Value-at-Risk analysis for long-term interest rate futures: Fat-tail and long memory in return innovations

This article uses the FIGARCH(1,d,1) models to calculate daily Value-at-Risk (VaR) for T-bond interest rate futures returns of long and short trading positions based on the normal, Student-t, and skewed Student-t innovations distributions. The empirical results show that based on Kupiec LR failure rate tests, in-sample and out-of-sample VaR values calculated using FIGARCH(1,d,1) model with skewed Student-t innovations are more accurate than those generated using traditional GARCH(1,1) models. Moreover, we find that the in-sample values of VaR are subject to a significant positive bias, as pointed out by Inui et al. [Inui, K., Kijima, M., Kitano, A., 2003. VaR is subject to a significant positive bias, working paper].     

Alternative statistical distributions for estimating value-at-risk: theory and evidence

A number of applications presume that asset returns are normally distributed, even though they are widely known to be skewed leptokurtic and fat-tailed and excess kurtosis. This leads to the underestimation or overestimation of the true value-at-risk (VaR). This study utilizes a composite trapezoid rule, a numerical integral method, for estimating quantiles on the skewed generalized t distribution (SGT) which permits returns innovation to flexibly treat skewness, leptokurtosis and fat tails. Daily spot prices of the thirteen stock indices in North America, Europe and Asia provide data for examining the one-day-ahead VaR forecasting performance of the GARCH model with normal, student??s t and SGT distributions. Empirical results indicate that the SGT provides a good fit to the empirical distribution of the log-returns followed by student??s t and normal distributions. Moreover, for all confidence levels, all models tend to underestimate real market risk. Furthermore, the GARCH-based model, with SGT distributional setting, generates the most conservative VaR forecasts followed by student??s t and normal distributions for a long position. Consequently, it appears reasonable to conclude that, from the viewpoint of accuracy, the influence of both skewness and fat-tails effects (SGT) is more important than only the effect of fat-tails (student??s t) on VaR estimates in stock markets for a long position.     

Aggregation of exponential smoothing processes with an application to portfolio risk evaluation

In this paper we propose a unified framework to analyse contemporaneous and temporal aggregation of a widely employed class of integrated moving average (IMA) models. We obtain a closed-form representation for the parameters of the contemporaneously and temporally aggregated process as a function of the parameters of the original one. These results are useful due to the close analogy between the integrated GARCH (1, 1) model for conditional volatility and the IMA (1, 1) model for squared returns, which share the same autocorrelation function. In this framework, we present an application dealing with Value-at-Risk (VaR) prediction at different sampling frequencies for an equally weighted portfolio composed of multiple indices. We apply the aggregation results by inferring the aggregate parameter in the portfolio volatility equation from the estimated vector IMA (1, 1) model of squared returns. Empirical results show that VaR predictions delivered using this suggested approach are at least as accurate as those obtained by applying standard univariate methodologies, such as RiskMetrics.     

The role of autoregressive conditional skewness and kurtosis in the estimation of conditional VaR

This paper investigates the role of high-order moments in the estimation of conditional value at risk (VaR). We use the skewed generalized t distribution (SGT) with time-varying parameters to provide an accurate characterization of the tails of the standardized return distribution. We allow the high-order moments of the SGT density to depend on the past information set, and hence relax the conventional assumption in conditional VaR calculation that the distribution of standardized returns is iid. The maximum likelihood estimates show that the time-varying conditional volatility, skewness, tail-thickness, and peakedness parameters of the SGT density are statistically significant. The in-sample and out-of-sample performance results indicate that the conditional SGT-GARCH approach with autoregressive conditional skewness and kurtosis provides very accurate and robust estimates of the actual VaR thresholds.     

Evaluating portfolio Value-at-Risk using semi-parametric GARCH models

In this paper we examine the usefulness of multivariate semi-parametric GARCH models for evaluating the Value-at-Risk (VaR) of a portfolio with arbitrary weights. We specify and estimate several alternative multivariate GARCH models for daily returns on the S&P 500 and Nasdaq indexes. Examining the within-sample VaRs of a set of given portfolios shows that the semi-parametric model performs uniformly well, while parametric models in several cases have unacceptable failure rates. Interestingly, distributional assumptions appear to have a much larger impact on the performance of the VaR estimates than the particular parametric specification chosen for the GARCH equations.     

Risk management under extreme events

This article presents two applications of extreme value theory (EVT) to financial markets: computation of value at risk (VaR) and cross-section dependence of extreme returns (i.e., tail dependence). We use a sample comprised of the United States, Europe, Asia, and Latin America. Our main findings are the following. First, on average, EVT gives the most accurate estimate of VaR. Second, tail dependence of paired returns decreases substantially when both heteroscedasticity and serial correlation are filtered out by a multivariate GARCH model. Both findings are in agreement with previous research in this area for other financial markets.     

Forecasting Value-at-Risk using high frequency data: The realized range model

Current studies on financial market risk measures usually use daily returns based on GARCH type models. This paper models realized range using intraday high frequency data based on CARR framework and apply it to VaR forecasting. Kupiec LR test and dynamic quantile test are used to compare the performance of VaR forecasting of realized range model with another intraday realized volatility model and daily GARCH type models. Empirical results of Chinese Stock Indices show that realized range model performs the same with realized volatility model, which performs much better than daily models.     

On the distribution and conditional heteroscedasticity in Taiwan stock prices

This paper studies the distribution and conditional heteroscedasticity in stock returns on the Taiwan stock market. Apart from the normal distribution, in order to explain the leptokurtosis and skewness observed in the stock return distribution, we also examine the Student-t, the Poisson–normal, and the mixed-normal distributions, which are essentially a mixture of normal distributions, as conditional distributions in the stock return process. We also use the ARMA (1,1) model to adjust the serial correlation, and adopt the GJR–generalized autoregressive conditional heteroscedasticity (GARCH (1,1)) model to account for the conditional heterscedasticity in the return process. The empirical results show that the mixed–normal–GARCH model is the most probable specification for Taiwan stock returns. The results also show that skewness seems to be diversifiable through portfolio. Thus the normal–GARCH or the Student-t–GARCH model which involves symmetric conditional distribution may be a reasonable model to describe the stock portfolio return process¹.     

Value-at-Risk: a multivariate switching regime approach

This paper analyses the application of a switching volatility model to forecast the distribution of returns and to estimate the Value-at-Risk (VaR) of both single assets and portfolios. We calculate the VaR value for 10 Italian stocks and a number of portfolios based on these stocks. The calculated VaR values are also compared with the variance–covariance approach used by JP Morgan in RiskMetrics™ and GARCH(1,1) models. Under backtesting, the VaR values calculated using the switching regime beta model are preferred to both other methods. The Proportion of Failure and Time Until First Failure tests [The Journal of Derivatives (1995) 73–84] confirm this result.     

Managing extreme risks in tranquil and volatile markets using conditional extreme value theory

Financial risk management typically deals with low-probability events in the tails of asset price distributions. To capture the behavior of these tails, one should therefore rely on models that explicitly focus on the tails. Extreme value theory (EVT)-based models do exactly that, and in this paper, we apply both unconditional and conditional EVT models to the management of extreme market risks in stock markets. We find conditional EVT models to give particularly accurate Value-at-Risk (VaR) measures, and a comparison with traditional (Generalized ARCH (GARCH)) approaches to calculate VaR demonstrates EVT as being the superior approach both for standard and more extreme VaR quantiles.     

基于Copula-GARCH-EVT的中国开放式基金投资组合风险度量

文章结合CARCH模型和EVT理论刻画了单个金融资产收益率的波动性和尾部分布,并将Copula函数和Monte Carlo技术应用于证券投资组合的VaR计算方法.通过对光大红利基金的实证研究,得到前十大重仓中单只股票及其投资组合的风险值,结果表明,基于Copula-GARCH-EVT的VaR方法具有重要的经济应用价值.     

Conditional higher order moments in metal asset returns

This study examines the role of higher order moments in the returns of four important metals, aluminium, copper, gold and silver, using the asymmetric GARCH (AGARCH) model with a conditional skewed generalized-t (SGT) distribution. Implications of higher order moments in metal returns are evaluated by comparing the performances of conditional value-at-risk measures obtained from the AGARCH models with SGT distributions to those obtained from the AGARCH models with normal and student-t distributions. With the exception of gold, the AGARCH model with the SGT distribution appears to have the best fit for all metals examined.     

Forecasting value-at-risk and expected shortfall using fractionally integrated models of conditional volatility: International evidence

The present study compares the performance of the long memory FIGARCH model, with that of the short memory GARCH specification, in the forecasting of multi-period value-at-risk (VaR) and expected shortfall (ES) across 20 stock indices worldwide. The dataset is composed of daily data covering the period from 1989 to 2009. The research addresses the question of whether or not accounting for long memory in the conditional variance specification improves the accuracy of the VaR and ES forecasts produced, particularly for longer time horizons. Accounting for fractional integration in the conditional variance model does not appear to improve the accuracy of the VaR forecasts for the 1-day-ahead, 10-day-ahead and 20-day-ahead forecasting horizons relative to the short memory GARCH specification. Additionally, the results suggest that underestimation of the true VaR figure becomes less prevalent as the forecasting horizon increases. Furthermore, the GARCH model has a lower quadratic loss between actual returns and ES forecasts, for the majority of the indices considered for the 10-day and 20-day forecasting horizons. Therefore, a long memory volatility model compared to a short memory GARCH model does not appear to improve the VaR and ES forecasting accuracy, even for longer forecasting horizons. Finally, the rolling-sampled estimated FIGARCH parameters change less smoothly over time compared to the GARCH models. Hence, the parameters' time-variant characteristic cannot be entirely due to the news information arrival process of the market; a portion must be due to the FIGARCH modelling process itself.     

Higher Moments and Exchange Rate Behavior

This paper uses 15‐minute exchange rate returns data for the six most liquid currencies (i.e., the Australian dollar, British pound, Canadian dollar, Euro, Japanese yen, and Swiss franc) vis‐à‐vis the United States dollar to examine whether a GARCH model augmented with higher moments (HM‐GARCH) performs better than a traditional GARCH (TG) model. Two findings are unraveled. First, the inclusion of odd/even moments in modeling the return/variance improves the statistical performance of the HM‐GARCH model. Second, trading strategies that extract buy and sell trading signals based on exchange rate forecasts from HM‐GARCH models are more profitable than those that depend on TG models.     

Varying the VaR for unconditional and conditional environments

Accurate forecasting of risk is the key to successful risk management techniques. Using the largest stock index futures from 12 European bourses, this paper presents VaR measures based on their unconditional and conditional distributions for single and multi-period settings. These measures underpinned by extreme value theory are statistically robust explicitly allowing for fat-tailed densities. Conditional tail estimates accounting for volatility clustering are obtained by adjusting the unconditional extreme value procedure with GARCH filtered returns. The conditional modelling results in iid returns allowing for the use of a simple and efficient multi-period extreme value scaling law. The paper examines the properties of these distinct conditional and unconditional trading models. The paper finds that the biases inherent in unconditional single and multi-period estimates assuming normality extend to the conditional setting.     

Empirical analysis of GARCH models in value at risk estimation

This paper studies seven GARCH models, including RiskMetrics and two long memory GARCH models, in Value at Risk (VaR) estimation. Both long and short positions of investment were considered. The seven models were applied to 12 market indices and four foreign exchange rates to assess each model in estimating VaR at various confidence levels. The results indicate that both stationary and fractionally integrated GARCH models outperform RiskMetrics in estimating 1% VaR. Although most return series show fat-tailed distribution and satisfy the long memory property, it is more important to consider a model with fat-tailed error in estimating VaR. Asymmetric behavior is also discovered in the stock market data that t-error models give better 1% VaR estimates than normal-error models in long position, but not in short position. No such asymmetry is observed in the exchange rate data.     

基于实现极差和实现波动率的中国金融市场风险测度研究

目前比较流行的金融市场风险价值研究一般采用日收益数据,并基于GARCH类模型进行估计和预测。本文利用沪深股指日内高频数据,分别通过ARFIMA模型和CARR模型对实现波动率和较新的实现极差建模,计算风险价值。通过对VaR的似然比和动态分位数等回测检验,实证分析了各种模型的VaR预测能力。结果显示,使用日内高频数据的实现波动率和实现极差模型的预测能力强于采用日数据的各种GARCH类模型。     

Uncertainty in Value-at-risk Estimates under Parametric and Non-parametric Modeling

This study evaluates a set of parametric and non-parametric value-at-risk (VaR) models that quantify the uncertainty in VaR estimates in form of a VaR distribution. We propose a new VaR approach based on Bayesian statistics in a GARCH volatility modeling environment. This Bayesian approach is compared with other parametric VaR methods (quasi-maximum likelihood and bootstrap resampling on the basis of GARCH models) as well as with non-parametric historical simulation approaches (classical and volatility adjusted). All these methods are evaluated based on the frequency of failures and the uncertainty in VaR estimates.Within the parametric methods, the Bayesian approach is better able to produce adequate VaR estimates, and results mostly in a smaller VaR variability. The non-parametric methods imply more uncertain 99%-VaR estimates, but show good performance with respect to 95%-VaRs.     

Asian Pacific Stock Market Volatility Modeling and Value at Risk Analysis

The potential for stock market growth in Asian Pacific countries has attracted foreign investors. However, higher growth rates come with higher risk. We apply value at risk (VaR) analysis to measure and analyze stock market index risks in Asian Pacific countries, exposing and detailing both the unique risks and system risks embedded in those markets. To implement the VaR measure, it is necessary to perform "volatility modeling" by mixture switch, exponentially weighted moving average (EWMA), or generalized autoregressive conditional heteroskedasticity (GARCH) models. After estimating the volatility parameters, we can calibrate the VaR values of individual and system risks. Empirically, we find that, on average, Indonesia and Korea exhibit the highest VaRs and VaR sensitivity, and currently, Australia exhibits relatively low values. Taiwan is liable to be in high-state volatility. In addition, the Kupiec test indicates that the mixture switch VaR is superior to delta normal VaR; the quadratic probability score (QPS) shows that the EWMA is inclined to underestimate the VaR for a single series, and GARCH shows no difference from GARCH t and GARCH generalized error distribution (GED) for a multivariate VaR estimate with more assets.