A copula–GARCH model for macro asset allocation of a portfolio with commodities

Many authors have suggested that the mean-variance criterion, conceived by Markowitz (The Journal of Finance 7(1):77–91, 1952), is not optimal for asset allocation, because the investor expected utility function is better proxied by a function that uses higher moments and because returns are distributed in a non-Normal way, being asymmetric and/or leptokurtic, so the mean-variance criterion cannot correctly proxy the expected utility with non-Normal returns. In Riccetti (The use of copulas in asset allocation: when and how a copula model can be useful? LAP Lambert, Saarbrücken 2010), a copula–GARCH model is applied and it is found that copulas are not useful for choosing among stock indices, but can be useful in a macro asset allocation model, that is, for choosing stock and bond composition of portfolios. In this paper I apply that copula–GARCH model for the macro asset allocation of portfolios containing a commodity component. I find that the copula model appears to be useful and better than the mean-variance one for the macro asset allocation also in presence of a commodity index, even if it is not better than GARCH models on independent univariate series, probably because of the low correlation of the commodity index returns to the stock, the bond and the exchange rate returns.     

Regional and copula estimation effects on EU and US energy equity portfolios

ABSTRACT

The aim of this paper is to investigate the regional interdependence structure of energy equities in the US and in the EU. Based on weekly stock prices of 28 big energy firms in the two regions from 2008 to 2019, we compare the efficiency of using bivariate or multivariate copulas to describe the dependence structure of energy equities. Furthermore, we investigate the impact of the choice between these two methods on the performance of energy equity portfolios. Our empirical results show that multivariate copulas, such as C-Vine, allow to better describe the dependence structure of energy equities. We also find that there is a stronger and more complex dependence structure among EU energy equities than among US energy equities. Our scenario analysis also shows that the dependence structure is stronger during the GFC while being weaker during the ESDC. More importantly, the correlation matrix obtained from the multivariate copula method allows to obtain optimal mean-CVaR portfolios with a higher performance than that from the bivariate copula method. More importantly, optimal portfolios constituted with multivariate copulas allow to reduce the portfolio’s sensitivity to oil prices.     

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.     

The time-varying and asymmetric dependence between crude oil spot and futures markets: Evidence from the Mixture copula-based ARJI–GARCH model

This paper designs a Mixture copula-based ARJI–GARCH model to simultaneously investigate the dynamic process of crude oil spot and futures returns and the time-varying and asymmetric dependence between spot and futures returns. The individual behavior of each market is modeled by the ARJI–GARCH process. The time-varying and asymmetric dependence is captured by the Mixture copula which is composed of the Gumbel copula and Clayton copula. Empirical results show three important findings. First, jumping behavior is an important process for each market. Second, spot and futures returns do not have the same jump process. Third, the tail dependence between spot and futures markets is time-varying and asymmetric with the magnitude of upper tail dependence being slightly weaker than that of lower tail dependence.     

A new test procedure for the choice of dependence structure in risk measurement: application to the US and UK stock market indices

The choice of an appropriate dependence structure in modelling multivariate risks is an important issue because different tail structure embedded in copula leads to a different capital requirement for the institution. We present how to select a well-specified dependence structure to given application data. Using a simple simulation technique, we develop a statistical test to assess the adequacy of a specific dependence structure. We examine the sensitivity of risk estimates to the choice of copulas using the S&P 500 and FTSE 100 stock indices.     

Directional time-varying partial correlation with the Gaussian copula–DCC–GARCH model

This article suggests a directional time-varying partial correlation based on the dynamic conditional correlation (DCC) method. A recent study proposed the copula DCC based on the vine structure. Due to the arbitrary variable selection, their method can produce unnecessary dependence in the multivariate structure, with extra economic and computational burdens. To overcome this limitation, we incorporate directional dependence by copula to track the causal relationship among multiple variables and then extend the copula bivariate DCC method to a directional time varying partial correlation in the multivariate structure. Our proposed method provides a reasonable and efficient conditional dependence structure, without the trial and error process. We offer an application of our method to the U.S. stock market as an illustrated example.     

Hedging and diversification across commodity assets

ABSTRACT

We investigate the conditional cross effects and volatility spillover between equity markets and commodity markets (oil and gold), Fama and French HML and SMB factors, volatility index (VIX) and bonds using different multivariate GARCH specifications considering the potential asymmetry and persistence behaviours. We analyse the dynamic conditional correlation between the US equity market and a set of commodity prices and risk factors to forecast the transmission of shock to the equity market firstly, and to determine and compare the optimal hedge ratios from the different models based on the hedging effectiveness of each model. Our findings suggest that all models confirm the significant returns and volatility spillovers. More importantly, we find that GO-GARCH is the best-fit model for modelling the joint dynamics of different financial variables. The results of the current study have implications for investors: (i) the equity market displays inverted dynamics with the volatility index suggesting strong evidence of diversification benefit; (ii) of the hedging assets gold appears the best hedge for the US equity market as it has a higher hedge effectiveness than oil and bonds over time; and (iii) despite these important results, a better hedge may be obtained by using well-selected firm sized and profitability-based portfolios.     

Can asymmetric conditional volatility imply asymmetric tail dependence?

In this article, we investigate two types of asymmetries, that is, the asymmetry of conditional volatility and the asymmetry of tail dependence in the crude oil markets. We employ the two different sample datasets in which each dataset covers the time period of stable and unstable oil prices, individually. A variety of different copulas and three asymmetric GARCH regression models are used in order to capture the two types of asymmetries. In particular, we extend the TBL-GARCH model proposed by Choi et al. (2012) to the asymmetric GARCH regression type model. The findings from the two different approaches are congruent, in that there is no asymmetry of tail dependence and no asymmetric conditional volatility in crude oil returns over the two different sample periods. Our study reconfirms the findings of Aboura and Wagner (2016) by showing that asymmetric conditional volatility relates to asymmetric tail dependence.     

The dependence structure between the Canadian stock market and the USD/CAD exchange rate: a copula approach

Abstract This paper investigates the dependence structure between the real Canadian stock returns and the real USD/CAD exchange rate returns, using the Symmetrized Joe‐Clayton (SJC) copula function. We estimate the SJC copula with monthly data over the period 1995:1 to 2006:12. Our results show significant asymmetric static and dynamic tail dependence between the real stock returns and the real exchange rate returns, such that the two returns are more dependent in the left than in the right tail of their joint distribution. We explain this asymmetric dependence in terms of an asymmetric interest rate policy by Canadian monetary authorities in response to changes in the real exchange rate during sub‐periods of falling and rising commodity prices.     

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。     

Tail dependence analysis of stock markets using extreme value theory

Financial risk modelling frequently uses the assumption of a normal distribution when considering the return series which is inefficient if the data is not normally distributed or if it exhibits extreme tails. Estimation of tail dependence between financial assets plays a vital role in various aspects of financial risk modelling including portfolio theory and hedging amongst applications. Extreme Value Theory (EVT) provides well established methods for considering univariate and multivariate tail distributions which are useful for forecasting financial risk or modelling the tail dependence of risky assets. The empirical analysis in this article uses nonparametric measures based on bivariate EVT to investigate asymptotic dependence and estimate the degree of tail dependence of the ASX-All Ordinaries daily returns with four other international markets, viz., the S&P-500, Nikkei-225, DAX-30 and Heng-Seng for both extreme right and left tails of the return distribution. It is investigated whether the asymptotic dependence between these markets is related to the heteroscedasticity present in the logarithmic return series using GARCH filters. The empirical evidence shows that the asymptotic extreme tail dependence between stock markets does not necessarily exist and rather can be associated with the heteroscedasticity present in the financial time series of the various stock markets.     

Conditional correlations in the returns on oil companies stock prices and their determinants

The identification of the forces that drive stock returns and the dynamics of their associated volatilities is a major concern in empirical economics and finance. This analysis is extremely important for determining optimal hedging strategies. This paper investigates the stock prices’ returns and their financial risk factors for several integrated oil companies, namely Bp (BP), Chevron-Texaco (CVX), Eni (ENI), Exxon-Mobil (XOM), Royal Dutch (RD) and Total-Fina Elf (TFE). We measure the actual co-risk in stock returns and their determinants “within” and “between” the different oil companies, using multivariate cointegration techniques in modelling the conditional mean, as well as multivariate GARCH models for the conditional variances. The distinguishing features of this paper are: (i) focus on the determinants of the market value of each company using the cointegrated VAR/VECM methodology; (ii) specification of the conditional variances of VECM residuals with the Constant Conditional Correlation (CCC) multivariate GARCH model of Bollerslev [(1990) Review of Economics and Statistics 72:498–505] and the Dynamic Conditional Correlation (DCC) multivariate GARCH model of Engle [(2002) Journal of Business and Economic Statistics 20:339–350]; (iii) discussion of the performance of optimal hedge ratios calculated with the DCC estimates. The “within” and “between” DCC indicate time-varying interdependence between stock return volatilities and their determinants. Moreover, DCC models are shown to produce more accurate hedging strategies.     

Sovereign bond market dependencies and crisis transmission around the eurozone debt crisis: a dynamic copula approach

We examine the dependency between the European government bond markets around the recent sovereign debt crisis. A dynamic copula approach is used to model the time-varying dependence structure of those government bond markets, evaluate the nature and strength of their dependencies over time, and gauge the transmission of the crisis shocks. Our results can be summarized as follows: i) the eurozone sovereign bond markets under consideration have a significant and positive dependence with the Greek and the EMU benchmark sovereign bond markets; ii) the dynamic-BB7 copula function best describes the dependence structure between these sovereign bond markets and provides evidence of asymmetric tail dependence; iii) the conditional probability of crisis transmission from Greece to other eurozone countries is higher than the other way around; and iv) Greece is the most vulnerable country when the eurozone entered into the sovereign debt crisis.     

Value-at-risk Trade-off and Capital Allocation with Copulas

This paper uses copula functions to evaluate tail probabilities and market risk trade-offs at a given confidence level, dropping the joint normality assumption on returns. Copulas enable one to represent distribution functions separating the marginal distributions from the association structure. We present an application to two stock market indices: for each market we recover the marginal probability distribution. We then calibrate copula functions and recover the joint distribution. The estimated copulas directly give the joint probabilities of extreme losses. Their level curves measure the trade-off between losses over different desks. This trade-off can be exploited for capital allocation and is shown to depend on fat tails.
(J.E.L.: C14, G19, G29).     

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.     

Portfolio Diversification Strategy Via Tail‐Dependence Clustering and ARMA‐GARCH Vine Copula Approach

This study proposes a diversified portfolio construction method based on the tail dependence between the financial assets and adopting both market prior information and the exports’ subject views. In this paper, tail‐dependence clustering was applied to divide candidate assets into different groups according to their tail dependence during the crisis period and the ARMA‐GARCH vine copula‐opinion pooling approach was applied to select the minimum Conditional Value‐at‐Risk portfolio according to the clustering results. The daily closed prices of the components of DAX 20 from 3 January 2006 to 20 December 2014 were studied to illustrate the methodology. The results reveal that more than 90% of 450 possible portfolios are modelled by D‐vine structure and Student's t‐copula dominates almost all the cases for pair copula selection. As Student's t‐copula captures the symmetric tail dependence, the 450 possible portfolios do not show stronger lower tail dependence than upper tail dependence. This study contributes by combining cluster analysis with portfolios selection. It uses vine copula to capture the dependence structure among assets. Finally, it offers a flexible method to describe market and offers a strategy to construct diversified portfolios by adding the investors’ information into portfolio selection procedure at the 1‐day forecast horizon.     

How to maximize golden opinions

In this article, we aim to model the level and structure of the dependence between the world's leading stock markets and those of the emerging market groups?–?Europe, Latin America and Far East. To this end we use a mixture model of Gaussian, Gumbel and Gumbel survival copulas. Our results indicate that none of the pairs of stock markets exhibit a right-tail dependence structure. All valid models exhibit a mixture of Gaussian and left-tail dependence structure. Our findings imply that Gaussian dependence structure is dominant in most of the models. The emerging equity markets in the European region exhibit the most significant dependence structure with the world leaders. Furthermore, most of the emerging equity markets have a significant dependence structure with the US stock market. We further compare our findings with the results of the conventional correlation coefficients and conclude the importance of using copula models in analysing the portfolio diversification opportunities. Our findings overall indicate two important remarks: First, the copula models reveal better indicators for global investors to establish a diversified portfolio; Second, international equity markets exhibit significant dependence, which leaves a smaller opportunity to benefit from international portfolio diversification.     

Pricing and Hedging Credit Derivatives with Copulas

In this paper, we apply a copula function pricing technique to the evaluation of credit derivatives, namely a vulnerable default put option and a credit switch. Also in this case, copulas enable one to separate the specification of marginal default probabilities from their dependence structure. Their use is based here on no–arbitrage arguments, which provide pricing bounds and easy–to–implement super–replication strategies.
At a second stage, we specify the copula function to be a mixture one. In this case, we obtain closed form prices and hedges, which we calibrate on real market data. For the sake of comparison, we add a Clayton calibration.
(J.E.L: G11, G12).     

The role of uncertainty in the term structure of interest rates: A GARCH-ATSM approach

This article examines the roles of uncertainties regarding various macro-variables in determining risk premiums of bond yields. We develop a multivariate GARCH-VAR to quantify uncertainties regarding inflation, real activities and monetary policy as time-varying conditional variances. We jointly estimate the multivariate GARCH and no-arbitrage bond pricing equations using a maximum likelihood method. The results indicate that the inflation uncertainty is the largest contributor to the dynamics of long-term yields since the 1980s, while the monetary policy uncertainty also plays noticeable roles.     

Relationship between oil price and exchange rate by FDA and copula

This article investigates the relationship between daily crude oil prices and exchange rates. Functional data analysis is used to show the clustering pattern of exchange rates and oil prices over the time period through high dimensional visualizations. We select exchange rates for important currencies related to crude oil prices by using the objective Bayesian variable selection method. The selected sample data exhibits non-normal distribution with fat tails and skewness. Under the non-normality of the return series, we use copula functions that do not require to assume the bivariate normality to consider marginal distribution. In particular, our study applies the popular and powerful statistical methods such as Gaussian copula partial correlations and Gaussian copula marginal regression. We find evidence of significant dependence for all considered pairs, except for the Mexican peso-Brent. Our empirical results also show that the rise in the West Texas Intermediate (WTI) oil price returns is associated with a depreciation of the US dollar.