Applications Of Evt In Financial Markets

Abstract

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

1 of 34

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.

2 of 34

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.

3 of 34

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:

4 of 34

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.

5 of 34

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

6 of 34

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.

7 of 34

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,

8 of 34

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:

9 of 34

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.

10 of 34

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:

11 of 34

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.

12 of 34

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.

13 of 34

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

14 of 34

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:

15 of 34

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:

16 of 34

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.

17 of 34

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.

18 of 34

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.

19 of 34

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

20 of 34

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:

21 of 34

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:


22 of 34

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:

23 of 34

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:

24 of 34

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

25 of 34

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:

26 of 34

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

27 of 34

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:


28 of 34

Shape Beta
Shape 0.02658 -6.342e-04
Beta -6.342e-04 3.123e-05


29 of 34

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

30 of 34

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”))

31 of 34

//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)

32 of 34

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.

33 of 34

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.

34 of 34

Applications Of Evt In Financial Markets

Recommended

Recommended

More Related Content

What's hot

What's hot (18)

Similar to Applications Of Evt In Financial Markets

Similar to Applications Of Evt In Financial Markets (20)

Applications Of Evt In Financial Markets