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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 compre hensively 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.

^{*,#}Corresponding authors.^{ }

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 fluctuations 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. [

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 and GED. Combined with CVaR, PM and CVaR-RAROC, the risk and performance of equity funds are comprehensively evaluated. The studies show that the comprehensive evaluation of the risk and performance of the equity funds can help investors, financial institutions and regulators to understand the risk and performance of the equity funds more comprehensively and effectively.

ARCH model is widely used to study the characteristics of financial asset return, such as peak and thick tail, volatility aggregation, asymmetry and so on. In real economic phenomena, high-order ARCH effects and lagging conditional variance often exist in economic time series, which increases the adaptiveness of conditional variance. At this time, the GARCH model is introduced to represent higher-order ARCH models. This paper establishes dynamic conditional mean equations and conditional Heteroskedasticity equations to study the volatility characteristics of equity funds. In the GARCH family model, the residual distributions are: normal 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.

In the 1970s, American statistician Box GEP and British statistician Jenkins GM [

μ t = ϕ 0 + ϕ 1 μ t − 1 + ⋯ + ϕ p μ t − p + ε t − θ 1 ε t − 1 − ⋯ − θ q ε t − q (1)

in which, { ε t } is a random sequence, μ t 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) was proposed by Bollerslov [

σ t 2 = ω + ∑ i = 1 p α i μ t − i 2 + ∑ j = 1 q β j σ t − j 2 (2)

in which, ω is a constant term, p and q are the maximum lag order of GARCH term and ARCH term; μ t is a residual term, and σ t is a conditional standard deviation of μ t .

In 1987, Engle, Lilien, and Robbins [

The expression of the GARCH-M model is:

{ y t = δ 0 x t + δ 1 f ( σ t 2 ) + μ t σ t 2 = ω + ∑ i = 1 p α i μ t − i 2 + ∑ j = 1 q β j σ t − j 2 (3)

in which, x t is a stationary random variable， f ( σ t 2 ) is a function of conditional variance σ t 2 , δ 1 is a risk premium coefficient, and C represents the effect of the predicted risk fluctuation on y t can be observed. When δ 1 > 0 , risk compensation γ 1 f ( σ t 2 ) > 0 , high return means high risk.

The threshold GARCH model (TGARCH model) was proposed by Zakoian [

{ σ t 2 = ω + ∑ i = 1 p α i μ t − 1 2 + ∑ j = 1 q β j σ t − 1 2 + γ μ t − 1 2 d t − 1 d t = { 0 , μ t ≥ 0 1 , μ t < 0 (4)

μ t ≥ 0 means good news and μ t < 0 means bad news. γ μ t − 1 2 d t − 1 is an asymmetric term, indicating that the influence of positive news and negative news on conditional variance is asymmetric. If γ ≠ 0 , it means that the impact of the shock response is asymmetric; if γ > 0 , Increase the leverage effect, and the shock will increase the fluctuation, and vice versa.

In 1991, Nelson [

log ( σ t 2 ) = ω + ∑ i = 1 q a i | μ t − i σ t − i − E ( μ t − i σ t − i ) | + ∑ j = 1 p β j log ( σ t − j 2 ) + ∑ k = 1 γ γ k μ t − k σ t − k (5)

in which, σ t 2 in logarithmic form guarantees that the value of σ t 2 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 γ k ≠ 0 , the impact of information is asymmetric; if γ k > 0 , the impact of good news is greater than the impact of negative news, and vice versa.

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 [

The calculation formula of VaR is expressed as:

VaR = P t − 1 Z α σ t (6)

in which, P t − 1 is the value of the asset on day t-1, Z α is the quantile of α at a given confidence level, and σ t is the standard deviation of conditions.

According to the definition, the CVaR expression based on the GARCH family model is:

CVaR = P t − 1 σ t 1 − c ∫ − ∞ − Z f ( q ) q d q (7)

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:

CVaR = − P t − 1 σ t ( 1 − c ) 2 π e − z 2 2 (8)

Under t-distribution, the specific calculation of CVaR is:

CVaR = P t − 1 σ t 1 − c d Γ ( ( d + 1 ) / 2 ) ( d − 1 ) π Γ ( d / 2 ) ( 1 + Z 2 d ) − d − 1 2 (9)

in which, Γ is the gamma function. d is the degree of freedom.

Under the GED, the specific calculation of CVaR is:

CVaR = − P t − 1 σ t 1 − c ∫ − ∞ − Z q q exp ( − 1 2 | q λ | d ) λ Γ ( 1 d ) 2 d + 1 d (10)

in which, λ = [ 2 ( − 2 / d ) Γ ( 1 / d ) Γ ( 3 / d ) ] 1 / 2 .

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:

DLC = | 1 N ∑ i + 1 N X i − 1 N ∑ i + 1 N CVaR i | (11)

in which, X i is the actual loss that exceeds VaR, DLC is the absolute value of the difference between the expected value of actual loss and the expected value of CVaR., N is the number of days of actual failure. The smaller the DLC, the closer the expected value of actual loss and the expected value of CVaR are, the CVaR measure the higher the accuracy, and vice versa.

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:

sp = R p ¯ − R f ¯ σ p (12)

in which, sp is the sharp value of the equity funds, R P ¯ is the average return rate of the equity funds, and R f ¯ 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. σ p 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 = R p ¯ − R f ¯ VaR (13)

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. [

PM = R p ¯ − R f ¯ CVaR (14)

RAROC (Risk Adjusted Return on Capital) is a risk-adjusted return on capital. It is a financial product indicator proposed by Banker Trust in the 1970s. It is the ratio of the rate of return on assets and the amount of value at risk during the sample period. Both benefits and risks are considered. Its expression is:

RAROC = ROC 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.

CVaR makes up for the shortcomings of VaR in measuring risk and has a better measurement effect. Tang Zhenpeng et al. [

CVaR − RAROC = ROC CVaR (16)

This paper selects the daily unit net value of five equity funds, namely Everbright Quantitative Stock (360001), E Fund Consumption Industry (110022), Yinhua Shenzhen Securities 100 graded (161812), Yinhua-Dow Jones 88 Index A (180003), China Merchants Shenzhen Stock Exchange 100 Index A (217016) , as the research object. The ARMA-GARCH family model is used to analyze the volatility characteristics of equity funds under t-distribution and GED. Combine CVaR, PM and CVaR-RAROC to comprehensively evaluate the risk and performance of equity funds. The data of this paper is from fund.eastmoney.com. The time span of the sample is from October 28, 2010 to May 17, 2019, with a total of 2090 observations. Convert the data to the rate of return and obtain 2089 observations. The calculation formula is as follows:

R t = ln P t − ln P t − 1 (17)

in which, R t is the daily rate of return of the equity funds,

The calculation formulas of standard deviation, skewness and kurtosis are as follows:

in which,

The basic statistics of the daily rate of return of the sample are shown in

From

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

From

Statistics | 360001 | 110022 | 161812 | 180003 | 217016 |
---|---|---|---|---|---|

Mean | 0.0000 | 0.0005 | −0.0002 | 0.00003 | 0.0000 |

median | 0.0005 | 0.0007 | 0.0000 | 0.0000 | 0.0000 |

Maximum | 0.0623 | 0.0783 | 0.0705 | 0.0606 | 0.0718 |

Minimum | −0.0824 | −0.0816 | −0.1148 | −0.0803 | −0.0928 |

Std.Dev | 0.0150 | 0.0146 | 0.0160 | 0.0136 | 0.0162 |

Skewness | −0.8237 | −0.4182 | −0.8329 | −0.4805 | −0.7410 |

Kurtosis | 7.4274 | 6.7209 | 7.9679 | 7.6036 | 7.6273 |

Jarque-Bera | 1942.390 | 1265.384 | 2389.714 | 1925.042 | 2054.928 |

Sample | ADF statistic | 1% level | 5% level | 10% level | Prob | Stationarity |
---|---|---|---|---|---|---|

360001 | −43.27519 | −3.433284 | −2.862722 | −2.567445 | 0.0000 | stable |

110022 | −34.73255 | −3.433284 | −2.862722 | −2.567445 | 0.0000 | stable |

161812 | −44.00777 | −3.433284 | −2.862722 | −2.567445 | 0.0001 | stable |

180003 | −34.41966 | −3.433284 | −2.862722 | −2.567445 | 0.0000 | stable |

217016 | −43.82818 | −3.433284 | −2.862722 | −2.567445 | 0.0001 | stable |

The autocorrelation of 5 samples was tested, and the autocorrelation coefficient AC and partial autocorrelation coefficient PAC were calculated. Both the methodology employed by EVIEWS or R results considered that 5 samples had autocorrelation.

The timing diagram of each sample is shown in

In which, the horizontal axis is the time axis, and the vertical axis is the rate of return. As can be seen from

According to the AIC criterion and the significance of the model coefficients, the optimal mean model of the samples is selected. The optimal average model of Everbright Quantitative Stock (360001) is ARMA (3, 3), the optimal average model of E Fund Consumption Industry (110022) is ARMA (0, 2), the optimal average model of Yinhua Shenzhen Securities 100 graded (161812) is ARMA (1, 1), Yinhua-Dow Jones 88 Index A (180003) is ARMA (3, 3), and the optimal mean model of the China Merchants Shenzhen Stock Exchange 100 Index A (217016) is ARMA (3, 3).

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

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.

Sample | F-statistic | Prob. F(1, 2086) | Obs*R-squared | Prob. Chi-Square (1) | ARCH effect |
---|---|---|---|---|---|

360001 | 117.3770 | 0.0000 | 111.2307 | 0.0000 | existence |

110022 | 141.9637 | 0.0000 | 133.0414 | 0.0000 | existence |

161812 | 55.96141 | 0.0000 | 54.55160 | 0.0000 | existence |

180003 | 33.77749 | 0.0000 | 33.27137 | 0.0000 | existence |

217016 | 111.4082 | 0.0000 | 105.8612 | 0.0000 | existence |

In determining the GARCH family model, this paper considers the t-distribution and the GED respectively, and through the AIC and SC criteria, after continuously trying the ARMA-GARCH, ARMA-GARCH-M, ARMA-TGARCH, and ARMA-EGARCH models, the final selection is made. The final models of different distributions are: for the t-distribution, the models are: ARMA (3, 3)-EGARCH (2, 1), ARMA (0, 2)-TGARCH (1, 2), ARMA (1, 1)-EARCH (1, 2), ARMA (3, 3)-GARCH-M (1, 2), ARMA (3, 3)-GARCH (1, 1); for GED, the models are: ARMA (3, 3)-GARCH (2, 1), ARMA (0, 2)-TGARCH (1, 2), ARMA (1, 1)-GARCH-M (1, 2), ARMA (3, 3)-GARCH (1, 2). ARMA (3, 3)-GARCH (1, 1).

Use formulas (9) and (10) to calculate CVaR as shown in

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 final model are ARMA (3, 3)-GARCH (2, 1)-GED, ARMA (0, 2)-TGARCH (1, 2)-GED, ARMA (1, 1)-EARCH (1, 2)-t, ARMA (3, 3)-GARCH (1, 2)-GED, ARMA (3, 3)-GARCH (1, 1)-GED.

The model parameters are shown in

The results in

LM Test is used to test the ARCH effect of the above models. The results in

Sample | Model | Days of Failure | Failure rate | DLC | |
---|---|---|---|---|---|

t-distribution | 360001 | ARMA (3, 3)-EGARCH (2, 1) | 70 | 0.0335 | 0.0217 |

110022 | ARMA (0, 2)-TGARCH (1, 2) | 46 | 0.022 | 0.0318 | |

161812 | ARMA (1, 1)-EARCH (1, 2) | 979 | 0.4686 | 0.0086 | |

180003 | ARMA (3, 3)-GARCH-M (1, 2) | 1028 | 0.4921 | 0.0094 | |

217016 | ARMA (3, 3)-GARCH (1, 1) | 974 | 0.4663 | 0.0120 | |

GED | 360001 | ARMA (3, 3)-GARCH (2, 1) | 117 | 0.056 | 0.0028 |

110022 | ARMA (0, 2)-TGARCH (1, 2) | 78 | 0.0374 | 0.0105 | |

161812 | ARMA (1, 1)-GARCH-M (1, 2) | 979 | 0.4686 | 0.0120 | |

180003 | ARMA (3, 3)-GARCH (1, 2) | 132 | 0.0632 | 0.0019 | |

217016 | ARMA (3, 3)-GARCH (1, 1) | 974 | 0.4663 | 0.0119 |

Sample | 360001 | 110022 | 161812 | 180003 | 217016 |
---|---|---|---|---|---|

Mean-variance equation | ARMA (3, 3)- GARCH (2, 1)-GED | ARMA (0, 2)- TGARCH (1, 2)-GED | ARMA (1, 1)- EARCH (1, 2)-t | ARMA (3, 3)- GARCH (1, 2)-GED | ARMA (3, 3)- GARCH (1, 1)-GED |

−0.6524 | - | −0.8660 | −0.6803 | −0.6731 | |

−0.8300 | - | - | −0.8395 | −0.8023 | |

−0.8584 | _ | - | −0.8036 | −0.8388 | |

0.6751 | 0.4481 | 0.9149 | 0.6771 | 0.6962 | |

0.7535 | −0.0930 | - | 0.7893 | 0.7654 | |

0.8758 | - | - | 0.8005 | 0.8848 | |

1.14E−06 | 7.19E−06 | −0.2436 | 8.66E−14 | 2.16E−11 | |

0.0332 | −0.0317 | −0.0165 | −0.0266 | 0.0529 | |

- | 0.0903 | 0.1539 | 0.0957 | - | |

1.2951 | 0.8747 | 0.9859 | 0.9281 | 0.9397 | |

−0.3340 | - | - | - | - | |

- | 0.0628 | −0.0371 | - | - |

Sample | F-statistic | Prob. F (1, 2086) | Obs * R-squared | Prob. Chi-Square (1) | ARCH effect |
---|---|---|---|---|---|

360001 | 0.3270 | 0.5675 | 0.3272 | 0.5673 | Non-existent |

110022 | 1.3938 | 0.2379 | 1.3942 | 0.2377 | Non-existent |

161812 | 0.8058 | 0.3695 | 0.8062 | 0.3692 | Non-existent |

180003 | 0.5391 | 0.4629 | 0.5395 | 0.4627 | Non-existent |

217016 | 1.3975 | 0.2373 | 1.3979 | 0.2371 | Non-existent |

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

As can be seen from

Use Formulas (11) and (13) to calculate PM and CVaR-RAROC, the results are as in

It can be seen from

Sample | CVaR’s maximum | CVaR’s minimum | CVaR’s mean | CVaR’s Std.DV | Ranking |
---|---|---|---|---|---|

360001 | 0.0962 | 0.0122 | 0.0284 | 0.0161 | 2 |

110022 | 0.1027 | 0.0111 | 0.0295 | 0.0172 | 1 |

161812 | 5.22E−05 | 2.23E−05 | 3.22E−05 | 5.67906E−06 | 5 |

180003 | 0.0850 | 0.0086 | 0.0236 | 0.0116 | 3 |

217016 | 3.39E−04 | 4.24E−05 | 1.00E−04 | 4.83002E−05 | 4 |

Sample | PM | Ranking | CVaR-RAROC | Ranking |
---|---|---|---|---|

360001 | −1.29067 | 2 | 0.001269 | 2 |

110022 | −0.10816 | 1 | 0.015962 | 1 |

161812 | −9.25943 | 4 | −0.04395 | 5 |

180003 | −1.55406 | 3 | 0.001086 | 3 |

217016 | −3.65E+02 | 5 | 1.46E−01 | 4 |

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”.

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 related financial products, and thus make poor financial decisions, which will harm the harmony of investors and society. The research of this paper can help investors understand the risk and performance of equity funds more comprehensively, so as to make accurate investment decisions, and provide decision-making basis and reference for financial institutions and regulatory departments.

National Natural Science Foundation of China (61703117); Guangxi Young and Middle-aged Teacher’s Basic Ability Improvement Project (2018ky0261).

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

Yang, J.L., Tang, G.Q., Yang, D.C. and Zhang, J.W. (2020) Risk Measurement and Performance Evaluation of Equity Funds Based on ARMA-GARCH Family Model. Open Journal of Statistics, 10, 325-340. https://doi.org/10.4236/ojs.2020.102022