Electricity Price Modeling for Profit at Risk Management

8/8/2019 Electricity Price Modeling for Profit at Risk Management

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ELECTRICITY PRICE MODELLING FOR PROFIT AT RISK

MANAGEMENT

JACOB LEMMING

SYSTEMS ANALYSIS DEPARTMENT

RIS NATIONAL LABORATORY, DK-4000 ROSKILDE

1. Introduction

In liberalized electricity markets power producers face a wide range of decisionproblems that require modelling of electricity prices as a crucial input. In thischapter we look at risk management decisions and specifically on how electricityprice modelling affects the optimal solution to a relatively simple risk managementdecision problem.

Risk management decisions are effected by different sources of risk throughoutthe modelling process as illustrated in figure 1. The decision problem must beformulated correctly in terms of objective, risk limits and input factors and eachinput factor must be modelled in a way that extracts as much relevant informationfrom available data as possible. The aim of this chapter is to analyze how thechoice of model for electricity prices affects the output from a simple risk manage-ment problem. The risk management problem is deliberately kept simple in order

to keep focus on the risk related to structural choices concerning electricity pricemodelling. Though the conclusions are based on this type of case study analysisthey provide valuable insight that can be generalized to other decision problems inthe electricity sector.

In the highly liquid and developed financial markets the literature on price mod-elling is extensive. To predict future price developments analysts use both technicalanalysis based on patterns in historical market price movements and fundamentalanalysis based on expectations about the development in the underlying marketprice drivers. Electricity markets worldwide are still in the development phase andnot surprisingly there exists conflicting views about the value of such modellingtools in electricity markets.

This chapter starts out by using a framework based on input data to discuss differ-ent price modelling techniques and their application to electricity price modellingin section 2. Section 3 describes the simple risk management problem used as a casestudy and section 4 reviews the construction of input parameter models focussingespecially on the electricity price in a model based on data from the Scandinavian1

1Strictly speaking Finland is not part of Scandinavia but is included as part of the Nordic

Nord Pool market.

1


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2 JACOB LEMMING

Parameters

Data

Input model risk

Decision model risk

Model output

Objective +

Constraints

Figure 1. Elements in the modelling process that affect the solu-

tion to risk management problems.

Nord Pool electricity market. Finally section 5 presents experimental results show-

ing how the solution to the risk management problem is affected by the choice ofinput parameter modelling, and section 6 concludes.

2. An overview of electricity price modelling methodologies based

on input data

A useful way of categorizing price models is to look at the data used to model inputparameters. In electricity markets one can distinguish between two main categoriesof data:

Market price data Fundamental data about market price drivers

With this distinction market price data includes both historical spot prices and de-rivative prices such as the forward curve. Fundamental data includes technical and

marked based information that can be used to construct the expected developmentof supply and demand curves in the market.

The distinction between fundamental data and market data sets the stage fortwo different approaches towards electricity price modelling. The first approachis based on econometric models such as Stochastic Differential Equations (SDE)known from financial theory. In such financial models market data is used to esti-mate a parametric structure adapted to fit the characteristics of electricity prices.


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ELECTRICITY PRICE MODELLING FOR PROFIT AT RISK MANAGEMENT 3

The motivation is that of technical analysis where patterns in market data is as-sumed to be the most valuable predictor of future prices. The second approachis based on a technical bottom-up modelling of the electricity system where dataabout supply, transmission, distribution and demand is used to model future mar-ket price dynamics. The motivation here is that of fundamental analysis where theexpected development in underlying price drivers is used construct expectationsabout future price developments.

2.1. Financial models: In financial price models the time dynamics of marketprices is driven by stochastic processes generally in the form of stochastic differen-tial equations and parameters are estimated using market data such as historicalspot and derivative prices.

As described in Clewlow & Strickland (1999b) most literature on financial pricemodels falls into one of two main categories. The first category of models describe

the spot price P(t) dynamics along with other key state variables using a set of sto-chastic processes. These processes are generally spilt into deterministic componentsf(t) modelling trends and cycles and a stochastic component S(t) modelling theuncertainty or distribution of prices. The second category of models use a similarset-up but focuses directly on the dynamic evolution of the entire forward pricecurve. The two approaches are interrelated as forward prices can be derived fromthe risk adjusted or risk neutral version of a spot price process provided that anexplicit solution to the stochastic differential equation governing the spot price cande obtained analytically (see Clewlow & Strickland (2000) for an example).

Applications of the spot price approach to electricity markets can be found inreferences such as Lucia & Schwartz (2002), De Jong & Huisman (2002), Pilipovic(1998), Deng (2000), Kellerhals (2001), Knittel & Roberts (2000) and Johnson &Barz (2000). References that apply the forward price approach to electricity pricinginclude Clewlow & Strickland (1999b), Koekebakker & Ollmar (2001), Clewlow &Strickland (1999a) and Joy (2000).

The main strength of financial models lies with the use of realized prices, whichinclude information about less tangible factors such as speculation, market powerand general trader phycology. The main weakness is the potential lack of predictivepower with historical data in developing markets such as the electricity market.

2.2. Fundamental models. Fundamental models constitutes a category of engi-neering models based on a technical bottom-up modelling of the production, trans-mission and consumption parts of the system. This form of modelling tends to be

rather complex since it requires an accurate description of a large technical systemincluding factors such as production capacity, production costs, transmission con-straints and consumption patterns. Modelling the development in average levels insuch factors over time is generally a data intensive task and the large amount ofstochastic fluctuations place an even higher demand on the amount and quality ofinput data required.

In fundamental models price spikes and seasonal variations are a direct result of


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4 JACOB LEMMING

movements in a set of underlying variables that must be modelled with a consid-erable degree of accuracy. The number of such underlying price driving variablesfar exceeds the number of variables used in any financial model. As a result notwo fundamental models will generally be based on the exact same data set makingthem much harder to verify and compare than financial models.

In addition to the complex technical modelling the main challenge with funda-mentally based price modelling lies with the translation of modelling results intocredible market price scenarios. The preferences of suppliers and consumers mustbe accurately reflected through the modelling, which is difficult task.

2.3. Combined approaches. The problems sketched above with financial andfundamental models combine to explain why practitioners often prefer models thatcombine the two model types. Fundamental data contains invaluable informationabout short-term weather related changes in supply and demand, which generallycannot be found in market data. In the very long-term market data will also tendto be insufficient, because the number of yearly observations required to statis-tically estimate how fluctuations in annual hydrological conditions (wet years vs.dry years) affects average annual prices would lack predictive power by the timeit became available. On the other hand market price data such as forward pricecurves represents important information about the comprised expectations and riskaversion of market players. This kind of data cannot be modelled from technicalfactors in any meaningful way.

An approach that combines the two types of models can be formed by using abottom-up model to construct price scenarios and then calibrate these such thatexpected prices in the set of scenarios fit the observed forward price curve in themarket2. If desired such calibration can include extrapolation of patterns found in

historical data or parameters inferred from other derivative prices e.g. volatilitiesfrom options.

3. A Profit at Risk risk management model

The primary goal of this chapter is to illustrate how the structural choices concern-ing electricity price modelling affects the output from to risk management decisionproblems in the electricity sector. To analyze such aspects we use a simple casestudy where a power producer wishes to hedge the annual Profit at Risk (PaR) of aportfolio with a physical long position in a thermal power plant and a short positionin an annual forward contract. In the absence of any start-up costs a power plantcan be seen as a series of spark spread options3 for each day in the annual timeperiod considered. With fixed variable costs V C these options can be simplifiedto a set of European call options on electricity. This effectively means that thedecision problem illustrates the effect of hedging a series of call options (a caption)to a certain PaR level using a forward contract.

2If arbitrage restrictions are imposed this approach is equivalent to the forward price curve

construction approach suggested in Fleten & Lemming (2003).3See e.g. XX or XX for a description of how power plants can be modelled as spark spread

options.


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The Profit at Risk (PaR) measure is a variant of the Value at Risk (VaR) measureknown from financial industry. PaR describes the worst expected operational profitto a specified confidence level over a given time period. In the above example anannual e 10 million 95% confidence PaR describes a setting where the power pro-ducer estimates that there is a 5% chance of incurring an annual profit of less thane 10 million on the portfolio.

In non-financial industries market value tends to be an insufficient statistic for cor-porate risk management decisions. This does not mean that maximizing of share-holder value is not the relevant optimization criteria, but rather that additionalfactors not reflected in market prices affects shareholder value. Especially in non-financial markets operational profit should be considered an important variablesbecause this form of liquidity can create value for shareholders. Unlike financialassets physical assets such as a power plant are not liquidly traded in any marketand cannot be sold on short notice to raise cash without large transaction costs.

Avoiding the cost of financial distress, the inability to pursue strategic investmentsor bankruptcy are all aspects that can create shareholder value in non-financialindustries and profit is therefore a relevant unit for risk measurement in the elec-tricity sector.

Speculation and hedging should generally be conducted as separate activities toavoid accidental speculation in a risk management context. As described in theprevious section forward prices can be seen as an estimate of expected spot pricesunder certain assumptions about the amount of hedgers and speculators in themarket. To avoid speculation the price scenarios used as input for PaR calculationshould be calibrated to fit any credible (liquidly traded) part of the electricity for-ward price curve. In this sense PaR comes to resemble VaR, however it correctlyexpands on the amount of information used in the model by including the effect of

additional risks factors not priced in the market e.g. volume risk and liquidity risk.

For annual profit risk management the average annual electricity price is an es-sential input variable. This does not imply however that there is no additional gainform high frequency modelling of input data such as an hourly or daily time reso-lution. The added value from such high frequency modelling comes from potentialrelations between electricity prices and other risk factors such as cost and volume.We choose a daily time resolution and model price and volume as the only two statedependent variables in calculation of profit P R as:

(1) P R(QF, t) = max [P(t) V C(t), 0] QP(t) + (P(t) F(T0

, T1

, T2

))QF

P(t) is the electricity spot price, V C(t) is the variable cost of operation, F(T0, T1, T2)is the price level defined in the forward contract for delivery during [ T1 T2] signedat some previous point in time T0. QP(t) is the volume produced at time t and QFis the volume of forward contracts with a positive value indicating a long position


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and a negative value indicating a short position4.

Risk measures such as PaR can be seen either as an objective or as a set of con-straints imposed by management. In either case the decision variable is the shortposition QF necessary to obtain some PaR level. In this chapter we consider botha case where the PaR value P V is maximized at a specified percentile as anobjective and a case where P V and are specified as exogenous constraints andthe objective is to minimize the forward position. This type of minimization prob-lem can be desirable for a number of reasons such as transaction costs T C(QF) orrisk/reward preferences directed towards staying open for potential large profits.Given a set of scenarios i I for spot price Ps(t) and production volume QiP(t)the problem minimization and maximization problems can be formulated as follows:

TC minimization with PaR constraints:

MinQF T C(QF)

s.t. 365t=1 P R

i(QF, t) P V qiM i I

Nii=1 q

i Ni

qs {0, 1} QF [1, 0]

PaR maximization:

MaxQF

P V

s.t.

365

t=1 P Ri(QF, t) P V q

iM i I

Nii=1 q

i Ni

qs {0, 1} QF [1, 0]

where P V is the Profit at Risk limit defined in the PaR measure with a confidencelevel . M is a suitably large constant that combined with a binary variable qi

equal to one allow a violation of the PaR level P V in a given scenario i. To en-sure that this happens only in a fraction of the total number of scenarios Ni

a second constraint limits the sum of the binary variables qi. In MC-simulationeach simulated scenario is weighted equally however if the problem where solvedusing probabilities these would be used to weigh the qi variables. P Rs(QF, t) is asdefined in equation 1.

4A short position implies a negative amount of assets and hence a position that profits from

decreasing prices. A short position in a forward contract means that the holder has agreed to to

sell the underlying product at the forward price at some specified time in the future.


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Both formulations yield mixed integer programming problems due to the non-convex structure of the PaR measure. Such problems are difficult to solve in thegeneral case where PaR as a function of the decision variable QF can have morethan one inflection point i.e. have local minima that are not global minima. Ingeneral we expect however that PaR will be a simple concave function of QF. Tounderstand this it is useful to look at the pay-off or profit diagrams for the portfolio.Figure 2 illustrates payoff-diagrams for the long position in the call option as curve1 (the open position in a power plant), the short position in the forward contract ascurve 2, and the combined portfolio with equal weights as curve 3 (the completelyhedged position where the amount of forward equal the amount of options).

The figure shows that a completely hedged portfolio (curve 3) always has a PaR of

VC

F-VC

F Price P(t)(NOK/MWh)

Long call option (Qp )

[curve 1]

Complete hedge

[curve 3]

Short forward (-Qf)

[curve 2]

Profit PR(NOK/MWh)

Partial hedge

[curve 4]

A B

F

Figure 2. Pay-off diagrams for portfolio combinations of a short

forward position and a long call option.

at least F V C at all levels and that the open position (curve 1) has a lowerprofit whenever the electricity price P(t) is below the forward price F. Unless morethan (1 )100% of all electricity price scenarios lie above the forward price F,which is highly unlikely if the forward price is an estimate of expected spot pricesthen PaR will be higher for the hedged portfolio than for the open portfolio.

By decreasing the amount of forward contracts shorted compared to the completelyhedged position i.e. QF < QP we obtain a payoff structure like curve 4. This


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8 JACOB LEMMING

curve is interesting because it shows that as the amount of forward contracts is de-creased from a complete hedge QF = QP towards zero PaR will rise until it reachesa maximum at the the point where the amount of scenarios electricity price scenar-ios in the interval [A,B] equals 100% of the total number of scenarios5. From thispoint on PaR will then decrease as QF approaches zero.

Depending on the distribution of electricity prices one can find special cases wherethis simply concave relationship between PaR and QF will not hold. However, asone can generally design to avoid such special cases quite easily we use this concaveproperty to solve the maximization problem (or minimization) with a simple onedimensional search procedure.

4. Modelling input parameters

This section describes in detail a base case financial model for the electricity pricealong with a set of simple models for volume, cost and dependencies between factors.

Though the main use of these models are as reference cases for sensitivity analysisin section 5, the detailed description of the modelling process for electricity pricesprovides valuable insight for the analysis of interaction between input models anddecision model.

4.1. Electricity price modelling. In the risk management case study describedabove profit is highly dependent on the developments in the electricity price. Sincethis parameter is also highly volatile it is most likely the input parameter wherestructural model choices will have the largest effect on the risk management prob-lem. Choosing a model structure for electricity prices is therefore crucial and valu-able insight is gained from a review of the choices involved in the constructionprocess.

Type of price model: Based on the input data framework described in section 2an initial decision involves the choice between either a fundamental, a financial ora combined approach. We choose a simple financial spot price model here, becausethis type of model allows for a more transparent model structure, use of input dataand reproducibility than fundamental or combined approaches.

As shown throughout the analysis in section 5 financial models are generally notthe most suitable approach for constructing input scenarios to risk managementproblems such as the case study used here. A combined approach would generallybe preferred for electricity market problems with this type of time horizon. Ourchoice of a financial model is therefore based more on the need for an illustrativeframework that can be reproduced by others than on the quality of financial modelsin this context. This does not mean that financial models cannot have value e.g as

a benchmark model or as a way of capturing market based effects that can be usedfor calibration of price scenarios.

5Notice as a special case that PaR will have a maxima at QF = QP if more than 100% of

all scenarios lie in the interval [VC,F]. Graphically this can be illustrated by noting that curve

3 cannot be tilted (as in curve 4) marginally without creating more than 100% profit scenarios

below F V C and hence decreasing PaR.


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Amount and type of input data used: To estimate parameters in the spotprice model we use historical prices from spot market on the Scandinavian NordPool power exchange. Choosing a cut of point for historical data must be basedon a qualitative evaluation of how the information value of the historical data setdeteriorates. We choose to include data from the time of writing 01-10-2002 andback to 01-01-1996 where a significant change occurred as the Nord Pool exchangewas expanded to include Sweden in addition to Norway. The choice of a spot pricemodel as opposed to a forward price model implies that forward prices are notrequired in the estimation process6.

Parametric structure: Electricity prices are driven by fundamental factors suchas temperature, hydrological conditions, demand patterns and plant availability.Even in financial models based solely on market data these factors serve as inspira-tion for the structural choices concerning parameters. In the scandinavian marketrising demand in the winter period combined with an annual hydrological cycle

where reservoirs of hydro power plants are filled during spring leads to a strongseasonal component. To model cyclic variations we follow the structure suggestedin Lucia & Schwartz (2002) with a deterministic component f(t) modelling bothseasonal variations and demand driven weekend effects combined and a stochasticcomponent S(t) including a mean reversion term characteristic for commodityprices. Using a well documented model is desirable here because it provides a cred-ible reference for the analysis of how parametric changes in the modelling affectsthe solution to a subsequent decision problem.

Electricity prices exhibit spikes as a result of extreme weather conditions or plantoutages. Though a jump-diffusion process with mean-reversion can be used to cap-ture such spike behavior an extremely high value of mean-reversion will generallybe required, which in turn will have the undesirable effect of removing all additional

stochastic variation in the data besides the jumps. We choose to instead to followthe approach of (De Jong & Huisman (2002)) by a adding a regime shift componentto the model, which encompasses but price spike and price jumps as special cases.

Negative prices can occur in electricity markets due to start/stopping costs of powerplants and other technical constraints. However, as our data is in the form of dailyaverages i.e. a sum of 24 hourly prices for a system price comprising all Scandina-vian price areas, negative electricity prices are not plausible. We therefore chooseto work directly with the logarithm of price Ln(Pt) with a base case model of theform:

(2) Ln(Pt) = f(t) + Ln(SNt ) ; if Tt = N

f(t) + Ln(SSt ) ; if Tt = S

6Forward prices can be used either in the estimation process using the relation under the risk

neutral measure described in section 2. Alternatively the data can be used for verification. For

simplicity no derivative prices is included in this study.


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where

Ln(SNt ) = (1 )ln(SNt1) + e

Nt ; e

Nt N(0, (

N)2)(3)

Ln(SSt ) = uS + eSt ; eSt N(0, (S)2)(4)

f(t) =

NDi=1

iD(i) +

NMi=1

iM(i)(5)

where SNt and SSt describe the stochastic part of the price process in respectively

the normal regime N and the spike regimes S. The parameters S and S representthe expected value and standard deviation of price spikes respectively, and eNt andeSt are independent noise components. M(i) and D(i) are 0-1 dummy variablesdescribing NM monthly and ND daily effects through parameters i and i. Thetwo stochastic processes SNt and S

St run in parallel and transition between the two

states Tt = N and Tt = S is governed by a transition matrix:

Tt|t1 =pNt|Nt1 pNt|St1pSt|Nt1 pst|St1

For simplicity we model regime shifts as spikes of a single day duration i.e. a shiftfrom the normal state to the spike state is always followed immediately by a shiftback to the normal regime i.e pSt|Nt1 = 1 and pSt|St1 = 0. With this formulationthe frequency of spikes Fspikes is equal to:

(6) Fspikes =pSt|Nt1

pNt|St1 +pSt|Nt1=

1

pNt|St1 + 1

Estimation: The parameters of the model (2-5) above can be estimated throughmaximum likelihood procedures as described in De Jong & Huisman (2002). How-ever, as noted in Clewlow & Strickland (2000) electricity prices tend to have severalspike components and maximum likelihood procedures tend to converge to high fre-

quency spike components in the data. This is undesirable with the model setupabove because the primary function of the spike regime should be to capture thelarge low frequency spikes that would not occur in the normal regime.

The times series of logarithmic spot prices used here (see figure 3) contains bothhigh and low frequency spike components of different duration and sign. Becauseof this complexity a modified version of the recursive filter used in Clewlow &Strickland (2000) was therefore constructed to estimate parameters of the spikeregime. These data points where subsequently removed and parameters in thenormal regime where then estimated on the remaining part of data using simplenon-linear regression.

In the normal regime of equation (2) the absolute difference of log prices dLn(Pt) is

normally distributed and Clewlow/Strickland therefore suggests an iteration pro-cedure where values more than 3 empirical standard deviations from the mean areremoved from the data in each iteration7. The standard deviation of the samplewithout these spikes is then recalculated and the procedure is repeated until con-vergence. As the choice of 3 standard deviations is somewhat arbitrary we choose

7In a normal distribution such observations would occur with a probability of less then 0.0003

i.e only one observation out of every 3333 observations


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instead to modify the spike criteria of the filter by using the difference between theobserved kurtosis in the sample K(sample) and the kurtosis of a normal distribu-tion K(normal) with the same mean and variance as the sample.

Table 1 shows the result of running this procedure on the data with a kurtosisbased stopping criteria defined as:

(7) K =K(sample)

K(normal)=

E[X4] 4E[X3]E[X] + 6E[X2]E[X]2 3E[X]4

3(E[X2] E[X]2)2< C

where E[XN] represents the Nth moment and the constant C terminates the re-cursive filter when the sample kurtosis is less than (C 1)100% higher than what anormal distribution would dictate. Optimally one would continue iteration with thefilter until C = 1, however this attempt to force normality upon data would lead tosimilar problems as the maximum likelihood estimation. A significant part of datawould simply be moved to the spike regime where the problem of non-normality

would be reencountered.

Table 1 shows the recursive iteration used to calculate an estimate of the spikefrequency Fspikes, the expected value of spikes

S and standard deviation of spikesS. Finally we illustrate the effect of removing spikes in figure 3 where the timeseries of historical spot prices is compared with the data series after spikes havebeen removed.

Nspikes Fspikes Espikes Sspikes K(normal) K(sample) K

5 0.002 0.910 0.216 2.77E-04 1.02E-03 3.682

10 0.004 0.749 0.226 2.25E-04 6.35E-04 2.818

15 0.006 0.671 0.215 2.02E-04 5.09E-04 2.514

20 0.008 0.628 0.203 1.92E-04 4.49E-04 2.33325 0.010 0.588 0.198 1.70E-04 3.62E-04 2.127

30 0.012 0.555 0.196 1.61E-04 3.28E-04 2.031

35 0.014 0.527 0.194 1.50E-04 2.93E-04 1.947

40 0.016 0.505 0.191 1.39E-04 2.58E-04 1.852

45 0.018 0.487 0.187 1.33E-04 2.38E-04 1.791

50 0.020 0.469 0.185 1.27E-04 2.19E-04 1.731

55 0.022 0.454 0.183 1.24E-04 2.11E-04 1.698

60 0.024 0.442 0.180 1.20E-04 2.04E-04 1.696

65 0.027 0.430 0.178 1.12E-04 1.84E-04 1.640

70 0.029 0.419 0.176 1.05E-04 1.70E-04 1.615

75 0.031 0.414 0.172 1.03E-04 1.71E-04 1.653

80 0.033 0.406 0.169 9.71E-05 1.53E-04 1.575

85 0.035 0.397 0.168 9.19E-05 1.42E-04 1.54990 0.037 0.391 0.166 8.93E-05 1.36E-04 1.524

95 0.039 0.383 0.164 8.58E-05 1.28E-04 1.498

Table 1. Iterations in recursive filter based on kurtosis criteria

with C = K = 1.5.


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I table 2 Nspikes represents the number of spikes removed and Fspikes is the result-ing frequency. Espikes =

S and Sspikes = S are respectively the mean value and

standard deviation of the Nspikes

price spikes removed.

Comments: Before testing the model we note that an important effect has beenleft out of the model. In an electricity market such as the Scandinavian there is a

2,500

3,000

3,500

4,000

4,500

5,000

5,500

6,000

6,500

01-01-1996 01-01-1997 02-01-1998 03-01-1999 04-01-2000 04-01-2001 05-01-2002

Time

ln[Price](NOK/MWh)

Daily prices

Daily prices jumps removed

Figure 3. Nord Pool spot price data 01/01/96 to 01/10/02 before

and after spikes are removed with a recursive filter.

significant stochastic variation in the annual price average due to the large amountof hydro power on the production size. The large variations in energy productionbetween dry and wet years has a large effect on the annual price as illustrated byan average system spot price on Nord Pool of 258 NOK/MWh (app. e 35/MWh)in the very dry year of 1996 compared to an average price of 106 NOK/MWh (app.e 15/MWh) in the very wet year of 2000. As mentioned earlier this characteristicis important for the annual risk management problem. To correctly estimate the

uncertainty of annual average prices one would however need a large amount ofrealized annual price averages spanning the entire possible range of hydrologicalconditions. This type of data set will generally not become available and we there-fore note that models based on market data will tend to lack an important sourceof information. This is not just problematic because of the annual time horizon.The fact that the distribution of wet and dry years is incorrectly represented inthe sample means that seasonal components and other model parameters will bebiased. The extent of this problem is examined in more detail in section 5


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4.2. Modelling volume risk. To model the volume risk associated with the longposition in a power plant we use a two dimensional transition matrix simulatinga Markovian 0-1 point process. The matrix is fully described by two probabilitiese.g. the probability p1|0 of a plant being unavailable in hour t given that it wasavailable at t 1 and the probability p0|1 of returning to availability state i t giventhat the plant was out in t 1.

Qt|t1 =

p1|1 p1|0p0|1 p0|0

4.3. Additional factors. Additional factors include variable costs and startupcosts. For simplicity we choose to model all costs as constants i.e variable costs V Cand exclude start-up costs. Both factors are easily included but are left out hereto keep focus on the effect of electricity price modelling in the risk managementproblem.

To model correlation between plant outages and price spikes the transition matricesQ for the volume process and T for the price process are simulated as dependent.The dependency is modelled by expanding the transition matrix T for prices to bea matrix of conditional probabilities depending not only on the previous price statebut also on the state of the volume process.

5. Experimental r esults

The framework described in the previous sections is now used to examine how thestructural modelling of electricity prices affects the optimal solution to the PaRrisk management problem. In particular we examine the effect of:

Parametric modelling of price spikes and forced outages (both as dependentand independent variables)

The general choice of a financial model for electricity price modelling inPaR risk management

As a base case we use the parameters shown in table 2 to generate electricity pricescenarios corresponding to the historically observed values for seasonal variationand price spikes8. Table 2 also include parameters for the volume process, costs (inNOK/MWh) and the PaR measure.

The base case model is solved both for the case with P V maximization and for thecase where QF is minimized given an exogenous limit on P V. Output in the formof P/L (profit/loss) histograms, the optimal amount of forward contracts QF, andthe optimal P V value in the maximization case (top figure) is illustrated in figure4. All simulations are based on I = 5000 scenarios each consisting of 365 dailyprices.

The effect of volume and price spike risk: To examine the effect of pricespikes and volume risk the base case minimization problem is solved for price spikefrequencies Fspikes in the interval [0, 0.1] and forced outage frequencies Foutages inthe interval [0, 0.15].

8Though the data contained both up and down spikes we model only up-spikes in this section

to avoid mixed effects.


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Input factor Parameter Value Parameter Value

Electricity price 0.0123 jul 4.921

N 0.0611

aug 4.937

sun 0.083 sep 4.958

sat 0.057 oct 4.967

jan 5.000 nov 4.997

feb 4.985 nov 4.997

mar 4.996 dec 5.017

apr 4.974 Fspikes 0.039

may 4.915 Espikes 0.383

jun 4.856 Sspikes 0.164

Volume process q1|0 0.07 q1|1 0.93

q0|1 0.90 q0|0 0.10

PaR 0.05 PV 6500

Variable cost V C 4.87

Forward prices F(T0, T1, T2) 4.92Table 2. Parameters for base case simulation. Significant param-

eter estimates based on the Nord Pool data set is indicated with

*.

P/L distribution for PaR maximization

0

0,5

1

1,5

2

2,5

3

3,5

4

4,5

5

5618

6341

7063

7786

8509

9232

9955

10677

11400

12123

12846

13569

14291

15014

15737

16460

17183

17905

18628

P/L

%

PaR(0.05) = 6684

QF = - 0.69

P/L distribution for QF minimization

with PaR(0.05) = 5500

0

1

2

3

4

5

6

7

8

4443

5960

7478

8995

10513

12030

13548

15065

16583

18100

19618

21135

22652

24170

25687

27205

28722

30240

31757

P/L

%

QF = - 0.39

Figure 4. Profit/Loss histograms for the minimization (bottom)

and maximization (top) problems in the base simulation.


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The frequency of price spikes should only affect the optimal amount of forwardcontracts to the extent that the distribution of annual price averages is changed.If the prices spike frequency is increased without any adjustments then the totalaverage price will also increase and the actual effect of price spikes will be distorted.To avoid this each simulation is adjusted so that the total average of all 365 5000daily prices equals the exogenously defined forward price F(T0, T1, T2)

9.

0

0,0

2

0,0

4

0,0

6

0,

08

0

0,0375

0,075

0,1125

0,15

0,25

0,3

0,35

0,4

0,45

0,5

Q_F

F(spikes)F(outages)

Figure 5. Optimal (minimization) short position in forward con-

tracts (positive value is a short position) for different combinations

of price spikes frequency and frequency of forced outages.

Figure 5 illustrates the results including the somewhat counterintuitive result thatthe required amount of forward contracts decreases as the number of spikes is in-creased. To understand this result we go back to the pay-off diagrams in figure 2.The worst case scenario in the open position (curve 1) is an annual profit of zero if

all prices in the series of 365 daily prices lie below the variable cost of production.As the frequency of price spikes increases the likelihood of such scenarios decreaseand as a result PaR tends to increase. The effect is illustrated in figure 6 where wecompare P/L histograms for the upper (Fspikes = 0.1) and lower limit (Fspikes = 0)

9What is actually done is that the sample mean is divided with the desired mean (exoge-

nous forward price) to obtain a regulation factor RF. All prices including the spikes are than

subsequently divided with this regulating factor.


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16 JACOB LEMMING

0

1

2

3

4

5

6

516

3120

5725

8329

10934

13538

16143

18748

21352

23957

26561

29166

31770

34375

36979

39584

42188

44793

47397

P/L

%

Spike frequency Fs=0.10PaR=1674

Spike frequency Fs=0PaR=884

Figure 6. P/L histograms for simulations with minimum or max-

imum level of price spikes.

used for price spikes in the case where there is no volume risk. In the minimiza-tion problem this general increase in PaR implies that a smaller amount of forwardcontracts is required to satisfy the exogenous constraint P V.

When volume risk is introduced a new factor will affect the optimal amount offorwards. When the plant experiences a forced outage the portfolio will consistonly of the forward position. If the electricity price is simultaneously above the for-ward price e.g. in case of a price spike, then the portfolio incurs a loss proportionalto the amount of forward contracts. Based on this we would expect the positive ef-fect that price spikes has on QF to decrease as volume risk increases. This is exactlywhat we observe in figure 7 where the decrease in the optimal amount of forwardsfrom the case Fspikes = 0 to the case Fspikes = 0.1 increases from (QF) = 0.102 to(QF) = 0.147 when volume risk is increased from Foutages = 0 to Foutages = 0.15.

Concerning the isolated effect of volume risk we see that the amount of forwardcontracts required to obtain the desired PaR value P V decreases a function ofvolume risk as expected. The largest decrease of 0.087 for the case without pricespikes is however modest compared to the relatively large variation 0 to 15% usedfor expected amount of forced outages.

Performing a similar set of simulations for the maximization problem we find no


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obvious trends in the amount of optimal contracts. However, as illustrated in figure7 the optimal PaR level follows the same pattern as in the minimization problem.

0

0,0

2

0,0

4

0,0

6

0,0

8

0,00

0,03

0,05

0,08

0,10

0,13

0,15

5000

5500

6000

6500

7000

7500

PaR

F(spikes)

F(outage)

Figure 7. Optimal (maximization) PaR level P V for different

combinations of price spikes frequency and frequency of forced out-ages.

The optimal PaR level decreases (in absolute terms) as the amount of volume riskincreases and increases10 as the amount of price spikes is increased.

To understand why no effect is found on the optimal amount of contracts in themaximization problem we illustrate PaR as a function of QF for a series of selectedsimulations in table 3.

Starting from the left in the table (comparing columns 1 and 2) we see that asthe frequency of spikes is increasing improved for smaller forward positions QF.Based on the argumentation made above this is to be expected since the likelihoodof a very low annual profits (due to 365 daily scenarios with low or zero profit)decreases as the frequency of spikes is increased. In the completely hedged positionQF = QP the increase in spike frequency has no effect because the portfolio profitis constant for all electricity prices above V C regardless of the frequency.

10An increase in the PaR level in absolute terms e.g. from 5000 to 7000 is a decrease in the

amount of risk exposure.


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18 JACOB LEMMING

1 2 3 4 5

QF Fs=0,Fo=0 Fs=0.1,Fo=0 Fs=0,Fo=0.15 Fs=0.1, Fo=0.15

= 0 = 1

0.00 813 1399 638 1133 571

0.05 1548 2388 1349 2067 1580

0.10 2281 3435 2101 2998 2312

0.15 3014 4090 2803 3594 2923

0.20 3695 4645 3379 4067 3365

0.25 4265 5117 3863 4544 3715

0.30 4760 5577 4265 5009 4078

0.35 5209 6079 4600 5330 4338

0.40 5567 6540 4942 5623 4398

0.45 5950 6849 5114 5706 4422

0.50 6274 7075 5217 5723 4313

0.55 6412 7265 5223 5703 4168

0.60 6562 7357 5122 5470 39070.65 6661 7368 4918 5202 3554

0.70 6669 7314 4594 4776 2917

0.75 6565 7137 4095 4070 2167

0.80 6405 6856 3422 3087 1122

0.85 6271 6541 2588 2020 30

0.90 5973 6087 1638 921 -1115

0.95 5520 5475 794 -255 -2202

1.00 4696 4564 -256 -1399 -3367

Table 3. PaR as a function of QF for upper and lower bound

cases in the set of simulations.

Looking at the effect of forced outages (comparing column 1 and 3) the table showstwo effects from increasing volume risk. First of all there is a general decreasein PaR and secondly there is a skewed effect that biases PaR more significantlydownwards at higher levels of QF. The skewed effect occurs as indicated above,because plant outages combined with prices above the forward price leads to lowprofit scenarios.

Columns 4 and 5 illustrate the simulations with largest values for both price spikeand outage frequencies both as uncorrelated and fully correlated variables. Againwe see a general and a skew effect this time caused by the introduction of correla-tion. Both effects occur because the number of simultaneous instances with highprices and outages increase. As the loss is generated solely by the forward contract

in such instances the negative effect increases with the size of the forward position.

The general use of a financial model based on market data for elec-

tricity price generation: Having examined the effect of parametric choices infinancial electricity price model we turn to the more general question of how thechoice of a financial model based on a specific sample market data affects the re-sults. The model used here is based on 6 years of data and as mentioned thereis a large variation between yearly average prices due to the large degree of hydro


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power. The 6 annual data points is clearly not a sufficient representation of thehydrological distribution. To examine the effect of rather small changes in the datasample a forecast for 2003 as a dry year with 1996 prices is added to the sample.After re-estimating the price model with this data the simulation is repeated for theminimization problem optimal solutions based on the new data set are comparedto those found in figure 5. Figure 8 illustrates the difference between the optimalsolution to the minimization problem with the original data set and the data set

0

0,

02

0,

04

0,

06

0,

08

0

0,0375

0,075

0,1125

0,15

0,05

0,1

0,15

0,2

0,25

Q_F

F(spikes)

F(outages)

Figure 8. Difference between optimal solution in the minimiza-

tion problem based on original sample v.s a sample with an addi-

tional dry year.

with an additional dry year added for 2003.

The figure shows that adding a single year of realistic data can have a signifi-cant effect. The optimal amount of forward contracts is approximately halved inall scenarios and the effect is larger than both the volume and price spike effect.This shows how the lack of market data for sufficiently accurate measurement ofvariations in the annual electricity price has a strong effect on the solution to therisk management problem. Moreover we see that in this case study the effect isso strong that structural choices about parametric modelling of the essential inputparameters such as electricity prices and volume risk is of minor importance.


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20 JACOB LEMMING

6. Conclusions

This chapter has illustrated how electricity price modelling affects the optimal solu-

tion to a risk management problem in the electricity sector. Though a specific casestudy has been used the critical approach towards choosing the type of price modeland parametric choices within a certain model category can however be generalizedto a broader context.

The analysis and experiments with the simple risk management problem showedthat an optimal PaR is generally reached at some intermediate level of forwardcontracts between a complete hedge of the power plant and an entirely open po-sition. The modelling of price spikes where seen to have a positive effect on PaRmaximization in the absence of volume risk. Similarly a positive effect was seenin the minimization problem in the sense that a PaR level can be reached with asmaller forward position. As volume risk increases the combination of volume riskand price spike risk lead to an oppositely directed negative effect on PaR depending

on the size of the forward position. Correlation between the factors was seen toincrease the negative effect of forward contracts on PaR though only moderatelycompared to the relatively large levels of volume risk and price spike risk examined.

Finally the historical data set used to estimate the parameters in a financial spotprice model was seen to affect the optimal solution significantly compared to para-metric choices concerning input parameter modelling i.e. price spike modelling andinclusion of volume risk. Based on this we conclude that the choice of model (finan-cial vs. fundamental) and data used for estimation of input parameters is crucialfor the output quality of such decision models.

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Electricity Price Modeling for Profit at Risk Management

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