Abstract

There is a substantial body of empirical research that has found the fund return distributions to exhibit pronounced peakiness, heavy tails, and skewness, deviating from a normal distribution. Addressing the limitations of the traditional Sharpe ratio, which assumes a normal distribution of returns and uses standard deviation to measure investment risk, this paper primarily employs the Value at Risk (VaR) based on Lp quantile to adjust excess returns of funds. This method offers superior robustness, is capable of capturing asymmetry and heavy-tailed characteristics, and is more flexible, providing a better description of the tail risk in fund returns. Empirical studies have shown that using the Sharpe ratio corrected with the Lp quantile is feasible for evaluating and ranking the performance of open-end funds.

Keywords

Sharpe Ratio, Expectile, Lp Quantile, VaR

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1. Introduction

As global financial markets rapidly develop, funds play a crucial role in wealth management for investors. There are two ways in which funds operate: open-end funds and closed-end funds. Open-end funds are the mainstream in the international fund market, offering significant advantages over closed-end funds in terms of incentive and constraint mechanisms, liquidity, transparency, and investment convenience. This is also why open-end funds occupy a prominent position in investors’ asset allocation. In this context, effectively evaluating the performance of funds is of great significance. From the perspective of investors, accurate fund performance evaluation helps them make more informed investment decisions. From the standpoint of fund managers, scientific performance evaluation methods can contribute to improving management effectiveness and enhancing market competitiveness.

Features of fund return distributions include skewness, heavy tails, and spikes, which are essential for investment analysis and risk assessment. Skewness describes the symmetry of the return distribution, with positive skewness usually implying potentially high returns, while negative skewness implies an increased risk of loss. Heavy tails indicate that extreme events are more likely to occur than conventionally expected, which puts investors at greater risk in the event of a market crash or economic crisis. In addition, the spike feature measures the level of return concentration, and a high spike may lead investors to misjudge risk. Currently, the research methods for evaluating fund performance mainly focus on the traditional Sharpe ratio [1], Treynor ratio [2], and Jensen’s alpha [3]. These three indicators are classic methods for measuring fund performance. This article primarily uses the Sharpe ratio to study fund performance evaluation. When applying the traditional Sharpe index to measure fund performance, it assumes that the sequence of fund returns follows a normal distribution. However, in reality, most financial time series data often do not conform to the normal distribution assumption, showing more characteristics of sharp peaks and thick tails. The standard deviation in the traditional Sharpe ratio only represents the volatility of returns, failing to adequately reflect the more critical tail risk in financial markets. Therefore, it cannot fully measure the true risk of the fund.

In response to the limitations of using standard deviation to measure the traditional Sharpe ratio, scholars have employed methods such as Value at Risk (VaR) to adjust the traditional Sharpe ratio for risk. Peng and Wu [4] studied the use of VaR and Conditional VaR to adjust the traditional Sharpe indicator for a more reasonable risk portrayal. To fully consider the skewness, peak, and heavy-tailed nature of return distributions, Shi et al. [5] and Yu et al. [6] respectively considered using the Asymmetric Laplace distribution and Skewed-t distribution to fit the return distribution, and then used the VaR values calculated based on these distributions and models to correct the Sharpe index. Additionally, Su and Zhou [7] used the Asymmetric Power Distribution (APD) to fit the distribution of fund returns, proving that the modified Sharpe ratio based on APD standard deviation and VaR is very feasible for application in fund ranking and evaluation. Their research indicates that the Sharpe index adjusted by VaR not only provides a more scientific evaluation of a fund’s overall returns, but also more accurately reflects the volatility characteristics of the fund’s returns, thereby better revealing the fund manager’s ability to control risk.

VaR is a widely used standardized tool for measuring financial risk, typically defined as the maximum potential loss at a given confidence level over a specific time period. It aids decision-makers in understanding the level of potential risk. Meanwhile, as a key risk management tool, VaR is widely applicable to various portfolios and market environments. In a diversified portfolio, VaR can conduct risk assessment for different asset classes such as stocks, bonds and derivatives, helping investors understand potential liquidity and credit risks. Under different market conditions, whether bull market, bear market or volatile market, VaR can effectively quantify potential losses and guide investors to adjust investment strategies to balance returns and risks. In addition, VaR also provides data support for risk management, helps develop hedging strategies, optimize capital allocation, and meet regulatory requirements, which enables investors and financial institutions to better identify and manage risks in a complex and volatile market environment, so as to make more informed decisions. However, traditional methods of measuring VaR, such as the variance-covariance method, historical simulation, and Monte Carlo simulation, each have their limitations. In response to these limitations, both domestically and internationally, numerous studies have adopted the expectile measure proposed by Newey and Powell [8] to estimate VaR. For instance, international researchers such as Kuan et al. [9], Taylor [10], and Kim and Lee [11] have, along with domestic scholars like Yao [12], Xie [13], and Hu et al. [14], developed different models based on expectile to explore the measurement of VaR for stocks or stock indices. These studies have confirmed that the expectile measure is more sensitive to risks in the tails. Particularly, in the research by Su and Zhou [15], it was shown that using the expectile measure to correct the traditional Sharpe ratio in the CARE model provides a more accurate assessment of extreme risk, better evaluating the performance of funds under downside risk.

Additionally, Chen [16] proposed the Lp quantile, a method that has gained popularity in recent years for estimating. The Lp quantile is obtained by minimizing the p-th power of the loss function. Especially when 1 < p < 2, the Lp quantile is favored due to its ability to balance the robustness of quantile and the efficiency of the expectile. Jiang et al. [17] provided the corresponding asymptotic estimation theory and proved its effectiveness. There have been numerous studies on Lp quantile regression in recent years, such as those by Usseglio-Carleve [18], Girard et al. [19], and Tang and Chen [20], among others (for more research on Lp quantile, see [21]-[24]). These studies have explored various estimators combined with Lp quantile regression methods, empirically demonstrating that estimators using Lp quantile regression are effective, highlighting the superiority of the Lp quantile regression method. Inspired by this, this article aims to use the Lp quantile regression estimate of VaR to replace the standard deviation in the denominator of the traditional Sharpe index, thereby correcting the traditional Sharpe index as a measure of fund performance.

The remainder of the article is organized as follows. In Section 2, a brief introduction is provided on the Lp quantile regression estimation method used in this article, as well as its application in conjunction with nonlinear models for measuring financial risk, VaR. Section 3 primarily conducts an empirical study on the performance of 22 sample funds based on the Sharpe ratio. The Lp quantile regression estimation method introduced in Section 2 is utilized to calculate the VaR values. Subsequently, the Sharpe ratio is adjusted based on these VaR values. The performance of the funds is then evaluated using both the traditional Sharpe ratio and the Sharpe ratio adjusted with VaR. Finally, Section 4 concludes and looks ahead to future directions of the article.

2. Lp Quantile Nonlinear Regression Model

Lp quantile regression is a relatively popular estimation method in recent years. Following the introduction of quantile regression and expectile regression by Koenker and Bassett [25] and Newey and Powell [8], Chen [16] proposed the Lp quantile regression method with p-values greater than 1. Jiang et al. [17] further studied Lp quantile regression, proposing a k-th power expectile regression method that lies between quantile and expectile regression, demonstrating that Lp quantile regression with p-values between 1 and 2 can better balance robustness and efficiency. Lin et al. [26] provided the asymptotic theoretical properties of the estimators for Lp quantile regression with p greater than 1 and less than or equal to 2, and their simulation results showed that in certain data scenarios, this method has higher asymptotic efficiency than ordinary quantile regression and expectile regression, providing new avenues for the study of financial risk measurement. In the latest research, Sun et al. [27] focused on the application of Lp quantile regression in the accurate estimation of VaR, proposed the conditional Lp quantile nonlinear autoregressive regression model (CAR-LP-quantile model), and showed that this method has strong effectiveness and advantages through simulation and empirical study. It can be seen that Lp quantile, as a risk measurement tool in financial risk management, provides greater flexibility and adaptability for risk assessment.

Let Y be a random variable, $\tau \in \left(0,1\right)$ , $p\in \left[1,2\right]$ , then the Lp quantile $L_{p,\tau }\left(Y\right)$ may be defined as the solution that minimizes the asymmetric p-th power loss function:

$L_{p,\tau }\left(Y\right)=\underset{x\in R}{\arg \min }E\left[|\tau -I\left(Y<x\right)|\cdot {\left|Y-x\right|}^p\right]$ .

The $\tau$ denotes the asymmetry level in the loss function. For p = 1 and p = 2, quantile and expectile can be incorporated into Lp quantiles. Additionally, Kim and Lee [11] introduced a non-linear expectile regression method that can be applied to the estimation of VaR and Expected Shortfall (ES). This paper combines the Lp quantile regression method with non-linear models for fund performance evaluation. Referencing model (2.3) from the research by Kim and Lee [11], the following Lp quantile non-linear regression model is considered:

$Y_t=f\left(Y_{t-1},Y_{t-2},\cdots ;{\beta }^0\right)+{\epsilon }_{t\tau }\equiv f_t\left({\beta }^0\right)+{\epsilon }_{t\tau },0<\tau <1,t\in \mathbb{Z}$ ,

where $f_t(\cdot )$ is a stochastic process parameterized by $\beta \in R^p$ , ${\beta }^o={\beta }^o\left(\tau ,p\right)$ is a true parameter, ${\epsilon }_{t\tau }$ are error terms satisfying ${\mu }_{{\epsilon }_{t\tau }|{\Omega }_t}\left(\tau \right)=0$ , and ${\Omega }_t$ denotes the information set available up to time t , say, $\sigma \left(Y_{t-1},Y_{t-2},\cdots \right)$ . To estimate the parameter ${\beta }^o$ , one needs to approximate $f_t\left(\beta \right)$ using the observation ${\tilde{f}}_t\left(\beta \right)$ from $Y_t$ . Therefore, consider:

${\tilde{f}}_t\left(\beta \right)=f\left(Y_{t-1},Y_{t-2},\cdots ,Y_1,0,\cdots ;\beta \right)$ ,

and estimate ${\beta }^o$ by the following equation:

${\widehat{\beta }}_n\left(\tau ,p\right)=\underset{\beta }{\arg \min }{n}^{-1}{\displaystyle \sum _{p=1.1}^2{\displaystyle \sum _{t=1}^n\left|\tau -I\left(Y_t<{\tilde{f}}_t\left(\beta \right)\right)\right|}{\left|Y_t-{\tilde{f}}_t\left(\beta \right)\right|}^p},$ (1)

where $\tau \in \left(0,1\right)$ , p is taken from 1.1 to 2 at intevals of 0.1.

In empirical studies of fund performance, the observation $Y_t$ is the daily return data of the fund. We consider using the following linear GARCH(1, 1) model to evaluate the performance of the Lp quantile nonlinear regression estimate:

$\begin{array}{l}{Y}_t=\sqrt{h_t}{\eta }_t,\\ h_t=1+\alpha Y_{t-1}^2+\gamma h_{t-1},\end{array}$

where ${\eta }_t$ is the error distribution, $\alpha$ and $\gamma$ are parameters, therefore the $\tau$ -th condition Lp quantile regression model is $Y_t=f_t\left(\tau \right)+{\epsilon }_{t\tau }=\sqrt{h_t}{\xi }_{\tau }+{\epsilon }_{t\tau }$ , ${\xi }_{\tau }$ is the Lp quantile of error distribution ${\eta }_t$ . The parameters ${\xi }_{\tau }$ , $\alpha$ and $\gamma$ of the model are estimated by the optimization function ${\widehat{\beta }}_n\left(\tau ,p\right)$ using the parameters estimated by the quasi-likelihood as initial values in R.

3. An Empirical Study on Fund Performance in China

3.1. Sample Selection and Data Sources

This article selects 22 open-end funds with complete data from January 1, 2017 to May 15, 2024 as the research object, each fund has 1785 samples. In empirical research, the entire sample is divided into two parts: one part is the in-sample data, which is used as an estimation sample, using the first 1000 data to estimate the model parameters; the other part is the out-of-sample data, which is used as a prediction sample or test sample, and the last 785 data are used for the prediction of out-of-sample VaR. All data in this article are derived from Eastmoney Choice Data.

3.2. Empirical Results and Analysis

3.2.1. Description of Basic Statistics

The paper initially presents the names and corresponding codes (see Table 1) of the 22 open-end funds, followed by providing basic statistics on their returns. Based on the analysis in Table 2, it is evident that, with regards to mean returns, all funds except Castrol Research Selected Mix and China Merchants CSI 300 Real Estate Equal-weighted Index A fund exhibit positive mean returns. Furthermore, the standard deviation of returns for all funds is greater than zero. In terms of skewness, Wanxia Xinli Flexible Allocation Hybrid and China Merchants CSI 300 real estate eq ual weight index A fund demonstrate positive skewness indicating right-skewness, whereas other funds display negative skewness implying left-skewness. Additionally, all fund return series exhibit high peaks and kurtosis values exceeding 3, suggesting that most sample funds have less dispersed return distributions with a more convex shape compared to a normal distribution. These findings indicate biased distributions with sharp peaks and fat tails for all fund returns.

Finally, the Jarque-Bera statistic is employed to test the normality assumption of the sample fund returns. The statistical values significantly surpassing the critical value at a significance level of 5% strongly reject the null hypothesis that daily return time series of sample funds follow a normal distribution.

Table 1. Names and codes of 22 funds.

Table 2. Descriptive statistics of open-end funds.

3.2.2. Computation of the Sharpe Ratio

Fund performance evaluation involves using historical data of fund operations to comprehensively assess the actual investment outcomes of the fund. The Sharpe ratio, in particular, is based on the Capital Market Line (CML) and assumes that returns follow a normal distribution. It adjusts total risk using standard deviation. The Sharpe ratio is a standardized metric used to evaluate fund performance, also known as the Sharpe Index. The traditional formula for calculating the Sharpe ratio is:

$SharpeRatio=\frac{E\left(R_P\right)-R_f}{{\sigma }_P}$

where $E\left(R_P\right)$ is the expected annualized return rate of the investment portfolio, $R_f$ is the annualized risk-free rate, and ${\sigma }_P$ is the standard deviation of the annualized return rate of the investment portfolio. According to the common practice of using the yield of national short-term government bonds as the risk-free rate, this article considers the one-year maturity yield of the China Bond Government Bond as of May 15, 2024, as the risk-free rate, which is 1.6134. To ensure that the data results of average returns, standard deviations, and risk-free rates for each fund are consistent over time, the annualized Sharpe ratio is calculated for each fund, excluding weekends and holidays, with the assumption that there are 252 trading days in a year. The following formula is used to convert the daily average returns and daily return standard deviations of each fund into annual average returns and annual return standard deviations. The basic steps for calculating the Sharpe ratio are as follows:

Firstly, obtain the latest net asset value per unit of the fund within the research period, and calculate the daily return rate:

$Rt=\frac{P_t-P_{t-1}}{P_{t-1}}$ .

According to the traditional Sharpe ratio calculation formula, the resulting daily yield needs to be annualized and denoted as $R_{annual}$ , then the annualized yield calculation formula is:

$R_{annual}={\left(1+\overline{R}\right)}^{252}-1$ ,

where $\overline{R}=\frac{1}{N}{\displaystyle \sum _{i=1}^{N}R{t}_i}$ , $\overline{R}$ denotes the mean of daily returns.

Secondly, calculate the corresponding annualized standard deviation, denoted as ${\sigma }_{annual}$ , with the formula:

${\sigma }_{annual}=\sqrt{252}\ast \sigma$ ,

where $\sigma =\sqrt{\frac{1}{N}{\displaystyle \sum _{i=1}^N{\left(R{t}_i-\overline{R}\right)}^2}}$ , $\sigma$ represents the standard deviation of daily returns.

Finally, the traditional Sharpe ratio formula is obtained as:

$Sharpe=\frac{R_{annual}-R_f}{{\sigma }_{annual}}$ . (2)

Due to the traditional Sharpe ratio’s assumption of normally distributed returns, which overlooks downside risk, it fails to fully measure the true risk of a fund. Therefore, we consider using VaR to adjust the traditional Sharpe ratio. For the calculation of the Sharpe ratio adjusted with VaR, using VaR values estimated through Lp quantile regression instead of the standard deviation in the traditional Sharpe ratio formula, we can derive the formula for the Sharpe ratio adjusted with VaR as follows:

$Sharp{e}_{VaR}=\frac{R_{annual}-R_f}{VaR}$ . (3)

There are several main implications for the Sharpe ratio modified by VaR: Firstly, more attention should be paid to tail wind, because the standard deviation mainly reflects the overall volatility, while VaR focuses more on extreme losses. By introducing VaR, investors can better understand the potential significant loss risk; Secondly, it is suitable for non-normal distribution. The traditional Sharpe ratio assumes that the return distribution is normal, while VaR can be better applied to non-normal distribution data, especially in the face of peak and fat-tailed distribution. In addition, the revised Sharpe ratio can be used as a tool for investment decision-making. It helps investors to consider not only profitability but also potential extreme risks when evaluating investment performance, which is especially important for risk-averse investors; Finally, the revised Sharpe ratio can be applied in practice. In actual risk management, the use of VaR as a risk measure can enable financial institutions and investors to carry out more effective capital allocation and risk hedging in the face of market crisis or potential loss.

In addition to this, the VaR values used to replace the standard deviation in the adjusted Sharpe ratio reflect different risk preferences and data characteristics when different regression methods are employed to estimate VaR. Compared to the expectile, the use of Lp quantile regression methods for estimating VaR in this paper is more advantageous. Firstly, the square loss function used in the expectile is highly sensitive to outliers, leading to estimation bias when extreme values or data noise are present. In contrast, the p loss function used in Lp quantile regression, where p is between 1 and 2, is more robust to outliers, making the estimated VaR values more reliable and thus improving the accuracy of the adjusted Sharpe ratio. Secondly, financial market data often exhibit asymmetric distribution characteristics, especially during extreme market fluctuations. Lp quantile regression allows for different degrees of punishment on the tails of the distribution, better capturing the asymmetric characteristics of the data. Moreover, financial data typically show a heavy-tailed phenomenon, with a higher probability of extreme returns. Lp quantile regression is more sensitive to data with heavier tails, providing a more conservative VaR estimate, which enhances the reliability of its application in risk management. Lastly, different p-values can be chosen based on varying market conditions and data characteristics to obtain the most optimal risk assessment model. While the square loss function is computationally simpler, its applicability is limited, especially in non-normal and heavy-tailed distributions, making Lp quantile regression more widely applicable due to its flexibility in practical applications.

Overall, using VaR as an alternative to standard deviation in Sharpe ratio calculations, VaR not only focuses on tail risk, but also applies to non-normal distributions, adding a stronger risk management perspective to the analysis of risk-adjusted returns, allowing investors to consider risk factors more fully when evaluating portfolio performance. Lp quantile regression is more robust, able to capture asymmetry and heavy-tailed distribution characteristics, and more flexible than expectile regression in estimating fund VaR. Therefore, using VaR based on Lp quantile estimation to modify Sharpe ratio can provide more accurate and reliable results, thus improving the quality of risk management and decision-making.

3.2.3. VaR Calculation

Before parameter estimation, we use Ljung-Box test method to test the correlation of the return series of 22 selected funds. The lag order is 10, and the p-value is obviously larger than the commonly used significance level of 0.05 or 0.01. This means that we cannot reject the null hypothesis that the autocorrelation of time series data is zero. Since there is no significant autocorrelation, consider using GARCH models to analyze this data set to demonstrate the rationality of volatility models.

Take the Jingshun Small and Medium Enterprise Board and Growth Enterprise Market Selected Stocks board as an example, use linear GARCH(1, 1) model to calculate VaR value, and calculate VaR value of other funds similarly. First, for a given quantile $\theta$ of 5%, when p takes different values from 1.1 to 2, we use the sample data of GEM selected stocks in Inveshun to obtain the corresponding $\tau$ estimates in the linear GARCH(1, 1) model. As shown in Table 3, under the given 5% quantile, the estimated values of 22 funds selected in this paper, such as Jingshun Small and Medium Enterprise Board and Growth Enterprise Market Selected Stocks, are given at different values $\tau$ of p. Then, the estimated values of parameters ${\xi }_{\tau },\alpha ,\gamma$ in linear GARCH(1, 1) model are −0.5383803, 0.5809303 and 0.8927805 by Lp quantile regression method of Equation (1). Finally, since the $\tau$ -th condition Lp quantile of $Y_t$ is given by:

$f_t\left(\tau \right)=\sqrt{h_t}{\xi }_{\tau }$ ,

where ${\xi }_{\tau }$ is the $\tau$ -th Lp quantile of the error distribution ${\eta }_t$ thus for a given 5% quantile the formula for calculating the VaR value corresponding to the Jingshun Small and Medium Enterprise Board and Growth Enterprise Market Selected Stocks data is:

$Va{R}_t\left(0.05\right)=\sqrt{h_t}{\xi }_{{\tau }_{0.05}}$ .

After obtaining the corresponding VaR value estimated by the Lp quantile regression of the selected equity funds of the Jingshun Small and Medium Enterprise Board and Growth Enterprise Market Selected Stocks, bring it into Equation (3) and calculate that the Sharpe ratio value based on the VaR correction is −1.6314, and the Sharpe ratio value of the remaining funds based on the VaR correction can be obtained similarly.

3.2.4. Fund Ranking Based on the Sharpe Ratio

Calculated the traditional Sharpe ratio results for 22 sample funds based on Equation (2). Then, using the example of the Jingshun Small and Medium Enterprise Board and Growth Enterprise Market Selected Stocks, obtained the Sharpe ratio values for the remaining 21 sample funds based on VaR correction at the optimal p, and recorded the VaR-corrected Sharpe ratio results for the expectile regression estimation at p = 2. As shown in Table 3, we compared the rankings of the three types of Sharpe ratios: traditional Sharpe ratio, expectile, and Lp expectile regression estimation based on VaR correction for the performance of open-end funds.

From the results in Table 4, it is evident that the ranking of the 22 funds based on the Sharpe ratio obtained from three methods shows significant differences. The number of funds ranked higher based on the Sharpe ratio adjusted for expectile and Lp quantile is 9 and 10, respectively, compared to the traditional Sharpe ratio calculation. This indicates that these funds have a better ability to control downside risk, and it also reflects the deficiency in the traditional Sharpe ratio where the standard deviation fails to capture tail risk.

Simultaneously, under the expectile and Lp quantile regression adjustments to the Sharpe ratio, the ranking of the adjusted values is largely similar. Six funds experienced changes in their rankings, with three funds, China New Silk Source Hybrid A, China Medicine ETF, and Dacheng CSI Dividend Index A, showing improved performance rankings after the Lp quantile method adjustment. This is due to the Lp quantile method’s more robust and effective approach to measuring VaR. Therefore, incorporating the Lp quantile method to measure VaR values for adjusting the Sharpe ratio holds significant value for investors and is both feasible and operable for fund managers.

Table 3. For a given 5% quantile, the linear GARCH(1, 1) model estimates the value of $\tau$ when p is taken with different values.

^aThe values in columns C1 to C10 represent estimated values of $\tau$ when p takes the values 1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7, 1.8, 1.9 and 2 for varying parameters in the linear GARCH model.

Table 4. Ranking of open-end funds based on 3 Sharpe ratios.

^aSharpe, EVaR-Sharpe, and LVaR-Sharpe represent the traditional Sharpe ratio, Sharpe ratio modified based on expectile, and Lp quantile, respectively.

4. Conclusion

This paper proposes a new method to calculate Sharpe ratio, which is based on Lp quantile regression to estimate VaR, and then uses estimated VaR instead of standard deviation to modify traditional Sharpe ratio as a new performance evaluation index. In order to measure VaR more accurately, we construct Lp quantile nonlinear regression model, which has better robustness, can capture asymmetry and heavy-tailed distribution characteristics, is more flexible, and can better describe the tail risk of fund return when estimating VaR value by Lp quantile. Therefore, using VaR based on Lp quantile estimation to modify Sharpe ratio can provide more accurate and reliable results, thus improving the quality of risk management and decision-making. The empirical results also show that our fund ranking method is effective and feasible. In addition, although this study provides a new idea for fund performance evaluation based on Lp quantile regression, the calculation is more complicated because this method selects the sum of multiple p-values as part of the loss function. In future research, we can choose the optimal p-value and different models to measure the risk in Lp quantile regression method, so as to analyze the performance of open-end funds in China more accurately and comprehensively.

Acknowledgements

This work is partly supported by the Graduate Textbook Construction Project of Sichuan University of Science and Engineering (Grant No. KA202011) and the Opening Project of Sichuan Province University Key Laboratory of Bridge Non-destruction Detecting and Engineering Computing (2024QYY02).

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

References