According to Jorion (2007), Value at Risk (VaR) is defined as the worst loss over a target horizon that will not be exceeded with a given level of confidence. VaR has been widely used for financial risk management since it is simple and intuitive in the sense that it summarizes the change in the value of a portfolio into a single number. Calculating VaR of a portfolio of assets is important in finance because it has been widely accepted that the investment in a portfolio of several assets has advantages over the investment in a single asset since Markowitz (1952). Different assets have different return distributions, and they are correlated in various ways. The key element of calculating VaR is to capture the relationship among assets in the portfolio appropriately.
In order to capture the relationship among assets and calculate VaR of the portfolio, multivariate distributions can be used. We may want to work with a univariate distribution from the history of returns from the same portfolio because the return of the portfolio itself is univariate. However, using a multivariate distribution for a collection of assets that make up the portfolio will give us more flexibility; for example, when we change the weight of the assets in a portfolio, we can handle the change more easily. Therefore, we would like to model the joint distribution of the asset returns with a proper multivariate distribution in this paper, especially with copulas.
Copulas are popular in modeling a joint distribution of several asset returns in finance. With copulas, we can construct multivariate distributions with different marginal distributions by separating the dependencies from marginal distributions. Also, there are many different copulas that can incorporate the proper dependence structure of the data. Embrechts
Elliptical copulas include Gaussian (normal) copula and Student’s
In this paper, we would like to search for the best copula in calculating the VaR of a portfolio of many assets. The critical point of this paper is to find out a good copula for portfolios with various dependence structures among assets. The dependence structure can be quite complicated in high dimensional data; therefore, we conjecture that vine copulas and hierarchical copulas perform better than single copulas. By comparing VaRs, we suggest a copula that performs better than other copulas in the simulation studies and real data applications. This study is to choose a proper copula function to reflect various dependence structures in a portfolio, and the performance is compared by VaR of the portfolio.
As for the marginal distributions, we can use different distributions for each asset in a portfolio. It has been known that the normal distribution is inappropriate to model the return distribution of financial assets. The return distributions of financial assets are slightly skewed and fat-tailed. There has been extensive research on finding good alternatives to the normal distribution in literature. For example, Venkataraman (1997) uses a quasi-Bayesian maximum likelihood estimation procedure, and Hull and White (1998) use a transform to multivariate normal distributions, which is updating schemes such as GARCH. Also, Eberlein and Keller (1995) use Hyperbolic distribution, and Madan
The remainder of the paper is organized as follows. Section 2 explains the model, VaR, NIG distribution, and copulas. Section 3 and Section 4 show the simulation results and real data applications, respectively. Finally, Section 5 concludes the paper.
Suppose we consider a portfolio of
Value at Risk (VaR) is a common measure of financial risk, especially the market risk, indicating the worst loss over a target horizon that will not be exceeded with a given level of confidence. Although various levels and target periods can be set, 1% and 5% probabilities and one day and ten day horizons are common. When a level of 1 −
Therefore, if we know the loss distribution of the given portfolio, VaR is easily obtained by the 100(1−
Normal inverse Gaussian (NIG) distribution, introduced by Barndorff-Nielsen (1997), is commonly used for return distributions of financial assets; for instance, Aas
where
For more details, refer to Schoutens (2003) or Barndorff-Nielsen
A copula is a function that combines several marginal distributions to form a joint distribution. The mathematical definition was initially introduced in Sklar (1959). Sklar’s Theorem says that any
for some copula
The copula is useful to estimate parameters when the dimension is large since the parameters of a multivariate distribution are estimated separately by each marginal distribution and a copula function. Using copulas, we can construct multivariate distributions with different marginal distributions. There are several types of copulas, such as elliptical copulas and Archimedean copulas. Readers may refer to Trivedi and Zimmer (2007) or Cherubini
The most commonly used elliptical copulas are Gaussian (normal) copula and Student’s
where
Similarly,
where
Elliptical copulas are commonly used in practice since its concept is simple and easy to understand.
Archimedean copulas are defined by a strictly decreasing convex generator function
where
Archimedean copulas are useful because they can reflect many dependence structures that are non-linear and non-symmetric. Kendall’s tau and Spearman’s rho are usually used by a measure of dependence under the non-linear relationship rather than the Pearson correlation coefficient. The relationship between the Archimedean copula and Kendall’s tau or Spearman’s rho in bivariate cases is given below. Kendall’s tau can be written as
and Spearman’s rho can be written as
This paper considers Clayton, Gumbel, and Frank copula among the many Archimedean copulas. Their generators and copula functions are given in Table 1.
The Archimedean copula is popular because it can model the dependence structure of any dimensions with one parameter. However, it would have limitations to reflect the actual dependence structure implied in high dimensions. In order to resolve this problem, several methods have been developed, such as vine copula and hierarchical copula, which use two or more parameters to model the copula function. Figure 1 shows the various methods of using copulas.
In order to mitigate the limitations of using a single parameter for reflecting dependence structure, vine copula was proposed by Aas
and
where
C-vine copula and D-vine copula are different classes of vine copulas. A C-vine copula is defined as
and a D-vine copula is defined as
The number of parameters to reflect the dependence structure is
Hierarchical copula is another way to ease the restriction that uses a single parameter to reflect dependencies. It is constructed using the composition of the generator functions. Again as an example, we consider a 4-dimensional multivariate distribution. Then the hierarchical copula (or fully nested Archimedean copula) is defined as (Okhrin and Ristig, 2014)
where
The number of parameters in a hierarchical copula model is
There is a caveat in the hierarchical copula in (
In this chapter, we will compare the performance of different copulas in calculating VaR through simulation experiments. We consider different combinations of correlation coefficients among log returns of assets to represent different dependence structures.
Random samples of four variables are generated, which represent daily log returns on four different assets. Since the return distributions of financial assets are commonly assumed to have heavier tails than the normal distribution, we generate random samples from a multivariate NIG distribution. Many research papers including Bølviken and Benth (2000) and Godin
In Case 1, the correlation coefficients of all pairs of assets are set to be similar. In Case 2, we suggest a hierarchical setup under which correlation coefficients of three pairs of assets are large, two pairs are medium, and one pair is relatively small. In Case 3, the correlation coefficients of all assets are random.
Weighted returns are calculated in order to make portfolio returns of four assets. The weights for each asset can have a variety of values, but they are set to be the same in this section. Although we fix the weights in the simulation studies and the real data applications, we can change them easily to construct different portfolios. To figure out the portfolio return distribution, we may fit the univariate distribution to the computed daily portfolio returns. One of the advantages of modeling the joint distribution of asset returns before computing the portfolio return is that changing the weights in the portfolio is easier.
The vine copula is affected by the order of combining marginal distributions and the families of the copula function. In this chapter, we use the order of combining marginal distributions according to the size of correlation coefficients and choose the copula function that minimizes the Akaike information criterion (AIC).
All parameters are estimated by the maximum likelihood estimation (MLE), and the levels for VaR are set from 95% to 99.5% by 0.5%.
In order to find out the best model to calculate VaR, we go through the following steps in each case. The true underlying distribution is assumed to be a multivariate NIG distribution.
Step 1: 10,000 random vectors of (
Step 2: For each vector obtained in Step 1, we calculate the weighted average of
Step 3: We estimate parameters of a copula model with NIG marginal distributions using the 10,000 random vectors generated in the same way as Step 1. The 4-dimensional multivariate NIG distribution in Step 1 serves as a true underlying distribution, and the copula model fitted in this step serves as a fitted distribution. Elliptical, vine, and hierarchical copulas are used.
Step 4: We create 10,000 random samples from the fitted copula model in Step 3. Then, we calculate the portfolio return with equal weights and the estimated VaR of a given level.
Step 5: Repeat the above steps 1,000 times to obtain the mean squared error (MSE) of VaR.
The results for each case given in Table 2 are in the following. In Tables 3
First, in case of elliptical copulas, Gaussian copula produces desirable results in Case 1, but the accuracy drops in Case 2 and Case 3 (Figures 2
Second, vine copulas show very accurate results in all cases. It seems that vine copulas can catch different dependence structures because they have the number of parameters equal to the number of pairs of variables. However, it does not directly imply that vine copulas will show good performance in prediction. To see the predictability of copula models, we will see their performance with test datasets (Tables 6
Finally, for the hierarchical copula, Gumbel, Clayton, and Frank copulas provide different results. The hierarchical copula has 3(=
Contrary to our expectations, we also found that the MSE does not grow as the dependence structure becomes more complex.
In many cases of calculating VaR, we would like to predict the loss level of the near future. For this purpose, backtesting has been used to examine the accuracy of the VaR prediction. Backtesting divides data into two pieces: ex-ante and ex-post. Models are fitted by using ex-ante data, and VaRs are calculated from models. Then, it is verified whether the estimated VaRs are accurate by using ex-post data. In order to improve the model accuracy, a rolling-window is used. Rolling-window means that ex-ante data and ex-post data move together as much as the number of windows. In this simulation, the number of windows is 1,000, and the size of each window is 250. Detailed steps of simulation experiments are as follows in each case of the correlation structure given in Table 2.
Step 1: 1,250 random vectors of (
Step 2: For each vector obtained in Step 1, we calculate the weighted average of
Step 3: We estimate parameters of a copula model with NIG marginal distributions using the first 250 random samples obtained in Step 1. Several different copula models are used.
Step 4: We create 10,000 random samples from the fitted copula model in Step 3. Then, we calculate the portfolio return with equal weights and VaR of a given level 1 −
Step 5: Check whether the 251st portfolio loss in Step 2 exceeds the estimated VaR in Step 4.
Step 6: Using the rolling-window method, we repeat Steps 3 to 5 1,000 times and compute the proportion of cases where the (
If the proportion is larger than
Thus, we want the violation rate, a proportion of portfolio losses in excess of VaR, as close to
The results are shown in Tables 6
The results are different from what we have seen in Section 3.2. In Section 3.2, we dealt with in-sample data so that there can be an overfitting problem. For instance, C-vine copula use 6(=
In this section, we investigate the performance of copula models for real data. We consider a portfolio that consists of Facebook, Amazon, Netflix, and Google (FANG), the four highest performing technology companies in the market. The daily log returns from January 2nd, 2013 to December 31st, 2017 are obtained from Yahoo Finance. The movements in log returns of stock prices are shown in Figure 8.
All of the log returns move around zero, with a relatively small variation for Google and a relatively large variation for Netflix. Table 9 provides some descriptive statistics and
It can be seen that log returns have positive skewness and high kurtosis. Especially, kurtosis is much larger than that of a normal distribution. Kolmogorov-Smirnov (KS) test also shows that the log returns do not follow a normal distribution.
In order to check the dependence structure in the portfolio, the pairwise Kendall’s tau values among the daily log returns of the four assets are computed in Table 10. Pearson correlation coefficients are also shown in parentheses. Amazon and Google have a relatively large Kendall’s tau value of 0.4501 and Pearson’s correlation coefficient of 0.5389, while Facebook and Netflix have a small Kendall’s tau value of 0.2743 and a Pearson’s correlation coefficient of 0.2859.
Normal distributions are not appropriate for marginal distributions in our copula model because of the fat tails of return distributions. Therefore, we instead use the normal inverse Gaussian (NIG) distribution, which is widely used for describing financial data with fat tails (Bølviken and Benth, 2000; Godin
In Figure 9, we see the empirical densities of log returns for each of the four assets and the densities of the NIG distribution with parameters estimated through MLE. Although the fitted NIG densities for Amazon and Netflix tend to be more peaked than empirical densities, the NIG distribution is well fitted to the real log returns. Also, when we see the goodness-of-fit of NIG distribution to the real log returns using the Kolmogorov-Smirnov test,
In this section, we compute VaR with various copula models and see which copula model gives the best performance in terms of the violation rate. We consider Gaussian,
The results in Table 12 and Figure 10 can be summarized as follows. First, the multivariate NIG distribution gives a better performance than the multivariate normal distribution, but their performances are not good. Gaussian copula and
Table 13 shows the
One thing to note is that the multivariate NIG distribution takes less computation time to calculate VaR compared to other copula models. Also, its performance is no worse than other copula models except the hierarchical Clayton copula. In that sense, the multivariate NIG distribution would be an alternative when we need a quick result. If we need an accurate result, the hierarchical Clayton copula would be more suitable.
Although not included in this paper, we checked the simulation results when the weights attached to four assets are changed from (0.25, 0.25, 0.25, 0.25) to (0.1, 0.2, 0.3, 0.4) and obtained similar results as those obtained above. We also considered all possible portfolios with three assets among FB, AMZN, NFLX, and GOOG. Again, the vine copula was the best in the in-sample performance and the hierarchical Clayton was the best in the out-of-sample performance.
As another example, we also tried Microsoft, Amazon, Google, and Apple (MAGA) to construct the portfolio within the same period as before. With MAGA, we obtain almost the same results as shown in Tables 14, 15, and Figure 11.
Copulas are one way of fitting a joint distribution to data, modeling the marginal distribution and the correlation structure separately. There are several types of copulas, including elliptical copula, vine copula, and hierarchical copula. In this paper, we want to investigate various copulas for modeling a multivariate distribution of returns of many assets in the financial market. We’re often interested in the portfolio of assets; therefore, the risk management of a portfolio is an important issue. If we find a good multivariate distribution that describes the returns of many assets well, we could easily calculate risk measures of portfolios with varying weights. We used the accuracy of VaR of the portfolio to see how each copula model performs. VaR of portfolio returns was computed with various copula functions in simulation experiments and real data applications. As a marginal distribution, we used a normal inverse Gaussian (NIG) distribution, which is widely used to model the asset returns in finance.
We expected that the performance of copula models might depend on the correlation structure of the underlying multivariate distribution. So we assumed three cases of the correlation structure in simulation and tried to find a more appropriate copula model than others in each case. In the simulation experiments of an in-sample performance, vine copulas showed the best performance in terms of the MSE of VaR regardless of the dependence structure. It was somehow expected because the number of parameters in the vine copulas is larger than other copulas. However, when it comes to out-of-sample performance, the backtesting results were no longer the best with vine copulas. We would explain this phenomenon by an overfitting problem because the model becomes too complex. Even though
The hierarchical Clayton copula was the best in prediction (out-of-sample performance) in both simulation studies and real data applications, regardless of the dependence structure of the underlying distribution. The number of parameters in the hierarchical copula is adequate to reflect the underlying dependence structure and not too big to make an overfitting problem. Also, it seems that Clayton copula works well because the Clayton copula is known to reflect left tail dependence.
In this paper, we analyzed the four dimensions. The larger the dimension, the harder it would be to establish the appropriate dependence structure and find a good copula model. In reality, diversified portfolios are common to mitigate the unsystematic risk that is inherent in a specific company or industry. Therefore, we would like to look at cases with larger dimensions as future work.
Common archimedean copulas
Range of | Bivariate copula | ||
---|---|---|---|
Clayton | [−1,∞)\{0} | ||
Gumbel | (− log( | [1,∞) | |
Frank | (−∞,∞) |
Pearson correlation coefficients for three different cases
Assets (1, 2) | Assets (1, 3) | Assets (1, 4) | Assets (2, 3) | Assets (2, 4) | Assets (3, 4) | |
---|---|---|---|---|---|---|
Case 1 | 0.73 | 0.75 | 0.71 | 0.72 | 0.75 | 0.73 |
Case 2 | 0.76 | 0.71 | 0.80 | 0.45 | 0.44 | 0.27 |
Case 3 | 0.80 | 0.68 | 0.49 | 0.38 | 0.16 | 0.08 |
MSE with various copulas in Case 1 (Modeling)
Elliptical | Vine | Hierarchical | |||||
---|---|---|---|---|---|---|---|
Gaussian | T | CVine | DVine | Gumbel | Clayton | Frank | |
95% | 0.0085 | 0.0089 | 0.0093 | 0.0095 | 0.0210 | 0.0242 | 0.0079 |
95.5% | 0.0098 | 0.0105 | 0.0106 | 0.0109 | 0.0283 | 0.0285 | 0.0114 |
96% | 0.0117 | 0.0121 | 0.0122 | 0.0123 | 0.0371 | 0.0348 | 0.0190 |
96.5% | 0.0132 | 0.0137 | 0.0138 | 0.0137 | 0.0491 | 0.0420 | 0.0334 |
97% | 0.0165 | 0.0156 | 0.0158 | 0.0160 | 0.0658 | 0.0526 | 0.0598 |
97.5% | 0.0231 | 0.0204 | 0.0212 | 0.0210 | 0.0924 | 0.0680 | 0.1090 |
98% | 0.0312 | 0.0283 | 0.0293 | 0.0278 | 0.1333 | 0.0913 | 0.1956 |
98.5% | 0.0477 | 0.0444 | 0.0429 | 0.0417 | 0.2036 | 0.1255 | 0.3635 |
99% | 0.0825 | 0.0733 | 0.0717 | 0.0709 | 0.3474 | 0.1908 | 0.7159 |
99.5% | 0.1758 | 0.1533 | 0.1564 | 0.1622 | 0.6912 | 0.3821 | 1.6435 |
MSE with various copulas in Case 2 (Modeling)
Elliptical | Vine | Hierarchical | |||||
---|---|---|---|---|---|---|---|
Gaussian | T | CVine | DVine | Gumbel | Clayton | Frank | |
95% | 0.0079 | 0.0120 | 0.0083 | 0.0085 | 0.0328 | 0.0180 | 0.0098 |
95.5% | 0.0094 | 0.0130 | 0.0098 | 0.0100 | 0.0425 | 0.0217 | 0.0156 |
96% | 0.0112 | 0.0143 | 0.0106 | 0.0109 | 0.0549 | 0.0253 | 0.0251 |
96.5% | 0.0135 | 0.0149 | 0.0120 | 0.0122 | 0.0713 | 0.0306 | 0.0411 |
97% | 0.0190 | 0.0175 | 0.0148 | 0.0155 | 0.0955 | 0.0406 | 0.0687 |
97.5% | 0.0265 | 0.0218 | 0.0181 | 0.0202 | 0.1300 | 0.0529 | 0.1142 |
98% | 0.0396 | 0.0282 | 0.0242 | 0.0267 | 0.1860 | 0.0721 | 0.1948 |
98.5% | 0.0614 | 0.0379 | 0.0336 | 0.0358 | 0.2716 | 0.1061 | 0.3340 |
99% | 0.1086 | 0.0659 | 0.0582 | 0.0596 | 0.4356 | 0.1758 | 0.6209 |
99.5% | 0.2537 | 0.1521 | 0.1399 | 0.1394 | 0.8587 | 0.3695 | 1.4181 |
MSE with various copulas in Case 3 (Modeling)
Elliptical | Vine | Hierarchical | |||||
---|---|---|---|---|---|---|---|
Gaussian | T | CVine | DVine | Gumbel | Clayton | Frank | |
95% | 0.0069 | 0.0114 | 0.0072 | 0.0070 | 0.0260 | 0.0086 | 0.0103 |
95.5% | 0.0082 | 0.0127 | 0.0083 | 0.0083 | 0.0346 | 0.0099 | 0.0161 |
96% | 0.0103 | 0.0143 | 0.0097 | 0.0098 | 0.0460 | 0.0114 | 0.0252 |
96.5% | 0.0132 | 0.0152 | 0.0113 | 0.0111 | 0.0594 | 0.0143 | 0.0387 |
97% | 0.0179 | 0.0171 | 0.0129 | 0.0134 | 0.0789 | 0.0177 | 0.0607 |
97.5% | 0.0260 | 0.0199 | 0.0163 | 0.0168 | 0.1081 | 0.0228 | 0.0968 |
98% | 0.0417 | 0.0264 | 0.0214 | 0.0228 | 0.1574 | 0.0288 | 0.1625 |
98.5% | 0.0693 | 0.0350 | 0.0306 | 0.0316 | 0.2356 | 0.0406 | 0.2757 |
99% | 0.1206 | 0.0533 | 0.0502 | 0.0509 | 0.3683 | 0.0696 | 0.4928 |
99.5% | 0.2493 | 0.1365 | 0.1246 | 0.1228 | 0.6852 | 0.1732 | 1.0513 |
Violation rate with various copulas in Case 1 (Prediction)
Elliptical | Vine | Hierarchical | |||||
---|---|---|---|---|---|---|---|
Gaussian | T | CVine | DVine | Gumbel | Clayton | Frank | |
95% | 0.057 | 0.060 | 0.060 | 0.059 | 0.064 | 0.057 | 0.060 |
95.5% | 0.053 | 0.057 | 0.053 | 0.055 | 0.060 | 0.047 | 0.056 |
96% | 0.048 | 0.049 | 0.048 | 0.048 | 0.055 | 0.043 | 0.052 |
96.5% | 0.043 | 0.045 | 0.045 | 0.044 | 0.051 | 0.039 | 0.047 |
97% | 0.040 | 0.038 | 0.037 | 0.038 | 0.044 | 0.033 | 0.044 |
97.5% | 0.036 | 0.033 | 0.031 | 0.033 | 0.040 | 0.026 | 0.042 |
98% | 0.027 | 0.027 | 0.023 | 0.027 | 0.032 | 0.021 | 0.036 |
98.5% | 0.020 | 0.018 | 0.017 | 0.016 | 0.028 | 0.015 | 0.032 |
99% | 0.015 | 0.013 | 0.013 | 0.014 | 0.019 | 0.010 | 0.024 |
99.5% | 0.006 | 0.005 | 0.006 | 0.003 | 0.012 | 0.004 | 0.016 |
Violation rate with various copulas in Case 2 (Prediction)
Elliptical | Vine | Hierarchical | |||||
---|---|---|---|---|---|---|---|
Gaussian | T | CVine | DVine | Gumbel | Clayton | Frank | |
95% | 0.051 | 0.054 | 0.053 | 0.060 | 0.062 | 0.048 | 0.056 |
95.5% | 0.049 | 0.051 | 0.048 | 0.053 | 0.056 | 0.044 | 0.053 |
96% | 0.044 | 0.047 | 0.043 | 0.047 | 0.054 | 0.041 | 0.047 |
96.5% | 0.042 | 0.042 | 0.042 | 0.045 | 0.047 | 0.037 | 0.044 |
97% | 0.039 | 0.037 | 0.038 | 0.041 | 0.045 | 0.030 | 0.043 |
97.5% | 0.032 | 0.031 | 0.031 | 0.036 | 0.040 | 0.026 | 0.040 |
98% | 0.028 | 0.026 | 0.026 | 0.027 | 0.036 | 0.022 | 0.035 |
98.5% | 0.021 | 0.019 | 0.021 | 0.021 | 0.028 | 0.019 | 0.029 |
99% | 0.018 | 0.018 | 0.017 | 0.021 | 0.023 | 0.014 | 0.025 |
99.5% | 0.013 | 0.009 | 0.009 | 0.010 | 0.019 | 0.009 | 0.019 |
Violation rate with various copulas in Case 3 (Prediction)
Elliptical | Vine | Hierarchical | |||||
---|---|---|---|---|---|---|---|
Gaussian | T | CVine | DVine | Gumbel | Clayton | Frank | |
95% | 0.053 | 0.060 | 0.055 | 0.068 | 0.063 | 0.051 | 0.057 |
95.5% | 0.050 | 0.051 | 0.050 | 0.061 | 0.056 | 0.048 | 0.052 |
96% | 0.047 | 0.048 | 0.047 | 0.055 | 0.052 | 0.043 | 0.047 |
96.5% | 0.043 | 0.045 | 0.042 | 0.049 | 0.047 | 0.039 | 0.046 |
97% | 0.038 | 0.038 | 0.037 | 0.043 | 0.044 | 0.033 | 0.044 |
97.5% | 0.033 | 0.031 | 0.030 | 0.037 | 0.039 | 0.028 | 0.037 |
98% | 0.026 | 0.025 | 0.024 | 0.032 | 0.034 | 0.021 | 0.034 |
98.5% | 0.022 | 0.020 | 0.019 | 0.024 | 0.027 | 0.018 | 0.029 |
99% | 0.017 | 0.015 | 0.015 | 0.018 | 0.020 | 0.015 | 0.020 |
99.5% | 0.013 | 0.010 | 0.009 | 0.013 | 0.015 | 0.007 | 0.018 |
Descriptive statistics of log returns and Kolmogorov-Smirnov test of normality
Mean | Sd | Skewness | Kurtosis | KS Test ( | |
---|---|---|---|---|---|
FB | 0.0015 | 0.0197 | 2.1503 | 29.8957 | 0.0000 |
AMZN | 0.0012 | 0.0182 | 0.3576 | 13.8721 | 0.0000 |
NFLX | 0.0021 | 0.0290 | 1.9922 | 29.3304 | 0.0000 |
GOOG | 0.0008 | 0.0137 | 1.7335 | 21.4751 | 0.0000 |
Pairwise Kendall’s tau among log return data (Pearson’s correlation coefficients are in parentheses)
FB | AMZN | NFLX | GOOG | |
---|---|---|---|---|
FB | 1 | 0.3821 | 0.2743 | 0.3990 |
AMZN | (0.4418) | 1 | 0.2990 | 0.4501 |
NFLX | (0.2859) | (0.3320) | 1 | 0.2816 |
GOOG | (0.4587) | (0.5389) | (0.3422) | 1 |
Maximum likelihood estimates of NIG distribution and
KS Test ( | |||||
---|---|---|---|---|---|
FB | 40.7240 | 2.1113 | 0.0145 | 0.0007 | 0.8819 |
AMZN | 43.9606 | 0.9009 | 0.0133 | 0.0009 | 0.5655 |
NFLX | 23.2511 | 4.5023 | 0.0163 | −0.0011 | 0.8559 |
GOOG | 64.0877 | 3.7463 | 0.0110 | 0.0002 | 0.9006 |
Violation rate with various distributions in FANG
Multivariate | Elliptical | Vine | Hierarchical | ||||||
---|---|---|---|---|---|---|---|---|---|
Normal | NIG | Gaussian | T | CVine | DVine | Gumbel | Clayton | Frank | |
95% | 0.053 | 0.052 | 0.053 | 0.053 | 0.053 | 0.052 | 0.064 | 0.050 | 0.055 |
95.5% | 0.049 | 0.049 | 0.051 | 0.051 | 0.046 | 0.048 | 0.060 | 0.045 | 0.053 |
96% | 0.043 | 0.044 | 0.045 | 0.045 | 0.043 | 0.045 | 0.054 | 0.042 | 0.052 |
96.5% | 0.041 | 0.038 | 0.042 | 0.041 | 0.038 | 0.039 | 0.053 | 0.033 | 0.048 |
97% | 0.036 | 0.037 | 0.035 | 0.035 | 0.035 | 0.033 | 0.048 | 0.028 | 0.045 |
97.5% | 0.030 | 0.029 | 0.033 | 0.031 | 0.029 | 0.028 | 0.044 | 0.026 | 0.040 |
98% | 0.028 | 0.023 | 0.028 | 0.026 | 0.025 | 0.024 | 0.038 | 0.022 | 0.035 |
98.5% | 0.025 | 0.022 | 0.023 | 0.023 | 0.020 | 0.020 | 0.032 | 0.015 | 0.028 |
99% | 0.021 | 0.017 | 0.020 | 0.016 | 0.015 | 0.015 | 0.026 | 0.013 | 0.025 |
99.5% | 0.018 | 0.011 | 0.013 | 0.012 | 0.012 | 0.012 | 0.019 | 0.007 | 0.022 |
The
Multivariate | Elliptical | Vine | Hierarchical | ||||||
---|---|---|---|---|---|---|---|---|---|
Normal | NIG | Gaussian | T | CVine | DVine | Gumbel | Clayton | Frank | |
95% | 0.67 | 0.77 | 0.67 | 0.67 | 0.67 | 0.77 | 0.05 | 1.00 | 0.47 |
95.5% | 0.55 | 0.55 | 0.37 | 0.37 | 0.88 | 0.65 | 0.03 | 1.00 | 0.23 |
96% | 0.63 | 0.53 | 0.43 | 0.43 | 0.63 | 0.43 | 0.03 | 0.75 | 0.06 |
96.5% | 0.31 | 0.61 | 0.24 | 0.31 | 0.61 | 0.50 | 0.00 | 0.73 | 0.03 |
97% | 0.28 | 0.21 | 0.37 | 0.37 | 0.37 | 0.58 | 0.00 | 0.71 | 0.01 |
97.5% | 0.33 | 0.43 | 0.12 | 0.24 | 0.43 | 0.55 | 0.00 | 0.84 | 0.01 |
98% | 0.09 | 0.51 | 0.09 | 0.19 | 0.28 | 0.38 | 0.00 | 0.66 | 0.00 |
98.5% | 0.02 | 0.09 | 0.05 | 0.05 | 0.22 | 0.22 | 0.00 | 1.00 | 0.00 |
99% | 0.00 | 0.04 | 0.01 | 0.08 | 0.14 | 0.14 | 0.00 | 0.36 | 0.00 |
99.5% | 0.00 | 0.02 | 0.00 | 0.01 | 0.01 | 0.01 | 0.00 | 0.40 | 0.00 |
Violation rate with various distributions in MAGA
Multivariate | Elliptical | Vine | Hierarchical | ||||||
---|---|---|---|---|---|---|---|---|---|
Normal | NIG | Gaussian | T | CVine | DVine | Gumbel | Clayton | Frank | |
95% | 0.054 | 0.052 | 0.055 | 0.056 | 0.053 | 0.055 | 0.061 | 0.049 | 0.056 |
95.5% | 0.050 | 0.050 | 0.051 | 0.051 | 0.049 | 0.051 | 0.059 | 0.043 | 0.054 |
96% | 0.046 | 0.048 | 0.046 | 0.047 | 0.044 | 0.046 | 0.055 | 0.038 | 0.049 |
96.5% | 0.040 | 0.040 | 0.041 | 0.041 | 0.039 | 0.040 | 0.050 | 0.034 | 0.048 |
97% | 0.037 | 0.034 | 0.037 | 0.037 | 0.035 | 0.036 | 0.047 | 0.032 | 0.041 |
97.5% | 0.035 | 0.030 | 0.034 | 0.033 | 0.031 | 0.032 | 0.039 | 0.028 | 0.039 |
98% | 0.034 | 0.027 | 0.030 | 0.029 | 0.027 | 0.029 | 0.035 | 0.023 | 0.035 |
98.5% | 0.031 | 0.020 | 0.025 | 0.023 | 0.022 | 0.023 | 0.032 | 0.016 | 0.030 |
99% | 0.026 | 0.015 | 0.018 | 0.017 | 0.014 | 0.015 | 0.026 | 0.010 | 0.028 |
99.5% | 0.015 | 0.006 | 0.009 | 0.005 | 0.007 | 0.008 | 0.017 | 0.003 | 0.018 |
The
Multivariate | Elliptical | Vine | Hierarchical | ||||||
---|---|---|---|---|---|---|---|---|---|
Normal | NIG | Gaussian | T | CVine | DVine | Gumbel | Clayton | Frank | |
95% | 0.57 | 0.77 | 0.47 | 0.39 | 0.67 | 0.47 | 0.12 | 0.88 | 0.39 |
95.5% | 0.45 | 0.45 | 0.37 | 0.37 | 0.55 | 0.37 | 0.04 | 0.76 | 0.18 |
96% | 0.34 | 0.21 | 0.34 | 0.27 | 0.53 | 0.34 | 0.02 | 0.74 | 0.16 |
96.5% | 0.40 | 0.40 | 0.31 | 0.31 | 0.50 | 0.40 | 0.02 | 0.86 | 0.03 |
97% | 0.21 | 0.47 | 0.21 | 0.21 | 0.37 | 0.28 | 0.00 | 0.71 | 0.05 |
97.5% | 0.06 | 0.33 | 0.08 | 0.12 | 0.24 | 0.17 | 0.01 | 0.55 | 0.01 |
98% | 0.00 | 0.13 | 0.04 | 0.06 | 0.13 | 0.06 | 0.00 | 0.51 | 0.00 |
98.5% | 0.00 | 0.22 | 0.02 | 0.05 | 0.09 | 0.05 | 0.00 | 0.80 | 0.00 |
99% | 0.00 | 0.14 | 0.02 | 0.04 | 0.23 | 0.14 | 0.00 | 1.00 | 0.00 |
99.5% | 0.00 | 0.66 | 0.11 | 1.00 | 0.40 | 0.22 | 0.00 | 0.33 | 0.00 |