Extreme changes in prices of electricity futures

Transcription

1 Isurace Marets ad Compaies: Aalyses ad Actuarial Computatios, Volume 2, Issue, 20 Roald Huisma (The Netherlads), Mehtap Kilic (The Netherlads) Extreme chages i prices of electricity futures Abstract The purpose of this paper is to aalyze the occurrece of extreme price chage i power delivery forward ad futures cotracts. The results idicate that the distributio of price chages are sigificatly fatter tailed tha a ormal distributio fuctio ad the authors discuss that ris maagers i the power idustry ca obtai better isight i the amout of ris their compaies face by applyig extreme value theory. Keywords: extreme value theory, electricity futures prices, eergy derivatives. Itroductio I this paper we focus o the occurrece of extreme price chages i power delivery forward or futures cotracts. These cotracts are traded o exchages worldwide ad eergy compaies use these cotracts to hedge themselves agaist maret ris. For istace, a eergy compay that eeds to deliver power to cliets i the year 20, ca buy a power futures cotract somewhere i 200 ad fixate the price agaist which it will purchase power for it s cliets. We refer to these cotracts as both power delivery forward ad futures cotracts i the remaider of this paper. The pricig of these cotracts is ot as straightforward as pricig futures cotracts o stocs, for istace. As discussed by Fama ad Frech (987) ad may others traders use the availability of storage capacity to valuate futures cotracts. A trader that sells a futures cotract ca mae his positio ris free by purchasig the commodity o the spot maret. As a result, the futures price should reflect the spot price of the commodity plus iterest forgoe, storage costs, ad a coveiece yield that reflects the value that ca be derived out of havig the commodity physically. Power is ot yet ecoomically storable ad, as a cosequece, the power futures prices reflect expectatios ad ris premiums (see Fama ad Frech, 987; Lucia ad Schwartz, 2002; Eydelad ad Wolyiec, 2003; ad Huisma, 2009 amog others). Power futures prices do ot ecessarily deped o the spot price of power ad therefore their price dyamics should be modeled as a stad-aloe process. Ris maagers i the power sector use these cotracts to actively maage maret ris. For istace, cosider a power compay that has agreed to deliver power agaist a fixed price to cliets i 20. Whe the compay will buy the power durig the delivery period 20 i the spot maret, it faces the ris that the average price, paid i the spot maret, is higher tha what is agreed with the cliets. By purchasig a power forward delivery cotract, the ris maager ca fixate the price agaist which the compay will purchase power i the maret i 20 ad by doig Roald Huisma, Mehtap Kilic, 20. so price ris is reduced. However, the timig of whe to purchase these forward cotracts is a difficult decisio. Oe ca buy such a cotract today or perhaps tomorrow whe prices might be lower. It depeds o the ris of a potetial price icrease that might occur betwee today ad tomorrow, whether a compay wats to purchase today or wait. I this paper, we focus o this price ris. We examie to what exted chages i power delivery futures prices ca be modeled usig a ormal distributio fuctio or whether aother method should be applied. We apply extreme value theory to assess the level of tail-fatess, i.e., the frequecy with which large price movemets occur, such that we ca observe whether these price chages ca be modeled usig a ormal distributio or ot. Berhardt et al. (2008) apply extreme value theory to estimate high quatiles dyamically for day-ahead electricity prices i Sigapore. Byström (2005) applies extreme value theory to model electricity prices o the NordPool maret, maig quatile (VaR) forecasts allowig both for fat tails ad timevaryig volatility. Both papers fid strog support for the existece of fat tails i day-ahead prices ad for the superior quatile estimates that extreme value theory produces. Re ad Giles (2007) preset a extreme value aalysis of daily Caadia crude oil prices ad fid strog support for fat tails. Although the amout of tail-fatess is examied i oil marets ad for day-ahead power prices, it has ever bee examied for chages i the price of power futures delivery cotracts. This is the goal set i this paper. We aalyze the occurrece of extreme price chage i power delivery forward ad futures cotracts. Our results idicate that the distributio of price chages are sigificatly fatter tailed tha a ormal distributio fuctio ad we discuss that ris maagers i the power idustry ca obtai better isight i the amout of ris their compaies face by applyig extreme value theory.. Extreme value theory Extreme value theory is a field withi statistics that deals with the frequecy with which extreme observatios occur. We follow Hull (2007) ad Huisma 2

2 Isurace Marets ad Compaies: Aalyses ad Actuarial Computatios, Volume 2, Issue, 20 (2009) i discussig extreme value theory ad we start with the ey result i extreme value theory foud by Gedeo (943). Suppose F () is the cdf of a variable : F () = Pr {V }. As extreme value theory focuses o the structure of the tail, cosider a value u that is a value of somewhere i the right tail of the distributio fuctio of. The probability that lies betwee u ad u + y equals F (u + y) F (u) for y > 0. Defie F u (y) as the probability that lies betwee u ad u + y coditioal o > u. Thus, F u (y) = Pr {u u + y > u}. Gedeo (943) shows that for large values for u, F u (y) coverges to the geeralized Pareto distributio for may probability distributio fuctios F (.). The geeralized Pareto distributio G, (y) is: y G, y. () Hull (2007) cotiues reasoig that the probability that > u + y give that > u, F u (y), the equals G, (y). Furthermore, the probability that > u is F (u). The ucoditioal probability that exceeds a value x, Pr { > x} equals: v x Prv uprv u x u Pr v u (2) Fu G, x u. This result of Gedeo (943) implies that may distributio fuctios follow a geeralized Pareto distributio i the tails. Whe we approximate F (u) by it s empirical couterpart u, where is the umber of observatios i the sample ad u is the umber of observatios that exceed the value u, equatio (2) ca be writte as: u Prv x G, x u (3) u x u. u If we ow set u = ad K, the we obtai what is called the power law: Pr X x Kx (4). We have ow formalized the mai ideas withi extreme value theory. Beyod a certai threshold, fattailed distributio fuctios exhibit power decay. The speed of decay is measured by i equatio (4). This parameter is called the tail-idex. The bigger is, i.e., the steeper the decay, the thier the tails become ad vice versa. The ormal distributio, beig a thi tailed distributio fuctio, exhibits expoetial decay, which is obtaied whe. The goal of this paper is to examie the tail structure of log-price chages of electricity forward prices. To do so, we estimate the tail-idex usig the procedure outlied i the followig paragraph... Estimatig the tail-idex. Let s focus o estimatig the tail-idex of the right tail of the distributio fuctio. Let be the umber of tail observatios that we iclude i the estimatio, such as the highest or lowest returs. Let x i be the i th order statistic, such that x i x i-. Hill (975) shows that the estimate of the iverse of the tail-idex,, for tail-observatios equals: l( x j ) l( x ), (5) j where is the total umber of observatios i the etire sample. How to select, the umber of tail observatios to iclude i the estimate? Iitially, researchers calculated estimates for for differet values for ad the state their coclusios i terms of the average result. Others tried to approximate the optimal by assumig that the data came from some distributio fuctio ad the select that would lead to the best results i a simulatio study. Examples of these approaches are (amog others) Jase ad de Vries (99), Koedij ad Kool (994), ad Kears ad Paga (997) who estimated the tail-idex for the returs distributios of exchage rates ad stocs. The results idicated fattails, but oe is left with the ucomfortable feelig that the tail-idex estimates suffer from a bias i choosig. Oe way to limit the ifluece of this bias is proposed by Huisma et al. (200), a method that we apply i this paper. It is a extesio of the Hill (975) estimator. Huisma et al. (200) observe that the expected value for the estimate equals plus some fuctio f () that depeds o : E f. (6) Huisma et al. (200) show for several distributio fuctios, amog them the Studet-t, that the fuctio f () is almost liear. They formulate the followig regressio equatio: 0, (7) ad the estimate for 0 is the a accurate estimate for. Basically, the Huisma et al. (200) 22

3 Isurace Marets ad Compaies: Aalyses ad Actuarial Computatios, Volume 2, Issue, 20 estimator combies iformatio from differet choices of to reduce the bias i the Hill (975) estimator. Still, a umber of observatios eeds to be chose, however Huisma et al. (200) show that the estimates of are ot that sesitive to wrog choice for. I this paper, we set as suggested by Huisma et al. (200). They applied their 4 method to chages i the values of exchage rates, fidig values for betwee 3 ad Data ad descriptive aalysis The primary data for this study cosists of daily forward closig prices for two marets, the Europea Eergy Exchage () i Germay ad the Nordic Power Exchage (NordPool), which is the sigle power maret for Norway, Demar, Swede ad Filad. The forward cotracts for the maret iclude the base- ad peaload delivery cotracts for the years of 2009, 200, ad These cotracts are traded for several years before delivery o the exchages. We limit ourselves to study the forward prices obtaied i the period betwee oe year before maturity util the last tradig day before delivery starts as commoly the ext-year s delivery cotract is the most liquid. Therefore, we study the prices as quoted i 2008 for the 2009 delivery cotracts, the prices quoted i 2009 for the 200 delivery cotracts ad the prices quoted i 200 for the 20 delivery period. Our dataset spas the tradig days betwee Jauary, 2008 through December 7, 200, havig approximately 250 daily forward price observatios per year. Table cotais descriptive statistics for the daily chages i the atural logarithms of the forward prices for the ad base- ad peaload forward cotracts. Table. Statistics for the daily log forward price chages observed Base Pea Base Pea Base Pea Mea Media Max Mi St.dev Sew Kurt We refer to Huisma et al. (200) for the weighted least squares method to estimate the tail-idex ad for the procedure to obtai stadard errors. 2 For istace, the baseload 2009 cotract ivolves the delivery of MW of power i ay hour of the caledar year 2009 ad the peaload 2009 cotract ivolves the delivery of MW of power i ay hour o weedays betwee 8 a.m. ad 8 p.m. i Base Pea Base Pea Base Pea Mea Media Max Mi St.dev Sew Kurt Table shows that the daily mea log-price chage was about i 200, or -0.%, for the 20 peaload delivery cotract. The maximum price chage was 3.8% o oe day ad the miimum was 2.5%. The daily log-price chages for the peaload 20 cotract were positively sewed, 0.696, ad exhibit excess urtosis of.253 (i excess of the ormal distributio fuctio) idicatig fatter tails tha a ormal distributio fuctio. All excess urtosis values are positive, which is a sig of fat tails i all years. O average, it seems that the tails of the distributio of log-price chages for cotracts are fatter tha for the distributio of log-prices chages i the. 3. Results This sectio shows the tail-idex estimates for the power delivery forward cotracts. Table 2 shows the tail-idex estimates for the baseload cotracts ad Table 3 shows those estimates for the peaload cotracts. Let s focus o the baseload results i Table 2 first. The estimate for the 2009 delivery cotract as traded o the is sigificatly differet from zero, beig with a stadard error of 0.8. This estimate yields a value of for. The left tail of the empirical distributio of log-price chages of the 2009 delivery cotract has a estimate of ad the right tail has a of 0.386, implyig that the right tail is fatter tha the left tail, i.e., more extreme positive tha egative price chages occurred for the 2009 delivery cotract. The first result we leared from Tables 2 ad 3 is that the tail-idex estimates (i terms of ) vary betwee.837 ad for the baseload cotracts ad betwee 2.46 ad for the baseload cotracts 3. For the peaload cotracts these values vary betwee 2.33 ad for the ad 3.05 ad for the cotracts. These levels are i lie with the tail-idex estimates as observed for returs o stocs ad exchage rates 4. The empirical distributio of log-price chages o power delivery forward cotract are clearly fatter tailed tha a ormal distributio. 3 We igore the estimate of here, which is perhaps due to some estimatio error. 4 See for istace Jase ad de Vries (99), Koedij ad Kool (994), Kears ad Paga (997), ad Huisma et al. (200). 23

4 Isurace Marets ad Compaies: Aalyses ad Actuarial Computatios, Volume 2, Issue, 20 The secod result we leared from these tables is that there is o clear relatio betwee the level of the tail-idex ad the specific maret i which the forward delivers. Neither it seems that there is a apparet differece i tail-idex values betwee the baseload ad peaload cotracts or betwee the left ad the right tail of the distributio. Table 2. Tail fatess estimates for ad forward base returs 2009 l r (0.8) 0.28 (3.027) (0.05) (0.04) (0.06) (0.5) l r 0.5 (0.02) (0.02) (0.003) (0.023) (0.443) 0.22 (0.9) l r (0.008) (0.029) (0.007) 0.66 (0.07) (0.058) Notes: Stadard errors are i parethesis. reflects the tailidex for both tails; l for the left tail, r for the right tail; is calculated as /. Table 3: Tail fatess estimates for ad forward pea returs 2009 l r (0.05) (0.97) 0.6 (2.433) (0.058) l r (0.099) (0.073) (0.070) (0.209) (0.023) (0.033) l r (0.0) 0.9 (0.04) (0.00) 0.8 (0.08) (0.032) (0.02) Notes: Stadard errors are i parethesis. reflects the tailidex for both tails; l for the left tail, r for the right tail; is calculated as /. Discussio ad cocludig remars I this paper, we have show that the empirical distributios of log-price chages (or returs) of power forward prices exhibit sigificat fat tails. This implies that extreme price movemets (both up ad dow) occur more frequetly tha what a ormal distributio fuctio would express. This is a result too importat to igore for ris maagers, for istace, as they caot use ormal distributios to calculate their ris measures or values for optios ad other derivative cotracts. If they would do so, they would uderestimate the level of ris. With this i mid they ca improve their estimates of value at ris, the average loss beyod the value at ris measure, or expected maximum losses usig extreme value theory. To see this, we briefly discuss i the extreme value theory way of calculatig value at ris, based o Huisma (2009). Suppose a ris maager lies to measure the 99% oeday Value at Ris (VaR) faced o a ope positio i the 20 power baseload delivery cotract o the. Per defiitio, the oe-day 99% VaR is that price icrease that is beig exceeded i oly % of all days. Let r be the oe-day percetage retur. The r 99% VaR 0.0. Pr (8) From equatio (3), we ca derive u VaR u Prr VaR 0.0. (9) Rewritig this yields 0.0 VaR u. u (0) Usig the estimates for we have obtaied before ad if we choose a proper value for u, we ca easily measure VaR usig extreme value theory. Hull (2007) suggests to set u equal to the 95% quatile of the distributio fuctio obtaied from historical u observatios, such that Still, the ris maager has to estimate. Huisma et al. (998) show a alterative way of calculatig VaR usig extreme value theory. They argue that the degrees of freedom i a Studet-t distributio, which is fattailed, equals the tail-idex. They apply extreme value theory to estimate the degrees of freedom ad the they read off the value at ris from the Studet-t distributio. The advatage of this method is that oe does ot eed to estimate i equatio (0) or to choose a proper value u. They show that their method provides better VaR estimates for stocs ad bods compared to VaR based o the ormal distributio. This paper shows that the price chages of power delivery forward cotracts are fatter tailed tha a ormal distributio fuctio. A ris maager i the power idustry ca, therefore, obtai more accurate ris calculatios by applyig extreme value theory tha by applyig measures based o ormal distributios. 24

5 Isurace Marets ad Compaies: Aalyses ad Actuarial Computatios, Volume 2, Issue, 20 Refereces. C. Berhardt, C. Klüppelberg, ad T. Meyer-Bradis (2008). Estimatig high quatiles for electricity prices by stable liear models, Joural of Eergy Marets, (), pp H.N.E. Byström (2005). Extreme value theory ad extremely large electricity price chages, Iteratioal Review of Ecoomics ad Fiace, 4, pp A. Eydelad ad K. Wolyiec (2003). Eergy ad power ris maagemet: ew developmets i modelig, pricig, ad hedgig, Joh Wiley & Sos, New Jersey, U.S.A. 4. E.F. Fama ad K.R. Frech (987). Commodity futures prices: some evidece o forecast power, premiums, ad the theory of storage, Joural of Busiess, 60 (), pp D.V. Gedeo (943). Sur la distributio limité du terme d ue série aléatoire. Aals of Mathematics, 44, pp B. Hill (975). A simple geeral appraoch to iferece about the tail of a distributio, The Aals of Mathematical Statistics, 3, pp R. Huisma (2009). A itroductio to models for the eergy marets, Ris boos. 8. R. Huisma, C.G. Koedij, ad R. Powall (998). Var-x: fat tails i fiacial ris maagemet, The Joural of Ris, (), pp R. Huisma, C.G. Koedij, C. Kool, ad F. Palm (200). Tail-idex estimates i small samples, Joural of Busiess ad Ecoomic Statistics, 9, pp J.C. Hull (2007). The power law. I Jo Daielsso, (ed.), The value at ris referece. Key issues i the implemetatio of maret ris. Ris boos.. D. Jase ad C. de Vries (99). O the frequecy of large stoc returs: puttig booms ad busts i perspective. Review of Ecoomics ad Statistics, 73, pp P. Kears ad A. Paga (997). Estimatig the desity tail idex for fiacial time series, Review of Ecoomics ad Statistics, 79, pp C.G. Koedij ad C. Kool (994). Tail estimates ad the ems target zoe, Review of iteratioal ecoomics, 2, pp J. Lucia ad E.S. Schwartz (2002). Electricity prices ad power derivatives: evidece from the Nordic power exchage, Review of Derivatives Research, 5, pp F. Re ad D.E. Giles (2007). Extreme value aalysis of daily Caadia crude oil prices, Techical Report EWP0708, Uiversity of Victoria. 25