Optimal selection of financial risk investment portfolio based on random matrix method

Abstract

The optimization of investment portfolio is the key to financial risk investment. In this study, the investment portfolio was optimized by removing the noise of covariance matrix in the mean-variance model. Firstly, the mean-variance model and noise in covariance matrix were briefly introduced. Then, the correlation matrix was denoised by KR method (Sharifi S, Grane M, Shamaie A) from random matrix theory (RMT). Then, an example was given to analyze the application of the method in financial stock investment portfolio. It was found that the stability of the matrix was improved and the minimum risk was reduced after denoising. The study of minimum risk under different M values and stock number suggested that calculating the optimal value of M and stock number based on RMT could achieve optimal financial risk investment portfolio result. It shows that RMT has a good effect on portfolio optimization and is worth promoting widely.

Keywords

Investment portfolio mean-variance noise random matrix theory

1. Introduction

The optimal selection of financial risk portfolio is a very important issue [1]. The portfolio model can predict the development of the stock market through the past data [2]. How to maximize the return under the minimum risk by rationally allocating the assets in the portfolio through the portfolio model has become a problem that more and more investors consider [3]. More and more methods have been applied in the optimal selection of investment portfolio, such as data mining [4], genetic algorithm [5] and neural network [6]. Based on the uncertainty of covariance matrix and the value of expected return of risky assets, Zheng and Liang [7] constructed the constrained tracking error of VaR portfolio optimization model in additional transaction cost and found that the model could obtain a good final rate of return, which was closer to the actual financial market. Tirea and Negru [8] designed a portfolio optimization model based on risk and trust and analyzed the impact of risk and market expectations through text mining and emotional analysis. Through the analysis of investment target data, Huang et al. [9] selected investment portfolio using data envelopment analysis (DEA) and found that the method was highly realizable. Currently the application of variance matrix model in the optimal selection of financial risk investment portfolio is seldom studied. To realize the optimal selection of investment portfolio, investment portfolio model was optimized by denoising covariance matrix in mean-variance model with KR method (Sharifi S, Grane M, Shamaie A) from random matrix theory (RMT), and the effectiveness of the method was verified through an example. The results showed that the method was effective. This work provide a guidance for the financial risk investment and is beneficial to the further application of mean-variance model in financial risk investment. Currently the commonly used denoising methods in RMT include LCPB (Laloux L, Cizeau P, Potters M, Bouchaud J P) and PG+ (Plerou V, Gopikrishnan P). Compared to those methods, KR method has significant advantages in risk and stability. KR method with higher stability and smaller risk has a better performance in optimization of investment portfolio.

2. Mean-variance model

With the maturity and development of the stock market, investors’ ideas have also changed. The traditional way of investment depends on the long-term experience and ability of investors, i.e., personal judgment, but with the expansion of the stock market, investment based on personal judgment is not only difficult, but also risky [10]. With the advancement of computer and mathematics and technologies, more and more investors have recognized the use of computer in processing a large number of sample information for investment decision-making. The current research directions of investment portfolio include:

(1)
Return and risk based measurement. In investment portfolio, the optimal allocation of asset can be realized by measuring the return and risk of asset. The risk can be measured by variance, lower semi-variance, absolute deviation and semi-absolute deviation. Risk measurement based on VaR model is also a good direction. The return of investment can be analyzed through fuzzy number, utility function and measure of skewness.
(2)
Different constraints based investment portfolio model. In investment portfolio, there are many constraint conditions. Under different constraint conditions, the mode of investment portfolio is also different. The constraint conditions considered currently include transaction fees, inflation, taxes and short selling. Establishing investment portfolio model based on different constraint conditions is also an important part of financial risk investment portfolio.
(3)
The optimal solution of investment portfolio model. The solution of investment portfolio is unstable, leading to large solution difficulty. Therefore researchers have made extensive and deep study on the optimal solution of model, for example, neural network, genetic algorithm [11], particle swarm algorithm and group decision method, which shows good performance in model optimal solution.

Markowitz first proposed and established the mean-variance model in 1952, which set a precedent for the portfolio investment [12]. With the successful application of the portfolio model in the stock market, more and more investors turn their attention to how to choose the appropriate method to construct and optimize the financial risk portfolio.

Mean-variance model includes the following hypothesis.

(1)
All the investment are risky.
(2)
Investors pursue for the maximum return at the same risk level and the minimum risk at the same return level.
(3)
Investors only make decisions based on risk and return.
(4)
Short selling is allowed and investment is infinitely divisible.
(5)
Rate of return is in normal distribution and linearly independent.

If an investor selected $n$ stocks, then

the rate of return was: $r_{i}(i=1,2,\ldots,n)$ ;

the random vector of rate of return was: $r=(r_{1},r_{2},\ldots,r_{n})$ ;

the vector of expected rate of return was: $\mu=(E(r_{1}),E(r_{2}),\ldots,E(r_{n}))$ ;

the covariance matrix was: $\Sigma=(\sigma_{ij})_{n\times n}$ , where $\sigma_{ij}$ stands for the covariance between the $i$ -th stock and $j$ -th stock.

The investment weight of the $i$ -th stock was: $\omega_{i}$ , satisfying $\sum_{i=1}^{n}{\omega_{i}=1}$ .

Suppose the given rate of return as $\mu_{0}$ , then the optimal model of minimized risk portfolio investment portfolio was:

$\displaystyle\mathop{\min}\limits_{\omega}\Phi_{\omega}=\omega^{\prime}\Sigma\omega$ $\displaystyle st.\left\{\begin{array}[]{l}\omega^{\prime}\mu=\mu_{0}\\ \sum\nolimits_{i=1}^{n}\omega_{i}=1\\ \omega_{i}\geqslant 0\\ \end{array}\right.$

Suppose the given risk as $\Sigma_{0}$ , then the optimal model of investment portfolio with the maximum return was:

$\displaystyle\mathop{\min}\limits_{\omega}\Phi_{\omega}=\omega^{\prime}\mu$ $\displaystyle st.\left\{\begin{array}[]{l}\omega^{\prime}\Sigma\omega=\Sigma_{% 0}\\ \sum\nolimits_{i=1}^{n}{\omega_{i}=1}\\ \omega_{i}\geqslant 0\\ \end{array}\right.$

Compared to other denoising methods, RMT method can determine the best dimension of model and has low technical difficulty and small limitation in application, which has a good application in different fields. According to the RMT [13], there is a lot of noise in the covariance matrix (correlation matrix), which will have a great impact on the optimization results of model. The noise in the correlation matrix can be determined by the distribution of eigenvalues of the random correlation matrix. The random correlation matrix is expressed as:

$\displaystyle R=\frac{1}{T}AA^{T},$

where $A$ stands for a $N\times T$ matrix, random variables with the same distribution (mean value: 0; variance: 1). Under the condition of $N,T\to\infty$ and $M=\frac{T}{N}$ , the probability density function of eigenvalue is:

$\displaystyle\rho(\lambda)=\left\{\begin{array}[]{ll}\frac{M}{2\pi}\frac{\sqrt% {(\lambda_{+}-\lambda)(\lambda-\lambda_{-})}}{\lambda}&\lambda_{-}\leqslant% \lambda\leqslant\lambda_{+}\\ 0&\textit{else}\\ \end{array}\right.$

where $\lambda_{\pm}$ stands for the top and bottom limitations of distribution interval of matrix eigenvalues, and its computational formula is $\lambda_{\pm}=1+\frac{1}{M}\pm 2\sqrt{\frac{1}{M}}$ .

Eigenvalues can be divided into noise and non-noise. The process of denoising is shown in Fig. 1.

Figure 1.
The matrix denoising process.

3. Denoising based on RMT-KR

There are many ways replacing noise eigenvalues with new eigenvalues. In this study, KR method [14] from RMT was used.

Suppose the eigenvalue of a $N\times N$ correlation matrix $M$ as $\Lambda=\{{y_{i}}\}_{i=1}^{N},y_{1}<y_{2}<\ldots<y_{N}$ , the matrix composed of the eigenvectors of $M$ as $E$ , the eigenvalue of noise as $\Lambda_{\textit{noisy}}=\{{y_{i}}\}_{i=1}^{n}$ , and the new eigenvalue which is used for replacing noise eigenvalue as $\Lambda_{\textit{new}}=\{{x_{i}}\}_{i=1}^{n}$ .

The eigenvalue obtained after denoising was $\Lambda_{\textit{filtered}}=\Lambda_{\textit{new}}\cup(\Lambda-\Lambda_{% \textit{noisy}})$ , where $\Lambda-\Lambda_{\textit{noisy}}=\{{\lambda\in\Lambda:\lambda\notin\Lambda_{% \textit{noisy}}}\}$ is useful information in the original correlation matrix besides noise, and $\{{y_{i}}\}_{i=n+1}^{N}$ is eigenvalue which contains information.

$\Lambda_{\textit{new}}=\{\lambda_{i}:\lambda_{i}=\bar{\Lambda}_{\textit{noisy}% }\forall i=1,2,\ldots,n\}$ is the set of substitute eigenvalues, which means that the original noise eigenvalue is replaced by the average value of noise eigenvalues.

The diagonal matrix of $\Lambda_{\textit{filtered}}$ is denoted as $D_{\textit{filtered}}$ , and the correlation matrix after denoising [15] is:

$\displaystyle M_{\textit{filtered}}=\textit{ED}_{\textit{filtered}}E^{-1}.$

4. Instance analysis

4.1 Denoising performance of RMT-KR

4.1.1 Stability of investment portfolio

One hundred A-share stocks of Shanghai Stock Exchange were randomly selected for constructing the investment portfolio. The weekly rate of return from June 1, 2012 to December 31, 2016 was taken as the actual rate of return. The correlation matrix was established and then denoised by RMT-KR method. Krzanowski stability [16] was used for representing the stability of eigenvector in the matrix, i.e., angle $\theta_{i}$ between eigenvector $c_{i}$ and maximum disturbance vector $c_{i}^{p}$ when eigenvalue $\lambda_{i}$ of the correlation matrix reached $\lambda_{i}^{p}$ after a change no larger than $\kappa(\kappa>0)$ .

$\displaystyle\cos\theta_{i}=\left\{\begin{array}[]{ll}\left({1+\frac{\kappa}{% \lambda_{i}-\lambda_{i-1}}}\right)^{-0.5}&\lambda_{i}^{p}<\lambda_{i}\\ \left({1+\frac{\kappa}{\lambda_{i+1}-\lambda_{i}}}\right)^{-0.5}&\lambda_{i}<% \lambda_{i}^{p}\\ \end{array}\right.,\quad\lambda_{1}<\lambda_{2}<,\ldots,<\lambda_{N}$

The larger the value of $\cos\theta_{i}$ , the smaller the value of $\theta_{i}$ and the more stable the value of $c_{i}$ . Krzanowski suggests $\kappa=\beta\lambda_{i},\beta=$ 0.1, 0.05 or 0.01.

The value of $\cos\theta_{i}$ of the matrix before and after denoising was calculated, and then the $\cos\theta$ diagram of the matrix was obtained, as shown in Fig. 2.

The black line in Fig. 2 shows cos $\theta$ before denoising. It was found that the stability of the matrix decreased with the increase of the order of the correlation matrix. The blue line in Fig. 2 shows cos $\theta$ after denoising by LCPB method. The red line in Fig. 2 shows cos $\theta$ after denoising by KR method. It was found that the stability of the matrix changed significantly after denoising treatment, and the denoising stability of KR method was superior to that of LCPB method, which indicated that RMT-KR method had a good denoising effect.

4.1.2 Minimized investment portfolio

Minimum investment risk means $\min\sum_{i,j=1}^{n}\omega_{i}\sigma_{ij}\omega_{j}$ on the premise of satisfying $\sum_{i=1}^{n}\omega_{i}=1$ , and the investment weight at that moment is:

$\displaystyle\omega_{i}=\frac{\sum_{j=1}^{n}{\sigma_{ij}^{-1}}}{\sum_{j,k=1}^{% n}\sigma_{jk}^{-1}}$

Four risk forms of investment portfolio with the minimum risk in two continuous holding periods were analyzed, as shown in Table 1.

Table 1
Four risk forms and their meanings

Risk form	Computational formula	Meaning
$R_{1}$	$R_{1}=\sum\nolimits_{i,j=1}^{n}{\omega_{i}(0)\sigma_{ij}(0)\omega_{j}(0})$	Risk after denoising in the first hold period
$R_{2}$	$R_{2}=\sum\nolimits_{i,j=1}^{n}{\omega_{i}(1)\sigma_{ij}(1)\omega_{j}(1})$	Risk in the first hold period
$R_{3}$	$R_{3}=\sum\nolimits_{i,j=1}^{n}{\omega_{i}(1)\sigma_{ij}(2)\omega_{j}(1})$	Risk in the second holding period
$R_{4}$	$R_{4}=\sum\nolimits_{i,j=1}^{n}{\omega_{i}(0)\sigma_{ij}(2)\omega_{j}(0})$	Risk after denoising in the second holding period

In Table 1, $\sigma_{ij}(0)$ stands for the denoising reconstruction covariance of the first phase of investment portfolio, and $\omega_{i}(0)$ stands for weight of asset.

Table 2

Investment portfolio with minimum risk

	$R_{1}$	$R_{2}$	$R_{3}$	$R_{4}$
Risk	0.2684	0.4025	0.4112	0.3266

Figure 2.

The rank-overlap (cos $\theta$ ) diagram of the matrix.

Table 2 shows that the minimum risk of the portfolio after RMT denoising was smaller than that without denoising. It was concluded from Table 2 and Fig. 2 that the denoising method could not only keep the matrix more stable, but also reduce the risk of investment portfolio.

4.2 Instance analysis of investment portfolio optimization

A-share stock of Shanghai Stock Exchange was selected as the sample. The weekly rate of return at June 2012 was regarded as the actual rate of return. The minimized risk portfolio of 50 stocks under the condition of $M=\frac{T}{N}=$ 1, 2, 3, 4, 5 and the minimized risk portfolio of 40, 50, 60, 70 and 80 stocks under the condition of $M=\frac{T}{N}=$ 1.5 were studied, and the results are shown in Fig. 3.

Figure 3.

The minimum risk of portfolio of 50 stocks under different values of M.

Figure 4.

The minimum risk of investment portfolio of different number of stocks when the value of M is 1.5.

With the increase of M value, the noise rate in the correlation matrix decreased from 98% to 66%. The risk after denoising was always less than the risk before denoising. When the value of M was 2, the predicted risk was the closest to the actual risk, which indicated that the most effective way to predict the investment risk in the next 50 weeks was to use data in 100 weeks, and the optimal value of M could be determined.

When the value of M was between 1 and 3, the portfolio risk after denoising was smaller than that before denoising, which indicated that denoising made the portfolio risk smaller. When the value of M was 4 or 5, the predicted risk was smaller than the actual risk, which indicated that the validity of historical information decreased with the increase of M value and the investment risk could not be accurately predicted.

Figure 5.

The efficient frontier of investment portfolio.

As the number of investment stocks increased, the noise in the correlation matrix increased gradually, from 85% to 92%. When the number of stocks was 60, the predicted risk was the closest to the actual risk, and the prediction effect of the model was the best, which showed that predicting the data in the following 90 weeks was highly accurate when the data in 90 weeks were taken for reference. When the number of stocks increased from 40 to 60, the predicted risk was larger than the actual risk; when the number of stocks increased from 60 to 80 stages, the predicted risk was smaller than the actual risk, indicating that there was an optimal stock number when the value of M remained unchanged and the risk might be overestimated if the number of stocks was smaller than the optimal number and underestimated if the number of stocks was larger than the optimal number.

The efficient frontier analysis of investment portfolio was carried out taking $M=$ 1.5 as an example, and the results are shown in Fig. 5.

Curve A in Fig. 5 represents the efficient frontier of the original correlation matrix prediction, curve B represents the efficient frontier of the prediction after denoising, curve C represents the efficient frontier obtained according to the original correlation matrix, and curve D represents the efficient frontier obtained from the correlation matrix after denoising. It was found that curve A was below curve B, indicating that the risk after denoising was significantly smaller than that before denoising. The comparison of the efficient frontier under different number of stocks suggested that the predicted and actual efficient frontier were the most similar when the number of stocks was 60, and the prediction effect at that moment was the best.

It was found from Figs 3–5 that establishing investment portfolio through RMT denoising and determining the best number of stocks and M value could obtain the optimal investment portfolio and achieve the largest benefit under the smallest risk.

5. Discussion

Investment portfolio optimization is the concern of many investors. The optimal selection of investment portfolio can help investors choose the appropriate number of assets and investment proportion, improve risk tolerance, and obtain greater benefits [17]. Markowitz’s mean-variance model has a good application in investment portfolio selection [18, 19], but some researchers have found that there are a lot of noise in the covariance matrix in the mean-variance model, which has a great impact on the accuracy of matrix prediction. RMT can effectively distinguish noise information from useful information in system, and has been widely used in fields such as biology and finance [20, 21]. In this study, the RMT based denoising method was applied in processing noise in covariance matrix. The results showed that the method was effective.

Firstly, the comparison of the stability of portfolio of 100 stocks suggested that the stability of the matrix changed significantly after RMT denoising, which verified the good denoising effect of the method, and it was consistent with the results of Daly et al. [22, 23]. Moreover, the denoising effect of KR method was significantly superior to LCPB method. The same result can be obtained from the comparison of the minimum risk value before and after denoising. The minimum risk value of the denoised matrix was smaller than that of the original one. It indicated that the matrix could not only ensure the stability of the matrix, but also reduce risks.

The analysis of the minimized risk portfolio of 50 stocks under different values of M showed that the noise rate in the portfolio decreased gradually with the increase of M value. When the value of M was 2, the predicted risk was closest to the actual risk, and the accuracy of risk prediction varied with the change of value of M. It showed that the optimal value of M could be determined by calculation. The analysis of the minimized risk portfolio of different number of stocks when the value of M was 1.5 showed that different number of stocks had an impact on the risk. It showed that the optimal number of stocks existed when the value of M was fixed and the optimization result of portfolio could be improved by RMT method. The analysis results of the portfolio efficient frontier sowed that the portfolio prediction effect was the best when the number of stocks was 60. Based on the above results, it was found that the optimal portfolio could be determined by determining the optimal number of stocks and M value in the RMT denoising-based portfolio optimization method.

This research is helpful for the further development of RMT denoising theory, promoting the research of portfolio optimization theory, helping investors make more correct decisions and promoting the healthy and stable development of the stock market. However, this research is still in its infancy, and there are still some problems to be solved, such as the applicability of the method in other financial investment such as foreign exchange and future goods and whether there is more effective RMT denoising method, which needs further study.

6. Conclusion

In this study, the RMT method was used to denoise the covariance matrix in the mean-variance matrix. RMT reduced the influence of noise in the matrix and optimized the financial risk portfolio. Instance analysis suggested that RMT could effectively maintain the stability of matrix and reduce risk. The application of RMT can obtain the minimized risk investment portfolio and has a good prospect in the stock market.

References

Bacanin

and Tuba

, Firefly algorithm for cardinality constrained mean-variance portfolio optimization problem with entropy diversity constraint, The Scientific World Journal 2014(1) (2014), 721521.

Hosseini

and Hamidi

, Common funds investment portfolio optimization with fuzzy approach, Procedia Economics & Finance 36 (2016), 96–107.

Marasović

P.B.

and Vukasović

, The impact of transaction costs on rebalancing an investment portfolio in portfolio optimization, International Journal of Social Education Economics & Management Engineering 9(3) (2015), 844–849.

Liao

S.H.

and Chou

S.Y.

, Data mining investigation of co-movements on the Taiwan and China stock markets for future investment portfolio, Expert Systems with Applications 40(5) (2013), 1542–1554.

Peng

and Zhang

, Investment portfolio optimization of power grid projects based on niche genetic algorithms, International Conference on Natural Computation, IEEE, 2016, pp. 1170–1175.

Liu

Dang

and Huang

, A one-layer recurrent neural network for real-time portfolio optimization with probability criterion, IEEE Transactions on Cybernetics 43(1) (2013), 14–23.

Zheng

and Liang

X.K.

, Constraint tracking error for investment portfolio optimization model and algorithm of VaR in additional transaction costs, Applied Mechanics & Materials 543–547 (2014), 1811–1816.

Tirea

and Negru

, Investment portfolio optimization based on risk and trust management, IEEE, International Symposium on Intelligent Systems and Informatics. IEEE, 2013, pp. 369–374.

Huang

C.Y.

Chiou

C.C.

T.H.

et al., An integrated DEA-MODM methodology for portfolio optimization, Operational Research 15(1) (2015), 115–136.

10.

Graham

J.R.

and Harvey

C.R.

, The long-run equity risk premium, Finance Research Letters 2(4) (2005), 185–194.

11.

Sukono

S.D.

Najmia

et al., Analysis of stock investment selection based on CAPM using covariance and genetic algorithm approach, IOP Conference Series: Materials Science and Engineering 332 (2018), 012046.

12.

Markowitz

H.M.

, Portfolio selection, Journal of Finance (7) (1952), 77–91.

13.

Baik

Deift

and Suidan

, Combinatorics and random matrix theory, Graduate Studies in Mathematics 172 (2016), 333–349.

14.

Sharifi

Crane

Shamaie

and Ruskin

, Random matrix theory for portfolio optimization: a stability approach, Physica A: Statistical Mechanics and Its Applications 335(3) (2003).

15.

Plerou

Gopikrishnan

Rosenow

Amaral

L.A.N.

and Stanley

H.E.

, A random matrix theory approach to financial cross-correlations, Physica A: Statistical Mechanics and Its Applications 287(3) (2000).

16.

Luo

Han

Gong

et al., Random matrix denoising method based on Monte Carlo simulation as amended, Journal of Computer Applications 36(9) (2016), 2642–2646.

17.

Mohan

Singh

J.G.

and Ongsakul

, Sortino Ratio Based Portfolio Optimization Considering EVs and Renewable Energy in Microgrid Power Market, IEEE Transactions on Sustainable Energy 8(1) (2016), 219–229.

18.

Nkeki

C.I.

, Optimal investment strategy with dividend paying and proportional transaction costs, Annals of Financial Economics 13(1) (2018), 1850001.

19.

Bijan

, Effective Stock selection and portfolio construction within us, international, and emerging markets, Frontiers in Applied Mathematics and Statistics 4 (2018), 17.

20.

Veraart

Fieremans

and Novikov

D.S.

, Diffusion MRI noise mapping using random matrix theory, Magnetic Resonance in Medicine 76(5) (2016), 1582–1593.

21.

Luo

Han

Chen

et al., Partial discharge detection and recognition in random matrix theory paradigm, IEEE Access PP(99) (2016), 1.

22.

Daly

Crane

and Ruskin

H.J.

, Random matrix theory filters and currency portfolio optimisation, Journal of Physics: Conference Series 221 (2010), 012003.

23.

Daly

Crane

and Ruskin

H.J.

, Random matrix theory filters in portfolio optimisation: A stability and risk assessment, Physica A 387(16-17) (2008), 4248–4260.

Optimal selection of financial risk investment portfolio based on random matrix method

Abstract

Keywords

1. Introduction

2. Mean-variance model

4. Instance analysis

4.1 Denoising performance of RMT-KR

4.1.1 Stability of investment portfolio

4.1.2 Minimized investment portfolio

Table 1 Four risk forms and their meanings

6. Conclusion

References

Table 1
Four risk forms and their meanings