Portfolio optimization with mixture vector autoregressive models

Obtaining reliable estimates of conditional covariance matrices is an important task of heteroskedastic multivariate time series. In portfolio optimization and financial risk management, it is crucial to provide measures of uncertainty and risk as accurately as possible. We propose using mixture vector autoregressive (MVAR) models for portfolio optimization. Combining a mixture of distributions that depend on the recent history of the process, MVAR models can accommodate asymmetry, multimodality, heteroskedasticity and cross-correlation in multivariate time series data. For mixtures of Normal components, we exploit a property of the multivariate Normal distribution to obtain explicit formulas of conditional predictive distributions of returns on a portfolio of assets. After showing how the method works, we perform a comparison with other relevant multivariate time series models on real stock return data.Comment: 19 pages, 9 figures, 2 table

Inhomogeneous Dependency Modelling with Time Varying Copulae

Measuring dependence in a multivariate time series is tantamount to modelling its dynamic structure in space and time. In the context of a multivariate normally distributed time series, the evolution of the covariance (or correlation) matrix over time describes this dynamic. A wide variety of applications, though, requires a modelling framework different from the multivariate normal. In risk management the non-normal behaviour of most financial time series calls for nonlinear (i.e. non-gaussian) dependency. The correct modelling of non-gaussian dependencies is therefore a key issue in the analysis of multivariate time series. In this paper we use copulae functions with adaptively estimated time varying parameters for modelling the distribution of returns, free from the usual normality assumptions. Further, we apply copulae to estimation of Value-at-Risk (VaR) of a portfolio and show its better performance over the RiskMetrics approach, a widely used methodology for VaR estimation.Value-at-Risk, time varying copula, adaptive estimation, nonparametric estimation.

Inhomogeneous dependence modelling with time varying copulae

Measuring dependence in a multivariate time series is tantamount to modelling its dynamic structure in space and time. In the context of a multivariate normally distributed time series, the evolution of the covariance (or correlation) matrix over time describes this dynamic. A wide variety of applications, though, requires a modelling framework different from the multivariate normal. In risk management the non-normal behaviour of most financial time series calls for non-Gaussian dependence. The correct modelling of non-Gaussian dependences is therefore a key issue in the analysis of multivariate time series. In this paper we use copulae functions with adaptively estimated time varying parameters for modelling the distribution of returns, free from the usual normality assumptions. Further, we apply copulae to estimation of Value-at-Risk (VaR) of portfolios and show their better performance over the RiskMetrics approach, a widely used methodology for VaR estimation

SFB 649 Discussion Paper 2006-075 Inhomogeneous Dependency Modelling with Time Varying

Measuring dependence in a multivariate time series is tantamount to modelling its dynamic structure in space and time. In the context of a multivariate normally distributed time series, the evolution of the covariance (or correlation) matrix over time describes this dynamic. A wide variety of applications, though, requires a modelling framework different from the multivariate normal. In risk management the non-normal behaviour of most financial time series calls for nonlinear (i.e. non-gaussian) dependency. The correct modelling of non-gaussian dependencies is therefore a key issue in the analysis of multivariate time series. In this paper we use copulae functions with adaptively estimated time varying parameters for modelling the distribution of returns, free from the usual normality assumptions. Further, we apply copulae to estimation of Value-at-Risk (VaR) of a portfolio and show its better performance over the RiskMetrics approach, a widely used methodology for VaR estimation. JEL classification: C 1

Investigating dynamic dependence using copulae

A general methodology for time series modelling is developed which works down from distributional properties to implied structural models including the standard regression relationship. This general to specific approach is important since it can avoid spurious assumptions such as linearity in the form of the dynamic relationship between variables. It is based on splitting the multivariate distribution of a time series into two parts: (i) the marginal unconditional distribution, (ii) the serial dependence encompassed in a general function , the copula. General properties of the class of copula functions that fulfill the necessary requirements for Markov chain construction are exposed. Special cases for the gaussian copula with AR(p) dependence structure and for archimedean copulae are presented. We also develop copula based dynamic dependency measures — auto-concordance in place of autocorrelation. Finally, we provide empirical applications using financial returns and transactions based forex data. Our model encompasses the AR(p) model and allows non-linearity. Moreover, we introduce non-linear time dependence functions that generalize the autocorrelation function

Modeling and Forecasting Realized Volatility

This paper provides a general framework for integration of high-frequency intraday data into the measurement forecasting of daily and lower frequency volatility and return distributions. Most procedures for modeling and forecasting financial asset return volatilities, correlations, and distributions rely on restrictive and complicated parametric multivariate ARCH or stochastic volatility models, which often perform poorly at intraday frequencies. Use of realized volatility constructed from high-frequency intraday returns, in contrast, permits the use of traditional time series procedures for modeling and forecasting. Building on the theory of continuous-time arbitrage-free price processes and the theory of quadratic variation, we formally develop the links between the conditional covariancematrix and the concept of realized volatility. Next, using continuously recorded observations for the Deutschemark Dollar and Yen / Dollar spot exchange rates covering more than a decade, we find that forecasts from a simple long-memory Gaussian vector autoregression for the logarithmic daily realized volatilities perform admirably compared to popular daily ARCH and related models. Moreover, the vector autoregressive volatility forecast, coupled with a parametric lognormal-normal mixture distribution implied by the theoretically and empirically grounded assumption of normally distributed standardized returns, gives rise to well-calibrated density forecasts of future returns, and correspondingly accurate quantile estimates. Our results hold promise for practical modeling and forecasting of the large covariance matrices relevant in asset pricing, asset allocation and financial risk management applications.

Innovations in dependence modelling for financial applications

The contribution of this thesis is in developing and investigating novel dependence modelling techniques in financial applications. Furthermore, the aim is to understand the key factors driving the dynamic nature of such dependence. When modelling the multivariate distribution of the returns associated to a portfolio of financial assets one is faced with a multitude of considerations and potential choices. For example, in the currency studies undertaken in this thesis suitably heavy-tailed marginal time series models are developed for the returns of each currency exchange rate, and then the multivariate dependence structure of the returns of multiple-currency baskets at each time instant is considered. These dependence relationships can be studied via numerous concordance measures such as correlation, rank correlations and extremal dependences. Such studies can be undertaken in a static or dynamic setting and either parametrically or non-parametrically. Another important aspect of financial time series is the enormous amount of financial data available for statistical analysis and financial econometrics that can be used to better understand economic and financial theories. In this thesis, the focus is on the influence of dependence structures in complex financial data in two asset classes: currencies and commodities. These are challenging data structures as they contain temporal serial dependence, cross dependence and term-structural dependences. Each of these forms of dependence are studied in this thesis in both parametric and non-parametric settings

Modeling and Forecasting Realized Volatility

This paper provides a general framework for integration of high-frequency intraday data into the measurement, modeling and forecasting of daily and lower frequency volatility and return distributions. Most procedures for modeling and forecasting financial asset return volatilities, correlations, and distributions rely on restrictive and complicated parametric multivariate ARCH or stochastic volatility models, which often perform poorly at intraday frequencies. Use of realized volatility constructed from high-frequency intraday returns, in contrast, permits the use of traditional time series procedures for modeling and forecasting. Building on the theory of continuous-time arbitrage-free price processes and the theory of quadratic variation, we formally develop the links between the conditional covariance matrix and the concept of realized volatility. Next, using continuously recorded observations for the Deutschemark/Dollar and Yen /Dollar spot exchange rates covering more than a decade, we find that forecasts from a simple long-memory Gaussian vector autoregression for the logarithmic daily realized volatitilies perform admirably compared to popular daily ARCH and related models. Moreover, the vector autoregressive volatility forecast, coupled with a parametric lognormal-normal mixture distribution implied by the theoretically and empirically grounded assumption of normally distributed standardized returns, gives rise to well-calibrated density forecasts of future returns, and correspondingly accurate quintile estimates. Our results hold promise for practical modeling and forecasting of the large covariance matrices relevant in asset pricing, asset allocation and financial risk management applications.

Bayesian estimation of a dynamic conditional correlation model with multivariate Skew-Slash innovations

Financial returns often present a complex relation with previous observations, along with a slight skewness and high kurtosis. As a consequence, we must pursue the use of flexible models that are able to seize these special features: a financial process that can expose the intertemporal relation between observations, together with a distribution that can capture asymmetry and heavy tails simultaneously. A multivariate extension of the GARCH such as the Dynamic Conditional Correlation model with Skew-Slashinnovations for financial time series in a Bayesian framework is proposed in the present document, and it is illustrated using an MCMC within Gibbs algorithm performed onsimulated data, as well as real data drawn from the daily closing prices of the DAX,CAC40, and Nikkei indicesWe acknowledge financial support by Spanish Ministry of Economy and Competition Grant ECO2012-38442