Tanzania Journal of Science 47(1): 194-203, 2021 ISSN 0856-1761, e-ISSN 2507-7961 © College of Natural and Applied Sciences, University of Dar es Salaam, 2021 Application of Guided Local Search (GLS) in Portfolio Optimization Collether John University of Dar es Salaam, College of Information and Communication Technology, Department of Electronics and Telecommunications, P. O. Box 33335, Dar es Salaam, Tanzania Email: [email protected]; [email protected] Received 11 Oct 2020, Revised 22 Dec 2020, Accepted 28 Dec 2020, Published Feb 2021 Abstract Portfolio optimization is a major activity in any operating business. Conventional portfolio optimization research makes simplifying assumptions; for example, they assume no constraint in how many assets one holds (cardinality constraint). They also assume no minimum and maximum holding sizes (holding size constraint). Once these assumptions are relaxed, conventional methods become inapplicable, and hence new methods are needed to tackle this challenge. Threshold Accepting is an established algorithm in the extended portfolio optimization problem. In this paper, an algorithm called Guided Local Search (GLS) is applied using an accurate and efficient designed hill climbing algorithm, named HC-C-R. GLS sitting on HC-C-R is for the purpose of solving the extended portfolio optimization problem. The improved hill climbing algorithm is tested on standard portfolio optimization problem. Results are compared (benchmarked) with the Threshold Accepting (TA) algorithm, a well-known algorithm for portfolio optimization and are also compared with its original algorithm HC-C-R. Results show that GLS sitting on HC-C-R is more effective than HC-C-R and the algorithms are more effective than TA. Keywords: Portfolio Optimization, Algorithm, Guided Local Search, GLS, Threshold Acceptance. Introduction Markowitz model to practical problems using The portfolio optimization problem is a the standard/traditional methods like quadratic problem concerning asset allocation and programming, strong assumptions and diversification for maximum return with simplifications of the real market situations minimum risk. The problem is to find the have to be made. portfolio weights, i.e. how to most Markowitz model considers what is termed appropriately distribute the initial wealth across as standard portfolio optimization. In the the available assets, in order to meet the standard portfolio optimization problem, the investor’s investment objectives and constraints taken into account are budget and constraints (Markowitz 1952, Markowitz 1959, no-short selling. In reality however, portfolio Meucci 2005, Maringer 2008). optimization has realistic constraints to be Markowitz (1952) came up with a incorporated, such as holding sizes, cardinality, parametric optimization model for the problem transaction costs, portfolio size or additional of asset allocation and diversification for requirements from investors and authorities. maximum return with minimum risk, which When these realistic constraints are added to has become the foundation for Modern portfolio optimization, the problem quickly Portfolio Theory (MPT) or Markowitz theory becomes too complex to be solvable by or Mean-Variance model. To apply the standard optimization methods. When the 194 http://journals.udsm.ac.tz/index.php/tjs www.ajol.info/index.php/tjs/ Tanz. J. Sci. Vol. 47(1) 2021 assumptions and simplifications of the real correlations. Markowitz (1952) introduced market situations are relaxed and realistic what is known as the mean-variance principle, constraints added, now we have an extended where future returns are regarded as random portfolio optimization problem. Here the numbers and expected value (mean) of the Markowitz solution and the conventional returns E(r) and their variance (whose square methods like quadratic programming become root is called standard deviation/ risk) capture inapplicable. Heuristic methods are usually all the information about the expected outcome used to deal with this extended portfolio and the likelihood and range of deviations from optimization problems (Dueck and Winker it (Markowitz 1952, Markowitz 1959). 1992, Streichert and Tamaka-Tamawaki 2006, Maringer 2008, Gilli and Kellezi 2000, Crama Objective function and Schyns 2003, Muralikrishna 2008). The In the standard Markowitz model, Equation (1), most established heuristic algorithm used in (2) and (3) under basic constraints (4) and (5), extended portfolio optimization problems being the goal is to maximize the expected return, R, Threshold Accepting (Maringer 2005, Winker while diminishing incurred risk, (measured and Maringer 2007, Gilli and Kellezi 2000, as standard deviation/variance) (Markowitz Winker 2001, Gilli and Schumann 2010, Gilli (1952). and Schumann 2012). The new algorithm Given return (Rp) of a portfolio and variance 2 proposed below is benchmarked with ( p) of portfolio, the equation to maximize is 2 Threshold Accepting algorithm under standard Max (.E (Rp) – (1- ). p) (1) portfolio optimization problem. Subject to Expected return: The objective 퐸(푅p) = ∑푖 푤푖 E(푅푖) (2) The objective of the research was to produce Portfolio return variance: more effective and more efficient heuristic 2 =∑ ∑ 푤 푤 (3) algorithm for the extended portfolio p 푖 푗 푖 푗 푖 푗 푖푗 optimization problem. In this research, a 푖푗 = 1 for i = j heuristic algorithm is designed, investigated Constraints: and then applied to portfolio optimization ∑푖 푤푖 = 1 (4) problem under some constraints of the market. 0 ≤ 푤푖 ≤ 1 (5) The produced algorithm is implemented in Where the expected return of each asset solving the standard portfolio optimization is 퐸(푅푖), each asset variance is 푖, and each problem. The problem is to find the portfolio asset weight is 푤푖. weights, i.e. how to distribute the initial wealth From the Equation (1), the trade-off across the available assets, in order to meet the between return (Rp) and risk (p) of portfolio is investor’s objectives and constraints. The reflected. The efficient line/frontier is then significance of the research lies in efficient identified by solving the above problem for portfolio selection/optimization and also in different values of (0, 1): If = 1 the model efficient investment management (Markowitz will search for the portfolio with highest 1959). possible return regardless of the variance. With = 0, the minimum variance portfolio (MVP) Modern Portfolio Theory (MPT) or will be identified. Higher values of put more Markowitz Theory or Markowitz Model emphasis on portfolio’s expected return and Markowitz’s standard portfolio optimization less on its risk. (Markowitz 1952). Equations model (Markowitz 1952, Markowitz 1959) is a (4) and (5) are the constraints on the weights mathematical framework for describing and that they must not exceed certain bounds. The assessing return and risk of a portfolio of most important constraints are budget and assets, using returns, volatilities and return constraints since they characterize the 195 John - Application of guided local search (GLS) in portfolio optimization main part of the portfolio problem (Di Tollo more widely for better solutions. The and Roli 2008). The return constraint is when probability of accepting an inferior point the investor requires a certain level of profit decreases over time, following a cooling from his investment with minimum risk. The schedule on the “temperature”. When the budget constraint is when the investor has to temperature falls to 0, Simulating Annealing invest all the money/capital in the portfolio. behaves exactly like hill climbing. It has been However, return constraints can only be applied for portfolio selection (Muralikrishna satisfied for a historical portfolio (Sharpe 2000, 2008), and with constraints and trading Korn 1997, Prigent 2007, Markowitz 1952 and restrictions according to Crama and Schyns Markowitz 1959). (2003). Threshold Accepting (TA) (Dueck and Characteristics of heuristic optimization Scheuer 1990 and Winker and Maringer 2007) techniques can be seen as a variation of simulated The core of heuristic methods is an iterative annealing, except that there is no introduction principle that includes stochastic elements in of temperature. Instead of accepting inferior generating new candidate solutions and/or in new points with a certain probability, it accepts deciding whether these replace their only the points that fall below a fixed predecessors while still incorporating some threshold. TA was originally proposed by mechanisms that prefer and encourage Dueck and Scheuer (1990) as a deterministic improvements (Maringer 2008, Winker and and faster variant of the original Simulated Maringer 2007, Glover and Kochenberger Annealing algorithm. As Threshold Accepting 2006, Voudouris et al. 2010). They seek to avoids the probabilistic acceptance calculations converge to the optimum in the course of the of simulated annealing, it may locate an iterated search. They are flexible and not so optimal value faster than the actual simulated restricted to certain forms of constraints (Gilli annealing technique. In Threshold Accepting and Kellezi 2000, Winker 2001, Gilli and algorithm, the best solution obtained depends Winker 2008, Gill et al. 2011). Heuristic on some parameters such as the initial techniques solve optimization problems by threshold value, the threshold decreasing rate repeatedly generating new solutions and testing and the number of permutations. The initial them. The stopping criterion of the heuristic threshold and threshold