VaR and ES forecasting with symmetric type models

Theα= 1% VaR and ES forecasting results with symmetric type models (GARCH, CARSR, CARR and CARSR2) are presented in this section. In order to include the 2008 global financial crisis period in the forecasting experiments, m = 1500 one-step-ahead volatility or conditional mean forecasts are calculated based on the 4 different models on 5 data sets, with fixed-size rolling sample n. Thus we have

n = N −m, given the size data set is N (number of obs. in Table 4.7). With CARSR model, we can estimate the signed range volatility σsr,t, then transform

it into an estimator of return volatility with ˆσr,t = √

0.811σsr,t. As to the CARR

and CARSR2 models, we can transform the conditional mean of range and signed range square into return volatility estimates, with ˆσr,t = 0.6006σR,t = 0.6006λR,t

and ˆσr,t = √

0.811σsr,t =

p

0.811λsr2,t. These volatility or conditional mean fore-

casts are employed to calculate VaR and ES.

To evaluate the performance of VaR and ES forecasts through different models, we employ the VaR violation rate (VRate) and ES violation rate (ESRate), which are presented in Equations (3.10) and (3.11).

As we discussed in Section4.5, four models (GARCH-Gaussian, CARSR-Gaussian, CARR-Exponential and CARSR2-Exponential) are on equal comparison ground, and they are employed in the 1% VaR and ES forecasting study, in order to see how the signed range type models work in the empirical study. Table 4.9 presents the VRate, for each model (estimated with ML and MCMC respectively) in each market. The expected VRate should be 1%: boxes represent the model in each market that has a VRate closest to that; bold indicates the model with VRate furthest away from expected.

To begin with, all the VRate values in Table4.9is clearly higher than 1%, because of the error distribution selection. Nevertheless, the VRate results already demon- strate the superiority of CARSR-G and CARSR2-Exp compared to GARCH-G and CARR-Exp, while results generated from ML and MCMC are quite close to each other, which is expected based on the simulation study. The mean VRate favours the CARSR2 model, and CARSR and CARSR2 both produced 1.67 % VRate considering the median of violations of 5 VaR forecasting series.

Furthermore, the ES tail risk forecasting results are shown in Table 4.10. As presented before, Gerlach and Chen (2016) illustrated that the quantile level that the 1% ES was estimated to fall was between 0.34% and 0.38%, depending on the error distribution selection. Still, we observe all the ESRate values are clearly higher than the expected violation rate. However, CARSR and CARSR2 generates the ES forecasts that have the ESRate closest to the nominal level among the 4 models, based on mean or median of violations of 5 ES forecasting series.

Lastly, 1500 ES forecasts by GARCH-G and CARSR-G for SP500 are plotted in Figure 4.12, which shows that ES forecasts from CARSR-G recovers faster than

that of GARCH-G during the 2008 GFC time period, meaning the extra efficiency can be gained with SR compared to return. In addition, Figure 4.12 presents the 1500 one-step-ahead ES forecasts by CARR-Exp and CARSR2-Exp with S&P500. It is shown that CARR estimates too low risk level, and leads to high VRate/α

ratio. This can be related to the simulation results that we observed in Section

4.3. The expected value of range is affected by the trading frequency. At 5- min frequency, we have σ2

r,t = 0.4265σR,t2 . Therefore, the theoretical coefficient

1/4log2 = 0.3607 might be too small in real-world data sets, thus provides much reduced risk level. However, the expected value of signed range is quite robust to the different trading frequencies, and it also considers the intra-day process compared to return. Therefore, CARSR type models provide the most satisfiable VaR and ES forecasting results, compared to that of GARCH or CARR type models.

Table 4.9: VRate for 5 data sets with 4 symmetric models estimated with

ML and MCMC.

Data sets S&P 500 AORD DAX FTSE Hang Seng Mean Median

GARCH-G-ML 2.67% 2.13% 2.07% 2.20% 1.47% 2.11% 2.13% GARCH-G-MCMC 2.67% 2.13% 2.07% 2.27% 1.53% 2.13% 2.13% CARSR-G-ML 2.67% 1.33% 1.67% 1.73% 1.47% 1.77% 1.67% CARSR-G-MCMC 2.67% 1.40% 1.67% 1.87% 1.53% 1.83% 1.67% CARR-Exp-ML 4.27% 4.20% 2.87% 2.73% 2.93% 3.40% 2.93% CARR-Exp-MCMC 4.20% 4.40% 3.20% 3.00% 3.27% 3.61% 3.27% CARSR2-Exp-ML 2.40% 1.13% 1.67% 1.87% 1.53% 1.72% 1.67% CARSR2-Exp-MCMC 2.33% 1.20% 1.67% 1.87% 1.53% 1.72% 1.67% Note:A box indicates the favored model based on mean or median VRate, whilst bold indicates the least favoured model.

Table 4.10: ESRate for 5 data sets with 4 symmetric models employing ML

and MCMC.

Data sets S&P 500 AORD DAX FTSE Hang Seng Mean Median

GARCH-G-ML 1.27% 1.20% 1.20% 1.27% 0.93% 1.17% 1.20% GARCH-G-MCMC 1.27% 1.13% 1.13% 1.33% 1.00% 1.17% 1.13% CARSR-G-ML 1.33% 0.67% 1.07% 0.93% 0.87% 0.97% 0.93% CARSR-G-MCMC 1.33% 0.67% 1.13% 1.00% 1.00% 1.03% 1.00% CARR-Exp-ML 3.00% 2.53% 1.33% 1.47% 1.87% 2.04% 1.87% CARR-Exp-MCMC 3.13% 2.40% 1.47% 1.33% 1.93% 2.05% 1.93% CARSR2-Exp-ML 1.33% 0.60% 1.07% 1.00% 1.00% 1.00% 1.00% CARSR2-Exp-MCMC 1.27% 0.60% 1.07% 1.00% 1.00% 0.99% 1.00% Note:A box indicates the favored model based on mean or median ESRate, whilst bold indicates the least favoured model.

1800 2000 2200 2400 2600 2800 3000 3200 3400 3600 −15 −10 −5 0 5 10 15 SP500 Return GARCH−G−ML CARSR−G−MCMC

Figure 4.11: 1500 S&P500 ES forecasts with GARCH-G-ML and CARSR-

Exp-MCMC. 1800 2000 2200 2400 2600 2800 3000 3200 3400 3600 −15 −10 −5 0 5 10 15 SP500 Return CARR−Exp−ML CARSR2−Exp−MCMC

Figure 4.12: 1500 SP500 ES forecasts with CARR-Exp-ML and CARSR2-