Risk Measurement and Performance Evaluation of Equity Funds Based on ARMA-GARCH Family Model

There are few comprehensive studies on risk measurement and performance evaluation of stock funds in China. This paper uses the ARMA-GARCH family model to analyze the volatility characteristics of equity funds under the t-distribution and Generalized error distribution (GED), and combines CVaR, PM (Second revised sharp ratio) and CVaR-RAROC (Revised RAROC) to comprehensively evaluate equity funds risk and performance. The empirical analysis of five equity funds in China from October 28, 2010 to May 17, 2019 shows that: Comprehensive evaluation of the risk and performance of equity funds can comprehensively and effectively examine the risks and returns of equity funds, helping investors, financial institutions and regulatory agencies to more fully understand the risks and performance of equity funds.


Introduction
Securities investment fund is a kind of collective investment tool which gathers small-scale funds and uses a variety of investment methods to carry out professional operation. It has been favored by investors since its birth and plays an important role in economic development. Equity funds are a kind of open-end fund. It is an investment fund with stocks as its investment object. The main functions of equity funds are to concentrate the small amount of investment of public investors into large-scale funds and invest in different stock combinations. Equity funds are the main institutional investor in the stock market. There are risks associated with investment returns. Equity funds have the greatest fluctua-tions in risk and return among all fund types. Studying its risks and performance is of great significance to investors, financial institutions and regulators.
At present, there have been many studies on the GARCH family model and the risk and performance measurement of China's open-end fund. Song Guanghui et al. [1] used the VaR and CVaR of GARCH family model to study the risk of Shibor. The research shows that the GED is better than the normal distribution and t-distribution, and the t-distribution hypothesis is not suitable to describe the dynamic characteristics of the logarithm yield of Shibor's weekly interest rate; CVaR model can effectively make up for the deficiency of VAR model and effectively measure the risk of actual loss. Wei Zhengyuan et al. [2] established a new realized GARCH-GED model. The empirical research results show that compared with the realized GARCH model under the assumption of normal distribution, the realized GARCH-GED model can better describe the leverage effect of volatility and improve the precision of tail risk to some extent. Zhao Zhenquan and Li Xiaozhou [3] use GARCH model to study the volatility of open-end fund' return rate, and use absolute VaR and RAROC index to comprehensively study the risk and return of open-end fund. Huang Chongzhen and Cao Qi [4] used the China Asset Shanghai-Shenzhen 300 ETF connection as an example to use the GARCH family model to study the risk of open-end funds. The research shows that the GARCH-M model under the t-distribution can better measure risk. Qi Yue and Sun Xinming [5] applied the method of copying the investment strategy of the fund for the first time to create a new investment portfolio and calculate its investment income. As a benchmark for fund performance evaluation, dynamically evaluate the fund's investment performance and investment behavior. Liu Junshan [6] pointed out that CVaR is better than VaR in nature, and proved that the CVaR index is consistent with the stochastic dominance theory. Zhu Fuyun, Zhou Ying, and Chen Yuan [7] The VaR method based on the EGARCH-GED model effectively characterizes the market risk of securities investment fund. Sharpe ratio and RAROC (risk-adjusted capital return) effectively evaluate the performance of securities investment fund. Tang Zhenpeng and Peng Wei [8] introduced CVaR into the RAROC model, which more accurately measures risk, improves accuracy, and provides a good performance reference index for fund investors. Huang Jinbo, Li Zhongfei, and Ding Jie [9] introduced CVaR into the Sharpe ratio model. This indicator overcomes the shortcomings of traditional Sharpe ratios that do not meet the monotonicity of stochastic dominance and does not consider high-order moment information. It can more accurately characterize risk-adjusted Return on assets.
Previous studies have shown that: 1) CVaR is more reasonable in measuring risk than VaR; 2) PM and CVaR-RAROC are more effective in measuring the performance of financial products; 3) The effects of t-distribution and GED are more effective than those of the normal distribution model. At present, there are few comprehensive studies on the risk measurement and performance evaluation of equity funds. This paper uses the ARMA-GARCH family model to analyze the volatility characteristics of equity funds under the conditions of t-distribution distribution, t-distribution, and GED. The research in this paper mainly studies the volatility characteristics and risk performance of equity funds in the context of t-distribution and GED.

ARMA Model
In the 1970s, American statistician Box GEP and British statistician Jenkins GM [10] proposed an autoregressive moving average model (ARMA model). The general ARMA model expression is: ε is a random sequence, t µ is the value of the current period. p and q are the lag orders of the autoregressive term (AR) and moving average term (MA), respectively.

GARCH (p, q) Model
GARCH (p, q) was proposed by Bollerslov [11] in 1985, and the expression is: in which, ω is a constant term, p and q are the maximum lag order of GARCH term and ARCH term; t µ is a residual term, and t σ is a conditional standard deviation of t µ .

GARCH-M Model
In 1987, Engle, Lilien, and Robbins [12] proposed the GARCH-M model, which The expression of the GARCH-M model is: δ is a risk premium coefficient, and C represents the effect of the predicted risk fluctuation on t y can be observed. When 1 0 δ > , risk compensation ( ) 2 1 0 t f γ σ > , high return means high risk.

TGARCH Model
The threshold GARCH model (TGARCH model) was proposed by Zakoian [13] and Glosten et al. [14] respectively, and its expression is:  γ > , Increase the leverage effect, and the shock will increase the fluctuation, and vice versa.

EGARCH Family Model
In 1991, Nelson [15] proposed an EGARCH model with guaranteed positive variance. Its expression is: in which, 2 t σ in logarithmic form guarantees that the value of 2 t σ is non-negative, and does not require that the coefficients on the right side of the equation be non-negative. The solution process is simpler. If 0 k γ ≠ , the impact of information is asymmetric; if 0 k γ > , the impact of good news is greater than the impact of negative news, and vice versa.

The Concept of VaR and CVaR
The VaR method (Value at Risk, referred to as VaR), known as the value-at-risk model, is often used in the risk management of financial institutions. It was proposed in 1993. The VaR model has been widely adopted by many financial institutions and has become the mainstream method for financial market risk measurement.
However, many empirical studies show that VaR method has its own insurmountable defects. Rockafeller and Uryasev [16] proposed Conditional Value at

Calculation of VaR and CVaR
The calculation formula of VaR is expressed as: in which, 1 t P − is the value of the asset on day t-1, Z α is the quantile of α at a given confidence level, and t σ is the standard deviation of conditions. According to the definition, the CVaR expression based on the GARCH family model is: c is the given significance level. Function ( ) f q is the probability density of the yield series. In the case of a normal distribution, the specific calculation of CVaR is: Under t-distribution, the specific calculation of CVaR is: in which, Γ is the gamma function. d is the degree of freedom.
Under the GED, the specific calculation of CVaR is: in which, After the value of CVaR is obtained, it is tested for validity, and the DLC is used to measure the actual loss over VaR. The definition of the statistic is:

Concepts and Calculations of SHARPE Ratio, Revised Sharpe Ratio, PM
The Sharpe Ratio, also known as the Sharpe Index, is one of three classic indicators that consider both returns and risks. The Sharpe ratio uses standard deviation to measure the risk of the returns of currency funds. The Sharpe ratio can be used as an important basis for fund performance evaluation only when considering the purchase of a certain fund among many funds. Therefore, the Sharpe ratio can be used as a standard for fund performance evaluation index. The calculation formula is: (12) in which, sp is the sharp value of the equity funds, P R is the average return rate of the equity funds, and f R is the risk-free interest rate. Under the current conditions of China's stock market, there is actually no uniform standard for the selection of the risk-free rate of return. Internationally, short-term government bond yields are generally used as market risk-free returns. Therefore, this paper uses the one-year Treasury bond rate (3.6661%) as the risk-free rate. p σ is the standard deviation of the return on equity funds. The larger the Sharpe ratio, the greater the return than the risk, the better the fund's performance.
There are certain limitations to using standard deviation as a risk indicator.
Revising the Sharpe Index solves this limitation and introduces VaR instead of standard deviation. Its expression is: VaR sp VaR VaR is the value-in-risk calculated based on the GARCH family model.
PM is the second revision of the Sharp Index, which is the introduction of CVaR instead of VaR by Golden Wave et al. [6].
PM CVaR ROC RAROC VaR = (15) in which, ROC is the expectation of return on assets. The larger the RAROC, the larger the ratio of benefits to risks, and the better the performance.

Selection and Processing of Sample Data
This The calculation formula is as follows: in which, t R is the daily rate of return of the equity funds, t P and 1 t P − are the unit net values of the day t and day t − 1, respectively. The data analysis of this paper is realized by R 3.5.1, Eviews 9 and MATLAB 2015b.

Basic Description Analysis
The calculation formulas of standard deviation, skewness and kurtosis are as follows: in which, std is the standard deviation, sk is the skewness, and kt is the kurtosis, t R is the average rate of return.
The basic statistics of the daily rate of return of the sample are shown in Table   1 below. tive. In addition, the kurtosis of the rate of return of the samples are all greater than 3, indicating that the samples are all fat tail distribution. In addition, the Jarque-Bera are all above 1000, and the adjoint probabilities are all close to 0.
Therefore, the sample's rate of return has a phenomenon of peaks and fat tails, and the samples do not obey the normal distribution.
An non stationary series does not have convergence. If the time series is not stable, applying it to the model will reduce the reliability of the model. This paper uses ADF statistics to test the stationarity of the rate of return. As shown in Table 2. From

Heteroskedasticity Test
The timing diagram of each sample is shown in Figure 1 below.
In which, the horizontal axis is the time axis, and the vertical axis is the rate of return. As can be seen from Figure 1  Autocorrelation test is carried out for the residual of mean model. The results show that there is no autocorrelation in the sample and the mean model is effective.
Perform a Heteroskedasticity test (Heteroskedasticity ARCH test) on the mean model, as shown in Table 3.
According to the ARCH test results, the Prob are all less than 0.05, indicating that the assumption of "there is no ARCH effect" is rejected at the significance level of 0.05. In addition, the residual sequence is known from the residual autocorrelation graph and partial autocorrelation graph. There is a high-order truncation phenomenon, and it is believed that there is a high-order ARCH effect in the rate of return sequence.

Selection of GARCH Family Models
In determining the GARCH family model, this paper considers the t-distribution and the GED respectively, and through the AIC and SC criteria, after conti- Use formulas (9) and (10) to calculate CVaR as shown in Table 4 below.
From the back test results, it is known that the GARCH model has a better effect of describing risks when GED is distributed. According to the principle that the smaller the DLC value is, the more accurate the model estimates the risk, the The model parameters are shown in Table 5.
The results in Table 5 show that the GARCH model can well fit the data of LM Test is used to test the ARCH effect of the above models. The results in Table 6 show that there is no ARCH effect in the residuals of the five models, indicating that the ARCH effect is well eliminated by each model.  Table 5. ARMA-GARCH family model corresponding parameters.

Calculation of Risk and Performance Based on GARCH Family Models
In the case of 95% confidence level, quantile and conditional standard deviation of the model are calculated by Eviews, and CVaR value is calculated by MATLAB according to Formula (9) and (10) as shown in Table 7.
As can be seen from Use Formulas (11) and (13) to calculate PM and CVaR-RAROC, the results are as in Table 8.
It can be seen from Table 8   top risk ranking, and the bottom performance ranking is also the bottom risk ranking. Therefore, the greater the risk of the equity funds, the greater the return, the better the performance. Among these 5 funds, investors who can accept high risks can choose E Fund Consumption Industry (110022) to obtain greater returns, and more conservative investors can choose funds that are more stable like the Y Yinhua-Dow Jones 88 Index A (180003).
Comparing risk ranking and performance ranking, it is found that the results of risk ranking and sharp ratio method are consistent. The ranking of comprehensive risk and performance found that the better the performance of equity funds, the higher the returns, the greater the risks. Equity funds have the characteristics of investment products "high return, high risk".

Conclusions
In this paper, the ARMA-GARCH model of the t-distribution and GED is established. Through CVaR back testing, it is found that under the GED, the model has a better effect of measuring risk. Under the 95% confidence level, the risk and performance indicators CVaR, PM, and CVaR-RAROC are calculated. The results show that the performance rankings of the Sharpe ratio method and the RAROC method are different. The performance ranking and risk ranking calculated by the CVaR-RAROC method Consistent. Comprehensive risk and performance ranking can be found that the higher the return, the greater the risk, there is a corresponding relationship of "high risk, high return" for equity funds.
To sum up, investment products have risks as well as returns. If investors have low financial literacy, they cannot fully collect and accurately identify the risks of