Innovative Applications of O.R.Portfolio problems with two levels decision-makers: Optimal portfolio selection with pricing decisions on transaction costs☆
Introduction
The classical model in portfolio optimization was originally proposed by Markowitz (1952). This model has served as the initial point for the development of modern portfolio theory. Over time, portfolio optimization problems have become more realistic, incorporating real-life aspects that make the resulting portfolios more cost-effective than the alternatives that do not consider them (Castro, Gago, Hartillo, Puerto, Ucha, 2011, Kolm, Tütüncü, Fabozzi, 2014, Lynch, Tan, 2011, Mansini, Ogryczak, Speranza, 2014, Mansini, Ogryczak, Speranza, 2015b). Transaction costs can be seen as one of these important actual features to be included in portfolio optimization. These costs are those incurred by the investors when buying and selling assets on financial markets, charged by the brokers, the financial institutions or the market makers playing the role of intermediary. Transaction costs usually include banks and brokers’ commissions, fees, etc. These commissions or fees have a direct impact on the portfolio, especially for individual or small investors, since they will determine the net returns, reducing them and decreasing also the budget available for future investments (Baule, 2010, Baumann, Trautmann, 2013, Liu, Loewenstein, 2002).
To the best of our knowledge, in the existing literature, transaction costs are assumed to be given (Davis, Norman, 1990, Korn, 1998, Lobo, Fazel, Boyd, 2007, Magill, Constantinides, 1976, Mansini, Ogryczak, Speranza, 2014, Mansini, Ogryczak, Speranza, 2015b, Morton and Pliska, 1995). They can be a fixed cost applied to each selected security in the portfolio; or a variable cost to be paid which depends on the amount invested in each security included in the portfolio (see e.g. (A. Valle, Meade, E. Beasley, 2014, Baule, 2010, Baumann, Trautmann, 2013, Kellerer, Mansini, Speranza, 2000, Mansini, Ogryczak, Speranza, 2014, Mansini, Ogryczak, Speranza, 2015b, Woodside-Oriakhi, Lucas, Beasley, 2013) and the references therein). This dependence can be proportional to the investment or given by a fixed cost that is only charged if the amount invested exceeds a given threshold, or by some other functional form (see e.g. Baule, 2010, Konno, Akishino, Yamamoto, 2005, Mansini, Ogryczak, Speranza, 2014, Mansini, Ogryczak, Speranza, 2015b, Thi, Moeini, Dinh, 2009 and the references therein). But in any case, unit transaction costs are known and predetermined in the optimization process. Nevertheless, it is meaningful to analyze the situations where transaction costs can be decision variables set by financial institutions so that they are trying to maximize its profit as part of the decision process that leads to optimal portfolios for the investors.
The portfolio optimization problem considered in this paper is based on a single-period model of investment and incorporates a transaction costs setting phase. We assume that there are two decision-makers involved in the situation: on the one hand, the investor and on the other hand, the broker specialist, market maker or financial institution (that we will call from now on, for simplicity broker–dealer). At the beginning of a period, an investor allocates his capital among various assets and during the investment period, each asset generates a random rate of return. Moreover, we consider that the broker–dealer can charge some unit transaction costs on the securities selected by the investor trying to maximize its benefits but anticipating the rational response of the investor. This is a pricing phase in which the broker–dealer decides on how much is going to charge to the investor for the traded securities. Considering unit transaction costs as a decision variable of the model is a novel element in portfolio optimization and this is one of the main contributions of this paper. Then, at the end of the period, the result for the investor is a variation of his capital (increased or decreased) which is measured by the weighted average of the individual rates of return minus commissions or fees. In addition, the result for the broker–dealer is the amount paid by the investor, which depends on the revcosts set on the traded securities included in the portfolio chosen by the investor.
Based on the structure of financial markets, we assume a hierarchical relationship between the parties involved in the portfolio problem, that is, we define a natural problem in which the broker–dealer sets the unit transaction costs first, trying to anticipate the rational response of the investor. This hierarchical analysis of the portfolio problem has not been addressed before and it is another contribution of our paper. Once the costs are fixed, the investor chooses his optimal portfolio. For the sake of completeness, we also analyze the case in which the investor chooses his portfolio first, and after that, the broker–dealer sets the transaction costs. In order to model these hierarchical structures, we use a bilevel optimization approach (see e.g. Bard, 2013, Colson, Marcotte, Savard, 2005, Labbé, Violin, 2016, Sinha, Pekka, Kalyanmoy, 2017). Furthermore, we consider a social welfare problem where both, broker–dealer and investor, cooperate to maximize their returns. We assume in the different problems that all economic or financial information is common knowledge and that all the decision-makers in the problem have access to it.
The contributions of this paper can be summarized as follows: (1) it incorporates for the first time the above hierarchical approaches with two-levels of decision-makers on portfolio optimization problems (the broker–dealer sets unit transaction costs trying to maximize its benefits, whereas the investor minimizes risk while ensuring a given expected return (Benati, 2003, Benati, 2014)); (2) it introduces transaction costs as decision variables controlled by the broker–dealer; and (3) it develops different bilevel programming formulations to obtain optimal solutions for the considered problems. This paper introduces new models for the bilevel portfolio optimization problem. As far as we know, bilevel models for the portfolio selection that set unit transaction costs as decision variables of the problem have not been considered in the literature before.
The rest of the paper is organized as follows. Section 2 states the preliminaries and the notation used throughout the paper. In Section 3, we present the problem in which the broker–dealer is the leader and we develop two different Mixed Integer Linear Programming (MILP) formulations to solve such problem. Section 4 introduces the investor-leader problem and develops a Linear Programming (LP) formulation for it. In the more general case where additional constraints are required on the portfolio selection, it is presented a convergent iterative algorithm based on an “ad hoc” decomposition of the model. Next, in Section 5, it is addressed a social welfare problem. There, we propose a MILP formulation and an algorithm based on Benders decomposition for solving it. Section 6 is devoted to reporting on the computational study of the different problems and solution methods discussed in the previous sections. Our results are based on data taken from Dow Jones Industrial Average. Finally, Section 7 concludes the paper.
Section snippets
Preliminaries
Let be the set of securities considered for an investment, B ⊆ N a subset of securities in which the broker–dealer can charge unit transaction costs to the investor and . In most cases, but there is no loss of generality to consider that B is a proper subset of N.
First, we assume that the broker–dealer can price security j ∈ B from a discrete set, with cardinality sj, of admissible costs, and the broker–dealer’s goal is to maximize its benefit. Further,
Bilevel broker-dealer-leader Investor-follower Portfolio Problem (BLIFP)
We start analyzing a hierarchical structure in the financial markets in which the broker–dealer sets the transaction costs first, and after that, the investor chooses his portfolio. Observe that in this situation, the problem faced from the investor point of view reduces to a portfolio selection, under the considered criterion, which in this case is to hedge against risk maximizing the average α-quantile of his smallest returns (CVaRα). Therefore, we study this situation from the point of view
Bilevel Investor-leader broker-dealer-follower Portfolio Problem (ILBFP)
For the sake of completeness, in this section, we consider the reverse situation to the one that has been analyzed in Section 3, i.e., a hierarchical structure in the financial market where the investor acts first and once his portfolio x is chosen the broker–dealer sets transaction costs. Although one could claim that this situation may be atypical in actual financial markets, we want to analyze this case from a theoretical point of view. Moreover, we wish to analyze its implications depending
The Maximum Social Welfare Problem (MSWP)
In some actual situations, the investor and the broker–dealer may have an incentive to work together to improve the social welfare of society. They can agree to cooperate and share risk and benefits to improve, in this way, their solutions by designing a joint strategy.
We also analyze this problem for the sake of completeness and to compare the performance of this situation where none of the parties has a hierarchical position over the other one. We think that even if the actual implementation
Computational study and empirical application
This section is devoted to reporting some numerical experiments conducted to: 1) compare the effectiveness of the methods proposed to solve the different problems; 2) analyze the form of the solutions within each model; and 3) compare the profiles of the solutions, in terms of net values for the broker–dealer and expected return for the investor, across the three defined problems.
The computational experiments were carried out on a personal computer with Intel(R) Core(TM) i7-2600 CPU, 3.40
Concluding remarks and extensions
We have presented three single-period portfolio optimization problems with transaction costs, considering two different decision-makers: the investor and the financial intermediary. Including the financial intermediaries (broker–dealers) as decision-makers leads to the incorporation of the transaction costs as decision variables in the portfolio selection problem. The action of both decision-makers was assumed to be hierarchical. We have considered the situations where each of these
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This research has been funded by the Spanish Ministry of Science and Technology project MTM2016-74983-C2-1-R.