A review of uncertain portfolio selection

Abstract

This paper reviews the theory of uncertain portfolio selection which uses human estimates as inputs and applies uncertainty theory to select portfolios. The difference of the uncertain portfolio selection theory from the stochastic portfolio theory is given and the necessity and conditions for using the theory are presented. Some basic works are introduced, including different types of mathematical risk measurements, their features, and some basic uncertain portfolio selection models and their theorems. Finally, future work for uncertain portfolio selection is discussed.

Keywords

Uncertain portfolio selection risk measurement uncertain measure uncertain variable uncertainty theory

1 Introduction

Portfolio selection has always been a hot topic which is concerned with optimal allocation of money to a variety of securities. The investors’ objective is to pursue the maximum investment return and in the meantime to control the investment risk. Though people talked about risk, before Markowitz [35] there was no definite mathematical way to describe it. In 1952, Markowitz proposed that we could treat security returns as random variables and use variance to measure portfolio risk and presented the famous mean-variance model. Since then a large number of important results have been achieved and the portfolio selection theory has been greatly developed.

Though probability theory is a powerful tool to solve portfolio selection problem, a lot of surveys found that in reality there are many situations where we do not input historical data of security returns to make investment decision. Instead, we use men’s estimates as inputs. Many people argue that neither history can repeat itself in security market in most cases nor men’s estimates are random in nature. With the development of fuzzy set theory [50, 51], scholars explored using fuzzy set theory to help make the decision. The first attempt in portfolio selection was made by Watada [48] in 1990’s. Since then fuzzy portfolio selection has been studied by many scholars, such as the work based on possibility measure, e.g., different fuzzy mean-variance models by Tanaka and Guo [45], Tanaka, Guo and Türksen [46], Arenas-Parra et al. [1], Carlsson et al. [5], León, Liern and Vercher [27], Zhang and Nie [52], Bilbao-Terol et al. [3], Zhang et al. [53], Gupta, Mehlawat and Saxena [7], etc., and the research based on credibility measure, e.g., credibility maximization model [11], mean-variance models [12], mean-variance-skewness models [28], mean-semivariance models [13], entropy optimization model [14], mean-risk curve model [15], fuzzy cross entropy models [40], mean-absolute deviation models [41], mean-absolute semi-deviation models [44], multi-period mean-entropy models [37] and multi-period mean-variance models [54], etc. These works broadened research thoughts. However, deep research found that paradox decision results might occur using fuzzy set theory [23, 33], which implies that it is unsuitable to use fuzzy set theory as the tool to handle portfolio selection problem.

With the introduction of uncertainty theory [30] and development of it [31, 32], Huang [16] initiated an uncertain portfolio selection theory in 2010 by employing the uncertainty theory systematically into the area. Since then, uncertain portfolio selection has attracted scholars’ attention and a fairly good number of achievements have been gained. According to the way to measure risk, we can classify these works into two types. One type is to provide solution methods based on indirect risk measurement. Variance, semivariance, semiabsolute deviation are the typical indirect risk measurements. Using them to measure risk, we have mean-variance model [18] and its extensions [2, 55], mean-semivariance model [18] and the extension [6], mean-semiabsolute deviation selection model [29] and mean-semiabsolute deviation adjustment model [42]. Since indirect risk measurement is not intuitive, it is hard for investors to give their tolerable risk level. To provide direct risk measurement and facilitate decision making, Huang proposed that we use the chance, expressed by uncertain measure, of portfolio return failing to reach the predetermined threshold level as an alternative risk measurement and presented a basic mean-chance model [16]. Following the mean-chance selection idea, Huang and Zhao [23] further provided a genetic algorithm considering transaction costs, diversification and single security investment limitation, and Huang and Di [22] proposed a portfolio optimization method in the presence of background risk. Besides, from different perspectives, Huang proposed risk curve [17] and risk index [19] as two new direct risk measurements. Risk curve gauges the chances of all the potential losses of a portfolio and risk index measures the average loss value of the portfolio. Furthermore, based on the risk index, more work has been done, e.g., uncertain portfolio adjusting method by Huang and Ying [21], and multi-period portfolio optimization method by Huang and Qiao [20].

This paper defines uncertain portfolio selection as a branch of portfolio selection theory which uses human estimates as inputs and applies uncertainty theory to select portfolios. Especially, we call Markowitz portfolio selection and its developments that use probability theory as the tool to select portfolios as stochastic portfolio selection. The author aims to review the development of uncertain portfolio selection along the development line of risk measurement rather than a simple collection of all the published papers in the area. The author hopes to clarify the necessity of research on it, review important available research results in the area, and look forward to the future research in this field.

The paper proceeds as follows. Section 2 will present the necessity of research on the uncertain portfolio selection. To facilitate understanding of the whole paper, we will first introduce some fundamentals of uncertainty theory in this section. Sections 3– 5will introduce several risk measurements and present some basic models based on them. Section 6 will discuss some future research directions. Finally, Section 7 will summarize the paper.

2 Why uncertain portfolio selection?

To better understand the answer to this question and the whole paper, let us first review some necessary knowledge about the uncertainty theory.

2.1 Uncertain variable and uncertainty theory

Uncertainty theory is a branch of mathematics that models human uncertainty. It has been developed based on the following four axioms.

Definition 1. Let Γ be a nonempty set, and ℒ a σ-algebra over Γ. Each element Λ ∈ ℒ is called an event. A set function ℳ {Λ} is called an uncertain measure if it satisfies the following three axioms [30]:

(Normality) ℳ {Γ} =1 .

(Duality) ℳ {Λ} + ℳ {Λ^c} =1 .

(Subadditivity) For every countable sequence of events {Λ_i}, we have $ℳ {⋃_{i = 1}^{\infty} Λ_{i}} \leq \sum_{i = 1}^{\infty} ℳ {Λ_{i}} .$

The triplet (Γ, ℒ, ℳ) is called an uncertainty space.

Uncertain measure is explained as the chance that a person believes an uncertain event will happen. If one believes that an uncertain event will surely happen, then the uncertain measure of the uncertain event is 1. On the contrary, if one does not believe at all that an uncertain event will happen, then the uncertain measure of the uncertain event is 0. The greater chance one believes an uncertain event will happen, the bigger the uncertain measure of the uncertain event is. If one believes that an uncertain event will happen with 80% chance, then the uncertain measure of the uncertain event is 0.8.

Duality axiom ensures that if a person believes an uncertain event will happen with chance α, we can deduce that the person thinks the opposite event will happen at chance 1 - α. For example, if an expert believes that the event of a stock price being no greater than 10 dollars will happen with 80% chance, then we will know that the expert believes the event of the stock price being higher than 10 dollars will occur with 20% chance. Since our human beings always judge two opposite events in duality way, duality axiom is in agreement with our thinking. In fact, duality axiom ensures that the developed uncertainty theory is consistent with the law of excluded middle and the law of contradiction.

Given chances that the two events occur, subadditivity helps to answer the question “what is the chance of the union of the two events”. Probability measure assumes that the chance of the union of the two disjoint events is the sum of the two chances, possibility measure assumes the chance of the union is the maximum of the two, while subadditivity axiom of the uncertain measure relaxes the assumption of probability to be that the chance of the union can be the one between the sum of the chances and the chances of the individuals. Besides, Liu [32] showed that subadditivity axiom is needed to ensure the measure to be reasonable because pathology would occur if subadditivity axiom were not assumed for any measures.

Though probability measure meets the above three axioms, it is not a special case of uncertainty theory because the product probability does not satisfy the fourth axiom of the uncertainty theory given below.

(iv) (Product Axiom) (Liu [31]) Let (Γ_k, ℒ_k, ℳ_k) be uncertainty spaces for k = 1, 2, ⋯ The product uncertain measure ℳ is an uncertain measure satisfying $ℳ {\prod_{k = 1}^{\infty} Λ_{k}} = ⋀_{k = 1}^{\infty} ℳ_{k} {Λ_{k}}$ where Λ_k are arbitrarily chosen events from ℒ_k for k = 1, 2, ⋯, respectively.

Definition 2. (Liu [30]) An uncertain variable is a measurable function ξ from an uncertainty space (Γ, ℒ, ℳ) to the set of real numbers, i.e., for any Borel set of B of real numbers, the set ${ξ \in B} = {γ \in Γ | ξ (γ) \in B}$ is an event.

An uncertainty distribution function is used to characterize an uncertain variable and is defined as follows.

Definition 3. (Liu [30]) The uncertainty distribution $Φ : R \to [0, 1]$ of an uncertain variable ξ is defined by $Φ (t) = ℳ {ξ \leq t} .$

For example, if an uncertain variable has the following linear uncertainty distribution, we call it the linear uncertain variable. $Φ (t) = {\begin{matrix} 0, if t < a \\ (t - a) / (b - a), if a \leq t \leq b \\ 1, if t > b . \end{matrix}$ For convenience, it is denoted in the paper by $ξ \sim L (a, b)$ where a < b.

An uncertain variable is called normal if it has the following normal uncertainty distribution $Φ (t) = {(1 + exp (\frac{π (e - t)}{\sqrt{3} σ}))}^{- 1}, t \in R,$ where e and σ are real numbers and σ > 0 . For convenience, it is denoted in the paper by $ξ \sim N (e, σ)$ .

Theorem 1. (Peng and Iwamura [39]) A function $Φ (t) : R \to [0, 1]$ is an uncertainty distribution if and only if it is a monotone increasing function except Φ (t) ≡0 and Φ (t) ≡1.

Definition 4. [31] The uncertain variables ξ₁, ξ₂, ⋯ , ξ_n are independent if $ℳ {⋂_{i = 1}^{n} (ξ_{i} \in B_{i})} = ⋀_{i = 1}^{n} ℳ {ξ_{i} \in B_{i}}$ for any Borel sets B₁, B₂, ⋯ , B_n .

The uncertain variables are always independent if they are defined on different uncertainty spaces. For example, if ξ₁ (γ₁) and ξ₂ (γ₂) are uncertain variables on the uncertainty spaces (Γ₁, ℒ₁, ℳ₁) and (Γ₂, ℒ₂, ℳ₂), respectively, then ξ₁ and ξ₂ are independent.

The operational law is given by Liu [32] as follows:

Theorem 2. (Liu [32]) Let ξ₁, ξ₂, ⋯ , ξ_n be independent uncertain variables with continuous and strictly increasing uncertainty distributions Φ₁, Φ₂, ⋯ , Φ_n, respectively. Let f (t₁, t₂, ⋯ , t_n) be strictly increasing with respect to t₁, t₂, ⋯ , t_n . Then $ξ = f (ξ_{1}, ξ_{2}, \dots, ξ_{n})$ is an uncertain variable with uncertainty distribution $Ψ (t) = sup_{f (t_{1}, t_{2}, \dots, t_{n}) = t} (min_{1 \leq i \leq n} Φ_{i} (t_{i})), t \in R$ (1) whose inverse function is

$\begin{matrix} Ψ^{- 1} (α) = f (Φ_{1}^{- 1} (α), Φ_{2}^{- 1} (α), \dots, Φ_{n}^{- 1} (α)), \\ 0 < α < 1 . \end{matrix}$ (2)

According to Equation (2) we can easily get that if $ξ_{i} \sim N (e_{i}, σ_{i}), i = 1, 2, \dots, n,$ are normal uncertain variables, then for λ_i > 0, i = 1, 2, ⋯ , n, $η = \sum_{i = 1}^{n} λ_{i} ξ_{i}$ is also a normal uncertain variable $η \sim N (\sum_{i = 1}^{n} λ_{i} e_{i}, \sum_{i = 1}^{n} λ_{i} σ_{i}) .$ And if $ξ_{i} \sim L (a_{i}, b_{i}), i = 1, 2, \dots, n,$ are linear uncertain variables, then for λ_i > 0, i = 1, 2, ⋯ , n, $η = \sum_{i = 1}^{n} λ_{i} ξ_{i}$ is also a linear uncertain variable $η \sim L (\sum_{i = 1}^{n} λ_{i} a_{i}, \sum_{i = 1}^{n} λ_{i} b_{i}) .$

To tell the size of an uncertain variable, Liu defined the expected value of uncertain variables.

Definition 5. (Liu [30]) Let ξ be an uncertain variable. Then the expected value of ξ is defined by $E [ξ] = \int_{0}^{\infty} ℳ {ξ \geq r} d r - \int_{- \infty}^{0} ℳ {ξ \leq r} d r$ (3) provided that at least one of the two integrals is finite.

It can be calculated that the expected value of the normal uncertain variable $ξ \sim N (e, σ)$ is E [ξ] = e, and the expected value of the linear uncertain variable $ξ \sim L (a, b)$ is E [ξ] = (a + b)/2 .

Theorem 3. (Liu [32]) Let ξ₁ and ξ₂ be independent uncertain variables with finite expected values. Then for any real numbers a₁ and a₂, we have $E [a_{1} ξ_{1} + a_{2} ξ_{2}] = a_{1} E [ξ_{1}] + a_{2} E [ξ_{2}] .$ (4)

Theorem 4. (Liu [32]) Let ξ be an uncertain variable with continuous and strictly increasing uncertainty distribution Φ . If its expected value exists, then $E [ξ] = \int_{0}^{1} Φ^{- 1} (α) d α .$ (5)

Definition 6. (Liu [30]) Let ξ be an uncertain variable with finite expected value e . Then the variance of ξ is defined by $V [ξ] = E [(ξ - e)^{2}] .$ (6)

Let ξ be an uncertain variable with uncertainty distribution Φ . Then $\begin{matrix} V [ξ] & = & \int_{0}^{+ \infty} ℳ {(ξ - e)^{2}} \geq t d t \\ = & \int_{0}^{+ \infty} ℳ {(ξ \geq e + \sqrt{t}) \cup (ξ \leq e - \sqrt{t})} d t \\ \leq & \int_{0}^{+ \infty} (ℳ {ξ \geq e + \sqrt{t}} + ℳ {ξ \leq e - \sqrt{t}}) d t \\ = & \int_{0}^{+ \infty} (1 - Φ (e + \sqrt{t}) + Φ (e - \sqrt{t})) d t . \end{matrix}$ In this case, it is always assumed that the variance is $V [ξ] = \int_{0}^{+ \infty} (1 - Φ (e + \sqrt{t}) + Φ (e - \sqrt{t})) d t .$ (7)

Theorem 5. (Yao [49]) Let ξ be an uncertain variable with a continuous and strictly increasing uncertainty distribution Φ and finite expected value e. Then $V [ξ] = \int_{0}^{1} (Φ^{- 1} (α) - e)^{2} d α .$ (8)

It can be calculated that the variance value of a normal uncertain variable $ξ \sim N (μ, σ)$ isV [ξ] = σ² .

For more expositions on uncertainty theory, the interested readers may consult the book [34].

2.2 Why uncertain portfolio selection?

In stochastic portfolio selection theory, the input of analysis is past data of the security returns. Do investors really do so in real life? To find the answer, we did a lot of surveys. In the surveys, we asked the question: If there is a lot of past return data about a stock, is all the past return data of the stock sufficient to attract you without knowing the name of the stock company? None of the respondents said “yes”. Then we asked them besides the past return data what other information they considered and how they selected portfolios. Though they did not consider exactly the same information, they did all estimate the stock prices according to the comprehensive information like financial statement and performances of the stock companies, the prospect of the industries the stock companies belong to, the government policies, the hot news, etc., and then make decision according to the estimations. Thus, we can see that in security market, in many cases investors do not believe that the past data of security returns can well reflect their future returns. Unexpected events occur now and then in the financial market, e.g., sudden announcement of interest rate drop by central bank, unexpected events of companies or issue of new regulation towards an industry by the government, etc. These complex factors make past data be difficult to reflect the future security returns, which can be transformed into the equivalent expression that investors do not have enough suitable historical data to get probability distributions of the stocks.

We know that the usage of probability is conditional upon there being enough suitable historical data such that probability distributions can be obtained via the past data. If no enough suitable past data are available, Kahneman and Tversky [26] found that people usually give too much weight to unlikely invents. This implies that the range of people’s estimations of security returns are usually wider than they can actually take. Then in this situation, can estimations still be treated as probability distributions and is it still suitable to use probability theory to help make decision? To answer this question, let us consider one example. Suppose there are twelve securities. The return rates of the first six ones distribute uniformly between 2 to 5 percent, and the other six ones distribute uniformly between 3 to 5 percent. If investors uniformly allocate their money to these twelve securities, it is obvious that the portfolio return rate cannot be greater than 5 percent. Let ξ_i denote the return rates of the securities i, i = 1, 2, ⋯ , 12, respectively. If there are abundant past data and history can repeat itself (which is the condition that probability theory should be used), then people can use uniformly distributed random variables to describe the security returns and get that Pr {(ξ₁ + ξ₂ + ⋯ + ξ₁₂)/12 ≥ 0.05} =0, which can help the investors make the correct decision. Unfortunately, many times people find that past data can hardly reflect the future returns of the securities. Therefore, rather than using historical data, people use experts’ or their own estimations of the security returns according to the past return data, the information of the companies, the industries and the relevant government polices, etc. As aforementioned reason, with no enough suitable historical data, people usually include in their estimations much wider range of values than the security returns can actually take. Suppose people estimate the return rates of the aforementioned first six securities to be uniformly between 2 to 14 percent and the other six ones to be uniformly between 2 to 16 percent. If people’s estimations are still treated as probability distributions, we can get by simulation (3000 times) that Pr {(ξ₁ + ξ₂ + ⋯ + ξ₁₂)/12 ≥ 0.05} =1.000000, which says that it is sure that the portfolio return rate will be equal to or greater than 5 percent. An event that is sure not to happen becomes sure to happen. This is dangerous because people will not be alert to and prepare for a sure to happen event and being off guard may lead to disastrous decision making. Then what if the estimations of the aforementioned first six security return rates are treated as linear uncertainty distributions on [2, 14] and the other six security return rates as linear uncertainty distributions on [2, 16]? Based on operational law of uncertain variables, we can infer that the chance, expressed by uncertain measure, of the event “the portfolio return rate equal to or greater than 5 percent”, denoted by A, is around 76.9%, i.e., $ℳ {A} ≃ 76.9 % .$ That is to say the portfolio return rate may still fail to reach 5 percent, which can alert the investors. Uncertainty theory will not make a sure not to happen event become a sure to happen event. Though the uncertain measure which is used to measure the chance of the event still deviates from the real case, it is caused by the great difference of the estimations from the real frequencies. Probability theory magnifies the estimation errors, but the uncertainty theory does not further magnify the estimation errors.

If random variables are not suitable to describe people’s estimations of security returns, is it suitable to describe the estimations by fuzzy variables? Let us first assume we can use fuzzy variables to describe people’s estimations of the security returns. Since security returns are fuzzy variables, they must have membership functions. Suppose that the membership function of a security return rate is as follows, $μ (r) = {\begin{matrix} 0, if r \leq 0.02 \\ \frac{r - 2}{6}, if 0.02 \leq r \leq 0.08 \\ 1, if 0.08 \leq r \leq 0.10 \\ \frac{16 - r}{6}, if 0.10 \leq r \leq 0.16 \\ 0, if r \geq 16 \end{matrix}$ (9) which is just the trapezoidal fuzzy variable (0.02, 0.08, 0.10, 0.16). For a fuzzy variable ξ, the definition of possibility measure is as follows, $Pos {ξ \in B} = sup_{r \in B} μ (r) .$ (10)

Then from Equations (9) and (10) we can easily infer that $Pos {“ the security return {rate}^{″} = 0.09} = 1 .$ (11) $Pos {“ the security return {rate}^{″} \neq 0.09} = 1 .$ (12)

It is easy to get from Equations (11) and (12) that the security return rate is exactly 9 percent and the event that the security return rate is not exactly 9 percent will happen equally likely. That is impossible! Judging from common sense, we can know the event that the security return rate is exactly 9 percent will happen with a far smaller chance than the event that the security return rate is not exactly 9 percent. From Equations (10) we can further infer that no matter what type of membership function the security return is regarded to be with, if membership function is continuous and its value of the return rate of 9 percent is 1, the same paradox will occur, which shows that the security return rate cannot be quantified by possibility measure, i.e., it cannot be described by fuzzy variables.

We propose that a suitable tool for a decision making problem should be self-consistent theoretically itself and in the meantime able to solve this type of problem best among other tools. Uncertainty theory meets the requirement in solving portfolio selection problem when inputs are people’s estimations rather than historical data.

3 Indirect risk measurements and the basicmodels on them

In uncertain portfolio selection, we treat the people’s estimations of security returns as uncertain variables and denote them by ξ_i, i = 1, 2, ⋯ , n, respectively. The investment proportions in the i-th securities are denoted by x_i for i = 1, 2, ⋯ , n, respectively.

3.1 Uncertain mean-variance model

Variance is first proposed as the portfolio risk by Markowitz in stochastic portfolio selection [35], and is the most popular and widely studied risk measurement in the field. The idea is that expected return should be regarded as the investment return and variance the investment risk because variance gives the average level that the investors can not achieve the expected return. Following the idea, Huang extended the Markowitz model to uncertain portfolio selection and proposed the basic uncertain mean-variance model [16] as follows, ${\begin{matrix} max E [x_{1} ξ_{1} + x_{2} ξ_{2} + \dots + x_{n} ξ_{n}] \\ subject to : \\ V [x_{1} ξ_{1} + x_{2} ξ_{2} + \dots + x_{n} ξ_{n}] \leq a \\ x_{1} + x_{2} + \dots + x_{n} = 1 \\ x_{i} \geq 0, i = 1, 2, \dots, n \end{matrix}$ (13) where E and V denote the expected value operator and variance of uncertain variables, respectively, and a the investors’ maximum tolerable variance level. In the mean-variance model, optimal portfolio is the one with the maximum expected return whose variance value is not greater than the preset level a .

3.2 Uncertain mean-semivariance model

When we use variance as the risk measurement, it implies that we believe the security returns are symmetrical because otherwise reducing variance may reduce much higher return from the expected value which is what investors welcome. When security returns are asymmetrical, semivariance is proposed as an alternative risk measurement. It follows the idea that expected deviation from the expected return should be regarded as risk but only gauges the lower return deviation from the expected return. Semivariance of the uncertain variable is defined by Huang as follows.

Definition 7. (Huang [18]) Let ξ be an uncertain variable with finite expected value e. Then the semivariance of ξ is defined by SV [ξ] = E [[(ξ - e) ^-] ²], where $(ξ - e)^{-} = {\begin{matrix} e - ξ, & if ξ \leq e \\ 0, & if ξ > e . \end{matrix}$ (14)

Substituting variance with semivariance, Huang developed uncertain mean-semivariance model as follows [18], ${\begin{matrix} max E [x_{1} ξ_{1} + x_{2} ξ_{2} + \dots + x_{n} ξ_{n}] \\ subject to : \\ SV [x_{1} ξ_{1} + x_{2} ξ_{2} + \dots + x_{n} ξ_{n}] \leq b \\ x_{1} + x_{2} + \dots + x_{n} = 1 \\ x_{i} \geq 0, i = 1, 2, \dots, n \end{matrix}$ (15) where SV is the semivariance of the uncertain variables, and a the maximum risk level the investors can tolerate.

When the uncertain variable ξ has continuous and strictly increasing uncertainty distribution Φ and finite expected value e, $\begin{matrix} SV [ξ] = \int_{0}^{+ \infty} ℳ {((ξ - e)^{-})^{2} \geq t} d t \\ = \int_{0}^{\infty} Φ (e - \sqrt{t}) d t . \end{matrix}$ (16)

Substituting $e - \sqrt{t}$ with x and t with (e - x) ² and integrating by parts, we get $SV [ξ] = \int_{0}^{\infty} Φ (e - \sqrt{t}) dt = \int_{0}^{β} (Φ^{- 1} (α) - e)^{2} d α .$ (17) where β is the root of Φ^-1 (β) = e.

Then from equations (8) and (17), we get the following theorem.

Theorem 6 Let ξ be an uncertain variable with continuous and strictly increasing uncertainty distribution funtion Φ. If Φ is symmetrical in terms of the expected value, then $SV [ξ] = \frac{1}{2} V [ξ] .$ (18)

It is seen from Theorem 6 that when the distributions of the security returns are symmetrical and a = 2b, mean-variance model (13) and mean-semivariance model (15) produces the same result. However, when the distributions of the security returns are asymmetrical, uncertain mean-semivariance model (15) is reasonable for choosing the optimal portfolio. Thus, the uncertain mean-semivariance model (15) is a development of the uncertain mean-variance model (13).

3.3 Calculation of expected return, variance, semivariance of portfolios

When uncertainty distributions Φ_i of uncertain security returns ξ_i are continuous and strictly increasing for i = 1, 2, ⋯ , n, respectively, according to Theorem 2, the uncertainty distribution Ψ of the portfolio return $\sum_{i = 1}^{n} x_{i} ξ_{i}$ can be obtained via $Ψ^{- 1} (α) = \sum_{i = 1}^{n} x_{i} Φ_{i}^{- 1} (α)$ where α ∈ [0, 1] . Then we can calculate expected return, variance, semivariance of the portfolio based on the uncertainty distribution Ψ and the inverse uncertainty distribution Ψ^-1 (α).

Calculation of expected return of portfolios

According to Theorem 4, the expected value of the uncertain portfolio return $\sum_{i = 1}^{n} x_{i} ξ_{i}$ can be calculated by $E [\sum_{i = 1}^{n} x_{i} ξ_{i}] = \sum_{j = 1}^{9999} \sum_{i = 1}^{n} x_{i} Φ_{i}^{- 1} (0.0001 j) / 9999 .$ (19)

Thus, the process for computing the expected value of $E [\sum_{i = 1}^{n} x_{i} ξ_{i}]$ can be designed as follows:

Method 1:

Step 1. Set e = 0 .

Step 2. Set j = 1 .

Step 3. Let $y_{j} = \sum_{i = 1}^{n} x_{i} Φ_{i}^{- 1} (0.0001 j) .$

Step 4. Let e = e + y_j .

Step 5. If j < 9999, let j = j + 1 . Then turn back to Step 3.

Step 6. Return e = e/9999 .

Please note that when we employ mean-variance models to select the optimal portfolio, the uncertainty distributions of the security returns should be symmetrical. Then, it is clear that in this case the expected return should be the value of Ψ^-1 (0.5) . Thus, when security returns are symmetrical, according to Theorem 2, we can obtain the expected value directly via $E [\sum_{i = 1}^{n} x_{i} ξ_{i}] = Ψ^{- 1} (0.5) = \sum_{i = 1}^{n} x_{i} Φ_{i}^{- 1} (0.5) .$ (20)

Calculation of variance

According to Equation (8), when security returns have continuous and strictly increasing uncertainty distributions Φ_i for i = 1, 2, ⋯ , n, respectively, the variance of the portfolio return $\sum_{i = 1}^{n} x_{i} ξ_{i}$ can be calculated via

$\begin{matrix} V [\sum_{i = 1}^{n} x_{i} ξ_{i}] \\ = \int_{0}^{1} (Ψ^{- 1} (α) - e)^{2} d α . \end{matrix}$ (21)

Then the process for computing the variance value $V [\sum_{i = 1}^{n} x_{i} ξ_{i}]$ is as follows,

Method 2:

Step 1. Use Method 1 or Equation (20) to get the expected portfolio return e .

Step 2. Set v = 0 .

Step 3. Set j = 1 .

Step 4. Let $y_{j} = (\sum_{i = 1}^{n} x_{i} Φ_{i}^{- 1} (0.0001 j) - e)^{2} .$

Step 5. Let v = v + y_j .

Step 6. If j < 9999, let j = j + 1 . Then turn back to Step 4.

Step 7. Return v = v/9999 .

3.4 Calculation of semivariance

According to Equation (16), the semivariance of the portfolio return $\sum_{i = 1}^{n} x_{i} ξ_{i}$ can be calculated via $SV [\sum_{i = 1}^{n} x_{i} ξ_{i}] = \int_{0}^{β} (Ψ^{- 1} (α) - e)^{2} d α$ (22) where e is the expected return of the uncertain portfolio. Thus, the semivariance value $SV [\sum_{i = 1}^{n} x_{i} ξ_{i}]$ can be got via the following process,

Method 3:

Step 1. Use Method 1 to calculate the expected portfolio return e .

Step 2. Set sv = 0 .

Step 3. Let K be the maximal positive integer such that $\sum_{i = 1}^{n} x_{i} Φ^{- 1} (0.0001 K) \leq e$ .

Step 4. Let $y_{j} = (\sum_{i = 1}^{n} x_{i} Φ^{- 1} (0.0001 j) - e)^{2}$ .

Step 5. Let sv = sv + y_j.

Step 6. If j < K, let j = j + 1 and turn back to Step 4.

Step 7. Return sv = sv/K .

After getting the expected return, variance, and semivariance values of the portfolio, we can integrate them into the genetic algorithm to find the optimal solutions of the uncertain mean-variance and mean-semivariance models [18].

4 Direct risk measurements and the basicmodels on them

4.1 Uncertain mean-chance model

In reality, sometimes, people are concerned about the chance of the event that portfolio return is lower than a threshold return. Therefore, Huang [16] proposed that we can use the chance of the event that portfolio return is low than a threshold return H as an investment risk measurement. Then the investors can require that the chance of the bad event, i.e., the portfolio return is lower than the threshold return H, must be small enough to be lower than the preset tolerable level β. When this risk control requirement is met, the optimal portfolio should be the one with the maximum expected return. The following mean-chance model reflects the selection idea. ${\begin{matrix} max E [ξ_{1} x_{1} + ξ_{2} x_{2} + \dots + ξ_{n} x_{n}] \\ subject to : \\ ℳ {ξ_{1} x_{1} + ξ_{2} x_{2} + \dots + ξ_{n} x_{n} \leq H} \leq β \\ x_{1} + x_{2} + \dots + x_{n} = 1 \\ x_{i} \geq 0, i = 1, 2, \dots, n \end{matrix}$ (23) where β is the preset tolerable chance level.

According to Theorems 2 and 3, when the uncertainty distributions of the security returns Φ_i, i = 1, 2, ⋯ , n, are continuous and strictly increasing, model (23) is equivalent to the followingmodel: ${\begin{matrix} max e_{1} x_{1} + e_{2} x_{2} + \dots + e_{n} x_{n} \\ subject to : \\ Φ_{1}^{-} (β) x_{1} + Φ_{2}^{-} (β) x_{2} + \dots + Φ_{n}^{-} (β) x_{n} \geq H \\ x_{1} + x_{2} + \dots + x_{n} = 1 \\ x_{i} \geq 0, i = 1, 2, \dots, n . \end{matrix}$ (24)

4.2 Uncertain mean-risk curve model

When we use the chance of the event that portfolio return is lower than a threshold return H as the investment risk measurement, we are concerned about just one sensitive bad event. However, if investors are conservative and cautious, they would like to know the chances of all the potential loss events and see if the chances of all those loss events are tolerable. To reflect this risk attitude, Huang [17] defined risk curve and proposed a mean-risk curvemodel.

Definition 8. (Huang [17]) Let ξ denote the uncertain return rate of a portfolio, and r_t the target return rate. Then the curve $R (r) = ℳ {r_{t} - ξ \geq r}, \forall r \geq 0$ (25) is called the risk curve of the portfolio.

It is seen that whenever the value of r_t - ξ is greater than zero, it means that the portfolio return is lower than the target return rate, which clearly can be understood as a loss. This is more easily to be understood if we set r_t = 0 . Therefore, {r_t - ξ ≥ r} for r ≥ 0 are the set of all potential loss events.

It is clear that the risk curve can be expressed in the form R (r) = ℳ {ξ ≤ r_t - r} . Thus, if we know the uncertainty distribution of the portfolio, we have the risk curve of the portfolio. From the monotonicity axiom of the uncertain measure we get that R (r) is a decreasing function with respect to r. That is, when the loss becomes bigger, the chance of the loss will become smaller. Then, with the investors’ confidence curve α (r) which gives the investors’ maximal tolerance towards the chance of each potential loss, it is clear that a portfolio is regarded to be safe if no part of its risk curve is above the confidence curve. From the safe portfolios, the optimal portfolio is the one with the maximum expected return. This selection idea is expressed by the following mean-risk curvemodel, ${\begin{matrix} max E [\sum_{i = 1}^{n} ξ_{i} x_{i}] \\ subject to : \\ ℳ {r_{t} - \sum_{i = 1}^{n} ξ_{i} x_{i} \geq r} \leq α (r), \forall r \geq 0 \\ \sum_{i = 1}^{n} x_{i} = 1 \\ x_{i} \geq 0, i = 1, 2, \dots, n . \end{matrix}$ (26)

It is seen that the loss event ${r_{t} - \sum_{i = 1}^{n} ξ_{i} x_{i} \geq r}$ can be transformed into the return event ${\sum_{i = 1}^{n} ξ_{i} x_{i} \leq r_{t} - r} .$ That is, loss equal to or greater than r level can be transformed in the expression that “return equal to or lower than r_t - r level. Therefore, confidence curve α (r) can also be understood as the tolerable chances of the events that portfolio return is equal to or lower than r_t - r level.

When security returns ξ_i have continuous and strictly increasing uncertainty distributions Φ_i for i = 1, 2, ⋯ , n, respectively, the mean-risk model (26) can be transformed into the following linear model, ${\begin{matrix} max \sum_{i = 1}^{n} x_{i} e_{i} \\ subject to : \\ \sum_{i = 1}^{n} x_{i} Φ_{i}^{- 1} (α (r)) \geq r_{t} - r, \forall r \geq 0 \\ x_{1} + x_{2} + \dots + x_{n} = 1 \\ x_{i} \geq 0, i = 1, 2, \dots, n . \end{matrix}$ (27)

Since H in model (23) is the threshold return and r_t in model (26) is the target return, H ∈ [r_t - a, r_t] should hold, where a is a big enough positive real number, we have β = α (H). Then it is easy to see that the optimal solution of the mean-risk curve model (26crisp) must be a feasible solution of the mean-chance model (23crisp). However, the optimal solution of the mean-chance model (23crisp) may not be a feasible solution of the mean-risk curve model (26crisp). Thus, the expected return of the mean-chance model (23crisp) will be equal to or higher than the mean-risk curve model (26crisp). This is because the mean-chance model (23crisp) controls only one loss event on risk curve, and so is more risky compared with the mean-risk curve model (26crisp). The higher expected portfolio return of mean-chance model (23crisp) is an exchange to tolerance to higher risk.

4.3 Uncertain mean-risk index model

To obtain an average level of the portfolio return below the target return rate, Huang [19] defined a risk index as follows:

Definition 9. (Huang [19]) Let ξ denote an uncertain return rate of a portfolio, and r_t the target return rate. Then the risk index of the portfolio is defined by $RI (ξ) = E [(r_{t} - ξ)^{+}],$ where E is the expected value operator of the uncertain variable and $(r_{t} - ξ)^{+} = {\begin{matrix} r_{t} - ξ, & if ξ \leq r_{t} \\ 0, & if ξ > r_{t} . \end{matrix}$ (28)

Let ξ be an uncertain portfolio return with uncertainty distribution Φ. Then

$\begin{matrix} RI (ξ) & = & E [(r_{t} - ξ)^{+}] \\ = & \int_{0}^{\infty} ℳ {r_{t} - ξ \geq r} d r \\ = & \int_{0}^{\infty} Φ (r_{t} - r) d r \\ = & \int_{- \infty}^{r_{t}} Φ (r) d r . \end{matrix}$ (29)

Theorem 7. (Huang [19]) Let ξ be an uncertain portfolio return with continuous and strictly increasing uncertainty distribution Φ. Then the risk index of the portfolio can be expressed as follows, $RI (ξ) = \int_{0}^{β} (r_{t} - Φ^{- 1} (α)) d α,$ (30) where β is the root of the equation Φ^-1 (β) = r_t .

Let c denote the maximum mean loss level below r_t that the investors can tolerate. Then it is clear that a portfolio is regarded safe if $RI (ξ) \leq c .$ (31) Thus, to pursue the maximum expected return among the safe portfolios, we have the following mean-risk index model, ${\begin{matrix} max E [ξ_{1} x_{1} + ξ_{2} x_{2} + \dots + ξ_{n} x_{n}] \\ subject to : \\ RI (ξ_{1} x_{1} + ξ_{2} x_{2} + \dots + ξ_{n} x_{n}) \leq c \\ x_{1} + x_{2} + \dots + x_{n} = 1 \\ x_{i} \geq 0, i = 1, 2, \dots, n \end{matrix}$ (32) where RI is the risk index of the portfolio defined as $\begin{matrix} RI (ξ_{1} x_{1} + ξ_{2} x_{2} + \dots + ξ_{n} x_{n}) \\ = E [(r_{t} - (ξ_{1} x_{1} + ξ_{2} x_{2} + \dots + ξ_{n} x_{n}))^{+}] . \end{matrix}$

According to Theorem 2 and the Equation (30), when security returns ξ_i has continuous and strictly increasing uncertainty distributions Φ_i for i = 1, 2, ⋯ , n, respectively, the risk index model (32) can be transformed into the following form, ${\begin{matrix} max x_{1} e_{1} + x_{2} e_{2} + \dots + x_{n} e_{n} \\ subject to : \\ β r_{t} - \int_{0}^{β} (\sum_{i = 1}^{n} x_{i} Φ_{i}^{- 1} (α)) d α \leq c \\ \sum_{i = 1}^{n} x_{i} Φ_{i}^{- 1} (β) = r_{t} \\ x_{1} + x_{2} + \dots + x_{n} = 1 \\ x_{i} \geq 0, i = 1, 2, \dots, n . \end{matrix}$ (33)

5 Uncertain portfolio selectionwith background risk

In the above sections it is all assumed that investors face only portfolio risk in the financial market. Yet in reality investors also face other sources of risk outside the financial market like those arising from variations in labor income, investments in real estate, and unexpected expenses related to health issues. The risk outside the financial market and cannot be hedged through portfolio diversification in the financial markets is called background risk [4]. The assets that are exposed to background risk is referred to as background assets and others as financial assets. Background assets are typically illiquid or nontradable, and it is practically impossible for investors to control background risk by adjusting these asset holdings in the short run. Researches have shown that the presence of background risk can affect investments because the investors concern the total risk instead of the sole portfolio risk. For example, Heaton and Lucas [9, 10] revealed that portfolio allocations were affected by labor and entrepreneurial risk. Rosen and Wu [43] showed that investors with bad health were more willing to allocate most of their wealth to the low risk assets instead of risky assets. Tsanakas [47] found that the presence of background risk made investors sensitive to the aggregation and size of risk. Research of Hara et al. [8] showed that background risk made investor more cautious. In stochastic portfolio selection, scholars have studied portfolio selection with background problems in different conditions, e.g., Menoncin [38], Baptista [4], Jiang [25], and Huang and Wang [24], etc. Considering that the background asset return is also given by human being’s estimations, Huang and Di [22] treated the background asset return as an uncertain variable and presented the uncertain mean-chance portfolio model with background risk. In their model, it is assumed that the background asset return rate has zero expected value. The assumption is made just for simplicity. The model and the conclusions about the model is also applicable in the situation where the expected value of the background asset return differs from zero.

Let r_b be the uncertain background asset return and H the investors’ threshold return. Then the uncertain mean-chance portfolio model with background risk is as follows, ${\begin{matrix} max E [ξ_{1} x_{1} + ξ_{2} x_{2} + \dots + ξ_{n} x_{n} + r_{b}] \\ subject to : \\ ℳ {ξ_{1} x_{1} + ξ_{2} x_{2} + \dots + ξ_{n} x_{n} + r_{b} \leq H} \leq β \\ x_{1} + x_{2} + \dots + x_{n} = 1 \\ x_{i} \geq 0, i = 1, 2, \dots, n \end{matrix}$ (34) where β is the preset tolerable chance level.

Let Ψ be the uncertainty distribution of the uncertain background asset return which is continuous and strictly increasing. According to Theorems 2 and 3, when the uncertainty distributions of the security returns Φ_i are continuous and strictly increasing, model (23b) is equivalent to the followingmodel, ${\begin{matrix} max e_{1} x_{1} + e_{2} x_{2} + \dots + e_{n} x_{n} \\ subject to : \\ \sum_{i = 1}^{n} Φ_{i}^{-} (β) x_{i} + Ψ^{-} (β) \geq H \\ x_{1} + x_{2} + \dots + x_{n} = 1 \\ x_{i} \geq 0, i = 1, 2, \dots, n . \end{matrix}$ (35)

Theorem 8. For a given threshold return level H and a tolerable chance level β (0 < β < 0.5), the expected return of the optimal portfolio with background risk is smaller than that without background risk.

The proof can be found in paper [22].

6 Future work

Though uncertain portfolio selection has gained some development, a lot of research studies need to be done. The author would like to divide the uncertain portfolio selection into new portfolio selection and the existing portfolio adjustment types. New portfolio selection means that investors have no security in hand and are going to select optimal portfolio from the candidate securities, and the existing portfolio adjustment means that investors already have a portfolio in hand, and are going to adjust the existing portfolio from the candidate securities. It is clear that the existing portfolio adjustment is the extension of the new portfolio selection. Therefore, the work the author points out on new portfolio selection can be studied on the existing portfolioadjustment.

Regarding the new portfolio selection, we can add investors’ specific requirements or the market limitations into the basic models and study the properties of the new problems. The investors’ specific requirements include requirements on liquidity, skewness and kurtosis, diversification, capital limitations on individual securities, etc. The popular market limitation is transaction cost and minimum transaction lot requirement. In this direction, Bhattacharyya et al. [2] have studied uncertain mean-variance-skewness model, Chen et al. [6] have considered diversification requirement based on the basic mean-semivariance model, and Huang and Zhao [23] have considered transaction costs, diversification and lower and higher capital limitation requirements on individual securities following the mean-chance selection idea. However, more work can be done in thedirection.

Besides, we can consider the background risk into the portfolio selection problem. Huang and Di [22] have initiated the study on this problem. However, they just studied the work in the framework of the basic mean-chance model. More work can be done based on the other basic models. Furthermore, investors’ specific requirements or the market limitations can also be considered in the uncertain portfolio selection with background risk problems.

Existing portfolio adjustment problem is an extension of the new portfolio selection. In the problem, not only the buying of the new securities and the more existing securities should be considered, selling of the existing securities also needs to be decided. This leads to the complexity of the problem. Following the mean-risk index selection idea, Huang and Ying [21] have studied a portfolio adjustment problem considering minimum transaction lot and investment capital limitations on individual securities. Yet studies in this area are still sparse and there is large researchspace.

So far, we have just talked about single period portfolio selection. In fact, multi-period uncertain portfolio selection is a direction which has great research potential. Of course, this kind of problem will be more challenging than single period selection, but is an interesting field to be exploited.

7 Summarization

This paper has reviewed an uncertain portfolio selection which is defined as a new branch of portfolio selection theory that uses experts’ estimations as inputs and uses uncertainty theory to select the portfolio. The necessity of studying uncertain portfolio selection has been provided. Along the development of indirect and direct risk measurements, different mathematical risk measurements of uncertain portfolio selection have been introduced and the basic models based on them have been provided. In addition, the difference and properties of the models have been documented. Furthermore, the future work on uncertain portfolio selection has been discussed.

Footnotes

Acknowledgments

This work was supported by Specialized Research Fund for the Doctoral Program of Higher Education No. 20130006110001 and the Fundamental Research Funds for the Central Universities No. 2302015FRF-BR-15-018A.

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