23-06-2015
Estimation of Value-at-Risk through Extreme Value Theory
With a description and comparison between several methods.Martin Kroon, 10359184 Bachelor Student Actuarial Science
FEB, University of Amsterdam
Subject proposed by dr. Sami Umut Can (s.u.can@uva.nl)
Abstract:
In this paper, I compare three methods for estimating the Value-at-Risk, a widely used risk measure in actuarial science and finance. These are the historical simulation method, the method that uses the translated Gamma distribution and the method that uses the ‘Extreme Value Theory’. From three simulated datasets, I found that the differences between the three methods weren’t significantly different (with the exception of special cases). Only when the dataset is skewed to the right (for example: Student’s-t distribution), I found that the Extreme Value Theory method gave bigger estimates than the other two methods.
Content
1.Introduction………..………....1
2. Theoretical background on Pickands-Balkema-de Haan theorem……….………….3
2.1 Estimating Value-at-Risk with application of Extreme Value Theory…………...5
2.2 Estimating Value-at-Risk using a translated Gamma distribution…………...….…6
3. The research design……….………...……...7
4. The historical simulation method ………...……….….9
4.1 The translated Gamma distribution method...………...………...………....11
4.2 The Extreme Value Theory method...………...……...14
4.3 Analysis of the results………...…………....18
5. The Conclusion………...………..…...20
Appendix
1. Introduction
Companies that manage insurance, reinsurance and also equity portfolios face great risks. If the stocks in an investment portfolio decrease in value or an insurance company has to pay out a lot of benefits in the same time period, it’s possible that those companies go bankrupt if they don’t have enough
reserves to cover these losses. Therefore, it’s necessary that a risk manager is able to estimate those risks with the help of different kinds of models and standards.
One of these standards is the Value-at-Risk, also shortened as VaR. This value explains the biggest loss you can incur with a given confidence level over a time period and it dates back to the 1980’s. It can be calculated with a number of methods. There is the Variance-Covariance method, the Monte Carlo simulation method described by Rouvinez (1997), the historical simulation method described by Pearson and Linsmeier (1996), the 'Extreme Value Theory' method first introduced by Schuerman, Diebold and Stroughair (1998) and finally you can use a translated Gamma distribution in order to calculate the Value-at-Risk. This paper discusses only the last three methods. At first there is the historical simulation method. This method starts with reorganising the historical losses from lowest to highest. These losses might be reported as negative values, but you make them positive in order to simplify your calculations (so the bigger the loss, the more positive it is). It then assumes that the historical data will repeat itself in the future. This assumption has to be made, otherwise it makes no sense to calculate a prediction standard. However, it’s not realistic because there are all kinds of events which can influence the future data, what causes different outcomes for the Value-at-Risk. After this is done, the risk manager puts the ordered data in a histogram and simply calculates a quantile estimate (depends on what quantile he prefers for the Value-at-Risk). This means that the probability of exceeding the loss at this quantile estimate (the Value-at-Risk), is equivalent to one minus this quantile.
Secondly you can use a translated Gamma distribution in order to calculate the Value-at-Risk. This method is often used in the insurance world for fitting a distribution function through a histogram with aggregated claims as for exampleDaykin, Pentikainen, and Pesonen (1994) did in their paper. At first you have to calculate the necessary parameters alpha, beta and 𝑥0 with the help of some simple formulas that depend on the mean, variance and skewness of the data. And then you can fit the best-fitting translated gamma density function through the histogram and read off its quantile estimate (again this is the Value-at-Risk for this method).
Finally you have the method that calculates the Value-at-Risk with the help of the ‘Extreme Value Theory’. In short, the theory helps to create a distribution function in the tail section (so where the extreme losses are). It has also been used in a lot of papers, so has for instance McNeil (1996) used it on Danish data from fire insurance companies and in McNeil (1999) it is used on data from the DAX (the German stock index). Keep in mind however that this theory hasn’t always found its application
in insurance. People used it for example to model extreme river stands, company losses or service times of machines (Reiss and Thomas, 2001, McNeil and Frey 2000).
For this last described method, you don’t have to use the entire ‘Extreme Value Theory’. You only have to use an important result from it, that is the Pickands-Balkema-de Haan theorem (Pickands 1975, Balkema & de Haan 1974). This result shows that for many kinds of distributions (for instance the Student’s t-distribution or the normal distribution), the values that exceed a certain pre-chosen value, a threshold, follow a GPD (Generalised Pareto Distribution). Knowing this, you can calculate the Value-at-Risk with some calculations which will be discussed later in this paper.
The disadvantage of all of these methods, is that they can give different outcomes for the Value-at-Risk. This can happen because different thresholds can give different estimates of the parameters of the ‘Extreme Value Theory’ method. As a consequence you can get entirely different quantile estimates in comparison with the historical simulation method and this makes it hard to decide for a risk manager what method he should use. Therefore, this paper examines the three previously described methods for calculating the Value-at-Risk and discusses when and why they differ significantly from each other.
This is done by analysing simulated data from a normal distribution, a chi-squared
distribution and a Student’s t-distribution. First a Generalised Pareto Distribution has been fitted to the series of data points that exceed a pre-chosen threshold (with the help of maximum likelihood, you can easily calculate the parameters 𝜉 and β of the Generalised Pareto Distribution). This way of modelling has been developed by Davison (1984) and Davison & Smith (1990). Then the Value-at-Risk has been calculated and finally been compared with the quantile estimates (Values-at-Risk) of the
historical simulation method and the translated Gamma distribution. These estimates of the historical simulation method can be acquired by looking into the histogram of the simulated data and the ones from the translated Gamma distribution can be calculated (you can find the method for doing this in chapter 2). This process has been done for different kinds of thresholds and for different kinds of simulation sizes in order to get a reliable comparison between the three methods.
In order to be able to answer the central question, you find a wide description of the
theoretical background on the used Pickands-Balkema-de Haan theorem in chapter 2. In this chapter a stepwise elaboration of the formulas from the used theorem is worked out in order to be able to build a model for the Value-at-Risk. Thereafter, you find the application of this theory on the Value-at-Risk, again a stepwise description of the used formulas is given, in order to explain the relationship between the Pickands-Balkema-de Haan theorem and the Value-at-Risk. And finally the simulation results are being analysed and discussed (with the help of histograms and tables) whereupon a conclusion with the answer to the central question is given.
2. Theoretical background on Pickands-Balkema-de Haan theorem
The Pickands-Balkema-de Haan theorem is a result of the Extreme Value Theory, but you can easily say that it is a full theory within a theory. In short, this result gives the asymptotic tail distribution of a random variable 𝑋which can be for instance a stock’s return, if the distribution of this variable is unknown (this is only valid for values above a pre-chosen threshold). Modelling in this way, so looking at the behaviour of values above a certain value (threshold) , is also known as Peaks over Threshold, shortened as POT (Pickands 1975, Balkema & de Haan 1974).
To start with the background information on the Pickands-Balkema-de Haan theorem, you first make some assumptions: First, a variable 𝑋 must have a cumulative distribution function (shortened as cdf): 𝐹(𝑥) = Pr(𝑋 ≤ 𝑥), where ‘Pr’ stands for probability. So this is the probability of 𝑋 being smaller than a value 𝑥. Finally, as already stated in the introduction, you consider the variable 𝑋 to be a positive number in order to make the histogram better readable and the calculations easier to
understand (So this is alsofor 𝑋 if it represents a loss).
Now you want to know what values of 𝑋you can find in the area between a threshold (let’s name it 𝑢) and infinity. That’s why you define the excess distribution, that is:
𝐹𝑢(𝑦) = Pr(𝑋 − 𝑢 ≤ 𝑦| 𝑋 > 𝑢) , with 𝑦 > 0.
If you transform this equation into words, you get the probability of 𝑋 − 𝑢 being smaller than a value 𝑦, given that 𝑋 must be bigger than the value𝑢. This expression can be simplified with the help of the probability theory.
This theory states that
Pr(𝐴|𝐵) =Pr(𝐴 ∩ 𝐵) Pr(𝐵)
So if you combine this with the equation above, you get (given that 𝑦 > 0): 𝐹𝑢(𝑦) = Pr(𝑋 − 𝑢 ≤ 𝑦|𝑋 > 𝑢) =Pr(𝑋 ≤ 𝑦 + 𝑢 ∩ 𝑋 > 𝑢)Pr(𝑋 > 𝑢) = 𝐹(𝑦 + 𝑢) − 𝐹(𝑢)
1 − 𝐹(𝑢) (1) Another result from the probability theory is that if you take a value 𝑢 (the threshold) big enough, it can be shown that 𝐹𝑢(𝑦) (for a lot of common continuous distributions such as: the normal distribution, the Student’s-t distribution, chi-squared distribution, etc.) will converge to a ‘special’ distribution. In formula form this means that
lim
𝑢→𝑥0sup |𝐹𝑢(𝑦) − 𝐺𝜉,𝛽(𝑦)| = 0,(2) 0 ≤ y < 𝑥0− u
(Embrechts, Klüpperberg & Mikosch, 1997)
As you can see, the supremum of the absolute difference between the two cdf’s becomes zero as the threshold 𝑢 value reaches a value 𝑥0(the smallest number for 𝑦 where the equation 𝐹(𝑦) = 1 becomes valid). This cumulative distribution function 𝐺𝜉,𝛽(𝑦) is known as the generalised Pareto distribution (shortened as GPD):
𝐺𝜉,𝛽(𝑦) = { 1 − (1 +𝜉𝑦 𝛽) −1/𝜉, 𝜉 ≠ 0 1 − exp(−𝛽𝑦), 𝜉 = 0(3) 𝛽 > 0, 𝑥 ≥ 0, 𝜉 ≥ 0 If 𝜉 < 0, then 0 ≤ 𝑥 ≤ −𝛽/𝜉.
Within this distribution you use two parameters, namely 𝛽 and 𝜉. They are respectively the scaling parameter and the shape parameter. The parameter 𝜉 determines which form the GPD takes.
There are three posibilities for this form: If 𝜉 > 0, 𝐺𝜉,𝛽(𝑦) is a Pareto distribution
if 𝜉 = 0, 𝐺𝜉,𝛽(𝑦) is a Exponential distribution
and if 𝜉 > 0 , 𝐺𝜉,𝛽(𝑦) becomes a Pareto type 2 distribution.
Note: riskmanagers often work with data that includes very high extreme values, therefore it’s common to use a pareto distribution in the tailsection, so they mostly have 𝜉 > 0.
Next, if you set 𝑥 equal to 𝑦 + 𝑢 (which is logical) and you combine that with formula (1), you get:
𝐹(𝑥)−𝐹(𝑢)
1−𝐹(𝑢) = 𝐹𝑢(𝑦) ⟺ 𝐹(𝑥) − 𝐹(𝑢) = 𝐹𝑢(𝑦)(1 − 𝐹(𝑢)) ⟺ 𝐹(𝑥) = (1 − 𝐹(𝑢))𝐹𝑢(𝑦) + 𝐹(𝑢).
Now you know from formula (3) that 𝐹𝑢(𝑦) converges (if 𝑢 is large enough) to 𝐺𝜉,𝛽(𝑦), so you get: 𝐹(𝑥) = (1 − 𝐹(𝑢))𝐺𝜉,𝛽(𝑦) + 𝐹(𝑢)𝑤𝑖𝑡ℎ𝐺𝜉,𝛽(𝑦) = 𝐹𝑢(𝑦)(4)
The cumulative distribution function 𝐹(𝑢) can easily being estimated, as it’s equal to the probability of 𝑋 being smaller than a threshold 𝑢. This means that if you count the number of values that exceed the value 𝑢 (let’s call it ℎ), you can estimate 𝐹(𝑢) with 𝑛−ℎ𝑛 . That is the number of values between 0 and 𝑢 divided by the total number of observations. So if you now substitute equation (3) with 𝜉 ≠ 0 into equation (4) you get:
Pr(𝑋 ≤ 𝑥) = 𝐹(𝑥) = 1 −ℎ 𝑛(1 + 𝜉 𝑥−𝑢 𝛽 ) −1 𝜉 , (5)
Where 𝜉, 𝛽 have to be estimated (see attachment 1).
These parameters 𝜉and 𝛽 can easily being estimated with the help of maximum likelihood. First you set up the loglikelihood function of formula (5), that is:
𝐿(𝛽, 𝜉|𝑥) = −ℎ log(𝛽) + ℎlog(ℎ 𝑛) − (1 + 1 𝜉) ∑ log(1 + 𝜉(𝑥𝑖− 𝑢) 𝛽 ℎ 𝑖=1 ),𝑖𝑓𝜉 ≠ 0
Note that the second term does not influence the estimated parameters, as it’s a constant term. So you can leave it out of the equation. Now you first take the derivative of the first equation with respect to 𝜉 and set this equal to 0 and then you do the same for 𝛽 (this has to be done numerically). Next you solve the two equations and you get your two maximum likelihood estimates.
2.1 Estimating Value-at-Risk with application of Extreme Value Theory
With the help of the results from the previous paragraph, you can make a model for the Value-at-Risk. To show this, you first consider that if you apply the historical simulation method for calculating the Value-at-Risk, you get as previously described in the introduction a value at a certain quantile. So this means that you have to look at the data in the histogram and calculate the following expression:
𝐹(𝑥) = Pr(𝑋 ≤ 𝑥) = 𝛼, with α being a desired quantile.
This means that you want 𝑋 to be smaller or equal to a value 𝑥 with a probability of 𝛼. Now you know with a probability of 1 − 𝛼 that the value 𝑥 can be exceeded.
This expression can also been written as:
𝑥 = 𝐹−1(𝛼) (6)
With 𝐹−1(𝛼) being the inverse function of the cumulative distribution function with as argument 𝛼. This is useful because now you can say that 𝑥 = 𝑉𝑎𝑅 and you get a function for the Value-at-Risk which can be calculated. So if you now combine formula (5) with (6) you get the following equation:
1 −ℎ 𝑛(1 + 𝜉 𝑥−𝑢 𝛽 ) −1 𝜉 = 𝛼, 𝑖𝑓1 −ℎ 𝑛< 𝛼 And if you solve this with respect to 𝑥, you get:
(1 + 𝜉𝑥 − 𝑢 𝛽 ) −1 𝜉 =𝑛 ℎ(1 − 𝛼) ⟺ (1 + 𝜉 𝑥 − 𝑢 𝛽 ) = ( 𝑛 ℎ(1 − 𝛼))−𝜉 ⟺ 𝑥 − 𝑢 𝛽 = (𝑛ℎ (1 − 𝛼))−𝜉 − 1 𝜉 ⟺ 𝑉𝑎𝑅 = 𝛽( 𝑛 ℎ (1 − 𝛼))−𝜉− 1 𝜉 + 𝑢 = 𝑢 + 𝛽 𝜉(( 1 − 𝛼 ℎ 𝑛 ) −𝜉 − 1),(7)
With again the estimated parameters 𝛽 and 𝜉, with threshold 𝑢and with ℎbeing the number of exceedances over 𝑢. Now you have a model to calculate the Value-at-Risk.
The parameters of the equation (7) are again calculated with the help of maximum likelihood in the way described earlier (see page 5), but they can also been calculated with the help of conditional Extreme Value Theory (shortened as EVT) . This EVT method makes use of the two step’s procedure of McNeil and Frey (2000)’s which is not used in this paper, because making a comparison between different methods from the ‘Extreme Value Theory’ is not a part of this paper.
On the other hand, it’s also possible to calculate the Value-at-Risk in another way. To show this, you first need to know what the expected shortfall is. That is:
𝐸𝑆1−𝛼= 𝑉𝑎𝑅1−𝛼+ 𝐸(𝑋 − 𝑉𝑎𝑅1−𝛼|𝑋 > 𝑉𝑎𝑅1−𝛼),
To put this formula in words, you get that the expected shortfall equals the expectation of the values 𝑥 that exceed your chosen threshold plus your Value-at-Risk. So briefly, it means that you calculate the value that you expect to get beyond the threshold 𝑢. This expression can then be written as:
𝑉𝑎𝑅1−𝛼 = 𝐸𝑆1−𝛼− 𝐸(𝑋 − 𝑉𝑎𝑅1−𝛼|𝑋 > 𝑉𝑎𝑅1−𝛼) Knowing the mean excess function, that is:
𝐸(𝑋 − 𝑉𝑎𝑅1−𝛼|𝑋 > 𝑉𝑎𝑅1−𝛼) = 𝑒(𝑉𝑎𝑅1−𝛼) =𝛽+𝜉(𝑉𝑎𝑅1−𝛼−𝑢)1−𝜉 (8) you can write the previous equation as:
𝑉𝑎𝑅1−𝛼= 𝐸𝑆1−𝛼−𝛽 + 𝜉(𝑉𝑎𝑅1−𝛼− 𝑢)
1 − 𝜉 ⟺ 𝑉𝑎𝑅1−𝛼= (1 − 𝜉) (𝐸𝑆1−𝛼−
𝛽 − 𝜉𝑢
1 − 𝜉 )(9) Again using the maximum likelihood method on the observations that exceed your threshold 𝑢, you get the estimated parameters𝛽 and 𝜉 and you will be able to calculate the Value-at-Risk (this method is not used in this paper, because it’s not very relevant for the goal of this thesis)
2.2
Estimating Value-at-Risk using a translated Gamma distribution
A translated Gamma distribution can be very helpful for data that is highly skewed to the right (skewness has to be higher than 0), which is mostly the case for insurance or financial market losses for example. However the data they use does not entirely follow a perfect Gamma distribution (this would be ideal). Therefore they permit a shift which is mostly denoted as 𝑥0(no relation to 𝑥0 that appears in expression (2)) and that is also where the name comes from. So if you have a variable 𝑋 which is Gamma distributed with parameters 𝛼, 𝛽 and with cumulative distribution function 𝐹(𝑥), it now becomes 𝑋 + 𝑥0 ( and again 𝑋 remains Gamma distributed) with a cumulative distribution function equal to 𝐹(𝑥 − 𝑥0).
In order to estimate the parameters 𝛼, 𝛽 and 𝑥0, you make sure that the first three moments (the expectation, the variance and the skewness) of 𝑋 + 𝑥0 equal the observed moments of the data. The first three moments of a translated Gamma distribution with parameters alpha, beta and 𝑥0 are as follows: 𝜇 = 𝑥0+𝛼 𝛽 𝜎2= 𝛼 𝛽2 γ = 2 √𝛼
So if you now rewrite the previous equations in terms of the parameters, you get: 𝛼 = 4
𝛽 =√𝛼 𝜎 𝑥0= 𝜇 −𝛼
𝛽
Now you can substitute the first equation into the second equation in order to make 𝛽 indepent of the other two parameters. If you do the same for the third equation you finally get:
𝛼 = 4 γ2 𝛽 = 2 γσ 𝑥0= 𝜇 −2𝜎 γ
And this makes sure that you can approximate the cdf F(x) of the data with a translated Gamma cdf which means that the following approximation must hold:
𝐹(𝑥) ≈ 1
Γ(α)∫ 𝑦1−𝛼𝛽𝛼 𝑥−𝑥0
0
𝑒−𝛽𝑦𝑑𝑦 For 𝑥 ≥ 0 and with Γ(α) being the Gamma function with as argument 𝛼.
So now knowing how the translated Gamma distribution works, you can make a model for the Value-at-Risk, which works quite the same as for the historical simulation method. You just have to estimate 𝑃(𝑋 ≤ 𝑥) = 𝜀 (with 𝜀 between 0 and 1), which in this case means, that you first calculate the parameters 𝛼, 𝛽 and 𝑥0 with the help of the skewness, the standard deviation and the mean of the data. And finally you solve the following equation for 𝑥 (the Value-at-Risk):
1
Γ(α)∫ 𝑦1−𝛼𝛽𝛼 𝑥−𝑥0
0
𝑒−𝛽𝑦𝑑𝑦 = 𝜀
3. The research design
The research in this paper is based on the models described earlier, as you want to estimate the Value-at-Risk. In order to do this and to reproduce the results, you can use the programming language R (see attachment 3 for the full R script)
First the data is simulated. This is done by pulling random values from a normal distribution, a chi-squared distribution and a Student’s t-distribution (this can be done for different amounts of observations, mostly you call this 𝑛). For the normal distribution you use a mean equal to 29 and a standard deviation of 5,2 and for the other two distributions you use 10 degrees of freedom with a non-negativity parameter of 20 (this has to be done in order to get the same results). Those arguments are used, because they widen the distributions (to make the data look extremer). In R it looks like this: set.seed(15647458)
a<-rnorm(n,mean=29,sd=5.2) #random normal distribution
c<-rt(n,10,ncp = 20) #random T-distribution
The set.seed(15647458) code has been used, because otherwise you get different random data if you want to reproduce the process (and different data gives you different results than the results stated in this paper).
On the other hand, the reason why these distributions are used, is because a normal distribution has a light tail whereas a Student’s t-distribution has a very heavy tail. So now a comparison can be made between the Value-at-risk estimates from different forms of distributions.
Next you want to estimate the parameters 𝛽 and 𝜉from the next formula:
𝑉𝑎𝑅 = 𝑢 +𝛽 𝜉(( 1 − 𝛼 ℎ 𝑛 ) −𝜉 − 1)
This is done with the help of a standard package in R called ‘evir’ (short for ‘extreme values in R’) which can easily been downloaded and installed. This package can estimate your parameters with the help of maximum likelihood and also gives the standard deviations, the number of exceedances, the data above the threshold and finally the variance-covariance matrix. After this is done you can calculate the VaR estimate above with the help of R (this can be done for different quantile values 𝛼). For example, you can use the following R code:
esVaR.b<- u+(est.par.b$par.ests[2]/est.par.b$par.ests[1])*((((n/nu.b)*(1-quantielen))^(-est.par.b$par.ests[1])-1)),
where est.par.b is vector with the output of the GPD function of the package ‘evir’, nu.b equals ℎ (the number of exceedances that appears in expression (7)) and quantielen equals a vector with different quantiles (in this paper you use the vector quantielen<-c(0.95,0.98,0.99)).
For the translated Gamma distribution you can use the following R code for calculating the necessary parameters:
alpha<-4/(skewness(.)^2) betha<-2/(skewness(.)*sd(.))
x.o<-mean(.)-(2*sd(.))/skewness(.)
where the function ‘skewness’ calculates the skewness of your data (this is a standard function from the package ‘moments’, which also can be easily installed in R) and the function ‘mean’ calculates the mean of the data.
After this is done , you can calculate the Value-at-Risk with the following R code: qgamma(c(0.95,0.98,0.99),alpha,betha)+x.o
where ‘qgamma’ calculates quantiles for the Gamma distribution. Of course you have to add 𝑥0 to this function, because otherwise it won’t be a shifted Gamma distribution.
In the end you want to compare the historical method with the method where you make use of the Extreme Value Theory or the translated Gamma distribution and to implement this in R you can use the following codes for example:
grafiek1<-hist(a,main="figuur 1: simulated normal distr.",xlab="waardes",breaks=100);
abline(v=quantile(a,c(0.95,0.98,0.99)),lty=2,col="blue") #historical method
estimated.VaR<-matrix(c(esVaR.a,esVaR.b,esVaR.c),nrow=3,ncol=3,byrow=TRUE). The first line gives a histogram of the simulated data from the normal distribution (this can also be done for the other two distributions in the same way). In this way you can show where the quantile estimates are (historical method). The other line gives a matrix with the estimated quantile values which were calculated with the help of the Extreme Value Theory. After this, you can compare them with each other (standard functions from the package ‘evir’ as, gpd.plot, gpd.sfall and gpd,q have also been used for this comparison). Moreover, now you can also answer the central question. Because knowing the outcomes for different kinds of thresholds and simulation sizes, you should be able to notice when one Value-at-Risk differs to much from the other estimated Value-at-Risk.
Moreover, I think that this difference will show up. As I think that for larger threshold values (the right amount will be shown in chapter 4), the ‘Extreme Value Theory ‘method will give too high quantile estimates. This should not be strange, because in order to plot a Generalised Pareto
distribution you use the maximum likelihood method. And this method needs data points, which it does not get when the threshold is too high (the higher the threshold, the lower the amount of data points).
4. The historical simulation method
To start, I have simulated the data as already mentioned from a normal distribution (with mean equal to 29 and standard deviation equal to 5.2), a chi-squared distribution (with 10 degrees of freedom and a non-negativity parameter of 20) and a Student’s t-distribution (with 10 degrees of freedom and a non-negativity parameter of 20) with a simulation size of 5.000, 100.000 and 250.000 . In order to simply demonstrate how the data is distributed I only show you the histograms for n=100.000 on the next page.
The figures 1,2 and 3 show the data from the three different distributions for n=100.000.
As you can see, the random normal distributed data has a normal left and a normal right tail. Its maximum data point is 50.00806. Whereas the simulated chi-squared and Student’s t-distributed data have a longer right tail (which is more common in the insurance and finance world). Their maximum data points are respectively 91.66818 and 134.24420.
In order to calculate the Value-at-Risk estimates with the historical simulation method, I use R to find the quantile estimates at the 95th, 98th and the 99th percentile for the simulation sizes of n=500, n=5.000 and for n=100.000 which are shown respectively in the tables 3,4 and 5 below.
95% 98% 99% normal distribution 37.06043 39.59314 40.83925 chi-squared distr. 47.31218 51.98701 56.25784
Student’s-t distr. 34.32778 37.73200 39.12439 Table 3: Value-at-Risk estimates of the historical simulation method (n=500).
95% 98% 99% normal distribution 37.63689 39.96297 41.16518 chi-squared distr. 48.19746 53.49571 57.27364 Student’s-t distr. 32.04732 36.44976 40.33723
table 4: Value-at-Risk estimates of the historical simulation method (n=5.000).
95% 98% 99% normal distribution 37.54505 39.71374 41.12865 chi-squared distr. 48.01441 53.55077 57.39935 Student’s-t distr. 32.04765 36.36022 39.71002
To visualize these results, I show you in figure 7 again the histograms from the figures 1,2 and 3, but now with the Value-at-Risk estimates. The blue lines you see represent these values. If you go from the left vertical line to the right vertical line, you get respectively the Value-at-Risk estimates at the 95th , 98th and the 99th percentile. So now I have shown you all the necessary results from the historical simulation method, which I use to make a comparison with the other two methods.
Figure 7: The histograms with the Value-at-risk estimates for n=100.000.
4.1 The translated Gamma distribution method
The translated Gamma distribution can be very helpful for calculating the Value-at-Risk. However as already mentioned before, your data must have a skewness that is higher than 0. So I first calculate the skewness of the data for 𝑛 = 500, 𝑛 = 5.000 and for 𝑛 = 100.000 (and for the different distributions). This gives for the chi-squared distribution:
N: Skewness:
500 0.6244504 5.000 0.6171329 100.000 0.5716668
for the Student’s-t distribution:
N: Skewness: 500 1.38925 5.000 1.604301 100.000 1.515198
And finally for the normal distribution:
This shows that you can use the translated Gamma distribution for the first two distributions for all three values of 𝑛, however this is not the case for the normal distribution. Its skewness is smaller than 0 for 𝑛 = 500 and for 𝑛 = 100.000. So I can’t use the translated gamma distribution for those simulation sizes and therefore neither for the comparison.
To show that you really get a fit, I fitted the translated Gamma distribution through the histograms that you get for 𝑛 = 5.000 (see figure 8). Again, I do this for 𝑛 = 5.000, because
otherwise I won’t have a fitted translated Gamma distribution for the normal distribution (the red lines represent the fit and a, b and c represent respectively the data from the normal distribution, the chi-squared distribution and the Student’s-t distribution).
Figure 8: the fitted translated Gamma distribution for n=5.000.
Now knowing that the translated Gamma distribution fits the data well (except for 𝑛 = 500 and 𝑛 = 100.000 when you use the normal distribution) , you should be able to calculate the
quantiles. As the quantiles still represent the Value-at-Risk estimates. Calculating these estimates can easily been done in R and this gives the following results and graphs for the three distributions:
The normal distribution
N: 𝛼: β: x0: 95%: 98%: 99%: 500 - - - - 5.000 16150.56 24.51427 -629.7227 37.65029 39.79063 41.22006 100.000 - - - - N: Skewness: 500 -0.04335362 5.000 0.01573752 100.000 -0.005729998
Figure 9: histogram of the normally distributed data (n=5.000) with translated Gamma distribution fit and VaR estimates (see table).
The chi-squared distribution
N: 𝛼: β: x0: 95%: 98%: 99%:
500 10.25803 0.3056649 -3.909107 48.53650 54.49442 58.69586 5.000 10.50274 0.3196634 -3.13067 47.98162 53.72707 57.77656 100.000 12.23979 0.3490486 -5.019688 47.99459 53.55745 57.46530
Figure 10: histogram of the chi-squared distributed data (n=500, n=5.000,n=100.000) with translated Gamma distribution fit and VaR estimates (see table).
The Student’s-t distribution
N: 𝛼: β: x0: 95%: 98%: 99%:
500 2.072522 0.2466863 13.56395 33.26917 37.73053 41.01901 5.000 1.554133 0.2178106 14.64425 33.01248 37.69769 41.18512 100.000 1.742293 0.2349594 14.23937 32.62415 37.09729 40.41373
Figure 11: histogram of the Student’s-t distributed data (n=500, n=5.000, n=100.000) with translated Gamma distribution fit and VaR estimates (see table).
Note:
The graphs in the figures 9, 10 and 11 look quite the same. This is because I only change the simulation size, so the quantile values should differ a tiny bit (together with 𝛼, 𝛽𝑎𝑛𝑑𝑥0)
The blue vertical lines represent the Value-at-Risk estimates at respectively the 95th, 98th and the 99th percentile.
So now again I got all the results that I need in order to make a reliable comparison between the historical simulation method and the Extreme Value theory method.
4.2 The Extreme Value Theory method
As already explained, you know that the ‘Extreme Value Theory’ depends on a threshold. For every change in this value, you get another estimated Value-at-Risk. Therefore I have to make use of different thresholds and different simulation sizes when I calculate the Value-at-Risk estimates.
At first, I take 𝑛 = 100.000 and I calculate for different thresholds the Value-at-Risk estimates again at the three percentiles. However you can’t use the same thresholds for the three different distributions. This is because the maxima of the datasets is not the same (you can’t choose 𝑢greater than you maximum) and 1 −ℎ
𝑛< 𝛼 must hold. So at first I use R to calculate the maxima of the datasets, this gives:
n: Normal distribution chi-squared distribution Student’s-t distribution 500 46.00052 74.20908 54.08786 5.000 46.54572 92.26558 64.85717 100.000 50.00806 91.66818 134.24420
Now knowing this, I may use the same thresholds for the normal distribution and for the chi-squared distribution (their maxima does not fluctuate much), however for the Student’s-t distribution I might have to use different thresholds. Firstly because this distribution has very extreme losses at its right tail end and secondly its maxima differs a lot for different values of 𝑛.
At first I begin with calculating the Value-at-Risk estimates for the normally distributed data set. As its maximum fluctuates around 50, I can use thresholds equal to 20, 30 and 35. For the simulation sizes 𝑛 = 500, 𝑛 = 5.000 and 𝑛 = 100.000, I get the results shown in the figures 12, 13 and 14 and in table 15.
Dataset: Normal distribution, n:100.000
threshold: 𝜉: β SE: 𝜎(𝜉) SE: 𝜎(β) 95%: 98%: 99%:
20 -0.4106431 12.3235353 2.385048e-06 2.385182e-06 41.08231 43.88198 45.40004 30 -0.2286553 4.6123726 2.105070e-06 1.273603e-02 37.79266 40.13258 41.60402 35 -0.1661872 3.0246026 0.00688316 0.03399528 32.18786 36.31878 39.73435
Dataset: chi-squared distribution, n:100.000
threshold: 𝜉: β: SE: 𝜎(𝜉) SE: 𝜎(β) 95%: 98%: 99%:
30 -0.1577922 9.9196679 0.001394749 0.04647883 48.63138 54.58604 58.55194 40 -0.1025104 7.3639603 0.006596289 0.07590675 47.98057 53.70542 57.69256 45 -0.08588358 6.64447849 0.00975506 0.09883311 47.96098 53.59178 57.56644
Figure 13: table, histograms and graphs of the chi-squared distribution dataset results for n=100.000.
Dataset: Student’s-t distribution, n:100.000
threshold: 𝜉: β: SE: 𝜎(𝜉) SE: 𝜎(β) 95%: 98%: 99%:
20 -0.03288415 5.42385446 0.002588756 0.02696214 32.55136 37.07451 40.40673 25 0.02954261 4.65684298 0.00631349 0.04292284 32.08391 36.60359 40.10480 30 0.07514123 4.46184437 0.01155733 0.07217179 31.99511 36.36970 39.88492
Figure 14: table, histograms and graphs of the Student’s-t distribution dataset results for n=100.000. The figures above show respectively tables with the important parameters (together with the Value-at-Risk estimates), histograms of the data with these estimates (the blue vertical lines) and finally the excess distributions fitted through the data points above different thresholds. This has now only been done for 𝑛 = 100.000. Hereafter I only give the tables for 𝑛 = 500 and for 𝑛 = 5.000, because the forms of the histograms and the excess distributions have quite the same shape (only the amount of data points that exceed the threshold should increase). After this is done, I should have enough results to make a comparison between all the methods. So finally I get the results shown in table 15 below.
Dataset: Normal distribution, n:500
threshold: 𝜉: β: SE: 𝜎(𝜉) SE: 𝜎(β) 95%: 98%: 99%:
20 -1.019352 26.503694 2.001553e-06 2.001553e-06 44.70787 45.49255 45.74992 30 -0.2450034 4.5726505 0.04145131 0.36165377 37.59639 39.82179 41.20276 35 -0.1533039 2.9283148 0.1132019 0.5093894 37.22146 39.43359 40.91227
Dataset: Normal distribution, n:5.000
threshold: 𝜉: β: SE: 𝜎(𝜉) SE: 𝜎(β) 95%: 98%: 99%:
20 -0.4965548 13.1936493 2.014942e-06 1.199943e-02 40.44006 42.68096 43.81357 30 -0.2774554 4.9287149 0.0130193 0.1210110 37.95529 40.15721 41.48805 35 -0.2105242 3.1692499 0.03216031 0.15926636 37.70692 39.87307 41.25541
Dataset: chi-squared distribution, n:500
threshold: 𝜉: β: SE: 𝜎(𝜉) SE: 𝜎(β) 95%: 98%: 99%:
30 -0.1519206 10.1478966 0.04822223 0.82623845 49.02174 55.23014 59.38472 40 -0.05463645 7.34886587 0.09947485 1.10309744 48.19512 54.362866 58.82770 45 0.1818984 5.1283560 0.2217764 1.3950241 47.37528 52.91943 57.77213
Dataset: chi-squared distribution, n:5.000
threshold: 𝜉: β: SE: 𝜎(𝜉) SE: 𝜎(β) 95%: 98%: 99%:
30 -0.1363425 9.8039850 0.01057188 0.22989768 48.57548 54.83878 59.08327 40 -0.08902152 7.50856469 0.02423993 0.32606688 47.98429 53.96577 58.17728 45 -0.04737972 6.42188570 0.03836586 0.40503931 47.953466 53.58640 57.68801
Dataset: Student’s-t distribution, n:500
threshold: 𝜉: β: SE: 𝜎(𝜉) SE: 𝜎(β) 95%: 98%: 99%:
20 -0.03858773 5.65594170 0.05638339 0.46071157 33.20959 37.84263 41.24016 25 -0.05397001 5.83428737 0.1009773 0.8114378 33.25834 38.07575 41.56495 30 -0.1246037 6.0390003 0.1179638 1.1272249 33.66617 38.49988 41.80681
Dataset: Student’s-t distribution, n:5.000
threshold: 𝜉: β: SE: 𝜎(𝜉) SE: 𝜎(β) 95%: 98%: 99%:
20 -0.02208357 5.42723564 0.0156319 0.1327479 32.81019 37.47653 40.94428 25 0.08040134 4.47536019 0.03110079 0.19213973 32.19992 37.00594 40.88417 30 0.1764369 4.1741883 0.06198092 0.32988376 32.07943 36.59561 40.53131
table 15: tables of all three datasets results for n=500 and n=5.000.
4.3 Analysis of the results
Now knowing all the necessary results for the three different values of 𝑛, I should be able to see significant differences between the three methods (if they exist). To begin, I divide the analysis into three fractions with a different 𝑛 for each section. At first, I start with 𝑛 = 100.000. As already shown in paragraph 4.1: I can’t use the translated Gamma distribution for the normal distribution for this 𝑛 value. Therefore, I can only compare the Extreme Value Theory method with the historical simulation method for this distribution. If you look at table 5 and figure 12, you see that the estimates reconcile with each other for 𝑢 = 30, but not exactly for the other two thresholds, but this is negligible.
Next I compare the estimates for the chi-squared distribution. The translated Gamma distribution method and the historical simulation method barely differ from each other (only a small bit at the 95th percentile). This also applies for the Extreme Value Theory method. Only this method also shows that the standard deviations of the estimated parameters grows when the threshold becomes bigger.
At last I have to compare the estimates for the Student’s –t distribution. Again, I see that the historical simulation method and the translated Gamma distribution method reconcile. Moreover, I see
that the Extreme Value Theory method gives matching estimates in comparison with the other two methods, as the threshold gets bigger (the standard deviations of the estimated parameters do not grow significantly as the threshold gets bigger).
In the next section, I begin looking at the results for 𝑛 = 500. Again, I still cannot use the translated Gamma distribution for the normal distribution as it does not exist. So if you look at the results from the historical simulation method and the Extreme Value Theory method, you see that they differ much from each other for this distribution. As you can see, you get matching estimates as the threshold gets bigger (together with bigger standard deviations of the estimated parameters).
Secondly, I compare the methods for the chi-squared distribution. The Extreme Value theory gives the same estimates for threshold value 40 as the translated Gamma distribution method. These estimates are however approximately 3 higher than the historical simulation method. Moreover, I see that when the threshold gets bigger, you get deviating estimates (with high standard deviations) with the Extreme Value Theory method. And this also applies for too low threshold values. Then you also get deviating estimates, but with low standard deviations of the estimated parameters.
Finally, I shall compare the estimates for the Student’s-t distribution. The historical simulation method and the translated Gamma distribution method give estimates which are approximately the same (small deviation at the 95th and 99th percentile). However, the Extreme Value Theory method gives Value-at-Risk estimates that barely fluctuate when the threshold gets bigger and they are all higher in comparison with the other two methods. Moreover, you see that the standard deviations of the estimated parameters does not increase significantly when the value 𝑢 increases.
For the last subsection, I finally make a comparison between the results you get for 𝑛 = 5.000. Again, I begin with the normal distribution. Now I can also use the translated Gamma distribution method for the normal distribution, as his skewness is bigger than 0 (this resolves into relatively high estimates for alpha, beta and x0). If I look at the results, I see that there is very little difference between the three methods. They differ approximately 1 or less between each other. Moreover, the standard errors of the estimated parameters of the Extreme Value Theory method are also relatively small, but again they get bigger as the threshold becomes higher.
Secondly, I look again at the chi-squared distribution for 𝑛 = 5.000. All the methods reconcile with each other more or less. I also see that when the threshold gets higher, you also get better
matching estimates in comparison with the other two methods.
And for the end of all the comparisons, I make a final comparison between the results for the Student’s-t distribution. For 𝑛 = 5.000 you see that you get a situation which is quite comparable with the Student’s-t distribution for 𝑛 = 500. Again I get an overestimate for the Value-at-Risk in
comparison with the other two methods. And I see that the standard deviations do not increase significantly as the threshold get bigger.
So for all the used values of 𝑛, I see that the Value-at-Risk estimates from the historical simulation method and the translated Gamma distribution method match. As regards to the Extreme Value Theory method, I see that its estimates also match quite often. However this is only the case when you take a big enough threshold. So this must be a consequence of formula 2. You must take a value 𝑢 big enough to make this equation valid.
As regards to the chi-squared distribution. I mentioned that the Extreme Value Theory method gives overestimated Value-at-Risk estimates in comparison with the historical simulation method. This is the result of the fact, that when thresholds get too big, you have too little data points above this value (see figure 15). Therefore, you are not able to calculate proper maximum likelihood estimates (the standard deviations of the estimated parameters are therefore also relatively high).
Finally I discus the differences between the results from the Student’s-t distribution. The Value-at-Risk estimates between on the one hand the Extreme Value Theory method and on the other hand the translated Gamma distribution method and the historical simulation method differ much (especially for 𝑛 = 500). This is because if you look at the histogram of the Student’s-t distribution, you see that its mass is stationed at the middle and at the left side. However, it also has a very large right tail (with a few data points in it). So if you estimate the Value-at-Risks with the historical simulation method and the translated Gamma distribution method, you get smaller estimates in comparison. This is because those methods just look where the quantiles are. But the Extreme Value Theory method fits an excess distribution and sees that it still has a few extreme data points. So therefore it does not tolerate too small Value-at-Risk estimates (it’s more carefully).
Note: the historical simulation method and the translated Gamma distribution method often give estimates which are almost the same, especially when 𝑛 gets bigger. This is because the second method fits a distribution through the dataset. So this means that you just replicate the dataset with an approximated density function (which should be as precise as possible). Therefore, you should also get Value-at-Risk estimates who are nearly the same (that is the intention).
5. The Conclusion
As risk managers, stockholders and all other people with financial products bear great risks, they need certain measures, models and standards in order to be able to understand the size and consequences of these risks. To help them a bit, I tried to expand the research on one of these standards, that is the Value-at-Risk. People used to estimate this value with common methods as the historical simulation method, the Monte Carlo simulation method and the variance-covariance method. But in this paper I tried two other methods for calculating the Value-at-Risk, namely the Extreme Value theory method and the translated Gamma distribution method. Moreover, I tried to compare them with the historical simulation method in order to answer the question whether there were significant differences between the Value-at-Risk estimates from all of those methods and why they showed up.
According to the results, I can state that the three methods mostly reconcile with each other, as the simulation size gets bigger. However, when you use too small thresholds for the Extreme Value Theory method (together with a small simulation size), you get deviating estimates for the Value-at-Risk in comparison with the ones from the other two methods. This is the results of formula 2: you use a too low 𝑢. On the other hand, when you use too high thresholds (also together with small simulation sizes), you also get deviating estimates for the Value-at-Risk which is the result of poor maximum likelihood usage (too little data points above the threshold).
Moreover, when the dataset is heavily tailed to the right, you get that the historical simulation method and the translated Gamma distribution method give lower estimates for the Value-at-Risk in comparison with the ones from the Extreme Value Theory method (when 𝑛 is small). This is the result of the fact that the last method estimates an excess distribution through the tail section of the dataset. It sees that , although there are a lot more smaller losses than bigger losses, that the bigger losses still form a danger. Therefore it gives relatively higher Value-at-Risk estimates in comparison with the ones from the other two methods (it’s more careful).
Basically, you can say that all of these differences disappear for all three distributions as the sample size gets bigger (see results for 𝑛 = 100.000). However it’s not realistic that you have this amount of data points in your dataset. Therefore, I have showed you all the differences you get when you simulation size is small (like in practice).
Finally, if people want to do more research on this subject, they could also use the hill method (this is another result from the Extreme Value Theory) for estimating the Value-at-Risk. Moreover, using other distributions as the normal Pareto and Pareto type 2 distribution, could also give nice results (Those also have heavy right tails).
Appendix
Attachment
I
Substitute1 − (1 +𝛽(𝑢)𝜉𝑦 )−1/𝜉 and 𝐹(𝑢) =𝑛−ℎ
𝑛 into𝐹(𝑥) = (1 − 𝐹(𝑢))𝐺𝜉,𝛽(𝑦) + 𝐹(𝑢), this gives the following equation: 𝐹(𝑥) = (1 −𝑛−ℎ
𝑛 ) (1 − (1 + 𝜉𝑦 𝛽) −1 𝜉 ) +𝑛−ℎ𝑛 = 1 − (1 +𝜉𝑦 𝛽) −1 𝜉 −𝑛−ℎ 𝑛 + 𝑛−ℎ 𝑛 (1 + 𝜉𝑦 𝛽) −1 𝜉 +𝑛−ℎ
𝑛 . You have that 𝑛−ℎ
𝑛 = 1 − ℎ
𝑛, so if you substitute this in the previous equation you get the desired equation: 1 −ℎ
𝑛(1 + 𝜉𝑦 𝛽) −1 𝜉.
Attachment
II
By definition: 𝐹(𝑥) = 1 −ℎ 𝑛(1 + 𝜉 𝑥−𝑢 𝛽 ) −1𝜉. So you know the cdf, but you want to know the pdf in
order to calculate the log likelihood. That is, 𝜕𝑥𝜕 𝐹(𝑥) = −𝑛𝜉𝛽ℎ𝜉 (1 + 𝜉𝑥−𝑢𝛽 ) −1
𝜉−1= 𝑓(𝑥). Now you take the logarithm of this equation and you get, log (𝜕
𝜕𝑥𝐹(𝑥)) = − log(𝛽) + log ( ℎ 𝑛) − ( 1 𝜉+ 1) log(1 + 𝜉𝑥−𝑢
𝛽 ) . Taking the summation of the log ( 𝜕
𝜕𝑥𝐹(𝑥)) gives: −ℎ log(𝛽) + ℎlog( ℎ 𝑛) − (1 + 1 𝜉) ∑ log(1 + 𝜉(𝑦𝑖−𝑢) 𝛽 ℎ
𝑖=1 ),𝑖𝑓𝜉 ≠ 0. Note that the summation of a constant term is equal to h times this term.
Attachment
III
R code:n<-500#amount of simulated values set.seed(15647458)
a<-rnorm(n,mean=29,sd=5.2) #random normal distribution
b<-rchisq(n, 10, ncp = 20) #random chi-squared distribution c<-rt(n,10,ncp = 20) #random T-distribution
summary<-round(matrix(c(n=length(a), mean=mean(a),
sd=sd(a),n=length(a), mean=mean(b), sd=sd(b),n=length(c), mean=mean(c), sd=sd(c)),nrow=3,ncol=3,byrow=TRUE), 2) dimnames(summary) = list(c("normal distribution", "chi-ssquared","student-t"),c("n", "mean", "sd"))
grafiek1<-hist(a,main="simulated normal distr.",xlab="values",breaks=100);
abline(v=quantile(a,c(0.95,0.98,0.99)),lty=2,col="blue") #historical method
grafiek2<-hist(b,main=" simulated chi-squared distr.",xlab="values",breaks=100);
abline(v=quantile(b,c(0.95,0.98,0.99)),lty=2,col="blue") #historical method
grafiek3<-hist(c,main="simulated Students-t distribution",xlab="values",breaks=100); abline(v=quantile(c,c(0.95,0.98,0.99)),lty=2,col="blue") #historical method quantiel<-matrix(c(quantile(a,c(0.95,0.98,0.99)),quantile(b,c(0.95,0.98,0.99)) ,quantile(c,c(0.95,0.98,0.99))),nrow=3,ncol=3,byrow=TRUE)
dimnames(quantiel) = list(c("normal distribution", "chi-squared","student-t"),c("95%", "98%", "99%"))
u<-30
nuna<-ifelse(a>u,1,0) #giving all the values above the threshold a value of 1
nunb<-ifelse(b>u,1,0) nunc<-ifelse(c>u,1,0)
nu.a<-sum(nuna) #Number of observations above the threshold nu.b<-sum(nunb) #Number of observations above the threshold nu.c<-sum(nunc) #Number of observations above the threshold est.par.a<-gpd(a, threshold = u, method = c("ml"),information = c("observed")) #estimate parameters (standard package 'evir') est.par.b<-gpd(b, threshold = u, method = c("ml"),information = c("observed")) #estimate parameters (standard package 'evir') est.par.c<-gpd(c, threshold = u, method = c("ml"),information = c("observed")) #estimate parameters (standard package 'evir') esVaR.a<-vector() #empty vectors
esVaR.b<-vector() esVaR.c<-vector() quantielen<-c(0.95,0.98,0.99) for (i in 1:3){ esVaR.a[i]<- u+(est.par.a$par.ests[2]/est.par.a$par.ests[1])*((((n/nu.a)*(1-quantielen[i]))^(-est.par.a$par.ests[1])-1)) #estimating VaR esVaR.b[i]<- u+(est.par.b$par.ests[2]/est.par.b$par.ests[1])*((((n/nu.b)*(1-quantielen[i]))^(-est.par.b$par.ests[1])-1)) esVaR.c[i]<- u+(est.par.c$par.ests[2]/est.par.c$par.ests[1])*((((n/nu.c)*(1-quantielen[i]))^(-est.par.c$par.ests[1])-1))} estimated.VaR<-matrix(c(esVaR.a,esVaR.b,esVaR.c),nrow=3,ncol=3,byrow=TRUE)
dimnames(estimated.VaR) = list(c("est. normal distribution", "est. chi-squared","est. student-t"),c("95%", "98%", "99%")) data.max<-c(max(a),max(b),max(c)) data.max estimated.VaR quantiel grafiek3<-hist(c,main="simulated Students-t distribution",xlab="values",breaks=100);
abline(v=esVaR.c,,lty=2,col="blue") #historical method o<-plot.gpd(est.par.c,xlog=F) #excess plot
gpd.q(o, 0.99, ci.type = "likelihood", ci.p = 0.80) gpd.sfall(o,0.99)
alpha<-4/(skewness(c)^2) betha<-2/(skewness(c)*sd(c))
x.o<-mean(c)-(2*sd(c))/skewness(c) q<-hist(c,prob=T,breaks=100)
curve(dgamma(x-x.o,alpha,betha), add=T, col="red", lwd=2)
abline(v=qgamma(c(0.95,0.98,0.99),alpha,betha)+x.o,lty=2,col="blue") #historical method
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