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This chapter presented a short introduction to some of the foundations of Modern Portfolio Theory (MPT) in the tradition of Harry Markowitz, James Tobin and William Sharpe. With respect to computability, these models have to rely on rather strict assumptions that are not always able to depict real market situations. Subsequent models try to include these missing aspects, yet suffer from other shortcomings as they usually have to make strong simplifications in other aspects in order to remain solvable.

In the main part of this contribution, too, the original MPT models will be enhanced to allow a more realistic study of portfolio management problems. Unlike other approaches in the literature, however, the trade-off between model complexity and its exact solvability will not be answered by “exchanging” one simplifying constraints for another, but by applying new solution methods and optimization techniques. The basic concepts of these methods will be presented in the following chapter.

In this chapter, some basic concepts of optimization in general and heuristic optimization methods in particular were introduced. The heuristics presented in this chapter differ significantly in various aspects: the varieties range from repeatedly modifying one candidate solution per iteration to whole populations of search agents each of them representing one candidate solution; from neighborhood search strategies to global search methods, etc. As diverse these methods are, as diverse are also their advantages and disadvantages: Simulated Annealing and Threshold Accepting are relatively easy to implement and are good general purpose methods, yet they tend to have problems when the search space is excessively large and has many local optima. Other methods such as Genetic Algorithms or Memetic Algorithms, on the other hand, are more complex and their implementation demands some experience with heuristic optimization, yet they can deal with more complicated and highly demanding optimization problems. Hence, there is not one best heuristic that would be superior to all other methods. It is rather a “different courses, different horses” situation where criteria such as the type of optimization problem, restrictions on computational time, experience with implementing different HO algorithms, the programming environment, the availability of toolboxes, and so on that influence the decision which heuristic to choose - or eventually lead to new or hybrid methods.

The following chapters of this contribution make use of heuristic optimization techniques for approaching problems, merely from the area portfolio management, that cannot be answered with traditional models. The diversity of the problems leads to the application of different methods as well as the introduction of a new hybrid approach. Though the main focus of these applications shall be on the financial implications that can be drawn from the results, there will also be some comparisons of these methods together with suggestions for enhancements.

In this chapter, a model for portfolio selection under fixed and/or proportional transaction costs together with non-negativity and integer constraints was presented and empirically studied on basis of DAX data. The major results from this study are that the presence of transaction costs might lead to significantly different results than for a perfect market and that the types of costs the investor occurs have different effects. Also, the optimal solution under transaction costs can not always be derived from the solution for frictionless markets.

Introducing transaction costs might have severe effects on the optimal portfolio structure. Even low fixed costs can lead to a substantial reduction in the number of different assets that ought to be included in a portfolio; the same is true for proportional costs or compound cost schemes. In due course, the asset weights might differ substantially. An investor facing transaction costs might therefore even have distinct advantages from individual stock selection over investing into a market fund — provided she does not simply try to replicate its weights but includes the relevant costs and additional constraints into the optimization process. Unlike claimed for a perfect market situation, in real world it might therefore be reasonable to hold a portfolio that deviates from the market portfolio even when all investors have homogeneous expectations. Another conclusion from these results is that investors might have advantages when they can invest in funds that are not just tracking the market portfolio, but also anticipate their clients’ transaction costs.

For various reasons, investors tend to hold a rather small number of assets. In this chapter, a method has been presented to approach the associated NP hard optimization problem of selecting the optimal set of assets under a given market situation and expectations. The main results from this empirical study are twofold: (i) the well known fact of decreasing marginal contribution to diversification is not only confirmed, but can be exploited by identifying those assets that, in combination, offer the highest risk premium; (ii) it has been shown that alternative rules, frequently found in practice, are likely to underperform as they offer solutions with risk premia lower than would be possible under the same constraints and market situations.

In this chapter a meta-heuristic was presented that basically combines principles from Simulated Annealing with evolutionary strategies and that uses additional modifications. Having applied this algorithm to the problem of portfolio selection when there are constraints on the number of different assets in the portfolio and non-negativity of the asset weights, we find this algorithm highly efficient and reliable. Furthermore, it is shown that the introduction of evolutionary principles has significant advantages.

The algorithm is flexible enough to allow for extensions in the optimization model by introducing additional constraints such as transaction costs, taxes, upper and/or lower limits for weights, alternative risk measures and distributions of returns, etc. First tests with such extensions led to promising results and supported the findings for the algorithm presented in this chapter.

The main findings from the empirical studies in this chapter are threefold: (i) for the used data the assumption of normal distribution can mostly be rejected and the empirical distribution is superior to the normal distribution; (ii) Value at Risk is a reliable measure of risk, in particular when used in conjunction with empirical distributions — and (iii) the opposite of the previous two points becomes true when Value at Risk under empirical distributions is used as an explicit constraint on portfolio risk in an optimization setting. Though empirical distributions are superior in describing the assets return and potential losses, this advantage is destroyed when included in the optimization process.

The reason for this can be found in the underlying concept: Value at Risk focuses on a single point of the distribution of returns or prices, namely the loss that will not be exceeded with a certain probability, but does not directly account for the magnitude of losses exceeding this limit. When VaR is estimated via empirical distributions based on historic simulation, as is done in this study, or Monte Carlo simulations, extreme losses might be ignored and assets, exhibiting these extreme losses, will readily be included in the portfolio when they have a sufficiently high expected return. Also, the use of empirical distributions encourages the optimizer to manipulate the asset weights in a way that losses close to the VaR limit are just small enough so that this limit is not exceeded and these losses will not count towards the shortfall probability. Both effects lead to severe under-estimating the actual risk of the portfolio.

Even when asset returns are not normally distributed, the assumption of this parametric distribution leads to more reliable risk estimations because extreme losses enter the optimization process via the volatility. When VaR is considered in the optimization process, parametric distributions might therefore be superior, despite the fact that their imprecision in measuring risk lead to imprecise estimates.

The results from this study suggest that empirical distributions should be used very reluctantly in VaR optimization, yet also that more research on the use of parametric distributions appears desirable.

In this chapter a meta-heuristic was presented that basically combines principles from Simulated Annealing with evolutionary strategies and that uses additional modifications. Having applied this algorithm to the problem of portfolio selection when there are constraints on the number of different assets in the portfolio and non-negativity of the asset weights, we find this algorithm highly efficient and reliable. Furthermore, it is shown that the introduction of evolutionary principles has significant advantages.

The algorithm is flexible enough to allow for extensions in the optimization model by introducing additional constraints such as transaction costs, taxes, upper and/or lower limits for weights, alternative risk measures and distributions of returns, etc. First tests with such extensions led to promising results and supported the findings for the algorithm presented in this chapter.

For various reasons, investors tend to hold a rather small number of assets. In this chapter, a method has been presented to approach the associated NP hard optimization problem of selecting the optimal set of assets under a given market situation and expectations. The main results from this empirical study are twofold: (i) the well known fact of decreasing marginal contribution to diversification is not only confirmed, but can be exploited by identifying those assets that, in combination, offer the highest risk premium; (ii) it has been shown that alternative rules, frequently found in practice, are likely to underperform as they offer solutions with risk premia lower than would be possible under the same constraints and market situations.