Portfolio selection under higher moments using fuzzy multi-objective linear programming

Abstract

Since asset returns have been recognized as not normally distributed, the avenue of research regarding portfolio higher moments soon emerged. To account for uncertainty and vagueness of portfolio returns as well as of higher moment risks, we proposed a new portfolio selection model employing fuzzy sets in this paper. A fuzzy multi-objective linear programming (MOLP) for portfolio optimization is formulated using marginal impacts of assets on portfolio higher moments, which are modelled by trapezoidal fuzzy numbers. Through a consistent centroid-based ranking of fuzzy numbers, the fuzzy MOLP is transformed into an MOLP that is then solved by the maximin method. By taking portfolio higher moments into account, the approach enables investors to optimize not only the normal risk (variance) but also the asymmetric risk (skewness) and the risk of fat-tails (kurtosis). An illustrative example demonstrates the efficiency of the proposed methodology comparing to previous portfolio optimization models.

Keywords

Portfolio selection higher moments fuzzy sets fuzzy multi-objective linear programming (FMOLP)marginal impacts

1 Introduction

Portfolio selection receives the careful attention of investors, and particularly after the seminal work of Markowitz [24]. The theory is primarily to maximize the expected utility and then derive investors’ preferences regarding investment risks. Several utility functions were suggested for investor risk measures such as quadratic, semi-quadratic, polynomial, power, or logarithmic utility functions. Since the real expected utility varies between investors, and changes over time, these utility functions only functionally represent utility in order to accommodate numerical implementations. Based on the preferences, say higher return and lower risk (variance), the standard mean-variance optimization (MVO) model was established that utilizes mean and (co)variance of asset returns. The fundamental lesson is that investors must take care not only of the realized returns, but also of the risk represented by the standard deviation of their portfolio return. By selecting portfolios on the efficient frontiers the investor would require a maximized expected return portfolio for any fixed standard deviation. As non-normality of asset returns has long been recognized, researchers have questioned the adequacy of the mean-variance criterion and investigated higher moments.

Consideration of higher moments therefore has emerged, but it results in complex multi-objective high order nonlinear programming especially when the number of assets increases. Evolutionary approaches have been usually deployed to deal with multi-objective problems but most of these algorithms involve computational burden due to the evolutionary iterations. In this paper, we suggest a method that uses multi-objective linear programming (MOLP) formulated based on the marginal impacts of assets. Moreover, because asset marginal impacts regarding variance, skewness and kurtosis before selecting portfolios are uncertain, they are characterized by fuzzy numbers. The MOLP thus is extended to a new fuzzy MOLP (FMOLP) to account for fuzzy coefficients.

In our previous study, Nguyen and Gordon-Brown [28], we modelled marginal impacts of assets by triangular fuzzy numbers. This approach of using triangular fuzzy numbers may limit the capability to represent uncertainty, whereas deployment of other kinds of fuzzy numbers would allow for different uncertainty representations to be facilitated. As an extension, we herein employ trapezoidal fuzzy numbers for modelling marginal impacts of assets. Furthermore, the constrained fuzzy analytical hierarchy process (FAHP) used in Nguyen and Gordon-Brown [28] is unable to deal with trapezoidal fuzzy numbers. The introduction of the aforementioned FMOLP, which is able to deal with trapezoidal fuzzy numbers, is also aimed to overcome the shortcoming of the FAHP.

The rest of the paper is organized as follows. The next section presents higher moment preferences and the marginal impacts of assets on portfolio higher moments. A comprehensive literature review of fuzzy approaches to portfolio selection is presented in Section 3. Modelling asset marginal impacts by trapezoidal fuzzy numbers will be addressed in Section 4. Section 5 is devoted to presentation and solution of the FMOLP. An illustrative example is presented in Section 6 followed by concluding remarks.

2 Higher moment preferences and marginal contributions

To measure the effect of higher moments on the asset allocation, it is necessary to consider a standard expected utility U (W) of an investor over the terminal wealth W. Let R = (R₁, …, R_n) ′ be the vector of rates of return of n risky assets, μ = E [R] = (μ₁, …, μ_n) ′ the expected return vector, w = (w₁, …, w_n) ′ the weight vector representing proportion of wealth allocated to various assets, and thus the terminal wealth is given by W = (1 + r_p) where r_p = w′R. Accordingly the first constraint $\sum_{i = 1}^{n} w_{i} = 1$ (1) assumes that the investor does not have access to a riskless asset, and another constraint $w_{i} \geq 0, \forall i$ (2) means short-selling is not allowed. In order to approximate the expected utility, a Taylor series expansion in terms of the wealth distribution is deployed $E [U (W)] = \int U (u) f (u) du$ (3) where f (u) is the probability distribution of terminal wealth. Consequently, the infinite Taylor series of the utility function is $U (W) = \sum_{k = 0}^{\infty} \frac{U^{(k)} (E (W))}{k!} {(W - E (W))}^{k}$ (4)

The expected utility is derived by applying the expectation operator to the above equation

$\begin{matrix} E [U (W)] & = & E [\sum_{k = 0}^{\infty} \frac{U^{(k)} (E (W))}{k!} {(W - E (W))}^{k}] \\ = & \sum_{k = 0}^{\infty} \frac{U^{(k)} (E (W))}{k!} E [{(W - E (W))}^{k}] \end{matrix}$ (5)

Clearly, the expected utility from an investment in risky assets depends on all central moments of the distribution of the terminal wealth. The infinite Taylor expansion is a solution for the expected utility but, is not possible for numerical implementations. A truncation on the first k orders of the infinite Taylor series is a reasonable approximation of the expected utility. What level of the infinite Taylor series should be truncated is intuitively interpreted (see e.g. [8, 16]). Whatever k is chosen, the resulting polynomial utility function is consistent with the findings of Scott and Horvath [32] that investors prefer higher odd, and lower even moments. We set out the fourth-order Taylor expansion that extends the conventional mean-variance method by including skewness and kurtosis aiming at a better approximation of the expected utility.

$\begin{matrix} E [U (W)] & \approx & U (E (W)) + \frac{1}{2!} U^{(2)} (E (W)) μ^{(2)} \\ + \frac{1}{3!} U^{(3)} (E (W)) μ^{(3)} \\ + \frac{1}{4!} U^{(4)} (E (W)) μ^{(4)} \end{matrix}$ (6) where μ⁽ⁿ⁾ is the nth-order centered moment: $μ^{(n)} = E ({(W - E (W))}^{n})$ (7)

Formulae of portfolio expected return, variance, skewness and kurtosis are defined: $μ_{p} = E [r_{p}] = w^{'} μ$ (8)

$σ_{p}^{2} = E [{(r_{p} - μ_{p})}^{2}] = E [{(W - E (W))}^{2}]$ (9) $s_{p}^{3} = E [{(r_{p} - μ_{p})}^{3}] = E [{(W - E (W))}^{3}]$ (10) $κ_{p}^{4} = E [{(r_{p} - μ_{p})}^{4}] = E [{(W - E (W))}^{4}]$ (11)

Note the difference between these definitions of skewness and kurtosis from the statistical definitions as standardised central higher moments $E [{(\frac{r_{p} - μ_{p}}{σ_{p}})}^{j}] for j = 3, 4$ . Then,

$\begin{matrix} E [U (W)] & \approx & U (E (W)) + \frac{1}{2!} U^{(2)} (E (W)) σ_{p}^{2} \\ + \frac{1}{3!} U^{(3)} (E (W)) S_{p}^{3} \\ + \frac{1}{4!} U^{(4)} (E (W)) κ_{p}^{4} \end{matrix}$ (12)

As with Scott and Horvath [32], investors’ expected utility depends positively on return and skewness and negatively on variance and kurtosis.

The brief co-moment matrix-based presentation of the portfolio return, variance, skewness and kurtosis introduced in Martellini and Ziemann [26] are adopted herein.

Define the (n, n) co-variance matrix as $M_{2} = E [(R - μ) (R - μ)^{'}] = {σ_{ij}}$ (13)

The (n, n²) co-skewness matrix $M_{3} = E [(R - μ) {(R - μ)}^{'} \otimes {(R - μ)}^{'}] = {S_{ijk}}$ (14)

The (n, n³) co-kurtosis matrix

$\begin{matrix} M_{4} & = & E [(R - μ) {(R - μ)}^{'} \otimes {(R - μ)}^{'} \otimes {(R - μ)}^{'}] \\ = & {κ_{ijkl}} \end{matrix}$ (15) where ⊗ stands for the Kronecker product, σ_ij, s_ijk and κ_ijkl are elements of the co-variance, co-skewness and co-kurtosis matrices respectively.

In detail:

$\begin{matrix} σ_{ij} & = & E [(R_{i} - μ_{i}) (R_{j} - μ_{j})] where \\ i, j = 1, \dots, n \end{matrix}$ (16)

$\begin{matrix} s_{ijk} & = & E [(R_{i} - μ_{i}) (R_{j} - μ_{j}) (R_{k} - μ_{k})] \\ where i, j, k = 1, \dots, n \end{matrix}$ (17)

$\begin{matrix} κ_{ijkl} = E [(R_{i} - μ_{i}) (R_{j} - μ_{j}) (R_{k} - μ_{k}) (R_{l} - μ_{l})] \\ where i, j, k, l = 1, \dots, n \end{matrix}$ (18)

Because of the symmetric property, not all of the elements of these matrices need to be computed. For instance, in case of the (n, n) co-variance matrix, only n (n + 1)/2 elements have to be computed. Similarly the (n, n²) co-skewness matrix requires n (n + 1) (n + 2)/6 different elements, and the (n, n³) co-kurtosis matrix needs n (n + 1) (n + 2) (n + 3)/24 different elements.

Given a portfolio weight vector w, the moments of the portfolio return: expected return, variance, skewness and kurtosis of the portfolio are now respectively computed as follow: $μ_{p} = \sum_{i = 1}^{n} w_{i} μ_{i} = w^{'} μ$ (19) $σ_{p}^{2} = \sum_{i = 1}^{n} \sum_{j = 1}^{n} w_{i} w_{j} σ_{ij} = w^{'} M_{2} w$ (20) $\begin{matrix} s_{p}^{3} & = & \sum_{i = 1}^{n} \sum_{j = 1}^{n} \sum_{k = 1}^{n} w_{i} w_{j} w_{k} s_{ijk} \\ = & w^{'} M_{3} (w \otimes w) \end{matrix}$ (21) $\begin{matrix} κ_{p}^{4} & = & \sum_{i = 1}^{n} \sum_{j = 1}^{n} \sum_{k = 1}^{n} \sum_{l = 1}^{n} w_{i} w_{j} w_{k} w_{l} κ_{ijkl} \\ = & w^{'} M_{4} (w \otimes w \otimes w) \end{matrix}$ (22)

The partial derivatives with respect to the weight vector w are: $\frac{\partial μ_{p}}{\partial w} = μ$ (23) $\frac{\partial μ_{p}^{2}}{\partial w} = 2 M_{2} w$ (24) $\frac{\partial s_{p}^{3}}{\partial w} = 3 M_{3} (w \otimes w)$ (25) $\frac{\partial κ_{p}^{4}}{\partial w} = 4 M_{4} (w \otimes w \otimes w)$ (26)

Note that expressions on the right hand side of Equations (23–26) are n × 1 vectors where their n elements are correspondent to the n asset classes. Conventionally, marginal impact of an asset is measured by the partial derivative of the portfolio higher moment with respect to the asset holding. Accordingly, the marginal contribution of asset i to the portfolio return, variance, skewness, and kurtosis is the i-th element of these partial derivative vectors, Equations (23–26), respectively. Marginal contribution expresses how much the portfolio higher moments will change with respect to a small change of the weight of an asset. The asset with higher marginal contribution will have more influence to the overall portfolio compared to the others.

With regard to the return criterion, the contribution of a given asset i to the whole portfolio return is obviously its expected return μ_i. However it is not straightforward with the portfolio variance, skewness or kurtosis. The marginal contribution of a given asset is not decreased (increased) to its own variance, kurtosis (skewness) but also takes account of its diversification potential in terms of co-variances, co-kurtosis (co-skewness) to other assets. This is an explanation against the possible argument that the evaluations of assets can be solely based on their own higher moments. Take an example of the portfolio’s variance:

$\begin{matrix} σ_{p}^{2} & = & w^{'} M_{2} w = \sum_{i = 1}^{n} \sum_{j = 1}^{n} w_{i} w_{j} σ_{ij} \\ = & \sum_{i = 1}^{n} w_{i}^{2} σ_{i}^{2} + 2 \sum_{i = 1}^{n - 1} \sum_{j = i + 1}^{n} w_{i} w_{j} σ_{ij} \end{matrix}$ (27)

The latter component of the above two-component decomposition relation represents the diversification effect of the overall portfolio variance. Manifestly, the portfolio variance encompasses not only the variance of individual assets but also takes into account the co-variance between assets. The partial derivative with respect to weight w_i of asset i is: $\frac{\partial σ_{p}^{2}}{\partial w_{i}} = 2 \sum_{j = 1}^{n} w_{j} σ_{ij}$ (28)

Equation (28) shows the variance marginal contribution of the asset i to the whole portfolio variance. The σ_ij factor can be realized from historical or simulation data but the w_j factor is still unknown in this stage of the allocation process. The same situation exists with skewness and kurtosis because marginal impacts of assets on portfolio higher moments also involve unknown portfolio weights.

Recall the two constraints imposed on the portfolio weights $\sum_{i = 1}^{n} w_{j} = 1$ and w_i ≥ 0, ∀ i, it is uncomplicated to find the minimum and maximum portfolio variance, skewness and kurtosis using constrained optimization solvers. The functions of the portfolio variance, skewness and kurtosis, presented in Equations (20–22), are derivative continuous multivariate functions with the second, third, and fourth orders respectively. The number of variables in these functions is in proportion to the number of assets in the portfolio. From minimum and maximum solutions, the portfolio weight vectors w = (w₁, …, w_n) ′ are exposed. For each of the criteria, i.e. variance, skewness and kurtosis, two weight vectors related to minimum and maximum circumstances are obtained. Whenever a weight vector w is known, marginal contributions of assets on the portfolio can be calculated by expressions on the right hand side of formulae (23)-(26). As a result, given a criterion, we can obtain marginal contributions of any assets in both minimum and maximum extremes. Let $\min_{mc}$ and $\max_{mc}$ be marginal contributions of assets in minimum and maximum cases respectively. The marginal contributions of each asset in extreme cases can be obtained but it is uncertain what is the exact contribution of an asset on the portfolio higher moments before the allocation process is accomplished (before choosing w). Note that for different criteria or different assets, $\min_{mc}$ is not necessarily less than $\max_{mc}$ or vice versa. This problem is approached using fuzzy numbers. Section 4 presents relevant fuzzy concepts and how to model these marginal contributions by fuzzy numbers using $\min_{m} c$ and $\max_{m} c$ of each asset. Before doing that, we present in the following section a broad review on portfolio selection models using fuzzy approaches proposed in the literature so far.

3 Fuzzy portfolio optimization literature review

Numerous fuzzy approaches have been proposed in the literature for portfolio optimization. For example, Arenas Parra et al. [2] proposed a goal programming problem where the goals and the constraints are fuzzy for portfolio selection. The model takes into account three criteria including return, risk and liquidity. As liquidity admits a high degree of subjectivity accounting for the knowledge of experts, the authors include in the model linguistic labels such as “near absolutely liquid”, “sufficient liquid” and “little liquid”. Fuzzy numbers are appropriately used to represent these natural expressions. The proposed model is capable of adjusting the risk preferences and deriving optimal portfolios corresponding to either risk adverse or risk seeking investors.

Lai et al. [17] on the other hand considered the uncertain returns of assets in capital markets as intervals and proposed a model for portfolio selection based on the risk semi-absolute deviation measure. The model is then can be transformed to a linear interval programming model with interval coefficients in both the objective functions and constraints. Based on two order relations between intervals, the non-inferior solutions of the proposed model can be found by solving a corresponding parametric linear programming problem.

Tiryaki and Ahlatcioglu [33] introduced a new method regarding fuzzy multi-criteria decision making (MCDM) for portfolio selection. Decision makers provide linguistic evaluations, which are expressed by fuzzy numbers, regarding assets with respect to a number of financial criteria including market value of firm-to-earnings before amortization, interest and taxes ratio, return on equity, dept/equity ratio, current ratio, market value/net sales ratio, and price/earnings ratio.

Bilbao-Terol et al. [5] represented financial beta value of each asset by fuzzy trapezoidal numbers based on not only the historical data series, but also expert’s subjective opinions. A fuzzy compromise programming model is then formulated to solve the portfolio selection problem. The solving approach requires the introduction of the fuzzy ideal solution concept, which is introduced based on soft preference and indifference relationships and on canonical representation of fuzzy numbers via α-cuts.

Ammar [1] investigated the solving methods for a fuzzy random multi-objective quadratic programming with fuzzy random coefficients in the objectives and constraints. Six quadratic programming models are formulated to represent the above problem using the concept of α-level set. Relations of the efficient solutions of fuzzy random quadratic multi-objective programming into series-optimal-solutions of relative fuzzy random quadratic programming were proved via some theorems. The solutions of random quadratic multi-objective programming hence are found under different conditions. The fuzzy portfolio optimization problem as a convex quadratic programming approach is investigated as an application of the problem.

In another approach, five criteria consisting of short term return, long term return, dividend, risk and liquidity were considered in a portfolio selection model proposed by Gupta et al. [11]. Semi-absolute deviation is defined as a risk whereas liquidity is measured by the degree of probability involved in the conversion of an investment into cash without any significant loss in value. Aspiration levels of both aggressive and conservative investors are modelled by using non-linear S-shape membership functions. The fuzzy set theory is incorporated into the semi-absolute deviation portfolio optimization model via fuzzy mathematical programming.

Li et al. [19] introduced a simulated annealing algorithm with memory and local search to solve fuzzy portfolio selection models. Expected value and variance of fuzzy returns are approximated by neural network and fuzzy simulation, which are integrated in a unique framework namely the hybrid intelligent algorithm. Alternatively, in Li and Xua [18], security returns in the future are estimated by incorporating statistical techniques and the experts’ judgement and experience are characterized by fuzzy random variables. A portfolio selection model based on the formulation of the mean-variance model is proposed in a hybrid uncertain environment. The λ-mean variance efficient frontiers representing optimal portfolios corresponding to pessimistic, neutral and optimistic financial situations are obtained. The advantage of the proposed model is its ability to generate different efficient frontiers according to the investor’s degree of optimism.

Chen and Huang [7] alternatively modelled future return rates and future risks of mutual funds by triangular fuzzy numbers and proposed a fuzzy model to optimize portfolios based on different confidence levels. Four indices including rates of return, standard deviation, turnover rate, and Treynor index are used to categorize a large number of equity mutual funds into several groups before running optimization models. Funds whose return rate is high, turnover rate is low and other indices are medium are classified as “stable funds” whereas funds that have high return rate, high Treynor index, and high turnover rate will be categorized as “short-swing funds” with a high level of risk. Allocation proportion for all clusters are derived from the optimization models, which either maximizes future expected return subject to a given future risk or minimize future risk subject to a required future expected return.

Hao and Liu [13] characterized security returns, which are not known with certainty, by triangular fuzzy random variables. Variance formulae for triangular fuzzy random variables are then derived and applied to two portfolio selection models to reduce them to non-linear programming problems. Genetic algorithm is employed to solve the nonlinear problems. Optimal portfolio solutions are verified by the Kuhn-Tucker conditions which demonstrate the effectiveness of the proposed methods.

On the other hand, the fusion between probabilistic and possibilistic future returns represented via fuzzy random variables was introduced in portfolio selection models proposed by Hasuike et al. [14]. Using chance constraints, and based on both stochastic and fuzzy programming approaches, two problems are formulated as nonlinear programming problems including possibility fractile optimization problem and possibility maximization problem. Probabilistic chance constraints, possibility measure and fuzzy goals are used to transform the proposed problems into equivalent deterministic quadratic programming problems which can be solved efficiently.

Duan and Stahlecker [9] modelled future returns of securities by fuzzy sets and derive optimal portfolios based on the maximal and minimal potential returns. These values are resulted from α-cuts of fuzzy sets, which are modelled in the ellipsoidal shape. Unique portfolio allocation solution is obtained through a simple optimization model, which allows short sales. The proposed approach also has an ability to include observed trends of returns or also experts’ subjective assessments in the portfolio decision. Alternatively, three classes of fuzzy expectation-spread models for portfolio optimization problems were developed by Wu and Liu [36]. In the models, the investment returns are characterized by fuzzy expected value of a portfolio whilst the investment risk is expressed by spread of a portfolio. Experimental results demonstrate the feasibility and effectiveness of the three models in some practical portfolio optimization.

Bhattacharyya et al. [4] utilized the concept of interval numbers in fuzzy set theory to extend the MVO model into the mean-variance-skewness model for portfolio selection. The model also considers transaction cost and some other criteria like short and long term returns, liquidity, dividends, number of assets in the portfolio and the maximum and minimum allowable capital invested in stocks. Portfolios are evaluated by considering not only variance risk but also the return and skewness augmentation. Fuzzy simulation and elitist genetic algorithm are integrated to solve the proposed problems. Experiments conducted in the study demonstrate the computational tractability and the effectiveness of the solving algorithms.

Liu et al. [22] considered several criteria including return, transaction cost, risk and skewness of portfolio in investigating the multi-period portfolio selection with fuzzy returns. A TOPSIS-compromised programming approach is introduced to convert the proposed multi-objective nonlinear programming models into corresponding single objective programming models, which can be solved by genetic algorithm.

Zhang et al. [38] introduced a possibilistic mean-semivariance-entropy model for multi-period portfolio selection problems by considering four criteria including return, risk, transaction cost and diversification degree of portfolio. Possibilistic mean-value is regarded as portfolio return whereas the lower possibilistic semivariance of the fuzzy return on asset is regarded as the risk.

Bermúdez et al. [3] represented the uncertainty of the returns on a given portfolio by trapezoidal fuzzy numbers. A fuzzy multi-objective portfolio selection problem is then proposed and solved by the genetic algorithm. The fitness function is constructed based on the value and the ambiguity of the fuzzy number that represents the uncertainty of its return. Non-dominated portfolios in the final generation of the genetic algorithm are assembled to build the efficient frontier. Different portfolios can be selected according to risk aversion attitude of investors (risky, medium risk or conservative).

Cadenas et al. [6] recently considered the vagueness of the investor’s preferences regarding risk criteria and proposed a fuzzy model for portfolio selection. An exact method to solve the fuzzy model is proposed by finding partially-feasible solutions involving slightly greater risk than that fixed by the decision maker. For medium-sized or large problems, the authors also proposed a solving hybrid meta-heuristic method, which integrates ideas from the simulated annealing technique and genetic algorithm. Alternatively, Yunusoglu and Selim (2012) developed a fuzzy rule based expert system to support portfolio managers in portfolio allocations. By taking investor’s preferences and risk profile, performance of the fuzzy system was shown superior to the benchmark index in most of investment situations.

Recently, there have been a number of more advanced approaches to portfolio optimization using the fuzzy logic technique, e.g. see Mehlawat and Gupta [27], Gupta et al. [12], Nguyen et al. [29], Liu and Zhang [23], and Xu et al. [37] for details.

4 Fuzzy modelling marginal impacts

4.1 Relevant fuzzy set concepts and notions

A fuzzy set A is defined by a membership function f_A (x) mapping from a universal set of concern X to a range from 0 to 1: f_A (x) : X → [0, 1]. For each x ∈ X, the value of f_A (x) expresses the degree (or grade) of membership of the element x of X in standard fuzzy set A.

α-cut is one of the most important concepts of fuzzy sets. Given a particular number α ∈ [0, 1] and a fuzzy set A defined on X, the α-cut of A, denoted by α_A, is a crisp set encompassing elements of X satisfying: α_A = {x|f_A (x) ≥ α}.

A fuzzy number is a special fuzzy set defined on the set of real numbers, $ℝ$ . A normal fuzzy number A is characterized for each $x \in ℝ$ by the canonical form: $f_{A} (x) = {\begin{matrix} f_{A}^{L} (x), a \leq x \leq b, \\ 1, b \leq x \leq c, \\ \begin{matrix} f_{A}^{R} (x), c \leq x \leq d, \\ 0 otherwise \end{matrix} \end{matrix}$ (29) where $a, b, c, d \in ℝ$ , a ≤ b ≤ c ≤ d, $f_{A}^{L} : [a, b] \to [0, 1]$ is an increasing real-valued function whereas $f_{A}^{R} : [c, d] \to [0, 1]$ is a real-valued decreasing function. The most widely used are normal trapezoidal fuzzy numbers denoted by A = (a, b, c, d) whose membership functions are piecewise linear: $f_{A} (x) = {\begin{matrix} (x - a) / (b - a), a \leq x \leq b, \\ 1, b \leq x \leq c, \\ (d - x) / (d - c), c \leq x \leq d, \\ 0 otherwise \end{matrix}$ (30)

Fuzzy logic in general or fuzzy set in particular has been successfully applied to various fields including those of the mathematical and computational economics [20 , 31]. The following presents a new method using fuzzy set in financial portfolio optimization.

4.2 Modelling marginal impacts by trapezoidal fuzzy numbers

The rational investor would prefer higher portfolio skewness but lower variance and kurtosis. An investment fund manager will try to allocate the portfolio wealth to satisfy or at least nearly satisfy the investor’s preference or utility. The portfolio weight vector should bring the investor an acceptable result in accordance with his/her expectation: maximum (as high as possible) skewness and minimum (as low as possible) variance and kurtosis. Hence, it is logical to design fuzzy numbers representing variance contributions of assets based mainly on their marginal contributions when the portfolio variance attains minimum. Similarly, contributions of assets with respect to the skewness (kurtosis) criterion will be characterized by fuzzy numbers that are designed towards skewness (kurtosis) marginal impacts in the maximum (minimum) context. In other words, maximum (minimum) is the preferred extreme regarding skewness (variance or kurtosis). With a preferred extreme (e.g. variance minimum), fuzzy numbers are designed so that the possibility of achieving (or at least nearly achieving) this extreme is higher than the possibility of achieving the other extreme (variance maximum). The proportion parameter ρ ∈ [0, 1] in the following equations represents the bias level towards the preferred extremes.

In portfolio optimization proposed herein, investors can flexibly state a preference weighting scheme concerning their attitude with regard to different criteria by the preference ratio series r = (r_r : r_v : r_s : r_k) where r_r, r_v, r_s and r_k are in that order the importance level of return, variance, skewness and kurtosis. For example, the scheme (1:1:2:1) indicates that the investor will focus more on portfolio skewness rather than return, variance and kurtosis. The scheme (3:0:1:3) says the investor greatly favours return and kurtosis, slightly favours skewness and pays no attention to variance. We explore some typical schemes exhibited in Table 1. The scheme (2:1:2:1) would be appropriate for young investors and contrastingly the scheme (1:2:1:2) is for conservative investors. The scheme (1:1:0:0) is equivalent to the conventional mean-variance approach.

Based on intended weighting schemes from investors, the parameters ρ in designing trapezoidal fuzzy numbers will be specified for each criterion. Let us define r_max = max (r_r, r_v, r_s, r_k). For any scheme (r_r : r_v : r_s : r_k), ρ in cases of return (denoted by ρ_r), variance (ρ_υ), skewness (ρ_s) and kurtosis (ρ_k) are specified as ρ_r = r_r/r_max, ρ_ν = r_ν/r_max, ρ_s = r_s/r_max, and ρ_k = r_k/r_max (see Table 1). So ρ_r = 1 if r_r = r_max implies most concern about return, or ρ_r = 0 if r_r = 0 implies no concern about return, and alike interpretations for ρ_v, ρ_s and ρ_k. When ρ = 1, the trapezoidal fuzzy number is of special triangular shape. There is no need for fuzzy modelling for the return criterion as return marginal impacts are crisp expected returns of stocks (23) but the parameter ρ_r will be useful in implementing the FMOLP presented in Section 5.

Assume a normal trapezoidal fuzzy number A_i (a_i, b_i, c_i, d_i) is to be constructed to stand for marginal impacts of asset ith with $\min_{m} c_{i}$ and $\max_{m} c_{i}$ being the marginal contributions in the minimum and maximum circumstances, so the support of A_i is a set {x|x∈ (a_i, d_i) } and the core of A_i is the set {x|x∈ (b_i, c_i) }.

Hence, $a_{i} = \min (\min_{m} c_{i}, \max_{m} c_{i})$ and $d_{i} = \max (\min_{m} c_{i}, \max_{m} c_{i})$ .

(a) Variance or kurtosis fuzzy numbers – illustrated byFig. 1

i) $If a_{i} = \min (\min_{m} c_{i}, max_{m} c_{i}) = \min_{m} c_{i} {andd}_{i} = \max (\min_{m} c_{i}, \max_{m} c_{i}) = \max_{m} c_{i} then$ : $\begin{matrix} b_{i} & = & \min_{m} c_{i} \end{matrix}$ (31I) $\begin{matrix} c_{i} & = & \frac{\min_{m} c_{i} + \max_{m} c_{i}}{2} \\ + ρ (\min_{m} c_{i} - \frac{\min_{m} c_{i} + \max_{m} c_{i}}{2}) \end{matrix}$ (31II)

With Equation (31II), we examine two cases: $\begin{matrix} ρ \to 0 \leftrightarrow c_{i} \to \frac{\min_{m} c_{i} + \max_{m} c_{i}}{2} \end{matrix}$ (31III) $\begin{matrix} ρ \to 1 \leftrightarrow c_{i} \to \min_{m} c_{i} \end{matrix}$ (31IV)

This circumstance is illustrated by Fig. 1a.

ii) $If a_{i} = \min (\min_{m} c_{i}, \max_{m} c_{i}) = \max_{m} c_{i} {andd}_{i} = \max (\min_{m} c_{i}, \max_{m} c_{i}) = \min_{m} c_{i} then$ (Fig. 1b). $\begin{matrix} b_{i} & = & \frac{\min_{m} c_{i} + \max_{m} c_{i}}{2} \\ + ρ (\min_{m} c_{i} - \frac{\min_{m} c_{i} + \max_{m} c_{i}}{2}) \end{matrix}$ (32I) $\begin{matrix} c_{i} & = & \min_{m} c_{i} \end{matrix}$ (32II) $\begin{matrix} ρ & \to & 0 \leftrightarrow b_{i} \to \frac{\min_{m} c_{i} + \max_{m} c_{i}}{2} \end{matrix}$ (32III) $\begin{matrix} ρ & \to & 1 \leftrightarrow b_{i} \to \min_{m} c_{i} \end{matrix}$ (32IV)

(b) Skewness fuzzy numbers – illustrated by Fig. 2

i) $If a_{i} = \min (\min_{m} c_{i}, \max_{m} c_{i}) = \min_{m} c_{i} {andd}_{i} = \max (\min_{m} c_{i}, \max_{m} c_{i}) = \max_{m} c_{i} then$ (Fig. 2a). $\begin{matrix} b_{i} & = & \frac{\min_{m} c_{i} + \max_{m} c_{i}}{2} \\ + ρ (\max_{m} c_{i} - \frac{\min_{m} c_{i} + \max_{m} c_{i}}{2}) \end{matrix}$ (33I) $\begin{matrix} c_{i} & = & \max_{m} c_{i} \end{matrix}$ (33II) $\begin{matrix} ρ \to 0 \leftrightarrow b_{i} \to \frac{\min_{m} c_{i} + \max_{m} c_{i}}{2} \end{matrix}$ (33III) $\begin{matrix} ρ \to 1 \leftrightarrow b_{i} \to \max_{m} c_{i} \end{matrix}$ (33IV)

ii) $If a_{i} = \min (\min_{m} c_{i}, \max_{m} c_{i}) = \max_{m} c_{i} {andd}_{i} = \max (\min_{m} c_{i}, \max_{m} c_{i}) = \min_{m} c_{i} then$ ( Fig. 2b ). $\begin{matrix} b_{i} & = & \max_{m} c_{i} \end{matrix}$ (34I) $\begin{matrix} c_{i} & = & \frac{\min_{m} c_{i} + \max_{m} c_{i}}{2} \\ + ρ (\max_{m} c_{i} - \frac{\min_{m} c_{i} + \max_{m} c_{i}}{2}) \end{matrix}$ (34II) $\begin{matrix} ρ & \to & 0 \leftrightarrow c_{i} \to \frac{\min_{m} c_{i} + \max_{m} c_{i}}{2} \end{matrix}$ (34III) $\begin{matrix} ρ & \to & 1 \leftrightarrow c_{i} \to \max_{m} c_{i} \end{matrix}$ (34IV)

The shapes of fuzzy numbers in case (a) and case (b) look similar, but the position of b_i, c_i and the differences at the pole values of these fuzzy numbers are worth noting.

Given that marginal impacts of assets on the portfolio can, technically, be modelled by fuzzy numbers, it remains for methods for evaluating such fuzzy numbers to be identified for application to this research.

4.3 Ranking fuzzy numbers

In order to evaluate fuzzy numbers, as with Nguyen and Gordon-Brown (2012), we suggest using their representative crisp numbers obtained via the centroid-based defuzzification method. Denoting $g_{A}^{L} (y) : [0, 1] \to [a, b]$ and $g_{A}^{R} (y) : [0, 1] \to [a, d]$ as the inverse functions of $f_{A}^{L} (x)$ and $f_{A}^{R} (x)$ , respectively. In the case of a trapezoidal fuzzy number, the functions $g_{A}^{L} (y)$ and $g_{A}^{R} (y)$ can be analytically expressed as $g_{A}^{L} (y) = a + (b - a) y$ and $g_{A}^{R} (y) = d + (d - c) y$ where 0 ≤ y ≤ 1.

The Wang et al. (2006) centroid formulae based on the general canonical form of a normal trapezoidal fuzzy number A are as follows:

$\begin{matrix} {\bar{x}}_{0} (A) & = & \frac{\int_{- \infty}^{+ \infty} {xf}_{A} (x) dx}{\int_{- \infty}^{+ \infty} f_{A} (x) dx} \\ = & \frac{\int_{a}^{b} {xf}_{A}^{L} (x) dx + \int_{b}^{c} xdx + \int_{c}^{d} {xf}_{A}^{R} (x) dx}{\int_{a}^{b} f_{A}^{L} (x) dx + \int_{b}^{c} dx + \int_{c}^{d} f_{A}^{R} (x) dx}, \end{matrix}$ (35)

${\bar{y}}_{0} (A) = \frac{\int_{0}^{1} y (g_{A}^{R} (y) - g_{A}^{L} (y)) dy}{\int_{0}^{1} (g_{A}^{R} (y) - g_{A}^{L} (y)) dy},$ (36) where the numerator $\int_{0}^{1} y (g_{A}^{R} (y) - g_{A}^{L} (y)) dy$ represents the weighted average of the area, whereas the denominator $\int_{0}^{1} (g_{A}^{R} (y) - g_{A}^{L} (y)) dy$ is the area of the trapezoid.

For trapezoidal fuzzy numbers A = [a, b, c, d], application of the above formulae leads to: ${\bar{x}}_{0} (A) = \frac{1}{3} [a + b + c + d - \frac{dc - ab}{(d + c) - (a + b)}]$ (37) ${\bar{y}}_{0} (A) = \frac{1}{3} [1 + \frac{c - b}{(d + c) - (a + b)}]$ (38)

Centroids on the horizontal axis are used as a basis to evaluate assets. If horizontal coordinates ${\bar{x}}_{0}$ of all assets in the portfolio are completely equal, then the vertical centroid coordinates ${\bar{y}}_{0}$ will be applied, though this situation seldom occurs in practice where the numbers of assets is large enough to support the kind of analysis demonstrated here. This also means that, in comparing fuzzy numbers, the representative location on the horizontal axis is more important than the average height [35]. In the next section this evaluating paradigm is used in application of FMOLP for portfolio allocation.

5 FMOLP for portfolio optimization

FMOLP involves models where objectives, usually conflicting, are simultaneously optimized and coefficients of the objective functions and constraints are fuzzy numbers. The model is generally described as follows: $\begin{matrix} Max & F (x) = Cx \\ x \in X \end{matrix}$ (39) $s . t . = {x \in R^{n} | Ax \leq B, x \geq 0}$ where C is a k × n matrix with fuzzy numbers C_ij characterized by membership function f_{C
_ij} (x) , A is an m × n matrix where each element A_ij represented by membership function f_{A
_ij} (x) , B is an m-vector, each element of which B_i is a fuzzy number with the membership function f_{B
_i} (x), and x is an n-vector of decision variables.

As marginal impacts of assets on portfolio higher moments have been modelled by fuzzy numbers, the following FMOLP is suggested for an optimal portfolio. For the sake of simplicity, let us define the set of constraints as $W = {w \in R^{n} | \sum_{i = 1}^{n} w_{i} = 1, w_{i} \geq 0, \forall i = 1, \dots, n}$

The portfolio optimization FMOLP is formulated: $FMOLP | \begin{matrix} Max & μ = \sum_{i = 1}^{n} w_{i} μ_{i} \\ Min & V = \sum_{i = 1}^{n} w_{i} V_{i} \\ Max & S = \sum_{i = 1}^{n} w_{i} S_{i} \\ Min & K = \sum_{i = 1}^{n} w_{i} K_{i} \\ s . t . & w \in W \end{matrix}$ (40) where V_i, S_i and K_i are fuzzy numbers respectively representing for variance, skewness and kurtosis marginal impacts of the ith asset on portfolio variance, skewness and kurtosis, and n is the number of assets.

An equivalent model is derived: $FMOLP | \begin{matrix} Max & {\begin{matrix} \sum_{i = 1}^{n} w_{i} μ_{i} \\ - \sum_{i = 1}^{n} w_{i} V_{i} \\ \sum_{i = 1}^{n} w_{i} S_{i} \\ - \sum_{i = 1}^{n} w_{i} K_{i} \end{matrix} \\ s . t . & w \in W \end{matrix}$ (41)

The FMOLP model presented in (41) can be solved using one of a vast number of methods available in the literature. The following present the maximin method introduced by Zimmermann [39] for solving (41).

With the consistent centroid-based ranking of fuzzy numbers in sub-section 4.3, all fuzzy coefficients are converted to crisp numbers and the FMOLP is transformed into the following solvable MOLP: $MOLP | \begin{matrix} Max & {\begin{matrix} F_{1} (w) = \sum_{i = 1}^{n} w_{i} μ_{i} \\ F_{2} (w) = - \sum_{i = 1}^{n} w_{i} v_{i} \\ F_{3} (w) = \sum_{i = 1}^{n} w_{i} s_{i} \\ F_{4} (w) = - \sum_{i = 1}^{n} w_{i} k_{i} \end{matrix} \\ s . t . & w \in W \end{matrix}$ (42) where υ_i, s_i and k_i are respectively x-centroids of fuzzy coefficients V_i, S_i and K_i.

The following details a method to solve the MOLP problem based on the maximin approach. The above MOLP includes four linear objective functions F_j (w) , ∀ j = 1, …, 4. The MOLP becomes: $MOLP | \begin{matrix} Max & F_{j} (w) \\ s . t . & w \in W \end{matrix}$ (43)

Solution of the above problem is equivalent to that of the following: $MOLP | \begin{matrix} Max & γ \\ \frac{F_{j} (w) - F_{j}^{\min}}{F_{j}^{\max} - F_{j}^{\min}} \geq γ * ρ_{j}, \\ s . t . & \forall j = 1, \dots, 4 \\ w \in W \end{matrix}$ (44) where $F_{j}^{\max} = F_{j} (w_{j}^{*}) = \max_{w \in W} F_{j} (w), F_{j}^{\min} = \min_{\forall k = 1, \dots, 4} (F_{j} (w_{k}^{*})), \forall j = 1, \dots, 4 . γ \in [0, 1]$ is the overall maximum satisfaction level, and ρ_j≡ 〈 ρ₁, ρ₂, ρ₃, ρ₄ 〉 ≡ 〈 ρ_r, ρ_υ, ρ_s, ρ_k 〉 as identified in sub-section 4.2 (ρ_j ≤ 1). The satisfaction levels for four criteria in (44) are thus different depending on the investor’s preference. The portfolio optimization now becomes a conventional linear programming (44) that can be solved by normal simplex method.

For the aim of comparisons, we also deploy in this research the portfolio optimization model under higher moment proposed by Jondeau and Rockinger [15], henceforth the JR method, and the FAHP approach proposed in Nguyen and Gordon-Brown [28]. In Nguyen and Gordon-Brown [28], we found that the second constrained FAHP method is superior to the first constrained FAHP method in terms of uncertainty of results. In this paper, we thus just compare results of the proposed FMOLP approach to those of the second constrained FAHP method. The summary of these methods are presented in the Appendix A and Appendix B respectively.

6 Illustrative experiment

Historical return data including nine stocks introduced by Markowitz [25] have been widely used for portfolio optimization experiments. The data span the period from 1937 to 1954 with 18 yearly observations for each stock. Table 2 represents moments of individual stocks where the return row is the arithmetic average rate of return whereas variance, skewness and kurtosis are respectively calculated using Equations (16–18) with i = j = k = l = 1, …, 9.

Prior to implementing FMOLP, co-variance, co-skewness and co-kurtosis matrices are constructed using Equations (13–15). Portfolio weight vectors when portfolio return and skewness get maximum, and variance and kurtosis get minimum are obtained using a constrained optimization solver with objective functions (19)-(22) and two portfolio weight constraints $\sum_{i = 1}^{n} w_{i} = 1$ and w_i ≥ 0, ∀ i = 1, …, 9. Among these four preferred optimal portfolio weights, weight vectors w at which portfolio skewness is lowest and variance and kurtosis are highest are identified. Then marginal impacts of each asset at extremes are computed using (24)–(26) for variance, skewness and kurtosis criteria. The return marginal impacts are the non-fuzzy expected returns of stocks.

Under each determined scheme, the parameter ρwill be indicated (Table 1) and fuzzy numbers representing marginal impacts with respect to variance, skewness and kurtosis criteria are designed for each stock. Tables 3–5 represent fuzzy numbers of marginal impacts under scheme (2:1:2:1) where $a, b, c, d and {\bar{x}}_{0}$ are parameters and x-centroids of trapezoidal fuzzy numbers respectively.

6.1 Portfolio allocations

Table 6 reports portfolio weights after carrying out eight typical schemes of the FMOLP method. For comparisons, we also report here the portfolios weights of the JR method and the FAHP in Tables 7 and 8 respectively.

A great similarity among portfolio allocations between the proposed approach and the JR method is recognized. Stock S₁and S₉ are almost not selected through different schemes in the proposed FMOLP approach and also different value of λ in the JR method. On the other hand, stocks S₆, S₇ and S₈ are chosen with large proportions in both methods.

Comparing between the proposed FMOLP approach and the FAHP method, we found there are some difference but also some similarities between them in portfolio allocations throughout eight investigated schemes. The FAHP portfolios are completely diversified whereas the FMOLP portfolios consist of just few stocks. The FMOLP method thus produces less diversified portfolios compared to the FAHP method.

In scheme (2:1:2:1), highest proportion stocks in the FAHP approach are S₆ and S₁. Stock S₆ is also selected with rather high proportion in the proposed FMOLP approach. Stock S₈ is very predominant in the FMOLP approach, but is just moderately chosen in the FAHP. In scheme (1:2:1:2), similar to the situations in scheme (2:1:2:1), some similarity and difference in portfolio selection are also recognized. Stock S₆ is preferred in both approaches whereas stock S₇ occupies very high percentage (67.2% ) in the FMOLP and moderately selected in the FAHP (11.8% ).

The difference between two approaches is found in scheme (4:3:2:1) where stocks S₇ and S₈ are selected in the proposed approach whereas stock S₅ and S₆ are chosen with highest proportions in the FAHP method. In scheme (1:1:0:0), the FMOLP approach does not select stock S₅ whereas the FAHP selects stock S₅ with the highest allocation (17.2% ) compared to other stocks. Two stocks S₇ and S₈ are in the stock basket of the proposed approach and they are quite preferred in the FAHP method.

In scheme (1:0:0:1), two methods demonstrate quite different selection. The FAHP selects S₁ and S₂ with large portfolio (16% and 14.9% respectively) whereas the proposed method picks only two stocks S₃ and S₄ with 63.7% and 36.3% respectively. Stocks S₃ and S₄ are just moderately selected in the FAHP portfolio. On the other hand, two methods show a pretty similar stock selection in scheme (1:0:1:0). The proposed FMOLP approach selects only two stocks S₆ and S₈ with 26.8% and 73.2% respectively. These two stocks are also selected in the FAHP with 14.7% and 15% respectively, which are the biggest allocations in this FAHP scheme. In scheme (0:0:1:1), two stocks S₆ and S₂ are selected in the proposed method (90.5% and 9.5% respectively) and they are also chosen with majorities in the FAHP method (18.9% and 14.1% respectively). The difference in this scheme is that the proposed FMOLP approach does not select the stock S₁ at all whereas the FAHP picks stock S₁ with the highest proportion (20.1% ). Alternatively, the difference is identified in scheme (1:0:0:0) where the proposed FMOLP method selects only stock S₅ (100% ) whereas the FAHP method selects S₅ with the biggest allocation (29.3% ).

6.2 Portfolio higher moments

Higher moments of the FMOLP portfolios are reported in Table 9 where the values in the γ column show the maximum satisfaction levels obtained during running model (44) for each scheme.

It is reasonably clear that when investors pay more attention on a particular criterion, FMOLP brings portfolios with more optimal values regarding that criterion. For example, scheme (1:0:0:0) (or 0:0:1:1) favours most (or least) the return criterion and in consequence resulting portfolios obtain highest (or lowest) returns among eight investigated schemes: 19.8% (or 5.6% ). Likewise, the FMOLP portfolio kurtosis at scheme (1:2:1:2) is less than that of the portfolio at the scheme (2:1:2:1) as the investor expresses most concern about the kurtosis risk in the scheme (1:2:1:2) whereas least concern about the kurtosis risk in the scheme (2:1:2:1). This demonstrates the strength of the proposed approach to handle the investor’s intention regarding risk preference via different schemes adapting to various kinds of investors (from conservative to risky investors).

The portfolio higher moments of the FAHP method and of the JR method are reported in Tables 10 and 11 respectively for comparisons.

In Nguyen and Gordon-Brown (2012), we recognized that performance in terms of higher moment optimization between JR method and the FAHP is comparable. The experimental results demonstrate the trade-offs within optimal criteria between the JR approach and the FAHP method. A method offers better (more optimal) value in a criterion then it is inferior to the other method regarding optimization of other criteria. The advantage of the FAHP over the JR method is that investor’s preferences regarding risks can be more explicitly expressed through different schemes.

From the experimental results reported in this paper, when comparing portfolios between the FMOLP and the JR method, we also found the same situations. The FMOLP approach and JR method show trade-offs in optimizing portfolio return and risk criteria.

For example, the portfolio (1:0:0:0) of the FMOLP approach obtains return at 19.8% , which is higher than that of the JR portfolio with λ= 10. The variance and kurtosis of the FMOLP (1:0:0:0) portfolio however are more optimal (at 0.128 and 0.037 respectively) than those of the JR portfolio with λ= 10 (0.147 and 0.060 respectively). Contrastingly, the skewness value of the FMOLP portfolio is less optimal than that of the JR portfolio: –0.003 compared to 0.041.

The other example regarding the trade-off efficiency between the FMOLP and the JR methods is in comparison between FMOLP scheme (2:1:2:1) portfolio and the JR portfolio with λ= 25. Two portfolios obtain the same return at 15.4% , but different risk values. The FMOLP portfolio has the variance and kurtosis less optimal than those of the JR portfolio: 0.093 versus 0.054 and 0.021 versus 0.006. In contrast, the FMOLP portfolio has the skewness much better than that of the JR portfolio: 0.015 of the FMOLP portfolio (scheme 2:1:2:1) compared to –0.002 of the JR portfolio (λ= 25).

Alternatively, comparing the portfolio FMOLP scheme (4:3:2:1) with the JR portfolio with λ= 25, the FMOLP portfolio gains higher return than that of the JR portfolio: 18.1% compared with 17.6% . This reasonably causes the variance and kurtosis of the FMOLP portfolio to be higher than those of the JR portfolio. The FMOLP however offers better skewness (higher value) compared to that of the JR portfolio: 0.025 versus –0.077 respectively.

Similar to comparing between the FMOLP method and the JR method, the trade-offs in optimizing portfolio return and risk criteria are also found when comparing between the FMOLP method with the FAHP (see Tables 9 and 10). With the same investigated scheme, the FMOLP portfolios almost obtain higher return and skewness values than those of the FAHP portfolios. In contrast, the variance and kurtosis of FMOLP portfolios are superior to those of the FAHP portfolios.

The significant difference between the FMOLP approach and the FAHP method shows in the possibility to obtain high portfolio returns through investigated schemes. The highest return in the FAHP portfolios is 16.9% which is obtained in the (1:0:0:0) scheme whereas this value is up to 19.8% in the (1:0:0:0) portfolio of the FMOLP method. The return range of the FAHP portfolios is much narrower than that of the FMOLP portfolios. FAHP portfolios can only obtain returns within the range from 10.3% to 16.9% whereas this range of the FMOLP portfolios is 5.6% to 19.8% . The return range of the FMOLP portfolios would satisfy various kinds from conservative to risky investors. Since the return range in the FAHP is narrow, the possibility to adapt various kinds of investors would be limited. The risky investors who require high return portfolios (congruent with high risk) will face challenges in employing the FAHP method. Obviously, the FMOLP approach proposed in this paper has better ability to handle investors’ risk preference. This advantage results from the combination between modelling marginal impacts by trapezoidal fuzzy numbers and deploying FMOLP in portfolio optimization. The limitation of the FAHP method is that it can only deal with triangular fuzzy numbers, which reduce the capability of handling uncertainty in financial environment. This limitation thus restricts performance of the FAHP portfolio optimization model. The proposed FMOLP approach in this paper, results in less diversifiedportfolios, but outperforms the FAHP approach proposed in Nguyen and Gordon-Brown (2012) in terms of uncertainty modelling and investor’s risk preference handling.

7 Conclusions

The traditional MVO only can be deployed relying on expert knowledge concerning exact asset returns and co-variances. It however would be infeasible when the number of assets increases and expert comprehension about various assets is uncertain, especially in a dynamic economic and financial environment. Deploying fuzzy programming systems able to capture uncertainty for portfolio optimization is thus essential. Through modelling asset marginal impacts by trapezoidal fuzzy numbers, this paper builds up a solvable FMOLP for portfolio selection taking into account higher moments.

Comparing to the JR method, it demonstrates that the proposed FMOLP methodology is effective in terms of portfolio higher moments and more flexible allowing investors to choose preferences regarding risk criteria. The FMOLP and JR methods show the trade-offs in optimizing portfolio return and higher moment risks. A method exhibiting a better optimizing performance in one criterion will show a worse optimizing performance in the other criterion and vice versa.

As the FAHP approach can only deal with triangular fuzzy numbers [28], the proposed FMOLP approach extends the possibility to utilize various kinds of fuzzy sets in fuzzy optimization. We demonstrated the use of the FMOLP method with trapezoidal fuzzy numbers in this paper. Other kinds of fuzzy numbers can be used to represent marginal impacts of assets and these fuzzy numbers can be utilized in the FMOLP approach easily. Given that the centroid-based ranking method can work with any kinds of fuzzy numbers, the proposed FMOLP model is always solvable.

Applying trapezoidal fuzzy numbers to represent marginal impacts is an appealing idea since it is impossible to indicate which stocks are more important than the others at the time of conducting the portfolio allocation. Again, fuzzy numbers in particular, or fuzzy logic in general provide an extremely helpful means to represent uncertainty and inexactness. Additionally, x-centroid based defuzzification used for fuzzy factors (variance, skewness and kurtosis) facilitates the integration between crisp factor (return) and fuzzy factors into a unique FMOLP framework where all criteria are satisfied at different levels depending on the preference weighting scheme of investors.

Further research would extend the FMOLP to incorporate other kinds of fuzzy numbers such as Gaussian, Cauchy, Laplace, Sinc, etc. rather than only trapezoidal shape. This is desirable because of the limitations in representing uncertainty with trapezoidal fuzzy numbers whereas other kinds of fuzzy numbers would allow for different uncertainty representations.

With the proposed FMOLP methodology, a dynamic portfolio model where portfolio weights are frequently reviewed (tick-by-tick) is a plausible and promising research direction. Furthermore, along with higher moments, alternative measures of risks such as liquidity risk, downside risk, Value-at-Risk, Expected Shortfall, etc. are also acknowledged as important factors in risky portfolio management. These quantitative criteria could be easily integrated in the FMOLP model by increasing the number of linear objective functions without changes of the solving method.

Footnotes

Appendix

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