Portfolio Selection with minimum transaction lots: an approach with dual expected utility

In this paper we analyse the portfolio selectionproblem with minimum transactionlots in the context of non-expected utility theory. We assume that the decisionmaker ranks the alternatives by using a specific DualExpectedUtility. This functionallows portfolio values less or equal a fixed benchmark tobe weighted inadifferent way from values greater than the fixedbenchmark. Under normallydistributedreturns and opportunechoice ofthe benchmark, the suggested approach leads to an NP-complete problemandhas the advantage ofusing mixed linear programming to obtainthe optimal portfolio. We also show resultsobtained by implementing the model on the Italian stock market. (keywords: dual expectedutility, portfolio selection, NP-completeness, linear programming with mixed variables)dual expected utility, portfolio selection, NP-completeness, linear programming with

GENERALIZED VECTOR RISK FUNCTIONS

The paper introduces a new notion of vector-valued risk function. Both deviations and expectation bounded coherent risk measures are defined and analyzed. The relationships with both scalar and vector risk functions of previous literature are discussed, and it is pointed out that this new approach seems to appropriately integrate several preceding point of view. The framework of the study is the general setting of Banach lattices and Bochner integrable vector-valued random variables. Sub-gradient linked representation theorems, as well as portfolio choice problems, are also addressed, and general optimization methods are presented. Finally, practical examples are provided.

Stability of the optimal reinsurance with respect to the risk measure

The optimal reinsurance problem is a classic topic in Actuarial Mathematics. Recent approaches consider a coherent or expectation bounded risk measure and minimize the global risk of the ceding company under adequate constraints. However, there is no consensus about the risk measure that the insurer must use, since every risk measure presents advantages and shortcomings when compared with others. This paper deals with a discrete probability space and analyzes the stability of the optimal reinsurance with respect to the risk measure that the insurer uses. We will demonstrate that there is a “stable optimal retention” that will show no sensitivity, insofar as it will solve the optimal reinsurance problem for many risk measures, thus providing a very robust reinsurance plan. This stable optimal retention is a stop-loss contract, and it is easy to compute in practice. A fast algorithm will be given and a numerical example presented.Optimal reinsurance, Risk measure, Sensitivity, Stable optimal retention, Stop-loss reinsurance

Stable solutions for optimal reinsurance problems involving risk measures.

The optimal reinsurance problem is a classic topic in actuarial mathematics. Recent approaches consider a coherent or expectation bounded risk measure and minimize the global risk of the ceding company under adequate constraints. However, there is no consensus about the risk measure that the insurer must use, since every risk measure presents advantages and shortcomings when compared with others. This paper deals with a discrete probability space and analyzes the stability of the optimal reinsurance with respect to the risk measure that the insurer uses. We will demonstrate that there is a ‘‘stable optimal retention’’ that will show no sensitivity, insofar as it will solve the optimal reinsurance problem for many risk measures, thus providing a very robust reinsurance plan. This stable optimal retention is a stop-loss contract, and it is easy to compute in practice. A fast linear time algorithm will be given and a numerical example presented.Optimal reinsurance; Risk measure; Sensitivity; Stable optimal retention; Stop-loss reinsurance;

Nonconvex optimization for pricing and hedging in imperfect markets.

The paper deals with imperfect financial markets and provides new methods to overcome many inefficiencies caused by frictions. Transaction costs are quite general and far from linear or convexo The concepts of pseudoarbitrage and efficiency are introduced and deeply analyzed by means of both scalar and vector optimization problems. Their optimality conditions and solutions yield strategies to invest and hedging portfolios, as well as bid-ask spread improvements. They also point out the role of coalitions when dealing with these markets. Several sensitivity results will permit us to show that a significant transaction costs reduction is very often feasible in practice, as well as to measure its effect on the general efficiency of the market. AII these findings may be especially important for many emerging and still illiquid spot or derivative markets (electricity markets, com odity markets, markets related to weather, infiation-linked or insurance-linked derivatives, etc.).Global optimization; Pseudoarbitrage; Spread reduction; Balance point;

Optimizing Measures of Risk: A Simplex-like Algorithm

The minimization of general risk or dispersion measures is becoming more and more important in Portfolio Choice Theory. There are two major reasons. Firstly, the lack of symmetry in the returns of many assets provokes that the classical optimization of the standard deviation may lead to dominated strategies, from the point of view of the second order stochastic dominance. Secondly, but not less important, many institutional investors must respect legal capital requirements, which may be more easily studied if one deals with a risk measure related to capital losses. This paper proposes a new method to simultaneously minimize several risk or dispersion measures. The representation theorems of risk measures are applied to transform the general risk minimization problem in a minimax problem, and later in a linear programming problem between infinite-dimensional Banach spaces. Then, new necessary and sufficient optimality conditions are stated and a simplex-like algorithm is developed. The algorithm solves the dual (and therefore the primal) problem and provides both optimal portfolios and their sensitivities. The approach is general enough and does not depend on any particular risk measure, but some of the most important cases are specially analyzed.Risk Measure. Deviation Measure. Portfolio Selection. Infinite-Dimensional Linear Programming. Simpl

Portfolio choice and optimal hedging with general risk functions: a simplex-like algorithm.

The minimization of general risk functions is becoming more and more important in portfolio choice theory and optimal hedging. There are two major reasons. Firstly, heavy tails and the lack of symmetry in the returns of many assets provokes that the classical optimization of the standard deviation may lead to dominated strategies, from the point of view of the second order stochastic dominance. Secondly, but not less important, many institutional investors must respect legal capital requirements, which may be more easily studied if one deals with a risk measure related to capital losses. This paper proposes a new method to simultaneously minimize several general risk or dispersion measures. The representation theorems of risk functions are applied to transform the general risk minimization problem in a minimax problem, and later in a linear programming problem between infinite-dimensional Banach spaces. Then, new necessary and sufficient optimality conditions are stated and a simplex-like algorithm is developed. The algorithm solves the dual problem and provides both optimal portfolios and their sensitivities. The approach is general enough and does not depend on any particular risk measure, but some of the most important cases are specially analyzed. A final real data numerical example illustrates the practical performance of the proposed methodology.Risk measures; Deviation measure; Portfolio selection; Infinite dimensional linear programming; Simplex like method;

Nonconvex optimization for pricing and hedging in imperfect markets

The paper deals with imperfect financial markets and provides new methods to overcome many inefficiencies caused by frictions. Transaction costs are quite general and far from linear or convexo The concepts of pseudoarbitrage and efficiency are introduced and deeply analyzed by means of both scalar and vector optimization problems. Their optimality conditions and solutions yield strategies to invest and hedging portfolios, as well as bid-ask spread improvements. They also point out the role of coalitions when dealing with these markets. Several sensitivity results will permit us to show that a significant transaction costs reduction is very often feasible in practice, as well as to measure its effect on the general efficiency of the market. AII these findings may be especially important for many emerging and still illiquid spot or derivative markets (electricity markets, com odity markets, markets related to weather, infiation-linked or insurance-linked derivatives, etc.).Partially funded by "Comunidad Autónoma de Madrid" and Spanish Ministry of Science and Education (ref: BEC2003-09067 -C04-03).Publicad

Mathematical methods in modern risk measurement: a survey

In the last ten years we have been facing the development on new approaches in Risk Measurement. The Coherent, Expectation Bounded, Convex, Consistent, etc. Risk Measures have been introduced and deeply studied, but there are many open problems that will have to be addressed in forthcoming research. The present paper attempts to summarize the achieved findings and the “State of the Art”, as well as their relationships with other Mathematical Fields, with special focus on other usual topics of Mathematical Finance.En los últimos diez años hemos asistido al desarrollo de nuevos enfoques en Medición de Riesgos. Las Medidas de Riesgo Coherentes, Acotadas por la Media, Convexas, Consistentes, etc., han sido introducidas y profundamente estudiadas, aunque siguen abiertos numerosos problemas que tendrán que ser abordados en investigaciones futuras. El presente artículo sintetiza los logros alcanzados y “El Estado Actual de la Cuestión”, así como las relaciones con otros campos de la Matemática, con atención especial a los temas cl´asicos de la Matem´atica Financiera.This research was partially supported by “Welzia Management SGIIC SA”, “RD Sistemas SA”, “Comunidad Autonoma de Madrid ´ ” (Spain), Grant s-0505/tic/000230, and “MEyC” (Spain), Grant SEJ2006-15401-C04Publicad

Polyhedral Coherent Risk Measures, Portfolio Optimization and Investment Allocation Problems

The class of polyhedral coherent risk measures that could be used in decision- making under uncertainty is studied. Properties of these measures and invariant operations are considered. Portfolio optimization problems on the return -risk ratio using these risk measures are analyzed. The developed mathematical technique allows to solve large-scale portfolio problems by standard linear programming methods as an example of applications, investment allocation problems under risk of catastrophic floods are considered