Optimization in finance : approaches for modeling and solving the multi-period loss offset problem in German income tax system

The major objective of this thesis is to study optimization techniques applied in financial planning. As financial optimization is a diverse field, we restrict our work onto tax planning. Our effort is directed towards studying the Loss Offset Problem which arises in German income tax system. The Loss Offset Problem deals with a situation where individuals or companies confront a loss in some financial years and profits in the years before and after the “loss years”. When such a situation occurs, it is allowed to divide a loss amount into two parts: the loss carry-backward and loss carry-forward. This will reduce the taxable income in other years, therefore reduces tax payments. The problem is of significant importance for a number of reasons. First, potentials for optimization procedures exist as there is a trade-off between the amount of loss to be carried back and forward. Second, from international perspective over the last several years, German loss offset regulations are still rather generous as many other countries do not allow a tax loss carry-backward at all. Besides, we consider two possible choices of taxation options in each period. The focus of this study is the multi-period scenario. As we will see, this hides many interesting dynamics in the interactive behavior of decisions. We formulate the mathematical model so as to optimize an objective function subject to appropriate constraints. The objective function itself is a discontinuous, non-linear, non-convex function with recursive characteristics, which makes the problem difficult to solve. In order to achieve this goal, we first study the complexity of the problem in two cases: a 3-period-model and a multiple-period-model and then apply optimization algorithms from Operations Research to search for solutions. We discuss several algorithms and their corresponding commonly mentioned application. We differentiate between exact and heuristic algorithms. An exact algorithm attempts to obtain the global optimal solution, no matter how long it takes for computational time. However, such approaches do not always work. For many practical problems in business, it is unlikely to acquire a global optimal solution in an acceptable amount of time. To the contrary, a heuristic algorithm may discover a very good feasible solution in a given number of iterations, but not necessarily the optimal solution for the specific problem being considered. To refine our analysis, both types of algorithms need to be adapted, applied, and analyzed under different scenarios of data setting.Gegenstand dieser Arbeit ist die Analyse der Optimierungstechniken in der Finanzplanung. Als Schwerpunkt gilt die betriebswirtschaftliche Steuerplanung, wobei die genaue Untersuchung auf das Verlustzuweisungsproblem eingeschränkt wird. Das Verlustzuweisungsproblem wird im deutschen Einkommensteuersystem formuliert und beschreibt die Situation, in der ein Steuerzahler einerseits finanzielle Gewinne in bestimmten Wirtschaftsperioden und andererseits Verluste in anderen Perioden erwirtschaftet. In so einer Konstellation erlaubt das deutsche Steuerrecht dem Steuerzahler, die Verluste in zwei Komponente aufzuteilen: den Verlustrücktrag und den Verlustvortrag. Diese Wahlmöglichkeiten führen dazu, dass das zu versteuernde Einkommen in den Gewinnperioden verringert wird und somit auch insgesamt die Steuerverpflichtungen. Die Formulierung des Problems sowie die Erarbeitung von passenden Lösungsverfahren sind aufgrund einer Vielzahl von Gründen nicht trivial. Erstens entstehen signifikante Optimierungspotenziale durch die Wahlmöglichkeiten bei der Zuweisung des Verlustes in den vorherigen und nachstehenden Perioden. Zweitens besteht der Vorteil, dass der Gesetzgeber in Deutschland – im Gegensatz zu den meisten anderen Ländern auf internationaler Ebene - einen Verlustrücktrag erlaubt. Darüber hinaus werden zwei unterschiedliche Besteuerungsalternativen, die in jeder einzelnen Periode ausgewählt werden müssen, berücksichtigt. Da das Problem einen multiperiodischen Charakter besitzt, verbergen sich interessante Wechselwirkungen in der Vielzahl der Entscheidungsmöglichkeiten dahinter. Für das Problem wird zunächst ein mathematisches Modell mit geeigneten Nebenbedingungen und einer Zielfunktion formuliert. Die analytische Komplexität entsteht durch eine nichtlineare, nichtstetige und nichtkonvexe Zielfunktion, die selbst ein rekursives Verhalten darstellt. In zwei Szenarien wird das Problem systematisch untersucht: ein 3 periodisches Modell und ein multiperiodisches Modell. Nach der Komplexitätsanalyse werden Algorithmen des Operations Research ausgewählt, auf das Problem angepasst und Lösungsverfahren erarbeitet. Dabei werden zwischen exakten und heuristischen Optimierungsverfahren unterschieden. Ein exaktes Suchverfahren findet das globale Optimum des Problems, jedoch kann der rechnerische Aufwand so hoch sein, dass keine Lösung in realistischer Zeit gefunden werden kann. Für viele Optimierungsprobleme in der Praxis ist es außerdem nicht notwendig, das absolute Optimum unbedingt zu erreichen. Ein heuristischer Algorithmus kann in solchen Situationen eine zulässige, zugleich sehr gute Lösung bei einer akzeptabler Laufzeit und geringem Aufwand berechnen. Bei der Analyse und Anwendung dieser beiden Gruppen von Algorithmen für das beschriebene Problem kamen verschiedenen Datenkonstellationen in Betracht

Optimization in finance : approaches for modeling and solving the multi-period loss offset problem in German income tax system

The major objective of this thesis is to study optimization techniques applied in financial planning. As financial optimization is a diverse field, we restrict our work onto tax planning. Our effort is directed towards studying the Loss Offset Problem which arises in German income tax system. The Loss Offset Problem deals with a situation where individuals or companies confront a loss in some financial years and profits in the years before and after the “loss years”. When such a situation occurs, it is allowed to divide a loss amount into two parts: the loss carry-backward and loss carry-forward. This will reduce the taxable income in other years, therefore reduces tax payments. The problem is of significant importance for a number of reasons. First, potentials for optimization procedures exist as there is a trade-off between the amount of loss to be carried back and forward. Second, from international perspective over the last several years, German loss offset regulations are still rather generous as many other countries do not allow a tax loss carry-backward at all. Besides, we consider two possible choices of taxation options in each period. The focus of this study is the multi-period scenario. As we will see, this hides many interesting dynamics in the interactive behavior of decisions. We formulate the mathematical model so as to optimize an objective function subject to appropriate constraints. The objective function itself is a discontinuous, non-linear, non-convex function with recursive characteristics, which makes the problem difficult to solve. In order to achieve this goal, we first study the complexity of the problem in two cases: a 3-period-model and a multiple-period-model and then apply optimization algorithms from Operations Research to search for solutions. We discuss several algorithms and their corresponding commonly mentioned application. We differentiate between exact and heuristic algorithms. An exact algorithm attempts to obtain the global optimal solution, no matter how long it takes for computational time. However, such approaches do not always work. For many practical problems in business, it is unlikely to acquire a global optimal solution in an acceptable amount of time. To the contrary, a heuristic algorithm may discover a very good feasible solution in a given number of iterations, but not necessarily the optimal solution for the specific problem being considered. To refine our analysis, both types of algorithms need to be adapted, applied, and analyzed under different scenarios of data setting.Gegenstand dieser Arbeit ist die Analyse der Optimierungstechniken in der Finanzplanung. Als Schwerpunkt gilt die betriebswirtschaftliche Steuerplanung, wobei die genaue Untersuchung auf das Verlustzuweisungsproblem eingeschränkt wird. Das Verlustzuweisungsproblem wird im deutschen Einkommensteuersystem formuliert und beschreibt die Situation, in der ein Steuerzahler einerseits finanzielle Gewinne in bestimmten Wirtschaftsperioden und andererseits Verluste in anderen Perioden erwirtschaftet. In so einer Konstellation erlaubt das deutsche Steuerrecht dem Steuerzahler, die Verluste in zwei Komponente aufzuteilen: den Verlustrücktrag und den Verlustvortrag. Diese Wahlmöglichkeiten führen dazu, dass das zu versteuernde Einkommen in den Gewinnperioden verringert wird und somit auch insgesamt die Steuerverpflichtungen. Die Formulierung des Problems sowie die Erarbeitung von passenden Lösungsverfahren sind aufgrund einer Vielzahl von Gründen nicht trivial. Erstens entstehen signifikante Optimierungspotenziale durch die Wahlmöglichkeiten bei der Zuweisung des Verlustes in den vorherigen und nachstehenden Perioden. Zweitens besteht der Vorteil, dass der Gesetzgeber in Deutschland – im Gegensatz zu den meisten anderen Ländern auf internationaler Ebene - einen Verlustrücktrag erlaubt. Darüber hinaus werden zwei unterschiedliche Besteuerungsalternativen, die in jeder einzelnen Periode ausgewählt werden müssen, berücksichtigt. Da das Problem einen multiperiodischen Charakter besitzt, verbergen sich interessante Wechselwirkungen in der Vielzahl der Entscheidungsmöglichkeiten dahinter. Für das Problem wird zunächst ein mathematisches Modell mit geeigneten Nebenbedingungen und einer Zielfunktion formuliert. Die analytische Komplexität entsteht durch eine nichtlineare, nichtstetige und nichtkonvexe Zielfunktion, die selbst ein rekursives Verhalten darstellt. In zwei Szenarien wird das Problem systematisch untersucht: ein 3 periodisches Modell und ein multiperiodisches Modell. Nach der Komplexitätsanalyse werden Algorithmen des Operations Research ausgewählt, auf das Problem angepasst und Lösungsverfahren erarbeitet. Dabei werden zwischen exakten und heuristischen Optimierungsverfahren unterschieden. Ein exaktes Suchverfahren findet das globale Optimum des Problems, jedoch kann der rechnerische Aufwand so hoch sein, dass keine Lösung in realistischer Zeit gefunden werden kann. Für viele Optimierungsprobleme in der Praxis ist es außerdem nicht notwendig, das absolute Optimum unbedingt zu erreichen. Ein heuristischer Algorithmus kann in solchen Situationen eine zulässige, zugleich sehr gute Lösung bei einer akzeptabler Laufzeit und geringem Aufwand berechnen. Bei der Analyse und Anwendung dieser beiden Gruppen von Algorithmen für das beschriebene Problem kamen verschiedenen Datenkonstellationen in Betracht

Development of evolutionary based techniques with applications to engineering.

Every possible problem can be considered to have a set of possible states by which amongst them, some are considered better than others by some chosen measure. It is the intention of optimisation to discover such states that perform better than all others for any given problem. It is an important tool within an array of subject areas, arguably all, in particular engineering, which tackles such applications as shape optimisation and industrial scheduling to name but a few. The aims of this work, are to increase the performance of the in-house general-purpose particle swarm optimiser designed at the department of engineering at Swansea University. This is to be achieved through its hybridisation with a local search, considering both solution refinement and early triggering mechanisms. In the discrete domain, an ant colony algorithm is to be chosen and evaluated by way of a parameter study and comparison against other leading ant colony algorithms made for the purpose of development for the future application to scheduling problems. Objectives are achieved through the increased refinement properties of the particle swarm optimiser with its hybridisation with local search. Additionally, an early switching mechanism is derived for the local search, resulting on average in a 20% reduction in the number of function evaluations required for constrained problems. With the highly unpredictable responses to unconstrained problems, only stagnation measures are derived. This study bridges the gap between the in-house optimiser and other hybrid particle swarm techniques available in the literature, resulting in competitive performance. An extensive literature review of ant colony identified the population-based ant colony algorithm (PACO) for further investigation. A detailed parameter study is conducted, resulting in the realisation of the strongly coupled parameters present. Following this, a hybrid off-line tuning method is devised, hybridising a simple particle swarm optimiser with the ant colony algorithm, resulting in an overall better performing algorithm. This indicated clear strengths in some cases over the more popular of ant colony algorithms

Understanding how Knowledge is exploited in Ant Algorithms

Centre for Intelligent Systems and their ApplicationsAnt algorithms were first written about in 1991 and since then they have been applied to many problems with great success. During these years the algorithms themselves have been modified for improved performance and also been influenced by research in other fields. Since the earliest Ant algorithms, heuristics and local search have been the primary knowledge sources. This thesis asks the question "how is knowledge used in Ant algorithms?" To answer this question three Ant algorithms are implemented. The first is the Graph based Ant System (GBAS), a theoretical model not yet implemented, and the others are two influential algorithms, the Ant System and Max-Min Ant System. A comparison is undertaken to show that the theoretical model empirically models what happens in the other two algorithms. Therefore, this chapter explores whether different pheromone matrices (representing the internal knowledge) have a significant effect on the behaviour of the algorithm. It is shown that only under extreme parameter settings does the behaviour of Ant System and Max-Min Ant System differ from that of GBAS. The thesis continues by investigating how inaccurate knowledge is used when it is the heuristic that is at fault. This study reveals that Ant algorithms are not good at dealing with this information, and if they do use a heuristic they must rely on it relating valid guidance. An additional benefit of this study is that it shows heuristics may offer more control over the exploration-exploitation trade-off than is afforded by other parameters. The second point where knowledge enters the algorithm is through the local search. The thesis looks at what happens to the performance of the Ant algorithms when a local search is used and how this affects the parameters of the algorithm. It is shown that the addition of a local search method does change the behaviour of the algorithm and that the strength of the method has a strong influence on how the parameters are chosen. The final study focuses on whether Ant algorithms are effective for driving a local search method. The thesis demonstrates that these algorithms are not as effective as some simpler fixed and variable neighbourhood search methods

Deception in ant colony optimization

SCOPUS: cp.kinfo:eu-repo/semantics/publishe

Theoretical and practical aspects of ant colony optimization

Combinatorial optimization problems are of high academical as well as practical importance. Many instances of relevant combinatorial optimization problems are, due to their dimensions, intractable for complete methods such as branch and bound. Therefore, approximate algorithms such as metaheuristics received much attention in the past 20 years. Examples of metaheuristics are simulated annealing, tabu search, and evolutionary computation. One of the most recent metaheuristics is ant colony optimization (ACO), which was developed by Prof. M. Dorigo (who is the supervisor of this thesis) and colleagues. This thesis deals with theoretical as well as practical aspects of ant colony optimization.* A survey of metaheuristics. Chapter 1 gives an extensive overview on the nowadays most important metaheuristics. This overview points out the importance of two important concepts in metaheuristics: intensification and diversification. * The hyper-cube framework. Chapter 2 introduces a new framework for implementing ACO algorithms. This framework brings two main benefits to ACO researchers. First, from the point of view of the theoretician: we prove that Ant System (the first ACO algorithm to be proposed in the literature) in the hyper-cube framework generates solutions whose expected quality monotonically increases with the number of algorithm iterations when applied to unconstrained problems. Second, from the point of view of the experimental researcher, we show through examples that the implementation of ACO algorithms in the hyper-cube framework increases their robustness and makes the handling of the pheromone values easier.* Deception. In the first part of Chapter 3 we formally define the notions of first and second order deception in ant colony optimization. Hereby, first order deception corresponds to deception as defined in the field of evolutionary computation and is therefore a bias introduced by the problem (instance) to be solved. Second order deception is an ACO-specific phenomenon. It describes the observation that the quality of the solutions generated by ACO algorithms may decrease over time in certain settings. In the second part of Chapter 3 we propose different ways of avoiding second order deception.* ACO for the KCT problem. In Chapter 4 we outline an ACO algorithm for the edge-weighted k-cardinality tree (KCT) problem. This algorithm is implemented in the hyper-cube framework and uses a pheromone model that was determined to be well-working in Chapter 3. Together with the evolutionary computation and the tabu search approaches that we develop in Chapter 4, this ACO algorithm belongs to the current state-of-the-art algorithms for the KCT problem.* ACO for the GSS problem. Chapter 5 describes a new ACO algorithm for the group shop scheduling (GSS) problem, which is a general shop scheduling problem that includes among others the well-known job shop scheduling (JSS) and the open shop scheduling (OSS) problems. This ACO algorithm, which is implemented in the hyper-cube framework and which uses a new pheromone model that was experimentally tested in Chapter 3, is currently the best ACO algorithm for the JSS as well as the OSS problem. In particular when applied to OSS problem instances, this algorithm obtains excellent results, improving the best known solution for several OSS benchmark instances. A final contribution of this thesis is the development of a general method for the solution of combinatorial optimization problems which we refer to as Beam-ACO. This method is a hybrid between ACO and a tree search technique known as beam search. We show that Beam-ACO is currently a state-of-the-art method for the application to the existing open shop scheduling (OSS) problem instances.Doctorat en sciences appliquéesinfo:eu-repo/semantics/nonPublishe

Theoretical and practical aspects of ant colony optimization

Combinatorial optimization problems are of high academical as well as practical importance. Many instances of relevant combinatorial optimization problems are, due to their dimensions, intractable for complete methods such as branch and bound. Therefore, approximate algorithms such as metaheuristics received much attention in the past 20 years. Examples of metaheuristics are simulated annealing, tabu search, and evolutionary computation. One of the most recent metaheuristics is ant colony optimization (ACO), which was developed by Prof. M. Dorigo (who is the supervisor of this thesis) and colleagues. This thesis deals with theoretical as well as practical aspects of ant colony optimization.* A survey of metaheuristics. Chapter 1 gives an extensive overview on the nowadays most important metaheuristics. This overview points out the importance of two important concepts in metaheuristics: intensification and diversification. * The hyper-cube framework. Chapter 2 introduces a new framework for implementing ACO algorithms. This framework brings two main benefits to ACO researchers. First, from the point of view of the theoretician: we prove that Ant System (the first ACO algorithm to be proposed in the literature) in the hyper-cube framework generates solutions whose expected quality monotonically increases with the number of algorithm iterations when applied to unconstrained problems. Second, from the point of view of the experimental researcher, we show through examples that the implementation of ACO algorithms in the hyper-cube framework increases their robustness and makes the handling of the pheromone values easier.* Deception. In the first part of Chapter 3 we formally define the notions of first and second order deception in ant colony optimization. Hereby, first order deception corresponds to deception as defined in the field of evolutionary computation and is therefore a bias introduced by the problem (instance) to be solved. Second order deception is an ACO-specific phenomenon. It describes the observation that the quality of the solutions generated by ACO algorithms may decrease over time in certain settings. In the second part of Chapter 3 we propose different ways of avoiding second order deception.* ACO for the KCT problem. In Chapter 4 we outline an ACO algorithm for the edge-weighted k-cardinality tree (KCT) problem. This algorithm is implemented in the hyper-cube framework and uses a pheromone model that was determined to be well-working in Chapter 3. Together with the evolutionary computation and the tabu search approaches that we develop in Chapter 4, this ACO algorithm belongs to the current state-of-the-art algorithms for the KCT problem.* ACO for the GSS problem. Chapter 5 describes a new ACO algorithm for the group shop scheduling (GSS) problem, which is a general shop scheduling problem that includes among others the well-known job shop scheduling (JSS) and the open shop scheduling (OSS) problems. This ACO algorithm, which is implemented in the hyper-cube framework and which uses a new pheromone model that was experimentally tested in Chapter 3, is currently the best ACO algorithm for the JSS as well as the OSS problem. In particular when applied to OSS problem instances, this algorithm obtains excellent results, improving the best known solution for several OSS benchmark instances. A final contribution of this thesis is the development of a general method for the solution of combinatorial optimization problems which we refer to as Beam-ACO. This method is a hybrid between ACO and a tree search technique known as beam search. We show that Beam-ACO is currently a state-of-the-art method for the application to the existing open shop scheduling (OSS) problem instances.Doctorat en sciences appliquéesinfo:eu-repo/semantics/nonPublishe