Estimation of tail-related risk measures for heteroscedastic financial time series: an extreme value approach

We propose a method for estimating Value at Risk (VaR) and related risk measures describing the tail of the conditional distribution of a heteroscedastic financial return series. Our approach combines pseudo-maximum-likelihood fitting of GARCH models to estimate the current volatility and extreme value theory (EVT) for estimating the tail of the innovation distribution of the GARCH model. We use our method to estimate conditional quantiles (VaR) and conditional expected shortfalls (the expected size of a return exceeding VaR), this being an alternative measure of tail risk with better theoretical properties than the quantile. Using backtesting of historical daily return series we show that our procedure gives better 1-day estimates than methods which ignore the heavy tails of the innovations or the stochastic nature of the volatility. With the help of our fitted models we adopt a Monte Carlo approach to estimating the conditional quantiles of returns over multiple-day horizons and find that this outperforms the simple square-root-of-time scaling method.     

典型事实、极值理论与金融市场动态风险测度研究

本文在金融市场典型事实约束下,运用ARFIMA模型对金融市场条件收益率建模,运用GARCH、GJR、FIGARCH、APARCH、FIAPARCH等5种模型对金融波动率进行建模,进而运用极值理论(EVT)对标准收益的极端尾部风险建模来测度各股市的动态风险,并用返回测试(Back-testing)方法检验模型的适应性。实证结果表明,总的来说,FIAPARCH-EVT模型对各个市场具有较强的适应性,风险测度能力较为优越。进一步,本文在ARFIMA-FIAPARCH模型下,假定标准收益分别服从正态分布(N)、学生t分布(st)、有偏学生t分布(skst)、广义误差分布(GED)共4种分布,对各股市的动态风险测度的准确性进行检验,并和EVT方法的测度结果进行对比分析。结果表明,EVT方法风险测度能力优于其他方法,有偏学生t分布假设下的风险测度模型虽然略逊于EVT方法,但也不失为一种较好的方法;ARFIMA-FI-APARCH-EVT不仅在中国大陆沪深股市表现最为可靠,而且在其他市场也表现出同样的可靠性。     

Dynamic relations of uncertainty expectations: a conditional assessment of implied volatility indices

This study examines the spillover effects in international financial markets with respect to implied volatility indices. The use of the latter as the basis of integration analysis means that we test market participants’ expectations and not the actual price fluctuations. The empirical analysis, which includes all publicly available implied volatility indices, employs the dynamic conditional correlation model of Engle (2002) and its findings suggest that there is significant integration of investors’ expectations about future uncertainty. Furthermore, by accounting for the dynamic volatility of implied volatility inter-dependencies, we are able to reveal possible shifts in conditional correlations of market expectations over time. More specifically, our findings show a slight increase in the conditional correlations for all the volatility indices under review over the years and prove that in periods of turbulence in the financial markets the conditional correlations across implied volatility indices increase.     

Backtesting extreme value theory models of expected shortfall

We use stock market data to analyze the quality of alternative models and procedures for forecasting expected shortfall (ES) at different significance levels. We compute ES forecasts from conditional models applied to the full distribution of returns as well as from models that focus on tail events using extreme value theory (EVT). We also apply the semiparametric filtered historical simulation (FHS) approach to ES forecasting to obtain 10-day ES forecasts. At the 10-day horizon we combine FHS with EVT. The performance of the different models is assessed using six different ES backtests recently proposed in the literature. Our results suggest that conditional EVT-based models produce more accurate 1-day and 10-day ES forecasts than do non-EVT based models. Under either approach, asymmetric probability distributions for return innovations tend to produce better forecasts. Incorporating EVT in parametric or semiparametric approaches also improves ES forecasting performance. These qualitative results are also valid for the recent crisis period, even though all models then underestimate the level of risk. FHS narrows the range of numerical forecasts obtained from alternative models, thereby reducing model risk. Combining EVT and FHS seems to be best approach for obtaining accurate ES forecasts.     

El efecto de la volatilidad del peso mexicano en los rendimientos y riesgo de la Bolsa Mexicana de Valores

The effect of heavy tails due to rare events and different levels of asymmetry associated with high volatility clustering in the emerging financial markets requires sophisticated models for statistical modelling of such stylized facts. This article applies extreme value theory (EVT) to quantify tail risk on the daily returns of Mexican stock market under aggregation of foreign exchange rate risk from January 1971 to December 2010. This study focuses on the maximum-block method and generalized extreme value distribution (GEVD) to model the asymptotic behavior of extreme returns in US dollars. The empirical results show that EVT-Based VaR measured at high confidence levels performs better than simulation historical and delta-normal VaR models on capturing fat-tails in the returns of highly volatile stock markets. Additionally, international investors holding long positions in Mexican stock market are more prone to experience larger potential losses than investors with short positions during local currency depreciation and financial crisis periods.     

Conditional VaR using EVT - Towards a planned margin scheme

This paper constructs a robust Value-at-Risk (VaR) measure for the Indian stock markets by combining two well-known facts about equity return time series — dynamic volatility resulting in the well-recognized phenomenon of volatility clustering, and non-normality giving rise to fat tails of the return distribution. While the phenomenon of volatility dynamics has been extensively studied using GARCH model and its many relatives, the application of Extreme Value Theory (EVT) is relatively recent in tracking extreme losses in the study of risk measurement. There are recent applications of Extreme Value Theory to estimate the unexpected losses due to extreme events and hence modify the current methodology of VaR. Extreme value theory (EVT) has been used to analyze financial data showing clear non-normal behavior. We combine the two methodologies to come up with a robust model with much enhanced predictive abilities. A robust model would obviate the need for imposing special ad hoc margins by the regulator in times of extreme volatility. A rule based margin system would increase efficiency of the price discovery process and also the market integrity with the regulator no longer seen as managing volatility.     

Volatility spillovers between oil prices and stock sector returns: Implications for portfolio management

In this article we take a recent generalized VAR-GARCH approach to examine the extent of volatility transmission between oil and stock markets in Europe and the United States at the sector-level. The empirical model is advantageous in that it typically allows simultaneous shock transmission in the conditional returns and volatilities. Insofar as volatility transmission across oil and stock sector markets is a crucial element for portfolio designs and risk management, we also analyze the optimal weights and hedge ratios for oil-stock portfolio holdings with respect to the results. Our findings point to the existence of significant volatility spillover between oil and sector stock returns. However, the spillover is usually unidirectional from oil markets to stock markets in Europe, but bidirectional in the United States. Our back-testing procedures, finally, suggest that taking the cross-market volatility spillovers estimated from the VAR-GARCH models often leads to diversification benefits and hedging effectiveness better than those of commonly used multivariate volatility models such as the CCC-GARCH of Bollerslev (1990), the diagonal BEKK-GARCH of Engle and Kroner (1995) and the DCC-GARCH of Engle (2002).     

Evaluating the impact of market reforms on Value-at-Risk forecasts of Chinese A and B shares

This paper analyses the time-varying conditional correlations between Chinese A and B share returns using the Dynamic Conditional Correlation (DCC) model of Engle [Engle, R.F. (2002), “Dynamic Conditional Correlation: A Simple Class of Multivariate Generalized Autoregressive Conditional Heteroskedasticity Models”, Journal of Business and Economic Statistics, 20, 339–350.]. The results show that the conditional correlations increased substantially following the B share market reform, whereby Chinese investors were permitted to purchase B shares. However, this increase in correlations was found to have begun well before the B share market reform. This result has significant implication relating to the structure of the information flow between the markets for the two classes of shares. Value-at-Risk (VaR) threshold forecasts are used to analyse the importance of accommodating dynamic conditional correlations between Chinese A and B shares, and thus reflects the impact of the changes in information flow on the risk evaluation of a diversified portfolio. The competing VaR forecasts are analysed using the Unconditional Coverage, Serial Independence and Conditional Coverage tests of Christoffersen [Christoffersen (1998), “Evaluating Interval Forecasts”, International Economic Review, 39, 841–862], and the Time Until First Failure Test of Kupiec [Kupiec, P.H., (1995), “Techniques for Verifying the Accuracy of Risk Measurements Models”, Journal of Derivatives, 73–84]. The results offer mild support for the DCC model over its constant conditional correlation counterpart.     

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.     

Effectiveness of copula-extreme value theory in estimating value-at-risk: empirical evidence from Asian emerging markets

A traditional Monte Carlo simulation using linear correlations induces estimation bias in measuring portfolio value-at-risk (VaR), due to the well-documented existence of fat-tail, skewness, truncations, and non-linear relations in return distributions. In this paper, we consider the above issues in modeling VaR and evaluate the effectiveness of using copula-extreme-value-based semiparametric approaches. To assess portfolio risk in six Asian markets, we incorporate a combination of extreme value theory (EVT) and various copulas to build joint distributions of returns. A backtesting analysis using a Monte Carlo VaR simulation suggests that the Clayton copula-EVT evinces the best performance regardless of the shapes of the return distributions, and that in general the copulas with the EVT provide better estimations of VaRs than the copulas with conventionally employed empirical distributions. These findings still hold in conditional-coverage-based backtesting. These findings indicate the economic significance of incorporating the down-side shock in risk management.     

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.     

A semi-parametric approach to risk management

Abstract

The benchmark theory of mathematical finance is the Black–Scholes–Merton (BSM) theory, based on Brownian motion as the driving noise process for stock prices. Here the distributions of financial returns of the stocks in a portfolio are multivariate normal. Risk management based on BSM underestimates tails. Hence estimation of tail behaviour is often based on extreme value theory (EVT). Here we discuss a semi-parametric replacement for the multivariate normal involving normal variance–mean mixtures. This allows a more accurate modelling of tails, together with various degrees of tail dependence, while (unlike EVT) the whole return distribution can be modelled. We use a parametric component, incorporating the mean vector μ and covariance matrix Σ, and a non-parametric component, which we can think of as a density on [0,∞), modelling the shape (in particular the tail decay) of the distribution. We work mainly within the family of elliptically contoured distributions, focusing particularly on normal variance mixtures with self-decomposable mixing distributions. We discuss efficient methods to estimate the parametric and non-parametric components of our model and provide an algorithm for simulating from such a model. We fit our model to several financial data series. Finally, we calculate value at risk (VaR) quantities for several portfolios and compare these VaRs to those obtained from simple multivariate normal and parametric mixture models.     

Estimation of tail‐related risk measures in the Indian stock market: An extreme value approach

The purpose of the study is to estimate tail-related risk measures using extreme value theory (EVT) in the Indian stock market. The study employs a two stage approach of conditional EVT originally proposed by McNeil and Frey (2000) to estimate dynamic Value at Risk (VaR) and expected shortfall (ES). The dynamic risk measures have been estimated for different percentiles for negative and positive returns. The estimates of risk measures computed under different quantile levels exhibit strong stability across a range of the selected thresholds, implying the accuracy and reliability of the estimated quantile based risk measures.     

Ethical and unethical investments under extreme market conditions

This study investigates the time-varying volatility and risk measures of ethical and unethical investments. We apply the Bayesian Markov-switching generalized autoregressive conditional heteroscedasticity (MS-GARCH) approach to compute the value-at-risk (VaR) and expected shortfall (ES) of ethical and unethical indices returns, which allows for detecting the differences between ethical and unethical investments. The innovative finding of our study is that ethical investments are less affected during global financial crises compared with unethical and conventional investments. The policy implication of this study is that investors should consider ethical investments as a hedging asset for their portfolios during extreme market conditions.     

Realizing the extremes: Estimation of tail-risk measures from a high-frequency perspective

This article applies realized volatility forecasting to Extreme Value Theory (EVT). We propose a two-step approach where returns are first pre-whitened with a high-frequency based volatility model, and then an EVT based model is fitted to the tails of the standardized residuals. This realized EVT approach is compared to the conditional EVT of McNeil & Frey (2000). We assess both approaches' ability to filter the dependence in the extremes and to produce stable out-of-sample VaR and ES estimates for one-day and ten-day time horizons. The main finding is that GARCH-type models perform well in filtering the dependence, while the realized EVT approach seems preferable in forecasting, especially at longer time horizons.     

Industry characteristics and financial risk contagion

This article proposes a new measure of tail risk spillover: the conditional coexceedance (CCX), defined as the number of joint occurrences of extreme negative returns in an industry, conditional on an extreme negative return in the financial sector. The empirical application provides evidence of significant volatility and tail risk spillovers from the financial sector to many real sectors in the U.S. economy from 2001 to 2011. These spillovers increase in crisis periods. The CCX in a given sector is positively related to its amount of debt financing and negatively related to its valuation and investment. Therefore, real economy sectors—which require relatively high debt financing and whose value and investment activity are relatively lower—are prime candidates for stock price volatility and depreciation in the wake of a financial sector crisis. Evidence also suggests that the higher the industry’s degree of competition, the stronger the tail risk spillover from the financial sector.     

Extreme risk spillover network: application to financial institutions

Using the CAViaR tool to estimate the value-at-risk (VaR) and the Granger causality risk test to quantify extreme risk spillovers, we propose an extreme risk spillover network for analysing the interconnectedness across financial institutions. We construct extreme risk spillover networks at 1% and 5% risk levels (which we denote 1% and 5% VaR networks) based on the daily returns of 84 publicly listed financial institutions from four sectors—banks, diversified financials, insurance and real estate—during the period 2006–2015. We find that extreme risk spillover networks have a time-lag effect. Both the static and dynamic networks show that on average the real estate and bank sectors are net senders of extreme risk spillovers and the insurance and diversified financials sectors are net recipients, which coheres with the evidence from the recent global financial crisis. The networks during the 2008–2009 financial crisis and the European sovereign debt crisis exhibited distinctive topological features that differed from those in tranquil periods. Our approach supplies new information on the interconnectedness across financial agents that will prove valuable not only to investors and hedge fund managers, but also to regulators and policy-makers.     

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.     

Return and Volatility Spillovers Between Large and Small Stocks in the UK

Abstract:  This paper investigates return and volatility spillover effects between the FTSE 100, FTSE 250 and FTSE Small Cap equity indices using the multivariate GARCH framework. We find that return and volatility transmission mechanisms between large and small stocks in the UK are asymmetric. In particular, there are significant spillover effects in both returns and volatility from the portfolios of larger stocks to the portfolios of smaller stocks. For volatility, there is also evidence of limited feedback from the portfolios of smaller stocks to the portfolios of larger stocks, although sub-period analysis suggests that this is to some extent period-specific. Simulation evidence shows that non-synchronous trading potentially explains some, but not all, of the spillover effects in returns, and that it explains none of the spillover effects in volatility. These results are consistent with a market in which information is first incorporated into the prices of large stocks before being impounded into the prices of small stocks.     

Forecasting daily variability of the S&P 100 stock index using historical,realised and implied volatility measurements

The increasing availability of financial market data at intraday frequencies has not only led to the development of improved volatility measurements but has also inspired research into their potential value as an information source for volatility forecasting. In this paper, we explore the forecasting value of historical volatility (extracted from daily return series), of implied volatility (extracted from option pricing data) and of realised volatility (computed as the sum of squared high frequency returns within a day). First, we consider unobserved components (UC-RV) and long memory models for realised volatility which is regarded as an accurate estimator of volatility. The predictive abilities of realised volatility models are compared with those of stochastic volatility (SV) models and generalised autoregressive conditional heteroskedasticity (GARCH) models for daily return series. These historical volatility models are extended to include realised and implied volatility measures as explanatory variables for volatility. The main focus is on forecasting the daily variability of the Standard & Poor's 100 (S&P 100) stock index series for which trading data (tick by tick) of almost 7 years is analysed. The forecast assessment is based on the hypothesis of whether a forecast model is outperformed by alternative models. In particular, we will use superior predictive ability tests to investigate the relative forecast performances of some models. Since volatilities are not observed, realised volatility is taken as a proxy for actual volatility and is used for computing the forecast error. A stationary bootstrap procedure is required for computing the test statistic and its p-value. The empirical results show convincingly that realised volatility models produce far more accurate volatility forecasts compared to models based on daily returns. Long memory models seem to provide the most accurate forecasts.