COMPARATIVE ANALYSIS OF ELECTRIC ENERGY RISK CHANGES IN SELECTED EUROPEAN REGIONS

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Studia Ekonomiczne. Zeszyty Naukowe Uniwersytetu Ekonomicznego w Katowicach ISSN 2083-8611 Nr 344 · 2017 Informatyka i Ekonometria 12

Alicja Ganczarek-Gamrot

University of Economics in Katowice Faculty of Informatics and Communication Department of Demography and Economic Statistics alicja.ganczarek-gamrot@ue.katowice.pl

COMPARATIVE ANALYSIS OF ELECTRIC ENERGY RISK CHANGES IN SELECTED EUROPEAN REGIONS

Summary: In the paper volatility of prices from selected European electric energy mar- kets was described using multivariate autoregressive models VAR-GARCH. Quotations from Polish TGE, European EEX, Nordic Nord Pool and center Europe OTE were used from January 2014 to October 2016. We made risk analysis of change in average daily prices and proposed a portfolio of electric energy contract. The risk of single contracts was estimated by Value-at-Risk (VaR). The risk of portfolio was estimated by Condi- tional Value-at-Risk (CVaR).

Keywords: VAR, DCC, VaR, CVaR, portfolio analysis.

JEL Classification: C5.

Introduction

The aim of the paper is to assess the risk of volatility in the price of electricity on the Polish Power Exchange (TGE) compared to the risk of changes in prices on the neighboring electricity markets. Markets such as Nord Pool operating in Sweden, Norway, Denmark, Finland, Lithuania, Latvia, Estonia, as well as markets coordinated by the technical operator OTE in the Czech Republic, Slo- vakia, Hungary and Romania and the European Electric Exchange (EEX) trading of electricity in Germany, Austria, France and Switzerland were considered.

A comparative study was carried out on the basis of average daily electricity prices [EUR/MWh] in the period from 01.01.2014 to 10.30.2016. The average daily price on day-ahead electricity markets quoted in [EUR/MWh] were repre-

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Alicja Ganczarek-Gamrot 8

sented by the index POLPX spot base (POLPX) [www 1] on TGE and by spot index ELIX base (ELIX) [www 2] on EEX. For the Nord Pool spot market which covers very diverse countries in terms of level and volatility of prices all the publicly available average prices were taken into account. These prices include [www 3]:

– SYS – average system price for the whole Nord Pool,

– SE1 (Luleå), SE2 (Sundsvall), SE3 (Stockholm), SE4 (Malmö) – average daily prices for Swedish subregions,

– FI – average daily price for Finland,

– DK1 (Aarhus), DK2 (Copenhagen) – average daily prices for Danish subregions,

– Oslo, Kr.sand, Bergen, Molde, Tr.heim, Tromsø – average daily prices for Norwegian subregions,

– EE – average daily price for Estonia, – LV – average daily price for Latvia, – LT – average daily price for Lithuania.

Prices for markets using the services operator OTE are represented by sym- bols:

– CR– average daily price for Czech Republic, – SR – average daily price for Slovakia, – HU – average daily price for Hungary, – RO – average daily price for Romania.

1. Introductory data analysis

Distributions of average electric energy prices from 01.01.2014 to 30.10.2016 for selected markets are represented by box-plots in Fig. 1.

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Table 1 cont.

1 2 3 4 5 6 Kr.sand 0,9356 0,0500 0,0071 0,7378 0,3441

Bergen 0,9225 0,0466 0,0136 0,7335 0,3423 Molde 0,9545 0,0760 0,0592 0,8456 0,1541 Tromsř 0,9289 0,0127 0,0539 0,8039 0,0374

EE 0,5368 0,2625 0,6211 0,6228 0,4557 LV 0,0349 0,3078 0,8948 0,4778 0,4197 LT 0,0369 0,2976 0,8967 0,4678 0,4195 POLPX 0,1238 0,4139 0,5017 0,1997 0,5632

CR 0,2507 0,8726 0,2612 0,3072 0,8536 SR 0,1916 0,8810 0,2899 0,2394 0,8738 HU –0,0599 0,8532 0,2498 0,1445 0,8065

RO –0,0326 0,7735 0,1840 0,0592 0,6905 ELIX 0,3573 0,8055 0,1806 0,3556 0,8121

Variance 8,9747 4,3505 3,0824 7,9058 5,1671

% of Variance 0,4487 0,2175 0,1541 0,3953 0,2584 Source: Own research.

Fig. 4. Loadings for prices and changes of prices with normalized varimax rotation Source: Own research.

Molde SE4

SE2 SE3 Kr.sandOslo BergenTromsř DK2 DK1 FIEE CRELIX

SR POLPX LVLT

HU RO

SE2SE4SE3 FI

DK1

DK2 Oslo Kr.sand Bergen

Molde Tromsř LV EE

LT POLPX

SR CR HU RO

ELIX

0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0

F1 0,0

0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0

F2

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Comparative analysis of electric energy risk changes… 13

2. Metodology

The following definition of Value-at-Risk (VaR) will be utilized in analysis of multivariate time series [Jajuga, 2000; Weron, Weron, 2000; Heilpern, 2011].

VaR is defined as such loss of value, which is not exceeded with the given prob- ability

α

at the given time period Δt, and it is expressed by the formula:

P(Y_t₊_Δ_t ≤ Y_t – VaR_α(Y)) =

α

, (2) where:

Yt – is a present value,

t

Yt₊_Δ – is arandom variable, value of in.

VaRα in the given time period Δt=1 (one day), may by written as a percentile of the order

α

of dynamics indicators:

P(ΔY_t≤ −VaR_α(ΔY_t)) =

α

, (3) where ΔY_t is a dynamics indicators.

For a single contract VaR_α(ΔY_t) is

α

– percentile of dynamics indicators:

VaRα(ΔY_t) = – F_ΔY⁻¹(

α

). (4) Using variance-covariance methods we obtain:

VaRα(ΔY_t) = – F⁻¹(

α

)⋅

σ

_Δ_Y +

μ

_Δ_Y, (5) where:

) (

α

F⁻1 – is a

α

-quantile of standarized distribution,

ΔY

σ

– is a standard deviation of ΔY_t,

ΔY

μ

– is an expected value of ΔY_t.

If we treat ΔY_t as a random realization of non-stationary process, then it may by expressed by the equation:

t =

Δ

Y

μ

_t_Δ_Y +

ε

_t, (6)

t Y t

t

σ ξ

ε

= _Δ , (7) where:

Y tΔ

μ

– a time-dependent expected value of dynamics indices described by SARIMA model,

Y tΔ

σ

– a time-dependent standard deviation of dynamics indices described by GARCH model,

(8)

ε

t – the vector of heteroscedastic residuals,

ξ

t – white noise.

Then equation (5) can by written in the form:

α

VaRt (ΔY_t) = – F⁻¹(

α

)⋅

σ

_t_Δ_Y +

μ

_t_Δ_Y. (8) For m-dimensional matrix of absolute differences ΔY_mt, VaR_t_α(ΔY_mt) can written analogically:

t t

mt

t Y F x

VaR_α( Δ ) =− ⁻¹(

α

) x^TH +x^Tμ , (9)

where:

Ht – variance-covariance matrix of multidimensional process of absolute differences,

μt – vector of expected values multidimensional process of absolute differences,

x – vector of weights of individual contracts.

In portfolio analysis a coherent risk measure CVaR (Conditional Value-at- -Risk), given by formula (10), is applied more often than VaR [Artzner et al., 1999; Rockafellar, Uryasev, 2000]:

) (

CVaR_α ΔY_tm =E{ΔY_tm_t|ΔY_tm≤VaR_t_α( ΔY_mt) _t}. (10) Linear and non-linear models of single and multi-dimensional are discussed in many works [Osińska, 2006; Zivot, Wang, 2006]. Their applications to the financial market, the commodity market or the electricity market are described by [Weron, 2006; Fiszeder, 2009; Trzpiot, 2010; Krężołek, 2015].

SARIMA vector models incorporating autocorrelation and periodicity may be employed to describe multivariate time series. The m-dimensional processes of ΔY_t can be written as m-SARIMA model (Seasonal Auto-Regressive Inte- grated Moving Average), (p, d, q) × (P, D, Q) with period s [Brockwell, Davis, 1996]:

mt

mt B B Y

Y = − − Δ

Δ^d_s (1 )^d(1 ^s)^D , (11) where:

B – shift operator, d – integration rank,

D – seasonal integration rank.

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If ε_t~D(0, H_t ), then we can write:

t 0,5 t

t H u

ε = , (12) where:

Ht – m×m dimensional conditional variance-covariance matrix,

ut – m×1-dimensional vector with zero expected value and unit variance- -covariance matrix.

Multivariate GARCH family comprises models of conditional covariance matrix H_t. These include the generalization of a 1-dimensional model GARCH to form a multi-dimensional model Veche (p, q) [Kraft, Engle, 1983] and BEKK (p, q, M) [Engle, Kroner, 1995], factor models K-factor GARCH [Engle, 1987]

and O-GARCH (1.1 m) [Alexander, Chibumba, 1996], as well as models of constant conditional correlation coefficients (CCC) [Bollerslev, 1990] and dynamic conditional correlation coefficients (DCC) [Engle, Sheppard, 2001; Engle, 2002; Tse, Tsui, 2002]. In this study, the following model of [Engle, 2002] is applied:

t t

t DΓD

H = _t , (13)

)

;...;

( )

;...;

( ₁₁⁻⁰_,^,⁵ ⁻⁰^,_,⁵ ₁₁⁻⁰_,^,⁵ ⁻⁰^,⁵_,

=diag q _t q_mm_t Q_tdiag q _t q_mm_t

Γ_t , (14)

where:

) ,..., ,

( ₁_t ₂_t _mt diag

σ σ σ

t =

D – is a diagonal matrix with dimensions m × m. El- ements of this matrix are estimated using univariate GARCH models,

Γt – matrix of dynamic conditional correlation coefficient, )

( _ijt

t = q

Q – symmetric, positive definite matrix of dimension m

×

m:

1 '

1

) 1

1

( −α−β ⁻+α ₋ ₋ +β ₋

= _t _t _t

t Q u u Q

Q ,

it it

uit

σ

= ε ,

Q− – unconditional variance matrix of u_t, β

α, – positive parameters, α +β <1^, m – dimension of time series.

The autocorrelation coefficient between two time series in Engle’ DCC model is:

) ))

1 )(((

) 1 ((

) 1 (

1 1 2 1 1

2

1 1 1

− −

−

β + β + β

− α

− β

+ α + β

− α

−

β + α

+ β

− α

= − ρ

jt jjt jjt it iit

iit

ijt jt it ijt

ijt q u q q u q

q u u

q . (15)

(10)

1

i

w

v

3 g 16

ing

with

whe

μ

0

valu xi –

3. E gain

U opt

h re

ere:

– t ue w

– th

Em In ns o

Usin tim

estri

: the with he s

mpir n th of p

g a mizat

icti

exp h eq shar

rica e c rice

a mo tion

ions

pec qua re o

al a ons es o

odif n pr

s:

cted al sh of m

ana stru on f

fica robl

d po hare mark

alys ctio five

atio lem

ortfo e fo ket

sis on o

ma on o m m

E

folio or ev

in p

of t arke

Ali

of th may b m x

Δ E(

o va very por

the ets (

icja

he c be min

>

x_i

ΔY_m

alue y m rtfol

por (Fig

a Ga

clas form n→

,

>0

∑

= 5 i

mt )

e be mark lio.

rtfo g. 4

ancz

ssic mu

→| C , i

∑

= 5

1

x

=

∑

i

efo ket)

lio 4): S

zare

cal M late CVa

= i

i = x

5

∑

1

= i

x

re p ),

(16 SE4

ek-G

Ma ed:

( aR Δ

1 K, 1,

iΔ x por

6), c 4, D

Gam arko

Ym

Δ 5 K,

i ≥ Y rtfol

con DK1

mrot

owit

mt) Y

5,

μ

≥ lio

ntrac , PO

t

tz [

|,

μ0, rec

cts OLP

[195

cons

on PX

59]

stru

abs , HU

mo

uctio

solu U, E

ode

on

ute ELI

el, t

(the

ave IX w

the

e ex

erag wer

foll

xpe

ge d re u

low

(16

ecte

dail used

w-

6)

ed

ly d.

(11)

F S

A f s t

F S Fig.

Sour

AC for son tilit

Fig.

Sour . 5.

ce: O

Fi F a Ind nalit

ty e

. 6.

ce: O Tim Own

ig.

and dex ty, a ffec

Aut Own

me s rese

5 sh PA

PO auto ct. D

toco rese

C

serie earch

how ACF OLP oco Dis

orre earch

Com

es o h.

ws t F se PX.

orrel trib

elati h.

mpa

f dy

the erie Co lati buti

on a arati

ynam

tim es an

ons on.

ions

and ive

mic

me s nd ider

In s ar

dis ana

s in

seri a h red ser re le

strib alysi

ndic

ies histo d dif ries epto

butio is of

ator

of s ogr ffer s of oku

on o f ele

rs fo

sele ram

renc f dif urtic

of P ectr

or se

ecte gra ces ffer c an

OL ric e

elec

ed i aph ch enc nd e

PX ener

cted

ncr h an ara ces exhi

dyn rgy r

d ma

rem nd q

cter we ibit

nam risk

arke

ment qua rize

ob fat

mics k cha

et

ts, F antil

ed b bserv

t tai

ind ang

Fig.

le-q by h

ved ils.

dicat ges…

. 6 quan

hig d clu

tors

…

– th ntil

h v uste

s he f le d vola erin

feat distr atili ng o

ture ribu ty, of v

1

es o utio

sea vola 7

of n a- a-

(12)

To estimate a vector of expected values in multidimensional process of dynamics indicators μ_t, in VaR_t_α ( ΔY_mt) , given by formula (9), SARIMA models are used. In Table 2 results of vector-SARIMA model estimation are presented.

Not all parameters differ significantly from zero. The best precision in SARIMA model estimation we obtained for time series from Sweden. In Table 3 the diagonal matrix D_t resulting from the first-step estimation of H_tby DCC model is presented. In Fig. 7 components of D_t matrix are presented. The highest estimated variances were obtained for POLPX index. The lowest variances were obtained for Danish dynamic indicator.

Table 2. Parameters of SARIMA(1,0,1)(1,1,1)7 models

Markets SE4 - 1 DK1 - 2 POLPX - 3 HU - 4 ELIX - 5 Parame-

ters value p-value value p-value value p-value value p-value value p-value p(1) 0,5879 0,0000 0,4948 0,0000 0,4031 0,0000 0,1806 0,0006 0,6098 0,0000 q(1) 0,9335 0,0000 0,8869 0,0000 0,8433 0,0000 0,6719 0,0000 0,8204 0,0000 Ps(1) 0,1351 0,0001 -0,0411 0,2496 0,0833 0,0212 -0,0692 0,0816 0,0044 0,9125 Qs(1) 0,9604 0,0000 0,8993 0,0000 0,9009 0,0000 0,7377 0,0000 0,8270 0,0000 MS 22,3890 30,1750 49,3720 47,5990 25,6540 Source: Own research.

Table 3. Parameters of first-step DCC model estimation– diagonal GARCH(1,1) models Markets SE4 - 1 DK1 - 2 POLPX - 3 HU - 4 ELIX - 5 Parameters value p-value value p-value value p-value value p-value value p-value ARCH(1) 0,1447 0,0351 0,0432 0,0147 0,4274 0,0003 0,2649 0,1075 0,1381 0,1503 GARCH(1) 0,8930 0,0000 0,9591 0,0000 0,7335 0,0000 0,8067 0,0000 0,8891 0,0000 Source: Own research.

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Fig. 7. Conditional variances of matrix Dt

Source: Own research.

In Table 4 the matrix Γ_t resulting from the second-step estimation of H_t by DCC model is presented. In Fig. 8 components of Γ_t matrix are presented.

Correlations are dynamic and positive but not strong. The strong dependence was observed only between dynamic indicators from Nord Pool (SE4 and DK1).

In Table 5 solutions of optimization task (16) are presented for different values of

α

. The greater level of risk we observed for short position (CVaR) than for long one. CVaR of portfolio means how much portfolio can average change next day bringing the lost. The most appropriate contract in every model it is SE4. But number k of exceed VaR in every portfolio is much greater than allowable. In every case share of k in number of observations – w is greater than

α

level. It means, that obtained portfolios’ quintiles are underestimated.

1 85 169 253 337 421 505 589 673 757 841 925 1009

0 10 20 30 40 50

CondVar_SE4

1 85 169 253 337 421 505 589 673 757 841 925 1009

0 10 20 30 40 50

CondVar_DK1

1 85 169 253 337 421 505 589 673 757 841 925 1009

0 10 20 30 40 50

CondVar_POLPX

1 85 169 253 337 421 505 589 673 757 841 925 1009

0 20 40

CondVar_HU

1 85 169 253 337 421 505 589 673 757 841 925 1009

0 10 20 30 40 50

CondVar_ELIX

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Table 4. Parameters of second-step DCC model estimation – parameters of Γ_tmatrix

Parameters value p-value

rho_21 0,7294 0,0000

rho_31 0,3409 0,0000

rho_41 0,1728 0,0000

rho_51 0,4179 0,0000

rho_32 0,3600 0,0000

rho_42 0,1880 0,0000

rho_52 0,5242 0,0000

rho_43 0,2138 0,0000

rho_53 0,3779 0,0000

rho_54 0,2885 0,0000

alpha 0,0216 0,0000

beta 0,9046 0,0000

df 5,2289 0,0000

Source: Own research.

1 85 169 253 337 421 505 589 673 757 841 925 1009 -0,1

0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8

CORR_SE4_DK1

1 85 169 253 337 421 505 589 673 757 841 925 1009 -0,1

0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8

CORR_SE4_POLPX

1 85 169 253 337 421 505 589 673 757 841 925 1009 -0,1

0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8

CORR_SE4_HU

1 85 169 253 337 421 505 589 673 757 841 925 1009 -0,1

0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8

CORR_SE4_ELIX

1 85 169 253 337 421 505 589 673 757 841 925 1009 -0,1

0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8

CORR_DK1_POLPX

1 85 169 253 337 421 505 589 673 757 841 925 1009 -0,1

0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8

CORR_DK1_HU

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Fig. 8. Conditional correlations between daily indexes on selected markets Source: Own research.

Table 5. Solutions of optimization task (16) Posi-

tion Port-

folios α Portfolios share xi Portfolios parameters SE4 DK1 POL

PX HU ELIX C

VaR μ ki wi

short

P1 0,001 0,970 0,030 0,000 0,000 0,000 –3,688 0,0143 115 0,112 P2 0,005 1,000 0,000 0,000 0,000 0,000 –3,437 0,0142 179 0,174 P3 0,01 1,000 0,000 0,000 0,000 0,000 –3,172 0,0143 207 0,201 P4 0,025 0,965 0,035 0,000 0,000 0,000 –2,837 0,0143 247 0,241 P5 0,05 0,93 0,04 0,03 0,000 0,000 –2,479 0,0139 279 0,272

long

P6 0,05 0,930 0,041 0,029 0,000 0,000 1,426 0,0139 323 0,315 P7 0,025 0,931 0,043 0,026 0,000 0,000 1,306 0,0139 271 0,264 P8 0,01 0,97 0,03 0,000 0,000 0,000 1,35 0,0143 210 0,205 P9 0,005 0,978 0,000 0,000 0,022 0,000 1,580 0,0139 172 0,168 P10 0,001 0,957 0,000 0,000 0,043 0,000 1,711 0,0135 111 0,108 Source: Own research.

1 85 169 253 337 421 505 589 673 757 841 925 1009 -0,1

0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8

CORR_DK1_ELIX

1 85 169 253 337 421 505 589 673 757 841 925 1009 -0,1

0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8

CORR_POLPX_HU

1 85 169 253 337 421 505 589 673 757 841 925 1009 -0,1

0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8

CORR_POLPX_ELIX

1 85 169 253 337 421 505 589 673 757 841 925 1009 -0,1

0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8

CORR_HU_ELIX

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Fig. 9 presented value of VaR0,001 for short and long position and value of P1. If we compare dynamic time series of VaR and P₁with means CVaR_0,001 =

= –3,6875 for short position and CVaR_0,999 = 1,82 for long one.

Fig. 9. VaR0,001 and P1 portfolio value Source: Own research.

Conclusion

Empirical results show that in neighboring markets, both distributions of electricity prices and absolute growth rates are characterized by similar distributions. Time series of prices, as well as increments show stronger correlations in neighboring markets. Correlation among changing prices is also dynamic across all markets. For the optimization criteria of the task (16), the lowest risk with relatively higher profit can be achieved primarily on contracts issued in Sweden and Denmark. Nevertheless, taking into account absolute increases, due to the negative prices, worse fit was obtained in terms of the number of excesses of estimated quantiles than for the relative growth.

VaR 0,001

P₁ 1 85 169 253 337 421 505 589 673 757 841 925 1009 -80

-60 -40 -20 0 20 40 60 80

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References

Alexander C., Chibumba A. (1996), Multivariate Orthogonal Factor GARCH, University of Sussex Discussion Papers in Mathematics.

Artzner P., Delbaen F., Eber J.M., Heath D. (1999), Coherent Measures of Risk, “Math- ematical Finance”, No. 9, pp. 203-228.

Bollerslev T. (1990), Modeling the Coherence in Short-Run Nominal Exchange Rates:

A Multivariate Generalized ARCH Approach, “Review of Economics and Statistics”, No. 72, pp. 498-505.

Brockwell P.J., Davis R.A. (1996), Introduction to Time Series and Forecasting, Springer–

–Verlag, New York.

Engle R.F. (1987), Multivariate ARCH with Factor Structures-Cointegraation in Vari- ance, Discussion Paper, No. 87, University of California, San Diego.

Engle R.F. (2002), Dynamic Conditional Correlation – A Simple Class of Multivariate GARCH Models, “Journal of Business and Economic Statistics”, No. 20, pp. 339-350.

Engle R.F., Kroner K.F. (1995), Multivariate Simultaneous Generalized ARCH, “Econ- ometric Theory”, No. 11, pp. 122-150.

Engle R.F., Sheppard J.R. (2001), Theoretical and Empirical Properties of Dynamic Conditional Correlation Multivariate GARCH, NAER, Working Paper, No. 8554.

Fiszeder P. (2009), Modele klasy GARCH w empirycznych badaniach finansowych, Wydawnictwo Naukowe UMK, Toruń.

Heilpern S.A. (2011), Aggregate Dependent Risks – Risk Measure Calculation, “Math- ematical Economics”, No. 7(14), pp. 108-122.

Jajuga K (2000), Ryzyko w finansach. Ujęcie statystyczne. Współczesne problemy badań statystycznych i ekonometrycznych, AE, Kraków.

Kraft D.F., Engle R.F. (1983), Autoregressive Conditional Heteroskedasticity in Multi- ple Time Series, Working Paper, UCSD, Department of Economics.

Krężołek D. (2015), Modele zmienności klasy GARCH oraz pomiar ryzyka – analiza porównawcza na rynku metali nieżelaznych i szlachetnych [in:] G. Trzpiot (ed.), Modelowanie wielowymiarowych struktur danych i analiza ryzyka, Wydawnictwo Uniwersytetu Ekonomicznego w Katowicach, pp. 82-100.

Markowitz H.M. (1959), Portfolio Selection. Efficient Diversification of Investments, Yale University Press, New Haven.

Osińska M. (2006), Ekonometria Finansowa, PWE, Warszawa.

Rockafellar R.T., Uryasev S. (2000), Optimization of Conditional Value-at-Risk, “Jour- nal of Risk”, No. 2, pp. 21-41.

Trzpiot G. (2010), Wielowymiarowe metody statystyczne w analizie ryzyka in- westycyjnego, PWE, Warszawa.

Tse Y.K., Tsui A.K.C. (2002), A Multivariate GARCH Model with Time-Varying Corre- lations, “Journal of Business & Economic Statistics”, No. 20, pp. 351-362.

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Weron A., Weron R. (2000), Giełda energii, Centrum Informacji Rynku Energii, Wrocław.

Weron R (2006), Modeling and Forecasting Electricity Loads and Prices, Wiley, Chichester.

Zivot E., Wang J. (2006), Modeling Financial Time Series with S-PLUS, Springer, New York.

[www 1] https://www.tge.pl (access: 30.10.2016).

[www 2] https://www.epexspot/ (access: 30.10.2016).

[www 3] http://www.nordpoolspot.com/ (access: 30.10.2016).

ANALIZA PORÓWNAWCZA RYZYKA ZMIANY CENY ENERGII ELEKTRYCZNEJ W WYBRANYCH REGIONACH EUROPY

Streszczenie: W pracy opisano za pomocą wielowymiarowych modeli autoregresyjnych VAR-GARCH procesy zmienności cen energii elektrycznej na wybranych europejskich rynkach energii elektrycznej. Na bazie notowań z towarowych rynków natychmiasto- wych: polskiej Towarowej Giełdy Energii (TGE), European Energy Exchange (EEX), skandynawskiej Nord Pool oraz krajów Europy środkowej Czech, Słowacji, Węgier i Rumunii, korzystających z usług czeskiego operatora (OTE) w okresie od stycznia 2014 do października 2016, przeprowadzono analizę ryzyka zmiany średniej dziennej ceny energii elektrycznej oraz zaproponowano portfel kontraktów na energię elektrycz- ną, minimalizując ryzyko straty w badanym okresie. Ryzyko straty pojedynczych kon- traktów estymowano za pomocą wartości zagrożonej VaR. Do optymalizacji portfela kontraktów wykorzystano warunkową wartość zagrożoną CVaR.

Słowa kluczowe: VAR, DCC, VaR, CVaR, analiza portfelowa.