3.3 Normalization and filtering of the observed data
4.1.2 Out-of-sample VaR backtesting results
The backtesting procedure conducted on the basis of the second half (another 200 days) of the available observed intraday data. The resulting failure rates with the corresponding Kupiec’s test p-values are presented in Table 2.
As we see the models with bigger T show better prediction ability in comparison with others, in particular the smallest quantiles are very well captured, the p-values 0.97 and 0.93 indicate a very strong result. This can be a consequence of the smaller amount of truncated trajectories, which leads to more flexible and accurate capability of capturing the quantiles.
Figures 1-4 present the exceedances within each of the models and con-sidered VaR-levels. All the figures display the findings of the previous table.
model/α 0.01 0.05 0.1 T = 1 0.0153 (0.4837) 0.0820 (0.0590) 0.1333 (0.1377) T = 2 0.0051 (0.4506) 0.0564 (0.6870) 0.1282 (0.2062) T = 3 0.0102 (0.9710) 0.0820 (0.0590) 0.1743 (0.0015) T = 4 0.0153 (0.4837) 0.0512 (0.9347) 0.1179 (0.4151)
Table 2: VaR performance of the IBM-Google portfolio. p-values of the Kupiec test in brackets.
It is clearly visible that the model does not exhibit a smooth quantile history, in turn responding very quickly to shocks. As on every step of the rolling window procedure the parameters of the model are re-estimated, the model possesses the ability to react rapidly to the changes. It is worth mentioning that increasing T leads to the smoothing of the behavior of the estimated VaR values, which reduces possible overestimation of the risk.
Figure 1: Exceedances plot forV aR(α) for the model withT = 1 on the IBM-Google portfolio. The dots represent the empirical portfolio value stated by the data. The three curves under the dots correspond to theα= 0.1 (yellow), α = 0.05 (green), α= 0.01 (blue)
Figure 2: Exceedances plot forV aR(α) for the model withT = 2 on the IBM-Google portfolio. The dots represent the empirical portfolio value stated by the data. The three curves under the dots correspond to theα= 0.1 (yellow), α = 0.05 (green), α= 0.01 (blue)
Figure 3: Exceedances plot forV aR(α) for the model withT = 3 on the IBM-Google portfolio. The dots represent the empirical portfolio value stated by the data. The three curves under the dots correspond to theα= 0.1 (yellow), α = 0.05 (green), α= 0.01 (blue)
Figure 4: Exceedances plot forV aR(α) for the model withT = 4 on the IBM-Google portfolio. The dots represent the empirical portfolio value stated by the data. The three curves under the dots correspond to theα= 0.1 (yellow), α = 0.05 (green), α= 0.01 (blue)
5 Conclusions
Based on assumptions of the form of the marginal tail integral and a L´evy copula family, we have introduced the model for intraday stock returns. The proposed model is dynamic, thus captures the time-varying aspects. Our empirical results demonstrate a good capability of the model for making the forecasts, in particular to predict the VaR of different levels.
The performed in this paper study shows that the more information we use for the estimation the smoother is the behavior of the simulated quantiles (VaR values) and the less becomes the amount of the simulated outliers.
However, using too much history makes the model very conservative and unable to react quickly to the incoming shocks and changes in economy, which is a serious problem, considering that we are dealing with the high frequency data. Thus, some payoff between these two features should be found in order to optimize the estimation-simulation procedure, maintaining a good capability of capturing the shocks. For the case of the intraday data it is often enough to use the observations of the last trading week.
As the obtained model allows for simulation of the arbitrary large amount of the price-paths of considered assets, one can use this property for predict-ing the value of any possible derivative of the underlypredict-ings. As well as VaR, the
systematic risk measures such as CoVaR (Adrian and Brunnermeier (2011)) could be easily calculated in order to measure the sensitivity of different financial institutions to the shocks in the economy.
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