Portfolio Decisions with Higher Order Moments

In this paper, we address the global optimization of two interesting nonconvex problems in finance. We relax the normality assumption underlying the classical Markowitz mean-variance portfolio optimization model and consider the incorporation of skewness (third moment) and kurtosis (fourth moment). The investor seeks to maximize the expected return and the skewness of the portfolio and minimize its variance and kurtosis, subject to budget and no short selling constraints. In the first model, it is assumed that asset statistics are exact. The second model allows for uncertainty in asset statistics. We consider rival discrete estimates for the mean, variance, skewness and kurtosis of asset returns. A robust optimization framework is adopted to compute the best investment portfolio maximizing return, skewness and minimizing variance, kurtosis, in view of the worst-case asset statistics. In both models, the resulting optimization problems are nonconvex. We introduce a computational procedure for their global optimization.Mean-variance portfolio selection, Robust portfolio selection, Skewness, Kurtosis, Decomposition methods, Polynomial optimization problems

Robust portfolio selection involving options under a “ marginal+joint ” ellipsoidal uncertainty set

AbstractIn typical robust portfolio selection problems, one mainly finds portfolios with the worst-case return under a given uncertainty set, in which asset returns can be realized. A too large uncertainty set will lead to a too conservative robust portfolio. However, if the given uncertainty set is not large enough, the realized returns of resulting portfolios will be outside of the uncertainty set when an extreme event such as market crash or a large shock of asset returns occurs. The goal of this paper is to propose robust portfolio selection models under so-called “ marginal+joint” ellipsoidal uncertainty set and to test the performance of the proposed models. A robust portfolio selection model under a “marginal + joint” ellipsoidal uncertainty set is proposed at first. The model has the advantages of models under the separable uncertainty set and the joint ellipsoidal uncertainty set, and relaxes the requirements on the uncertainty set. Then, one more robust portfolio selection model with option protection is presented by combining options into the proposed robust portfolio selection model. Convex programming approximations with second-order cone and linear matrix inequalities constraints to both models are derived. The proposed robust portfolio selection model with options can hedge risks and generates robust portfolios with well wealth growth rate when an extreme event occurs. Tests on real data of the Chinese stock market and simulated options confirm the property of both the models. Test results show that (1) under the “ marginal+joint” uncertainty set, the wealth growth rate and diversification of robust portfolios generated from the first proposed robust portfolio model (without options) are better and greater than those generated from Goldfarb and Iyengar’s model, and (2) the robust portfolio selection model with options outperforms the robust portfolio selection model without options when some extreme event occurs

Robust and Multi-objective Portfolio Selection

In this thesis, robust and multi-objective portfolio selection problem will be studied. New models and computational algorithms will be developed to solve the proposed models. In particularly, we have studied multi-objective portfolio selection with inexact information on investment return and covariance matrix. The problems have been transformed into easily solvable problems through theoretical analysis. Numerical experiments are presented to validate the methods

Bio-Inspired Machine Learning for Distributed Confidential Multi-Portfolio Selection Problem

The recently emerging multi-portfolio selection problem lacks a proper framework to ensure that client privacy and database secrecy remain intact. Since privacy is of major concern these days, in this paper, we propose a variant of Beetle Antennae Search (BAS) known as Distributed Beetle Antennae Search (DBAS) to optimize multi-portfolio selection problems without violating the privacy of individual portfolios. DBAS is a swarm-based optimization algorithm that solely shares the gradients of portfolios among the swarm without sharing private data or portfolio stock information. DBAS is a hybrid framework, and it inherits the swarm-like nature of the Particle Swarm Optimization (PSO) algorithm with the BAS updating criteria. It ensures a robust and fast optimization of the multi-portfolio selection problem whilst keeping the privacy and secrecy of each portfolio intact. Since multi-portfolio selection problems are a recent direction for the field, no work has been done concerning the privacy of the database nor the privacy of stock information of individual portfolios. To test the robustness of DBAS, simulations were conducted consisting of four categories of multi-portfolio problems, where in each category, three portfolios were selected. To achieve this, 200 days worth of real-world stock data were utilized from 25 NASDAQ stock companies. The simulation results prove that DBAS not only ensures portfolio privacy but is also efficient and robust in selecting optimal portfolios

Robust optimisation and its application to portfolio planning

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.Decision making under uncertainty presents major challenges from both modelling and solution methods perspectives. The need for stochastic optimisation methods is widely recognised; however, compromises typically have to be made in order to develop computationally tractable models. Robust optimisation is a practical alternative to stochastic optimisation approaches, particularly suited for problems in which parameter values are unknown and variable. In this thesis, we review robust optimisation, in which parameter uncertainty is defined by budgeted polyhedral uncertainty sets as opposed to ellipsoidal sets, and consider its application to portfolio selection. The modelling of parameter uncertainty within a robust optimisation framework, in terms of structure and scale, and the use of uncertainty sets is examined in detail. We investigate the effect of different definitions of the bounds on the uncertainty sets. An interpretation of the robust counterpart from a min-max perspective, as applied to portfolio selection, is given. We propose an extension of the robust portfolio selection model, which includes a buy-in threshold and an upper limit on cardinality. We investigate the application of robust optimisation to portfolio selection through an extensive empirical investigation of cost, robustness and performance with respect to risk-adjusted return measures and worst case portfolio returns. We present new insights into modelling uncertainty and the properties of robust optimal decisions and model parameters. Our experimental results, in the application of portfolio selection, show that robust solutions come at a cost, but in exchange for a guaranteed probability of optimality on the objective function value, significantly greater achieved robustness, and generally better realisations under worst case scenarios

Case: Peatland Selection

The importance of environmental decision making is growing. Private companies and public organizations are facing decisions involving multiple objectives. In particular, focusing solely on financial objectives is no longer enough but taking into account the environmental, social and political objectives is needed. The methods used to solve these environmental problems have been based on heuristic approaches. However, these methods lack the capability to provide optimal solutions as most of the environmental decisions are portfolio selection problems. Robust Portfolio Modeling (RPM) is a decision analysis method that combines mathematical optimization in portfolio selection to incomplete preference information. This incomplete information is common in environmental decision making which includes multiple stakeholders with conflicting views. However, RPM has not been applied before to real-life environmental cases. This thesis will first explore the characteristics of environmental decision making, secondly go through different methods used in environmental decision making and finally apply RPM methodology into peatland selection case. The results of RPM are then compared to the results of the heuristic YODA method previously used in the same peatland selection case. Results indicate that RPM and YODA select highly different type of peatlands. RPM takes better into account the cumulative effects related to portfolio selection than YODA. Therefore, it is argued that RPM might be suitable for environmental decision making

Robust optimisation and its application to portfolio planning

Decision making under uncertainty presents major challenges from both modelling and solution methods perspectives. The need for stochastic optimisation methods is widely recognised; however, compromises typically have to be made in order to develop computationally tractable models. Robust optimisation is a practical alternative to stochastic optimisation approaches, particularly suited for problems in which parameter values are unknown and variable. In this thesis, we review robust optimisation, in which parameter uncertainty is defined by budgeted polyhedral uncertainty sets as opposed to ellipsoidal sets, and consider its application to portfolio selection. The modelling of parameter uncertainty within a robust optimisation framework, in terms of structure and scale, and the use of uncertainty sets is examined in detail. We investigate the effect of different definitions of the bounds on the uncertainty sets. An interpretation of the robust counterpart from a min-max perspective, as applied to portfolio selection, is given. We propose an extension of the robust portfolio selection model, which includes a buy-in threshold and an upper limit on cardinality. We investigate the application of robust optimisation to portfolio selection through an extensive empirical investigation of cost, robustness and performance with respect to risk-adjusted return measures and worst case portfolio returns. We present new insights into modelling uncertainty and the properties of robust optimal decisions and model parameters. Our experimental results, in the application of portfolio selection, show that robust solutions come at a cost, but in exchange for a guaranteed probability of optimality on the objective function value, significantly greater achieved robustness, and generally better realisations under worst case scenarios.EThOS - Electronic Theses Online ServiceGBUnited Kingdo