Measuring value-at-risk for intraday electricity prices.

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NIVERSITY OF

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MSTERDAM

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MSTERDAM

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USINESS

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CHOOL

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ASTER IN

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NTERNATIONAL

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INANCE

Master Thesis

Measuring Value-at-Risk for Intraday Electricity Prices

Andreea Tofan

September 2014

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German Power Set for More Spikes in Winter on Supply Risk1

by Rachel Morrison, Bloomberg, August 29th₂₀₁₄

“Price swings in Germany’s intraday power market are set to increase this winter as unpredictable renewable energy makes up for lower capacity caused by closings of fossil-fuel plants, according to Danske Commodities A/S.

Hourly and 15-minute power prices in Germany and France spiked above 300 euros ($395) a megawatt-hour 12 times so far this year, down from 130 times in 2013, as demand declined during Europe’s mildest winter for seven years. Falling long-term prices have led RWE AG and EON SE, Germany’s two biggest utilities, to pledge to close power plants with 16.7 gigawatts of capacity through March 2017. One gigawatt can supply about 2 million European homes. “Plants shutting definitely has an effect on the intraday markets,” said Bo Palmgren, head of intraday at Danske Commodities, which buys and sells power in more than 30 nations. “It should lead to more price spikes. The traditional producers are going away, and that means renewable producers will have much more impact on the intraday market. They are traditionally volatile,” he said yesterday by phone from Aarhus, Denmark. Colder weather in winter usually encourages demand for power, pushing up short-term prices. A mild winter compounded with an economic slowdown means European power demand is set to decrease 2.9 percent this year, according to Societe Generale SA. A drive by Germany, Europe’s biggest economy, to almost double power output from renewables by 2035 has made supply more difficult to predict. ..”

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consumers (Moreno et al. 2012). Cross country, power markets have developed similarly causing energy trading to thrive in traded products and types of markets the trading activities occur. The range allows market participants to take physical delivery or exchange financial contracts on long or short term markets. In this newly created environment, the concept of risk management translates into the need to monitor and better predict the probability of risky events to safeguard portfolio losses regardless of future scenarios.

Bearing in mind that demand is inelastic for necessity goods, in this case electricity, minor changes in the supply forecasted in the day ahead market can cause major price changes in the intraday market. With spot prices showing sudden and very large jumps to extreme levels due to unexpected increases in demand, shortfall in supply and failures of transmission infrastructure (Geman & Roncorni, 2006), it is becoming more relevant to understand the electricity price behavior and the magnitude of losses a portfolio can incur in a predefined timeframe. One tool which is broadly used in financial institutions and corporations to assess risk is Value at Risk. Developed by J.P. Morgan in an attempt to summarize a portfolio’s risk with one number and released to the worldwide audience in 1994, Value at Risk answers the question how much one can lose with a given probability over a certain time horizon.

The existing approaches for VaR estimation may be classified into three main categories:

non-parametric, non-parametric, and semi-parametric methods. In this paper, a model of each category will be

used to forecast VaR since in the case of intraday electricity markets where demand for electricity is highly inelastic to changes in price in the short run and prices exhibit effects as multiple seasonality, a high degree of mean reversion, large volatility and high volatility persistence, frequent price jumps and short-lived spikes it becomes more interesting to examine which of the existing models meet an accepted accuracy VaR level.

As a result, value at risk will be estimated with the historical simulation approach part of non-parametric models followed by GARCH (1, 1) model based on an econometric model for volatility dynamics and lastly with the semi-parametric extreme value theory approach which models only the tails of the return distribution. With VaR estimations being related to the tails of a probability distribution, techniques from EVT may be particularly more effective in forecasting VaR than other models focusing on the entire distribution of the sample.

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This study aims to analyze the hourly intraday electricity prices from EEX, the German power exchange, for 2012- 2013 in order to conclude which method from the three categories mentioned provides best estimates. Intraday electricity prices are electricity prices as observed on the intraday market, a market where trading occurs from thirty-six hours up to forty-five minutes before actual delivery. Back testing of the by models will be done on the first half year of 2014. The comparison will be extended to incorporate results at different confidence levels and specifically at 95% and 99% confidence intervals.

In the light of the occurrences during and after the financial crisis from 2008, this investigation is of importance because companies incur a greater probability of default if risks are not adequately managed. Hence, a discussion on which tool is best to measure value at risk will, the least, create awareness for utilities and/or trading institutions. Moreover, this study differs from others by performing on analysis on intraday electricity prices. This is an area which has been studied in a limited number of papers due to the low availability of data to the public as opposed to the day-ahead market.

The paper is organized as follows. Introduction chapter is followed up by an explanatory chapter describing the design, supply and demand together with stylized characteristics of the electricity market and prices. In the 2nd_{chapter, literature on previous studies is reviewed for previous}

employed models in calculating VaR and electricity price behavior. Theoretical research hypotheses about the relations between the accuracy of VaR measurements and the distribution of return fitted for calculation and characteristics of intraday markets are then worked out. Chapter three introduces the methodology while chapter four gives an overview of the data used in the empirical analysis, construct measurement, results and back testing. In chapter five, the empirical findings, results and hypotheses on the VaR models are concluded.

1.1 Design of electricity markets 1.1.1 The intraday market for electricity

Electricity trading assumes financial settlement until the traded product is delivered. In other words, the term, day-ahead and intraday markets are financially settled markets while the balancing market is the only physical market.

As soon as the traded electricity reaches delivery the open position holder needs to ensure that the traded flows correspond to the actual delivered flows. This is driven by the below market characteristics:

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 Electricity must be transmitted though the transmission network.  Electricity storage in large quantities is not yet possible.

 Demand and supply must be balanced at every instant in every part of the network.  In real time operations the price elasticity of electricity consumption is nil.

Table 1

The table explains broadly the division of the electricity market by means of clearing venues (organized venue or bilateral agreements) and possible delivery periods of electricity.

Centralized vs Decentralized Delivery Definition

1. Exchange a) Short term

b) Term

Exchange traded with delivery periods varying from day ahead to intraday.

Exchange traded with delivery periods varying from a week up to 3 years in the future.

2. OTC a) Short term

b) Term

Contracts traded outside the exchange (where prices and volumes and not made public) with delivery periods varying from day ahead to intraday.

Contracts traded outside the exchange with delivery periods varying from a week up to 3 years in the future. The short term market is organized in day ahead, intraday and balancing markets.

In the day-ahead market, energy with physical delivery on the next day is traded anonymously on a trading platform or OTC. On the exchange, day-ahead auctions are organized in which market participants express their willingness to sell or buy energy at a certain price for each of the 24hrs of the next day before the gate closure, usually 12PM. The exchange then sorts all available offers and bids and the hourly day-ahead prices and market clearing volumes are determined by matching forecasted demand and aggregated supply in each delivery hour.

After the gate closure of the day-ahead market until physical delivery, trading continues from three PM onwards on the intraday market, as it becomes clearer whether the traded flows will be the actual delivered flows due to updated meteorological forecasts, information of turbines' availability and information about previous market results.

Shortly before delivery, the Transmission System Operators (TSO’s) take over the responsibility for all remaining imbalances forming the imbalance market. To ensure the constant equilibrium between physical demand and supply of the non-storable good electricity in real time, the TSOs make use of pre-contracted balancing services or voluntary offered capacity. A balance responsible party’s balance group whose net production or consumption deviates from the previously scheduled values will be balanced in real time by the TSO. This way, market participants can close their open positions through the balancing services but will always try to avoid this option because the market is designed in a way that the use of balancing services is often more expensive than self-balancing on the intraday market and the TSOs may penalize the

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occurrence of too many imbalances of one market participant with the abrogation of his balancing contract.

The focus of the paper is to examine specifically the intraday market as observed on the exchange. This is limited only to exchange traded deals as access to data on bilateral deals is scarce. The market design of a typical sequence of electricity markets is summarized in Figure 1.

Figure 1

The figure captures the electricity trading sequence by moving from the left side towards right. An hourly open position on d-2 for delivery day d can be exited on d-1 of delivery d through an organized action or a bilateral trade on the day-ahead market. From 3PM on day d-1 until 45 min of delivery on day d exiting the position is only possible on the intraday market. If position is not balanced at the moment of delivery, it creates network imbalances which can result in penalties from the TSO.

1.1.2 Supply and demand in the electricity market

Typically, the power portfolio is made up of a range of power technologies: wind, nuclear, combined heat and power plants (CHP’s), and condensing plants and gas turbines. The ordering of the power supply of each of these players depends on the amount of power they can supply and the cost of this power. The supply and demand curves1_{are represented in the figure below.}

In the power market, the supply curve is called the ‘merit order curve’.

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Figure 2

The figure shows a typical example of an annual supply and demand curve. In a power market, the supply curve is called the ‘merit order curve’. Such curves go from the least expensive to the most expensive units and present the costs and capacities of all generators. Each unit is shown as a step in the ‘curve’.

The bids from nuclear and wind power enter the supply curve at the lowest level, due to their low marginal costs, followed by combined heat and power plants, while condensing plants are those with the highest marginal costs of power production. The electricity price is formed at the point where supply and the marginal cost of the asset class meet.

1.1.3 Stylized characteristics of the electricity prices

Due to the design of the physical electricity market – no storability, constant balancing between generation and consumption, transmission of the electricity though the TSO at all times and inelastic price of consumption, short term electricity prices show several characteristics summarized below:

Seasonality

The seasonal component in electricity prices is more pronounced than in any other commodity and several different seasonal patterns can be found in electricity prices during the course of a day, week and year. They mainly arise due to changing level of business activities or climate conditions, such as temperature or the number of daylight hours. In addition, there is the continuous real time balancing needs of electricity supply and demand that follows the cyclical demand of the seasons. Thus, the seasonal fluctuations in demand and supply translate into the seasonal behavior of spot electricity prices.

Volatility

Another stylized fact of electricity short term prices is the unusually high volatility that is unprecedented in any other financial or other commodity markets. It is not unusual to observe annualized volatilities of more than 1000% on hourly spot prices. The high volatility can be traced back to storage, capacity and transmission problems and the need for markets to be balanced in real time. Inventories cannot be used to smooth price fluctuations. Temporary

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demand and supply imbalances in the market are difficult to correct in the short-term. As a result price movements in electricity markets are more extreme than in other commodity markets. Mean reversion

Besides seasonality, electricity short term prices are regarded to be mean reverting, (Schwartz, 1997). The form of mean reversion observed in electricity markets is very strong what can be explained by the markets fundamentals. When there is an increase in demand generation assets with higher marginal costs will enter the market on the supply side, pushing prices higher. When demand returns to normal levels, these generation assets with relatively high marginal costs will be turned off and prices will fall. This rational operating policy for the employment of generation assets supports the assumption of mean reversion in electricity spot prices. Further, the determinants of demand like weather and climate are cyclical as well.

Jumps and spikes

Short term electricity prices exhibit infrequent, but large spikes or jumps. Price jumps are explained by sudden outages or failures in the power grid and lead to a large increase in prices in a very short amount of time. Conversely, spikes are interpreted as the result of a sudden increase in demand and when demand reaches the limit of available capacity, the electricity prices exhibit positive price spikes. In periods of lower demand, electricity prices fall. Due to the operating cost or constraints of generators, who cannot adjust to the new demand level, also negative price spikes can occur. This is discussed in detail by Nicolosi (2010) who assesses the flexibility of the German power market and discusses specifically the occurrence of 19 events with significant negative prices. Primarily, the non-economic storage possibilities of large amounts and unit commitment, in combination with very limited flexibility of demand, lead to the occurrence of bids below variable costs, even negative prices on the short term. To exemplify, in circumstances of low demand, high wind in feed and running power plants, if a power plant needs to ramp-down, additional costs occur for the later ramp-up, therefore the opportunity costs are integrated in the bid to avoid the ramp down and produce, even though prices do not cover short term variable costs.

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2. Literature Review and Hypotheses Development 2.1 Value at Risk Models

A broad classification of value at risk models can be made into parametric, non-parametric and

semi-parametric methods. To begin with, the non-semi-parametric models make no arbitrary assumptions

around the distribution of returns. However, the historical simulation methodology which falls under this category infers that past returns can be used to construct a distribution for future returns. More, while the historical simulation approach is very easy to implement and does not depend on parametric assumptions on the distribution of the return portfolio by accommodating wide tails, skewness and any other non-normal features in financial observations, the results computed are dependent on the data set used which leads to underestimation or overestimation of risk.

Conversely, the parametric approach implies that a specific distribution for returns must be assumed, with the normal distribution being a common choice. In an energy application, the choice of a distribution is particularly challenging to delimit since there is no empirical distribution defined that fits returns exhibiting high volatility and price jumps representative for electricity prices.

Lastly, extreme value theory falls under semi-parametric models and simulates only the tails of the return distribution. Since VaR estimations are only related to the tails of a probability distribution, techniques from EVT may be particularly effective.

Comparisons between VaR models have been discussed in various studies in stock and foreign exchange markets. To exemplify, Abad and Benito (2013) investigated international stock indexes on stable and volatile periods and compared several VaR methods as historical simulation, Monte Carlo simulation, parametric methods and extreme value theory. The results of the paper highlight that these techniques perform better in a stable than a volatile period. Also, outcomes depend on the volatility model used to estimate the standard deviation of the returns portfolio and the assumptions made around the distribution of the returns portfolio. The paper further concludes that VaR results obtained with parametric models are subject to improvement if other distributions are considered, such as the student t-distribution or skewed distribution and that this method estimates VaR at least as well as other VaR methods that have been developed recently, such as the models based on extreme value theory.

More, several studies of Angelidis et al. (2007), Danielsson and de Vries (2000) have reported that the historical simulation approach produces inaccurate VaR estimates in application to

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international stock indexes. In comparison with other recently developed methodologies such as a mixture of parametric and non-parametric statistical procedures in the historical simulation filtered, conditional extreme value theory and parametric approaches, historical simulation provides a very poor VaR estimate.

In the case of intraday electricity markets where demand for electricity is highly inelastic to changes in price in the short run and prices exhibit effects as multiple seasonality, a high degree of mean reversion, large volatility and high volatility persistence, frequent price jumps and short-lived spikes, an inverse leverage effect, stationarity at both the price and the squared price level, and (possibly) long memory at the price level (Bierbrauer et al., 2007, Byström, 2005, Crespo Cuaresma et al., 2004, Haldrup and Nielsen, 2006, Higgs and Worthington, 2008, Huisman et al., 2007,Knittel and Roberts, 2005, Seifert and Uhrig-Homburg, 2007 and Weron and Misiorek, 2008) it becomes more interesting to examine which model meets an accepted accuracy VaR level.

Extreme value theory approach models only the tails of the distribution. According to Dacorogna et al., (1995); Longin, (2000), from the different approaches - historical simulation approach, GARCH models and extreme value theory - for estimating VaR, extreme value theory could potentially perform better in predicting unexpected extreme changes because of its direct focus on the tails of the sample distribution. This is supported also by Hung et al. (2008) who found that heavy tailed distributions are more suitable for energy commodities, particularly VaR calculation. Additionally, Byström (2005) employed a GARCH-EVT model to electricity prices on the Swedish exchange Nord Pool to investigate the tails of the price change distribution. The results of the study indicate that GARCH-EVT produces more accurate estimates of extreme tails than a pure GARCH model. Similarly, Chan and Gray (2006) compute VaR by means of extreme value theory (EVT) and adjust the volatility in the time series with a GARCH model on a data sample of five international power markets and find the EVT-based model is a useful technique in forecasting VaR in electricity markets.

The above discussion leads to the following testable hypotheses 1a and 1b.

Hypothesis 1a: VaR estimates obtained with the EVT method are more accurate than the other

tested models.

Hypothesis 1b: VaR estimates obtained with the EVT method are not more accurate than the other

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The hypotheses are tested on four different portfolios to capture the effect of seasonality (winter, summer portfolio) for either peak of off peak hours of the week. Therefore the portfolio combinations are winter off peak, winter peak, summer off peak, summer peak. In the hypothesis, more accurate refers to how well the realized violations of the model fit the expected violations.

2.2 Characteristics of the Intraday Electricity Market

The main use of intraday markets is to allow for adjustment of infeasible schedules resulting from spot markets with simplified designs, changing weather conditions and outages.

Studies on liquidity of the intraday electricity markets (Weber 2010) highlight the existence of low intraday liquidity in terms of trading volume. By definition, liquidity in the intraday electricity market is the ability of easily and immediately buying or selling electricity without causing a significant movement in the price. Brunnermeier and Pedersen (2008) highlight the link between market liquidity, the ease at which assets are traded, and funding liquidity, the ease at which the necessary funding can be obtained. In their model, funding and market liquidity mutually reinforce each other, leading to downward spirals when bad shocks occur. As a result, liquidity has an eﬀect on asset price volatility. Also Longin (2000) and Bali (2000) conclude that volatility measures based on asset return distributions cannot produce accurate measures of market risk during volatile periods.

Although, intraday markets allow market participants to trade during the day of operation, there are difficulties that market participants have in finding each other, which is the result of low liquidity, due to the fact that most of the generation units prefer to commit their units in advance to consider start-up costs and plan the operation of their units and therefore there is low availability of electricity. In addition, market participants trade to speculate on prices or in order to close open positions or to optimize the short term operation of flexible power plants and therefore the size of the market impacts directly the liquidity and electricity prices spikes.

Electricity prices are determined by supply and demand, which fluctuate cyclically throughout seasons. Nevertheless, in the context of spot electricity prices it becomes difficult to distinguish seasonality due to the abrupt and unanticipated extreme changes known as spikes. More, according to Ullrich (2012), electricity prices have intraday patterns that vary by day of week and time of year. Karakatsani and Bunn (2008) and Klüppelberg et al. (2010) show that the daily spot prices in electricity markets exhibit mean reverting behaviour, high volatility, price spikes and jump clustering at both intraday and daily levels. Additionally, Hagemann et al (2013) show that

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time of the day and day of the week effects have an impact on intraday liquidity. The trading day in the electricity setting is split between off-peak hours which are the hours from midnight to eight a.m. and from eight p.m. to midnight and show a lower liquidity than the peak hours (eight a.m. to eight p.m.). Similarly, liquidity was found to be low on Sundays when trading activities are limited due to, among others, lower demand.

Ways of measuring liquidity have been presented in several studies which summarize that liquidity has different dimensions and there are: tightness of the market or bid-ask spread (Kyle, 1985; Amihud and Mendelson, 1986), resiliency – the rate at which the prices in the limit order book bounce back at competitive level after liquidity shocks are felt (Kyle, 1985), market depth as the size of an order flow innovation leading to a change in prices at a given moment (Kyle, 1985), delay and search costs incurred when a trader delays a trade in order to achieve better prices than those currently quoted (Amihud and Mendelson, 1991) and trading activity which consists of trading volume and the number of trades executed per delivery hour. Trading volume can be defined as the absolute amount of Megawatt hours (MWh) electricity bought or sold per trading hour. In this paper, liquidity will be measured in trading volume.

Hypothesis 2a: VaR models are more accurate in periods of high liquidity (high trading volume). Hypothesis 2b: VaR models are not more accurate in periods of high liquidity (high trading

volume).

To scale the periods from most to least liquid, the trading volume will be of each period. The period with highest observed volume will be considered as most liquid.

3. Methodology

3.1 Historical Simulation

The advantages and disadvantages of the Historical Simulation have been well documented by Down (2002). Advantages are that the method is very easy to implement, and as this approach does not depend on parametric assumptions on the distribution of the return portfolio, it can accommodate wide tails, skew ness and any other non-normal features. The biggest potential weakness of this approach is that its results are completely dependent on the data set. If the data period is unusually quiet, Historical Simulation will often underestimate risk and if our data

period is unusually volatile, Historical Simulation will often overestimate it. This method looks at the returns of the asset in each portfolio, intraday electricity prices, to

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generate the portfolio loss or gain. Given a dataset of n days return, the return of the portfolio dependent on investment on day i is calculated as per the below:

, is the price of the asset on a day i for a long position and

, for a short position.

The portfolio VaR is then estimated from the maximum loss in the distribution of portfolio returns, associated with the required statistic likelihood percentile.

3.2 GARCH (1, 1)

The generalized autoregressive conditional heterocedasticity model, or GARCH, is a VaR methodology that considers volatility clustering and incorporates mean reversion (Bollerslev, 1986). This approach takes account of volatility changes in a natural and intuitive way and produces VaR estimates that incorporate more current information. The model is a weighted average of past squared residuals, but it has declining weights that never go completely to zero. It gives parsimonious models that are easy to estimate and, even in its simplest form, has proven successful in predicting conditional variances.

VaR is defined by the below formula:

_{where is the confidence level, is the standard deviation of the}

portfolio change over the time horizon and is the inverse cumulative normal distribution In GARCH (1, 1) we assign weights to all parameters used:

, the variance of the market variable on a day n, as estimated at the end of day n-1

, the squared return of the market variable on day n-1 and, , the variance of the market variable on day n-1

The parameters (weights) used . Setting and .

Here the parameters will be estimated with maximum likelihood method. ∑

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3.3 Extreme Value Theory

When modelling the maxima of a random variable, extreme value theory plays the same fundamental role as the central limit theorem plays when modelling sums of random variables. In both cases, the theory tells us what the limiting distributions are.

Generally there are two related ways of identifying extremes in real data. Let us consider a random variable representing daily losses or returns. The first approach considers the maximum the variable takes in successive periods, for example months or years. These selected observations constitute the extreme events, also called block (or per period) maxima. In the left panel of Figure 3, the observations and represent the block maxima for four

periods of three observations each.

Fig. 3 Block-maxima (left panel) and excesses over a threshold (right panel).

The second approach focuses on the realizations exceeding a given (high) threshold. The observations and in the right panel of Figure 3, all exceed the threshold

and constitute extreme events.

The block maxima method is the traditional method used to analyse data with seasonality as for instance hydrological data. However, the threshold method uses data more efficiently and, for that reason, seems to become the method of choice in recent applications.

In the following subsections, the fundamental theoretical results underlying the block maxima and the threshold method are presented.

3.3.1 Distribution of Maxima

The limit law for the block maxima, which we denote by , with the size of the subsample (block), is given by the following theorem:

Theorem 1 (Fisher and Tippett (1928), Gnedenko (1943)) Let be a sequence of i.i.d. random

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→ ,then H belongs to one of the three standard extreme value distributions: Frèchet: { Weibull: { Gumbel: Figure 4

The figure captures the densities for the Frèchet, Weibull and Gumbel functions. The Frèchet distribution has a polynomially decaying tail and suits well heavy tailed distributions. The exponentially decaying tails of the Gumbel distribution characterize thin tailed distributions. The Weibull distribution is the asymptotic distribution of finite endpoint distributions.

Jenkinson (1955) and von Mises (1954) suggested the following one-parameter representation {

of these three standard distributions, with such that . This generalization, known as the generalized extreme value (GEV) distribution, is obtained by setting 1 for the Frèchet distribution, for the Weibull distribution and by interpreting the Gumbel distribution as the limit case for .

A corresponds to thin-tailed distributions such as Gumbel, normal, exponential, gamma and lognormal, whose tails decay exponentially. A corresponds to finite distributions such as the uniform and beta distributions. In most financial applications, the data exhibit heavy-tails suggesting .

Values of reflect heavy-tailed distributions. In each power market, the estimate is positive and statistically significantly different from zero, suggesting that the right tail of the distribution of standardized residuals is characterized by the Frèchet distribution.

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As in general, the type of limiting distribution of the sample maxima is not known in advance, the generalized representation is particularly useful when maximum likelihood estimates have to be computed. Moreover the standard GEV defined above is the limiting distribution of normalized extreme. Given that in practice the true distribution of the returns is unknown and, as a result, the norming constants and are unknown, the three parameter specification of the GEV is used. ( ) {

This is the limiting distribution of the un normalized maxima. The two additional parameters and are the location and the scale parameters representing the unknown norming constants. The quantities of interest are not the parameters themselves, but the quantiles, also called return levels, of the estimated GEV:

₍ ₎

Substituting the parameters , and by their estimates ̂, ̂ and ̂, the below is obtained.

̂ {

̂ ̂ ( ( ) ̂_{) ̂}

̂ ̂ ̂

A value of ̂ of 7 means that the maximum loss observed during a period of one year will

exceed 7% once in ten years on average.

3.3.2 Distribution of Exceedances

An alternative approach, called the peak over threshold (POT) method, is to consider the distribution of exceedances over a certain threshold. Our problem is illustrated in Figure 3 where we consider an (unknown) distribution function of a random variable . We are interested in estimating the distribution function of values of above a certain threshold .

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The figure presents the distribution function F and conditional distribution function .

The distribution function is called the conditional excess distribution function and is defined as , where is a random variable, is a given threshold, are the excesses and is the right endpoint of . We verify that can be written in terms of , i.e.

The realizations of the random variable lie mainly between 0 and and therefore the estimation of in this interval generally poses no problems. The estimation of the portion however might be difficult as we have in general very little observations in this area. At this point EVT can prove very helpful as it provides us with a powerful result about the conditional excess distribution function which is stated in the following theorem:

Theorem 2 (Pickands (1975), Balkema and de Haan (1974)) For a large class of underlying distributions

F the conditional excess distribution function , for u large, is well approximated by

→ , where

{ ( )

For is the so called generalised Pareto distribution (GPD). If is defined as , the GPD can also be expressed as a function of i.e.

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Figure 6

The figure illustrates the shape of the generalized Pareto distribution when , called the shape

parameter or tail index, takes a negative, a positive and a zero value. The scaling parameter is kept equal to one.

The tail index gives an indication of the heaviness of the tail, the larger , the heavier the tail. As, in general, one cannot fix an upper bound for financial losses, only distributions with shape parameter are suited to model financial return distributions.

Assuming a GPD function for the tail distribution, analytical expressions for and can be defined as a function of GPD parameters. Isolating from (1) ( ) and replacing by the GPD and by the estimate

, where is the total number of observations and the number of observations above the threshold , we obtain ̂ ( ( _̂̂ ) ̂) which

simplifies to ̂ ( _̂̂ ̂). Inverting the above for a given probability

gives ̂ ̂_̂ ( ) ̂ .

In POT method GPD is fitted to the excess distribution (value above threshold ) by MLE and the confidence interval estimates are calculated by profile likelihood and then the estimates for VaR are calculated.

4. Empirical Analysis

4.1 Data

To assess the relative ability of a number of alternate approaches to accurately measure VaR in electricity markets, the full data sample is divided per portfolio (07:00 winter off peak, 18:00 winter peak, 07:00 summer off peak and 18:00 summer peak) into an in-sample period on which

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model estimates are based, and an out-of-sample period over which VaR performance is measured. The portfolios contain electricity returns for the specific product contained in the naming of the portfolio for the in sample period. The starting investment (amount) is in each portfolio is €1.

Table 2

The table records summary statistics for all four portfolios. The statistics comprise the minimum portfolio value, median, mean and maximum. More, standard deviation, skewness and kurtosis and the Ljung-Box Q-test for residual autocorrelation (at  = 5%) are analysed. For the Ljung-Box Q-test, only the first portfolio indicated a failure to reject the null of no autocorrelation in the residuals.

07:00 Winter off-peak 18:00 Winter peak 07:00 Summer off-peak 18:00 Summer peak

In sample Start date 01.01.2012 01.01.2012 01.04.2012 01.04.2012 End date 31.12.2013 31.12.2013 30.09.2013 30.09.2013 # observations 364 260 365 260 Min -6.6704 -0.6576 -3.3500 -0.6661 Median -0.0401 0.0163 -0.0700 -0.0029 Mean 0.7227 0.0539 0.2332 0.0404 Max. 113.3056 1.9921 10.9900 2 Std. dev. 6.6115 0.3654 1.1991 0.3072 Skewness 14.4480 1.6713 4.0201 1.9111 Kurtosis 236.5623 8.8059 28.1424 11.1060 Ljung-Box Q 0.9446(0) 57.6422(1) 194.0915(1) 50.4160(1) Out of sample Start date 01.01.2014 01.01.2014 01.04.2014 01.04.2014 End date 31.03.2014 31.03.2014 30.06.2014 30.06.2014 # observations 90 64 91 65 Min -4.7563 -0.4741 -0.8600 -0.6568 Median -0.0628 -0.0279 -0.0400 -0.0032 Mean 0.8264 0.0384 0.2420 0.0670 Max. 37.8309 0.9882 12.6600 2.1710 Std. dev. 5.2163 0.3032 1.4569 0.4572 Skewness 5.7207 1.3214 7.0000 2.2753 Kurtosis 37.5548 4.7688 59.4540 10.3413 Ljung-Box Q 30.3014(0) 20.5493(0) 24.3856(0) 31.4104(0)

Each portfolio holds a different n number of observations because of the seasonal and product differences. Peak runs from eight a.m. until eight p.m. from Monday through Friday. Off peak runs from eight p.m. until eight a.m. from Monday through Friday and the whole day in the weekend.

The skewness factor indicates asymmetry and deviation from the normal distribution and is greater than 0 in all portfolios. The conclusion to be drawn from this is intraday prices follow right skewed distributions with most values being concentrated on left of the mean, with extreme values to the right. Kurtosis is all greater than 3, and so they follow distributions sharper than the normal distribution, with values concentrated around the mean and thicker tails. This means high probability for extreme values.

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Figure 7

The figure graphs intraday price returns for the full sample. QQ plots show the full sample data versus the Standard Normal for each portfolio.

07:00 Winter Off peak returns 07:00 Winter Offpeak QQ-plot

18:00 Winter peak returns 18:00 Winter peak QQ-plot

07:00 Summer Off peak returns 07:00 Summer Offpeak QQ-plot

18:00 Summer peak returns 18:00 Summer peak QQ-plot

-20,00 0,00 20,00 40,00 60,00 80,00 100,00 120,00 2-1-2012 11-10-2012 19-1-2013 29-10-2013 6-2-2014 -3 -2 -1 0 1 2 3 0 20 40 60 80 100 120 140 160 180 -1,00 -0,50 0,00 0,50 1,00 1,50 2,00 2,50 3-1-2012 20-11-2012 9-10-2013 26-2-2014 -3 -2 -1 0 1 2 3 0 50 100 150 200 250 -6,00 -4,00 -2,00 0,00 2,00 4,00 6,00 8,00 10,00 12,00 14,00 2-4-2012 11-7-2012 19-4-2013 28-7-2013 6-5-2014 -20-3 -2 -1 0 1 2 3 0 20 40 60 80 100 -1,00 -0,50 0,00 0,50 1,00 1,50 2,00 2,50 3-4-2012 21-8-2012 9-7-2013 27-5-2014 -3 -2 -1 0 1 2 3 0 10 20 30 40 50 60 70 80 90 100

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The statistics in Table 3 together with Figure 7 demonstrate the defining characteristics of electricity markets: high volatility, occasional extreme movements, volatility clustering and fat tailed distributions. A QQ plot displays the quantile - quantile plot of the sample quantiles of each portfolio versus theoretical quantiles from a normal distribution. These descriptive statistics and plots further motivate the exploration of the alternative approaches to the standard normal distribution of measuring VaR described in the previous chapter.

Both the left and the right tail of the return distribution will be considered. The reason is that the left tail represents losses for an investor with a long position on the portfolio, whereas the right tail represents losses for an investor being short on the portfolio.

4.2 Model Application

Firstly the data for intraday electricity prices available from January 2012 until July 2014 is split into winter and summer periods.

 Summer is defined to include month April - September.  Winter is defined to include month October - March.

Further, within these two periods peak and off-peak periods are identified.

 Peak refers to prices for intraday electricity for weekdays for eight am to eight pm.  Off peak refers to intraday electricity prices all other hours of the week.

Historical simulation

To calculate value at risk with the historical approach, the 99% and 95% percentile is taken over each portfolio’s predefined number of observations . Doing this for the left and right tail will estimate the value at risk with a confidence interval = 5% and 1% respectively.

GARCH (1,1)

To calculate value at risk with this approach, first it is needed to estimate the returns, lagged returns, variance and ultimately volatility with the help of the parameters. Following to this, value at risk is calculated for and by multiplying the standard deviation of the portfolio change over the time horizon with the inverse cumulative normal distribution defined in chapter 3.

Peaks over threshold

The implementation of the Peak over Threshold method involves selecting the threshold , fit the GPD function to the exceedances over and then compute the parameter estimates and

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value at risk. Similarly to the historical simulation, the predefined number of observations , will

be used in each estimation of the value at risk or parameters need for the calculation.

Theory says that should be high in order to satisfy Theorem 2, but the higher the threshold, the fewer observations are left for the estimation of the parameters of the tail distribution function. So far, no automatic algorithm with satisfactory performance for the selection of the threshold is available. The issue of determining the fraction of data belonging to the tail is treated by Danielsson et al. (2001), Danielsson and de Vries (1997) and Dupuis (1998) among others. However these references do not provide a clear answer to the question of which method should be used. A graphical tool that is very helpful for the selection of the threshold is the sample mean excess plot defined by the points

, where ∑ ( )

{ } and

is the number of observations exceeding the threshold .

Further, it is known that the distribution of the observations above the threshold in the tail should be a generalized Pareto distribution (GPD). Different methods can be used to estimate the parameters of the GPD, however in the current paper the maximum likelihood estimation method has been used.

The maximum likelihood function can be described by introducing a sample { } for which the log-likelihood function for the GPD is the logarithm of the joint density of the observations. ⁄ { ∑ ∑

Values ̂ and ̂ that maximize the log-likelihood function are computed for the sample defined by the observations exceeding the threshold . The calculation of the likelihood function and maximizing parameters ̂ and ̂ is done in MATLAB.

4.3 Model estimates

The following paragraphs present and relate the value at risk results obtained with the historical simulation and peaks over threshold method to the realized values.

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Historical simulation

Figure 8

The figure is a combination of four graphs which show the daily portfolio changes in value for intraday electricity prices marked with the compound black line. Above the zero line, the red compound and dash line show the value at risk values in the right tail (losses for an investor being short) for 99% and 95% respectively. Below the zero line, the dash grey line represents the 99% and the compound grey line the 95% value at risk for the left tail (losses for an investor being long).

07:00 winter off peak 18:00 winter peak

07:00 summer off peak 18:00 summer peak

GARCH (1, 1)

The performance of the GARCH model strongly depends on the assumption of returns distribution. Overall, under a normal distribution, the VaR estimates are not very accurate and GARCH effects seemed insignificant or the model failed to converge. Early calculations of VaR with GARCH (1, 1) approach that assumed normality have been abandoned since normality was not approved analytically in the statistical description of the tails for each portfolio.

Peaks over Threshold

Moving to peaks over threshold method, Figure 9 shows the sample mean excess plots corresponding to the portfolios analyzed. An upward linear trend indicates a positive shape parameter for the GPD, a horizontal linear trend indicates a GPD with ≈0 and a linear downward trend can be interpreted as GPD with negative . The figure portrays that for portfolio 18:00 winter peak and 18:00 summer peak is slightly negative in the right tail. For

1 Jan 2014-5 1 Feb 2014 1 Mar 2014 30 Mar 2014 0 5 10 15 20 25 30 35 40

1 Jan 2014-1 3 Feb 2014 3 Mar 2014 30 Mar 2014 -0.5 0 0.5 1 1.5 2

1 Apr 2014-2 1 May 2014 1 Jun 2014 30 Jun 2014 0 2 4 6 8 10 12 14

1 Apr 2014-1 1 May 2014 2 Jun 2014 30 Jun 2014 -0.5 0 0.5 1 1.5 2 2.5

- - - - 99% VaR right tail - - - - 99% VaR left tail

--- 95% VaR right tail --- 95% VaR left tail

- - - - 99% VaR right tail - - - - 99% VaR left tail

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portfolio 18:00 winter peak tends to be negative in the left tail. All other portfolio describe a positive .

Figure 9

The figure is a representation of the sample mean excess plot for the left and right tail determination. (a) 07:00 winter off peak left tail threshold = 0.729 (b) right tail threshold = 2.779

(a) 18:00 winter peak left tail threshold = 0.428 (b) right tail threshold = 0.688

(a) 07:00 summer off peak left tail threshold = 0.588 (b) right tail threshold = 2.252

(a) 18:00 summer peak left tail threshold = 0.408 (b) right tail threshold = 0.578

-1200 -100 -80 -60 -40 -20 0 20 20 40 60 80 100 120 u u -10 -5 0 5 10 15 20 25 30 35 40 0 10 20 30 40 50 60 70 80 u u -2 -1.5 -1 -0.5 0 0.5 1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 u u -1 -0.5 0 0.5 1 1.5 2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 u u -14 -12 -10 -8 -6 -4 -2 0 2 4 0 2 4 6 8 10 12 14 u u -4 -2 0 2 4 6 8 10 12 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 u u -2.50 -2 -1.5 -1 -0.5 0 0.5 1 0.5 1 1.5 2 2.5 u u -1 -0.5 0 0.5 1 1.5 2 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 u u

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Values ̂ and ̂ that maximize the log-likelihood function are computed for the sample defined by the observations exceeding the threshold . The GPD is defined for , however the maximum likelihood is problematic when . To ensure that the parameters estimates remain in the appropriate region during the iterative estimations algorithm used by MATLAB, lower bounds and upper bounds are defined: and .

Table 3

The table records intervals of the parameters psi and beta estimated by maximizing the log-likelihood function.

Winter off-peak Winter peak Summer off-peak Summer peak

Right tail [0.8984, 1.2576] [-0.4679, 0.0126] [0.2132, 0.4396] [-0.5000, 0.2516] [1.8507, 2.5774] [0.3947, 0.7696] [1.1390, 1.5865] [0.3596, 0.9663] Left tail [0.6909, 0.9066] [-0.5000, -0.3023] [0.6720, 0.7662] [0.8959, 1.6228] [0.2145, 0.3330] [0.0802, 0.1146] [0.0898, 0.1080] [0.0096, 0.0370] Figure 10

Figure 10 aligns the parameters ̂ and ̂ for which the log-likelihood function is maximized for 07:00 winter off peak left tail.

Having estimated the threshold , shape and location parameters, and , it is possible to estimate the value at risk based on the peaks over threshold method.

0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 0.2 0.22 0.24 0.26 0.28 0.3 0.32 0.34  

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Figure 11

The figure is a combination of four graphs which show the daily portfolio changes in value for intraday electricity prices marked with the compound black line. Above the zero line, the blue compound and dash line show the peaks over threshold value at risk values in the right tail (losses for an investor being short) for 99% and 95% respectively. Below the zero line, the dash brown line represents the 99% and the compound brown line the 95% value at risk for the left tail (losses for an investor being long).

4.4 Back testing

In this section the actual measurement of VaR proceeds will take place as follows. On the first day of the out-of-sample period, the most recent number of observations returns is used to estimate parameters and value at risk under the considered approaches of HS and EVT. From the parameter estimates, the next-day VaR is estimated using the different methods described in chapter 3. If the realized next-day returns exceed the estimated value at risk values, this is labelled a violation. Moving to time , the estimation procedure is rolled forward one day and repeated. Note that the size of the estimation window is kept constant and simply rolled forward one day at a time, thus ensuring that model estimates are not based on stale data.

1 Jan 2014-5 1 Feb 2014 1 Mar 2014 30 Mar 2014 0 5 10 15 20 25 30 35 40

1 Jan 2014-1 3 Feb 2014 3 Mar 2014 30 Mar 2014 -0.5 0 0.5 1 1.5 2

1 Apr 2014-2 1 May 2014 1 Jun 2014 30 Jun 2014 0 2 4 6 8 10 12 14

1 Apr 2014-1 1 May 2014 2 Jun 2014 30 Jun 2014 -0.5 0 0.5 1 1.5 2 2.5

- - - - 99% VaR right tail - - - -99% VaR left tail

--- 95% VaR right tail ---95% VaR left tail

- - - - 99% VaR right tail - - - -99% VaR left tail

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The simplest backtest consist of counting the number of exceptions (losses larger than estimated VaR) for a given period and comparing to the expected number for the chosen confidence interval.

A more rigorous way to perform the backtesting analysis is to determine the accuracy of the model by constructinga binomial test for the success of these VaR forecasting models based on the number of violations. The test based on violations counts only two possible (binomial) outcomes: a violation or no violation. If  is the quantile for VaR (95% and 99%) the estimated number of violations is given by . A two-sided binomial test will be calculated to compare the null hypothesis against the alternative that the method has prediction errors and it underestimates (too many violations) or overestimates (too few violations) the conditional quantile.

Table 4 gives the back test statistics for the models along with the two-sided -value. A commonly used confidence level in a statistical test is 5%. A -value smaller than shows the rejection of the alternate hypothesis (that the method has prediction errors) and hence is significant. A high -value shows that the model is not rejected.

Table 4

The table details the out-of-sample VaR violations for all competing models. A violation occurs if the realized empirical return exceeds the predicted VaR on a particular day.

07:00 winter off peak 18:00 winter peak 07:00 summer off peak 18:00 summer peak

Total trials 94 64 95 65 =5% Expected 4.5 3.2 4.55 3.25 Right tail HS 6 (0.33) 4 (0.40) 3 (0.86) 5 (0.22) EVT 6 (0.33) 5 (0.22) 2 (0.95) 5 (0.22) Left tail HS 10 (0.02) 1 (0.96) 2 (0.95) 5 (0.22) EVT 10 (0.02) 1 (0.98) 3 (0.86) 5 (0.22) =1% Expected 0.9 0.64 0.91 0.65 Right tail HS 2 (0.24) 0 (0.47) 1 (0.62) 2 (0.14) EVT 2 (0.24) 0 (0.47) 1 (0.62) 2 (0.14) Left tail HS 3 (0.07) 0 (0.47) 1 (0.62) 1 (0.48) EVT 3 (0.07) 0 (0.47) 0 (0.62) 3 (0.03)

Bold indicates significant at 5% or greater.

Under the Hypothesis 1a it was stated that VaR estimates obtained with the EVT method are more accurate than the other tested models, in this case the historical simulation. Table 4 suggests that the models employed, historical simulation and peaks over threshold part of the EVT, are valid but give different results depending on the portfolio and tail.

At =5% for the right tail of the portfolios, both models are approved by the -value. Moving to the left tail for the same confidence level, the models are again approved by the -value except

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for the 07:00 winter off peak portfolio. In this case, even when the -value does not approve none of the methods, historical simulation and peaks over threshold perform similarly.

Looking at =1%, both models give the same result apart for left tail 18:00 summer peak. Here the historical simulation is a valid model whereas the EVT model is rejected.

The conclusion is that both models permit a value at risk estimation and provide results in the same line for almost all portfolios for both left and right tail. However, the results for the 18:00 summer peak summarize that the historical simulation works slightly better for this portfolio. Although number of realized violations is consistent across methodologies, it is worth looking at the deviations between different VaR models. Figure 12 outlines that for the 18:00 winter off peak portfolio, the deviation between the VaR level estimated with the HS and EVT at 99% level is significantly great. The explanation for this is directly linked to the return observed on first day of the portfolio (113% return – Figure 7). Looking at the other portfolios the deviation is smaller and fluctuating around zero.

Figure 12

The results obtained in the previous chapter together with summary statistics for these portfolios reveal that electricity returns are characterized by extremely high levels of skewness and kurtosis, high variance and an extreme range (especially portfolio 18:00 winter off peak). Under such

1 Jan 2014-2 30 March 2014 0 2 4 6 8 10 12 14 16 18 HS - EVT 99% HS - EVT 95% HS - EVT 5% HS - EVT 1% 1 Jan 2014 30 Mar 2014 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 HS - EVT 99% HS - EVT 95% HS - EVT 5% HS - EVT 1% 1 April 2014-0.4 30 June 2014 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 HS - EVT 99% HS - EVT 95% HS - EVT 5% HS - EVT 1% 1 April 2014-0.2 30 June 2014 -0.1 0 0.1 0.2 0.3 0.4 0.5 HS - EVT 99% HS - EVT 95% HS - EVT 5% HS - EVT 1%

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conditions and taking into account the deviations between the VAR levels presented in Figure 12, it can be concluded that although both models perform well, a sophisticated model like EVT which explicitly models the tails of the return distribution is better-equipped to produce accurate VaR forecasts during periods of extreme (113% increase on price) and normal events.

Hypothesis 2a argues that VaR models are more accurate in periods of high liquidity. The liquidity

measure applied this paper is the trading volume. To assess the validity of this statement the portfolios analyzed are ranked in Table 5 from the most to the least liquid in function of the volume traded per portfolio during the full sample.

Table 5

The table details and ranks the average traded volumes in Megawatt per hour for each portfolio. The volumes are taken for the full sample.

07:00 winter off peak 18:00 winter peak 07:00 summer off peak 18:00 summer peak

Volume traded MWh 1998.41 3526.92 1933.39 3451.47

Rank 3 1 4 2

Table 5 ranks 18:00 winter peak portfolio as most liquid of the four in terms of volumes traded. Assessing table 4 and 5, at , both models employed in estimating value at risk are valid with the exception of the 3rd_{ranked most liquid portfolio on the left tail. At , the}

historical simulation model is approved for all portfolios while the peaks over threshold method performs well with the exception of the 2nd_{most liquid ranked portfolio. Overall, the most liquid}

and least liquid portfolios from the above ranking have shown effective results for the models used. However, to draw binding conclusions around any correlations between liquidity and good performance of the forecasting model, more dimensions of liquidity should be taken into account.

5. Conclusion

The paper conducts an extensive value at risk analysis on the intraday electricity prices captured by the German EPEX Spot for the year 2012-2013 and illustrates how historical simulation and extreme value theory can be used to model risk levels for different portfolios.

The analysis confirmed that historical simulation is a reliable model to implement when calculating market risk. Also, seeing that intraday electricity prices exhibit unusual volatility, orders of magnitude higher than financial assets and other commodities, conventional price volatility models derive erroneous results (e.g. GARCH), whereas semi-parametric models of the type extreme value theory give more realistic value at risk estimates in extreme events.

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The performance of the models cannot be linked to a degree of liquidity when liquidity is only measured by trading volume. To tackle this, additional liquidity dimension should be considered when addressing liquidity such as tightness of the market or bid-ask spread. More, the dataset only includes the exchange traded deals leaving out an unknown portion represented by the bilateral trades.

Overall, although the number of realized violations are equal for most portfolio and both models work well, the value at risk estimations done with the historical simulation are much higher than the approximations obtained with the extreme value theory for portfolio 07:00 winter off peak. (see Figure 8 and Figure 11 on the value at risk levels).

5.1 Limitations of the study

The main limitation of the research is represented by the fact our full sample contains intraday electricity prices as observed on the exchange. This leaves out an unknown area covered in bilateral trading prices which can possible smooth the noise in the observed prices on the exchange.

5.2 Potential avenues for future research

This study’s goal is to calculate value at risk with different models for intraday electricity prices without including expected shortfall which is the expected loss given the breach of a particular quantile. However, looking at portfolio 07:00 winter off peak at α=5% on the right tail, the value at risk estimated with the historical simulation approach is much higher than the estimate generated with EVT but both models register the exact amount of realized violations. Here it becomes evident that expected shortfall is a better way of capturing risk and therefore should be considered for future research.

Secondly, the performance of the models cannot be linked to a degree of liquidity when liquidity is only measured by trading volume. To challenge this in future research, additional liquidity dimensions should be considered such as tightness of the market or bid-ask spread besides trading volume.

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5.3 Practical implications

The most evident practical implication of the study emphasizes that an open long positions on any of the portfolios bears higher risk than holding a short position, especially when investing in portfolio 07:00 winter off peak.

5.4 Acknowledgements

I would like to thank my supervisor Chris Florackis for his encouragement and advice during the thesis writing but also the master period. More, I would also like to express my gratitude to David Plomp, senior risk analyst at Vattenfall Energy Trading who has been challenging many of the assumptions mentioned in this paper.

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Measuring value-at-risk for intraday electricity prices.

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Master Thesis

Measuring Value-at-Risk for Intraday Electricity Prices

Andreea Tofan

Table of Contents