Keywords

1 Introduction

Portfolio optimization (PO) problem has been a practically important challenge and issue in financial markets ((Markowitz 1952); (Lobo et al. 2007); (Kalayci et al. 2019); (Ahmadi-Javid and Fallah-Tafti 2019)). PO problem is concerned with choosing an optimal and efficient portfolio strategy that can strike a trade-off between maximizing investment return and minimizing investment risk ((Konno and Suzuki 1995); (Cesarone et al. 2013); (Björk et al. 2014); (Guo et al. 2019); (Salah et al. 2020)). The important point that should be considered for modeling investment PO problem is the uncertainty of financial data ((Ghahtarani and Najafi 2013, 2018); (Huang 2017); (Peykani and Mohammadi 2018); (Li et al. 2019); (Peykani et al. 2020)). It should be noted that for proposing uncertain portfolio optimization (UPO) models, according to the nature and type of uncertainty, the popular uncertain programming approaches such as stochastic optimization (SO) ((Land et al. 1993); (Sueyoshi 2000); (Cooper et al. 2002); (Wu et al. 2013); (Zha et al. 2016)), fuzzy optimization (FO) ((Zadeh 1978); (Azadeh and Kokabi 2016); (Peykani et al. 2018a, 2019a, 2019b, 2021); (Peykani and Mohammadi 2018); (Seyed Esmaeili et al. 2019); (Peykani and Gheidar-Kheljani 2020)), and robust optimization (RO) ((Soyster 1973); (Ben-Tal and Nemirovski 2000); (Bertsimas and Sim 2004); (Ghassemi et al. 2017); (Namakshenas et al. 2017); (Ghassemi 2019); (Namakshenas and Pishvaee 2019); (Peykani and Roghanian 2015); (Peykani and Mohammadi 2018); (Peykani et al. 2018, 2019, 2019d; Peykani et al. 2020)) can be utilized.

The goal of this paper is to present the possibilistic portfolio optimization (PPO) model that is capable to be implemented under fuzzy data. It should be explained that a possibilistic programming approach is employed for dealing with the uncertainty and ambiguity of financial data. The possibilistic programming is an applicable and powerful approach for dealing with the epistemic uncertainty that is caused by the absence or lack of knowledge about the exact value of model parameters in fuzzy mathematical programming (FPP) (Naderi et al., 2016). Notably, to demonstrate the applicability of the proposed fuzzy portfolio optimization model, the PPO approach is implemented in a Tehran stock exchange (TSE).

The rest of this paper is organized as follows. The modeling of the deterministic portfolio optimization model will be explained in Sect. 2. Then, the possibilistic portfolio optimization model using possibilistic programming will be proposed in Sect. 3. The proposed PPO model is employed for the sample test from the Tehran stock exchange in Sect. 4. Finally, conclusions, as well as some future research directions, will be introduced in Sect. 5.

2 Deterministic Portfolio Optimization Model

In this section, the deterministic portfolio optimization (DPO) model will be introduced. It should be noted that three aspects of stock including the rate of return, the systematic risk, and the non-systematic risk are considered in the DPO model. The nomenclatures of the paper are introduced as follows:

\(j\)

the indices of stocks \(j = 1,...,n\)

\(t\)

the indices of periods \(t = 1,...,T\)

\(R_{E}\)

the expected return of portfolio

\(R_{j}\)

the average return of jth stock

\(R_{tj}\)

the return of jth stock in tth period

\(R_{M}\)

the return of the market

\(\sigma_{M}^{2}\)

the variance of the market

\(\beta_{E}\)

the expected beta of the portfolio

\(\beta_{j}\)

the beta of jth stock

\(A_{j}\)

the minimum level of total fund which can be invested in the jth stock

\(B_{j}\)

the maximum level of total fund which can be invested in the jth stock

\(\varpi_{j}\)

the weight of jth stock in the portfolio

\(\Omega_{t}\)

the value of the non-systematic risk of the portfolio in tth period

\(\tau_{j}\)

a binary variable which will be one if jth stock is selected and zero otherwise

Investment risk can be decomposed into two components including systematic risk and non-systematic risk. Systematic risk includes that part of the risk which depends on market variability and is unavoidable.

Beta \((\beta )\) sensitivity coefficient is one of the most popular systematic risk measures. The beta coefficient describes the sensitivity of the share return to the market portfolio return. In other words, beta is a measure of the volatility of share with the overall market and is obtained from Eq. (1) as follows:

$$\beta = \frac{{Cov\left( {R_{j} ,{ }R_{M} } \right)}}{{\sigma _{M}^{2} }}$$
(1)

Non-systematic risk indicates that a part of the investment risk can be eliminated by diversification. The absolute deviation (AD) is a non-systematic risk measure introduced by Konno & Yamazaki ((Konno and Yamazaki 1991)) for the first time. The definition of absolute deviation is as given in Eq. (2):

$$AD = \left| {R_{j} - R_{E} } \right| = \left\{ \begin{gathered} R_{j} - R_{E} { }if{ }R_{j} > R_{E} { }; \hfill \\ R_{E} - R_{j} { }if{ }R_{j} \le R_{E} { }. \hfill \\ \end{gathered} \right.$$
(2)

Now, the mean-absolute deviation-beta (MADB) model for portfolio optimization problem is presented as Model (3):

$${\text{Min}}\frac{1}{{\text{T}}}\sum\limits_{{t = 1}}^{T} {\Omega _{t} }$$
(3)
$${\text{S.t.}}\sum\limits_{{j = 1}}^{n} {R_{j} \varpi _{j} } \ge R_{E}$$
$$R_{E} - \sum\limits_{{j = 1}}^{n} {R_{{tj}} \varpi _{j} } \le \Omega _{t} { },{ }\forall t$$
$$\sum\limits_{{j = 1}}^{n} {R_{{tj}} \varpi _{j} } - R_{E} \le { }\Omega _{t} { },{ }\forall t$$
$$\sum\limits_{{j = 1}}^{n} {\beta _{j} \varpi _{j} } \le \beta _{E}$$
$$\sum\limits_{{j = 1}}^{n} {\varpi _{j} } = 1$$
$$A_{j} \tau _{j} \le \varpi _{j} \le B_{j} \tau _{j} { },{ }\forall j$$
$$\tau _{j} \in \{ 0,1\} { },{ }\forall j$$
$$\Omega _{t} \ge 0{ },{ }\forall t$$
$$\varpi _{j} \ge 0{ },{ }\forall j$$

It should be noted that in Model (3), \(A_{j}\) and \(B_{j}\) are the lower and the upper bounds, respectively, for each stock and \(\varpi_{j}\) is the portion of the investment portfolio that is assigned to each stock. Accordingly, the limit of investment for each stock as a common financial market constraint is considered in the proposed portfolio optimization model.

3 Possibilistic Portfolio Optimization Model

In this section, the possibilistic portfolio optimization (PPO) model under fuzzy data will be presented. It should be explained that the return and the beta have a trapezoidal fuzzy distribution \(\tilde{R}{ }(R^{1} ,R^{2} ,R^{3} ,R^{4} )\) and \(\tilde{\beta }{ }(\beta^{1} ,\beta^{2} ,\beta^{3} ,\beta^{4} )\) with the condition of \(R^{1} < R^{2} < R^{3} < R^{4}\) and \(\beta^{1} < \beta^{2} < \beta^{3} < \beta^{4}\). Now, a possibilistic programming approach and chance-constrained programming (CCP) will be applied to deal with fuzzy data in the MADB model as follows:

$${\text{Min}}\frac{1}{{\text{T}}}\sum\limits_{{t = 1}}^{T} {\Omega _{t} }$$
(4)
$${\text{S.t.}}\,{\text{Pos}}\left\{ {\sum\limits_{{j = 1}}^{n} {\tilde{R}_{j} \varpi _{j} } \ge R_{E} } \right\} \ge \delta$$
$${\text{Pos}}\left\{ {R_{E} - \sum\limits_{{j = 1}}^{n} {\tilde{R}_{{tj}} \varpi _{j} } \le \Omega _{t} } \right\} \ge \delta { },{ }\forall t$$
$${\text{Pos}}{ }\left\{ {\sum\limits_{{j = 1}}^{n} {\tilde{R}_{{tj}} \varpi _{j} } - R_{E} \le { }\Omega _{t} } \right\} \ge \delta { },{ }\forall t$$
$${\text{Pos}}{ }\left\{ {\sum\limits_{{j = 1}}^{n} {\tilde{\beta }_{j} \varpi _{j} } \le \beta _{E} } \right\} \ge \delta$$
$$\sum\limits_{{j = 1}}^{n} {\varpi _{j} } = 1$$
$$A_{j} \tau _{j} \le \varpi _{j} \le B_{j} \tau _{j} { },{ }\forall j$$
$$\tau _{j} \in \{ 0,1\} { },{ }\forall j$$
$$\Omega _{t} \ge 0{ },{ }\forall t$$
$$\varpi _{j} \ge 0{ },{ }\forall j$$

Then, by applying the possibility measure and chance-constrained programming, converting fuzzy chance-constraints into their equivalent crisp ones in one special confidence level \((\delta )\) is done as follows:

\({\text{Min}}{ }\frac{{1}}{{\text{T}}} \, \sum\limits_{t = 1}^{T} { \, \Omega_{t} }\)

(5)

\({\text{S.t.}}\sum\limits_{j = 1}^{n} {\left( {\left( \delta \right)R_{j}^{3} + \left( {1 - \delta } \right)R_{j}^{4} } \right)\varpi_{j} } \ge R_{E}\)

 

\(R_{E} - \sum\limits_{j = 1}^{n} {\left( {\left( \delta \right)R_{tj}^{3} + \left( {1 - \delta } \right)R_{tj}^{4} } \right)\varpi_{j} } \le \Omega_{t} { },{ }\forall t\)

 

\(\sum\limits_{j = 1}^{n} {\left( {\left( {1 - \delta } \right)R_{tj}^{1} + \left( \delta \right)R_{jt}^{2} } \right)\varpi_{j} } - R_{E} \le { }\Omega_{t} { },{ }\forall t\)

 

\(\sum\limits_{j = 1}^{n} {\left( {\left( {1 - \delta } \right)\beta_{j}^{1} + \left( \delta \right)\beta_{j}^{2} } \right)\varpi_{j} } \le \beta_{E}\)

 

\(\sum\limits_{j = 1}^{n} {\varpi_{j} } = 1\)

 

\(A_{j} \tau_{j} \le \varpi_{j} \le B_{j} \tau_{j} { },{ }\forall j\)

 

\(\tau_{j} \in \{ 0,1\} { },{ }\forall j\)

 

\(\Omega_{t} \ge 0{ },{ }\forall t\)

 

\(\varpi_{j} \ge 0{ },{ }\forall j\)

 

Finally, the possibilistic mean-absolute deviation-beta (PMADB) model is proposed as Model (5) that can be employed by investors for portfolio optimization under fuzzy data and linguistic variables.

4 Experimental Results

In this section, the possibilistic portfolio optimization model will be implemented for a real-world case study from the Tehran stock exchange. Accordingly, the data set for five stocks are extracted from TSE. Tables 1, 2, 3, 4, and 5 show the data set for beta and return of five stocks under trapezoidal fuzzy number:

Table 1 Fuzzy data set for beta
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Table 2 Fuzzy data set for monthly return—first period
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Table 3 Fuzzy Data Set for Monthly Return—Second Period
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Table 4 Fuzzy data set for monthly return—third period
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Now, after collecting data, the possibilistic mean-absolute deviation-beta model will be run. The results of the PMADB model that is presented in Model (5) for five confidence levels including 0, 25, 50, 75, and 100% are introduced in Table 6:

Table 5 Fuzzy data set for monthly return—average
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Table 6 The results of the PMADB model
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As can be seen in Table 6, by increasing the confidence level from 0 to 100%, the objective functions including mean, absolute deviation, and beta get worse. Also, illustrative results show that the proposed PMADB model is effective for portfolio optimization in the presence of fuzzy data.

5 Conclusions

In this study, an uncertain portfolio optimization model is presented that is capable to be used under fuzzy environment. It should be explained that in the proposed portfolio optimization model, three objectives including mean, absolute deviation, and beta as well as investment constraint are considered. Also, the possibilistic programming and chance-constrained programming approaches are employed to deal with uncertainty. For future studies, data envelopment analysis approach (Seyed Esmaeili, 2014; Peykani et al., 2018c; Peykani and Mohammadi , 2018, 2019, 2020; Seyed Esmaeili and Rostamy-Malkhalifeh , 2018), machine learning models (Park et al., 2014; Ban et al., 2018; Shahhosseini et al., 2019, 2020, 2019; Paiva et al., 2019), and game theory ( Migdalas, 2002; Sadeghi and Zandieh, 2011; Esmaeili et al., 2015) can be applied for presenting investment portfolio optimization approach.