Non-Negative Equilibrium Prices and Market Portfolio Under Minimax Diversification with Non-Homogeneous Investors

Abstract

Market equilibrium is characterized by a state wherein aggregate demand equals aggregate supply for all assets, a condition arising from consumers maximizing utility within budget constraints and producers maximizing profits. This paper investigates a financial market populated by non-homogeneous investors who may employ heterogeneous deviation measures to formulate their risk functions. By integrating the minimax risk diversification principle with the framework of individual utility maximization, we analytically derive the master fund for each investor. Furthermore, we establish the necessary and sufficient conditions for the existence of a unique non-negative equilibrium price system for risky assets and provide its explicit formula. A key finding is that the market portfolio is a convex combination of all investors’ master funds.

1. Introduction

Optimizing resource allocation constitutes a fundamental function of market economies. Well-structured portfolios are believed to enhance the efficiency of capital utilization and promote the healthy and stable development of capital market. The problem of portfolio selection, regarded as one of the core topics in modern finance, involves the study of providing theoretical foundations and methodologies for scientific decision-making to enterprises and individual investors. This research is considered to be beneficial for enabling rational investment choices in complex and dynamic financial environments. From a mathematical perspective, the portfolio selection problem can be formulated as a constrained optimization problem, with the measurement of investment risk for modeling purposes being recognized as one of its central aspects.

Markowitz [1] utilized two statistical measures, mean and variance, to quantify returns and risks of various risky assets, thereby providing a solution to the fundamental question of how investors should allocate funds among available investment options. When a risk-free asset is available, the mean-variance Capital Asset Pricing Model (CAPM), developed by Sharpe [2], Lintner [3], and Mossin [4], emerges as an equilibrium model. As various alternative risk measures have been progressively introduced, equilibrium capital asset pricing models based on these distinct risk metrics have also been developed. Examples include the mean-semivariance CAPM (Hogan and Warren [5]), the mean-lower partial moment CAPM (Bawa and Lindenberg [6]), the mean-absolute deviation CAPM (Konno and Shirakawa [7]), the mean-reward risk CAPM (Giorgi and Post [8]), and the mean-general deviation measure CAPM (Rockafellar et al. [9]).

Konno and Shirakawa [10] observe that much of the work in the field of equilibrium market capital asset pricing models has focused on the separation theorem and “beta”-like relationships. However, they suggest that explicit formulas for equilibrium prices and the conditions under which they exist hold significant importance for understanding the equilibrium and stability of capital market. Deng et al. [11] developed a portfolio selection model under the assumption of varying expected returns within an estimation interval and a fixed covariance matrix. Within this framework, a sufficient condition for the existence and uniqueness of a non-negative equilibrium price system is provided and an explicit formula for such a price system is obtained. Farahat and Perakis [12] demonstrated the existence and uniqueness of market equilibrium prices for consumers with quadratic utility functions, providing insights into how these prices respond to market structure and cost changes.

Inspired by the market equilibrium model with non-homogeneous investors in Konno and Suzuki [13], this paper extends the single-investor portfolio selection framework of Meng et al. [14]. Specifically, we generalize their model, which considers only risky assets, to a comprehensive market equilibrium model. This extension incorporates a risk-free asset and analyzes a market composed of non-homogeneous investors, thereby addressing the methodological limitation of prior studies of Meng et al. [14]. In this study, investors can use different derivation measures to construct their own risk functions. Applying the minimax risk diversification principle alongside the principle of maximizing individual utility, the master fund for each investor is analytically derived. Although short selling of risky assets is not restricted, our rigorous demonstration shows that market equilibrium is achievable only if the risk-free rate is below the expected return rates of all risky assets. Consequently, in this scenario, rational investors refrain from short selling risky assets. Additionally, the necessary and sufficient condition for the existence of a unique non-negative equilibrium price system for risky assets is provided, along with its explicit formula. Finally, we demonstrate that the market portfolio constitutes a convex combination of these master funds, establishing a clear link between individual- and market-level optimization.

The rest of this paper is organized as follows: Section 2 outlines the fundamental assumptions of the capital asset market while incorporating the risk functions utilized by investors. Section 3 is dedicated to examining the optimization problem specific to individual investors. Section 4 provides the proof for the existence of a non-negative equilibrium price system. Section 5 concludes this paper.

For a random variable X, we denote by $E\left(X\right)$ the mathematical expectation of X throughout the paper. The space ${\mathcal{L}}^2(\mathrm{\Omega })={\mathcal{L}}^2(\mathrm{\Omega },\mathcal{F},\mathcal{P})$ is the set of all random variables defined on the probability space $(\mathrm{\Omega },\mathcal{F},\mathcal{P})$ for which the second moment is finite, i.e., as follows:

$${\mathcal{L}}^2(\mathrm{\Omega }):=\left\{X:\mathrm{\Omega }\to \mathcal{R}\mid E\left[\right|X{|}^2]<\infty \right\}.$$

In particular, the space ${\mathcal{L}}^2(\mathrm{\Omega })$ contains all constant random variables, $X\equiv \mathcal{C}$. The letter $\mathcal{C}$ will always stand for a constant in the real numbers $\mathcal{R}$, and any (in)equalities between random variables are to be viewed in the sense of holding almost surely. We adopt the notion that

$$X=X_{+}-X_{-}\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h} X_{+}=max\{0,X\},X_{-}=max\{0,-X\}.$$

2. The Capital Asset Market

We consider in this paper a capital market consisting of m investors $I_i(i=1,2,\dots ,m)$ with n risky assets $S_j(j=1,2,\dots ,n)$ and one risk-less asset $S_0$. The following notations will be employed:

$r_0$   The risk-free rate;

$R_j$   The random return rate of $S_j(j=1,2,\dots ,n)$ per period;

$r_j$   The expected return rate of $S_j(j=1,2,\dots ,n)$ per period;

$z_{ij}^0$   the units of asset $S_j(j=0,1,2,\dots ,n)$ owned by $I_i$ before the transaction;

$z_{ij}$   The units of asset $S_j(j=0,1,2,\dots ,n)$ owned by $I_i$ after the transaction;

$p_j$   The unit price of $S_j(j=0,1,2,\dots ,n)$ at the time of transaction.

In order to keep the statement simple, denote

$$R={(R_1,R_2,\dots ,R_n)}^T,r={(r_1,r_2,\dots ,r_n)}^T,e={(1,1,\dots ,1)}^T\in {\mathcal{R}}^n.$$

As usual, we will assume the market satisfies several conditions commonly imposed in financial economics.

Assumption 1.

No casts and no taxes are associated with transactions. All assets are infinitely divisible and have no limitation on short-sales.

Assumption 2.

All investors participate in the market with non-negative initial endowments. For each investor $I_i$, the initial holdings satisfy

$$z_{ij}^0\ge 0,j=0,1,2,\dots ,n,$$

with at least one $j_0\in 0,1,2,\dots ,n$ such that $z_{i{j}_0}^0>0$.

Assumption 3.

The total supply of any risky asset $S_j$ in the market is positive, i.e., as follows:

$$z_j^0:=\sum _{i=1}^m{z}_{ij}^0>0,j=1,\dots ,n$$

Assumption 4.

$p_0=1$ and $p_j\ge 0$ for all $j=1,\dots ,n$.

Assumption A4 imposes that the value of investor $I_i$’s initial endowments is

$$w_i^0:=z_{i0}^0+\sum _{j=1}^n{p}_j{z}_{ij}^0.$$

(1)

Because no money is added into or withdrawn from the market during the transaction, the self-financing condition is satisfied, i.e.,as follows:

$$z_{i0}+\sum _{j=1}^n{p}_j{z}_{ij}=w_i^0,i=1,\dots ,m.$$

(2)

Moreover, market clearing implies

$$\sum _{i=1}^m{z}_{ij}=z_j^0,j=1,\dots ,n.$$

(3)

Remark 1.

$w_i^0>0$ by Assumptions 2 and 4.

Let $x_{ij}$ be the amount invested in risky asset $S_j$

$$x_{ij}:={\displaystyle \frac{p_j{z}_{ij}}{w_i^0}},j=0,1,\dots ,n.$$

(4)

Then, a risky portfolio for investor $I_i$ corresponds to a vector $x_i={(x_{i1},\dots ,x_{in})}^T\in {\mathcal{R}}^n$.

Remark 2.

According to the self-financing condition (2), it can be seen that

$$x_{i0}+e^T{x}_i=1.$$

Meanwhile, $x_{i0}{r}_0+R^T{x}_i$ is the random return rate of investor $I_i$, and $x_{i0}{r}_0+r^T{x}_j$ is the expected return rate of investor $I_i$.

Subsequently, we will incorporate the risk functions used by investors. For a random variable X, we use $\mathcal{D}\left(X\right)$ to denote the general deviation measure of X in the sense of [15]

Definition 1.

A deviation measure is a function $\mathcal{D}:{\mathcal{L}}^2(\mathrm{\Omega })\to [0,\infty ]$ that satisfies

(D1) 

$\mathcal{D}(X+C)=\mathcal{D}\left(X\right)$ for all X and constants C;

(D2) 

$\mathcal{D}\left(0\right)=0$ and $\mathcal{D}\left(\lambda X\right)=\lambda \mathcal{D}\left(X\right)$ for all X and $\lambda >0$;

(D3) 

$\mathcal{D}(X+X^{\prime })\le \mathcal{D}\left(X\right)+\mathcal{D}\left(X^{\prime }\right)$ for all X and $X^{\prime }$;

(D4) 

$\mathcal{D}\left(X\right)\ge 0$ for all X, with $\mathcal{D}\left(X\right)>0$ for nonconstant X.

Axiom $\left(\mathbf{D}\mathbf{1}\right)$ is equivalent to $\mathcal{D}\left(X\right)=\mathcal{D}(X-E(X\left)\right)$ for all X. Axiom $\left(\mathbf{D}\mathbf{2}\right)$ is positive homogeneity. The combination of axioms $\left(\mathbf{D}\mathbf{2}\right)$ and $\left(\mathbf{D}\mathbf{3}\right)$ is the property known as sublinearity. It implies that $\mathcal{D}$ is a convex functional on ${\mathcal{L}}^2(\mathrm{\Omega })$. Axiom $\left(\mathbf{D}\mathbf{4}\right)$ means $\mathcal{D}\left(X\right)=0$ for constant X, whereas $\mathcal{D}\left(X\right)>0$ for nonconstant X. Some commonly used deviation measures are as follows:

(a) 

The standard deviation $\mathcal{D}\left(X\right):=\sqrt{E{(E\left(X\right)-X)}^2}$;

(b) 

The absolute deviation $\mathcal{D}\left(X\right):=E\left(\right|X-E\left(X\right)\left|\right)$;

(c) 

The lower semi-absolute deviation $\mathcal{D}\left(X\right):=E{(E\left(X\right)-X)}_{+}$;

(d) 

The standard lower semi-deviation $\mathcal{D}\left(X\right):=\sqrt{E{(E\left(X\right)-X)}_{+}^2}$;

(e) 

The CVaR-deviation $\mathcal{D}\left(X\right):={\mathrm{CVaR}}_{\alpha }(X-E\left(X\right))$ for some $\alpha \in (0,1)$.

More derivation measures and their properties can be found in Rockafellar et al. [15].

Consider a deviation measure ${\mathcal{D}}_i$ chosen by investor $I_i$. It is natural to use ${\mathcal{D}}_i\left(x_{i0}{r}_0+R^T{x}_i\right)$ as a risk metric for the portfolio $x_i$, following the approach of Rockafellar et al. [16]. However, motivated by Cai et al. [17] and Meng et al. [14], we employ an $l_{\infty }$ norm-based risk function for the component risks. Specifically, throughout this paper, the risk function for investor $I_i$’s portfolio $x_i$ is given by

$$\begin{array}{cc}{\omega }_{{\mathcal{D}}_i}\left(x_i\right)\hfill & :=\left|\right|{\mathcal{D}}_i\left(x_{i0}{r}_0\right),{\mathcal{D}}_i\left(x_{i1}{R}_1\right),{\mathcal{D}}_i\left(x_{i2}{R}_2\right),\dots ,{\mathcal{D}}_i\left(x_{in}{R}_n\right){\left|\right|}_{\infty }\hfill \\ \hfill & {\displaystyle =\underset{1\le j\le n}{max}{\mathcal{D}}_i\left(x_{ij}{R}_j\right).}\hfill \end{array}$$

(5)

This formulation explicitly manages concentration risk by focusing on the largest deviation of any single position. Consequently, an investor minimizing this function is motivated to diversify holdings to mitigate the impact of a potential adverse outcome in any one asset. This is the general $l_{\infty }$ risk function introduced in Meng et al. [14]. It is worth noting that the deviation measure ${\mathcal{D}}_t$ chosen by investor $I_t$ may differ from the deviation measure ${\mathcal{D}}_i$ chosen by investor $I_i$. As a result, for the same portfolio, $y\in {\mathcal{R}}^n$, ${\omega }_{{\mathcal{D}}_t}\left(y\right)$ might not equal ${\omega }_{{\mathcal{D}}_i}\left(y\right)$. This indicates that investors in the market are non-homogeneous.

Remark 3.

Due to axioms $\left(\mathbf{D}\mathbf{3}\right)$ and $\left(\mathbf{D}\mathbf{4}\right)$ of ${\mathcal{D}}_i$, we have

$${\mathcal{D}}_i\left(x_{i0}{r}_0+R^T{x}_i\right)\le {\mathcal{D}}_i\left(x_{i0}{r}_0\right)+\sum _{j=1}^n{\mathcal{D}}_i\left(x_{ij}{R}_j\right)\le n{\omega }_{{\mathcal{D}}_i}\left(x_i\right).$$

While ${\omega }_{{\mathcal{D}}_i}\left(x_i\right)$ directly manages concentration risk by focusing on the single largest position, the above inequality shows it also provides a reliable bound on the total portfolio risk. Consequently, minimizing ${\omega }_{{\mathcal{D}}_i}\left(x_i\right)$ not only controls for worst-case exposure but also ensures the overall portfolio risk remains small, all while preserving the analytical tractability essential for deriving explicit equilibrium solutions.

Remark 4.

Due to the positive homogeneity of ${\mathcal{D}}_i$, we have

$${\mathcal{D}}_i\left(x_{ij}{R}_j\right)=\left\{\begin{array}{cc}{x}_{ij}{\mathcal{D}}_i\left(R_j\right)\hfill & \hfill if x_{ij}\ge 0,\\ -x_{ij}{\mathcal{D}}_i(-R_j)\hfill & \hfill if x_{ij}<0.\end{array}\right.$$

Thus, the general $l_{\infty }$ risk function of investor $I_i$ can be rewritten as

$${\displaystyle {\omega }_{{\mathcal{D}}_i}\left(x_i\right)=\underset{1\le j\le n}{max}max\left\{x_{ij}{\underline{q}}_{ij},-x_{ij}{\overline{q}}_{ij}\right\},}$$

where ${\underline{q}}_{ij}:={\mathcal{D}}_i\left(R_j\right)$ and ${\overline{q}}_{ij}:={\mathcal{D}}_i(-R_j)$. That is, this risk function permits the independent assessment of both upside and downside risks, all while preserving a linear framework, thereby facilitating the resolution of portfolio optimization models.

The remainder of this section will be dedicated to showcasing some of the beneficial properties of ${\omega }_{{\mathcal{D}}_i}\left(x_i\right)$. Nevertheless, beforehand, it is imperative to delineate two foundational assumptions upon which this paper is predicated.

Assumption 5.

$r_1\le r_2\le \dots \le r_{n-1}\le r_n$ and these expected return rates are not all the same.

Remark 5.

Following Meng et al. [14], it is necessary to use Assumption 5 to define the trading strategy of shorting lower-return assets to fund higher-return ones. The model is robust to estimation errors, provided they do not alter the underlying ranking of the assets.

Assumption 6.

$0<{\underline{q}}_{ij}<\infty$ and $0<{\overline{q}}_{ij}<\infty$ for all $i=1,2,\dots ,m$ and $j=1,2,\dots ,n$.

Remark 6.

Assumption 6 ensures the risk function is well-defined and excludes pathological cases, such as non-finite deviation measures like the lower-range deviation, which could arise from heavy-tailed distributions.

Assumption 5 ensures the richness of price–gain combinations as shown in the following proposition.

Proposition 1.

For every choice of ${(\xi ,\mu )}^T\in {\mathcal{R}}^2$, there exists a portfolio $x_i\in {\mathcal{R}}^n$ with price $e^T{x}_i=\xi$ and expected return $r^T{x}_i=\mu$.

Proof. 

Assumption 5 indicates that the n-dimensional vector e is linearly independent of r. Therefore, the coefficient matrix of the linear equation system

$$\left(\begin{array}{c}{e}^T\\ r^T\end{array}\right)x_i=\left(\begin{array}{c}\xi \\ \mu \end{array}\right),$$

(6)

has full rank rows, and then $rank\left(\begin{array}{c}{e}^T\\ r^T\end{array}\right)=rank\left(\begin{array}{cc}\begin{array}{c}{e}^T\\ r^T\end{array}& \begin{array}{c}\xi \\ \mu \end{array}\end{array}\right)$. This indicates that the linear equation system (6) must have a solution. □

Combining Assumption 6 and Definition 1, we will show some beneficial properties of ${\omega }_{{\mathcal{D}}_i}\left(x_i\right)$.

Proposition 2.

The risk function ${\omega }_{{\mathcal{D}}_i}$ is finite and convex on ${\mathcal{R}}^n$ with the following properties:

(a)

${\omega }_{{\mathcal{D}}_i}\left(0\right)=0$ whereas ${\omega }_{{\mathcal{D}}_i}\left(x_i\right)>0$ for $x_i\ne 0$.

(b)

${\omega }_{{\mathcal{D}}_i}\left(\lambda x_i\right)=\lambda {\omega }_{{\mathcal{D}}_i}\left(x_i\right)$ for all $x_i\in {\mathcal{R}}^n$ when $\lambda >0$.

(c)

${\omega }_{{\mathcal{D}}_i}(x_i+x_i^{\prime })\le {\omega }_{{\mathcal{D}}_i}\left(x_i\right)+{\omega }_{{\mathcal{D}}_i}\left(x_i^{\prime }\right)$ for all $x_i,x_i^{\prime }\in {\mathcal{R}}^n$.

(d)

$\{x_i\mid {\omega }_{{\mathcal{D}}_i}\left(x_i\right)\le \delta \}$ is a bounded set for every $\delta \ge 0$.

Actually, ${\omega }_{{\mathcal{D}}_i}$ is Lipschitzian and particular uniformly continuous relative to ${\mathcal{R}}^n$.

Proof. 

Properties (a), (b), and (c) can be obtained directly from axioms $\left(\mathbf{D}\mathbf{2}\right)$ and $\left(\mathbf{D}\mathbf{3}\right)$. Properties (b) and (c) imply the convexity of ${\omega }_{{\mathcal{D}}_i}$. Moreover, Assumption 6 guarantees the finiteness of ${\omega }_{{\mathcal{D}}_i}$. Hence, it is Lipschitzian and particular uniformly continuous relative to ${\mathcal{R}}^n$, since

$$\underset{\lambda \to \infty }{lim\; inf}{\omega }_{{\mathcal{D}}_i}\left(\lambda x_i\right)/\lambda ={\omega }_{{\mathcal{D}}_i}\left(x_i\right)<\infty \forall x_i\in {\mathcal{R}}^n,$$

(cf. Corollary 10.5.1 of Rockafellar [18]). Meanwhile, $\{x_i\mid {\omega }_{{\mathcal{D}}_i}\left(x_i\right)\le 0\}=\left\{0\right\}$ by property (a). Therefore, $\{x_i\mid {\omega }_{{\mathcal{D}}_i}\left(x_i\right)\le \delta \}$ is bounded for every $\delta \ge 0$ (cf. Corollary 8.7.1 of Rockafellar [18]). This completes the proof. □

While our risk measure ${\omega }_{{\mathcal{D}}_i}\left(x_i\right)$ does not explicitly account for inter-asset correlations, this simplification enhances computational tractability for large-scale problems where estimating a full covariance matrix is infeasible. Crucially, the following theorem establishes that our measure is mathematically equivalent to any standard portfolio-level deviation measure, thereby ensuring its economic validity.

Theorem 1.

Assume that the risky asset returns are linearly independent, or equivalently, that no non-zero portfolio $x_i\in {\mathcal{R}}^n\setminus \left\{0\right\}$ yields a riskless return $x_i^{T}R$. Then, for any deviation measure ${\mathcal{D}}_i$, there exist constants $a_1\ge a_2>0$ such that

$$a_2{\omega }_{{\mathcal{D}}_i}\left(x_i\right)\le {\mathcal{D}}_i\left(x_i^{T}R\right)\le a_1{\omega }_{{\mathcal{D}}_i}\left(x_i\right)\forall x_i\in {\mathcal{R}}^n.$$

Proof. 

For any non-zero $x_i\in {\mathcal{R}}^n$, define $f_{{\mathcal{D}}_i}\left(x_i\right):={\mathcal{D}}_i\left(x_i^{T}R\right)$. The ratio ${\displaystyle \frac{f_{{\mathcal{D}}_i}\left(x_i\right)}{{\omega }_{{\mathcal{D}}_i}\left(x_i\right)}}$ is a continuous function on the compact set $S^n:=\{x\in {\mathcal{R}}^n\mid {\parallel x\parallel }_2=1\}$ (by Proposition 2 [16] (Proposition 4)). By the extreme value theorem, this function attains its minimum and maximum on $S^n$. Thus, there exist $x_{min},x_{max}\in S^n$ such that

$$a_2\le {\displaystyle \frac{f_{{\mathcal{D}}_i}\left(x_i\right)}{{\omega }_{{\mathcal{D}}_i}\left(x_i\right)}}\le a_1\forall x_i\in S^n,$$

where we define $a_1:={\displaystyle \frac{f_{{\mathcal{D}}_i}\left(x_{max}\right)}{{\omega }_{{\mathcal{D}}_i}\left(x_{max}\right)}}$ and $a_2:={\displaystyle \frac{f_{{\mathcal{D}}_i}\left(x_{min}\right)}{{\omega }_{{\mathcal{D}}_i}\left(x_{min}\right)}}$. Since $x_{min}\ne 0$, we have ${\omega }_{{\mathcal{D}}_i}\left(x_{min}\right)>0$ by Proposition 2(a). Furthermore, the non-redundancy assumption implies $f_{{\mathcal{D}}_i}\left(x_{min}\right)>0$ (see [16], Proposition 4(a)), which ensures $a_2>0$. The positive homogeneity of both $f_{{\mathcal{D}}_i}$ and ${\omega }_{{\mathcal{D}}_i}$ extends the above inequality to all $x_i\in {\mathcal{R}}^n\setminus \left\{0\right\}$:

$$a_2\le {\displaystyle \frac{f_{{\mathcal{D}}_i}\left(x_i\right)}{{\omega }_{{\mathcal{D}}_i}\left(x_i\right)}}\le a_1\forall x_i\in {\mathcal{R}}^n\setminus \left\{0\right\}.$$

The case of $x_i=0$ is trivial, as both sides of the target inequality are zero by axioms $\left(\mathbf{D}\mathbf{1}\right)$ and $\left(\mathbf{D}\mathbf{2}\right)$ and Proposition 2(a). This completes the proof. □

3. Utility Maximization of an Individual Investor

This section assumes each investor $I_i$, as a price taker and exchanges assets from initial endowments to maximize their utility $U_i$.

Assumption 7.

$U_i$ is a function of μ and ρ, where μ is the expected rate of return of the portfolio and ρ is the investment risk. Also, $U_i$ satisfies the following conditions.

$${\displaystyle \frac{\partial U_i(\mu ,\rho )}{\partial \mu }}>0,{\displaystyle \frac{\partial U_i(\mu ,\rho )}{\partial \rho }}<0.$$

As can be seen from the previous section, $x_{i0}{r}_0+r^T{x}_j$ is the expected return rate of investor $I_i$, and ${\displaystyle {\omega }_{{\mathcal{D}}_i}\left(x_i\right)=\underset{1\le j\le n}{max}{\mathcal{D}}_i\left(x_{ij}{R}_j\right)}$ is the risk function of investor $I_i$. Therefore, the individual utility maximization problem faced by investor $I_i$ can be expressed as

$$\begin{array}{ccc}\underset{x_i\in {\mathcal{R}}^n}{maximize}\hfill & \hfill & U_i\left(x_{i0}{r}_0+r^T{x}_i,{\omega }_{{\mathcal{D}}_i}\left(x_i\right)\right).\hfill \end{array}$$

(7)

Next, we will show that ${\omega }_{{\mathcal{D}}_i}\left(x_i\right)$ can be represented as a function of ${\mathrm{\Delta }}_i$, satisfying

$$x_{i0}{r}_0+r^T{x}_i\ge r_0+{\mathrm{\Delta }}_i.$$

(8)

For this reason, consider the following individual optimization problem of investor $I_i$

$$\begin{array}{ccc}\mathrm{m}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{z}\mathrm{e}\hfill & \hfill & {\omega }_{{\mathcal{D}}_i}\left(x_i\right)\hfill \\ \mathrm{s}\mathrm{u}\mathrm{b}\mathrm{j}\mathrm{e}\mathrm{c}\mathrm{t} \mathrm{t}\mathrm{o}\hfill & \hfill & x_{i0}+e^T{x}_i=1,\hfill \\ \hfill & \hfill & x_{i0}{r}_0+r^T{x}_i\ge r_0+{\mathrm{\Delta }}_i.\hfill \end{array}$$

(9)

or equivalently

$$\begin{array}{ccc}\mathrm{m}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{z}\mathrm{e}\hfill & \hfill & {\omega }_{{\mathcal{D}}_i}\left(x_i\right)\hfill \\ \mathrm{s}\mathrm{u}\mathrm{b}\mathrm{j}\mathrm{e}\mathrm{c}\mathrm{t} \mathrm{t}\mathrm{o}\hfill & \hfill & {(r-r_{0}e)}^T{x}_i\ge {\mathrm{\Delta }}_i.\hfill \end{array}$$

(10)

Throughout the paper, we denote by ${\omega }_{{\mathrm{\Delta }}_i}^{*}$ the optimal value, and by $S_{{\mathrm{\Delta }}_i}$ the optimal solution set, i.e., the minimizing vectors $x_i$, of this problem.

For a rational investor, the reason why they are willing to take risks is to obtain the returns that a risk-free asset cannot achieve. That is to say, this optimization problem is reasonable only when ${\mathrm{\Delta }}_i>0$, as the following proposition states.

Proposition 3.

No matter what the choice of ${\mathrm{\Delta }}_i$, the optimal solution set $S_{{\mathrm{\Delta }}_i}$ is always nonempty, convex, and compact. Moreover

(i) 

when ${\mathrm{\Delta }}_i\le 0$, ${\omega }_{{\mathrm{\Delta }}_i}^{*}=0$ and $S_{{\mathrm{\Delta }}_i}=\left\{0\right\};$

(ii) 

when ${\mathrm{\Delta }}_i>0$, the gain constraint is active and ${\omega }_{{\mathrm{\Delta }}_i}^{*}>0$ with

$${\omega }_{\theta {\mathrm{\Delta }}_i}^{*}=\theta {\omega }_{{\mathrm{\Delta }}_i}^{*},S_{\theta {\mathrm{\Delta }}_i}=\theta S_{{\mathrm{\Delta }}_i},$$

for every $\theta >0$.

Proof. 

As shown in Proposition 1, Assumption 5 ensures the richness of price–gain combinations. Therefore the constraint in problem (10) can be satisfied regardless of the choice of $r_0$ and ${\mathrm{\Delta }}_i$. In addition, ${\omega }_{{\mathrm{\Delta }}_i}^{*}$ is finite because of the finiteness of $\omega$. Then for every $\delta >{\omega }_{{\mathrm{\Delta }}_i}^{*}$

$$\{x_i\mid \omega \left(x_i\right)\le \delta ,{(r-r_{0}e)}^T{x}_i\ge {\mathrm{\Delta }}_i\}\ne \varnothing .$$

$\{x_i\mid \omega \left(x_i\right)\le {\delta }_i\}$ is bounded in addition to being closed and convex because of the continuity and convexity of $\omega$ by Proposition 2, while $\{x_i\mid {(r-r_{0}e)}^T{x}_i\ge {\mathrm{\Delta }}_i\}$ is a closed half space. Therefore, $\{x_i\mid \omega \left(x_i\right)\le {\delta }_i,{(r-r_{0}e)}^T{x}_i\ge {\mathrm{\Delta }}_i\}$ is also convex and compact. Thus, the optimal solution set

$${\displaystyle S_{{\mathrm{\Delta }}_i}=\bigcap _{\delta >{\omega }_{{\mathrm{\Delta }}_i}^{*}}\{x_i\mid \omega \left(x_i\right)\le {\delta }_i,{(r-r_{0}e)}^T{x}_i\ge {\mathrm{\Delta }}_i\},}$$

is also nonempty, convex, and compact.

${\omega }_{{\mathrm{\Delta }}_i}^{*}=0$ and $S_{{\mathrm{\Delta }}_i}=\left\{0\right\}$ for ${\mathrm{\Delta }}_i\le 0$ can be verified by Proposition 2 (a) and the fact that $x_i=0$ satisfies the constraint, while ${\omega }_{{\mathrm{\Delta }}_i}^{*}>0$ for ${\mathrm{\Delta }}_i>0$. Moreover, when $x_i\in S_{{\mathrm{\Delta }}_i}$ for ${\mathrm{\Delta }}_i>0$, we have $\omega \left(x_i\right)={\omega }_{{\mathrm{\Delta }}_i}^{*}$ and ${(r-r_{0}e)}^T{x}_i\ge {\mathrm{\Delta }}_i$. Then ${(r-r_{0}e)}^T\left(\theta x_i\right)\ge \left(\theta {\mathrm{\Delta }}_i\right)$ for any $\theta >0$. Assume ${\omega }_{\theta {\mathrm{\Delta }}_i}^{*}<\omega \left(\theta x_i\right)$, then there exists $x_i^{\prime }\in {\mathcal{R}}^n$ such that $\omega \left(x_i^{\prime }\right)={\omega }_{\theta {\mathrm{\Delta }}_i}^{*}$. Thus $\omega \left(x_i^{\prime }\right)<\omega \left(\theta x_i\right)$ and $\theta \omega (x_i^{\prime }/\theta )<\theta \omega \left(x_i\right)$ by the positive homogeneity of $\omega$, which contradicts that $x_i\in S_{{\mathrm{\Delta }}_i}$. Therefore, ${\omega }_{\theta {\mathrm{\Delta }}_i}^{*}=\omega \left(\theta x_i\right)=\theta {\omega }_{{\mathrm{\Delta }}_i}^{*}$, which shows $\{\theta x_i\mid x_i\in S_{{\mathrm{\Delta }}_i}\}\subset S_{\theta {\mathrm{\Delta }}_i}.$ Moreover, assume there exists $x_i^{\prime \prime }\in S_{\theta {\mathrm{\Delta }}_i}\setminus \{\theta x_i\mid x_i\in S_{{\mathrm{\Delta }}_i}\}$. It is noted that $\omega \left(x_i^{\prime \prime }\right)={\omega }_{\theta {\mathrm{\Delta }}_i}^{*}=\theta {\omega }_{{\mathrm{\Delta }}_i}^{*}$, thus $\omega (x_i^{\prime \prime }/\theta )={\omega }_{{\mathrm{\Delta }}_i}^{*}$, which contradicts that $x_i^{\prime \prime }\notin \{\theta x_i\mid x_i\in S_{{\mathrm{\Delta }}_i}\}$.

It remains to show the gain constraint is active when ${\mathrm{\Delta }}_i>0$. Assume there exists $x_i\in {\mathcal{R}}^n$ such that ${(r-r_{0}e)}^T{x}_i>{\mathrm{\Delta }}_i$. Then there exists $\eta \in (0,1)$ such that ${(r-r_{0}e)}^T\left(\eta x_i\right)>{\mathrm{\Delta }}_i$ and $\omega \left(\eta x_i\right)=\eta \omega \left(x_i\right)<\omega \left(x_i\right)$. This is incompatible with $x_i$ being optimal. This completes the proof. □

According to Proposition 3, let

$$U_i^{*}\left({\mathrm{\Delta }}_i\right):=U_i\left(r_0+{\mathrm{\Delta }}_i,{\omega }_{{\mathrm{\Delta }}_i}^{*}\right),$$

and let

$${\mathrm{\Delta }}_i^{*}:=arg\; max\left\{U_i^{*}\left({\mathrm{\Delta }}_i\right)\mid {\mathrm{\Delta }}_i>0\right\}.$$

Assumption 8.

Each investor $I_i$ possesses a unique and finite optimal target excess return, denoted by ${\mathrm{\Delta }}_i^{*}$.

The reasonableness of Assumption A8 is not merely a technical requirement but a natural consequence of standard economic behavior. In what follows, we verify this property for two canonical utilities. In both cases, the optimization process yields a unique and finite optimal target excess return ${\mathrm{\Delta }}_i^{*}$.

Example 1.

Consider the linear utility with a quadratic risk penalty for investor $I_i$:

$$U_i^{*}\left({\mathrm{\Delta }}_i\right)=(r_0+{\mathrm{\Delta }}_i)-{\displaystyle \frac{a_i}{2}}{\left({\omega }_{{\mathrm{\Delta }}_i}^{*}\right)}^2,$$

(11)

where $a_i>0$ is the coefficient of risk aversion. To determine the optimal ${\mathrm{\Delta }}_i^{*}$, we set the first-order condition to zero:

$${\displaystyle \frac{d{U}_i^{*}}{d{\mathrm{\Delta }}_i}}=1-a_i{\omega }_{{\mathrm{\Delta }}_i}^{*}{\displaystyle \frac{d{\omega }_{{\mathrm{\Delta }}_i}^{*}}{d{\mathrm{\Delta }}_i}}=0.$$

(12)

To evaluate the derivative ${\displaystyle \frac{d{\omega }_{{\mathrm{\Delta }}_i}^{*}}{d{\mathrm{\Delta }}_i}}$, we leverage the property from Proposition 3(ii), ${\omega }_{\theta {\mathrm{\Delta }}_i}^{*}=\theta {\omega }_{{\mathrm{\Delta }}_i}^{*}$ for all $\theta >0$. Differentiating this identity with respect to θ yields

$${\displaystyle \frac{d{\omega }_{\theta {\mathrm{\Delta }}_i}^{*}}{d\left(\theta {\mathrm{\Delta }}_i\right)}}·{\mathrm{\Delta }}_i={\omega }_{{\mathrm{\Delta }}_i}^{*}.$$

Setting $\theta =1$ establishes the fundamental relationship

$${\displaystyle \frac{d{\omega }_{{\mathrm{\Delta }}_i}^{*}}{d{\mathrm{\Delta }}_i}}={\displaystyle \frac{{\omega }_{{\mathrm{\Delta }}_i}^{*}}{{\mathrm{\Delta }}_i}}.$$

(13)

Substituting (13) into the first-order condition (12) gives

$$1-a_i{\omega }_{{\mathrm{\Delta }}_i}^{*}\left({\displaystyle \frac{{\omega }_{{\mathrm{\Delta }}_i}^{*}}{{\mathrm{\Delta }}_i}}\right)=0.$$

(14)

Solving for ${\mathrm{\Delta }}_i$ yields the unique optimal value, implicitly defined by ${\mathrm{\Delta }}_i^{*}=a_i{\left({\omega }_{{\mathrm{\Delta }}_i^{*}}^{*}\right)}^2$.

Example 2.

Let the utility of investor $I_i$ be given by a power utility function with an explicit risk penalty:

$$U_i^{*}\left({\mathrm{\Delta }}_i\right)={\displaystyle \frac{{\mathrm{\Delta }}_i^{1-{\gamma }_i}}{1-{\gamma }_i}}-{\displaystyle \frac{b_i}{2}}{\left({\omega }_{{\mathrm{\Delta }}_i}^{*}\right)}^2,$$

(15)

where ${\gamma }_i\ne 1$ governs the curvature of the utility of gains, and $b_i>0$ determines the penalty for portfolio risk. The optimal ${\mathrm{\Delta }}_i^{*}$ is found by setting the first-order condition to zero:

$${\displaystyle \frac{d{U}_i^{*}}{d{\mathrm{\Delta }}_i}}={\mathrm{\Delta }}_i^{-{\gamma }_i}-b_i{\omega }_{{\mathrm{\Delta }}_i}^{*}{\displaystyle \frac{d{\omega }_{{\mathrm{\Delta }}_i}^{*}}{d{\mathrm{\Delta }}_i}}=0.$$

(16)

Substituting the fundamental relationship (13)

$${\mathrm{\Delta }}_i^{-{\gamma }_i}-b_i{\omega }_{{\mathrm{\Delta }}_i}^{*}\left({\displaystyle \frac{{\omega }_{{\mathrm{\Delta }}_i}^{*}}{{\mathrm{\Delta }}_i}}\right)=0,$$

(17)

which simplifies to

$${\mathrm{\Delta }}_i^{-{\gamma }_i}=b_i{\displaystyle \frac{{\left({\omega }_{{\mathrm{\Delta }}_i}^{*}\right)}^2}{{\mathrm{\Delta }}_i}}.$$

(18)

Rearranging to solve for ${\mathrm{\Delta }}_i$ gives the unique optimal value:

$${\mathrm{\Delta }}_i^{*}={\left(b_i{\left({\omega }_{{\mathrm{\Delta }}_i^{*}}^{*}\right)}^2\right)}^{{\displaystyle \frac{1}{1-{\gamma }_i}}}.$$

(19)

For a risk-averse investor (${\gamma }_i>0$), this solution is unique, positive, and finite.

Based on the above discussion and Equation (4), we have obtained the following theorem.

Theorem 2.

Let $x_i^{*}\in S_{{\mathrm{\Delta }}_i^{*}}$ be the optimal portfolio of investor $I_i$, then $z_{ij}^{*}$, the optimal units of asset $S_j$ owned by $I_i$ after exchange, satisfies the relation

$$p_j{z}_{ij}^{*}=x_{ij}^{*}{w}_i^0,j=1,2,\dots ,n.$$

4. Equilibrium Relation

The short selling of risky assets is not restricted in this paper. Nevertheless, the subsequent proposition indicates that market equilibrium can only be achieved if the risk-free rate is lower than the expected return rates of all risky assets. Consequently, in this scenario, rational investors will not short sell risky assets.

Proposition 4.

Only when $r_0\le r_1$ can the market clearing condition (3) be satisfied. Meanwhile, let

$${\overline{x}}_i:={\left(1/{\underline{q}}_{i1},1/{\underline{q}}_{i2},\dots ,1/{\underline{q}}_{in}\right)}^T,$$

(20)

and let ${\overline{\mathrm{\Delta }}}_i:={(r-r_{0}e)}^T{\overline{x}}_i>0$. Then, $\left({\mathrm{\Delta }}_i^{*}/{\overline{\mathrm{\Delta }}}_i\right){\overline{x}}_i\in S_{{\mathrm{\Delta }}_i^{*}}$ with

$$p_j{z}_{ij}^{*}={\displaystyle \frac{{\mathrm{\Delta }}_i^{*}}{{\overline{\mathrm{\Delta }}}_i{\underline{q}}_{ij}}}{w}_i^0,i=1,2,\dots ,m,j=1,2,\dots ,n.$$

(21)

Proof. 

Let ${\overline{S}}_i:=\underset{\{x_i\in {\mathcal{R}}^n\mid {\omega }_{{\mathcal{D}}_i}\left(x_i\right)\le 1\}}{arg\; max}{(r-r_{0}e)}^T{x}_i$. According to Remark 4

$${\displaystyle \{x_i\in {\mathcal{R}}^n\mid {\omega }_{{\mathcal{D}}_i}\left(x_i\right)\le 1\}=\{x\in {\mathcal{R}}^n\mid -\frac{1}{{\overline{q}}_{ij}}\le x_{ij}\le \frac{1}{{\underline{q}}_{ij}}\forall j\}.}$$

Thus, $x_i\in {\overline{S}}_i$ if and only if

$$x_{ij}=\left\{\begin{array}{cc}{\displaystyle \frac{1}{{\underline{q}}_{ij}}}\hfill & if{r}_j>r_0,\hfill \\ {\displaystyle -\frac{1}{{\overline{q}}_{ij}}}\hfill & if{r}_j<r_0,\hfill \end{array}\right.$$

(22)

and

$$x_{ij}\in \left[-{\displaystyle \frac{1}{{\overline{q}}_{ij}}},{\displaystyle \frac{1}{{\underline{q}}_{ij}}}\right]if{r}_j=r_0.$$

(23)

Take any ${\overline{x}}_i\in {\overline{S}}_i$. Assuming ${\overline{x}}_i\in {\overline{S}}_i$ with ${\omega }_{{\mathcal{D}}_i}\left({\overline{x}}_i\right)<1$, then $r-r_{0}e=0$ by optimality. This contradicts Assumption 5, and then ${\omega }_{{\mathcal{D}}_i}\left({\overline{x}}_i\right)=1$.

Next, we will show ${\overline{x}}_i\in S_{{\overline{\mathrm{\Delta }}}_i}$ for ${\overline{\mathrm{\Delta }}}_i:={(r-r_{0}e)}^T{\overline{x}}_i>0$. Assume ${\overline{x}}_i\notin S_{{\overline{\mathrm{\Delta }}}_i}$, then there exists $y\in {\mathcal{R}}^n$ with ${(r-r_{0}e)}^{T}y={\overline{\mathrm{\Delta }}}_i$ such that $0<{\omega }_{{\mathcal{D}}_i}\left(y\right)<{\omega }_{{\mathcal{D}}_i}\left({\overline{x}}_i\right)=1$. Thus, there exists $\eta >1$ such that

$${(r-r_{0}e)}^T\left(\eta y\right)>{(r-r_{0}e)}^T{\overline{x}}_i,$$

and ${\omega }_{{\mathcal{D}}_i}\left(\eta y\right)=\eta {\omega }_{{\mathcal{D}}_i}\left(y\right)=1$ by Proposition 2(b). This is incompatible with ${\overline{x}}_i\in {\overline{S}}_i$ and then ${\overline{x}}_i\in S_{{\overline{\mathrm{\Delta }}}_i}$.

Consequently, $\left({\mathrm{\Delta }}_i^{*}/{\overline{\mathrm{\Delta }}}_i\right){\overline{x}}_i\in S_{{\mathrm{\Delta }}_i^{*}}$ by Proposition 3(ii). Moreover

$$p_j{z}_{ij}^{*}=\left({\mathrm{\Delta }}_i^{*}/{\overline{\mathrm{\Delta }}}_i\right){\overline{x}}_{ij}{w}_i^0,i=1,2,\dots ,m,j=1,2,\dots ,n,$$

by Theorem 2. Assume $r_1<r_0$, then ${\overline{x}}_{i1}=-1/{\overline{q}}_{i1}$ for all $i=1,2,\dots ,m$ by (22). Then

$$z_{i1}^{*}=-{\displaystyle \frac{{\mathrm{\Delta }}_i^{*}}{p_1{\overline{\mathrm{\Delta }}}_i{\overline{q}}_{i1}}}{w}_i^0<0,i=1,2,\dots ,m,$$

and then ${\sum }_{i=1}^m{z}_{i1}^{*}<0$ while $z_1^0>0$. At this time, the market cannot be cleared. On the contrary, when $r_0\le r_1$

$${\overline{x}}_i={\left(1/{\underline{q}}_{i1},1/{\underline{q}}_{i2},\dots ,1/{\underline{q}}_{in}\right)}^T\in {\overline{S}}_i,$$

by (22) and (23). This completes the proof. □

Remark 7.

The condition $r_0\le r_1$ is a fundamental requirement for market viability. Economically, it ensures that every risky asset offers a risk premium over the risk-free alternative. If this condition fails, i.e., $r_0>r_1$, the risky asset with the lowest expected return would be unattractive to all rational investors. As the above proof shows, this would lead to a universal short position in that asset, making it impossible to clear the market with non-negative prices. Thus, the condition prevents a market breakdown.

Throughout the following discussion, we maintain the assumption that $r_0\le r_1$. In view of (21), we have

$$\sum _{i=1}^m{p}_j{z}_{ij}^{*}=\sum _{i=1}^m{\displaystyle \frac{{\mathrm{\Delta }}_i^{*}}{{\overline{\mathrm{\Delta }}}_i{\underline{q}}_{ij}}}{w}_i^0,j=1,2,\dots ,n.$$

Combined with Equation (1), it can be seen that

$$\sum _{i=1}^m{p}_j{z}_{ij}^{*}=\sum _{i=1}^m{\displaystyle \frac{{\mathrm{\Delta }}_i^{*}}{{\overline{\mathrm{\Delta }}}_i{\underline{q}}_{ij}}}\left(z_{i0}^0+\sum _{k=1}^n{p}_k{z}_{ik}^0\right),j=1,2,\dots ,n.$$

Furthermore, market clearing condition (3) indicates

$$p_j{z}_j^0=\sum _{i=1}^m{\displaystyle \frac{{\mathrm{\Delta }}_i^{*}}{{\overline{\mathrm{\Delta }}}_i{\underline{q}}_{ij}}}\left(z_{i0}^0+\sum _{k=1}^n{p}_k{z}_{ik}^0\right),j=1,2,\dots ,n.$$

(24)

Let

$$\alpha =\sum _{i=1}^m\sum _{j=1}^n{\displaystyle \frac{{\mathrm{\Delta }}_i^{*}}{{\overline{\mathrm{\Delta }}}_i{\underline{q}}_{ij}}}{\displaystyle \frac{z_{ij}^0}{z_j^0}}.$$

(25)

Theorem 3.

The system of Equation (24) has a unique non-negative solution

$$p_j^{*}={\displaystyle \frac{1}{1-\alpha }}{\displaystyle \frac{1}{z_j^0}}\sum _{i=1}^m{\displaystyle \frac{{\mathrm{\Delta }}_i^{*}}{{\overline{\mathrm{\Delta }}}_i{\underline{q}}_{ij}}}{z}_{i0}^0,j=1,2,\dots ,n,$$

if and only if $\alpha <1$.

Proof. 

Let

$$c_j=\sum _{j=1}^m{\displaystyle \frac{{\mathrm{\Delta }}_i^{*}}{{\overline{\mathrm{\Delta }}}_i{\underline{q}}_{ij}}}{z}_{ij}^0,j=0,1,2,\dots ,n,$$

and let

$$d_j={\displaystyle \frac{1}{z_j^0}},j=1,2,\dots ,n.$$

Denote

$$c={(c_1,c_2,\dots ,c_n)}^T,d={(d_1,d_2,\dots ,d_n)}^T,p={(p_1,p_2,\dots ,p_n)}^T.$$

Then Equation (25) can be rewritten as

$$\alpha =\sum _{j=1}^n\left(\sum _{i=1}^m{\displaystyle \frac{{\mathrm{\Delta }}_i^{*}}{{\overline{\mathrm{\Delta }}}_i{\underline{q}}_{ij}}}{z}_{ij}^0\right){\displaystyle \frac{1}{z_j^0}}=c^{T}d,$$

and system (24) can be rewritten as

$$p_j={\displaystyle \frac{1}{z_j^0}}\sum _{i=1}^m{\displaystyle \frac{{\mathrm{\Delta }}_i^{*}}{{\overline{\mathrm{\Delta }}}_i{\underline{q}}_{ij}}}\left(z_{i0}^0+\sum _{k=1}^n{p}_k{z}_{ik}^0\right)=d_j{c}_0+d_j\left(c^{T}p\right),j=1,2,\dots ,n,$$

or as the vector form

$$p=d{c}_0+d\left(c^{T}p\right),$$

as follows

$$\left(I-d{c}^T\right)p=c_{0}d,$$

(26)

where I is the unit matrix. Therefore, when $\alpha \ne 1$, the system of Equation (24) has the following unique solution:

$$p={\left(I-d{c}^T\right)}^{-1}d{c}_0=\left(I+{\displaystyle \frac{d{c}^T}{1-\alpha }}\right)d{c}_0={\displaystyle \frac{1}{1-\alpha }}d{c}_0.$$

(27)

Moreover, in view of Assumptions 2 and 3, we have $c\ge 0$, $c_0\ge 0$, and $d>0$, and then $\alpha =c^{T}d\ge 0$. Then, the unique solution (27) is non-negative if and only if $\alpha <1$.

On the contrary, by Farkas Lemma [19], the system (26) has a non-negative solution if and only if for any $y\in {\mathcal{R}}^n$ with

$$\left(I-c{d}^T\right)y\le 0,$$

(28)

it holds that

$$d^{T}y\le 0.$$

(29)

If $\alpha \ge 1$, then $\left(I-c{d}^T\right)c=(1-\alpha )c\le 0$ i.e. $y=c$ satisfies (28). However

$$d^{T}c=\alpha \ge 1>0.$$

As a consequence, system (26) admits no non-negative solution when $\alpha \ge 1$. This completes the proof. □

Remark 8.

The parameter α can be interpreted as a measure of the market’s overall “excess demand pressure.” If $\alpha \ge 1$, the aggregate demand from investors, when scaled to their optimal risk levels, would equal or exceed the total supply of assets. In this scenario, the system of market-clearing equations has no finite, non-negative price solution. Economically, this implies a state of hyper-demand or a shortage of assets, leading to price instability or the absence of a finite equilibrium. Therefore, $\alpha <1$ is the necessary and sufficient condition for a stable, finite equilibrium where aggregate demand can be satisfied by the available supply.

Finally, we will present the expression for the market portfolio. Let

$$x_i^M={\displaystyle \frac{{\overline{x}}_i}{e^T{\overline{x}}_i}},i=1,2,...,m,$$

(30)

in which ${\overline{x}}_i$ is given by (20). Then

$${\mathrm{\Delta }}_i^M:={(r-r_{0}e)}^T{x}_i^M={\displaystyle \frac{{\overline{\mathrm{\Delta }}}_i}{e^T{\overline{x}}_i}},$$

(31)

and $({\mathrm{\Delta }}_i^{*}/{\mathrm{\Delta }}_i^M)x_i^M=({\mathrm{\Delta }}_i^{*}/{\overline{\mathrm{\Delta }}}_i){\overline{x}}_i\in S_{{\mathrm{\Delta }}_i^{*}}$ by Proposition 4. This means that investor $I_i$ holds the combination of positive multiple of the portfolio $x_i^M$ and the risk-less asset after the transaction. We call $x_i^M$ the master fund of investor $I_i$. It is important to note that the composition of this master fund is inherently dependent on the specific deviation measure ${\mathcal{D}}_i$ chosen by investor $I_i$, reflecting their individual risk assessment. Note that, due to the non-homogeneous investors, $x_i^M$ is not the market portfolio. In fact, the market portfolio $y^M$ can be defined by

$$y_j^M:={\displaystyle \frac{p_j^{*}{z}_j^0}{{\sum }_{k=1}^n{p}_k^{*}{z}_k^0}},j=1,2,\dots ,n.$$

(32)

Let

$${\theta }_i:={\displaystyle \frac{{\mathrm{\Delta }}_i^{*}}{{\mathrm{\Delta }}_i^M}}·{\displaystyle \frac{w_i^0}{{\sum }_{k=1}^n{p}_k^{*}{z}_k^0}}>0,i=1,2,\dots ,m.$$

(33)

The following proposition provides the relationship between the market portfolio and master funds.

Proposition 5.

The market portfolio $y^M$ is a convex combination of master funds. Specifically

$$y^M=\sum _{i=1}^m{\theta }_i{x}_i^M.$$

(34)

Proof. 

Combining with Equations (20), (21), (30), (31), and (33), we have

$$\begin{array}{cc}{\displaystyle \sum _{i=1}^m{\theta }_i{x}_{ij}^M}& {\displaystyle =\sum _{i=1}^m\left(\frac{{\mathrm{\Delta }}_i^{*}}{{\mathrm{\Delta }}_i^M}·\frac{w_i^0}{{\sum }_{k=1}^n{p}_k^{*}{z}_k^0}\right)·\frac{1}{{\underline{q}}_{ij}}·\frac{1}{e^T{\overline{x}}_i}}\hfill \\ & {\displaystyle =\frac{1}{{\sum }_{k=1}^n{p}_k^{*}{z}_k^0}\sum _{i=1}^m\frac{{\mathrm{\Delta }}_i^{*}}{\left(e^T{\overline{x}}_i\right){\mathrm{\Delta }}_i^M}·\frac{w_i^0}{{\underline{q}}_{ij}}}\hfill \\ & {\displaystyle =\frac{1}{{\sum }_{k=1}^n{p}_k^{*}{z}_k^0}\sum _{i=1}^m\frac{{\mathrm{\Delta }}_i^{*}}{{\overline{\mathrm{\Delta }}}_i}·\frac{w_i^0}{{\underline{q}}_{ij}}}\hfill \\ & {\displaystyle =\frac{1}{{\sum }_{k=1}^n{p}_k^{*}{z}_k^0}\sum _{i=1}^m{p}_j^{*}{z}_{ij}^{*}}\hfill \\ & {\displaystyle =\frac{p_j^{*}{z}_j^0}{{\sum }_{k=1}^n{p}_k^{*}{z}_k^0}=y_j^M.}\hfill \end{array}$$

Since ${\sum }_{j=1}^n{y}_j^M=1$ and ${\sum }_{j=1}^n{x}_{ij}^M=1$, we have ${\sum }_{i=1}^m{\theta }_i=1$ for ${\theta }_i>0,i=1,2,\dots ,m$. This completes the proof. □

Remark 9.

In view of Proposition 5, the weight ${\theta }_i$ represents the proportion of total market wealth allocated to investor i’s optimal risky portfolio. As defined in Equation (33), ${\theta }_i$ is proportional to the investor’s initial wealth $w_i^0$ and their scaled risk appetite ${\mathrm{\Delta }}_i^{*}/{\mathrm{\Delta }}_i^M$. Economically, this weight signifies an investor’s influence in the market, determined by their financial resources and risk-taking behavior. Consequently, the market portfolio is not a single entity but a convex combination of distinct master funds $x_i^M$, each tailored to a specific investor’s risk profile ${\underline{q}}_{ij}$ and aggregated according to its corresponding market influence.

5. Conclusions

In this paper, we analyzed a market with non-homogeneous investors who utilized different deviation measures to construct their individual risk functions. Although the short selling of risky assets was not restricted by assumption, our rigorous analysis demonstrated that a market equilibrium was achievable only when the risk-free rate was lower than the expected return rates of all risky assets. Consequently, under this condition, rational investors were incentivized to refrain from short selling risky assets. Moreover, we provided the necessary and sufficient condition for the existence of a unique non-negative equilibrium price system for risky assets and derived its explicit formula. Finally, we proved that the market portfolio constituted a convex combination of the master funds of these non-homogeneous investors.