Applications Of Evt In Financial Markets

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Extreme Value Theory (EVT) has emerged as an important statistical

discipline for the applied sciences. It is useful because it provides techniques for

estimating models that predict events occurring at extremely low probabilities.

This paper describes EVT and tools such as quantile plots and mean excess

plots used to determine the appropriateness of EVT for modeling given data.

EVT techniques are then applied to model daily return of stock of three large

companies: IBM, Ford and Nortel. The results show that Generalized Pareto

Distribution (GPD) can appropriately model extreme daily returns, particularly

extreme daily losses. Finally the parameters of the appropriate GPD are

estimated, and Value-at-Risk (VaR) and Expected Shortfall, the two key risk

measures used by industry practitioners, are calculated based on the estimated

GPD.

Keywords: Extreme Value Theory, Generalized Pareto Distribution, Value-at-

Risk, Expected Shortfall

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Introduction, Scope & Purpose

EVT has been successfully applied in engineering, biology, meteorology,

insurance and a myriad of other applied sciences1. This paper will focus on

Applications of EVT in Financial Markets.

It has been noted that EVT’s application to extreme risk in financial

markets maybe motivated by the problem of daily determination of VaR for

losses incurred due to adverse market movements. Risk Managers are

interested in describing the tail of a loss distribution and measuring the expected

size of a loss that exceeds VaR2. Until recently, most parametric methods used

the Normal distribution to estimate VaR. However, under the assumption of

normality, the risk of high quantiles is underestimated, especially for the fat-tailed

series common in financial data.

This paper outlines the theoretical underpinnings of EVT and works

through examples illustrating how EVT can be applied to financial data. The first

part of this paper introduces classical EVT, models for maxima/minima and

threshold models. However, modeling the maxima or minima of financial data is

of little value to risk managers. Instead, a threshold model based on the

Generalized Pareto Distribution (GPD) is argued to be most suited for risk

managers because it can be used to model the tail of a loss distribution3. Thus,

the second part of this paper models daily returns price for IBM, Ford and Nortel

stocks using the Generalized Pareto Distribution.

1 Coles, Stuart. An Introduction to Statistical Modeling of Extreme Values.

2 McNeil, A. “Extreme Value Theory for Risk Managers”. Pg 1-2

3 Ibid.

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Theoretical Underpinnings4

Classical EVT and Models

The fundamental model of Extreme Value Theory is based on the

behavior of Mn where:

Mn = max{X1, …, Xn}

and Xi are independent identically distributed (iid) random variables. In theory,

the distribution of Mn can be easily derived if the distribution of Xi are known

because if F(z) is the distribution of the Xi, then

FMn(z) = P(Mn ≤ z) = P(X1 ≤ z, …, Xn ≤ z) = P(X1 ≤ z)…P(Xn ≤ z) = [F(z) ]n

In practice this is not possible because the distributions of the Xi are not

usually known. One possible solution to this problem is to estimate F(z) based on

observed values and then to derive FMn(z). However this approach is problematic

because any estimation involves errors and small errors in estimating F(z) would

lead to large errors in the estimation of FMn(z).

Another solution is to accept F(z) as unknown and then to try and find a

family of functions that model FMn based only on extreme (maximal) data. The

arguments to justify this method are analogous to the justifications underlying the

Central Limit Theorem. Pursuing this method further, we consider:

Mn* = (Mn – bn)/an where {an > 0} and {bn} are constants

Appropriate choices of bn and an stabilize location and scale of Mn* as n � ∞. All

possible limit distributions for Mn* are given by the Extremal Types Theorem:

4 Adapted from book by Stuart Coles. See references.

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In the above distribution, b is the location parameter, a is the scale

parameter and α is the shape parameter. The type I, type II and type III models

above are known as the Gumbel, Frechet, and Weibull distributions respectively.

This theorem says that regardless of the distribution of the Xi, if we normalize Mn

to Mn*, then the distribution of Mn* is of only 3 possible types.

These three types of extreme models have distinct forms of behavior

corresponding to different forms of the tail distribution of the Xi. Application of

EVT requires choosing one of the three models to estimate parameters. But this

raises two important problems. Firstly, how do you know which model type to use?

A technique is needed to choose the appropriate model type for given data.

Secondly, once the decision of model type has been made, all subsequent

inferences will rest on the assumption that the decision of model type was correct

and thus will not allow for any uncertainty in that decision.

These problems are solved by combing the three model forms into the

Generalized Extreme Value Distribution:

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In this combined model, µ is the location parameter, σ is the scale

parameter and ξ is the shape parameter. When ξ > 0, the GEV distribution

corresponds to the Frechet distribution. When ξ < 0, the GEV distribution

corresponds to the Weibull distribution. The case of ξ = 0 can be interpreted as

taking the limit as ξ�0 and this corresponds to the Gumbel distribution.

By doing statistical inference on ξ, the two problems associated with

choosing model types are solved. The data itself now determines the most

appropriate type of tail behavior. Furthermore, uncertainty in estimating ξ

corresponds to uncertainty in choosing the correct model type.

At this point we stop to consider that our original problem was to model Mn,

not Mn*. In practice, the constants an and bn may not be known. However, we

have shown that P(Mn* ≤ z) = P((Mn - bn)/ an ≤ z) ≈ G(z). Then for large n, we can

write

P(Mn≤ z) ≈ G((z - bn)/ an) = G*(z)

where G*(z) is also a member of the GEV family of distributions. Since the

parameters of the distribution have to be estimated anyways, it is irrelevant that

the parameters for G(z) are different than the parameters for G*(z) for a given

data set.

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By the above argument, an approach to modeling extremes of Xi is

developed. First all the data is blocked into a sequence of observations of size n

(n is large). From each block, we can derive block maxima: Mn 1, …, Mn m where

Mn i is the block maxima from the block i of size n. The GEV can now be fitted

using Mn 1, …, Mn m as data. Often blocks are chosen according to time periods of

1 year.

Block Minima

At this point let us note that the GEV distribution also provides asymptotic

models for minima. Given data z1,…,zn , we can simply maximize -z1,…,-zn.

Inference on GEV distribution:

The aforementioned method for implementing the GEV distribution

required us to divide the data into equal size blocks and fit to the set of block

maxima. However choosing block size is always a trade off between bias and

variance. Overly small block sizes lead to bias because approximation by the

GEV model is poor. Overly large block sizes generate fewer block maxima and

thus lead to large estimation variance. For time series data sets, block sizes of

one year are commonly chosen because this usually makes plausible the

assumption that block maxima have common distribution.

Now we consider Z1 ,…,Zn where Zi are iid block maxima from a GEV

distribution whose parameters need estimation. The Zi are independent even if

the Xi are not independent (as in the case of most time series). Likelihood-based

estimation provides one effective method of estimating these parameters. The

log-likelihood for the GEV when ξ ≠ 0 is

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The log-likelihood for the GEV when ξ = 0 (corresponding to the Gumbel distribution) is:

For any given data set, these log likelihood functions can be maximized

using numerical methods. The estimators derived from this method can be

assumed to be approximately multivariate normal and unbiased. Confidence

intervals and other inferences follow from this assumption of normality. Model

checking can be done by plotting the empirical distribution function evaluated at

zi for ordered block maxima against the model evaluated with the estimates. A

good fit will produce a linear graph lying close to the line y = x.

Threshold Models

In our quest to model extreme events, we may be given an entire time

series of daily observations. Better use of this data is made by avoiding blocking.

If X1, X2, … are iid, then extreme events can be defined as those Xi that exceed

some high threshold u. In modeling extreme events, we are interested in the

conditional probability of X-u given X > u. Theory tells us that if block maxima of

the data have an approximate GEV distribution, then this conditional probability

can be approximated by the Generalized Pareto Distribution.

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Moreover, the parameters of the GPD (modeling excess over threshold)

are uniquely determined by the parameters of the associated GEV model

regardless of the block size. This is because the shape parameter ξ is

independent of block size. Also β = σ + ξ(u – µ) where σ and µ are from the

associated GEV distribution. Changing block size n in the associated model

adjusts σ and µ in a self-compensating way so that β remains constant.

Modeling Threshold Excesses

Naturally the first step in modeling extreme data within the framework of

threshold models is to choose an appropriate threshold. Choice of threshold is

analogous to the problem of choosing the appropriate block size in the GEV

model. If the threshold is set too low, then the data beyond the threshold will

deviate significantly from the GPD. On the other hand, if the threshold is set too

high, there will not be enough data to estimate the model, and a high variance

will result. So we must choose as low a threshold as possible provided that the

GPD is still a reasonable approximation for excesses beyond the threshold. One

method to determine the threshold is to create the mean excess function plot.

Theory tells us that if Y has GPD, then E(Y) = β/(1- ξ). Since the conditional

distribution of excesses beyond threshold is approximated by GPD, we know

from theory that for a threshold u0:

E( X– u0 | X > u0 ) = β(u0)/(1- ξ)

where β(u0) is the value of β corresponding to the threshold u0. However, if the

excess beyond u0 can be modeled with GPD, then so can excess beyond any

other threshold u>u0. It has been shown that for u >u0,

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E( X– u | X > u ) = (β(u0) + ξu) / (1- ξ)

This is linear in u. Therefore we expect that for thresholds beyond which the

excesses follow a GPD, the conditional mean will be a linear function. Thus we

have the following method to determine the threshold. : Let x1, …, xn be the data

to be modeled.

1. Order the data: x(1),…, x(n)

2. For each u in { x(1),…, x(n) } calculate the sample mean of the difference

between the x’s and u for all x’s > u. In other words, calculate and plot:

3. Identify the point beyond which this graph is approximately linear and

choose that as the threshold for the model. Ensure that there are sufficient

points beyond the chosen threshold to make meaningful inferences.

Estimating Parameters

Once the threshold is chosen, likelihood techniques can be used to

estimate parameters for the GPD model while considering only the data that lies

beyond the chosen threshold. Let y1,…, yk be the excesses of a threshold u (so yi

= xi – u for xi > u). The log likelihood function for the GPD in the case where ξ ≠ 0

is:

ℓ(β,ξ) = -kln(β) – (1 + 1/ξ)∑ ln(1 + ξyi/β)

provided that (1 + ξyi/β) > 0 for i = 1,…,k.

In the case of ξ = 0, the log likelihood function is:

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ℓ(β) = -k*ln(β) – (1/β)∑ yi

These log likelihood functions can be numerically maximized and the maximum

likelihood estimate (mle) for β, ξ can be found.

Model Checking

The models validity can be checked with probably plots, quantile plots and

density plots. For a threshold u and threshold excesses y(1),…, y(k), the probability

plot (for i = 1,…, k) consists of the pairs

{i/(k+1), 1 – (1 + ξy(i)/β)-1/ξ }using the mle for ξ

The quantile plot consists of the pairs

{H-1(i/(k+1)), y(i)} for i= 1,…, k

where H-1(t) = u + β/ξ[t-ξ – 1] using the mle for β and ξ.

If the GPD is a good fit, the probability plot and the quantile plot will be

approximately linear. Also the density function of the fitted GPD can be

compared to a histogram of the threshold excesses.

Estimating VaR and Expected Shortfall5

As mentioned in the introduction, a major use of EVT in risk management

is to characterize the tail of a loss distribution using VaR and Expected Shortfall.

VaR is a high quantile of a distribution of losses and can represent an upper

bound for losses that is exceeded only rarely. Expected Shortfall is expected size

of a loss that exceeds VaR.

5 McNeil, A. “Extreme Value Theory for Risk Managers”. Pg 2-3.

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Using historical simulation and maximum likelihood estimates of the

parameters of GPD, the following tail estimator has been derived:

where Nu is the number of data points that exceed the threshold u and n is the

total number of data points. The VaR estimate for a probability q is calculated by

inverting the tail estimation of the above formula to give:

The expected shortfall is related to VaR by the following formula:

where the second term is the mean of the excess distribution over the threshold.

In practice, these can be calculated using the ‘riskmeasures’ function of the EVIR

package in R.

Application of EVT

We now turn to three examples that illustrate how to fit the GPD to

financial data and produce a model for extremes beyond a threshold. The raw

data for these examples are the historical prices of IBM, Ford, and Nortel stocks

which were downloaded from:

IBM stock: http://finance.yahoo.com/q/hp?s=IBM

Ford stock: http://finance.yahoo.com/q/hp?s=F

Nortel stock: http://finance.yahoo.com/q/hp?s=NT

The data analyzed however are historical daily returns price. The returns

price for each day was calculated by the formula:

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Returns price = (today’s price – yesterday’s price)/yesterday’s price

Clearly, the returns price is a measure of daily gain or loss in stock price

regardless of the actual stock price. Modeling extremes of returns price is useful

because it can help risk managers determine what the maximum gain or more

importantly, maximum loss that can be incurred in one day.

Historical Daily Returns Prices of IBM

The time series for IBM ranged between January 2 1980 and March 26

2004. The first step in analyzing this data is to see if it can be satisfactorily

modeled by the Normal distribution. We do this by creating a normal quantile plot

and look for linearity:

Notice that the data near the endpoints deviates from linearity significantly.

This implies that the true distribution of the data is fat tailed and so the extremes

are not normally distributed. This provides us with the impetus to model the

extremes using EVT.

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Now we must determine the threshold beyond which we can define the

data as “extreme”. To do this, we must first construct a plot of the mean excess

function. Then we try to determine a high enough point beyond which the plot

looks linear but at the same time provides sufficient points for inference.

Furthermore, we will omit the three largest losses because they tend to distort

the plot6. The mean excess function for different possible thresholds is:

From this plot we can estimate the threshold to be approximately 0.06 because

the data seems to kink downward at this point. This represents the threshold for

daily gain in stock prices. Now we can fit the GPD to data beyond this threshold

by using the gpd function in the EVIR software. We get the following results using

likelihood methods of estimation:

Total Number of Data Points: 6118

Chosen Threshold: 0.06

Number of Points Exceeding Threshold: 44

6 As mentioned in McNeil, A. & Saladin, T. “The Peaks over Threshold Method for Estimating High

Quantiles of Loss Distributions” Departement Mathematik ETH Zentrum.

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Approximate Percentile at which threshold is located: 99th percentile

Parameter Estimate for Shape (ξ): -0.1376

Parameter Estimate for β: 0.0253

Variance-Covariance Matrix:

Shape Beta

Shape 0.0631 -0.0017

Beta -0.0017 5.406e-05

Now we must do diagnostic checks to see if the model is a good fit. The quantile

plot of the residuals is:

Since this quantile plot is approximately linear, we conclude that the GPD is a

good fit for this data. Thus extreme values (beyond 0.06) can be modeled by:

G(x) = 1 – (1 + (-0.1376)x/(0.0253))1/(0.1376)

We have just modeled the extremes for daily gain of IBM stock prices.

However, in many financial situations, we have greater concern with negative

values for daily returns because that implies that a loss is incurred. In order to

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analyze extreme negative values, we need to multiply the given data by -1 and

repeat the procedure outlined above.

We will now model the negative daily returns for IBM stock prices. The

modified mean excess function is:

From this plot we choose can choose 0.05 as a threshold. So -0.05 represents

the threshold for daily loss. Now we can fit the GPD to data beyond this threshold

by using the gpd function in the EVIR software. We get the following results using

likelihood methods of estimation:

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Parameter Estimate for Shape (ξ): 0.4689



Shape Beta

Shape 0.0566 -0.0005

Beta -0.0005 1.169e-05


plot of the residuals

is:

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Since this quantile plot is approximately linear, we conclude that the GPD

is a good fit for this data. Thus the negative extreme values (beyond -0.025) can

be modeled by:

G(x) = 1 – (1 + (0.4689)x/(0.0128))-1/(0.4689)

In risk management, we are interested in estimates of VaR and Expected

Shortfall for different p-values of this model. In statistical language, VaR is simply

a quantile estimate7. These are easily found for the above model using the EVIR

software:

p-value Estimate of VaR Estimate of Expected Shortfall

0.99 0.04658902 0.0676375

0.999 0.09292915 0.1548897

0.9999 0.22934078 0.4117342

0.99999 0.63089623 1.1678084

Historical Daily Returns Prices of Ford

Historical Prices of Ford Stocks was downloaded from Yahoo Finance.

The time series ranged from January 2 1987 to March 26 2004. The returns price

was calculated and forms the data for the present analysis. To see if the data has

a fat tail, we plot the normal quantiles for this data:

7 McNeil, A. “Extreme Value Theory for Risk Managers” pg 7.

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We see that the tail curves slightly away from the straight line, indicating

deviation from normality. For confirmation, we also plot the empirical distribution

function of the data on the log-log scale. A straight line on the double log scale

implies Pareto tail behavior8:

8 EVIR help document.

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We see that the tail is approximately linear. So we are now justified in fitting

the GPD to the tails. As before, we must first find the threshold by plotting the

Mean Excess function. Furthermore, we will omit the three largest losses

because they tend to distort the plot9:

9 As mentioned in McNeil, A. & Saladin, T. “The Peaks over Threshold Method for Estimating High

Quantiles of Loss Distributions” Departement Mathematik ETH Zentrum.

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We see that the graph is approximately linear beyond 0.06; so we will choose

this as our threshold. This represents the threshold for maximum possible gain

per day. Now we will fit the GPD for the data beyond this threshold using the gpd

function of the EVIR package. We get the following results using likelihood

methods of estimation:








Shape Beta

Shape 0.03637 -0.0003453

Beta -0.0003453 1.306e-05

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Now we look at the quantile plot of the residuals to see if the GPD model is a good fit:

The quantile plot seems to be approximately linear, indicating that we have found

a good fit for this data. So we conclude that negative extreme values (beyond

0.03) can be modeled by:

G(x) = 1 – (1 + (0.4607)x/(0.0158))-1/(0.4607)

The above model corresponds to daily gain of Ford stock prices. In order

to analyze extreme negative values, we need to multiply the given data by -1

and repeat the procedure outlined above. We will now model the negative daily

returns for Ford stock prices. The last 3 data points in the mean excess function

are omitted as before. The mean excess function is now:

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From this plot we can estimate the threshold to be approximately 0.05. So

-0.05 represents the threshold for daily loss. Now we can fit the GPD to data

beyond this threshold by using the GPD function in the EVIR software. We get

the following results using likelihood methods of estimation:







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Shape Beta

Shape 0.02022 -1.619e-04

Beta -1.619e-04 4.577e-06




good fit for this data. Thus negative extreme values (beyond -0.05) can be

modeled by:

G(x) = 1 – (1 + (0.2646)x/(0.01145))-1/(0.2646)


Shortfall for different p-values of this model. These are easily found for the above

model using the EVIR software:

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p-values Estimate of VaR Estimate of Expected Shortfall

0.99 0.05343 0.0702

0.999 0.09260 0.1235

0.9999 0.1646 0.2214

0.99999 0.2971 0.4015

Historical Daily Returns Prices of Nortel

Historical Prices of Nortel stocks were downloaded from Yahoo Finance.

The time series ranged from December 16 1991 to March 26 2004. The returns

price was calculated and forms the data for the present analysis. To see if the

data has a fat tail, we plot the normal quantiles for this data:

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Notice that the data near the endpoints deviates significantly from linearity.

This implies that the true distribution of the data is a fat tailed distribution and so

the extremes are not normally distributed. This provides us with the impetus to

model the extremes using EVT.

To choose the threshold we now consider the plot of the mean excess

function:

From this plot we estimate the threshold to be approximately 0.105. This

represents the threshold for the maximum possible gain everyday. Now we can

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fit the GPD to data beyond this threshold by using the gpd function in the EVIR

software. We get the following results using likelihood methods of estimation:





Parameter Estimate for Shape (ξ): -0.1379



Shape Beta

Shape 0.0329 -0.0017

Beta -0.0017 0.0001349

Now consider the following quantile plot of the residual to verify whether the

aforementioned model is a good fit for the data:

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The quantile plot seems to be approximately linear, indicating that we have found

a good fit for this data. Thus we conclude that negative extreme values (beyond

0.105) can be modeled by:

G(x) = 1 – (1 + (-0.1379)x/(0.04853))-1/(-0.1379)

The above model corresponds to the extremes for daily gain of Nortel

stock prices. In order to analyze extreme negative values, we need to multiply

the given data by -1 and repeat the procedure outlined above. We will now model

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the negative daily returns for Nortel stock prices. The modified mean excess

function is:

From this plot we can estimate the threshold to be approximately 0.07. So -0.07

represents the threshold for daily loss. Now we can fit the GPD to data beyond

this threshold by using the gpd function in the EVIR software. We get the

following results using likelihood methods of estimation:




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Shape Beta

Shape 0.02658 -6.342e-04

Beta -6.342e-04 3.123e-05



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good fit for this data. So we conclude that negative extreme values (beyond

-0.03) can be modeled by:

G(x) = 1 – (1 + (0.2601)x/(0.0285))-1/(0.2601)


Shortfall for different p-values of this model. These are also easily found for the

above model using the EVIR software:

p-values Estimate of VaR Estimate of Expected

Shortfall

0.99 0.09933 0.1482

0.999 0.2132 0.3021

0.9999 0.4206 0.5823

0.99999 0.7980 1.092

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R Software and Code Used to Model Data

First of all, the EVIR and EVD packages must be downloaded and

installed into the R program in order to utilize functions pertaining to EVT. These

can be procured for free from the following website:

http://www.maths.lancs.ac.uk/~stephena/software.html

The following is a generic version of the specific code used to do model the three

cases above.

//Imports the EVD and EVIR packages needed for modeling.

>library(evd)

>library(evir)

//Reads the file “data.txt” containing data to be modeled.

>data = scan(“data.txt”)

//Plots the sample normal quantiles against the theoretical normal quantiles

>qqnorm(data)

//Adds a line to the normal QQ-plot.

>qqline(data, col=2)

//Plots empirical distribution of data on a log-log scale

>emplot(data, alog = “xy”, labels=TRUE)

//Creates the plot of the mean excess function for the data.

>meplot(data, omit=3, labels=TRUE)

//Fits the GPD for data beyond threshold t, and uses maximum likelihood method to

//estimate parameters.

>FittedData=gpd(data, threshold = t, method = c(“ml”))

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//Displays the details of the GPD fit to the data.

>FittedData

//Provides 4 different plots to assess the fit of the GPD model. The user can choose the

//requisite plot from a menu.

>plot.gpd(FittedData, labels=TRUE)

//Once data is fitted, calculates estimates of quantiles and expected shortfall for the model

//for a given vector of probability levels p.

>riskmeasures(FittedData, p)

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Conclusion

EVT is here to stay as a technique for the risk managers toolkit. Whenever

the tails of probability distributions are of interest, it is natural to consider applying

the theoretically supported techniques of EVT. Methods based around the

assumptions of normal distributions are likely to underestimate tail risk. Although

not perfect, EVT provides the best available models to predict extreme events.

In the second part of the study, it was shown that EVT can be used to

successfully model the daily returns of the stock prices. It illustrates how EVT can

be used as a day-to-day exploratory risk management tool.

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References

1. Bensalah, Younes. (November 2000) “Steps in Applying Extreme Value

Theory to Finance: A Review” Bank of Canada Working Paper 2000-20.

2. Coles, Stuart. An Introduction to Statistical Modelling of Extreme Values.

London: Springer, 2001.

3. Embrechts P., Klüppelberg C., & Mikosch T. Modelling Extremal Events

for Insurance and Finance. Heidelberg: Springer-Verlag, 1999.

4. McNeil, A. & Saladin, T. (April 24, 1997), “The Peaks over Threshold

Method for Estimating High Quantiles of Loss Distributions” Departement

Mathematik ETH Zentrum.

5. McNeil, A. (May 17, 1999), “Extreme Value Theory for Risk Managers”

Departement Mathematik ETH Zentrum.

6. Stephenson, A. (2003), “EVD Documentation”. Documentation for

Extreme Value Distributions package for R Statistical Program.

(http://www.maths.lancs.ac.uk/~stephena/software.html)

7. Stephenson, A. (2003), “EVD Documentation”. Documentation for

Extreme Value Distributions package for R Statistical Program.


8. Stephenson, A. (2002), “EVIR”. Documentation for Extreme Value In R

package for R Statistical Program.


Applications Of Evt In Financial Markets

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