Energy Price Risk Modelling

Transcription

1 MSc Finance Louise Daugaard Jensen & Margrethe Nhu Le Pham Supervised by Asger Lunde Energy Price Risk Modelling Forecasting energy spot prices and measuring cash flow risk September 2013 School of Business and Social Sciences, Aarhus University

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3 Abstract This thesis has the purpose of creating a fundament for cash flow management with respect to energy price risk for a North European energy company. To accomplish this object, the paper examines the relationship between North European oil, gas, and coal spot prices in order to formulate a price model applicable for forecasting energy prices in risk modelling. Tests for unit root and cointegration is carried out, and evidence suggests that all three energy commodities share one cointegration relationship with oil and coal being the independent leaders of gas. A VECM estimated upon the derived cointegration relationship is found to perform quite well in back-testing, indicating its applicability as an energy price forecasting tool. The paper further examines the cash flow management tool Cash-Flow-at-Risk (CFaR) and its applicability in measuring cash flow risk from energy price fluctuations. In a practical example of a utility unit it is suggested that a CFaR model with a three-year horizon, 95% confidence level, and oil and gas prices as risk factors, can measure cash flow risk appropriately. This method can be applied to all managerial levels of risk management and can even serve as input to other aspects of the cash flow management task for example by serving as a what-if analysis tool.

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5 Preface This thesis concludes our Master of Science education within Finance from Aarhus University. The topic is motivated by our personal interest in econometric modelling and risk management; while an energy company is chosen as case framework for our study, since the energy sector encounters advanced contemporary challenges within price modelling as well as risk management. We would like to acknowledge the persons who have made it possible for us to complete this thesis. We would like to thank Asger Lunde from Aarhus University, our supervisor, for guidance throughout the thesis-writing process. We especially appreciate his guidance on econometric modelling of energy prices. We would also like to thank Christian Præstegård Sørensen, Lead Risk Analyst, for providing us with insight in price modelling in the framework of an energy company, the energy market, and the challenges an energy company might face. Furthermore, we would like to thank the energy company DONG Energy for providing us with historic data on the relevant energy prices. Lastly, we would like to thank our friends and family for their patience and understanding throughout this process. We hope you enjoy reading our thesis.

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7 Energy Price Risk Modelling Table of content 1 Introduction Problem statement Focus and delimitations Research question focus Geographic area and energy commodities Risk types Structure PART I PART II PART I - Energy Price Modelling 2 Literature review Fundamental energy price models Mean-reverting energy price models Cointegration energy price models Findings in the literature Energy price analysis Energy market integration Energy price drivers Initial data analysis Data description Raw data analysis Treated data analysis Page 5 of 129

8 Table of content 4 Energy price modelling Treatment of data Outlier treatment Seasonality adjustment Price conversion Theory on stationarity Stationarity and non-stationarity Tests for stationarity Empirical analysis of stationarity Theory on cointegration Cointegration Error correction model Vector error correction model Tests for cointegration Empirical analysis of cointegration The final price model Forecasting prices Testing the model Test of assumptions Parameter stability test Back-testing Back-test A Back-test B Back-test C Conclusion PART I Page 6 of 129

9 Energy Price Risk Modelling PART II - Modelling CFaR 7 Managing cash flow risk Cash flow management Benefits of cash flow management CFaR CFaR in practice Constructing the CFaR model Define area of interest Metric specification Exposure mapping Risk factor forecasting Scenario generation Risk metric computation Interpretation of CFaR results Example of CFaR application Background of the renegotiation problem CFaR as input to decision making Conclusion PART II Conclusion References Appendix A: Initial treatment of data Empirical returns Test for seasonality in gas prices Appendix B: Empirical cointegration analysis Empirical analysis of stationarity Inspection of autocorrelation functions Page 7 of 129

10 Table of content 13.2 Johansen s test Lag length Rank test Hypotheses testing Sequential elimination of regressors Oil Gas Coal Appendix C: Forecasting Historical residuals Confidence levels for forecasted ln prices Appendix D: Testing the model Test of assumptions ADF assumption ML assumption Trace and Max-test assumption LR test assumptions OLS assumptions T-test assumptions Page 8 of 129

11 Energy Price Risk Modelling 1 Introduction Producing, trading, and distributing energy are the main activities of an energy company and these activities all have operating cash flows that are highly dependent on the price of energy commodities. Cash flows tie the energy company together by supporting all aspects of activities both internally and externally. The cash flows are, in a sense, the blood that gives life to the corporation, and that is why uncertainty and instability in cash flows can lead to inefficient operations. To administer the produced, supplied and demanded energy, an energy company needs to perform a substantial amount of transactions in the energy market, and all cash flow from these activities depend on price fluctuation, which inevitably will impose risk to the company. Energy price risk is an inherent part of being an energy company, and movements in energy prices can be value creating as well as harmful, since they affect both in-coming and out-going cash flows. Therefore, risk managers strive to optimise the balance between earning opportunities and associated risk in line with traditional portfolio theory. A crucial element in cash flow management is the task of constructing a qualified risk metric with the ability to forecast future price movements in the energy market. However, it is not a trivial task to forecast energy prices over a long horizon. Energy prices are driven by traditional market forces of supply and demand, like any other commercial asset, but they operate under physical constraints imposed by production/extraction, storage, transportation, and usage of the commodity. Therefore, energy prices often show irregular movements with large volatilities when friction between supply, demand, and the physical settings occur. Additionally, the role that energy commodities play for households and industry creates a complex relationship between the different energy commodities. Constructing the framework for cash flow risk management for an energy company is a complicated task. Both the practical and the theoretical field have focused on this task for Page 9 of 129

12 1 Introduction quite some years now, and the resulting toolbox contains a selection of highly qualified methods; both for price modelling and risk management. The difficulty lies within choosing the appropriate tool for the purpose from the selection of tools, which all have different properties and applications, and then tailoring the chosen method to the energy company and its risk profile. 1.1 Problem statement The main activities of an energy company have operating cash flows that are highly dependent on often volatile energy prices. Therefore, managing cash flow risk is a crucial task that needs to be addressed. However, this is a very comprehensive task with many aspects to it. The objective of this thesis is to provide a solid foundation for the management of cash flow with respect to energy price risk in an energy company. An important element in a foundation for risk management is the measuring of risk; an element that is dependent on the essential task of forecasting future energy prices. Therefore, in order to construct at solid foundation for managing cash flow with respect to energy price risk, this thesis will investigate the following two research questions: a) How can an energy company forecast energy prices with respect to managing cash flows? b) How can an energy company measure cash flow risk originating from energy price fluctuations? Throughout this thesis, the two research questions will be investigated from the views of both academics and practitioners, and a practical solution, that incorporates both fields, will be proposed for each question. The first research question is approached in a contemporary and empirical study of North European oil, gas and coal commodities. It will be addressed by first performing an initial analysis of energy prices, in order to gain insight in the drivers that cause the movements and complex relationship between energy prices. With support from the econometric theory, we will argue for a cointegration method of modelling the energy price processes. We will then perform the relevant empirical analysis and construct an energy price model with the purpose of forecasting energy prices. Finally, we will validate our findings and test the performance of the model as a forecasting tool. The second research question is approached in an exemplified framework based on the premises from realistic business operations of a North European energy company. Firstly, we Page 10 of 129

13 Energy Price Risk Modelling discuss cash flow management and the benefits of it from a corporate perspective. Supported by theory, we will argue for how a CFaR model can measure cash flow risk from energy price fluctuations. Secondly, as a practical example we will, based on the forecasted prices from the first research question, illustrate the applicability of the method by building a CFaR model for a business unit of an energy company. We interpret the results and applications of using CFaR in cash flow management. Finally, we will extend the example to include the application of the CFaR model in decision making. By solving these two research questions and constructing practical solutions, we believe that this thesis creates a good foundation for a solid cash flow management framework. 1.2 Focus and delimitations Research question focus The two research questions that this thesis will answer relates to two different fields of study: Econometric time series modelling and risk management. These fields have very different depths to them and different degrees of generalisability and constructing practical solutions require data of different amounts and availability. Consequently we chose to focus mostly on the first research question, which is the price model. The price model will be built from empirical market data, which have been kindly lent to us, and from well-founded econometric theory. The results will be comparable to findings in the literature from academics and practitioners. We do not have access to the needed data, in order to construct a empiric CFaR model, and the CFaR model does not have the same theoretical depth as the price model. Furthermore, a CFaR model needs to be tailored to the individual company and reflect its business area and risk complexity. We do not have the sufficient time or information required to build a complete CFaR model; instead we need to make simplifying assumptions about the complexity and risk profile of the company in order to illustrate the application of CFaR. Therefore our investigation of the second research question will be based mostly on practitioners guides and arguments. In conclusion, due to the fact that we have the possibility of performing an empirical and thorough analysis of our price model, that is comparable to existing literature and that does need simplifying assumptions, we choose to put more weight on the first research question of our problem statement than on our second research question Geographic area and energy commodities This study will be based on the North European energy markets. Energy companies in different geographic areas operate under different circumstances due to the physical availability of energy commodities in the area and other factors such as regulation and Page 11 of 129

14 1 Introduction competition. Therefore, we need to define our geographic area in order to analyse the appropriate energy prices, energy types, setup, and surroundings. A similar study could be performed in another geographical area, but due to personal interests we have chosen our home market. Despite the fact that an energy company is exposed to energy price risk from many types of energy commodities, we choose only to include oil, gas, and coal in our thesis. The reason for our choice is that these are the three energy commodities that pose the main risk to a typical energy company in Northern Europe. Hence, we have excluded CO2-emission allowances and input in renewable energy production such as biomass and wood pellets. Electricity is also excluded even though this commodity poses great risk to cash flows and is of high importance to an energy company. We did a preliminary empirical investigation on power prices, comprised of unit root tests, which showed that power, in our geographical area, is unsuitable for modelling using cointegration. Alternatively power prices can be modelled in another framework using other methods, but due to time limit and scope of interest this is left out Risk types An energy company has many sources of risk. We only focus on energy price risk, and other types of risk such as financial risk, exchange rate risk, credit risk, operational risk, and business risk, are left out. A competent enterprise-wide risk management system would consolidate the management of all these risks, but due to the time limit, scarce information, and scope of interest, these other types of risk are excluded from this thesis. 1.3 Structure This thesis is contains two parts: PART I Energy price analysis and modelling and PART II Modelling CFaR PART I The objective of PART I is to give an answer to the first research question: How can an energy company forecast energy prices with respect to managing cash flows? PART I is initiated with Chapter 2 which contains a literature review. Here we review the different methodologies and findings within energy price modelling and elaborate on the cointegration literature. This chapter serves to place our research and selected method in a broader perspective. Page 12 of 129

15 Energy Price Risk Modelling Chapter 3 contains an analysis on energy prices from an energy market perspective. It consists of an introduction to energy market integration, energy price drivers, and a graphical analysis of empirical data. We close chapter 3 by formulating hypotheses on the energy price relationship. These hypotheses create basis for the following price modelling and advocate for modelling using a cointegration approach in chapter 4. Chapter 4 contains the energy price modelling. Initially, the empirical data is treated and prepared for econometric analysis. Next, theory regarding stationarity is presented and empirical tests are performed, since price processes being stationary or non-stationary is a determining factor for the choice of price modelling approach. Then we present the theory on cointegration, the Vector-Error-Correction-Model (VECM) representation, and Johansen s approach for testing for cointegration. In the empirical analysis of data we use Johansen s approach to determine number of cointegration relationships between the three price series and to support the hypotheses we formulated in chapter 3. In the end we estimate a final model via the VECM representation and accomplish the purpose of PART I by forecasting energy prices. In chapter 5 we validate the final model and its forecasting performance by testing the model in various ways. We test the underlying assumptions of the statistical tests that we perform, and we test the model estimation procedures. We also investigate the stability of the parameters of the model, and we test the forecasting performance of the model via backtesting. We end PART I with chapter 6 which is a conclusion on our constructed energy price model, its forecasting performance, and its applicability PART II The objective of PART II is to give an answer to the second research question: How can an energy company measure cash flow risk originating from energy price fluctuations? PART II is initiated with chapter 7. Here the aim of cash flow management and the theory of CFaR are presented. We argue for the benefits of managing cash flows and how CFaR can contribute as a risk management tool. Chapter 8 contains practical examples of the construction of CFaR. First we build a CFaR model and perform the necessary steps in the process of constructing CFaR. We tailor the model to a realistic example and interpret the results of the model. To illustrate the applicability of the results, we apply the CFaR model in a decision making situation. Page 13 of 129

16 1 Introduction We close PART II with chapter 9 where we conclude on the results and applications of using CFaR in cash flow management. The thesis ends with an overall conclusion in chapter 10 where we summarise the answers to our problem statement. Page 14 of 129

17 PART I Energy price analysis and modelling The objective of PART I is to answer the first research question: How can an energy company forecast energy prices with respect to managing cash flows? The research question is approached in a contemporary and empirical study of North European oil, gas and coal commodities. PART I leads you through a literature review on the different fields within energy price modelling, an analysis on energy prices from an energy market and price driver perspective, and graphical inspections of historical price data. This results in hypotheses on energy price relations that, together with econometric analyses and modelling procedures, lead to the formulation of a final price model. The final model is in the end subject to validation of the reliability and subject to a test on performance as a forecasting instrument. We close PART I by concluding on our findings regarding the research question.

18 2 Literature review 2 Literature review Energy price movements have not been analysed extensively in the literature until more recent times. Before the 1990s most energy prices were heavily regulated by governmental entities and therefore not very relevant to analyse from a statistical point of view. When the liberalization processes began, first in the US and then in EU, the interest in energy price modelling increased and likewise did the application of contemporary theories and the evolution of new methodologies. Today energy prices are thoroughly studied and when it comes to modelling the price processes we find that the prevailing methodologies can be divided into three branches of energy price models: Fundamental models, mean-reverting models, and cointegration models. All three branches often consider stochastic fluctuations in the price series and also co-movements between the prices of energy commodities from different regional areas or different commodity types. Co-movements between energy prices (cross-region or cross-commodity) is of high interest to the energy companies due to the direct effect price spreads may have on their profits. We can reveal that our later analysis will lead us to apply price modelling methods founded on the cointegration type of price models, why we will review the literature on this branch thoroughly. However, we also believe that the other branches are important to review; first of all in order to grasp the price modelling landscape and to put perspective on our choice of modelling approach, but also because of the recent emergence of the so-called hybrid models that advocate for the combination of elements from the different branches. Therefore, we will now present a short review of the literature on fundamental and mean-reverting models before elaborating on the literature on cointegrated price models. Page 16 of 129

19 Energy Price Risk Modelling 2.1 Fundamental energy price models Fundamental energy price models cover a range of price models that all have one thing in common: They all include external factors to model energy price processes. By external factors we mean for example economic factors such as the interest rate or the convenience yield, or physical factors such as the weather or storage capacity. The fundamental models can also include internal factors which is the price itself. The models are typically based on regressions and can furthermore include stochastic processes - for example mean-reversion or nonstationary processes. Gibson and Schwartz (1990) were pioneers within the field of energy price modelling. They priced oil contingent claims via a two-factor model; where the first factor was the oil price itself, modelled as a non-stationary stochastic element described by a Geometric Brownian Motion (GBM), and the second factor was a convenience yield. Later Schwartz (1997) went on to modify Gibson and Schwartz s model and proposed a threefactor model to price claims on oil, copper, and gold. Besides the two factors from before, the model now also included an interest rate factor. A major difference between this model and the one proposed by Gibson and Schwartz (1990) was that the price was assumed no longer to follow a non-stationary GBM. Instead it followed the Ornstein-Uhlenbeck version of the GBM which is mean-reverting. 2.2 Mean-reverting energy price models The mean-reverting models cover a range of price models, where the stochastic and fluctuating prices are continuously dragged towards their mean or equilibrium level. The models often include lagged price variables, and mean-reversion is maintained by a speed of adjustment parameter. This type of model can also be combined with stochastic jump diffusion components to encounter the price peaks characterising many energy prices. In the mean-reverting framework, co-movements in prices are often incorporated through covariances and traditional correlation principles. Gibson and Swartz (1990), Brennan (1991), Seguin and Smoller (1995), and Ross (1995) all find evidence that energy spot prices follow mean-reverting (Clewlow & Strickland, 2000). Pilipovic (2007) presented two acknowledged Pilipovic Models, which have been applied for most types of energy commodities: A one-factor and a two-factor model. The one-factor model is a mean reverting GBM process and the two-factor model is extended to include a stochastic long-term equilibrium level that follows a correlated GBM. Pindyck (1999) analysed the price of oil, gas, and coal, and he found by means of unit root tests that prices indeed are mean- Page 17 of 129

20 2 Literature review reverting, but that the rate of mean-reversion is very slow. Additionally, he found that the mean the prices revert to in itself is fluctuating. This lead to Pindyck s final price model which also was an Ornstein-Uhlenbeck version of the GBM, but this model allowed for fluctuations in both the level and the slope of the trend. Around the same time, with inspiration from jump diffusion models, Clewlow and Strickland (2000) proposed a rewarded price model which included a jump process to capture the behaviour often seen in energy prices; a behaviour that cannot be modelled by mean-reversion or by jump diffusion models alone. 2.3 Cointegration energy price models As opposed to mean-reverting processes, which are stationary, non-stationary processes wanders and do not return to a long-term mean. If non-stationary price series co-move and have a linear relationship that is stationary, they are said to be cointegrated. In the existing literature on cointegration among energy prices two types of cointegration is investigated: The first is cointegration between different types of commodity (cross-commodity) and the second is cointegration between the same commodities but from different geographic or regional areas (cross-region). First, we present the findings of cross-commodity cointegration. Pioneers within this type of cointegration are Yucel and Guo (1994). They analysed the US prices of oil, gas, and coal from 1947 to 1990 and concluded by means of Johansen s approach, that from 1947 to 1974 a cointegration relationship existed between oil and gas, and for the remaining period a cointegration relationship existed between all three prices. Bachmeier and Griffin (2010) also compared the US price of oil, gas, and coal but for the period of 1991 to They found by means of Engle and Granger s two-step procedure that the commodities are only weakly cointegrated, and that the presence of cointegration depends on the geographic area under study. Serletis and Herbert (1999) investigated the relationship between oil, gas, and power in the US and found, by means of Engle and Granger s method, evidence of cointegration between oil and gas but not between oil, gas, and power. Literature on cross-region cointegration includes Ghouri (2006) who investigated the oil and gas markets of USA, UK, and Japan. He found cointegration in both oil and gas across regions by means of Johansen s test. Asche, Osmundsen, and Tveterås (2002) tested for market integration between Norwegian, Dutch, and Russian gas prices and they found evidence of cointegration, also by means of Johansen s test, and furthermore that the Law of One Price (LOP) and the Generalised Composite Commodity Theory (GCCT) hold. Page 18 of 129

21 Energy Price Risk Modelling Besides modelling using cointegration alone, some authors have added exogenous variables to the models. Villar and Joutz (2006) examined US oil and gas and applied Johansen s test for cointegration while adding exogenous variables such as storage, seasonality, and other transitory shocks. They found that gas prices follow oil prices, and that the cointegration relationship exhibits a positive time trend. Brown and Yucel (2008) followed the work of Villar and Joutz but included weather and shutdown of production in the Gulf of Mexico. They, as well, found that gas prices follow oil prices. Hartley, B Medlock, and Rosthal (2008) also followed Villar and Joutz, but instead of a time trend they considered a technology trend; a link between oil and gas was also the final result. To the best of our knowledge, no authors have performed a study on the same commodities, geographic area and time period like us. The articles most relevant for our study on cointegration between oil, gas, and coal in Northern Europe are reviewed in the following: Asche, Osmundsen, and Sandsmark (2006) investigated the decoupling of energy prices caused by a deregulation in gas markets. This deregulation serves to liberalise the gas markets which, among other things, may lead to a pricing of gas contracts independent of oil prices. In this study they first investigated the degree of integration in the UK market for natural gas, crude oil, and power. The period of interest was the interim period after the deregulation of the gas market in 1995 and until the opening of the Interconnector in By use of Johansen s test for cointegration they concluded that the three energy commodities were highly integrated in the period of 1995 to 1998, that LOP and GCCT hold, and that oil is the leading energy commodity. Second, they investigated the period after the opening of the Interconnector, and found that prices were no longer cointegrated, which is in line with their expectations of a decoupling due to deregulation. Contradictory findings were present in a study by Panagiotidis and Rutledge s (2007), who also investigated the decoupling in UK oil and gas market from 1996 to Panagiotidis and Rutledge deployed a more thorough analysis of the energy prices by using three different methodologies and, furthermore, also by allowing for structural breaks. Via Johansen s test for cointegration, a non-parametric test for cointegration by Breitung, and recursive techniques Panagiotidis and Rutledge found evidence of cointegration between UK gas and oil in the whole period. In fact they found a very strong relationship in the period of right after the opening of the Interconnector, which is contradictory to the findings of Asche, Osmundsen, and Sandsmark (2006). Panagiotidis and Rutledge concluded that this casts doubt on the EU Commission s effort to liberalise the EU gas markets, since a deregulation according to theory should lead to a weakening of a linkage between oil and gas. Page 19 of 129

22 2 Literature review 2.4 Findings in the literature Findings of cointegration between energy prices and findings of mean reversion are almost equally represented in the literature. Most analysts explain that the different findings are results of the choice of energy commodity, geographic area, and time period under investigation, and they conclude that there is no final answer to whether all energy prices are mean-reverting or non-stationary. In the reviewed literature, we also find many examples of cointegration between oil and gas: Yucel and Guo (1994), Bachmeier and Griffin (2010), Villar and Joutz (2006), Asche, Osmundsen, and Sandsmark (2006), and Panagiotidis and Rutledge (2007) all find evidence of oil and gas being cointegrated with oil being the leading price. Yucel and Guo (1994) and Bachmeier and Griffin (2010), furthermore, find cointegration between oil and coal and that oil is the price leader. While a link between oil and gas prices is acknowledged widely, there are still discussions about how a current transition in the gas markets will affect the link between the two commodities prices. Asche, Osmundsen, and Sandsmark (2006) found that the EU liberalisation decouples oil and gas; while Panagiotidis and Rutledge (2007) found that liberalisation enforce such link. This contradiction may arise for different reasons. Barcella (1999) investigated oil and gas coupling in the liberalised US market, and concluded that the coupling of oil and gas not only comes from the oil-dependent pricing of gas, which will dilute with liberalisation, but also from underlying economic factors and substitution. This means that liberalisation does not necessarily lead to a decoupling, since other reasons for a linkage may prevail. Now that we have reviewed the literature on energy price models, we will continue with an initial energy price analysis in order create foundation for further empirical investigation in the form of a formulation of hypotheses. Page 20 of 129

23 Energy Price Risk Modelling 3 Energy price analysis This chapter contains an initial energy price analysis with the purpose of formulating hypotheses regarding energy price relationships the in North European energy markets. These hypotheses serve as basis for the price modelling in chapter 4. When modelling energy prices as an input to risk modelling, it is very important to take into account whether the prices, which the company are exposed to, are interrelated. Therefore the energy price analysis is initiated with a presentation of the idea of market integration. After the presentation of market integration, we will discuss various underlying drivers that may cause markets to be integrated. Next we perform a graphical analysis of the empirical price data to further examine the price relationship. The presentation of market integration, price drivers, and the graphical data analysis serve as foundation for the formulation of hypotheses regarding energy price relationships, which will be presented at the end of this chapter. 3.1 Energy market integration Market integration describes a situation where prices of goods from different suppliers or of different commodity type move in the same direction, and price differentials are only present if there are differences in transportation costs or in the quality of the goods (Asche et al., 2002). A number of alternative market integration definitions exist. In the context of energy market integration we find George J. Stigler s definition from 1969 suitable for our thesis. Stigler defines the integrated market as, the area within which the price of goods tends to uniformity, allowances being made for transportation costs, (Asche et al., 2002). In this definition Stigler refers to what we call cross-region integration, but the definition can without Page 21 of 129

24 3 Energy price analysis loss of generality be extended to include cross-commodity integration as well - meaning integration across different energy commodities such as integration between oil, gas, and coal. The basic relationship between prices under investigation is then: Eq. 1 where is a constant term that captures the transportation costs and quality differences, and quantifies the linear relationship between prices. If there is no relationship between the two prices, if then there is a relationship but the goods are imperfect substitutes, and if the two goods are perfect substitutes. The latter indicates that the Law of One Price (LOP) holds (Asche et al., 2002). If the prices are integrated ( 0) then in accordance with Stigler s definition, there exists one common market for the n prices under investigation instead of n independent markets. Before 1990 the European energy markets were far from integrated, since the energy sector was organised as state owned monopolies. The governments regulated the prices for the nonsubstitutable energy commodities and stabilised the fluctuating production, storage, and distribution costs with the intention of protecting the consumers. The highly regulated and clearly separated European energy markets were not very efficient and the prices were high (Rotaru, 2013). In the mid 1990 s EU began a liberalisation process of the regulated gas and electricity markets throughout Europe, which in the end should result in full market integration. The purpose was to ensure low prices and supply security. The process contained three main elements: 1) Prices should no longer be regulated by authorities but follow supply and demand, 2) consumers should be able to choose among suppliers, and 3) ownership of generation companies and network operators should no longer be bundled. The process also involved initiatives to entail cross-region transactions both infrastructural and commercial (European Monitoring Centre of Change, 2008). As can be seen from these initiatives, it is mostly cross-region market integration that is a consequence of liberalisation. The current situation in EU is that some countries are fully liberalised regarding all types of energy commodities, this for example goes for the UK, while other countries, such as Romania, lack behind on the liberalisation process (European Monitoring Centre of Change, 2008). This obviously affects market integration, and that is why the degree of market integration in Europe differs and is very dependent on the individual country. In Northern Europe, which is our area of interest, all commonly traded energy commodities are generally considered integrated across regions, while cross-commodity integration is still a field characterised by different findings and opinions. Page 22 of 129

25 Energy Price Risk Modelling Energy price drivers The underlying causes of market integration stem from substitution, distribution, and interdependency in pricing of contracts among energy commodities (Blanco & Pierce, 2012) and (Asche et al., 2002). If these causes are absent or somewhat inefficient, markets are unlikely to be integrated. A good starting point when modelling energy prices is therefore to look at the forces that drive energy prices. It is ultimately these drivers that determine whether market integration exists. In this section we review the drivers behind the North European oil, gas, and coal prices. Integration is enhanced if energy commodities can substitute each other. Gas and coal are close substitutes, since they both can serve as input in power production and heat generation (American Petroleum Institute, 2013). In Northern Europe most newly built power plants, and even some old plants, are being designed to fuel-switch between gas and coal as input in heat and electricity production. Therefore, if gas prices are relatively high, energy companies will buy coal to fuel the power plants. In this case the coal price will rise due to an increase in demand until supply and demand equilibrium is established. This enforces the link between gas and coal. Oil also serves as a substitute, although to a lesser degree, since oil is still used for heating of private homes by oil-fired boilers and has traditionally also been used for electricity production as well. The main consumption of oil, however, goes to transportation where there are no adequate substitutes (American Petroleum Institute, 2013). When the energy commodities are substitutes, factors like weather conditions, trade embargoes, and supply chain malfunctions will directly or indirectly affect the demand for all substitutable energy commodities. Increased possibilities for distribution, transportation, and transaction also enhance integration between energy markets mostly cross-regional but also cross-commodity, as efficient distribution channels enhance substitution possibilities e.g. enables a fast switch between substitutes. This is because transportation costs, distribution difficulties, and trade barriers hinder exchange of commodities. The European energy infrastructure has through the liberalisation process expanded both in range and capacity especially when it comes to the Liquefied Natural Gas (LNG) pipelines: The opening of the Interconnector between Belgium and UK and more pipelines under construction, e.g. Turkey-Greece-Italy, enable more crossborder transaction of gas. The recent infrastructure expansion has even led to an almost integrated European gas market. The European oil and coal markets are already considered integrated across regions, perhaps even globally integrated, due to their more portable substance. The liberalisation process has also brought an increase in commercial energy trading, which further enhances integration, since more mature markets enable transactions and eliminate arbitrage. Page 23 of 129

26 3 Energy price analysis Interdependency in pricing of contracts also leads to market integration. Gas has, historically, mainly been traded in long-term contracts, where the price of the gas volume was expressed in term of an oil price, due to the co-production of the two commodities and in lack of a commercial gas market. In recent years, the maturing of an independent and sizeable gas market has led to renegotiations of gas contracts and a-delink of gas and oil (Panagiotidis & Rutledge, 2007). However, many of the oil-linked gas contracts are still present and even prolonged due to the bargaining power of the market participants (Asche et al., 2002). Factors of more fundamental character may also have common and linking effects on energy commodities. These factors can for example be political, macroeconomic, meteorological, environmental, or technological and affect the general demand and price level of all energy commodities instead of any specific energy commodity (Amadeo, 2012) and (Asche et al., 2002). For example many practitioners and academics argue that price fluctuations in oil affect all other energy commodity prices. This is because oil is the most consumed energy source and the price is a signal of global demands for energy (Amadeo, 2012). Therefore, oil is considered the global market leader of all energy commodities; however, regionally the monopolistic power of oil can be discussed. To sum up we see that energy commodities may share common drivers and be subject to the same supply and demand factors which can lead to integration. In the next section we will therefore make an initial analysis of the energy prices graphically in order to present hypotheses about potential relationships between them. 3.2 Initial data analysis First we present the empirical data on the energy commodities of interest. An initial graphical data analysis of the empirical price series should indicate whether the prices may be integrated and how. With support from the findings in section this section closes with formulation of hypotheses, which will be basis for the proceeding econometric analysis in chapter Data description The data we use for our empirical price analysis is kindly lent to us from the Danish energy company DONG Energy. We have received price data on Brent crude oil, NBP natural gas, and API2 coal. In accordance with the prevailing market opinion, DONG Energy has confirmed to us, that these products are considered the leading products with regards to the three energy commodities: Oil, gas, and coal in Northern Europe. This means that the prices already are assumed to be cointegrated cross-regional, since one benchmark price can represent the price Page 24 of 129

27 Energy Price Risk Modelling of e.g. all oil prices in Northern Europe. We believe that this is a valid assumption, since our discussion on price drivers in showed that the recent infrastructure expansion has led to cross-regional integration in the North European energy markets. Also received are foreign exchange rates between DKK and the currency of the energy products 1. More information about the data is presented in Table 3.2.1, and when we from now on use the terms oil, gas, and coal we will refer to these specific energy products from the table, unless otherwise is clearly stated. Table Data description Energy commodity Product Unit Description Source History Oil Brent USD/bbl Raw crude oil index in Northern Europe ICE ; Gas NBP Pence/therm UK Natural Gas index Heren ; Coal API2 USD/tonn Rotterdam Coal index Argus ; FX USDDKK USD/DKK USD/DKK Nationalbanken ; FX GBPDKK GBP/DKK GBP/DKK Nationalbanken ; All energy prices are wholesale spot prices. At this price stage, in the energy supply chain, most transactions are made and here competition is at its highest, this leads to the most liquid and market-forces-efficient commodity prices. The wholesale price of coal is obtained by Argus from the OTC market, since coal does not have a sufficient liquid exchange market; why, the majority of coal trading is OTC. The remaining prices are observed from the respective commodity exchanges, since their spot markets are considered adequate for assessment of the prevailing prices. The prices are monthly averages of daily closing prices. Monthly sampling is a suitable sampling frequency, when the purpose of the price analysis is to forecast prices as input to cash flow management. Higher frequency (e.g. daily or weekly) contains too much noise which can blur the long-term relationship. Furthermore, monthly sampled price data is appropriate when assessing e.g. production and utility activities, since most contracts are settled at 1 In this thesis we only use exchange rates to convert the energy prices into a common currency. Exchange rates could further have been investigated for usage in a Cash-Flow-at-Risk model; however, we do not investigate this due to the limitations of the thesis. Page 25 of 129

28 3 Energy price analysis monthly averages. Also, the coal price is only available at monthly frequency for the first many years of the data history. Missing daily prices are ignored. All data is regarded as reliable and free from errors, because they consist of monthly averages of daily prices that all enter into the daily risk surveillance in DONG Energy s risk department. Therefore, all data is validated and corrected if faulty. The length of data history is determined by the availability and existence of data Raw data analysis Here we perform a graphical inspection of the raw energy price data. What we wish to look for are general tendencies in the price series that can give rise to a later formulation of hypotheses on cross-commodity integration. Figure Oil price Figure shows the monthly oil prices over the whole sampling period from September 1996 to December The price is increasing through the period with only short periods of consecutive decreases. The price ranges from a minimum of 9.88 USD/BBL in December 1998 to a maximum of USD/BBL in July 2008, which is more than twelve times the price in The rapid increase in 2007 to mid-2008 can, according to Kahn (2009), be explained by extraordinary behaviour in the macroeconomic supply and demand drivers such as the economic upturn, and by speculations on the energy derivatives market that lead to a price bubble the first half of According to Khan (2009), if the bubble had not been there the maximum price of oil would have been around USD/BBL in that period (this should be compared to the empirical maximum price of USD/BBL in July 2008). Macroeconomic price drivers also explain the decline in prices that happened in the second half of 2008, where the recession lowered the demand for oil, as the economy in general was in recession. At the same time new oil reserves were discovered which further lowered the prices (Khan, 2009). In Page 26 of 129

29 Energy Price Risk Modelling the aftermath of 2008 the oil prices regained strength and therefore shows a steady increase from 2009 and onwards. Figure also shows the high volatility that characterises oil, which in particular is manifested by the extreme fluctuations in 2008 and large fluctuations in From the time series it seems that the volatility has been higher in more recent years. The current market expectation is that the oil price, like the price for all other energy commodities, will continue to increase due to a growing demand and uncertainty in future supply, and that the volatility will remain high. Figure Gas price Figure shows the monthly prices for gas. We observe that the price for gas, like the price for oil, can be characterised as increasing throughout the period. Gas experiences an almost eight times increase in price from a minimum of 7.94 Pence/Thmin September 1999 to a price of Pence/Thmin September Again the general market opinion is that the price increase is caused by an increase in demand due to the economic upturn. But gas may also increase due to the contractual link between gas and oil prices, already mentioned in 3.1.1, that to some extend forces gas to follow oil. It could seem as if gas follows oil but with a time lag, and price changes happen in smaller steps. We will investigate this further when we have converted the prices into comparable sizes, so that we can graph them simultaneously. The gas prices seem far more volatile than the price of oil, and there are relatively long periods with extreme price peaks. Furthermore, it seems that the gas prices experience a seasonal pattern with higher prices in the cold months and lower prices in the warm months. We did not see a comparable seasonality in the oil price. Page 27 of 129

30 3 Energy price analysis Figure Coal price As can be seen from Figure 3.2.3, coal prices also increase throughout the period of September 1996 to December but of lesser magnitude than oil and gas. The coal price ranges from a minimum of in July 1999 to a maximum of in August This corresponds to a six fold increase; thereby, coal price has a narrower price span than the oil and gas prices. The extreme changes in oil and gas prices in 2008 also recur for coal; however, it is worth mentioning that the peak does not occur in the same month for the three energy commodities, and it seems both gas and coal react with a few months lag to oil. We also see that the volatility for coal seems less than for gas and oil, which appears reasonable considering the good transportation and storage opportunities for coal, lowering its proneness to react to supply and demand imbalances. Furthermore, we do not see any clear sign of seasonality in the coal prices. Lastly, we observe that while oil and gas prices have been at a relatively high level in more recent times (2011 and onwards), the level of coal cannot be considered strikingly high at the end of our time series. In the three energy prices series presented above, we see common features that lead us to suspect cross-commodity integration of the prices. We observe that oil, gas, and coal have all been increasing through the period, and they all have common periods of rather extreme price levels. Especially the price increase around fall 2008, and subsequent price drop, seems common for the three energy commodities, although, gas and coal drops with a time lag compared to oil. In order to investigate how the prices react in relation to each other and to get a clearer indication of who leads who, we need to convert the prices into comparable sizes Treated data analysis Here we present the treated empirical data in order to make a graphical analysis on the relations between the three energy price series. By treated we mean that the data is adjusted for outliers, seasonality, and that they are converted to ln DKK prices. Please refer to 4.1 for Page 28 of 129

31 Energy Price Risk Modelling an explanation of the treatment. The treatment makes the prices comparable, and that is why we can depict them in one single graph in Figure below. Figure Treated oil, gas, and coal prices From Figure above it seems that the oil, gas, and coal prices follow each other rather closely throughout the years. The relationship appears to be strong and consistent despite short-run fluctuations, and it seems as if gas and coal react with a time lag to the oil price movements. We see that for the first years up until 2002 coal seems to react to movements in the oil price: When oil prices decline in the first period and then increase and decrease again, coal prices also move in these directions but with a delay. But from 2002 and onwards it is a somewhat mixed picture of whether oil or coal makes the initial move. The relationship between oil and gas on the other hand is clearer: Oil moves first and then gas follows with a delay the whole period. Therefore, one could state the hypothesis that oil, gas, and coal are all integrated, and that oil is the independent price leader which both gas and coal follow. This hypothesis is consistent with the market opinions presented in section 3.1.1, regarding price drivers, declaring oil as the ultimate leader of all other energy commodities. Therefore, we present our first hypothesis on energy prices as follows: Hypothesis 1: Oil, gas, and coal are all cointegrated with oil being the independent leader of both gas and coal. Alternatively, from looking at the empirical price movements, one might argue that from 2002 and onwards both oil and coal jointly lead the price of gas. It seems that gas often lies between both prices, and that it reacts to both their movements but with a delay. This gives rise to a second hypothesis that oil and coal are both independent leaders of gas, and this hypothesis is also consistent with the findings in section Here we stated that gas and oil are connected Page 29 of 129

32 3 Energy price analysis via the long-term contracts, and gas and coal are connected since they are direct substitutes in power and heat generation. Therefore, we present our second hypothesis on energy prices as follows: Hypothesis 2: Oil, gas, and coal are all cointegrated with oil and coal being the independent leaders of gas. The energy price analysis of this chapter has resulted in two hypotheses on energy price integration. Based on these market hypotheses, we will conduct a thorough econometric analysis in order to investigate whether there indeed is an integration relationship between the prices. Page 30 of 129

33 Energy Price Risk Modelling 4 Energy price modelling From our energy price analysis in the previous chapter, we proposed two hypotheses both stating that oil, gas and, coal prices share a cointegration relationship. The aim of this chapter is, therefore, to conduct a thorough econometric analysis of the prices in order to test the hypotheses. In the end, we should derive a price model that can be used for forecasting of energy prices for cash flow risk assessment. This chapter serves as the primary element in answering our first research question: How can an energy company forecast energy prices with respect to managing cash flows? This chapter starts by a description of the treatment of the empirical data, which is performed in order to cleanse and prepare data for econometric analyses. Next we present the theory on stationarity and conduct an empirical analysis hereof, since it is important for the choice of methodology to find out if the price processes are stationary or non-stationary. Then the theory on cointegration is presented, and an empirical analysis is performed with the aim of finding support for the market hypotheses we proposed in chapter 3. We close this chapter by presenting the findings of our analysis in a final model of the three energy price series and by applying the model for forecasting. 4.1 Treatment of data In this section we prepare the data for further analysis. There are generally two reasons why our original data needs to be treated before an analysis: First of all, we need to cleanse the data for elements in the price series that we do not wish to model or elements that may hamper the analysis e.g. outliers and seasonality. Second of all, we need to transform the price series in order to compare them with each other and to be able to apply econometric procedures. Page 31 of 129

34 4 Energy price modelling Outlier treatment The purpose of modelling energy prices is, in our case, to forecast energy prices to use them in a CFaR model. A further elaboration of CFaR is found in PART II of this thesis. For now it is worth noticing that the CFaR model is a risk management tool that estimates risk under normal market conditions, and it is not the purpose of the CFaR model to predict prices under irregular market conditions. There is other types of risk management tools that is appropriate for the management of extreme market conditions. Therefore, we do not need to model very extreme price scenarios, and including extreme observation in our data may hamper the analysis, as these extreme observations are often driven by severe market irregularities that can be contradictory to the natural relationship of the prices. With that in mind we look at the price series presented above and see few very extreme observations, which might not be the results of normal market conditions. We see a large price jump in oil and coal in July 2008, and periods of unusual high prices for gas in the winter and perhaps also in most of These observations are forced by unusual events: In 2008 there was the outburst of the financial crisis, where economic uncertainty and speculations drove the oil prices to extreme levels (Khan, 2009). Simultaneously, with the financial crisis in July 2008, there also was a surge in coal prices caused by the unusual high demand for coal from especially China (Flood, 2008). In November 2005 there was a fire at the main gas storage facility in the UK combined with severe cold weather leading to unusual high gas prices (Simmons, Horlings, & Cronshaw, 2006). Unusual events caused by severe market irregularities can disturb the results of our model of normal price behaviour and can in our modelling framework bias our results (Asche et al., 2006). If a single event changes the results of our econometric analyses notably, then we deem that this observation is an outlier and should be removed 2. We end up with treating the gas price in winter and the coal price in July 2008, as these events affects the properties of the time series strikingly. Both events are outlier treated by interpolating the prices. In Appendix A the empirical returns are presented before and after outlier treatment (OT), where it is obvious that the treatment has a great effect on the distribution of the returns, especially for gas Seasonality adjustment As mentioned, the gas prices exhibit seasonality. Seasonality is a characteristic of gas that is not present in the other energy prices. Therefore we wish to deseasonalise the gas prices in order to eliminate movements that can obscure the inference on the relation to other time 2 By changes in results we mean weakening of non-stationarity, weakening of the cointegration relationship, or less normally distributed residuals or returns. Page 32 of 129

35 Energy Price Risk Modelling series 3. The X12-ARIMA seasonal adjustment (SA) routine in the statistical software package EViews shows that seasonality is significant for gas, and we adjust the time series accordingly. Test results and historical seasonality factors used to adjust are presented in Output in Appendix A, which presents a selection of the generated output from the EViews routine Price conversion Finally, in order to compare prices and establish their relationship we convert all prices into one common currency. The prices were provided to us in US dollars and British pence, and we use the historical exchange rates to convert to DKK and øre. Furthermore, we also convert the prices to their natural logarithms (ln). This conversion enables us to use econometric analytical methods which are built around ln prices. From now on, when referring to prices we will mean ln prices unless otherwise explicitly stated. Now our data is ready for analysis but before proceeding to the empirical investigations we cover the theory behind stationarity. 4.2 Theory on stationarity When investigating potential relationships between price series, it is of uttermost importance to distinguish between whether the series are stationary or non-stationary. The two types of processes call for different modelling method, and applying an unsuitable method often leads to incorrect findings. Moreover the presence of stationarity or non-stationarity is crucial for the existence of cointegration. This section will cover the difference between stationary and non-stationary price series, why different analysis methods are needed for the two types of price processes, and how this relates to cointegration Stationarity and non-stationarity Stationarity implies that the stochastic process of a price series is mean-reverting and memoryless meaning that the process is dragged towards a mean and that a shock in one period will die out and have no effect on the price in later periods. A simple and well-known version of a stationary process is the white noise process. A stationary time series is one whose probability distributions are stable over time (Wooldridge, 2008). Stationarity comes both in a strict and a weak form. Strictly stationary 3 When the gas prices are simulated later in this thesis the seasonal pattern is added back onto each individual gas price path. Page 33 of 129

36 4 Energy price modelling processes have for all sets of intermediate sequences joint probability distributions that are independent of time and space. This also means that all moments of the distribution are independent of time and space. Weakly stationary (covariance-stationary) processes requires that mean, variance, and autocovariances are unaffected by a change of time origin (Enders, 2010). Thus weak stationarity only focus on the first two moments of the stochastic process; and requires that the mean and variance is constant over time, and that the covariance matrix between and only depends on and not (Wooldridge, 2008) 4. When we refer to stationarity, in this paper, it is in the weak form. A stationary process is said to be weakly dependent if and are almost independent for large meaning that goes to zero sufficiently quickly for approaching infinity 5. This is the memoryless property of the stationary process and it is very important, since weak dependency implies that the law of large numbers (LLN) and the central limit theorem (CLT) hold; and therefore standard inference techniques can be used in regression analyses (Wooldridge, 2008). A very common standard inference technique, for example, is the Ordinary Least Squares (OLS) estimation procedure. Time series that are not weakly dependent do general not satisfy LLN and CLT; therefore, standard regression techniques like OLS cannot be applied. The non-stationary stochastic process wanders, does not revert to a mean, and a shock in one period will persist and never die out (or dies out very slowly). The random walk is a simple and well-known version of a non-stationary process: Eq. 2 Unlike the stationary process, the non-stationary process does not have stable probability distributions over time meaning that the moments may change between periods. The fact that the moments, and the second moment in particular, are not constant for a non-stationary process is crucial to consider in risk modelling. In risk modelling it is critical to model with regards to the volatilities and co-movements between prices the company is exposed to. Assuming stable variances and covariances, when in fact they are not, may lead to dangerous underestimations of risk. 4 Notice that strict stationarity does not necessarily imply weak stationarity, as finite variance is not assumed for strict stationarity. Likewise, weak stationarity usually does not imply strict stationarity, since higher moments may depend on time. 5 This is a very vague description of weak dependency. Weak dependency cannot be formally defined because there is no definition that covers all cases of interest (Wooldridge, 2008). Page 34 of 129

37 Energy Price Risk Modelling Non-stationary processes 6, like the random walk presented above, are not weakly dependent, as it can be shown that if then: Eq. 3 The correlation depends on, and no matter the size of one can always set so that the correlation does not converge to zero. Hence, the random walk process is not weakly dependent, and the usual asymptotic theories do not hold (Wooldridge, 2008). A non-stationary process may become stationary after differencing. If a process becomes stationary after differencing times it is integrated of order and denoted an process. A non-stationary process that becomes stationary after first differencing is integrated of order one, denoted. A stationary process is denoted, because it needs to be differenced times to become stationary. Once a series is stationary the standard regression techniques are valid. However, a differenced series only contain information about changes in prices, and important information of the price in its level is lost. This is very inconvenient when investigating relationships between multiple series, as those relationships may be established in levels. Granger and Newbold showed in 1974 that when applying the standard regression technique OLS in the multi-variable framework in order to investigate relationships between variables, then if the variables are non-stationary applying OLS potentially leads to very serious misconclusions (Verbeek, 2004). Granger and Newbold showed by simulation, that when analysing the relationship between two independent random walks and in the following simple regression: Eq. 4, the OLS regression often yields a statistically significant t-statistics for and an unusual high, even though the series are independent by construction and are not related. This is the so-called spurious regression problem that can occur when using standard regression techniques on series that do not satisfy the inference assumptions; OLS will then detect correlation between the variables, even if they are completely unrelated, simply because of a similar rising trend over time (Verbeek, 2004). The problem is that when and are both non-stationary, then will also be nonstationary. But there is a special case where a linear combination of two non-stationary processes, and, is and thus stationary. If such stationary linear combination exists 6 Notice that a non-stationary series can be weakly dependent. This is for example the case for the trend-stationary processes. Page 35 of 129

38 4 Energy price modelling between the two non-stationary series, the series are said to be cointegrated. When are cointegrated Eq. 4 can be estimated using standard regression techniques. and We will thoroughly treat cointegration in section 4.4 and 4.5, but for now it is important to realise that we need to know whether the series are stationary or non-stationary in order to analyse our empirical prices with respect to the relationships between them. Moreover, if the series are non-stationary then in order for them to be cointegrated, all series need to be integrated of same order (for example all processes) and a linear combination of them that is stationary must exist. The next section covers an approach for determining whether all price series are Tests for stationarity Testing for stationarity can be performed in several ways. A common approach is testing for the presence of a unit root in the process. A unit root process is an example of a non-stationary process, where the correlation between observations does not converge to zero and weak dependency does not apply. The term unit root comes from the situation when a process has a root that is exactly equal to one. The model in Eq. 5, for example, has a unit root when. Eq. 5 If a process contains a unit root all previous shocks in the past have a permanent effect on the outcome in the current period (Verbeek, 2004). But if the model does not have a unit root, that is when, is a stable, correlations between observations tend to zero, weakly dependency holds, and previous shocks are not permanent. Testing for unit root is the same as testing whether a process is, because if a process has a unit root it also means that the process becomes stationary after differencing once (Wooldridge, 2008). The combined hypothesis for testing for unit root can be set up as: There are several tests for unit root and we apply two of the most acknowledged ones. First, we apply the Augmented Dickey-Fuller (ADF) test proposed by Dickey and Fuller in The ADF test tests for unit root in an autoregressive model. It includes lags to account for the dynamics in the process and in order to make sure that there is no autocorrelation in the error terms (Verbeek, 2004). Adding additional lags hurts the sample power of the test, why choosing the number of lags can be seen as a trade-off between fit, measured by the Page 36 of 129

39 Energy Price Risk Modelling loglikelihood value, and parsimony, as measured by the number of parameters (Verbeek, 2004). Common selection criteria are: Akaike Information Criterion (AIC), Schwartz information criterion (BIC) and Hannan-Quinn information criterion (HQ). The criteria differ in the penalty they impose when including an extra parameter, therefore, we apply all three criteria in hope to find a robust conclusion that all criteria can support. In case of different results we will lean towards BIC as that criterion, according to Verbeek (2004), is most likely to choose the true model when the true number of parameters is low. Under is nonstationary and thus the CLT does not hold and asymptotic standard normal distributions for the t-statistic cannot be used. Therefore, an alternative distribution of the t-statistic under the null, known as the Dickey-Fuller (DF) distribution, is applied for identification of critical values (Enders, 2010). The second test we apply, which is sometimes referred to as a nonparametric test for unit root, was proposed by Phillips and Perron (PP) in The PP test does not add extra lags to the regression but instead adjusts the t-test statistic to allow for potential autocorrelation in the error terms. According to Verbeek (2004), Monte Carlo studies do not point towards a preferable test of the two, therefore we apply both tests. For both types of unit root tests we include a constant and a deterministic time trend, because the upward rise in the prices from the graphical inspection indicates that there might be a drift plus a deterministic time trend in our series. This is thereby a test for trend-stationarity. If a deterministic trend is present, then it is important to include it when testing for unit-root stationarity, else a faulty acceptance of the unit root hypothesis can be caused by the time trend and not by non-stationarity. In the following section we perform the empirical analyses of stationarity to investigate whether our energy prices are processes. 4.3 Empirical analysis of stationarity In this section we examine whether the three price series can be described by unit root processes and therefore are. We recall that in a stationary process, a shock in time will die out and not persist in the long run, as the correlation between observation and tends to zero. This is opposite for a non-stationary process, in which the shock will persist and never completely die out, since the correlation does not asymptotically tend to zero when time passes (c.f. Eq. 3). Therefore, before performing the statistical unit root tests presented in the previous section, we start out with an initial graphical inspection of the autocorrelation functions (ACFs). Page 37 of 129

40 4 Energy price modelling In Output in Appendix B the ACFs are shown for our three price series in levels, and in Output the ACFs for the series in first differences are shown. If the prices are integrative of order one, we expect the series in levels to have significant autocorrelations throughout all lags, while we expect for the differenced series that autocorrelations die out over time. For all three prices in levels we see that the autocorrelations are all significantly different from zero and decay very slowly this indicates that prices are non-stationary in levels. A similar inspection of the autocorrelation functions after first differencing shows, that the autocorrelations are much smaller at all lags and we do not see the same persistance as before. When looking at the autocorrelation in the differenced oil series, we see that all p- values are insignificant, why oil seems to be integrated of order one. The differenced gas series still has some significant lags but seem altogether to be integrated of order one. For th differenced coal series, on the other hand, all lags are significant, which indicates that the series are not weakly dependent and integrated of order one. Note that an inspection of the ACFs is not a formal test for non-stationarity or order of integration, and the unit root tests are the acknowledged statistical methods, which we therefore will proceed with. As presented in 4.2.2, we use the Augmented Dickey-Fuller test, with the lag selection criteria AIC, SIC, and HQ, and the Phillips-Perron test to test for unit root. We perform the tests on the three price series both in levels and in first differences in order to arrive to robust conclusions. If our price series have a unit root in levels but not in first differences, we can infer that they are. The results are presented in Table From the results in Table 4.3.1, we deduce that we cannot reject the hypothesis of a presence of unit root in the oil price in levels, and we reject the hypothesis when oil prices are in first differences. Therefore, we cannot reject that oil is. For the gas price series, we reject the null of a presence of unit root in levels when applying the traditional 5% significance level. However, if we use a 1% significance level, then we cannot reject the null hypothesis on gas price in levels. If we recall the inspection of the ACF, we saw that gas in levels have significant autocorrelation throughout all presented lags, this help the conclusion of unit root in levels. In the differenced gas price we reject unit root at both significance levels. Therefore, we say we cannot reject that gas is, however, we keep in mind that the conclusion is sensitive to choice of significance level. For the coal price the ADF test based on AIC shows contradictory results compared to the other three tests. When considering the high number of included lags (11) and Verbeek s (2004) statement that AIC often includes too many lags, we choose to disregard this particular result. The remaining three tests on coal prices in levels cannot reject the null of a presence of unit root, while all four tests reject the null in the differenced coal price. Therefore, we also cannot rejects that coal is. Page 38 of 129

41 Energy Price Risk Modelling Table Unit root test results Unit root test on levels Including constant and linear trend Unit root test on first differences Including constant and linear trend Test Lag length/ Lag length/ t-statistic P-value % Bandwidth Bandwidth t-statistic P-value % Oil ADF AIC ** ADF SIC ** ADF HQ ** PP ** Gas ADF AIC ** ** ADF SIC * ** ADF HQ * ** PP * ** Coal ADF AIC ** ** ADF SIC ** ADF HQ ** PP ** * Significant at 5% ** Significant at 1% In this analysis we work under the paradigm that our price series either follow stationary or non-stationary processes. This distinction is unfortunately not always evident for most empirical price series. Energy prices can show sub-periods of both stationarity and nonstationarity. However, we do not see any clear signs of time delimited regimes common for all three price series and also we are not interested in modelling specific time periods or regimes in the energy markets 7. Instead, we are interested in deriving at a model that is stable across normal and smaller market events; hence, we need a model that will persist throughout the time period. We therefore conclude that oil, gas, and coal can all be well approximated by processes that are integrated of order one, and therefore it is valid to proceed with our analysis using a cointegration approach to investigate possible relationships between the three energy prices. 7 If one is interested in investigating regime shifts in the energy prices, Chow s test for structural breaks could be applied. Page 39 of 129

42 4 Energy price modelling 4.4 Theory on cointegration This section presents the theory on cointegration. Firstly, we present the concept of cointegration based on Engle and Granger s formalisation, which descents from the idea of market integration. Secondly, we treat the Error Correction Model (ECM), which is a representation of the dynamics in a bivariate cointegrated relationship. Thirdly, we extend the theory to the more general Vector Error Correction Model (VECM), which is a representation of the dynamics in a multivariate cointegrated system. Finally, we close this theory section with an elaboration on Johansen s approach to test for cointegration, and how this approach enables hypotheses testing Cointegration Stigler s definition of market integration from 1969, which we presented in section 3.1, is evidently extendable to Engle and Granger s (1987) formalisation of cointegration, and testing for market integration is today equivalent to testing for cointegration. The relationship under investigation is still the equation below, which is a reproduction of Eq. 1: Eq. 6 Before the late 1980s relationships like Eq. 6 have been estimated using traditional econometric methods like ordinary least squares (OLS) and correlations have been used to describe relationships between variables. However, as we learnt from section 4.2 on stationarity, when the variables are non-stationary, like our energy prices are, the fundamental inference theory breaks down causing the traditional econometric tools to be nonapplicable (Asche et al., 2002). Instead Engle and Granger (1987) pointed at cointegration analyses for the investigation of relationships between non-stationary series and today that is the common methodology. Cointegration describes a situation where two or more price series follow a common trend; i.e. move together like the market integration theory suggests. More precisely for non-stationary series: When there exists a linear combination of two or more of integrated time series that is stationary then the series are said to be cointegrated (Enders, 2010). The series can then be interpreted as co-dependent in levels, and even though the prices may diverge in the short-run the relationship will prevail in the long-run. This long-run equilibrium can be expressed by the following equation: Eq. 7 Page 40 of 129

43 Energy Price Risk Modelling with being a stationary series of residuals (Asche et al., 2002). When the system is in its long-run equilibrium, and within this framework any deviation from the long-run relationship, i.e., must be temporary (Enders, 2010). Eq. 6 describes a situation where prices adjust immediately, however, often there will be response time between the prices, and the adjustment pattern can then be seen as more dynamic. By introducing price lags in the equation this response time can be accounted for. Adding lags to the short-run relationship does not change the form of the long-term relationship, which is still represented by Eq Error correction model To maintain the long-run equilibrium there must be an error correcting mechanism between the variables, which adjusts the deviations so that they on average equal zero. This mechanism works in the short-run to establish the long-run relationship. According to Granger Engle and Granger (1987) an Error Correction Model (ECM), which can be seen as an extension of an AutoRegressive (AR) model, can capture both the long-run relationship and the error correction mechanism and allows us to study the short-term dynamics, or changes, in a dependent variable,, in a cointegration relationship: Eq. 8, where the constant corresponds to a linear trend in levels, is the lagged effect of changes in the independent variable, and is the error correction mechanism. The error correcting mechanism can further be separated into which is the speed-of-adjustment parameter, and which is the cointegration equation (CE), where is the cointegration vector. This means that the short term dynamics of the dependent variable on the left-hand side is determined by a constant term, a short-run effect from the lagged independent variable, and a short-run effect from any disequilibrium in the CE, where the speed-of-adjustment parameter decides by how much the CE affects the left-hand side. Hence, the speed-of-adjustment parameter makes sure that the long-run relationship is maintained. The ECM can without loss of generality include more lags of both and Vector error correction model Cointegration can also exist between more than two variables. In a multivariate cointegrated system of variables there can exist up to cointegration relationships. For example 8 The inclusion of more than one lag might seem contradictory to the fact that the prices are. However, as discussed earlier, being means that the series becomes weakly dependent after differentiating once - indicating that the autocorrelation tends zero for increasingly distant observations. Therefore processes can still have some degree of autocorrelation in the nearest lags. Page 41 of 129

44 4 Energy price modelling in a cointegrated system of the three variables,, and, cointegration can exist between and and also between and resulting in two cointegration relationships. One common cointegration relationship between all three variables can also be present, or one common cointegration relationship between two of the variables, leaving the last variables unintegrated. The ECM from the previous section can then be extended to a Vector Error Correction Model (VECM), a system with variables and cointegration relationships, which is an extension of the well-known Vector AutoRegressive (VAR) model: Eq. 9 is a vector of variables, is a vector of constants, which again corresponds to linear deterministic trends in levels, and is a x matrix holding the coefficients of the lagged variables. and are matrices of a dimension where holds the speed-of-adjustment parameters and is the cointegration matrix (CM), which holds the cointegration vectors. is a -dimensional vector of white noise terms with covariance matrix. Eq. 9 could also be extended with a vector of constant terms inside the CM, which corresponds to having a constant price difference in the cointegrated variables. Trend terms could as well be included both outside and/or inside the CM. A trend term outside the CM means that the series have a quadratic deterministic trend in levels, which is unlikely for financial price series. A trend inside the CM means that the cointegration relationship will diverge linearly increasing. Before modelling the VECM one should inspect the data graphically and make a priori assumptions to decide on which constants and trends to include in the model (Verbeek, 2004). In a cointegration relationship the variables share common trends that link them together in the long run. According to Stock and Watson (1988), if there are cointegration relationships in a system of variables, then there must be stochastic trends (Asche et al., 2006). These stochastic trends can be manifested by a common external factor, the specific relationship between the involved variables, or simply by one of the involved variables. There is no method to determine whether the stochastic trend indeed is externally or internally driven within the cointegration relationship. However, for interpretational reasons it makes sense to appoint one of the cointegrated variables the leader of the stochastic trend, and hopefully this will make economic sense. Weak exogeneity tests can be used to determine whether a variable is dependent or independent, and many practitioners interpret weakly exogenous variables as being leaders of the cointegration relationship even if one cannot be sure the real leader is not an omitted external factor. Page 42 of 129

45 Energy Price Risk Modelling Tests for cointegration With regards to testing for cointegration and modelling the VECM, two approaches are widely applied in the literature. The first was introduced by Engle and Granger in 1987 and later Johansen presented an approach in We prefer Johansen s approach, since this framework allows for hypothesis testing on the cointegration relationships an element we will elaborate on later in this section. Furthermore, when testing for cointegration in the multivariate framework Johansen s approach, unlike Engle and Granger s approach, can identify more than one cointegration relationships, if such exists (Enders, 2010). Johansen s approach is based on the VECM representation in Eq. 9 and contains the following steps: 1) Estimation of in the VECM using maximum likelihood (ML) 2) Identification of number of cointegration relationships via rank test 3) Identification of the most significant cointegration vector(s) and estimation of 4) Tests of restrictions 5) Estimation of the final VECM. STEP 1 contains an initial ML estimation of the VECM and the results are used in step 2. In order to estimate the full VECM from Eq. 9 needs to be specified. This entails determining whether or not to include constants and time trends outside and/or inside the CM and how many lags to include. Whether or not to include constants and/or time trends should be dependent on the empirical prices and on economic sense. The number of lags should be chosen as at trade-off between removing autocorrelation in residuals and parsimony. When the model is specified, the parameters are estimated using ML, and the resulting is used for further analysis (Enders, 2010). STEP 2 contains a hypothesis test upon to identify its rank. This is equivalent to finding the number of columns in, which is also the number of cointegration vectors and thereby cointegration relationships. In the case of cointegration the long-run matrix has a reduced rank of, which means that there exists independent linear combinations of the variables that are stationary. Notice that there cannot exist more than cointegration relationships, because if independent linear combinations produces stationary series, then all variables must be stationary themselves and there is no cointegration. Likewise if has rank of zero then there exists no stationary combination of the variables and there is no cointegration (Verbeek, 2004). The rank of an x matrix is equal to the number of non-zero eigenvalues (Enders, 2010). Therefore, Johansen s test identifies the number of non-zero eigenvalues in. If an eigenvalue, denoted, is equal to zero then ( ) where ; this is used in two asymptotically equivalent likelihood ratio tests called trace test and maximum eigenvalue test (Verbeek, 2004). Page 43 of 129

46 4 Energy price modelling The trace test tests the hypothesis against the alternative and the teststatistic is: ( ) As can be seen, it tests whether the smallest eigenvalues are different from zero where the eigenvalues are lined up so that. For example if we want to test whether the rank is one in our three variable case, then if the second smallest ( ) and smallest ( ) eigenvalues are equal to zero, indicating that the rank is one, the teststatistic will become zero and the p-value very high. This means that the null is not rejected, which also means that we cannot reject that there is at most one cointegration relationship. The maximum eigenvalue test has the same null hypothesis but the alternative is more restrictive: and the test-statistic is: Eq. 11 ( ) This test statistic is based on the th largest eigenvalue. For example if we again want to test if the rank is equal to one then the second largest eigenvalue is plugged into the formula. If this eigenvalue is equal to zero we will get a very high p-value and we cannot reject the null arriving at the same conclusion that there is at most one cointegration relationship. Although the two test statistics are likelihood ratios they are not chi-square distributed, but instead they follow the multivariate form of the Dickey-Fuller distribution (Verbeek, 2004). Performing tests on is actually a multivariate generalization of the Dickey-Fuller test, since Eq. 9 is equivalent to the augmented Dickey-Fuller equation. In STEP 3 we have determined the rank of and we can proceed with the identification of the most significant cointegration vector(s) and re-estimate the speed-of-adjustment parameters via ML while keeping. The Johansen approach has the advantage, that it performs a joint test and estimation. This enables inclusion of interactions between all variables in the system. However, the estimated results of and are not uniquely identified, since different combinations of and can produce the same matrix. Therefore, the cointegration vectors in have to be normalised. Ideally in the case where one variable in a vector prudently can be appointed the leader, the other beta coefficients are normalised upon the beta coefficient of the leader. This normalisation contributes to the economic interpretation of the system. Page 44 of 129

47 Energy Price Risk Modelling If one is only interested in determining whether there is cointegration the cointegration analysis can end here. But if one is interested in testing market hypotheses or modelling the system of variables then step 4 and 5 should also be executed. STEP 4 involves imposing of restrictions in order to conduct hypothesis testing on the cointegration relationship. Imposing restrictions on the parameters may reveal a cointegration relationship that makes more economically sense (Verbeek, 2004). Performing significance tests on the elements in the cointegration vectors (the columns of ) by using a standard t-test is invalid as this requires that the residuals are serially uncorrelated and cross-correlation between variables must be zero. Especially the latter requirement cannot be fulfilled for cointegrated variables (Enders, 2010). Therefore, we test hypotheses by imposing restrictions on or on either or. If a restriction is not in correspondence with the cointegration relationship, the rank of may change. Thereby, the restriction test basically compares the number of cointegration vectors in the unrestricted and restricted model. If a cointegration vector has diminished the restrictions are not in correspondence with the prices relationship (Enders, 2010). The restriction test is a likelihood ratio test based on the following test-statistic: [ ( ) ( )] which follows a chi-square distribution with degrees of freedom given by the number of restrictions imposed. As can be seen from the equation the test compares eigenvalues of the unrestricted model with eigenvalues of the restricted model. The intuition is that if the cointegration vector(s) is to remain, then all values of and should be near equivalent. If there are large differences between the two models this will yield a large test statistic, which indicates that the restriction is binding and invalid (Enders, 2010). For instance if we want to test a hypothesis about the cointegration relationship between variables, we need to impose restrictions on the matrix in Eq. 13. We impose one restriction, which is that the third variable is excluded from the cointegration relationship. We also choose to normalise upon the first variable and set the second variable to be equal to -1. These implementations are represented in Eq. 14. Note that with this restriction we impose here, we actually tests whether the Law of One Price holds for the first and second variable, which is the ultimate case of market integration. Eq. 13 Eq. 14 Page 45 of 129

48 4 Energy price modelling If we want to investigate which variables respond to deviations in the cointegration relationship we can perform tests on the adjustment parameters in. Testing on the alphas is the same as testing whether a variable responds to discrepancies from the long-run equilibrium relationship. If is equal to zero it is said that is weakly exogenous and independent of the cointegration relationship. In this thesis we interpret weak exogeneity as an indication that the variable is a manifestation of a common trend. If the speed-ofadjustment parameter of a variable is equal to zero and the variable is a part of the cointegration relationship then we appoint the variable as price leader of all other variables that do respond to deviations from the long-run equilibrium 9. In STEP 5 the final VECM is derived. The full VECM from Eq. 9 is re-estimated with the imposed restrictions, and the final model specification can be derived by eliminating insignificant regressors from the full VECM. Standard regression techniques such as OLS and inferences based on t-tests can be applied in this step on all the stationary regressors (i.e. all but the CE parameters). Having gone through the most essential theory behind cointegration and the Johansen approach we continue with an empirical cointegration analysis on our data. 4.5 Empirical analysis of cointegration From earlier analysis we know that our three empirical price series oil, gas, and coal can all be considered processes, why a cointegration approach is appropriate to investigate potential relationships between them. We use Johansen s approach and follow the five steps covered in the previous section. STEP 1 Estimation of VECM The relationship between the three price series; oil, gas, and coal can be represented in a VECM. A VECM can include different components; hence, we need to decide upon the form of the VECM before we can estimate it using ML. Based upon the graphical inspection of the empirical price series we made in section 3.2, we assume that there is a trend in each of the three price series. Therefore, we include a constant term in the VECM. Please recall that a constant in the VECM model corresponds to a trend in levels. We also place a constant inside the CM because we assume that if the prices are 9 In the literature there are many interpretations and definitions of the term price leader. For example price leadership can be equal to Granger causality or it can be applied to a price from purely economic arguments. We choose to define price leadership as stated above. Page 46 of 129

49 Energy Price Risk Modelling cointegrated the difference in prices will not deviate around zero but around a constant level that is equal to the differences in price levels. We choose not to include any trend term outside or inside the CE. We do not include a trend term inside the CE, because there empirically does not seem to have been an increasing distance between the series over time, why we do not expect the prices to diverge with a trend in the future. A time-increasing distance between prices would not make any economic sense, considering the price drivers and especially the substitution possibility, which makes a persistent price divergence unrealistic. We do not place a trend outside the cointegration relationship either, because this would require a quadratic trend term in levels and that is very rare and almost meaningless when regarding price series. An important assumption of the rank tests is that zero autocorrelation appears in the residuals from the VECM, because the test essentially is a multivariate Dickey Fuller test. Adding lags of the dependant variables to the system removes the autocorrelation in the residuals. In Appendix B, Output a joint test on number of lags to include in the trivariate system is shown. According to the likelihood ratio test and based upon different criteria (AIC, SIC etc.) all tests point towards a lag length of two. Furthermore, in Output to Output we perform a graphical inspection of the autocorrelation functions of all prices with one, two, and three lags. Also the Breusch-Godfrey serial correlation LM test is performed on the residuals. We find that a reasonable number of lags to include in the individual price models, in order to remove serial correlation while keeping the number of lags to a minimum is: One for oil, one for gas, and two for coal. We cannot compute the VECM with different lags included for the individual prices, therefore, we include two lags for all three prices. Having established the specification of the VECM we can now estimate VECM via ML: of the following Eq. 15 [ ] [ ] [ ] [ ] [ ] [ ] where [ ] [ ] We conduct our analyses using the EViews software. The Johansen routine in Eviews estimates the VECM simultaneously with performing the rank tests. Therefore, we do not present the estimated VECM in this step. Page 47 of 129

50 4 Energy price modelling STEP 2 Rank test If cointegration exists in the VECM presented above then the rank of the long-run matrix will be reduced. More specifically the rank will be one if there is one cointegration relationship between the three variables, two if there are two cointegration relationships, and if the rank is zero or three, there is no cointegration in the system. The test results for the trace and max eigenvalue tests are given in Table below. Table Johansen s rank test Hypothesized no. of CE(s) Eigenvalue Test Statistic Critical Value P-value Trace test r = % r % r % Max eigenvalue test r = % r = % r = % Recall that the null hypotheses for the trace test is and the alternative is. The trace test rejects the null of a rank of zero with a p-value of 0.57%, but with a p value of 9.73% we cannot reject the null of at most one cointegration relationship. The maximum eigenvalue test has the same null hypothesis, however, the alternative is more strict and looks the following. It also rejects the null of zero rank with a p-value of 2.09% and this means that the alternative hypothesis of one cointegration relationship ( ) is more likely. With a p-value of 10.83% the test cannot not reject the null, hence, we cannot reject that there is at most one cointegration relationship. Because both tests cannot reject the null hypothesis of a rank of one, we accept the hypothesis that there is one cointegration relationship between the three energy prices. STEP 3 Identification of Based on a rank of one the most significant cointegration vector of the CM is determined to form a single vector and the vector is isolated in the equation. The result can be seen below: Eq. 16 [ ] Page 48 of 129

51 Energy Price Risk Modelling Coal has been randomly normalised upon. With these parameters we can now proceed with hypotheses testing in order to arrive at a more meaningful interpretations of the cointegration relationship. STEP 4 Imposing and testing restrictions As already stated in section 4.4 the results of our analyses up until now only spans a cointegration space and and are not uniquely identified. By imposing restrictions on this space we can deduct results that can be more meaningful from an economic perspective and provide alternative interpretations. In chapter 3 we formulated two hypotheses based on the price drivers and a graphical inspection of the oil, gas, and coal prices. In the following we search to find support for these hypotheses. The first hypothesis that we want to investigate is: Hypothesis 1: Oil, gas, and coal are all cointegrated with oil being the independent leader of both gas and coal. This formulation can be tested by imposing following restrictions on the cointegration space: [ ] The alternative hypothesis is the unrestricted from Eq. 16. We test the restriction by the LR test covered previously, where not rejecting the null means that the number of cointegration relationships is unchanged and the restriction can be accepted. Note that we have normalised upon because we assume that this is the leading price. is based on one restriction on while leaving all other parameters unchanged, hence, we test whether oil is the price leader of gas and coal. This also means that only gas and coal react to any disequilibrium between the three prices. The test result can be seen in Output in Appendix B, where we find that the null hypothesis cannot be rejected with a p- value of 96.2% why we cannot reject that the restrictions are not binding. Therefore, we find that the statistical test supports Hypothesis 1. The second hypothesis that we want to investigate is: Hypothesis 2: Oil, gas, and coal are all cointegrated with oil and coal being the independent leaders of gas. Page 49 of 129

52 4 Energy price modelling This formulation can be tested by imposing following restrictions on the cointegration space: [ ] Now we have imposed the restriction that both and equals zero so this test is more restrictive than the former. What we test for is that both oil and coal are price leaders of gas. The test output can be seen in Output We find that the hypothesis cannot be rejected with a p-value of 72.2%, and therefore we again cannot reject that these restrictions are not binding. This supports Hypothesis 2. The p-value is lower than the p-value regarding hypotheses 1 of 96.2% indicating that we are less willing to not reject the null. It is the conventional procedure to normalise upon the market leader, however, in this hypothesis we have two leaders. Therefore, in order not to misinterpret a normalisation upon oil as an indication that it leads both gas and coal, we choose to normalise upon gas instead. A property of the Johansen approach is that the choice of variable to normalise upon does not change the conclusion, since the cointegration space remains intact regardless of choice of normalisation procedure. In conclusion, we find statistical evidence for both Hypothesis 1 and Hypothesis 2, as both tests have very high p-values. Still we prefer to continue with Hypothesis 2. This is because we imposed two restrictions instead of one but only experienced a decline in p-value of 24.0%, and a p-value of 72.2% is still very high. We say that Hypothesis 2 is more likely, and the specified with the restrictions from then becomes: Eq. 17 [ ] [ ] STEP 5 Estimation of the final VECM Now that we have imposed our restrictions and specified we can estimate the rest of the VECM. We estimate the coefficients using OLS while we maintain the parameters from Eq. 17. After the OLS regression we eliminate all statistically insignificant regressors using a sequential elimination based on standard t-tests. Throughout the elimination we test for autocorrelation and heteroscedasticity in residuals since these are important assumptions that need to be fulfilled for the t-tests to be valid. Outputs can be found in Appendix B section 13.3 and the final model is presented in the next section. Page 50 of 129

53 Energy Price Risk Modelling 4.6 The final price model After imposing the restrictions relevant to our chosen market hypothesis and performing a sequential elimination of insignificant regressors, we arrive at the final price model, which we present in its VECM representation below: Eq. 18 [ ] [ ] [ ] [ ] [ ] [ ] [ ] Writing out the individual price equations in the vector system we get: Box 1: The final price model where ([ ] [ ]) From the final model we see that all three prices are interconnected both through CE, through lags in their respective equation, and through the covariances in the residuals. If we look at the equation for oil we see the price is weakly exogenous, because the speed-ofadjustment parameter is equal to zero. This means that the oil price does not respond to deviations from the long-run equilibrium. The change in oil price can be expressed by the constant term of, which corresponds to a positive deterministic trend in levels, the lagged change in gas price, and a stochastic element. Therefore, the final equation for oil is equivalent to a random walk with a drift. The included lagged gas price is contradictory to what we expected from section on price drivers and to the general findings in the literature. Here we find indication that gas contracts are still linked to oil prices, hence, we expect that oil prices feed into gas prices but not the other way around. Panagiotidis and Rutledge (2007) encountered the same unexpected relationship between oil and gas and states that a reason why oil may respond to changes in gas prices, is that the Page 51 of 129

54 4 Energy price modelling expectations to the link between oil and gas can reflect in a mechanism where gas prices feed into oil prices. The final equation for gas is a form of error correction model with a drift. The equation is comprised of a constant term of, which again corresponds to a deterministic positive trend in levels, and a stochastic element. Furthermore, the equation holds the error correction term, which means that the gas price adjusts to deviations from the longrun equilibrium in CE with a speed-of-adjustment of. Disregarding the constant term and the stochastic element, this means that every time then in the next period gas will respond with factor of. Note that the negative sign of combined with the negative signs of and in the CE, makes sure that if oil or coal increases then the gas price will follow in same direction the next period (ceteris paribus and disregarding the constant term and the stochastic element in gas). Just like for oil, the equation for coal is equivalent to a random walk with a drift. This means that the coal price is weakly exogenous and does not respond to deviations in the long-run equilibrium. Changes in coal prices are determined by a small constant term, its own lag, a lagged oil price, and a stochastic element. The constant term is smaller for coal than for oil and gas. That is in line with our expectations, because we saw an increasing time trend in all three prices in Figure 3.2.4, however, the increase in coal prices was smaller. That the coal equation holds a lagged price change of oil can be interpreted as a result of the substitution opportunity that exists between coal and oil or more likely as an indication of oil being the overall global leader of the energy market. The stochastic elements of the model can be described by the distribution of the residuals. The covariance matrix is generated from the empirical residuals fitted to a normal distribution with zero mean, since normal distributed residuals is an assumption of the VECM model. From the covariance matrix we see that gas has the highest variance while coal has the lowest. This is in line with our expectations based on the graphical inspection. We also see that the strongest covariance is between oil and gas, which is not surprising when looking at the relationship between the two prices historically, and when considering that they both have relatively high volatilities. The weakest covariance is between oil and coal. The CE consists of the lagged price of oil, gas, and coal and also a constant term. We see that oil and coal affect the CE almost equally when normalising upon gas. This can be interpreted as if gas is evenly driven by the other two energy commodities, and in the long run gas will therefore be expected to lie between the two prices. The constant term of reflects a longterm difference in price between the three of commodities, and this may be caused by Page 52 of 129

55 Energy Price Risk Modelling production cost, transportation cost, or quality differences (e.g. amount of energy contained) which is also suggested by Stigler in his definition of market integration in 3.1. In Figure the historical CE is depicted based on the empirical prices. We see that CE fluctuates but tends to return to zero, which shows that there is a stable long-run relationship between the three price series. This CE time series, which is a linear combination of the three non-stationary price series, can clearly be regarded as stationary based on this graph. From 2000 to 2002 the CE was positive throughout a long period; this was a period where oil and coal showed subsequently decreases and the gas price dragged behind, restoring the equilibrium with a delay. From 2009 to 2012 CE was very negative due to the big increase in oil and coal prices where gas again adjusted with a delay. Figure Cointegration Equation Lastly, our findings, that oil and coal wanders and that gas adjusts to CE, is in line with what we expected: When one cointegration relationship exists between the three prices then two stochastic trends must drive the system (cf. the discussion in 4.4.3). In more technical terms this is because one of the commodities prices must be expressed by a linear combination of the other two. In this case gas is expressed by a linear combination of both oil and coal in the CE, then oil and coal can be interpreted as the being the stochastic trends. In conclusion, our analysis shows that we can agree to Hypothesis 2 that oil, gas, and coal are all cointegrated with oil and coal being the independent leaders of gas Forecasting prices The purpose of deriving an econometric model describing the energy prices is to be able to predict the prices in the future. Now that we have the final model we can forecast future price movements. Page 53 of 129

56 4 Energy price modelling We do not expect our econometric model to perform perfectly with regards to accuracy and prediction due to the stochastic elements in the model. Therefore, in order to take into account the stochastic nature of the energy price processes, we forecast the prices using numerous simulations to generate a fan of possible future price scenarios upon which we derive confidence levels. By Monte Carlo simulation we forecast 1000 price scenarios of the three stochastic price processes in Box 1. Each of the 1000 price scenarios contains three price paths one for each energy commodity. We forecast three years into the future month-by-month and a joint simulation of the three price series maintains the cointegration relationship and the complex dependency structure between the model variables. The stochastic elements of the price equations, which are the residuals, need to be generated for the Monte Carlo simulation. By observing the historical residuals derived from our final VECM model and the historical prices (c.f. Figure to Figure in Appendix C) we find that we cannot adequately describe the historical residuals with a simple parametric distribution e.g. a normal distribution with fitted mean and variance. The distributions of the residuals all resemble bell shapes but with skewness and excess kurtosis. The empirical residual series also show periods of both time varying volatility and heteroscedasticity, and Jarque-Bera tests reject normal distribution in residuals for oil and coal. Therefore, we cannot assume a parametric distribution of the empirical residuals 10. We find that a bootstrapping method is more appropriate for generating the stochastic residuals needed for simulation. By bootstrapping then the stochastic future residuals become a resample (with replacement) of the historical residuals. When we use the empirical residuals we maintain the empirical distribution and by sampling them in groups we also maintain the relationship between the price series meaning that for each month-by-month iteration we draw the three residuals simultaneously from the same random point of time in history. Thereby we preserve the historical cross-correlation between the price series, and since we do not have autocorrelation in our model (c.f. Appendix D) we can draw the grouped residuals randomly from the history. We jointly simulate the stochastic price processes 1000 times and as a last adjustment we convert the prices from ln to DKK and add the seasonality back to the gas price paths. We use the X12-ARIMA seasonal adjustment procedure to forecast the monthly season-factors for 2013 (c.f. Output ) and assume same pattern for 2014 and Figure is an example of one random price scenario out of the 1000 generated scenarios. 10 We would like to have performed a thorough analysis of the residuals by modelling future residuals via econometric techniques. Due to the time constraints of this thesis we leave this part for further research. Page 54 of 129

57 Energy Price Risk Modelling Figure Price scenario i=42 To extract the relevant information from the 1000 scenarios, we separate the three prices and sort the individual prices within each commodity according to size for each time point to get the confidence levels for the price movements. The confidence levels for the three commodities are depicted in Figure to Figure below, and in Appendix C: Forecasting, you find the corre-sponding graphs in ln prices. Figure Confidence levels for the oil price Jan2013-Dec2015 Figure Confidence levels for the gas price Jan2013-Dec2015 Figure Confidence levels for the coal price Jan2013-Dec2015 Page 55 of 129

58 4 Energy price modelling Note that when we refer to the 95% confidence level line in any of the graphs (the bottom line) it means that 5% of the simulated prices fall below the line and 95% lie above. Likewise, 10% of the simulated prices will lie outside the range delimited by the lower confidence level (95%) and upper confidence level (5%). The statistical inference interpretation of the 95% confidence level line, is that we are 95% confidence that the true commodity price will lie above the line. Figure depicts the confidence levels for the forecasted oil price scenarios. We observe that the range covered by the confidence levels increases steadily with time, which reflects the stochastic element of the price process. The distribution has a right (positive) skew indicating a possibility for large positive increases. The right skew comes from the ln conversion: If the residuals are symmetric in ln this corresponds to a positive (exponential) skew in DKK. At the end of the three-year horizon the range between the 95% and 5% confidence level is [311;1992] = 1681 DKK/bbl. This is a wide range and indicates that the oil price over the next three years can decrease with 53% or increase with 201% relative to December 2012 where the price was 662 DKK/bbl. Although this range might seem broad it is definitely not unrealistic. To draw an empirical comparison we identify the historical observations between which the oil price has increased most over a three-year horizon: In May 2001 the oil price reached 243 DKK/bbl and compared to the oil price three years before of 82 DKK/bbl this gives an increase of 196%. If this empirical increase of 196% would recur three years from December 2012 it would result in a price of 1960 DKK/bbl in December 2015, which is just inside the confidence levels. Figure depicts the confidence levels for the forecasted gas price scenarios. Again the range between the levels widens with time, and the price scenarios are right skewed just like for oil. Now we also observe a seasonality pattern in the gas price, and that gas is more expensive during the cold months and less expensive during the warm months. At the end of the three-year horizon the range between the 95% and 5% confidence level is [248;1434] = 1186 øre/therm. This is a relative narrower range, and it indicates that the gas price can decrease with 56% or increase with 154%. If we again make a comparison to the historical prices, we see that the gas price in September 2012 was 565 øre/therm, and relative to the price three years before of 168 øre/thmthe price increased with 236%. Again we see that although we have a wide range it is definitely not unrealistic over a three-year horizon. Some might argue that the forecasted price range is too narrow because it only predicts a 154% increase when a 236% increase can be seen in the past. Then we have to remember that 10% of the true observations actually could lie outside the confidence levels, therefore, the forecasted range may not be too narrow. Alternatively higher confidence levels can be set for a more conservative picture of the price movements. In our case the price model we have derived is not intended to forecast extreme events, therefore, our levels run from 5% to 95%. Page 56 of 129

59 Energy Price Risk Modelling Figure depicts the confidence levels for forecasted coal scenarios. Again the range increases with time and we also have a distinct right skew indicating a possibility for large increases. At the end of the three-year horizon the range between the 95% and 5% confidence level is [209;1462] = 1253 DKK/t. The range indicates that the price can decrease with 59% or increase with 186% relative to the December 2012 coal price of 512 DKK/t. To draw a comparison to the largest three-year increase in empirical prices, we saw a coal price of 962 DKK/t in August 2008 and a price of 357 DKK/t three years before that is an increase of 170%, which is lower than the forecasted 186%. If we extract the 50 percentile price paths for each energy commodity, we get the median of forecasted prices. These medians we refer to as being the most likely and they are depicted in Figure In Figure we also present the median of ln prices without seasonality in gas because it allows for a better investigation of the price dynamics. Figure Median of forecasted prices Jan2013- Dec2015 Figure Median of forecasted ln prices Jan2013-Dec2015 without season in gas The median prices express the price movement suggested by the final price model in Box 1. As already mentioned in 4.6 we expect the prices to increase with linear trends due to the constant term in the individual price equations, and that the gas price responds to deviations in the long-run equilibrium between the three prices. The final equation for oil is a type of random walk with drift, therefore, we expect the price to increase in accordance with the drift term and nothing else (the effects from the lagged gas price change and the residuals are zero on average 11 ). This increase can be seen from Figure Over this three-year horizon the constant term of in the oil equation leads to an increase from 622 DKK/bbl in December 2012 to 794 DKK/bbl in December 2015, and this corresponds to 8.5% per year The effect from the lagged gas price only asymptotically tends towards zero. 12 Calculated as compounding interest. Page 57 of 129

60 4 Energy price modelling The final equation for gas is an error correction model with a drift. If we started forecasting from a time point where CE was in equilibrium, then we would expect the gas price to increase in accordance with the constant term of (the effects from CE and the residuals are zero on average). However, if we look at the CE value in December 2012 in Figure 4.6.1, which is the starting point for our forecast, we see that the system is in disequilibrium. Therefore, gas responds to this deviation from the long-run equilibrium by decreasing in the first period of 2013, which can be seen from Figure It takes until mid-2013 to restore the equilibrium and from there on; it is the drift that drives the gas movements. In general the effect of CE deviations on gas price is much stronger than the trend, why the development is mainly driven by CE if it is in disequilibrium. This can also be seen from the constant term of 0.008, which is much smaller than the speed-of-adjustment parameter of Therefore, over this 3 year horizon the median decreases from 609 øre/thmin December 2012 to 590 øre/thmin December 2015, which corresponds to a yearly decrease of 1.1%. The final model for coal is also a type of random walk with drift, why we expect the median to increase in accordance with the constant term of (the effects from the lagged price changes and the residual are on average zero). Over this 3 year horizon the constant term leads to an increase in the median from 512 DKK/tonn in December 2012 to 554 DKK/tonn in December 2015, which corresponds to a yearly increase of 2.7%. Therefore, the movements in median prices support our expectations as we see an increasing oil price, a less increasing coal price, and a gas price that adjusts to the relationship between the three prices. This forecast aligns with the general opinion of market experts about continuously increasing energy prices in the future. Finally, we have forecasted CE based on the medians of the forecasted prices. Figure depicts the level of CE in a three-year forecast. Figure Forecasted CE Jan2013-Dec2015 Page 58 of 129

61 Energy Price Risk Modelling By comparing the median of ln gas prices from Figure to the CE graph from Figure 4.6.8, we observe that the gas price drops in Q1 and Q2 in 2013 is a consequence of CE imbalances in the preceding period. The plot shows that CE starts by being positive but adjusts to zero rather fast. The rapid adjustment is a result of the speed-of-adjustment parameter of in front of the CE in the equation for gas. After the adjustment CE fluctuates around the equilibrium level in the remaining period, and the small fluctuations represent the long-run relationship caused by the stochastic elements of the three prices. Compared to the historical CE we saw in Figure 4.6.1, the forecasted CE is rather smooth (notice the different scales on the vertical axes) which is a natural result when computing the CE from the median prices. We have now performed an analysis of the empirical energy prices and derived a final price model from which we have forecasted future energy prices. Before applying these forecasted prices in a forthcoming risk assessment of future cash flows, the next section will test the final price model in various ways to assess the model and its applicability. Page 59 of 129

62 5 Testing the model 5 Testing the model As a final element in our analysis this chapter serves to validate the derived model and to assess its stability, performance, and applicability. First, an investigation of assumptions from the performed tests and estimation procedures is conducted in order to validate the final model specification; i.e. we use statistical tests to support the content of the final price model. Second, the stability of the parameters, when adding new data, is studied to assess the consistency of the model. Last, back-tests are performed to assess the forecasting performance and applicability of the model. The out-of-sample method to test the model is the most crucial, and back-testing is the best method to uncover potential model misspecifications and shortcomings regarding its forecasting purpose. Therefore, the general applicability of the model can be assessed from back-testing. 5.1 Test of assumptions In our price analysis and modelling we have performed a range of tests and used estimation procedures which all rely on underlying assumptions. Tests of assumptions is important to our price model, because unfulfilled assumptions may lead to biased or inconsistent estimators or invalid test results which may lead to a poor and in worst case misspecified model. To validate the test and estimation results from chapter 4, this section will investigate the most important assumptions of the procedures. The test and estimation procedures we have used are all related to: The determining of order of integration of the price series, Johansen s test for cointegration, and the estimation of the final VECM. One or several statistical tests or estimation procedures are performed in each case, and we wish to investigate whether the assumptions are fulfilled. A complete overview of the three cases, statistical tests, estimation procedures, and assumptions are given in Table Page 60 of 129

63 Energy Price Risk Modelling in Appendix D. It is especially worth noticing that the table provides answers to whether or not an assumption is fulfilled. Following the table we have shown all the necessary graphs and tests that we have employed to check if the assumptions are fulfilled, therefore, for a thorough description of the tests of assumptions see Appendix D. Here we will only comment on the problematic assumptions. One problematic assumption is related to Johansen s approach, because the approach uses the Maximum Likelihood (ML) procedure to estimate initial parameters in the VECM model whereupon the actual test for cointegration is performed. The ML procedure requires the residuals in the full VECM to be normal, independent, and identically distributed (NIID). If the residuals are NIID then the likelihood function is correctly specified, and under weak regularity conditions it can be shown that the maximum likelihood estimators are consistent, asymptotically efficient, and asymptotically normal (Verbeek, 2004). By testing for normality in the residuals we find that this assumption is unfulfilled, since Jarque-Bera test rejects normality in the residuals from oil and coal (c.f. Output to Output ). As pointed out in the purpose of the ML estimation is to estimate and base the rank test upon this estimation. Since we have a clear and non-sensitive conclusion on the rank test we deduce that potential inconsistency in the ML estimators will not affect the conclusion from the test. This deduction is also based on the fact that we have a sample size of 194, which should improve the asymptotic properties of the ML procedure. Also, the final VECM parameters outside CE are not based on this ML estimation but re-estimated using OLS. Therefore we presume that this unfulfilled assumption does not have notably negative consequences for our modelling 13. Another cause for concern arises from one of the assumptions related to the estimation of the final VECM where we use OLS. The OLS procedure requires, among other assumptions, that the residuals have zero conditional mean in order for the OLS estimator to be unbiased (Wooldridge, 2008). We have not statistically tested this assumption, but we find that it is highly unlikely that the residuals have zero conditional mean, since this would imply that omitted variables, e.g. weather conditions, do not affect the dependent and explanatory variables in the equations. If the estimators are biased it means that the expected value of the estimators are different from the true values. However, if the biasedness is limited we assume that it may not have severely negative consequences for the application of our model, since it is used for forecasting of confidence levels based on numerous stochastic simulations and it is not used to make predictions based on single simulations. Another problematic assumption stems from the specification of the final equations, since we use t-tests to eliminate insignificant regressors in the VECM. The validity of the t-tests is, 13 Alternatively if NIID residuals cannot be assumed for our VECM one could formulate another model type: A Conditional Cointegrated VAR where the NIID assumption is not present. However, that is out of scope for this thesis. Page 61 of 129

64 5 Testing the model among others, dependent on the assumption that the residuals from the restricted VECM are NIID (Wooldridge, 2008). From Output to Output we again see that the oil and coal equations are rejected as having normally distributed residuals both in the full restricted, VECM and in the final VECM. This may result in invalid t-tests. We have a sample size of 194, which helps on the inference properties of the test statistic asymptotically, however, this unfulfilled assumption is still a cause for concern. An invalid t-test may lead to faulty conclusions on eliminating or accepting regressors which may result in a misspecified final price model. The back-tests performed later in this section, should reveal if severe misspecifications are present. For that reason, we continue with the final price model based on an elimination of regressors using t-tests and assume that, unless our back-test performs poorly, we do not have severe misspecifications in our model. In conclusion, in Appendix D we find that most assumptions for the tests and estimation procedures we have performed are fulfilled. Although we have a few assumptions that are unfulfilled, we do not expect that they invalidate our final VECM model, and we do not expect problems when applying the model for forecasting however we will keep the issues in mind. 5.2 Parameter stability test Parameter stability is very important in order to have a reliable and consistent model. In order to continuously apply the model for price forecasting is cash flow management, the model needs to be re-estimated as time passes (perhaps yearly). Therefore the model should not be too sensitive to the inclusion of new data. For that reason, it should produce stable parameters when the sample is prolonged. To investigate the stability of the parameters we re-estimate the model based on a subsample of the prices that goes from 1996 to Then we re-estimate the parameters again after adding an extra year of data at a time up until the full sample size has been included. Because we are interested in the sensitivity to an extension of the sample, we have overlapping samples for the parameters we wish to compare. We find no statistical method of testing if the parameters are stable, since this requires a comparison of parameters from non-overlapping subsamples. Therefore, in order to investigate the stability of the parameters we make a simple judgement of the fluctuations in the yearly re-estimated parameters. Besides the level of general fluctuations it is also interesting to see if the parameters are more sensitive to an Page 62 of 129

65 Energy Price Risk Modelling inclusion of non-normal time periods e.g. the financial crisis in The parameter estimates are presented in Table Table Parameter stability Dep. variable Indep. variable Mean Std. dev Judging from Table we find that the VECM parameters seem quite stable when including additional years to the sample. The level of fluctuations does not seem alarming, and there are no extreme parameter estimates in any of the years. To conclude on parameter stability, we find that our model seems reliable and consistent; therefore, the sample can be extended with future observations and applied for continuous forecasting. 5.3 Back-testing The most important test for us to conduct on our price model is back-testing also referred to as an integrated forecast analysis. A model may have a good statistical fit and be able to explain variations in-sample, but this could be a result of overfitting and lead to poor out-ofsample performance regarding forecasting, prediction, and accuracy. Because the final purpose of our price model is forecasting, back-testing is very important since this directly tests forecasting performance. Due to the application purpose of the model, we are less interested in 14 In the table we have only re-estimated the parameters. We have not performed the rank test, restrictions test, or sequential elimination of regressors. Hence, we maintain the model specification without question, as we wish to investigate the stability of the parameters in this given model. Another type of stability test could be to test the stability of the entire model specification requiring re-estimation of everything. This type of stability test is left for further research. Page 63 of 129

66 5 Testing the model the accuracy in predictions and more interested in the range of possible futures price scenarios. Therefore, we use the same simulation procedure as in section and generate confidence levels for the future prices. In a back-test a subsample of empirical data is used to forecast subsequent years. The forecast should only cover years where one has the actual observations, because the forecasted results are compared to these actual observed values (Wooldridge, 2008). We perform three back-tests by forecasting the prices in three subsequent years based on these subsamples: A: Sep Dec2007, B: Sep Dec2008, and C: Sep Dec2009. In Figure the time-cuts can be seen, where the vertical lines indicate from left to right when the sampling period ends for back-test A, B, and C respectively. Figure Time-cuts for back-tests We have chosen these three subsamples so that the forecasted periods are very recent, and because we need sufficiently large subsamples to estimate our model upon. Additionally, the financial crisis in 2008 had a massive impact on the energy prices, why it is interesting to see the forecasting performance during this crisis. Therefore, subsample A stops in Dec2007 which is before the financial crisis meaning we will forecast during the crisis and the aftermath. Subsample B stops in Dec2008 which is at the bottom of the crisis so that we forecast the aftermath in And finally subsample C stops in Dec2009 after the aftermath of the crisis in order to forecast more normal movements. It is very difficult to forecast price movements during a crisis - especially when you do not expect the crisis to occur. Stress-testing or extreme shortfall tools, can be used to forecast prices under such extreme events. Our model on the other hand is not build to forecast extreme price movements, and we do not expect our model to perform well during the crisis; however, it is still very interesting to observe how it performs. Therefore, back-test C is the real test on our model since there are no extreme events happening and in this case our model should perform well if it is an adequate model. Page 64 of 129

67 Energy Price Risk Modelling For all tree back-tests we forecast three years ahead by producing confidence levels from 1000 simulated price scenarios just like in section When interpreting the graphs in the following it should be noted that we still operate with confidence level lines. Because the actual prices (black line) can be interpreted as one simulated price path, it is still consistent with the model if all, several, or none of the actual prices lie outside the outermost confidence levels. This is because 10% of the simulated prices fall outside the range delimited by the lower (95%) and upper (5%) confidence levels. Note that we forecast the ln prices instead of DKK prices, and we have not added seasonality to the gas prices. This is because the back-test is a test on the final VECM model and not a test on price conversion and seasonality adjustment. Including these elements would definitely also be interesting in another back-test, but because we only wish to test the VECM model adding these elements induce noise to the analysis Back-test A Figure Back-test oil prices based on Figure Back-test gas prices based on Figure Back-test coal prices based on Page 65 of 129

68 5 Testing the model In back-test A we forecast the period of , which includes the financial crisis and the aftermath of the crisis. As presumed the results yield poor forecasting performance for our VECM, because our model cannot forecast the large drop in oil price in Q or the decrease in gas and coal prices in Therefore, it clearly overestimates the price for oil, gas, and coal where the overestimation for gas and coal happens over a longer period because the actual prices only slowly recover. The large price drops during 2008 and 2009 are simply too un-normal for the model to forecast if it is based on the subsample of Furthermore, we observe that the actual prices suffer from strong autocorrelations in this period, as negative returns follow negative returns, and positive return follows positive returns for several periods. Our model is not built for modelling changing autocorrelations either Back-test B Figure Back-test oil prices based on Figure Back-test gas prices based on Figure Back-test coal prices based on In back-test B we forecast the period of , which is a period that starts where the energy prices were around their bottom values and later recovered in the aftermath period. Page 66 of 129

69 Energy Price Risk Modelling Our model performs well for gas and coal where the actual prices lie within the confidence levels. The forecast for gas even predicts the decreasing prices in the first part of With regards to forecasting oil prices, our model performs quite poorly as the actual prices only lies within the 5% confidence level line to start with and then outside for the remaining period; where the model consequently underestimates the oil price. This is because the sampling period stops at a time point where oil price was at an extreme minimum, and our model fails to predict the immediate gain the oil price experienced in the aftermath. A market expert might have recognised that oil prices were extremely low in December 2008, when standing in the moment, and would probably expect it to rise again. Therefore, adjusting the model with expert knowledge would perhaps be relevant in this case. For example an expert could replace the actual price in December 2008 with a less extreme observation and base the forecast on that point of origin instead Back-test C Figure Back-test oil prices based on Figure Back-test gas prices based on Figure Back-test coal prices based on Page 67 of 129

70 5 Testing the model In back-test C we forecast the period of , which a period of more normal market conditions. When forecasting this period our model performs quite well for all three energy prices, and none of the actual prices lie outside the confidence levels. To summarize: In back-test A we experience that our model performs poor when forecasting the financial crisis. This is not surprising as the actual prices in that period experienced both extreme movements and unusual autocorrelation. In back-test B we find that our model and its forecasting abilities are very sensitive to point of origin, and forecasting from an extreme starting point may affect the entire forecast. In back-test C we find that our model performs quite well when forecasting during normal periods. The shortcomings of back-test A and B are natural consequences of the fact that our model is built in retrospect, and that forecasting energy prices accurately can only happen in the rare case that history repeats itself. Therefore, results from this price model should, like any other econometric and statistical tool, be adjusted by expectations to the future. The model can for example be extended to include forward energy prices, which represent expectations to the future. Alternatively market experts could adjust the model according to their professional opinions. This adjustment can either be directly implemented in the model by changing the data input or parameters in the VECM or indirectly by combining the result with experts opinions. In back-test C we experience that our model performs fairly well when forecasting periods that are not extreme. From this we find support that our model does not suffer from severe misspecification or bias. And, as mentioned earlier, the simulation framework allows that all, some, or none of the actual prices lie outside the confidence levels, while still consistent with the model. Therefore, outliers from back-test A and B does not invalidate our model per se. In conclusion, we find that our model performs well in the back-tests, and that the model is applicable for forecasting future energy prices in markets that are not extreme. We close Chapter 5 by concluding that in all tests we have performed on our final model regarding underlying assumption, parameter stability, and back-tests we have not found evidence of substantial bias, invalidation, misspecification, or poor performance. In the backtests we even saw the consequences of shortcomings of the model. These shortcomings are expected and do not cause any concern as long as the user of this model is aware of its applicability and limitations. With this in mind, we find that our model is adequate for its purpose of forecasting energy prices for cash flow management. Page 68 of 129

71 Energy Price Risk Modelling 6 Conclusion PART I The objective of PART I was to give an answer to the first research question of the problem statement: How can an energy company forecast energy prices with respect to managing cash flows? The research question was approached in a contemporary and empirical study of North European oil, gas, and coal commodities. We started by reviewing the existing literature on energy price modelling. The purpose of this review was to place our study in the landscape of energy price modelling and to investigate findings from any comparable studies. We found that the energy price modelling methodologies generally can be divided into three types: Fundamental models, mean-reverting models, and cointegration models, where our study falls into the last type of methodology. Further investigating cointegration literature, we found that the findings in these studies can depend on the commodities, region, and time period investigated. We also did not find any study directly comparable to our study, but the most similar studies generally find cointegration between oil, gas and, coal. We continued with an initial analysis of energy prices. The analysis focused on potential interrelationships between energy prices, because such co-movements are important to incorporate in risk models. Here we found that oil, gas, and coal, in Northern Europe, all share common price drivers why there was reason to investigate whether market integration among these commodities exists. The graphical analyses on the empirical prices confirmed that although the three price series seem to wander stochastically, they have strong co-movements and may be interrelated. Based on this initial analysis we found reason to suggest that our three prices are interrelated, but we did not see a clear indication of how such relationship manifests itself. Therefore, we closed the chapter by formulating two hypotheses on cointegration relationships between the three energy prices which were: Hypothesis 1: Oil, gas, and coal are all cointegrated with oil being the independent leader of both gas and coal, Page 69 of 129

72 6 Conclusion PART I and Hypothesis 2: Oil, gas, and coal are all cointegrated with oil and coal being the independent leaders of gas. These two hypotheses created basis for the subsequent energy price modelling. In the energy price modelling process we first investigated stationarity. From the theory we learned that it is a prerequisite for cointegration that the price series are non-stationary and that they are integrated of same order. From the empirical analysis of stationarity we found that our three energy price series can be considered processes, which made it possible for us to continue the price modelling using cointegration techniques. We continued with the theory on cointegration, where we described how cointegration can be interpreted and how the dynamics of cointegrated processes can be expressed in an error correction framework. We also described how cointegration can be tested for in a multivariate case via Johansen s approach. Performing the empirical cointegration analysis, we found by means of rank tests that our three prices share one cointegration relationship. With a rank of one we proceeded to impose restrictions to test the two hypotheses presented in the previous chapter. We concluded that we are more prone to accept hypotheses 2: Oil, gas, and coal are all cointegrated with oil and coal being the independent leaders of gas. Based on these restrictions we modelled the VECM and eliminated all insignificant regressors. This resulted in a final price model where the gas price follows a form of an error correction process and the oil and coal processes are equivalent to random walks. As a last element in constructing an energy price model, we tested the model in various ways to assess its validity, stability, and applicability as a forecasting model. We investigated its validity by examining the assumptions behind the statistical tests and estimation procedures applied. Despite a few unfulfilled assumptions, we concluded that this has not led to an invalid or misspecified final model. Parameter stability was also investigated and they were considered quite stable and insensitive to an extension of the sample. Most importantly the final model performed well in back-tests; therefore, we find the forecasting ability of the final model satisfying. In conclusion, within the framework of our study, we find that an energy company can forecast energy prices, with respect to managing cash flows, by means of a VECM model where one cointegration relationship is shared by all three commodities and where oil and coal lead gas prices. This method produces a model that forecasts price scenarios that are suitable in a Cash-Flow-at-Risk model for assessment of energy price risk. Page 70 of 129

73 PART II Modelling CFaR The main objective of Part II is to answer the second research question of the problem statement: How can an energy company measure cash flow risk originating from energy price fluctuations? The research question is investigated in an exemplified framework based on the premises from realistic business operations of a North European energy company. PART II first leads you through a general discussion of the importance and benefits of cash flow management in a corporation. The CFaR model is then proposed as a tool to measure market risk in cash flows, and the theory behind the model is presented. PART II continues with a practical example where we illustrate and interpret how a CFaR model can measure energy price risk. Lastly, the applicability of the model as a more active cash flow management tool is illustrated in a what-if analysis. We close PART II by concluding on our findings regarding the research question.

74 7 Managing cash flow risk 7 Managing cash flow risk In this chapter we will discuss the role of cash flow management in the corporate environment. From a general perspective we will discuss the benefits of performing cash flow management by providing examples of the elements and operations this undertaking often addresses. A key element of cash flow management is the task of measuring risk; therefore, the theoretical perspective will focus on a tool that is widely applied in corporative cash flow management and that is the CFaR model. This model is specifically developed to measure cash flow risk in the corporate environment. 7.1 Cash flow management Cash flow is the movement of cash in a company. Cash flows arise from operating, investing, or financing activities, where cash outflows can stem from expenses or investments while cash inflows can stem from sales income or funding. Cash flow can be any operating cash flow from the overall EBITDA to a specific sales income from a certain business unit. Overall, cash flow can be thought of as the blood that runs through the corporate body, and without cash flow the company cannot exist. Cash flow management, cash flow risk management, and risk management are all overlapping terms with no strict separation or definitions. In this thesis we will mainly use the term cash flow management and define it as: The task of identifying, measuring, monitoring, forecasting, and managing cash flows and communicating clearly about the process (Lee, 1999) and (Jorion, 2007). Sometimes we will also use the term cash flow risk management which is merely managing cash flow with focus on avoiding or reducing adverse consequences of internal or external events a field within risk management (Jorion, 2007). Note that managing risk is not all about eliminating risk, but it is about finding the right balance Page 72 of 129

75 Energy Price Risk Modelling between risk prevention and proactive value generation (Andrén, Jankensgård, & Oxelheim, 2005). Cash flow management is never a one-way-fits-all solution and this needs to be tailored to the specific company, the specific business unit, the specific area, and the specific risk types. Moreover, cash flow management can exist on all corporate levels: From the very specific departments, accounts, or portfolios to the overall corporate figure like EBITDA. It can therefore be an immense and continuous process to manage cash flow, and a competent risk governance system would entail a cash flow management system consolidating all focus areas, risk types, and levels of cash flow management Benefits of cash flow management The benefits of managing cash flow are limitless, and a non-exhaustive list of reasons why a company should perform cash flow management is: Control and assessment of profit and loss: First and foremost cash flow management helps measuring and controlling the flow of cash and cash equivalents that results in the profit and loss of a business (Stolowy & Lebas, ). Measurement of cash flows gives the company an insight into its health and competitiveness, and they can design strategies to optimize operations or avoid losses and uncertainties due to unfavourable events. It is fundamental for a company to measure and control their profitability or possible losses of their operations in order to be a going concern. Liquidity: Knowing, understanding, and controlling the cash flows help avoid shortages of cash or cash equivalents needed to fulfil short-term obligations (Stolowy & Lebas, ). Even a financial sound company needs to make sure it has enough liquidity to meet payments to e.g. creditors or in other words that it has sufficient working capital. If payments are delayed it might result in charges, delayed delivery, or contractual deteriorations in the future. Furthermore, certainty and stability in working capital releases more capital for other activities creating a more efficient business. Solvency: Cash flow management is also very important for the company to be able to maintain and assess solvency to meet any long-term financial obligations. This is in order to avoid financial distress and in the worst case bankruptcy (Stolowy & Lebas, ). Large companies have a regulatory-required capital buffer to protect the solvency of the company from adverse events and volatility in general. If the buffer is not sufficiently large the company may experience financial distress and stakeholders are often highly sensitive to such signs of distress. Adverse reactions may be bad credit ratings, devaluation of stock prices, or in general bad image all leading to severe costs for the company. Furthermore, effective management of Page 73 of 129

76 7 Managing cash flow risk cash flows can reduce the size of the needed/required buffer releasing free capital for investments or other activities. Investments: Cash should, as stated earlier, not only be available to meet any obligations but also for financing of investment activities such as new projects. Lowering volatility in cash flows makes capital budgeting plans for new investments less uncertain, and therefore analysing the profitability becomes more precise. A project s or investment s profitability is often assessed by the net present value (NPV) of future cash flows, and lowering uncertainty regarding future cash flows provides more reliable NPVs. This enables managers to make better and more informed investment decisions. Company valuation: From a company s financial report the cash flow statement is a key element used by analysts and investors to value the company (Lee, 1999). The cash flow statement signals the company s health and growth opportunities and competent cash flow management will improve these signals through a stabilisation of cash flows. Better signals provide more attractive funding and contracting terms as it shows the company s ability to fulfil future obligations. This will lower the cost of capital which is a key figure used by analysts and investors to value a company; a lower cost of capital yields a higher firm value. Furthermore, credit rating agencies interpret having surplus inactive cash as a sign of the company having no investment opportunities; this may lead to poor growth potential valuations and to lower credit ratings. From this non-exhaustive list of benefits it should be obvious why companies should perform cash flow management especially when cash flows are volatile and uncertain. One of the important tasks of cash flow management is the measuring of risk (cf. the definition in the start of this section); therefore, in the next section we present CFaR which is a cash flow management tool developed for exactly that. 7.2 CFaR Basically cash flow management is about enhancing transparency and reducing uncertainty and a key element in the process is the ability to measure risk. This section presents the theory on CFaR and how this tool measures risk. An elaboration on how to construct a CFaR model in practice follows in chapter 8. CFaR is a risk metric that has the purpose of quantifying cash flow risk related to adverse movements in specific market factors. CFaR can be expressed as: Page 74 of 129

77 Energy Price Risk Modelling The maximum shortfall of net cash generated, relative to a specified target, which could be experienced due to the impact of market risk on a specific set of exposures, for a specific reporting period, and confidence level (Lee, 1999). From this definition risk, in the CFaR framework, is interpreted as shortfall of cash flow relative to the targeted. Notice that the risk is measured in cash flows, while other popular market risk metrics, e.g. Value-at-Risk (VaR), interpret the risk as a negative change in market value of assets. It makes sense to measure risk in terms of cash flow, when transactions or contracts are not traded in the commercial markets or when commercially traded assets are illiquid and have long horizons. In such case, it can be difficult to assign a sensible market value. As an example: An energy company s strategy and overall production is planned several years into the future. The company can forecast cash flows streams from the production plan, but they cannot assign the transactions or contracts reliable market values, as the energy markets are not efficient and liquid enough beyond a certain time horizon of three to five years. Additionally, even if they could be assessed at market values, measuring risk in terms of market value over long horizons would still not be preferable, as continuously fluctuations most likely would result in an overestimation of the risk or at least serve as noise (Strickland, 2012). CFaR furthermore measures risk as a shortfall relative to a specified target, where the target level can be set as expectations, strategic goals, or analytically estimated forecasts. This means that CFaR measures instability and not necessarily losses which is typical for other risk metrics. This is because the target does not need to be the mean or zero (such as in VaR) but can even be negative; in the short term a company can budget with deficits but still survive. Over long horizons, on the other hand, a company needs positive net cash flow of a certain amount. Like many other risk metrics, CFaR measures risk over a specific reporting period. However, CFaR often has a longer horizon than most other metrics. An energy company s cash flows are determined by long-term production plans, investments, contracts, and obligations all have long term prospective and are hard to negotiate. For these to be orderly managed and to be able to perform risk mitigation this requires a long horizon. In general the core assets of an energy company are very illiquid why it does not make sense to measure their risk on e.g. a daily basis; instead, a horizon of one to five years is recommended (Strickland, 2012). CFaR also estimates risk within a chosen confidence level. The CFaR metric is based on an estimated distribution of the future cash flows, where the lower tail indicates the risk. The level of confidence is chosen by the risk manager and typical levels are 99%, 95%, or 90%. If we for example identify a CFaR of 5 M DKK using a 95% confidence level, then we are 95% Page 75 of 129

78 7 Managing cash flow risk confident that the shortfall of net cash will not be greater than 5 M DKK. In this sense, CFaR measures risk under normal market conditions and only tells us what happens with 95% certainty. We do not know how bad things can go the last 5% of the times, and other risk measures, such as the stress testing or extreme shortfall models, will be necessary to apply here instead. Figure depicts how the CFaR metric is derived from a cash flow distribution. The figure shows the distribution of forecasted net cash flows from a specific portfolio aggregated over a time horizon e.g. three years. If we set the target at 8 M DKK, choose a 95% confidence level, and observe a cash flow of 3 M DKK at the 95% confidence limit line this gives us a CFaR of 8 3 M DKK = 5 M DKK. We can now say, We are 95% certain that we will not be short of more than 5 M DKK in net cash flow, during the next three-year period. Figure Illustration of CFaR Frequency percent target CF 95% confidence limit Source: Own contribution CFaR Net cash flow In order to get a reliable CFaR metric you need to estimate the cash flow distribution above for the time period of interest and that is the real challenge of CFaR. Constructing a cash flow distribution involves several overall steps each with many options and custom tailoring to the portfolio or company in question. In the next chapter we elaborate on these steps, when we in a practical example construct a CFaR model. Page 76 of 129

79 Energy Price Risk Modelling 8 CFaR in practice Until now we have covered cash flow management and CFaR from a theoretical and general perspective. Now that we have the foundation, we will narrow our focus to management of cash flow uncertainty driven by energy price risk. In the first section, using a fictive case as a practical example, we will illustrate how CFaR can be constructed by tailoring the model to a specific area of interest. Next we will interpret the results from our CFaR model and the inherent risk assessment. As a last element in the practical example, we will illustrate the applicability of a CFaR model as a more active cash flow management tool by applying a what-if analysis. 8.1 Constructing the CFaR model We will follow these steps when building our CFaR model: 1) Define area of interest 2) Metric specification 3) Exposure mapping 4) Risk factor forecasting 5) Scenario generation 6) Risk metric computation. With inspiration from different authors (Lee, 1999) and (Strickland, 2012), we have designed these steps specifically to our case example. The overall process of constructing a CFaR model is generic but the division into steps vary as there is no one-wayfits-all procedure and all models need to be tailored specifically to the company and area of interest Define area of interest This first step is one we have added to our procedure that our sources have not covered explicitly. Often the area of interest is given by the nature of the business or is decided upon by higher management levels. We would like to advocate for including this initial step in the process, in order to make sure that there is an alignment between what we wan to measure and what the CFaR metric actually measures. In our practical example we will investigate a Page 77 of 129

80 8 CFaR in practice utility unit. We have chosen this business unit because it is very exposed towards energy market risk and because it manifests a current risk management challenge, which most energy companies face the oil-gas spread. An energy company often has substantial utility service obligations. This means that if the energy company cannot extract and produce the amount of energy in demand from retail and wholesale customers, it needs to acquire the shortage amount in the wholesale markets for resale on the retail markets. In regards to gas, this is a main business activity for a North European energy company due to the limited amounts of gas in the area and a relatively high demand caused by thermal energy production and heating. This demand is rather stable and continuous in the long run, why the energy companies hold long-term purchase agreements to meet the demand. However, as discussed earlier, the long-term gas purchase agreements are often oil-indexed, while the short-term to medium-term resale in the retail markets are often gas hub-indexed. Thereby, the utility unit purchase at an oil-indexed price and resells at a gasindexed price leading to a short position in oil and a long position in gas. The resulting energy risk exposure is considered a spread risk, since it is the price difference between two different energy products, and this spread determines the profitability of the business unit. A gas purchase agreement encompasses many elements and terms. In order to reduce unnecessary complexity, we reduce all the contracts into one single oil-gas spot relationship based on Brent crude oil and NBP natural gas. Most contracts would be indexed on a three months lagged oil price and the typical type of oil products to index upon would be fuel oil rather than crude oil (Dragomanovich, 2012). However, due to extent of this thesis, we impose simplifications and express the average price relationship in portfolio of contracts to be: Eq. 19, where is a constant determining the oil-indexation of the gas contract. We call the contract fair when the equality holds, when it does not hold (due to fluctuating prices) the risk the energy company is exposed to is the spread between the left and the right side of the equation. To be able to construct an asset portfolio in terms of positions, we have translated Eq. 19 into a relationship between quantities so that the relationship between gas and oil in amounts becomes: Eq. 20, Here we have set. This relationship, which sums up all contracts in the portfolio, will obviously change with expiration of contracts, signing of new contracts, or with renegotiation of current contracts, however, in our example we assume that the relationship is Page 78 of 129

81 Energy Price Risk Modelling constant throughout the investigated horizon. The positions of the asset portfolio can be seen in Table Table Utility portfolio Quarter Oil (T BBL) Gas (T Therms) Q Q Q Q Q Q Q Q Q Q Q Q As it appears fromtable there is a seasonal pattern in the amounts, which reflects the higher demand for gas in the winters for electricity and heating. We assume constant demand and seasonality pattern in the near future, no storage, and that everything is sold at the monthly average price. The area described is very specific and delimited as opposed to if the area of interest for example was the net cash flow EBITDA. Hence, our area of interest only represents a specific level of cash flow management which could potentially be a part of a greater consolidated cash flow management system for enterprise wide risk governance Metric specification The CFaR needs to be measured within a specific time horizon and confidence level. The confidence level for our CFaR is set to 95% since this is the conventional level in the existing literature. The time horizon of a CFaR needs to match our business unit and allow for an orderly management and potential mitigation of risks. The assets (the purchase agreements and utility obligations) in the utility unit can be rather illiquid, and therefore, it can take a long time to divest or hedge the contracts orderly. Furthermore, a contract might generate negative cash flows over a period while it still remains a profitable asset in the long run. This argues for a rather long horizon for CFaR (Strickland, 2012). But even if the asset might live many years Page 79 of 129

82 8 CFaR in practice into the future (a purchase agreement can be up to 30 years), the North European energy market is not very liquid beyond three to five years making it impossible for energy companies to properly hedge beyond that period. Based on these considerations we set the horizon to be three years Exposure mapping In this step, we identify the exposures of our cash flows. Our portfolio consists of positions, i.e. quantities of oil and gas, and we expect to turn those positions into cash flows by selling them in the energy markets. The energy prices then become our risk factors, because the cash flows of the portfolio depend on the future energy prices. Commodity prices are typical risk factors for corporate risk models. Other typical risk factors include exchange rates, interest rates, growth rates, demand rates etc. (Lee, 1999). The relevant risk factors depend on the area of interest. To establish how our cash flows depend on the risk factors, we need to construct an exposure map stating the relationship between cash flows, positions, and risk factors. In our example with constant positions, we have a linear relationship between positions and risk factors and the exposure mapping is simply a multiplication of the energy prices by the quantities of our portfolio. The exposure map can be expressed as follows: Eq. 21 where stands for quantity, for price, and in our case goes from 1 to 36 months as we look three years ahead Risk factor forecasting In this step we forecast the energy prices. No single forecast is perfect; therefore, the CFaR approach is based upon simulations of several forecast scenarios in order to generate a distribution of possible outcomes. Forecasting and simulating the risk factors, can be a very complicated and advanced process especially if no parametric assumption can be made. For some risk factors it can be assumed that their movements approximately follow a well-defined distribution such as the normal distribution. However, the energy prices relevant to our CFaR is not well described by a parametric distribution, therefore, we performed an econometric time series analysis in PART I to estimate a price model. As a reminder to the reader, the analysis resulted in this final price model: Page 80 of 129

83 Energy Price Risk Modelling Box 2 Final price model 15 where Based on the price model and bootstrapped historical residuals, we simulate 1000 price scenarios via Monte Carlo simulation. We refer to section for a presentation of the forecasted price scenarios Scenario generation Now that we have simulated 1000 price scenarios, we can generate 1000 cash flow scenarios simply by using equation Eq. 21 and the information on the quantities of oil and gas in the utility portfolio. Some practitioners would now go directly to the risk metric computation in the next step. Inspired by Strickland (2012) we expand this step with graphs of confidence levels over time of the cash flow scenarios. Here we investigate the cash flow scenarios further as they, when arranged properly, contribute with valuable information for the risk managers. Instead of creating a single cash flow distribution, the generated cash flow scenarios are sorted with respect to size. From the sorted cash flows it is possible to identify the percentiles for each time bucket which provides us with confidence levels for the possible cash flows at each time step of monthly granularity. This analysis gives risk manager information about the evolution of risk throughout the horizon. We refer to Figure in section 8.2 for an illustration of the graph, where we also interpret the other results of the CFaR model Risk metric computation This last step involves the generation of the aggregated cash flow distribution and identification of the final CFaR metric. For each of the 1000 cash flow scenarios generated in the previous step, the cash flows are aggregated across time buckets and presented in a net cash flow distribution. The aggregation across months ensures that CFaR is measured as a total shortfall for the entire period of three years. The CFaR metric was defined as: The maximum shortfall of net cash generated, relative to a specified target, which could be experienced due to the impact of market risk on a specific set 15 Please note that the equation for the coal price is not redundant, as the oil and gas prices depend upon it. Page 81 of 129

84 8 CFaR in practice of exposures, for a specific reporting period, and confidence level. This can now be identified as the target level cash flow minus the value at the confidence limit line. We set the target to be the mean, and the CFaR metric thereby measures risk as a deviation from the expected. Alternatively, the target could be set by management to reflect strategic goals for the relevant horizon. The results can be seen in Figure and Table in the next section. 8.2 Interpretation of CFaR results The aggregated cash flow distribution over the three-year horizon can be seen in Figure and the descriptive statistics in Table Figure Distribution of aggregated cash flows Jan2013-Dec2015 Table Descriptive statistics for aggregated cash flows CFaR Mean Std Dev Skewness Ex. kurtosis Range 95% CL 50% CL 4, , ,104-4, From Figure and Table 8.2.1, we see that CFaR is 4,618 T DKK. This is computed from the expected cash flow of -108 T DKK minus the 95% worst scenario of -4,726 T DKK. Thus we can say that, We are 95% certain that the utility unit will not be short of more than 4,618 T DKK in net cash flow, relative to the expected -108 T DKK, during the next three-year period. We notice that the expected cash flow is negative, which means that under the current purchase agreements this business unit is expected to be deficit generating over the next three years. If this expectation does not match the strategic goal of the utility unit then it might be necessary to renegotiate the underlying contracts, as they directly affect the profitability of the unit. In the next section we will elaborate on how CFaR can be used as a decision tool for the renegotiation process. Page 82 of 129

85 Energy Price Risk Modelling If we look at the shape of the distribution and its descriptive statistics, we see that it is almost bell-shaped except for the slightly negative skewness of and an excess kurtosis of leading to fatter tails and a higher peak than for a normal distribution. The near bell shaped distribution indicates that the utility unit has noticeable amounts of both downside and upside risks. The bell shape is expected because we subtract the right skewed oil distribution from the right skewed gas distribution. The CFaR metric of 4,618 T DKK should match the risk appetite of the company. If the company cannot endure a cash flow that is 4,618 T DKK lower than expected, or if the company is unwilling to take the possible negative consequences, such as financial distress costs, bad credit ratings, and increased cost of capital, it should mitigate risk until an acceptable risk level is reached. Looking at the cash flow scenario paths or confidence level lines in Figure might help us in the decision on how to mitigate risk. Figure Cash flow confidence levels In Figure 8.2.2, which we generated in step 5, we see the confidence level for cash flow distributions in each time step. We see that the range of possible scenarios evolve in a seasonal pattern and increase with time. The strong seasonal pattern is the result of the combination of high demand and a high price level with increased volatility during the cold months, while the increase over time stems from the time-effect of uncertainty. This figure reveals the great risk exposure in the winter periods, which can help risk managers design a strategy that focus on hedging in these periods. An effective hedging strategy might aim at the high risk periods in Q1 for all three years and such mitigation could be conducted by hedging with financial instruments such as forwards, futures, put options, floor instruments, spread options, swaps, and many other financial instruments from the more exotic category. Now we have presented and interpreted the results of a CFaR model. These results combined with the company s established risk strategy should help risk managers decide whether they Page 83 of 129

86 8 CFaR in practice would like to mitigate risk. Active risk management questions such as when, where, and how the risk mitigation should be conducted are not straightforward to answer from the results we have presented so far. The CFaR number can with the company s established risk strategy help risk managers decide whether they would like to mitigate the cash flow at risk. The cash flow scenario paths give us a better indication on when we should mitigate. And if we dive into the underlying calculations, which are not presented in this thesis, we can identify which commodities that are the largest contributor to risk, and this information can give us valuable information on where to mitigate. The CFaR model can also contribute to the decision on how to mitigate. We will elaborate on this in the next section, where we expand our example with an illustration of how the CFaR model can be used for what-if analysis. 8.3 Example of CFaR application The ultimate goal of a CFaR-model is to estimate the CFaR metric; the isolated number representing the riskiness of the studied area. However, CFaR is not merely a method of quantifying risk. It is also a flexible tool that can be applied for many aspects of cash flow management and serves as input in high level managerial decision processes. Examples of the broad applicability of the model entails: Creating basis for risk communication by presenting risk in an easy understandable formulation, from a method that can be applied and consolidated throughout all managerial levels of interest. Supporting risk governance by setting risk targets and risk limits for managerial levels and by setting ladders of intervention for breaching these limits. Supporting decisions such as hedging strategies, production plans, or negotiation strategies by serving as a CFaR based what-if analysis tool for managers to make informed decisions on where, when, and how to act and mitigate. Based on the current renegotiation problem in the energy sector, we will illustrate how the CFaR model can help managers make more informed decisions by applying a what-if analysis to our utility portfolio Background of the renegotiation problem European energy companies with utility service obligations often hold large amounts of longterm gas purchase agreement with oil-indexed prices. About half of Europe s gas supply comes from long-term contracts, mostly from Russian Gazprom and Norwegian Statoil (Dragomanovich, 2012), and about 70% of EU-15 s gas supply was in 2010 oil-indexed (Molloy, 2011). Thus, many utility companies purchase at an oil price and sell to consumers at a gas price. Page 84 of 129

87 Energy Price Risk Modelling This contractual link between oil and gas has recently become quite problematic for the utility companies. Since 2009 there has been a persistent and larger-than-usual gap between the two energy prices and this is expected to last or even increase (Molloy, 2011). In Figure the historical spread between Brent crude oil (DKK/bbl) and NBP natural gas (øre/thm) is depicted. Figure Historical spreads between oil and gas prices As can be seen from Figure 8.3.1, the oil-gas spread has been relatively large and persistent most of the period from Q until end of Practitioners argue that the large gap between the prices stems from market changes for the two commodities: Brent crude oil, being a globally integrated commodity, is rising due to a global increase in demand especially driven by the demand growth from Asia. Additionally, when the oil supply from Libya was cut in April 2011, due to a political embargo, the global oil price increased even further (Molloy, 2011). NBP gas on the other hand does not operate in a fully global market, and an ample supply of gas in the European market has dampened the prices. The abundant supply have three causes: 1) Decrease in European demand caused by the deteriorating economic situation in Europe and the political disabling of CO2 emission allowances an initiative launched to enhance gas driven power generation 2) Increase in gas supply to Europe from earlier US gas suppliers, e.g. Qatar, due to opening of new liquefied natural gas facilities and the expansion of shale gas in the US 3) The take-or-pay condition of most gas purchase agreements which gives the European gas buyer limited volume flexibility as penalties are charged if the agreed upon volume is not taken, which means they cannot adapt supply to demand (Molloy, 2011) and (Dragomanovich, 2012). This situation is obviously a large problem for the utility companies, as they lose money and most likely will continue to for several years according to market predictions and the length of the contracts. However, many practitioners argue that the negotiation problem not only is a Page 85 of 129

88 8 CFaR in practice high priority because of the current adverse spread between the prices but also because of the general risk aspect of the spread risk. This oil-gas spread risk is from a risk manager s perspective an unnatural risk: It makes sense to have your exposure to the underlying product you re trading rather than another product, says an anonymous gas broker to Ned Molloy (Molloy, 2011). Therefore, from a risk perspective, the contracts could be and should be gas-priced. This would give a natural relationship between the business activity and the exposure, as the spread risk would now be a spread between different gas indices. A gas-to-gas spread would most likely be less uncertain and less volatile compared to an oil-gas spread. The conflict of how to price the contracts have been going on for more than 15 years and, despite conflicting opinions, the majority of practitioners expectation is that the gas contracts eventually will end up being gas-indexed rather than oil-indexed (Dragomanovich, 2012). This transition has been long and on-going for years and will continue years to come. This is because the suppliers are unwilling to give up profit, and because of the need of an adequate alternative to base the contracts on. The alternative should be the EU-hub gas spot markets, but there is disagreement on whether the markets are sufficiently liquid and mature enough to provide a basis on which to structure and price contracts (Molloy, 2011). Alexander Burgansky, analyst at Otkritie Bank in Moscow, said that: Falling indigenous gas production, low availability of alternative gas suppliers including liquefied natural gas (LNG), limited infrastructure and low spot trading volumes create conditions for the oil-price link to remain the contract model of choice for both buyers and sellers of gas for the foreseeable future (Dragomanovich, 2012). On the other hand, an anonymous London-based gas broker said that: One of the reasons people are happy now to settle against the gas indices is that they are now reliable and based upon a liquid pool of gas, such that it makes sense for people to index against them rather than a bundle of other energy products. I can t think of any logical reason why you would now settle a long-term gas contract against oil (Molloy, 2011). Whether the gas markets are ready to support a full transition from oil to gas-indexation is mostly relevant for the future entering of contracts. For the negotiation of current contract the focus is on the degree of oil-indexation and terms of flexibility. Now that we have presented the renegotiation problem we will illustrate how an energy company can use CFaR to help decide which elements to focus on in the negotiation. Page 86 of 129

89 Energy Price Risk Modelling CFaR as input to decision making When it comes to the renegotiation of contracts there are many elements that can be subject to negotiation, and there are many contracts with various compositions. Therefore, there can be many negotiation strategies for a utility company to follow. Possible strategies could be a) to negotiate a direct price reduction by lowering the oil-indexation, b) to negotiate that a part of the oil-indexation changes to gas-indexation, c) to negotiate more flexibility in the take-or-pay contracts by reducing charges for non-taken amounts, d) to shorten the contracts and replace with gas purchased in the gas spot markets, or e) to reduce the contractual amounts and replace with gas purchased in the gas spot markets. The CFaR model can help the company make decisions about which strategy, or combination of strategies, to pursue. For each of the considered strategies the risk manager should estimate the worst case, base case, and best case of possible negotiation outcomes; then the CFaR model can be used in a what-if analysis of the possible outcomes. A comparison of the results from the different outcomes will help the managers make an informed decision. We illustrate the what-if analysis by investigating strategy a): Lowering the oil-indexation. When we presented the utility portfolio in section 8.1.1, we determined that the average gas contract price of the entire portfolio was 0.80% of the oil price so that. Now, let us assume that the risk manager estimates that the base scenario is, that the contracts can be lowered in oil-index resulting in a new average portfolio index of. With everything else being equal, the base scenario of the renegotiation is illustrated in Table 8.3.1, and the new cash flow distribution and descriptive statistics are presented in Figure and Table Table Base scenario of renegotiation Quarter Oil (T BBL) Gas (T Therms) Q Q Q Q Q Q Q Q Q Q Q Q Page 87 of 129

90 8 CFaR in practice Figure Distribution of aggregated cash flows from base scenario of renegotiation Table Descriptive statistics for aggregated cash flows base scenario CFaR Mean Std Dev Skewness Ex. kurtosis Range 95% CL 50% CL 4,166 1,481 2, ,517-2,685 1,538 From Figure and Table we see that the expected cash flow is now positive and it has increased from -108 T DKK to 1,481 T DKK as a result of the renegotiation. The CFaR is reduced from 4,618 T DKK to 4,166 T DKK, while the shape of the distribution has not changed notably. Based on this what-if analysis the risk manager knows that he can expect a positive cash flow from this strategy and that CFaR is slightly reduced. This information can be valuable in that the risk manager now knows the expected effect of this strategy. For a full analysis, the worst, base and best case scenarios for all considered strategies should be compared and the outcomes of each analysis should serve as information in the decision making. However, this example ends here, as we believe we have illustrated how CFaR can be used as input to decision making. With this case example of how CFaR can be constructed and how the results can be interpreted, we believe we have uncovered how CFaR can be used to measure energy price risk and also how this information can be used in decision making. Therefore, we end this part here and conclude upon our findings in next section. Page 88 of 129

91 Energy Price Risk Modelling 9 Conclusion PART II The main objective of Part II was to answer the second research question of the problem statement: How can an energy company measure cash flow risk originating from energy price fluctuations? The research question was approached in an exemplified framework of a North European energy company. In order to answer this question we first discussed the importance of cash flows from a general point of view. We argued that the cash flows can be considered the blood of the corporate body, and adequate management can contribute with numerous benefits all from efficient operations and improved external relationships. Then we presented the risk management tool, Cash-Flow-at-Risk (CFaR), from a theoretical point of view and how it is defined as, the maximum shortfall of net cash generated, relative to a specified target, which could be experienced due to the impact of market risk on a specific set of exposures, for a specific reporting period and confidence level. Thus, the CFaR model measures risk in terms of cash flow uncertainty and quantify it as the difference between the targeted net cash flow and the e.g. 95% worst net cash flow, from a distribution of forecasted net cash flows. The theory emphasised that a CFaR model needs to be tailored to the specific area of interest and there is no on-way-fits-all solution. In the example of an energy company s utility unit, we gave a more precise and tangible answer to how the energy company can measure risk. Here we executed the steps in constructing a CFaR model on the premises of a North European energy company. While performing the steps, we discussed how the interpretation of risk as uncertainty is cash flows, is appropriate for energy companies, as their cash flows are highly exposed to energy price fluctuations and that the relatively long horizon of the CFaR suits the illiquidity of energy assets. We illustrated how seemingly complex and immeasurable Page 89 of 129

92 9 Conclusion PART II risk from different activities and sources can be mapped onto specified risk factors in order to create transparency in the risk picture of a business unit. The challenging task of forecasting risk factors in order to generate the net cash flow distribution, was managed by employing the price model from PART I, which is developed for the exact purpose of serving as input to this CFaR model. When interpreting the findings we deduced that if the CFaR results do not align with the strategic goal and risk appetite of the business unit, then mitigation action should be considered or the strategic goals adjusted in order to align the measured and the wanted. In this way, the CFaR model can help decide whether risk mitigation is needed. Lastly, we discussed how the CFaR model is a flexible and applicable tool that can serve many aspects of the cash flow management task. The summative and easily understandable form in presenting risk creates a good foundation for understanding and communicating risk. Furthermore, it can serve to help make decision on where, when and how to mitigate in various ways, e.g. by indicating target levels and critical limits for riskiness, or by serving as a what-if-analysis tool. Based on the contemporary problem of an increased oil-gas spread that dilutes the value of utility activities, we illustrated how the CFaR model can be used in the renegotiation process in a what-if-analysis by assessing the expected outcome of a specific negotiation strategy. In conclusion, we find that an energy company can measure cash flow risk originating from energy price fluctuations by constructing a CFaR model. The model should incorporate the energy prices to which the cash flow of the investigated portfolio is exposed and measure risk over a horizon that allows for orderly mitigation of potential undesirable risk. Page 90 of 129

93 Energy Price Risk Modelling 10 Conclusion Energy price risk is an inherent part of being an energy company. The main activities of producing, trading, and distributing energy all have operating cash flows that are highly dependent on the price of energy commodities. Energy prices react promptly to imbalances in supply and demand because of the inelastic demand for commodities. Therefore, energy prices are characterized by constant fluctuations and occasionally extreme price jumps. This makes management of cash flows with respect to energy price risk an essential, although very comprehensive, task. The purpose of this thesis was to create a solid foundation for cash flow management that encompasses the very crucial element of measuring cash flow risk. The ability to measure cash flow risk from energy price fluctuations is determined by the ability to forecast energy prices. Therefore the aim of the thesis manifested itself in the following two research questions: a) How can an energy company forecast energy prices with respect to managing cash flows? b) How can an energy company measure cash flow risk originating from energy price fluctuations? From a contemporary study of the underlying energy price drivers and a graphical inspection of the empirical prices of Brent crude oil, NBP natural gas, and API2 coal index, we found that the energy prices share common drivers and seem to co-move over time. This gave us reason to hypothesize that the three energy prices are cointegrated. From an empirical econometric study including unit root tests and cointegration analysis, we found that a hypothesis of one common cointegration relationship between the three energy commodities can be accepted, and that oil and coal prices are independent leaders of the gas prices. The implication of this Page 91 of 129

94 10 Conclusion relationship is that oil and coal prices will wander independently, while the gas price will adjust to deviations from the cointegration relationships and in the long run place itself between the oil and coal price. We employed a VECM specification to formalise the short and long-term dynamics of the three energy prices, and additional tests demonstrated that the estimated model is applicable for the purpose of forecasting energy prices as input to cash flow management. From theory and a practical example of a utility unit, we found that the CFaR model can be applied for measuring cash flow risk from energy price fluctuations in the energy company. By performing the steps of constructing a CFaR model we illustrated how the company can apply the tool and measure risk. Moreover, when interpreting the results we discussed how risk mitigation decisions may be derived from the elements of the CFaR model. Furthermore, a what-if analysis served to illustrate how the CFaR model is applicable in other aspects of cash flow management tasks. The solutions we have derived to the problem statement compose a foundation for cash flow risk measurement and management. Arriving at a complete cash flow risk management solution can be a never ending process of improving and extending the tools. Directions for further improvements regarding the price model could entail econometric modelling of residuals or an inclusion of forward prices, spread instrument prices, implied volatilities, or other variables in the VECM model. Relevant improvements of the CFaR model could entail modelling of foreign exchange rates and including these as risk factors, eliminating the simplifying assumption of the content of the utility portfolio, or modelling with regards to nonconstant supply and demand of the energy commodities. The need for a more advanced model should be a trade-off between cost of extensions and savings from the benefits of an improved model that may lead to more efficient business operations. However, there are pitfalls of having very advanced models, as they can be difficult to understand while creating a false sense of security. Therefore, we close this thesis with a quote of Paul Carrett who is often cited when dealing with risk models: Better to have a simple model backed by excellent people than the other way around. Page 92 of 129

95 Energy Price Risk Modelling 11 References Amadeo, K. (2012). Crude oil price definition. Retrieved 04/24, 2013, from American Petroleum Institute. (2013). Facts about fossil fuels. Retrieved 05/01, 2013, from Andrén, N., Jankensgård, H., & Oxelheim, L. (2005). Exposure-based cash-flow-at-risk: An alternative to VaR for industrial companies. Journal of Applied Corporate Finance, 17(3), Asche, F., Osmundsen, P., & Sandsmark, M. (2006). The UK market for natural gas, oil and electricity: Are the prices decoupled? The Energy Journal - LA English, 27(2), Asche, F., Osmundsen, P., & Tveterås, R. (2002). European market integration for gas? volume flexibility and political risk. Energy Economics, 24(3), Bachmeier, L. J., & Griffin, J. M. (2010). Testing for market integration: Crude oil, coal, and natural gas. The Energy Journal, 27(2), Barcella, M. L. (1999). The pricing of gas - mary lashley barcella discusses the US scene. Oxford Energy Forum, May Blanco, C., & Pierce, M. (2012). Analysing common processes used to model energy prices. Energy Risk, (06) Brown, S. P., & Yucel, M. K. (2008). What drives natural gas prices? The Energy Journal, 29(2), Clewlow, L., & Strickland, C. (2000). Understanding and analyzing spot prices. Energy derivatives: Pricing and risk management (2nd ed., ). London: Lacima Corp. Page 93 of 129

96 11 References Dragomanovich, V. (2012). Growing pressure to de-link european oil and gas prices. Retrieved 05/20, 2013, from Enders, W. (2010). Applied econometric time series (3rd ed.). Hoboken, N.J.: John Wiley & Sons, Inc. Engle, R. F., & Granger, C. W. J. (1987). Co-integration and error correction: Representation, estimation, and testing. Econometrica, 55(2), European Monitoring Centre of Change. (2008). Trends and drivers of change in the european energy sector. (Mapping Report). Dublin: European Foundation for the Improvement of Living and Working Conditions. Flood, C. (2008). Coal prices surge as investor interest grows. Retrieved 4/17, 2013, from Ghouri, S. S. (2006). Forecasting natural gas prices using cointegration technique. OPEC Review, 30(4), Gibson, R., & Schwartz, E. S. (1990). Stochastic convenience yield and the pricing of oil contingent claims. The Journal of Finance, 45(3), Hartley, P. R., Medlock III, K. B., & Rosthal, J. E. (2008). The relationship of natural gas to oil prices. The Energy Journal - LA English, 29(3), Jorion, P. (2010). Risk management. Annual Review of Financial Economics, 2(1), Khan, M. S. (2009). The 2008 oil price "bubble". (Policy Brief No. PB09-19). Washington DC: Peterson Institute for International Economics. Lee, A. Y. (1999). CorporateMetrics: The benchmark for corporate risk management. (Tech. Rep). United States: RiskMetrics Group. Molloy, N. (2011). Are the days of oil/gas indexation numbered? Retrieved 05/08, 2013, from Panagiotidis, T., & Rutledge, E. (2007). Oil and gas markets in the UK: Evidence from a cointegrating approach. Energy Economics, 29(2), Pilipovic, D. (2007). Spot price behavior. In GARP (Ed.), Energy risk: Valuing and managing energy derivatives (2nd ed., ) McGraw-Hill Companies inc. Pindyck, R. S. (1999). The long-run evolution of energy prices. The Energy Journal, 20(2), Page 94 of 129

97 Energy Price Risk Modelling Rotaru, D. V. (2013). A glance at the european energy market liberalization. CES Working Papers, 5(1), Schwartz, E. S. (1997). The stochastic behavior of commodity prices: Implications for valuation and hedging. The Journal of Finance, 52(3), Serletis, A., & Herbert, J. (1999). The message in north american energy prices. Energy Economics, 21(5), Simmons, D., Horlings, H., & Cronshaw, I. (2006). Natural gas market review (Market Review). France: International Energy Agency. Stock, J. H., & Watson, M. W. (1988). Testing for common trends. Journal of the American Statistical Association, 83(404), Stolowy, H., & Lebas, M. J. ( ). Financial accounting and reporting : A global perspective (2. edition, reprint ed.). London: Thomson Learning. Strickland, C. (2012). Enterprise-wide risk management: The power of cashflow-based metrics. Retrieved 05/01, 2013, from Verbeek, M. (2004). A guide to modern econometrics (2. ed. ed.). Chichester: John Wiley. Villar, J. A., & Joutz, F. L. (2006). The relationship between crude oil and natural gas prices. Energy Information Administration, Office of Oil and Gas, Wooldridge, J. M. (2008). Introductory econometrics: A modern approach (4. ed ed.). Cincinnati, Ohio: South-Western. Yucel, M. K., & Guo, S. (1994). Fuel taxes and cointegration of energy prices. Contemporary Economic Policy, 12(3), Page 95 of 129

98 12 Appendix A: Initial treatment of data 12 Appendix A: Initial treatment of data 12.1 Empirical returns Empirical returns before and after outlier treatments Figure Oil price returns Figure Gas price returns Figure Outlier treated gas price returns Page 96 of 129

99 Energy Price Risk Modelling Figure Coal price returns Figure Outlier treated coal price returns Test for seasonality in gas prices Note that the output below only presents selected parts of the full output generated by the X12-ARIMA routine in EViews. Full output is available on the attached CD-ROM. Output Part of test results for seasonality in gas U. S. Department of Commerce, U. S. Census Bureau X-12 monthly seasonal adjustment Method, Release Version This method modifies the X-11 variant of Census Method II by J. Shiskin A.H. Young and J.C. Musgrave of February, and the X-11-ARIMA program based on the methodological research developed by Estela Bee Dagum, Chief of the Seasonal Adjustment and Time Series Staff of Statistics Canada, September, D 8.A F-tests for seasonality Test for the presence of seasonality assuming stability. Sum of Dgrs.of Mean Squares Freedom Square F-Value Between months ** Residual Total **Seasonality present at the 0.1 per cent level. Nonparametric Test for the Presence of Seasonality Assuming Stability Kruskal-Wallis Degrees of Probability Page 97 of 129

100 12 Appendix A: Initial treatment of data Statistic Freedom Level % Seasonality present at the one percent level. Moving Seasonality Test Sum of Dgrs.of Mean Squares Freedom Square F-value Between Years ** Error **Moving seasonality present at the one percent level. COMBINED TEST FOR THE PRESENCE OF IDENTIFIABLE SEASONALITY IDENTIFIABLE SEASONALITY PRESENT D 10 Final seasonal factors From 1996.Sep to 2012.Dec Observations 196 Seasonal filter 3 x 5 moving average Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec AVGE Page 98 of 129

101 Energy Price Risk Modelling AVGE Table Total Mean Std. Dev Min Max D 10.A Final seasonal component forecasts From 2013.Jan to 2013.Dec Observations Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec AVGE Page 99 of 129

102 13 Appendix B: Empirical cointegration analysis 13 Appendix B: Empirical cointegration analysis 13.1 Empirical analysis of stationarity Inspection of autocorrelation functions Output Autocorrelation functions for oil, gas and coal in levels Page 100 of 129

103 Energy Price Risk Modelling Output Autocorrelation functions for oil, gas, and coal in first difference 13.2 Johansen s test Lag length Output Joint selection of lag length in VAR The different criteria point towards two lags. Page 101 of 129

104 13 Appendix B: Empirical cointegration analysis Output Autocorrelation functions for oil Output Test for autocorrelation on oil AR(1) The graph indicates that one lag is sufficient to remove the autocorrelation, and the LM test does not reject the null of no serial correlation (both the F-statistic and Chi-squared value do not reject the null-hypothesis). Hence we chose one lag for oil. Output Autocorrelation functions for gas Output Test for autocorrelation on gas AR(1) We also choose one lag for gas in that both p-values based on the F-test and Chi-square are above 5%. Page 102 of 129

105 Energy Price Risk Modelling Output Autocorrelation functions for coal Output Test for autocorrelation on coal AR(1) After one lag there is still autocorrelation in the residuals because the null of zero autocorrelation is rejected. Output Test for autocorrelation on coal AR(2) It is unclear from the graph whether autocorrelation disappears after lag 1-3 but the LM-test does not reject the null hypothesis after introducing two lags. Therefore we need two lags to remove the serial correlation in the residuals. Page 103 of 129

106 13 Appendix B: Empirical cointegration analysis Rank test Output Johansen s test for rank Date: 07/31/13 Time: 09:20 Sample (adjusted): 1996M M12 Included observations: 194 after adjustments Trend assumption: Linear deterministic trend Series: LN_COAL LN_GAS LN_OIL Lags interval (in first differences): 1 to 1 Unrestricted Cointegration Rank Test (Trace) Hypothesized Trace 0.05 No. of CE(s) Eigenvalue Statistic Critical Value Prob.** None * At most At most Trace test indicates 1 cointegrating eqn(s) at the 0.05 level * denotes rejection of the hypothesis at the 0.05 level **MacKinnon-Haug-Michelis (1999) p-values Unrestricted Cointegration Rank Test (Maximum Eigenvalue) Hypothesized Max-Eigen 0.05 No. of CE(s) Eigenvalue Statistic Critical Value Prob.** None * At most At most Max-eigenvalue test indicates 1 cointegrating eqn(s) at the 0.05 level * denotes rejection of the hypothesis at the 0.05 level **MacKinnon-Haug-Michelis (1999) p-values Unrestricted Cointegrating Coefficients (normalized by b'*s11*b=i): LN_COAL LN_GAS LN_OIL Unrestricted Adjustment Coefficients (alpha): D(LN_COAL) D(LN_GAS) D(LN_OIL) Cointegrating Equation(s): Log likelihood Normalized cointegrating coefficients (standard error in parentheses) LN_COAL LN_GAS LN_OIL ( ) ( ) Page 104 of 129

107 Energy Price Risk Modelling Adjustment coefficients (standard error in parentheses) D(LN_COAL) ( ) D(LN_GAS) ( ) D(LN_OIL) ( ) 2 Cointegrating Equation(s): Log likelihood Normalized cointegrating coefficients (standard error in parentheses) LN_COAL LN_GAS LN_OIL ( ) ( ) Adjustment coefficients (standard error in parentheses) D(LN_COAL) ( ) ( ) D(LN_GAS) ( ) ( ) D(LN_OIL) ( ) ( ) From the output the test point towards one cointegration relationship in that the Trace test cannot reject that there is at most one cointegration relationship and the Maximum eigenvalue test-statistics cannot reject a rank equal to one Hypotheses testing Output and Vector Error Correction Estimates Date: 07/30/13 Time: 17:41 Sample (adjusted): 1996M M12 Included observations: 194 after adjustments Standard errors in ( ) & t-statistics in [ ] Cointegration Restrictions: A(1,1)=0 B(1,1)=1 Convergence achieved after 5 iterations. Restrictions identify all cointegrating vectors LR test for binding restrictions (rank = 1): Chi-square(1) Probability Cointegrating Eq: CointEq1 LN_OIL(-1) LN_GAS(-1) ( ) [ ] Output , and Vector Error Correction Estimates Date: 07/30/13 Time: 17:43 Sample (adjusted): 1996M M12 Included observations: 194 after adjustments Standard errors in ( ) & t-statistics in [ ] Cointegration Restrictions: A(1,1)=0 A(3,1)=0 B(1,2)=1 Convergence achieved after 4 iterations. Restrictions identify all cointegrating vectors LR test for binding restrictions (rank = 1): Chi-square(2) Probability Cointegrating Eq: CointEq1 LN_OIL(-1) ( ) [ ] Page 105 of 129

108 13 Appendix B: Empirical cointegration analysis LN_COAL(-1) ( ) [ ] C Error Correction: D(LN_OIL) D(LN_GAS) D(LN_COAL) CointEq ( ) ( ) ( ) [ NA] [ ] [ ] D(LN_OIL(-1)) ( ) ( ) ( ) [ ] [ ] [ ] D(LN_GAS(-1)) ( ) ( ) ( ) [ ] [ ] [ ] D(LN_COAL(-1)) ( ) ( ) ( ) [ ] [ ] [ ] C ( ) ( ) ( ) [ ] [ ] [ ] R-squared Adj. R-squared Sum sq. resids S.E. equation F-statistic Log likelihood Akaike AIC Schwarz SC Mean dependent S.D. dependent Determinant resid covariance (dof adj.) 4.27E-07 Determinant resid covariance 3.95E-07 Log likelihood Akaike information criterion Schwarz criterion LN_GAS(-1) LN_COAL(-1) ( ) [ ] C Error Correction: D(LN_OIL) D(LN_GAS) D(LN_COAL) CointEq ( ) ( ) ( ) [ NA] [ ] [ NA] D(LN_OIL(-1)) ( ) ( ) ( ) [ ] [ ] [ ] D(LN_GAS(-1)) ( ) ( ) ( ) [ ] [ ] [ ] D(LN_COAL(-1)) ( ) ( ) ( ) [ ] [ ] [ ] C ( ) ( ) ( ) [ ] [ ] [ ] R-squared Adj. R-squared Sum sq. resids S.E. equation F-statistic Log likelihood Akaike AIC Schwarz SC Mean dependent S.D. dependent Determinant resid covariance (dof adj.) 4.27E-07 Determinant resid covariance 3.95E-07 Log likelihood Akaike information criterion Schwarz criterion When testing H1: With a p-value of 96.2% we cannot reject the null. When testing H2: With a p-value of 72.2% we cannot reject the null. Page 106 of 129

109 Energy Price Risk Modelling 13.3 Sequential elimination of regressors Oil The full equation for oil after imposing the H2 restrictions: D(LN_OIL) = C(2)*D(LN_OIL(-1)) + C(3)*D(LN_GAS(-1)) + C(4)*D(LN_COAL(-1)) + C(5) Output Test for heteroscedasticity in full oil equation Heteroskedasticity Test: Breusch-Pagan-Godfrey F-statistic Prob. F(6,187) Obs*R-squared Prob. Chi-Square(6) Scaled explained SS Prob. Chi-Square(6) Test Equation: Dependent Variable: RESID^2 Method: Least Squares Date: 07/30/13 Time: 18:16 Sample: 1996M M12 Included observations: 194 Variable Coefficient Std. Error t-statistic Prob. C LN_OIL(-1) LN_OIL(-2) LN_GAS(-1) LN_GAS(-2) LN_COAL(-1) LN_COAL(-2) R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood Hannan-Quinn criter F-statistic Durbin-Watson stat Prob(F-statistic) The null of homoscedasticity is rejected, thus we have heteroscedasticity in the residuals that we need to adjust for by using White standard errors in the t-tests. Output Test for autocorrelation in full oil equation Breusch-Godfrey Serial Correlation LM Test: F-statistic Prob. F(2,188) Obs*R-squared Prob. Chi-Square(2) Page 107 of 129

110 13 Appendix B: Empirical cointegration analysis Test Equation: Dependent Variable: RESID Method: Least Squares Date: 07/30/13 Time: 18:23 Sample: 1996M M12 Included observations: 194 Presample missing value lagged residuals set to zero. Variable Coefficient Std. Error t-statistic Prob. C(2) C(3) C(4) C(5) RESID(-1) RESID(-2) R-squared Mean dependent var 4.69E-18 Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood Hannan-Quinn criter F-statistic Durbin-Watson stat Prob(F-statistic) The null of no autocorrelation cannot be rejected. Output Full oil equation using White s standard errors Dependent Variable: D(LN_OIL) Method: Least Squares Date: 07/30/13 Time: 18:24 Sample (adjusted): 1996M M12 Included observations: 194 after adjustments White heteroskedasticity-consistent standard errors & covariance D(LN_OIL) = C(2)*D(LN_OIL(-1)) + C(3)*D(LN_GAS(-1)) + C(4) *D(LN_COAL(-1)) + C(5) Coefficient Std. Error t-statistic Prob. C(2) C(3) C(4) C(5) R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood Hannan-Quinn criter F-statistic Durbin-Watson stat Prob(F-statistic) Page 108 of 129

111 Energy Price Risk Modelling Output C4 is eliminated Dependent Variable: D(LN_OIL) Method: Least Squares Date: 07/30/13 Time: 18:25 Sample (adjusted): 1996M M12 Included observations: 194 after adjustments White heteroskedasticity-consistent standard errors & covariance D(LN_OIL) = C(2)*D(LN_OIL(-1)) + C(3)*D(LN_GAS(-1)) + C(5) Coefficient Std. Error t-statistic Prob. C(2) C(3) C(5) R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood Hannan-Quinn criter F-statistic Durbin-Watson stat Prob(F-statistic) Output C2 is eliminated Dependent Variable: D(LN_OIL) Method: Least Squares Date: 07/31/13 Time: 09:36 Sample (adjusted): 1996M M12 Included observations: 194 after adjustments White heteroskedasticity-consistent standard errors & covariance D(LN_OIL) = C(3)*D(LN_GAS(-1))+ C(5) Coefficient Std. Error t-statistic Prob. C(3) C(5) R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood Hannan-Quinn criter F-statistic Durbin-Watson stat Prob(F-statistic) We cannot eliminate any more parameters thus; this is our final equation for oil: D(LN_OIL) = *D(LN_GAS(-1)) Page 109 of 129

112 13 Appendix B: Empirical cointegration analysis Output Test for heteroscedasticity Heteroskedasticity Test: Breusch-Pagan-Godfrey F-statistic Prob. F(2,191) Obs*R-squared Prob. Chi-Square(2) Scaled explained SS Prob. Chi-Square(2) Test Equation: Dependent Variable: RESID^2 Method: Least Squares Date: 07/31/13 Time: 09:37 Sample: 1996M M12 Included observations: 194 Variable Coefficient Std. Error t-statistic Prob. C LN_GAS(-1) LN_GAS(-2) R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood Hannan-Quinn criter F-statistic Durbin-Watson stat Prob(F-statistic) The null of homoscedasticity cannot be rejected meaning we do not have problems with heteroscedasticity Output Test for autocorrelation Breusch-Godfrey Serial Correlation LM Test: F-statistic Prob. F(2,190) Obs*R-squared Prob. Chi-Square(2) Test Equation: Dependent Variable: RESID Method: Least Squares Date: 07/31/13 Time: 09:38 Sample: 1996M M12 Included observations: 194 Presample missing value lagged residuals set to zero. Variable Coefficient Std. Error t-statistic Prob. C(3) C(5) 8.97E RESID(-1) RESID(-2) R-squared Mean dependent var 3.47E-18 Page 110 of 129

113 Energy Price Risk Modelling Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood Hannan-Quinn criter F-statistic Durbin-Watson stat Prob(F-statistic) The null of no autocorrelation cannot be rejected Gas The full equation for gas after imposing H2 restrictions is: D(LN_GAS) = C(6)*( *LN_OIL(-1) + LN_GAS(-1) *LN_COAL(-1) ) + C(7)*D(LN_OIL(-1)) + C(8)*D(LN_GAS(-1)) + C(9)*D(LN_COAL(-1)) + C(10) Output Test for heteroscedasticity in full gas equation Heteroskedasticity Test: Breusch-Pagan-Godfrey F-statistic Prob. F(6,187) Obs*R-squared Prob. Chi-Square(6) Scaled explained SS Prob. Chi-Square(6) Test Equation: Dependent Variable: RESID^2 Method: Least Squares Date: 07/30/13 Time: 18:37 Sample: 1996M M12 Included observations: 194 Variable Coefficient Std. Error t-statistic Prob. C LN_OIL(-1) LN_GAS(-1) LN_COAL(-1) LN_OIL(-2) LN_GAS(-2) LN_COAL(-2) R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood Hannan-Quinn criter F-statistic Durbin-Watson stat Prob(F-statistic) The null of homoscedasticity cannot be rejected meaning we have no problem with heteroscedasticity. Page 111 of 129

114 13 Appendix B: Empirical cointegration analysis Output Test for autocorrelation in full gas equation Breusch-Godfrey Serial Correlation LM Test: F-statistic Prob. F(2,187) Obs*R-squared Prob. Chi-Square(2) Test Equation: Dependent Variable: RESID Method: Least Squares Date: 07/30/13 Time: 18:38 Sample: 1996M M12 Included observations: 194 Presample missing value lagged residuals set to zero. Variable Coefficient Std. Error t-statistic Prob. C(6) C(7) C(8) C(9) C(10) RESID(-1) RESID(-2) R-squared Mean dependent var -3.28E-18 Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood Hannan-Quinn criter F-statistic Durbin-Watson stat Prob(F-statistic) The null of no autocorrelation cannot be rejected. Output Full equation for gas Dependent Variable: D(LN_GAS) Method: Least Squares Date: 07/30/13 Time: 18:37 Sample (adjusted): 1996M M12 Included observations: 194 after adjustments D(LN_GAS) = C(6)*( *LN_OIL(-1) + LN_GAS(-1) *LN_COAL(-1) ) + C(7)*D(LN_OIL( -1)) + C(8)*D(LN_GAS(-1)) + C(9)*D(LN_COAL(-1)) + C(10) Coefficient Std. Error t-statistic Prob. C(6) C(7) C(8) C(9) C(10) R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Page 112 of 129

115 Energy Price Risk Modelling Sum squared resid Schwarz criterion Log likelihood Hannan-Quinn criter F-statistic Durbin-Watson stat Prob(F-statistic) Output C7 is eliminated Dependent Variable: D(LN_GAS) Method: Least Squares Date: 07/31/13 Time: 09:44 Sample (adjusted): 1996M M12 Included observations: 194 after adjustments D(LN_GAS) = C(6)*( *LN_OIL(-1) + LN_GAS(-1) *LN_COAL(-1) ) + C(8) *D(LN_GAS(-1)) + C(9)*D(LN_COAL(-1)) + C(10) Coefficient Std. Error t-statistic Prob. C(6) C(8) C(9) C(10) R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood Hannan-Quinn criter F-statistic Durbin-Watson stat Prob(F-statistic) Output C9 is eliminated Dependent Variable: D(LN_GAS) Method: Least Squares Date: 07/31/13 Time: 09:46 Sample (adjusted): 1996M M12 Included observations: 194 after adjustments D(LN_GAS) = C(6)*( *LN_OIL(-1) + LN_GAS(-1) *LN_COAL(-1) ) + C(8) *D(LN_GAS(-1)) + C(10) Coefficient Std. Error t-statistic Prob. C(6) C(8) C(10) R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood Hannan-Quinn criter F-statistic Durbin-Watson stat Prob(F-statistic) Page 113 of 129

116 13 Appendix B: Empirical cointegration analysis Output C8 is eliminated Dependent Variable: D(LN_GAS) Method: Least Squares Date: 07/31/13 Time: 09:47 Sample (adjusted): 1996M M12 Included observations: 195 after adjustments D(LN_GAS) = C(6)*( *LN_OIL(-1) + LN_GAS(-1) *LN_COAL(-1) ) + C(10) Coefficient Std. Error t-statistic Prob. C(6) C(10) R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood Hannan-Quinn criter F-statistic Durbin-Watson stat Prob(F-statistic) This is the final equation for gas: D(LN_GAS) = *( *LN_OIL(-1) + LN_GAS(-1) *LN_COAL(-1) ) Output Test for heteroscedasticity Heteroskedasticity Test: Breusch-Pagan-Godfrey F-statistic Prob. F(3,191) Obs*R-squared Prob. Chi-Square(3) Scaled explained SS Prob. Chi-Square(3) Test Equation: Dependent Variable: RESID^2 Method: Least Squares Date: 07/31/13 Time: 09:48 Sample: 1996M M12 Included observations: 195 Variable Coefficient Std. Error t-statistic Prob. C LN_OIL(-1) LN_GAS(-1) LN_COAL(-1) R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood Hannan-Quinn criter Page 114 of 129

117 Energy Price Risk Modelling F-statistic Durbin-Watson stat Prob(F-statistic) The null of homoscedasticity cannot be rejected. Output Test for autocorrelation Breusch-Godfrey Serial Correlation LM Test: F-statistic Prob. F(2,191) Obs*R-squared Prob. Chi-Square(2) Test Equation: Dependent Variable: RESID Method: Least Squares Date: 07/31/13 Time: 09:49 Sample: 1996M M12 Included observations: 195 Presample missing value lagged residuals set to zero. Variable Coefficient Std. Error t-statistic Prob. C(6) C(10) 5.23E RESID(-1) RESID(-2) R-squared Mean dependent var -3.06E-18 Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood Hannan-Quinn criter F-statistic Durbin-Watson stat Prob(F-statistic) The null of no autocorrelation cannot be rejected Coal The full equation for coal after imposing H2 restrictions is: D(LN_COAL) =C(12)*D(LN_OIL(-1)) + C(13)*D(LN_GAS(-1)) + C(14)*D(LN_COAL(-1)) + C(15) Output Test for heteroscedasticity in full coal equation Heteroskedasticity Test: Breusch-Pagan-Godfrey F-statistic Prob. F(6,187) Obs*R-squared Prob. Chi-Square(6) Scaled explained SS Prob. Chi-Square(6) Test Equation: Dependent Variable: RESID^2 Page 115 of 129

118 13 Appendix B: Empirical cointegration analysis Method: Least Squares Date: 07/30/13 Time: 18:32 Sample: 1996M M12 Included observations: 194 Variable Coefficient Std. Error t-statistic Prob. C LN_OIL(-1) LN_OIL(-2) LN_GAS(-1) LN_GAS(-2) LN_COAL(-1) LN_COAL(-2) R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood Hannan-Quinn criter F-statistic Durbin-Watson stat Prob(F-statistic) The null of homoscedasticity is rejected, thus we have heteroscedasticity that we need to adjust for by using White standard errors for t-tests. Output Test for autocorrelation in full coal equation Breusch-Godfrey Serial Correlation LM Test: F-statistic Prob. F(2,188) Obs*R-squared Prob. Chi-Square(2) Test Equation: Dependent Variable: RESID Method: Least Squares Date: 07/30/13 Time: 18:33 Sample: 1996M M12 Included observations: 194 Presample missing value lagged residuals set to zero. Variable Coefficient Std. Error t-statistic Prob. C(12) C(13) C(14) C(15) RESID(-1) RESID(-2) R-squared Mean dependent var -1.14E-18 Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood Hannan-Quinn criter F-statistic Durbin-Watson stat Page 116 of 129

119 Energy Price Risk Modelling Prob(F-statistic) The null of no autocorrelation is rejected. Output Full coal equation using White s standard errors Dependent Variable: D(LN_COAL) Method: Least Squares Date: 07/30/13 Time: 18:33 Sample (adjusted): 1996M M12 Included observations: 194 after adjustments White heteroskedasticity-consistent standard errors & covariance D(LN_COAL) =C(12)*D(LN_OIL(-1)) + C(13)*D(LN_GAS(-1)) + C(14) *D(LN_COAL(-1)) + C(15) Coefficient Std. Error t-statistic Prob. C(12) C(13) C(14) C(15) R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood Hannan-Quinn criter F-statistic Durbin-Watson stat Prob(F-statistic) Output C13 eliminated Dependent Variable: D(LN_COAL) Method: Least Squares Date: 07/31/13 Time: 09:57 Sample (adjusted): 1996M M12 Included observations: 194 after adjustments White heteroskedasticity-consistent standard errors & covariance D(LN_COAL) =C(12)*D(LN_OIL(-1)) + C(14)*D(LN_COAL(-1)) + C(15) Coefficient Std. Error t-statistic Prob. C(12) C(14) C(15) R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood Hannan-Quinn criter F-statistic Durbin-Watson stat Prob(F-statistic) Page 117 of 129

120 13 Appendix B: Empirical cointegration analysis This is the final equation for coal: D(LN_COAL) = *D(LN_OIL(-1)) *D(LN_COAL(-1)) Output Test for heteroscedasticity Heteroskedasticity Test: Breusch-Pagan-Godfrey F-statistic Prob. F(4,189) Obs*R-squared Prob. Chi-Square(4) Scaled explained SS Prob. Chi-Square(4) Test Equation: Dependent Variable: RESID^2 Method: Least Squares Date: 07/31/13 Time: 09:58 Sample: 1996M M12 Included observations: 194 Variable Coefficient Std. Error t-statistic Prob. C LN_OIL(-1) LN_OIL(-2) LN_COAL(-1) LN_COAL(-2) R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood Hannan-Quinn criter F-statistic Durbin-Watson stat Prob(F-statistic) The null of homoscedasticity is rejected meaning we have a problem with heteroscedasticity in the residuals from the final equation for coal. Output Test for autocorrelation Breusch-Godfrey Serial Correlation LM Test: F-statistic Prob. F(2,189) Obs*R-squared Prob. Chi-Square(2) Test Equation: Dependent Variable: RESID Method: Least Squares Date: 07/31/13 Time: 09:58 Sample: 1996M M12 Included observations: 194 Presample missing value lagged residuals set to zero. Variable Coefficient Std. Error t-statistic Prob. C(12) Page 118 of 129

121 Energy Price Risk Modelling C(14) C(15) RESID(-1) RESID(-2) R-squared Mean dependent var -5.72E-19 Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood Hannan-Quinn criter F-statistic Durbin-Watson stat Prob(F-statistic) The null of no autocorrelation cannot be rejected. Page 119 of 129

122 14 Appendix C: Forecasting 14 Appendix C: Forecasting 14.1 Historical residuals Figure : Historical residuals for oil. Plot and histogram Figure : Historical residuals for gas. Plot and histogram Page 120 of 129

123 Energy Price Risk Modelling Figure : Historical residuals for coal. Plot and histogram 14.2 Confidence levels for forecasted ln prices Figure Confidence levels for ln oil Figure Confidence levels for ln gas (without season) Figure Confidence levels for ln coal Page 121 of 129

124 15 Appendix D: Testing the model 15 Appendix D: Testing the model 15.1 Test of assumptions The table below gives an overview of the most important assumptions and requirements we encounter when applying the statistical test and estimation procedures. Subsequent to this table is an investigation of each assumption arranged after the test/procedure it belongs to. Table Overview of assumptions Application Purpose Test/procedure Assumptions/requirements Satisfied? Cause for concern? Order of integration Test for unit root Augmented Dickey Fuller test (ADF) Johansen Initial Maximum Approach estimation Likelihood of VECM estimation (ML) Rank test Trace and Maximum eigenvalue tests Test on Likelihood Ratio restrictions test (LR) Estimate Estimate Ordinary Least final VECM VECM Square parameters estimation (OLS) No autocorrelation in residuals from AR(p) Yes No Normal, independent, and No Yes identically distributed residuals No autocorrelation in residuals Yes No in full VECM Hierarchical nested models Yes No Normal, independent, and No identically distributed residuals Linearity in parameters from Yes Yes restricted VECM Zero conditional mean in No residuals from restricted VECM No perfect collinearity Yes Page 122 of 129

125 Energy Price Risk Modelling between variables in restricted VECM Eliminate T-test Homoscedastic residuals in No Yes insignificant each VECM parameters No autocorrelation in each Yes VECM Normal, independent, and No identically distributed residuals in each VECM ADF assumption No autocorrelation in residuals from AR(p) The unit root test procedure adds lags in order to remove autocorrelation. Therefore the procedure automatically makes sure the assumption is fulfilled. The number of lags included can be seen in Table ML assumption The crucial assumption for ML estimation is that the distributional assumptions for the residuals are correct. A common distributional assumption, and the only one we apply here, is that residuals are. If this assumption is correct the ML estimates are consistent and asymptotically normal (Verbeek, 2004). Normal, independent, and identically distributed residuals Below we show normality tests on the full and unrestricted price equations. Output Normality test on residuals from full oil equation Page 123 of 129

126 15 Appendix D: Testing the model Output Normality test on residuals from full gas equation Output Normality test on residuals from full coal equation We find that oil and coal, with p-values of 0.81% and 0.00% from the Jarque-Bera test, are rejected as having normally distributed residuals. Gas can with a p-value of 9.52% not be rejected as having normally distributed residuals. Testing if the residuals are independently distributed is left out because removal of the serial correlation by adding two lags is assumed to be sufficient to remove the dependency between prices in different time periods Trace and Max-test assumption The trace and maximum eigenvalue tests for rank are actually likelihood ratio tests with a test statistic that follows the Dickey-Fuller distribution instead of the usual chi-square distribution. Therefore, the crucial assumption is similar to the one in the ADF test; that there is no autocorrelation in the model s residuals in order to have valid test results (Verbeek, 2004). This time the model is our full VECM. No autocorrelation in residuals in full VECM Below are Breusch-Godfrey test for autocorrelation performed on residuals from the oil, gas, and coal equations. Page 124 of 129

127 Energy Price Risk Modelling Output Test for autocorrelation in full equation for oil Output Test for autocorrelation in full equation for gas Output Test for autocorrelation in full equation for coal The test cannot reject zero autocorrelation in any of the residuals series. The assumption is thereby satisfied LR test assumptions Hierarchical nested models The LR test is only valid for comparison of alternative models if used to compare hierarchically nested models, which is an assumption our two versions of the same VECM model fulfils. Page 125 of 129

128 15 Appendix D: Testing the model Normally distributed residuals in full VECM The relevant test for normality in residuals from the full VECM is performed above as an assumption for ML estimation. The assumption is, as recalled, not satisfied which may cause the LR test to be invalid. However, the LR test has good large sample properties why a sample size of 194 should be sufficient for us to not be concerned about the validity of the test results OLS assumptions The following three assumptions need to be fulfilled in order for the OLS estimates to be unbiased (Wooldridge, 2008). Linearity in parameters from restricted VECM We plot the residuals on the predicted values for the final equations of all three prices. Output Fitted values and residuals of oil Output Fitted values and residuals of gas Output Fitted values and residuals of coal Page 126 of 129