Constraint Programming Applied to Portfolio Design Problem

This chapter introduces Constraint Programming (CP) approaches for solving efficiently a ðnancial portfolio design problem. The CP includes powerful techniques for modeling and solving complex problems. Symmetry breaking coming firstly from CP has proved its efficiency in minimizing CPU times when the problem is symmetric. The authors have adopted CP techniques to model the problem in a constraints system to capitalize on the flexibility of the CP paradigm and to take into consideration the symmetric aspect of the problem. The authors propose different CP models and different hybridizations of symmetry breaking techniques to tackle the problem. Experimental results on non-trivial instances of the problem show the effectiveness of the CP approach.

Financial Engineering and Portfolio Design Problem

Any financial institution is in charge of assigning to a client's portfolio a set of assets in a reliable way by minimizing the risk of loss and maximizing gain. All portfolios should share in an equitable way risky assets and safe assets with respect to a financial structured product such as CDO2 (collateralized debt obligation squared). Realizing a balanced portfolio requires a good level of diversification on the chosen assets. Many works have made proposals for modeling and solving the (OPD) problem, but each one has taken into account specific types of risks and cases. However, in this chapter, the authors introduce the problem in a general way by using the CDO2 structure. The authors focus in this chapter on the basic and useful notions of financial engineering, followed by a description of the financial portfolio structure CDO2, the most used and structured financial product. The chapter introduces the financial portfolio optimization problem through the CDO2 structure, the effect of the diversification on the efficiency of the financial portfolio.

Global Results Synthesis

This chapter provides a global synthesis of the realized results by applying exact and approximate approaches on the portfolio design (PD) problem. The authors introduce an experimental analysis of best approaches based on linear programming and constraint programming techniques, according to the CPU time. Next, a global experiment synthesis of the best approximate approaches based on Simulated Annealing, IDWalk, Tabu Search, GWW, and VNS is realized according to the number of success and the CPU time. First results show that constraint programming with breaking all the detected symmetries is the best as an exact approach, VNS combined with simulated annealing is effective on non-trivial instances of the problem, and simulated annealing is the most effective as a simple local search.

VNS Metaheuristics to Solve a Financial Portfolio Design Problem

This chapter introduces a VNS-based local search for solving efficiently a financial portfolio design problem described in Chapter 1 and modeled in Chapter 3. The mathematical model tackled is a 0-1 quadratic model. It is well known that exact solving approaches on large instances of this kind of model are costly. The authors have proposed local search approaches to solve the problem, and the efficiency of this type of method has been proved. This chapter shows that the matricial 0-1 model of the problem enables specialized VNS algorithms by taking into account the particular structure of the financial problem considered. First experiments show that VNS with simulated annealing is effective on non-trivial instances of the problem.

Simple and Population Local Search Approaches for Portfolio Design Problem

This chapter introduces a local search optimization technique for solving efficiently a ðnancial portfolio design problem that consists of assigning assets to portfolios, allowing a compromise between maximizing gains, and minimizing losses. This practical problem appears usually in ðnancial engineering, such as in the design of CDO-squared portfolios. This problem has been modeled by Flener et al., who proposed an exact method to solve it. It can be formulated as a quadratic program on the (0,1) domain. It is well known that exact solving approaches on difficult and large instances of quadratic integer programs are known inefficient. That is why the authors have adopted local search methods, namely simple local search and population local search. They propose neighborhood and evaluation functions specialized on this problem. To boost the local search process, they propose also a greedy algorithm to start the search with an optimized initial configuration. Experimental results on non-trivial instances of the problem show the effectiveness of the incomplete approach.

Resolution Methods

The aim of this chapter is to introduce the different notions of the techniques used to solve the portfolio design problem. These techniques can be divided into two exact (or complete) methods and approached (or incomplete) methods. In the first part, the authors provide the exact approaches, namely linear programming and constraint programming, as well as the techniques of symmetry breaking, the modeling notions, and the different solving algorithms. The second part concerns approached methods, namely Simulated Annealing, IDWalk, Tabu Search, GWW, and Variable Neighborhood Search, including the techniques of studying the performance profiles of a method.

Linear Integer Programming Techniques for Portfolio Design Problem

This chapter introduces Integer Linear Programming (ILP) approaches for solving efficiently a ðnancial portfolio design problem. The authors proposed a matricial model in Chapter 3, which is a mathematical quadratic model. A linearization step is considered necessary to apply linear programming techniques. The corresponding matricial model shows clearly that the problem is strongly symmetrical. The row and column symmetries are easily handled by adding a negligible number of new constraints. The authors propose two linear models, which are given in detail and proven. These models represent the problem as linear constraint systems with 0-1 variables, which will be implemented in ILP solver. Experimental results in non-trivial instances of portfolio design problem are given.

Portfolio Design Models

This chapter applies different models to the financial portfolio design problem that affect assignment of assets to portfolios subject to a compromise between maximizing gains and minimizing losses. This practical problem appears in financial engineering, such as in the design of a CDO Squared portfolio. The aim of the authors is to propose and to solve a general model corresponding to the problem, within well classified assets. The authors express the diversification problem through a panoply of models such as the set model, matricial model, and MiniZinc model. These models represent an optimized problem of building efficient financial portfolios by maximizing the diversification rate. As long as the diversification rate is increased, the profit is increased, and the risk rate is decreased.