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

Risk measures for direct real estate investments with non-normal or unknown return distributions

The volatility of returns is probably the most widely used risk measure for real estate. This is rather surprising since a number of studies have cast doubts on the view that volatility can capture the manifold risks attached to properties and corresponds to the risk attitude of investors. A central issue in this discussion is the statistical properties of real estate returns—in contrast to neoclassical capital market theory they are mostly non-normal and often unknown, which render many statistical measures useless. Based on a literature review and an analysis of data from Germany we provide evidence that volatility alone is inappropriate for measuring the risk of direct real estate. We use a unique data sample by IPD, which includes the total returns of 939 properties across different usage types (56% office, 20% retail, 8% others and 16% residential properties) from 1996 to 2009, the German IPD Index, and the German Property Index. The analysis of the distributional characteristics shows that German real estate returns in this period were not normally distributed and that a logistic distribution would have been a better fit. This is in line with most of the current literature on this subject and leads to the question which indicators are more appropriate to measure real estate risks. We suggest that a combination of quantitative and qualitative risk measures more adequately captures real estate risks and conforms better with investor attitudes to risk. Furthermore, we present criteria for the purpose of risk classification

Topics in portfolio management

In this thesis, two topics in portfolio management have been studied: utility-risk portfolio selection and a paradox in time consistency in mean-variance problem. The first topic is a comprehensive study on utility maximization subject to deviation risk constraints. Under the complete Black-Scholes framework, by using the martingale approach and mean-field heuristic, a static problem including a variational inequality and some constraints on nonlinear moments, called Nonlinear Moment Problem, has been obtained to completely characterize the optimal terminal payoff. By solving the Nonlinear Moment Problem, the various well-posed mean-risk problems already known in the literature have been revisited, and also the existence of the optimal solutions for both utility-downside-risk and utility-strictly-convex-risk problems has been established under the assumption that the underlying utility satisfies the Inada Condition. To the best of our knowledge, the positive answers to the latter two problems have long been absent in the literature. In particular, the existence of an optimal solution for utility-semivariance problem, an example of the utility-downside-risk problem, is in substantial contrast to the nonexistence of an optimal solution for the mean-semivariance problem. This existence result allows us to utilize semivariance as a risk measure in portfolio management. Furthermore, it has been shown that the continuity of the optimal terminal wealth in pricing kernel, thus the solutions in the binomial tree models converge to the solution in the continuous-time Black-Scholes model. The convergence can be applied to provide a numerical method to compute the optimal solution for utility-deviation-risk problem by using the optimal portfolios in the binomial tree models, which are easily computed; such numerical algorithm for optimal solution to utility-risk problem has been absent in the literature. In the second part of this thesis, a paradox in time consistency in mean-variance has been established. People often change their preference over time, so the maximizer for current preference may not be optimal in the future. We call this phenomenon time inconsistency or dynamic inconsistency. To manage the issues of time inconsistency, a game-theoretic approach is widely utilized to provide a time-consistent equilibrium solution for dynamic optimization problem. It has been established that, if investors with mean-variance preference adopt the equilibrium solutions, an investor facing short-selling prohibition can acquire a greater objective value than his counterpart without the prohibition in a buoyant market. It has been further shown that the pure strategy of solely investing in bond can sometimes simultaneously dominate both constrained and unconstrained equilibrium strategies. With numerical experiments, the constrained investor can dominate the unconstrained one for more than 90% of the time horizon. The source of paradox is rooted from the nature of game-theoretic approach on time consistency, which purposely seeks for an equilibrium solution but not the ultimate maximizer. Our obtained results actually advocate that, to properly implement the concept of time consistency in various financial problems, all economic aspects should be critically taken into account at a time.Open Acces

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

The risk parity approach to asset allocation

Thesis (MSc)--Stellenbosch University, 2014.ENGLISH ABSTRACT: We consider the problem of portfolio's asset allocation characterised by risk and return. Prior to the 2007-2008 financial crisis, this important problem was tackled using mainly the Markowitz mean-variance framework. However, throughout the past decade of challenging markets, particularly for equities, this framework has exhibited multiple drawbacks. Today many investors approach this problem with a 'safety first' rule that puts risk management at the heart of decision-making. Risk-based strategies have gained a lot of popularity since the recent financial crisis. One of the 'trendiest' of the modern risk-based strategies is the Risk Parity model, which puts diversification in terms of risk, but not in terms of dollar values, at the core of portfolio risk management. Inspired by the works of Maillard et al. (2010), Bruder and Roncalli (2012), and Roncalli and Weisang (2012), we examine the reliability and relationship between the traditional mean-variance framework and risk parity. We emphasise, through multiple examples, the non-diversification of the traditional mean-variance framework. The central focus of this thesis is on examining the main Risk-Parity strategies, i.e. the Inverse Volatility, Equal Risk Contribution and the Risk Budgeting strategies. Lastly, we turn our attention to the problem of maximizing the absolute expected value of the logarithmic portfolio wealth (sometimes called the drift term) introduced by Oderda (2013). The drift term of the portfolio is given by the sum of the expected price logarithmic growth rate, the expected cash flow, and half of its variance. The solution to this problem is a linear combination of three famous risk-based strategies and the high cash flow return portfolio.AFRIKAANSE OPSOMMING: Ons kyk na die probleem van batetoewysing in portefeuljes wat gekenmerk word deur risiko en wins. Voor die 2007-2008 finansiele krisis, was hierdie belangrike probleem deur die Markowitz gemiddelde-variansie raamwerk aangepak. Gedurende die afgelope dekade van uitdagende markte, veral vir aandele, het hierdie raamwerk verskeie nadele getoon. Vandag, benader baie beleggers hierdie probleem met 'n 'veiligheid eerste' reël wat risikobestuur in die hart van besluitneming plaas. Risiko-gebaseerde strategieë het baie gewild geword sedert die onlangse finansiële krisis. Een van die gewildste van die moderne risiko-gebaseerde strategieë is die Risiko- Gelykheid model wat diversifikasie in die hart van portefeulje risiko bestuur plaas. Geïnspireer deur die werke van Maillard et al. (2010), Bruder and Roncalli (2012), en Roncalli and Weisang (2012), ondersoek ons die betroubaarheid en verhouding tussen die tradisionele gemiddelde-variansie raamwerk en Risiko- Gelykheid. Ons beklemtoon, deur middel van verskeie voorbeelde, die niediversifikasie van die tradisionele gemiddelde-variansie raamwerk. Die sentrale fokus van hierdie tesis is op die behandeling van Risiko-Gelykheid strategieë, naamlik, die Omgekeerde Volatiliteit, Gelyke Risiko-Bydrae en Risiko Begroting strategieë. Ten slotte, fokus ons aandag op die probleem van maksimering van absolute verwagte waarde van die logaritmiese portefeulje welvaart (soms genoem die drif term) bekendgestel deur Oderda (2013). Die drif term van die portefeulje word gegee deur die som van die verwagte prys logaritmiese groeikoers, die verwagte kontantvloei, en die helfte van die variansie. Die oplossing vir hierdie probleem is 'n lineêre kombinasie van drie bekende risiko-gebaseerde strategieë en die hoë kontantvloei wins portefeulje

Sequential Machine learning Approaches for Portfolio Management

Cette thèse envisage un ensemble de méthodes permettant aux algorithmes d'apprentissage statistique de mieux traiter la nature séquentielle des problèmes de gestion de portefeuilles financiers. Nous débutons par une considération du problème général de la composition d'algorithmes d'apprentissage devant gérer des tâches séquentielles, en particulier celui de la mise-à-jour efficace des ensembles d'apprentissage dans un cadre de validation séquentielle. Nous énumérons les desiderata que des primitives de composition doivent satisfaire, et faisons ressortir la difficulté de les atteindre de façon rigoureuse et efficace. Nous poursuivons en présentant un ensemble d'algorithmes qui atteignent ces objectifs et présentons une étude de cas d'un système complexe de prise de décision financière utilisant ces techniques. Nous décrivons ensuite une méthode générale permettant de transformer un problème de décision séquentielle non-Markovien en un problème d'apprentissage supervisé en employant un algorithme de recherche basé sur les K meilleurs chemins. Nous traitons d'une application en gestion de portefeuille où nous entraînons un algorithme d'apprentissage à optimiser directement un ratio de Sharpe (ou autre critère non-additif incorporant une aversion au risque). Nous illustrons l'approche par une étude expérimentale approfondie, proposant une architecture de réseaux de neurones spécialisée à la gestion de portefeuille et la comparant à plusieurs alternatives. Finalement, nous introduisons une représentation fonctionnelle de séries chronologiques permettant à des prévisions d'être effectuées sur un horizon variable, tout en utilisant un ensemble informationnel révélé de manière progressive. L'approche est basée sur l'utilisation des processus Gaussiens, lesquels fournissent une matrice de covariance complète entre tous les points pour lesquels une prévision est demandée. Cette information est utilisée à bon escient par un algorithme qui transige activement des écarts de cours (price spreads) entre des contrats à terme sur commodités. L'approche proposée produit, hors échantillon, un rendement ajusté pour le risque significatif, après frais de transactions, sur un portefeuille de 30 actifs.This thesis considers a number of approaches to make machine learning algorithms better suited to the sequential nature of financial portfolio management tasks. We start by considering the problem of the general composition of learning algorithms that must handle temporal learning tasks, in particular that of creating and efficiently updating the training sets in a sequential simulation framework. We enumerate the desiderata that composition primitives should satisfy, and underscore the difficulty of rigorously and efficiently reaching them. We follow by introducing a set of algorithms that accomplish the desired objectives, presenting a case-study of a real-world complex learning system for financial decision-making that uses those techniques. We then describe a general method to transform a non-Markovian sequential decision problem into a supervised learning problem using a K-best paths search algorithm. We consider an application in financial portfolio management where we train a learning algorithm to directly optimize a Sharpe Ratio (or other risk-averse non-additive) utility function. We illustrate the approach by demonstrating extensive experimental results using a neural network architecture specialized for portfolio management and compare against well-known alternatives. Finally, we introduce a functional representation of time series which allows forecasts to be performed over an unspecified horizon with progressively-revealed information sets. By virtue of using Gaussian processes, a complete covariance matrix between forecasts at several time-steps is available. This information is put to use in an application to actively trade price spreads between commodity futures contracts. The approach delivers impressive out-of-sample risk-adjusted returns after transaction costs on a portfolio of 30 spreads