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.     

“金砖四国”股票市场间相依结构分析

金融市场间的相依关系及其结构分析是金融风险测度、资产组合管理等金融理论和实务中的一个重要问题,而基于线性相关系数是难以正确刻画非线性条件下金融资产间的相关结构,特别是尾相依关系。为研究"金砖四国"新兴股票市场间的相依结构,文章构建了一个混合Copula模型,对"金砖四国"股票市场间的相依结构进行建模分析,并将结果与单一阿基米德Copula模型进行了比较,表明混合Copula模型既能保留单一Copula模型的特性,更能灵活、全面地刻画变量间的相依关系。实证研究的结果表明:在样本数据期间,"金砖四国"股票市场间存在非对称相关关系,相依结构上存在差异;中国上证综指波动较为剧烈,与其它三国指数间以下尾相依为主;巴西、印度、俄罗斯三国股指间的相依结构相似,联动相关性相近,并无显著的下尾相依关系。     

How are Africa's emerging stock markets related to advanced markets? Evidence from copulas

The finance literature provides ample evidence that diversification benefits hinges on dependence between assets returns. A notable feature of the recent financial crisis is the extent to which assets that had hitherto moved mostly independently suddenly moved together resulting in joint losses in most advanced markets. This provides grounds to uncover the relative potential of African markets to provide diversification benefits by means of their correlation with advanced markets. Therefore, we examine the dependence structure between advanced and emerging African stock markets using copulas. Several findings are documented. First, dependence is time-varying and weak for most African markets, except South Africa. Second, we find evidence of asymmetric dependence, suggesting that stock return comovement varies in bearish and bullish markets. Third, extreme downward stock price movements in the advanced markets do not have significant spillover effects on Africa’s emerging stock markets. Our results, implying that African markets, with the exception of South Africa, are immune to risk spillover from advanced markets, improves the extant literature and have implications for portfolio diversification and risk management.     

Quantile spillovers and dependence between Bitcoin,equities and strategic commodities

This paper has two aims. We first examine the dynamic spillovers between Bitcoin and 12 developed equities, gold, and crude oil for different market conditions using a Bayesian Time-Varying Parameter Vector Autoregressive (TVP-VAR) model with daily spot prices. Our econometric approach enables us to capture the left and right tails as well as the shoulders of the return distribution corresponding to volatility spillovers under the bear, normal, and bull market states among these financial assets. We quantify and trace the dependence and directional predictability from Bitcoin to other assets using the sample cross-quantilogram. Our key findings offer convincing evidence of time variation in the level of volatility. Spillovers between Bitcoin and other financial assets intensify during extreme global market conditions. Secondly, results from the cross-quantilogram indicate strong dependence and positive directional predictability between Bitcoin and most equities and crude oil when market returns are bullish. However, during the bearish market period, there is negative dependence and predictability from Bitcoin to stocks in Finland, the Netherlands, the U.S.A, and the crude oil market only. This implies that Bitcoin can act as a hedge to stocks in Finland, the Netherlands, the U.S.A, and the crude oil market. However, insignificant dependence and directional predictability from Bitcoin to the remaining assets indicate that Bitcoin may act as a safe-haven to these assets during bearish markets. Our findings hold important implications for both international investors and portfolio managers who consider Bitcoin as part of their portfolio diversification and other investment strategies.     

Global and Extreme Dependence Between Investor Sentiment and Stock Returns in European Markets

The authors investigate the global and extreme dependence structure between investor sentiment and stock returns in 7 European stock markets (Belgium, France, Germany, Greece, the Netherlands, Portugal, and the UK), over the period 1985–2015. Global dependence refers to the correlation of changes in sentiment and stock returns over the whole range of these 2 variables, and extreme dependence refers to the local correlation of high (i.e. asymptotic) changes in sentiment and high stock returns. Using copula models and a bootstrap procedure, 6 statistical tests are performed for this purpose. Among the results of the tests, the authors highlight those that provide evidence of contemporaneous lower extreme dependence and contemporaneous upper extreme independence between sentiment and returns. As policy implications, these results suggest that financial stability can be promoted if regulators consider the impact of their decisions on investor sentiment. Also, the results seem to support the arguments in favor of short selling ban during turmoil periods. Finally, overall, the results are relevant for both investors and regulators and reinforce the importance of considering investor sentiment to better understand the behavior of financial markets.     

STABILITY OF THAI BAHT: TALES FROM THE TAILS

We demonstrate how the EVT‐based signalling approach for currency crises can be applied to an individual country with a small sample size. Using Thai historical data, first, we study the tail characteristics of the distributions of two Thai baht instability measures and 21 economic fundamentals. Then, we test asymptotic dependence between the currency instability measures and lagged economic fundamentals. Empirically, we find that the distributions of both currency instability measures and economic variables are heavy tailed. Assuming a normal distribution for the variables tends to underestimate the probability of extreme events. Furthermore, most of the economic variables which are usually used as signalling indicators for currency crises are asymptotically independent of the currency instability measures. Signals issued by these variables are thus not reliable. Nevertheless, the non‐parametric EVT approach facilitates the selection of economic indicators with credible signals and high crisis prediction success.     

Modeling the dependence structure between default risk premium,equity return volatility and the jump risk: Evidence from a financial crisis

This paper investigates the dependence structure between default risk premium, equity return volatility and jump risk in the equity market before and during the subprime crisis. Using iTraxx CDS index spreads from Japanese and Australian markets, the paper models the different relationships that can exist in different ranges of behavior. We consider several Archimedean copula models with different tail dependence structures, namely, Gumbel, Clayton, Frank, AMH and Joe copulas. Although the dramatic change in the levels of the iTraxx CDS index, we find strong evidence that the dependence structure between CDS and stock market conditions is asymmetric and orienting toward the upper side. In addition, we find that the Japanese CDS market is more sensitive to the stock return volatility than the jump risk and the magnitude of this sensitivity is related to the market circumstances. However, Australian CDS market is more sensitive to the jump risk than stock return volatility before and during the financial crisis. This result has important implications for both global financial stability and default risk management. Specifically, the heterogeneity of markets, coupled with the diversity in the risk exposures cause the default risk premium and equity markets to exhibit different levels of sensitivity.     

An empirical re-examination of extreme tail behavior: testing the assumptions of the power laws and the generalized Pareto distribution on the financial series

This study investigates whether the power laws and the associated generalized Pareto distribution (GPD) exist in the extreme tail behavior of financial return series. We include 10 series of five major financial categories over the period 1971–2018 for empirical analysis. For the former assumption, we test three representative power-law distributions. For the latter, we employ an innovative bootstrap goodness-of-fit test of GPD modeling. We also discuss the relationship between both assumptions. The empirical outcomes indicate that both assumptions do not necessarily hold for all tail series due to the outlying observations. The rejection of the power laws assumption leads to the rejection of the GPD assumption. This rejection does not promise the non-rejection of power laws either. However, the non-rejection of either assumption does not imply non-rejection of the other assumption. Power-law distribution and exponential distribution outperform log-normal distribution in tail fitting. GPD fits better at the 1% quantile level than at the 5% level. Overall, we need to acknowledge the considerable gap between the goodness-of-fit testing outcomes of both the power laws and GPD assumptions.     

Time-varying dependence structures of equity markets of China,ASEAN and the USA

Popular time-varying Copulas are used to analyse the dependence structure between the CSI 300 index return, the S&P 300 index return and the Association of South East Asian Nations (ASEAN) 80 index return. Results show that these three types of stock index returns have obvious time-varying characteristics. The US sub-prime mortgage crisis has strengthened the correlation among the three-stock index returns, whereas the dependence between China and the ASEAN stock markets is more sensitive to the financial crisis. The time-varying features of the extreme dependence risk between China-ASEAN and China-US are very different.     

Portfolio management with tail dependence

Many publications, that treated with Portfolio Management, were devastating for all asset allocation models in the context of portfolios. The elimination of extreme events (asymmetric or tail dependence) during the portfolio construction process can reduce the skills of asset managers to reduce risk through diversification. The copula theory allows us to calculate an alternative to measure the dependence of extreme events in assets through the index lower tail dependence. We check that the strategies with tail dependence overcame Talmud rule, the Markowitz model and the model of Tu and Zhou by simulating 1,000 portfolios with 3, 5, 10 and 20 randomly selected assets from DJIA for the period 03/1990 until 12/2016. We conclude that models of tail dependence and Markowitz had more performance ex-ante than Talmud and the Tu and Zhou model for portfolios with 3, 5, 10 and 20 assets. Tail dependence models overcome Markowitz, in terms of cumulative return, in over 60% of months considered in the analysis. The results indicate that the Talmud rule should be discarded in a context of constructing portfolios with individual stocks ahead strategies with tail dependence.     

Overnight borrowing, interest rates and extreme value theory

We examine the dynamics of extreme values of overnight borrowing rates in an inter-bank money market before a financial crisis during which overnight borrowing rates rocketed up to (simple annual) 4000 percent. It is shown that the generalized Pareto distribution fits well to the extreme values of the interest rate distribution. We also provide predictions of extreme overnight borrowing rates using pre-crisis data. The examination of tails (extreme values) provides answers to such issues as to what are the extreme movements to be expected in financial markets; is there a possibility for even larger movements and, are there theoretical processes that can model the type of fat-tails in the observed data? The answers to such questions are essential for proper management of financial exposures and laying ground for regulations.     

Estimating the Marginal Contribution to Systemic Risk by A CoVaR‐model Based on Copula Functions and Extreme Value Theory

In this paper, we quantify the contribution to systemic risk of a single financial institution by utilizing a analytical framework based on the principles of Extreme Value Theory (EVT) for modelling the marginal distributions and on the properties of copula functions for describing the dependence structure between the financial system and the single financial institution. Among the several systemic risk measures proposed nowadays by academics and estimated by public data, we choose to adopt as systemic risk metric the Conditional Value‐at‐Risk (CoVaR). We select a co‐risk measure like the CoVaR because of its macro‐dimension that allows us to integrate the dependence structure of the single financial institution and of the whole financial system in the systemic risk measurement. While the copula functions have been utilized in some pioneer studies on this area, the EVT principles have not yet been implemented in such a context of systemic risk contribution measurement.     

Risk assessment of oil price from static and dynamic modelling approaches

The price gap between West Texas Intermediate (WTI) and Brent crude oil markets has been completely changed in the past several years. The price of WTI was always a little larger than that of Brent for a long time. However, the price of WTI has been surpassed by that of Brent since 2011. The new market circumstances and volatility of oil price require a comprehensive re-estimation of risk. Therefore, this study aims to explore an integrated approach to assess the price risk in the two crude oil markets through the value at risk (VaR) model. The VaR is estimated by the extreme value theory (EVT) and GARCH model on the basis of generalized error distribution (GED). The results show that EVT is a powerful approach to capture the risk in the oil markets. On the contrary, the traditional variance–covariance (VC) and Monte Carlo (MC) approaches tend to overestimate risk when the confidence level is 95%, but underestimate risk at the confidence level of 99%. The VaR of WTI returns is larger than that of Brent returns at identical confidence levels. Moreover, the GED-GARCH model can estimate the downside dynamic VaR accurately for WTI and Brent oil returns.     

Asymmetric downside and upside co-movements between stock and REIT markets

This article aims to investigate whether stock and real estate investment trust (REIT) returns have asymmetric downside and upside co-movements. The investigation is carried out by employing a mixture copula with Markov-switching coefficients for weight, upper tail dependence and lower tail dependence. Empirical result demonstrates that an asymptotic independence and a positive relationship, which has greater dependence in the left tail than in the right tail, coexist. The empirical result provides useful information for making portfolio decisions. In the independent state, the inclusion of stock and REIT indices in a portfolio builds a diversified portfolio. However, a portfolio with stock and REIT assets cannot get the benefit of risk reduction in the positive dependence state.     

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.     

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.     

股市收益率与波动性长期记忆效应的实证研究

股票市场长期记忆效应问题是近来金融实证研究的一个热点.多数的研究集中在收益率长期相关性的考察上,较少有对波动率序列的研究.然而,波动率的长期记忆性不仅会导致金融市场上的波动持久性特征,而且将对波动率的预测与衍生证券定价产生重要的影响.基于此,本文通过修正的R/S分析与ARFIMA模型对我国股市收益率及其波动性的长期相关性进行了实证研究.结果表明:中国股市具有显著的非线性特征,虽然收益率序列的自相关性较弱,但波动性序列却表现出显著的长期记忆效应.这一结论将为研究股票价格行为特征与金融经济学理论提供新的方向.     

Tail dependence and diversification benefits in emerging market stocks: an extreme value theory approach

This article examines tail dependence, the benefits of diversification and the relation between the two for emerging stock markets. We find most emerging equity markets are independent in limiting joint extremes. However, the dependence in finite levels of extremes is still much stronger than the dependence implied by multivariate normality. Therefore, simple correlation analysis can lead to gross underestimation of the chances of joint crashes in multiple markets. Assuming risk-averse investors guarding against extreme losses, diversification benefits are measured for each two-country optimal portfolio by the reduction in quantile risk measures such as value-at-risk and expected shortfall relative to an undiversified portfolio. It is shown that tail dependence measures developed from multivariate extreme value theory are negatively related to diversification benefits and more importantly can explain diversification benefits better than the correlation coefficient at the most extreme quantiles.     

Modelling the spreading process of extreme risks via a simple agent-based model: Evidence from the China stock market

This paper focuses on investigating financial asset returns' extreme risks, which are defined as the negative log-returns over a certain threshold. A simple agent-based model is constructed to explain the behavior of the market traders when extreme risks occur. We consider both the volatility clustering and the heavy tail characteristics when constructing the model. Empirical study uses the China securities index 300 daily level data and applies the method of simulated moments to estimate the model parameters. The stationarity and ergodicity tests provide evidence that the proposed model is good for estimation and prediction. The goodness-of-fit measures show that our proposed model fits the empirical data well. Our estimated model performs well in out-of-sample Value-at-Risk prediction, which contributes to the risk management.     

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.