Uncertain random portfolio optimization models based on value-at-risk

Abstract

This paper studies a portfolio optimization problem in which some candidate securities possess sufficient transaction data and the others are newly listed and lack enough data. Their corresponding returns are assumed to be random variables and uncertain variables, respectively. Accordingly, the total return on a portfolio becomes an uncertain random variable. In this paper, we first define value-at-risk of uncertain random variable and discuss its mathematical properties as well as numerical solution procedure. Then we employ it to measure the risk associated with uncertain random returns and formulate the corresponding portfolio optimization models with uncertain random returns. An active-set method is used to solve the proposed models and a numerical example is given to illustrate its application.

Keywords

Uncertain random variable value-at-risk portfolio optimization uncertain measure mean-VaR model

1 Introduction

Portfolio optimization problem is to determine an optimal allocation among different securities, which is originated from the mean-variance model of Markowitz [18] in 1952. In this model, the return of security is considered as random variable, and the investment return and risk are measured by expected value and variance, respectively. Though variance is a popular risk measure, it is sometimes not convenient to be used in practice. Therefore, many researchers introduced some other risk measures and formulated the related models such as mean-semivariance model [17], mean-variance-skewness model [9], the model based on value-at-risk (VaR) [10], the model based on conditional value-at-risk (CVaR) [25], mean-semivariance-CVaR model [19], mean-risk curve model [6] and so forth.

Essentially, VaR is the lower α-percentile of probability distribution of random return for a given confidence level α from (0, 1). Compared to variance, VaR can well measure the downside risk. Nowadays, VaR has become a popular method to evaluate the risk in financial market, especially in Basel Accord. A large number of works are devoted to the risk analysis associated with VaR. For example, Krejic et al. [10] studied portfolio optimization under VaR risk measure by taking into account transaction costs. Vladimir et al. [23] introduced a mean-VaR model by estimating VaR by a univariate GARCH volatility model and Shawkat et al. [3] applied VaR to the downside risk management and optimal portfolios for precious metals, oil and stocks.

In the case with sufficient historical data, it is reasonable to regard security returns as random variables. However, there are sometimes lack of enough data for the newly listed securities. An alternative way is to invite domain experts to evaluate the returns, which can be described by uncertain variables [13]. Following the idea, Qin et al. [21] first formulated mean-variance model for uncertain portfolio optimization, in which security returns are assumed to be uncertain variables. Further, Huang [5] introduced a risk curve and developed a mean-risk model. In addition, Huang [4] proposed a risk index as a safe criterion and developed a mean-risk index model and then extended it to the multi-period case [7] and the adjusting problem [8], respectively. Liu and Qin [16] defined the semiabsolute deviation for uncertain returns, which is also applied to interval analysis [11] and the adjusting problem [22]. Other uncertain optimization problems were also discussed such as railway transportation planning [1] and uncertain graph [2].

This paper will consider a hybrid case in which some existing securities have been listed for a long period, whose returns are described by random variables, and the others are newly listed with insufficient data, whose returns are described by uncertain variables subject to experts’ evaluations. Qin [20] first investigated such a problem with simultaneous appearance of random and uncertain returns and presented a mean-variance model by using uncertain random variable [14] to handle the total return on a portfolio. Other uncertain random optimization problems were also studied such as shortest path problem [24]. Different from Qin [20], this paper will introduce VaR to evaluate the risk associated with uncertain random returns and further formulate mean-VaR models for uncertain random portfolio optimization.

The rest of the paper is structured as follows. In Section 2, we review the preliminaries consisting of several basic concepts in uncertainty theory. In Section 3, we define VaR for uncertain random variable and discuss its mathematical properties as well as numerical solution procedure. Section 4 formulates uncertain random mean-VaR models and then employs active-set method to solve the proposed model. In Section 5, a numerical example is discussed to illustrate the application of mean-VaR models. Finally, Section 6 gives some concluding remarks.

2 Preliminaries

In this section, we review some concepts and theorems in uncertainty theory. A basic concept is uncertain measure which was proposed by Liu [13] to indicate the belief degree than an event happens. Let Γ be a nonempty set and ℒ be a σ-algebra on it. A set function ℳ : ℒ → [0, 1] is called an uncertain measure if it satisfies: (1) ℳ {Γ} =1 for the universal set Γ; (2) ℳ {Λ} + ℳ {Λ^c} =1 for any event Λ ∈ ℒ; (3) For every countable sequence of events Λ₁, Λ₂, ⋯ , we have $ℳ {⋃_{i = 1}^{\infty} Λ_{i}} \leq \sum_{i = 1}^{\infty} ℳ {Λ_{i}} .$ The triple (Γ, ℒ, ℳ) is called an uncertainty space. Further, Liu [12] also defined the product uncertain measure.

The second concept is uncertain variable which is proposed by Liu [13] to describe a quantity with uncertainty. An uncertain variable τ is a measurable function from an uncertainty space (Γ, ℒ, ℳ) to the set of real numbers, i.e., for any Borel set B of real numbers, the set {τ ∈ B} = {γ ∈ Γ|τ (γ) ∈ B} is an event. For $x \in R$ , the function Φ (x) = ℳ {τ ≤ x} is called the uncertainty distribution of τ. Further, Φ (x) is said to be regular if it is continuous and strictly increasing with respect to x at which 0 < Φ (x) <1, and $lim_{x \to - \infty} Φ (x) = 0, lim_{x \to + \infty} Φ (x) = 1 .$ The inverse function Φ^-1 (α) is called the inverse uncertainty distribution of τ if it exists and is unique for each α ∈ (0, 1). The commonly used uncertain variables are linear, zigzag and normal ones, whose uncertainty distributions can be found in Liu’s book [12].

The third concept is expected value. The expected value of an uncertain variable τ was defined by Liu [13] as $E [τ] = \int_{0}^{+ \infty} ℳ {τ \geq x} d x - \int_{- \infty}^{0} ℳ {τ \leq x} d x$ provided that at least one of the two integrals exists. Further, Liu [12] proved that if τ has a regular uncertainty distribution and a finite expected value, then $E [τ] = \int_{0}^{1} Φ^{- 1} (α) d α .$

The fourth concept is uncertain random variable defined by Liu [14], which is to describe a complex system with not only uncertainty but also randomness. Let (Γ, ℒ, ℳ) be an uncertainty space, and $(Ω, P, Pr)$ a probability space. The product $(Γ, ℒ, ℳ) \times (Ω, P, Pr)$ is called a chance space. Let $Θ \in ℒ \times P$ . The chance measure of uncertain random event Θ is defined by Liu [14] as $\begin{matrix} Ch {Θ} \\ = \int_{0}^{1} Pr {ω \in Ω | ℳ {γ \in Γ | (γ, ω) \in Θ} \geq r} d r . \end{matrix}$

An uncertain random variable is a function ξ from the chance space $(Γ, ℒ, ℳ) \times (Ω, P, Pr)$ to the set of real numbers, i.e., ξ ∈ B is an event in $ℒ \times P$ for any Borel set B. Its chance distribution is defined by Φ (x) = Ch {ξ ≤ x} for $x \in R$ , and its expected value is defined by $E [ξ] = \int_{0}^{+ \infty} Ch {ξ \geq r} d r - \int_{- \infty}^{0} Ch {ξ \leq r} d r$ provided that at least one of the two integrals is finite. Note that random variable and uncertain variable are its special case. Furthermore, if η is a random variable and τ is an uncertain variable, then the sum η + τ and the product ητ are both uncertain random variables.

Let $f : R^{n} \to R$ be a measurable function, and ξ₁, ξ₂, ⋯ , ξ_n uncertain random variables on a chance space $(Γ, ℒ, ℳ) \times (Ω, P, Pr)$ . Then ξ = f (ξ₁, ξ₂, ⋯ , ξ_n) is proved by Liu [15] to be an uncertain random variable determined by $ξ (γ, ω) = f (ξ_{1} (γ, ω), ξ_{2} (γ, ω), \dots, ξ_{n} (γ, ω))$ for all (γ, ω) ∈ Γ × Ω.

3 Uncertain random value-at-risk

Let ξ represent the loss of a portfolio, which is assumed to be an uncertain random variable in this section. We first define VaR of uncertain random variable by chance measure and then discuss its mathematical properties as well as the numerical solution procedure.

Definition 1. Let ξ be an uncertain random variable defined on a chance space $(Θ, L, ℳ) \times (Ω, P, Pr)$ . The VaR of ξ with confidence level α ∈ (0, 1) is defined as ${VaR}_{α} (ξ) = sup {λ | Ch {ξ \geq λ} \geq 1 - α} .$ (1)

Remark 1. If ξ degenerates into a random variable, then it follows that Ch {ξ ≥ λ} = Pr {ξ ≥ λ} and further ${VaR}_{α} (ξ) = sup {λ | Pr {ξ \geq λ} \geq 1 - α}$ is satisfied, which is exactly the VaR in stochastic environment. Similarly, if ξ degenerates into an uncertain variable, then it follows that Ch {ξ ≥ λ} = ℳ {ξ ≥ λ} and further ${VaR}_{α} (ξ) = sup {λ | ℳ {ξ \geq λ} \geq 1 - α}$ is satisfied, which is the VaR in uncertain environment.

Theorem 1. Let ξ be an uncertain random variable representing the loss of a portfolio. Then VaR_α (ξ) is the largest potential loss at the confidence level α ∈ (0, 1).

Proof. Assume that there is a value λ′ > VaR_α (ξ) such that Ch {ξ < λ′} = α. On account of the self-duality of chance measure, we have $Ch {ξ \geq λ^{'}} = 1 - Ch {ξ < λ^{'}} = 1 - α .$

It follows from Definition 1 that λ′ ≤ VaR_α (ξ) which leads to a contradiction. The theorem is completed.

Remark 2. Let ξ be an uncertain random variable representing the return of a portfolio. Then the VaR of ξ with confidence level α ∈ (0, 1) is defined as ${VaR}_{α} (ξ) = - inf {λ | Ch {ξ \leq λ} \geq 1 - α} .$ (2)

Similar to Theorem 1, VaR_α (ξ) is the opposite of minimum potential return of the portfolio at the confidence level α ∈ (0, 1).

Example 1. We consider a function defined as $ξ = {\begin{matrix} τ_{1} = L (- 2, 1.5), & with probability 0.7 \\ τ_{2} = L (- 1, 2), & with probability 0.3, \end{matrix}$ (3) in which $L (a, b)$ is a linear uncertain variable. Obviously, ξ is an uncertain random variable. First, according to the uncertain measure, we have $ℳ {τ_{1} \geq λ} = {\begin{matrix} 1, & if λ \leq - 2 \\ (3 - 2 λ) / 7, & if - 2 \leq λ \leq 1.5 \\ 0, & if λ \geq 1.5 \end{matrix}$ and $ℳ {τ_{2} \geq λ} = {\begin{matrix} 1, & if λ \leq - 1 \\ (2 - λ) / 3, & if - 1 \leq λ \leq 2 \\ 0, & if λ \geq 2 . \end{matrix}$

Then it follows from the definition of chance measure that

$\begin{matrix} Ch {ξ \geq λ} \\ = 0.7 \cdot ℳ {τ_{1} \geq λ} + 0.3 \cdot ℳ {τ_{2} \geq λ} \\ = {\begin{matrix} 1, & if λ \leq - 2 \\ (3 - λ) / 5, & if - 2 \leq λ \leq - 1 \\ (5 - 3 λ) / 10, & if - 1 \leq λ \leq 1.5 \\ (2 - λ) / 10, & if 1.5 \leq λ \leq 2 \\ 0, & if λ \geq 2 . \end{matrix} \end{matrix}$ (4)

Thus, according to Equation (1), the VaR of ξ is ${VaR}_{α} (ξ) = {\begin{matrix} 10 α - 8, & if 0.95 \leq α < 1.00 \\ (10 α - 5) / 3, & if 0.20 \leq α \leq 0.95 . \end{matrix}$ (5)

Theorem 2. (Positive Homogeneity) Let ξ be an uncertain random variable representing the loss of a portfolio, and α ∈ (0, 1) a given confidence level. Then, for any a ≥ 0, we have ${VaR}_{α} (a ξ) = a {VaR}_{α} (ξ) .$ (6)

Proof. The conclusion obviously holds when a = 0. In the case of a > 0, it follows from Definition 1 that $\begin{matrix} {VaR}_{α} (a ξ) & = & sup {λ | Ch {a ξ \geq λ} \geq 1 - α} \\ = & a sup {λ / a | Ch {ξ \geq λ / a} \geq 1 - α} \\ = & a {VaR}_{α} (ξ) . \end{matrix}$

The theorem is completed.

Theorem 3. (Translation Invariance) Let ξ be an uncertain random variable representing the loss of a portfolio, and α ∈ (0, 1) a given confidence level. Then, for any real number b, we have ${VaR}_{α} (ξ + b) = {VaR}_{α} (ξ) + b .$ (7)

Proof. It follows from Definition 1 that $\begin{matrix} {VaR}_{α} (ξ + b) \\ = sup {λ | Ch {ξ + b \geq λ} \geq 1 - α} \\ = sup {λ - b | Ch {ξ \geq λ - b} \geq 1 - α} + b \\ = {VaR}_{α} (ξ) + b . \end{matrix}$

The theorem is completed.

Theorem 4. Let ξ be an uncertain random variable representing the loss of a portfolio, and α ∈ (0, 1) a given confidence level. Then VaR_α (ξ) is an increasing and right-continuous function with respect to α.

Proof. For any α₁ < α₂, we have 1 - α₂ < 1 - α₁. Since Ch {ξ ≥ λ} ≥1 - α₁ > 1 - α₂, on account of Definition 1, VaR_{α
₁} (ξ) ≤ VaR_{α
₂} (ξ).

Next, we will verify the right-continuity of VaR_α (ξ) with respect to α. Let {α_i} be an arbitrary sequence of positive numbers which is monotonically decreasing and tends to α, i.e., {α_i} ↓ α. Then {VaR_{α
_i} (ξ)} is a decreasing sequence, and if the limitation of the sequence is exactly VaR_α (ξ), then the right-continuity is proved. Otherwise, there is a number λ^* such that $lim_{i \to \infty} {VaR}_{α_{i}} (ξ) > λ^{*} > {VaR}_{α} (ξ) .$

That is to say Ch {ξ ≥ λ^*} ≥1 - α_i for each i. Letting i→ ∞, we get Ch {ξ ≥ λ^*} ≥1 - α or λ^* ≤ VaR_α (ξ). The contradiction proves the right-continuity of VaR_α (ξ) with respect to α. The theorem is completed.

Remark 3. It is known that VaR in stochastic environment does not have subadditivity. Since stochastic VaR is a special case of uncertain random VaR, it is evident that the subadditivity for VaR does not hold in uncertain random environment as well.

Remark 4. If ξ is an uncertain random variable representing the loss of a portfolio, whose chance distribution Φ (u) is an increasing and continuous function on (0, 1), then its VaR is the unique solution of the equation Φ (u) = α on account of Theorem 4. That is to say, VaR_α (ξ) = Φ^-1 (α).

We consider several uncertain random variables consisting of a random variable and an uncertain variable. Three examples will be given and the corresponding chance distributions turn out to be continuous. Assume that η ∼ N (μ, σ²) is a normal random variable with the cumulative distribution function Ψ (u). Denote I_{·} by the indicator function of set {·}.

Example 2. Let $τ \sim L (a, b)$ be an uncertain variable with a linear uncertainty distribution ϒ (u). Then, the sum ξ = η + τ is naturally an uncertain random variable. The chance distribution of ξ is determined by $\begin{matrix} Φ (u) \\ = \int_{- \infty}^{\infty} ϒ (u - w) d Ψ (w) \\ = \int_{- \infty}^{\infty} (\frac{u - w - a}{b - a} I_{{a \leq u - w \leq b}} + I_{{u - w \geq b}}) d Ψ (w) \\ = \frac{1}{\sqrt{2 π} σ} \int_{- \infty}^{u - b} exp (- \frac{(w - μ)^{2}}{2 σ^{2}}) d w \\ - \frac{1}{\sqrt{2 π} σ} \int_{u - b}^{u - a} \frac{w - u + a}{b - a} exp (- \frac{(w - μ)^{2}}{2 σ^{2}}) d w, \end{matrix}$ which is a continuous function with respect to u.

Example 3. Let $τ \sim Z (a, b, c)$ be an uncertain variable with a zigzag uncertainty distribution ϒ (u). The sum ξ = η + τ has the following chance distribution,

$\begin{matrix} Φ (u) \\ = \int_{- \infty}^{\infty} \frac{u - w - a}{2 b - 2 a} I_{{a \leq u - w \leq b}} d Ψ (w) \\ + \int_{- \infty}^{\infty} \frac{u - w + c - 2 b}{2 c - 2 b} I_{{b \leq u - w \leq c}} d Ψ (w) \\ + \int_{- \infty}^{\infty} I_{{u - w \geq c}} d Ψ (w) \\ = \frac{1}{\sqrt{2 π} σ} \int_{u - b}^{u - a} \frac{u - w - a}{2 b - 2 a} exp (- \frac{(w - μ)^{2}}{2 σ^{2}}) d w \\ + \frac{1}{\sqrt{2 π} σ} \int_{u - c}^{u - b} \frac{u - w + c - 2 b}{2 c - 2 b} exp (- \frac{(w - μ)^{2}}{2 σ^{2}}) d w \\ + \frac{1}{\sqrt{2 π} σ} \int_{- \infty}^{u - c} exp (- \frac{(w - μ)^{2}}{2 σ^{2}}) d w \end{matrix}$

which is also continuous with respect to u.

Example 4. Let $τ \sim N (e, δ)$ be an uncertain variable with a normal uncertainty distribution ϒ (u). Similarly, the chance distribution of ξ = η + τ is $Φ (u) = \frac{1}{\sqrt{2 π} σ} \int_{- \infty}^{\infty} \frac{exp (- \frac{(w - μ)^{2}}{2 σ^{2}})}{1 + exp (\frac{π (e - u + w)}{\sqrt{3} δ})} d w$ (8) which is a continuous function of u.

Examples 2, 3 and 4 imply that Φ (u) is a nonlinear function of u. Thus it is difficult to obtain the analytical expression of VaR. Next we introduce the secant method to compute VaR_α (ξ), which is just Φ^-1 (α).

Algorithm 1 The secant method for solving VaR

Input: input parameters α (confidence level), ɛ (a predetermined precision) and u₁, u₂ (initial values)

Output: output VaR_α (ξ)

1: Set F (u) = Φ (u) - α and calculate F (u₁) and F (u₂)

2: while F (u₂) > ɛ do

3: Set $u_{3} = u_{2} - \frac{F (u_{2}) (u_{2} - u_{1})}{F (u_{2}) - F (u_{1})}$

4: Set u₁ = u₂ and u₂ = u₃

5: Calculate F (u₂)

6: end while

7: Set VaR_α (ξ) = u₂

8: return VaR_α (ξ)

Example 5. Without loss of generality, we set μ = 0.005, σ = 0.07, e = 0.002 and δ = 0.11 in Example 4. Thus Equation (8) becomes $Φ (u) = \frac{1}{0.07 \sqrt{2 π}} \int_{- \infty}^{\infty} \frac{exp (- \frac{(w - 0.005)^{2}}{0.0098})}{1 + exp (\frac{π (0.002 - u + w)}{\sqrt{3} \times 0.11})} d w .$

Setting predetermined precision ɛ = 1 ×10^-4, we run Algorithm 1 with different confidence levels and the results are shown in Table 1. It can be seen that the value of VaR_α (ξ) increases with the increase of α.

Table 1

The VaRs of ξ with different confidence levels α

Confidence level α	0.90	0.95	0.97	0.99
VaR_α (ξ)	0.7217	0.8174	0.8556	0.8939

4 Uncertain random mean-VaR models

In this section, we first state the uncertain random portfolio optimization problem and then establish the corresponding mean-VaR models. Finally, we introduce an active-set method to solve these models.

Assume that there are m existing risky securities which have enough historical data and n newly listed ones with insufficient data. Let η_i be a random variable representing the future return of the ith existing risky security with expected value μ_i and cumulative distribution function Ψ_i for i = 1, 2, ⋯ , m. And let τ_j be an uncertain variable representing the future return of the jth newly listed risky security with expected value ν_j and uncertainty distribution ϒ_j for j = 1, 2, ⋯ , n. Denote σ_ij by the covariance of random returns η_i and η_j. Finally, let x_i be the holding proportion of the ith existing security and y_j the holding proportion of the jth newly listed security, for i = 1, 2, ⋯ , m and j = 1, 2, ⋯ , n.

For sake of convenience, we will use the following vectors in the rest, η = (η₁, η₂, ⋯ , η_m) ′ : the random vector of future returns of the existing securities; x = (x₁, x₂, ⋯ , x_m) ′ : the holding proportion vector of the existing securities; τ = (τ₁, τ₂, ⋯ , τ_n) ′ : the uncertain vector of future returns of the newly listed securities; y = (y₁, y₂, ⋯ , y_n) ′ : the holding proportion vector of the newly listed securities; μ = (μ₁, μ₂, ⋯ , μ_m) ′ : the expected vector of η ; ν = (ν₁, ν₂, ⋯ , ν_n) ′ : the expected vector of τ . Here ^′ is the matrix transpose symbol.

A portfolio vector of all the candidate securities can be represented as $(x^{'}, y^{'})^{'} = (x_{1}, x_{2}, \dots, x_{m}, y_{1}, y_{2}, \dots, y_{n})^{'},$ which yields the total return $x^{'} η + y^{'} τ = \sum_{i = 1}^{m} x_{i} η_{i} + \sum_{j = 1}^{n} y_{j} τ_{j} .$

Obviously, the total return is an uncertain random variable since which is the sum of random variables and uncertain variables.

We replace variance with VaR to measure the risk associated with each portfolio. If an investor wants to maximize the expected return at a given risk level, then we can formulate the following mean-VaR model for uncertain random portfolio optimization problem, ${\begin{matrix} \max & E [x^{'} η + y^{'} τ] \\ s . t . & {VaR}_{α} (x^{'} η + y^{'} τ) = λ_{0} \\ \sum_{i = 1}^{m} x_{i} + \sum_{j = 1}^{n} y_{j} = 1 \\ x_{i} \geq 0, y_{j} \geq 0, i = 1, \dots, m; j = 1, \dots, n \end{matrix}$ (9) in which λ₀ is a tolerable risk level. The first constraint ensures that the portfolio will be chosen only from the feasible ones whose VaRs are equal to λ₀.

As an alternative of Model (9), we may minimize the risk of investment with an acceptable return level and formulate another mean-VaR model as follows, ${\begin{matrix} \min & {VaR}_{α} (x^{'} η + y^{'} τ) \\ s . t . & E [x^{'} η + y^{'} τ] = μ_{0} \\ \sum_{i = 1}^{m} x_{i} + \sum_{j = 1}^{n} y_{j} = 1 \\ x_{i} \geq 0, y_{j} \geq 0, i = 1, \dots, m; j = 1, \dots, n \end{matrix}$ (10) where μ₀ is an acceptable return level. The first constraint requires that the portfolio will be chosen only from the feasible ones whose expected returns are equal to μ₀.

In order to simplify Models (9) and (9VaR), we need the following assumptions which were also used in Qin [20]. The random vector η has a multivariate normal distribution with the probability density function $\begin{matrix} ψ_{η} (w) & = & \frac{1}{\sqrt{(2 π)^{m}} | Σ |} \\ exp (- \frac{1}{2} (w - μ)' Σ^{- 1} (w - μ)), \end{matrix}$ in which Σ = (σ_ij) _m×m is the covariance matrix of η . And uncertain returns τ₁, τ₂, ⋯ , τ_n are independent in the sense of uncertain measure [12]. On the one hand, it follows from the former assumption that x ′ η is a normal random variable with expected value μ ′ x and probability density function $ψ (w) = \frac{1}{\sqrt{2 π} σ (x)} exp (- \frac{(w - μ^{'} x)^{2}}{2 σ^{2} (x)}),$ where $σ (x) = \sqrt{x' Σ x}$ , and σ² ( x ) = (σ ( x )) ². On the other hand, it follows from the latter assumption that E [ y ′ τ ] = y ′ ν . Thus, the expected value of total return x ′ η + y ′ τ is equal to $x^{'} μ + y^{'} ν = \sum_{i = 1}^{m} x_{i} μ_{i} + \sum_{j = 1}^{n} y_{j} ν_{j} .$

Note that, in Model (9VaR), the constraints are all linear and objective function is a nonlinear implicit function. We may employ active-set method to solve it. Next we first transform this model into a standard form. Denote b = (b₁, b₂, ⋯ , b_m+n+2) ′ = (μ₀, 1, 0, 0, ⋯ , 0) ′ and A ₁ = ( μ ′, ν ′) ′. Here A ₂ is an all ones vector of size m + n and A _j+2 = e _j for j = 1, 2, ⋯ , m + n in which e _j is the ith column of the identify matrix. Let p = ( x ′, y ′) ′. Then Model (9VaR) is rearranged as follows, ${\begin{matrix} \min & {VaR}_{α} (x^{'} η + y^{'} τ) \\ s . t . & a_{i}^{'} p = b_{i}, i = 1, 2 \\ a_{i}^{'} p \leq b_{i}, i = 3, 4, \dots, m + n + 2 . \end{matrix}$ (11)

Next we consider the model with equality constraints since a key step in the active-set method is to solve it. Record 𝒜 as the active set of each iteration, and the corresponding equality problem is as follows, ${\begin{matrix} \min & {VaR}_{α} (x^{'} η + y^{'} τ) \\ s . t . & a_{i}^{'} p = b_{i}, i \in 𝒜 . \end{matrix}$ (12)

For given 𝒜, we denote A ^* = ( A _j) _j∈𝒜, and $b^{*} = (b_{j})_{j \in 𝒜}^{'}$ . For example, if 𝒜 = {1, 2, 4, 7}, then A ^* = ( A ₁, A ₂, A ₄, A ₇) and b ^* = (b₁, b₂, b₄, b₇) ′. Let S , T be two matrixes such that ( A ^*) ′ S = E and ( A ^*) ′ T = 0 , in which E is the identify matrix. Since ( S , T ) is nonsingular, p = Sb ^* is a special solution of the equation ( A ^*) ′ p = b ^*. The other feasible solutions can be expressed as p = Sb ^* + Tl , in which l is an arbitrary column vector. In this way, it is natural to transform Model (12) into the following unconstrained problem (13), $\begin{matrix} \min & ψ (l) : = {VaR}_{α} (p^{'} ξ) . \end{matrix}$ (13) in which p = Sb ^* + Tl and ξ = ( η ′, τ ′) ′ .

Quasi-Newton method is an effective method to solve an unconstrained optimization problem, which is applied to seek the optimal solution of Model (13).

The above statement provides the solution procedure of Model (11). For each updated active set 𝒜, we solve Model (12) until we find an optimal portfolio satisfying the accuracy requirement. We write ℐ = {3, 4, ⋯ , m + n + 2}. The detailed solution procedure is summarized as Algorithm 2.

Algorithm 2 Active-set method for solving Model (11)

Input: input parameters p ⁽⁰⁾ (initial value) and 𝒜 (active set)

Output: output optimal portfolio p ^*

1: Set k = 0

2: while 1 do

3: if p ^(k) solves the equality problem (12) then

4: compute its Lagrange multiplier λ ^(k);

5: find w to solve $\min {λ_{i}^{(k)} | i \in ℐ \cap 𝒜}$ ;

6: if $λ_{w}^{(k)} \geq 0$ then

7: terminate with p ^* = p ^(k),

8: else

9: remove w from 𝒜;

10: end if

11: end if

12: solve the equality problem (12) to obtain the optimal solution q ^(k);

13: solve the problem $\max {d | d > 0, a_{i}^{'} p^{(k)} - b_{i} + d a_{i}^{'} (q^{(k)} - p^{(k)}) \leq 0, a_{i}^{'} p^{(k)} > 0, \forall i \notin 𝒜}$ and the optimal solution is recorded as ${\bar{d}}_{k} = (b_{j} - a_{j}^{'} p^{(k)}) / a_{j}^{'} (q^{(k)} - p^{(k)})$ ;

14: set $d_{k} = \max (1, {\bar{d}}_{k})$ and p ^(k+1) = p ^(k) + d_k ( q ^(k) - p ^(k))

15: if d_k < 1 then

16: add j to 𝒜;

17: end if

18: set k = k + 1;

19: end while

20: return p ^*

5 A numerical example

In this section, we apply Model (9VaR) to a numerical example with 20 securities. The existing 15 securities are arbitrarily chosen from Shanghai Stock Exchange (SSE), whose monthly return rates are collected from January 1, 2010 to June 30, 2016. Their codes and sample means are shown in Table 2 and the corresponding covariance matrix is shown in Table 3. Assume that there are five newly listed securities whose monthly return rates are regarded as zigzag uncertain variables subject to experts’ evaluations. The simulated data are shown in Table 4.

Table 2
The codes and sample means of 15 existing securities chosen from SSE

No. 1 2 3 4 5

Code 600004 600050 600066 600089 600100

Mean (%) 0.50 –0.21 0.47 –0.70 0.87

No. 6 7 8 9 10

Code 600111 600126 600127 600139 600196

Mean (%) 0.18 0.56 0.12 0.38 0.41

No. 11 12 13 14 15

Code 600201 600238 600338 600400 600600

Mean (%) 2.14 0.70 2.57 1.47 –0.12

No.	1	2	3	4	5
Code	600004	600050	600066	600089	600100
Mean (%)	0.50	–0.21	0.47	–0.70	0.87
No.	6	7	8	9	10
Code	600111	600126	600127	600139	600196
Mean (%)	0.18	0.56	0.12	0.38	0.41
No.	11	12	13	14	15
Code	600201	600238	600338	600400	600600
Mean (%)	2.14	0.70	2.57	1.47	–0.12

Table 3

The sample covariance matrix of the 15 securities from SSE (10^-4)

51.8	45.0	17.8	51.2	56.2	38.7	24.7	27.1	29.3	28.8	41.0	47.0	22.3	52.7	19.6
45.0	145.1	36.7	69.2	59.8	45.4	18.3	34.0	26.7	29.0	22.7	–9.8	46.5	46.4	20.7
17.8	36.7	92.4	32.1	22.6	55.3	33.1	31.2	30.2	2.0	–7.0	6.8	18.3	23.9	9.7
51.2	69.2	32.1	126.9	76.1	70.8	35.4	39.0	45.7	43.8	53.2	28.7	27.4	79.4	28.3
56.2	59.8	22.6	76.1	206.9	74.2	17.0	32.1	58.9	73.6	102.6	86.7	37.4	93.8	32.1
38.7	45.4	55.3	70.8	74.2	221.1	31.8	32.2	71.8	19.9	30.9	22.3	59.2	42.6	24.1
24.7	18.3	33.1	35.4	17.0	31.8	147.6	35.5	16.8	–1.3	6.9	29.1	–37.7	48.6	–1.3
27.1	34.0	31.2	39.0	32.1	32.2	35.5	101.2	26.5	23.7	20.8	22.4	27.7	54.2	9.7
29.3	26.7	30.2	45.7	58.9	71.8	16.8	26.5	233.5	28.0	35.2	60.2	–14.4	71.7	–1.1
28.8	29.0	2.0	43.8	73.6	19.9	–1.3	23.7	28.0	89.3	50.5	44.4	23.0	49.2	32.7
41.0	22.7	–7.0	53.2	102.6	30.9	6.9	20.8	35.2	50.5	164.7	77.9	26.9	63.6	36.1
47.0	–9.8	6.8	28.7	86.7	22.3	29.1	22.4	60.2	44.4	77.9	204.9	18.5	65.8	29.0
22.3	46.5	18.3	27.4	37.4	59.2	–37.7	27.7	–14.4	23.0	26.9	18.5	302.6	38.7	16.4
52.7	46.4	23.9	79.4	93.8	42.6	48.6	54.2	71.7	49.2	63.6	65.8	38.7	218.5	21.8
19.6	20.7	9.7	28.3	32.1	24.1	–1.3	9.7	–1.1	32.7	36.1	29.0	16.4	21.8	42.1

Next we employ Algorithm 2 to solve Model (9VaR) in which the parameters are presented in Tables 2–4. Set the iterative initial value p ⁽⁰⁾ = 0 , initial active set 𝒜 = {1, 2} and the confidence level α = 0.99. The equality problem (12) will be first solved. Then the iterative step d_k is computed, and p ⁽⁰⁾ and 𝒜 are updated. If p is exactly the solution of Model (12) with the new 𝒜 and satisfies all the constraints, then it is returned as the optimal portfolio. Otherwise, a new iteration is began by solving Model (12) with an updated 𝒜. For each given return level μ₀, we run Algorithm 2 and get an optimal portfolio. The computational results are shown in Table 5. The last line shows the value of VaR for each corresponding optimal portfolio. It is obvious that the value of VaR increases with the increase of μ₀.

Table 4

Uncertain returns and their expected values of simulated 5 newly listed securities

No.	Uncertain return	Expected value (%)
τ ₁	𝒵 (-20, 0, 18)	–0.50
τ ₂	𝒵 (-30, 5, 25)	1.25
τ ₃	𝒵 (-20, 0, 20)	0
τ ₄	𝒵 (-15, 1.9, 22)	0.80
τ ₅	𝒵 (-32, 4, 20)	–1.00

Table 5

The investment proportions in the optimal portfolio of Model (9VaR) when α = 0.99

μ₀ (%)	0.7	0.8	1.0	1.4	1.7	1.8	2.1	2.2	2.3	2.4	2.5
600004	5.25	6.58	8.45	5.57	0	0	0	0	0	0	0
600066	5.21	12.23	9.11	5.37	8.94	8.54	0	0	0	0	0
600089	3.72	0	0	0	0	0	0	0	0	0	0
600100	5.72	0	0	0	0	0	0	0	0	0	0
600126	5.33	15.59	10.45	6.01	13.91	13.47	0	0	0	0	0
600127	4.76	0	0	0	0	0	0	0	0	0	0
600139	5.1	4.22	6.17	4.8	0	0	0	0	0	0	0
600196	5.13	4.41	9.33	4.98	0	0	0	0	0	0	0
600201	7.34	9.79	14.01	16.61	34.54	34.22	46.09	51.37	54.48	39.83	16.26
600238	5.51	6.15	3.6	6.94	0	0	0	0	0	0	0
600338	11.23	10.04	14.96	19.45	22.87	23.6	32.55	37.18	42.65	60.17	83.74
600400	6.48	2.21	4.67	12.06	2.25	2.12	3.86	0	0	0	0
600600	4.46	20.09	8.92	0	0	0	0	0	0	0	0
τ ₁	3.97	0	0	0	0	0	0	0	0	0	0
τ ₂	10.55	2.55	3.45	10.61	17.49	18.05	17.5	11.45	2.87	0	0
τ ₃	4.61	0	0	0	0	0	0	0	0	0	0
τ ₄	5.63	6.14	6.88	7.6	0	0	0	0	0	0	0
τ ₅	0	0	0	0	0	0	0	0	0	0	0
VaR	0.6089	0.6101	0.6278	0.6321	0.6489	0.6600	0.6785	0.6833	0.6941	0.7127	0.7481

6 Conclusions

This paper was devoted to a hybrid portfolio optimization problem with existing risky securities and newly listed ones. Uncertain random VaR was introduced to evaluate the risk associated with individual security and the portfolio of securities. Its mathematical properties were discussed in detail and numerical solution procedure was given for continuous chance distribution. The mean-VaR models were formulated and active-set method was introduced to solve these modells. Finally a numerical example was given to illustrate the application of the proposed approach.

Footnotes

Acknowledgments

This work was supported in part by National Natural Science Foundation of China (Nos. 71371019), and in part by the Fundamental Research Funds for the Central Universities (No. YWF-16-BJY-20).

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