APT Model for Electricity Prices on the Day Ahead Market of the Polish Power Exchange

A C T A U N I V E R S I T A T I S L O D Z I E N S I S

F O L IA O E C O N O M IC A 194, 2005

A l ic j a G a n c z a r e k *

APT M O D EL FOR ELECTRICITY PRICES O N TH E DAY AH EAD MARKET OF T H E PO L ISH POW ER EXCH ANG E

Abstract

In th is p a p e r we p resen ted the m odel o f the d ep en d en ce o f the electricity price on m ac ro eco n o m ic facto rs such as changes in the d o lla r price, the D eu tsch e m ark price, th e rate o f in flatio n , the ra te o f un em p lo y m en t, price changes in th e m ining in d u stry , the p ro d u c tio n o f the m a n u fa c tu rin g sector, the o u tp u t o f the m ining in d u stry and w eath er co n d itio n s. T he aim o f th is article w as the em pirical verification o f the price m odel o n the D a y A head M a rk et (D A M ) o f the P olish P ow er E xchange in 2001 based o n the prin cip al c o m p o n e n ts m eth o d .

T h e resu lts w ere c o m p ared w ith the results fo r the A P T m odel, selected by m eans o f the g rap h analysis m eth o d an d the o p tim u m choice m eth o d p ro p o se d by Z. H ellw ig. T h e aim o f this w o rk w as to choose the best m odel for the descrip tio n o f p rice tren d s o n the D A M .

Key words: T h e D ay A head M a rk et, A rbitrage Pricing T h eo ry , factors analysis, the principal c o m p o n e n ts, eigenvalue, eigenvector, the g ra p h analysis m eth o d , an d the o p tim u m choice m eth o d p ro p o se d by Z. H ellw ig

I. IN T R O D U C T IO N

T h e D ay A h ead M a rk e t (D A M ) w as th e first m a rk e t, w hich was established on the Polish Pow er Exchange. T his w hole-day m a rk e t consists o f tw enty-four separate, independent m arkets, w here p a rtic ip a n ts can freely buy and sell electricity. T h e b reak th ro u g h in the developm ent o f the Polish Pow er Exchange was m ade on 1st July 2000, w hen the first tran saction s were com pleted on the D ay A head M arket.

T h e energy m a rk e t is no t a neutral isolated stru ctu re. In this p ap e r we a tte m p t to answ er the question, how the electricity price depends on the

m acro eco n o m ic factors. T o exam ine these dependencies wc use the APT m odel. W hile selecting the variables for the m odel o f price changes we use th ree m ethods: the factors analysis, the optim um choice m eth o d proposed by Z. H ellwig and the grap h analysis m ethod.

II. T H E A P T M O D E L

T h e A rb itra g e Pricing T heory is based on the ‘o ne p rice’ rule. T he differences in prices o f the same goods can be affected both by system atic risk factors and accidental risk factors. W e take into accou nt th e follow ing items:

R, - ra te o f re tu rn from a price,

E (R ,) - expected ra te o f return from a price, f it - ra te o f re tu rn o f Ith system atic risk factor,

/J; — i — 1 , . . . , k sensitivity coefficient o f the rate o f re tu rn against the ith system atic risk factor,

- accidental risk factor, with the m ean Е (£() = 0 and the covariance D (£t) = a 2, < reR + .

In this p ap e r we have applied the A P T m odel:

R f = E(R ,) + ß i f if + ß d 21 + ••• + ßkfkt+ £(> t = l , . . . , n . (1) T h e e q u a tio n presented in exam ple (1) can be expressed w ith th e m atrix eq u a tio n . We notice:

R - ( n x 1) - vector o f rates o f re tu rn o f prices,

E ( R ) - ( n x 1) - vector o f expected value rates o f re tu rn o f prices, F — (n x k) - m atrix o f rates o f re tu rn o f system atic facto rs o f risk, w here E(F) = 0, E(F F r ) = I,

ß — (k x 1) - vector of coefficients sensitivity o f rates o f return o f systematic facto rs o f risk,

Ł,— (n x 1) - vector o f e rro r term s with n - no rm al d istrib u tio n E(£) = 0, D ( 0 = ff2I, <76R + .

W ith the supp lem entary assum ptions:

E(£ F r ) = 0, (2)

E[R — E(R)] = 0, (3)

T he price arb itra g e will n o t be possible w hen the n um b er o f values grows. So we can w rite this m odel with the equation:

Rt =

+

+ •••

+

t=l,...,n.

IIL FACTOR ANALYSIS

ln the m odel we use the variables which wc can observe and m easure. These variables often depend on o th er factors which we c a n n o t im m ediately describe. T hese factors are called latent variables o r factors. T o separate these facto rs we use the facto r analysis. In this p ap e r we sep arate the laten t variables, which we have applied, m ak in g use o f the principal com ponents m eth o d . We notice:

F = —, f i ] T - vector o f o bservational variables, X = [Xy, X 2, —, -X*]T - vector o f principal co m po nen ts, Л = [at , a 2, ..., a j - o rth o g o n al and norm ality m atrix. T hen wc can express the X vector o f tran sfo rm atio n

X = ATF. (6)

T h e X j principal co m ponents have the follow ing properties:

D 2( X 1) > D 2( X 2) > . . . > D 2( X k), (7)

I D 2( / j ) = £ D2(Xj ). (8)

J= i J = i

Wc determ ine X j m aking use o f eigenvalues and eigenvectors o f the covariance m atrix o f f j o bservational variables. I f we define the ^ -e ig en v alu e o f the / * eigenvector o f the covariance m atrix, then

D 2( X j ) = lj, y = l , . . „ /с. (9) E q u a tio n (8) m eans th a t all the observational variables are described by all principal com ponents and by eigenvalue (9). If w, denotes th e contribution o f X j to th e explan atio n o f observational variables, we can write:

= j = l , . . . , k . (10)

In the m odel we use only these principal com ponents w hich have th e biggest share in explaining the variance o f observational variables. In the paper we use tw o criteria to determ ine the num ber o f principal com p onents. T he first one, developed by K eiser, takes into accou nt only these principal co m ponents, eigenvalues o f which are close to one o r higher th a n one. T he second criterion is proposed by C attcl. It elim inates these principal co m ­ ponents, eigenvalues o f which decrease very slightly (Ostasiewicz, (ed.) 1999).

IV. E M P IR IC A L A N A L Y S IS

In o rd e r to verify the A PT m odel for the D A M electricity price in 2001 we have used m onthly changes in m acroeconom ic facto rs (T ab. 1).

T abic 1. D escrip tio n o f sym bols and d efinition o f variables

Sym bol

o f variable N a m e o f variable D escrip tio n o f v ariab le

R C , C h an g e in elect­

ricity price

M o n th ly average D A M price o f electricity (C,) in PLN

R C , = — --- —

c , - ,

RW , C hange in electri­ city volum e

M o n th ly average D A M electricity volum e (W,) in M W h

RW . — — ---— Щ -г f u C hange in the r a ­ te o f in flatio n R ate o f inflation (1%) f u = / /I * » /2, C hange in the r a ­ te o f u n em p lo y ­ m ent

R ate o f u nem ploym ent (w % )

f 2l = " /1 0 0

/3, Price changes in the m ining indus­ try

C oefficient o f prices o f sold p ro d u c tio n in th e m ining industry (G K t)

U - G K J 1 0 0 - 1

/4, Price changes in th e m a n u f a c tu ­ rin g in d u stry

C oefficient o f prices o f sold p ro d u c tio n in the m an u fa ­ ctu rin g in d u stry (PP,)

Д , = PPJIQQ - 1 /5, Price changes in

the in d u stry p ro ­ d u cin g and deli­ v e rin g en erg y , gas and w ater

Coefficient o f prices o f sold p ro d u c tio n in th e industry p ro d u cin g and delivering energy, gas an d w ater (EGW,)

T ab le 1. (contd.) Sym bol

o f variable N am e o f variable D escrip tio n o f v ariab le

f u C h an g e o f d o lla r price N B P D, d o lla r price in P L N . d. - d, . , 61 “ D f i , C h an g e o f D e u ­

tsche m ark price

N B P D M , D eutsche m ark price in PL N

D M , - D M , .

r = J. — — J n ł

/ „ C h a n g e in the o u tp u t o f the m i­ n in g in d u stry

O u tp u t o f the m ining in d u stry (W K ,) in th o u sa n d s o f tonnes W K , - W K , . f _ ' ' - 1 W K ,_ , / * T em p era tu re change

A verage tem p eratu re n o te d in W arsaw ( T t) in °C

T — T

r t 1 t - 1

T_{1 r - 1}

/ 1 Of C hange in cloudi­ ness

A verage cloudiness no ted in W arsaw (Z ,) in octanes

J 101 у

i - i

f u , C h a n g e in s u n ­

lig h t exposition

A verage sunlight ex position n o ted in W arsa w (I/,) in h

f i U ~ U , - i

F o r these variables we use the facto r analysis. In T ab le 2 we present the eigenvalue o f the covariance m atrix for these variables.

Table 2. E igenvalue o f facto rs

N u m b e r o f fa c to r

X J

E igenvalue % to ta l variance C um ulative C um u lativ e variance in % 1 3.31 27.57 3.31 27.57 2 2.66 22.13 5.96 49.69 3 1.96 16.29 7.92 65.98 4 1.34 11.19 9.26 77.18 5 0.95 7.92 10.21 85.10 6 0.70 5.85 10.91 90.95 7 0.48 4.01 11.39 94.96 8 0.43 3.59 11.83 98.55 9 0.10 0.83 11.93 99.38 10 0.06 0.51 11.99 99.89 11 0.01 0.11 12.00 100.00

Figure 1. P lot o f eigenvalues

B ased on C a tte l’s criterion we use the first eight principal com ponents. W e in co rp o rate these m acroeconom ic variables which have the greatest loadings in relatio n to latent variables into the A PT m odel.

T able 3. F a c to r load in g s (varim ax n o rm alized )

V ariable F a c to r F a c to r * 2 F a c to r * 3 F a c to r F a c to r * 5 F a c to r * 6 F a c to r * 7 F a c to r 0.06 0.01 -0 .9 5 -0.01 0.01 -0 .1 5 0.11 -0.23 f u 0.17 0.14 -0.11 0.04 0.17 -0 .9 4 -0 .0 9 -0 .0 9 f i , 0.12 -0 .8 5 0.06 -0 .1 8 0.04 0.08 0.33 0.31 3* -0 .7 5 -0 .1 2 -0 .2 0 -0 .2 5 -0 .3 5 0.31 0.16 -0.02 4 f -0 .8 6 0.13 0.34 0.04 0.11 0.01 -0.21 0.19 51 0.02 0.17 0.00 0.97 0.06 0.08 0.12 -0 .0 9 6* -0 .2 7 0.00 0.27 0.43 0.03 0.69 -0 .2 6 0.01 I t -0 .7 6 -0 .0 6 -0 .0 9 0.18 0.10 0.56 -0 .0 4 -0 .2 0 8( 0.04 0.16 -0 .2 7 0.09 0.18 -0 .0 4 0.14 -0 .9 2 9 f -0 .0 7 0.22 0.12 -0.11 0.08 0.01 -0 .9 5 0.13 1 Of 0.03 -0 .1 2 0.02 -0 .0 4 -0 .9 7 0.09 0.06 0.17 M i l 0.15 0.78 0.06 0.12 0.54 -0 .1 4 -0 .0 4 0.08 B old - A lp h a reliabilities at a = 0,05.

W e have built the A PT m odel on the ground o f K eiser’s criterion. We have selected the follow ing variables as the first fo u r p rincipal com ponents: change in electricity volum e, change in the ra te o f in flation, change in the

in the m an u fa c tu rin g industry, price changes in the industry pro du cin g and delivering energy, gas and w ater, change in the D eutsch e m a rk price, change in the o u tp u t o f the m ining industry. T his m odel is insignificant w ith alph a reliabilities o f a = 0.05 based on the F isher-S nedecor test.

T h e m odel built based on the first principal co m p o n en t, is insignificant. If we increase the n um ber o f principal co m p onents, we o b tain insignificant m odels, to o . In the next step we analyse only these factors, which have the absolute value o f loadings higher than 0.8. T his way the m odel comprises: change in electricity volum e, change in the ra te o f unem ploym ent, price changes in the m anufacturing industry, price changes in the industry producing and delivering energy, gas and w ater, change in cloudiness, change in the rate o f inflation, change in the outp u t o f the m ining industry, and tem perature change. W c use th e o rd in ary least squares m eth o d (O LS) to estim ate the param eters o f the m odel:

RC, = a0 + a , / i , -I- a2/ 2( + a 4/ 4( -I- a sf 5, + a 8/ 8, + a 9/ 9l + a 10/ 10( + a ^ R W , + £(, (ID W ith alp h a reliabilities a t a — 0,05 o f T -statistics and F -statistics we receive the m odel:

R C t = 0.272 - 1.7882( - 1.850/4I + 0.855f St- 0.260 /8t - 0 .0 2 0 /1O( + 0 .0 3 2 Я ^ (0.067) (0.408) (0.306) (0.120) (0.018) (0.006) (0.002)

( 12) with change in the ra te o f unem ploym ent, price changes in the m anufactu ring industry, price changes in the industry produ cin g and delivering energy, gas and w ater, change in the o u tp u t o f the m ining industry, change in cloudiness, and change in electricity volume.

In the econom etric m odel we use the m ethod which is proposed by Hellw ig (G an czarek, 2002). We choose the variable co m b in atio n with three variables: change in electricity volume, price changes in the industry producing and delivering energy, gas and w ater and change in the D eutsch e m ark price. T h e integral capacity o f this com bination is the greatest o f all possible co m b in atio n s o f the variables from T ab le 1.

H = 0,5545.

We use the o rd in ary least squares estim ate p aram eters o f th e m odel for this com binations:

and we receive:

R C , = — 0.016 + 0.762/S( — 0.236/-7, + 0.023RW,. „

(0.008) (0.564) (0.194) (0.007) 1 ’

In the third m odel we select the variables based on the g rap h analysis m eth od and receive the follow ing variables: change in the ra te o f un em p­ loym ent, price changes in the m ining industry, the m a n u fa ctu rin g industry, and the industry producing and delivering energy, gas an d w ater, as well as ch ange in the o u tp u t in the m ining industry (G an czarek 2002). T h e A PT m odel fo r electricity prices can described w ith the eq uation :

RC, = ß 0 + ß z f 21 + ß j it + ßbfbt + ß s f i l + ß s f s i + ß l2RW, + (15) W e use the ordin ary least squares estim ate p aram eters for this m odel and w ith a lp h a reliabilities a t a = 0,05 o f T-statistics an d F -statistics we receive:

R C t = 0.219 - 1.455/2t + 0.8 0 6 /5t —0.23 4/8, + 0.034J?Wi. , (0.072) (0.444) (0.522) (0.022) (0.003) U '

V. C O M P A R IS O N O F T H E E F F E C T IV E N E S S O F T H E M O D E L S

W e com p are the three m odels with the changes in electricity price listed on the D ay A head M ark et in 2001 and the results o f the analysis are presented below.

T able 4. C o m p ariso n o f the values o f changes in electricity price o n the D a y A head M a rk e t in 2001 with the values o f the presented m odels

M o n th C h an g e in price on D A M F a c to r analysis (12) H ellw ig’s m ethod (14) G ra p h analysis (16) T he residuals o f m odel (12) T h e resid u als o f m odel (14) T he residuals o f m odel (16) 1 0.0355 0.0380 0.0444 0.0379 -0.0025 -0 .0 0 8 9 -0.0024 2 0.0088 0.0095 -0.0119 0.0055 -0.0007 0.0207 0.0033 3 0.0069 0.0056 0.0212 0.0079 0.0012 -0.0143 -0 .0 0 1 0 4 0.0248 0.0268 0.0031 0.0281 -0.0019 0.0217 -0.0033 5 -0 .0 1 3 5 -0.0168 -0.0074 -0.0150 0.0033 -0.0061 0.0015 6 -0 .0 1 4 2 -0.0141 -0.0191 -0.0155 -0.0001 0.0050 0.0014 7 -0 .0 0 3 0 -0 .0 0 2 6 -0.0045 -0.0054 -0.0004 0.0014 0.0024 8 -0 .0 1 9 4 -0 .0 2 0 2 -0.0274 -0.0257 0.0009 0.0080 0.0063 9 -0 .0 5 3 2 -0.0523 -0.0349 -0.0442 -0.0009 -0.0183 -0.0090 10 -0.0433 -0.0428 -0.0055 -0.0431 -0.0004 -0.0378 -0 .0 0 0 2 11 0.0402 0.0362 0.0195 0.0405 0.0040 0.0207 -0.0002 12 -0 .0 0 5 0 -0 .0 0 2 5 -0.0128 -0.0062 -0.0025 0.0078 0.0012

T abic 5. E rro rs o f the m odels

Sym bol o f variable T h e A F I' m odel (12) F a c to r analysis T h e A P T m odel (24) M odel o f H ellw ig T h e A P T m odel (16) M odel o f analysis o f g raph R w 0.997 0.777 0.991 R 2 0.994 0.605 0.982 S u 0.004 0.021 0.006 D u rb in -W a tso n 1.386 1.470 1.450

--- Change in price on DAM — — Model of factor analysis ... Model of Hellwig --- Model of graph analysis

Figure 2. V alues o f em pirical and theoretical changes in electricity price on D A M

F igure 2 show s th a t the A P I' m odel (12), w hich is based on the factor analysis, describes changes in the electricity price on D A M in 2001 m ost accurately. T h e stan d ard deviation (S u) o f this m odel is sm aller th an in o th er m odels. T h e determ ination coefficient R 2 = 0.994 m eans th a t this m odel describes the price change on D A M with the accuracy o f 99.4% . But we should no t com pare the determ ination coefficients ( R 2) and m u lti­ c o rrelatio n coefficient (R w), because the m odels have a different num ber o f variables. D u rb in -W atso n test has the lowest value for th e m odel (12), so this m odel has the highest au to co rrelatio n o f residuals.

In this p ap e r we have presented three m ethods o f selecting variables for econom etric m odels and we developed three different m odels. In m y opinion, the m odel which we developed with the principal co m pon ents m ethod is the best. B ut the selection depends on individual preferences and can be different.

R E F E R E N C E S

B arczak A .S ., B iolik J. (1999), P odstaw y ekonom etrii, A E , K atow ice. C h o w G .C . (1995), E konom etria, P W N , W arszaw a.

G a n cz a rek A. (2002), P rzew idyw anie k ształto w an ia się cen energii elektrycznej n a R D N polskiej giełdy energii, [in:] M odelowanie procesów ekonom icznych, W S H -S G G W , Kicl- ce W arszaw a, 21-28.

G a n cz a rek A . (2002), W eryfikacja em piryczna ceny energii elektrycznej n a R D N G iełdy Energii SA , [in:] M e to d y ilościowe w badaniach ekonom icznych, S G G W , W arszaw a, 64-72. Jaju g a K . (1993), S ta ty sty cz n a analiza wielowymiarowa, P W N , W arszaw a.

M ielczarski W. (2000), R yn e k energii elektrycznej. W ybrane a sp ekty techniczne i ekonom iczne, A gencja R y n k u Energii SA, W arszaw a.

O stasiew icz W. (od.) (1999), S ta ty sty czn e m etody analizy danych, A E , W rocław .

Alicja Ganczarek

M O D E L A P T D LA C E N Y E N E R G II E L E K T R Y C Z N E J N A R D N G IE Ł D Y E N E R G II SA

Streszczenie

W p ra cy przedstaw iliśm y m odel zależności zm iany ceny energii elektrycznej od czynników m ak ro ek o n o m iczn y ch , takich ja k zm iany: k u rsu d o lara , k u rsu m ark i, inflacji, b ezrobocia, cen p ro d u k c ji w górnictw ie, kopalnictw ie o ra z p rzetw órstw ie przem ysłow ym , w ydobyciu węgla k a m ien n eg o o ra z czynników pogod o w y ch . P rzedm iotem b a d a ń jes t e m p iry cz n a w eryfikacja m o d elu ceny n a R D N G iełdy E nergii SA w 2001 r. z w y k o rzy stan iem m eto d y głów nych składow ych.

O trzy m an e w yniki sk o n fro n to w aliśm y z w ynikam i uzyskanym i d la m odelu A P T , w któ ry m d o d o b o ru sk ładow ych m odelu zastosow aliśm y m eto d ę analizy grafów i m eto d ę o p tym alnego w yboru p re d y k ta n t zap ro p o n o w an y ch przez Z. Hellw iga. Celem tej p racy jest w yłonienie m odelu efektyw niej opisującego k ształto w an ie się cen n a R D N .