Minimal Investment Risk with Cost and Return Constraints: A Replica Analysis

TL;DR

This paper applies replica analysis to portfolio optimization with distinct asset costs and fund size considerations, deriving a macroscopic theory and validating it through numerical experiments.

Contribution

It introduces a novel replica analysis approach to account for asset-specific costs and fund size effects in risk minimization.

Findings

Derived a macroscopic theory for constrained portfolio optimization

Validated the theoretical results with numerical experiments

Highlighted the impact of cost and return constraints on optimal investment

Abstract

Previous studies into the budget constraint of portfolio optimization problems based on statistical mechanical informatics have not considered that the purchase cost per unit of each asset is distinct. Moreover, the fact that the optimal investment allocation differs depending on the size of investable funds has also been neglected. In this paper, we approach the problem of investment risk minimization using replica analysis. This problem imposes cost and return constraints. We also derive the macroscopic theory indicated by the optimal solution and confirm the validity of our proposed method through numerical experiments.

Equations

$$N\times C$$

$$N\times R$$

$$\mathcal{W}$$

$$\mathcal{H}(\vec{w})$$

$$J_{ij}$$

$${\vec{w}}^{\ast }$$

$$\mathcal{L}=\frac{1}{2}{\vec{w}}^{\mathrm{T}}J\vec{w}+\theta (NR-{\vec{w}}^{\mathrm{T}}\vec{r})+k(NC-{\vec{w}}^{\mathrm{T}}\vec{c}).$$

$$\epsilon$$

$$k^{\ast }$$

$${\theta }^{\ast }$$

$${\vec{w}}^{\ast }$$

$$Z$$

$$\epsilon$$

$$\varphi$$

$$\psi (n)$$

$$⟨f(r,c,v)⟩$$

$$q_{sab}$$

$$\psi (n)$$

$$\varphi$$

$${\chi }_s$$

$$q_s$$

$${\tilde{\chi }}_s$$

$${\tilde{q}}_s$$

$$\theta$$

$$k$$

$$\epsilon$$

$$R_0$$

$$V$$

$$\varphi$$

$$\epsilon$$

$$Z$$

$$\log Z$$

$$\varphi$$

$$\underset{N\to \infty }{\lim }\frac{1}{N}{\vec{r}}^{\mathrm{T}}{J}^{-1}\vec{r}$$

Full text

1 Introduction

Extant literature in the domain of operations research has analyzed annealed disordered systems in the context of spin glass theory against portfolio optimization problems such as budget constrained investment risk minimization problems and risk constrained expected return maximization[1, 2]. However, the investment information sought by investors is actually the optimal portfolio in the quenched disordered system of the investment market. Thus, in recent years, researchers have actively analyzed these portfolio optimization problems using statistical mechanical informatics represented by random matrix theory, replica analysis, and the belief propagation method[3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23]. Through these studies, it is possible to analyze the quenched disordered system of the investment market, which was hitherto difficult to analyze by applying the well-used analysis methods of operations research. These studies in cross-disciplinary research fields also could analyze the mathematical structure of the minimum investment risk, the concentrated investment, and the maximum expected return[11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23].

However, although the budget constraint is used as a representative constraint condition in portfolio optimization problems approached using statistical mechanical informatics, we impose the strong assumption that purchase costs per unit of each asset are the same for all assets. Furthermore, previous studies used problem settings relevant to operations research; thus, we focus on the investment ratio of each asset as a decision variable, regardless of investment fund size. However, due to the size of working capital in actual investment contexts, the optimal investment strategy of individual investors with low working capital and the optimal investment strategy of institutional investors with sufficiently large working capital are different.

Therefore, in the present paper, we improve on the analytical approaches of previous works that utilized statistical mechanical informatics and discuss the investment risk minimization problem imposing cost and return constraints by using replica analysis. We also derive the macroscopic theory satisfied by the optimal portfolio.

The remainder of the paper is organized as follows. The next two sections describe the model setting and the replica analysis used to solve the portfolio optimization problem imposing constraints of initial cost and final return. Section 4 discusses the optimal portfolio in several situations and the macroscopic relations of the optimal solution. Numerical experiments confirm the validity of our proposed method based on replica analysis. The final section offers a summary and discusses potential future work in this domain.

2 Model setting

Let us consider a situation whereby $N$ assets are invested for $p$ periods in a steady trading market. Similar to the related literature by using replica analysis, we assume that no short selling regulation is imposed on the investment market. We denote the portfolio of asset $i(=1,2\cdots,N)$ as $w_{i}\in{\bf R}$, such that the vector $\vec{w}=(w_{1},w_{2},\cdots,w_{N})^{\rm T}\in{\bf R}^{N}$ describes the portfolio of $N$ assets, where the notation ${\rm T}$ represents the transpose of the vector and/or matrix. Moreover, the purchase cost per unit of asset $i$ at the initial investment period is expressed by $c_{i}$ and the return per unit of asset $i$ at period $\mu(=1,2,\cdots,p)$ is represented by $\bar{x}_{i\mu}$. We also assume that each return is independently distributed and the mean $E[\bar{x}_{i\mu}]=r_{i}$ and variance of the return $V[\bar{x}_{i\mu}]=v_{i}$ are known. Next, we assume that portfolio $\vec{w}$ imposes the cost constraint in Eq. (1) and the return constraint in Eq. (2);

[TABLE]

where $NC$ is the initial budget at the initial investment period and $NR$ is the final return at the last investment period.

The coefficients $C$ and $R$ denote the unit cost per asset and the unit return per asset, respectively. In practice, since the purchase costs per unit of each asset $c_{i}$ do not always coincide, in this paper, we do not apply the budget constraint used in previous studies $\sum_{i=1}^{N}w_{i}=N$, but we apply the cost constraint in Eq. (1). Thus, the feasible subspace of portfolio $\vec{w}$, ${\cal W}\subseteq{\bf R}^{N}$, is defined by

[TABLE]

where $\vec{c}=(c_{1},c_{2},\cdots,c_{N})^{\rm T}\in{\bf R}^{N}$ and $\vec{r}=(r_{1},r_{2},\cdots,r_{N})^{\rm T}\in{\bf R}^{N}$ are used.

From this, the investment risk of portfolio $\vec{w}$, ${\cal H}(\vec{w})$, is as follows:

[TABLE]

The $i,j$th component of Wishart matrix $J=\left\{J_{ij}\right\}\in{\bf R}^{N\times N}$, $J_{ij}$, is given by

[TABLE]

where in Eq. (5) the modified return $x_{i\mu}=\bar{x}_{i\mu}-r_{i}$ is already used, its mean and variance are $E[x_{i\mu}]=0$ and $V[x_{i\mu}]=v_{i}$, respectively. Thus, the optimal portfolio of the portfolio optimization problem that we discuss $\vec{w}^{*}$ is described as

[TABLE]

We accept $p>N$ herein since the optimum can be uniquely determined.

This portfolio optimization problem can be solved by using the extremum of the following Lagrange multiplier function ${\cal L}$:

[TABLE]

That is, from the extremum of ${\cal L}$, $\frac{\partial\cal L}{\partial w_{i}}=\frac{\partial\cal L}{\partial k}=\frac{\partial\cal L}{\partial\theta}=0$, the optimal $\vec{w}^{*}=\arg\mathop{\min}_{\vec{w}\in{\cal W}}{\cal H}(\vec{w})$ is derived. Then, the minimal investment risk per asset $\varepsilon=\frac{1}{N}{\cal H}(\vec{w}^{*})$ is obtained:

[TABLE]

where Eqs. (9)–(11) are used:

[TABLE]

It transpires that the optimal portfolio $\vec{w}^{*}$ is dependent on the initial cost $C$ and the final return $R$ from Eqs. (9)–(11). It is also the case that the optimal investment strategy is a function of the size of working capital and the target figure. If we can assess $\vec{c}^{\rm T}J^{-1}\vec{c},\vec{c}^{\rm T}J^{-1}\vec{r},\vec{r}^{\rm T}J^{-1}\vec{r}$, using Eq. (8), the minimal investment risk per asset $\varepsilon$ is calculated. However, in general, it is computationally onerous to solve for the inverse matrix $J^{-1}$ of the regular matrix $J\in{\bf R}^{N\times N}$ as $N$ increases. Therefore, we discuss the portfolio optimization problem using replica analysis which can resolve the minimal investment risk per asset $\varepsilon$ without directly solving for the inverse matrix $J^{-1}$.

3 Replica analysis

Following an analytical procedure based on statistical mechanical informatics, we discuss an optimization problem that has a Hamiltonian of the investment system defined in Eq. (4). Then the partition function $Z$ of the inverse temperature $\beta(>0)$ of the canonical ensemble is defined as

[TABLE]

where $k,\theta$ are the variables related to the constraints in Eqs. (1) and (2). From this, the minimal investment risk per asset $\varepsilon$ is solved from the following thermodynamic relation:

[TABLE]

where it is well-known that the minimal investment risk per asset $\varepsilon$ holds if the following self-averaging property is used:

[TABLE]

In general, it is cumbersome to directly evaluate the configuration average of $\log Z$ over return matrix $X$, $E[\log Z]$. Since it is comparatively easy to execute $E[Z^{n}],n\in{\bf Z}$ using replica analysis in the limit that the number of assets $N$ is sufficiently large,

[TABLE]

is analytically evaluated where the period ratio $\alpha=p/N\sim O(1)$, the order parameters $\vec{\theta}=(\theta_{1},\theta_{2},\cdots,\theta_{n})^{\rm T}\in{\bf R}^{n}$, $\vec{k}=(k_{1},k_{2},\cdots,k_{n})^{\rm T}\in{\bf R}^{n}$, $Q_{s}=\left\{q_{sab}\right\}\in{\bf R}^{n\times n}$ $\tilde{Q}_{s}=\left\{\tilde{q}_{sab}\right\}\in{\bf R}^{n\times n}$, the identity matrix $I\in{\bf R}^{n\times n}$, and constant vector $\vec{e}=(1,1,\cdots,1)^{\rm T}\in{\bf R}^{n}$ are already used. Moreover, the notation

[TABLE]

is employed. Further, the notation ${\rm Extr}_{z}g(z)$ denotes the extremum of $g(z)$ by $z$, and $\tilde{q}_{sab}$ is the auxiliary order parameter of

[TABLE]

Here we assume the replica symmetry solution. Then, $\theta_{a}=\theta,k_{a}=k$, $q_{saa}=\chi_{s}+q_{s}$, ${q}_{sab}=q_{s}$, $\tilde{q}_{saa}=\tilde{\chi}_{s}-\tilde{q}_{s}$, $\tilde{q}_{sab}=-\tilde{q}_{s},(a\neq b)$ are set; thus,

[TABLE]

is replaced where ${\rm Extr}$ is abbreviated. From this, $\phi=\lim_{n\to 0}\frac{\partial\psi(n)}{\partial n}$ is summarized as follows:

[TABLE]

Moreover, from the extremum condition of Eq. (19),

[TABLE]

are obtained. From these results, using the identical equation in Eq. (13), $\varepsilon=-\lim_{\beta\to\infty}\frac{\partial\phi}{\partial\beta}$, the minimal investment risk per asset $\varepsilon$ is summarized as

[TABLE]

from $\frac{\partial\phi}{\partial\beta}=-\frac{\alpha\chi_{s}}{2(1+\beta\chi_{s})}-\frac{\alpha q_{s}}{2(1+\beta\chi_{s})^{2}}$, where

[TABLE]

are used.

4 Discussion

In the case where the only portfolio constraint concerns cost, $\phi=\lim_{N\to\infty}\frac{1}{N}E[\log Z]$ is

[TABLE]

then, the minimal investment risk per asset $\varepsilon=-\lim_{\beta\to\infty}\frac{\partial\phi}{\partial\beta}$ is

[TABLE]

When $C=c_{i}=1$, the result already available in the literature $\frac{\alpha-1}{2\left\langle v^{-1}\right\rangle}$ is derived[16]. Moreover, to compare Eqs. (26) and (30), when the return coefficient $R=R_{0}$, that is, when the weighted average of the revenue growth rate of asset $i$, $\frac{r_{i}}{c_{i}}$ is equal to the revenue growth rate of the portfolio $\frac{R}{C}$, $\frac{R}{C}=\frac{\left\langle v^{-1}c^{2}\frac{r}{c}\right\rangle}{\left\langle v^{-1}c^{2}\right\rangle}$, it is possible to resolve the minimal investment risk per asset under the cost constraint.

Next, let us compare results according to replica analysis and the Lagrange multiplier method that solves three moments $\frac{1}{N}\vec{r}^{\rm T}J^{-1}\vec{r}$, $\frac{1}{N}\vec{r}^{\rm T}J^{-1}\vec{c}$, $\frac{1}{N}\vec{c}^{\rm T}J^{-1}\vec{c}$. We consider the following partition function:

[TABLE]

It is straightforward to calculate the integral of the partition function,

[TABLE]

Furthermore, $\log Z$ holds the self-averaging property, $\phi=\lim_{N\to\infty}\frac{1}{N}E[\log Z]$ is solved, and from the derivative function with respect to $\theta,k$, we can solve the three moments. From the assumption of the replica symmetry solution,

[TABLE]

is summarized. Thus, from the extrema, $\chi_{s}=\frac{1}{\alpha-1}$, $q_{s}=\frac{\alpha}{(\alpha-1)^{3}}\left\langle v^{-1}(ck+r\theta)^{2}\right\rangle$, $\tilde{\chi}_{s}=\alpha-1$, $\tilde{q}_{s}=\frac{1}{\alpha-1}\left\langle v^{-1}(ck+r\theta)^{2}\right\rangle$ are obtained. Substituting these into Eq. (33),

[TABLE]

is derived. From this,

[TABLE]

are obtained. We substitute these into Eq. (8); then it transpires that this is consistent with the result in Eq. (26).

The Sharpe ratio, which is defined by the return per unit risk, $S=S(R)$ is given by

[TABLE]

Then, the maximal Sharpe ratio in the range of $R\geq C$ is at $R^{*}=\arg\mathop{\max}_{R\geq C}S(R)=\frac{V}{R_{0}-C}C^{2}+R_{0}$. Moreover, the maximum and minimum of the minimal investment risk per asset $\varepsilon=\varepsilon(R)$ are at $R_{\min}=\arg\mathop{\min}_{R\geq C}\varepsilon(R)=R_{0}$ and $R_{\max}=\arg\mathop{\max}_{R\geq C}\varepsilon(R)=\infty$, respectively. The squares of the Sharpe ratio are assessed as

[TABLE]

We obtain the following Pythagorean theorem of the Sharpe ratio:

[TABLE]

Similar to what has been reported in the extant literature, this theorem is not dependent on $\alpha,C$ and the probabilities of $r_{i},c_{i},v_{i}$. Further, the investment risk is summarized with respect to $C$:

[TABLE]

From this, the maximal Sharpe ratio in the range of $C\leq R$ is at $C^{*}=\arg\mathop{\max}_{C\leq R}S(C)=\frac{V_{r}}{C_{0}-R}R^{2}+C_{0}$. Moreover, the maximum and minimum of the minimal investment risk per asset $\varepsilon=\varepsilon(C)$ are at $C_{\min}=\arg\mathop{\min}_{C\leq R}\varepsilon(C)=C_{0}$ and $C_{\max}=\arg\mathop{\max}_{C\leq R}\varepsilon(C)=-\infty$, respectively. The squares of the Sharpe ratio are calculated as

[TABLE]

We also obtain the following Pythagorean theorem of the Sharpe ratio:

[TABLE]

Next, let us compare the result of the annealed disordered investment system. Applying a well-used analytical procedure of operations research,the minimal expected investment risk per asset $\varepsilon_{\rm OR}=\lim_{N\to\infty}\frac{1}{N}\mathop{\min}_{\vec{w}\in{\cal W}}E[{\cal H}(\vec{w})]$ is

[TABLE]

where $E[{\cal H}(\vec{w})]=\frac{\alpha}{2}\sum_{i=1}^{N}v_{i}w_{i}^{2}$ is used. From this, the proportion between the minimal expected investment risk per asset derived by operations research $\varepsilon_{\rm OR}$ and the minimal investment risk per asset $\varepsilon$, the opportunity loss $\kappa=\frac{\varepsilon_{\rm OR}}{\varepsilon}$, is solved as

[TABLE]

From this result, when $\alpha$ is close to 1, since the opportunity loss $\kappa$ is increasing, that is, since $\kappa=1$ is not satisfied, unfortunately, the portfolio which can minimize the expected investment risk $E[{\cal H}(\vec{w})]$, $\vec{w}_{\rm OR}=\arg\mathop{\min}_{\vec{w}\in{\cal W}}E[{\cal H}(\vec{w})]$, cannot minimize the investment risk ${\cal H}(\vec{w})$. Moreover, it transpires that the opportunity loss $\kappa$ is not dependent on $R,C$ and the probabilities of $r_{i},c_{i},v_{i}$.

5 Numerical experiments

Here we focus on the case where the mean and square mean of the return $\bar{x}_{i\mu}$ are represented by $E[\bar{x}_{i\mu}]=r_{i}$ and $E[\bar{x}_{i\mu}^{2}]=(h_{i}+1)r_{i}^{2}$, respectively. From this, the variance of the modified return $x_{i\mu}=\bar{x}_{i\mu}-E[\bar{x}_{i\mu}]$ is described by $v_{i}=V[x_{i\mu}]=h_{i}r_{i}^{2}$. Moreover, we set the relation between the purchase cost per unit of asset $i$, $c_{i}$, and the mean return $r_{i}$, $c_{i}=r_{i}z_{i}$, where $h_{i},z_{i}$ are independently distributed and non-negative. Then we assume that $r_{i},h_{i}$ are distributed by the bounded Pareto distribution. These density functions $f_{r}(r_{i}),f_{h}(h_{i})$ are defined as

[TABLE]

where $c_{r},c_{h}>0$ are exponentials of the bounded Pareto distributions. Moreover, $z_{i}$ is distributed uniformly with $0\leq z_{i}\leq 1$.

From this numerical setting, the analytical procedure in the numerical experiments is organized as follows:

Step 1

Assign randomly $r_{i},h_{i}$ with the bounded Pareto distributions and evaluate the variance $v_{i}(=h_{i}r_{i}^{2})$. Moreover, using $z_{i}$, which is distributed uniformly with $0\leq z_{i}\leq 1$, $c_{i}=r_{i}z_{i}$.

Step 2

Assign the return $\bar{x}_{i\mu}$ with the Gaussian distribution $N(r_{i},v_{i})$, then the modified return $x_{i\mu}=\bar{x}_{i\mu}-E[\bar{x}_{i\mu}]$ is assessed. Moreover, return matrix $X=\left\{\frac{x_{i\mu}}{\sqrt{N}}\right\}\in{\bf R}^{N\times p}$ is set.

Step 3

Solve Wishart matrix $J=XX^{\rm T}\in{\bf R}^{N\times N}$ and its inverse matrix $J^{-1}$.

Step 4

$\vec{c}^{\rm T}J^{-1}\vec{c},\vec{c}^{\rm T}J^{-1}\vec{r},\vec{r}^{\rm T}J^{-1}\vec{r}$ are calculated.

Step 5

Using Eqs. (8) and (38), we assess $\varepsilon$ and $S$.

Setting $N=1000,p=2000,(\alpha=p/N=2),C=1,l_{r}=l_{h}=1,u_{r}=u_{h}=2,c_{r}=c_{h}=2$, we average the minimal investment risk per asset and the Sharpe ratio over 100 trials and compare it with results based on replica analysis. Fig. 1(a) represents the return coefficient $R$ and the minimal investment risk $\varepsilon$. Fig. 1(b) represents the return coefficient $R$ and Sharpe ratio $S$. The markers with error bars are the result of numerical experiments and the solid line is the result of replica analysis. The dotted line in Fig. 1(a) is the minimum of the minimal investment risk $\varepsilon(R_{\min})$ and the dotted line in Fig. 1(b) is the maximum of Sharpe ratio $S(R^{*})$. From both figures, it is concluded that the results of replica analysis and the numerical experiments are consistent.

6 Conclusion

We have improved on the analytical methods in the extant literature and discussed the investment risk minimization problem imposing cost and return constraints. In the budget constraint used in previous studies, the purchase cost per unit of each asset is not considered in detail. The investment risk minimization problem in previous studies has focused on considering the investment ratio as a portfolio (or rendering the purchase cost identical) without considering the purchase cost in actual investment market contexts. Because the optimal investment strategy varies depending on the size of working capital and the target figure, this study investigated the investment risk minimization problem imposing the cost constraint at the initial investment period and the return constraint at the final investment period by using replica analysis, with consideration of purchase cost, initial cost, and final return. The results suggest that the minimal investment risk per asset can be expressed as a function of the initial cost and final return. We compared the minimal investment risk derived by our proposed method with the minimal expected investment risk derived by an analytical method common to operations research and confirmed that the minimal investment risk is always lower than the minimal expected investment risk. We succeeded in deriving the opportunity loss from both results. We also confirmed that the Pythagorean theorem of the Sharpe ratio holds given the relationship between the maximum value of the Sharpe ratio corresponding to the minimum and maximum values of the minimal investment risk for the revenue coefficient and the cost coefficient. Finally, we show that the results derived by our proposed method are consistent with results from numerical experiments.

The Pythagorean theorem of the Sharpe ratio and the opportunity loss are macroscopic relations that do not depend on the distribution according to purchase cost or final return. It would be fruitful for future research to explore the generality of this finding. It would also be useful to investigate whether other macroscopic relations hold in comparable contexts.

Acknowledgements

The author is grateful for discussions with D. Tada and I. Suzuki. This work was partially supported by Grants-in-Aid Nos. 15K20999, 17K01260, and 17K01249; Research Project of the Institute of Economic Research Foundation at Kyoto University; and Research Project No. 4 of the Kampo Foundation.