بازنمونه گیری زمان مهر برای بهینه سازی سبد سرمایه گذاری تکاملی قدرتمند

Traditional mean–variance financial portfolio optimization is based on two sets of parameters, estimates for the asset returns and the variance–covariance matrix. The allocations resulting from both traditional methods and heuristics are very dependent on these values. Given the unreliability of these forecasts, the expected risk and return for the portfolios in the efficient frontier often differ from the expected ones. In this work we present a resampling method based on time-stamping to control the problem. The approach, which is compatible with different evolutionary multiobjective algorithms, is tested with four different alternatives. We also introduce new metrics to assess the reliability of forecast efficient frontiers.

Asset allocation is one of the core topics in financial management. The search for the optimal choice of financial assets to be included in a portfolio has been the subject of research for a long time and it is one of the most active lines in finance. The academic literature on this subject is very large and mostly based on the work of Markowitz, 1952 and Markowitz, 1959. Under this framework, the problem is introduced as a multiobjective optimization problem where the investor tries to figure out the weight that each of the investment alternatives should carry in the portfolio. The target of this investor would be both minimizing risk and maximizing return at the same time. The solution to the problem of optimizing for these two objectives in conflict defines a set of solutions called the efficient frontier. This Pareto front consists on portfolios that are neither better or worse than the rest. For each level of risk or return, there is no better alternative in terms of the other objective. This makes the election of one of them a choice to be made by investors according to the way they weight risk and rewards. The basic version of the problem can be solved using Quadratic Programming (QP). Unfortunately, this approach is built on a set of assumptions that are unlikely to hold in the real world. The quest for alternatives has driven attention to metaheuristics that might not suffer this limitation. This is the reason why the framework of evolutionary computation is getting traction on this area. The solutions based on metaheuristics mostly fall into one this two categories: solutions that transform the multiobjective problem into a single-objective form, and those who deal with it using a mutiobjective approach. Among the first group, we would mention the work by Soleimani, Golmakani, and Salimi (2009). These authors use a genetic algorithm and extend the mentioned classic mean–variance optimization model and consider constraints on transaction costs, round lots or cardinality. They tackle the multiobjective nature of the problem minimizing the risk objective while setting a range of minimum acceptable returns in the constraints. Chiranjeevi and Sastry (2007) use another popular approach to transform the multiobjective problem into a version that can be handled by single-objective algorithms. Instead of keeping one of the objectives on the objective function and the other in the constraint, they consider the objectives in the fitness function weighting them. In this particular case, they manage five objectives that are obtained breaking down the basic two. Chang, Yang, and Chang (2009) suggest a similar solution and optimize a function that weights risk and return using a risk aversion parameter. Zhu, Wang, Wang, and Chen (2011) target a popular metric, the Sharpe Ratio, that combines both elements into a single expression. The development of multiobjective evolutionary algorithms (MOEAs) has resulted into a number researchers exploring their performance in this area. Among these we could mention Skolpadungket, Dahal, and Harnpornchai (2007), who test a set of multi-objective algorithms (VEGA, SPEA2, NSGA-II etc.) on a constrained version of the two objective problem. More recently, Anagnostopoulos and Mamanis (2011) compare the performance of different multiobjective algorithms, and Deb, Steuer, Tewari, and Tewari (2011) introduce a customized hybrid version of NSGA-II to tackle the problem. Finally, we will mention the work of Radziukyniene and Xilinskas (2008), where authors compare FastPGA, MOCELL, AbYSS, and NSGA-II on both of the basic problems, and an extended version that considers the dividend yield as a third objective. Despite of the amount of research on portfolio optimization, there are still open issues. A key one is the robustness of results provided by algorithms. Among the most important factors that asset managers have to consider when evaluating the results provided by any of the above-mentioned methods, there is reliability. Very often, the expected efficient frontier lies far from the real one. This is due to the fact that the estimates for the expected risk and returns for the assets in the solution, and the portfolios derived from them, are very inaccurate. Understandably, this results on mistrust by some practitioners. The search for solutions for this problem has cleared the way for the field of robust portfolio optimization. It is in this area where we focus our contribution. We introduce a time-stamping method to control the population that enhances the reliability of the solutions provided by MOEAs. The process of optimizing the risk and return of a portfolio relies on two parameters: the estimates for the expected asset returns and the variance–covariance matrix. The values of these parameters are usually based on past data and they might be inaccurate due to, for instance, the presence of outliers. In this context, there are several potential ways to tackle the problem. The two most prevalent alternatives found in the literature rely on either putting an emphasis on having robust estimates for the parameters, or managing the optimization process itself. The first one usually tries to filter the estimates to control, for instance, the influence of extreme past events on their computation (Perret-Gentil et al., 2005). The authors focusing on the second alternative design approaches handle uncertainty in the parameters during the optimization process (Pflug and Wozabal, 2007 and Tütüncü and Koenig, 2004). The alternative suggested in this paper falls in the latter category. We will enhance the solutions of MOEAs testing the population for different values for the parameters, and selecting the portfolios that consistently offer a good performance. Optimizing for a single scenario, a set of expected asset returns and the use of a single variance–covariance matrix, bears the risk of getting solutions that might be extremely sensitive to deviations. This could potentially be a problem as it is almost certain that the estimates will not be accurate. We have to bear in mind that having perfect estimates for the expected returns for instance, implies that we can make perfect predictions for future prices, which is highly unlikely. For this reason we consider that assessing the candidate solutions in different likely scenarios and favoring those that consistently offer a good performance, might be a promising approach. The requirement of consistency is key. In order to achieve it, we introduce a time-stamping mechanism that, together with performance in terms of risk and return, will drive the evolution process to find robust and stable solutions. The approach introduced in this paper is related to alternatives based on resampling (Ruppert, 2006 and Shiraishi, 2008). The most comparable among the traditional approaches is the one described by Idzorek (2006). This author suggests using combining traditional QP with Monte Carlo simulation to derive a set of fronts that are merged into a single solution at a later stage. This idea is very interesting but, unfortunately, the approach suffers the shortcomings of QP, namely, the ability to deal with real-world constraints. This is the reason why we feel that adapting the idea to the framework of MOEAs is very promising, as they do not suffer from this limitation. There is a previous effort based of evolutionary algorithms along the lines of this work, but is based on a very simple resampling approach that optimizes for a different scenario in each generation (Garcia, Quintana, Galvan, & Isasi, 2011). The time-stamping mechanism that we introduce in this work is based on the previous one extending the problem with a third implicit objective that favors solutions that are consistently reliable. As we will see in the experimental section, this results on significantly higher robustness. The proposed approach is compatible with a wide array of MOEAs. Given their different nature and behavior, we will test the approach on a set of popular algorithms. Specifically, the experimental section will consider NSGA-II, SPEA2, SMPSO and GDE3. NSGA-II (Deb, Pratap, Agarwal, & Meyarivan, 2002) is one the most referenced algorithms in the field of multiobjective optimization. This one, together with SPEA2 (Zitzler, Laumanns, & Thiele, 2001) have been confirmed (Skolpadungket et al., 2007) to offer good performance in portfolio optimization. Apart from the mentioned two, we consider GDE3 (Kukkonen & Lampinen, 2005), a differential evolution strategy, and SMPSO, (Nebro et al., 2009) a multiobjective algorithm based on particle swarm optimization. In the context of multiobjective problems, an important issue is the metric used to evaluate the solutions. It is generally admitted that there is no ideal single metric that should be used to evaluate different objectives simultaneously in every circumstance. The metrics that are most commonly used in this field, such as Hypervolume, Spread or SetCoverage are not appropriate indicators of stability in this context. For this reason, we define a set of metrics that capture different aspects of robustness in efficient frontiers. These, Estimation Error, Stability, Extreme Risk and Unrealized Returns draw on the basic principle that the expected risk and returns for the portfolios in the solution should be close to the observed ones ex-post. The rest of the paper is organized as follows. First, we make a formal introduction to the financial portfolio optimization problem. After, we describe in detail the evolutionary approach proposed in this work. This section includes a brief description of the MOEAs, the solution encoding and the fitness mechanism in order to find robust and stable portfolios. Next, the different metrics used in this work to evaluate robustness of solutions are described. That will be followed by the experimental results and, finally, there will be a section devoted to summary and conclusions.

Portfolio optimization represents one of the main topics in financial research. The basic framework targets the combination of financial assets to be included in a portfolio in order to optimize the balance of risk and return. This process is usually based on two parameters: the estimates for the expected asset returns, and the variance–covariance matrix. Quadratic Programming can be used to solve the basic version of this problem. However, in the real world there are some constraints that cannot be easily tackled by traditional techniques. This is the reason why evolutionary multiobjective algorithms (MOEAs), that do not present these kind of limitations, are getting traction in the domain. One of the main problems that portfolio managers face is the uncertainty regarding the expected frontier derived from their forecasts for future returns. Very often, it lies far from the actual one, resulting in inaccurate forecast risk/return profiles for the portfolios. Our work is focused on this point, adding robustness to the results by preventing optimizing for a single expected scenario that may produce solutions that are hyper-specialized and might be extremely sensitive to likely deviations. We handle the uncertainty in these parameters during the optimization process testing the population for different values of parameters and selecting the portfolios that consistently offer a good performance. We also use a time-stamping mechanism that, in collaboration with performance in terms of risk and return, drives the evolution process to find robust and stable solutions. The assessment of the robustness of the efficient frontiers resulting from the process requires metrics that differ from the most traditional ones. For this reason, we introduce a set of four metrics “Estimation Error”, “Stability”, “Extreme Risk” and “Unrealized Returns”. The approach was tested through experimentation using four popular MOEAs (NSGA-II, SPEA2, SMPSO and GDE3). The results were compared in both the standard and the time-stamped resampled versions of the algorithms. They were tested on a sample of monthly returns for eight indexes representing different broad investment categories including stock, bonds, etc. The results show that the suggested approach enhances significantly the reliability of solutions for all algorithms. The time-stamped resampled versions improved the results across all the considered MOEAs. The improvements got up to 67.53% in “Stability”, 54.93% in “Estimation Error”, 52.92% in “Extreme Risk” and 33.32% in “Unrealized Returns”. These improvements were higher when standard MOEA provided worse results in terms of the metrics. We also observed that the R + T version of SMPSO tended to outperform the alternatives. Even though these results are good and promising in terms of robustness, there are several issues left open that could lead to future extensions of this work. Among them, the analysis of the performance of this robust approach on other MOEAs or the scalability of the results with different number of investment alternatives.