E MPIRICAL A NALYSIS

4.3.1. Tail index

From precious analysis, the daily returns of S&P 500 and Nasdaq Index are fat-tailed distributed. The thickness of tail distributed can be measured by the tail index. In this section, we use the revised Hill’s estimator to estimate the tail index and capture the tail shape of these two indices.

Our purpose in this research is to find a precise method to evaluate downside risks. As a

result, the left tail of observations is used to estimate tail index and measure the VaR. We use rolling sample method (Brooks, 2002) with 250 days to estimate the tail index and VaR of day t+1. Table 4 shows that the returns of both S&P 500 and Nasdaq Index for the whole sample period (1997~2003) are in existence of fat-tail property. The estimates of β₀ are bigger than 0 and between 0.1033 to 0.3036. According to Koedijk, Schafgans, and de Vries (1990), the distribution of these returns is fatter than normal distribution and belongs to student-t distribution. In addition, the estimated tail index of RR_30m and RR_5m are the smallest two among other models. This may imply that more problems of heteroscedasticity and volatility-clustered situation can be solved by using realized range method to standardize.

4.3.2. Comparison of VaR models

From precious discussion, the downside risks can be captured by tail index. In this section, the result of using variance-covariance and extreme value theory to measure VaR is presented. The average of estimated VaR for all sample period and individual years are listed in Table 5. Moreover, we use backtesting method to compare the ability of forecasting VaR in different models. For detailed analysis, many criteria of comparing VaR are applied and classified to three different dimensions. We focus on the result if it is better to evaluate VaR using realized range models and the comparison of VaR-normal with VaR-x models. In this research, we use rolling sample method and the length of each rolling sample is 250 days.

VaR estimate of the next day is based on the precious 250 prices. By this approach, 1512 volatility and VaR estimates are obtained. Figure 2 and 3 show daily returns and VaR-normal estimates of these conditional volatility models for S&P 500 and Nasdaq Index, respectively.

Figure 4 and 5 present daily returns and VaR-x estimates of these conditional volatility models for S&P 500 and Nasdaq Index, respectively. The biases of return-based models are obvious in evaluating VaR-x. The detailed analysis of comparing variance-covariance method by normal distribution with extreme value theory of VaR-x model is discussed below.

The models included in our research are described as follows. The first part is the models assumed to be normal distribution of financial returns in different conditional volatility process: EWMA-normal, GARCH-normal, CARR-normal, RR_30m-normal, and RR_5m-normal models. The second part is the models combining VaR-x and extreme value theory in different conditional volatility process: EWMA-VaR-x, GARCH-VaR-x, CARR-VaR-x, RR_30m-VaR-x, and RR_5m-VaR-x models. We use the same order of models in each rolling sample. Table 6 presents the number of failures in these ten conditional VaR models. In 95% confidence level for S&P 500, the failure number of EWMA-normal model and RR_30m-VaR-x model (77 and 75, respectively) are the most closest to the theoretical number (76). The third and fourth better ones are GARCH-normal and RR_5m-VaR-x models in sequence. As for Nasdaq Index, GARCH-normal model is the best one. The second best models are EWMA-normal, RR_5m-normal and CARR-VaR-x models. We conclude that it is better to evaluate risks under normal-distributed assumption in 95% confidence level. In 97.5% confidence level for S&P 500, RR_30m-VaR-x model performs the best.

RR_30m-normal, RR_5m-VaR-x and GARCH-VaR-x models are the next. For Nasdaq Index, EWMA-normal is the best one. The second and third better ones are GARCH-normal and GARCH-VaR-x models in sequence. In 99% confidence level for S&P 500, the closest one is CARR-VaR-x model. The next are GARCH-normal, CARR-normal, RR_5m-normal and RR_30m-VaR-x models. As for Nasdaq Index, the five normal ones and GARCH-VaR-x model perform nearly. As a whole, the normal-distributed models and the extreme value ones perform about the same in 97.5% and 99% level. In addition, RR_30m-VaR-x is the most precise model in different confidence level for S&P 500. EWMA-normal and RR_5m-VaR-x are the second better. For Nasdaq Index, EWMA-normal and GARCH-normal are the best.

RR_5m-normal and GARCH-VaR-x are the second ones. In empirical research of the failure number, realized range model performs better for S&P 500 than Nasdaq Index. In detail, realized range models with extreme value theory are more proper for S&P 500 and those with

normal distribution are more suitable for Nasdaq Index.

Except for the number of failures, there are other testing dimensions to evaluate VaR models. When computing equation of some criteria, like MRB in equation (41), RMSRB in equation (42), LRuc in equation (45), MRSB in equation (49) and error efficiency in equation (50), the number of sample days T is 1512 and the number of VaR models N equals to 10. The backtesting results of VaR models are listed in Table 7 to 12. Considering the conservatism of VaR models of S&P500 in 95% level in Table 7, the average of all models’ mean relative bias (MRB) are from -0.0657 to 0.0537. EWMA-VaR-x, GARCH-VaR-x, CARR-normal, and RR_5m-normal models are more conservative because of its large MRB in sequence.

According to root mean squared relative bias, CARR-normal, RR_5m-normal, RR_30m-normal model, RR_5m-VaR-x and GARCH-normal are the least divergent method.

The range-based models with normal distribution perform more conservative than others in 95% confident interval.

In the accuracy analysis of VaR models, the comparing result of the binary loss functions (BLF) is as same as the number of failures discussed above. RR_30m-VaR-x, EWMA-normal, GARCH-normal and RR_5m-VaR-x are the most accurate in sequence. When discussing the mean excess, RR_5m-VaR-x, GARCH-normal, EWMA-VaR-x and RR_30m-normal model are the smallest ones in sequence in 95% percentile. In regard to the LR test of unconditional coverage, independence, and conditional coverage, all models’ assumption are not rejected. In other words, all VaR models pass the statistic test. According to MOC criterion, the result is that EWMA-normal and RR_30m-VaR-x models are the best ones. GARCH-normal and RR_5m-VaR-x are the third and fourth models in 95% percentile. As a whole for accuracy test, GARCH-normal, RR_5m-VaR-x, RR_30m-VaR-x and EWMA-normal models perform well on accuracy test in sequence.

For the efficiency of VaR models, the best two efficient models are RR_30m-VaR-x and

RR_5m-VaR-x models. The next two are RR_5m-normal and RR_30m-normal models. The realized range model performs efficient among others. To summarize, the whole performance of realized range models with normal distritubion is close to EWMA-normal and GARCH-normal. However, realized range with VaR-x models dominate over others.

RR_5m-VaR-x model is the best one.

As for Nasdaq in Table 8, the average of all models’ mean relative bias (MRB) are from -0.0989 to 0.0288. RR_5m-normal, CARR-normal and RR_30m-normal models are the most conservative ones according to large MRB. For root mean squared relative bias, RR_5m-normal, RR_30m-normal and CARR-VaR-x models are the least divergent method.

The range-based models with normal distribution perform more conservative than others. In the accuracy analysis of VaR models, GARCH-normal, EWMA-normal, RR_5m-normal and CARR-VaR-x can produce more accurate value according to BLF. As to mean excess, the four realized range models perform well. As same as S&P500, all LR tests are passed. In regard to MOC, EWMA-normal, GARCH-normal, RR_5m-VaR-x and RR_30m-VaR-x models are closest to 1. Realized-range-based VaR-x models and return-based normal models perform about the same in accuracy. As for efficiency test, CARR-VaR-x and RR_30m-VaR-x models are the most efficient ones. To conclude the result of Table 8, CARR-VaR-x, RR_5m-normal, EWMA-normal and RR_30m-VaR-x are the top four models for Nasdaq Index in 95%

percentile.

Considering the conservatism of VaR models of S&P500 in 97.5% level in Table 9, GARCH-VaR-x and EWMA-VaR-x models produce largest number in MRB. As for RMSRB, the value of CARR-normal, RR_5m-VaR-x, RR_5m-normal and RR_30m-normal are the smallest in sequence. Considering the accuracy, RR_30m-VaR-x is the most precise model.

RR_30-normal, RR_5m-VaR-x and GARCH-VaR-x models perform well, too. For the efficiency test, CARR-normal, RR_5m-normal and RR_30m-normal models have the smallest

value of MRSB and error efficiency. As a whole, realized range models perform much better than return-based and CARR models. RR_30m-normal and RR_5m-VaR-x models are the best two on estimating VaR of S&P 500 in 97.5% confidence interval.

As for Nasdaq Index in 97.5% level in Table 10, CARR-VaR-x model is the most conservative one. In regard to accuracy, EWMA-normal, GARCH-normal, RR_30m-normal and GARCH-VaR-x models perform much better than others. For efficiency, CARR-normal, RR_30m-normal and RR_5m-normal models are more efficient. To summarize the result in Table 10, EWMA-normal and RR_30m-normal are the best models. CARR-normal and RR_5m-normal models are the next best ones.

Table 11 shows the result of S&P 500 in 99% level. For conservatism test, EWMA-VaR-x and GARCH-VaR-x models are the most conservative and divergent ones. As to accuracy, RR_5m-normal is the most precise model. Other range-based models also perform well. According to efficiency, GARCH-normal, EWMA-normal, RR_30m-normal and RR_5m-normal models produce smaller value of MRSB and error efficiency.

RR_5m-normal and RR_30m-normal model are the most proper ones to evaluate the risks of S&P 500 in 99% confident interval. In addition, models with normal distribution perform better than the VaR-x models.

Last, the result of Nasdaq in 99% level is presented in Table 12. As for conservatism test, EWMA-VaR-x and GARCH-VaR-x models are the most conservative and divergent ones. For accuracy test, RR_5m-normal and RR_30m-normal models produce more precise estimates.

According to efficiency test, EWMA-normal, CARR-normal and RR_30m-normal models perform well. As a whole, the performance of RR_30-normal, RR_5m-normal and CARR-normal is much better than others. Moreover, normal-distributed models are better than extreme value theory.

Various criteria are used to show different importance of VaR dimensions. Meanwhile, accuracy is more popular and important among these three dimensions. Users of VaR models often focus on the difference between failure rate and theoretical rate first. Moreover, financial institutions always don’t want to be conservative because they have to spend more costs to reach the restriction of required minimum capital. In practical, conservatism and efficiency seem to be paid less attention. To conclude, realized range models can produce better evaluation of financial risks. In 95% confident interval, realized range models with normal distribution perform as well as other normal models. In regard to VaR-x models, range-based ones are better than return-base method. As the increasing of percentile, realized range models dominate over return-based models and CARR model. Moreover, in 99%

confident level, RR_30m-normal and RR_5m-normal models are sufficient to capture the downside risks. It may imply that realized range models with normal distribution can evaluate VaR much better than VaR-x models in high confident level.