Keywords

1 Introduction

A number of authors(e.g., [1–5] ) have proposed to select stock portfolios on the basis of the first three moments of return distributions, rather than the first two (mean and variance) proposed by Markowitz [6] in 1952. The third moment of return distribution is called skewness. Researchers interested in skewness believe investors should prefer positive skewness. All else constant, they should prefer portfolios with a larger probability of very large payoffs. This is not only logical, but also consistent with some empirical evidence that investors exhibit this preference. If the three moments are important to the investor, then the portfolio problem is represented in three-dimensional space with mean on one axis, variance on the second, and skewness on the third. The efficient set would be the outer shell of the feasible set with maximum mean, minimum variance, and maximum skewness. However, it is evident that the measures of a return distribution, mean, variance, and skewness cannot form a complete design about the distribution. In addition to these measures, we should consider one more measure which Prof. Karl Pearson calls the Convexity of a curve or Kurtosis. Kurtosis enables us to have an idea about the flatness or peakness of the curve. It is measured by the coefficient \(\beta _{2}\) or its derivation \(\gamma _{2}\) given by \(\beta _{2} = \left( \frac{\mu _{4}}{\mu _{2}^2}\right) \), \(\gamma _{2} = \beta _{2} - 3\) [\(\mu _{i}\) being the \(i\)th order moment]. Curve which is neither flat nor peaked is called the normal curve or mesokurtic curve and for such a curve \(\beta _{2} =3\), i.e., \(\gamma _{2} =0\). Curve which is flatter than the normal curve is known as platykurtic and for such a curve, \(\beta _{2} <3\), i.e.,\(\gamma _{2} <0\). Curve which is more peaked than the normal curve is known as leptokurtic and for such a curve \(\beta _{2} >3\), i.e., \(\gamma _{2} >0\). The high kurtosis (fat tails) in return distribution suggests that periods of stability are interspersed by rapid change.

Two distributions may have the same average, dispersion, and skewness, yet in one there may be high concentration of values near the mode, showing a sharper peak in frequency curve than the other. The classical capital market theory, like the bulk of economics, is based on the equilibrium system articulated so well by Alfred Marshall, the father of modern economics in the 1890s. This view is based on the idea that economics is like Newtonian physics, with well-defined cause–effect relationships.

Empirical evidence suggests that the classical capital market theory falls short in the following ways:

  • The distribution of stock returns exhibit a high degree of kurtosis. This means that the tails of the distribution are fatter and the mean of the distribution is higher than what is predicted by a normal distribution. In other words, it means that periods of relatively modest changes are interspersed with periods of booms and busts.

  • Financial returns are predictable to some extent.

  • Risk and return are not related in a linear manner.

  • Investors are prone to make systematic errors in their judgment and trade excessively.

The mean–variance decision criterion by Markowitz [6] is inadequate for allocating wealth when we deal with the funds to be invested in the stock market. Not only are the return distributions asymmetric and leptokurtic, they also display significant coskewness and cokurtosis with the return of other asset classes due to the option-like features of alternative investments. Different approaches have been developed in the financial literature to incorporate the individual preferences for higher order moments into the optimal security allocation problem. Davies et al. [7] and Berenyi ([8, 9]) use the goal programming approach to determine the set of the mean–variance–skewness–kurtosis efficient funds of hedge funds.

Different from [10–12], after recalling the definition of mean, variance, semi-variance, and skewness, this paper considers the kurtosis for portfolio selection with possibilitic fuzzy risk factors. Several empirical studies show that portfolio returns have fat tails. Generally, investors would prefer a portfolio return with smaller kurtosis which indicates the leptokurtosis (fat tails or thin tails) when the mean value, the variance, and the asymmetry are the same. The paper is organized as follows: In Sect. 2, we recall the weighted possibilistic measures of means, variance, skewness of a trapezoidal fuzzy variable. Then we introduce the possibilistic measure of kurtosis for a trapezoidal fuzzy number. In Sect. 3, we have proposed a tetra-objective optimization model for portfolio selection problems. In Sect. 4, we discuss Zimmerman’s goal programming method for multiple objective optimization. In Sect. 5, a case study has been done to illustrate our model. In Sect. 6, some concluding remarks are specified.

2 Weighted Possibilistic Measures of Mean, Variance, and Skewness of Trapezoidal Fuzzy Numbers

In this section some basic ideas of fuzzy sets and possibilistic measures of fuzzy sets are discussed. We also introduce possibilistic measure of fourth-order moment followed by possibilistic measure of kurtosis for trapezoidal fuzzy numbers.

Definition 1

A fuzzy set \(\tilde{A}\) in \(U\subset \mathrm I\! \mathrm R\), where \(\mathrm I\! \mathrm R\) is the set of all real numbers, is an ordered paired set \(\tilde{A}=\{(x,\mu _{\tilde{A}}(x)):x\in \mathrm I\! \mathrm R\}\), where \(\mu _{\tilde{A}}(x)\) is the membership function of x and \(0\le \mu _{\tilde{A}}(x)\le 1\).

Definition 2

An \(\alpha \)-cut of a fuzzy set \(\tilde{A}\) is a crisp set \(\tilde{A}_\alpha \) that contains all the elements in \(U\) and that has membership values in \(\tilde{A}\) greater than or equal to \(\alpha \), i.e., \(\tilde{A}_{\alpha } =\{x\in U:\mu _{\tilde{A}}(x)\ge \alpha \}\).

Definition 3

A fuzzy number \(\tilde{A}=(a,b,c,d)\) is called a trapezoidal fuzzy number (Tr.F.N.) with core \([b,c]\) if its membership function has the following form:

$$\begin{aligned} \mu _{\tilde{A}}(x)= \left\{ \begin{array}{ll}\frac{x-a}{b-a} \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, for\, x \in [a, b] \nonumber \\ 1 \ \ \ \ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, for\, x \in [b,c] \nonumber \\ \frac{d-x}{d-c} \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, for\, x \in [c,d]\nonumber \\ 0 \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, otherwise. \end{array} \right. \end{aligned}$$

Its \(\alpha \)-level sets are .

Definition 4

Let be a \(\alpha \)-cut of a fuzzy number \(\tilde{A}_\alpha \) and \(f(\alpha )\) be a weighted function. Also, let \(D_{L}\) and \(D_{U}\) be two real numbers such that \(D_{L}\le D_{U}\). Then nth weighted double possibilistic moments of fuzzy number \(\tilde{A}\) about points \(D_{L}\) and \(D_{U}\) are defined as:

If \(D_{L}\) = \(D_{U}\) = \(m(\tilde{A})\), where \(m(\tilde{A})\) is the possibilistic mean of the fuzzy number \(\tilde{A}\) and \(m(\tilde{A})\) is given by . If \(f(\alpha )=2\alpha \), then

The second, third, and fourth probalistic moments are, respectively, given as

Definition 5

The weighted possibilistic skewness (WPS) of the fuzzy number \(\tilde{A}\) is defined by \(\gamma _{1}=\frac{M_{3}(\tilde{A})}{\left( \sqrt{M_{2}(\tilde{A})}\right) ^{3}}\).

Definition 6

The weighted possibilistic kurtosis (WPK) of the fuzzy number \(\tilde{A}\) is defined by \(\gamma _{2}=\frac{M_{4}(\tilde{A})}{\left( \sqrt{M_{2}(\tilde{A})}\right) ^{ 4}}\).

Theorem 1

Let \(\tilde{A}=(a,b,c,d)\) be a trapezoidal fuzzy number. Then the weighted possibilistic mean, variance, and skewness of \(\tilde{A}\) are, respectively, given by

\(\textit{WPM}=\frac{1}{6}[a+2(b+c)+d]\)

\(\textit{WPV}=\frac{1}{36}[2(a^2+d^2)+5(b^2+c^2)+2(ab+cd-da)-4(ac+bd)-8bc]\)

\(\textit{WPS}=\frac{1}{5}[19(a^3+d^3)+26(b^3+c^3)-15ad(a+d)-30bc(b+c)+60bc.\)

\((a+d)+30ad(b+c)-12(a^2b+cd^2)-30(a^2c+bd^2)-33(ab^2+c^2d)\)

\(-15(a^2c+b^2d)]/[2(a^2+ d^2)+5(b^2+c^2)+2(ab+cd-da)-4(ac+bd)-8bc]^{\frac{3}{2}}\)

Proof

For proof refer to Battacharyya et al. [4].

Theorem 2

Let \(\tilde{A}=(a,b,c,d)\) be a trapezoidal fuzzy number. Then the weighted possibilistic kurtosis of \(\tilde{A}\) is given as

\(\textit{WPK}=1296[\frac{1}{72}(b-a)^2(d-c)^2+\frac{3}{8}b^2c^2-\frac{1}{6}bc[(b-c)^2+(b-a)(d-c)\)

\(+(d-c)^2]-\frac{1}{4}bc[b^2+c^2-(b-c)(b-a+d-c)]-\frac{1}{18}(b-a)(b-c)\)

\((d-c)(b-a+d-c)+\frac{5}{432}[(b-a)^4+(d-c)^4]+\frac{1}{16}(b^4+c^4)+\frac{1}{12}\)

\((b^2+c^2)[(b-a)^2+(b-a)(d-c)+(d-c)^2]-\frac{1}{12}(b^3-c^3)(b-a+d-c)\)

\(+\frac{2}{135}(b-a)(d-c)[(b-c)^2+((d-c)^2]/[2(a^2+d^2)+5(b^2+c^2)+2\)

\((ab+cd-da)-4(ac+bd)-8bc]^2\).

Proof

We have \(\tilde{A}=(a,b,c,d)\) to be a trapezoidal fuzzy number. Its \(\alpha \)-level sets are . Then the weighted probabilistic fourth-order moment is given as

\(=\int _{0}^{1}\alpha [(a+(b-a)\alpha -\frac{1}{6}\{a+2(b+c)+d\})^{4}+(d-(d-c)\alpha -\frac{1}{6}\{a+2\)

\((b+c)+d\})^{4}]d\alpha \)

\(=\frac{1}{72}(b-a)^2(d-c)^2+\frac{3}{8}b^2c^2-\frac{1}{6}bc[(b-c)^2+(b-a)(d-c)+(d-c)^2]-\frac{1}{4}bc[b^2+c^2-(b-c)(b-a+d-c)]-\frac{1}{18}(b-a)(b-c)(d-c)(b-a+d-c)+\frac{5}{432}[(b-a)^4+(d-c)^4]+\frac{1}{16}(b^4+c^4)+\frac{1}{12}(b^2+c^2)[(b-a)^2+(b-a)(d-c)+\)

\((d-c)^2]-\frac{1}{12}(b^3-c^3)(b-a+d-c)+\frac{2}{135}(b-a)(d-c)[(b-c)^2+((d-c)^2]\)

By definition \(\mathrm{WPK}=\gamma _{2}=\frac{M_{4}(\tilde{A})}{ (\sqrt{M_{2}(\tilde{A})})^{ 4}}\).

Hence the result follows.

3 Weighted Possibilistic Mean–Variance–Skewness–Kurtosis Models for Portfolio Selection

Empirically, it is evident that return from stocks is not fixed, rather range bound. Analyzing the ranges of different stocks we can find that returns can be considered as fuzzy numbers. Let \(\tilde{r}_{i}\) be a fuzzy number representing the return of \(i\)th security. Let \(x_{i}\) be the portion of the total capital invested in security \( i, i=1, 2,\ldots ,n\). Then \(\frac{p_{i}+d_{i}-p_{i}'}{p_{i}}\) , where \(p_{i}\), \(p_{i}'\), \(d_{i}\) are, respectively, closing price at previous year, closing price at next year, and dividend paid for \(i\)th security calculates a particular return.

Theorem 3

Let \(\tilde{r}_{i}=(a_{i},b_{i},c_{i},d_{i})\) be independent trapezoidal fuzzy numbers and \(\varvec{\tilde{r}}=(\tilde{r}_{1},\tilde{r}_{2},\ldots ,\tilde{r}_{n})\) \(n\) component row vector and \(\varvec{\tilde{x}}=\left( x_{1},x_{2},\ldots ,x_{n}\right) '\) \(n\) component column vector. The weighted possibilistic mean, variance, skewness, and kurtosis of fuzzy number are, respectively, given as

\(E=E(\varvec{\tilde{r}x)}=\frac{1}{6} \left[ \sum _{i=1}^{n}a_{i}x_{i}+2\left( \sum _{i=1}^{n}b_{i}x_{i}+\sum _{i=1}^{n}c_{i}x_{i}\right) +\sum _{i=1}^{n}d_{i}x_{i}\right] \)

\(V=V(\varvec{\tilde{r}x)}=\frac{1}{36} [2((\sum _{i=1}^{n}a_{i}x_{i})^2+(\sum _{i=1}^{n}d_{i}x_{i})^2)+5((\sum _{i=1}^{n}b_{i}x_{i})^2\)

\(+(\sum _{i=1}^{n}c_{i}x_{i})^2)+2((\sum _{i=1}^{n}a_{i}x_{i})(\sum _{i=1}^{n}b_{i}x_{i})+ (\sum _{i=1}^{n}c_{i}x_{i})(\sum _{i=1}^{n}d_{i}x_{i})-\)

\((\sum _{i=1}^{n}d_{i}x_{i})(\sum _{i=1}^{n}a_{i}x_{i}))-4((\sum _{i=1}^{n}a_{i}x_{i})(\sum _{i=1}^{n}c_{i}x_{i})+(\sum _{i=1}^{n}b_{i}x_{i})\)

\((\sum _{i=1}^{n}d_{i}x_{i}))-8(\sum _{i=1}^{n}b_{i}x_{i}) (\sum _{i=1}^{n}c_{i}x_{i})]\)

\(S=S(\varvec{\tilde{r}x)}= \frac{1}{5}[19((\sum _{i=1}^{n}a_{i}x_{i})^3+(\sum _{i=1}^{n}d_{i}x_{i})^3)+26((\sum _{i=1}^{n}b_{i}x_{i})^3+ (\sum _{i=1}^{n}c_{i}x_{i})^3)-15(\sum _{i=1}^{n}a_{i}x_{i})(\sum _{i=1}^{n}d_{i}x_{i}) ((\sum _{i=1}^{n}a_{i}x_{i})+(\sum _{i=1}^{n}d_{i}x_{i}))-30 (\sum _{i=1}^{n}b_{i}x_{i})(\sum _{i=1}^{n}c_{i}x_{i}) ((\sum _{i=1}^{n}b_{i}x_{i})+(\sum _{i=1}^{n}c_{i}x_{i}))+60(\sum _{i=1}^{n}b_{i}x_{i}) (\sum _{i=1}^{n}c_{i}x_{i}) ((\sum _{i=1}^{n}a_{i}x_{i})+(\sum _{i=1}^{n}d_{i}x_{i})) +30(\sum _{i=1}^{n}a_{i}x_{i})(\sum _{i=1}^{n}d_{i}x_{i})((\sum _{i=1}^{n}b_{i}x_{i})+(\sum _{i=1}^{n}c_{i}x_{i}))-12((\sum _{i=1}^{n}a_{i}x_{i})^2(\sum _{i=1}^{n} b_{i}x_{i})+(\sum _{i=1}^{n}c_{i}x_{i})(\sum _{i=1}^{n}d_{i}x_{i})^2)-30(\!(\sum _{i=1}^{n}a_{i}x_{i}\!)^2 (\sum _{i=1}^{n}c_{i}x_{i})+(\sum _{i=1}^{n}b_{i}x_{i}) (\sum _{i=1}^{n}d_{i}x_{i})^2)-33((\sum _{i=1}^{n}a_{i}x_{i})(\sum _{i=1}^{n}b_{i}x_{i})^2+(\sum _{i=1}^{n} c_{i}x_{i})^2(\sum _{i=1}^{n}d_{i}x_{i}))-15 ((\sum _{i=1}^{n}a_{i}x_{i})^2(\sum _{i=1}^{n}c_{i}x_{i})+(\sum _{i=1}^{n}b_{i}x_{i})^2(\sum _{i=1}^{n}d_{i}x_{i}))] / [2((\sum _{i=1}^{n}a_{i}x_{i})^2+(\sum _{i=1}^{n}d_{i}x_{i})^2) +5((\sum _{i=1}^{n}b_{i}x_{i})^2+(\sum _{i=1}^{n}c_{i}x_{i})^2)+2((\sum _{i=1}^{n}a_{i}x_{i})(\sum _{i=1}^{n}b_{i}x_{i})+(\sum _{i=1}^{n}c_{i}x_{i})(\sum _{i=1}^{n}d_{i}x_{i}) -(\sum _{i=1}^{n}d_{i}x_{i})(\sum _{i=1}^{n}a_{i}x_{i}))-4((\sum _{i=1}^{n}a_{i}x_{i}) (\sum _{i=1}^{n}c_{i}x_{i})+(\sum _{i=1}^{n}b_{i}x_{i})(\sum _{i=1}^{n}d_{i}x_{i}))-8(\sum _{i=1}^{n}b_{i}x_{i})(\sum _{i=1}^{n}c_{i}x_{i})]^{\frac{3}{2}}\)

\(K=K(\varvec{\tilde{r}x)}=1296[\frac{1}{72}((\sum _{i=1}^{n}b_{i}x_{i})-(\sum _{i=1}^{n}a_{i}x_{i}))^2((\sum _{i=1}^{n}d_{i}x_{i})-(\sum _{i=1}^{n}c_{i}x_{i}))^2+\frac{3}{8}(\sum _{i=1}^{n}b_{i}x_{i})^2(\sum _{i=1}^{n}c_{i}x_{i})^2-\frac{1}{6}(\sum _{i=1}^{n}b_{i}x_{i})(\sum _{i=1}^{n}c_{i}x_{i})[((\sum _{i=1}^{n}b_{i}x_{i})-(\sum _{i=1}^{n}c_{i}x_{i}))^2+((\sum _{i=1}^{n}b_{i}x_{i})-(\sum _{i=1}^{n}a_{i}x_{i}))((\sum _{i=1}^{n}d_{i}x_{i})-(\sum _{i=1}^{n}c_{i}x_{i}))+((\sum _{i=1}^{n}d_{i}x_{i})-(\sum _{i=1}^{n}c_{i}x_{i}))^2]-\frac{1}{4}(\sum _{i=1}^{n}b_{i}x_{i})(\sum _{i=1}^{n}c_{i}x_{i})[(\sum _{i=1}^{n}b_{i}x_{i})^2+(\sum _{i=1}^{n}c_{i}x_{i})^2-((\sum _{i=1}^{n}b_{i}x_{i})-(\sum _{i=1}^{n}c_{i}x_{i}))((\sum _{i=1}^{n}b_{i}x_{i})-(\sum _{i=1}^{n}a_{i}x_{i})+(\sum _{i=1}^{n}d_{i}x_{i})-(\sum _{i=1}^{n}c_{i}x_{i}))]-\frac{1}{18}((\sum _{i=1}^{n}b_{i}x_{i})-(\sum _{i=1}^{n}a_{i}x_{i}))((\sum _{i=1}^{n}b_{i}x_{i})-(\sum _{i=1}^{n}c_{i}x_{i}))((\sum _{i=1}^{n}d_{i}x_{i})-(\sum _{i=1}^{n}c_{i}x_{i}))((\sum _{i=1}^{n}b_{i}x_{i})-(\sum _{i=1}^{n}a_{i}x_{i})+(\sum _{i=1}^{n}d_{i}x_{i})-(\sum _{i=1}^{n}c_{i}x_{i}))+\frac{5}{432}[((\sum _{i=1}^{n}b_{i}x_{i})-(\sum _{i=1}^{n}a_{i}x_{i}))^4+((\sum _{i=1}^{n}d_{i}x_{i})-(\sum _{i=1}^{n}c_{i}x_{i}))^4]+\frac{1}{16}((\sum _{i=1}^{n}b_{i}x_{i})^4+(\sum _{i=1}^{n}c_{i}x_{i})^4)+\frac{1}{12}((\sum _{i=1}^{n}b_{i}x_{i})^2+(\sum _{i=1}^{n}c_{i}x_{i})^2)[((\sum _{i=1}^{n}b_{i}x_{i})-(\sum _{i=1}^{n}a_{i}x_{i}))^2+((\sum _{i=1}^{n}b_{i}x_{i})-(\sum _{i=1}^{n}a_{i}x_{i}))((\sum _{i=1}^{n}d_{i}x_{i})-(\sum _{i=1}^{n}c_{i}x_{i}))+((\sum _{i=1}^{n}d_{i}x_{i})-(\sum _{i=1}^{n}c_{i}x_{i}))^2]-\frac{1}{12}((\sum _{i=1}^{n}b_{i}x_{i})^3-(\sum _{i=1}^{n}c_{i}x_{i})^3)((\sum _{i=1}^{n}b_{i}x_{i})-(\sum _{i=1}^{n}a_{i}x_{i})+(\sum _{i=1}^{n}d_{i}x_{i})-(\sum _{i=1}^{n}c_{i}x_{i}))+\frac{2}{135}((\sum _{i=1}^{n}b_{i}x_{i})-(\sum _{i=1}^{n}a_{i}x_{i}))((\sum _{i=1}^{n}d_{i}x_{i})-(\sum _{i=1}^{n}c_{i}x_{i}))[((\sum _{i=1}^{n}b_{i}x_{i})-(\sum _{i=1}^{n}c_{i}x_{i}))^2+(((\sum _{i=1}^{n}d_{i}x_{i})-(\sum _{i=1}^{n}c_{i}x_{i}))^2]] /[2((\sum _{i=1}^{n}a_{i}x_{i})^2+(\sum _{i=1}^{n}d_{i}x_{i})^2)+5((\sum _{i=1}^{n}b_{i}x_{i})^2+(\sum _{i=1}^{n}c_{i}x_{i})^2)+2((\sum _{i=1}^{n}a_{i}x_{i})(\sum _{i=1}^{n}b_{i}x_{i})+(\sum _{i=1}^{n}c_{i}x_{i})(\sum _{i=1}^{n}d_{i}x_{i})-(\sum _{i=1}^{n}d_{i}x_{i})(\sum _{i=1}^{n}a_{i}x_{i}))-4((\sum _{i=1}^{n}a_{i}x_{i})(\sum _{i=1}^{n}c_{i}x_{i})+(\sum _{i=1}^{n}b_{i}x_{i}\!)(\sum _{i=1}^{n}d_{i}x_{i}))-8(\sum _{i=1}^{n}b_{i}x_{i})(\sum _{i=1}^{n}c_{i}x_{i})]^2\).

Proof

We have

\(\varvec{\tilde{r}x}=\tilde{r}_{1}x_{1}+\tilde{r}_{2}x_{2}+\cdots +\tilde{r}_{n}x_{n}\)

\(= (\sum _{i=1}^{n}a_{i}x_{i},\sum _{i=1}^{n}b_{i}x_{i},\sum _{i=1}^{n}c_{i}x_{i},\sum _{i=1}^{n}d_{i}x_{i})\)

\(=(a,b,c,d)\) (say),

where \(a=\sum _{i=1}^{n}a_{i}x_{i},\,\,b=\sum _{i=1}^{n}b_{i}x_{i},\,\,c=\sum _{i=1}^{n}c_{i}x_{i},\,\,d=\sum _{i=1}^{n}d_{i}x_{i}\)

Hence, proof of the theorem immediately follows from Theorems 1 and 2.

3.1 Proposed Multi-Objective Optimization Model

In this section we have proposed a multi-objective optimization model consisting of four objectives, viz., maximization of return(E), minimization of risk (variance)(V), maximization of skewnwss (S), and minimization of kurtosis subject to the constraint that the sum of all portions of shares is equal to one.

$$\begin{aligned} \left. \begin{array}{l} {Maximize}\,\,\tilde{E}(\varvec{\tilde{r}x})=\tilde{E}(\tilde{r}_{1}x_{1}+\tilde{r}_{2}x_{2}+\cdots +\tilde{r}_{n}x_{n})\\ {Minimize} \,\, \tilde{V}(\varvec{\tilde{r}x})=\tilde{V}(\tilde{r}_{1}x_{1}+\tilde{r}_{2}x_{2}+\cdots +\tilde{r}_{n}x_{n})\\ {Maximize} \,\, \tilde{S}(\varvec{\tilde{r}x})=\tilde{S}(\tilde{r}_{1}x_{1}+\tilde{r}_{2}x_{2}+\cdots +\tilde{r}_{n}x_{n})\\ {Minimize} \,\, \tilde{ K}(\varvec{\tilde{r}x})=\tilde{K}(\tilde{r}_{1}x_{1}+\tilde{r}_{2}x_{2}+\cdots +\tilde{r}_{n}x_{n})\\ \,\, x_{1}+x_{2}+\cdots +x_{n}=1\\ \,\, x_{i}\ge 0, i=1, 2,\ldots ,n. \end{array} \right\} \end{aligned}$$
(1)

4 Solution Methodology: Zimmerman’s Fuzzy Goal Programming

A general multi-objective nonlinear programming problem is of the following form:

$$\begin{aligned} \left. \begin{array}{l} max/min[f_{1}(x),f_{2}(x),\ldots ,f_{K}(x)]\\ subject\,\ to\\ x\in X=\{x:g_{s}(x)(\le , =, \ge )0, s=1, 2, \ldots , m\} \end{array} \right. \end{aligned}$$
(2)

where \(f_{a}(x)\) are objective functions for maximization, \(a\in A\) and \(f_{b}(x)\) are objective functions for minimization \(b\in B\), A, B being two exhaustive subsets of the index set 1, 2, ...K and x being the decision variable. It is noted that all functions \(f_{k}(x)\) and \(g_{i}(x)\) (\(k = 1, 2,\ldots \mathrm{K}\) and \(i = 1, 2,\ldots \mathrm{m}\)) can be linear or nonlinear. In the past two decades, many fuzzy programming techniques have been developed for solving multi-objective optimization problems. In this area, Zimmermann [13] first shows that fuzzy programming technique can be used satisfactorily to solve the multi-objective programming problem using maxmin operator of Bellman and Zadeh [14].

The steps of the fuzzy programming technique are as follows:

Step 1. Each objective function \(f_{k}(x)\) of the MOP problem is optimized separately subject to the constraints of the problem. Let these optimum values be \(f_{k}^*(x) (k = 1, 2,\ldots K)\).

Step 2. For each optimal solution of the K single-objective programming problem solved in step 1, find the value of the remaining objective functions and construct a payoff matrix of order \(K\times K\) as given in Table 1.

Step 3. Evaluate

\(f_{k}^{L}=Min\{f_{k}(x^{1}),f_{k}(x^{2}),\ldots ,f_{k}(x^{K})\}\) and \(f_{k}^{U}=Max\{f_{k}(x^{1}),f_{k}(x^{2}),\ldots , f_{k}(x^{K})\}\) for all \(k=1,2,\ldots ,K\).

Table 1 Payoff matrix
Full size table

Step 4. Form the membership functions \(\mu _{f_{i}}(x),\,\ i\in A\) and \(\mu _{f_{j}}(x),\,\ j\in B\), respectively, for the maximization objective functions \(\mu _{f_{i}}(x),\,\ i\in A\) and minimization objective function \(\mu _{f_{j}}(x),\,\ j\in B\), where \(A U B = \{1,2,\ldots ,K\}\) in the linear form as follows.

$$\begin{aligned} \mu _{f_{i}}(x)=\left\{ \begin{array}{lllll} 1 \ \ \ \ \ \ \ \ \ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{ if } \,\,f_{i}(x) > f_{i}^{U} \\ \frac{f_{i}(x)-f_{i}^{L}}{f_{i}^{U}-f_{i}^{L}} \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{ if } \,\,f_{i}^{L} \le f_{i}(x) \le f_{i}^{U}, {for\,\, all}\,\, i\in A \\ 0 \ \ \ \ \ \ \ \ \ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{ if } \,\,f_{i}(x) < f_{i}^{L} \nonumber \\ \end{array} \right. \end{aligned}$$
$$\begin{aligned} \mu _{f_{j}}(x)=\left\{ \begin{array}{lllll} 1 \ \ \ \ \ \ \ \ \ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{ if } \,\,f_{i}(x) > f_{i}^{U} \\ \frac{f_{j}^{U}-f_{j}(x)}{f_{j}^{U}-f_{j}^{L}} \,\,\,\,\,\,\,\,\,\,\,\,\,\text{ if } \,\,f_{j}^{L} \le f_{j}(x) \le f_{j}^{U}, {for\,\, all}\,\, j\in B \\ 0 \ \ \ \ \ \ \ \ \ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{ if }\,\,f_{j}(x) < f_{j}^{L} \nonumber \\ \end{array} \right. \end{aligned}$$

Step 5. Using the above membership functions formulate and solve the crisp nonlinear programming model following the methods due to Zimmermann [13].

4.1 Zimmermann’s Model

If \(w_{1}\), \(w_{2}\), \(w_{3}\) and \(w_{4}\) are the intuitive crisp weights for the portfolio mean(E), variance(V), skewness(S) , and kurtosis(K), respectively, then for different models the problem (1) can be formulated as follows:

$$\begin{aligned} \left\{ \begin{array}{ll} Maximize\ \ \ \alpha \\ such \ \ that\\ w_{1}(\frac{E-E^{U}}{E^{U}-E^{L}})=\alpha , \ w_{2}(\frac{V^{U}-V}{V^{U}-V^{L}})=\alpha , \ w_{3}(\frac{S-S^{U}}{S^{U}-S^{L}})=\alpha , \ w_{4}(\frac{K^{U}-K}{K^{U}-K^{L}})=\alpha \\ x\in X\\ 0\le \alpha \le 1, \ \ w_{1}+w_{2}+w_{3}+w_{4}=1 \end{array} \right. \end{aligned}$$
(3)

Zimmerman’s fuzzy goal programming is a pre-emptive fuzzy goal programming method where the priorities of the goals are considered to be the same (e.g., \(\alpha \)).

5 Case Study: Bombay Stock Exchange (BSE)

In this section we apply the proposed portfolio selection models on the data set extracted from the Bombay Stock Exchange (BSE). BSE is the oldest stock exchange in Asia with a rich heritage of over 133 years of existence. What is now popularly known as BSE was established as The Native Share & Stock Brokers’ Association in 1875. It is the first stock exchange in India which obtained permanent recognition (in 1956) from the Government of India under the Securities Contracts (Regulation) Act (SCRA) 1956. With demutualization, the stock exchange has two of the world’s prominent exchanges, Deutsche Borse and Singapore Exchange, as its strategic partners. Today, BSE is the world’s number one exchange in terms of the number of listed companies and the world’s fifth in handling of transactions through its electronic trading system. The companies listed on BSE command a total market capitalization of USD trillion 1.06 as of July 2009. The BSE Index, SENSEX, is India’s first and most popular stock market benchmark index. Sensex is tracked worldwide. It constitutes 30 stocks representing 12 major sectors. It is constructed on a free-float methodology, and is sensitive to market movements and market realities. Apart from the SENSEX, BSE offers 23 indices, including 13 sectoral indices. We have taken monthly share price data for 60 months (March 2003–February 2008) of just five companies which are included in Bombay Stock Exchange (BSE) index. Though any finite number of stocks can be considered, we have taken only five stocks to reduce the complexity of representation.

Table 2 Fuzzy returns of stocks listed at Bombay Stock Exchange (BSE)
Full size table

5.1 Example

Let us consider the following multi-objective portfolio selection problem.

$$\begin{aligned} \left. \begin{array}{l} Maximize \,\,\tilde{ E}(\varvec{\tilde{r}x})=\tilde{E}(\tilde{r}_{1}x_{1}+\tilde{r}_{2}x_{2}+\tilde{r}_{3}x_{3}+\tilde{r}_{4}x_{4}+\tilde{r}_{5}x_{5})\\ Minimize \,\, \tilde{ V}(\varvec{\tilde{r}x})=\tilde{V}(\tilde{r}_{1}x_{1}+\tilde{r}_{2}x_{2}+\tilde{r}_{3}x_{3}+\tilde{r}_{4}x_{4}+\tilde{r}_{5}x_{5})\\ Maximize \,\, \tilde{ S}(\varvec{\tilde{r}x})=\tilde{S}(\tilde{r}_{1}x_{1}+\tilde{r}_{2}x_{2}+\tilde{r}_{3}x_{3}+\tilde{r}_{4}x_{4}+\tilde{r}_{5}x_{5})\\ Minimize \,\, \tilde{ K}(\varvec{\tilde{r}x})=\tilde{K}(\tilde{r}_{1}x_{1}+\tilde{r}_{2}x_{2}+\tilde{r}_{3}x_{3}+\tilde{r}_{4}x_{4}+\tilde{r}_{5}x_{5})\\ \,\, x_{1}+x_{2}+x_{3}+x_{4}+x_{5}=1\\ \,\, x_{i}\ge 0,\,\, i=1,2,3,4,5. \end{array} \right\} \end{aligned}$$
(4)

The Zimmermann’s model for multi-objective decision-making(as described in Sect. 4 is used to solve the example. Using Theorem 3 and using the data given in Table 2, we can find the payoff matrix (Table 3).

Table 3 Payoff matrix on the basis of data in Table 2
Full size table

Applying Zimmermann’s method we obtain the following solution:

$$\begin{aligned}&\mathrm{E}=0.039377, \mathrm{V}=0.0004147, \mathrm{S}=0.000003345, \mathrm{K}=1.142835 \\&\qquad \quad x_{1}=0.00, x_{2}=0.33, x_{3}=0.27, x_{4}=0.40, x_{5}=0.00. \end{aligned}$$

6 Conclusion

In this paper, we have used the fuzzy possibilistic measure of kurtosis to model a new possibilistic mean–variance–skewness–kurtosis stock portfolio selection model. Zimmerman’s fuzzy goal programming method is applied to convert the tetra-objective programming problem into a single-objective programming problem. Data of 60 months from BSE of five stocks are used for testing the effectiveness of the proposed model. In the future we will apply the model on a larger data set. We will also compare the proposed model with other established models of portfolio selection problem. For simulation, genetic algorithm or ant colony optimization algorithm can be used.