Abstract. Mean-variance portfolio analysis provided the first quantitative treatment of the tradeoff between profit and risk. We describe in detail the interplay between objective and constraints in a number of single-period variants, including semivariance models. Particular emphasis is laid on avoiding the penalization of overperformance. The results are then used as building blocks in the development and theoretical analysis of multiperiod models based on scenario trees. A key property is the possibility of removing surplus money in future decisions, yielding approximate downside risk minimization.

Introduction There has been a controversy in this journal about using downside risk measures in portfolio analysis. The downside risk measures supposedly are a major improvement over traditional portfolio theory. That is where the battle lines clashed when Rom and Ferguson (1993, 1994b) and Kaplan and Siegel (1994a, 1994b) engaged in a "tempest in a teapot". I should confess that I am strong supporter of downside risk measures and have used them in my teaching, research and software for the past two decades. Therefore, you should keep that bias in mind as you read this article. One of the best means to understand a concept is to study the history of its development. Understanding the issues facing researchers during the development of a concept results in better knowledge of the concept. The purpose of this paper is to provide an understanding of the measurement of downside risk. First, it helps to define terms. Portfolio theory is the application of decision-making tools unde

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In this paper we deal with sensitivity analysis in convex quadratic programming, without making assumptions on nondegeneracy, strict convexity of the objective function, and the existence of a strictly complementary solution. We show that the optimal value as a function of a right--hand side element (or an element of the linear part of the objective) is piecewise quadratic, where the pieces can be characterized by maximal complementary solutions and tripartitions. Further, we investigate differentiability of this function. A new algorithm to compute the optimal value function is proposed. Finally, we discuss the advantages of this approach when applied to mean--variance portfolio models.

In this chapter we describe the optimal set approach for sensitivity analysis for LP. We show that optimal partitions and optimal sets remain constant between two consecutive transition-points of the optimal value function. The advantage of using this approach instead of the classical approach (using optimal bases) is shown. Moreover, we present an algorithm to compute the partitions, optimal sets and the optimal value function. This is a new algorithm and uses primal and dual optimal solutions. We also extend some of the results to parametric quadratic programming, and discuss differences and resemblances with the linear programming case.

The Solution of a Class of Limited Diversification Portfolio Selection Problems by Gwyneth Owens Butera A branch-and-bound algorithm for the solution of a class of mixed-integer nonlinear programming problems arising from the field of investment portfolio selection is presented. The problems in this class are characterized by the inclusion of the fixed transaction costs associated with each asset, a constraint that explicitly limits the number of distinct assets in the selected portfolio, or both. Modeling either of these forms of limiting the cost of owning an investment portfolio involves the introduction of binary variables, resulting in a mathematical programming problem that has a nonconvex feasible set. Two objective functions are examined in this thesis; the first is a positive definite quadratic function which is commonly used in the selection of investment portfolios. The second is a convex function that is not continuously differentiable; this objective function, although not...

In sensitivity analysis one wants to know how the problem and the optimal solutions change under the variation of the input data. We consider the case when variation happens in the right hand side of the constraints and/or in the linear term of the objective function. We are interested to find the range of the parameter variation in Convex Quadratic Optimization (CQO) problems where the support set of a given primal optimal solution remains invariant. This question has been first raised in Linear Optimization (LO) and known as Type II (so called Support Set Invariancy) sensitivity analysis. We present computable auxiliary problems to identify the range of parameter variation in support set invariancy sensitivity analysis for CQO. It should be mentioned that all given auxiliary problems are LO problems and can be solved by an interior point method in polynomial time. We also highlight the differences between characteristics of support set invariancy sensitivity analysis for LO and CQO.

For the standard Mean-Variance model for portfolio selection with linear constraints, there are several algorithms that can efficiently compute both a single point on the Pareto front and even the whole front. Unfortunately, commonly used constraints (e.g. cardinality constraints or buy-in thresholds) result in the optimization problem to become intractable by standard algorithms. In this paper, two paradigms to deal with this kind of constraint are presented and their advantages and disadvantages are highlighted.

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The problem of portfolio selection is a standard problem in financial engineering and has received a lot of attention in recent decades. Classical mean-variance portfolio selection aims at simultaneously maximizing the expected return of the portfolio and minimizing portfolio variance. In the case of linear constraints, the problem can be solved efficiently by parametric quadratic programming (i.e., variants of Markowitz ’ critical line algorithm). However, there are many real-world constraints that lead to a non-convex search space, e.g. cardinality constraints which limit the number of different assets in a portfolio, or minimum buy-in thresholds. As a consequence, the efficient approaches for the convex problem can no longer be applied, and new solutions are needed. In this paper, we propose to integrate an active set algorithm optimized for portfolio selection into a multi-objective evolutionary algorithm (MOEA). The idea is to let the MOEA come up with some convex subsets of the set of all feasible portfolios, solve a critical line algorithm for each subset, and then merge the partial solutions to form the solution of the original non-convex problem. We show that the resulting envelope-based MOEA significantly outperforms existing MOEAs. 1 1