Portfolio Selection Models And Algorithms Based On Multiple Measures

The volatility of financial assets induces the uncertainty in the financial market. In generalinvestors make investment between uncertain returns and risk. Thus the key concept in finan-cial decision analysis research is uncertainty research. The main uncertainty in an event can beexpressed in two different forms: random uncertainty and fuzzy uncertainty. Under random un-certainty, Markowitz proposed the mean-variance portfolio theory, which is the beginning of thesystemized quantitative study in financial investment. The mean-variance methodology formsthe core foundation of portfolio theory and becomes an important tool for investment manage-ment and decision making. However, for general mean-variance portfolio selection problemswith general inequality linear constraints, it is often impossible to obtain a closed form solutionand the explicit form of efficient frontier. Therefore, the explicit determination of the efficientasset holdings for some particular problems gives insight into the nature of the efficient fron-tier. It offers a basic idea to further discuss the structure of the efficient frontier. Furthermore,an abundance of empirical studies have confirmed that people exhibit more sensitivity towardlosses than they do toward gains. It is more practical to develop an asymmetric measure wherelosses are weighted differently from gains and satisfies weak loss aversion. In addition, Ambi-guity is a kind of things with objective attributes. Especially in the field of subjective cognitive,the application of fuzziness is wider than the randomness. In order to deal with different formsof fuzzy uncertainty in the securities market, we discuss portfolio selection problems based onpossibility theory and credibility theory. Moreover, we establish a theoretical framework forfuzzy portfolio selection research.The main results in this thesis are based on the following four aspects: discussing effectivefrontierforthesingle-indexportfolioselectionproblemandloss-aversionportfolioselectionun-der random uncertainty, discussing the single-period possibilistic portfolio selection problemswith different kinds of constraints, discussing two-period possibilistic and credibilistic portfo-lio adjusting problems and discussing multi-period possibilistic portfolio selection problems.Therefore, the main innovations are listed in the following four aspects:First, we discuss portfolio selection problem and the related algorithm with multiple mea-sures based on the random uncertainty. We obtained the closed form solution and explicit form of efficient frontier for the single-index problems with the budget constraints and lower or up-per bound constraints. It is shown that there are only asset quantities of parametric intervals.And the efficient frontier can be traced out in a monotonic fashion whereby assets reach to andremain at their boundaries in order of their expected returns. With an additional riskless asset,we show the Capital Market Line meets the efficient frontier for the risky assets only in the partcorresponding to its first parametric interval. In addition, we discuss the loss aversion portfolioselection problem with bilinear utility function, in which the CVaR model is a special case. Be-cause the bilinear utility function is non-differentiable, we transform the original problem to ahigher dimensional linear programming model. Moreover, degeneracy is a common occurrenceand does indeed result in cycling. We develop an active set degeneracy resolution method forthehigherdimensionalproblemonlywithassetquantitiesworkingmatrix. Theefficientfrontierstructure for single-index model is the extension of the result obtained by Best and Hlouskova.And there are little research considering the influence of the degeneracy in portfolio selection.Second, we discuss single-period portfolio selection problems with different constraintsbased on the possibility theory. At the beginning we proposed a new definition for weightedaverage of crisp possibilistic mean value, which measure an investor’s evaluation on riskyassets returns. Then we propose the single-period possibilistic portfolio selection models withhighestutilityscoreforboundedconstraintsanddevelopasequentialminimizationoptimizationalgorithm, which has linearly convergence. After that we discuss the problem with the generaltransaction costs and develop a comprehensive leaning partial swarm optimization algorithm.Moreover, we define a non-convex non-concave transaction cost function. In addition, based oncrisp possibilistic mean value, we define a general possibilistic lower partial moments, whichmeasuresdownsidelossesratherthanupsidegains. Undertheassumptionthattheassets’returnsareLR-typefuzzy variable, weobtaintheformulation of the possibilistic lower partial momentsandestablishsomemodels. Therearefewstudiesonsingle-periodfuzzyportfolioselectionwiththe constraints mentioned above.Third, we discuss two-period portfolio adjusting problems based on the possibility andcredibility theory. Based on possibility theory, using the weighted average of crisp possibilisticmean value and first type of possibilistic variance measures, we develop a possibilistic portfolioadjusting model with highest utility score, which accounts for an investor’s evaluation on future returns. Moreover,usingcrisppossibilisticmeanvalueandsecondtypeofpossibilisticvariance,we develop a risk aversion possibilistic portfolio adjusting model. We develop a correspond-ing sequential minimization optimization algorithm for each proposed problem. In addition,we give a simple method to estimate trapezoidal and triangular possibility distribution. Basedon credibility theory, by using credibilistic mean value and credibilistic variance, we propose acredibilistic portfolio adjusting model. Under the assumption that the returns of risky assets aretriangular variables, we present the crisp form for the proposed model and develop a sequen-tial quadratic programming algorithm. After that, we discuss the impact of new added assetsincluding risky and riskless assets. Then we design a quadratic programming active set methodto solve for the optimal solution. There is little research on two-period fuzzy portfolio adjustingproblem. The discussion on two-period portfolio adjusting problems based on different kinds ofmeasures and constraints offers basic theory and helps investors making decision in constantlychanging securities market.Forth, we discuss multi-period portfolio selection problems based on the possibility theo-ry. There is little research on multi-period fuzzy portfolio selection problems. Most of them arefocus on single period. Based on the central value concept, we develop general expressions ofcentral value and dispersion for multi-period portfolios. We furthermore propose the measuresof possibilistic mean value and variance for multi-period portfolios and obtain the possibilisticmulti-period portfolio selection model. Given symmetric triangular fuzzy variables, we respec-tively obtain the crisp forms for the proposed models and develop a partial swarm optimization.Furthermore, we extend the result to the multi-period portfolio selection problems with transac-tion costs and obtain the corresponding model and algorithm.