HELSINKI UNIVERSITY OF TECHNOLOGY SYSTEMS ANALYSIS LABORATORY DEPARTMENT OF ENGINEERING PHYSICS AND MATHEMATICS MAT-2.108 INDEPENDENT RESEARCH PROJECT IN APPLIED MATHEMATICS

Extreme Value Theory in Risk Management for Electricity Market

6.5.2007

Kimmo Lehikoinen 58016L

1 Introduction ........................................................................................................................ 1

2 Electricity Market Structure ............................................................................................... 3

2.1 Electricity Price Formation ........................................................................................ 3

2.2 Conventional Spot Price Forecasting Models ............................................................ 5

3 Extreme Value Theory ....................................................................................................... 8

3.1 Outline of Extreme Value Theory.............................................................................. 8

3.2 Peaks Over Thresholds............................................................................................. 10

3.3 Other EVT Methods ................................................................................................. 12

3.4 Validity Assessment................................................................................................. 13

3.4.1 Quantile-Quantile Plot...................................................................................... 13

3.4.2 Mean Excess Plot ............................................................................................. 14

3.4.3 Other Validation Methods................................................................................ 15

4 Electricity Risk Management ........................................................................................... 16

4.1 Forecasting Electricity Prices................................................................................... 16

4.2 Application of EVT to Risk Management................................................................ 18

4.2.1 Extreme Value Theory and Value-at-Risk ....................................................... 18

4.2.2 Other Extreme Value Theory Risk Management Applications ....................... 20

5 Discussion and Conclusion .............................................................................................. 22

6 References ........................................................................................................................ 24

1

1 Introduction

Traditionally electricity distribution has been highly regulated or monopolized market around

the world, but since the 1990s the market structure has changed dramatically. The power

sector in many European countries and in North and South America has undergone reforms

that have liberated the markets to allow competitive electricity prices. The market efficiency

implies prices based more on the law of demand and supply and marginal costs of production,

as opposed to the pre-reform era when electricity prices mostly reflected the governmental

social and industrial policy. However, the liberalization of the electricity markets resulted to a

new element of risk through price uncertainty, which is a significant factor due to the nature

of electricity as a product. (Bunn and Karakatsani, 2003)

Electricity wholesale markets are very liquid compared to the markets of many other

commodities, but electricity is effectively more like a service than a commodity by nature.

This is due to the physical limitations related to the production, consumption, and delivery of

electricity, which make electricity an “instantaneous” product. The instantaneity is due to the

fact that electricity can not be stored1, but the production and consumption need to be

perfectly synchronized. Thus, the demand and supply have to be in perfect balance at all

times, which in turn affects the electricity prices considerably, as sudden large peaks and

drops in demand and supply are reflected directly to the prices. Accordingly, the electricity

market is known to exhibit exceptionally high price volatility and a substantial number of

extreme prices in the efficient electricity markets (Byström, 2005).

The extreme electricity price changes expose the producers and retailers to significant risks,

which they can hedge provided they have good enough price modeling tools and they can

forecast the price movements well enough. There exists a wide variety of mathematical

approaches to price modeling and forecasting, but often they perform poorly in case of

extreme events. Extreme Value Theory (EVT) can be used to capture extremities better than

conventional time series models (Byström, 2005), as it concentrates on the observations that

exceed certain limit; the attention focuses on the tail of the price distribution. The main

objective of EVT is to provide asymptotic models that can be explicitly used to model the

tails of a distribution, whereas the theory does not attempt to model the entire distribution.

1 Electricity storage in small proportions is physically possible, but not in bulk. In consideration of the whole electricity market, the amounts that could be stored are of no significance.

2

The foundations of EVT are based on the publication by Fisher and Tippett (1928) on the

extreme value theorem by Gnedenko (1943), but there has been an abundance of research on

EVT recently.

This paper introduces the characteristics of an efficient electricity market, and discusses the

factors affecting the electricity prices. The complexity of electricity price forecasting is

considered with different existing models for it, as well as implications of inaccurate forecasts

are presented. Extreme Value Theory properties are reviewed and explored, and the intuitive

interpretations of the theory are considered. The applications of EVT to forecasting extreme

changes in electricity prices are introduced, and the uses of EVT in sound risk management

are acknowledged. The major objective of this paper is also to conduct a literature review of

the research on EVT in managing electricity price related risks. This provides risk managers

with additional insight on the EVT and how it can be used to better anticipate and hedge risks

that stem from electricity markets.

Section 2 discusses the structure of the electricity markets in more detail. The price formation

and the factors behind the prices are considered, and also the most common models for

forecasting spot prices are reviewed. An overview of the Extreme Value Theory is carried out

in section 3, and the Peaks Over Thresholds method along with other EVT methods are

introduced. Section 4 concentrates on the EVT-based risk management applications in the

electricity markets. Section 5 discusses the usage and benefits of EVT in electricity market

risk management, and concludes.

3

2 Electricity Market Structure

The electricity markets in this paper are assumed to be fully liberalized, so that they are not

regulated and that perfect competition prevails. Under perfect competition the prices are

determined entirely by demand and supply, and the prices reflect marginal costs. However,

the emergent power exchanges are somewhat incomplete, and most of the reformed electricity

markets can not be regarded as fully liquid in the financial sense (Bunn and Karakatsani,

2003). Still Bunn and Karakatsani (2003) argue that electricity price models are far richer in

structure than the models in other commodities. Ultimately, the large electricity consumers

and producers are mainly concerned about the high and unexpected increases and decreases in

electricity prices and loads, respectively. The interest in more accurate risk management and

worst-case scenario evaluation stresses the importance of superior quality price forecasting

models (Byström, 2005), for which Extreme Value Theory has produced good results.

Conventional forecasting models are criticized and many authors acknowledge that accurate

price forecasts are difficult to come of in electricity markets (see for example the results by

Green and Newbery, 1992; Skantze et al., 2000; Day et al., 2002).

2.1 Electricity Price Formation

The electricity exchanges (e.g. German EEX, Scandinavian Nordpool2) quote electricity spot

prices as well as derivatives prices for futures, forwards and options, so the type of price

needs to be determined for the considerations. Typically, the spot prices function both as an

indicator of the current price and as an underlying parameter for the derivative prices. Hence,

electricity prices are considered as true spot prices in the following analysis, although they are

in practice short-term futures prices because spot markets are defined on hourly or half-hourly

intervals3. In practice, the market equilibrium spot prices are determined one day ahead by

matching the demand and supply curves, which are derived from the bid and offer prices

given by the market participants (Vehviläinen and Pyykkönen, 2005).

The instantaneous nature of electricity trading suggests spot price formation to be simple

based on only the demand and supply at all times. However, the spot price time series exhibits

2 Nordpool was established in 1996 and it is the first multinational power exchange. 3 Most exchanges quote hourly spot prices (e.g. Nordpool in Scandinavia, EEX in Germany, and APX Group in the UK, the Netherlands and Belgium), while some use half-hourly spot price intervals (e.g. NEMMCO in Australia).

4

larger fluctuation than what is expected from the demand as observed by Bunn and

Karakatsani (2003) and Byström (2005). Thus, it can be concluded that the spot price time

series is very rich in structure and shows more complex behavior than reflection of marginal

costs of production. Many authors have also analyzed country or area specific electricity price

data, and it is obvious that price formation has its own area specific factors present in each

electricity market (see for example Green and Newbery, (1992); Bunn and Karakatsani

(2003); Bunn (2004); Byström (2005); Vehviläinen and Pyykkönen (2005)). Typical

characteristics of electricity spot prices are strong seasonality effects on intra-day, weekly,

yearly, and other levels, rapidly reverting extreme price spikes, and distributional skewness

and leptokurtosis. The prices are also prone to calendar effects, and there is a strong evidence

of heteroscedastic volatility and mean-reversion effects across the international electricity

markets (Escribano et al., 2002).

The most relevant elements that have impact on the characteristics of price formation are

discussed by Bunn and Karakatsani (2003). One of these elements is that the market-clearing

price at different demand levels will be set by different plants that differ in technology and

efficiency, and the changing efficiency causes variability to the prices due to different plant

operation costs. Another plant-based factor intensifying the price peaks is the existence of

peakload power plants that are for answering to the peaks in demand, and accordingly their

operation costs, which are reflected to the price, are much higher than those of a baseload

power plant. The spot price fluctuation is also explained by congestions in the transmission

grid, plant failures, and unanticipated demand changes. In addition, the power markets are not

efficient in reality, as few dominant players typically cause them to function like an oligopoly,

because they can have impact on the prices. This indicates price distortion compared to

perfect competition, and the prices are higher and more volatile than they would be without

these effects (Bunn and Karakatsani, 2003).

The strong seasonality factor in the electricity spot prices is strongly dependent on the

geographical area, because the level of temperature has a substantial impact on the demand of

electricity and the means of power production varies by areas and climates significantly.

Seasonality results mostly from the fact that electricity is a non-storable commodity. The

regional temperature effects reflect in the price when for example the power needs for heating

increase substantially in the winter months in the Nordic markets (Vehviläinen and

Pyykkönen, 2005). A similar effect of power use increase can be seen in other parts of the

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world during summer, when the use of air-conditioning builds up (Byström, 2005). Byström

(2005) proposes the short-term seasonality (i.e. over a day or a week) to be directly related to

the level of demand, and he states demand of electricity to be higher during the day than

during the night as well as it being higher during the week than during the weekend.

Vehviläinen and Pyykkönen (2005) describe the distribution of power production in the

Nordic market as follows. The hydro-based production covers over half of the annual

production, while approximately one fourth is generated with nuclear power, and the rest is a

mixture of production types (e.g. wind power, solar power, industrial and municipal combined

heat and power). In other markets, the production structure can be totally different. It is

obvious that the forces of nature have impact on the power production capabilities (e.g. a dry

summer can cause depletion in the hydro-based production), and they can fluctuate the prices.

2.2 Conventional Spot Price Forecasting Models

Conventionally, the price forecasting models can be roughly divided into two categories,

which are referred to as statistical and fundamental models (Vehviläinen and Pyykkönen,

2005). Statistical models are determined by a set of parameters, which describe the price

formation process. The prices are modeled directly and the parameters are estimated from

historical data. Fundamental models are competitive electricity market equilibrium models, in

which the prices are obtained for certain expected marginal costs of production and expected

power consumption. Vehviläinen and Pyykkönen (2005) propose that the statistical models

function best for short-term forecasting due to the complexity of modeling the long-term

dynamics, for which the relatively small set of parameters is often insufficient. They criticize

fundamental models to be too burdensome to use and to require too comprehensive a data set,

which may not always be available. Vehviläinen and Pyykkönen (2005) also introduce a

stochastic factor model, which combines the advantageous features from both modeling

approaches.

An extensive research on the existing methodologies for electricity price forecasting is made

by Bunn and Karakatsani (2003). They find on the basis of comparative research in several

deregulated markets that a price forecasting model should at least have two crucial features,

the jump-diffusion component and the mean-reversion component. For a more detailed

description, Merton (1976) introduces the random walk jump-diffusion model, and Johnson

and Barz (1998) establish the mean-reversion property. This type of model assumes necessary

6

characteristics of financial price dynamics, but contradicts empirical evidence (Bunn and

Karakatsani, 2003). Different stochastic models for spot price forecasting include a random-

walk model, a mean-reversion model, and a mean-reversion model with jumps, which can be

defined through constant (e.g. Poisson-process) or time-varying (e.g. GARCH-model)

parameters. Bunn and Karakatsani (2003) conclude the fundamental models, which they call

parsimonious, to be incapable of electricity price modeling and they prefer statistical models,

which they call structural models, instead. They provide an extensive list of relevant authors,

who have contributed to structural models, and compare parametric and non-parametric

models. An example of a parametric model is a simple regression model to lagged price and

demand as suggested by Nogales et al. (2002), while non-parametric models consist of such

approaches as neural networks and genetic algorithms. Bunn and Karakatsani (2003) list some

research done on neural networks in real electricity markets in different regions and they

conceive the flexibility of the approach, but still conclude the non-parametric approaches to

be insufficient for risk management purposes.

The different fundamental factors that affect the spot prices are categorized into Table 1, as

they have been defined by Bunn and Karakatsani (2003). The categorization can provide

additional insight to the variance decomposition for the different structural effect factors.

Furthermore, Table 1 provides a comprehensive list of factors whose effect to electricity spot

prices should be taken into account.

Table 1. Factors that have impact on spot prices. (Bunn and Karakatsani, 2003)

Structural Effect Influencing Factors

Market MechanismDemand Polynomial, Capacity Margin, Fuel Prices, Demand Slope, Demand Curvature, Demand Volatility, Forward Prices

Market StructureMargin, Concentration of Indices, Ratio of Margin to Demand Forecast

Non-Strategic Uncertainties Demand Forecast ErrorEfficiency Variables Trading Volume, Availability of Indices

Behavioral VariablesLagged Price, Lagged Daily Average Price, Price Volatility, Demand Volatility, Lagged Price Spread, Time

Time Effects Daily, Weekly, Seasonal

The authors conclude the parametric structural approach to be suitable for spot price

forecasting, but list the following unresolved modeling issues, which can cause problems:

• The set of factors that affect prices, for example economic fundamentals, production

constraints, strategic behavior, observed risks, forward contracting, trading

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inefficiencies and market design effects, can be incomplete. Also the magnitude,

relative importance and intra-day variation of these impacts is not fully captured.

• The modeling of non-linearities and dynamics of structural effects should be

improved.

• Volatility response to fundamental properties and price shocks can be incomplete, and

the sources of residual uncertainty, when the structural effects are removed, require

research.

• How to capture the market drivers of regime switching and the characteristics of

market cycles and effects of plant failures.

• It is difficult to capture the impact of electricity prices to other commodity prices,

which in turn affect electricity prices. The occasional congestion problems in the

power networks need further research about how they would be modeled.

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3 Extreme Value Theory

The mathematical foundation of Extreme Value Theory is the class of extreme value limit

laws, which were first derived heuristically by Fisher and Tippett (1928). Later Gnedenko

(1943) derived the same laws from a rigorous standpoint. However, EVT is adopted for risk

management and for financial market calculations only recently, although the theory has

longer roots in insurance applications. A comprehensive introduction to EVT is given by

Embrechts et al. (1997) and by Falk et al. (1994), while a lighter overview of the theory is

given by McNeil and Frey (2000). For a thorough understanding, especially Embrechts et al.

(1997) is worth reviewing. In the context of this paper, mostly one-dimensional EVT is

considered, but for the multivariate analysis Smith (2000) provides a good introduction.

McNeil et al. (2005) provide a very recent and well formulated introduction to EVT.

3.1 Outline of Extreme Value Theory

The setting for the outline of EVT begins with an assumption that we have a sequence of

independent and identically distributed (i.i.d.) observations 1, , nX X… from an unknown

distribution F . The distribution function is ( ) PrF x X x= ≤ and we define

1max , ,n nM X X= … , whose behavior is under investigation when n →∞ 4. The i.i.d.

condition of the sequence 1, , nX X… implies that ( )F x converges towards zero or unity

when n approaches infinity (de Rozario, 2002). Let 0x be the right, or upper, endpoint of

distribution ( )F x , for which we have ( ) 0 sup : 1x x F x= ∈ < ≤ ∞ .

The previous analysis does not reveal anything about the distribution of nM for large n , but

Gnedenko (1943) proved that there exist constants ,n na b ∈ , for which , 0n na b > , so that we

can find a non-degenerate limiting distribution ( )G x such that

( ) ( )lim Pr limn nn nn n

n

M b x F a x b G xa→∞ →∞

⎧ ⎫−≤ = + =⎨ ⎬

⎩ ⎭, (1)

4 This consideration is for the use of EVT in case of maxima, but an analogous consideration can be performed for minima.

9

which holds for each x . It can be now stated that ( )F MDA G∈ , which means that F is in

the maximum domain of attraction (MDA) of an extreme value distribution ( )G x . Fisher and

Tippett (1928) gave first proof that ( )G x is one of the three types of distributions: Frechet,

Weibull or Gumbel. Embrechts et al. (1997) formulate these three fundamental extreme value

limit laws mathematically so that ( )G x is one of the following distribution types:

Type I (Fréchet): ( ) ( )( )0 , 0

, 0x

xx

xeαα −−

⎧ ≤⎪Φ = ⎨ >⎪⎩ (2)

Type II (Weibull): ( )( )( ) , 0

, 01

x xexx

α

α

− −⎧ ≤⎪Ψ = ⎨ >⎪⎩ (3)

Type III (Gumbel): ( )( )( )xe

x e−−

Λ = (4)

where 0α > and x∈ . These three types of limit laws can be combined into a single

generalized extreme value distribution (GEV), which is originally proposed by von Mises

(1936). The GEV distribution function is the following:

( )1/

1 x

G x e

ξµξσ

ξ

−

+

⎛ ⎞−⎛ ⎞⎜ ⎟− +⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠= , (5)

where ( )max 0,y y+ = , ,µ ξ ∈ and 0σ > . Parameter ξ is known as the tail index and it

determines the three fundamental types, because (5) reduces to type I when 0ξ > and

1/α ξ= , to type II when 0ξ < and 1/α ξ= − , and to type III when 0ξ = .

The approach of classic EVT is to fit one of the extreme value limit laws, (2)-(4), to the

annual maxima of a series. Alternatively, the GEV can be directly fitted to the sample

maxima. When the necessary distribution parameters are estimated, equation (5) can then be

used to forecast extreme events. However, from modern perspective, the classic approach is

too narrow to be applied to the existing range of different extreme value problems (Smith,

2000). The quantile estimates from the classic approach are also unreliable (Lauridsen, 2000).

10

3.2 Peaks Over Thresholds

A more modern and popular EVT method than classic EVT is known as peaks over thresholds

(POT) or excesses over thresholds (see for example Embrechts et al., 1997; McNeil and

Saladin, 1997; Embrechts et al., 1999; McNeil and Frey, 2000; Smith, 2000; de Rozario,

2002). The setting in this method is such that we are only interested in observations that

exceed certain high threshold, u . Now the distribution function of excesses is conditioned on

the observations to be higher than u 5, and the distribution function of the excesses over the

threshold can be defined by

( ) ( ) ( )( )

Pr1u

F x u F uF x X u x X u

F u+ −

= − ≤ > =−

, (6)

for 00 x x u≤ ≤ − . The interpretation of (6) is that it tells the probability of an excess over the

threshold but by no more than amount x (McNeil and Saladin, 1997).

A useful distribution in modeling the excesses is the generalized Pareto distribution (GPD),

which can be expressed as

( )( )

,

1/

/

1 1 0

1 0x

xH x

eξ σ

ξ

σ

ξ ξσ

ξ

−

−

⎧ ⎛ ⎞− + ≠⎪ ⎜ ⎟= ⎝ ⎠⎨⎪ − =⎩

, (7)

where the scale parameter is 0σ > , and 0x ≥ when 0ξ > or 0 /x σ ξ≤ ≤ − when 0ξ < . The

GPD is a generalization in the same sense as the GEV, (5), because it reduces to other

distributions for certain values of the tail index ξ . In case of 0ξ < the GPD subsumes to the

uniform distribution, if 0ξ = the GPD equals to the exponential distribution, and the GPD

reduces to the usual Pareto distribution when 0ξ > (McNeil and Saladin, 1997). A location

parameter µ can be added to the GPD, when the family is extended and the GPD is defined

to be ( ) ( ), ,H x H xξ σ ξ σ µ= − . Now it can be seen that there is a fundamental relationship

between the GEV, (5), and the GPD, (7), because the tail index ξ is identical and there is a

5 In an analogous way to the earlier consideration of EVT, the POT method can be used for minima instead of maxima.

11

direct mathematical relationship between the scale σ and location µ parameters between the

two distributions (Davison and Smith, 1990).

The choice of the generalized Pareto distribution is supported by the Pickands (1975),

Balkema and de Hann (1974) theorem, which tells that the distribution of excesses can be

approximated by the GPD for sufficiently high thresholds u . Formally the theorem can be

written as

( ) ( )0 0

,0

lim sup 0uu x x x uF x H xξ σ→ ≤ ≤ −

− = (8)

if and only if uF is in the MDA of the extreme value distribution ( ),H xξ σ . The Pickands,

Balkema and de Haan theorem proves that we can set ( ) ( ),uF x H xξ σ= in equation (6), and

thus only parameters σ and ξ need to be estimated. The result from the Pickand, Balkema

and de Haan theorem is that we have the estimate:

( ) ( ) ( )( )ˆ ˆ,

ˆ ˆˆ1

F x u F uH x

F uξ σ

+ −=

− x u> . (9)

The POT method is used in practice so that a threshold value is chosen and the GPD is fitted

accordingly to the sample data of extreme observations that exceed the threshold. The

distribution parameter estimates can be derived for example by maximum likelihood

estimation (MLE)6. The parameters that need to be estimated for the GPD are the tail index ξ

and the scale σ , of which the tail index is the most crucial one as it sets the rate how quickly

the tails of ( ),H xξ σ decay away. Although the POT method provides good forecast results

(McNeil, 1997; de Rozario, 2000; Byström, 2005), the ultimate problem of the method is the

selection of the correct threshold, so that it is as high as possible with still a sufficient sample

data to be used for fitting the GPD (Embrechts et al., 1997). Tools for evaluation of goodness

of the fit are introduced later in this paper.

6 The maximum likelihood estimation is commonly used and straightforward method for maximizing the fit of a model to the sample data. De Rozario (2000) gives a compact introduction to MLE.

12

3.3 Other EVT Methods

In addition to the selection of the threshold, the open issue in the POT method is the treatment

of time series dependence. Smith (2000) proposes that the POT method is incapable of

handling stochastic volatility in financial time series. He claims a joint distribution of k

largest-order or smallest-order statistics from the classic annual maximum approach to better

respond to stochastic volatility. Smith (2000) extends the POT method to two dimensions to

capture the stochastic volatility effects so that the excess times and excess values are viewed

as a two-dimensional point process. If this process is stationary and there are no clusters

asymptotically, he concludes that the limiting form distribution for the process is non-

homogenous Poisson and the following equation holds:

( ) ( )1/

2 1 1 xA t tξµξ

σ

−

+

−⎛ ⎞Λ = − +⎜ ⎟⎝ ⎠

, (10)

where ( )1 2,t t is the interval under consideration. Smith (2000) extends the multivariate

analysis and derives the generalization of (10), which can be expressed by

( )2

1

1/

1tt

tt t

t t

xA dtξ

µξσ

−

+

⎛ ⎞−Λ = +∫ ⎜ ⎟

⎝ ⎠, (11)

where the extreme value parameters tξ , tµ and tσ are assumed to change periodically

according to a random change-point process. The distribution ( )t AΛ can then be fitted into

the sample data taking into account the excess times, and this approach should account for

stochastic volatility as well. However, the application of the time-dependent methods can be

computationally intensive.

Smith and Weissman (1999) introduce a family of multivariate maxima of moving maxima

(M4) processes that can be used for extreme event forecasting. They state that under fairly

general conditions a large number of extreme events in multivariate time series can be fitted

into one form of the M4 family. Specifically the multivariate maxima of moving maxima is

captured into equation

,1max maxij lkj l i kl k

M a X −≥ −∞< <∞= , (12)

13

where ,l i kX − consists of two vectors of independent unit Fréchet (2) random variables and lkja

are constants that form a positive linear combination

1

1lkjl k

a∞

==∑∑ 1,...,j p∀ = , 0lkja ≥ , (13)

and p determines the length of the time series sample.

3.4 Validity Assessment

The EVT has a variety of different approaches to model a certain extreme value sample.

Although the methods differ from each other by the underlying distribution, EVT approaches

are still parametric, as the EVT model is fitted into the sample of extreme outcomes. This

necessitates the validation of the method in use before it can be used in for example extreme

event forecasting. There are several widely adopted and well-functioning validation tools,

which are introduced in this section. These diagnostic tests concentrate on evaluating whether

the model assumptions are satisfied in practice, and the tests are complementary to each other.

3.4.1 Quantile-Quantile Plot

The quantile-quantile (QQ) plot is a graphical representation of the quantiles of the model

distribution against the ordered sample data. The sample data can be interpreted as an

empirical distribution, so the QQ-plot is an assessment of the correspondence between the

estimated model distribution and the data. This plot is a plausibility test of the GPD or another

EVT distribution for modeling excesses, and if the points of the plot form approximately a

straight line this indicates that the model is correctly specified (de Rozario, 2002). Assuming

the GPD is fitted, the plot consists of points

,1

,1, 1,...,

1k nn kX H k n

nξ σ−

⎧ ⎫⎛ ⎞− +⎪ ⎪⎛ ⎞ =⎨ ⎬⎜ ⎟⎜ ⎟+⎝ ⎠⎝ ⎠⎪ ⎪⎩ ⎭, (14)

where ,k nX is k th order statistic (e.g. ( )1, 1max ,...,n nX X X= and ( ), 1min ,...,n n nX X X= ).

The QQ-plot not forming a straight line indicates malformation in the GPD model fit with the

chosen parameters. In case of concave curvature in the QQ-plot, the GPD underestimates the

empirical distribution tail (i.e. sample data has heavy tails), and in case of convex curvature in

14

the QQ-plot, the tail is overestimated (i.e. sample data has short tails). Generally, the clearer

the indication of the QQ-plot is the more data points are used. The ease of use and intuitive

representation of the QQ-plot make it a useful tool. The QQ-plot can also be used to assess

the presence of heavy tails in the data, which contradicts the assumption of normality.

3.4.2 Mean Excess Plot

Another popular graphical diagnostic tool is the mean excess plot, which is a plot of mean of

all excess values over a threshold u against the threshold u itself. The mean excess plot

gives an indication of the goodness of fit of the GPD or some other EVT distribution like the

QQ-plot. As the selection of a proper threshold u is one of the crucial factors in EVT, the

mean excess plot also provides guidance for the suitable threshold. The mean excess plot

consists of points

( )( ) , 1,, n n nu e u X u X< < , (15)

where u is the threshold and ( )e u is the mean excess function, which is defined as

( )e u E X u X u⎡ ⎤= − >⎣ ⎦ . (16)

The mean excess function describes the expected overshoot of a threshold given it has been

breached, and in case of the GPD (16) has the exact form (de Rozario, 2002)

( )1

ue u σ ξξ

+=

−. (17)

The empirical estimate of the mean excess function is given by the sample mean excess

function (McNeil, 1997):

( ) ( )

1

11 i

ni i

n ni X u

X ue u = +

= >

−∑=

∑`, (18)

where 1iX u> is an indicator function that evaluates to 1 if iX u> and 0 otherwise, and ( )+⋅ is

as in (5). Thus, (18) is the sum of excesses over threshold divided by the number of excesses.

From (17) it can be concluded that the mean excess plot should be approximately a straight

line with slope ( )/ 1ξ ξ− .

15

The mean excess plot can be difficult to interpret because there are few excesses for large u ,

but it is readily used to detect significant shifts in slope trend at lower thresholds. Embrechts

et al. (1997) give a good explanation of the interpretation of the mean excess plots. The

upward sloping trend in the mean excess plot indicates a heavy tailed behavior, as the

downward sloping behavior is explained by a short tailed distribution, and a horizontal line is

linked to an exponential distribution (Embrechts et al., 1999). If the mean excess plot follows

a reasonably straight line with a positive gradient above a certain threshold u , this indicates

that the sample data follows the GPD with positive tail index.

3.4.3 Other Validation Methods

The two above introduced plots are the main diagnostics that are often used in validation of

the parameterization of the fitted GPD to the extreme data. Some other diagnostic tools that

do not directly assess the validity of the fit of the EVT distribution to the data have also been

developed.

A helpful tool both in validation of the fit and in selection of the threshold for the fitting is the

tail index ξ plot against the threshold u . This plot represents the maximum likelihood

estimates of ξ against either u or the number of excesses. In effect, the graphical

representation can point out which threshold value to select according to whether and where

the slope stabilizes. It can also give indication of the tail index, or shape parameter, ξ , which

is the key parameter in EVT, if the threshold is already chosen and fixed.

Embrechts et al. (1999) point out in their study the usefulness of the Hill-estimator, which can

be used in detection of heavy tails and in estimation of the tail index. Embrechts et al. (1997)

provide a proper introduction to the Hill-estimator. In practice, the Hill plot consists of data

points

( ) ( ) 1,

ˆ, 0 1, ,n kθ ξ θ θ θ ξ σ− ≤ ≤ = , (19)

where , , ,1

1ˆ log logk

n k j n k nj

X Xk

ξ=

= −∑ is the Hill-estimator and it is consistent to 2κ for the

ARCH process (see Embrechts et al., 1999). Dowd (1999) recognizes the good asymptotic

properties and ease of estimation of the Hill-estimator as benefits, although it is sensitive to

k .

16

4 Electricity Risk Management

Market risk arising from positions in electricity market products can cause significant losses,

especially if the positions are not hedged properly. Trading can be executed entirely back-to-

back, where all the long positions are covered by exactly similar short positions and vice

versa. However, this kind of trading can be expensive in practice, and result in losing

lucrative profits, which can be obtained by bearing some amount of risk. Other market risk

can arise, for example from foreign exchange rates and interest rates instead of commodity

prices (i.e. electricity prices), when the commodity risk is entirely hedged but there is cash

flow timing or currency mismatch between the long and short positions. This paper considers

mainly conventional trading positions, where there is also some commodity risk involved

from open positions. These positions often entail high market risk and price forecasting

accuracy and other risk management tools are of high importance.

The EVT can be used in market risk management stemming from investments in electricity

contracts. The earlier mentioned conventional EVT methods and especially the POT method

are useful in extreme price forecasting. These forecasts in turn can be used for developing

robust hedging strategies.

This section examines how Value-at-Risk (VaR) type risk measures, which are extensively

used in market risk management (Dempster, 2002), can be combined with the EVT. This

provides a different approach to calculate VaR. This section also examines GARCH

integrated EVT models that are used in electricity market risk management applications.

4.1 Forecasting Electricity Prices

The pricing and hedging of electricity derivatives are linked to each other and they are

typically more complex than conventional financial markets. Byström (2005) claims that

better modeling of the price distribution tails in the electricity market would improve pricing

of certain electricity derivatives (e.g. look-back options) compared to the conventional

normality based pricing model. A good overview of conventional electricity pricing models is

provided by Bunn and Karakatsani (2003). The complexity in electricity derivatives pricing

arises in the first place from the extreme behavior, substantial non-normality and large

volatility in electricity spot prices (Byström, 2005). This complexity implies higher risk

17

because of the related uncertainties and difficulties with calculating the sensitivities of the

price to different factors.

An EVT based electricity pricing model, which can be used for risk management purposes

also, is developed by Byström (2005). He introduces a combined time series and EVT model,

where the extreme observations are filtered by a time series process, and the POT method is

applied on the filtered data following McNeil and Frey (2000) closely. The method is a basic

AR(1,24,168)-GARCH(1,1) model (Byström, 2005):

0 1 1 2 24 3 1682 2 2 2

0 1 1 1 2 1

t t t t t t

t t t t

r t r rβ β β β σ η

σ φ φσ η φ σ− − −

− − −

= + + + +

= + +, (20)

where ( )~ 0,1t Nη or ( )~t tη ν , tr is a one period return at t , iβ and jφ are regression

constants, and 2tσ is the conditional variance. Now the time series model residuals tη are

modeled by the POT method using (9), where ( )ˆ1 F u− can be written as ( ) /un N n− , where

uN is the number of observations above threshold u and n is the total amount of

observations. Combination of these estimates with ( )ˆ ˆ, 0H x

ξ σ ξ ≠, we get from (9) (Byström,

2005):

( )ˆ1/ˆˆ 1 1

ˆuNF x u x

n

ξξσ

−⎛ ⎞

+ = − +⎜ ⎟⎜ ⎟⎝ ⎠

(21)

By setting x u y+ = and ( )ˆ 1F y α= − in (21), we get by inverting equation (21) an

expression for the unconditional tail quantiles:

1u

ny uN

ξ

ασ αξ

−⎛ ⎞⎛ ⎞⎜ ⎟= + −⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

(22)

Now Byström (2005) suggests that the conditional tail quantiles of the original return

distribution can be estimated from:

, 0 1 1 2 24 3 168t t t t ty r r r yα αβ β β β σ− − −= + + + + . (23)

18

This approach is one of the EVT based price forecasting methods. Byström (2005) provides

an extensive empirical study about the accuracy of the introduced method and he finds the

method to produce accurate estimates. The time series methods are inadequate for price

forecasting compared to the time series integrated EVT method introduced above, but

Byström (2005) concludes that the further inclusion of seasonality parameters into (20)

improves the accuracy of the price forecasts even more. He also states that none of the models

developed by him captures all the dynamics of electricity price changes.

4.2 Application of EVT to Risk Management

The foundation of the EVT risk management methods are based on the three introduced EVT

limit laws and often on the POT method especially. Generally, VaR is the most typical

implementation of a risk model using EVT. Embrechts et al. (1999) criticize VaR for it being

incoherent risk measure and they propose the usage of coherent Conditional Value-at-Risk

(CVaR), which is also known as Expected Shortfall (ES) and Expected Tail Loss (ETL),

instead. Longin (2000) discusses some relevant EVT models as well.

4.2.1 Extreme Value Theory and Value-at-Risk

Value-at-Risk is extensively used tool in risk management around the financial world

(Dempster, 2002), and it compactly portrays the risk in a portfolio or in a single asset into a

single figure. This measure is the worst expected potential loss in the market value of the

asset or the portfolio over a specified period of time with a given level of confidence.

Formally VaR is defined as

( ) ( )1t tVaR Fα α−= − Ω , (24)

where 1F − is the inverse of a given profit and loss (P/L) distribution F and is interpreted as

the quantile function F , α is the quantile of the inverse distribution function and tΩ is the

information set up to time t . Thus, VaR corresponds to the α th quantile of the P/L

distribution and can be expressed equivalently as

( )( )t t tF r VaR α α< − Ω = , (25)

19

where tr is the return in the P/L distribution. The negative sign in (24) and in (25) is due to

the fact that VaR is expressed as a positive figure describing the amount of loss by

convention.

There exists a wide variety of different approaches to calculating VaR as it is not model

dependent by definition. The EVT based VaR can be derived on the basis of already obtained

equation (21), as it can be written as

( ) ( )ˆ1/ˆˆ 1 1

ˆuNF x x u

n

ξξσ

−⎛ ⎞

= − + −⎜ ⎟⎜ ⎟⎝ ⎠

. (26)

Seeing the connection between (25) and (26), we can set ( )F x α= and solve for x by

inverting ( )F x , which gives

ˆ

ˆ1ˆ

u

nx uN

ξσ αξ

−⎛ ⎞⎛ ⎞⎜ ⎟= + −⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠. (27)

Now the definition of VaR (24) can be applied by remembering that (27) is the inverse of

( )F x , which results

ˆ

ˆ1ˆt

u

nVaR uN

ξσ αξ

−⎛ ⎞⎛ ⎞⎜ ⎟= − − −⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠, (28)

where VaR being a negative risk measure has been taken into account.

VaR can be readily measured with the means of extreme value theory and it can be applied to

electricity market risk management. The benefit of EVT based VaR is that it differs

fundamentally from the conventional approaches of VaR (e.g. delta-normal VaR, historical

simulation VaR etc.) and can capture extreme risks better. Historical evidence shows the

extreme nature of electricity prices and accordingly risk (de Rozario, 2002; Bunn and

Karakatsani, 2003; Byström, 2005), which indicates a strong need for a risk management

methodology that captures the extreme risks efficiently. The introduced EVT VaR is a tool for

this extreme environment and it can perform better than the conventional approaches of VaR

20

under extreme conditions. De Rozario (2002) and Byström (2005) provide empirical analyses

of the performance of EVT and the conventional approaches in risk management.

The selection of an appropriate threshold for EVT methods is the major problem in risk

management applications. Hall and Welsh (1985) provide an adaptive rule for the threshold

selection building on top of the goodness of fit procedure developed by Pickands (1975).

Embrechts et al. (1997) suggest that the estimates of ξ and σ are calculated using several

threshold values and then the goodness of the estimates is inspected and the best threshold is

chosen based on this. Thus, the selection of the threshold value is very data specific. Smith

(2000) provides a graphical tool for a proper threshold selection.

4.2.2 Other Extreme Value Theory Risk Management Applications

The EVT based price forecasting methods and Value-at-Risk are widely used EVT based risk

management methods for the electricity market. However, some other tools have also been

developed. McNeil (1999) discusses Block Maxima Models (BMM) for risk management and

provides some insight to Multivariate EVT (MEVT) in risk management.

McNeil (1999) outlines the use of BMM in risk management and describes the usage of these

models intuitively. The idea is somewhat similar to application of the three EVT limit laws

(2)-(4). In the BMM approach, the series of observations 1, , nX X… is divided into k blocks

of essentially equal size. Then block maxima are selected according to

( )1 2max , ,...,j j j jh hM X X X= , which is in effect the maximum of the block j that consists of h

observations. When all k blocks have a maximum, the GEV (equation (5)) is fitted to the

block maxima data 1 2, ,..., kh h hM M M . The fitted GEV can now be used for electricity price

forecasting or generating the estimated return distribution for a given portfolio of electricity

derivatives. Embrechts et al. (1997) provide also a detailed description of the block maxima

modeling theory and how to use BMM in practice.

Multivariate Extreme Value Theory (MEVT) is an extension to univariate EVT, which is the

main concern in this paper. However, MEVT can be useful in more complex settings,

especially if several risk factors show extreme behavior. Dempster (2002) provides a detailed

description of the application of MEVT and the theoretical framework. One way of using

MEVT is by combining univariate EVT with copulas. Copulas can be thought of functions

21

that map the values in a unit hypercube to values in the unit interval. In mathematical terms,

copula C can be written as

( ) ( ) ( )( )1 1 1,..., ,...,k k kF x x C F x F x= , (29)

where C is said to be the copula of F .

MEVT tools can be devised using different copulas to obtain multivariate distribution

functions. One possible MEVT approach is to take the POT tail estimate function (26) for

different risk factors i , when we have tail estimators of the form

( ) ( )ˆ1/ˆˆ 1 1

ˆ

i

iu ii i i i

i

NF x x u

n

ξξσ

−⎛ ⎞

= − + −⎜ ⎟⎜ ⎟⎝ ⎠

. (30)

The multivariate tail estimate function can now be obtained through a copula. McNeil (1999)

shows the multivariate tail estimate function in the simplest case of two risk factors, when

1, 2i∈ . The joint tail of the two risk factors can then be expressed as

( ) ( ) ( )( )1 2 1 1 2 2ˆ ˆ ˆ, ,GuF x x C F x F xβ= , (31)

where GuCβ is the Gumbel copula, which has a closed form representation

( ) ( ) ( )( )1/ 1/1 2log log

1 2,y yGuC y y e

ββ β

β

− − + −= 0 1β< ≤ . (32)

Copulas do not often have a simple closed form solution, but they must be written in other

ways. However, they are useful in MEVT applications. More about different copulas and their

usage can be found in Dempster (2002). In electricity market, the electricity price and load

have extreme events and thus MEVT may be of significant use. Also MEVT could be

beneficial in case of a pricing and risk management of a complex financial derivative, which

has another extreme risk factor included.

22

5 Discussion and Conclusion

The electricity market is more complex in terms of price formation than other more

conventional financial markets. The complexity arises from the fact that in practice electricity

can not be stored and the price formation is affected by a several factors, such as temperature

or electricity consumption patterns, which are not standard across the markets and which

effects are clear. As a result the load and price of electricity have spikes. These extreme

events mean that the risk involved in invested positions of electricity spot and derivative

contracts can be significant. Due to the extreme nature of the electricity prices, conventional

risk management methods may not produce good enough results and accuracy for having

reliable risk management practices. The Extreme Value Theory provides tools and solutions

for risk management in the electricity market.

The main objective of this paper is to give a clear picture how the EVT can be used in the

electricity market risk management. Therefore the foundations of EVT are considered in

detail. The three extreme limit laws and the generalized extreme value distribution lay the

outline for EVT. In practice, the most widely used EVT method is the peaks over thresholds

(POT) method. This method is considered as the main univariate EVT model.

The purpose of this paper is also to serve as a literature review on the extreme value theory in

electricity market risk management, and thus the references in this paper provide a collection

of relevant papers to EVT, which give additional insight to the theory and application of EVT.

The majority of academic research on EVT is conducted by relatively small number of

researchers and the work of these authors should be followed in the future for relevant

development of the EVT. Especially authors like Paul Embrechts, Alexander McNeil, Richard

Smith, Rüdiger Frey, Francois Longin and Hans Byström are relevant names in the

conjunction with the EVT applications in the electricity market.

The EVT methods are based on the idea of fitting certain distribution to the data of extreme

observations. The goodness of fit depends on the estimated parameters that define the shape

and behavior of the fitted distribution function. This indicates EVT to be case specific, which

emphasizes the role of backtesting in the result validation. It is extremely important to

backtest the fitted distribution function and to verify that the results comply with real world

outcomes (Embrechts et al., 1997; McNeil, 1999; McNeil and Frey, 2000; de Rozario, 2002).

23

The tools required for actual electricity risk management are the price forecasting methods

that produce reliable forecasts and accurate risk management methods that display the level of

risk in a portfolio of given electricity contracts. These tools are considered in this paper with

the main concern on the latter. A good price forecasting method, which uses an AR-GARCH

filtering with EVT component applied afterwards, is proposed by Byström (2005). This

method is found to produce reliable forecasts in practice, when Nordic Nordpool electricity

data is used (Byström, 2005). Value-at-Risk (VaR) is considered as the risk measure for a

portfolio of electricity derivatives. Other risk measure is suitable as long as the EVT is

adjusted to it. VaR is chosen in this paper as it is extensively used in practice. The risk

management application section covers briefly block maxima modeling as an alternative to

the conventional extreme value limit laws. Multivariate EVT is also considered in case other

extreme risk factors need to be modeled simultaneously.

Other risk measures than VaR (e.g. expected shortfall) can be modeled using EVT with some

small changes. The fundamentals of EVT still remain the same. For example, maximum loss

based scenario analysis can be easily carried out after fitting the generalized extreme value

distribution to the sample data or by applying the POT method. This paper points the future

research to concentrate on producing more standardized methods for estimating the EVT

distribution functions, so that the emphasis on the evaluation of goodness of the fit can be

reduced. The research by Byström (2005) also suggests further development of combined

time series and EVT methods for price estimation. Future research can concentrate on

applying the EVT to other areas than electricity market, for example credit risk modeling.

The overall purpose of this paper is to give a good overview of the extreme value theory and

to present the specialties of the electricity market. Understanding can be deepened through the

extensive references in this paper. This understanding is used to employ the tools from the

EVT to manage electricity market risk. The EVT risk management methods introduced in this

paper can be used in practical setting to manage risk related to electricity market spot and

derivative positions. Both univariate and multivariate approaches are considered in order to be

able to handle even the most complex situations.

24

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