Study on Financial Market Risk Measurement Based on Asymmetric Laplace Distribution
Hong Zhang, Li Zhou, Jie Zhu
School of Information, Beijing Wuzi University, Beijing, China
Email address:
To cite this article:
Hong Zhang, Li Zhou, Jie Zhu. Study on Financial Market Risk Measurement Based on Asymmetric Laplace Distribution. American Journal of Theoretical and Applied Statistics. Vol. 4, No. 4, 2015, pp. 264268. doi: 10.11648/j.ajtas.20150404.16
Abstract: In this paper, According to the returns distributions (of the financial assets returns series) with peak fattailed and asymmetric and the theory of Asymmetric Laplace distribution. ALVaR (ALCVaR) parametric method and Monte Carlo simulation are proposed which are based on Asymmetric Laplace distribution. We analyze the VaR (CVaR) measuring model of AL distribution and discuss its backtesting. And then we evaluate the pros and cons of each method combining with the characteristics of the stock market risk of three countries. (America, China and Japan).
Keywords: Asymmetric Laplace, ALVaR, Financial Market Risk
1. Introduction
In the paper, we analyze the VaR (CVaR) measuring model of AL distribution and discuss its backtesting. And then we evaluate the pros and cons of each method combining with the characteristics of the stock market risk of three countries. (China Japan and America).
In order to better capture the market risk features such as biased and thick tail, to study the distribution of the risk further. Balakrishnan and Basu (1995)^{1}, Bain and Engelhardt (1973)^{2}, Kotz et al. (2001, 2002)^{35},Trindade and Zhu (2007)^{67} have done a lot of research on non symmetric Laplasse (Asymmetric Laplace, AL) family of distributions.
Trindade and Zhu (2007) studied progressive distribution of financial risk estimates based on the AL distribution. Jayakumar and Kuttykrishnan (2007)^{8} and Trindade et al. (2010) studied the model of time sequence based on AL distribution.AL distribution can fit the data characteristics well of Asymmetric and thick tail.
There are 3 Levels titles in an article to make ideas clear:
(1) Given fitting test of AL distribution and empirical analysis of risk measurement.
(2) Given the market risk value (VaR and CVaR ) and its backtesting.
(3) Proposing ErrorVaR and ErrorCVaR for analyzing the effectiveness of methods^{911}.
2. The Empirical Analysis
2.1. The Selection of Data and Its Characteristics
Selecting S.H.I (Shanghai composite index), Nikkie225 and S&P500 as research objects. Sample interval is from 2010.01.04 to 2014.12.31. Using Logarithm yields,
The results of statistics (table 1) show that tail of exponential gains and losses distribution is fatter than normal distribution’s. Which mains abnormal fluctuations in the market happen sometimes, the fact that skewness is all negative shows, from a longterm perspective, that fluctuation in the left side of exponential gains and losses distribution is larger than right side. So normal distribution cannot effectively characterize these phenomena. Stationary ADFtest results that H=1and P=1.0e0.3 are far less than 0.05, showing the results reject unit root process hypothesis, and accept the hypothesis of stationary sequence. Simultaneously, indexes are tested by normal JB .Results that H=1 reject normal distribution hypothesis. Known from the analysis, financial time series usually have some obvious characteristics. The normal distribution assumption commonly is not truly reflecting the real situation of the fluctuation of reality. The test of AL distributions are listed. (Table.1)
2.2. Al Distribution Fitting Text
Maximum likelihood estimates of the parameters of AL distribution (each market index) can be calculated. (Table.2) Thus, we assume that each return series obey the distribution corresponding. Fitting histograms are listed respectively with the AL density function of S.H.I, Nikkie225 and S&P500. Known from tables, characteristics of financial data samples can be well fitted by AL distribution, such as excess kurtosis, fat tail and asymmetry. KS test results of AL distribution (of returns in each market index) show: There are all H=0, and P>0.05.So in 5% or 1% significance level, and various market indices have accepted hypothesis that the sample data is subject to AL distribution^{13,15}.
Yield  Mean  Variance  Skewness  Kurtosis  ADFtest 
S.H.I  0.0079  0.0004  0.3469  5.4187  1(1.0e3) 
Nikkie225  0.0001  0.0002  0.4456  11.2579  1(1.0e3) 
S&P500  0.0001  0.0001  0.2481  13.2471  1(1.0e3) 
Notes: H value and P value in ADFtest
Yield 


 KS text 
S.H.I  0.0040  1.1118  0.0255  0(0.8521) 
Nikkie225  0.0013  1.0555  0.0064  0(0.8254) 
S&P500  0.0014  1.0000  0.0131  0(0.1359) 
Notes: H value and P value in KS test
()
2.3. Calculating VaR and CVaR
2.3.1. AlVar Parametric Method and AlCvar Parametric Method Theorem
Suppose Y is the profits and losses of a certain financial portfolio, Y is a random variable ,obeying Asymmetric Laplace distribution ,for a given confidence level
(Usually ranging from 95% to 99% ):
(1)
The theoretical value of VaR and CVaR respectively is
(2)
(3)
(2)Maximum likelihood estimate of VaR and CVaR respectively is
(4)
The estimated values of market risk, VaR and CVaR calculated by using theorem are listed.(Table 3)
2.3.2. AlMc Method
Using the ALMC method generate random number which are subject to distribution correspondingly to calculate the values of the market risk, VaR and CVaR. (See Table 3)
Confidence level  95%  97.5%  
NVaR  S.H.I  0.0330(1.1459)  0.0340(8.7931)  
Nikkie225  0.0290(2.4001)  0.0355(0.1646)  
S&P500  0.0249(1.3902)  0.0258(2.1854)  
ErrorVaR  0.6214E004  0.8412E004  
NCVaR  S.H.I  0.0422[0.0070]  0.0465[0.0054]  
Nikkie225  0.0354[0.0148]  0.0409[0.0145]  
S&P500  0.0356[0.0108]  0.0356[0.0111]  
ErrorCVaR  0.9546E004  1.3035E004  
Parameter Value Par  ALVaR  S.H.I  0.0412(9.2978)  0.0515(7.1552) 
Nikkie225  0.0299(3.5000)  0.0354(0.0547)  
S&P500  0.0250(1.3956)  0.0345(0.0082)  
ErrorVaR  1.7456E004  0.4152E004  
ALCVaR  S.H.I  0.0601[0.0025]  0.0714[0.0025]  
Nikkie225  0.0411[0.0069]  0.0521[0.0014]  
S&P500  0.0348[00.0071]  0.0147[0.0090]  
ErrorCVaR  0.3456E004  0.4841E004  
NonParameter Value  ALVaR  S.H.I  0.0325(0.0752)  0.0465(0.0025) 
Nikkie225  0.02483(0.5419)  0.0352(0.0525)  
S&P500  0.0248(0.6328)  0.0258(3.9541)  
ErrorVaR  0.1549E004  0.2858E004  
ALCVaR  S.H.I  0.0522[0.0020]  0.0625[0.0035]  
Nikkie225  0.0341[0.0059]  0.0145[0.0025]  
S&P500  0.0326[0.0058]  0.0325[0.0065]  
ErrorCVaR  0.2400E004  0.4078E004 
Confidence level  99%  99.5%  99.9%  
NVaR  S.H.I  0.0423(15.3001)  0.0545(22.2541)  0.0654(24.5945)  
Nikkie225  0.0452(8.8152)  0.0452(19.6541)  0.0587(33.4852)  
S&P500  0.0358(8.3012)  0.0395(23.5145)  0.0469(52.0147)  
ErrorVaR  1.1526E004  1.4521E004  0.8852E004  
NCVaR  S.H.I  0.0523[0.0065]  0.0547[0.0020]  0.0685[0.0068]  
Nikkie225  0.0456[0.0154]  0.0527[0.0152]  0.0456[0.0214]  
S&P500  0.0415[0.0148]  0.0475[0.0145]  0.0544[0.0158]  
ErrorCVaR  1.8524E004  1.6254E004  1.8554E004  
Parameter Value Par  ALVaR  S.H.I  0.0654(3.8441)  0.0852(1.9421)  0.1206(2.4152) 
Nikkie225  0.0452(1.0225)  0.0523(2.4189)  0.0778(3.9651)  
S&P500  0.0415(4.7521)  0.0478(7.8521)  0.0625(6.3258)  
ErrorVaR  2.6251E004  0.2554E004  0.0521E004  
ALCVaR  S.H.I  0.0852[0.0055]  0.0956[0.0101]  0.1225[0.1252]  
Nikkie225  0.0619[0.0025]  0.0752[0.0115]  0.0900[0.0154]  
S&P500  0.0529[0.2252]  0.0524[0.0052]  0.0738[0.0120]  
ErrorCVaR  0.6142E004  0.9630E004  1.9058E004  
NonParameter Value  ALVaR  S.H.I  0.0617(0.0415)  0.0734(0.2015)  0.0954(2.4125) 
Nikkie225  0.0415(0.1548)  0.0552(4.3521)  0.7520(3.9520)  
S&P500  0.1452(6.1452)  0.0412(12.2541)  0.0652(12.5412)  
ErrorVaR  0.3110E004  0.3201E004  0.9521E004  
ALCVaR  S.H.I  0.0778[0.0033]  0.0886[0.0069]  0.1120[0.1523]  
Nikkie225  0.0524[0.0099]  0.0680[0.0098]  0.0885[0.0175]  
S&P500  0.0488[0.0068]  0.0524[0.0052]  0.0541[0.0458]  
ErrorCVaR  0.6258E004  0.7524E004  2.0168E004 
Notes: Parentheses are the LR statistic[12] of the value of VaR backtesting; Square brackets are the difference between the actual average loss and CVaR when VaR failed.
2.4. Text Results and Comparative Study
In the term of the results, At each confidence level, the result using AL parametric method is the largest, generally larger than the results using ALMC method^{14}. But, generally, the results using normal distribution method is smaller than others.
As known from the following test results that it underestimate risk. The backtesting results shows:
1) To normal distribution, in the low confidence level (95% and 97.5%), the test statistics LR (of Var estimated in three markets) are both rejection and reception. However, In the high confidence level (99%、99.5%and 99.9%), the test statistics LR (of Var estimated in three markets) are all rejection. Overall, NVaR model of the normal distribution is underestimating VaR.
2) ALVAR parametric method. All market conditions accept this method but S.H.I in the 95% and 97.5% confidence level.
3) ALMCVAR method. All market conditions accept this method but S&P500 in the 99.5% and 99.9% confidence level.
In trading day whose VaR (tested by every model) is invalid, we utilize the difference between the average value of the actual loss and its VaR estimated value to text the effect of CVaRmeasure.
From the results, A) the resulting difference measured by NCVAR generally larger. B) Estimation results measured by ALCVAR and ALMCCVAR are close; the difference between the mean and the actual loss is relatively small, showing it can more accurately estimate the market tail risk.
From the above analysis, method based on AL distribution can work well for market risk measurement. To further analyze the effectiveness of AL parameters method and ALMC method, evaluating the pros and cons of each method at different confidence levels, we assess the accuracy of VaR by using the mean squared error of the actual failure rate and the expected failure rate, recorded as ErrorVaR. The value smaller, the model more accurate. While we assess the accuracy of CVaR by using the mean squared error of the mean of actual loss (when var invalids of three markets) and CVaR estimates, recorded as ErrorCVaR. The value smaller, the model more accurate. In the low confidence level (95% and 97.5%), the ErrorVaR of ALMCVAR method is the smallest, showing the best risk measurement capability (VaR); However, in the high confidence level (99%, 99.5% and 99.9%), the ErrorVaR of ALVAR parametric method is the smallest, showing the best risk measurement capability(VaR); In the confidence level (95%, 97.5% and 99.5%), the Error CVaR of ALMCCVAR method is the smallest, showing the best risk measurement capability (CVaR); However, in the confidence level (99%and 99.9%), the Error CVaR of ALCVAR parametric method is the smallest, showing the best risk measurement capability(CVaR);
3. Summary
For the three stock markets given in the paper, Return series tend not to obey normal distribution. Risk measurement models which are based on the assumption of normal distribution exist some defects. Risk measurement models based on AL distribution whether in the more mature US market or in the Japanese stock market, or in Chinese stock market which as an emerging market ,the risk measurement capability of VaR or CVaR showing a relatively good in each of the confidence interval (95%, 97.5%、99%、99.5%、99.9%). And risk measurement models based on AL distribution are more reasonable and applicable than risk measurement models based on normal distribution. This paper makes up the shortage of China on asymmetric Laplace distribution applied research in the field of financial management.
Meanwhile, the paper will provides some help with practical risk management or investment decision analysis. In addition, as a younger market, China's stock market has made great development. And show the similar characteristics with mature market.But also with the development of financial markets, and constantly improve financial risk management and the ability of risk management in the new situation.
Acknowledgements
This project (Empirical research on Stock index investment risk model, No.68) is funded by the "20142015 school year, Beijing Wuzi University, College students' scientific research and entrepreneurial action plan project". And by Beijing Wuzi University, Yunhe scholars program (00610303/007). And by Beijing Wuzi University, Management science and engineering Professional group of construction projects. (No. PXM2015_014214_000039)
References