Polynomial Goal Programming Research Articles

Optimization investment portfolio of ESG capital market in Indonesia: An investigation of Polynomial Goal Programming based on higher moments

The development of capital market investors in Indonesia has experienced rapid growth, leading to increasingly volatile movements in the domestic market. Companies or investment managers must form asset portfolios with optimal expected returns and measurable risks to address this issue. This study aims to analyze the potential combinations that can be formed using ESG Capital Market Indonesia 2022 and safe-haven instruments, utilizing the Markowitz theory and Polynomial Goal Programming (PGP) method. This research uses secondary data, and the research variables used are ESG Capital Market Indonesia 2022 and Gold from January 2007 to December 2022. Based on the previous research, the formation of portfolio combinations using safe-haven instruments, employing the Markowitz theory and Polynomial Goal Programming (PGP) method can provide better investment portfolio combinations during crises. Furthermore, the performance of each formed combination is measured using the Sharpe Ratio. The findings indicate that Markowitz's theory can generate better performance than the PGP method. However, it exhibits suboptimal skewness and kurtosis values compared to the PGP method.

Hedging longevity risk under non-Gaussian state-space stochastic mortality models: A mean-variance-skewness-kurtosis approach

Longevity risk has recently become a high profile risk among insurers and pension plan sponsors. One way to mitigate longevity risk is to build a hedge using derivatives that are linked to mortality indexes. Longevity hedging methods are often based on the normality assumption, considering only the variance but no other (higher) moments. However, strong empirical evidence suggests that mortality improvement rates are driven by asymmetric and fat-tailed distributions, so that existing longevity hedging methods should be expanded to incorporate higher moments. This paper fills the gap by adopting a mean-variance-skewness-kurtosis approach based on non-Gaussian extensions of commonly-used stochastic mortality models, formulated in a state-space setting. On the basis of a general representation of these models, the authors derive (approximate) analytical expressions for the moments of the present values of the hedging instruments and the liability being hedged. These expressions are then integrated with a polynomial goal programming model, from which the optimal hedge portfolio is identified. Finally, the paper demonstrates the theoretical results with a real mortality data set and a range of hedger preferences.

Higher moment of portfolio optimization with Polynomial Goal Programming approach

The mean-variance (MV) model has been introduced in portfolio optimization to minimize the risk and achieve the target rate of return in the investment. However, the higher moment skewness and kurtosis are not considered in this model. The investors prefer portfolio with high skewness value and low kurtosis value so that the probability of getting extreme negative rates of return will be reduced. Therefore, the MV model has been extended to the mean-variance-skewness-kurtosis (MVSK) model by incorporating the skewness and kurtosis factor. The objective of this study is to construct the optimal portfolio of the MVSK model by using the polynomial goal programming (GP) approach. The data of this study comprises technology companies that listed in Malaysian stock market. In the fourth industrial revolution, technology companies play an important role in the development of a country. The results of this study show that the optimal portfolio of MVSK model outperforms the MV model by giving higher portfolio skewness value and lower portfolio kurtosis value. This study is significant because the investors can maximize the portfolio skewness value and minimize the portfolio kurtosis value with the MVSK model.

Credibilistic variance and skewness of trapezoidal fuzzy variable and mean–variance–skewness model for portfolio selection

The fuzzy set theory is widely used to describe the uncertainty of financial markets in modern portfolio selection problems. In this study, the credibility theory (a popular branch of the fuzzy set theory) is applied to extend Markowitz’s mean–variance portfolio selection model into mean–variance–skewness portfolio selection model. For this, the returns on risky stocks are regarded as trapezoidal fuzzy variables. The credibilistic skewness is obtained originally for trapezoidal fuzzy variable, and the incorporation of the credibilistic variance of trapezoidal fuzzy variable enables the estimation of risk on portfolio return. To solve the proposed multi-objective optimization problem, a polynomial goal programming approach is suggested in this study. In addition, a dominant numerical analysis of the proposed work and its comparison with existing works are presented.

Gainers and losers with higher order portfolio risk optimization

There is a large body of knowledge concerning the cointegration of international stock markets. In light of such a concept, this study attempts to provide an answer to the international portfolio optimization problem. It is likely that investors’ preferences for mean, variance, skewness and kurtosis may vary over time. One investor may choose to give more weight to profits, while another may choose variance or skewness over return. To achieve multi-objective optimization, the study uses the polynomial goal programming (PGP) model. It is the first study of its kind to consider the closing stock indices of BRICS (Brazil, Russia, India, China and South Africa) economies from January 2010 to December 2016, by assigning various weights to individual BRICS countries under an alternative MVSK (mean, variance, skewness, kurtosis) framework. It notes that two cornerstones of BRICS (India and China) show an investment pattern such that when potential investors approach India as a favourable country to invest in stocks, Chinese portfolio returns move the other way, and vice versa.

Polynomial goal programming and particle swarm optimization for enhanced indexation

Enhanced indexation is an investment strategy that aims to generate moderate and consistent excess returns with respect to a tracked benchmark index. In this work, we introduce an optimization approach where the risk of under-performing the benchmark is separated from the potential over-performance, and the Sharpe ratio measures the profitability of the active management. In addition, a cardinality constraint controls the number of active positions in the portfolio, while a turnover threshold limits the transaction costs. We adopt a polynomial goal programming approach to combine these objectives with the investor’s preferences. An improved version of the particle swarm optimization algorithm with a novel constraint-handling mechanism is proposed to solve the optimization problem. A numerical example, where the Euro Stoxx 50 Index is used as the benchmark, shows that our method consistently produces larger returns, with reduced costs and risk exposition, than the standard indexing strategies over a 10-year backtesting period.

A polynomial goal programming approach for intuitionistic fuzzy portfolio optimization using entropy and higher moments

Although there are myriad works that deal with fuzzy portfolio optimization, there is a dearth of research that deals with the problem in an intuitionistic fuzzy environment. So, taking the route less travelled, we propose two intuitionistic fuzzy portfolio selection models for optimistic and pessimistic scenarios, respectively. For this purpose, we take into account four objectives, namely return, variance, skewness and entropy, along with some realistic constraints such as a cardinality constraint, a contingent constraint, and complete capital utilization. Also, short selling is prohibited. Another popular constraint, the “floor and ceiling” constraint, is presented and employed in the form of a flexible constraint by considering the confidence interval in the model. The membership and non-membership of the objectives are modelled using the extreme values of the four objectives. The proposed approach provides avenues for the inclusion and minimization of the hesitation degree into the decision making, thereby, resulting in a significantly better portfolio. The proposed model enables the decision makers to obtain a variety of results through the use of several schemes that can be postulated (customized) according to the decision makers’ preferences, and from which, results best suited to their preferences, can be chosen. A numerical illustration with eight different schemes is presented to demonstrate the virtues of the proposed model.

Performance of Different Risk Indicators in a Multi-Period Polynomial Portfolio Selection Problem Based on the Credibility Measure.

In this paper, we study the portfolio selection problem considering transaction costs under multiple periods. For non-professional investors, it is a critical factor to choose an appropriate model among multiple portfolio selection models in investment. Based on the credibility measure, we formulate a multi-period polynomial portfolio selection model to gather the risk indicators involving variance, semi-variance, entropy, and semi-entropy, helping investors bet on assets. According to the polynomial goal programming (PGP) approach, investors can conquer the fields by combining apposite indicators to build appropriate models. Subsequently, an adjusted genetic algorithm on the foundation of the penalty function is designed to obtain the optimal solution of this multi-period model. The results indicate that the PGP method is suitable for investors to choose the model and assigns the proper models to investors with different risk preferences.

A polynomial goal programming model for portfolio optimization based on entropy and higher moments

Portfolio selection is a critical factor in investment. Having considered a number of risky assets, fund managers must choose the optimum portfolio. Stock values can be affected by different types of events such as governmental crises, economic turmoil and industrial improvements. Due to the vague nature of these events, it is difficult to estimate the future value of stocks. However, Markowitz's Modern portfolio theory, which is principally focused on portfolio risk, has introduced a novel model for stock diversification. According to this approach, an investor can decrease portfolio risk basically by holding mixtures of assets that are not highly positively correlated. Meanwhile, an efficient portfolio can only be obtained by focusing on return and risk simultaneously. When the normality assumption of return series of assets are not valid, higher moments can also be added to ensure the efficiency of the Markowitz model. In this study, a new approach for polynomial goal programming which is based on a mean-variance-skewness-kurtosis-entropy model is proposed. Two real data sets were used in experiments to verify the effectiveness of the proposed model. Based on a variety of portfolio optimization model, the out-of-sample performance of two entropy measures- Shannon entropy and Gini-Simpson entropy- were compared in the portfolio selection. The results indicate that the proposed approach is well-suited especially for portfolio models with higher moments.

Portfolio Optimization using Higher Order Moments of the Stocks Returns Distribution: The Case of Bucharest Stock Exchange

The Modern Portfolio Theory, based on Markowitz’s (1952) work, propose a portfolio selection that consider only the first two moments from a time series of returns. In spite of the popularity of Markowitz’s portfolio selection, many critiques have been emerging throughout the years. All this critics are about the hypothesis that Modern Portfolio Theory uses in order to get the equilibrium on capital markets constrain like the absence of transactions cost and assets financial efficiency. The aim of this paper is to use higher return moments such as skewness and kurtosis for portfolio selection. A series of theoretical papers pointed out that portfolio with excess skewness and smaller kurtosis are preferred by individual investors. Using polynomial goal programming we make a comparison of two different strategies of portfolio selection bases on Bucharest Stock Exchange quotes. Intrinsic reusable preference parameters for higher order moments have resulted with respect to BSE shares. Shares having returns with low sensitivity to the market evolution get to be the most selected ones.

Polynomial goal programming and the implicit higher moment preferences of US institutional investors in hedge funds

Polynomial goal programming (PGP) is a flexible method that allows investor preferences for different moments of the return distribution of financial assets to be included in the portfolio optimization. The method is intuitive and particularly suitable for incorporating investor preferences in higher moments of the return distribution. However, until now, PGP has not been able to meet its full potential because it requires quantification of “real” preference parameters towards those moments. To date, the chosen preference parameters have been selected somewhat “arbitrarily”. Our goal is to calculate implied sets of preference parameters using investors’ choices of and the importance they attribute to risk and performance measures. We use three groups of institutional investors—pension funds, insurance companies, and endowments—and derive implied sets of preference parameters in the context of a hedge fund portfolio optimization. To determine “real” preferences for the higher moments of the portfolio return distribution, we first fit implied preference parameters so that the PGP optimal portfolio is identical to the desired hedge fund portfolio. With the obtained economically justified sets of preference parameters, the well-established PGP framework can be employed more efficiently to derive allocations that satisfy institutional investor expectations for hedge fund investments. Furthermore, the implied preference parameters enable fund of hedge fund managers and other investment managers to derive optimal portfolio allocations based on specific investor expectations. Moreover, the importance of individual moments, as well as their marginal rates of substitution, can be assessed.

A new higher moment portfolio optimisation model with conditional value at risk

Variance is a common risk measure used for constructing portfolios. However, variance strictly depends on the assumptions that the returns of assets are not normally distributed or investor’s function is quadratic. Moreover, variance not only penalises the downside deviations below the mean return but also the upside deviation, variance, thus, does not match investor’s desire to maximise the upside deviation and minimise the downside deviation. The objective of this paper is to propose a new four moment mean-conditional-value-at-risk-skewness-kurtosis model and empirically test the model. In this proposed model, variance is replaced with conditional value at risk as the risk measure. The polynomial goal programming method is used in this study as it is flexible to incorporate different degree of investor’s preference on mean, skewness and kurtosis. Results of this study demonstrate that the mean-CVaR-skewness-kurtosis model gives higher mean return and skewness and provides better performance than the mean-variance-skewness-kurtosis model for all combinations of degree of preferences. This implies that CVaR is a better risk measure than variance in portfolio optimisation.

Portfolio selection with skewness: A comparison of methods and a generalized one fund result

This contribution compares existing and newly developed techniques for geometrically representing mean–variance–skewness portfolio frontiers based on the rather widely adapted methodology of polynomial goal programming (PGP) on the one hand and the more recent approach based on the shortage function on the other hand. Moreover, we explain the working of these different methodologies in detail and provide graphical illustrations in relation to the goal programming literature in operations research. Inspired by these illustrations, we prove two new results: a formal relation between both approaches and a generalization of the well-known one fund separation theorem from traditional mean–variance portfolio theory.

Portfolio Optimization with Asian Hedge Funds: A Comparison between the Mean - Variance and the Polynomial Goal Progamming Methods

This study attempts to construct optimal portfolios of Asian hedge funds (AHFs) by employing both the traditional mean - variance (MV) method and the polynomial goal programming (PGP) method. In addition, further issues such as the existence of optimal number of funds and the contribution of a specific fund characteristic to the performance of a portfolio of AHFs are also investigated. From the Eurekahedge database, a sample consisting of 95 AHFs was selected for this study, covering the period of 1st January 2000 - 31st May 2009. Results confirmed that (1) diversification benefits exist in AHFs, (2) optimal number of funds exist for a portfolio of AHFs, (3) many portfolios of AHFs with specific fund characteristics achieved high probabilities of being either the first or the second best-performing portfolios at various levels of standard deviations, and (4) the PGP method was shown to be more effective than the MV method as it enables different efficient frontiers to be constructed according to individual investors’ risk preferences.

Higher moment portfolio management with downside risk

Variance is the common risk measure that has been used in portfolio optimization since the introduction of the mean-variance model. However, the mean-variance model penalizes not only the downside deviation but also the upside deviation. The upside deviation is desirable to the investors. The objective of this study is to compare the compositions and performances of optimal portfolios by replacing variance with lower partial moment. The lower partial moment is the downside risk that focuses on the deviation below the specified target rate of return which better matches investor's perception towards risk. Incorporating skewness, this study employs the polynomial goal programming method. This method is flexible to incorporate different degree of investor's preference for mean return and skewness. Skewness is important because increasing skewness reduces the probability of getting negative rates of return. The empirical results demonstrate that the optimal portfolios compositions base on lower partial moment are different to the optimal portfolios compositions based on variance. Furthermore, at the same level of downside risk, optimal portfolios that based on lower partial moment are found to give higher expected return and skewness than the optimal portfolios that are based on variance. This implies that the optimal portfolios that are based on lower partial moment dominate those based on variance at the same level of downside risk. Lower partial moment is a more appropriate risk measure for the investors because it only penalizes the downside deviation.

Performance Analysis of a Collateralized Fund Obligation (CFO) Equity Tranche

This article examines the performance of the junior tranche of a Collateralized Fund Obligation (CFO), i.e. the residual claim (equity) on a securitized portfolio of hedge funds. We use a polynomial goal programming model to create optimal portfolios of hedge funds, conditional to investor preferences and diversification constraints (maximum allocation per strategy). For each portfolio we build CFO structures that have different levels of leverage, and analyze both the stand alone performance as well as potential diversification benefits (low systematic risk exposures) of investing in the Equity Tranche of these structures. We find that the unconstrained mean-variance portfolio yields a high performance, but greater exposure to systematic risk. We observe the exact opposite picture in the case of unconstrained optimization where a skewness bias is added, thus proving the existence of a trade-off between stand alone performance and low exposure to systematic risk factors. We provide evidence that leveraged exposure to these hedge fund portfolios through the structuring of CFOs creates value for the Equity Tranche investor.

Performance analysis of a collateralized fund obligation (CFO) equity tranche

This article examines the performance of the junior tranche of a collateralized fund obligation (CFO), i.e. the residual claim (equity) on a securitized portfolio of hedge funds. We use a polynomial goal programming model to create optimal portfolios of hedge funds, conditional to investor preferences and diversification constraints (maximum allocation per strategy). For each portfolio, we build CFO structures that have different levels of leverage, and analyze both the stand-alone performance as well as potential diversification benefits (low systematic risk exposures) of investing in the equity tranche of these structures. We find that the unconstrained mean-variance portfolio yields a high performance, but greater exposure to systematic risk. We observe the exact opposite picture in the case of unconstrained optimization, where a skewness bias is added, thus proving the existence of a trade-off between stand-alone performance and low exposure to systematic risk factors. We provide evidence that leveraged exposure to these hedge fund portfolios through the structuring of CFOs creates value for the equity tranche investor, even during the recent financial crisis.

Fund of hedge funds portfolio selection: A multiple-objective approach

This paper develops a technique for fund of hedge funds to allocate capital across different hedge fund strategies and traditional asset classes. Our adaptation of the polynomial goal programming optimisation method incorporates investor preferences for higher return moments, such as skewness and kurtosis, and provides computational advantages over rival methods. We show how optimal allocations depend on the interaction between strategies, as measured by covariance, co-skewness and co-kurtosis. We also demonstrate the importance of constructing ‘like for like’ representative portfolios that reflect the investment opportunities available to different-sized funds. Our empirical results reveal the importance of equity market neutral funds as volatility and kurtosis reducers and of global macro funds as portfolio skewness enhancers.

Polynomial Goal Programming and the Implicit Higher Moment Preferences of U.S. Institutional Investors in Hedge Funds

Polynomial goal programming (PGP) has been successfully introduced in the hedge fund allocation literature. The method is intuitive, and flexible enough to incorporate investor preferences in higher moments of the return distribution. However, until now, PGP could not be used to its full effectiveness because it requires quantification of real preference parameters toward those moments. Previously, those parameters were chosen without elaborate research. Our goal in this paper is to identify implied sets of preference parameters by using investors' choice of and the importance they attribute to risk and performance measures. We use three groups of institutional investors, pension funds, insurance companies, and endowments, and we derive implied sets of preference parameters in the context of a hedge fund portfolio optimization. To extract implied investor preference parameters, we determine optimal hedge fund portfolios according to the criteria set by those fund of hedge fund sponsors. We then fit implied preference parameters so that the PGP optimal portfolio is identical to the desired hedge fund portfolio. We can thus determine real preferences for the higher moments of the portfolio return distribution. With the obtained economically justified sets of preference parameters, we can use the well established PGP framework more efficiently to derive allocations that satisfy institutional investor expectations for hedge fund investments. Furthermore, the implied preference parameters enable fund of hedge fund managers and other managers to combine different hedge fund strategies optimally according to investor expectations. Finally, we can assess the importance of individual moments, as well as their marginal rates of substitution.

Hedge funds and higher moment portfolio selection

Two key tasks faced by industry practitioners are that of asset allocation and that of strategy allocation within the hedge fund universe. It is well documented (see Cvitani’c et al.,1 Agarwal and Naik,2 Amenc and Martellini3 and Amin and Kat4) that hedge funds are marked by their heterogeneity and unusual statistical properties. This makes the use of conventional methods of portfolio construction subject to question and necessitates the investigation of a more sophisticated approach to inform the construction of appropriate and efficient portfolios. This paper compares and evaluates the results of two related optimisation procedures. First, the classic mean- variance portfolio optimisation of Markowitz5 and secondly a recent approach introduced by Davies et al.6 utilising Polynomial Goal Programming (PGP) to optimise portfolios return distributions for higher moments to include mean, variance, skewness and kurtosis for a given set of investor preferences. This comparison will be presented in the context of both a fund-of-hedge-fund strategy allocation as well as the asset allocation problem of what proportion to allocate to hedge funds in a balanced portfolio.