Robust Portfolio Selection Model with Random Fuzzy Returns Based on Arbitrage Pricing Theory and Fuzzy Reasoning Method

Chapter
Part of the Lecture Notes in Electrical Engineering book series (LNEE, volume 186)

Abstract

This paper considers a robust-based random fuzzy mean-variance portfolio selection problem using a fuzzy reasoning method, particularly a single input type fuzzy reasoning method. Arbitrage Pricing Theory (APT) is introduced as a future return of each security, and each factor in APT is assumed to be a random fuzzy variable whose mean is derived from a fuzzy reasoning method. Furthermore, under interval inputs of fuzzy reasoning method, a robust programming approach is introduced in order to minimize the worst case of the total variance. The proposed model is equivalently transformed into the deterministic nonlinear programming problem, and so the solution steps to obtain the exact optimal portfolio are developed.

Keywords

Portfolio selection problem Arbitrage pricing theory (APT) Random fuzzy programming Fuzzy reasoning method Robust programming Equivalent transformation Exact solution algorithm 

1 Introduction

The decision of optimal asset allocation among various securities is called portfolio selection problem, and it is one of the most important research themes in investment and financial research fields since the mean-variance model was proposed by Markowitz [20]. Then, after this outstanding research, numerous researchers have contributed to the development of modern portfolio theory (cf. Elton and Gruber [1], Luenberger [19]), and many researchers have proposed several types of portfolio models extending Markowitz model; mean-absolute deviation model (Konno [13], Konno et al. [14]), safety-first model [1], Value at Risk and conditional Value at Risk model (Rockafellar and Uryasev [23]), etc. As a result, nowadays it is common practice to extend these classical economic models of financial investment to various types of portfolio models because investors correspond to present complex markets. In practice, many researchers have been trying different mathematical approaches to develop the theory of portfolio model. Particularly, Capital Asset Pricing Model (CAPM), which is a single factor model proposed by Sharpe [25], Lintner [16] and Mossin [22], has been one of the most useful tools in the investment fields and also used in the performance measure of future returns for portfolios and the asset pricing theory. Hasuike et al. [5] proposed a CAPM-based robust random fuzzy mean-variance model. However, CAPM is a single factor model, and some researchers showed that real economic phenomena were different from the result of CAPM. Therefore, in this paper, we extend the previous model [5] to an uncertain portfolio model based on Arbitrage Pricing Theory (APT) which is a multi-factor model proposed by Ross [24]. APT is one of the most general theories of asset pricing that holds that expected security returns are modeled as a linear function of various macro-economic factors or theoretical market indices, where sensitivity to changes in each factor is represented by a factor-specific beta coefficient.

In such previous researches, expected future return and variance of each asset are assumed to be known. Then, in previous many studies in the sense of mathematical programming for the investment, future returns are assumed to be continuous random variables according to normal distributions. However, investors receive ineffective information from the real markets and economic analysts, and ambiguous factors usually exist in it. Furthermore, investors often have the subjective prediction for future markets which are not derived from the statistical analysis of historical data, but their long-term experiences of investment. Then, even if investors hold a lot of information from the real market, it is difficult that the present or future random distribution of each asset is strictly set. Consequently, we need to consider not only random conditions but also ambiguous and subjective conditions for portfolio selection problems.

As recent studies in mathematical programming, some researchers have proposed various types of portfolio models under randomness and fuzziness. These portfolio models with probabilities and possibilities are included in stochastic programming problems and fuzzy programming problems, respectively, and there are some basic studies using stochastic programming approaches, goal programming approaches, and fuzzy programming approaches to deal with ambiguous factors as fuzzy sets (Inuiguchi and Ramik [9], Leon et al. [15], Tanaka and Guo [27], Tanaka et al. [28], Vercher et al. [29], Watada [30]). Furthermore, some researchers have proposed mathematical programming problems with both randomness and fuzziness as fuzzy random variables (for instance, Katagiri et al. [11, 12]). In the studies [11, 12], fuzzy random variables were related with the ambiguity of the realization of a random variable and dealt with a fuzzy number that the center value occurs according to a random variable. On the other hand, future returns may be dealt with random variables derived from the statistical analysis, whose parameters are assumed to be fuzzy numbers due to the decision maker’s subjectivity, i.e., random fuzzy variables which Liu [17] defined. There are a few studies of random fuzzy programming problem (Hasuike et al. [3, 4], Huang [8], Katagiri et al. [10]). Most recently, Hasuike et al. [4] proposed several portfolio selection models including random fuzzy variables and developed the analytical solution method.

However, in [4], each membership function of fuzzy mean values of future returns was set by the investor, but the mathematical detail of setting the membership function is not obviously given. Of course, it is also important to determine the fuzzy mean values of future returns with the investor’s long-term experiences and economical analysts’ effective information. Therefore, in order to involve the necessary information into mean values of future returns mathematically, we introduce a fuzzy inference or reasoning method based on fuzzy if-then rules. The fuzzy reasoning method is the most important approach to extract and decide effective rules under fuzziness mathematically. Since outstanding studies of Mamdani [21] and Takagi and Sugeno [24], many researchers have extended these previous approaches, and proposed new fuzzy reasoning methods. Particularly, we focus on a single input type fuzzy reasoning method proposed by Hayashi et al. [6, 7]. This method sets up rule modules to each input item, and the final inference result is obtained by the weighted average of the degrees of the antecedent part and consequent part of each rule module. Nevertheless this approach is one of the simplest mathematical approaches in fuzzy reasoning methods, the final inference result is similar to the other standard approaches. Therefore, in this paper, we proposed a random fuzzy mean-variance model introducing APT-based future returns and Hayashi’s single input type fuzzy reasoning method for the mean value of market portfolio of APT.

The proposed random fuzzy mean-variance model is not formulated as a well-defined problem due to fuzziness, we need to set some certain optimization criterion so as to transform into well-defined problems. In this paper, assuming the interval values as a special case of fuzzy numbers and introducing the concept of robust programming, we transform the main problem into a robust programming problem. Recently, the robust optimization problem becomes a more active area of research, and there are some studies of robust portfolio selection problems determining optimal investment strategy using the robust approach (For example, Goldfarb and Iyengar [2], Lobo [18]). In robust programming, we obtain the exact optimal portfolio.

This paper is organized in the following way. In Sect. 2, we introduce mathematical concepts of random fuzzy variables, Arbitrage Pricing Theory, and a single input type fuzzy reasoning method. In Sect. 3, we propose a random fuzzy portfolio selection problem with mean values derived from the fuzzy reasoning method. Performing the deterministic equivalent transformations, we obtain a fractional programming problem with one variable. Finally, in Sect. 4, we conclude this paper.

2 Mathematical Definition and Notation

In many existing studies of portfolio selection problems, future returns are assumed to be random variables or fuzzy numbers. However, since there are few studies of them treated as APT with random fuzzy variables and fuzzy reasoning method, simultaneously. Therefore, in this section, we explain definitions and mathematical formulations of random fuzzy variable, APT, and single input type fuzzy reasoning method proposed by Hayashi et al. [6, 7].

2.1 Random Fuzzy Variables

First of all, we introduce a random fuzzy variables defined by Liu [17] as follows.

Definition 1 (Liu [17])

A random fuzzy variable is a function \( \xi \) from a collection of random variables R to [0, 1]. An n-dimensional random fuzzy vector \( \xi = \left( {\xi_{1} ,\xi_{2} , \ldots ,\xi_{n} } \right) \) is an n-tuple of random fuzzy variables \( \xi_{1} ,\xi_{2} , \ldots ,\xi_{n} \).

That is, a random fuzzy variable is a fuzzy set defined on a universal set of random variables. Furthermore, the following random fuzzy arithmetic definition is introduced.

Definition 2 (Liu [17])

Let \( \xi_{1} ,\xi_{2} , \ldots ,\xi_{n} \) be random fuzzy variables, and \( f:R^{n} \to R \) be a continuous function. Then, \( \xi = f\left( {\xi_{1} ,\xi_{2} , \ldots ,\xi_{n} } \right) \) is a random fuzzy variable on the product possibility space \( (\Uptheta ,P(\Uptheta ),{\text{Pos}}) \), defined as \( \xi \left( {\theta_{ 1} ,\theta_{2} , \ldots ,\theta_{n} } \right) = f\left( {\xi_{1} \left( {\theta_{1} } \right),\,\xi_{2} \left( {\theta_{2} } \right), \ldots ,\xi_{n} \left( {\theta_{n} } \right)} \right) \)for all \( \left( {\theta_{1} ,\theta_{2} , \ldots ,\theta_{n} } \right) \in \Uptheta \).

From these definitions, the following theorem is derived.

Theorem 1 (Liu [17])

Let \( \xi_{i} \) be random fuzzy variables with membership functions\( \mu_{i} \), i = 1, 2,…,n, respectively, and\( f:R^{n} \to R \) be a continuous function. Then, \( \xi = f\left( {\xi_{1} ,\xi_{2} , \ldots ,\xi_{n} } \right) \) is a random fuzzy variable whose membership function is\( \mu \left( \eta \right) = \mathop {\sup }\limits_{{\eta_{i} \in R_{i} ,1 \le i \le n}} \left\{ {\mathop {\hbox{min} }\limits_{1 \le i \le n} \,\mu_{i} \left( {\eta_{i} } \right)|\eta = f\left( {\eta_{1} ,\eta_{2} , \ldots ,\eta_{n} } \right)} \right\} \) for all \( \eta \in R \), where\( R = \left\{ {f\left( {\eta_{1} ,\eta_{2} , \ldots ,\eta_{n} } \right)\,|\,\eta_{i} \, \in \,R_{i} ,i = 1,2, \ldots ,n} \right\} \).

2.2 Arbitrage Pricing Theory

As a useful pricing theory in portfolio models, Capital Asset Pricing Model (CAPM) proposed by Sharpe [25], Lintner [16], and Mossin [22] is well-known and used in many practical investment cases by not only researchers but also practical investors. However, CAPM is a single factor model, and some researchers showed that real economic phenomena were different from the result of CAPM. Therefore, in this paper, we focus on Arbitrage Pricing Theory (APT) which is a multi-factor model proposed by Ross [24] as follows:
$$ r_{j} = a_{0j} + \sum\limits_{i = 1}^{m} {a_{ij} f_{i} } $$
where each \( f_{i} ,\left( {i = 1,2, \ldots ,m} \right) \) is the value of factors related to the economic and financial market such as diversified stock indexes and surprises in inflation and GNP. Then, \( a_{0j} \) and \( a_{ij} ,\left( {i = 1,2, \ldots ,m} \right) \) are inherent values derived from historical data in investment fields. This model includes the CAPM in the case that \( a_{ij} = 0,\left( {i = 2, \ldots ,m} \right) \) and \( f_{1} = r_{m} \). Therefore, the APT is the more extended and versatile pricing model than CAPM.

However, in the case that the decision maker predicts the future return using APT, it is obvious that each factor \( f_{i} \) also occurs according to a random distribution with the investor’s subjectivity. Therefore, in these situations, we propose a random fuzzy APT model. In this model we assume that \( \bar{\tilde{f}}_{i} \) is a random fuzzy variable, and the “dash above” and “wave above”, i.e., “-” and “~”, denote randomness and fuzziness of the coefficients, respectively. In this paper, \( \bar{\tilde{f}}_{i} \) occurs according to a random distribution with fuzzy mean value \( \tilde{f}_{i} \) and constant variance \( \sigma_{i}^{2} \). We assume that each factor is independent of each other. To simplify, we also assume that each fuzzy expected return \( \tilde{f}_{i} \) is an interval values \( \tilde{f}_{i} = [f_{i}^{L} ,f_{i}^{U} ] \) derived from a fuzzy reasoning method in the next subsection.

2.3 Single Input Type Fuzzy Reasoning Method

Many researchers have proposed various fuzzy inference and reasoning methods based on or extending Mamdani [21] or Takagi and Sugeno’s [26] outstanding studies. In this paper, as a mathematically simple approach, we introduce a single input type fuzzy reasoning method proposed by Hayashi et al. [6, 7]. In this method, we consider the following K rule modules:

Rule-k: \( \left\{ {\zeta_{ik} = A_{s}^{ik} \, \to \,f_{ik} = f_{s}^{ik} } \right\}_{s = 1}^{{S_{k} }} ,\left( {i = 1,2, \ldots ,m,\,k = 1,2, \ldots ,K} \right) \)

where \( \zeta_{ik} \) and \( f_{ik} \) are the ith input and consequent data, respectively. Then, \( f_{s}^{ik} \) is the real value of output for the consequent part. \( A_{s}^{ik} \) is the fuzzy set of the sth rule of the Rules-k, and \( S_{k} \) is the total number of membership function of \( A_{s}^{ik} \). The degree of the antecedent part in the sth rule of Rules-k is obtained as \( h_{s}^{ik} = A_{s}^{ik} (\zeta_{ik}^{0} ) \). In Hayashi’s single input type fuzzy reasoning method, the inference result \( f_{i}^{0} \) is calculated as follows:
$$ f_{i}^{0} = \frac{{\sum\limits_{s = 1}^{{S_{1} }} {h_{s}^{i1} f_{s}^{i1} } + \cdots + \sum\limits_{s = 1}^{{S_{K} }} {h_{s}^{iK} f_{s}^{iK} } }}{{\sum\limits_{s = 1}^{{S_{1} }} {h_{s}^{i1} } + \cdots + \sum\limits_{s = 1}^{{S_{K} }} {h_{s}^{iK} } }} = \frac{{\sum\limits_{k = 1}^{K} {\sum\limits_{s = 1}^{{S_{k} }} {h_{s}^{ik} f_{s}^{ik} } } }}{{\sum\limits_{k = 1}^{K} {\sum\limits_{s = 1}^{{S_{k} }} {h_{s}^{ik} } } }} $$
(1)
Particularly, if membership functions of all fuzzy sets \( A_{s}^{ik} \) are triangle fuzzy numbers, formula (1) is a linear fractional function on input column vector \( \zeta_{i}^{0} \). In this paper, using this fuzzy reasoning method, we obtain the mean value of each factor \( f_{i} \). We assume that input column vector \( \zeta_{i}^{0} \) means important financial and social factors to decide the mean value of market portfolio. However, it is difficult to set input column vector \( \zeta_{i}^{0} \) as a constant value vector. Therefore, we set each input \( \zeta_{ik}^{0} \) as an interval value, and in order to obtain the maximum and minimum values of \( f_{i} \) under interval values of \( \left[ {\underline{\zeta }_{ik}^{0} ,\overline{\zeta }_{ik}^{0} } \right] \), we introduce the following mathematical programming:
$$ \begin{gathered} {\text{Maximize(Minimize) }}\frac{{\sum\limits_{k = 1}^{K} {\sum\limits_{s = 1}^{{S_{k} }} {h_{s}^{ik} f_{s}^{ik} } } }}{{\sum\limits_{k = 1}^{K} {\sum\limits_{s = 1}^{{S_{k} }} {h_{s}^{ik} } } }} \hfill \\ {\text{subject to }}\underline{\zeta }_{ik}^{0} \, \le \,\zeta_{ik}^{0} \, \le \,\overline{\zeta }_{ik}^{0} ,k = 1,2, \ldots ,K \hfill \\ \end{gathered} $$
(2)

Each problem is a fractional linear programming problem under triangle fuzzy numbers \( A_{s}^{ik} \), and so we obtain the optimal solutions. Let \( f_{i}^{U} \) and \( f_{i}^{L} \) be the optimal solution maximizing and minimizing the object, respectively.

3 Formulation of Portfolio Selection Problem with Random Fuzzy Returns

The previous studies on random and fuzzy portfolio selection problems often have considered standard mean-variance model or safety first models introducing probability or fuzzy chance constraints based on modern portfolio theories (e.g. Hasuike et al. [4]). However, there is no study to the random fuzzy mean variance model using the fuzzy reasoning method to obtain the interval mean value of market portfolio. Therefore, in this paper, we extend the previous random fuzzy mean-variance model to a robust programming-based model using the fuzzy reasoning method.

First, we deal with the following most simple portfolio selection problem involving the random fuzzy variable based on the standard asset allocation problem to maximize total future returns:
$$ \begin{gathered} {\text{Maximize }}\sum\limits_{j = 1}^{n} {\bar{\tilde{r}}_{j} x_{j} } \hfill \\ {\text{subject to }}\sum\limits_{j = 1}^{n} {x_{j} } = 1, \, x_{j} \, \ge \,0, \, j = 1,2, \ldots ,n \hfill \\ \end{gathered} $$
(3)
where the notation of parameters used in this paper is as follows:
\( \bar{\tilde{r}} \)

Future return of the jth financial asset assumed to be a random fuzzy variable, whose fuzzy expected value is \( \tilde{m}_{j} \) and variance-covariance matrix is V, respectively. Then, we denote randomness and fuzziness of the coefficients by the “dash above” and “wave above”, i.e., “-” and “~”, respectively.

rG

Target total return

n

Total number of securities

xj

Budgeting allocation to the jth security

In [4], we consider several models and solution approaches based on standard safety-first models of portfolio selection problems. However, in order to solve the previous models analytically, we must assume that each return occurs according to the normal distributions in the sense of randomness. This assumption is a little restricted. Therefore, in this paper, we do not assume certain random distributions for future returns. Alternatively, we introduce the following portfolio model minimizing the worst total variance, i.e., maximizing the total variance, as a robust portfolio model:
$$ \begin{gathered} {\text{Minimize }}\mathop { \hbox{max} }\limits_{{f_{i} \in [f_{i}^{L} ,f_{i}^{U} ]}} \left\{ {\frac{1}{2}V\left( {\sum\limits_{j = 1}^{n} {\bar{\tilde{r}}_{j} x_{j} } } \right)} \right\} \hfill \\ {\text{subject to }}E\left( {\sum\limits_{j = 1}^{n} {\bar{\tilde{r}}_{j} x_{j} } } \right) \ge r_{G} , \hfill \\ \, \sum\limits_{j = 1}^{n} {x_{j} } = 1, \, x_{j} \, \ge \,0, \, j = 1,2, \ldots ,n \hfill \\ \end{gathered} $$
(4)
By assuming this robust programming problem, the investor may be able to avoid the latent risk including the worst case of future return. In order to solve this problem, we fix the value of interval mean value of random fuzzy market portfolio \( \bar{\tilde{f}}_{i} \) as \( \bar{f}_{i} (\omega ) \in [f_{i}^{L} ,f_{i}^{U} ] \), i.e., each future return is also only a random variable as \( \bar{r}_{j} (\omega ) = a_{0j} + \sum\limits_{i = 1}^{m} {a_{ij} \bar{f}_{i} (\omega )} \). Therefore, problem (4) is transformed into the following standard mean-variance portfolio model:
$$ \begin{gathered} {\text{Minimize }}\frac{ 1}{ 2}V\left( {\sum\limits_{j = 1}^{n} {\bar{r}_{j} \left( \omega \right)x_{j} } } \right) \hfill \\ {\text{subject to }}E\left( {\sum\limits_{j = 1}^{n} {\bar{r}_{j} \left( \omega \right)x_{j} } } \right)\, \ge \,r_{G} , \hfill \\ \, \sum\limits_{j = 1}^{n} {x_{j} } = 1,x_{j} \, \ge \,0,\,j = 1,2, \ldots ,n \hfill \\ \, \bar{r}_{j} (\omega ) = a_{0j} \, + \,\sum\limits_{i = 1}^{m} {a_{ij} \bar{f}_{i} (\omega )} \hfill \\ \end{gathered} $$
(5)

This problem is a convex quadratic programming problem due to positive definite matrix, and so we obtain the exact optimal portfolio by using the following steps in nonlinear programming.

First we introduce the Lagrange function for problem (5) as follows:
$$ L = \frac{1}{2}V\left( {\sum\limits_{j = 1}^{n} {\bar{r}_{j} \left( \omega \right)x_{j} } } \right) + \lambda \left( {r_{G} - E\left( {\sum\limits_{j = 1}^{n} {\bar{r}_{j} \left( \omega \right)x_{j} } } \right)} \right) + \xi \left( {1 - \sum\limits_{j = 1}^{n} {x_{j} } } \right) $$
(6)
where \( \lambda \) and \( \xi \) are Lagrange multiples. Variance \( V\left( {\sum\limits_{j = 1}^{n} {\bar{r}_{j} \left( \omega \right)x_{j} } } \right) \) and \( E\left( {\sum\limits_{j = 1}^{n} {\bar{r}_{j} \left( \omega \right)x_{j} } } \right) \) are equivalently transformed into the following forms from the assumption of APT in Sect. 2.2:
$$ \left\{ \begin{gathered} V\left( {\sum\limits_{j = 1}^{n} {\bar{r}_{j} \left( \omega \right)x_{j} } } \right) = \sum\limits_{j = 1}^{n} {\hat{\sigma }_{j}^{2} x_{j}^{2} } \hfill \\ \hat{\sigma }_{j}^{2} = \sum\limits_{i = 1}^{m} {a_{ij}^{2} \sigma_{i}^{2} } \hfill \\ \end{gathered} \right., \, \left\{ \begin{gathered} E\left( {\sum\limits_{j = 1}^{n} {\bar{r}_{j} \left( \omega \right)x_{j} } } \right) = \sum\limits_{j = 1}^{n} {\hat{r}_{j} \left( \omega \right)x_{j} } \hfill \\ \hat{r}_{j} \left( \omega \right) = a_{0j} + \sum\limits_{i = 1}^{m} {a_{ij} \bar{f}_{i} \left( \omega \right)} \hfill \\ \end{gathered} \right. $$
Therefore, by using Karush-Kuhn-Tucker (KKT) condition, we obtain the following equation on each variable \( x_{j} \):
$$ \begin{gathered} \frac{\partial L}{{\partial x_{j} }} = \hat{\sigma }_{j}^{2} x_{j} \, - \,\lambda \hat{r}_{j} \left( \omega \right)\, - \,\xi = 0,\left( {j = 1,2, \ldots ,n} \right) \hfill \\ \sum\limits_{j = 1}^{n} {\hat{r}_{j} \left( \omega \right)x_{j} } = r_{G} , \, \sum\limits_{j = 1}^{n} {x_{j} } = 1 \hfill \\ \end{gathered} $$
(7)
In order to solve equations derived from KKT condition, we set the vector notation, and obtain the solution of \( x \) as follows:
$$ \begin{gathered} {\mathbf{V}}\text{x} - \lambda \hat{r}\left( \omega \right) - \xi I = {\mathbf{0}} \hfill \\ \Leftrightarrow \, {\mathbf{V}}\text{x} - {\mathbf{A}}^{t} \left( \begin{gathered} \lambda \hfill \\ \xi \hfill \\ \end{gathered} \right) = {\mathbf{0}}, \hfill \\ \, \left( {{\mathbf{V}} = \left( {\begin{array}{*{20}c} {\hat{\sigma }_{1}^{2} } & {} & {\mathbf{0}} \\ {} & \ddots & {} \\ {\mathbf{0}} & {} & {\hat{\sigma }_{n}^{2} } \\ \end{array} } \right), \, {\mathbf{A}} = \left( {\begin{array}{*{20}c} {\hat{r}_{1} \left( \omega \right)} & \cdots & {\hat{r}_{n} \left( \omega \right)} \\ 1 & \cdots & 1 \\ \end{array} } \right)} \right) \hfill \\ \Leftrightarrow \, x = {\mathbf{V}}^{ - 1} {\mathbf{A}}^{t} \left( \begin{gathered} \lambda \hfill \\ \xi \hfill \\ \end{gathered} \right), \, \left( {{\mathbf{V}} = \left( {\begin{array}{*{20}c} {{1 \mathord{\left/ {\vphantom {1 {\hat{\sigma }_{1}^{2} }}} \right. \kern-0pt} {\hat{\sigma }_{1}^{2} }}} & {} & {\mathbf{0}} \\ {} & \ddots & {} \\ {\mathbf{0}} & {} & {{1 \mathord{\left/ {\vphantom {1 {\hat{\sigma }_{n}^{2} }}} \right. \kern-0pt} {\hat{\sigma }_{n}^{2} }}} \\ \end{array} } \right)} \right) \hfill \\ \end{gathered} $$
(8)
By substituting this solution into second and third equations in KKT condition (7), we obtain the following optimal values of Lagrange multiples:
$$ \begin{gathered} {\mathbf{AV}}^{ - 1} {\mathbf{A}}^{t} \left( \begin{gathered} \lambda \hfill \\ \xi \hfill \\ \end{gathered} \right) = \left( {\begin{array}{*{20}c} {r_{G} } \\ 1 \\ \end{array} } \right) = \hat{\varvec{r}}_{G} \hfill \\ \Leftrightarrow \, \left( \begin{gathered} \lambda \hfill \\ \xi \hfill \\ \end{gathered} \right) = \left( {{\mathbf{AV}}^{ - 1} {\mathbf{A}}^{t} } \right)^{ - 1} \hat{\varvec{r}}_{G} \hfill \\ \end{gathered} $$
(9)
Consequently, we obtain the optimal portfolio \( \varvec{x}^{*} \) and the objective value \( \sum\limits_{j = 1}^{n} {\hat{\sigma }_{j}^{2} \left( {x_{j}^{*} } \right)^{2} } = \left( {\varvec{x}^{*} } \right)^{t} {\mathbf{V}\varvec{}x}^{*} \) as follows:
$$ \begin{aligned} \varvec{x}^{*} = & {\mathbf{V}}^{ - 1} {\mathbf{A}}^{t} \left( {{\mathbf{AV}}^{ - 1} {\mathbf{A}}^{t} } \right)^{ - 1} \hat{\varvec{r}}_{G} \\ \left( {\varvec{x}^{*} } \right)^{t} {\mathbf{V}}\varvec{x}^{*} = & \left( {\hat{\varvec{r}}_{G}^{t} \left( {{\mathbf{AV}}^{ - 1} {\mathbf{A}}^{t} } \right)^{ - 1} {\mathbf{AV}}^{ - 1} } \right){\mathbf{V}}\left( {{\mathbf{V}}^{ - 1} {\mathbf{A}}^{t} \left( {{\mathbf{AV}}^{ - 1} {\mathbf{A}}^{t} } \right)^{ - 1} \hat{\varvec{r}}_{G} } \right) \\ \, = & \hat{\varvec{r}}_{G}^{t} \left( {{\mathbf{AV}}^{ - 1} {\mathbf{A}}^{t} } \right)^{ - 1} {\mathbf{AV}}^{ - 1} {\mathbf{A}}^{t} \left( {{\mathbf{AV}}^{ - 1} {\mathbf{A}}^{t} } \right)^{ - 1} \hat{\varvec{r}}_{G} \\ \, = & \hat{\varvec{r}}_{G}^{t} \left( {{\mathbf{AV}}^{ - 1} {\mathbf{A}}^{t} } \right)^{ - 1} \hat{\varvec{r}}_{G} \\ \end{aligned} $$
(10)
From \( x^{*} \) and the optimal objective value, we secondly consider a robust programming-based portfolio model, i.e. the worst case of the total variance:
$$ \begin{gathered} {\text{Maximize }}\hat{\varvec{r}}_{G}^{t} \left( {{\mathbf{AV}}^{ - 1} {\mathbf{A}}^{t} } \right)^{ - 1} \hat{\varvec{r}}_{G} \hfill \\ {\text{subject to }}f_{i} \in [f_{i}^{L} ,f_{i}^{U} ], \, \left( {i = 1,2, \ldots ,m} \right) \hfill \\ \end{gathered} $$
(11)
In objective function \( \hat{\varvec{r}}_{G}^{t} \left( {{\mathbf{AV}}^{ - 1} {\mathbf{A}}^{t} } \right)^{ - 1} \hat{\varvec{r}}_{G} \), inverse matrix \( \left( {{\mathbf{AV}}^{ - 1} {\mathbf{A}}^{t} } \right)^{ - 1} \) is calculated as the following form:
$$ {\mathbf{AV}}^{ - 1} {\mathbf{A}}^{t} = \left( {\begin{array}{*{20}c} {\sum\limits_{j = 1}^{n} {\frac{{\hat{r}_{j}^{2} (\omega )}}{{\hat{\sigma }_{j}^{2} }}} } & {\sum\limits_{j = 1}^{n} {\frac{{\hat{r}_{j} (\omega )}}{{\hat{\sigma }_{j}^{2} }}} } \\ {\sum\limits_{j = 1}^{n} {\frac{{\hat{r}_{j} (\omega )}}{{\hat{\sigma }_{j}^{2} }}} } & {\sum\limits_{j = 1}^{n} {\frac{1}{{\hat{\sigma }_{j}^{2} }}} } \\ \end{array} } \right) $$
$$ \left\{ \begin{gathered} \left( {{\mathbf{AV}}^{ - 1} {\mathbf{A}}^{t} } \right)^{ - 1} = \frac{1}{\Updelta (r)}\left( {\begin{array}{*{20}c} {\sum\limits_{j = 1}^{n} {\frac{1}{{\hat{\sigma }_{j}^{2} }}} } & { - \sum\limits_{j = 1}^{n} {\frac{{\hat{r}_{j} (\omega )}}{{\hat{\sigma }_{j}^{2} }}} } \\ { - \sum\limits_{j = 1}^{n} {\frac{{\hat{r}_{j} (\omega )}}{{\hat{\sigma }_{j}^{2} }}} } & {\sum\limits_{j = 1}^{n} {\frac{{\hat{r}_{j}^{2} (\omega )}}{{\hat{\sigma }_{j}^{2} }}} } \\ \end{array} } \right) \hfill \\ \Updelta (\varvec{r}) = \left( {\sum\limits_{j = 1}^{n} {\frac{1}{{\hat{\sigma }_{j}^{2} }}} } \right)\left( {\sum\limits_{j = 1}^{n} {\frac{{\hat{r}_{j}^{2} (\omega )}}{{\hat{\sigma }_{j}^{2} }}} } \right) - \left( {\sum\limits_{j = 1}^{n} {\frac{{\hat{r}_{j} (\omega )}}{{\hat{\sigma }_{j}^{2} }}} } \right)^{2} \hfill \\ \end{gathered} \right. $$
(12)
Therefore, objective function \( \hat{\varvec{r}}_{G}^{t} \left( {{\mathbf{AV}}^{ - 1} {\mathbf{A}}^{t} } \right)^{ - 1} \hat{\varvec{r}}_{G} \) is also calculated as follows:
$$ \begin{gathered} \hat{\varvec{r}}_{G}^{t} \left( {{\mathbf{AV}}^{ - 1} {\mathbf{A}}^{t} } \right)^{ - 1} \hat{\varvec{r}}_{G} \hfill \\ = \frac{1}{\Updelta (r)}\left( {\sum\limits_{j = 1}^{n} {\frac{{\hat{r}_{j}^{2} (\omega )}}{{\hat{\sigma }_{j}^{2} }}} - 2r_{G} \sum\limits_{j = 1}^{n} {\frac{{\hat{\varvec{r}}_{j} (\omega )}}{{\hat{\sigma }_{j}^{2} }}} + r_{G}^{2} \sum\limits_{j = 1}^{n} {\frac{1}{{\hat{\sigma }_{j}^{2} }}} } \right) \hfill \\ \end{gathered} $$
(13)
Consequently, problem (11) is equivalently transformed into the following problem:
$$ \begin{gathered} {\text{Maximize }}\frac{1}{\Updelta (\varvec{r})}\left( {\sum\limits_{j = 1}^{n} {\frac{{r_{j}^{2} }}{{\hat{\sigma }_{j}^{2} }}} - 2r_{G} \sum\limits_{j = 1}^{n} {\frac{{r_{j} }}{{\hat{\sigma }_{j}^{2} }}} + r_{G}^{2} \sum\limits_{j = 1}^{n} {\frac{1}{{\hat{\sigma }_{j}^{2} }}} } \right) \hfill \\ {\text{subject to }}r_{j} = a_{0j} + \sum\limits_{i = 1}^{m} {a_{ij} f_{i} } , \, f_{i} \in [f_{i}^{L} ,f_{i}^{U} ], \, \left( {i = 1,2, \ldots ,m} \right) \hfill \\ \, \left( {{\text{where }}\Updelta (\varvec{r}) = \left( {\sum\limits_{j = 1}^{n} {\frac{1}{{\hat{\sigma }_{j}^{2} }}} } \right)\left( {\sum\limits_{j = 1}^{n} {\frac{{\hat{r}_{j}^{2} (\omega )}}{{\hat{\sigma }_{j}^{2} }}} } \right) - \left( {\sum\limits_{j = 1}^{n} {\frac{{\hat{r}_{j} (\omega )}}{{\hat{\sigma }_{j}^{2} }}} } \right)^{2} } \right) \hfill \\ \end{gathered} $$
(14)
In this problem, we substitute \( r_{j} = a_{0j} + \sum\limits_{i = 1}^{m} {a_{ij} f_{i} } \) derived from APT, and the numerator and denominator of objective function are calculated as follows:
$$ \begin{aligned} & \sum\limits_{j = 1}^{n} {\frac{{r_{j}^{2} }}{{\hat{\sigma }_{j}^{2} }}} - 2r_{G} \sum\limits_{j = 1}^{n} {\frac{{r_{j} }}{{\hat{\sigma }_{j}^{2} }}} + r_{G}^{2} \sum\limits_{j = 1}^{n} {\frac{1}{{\hat{\sigma }_{j}^{2} }}} \\ & \, = \sum\limits_{h = 1}^{m} {\sum\limits_{i = 1}^{m} {\sum\limits_{j = 1}^{n} {\frac{{a_{hj} a_{ij} f_{h} f_{i} }}{{\hat{\sigma }_{j}^{2} }}} } } - 2\sum\limits_{i = 1}^{m} {\sum\limits_{j = 1}^{n} {\left( {\frac{{a_{ij} f_{i} }}{{\hat{\sigma }_{j}^{2} }}r_{G} + \frac{{a_{0j} a_{ij} f_{i} }}{{\hat{\sigma }_{j}^{2} }}} \right)} } + const. \\ & \, = \sum\limits_{h = 1}^{m} {\sum\limits_{i = 1}^{m} {\sum\limits_{j = 1}^{n} {p_{hij}^{(1)} f_{h} f_{i} } } } - \sum\limits_{i = 1}^{m} {\sum\limits_{j = 1}^{n} {p_{ij}^{(2)} f_{i} } } + p_{c} \\ \end{aligned} $$
$$ \begin{aligned} \Updelta (\varvec{r}) = & \left( {\sum\limits_{j = 1}^{n} {\frac{1}{{\hat{\sigma }_{j}^{2} }}} } \right)\left( {\sum\limits_{h = 1}^{m} {\sum\limits_{i = 1}^{m} {\sum\limits_{j = 1}^{n} {\frac{{a_{hj} a_{ij} f_{h} f_{i} }}{{\hat{\sigma }_{j}^{2} }}} } } } \right) - \left( {\sum\limits_{j = 1}^{n} {\frac{{a_{0j} + \sum\limits_{i = 1}^{m} {a_{ij} f_{i} } }}{{\hat{\sigma }_{j}^{2} }}} } \right)^{2} \\ \, & + \left( {\sum\limits_{j = 1}^{n} {\frac{1}{{\hat{\sigma }_{j}^{2} }}} } \right)\left( {\sum\limits_{i = 1}^{m} {\sum\limits_{j = 1}^{n} {\frac{{2a_{0j} a_{ij} f_{i} }}{{\hat{\sigma }_{j}^{2} }}} } } \right) + const. \\ \, = & \sum\limits_{h = 1}^{m} {\sum\limits_{i = 1}^{m} {\sum\limits_{j = 1}^{n} {q_{hij}^{(1)} f_{h} f_{i} } } } - \sum\limits_{i = 1}^{m} {\sum\limits_{j = 1}^{n} {q_{ij}^{(2)} f_{i} } } + q_{c} \\ \end{aligned} $$
(15)
Therefore, the main problem is represented as the following mathematical programming problem with variable \( f_{i} ,\left( {i = 1,2, \ldots, m} \right) \):
$$ \begin{gathered} {\text{Maximize }}\frac{{\sum\limits_{h = 1}^{m} {\sum\limits_{i = 1}^{m} {\sum\limits_{j = 1}^{n} {p_{hij}^{(1)} f_{h} f_{i} } } } \, - \,\sum\limits_{i = 1}^{m} {\sum\limits_{j = 1}^{n} {p_{ij}^{(2)} f_{i} } } + p_{c} }}{{\sum\limits_{h = 1}^{m} {\sum\limits_{i = 1}^{m} {\sum\limits_{j = 1}^{n} {q_{hij}^{(1)} f_{h} f_{i} } } } \, - \,\sum\limits_{i = 1}^{m} {\sum\limits_{j = 1}^{n} {q_{ij}^{(2)} f_{i} } } + q_{c} }} \hfill \\ {\text{subject to }}f_{i} \in [f_{i}^{L} ,f_{i}^{U} ],\left( {i = 1,2, \ldots ,m} \right) \hfill \\ \end{gathered} $$
(16)

This problem is a nonlinear fractional programming problem, and so it is generally difficult to obtain the optimal solution. However, this problem has variables \( f_{i} ,\left( {i = 1,2, \ldots, m} \right) \), and so we can use the quadratic convex programming approaches, and obtain the strict optimal portfolio.

4 Conclusion

In this paper, we have proposed a robust-based mean-variance portfolio selection problem with random fuzzy APT using a single input type fuzzy reasoning method. In order to deal with each factor in APT as a random interval variable, and to perform the deterministic equivalent transformations, the proposed model has been nonlinear programming problem with only one variable. Therefore, we have obtained the exact optimal portfolio using standard nonlinear programming approaches.

As future studies, we need to develop the solution algorithm in cases of general fuzzy numbers including interval values. Furthermore, we also need to consider random fuzzy portfolio models derived from not only a single input type fuzzy reasoning method but also more general fuzzy reasoning methods.

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Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  • Takashi Hasuike
    • 1
  • Hideki Katagiri
    • 2
  • Hiroshi Tsuda
    • 3
  1. 1.Graduate School of Information Science and TechnologyOsaka UniversitySuitaJapan
  2. 2.Graduate School of EngineeringHiroshima UniversityHigashi-HiroshimaJapan
  3. 3.Department of Mathematical Sciences, Faculty of Science and EngineeringDoshisha UniversityKyotanabeJapan

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