Value-at-Risk Forecasting of Chinese Stock Index and Index Future Under Jumps, Permanent Component, and Asymmetric Information

ABSTRACT

This article investigates the performance of time series models considering the jumps, permanent component of volatility, and asymmetric information in predicting value-at-risk (VaR). We use evaluation statistics including size and variability, accuracy, and efficiency to determine some suitable VaR measures for the Chinese stock index and its futures. The results reveal that models with jumps can provide VaR series that are less average conservative and have higher variability. Furthermore, additional considering the permanent component of volatility and asymmetric effect can induce more accurate and efficient risk measure in the long and short positions of the stock index and its futures.

KEY WORDS:

• asymmetric information
• jump
• permanent component
• value-at-risk

Notes

1. In the Chinese spot stock market, margin trading can be feasible to investment institutions and professional investors, or even qualified personal investors. At the end of 2012, there were 174 stocks from the basket of the HS300 index and eight ETF funds related to the HS300 index available for margin trading. The total amount of margin trading at the end of 2012 reached approximately RMB 89.5 billion.

2. We began to study the VaR of the stock index and index futures from the beginning of 2013. The RESET database only provided the stock index future data from 2006 to 2012. We censored the beginning part of the simulated data, which may contain weird dynamics, so the stock index futures data is from March 30, 2007 to December 20, 2012. Prior to March 2010, the index futures market was simulated with a large amount of active traders; most of them were institutional investors and professional investors. After that, the real trading system began to run.

The main market force is the institutional investors and professional investors. Hence, we can infer that their behaviors are similar and consistent before and after March 2010. We combine the trading data of these consecutive two periods together based on the assumption that rational investors learn from the former trading experience and information by the Bayes rule. The learning rule is implied by the estimating formula (A.3) in Part B of the Appendix. To compare the VaRs between the stock index and index futures, we use the same sample period. Since we use the rolling window method to do the VaR forecasting, the evaluation results are robust as to whether to extend the data period or not.

3. The AIC method is only applied to the in-sample data here. It may induce different specifications of the lag orders of the ARMA dynamics and the GARCH dynamics for the out-of-sample data after the rolling window. This problem is trivial in this article. The specification of ARMA(1,1) and GARCH(1,1) is fixed for the GARCH-JUMP type models. As such, we focus on investigating the impacts of the three characteristics (jumps, the asymmetric effects on the volatility dynamics and jumps, and the long-term trend of the volatility) on the VaR estimates.

4. We follow the Chan and Maheu (2002) method to set the maximum jump number. We find that the log-likelihood value of the MLE reaches a ceiling at the specification of ten jumps a day. For instance, when we set the value at twenty jumps per day, the log-likelihood value after the estimation changed very little from that of the ten jumps case. Therefore, the maximum jump number is set to ten.

5. We list the maximum likelihood estimation method in Part B of the Appendix. The details of the maximum likelihood estimation method are located in Chan and Maheu (2002) and Maheu and McCurdy (2004).

6. Although the Ljung-box test results of the jump intensity residuals are not significant in the case of using the stock index return to estimate the models of ARJI, ARJI-TREND, and GARJI, the model specification test of the whole models tends to prefer these models to the GARCH-JUMP model.

7. We should note that there is no distribution assumption on several parameters, such as ${\lambda }_0$ and in the models with a dynamic jump (ARJI, ARJI-TREND, and GARJI). Hence, the log-likelihood or likelihood ratio test cannot apply here to compare the model specifications among the models with a dynamic jump and without the dynamic jump strictly. Here, we only use it as a raw instrument to compare the different models. The former literature on the GARCH-JUMP type model prefers to use the goodness-of-fit test to do the model specification test. Thus, we borrow the robust test method from Hong and Li (2005). The test statistics are listed in Part A of the Appendix. For the details of the test method, the reader can refer to their article.

Additional information

Funding

The project is supported by National Nature Science Foundation of China (NSFC71301131 and NSFC71403192) and Humanities and Social Sciences Foundation of Ministry of Education of China (4YJC790130).