Dependence estimation and controlled CVaR portfolio optimization of a highly kurtotic Australian mining sample of stocks

The drivers of mining stock prices are known to be several. Sharp spikes on the stocks return distribution have been linked to the presence of unusually high volatility signifying the presence of high levels of kurtosis. The accurate measurement of the stocks’ underlying co-movements for more accurate CVaR portfolio optimization requires, therefore, the utilization of sophisticated and specific-specialized techniques which could adequately capture and model these characteristics. Here this issue is addressed by applying statistical-graphical models for dependence estimation. Twenty mining stocks, out of the 801 listed in the ASX as of December 2012, have been selected for the analysis under the criteria of satisfying the eight years trading period sought, having very weak or no autocorrelation of residuals and displaying the highest kurtosis. Models’ estimations of dependence are compared and inserted into a differential evolution algorithm for non-convex global optimization in order to conduct risk controlled CVaR portfolio optimization (Ardia, Boudt, Carl, Mullen & Peterson, 2011) and be able to identify the one yielding the highest portfolio return. The findings are of relevance in portfolio allocation and portfolio risk management. Energy and mining stock markets are subjected to numerous price drivers holding complex relationships. The dynamics of production and consumption based on seasonality features, transportation and storage, weather conditions, commodity price fluctuations, currency changes, market confidence and expectations, trading speculations and the domestic and international states of the economy impact mining stock prices in particular and unobvious ways reflected in high volatility with sudden spikes in the stock’s return distribution (Pilipovic, 1998). The generation of accurate measurements of the dependence matrix of mining stock’s return series is therefore both, a non-trivial task due to the hard to decipher characteristics present in return series suffering from high levels of kurtosis (Carvalho, Lopes & Aguilar, 2010) and, a crucial element in portfolio optimization and portfolio risk management. The use of graphical techniques in this study is justified on the basis of their utility and suitableness. Graphical models such as pair c-vine copulas, the graphical lasso and adaptive graphical lasso provide, for instance, the visualization and flexibility to represent a problem in a more simplified and dissected form (Lauritzen, 1996). Graphs also appear to be naturally adequate to express the interaction of variables and thus facilitate the analysis of their dependency. The models of dependence estimation and CVaR portfolio optimization, on the other hand, are desirable due to mathematical and statistical framework they provide which may lead to satisfactory results and, their apparent ability to overcome the flaws (i.e. standardized model application to all joint distributions, restrictive and deterministic linear and monotonic modelling functions as in the Pearson and Spearman) traditional measures display when dealing with highly kurtotic data, joint distributions with stronger dependence in the tails and controlled risk non-convex portfolio optimization problems. Findings indicate that the highest portfolio returns are generated by inserting the covariance output matrix from the student-t copula into the differential optimization algorithm and, the student-t copula fitting with separate modelling of the marginal distributions appears to be the most desirable modelling choice. The portfolio return by the adaptive graphical lasso is lower than that of the student-t and is followed by the Gaussian pair c-vine copula. The regular graphical lasso produced the lowest portfolio return and the covariance matrix values were higher for models producing the highest portfolio returns implying that the models generating the lowest portfolio returns underestimated the dependence of the assets. The implications of the findings suggest that specific modelling of each marginal distribution, as compared to modelling based on a Gaussian framework, may lead to an edge in the estimations due to the distribution differences encountered on each marginal. Furthermore, the ability of the model to capture dependence in the tails, as it is the case of the student-t copula, does provide a modelling advantage too. This paper appears to be the first one in, comparing the portfolio performance of the models of dependence estimation in the context of controlled CVaR, applying the models treated to a highly kurtotic mining sample of stocks from the Australian market and modelling separately the distribution of the marginals when fitting the student-t copula

Sparse Models in High-Dimensional Dependence Modelling and Index Tracking

This thesis is divided into two parts. The first part proposes parsimonious models to the vine copula. The second part is devoted to the index tracking problem. Vine copulas provide a flexible tool to capture asymmetry in modelling multivariate distributions. Nevertheless, the computational expense of its flexibility increases exponentially as the dimension of the joint distribution grows. To alleviate this issue, the simplifying assumption (SA) is commonly adopted in special applications of vine copula models. In order to relax SA, Chapter 2 proposes generalized linear models (GLMs) to model parameters in conditional bivariate copulas. In the spirit of the principle of parsimony, a regularization methodology is developed to control the number of parameters. This leads to sparse vine copula models. The conventional vine copula with the SA, the proposed GLM-based vine copula and the sparse vine copula are applied to several financial datasets. Empirical results show that the proposed models in this chapter outperform the one with SA significantly in terms of the Bayesian information criterion. Index tracking is a dominant method among passive investment strategies. It attempts to reproduce the return of stock-market indices. Chapter 3 focuses on selecting stocks to construct tracking portfolios. In order to do that, principal component analysis (PCA) is applied via a two-step procedure. In the first step, the index return is expressed as a function of the principal components (PCs) of stock returns, and a subset of PCs is selected according to Sobol's total sensitivity index. In the second step, a subset of stocks, which is most "similar" to those selected PCs, is detected. This similarity is measured by Yanai's generalized coefficient of determination, the distance correlation, or Heller-Heller-Gorfine test statistics. Given selected stocks, their weights in the tracking portfolio can be determined by minimizing a specific tracking error. Compared with existing methods, constructing tracking portfolios based on stocks selected by this PCA-based method is more computationally efficient and comparably effective at minimizing the tracking error. When the number of index components is large, it is too computationally demanding to apply methods in Chapter 3 or most of existing methods, such as those relying on mixed-integer quadratic programming. In Chapter 4, factor models are used to describe stock returns. Under this assumption, the tracking error is partitioned into two parts: one depends on common economic factors, and the other depends on idiosyncratic risks. According to this partition, a 2-stage method is introduced to construct tracking portfolios by minimizing the tracking error. Stage 1 relies on a mixed-integer linear program to identify stocks that are able to reduce factors' impacts on the tracking error, and Stage2 determines weights of identified stocks by minimizing the tracking error. This 2-stage method efficiently constructs tracking portfolios benchmarked to indices with thousands of components. It reduces out-of-sample tracking errors significantly. In Chapter 5, the index tracking problem is solved by repeatedly solving one-period tracking problems. Each one-period tracking strategy is determined by a quadratic optimization with the L-1 regularization on asset weights. This formulation considers transaction costs and other practical constraints. Since the true joint distribution of financial returns is usually unknown, we solve one-period tracking problems under empirical distributions. With the L-1 regularization on asset weights, our one-period tracking strategy enjoys persistent properties in the high-dimensional setting. More specifically, the variable number d=d(n)=O(n^ α), where n is the sample size and α>1. Simulation studies are carried out to support our one-period tracking strategy's performance with finite samples. Applications on real financial data provide evidence that, in dealing with one-period tracking, this tracking strategy outperforms the L-q penalty tracking method in terms of tracking performance and computational efficiency. In terms of multi-period tracking, this proposed method outperforms the full-replication strategy

Approximating non-Gaussian Bayesian networks using minimum information vine model with applications in financial modelling

Many financial modeling applications require to jointly model multiple uncertain quantities to presentmore accurate, near future probabilistic predictions. Informed decision making would certainly benefitfrom such predictions. Bayesian networks (BNs) and copulas are widely used for modeling numerousuncertain scenarios. Copulas, in particular, have attracted more interest due to their nice property ofapproximating the probability distribution of the data with heavy tail. Heavy tail data is frequentlyobserved in financial applications. The standard multivariate copula suffer from serious limitations whichmade them unsuitable for modeling the financial data. An alternative copula model called the pair-copulaconstruction (PCC) model is more flexible and efficient for modeling the complex dependence of finan-cial data. The only restriction of PCC model is the challenge of selecting the best model structure. Thisissue can be tackled by capturing conditional independence using the Bayesian network PCC (BN-PCC).The flexible structure of this model can be derived from conditional independences statements learnedfrom data. Additionally, the difficulty of computing conditional distributions in graphical models for non-Gaussian distributions can be eased using pair-copulas. In this paper, we extend this approach furtherusing the minimum information vine model which results in a more flexible and efficient approach inunderstanding the complex dependence between multiple variables with heavy tail dependence andasymmetric features which appear widely in the financial applications