Prospect theory-based formulation of chance constrained portfolio optimization problem using loan

Abstract

Portfolio optimization that allows the borrowed money from a loan to be invested in risk assets has been formulated as a chance constrained problem. In this paper, in order to reflect investor preferences called “values” in the solution, or the portfolio, prospect theory is used to expand the chance constrained problem that maximizes the profit. The new portfolio optimization problem maximizes the value instead of the profit. In order to get a deeper understanding of the characteristics of the new portfolio optimization problem, the fitness landscape analysis method using Convex-Hull Mapping (CHM) is employed. Furthermore, since the new portfolio optimization problem is usually non-convex, the multi-start local search method with CHM is used to find the solution. Finally, numerical experiments on artificial and actual asset data sets show that the solution of the new portfolio optimization problem is more acceptable to investors than the solution of the conventional one.

Keywords:

• Portfolio optimization problem
• chance constrained problem
• optimization method
• constraint-handling
• prospect theory

1. Introduction

Portfolio optimization is the process of determining the best proportion of investment in different assets according to some objective. Since portfolio optimization is one of the most challenging problems in the field of finance, a large number of works about portfolio optimization have been reported [1–11]. In these works, in order to maximize profits such as expected return or minimize costs such as financial risk, portfolio optimization has been formulated as various optimization problems.

In recent years, the economic stimulus package known as “low-interest monetary policy” has kept interest rates of bank deposits and loans very low in Japan. Thus, we have been studying the portfolio optimization that allows the borrowed money from a loan to be invested in risk assets. In our previous work [12,13], the portfolio optimization using loan has been formulated as a chance constrained problem that maximizes the profit. Then it has been proven that the portfolio optimization problem using loan is the convex optimization problem, and the optimal solution, or the best portfolio, is either not to borrow money at all or to borrow money as much as possible.

Prospect theory is a theory of behavioural economics and decision making that was developed by two psychologists: Kahneman and Tversky [14,15]. The theory provides the best available description of how people evaluate risk under uncertainties. Therefore, prospect theory has been used to formulate several portfolio optimization problems [16–21]. Prospect theory can be summarized by two functions: the value function and the probability weighting function. The former provides a nonlinear mapping from profits to human preferences called “values.” The latter provides a mapping from objective probabilities to weighted probabilities.

By using the two functions of prospect theory, this paper expands the conventional portfolio optimization problem using loan [12,13] to reflect investor preferences, or “values,” in the solution. Specifically, the new portfolio optimization problem using loan maximizes the value, while the conventional one maximizes the profit.

In order to get a deeper understanding of the characteristics of the new portfolio optimization problem in comparison with the conventional one, two methods previously proposed by the authors are used. The first one is the fitness landscape analysis method using Convex-Hull Mapping (CHM) [22]. The second one is the multi-start local search method with CHM [23]. The constraint-handling technique called CHM [24] generates a set of feasible solutions randomly. By using the two methods, the relationship between the profit and the value of the portfolio using loan is examined on artificial and actual asset data sets. As a result, it is revealed that the solution of the new portfolio optimization problem is more acceptable to investors.

This paper is an extended version of the paper presented in the SICE Annual Conference 2023 [22] and differs from the paper in the following four points.

1. The advantages of the proposed prospect theory-based portfolio optimization problem over traditional prospect theory-based ones are discussed.

2. The effects of the interest rate and upper limit of the loan on the solution of the new prospect theory-based portfolio optimization problem are evaluated.

3. The multi-start local search method [23] is addressed in detail.

4. The proposed methodology is applied to actual asset data of Japanese stock market (TOPIX Large 70) and validated.

The goal of this paper is to clarify that prospect theory can be introduced as a worth-while modification in the conventional portfolio optimization problem using loan. Specifically, the conventional solution that maximizes the profit of the portfolio may diminish its value. Such a solution would not be acceptable to risk-averse investors and would not lead to their investment behaviour. On the other hand, the new solution that maximizes the value of the portfolio always increases its profit as well.

The remainder of this paper is organized as follows. Previous studies about prospect theory-based portfolio optimization problems are presented in Section 2. Then the uniqueness and advantages of the proposed problem formulation based on prospect theory are discussed. Portfolio optimization problem using loan is formulated as a chance constrained problem that maximizes the profit in Section 3. Then the chance constrained problem is transformed into a deterministic optimization problem without the chance constraint in Section 4. Section 5 explains the two functions of prospect theory: the value one and the probability weighting one. Section 6 formulates the prospect theory-based portfolio optimization problem using loan that maximizes the value of the portfolio instead of the profit. Section 7 explains the fitness landscape analysis method and the multi-start local search method using CHM. The results of numerical experiments on artificial asset data are shown in Section 8. The proposed methodology is applied to actual asset data from Japanese stock market and validated in Section 9. Section 10 concludes this paper and discusses future work.

2. Related works

Portfolio optimization is the process of determining the best portion of investment in different risk assets according to some objective. As mentioned above, since portfolio optimization is one of the most challenging problems in the field of finance [8], a large number of problem formulations have been reported [1–11]. Generally speaking, the traditional problem formulations of portfolio optimization can be divided into two groups: deterministic formulations and stochastic formulations.

The best-known Markowitz's model [1] is deterministic. In Markowitz's model, the risk of the portfolio is evaluated by the variance of returns. Then the risk is minimized under the constraint about the expected return. Considering the trade-off between the risk and the return, Markowitz's model has been expanded in various ways and many deterministic formulations of portfolio optimization have been reported [5,7]. Furthermore, in order to solve deterministic portfolio optimization problems, not only optimization algorithms of mathematical programming but also heuristic methods such as evolutionary algorithms [6,11,24] have been used effectively.

Kataoka's model [2] is the basic formulation of stochastic ones. In Kataoka's model, portfolio optimization is formulated as a chance constrained problem. Then the risk of the portfolio is evaluated by the failure probability that the return falls below a desired value. By assessing risk in terms of probability rather than variance, the investor can get a concrete picture of risk. Furthermore, the investor can compare multiple portfolios with different failure probabilities. On the other hand, due to chance constraints, it is usually hard to solve stochastic portfolio optimization problems.

We have expanded Kataoka's model to formulate the portfolio optimization using loan as a chance constrained problem [12,13]. Even though various problem formulations have been reported for portfolio optimization, authors' model that allows the borrowed money from a loan to be invested in risk assets is very unique.

Chance constrained problems, which are also referred to as probabilistic constrained problems, have been studied in the field of stochastic programming for many years [25]. In fact, many real-world optimization problems that consider uncertainties to obtain realistic solutions have been formulated as chance constrained problems [26].

Chance constrained problems can be divided into two groups depending on how uncertainty is given. In the first group, uncertainty is described by a mathematical model. Well-known probability distributions, such as normal distribution, are often used for the mathematical model [27,28]. Knowledge of the mathematical model is used effectively not only in the problem formulation but also for solving chance constrained problems. In the second group, uncertainty is given by a set of samples, also called scenarios. The set of samples is usually generated randomly using the Monte Carlo method [29–32]. In real-world optimization problems, observed data are often used as samples [26,33]. In the second group of chance constrained problems, the number of samples is important to guarantee the accuracy of the solution theoretically [34].

Chance constraints are also classified into two groups: individual chance constraints and joint chance constraints. For multiple constraints with uncertainty, the individual chance constrained problem specifies the probability that each of constraints will be satisfied separately. On the other hand, the joint chance constrained problem specifies the probability that all constraints will be satisfied simultaneously. Generally speaking, the joint chance constrained problem is far more difficult than the individual chance constrained problem. Therefore, heuristic methods such as evolutionary algorithms are also used for solving joint chance constrained problems practically [35,36].

The chance constrained problem targeted in this paper specifies the probability that only one constraint will be satisfied. The mathematical model of uncertainty is also given by a normal distribution. That is because the normal distribution is widely used as a mathematical model of asset returns in the field of financial engineering [3]. As a result, by using the distribution function of the standard normal distribution, the chance constrained problem is transformed into a simple deterministic equivalent problem [25]. Then an existing mathematical programming solver is used to obtain the solution of the chance constrained problem targeted in this paper. However, in terms of applications of chance constrained optimization, the proposed problem formulation taking into account human preferences will be applicable to many real-world problems other than the portfolio optimization. Furthermore, probability weighting based on prospect theory has not been reported to the beat of the authors' knowledge.

As mentioned above, prospect theory is an essential theory of behavioural economics and decision making that was developed by two psychologists in 1979 [14]. More than 40 years later, prospect theory is still widely viewed as the best available description of how people evaluate risk. Even though prospect theory has been employed in many applications where attitudes toward risk play a central role [37–39], the theory has been most extensively applied to the fields of finance and insurance [40].

In fact, several problem formulations based on prospect theory have been reported for portfolio optimization [17–21]. The only commonality among the conventional works is that two functions of prospect theory, namely the value function and the probability weighting function, are used in their problem formulations. In portfolio optimization problems, asset returns are regarded as random variables. The return of the portfolio is also a random variable. Some prospect theory-based formulations modify each of asset returns by the two functions [21], while others modify the return of the portfolio by them [20]. Then the investor's utility called “prospect” is calculated from the products of values and weighted probabilities in various ways.

There are three drawbacks to the traditional prospect theory-based formulations of portfolio optimization: (1) It is not realistic for an investor to fully understand and weight the probability distribution of asset returns; (2) The size of the calculated prospect is not directly related to the investor's investment behaviour; (3) Since the optimal solution is uniquely determined as with deterministic portfolio optimization problems, the investor cannot compare multiple solutions with different risks.

The prospect theory-based portfolio optimization problem proposed in this paper is derived from the above chance constrained problem. Therefore, the investor can choose any failure probability that is worth investing in. Then only the acceptable failure probability is converted by the probability weighting function. Therefore, the investor does not need to reassess the probabilities of many asset returns. Since the borrowed money from a loan is invested in risk assets, the loan amount is valuated properly as a loss by the value function. On the other hand, “prospect” is not calculated explicitly in the proposed prospect theory-based portfolio optimization problem.

3. Portfolio optimization using loan

3.1. Definition of portfolio and loan

We invest money in n risk assets. Let $x_i\in \mathbb{R}$, $i=1,\dots ,n$ be the amount of money invested in i-asset. The portfolio is defined as $\mathit{x}=(x_1,\dots ,x_n)$. Since none of the assets in the portfolio $\mathit{x}\in {\mathbb{R}}^n$ will be sold over a single-period, $x_i\ge 0$, $i=1,\dots ,n$ holds. Incidentally, such a portfolio $\mathit{x}\in {\mathbb{R}}^n$ is called the long-only one [4]. Let $x_0\in \mathbb{R}$, $x_0\le 0$ be an amount of borrowing money from a loan. The upper limit of the loan is $m\in \mathbb{R}$, m>0. The loan is not used if $x_0=0$. The loan is used up to the limit if $x_0=-m$. In this paper, all amounts are normalized by own funds. Therefore, constraints for the portfolio $\mathit{x}=(x_1,\dots ,x_n)\in {\mathbb{R}}^n$ considering the loan amount $x_0\in \mathbb{R}$ are (1) $(\begin{array}{l}{x}_0+x_1+x_2+\cdots +x_n=1,\\ -m\le x_0\le 0,0\le x_i,i=1,\dots ,n.\end{array}$(1) For example, if we have 1000USD as own funds to invest, $x_i=0.2$ and $x_i=1.3$ correspond, respectively, to 200USD and 1300USD. If the upper limit of the loan is given as m = 2, we can borrow up to 2000USD. Furthermore, if the loan amount is $x_0=-2$, the total investment will increase to 3000USD.

From constraints in (1(1) $(\begin{array}{l}{x}_0+x_1+x_2+\cdots +x_n=1,\\ -m\le x_0\le 0,0\le x_i,i=1,\dots ,n.\end{array}$(1) ), the loan amount $x_0\in \mathbb{R}$ for the portfolio $\mathit{x}\in {\mathbb{R}}^n$ is (2) $x_0=1-\text{𝟙}{\mathit{x}}^T$(2) where $\text{𝟙}=(1,\dots ,1)\in {\mathbb{R}}^n$ is a constant vector.

One unit investment in i-asset yields a return ${\xi }_i\in \mathbb{R}$. From the efficient-market hypothesis [3], we can suppose that each of asset returns ${\xi }_i\in \mathbb{R}$, $i=1,\dots ,n$ is normally distributed with mean ${\mu }_i$ and variance ${\sigma }_i^2$ as (3) ${\xi }_i\sim \text{Normal}({\mu }_i,{\sigma }_i^2).$(3) If the mean of asset returns in (3(3) ${\xi }_i\sim \text{Normal}({\mu }_i,{\sigma }_i^2).$(3) ) is ${\mu }_i\le 0$ for an asset, the asset is not worth investing in. Therefore, we assume that ${\mu }_i>0$, $i=1,\dots ,n$ hold. From (3(3) ${\xi }_i\sim \text{Normal}({\mu }_i,{\sigma }_i^2).$(3) ), the vector $\mathit{\xi }=({\xi }_1,\dots ,{\xi }_n)\in {\mathbb{R}}^n$ of ${\xi }_i\in \mathbb{R}$ obeys a multivariable normal distribution as (4) $\mathit{\xi }\sim \text{Normal}(\mathit{\mu },\mathit{C})$(4) where the mean is given as $\mathit{\mu }=({\mu }_1,\dots ,{\mu }_n)\in {\mathbb{R}}^n$.

The covariance matrix $\mathit{C}$ in (4(4) $\mathit{\xi }\sim \text{Normal}(\mathit{\mu },\mathit{C})$(4) ) is derived as follows. From standard deviations ${\sigma }_i$, $i=1,\dots ,n$ in (3(3) ${\xi }_i\sim \text{Normal}({\mu }_i,{\sigma }_i^2).$(3) ), the diagonal matrix $\mathit{D}$ is defined as (5) $\mathit{D}=\left(\begin{array}{cccc}{\sigma }_1& 0& \cdots & 0\\ 0& {\sigma }_2& \cdots & 0\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \cdots & {\sigma }_n\end{array}\right).$(5) From correlation coefficients ${\rho }_{\mathrm{ij}}\in [-1,1]$, $i,j=1,\dots ,n$ between ${\xi }_i$ and ${\xi }_j$, the coefficient matrix $\mathit{R}$ is defined as (6) $\mathit{R}=\left(\begin{array}{cccc}1& {\rho }_{12}& \cdots & {\rho }_{1n}\\ {\rho }_{21}& 1& \cdots & {\rho }_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ {\rho }_{n1}& {\rho }_{n2}& \cdots & 1\end{array}\right).$(6) From $\mathit{D}$ in (5(5) $\mathit{D}=\left(\begin{array}{cccc}{\sigma }_1& 0& \cdots & 0\\ 0& {\sigma }_2& \cdots & 0\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \cdots & {\sigma }_n\end{array}\right).$(5) ) and $\mathit{R}$ in (6(6) $\mathit{R}=\left(\begin{array}{cccc}1& {\rho }_{12}& \cdots & {\rho }_{1n}\\ {\rho }_{21}& 1& \cdots & {\rho }_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ {\rho }_{n1}& {\rho }_{n2}& \cdots & 1\end{array}\right).$(6) ), the covariance matrix $\mathit{C}$ in (4(4) $\mathit{\xi }\sim \text{Normal}(\mathit{\mu },\mathit{C})$(4) ) is obtained as (7) $\mathit{C}=\mathit{DRD}.$(7) Let $L\in \mathbb{R}$, $L\ge 0$ be the fixed interest rate of the loan. Then, from the loan amount $x_0\in \mathbb{R}$ in (2(2) $x_0=1-\text{𝟙}{\mathit{x}}^T$(2) ) and $\mathit{\xi }\in {\mathbb{R}}^n$ in (4(4) $\mathit{\xi }\sim \text{Normal}(\mathit{\mu },\mathit{C})$(4) ), the return of the portfolio $\mathit{x}\in {\mathbb{R}}^n$ is (8) $r(\mathit{x},\mathit{\xi })=\mathit{\xi }{\mathit{x}}^T+L{x}_0=\mathit{\xi }{\mathit{x}}^T+L(1-\text{𝟙}{\mathit{x}}^T)$(8) where the return $r(\mathit{x},\mathit{\xi })\in \mathbb{R}$ does not contain $x_0\in \mathbb{R}$ explicitly.

The return in (8(8) $r(\mathit{x},\mathit{\xi })=\mathit{\xi }{\mathit{x}}^T+L{x}_0=\mathit{\xi }{\mathit{x}}^T+L(1-\text{𝟙}{\mathit{x}}^T)$(8) ) also obeys a normal distribution as (9) $r(\mathit{x},\mathit{\xi })\sim \text{Normal}\left(\mu \right(\mathit{x}),\sigma (\mathit{x}{)}^2)$(9) where $\mu \left(\mathit{x}\right)=\mathit{\mu }{\mathit{x}}^T+L{x}_0=\mathit{\mu }{\mathit{x}}^T+L(1-\text{𝟙}{\mathit{x}}^T)$ and ${\sigma }^2\left(\mathit{x}\right)=\mathit{xC}{\mathit{x}}^T$.

3.2. Formulation of POL

From among various problem formulations contrived for the portfolio optimization [5,8], we have adopted a stochastic one known as Kataoka's model [2]. In other words, Portfolio Optimization problem using Loan (POL) [12,13] is an expanded variant of Kataoka's model.

The risk of the portfolio $\mathit{x}\in {\mathbb{R}}^n$ is evaluated by the probability that the return $r(\mathit{x},\mathit{\xi })$ in (8(8) $r(\mathit{x},\mathit{\xi })=\mathit{\xi }{\mathit{x}}^T+L{x}_0=\mathit{\xi }{\mathit{x}}^T+L(1-\text{𝟙}{\mathit{x}}^T)$(8) ) falls below a desired value $\gamma \in \mathbb{R}$. Let $Pr\left(\mathcal{A}\right)$ be the probability that event $\mathcal{A}$ occurs. Let $\alpha \in (0,0.5)$ be the failure probability that is acceptable to the investor. Then POL is formulated as a chance constrained problem: (10) $[\begin{array}{rl}max& \gamma \\ \text{sub. to}& Pr\left(r\right(\mathit{x},\mathit{\xi })\le \gamma )\le \alpha ,\\ & x_1+x_2+\cdots +x_n=\text{𝟙}{\mathit{x}}^T\ge 1,\\ & x_1+x_2+\cdots +x_n=\text{𝟙}{\mathit{x}}^T\le m+1,\\ & 0\le x_i,i=1,\dots ,n\end{array}$(10) where the loan amount $x_0\in [-m,0]$ is eliminated from the constraints in (1(1) $(\begin{array}{l}{x}_0+x_1+x_2+\cdots +x_n=1,\\ -m\le x_0\le 0,0\le x_i,i=1,\dots ,n.\end{array}$(1) ) by using the portfolio $\mathit{x}\in {\mathbb{R}}^n$ as shown in (2(2) $x_0=1-\text{𝟙}{\mathit{x}}^T$(2) ).

4. Maximization of profit

4.1. Formulation of POL_P

It is hard to solve POL in (10(10) $[\begin{array}{rl}max& \gamma \\ \text{sub. to}& Pr\left(r\right(\mathit{x},\mathit{\xi })\le \gamma )\le \alpha ,\\ & x_1+x_2+\cdots +x_n=\text{𝟙}{\mathit{x}}^T\ge 1,\\ & x_1+x_2+\cdots +x_n=\text{𝟙}{\mathit{x}}^T\le m+1,\\ & 0\le x_i,i=1,\dots ,n\end{array}$(10) ) due to the chance constraint. Therefore, POL is transformed into a deterministic problem called POL for maximizing Profit (POL_P).

Since the return $r(\mathit{x},\mathit{\xi })$ of the portfolio $\mathit{x}\in {\mathbb{R}}^n$ is a random variable following the normal distribution in (9(9) $r(\mathit{x},\mathit{\xi })\sim \text{Normal}\left(\mu \right(\mathit{x}),\sigma (\mathit{x}{)}^2)$(9) ), the chance constraint in (10(10) $[\begin{array}{rl}max& \gamma \\ \text{sub. to}& Pr\left(r\right(\mathit{x},\mathit{\xi })\le \gamma )\le \alpha ,\\ & x_1+x_2+\cdots +x_n=\text{𝟙}{\mathit{x}}^T\ge 1,\\ & x_1+x_2+\cdots +x_n=\text{𝟙}{\mathit{x}}^T\le m+1,\\ & 0\le x_i,i=1,\dots ,n\end{array}$(10) ) is standardized as (11) $Pr(\frac{r(\mathit{x},\mathit{\xi })-\mu \left(\mathit{x}\right)}{\sigma \left(\mathit{x}\right)}\le \frac{\gamma -\mu \left(\mathit{x}\right)}{\sigma \left(\mathit{x}\right)})\le \alpha .$(11) The probability in (11(11) $Pr(\frac{r(\mathit{x},\mathit{\xi })-\mu \left(\mathit{x}\right)}{\sigma \left(\mathit{x}\right)}\le \frac{\gamma -\mu \left(\mathit{x}\right)}{\sigma \left(\mathit{x}\right)})\le \alpha .$(11) ) is also written as (12) $\mathrm{\Phi }\left(\frac{\gamma -\mu \left(\mathit{x}\right)}{\sigma \left(\mathit{x}\right)}\right)\le \alpha$(12) where $\mathrm{\Phi }:\mathbb{R}\to [0,1]$ is the Cumulative Distribution Function (CDF) of the standard normal distribution.

From (12(12) $\mathrm{\Phi }\left(\frac{\gamma -\mu \left(\mathit{x}\right)}{\sigma \left(\mathit{x}\right)}\right)\le \alpha$(12) ), the deterministic equivalent problem of POL in (10(10) $[\begin{array}{rl}max& \gamma \\ \text{sub. to}& Pr\left(r\right(\mathit{x},\mathit{\xi })\le \gamma )\le \alpha ,\\ & x_1+x_2+\cdots +x_n=\text{𝟙}{\mathit{x}}^T\ge 1,\\ & x_1+x_2+\cdots +x_n=\text{𝟙}{\mathit{x}}^T\le m+1,\\ & 0\le x_i,i=1,\dots ,n\end{array}$(10) ) is (13) $[\begin{array}{rl}max& \gamma \\ \text{sub. to}& \mu \left(\mathit{x}\right)+{\mathrm{\Phi }}^{-1}\left(\alpha \right)\sigma \left(\mathit{x}\right)\ge \gamma ,\\ & x_1+x_2+\cdots +x_n=\text{𝟙}{\mathit{x}}^T\ge 1,\\ & x_1+x_2+\cdots +x_n=\text{𝟙}{\mathit{x}}^T\le m+1,\\ & 0\le x_i,i=1,\dots ,n.\end{array}$(13) By injecting the soft constraint with $\gamma \in \mathbb{R}$ into the objective function $\gamma \in \mathbb{R}$ in (13(13) $[\begin{array}{rl}max& \gamma \\ \text{sub. to}& \mu \left(\mathit{x}\right)+{\mathrm{\Phi }}^{-1}\left(\alpha \right)\sigma \left(\mathit{x}\right)\ge \gamma ,\\ & x_1+x_2+\cdots +x_n=\text{𝟙}{\mathit{x}}^T\ge 1,\\ & x_1+x_2+\cdots +x_n=\text{𝟙}{\mathit{x}}^T\le m+1,\\ & 0\le x_i,i=1,\dots ,n.\end{array}$(13) ), we define POL_P as (14) $[\begin{array}{rl}max& {\gamma }_{\alpha }\left(\mathit{x}\right)=\mu \left(\mathit{x}\right)+{\mathrm{\Phi }}^{-1}\left(\alpha \right)\sigma \left(\mathit{x}\right)\\ \text{sub. to}& x_1+x_2+\cdots +x_n=\text{𝟙}{\mathit{x}}^T\ge 1,\\ & x_1+x_2+\cdots +x_n=\text{𝟙}{\mathit{x}}^T\le m+1,\\ & 0\le x_i,i=1,\dots ,n\end{array}$(14) where $\mu \left(\mathit{x}\right)=\mathit{\mu }{\mathit{x}}^T+L(1-\text{𝟙}{\mathit{x}}^T)$ and ${\sigma }^2\left(\mathit{x}\right)=\mathit{xC}{\mathit{x}}^T$ are given in (9(9) $r(\mathit{x},\mathit{\xi })\sim \text{Normal}\left(\mu \right(\mathit{x}),\sigma (\mathit{x}{)}^2)$(9) ).

The optimal solution of POL_P in (14(14) $[\begin{array}{rl}max& {\gamma }_{\alpha }\left(\mathit{x}\right)=\mu \left(\mathit{x}\right)+{\mathrm{\Phi }}^{-1}\left(\alpha \right)\sigma \left(\mathit{x}\right)\\ \text{sub. to}& x_1+x_2+\cdots +x_n=\text{𝟙}{\mathit{x}}^T\ge 1,\\ & x_1+x_2+\cdots +x_n=\text{𝟙}{\mathit{x}}^T\le m+1,\\ & 0\le x_i,i=1,\dots ,n\end{array}$(14) ) is the portfolio ${\mathit{x}}^{†}\in {\mathbb{R}}^n$ that maximizes the profit ${\gamma }_{\alpha }\left({\mathit{x}}^{†}\right)\in \mathbb{R}$ under a given $\alpha \in (0,0.5)$. From the mean $\mu \left(\mathit{x}\right)$ in (9(9) $r(\mathit{x},\mathit{\xi })\sim \text{Normal}\left(\mu \right(\mathit{x}),\sigma (\mathit{x}{)}^2)$(9) ), the loan amount $x_0\in [-m,1]$ is included in the profit ${\gamma }_{\alpha }\left(\mathit{x}\right)$. Therefore, the investor may liquidate the debt and still make a profit if ${\gamma }_{\alpha }\left({\mathit{x}}^{†}\right)>0$. On the other hand, in the case of ${\gamma }_{\alpha }\left({\mathit{x}}^{†}\right)\le 0$, the portfolio ${\mathit{x}}^{†}\in {\mathbb{R}}^n$ will not be adopted by the investor.

4.2. Characteristics of POL_P

Since all the constraints of POL_P in (14(14) $[\begin{array}{rl}max& {\gamma }_{\alpha }\left(\mathit{x}\right)=\mu \left(\mathit{x}\right)+{\mathrm{\Phi }}^{-1}\left(\alpha \right)\sigma \left(\mathit{x}\right)\\ \text{sub. to}& x_1+x_2+\cdots +x_n=\text{𝟙}{\mathit{x}}^T\ge 1,\\ & x_1+x_2+\cdots +x_n=\text{𝟙}{\mathit{x}}^T\le m+1,\\ & 0\le x_i,i=1,\dots ,n\end{array}$(14) ) are linear, the feasible region of POL_P is given by a convex polyhedron. Figure 1 illustrates the feasible region of POL_P for the case of n = 2. The feasible region is shown by the gray area between two hyper-planes. If a portfolio $\mathit{x}\in {\mathbb{R}}^n$ does not use the loan at all as $x_0=0$, it exists on the lower hyper-plane: $\text{𝟙}{\mathit{x}}^T=1$. On the other hand, if a portfolio $\mathit{x}\in {\mathbb{R}}^n$ uses the loan up to the limit as $x_0=-m$, it exists on the upper hyper-plane: $\text{𝟙}{\mathit{x}}^T=m+1$.

Figure 1. Feasible region of POL_P for the case of n = 2.

Theorem 4.1

The objective function ${\gamma }_{\alpha }\left(\mathit{x}\right)$ of POL_P in (14(14) $[\begin{array}{rl}max& {\gamma }_{\alpha }\left(\mathit{x}\right)=\mu \left(\mathit{x}\right)+{\mathrm{\Phi }}^{-1}\left(\alpha \right)\sigma \left(\mathit{x}\right)\\ \text{sub. to}& x_1+x_2+\cdots +x_n=\text{𝟙}{\mathit{x}}^T\ge 1,\\ & x_1+x_2+\cdots +x_n=\text{𝟙}{\mathit{x}}^T\le m+1,\\ & 0\le x_i,i=1,\dots ,n\end{array}$(14) ) is concave, In other words, $-{\gamma }_{\alpha }\left(\mathit{x}\right)$ is convex [12].

Proof.

See Appendix 1.

From Theorem 4.1, POL_P in (14(14) $[\begin{array}{rl}max& {\gamma }_{\alpha }\left(\mathit{x}\right)=\mu \left(\mathit{x}\right)+{\mathrm{\Phi }}^{-1}\left(\alpha \right)\sigma \left(\mathit{x}\right)\\ \text{sub. to}& x_1+x_2+\cdots +x_n=\text{𝟙}{\mathit{x}}^T\ge 1,\\ & x_1+x_2+\cdots +x_n=\text{𝟙}{\mathit{x}}^T\le m+1,\\ & 0\le x_i,i=1,\dots ,n\end{array}$(14) ) is the convex optimization problem. Therefore, the global optimal solution ${\mathit{x}}^{†}\in {\mathbb{R}}^n$ of POL_P can be obtained by the local search method. The optimal solution ${\mathit{x}}^{†}\in {\mathbb{R}}^n$ of POL_P may or may not use the loan. Furthermore, the loan is used up to the limit if it is used as shown in Theorem 4.2.

Theorem 4.2

The optimal solution ${\mathit{x}}^{†}\in {\mathbb{R}}^n$ of POL_P uses the loan up to the limit if it uses the loan. Specifically, the loan amount $x_0=1-\text{𝟙}{\mathit{x}}^{†T}$ in (2(2) $x_0=1-\text{𝟙}{\mathit{x}}^T$(2) ) becomes either $x_0=0$ or $x_0=-m$ for ${\mathit{x}}^{†}\in {\mathbb{R}}^n$ [13].

Proof.

See Appendix 2.

5. Prospect theory

Prospect theory provides the best available description of how people evaluate risk under uncertainties. Specifically, prospect theory can be summarized by two functions: the value function and the probability weighting function [14,15].

5.1. Value function

The value function provides a nonlinear mapping from profits to human values. Profits $z\in \mathbb{R}$ are called gains if $z\ge 0$. Profits $z\in \mathbb{R}$ are called losses if z<0. Then the value function is defined by a two-part power function as (15) $\upsilon \left(z\right)=\{\begin{array}{ll}{z}^a& \text{if}z\ge 0\\ -\lambda (-z{)}^b& \text{if}z<0\end{array}$(15) where $0<a\le b\le 1$ and $\lambda \ge 1$ [15].

The value function in (15(15) $\upsilon \left(z\right)=\{\begin{array}{ll}{z}^a& \text{if}z\ge 0\\ -\lambda (-z{)}^b& \text{if}z<0\end{array}$(15) ) represents three elements of prospect theory:

1. Reference dependence,

2. Diminishing sensitivity,

3. Loss aversion.

First, gains and losses are measured relative to some reference point. The reference point is given as z = 0 in (15(15) $\upsilon \left(z\right)=\{\begin{array}{ll}{z}^a& \text{if}z\ge 0\\ -\lambda (-z{)}^b& \text{if}z<0\end{array}$(15) ). Second, people are generally risk averse in gains but risk seeking in losses. Hence, the value function is concave in the region of gains ($z\ge 0$) but convex in the region of losses (z<0). Third, people are more sensitive to losses than gains. Therefore, the parameter $\lambda \ge 1$ in (15(15) $\upsilon \left(z\right)=\{\begin{array}{ll}{z}^a& \text{if}z\ge 0\\ -\lambda (-z{)}^b& \text{if}z<0\end{array}$(15) ) is called “loss aversion.” Figure 2 shows the value function in (15(15) $\upsilon \left(z\right)=\{\begin{array}{ll}{z}^a& \text{if}z\ge 0\\ -\lambda (-z{)}^b& \text{if}z<0\end{array}$(15) ) with a = 0.6, b = 0.6, and $\lambda =2.0$.

Figure 2. Value function in (15(15) $\upsilon \left(z\right)=\{\begin{array}{ll}{z}^a& \text{if}z\ge 0\\ -\lambda (-z{)}^b& \text{if}z<0\end{array}$(15) ) with a = 0.6, b = 0.6, and $\lambda =2.0$.

5.2. Probability weighting function

The fourth element of prospect theory is probability weighting. According to prospect theory, people do not evaluate uncertain profits by their objective probabilities but rather by weighted probabilities. More specifically, people generally overestimate low probabilities and underestimate high probabilities.

The probability weighting function provides a nonlinear mapping from objective probabilities $\theta \in (0,1)$ to weighted ones $\omega \left(\theta \right)\in (0,1)$. In order to represent the above human characteristics, the probability weighting function is defined as (16) $\omega \left(\theta \right)=\frac{{\theta }^c}{({\theta }^c+(1-\theta {)}^c{)}^{1/c}}$(16) where $0<c\le 1$. If c = 1 then $\omega \left(\theta \right)=\theta$ holds [15].

Figure 3 shows the probability weighting function with c = 0.6.

Figure 3. Probability weighting function in (16(16) $\omega \left(\theta \right)=\frac{{\theta }^c}{({\theta }^c+(1-\theta {)}^c{)}^{1/c}}$(16) ) with c = 0.6.

6. Maximization of value

6.1. Gain and loss of portfolio

By using the probability weighting function in (16(16) $\omega \left(\theta \right)=\frac{{\theta }^c}{({\theta }^c+(1-\theta {)}^c{)}^{1/c}}$(16) ), the failure probability $\alpha \in (0,0.5)$ is weighted as $\omega \left(\alpha \right)$. Then the profit ${\gamma }_{\alpha }\left(\mathit{x}\right)\in \mathbb{R}$ in (14(14) $[\begin{array}{rl}max& {\gamma }_{\alpha }\left(\mathit{x}\right)=\mu \left(\mathit{x}\right)+{\mathrm{\Phi }}^{-1}\left(\alpha \right)\sigma \left(\mathit{x}\right)\\ \text{sub. to}& x_1+x_2+\cdots +x_n=\text{𝟙}{\mathit{x}}^T\ge 1,\\ & x_1+x_2+\cdots +x_n=\text{𝟙}{\mathit{x}}^T\le m+1,\\ & 0\le x_i,i=1,\dots ,n\end{array}$(14) ) is expanded to (17) ${\hat{\gamma }}_{\alpha }\left(\mathit{x}\right)=\mathit{\mu }{\mathit{x}}^T+L(1-\text{𝟙}{\mathit{x}}^T)+{\mathrm{\Phi }}^{-1}\left(\omega \right(\alpha \left)\right)\sigma \left(\mathit{x}\right).$(17)

Theorem 6.1

The probability weighting function $\omega \left(\theta \right)$ in (16(16) $\omega \left(\theta \right)=\frac{{\theta }^c}{({\theta }^c+(1-\theta {)}^c{)}^{1/c}}$(16) ) returns the weighted probability within the range $\omega \left(\alpha \right)\in (0,0.5)$ for the failure probability given from the range $\alpha \in (0,0.5)$.

Proof.

The weighted probability $\omega \left(\theta \right)$ for $\theta =0.5$ is (18) $\omega \left(0.5\right)=\frac{{0.5}^c}{(0,5^c+(1-0.5{)}^c{)}^{1/c}}={0.5}^h$(18) where $h=(c^2-c+1)/c=1+(c-1{)}^2/c$.

From c>0 in (16(16) $\omega \left(\theta \right)=\frac{{\theta }^c}{({\theta }^c+(1-\theta {)}^c{)}^{1/c}}$(16) ), h>1 holds in (18(18) $\omega \left(0.5\right)=\frac{{0.5}^c}{(0,5^c+(1-0.5{)}^c{)}^{1/c}}={0.5}^h$(18) ). Thus, we obtain (19) $\omega \left(0.5\right)={0.5}^h<\mathrm{0.5.}$(19) Since the probability weighting function $\omega \left(\theta \right)$ in (16(16) $\omega \left(\theta \right)=\frac{{\theta }^c}{({\theta }^c+(1-\theta {)}^c{)}^{1/c}}$(16) ) is an increasing one, it returns the weighted probability $\omega \left(\alpha \right)\in (0,0.5)$ for the failure probability $\alpha \in (0,0.5)$.

The profit ${\hat{\gamma }}_{\alpha }\left(\mathit{x}\right)$ in (17(17) ${\hat{\gamma }}_{\alpha }\left(\mathit{x}\right)=\mathit{\mu }{\mathit{x}}^T+L(1-\text{𝟙}{\mathit{x}}^T)+{\mathrm{\Phi }}^{-1}\left(\omega \right(\alpha \left)\right)\sigma \left(\mathit{x}\right).$(17) ) is divided into two parts as (20) $(\begin{array}{rl}& {\hat{\gamma }}_{\alpha }\left(\mathit{x}\right)=z_g\left(\mathit{x}\right)+z_s\left(\mathit{x}\right)\\ & z_g\left(\mathit{x}\right)=\mathit{\mu }{\mathit{x}}^T\\ & z_s\left(\mathit{x}\right)=L(1-\text{𝟙}{\mathit{x}}^T)+{\mathrm{\Phi }}^{-1}\left(\omega \right(\alpha \left)\right)\sigma \left(\mathit{x}\right).\end{array}$(20) From ${\mu }_i>0$ and $x_i\ge 0$, $i=1,\dots ,n$, we can see that $z_g\left(\mathit{x}\right)\ge 0$ holds in (20(20) $(\begin{array}{rl}& {\hat{\gamma }}_{\alpha }\left(\mathit{x}\right)=z_g\left(\mathit{x}\right)+z_s\left(\mathit{x}\right)\\ & z_g\left(\mathit{x}\right)=\mathit{\mu }{\mathit{x}}^T\\ & z_s\left(\mathit{x}\right)=L(1-\text{𝟙}{\mathit{x}}^T)+{\mathrm{\Phi }}^{-1}\left(\omega \right(\alpha \left)\right)\sigma \left(\mathit{x}\right).\end{array}$(20) ). Therefore, $z_g\left(\mathit{x}\right)\in \mathbb{R}$ is the gain of the portfolio $\mathit{x}\in {\mathbb{R}}^n$. On the other hand, from Theorem 6.1, ${\mathrm{\Phi }}^{-1}\left(\omega \right(\alpha \left)\right)<0$ holds for $\alpha \in (0,0.5)$. Furthermore, from $\text{𝟙}{\mathit{x}}^T\ge 1$, $z_s\left(\mathit{x}\right)<0$ holds in (20(20) $(\begin{array}{rl}& {\hat{\gamma }}_{\alpha }\left(\mathit{x}\right)=z_g\left(\mathit{x}\right)+z_s\left(\mathit{x}\right)\\ & z_g\left(\mathit{x}\right)=\mathit{\mu }{\mathit{x}}^T\\ & z_s\left(\mathit{x}\right)=L(1-\text{𝟙}{\mathit{x}}^T)+{\mathrm{\Phi }}^{-1}\left(\omega \right(\alpha \left)\right)\sigma \left(\mathit{x}\right).\end{array}$(20) ). Therefore, $z_s\left(\mathit{x}\right)\in \mathbb{R}$ is the loss of the portfolio $\mathit{x}\in {\mathbb{R}}^n$.

6.2. Formulation of POL_V

According to prospect theory, POL_P in (14(14) $[\begin{array}{rl}max& {\gamma }_{\alpha }\left(\mathit{x}\right)=\mu \left(\mathit{x}\right)+{\mathrm{\Phi }}^{-1}\left(\alpha \right)\sigma \left(\mathit{x}\right)\\ \text{sub. to}& x_1+x_2+\cdots +x_n=\text{𝟙}{\mathit{x}}^T\ge 1,\\ & x_1+x_2+\cdots +x_n=\text{𝟙}{\mathit{x}}^T\le m+1,\\ & 0\le x_i,i=1,\dots ,n\end{array}$(14) ) will be expanded to POL for maximizing Value (POL_V). By using the value function in (15(15) $\upsilon \left(z\right)=\{\begin{array}{ll}{z}^a& \text{if}z\ge 0\\ -\lambda (-z{)}^b& \text{if}z<0\end{array}$(15) ), the value of the gain $z_g\left(\mathit{x}\right)\ge 0$ in (20(20) $(\begin{array}{rl}& {\hat{\gamma }}_{\alpha }\left(\mathit{x}\right)=z_g\left(\mathit{x}\right)+z_s\left(\mathit{x}\right)\\ & z_g\left(\mathit{x}\right)=\mathit{\mu }{\mathit{x}}^T\\ & z_s\left(\mathit{x}\right)=L(1-\text{𝟙}{\mathit{x}}^T)+{\mathrm{\Phi }}^{-1}\left(\omega \right(\alpha \left)\right)\sigma \left(\mathit{x}\right).\end{array}$(20) ) can be evaluated as (21) ${\upsilon }_g\left(\mathit{x}\right)=\upsilon \left(z_g\right(\mathit{x}\left)\right)=\left(z_g\right(\mathit{x}){)}^a.$(21) On the other hand, by using the value function in (15(15) $\upsilon \left(z\right)=\{\begin{array}{ll}{z}^a& \text{if}z\ge 0\\ -\lambda (-z{)}^b& \text{if}z<0\end{array}$(15) ), the value of the loss $z_s\left(\mathit{x}\right)<0$ in (20(20) $(\begin{array}{rl}& {\hat{\gamma }}_{\alpha }\left(\mathit{x}\right)=z_g\left(\mathit{x}\right)+z_s\left(\mathit{x}\right)\\ & z_g\left(\mathit{x}\right)=\mathit{\mu }{\mathit{x}}^T\\ & z_s\left(\mathit{x}\right)=L(1-\text{𝟙}{\mathit{x}}^T)+{\mathrm{\Phi }}^{-1}\left(\omega \right(\alpha \left)\right)\sigma \left(\mathit{x}\right).\end{array}$(20) ) can be evaluated as (22) ${\upsilon }_s\left(\mathit{x}\right)=\upsilon \left(z_s\right(\mathit{x}\left)\right)=-\lambda (-z_s(\mathit{x}){)}^b.$(22) From (21(21) ${\upsilon }_g\left(\mathit{x}\right)=\upsilon \left(z_g\right(\mathit{x}\left)\right)=\left(z_g\right(\mathit{x}){)}^a.$(21) ) and (22(22) ${\upsilon }_s\left(\mathit{x}\right)=\upsilon \left(z_s\right(\mathit{x}\left)\right)=-\lambda (-z_s(\mathit{x}){)}^b.$(22) ), the value of the portfolio $\mathit{x}\in {\mathbb{R}}^n$ is defined as (23) ${\upsilon }_{\alpha }\left(\mathit{x}\right)={\upsilon }_g\left(\mathit{x}\right)+{\upsilon }_s\left(\mathit{x}\right).$(23) From POL_P in (14(14) $[\begin{array}{rl}max& {\gamma }_{\alpha }\left(\mathit{x}\right)=\mu \left(\mathit{x}\right)+{\mathrm{\Phi }}^{-1}\left(\alpha \right)\sigma \left(\mathit{x}\right)\\ \text{sub. to}& x_1+x_2+\cdots +x_n=\text{𝟙}{\mathit{x}}^T\ge 1,\\ & x_1+x_2+\cdots +x_n=\text{𝟙}{\mathit{x}}^T\le m+1,\\ & 0\le x_i,i=1,\dots ,n\end{array}$(14) ) and (23(23) ${\upsilon }_{\alpha }\left(\mathit{x}\right)={\upsilon }_g\left(\mathit{x}\right)+{\upsilon }_s\left(\mathit{x}\right).$(23) ), POL_V is formulated as (24) $[\begin{array}{rl}max& {\upsilon }_{\alpha }\left(\mathit{x}\right)={\upsilon }_g\left(\mathit{x}\right)+{\upsilon }_s\left(\mathit{x}\right)\\ \text{sub. to}& x_1+x_2+\cdots +x_n=\text{𝟙}{\mathit{x}}^T\ge 1,\\ & x_1+x_2+\cdots +x_n=\text{𝟙}{\mathit{x}}^T\le m+1,\\ & 0\le x_i,i=1,\dots ,n.\end{array}$(24) The optimal solution of POL_V in (24(24) $[\begin{array}{rl}max& {\upsilon }_{\alpha }\left(\mathit{x}\right)={\upsilon }_g\left(\mathit{x}\right)+{\upsilon }_s\left(\mathit{x}\right)\\ \text{sub. to}& x_1+x_2+\cdots +x_n=\text{𝟙}{\mathit{x}}^T\ge 1,\\ & x_1+x_2+\cdots +x_n=\text{𝟙}{\mathit{x}}^T\le m+1,\\ & 0\le x_i,i=1,\dots ,n.\end{array}$(24) ) is the portfolio ${\mathit{x}}^{\star }\in {\mathbb{R}}^n$ that maximizes the value ${\upsilon }_{\alpha }\left({\mathit{x}}^{\star }\right)$ for the investment under a given $\alpha \in (0,0.5)$. However, in the case of ${\upsilon }_{\alpha }\left({\mathit{x}}^{\star }\right)\le 0$, the portfolio ${\mathit{x}}^{\star }\in {\mathbb{R}}^n$ will not be adopted by the investor.

The objective functions of POL_V in (24(24) $[\begin{array}{rl}max& {\upsilon }_{\alpha }\left(\mathit{x}\right)={\upsilon }_g\left(\mathit{x}\right)+{\upsilon }_s\left(\mathit{x}\right)\\ \text{sub. to}& x_1+x_2+\cdots +x_n=\text{𝟙}{\mathit{x}}^T\ge 1,\\ & x_1+x_2+\cdots +x_n=\text{𝟙}{\mathit{x}}^T\le m+1,\\ & 0\le x_i,i=1,\dots ,n.\end{array}$(24) ) and POL_P in (14(14) $[\begin{array}{rl}max& {\gamma }_{\alpha }\left(\mathit{x}\right)=\mu \left(\mathit{x}\right)+{\mathrm{\Phi }}^{-1}\left(\alpha \right)\sigma \left(\mathit{x}\right)\\ \text{sub. to}& x_1+x_2+\cdots +x_n=\text{𝟙}{\mathit{x}}^T\ge 1,\\ & x_1+x_2+\cdots +x_n=\text{𝟙}{\mathit{x}}^T\le m+1,\\ & 0\le x_i,i=1,\dots ,n\end{array}$(14) ) are the same as ${\upsilon }_{\alpha }\left(\mathit{x}\right)={\gamma }_{\alpha }\left(\mathit{x}\right)$ if $a=b=\lambda =c=1$ holds for the functions in (15(15) $\upsilon \left(z\right)=\{\begin{array}{ll}{z}^a& \text{if}z\ge 0\\ -\lambda (-z{)}^b& \text{if}z<0\end{array}$(15) ) and (16(16) $\omega \left(\theta \right)=\frac{{\theta }^c}{({\theta }^c+(1-\theta {)}^c{)}^{1/c}}$(16) ). Therefore, POL_V in (24(24) $[\begin{array}{rl}max& {\upsilon }_{\alpha }\left(\mathit{x}\right)={\upsilon }_g\left(\mathit{x}\right)+{\upsilon }_s\left(\mathit{x}\right)\\ \text{sub. to}& x_1+x_2+\cdots +x_n=\text{𝟙}{\mathit{x}}^T\ge 1,\\ & x_1+x_2+\cdots +x_n=\text{𝟙}{\mathit{x}}^T\le m+1,\\ & 0\le x_i,i=1,\dots ,n.\end{array}$(24) ) is an expanded variant of POL_P in (14(14) $[\begin{array}{rl}max& {\gamma }_{\alpha }\left(\mathit{x}\right)=\mu \left(\mathit{x}\right)+{\mathrm{\Phi }}^{-1}\left(\alpha \right)\sigma \left(\mathit{x}\right)\\ \text{sub. to}& x_1+x_2+\cdots +x_n=\text{𝟙}{\mathit{x}}^T\ge 1,\\ & x_1+x_2+\cdots +x_n=\text{𝟙}{\mathit{x}}^T\le m+1,\\ & 0\le x_i,i=1,\dots ,n\end{array}$(14) ).

From Theorem 4.1, the profit ${\gamma }_{\alpha }\left(\mathit{x}\right)$ in (14(14) $[\begin{array}{rl}max& {\gamma }_{\alpha }\left(\mathit{x}\right)=\mu \left(\mathit{x}\right)+{\mathrm{\Phi }}^{-1}\left(\alpha \right)\sigma \left(\mathit{x}\right)\\ \text{sub. to}& x_1+x_2+\cdots +x_n=\text{𝟙}{\mathit{x}}^T\ge 1,\\ & x_1+x_2+\cdots +x_n=\text{𝟙}{\mathit{x}}^T\le m+1,\\ & 0\le x_i,i=1,\dots ,n\end{array}$(14) ) is concave. On the other hand, the value function $\upsilon \left(z\right)$ in (15(15) $\upsilon \left(z\right)=\{\begin{array}{ll}{z}^a& \text{if}z\ge 0\\ -\lambda (-z{)}^b& \text{if}z<0\end{array}$(15) ) is convex for $z<0$. Thus, the objective function ${\upsilon }_{\alpha }\left(\mathit{x}\right)$ of POL_V in (24(24) $[\begin{array}{rl}max& {\upsilon }_{\alpha }\left(\mathit{x}\right)={\upsilon }_g\left(\mathit{x}\right)+{\upsilon }_s\left(\mathit{x}\right)\\ \text{sub. to}& x_1+x_2+\cdots +x_n=\text{𝟙}{\mathit{x}}^T\ge 1,\\ & x_1+x_2+\cdots +x_n=\text{𝟙}{\mathit{x}}^T\le m+1,\\ & 0\le x_i,i=1,\dots ,n.\end{array}$(24) ) is usually neither concave nor convex. It means that POL_V is the non-convex optimization problem which may possibly has multiple local optimal solutions.

7. Optimization method

7.1. Convex-Hull mapping

Convex-Hull Mapping (CHM) [24] can generate feasible solutions $\mathit{x}\in {\mathbb{R}}^n$ randomly for a sort of constrained optimization problems. Let $\mathit{X}\subseteq {\mathbb{R}}^n$ be the feasible region of the constrained optimization problem. We assume that the feasible region is a convex polyhedron and all vertexes of $\mathit{X}\subseteq {\mathbb{R}}^n$ are known.

Let ${\hat{\mathit{x}}}_j\in {\mathbb{R}}^n$, $j=1,\dots ,p$ be the set of the vertexes of the feasible region $\mathit{X}\subseteq {\mathbb{R}}^n$. Then CHM generates a feasible solution $\mathit{x}\in \mathit{X}$ randomly as follows:

Step 1:	Generate uniformly distributed random numbers $e_j\in [0,1]$, $j=1,\dots ,q$ where q is determined from the number of vertexes p as q = p−1.
Step 2:	Sort $e_j\in [0,1]$ in ascending order as (25) $0\le {\tilde{e}}_1\le \cdots \le {\tilde{e}}_j\le \cdots \le {\tilde{e}}_q\le 1.$(25)
Step 3:	Get coefficients ${\delta }_j\in [0,1]$, $j=1,\dots ,p$ as (26) $(\begin{array}{rl}& {\delta }_1={\tilde{e}}_1\\ & {\delta }_j={\tilde{e}}_j-{\tilde{e}}_{j-1},j=2,\dots ,p-1\\ & {\delta }_p=1-{\tilde{e}}_q.\end{array}$(26)
Step 4:	Compose a feasible solution $\mathit{x}\in \mathit{X}$ as (27) $\mathit{x}=\sum _{j=1}^p{\delta }_j{\hat{\mathit{x}}}_j$(27) where ${\delta }_1+\cdots +{\delta }_j+\cdots +{\delta }_p=1$ holds.

7.2. Landscape analysis method

Over past few decades, as a powerful analytical tool to reveal the structure of the decision variable space, the fitness landscape analysis has been widely concerned and utilized for all kinds of optimization areas [41].

We have proposed the fitness landscape analysis method using CHM for POL_P and POL_V [22]. The feasible regions of POL_P and POL_V are the same. The feasible region $\mathit{X}\subseteq {\mathbb{R}}^n$ is given by a convex polyhedron as shown in Figure 1. In the proposed landscape analysis method, a large number of feasible solutions ${\mathit{x}}_k\in \mathit{X}$ are generated randomly by CHM. Then these solutions ${\mathit{x}}_k\in \mathit{X}$ are mapped to the objective function space $\left({\gamma }_{\alpha }\right({\mathit{x}}_k),{\upsilon }_{\alpha }({\mathit{x}}_k\left)\right)\in {\mathbb{R}}^2$. Therefore, we can estimate the structure of the feasible region from the objective function space.

The same feasible region $\mathit{X}\subseteq {\mathbb{R}}^n$ in POL_P and POL_V can be represented by the following p = 2n vertexes ${\hat{\mathit{x}}}_j\in {\mathbb{R}}^n$, $j=1,\dots ,p$: (28) $(\begin{array}{rl}{\hat{\mathit{x}}}_1& =(m+1,0,0,\dots ,0,0,0)\\ {\hat{\mathit{x}}}_2& =(0,m+1,0,\dots ,0,0,0)\\ & ⋮\\ {\hat{\mathit{x}}}_{n-1}& =(0,0,0,\dots ,0,m+1,0)\\ {\hat{\mathit{x}}}_n& =(0,0,0,\dots ,0,0,m+1)\\ {\hat{\mathit{x}}}_{n+1}& =(1,0,0,\dots ,0,0,0)\\ & ⋮\\ {\hat{\mathit{x}}}_{2n-1}& =(0,0,0,\dots ,0,1,0)\\ {\hat{\mathit{x}}}_{2n}& =(0,0,0,\dots ,0,0,1)\end{array}$(28) where $m\in \mathbb{R}$, m>0 is the upper limit of the loan.

From all vertexes ${\hat{\mathit{x}}}_j\in {\mathbb{R}}^n$ in (28(28) $(\begin{array}{rl}{\hat{\mathit{x}}}_1& =(m+1,0,0,\dots ,0,0,0)\\ {\hat{\mathit{x}}}_2& =(0,m+1,0,\dots ,0,0,0)\\ & ⋮\\ {\hat{\mathit{x}}}_{n-1}& =(0,0,0,\dots ,0,m+1,0)\\ {\hat{\mathit{x}}}_n& =(0,0,0,\dots ,0,0,m+1)\\ {\hat{\mathit{x}}}_{n+1}& =(1,0,0,\dots ,0,0,0)\\ & ⋮\\ {\hat{\mathit{x}}}_{2n-1}& =(0,0,0,\dots ,0,1,0)\\ {\hat{\mathit{x}}}_{2n}& =(0,0,0,\dots ,0,0,1)\end{array}$(28) ), CHM generates a set of feasible solutions ${\mathit{x}}_k\in \mathit{X}$ to analyse the overall structure of the feasible region $\mathit{X}\subseteq {\mathbb{R}}^n$. Figure 4 shows a set of 100 feasible solutions ${\mathit{x}}_k\in \mathit{X}$, $k=1,\dots ,100$ generated by CHM in the feasible region $\mathit{X}\subseteq {\mathbb{R}}^n$ with n = 2 and m = 2. From Figure 4, we can see that all solutions ${\mathit{x}}_k\in \mathit{X}$ are uniformly distributed in the feasible region $\mathit{X}\subseteq {\mathbb{R}}^2$.

Figure 4. Feasible solutions generated by CHM.

From (2(2) $x_0=1-\text{𝟙}{\mathit{x}}^T$(2) ), the feasible solution $\mathit{x}\in \mathit{X}$, or the portfolio $\mathit{x}\in \mathit{X}$, using the loan totalling $x_0\in [-m,0]$ exists on the hyper-plane: (29) $\text{𝟙}{\mathit{x}}^T=1-x_0.$(29) CHM can also generate a set of feasible solutions ${\mathit{x}}_k\in \mathit{X}$ randomly on the hyper-plane in (29(29) $\text{𝟙}{\mathit{x}}^T=1-x_0.$(29) ) from the following p = n vertexes ${\hat{\mathit{x}}}_j\in {\mathbb{R}}^n$, $j=1,\dots ,p$: (30) $(\begin{array}{rl}{\hat{\mathit{x}}}_1& =(1-x_0,0,0,\dots ,0,0,0)\\ {\hat{\mathit{x}}}_2& =(0,1-x_0,0,\dots ,0,0)\\ & ⋮\\ {\hat{\mathit{x}}}_{n-1}& =(0,0,0,\dots ,0,1-x_0,0)\\ {\hat{\mathit{x}}}_n& =(0,0,0,\dots ,0,0,1-x_0).\end{array}$(30) The hyper-plane in (29(29) $\text{𝟙}{\mathit{x}}^T=1-x_0.$(29) ) is a cross-section of the feasible region. Therefore, the structure of the feasible region $\mathit{X}\subseteq {\mathbb{R}}^n$ can be also analysed by a large number of solutions ${\mathit{x}}_k\in \mathit{X}$ generated on hyper-planes with various values of $x_0\in [-m,0]$.

7.3. Multi-start local search method

Since POL_V in (24(24) $[\begin{array}{rl}max& {\upsilon }_{\alpha }\left(\mathit{x}\right)={\upsilon }_g\left(\mathit{x}\right)+{\upsilon }_s\left(\mathit{x}\right)\\ \text{sub. to}& x_1+x_2+\cdots +x_n=\text{𝟙}{\mathit{x}}^T\ge 1,\\ & x_1+x_2+\cdots +x_n=\text{𝟙}{\mathit{x}}^T\le m+1,\\ & 0\le x_i,i=1,\dots ,n.\end{array}$(24) ) is the non-convex optimization problem, the solution obtained by the local search method depends on the initial solution. Therefore, in order to find a solution as close as possible to the optimal solution ${\mathit{x}}^{\star }\in \mathit{X}$ of POL_V, the multi-start local search method uses a set of various initial solutions generated by CHM [23].

Let K be the number of initial solutions. The algorithm of the multi-start local search method is described as follows:

Step 1:	From the vertexes ${\hat{\mathit{x}}}_j\in {\mathbb{R}}^n$, $j=1,\dots ,p$ in (28(28) $(\begin{array}{rl}{\hat{\mathit{x}}}_1& =(m+1,0,0,\dots ,0,0,0)\\ {\hat{\mathit{x}}}_2& =(0,m+1,0,\dots ,0,0,0)\\ & ⋮\\ {\hat{\mathit{x}}}_{n-1}& =(0,0,0,\dots ,0,m+1,0)\\ {\hat{\mathit{x}}}_n& =(0,0,0,\dots ,0,0,m+1)\\ {\hat{\mathit{x}}}_{n+1}& =(1,0,0,\dots ,0,0,0)\\ & ⋮\\ {\hat{\mathit{x}}}_{2n-1}& =(0,0,0,\dots ,0,1,0)\\ {\hat{\mathit{x}}}_{2n}& =(0,0,0,\dots ,0,0,1)\end{array}$(28) ), generate a set of feasible initial solutions ${\mathit{x}}_k\in \mathit{X}$, $k=1,\dots ,K$ randomly by using CHM.
Step 2:	Apply the local search method to each of the feasible initial solutions ${\mathit{x}}_k\in \mathit{X}$, $k=1,\dots ,K$. Then obtain K local optimal solutions ${\mathit{x}}_k^{\star }\in \mathit{X}$.
Step 3:	Select the best solution ${\mathit{x}}^{\star }\in \mathit{X}$ from among the set of local optimal solutions ${\mathit{x}}_k^{\star }\in \mathit{X}$, $k=1,\dots ,K$.

Since any local search method can be used in Step 2 of the above optimization method, Sequential Quadratic Programming (SQP) of MATLAB solver [42] is used in this paper. The multi-start local search method is equally valid for POL_P in (14(14) $[\begin{array}{rl}max& {\gamma }_{\alpha }\left(\mathit{x}\right)=\mu \left(\mathit{x}\right)+{\mathrm{\Phi }}^{-1}\left(\alpha \right)\sigma \left(\mathit{x}\right)\\ \text{sub. to}& x_1+x_2+\cdots +x_n=\text{𝟙}{\mathit{x}}^T\ge 1,\\ & x_1+x_2+\cdots +x_n=\text{𝟙}{\mathit{x}}^T\le m+1,\\ & 0\le x_i,i=1,\dots ,n\end{array}$(14) ) because the feasible regions of POL_P and POL_V are the same.

8. Numerical experiments

8.1. Artificial data of asset returns

By using artificial data of n = 4 asset returns, the characteristics of POL_V are clarified in comparison with POL_P. Table 1 shows the mean ${\mu }_i$ and standard deviation ${\sigma }_i$ of each asset return ${\xi }_i$, $i=1,\dots ,n$. Table 2 shows the correlation coefficient ${\rho }_{\mathrm{ij}}\in [-1,1]$ between ${\xi }_i$ and ${\xi }_j$. The values of ${\mu }_i$, ${\sigma }_i$, and ${\rho }_{\mathrm{ij}}$ are used for $\mathit{\mu }$ and $\mathit{C}$ in (4(4) $\mathit{\xi }\sim \text{Normal}(\mathit{\mu },\mathit{C})$(4) ).

Table 1. Mean ${\mu }_i$ and standard deviation ${\sigma }_i$ of asset return ${\xi }_i$.

	${\xi }_1$	${\xi }_2$	${\xi }_3$	${\xi }_4$
${\mu }_i$	0.05	0.06	0.07	0.08
${\sigma }_i$	0.10	0.20	0.15	0.25

Table 2. Correlation coefficient ${\rho }_{\mathrm{ij}}$ between ${\xi }_i$ and ${\xi }_j$, $i,j=1,\dots ,4$.

${\rho }_{\mathrm{ij}}$	${\xi }_1$	${\xi }_2$	${\xi }_3$	${\xi }_4$
${\xi }_1$	1.0	$-0.7$	0.1	$-0.4$
${\xi }_2$	$-0.7$	1.0	$-0.5$	0.2
${\xi }_3$	0.1	$-0.5$	1.0	$-0.3$
${\xi }_4$	$-0.4$	0.2	$-0.3$	1.0

For the value function in (15(15) $\upsilon \left(z\right)=\{\begin{array}{ll}{z}^a& \text{if}z\ge 0\\ -\lambda (-z{)}^b& \text{if}z<0\end{array}$(15) ) and the probability weighting function in (16(16) $\omega \left(\theta \right)=\frac{{\theta }^c}{({\theta }^c+(1-\theta {)}^c{)}^{1/c}}$(16) ), we adopt the parameter values proposed by Tversky and Kahneman in their paper [15], as listed in Table 3. In fact, the parameter values in Table 3 have been widely used in prospect theory-based portfolio optimization problems [18,19]. That is because these values have been theoretically and empirically verified [20].

Table 3. Parameter values of functions in (15(15) $\upsilon \left(z\right)=\{\begin{array}{ll}{z}^a& \text{if}z\ge 0\\ -\lambda (-z{)}^b& \text{if}z<0\end{array}$(15) ) and (16(16) $\omega \left(\theta \right)=\frac{{\theta }^c}{({\theta }^c+(1-\theta {)}^c{)}^{1/c}}$(16) ).

a	b	λ	c
0.88	0.88	2.25	0.69

8.2. Fitness landscape analysis

By using CHM with vertexes in (28(28) $(\begin{array}{rl}{\hat{\mathit{x}}}_1& =(m+1,0,0,\dots ,0,0,0)\\ {\hat{\mathit{x}}}_2& =(0,m+1,0,\dots ,0,0,0)\\ & ⋮\\ {\hat{\mathit{x}}}_{n-1}& =(0,0,0,\dots ,0,m+1,0)\\ {\hat{\mathit{x}}}_n& =(0,0,0,\dots ,0,0,m+1)\\ {\hat{\mathit{x}}}_{n+1}& =(1,0,0,\dots ,0,0,0)\\ & ⋮\\ {\hat{\mathit{x}}}_{2n-1}& =(0,0,0,\dots ,0,1,0)\\ {\hat{\mathit{x}}}_{2n}& =(0,0,0,\dots ,0,0,1)\end{array}$(28) ) and (30(30) $(\begin{array}{rl}{\hat{\mathit{x}}}_1& =(1-x_0,0,0,\dots ,0,0,0)\\ {\hat{\mathit{x}}}_2& =(0,1-x_0,0,\dots ,0,0)\\ & ⋮\\ {\hat{\mathit{x}}}_{n-1}& =(0,0,0,\dots ,0,1-x_0,0)\\ {\hat{\mathit{x}}}_n& =(0,0,0,\dots ,0,0,1-x_0).\end{array}$(30) ), a huge number of solutions ${\mathit{x}}_k\in \mathit{X}$ are generated within the feasible region $\mathit{X}\subseteq {\mathbb{R}}^n$ and on the hyper-plain in (29(29) $\text{𝟙}{\mathit{x}}^T=1-x_0.$(29) ). Then pairs of profits and values, or points $\left({\gamma }_{\alpha }\right({\mathit{x}}_k),{\upsilon }_{\alpha }({\mathit{x}}_k\left)\right)$, are evaluated for them.

Firstly, the interest rate and upper limit of the loan are fixed as L = 0.01 and m = 2. From Theorem 4.2, the optimal solution of POL_P uses the loan up to the limit if it uses the loan. Therefore, we are interested in the loan amount spent by the solution of POL_V. Since the size of the feasible region $\mathit{X}\subseteq {\mathbb{R}}^n$ is proportional to the upper limit of the loan $m\in \mathbb{R}$, we have chosen m = 2 to assess a sufficiently large area. The failure probability $\alpha \in (0,0.5)$ is chosen as $\alpha =0.15$, 0.35. Then, by using CHM with vertexes in (28(28) $(\begin{array}{rl}{\hat{\mathit{x}}}_1& =(m+1,0,0,\dots ,0,0,0)\\ {\hat{\mathit{x}}}_2& =(0,m+1,0,\dots ,0,0,0)\\ & ⋮\\ {\hat{\mathit{x}}}_{n-1}& =(0,0,0,\dots ,0,m+1,0)\\ {\hat{\mathit{x}}}_n& =(0,0,0,\dots ,0,0,m+1)\\ {\hat{\mathit{x}}}_{n+1}& =(1,0,0,\dots ,0,0,0)\\ & ⋮\\ {\hat{\mathit{x}}}_{2n-1}& =(0,0,0,\dots ,0,1,0)\\ {\hat{\mathit{x}}}_{2n}& =(0,0,0,\dots ,0,0,1)\end{array}$(28) ), a set of ${10}^3$ solutions ${\mathit{x}}_k\in \mathit{X}$ is generated randomly in the feasible region $\mathit{X}\subseteq {\mathbb{R}}^4$. Figure 5 shows points $\left({\gamma }_{\alpha }\right({\mathit{x}}_k),{\upsilon }_{\alpha }({\mathit{x}}_k\left)\right)\in {\mathbb{R}}^2$ evaluated for the set of solutions ${\mathit{x}}_k\in \mathit{X}$. As described below, the solutions of POL_P and POL_V are obtained, respectively, by the multi-start local search method. Figure 5 also shows the points evaluated for the best solutions of POL_P (×) and POL_V (•).

Figure 5. Profits ${\gamma }_{\alpha }\left({\mathit{x}}_k\right)$ and values ${\upsilon }_{\alpha }\left({\mathit{x}}_k\right)$ of solutions ${\mathit{x}}_k\in \mathit{X}$ in feasible region and the best solutions of POL_P and POL_V. The interest rate and upper limit of the loan are fixed as L = 0.01 and m = 2. (a) $\alpha =0.15$ (b) $\alpha =0.35$.

Furthermore, the loan amount is chosen as $x_0=-2$, 0 for vertexes ${\hat{\mathit{x}}}_k\in {\mathbb{R}}^4$ in (30(30) $(\begin{array}{rl}{\hat{\mathit{x}}}_1& =(1-x_0,0,0,\dots ,0,0,0)\\ {\hat{\mathit{x}}}_2& =(0,1-x_0,0,\dots ,0,0)\\ & ⋮\\ {\hat{\mathit{x}}}_{n-1}& =(0,0,0,\dots ,0,1-x_0,0)\\ {\hat{\mathit{x}}}_n& =(0,0,0,\dots ,0,0,1-x_0).\end{array}$(30) ). Then a set of ${10}^3$ solutions ${\mathit{x}}_k\in \mathit{X}$ is generated randomly by CHM on each of the two hyper-planes in (29(29) $\text{𝟙}{\mathit{x}}^T=1-x_0.$(29) ). Figure 6 shows points $\left({\gamma }_{\alpha }\right({\mathit{x}}_k),{\upsilon }_{\alpha }({\mathit{x}}_k\left)\right)\in {\mathbb{R}}^2$ evaluated for the feasible solutions ${\mathit{x}}_k\in \mathit{X}$ on the two hyper-planes.

Figure 6. Profits ${\gamma }_{\alpha }\left({\mathit{x}}_k\right)$ and values ${\upsilon }_{\alpha }\left({\mathit{x}}_k\right)$ of solutions ${\mathit{x}}_k\in \mathit{X}$ on two hyper-planes in (29(29) $\text{𝟙}{\mathit{x}}^T=1-x_0.$(29) ) with $x_0=-2$ and $x_0=0$. The interest rate and upper limit of the loan are fixed as L = 0.01 and m = 2. (a) $\alpha =0.15$ (b) $\alpha =0.35$.

From Figures 5 and 6, values ${\upsilon }_{\alpha }\left({\mathit{x}}_k\right)$ are generally proportional to profits ${\gamma }_{\alpha }\left({\mathit{x}}_k\right)$. In other words, the greater the profit, the greater the value. However, we can see the conflict between the two best solutions to $\alpha =0.15$ in Figure 5(a). By the way, the range of profits ${\gamma }_{\alpha }\left({\mathit{x}}_k\right)$ is wider for $\alpha =0.15$ rather than for $\alpha =0.35$. That is because $0>{\mathrm{\Phi }}^{-1}\left(0.35\right)>{\mathrm{\Phi }}^{-1}\left(0.15\right)$ holds for the profit ${\gamma }_{\alpha }\left(\mathit{x}\right)$ in (14(14) $[\begin{array}{rl}max& {\gamma }_{\alpha }\left(\mathit{x}\right)=\mu \left(\mathit{x}\right)+{\mathrm{\Phi }}^{-1}\left(\alpha \right)\sigma \left(\mathit{x}\right)\\ \text{sub. to}& x_1+x_2+\cdots +x_n=\text{𝟙}{\mathit{x}}^T\ge 1,\\ & x_1+x_2+\cdots +x_n=\text{𝟙}{\mathit{x}}^T\le m+1,\\ & 0\le x_i,i=1,\dots ,n\end{array}$(14) ).

Secondly, the interest rate and upper limit of the loan are fixed as L = 0.04 and m = 2. The failure probability $\alpha \in (0,0.5)$ is chosen as $\alpha =0.15$, 0.35. In the same way with Figure 5, Figure 7 shows points $\left({\gamma }_{\alpha }\right({\mathit{x}}_k),{\upsilon }_{\alpha }({\mathit{x}}_k\left)\right)$ evaluated for the solutions ${\mathit{x}}_k\in \mathit{X}$ in the feasible region. Figure 7 also shows points evaluated for the best solutions of POL_P (×) and POL_V (•). In the same way with Figure 6, Figure 8 shows points evaluated for the solutions ${\mathit{x}}_k\in \mathit{X}$ on the two hyper-planes.

Figure 7. Profits ${\gamma }_{\alpha }\left({\mathit{x}}_k\right)$ and values ${\upsilon }_{\alpha }\left({\mathit{x}}_k\right)$ of solutions ${\mathit{x}}_k\in \mathit{X}$ in feasible region and the best solutions of POL_P and POL_V. The interest rate and upper limit of the loan are fixed as L = 0.04 and m = 2. (a) $\alpha =0.15$ (b) $\alpha =0.35$.

Figure 8. Profits ${\gamma }_{\alpha }\left({\mathit{x}}_k\right)$ and values ${\upsilon }_{\alpha }\left({\mathit{x}}_k\right)$ of solutions ${\mathit{x}}_k\in \mathit{X}$ on two hyper-planes in (29(29) $\text{𝟙}{\mathit{x}}^T=1-x_0.$(29) ) with $x_0=-2$ and $x_0=0$. The interest rate and upper limit of the loan are fixed as L = 0.04 and m = 2. (a) $\alpha =0.15$ (b) $\alpha =0.35$.

The distributions of points in Figures 7 and 8 are almost the same as those of Figures 5 and 6, respectively. However, we can see the conflict between the two best solutions to $\alpha =0.35$ in Figure 7(b), not in Figure 7(a) of $\alpha =0.15$.

8.3. Solutions of POL_P and POL_V

The solutions of POL_P in (14(14) $[\begin{array}{rl}max& {\gamma }_{\alpha }\left(\mathit{x}\right)=\mu \left(\mathit{x}\right)+{\mathrm{\Phi }}^{-1}\left(\alpha \right)\sigma \left(\mathit{x}\right)\\ \text{sub. to}& x_1+x_2+\cdots +x_n=\text{𝟙}{\mathit{x}}^T\ge 1,\\ & x_1+x_2+\cdots +x_n=\text{𝟙}{\mathit{x}}^T\le m+1,\\ & 0\le x_i,i=1,\dots ,n\end{array}$(14) ) and POL_V in (24(24) $[\begin{array}{rl}max& {\upsilon }_{\alpha }\left(\mathit{x}\right)={\upsilon }_g\left(\mathit{x}\right)+{\upsilon }_s\left(\mathit{x}\right)\\ \text{sub. to}& x_1+x_2+\cdots +x_n=\text{𝟙}{\mathit{x}}^T\ge 1,\\ & x_1+x_2+\cdots +x_n=\text{𝟙}{\mathit{x}}^T\le m+1,\\ & 0\le x_i,i=1,\dots ,n.\end{array}$(24) ) are obtained, respectively, by the multi-start local search method using K = 30 initial solutions.

Firstly, the interest rate and upper limit of the loan are fixed as L = 0.01 and m = 2. Figure 9 shows the profits ${\gamma }_{\alpha }\left(\mathit{x}\right)$ achieved, respectively, by the solutions of POL_P (×) and POL_V (•). The horizontal axis of Figure 9 is the failure probability $\alpha \in (0,0.5)$. Similarly, Figure 10 shows the values ${\upsilon }_{\alpha }\left(\mathit{x}\right)$ achieved by the two kinds of solutions. Figure 11 shows the loan amounts $x_0\in [-m,0]$ spent by the two kinds of solutions. Incidentally, Figure 9 shows the efficient frontier of modern portfolio theory [8], i.e. the trade-off between the profit and the risk measured by $\alpha \in (0,0.5)$.

Figure 9. Profits ${\gamma }_{\alpha }\left(\mathit{x}\right)$ achieved by solutions of POL_P and POL_V with L = 0.01 and m = 2.

Figure 10. Values ${\upsilon }_{\alpha }\left(\mathit{x}\right)$ achieved by solutions of POL_P and POL_V with L = 0.01 and m = 2.

Figure 11. Loan amounts $x_0$ spent by solutions of POL_P and POL_V with L = 0.01 and m = 2.

From Figures 9 and 10, both the profit and the value increase in proportion to the failure probability. We can also confirm the difference between the two kinds of solutions, especially when the failure probability is small: $\alpha \le 0.25$. Specifically, the solution of POL_P is better than the solution of POL_V in terms of profit. The solution of POL_V is better than the solution of POL_P in terms of value. On the other hand, there is no clear difference between the two kinds of solutions when the failure probability is large: $\alpha \ge 0.3$. From Figure 11, we can see that the inconsistency between the profit and the value is caused by the difference in loan amounts. The solution of POL_V does not use the loan ($x_0=0$) if $\alpha \le 0.25$. On the other hand, the solution of POL_P uses the loan up to the limit ($x_0=-2$) for every $\alpha \in (0,0.5)$.

Secondly, the interest rate and upper limit of the loan are fixed as L = 0.04 and m = 2. In the same way as Figure 9 through Figure 11, Figure 12 shows the profits ${\gamma }_{\alpha }\left(\mathit{x}\right)$ achieved by the solutions of POL_P (×) and POL_V (•) for failure probabilities $\alpha \in (0,0.5)$. Figure 13 shows the values ${\upsilon }_{\alpha }\left(\mathit{x}\right)$ achieved by the two kinds of solutions. Figure 14 shows the loan amounts $x_0\in [-m,0]$ spent by the two kinds of solutions.

Figure 12. Profits ${\gamma }_{\alpha }\left(\mathit{x}\right)$ achieved by solutions of POL_P and POL_V with L = 0.04 and m = 2.

Figure 13. Values ${\upsilon }_{\alpha }\left(\mathit{x}\right)$ achieved by solutions of POL_P and POL_V with L = 0.04 and m = 2.

Figure 14. Loan amounts $x_0$ spent by solutions of POL_P and POL_V with L = 0.04 and m = 2.

From Figures 12 and 13, we can confirm the difference between the two kinds of solutions, especially when the failure probability is large. However, there is no clear difference between the two kinds of solutions when the failure probability is small. From Figure 14, we can see that the inconsistency between the profit and the value is caused by the difference in loan amounts. The solution of POL_V does not use the loan ($x_0=0$) at all. On the other hand, the solution of POL_P uses the loan up to the limit ($x_0=-2$) if the failure probability is large. By using the loan, the profit of the solution of POL_P increases in Figure 12. However, the value of the same solution of POL_P drops significantly and becomes negative in Figure 13. That is because the value function in (15(15) $\upsilon \left(z\right)=\{\begin{array}{ll}{z}^a& \text{if}z\ge 0\\ -\lambda (-z{)}^b& \text{if}z<0\end{array}$(15) ) evaluates the loss much larger than the gain.

If the value ${\upsilon }_{\alpha }\left(\mathit{x}\right)$ of the solution $\mathit{x}\in {\mathbb{R}}^n$ is negative, the investor will not want to use the portfolio $\mathit{x}\in {\mathbb{R}}^n$ even if it is profitable. From Figures 10 and 13, we can see that the solutions of POL_P are not desired by the investor in many cases.

8.4. Characteristics of POL_V

By using the multi-start local search method with K = 30 initial solutions, the best solutions ${\mathit{x}}^{\star }\in \mathit{X}$ of POL_V in (24(24) $[\begin{array}{rl}max& {\upsilon }_{\alpha }\left(\mathit{x}\right)={\upsilon }_g\left(\mathit{x}\right)+{\upsilon }_s\left(\mathit{x}\right)\\ \text{sub. to}& x_1+x_2+\cdots +x_n=\text{𝟙}{\mathit{x}}^T\ge 1,\\ & x_1+x_2+\cdots +x_n=\text{𝟙}{\mathit{x}}^T\le m+1,\\ & 0\le x_i,i=1,\dots ,n.\end{array}$(24) ) are obtained under several conditions. Then the characteristics of POL_V are clarified through the analysis of those solutions.

Firstly, in order to assess the effect of the interest rate of the loan $L\in \mathbb{R}$ on the solution of POL_V, the upper limit of the loan is fixed as m = 2. Then the solutions of POL_V are obtained for L = 0, 0.01, and 0.04. Figure 15 shows the profits ${\gamma }_{\alpha }\left({\mathit{x}}^{\star }\right)$ achieved by the solutions of POL_V with three different interest rates. The horizontal axis of Figure 15 is the failure probability $\alpha \in (0,0.5)$. Similarly, Figure 16 shows the values ${\upsilon }_{\alpha }\left({\mathit{x}}^{\star }\right)$ achieved by the three kinds of solutions of POL_V. Figure 17 shows the loan amounts $x_0\in [-m,0]$ spent by the three kinds of solutions of POL_V.

Figure 15. Profits ${\gamma }_{\alpha }\left({\mathit{x}}^{\star }\right)$ achieved by solutions of POL_V with m = 2 and L = 0, 0.01, 0.04.

Figure 16. Values ${\upsilon }_{\alpha }\left({\mathit{x}}^{\star }\right)$ achieved by solutions of POL_V with m = 2 and L = 0, 0.01, 0.04.

Figure 17. Loan amounts $x_0$ spent by solutions of POL_V with m = 2 and L = 0, 0.01, 0.04.

Secondly, in order to assess the effect of the upper limit of the loan $m\in \mathbb{R}$ on the solution of POL_V, the interest rate of the loan is fixed as L = 0.01. Then the solutions of POL_V are obtained for m = 1, 2, and 4. Figure 18 shows the profits ${\gamma }_{\alpha }\left({\mathit{x}}^{\star }\right)$ achieved by the solutions of POL_V with three different loan limits. The horizontal axis of Figure 18 is the failure probability $\alpha \in (0,0.5)$. Figure 19 shows the values ${\upsilon }_{\alpha }\left({\mathit{x}}^{\star }\right)$ achieved by the three kinds of solutions of POL_V. Figure 20 shows the loan amounts $x_0\in [-m,0]$ spent by the three kinds of solutions of POL_V.

Figure 18. Profits ${\gamma }_{\alpha }\left({\mathit{x}}^{\star }\right)$ achieved by solutions of POL_V with L = 0.01 and m = 1, 2, 4.

Figure 19. Values ${\upsilon }_{\alpha }\left({\mathit{x}}^{\star }\right)$ achieved by solutions of POL_V with L = 0.01 and m = 1, 2, 4.

Figure 20. Loan amounts $x_0$ spent by solutions of POL_V with L = 0.01 and m = 1, 2, 4.

From Figures 15 and 18, the profit of the solution of POL_V increases in proportion to the failure probability. Furthermore, the profit increases rapidly when the loan is used. From Figure 15, the smaller the interest rate, the greater the profit. From Figure 18, the greater the loan amount, the greater the profit.

From Figures 16 and 19, the value of the solution of POL_V increases in proportion to the failure probability as well as the profit. The greater the profit, the greater the value. However, the value does not increase rapidly even if the loan is used. The increase in values is very moderate compared to the increase in profits.

From Figures 17 and 20, the solution of POL_V uses the loan up to the limit ($x_0=-m$) when it uses the loan. At that time, the solution of POL_V exists on the upper hyper-plane: $\text{𝟙}{\mathit{x}}^T=m+1$. In fact, we have confirmed experimentally that the loan is used up to the limit even if the upper limit is given as m = 10.

Figure 17, we can see that the interest rate $L\in \mathbb{R}$ determines whether the loan is used or not. The loan with L = 0.04 is not used for any $\alpha \in (0,0.5)$. On the other hand, the loan with L = 0 is used for all $\alpha \in (0,0.5)$ except $\alpha =0.1$.

9. Real world application

9.1. Actual data of asset returns

The proposed methodology is applied to historical asset data of Japanese stock market. First of all, in order to obtain the multivariable normal distribution in (4(4) $\mathit{\xi }\sim \text{Normal}(\mathit{\mu },\mathit{C})$(4) ), or the mathematical model of asset returns, we used 60 monthly data (from January 2018 to December 2022) of TOPIX (Tokyo Stock Price Index) Large 70 published by Toyo Keizai Inc. From 70 assets included in TOPIX Large 70, we selected top n = 30 assets in descending order of the mean ${\mu }_i\in \mathbb{R}$ of asset return ${\xi }_i\in \mathbb{R}$ in (3(3) ${\xi }_i\sim \text{Normal}({\mu }_i,{\sigma }_i^2).$(3) ).

Figure 21 shows the means ${\mu }_i\in \mathbb{R}$ of the top 30 asset returns. Even though profits from investments include capital gains and income gains, income gains are not taken into account in the means in Figure 21. Therefore, we have added the average income gain of Japanese stocks, i.e. 0.02, uniformly to all means in Figure 21. Figure 22 also shows the standard deviations ${\sigma }_i\in \mathbb{R}$ of the top 30 asset returns. By using the Kolmogorov–Smirnov test [43], we have verified the normality of the top 30 asset returns ${\xi }_i\in \mathbb{R}$ with significance level: 0.05. As a result, we have confirmed that every asset return ${\xi }_i\in \mathbb{R}$ can be regarded as following a normal distribution.

Figure 21. Means ${\mu }_i\in \mathbb{R}$ of top 30 asset returns ${\xi }_i\in \mathbb{R}$, $i=1,\dots ,30$.

Figure 22. Standard Deviations (SD) ${\sigma }_i\in \mathbb{R}$ of top 30 asset returns ${\xi }_i\in \mathbb{R}$, $i=1,\dots ,30$.

9.2. Performance of optimization method

The multi-start local search method is used to solve POL_P in (14(14) $[\begin{array}{rl}max& {\gamma }_{\alpha }\left(\mathit{x}\right)=\mu \left(\mathit{x}\right)+{\mathrm{\Phi }}^{-1}\left(\alpha \right)\sigma \left(\mathit{x}\right)\\ \text{sub. to}& x_1+x_2+\cdots +x_n=\text{𝟙}{\mathit{x}}^T\ge 1,\\ & x_1+x_2+\cdots +x_n=\text{𝟙}{\mathit{x}}^T\le m+1,\\ & 0\le x_i,i=1,\dots ,n\end{array}$(14) ) and POL_V in (24(24) $[\begin{array}{rl}max& {\upsilon }_{\alpha }\left(\mathit{x}\right)={\upsilon }_g\left(\mathit{x}\right)+{\upsilon }_s\left(\mathit{x}\right)\\ \text{sub. to}& x_1+x_2+\cdots +x_n=\text{𝟙}{\mathit{x}}^T\ge 1,\\ & x_1+x_2+\cdots +x_n=\text{𝟙}{\mathit{x}}^T\le m+1,\\ & 0\le x_i,i=1,\dots ,n.\end{array}$(24) ), which are defined with the above mathematical model. Since the set of initial solutions are generated randomly, the solution obtained by the multi-start local search method depends not only on the number of initial solutions, but also on the random seed. Of course, the solution also depends on the failure probability $\alpha \in (0,0.5)$. In order to determine an appropriate number of initial solutions K, we have statistically evaluated the effect of K on the solution of POL_V with L = 0.001 and m = 2.

The multi-start local search method using MATLAB solver [42] is applied to POL_V with K = 10, 30, 90 and $\alpha =0.15$, 0.35. The local optimality and feasibility of the solution obtained by MATLAB solver are guaranteed based on Karush–Kuhn–Tucker (KKT) conditions. For each case, 20 best solutions ${\mathit{x}}^{\star }\in {\mathbb{R}}^n$ are obtained by the proposed method with different random seeds. The mean and standard deviation (within parentheses) of the values ${\upsilon }_{\alpha }\left({\mathit{x}}^{\star }\right)\in \mathbb{R}$ of them are summarized in Table 4.

Table 4. Values ${\upsilon }_{\alpha }\left({\mathit{x}}^{\star }\right)\in \mathbb{R}$ of best solutions ${\mathit{x}}^{\star }\in {\mathbb{R}}^n$ obtained by multi-start local search method for the number of initial solutions.

K	10	30	90
$\alpha =0.15$	$-3.79695\times {10}^{-2}$	$-3.79695\times {10}^{-2}$	$-3.79695\times {10}^{-2}$
	($5.07744\times {10}^{-13}$)	($1.86414\times {10}^{-13}$)	($1.01382\times {10}^{-13}$)
$\alpha =0.35$	$6.74860\times {10}^{-3}$	$6.74860\times {10}^{-3}$	$6.74860\times {10}^{-3}$
	($1.97847\times {10}^{-12}$)	($6.80042\times {10}^{-13}$)	($3.34231\times {10}^{-13}$)

For each of $\alpha \in \{0.15,0.35\}$ in Table 4, there is no difference in the mean values due to difference in the figures of K. The standard deviations of ${\upsilon }_{\alpha }\left({\mathit{x}}^{\star }\right)$ are also very small. From Table 4, the best solutions ${\mathit{x}}^{\star }\in {\mathbb{R}}^n$ obtained by multi-start local search method seem to converge sufficiently to the global optimal solution of POL_V.

The computation time for the multi-start local search method is proportional to the number of initial solutions. From Table 4, we adopt K = 30 for the method because it seems to be sufficient for a practical analysis of the solution of POL_V.

9.3. Comparison of POL_P and POL_V

In both of POL_P and POL_V, the upper limit of the loan is fixed as m = 2. For the functions in (15(15) $\upsilon \left(z\right)=\{\begin{array}{ll}{z}^a& \text{if}z\ge 0\\ -\lambda (-z{)}^b& \text{if}z<0\end{array}$(15) ) and (16(16) $\omega \left(\theta \right)=\frac{{\theta }^c}{({\theta }^c+(1-\theta {)}^c{)}^{1/c}}$(16) ), the parameter values in Table 3 are used.

Firstly, the interest rate of the loan is fixed as L = 0.001. Figure 23 shows the profits ${\gamma }_{\alpha }\left(\mathit{x}\right)$ achieved, respectively, by the solutions of POL_P (×) and POL_V (•). The horizontal axis of Figure 23 indicates the failure probability $\alpha \in (0,0.5)$. Similarly, Figure 24 shows the values ${\upsilon }_{\alpha }\left(\mathit{x}\right)$ achieved by the two kinds of solutions. Figure 25 shows the loan amounts $x_0\in [-m,0]$ spent by the two kinds of solutions.

Figure 23. Profits ${\gamma }_{\alpha }\left(\mathit{x}\right)$ achieved by solutions of POL_P and POL_V with L = 0.001 and m = 2.

Figure 24. Values ${\upsilon }_{\alpha }\left(\mathit{x}\right)$ achieved by solutions of POL_P and POL_V with L = 0.001 and m = 2.

Figure 25. Loan amounts $x_0$ spent by solutions of POL_P and POL_V with L = 0.001 and m = 2.

From Figure 25, the solution of POL_P uses the loan for $\alpha \ge 0.2$, while the solution of POL_V uses the loan for $\alpha \ge 0.3$. From Figures 23 and 25, the use of loan clearly increases the profits of the solutions of POL_P and POL_V.

From Figures 23 and 24, we can see the inconsistency between the profit and the value when the failure probability is large: $\alpha \ge 0.2$. Especially, the value of the solution of POL_P drops significantly in Figure 24 when the solution uses the loan in Figure 25. In order to analyse the phenomenon observed in Figure 24, we have examined the components of the values achieved by the two kinds of solutions.

Figure 26 shows the gain ${\upsilon }_g\left({\mathit{x}}^{†}\right)$, loss ${\upsilon }_s\left({\mathit{x}}^{†}\right)$, and value ${\upsilon }_{\alpha }\left({\mathit{x}}^{†}\right)={\upsilon }_g\left({\mathit{x}}^{†}\right)+{\upsilon }_s\left({\mathit{x}}^{†}\right)$ for the solution ${\mathit{x}}^{†}\in {\mathbb{R}}^n$ of POL_P in Figure 24. Figure 27 also shows the gain ${\upsilon }_g\left({\mathit{x}}^{\star }\right)$, loss ${\upsilon }_s\left({\mathit{x}}^{\star }\right)$, and value ${\upsilon }_{\alpha }\left({\mathit{x}}^{\star }\right)={\upsilon }_g\left({\mathit{x}}^{\star }\right)+{\upsilon }_s\left({\mathit{x}}^{\star }\right)$ for the solution ${\mathit{x}}^{\star }\in {\mathbb{R}}^n$ of POL_V in Figure 24. From Figures 26 and 27, both gains and losses increase sharply when the loan is used in Figure 25. From Figure 26, the increase in loss is greater than the increase in gain for the solution of POL_P. That is because the value function in (15(15) $\upsilon \left(z\right)=\{\begin{array}{ll}{z}^a& \text{if}z\ge 0\\ -\lambda (-z{)}^b& \text{if}z<0\end{array}$(15) ) evaluates the loss much larger than the gain. Furthermore, the solution of POL_P uses the loan even if the failure probability is small. As a result, the value ${\upsilon }_{\alpha }\left({\mathit{x}}^{†}\right)$ for the solution ${\mathit{x}}^{†}\in {\mathbb{R}}^n$ of POL_P drops significantly in Figure 24.

Figure 26. Gain ${\upsilon }_g\left({\mathit{x}}^{†}\right)$, loss ${\upsilon }_s\left({\mathit{x}}^{†}\right)$, and value ${\upsilon }_{\alpha }\left({\mathit{x}}^{†}\right)$ for solution of POL_P in Figure 24.

Figure 27. Gain ${\upsilon }_g\left({\mathit{x}}^{\star }\right)$, loss ${\upsilon }_s\left({\mathit{x}}^{\star }\right)$, and value ${\upsilon }_{\alpha }\left({\mathit{x}}^{\star }\right)$ for solution of POL_V in Figure 24.

Figure 28 shows the profits in Figure 23 and the values in Figure 24 for the solutions of POL_P and POL_V. From Figure 28, the value ${\upsilon }_{\alpha }\left({\mathit{x}}^{\star }\right)$ for the solution ${\mathit{x}}^{\star }\in {\mathbb{R}}^n$ of POL_V increases in proportion to the profit. On the other hand, the value ${\upsilon }_{\alpha }\left({\mathit{x}}^{†}\right)$ for the solution ${\mathit{x}}^{†}\in {\mathbb{R}}^n$ of POL_P does not increase in proportion to the profit. Therefore, the solution of POL_P creates a conflict between the profit and the value.

Figure 28. Profits and values for solutions of POL_P and POL_V with L = 0.001 and m = 2.

Secondly, the interest rate of the loan is raised and fixed as L = 0.004. In the same way as Figures 23 through 28, Figures 29 through 34 compare the solution of POL_V (•) with the solution of POL_P (×).

Figure 29. Profits ${\gamma }_{\alpha }\left(\mathit{x}\right)$ achieved by solutions of POL_P and POL_V with L = 0.004 and m = 2.

Figure 31 shows the loan amounts $x_0\in [-m,0]$ spent by the two kinds of solutions. From Figure 31, the solution ${\mathit{x}}^{\star }\in {\mathbb{R}}^n$ of POL_V does not use the loan at all. Therefore, the increase of the profit ${\gamma }_{\alpha }\left({\mathit{x}}^{\star }\right)$ in Figure 29 is moderate for the failure probability $\alpha \in (0,0.5)$. The solution ${\mathit{x}}^{†}\in {\mathbb{R}}^n$ of POL_P uses the loan up to the limit ($x_0=-2$) for $\alpha \ge 0.2$. As a result, the profit ${\gamma }_{\alpha }\left({\mathit{x}}^{†}\right)$ rises sharply in Figure 29. However, the value ${\upsilon }_{\alpha }\left({\mathit{x}}^{†}\right)$ drops significantly in Figure 30 when the loan is used in Figure 31. The loss for the solution of POL_L in Figure 32 is much larger than the loss for the solution of POL_V in Figure 33.  Furthermore, the loss is much larger than the gain in Figure 32.

Figure 30. Values ${\upsilon }_{\alpha }\left(\mathit{x}\right)$ achieved by solutions of POL_P and POL_V with L = 0.004 and m = 2.

Figure 31. Loan amounts $x_0$ spent by solutions of POL_P and POL_V with L = 0.004 and m = 2.

Figure 34 shows the profits in Figure 29 and the values in Figure 30 for the solutions of POL_P and POL_V. From Figure 34, the value ${\upsilon }_{\alpha }\left({\mathit{x}}^{\star }\right)$ for the solution ${\mathit{x}}^{\star }\in {\mathbb{R}}^n$ of POL_V increases in proportion to the profit. On the other hand, the value ${\upsilon }_{\alpha }\left({\mathit{x}}^{†}\right)$ for the solution ${\mathit{x}}^{†}\in {\mathbb{R}}^n$ of POL_P does not increase in proportion to the profit. Therefore, the solution of POL_P creates a conflict between the profit and the value.

Figure 32. Gain ${\upsilon }_g\left({\mathit{x}}^{†}\right)$, loss ${\upsilon }_s\left({\mathit{x}}^{†}\right)$, and value ${\upsilon }_{\alpha }\left({\mathit{x}}^{†}\right)$ for solution of POL_P in Figure 30.

Figure 33. Gain ${\upsilon }_g\left({\mathit{x}}^{\star }\right)$, loss ${\upsilon }_s\left({\mathit{x}}^{\star }\right)$, and value ${\upsilon }_{\alpha }\left({\mathit{x}}^{\star }\right)$ for solution of POL_V in Figure 30.

Figure 34. Profits and values for solutions of POL_P and POL_V with L = 0.004 and m = 2.

A portfolio in which either the profit or the value is negative will not be adopted by investors. Since the value of the solution of POL_P is negative for any $\alpha \in (0,0.5)$ in both Figures 28 and 34, we can conclude that the solution of POL_V is more acceptable to investors than the solution of POL_P. We can also say that the use of the loan with lower interest rate is effective to increase the value of the portfolio.

10. Conclusion

The portfolio optimization problem that allows the borrowed money from a loan to be invested in risk assets has been formulated as a chance constrained problem [12].

First of all, the chance constrained problem is transformed into a deterministic optimization problem called POL_P. The optimal solution of POL_P is the portfolio that maximizes the profit under a given failure probability $\alpha \in (0,0.5)$.

In order to reflect investor preferences called “values” in the portfolio, POL_P has been expanded to POL_V based on prospect theory. Prospect theory [14,15] can be summarized by two functions, namely, the value function and the probability weighting function. Therefore, the failure probability $\alpha \in (0,0.5)$ is transformed into the weighted one by the probability weighting function. Then the profit of the portfolio is also transformed into the value by the value function. The optimal solution of POL_V is the portfolio that maximizes the value under a given failure probability.

CHM proposed by authors [24] can generate a set of solutions randomly in the feasible regions of POL_P and POL_V. In order to have an in-depth understanding of the characteristics of POL_V in comparison with POL_P, the fitness landscape analysis method using CHM [22] is employed. Furthermore, it is shown mathematically that POL_V is the non-convex optimization problem. Therefore, the multi-start local search method with CHM [23] is also used to solve POL_V effectively.

Through the numerical experiments conducted on artificial and actual asset data sets, the solution of POL_V is compared with the solution of POL_P. As a result, the characteristics of POL_V are revealed as follows:

• Even if the solution ${\mathit{x}}^{†}\in {\mathbb{R}}^n$ of POL_P uses the loan to maximize the profit, the solution ${\mathit{x}}^{\star }\in {\mathbb{R}}^n$ of POL_V does not always use the loan to maximize the value in the same condition. As a result, the conflict between the profit and the value, i.e. ${\gamma }_{\alpha }\left({\mathit{x}}^{†}\right)>{\gamma }_{\alpha }\left({\mathit{x}}^{\star }\right)$ and ${\upsilon }_{\alpha }\left({\mathit{x}}^{†}\right)<{\upsilon }_{\alpha }\left({\mathit{x}}^{\star }\right)$, is observed in many cases.

• Whether or not the solution of POL_V uses the loan depends on the failure probability $\alpha \in (0,0.5)$ and the interest rate $L\in \mathbb{R}$ of the loan. Since the solution of POL_V is risk averse, it does not use the loan if the failure probability is small. Also, the loan is not used by the solution if the interest rate is high.

• From Figures 16 and 19, the loan is effectively used by the solution ${\mathit{x}}^{\star }\in {\mathbb{R}}^n$ of POL_V to increase the value ${\upsilon }_{\alpha }\left({\mathit{x}}^{\star }\right)$ if the conditions are right.

• From Theorem 4.1, the solution of POL_P uses the loan up to the limit if it uses the loan. From Figures 17 and 20, it is experimentally confirmed that the solution of POL_V also uses the loan up to the limit if it uses the loan.

• From Figures 28 and 34, the solution ${\mathit{x}}^{\star }\in {\mathbb{R}}^n$ of POL_V is more practical than the solution ${\mathit{x}}^{†}\in {\mathbb{R}}^n$ of POL_P. That is because the value ${\upsilon }_{\alpha }\left({\mathit{x}}^{†}\right)$ does not increase in proportion to the profit ${\gamma }_{\alpha }\left({\mathit{x}}^{†}\right)$ and decreases in some cases, while the value ${\upsilon }_{\alpha }\left({\mathit{x}}^{\star }\right)$ always increases in proportion to the profit ${\gamma }_{\alpha }\left({\mathit{x}}^{\star }\right)$.

From the last characteristic of POL_V, we can say that POL_V offers a realistic portfolio acceptable to risk-averse investors rather than the conventional POL_P.

In this paper, the mathematical model of asset returns is provided by a normal distribution. However, the normality of data on asset returns is not always guaranteed. Therefore, in future work, the prospect theory-based portfolio optimization problem using loan should be formulated as a sample-based chance constrained problem and solved with advanced methods [29–32]. Furthermore, we would like to theoretically prove the properties of POL_V that we have experimentally clarified in this paper.

Acknowledgments

We are grateful for the valuable comments and suggestions received from anonymous reviewers. These contributed highly to the improvement of this paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by JSPS KAKENHI Grant Numbers 21K12060,22K12187.

Notes on contributors

Kiyoharu Tagawa

Kiyoharu Tagawa received his Ph. D. degree from Kobe University, Japan, in 1997. He is a professor in Kindai University, Japan since 2007. His current research interests include system modeling, decision making, optimization methods under uncertainties, and their applications to real-world problems. He is a member of SICE, IEEJ, IPSJ, and IEEE.

Yukiko Orito

Yukiko Orito received her Ph.D. degree from Tokyo Metropolitan Institute of Technology in 2003. She is an associate professor in College of Engineering, Tamagawa University, Japan since 2022. Her current research interests are in evolutionary algorithms and its applications to engineering, finance, and economics. She is a member of IEEJ, JSEC, IPSJ and IEEE.

Appendices

Appendix 1.

Proof of Theorem 4.1

Theorem 4.1 is proved from Lemma A.1.

Lemma A.1

The standard deviation $\sigma \left(\mathit{x}\right)$ in (9(9) $r(\mathit{x},\mathit{\xi })\sim \text{Normal}\left(\mu \right(\mathit{x}),\sigma (\mathit{x}{)}^2)$(9) ) is convex.

Proof.

Since the covariance matrix $\mathit{C}$ in (7(7) $\mathit{C}=\mathit{DRD}.$(7) ) is positive semi-define, the standard deviation $\sigma \left(\mathit{x}\right)$ in (9(9) $r(\mathit{x},\mathit{\xi })\sim \text{Normal}\left(\mu \right(\mathit{x}),\sigma (\mathit{x}{)}^2)$(9) ) can be decomposed as (A1) $\sigma \left(\mathit{x}\right)=\sqrt{\mathit{xC}{\mathit{x}}^T}=\sqrt{\mathit{xA}{\mathit{A}}^T{\mathit{x}}^T}=\sqrt{\mathit{y}{\mathit{y}}^T}$(A1) where $\mathit{C}=\mathit{A}{\mathit{A}}^T$ and $\mathit{y}=\mathit{xA}\in {\mathbb{R}}^n$.

From (A1(A1) $\sigma \left(\mathit{x}\right)=\sqrt{\mathit{xC}{\mathit{x}}^T}=\sqrt{\mathit{xA}{\mathit{A}}^T{\mathit{x}}^T}=\sqrt{\mathit{y}{\mathit{y}}^T}$(A1) ), $\sigma \left(\mathit{x}\right)$ is a norm. Therefore, the norm $\sigma \left(\mathit{x}\right)$ meets the triangle inequality for any $\eta \in [0,1]$ as (A2) $\sigma \left(\eta \mathit{x}\right)+\sigma \left(\right(1-\eta \left)\mathit{v}\right)\ge \sigma (\eta \mathit{x}+(1-\eta \left)\mathit{v}\right)$(A2) where $\mathit{v}\in {\mathbb{R}}^n$ and $\mathit{x}\ne \mathit{v}$.

From (A1(A1) $\sigma \left(\mathit{x}\right)=\sqrt{\mathit{xC}{\mathit{x}}^T}=\sqrt{\mathit{xA}{\mathit{A}}^T{\mathit{x}}^T}=\sqrt{\mathit{y}{\mathit{y}}^T}$(A1) ), the first term of the left side of (A2(A2) $\sigma \left(\eta \mathit{x}\right)+\sigma \left(\right(1-\eta \left)\mathit{v}\right)\ge \sigma (\eta \mathit{x}+(1-\eta \left)\mathit{v}\right)$(A2) ) can be transformed as (A3) $\sigma \left(\eta \mathit{x}\right)=\sqrt{\eta \mathit{y}(\eta \mathit{y}{)}^T}=\eta \sqrt{\mathit{y}{\mathit{y}}^T}=\mathrm{\eta \sigma }\left(\mathit{x}\right).$(A3) The second term of the left side of (A2(A2) $\sigma \left(\eta \mathit{x}\right)+\sigma \left(\right(1-\eta \left)\mathit{v}\right)\ge \sigma (\eta \mathit{x}+(1-\eta \left)\mathit{v}\right)$(A2) ) can be transformed in the same way with (A3(A3) $\sigma \left(\eta \mathit{x}\right)=\sqrt{\eta \mathit{y}(\eta \mathit{y}{)}^T}=\eta \sqrt{\mathit{y}{\mathit{y}}^T}=\mathrm{\eta \sigma }\left(\mathit{x}\right).$(A3) ). Therefore, (A2(A2) $\sigma \left(\eta \mathit{x}\right)+\sigma \left(\right(1-\eta \left)\mathit{v}\right)\ge \sigma (\eta \mathit{x}+(1-\eta \left)\mathit{v}\right)$(A2) ) is written as (A4) $\mathrm{\eta \sigma }\left(\mathit{x}\right)+(1-\eta )\sigma \left(\mathit{v}\right)\ge \sigma (\eta \mathit{x}+(1-\eta \left)\mathit{v}\right).$(A4) From (A4(A4) $\mathrm{\eta \sigma }\left(\mathit{x}\right)+(1-\eta )\sigma \left(\mathit{v}\right)\ge \sigma (\eta \mathit{x}+(1-\eta \left)\mathit{v}\right).$(A4) ), $\sigma \left(\mathit{x}\right)$ is convex.

From ${\gamma }_{\alpha }\left(\mathit{x}\right)=\mu \left(\mathit{x}\right)+{\mathrm{\Phi }}^{-1}\left(\alpha \right)\sigma \left(\mathit{x}\right)$ in (14(14) $[\begin{array}{rl}max& {\gamma }_{\alpha }\left(\mathit{x}\right)=\mu \left(\mathit{x}\right)+{\mathrm{\Phi }}^{-1}\left(\alpha \right)\sigma \left(\mathit{x}\right)\\ \text{sub. to}& x_1+x_2+\cdots +x_n=\text{𝟙}{\mathit{x}}^T\ge 1,\\ & x_1+x_2+\cdots +x_n=\text{𝟙}{\mathit{x}}^T\le m+1,\\ & 0\le x_i,i=1,\dots ,n\end{array}$(14) ), (A5) $\begin{array}{rl}& \eta {\gamma }_{\alpha }\left(\mathit{x}\right)+(1-\eta ){\gamma }_{\alpha }\left(\mathit{v}\right)-{\gamma }_{\alpha }(\eta \mathit{x}+(1-\eta \left)\mathit{v}\right)\\ & ={\mathrm{\Phi }}^{-1}\left(\alpha \right)\left(\mathrm{\eta \sigma }\right(\mathit{x})+(1-\eta \left)\sigma \right(\mathit{v})-\sigma (\eta \mathit{x}+(1-\eta )\mathit{v}\left)\right).\end{array}$(A5) From (A4(A4) $\mathrm{\eta \sigma }\left(\mathit{x}\right)+(1-\eta )\sigma \left(\mathit{v}\right)\ge \sigma (\eta \mathit{x}+(1-\eta \left)\mathit{v}\right).$(A4) ) in Lemma A.1 and ${\mathrm{\Phi }}^{-1}\left(\alpha \right)<0$ for $\alpha \in (0,0.5)$, the right side of (A5(A5) $\begin{array}{rl}& \eta {\gamma }_{\alpha }\left(\mathit{x}\right)+(1-\eta ){\gamma }_{\alpha }\left(\mathit{v}\right)-{\gamma }_{\alpha }(\eta \mathit{x}+(1-\eta \left)\mathit{v}\right)\\ & ={\mathrm{\Phi }}^{-1}\left(\alpha \right)\left(\mathrm{\eta \sigma }\right(\mathit{x})+(1-\eta \left)\sigma \right(\mathit{v})-\sigma (\eta \mathit{x}+(1-\eta )\mathit{v}\left)\right).\end{array}$(A5) ) is not positive. Hence, we obtain (A6) $\eta {\gamma }_{\alpha }\left(\mathit{x}\right)+(1-\eta ){\gamma }_{\alpha }\left(\mathit{v}\right)\le {\gamma }_{\alpha }(\eta \mathit{x}+(1-\eta \left)\mathit{v}\right).$(A6) From (A6(A6) $\eta {\gamma }_{\alpha }\left(\mathit{x}\right)+(1-\eta ){\gamma }_{\alpha }\left(\mathit{v}\right)\le {\gamma }_{\alpha }(\eta \mathit{x}+(1-\eta \left)\mathit{v}\right).$(A6) ), the profit ${\gamma }_{\alpha }\left(\mathit{x}\right)$ in (14(14) $[\begin{array}{rl}max& {\gamma }_{\alpha }\left(\mathit{x}\right)=\mu \left(\mathit{x}\right)+{\mathrm{\Phi }}^{-1}\left(\alpha \right)\sigma \left(\mathit{x}\right)\\ \text{sub. to}& x_1+x_2+\cdots +x_n=\text{𝟙}{\mathit{x}}^T\ge 1,\\ & x_1+x_2+\cdots +x_n=\text{𝟙}{\mathit{x}}^T\le m+1,\\ & 0\le x_i,i=1,\dots ,n\end{array}$(14) ) is concave.

Appendix 2.

Proof of Theorem 4.2

Theorem 4.2 is proved from Lemma A.2.

Lemma A.2

Let ${\mathit{x}}^{\text{‡}}\in {\mathbb{R}}^n$ be the optimal solution of POL_P under the condition that it does not use the loan: $x_0=0$. The optimal solution ${\mathit{x}}^{\text{‡}}\in {\mathbb{R}}^n$ is improved by borrowing money from the loan if the interest rate of the loan $L\in \mathbb{R}$ meets the condition: (A7) ${\gamma }_{\alpha }\left({\mathit{x}}^{\text{‡}}\right)>L\ge 0$(A7) where ${\gamma }_{\alpha }\left({\mathit{x}}^{\text{‡}}\right)=\mathit{\mu }{\mathit{x}}^{\text{‡}T}+{\mathrm{\Phi }}^{-1}\left(\alpha \right)\sigma \left({\mathit{x}}^{\text{‡}}\right)$ and $x_0=1-\text{𝟙}{\mathit{x}}^{\text{‡}T}=0$.

Proof.

Let us think of a new solution $\mathit{v}=\kappa {\mathit{x}}^{\text{‡}}$, $\kappa >1$ for POL_P. The new solution $\mathit{v}\in {\mathbb{R}}^n$ borrows money from the loan as (A8) $v_0=1-\text{𝟙}{\mathit{v}}^T=1-\kappa \text{𝟙}{\mathit{x}}^{\text{‡}T}=1-\kappa <0$(A8) where $v_0\in \mathbb{R}$ is the loan amount of $\mathit{v}\in {\mathbb{R}}^n$.

Incidentally, there is no need to consider the case of $\kappa <1$ for the solution $\mathit{v}=\kappa {\mathit{x}}^{\text{‡}}$. That is because such a solution $\mathit{v}\in {\mathbb{R}}^n$ is not feasible as $v_0=1-\text{𝟙}{\mathit{v}}^T>0$.

The profit of the new solution $\mathit{v}\in {\mathbb{R}}^n$ is

(A9) $\begin{array}{rl}{\gamma }_{\alpha }\left(\mathit{v}\right)& =\mathit{\mu }{\mathit{v}}^T+L(1-\text{𝟙}{\mathit{v}}^T)+{\mathrm{\Phi }}^{-1}\left(\alpha \right)\sigma \left(\mathit{v}\right)\\ & =\kappa \mathit{\mu }{\mathit{x}}^{\text{‡}T}+L(1-\kappa \text{𝟙}{\mathit{x}}^{\text{‡}T})+\kappa {\mathrm{\Phi }}^{-1}\left(\alpha \right)\sigma \left({\mathit{x}}^{\text{‡}}\right)\\ & =\kappa \mathit{\mu }{\mathit{x}}^{\text{‡}T}+L(1-\kappa \text{𝟙}{\mathit{x}}^{\text{‡}T})+\kappa {\mathrm{\Phi }}^{-1}\left(\alpha \right)\sigma \left({\mathit{x}}^{\text{‡}}\right)+\mathrm{\kappa L}-\mathrm{\kappa L}\\ & =\kappa (\mathit{\mu }{\mathit{x}}^{\text{‡}T}+L(1-\text{𝟙}{\mathit{x}}^{\text{‡}T})+{\mathrm{\Phi }}^{-1}(\alpha \left)\sigma \right({\mathit{x}}^{\text{‡}}\left)\right)+L-\mathrm{\kappa L}\\ & =\kappa {\gamma }_{\alpha }\left({\mathit{x}}^{\text{‡}}\right)+L(1-\kappa )\end{array}$(A9) where $\sigma \left(\mathit{v}\right)=\sigma \left(\kappa {\mathit{x}}^{\text{‡}}\right)=\mathrm{\kappa \sigma }\left({\mathit{x}}^{\text{‡}}\right)$.

From (A9(A9) $\begin{array}{rl}{\gamma }_{\alpha }\left(\mathit{v}\right)& =\mathit{\mu }{\mathit{v}}^T+L(1-\text{𝟙}{\mathit{v}}^T)+{\mathrm{\Phi }}^{-1}\left(\alpha \right)\sigma \left(\mathit{v}\right)\\ & =\kappa \mathit{\mu }{\mathit{x}}^{\text{‡}T}+L(1-\kappa \text{𝟙}{\mathit{x}}^{\text{‡}T})+\kappa {\mathrm{\Phi }}^{-1}\left(\alpha \right)\sigma \left({\mathit{x}}^{\text{‡}}\right)\\ & =\kappa \mathit{\mu }{\mathit{x}}^{\text{‡}T}+L(1-\kappa \text{𝟙}{\mathit{x}}^{\text{‡}T})+\kappa {\mathrm{\Phi }}^{-1}\left(\alpha \right)\sigma \left({\mathit{x}}^{\text{‡}}\right)+\mathrm{\kappa L}-\mathrm{\kappa L}\\ & =\kappa (\mathit{\mu }{\mathit{x}}^{\text{‡}T}+L(1-\text{𝟙}{\mathit{x}}^{\text{‡}T})+{\mathrm{\Phi }}^{-1}(\alpha \left)\sigma \right({\mathit{x}}^{\text{‡}}\left)\right)+L-\mathrm{\kappa L}\\ & =\kappa {\gamma }_{\alpha }\left({\mathit{x}}^{\text{‡}}\right)+L(1-\kappa )\end{array}$(A9) ), the difference between two profits is (A10) $\begin{array}{rl}{\gamma }_{\alpha }\left(\mathit{v}\right)-{\gamma }_{\alpha }\left({\mathit{x}}^{\text{‡}}\right)& =(\kappa -1){\gamma }_{\alpha }\left({\mathit{x}}^{\text{‡}}\right)+L(1-\kappa )\\ & =(\kappa -1)\left({\gamma }_{\alpha }\right({\mathit{x}}^{\text{‡}})-L).\end{array}$(A10) From (A10(A10) $\begin{array}{rl}{\gamma }_{\alpha }\left(\mathit{v}\right)-{\gamma }_{\alpha }\left({\mathit{x}}^{\text{‡}}\right)& =(\kappa -1){\gamma }_{\alpha }\left({\mathit{x}}^{\text{‡}}\right)+L(1-\kappa )\\ & =(\kappa -1)\left({\gamma }_{\alpha }\right({\mathit{x}}^{\text{‡}})-L).\end{array}$(A10) ) and $\kappa >1$, ${\gamma }_{\alpha }\left(\mathit{v}\right)>{\gamma }_{\alpha }\left({\mathit{x}}^{\text{‡}}\right)$ holds if the condition in (A7(A7) ${\gamma }_{\alpha }\left({\mathit{x}}^{\text{‡}}\right)>L\ge 0$(A7) ) is satisfied.

Let $\mathit{x}\in {\mathbb{R}}^n$ be a solution of POL_P that uses the loan. From Lemma A.2, (A11) ${\gamma }_{\alpha }\left(\mathit{x}\right)>{\gamma }_{\alpha }\left({\mathit{x}}^{\text{‡}}\right)>L.$(A11) Let us think of a new solution $\mathit{v}=\kappa \mathit{x}$, $\kappa >1$ for POL_P. The loan amount $v_0\in \mathbb{R}$ of the new solution $\mathit{v}\in {\mathbb{R}}^n$ is (A12) $v_0=1-\text{𝟙}{\mathit{v}}^T=1-\kappa \text{𝟙}{\mathit{x}}^T<1-\text{𝟙}{\mathit{x}}^T=x_0$(A12) where $x_0\in \mathbb{R}$ is the loan amount of $\mathit{x}\in {\mathbb{R}}^n$.

From (A12(A12) $v_0=1-\text{𝟙}{\mathit{v}}^T=1-\kappa \text{𝟙}{\mathit{x}}^T<1-\text{𝟙}{\mathit{x}}^T=x_0$(A12) ), the new solution $\mathit{v}\in {\mathbb{R}}^n$ borrows much money than $\mathit{x}\in {\mathbb{R}}^n$.

The profit of the new solution $\mathit{v}\in {\mathbb{R}}^n$ is (A13) $\begin{array}{rl}{\gamma }_{\alpha }\left(\mathit{v}\right)& =\mathit{\mu }{\mathit{v}}^T+L(1-\text{𝟙}{\mathit{v}}^T)+{\mathrm{\Phi }}^{-1}\left(\alpha \right)\sigma \left(\mathit{v}\right)\\ & =\kappa \mathit{\mu }{\mathit{x}}^T+L(1-\kappa \text{𝟙}{\mathit{x}}^T)+\kappa {\mathrm{\Phi }}^{-1}\left(\alpha \right)\sigma \left(\mathit{x}\right)\\ & =\kappa \mathit{\mu }{\mathit{x}}^T+L(1-\kappa \text{𝟙}{\mathit{x}}^T)+\kappa {\mathrm{\Phi }}^{-1}\left(\alpha \right)\sigma \left(\mathit{x}\right)+\mathrm{\kappa L}-\mathrm{\kappa L}\\ & =\kappa (\mathit{\mu }{\mathit{x}}^T+L(1-\text{𝟙}{\mathit{x}}^T)+{\mathrm{\Phi }}^{-1}(\alpha \left)\sigma \right(\mathit{x}\left)\right)+L-\mathrm{\kappa L}\\ & =\kappa {\gamma }_{\alpha }\left(\mathit{x}\right)+L(1-\kappa ).\end{array}$(A13) From (A13(A13) $\begin{array}{rl}{\gamma }_{\alpha }\left(\mathit{v}\right)& =\mathit{\mu }{\mathit{v}}^T+L(1-\text{𝟙}{\mathit{v}}^T)+{\mathrm{\Phi }}^{-1}\left(\alpha \right)\sigma \left(\mathit{v}\right)\\ & =\kappa \mathit{\mu }{\mathit{x}}^T+L(1-\kappa \text{𝟙}{\mathit{x}}^T)+\kappa {\mathrm{\Phi }}^{-1}\left(\alpha \right)\sigma \left(\mathit{x}\right)\\ & =\kappa \mathit{\mu }{\mathit{x}}^T+L(1-\kappa \text{𝟙}{\mathit{x}}^T)+\kappa {\mathrm{\Phi }}^{-1}\left(\alpha \right)\sigma \left(\mathit{x}\right)+\mathrm{\kappa L}-\mathrm{\kappa L}\\ & =\kappa (\mathit{\mu }{\mathit{x}}^T+L(1-\text{𝟙}{\mathit{x}}^T)+{\mathrm{\Phi }}^{-1}(\alpha \left)\sigma \right(\mathit{x}\left)\right)+L-\mathrm{\kappa L}\\ & =\kappa {\gamma }_{\alpha }\left(\mathit{x}\right)+L(1-\kappa ).\end{array}$(A13) ), the difference between two profits is (A14) $\begin{array}{rl}{\gamma }_{\alpha }\left(\mathit{v}\right)-{\gamma }_{\alpha }\left(\mathit{x}\right)& =(\kappa -1){\gamma }_{\alpha }\left(\mathit{x}\right)+L(1-\kappa )\\ & =(\kappa -1)\left({\gamma }_{\alpha }\right(\mathit{x})-L).\end{array}$(A14) From (A11(A11) ${\gamma }_{\alpha }\left(\mathit{x}\right)>{\gamma }_{\alpha }\left({\mathit{x}}^{\text{‡}}\right)>L.$(A11) ) and $\kappa >1$, ${\gamma }_{\alpha }\left(\mathit{v}\right)>{\gamma }_{\alpha }\left(\mathit{x}\right)$ holds for (A14(A14) $\begin{array}{rl}{\gamma }_{\alpha }\left(\mathit{v}\right)-{\gamma }_{\alpha }\left(\mathit{x}\right)& =(\kappa -1){\gamma }_{\alpha }\left(\mathit{x}\right)+L(1-\kappa )\\ & =(\kappa -1)\left({\gamma }_{\alpha }\right(\mathit{x})-L).\end{array}$(A14) ). Therefore, we can increase the profit ${\gamma }_{\alpha }\left(\mathit{x}\right)$ of $\mathit{x}\in {\mathbb{R}}^n$ by borrowing money as much as possible.

Contrary to $\kappa >1$, if we think of a new solution $\mathit{v}=\kappa \mathit{x}$, $\kappa <1$ in (A12(A12) $v_0=1-\text{𝟙}{\mathit{v}}^T=1-\kappa \text{𝟙}{\mathit{x}}^T<1-\text{𝟙}{\mathit{x}}^T=x_0$(A12) ), the new solution $\mathit{v}\in {\mathbb{R}}^n$ borrows less money than $\mathit{x}\in {\mathbb{R}}^n$ as $0\ge v_0>x_0$. Furthermore, from (A11(A11) ${\gamma }_{\alpha }\left(\mathit{x}\right)>{\gamma }_{\alpha }\left({\mathit{x}}^{\text{‡}}\right)>L.$(A11) ) and $\kappa <1$, ${\gamma }_{\alpha }\left(\mathit{v}\right)<{\gamma }_{\alpha }\left(\mathit{x}\right)$ holds for (A14(A14) $\begin{array}{rl}{\gamma }_{\alpha }\left(\mathit{v}\right)-{\gamma }_{\alpha }\left(\mathit{x}\right)& =(\kappa -1){\gamma }_{\alpha }\left(\mathit{x}\right)+L(1-\kappa )\\ & =(\kappa -1)\left({\gamma }_{\alpha }\right(\mathit{x})-L).\end{array}$(A14) ). Therefore, we can confirm that the optimal solution of POL_P uses the loan up to the limit if it uses the loan.