the VaR estimate. Finally, via in-sample VaR forecasts in regard to the low level, it is found that long memory is important for the non-GFC period, whereas the distribution effect is crucial for the GFC period. On the other hand, with regard to the high level, the distribution effect is crucial for both the non-GFC and the GFC period. These results seem to be consistent with those found in the out-of-sample VaR forecasts. In accordance with these results, several important policy implications are proposed in this study.     

Time-varying higher-order conditional moments and forecasting intraday VaR and Expected Shortfall

We estimate several GARCH- and Extreme Value Theory (EVT)-based models to forecast intraday Value-at-Risk (VaR) and Expected Shortfall (ES) for S&P 500 stock index futures returns for both long and short positions. Among the GARCH-based models we consider is the so-called Autoregressive Conditional Density (ARCD) model, which allows time-variation in higher-order conditional moments. ARCD model with time-varying conditional skewness parameter has the best in-sample fit among the GARCH-based models. The EVT-based model and the GARCH-based models which take conditional skewness and kurtosis (time-varying or otherwise) into account provide accurate VaR forecasts. ARCD model with time-varying conditional skewness parameter seems to provide the most accurate ES forecasts.     

Semi-parametric dynamic asymmetric Laplace models for tail risk forecasting,incorporating realized measures

This paper extends the joint Value-at-Risk (VaR) and expected shortfall (ES) quantile regression model of Taylor (2019), by incorporating a realized measure to drive the tail risk dynamics, as a potentially more efficient driver than daily returns. Furthermore, we propose and test a new model for the dynamics of the ES component. Both a maximum likelihood and an adaptive Bayesian Markov chain Monte Carlo method are employed for estimation, the properties of which are compared in a simulation study. The results favour the Bayesian approach, which is employed subsequently in a forecasting study of seven financial market indices. The proposed models are compared to a range of parametric, non-parametric and semi-parametric competitors, including GARCH, realized GARCH, the extreme value theory method and the joint VaR and ES models of Taylor (2019), in terms of the accuracy of one-day-ahead VaR and ES forecasts, over a long forecast sample period that includes the global financial crisis in 2007–2008. The results are favorable for the proposed models incorporating a realized measure, especially when employing the sub-sampled realized variance and the sub-sampled realized range.     

Forecast combinations for value at risk and expected shortfall

Combining provides a pragmatic way of synthesising the information provided by individual forecasting methods. In the context of forecasting the mean, numerous studies have shown that combining often leads to improvements in accuracy. Despite the importance of the value at risk (VaR), though, few papers have considered quantile forecast combinations. One risk measure that is receiving an increasing amount of attention is the expected shortfall (ES), which is the expectation of the exceedances beyond the VaR. There have been no previous studies on combining ES predictions, presumably due to there being no suitable loss function for ES. However, it has been shown recently that a set of scoring functions exist for the joint estimation or backtesting of VaR and ES forecasts. We use such scoring functions to estimate combining weights for VaR and ES prediction. The results from five stock indices show that combining outperforms the individual methods for the 1% and 5% probability levels.     

Forecasting value at risk with intra-day return curves

Methods for incorporating high resolution intra-day asset price data into risk forecasts are being developed at an increasing pace. Existing methods such as those based on realized volatility depend primarily on reducing the observed intra-day price fluctuations to simple scalar summaries. In this study, we propose several methods that incorporate full intra-day price information as functional data objects in order to forecast value at risk (VaR). Our methods are based on the recently proposed functional generalized autoregressive conditionally heteroscedastic (GARCH) models and a new functional linear quantile regression model. In addition to providing daily VaR forecasts, these methods can be used to forecast intra-day VaR curves, which we considered and studied with companion backtests to evaluate the quality of these intra-day risk measures. Using high-frequency trading data from equity and foreign exchange markets, we forecast the one-day-ahead daily and intra-day VaR with the proposed methods and various benchmark models. The empirical results suggested that the functional GARCH models estimated based on the overnight cumulative intra-day return curves exhibited competitive performance with benchmark models for daily risk management, and they produced valid intra-day VaR curves.     

Forecasting risk measures using intraday data in a generalized autoregressive score framework

A new framework for the joint estimation and forecasting of dynamic value at risk (VaR) and expected shortfall (ES) is proposed by our incorporating intraday information into a generalized autoregressive score (GAS) model introduced by Patton et al., 2019 to estimate risk measures in a quantile regression set-up. We consider four intraday measures: the realized volatility at 5-min and 10-min sampling frequencies, and the overnight return incorporated into these two realized volatilities. In a forecasting study, the set of newly proposed semiparametric models are applied to four international stock market indices (S&P 500, Dow Jones Industrial Average, Nikkei 225 and FTSE 100) and are compared with a range of parametric, nonparametric and semiparametric models, including historical simulations, generalized autoregressive conditional heteroscedasticity (GARCH) models and the original GAS models. VaR and ES forecasts are backtested individually, and the joint loss function is used for comparisons. Our results show that GAS models, enhanced with the realized volatility measures, outperform the benchmark models consistently across all indices and various probability levels.     

Modelling and forecasting value at risk and expected shortfall for GCC stock markets: Do long memory,structural breaks,asymmetry, and fat-tails matter?

This paper addresses the question whether dual long memory (LM), asymmetry and structural breaks in stock market returns matter when forecasting the value at risk (VaR) and expected shortfall (ES) for short and long trading positions. We answer this question for the Gulf Cooperation Council (GCC) stock markets. Empirically, we test the occurrence of structural breaks in the GCC return data using the Inclan and Tiao (1994)’s algorithm and we check the relevance of LM using Shimotsu (2006) procedure before estimating the ARFIMA-FIGARCH and ARFIMA-FIAPARCH models with different innovations’ distributions and computing VaR and ES. Our results show that all the GCC market's volatilities exhibit significant structural breaks matching mainly with the 2008–2009 global financial crises and the Arab spring. Also, they are governed by LM process either in the mean or in the conditional variance which cannot be due to the occurrence of structural breaks. Furthermore, the forecasting ability analysis shows that the FIAPARCH model under skewed Student-t distribution turn out to improve substantially the VaR and the ES forecasts.     

利用GARCH-M类模型和极值理论对上证综指的研究

金融时间序列具有分布的厚尾性、波动的集聚性等特征,传统的方法难以准确的度量其风险。文中运用一种新的估计VaR和ES的方法,即采取两阶段法。首先用GARCH-M类模型(GARCH-M、EGARCH-M和TGARCH-M)拟和原始收益率数据,得到残差序列;第二步用极值分析的方法分析的尾部,最后得到收益率序列的动态VaR和ES。最后对三个模型的计算结果进行比较。     

Nonparametric expected shortfall forecasting incorporating weighted quantiles

A new semi-parametric expected shortfall (ES) estimation and forecasting framework is proposed. The proposed approach is based on a two-step estimation procedure. The first step involves the estimation of value at risk (VaR) at different quantile levels through a set of quantile time series regressions. Then, the ES is computed as a weighted average of the estimated quantiles. The quantile weighting structure is parsimoniously parameterized by means of a beta weight function whose coefficients are optimized by minimizing a joint VaR and ES loss function of the Fissler–Ziegel class. The properties of the proposed approach are first evaluated with an extensive simulation study using two data generating processes. Two forecasting studies with different out-of-sample sizes are then conducted, one of which focuses on the 2008 Global Financial Crisis period. The proposed models are applied to seven stock market indices, and their forecasting performances are compared to those of a range of parametric, non-parametric, and semi-parametric models, including GARCH, conditional autoregressive expectile (CARE), joint VaR and ES quantile regression models, and a simple average of quantiles. The results of the forecasting experiments provide clear evidence in support of the proposed models.     

Value at risk models for volatile emerging markets equity portfolios

This paper investigates the issue of market risk quantification for emerging and developed market equity portfolios. A very wide spectrum of popular and widely used in practice Value at Risk (VaR) models are evaluated and compared with Extreme Value Theory (EVT) and adaptive filtered models, during normal, crises, and post-crises periods. The results are interesting and indicate that despite the documented differences between emerging and developed markets, the most successful VaR models are common for both asset classes. Furthermore, in the case of the (fatter tailed) emerging market equity portfolios, most VaR models turn out to yield conservative risk forecasts, in contrast to developed market equity portfolios, where most models underestimate the realized VaR. VaR estimation during periods of financial turmoil seems to be a difficult task, particularly in the case of emerging markets and especially for the higher loss quantiles. VaR models seem to be affected less by crises periods in the case of developed markets. The performance of the parametric (non-parametric) VaR models improves (deteriorates) during post-crises periods due to the inclusion of extreme events in the estimation sample.     

Has the Basel Accord improved risk management during the global financial crisis?

The Basel II 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 or more risk models to measure Value-at-Risk (VaR). The risk estimates of these models are used to determine capital requirements and associated capital costs of ADIs, depending in part on the number of previous violations, whereby realised losses exceed the estimated VaR. In this paper we define risk management in terms of choosing from a variety of risk models, and discuss the selection of optimal risk models. A new approach to model selection for predicting VaR is proposed, consisting of combining alternative risk models, and we compare conservative and aggressive strategies for choosing between VaR models. We then examine how different risk management strategies performed during the 2008–09 global financial crisis. These issues are illustrated using Standard and Poor's 500 Composite Index.     

EVT and tail-risk modelling: Evidence from market indices and volatility series

Value-at-Risk (VaR) has become the universally accepted risk metric adopted internationally under the Basel Accords for banking industry internal control, capital adequacy and regulatory reporting. The recent extreme financial market events such as the Global Financial Crisis (GFC) commencing in 2007 and the following developments in European markets mean that there is a great deal of attention paid to risk measurement and risk hedging. In particular, to risk indices and attached derivatives as hedges for equity market risk. The techniques used to model tail risk such as VaR have attracted criticism for their inability to model extreme market conditions. In this paper we discuss tail specific distribution based Extreme Value Theory (EVT) and evaluate different methods that may be used to calculate VaR ranging from well known econometrics models of GARCH and its variants to EVT based models which focus specifically on the tails of the distribution. We apply Univariate Extreme Value Theory to model extreme market risk for the FTSE100 UK Index and S&P-500 US markets indices plus their volatility indices. We show with empirical evidence that EVT can be successfully applied to financial market return series for predicting static VaR, CVaR or Expected Shortfall (ES) and also daily VaR and ES using a GARCH(1,1) and EVT based dynamic approach to these various indices. The behaviour of these indices in their tails have implications for hedging strategies in extreme market conditions.     

Decay factor optimisation in time weighted simulation — Evaluating VaR performance

We propose an optimisation approach for determining the optimal decay factor in time weighted (BRW) simulation. The backtesting of the BRW simulation, which involves different decay factors, together with a broad range of competing VaR models, has been performed on a sample of seven stock indexes and two commodities: gold and WTI oil. The results obtained show that the BRW simulation with an optimised decay factor relative to the Lopez (1998) size-adjusted function is among the best performing VaR models, second only to the conditional extreme value approach (McNeil & Frey, 2000). The optimised decay factors are sufficiently stable over time, giving economic justification to the optimisation because they do not change over longer time periods. Unlike most of the VaR models tested, in the large majority of cases, the optimised BRW model passes the Basel II criteria but yields significantly lower VaR forecasts than the extreme value approaches, thus resulting in a lower idle capital, i.e. lower costs.     

Downside risk management and VaR-based optimal portfolios for precious metals,oil and stocks

Value-at-Risk (VaR) is used to analyze the market downside risk associated with investments in six key individual assets including four precious metals, oil and the S&P 500 index, and three diversified portfolios. Using combinations of these assets, three optimal portfolios and their efficient frontiers within a VaR framework are constructed and the returns and downside risks for these portfolios are also analyzed. One-day-ahead VaR forecasts are computed with nine risk models including calibrated RiskMetrics, asymmetric GARCH type models, the filtered Historical Simulation approach, methodologies from statistics of extremes and a risk management strategy involving combinations of models. These risk models are evaluated and compared based on the unconditional coverage, independence and conditional coverage criteria. The economic importance of the results is also highlighted by assessing the daily capital charges under the Basel Accord rule. The best approaches for estimating the VaR for the individual assets under study and for the three VaR-based optimal portfolios and efficient frontiers are discussed. The VaR-based performance measure ranks the most diversified optimal portfolio (Portfolio #2) as the most efficient and the pure precious metals (Portfolio #1) as the least efficient.     

Does Forecast Combination Improve Norges Bank Inflation Forecasts?*

We develop a system that provides model‐based forecasts for inflation in Norway. We recursively evaluate quasi out‐of‐sample forecasts from a large suite of models from 1999 to 2009. The performance of the models are then used to derive quasi real time weights that are used to combine the forecasts. Our results indicate that a combination forecast improves upon the point forecasts from individual models. Furthermore, a combination forecast outperforms Norges Bank's own point forecast for inflation. The beneficial results are obtained using a trimmed weighted average. Some degree of trimming is required for the combination forecasts to outperform the judgmental forecasts from the policymaker.     

Evaluating the accuracy of value-at-risk forecasts: New multilevel tests

We propose independence and conditional coverage tests which are aimed at evaluating the accuracy of Value-at-Risk (VaR) forecasts from the same model at different confidence levels. The proposed procedures are multilevel tests, i.e., joint tests of several quantiles corresponding to different confidence levels. In a comprehensive Monte Carlo exercise, we document the superiority of the proposed tests with respect to existing multilevel tests. In an empirical application, we illustrate the implementation of the tests using several VaR models and daily data for 15 MSCI world indices.     

An approximate long-memory range-based approach for value at risk estimation

This paper proposes new approximate long-memory VaR models that incorporate intra-day price ranges. These models use lagged intra-day range with the feature of considering different range components calculated over different time horizons. We also investigate the impact of the market overnight return on the VaR forecasts, which has not yet been considered with the range in VaR estimation. Model estimation is performed using linear quantile regression. An empirical analysis is conducted on 18 market indices. In spite of the simplicity of the proposed methods, the empirical results show that they successfully capture the main features of the financial returns and are competitive with established benchmark methods. The empirical results also show that several of the proposed range-based VaR models, utilizing both the intra-day range and the overnight returns, are able to outperform GARCH-based methods and CAViaR models.     

VaR、ES与一致性风险测度

基于分位数的VaR(风险价值)不具有一致性,可能误导投资组合优化和风险管理,ES(预期短缺)测度克服了这一缺点。谱测度和失真风险测度更具一般性,考虑了投资者风险厌恶对风险测度的影响,其中VaR和ES均为其特例。从实用性看,ES仍是业界普遍采用的方法。     

EVALUATING MACROECONOMIC FORECASTS: A CONCISE REVIEW OF SOME RECENT DEVELOPMENTS

Macroeconomic forecasts are frequently produced, widely published, intensively discussed, and comprehensively used. The formal evaluation of such forecasts has a long research history. Recently, a new angle to the evaluation of forecasts has been addressed, and in this review we analyze some recent developments from that perspective. The literature on forecast evaluation predominantly assumes that macroeconomic forecasts are generated from econometric models. In practice, however, most macroeconomic forecasts, such as those from the IMF, World Bank, OECD, Federal Reserve Board, Federal Open Market Committee (FOMC), and the ECB, are typically based on econometric model forecasts jointly with human intuition. This seemingly inevitable combination renders most of these forecasts biased and, as such, their evaluation becomes nonstandard. In this review, we consider the evaluation of two forecasts in which: (i) the two forecasts are generated from two distinct econometric models; (ii) one forecast is generated from an econometric model and the other is obtained as a combination of a model and intuition; and (iii) the two forecasts are generated from two distinct (but unknown) combinations of different models and intuition. It is shown that alternative tools are needed to compare and evaluate the forecasts in each of these three situations. These alternative techniques are illustrated by comparing the forecasts from the (econometric) Staff of the Federal Reserve Board and the FOMC on inflation, unemployment, and real GDP growth. It is shown that the FOMC does not forecast significantly better than the Staff, and that the intuition of the FOMC does not add significantly in forecasting the actual values of the economic fundamentals. This would seem to belie the purported expertise of the FOMC.     

Using forecast evaluation to improve the accuracy of the Greenbook forecast

Recently, Patton and Timmermann (2012) proposed a more powerful kind of forecast efficiency regression at multiple horizons, and showed that it provides evidence against the efficiency of the Fed’s Greenbook forecasts. I use their forecast efficiency evaluation to propose a method for adjusting the Greenbook forecasts. Using this method in a real-time out-of-sample forecasting exercise, I find that it provides modest improvements in the accuracies of the forecasts for the GDP deflator and CPI, but not for other variables. The improvements are statistically significant in some cases, with magnitudes of up to 18% in root mean square prediction error.