Research Of VAR And CVaR In Stock Index Returns Under Bilateral Gamma Distribution

This article utilized gamma distribution to make a research about the returns of stock index.We use three kinds of indexes in United States stock market to conduct gamma distribution’s fitting.These three kinds of index are standard & poor’s 500 index,the Dow Jones industrial average index and the NASDAQ index.We got bilateral gamma distribution of stock-index,and passed the Kolmogorov-Smirnov test.On this basis,we conduct some simple probability inference of these indexes.In addition,we applied the bilateral gamma distribution to value at risk(VaR)and conditional value at risk(CVaR)and calculate the VaR and CVaR of the three major U.S.stock indexes.The main contents are as follows:1.Using the kurtosis,skewness,Jarque-Bera statistic,and Q-Q chart to analyze the limitations of the normal distribution when characterizing daily yield of stock-index.2.This paper introduces the theory knowledge of gamma distribution and used the maximum likelihood estimation to estimate gamma distribution..3.We utilized three kinds of indexes in United States stock market to conduct gamma distribution’s fitting.We get bilateral gamma distribution of stock-index,and passed the Kolmogorov-Smirnov test.The results show that the yield data of stock-index obey bilateral gamma distribution.4.Use the statistical characteristics of bilateral yields of stock-index to deduce the characteristics of overall yields,and compare the theoretical value with the actual value.We found that they are very close.In addition,we conduct some simple probability inference of stock-index,giving the probability of overall yield greater than or less than one value,and the probability in a certain range.By the way,we also consider the impacts of volume.5.We applied the bilateral gamma distribution to value at risk(VaR)and conditional value at risk(CVaR)and calculate the VaR and CVaR of three major U.S.stock indexes.And then,we conduct backing test of VaR and CVaR.The results are very good.We can say that the VaR model and CVaR model based on the bilateral gamma distribution are suitable for measuring the risk and the tail risk of the stock market.