Forecasting High‐Frequency Risk Measures

This article proposes intraday high‐frequency risk (HFR) measures for market risk in the case of irregularly spaced high‐frequency data. In this context, we distinguish three concepts of value‐at‐risk (VaR): the total VaR, the marginal (or per‐time‐unit) VaR and the instantaneous VaR. Since the market risk is obviously related to the duration between two consecutive trades, these measures are completed with a duration risk measure, i.e. the time‐at‐risk (TaR). We propose a forecasting procedure for VaR and TaR for each trade or other market microstructure event. Subsequently, we perform a backtesting procedure specifically designed to assess the validity of the VaR and TaR forecasts on irregularly spaced data. The performance of the HFR measure is illustrated in an empirical application for two stocks (Bank of America and Microsoft) and an exchange‐traded fund based on Standard & Poor's 500 index. We show that the intraday HFR forecasts capture accurately the volatility and duration dynamics for these three assets. Copyright © 2015 John Wiley & Sons, Ltd.     

Single‐index and portfolio models for forecasting value‐at‐risk thresholds

The variance of a portfolio can be forecast using a single index model or the covariance matrix of the portfolio. Using univariate and multivariate conditional volatility models, this paper evaluates the performance of the single index and portfolio models in forecasting value‐at‐risk (VaR) thresholds of a portfolio. Likelihood ratio tests of unconditional coverage, independence and conditional coverage of the VaR forecasts suggest that the single‐index model leads to excessive and often serially dependent violations, while the portfolio model leads to too few violations. The single‐index model also leads to lower daily Basel Accord capital charges. The univariate models which display correct conditional coverage lead to higher capital charges than models which lead to too many violations. Overall, the Basel Accord penalties appear to be too lenient and favour models which have too many violations. Copyright © 2008 John Wiley & Sons, Ltd.     

The Role of High‐Frequency Intra‐daily Data,Daily Range and Implied Volatility in Multi‐period Value‐at‐Risk Forecasting

This paper assesses the informational content of alternative realized volatility estimators, daily range and implied volatility in multi‐period out‐of‐sample Value‐at‐Risk (VaR) predictions. We use the recently proposed Realized GARCH model combined with the skewed Student's t distribution for the innovations process and a Monte Carlo simulation approach in order to produce the multi‐period VaR estimates. Our empirical findings, based on the S&P 500 stock index, indicate that almost all realized and implied volatility measures can produce statistically and regulatory precise VaR forecasts across forecasting horizons, with the implied volatility being especially accurate in monthly VaR forecasts. The daily range produces inferior forecasting results in terms of regulatory accuracy and Basel II compliance. However, robust realized volatility measures, which are immune against microstructure noise bias or price jumps, generate superior VaR estimates in terms of capital efficiency, as they minimize the opportunity cost of capital and the Basel II regulatory capital. Copyright © 2013 John Wiley & Sons, Ltd.     

Evaluating predictive performance of value‐at‐risk models in emerging markets: a reality check

We investigate the predictive performance of various classes of value‐at‐risk (VaR) models in several dimensions—unfiltered versus filtered VaR models, parametric versus nonparametric distributions, conventional versus extreme value distributions, and quantile regression versus inverting the conditional distribution function. By using the reality check test of White (2000), we compare the predictive power of alternative VaR models in terms of the empirical coverage probability and the predictive quantile loss for the stock markets of five Asian economies that suffered from the 1997–1998 financial crisis. The results based on these two criteria are largely compatible and indicate some empirical regularities of risk forecasts. The Riskmetrics model behaves reasonably well in tranquil periods, while some extreme value theory (EVT)‐based models do better in the crisis period. Filtering often appears to be useful for some models, particularly for the EVT models, though it could be harmful for some other models. The CaViaR quantile regression models of Engle and Manganelli (2004) have shown some success in predicting the VaR risk measure for various periods, generally more stable than those that invert a distribution function. Overall, the forecasting performance of the VaR models considered varies over the three periods before, during and after the crisis. Copyright © 2006 John Wiley & Sons, Ltd.     

Market risk management of banks: implications from the accuracy of Value‐at‐Risk forecasts

This paper adopts the backtesting criteria of the Basle Committee to compare the performance of a number of simple Value‐at‐Risk (VaR) models. These criteria provide a new standard on forecasting accuracy. Currently central banks in major money centres, under the auspices of the Basle Committee of the Bank of International settlement, adopt the VaR system to evaluate the market risk of their supervised banks. Banks are required to report VaRs to bank regulators with their internal models. These models must comply with Basle's backtesting criteria. If a bank fails the VaR backtesting, higher capital requirements will be imposed. VaR is a function of volatility forecasts. Past studies mostly conclude that ARCH and GARCH models provide better volatility forecasts. However, this paper finds that ARCH‐ and GARCH‐based VaR models consistently fail to meet Basle's backtesting criteria. These findings suggest that the use of ARCH‐ and GARCH‐based models to forecast their VaRs is not a reliable way to manage a bank's market risk. Copyright © 2002 John Wiley & Sons, Ltd.     

Incorporating higher moments into value‐at‐risk forecasting

Value‐at‐risk (VaR) forecasting generally relies on a parametric density function of portfolio returns that ignores higher moments or assumes them constant. In this paper, we propose a simple approach to forecasting of a portfolio VaR. We employ the Gram‐Charlier expansion (GCE) augmenting the standard normal distribution with the first four moments, which are allowed to vary over time. In an extensive empirical study, we compare the GCE approach to other models of VaR forecasting and conclude that it provides accurate and robust estimates of the realized VaR. In spite of its simplicity, on our dataset GCE outperforms other estimates that are generated by both constant and time‐varying higher‐moments models. Copyright © 2009 John Wiley & Sons, Ltd.     

A Study of Value‐at‐Risk Based on M‐Estimators of the Conditional Heteroscedastic Models

In this paper, we investigate the performance of a class of M‐estimators for both symmetric and asymmetric conditional heteroscedastic models in the prediction of value‐at‐risk. The class of estimators includes the least absolute deviation (LAD), Huber's, Cauchy and B‐estimator, as well as the well‐known quasi maximum likelihood estimator (QMLE). We use a wide range of summary statistics to compare both the in‐sample and out‐of‐sample VaR estimates of three well‐known stock indices. Our empirical study suggests that in general Cauchy, Huber and B‐estimator have better performance in predicting one‐step‐ahead VaR than the commonly used QMLE. Copyright © 2011 John Wiley & Sons, Ltd.     

Forecasting value‐at‐risk with a parsimonious portfolio spillover GARCH (PS‐GARCH) model

Accurate modelling of volatility (or risk) is important in finance, particularly as it relates to the modelling and forecasting of value‐at‐risk (VaR) thresholds. As financial applications typically deal with a portfolio of assets and risk, there are several multivariate GARCH models which specify the risk of one asset as depending on its own past as well as the past behaviour of other assets. Multivariate effects, whereby the risk of a given asset depends on the previous risk of any other asset, are termed spillover effects. In this paper we analyse the importance of considering spillover effects when forecasting financial volatility. The forecasting performance of the VARMA‐GARCH model of Ling and McAleer (2003), which includes spillover effects from all assets, the CCC model of Bollerslev (1990), which includes no spillovers, and a new Portfolio Spillover GARCH (PS‐GARCH) model, which accommodates aggregate spillovers parsimoniously and hence avoids the so‐called curse of dimensionality, are compared using a VaR example for a portfolio containing four international stock market indices. The empirical results suggest that spillover effects are statistically significant. However, the VaR threshold forecasts are generally found to be insensitive to the inclusion of spillover effects in any of the multivariate models considered. Copyright © 2008 John Wiley & Sons, Ltd.     

Using CAViaR Models with Implied Volatility for Value‐at‐Risk Estimation

This paper proposes value‐at risk (VaR) estimation methods that are a synthesis of conditional autoregressive value at risk (CAViaR) time series models and implied volatility. The appeal of this proposal is that it merges information from the historical time series and the different information supplied by the market's expectation of risk. Forecast‐combining methods, with weights estimated using quantile regression, are considered. We also investigate plugging implied volatility into the CAViaR models—a procedure that has not been considered in the VaR area so far. Results for daily index returns indicate that the newly proposed methods are comparable or superior to individual methods, such as the standard CAViaR models and quantiles constructed from implied volatility and the empirical distribution of standardised residuals. We find that the implied volatility has more explanatory power as the focus moves further out into the left tail of the conditional distribution of S&P 500 daily returns. Copyright © 2012 John Wiley & Sons, Ltd.     

The impact of parameter and model uncertainty on market risk predictions from GARCH‐type models

We study the effect of parameter and model uncertainty on the left‐tail of predictive densities and in particular on VaR forecasts. To this end, we evaluate the predictive performance of several GARCH‐type models estimated via Bayesian and maximum likelihood techniques. In addition to individual models, several combination methods are considered, such as Bayesian model averaging and (censored) optimal pooling for linear, log or beta linear pools. Daily returns for a set of stock market indexes are predicted over about 13 years from the early 2000s. We find that Bayesian predictive densities improve the VaR backtest at the 1% risk level for single models and for linear and log pools. We also find that the robust VaR backtest exhibited by linear and log pools is better than the backtest of single models at the 5% risk level. Finally, the equally weighted linear pool of Bayesian predictives tends to be the best VaR forecaster in a set of 42 forecasting techniques.     

Backtesting Value‐at‐Risk: A Generalized Markov Test

Testing the validity of value‐at‐risk (VaR) forecasts, or backtesting, is an integral part of modern market risk management and regulation. This is often done by applying independence and coverage tests developed by Christoffersen (International Economic Review, 1998; 39(4), 841–862) to so‐called hit‐sequences derived from VaR forecasts and realized losses. However, as pointed out in the literature, these aforementioned tests suffer from low rejection frequencies, or (empirical) power when applied to hit‐sequences derived from simulations matching empirical stylized characteristics of return data. One key observation of the studies is that higher‐order dependence in the hit‐sequences may cause the observed lower power performance. We propose to generalize the backtest framework for VaR forecasts, by extending the original first‐order dependence of Christoffersen to allow for a higher‐ or kth‐order dependence. We provide closed‐form expressions for the tests as well as asymptotic theory. Not only do the generalized tests have power against kth‐order dependence by definition, but also included simulations indicate improved power performance when replicating the aforementioned studies. Further, included simulations show much improved size properties of one of the suggested tests. Copyright © 2017 John Wiley & Sons, Ltd.     

Measuring large‐scale market responses and forecasting aggregated sales: Regression for sparse high‐dimensional data

In this article, we propose a regression model for sparse high‐dimensional data from aggregated store‐level sales data. The modeling procedure includes two sub‐models of topic model and hierarchical factor regressions. These are applied in sequence to accommodate high dimensionality and sparseness and facilitate managerial interpretation. First, the topic model is applied to aggregated data to decompose the daily aggregated sales volume of a product into sub‐sales for several topics by allocating each unit sale (“word” in text analysis) in a day (“document”) into a topic based on joint‐purchase information. This stage reduces the dimensionality of data inside topics because the topic distribution is nonuniform and product sales are mostly allocated into smaller numbers of topics. Next, the market response regression model for the topic is estimated from information about items in the same topic. The hierarchical factor regression model we introduce, based on canonical correlation analysis for original high‐dimensional sample spaces, further reduces the dimensionality within topics. Feature selection is then performed on the basis of the credible interval of the parameters' posterior density. Empirical results show that (i) our model allows managerial implications from topic‐wise market responses according to the particular context, and (ii) it performs better than do conventional category regressions in both in‐sample and out‐of‐sample forecasts.     

A decision rule to minimize daily capital charges in forecasting value‐at‐risk

Under the Basel II Accord, banks and other authorized deposit‐taking institutions (ADIs) have to communicate their daily risk estimates to the monetary authorities at the beginning of the trading day, using a variety of value‐at‐risk (VaR) models to measure risk. Sometimes the risk estimates communicated using these models are too high, thereby leading to large capital requirements and high capital costs. At other times, the risk estimates are too low, leading to excessive violations, so that realized losses are above the estimated risk. In this paper we analyze the profit‐maximizing problem of an ADI subject to capital requirements under the Basel II Accord as ADIs have to choose an optimal VaR reporting strategy that minimizes daily capital charges. Accordingly, we suggest a dynamic communication and forecasting strategy that responds to violations in a discrete and instantaneous manner, while adapting more slowly in periods of no violations. We apply the proposed strategy to Standard & Poor's 500 Index and show there can be substantial savings in daily capital charges, while restricting the number of violations to within the Basel II penalty limits. Copyright © 2009 John Wiley & Sons, Ltd.     

A non‐linear dynamic model for multiplicative seasonal‐trend decomposition

A non‐linear dynamic model is introduced for multiplicative seasonal time series that follows and extends the X‐11 paradigm where the observed time series is a product of trend, seasonal and irregular factors. A selection of standard seasonal and trend component models used in additive dynamic time series models are adapted for the multiplicative framework and a non‐linear filtering procedure is proposed. The results are illustrated and compared to X‐11 and log‐additive models using real data. In particular it is shown that the new procedures do not suffer from the trend bias present in log‐additive models. Copyright © 2002 John Wiley & Sons, Ltd.     

Exponentially Smoothing the Skewed Laplace Distribution for Value‐at‐Risk Forecasting

Value‐at‐risk (VaR) is a standard measure of market risk in financial markets. This paper proposes a novel, adaptive and efficient method to forecast both volatility and VaR. Extending existing exponential smoothing as well as GARCH formulations, the method is motivated from an asymmetric Laplace distribution, where skewness and heavy tails in return distributions, and their potentially time‐varying nature, are taken into account. The proposed volatility equation also involves novel time‐varying dynamics. Back‐testing results illustrate that the proposed method offers a viable, and more accurate, though conservative, improvement in forecasting VaR compared to a range of popular alternatives. Copyright © 2013 John Wiley & Sons, Ltd.     

International Evidence on GFC‐Robust Forecasts for Risk Management under the Basel Accord

A risk management strategy designed to be robust to the global financial crisis (GFC), in the sense of selecting a value‐at‐risk (VaR) forecast that combines the forecasts of different VaR models, was proposed by McAleer and coworkers in 2010. The robust forecast is based on the median of the point VaR forecasts of a set of conditional volatility models. Such a risk management strategy is robust to the GFC in the sense that, while maintaining the same risk management strategy before, during and after a financial crisis, it will lead to comparatively low daily capital charges and violation penalties for the entire period. This paper presents evidence to support the claim that the median point forecast of VaR is generally GFC robust. We investigate the performance of a variety of single and combined VaR forecasts in terms of daily capital requirements and violation penalties under the Basel II Accord, as well as other criteria. In the empirical analysis we choose several major indexes, namely French CAC, German DAX, US Dow Jones, UK FTSE100, Hong Kong Hang Seng, Spanish Ibex 35, Japanese Nikkei, Swiss SMI and US S&P 500. The GARCH, EGARCH, GJR and RiskMetrics models as well as several other strategies, are used in the comparison. Backtesting is performed on each of these indexes using the Basel II Accord regulations for 2008–10 to examine the performance of the median strategy in terms of the number of violations and daily capital charges, among other criteria. The median is shown to be a profitable and safe strategy for risk management, both in calm and turbulent periods, as it provides a reasonable number of violations and daily capital charges. The median also performs well when both total losses and the asymmetric linear tick loss function are considered Copyright © 2012 John Wiley & Sons, Ltd.     

Forecasting volatility with support vector machine‐based GARCH model

Recently, support vector machine (SVM), a novel artificial neural network (ANN), has been successfully used for financial forecasting. This paper deals with the application of SVM in volatility forecasting under the GARCH framework, the performance of which is compared with simple moving average, standard GARCH, nonlinear EGARCH and traditional ANN‐GARCH models by using two evaluation measures and robust Diebold–Mariano tests. The real data used in this study are daily GBP exchange rates and NYSE composite index. Empirical results from both simulation and real data reveal that, under a recursive forecasting scheme, SVM‐GARCH models significantly outperform the competing models in most situations of one‐period‐ahead volatility forecasting, which confirms the theoretical advantage of SVM. The standard GARCH model also performs well in the case of normality and large sample size, while EGARCH model is good at forecasting volatility under the high skewed distribution. The sensitivity analysis to choose SVM parameters and cross‐validation to determine the stopping point of the recurrent SVM procedure are also examined in this study. Copyright © 2009 John Wiley & Sons, Ltd.     

Backtesting Aggregate Risk

This paper proposes the implementation of a VaR backtesting procedure able to overcome the subadditivity property failure of value‐at‐risk (VaR). More precisely, we propose the implementation of a multivariate portmanteau test statistic of Ljung–Box type applied to hits collected from several trading desks or divisions at once. Simulation \exercises illustrate that this method is testing for aggregate risk, accurately accounting both for diversification (negative hit cross‐correlation) and contagion/risk spillovers (positive hit cross‐correlation). An application using profit and loss and VaR data collected for two international major banks illustrates how our proposed backtesting procedure performs in a realistic environment. Copyright © 2015 John Wiley & Sons, Ltd.     

A Neuro‐wavelet Model for the Short‐Term Forecasting of High‐Frequency Time Series of Stock Returns

We propose a wavelet neural network (neuro‐wavelet) model for the short‐term forecast of stock returns from high‐frequency financial data. The proposed hybrid model combines the capability of wavelets and neural networks to capture non‐stationary nonlinear attributes embedded in financial time series. A comparison study was performed on the predictive power of two econometric models and four recurrent neural network topologies. Several statistical measures were applied to the predictions and standard errors to evaluate the performance of all models. A Jordan net that used as input the coefficients resulting from a non‐decimated wavelet‐based multi‐resolution decomposition of an exogenous signal showed a consistent superior forecasting performance. Reasonable forecasting accuracy for the one‐, three‐ and five step‐ahead horizons was achieved by the proposed model. The procedure used to build the neuro‐wavelet model is reusable and can be applied to any high‐frequency financial series to specify the model characteristics associated with that particular series. Copyright © 2013 John Wiley & Sons, Ltd.     

Forecasting VaR models under Different Volatility Processes and Distributions of Return Innovations

This paper provides clear‐cut evidence that the out‐of‐sample VaR (value‐at‐risk) forecasting performance of alternative parametric volatility models, like EGARCH (exponential general autoregressive conditional heteroskedasticity) or GARCH, and Markov regime‐switching models, can be considerably improved if they are combined with skewed distributions of asset return innovations. The performance of these models is found to be similar to that of the EVT (extreme value theory) approach. The performance of the latter approach can also be improved if asset return innovations are assumed to be skewed distributed. The performance of the Markov regime‐switching model is considerably improved if this model allows for EGARCH effects, for all different volatility regimes considered. Copyright © 2014 John Wiley & Sons, Ltd.