## Report in Wirtschaftsmathematik (WIMA Report)

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#### Year of publication

- 1999 (49) (remove)

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#### Keywords

- Location Theory (5)
- Multicriteria Optimization (3)
- Algebraic Optimization (2)
- Combinatorial Optimization (2)
- Geometrical Algorithms (2)
- Algebraic optimization (1)
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- 58
- Portfolio management and market risk quantification using neural networks (1999)
- We discuss how neural networks may be used to estimate conditional means, variances and quantiles of nancial time series nonparametrically. These estimates may be used to forecast, to derive trading rules and to measure market risk.

- 57
- On value preserving and growth optimal portfolios (1999)
- In a discrete-time financial market setting, the paper relates various concepts introduced for dynamic portfolios (both in discrete and in continuous time). These concepts are: value preserving portfolios, numeraire portfolios, interest oriented portfolios, and growth optimal portfolios. It will turn out that these concepts are all associated with a unique martingale measure which agrees with the minimal martingale measure only for complete markets.

- 55
- Some Applications of Impulse Control in Mathematical Finance (1999)
- We consider three applications of impulse control in financial mathematics, a cash management problem, optimal control of an exchange rate, and portfolio optimisation under transaction costs. We sketch the different ways of solving these problems with the help of quasi-variational inequalities. Further, some viscosity solution results are presented.

- 54
- Value Preserving Strategies and a General Framework for Local Approaches to Optimal Portfolios (1999)
- We present some new general results on the existence and form of value preserving portfolio strategies in a general semimartingale setting. The concept of value preservation will be derived via a mean-variance argument. It will also be embedded into a framework for local approaches to the problem of portfolio optimisation.

- 53
- Multicriteria Ordered Weber Problems (1999)
- In this paper we deal with the determination of the whole set of Pareto-solutions of location problems with respect to Q general criteria.These criteria include median, center or cent-dian objective functions as particular instances.The paper characterizes the set of Pareto-solutions of a these multicriteria problems. An efficient algorithm for the planar case is developed and its complexity is established. Extensions to higher dimensions as well as to the non-convexcase are also considered.The proposed approach is more general than the previously published approaches to multi-criteria location problems and includes almost all of them as particular instances.

- 51
- A Fuzzy Programming Approach to Multicriteria Facility Location Problems (1999)
- Facility Location Problems are concerned with the optimal location of one or several new facilities, with respect to a set of existing ones. The objectives involve the distance between new and existing facilities, usually a weighted sum or weighted maximum. Since the various stakeholders (decision makers) will have different opinions of the importance of the existing facilities, a multicriteria problem with several sets of weights, and thus several objectives, arises. In our approach, we assume the decision makers to make only fuzzy comparisons of the different existing facilities. A geometric mean method is used to obtain the fuzzy weights for each facility and each decision maker. The resulting multicriteria facility location problem is solved using fuzzy techniques again. We prove that the final compromise solution is weakly Pareto optimal and Pareto optimal, if it is unique, or under certain assumptions on the estimates of the Nadir point. A numerical example is considered to illustrate the methodology.

- 50
- Robustness in the Pareto-solutions for the Multicriteria Weber Location Problem (1999)
- In this paper a new trend is introduced into the field of multicriteria location problems. We combine the robustness approach using the minmax regret criterion together with Pareto-optimality. We consider the multicriteria Weber location problem which consists of simultaneously minimizing a number of weighted sum-distance functions and the set of Pareto-optimal locations as its solution concept. For this problem, we characterize the Pareto-optimal solutions within the set of robust locations for the original weighted sum-distance functions. These locations have both the properties of stability and non-domination which are required in robust and multicriteria programming.

- 49
- Optimal portfolio management using neural networks - a case study (1999)
- Neural networks are now a well-established tool for solving classification and forecasting problems in financial applications (compare, e.g., Bol et al., 1996, Evans, 1997, Rehkugler and Zimmermann, 1994, Refenes 1995, and Refenes et al. 1996a) though many practioners are still suspicious against too evident success stories. One reason may be that the construction of an appropriate network which provides a reasonable solution to a complex data-analytic problem is rarely made explicit in the literature. In this paper, we try to contribute to filling this gap by discussing in detail the problem of dynamically allocating capital to various components of a currency portfolio in such a manner that the average gain will be larger than for certain benchmark portfolios. We base our solution on feedforward neural networks which are constructed employing various statistical model selection procedures described in, e.g., (Anders, 1997, or Refenes et al., 1996b). Neural networks which are used as the basis of trading strategies in finance should be assessed differently than in technical applications. The task is not to construct a network which provides good forecasts with respect to mean-square error of some quantities of interest or to provide good approximation of some given target values, but to achieve a good performance in economic terms. For portfolio allocation, the main goal is to achieve on the average a large return combined with a small risk. Therefore, we do not consider forecasts of the foreign exchange (FX-) rate time series using neural networks, but we try to get the allocation directly as the output of a network. Furthermore, we do not minimize some estimation or prediction error, but we try to maximize an economically meaningful performance measure, the risk-adjusted return, directly (compare also Heitkamp, 1996). In the subsequent chapter, we describe the details of the portfolio allocation problem. The following two chapters provide some technical information on how the networks were fitted to the available data and how the network inputs and outputs were selected. In chapter 5, finally, we discuss the promising results.

- 48
- Local Search Algorithms for the k-Cardinality Tree Problem (1999)
- In this paper we deal with an NP-hard combinatorial optimization problem, the k-cardinality tree problem in node weighted graphs. This problem has several applications , which justify the need for efficient methods to obtain good solutions. We review existing literature on the problem. Then we prove that under the condition that the graph contains exactly one trough, the problem can be solved in ploynomial time. For the general NP-hard problem we implemented several local search methods to obtain heuristics solutions, which are qualitatively better than solutions found by constructive heuristics and which require significantly less time than needed to obtain optimal solutions. We used the well known concepts of genetic algorithms and tabu search with useful extensions. We show that all the methods find optimal solutions for the class of graphs containing exactly one trough. The general performance of our methods as compared to other heuristics is illustrated by numerical results.