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Analytical Shortcuts to Multiple-Objective Portfolio Optimization: Investigating the Non-Negativeness of Portfolio Weight Vectors of Equality-Constraint-Only Models and Implications for Capital Asset Pricing Models

Department of Financial Management, Business School, Nankai University, 94 Weijin Road, Tianjin 300071, China

Department of Investing, School of Economics, Tianjin Normal University, 393 Binshuixi Road, Tianjin 300071, China

Author to whom correspondence should be addressed.

Mathematics 2024, 12(24), 3946; https://doi.org/10.3390/math12243946

Submission received: 9 November 2024 / Revised: 4 December 2024 / Accepted: 9 December 2024 / Published: 15 December 2024

(This article belongs to the Special Issue Mathematical Models and Applications in Finance)

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Abstract

Computing optimal-solution sets has long been a topic in multiple-objective optimization. Despite substantial progress, there are still research limitations in the multiple-objective portfolio optimization area. The optimal-solution sets’ structure is barely known. Public-domain software for even three objectives is absent. Alternatively, researchers scrutinize equality-constraint-only models and analytically resolve them. Within this context, this paper extends these analytical methods for nonnegative constraints and thus theoretically contributes to the literature. We prove the existence of positive elements and negative elements for the optimal-solution sets. Practically, we prove that non-negative subsets of the optimal-solution sets can exist. Consequently, the possible existence endorses these analytical methods, because researchers bypass mathematical programming, analytically resolve, and pinpoint some non-negative optima. Moreover, we elucidate these analytical methods’ alignment with capital asset pricing models (CAPMs). Furthermore, we generalize for k-objective models. In conclusion, this paper theoretically reinforces these analytical methods and hints the optimal-solution sets’ structure for multiple-objective portfolio optimization.

Keywords:

multiple-objective optimization; portfolio selection; capital asset pricing models (CAPMs); multiple-objective portfolio selection; analytical methods

MSC:

90C29; 91G10; 91G30

1. Introduction

We delineate a Graphical Abstract to initially manifest this paper’s theme and originality. It horizontally contains

The research for (traditional) portfolio selection and capital asset pricing models (CAPM) in the left part;
The originality for multiple-objective portfolio selection and CAPM in the right part.

It vertically contains six gradually shaded panels for research progression as follows:

Panel A: Traditional portfolio selection and multiple-objective portfolio selection;
Panel B: Efficient sets of (2) and (5);
Panel C: Proving the existence of positive elements and negative elements for $E_{(5)}$ ;
Panel D: Proving the possible existence of a non-negative subset of $E_{(5)}$ ;
Panel E: Implications for capital asset pricing models;
Panel F: General k-objective portfolio selection models.

We will elaborate the graphical abstract in the following subsections. We will explicitly refer to it by stating that, for instance, “Particularly in the left part of Panel A of Graphical Abstract, we depict (2)”.

1.1. Portfolio Selection and Capital Asset Pricing Models

Markowitz [1] (p. 6) accentuates both risk and return. Markowitz [2] (p. 83) originates portfolio selection as the following two-objective optimization (Steuer [3] illuminates multiple-objective optimization):

\{\begin{matrix} min z_{1} = x^{T} Σ x, \\ max z_{2} = x^{T} μ_{2}, \\ subject to x \in S, \end{matrix}

(1)

where for n stocks,

Σ

is a covariance matrix of stock returns:

μ_{2}

is a vector of expected stock returns: subscript 2 is an identifier for

μ_{2}

, because we will initiate

μ_{3}

later: a portfolio is fixed by its weight vector

x

z_{1}

and

z_{2}

, respectively, measure the portfolio return’s variance and expectation; and

S \subset R^{n}

is a feasible region: Scholars usually assume that S is formed by linear constraints. The optimality of (1) in

(z_{1}, z_{2})

space is called an efficient frontier (For notations, normal symbols (e.g., n) denote scalars. Bold-face symbols (e.g.,

Σ

μ_{2}

) denote matrices or vectors.).

Sharpe [4] and Merton [5] dissect the following equality-constraint-only model:

\{\begin{matrix} min z_{1} = x^{T} Σ x, \\ max z_{2} = x^{T} μ_{2}, \\ subject to 1^{T} x = 1, \end{matrix}

(2)

where

1

is a vector of ones. Particularly in the left part of Panel A of Graphical Abstract, we depict (2). Merton [5] analytically derives the efficient frontier’s

x

set as a one-dimensional cone. Particularly in the left part of Panel B, we depict the cone with expression (8). Sharpe [4] further anchors the market portfolio (as defined by Bodie et al. [6] (p. 109)) and originates their capital asset pricing model by (2).

However, (2) allows negative elements in

x

, so the market portfolio can carry negative elements. Brennan and Lo [7] deduce that the capital asset pricing model of Sharpe [4] is inconsistent with the negative elements.

Markowitz and Todd [8] (p. 3) also dispute the negative elements and launch the following standard portfolio selection model:

\{\begin{matrix} min z_{1} = x^{T} Σ x, \\ max z_{2} = x^{T} μ_{2}, \\ subject to 1^{T} x = 1, \\ x \geq 0, \end{matrix}

(3)

where

0

is a vector of zeros.

1.2. Rise of Multiple-Objective Portfolio Selection

Markowitz [9] (pp. 471 and 476) observes extra objectives after his feat (1). The same is true in Sharpe [10]. Fama [11] (pp. 445–447) and Cochrane [12] (pp. 1081–1082) aim to use multiple factors for capital asset pricing models. Harvey and Siddique [13] investigate skewness. Lo et al. [14] watch liquidity. Pedersen et al. [15] treat ESG. Jayasekara et al. [16] and Qi and Steuer [17] optimize the portfolio selection.

Hirschberger et al. [18], Utz et al. [19], Qi et al. [20], Qi and Steuer [21], and Utz and Steuer [22] extend (1) and propose a multiple-objective portfolio selection as follows:

\{\begin{matrix} min z_{1} = x^{T} Σ x, \\ max z_{2} = x^{T} μ_{2}, \\ ⋮ \\ max z_{k} = x^{T} μ_{k}, \\ subject to x \in S, \end{matrix}

(4)

where k is the number of objectives.

μ_{3} \dots μ_{k}

are vectors of general stock expected objectives (e.g., ESG).

z_{3} \dots z_{k}

measures the general portfolio expected objectives. Overall, Aouni et al. [23] and La Torre et al. [24] offer surveys.

Expressly, Qi et al. [20] extend (2) as follows:

\{\begin{matrix} min z_{1} = x^{T} Σ x, \\ max z_{2} = x^{T} μ_{2}, \\ max z_{3} = x^{T} μ_{3}, \\ subject to 1^{T} x = 1 . \end{matrix}

(5)

Particularly in the right part of Panel A of Graphical Abstract, we depict (5). Qi et al. [20] analytically derive the efficient frontier’s

x

set as a two-dimensional cone. Particularly in the right part of Panel B, we depict the cone with expression (12).

1.3. Research Limitations of Multiple-Objective Portfolio Selection (4)

Despite the rise, there still exist the following research limitations of multiple-objective portfolio selection:

1.3.1. Research Limitations in Proving the Structure of the Optimal-Solution Sets

Bank et al. [25] profoundly investigate parametric quadratic programming and prepare the resolution methods for (4). Best [26], Goh and Yang [27], Hirschberger et al. [18], Jayasekara et al. [28], and Jayasekara et al. [16] propose their parametric quadratic programming algorithms.

However, the researchers do not specifically prove the optimal-solution sets’ structure. Explicitly, Best [26] and Goh and Yang [27] execute active-set algorithms for (4) but do not prove the structure. Hirschberger et al. [18] deploy the framework of Bank et al. [25] and exhaustively gain the optimal-solution sets but do not prove the structure. Jayasekara et al. [28] and Jayasekara et al. [16] compare optimization methods and develop generalized scalarization algorithms but do not prove the structure.

Moreover, the researchers do not equip public-domain software, so the community optimization is still absent. To the best of our knowledge, Hirschberger et al. [18] code the only software for (4) but for just three objectives. Utz et al. [29], Utz et al. [19], and Utz and Steuer [22] channel the software. Unfortunately, the software is still private.

1.3.2. Research Limitations in Investigating the Non-Negativeness of the Optimal-Solution Sets

Due to the previous subsubsection’s limitations, the non-negativeness of the optimal-solution sets is still unfathomed. Hirschberger et al. [18] (p. 181) tentatively document the optimization results of merely five problems but do not disclose the non-negativeness. Jayasekara et al. [28] (pp. 194–199) illustrate by two problems with three stocks and three objectives.

1.3.3. Research Limitations in Implicating the Capital Asset Pricing Models

The research for capital asset pricing models by (4) is scant. Manifestly, Aouni et al. [23] and La Torre et al. [24] neither survey the research nor propose it as future directions. Especially, Ingersoll [30] meditates skewness and infers capital asset pricing models with skewness. Qi [31] classifies minimum-variance surfaces. Qi et al. [32] discover zero-covariance-portfolio curves (Promisingly, scholars (e.g., Harvey and Siddique [13]) typically probe capital asset pricing models with skewness and kurtosis by utility functions (instead of portfolio selection). In this paper, we trail Markowitz [2] and Sharpe [4] and proceed by multiple-objective portfolio selection).

1.4. Originality of This Paper: Investigating the Non-Negativeness, Justifying the Analytical Shortcuts, Implicating the Capital Asset Pricing Models for (5), and Generalizing

In contrast to parametric quadratic programming, Merton [5], Qi et al. [20], Qi and Steuer [21], and Qi and Steuer [17] presume equality constraints only and analytically derive efficient frontiers’

x

sets. The key advantage of these analytical methods is bypassing mathematical programming and readily optimizing in formulae (We will deliberate these methods in Section 2). In this paper, we advance (5) and its extension and establish the originality as follows:

1.4.1. Investigating the Possible Existence of a Non-Negative Subset of the Optimal-Solution Set in the Absence of Public-Domain Software

In theorems, we prove the existence of positive elements and negative elements for the optimal-solution set of (5). Particularly in the right part of Panel C of Graphical Abstract, we depict its existence. We prove that there can exist a non-negative subset of the optimal-solution set of (5). We define the subset as follows: Firstly, it is a subset of the optimal-solution set of (5). Secondly, for any

x

of the subset, all elements of

x

are non-negative. The nonnegativeness is essential for the capital asset pricing models of (4) (as highlighted by Brennan and Lo [7]). Particularly in the right part of Panel D, we depict the possible existence as a red portion.

1.4.2. Consequently Justifying the Analytical Shortcuts for Optimizing (4)

Consequently, by the possible existence and (12), scholars can straightforwardly analytically derive some optima of the following standard multiple-objective portfolio selection model (as an extension of (3)):

\{\begin{matrix} min z_{1} = x^{T} Σ x, \\ max z_{2} = x^{T} μ_{2}, \\ max z_{3} = x^{T} μ_{3}, \\ subject to 1^{T} x = 1, \\ x \geq 0 . \end{matrix}

(6)

Typically, scholars must deploy mathematical programming to obtain the optima of (6). We call the readily analytical derivation as analytical shortcuts.

1.4.3. Implicating the Capital Asset Pricing Models

As implications, we harness convex sets and prove the effectiveness of (5) for the capital asset pricing models. Particularly in the right part of Panel E of Graphical Abstract, we depict the effectiveness.

1.4.4. Generalizing for k-Objective Models

We generalize for the following k-objective models:

\{\begin{matrix} min z_{1} = x^{T} Σ x, \\ max z_{2} = x^{T} μ_{2}, \\ ⋮ \\ max z_{k} = x^{T} μ_{k}, \\ subject to 1^{T} x = 1 . \end{matrix}

(7)

Particularly in the right part of Panel F of Graphical Abstract, we depict the generalization.

1.5. Paper Structure

The rest of this paper is organized as follows: We review multiple-objective optimization and portfolio optimization in Section 2. For (5), we prove the optimal solution set’s properties: the existence of positive elements and negative elements for

Δ_{2}

Δ_{3}

, and

x_{m v}

, and for

E_{(5)}

in the limit in Section 3. We prove the possible existence of a non-negative subset of the optimal-solution set in Section 4. We illustrate in Section 5. We enumerate the implications for capital asset pricing models in Section 6. We generalize and deliberate this paper’s advantage and disadvantage in Section 7. We conclude in Section 8. As technicalities, we compare major portfolio optimization methods in Appendix A.

2. Theoretical Background: Multiple-Objective Optimization and Portfolio Optimization

2.1. Multiple-Objective Optimization

Multiple-objective optimization can be modelled as follows:

\{\begin{matrix} max z_{1} = f_{1} (x), \\ ⋮ \\ max z_{k} = f_{k} (x), \\ subject to x \in S, \end{matrix}

where

x \in R^{n}

is a decision vector in decision space. k is the number of objectives.

f_{1} (x) \dots f_{k} (x)

are objective functions.

z = {[\begin{matrix} z_{1} & \dots & z_{k} \end{matrix}]}^{T}

is a criterion vector in criterion space.

S \subset R^{n}

is a feasible region in decision space.

Z = {z ∣ x \in S}

is the feasible region in criterion space.

Definition 1.

For

\bar{z} \in Z

and

z \in Z

, the fact that

\bar{z}

dominates

z

is defined as

{\bar{z}}_{1} \geq z_{1}, \dots, {\bar{z}}_{k} \geq z_{k}

with at least one strict inequality.

Definition 2.

The fact that

\bar{z} \in Z

is nondominated is defined as such means that a

z \in Z

such that

z

dominates

\bar{z}

. Then if

\bar{x} \in S

is an inverse image of

\bar{z}

(i.e.,

\bar{z} = [\begin{matrix} f_{1} (\bar{x}) \\ ⋮ \\ f_{k} (\bar{x}) \end{matrix}]

\bar{x}

is efficient.

One purpose of multiple-objective optimization is to total the following:

Set of efficient decision vectors (as efficient set);
Set of non-dominated criterion vectors (as nondominated set).

For terminologies, Markowitz [2] specifies the optimality of (1) in

(z_{1}, z_{2})

space as an efficient frontier. Contrarily, we respect Definition 2 and reserve “nondominated” for criterion space and “efficient” for decision space. Henceforth, we term the previous “optimal-solution set” (e.g., in Section 1) as efficient set.

2.2. Portfolio Optimization by Analytical Methods

2.2.1. Analytically Resolving (2)

Merton [5] assumes as follows:

Assumption 1.

Σ is positive definite (and thus invertible).

Assumption 2.

μ_{2}

and

1

are linearly independent.

Merton [5] analytically derives the efficient set of (2) as follows:

\begin{matrix} E_{(2)} = {x \in R^{n} | x = x_{m v} + λ_{2} Δ_{2}, λ_{2} \in [0, \infty)}, \end{matrix}

(8)

where

\begin{matrix} x_{m v} & = \frac{1}{f} Σ^{- 1} 1 as the minimum-variance portfolio of (2), \end{matrix}

(9)

\begin{matrix} Δ_{2} & = Σ^{- 1} μ_{2} - \frac{c}{f} Σ^{- 1} 1 . \end{matrix}

(10)

Merton [5] introduces symbols f and c as follows:

\begin{matrix} c = μ_{2}^{T} Σ^{- 1} 1 f = 1^{T} Σ^{- 1} 1 . \end{matrix}

(11)

Merton [5] identifies

Δ_{2}

as unequal to

0

. Therefore,

E_{(2)}

is a one-dimensional cone in

R^{n}

2.2.2. Analytically Resolving (5)

Qi et al. [20] extend (2) to (5) and additionally assume the following:

Assumption 3.

μ_{2}

μ_{3}

, and

1

are linearly independent.

Qi et al. [20] analytically derive the efficient set of (5) as follows:

\begin{matrix} E_{(5)} = {x \in R^{n} | x = x_{m v} + λ_{2} Δ_{2} + λ_{3} Δ_{3}, λ_{2} \in [0, \infty), λ_{3} \in [0, \infty)}, \end{matrix}

(12)

where

\begin{matrix} x_{m v} & = \frac{1}{f} Σ^{- 1} 1 as the minimum-variance portfolio of (5), \end{matrix}

(13)

\begin{matrix} Δ_{2} & = Σ^{- 1} μ_{2} - \frac{c}{f} Σ^{- 1} 1, \end{matrix}

(14)

\begin{matrix} Δ_{3} & = Σ^{- 1} μ_{3} - \frac{e}{f} Σ^{- 1} 1 . \end{matrix}

(15)

By (11), Qi et al. [20] restate or introduce the following symbols:

\begin{matrix} C \equiv {[\begin{matrix} a & b & c \\ b & d & e \\ c & e & f \end{matrix}]}_{3 \times 3} \equiv [\begin{matrix} μ_{2}^{T} Σ^{- 1} μ_{2} & μ_{2}^{T} Σ^{- 1} μ_{3} & μ_{2}^{T} Σ^{- 1} 1 \\ μ_{2}^{T} Σ^{- 1} μ_{3} & μ_{3}^{T} Σ^{- 1} μ_{3} & μ_{3}^{T} Σ^{- 1} 1 \\ μ_{2}^{T} Σ^{- 1} 1 & μ_{3}^{T} Σ^{- 1} 1 & 1^{T} Σ^{- 1} 1 \end{matrix}] . \end{matrix}

(16)

Qi et al. [20] prove

f \neq 0

, so

\frac{1}{f}

in (14) and (15) is well defined. Qi et al. [20] prove

C

in (16) as positive definite and invertible.

E_{(5)}

is generated by

Δ_{2}

and

Δ_{3}

and translated by

x_{m v}

. We depict

E_{(5)}

in the right part of Figure 1A. Qi et al. [20] (p. 171) prove the independence between

Δ_{2}

and

Δ_{3}

as follows:

Lemma 1.

Δ_{2}

and

Δ_{3}

are linearly independent.

Moreover, Qi et al. [20] prove the minimum-variance surface of (5) as a paraboloid. For instance, we depict a paraboloid

z_{1} = z_{2}^{2} + 2 z_{3}^{2} + 3 z_{2} z_{3} + 4 z_{2} + 5 z_{3} + 9

in the left part of Figure 1A. Therefore, the nondominated set of (5) is a paraboloidal segment.

Analytical methods are based on calculus and linear algebra and thus understandable. However, analytical methods fit models with equality constraints only. Thus, unpractically unbounded portfolio weight vectors are allowed. Nevertheless, with almost all the results in formulae, analytical methods bypass the need of mathematical programming and bring convenience in research and instruction. For instance, Huang and Litzenberger [33] (p. 60) endorse (2).

2.3. Portfolio Optimization by Other Methods

As technicalities, we report other portfolio optimization methods and compare them in Appendix A.

3. Proving Properties of the Efficient Set of (5): The Existence of Positive Elements and Negative Elements for $Δ_{2}$ , $Δ_{3}$ , and $x_{mv}$ and for $E_{(5)}$ in Limit

We prove the following properties of

E_{(5)}

in (12):

$x_{m v}$ always contains at least one positive element.
$Δ_{2}$ always contains at least one positive element and at least one negative element.
$Δ_{3}$ always contains at least one positive element and at least one negative element.
$E_{(5)}$ always contains some $x$ with at least one positive element and at least one negative element as either $λ_{2}$ or $λ_{3}$ approaches infinity.

3.1. Proving the Existence of Positive Elements for $x_{m v}$

Theorem 1.

x_{m v}

in (13) satisfies the following condition:

\begin{matrix} 1^{T} x_{m v} = 1 . \end{matrix}

(17)

Therefore,

x_{m v}

contains at least one positive element.

Proof.

We gauge

1^{T} x_{m v}

by substituting

x_{m v}

in (13) and maneuvering (16) as follows:

\begin{matrix} 1^{T} x_{m v} = 1^{T} \frac{1}{f} Σ^{- 1} 1 = \frac{1}{f} 1^{T} Σ^{- 1} 1 = \frac{1}{f} f = 1 . \end{matrix}

The result above matches (17).

Suppose all elements of

x_{m v}

are non-positive (i.e.,

\leq 0

). Then, we infer

1^{T} x_{m v} \leq 0

. However,

1^{T} x_{m v} \leq 0

contradicts (17). Therefore, the supposition is incorrect.

x_{m v}

thus contains at least one positive element. Theorem 1 is proved. □

3.2. Proving the Existence of Positive Elements and Negative Elements for $Δ_{2}$

Theorem 2.

Δ_{2}

in (14) satisfies the following condition:

\begin{matrix} 1^{T} Δ_{2} = 0 . \end{matrix}

(18)

Therefore,

Δ_{2}

contains at least one positive element, at least one negative element, and at most

n - 2

zero elements.

Proof.

We gauge

1^{T} Δ_{2}

by substituting

Δ_{2}

in (14) and maneuvering (16) as follows:

\begin{matrix} 1^{T} Δ_{2} & = 1^{T} (Σ^{- 1} μ_{2} - \frac{c}{f} Σ^{- 1} 1) \\ = {(μ_{2}^{T} Σ^{- 1} 1)}^{T} - \frac{c}{f} 1^{T} Σ^{- 1} 1 \\ = c^{T} - c = 0 . \end{matrix}

The result above matches (18).

By Lemma 1, we deduce

Δ_{2} \neq 0

. Therefore,

Δ_{2}

contains at least one nonzero element.

Suppose that

Δ_{2}

contains only negative elements and zero elements. We then decide

1^{T} Δ_{2} < 0

. However,

1^{T} Δ_{2} < 0

contradicts (18). Therefore, the supposition is incorrect. We conclude that

Δ_{2}

contains at least one positive element.

Likewise, suppose that

Δ_{2}

contains only positive elements and zero elements. We then decide

1^{T} Δ_{2} > 0

. However,

1^{T} Δ_{2} > 0

contradicts (18). Therefore, the supposition is incorrect. We conclude that

Δ_{2}

contains at least one negative element.

By at least one positive element, at least one negative element, and n dimensions of

Δ_{2}

, we conclude that

Δ_{2}

contains at most

n - 2

zero elements. Theorem 2 is proved. □

3.3. Proving the Existence of Positive Elements and Negative Elements for $Δ_{3}$

Theorem 3.

Δ_{3}

in (15) satisfies the following condition:

\begin{matrix} 1^{T} Δ_{3} = 0 . \end{matrix}

(19)

Therefore,

Δ_{3}

contains at least one positive element, at least one negative element, and at most

n - 2

zero elements.

Proof.

We gauge

1^{T} Δ_{3}

by substituting

Δ_{3}

in (15) and maneuvering (16) as follows:

\begin{matrix} 1^{T} Δ_{3} & = 1^{T} (Σ^{- 1} μ_{3} - \frac{e}{f} Σ^{- 1} 1) \\ = {(μ_{3}^{T} Σ^{- 1} 1)}^{T} - \frac{e}{f} 1^{T} Σ^{- 1} 1 \\ = e^{T} - e = 0 . \end{matrix}

The result above matches (19).

We prove the rest part of Theorem 3 by consulting the proof of Theorem 2 and by replacing

Δ_{2}

with

Δ_{3}

. Theorem 3 is proved. □

3.4. Proving the Existence of Positive Elements and Negative Elements for $E_{(5)}$ in Limit

Theorem 4.

When

λ_{2}

approaches infinity and

λ_{3}

is fixed,

E_{(5)}

always contains some

x

with at least one positive element and at least one negative element.

Proof.

By Theorem 2, we presume the first element of

Δ_{2}

as positive and the second element as negative as follows:

\begin{matrix} Δ_{2} = {[\begin{matrix} + \\ - \\ δ_{23} \\ ⋮ \\ δ_{2 n} \end{matrix}]}_{n \times 1}, \end{matrix}

(20)

where

δ_{23} \dots δ_{2 n}

are other elements of

Δ_{2}

. We substitute (20) into (12) as follows:

\begin{matrix} x = x_{m v} + λ_{2} Δ_{2} + λ_{3} Δ_{3}, \\ [\begin{matrix} x_{1} \\ x_{2} \\ x_{3} \\ ⋮ \\ x_{n} \end{matrix}] = [\begin{matrix} x_{m v 1} \\ x_{m v 2} \\ x_{m v 3} \\ ⋮ \\ x_{m v n} \end{matrix}] + λ_{2} [\begin{matrix} + \\ - \\ δ_{23} \\ ⋮ \\ δ_{2 n} \end{matrix}] + λ_{3} [\begin{matrix} δ_{31} \\ δ_{32} \\ δ_{33} \\ ⋮ \\ δ_{3 n} \end{matrix}], \\ lim_{λ_{2} \to \infty} x_{1} = \infty lim_{λ_{2} \to \infty} x_{2} = - \infty, \end{matrix}

where

x_{1} \dots x_{n}

are elements of

x

x_{m v 1} \dots x_{m v n}

are elements of

x_{m v}

δ_{31} \dots δ_{3 n}

are elements of

Δ_{3}

. Theorem 4 holds by the two limits above. □

Similarly, we regain the

x

by exchanging

λ_{2}

and

λ_{3}

in the following theorem:

Theorem 5.

When

λ_{2}

is fixed and

λ_{3}

approaches infinity,

E_{(5)}

always contains some

x

with at least one positive element and at least one negative element.

Proof.

We prove Theorem 5 by consulting the proof of Theorem 4 and exchanging

λ_{2}

and

λ_{3}

and exchanging

Δ_{2}

and

Δ_{3}

. □

4. Proving the Possible Existence of a Non-Negative Subset of the Efficient Set of (5)

4.1. Proving the Possible Existence of a Non-Negative Subset

We prove the possible existence as follows:

Theorem 6.

E_{(5)}

can contain some

x

with all elements as non-negative.

Proof.

By Theorem 1, we presume the first element of

x_{m v}

as positive and the other elements as non-negative as follows:

\begin{matrix} x_{m v} = {[\begin{matrix} x_{m v 1} \\ x_{m v 2} \\ x_{m v 3} \\ ⋮ \\ x_{m v n} \end{matrix}]}_{n \times 1} = {[\begin{matrix} + \\ \geq 0 \\ \geq 0 \\ ⋮ \\ \geq 0 \end{matrix}]}_{n \times 1} . \end{matrix}

(21)

By Theorems 2 and 3, we presume the first element of

Δ_{2}

and

Δ_{3}

as positive, the second element as negative, and the other elements as non-negative as follows:

\begin{matrix} Δ_{2} = {[\begin{matrix} δ_{21} \\ δ_{22} \\ δ_{23} \\ ⋮ \\ δ_{2 n} \end{matrix}]}_{n \times 1} = {[\begin{matrix} + \\ - \\ \geq 0 \\ ⋮ \\ \geq 0 \end{matrix}]}_{n \times 1} Δ_{3} = {[\begin{matrix} δ_{31} \\ δ_{32} \\ δ_{33} \\ ⋮ \\ δ_{3 n} \end{matrix}]}_{n \times 1} = {[\begin{matrix} + \\ - \\ \geq 0 \\ ⋮ \\ \geq 0 \end{matrix}]}_{n \times 1} . \end{matrix}

(22)

We substitute (21) and (22) into (12) as follows:

\begin{matrix} x = x_{m v} + λ_{2} Δ_{2} + λ_{3} Δ_{3}, \\ [\begin{matrix} x_{1} \\ x_{2} \\ x_{3} \\ ⋮ \\ x_{n} \end{matrix}] = [\begin{matrix} + \\ \geq 0 \\ \geq 0 \\ ⋮ \\ \geq 0 \end{matrix}] + λ_{2} [\begin{matrix} + \\ - \\ \geq 0 \\ ⋮ \\ \geq 0 \end{matrix}] + λ_{3} [\begin{matrix} + \\ - \\ \geq 0 \\ ⋮ \\ \geq 0 \end{matrix}] = [\begin{matrix} \geq 0 \\ \geq 0 \\ \geq 0 \\ ⋮ \\ \geq 0 \end{matrix}] . \end{matrix}

Theorem 6 is proved (Our proof is based on specific presumptions. Of course, there exist numerous other situations. Our goal is to prove the possible existence. Therefore, we do not list all the situations, and the listing is nearly impossible). □

We render the following example for Theorem 6:

Example 1.

n = 3

x_{m v} = [\begin{matrix} \frac{1}{2} \\ \frac{1}{2} \\ 0 \end{matrix}]

Δ_{2} = [\begin{matrix} \frac{1}{4} \\ - \frac{1}{4} \\ 0 \end{matrix}]

Δ_{3} = [\begin{matrix} \frac{1}{4} \\ - \frac{1}{2} \\ \frac{1}{4} \end{matrix}]

λ_{2} = 1

λ_{3} = \frac{1}{2}

. By (12), we gain the non-negative

x

as follows:

x = [\begin{matrix} x_{1} \\ x_{2} \\ x_{3} \end{matrix}] = [\begin{matrix} \frac{1}{2} \\ \frac{1}{2} \\ 0 \end{matrix}] + 1 [\begin{matrix} \frac{1}{4} \\ - \frac{1}{4} \\ 0 \end{matrix}] + \frac{1}{2} [\begin{matrix} \frac{1}{4} \\ - \frac{1}{2} \\ \frac{1}{4} \end{matrix}] = [\begin{matrix} \frac{7}{8} \\ 0 \\ \frac{1}{8} \end{matrix}] .

Theorem 6 modulates the possible existence. We actually prove just one scenario for the existence. Of course, there exist other scenarios. We enlist another scenario as follows:

Example 2.

n = 3

x_{m v} = [\begin{matrix} 1 \\ - \frac{1}{2} \\ \frac{1}{2} \end{matrix}]

Δ_{2} = [\begin{matrix} - \frac{1}{9} \\ \frac{1}{3} \\ - \frac{2}{9} \end{matrix}]

Δ_{3} = [\begin{matrix} \frac{1}{9} \\ \frac{4}{9} \\ - \frac{5}{9} \end{matrix}]

λ_{2} = 1

λ_{3} = \frac{1}{2}

. By (12), we gain the non-negative

x

as follows:

x = [\begin{matrix} x_{1} \\ x_{2} \\ x_{3} \end{matrix}] = [\begin{matrix} 1 \\ - \frac{1}{2} \\ \frac{1}{2} \end{matrix}] + 1 [\begin{matrix} - \frac{1}{9} \\ \frac{1}{3} \\ - \frac{2}{9} \end{matrix}] + \frac{1}{2} [\begin{matrix} \frac{1}{9} \\ \frac{4}{9} \\ - \frac{5}{9} \end{matrix}] = [\begin{matrix} \frac{17}{18} \\ \frac{1}{18} \\ 0 \end{matrix}] .

We prove the possible non-negative subset as bounded in the following theorem:

Theorem 7.

If a non-negative subset of Theorem 6 exists, the subset is bounded.

Proof.

Suppose the subset as unbounded. By Theorem 4, some

x

of the subset contains at least one positive element and at least one negative element, when

λ_{2}

approaches infinity and

λ_{3}

is fixed. However by Theorem 6, the

x

contains all nonnegative elements. The contradiction emerges. Therefore, the supposition is incorrect. The subset is thus bounded. □

4.2. Justifying the Analytical Shortcuts to Multiple-Objective Portfolio Optimization (4)

We justify analytically resolving (5) (as analytical shortcuts) through the following aspects:

By the research of Merton [5], Qi et al. [20], Qi and Steuer [21], Qi and Steuer [17], and Salas-Molina et al. [34], scholars straightforwardly analytically resolve equality-constraint-only model series even with k objectives (7). In contrast, practically resolving (4) for $k > 3$ is still impossible.
By Theorem 6, scholars can still straightforwardly analytically resolve (5), screen the efficient set, bypass mathematical programming, and pinpoint some efficient $x$ of (6). Moreover, we will argue in Section 7 that scholars can still straightforwardly analytically resolve (7) and pinpoint some efficient $x$ of the following standard multiple-objective portfolio selection model (as an extension of (6)):

$\{\begin{matrix} min z_{1} = x^{T} Σ x, \\ max z_{2} = x^{T} μ_{2}, \\ ⋮ \\ max z_{k} = x^{T} μ_{k}, \\ subject to 1^{T} x = 1, \\ x \geq 0 . \end{matrix}$
In Section 6, we will argue that the efficient set of (5) is convex and that (5) is more consistent with capital asset pricing models than (6) is.

5. Illustrations

5.1. Checking the Existence of Non-Negative Subsets of the Efficient Sets of (5) and Inheriting the Samples

Qi [35] samples US stocks, explores five-stock portfolio selection up to 1800-stock portfolio selection in 312 problems, and reports the nondominated set’s properties for the following model:

\{\begin{matrix} min z_{1} = x^{T} Σ x, \\ max z_{2} = x^{T} μ_{2}, \\ subject to 1^{T} x = 1, \\ d^{T} x \geq t_{d}, \\ 0 \leq x \leq υ, \end{matrix}

(23)

where

d^{T} x \geq t_{d}

is a constraint for dividend yields.

υ

is an upper bound for

x

Qi [35] systematically samples, explores, and reports. A natural future direction is to inherit the samples and pragmatically check the existence of nonnegative subsets of the efficient sets of (5).

5.2. Searching a Non-Negative Subset

Because we have already theoretically proved the properties of the efficient set of (5) and the possible existence of a non-negative subset in Section 3 and Section 4, we depict a flow chart for searching a non-negative subset in Figure 2.

Overall, the searching scheme is direct. We discretize the ranges for

λ_{2}

and

λ_{3}

, assign

λ_{2}

and

λ_{3}

by nested loops, and check the nonnegativeness of all elements of

x

in (12).

5.3. Results for One Five-Stock Problem

For instance, we select the following five stocks with ticker symbols, namely MSFT, TROW, HON, EMC, and AKN from 1 January 2006 to 31 December 2010.

We sample the monthly returns without dividends (Data source: the Center for Research in Security Prices, Wharton Research Data Services, https://wrds-www.wharton.upenn.edu, 15 February 2017). We compute the sample covariance matrix of the monthly returns and assume the matrix to be

Σ

. We compute the sample mean vector of the monthly returns and assume that vector is

μ_{2}

In order to measure liquidity, we also sample the stocks’ monthly bid-ask spread ratio (

\frac{ask price - bid price}{price}

) of Bodie et al. [6] (pp. 30, 293). We compute the sample mean vector of the monthly ratios and assume the vector as

μ_{3}

. Because the ratios negatively relate to liquidity (i.e., low bid-ask spread ratios for high liquidity), we multiply

μ_{3}

by −1 for maximizing the expected liquidity. We list the parameters for (5) as follows:

\begin{matrix} Σ = [\begin{matrix} 0.0064 & 0.0055 & 0.0032 & 0.0034 & 0.0016 \\ 0.0055 & 0.0108 & 0.0055 & 0.0046 & 0.0041 \\ 0.0032 & 0.0055 & 0.0066 & 0.0037 & 0.0070 \\ 0.0034 & 0.0046 & 0.0037 & 0.0074 & 0.0035 \\ 0.0016 & 0.0041 & 0.0070 & 0.0035 & 0.0356 \end{matrix}] μ_{2} = [\begin{matrix} 0.0043 \\ 0.0144 \\ 0.0094 \\ 0.0125 \\ 0.0238 \end{matrix}] μ_{3} = [\begin{matrix} - 0.0004 \\ - 0.0010 \\ - 0.0008 \\ - 0.0012 \\ - 0.0054 \end{matrix}] . \end{matrix}

By the parameters, we exploit (13)–(15) and obtain as follows:

\begin{matrix} x_{m v} = [\begin{matrix} 0.4810 \\ - 0.1659 \\ 0.4081 \\ 0.2638 \\ 0.0131 \end{matrix}] Δ_{2} = [\begin{matrix} - 1.9092 \\ 1.6769 \\ - 1.1863 \\ 0.9105 \\ 0.5082 \end{matrix}] Δ_{3} = [\begin{matrix} 0.1126 \\ - 0.1083 \\ 0.2546 \\ - 0.0925 \\ - 0.1665 \end{matrix}] . \end{matrix}

We locate the efficient set of this problem by (12) as follows:

{x \in R^{5} | x = [\begin{matrix} 0.4810 \\ - 0.1659 \\ 0.4081 \\ 0.2638 \\ 0.0131 \end{matrix}] + λ_{2} [\begin{matrix} - 1.9092 \\ 1.6769 \\ - 1.1863 \\ 0.9105 \\ 0.5082 \end{matrix}] + λ_{3} [\begin{matrix} 0.1126 \\ - 0.1083 \\ 0.2546 \\ - 0.0925 \\ - 0.1665 \end{matrix}], λ_{2} \in [0, \infty), λ_{3} \in [0, \infty)} .

(24)

We discover one non-negative subset of (24). The subset contains at least the following

x

with

λ_{2} = 0.1010

and

λ_{3} = 0.0210

x = [\begin{matrix} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \\ x_{5} \end{matrix}] = [\begin{matrix} 0.2905 \\ 0.0012 \\ 0.2936 \\ 0.3538 \\ 0.0609 \end{matrix}] .

5.4. Results for a Batch of Ten Five-Stock Problems: Locating Six Non-Negative Subsets of Six Problems

We extend the computation style of the previous subsubsection to the ten five-stock problems of Qi [35] (pp. 1680–1681). We discover six non-negative subsets of the efficient sets of six problems out of the ten problems. We report the results in Table 1.

We report which problems (i.e., problems 01–10) are in the first row (Data available in a publicly accessible repository: The original data presented in the study are openly available in Mendeley Data at https://data.mendeley.com/datasets/ct532ygrpf/1 (accessed on 17 September 2024)). We specify which subperiods (i.e., 1 January 2006 to 31 December 2010 for problems 01 to 05; and 1 January 2011 to 31 December 2015 for problems 06 to 10) in the second row. We report which five stocks for a problem in the third row. We report a non-negative

x

for the problem with a non-negative subset and a vector of/for the problem without a non-negative subset.

5.5. Instant Computational Times by the Analyticity

By the analyticity for (12)–(15) and the straight forward computation for Figure 2, we instantly accomplish the computation.

6. Implications for Capital Asset Pricing Models

In this section, we inspect and enhance the model’s anchorage from a new aspect. We harness convex sets and prove that (5) with its efficient set as convex enjoys more alignment with capital asset pricing models than (6) does, although (6) is more practical than (5) is. Fama and French [36], Perold [37], and Levy [38] systematically review capital asset pricing models. Bodie et al. [6] (pp. 286–290) deliberate and criticize the assumptions. Their arguments typically hinge on empirical analyses rather than theoretical analyses (especially efficient set properties and convex sets).

6.1. Commencing Convex Sets for Capital Asset Pricing Models and Asserting the Market Portfolio as a Convex Combination and Thus Efficient

Lay et al. [39] (pp. 503–505) expose the following definitions and theorem for convex combinations and convex sets:

Definition 3.

A convex combination of points

x_{1} \dots x_{k} \in R^{n}

is defined as a linear combination of the form

α_{1} x_{1} + α_{2} x_{2} + \dots + α_{k} x_{k}

such that

α_{1} + α_{2} + \dots + α_{k} = 1

and

α_{1} \geq 0, \dots α_{k} \geq 0

Definition 4.

That a set S isconvexis defined as that if for any two points in S, the line segment of that two points is contained in S.

Theorem 8.

A set S is convex if and only if every convex combination of points of S lies in S.

We depict a convex set in Figure 3A. Because Bodie et al. [6] (p. 277) claim that market portfolio is “the aggregation of all investors’ risky portfolios”, we formulate the portfolio by convex combinations as follows:

Assumption 4.

The market portfolio is a convex combination of all investors’ portfolios.

For capital asset pricing models, we propose a sufficient condition for the market portfolio as efficient as follows:

Theorem 9.

For capital asset pricing models on the basis of portfolio selection (1), a sufficient condition for such an efficient market portfolio is given as follows:

The efficient set of (1) is convex;
All investors hold efficient portfolios.

The condition is sufficient but not necessary for such an efficient market portfolio. Namely, cases where the market portfolio is efficient can exist but the efficient set of (1) is nonconvex.

Proof.

The proof is a direct application of Theorem 8, because the market portfolio acts as a convex combination of all investors’ portfolios and all investors hold efficient portfolios.

Next, we provide a counter-example for the non-necessary argument. We depict an efficient set in Figure 3B. The set is nonconvex by Theorem 10 which will be introduced later. Suppose that there exist only two investors and their portfolios

x_{1}

and

x_{2}

are efficient. Because

x_{1}

and

x_{2}

lie on the same linear segment of the efficient set and the linear segment is convex, the market portfolio

x_{m}

is also efficient. □

Bodie et al. [6] (pp. 345–357), Lim and Brooks [40], and Bock and Geissel [41] review the market’s inefficiency and typically institute empirical approaches (instead of theoretical approaches).

6.2. Favoring (2) Rather than (3) for Capital Asset Pricing Models by Portfolio Selection

Markowitz and Todd [8] (p. 176) resolve (1) by parametric quadratic programming and prove the efficient set’s structure in the following theorem:

Theorem 10.

The efficient set of (1) is piecewise composed of connected linear segments.

Theorem 10 holds for (3). We depict an efficient set of (3) in Figure 3C. The set consists of two segments. However, the set is not convex, because we pick

x_{1}

and

x_{2}

on the two segments and the convex combination (as the thick red line) is not in the set.

In contrast, in Figure 3D, we promptly observe that the efficient set of (2) is convex as follows:

Theorem 11.

The efficient set of (2) in (8) is convex.

By Theorems 9 and 11, the market portfolio for (2) is efficient. Therefore, (2) enjoys more alignment with capital asset pricing models than (3) does.

6.3. Favoring (5) Rather than (6) for Capital Asset Pricing Models by Multiple-Objective Portfolio Selection

Hirschberger et al. [18] resolve (4) with three objectives and suggest that the efficient set of (4) is piecewise composed of connected linear segments. The suggestion holds for (6). We depict an efficient set of (6) in Figure 3E. The set consists of two segments. However, the set is not convex, because we pick

x_{1}

and

x_{2}

on the two segments and the convex combination (as the thick red line) is not in the set.

In contrast, in Figure 3F, we promptly observe the efficient set of (5) as convex as follows:

Theorem 12.

The efficient set of (5) in (12) is convex.

By Theorems 9 and 12, the market portfolio for (5) is efficient. Therefore, (5) enjoys more alignment with capital asset pricing models than (6) does.

7. General $k$ -Objective Portfolio Selection (7)

7.1. The General Model

We can generalize Section 3 and Section 4 as follows:

Assumption 5.

μ_{2} \dots μ_{k}

and

1

are linearly independent.

Qi and Steuer [21] concentrate on (7) and analytically derive the efficient set of (7) as follows:

\begin{matrix} E_{(7)} = {x \in R^{n} | x = x_{m v} + λ_{2} Δ_{2} + \dots + λ_{k} Δ_{k}, λ_{2} \dots λ_{k} \in [0, \infty)}, \end{matrix}

(25)

where

\begin{matrix} x_{m v} & = \frac{1}{f} Σ^{- 1} 1 as the minimum-variance portfolio of (7), \end{matrix}

(26)

\begin{matrix} Δ_{i} & = \frac{1}{2} (Σ^{- 1} μ_{i} - \frac{1}{f} (1^{T} Σ^{- 1} μ_{i}) Σ^{- 1} 1), i = 2 \dots k . \end{matrix}

(27)

E_{(7)}

is a

(k - 1)

-dimensional cone (i.e., generated by

Δ_{2} \dots Δ_{k}

and translated by

x_{m v}

). We could follow the pattern of Section 3 and Section 4 and extend Theorems 1–7.

7.2. Model Applications for (7) and Future Directions

Lo et al. [14] watch liquidity. Pedersen et al. [15] treat ESG. We extend their models as follows:

\{\begin{matrix} min z_{1} = x^{T} Σ x, \\ max z_{2} = x^{T} μ_{2}, \\ max z_{3} = x^{T} μ_{3} for expected liquidity, \\ max z_{4} = x^{T} μ_{4} for environment of ESG, \\ max z_{5} = x^{T} μ_{5} for social of ESG, \\ max z_{6} = x^{T} μ_{6} for governance of ESG, \\ subject to 1^{T} x = 1 . \end{matrix}

(28)

We follow Markowitz and Todd [8] (p. 3) and launch the following standard portfolio selection model for (28):

\{\begin{matrix} min z_{1} = x^{T} Σ x, \\ max z_{2} = x^{T} μ_{2}, \\ max z_{3} = x^{T} μ_{3} for expected liquidity, \\ max z_{4} = x^{T} μ_{4} for environment of ESG, \\ max z_{5} = x^{T} μ_{5} for social of ESG, \\ max z_{6} = x^{T} μ_{6} for governance of ESG, \\ subject to 1^{T} x = 1, \\ x \geq 0 . \end{matrix}

(29)

Few scientists have tried to crack (29) with six objectives. For instance, Goh and Yang [27] numerically cracked three-objective problems. So did Jayasekara et al. [28] and Jayasekara et al. [16]. Through this paper, scientists can try. Moreover, we can also enrich (29) and contemplate the constraints of (23) as follows:

\{\begin{matrix} min z_{1} = x^{T} Σ x, \\ max z_{2} = x^{T} μ_{2}, \\ max z_{3} = x^{T} μ_{3} for expected liquidity, \\ max z_{4} = x^{T} μ_{4} for environment of ESG, \\ max z_{5} = x^{T} μ_{5} for social of ESG, \\ max z_{6} = x^{T} μ_{6} for governance of ESG, \\ subject to 1^{T} x = 1, \\ d^{T} x \geq t_{d}, \\ 0 \leq x \leq υ . \end{matrix}

7.3. This Paper’s Computational Advantage and Disadvantage

We explicitly compare this paper’s computational fashions against those of relevant publications. We have already stated this paper’s advantage in the abstract and introduction. The disadvantage is that the possible existence of non-negative subsets may not always occur. We have already appraised and criticized the research of Goh and Yang [27], Hirschberger et al. [18], Jayasekara et al. [28], and Jayasekara et al. [16].

Moreover, Qi et al. [20], Qi and Steuer [21], and Qi and Steuer [17] extend (2) to (5), analytically derive the efficient set, and prove the efficient set’s and nondominated set’s structure. However, they do not envision expanding their derivation for non-negative constraints. In contrast, we relatively achieve the expansion in this paper.

8. Conclusions

In the multiple-objective portfolio optimization area, the efficient sets’ structure is barely known. Public-domain software for even three objectives is absent. Accordingly, we reinforce the analytical methods in the form of theorems and formulations rather than empirical analyses in the form of hypotheses. Of course, we fully respect the empirical analysis fashion. We prove the existence of positive elements and negative elements for the efficient sets. Practically, we prove that there can exist non-negative subsets. By the possible existence, researchers bypass mathematical programming, and then analytically resolve and pinpoint some non-negative efficient solutions. Moreover, we elucidate these analytical methods’ advantages for capital asset pricing models. Furthermore, we generalize for the k-objective models.

Inspired by Markowitz [2] and Sharpe [4], researchers are progressing the multiple-objective portfolio selection and the capital asset pricing models.

Author Contributions

Conceptualization, writing, and funding acquisition, Y.Q.; methodology and validation, Y.W.; formal analysis and visualization, J.H.; and resources and supervision and project administration, Y.Z. All authors have read and agreed to the published version of the manuscript.

Funding

This research received no external funding.

Data Availability Statement

Data available in a publicly accessible repository. The original data presented in the study are openly available in Mendeley Data at https://data.mendeley.com/datasets/ct532ygrpf/1 (accessed on 17 September 2024).

Acknowledgments

The authors would very much like to thank the three anonymous referees for their highly constructive comments. The first author appreciates the constant direction of Ralph E. Steuer at the University of Georgia, USA.

Conflicts of Interest

The authors declare no conflicts of interest.

Abbreviations

CAPM	Capital Asset Pricing Models

Appendix A. Comparing Methods to Resolve (4)

In addition to analytical methods (as reviewed in Section 2.2), scholars also exploit other methods below. We briefly compare the methods’ results, advantage, and disadvantage in Figure 1. The left and right parts of Figure 1 illuminate the criterion space and decision space (

R^{n}

) of (4), respectively (Because

R^{n}

is not directly visible, Figure 1 serves as intuitional guidance.).

Appendix A.1. Parametric Quadratic Programming

Markowitz [42] and Markowitz and Todd [8] employ parametric quadratic programming to crack (1). Hirschberger et al. [18] ascertain the nondominated set of (4) with

k = 3

and suggest that the set is piecewise made up by connected paraboloidal segments. For example, a set consists of two segments in the left part of Figure 1B. The lower segment is a portion of paraboloid

z_{1} = z_{2}^{2} + 5 z_{3}^{2} + 6 z_{2} z_{3} + 8 z_{2} + 8 z_{3} + 2

. The upper segment is a portion of paraboloid

z_{1} = z_{2}^{2} + 2 z_{3}^{2} + 3 z_{2} z_{3} + 4 z_{2} + 4 z_{3} + 8

. Hirschberger et al. [18] suggest that the efficient set is piecewise made up by connected linear segments. For example, a set consists of two segments in the right part of Panel B.

However, parametric quadratic programming is typically difficult to understand and practically solves (4) with

k \leq 3

only.

Appendix A.2. Repetitive Quadratic Programming

By repetitive quadratic programming, scientists easily acquire isolated points (e.g.,

z_{1}

…

z_{3}

and

x_{1}

…

x_{3}

in Figure 1C). However in discrete approximations, the points act as only incomplete information of the nondominated set and efficient set and thus can not reveal the sets’ structure (e.g., the paraboloidal structure of the nondominated set for (5)). Moreover, Qi [35] demonstrates difficulties in presetting the parameters.

Blay [43], Kritzman [44], and Statman [45] review Markowitz’s far-reaching contributions to the theory and practice of the biobjective portfolio selection.

Appendix A.3. Heuristic Methods

Heuristic methods can handle additional nonlinear formulations and thus become important provisional tools. However, heuristic methods inherently provide suboptimal solutions. For example, the methods obtain

z_{4}

…

z_{6}

and

x_{4}

…

x_{6}

in Figure 1D.

z_{4}

…

z_{6}

are lower than the corresponding optimality. Moreover, the methods also suffer from the disadvantage of repetitive quadratic programming.

Leung and Wang [46] employ neurodynamic approaches and metaheuristic optimization algorithms to solve minimax and biobjective optimization problems. Zhao et al. [47] propose a multiple populations co-evolutionary particle swarm optimization algorithm for multi-objective cardinality-constrained portfolio optimization problems. Leung and Wang [48] further present a collaborative neurodynamic optimization approach for cardinality-constrained portfolio selection.

References

Markowitz, H.M. Portfolio Selection: Efficient Diversification in Investments, 1st ed.; John Wiley & Sons: New York, NY, USA, 1959. [Google Scholar]
Markowitz, H.M. Portfolio Selection. J. Financ. 1952, 7, 77–91. [Google Scholar]
Steuer, R.E. Multiple Criteria Optimization: Theory, Computation, and Application; John Wiley & Sons: New York, NY, USA, 1986. [Google Scholar]
Sharpe, W.F. Capital asset prices: A theory of market equilibrium. J. Financ. 1964, 19, 425–442. [Google Scholar]
Merton, R.C. An analytical derivation of the efficient portfolio frontier. J. Financ. Quant. Anal. 1972, 7, 1851–1872. [Google Scholar] [CrossRef]
Bodie, Z.; Kane, A.; Marcus, A.J. Investments, 12th ed.; McGraw-Hill Education: New York, NY, USA, 2021. [Google Scholar]
Brennan, T.J.; Lo, A.W. Impossible frontiers. Manag. Sci. 2010, 56, 905–923. [Google Scholar] [CrossRef]
Markowitz, H.M.; Todd, G.P. Mean-Variance Analysis in Portfolio Choice and Capital Markets; Frank J. Fabozzi Associates: New Hope, PA, USA, 2000. [Google Scholar]
Markowitz, H.M. Foundations of portfolio selection. J. Financ. 1991, 46, 469–477. [Google Scholar] [CrossRef]
Sharpe, W.F. Optimal Portfolios Without Bounds on Holdings, Online Lecture; Graduate School of Business, Stanford University: Stanford, CA, USA, 2001; Available online: https://web.stanford.edu/~wfsharpe/mia/opt/mia_opt2.htm (accessed on 1 September 2024).
Fama, E.F. Multifactor portfolio efficiency and multifactor asset pricing. J. Financ. Quant. Anal. 1996, 31, 441–465. [Google Scholar] [CrossRef]
Cochrane, J.H. Presidential address: Discount rates. J. Financ. 2011, 66, 1047–1108. [Google Scholar] [CrossRef]
Harvey, C.R.; Siddique, A. Conditional skewness in asset pricing tests. J. Financ. 2000, 55, 1263–1296. [Google Scholar] [CrossRef]
Lo, A.W.; Petrov, C.; Wierzbicki, M. It’s 11pm—Do you know where your liquidity is? The mean-variance-liquidity frontier. J. Investig. Manag. 2003, 1, 55–93. [Google Scholar]
Pedersen, L.H.; Fitzgibbons, S.; Pomorski, L. Responsible investing: The ESG-efficient frontier. J. Financ. Econ. 2021, 142, 572–597. [Google Scholar] [CrossRef]
Jayasekara, P.L.; Pangia, A.C.; Wiecek, M.M. On solving parametric multiobjective quadratic programs with parameters in general locations. Ann. Oper. Res. 2023, 320, 123–172. [Google Scholar] [CrossRef]
Qi, Y.; Steuer, R.E. An analytical derivation of properly efficient sets in multi-objective portfolio selection. Ann. Oper. Res. 2024. forthcoming. [Google Scholar] [CrossRef]
Hirschberger, M.; Steuer, R.E.; Utz, S.; Wimmer, M.; Qi, Y. Computing the nondominated surface in tri-criterion portfolio selection. Oper. Res. 2013, 61, 169–183. [Google Scholar] [CrossRef]
Utz, S.; Wimmer, M.; Steuer, R.E. Tri-criterion modeling for constructing more-sustainable mutual funds. Eur. J. Oper. Res. 2015, 246, 331–338. [Google Scholar] [CrossRef]
Qi, Y.; Steuer, R.E.; Wimmer, M. An analytical derivation of the efficient surface in portfolio selection with three criteria. Ann. Oper. Res. 2017, 251, 161–177. [Google Scholar] [CrossRef]
Qi, Y.; Steuer, R.E. On the analytical derivation of efficient sets in quad-and-higher criterion portfolio selection. Ann. Oper. Res. 2020, 293, 521–538. [Google Scholar] [CrossRef]
Utz, S.; Steuer, R.E. Empirical analysis of the trade-offs among risk, return, and climate risk in multi-criteria portfolio optimization. Ann. Oper. Res. 2024. forthcoming. [Google Scholar] [CrossRef]
Aouni, B.; Doumpos, M.; Pérez-Gladish, B.; Steuer, R.E. On the increasing importance of multiple criteria decision aid methods for portfolio selection. J. Oper. Res. Soc. 2018, 69, 1525–1542. [Google Scholar] [CrossRef]
La Torre, D.; Boubaker, S.; Pérez-Gladish, B.; Zopounidis, C. Preface to the special issue on multidimensional finance, insurance, and investment. Int. Trans. Oper. Res. 2023, 30, 2137–2138. [Google Scholar] [CrossRef]
Bank, B.; Guddat, J.; Klatte, D.; Kummer, B.; Tammer, K. Non-Linear Parametric Optimization; Birkhäuser: Basel, Switzerland, 1983. [Google Scholar]
Best, M.J. An algorithm for the solution of the parametric quadratic programming problem. In Proceedings of the Applied Mathematics and Parallel Computing: Festschrift for Klaus Ritter; Fischer, H., Riedmüller, B., Schäffler, S., Eds.; Physica: Heidelberg, Germany, 1996; pp. 57–76. [Google Scholar]
Goh, C.; Yang, X. Analytic efficient solution set for multi-criteria quadratic programs. Eur. J. Oper. Res. 1996, 92, 166–181. [Google Scholar] [CrossRef]
Jayasekara, P.L.; Adelgren, N.; Wiecek, M.M. On convex multiobjective programs with application to portfolio optimization. J. Multi-Criteria Decis. Anal. 2019, 27, 189–202. [Google Scholar] [CrossRef]
Utz, S.; Wimmer, M.; Hirschberger, M.; Steuer, R.E. Tri-criterion inverse portfolio optimization with application to socially responsible mutual funds. Eur. J. Oper. Res. 2014, 234, 491–498. [Google Scholar] [CrossRef]
Ingersoll, J. Multidimensional security pricing. J. Financ. Quant. Anal. 1975, 10, 785–798. [Google Scholar] [CrossRef]
Qi, Y. Classifying the minimum-variance surface of multiple-objective portfolio selection for capital asset pricing models. Ann. Oper. Res. 2022, 311, 1203–1227. [Google Scholar] [CrossRef]
Qi, Y.; Qi, Z.; Zhang, S.; Wang, Y. Discovering zero-covariance-portfolio curves for capital asset pricing models of multiple-objective portfolio selection. J. Ind. Manag. Optim. 2024. forthcoming. [Google Scholar] [CrossRef]
Huang, C.; Litzenberger, R.H. Foundations for Financial Economics; Prentice Hall: Englewood Cliffs, NJ, USA, 1988. [Google Scholar]
Salas-Molina, F.; Pla-Santamaria, D.; Rodriguez-Aguilar, J.A. An analytic derivation of the efficient frontier in biobjective cash management and its implications for policies. Ann. Oper. Res. 2023, 328, 1523–1536. [Google Scholar] [CrossRef]
Qi, Y. Parametrically computing efficient frontiers of portfolio selection and reporting and utilizing the piecewise-segment structure. J. Oper. Res. Soc. 2020, 71, 1675–1690. [Google Scholar] [CrossRef]
Fama, E.F.; French, K.R. The Capital Asset Pricing Model: Theory and Evidence. J. Econ. Perspect. 2004, 18, 25–46. [Google Scholar] [CrossRef]
Perold, A.F. The Capital Asset Pricing Model. J. Econ. Perspect. 2004, 18, 3–24. [Google Scholar] [CrossRef]
Levy, H. The CAPM is Alive and Well: A Review and Synthesis. Eur. Financ. Manag. 2010, 16, 43–71. [Google Scholar] [CrossRef]
Lay, D.C.; Lay, S.R.; McDonald, J.J. Linear Algebra and Its Applications, 6th ed.; Pearson Education Limited: London, UK, 2022. [Google Scholar]
Lim, K.P.; Brooks, R. The evolution of stock market efficiency over time: A survey of the empirical literature. J. Econ. Surv. 2011, 25, 69–108. [Google Scholar] [CrossRef]
Bock, J.; Geissel, S. Evolution of stock market efficiency in Europe: Evidence from measuring periods of inefficiency. Financ. Res. Lett. 2024, 62, 1–8. [Google Scholar] [CrossRef]
Markowitz, H.M. The optimization of a quadratic function subject to linear constraints. Nav. Res. Logist. Q. 1956, 3, 111–133. [Google Scholar] [CrossRef]
Blay, K.A. From portfolio selection to portfolio choice: Remembering Harry Markowitz. J. Portf. Manag. 2024, 50, 45–58. [Google Scholar] [CrossRef]
Kritzman, M. A Tribute to Harry Markowitz. J. Portf. Manag. 2024, 50, 39–44. [Google Scholar] [CrossRef]
Statman, M. Harry Markowitz’s two intellectual children: Mean–variance and behavioral portfolio theories. J. Portf. Manag. 2024, 50, 24–29. [Google Scholar] [CrossRef]
Leung, M.F.; Wang, J. Minimax and biobjective portfolio selection based on collaborative neurodynamic optimization. IEEE Trans. Neural Netw. Learn. Syst. 2021, 32, 2825–2936. [Google Scholar] [CrossRef] [PubMed]
Zhao, H.; Chen, Z.G.; Zhan, Z.H.; Kwong, S.; Zhang, J. Multiple populations co-evolutionary particle swarm optimization for multi-objective cardinality constrained portfolio optimization problem. Neurocomputing 2021, 430, 58–70. [Google Scholar] [CrossRef]
Leung, M.F.; Wang, J. Cardinality-constrained portfolio selection based on collaborative neurodynamic optimization. Neural Netw. 2022, 145, 68–79. [Google Scholar] [CrossRef] [PubMed]

Figure 1. Comparing methods to resolve (4) with the criterion space in the left part and decision space respectively in the right part.

Figure 2. A flow chart of searching a non-negative subset.

Figure 3. A convex set and efficient sets of (3), (2), (6) and (5) in

R^{n}

Figure 3. A convex set and efficient sets of (3), (2), (6) and (5) in

R^{n}

Table 1. Discovering six non-negative subsets of the efficient sets of six problems out of the ten problems.

Problems		01	02	03	04	05	06	07	08	09	10
Subperiods		1 January 2006 to 31 December 2010					1 January 2011 to 31 December 2015
Ticker symbols		$[\begin{matrix} A E P I \\ J J S F \\ P L X S \\ R M C F \\ O R C L \end{matrix}]$	$[\begin{matrix} M S F T \\ T R O W \\ H O N \\ E M C \\ A K N \end{matrix}]$	$[\begin{matrix} T E C D \\ R G E N \\ F U N D \\ B W I N B \\ A V A N \end{matrix}]$	$[\begin{matrix} S I G M \\ C Y \\ B C P C \\ A M A G \\ T C B \end{matrix}]$	$[\begin{matrix} A S T E \\ N A V G \\ W E R N \\ S K Y W \\ R E F R \end{matrix}]$	$[\begin{matrix} A E P I \\ J J S F \\ P L X S \\ R M C F \\ O R C L \end{matrix}]$	$[\begin{matrix} M S F T \\ T R O W \\ H O N \\ E M C \\ A K N \end{matrix}]$	$[\begin{matrix} T E C D \\ R G E N \\ F U N D \\ B W I N B \\ A V A N \end{matrix}]$	$[\begin{matrix} S I G M \\ C Y \\ B C P C \\ A M A G \\ T C B \end{matrix}]$	$[\begin{matrix} A S T E \\ N A V G \\ W E R N \\ S K Y W \\ R E F R \end{matrix}]$
$x$	$[\begin{matrix} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \\ x_{5} \end{matrix}]$	$[\begin{matrix} / \\ / \\ / \\ / \\ / \end{matrix}]$	$[\begin{matrix} 0.2905 \\ 0.0012 \\ 0.2936 \\ 0.3538 \\ 0.0609 \end{matrix}]$	$[\begin{matrix} 0.0590 \\ 0.1816 \\ 0.3177 \\ 0.4122 \\ 0.0295 \end{matrix}]$	$[\begin{matrix} 0.0493 \\ 0.1770 \\ 0.3780 \\ 0.0667 \\ 0.3290 \end{matrix}]$	$[\begin{matrix} 0.0515 \\ 0.5270 \\ 0.3981 \\ 0.0025 \\ 0.0209 \end{matrix}]$	$[\begin{matrix} 0.0342 \\ 0.5956 \\ 0.0212 \\ 0.2127 \\ 0.1363 \end{matrix}]$	$[\begin{matrix} / \\ / \\ / \\ / \\ / \end{matrix}]$	$[\begin{matrix} / \\ / \\ / \\ / \\ / \end{matrix}]$	$[\begin{matrix} 0.0218 \\ 0.0186 \\ 0.3079 \\ 0.0658 \\ 0.5859 \end{matrix}]$	$[\begin{matrix} / \\ / \\ / \\ / \\ / \end{matrix}]$

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Qi, Y.; Wang, Y.; Huang, J.; Zhang, Y. Analytical Shortcuts to Multiple-Objective Portfolio Optimization: Investigating the Non-Negativeness of Portfolio Weight Vectors of Equality-Constraint-Only Models and Implications for Capital Asset Pricing Models. Mathematics 2024, 12, 3946. https://doi.org/10.3390/math12243946

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Qi Y, Wang Y, Huang J, Zhang Y. Analytical Shortcuts to Multiple-Objective Portfolio Optimization: Investigating the Non-Negativeness of Portfolio Weight Vectors of Equality-Constraint-Only Models and Implications for Capital Asset Pricing Models. Mathematics. 2024; 12(24):3946. https://doi.org/10.3390/math12243946

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Qi, Yue, Yue Wang, Jianing Huang, and Yushu Zhang. 2024. "Analytical Shortcuts to Multiple-Objective Portfolio Optimization: Investigating the Non-Negativeness of Portfolio Weight Vectors of Equality-Constraint-Only Models and Implications for Capital Asset Pricing Models" Mathematics 12, no. 24: 3946. https://doi.org/10.3390/math12243946

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Qi, Y., Wang, Y., Huang, J., & Zhang, Y. (2024). Analytical Shortcuts to Multiple-Objective Portfolio Optimization: Investigating the Non-Negativeness of Portfolio Weight Vectors of Equality-Constraint-Only Models and Implications for Capital Asset Pricing Models. Mathematics, 12(24), 3946. https://doi.org/10.3390/math12243946

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Analytical Shortcuts to Multiple-Objective Portfolio Optimization: Investigating the Non-Negativeness of Portfolio Weight Vectors of Equality-Constraint-Only Models and Implications for Capital Asset Pricing Models

Abstract

1. Introduction

1.1. Portfolio Selection and Capital Asset Pricing Models

1.2. Rise of Multiple-Objective Portfolio Selection

1.3. Research Limitations of Multiple-Objective Portfolio Selection (4)

1.3.1. Research Limitations in Proving the Structure of the Optimal-Solution Sets

1.3.2. Research Limitations in Investigating the Non-Negativeness of the Optimal-Solution Sets

1.3.3. Research Limitations in Implicating the Capital Asset Pricing Models

1.4. Originality of This Paper: Investigating the Non-Negativeness, Justifying the Analytical Shortcuts, Implicating the Capital Asset Pricing Models for (5), and Generalizing

1.4.1. Investigating the Possible Existence of a Non-Negative Subset of the Optimal-Solution Set in the Absence of Public-Domain Software

1.4.2. Consequently Justifying the Analytical Shortcuts for Optimizing (4)

1.4.3. Implicating the Capital Asset Pricing Models

1.4.4. Generalizing for k-Objective Models

1.5. Paper Structure

2. Theoretical Background: Multiple-Objective Optimization and Portfolio Optimization

2.1. Multiple-Objective Optimization

2.2. Portfolio Optimization by Analytical Methods

2.2.1. Analytically Resolving (2)

2.2.2. Analytically Resolving (5)

2.3. Portfolio Optimization by Other Methods

3. Proving Properties of the Efficient Set of (5): The Existence of Positive Elements and Negative Elements for Δ 2 , Δ 3 , and x mv and for E ( 5 ) in Limit

3.1. Proving the Existence of Positive Elements for x m v

3.2. Proving the Existence of Positive Elements and Negative Elements for Δ 2

3.3. Proving the Existence of Positive Elements and Negative Elements for Δ 3

3.4. Proving the Existence of Positive Elements and Negative Elements for E ( 5 ) in Limit

4. Proving the Possible Existence of a Non-Negative Subset of the Efficient Set of (5)

4.1. Proving the Possible Existence of a Non-Negative Subset

4.2. Justifying the Analytical Shortcuts to Multiple-Objective Portfolio Optimization (4)

5. Illustrations

5.1. Checking the Existence of Non-Negative Subsets of the Efficient Sets of (5) and Inheriting the Samples

5.2. Searching a Non-Negative Subset

5.3. Results for One Five-Stock Problem

5.4. Results for a Batch of Ten Five-Stock Problems: Locating Six Non-Negative Subsets of Six Problems

5.5. Instant Computational Times by the Analyticity

6. Implications for Capital Asset Pricing Models

6.1. Commencing Convex Sets for Capital Asset Pricing Models and Asserting the Market Portfolio as a Convex Combination and Thus Efficient

6.2. Favoring (2) Rather than (3) for Capital Asset Pricing Models by Portfolio Selection

6.3. Favoring (5) Rather than (6) for Capital Asset Pricing Models by Multiple-Objective Portfolio Selection

7. General k -Objective Portfolio Selection (7)

7.1. The General Model

7.2. Model Applications for (7) and Future Directions

7.3. This Paper’s Computational Advantage and Disadvantage

8. Conclusions

Author Contributions

Funding

Data Availability Statement

Acknowledgments

Conflicts of Interest

Abbreviations

Appendix A. Comparing Methods to Resolve (4)

Appendix A.1. Parametric Quadratic Programming

Appendix A.2. Repetitive Quadratic Programming

Appendix A.3. Heuristic Methods

References

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3. Proving Properties of the Efficient Set of (5): The Existence of Positive Elements and Negative Elements for $Δ_{2}$ , $Δ_{3}$ , and $x_{mv}$ and for $E_{(5)}$ in Limit

3.1. Proving the Existence of Positive Elements for $x_{m v}$

3.2. Proving the Existence of Positive Elements and Negative Elements for $Δ_{2}$

3.3. Proving the Existence of Positive Elements and Negative Elements for $Δ_{3}$

3.4. Proving the Existence of Positive Elements and Negative Elements for $E_{(5)}$ in Limit

7. General $k$ -Objective Portfolio Selection (7)