VaR and ES for linear portfolios with mixture of generalized Laplace distributions risk factors

In this paper, we propose an explicit estimation of Value-at-Risk (VaR) and Expected Shortfall (ES) for linear portfolios when the risk factors change with a convex mixture of generalized Laplace distributions (M-GLD). We introduce the dynamics Delta-GLD-VaR, Delta-GLD-ES, Delta-MGLD-VaR and Delta-MGLD-ES, by using conditional correlation multivariate GARCH. The generalized Laplace distribution impose less restrictive assumptions during estimation that should improve the precision of the VaR and ES through the varying shape and fat tails of the risk factors in relation with the historical sample data. We also suggested some areas of application to measure price risk in agriculture, risk management and financial portfolio optimization.     

On the distribution and conditional heteroscedasticity in Taiwan stock prices

This paper studies the distribution and conditional heteroscedasticity in stock returns on the Taiwan stock market. Apart from the normal distribution, in order to explain the leptokurtosis and skewness observed in the stock return distribution, we also examine the Student-t, the Poisson–normal, and the mixed-normal distributions, which are essentially a mixture of normal distributions, as conditional distributions in the stock return process. We also use the ARMA (1,1) model to adjust the serial correlation, and adopt the GJR–generalized autoregressive conditional heteroscedasticity (GARCH (1,1)) model to account for the conditional heterscedasticity in the return process. The empirical results show that the mixed–normal–GARCH model is the most probable specification for Taiwan stock returns. The results also show that skewness seems to be diversifiable through portfolio. Thus the normal–GARCH or the Student-t–GARCH model which involves symmetric conditional distribution may be a reasonable model to describe the stock portfolio return process¹.     

Analyzing hedging strategies for fixed income portfolios: A Bayesian approach for model selection

During the recent European sovereign debt crisis, returns on EMU government bond portfolios experienced substantial volatility clustering, leptokurtosis and skewed returns as well as correlation spikes. Asset managers invested in European government bonds had to derive new hedging strategies to deal with changing return properties and higher levels of uncertainty. In this environment, conditional time series approaches such as GARCH models might be better suited to achieve a superior hedging performance relative to unconditional hedging approaches such as OLS. The aim of this study is to test innovative hedging strategies for EMU bond portfolios for non-crisis and crisis periods. We analyze single and composite hedges with the German Bund and the Italian BTP futures contracts and evaluate the hedging effectiveness in an out-of-sample setting. The empirical analysis includes OLS, constant conditional correlation (CCC), and dynamic conditional correlation (DCC) multivariate GARCH models. We also introduce a Bayesian composite hedging strategy, attempting to combine the strengths of OLS and GARCH models, thereby endogenizing the dilemma of selecting the best estimation model. Our empirical results demonstrate that the Bayesian composite hedging strategy achieves the highest hedging effectiveness and compares particularly favorable to OLS during the recent sovereign debt crisis. However, capturing these benefits requires low transactions cost and efficiently functioning futures markets.     

Positivity constraints on the conditional variances in the family of conditional correlation GARCH models

In this article, we derive a set of necessary and sufficient conditions for positivity of the vector conditional variance equation in multivariate GARCH models with explicit modelling of conditional correlation. These models include the constant conditional correlation GARCH model of Bollerslev [1990. Review of Economics and Statistics 72, 498–505] and its extensions. Under the new conditions, it is possible to introduce negative volatility spillovers in the model. An empirical example illustrates usefulness of having such conditions in practice.     

Estimating GARCH models using support vector machines*

Abstract

Support vector machines (SVMs) are a new nonparametric tool for regression estimation. We will use this tool to estimate the parameters of a GARCH model for predicting the conditional volatility of stock market returns. GARCH models are usually estimated using maximum likelihood (ML) procedures, assuming that the data are normally distributed. In this paper, we will show that GARCH models can be estimated using SVMs and that such estimates have a higher predicting ability than those obtained via common ML methods.     

International stock markets interactions and conditional correlations

This paper investigates the transmission of price and volatility spillovers across the US and European stock markets in bivariate combinations. The framework used encompasses the most popular multivariate GARCH models, with News Impact Surfaces employed for interpretation. By using synchronous data the dynamic conditional correlation model (Engle, R., 2002. Dynamic conditional correlation: a simple class of multivariate GARCH models. Journal of Business and Economic Statistics 20, 339–350) is found to best capture the relationships for over half of the bivariate combinations of markets. Other findings include volatility spillovers from the US to European markets, and a reverse spillover. In addition, the magnitude of the correlation between markets is higher not only for negative shocks in both markets, but also when a combination of shocks of opposite signs occurs.     

Covariance dependent kernels,a Q-affine GARCH for multi-asset option pricing

This paper introduces a class of multivariate GARCH models that extends the existing literature by explicitly modeling correlation dependent pricing kernels. A large subclass admits closed-form recursive solutions for the moment generating function under the risk-neutral measure, which permits efficient pricing of multi-asset options. We perform a full calibration to three bivariate series of index returns and their corresponding volatility indexes in a joint maximum likelihood estimation. The results empirically confirm the presence of correlation dependance in addition to the well known variance dependance in the pricing kernel. The model improves both the overall likelihood and the VIX-implied likelihoods, with a better fitting of marginal distributions, e.g., 15% less error on one-asset option prices. The new degree of freedom is also shown to significantly impact the shape of marginal and joint pricing kernels, and leads to up to 53% differences for out-of-the-money two-asset correlation option prices.     

An international analysis of REITs and stock portfolio management based on dynamic conditional correlation models

This paper analyzes the volatility spillovers and asymmetry between REITs and stock prices for nine countries (Australia, Belgium, Germany, Italy, Japan, The Netherlands, Singapore, the United Kingdom, and the United States) using eight different multivariate GARCH models. We also analyze the optimal weights, hedging effectiveness, and hedge ratios for REIT-stock portfolio holdings with respect to the results. The empirical results indicate that dynamic conditional correlation (DCC) models provide a better fit than the constant conditional correlation models. The DCC with volatility spillovers and asymmetry (DCC-SA) model provides a better fit than the other multivariate GARCH models. The DCC-SA model also provides the best hedging effectiveness for all pairs of REIT-stock assets. More importantly, this result holds for all cases and for all models that we consider, which means that by taking spillover and asymmetry into consideration, hedging effectiveness can be vastly improved.     

A Closer Look at the Relation between GARCH and Stochastic Autoregressive Volatility

We show that, for three common SARV models, fitting a minimummean square linear filter is equivalent to fitting a GARCH model.This suggests that GARCH models may be useful for filtering,forecasting, and parameter estimation in stochastic volatilitysettings. To investigate, we use simulations to evaluate howthe three SARV models and their associated GARCH filters performunder controlled conditions and then we use daily currency andequity index returns to evaluate how the models perform in arisk management application. Although the GARCH models produceless precise forecasts than the SARV models in the simulations,it is not clear that the performance differences are large enoughto be economically meaningful. Consistent with this view, wefind that the GARCH and SARV models perform comparably in testsof conditional value-at-risk estimates using the actual data.     

Currency futures market in India: an empirical analysis of market efficiency and volatility

As the Indian currency futures market has been in existence for over 7 years, this paper analyses the effectiveness of the 1-month USD/INR currency futures rates in predicting the expected spot rate. The volatility of the USD/INR spot returns was also analysed. Modelling volatility of the USD/INR spot rate using a generalized autoregressive conditional heteroskedasticity (GARCH) and exponential generalized autoregressive conditional heteroskedasticity (EGARCH) model indicated the presence of volatility clustering. Using multivariate GARCH models such as the constant conditional correlation and dynamic conditional correlation, signs of a volatility spillover between the USD/INR spot and currency futures market were also observed.     

Risk forecasting in (T)GARCH models with uncorrelated dependent innovations

(G)ARCH-type models are frequently used for the dynamic modelling and forecasting of risk attached to speculative asset returns. While the symmetric and conditionally Gaussian GARCH model has been generalized in a manifold of directions, model innovations are mostly presumed to stem from an underlying IID distribution. For a cross section of 18 stock market indices, we notice that (threshold) (T)GARCH-implied model innovations are likely at odds with the commonly held IID assumption. Two complementary strategies are pursued to evaluate the conditional distributions of consecutive TGARCH innovations, a non-parametric approach and a class of standardized copula distributions. Modelling higher order dependence patterns is found to improve standard TGARCH-implied conditional value-at-risk and expected shortfall out-of-sample forecasts that rely on the notion of IID innovations.     

Stochastic volatility and time-varying country risk in emerging markets

This study suggests an alternative method to estimate time-varying country risk. We first apply a new multivariate stochastic volatility (SV) model to a set of emerging stock markets. To estimate the SV model, we use a Bayesian Markov chain Monte Carlo simulation procedure. By applying the deviance information criterion, we show that the new model performs well relative to alternative multivariate SV models. We then compute the conditional betas for the different markets and compare the results with an often-used procedure based on multivariate GARCH models. We show that the new multivariate SV model more accurately captures the time-varying nature of country risk. The conditional betas show signs of large variations, indicating the importance of taking time-varying country risk into consideration when managing emerging market portfolios.     

Bayesian estimation of truncated data with applications to operational risk measurement

Data insufficiency and reporting threshold are two main issues in operational risk modelling. When these conditions are present, maximum likelihood estimation (MLE) may produce very poor parameter estimates. In this study, we first investigate four methods to estimate the parameters of truncated distributions for small samples—MLE, expectation-maximization algorithm, penalized likelihood estimators, and Bayesian methods. Without any proper prior information, Jeffreys’ prior for truncated distributions is used. Based on a simulation study for the log-normal distribution, we find that the Bayesian method gives much more credible and reliable estimates than the MLE method. Finally, an application to the operational loss severity estimation using real data is conducted using the truncated log-normal and log-gamma distributions. With the Bayesian method, the loss distribution parameters and value-at-risk measure for every cell with loss data can be estimated separately for internal and external data. Moreover, confidence intervals for the Bayesian estimates are obtained via a bootstrap method.     

Asian Pacific Stock Market Volatility Modeling and Value at Risk Analysis

The potential for stock market growth in Asian Pacific countries has attracted foreign investors. However, higher growth rates come with higher risk. We apply value at risk (VaR) analysis to measure and analyze stock market index risks in Asian Pacific countries, exposing and detailing both the unique risks and system risks embedded in those markets. To implement the VaR measure, it is necessary to perform "volatility modeling" by mixture switch, exponentially weighted moving average (EWMA), or generalized autoregressive conditional heteroskedasticity (GARCH) models. After estimating the volatility parameters, we can calibrate the VaR values of individual and system risks. Empirically, we find that, on average, Indonesia and Korea exhibit the highest VaRs and VaR sensitivity, and currently, Australia exhibits relatively low values. Taiwan is liable to be in high-state volatility. In addition, the Kupiec test indicates that the mixture switch VaR is superior to delta normal VaR; the quadratic probability score (QPS) shows that the EWMA is inclined to underestimate the VaR for a single series, and GARCH shows no difference from GARCH t and GARCH generalized error distribution (GED) for a multivariate VaR estimate with more assets.     

An Assessment of Empirical Model Performance When Financial Market Transactions are Observed at Different Data Frequencies: An Application to East Asian Exchange Rates

This paper compares the performance of alternative models of east Asian exchange rates at different data frequencies. Selected models employ different specifications of the conditional variance and the conditional error distribution. Conditional variance specifications include: homoscedasticity, GARCH, LGARCH, and EGARCH. Conditional error distribution specifications include normal and Student t. The best exchange rate model specification is clearly conditional on data frequency. Higher frequency (daily, weekly) data commonly exhibit characteristics that demand more sophisticated estimation methods than analysts commonly employ. These characteristics generally vanish at lower (monthly, quarterly) frequencies. Overall we find significant benefit from accommodating heteroscedasticity and leptokurtic properties of the conditional distribution as data frequency increases. Using a likelihood ratio test we compare the relative gain from addressing heteroscedasticity (through use of GARCH models) versus accommodation of leptokurtosis. This comparison suggests that the gains from correct specification of the conditional distribution dominate those obtained from addressing problems of heteroscedasticity.     

An economic evaluation of stock–bond return comovements with copula-based GARCH models

Owing to their importance in asset allocation strategies, the comovements between the stock and bond markets have become an increasingly popular issue in financial economics. Moreover, the copula theory can be utilized to construct a flexible joint distribution that allows for skewness in the distribution of asset returns as well as asymmetry in the dependence structure between asset returns. Therefore, this paper proposes three classes of copula-based GARCH models to describe the time-varying dependence structure of stock–bond returns, and then examines the economic value of copula-based GARCH models in the asset allocation strategy. We compare their out-of-sample performance with other models, including the passive, the constant conditional correlation (CCC) GARCH and the dynamic conditional correlation (DCC) GARCH models. From the empirical results, we find that a dynamic strategy based on the GJR-GARCH model with Student-t copula yields larger economic gains than passive and other dynamic strategies. Moreover, a less risk-averse investor will pay higher performance fees to switch from a passive strategy to a dynamic strategy based on copula-based GARCH models.     

Multivariate Models of Equity Returns for Investment Guarantees Valuation

Abstract

In this paper we investigate the valuation of investment guarantees in a multivariate (discrete-time) framework. We present how to build multivariate models in general, and we survey the most important multivariate GARCH models. A direct multivariate application of regime-switching models is also discussed, as is the estimation of these models using maximum likelihood and their comparison in a multivariate setting. The computation of the CTE provision is further presented. We have estimated the models with a multivariate dataset (Canada, United States, United Kingdom, and Japan), and we compared the quality of their fit using multiple criteria and tests. We observe that multivariate GARCH models provide a better overall fit than regime-switching models. However, regime-switching models appropriately represent the fat tails of the returns distribution, which is where most GARCH models fail. This leads to significant differences in the value of the CTE provisions, and, in general, provisions computed with regime-switching models are higher. Thus, the results from this multivariate analysis are in line with what was obtained in the literature of univariate models.     

Asset Allocation by Variance Sensitivity Analysis

This article provides a solution to the curse of dimensionalityassociated to multivariate generalized autoregressive conditionallyheteroskedastic (GARCH) estimation. We work with univariateportfolio GARCH models and show how the multivariate dimensionof the portfolio allocation problem may be recovered from theunivariate approach. The main tool we use is "variance sensitivityanalysis," the change in the portfolio variance induced by aninfinitesimal change in the portfolio allocation. We suggesta computationally feasible method to find minimum variance portfoliosand estimate full variance-covariance matrices. An applicationto real data portfolios implements our methodology and comparesits performance against that of selected popular alternatives.     

Persistence and Kurtosis in GARCH and Stochastic Volatility Models

This article shows that the relationship between kurtosis, persistenceof shocks to volatility, and first-order autocorrelation ofsquares is different in GARCH and ARSV models. This differencecan explain why, when these models are fitted to the same series,the persistence estimated is usually higher in GARCH than inARSV models, and, why gaussian ARSV models seem to be adequate,whereas GARCH models often require leptokurtic conditional distributions.We also show that introducing the asymmetric response of volatilityto positive and negative returns does not change the conclusions.These results are illustrated with the analysis of daily financialreturns.