Multiobjective Approach to Portfolio Optimization in the Light of the Credibility Theory

[EN] The present research proposes a novel methodology to solve the problems faced by investors who take into consideration different investment criteria in a fuzzy context. The approach extends the stochastic mean-variance model to a fuzzy multiobjective model where liquidity is considered to quantify portfolio's performance, apart from the usual metrics like return and risk. The uncertainty of the future returns and the future liquidity of the potential assets are modelled employing trapezoidal fuzzy numbers. The decision process of the proposed approach considers that portfolio selection is a multidimensional issue and also some realistic constraints applied by investors. Particularly, this approach optimizes the expected return, the risk and the expected liquidity of the portfolio, considering bound constraints and cardinality restrictions. As a result, an optimization problem for the constraint portfolio appears, which is solved by means of the NSGA-II algorithm. This study defines the credibilistic Sortino ratio and the credibilistic STARR ratio for selecting the optimal portfolio. An empirical study on the S&P100 index is included to show the performance of the model in practical applications. The results obtained demonstrate that the novel approach can beat the index in terms of return and risk in the analyzed period, from 2008 until 2018.García García, F.; González-Bueno, J.; Guijarro, F.; Oliver-Muncharaz, J.; Tamosiuniene, R. (2020). Multiobjective Approach to Portfolio Optimization in the Light of the Credibility Theory. Technological and Economic Development of Economy (Online). 26(6):1165-1186. https://doi.org/10.3846/tede.2020.13189S11651186266Acerbi, C., & Tasche, D. (2002). On the coherence of expected shortfall. Journal of Banking & Finance, 26(7), 1487-1503. doi:10.1016/s0378-4266(02)00283-2Ahmed, A., Ali, R., Ejaz, A., & Ahmad, I. (2018). 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An Evolutionary Approach to Multistage Portfolio Optimization

Portfolio optimization is an important problem in quantitative finance due to its application in asset management and corporate financial decision making. This involves quantitatively selecting the optimal portfolio for an investor given their asset return distribution assumptions, investment objectives and constraints. Analytical portfolio optimization methods suffer from limitations in terms of the problem specification and modelling assumptions that can be used. Therefore, a heuristic approach is taken where Monte Carlo simulations generate the investment scenarios and' a problem specific evolutionary algorithm is used to find the optimal portfolio asset allocations. Asset allocation is known to be the most important determinant of a portfolio's investment performance and also affects its risk/return characteristics. The inclusion of equity options in an equity portfolio should enable an investor to improve their efficient frontier due to options having a nonlinear payoff. Therefore, a research area of significant importance to equity investors, in which little research has been carried out, is the optimal asset allocation in equity options for an equity investor. A purpose of my thesis is to carry out an original analysis of the impact of allowing the purchase of put options and/or sale of call options for an equity investor. An investigation is also carried out into the effect ofchanging the investor's risk measure on the optimal asset allocation. A dynamic investment strategy obtained through multistage portfolio optimization has the potential to result in a superior investment strategy to that obtained from a single period portfolio optimization. Therefore, a novel analysis of the degree of the benefits of a dynamic investment strategy for an equity portfolio is performed. In particular, the ability of a dynamic investment strategy to mimic the effects ofthe inclusion ofequity options in an equity portfolio is investigated. The portfolio optimization problem is solved using evolutionary algorithms, due to their ability incorporate methods from a wide range of heuristic algorithms. Initially, it is shown how the problem specific parts ofmy evolutionary algorithm have been designed to solve my original portfolio optimization problem. Due to developments in evolutionary algorithms and the variety of design structures possible, a purpose of my thesis is to investigate the suitability of alternative algorithm design structures. A comparison is made of the performance of two existing algorithms, firstly the single objective stepping stone island model, where each island represents a different risk aversion parameter, and secondly the multi-objective Non-Dominated Sorting Genetic Algorithm2. Innovative hybrids of these algorithms which also incorporate features from multi-objective evolutionary algorithms, multiple population models and local search heuristics are then proposed. . A novel way is developed for solving the portfolio optimization by dividing my problem solution into two parts and then applying a multi-objective cooperative coevolution evolutionary algorithm. The first solution part consists of the asset allocation weights within the equity portfolio while the second solution part consists 'ofthe asset allocation weights within the equity options and the asset allocation weights between the different asset classes. An original portfolio optimization multiobjective evolutionary algorithm that uses an island model to represent different risk measures is also proposed.Imperial Users onl

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

Twenty years of linear programming based portfolio optimization

a b s t r a c t Markowitz formulated the portfolio optimization problem through two criteria: the expected return and the risk, as a measure of the variability of the return. The classical Markowitz model uses the variance as the risk measure and is a quadratic programming problem. Many attempts have been made to linearize the portfolio optimization problem. Several different risk measures have been proposed which are computationally attractive as (for discrete random variables) they give rise to linear programming (LP) problems. About twenty years ago, the mean absolute deviation (MAD) model drew a lot of attention resulting in much research and speeding up development of other LP models. Further, the LP models based on the conditional value at risk (CVaR) have a great impact on new developments in portfolio optimization during the first decade of the 21st century. The LP solvability may become relevant for real-life decisions when portfolios have to meet side constraints and take into account transaction costs or when large size instances have to be solved. In this paper we review the variety of LP solvable portfolio optimization models presented in the literature, the real features that have been modeled and the solution approaches to the resulting models, in most of the cases mixed integer linear programming (MILP) models. We also discuss the impact of the inclusion of the real features

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

An asset and liability management model incorporating uncertainty

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.Asset and Liability Management (ALIvI) is a well-established method, which enables companies to match future liabilities with future cash flow streams of assets. The first stage is to develop a deterministic model with forecast cash flow streams. In reality this can lead to results that are often volatile to deviations of future cash flows from their predicted values. There are two main stages to this problem. Firstly, there is the issue of representing the future uncertainties. To this end we have developed a scenario generator that forecasts alternative realizations of future cash flows streams of different assets using alternative scenarios about a financial Index and the Capital Asset Pricing Model (CAPM). Considering this with the deterministic model leads to the creation of ALM models which incorporate uncertainty. Having represented the uncertainty, we use an optimisation model to generate the current decisions concerning acquisition and disposal of assets. This model is a two stage stochastic programming model that aims to achieve targeted cash flows for each future year. Risk is represented in the form of assigning shares to different risk groups. In this thesis we describe our models of randomness and how they are captured in the two-stage stochastic programming model. We compare our model to a mean-variance representation. Both models are simulated through time. Backtesting is used to investigate the quality of both approaches

Modelling Financial Data and Portfolio Optimization Problems

This doctoral dissertation in management science, entitled “Modelling Financial Data and Portfolio Optimization Problems”, consists of two independent parts, whose unifying theme is the construction and solution of mathematical programming models motivated by portfolio selection problems. As such, this work is located at the interface of operations research and of finance. It draws heavily on techniques and theoretical results originating in both disciplines. The first part of the dissertation (Chapter 2) deals with an extension of Markowitz model and takes into account some of the side-constraints faced by a decision-maker when composing an investment portfolio, viz. lower and upper bounds on the quantities traded, and upper bounds on the number of assets included in the portfolio. We focus on the algorithmic difficulties raised by this model and we describe an original simulated annealing heuristic for its solution. The second (and largest) part of the thesis deals with a new multiperiod model for the optimization of a portfolio of options linked to a single index (Chapters 4-10). The objective of the model is to maximize the expected return of the portfolio under constraints limiting its value-at-risk. The model contains several interesting features, like the possibility to rebalance the portfolio with options introduced at the start of each period, explicit consideration of transaction costs, realistic pricing of options, consideration of advanced probability models to represent the future, etc. Some deep theoretical results from the financial literature are exploited in order to enrich the model and to extend its applicability. In particular, several available schemes for the generation of scenarios and for option pricing have been critically examined, and the most appropriate ones have been implemented. Furthermore, several optimization approaches (heuristic or exact procedures) have also been developed, implemented and tested. The models investigated in the dissertation bear on very different portfolio problems, draw on separate streams of scientific literature, and are handled by distinct algorithmic techniques. Therefore, the corresponding parts of the dissertation are fully independent, and each part contains its own specific introduction and literature review

A stochastic programming model for dynamic portfolio management with financial derivatives

Stochastic optimization models have been extensively applied to financial portfolios and have proven their effectiveness in asset and asset-liability management. Occasionally, however, they have been applied to dynamic portfolio problems including not only assets traded in secondary markets but also derivative contracts such as options or futures with their dedicated payoff functions. Such extension allows the construction of asymmetric payoffs for hedging or speculative purposes but also leads to several mathematical issues. Derivatives-based nonlinear portfolios in a discrete multistage stochastic programming (MSP) framework can be potentially very beneficial to shape dynamically a portfolio return distribution and attain superior performance. In this article we present a portfolio model with equity options, which extends significantly previous efforts in this area, and analyse the potential of such extension from a modeling and methodological viewpoints. We consider an asset universe and model portfolio set-up including equity, bonds, money market, a volatility-based exchange-traded-fund (ETF) and over-the-counter (OTC) option contracts on the equity. Relying on this market structure we formulate and analyse, to the best of our knowledge, for the first time, a comprehensive set of optimal option strategies in a discrete framework, including canonical protective puts, covered calls and straddles, as well as more advanced combined strategies based on equity options and the volatility index. The problem formulation relies on a data-driven scenario generation method for asset returns and option prices consistent with arbitrage-free conditions and incomplete market assumptions. The joint inclusion of option contracts and the VIX as asset class in a dynamic portfolio problem extends previous efforts in the domain of volatility-driven optimal policies. By introducing an optimal trade-off problem based on expected wealth and Conditional Value-at-Risk (CVaR), we formulate the problem as a stochastic linear program and present an extended set of numerical results across different market phases, to discuss the interplay among asset classes and options, relevant to financial engineers and fund managers. We find that options’ portfolios and trading in options strengthen an effective tail risk control, and help shaping portfolios returns’ distributions, consistently with an investor's risk attitude. Furthermore the introduction of a volatility index in the asset universe, jointly with equity options, leads to superior risk-adjusted returns, both in- and out-of-sample, as shown in the final case-study