Enriqueta Vercher

Fuzzy portfolio optimization under downside risk measures

This paper presents two fuzzy portfolio selection models where the objective is to minimize the downside risk constrained by a given expected return. We assume that the rates of returns on securities are approximated as LR-fuzzy numbers of the same shape, and that the expected return and risk are evaluated by interval-valued means. We establish the relationship between those mean-interval definitions for a given fuzzy portfolio by using suitable ordering relations. Finally, we formulate the portfolio selection problem as a linear program when the returns on the assets are of trapezoidal form.

Viability of infeasible portfolio selection problems: A fuzzy approach☆

Abstract This paper deals with fuzzy optimization schemes for managing a portfolio in the framework of risk–return trade-off. Different models coexist to select the best portfolio according to their respective objective functions and many of them are linearly constrained. We are concerned with the infeasible instances of such models. This infeasibility, usually provoked by the conflict between the desired return and the diversification requirements proposed by the investor, can be satisfactorily avoided by using fuzzy linear programming techniques. We propose an algorithm to repair infeasibility and we illustrate its performance on a numerical example.

A multi-objective genetic algorithm for cardinality constrained fuzzy portfolio selection

This paper presents a new procedure that extends genetic algorithms from their traditional domain of optimization to fuzzy ranking strategy for selecting efficient portfolios of restricted cardinality. The uncertainty of the returns on a given portfolio is modeled using fuzzy quantities and a downside risk function is used to describe the investors aversion to risk. The fitness functions are based both on the value and the ambiguity of the trapezoidal fuzzy number which represents the uncertainty on the return. The soft-computing approach allows us to consider uncertainty and vagueness in databases and also to incorporate subjective characteristics into the portfolio selection problem. We use a data set from the Spanish stock market to illustrate the performance of our approach to the portfolio selection problem.

A spreadsheet modeling approach to the Holt–Winters optimal forecasting

Abstract The objective of this paper is to determine the optimal forecasting for the Holt–Winters exponential smoothing model using spreadsheet modeling. This forecasting procedure is especially useful for short-term forecasts for series of sales data or levels of demand for goods. The non-linear programming problem associated with this forecasting model is formulated and a spreadsheet model is used to solve the problem of optimization efficiently. Also, a spreadsheet makes it possible to work in parallel with various objective functions (measures of forecast errors) and different procedures for calculating the initial values of the components of the model. Using a scenario analysis, the set of local minima obtained may be visualized. We have solved some examples in order to illustrate this approach.

Optimality conditions for nondifferentiable convex semi-infinite programming

This paper gives characterizations of optimal solutions to the nondifferentiable convex semi-infinite programming problem, which involve the notion of Lagrangian saddlepoint. With the aim of giving the necessary conditions for optimality, local and global constraint qualifications are established. These constraint qualifications are based on the property of Farkas-Minkowski, which plays an important role in relation to certain systems obtained by linearizing the feasible set. It is proved that Slaters qualification implies those qualifications.

Solving a class of fuzzy linear programs by using semi-infinite programming techniques☆

This paper deals with a class of Fuzzy Linear Programming problems characterized by the fact that the coefficients in the constraints are modeled as LR-fuzzy numbers with different shapes. Solving such problems is usually more complicated than finding a solution when all the fuzzy coefficients have the same shape. We propose a primal semi-infinite algorithm as a valuable tool for solving this class of Fuzzy Linear programs and, we illustrate it by means of several examples.

Evolutionary multi-objective optimization algorithms for fuzzy portfolio selection

Graphical abstractDisplay Omitted HighlightsWe consider a constrained three-objective optimization portfolio selection problem.We solve the problem by means of evolutionary multi-objective optimization.New mutation, crossover and reparation operators are designed for this problem.They are tested in several algorithms for a data set from the Spanish stock market.Results for two performance metrics reveal the effectiveness of the new operators. In this paper, we consider a recently proposed model for portfolio selection, called Mean-Downside Risk-Skewness (MDRS) model. This modelling approach takes into account both the multidimensional nature of the portfolio selection problem and the requirements imposed by the investor. Concretely, it optimizes the expected return, the downside-risk and the skewness of a given portfolio, taking into account budget, bound and cardinality constraints. The quantification of the uncertain future return on a given portfolio is approximated by means of LR-fuzzy numbers, while the moments of its return are evaluated using possibility theory. The main purpose of this paper is to solve the MDRS portfolio selection model as a whole constrained three-objective optimization problem, what has not been done before, in order to analyse the efficient portfolios which optimize the three criteria simultaneously. For this aim, we propose new mutation, crossover and reparation operators for evolutionary multi-objective optimization, which have been specially designed for generating feasible solutions of the cardinality constrained MDRS problem. We incorporate the operators suggested into the evolutionary algorithms NSGAII, MOEA/D and GWASF-GA and we analyse their performances for a data set from the Spanish stock market. The potential of our operators is shown in comparison to other commonly used genetic operators and some conclusions are highlighted from the analysis of the trade-offs among the three criteria.

Holt–Winters Forecasting: An Alternative Formulation Applied to UK Air Passenger Data

Abstract This paper provides a formulation for the additive Holt–Winters forecasting procedure that simplifies both obtaining maximum likelihood estimates of all unknowns, smoothing parameters and initial conditions, and the computation of point forecasts and reliable predictive intervals. The stochastic component of the model is introduced by means of additive, uncorrelated, homoscedastic and Normal errors, and then the joint distribution of the data vector, a multivariate Normal distribution, is obtained. In the case where a data transformation was used to improve the fit of the model, cumulative forecasts are obtained here using a Monte-Carlo approximation. This paper describes the method by applying it to the series of monthly total UK air passengers collected by the Civil Aviation Authority, a long time series from 1949 to the present day, and compares the resulting forecasts with those obtained in previous studies.

A decision support system methodology for forecasting of time series based on soft computing

Exponential procedures are widely used as forecasting techniques for inventory control and business planning. A number of modifications to the generalized exponential smoothing (Holt-Winters) approach to forecasting univariate time series is presented, which have been adapted into a tool for decision support systems. This methodology unifies the phases of estimation and model selection into just one optimization framework which permits the identification of robust solutions. This procedure may provide forecasts from different versions of exponential smoothing by fitting the updated formulas of Holt-Winters and selects the best method using a fuzzy multicriteria approach. The elements of the set of local minima of the non-linear programming problems allow us to build the membership functions of the conflicting objectives. It is compared to other forecasting methods on the 111 series from the M-competition.

A Possibilistic Mean-Downside Risk-Skewness Model for Efficient Portfolio Selection

This paper presents a new possibilistic model for the portfolio selection problem. The uncertainty of future returns on a given portfolio is modeled using LR-fuzzy numbers. Some possibilistic moments are considered to measure the risk of and return on the investment. Since the joint possibility distribution of the returns on the assets is unknown, we consider the returns on a given portfolio as the historical dataset instead of considering the individual returns on the assets as the dataset. We introduce a coefficient of possibilistic skewness in order to incorporate a measurement of the asymmetry of the fuzzy return on a given portfolio. We solve the multi-objective optimization problems that are associated with the possibilistic mean-downside risk-skewness model by using an evolutionary procedure to generate efficient portfolios. The procedure provides different patterns of investment, whose portfolios meet the explicit restrictions imposed by the investor. Thus, from among the points in the efficient frontier, the investor may select a portfolio that optimizes an economically meaningful objective function. The performance of this approach is tested using a dataset of assets from the Spanish stock market.