GO-GJRSK Model with Application to Higher Order Risk-Based Portfolio

Abstract

There are three distinguishing features in the financial time series, such as stock prices, are as follows: (1) Non-normality, (2) serial correlation, and (3) leverage effect. All three points need to be taken into account to model the financial time series. However, multivariate financial time series modeling involves a large number of stocks, with many parameters to be estimated. Therefore, there are few examples of multivariate financial time series modeling that explicitly deal with higher-order moments. Furthermore, there is no multivariate financial time series model that takes all three characteristics above into account. In this study, we propose the generalized orthogonal (GO)-Glosten, Jagannathan, and Runkle GARCH (GJR) model which extends the GO-generalized autoregressive conditional heteroscedasticity (GARCH) model and incorporates the three features of the financial time series. We confirm the effectiveness of the proposed model by comparing the performance of risk-based portfolios with higher-order moments. The results show that the performance with our proposed model is superior to that with baseline methods, and indicate that estimation methods are important in risk-based portfolios with higher moments.

1. Introduction

The financial time series is continually brought to our attention. Daily news reports inform us for instance of the latest stock market index values, currency exchange rates, oil prices, and interest rates. It is often desirable to monitor their price behaviors frequently and to try to understand the probable development of the prices in the future. Institutional and individual investors involved in international trades can all benefit from a deeper understanding of price behaviors. Therefore, modeling and understanding the financial time series is a very important task.

There are three distinguishing features in the financial time series, such as stock prices, as follows: (1) Non-normality, (2) serial correlation, and (3) leverage effect. (1) Non-normality means that the returns of financial time series does not follow the normal distribution because of negative skewness and large kurtosis [1,2]. Therefore, we need to model not only mean and variance but also higher-order moments such as skewness and kurtosis. (2) The serial correlation of return is known as momentum or reversal effect. The momentum [3] (reversal [4]) effect means that stock returns with a large (small) return in the past will continue to increase (decrease) during a certain period thereafter. In the case of volatility, the serial correlation is widely known as the volatility clustering [5]. It is a phenomenon that large changes in returns tend to cluster together, resulting in the persistence of the amplitudes of returns. The volatility clustering has intrigued many researchers and oriented in a major way the development of financial time series models. For example, the autoregressive conditional heteroscedasticity (ARCH) [5] and the generalized ARCH (GARCH) model [6] are intended primarily to model this phenomenon. The authors of [7] demonstrated that skewness and kurtosis also have serial correlations, and they developed the GARCH with skewness and kurtosis (GARCHSK) model to capture these correlations of skewness and kurtosis. (3) The leverage effect corresponds to a negative correlation between returns and volatility. The Glosten, Jagannathan, and Runkle GARCH (GJR) model [8] is a representative model that captures the leverage effect. Recently, Many studies [9,10] demonstrated a negative correlation between returns, and skewness and kurtosis.

We need to take all three points into account to model the financial time series. However, when considering multivariate financial time series modeling involving a large number of stocks, there are many parameters to be estimated. Even when we only model the time-varying covariance matrix, it is very difficult to estimate it completely [11]. The generalized orthogonal GARCH (GO-GARCH) model [12] is proposed to estimate the time-varying covariance matrix using independent component analysis (ICA) to reduce the parameters to be estimated.

Therefore, there are few examples of multivariate financial time series modeling that explicitly deal with higher-order moments [13]. Furthermore, there is no multivariate financial time series that takes all three characteristics above into account.

In this study, we propose the GO-GJRSK model which extends the GO-GARCH and incorporates the three features of the financial time series. To define GO-GJRSK model, we first introduce a univariate GJR with skewness and kurtosis (GJRSK) model which is the GARCHSK model with a leverage effect. Furthermore, we extend the GJRSK model to a multivariate model with ICA.

Our proposed method describes the conditional volatility (skewness and kurtosis) in a straightforward form: The linear sum of the previous time volatility (skewness and kurtosis) and the shock with leverage effect. As with the GO-GARCH model, the estimation of the GO-GJRSK model is also easy when the number of stocks is large enough.

We confirm the effectiveness of the proposed methodology by comparing the performance of risk-based portfolios with higher-order moments. We use a representative benchmark dataset to test whether differences in estimation methods improve the performance of risk-based portfolios.

The remainder of the paper is organized as follows. Section 2 gives a brief explanation of risk-based portfolios and the volatility modeling methods. Section 3 describes our proposed GO-GJRSK model, which consists of an ICA and GJRSK model. Section 4 experiments with various risk-based portfolios with different estimation methods. Lastly, Section 5 concludes the paper.

2. Related Work

2.1. Risk Based Portfolio

Portfolio construction is one of the most important tasks in finance. Mean Variance Optimization (MVO) proposed by [14] has been the most used in practice for portfolio construction and asset allocation for a long time. The MVO is still a very important framework for investment decision making today. The MVO provides economical justification because it is not only simple and practically tractable but also because it is consistent with the expected utility maximization principle. The goal of MVO is to find portfolios that optimally diversify risk without reducing expected return and to facilitate portfolio construction.

However, there are some practical difficulties with the MVO. First, it is very difficult to estimate the expected return as inputs of the MVO. In addition, the MVO is referred to as “Error Maximization” [15,16]. The reason is that even small changes in the estimated expected returns can result in huge changes in the whole portfolio structure.

Therefore, risk-based portfolio construction methods, which do not require the expected return, which is difficult to estimate, but are based only on risk, are attracting attention as an alternative to the MVO. Minimum variance portfolio [17], risk parity portfolio [18], and maximum diversification portfolio [19] are known as the typical risk-based portfolio construction methods.

In fact, many empirical studies of equity portfolios and asset allocations [17,18,19,20] showed that the risk-based portfolio construction methods perform better than the MVO and market capitalization-weighted portfolios. In [21], it was demonstrated that risk-based portfolios are not very sensitive to the accuracy of estimating covariance matrices, unlike MVO.

More recently, the covariance matrix has been estimated using Gaussian process latent variable models [22], and Student’s t-process latent variable models [23] for the minimum variance portfolio. And new risk parity portfolios are proposed by using the Hilbert transform to make the covariance matrix complex [24] or extended to quaternions [25].

However, these risk-based portfolio construction methods capture risk only in terms of covariance matrix (second moments) and do not capture the non-normality, i.e., skewness and kurtosis. Therefore, risk-based portfolios with higher-order moments have been proposed and their effectiveness has been confirmed [13,26,27]. We examine whether the estimation method improves the performance of risk-based portfolios with higher moments.

2.2. Volatility Modeling

Volatility modeling is important for risk management and the portfolio selection. In [5], it was proposed that the ARCH model could model the serial correlation of volatility. In [6], the GARCH model was proposed, which is a generalization of the ARCH model. The GARCH model assumes that the volatility at time t is formulated with only variables whose values are already known in time $t-1$. The GARCH model has been applied to the empirical analysis of many asset prices because the parameters can easily be estimated by the maximum likelihood method. The GARCH model has been modified and extended in a number of ways. An important aspect of improving the GARCH model is the introduction of a leverage effect.

The GJR model [8] is a representative model that incorporates a leverage effect. In another direction of extension of GARCH model, León et al. [7] proposed the GARCHSK model, which also incorporates conditional skewness and conditional kurtosis variations. The feature of the GARCHSK model is that the conditional skewness and conditional kurtosis variations can be explicitly captured with an easy-to-understand structure equivalent to the GARCH model, and the easy to estimate.

Furthermore, extensions to multivariate models are also proposed. In order to efficiently parameterize multivariate GARCH, some methods have been developed based on univariate GARCH models in the following two directions. One is to construct the conditional covariance matrix by explicitly modeling both the volatility and conditional correlation matrix. For example, Engle [28] proposed the dynamic conditional correlation (DCC) model. The other direction is to exploit the idea of factor models to construct multivariate GARCH models. It is widely believed in finance that there exist some common factors, which may be determined empirically or constructed by some statistical methods, driving the evolution of the return series [22,23,29,30]. This approach has commonly been used in much research to model the conditional covariance of financial time series due to its feasibility in estimating large covariance matrices.

Orthogonal GARCH (OGARCH) was first proposed by [31]. The observed time series can be linearly transformed into a set of uncorrelated time series using a principal component analysis. In [12], the concept of ICA is applied to propose the GO-GARCH model for volatility modeling. It consists of a set of conditionally uncorrelated univariate GARCH and a linear map that allows the linkage between these components and observed data. Furthermore, a modified GO-GARCHSK model [32] that incorporates the GO-GARCH model with conditional skewness and conditional kurtosis variation has been proposed. However, there are no multivariate models that have a conditional skewness and kurtosis variation and leverage effect.

Hence, as a multivariate extension of the GJRSK, we propose GO-GJRSK model that has higher order moment variations and a leverage effect (

Table 1. Comparison between our proposed and the models of related work.

	Non-Normality	Higher Moment	Leverage Effect
GO-GARCH [12]	〇
GO-GARCHSK [32]	〇	〇
GO-GJRSK (Our proposed)	〇	〇	〇

).

3. GO-GJRSK Model

3.1. Univariate GJRSK Model

We first define a univariate GJRSK model. This model introduces leverage effects in the GARCHSK model. Here, $r_t$ is the asset return, $h_t$ is conditional volatility, $s_t$ is conditional skewness, and $k_t$ is conditional kurtosis at time t. The formulation of the GJRSK model is given as follows.

$$\begin{array}{cc}\hfill & r_t={\alpha }_0+{\alpha }_1{r}_{t-1}+{\epsilon }_t\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill & h_t={\beta }_0+{\beta }_1{\epsilon }_{t-1}^2+{\beta }_2{h}_{t-1}+{\beta }_3{\epsilon }_{t-1}^2{I}_{\{{\eta }_{t-1}<0\}}\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill & s_t={\gamma }_0+{\gamma }_1{\eta }_{t-1}^3+{\gamma }_2{s}_{t-1}+{\gamma }_3{\eta }_{t-1}^3{I}_{\{{\eta }_{t-1}<0\}}\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill & k_t={\delta }_0+{\delta }_1{\eta }_{t-1}^4+{\delta }_2{k}_{t-1}+{\delta }_3{\eta }_{t-1}^4{I}_{\{{\eta }_{t-1}<0\}}\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill & {\eta }_t=h_t^{-\frac{1}{2}}{\epsilon }_t\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill & {\eta }_t|I_{t-1}\sim g(0,1,s_t,k_t)\hfill \end{array}$$

(6)

Here ${\alpha }_0,{\alpha }_1$ are parameters of the AR model and ${\beta }_i,{\gamma }_i,{\delta }_i,i=0,1,2,3$ are parameters of the GJRSK model. $I_A$ is an indicator function that returns 1 if A is true and 0 otherwise. g is a probability density function with mean 0, variance 1, skewness $s_t$, and kurtosis $k_t$.

As with GARCHSK model, the probability density function $g(0,1,s_t,k_t)$ of the GJRSK model can be given by Gram-Charlier expansion using Chebyshev-Hermite polynomials as follows:

$$\begin{array}{cc}\hfill & g\left({\eta }_t\right|I_{t-1})=\frac{\varphi \left({\eta }_t\right){\psi }^2\left({\eta }_t\right)}{{\mathsf{\Gamma }}_t},\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill & \varphi \left({\eta }_t\right)=\frac{1}{\sqrt{2\pi h_t}}exp({\eta }_t^2-h_t),\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill & \psi \left({\eta }_t\right)=1+\frac{s_t}{3!}({\eta }_t^3-3{\eta }_t)+\frac{k_t-3}{4!}({\eta }_t^4-6{\eta }_t^2+3),\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill & {\mathsf{\Gamma }}_t=1+\frac{s_t^2}{3!}+\frac{{(k_t-3)}^2}{4!}.\hfill \end{array}$$

(10)

The probability density function of ${\epsilon }_t$ is $f\left({\eta }_t\right|I_{t-1})=h_t^{\frac{1}{2}}g\left({\eta }_t\right|I_{t-1})$ because ${\eta }_t=h_t^{-\frac{1}{2}}{\epsilon }_t$. Therefore, the log-likelihood function $l_t$ without the constant term is obtained as:

$$l_t=-\frac{1}{2}ln{h}_t-\frac{1}{2}{\eta }_t^2+ln\left({\psi }^2\left({\eta }_t\right)\right)-{\mathsf{\Gamma }}_t.$$

(11)

We can estimate each parameter of the GJRSK model by maximizing log-likelihood function $l_t$.

3.2. GO-GJRSK Model

In this section, we introduce the GO-GJRSK model. We define the co-skewness matrix $M_3$ and co-kurtosis matrix $M_4$. Let co-skewness of assets $i,j,k$ be $s_{ijk}$ and $S_{ijk}$ be the matrix that can be made by fixing i of $s_{ijk}$. The co-skewness matrix $M_3$ is defined as follows.

$$\begin{array}{cc}\hfill & S_{ijk}=\left(\begin{array}{ccc}{s}_{i11}& \dots & s_{i1k}\\ ⋮& \ddots & ⋮\\ s_{ij1}& \dots & s_{ijk}\end{array}\right)\hfill \\ \hfill & M_3=[S_{1jk},S_{2jk},\cdots ,S_{Njk}]\hfill \end{array}$$

(12)

Let co-kurtosis of assets $i,j,k,l$ be $k_{ijkl}$ and $K_{ijkl}$ be the matrix that can be made by fixing $i,j$ of $k_{ijk}$. The co-kurtosis matrix $M_4$ is defined as follows:

$$\begin{array}{cc}\hfill & K_{ijkl}=\left(\begin{array}{ccc}{k}_{ij11}& \dots & k_{ij1k}\\ ⋮& \ddots & ⋮\\ k_{ijk1}& \dots & k_{ijkl}\end{array}\right)\hfill \\ \hfill & M_4=[K_{11kl},\cdots ,K_{1Nkl}|K_{21kl},\cdots ,K_{2Nkl}|,\cdots ,|K_{N1kl},\cdots ,K_{NNkl}].\hfill \end{array}$$

(13)

The GO-GJRSK model assumes that ${\epsilon }_t\in R^n$ is defined by mixing matrix $Z\in R^{n\times n}$ and n independent factors $y_t={(y_{1,t},\cdots ,y_{n,t})}^T\in R^n$. When the n assets returns are $r_t\in R^n$, the GO-GJRSK model can be written as follows:

$$\begin{array}{cc}\hfill & r_t=A_0+A_1{r}_{t-1}+{\epsilon }_t,\hfill \end{array}$$

(14)

$$\begin{array}{cc}\hfill & {\epsilon }_t=Z{y}_t,\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill & {\eta }_{i,t}={m_{i,t}^H}^{-\frac{1}{2}}{y}_{i,t},\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill & {\eta }_{i,t}|I_{t-1}\sim g(0,1,m_{i,t}^S,m_{i,t}^K),\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill & m_{i,t}^H={\beta }_{i,0}+{\beta }_{i,1}{y}_{i,t-1}^2+{\beta }_{i,2}{m}_{i,t-1}^H+{\beta }_{i,3}{y}_{i,t-1}^2{I}_{\{{\eta }_{i,t-1}<0\}},\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill & m_{i,t}^S={\gamma }_{i,0}+{\gamma }_{i,1}{\eta }_{i,t-1}^3+{\gamma }_{i,2}{m}_{i,t-1}^S+{\gamma }_{i,3}{\eta }_{i,t-1}^3{I}_{\{{\eta }_{i,t-1}<0\}},\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill & m_{i,t}^K={\delta }_{i,0}+{\delta }_{i,1}{\eta }_{i,t-1}^4+{\delta }_{i,2}{m}_{i,t-1}^K+{\delta }_{i,3}{\eta }_{i,t-1}^4{I}_{\{{\eta }_{i,t-1}<0\}},\hfill \end{array}$$

(20)

where $A_0\in R^n$ and $A_1\in R^{n\times n}$ are parameters of the VAR model and ${\beta }_{i,j},{\gamma }_{i,j},{\delta }_{i,j},i=1,\cdots ,n$, $j=0,\cdots ,3$ are parameters of the GJRSK model. The GO-GJRSK model follows a probability density function $p(0,H_t,S_t,K_t)$ with a conditional covariance matrix $H_t$, a co-skewness matrix $S_t$, and a co-kurtosis matrix $K_t$. Here, the following relationship holds:

$$\begin{array}{cc}\hfill & H_t=Z{M}_t^H{Z}^T\hfill \end{array}$$

(21)

$$\begin{array}{cc}\hfill & S_t=Z{M}_t^S(Z^T\otimes Z^T)\hfill \end{array}$$

(22)

$$\begin{array}{cc}\hfill & K_t=Z{M}_t^K(Z^T\otimes Z^T\otimes Z^T)\hfill \end{array}$$

(23)

$$\begin{array}{cc}\hfill & M_t^H={\mathrm{diag}}^H\{m_{1,t}^H,\cdots ,m_{n,t}^H\}\hfill \end{array}$$

(24)

$$\begin{array}{cc}\hfill & M_t^S={\mathrm{diag}}^S\{m_{1,t}^S,\cdots ,m_{n,t}^S\}\hfill \end{array}$$

(25)

$$\begin{array}{cc}\hfill & M_t^K={\mathrm{diag}}^K\{m_{1,t}^K,\cdots ,m_{n,t}^K\}.\hfill \end{array}$$

(26)

The operator ⊗ represents the Kronecker product. ${\mathrm{diag}}^H(e_1,\cdots ,e_n)$ makes a $(n\times n)$ sparse covariance matrix with elements $e_i$ when $i=j$ and zero otherwise. ${\mathrm{diag}}^S(e_1,\cdots ,e_n)$ makes a $(n\times n^2)$ sparse co-skewness matrix with elements $e_i$ when $i=j=k$ and zero otherwise. ${\mathrm{diag}}^K(e_1,\cdots ,e_n)$ makes a $(n\times n^3)$ sparse co-kurtosis matrix with elements $e_i$ when $i=j=k=l$ and zero otherwise. Using this relationship, we can obtain the conditional covariance matrix $H_t$, the co-skewness matrix $S_t$, and the co-kurtosis matrix $K_t$.

3.3. Estimation Procedure

The parameters of GO-GJRSK model are estimated by the following procedure.

• Step1 VAR Model:
  First, we estimate the vector autoregression model (VAR) model in Equation (15). After estimating the parameter of VAR model, the residuals ${\epsilon }_t$ have been collected for further GO-GJRSK modeling.
• Step2 Independent Component Analysis:
  ICA is a signal processing method to express a set of random variables as linear combinations of statistically independent component variables [33]. If $yi,t$ and $y_{j,t},(i\ne j)$ are mutually statistically independent, and mixing matrix Z exists, such that:

  $${\epsilon }_t=Z{y}_t$$

  (27)

  then $y_t$ are n independent component of the residuals ${\epsilon }_t$. The mixing matrix Z can be solved by fast fixed-point algorithm given by [34].
• Step3 Estimate Univariate GJRSK Model:
  Once the independent components $y_t$ are obtained, the conditional variance, conditional skewness, and conditional kurtosis of independent components can be estimated by the univariate GJRSK model. The parameters of the univariate GARCHSK model can be estimated by maximizing the log-likelihood function in Equation (11).

4. Experiment

In this section, we report the results of our empirical studies with well-known benchmarks. We compared the out-of-sample performance among several portfolio strategies to check the effectiveness of our GO-GJRSK model.

4.1. Dataset

In the experiments, we used well-known academic benchmarks called Fama and French (FF) datasets (https://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html) to ensure the reproducibility of the experiment. This FF dataset is public and is readily available to anyone. The FF datasets have been recognized as standard datasets and heavily adopted in finance research because of their extensive coverage of asset classes and very long historical data series. We used the FF17 dataset and FF25 dataset. For example, the FF17 dataset contains monthly returns of 17 portfolios representing different industrial sectors while the FF25 dataset includes 25 portfolios formed on the basis of size and book-to-market ratio. We used both datasets as monthly data from January 1989 to December 2018.

4.2. Higher Order Risk-Based Portfolio

Here, we formulate the risk-based portfolio construction methodology in the experiment. In this study, two risk-based portfolios with higher-order moments are used to confirm the effectiveness of our proposed model. One is a minimum variance portfolio and the other is a risk parity portfolio.

Let $w={(w_1,\cdots ,w_n)}^{\mathrm{T}}$ be the weights of each of the n assets, $M_2$ is the covariance matrix, $M_3$ is the co-skewness matrix, and $M_4$ is the co-kurtosis matrix. Then, the portfolio variance ${\sigma }_P^2$, skewness $s_P^3$, and kurtosis $k_P^4$ can be expressed as follows [35]:

$$\begin{array}{cc}\hfill & {\sigma }_P^2=w^{\mathrm{T}}{M}_{2}w\hfill \end{array}$$

(28)

$$\begin{array}{cc}\hfill & s_P^3=w^{\mathrm{T}}{M}_3(w\otimes w)\hfill \end{array}$$

(29)

$$\begin{array}{cc}\hfill & k_P^4=w^{\mathrm{T}}{M}_4(w\otimes w\otimes w)\hfill \end{array}$$

(30)

We then use these portfolio characteristics to extend the minimum variance portfolio and the risk-based portfolio to include higher-order moments.

4.3. Minimum Variance Portfolio with Higher Moments

The minimum variance portfolio is extended with the skewness in Equation (29) and kurtosis in Equation (30) as follows [26]:

$$\begin{array}{cc}\hfill & \underset{w}{min}{\lambda }_1{\sigma }_P^2-{\lambda }_2{s}_P^3+{\lambda }_3{k}_P^4,\hfill \\ \hfill & s.t.\sum _{i=1}^N{w}_i=1,w_i>0,\hfill \end{array}$$

(31)

where ${\lambda }_i>0,i=1,2,3$ are weights to account for the trade-offs between the moments of each order. Here, if ${\lambda }_1=1,{\lambda }_2=0,{\lambda }_3=0$, Equation (31) is the minimum variance portfolio.

4.4. Risk Parity Portfolio with Higher Moments

The extension of risk parity portfolios to higher-order moments is discussed in [27]. First, as in the risk parity portfolio, we define from the second to fourth order marginal risk contribution ($MRC$). $MR{C}_i,i=2,3,4$ are derived by differentiating the portfolio variance of Equation (28), skewness of Equation (29), and the kurtosis of Equation (30) by the weight w, respectively.

$$\begin{array}{cc}\hfill & MR{C}_2=\frac{1}{2}\frac{\partial {\sigma }_P^2}{\partial w}=M_{2}w\hfill \end{array}$$

(32)

$$\begin{array}{cc}\hfill & MR{C}_3=\frac{1}{3}\frac{\partial s_P^3}{\partial w}=M_3(w\otimes w)\hfill \end{array}$$

(33)

$$\begin{array}{cc}\hfill & MR{C}_4=\frac{1}{4}\frac{\partial k_P^3}{\partial w}=M_4(w\otimes w\otimes w).\hfill \end{array}$$

(34)

Then, we can decompose the portfolio variance, skewness and kurtosis using $MM{C}_i$ as follows:

$$\begin{array}{cc}\hfill & {\sigma }_P^2=\sum _{i=1}^N{w}_i\times MR{C}_{2,i}=w^{\mathrm{T}}MR{C}_2\hfill \end{array}$$

(35)

$$\begin{array}{cc}\hfill & s_P^3=\sum _{i=1}^N{w}_i\times MR{C}_{3,i}=w^{\mathrm{T}}MR{C}_3\hfill \end{array}$$

(36)

$$\begin{array}{cc}\hfill & k_P^4=\sum _{i=1}^N{w}_i\times MR{C}_{4,i}=w^{\mathrm{T}}MR{C}_4.\hfill \end{array}$$

(37)

We define the contribution of the moments ($RM$) by dividing Equations (35)–(37) by each portfolio moment, respectively.

$$\begin{array}{cc}\hfill & R{M}_{2,i}=\frac{w_i\times MR{C}_{2,i}}{{\sigma }_P^2}\hfill \end{array}$$

(38)

$$\begin{array}{cc}\hfill & R{M}_{3,i}=\frac{w_i\times MR{C}_{3,i}}{s_P^3}\hfill \end{array}$$

(39)

$$\begin{array}{cc}\hfill & R{M}_{4,i}=\frac{w_i\times MR{C}_{4,i}}{k_P^4}\hfill \end{array}$$

(40)

Based on the above, the risk parity portfolio is extended as follows [27]:

$$\begin{array}{cc}\hfill & \underset{w}{min}{\lambda }_1{f}_1\left(w\right)+{\lambda }_2{f}_2\left(w\right)+{\lambda }_3{f}_3\left(w\right)\hfill \\ \hfill & s.t.\sum _{i=1}^N{w}_i=1,w_i>0,\hfill \\ \hfill & f_1\left(w\right)=\sum _{i=1}^N\sum _{j=1}^N{\left(M{C}_{2,i}-M{C}_{2,j}\right)}^2,\hfill \\ \hfill & f_2\left(w\right)=\sum _{i=1}^N\sum _{j=1}^N{\left(M{C}_{3,i}-M{C}_{3,j}\right)}^2,\hfill \\ \hfill & f_3\left(w\right)=\sum _{i=1}^N\sum _{j=1}^N{\left(M{C}_{4,i}-M{C}_{4,j}\right)}^2,\hfill \end{array}$$

(41)

where ${\lambda }_i>0,i=1,2,3$ are weights to account for the trade-offs among the $M{C}_i$. Here, if ${\lambda }_1=1,$${\lambda }_2=0,{\lambda }_3=0$, Equation (41) is the risk parity portfolio.

4.5. Experimental Settings

In our empirical studies, the tested portfolios have the following meanings:

• “EW” stands for equally-weighted portfolio [36];
• “MV” stands for minimum-variance portfolio [14];
• “RP” stands for risk parity portfolio [18];
• “HMV” stands for higher order minimum-variance portfolio [26]. Here, ${\lambda }_1=1,{\lambda }_2=1,{\lambda }_3=1$ in Equation (31);
• “HRP” stands for higher order risk parity portfolio [27]. Here, ${\lambda }_1=1,{\lambda }_2=1,{\lambda }_3=1$ in Equation (41).

The following method is used to estimate the covariance, co-skewness, and co-kurtosis matrix:

• “MLE” stands for maximum likelihood estimation of covariance, co-skewness and co-kurtosis matrix;
• “GO” stands for covariance matrix in GO-GARCH model [12];
• “GO-SK” stands for covariance, co-skewness and co-kurtosis matrix in GO-GARCHSK model [32];
• “GO-GJRSK” stands for covariance, co-skewness, and co-kurtosis matrix in our proposed GO-GJRSK model.

We used the latest 10 years (120 months) to compute for the parameter estimation, and used each dataset from July 1936 to March 2020, as the out-of-sample period. Each portfolio was updated by sliding one-month-ahead.

We performed our experiment with R language.

4.6. Performance Measures

We compared the out-of-sample performance of the portfolios. In evaluating the portfolio strategy, we used the following measures.

Here, the portfolio return $R_t$ at time t is defined as:

$$R_t=\sum _{i=1}^n{r}_{i,t}{w}_{i,t-1},$$

(42)

where $r_{i,t}$ is the return of i asset at time t, $w_{i,t-1}$ is the weight of i asset in the portfolio at time $t-1$, and n is the number of assets.

We evaluated the portfolio strategy using its annualized return (AR), the downside risk as the standard deviation of negative return (DR), and the return/downside risk (R/R) (return divided by downside risk) as the portfolio strategy. As a measure of risk, standard deviation is usually used [14]. However, the standard deviation as a risk measure has several drawbacks. It punishes both upside and downside deviations of return. Moreover, the variance is justified only when the joint distribution of asset returns is symmetric. In this study, we assume that the asset returns are asymmetric and non-normal as several empirical study shows. On the other hand, DR only penalizes the downside deviations of returns. Hence, it is intuitively much more appealing.

$$\begin{array}{cc}\hfill \mathrm{AR}& =\frac{12}{T}\sum _{t=1}^T{R}_t\hfill \end{array}$$

(43)

$$\begin{array}{cc}\hfill \mathrm{DR}& =\sqrt{\frac{12}{T-1}{\sum }_{t=1}^T{(R_t\times 1_{\{R_t\le 0\}})}^2}\hfill \end{array}$$

(44)

$$\begin{array}{cc}\hfill \mathrm{R}/\mathrm{R}& =\mathrm{AR}/\mathrm{DR}\hfill \end{array}$$

(45)

Here, $\mu =(1/T){\sum }_{t=1}^T{R}_t$ is the average return of the portfolio.

4.7. Results

The results are shown in

Table 2. AR, DR, and R/R for an EW, MV, RP, HMV, and HRP portfolios using three methods. In the rows labeled AR, DR, and R/R, the best results appear in bold for MV, HMV portfolios, and RP, HRP portfolios.

FF17	EW	MV		HMV			RP		HRP
		MLE	GO	MLE	GO-SK	GO-GJRSK	MLE	GO	MLE	GO-SK	GO-GJRSK
AR [%]	11.31	10.69	10.26	11.24	11.02	12.03	11.36	11.36	11.35	11.01	12.37
DR [%]	12.06	9.17	9.07	11.55	11.81	9.47	11.52	11.52	11.53	11.64	10.61
R/R	0.94	1.17	1.13	0.97	0.93	1.27	0.99	0.99	0.98	0.95	1.17
FF25	EW	MV		HMV			RP		HRP
		MLE	GO	MLE	GO-SK	GO-GJRSK	MLE	GO	MLE	GO-SK	GO-GJRSK
AR [%]	13.20	11.58	11.69	12.83	12.45	13.81	13.06	13.07	12.97	12.71	14.05
DR [%]	14.34	10.94	10.82	13.91	14.32	11.54	13.89	13.89	13.88	14.20	11.38
R/R	0.92	1.06	1.08	0.92	0.87	1.20	0.94	0.94	0.93	0.90	1.23

. For both FF17 and FF25, the AR and R/R of HMV portfolio using the GO-GJRSK model are the highest in MV and HMV portfolios. The DR of MV portfolio using the GO-GARCH model has the lowest in MV and HMV portfolios.

Furthermore, interestingly, extending MV to HMV using a simple MLE does not improve R/R. Furthermore, when the GO-GARCHSK model is used in HMV, the R/R is subordinated to the HMV using MLE.

On the other hand, for both FF17 and FF25, HRP using the GO-GJRSK model gives the best results in terms of AR, DR, and R/R in RP and HRP portfolios. As in the result of MV portfolio, extending RP to HRP with a simple MLE does not improve R/R. This is a different result from the previous study [27]. Furthermore, when the GO-GARCHSK model is used in HRP, the R/R is inferior to the HMV using MLE. As in [21], risk-based portfolios without higher moments have little impact on performance due to differences in estimation methods.

We can confirm the effectiveness of our proposed GO-GJRSK model for simple MLE and GO-GARCHSK.

5. Conclusions

In this study, we proposed the GO-GJRSK model which extends the GO-GARCH and incorporates the three features of a financial time series: (1) non-normality, (2) serial correlation, and (3) leverage effect.

We confirmed the effectiveness of the proposed method by comparing the performance of risk-based portfolios with higher-order moments. The results showed that the performance of the proposed method was superior to that of the MLE and GO-GARCHSK. We showed that extending MV to HMV and RP to HRP with a simple MLE did not improve R/R. This result indicated that estimation methods are important in risk-based portfolios with higher moments as with MVO.

A future challenge is to extend our model to stochastic volatility, skewness, and kurtosis models because stochastic volatility model might provide a better alternative to the GARCH models [37].