A semiparametric approach to value-at-risk,expected shortfall and optimum asset allocation in stock–bond portfolios

This paper investigates stock–bond portfolios' tail risks such as value-at-risk (VaR) and expected shortfall (ES), and the way in which these measures have been affected by the global financial crisis. The semiparametric t-copulas adequately model stock–bond returns joint distributions of G7 countries and Australia. Empirical results show that the (negative) weak stock–bond returns dependence has increased significantly for seven countries after the crisis, except for Italy. However, both VaR and ES have increased for all eight countries. Before the crisis, the minimum portfolio VaR and ES were achieved at an interior solution only for the US, the UK, Australia, Canada and Italy. After the crisis, the corner solution was found for all eight countries. Evidence of “flight to quality” and “safety first” investor behaviour was strong, after the global financial crisis. The semiparametric t-copula adequately forecasts the outer-sample VaR. These findings have implications for global financial regulators and the Basel Committee, whose central focus is currently on increasing the capital requirements as a consequence of the recent global financial crisis.     

Long-Memory,Asymmetry and Fat-Tailed GARCH Models in Value-at-Risk Estimation: Empirical Evidence from the Global Real Estate Markets

In this paper, we address the question of whether long memory, asymmetry, and fat-tails in global real estate markets volatility matter when forecasting the two most popular measures of risk in financial markets, namely Value-at-risk (VaR) and Expected Shortfall (ESF), for both short and long trading positions. The computations of both VaR and ESF are conducted with three long memory GARCH-class models including the Fractionally Integrated GARCH (FIGARCH), Hyperbolic GARCH (HYGARCH), and Fractionally Integrated Asymmetric Power ARCH (FIAPARCH). These models are estimated under three alternative innovation’s distributions: normal, Student, and skewed Student. To test the efficacy of the forecast, we employ various backtesting methodologies. Our empirical findings show that considering for long memory, fat-tails, and asymmetry performs better in predicting a one-day-ahead VaR and ESF for both short and long trading positions. In particular, the forecasting ability analysis points out that the FIAPARCH model under skewed Student distribution turns out to improve substantially the VaR and ESF forecasts. These results may have several potential implications for the market participants, financial institutions, and the government.

     

A semi-parametric approach to estimating the operational risk and Expected Shortfall

In this article, we propose improvements to the peak-over-threshold (POT) method and apply this improved method for modelling US business operational losses and estimating operational risks (ORs). In the widely used traditional POT method, the generalized Pareto distribution (GPD) is fitted to severity losses, while an empirical distribution is fitted to small to medium losses. Then, the Expected Loss and the 99.9% operational value-at-risk (OpVaR) are estimated. Additionally, the Expected Shortfall (ES) – a coherent risk measure – is estimated in this article as an alternative to OpVaR. These risk measures constitute the levels of regulatory and economic capitals to cover risks. With the improved POT method, the risks can be estimated more accurately than with the traditional POT method. The results indicate that the OpVaR are much lower than the ES and that the larger the tail losses the greater the difference between these two risk measures. Our findings imply that the ES would provide higher levels of capitals to cover risks than would the OpVaR, particularly during crises, and they have implications for the efficient OR management and regulators.     

Conditional volatility and correlations of weekly returns and the VaR analysis of 2008 stock market crash

Modelling of conditional volatilities and correlations across asset returns is an integral part of portfolio decision making and risk management. Over the past three decades there has been a trend towards increased asset return correlations across markets, a trend which has been accentuated during the recent financial crisis. We shall examine the nature of asset return correlations using weekly returns on futures markets and investigate the extent to which multivariate volatility models proposed in the literature can be used to formally characterize and quantify market risk. In particular, we ask how adequate these models are for modelling market risk at times of financial crisis. In doing so we consider a multivariate t version of the Gaussian dynamic conditional correlation (DCC) model proposed by Engle (2002), and show that the t-DCC model passes the usual diagnostic tests based on probability integral transforms, but fails the value at risk (VaR) based diagnostics when applied to the post 2007 period that includes the recent financial crisis.     

Score-driven Markov-switching EGARCH models: an application to systematic risk analysis

We introduce new Markov-switching (MS) dynamic conditional score (DCS) exponential generalized autoregressive conditional heteroscedasticity (EGARCH) models, to be used by practitioners for forecasting value-at-risk (VaR) and expected shortfall (ES) in systematic risk analysis. We use daily log-return data from the Standard & Poor’s 500 (S&P 500) index for the period 1950–2016. The analysis of the S&P 500 is useful, for example, for investors of (i) well-diversified US equity portfolios; (ii) S&P 500 futures and options traded at Chicago Mercantile Exchange Globex; (iii) exchange traded funds (ETFs) related to the S&P 500. The new MS DCS-EGARCH models are alternatives to of the recent MS Beta-t-EGARCH model that uses the symmetric Student’s t distribution for the error term. For the new models, we use more flexible asymmetric probability distributions for the error term: Skew-Gen-t (skewed generalized t), EGB2 (exponential generalized beta of the second kind) and NIG (normal-inverse Gaussian) distributions. For all MS DCS-EGARCH models, we identify high- and low-volatility periods for the S&P 500. We find that the statistical performance of the new MS DCS-EGARCH models is superior to that of the MS Beta-t-EGARCH model. As a practical application, we perform systematic risk analysis by forecasting VaR and ES.

Abbreviation Single regime (SR); Markov-switching (MS); dynamic conditional score (DCS); exponential generalized autoregressive conditional heteroscedasticity (EGARCH); value-at-risk (VaR); expected shortfall (ES); Standard & Poor's 500 (S&P 500); exchange traded funds (ETFs); Skew-Gen-t (skewed generalized t); EGB2 (exponential generalized beta of the second kind); NIG (normal-inverse Gaussian); log-likelihood (LL); standard deviation (SD); partial autocorrelation function (PACF); likelihood-ratio (LR); ordinary least squares (OLS); heteroscedasticity and autocorrelation consistent (HAC); Akaike information criterion (AIC); Bayesian information criterion (BIC); Hannan-Quinn criterion (HQC).     

How does issuing contingent convertible bonds improve bank's solvency? A Value-at-Risk and Expected Shortfall approach

This paper examines how issuing an innovative financial instrument called contingent convertible bond (CoCo) may enhance bank's solvency in comparison to issuing a conventional bond. CoCos convert automatically into common equity or have a principal write-down when bank's regulatory capital fails to meet a predetermined level. They have been invented and put into legislation with an objective to absorb losses thus preventing institutions from bankruptcy. From the standpoint of an issuer CoCos bring about two counter effects regarding his solvency: on one hand they recapitalize a bank approaching insolvency on the other hand CoCos pay much higher coupon comparing to conventional bonds. In our model a bank has two funding alternatives: either to issue CoCos or conventional bonds. We measure issuer's default risk using the concept of Value-at-Risk (VaR) and Expected Shortfall (ES). We conclude that CoCos have the potential to strengthen the resilience of the issuer on the condition that the probability of conversion triggering is higher than the VaR's significance level. Our findings can be helpful to the policymakers and banks to better understand the impact of CoCos on issuer's solvency.     

Mixed data sampling expectile regression with applications to measuring financial risk

To avoid information loss or measurement error in traditional methods dealing with mixed frequency data, we develop a novel mixed data sampling expectile regression (MIDAS-ER) model to measure financial risk. We construct the MIDAS-ER model by introducing a MIDAS structure into expectile regressions. This enables us to perform an expectile regression on raw mixed frequency data directly. We apply the proposed MIDAS-ER model to estimate two popular financial risk measures, namely, Value at Risk and Expected Shortfall, with both simulated data and four stock indices, and compare the model's performance with those of several popular models. The outstanding performance of our model demonstrates that high-frequency information helps to improve the accuracy of risk measurement. In addition, the numerical results also imply that our model can be a significant tool for risk-averse investors to control risk losses and for financial institutions to implement robust risk management.     

Energy portfolio risk management using time-varying extreme value copula methods

This work is concerned with the statistical modeling of the dependence structure between three energy commodity markets (WTI crude oil, natural gas and heating oil) using the concept of copulas and proposes a method for estimating the Value at risk (VaR) of energy portfolio based on the combination of time series models with models of the extreme value theory before fitting a copula. Each return series is modeled by AR-(FI) GARCH univariate model. Then, we fit the GPD distribution to the tails of the residuals to model marginal residuals distributions. The extreme value copula to the iid residuals is fitted and we simulate from it to construct N portfolios and estimate VaR. As a first step, the method is applied to a two-dimensional energy portfolio. In second step, we extend method in trivariate context to measure VaR of three-dimensional energy portfolio. Dependences between residuals are modeled using a trivariate nested Gumbel copulas. Methods proposed are compared with various univariate and multivariate conventional VaR methods. The reported results demonstrate that GARCH-t, conditional EVT and FIGARCH extreme value copula methods produce acceptable estimates of risk both for standard and more extreme VaR quantiles. Generally, copula methods are less accurate compared with their predictive performances in the case of portfolio composed of exchange market indices.     

两类混合藤Copula模型的比较研究——基于外汇资产投资组合VaR的研究

本文以中国外汇市场上四种主要外汇资产的投资组合作为研究对象,基于Pair Copula高维建模思想,分别建立了两类能真实反映资产组合相关结构差异性的混合藤Copula模型,即混合C藤和混合D藤Copula模型。两类混合藤Copula模型,对传统的藤Copula模型作了进一步的改进,是通过一定的选择标准,确定模型中每个Pair Copula函数的最优函数族,这样可以使得所建立的模型既能考虑资产组合维数的影响,又能捕捉到组合内部各资产相关结构的差异性。为了得到较优的风险分析模型,在实证研究中,将两类模型在资产组合VaR计算精度方面进行比较。     

Forecasting the volatility of S&P depositary receipts using GARCH-type models under intraday range-based and return-based proxy measures

We employ four various GARCH-type models, incorporating the skewed generalized t (SGT) errors into those returns innovations exhibiting fat-tails, leptokurtosis and skewness to forecast both volatility and value-at-risk (VaR) for Standard & Poor's Depositary Receipts (SPDRs) from 2002 to 2008. Empirical results indicate that the asymmetric EGARCH model is the most preferable according to purely statistical loss functions. However, the mean mixed error criterion suggests that the EGARCH model facilitates option buyers for improving their trading position performance, while option sellers tend to favor the IGARCH/EGARCH model at shorter/longer trading horizon. For VaR calculations, although these GARCH-type models are likely to over-predict SPDRs' volatility, they are, nevertheless, capable of providing adequate VaR forecasts. Thus, a GARCH genre of model with SGT errors remains a useful technique for measuring and managing potential losses on SPDRs under a turbulent market scenario.     

股票资产组合VaR研究基于混合D藤Copula模型

本文在对上证市场五种股票资产组合的风险分析中以VaR作为风险度量指标,采用基于Pair Copula高维建模理论的混合D藤Copula模型,建立了反应多个资产组合相关结构的联合分布模型。该模型对传统D藤Copula建模方法作了进一步的改进,通过一定的选择标准,确定了D藤中每个Pair Copula函数的最优函数族,这样使得所建立的模型不仅考虑到了资产维数的影响,而且还能捕捉到组合内部因子间相关结构的差异性,从而改进后的模型能更好地描述资产组合的相关结构,并且能更精确地反映资产组合收益的实际分布。最后,以混合D藤Copula模型为基础,利用Monte Carlo方法计算了上证市场五种股票资产组合的VaR,并通过实证研究进一步证明了该模型的有效性。     

A new approach to modeling co-movement of international equity markets: evidence of unconditional copula-based simulation of tail dependence

Analyzing equity market co-movements is important for risk diversification of an international portfolio. Copulas have several advantages compared to the linear correlation measure in modeling co-movement. This paper introduces a copula ARMA-GARCH model for analyzing the co-movement of international equity markets. The model is implemented with an ARMA-GARCH model for the marginal distributions and a copula for the joint distribution. After goodness of fit testing, we find that the Student’s t copula ARMA(1,1)-GARCH(1,1) model with fractional Gaussian noise is superior to alternative models investigated in our study where we model the simultaneous co-movement of nine international equity market indexes. This model is also suitable for capturing the long-range dependence and tail dependence observed in international equity markets. Rachev’s research was supported by grants from Division of Mathematical, Life and Physical Science, College of Letters and Science, University of California, Santa Barbara, and the Deutschen Forschungsgemeinschaft (DFG). Sun’s research was supported by grants from the Deutschen Forschungsgemeinschaft (DFG) and Chinese Government Award for Outstanding Ph.D Students Abroad 2006, No. 2006-180. Kalev’s research was supported with a NCG grant from the Faculty of Business and Economics, Monash University. Data are supplied by Securities Industry Research Center of Asia-Pacific (SIRCA) on behalf of Reuters. The constructive comments of two anonymous referees, the Associate Editor, A.S. Wirjanto, and the Editor-in-charge, Baldev Raj, are gratefully acknowledged. The reviewers and editors are not responsible for any residual errors and omissions.     

Full versus quasi MLE for ARMA-GARCH models with infinitely divisible innovations

We compare the backtesting performance of ARMA-GARCH models with the most common types of infinitely divisible innovations, fit with both full maximum likelihood estimation (MLE) and quasi maximum likelihood estimation (QMLE). The innovation types considered are the Gaussian, Student’s t, α-stable, classical tempered stable (CTS), normal tempered stable (NTS) and generalized hyperbolic (GH) distributions. In calm periods of decreasing volatility, MLE and QMLE produce near identical performance in forecasting value-at-risk (VaR) and conditional value-at-risk (CVaR). In more volatile periods, QMLE can actually produce superior performance for CTS, NTS and α-stable innovations. While the t-ARMA-GARCH model has the fewest number of VaR violations, rejections by the Kupeic and Berkowitz tests suggest excessively large forecasted losses. The α-stable, CTS and NTS innovations compare favourably, with the latter two also allowing for option pricing under a single market model.     

Measuring and Optimizing Portfolio Credit Risk: A Copula‐based Approach*

In this work, we present a methodology for measuring and optimizing the credit risk of a loan portfolio taking into account the non‐normality of the credit loss distribution. In particular, we aim at modelling accurately joint default events for credit assets. In order to achieve this goal, we build the loss distribution of the loan portfolio by Monte Carlo simulation. The times until default of each obligor in portfolio are simulated following a copula‐based approach. In particular, we study four different types of dependence structure for the credit assets in portfolio: the Gaussian copula, the Student's t‐copula, the grouped t‐copula and the Clayton n‐copula (or Cook–Johnson copula). Our aim is to assess the impact of each type of copula on the value of different portfolio risk measures, such as expected loss, maximum loss, credit value at risk and expected shortfall. In addition, we want to verify whether and how the optimal portfolio composition may change utilizing various types of copula for describing the default dependence structure. In order to optimize portfolio credit risk, we minimize the conditional value at risk, a risk measure both relevant and tractable, by solving a simple linear programming problem subject to the traditional constraints of balance, portfolio expected return and trading. The outcomes, in terms of optimal portfolio compositions, obtained assuming different default dependence structures are compared with each other. The solution of the risk minimization problem may suggest us how to restructure the inefficient loan portfolios in order to obtain their best risk/return profile. In the absence of a developed secondary market for loans, we may follow the investment strategies indicated by the solution vector by utilizing credit default swaps.     

Fuzzy value-at-risk and expected shortfall for portfolios with heavy-tailed returns

This paper is concerned with linear portfolio value-at-risk (VaR) and expected shortfall (ES) computation when the portfolio risk factors are leptokurtic, imprecise and/or vague. Following Yoshida (2009), the risk factors are modeled as fuzzy random variables in order to handle both their random variability and their vagueness. We discuss and extend the Yoshida model to some non-Gaussian distributions and provide associated ES. Secondly, assuming that the risk factors' degree of imprecision changes over time, original fuzzy portfolio VaR and ES models are introduced. For a given subjectivity level fixed by the investor, these models allow the computation of a pessimistic and an optimistic estimation of the value-at-risk and of the expected shortfall. Finally, some empirical examples carried out on three portfolios constituted by some chosen French stocks, show the effectiveness of the proposed methods.     

中国股票市场ES和VaR的实证比较分析

以我国股票收益率为研究对象,分别在正态分布和非正态稳定分布条件下对ES和VaR的凸性、次可加性和有效性进行了实证比较分析,发现:在非正态稳定分布条件下VaR不满足凸性和次可加性,ES满足凸性和次可加性,在正态分布条件下VaR和ES都满足凸性和次可加性;在两种分布条件下ES的有效性都高于VaR的有效性,而在非正态稳定分布条件下ES的优势更加明显。由于本文的收益率分布拟合检验表明我国的股票收益率服从非正态稳定分布,所以在我国股票市场上ES是比VaR更好的风险度量。     

Analyzing the dependence structure of various sectors in the Brazilian market: A Pair Copula Construction approach

In this paper we estimate the dependence structure between economic sectors in the Brazilian financial market through Pair Copula Construction. We use daily data from indices which represent telecommunications, energy, industrials, consumer, financial, basic materials and real estate sectors in BM&F/Bovespa. Results indicate predominance of student's t copula in structure. BB1, BB7, BB8, Frank and Joe copulas also fit into some relationships. Regarding dependence, tail measures obtain relevant values in most relationships. Lower tail dependence exceeds absolute, measured by Kendall's Tau, and upper tail in many cases, reflecting the asymmetry in some relationships. Further, in order to give robustness to these results, we forecast daily Value at Risk, considering distinct significance levels, of a portfolio composed of studied sectors through the estimated structure. Results allow one to conclude that VaR predictions are correct. These results permit business industry participants to construct portfolios with assets of these sectors under a proper diversification structure. Moreover, from an international point of view, investors who are interested in diversification could perform more sophisticated strategies in this country rather than simply trading the index.     

Tail risk in emerging markets of Southeastern Europe

This article examines the left-tail behaviour of returns on stocks in Southeastern Europe (SEE). We apply conditional extreme value theory (EVT) approach on daily returns of six stock market indices from SEE between 2004 and 2013. Predictive performance of value-at-risk (VaR) and expected shortfall (ES) based on EVT is compared against several alternatives, such as historical simulation and analytical approach based on GARCH with a single conditional distribution. Model backtesting with daily returns shows that EVT-based models provide more reliable VaR and ES forecasts than the alternative models in all six markets. Unlike the alternatives, the EVT-based models cannot be rejected as VaR confidence level is increased. This emphasizes the importance of extreme events in SEE markets and indicates that the ability of a model to capture volatility clustering accurately is not sufficient for a correct assessment of risk in these markets.     

Forecasting the volatility of S&P depositary receipts using GARCH-type models under intraday range-based and return-based proxy measures

We employ four various GARCH-type models, incorporating the skewed generalized t (SGT) errors into those returns innovations exhibiting fat-tails, leptokurtosis and skewness to forecast both volatility and value-at-risk (VaR) for Standard & Poor's Depositary Receipts (SPDRs) from 2002 to 2008. Empirical results indicate that the asymmetric EGARCH model is the most preferable according to purely statistical loss functions. However, the mean mixed error criterion suggests that the EGARCH model facilitates option buyers for improving their trading position performance, while option sellers tend to favor the IGARCH/EGARCH model at shorter/longer trading horizon. For VaR calculations, although these GARCH-type models are likely to over-predict SPDRs' volatility, they are, nevertheless, capable of providing adequate VaR forecasts. Thus, a GARCH genre of model with SGT errors remains a useful technique for measuring and managing potential losses on SPDRs under a turbulent market scenario.