Scalable sparse covariance estimation via self-concordance

We consider the class of convex minimization problems, composed of a self-concordant function, such as the $\log\det$ metric, a convex data fidelity term $h(\cdot)$ and, a regularizing -- possibly non-smooth -- function $g(\cdot)$. This type of problems have recently attracted a great deal of interest, mainly due to their omnipresence in top-notch applications. Under this \emph{locally} Lipschitz continuous gradient setting, we analyze the convergence behavior of proximal Newton schemes with the added twist of a probable presence of inexact evaluations. We prove attractive convergence rate guarantees and enhance state-of-the-art optimization schemes to accommodate such developments. Experimental results on sparse covariance estimation show the merits of our algorithm, both in terms of recovery efficiency and complexity.Comment: 7 pages, 1 figure, Accepted at AAAI-1

Optimal risk minimization of Australian energy and mining portfolios under multiple measures of risk

Australia’s 2000’s decade saw the sharpest rise in mining investments arising from developing Asian emerging economies’ high demand for commodities like coal, iron ore, nickel, oil and gas which drove up prices to a historic level (Connolly & Orsmond, 2011). As of December 2012, 39 % and 9 % of the Australian Securities Exchange’s stocks were of the mining (coal and uranium stocks are included in this category) and energy (e.g. oil, gas and renewable energy stocks) sectors respectively, and investors recently have been considering separate portfolio positions in energy and mining stocks (Jennings, 2010). Facts of these nature set the stage for the task of selecting an optimal portfolio of stock securities where the fundamental questions faced by every investor, individual or institutional, are: a) what is the optimal point in time to go long in the investment position?, b) what are the optimal amounts to invest in every asset of a portfolio? and, c) when is the optimal time to short the portfolio investment position? The focus of the present study is on b) within a one period ahead forecast scenario. Understanding the price and volatility movements of stock securities taking as a basis of study their own dynamics and co-dynamics is a complex task that may be better addressed through a multilateral modelling approach. This paper, in this regard, departs from a single model application by fitting multiple risk measures to the optimization of four portfolios each consisting of 20 ASX’s stocks from the gold, iron ore-nickel, uranium-coal and oil-gas sectors. The five risk measures compared are: the variance, mean absolute deviation (MAD), minimizing regret (Minimax), conditional value at risk (CVaR), and conditional drawdown at risk (CDaR), where the last two are threshold based measures. The risk measure parameters are input into meanvariance quadratic (QP) and differential evolution (DE) portfolio problem specifications. Accurate estimations of the underlying interaction of stocks return series is a crucial element in portfolio allocation and portfolio risk management and frequentist traditional measures of dependence are rather inadequate. Here, with the objective of achieving more accuracy in the estimation of the dependence matrix, a Gaussian pair c-vine copula (PC), the regular graphical lasso (RL) and adaptive graphical lasso (AL) are fitted. Possible advantages from using these recently proposed and sophisticated techniques under model specifications where the covariance matrix is the measure of risk are indicated. The main objectives of the present study are to calculate the optimal weights to be invested in every stock of the portfolios making use of linear and nonlinear model specifications and the risk measures suggested, analyse the weight allocation differences and seek portfolio optimization advantages from using pair vine copulas and the graphical lasso in the estimation of dependence. The present multimodal approach is, therefore, expected to be more robust and as a consequence, provide more complete information that could serve for improved decision making on portfolio selection, allocation and rebalancing. Research questions are answered based on the analysis of gold portfolio outcome values, only. Findings indicate that CDaR is an important risk measure to be considered, along with other measures of risk when optimizing portfolios of stocks and no single measure of risk is suggested alone. The Gaussian pair cvine copula through the use of one different parameter in the modelling of every pair of variables’ joint distribution appears to be more sensitive in capturing data’s distribution characteristics. The adaptive graphical lasso also appears to be more perceptive when grasping the signal of the underlying interaction of the stocks. Therefore, valuable information could be drawn and inferred from applying multiple risk measures and sophisticated statistical techniques for their estimation. The weight allocation from threshold risk measures such as CVar and DaR and Minimax clearly differs from the rest. The models identified stocks with high return relative to risk and vice versa. The originality of the present study lies on the sectors of application and its multi-model nature

Optimal Portfolio Using Factor Graphical Lasso

Graphical models are a powerful tool to estimate a high-dimensional inverse covariance (precision) matrix, which has been applied for a portfolio allocation problem. The assumption made by these models is a sparsity of the precision matrix. However, when stock returns are driven by common factors, such assumption does not hold. We address this limitation and develop a framework, Factor Graphical Lasso (FGL), which integrates graphical models with the factor structure in the context of portfolio allocation by decomposing a precision matrix into low-rank and sparse components. Our theoretical results and simulations show that FGL consistently estimates the portfolio weights and risk exposure and also that FGL is robust to heavy-tailed distributions which makes our method suitable for financial applications. FGL-based portfolios are shown to exhibit superior performance over several prominent competitors including equal-weighted and Index portfolios in the empirical application for the S&P500 constituents.Comment: 71 pages, 10 figures, 5 table