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.     

基于极值分布理论的VaR与ES度量

本文应用极值分布理论对金融收益序列的尾部进行估计,计算收益序列的在险价值VaR和预期不足ES来度量市场风险。通过伪最大似然估计方法估计的GARCH模型对收益数据进行拟合,应用极值理论中的GPD对新息分布的尾部建模,得到了基于尾部估计产生收益序列的VaR和ES值。采用上证指数日对数收益数据为样本,得到了度量条件极值和无条件极值下VaR和ES的结果。实证研究表明:在置信水平很高(如99%)的条件下,采用极值方法度量风险值效果更好。而置信水平在95%下,其他方法和极值方法结合效果会很好。用ES度量风险能够使我们了解不利情况发生时风险的可能情况。     

Model choice and value-at-risk estimation

Value at risk (VaR) is a commonly used tool to measure market risk. In this paper, we discuss the problems of model choice and VaR performance. The VaRs of daily returns of the Shanghai and Shenzhen indexes are calculated using equally weighted moving average (EQMA), exponentially weighted moving average (EWMA), GARCH(1,1), empirical density estimation method, and the Pareto-type extreme-value distribution methods. Considering the length of the window and the requirement for adequate capital, back testing indicates that the Pareto-type extreme-value distribution method reflects the real market risk more accurately than the other models.     

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.     

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.     

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.     

Fat tails,VaR and subadditivity

Financial institutions rely heavily on Value-at-Risk (VaR) as a risk measure, even though it is not globally subadditive. First, we theoretically show that the VaR portfolio measure is subadditive in the relevant tail region if asset returns are multivariate regularly varying, thus allowing for dependent returns. Second, we note that VaR estimated from historical simulations may lead to violations of subadditivity. This upset of the theoretical VaR subadditivity in the tail arises because the coarseness of the empirical distribution can affect the apparent fatness of the tails. Finally, we document a dramatic reduction in the frequency of subadditivity violations, by using semi-parametric extreme value techniques for VaR estimation instead of historical simulations.     

Modified maximum spacings method for generalized extreme value distribution and applications in real data analysis

This paper analyzes weekly closing price data of the S&P 500 stock index and electrical insulation element lifetimes data based on generalized extreme value distribution. A new estimation method, modified maximum spacings (MSP) method, is proposed and obtained by using interior penalty function algorithm. The standard error of the proposed method is calculated through Bootstrap method. The asymptotic properties of the modified MSP estimators are discussed. Some simulations are performed, which show that the proposed method is not only available for the whole shape parameter space, but is also of high efficiency. The benchmark risk index, value at risk (VaR), is evaluated according to the proposed method, and the confidence interval of VaR is also calculated through Bootstrap method. Finally, the results are compared with those derived by empirical calculation and some existing methods.     

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.     

Dynamic correlation multivariate stochastic volatility with latent factors

Modeling the correlation structure of returns is essential in many financial applications. Considerable evidence from empirical studies has shown that the correlation among asset returns is not stable over time. A recent development in the multivariate stochastic volatility literature is the application of inverse Wishart processes to characterize the evolution of return correlation matrices. Within the inverse Wishart multivariate stochastic volatility framework, we propose a flexible correlated latent factor model to achieve dimension reduction and capture the stylized fact of ‘correlation breakdown’ simultaneously. The parameter estimation is based on existing Markov chain Monte Carlo methods. We illustrate the proposed model with several empirical studies. In particular, we use high‐dimensional stock return data to compare our model with competing models based on multiple performance metrics and tests. The results show that the proposed model not only describes historic stylized facts reasonably but also provides the best overall performance.     

基于AEPD分布和ALD分布的VaR模型

本文以成熟市场和新兴市场的六个主要的市场指数为例,将更精确反映金融资产收益率典型事实的AEPD分布和ALD分布运用于股票市场VaR的度量。并与其它常见的非参、半参和参数法VaR模型进行全面比较。实证表明,对于参数法模型,误差项服从ALD分布和正态分布的GARCH族模型分别当且仅当在度量低分位数和高分位数水平下的VaR值时表现优异;而误差项服从AEPD分布的GARCH族模型在度量各种分位数水平下的VaR值时均取得不错的效果。另外对于CAViaR模型,它们在度量VaR时与参数法中表现最好的AR-GJR-GARCH-AEPD(ALD)两个模型效果相当。     

Analysis of high dimensional multivariate stochastic volatility models

This paper is concerned with the Bayesian estimation and comparison of flexible, high dimensional multivariate time series models with time varying correlations. The model proposed and considered here combines features of the classical factor model with that of the heavy tailed univariate stochastic volatility model. A unified analysis of the model, and its special cases, is developed that encompasses estimation, filtering and model choice. The centerpieces of the estimation algorithm (which relies on MCMC methods) are: (1) a reduced blocking scheme for sampling the free elements of the loading matrix and the factors and (2) a special method for sampling the parameters of the univariate SV process. The resulting algorithm is scalable in terms of series and factors and simulation-efficient. Methods for estimating the log-likelihood function and the filtered values of the time-varying volatilities and correlations are also provided. The performance and effectiveness of the inferential methods are extensively tested using simulated data where models up to 50 dimensions and 688 parameters are fit and studied. The performance of our model, in relation to various multivariate GARCH models, is also evaluated using a real data set of weekly returns on a set of 10 international stock indices. We consider the performance along two dimensions: the ability to correctly estimate the conditional covariance matrix of future returns and the unconditional and conditional coverage of the 5% and 1% value-at-risk (VaR) measures of four pre-defined portfolios.     

Risk management of precious metals

This paper examines volatility and correlation dynamics in price returns of gold, silver, platinum and palladium, and explores the corresponding risk management implications for market risk and hedging. Value-at-Risk (VaR) is used to analyze the downside market risk associated with investments in precious metals, and to design optimal risk management strategies. We compute the VaR for major precious metals using the calibrated RiskMetrics, different GARCH models, and the semi-parametric Filtered Historical Simulation approach. The best approach for estimating VaR based on conditional and unconditional statistical tests is documented. The economic importance of the results is highlighted by assessing the daily capital charges from the estimated VaRs.     

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.     

Forecasting the volatility of asset returns: The informational gains from option prices

A new class of forecasting models is proposed that extends the realized GARCH class of models through the inclusion of option prices to forecast the variance of asset returns. The VIX is used to approximate option prices, resulting in a set of cross-equation restrictions on the model’s parameters. The full model is characterized by a nonlinear system of three equations containing asset returns, the realized variance, and the VIX, with estimation of the parameters based on maximum likelihood methods. The forecasting properties of the new class of forecasting models, as well as a number of special cases, are investigated and applied to forecasting the daily S&P500 index realized variance using intra-day and daily data from September 2001 to November 2017. The forecasting results provide strong support for including the realized variance and the VIX to improve variance forecasts, with linear conditional variance models performing well for short-term one-day-ahead forecasts, whereas log-linear conditional variance models tend to perform better for intermediate five-day-ahead forecasts.     

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.     

Parametric Value-at-Risk analysis: Evidence from stock indices

We evaluate the performance of several volatility models in estimating one-day-ahead Value-at-Risk (VaR) of seven stock market indices using a number of distributional assumptions. Because all returns series exhibit volatility clustering and long range memory, we examine GARCH-type models including fractionary integrated models under normal, Student-t and skewed Student-t distributions. Consistent with the idea that the accuracy of VaR estimates is sensitive to the adequacy of the volatility model used, we find that AR (1)-FIAPARCH (1,d,1) model, under a skewed Student-t distribution, outperforms all the models that we have considered including widely used ones such as GARCH (1,1) or HYGARCH (1,d,1). The superior performance of the skewed Student-t FIAPARCH model holds for all stock market indices, and for both long and short trading positions. Our findings can be explained by the fact that the skewed Student-t FIAPARCH model can jointly accounts for the salient features of financial time series: fat tails, asymmetry, volatility clustering and long memory. In the same vein, because it fails to account for most of these stylized facts, the RiskMetrics model provides the least accurate VaR estimation. Our results corroborate the calls for the use of more realistic assumptions in financial modeling.     

The structure of dynamic correlations in multivariate stochastic volatility models

This paper proposes two types of stochastic correlation structures for Multivariate Stochastic Volatility (MSV) models, namely the constant correlation (CC) MSV and dynamic correlation (DC) MSV models, from which the stochastic covariance structures can easily be obtained. Both structures can be used for purposes of determining optimal portfolio and risk management strategies through the use of correlation matrices, and for calculating Value-at-Risk (VaR) forecasts and optimal capital charges under the Basel Accord through the use of covariance matrices. A technique is developed to estimate the DC MSV model using the Markov Chain Monte Carlo (MCMC) procedure, and simulated data show that the estimation method works well. Various multivariate conditional volatility and MSV models are compared via simulation, including an evaluation of alternative VaR estimators. The DC MSV model is also estimated using three sets of empirical data, namely Nikkei 225 Index, Hang Seng Index and Straits Times Index returns, and significant dynamic correlations are found. The Dynamic Conditional Correlation (DCC) model is also estimated, and is found to be far less sensitive to the covariation in the shocks to the indexes. The correlation process for the DCC model also appears to have a unit root, and hence constant conditional correlations in the long run. In contrast, the estimates arising from the DC MSV model indicate that the dynamic correlation process is stationary.     

Forecasting value at risk and expected shortfall with mixed data sampling

I propose applying the Mixed Data Sampling (MIDAS) framework to forecast Value at Risk (VaR) and Expected shortfall (ES). The new methods exploit the serial dependence on short-horizon returns to directly forecast the tail dynamics of the desired horizon. I perform a comprehensive comparison of out-of-sample VaR and ES forecasts with established models for a wide range of financial assets and backtests. The MIDAS-based models significantly outperform traditional GARCH-based forecasts and alternative conditional quantile specifications, especially in terms of multi-day forecast horizons. My analysis advocates models that feature asymmetric conditional quantiles and the use of the Asymmetric Laplace density to jointly estimate VaR and ES.     

Variance-Gamma and Normal-Inverse Gaussian models: Goodness-of-fit to Chinese high-frequency index returns

In this study Variance-Gamma (VG) and Normal-Inverse Gaussian (NIG) distributions are compared with the benchmark of generalized hyperbolic distribution in terms of their fit to the empirical distribution of high-frequency stock market index returns in China. First, we estimate the considered models in a Markov regime switching framework for the identification of different volatility regimes. Second, the goodness-of-fit results are compared at different time scales of log-returns. Third, the goodness-of-fit results are validated through bootstrapping experiments. Our results show that as the time scale of log-returns decrease NIG model outperforms the VG model consistently and the difference between the goodness-of-fit statistics increase. For high-frequency Chinese index returns, NIG model is more robust and provides a better fit to the empirical distributions of returns at different time scales.