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

## 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

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

*i*th 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

*s*th 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

*s*th 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:

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.

- \( \bar{\tilde{r}} \)
Future return of the

*j*th 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.*r*_{G}Target total return

*n*Total number of securities

*x*_{j}Budgeting allocation to the

*j*th security

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.

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.

### References

- 1.Elton EJ, Gruber MJ (1995) Modern portfolio theory and investment analysis. Wiley, New YorkGoogle Scholar
- 2.Goldfarb D, Iyengar G (2003) Robust portfolio selection problems. Math Operat Res 28:1–38MathSciNetMATHCrossRefGoogle Scholar
- 3.Hasuike T, Katagiri H, Ishii H (2009) Multiobjective random fuzzy linear programming problems based on the possibility maximization model. J Adv Comput Intell Intellt Inform 13(4):373–379Google Scholar
- 4.Hasuike T, Katagiri H, Ishii H (2009) Portfolio selection problems with random fuzzy variable returns. Fuzzy Sets Syst 160:2579–2596MathSciNetMATHCrossRefGoogle Scholar
- 5.Hasuike T, Katagiri H, Tsuda H (2012) Robust-based random fuzzy mean-variance model using a fuzzy reasoning method. In: Lecture notes in engineering and computer science: proceedings of the international multi conference of engineers and computer scientists 2012, IMECS 2012, 14–16 March, 2012, Hong Kong, pp 1461–1466Google Scholar
- 6.Hayashi K, Otsubo A, Shiranita K (1999) Realization of nonlinear and linear PID control using simplified direct inference method. IEICE Trans Fundam (Japanese Edition), J82-A(7), pp 1180–1184Google Scholar
- 7.Hayashi K, Otsubo A, Shirahata K (2001) Improvement of conventional method of PI fuzzy control. IEICE Trans Fundam E84-A(6), pp 1588–1592Google Scholar
- 8.Huang X (2007) Two new models for portfolio selection with stochastic returns taking fuzzy information. Eur J Oper Res 180:396–405MATHCrossRefGoogle Scholar
- 9.Inuiguchi M, Ramik J (2000) Possibilisitc linear programming: A brief review of fuzzy mathematical programming and a comparison with stochastic programming in portfolio selection problem. Fuzzy Sets Syst 111:3–28MathSciNetMATHCrossRefGoogle Scholar
- 10.Katagiri H, Hasuike T, Ishii H, Nishizaki I (2008) Random fuzzy programming models based on possibilistic programming. In:Proceedings of the 2008 IEEE international conference on systems, man and cybernetics (to appear)Google Scholar
- 11.Katagiri H, Ishii H, Sakawa M (2004) On fuzzy random linear knapsack problems. CEJOR 12:59–70MathSciNetMATHGoogle Scholar
- 12.Katagiri H, Sakawa M, Ishii H (2005) A study on fuzzy random portfolio selection problems using possibility and necessity measures. Scientiae Mathematicae Japonocae 65:361–369MathSciNetGoogle Scholar
- 13.Konno H, Yamazaki H (1991) Mean-absolute deviation portfolio optimization model and its applications to Tokyo stock market. Manage Sci 37:519–531CrossRefGoogle Scholar
- 14.Konno H, Shirakawa H, Yamazaki H (1993) A mean-absolute deviation-skewness portfolio optimization model. Ann Oper Res 45:205–220MathSciNetMATHCrossRefGoogle Scholar
- 15.Leon RT, Liern V, Vercher E (2002) Validity of infeasible portfolio selection problems: fuzzy approach. Eur J Oper Res 139:178–189MATHCrossRefGoogle Scholar
- 16.Lintner BJ (1965) Valuation of risky assets and the selection of risky investments in stock portfolios and capital budgets. Rev Econ Stat 47:13–37CrossRefGoogle Scholar
- 17.Liu B (2002) Theory and practice of uncertain programming. Physica Verlag, HeidelbergGoogle Scholar
- 18.Lobo MS (2000) Robust and convex optimization with applications in finance. Doctor thesis of the department of Electrical engineering and the committee on graduate studies, Stanford University, StanfordGoogle Scholar
- 19.Luenberger DG (1997) Investment science, Oxford University Press, OxfordGoogle Scholar
- 20.Markowitz HM (1952) Portfolio selection. J Financ 7(1):77–91Google Scholar
- 21.Mamdani EH (1974) Application of fuzzy algorithms for control of simple dynamic plant. Proc IEE 121(12):1585–1588Google Scholar
- 22.Mossin J (1966) Equilibrium in capital asset markets. Econometrica 34(4):768–783CrossRefGoogle Scholar
- 23.Rockafellar RT, Uryasev S (2000) Optimization of conditional value-at-risk. J Risk 2(3):1–21Google Scholar
- 24.Ross S (1976) The arbitrage theory of capital asset pricing. J Econ Theory 13(3):341–360CrossRefGoogle Scholar
- 25.Sharpe WF (1964) Capital asset prices: A theory of market equivalent under conditions of risk. J Financ 19(3):425–442MathSciNetGoogle Scholar
- 26.Takagi T, Sugeno M (1985) Fuzzy identification of systems and its applications to modeling and control. IEEE Trans Syst Man Cybern SMC-15(1), pp 116–132Google Scholar
- 27.Tanaka H, Guo P (1999) Portfolio selection based on upper and lower exponential possibility distributions. Eur J Oper Res 114:115–126MATHCrossRefGoogle Scholar
- 28.Tanaka H, Guo P, Turksen IB (2000) Portfolio selection based on fuzzy probabilities and possibility distributions. Fuzzy Sets Syst 111:387–397MathSciNetMATHCrossRefGoogle Scholar
- 29.Vercher E, Bermúdez JD, Segura JV (2007) Fuzzy portfolio optimization under downside risk measures. Fuzzy Sets Syst 158:769–782MATHCrossRefGoogle Scholar
- 30.Watada J (1997) Fuzzy portfolio selection and its applications to decision making. Tatra Mountains Math Pub 13:219–248MathSciNetMATHGoogle Scholar