In the traditional time series modeling and portfolio design, the data is usually conveniently assumed to follow the multivariate Gaussian distribution. But it has been empirically recognized that the real-world data is rarely Gaussian distributed. The collected data usually exhibit significantly heavier tail than the Gaussian distribution. The traditional statistical analysis methods based on the Gaussian assumption are no longer applicable, and new efficient statistical analysis methods for heavy-tailed data need to be designed. Besides, it has been recognized that, some heuristic portfolios may outperform the theory-based portfolios, which are initially designed from a reasonable and rigorously theoretical motivation. There is a lack of explanation for such surprising performance of...[ Read more ]

In the traditional time series modeling and portfolio design, the data is usually conveniently assumed to follow the multivariate Gaussian distribution. But it has been empirically recognized that the real-world data is rarely Gaussian distributed. The collected data usually exhibit significantly heavier tail than the Gaussian distribution. The traditional statistical analysis methods based on the Gaussian assumption are no longer applicable, and new efficient statistical analysis methods for heavy-tailed data need to be designed. Besides, it has been recognized that, some heuristic portfolios may outperform the theory-based portfolios, which are initially designed from a reasonable and rigorously theoretical motivation. There is a lack of explanation for such surprising performance of the heuristic portfolios. The main purpose of this thesis is to develop efficient algorithms for heavy-tailed vector autoregressive (VAR) modeling with missing data and high-order portfolio (which is able to handle heavy-tailed returns) optimization, and to explain the heuristic portfolios from the perspective of robust portfolio optimization.

The VAR models provide a significant tool for multivariate time series analysis. Owing to the mathematical simplicity, existing works on VAR modeling are rigidly inclined towards the multivariate Gaussian distribution. However, heavy-tailed distributions are suggested more reasonable for capturing the real-world phenomena, like the presence of outliers and a stronger possibility of extreme values. Furthermore, missing values in observed data is a real problem, which typically happens during the data observation or recording process. In this thesis, we propose an algorithmic framework to estimate the parameters of a VAR model with heavy-tailed Student’s t distributed innovations from incomplete data based on the stochastic approximation expectation maximization algorithm coupled with a Markov Chain Monte Carlo procedure. We propose two fast and computationally cheap Gibbs sampling schemes. The algorithms developed are effective in capturing the heavy-tailed phenomenon and being robust against outliers and missing data. Extensive experiments with both synthetic data and real financial data corroborate our claims.

The first moment and second central moments of the portfolio return, a.k.a. mean and variance, have been widely employed to assess the expected profit and risk of the portfolio. Investors pursue higher mean and lower variance when designing the portfolios. The two moments can well describe the distribution of the portfolio return when it follows the Gaussian distribution. The asymmetry and the heavy-tailedness, which are properties of empirical data, are characterized by the third and fourth central moments, i.e., skewness and kurtosis, respectively. Higher skewness and lower kurtosis are preferred to reduce the probability of extreme losses. However, incorporating high-order moments in the portfolio design is very difficult due to their non-convexity and rapidly increasing computational cost with the dimension. In this thesis, we propose a very efficient and convergence-provable algorithm framework based on the successive convex approximation (SCA) algorithm to solve high-order portfolios. The efficiency of the proposed algorithm framework is demonstrated by the numerical experiments.

The heuristic 1/N (i.e., equally weighted) portfolio and heuristic quintile portfolio are both popular simple strategies in financial investment. In the 1/N portfolio, a fraction of 1/N of the wealth is allocated to each of the N available assets. In the quintile portfolio, first the assets are sorted according to some characteristics, e.g., expected returns, and then the strategy equally longs the top 20% (i.e., top quintile) and perhaps shorts the bottom 20% (i.e., bottom quintile). Although they have been criticized for their lack of mathematical justification when proposed by practitioners, they have shown great advantage over more sophisticated portfolios in terms of stable performance and easy deployment. In this thesis, we reinterpret the 1/N and quintile portfolios as solutions to a mathematically sound robust portfolio optimization under different levels of robustness level in the stocks’ characteristics.