Estimation of Value-at-Risk through
Extreme Value Theory
Basile Berg
Afstudeerscriptie voor de
Bachelor Actuari¨ele Wetenschappen Universiteit van Amsterdam
Faculteit Economie en Bedrijfskunde Amsterdam School of Economics Auteur: Basile Berg
Studentnr: 10995501
Email: basileberg@upcmail.nl Datum: June 26, 2018
Estimating Value-at-Risk — Basile Berg iii
Abstract
In this thesis the generalized Pareto distributon is estimated by maximum likelihood and probability weighted moments, wherein the threshold is estimated to be 600, although choosing the threshold at 0 would also be possible. After the GPD is estimated, the value at risk is calculated for the two different methods of estimation, which is then compared to each other and an empirically estimated value at risk. The maximum likelihood was the best estimator of the real number of exceedances, while the probability weighted moments method estimated the value at risk the closest. Because the value at risk is used in insurance data, and an underestimation of the risk could lead to bankruptcy, the probability weighted moments method is the best method to estimate the quantiles.
Keywords Value-at-Risk, Probability weighted moments, Quantile estimation, Extreme value theory, Generalized pareto distribution, Peaks-over-threshold
Acknowledgements v
1 Introduction 1
2 Extreme Value Theory 3
2.1 Generalized Pareto distribution . . . 3
2.2 Estimating truncated distribution . . . 4
2.3 Estimating the parameters . . . 4
2.4 Value-at-Risk . . . 6
2.5 Binomial distribution. . . 6
3 Data Analysis 8 4 Results & Evaluation 9 4.1 Fitting the data. . . 9
4.2 Overall fit of the different models . . . 12
4.3 Parameter and Value-at-Risk estimations . . . 13
5 Conclusion 17
References 19
Acknowledgements
Helping me write this thesis were some of the nicest people I know, for a starter my parents, who helped me write this thesis by being supportive and feeding me when I didn’t want to cook for myself. Also, I would like to thank all my friends who I could always go to with any of my questions and problems. Last but not least, I would like to thank Umut, who supported me throughout the whole thesis and kept believing in the success of this thesis even when I didn’t know if the thesis would be finished in time.
Chapter 1
Introduction
For insurance companies it is important to know what the chances are of getting an ex-tremely high claim, because an exex-tremely high claim can put an insurance company out of business. It is also important to know what the expected value is of those extremely high claims. To estimate these statistics the extreme value theory is used.
The extreme value theory is a theory first used to predict the height of rivers in the Netherlands, so the probability of a dike breach could be calculated. Beirlant and Teugels (1992) and Kl¨uppelberg (1993) are the first who linked the extreme value theory to in-surance losses. For this thesis the important results in the extreme value theory are the Pickands-Balkema-de Haan theorem (Balkema & De Haan 1974, Pickands 1975), which proves that the distribution above a high threshold follow the generalized Pareto dis-tribution, and Davison (1984) in which the generalized Pareto distribution is used with real data for the first time. All models based on a distribution above a threshold are called Peaks over Threshold (POT) models. Besides POT models, there are also block maxima models. These models give an estimation of the maximum value in a certain period (McNeil 1998). Although both models are useful for insurance companies, this thesis will focus on POT models.
In the POT models there are two ways of analysing the data, with semi parametric models and with parametric models. The semi parametric models are based on the Hill estimator (Hill 1975). This estimator does not require a distribution function to draw inference about the behavior of a distribution function. The parametric models use the Pickands-Balkema-de Haan theorem and base there model on the generalized Pareto distribution. Both models are theoretically correct and choosing between these models is a matter of taste. For this thesis the parametric models are chosen, because these are
somewhat easier to compute.
To estimate the threshold using parametric models is something McNeil (1997) has al-ready done. In his article the losses are all the fire damages in Denmark, above one million danish kronen (DKK) in the period 1980-1990. The article uses the generalized Pareto distribution to estimate the density function above the threshold. Thereafter it uses a Maximum Likelihood model to estimate the parameter values in this model. But the maximum likelihood estimation is not the only model that can be used to estimate the generalized Pareto distribution. Hosking & Wallis (1987) argue that the Probability Weighted Moments model is a better model to estimate quantiles and pa-rameters for the generalized Pareto distribution, given the data set has less than 100 data points. For extreme value theory the data set is almost always small, therefore the probability weighted moments model might be better at estimating the parameters and quantiles, when the extreme value theory is used.
Taking all this into consideration, the main objective of this thesis is to estimate the optimal Value-at-Risk in a data set of business interruption losses by means of an em-pirical research. In this objective the Maximum Likelihood and Probability Weighted Moments estimations of the generalized Pareto distribution are both used and compared to get an optimal estimation.
Chapter 2
Extreme Value Theory
For the Extreme value Theorem it is assumed that the data is independently identically distributed (IID). It is not important what kind of data is used, as long as the IID assumption is satisfied. Some examples of the data used are operational losses, insurance claims and financial returns.
2.1
Generalized Pareto distribution
Gξ,β(x) = { 1 − (1 + ξ x β) −1 ξξ 6= 01 − e −x β ξ = 0
If IID holds, Pickands (1975) proofs that, asymptotically, above a certain high threshold the distribution of the data above this threshold follows the generalized Pareto distri-bution. This distribution function has two different parameters. The shape parameter ξ and the scale parameter β. In the generalized Pareto distribution the scale parameter β must be greater then 0. Further x ≥ 0 when ξ ≥ 0 and −β/ξ ≥ x ≥ 0 if ξ < 0 must hold as well.
The shape parameter ξ is an indicator for the distribution in the extremes of the data set. If the ξ is greater then 0, the distribution is heavy-tailed (McNeil 1999). If ξ > 0 the generalized Pareto distribution has a distribution that is a reparameterized version of the normal Pareto distribution. The Pareto distribution is a distribution which was his-torically used in models to predict large losses. The Pareto distribution is heavy-tailed. If ξ = 0 the generalized Pareto distribution has the same distribution as the exponen-tial distribution. This distribution is not heavy-tailed. For ξ < 0 the generalized Pareto distribution has the same distribution as the Weibull distribution, this distribution is also heavy-tailed.
Heavy-tailedness is a favorable properly for a distribution in the extreme value theory, because the empirical losses in insurance data are almost always heavy-tailed (McNeil 1999). So if the distribution is heavy-tailed it is a better representation of the data.
2.2
Estimating truncated distribution
In insurance data, only claims are shown and not the data of a zero claim, therefore the data is always truncated at zero. But a truncated distribution is not the same as the distribution of the non trunctated data, the truncated cumulative density function (CDF) truncated at u is divined as:
Fu(x) = P [x − u ≤ y|x > u]
where y is between 0 and the largest overshoot of the threshold (Balkema de Haan 1974, Pickands 1975). This CDF is the probability of overshooting the threshold with a certain value y, given that the value x is larger than the threshold. It is also possible to write the truncated CDF as a combination of the underlying CDF.
P [x − u ≤ y|x > u] = P [y + u ≤ x] − P [u ≤ x] 1 − P [u ≤ x] =
F (y + u) − F (u) 1 − F (u)
The Pickands-Balkema-de Haan theorem is about the truncated CDF and states that if the threshold is sufficiently high, the truncated CDF will follow the generalized Pareto distribution (Balkema de Haan 1974, Pickands 1975).
Fu(x) = Gξ,β(x)
2.3
Estimating the parameters
Because the distribution of the data is now known, it is possible to estimate the pa-rameters of the generalized Pareto distribution in the data set. Fitting the data to the model is possible in two different ways (McNeil 1997). The first possibility is to use maximum likelihood. This method is based on the density function of the distribution and minimizes the distance between the empirical data and the estimated value. The maximum likelihood estimators of the generalized Pareto distribution can be found by maximizing the log-likelihood function.
Estimating Value-at-Risk — Basile Berg 5
With f(x) the density function of the generalized Pareto function. This density function is the derivative of the cumulative density function divined in the chapter about the generalized Pareto distribution. The density function simplifies to:
f (x) { 1β(1 + ξx β) −1/ξξ 6= 01 βe −x β ξ = 0
With the density function it is possible to maximize the log-likelihood function.
log(L) = {Pn i=1log( 1 β(1 + ξ xi β) −1/ξ−1 )ξ 6= 0 n X i=1 log(1 βe −xi β )ξ = 0
The second possibility is the probability weighted moments. The advantage of the prob-ability weighted moments model over the maximum likelihood model is that in small sample estimations, the probability weighted moments is reliable and the maximum like-lihood is not (Hosking, Wallis, and Wood 1985). In the probability weighted moments method, the moments are used to estimate the parameters of the generalized Pareto distribution. Where the moments are divined as:
Mp,r,s= E[xpF (x)r(1 − F (x))s]
Where p, r and s are real numbers. In Hosking, Wallis, & Wood (1985) it is proposed to look at βr = M1,r,0 to estimate the parameters in the generalized Pareto distribution.
This βr is estimated by br= n1Pnj=1
(j−1)(j−2)...(j−r)
(n−1)(n−2)...(n−r)xj with x as the ordered statistic
(Landwehr, Matalas, & Wallis 1979). From this estimator it is possible to produce mul-tiple equations, from which the parameters in the generalized Pareto distribution can be estimated. The main disadvantages of the probability weighted moments model is its inefficiency, while the maximum likelihood model is the most efficient estimation. Concluding, the best way to estimate the extreme value models is by fitting the general-ized Pareto distribution to data above a high enough threshold. There are two different models to estimate the parameters in the generalized Pareto distribution. The maxi-mum likelihood and the probability weighted moments models. Both have their own disadvantages. Maximum likelihood is asymptotically correct, but has a slow conversion rate and the probability weighted moments is inefficient. Unfortunately, it is not clear which of the disadvantages is the least restricting in extreme value theory, therefore both models will be fitted to the data.
2.4
Value-at-Risk
It is important to check the fit of the generalized Pareto distribution for the different methods, this is done by using the Value-at-Risk (VaR). The VaR is a quantile estimation based on the cumulative density function (McNeil 1999), to calculate the VaR, the cumulative density function is inverted. For this thesis, the VaR of the generalized Pareto distribution is used, the generalized Pareto distribution only exists above a certain threshold, therefore the cumulative density distribution of the whole data set is not yet known. The cumulative density function of the whole data set can be split into two parts, the first part where the x is below the threshold and the second part where the x is above the threshold (McNeil 1999). If this is used the formula simplifies to:
F (x) = (1 − F (u))Fu(x − u) + F (u)
This is the cumulative density function for the whole data set, but only for the part above the threshold. For some more simplifications the cumulative density function below the threshold is assumed to be equal to the historical estimator (n − Nu)/n,
where n is the total amount of data points and Nu is the amount of data points which
exceed the threshold (McNeil 1999). If this is used the cumulative density function of the generalized Pareto distribution above the threshold is equal to:
ˆ F = 1 − Nu n (1 + ˆξ x − u ˆ β ) −1/ ˆξ
with ˆξ and ˆβ the estimated parameters from the generalized Pareto distribution. From this cumulative density function, the VaR can be calculated by inverting the formula to get an expression in x, in this formula, the q the quantile for which the VaR is calculated, while all the other parameters are still the same as in the cumulative density function.
ˆ V arq = u + ˆ β ˆ ξ(( n Nu (1 − q))− ˆξ− 1)
2.5
Binomial distribution
The binomial distribution is about the number of successes in a sequence, with a fixed probability of success for all the individual trials. An often given example is tossing a coin multiple times, where the probability of success is the probability of landing on heads. This has a fixed probability, specifically 50% and the amount of tosses is the length of the sequence. Because this trial gives an integer outcome, the binomial distribution has a discreet probability function called the probability mass function, which estimates the
Estimating Value-at-Risk — Basile Berg 7
probability of an integer value coming from the corresponding binomial distribution. This probability mass function is given by:
f (k) = P [X = k] = n k
!
Data Analysis
To be able to do the analyses, some kind of data is required, for this thesis, the data is from French business interruption losses between the years 1985 and 2000. The data is collected by APSAD (assemblee pleniere des societes d’assurances de dommages), the French insurance union (Zajdenweber 1996). The data collected is all the losses above 100.000 French Franks and is corrected for inflation, this is done by discounting all the losses to 2007, in total there are 2387 losses.
In the results and evaluation, the generalized Pareto distribution will be fitted to the first part of the data, which will be called the estimation data from now on, using the different methods discussed in the theory. The two methods used to estimate the generalized Pareto distribution are maximum likelihood and probability weighted moments. After the generalized Pareto distribution is estimated with the two different methods, the VaR is estimated for the 95% quantile, 99% quantile and 99.5% quantile, the VaR is also empirically estimated for the three quantiles. Thereafter the VaR estimations are used as quantile estimations in the second part of the data, which will be called the test data, the fit of the three different models is checked by the number of exceedances. Every datapoint is a Bernoulli trial with a the probability of overshooting the VaR corresponding to the quantile, for the 95% quantile the probability is 5%, the 99% quantile has a probability of 1% and the 99.5% quantile has a probability of 0.5%. If a Bernoulli trial is done for multiple values in a row, the trial becomes a binomial distribution, this is a well known distribution, from where it is possible to judge the fit of the different models.
Chapter 4
Results & Evaluation
4.1
Fitting the data
For the analysis of data sets, it is important that the data is uncorrelated, to check if the data is uncorrelated the Ljung-Box test for one lag term is done. This test checks if there is any correlation between all the consecutive data points, where the H0 is defined
as no correlation between the consecutive data points and for Ha the consecutive data
points are correlated (Ljung Box 1978). The test statistic is:
QLB = n(n + 2) h X j=1 ˆ ρ2(j)(n − j)
the ρ is the auto correlation, h is the degree of freedom and n is the amount of data points in the data set. The QLB is chi-squared distributed with h degrees of freedom,
with this information it is possible to test the data set.
For this thesis the correlation between the consecutive data points are checked and therefore the degree of freedom is one. If the test is applied to the data set, the QLB
is equal to 0.066024, this corresponds with a probability value of 0.7972 and therefore does not reject the H0. Looking at the time series plot, the assumption of uncorrelated
data seems reasonable, since there isn’t any clustering of high values, because there is no indication of serial correlation from the time series plot and the H0 of the Ljung-Box
test was not rejected, the data set is from now on assumed to be uncorrelated.
As said in the data analyses, the data will be split exactly in half, creating two data sets with approximately the same underlying distribution. Because there are 2387 data points the data can not be split into two even data sets, therefore the choice is made to set the first data set to be one value shorter than the second data set. This creates an estimation data set with 1193 values, and a test data set with 1194 values. While
Figure 4.1: Time series plot
the estimation data set is used to do the analyses, the test data set is used to compare the predicted VaR from the different methods. If the QQ-plot of the estimation data is
Figure 4.2: quantile-quantile plot of the estimation data
analyzed, it becomes clear that the empirical QQ-line is convex, while the QQ-line of the exponential distribution is straight, this convexity is a clear sign of heavy-tailedness in the empirical distribution (McNeil 1997).
A further indication of heavy-tailedness is apparent in the histogram, where there are some clear outliers at approximately 2500 and 3300, these outliers would only be
rea-Estimating Value-at-Risk — Basile Berg 11
Figure 4.3: Histogram of the estimation data set
sonable in a heavy-tailed distribution. The tail also seems stretch out at around 1000, this is a further indication of heavy-tailedness as normal tailed distribution, in general, do not have such stretched out tails.
As it is now reasenable to assume that the underlying distribution of the estimation data set is indeed heavy-tailed, the generalized Pareto distribution can be fitted to the data. The generalized Pareto distribution only describes the data above a certain threshold, to estimate where the threshold should be, it is easiest to look at the mean excess plot. As stated in the theory, if the mean excess plot is linear from a certain point onward, the data follows the generalized Pareto distribution.
The mean excess plot seems to be fairly linearly increasing from start to finish, but the line starts to drop off a bit around 500 and starts increasing again at approximately 600, therefore the threshold is chosen to be 600, while choosing the threshold at zero would also be possible. This is reasonable because the data set was already truncated at 100.000 French Franks when the data was collected.
To further investigate the threshold, the shape parameter is plotted against the number of exceedances. In choosing a threshold, there is a trade-off between the asymptotically valid theory, which doesn’t hold if the threshold is chosen too low and a high variance if the threshold is chosen too high (McNeil 1997). In figure 4.5 it seems apparent that the variance increases dramatically around 600, while the shape parameter is to some extend stable until 600, therefore the threshold of 600 is a good fit for our data and this
Figure 4.4: Mean excess plot of the estimation data set
threshold balances the trade-off quite well.
4.2
Overall fit of the different models
In this section the fit of generalized Pareto distribution is discussed, this is done for the maximum likelihood method and the probability weighted moments method. The comparison is done on the estimation data set to see how good the different methods fit to the data set they are based upon.
To compare the fit of the different models the truncated cumulative density functions are plotted against the empirical truncated cumulative density function. The truncation is at 600, the threshold chosen in the previous section. Figure 4.6 only uses the values above this threshold, but the threshold is subtracted off the data and therefore the plot starts at 0. The truncated cumulative density functions estimated with maximum likeli-hood and probability weighted moments are almost identical and both models follow the truncated empirical cumulative density function closely. The truncated empirical cumu-lative density function estimated by the two methods does underestimate the probability of the last few high values, which is not favorable in insurance data because these high values have the biggest impact on the companies.
Estimating Value-at-Risk — Basile Berg 13
Figure 4.5: Maximum likelihood estimation of the shape parameter with an increasingly high threshold
4.3
Parameter and Value-at-Risk estimations
In the section about fitting the data, the shape parameter is plotted against the thresh-old this gives a good insight in what the shape parameter is and how it is impacted by the threshold. In this section the shape parameters obtained under the two different methods is analyzed and compared. After the shape parameter is analyzed, the maxi-mum likelihood estimation of the VaR is compared to the VaR estimation of probability weighted moments and the empirical VaR.
The shape parameter is estimated with a threshold of 600, there are 82 data points estimation of the parameters
Parameter Maximum likelihood Probability weighted moments
ξ 0.612595 0.4964017
β 342 372
Table 4.1: Parameter estimations of the two methods
above this threshold in the estimation data set. The estimation of the shape parame-ter with maximum likelihood and probability weighted moments are both above zero and, as mentioned in the theory, a shape parameter above zero means that the dis-tribution is heavy-tailed, and therefore is also in agreement with the expectation from the section about fitting the data. The VaR is estimated with the generalized Pareto
Figure 4.6: truncated CDF of the different models in the estimation data set
estimation of Value-at-Risk
Quantile Maximum likelihood Probability weighted moments empirical test data set
95% 720 728 696 737
99% 1860 1803 2298 1864
99.5% 2821 2605 3018 2684
Table 4.2: Quantile estimations and real quantiles
distribution estimated by maximum likelihood and probability weighted moments, but also empirically, the empirical VaR is the corresponding quantile in the estimation data set and the test data set is the corresponding quantile estimation in the test data set. The real VaR estimations for the 95% quantile are all close to 700, but the empirical estimation is a bit lower than the VaR estimation done by the maximum likelihood and probability weighted moments methods, while these two methods are almost the same. The real VaR is 737 and is close to all of estimations of VaR, but closest to the probability weighted moments estimation. For the 99% quantile the VaR estimated by maximum likelihood and probability weighted moments are again close, but the prob-ability weighted moments is a bit lower than the maximum likelihood estimation. The empirically estimated VaR for the 99% quantile however, is a lot higher at almost 2300, while both maximum likelihood and probability weighted moments lay around 1800. The real VaR is 1864, this is closest to the maximum likelihood estimation, but also
Estimating Value-at-Risk — Basile Berg 15
near to the probability weighted moments estimation, whereas the empirically estimated VaR is a lot further away from the real VaR. In the estimation of the 99.5% quantile, the differences are a lot bigger. The estimations done with probability weighted moments is 2605 and is therefore closest to the real VaR of 2684, while the maximum likelihood estimation of the VaR is also close to the real VaR, but the estimation is still 150 away from the real VaR. The empirical VaR, however is not so close to the real VaR and differs almost 350.
number of exceedances (probability of coming from the corresponding distribution) Quantile Maximum likelihood Probability weighted moments empirical test value
95% 61 (0.05159539) 61 (0.05159539) 62 (0.04962443) 60 (-) 99% 12 (0.11492944) 13 (0.10555291) 9 (0.08868619) 12 (-) 99.5% 6 (0.16101601) 7 (0.13732019) 6 (0.16101601) 6 (-) Table 4.3: exceedances and likelihood of the number of exceedances coming from the corresponding quantile
The number of exceedances is based upon the VaR estimation, if a data point is higher than the VaR, it is an exceedance. For the test data, the real quantile is used as the VaR, this is done to be able to compare the VaR of the different methods to the test data. If the number of exceedances is analyzed, all the differences become significantly smaller. For the 95% quantile, the maximum likelihood estimation gives 61 exceedances, this is the same as the probability weighted moments predicts, while the empirical estimation predicts 62 exceedances. The real number of exceedances is 60, therefore all the estimations are close to the real value. This is also visible from the corresponding probabilities, which all are around 5%, because the probability of a point exceeding the VaR is 5%, the variation is a lot higher and the probability density is therefore more spread out. In the 99% quantile, the differences are a lot bigger, especially the empirical estimation is a lot lower than all the other estimations. The empirical estimation is 9, while the estimation done with maximum likelihood is 12 and with probability weighted moments is 13, which lay significantly closer to the real value of 12 than the empirical estimation. This is also clear from the probabilities, where the maximum likelihood estimation has a 11.5% probability of coming from the binomial distribution with a probability of success of 1%, the probability weighted moments has a probability of 10.6 % of coming from the distribution, but the empirical estimation has a probability of 8.9% of being from the binomial distribution with a probability of success of 1%. For
the 99.5% quantile the estimations are a lot closer than the 99% quantile estimation, the maximum likelihood and empirically estimated number of exceedances are both 6, while the probability weighted moments estimates 7 exceedances. In the test data there are 6 exceedances in the 99.5% quantile, therefore the probability weighted moments is the only estimator which did not correctly predict the number of exceedances. But if the probabilities are compared the estimation of probability weighted moments is not that far off in its prediction. The probability of getting 7 exceedances in the binomial trial with probability of success of 0.5% is 13.7% while the probability of getting 6 is 16.1%.
Chapter 5
Conclusion
This thesis analyzed the VaR estimations of three different methods, two of these mod-els are estimated with the generalized Pareto distribution and the third is an empirical estimation on the basis of a data set. The generalized Pareto distribution was analyzed first, this distribution is only valid above a certain threshold, in this thesis the threshold was estimated to be 600. Because the generalized Pareto distribution and VaR are now estimated correctly, it is now possible to accurately draw a conclusion about the fit of the generalized Pareto distribution in the data set and the exactness of the three VaR estimations. There were multiple descriptive plots used to check the fit of the generalized Pareto distribution, the conclusion of all these separate plots is that the underlying dis-tribution should be heavy-tailed. This is in agreement with the estimation of the shape parameter done by maximum likelihood and probability weighted moments, both are positive and a positive shape parameter gives a reparameterized version of the Pareto distribution, which is a heavy-tailed distribution. The mean excess plot is used to deter-mine the threshold, in this plot, the points lay on an approximately straight line from the start, but still has a slight drop off at around 500 till 600 and therefore the choice is made to set the threshold to 600.
For the best estimation of VaR, the binomial distribution is used to estimate the prob-ability of the number of exceedances coming from the corresponding quantile. This is done for the 95%, 99% and 99.5% quantile. For the 95% quantile the maximum like-lihood and probability weighted moments estimations had the highest probability of coming from the binomial distribution with probability of success of 5%. For the 99%, the maximum likelihood estimation has the highest probability of coming from the bi-nomial distribution with probability of success of 1%. Finally, for the 99.5% quantile,
the maximum likelihood and the empirical estimations have the highest probability of coming from a binomial distribution with probability of success of 0.5%. Because the maximum likelihood estimation is closest in all of the quantile estimations, maximum likelihood VaR is the best predictor of the quantile. But for an insurer, a lower estima-tion might be more useful, because an underestimaestima-tion of the risk of the high values can easily lead to bankruptcy. Therefore it is also important to look at the estimation of the VaR. The VaR estimation of probability weighted moments is closest for 2 out of the 3 quantiles, while the maximum likelihood is closest for the 99% quantile, but the VaR estimated with probability weighted moments is the lowest for the two higher quantiles. Because the probability weighted moments model has a low estimation of the VaR, which is desirable, and the probabilities of the probability weighted moments are also really close the maximum likelihood probabilities, the probability weighted moments is the estimator to use for insurers. Since the data is insurance data and the proba-bility weighted moments is the most conservative estimation, the probaproba-bility weighted moments method is the best method in this thesis.
Estimating Value-at-Risk — Basile Berg 19
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