Bartosz Sawik

Downside Risk Approach for Multi-Objective Portfolio Optimization

This paper presents a multi-objective portfolio model with the expected return as a performance measure and the expected worst-case return as a risk measure. The problems are formulated as a triple-objectivemixed integer program. One of the problem objectives is to allocate the wealth on different securities to optimize the portfolio return. The portfolio approach has allowed the two popular in financial engineering percentile measures of risk, value-at-risk (VaR) and conditional valueat- risk (CVaR) to be applied. The decision maker can assess the value of portfolio return and the risk level, and can decide how to invest in a real life situation comparing with ideal (optimal) portfolio solutions. The concave efficient frontiers illustrate the trade-off between the conditional value-at-risk and the expected return of the portfolio. Numerical examples based on historical daily input data from the Warsaw Stock Exchange are presented and selected computational results are provided. The computational experiments show that the proposed solution approach provides the decision maker with a simple tool for evaluating the relationship between the expected and the worst-case portfolio return.

LEXICOGRAPHIC AND WEIGHTING APPROACH TO MULTI-CRITERIA PORTFOLIO OPTIMIZATION BY MIXED INTEGER PROGRAMMING

This chapter presents the portfolio optimization problem formulated as a multi-criteria mixed integer program. Weighting and lexicographic approach are proposed. The portfolio selection problem considered is based on a single-period model of investment. An extension of the Markowitz portfolio optimization model is considered, in which the variance has been replaced with the Value-at-Risk (VaR). The VaR is a quantile of the return distribution function. In the classical Markowitz approach, future returns are random variables controlled by such parameters as the portfolio efficiency, which is measured by the expectation, whereas risk is calculated by the standard deviation. As a result, the classical problem is formulated as a quadratic program with continuous variables and some side constraints. The objective of the problem considered in this chapter is to allocate wealth on different securities to maximize the weighted difference of the portfolio expected return and the threshold of the probability that the return is less than a required level. The auxiliary objectives are minimization of risk probability of portfolio loss and minimization of the number of security types in portfolio. The four types of decision variables are introduced in the model: a continuous wealth allocation variable that represents the percentage of wealth allocated to each asset, a continuous variable that prevents the probability that return of investment is not less than required level, a binary selection variable that prevents the choice of portfolios whose VaR is below the minimized threshold, and a binary selection variable that represents choice of stocks in which capital should be invested. The results of some computational experiments with the mixed integer programming approach modeled on a real data from the Warsaw Stock Exchange are reported.

Conditional Value-at-Risk Vs. Value-at-Risk to Multi-Objective Portfolio Optimization

This chapter presents a multi-criteria portfolio model with the expected return as a performance measure and the expected worst-case return as a risk measure. The problems are formulated as a single-objective linear program, as a bi-objective linear program, and as a triple-objective mixed integer program. The problem objective is to allocate the wealth on different securities to optimize the portfolio return. The portfolio approach has allowed the two popular financial engineering percentile measures of risk, value-at-risk (VaR) and conditional value-at-risk (CVaR) to be applied. The decision-maker can assess the value of portfolio return, the risk level, and the number of assets, and can decide how to invest in a real-life situation comparing with ideal (optimal) portfolio solutions. The concave efficient frontiers illustrate the trade-off between the conditional value-at-risk and the expected return of the portfolio. Numerical examples based on historical daily input data from the Warsaw Stock Exchange are presented and selected computational results are provided. The computational experiments prove that both proposed linear and mixed integer programming approaches provide the decision-maker with a simple tool for evaluating the relationship between the expected and the worst-case portfolio return.

Triple-objective models for portfolio optimisation with symmetric and percentile risk measures

The purpose of this paper is to compare three different triple-criteria portfolio optimisation models. The first model is constructed with the use of percentile risk measure value-at-risk and solved by mixed integer programming. The second one is constructed with the use of percentile risk measure conditional value-at-risk and solved by linear programming. The third one is constructed with the use of a symmetric measure of risk - variance of return - as in the Markowitz portfolio and solved by quadratic programming. Cardinality constraints are formulated in all models that limit the number of assets selected in the portfolio. Computational experiments are conducted for triple-criteria portfolio stock exchange investments. The results obtained prove that the triple-objective portfolio optimisation models with value-at-risk and conditional value-at-risk could be used to shape the distribution of portfolio returns. The decision maker can assess the value of portfolio return, the risk level and number of assets chosen in the portfolio, and can decide how to invest in a real life situation comparing with ideal (optimal) portfolio solutions. The proposed scenario-based portfolio optimisation problems under uncertainty, formulated as a triple-objective linear, mixed integer or quadratic program are solved using commercially available software (AMPL/CPLEX) for mathematical programming.

A single and triple-objective mathematical programming models for assignment of services in a healthcare institution

This paper presents three multi-objective mathematical programming models by mixed integer programming for optimal allocation of workers among supporting services in a hospital. The services include logistics, inventory management, financial management, operations management, medical analysis, etc. The optimality criteria of the problem are minimisation of operational costs of supporting services subject to some specific constraints. The constraints represent specific conditions for resource allocation in a hospital. The overall problem is formulated as a multi-criteria assignment model, where the decision variables represent the assignment of people to various jobs. Numerical examples are presented and some computational results modelled on a real data from a hospital in Lesser Poland are reported.

Optimal assignment of workers to supporting services in a hospital

Optimal assignment of workers to supporting services in a hospital Supporting services play an important role in health care institutions such as hospitals. This paper presents an application of operations research model for optimal allocation of workers among supporting services in a public hospital. The services include logistics, inventory management, financial management, operations management, medical analysis, etc. The optimality criterion of the problem is to minimize operations costs of supporting services subject to some specific constraints. The constraints represent specific conditions for resource allocation in a hospital. The overall problem is formulated as an integer program in the literature known as the assignment problem, where the decision variables represent the assignment of people to various jobs. The results of some computational experiments modeled on a real data from a selected Polish hospital are reported.

Multi-Objective Traveling Salesman and Transportation Problems with Environmental Aspects

Abstract The purpose of this chapter is to solve multi-objective formulation for traveling salesman and transportation problems. Computations are based on real data for the road freight transportation of a Spanish company. The company was selected because of its importance in Spanish economy and market. This company is important in the whole country; however, it has its higher importance in the northern part of Spain. The requirements for these models are the minimization of total distance and the CO2 emissions. To achieve this, it is required to know and carry out the minimization of the total distance traveled by the trucks during the deliveries. The deliveries are going to be executed between the different locations, nodes, in the region, and Elorrio, where the depot is situated. The data have been used to decide the best route in order to obtain a minimization of cost for the company. As it was mentioned earlier, the problems are focused on the reduction of the amount of CO2 emissions and minimization of total distance; by studying different parameters, the best solutions of route transportation have been obtained. The software used to solve these models is CPLEX solver with AMPL programming language.

Risk Scoring Models for Trade Credit in Small and Medium Enterprises

Trade credit refers to providing goods and services on a deferred payment basis. Commercial credit management is a matter of great importance for most small and medium enterprises (SMEs), since it represents a significant portion of their assets. Commercial lending involves assuming some credit risk due to exposure to default. Thus, the management of trade credit and payment delays is strongly related to the liquidation and bankruptcy of enterprises. In this paper we study the relationship between trade credit management and the level of risk in SMEs. Despite its relevance for most SMEs, this problem has not been sufficiently analyzed in the existing literature. After a brief review of existing literature, we use a large database of enterprises to analyze data and propose a multivariate decision-tree model which aims at explaining the level of risk as a function of several variables, both of financial and non-financial nature. Decision trees replace the equation in parametric regression models with a set of rules. This feature is an important aid for the decision process of risk experts, as it allows them to reduce time and then the economic cost of their decisions.

Selected Multi-Criteria Green Vehicle Routing Problems

Abstract The purpose of this chapter is to optimize multi-criteria formulation for green vehicle routing problems by mixed integer programming. This research is about the road freight transportation of a Spanish company of groceries. This company has more power in the north of Spain and hence it was founded there. The data used for the computational experiments are focused in the northern region of Spain. The data have been used to decide the best route in order to obtain a minimization of costs for the company. The problem focused on the distance traveled and the altitude difference; by studying these parameters, the best solution of route transportation has been made. The software used to solve this model is CPLEX solver with AMPL programming language. This has been helpful to obtain the results for the research and some conclusions have been obtained from them.