Dietmar Maringer

Index tracking with constrained portfolios

Passive portfolio management strategies, such as index tracking, are popular in the industry, but so far little research has been done on the cardinality of such a portfolio, i.e. on how many different assets ought to be included in it. One reason for this is the computational complexity of the associated optimization problems. Traditional optimization techniques cannot deal appropriately with the discontinuities and the many local optima emerging from the introduction of explicit cardinality constraints. More recent approaches, such as heuristic methods, on the other hand, can overcome these hurdles. This paper demonstrates how one of these methods, differential evolution, can be used to solve the constrained index-tracking problem. We analyse the financial implication of cardinality constraints for a tracking portfolio using an empirical study of the Down Jones Industrial Average. We find that the index can be tracked satisfactorily with a subset of its components and, more important, that the deviation between computed actual tracking error and the theoretically achievable tracking error out of sample is negligibly affected by the portfolios cardinality. Copyright

Global optimization of higher order moments in portfolio selection

We discuss the global optimization of the higher order moments of a portfolio of financial assets. The proposed model is an extension of the celebrated mean variance model of Markowitz. Asset returns typically exhibit excess kurtosis and are often skewed. Moreover investors would prefer positive skewness and try to reduce kurtosis of their portfolio returns. Therefore the mean variance model (assuming either normally distributed returns or quadratic utility functions) might be too simplifying. The inclusion of higher order moments has therefore been proposed as a possible augmentation of the classical model in order to make it more widely applicable. The resulting problem is non-convex, large scale, and highly relevant in financial optimization. We discuss the solution of the model using two stochastic algorithms. The first algorithm is Differential Evolution (DE). DE is a population based metaheuristic originally designed for continuous optimization problems. New solutions are generated by combining up to four existing solutions plus noise, and acceptance is based on evolutionary principles. The second algorithm is based on the asymptotic behavior of a suitably defined Stochastic Differential Equation (SDE). The SDE consists of three terms. The first term tries to reduce the value of the objective function, the second enforces feasibility of the iterates, while the third adds noise in order to enable the trajectory to climb hills.

MOEA/D with NBI-style Tchebycheff approach for portfolio management

MOEA/D is a generic multiobjective evolutionary optimization algorithm. MOEA/D needs a approach to decompose a multiobjective optimization problem into a number of single objective optimization problems. The commonly-used weighted sum approach and the Tchebycheff approach may not be able to handle disparately scaled objectives. This paper suggests a new decomposition approach, called NBI-style Tchebycheff approach, for MOEA/D to deal with such objectives. A portfolio management MOP has been used as an example to test the effectiveness of MOEA/D with NBI-style Tchebycheff approach.

Lower bounds and stochastic optimization algorithms for uniform designs with three or four levels

New lower bounds for three- and four-level designs under the centered L 2 -discrepancy are provided. We describe necessary conditions for the existence of a uniform design meeting these lower bounds. We consider several modifications of two stochastic optimization algorithms for the problem of finding uniform or close to uniform designs under the centered L 2 -discrepancy. Besides the threshold accepting algorithm, we introduce an algorithm named balance-pursuit heuristic. This algorithm uses some combinatorial properties of inner structures required for a uniform design. Using the best specifications of these algorithms we obtain many designs whose discrepancy is lower than those obtained in previous works, as well as many new low-discrepancy designs with fairly large scale. Moreover, some of these designs meet the lower bound, i.e., are uniform designs.

Applications of Heuristics in Finance

Having the optimal solution for a given problem is crucial in the competitive world of finance; finding this optimal solution, however, is often an utter challenge. Even if the problem is well defined and all necessary data are available, it is not always well behaved: rather simple constraints are often enough to prohibit closed form solutions or evade the reliable application of standard numerical solutions. A common way to avoid this difficulty is to restate the problem: restricting and cumbersome constraints are relaxed and simplifying assumptions are introduced until the revised problem is approachable with the available methods, and are afterwards superimposed onto the solution for the simplified problem. Unfortunately, subtleties as well as central properties of the initial problem can be lost in this process, and results assumed to be ideal can actually be far away from the true optimum.

Constrained Index Tracking under Loss Aversion Using Differential Evolution

Index tracking is concerned with forming a portfolio that mimics a benchmark index as closely as possible. Traditionally, this implies that the returns between the index and the portfolio should differ as little as possible. However, investors might happily accept positive deviations (ie, returns higher than the index’s) while being particularly concerned with negative deviations. In this chapter, we model these preferences by introducing loss aversion to the index tracking problem and analyze the financial implications based on a computational study for the US stock market. In order to cope with this demanding optimization problem, we use Differential Evolution and investigate some calibration issues.

The Threshold Accepting Optimisation Algorithm in Economics and Statistics

Threshold Accepting (TA) is a powerful optimisation heuristic from the class of evolutionary algorithms. Using several examples from economics, econometrics and statistics, the issues related to implementations of TA are discussed and demonstrated. A problem specific implementation involves the definition of a local structure on the search space, the analysis of the objective function and of constraints, if relevant, and the generation of a sequence of threshold values to be used in the acceptance-rejection-step of the algorithm. A routine approach towards setting these implementation specific details for TA is presented, which will be partially data driven. Furthermore, fine tuning of parameters and the cost and benefit of restart versions of stochastic optimisation heuristics will be discussed

Finding the relevant risk factors for asset pricing

The arbitrage pricing theory shows that in a complete market, an asset can be priced by first identifying and pricing indices and then replicating the asset with a sufficiently large number of these indices. In practice, however, it appears favorable to focus on a small set of indices that captures most of an assets price movements. The selection of factors is therefore crucial for the models quality. Ideally, the selection is based on fundamental and economically reasonable relationships between asset and factors. If gathering or implementing these economic fundamentals is too costly or impossible, a selection based on statistical grounds might be considered. This concept is applied to the stocks of the S&P 100 where for each stock the bundle of 5 out of 103 MSCI indices is chosen that statistically explains most of the volatility. The optimization is done with a heuristic search method, memetic algorithms, which basically combines simulated annealing and evolutionary principles. Our results indicate that stock prices are merely influenced by industry factors, whereas country or regional indices seem to have less effect on asset returns.

Risk Preferences and Loss Aversion in Portfolio Optimization

Traditionally, portfolio optimization is associated with finding the ideal trade-off between return and risk by maximizing the expected utility. Investors’ preferences are commonly assumed to follow a quadratic or power utility function, and asset returns are often assumed to follow a Gaussian distribution. Investment analysis has therefore long been focusing on the first two moments of the distribution, mean and variance. However, empirical asset returns are not normally distributed, and neither utility function captures investors’ true attitudes towards losses. The impact of these specification errors under realistic assumptions is investigated. As traditional optimization techniques cannot deal reliably with the extended problem, the use of a heuristic optimization approach is suggested. It is found that loss aversion has a substantial impact on what investors consider to be an efficient portfolio and that mean-variance analysis alone can be utterly misguiding.

Metaheuristics for the Index Tracking Problem

Passive portfolio management strategies such as Index Tracking have gained considerable attention the industry recently which has also sparked some research interest. Index Tracking Problems are often complex in nature because of constraints on the portfolio structure and the chosen measure for the Tracking Error, quickly defying the use of traditional deterministic methods. Heuristic search and optimizationmethods, on the other hand, can dealwith this complexity and therefore appear as ideal choice to solve this class of problems. This contribution describes the Index Tracking Problem together with typical constraint encountered in practice, and illustrates how metaheuristic methods can be employed.