Theory of VaR value at risk is refers to under certain confidence level, asset portfolio or portfolio in the future a period of time may be the biggest loss. That is:

where, Δp is assets in the holding period of loss; VaR is the confidence level c of value at risk; C is the confidence level.

To simplify the calculation, the commonly used GARCH model to measure the volatility, and the sequence of assets normal distribution is assumed. By type (1), when the confidence level c, asset holding period after standardization, VaR value can be obtained, namely the asset return sequence of the left tail maximum loss:

where type of P₀ is initial value of the assets, sigma for variance, Ζ_c is confidence level, T is asset holding period.

After calculating the VaR value, it is necessary to estimate the results of testing. Failure Test is the most commonly used test after testing, it is by comparing the frequency of the actual loss more than VaR and the upper limit of under a certain confidence level is close to or equal, which to determine the effectiveness of the VaR model. If the model is valid, the failure rate of the model should be equal to the preset VaR confidence (1 ? c), the difference between (1 ? c) if the failure rate is bigger, showing that model is not appropriate. Assumes that the confidence level for c, actual investigation the number of days for T, failed days to N, the failure rate for p(=N/T), such failure rate is subject to a binomial distribution, expecting probability is p^*, the original hypothesis for H0: p = p^*., alternative hypothesis for the H1: p ≠ p^*, whether to reject null hypothesis test failure frequency. Kupiec (1995) proposed that the likelihood ratio test is adopted to zero hypothesis test, and the likelihood ratio equation is:

Of formula (3) under the null hypothesis conditions LR statistic distribution of degrees of freedom is 1.

The general expression GARCH (q, p) model is:

where for the return series, for the residuals, for the conditional variance, for independent.

Zakoian, et al. (1990), based on this, advanced the TARCH or Threshold ARCH Threshold (ARCH) model, the conditional variances into:

Nelson (1990) proposed the allowed and is more flexible than quadratic equation index GARCH model of the relationship between the EGARCH model conditional variances into:

Ding et al. (1993) proposed PARCH model, conditional variance equation form PARCH model is:

Thus, we calculate the conditional variance, and then to predict earnings and variance, using predicted income and the corresponding probability distribution and variance under quantile can be substituted into (2) determine the future VaR value of T time.

In GARCH model residual distribution usually have three: a normal (Gaussian) distribution, student t-distribu- tion and generalized error distribution (Generalized Error Distribution, GED). Nelson and Hamilton, who made ​​use of the student t-distribution and generalized error distribution to reflect the characteristics of its tail spikes. Student t-distribution probability density function is:

where Γ(・) is the Gamma function, v is the degree of freedom, when v approaches ∞, t-distribution converges to the normal distribution. GED distribution probability density function is:

There are many normal test methods; the simplest method is to test skewness and kurtosis test. Using Eviews, analysis of the copper index logarithmic normality test results was shown in Figure 1.

As can be seen from the figure above, there is a certain day yield characteristics of normality, but it does not obey the standard normal distribution. Its average daily yield of 0.000256, −0.047806 skewness, kurtosis is 4.118310 3 larger than the standard normal distribution, showing the characteristics of a fat tail, which is more in line with theoretical estimates. The JB is much greater than 73.53960 significant level of 0.05 quantile 5.991, so the distribution is rejected the null hypothesis of normality. Logarithmic returns are not normally distributed, and have a left side, a fat tail nature.

With method of unit root test of time series, the results are shown in Table 3 below.

Table 3 shows, the ADF test day yield index futures time series t statistic −12.47478 less than 1% of the critical value −3.467633. Therefore, we can reject the null hypothesis at least at the 99% confidence level, the sequence does not exist unit root, is smooth.

Considering the validity of the model, autocorrelation test was carried out on the yield. Through Eviews autocorrelation test of the data, we get the result. Because of the autocorrelation coefficient and partial correlation coefficient is not 0, significantly and Q statistic is significant, so can get its residual error sequence exists ARCH effect. So using GARCH (1, 1) model to estimate, usually model assumes that the residuals obey normal distribution, but in practice, gains sequences often there is a peak, thick tail, namely the distribution of returns on kurtosis kurtosis is high than the standard normal distribution. Distribution in thick tail risk managers are particularly worried about, because it means that the frequency of the excess loss is higher than the predictions of a normal distribution. Have in front of the logarithm yield of normality test, it is concluded that yield sequence is not normal distribution, so we used GARCH (1, 1) model to estimate, the results are shown in the Table 4.

Before the establishment of the GARCH model, using AIC information criterion repeated tests, AIC get minimum values when the lag order (p, q) is (1,1), so the following models are GARCH (1,1) class model. We’ve already come to mean yield of 0.000256, the table can be combined to build GARCH model equations yields:

Mean equation:

Variance equation:

And then the residual model are heteroscedasticity LM test, the conclusion cannot reject the null hypothesis that the model does not consider the presence of residual heteroskedasticity and β values greater than 0.6, the

return value of the coefficient α is less than 0.25, you can explain this equation has strong explanatory power, can better describe the copper index yield heteroskedasticity phenomenon.

GARCH model with conditional variance to measure the futures market, which is calculated as follows:

Using Eviews, GARCH conditional variance sequences can be generated, the conditions of the conditional variance standard deviation obtained sequence evolution sequence, the figure is the standard deviation of the conditions GARCH model sequences, can be clearly seen from Figure 2, when the conditional variance has obvious degeneration, that change over time and depend on past values. With conditional VaR instead of the standard deviation obtained in the standard deviation is calculated to obtain statistics VaR at 95% confidence level, the following Table 5.

From the table above shows that the biggest risk income loss can reach 5.578978%, the smallest risk loss is 1.426879%, average risk loss is 3.698741%, the VaR of the standard deviation is 1.685201%.

Under 5% significance level, the failure of the number of days should not be more than 9 days, otherwise the risk has the potential to underestimate. Test results show that the failure rate is far less than 5%, says the VaR better estimate the risk.

Through the above empirical analysis, we can find that VaR method based on the GARCH model can well fitting value at risk of Shanghai copper futures in our country. We can consider it as one of the important means of futures risk management in our country, and with reference to their corresponding risk early warning system.