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.

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:

${\hat{\beta }}_n\left(\tau ,p\right)=\underset{\beta }{\arg \min }{n}^{-1}{\displaystyle \underset{p=1.1}{\overset{2}{\sum }}{\displaystyle \underset{t=1}{\overset{n}{\sum }}\left|\tau -I\left(Y_t<{\tilde{f}}_t\left(\beta \right)\right)\right|}{\left|Y_t-{\tilde{f}}_t\left(\beta \right)\right|}^p},$ (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 ${\hat{\beta }}_n\left(\tau ,p\right)$ using the parameters estimated by the quasi-likelihood as initial values in R.

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.

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).