Analysis of risk measures in multiobjective optimization portfolios with cardinality constraint/Analise de risco em otimizacao multiobjetivo de carteiras com restricao de cardinalidade.
| Autor | Cardoso, Rodrigo T.N. |
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Introduction
The financial market generally allows investors to obtain higher profits, with the counterpoint of being exposed to greater risks. Therefore, investors are interested in simultaneously maximizing profits and minimizing risks. Using mathematical and computational models is proven to be decisively helpful in achieving optimal investments in stock markets. Markowitz (1952) was the first to establish a model based on diversification, with the premise that stock prices are normally distributed. In his work, the return of a portfolio is considered as the weighting of the expected values of the individual assets' returns, and the risk is considered as the variance or dispersion of the assets returns series in relation to the expected value (Nawrocki, 1999).
In portfolio selection, the higher the return, the greater the risk incurred, and there are two conflicting objectives. Thus, the investor must make a tradeoff between risk and return. Therefore, a multiobjective model is considered here, which presents solutions of compromise between risk and return as the final answer. The model also considers a cardinality constraint, with a given minimum and maximum quantity of assets to be included in the portfolio. The inclusion of the cardinality constraint greatly increases the algorithmic complexity of the solution (Cheng & Gao, 2015), so that the use of computational techniques as evolutionary algorithms is advisable to ensure good solutions in a reasonable time.
A risk measure can be defined as a function that associates to each distribution a number which describes its riskiness (Roman & Mitra, 2009). The study of new measures of risk is important since several financial catastrophes have shown the need to better evaluate the risk of investments. In fact, several risk models have been proposed taking into account that the return distributions may have heavy tails and fluctuations in the second moment of prices (Rosario, Mantegna & Stanley, 2000).
This work considers the comparison of the following risk measures in the Brazilian market case: variance, Exponentially Weighted Moving Average (EWMA), Generalized Autoregressive Conditional Heterosedasticity (GARCH), semivariance, Value at Risk (VaR), and Conditional Value at Risk (CVaR).
In fact, the variance as a measure of risk, as proposed by Markovitz, is subject to many criticisms. For example, it considers the price variation as the risk and, thus, returns below and above expectations are equally treated. Given this scenario, downside risk measures, such as VaR and CVaR, which consider just the risk from the perspective of losses, have gained ground over the years (Guo, Chan, Wong & Zhu, 2019). In view of that, risk aversion is one of the premises of this work, assuming that when the return is constant, investors always choose the lowest risk portfolio. Furthermore, some articles have also dealt with traditional volatility measures, such as EWMA and GARCH, as the risk measure in portfolio selection problems (Sahamkhadam, Stephan & Ostermark, 2018; Tang & Do, 2019).
Portfolio optimization problems with cardinality constraints cannot be solved by global techniques from discrete optimization (Burdakov, Kanzow & Schwartz, 2016). Furthermore, genetic algorithms have been used in the multiobjective portfolio optimization literature, both because they represent a good alternative in the face of hard optimization problems and also because they simultaneously generate several nondominated solutions in each execution run (Ferreira, Barroso, Hanaoka, Paiva & Cardoso, 2017).
To solve each biobjective risk-return optimization problem with cardinality constraint, two standard genetic algorithms are proposed here: based on Non-dominated Sorting Genetic Algorithm (NSGA-II) (Deb, Pratap, Agarwal & Meyarivan, 2002) and based on Strength Pareto Evolutionary Algorithm (SPEA2) (Zitzler, Laumanns, Thiele et al., 2001). For both algorithms, specific operators are considered, according to the nature of the problem. Statistical tests are performed to evaluate the consistency of the return, risk, drawdown, and drawup of each risk measure.
Some related works also consider a comparison with different risk measures and optimization algorithms. For a mono-objective optimization problem, Gaivoronski, Krylov & Van der Wijst (2005) consider variance, downside variance, downside absolute deviation, VaR, and CVaR, as a risk measure, optimizing with rebalancing. Paiva, Alves, Silva & Hanaoka (2014) compare variance and semivariance and show that semivariance yields better results in an out-of-sample analysis. Araujo & Montini (2015) compare variance, the root of semivariance and CVaR and show that CVaR is best in most scenarios evaluated. Bertsimas & Takeda (2015) study minimizing coherent risk measures under a norm equality constraint using a robust optimization formulation. More recently, Branda, Bucher, (Cervinka & Schwartz (2018) consider VaR and CVaR for the investment problems formulated as a nonlinear programming problem with cardinality constraints.
Comparing the performance of multiobjective algorithms, considering the variance as the risk measure, Anagnostopoulos & Mamanis (2011) show that algorithms SPEA2, NSGA-II, and e-MOEA performed better. Deb, Steuer, Tewari & Tewari (2011) propose a new version of NSGA-II with local search in a portfolio optimization problem with cardinality constraint. Considering just CVaR as the risk measure, ? show that NSGA-II performs better in a problem with cardinality constraint. Other algorithms have also been used in real-world portfolio optimization, such as those based on differential evolution (Krink & Paterlini, 2011).
Synthetically, this article proposes:
* a comparison of optimizing with cardinality constraint and several risk measures: variance, semivariance, EWMA, GARCH, VaR with two approaches, under normal distribution of returns and their robust counterparts under moment conditions, and CVaR, considering two multiobjective genetic algorithms, one based on NSGA-II and another based on SPEA2, in an in-sample analysis; and
* a comparison of optimizing with these risk measures in an out-of-sample analysis, performing a realistic case study, with rebalancing, considering fifty-three shares of B3, the Brazilian stock market, between the years 2012 to 2015.
The in-sample study shows that the algorithm SPEA2 performed better. With this algorithm, the relationship between the cardinality of the portfolio and each risk measured is evaluated, considering three cardinality intervals. Finally, the Pareto fronts obtained by each risk measure are compared, and the optimal portfolios of the greater ratio of risk and return as well. In the out-of-sample analysis, the stock marketing trades show the superiority of the risk measures based on quantiles, CVaR, and VaR, because they provide better monthly and accumulated returns and they are able to reduce negative returns scenarios, promoting a reduction of drawdown risk without limiting the financial drawup.
The article is organized as follow. Section 2 shows the theoretical basis of the considered risk measures. Section 3 shows the model and the proposed algorithms to solve the problem, and explains the metrics chosen to compare the algorithms. Section 4 presents the methodological details of the comparison between the two proposed algorithms (the in-sample analysis) and the behavior of the best algorithm in a stock market simulation with the comparison of the risk measures (the out-of-sample analysis). Section 5 shows and discusses the results. Finally, Section 6 summarizes and analyzes the implications of the results.
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Risk measures
Risk models can be divided into two categories: dispersion measures and measures based on quantiles (Roman & Mitra, 2009). Dispersion measures are based on a targeted return and generate only positive values. They can be divided into symmetrical and asymmetrical measures. The symmetrical measures evaluate the returns dispersion treating positive and negative values in the same way, while asymmetrical ones capture the risk considering the downside risk, the undesired returns. The dispersion measures considered here are variance, EWMA, semivariance, and GARCH. Measures based on quantiles quantify the risk by targeting the severity of losses, focusing on the left tail of the returns probability distribution. The measures based on quantiles considered here are VaR and CVaR.
2.1 Variance
The formula of variance is expressed by:
[[sigma].sup.2] = [T.summation over (t=1)] [([r.sup.t.sub.i] - [mu]).sup.2]/T, (1)
where [r.sup.t.sub.i] is the returns series of asset i in time t, [mu] the mean of returns, and T the number of observations in the series of returns. The variance is used in the popular Markowitz model expressed in terms of covariance between assets.
2.2 Semivariance
Semivariance was considered by Markowitz since 1959 as a more adequate risk measure, but the meanvariance model has gained more space for its ease of implementation. The semivariance can be treated as a particular case of the Lower Partial Moment (LPM) family, a model that considers returns smaller than a target return or minimum acceptable return, which characterizes the model as based on downside risk.
The semivariance with respect to a benchmark B, chosen by the investor, can be expressed by
SV = E[[min([r.sub.i] -B,0)].sup.2] = [1/T] [T.summation over (t=1)] [[min([r.sub.i,t] - B,0)].sup.2], (2)
where [r.sub.i,t] t is the return of assets i in time t. The semivariance based in semi-covariance expressed in Eq. (2) is than defined as follows:
SV = [n.summation over (i=1)] [n.summation over (j=1)][x.sub.i][x.sub.j][[summation].sub.ijB], (3)
where [x.sub.i] and [x.sub.j] are the weights of assets i and j in the portfolio.
2.3 EWMA
The Exponentially Weighted Moving Average model (EWMA), proposed by Riskmetrics[TM], calculates the conditional volatility at the time t, that is, it assumes that...
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