Mean-entropy uncertain portfolio with risk curve and total mental accounts under multiple background risks

Abstract

In the previous uncertain portfolio literature on background risk and mental account, only a general background risk and a few kinds of mental accounts were considered. Based on the above limitations, on the one hand, the multiple background risks are defined by linear weighting of different background asset risks in this paper; on the other hand, the total nine kinds of mental accounts are comprehensively considered. Especially, the risk curve is regarded as the risk measurement of different mental accounts for the first time. Under the framework of uncertainty theory, a novel mean-entropy portfolio model with risk curve and total mental accounts under multiple background risks is constructed. In addition, transaction fees, chance constraint, upper and lower limits and initial wealth constraints are also considered in our proposed model. In theory, the equivalent forms of the models with different uncertainty distributions (general, normal and zigzag) are presented by three theorems. Simultaneously, the corresponding concrete expressions of risk curves are obtained by another three theorems. In practice, two numerical examples verify the feasibility and effectiveness of our proposed model. Finally, we can obtain the following unique and meaningful findings: (1) investors will underestimate the potential risk if they ignore the existence of multiple background risks; (2) with the increase of the return threshold, the return of the sub-portfolio will inevitably increase, but investors also bear the risk that the risk curve is higher than the confidence curve at this time.

Keywords

Uncertainty theory mental account background risk entropy risk curve

1 Introduction

The portfolio problem is how to allocate securities reasonably, so as to achieve the goal of minimizing the risk under a given expected return or maximizing the expected return under a given risk. In 1952, Markowitz [1] opened the era of quantifying the return and risk of portfolio with expectation value and variance, which was the beginning and core of modern portfolio theory. Due to the limitation of using the variance of portfolio return to measure risk, researchers had carried out further research on other risk measures under the framework of probability theory, for example, Krejic, et al. [2] proposed VaR model, Najafi and Mushakhian [3] proposed mean-semi variance-CVaR model, Tong, et al. [4] proposed CVaR model. In addition, the latest research on portfolio is emerging in an endless stream. For example, Cacador et al. [5] constructed a relatively robust portfolio model integrating minimax regret method. Jia et al. [6] established a mean multifractal detrended cross correlation analysis portfolio model. Ziakas and Getz [7] reviewed the evolution of event portfolio and summarized the significance of event portfolio for interdisciplinary field.

In the above portfolio studies, security return was regarded as random variable and the distribution of return was estimated by historical data. However, due to the complexity of the investment environment, the parameters of stochastic returns may not be accurately estimated. Experts began to use historical information and subjective beliefs to estimate the security return, which was regarded as fuzzy variables. Therefore, fuzzy set theory [8] was widely used in the field of portfolio. For example, Li and Yi [9] proposed a new trapezoidal fuzzy number and applied it to the fuzzy mean-variance model and mean-variance-skewness model to optimize asset allocation. Ramli and Jaaman [10] proposed seven extended mean-variance methods for portfolio selection models based on possible returns and possible risks. Chen et al. [11] constructed a two-stage fuzzy portfolio selection model with transaction costs. In the real financial market, there are some cases of historical data failure or lack of historical data. For example, after the impact of emergencies, the historical data of securities have lost their original reference value, and the new listed stocks have no historical data. In this scenario, the method of predicting the security return based on historical data is not feasible, so it is necessary to use the experience and confidence of experts to estimate the security return. Several studies in [12, 13] had shown paradoxical and counter-intuitive results if random or fuzzy variables are used to describe subjectively estimated security return. In view of this phenomenon, Liu [14] proposed the uncertainty theory, and proved that the confidence degree satisfies the four axioms of the uncertainty theory, which means that we can study the portfolio when the security return is subjectively judged by experts according to the uncertainty theory. After the uncertainty theory of axiomatic system was put forward, the research of uncertainty theory has attracted the attention of many scholars. At present, uncertainty theory has been applied to economics and finance in [15, 16]. In the field of portfolio, Huang [17] systematically combined portfolio selection with uncertainty theory. After that, Huang put forward risk curve [18] and risk index [19] as new risk measurement indicators. In addition, Zhang et al. [20] proposed the mean-variance-chance uncertain portfolio model, and Li et al. [21] proposed the mean-variance mixed integer nonlinear dynamic uncertain portfolio model.

Entropy, which was proposed by Shannon [22] in 1948, is used to describe the uncertainty of random variables. Then this concept has been widely used in many fields. In the field of uncertainty, Liu [14] proposed the entropy of uncertain variables, which is used to measure the degree of uncertainty of uncertain variables and express the difficulty of accurately estimating an uncertain variable. In the framework of uncertain portfolio, scholars also pay attention to the expansion of entropy. Chen [23] used entropy to measure the degree of portfolio diversification, and constructed the mean-variance-entropy model. Ning et al. [24] explored the mean-variance uncertain portfolio selection problem constrained by triangular entropy. Sajedi and Yari [25] defined υ-order entropy and cross entropy, applied them to the mean-variance portfolio selection model.

When making investment decisions, investors bear not only the endogenous risk of portfolio, but also the risk from background assets, which is called background risk [26], which mainly include labor income fluctuation risk, housing risk, physical and mental health risk. Fan and Zhao [27] found that the background risk of financial asset investment includes the health status of personal characteristics. Cardak and Wilkins [28] investigated the income and labor dynamics of new households as important factors that constitute background risk in asset allocation.

Some scholars grasp this neglected risk and carry out much research in the field of portfolio. For example, Huang and Di [29] proposed an uncertain portfolio model with background risk, and compared the optimal portfolio with and without background risk. Huang and Yang [30] discussed how background risk affects individual investment decision-making, and studied the characteristics of portfolio efficient frontier in the presence of background risk. Hara et al. [31] found that the presence of background risk can improve the prudence of investment, and investors tend to choose safer assets. These studies show that background risk has a significant impact on investors’ decision-making, and when it is considered, investors’ aversion to risk will increase.

Risk curve is a new portfolio security criterion proposed by Huang [18]. The risk curve provides all possible loss information, and the confidence curve represents the tolerance of loss. If the risk curve of the portfolio is lower than the confidence curve, it means that the portfolio is safe. On the contrary, if the risk curve is higher than the confidence curve, the portfolio is risky. Therefore, the risk curve is used as a risk measure in this paper. In recent years, scholars studied the model of risk curve and portfolio. For example, Zhao and Bai [32] proposed an uncertain mean-risk model with background risk. Zhao et al. [33] proposed a mean-risk-skewness model for portfolio selection under uncertain environment. Rouhollah et al. [34] defined the risk curve under uncertain random environment, and proposed the mean-risk model.

In the above research on portfolio theory, the prediction of return is only for the same goal, but in real financial markets, people usually divide money into different types of management and budget. The existence of mental accounts will lead to multiple contradictory goals. Attitudes towards risks and returns will vary depending on the mental accounts. In order to solve such problems, Das et al. [35] combined mental accounts with the mean-variance portfolio model for the first time and established different mental accounts for different goals. In recent years, research on mental accounts and portfolios has gradually become popular. Momen et al. [36] proposed the integrated mental account model, which is superior to the mental account model in terms of effective boundary and utility function. Alexander et al. [37] aggregated the moments of the endogenous asset return distribution into a positive model under the framework of mental account. Chang et al. [38] proposed an uncertain nonlinear multi-period portfolio model, which considered mental accounts and some realistic constraints. Xue et al. [39] proposed an uncertain portfolio model, which considered mental accounts, background risks and realistic constraints. Moreover, Li et al. [40] studied the classification structure of mental accounts.

This article is inspired by two articles, Zhai [32] and Xue et al. [39]. Zhai [32] discussed the problem of considering the mean-risk portfolio selection of background risk, but didn’t combine mental account. Xue et al. [39] studied an uncertain portfolio with background risk and mental account under realistic constraints, but didn’t combine the risk curve. What’s more, in the past literature on background risk, only a general background risk was considered. Based on this, this paper assumes that investors are faced with multiple background risks, and uses the linear weighting of multiple background asset returns to measure the multiple background risks. Since the risk curve measures all possible losses of portfolio, we use risk curves as risk measures for different mental accounts, then a mean-entropy uncertain portfolio with risk curve and total mental accounts under multiple background risks is constructed. In addition, the model also considers chance constraint, upper and lower bound and initial wealth constraint.

The main contributions are as follows. Firstly, the risk curve is combined with mental accounts for the first time, and the confidence curve is set according to the tolerance of nine kinds of mental accounts for loss. Secondly, the multiple background risks are defined by linear weighting of different background asset risks. Thirdly, we construct the mean-entropy uncertain portfolio with risk curve and total mental accounts under the multiple background risks, and further deduce the equivalent form of the model when the security return and background asset return obey general, normal and zigzag uncertainty distributions. Fourthly, the specific expression of risk curve under the optimal solution of the model is derived, in addition, the position relationship between risk curve and confidence curve under different mental accounts is also discussed.

The structure of paper is as follows. The basic knowledge is introduced in Section 2. Section 3 constructs a mean-entropy uncertain portfolio with risk curve and total mental accounts under the multiple background risks. Section 4 deduces the equivalent form of the model and the specific expression of the risk curve. Section 5 verifies the feasibility of the model through numerical examples. Section 6 is the comparison of models and sensitivity analysis. Finally, Section 7 summarizes the main work and concludes this paper.

2 Preliminaries

2.1 Uncertainty theory

Definition 1. [14] Let L be a σ-algebra over a nonempty set Ω. Each element Γ ∈ L is called an event. A set function M : L → [0, 1] is called an uncertain measure if it satisfies the following four axioms:

(Normality) M {Ω} =1;

(Self-Duality) M {Γ} + M {Γ^c} =1;

(Countable Subadditivity) For every countable sequence of events {Γ_i}, we have $M {⋃_{i = 1}^{\infty} Γ_{i}} ⩽ \sum_{i = 1}^{\infty} M {Γ_{i}}$ ;

(Product Measure) For uncertainty spaces (Ω_i, L_i, M_i) , i = 1, 2, … the product uncertain measure is $M {\prod_{i = 1}^{\infty} Γ_{i}} = min_{1 ⩽ i < \infty} M_{i} {Γ_{i}}$ , where Γ_i are arbitrarily chosen events from L_i for i = 1, 2, …, and it satisfies M {Γ₁} ⩽ M {Γ₂} for any events Γ₁ ⊂ Γ₂.

Definition 2. [14] The uncertainty distribution Φ : R → [0, 1] of an uncertain variable ξ is defined by Φ (x) = M {ξ ⩽ x} for any real number x. If ξ has a regular uncertainty distribution Φ (x), then the inverse function Φ^-1 (α) is said to be the inverse uncertainty distribution of ξ.

Definition 3. [14] An uncertain variable ξ is called zigzag if it has a zigzag uncertainty distribution $Φ (x) = {\begin{matrix} 0, if x ⩽ a, \\ \frac{x - a}{2 (b - a)}, if a ⩽ x ⩽ b, \\ \frac{x + c - 2 b}{2 (c - b)}, if b ⩽ x ⩽ c, \\ 1, if x ⩾ c . \end{matrix}$ (1) denoted by Z (a, b, c) where a, b, c are real numbers with a < b < c. The inverse uncertainty distribution of zigzag uncertain variable Z (a, b, c) is $Φ^{- 1} (α) = {\begin{matrix} (1 - 2 α) a + 2 α b, if 0 ⩽ α < 0.5, \\ (2 - 2 α) b + (2 α - 1) c, if 0.5 ⩽ α ⩽ 1 . \end{matrix}$ (2)

Definition 4. [14] An uncertain variable ξ is called normal if it has a normal uncertainty distribution $Φ (x) = (1 + exp (\frac{π (e - x)}{\sqrt{3} σ}))^{- 1}, x \in R$ (3) denoted by N (e, σ) where e and σ are real numbers with σ > 0. The inverse uncertainty distribution of normal uncertain variable N (e, σ) is $Φ^{- 1} (α) = e + \frac{σ \sqrt{3}}{π} ln \frac{α}{1 - α}, 0 < α < 1$ (4)

Definition 5. [14] An uncertainty distribution Φ (x) is called regular if it is a continuous and strictly increasing function with respect to x at which 0 < Φ (x) <1, and $lim_{x \to - \infty} Φ (x) = 0, lim_{x \to + \infty} Φ (x) = 1$ . It is seen that normal uncertainty distribution and zigzag uncertainty distribution are regular.

Lemma 1. [14] Let ξ₁, …, ξ_n be independent uncertain variables with regular uncertainty distribution functions Φ₁, …, Φ_n. Let f (r₁, …, r_n) be strictly increasing with respect to r₁, …, r_n. Then ξ = f (ξ₁, …, ξ_n) is an uncertain variable whose inverse uncertainty distribution function is $Ψ^{- 1} (α) = f (Φ_{1}^{- 1} (α), \dots, Φ_{n}^{- 1} (α)), 0 < α < 1$ (5)

Definition 6. [14] Let ξ be an uncertain variable. Then the expected value of ξ is defined by

$E [ξ] = \int_{0}^{+ \infty} M {ξ ⩾ r} d r - \int_{- \infty}^{0} M {ξ ⩽ r} d r$ (6) provided that at least one of the two integrals are finite.

Lemma 2. [41] Let ξ₁ and ξ₂ be independent uncertain variables with finite expected values. Then for any real numbers a₁ and a₂, we have

$E [a_{1} ξ_{1} + a_{2} ξ_{2}] = a_{1} E [ξ_{1}] + a_{2} E [ξ_{2}]$ (7)

Definition 7. [41] Suppose that ξ is an uncertain variable with uncertainty distribution Φ. Then its entropy is defined by

$H [ξ] = \int_{- \infty}^{+ \infty} S (Φ (x)) d x$ (8)

Where S (t) = - t ln t - (1 - t) ln(1 - t) .

Lemma 3. [41] Let ξ be an uncertain variable with regular uncertainty distribution Φ. Then

$H [ξ] = \int_{0}^{1} Φ^{- 1} (α) ln \frac{α}{1 - α} d α$ (9)

Lemma 4. [41] Let ξ be an uncertain variable, and let c be a real number. Then

$H [ξ + c] = H [ξ]$ (10)

That is, the entropy is invariant under arbitrary translations.

Lemma 5. [41] Let ξ₁ and ξ₂ be independent uncertain variables. Then for any real numbers a₁ and a₂, we have $H [a_{1} ξ_{1} + a_{2} ξ_{2}] = | a_{1} | H [ξ_{1}] + | a_{2} | H [ξ_{2}]$ (11)

Lemma 6. [41] Let ξ be a zigzag uncertain variable Z (a, b, c). Then its entropy is

$H [ξ] = \frac{c - a}{2}$ (12)

Lemma 7. [41] Let ξ be a normal uncertain variable N (e, σ). Then its entropy is

$H [ξ] = \frac{π σ}{\sqrt{3}}$ (13)

Lemma 8. [41] Assume that ξ₁ and ξ₂ be independent zigzag uncertain variables Z (a₁, b₁, c₁) and Z (a₂, b₂, c₂), respectively. Then the sum ξ₁ + ξ₂ is also a zigzag uncertain variable Z (a₁ + a₂, b₁ + b₂, c₁ + c₂), i.e.,

$\begin{matrix} Z (a_{1}, b_{1}, c_{1}) + Z (a_{2}, b_{2}, c_{2}) \\ = Z (a_{1} + a_{2}, b_{1} + b_{2}, c_{1} + c_{2}) \end{matrix}$ (14)

The product of a zigzag uncertain variable Z (a, b, c) and a scalar number ω > 0 is also a zigzag uncertain variable Z (ωa, ωb, ωc), i.e., $ω Z (a, b, c) = Z (ω a, ω b, ω c)$ (15)

Lemma 9. [41] Let ξ₁ and ξ₂ be independent normal uncertain variables N (e₁, σ₁) and N (e₂, σ₂), respectively. Then the sum ξ₁ + ξ₂ is also a normal uncertain variable N (e₁ + e₂, σ₁ + σ₂), i.e., $N (e_{1}, σ_{1}) + N (e_{2}, σ_{2}) = N (e_{1} + e_{2}, σ_{1} + σ_{2})$ (16)

The product of a normal uncertain variable N (e, σ) and a scalar number ω > 0 is also a normal uncertain variable N (ωe, ωσ), i.e., $ω N (e, σ) = N (ω e, ω σ)$ (17)

2.2 Mental account

Mental account is a psychological process of recording, coding, classifying and valuing wealth. Li et al. [40] believed that people’s mental account system is a relatively stable classification structure as shown in Table 1. This paper, based on the 9 kinds of sub-accounts in Table 1, adds the total mental accounts to the portfolio model.

Table 1
Division of mental accounts

Account Number Sub-account

income account 1 regular income account

2 additional income account

3 operating income account

storage account 4 secure storage account

5 risk storage account

expense account 6 living expenses account

7 family and personal development expenses account

8 emotional maintenance expense account

9 leisure expense account

Account	Number	Sub-account
income account	1	regular income account
	2	additional income account
	3	operating income account
storage account	4	secure storage account
	5	risk storage account
expense account	6	living expenses account
	7	family and personal development expenses account
	8	emotional maintenance expense account
	9	leisure expense account

3 Mean-entropy uncertain portfolio with risk curve and total mental accounts under multiple background risks

In order to fully consider the mental accounts of investors in real life, this paper assumes that investors divide their portfolios into sub-portfolios in nine mental accounts and make decisions on N securities in each mental account. All mental accounts are shown in Table 1. In different mental accounts, the pursuit of profit and tolerance of loss are different, and investors will face multiple background risks. In the process of financial transactions, there are usually inevitable transaction fees. In order to ensure that the portfolio is safe enough to avoid the phenomenon of centralized investment, investors have their own preference for the upper and lower limits of shareholding. We assume that the security return and the background asset return are described by uncertain variables, and parameters are estimated by experts rather than historical data.

In order to explain the portfolio in detail, we will use the following notations when modeling, as shown in Table 2.

Table 2
Notation and Illustration

Notation Illustration

i = 1, 2, …, N security

k = 1, 2, …, 9 mental account

j = 1, 2, …, J background asset

C _i transaction cost rate of security i

w _k initial wealth within account k

x _ki investment proportion of security i within account k

ξ _i security return i

r _kj background asset return j within account k

λ _kj weight of the investor facing background risk j within account k

r _f risk-free interest rate

A _k return threshold within account k

δ _k uncertainty measure threshold within account k

l _ki lower bound of security i within account k

L _ki upper bound of security i within account k

γ _k proportion of initial wealth within account k to total wealth

Notation	Illustration
i = 1, 2, …, N	security
k = 1, 2, …, 9	mental account
j = 1, 2, …, J	background asset
C _i	transaction cost rate of security i
w _k	initial wealth within account k
x _ki	investment proportion of security i within account k
ξ _i	security return i
r _kj	background asset return j within account k
λ _kj	weight of the investor facing background risk j within account k
r _f	risk-free interest rate
A _k	return threshold within account k
δ _k	uncertainty measure threshold within account k
l _ki	lower bound of security i within account k
L _ki	upper bound of security i within account k
γ _k	proportion of initial wealth within account k to total wealth

Among them, ξ_i, i = 1, 2, . . . , N and r_kj, j = 1, 2, . . . , J are independent uncertain variables. ξ_i and r_kj are also independent.

3.1 Objective functions

Since background assets have different characteristics from financial assets, i.e., illiquidity and non-hedgeability. In the past literature, the common method is to use a parameter r_b to display all the background assets return in real life. However, there is a big gap between different background assets, such as income fluctuation risk and housing risk, which are inconsistent for investors. Therefore, in this paper, the multiple background risks are defined by linear weighting of different background asset risks. The background asset return r_kb within account k is characterized by the linear weighting of J background asset returns, that is, $r_{kb} = \sum_{j = 1}^{J} λ_{kj} r_{kj}$ , where $\sum_{j = 1}^{J} λ_{kj} = 1, λ_{kj} > 0$ . Investors can adjust the weight of their exposure to different background risks according to their own circumstances. In addition, we assume that the expected values of all background asset returns are zeros, which is consistent with most assumptions on background risks [42], indicating that investors are concerned about the background risks brought by background assets. Considering the multiple background risks and transaction costs, the specific expression of return ξ_k within account k is as follows. $ξ_{k} = \sum_{i = 1}^{N} x_{ki} ξ_{i} + \sum_{j = 1}^{J} λ_{kj} r_{kj} - \sum_{i = 1}^{N} C_{i} x_{ki}$ (18)

3.1.1 Maximum mean

Investors pursue the maximum investment return, therefore, they can maximize the expected value of portfolio return. In the mental account k, the first objective function is $max E [\sum_{i = 1}^{N} x_{ki} ξ_{i} + \sum_{j = 1}^{J} λ_{kj} r_{kj} - \sum_{i = 1}^{N} C_{i} x_{ki}]$ (19) where E is the expected operator in uncertainty theory.

3.1.2 Minimizing entropy

In the theory of uncertainty [41], entropy can be used as a measure of uncertainty to show the difficulty of accurately estimating an uncertain variable. The greater the entropy of an uncertain variable, the more difficult it is to judge the exact value of the uncertain variable. Obviously, investors want the degree of uncertainty in portfolio return to be as small as possible, so that they can expect that the smaller the entropy is, the more accurate the estimate of portfolio return. Therefore, investors can minimize the entropy of portfolio return. In the mental account k, the second objective function is $min H [\sum_{i = 1}^{N} x_{ki} ξ_{i} + \sum_{j = 1}^{J} λ_{kj} r_{kj} - \sum_{i = 1}^{N} C_{i} x_{ki}]$ (20) where H is the entropy operator in uncertainty theory.

3.2 Constraints

3.2.1 Risk curve constraint

Definition 8. Let ξ be a security return, and r_f be the risk-free interest rate. Then the curve $R (r) = M {r_{f} - ξ ⩾ r}, \forall r ⩾ 0$ (21) is called the risk curve of the security, where M is the uncertainty measure.

When the portfolio return is even less than the risk-free interest rate, it indicates that losses have occurred. For possible losses, investors have the maximum tolerance for losses, which is called the confidence curve α (r). The confidence curve has the form of linear function, piecewise function and power function. Among them, the linear function form does not well reflect the investor’s tolerance for loss, because as the loss increases, the tolerance change tends to be gentle. It is more reasonable to believe that the confidence curve presents a marginal decreasing function form. In addition, there is a turning point in the form of piecewise function, and the curve is not smooth enough. The form of power function satisfies the marginal decrease and the curve is smooth enough. Therefore, in this paper, the confidence curve in the form of power function is used.

Definition 9. [18] The confidence curve in the form of power function is $α (r) = \frac{β}{(r + 1)^{η}}, r ⩾ 0$ (22) where β is a positive number and η is a positive integer.

When the risk curve of the portfolio is lower than the confidence curve, then the portfolio is safe. On the contrary, if the risk curve is higher than the confidence curve, then the portfolio is risky. The risk curve within account k is as follows.

$\begin{matrix} R (r) = & M {r_{f} - (\sum_{i = 1}^{N} x_{ki} ξ_{i} + \sum_{j = 1}^{J} λ_{kj} r_{kj} - \sum_{i = 1}^{N} C_{i} x_{ki}) ⩾ r}, \\ \forall r ⩾ 0 \end{matrix}$ (23)

where M is the uncertainty measure. In the account k, the confidence curve set by investors is α_k (r). The risk curve of the optimal portfolio should be lower than the given confidence curve.

Therefore, the risk curve constraint is as follows.

$\begin{matrix} M {r_{f} - (\sum_{i = 1}^{N} x_{ki} ξ_{i} + \sum_{j = 1}^{J} λ_{kj} r_{kj} - \sum_{i = 1}^{N} C_{i} x_{ki}) ⩾ r} \\ ⩽ α_{k} (r), \forall r ⩾ 0 \end{matrix}$ (24)

3.2.2 Chance constraint

In order to reflect the risk conflict between different objectives in the mental account, the return threshold A_k and the uncertainty measure threshold δ_k are set in the account k. In mathematical language, it is expressed as M {ξ ⩾ A_k} ⩾ δ_k, it means that A_k is the lower limit of the expected security return and δ_k is the lower limit of uncertainty. Therefore, the chance constraint within account k is $M {\sum_{i = 1}^{N} x_{ki} ξ_{i} + \sum_{j = 1}^{J} λ_{kj} r_{kj} - \sum_{i = 1}^{N} C_{i} x_{ki} ⩾ A_{k}} ⩾ δ_{k}$ (25) where M is the uncertainty measure.

3.2.3 Upper and lower bound constraints

In order to control the appropriate proportion of investment and prevent the phenomenon of centralized investment. Investors can set the upper and lower limits of the shareholding ratio for each kind of securities, so as to achieve the purpose of decentralized investment to some extent. It can be expressed as $l_{ki} ⩽ x_{ki} ⩽ L_{ki}$ (26)

3.2.4 Initial wealth constraint

When making investment decision in the account k, the amount of securities purchased can be expressed as $\sum_{i = 1}^{N} w_{k} x_{ki}$ . The sum of transaction costs can be expressed as $\sum_{i = 1}^{N} C_{i} w_{k} x_{ki}$ . Since investors plan to purchase securities with fixed initial wealth in each mental account, the amount of securities purchased and transaction costs should not exceed the initial wealth of account k. Therefore, the initial wealth constraint is as follows. $\sum_{i = 1}^{N} w_{k} x_{ki} + \sum_{i = 1}^{N} C_{i} w_{k} x_{ki} ⩽ w_{k}$ (27)

3.2.5 Coefficient constraint of multiple background risks

In the definition of multiple background risks, λ_kj represents weight of the investor facing background risk j within account k, and the sum of weights is 1. λ_kj is required to be bigger than 0, which is used in subsequent theoretical proof. Therefore, the coefficient constraint of multiple background risks is as follows. $\sum_{j = 1}^{J} λ_{kj} = 1, λ_{kj} > 0$ (28)

3.3 Establishment of model

The goal of investors is to choose a portfolio which can maximize the expected return and minimize the entropy under the proposed constraints. Under the framework of uncertainty theory, the portfolio in account k is as follows.

${\begin{matrix} max E [\sum_{i = 1}^{N} x_{ki} ξ_{i} + \sum_{j = 1}^{J} λ_{kj} r_{kj} - \sum_{i = 1}^{N} C_{i} x_{ki}] \\ min H [\sum_{i = 1}^{N} x_{ki} ξ_{i} + \sum_{j = 1}^{J} λ_{kj} r_{kj} - \sum_{i = 1}^{N} C_{i} x_{ki}] \\ s . t . M {r_{f} - [\sum_{i = 1}^{N} x_{ki} ξ_{i} + \sum_{j = 1}^{J} λ_{kj} r_{kj} - \sum_{i = 1}^{N} C_{i} x_{ki}] ⩾ r} \\ ⩽ α_{k} (r), \forall r ⩾ 0 \\ M {[\sum_{i = 1}^{N} x_{ki} ξ_{i} + \sum_{j = 1}^{J} λ_{kj} r_{kj} - \sum_{i = 1}^{N} C_{i} x_{ki}] ⩾ A_{k}} ⩾ δ_{k} \\ l_{ki} ⩽ x_{ki} ⩽ L_{ki} \\ \sum_{i = 1}^{N} w_{k} x_{ki} + \sum_{i = 1}^{N} C_{i} w_{k} x_{ki} ⩽ w_{k} \\ \sum_{j = 1}^{J} λ_{kj} = 1, λ_{kj} > 0 \end{matrix}$ (29)

The bi-objective model (29) is converted into a single objective model (30) through the adjustment parameter θ, indicating the preference of investors for the objective functions.

${\begin{matrix} min θ H [\sum_{i = 1}^{N} x_{ki} ξ_{i} + \sum_{j = 1}^{J} λ_{kj} r_{kj} - \sum_{i = 1}^{N} C_{i} x_{ki}] - (1 - θ) E [\sum_{i = 1}^{N} x_{ki} ξ_{i} + \sum_{j = 1}^{J} λ_{kj} r_{kj} - \sum_{i = 1}^{N} C_{i} x_{ki}] \\ s . t . M {r_{f} - [\sum_{i = 1}^{N} x_{ki} ξ_{i} + \sum_{j = 1}^{J} λ_{kj} r_{kj} - \sum_{i = 1}^{N} C_{i} x_{ki}] ⩾ r} ⩽ α_{k} (r), \forall r ⩾ 0 \\ M {[\sum_{i = 1}^{N} x_{ki} ξ_{i} + \sum_{j = 1}^{J} λ_{kj} r_{kj} - \sum_{i = 1}^{N} C_{i} x_{ki}] ⩾ A_{k}} ⩾ δ_{k} \\ l_{ki} ⩽ x_{ki} ⩽ L_{ki} \\ \sum_{i = 1}^{N} w_{k} x_{ki} + \sum_{i = 1}^{N} C_{i} w_{k} x_{ki} ⩽ w_{k} \\ \sum_{j = 1}^{J} λ_{kj} = 1, λ_{kj} > 0 \end{matrix}$ (30)

Model (30) only considers the portfolio model of account k. Through comprehensive consideration of nine kinds of mental accounts, including regular income account, additional income account, operating income account, secure storage account, risk storage account, living expenses account, family and personal development expenses account, emotional maintenance expense account, and leisure expense account, we seek to maximize the return and minimize the entropy of the aggregate portfolio. Note expected return E_k and entropy H_k solved by the k-th mental account model (30). By linearly weighting the solutions obtained from the sub-portfolio models of the nine mental accounts, we can obtain the expected return $\sum_{k = 1}^{9} γ_{k} E_{k}$ and entropy $\sum_{k = 1}^{9} γ_{k} H_{k}$ of the aggregate portfolio.

4 Equivalent forms of model and expression of risk curve

This section mainly discusses the concrete equivalent forms of uncertain variables in model (30) which obey general, normal and zigzag uncertainty distribution. When the optimal investment proportion has been solved by the model, the corresponding concrete expression of risk curve is given. Here are six theorems and their proofs.

4.1 Equivalent forms of model

Theorem 1. Suppose the security returns ξ_i, i = 1, . . . , N have regular uncertainty distribution functions Φ_i, background asset returns r_kj, j = 1, . . . , J have regular uncertainty distribution functions Θ_j. ξ_i, r_kj are independent of each other. Then we can convert the model (30) into the following form.

${\begin{matrix} min θ [\sum_{i = 1}^{N} x_{ki} H (ξ_{i}) + \sum_{j = 1}^{J} λ_{kj} H (r_{kj})] \\ - (1 - θ) [\sum_{i = 1}^{N} x_{ki} E (ξ_{i}) + \sum_{j = 1}^{J} λ_{kj} E (r_{kj}) - \sum_{i = 1}^{N} C_{i} x_{ki}] \\ s . t . \sum_{i = 1}^{N} [C_{i} - Φ_{i}^{- 1} (α_{k} (r))] x_{ki} ⩽ r \\ + \sum_{j = 1}^{J} λ_{kj} Θ_{j}^{- 1} (α_{k} (r)) - r_{f}, \forall r ⩾ 0 \\ \sum_{i = 1}^{N} [C_{i} - Φ_{i}^{- 1} (1 - δ_{k})] x_{ki} ⩽ \sum_{j = 1}^{J} λ_{kj} Θ_{j}^{- 1} (1 - δ_{k}) - A_{k} \\ l_{ki} ⩽ x_{ki} ⩽ L_{ki} \\ \sum_{i = 1}^{N} w_{k} x_{ki} + \sum_{i = 1}^{N} C_{i} w_{k} x_{ki} ⩽ w_{k} \\ \sum_{j = 1}^{J} λ_{kj} = 1, λ_{kj} > 0 \end{matrix}$ (31)

Proof. Since ξ_i, i = 1, 2, . . . , N, r_kj, j = 1, 2, . . . , J are independent of each other, according to Lemma 2, the expected operator obeys linear operation in independent uncertain variables. We have

$\begin{matrix} E [\sum_{i = 1}^{N} x_{ki} ξ_{i} + \sum_{j = 1}^{J} λ_{kj} r_{kj} - \sum_{i = 1}^{N} C_{i} x_{ki}] \\ = \sum_{i = 1}^{N} x_{ki} E (ξ_{i}) + \sum_{j = 1}^{J} λ_{kj} E (r_{kj}) - \sum_{i = 1}^{N} C_{i} x_{ki} . \end{matrix}$ (32)

According to Lemma 4, entropy satisfies translation invariance, and $\sum_{i = 1}^{N} C_{i} x_{ki}$ is a constant, then we have

$\begin{matrix} H [\sum_{i = 1}^{N} x_{ki} ξ_{i} + \sum_{j = 1}^{J} λ_{kj} r_{kj} - \sum_{i = 1}^{N} C_{i} x_{ki}] \\ = H [\sum_{i = 1}^{N} x_{ki} ξ_{i} + \sum_{j = 1}^{J} λ_{kj} r_{kj}], \end{matrix}$ (33) according to Lemma 5, the entropy operator obeys the linear operation in the independent uncertain variable, and the coefficient satisfies x_ki > 0, λ_kj > 0, we have

$\begin{matrix} H [\sum_{i = 1}^{N} x_{ki} ξ_{i} + \sum_{j = 1}^{J} λ_{kj} r_{kj}] \\ = \sum_{i = 1}^{N} x_{ki} H (ξ_{i}) + \sum_{j = 1}^{J} λ_{kj} H (r_{kj}) . \end{matrix}$ (34)

As for constrained condition

$\begin{matrix} M {r_{f} - (\sum_{i = 1}^{N} x_{ki} ξ_{i} + \sum_{j = 1}^{J} λ_{kj} r_{kj} - \sum_{i = 1}^{N} C_{i} x_{ki}) ⩾ r} \\ ⩽ α_{k} (r), \forall r ⩾ 0, \end{matrix}$ (35) move the term in the brackets of the uncertain measure and the constant term to the right, and then

$\begin{matrix} M {\sum_{i = 1}^{N} x_{ki} ξ_{i} + \sum_{j = 1}^{J} λ_{kj} r_{kj} - \sum_{i = 1}^{N} C_{i} x_{ki} ⩽ r_{f} - r} \\ ⩽ α_{k} (r), \forall r ⩾ 0 . \end{matrix}$ (36)

Suppose that the distribution function of $\sum_{i = 1}^{N} x_{ki} ξ_{i} + \sum_{j = 1}^{J} λ_{kj} r_{kj} - \sum_{i = 1}^{N} C_{i} x_{ki}$ is Ψ, the above formula means Ψ (r_f - r) ⩽ α_k (r) , ∀ r ⩾ 0, that is, Ψ (r_f - r) ⩽ Ψ (Ψ^-1 (α_k (r))) , ∀ r ⩾ 0. The distribution function Ψ is an increasing function, then we have r_f - r ⩽ Ψ^-1 (α_k (r)) , ∀ r ⩾ 0. According to Lemma 1, $\sum_{i = 1}^{N} x_{ki} ξ_{i} + \sum_{j = 1}^{J} λ_{kj} r_{kj} - \sum_{i = 1}^{N} C_{i} x_{ki}$ is a strict increasing function of independent uncertain variable ξ_i, r_kj, thus

$\begin{matrix} \sum_{i = 1}^{N} x_{ki} Φ_{i}^{- 1} (α_{k} (r)) + \sum_{j = 1}^{J} λ_{kj} Θ_{j}^{- 1} (α_{k} (r)) \\ - \sum_{i = 1}^{N} C_{i} x_{ki} ⩾ r_{f} - r, \forall r ⩾ 0, \end{matrix}$ (37) by moving the term without x_ki to the right of inequality and changing the direction of inequality, we can deduce:

$\begin{matrix} \sum_{i = 1}^{N} [C_{i} - Φ_{i}^{- 1} (α_{k} (r))] x_{ki} \\ ⩽ r + \sum_{j = 1}^{J} λ_{kj} Θ_{j}^{- 1} (α_{k} (r)) - r_{f}, \forall r ⩾ 0 . \end{matrix}$ (38)

According to Definition 1, uncertain measures satisfy self-duality, so

$\begin{matrix} M {\sum_{i = 1}^{N} x_{ki} ξ_{i} + \sum_{j = 1}^{J} λ_{kj} r_{kj} - \sum_{i = 1}^{N} C_{i} x_{ki} ⩾ A_{k}} \\ = 1 - M {\sum_{i = 1}^{N} x_{ki} ξ_{i} + \sum_{j = 1}^{J} λ_{kj} r_{kj} - \sum_{i = 1}^{N} C_{i} x_{ki} ⩽ A_{k}}, \end{matrix}$ (39) based on the second constraint in the model, then

$1 - M {\sum_{i = 1}^{N} x_{ki} ξ_{i} + \sum_{j = 1}^{J} λ_{kj} r_{kj} - \sum_{i = 1}^{N} C_{i} x_{ki} ⩽ A_{k}} ⩾ δ_{k},$ (40) $M {\sum_{i = 1}^{N} x_{ki} ξ_{i} + \sum_{j = 1}^{J} λ_{kj} r_{kj} - \sum_{i = 1}^{N} C_{i} x_{ki} ⩽ A_{k}} ⩽ 1 - δ_{k},$ (41)

suppose the distribution function of $\sum_{i = 1}^{N} x_{ki} ξ_{i} + \sum_{j = 1}^{J} λ_{kj} r_{kj} - \sum_{i = 1}^{N} C_{i} x_{ki}$ is Ψ, the above formula meansΨ (A_k) ⩽1 - δ_k, Ψ (A_k) ⩽ Ψ (Ψ^-1 (1 - δ_k)). Since distribution function Ψ is an increasing function, we have A_k ⩽ Ψ^-1 (1 - δ_k). According to Lemma 1, $\sum_{i = 1}^{N} x_{ki} ξ_{i} + \sum_{j = 1}^{J} λ_{kj} r_{kj} - \sum_{i = 1}^{N} C_{i} x_{ki}$ is a strictly increasing function of independent uncertain variable ξ_i, r_kj, so

$\sum_{i = 1}^{N} x_{ki} Φ_{i}^{- 1} (1 - δ_{k}) + \sum_{j = 1}^{J} λ_{kj} Θ_{j}^{- 1} (1 - δ_{k}) - \sum_{i = 1}^{N} C_{i} x_{ki} ⩾ A_{k},$ (42)

by moving the term without x_ki to the right of inequality and changing the direction of inequality, we can deduce:

$\begin{matrix} \sum_{i = 1}^{N} [C_{i} - Φ_{i}^{- 1} (1 - δ_{k})] x_{ki} \\ ⩽ \sum_{j = 1}^{J} λ_{kj} Θ_{j}^{- 1} (1 - δ_{k}) - A_{k} . \end{matrix}$ (43)

The theorem is proved.

Theorem 2. Suppose the security returns ξ_i, i = 1, 2, . . . , N and the background asset returns r_kj, j = 1, 2, . . . , J are all normal uncertain variables, i.e.,ξ_i ∼ N (e_i, σ_i), r_kj ∼ N (0, ρ_kj), then the model (31) can be transformed into the following form.

${\begin{matrix} min θ [\sum_{i = 1}^{N} \frac{π σ_{i}}{\sqrt{3}} x_{ki} + \sum_{j = 1}^{J} λ_{kj} \frac{π ρ_{kj}}{\sqrt{3}}] \\ - (1 - θ) [\sum_{i = 1}^{N} x_{ki} e_{i} - \sum_{i = 1}^{N} C_{i} x_{ki}] \\ s . t . \sum_{i = 1}^{N} [C_{i} - (e_{i} + \frac{σ_{i} \sqrt{3}}{π} ln \frac{α_{k} (r)}{1 - α_{k} (r)})] x_{ki} ⩽ r \\ + \sum_{j = 1}^{J} λ_{kj} (\frac{ρ_{kj} \sqrt{3}}{π} ln \frac{α_{k} (r)}{1 - α_{k} (r)}) - r_{f}, \forall r ⩾ 0 \\ \sum_{i = 1}^{N} [C_{i} - (e_{i} + \frac{σ_{i} \sqrt{3}}{π} ln \frac{1 - δ_{k}}{δ_{k}})] x_{ki} \\ ⩽ \sum_{j = 1}^{J} λ_{kj} (\frac{ρ_{kj} \sqrt{3}}{π} ln \frac{1 - δ_{k}}{δ_{k}}) - A_{k} \\ l_{ki} ⩽ x_{ki} ⩽ L_{ki} \\ \sum_{i = 1}^{N} w_{k} x_{ki} + \sum_{i = 1}^{N} C_{i} w_{k} x_{ki} ⩽ w_{k} \\ \sum_{j = 1}^{J} λ_{kj} = 1, λ_{kj} > 0 \\ 0 < α_{k} (r) < 1 \\ 0 < δ_{k} < 1 \end{matrix}$ (44)

Proof. Since ξ_i ∼ N (e_i, σ_i) , i = 1, 2, . . . , N and r_kj ∼ N (0, ρ_kj) , j = 1, 2, . . . , J, according to Definition 6, the expected value of normal uncertainty distribution ξ ∼ N (e, σ) is E [ξ] = e, so $E [ξ_{i}] = e_{i}, E [r_{kj}] = 0$ (45)

According to Lemma 7, the entropy of normal uncertainty distribution ξ ∼ N (e, σ) is $H [ξ] = \frac{π σ}{\sqrt{3}}$ , so $H [ξ_{i}] = \frac{π σ_{i}}{\sqrt{3}}, H [r_{kj}] = \frac{π ρ_{i}}{\sqrt{3}} .$ (46)

According to Definition 4, The inverse uncertainty distribution of normal uncertain variable N (e, σ) is

$Φ^{- 1} (α) = e + \frac{σ \sqrt{3}}{π} ln \frac{α}{1 - α}, 0 < α < 1,$ (47) then ξ_i and r_kj have inverse uncertainty distribution

$\begin{matrix} Φ_{i}^{- 1} (α) & = e_{i} + \frac{σ_{i} \sqrt{3}}{π} ln \frac{α}{1 - α}, Θ_{j}^{- 1} (α) \\ = \frac{ρ_{kj} \sqrt{3}}{π} ln \frac{α}{1 - α}, 0 < α < 1 . \end{matrix}$ (48)

In the model, when 0 < α_k (r) <1, 0 < δ_k < 1, , we have

$\begin{matrix} Φ_{i}^{- 1} (α_{k} (r)) & = e_{i} + \frac{σ_{i} \sqrt{3}}{π} ln \frac{α_{k} (r)}{1 - α_{k} (r)}, Θ_{j}^{- 1} (α_{k} (r)) \end{matrix}$

$= \frac{ρ_{kj} \sqrt{3}}{π} ln \frac{α_{k} (r)}{1 - α_{k} (r)},$ (49)

$\begin{matrix} Φ_{i}^{- 1} (1 - δ_{k}) & = e_{i} + \frac{σ_{i} \sqrt{3}}{π} ln \frac{1 - δ_{k}}{δ_{k}}, Θ_{j}^{- 1} (1 - δ_{k}) \\ = \frac{ρ_{kj} \sqrt{3}}{π} ln \frac{1 - δ_{k}}{δ_{k}} . \end{matrix}$ (50)

Thus, the theorem is proved.

Theorem 3. Suppose the security returns ξ_i, i = 1, 2, . . . , N are all zigzag uncertain variables ξ_i ∼ Z (a_i, b_i, c_i), the background asset returns r_kj, j = 1, 2, . . . , J are all zigzag uncertain variables r_kj ∼ Z (- ρ_kj, 0, ρ_kj), Then the model (31) can be transformed into the following form.

When 0 ⩽ α_k (r) <0.5, 0 ⩽ δ_k < 0.5, ${\begin{matrix} min θ [\sum_{i = 1}^{N} x_{ki} \frac{c_{i} - a_{i}}{2} + \sum_{j = 1}^{J} λ_{kj} ρ_{kj}] - (1 - θ) [\sum_{i = 1}^{N} x_{ki} \frac{a_{i} + 2 b_{i} + c_{i}}{4} - \sum_{i = 1}^{N} C_{i} x_{ki}] \\ s . t . \sum_{i = 1}^{N} [C_{i} - (1 - 2 α_{k} (r)) a_{i} - 2 α_{k} (r) b_{i}] x_{ki} ⩽ r + \sum_{j = 1}^{J} λ_{kj} (- ρ_{kj} (1 - 2 α_{k} (r))) - r_{f}, \forall r ⩾ 0 \\ \sum_{i = 1}^{N} [C_{i} - (2 - 2 (1 - δ_{k})) b_{i} - (2 (1 - δ_{k}) - 1) c_{i}] \\ x_{ki} ⩽ \sum_{j = 1}^{J} λ_{kj} ρ_{kj} (2 (1 - δ_{k}) - 1) - A_{k} \\ l_{ki} ⩽ x_{ki} ⩽ L_{ki} \\ \sum_{i = 1}^{N} w_{k} x_{ki} + \sum_{i = 1}^{N} C_{i} w_{k} x_{ki} ⩽ w_{k} \\ \sum_{j = 1}^{J} λ_{kj} = 1, λ_{kj} > 0 \end{matrix}$ (51)

When 0.5 ⩽ α_k (r) ⩽1, 0.5 ⩽ δ_k ⩽ 1, ${\begin{matrix} min θ [\sum_{i = 1}^{N} x_{ki} \frac{c_{i} - a_{i}}{2} + \sum_{j = 1}^{J} λ_{kj} ρ_{kj}] \\ - (1 - θ) [\sum_{i = 1}^{N} x_{ki} \frac{a_{i} + 2 b_{i} + c_{i}}{4} - \sum_{i = 1}^{N} C_{i} x_{ki}] \\ s . t . \sum_{i = 1}^{N} [C_{i} - (2 - 2 α_{k} (r)) b_{i} - (2 α_{k} (r) - 1) c_{i}] x_{ki} \\ ⩽ r + \sum_{j = 1}^{J} λ_{kj} ρ_{kj} (2 α_{k} (r) - 1) - r_{f}, \forall r ⩾ 0 \\ \sum_{i = 1}^{N} [C_{i} - (1 - 2 (1 - δ_{k})) a_{i} - 2 (1 - δ_{k}) b_{i}] x_{ki} \\ ⩽ \sum_{j = 1}^{J} λ_{kj} (- ρ_{kj} (1 - 2 (1 - δ_{k}))) - A_{k} \\ l_{ki} ⩽ x_{ki} ⩽ L_{ki} \\ \sum_{i = 1}^{N} w_{k} x_{ki} + \sum_{i = 1}^{N} C_{i} w_{k} x_{ki} ⩽ w_{k} \\ \sum_{j = 1}^{J} λ_{kj} = 1, λ_{kj} > 0 \end{matrix}$ (52)

Proof. Since ξ_i ∼ Z (a_i, b_i, c_i) and r_kj ∼ Z (- ρ_kj, 0, ρ_kj), according to Definition 6, the expected value of the zigzag uncertain distribution ξ ∼ Z (a, b, c) is $E [ξ] = \frac{a + 2 b + c}{4}$ , so $E [ξ_{i}] = \frac{a_{i} + 2 b_{i} + c_{i}}{4}, E [r_{kj}] = 0 .$ (53)

According to Lemma 6, the entropy of the zigzag uncertain distribution ξ ∼ Z (a, b, c) is $H [ξ] = \frac{c - a}{2}$ , so $H [ξ_{i}] = \frac{c_{i} - a_{i}}{2}, H [r_{kj}] = ρ_{kj} .$ (54)

According to Definition 3, the inverse uncertainty distribution of zigzag uncertain variable Z (a, b, c) is $Φ^{- 1} (α) = {\begin{matrix} (1 - 2 α) a + 2 α b, if 0 ⩽ α < 0.5, \\ (2 - 2 α) b + (2 α - 1) c, if 0.5 ⩽ α ⩽ 1 . \end{matrix}$ (55) so ξ_i and r_kj have inverse uncertainty distribution $Φ_{i}^{- 1} (α) = {\begin{matrix} (1 - 2 α) a_{i} + 2 α b_{i}, if 0 ⩽ α < 0.5, \\ (2 - 2 α) b_{i} + (2 α - 1) c_{i}, if 0 . 5 ⩽ α ⩽ 1 . \end{matrix}$ (56) $Θ_{j}^{- 1} (α) = {\begin{matrix} - ρ_{kj} (1 - 2 α), if 0 ⩽ α < 0.5, \\ ρ_{kj} (2 α - 1), if 0 . 5 ⩽ α ⩽ 1 . \end{matrix}$ (57)

In the specific model, we have

$\begin{matrix} Φ_{i}^{- 1} (α_{k} (r)) \\ = {\begin{matrix} (1 - 2 α_{k} (r)) a_{i} + 2 α_{k} (r) b_{i}, if 0 ⩽ α_{k} (r) < 0.5, \\ (2 - 2 α_{k} (r)) b_{i} + (2 α_{k} (r) - 1) c_{i}, if 0 . 5 ⩽ α_{k} (r) ⩽ 1 . \end{matrix} \end{matrix}$ (58)

$\begin{matrix} Φ_{i}^{- 1} (1 - δ_{k}) \\ = {\begin{matrix} (2 - 2 (1 - δ_{k})) b_{i} + (2 (1 - δ_{k}) - 1) c_{i}, if 0 ⩽ δ_{k} < 0.5, \\ (1 - 2 (1 - δ_{k})) a_{i} + 2 (1 - δ_{k}) b_{i}, if 0 . 5 ⩽ δ_{k} ⩽ 1 . \end{matrix} \end{matrix}$ (59) $Θ_{j}^{- 1} (α_{k} (r)) = {\begin{matrix} - ρ_{kj} (1 - 2 α_{k} (r)), if 0 ⩽ α_{k} (r) < 0.5, \\ ρ_{kj} (2 α_{k} (r) - 1), if 0 . 5 ⩽ α_{k} (r) ⩽ 1 . \end{matrix}$ (60) $Θ_{j}^{- 1} (1 - δ_{k}) = {\begin{matrix} ρ_{kj} (2 (1 - δ_{k}) - 1), if 0 ⩽ δ_{k} < 0.5, \\ - ρ_{kj} (1 - 2 (1 - δ_{k})), if 0 . 5 ⩽ δ_{k} ⩽ 1 . \end{matrix}$ (61)

Then the theorem is proved.

4.2 Expression of risk curve

Theorem 4. When the optimal investment proportion of model (31) is known, the expression of the risk curve is $R (r) = Ω (r_{f} - r + \sum_{i = 1}^{N} C_{i} x_{ki}), \forall r ⩾ 0$ (62) where Ω is the distribution function of $\sum_{i = 1}^{N} x_{ki} ξ_{i} + \sum_{j = 1}^{J} λ_{kj} r_{kj}$ .

Proof. The risk curve of the k-th mental account is

$\begin{matrix} R (r) = & M {r_{f} - (\sum_{i = 1}^{N} x_{ki} ξ_{i} + \sum_{j = 1}^{J} λ_{kj} r_{kj} - \sum_{i = 1}^{N} C_{i} x_{ki}) ⩾ r}, \\ \forall r ⩾ 0, \end{matrix}$ (63)

move the constant term to the right, and assume that the distribution function of $\sum_{i = 1}^{N} x_{ki} ξ_{i} + \sum_{j = 1}^{J} λ_{kj} r_{kj}$ is Ω, then

$\begin{matrix} R (r) = & M {\sum_{i = 1}^{N} x_{ki} ξ_{i} + \sum_{j = 1}^{J} λ_{kj} r_{kj} ⩽ r_{f} - r + \sum_{i = 1}^{N} C_{i} x_{ki}} \\ = Ω (r_{f} - r + \sum_{i = 1}^{N} C_{i} x_{ki}), \forall r ⩾ 0 . \end{matrix}$ (64)

The theorem is proved.

Theorem 5. When the optimal investment proportion of model (44) is known, the expression of the risk curve is

$\begin{matrix} R (r) = & (1 + exp (\frac{π ((\sum_{i = 1}^{N} x_{ki} e_{i}) - (r_{f} - r + \sum_{i = 1}^{N} C_{i} x_{ki}))}{\sqrt{3} (\sum_{i = 1}^{N} x_{ki} σ_{i} + \sum_{j = 1}^{J} λ_{kj} ρ_{kj})}))^{- 1}, \\ \forall r ⩾ 0 \end{matrix}$ (65)

Proof. Since ξ_i ∼ N (e_i, σ_i) and r_kj ∼ N (0, ρ_kj) are independent uncertain variables, and x_ki, λ_kj > 0, according to Lemma 9, then $\sum_{i = 1}^{N} x_{ki} ξ_{i} + \sum_{j = 1}^{J} λ_{kj} r_{kj} \sim N (\sum_{i = 1}^{N} x_{ki} e_{i}, \sum_{i = 1}^{N} x_{ki} σ_{i} + \sum_{j = 1}^{J} λ_{kj} ρ_{kj})$ , its distribution function is

$\begin{matrix} Ω (x) = & (1 + exp (\frac{π ((\sum_{i = 1}^{N} x_{ki} e_{i}) - x)}{\sqrt{3} (\sum_{i = 1}^{N} x_{ki} σ_{i} + \sum_{j = 1}^{J} λ_{kj} ρ_{kj})}))^{- 1}, \\ x \in R . \end{matrix}$ (66)

According to Theorem 4, the expression of the risk curve is

$\begin{matrix} R (r) = Ω (r_{f} - r + \sum_{i = 1}^{N} C_{i} x_{ki}) \\ = (1 + exp (\frac{π ((\sum_{i = 1}^{N} x_{ki} e_{i}) - (r_{f} - r + \sum_{i = 1}^{N} C_{i} x_{ki}))}{\sqrt{3} (\sum_{i = 1}^{N} x_{ki} σ_{i} + \sum_{j = 1}^{J} λ_{kj} ρ_{kj})}))^{- 1}, \\ \forall r ⩾ 0 . \end{matrix}$ (67)

The theorem is proved.

Theorem 6. When the optimal investment proportion of model (51) and (52) is known, the expression of risk curve is shown below. $R (r) = {\begin{matrix} 1, if r ⩽ r_{1}, \\ \frac{r_{f} - r + \sum_{i = 1}^{N} C_{i} x_{ki} + \sum_{i = 1}^{N} x_{ki} c_{i} + \sum_{j = 1}^{J} λ_{kj} ρ_{kj} - 2 \sum_{i = 1}^{N} x_{ki} b_{i}}{2 (\sum_{i = 1}^{N} x_{ki} c_{i} + \sum_{j = 1}^{J} λ_{kj} ρ_{kj} - \sum_{i = 1}^{N} x_{ki} b_{i})}, if r_{1} ⩽ r ⩽ r_{2}, \\ \frac{r_{f} - r + \sum_{i = 1}^{N} C_{i} x_{ki} - (\sum_{i = 1}^{N} x_{ki} a_{i} + \sum_{j = 1}^{J} (- λ_{kj} ρ_{kj}))}{2 (\sum_{i = 1}^{N} x_{ki} b_{i} - (\sum_{i = 1}^{N} x_{ki} a_{i} + \sum_{j = 1}^{J} (- λ_{kj} ρ_{kj})))}, if r_{2} ⩽ r ⩽ r_{3}, \\ 0, if r ⩾ r_{3} . \end{matrix}$ (68) where $r_{1} = r_{f} + \sum_{i = 1}^{N} C_{i} x_{ki} - \sum_{i = 1}^{N} x_{ki} c_{i} - \sum_{j = 1}^{J} λ_{kj} ρ_{kj}, r_{2} = r_{f} + \sum_{i = 1}^{N} C_{i} x_{ki} - \sum_{i = 1}^{N} x_{ki} b_{i}, r_{3} = r_{f} + \sum_{i = 1}^{N} C_{i} x_{ki} + \sum_{j = 1}^{J} λ_{kj} ρ_{kj} - \sum_{i = 1}^{N} x_{ki} a_{i}$ .

Proof. Since ξ_i ∼ Z (a_i, b_i, c_i) and r_kj ∼ Z (- ρ_kj, 0, ρ_kj) are independent uncertain variables. What’s more, x_ki > 0, λ_j > 0. According to Lemma 8, then

$\begin{matrix} \sum_{i = 1}^{N} x_{ki} ξ_{i} + \sum_{j = 1}^{J} λ_{kj} r_{kj} \sim Z (\sum_{i = 1}^{N} x_{ki} a_{i} \\ + \sum_{j = 1}^{J} (- λ_{kj} ρ_{kj}), \sum_{i = 1}^{N} x_{ki} b_{i}, \sum_{i = 1}^{N} x_{ki} c_{i} + \sum_{j = 1}^{J} λ_{kj} ρ_{kj}) . \end{matrix}$ (69)

Its distribution function is

$Ω (x) = {\begin{matrix} 0, if x ⩽ \sum_{i = 1}^{N} x_{ki} a_{i} + \sum_{j = 1}^{J} (- λ_{kj} ρ_{kj}), \\ \frac{x - (\sum_{i = 1}^{N} x_{ki} a_{i} + \sum_{j = 1}^{J} (- λ_{kj} ρ_{kj}))}{2 (\sum_{i = 1}^{N} x_{ki} b_{i} - (\sum_{i = 1}^{N} x_{ki} a_{i} + \sum_{j = 1}^{J} (- λ_{kj} ρ_{kj})))}, if \sum_{i = 1}^{N} x_{ki} a_{i} + \sum_{j = 1}^{J} (- λ_{kj} ρ_{kj}) ⩽ x ⩽ \sum_{i = 1}^{N} x_{ki} b_{i}, \\ \frac{x + \sum_{i = 1}^{N} x_{ki} c_{i} + \sum_{j = 1}^{J} λ_{kj} ρ_{kj} - 2 \sum_{i = 1}^{N} x_{ki} b_{i}}{2 (\sum_{i = 1}^{N} x_{ki} c_{i} + \sum_{j = 1}^{J} λ_{kj} ρ_{kj} - \sum_{i = 1}^{N} x_{ki} b_{i})}, if \sum_{i = 1}^{N} x_{ki} b_{i} ⩽ x ⩽ \sum_{i = 1}^{N} x_{ki} c_{i} + \sum_{j = 1}^{J} λ_{kj} ρ_{kj}, \\ 1, if x ⩾ \sum_{i = 1}^{N} x_{ki} c_{i} + \sum_{j = 1}^{J} λ_{kj} ρ_{kj} . \end{matrix}$ (70)

According to Theorem 4, the expression of the risk curve is

$\begin{matrix} R (r) = Ω (r_{f} - r + \sum_{i = 1}^{N} C_{i} x_{ki}) \\ = {\begin{matrix} 0, if r_{f} - r + \sum_{i = 1}^{N} C_{i} x_{ki} ⩽ f_{1}, \\ \frac{r_{f} - r + \sum_{i = 1}^{N} C_{i} x_{ki} - (\sum_{i = 1}^{N} x_{ki} a_{i} + \sum_{j = 1}^{J} (- λ_{kj} ρ_{kj}))}{2 (\sum_{i = 1}^{N} x_{ki} b_{i} - (\sum_{i = 1}^{N} x_{ki} a_{i} + \sum_{j = 1}^{J} (- λ_{kj} ρ_{kj})))}, if f_{1} ⩽ r_{f} - r + \sum_{i = 1}^{N} C_{i} x_{ki} ⩽ f_{2}, \\ \frac{r_{f} - r + \sum_{i = 1}^{N} C_{i} x_{ki} + \sum_{i = 1}^{N} x_{ki} c_{i} + \sum_{j = 1}^{J} λ_{kj} ρ_{kj} - 2 \sum_{i = 1}^{N} x_{ki} b_{i}}{2 (\sum_{i = 1}^{N} x_{ki} c_{i} + \sum_{j = 1}^{J} λ_{kj} ρ_{kj} - \sum_{i = 1}^{N} x_{ki} b_{i})}, if f_{2} ⩽ r_{f} - r + \sum_{i = 1}^{N} C_{i} x_{ki} ⩽ f_{3}, \\ 1, if r_{f} - r + \sum_{i = 1}^{N} C_{i} x_{ki} ⩾ f_{3} . \end{matrix} \end{matrix}$ (71)

where $f_{1} = \sum_{i = 1}^{N} x_{ki} a_{i} + \sum_{j = 1}^{J} (- λ_{kj} ρ_{kj}), f_{2} = \sum_{i = 1}^{N} x_{ki} b_{i}, f_{3} = \sum_{i = 1}^{N} x_{ki} c_{i} + \sum_{j = 1}^{J} λ_{kj} ρ_{kj}$ . The theorem is proved.

5 Numerical examples

In order to verify the feasibility of the model, we give numerical examples in this section. Suppose that investors set up the 4^th, 5^th, and 9^th mental accounts, which are respectively secure storage account, risk storage account, and leisure expense account, while the remaining mental accounts are not set. The initial wealth and the proportion of total investment wealth are shown in Table 3.

Table 3
The initial wealth of accounts and the proportion of total investment wealth

k Mental account w _k γ _k

1 regular income account 0 0

2 additional income account 0 0

3 operating income account 0 0

4 secure storage account 20000 0.2

5 risk storage account 50000 0.5

6 living expenses account 0 0

7 family and personal development expenses account 0 0

8 emotional maintenance expense account 0 0

9 leisure expense account 30000 0.3

k	Mental account	w _k	γ _k
1	regular income account	0	0
2	additional income account	0	0
3	operating income account	0	0
4	secure storage account	20000	0.2
5	risk storage account	50000	0.5
6	living expenses account	0	0
7	family and personal development expenses account	0	0
8	emotional maintenance expense account	0	0
9	leisure expense account	30000	0.3

In mental accounts, investors have different attitudes towards risk and individual tolerance for losses. In the secure storage account, investors are risk averse and have little tolerance for losses; in the leisure expense account, investors are risk neutral and have moderate tolerance for losses; in the risk storage account, the investors are risk preference and have a greater tolerance for losses.

5.1 Normal uncertainty distribution

Assume that investors invest in eight securities in secure storage account (k = 4), risk storage account (k = 5), and leisure expense account (k = 9). The security returns follow normal uncertain distribution ξ_i ∼ N (e_i, σ_i) , i = 1, 2, . . . , 8. The data comes from [39], as shown in Table 4. It is assumed that investors in three kinds of mental accounts are exposed to two background risks: income and housing. Among them, the income background asset return obeys the uncertain distribution r_k1 ∼ N (0, 0.01), and the housing background asset return obeys the uncertainty distribution r_k2 ∼ N (0, 0.02). Investors believe that the two background risk weights are equal, that is λ_k1 = λ_k2 = 0.5. In the three kinds of mental accounts, the lower limit of the securities investment ratio is l_ki = 0.01, and the upper limit is L_ki = 0.8. In the securities market, assume that the monthly risk-free interest rate r_f is 0.01, and the transaction fee rate for buying security i is C_i = 0.08%.

Table 4
Normal uncertain return rates of 8 securities [39]

i 1 2 3 4 5 6 7 8

e _i 0.0940 0.0560 0.3136 0.1638 0.0712 0.1310 0.1824 0.1650

σ _i 0.1060 0.0500 0.7760 0.2280 0.0920 0.1860 0.3960 0.1880

i	1	2	3	4	5	6	7	8
e _i	0.0940	0.0560	0.3136	0.1638	0.0712	0.1310	0.1824	0.1650
σ _i	0.1060	0.0500	0.7760	0.2280	0.0920	0.1860	0.3960	0.1880

The confidence curve represents the investor’s tolerance for losses in the mental accounts. The risk curve of a safe portfolio is lower than the confidence curve. In real life, many investors only care about 0 < α_k (r) <0.5[32], since α_k (r) ⩾0.5 means greater tolerance for losses. According to the different tolerances of the mental accounts for losses, we set three corresponding confidence curves under the mental accounts, the expressions being as follows.

$\begin{matrix} α_{4} (r) = \frac{0.49}{(r + 1)^{9}}, α_{5} (r) = \frac{0.49}{(r + 1)^{2}}, \\ α_{9} (r) = \frac{0.49}{(r + 1)^{4}}, r ⩾ 0 \end{matrix}$ (72)

Obviously 0 < α₄ (r) , α₅ (r) , α₉ (r) <0.5, the three confidence curves are shown in Fig. 1. α₄ (r) is the confidence curve in the secure storage account, which means that the space of the safe portfolio in the account is the smallest; α₅ (r) is the confidence curve in the risk storage account, at this time, the space of the safe portfolio is the largest; α₉ (r) is the confidence curve in the leisure expense account, and the space of the safe portfolio is moderate.

Fig. 1

Confidence curves in (72).

5.1.1 Results of secure storage account (k = 4)

In this account, let the return threshold A₄ = 0.1, uncertainty measure threshold δ₄ = 0.2. The confidence curve uses α₄ (r) to represent the tolerance for losses in this account. Investors think that the two objective functions are equally important, so the adjustment parameter takes θ = 0.5. After sorting out the above data, the linear model (44) is solved by MATLAB2017a, then the optimal investment proportion, expected value and entropy in the secure storage account are obtained. The results are shown in Table 5.

Table 5
Sub-portfolio (k = 4)

x ₄₁ x ₄₂ x ₄₃ x ₄₄ x ₄₅ x ₄₆ x ₄₇ x ₄₈ E ₄ H ₄

0.0100 0.6670 0.0100 0.0100 0.0100 0.0100 0.0100 0.0100 4.8% 0.1235

x ₄₁	x ₄₂	x ₄₃	x ₄₄	x ₄₅	x ₄₆	x ₄₇	x ₄₈	E ₄	H ₄
0.0100	0.6670	0.0100	0.0100	0.0100	0.0100	0.0100	0.0100	4.8%	0.1235

According to Theorem 5, the risk curve R (r) corresponding to the optimal investment proportion is shown in Fig. 2. We can obtain the following information from Fig. 2.

It can be seen that when r ∈ [0, 1], the risk curve is lower than the confidence curve α₄ (r), obviously lower than the confidence curve α₅ (r) and α₉ (r), indicating that the portfolio is safe at this time.

The risk curve of secure storage account is lower than all confidence curves. This phenomenon is consistent with the starting point of risk aversion and conforms to the investment psychology of this account.

Fig. 2

Comparison of confidence curves and risk curve at k = 4.

5.1.2 Results of risk storage account (k = 5)

In this account, investors think that the two objective functions are equally important, so the adjustment parameter takes θ = 0.5. Since it is in risky storage account, let the return threshold A₅ = 0.3, uncertainty measure threshold δ₅ = 0.4, and take confidence curve α₅ (r). Organize the above data, then solve the linear model (44) through MATLAB2017a to obtain the optimal investment ratio, expected mean and entropy in the risk storage account. The results are shown in Table 6.

Table 6
Sub-portfolio (k = 5)

x ₅₁ x ₅₂ x ₅₃ x ₅₄ x ₅₅ x ₅₆ x ₅₇ x ₅₈ E ₅ H ₅

0.0100 0.0100 0.3344 0.0100 0.0100 0.0100 0.0100 0.6048 21.09% 0.7233

x ₅₁	x ₅₂	x ₅₃	x ₅₄	x ₅₅	x ₅₆	x ₅₇	x ₅₈	E ₅	H ₅
0.0100	0.0100	0.3344	0.0100	0.0100	0.0100	0.0100	0.6048	21.09%	0.7233

According to Theorem 5, the risk curve R (r) under the optimal investment ratio is shown in Fig. 3. We can obtain the following information from Fig. 3.

Fig. 3

Comparison of confidence curves and risk curve at k = 5.

It can be seen that when r ∈ [0, 1], the risk curve R (r) is completely lower than the confidence curve α₅ (r), indicating that the portfolio is safe under the constraints of the confidence curve α₅ (r).

However, part of the risk curve is higher than the confidence curves α₄ (r) and α₉ (r), it means that if the investor’s confidence curves are α₄ (r) and α₉ (r), then this portfolio may be at risk.

The risk curve of risk storage account is higher than α₄ (r) and α₉ (r), which indicates that the portfolio does not meet the tolerance for losses of safe storage account and leisure expense account. This phenomenon is also in line with the characteristics of risk storage account, which is risk preference.

5.1.3 Results of leisure expense account (k = 9)

In this account, take the adjustment parameter θ = 0.5. Let the return threshold A₉ = 0.15, uncertainty measure threshold δ₉ = 0.4. At this time, investors’ tolerance for possible losses is medium, so the confidence curve α₉ (r) is used. Then solve the linear model (44) through MATLAB2017a to obtain the optimal investment proportion, expected mean and entropy in the leisure expense account. The results are shown in Table 7.

Table 7
Sub-portfolio (k = 9)

x ₉₁ x ₉₂ x ₉₃ x ₉₄ x ₉₅ x ₉₆ x ₉₇ x ₉₈ E ₉ H ₉

0.0100 0.4329 0.0100 0.0100 0.0100 0.0100 0.0100 0.5063 11.65% 0.2715

x ₉₁	x ₉₂	x ₉₃	x ₉₄	x ₉₅	x ₉₆	x ₉₇	x ₉₈	E ₉	H ₉
0.0100	0.4329	0.0100	0.0100	0.0100	0.0100	0.0100	0.5063	11.65%	0.2715

According to Theorem 5, The risk curve R (r) corresponding to the optimal investment ratio is shown in Fig. 4. We can obtain the following information from Fig. 4.

Fig. 4

Comparison of confidence curves and risk curve at k = 9.

It can be seen that when r ∈ [0, 1], the risk curve R (r) is completely lower than the confidence curves α₅ (r) and α₉ (r), indicating that the portfolio is safe under the constraints of the confidence curve α₉ (r).

However, part of the risk curve is higher than the confidence curve α₄ (r), indicating that if the investor’s confidence curve is α₄ (r), then this portfolio may be at risk.

The risk curve of leisure expense account is higher than α₄ (r), which indicates that the portfolio does not meet the tolerance of loss of safe storage account. This phenomenon is also in line with the characteristics of leisure expense account, which is risk neutral.

Finally, by proportion γ_k and formula $\sum_{k = 1}^{9} γ_{k} E_{k}, \sum_{k = 1}^{9} γ_{k} H_{k}$ , we weight the expected return and entropy of each account. Finally, the expected return and entropy of the aggregate portfolio are obtained when the security return and the background asset return follow normal uncertain distribution. The results are shown in Table 8.

Table 8

Optimal sub-portfolios and aggregate portfolio

k	1	2	3	4	5	6	7	8	9	Aggregate
γ _k	0	0	0	0.2	0.5	0	0	0	0.3	1
E _k	0	0	0	4.8%	21.09%	0	0	0	11.65%	15.00%
H _k	0	0	0	0.1235	0.7233	0	0	0	0.2715	0.4678

We can know the following two pieces of information from Table 8:

When k = 4, it represents the mental account of risk aversion, the income and entropy are both the smallest at this time; k = 5 represents the mental account of risk preference, at this time both the return and entropy are the largest; k = 9 represents mental account of risk neutral, the return and entropy are medium at this time.

In different mental accounts, the higher the return, the higher the entropy, which shows that if the return is higher, it is more difficult to accurately estimate the uncertain variable. From another perspective, it can also be understood as the greater the portfolio risk.

5.2 Zigzag uncertainty distribution

Assume that investors invest in eight securities in secure storage account (k = 4), risk storage account (k = 5), and leisure expense account (k = 9). The security returns follow zigzag uncertain distribution ξ_i ∼ Z (a_i, b_i, c_i) , i = 1, 2, . . . , 8; The data comes from [32], as shown in Table 9. It is assumed that investors in three kinds of mental account are exposed to two background risks: income and housing. Among them, the income background asset return obeys the uncertain distribution r_k1 ∼ Z (-0.01, 0, 0.01), and the housing background asset return obeys the uncertainty distribution r_k2 ∼ Z (-0.02, 0, 0.02). Investors believe that the two background risk weights are equal, that is λ_k1 = λ_k2 = 0.5. In the three kinds of mental accounts, the lower limit of the securities investment ratio is l_ki = 0.01, and the upper limit is L_ki = 0.8. In the securities market, assume that the monthly risk-free interest rate r_f is 0.01, and the transaction fee rate for buying security i is C_i = 0.08%.

Table 9
Zigzag uncertain return rates of 8 securities [32]

i 1 2 3 4 5 6 7 8

a _i –0.16 –0.13 –0.13 –0.18 –0.14 –0.17 –0.24 –0.14

b _i –0.01 0.01 0.03 0.02 0.00 0.00 0.02 0.03

c _i 0.14 0.19 0.20 0.16 0.18 0.33 0.45 0.21

i	1	2	3	4	5	6	7	8
a _i	–0.16	–0.13	–0.13	–0.18	–0.14	–0.17	–0.24	–0.14
b _i	–0.01	0.01	0.03	0.02	0.00	0.00	0.02	0.03
c _i	0.14	0.19	0.20	0.16	0.18	0.33	0.45	0.21

In reality, many investors only care about 0 < α_k (r) <0.5[32], so when the return follows zigzag uncertainty distribution, we mainly choose the model (51) to experiment. According to the risk attitude of different accounts, we set three confidence curves the same as in Section 5.1. In the secure storage account (k = 4), set A₄ = 0.08, δ₄ = 0.2, and the confidence curve adopts α₄ (r). In the risk storage account (k = 5), set A₅ = 0.20, δ₅ = 0.25, and the confidence curve adopts α₅ (r). In the leisure expense account (k = 9), set A₉ = 0.15, δ₉ = 0.25, and the confidence curve adopts α₉ (r). Under the three accounts, let θ = 0.5. Organize the above data and solve the linear model (51) through MATLAB2017a to obtain the optimal investment ratio, expected mean and entropy of the three accounts. The results are shown in Table 10.

Table 10

Optimal sub-portfolios

x ₄₁	x ₄₂	x ₄₃	x ₄₄	x ₄₅	x ₄₆	x ₄₇	x ₄₈	E ₄	H ₄
0.0100	0.0100	0.4635	0.0100	0.0100	0.0100	0.0100	0.0100	1.62%	0.0756
x ₅₁	x ₅₂	x ₅₃	x ₅₄	x ₅₅	x ₅₆	x ₅₇	x ₅₈	E ₅	H ₅
0.0100	0.0100	0.0794	0.0100	0.0100	0.0698	0.8000	0.0100	5.51%	0.2997
x ₉₁	x ₉₂	x ₉₃	x ₉₄	x ₉₅	x ₉₆	x ₉₇	x ₉₈	E ₉	H ₉
0.0100	0.0100	0.7312	0.0100	0.0100	0.0100	0.2080	0.0100	3.69%	0.1881

After obtaining the optimal investment ratio, we draw the risk curves as shown in Fig. 5 according to Theorem 6. We can obtain the following information from Fig. 5.

Fig. 5

Comparison of confidence curves and risk curves when returns are zigzag uncertain variables.

R₄ (r) represents the risk curve under the optimal investment ratio of the secure storage account, which is lower than the confidence curve α₄ (r), indicating that this portfolio is safe under the condition of the confidence curve α₄ (r).

R₅ (r) represents the risk curve under the optimal investment ratio of the risk storage account, which is lower than the confidence curve α₅ (r), indicating that the investment portfolio at this time is safe under the conditions of the confidence curve α₅ (r), but it is risky under the confidence curves α₄ (r) , α₉ (r).

R₉ (r) represents the risk curve under the optimal investment ratio of the leisure expense account, which is lower than the confidence curve α₅ (r) , α₉ (r), indicating that the investment portfolio at this time is safe, but it is risky under the condition of the confidence curve α₄ (r).

Finally, by proportion γ_k and formula $\sum_{k = 1}^{9} γ_{k} E_{k}, \sum_{k = 1}^{9} γ_{k} H_{k}$ , we weight the expected return and entropy of each account. Finally, the expected return and entropy of the aggregate portfolio are obtained when the security return and the background asset return follow zigzag uncertain distribution. The results are shown in Table 11.

Table 11

Optimal sub-portfolios and aggregate portfolio

k	1	2	3	4	5	6	7	8	9	Aggregate
γ _k	0	0	0	0.2	0.5	0	0	0	0.3	1
E _k	0	0	0	1.62%	5.51%	0	0	0	3.69%	4.19%
H _k	0	0	0	0.0756	0.2997	0	0	0	0.1881	0.2214

From Table 11 we can know the following two pieces of information, which are similar to Table 8.

Both returns and entropy are minimum when k = 4, because this is solved in the risk averse mental account; returns and entropy are maximum when k = 5, because the mental account is risk-averse at this point; and returns and entropy are medium when k = 9, precisely because the mental account is risk-neutral.

When returns are zigzag uncertain variables, the same conclusion is reached as when returns are normally uncertain distributions. When the return is higher the entropy is also higher. From another point of view, it can also be understood that the greater the return, the greater the risk of the portfolio.

6 Comparison and sensitivity analysis

6.1 Comparison

6.1.1 Comparison with other portfolio studies

Table 12 illustrates the comparison between our paper and other portfolio studies under the uncertain theory framework. Our paper considers the bi-objective functions of mean and entropy, and discusses total mental accounts and multiple background risks. In the model, we set the corresponding confidence curve according to the mental account’s tolerance for losses. The relationship between risk curve and confidence curve is discussed when the returns obey normal and zigzag uncertainty distribution respectively.

Table 12
Comparison with other portfolio studies under the framework of uncertainty theory

Condition [30] [32] [33] [39] Our

(Huang, 2020) (Zhai, 2018) (Zhai, 2018) (Xue, 2019) paper

Objective function-maximum mean √ √ √ √ √

Objective function-minimizing entropy × × × × √

Background risk √ √ × √ √

Multiple background risks × × × × √

Risk curve × √ √ × √

Number of mental accounts 0 0 0 2 9

Normal uncertainty distribution √ √ √ √ √

Zigzag uncertainty distribution × √ √ × √

Condition	[30]	[32]	[33]	[39]	Our
Objective function-maximum mean	√	√	√	√	√
Objective function-minimizing entropy	×	×	×	×	√
Background risk	√	√	×	√	√
Multiple background risks	×	×	×	×	√
Risk curve	×	√	√	×	√
Number of mental accounts	0	0	0	2	9
Normal uncertainty distribution	√	√	√	√	√
Zigzag uncertainty distribution	×	√	√	×	√

6.1.2 Comparison of models with and without multiple background risks

In order to explore the impact of multiple background risks on the portfolio, we make the following comparison. When the standard deviation of the background asset return is ρ_kj = 0, j = 1, . . . , J, then the model (44) degenerates to the model (73) without considering the multiple background risks. ${\begin{matrix} min θ [\sum_{i = 1}^{N} \frac{π σ_{i}}{\sqrt{3}} x_{ki}] \\ - (1 - θ) [\sum_{i = 1}^{N} x_{ki} e_{i} - \sum_{i = 1}^{N} C_{i} x_{ki}] \\ s . t . \sum_{i = 1}^{N} [C_{i} - (e_{i} + \frac{σ_{i} \sqrt{3}}{π} ln \frac{α_{k} (r)}{1 - α_{k} (r)})] x_{ki} \\ ⩽ r - r_{f}, \forall r ⩾ 0 \\ \sum_{i = 1}^{N} [C_{i} - (e_{i} + \frac{σ_{i} \sqrt{3}}{π} ln \frac{1 - δ_{k}}{δ_{k}})] x_{ki} ⩽ - A_{k} \\ l_{ki} ⩽ x_{ki} ⩽ L_{ki} \\ \sum_{i = 1}^{N} w_{k} x_{ki} + \sum_{i = 1}^{N} C_{i} w_{k} x_{ki} ⩽ w_{k} \\ 0 < α_{k} (r) < 1, 0 < δ_{k} < 1 \end{matrix}$ (73)

We compared it with the experiment in Section 5.1. When other parameters remain unchanged and the standard deviation of income and housing background asset returns are zero (ρ_k1 = ρ_k2 = 0), then through solving the model (73), the comparative results are shown in Table 13. As can be seen from Table 13, when the multiple background risks are not considered, the return of the aggregate portfolio is 15.3% >15%, the entropy of the aggregate portfolio is 0.4510 < 0.4678. Disregarding the multiple background risks in the model will increase the return and decrease the entropy of the portfolio. If multiple background risks are real but ignored, this will cause investors to ignore potential unknown risks and make some wrong judgments. Multiple background risks do affect portfolio selection.

Table 13

Optimal sub-portfolios and aggregate portfolio

Parameter	E ₄	H ₄	E ₅	H ₅	E ₉	H ₉	E	H
ρ_k1 = ρ_k2 = 0	0.0547	0.1074	0.2126	0.7089	0.1192	0.2503	0.1530	0.4510
ρ_k1 = 0.01, ρ_k2 = 0.02	0.0480	0.1235	0.2109	0.7233	0.1165	0.2715	0.1500	0.4678

6.2 Sensitivity analysis

6.2.1 Impact of multiple background risks on portfolio

In order to observe the impact of multiple background risks on portfolio risk, we carry out further experiments in section 5.1.1. When other parameters remain unchanged and only the parameters ρ₄₁, ρ₄₂ increase, we draw the corresponding risk curve as shown in Fig. 6. Then we can obtain the following information from Fig. 6.

Fig. 6

Comparison of confidence curve and risk curves with the increase of standard deviation.

With the increase of ρ₄₁, ρ₄₂, the risk curve moves upward gradually.

When ρ₄₁ = 0.03, ρ₄₂ = 0.04, the risk curve is higher than the confidence curve α₄ (r). With the increase of ρ₄₁, ρ₄₂, the part higher than the confidence curve will increase.

If investors ignore the existence of multiple background risks, the original risky portfolio will be considered safe. This is a potential risk.

The addition of multiple background risks covers all the background risks in the portfolio model. The above also shows that it is necessary to consider the multiple background risks, and investors can personalize the multiple background risks according to their actual situation.

6.2.2 Impact of return threshold on portfolio

In order to observe the impact of return threshold on portfolio risk, we carry out further experiments in Section 5.1. When the other parameters remain unchanged, the influence of the change of return threshold A_k, k = 4, 5, 9 on the sub-portfolio is discussed.

In Section 5.1.1, around the original data A₄ = 0.1, four points near 0.1 are selected. The results of model solution are shown in Table 14. It can be found that when the return threshold A₄ increases, the return of the sub-portfolio increases, and the entropy also increases. In addition, we draw the corresponding risk curve as shown in Fig. 7. Then we can obtain the following information.

Table 14
Sub-portfolio with the increase of return threshold A₄

A ₄ 0 0.05 0.1 0.15 0.2

E ₄ 0.0117 0.0184 0.0480 0.0754 0.1008

H ₄ 0.0639 0.0749 0.1235 0.1771 0.2354

A ₄	0	0.05	0.1	0.15	0.2
E ₄	0.0117	0.0184	0.0480	0.0754	0.1008
H ₄	0.0639	0.0749	0.1235	0.1771	0.2354

Fig. 7

Comparison of confidence curve and risk curves with the increase of return threshold A₄.

With the increase of A₄, the risk curve of the optimal portfolio will rotate anticlockwise;

When A₄ = 0.15, part of the risk curve is higher than the confidence curve α₄ (r);

When A₄ = 0.2, the part of the risk curve higher than the confidence curve increases.

In Section 5.1.2, around the original data A₅ = 0.3, four points near 0.3 are selected. The solution results of the model are shown in Table 15. It can be found that the mean value and entropy of sub-portfolio are directly proportional to A₅. Then we draw the corresponding risk curve as shown in Fig. 8. Then we can obtain the following information.

Table 15

Sub-portfolio with the increase of return threshold A₅

A ₅	0.2	0.25	0.3	0.35	0.4
E ₅	0.1552	0.1843	0.2109	0.2374	0.2639
H ₅	0.3636	0.5329	0.7233	0.9138	1.1042

Fig. 8

Comparison of confidence curve and risk curves with the increase of return threshold A₅.

With the increase of A₅, the risk curve of the optimal portfolio will move upward;

When A₅ = 0.35, the risk curve is very close to the confidence curve α₅ (r);

When A₅ = 0.4, part of the risk curve is above the confidence curve.

In Section 5.1.3, around the original data A₉ = 0.15, four points near 0.15 are selected. The model solution results are shown in Table 16. It can be found that the mean value and entropy of sub-portfolio are directly proportional to A₉. Therefore, we draw the corresponding risk curve as shown in Fig. 9. Then we can obtain the following information.

Table 16

Sub-portfolio with the increase of return threshold A₉

A ₉	0.05	0.1	0.15	0.2	0.25
E ₉	0.0370	0.0776	0.1165	0.1552	0.1843
H ₉	0.1054	0.1820	0.2715	0.3636	0.5329

Fig. 9

Comparison of confidence curve and risk curves with the increase of return threshold A₉.

With the increase of A₉, the risk curve of the optimal portfolio will gradually move up in the middle and tail;

When A₉ = 0.25, the risk curve is partially higher than the confidence curve.

Through the above analysis, we can know that the increase of the return threshold will inevitably increase the return of the sub-portfolio, but at the same time, it also faces the risk that the risk curve is higher than the confidence curve. According to the sensitivity analysis, investors can choose the optimal return threshold under different mental accounts.

7 Conclusion

This paper discusses the portfolio selection problem when the security return and background asset return are regarded as uncertain variables. In this paper, risk curve and nine kinds of common mental accounts are applied to securities investment, and the confidence curve is used to reflect the tolerance of different mental accounts for loss. In real life, investors face a variety of background risks, so this paper takes the linear weighting of different background asset risks as multiple background risks, which are added to the model. In addition, considering transaction cost, opportunity constraint, upper and lower bound constraints and initial wealth constraint, we construct a mean-entropy uncertain portfolio model with risk curve and total mental accounts. On this basis, we give the exact equivalent form of the model when returns follow the general, normal and zigzag uncertainty distribution, which are shown by Theorems 1, 2 and 3. Furthermore, the concrete expressions of the corresponding risk curve are derived, which are shown by Theorems 4, 5 and 6. Numerical examples and analysis results show that the model is effective and feasible. The results and figures reveal that the risk curve can directly describe the investors’ attitude towards risk. The combination of risk curve and mental account is of practical significance. Finally, Section 6 explains that the setting of multiple background risks covers all possible situations of background risks. With the increase of multiple background risks, the risk curve will gradually become higher than the confidence curve. If investors ignore the existence of multiple background risks, the original risky portfolio will be considered safe. In addition, the increase in the return threshold will inevitably increase the expected return of the portfolio, but at the same time it will also face the risk that the risk curve is higher than the confidence curve. From the perspective of risk curve, this paper describes the necessity of multiple background risks and the influence of return threshold, which can bring novel ideas to and visual impact on investors.

It is inevitable that this paper still has some limitations, which is also the subject that we want to continue to explore in the future. For example, (1) multiple background risk is defined by linear weighting in this paper, but the form of multiple background risk can still be discussed, such as the form of quadratic function and power function. (2) The method of solving the model in this paper is the linear programming method. In our future work, we will take nonlinear constraints into the model, such as cardinality constraints, industry constraints, and further design hybrid intelligent algorithms to solve them.

Footnotes

Acknowledgments

This research was supported by the “Humanities and Social Sciences Research and Planning Fund of the Ministry of Education of China, No. 18YJAZH014-x2lxY9180090”, “Natural Science Foundation of Guangdong Province, No. 2019A1515011038”, “Guangdong Province Characteristic Innovation Project of Colleges and Universities, No. 2019GKTSCX023”, “Soft Science of Guangdong Province, No. 2018A070712006, 2019A101002118”. The authors are highly grateful to the referees and editor in-chief for their very helpful comments.

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Mean-entropy uncertain portfolio with risk curve and total mental accounts under multiple background risks

Abstract

Keywords

1 Introduction

2 Preliminaries

2.1 Uncertainty theory

3.2.1 Risk curve constraint

4.1 Equivalent forms of model

Table 4 Normal uncertain return rates of 8 securities [39] i 1 2 3 4 5 6 7 8 e i 0.0940 0.0560 0.3136 0.1638 0.0712 0.1310 0.1824 0.1650 σ i 0.1060 0.0500 0.7760 0.2280 0.0920 0.1860 0.3960 0.1880

Table 5 Sub-portfolio (k = 4) x 41 x 42 x 43 x 44 x 45 x 46 x 47 x 48 E 4 H 4 0.0100 0.6670 0.0100 0.0100 0.0100 0.0100 0.0100 0.0100 4.8% 0.1235

Table 6 Sub-portfolio (k = 5) x 51 x 52 x 53 x 54 x 55 x 56 x 57 x 58 E 5 H 5 0.0100 0.0100 0.3344 0.0100 0.0100 0.0100 0.0100 0.6048 21.09% 0.7233

Table 7 Sub-portfolio (k = 9) x 91 x 92 x 93 x 94 x 95 x 96 x 97 x 98 E 9 H 9 0.0100 0.4329 0.0100 0.0100 0.0100 0.0100 0.0100 0.5063 11.65% 0.2715

Table 9 Zigzag uncertain return rates of 8 securities [32] i 1 2 3 4 5 6 7 8 a i –0.16 –0.13 –0.13 –0.18 –0.14 –0.17 –0.24 –0.14 b i –0.01 0.01 0.03 0.02 0.00 0.00 0.02 0.03 c i 0.14 0.19 0.20 0.16 0.18 0.33 0.45 0.21

6.1 Comparison

6.1.1 Comparison with other portfolio studies

6.2.1 Impact of multiple background risks on portfolio

Table 14 Sub-portfolio with the increase of return threshold A4 A 4 0 0.05 0.1 0.15 0.2 E 4 0.0117 0.0184 0.0480 0.0754 0.1008 H 4 0.0639 0.0749 0.1235 0.1771 0.2354

Footnotes

Acknowledgments

References

Table 4
Normal uncertain return rates of 8 securities [39]

i 1 2 3 4 5 6 7 8

e _i 0.0940 0.0560 0.3136 0.1638 0.0712 0.1310 0.1824 0.1650

σ _i 0.1060 0.0500 0.7760 0.2280 0.0920 0.1860 0.3960 0.1880

Table 5
Sub-portfolio (k = 4)

x ₄₁ x ₄₂ x ₄₃ x ₄₄ x ₄₅ x ₄₆ x ₄₇ x ₄₈ E ₄ H ₄

0.0100 0.6670 0.0100 0.0100 0.0100 0.0100 0.0100 0.0100 4.8% 0.1235

Table 6
Sub-portfolio (k = 5)

x ₅₁ x ₅₂ x ₅₃ x ₅₄ x ₅₅ x ₅₆ x ₅₇ x ₅₈ E ₅ H ₅

0.0100 0.0100 0.3344 0.0100 0.0100 0.0100 0.0100 0.6048 21.09% 0.7233

Table 7
Sub-portfolio (k = 9)

x ₉₁ x ₉₂ x ₉₃ x ₉₄ x ₉₅ x ₉₆ x ₉₇ x ₉₈ E ₉ H ₉

0.0100 0.4329 0.0100 0.0100 0.0100 0.0100 0.0100 0.5063 11.65% 0.2715

Table 9
Zigzag uncertain return rates of 8 securities [32]

i 1 2 3 4 5 6 7 8

a _i –0.16 –0.13 –0.13 –0.18 –0.14 –0.17 –0.24 –0.14

b _i –0.01 0.01 0.03 0.02 0.00 0.00 0.02 0.03

c _i 0.14 0.19 0.20 0.16 0.18 0.33 0.45 0.21

Table 14
Sub-portfolio with the increase of return threshold A₄

A ₄ 0 0.05 0.1 0.15 0.2

E ₄ 0.0117 0.0184 0.0480 0.0754 0.1008

H ₄ 0.0639 0.0749 0.1235 0.1771 0.2354