copula-models in the electric power industry

University of St.Gallen – Graduate School of Business Administration, Economics, Law and Social Sciences (HSG)

Copula-Models in the

Electric Power Industry

Master‟s Thesis

Pascal Fischbach

Rislenstrasse 11

8590 Romanshorn

Tel.: 071/ 463 42 06

[email protected]

Supervisor: Prof. Dr. Karl Frauendorfer

Submitted on: 16 August 2010

Abstract

I

ABSTRACT

As a consequence of the ongoing deregulation process of electricity markets, prices for elec-

tricity are now determined by the forces of supply and demand. The non-storability feature of

electricity thereby has a crucial impact on the specific characteristics of spot power prices

that differ from most other assets. In order to cope with the correspondingly high degree of

uncertainty, providers of trading platforms, such as for instance the EEX, have introduced a

variety of spot and derivatives products. Clearly, the companies within the electric power

industry are interested in the dependence structure of these contracts. The present thesis

therefore applies bivariate Gaussian, t-, Gumbel, Clayton and Frank copula models to vari-

ous return series of spot and futures contracts traded at the EEX. The majority of the results

thereby suggest that the t-copula provides an adequate representation of the empirical de-

pendence structures.

Table of Contents

II

TABLE OF CONTENTS

List of Figures .................................................................................................................... IV

List of Tables ....................................................................................................................... V

List of Abbreviations and Symbols ................................................................................... VI

1 Introduction .................................................................................................................. 1

1.1 Presentation of the Problem and its Relevance ................................................... 1

1.2 Goal Setting and Delimitation .............................................................................. 1

1.3 Methodology and Structure ................................................................................. 2

2 The Electric Power Industry ........................................................................................ 4

2.1 Trading in Power at the European Energy Exchange (EEX) ............................... 4

2.1.1 The European Energy Market ...................................................................... 4

2.1.2 The Holding Structure of the EEX and its sub-markets ................................. 5

2.1.3 Power Products and Trading Processes at the EEX ..................................... 7

2.1.3.1 Trading Spot Power Contracts at the EEX ................................................... 7

2.1.3.2 Trading Power Derivatives Contracts at the EEX ......................................... 9

2.1.3.3 Trading Volumes and Over the Counter Trading ........................................ 12

2.2 Characteristics of Electricity Prices ................................................................... 13

2.2.1 Spot Price Characteristics .......................................................................... 15

2.2.1.1 Seasonalities .............................................................................................. 15

2.2.1.2 Mean Reversion ......................................................................................... 16

2.2.1.3 Jumps, Spikes and exceptionally high Volatility .......................................... 17

2.2.2 Price Characteristics of Futures Contracts ................................................. 17

3 Dependence: Linear Correlation, Copulas and Measures of Association ...............19

3.1 Theoretical Background of Copulas and Dependence ...................................... 19

3.1.1 Pearson‟s Linear Correlation ...................................................................... 19

3.1.2 Copulas ...................................................................................................... 22

3.1.2.1 Preliminaries .............................................................................................. 22

3.1.2.2 Definition of Copula, Sklar‟s Theorem and Basic Properties ....................... 24

3.1.2.3 Fundamental Copulas ................................................................................ 28

3.1.2.4 Elliptical and Archimedean Copulas ........................................................... 28

3.1.3 Measures of Association ............................................................................ 34

3.1.3.1 Kendall‟s Tau ............................................................................................. 36

3.1.3.2 Spearman‟s Rho ........................................................................................ 36

3.1.3.3 Tail Dependence ........................................................................................ 39

3.2 Fitting Copulas to Data ...................................................................................... 40

3.2.1 Estimation of Copulas ................................................................................ 40

Table of Contents

III

3.2.1.1 Full Maximum Likelihood Approach (FML) ................................................. 41

3.2.1.2 Inference Method for Margins (IFM) ........................................................... 41

3.2.1.3 Canonical Maximum Likelihood Approach (CML) ....................................... 42

3.2.1.4 Calibration with Kendall‟s tau and Spearman‟s rho ..................................... 43

3.2.1.5 Nonparametric Method ............................................................................... 43

3.2.2 Goodness of Fit Tests for Copulas ............................................................. 44

4 Applying Copulas to the Electric Power Industry .....................................................47

4.1 General Remarks about the Estimation Procedure ........................................... 48

4.2 Phelix and Swissix Spot Analysis ...................................................................... 50

4.2.1 Data Set and Descriptive Statistics ............................................................ 50

4.2.2 Estimation of the Marginals ........................................................................ 57

4.2.3 Phelix vs. Swissix ....................................................................................... 58

4.2.4 Base vs. Peak ............................................................................................ 61

4.3 Phelix Year Futures Analysis ............................................................................ 62



4.3.3 Base vs. Peak ............................................................................................ 66

4.3.4 2010 vs. 2011 ............................................................................................ 67

4.4 Further Analysis involving various Phelix Futures Contracts ............................. 69



4.4.3 Different Time to Delivery ........................................................................... 74

4.4.4 Different Delivery Period ............................................................................ 75

4.4.5 Spot vs. Futures ......................................................................................... 76

4.5 Summary of the Results across all Parts of the Analysis ................................... 77

5 Conclusion ...................................................................................................................79

Appendix A: Additional Figures ........................................................................................81

Appendix B: Additional Tables ........................................................................................ 107

Appendix C: Codes for Matlab and R .............................................................................. 113

Bibliography ..................................................................................................................... 117

Declaration of Authorship................................................................................................ 122

List of Figures

IV

LIST OF FIGURES

Figure 1 Central West European electricity market as of 2010 ........................................ 5

Figure 2 Submarkets of the EEX ..................................................................................... 6

Figure 3 Fréchet-Hoeffding bounds for C(u,v) ................................................................27

Figure 4 Gaussian and t-copula densities ......................................................................31

Figure 5 Gumbel, Clayton and Frank copula densities ...................................................33

Figure 6 Relation between Kendall‟s tau and Spearman‟s rho .......................................38

Figure 7 Seasonalities during the day, week and year ...................................................52

Figure 8 Autocorrelation function for daily log returns of spot products ..........................53

Figure A1 Historical movement of the prices and log returns of the various time series of

the Phelix and Swissix spot data set. ...............................................................82

Figure A2 Histogram of the various return series of the Phelix and Swissix spot data set 83

Figure A3 Phelix Day Base vs. Swissix Day Base point clouds ........................................84

Figure A4 Phelix Day Peak vs. Swissix Day Peak point clouds ........................................85

Figure A5 Phelix Hourly vs. Swissix Hourly point clouds ..................................................86

Figure A6 Phelix Day Base vs. Phelix Day Peak point clouds ..........................................87

Figure A7 Swissix Day Base vs. Swissix Day Peak point clouds ......................................88

Figure A8 Historical movement of the prices and log returns of the various time series of

the Phelix Year Futures data set. .....................................................................89

Figure A9 Histogram of the various Phelix Year Futures return series ..............................90

Figure A10 Phelix Jan 2010 Base vs. Phelix Jan 2010 Peak point clouds ..........................91

Figure A11 Phelix Jan 2011 Base vs. Phelix Jan 2011 Peak point clouds ..........................92

Figure A12 Phelix Jan 2010 Base vs. Phelix Jan 2011 Base point clouds ..........................93

Figure A13 Phelix Jan 2010 Peak vs. Phelix Jan 2011 Peak point clouds ..........................94

Figure A14 Historical movement of the synthetic return series ...........................................95

Figure A15 Histogram of the various synthetic return series ...............................................97

Figure A16 1 Month ahead vs. 2 Months ahead point clouds .............................................98

Figure A17 1 Quarter ahead vs. 2 Quarters ahead point clouds .........................................99

Figure A18 1 Year ahead vs. 2 Years ahead point clouds ................................................ 100

Figure A19 1 Month ahead vs. 1 Quarter ahead point clouds ........................................... 101

Figure A20 1 Month ahead vs. 1 Year ahead point clouds ............................................... 102

Figure A21 1 Quarter ahead vs. 1 Year ahead point clouds ............................................. 103

Figure A22 Spot vs. 1 Month ahead point clouds ............................................................. 104

Figure A23 Spot vs. 1 Quarter ahead point clouds ........................................................... 105

Figure A24 Spot vs. 1 Year ahead point clouds ................................................................ 106

List of Tables

V

LIST OF TABLES

Table 1 Characteristics of spot and derivatives contracts traded at the EEX .....................11

Table 2 Trading volume in the spot and derivatives power markets of the EEX .................12

Table 3 Copula families, generator functions and permissible parameter ranges ..............29

Table 4 Relation between the copula parameter and 𝜌𝜏 , 𝜌𝑠, 𝜆𝑢 and 𝜆𝑙 ..............................38

Table 5 Descriptive statistics of the Phelix and Swissix spot data set ................................55

Table 6 Marginal parameter estimates for the Phelix and Swissix spot data set ................58

Table 7 Copula parameter estimates (Phelix vs. Swissix) for the Phelix and Swissix spot

return series based on non-central t-distributed marginals ...................................59

Table 8 Copula parameter estimates (Base vs. Peak) for the Phelix and Swissix spot return

series based on non-central t-distributed marginals ..............................................62

Table 9 Descriptive statistics of the Phelix Year Futures data set ......................................64

Table 10 Marginal parameter estimates for the Phelix Year Futures data set ......................66

Table 11 Copula parameter estimates (Base vs. Peak) for the Phelix Year Futures return


Table 12 Copula parameter estimates (2010 vs. 2011) for the Phelix Year Futures return


Table 13 Descriptive statistics of the synthetic time series ..................................................71

Table 14 Marginal parameter estimates for the synthetic return series ................................72

Table 15 Correlation matrices for the synthetic return series ...............................................73

Table 16 Copula parameter estimates (different time to delivery) for the synthetic return


Table 17 Copula parameter estimates (different delivery period) for the synthetic return


Table 18 Copula parameter estimates (Spot vs. Futures) for the synthetic return series

based on non-central t-distributed marginals ........................................................76

Table B3 Copula parameter estimates (Base vs. Peak) for the Phelix Year Futures return

series based on empirically distributed marginals ............................................... 108

Table B4 Copula parameter estimates (2010 vs. 2011) for the Phelix Year Futures return


Table B5 Copula parameter estimates (different time to delivery) for the synthetic return


Table B6 Copula parameter estimates (different delivery period) for the synthetic return

series based on non-central t-distributed marginals ............................................ 111

Table B7 Copula parameter estimates (Spot vs. Futures) for the synthetic return series

based on empirically distributed marginals ......................................................... 112

List of Abbreviations and Symbols

VI

LIST OF ABBREVIATIONS AND SYMBOLS

10B Phelix Jan 2010 Base time series

10P Phelix Jan 2010 Peak time series

11B Phelix Jan 2011 Base time series

11P Phelix Jan 2011 Peak time series

1M 1 Month ahead time series

1Q 1 Quarter ahead time series

1Y 1 Year ahead time series

2M 2 Months ahead time series

2Q 2 Quarters ahead time series

2Y 2 Years ahead time series

AIC Akaike information criterion

APG Austrian Power Grid (Austrian TSO)

BIC Bayesian information criterion

cdf Cumulative distribution function

CML Canonical maximum likelihood method

𝐶(𝑢1 ,… ,𝑢𝑑) 𝑑-dimensional copula function

𝐶M Comonotonicity copula

𝐶𝑛 Empirical copula

𝐶𝑊 Countermonotonicity copula

𝐶Π Independence copula

𝐶𝜃𝐶𝑙 Clayton copula (with parameter 𝜃)

𝐶𝜃𝐹𝑟 Frank copula (with parameter 𝜃)

𝐶Σ𝐺𝑎 Gaussian copula (with parameter Σ)

𝐶𝜃𝐺𝑢 Gumbel copula (with parameter 𝜃)

𝐶Σ ,ν𝑡 t-copula (with parameters Σ and 𝜈)

𝑐(𝑢1 ,… ,𝑢𝑑) Density of a 𝑑-dimensional copula function

𝐷𝐴𝐷 Anderson-Darling distance measure

𝐷𝐶𝑣𝑀 Cramer-von-Mises distance measure

𝐷𝐼𝐴𝐷 Integrated Anderson-Darling distance measure

𝐷𝐾𝑆 Kolmogorov-Smirnov distance measure

ECC European Commodity Clearing AG

EEX European Energy Exchange AG

ENBW Energie Baden-Württemberg Transportnetze AG (German TSO)

ENDEX European Energy Derivatives Exchange N.V.

EU European Union

List of Abbreviations and Symbols

VII

EUR Euro

FML Full maximum likelihood method

𝐹 𝑥1 ,… , 𝑥𝑛 𝑛-dimensional multivariate (joint) distribution function

𝐹𝑖 𝑥𝑖 𝑖-th marginal distribution function (𝑖-th marginal)

𝐹 𝑗 𝑥𝑗 𝑗-th empirical marginal distribution function

𝐹𝑗nct 𝑥𝑗 ;𝛼 𝑗

nct 𝑗-th non-central t-distributed marginal distribution (with parameter 𝛼 𝑗nct)

𝑓 𝑥1 ,… , 𝑥𝑛 n-dimensional multivariate (joint) density function

IFM Inference for margins method

ln 𝐿 ∙ Maximum log likelihood function

MW Megawatt

MWh Megawatt hour

OTC Over the counter

pdf Probability density function

PhB Phelix Day Base time series

PhH Phelix Hourly time series

PhP Phelix Day Peak time series

RTE Réseau de Transport d‟Electricité (French TSO)

RWE Rheinisch-Westfälisches Elektrizitätswerk AG (German TSO)

𝑟𝑠 𝑋𝑖 ,𝑋𝑗 Estimator of Spearman‟s rank correlation coefficient (Spearman‟s rho)

𝑟𝜏 𝑋𝑖 ,𝑋𝑗 Estimator of Kendall‟s rank correlation coefficient (Kendall‟s tau)

SGD Swissgrid (Swiss TSO)

Sp Spot time series

SwB Swissix Day Base time series

SwH Swissix Hourly time series

SwP Swissix Day Peak time series

TSO Transmission system operator

𝑇← Generalized inverse

TWh Terawatt hour

VE Vattenfall Europe AG (German TSO)

𝜅𝑋1 ,𝑋2 Measure of association

𝜆𝑙 𝑋1,𝑋2 Lower tail dependence coefficient

𝜆𝑢 𝑋1,𝑋2 Upper tail dependence coefficient

𝜌 𝑋1 ,𝑋2 Pearson‟s linear correlation coefficient

𝜌𝑠 𝑋1 ,𝑋2 Spearman‟s rank correlation coefficient (Spearman‟s rho)

𝜌𝜏 𝑋1 ,𝑋2 Kendall‟s rank correlation coefficient (Kendall‟s tau)

𝜓 Generator function of an Archimedean copula

Introduction

1

1 INTRODUCTION

1.1 Presentation of the Problem and its Relevance

With the initiation of the deregulation process of electricity markets in the last decades, com-

panies within the electric power industry are currently not only subject to volume uncertain-

ties in demand but also to a high degree of uncertainty in electricity prices (He, 2007, p.26).

The market participants are hence exposed to an unprecedented amount of financial risk that

requires a considerate risk management (Lemming, 2003, p.13). On one side, power pro-

ducers generate electricity by operating various types of power plants to supply wholesale

markets with electricity. On the other side, power providers engage in wholesale trading ac-

tivities in order to provide their customers with electricity. They sell respectively buy electricity

in accordance with projections about their electricity output respectively need. Generally, the

loads that are rather certain to occur are sold respectively purchased in advance on the

power derivatives market, while the more uncertain components of their portfolios are traded

at short notice on spot markets (Lichtblick AG, 2008, p.26). In this sense, the analysis of the

dependence structure between the various power products constitutes an integral part of risk

management. The application of adequate financial risk management tools is thereby of par-

ticular importance, as electricity prices and their respective returns show some characteris-

tics, which crucially differentiate them from the prices and returns of other assets. In fact, the

usual simplifying assumption that asset returns follow a normal distribution is not sensible

within the context of electricity prices. But this also implies that describing the dependence

structures between various random vectors of electricity price returns via a multivariate nor-

mal distribution and Pearson‟s linear correlation coefficient is not appropriate. Consequently,

more sophisticated concepts to describe these dependence structures are required.

1.2 Goal Setting and Delimitation

Copulas are a tool to capture stochastic dependence of random variables in a far more com-

plex way than with Pearson‟s linear correlation coefficient. It is important to recognize that

the latter only fully characterizes the dependence structure of a set of random variables in

the case that they are elliptically distributed. By contrast, the application of copula models is

sensible under any kind of distribution. Furthermore, copulas are able to model a broad

range of distinct dependence structures, including cases of upper and lower tail dependence,

whereas stochastic modelling on the basis of a multivariate normal distribution would imply

no tail dependence at all. In this sense, copula models allow to account for joint extreme ob-

servations either to the upside, to the downside, or in both directions. Thus, the application of

Introduction

2

copula models clearly provides an opportunity to model a specific dependence structure in a

way that is closer to the empirically observable reality than with conventional methods.

The intention of this thesis is to gain insight into the pair wise dependence structure of the

return series of various spot and futures power products. In particular, with respect to the

spot products, the dependence between Phelix and Swissix contracts and the dependence

between base and peak load contracts shall be analyzed. Moreover, Phelix Year Futures are

investigated in order to conclude on the dependence between base and peak load futures

contracts as well as between futures products with a different time of delivery. Finally, a fur-

ther analysis involving synthetic return series for Phelix futures contracts with various deli-

very periods and times to delivery is employed.

There exists a variety of power exchanges throughout Europe. The focus of the present the-

sis, however, is set on a range of products traded at the European Energy Exchange (EEX).

Furthermore, additional products are available for trading at the EEX that are not part of the

present analysis. This includes, for instance, various intraday products, contracts with deliv-

ery in the French market area, or Phelix options. Delimitations also have to be taken with

respect to the specific copula functions under analysis. Whereas this thesis concentrates on

the most basic and well-known copula families (i.e. Gaussian, t-, Gumbel, Clayton and

Frank), subsequent research could take into account further, more complex copula families.

Finally, while this thesis is self-contained in the sense that it attempts to capture all relevant

theory and background information with regard to copulas and electricity products as well as

prices, some basic concepts within the area of finance and statistics are nevertheless as-

sumed to be prerequisites.

1.3 Methodology and Structure

The subsequent Section 2 will commence this thesis by providing a short overview of the

electric power industry. In particular, Section 2.1 describes the activities of trading electricity

at the EEX. This will include a presentation of the EEX as a trading platform and a descrip-

tion of the various power products traded at the EEX. Section 2.2 continues by elaborating

on the characteristics of electricity prices. Section 3 will then shift the focus towards a more

statistical context by studying concepts of stochastic dependence. Firstly, Section 3.1 will put

forth the basic concepts behind the theory of copulas and various dependence measures.

Thereafter, Section 3.2 enters into the discussion of the problem of fitting copulas to a given

set of data, covering both the topics of estimating the parameters and testing the goodness

of fit of various model specifications. While Section 2 and Section 3 mainly involve a review

of the trading possibilities in power at the EEX, the specific characteristics of electricity prices

and the relevant body of copula literature, they will together provide the required theoretical

Introduction

3

background in order to proceed with the empirical investigation in Section 4. Specifically,

Section 4.2, 4.3 and 4.4 will cover a Phelix and Swissix spot analysis, a Phelix Year Futures

analysis and a further analysis involving various Phelix futures contracts, respectively. Prior

to the presentation of the respective results, the general procedure applied throughout these

sub-analyses is presented in Section 4.1. Within each subsection, first the data set under

consideration is presented and corresponding descriptive statistics are provided. Following

this, the marginal distributions are estimated in order to subsequently present the results of

applying copula models onto the various time series of electricity prices (respectively returns)

in compliance with the theoretical considerations presented in Section 3. Section 5 finally

concludes this paper.

The Electric Power Industry

4

2 THE ELECTRIC POWER INDUSTRY

With the ongoing deregulation in the electric power industry, electricity markets have expe-

rienced substantial changes. Electricity prices are no longer subject to government decisions,

but rather result from the trading activities at power exchanges, driven by the forces of supply

and demand. Due to the non-existence of decent storage capabilities, electricity as a com-

modity strongly differs from other financial assets. On one side, this has an impact on the

processes of trading electricity at the various power exchanges. For example, trading is

possible in hourly contracts as well as block contracts such as peak or base loads. This is

clearly a distinguishing feature of electricity markets compared to financial markets, where

standard spot products are not subject to the specification of a delivery period. On the other

side, electricity prices exhibit certain characteristics, which differentiate them from the prices

of common financial assets. According to Blöchlinger (2008), Borchert et al. (2006), Weber

(2005) and Weron (2005) this includes seasonal patterns, mean reversion, spikes and jumps,

and, as a result, incomparably high volatility. In the following sections both aspects, firstly,

regarding electricity trading and secondly, regarding the characteristics of electricity prices,

are discussed in more detail.

2.1 Trading in Power at the European Energy Exchange (EEX)

2.1.1 The European Energy Market

Since 1998, when the liberalization of European energy markets was initiated, a variety of

power exchanges has been established in Europe. While competition on the level of power

supply is of advantage for end consumers, the establishment of a single European electricity

market on the wholesale level has been a major aim of the liberalization process (EEX AG &

Powernext SA, 2008a, p.2). Meeus and Belmans (2008, pp.5-10) adhere that the price dif-

ferences across European countries are still large and that Europe is still far away from an

integrated Europe-wide market for electricity. Nevertheless, some promising regional mar-

kets have originated that have a potential of being pathbreaking in the process of integration.

In 2004, the European Commission recognized that such regional markets may provide an

inevitable interim stage in achieving the ambitious goal of a single European electricity mar-

ket (EEX AG & Powernext SA, 2008a, p.2). As a consequence thereof, the formation of re-

gional markets was actively promoted by the commission. Since then, major attempts to a

continuing integration of regional electricity markets have been made in recent years. These

include the continuing integration of energy markets in Central Western Europe through the

recent cooperation of the German EEX and the French Powernext. Figure 1 illustrates the

fact that, with the inclusion of Germany, France, Austria and Switzerland, the cooperation


5

covers a geographical area that constitutes more than one third of the European electricity

consumption (EEX AG & Powernext SA, 2008c, p.8). As stated by EU Energy Commissioner

Andris Piebalgs, this “means great progress for the European electricity market” and

represents an “important step on the path towards a fair and uniform price for Europe” (cited

in EEX AG & Powernext SA (2008b, p.1,4). According to EEX AG and Powernext SA (2008c,

pp.3,6,19) the benefits of this cooperation are an increased security of supply, enhanced

competition on the level of supply, price convergence, centralized and increased liquidity,

lower transaction costs through harmonization of trading and settlement, integrated clearing

over several markets, and a facilitated governance of market splitting and coupling projects.

Figure 1 Central West European electricity market as of 2010. The Central West European

electricity market comprises the products traded on the German EEX AG and the French

Powernext SA. In particular, it involves spot and futures markets for Germany, Austria and

France and the spot market for Switzerland. Source: EEX AG & Powernext SA, 2008c, p.18.

Note, however, that it is neither in the scope of this thesis to extensively describe the various

regional power markets within Europe nor to analyze their integration processes. Rather, the

aim of this thesis is to analyze the dependence structure between return series of the prices

of spot power contracts with delivery within Germany/Austria (Phelix) and Switzerland (Swis-

six) and various futures contracts on the Phelix. Since all these products are traded on the

EEX (respectively on various subsidiaries and joint ventures thereof), the following subsec-

tion will focus on examining the EEX and its group structure in a more elaborate way. The

intention is to subsequently provide an overview of the trading possibilities in power at the

EEX. This will include a detailed specification of the products that are of relevance in the em-

pirical part of this thesis.

2.1.2 The Holding Structure of the EEX and its sub-markets

The Leipzig based European Energy Exchange AG is the leading energy exchange in Conti-

nental Europe both in terms of trading participants and turnover. Originating from the 2002


6

merger of the two German power exchanges located in Frankfurt and Leipzig, the EEX has

established itself as a leading operator of market platforms for trading in power, natural gas,

emission rights and coal (EEX AG, 2010, p.1). The particular corporate structure together

with the adoption of an open business model where the spin-offs are able to form partner-

ships and co-operations with other power exchanges across Europe constitutes a major con-

tribution with regard to the integration of European energy markets. In 2006, as a first step in

this process, the EEX outsourced the clearing activities into a subsidiary named European

Commodity Clearing AG (ECC). In the same year, ECC started to cooperate with the Dutch

European Energy Derivatives Exchange N.V. (ENDEX). Today, ECC provides clearing and

settlement services for all products traded on the EEX and its partner exchanges, such as

the Powernext SA or the ENDEX. Further spin-offs took place in 2007 and 2008, resulting in

the transfer of the spot and derivatives trading activities into separate entities. The EEX now

offers a market place on several distinct sub-markets: EPEX Spot Market, EEX Spot Market

and EEX Derivatives Market. Figure 2 provides an overview of these sub-markets, the prod-

ucts traded on each of them and the respective subsidiaries of the EEX.

Figure 2 Submarkets of the EEX. The submarkets of the EEX comprise the EPEX Spot Market, the

EEX Spot Market, the EEX Power Derivatives Market and the EEX Derivatives Market. Source:

http://www.eex.com

The EPEX Spot SE, established in late 2008, is a joint venture of the EEX and the French

Powernext SA, each of them holding a 50% stake in the Paris based company (EEX AG &

Powernext SA, 2008b, p.5). Both companies transferred their entire spot power trading activi-

ties into the newly founded company that is now offering products on a day-ahead basis for

France, Germany/Austria (Phelix) and Switzerland (Swissix) and intraday markets for France

and Germany. In the cooperation with the Powernext SA, the EEX also agreed on the crea-

tion of the EEX Power Derivatives GmbH, which offers trading in German and French power

derivatives. Powernext which contributed with the injection of its trading platform for French


7

power futures holds 20% in this joint venture, while the EEX holds the remaining 80% (EEX

AG & Powernext SA, 2008b, p.5). As illustrated in Figure 2, the product range of the EEX

further includes day ahead spot trading in natural gas and emission rights and derivative con-

tracts in natural gas, emission rights and coal (EEX AG, 2010, p.3).

2.1.3 Power Products and Trading Processes at the EEX

The fact that electricity is a commodity which must be consumed immediately after being

produced together with the fact that electricity is a non-storable good has a decisive influence

on how power products are specified for trading purposes. In particular, an important feature

of power products, compared to most other commodities, is the necessity to determine a

delivery period during which electricity is delivered at a constant rate. In its most basic form,

a contract may involve the constant delivery of a certain amount of power (e.g. 1 MW) over a

certain period of time (e.g. a single hour on a specific day in the future), leading to the total

contract volume measured in energy units (e.g. 1 MWh in our case). Apparently, it is possible

to acquire a certain amount of power during a delivery period which extends one hour by

purchasing a portfolio of these hourly contracts. For instance, suppose that we buy 24 hourly

contracts, one for each hour of the day, to get a constant delivery of power during an entire

day. In general, however, it is not necessary to purchase these hourly contracts individually,

as power exchanges offer the possibility to buy electricity on a block basis. A block order

thereby covers a constant delivery over several consecutive hours where the individual hours

depend on each other with respect to their execution, in the sense that all or none of the

hours are executed (EPEX Spot SE, 2010, p.8). A base load contract, for instance, covers a

constant delivery from hour 1 to hour 24 on any day of the week. The counterpart to the base

load is the peak load that covers 12 hours of constant delivery from 8 a.m. to 8 p.m. (i.e.

hours 9 to 20). Furthermore, for a delivery period of more than one week, a peak load con-

tract typically covers only the days from Monday to Friday. The price of such a contract,

measured in MWh, is straightforward, as it is the arithmetic average over the prices of the

underlying hourly contracts. Thus, as no separate pricing takes place, block contracts can-

not, in a strict sense, be seen as distinct products (EPEX Spot SE, 2009, p.5). But clearly,

any other price would result in arbitrage opportunities. According to the European Federation

of Energy Traders (2008, p.38), base and peak load contracts have become prevalent as

standard contracts in power trading mainly due to the fact that a low number of contracts

allows for high liquidity while still providing an acceptable mapping of the typical demand load

pattern.

2.1.3.1 Trading Spot Power Contracts at the EEX

Spot power trading encompasses the physical delivery of power on a short-term basis. Ac-

cording to Wenzel (2007, p.14), spot trading mainly serves for balancing short-term devia-


8

tions in the purchase and selling portfolios of the market participants. Wenzel (2007, p.14)

further addresses the fact that risk and price expectations have an influence on whether

larger or smaller parts of one‟s strategy are transacted on the spot segment. The EEX offers

contracts on two distinct trading platforms within the spot market segment, namely conti-

nuous trading on the intraday market and closed auction trading on a day-ahead basis1. Both

are offered via the EEX and the Powernext SA‟s joint venture EPEX Spot SE. An in-depth

representation of the concrete trading processes at the EPEX Spot SE can be found in EEX

AG (2008a), EEX AG (2010), EPEX Spot SE (2009) and EPEX Spot SE (2010), on which

also the following remarks are based on.

Through its continuous intraday trading, the EEX offers its market participants a platform to

buy and sell power at very short notice, with delivery taking place on the same (or the next)

trading day. In general, the order book is open within intraday trading, so that price and vo-

lume information is visible to the market participants. Each offer is thus immediately checked

for executability with a matching offer. With regard to the place of delivery, intraday trading is

divided into the market areas France and Germany. It is important to note that the trading

specifications correspondingly vary up to a certain extent. German and French intraday trad-

ing comprises the individual hourly contracts of the current day, which can be traded until 75

respectively 60 minutes before the start of the corresponding delivery period. Trading in the

hourly (and block) contracts of the following day is possible from 3 p.m., respectively 11.30

a.m. onwards. Besides hourly contracts, intraday trading involves base and peak load con-

tracts. French intraday trading further allows for a number of other standardized block con-

tracts and any user-defined block contracts, while German intraday trading only provides the

latter. Trading on the German intraday market takes place continuously 24/7, whereas the

trading hours of intraday trading for France are confined to between 7.30 a.m. and 11 p.m.

Day-ahead auction trading is offered for the market areas France, Germany/Austria (Phelix)

and Switzerland (Swissix) and encompasses the trading of power contracts for individual

hours of the next day as well as various blocks thereof. Phelix hereby stands for Physical

Electricity Index which, according to the EEX AG (2010, p.6), provides market participants

both on and off the exchange with a reference price for power traded on the wholesale spot

market in Continental Europe. The Phelix is calculated daily as the arithmetic average of the

auction prices of the hours 1 to 24 (Phelix Day Base), respectively of the hours 9 to 20 (Phe-

lix Day Peak) for the market area Germany/Austria without taking into account any power

transmission bottlenecks. The Swiss Electricity Index, abbreviated as Swissix, represents the

1 Note that the EPEX Spot SE actually offers a third trading platform for spot power contracts, which consists of continuous

trading of electricity block contracts with delivery on the following day within the French transmission system (cf. EEX AG,

2010, p.9). As it combines day-ahead with continuous trading, it basically represents a combination of the two previously men-

tioned platforms.


9

corresponding index for the Swiss market area (EEX AG, 2009b, p.5). Like the Phelix, it is

calculated and published daily on a base and peak load basis under the notion Swissix Day

Base and Swissix Day Peak, respectively. Both indexes are also calculated on a monthly

basis. As we will see later, the monthly Phelix represents the settlement price of the Phelix

power futures. Similar to intraday trading, market participants can make bids for the hourly

contracts, base and peak load block bids and other standardized and user-specific block

contracts with delivery on the following day. However, contrary to intraday trading, the price

is determined in an auction procedure. Orders for all contracts can be entered into the sys-

tem starting fourteen days before their respective delivery period begins (Phelix and Swis-

six), respectively on the Wednesday preceding the week when delivery takes place (French

day-ahead). According to the EPEX Spot SE (2010, p.6), the order book then remains open

24 hours per day until the pricing process is initiated on the exchange trading day before

delivery, timed at 11 a.m., 12 p.m. (noon) and 10.30 a.m., for the French, German/Austrian

and Swiss market area, respectively. The bids are hereby aggregated to demand and supply

curves, with the intersection providing the market clearing price and volume.

2.1.3.2 Trading Power Derivatives Contracts at the EEX

Due to the incomparably high uncertainty related to spot power prices (cf. Section 2.2.1),

market participants typically try to cover only a minor part of their portfolio with spot products

(Wenzel, 2007, p.14). In particular, those parts of their portfolio which are not subject to un-

certainties in demand can be purchased or sold in advance on the long-term derivatives

market (Lichtblick AG, 2008, p.26). A potential use of futures contracts further lies in the sup-

port of risk management by providing hedging possibilities against future price risk (EEX AG,

2010, p.13). Clearly, a long position in a derivatives contract can serve as a hedge against

increasing prices, while a short position provides an insurance against declining prices. Be-

sides, market participants who primarily act as speculators or arbitrageurs may likewise

heavily rely on power derivatives (EEX AG, 2008b, p.5). Finally, Wenzel (2007, p.14) also

points out that while the spot power market provides market participants with information

about the current price, forwards and futures contracts disclose the market‟s expectations

about future spot price movements.

The EEX Power Derivatives GmbH offers derivatives contracts both with an unconditional

and a conditional nature. Futures (as well as forwards) are unconditional contracts, in the

sense that they comprise the obligation to buy (i.e. long position) or sell (i.e. short position) a

predefined underlying at a certain point of time in the future at a price specified today. Note

that a futures contract usually refers to a standardized contract, usually traded on an ex-

change, while a forward contract represents a non-standardized contract, for instance traded

over the counter. Options on the other side are conditional contracts, as they convey the right


10

but not the obligation to buy (i.e. call option) or sell (i.e. put option) a predefined underlying at

maturity (i.e. European option) or until maturity (i.e. American option) of the contract at a

price specified today. Regarding the subsequent remarks, we shall refer to EEX AG (2008b),

EEX AG (2010) and EEX Power Derivatives GmbH (2009).

The Power futures that are tradable at the EEX Power Derivatives GmbH can be subdivided

into French and German futures as well as Phelix futures contracts. French and German

power futures involve the delivery of electricity on a base or peak load basis into the respec-

tive market area during the entire delivery period in the future, where the delivery period can

be chosen to be a month, a quarter or a year2. Moreover, contracts are traded for the current

and the next six months, the next seven quarters and the next six years. French/German

power futures imply a physical fulfillment as mandatory. Year and quarter futures are traded

until three days before the delivery period begins, where they cascade into the respective

quarter or month futures. Thus, no direct physical delivery takes place at this stage, as the

contracts are replaced by contracts of the next lower delivery period. For the month futures,

however, partial physical deliveries will be taking place on each day during the entire delivery

month. The corresponding final settlement price, which is determined on the last trading day,

stays constant over the whole settlement period and represents the basis for the payments.

The contract volume of the respective month futures contract is thereby reduced with the

execution of every partial delivery. As a result, the month futures contract can still be traded

during the delivery month until the final partial delivery has taken place.

Besides a different underlying, Phelix futures differ from the above mentioned futures con-

tracts mainly with respect to their fulfillment, which is of a financial nature. Furthermore, con-

tracts are traded for the current and the next nine months (Phelix Month Futures), the next

eleven quarters (Phelix Quarter Futures) and the next six years (Phelix Year Futures). Year

and quarter futures cascade in the same way as the French/German power futures and the

settlement price is established in a similar way too, based on the value of the corresponding

monthly Phelix index. During the delivery month, however, the futures contract will be settled

financially by accounting for the daily variation margin. Alternatively, the market participants

can also opt for a physical delivery at the EPEX Spot SE on the day-ahead auction segment

for the market area Germany/Austria (Phelix).

2 For a monthly base respectively peak load contract this corresponds to between 672 (= 28 x 24) and 744 (= 31 x 24), respec-

tively 240 (= (30 – 5 x 2) x 12) to 276 (= (31 – 4 x 2) x 12) hours. A quarterly contract covers 2160 (= (31 + 28 + 31) x 24) to

2208 (= (31 + 31 + 30) x 24), respectively 768 (= (31 + 28 + 31 – 13 x 2) x 12) to 792 (= (31 + 31 + 30 – 13 x 2) x 12) hours.

Lastly, for a yearly base respectively peak load contract, the number of hours of delivery is between 8760 (= 365 x 24) and

8784 (= 366 x 24), respectively between 3120 (= (365 – 53 x 2) x 12) and 3132 (= (365 – 52 x 2) x 12).


11

Ta

ble

1 C

hara

cte

ristics o

f spot

and d

erivatives c

ontr

acts

tra

ded a

t th

e E

EX

. T

he t

able

com

pare

s t

he m

ark

et

are

as G

erm

any a

nd F

rance f

or

the intr

aday s

egm

ent

and G

erm

any/A

ustr

ia,

Fra

nce a

nd

Sw

itzerla

nd f

or

the d

ay-a

head s

egm

ent. F

urt

herm

ore

, F

rench a

nd G

erm

an p

ow

er

futu

res,

Phelix

futu

res a

nd P

helix

optio

ns a

re i

nclu

ded in t

he c

om

parison.

Based o

n:

htt

p:/

/ww

w.e

pexspot.com

, E

EX

A

G (

2008a),

EE

X A

G (

2008b),

EE

X A

G (

2010),

EP

EX

Spot S

E (

2009),

EP

EX

Spot

SE

(2010),

EE

X P

ow

er

Derivatives G

mb

H (

2009).

Op

tio

ns

Phelix

Optio

ns

Contin

uous

1 M

W

EU

R 0

.001/M

Wh

Corr

espondin

g

Phelix

base lo

ad

or

peak lo

ad f

u-

ture

s c

ontr

act

Base a

nd p

eak

load c

ontr

acts

, next

5 m

onth

s,

6

quart

ers

and 3

years

Exerc

isem

ent on

the last tr

adin

g

day

8.2

5 a

m –

4 p

m,

year-

round (

ex-

cept

week-e

nds &

sta

tuto

ry h

olid

ays)

Fin

ancia

l (P

hys.)

- Yes

Fu

ture

s

Phelix

Futu

res

Contin

uous

1 M

W

EU

R 0

.01/M

Wh

Phelix

base lo

ad

or

Phelix

peak

load o

ver

the

respective d

eli-

very

perio

d

Base a

nd p

eak

load c

ontr

acts

, curr

ent

and n

ext 9

mo

nth

s,

11 q

uar-

ters

and 6

years

3 d

ays b

efo

re

deliv

ery

(quart

er

and y

ear)

, m

onth

futu

res c

an s

till

be

traded d

urin

g

deliv

ery

mo

nth

8.2

5 a

m –

4 p

m,

year-

round (

ex-

cept

week-e

nds &

sta

tuto

ry h

olid

ays)

Fin

ancia

l (P

hys.)

- Yes

Fre

nch/ G

erm

an

Pow

er

Fu

ture

s

Contin

uous

1 M

W

EU

R 0

.01/M

Wh

Ele

ctr

icity w

ith

deliv

ery

over

the

respective d

eli-

very

perio

d in

to

the a

rea o

f th

e

Fre

nch/ G

erm

an

TS

Os

Base a

nd p

eak

load c

ontr

acts

, curr

ent

and n

ext 6

mo

nth

s,

7 q

ua

r-

ters

and 6

years

3 d

ays b

efo

re

deliv

ery

(quart

er

and y

ear)

, m

onth

futu

res c

an s

till

be

traded d

urin

g

deliv

ery

mo

nth

8.2

5 a

m –

4 p

m,

year-

round (

ex-

cept

week-e

nds &

sta

tuto

ry h

olid

ays)

Physic

al

RT

E/A

mp

rio

n

Yes

Sp

ot

da

y-a

he

ad

au

cti

on

tra

din

g

Sw

itzerla

nd

(Sw

issix

)

Daily

auctio

n

0.1

MW

EU

R 0

.1/M

Wh

Ele

ctr

icity tra

ded

for

deliv

ery

on t

he

next

day

Indiv

idual hours

,

base a

nd p

eak

load,

oth

er

sta

n-

dard

blo

cks,

user-

defin

ed b

locks

Auctio

n a

t 10.3

0

am

Ord

er

book o

pen

24h,

year-

round

Physic

al

SG

D

No

Germ

any/ A

ustr

ia

(Phelix

)

Daily

auctio

n

0.1

MW

EU

R 0

.1/M

Wh

(negative p

rices

auth

orized)

Ele

ctr

icity tra

ded

for

deliv

ery

on t

he

next

day

Indiv

idual hours

,

base a

nd p

eak

load,

oth

er

sta

n-

dard

blo

cks,

user-

defin

ed b

locks

Auctio

n a

t 12 p

m

Ord

er

book o

pen

24h,

year-

round

Physic

al

RW

E, E

ON

, V

E,

EN

BW

, A

PG

No

Fra

nce

Daily

auctio

n

1 M

W

EU

R 0

.01/M

Wh

Ele

ctr

icity tra

ded

for

deliv

ery

on t

he

next

day

Indiv

idual hours

,

base a

nd p

eak

load,

oth

er

sta

n-

dard

blo

cks,

user-

defin

ed b

locks

Auction a

t 11 a

m

Ord

er

book o

pen

24h,

year-

round

Physic

al

RT

E

No

Sp

ot

intr

ad

ay c

on

tin

uo

us t

rad

ing

Germ

any

Contin

uous

0.1

MW

EU

R 0

.01/M

Wh

(negative p

rices

auth

orized)

Ele

ctr

icity tra

ded

for

deliv

ery

on t

he

sam

e d

ay a

nd,

from

3 p

m, all

hours

of th

e n

ext

day

Indiv

idual hours

,

base a

nd p

eak

load c

ontr

acts

,

oth

er

sta

ndard

blo

cks, user-

defin

ed b

locks

75 m

inute

s b

efo

re

deliv

ery

Contin

uous

(24/7

), y

ear-

round

Physic

al

RW

E, E

ON

, V

E,

EN

BW

Yes

Fra

nce

Contin

uous

1 M

W

EU

R 0

.01/M

Wh

Ele

ctr

icity tra

ded

for

deliv

ery

on t

he

sam

e d

ay a

nd,

from

11.3

0 a

m,

all

hours

of th

e n

ext

day

Indiv

idual hours

,

base a

nd p

eak

load c

ontr

acts

,

user-

defin

ed

blo

cks

60 m

inute

s b

efo

re

deliv

ery

7.3

0 a

m –

11 p

m,

year-

round

Physic

al

RT

E

No

Tra

din

g p

roc

ed

ure

Siz

e (

min

. vo

lum

e in

cre

men

t)

Tic

k (

min

. p

rice in

cre

men

t)

Un

de

rly

ing

Co

ntr

acts

(lo

ad

pro

file

,

de

livery

pe

rio

d a

nd

tim

e t

o

de

livery

)

Tra

din

g o

f in

div

idu

al

co

ntr

acts

un

til

Tra

din

g h

ou

rs

Sett

lem

en

t

Pla

ce o

f d

elivery

Cle

ari

ng

of

OT

C

tran

sacti

on

s


12

In addition to French/German power futures and Phelix futures, the EEX also offers trading in

Phelix options. The underlying security is a Phelix futures contract, either on a base or peak

load basis. With respect to the delivery period of the underlying futures contract, monthly,

quarterly and yearly Phelix options exist. Moreover, Phelix options are of the European type

and can thereby only be exercised by the buyer of the option on the last trading day. After

being exercised, Phelix options are fulfilled by opening a position in corresponding Phelix

futures at the given exercise price.

The main specifications of the various spot and forward contracts described above are sum-

marized in Table 1 to provide a direct comparison of the various product categories across

contract type and market area.

Finally, note that the EEX has recently introduced two product innovations, namely Phelix

Week Futures and Phelix futures contracts with a delivery during the off-peak hours of the

respective delivery period. Hence, these products extend the product range of the EEX with

respect to the delivery period as well as with respect to the load profile of the contracts.

2.1.3.3 Trading Volumes and Over the Counter Trading

Table 2 lists the main product categories traded at the EEX together with their respective

trading volumes in 2008 and 2009. The figures indicate towards the relative importance of

derivatives contracts in comparison with spot contracts, where the prior constitute more than

80% of the total trading volume. Within the spot market segment, the volume of day ahead

trading clearly exceeds that of intraday trading with the latter presenting merely 5% or even

less of the total spot volume traded.

Table 2 Trading volume in the spot and derivatives power markets of the EEX (all numbers in

TWh). Based on: EEX AG (2009a, pp.80-81), EPEX Spot SE and EEX Power Derivatives

GmbH (2010).

2009

2008

(incl. Powernext)

2008

(excl. Powernext)

Spot Volume 203.0 206.3 154.0

Day ahead Auction 196.3 203.3 151.7

Germany/Austria 135.6 145.6 145.6

France 52.7 51.6 -

Switzerland 8.0 6.1 6.1

Intraday Market 6.7 3.0 2.3

Germany 5.7 2.3 2.3

France 1.0 0.7 -

Power Derivatives 1025.0 1165.0 1165.0

OTC Clearing 739.7 886.6 886.6

Exchange Trading 285.3 278.4 278.4

Total Trading Volume 1228.0 1371.3 1319.0


13

Besides the categorization into spot and derivatives trading, the wholesale market for trading

in power can also be classified into exchange based and over the counter (OTC) trading.

Basically, trading over the counter allows for custom-tailored contracts (i.e. forwards), whe-

reas on an exchange only certain standardized products (i.e. futures) are offered. However,

Wenzel (2007, p.13) points out that the exchange based and OTC market have assimilated

more and more so that comparable products are now being traded on both platforms. In or-

der to prevent the emergence of price differences and thus of arbitrage possibilities, market

participants operating in the OTC market closely follow the prices given through the trading

activities on the exchange (cf. Lichtblick AG, 2008, p.24). In the following analysis, we will

hence focus on the prices of futures contracts, assuming that the prices of futures and for-

ward contracts are approximately equal. According to RWE (2009, pp.2-3) one of the great-

est differences is that forward contracts traded over the counter generally allow for physical

or financial fulfillment while exchange traded futures (i.e. Phelix futures) mainly involve finan-

cial fulfillment. It is important to recognize that the EEX does not only provide clearing and

settlement for exchange traded products but also offers clearing services for certain OTC

transactions (compare last row in Table 1). The corresponding trading volume for power de-

rivatives is presented in Table 2, amounting to 886.6 TWh in 2008 and 739.7 TWh in 2009.

Hence, the share of OTC trading amounts to above 70%, leaving the volume of exchange

traded contracts at roughly 30%. According to RWE (2009, pp.1-2), in 2008 further 2705

TWh were traded in Germany apart from the EEX on electronic trading facilities. The impor-

tance of the EEX is nevertheless undisputable as the prices implied by the exchange act as a

reference prices also off the exchange.

2.2 Characteristics of Electricity Prices

Commodity markets generally allow for the build-up of an inventory so that shortages and

surpluses in a certain commodity can be compensated by corresponding adjustments in the

inventory. According to Borchert et al. (2006, p.51) this has the favorable effect that sudden

changes in supply or demand have only a limited effect on the price of the respective com-

modity. Electricity, however, has physical attributes that crucially differentiate it from other

commodities. Most importantly, electricity must be consumed immediately, i.e. it is a non-

storable good. The absence of efficient storing possibilities has the consequence that supply

and demand imbalances directly push through onto the market prices of electricity, leading to

the enormous price fluctuations generally observed in spot power markets. At this place, it is

worth mentioning, though, that some restricted possibilities for storing electricity do exist. For

instance, managers of hydro storage power plants are faced with the task of managing the

water level of the reservoirs by deciding on the timing of pumping and turbining activities.

Since the water in the reservoir represents a storable commodity that can be used to instan-


14

taneously produce electricity by activating the turbines, the storability property inherent to

water is to some extent transferred to electricity (cf. He, 2007, p.38 and Weber, 2005, p.15).

Indeed, managers of hydro storage power plants may decide to pump water into the reser-

voirs using electricity at one point in time (typically when electricity prices are low) in order to

make use of it to operate the turbines and produce electricity at a later point in time (when

prices have risen above a certain level). It is hence not surprising that in hydro-dominated

systems, as for instance in the Nordic power market, electricity prices behave much more

like other commodity prices, i.e. they are considerably less exposed to price fluctuations in

the short-term (Weber, 2005, p.15). However, considering that in most countries hydro sto-

rage power plants are rather scarce and that pumping generally leads to an energy loss of

approximately 30%, it is reasonable to say that electricity is not storable, at least not in an

adequately efficient and conventional way and at sufficiently large volumes (Burger et al.,

2003, p.2).

A further implication of the non-storability feature of electricity is the requirement that power

supply and demand must exactly equal each other at any location and at each point in time

(Borchert et al., 2006, p.51). Any shortfalls or surpluses that occur through imbalances in

supply and demand have the potential to destabilize the entire electricity grid, with the result-

ing frequency and voltage fluctuations being capable of inflicting serious damage onto gen-

eration and transmission equipment (Lemming, 2003, p.3). The situation is aggravated by the

fact that the interregional exchange of excess units is limited due to the grid dependence of

electricity transmission and various bottlenecks (Borchert et al., 2006, p.51). Additionally,

electricity is an important input factor in many domestic and industrial processes so that elec-

tricity demand is highly inelastic in the short-term, preventing any balancing attempts from

the demand side (He, 2007, p.56). It is hence in the duty of the grid operators to ensure that

the amount of power being produced exactly equals the amount being consumed. Transmis-

sion system operators such as Swissgrid (cf. http://www.swissgrid.ch) therefore have to bal-

ance out unforeseen fluctuations in production and consumption in order to ensure a secure

supply of electricity at a constant frequency. For this to be achieved, power plants and other

suppliers of electricity may be required to increase or decrease the volume of energy that

they inject into or withdraw from the system.

Briefly worded, the characteristics of electricity prices are to a large extent driven by this in-

terrelation between non-storability of electricity on one side and the requirement of instanta-

neous equilibrium of power supply and demand on the other side. Various studies have ex-

amined electricity prices and their characteristics. For instance, Borchert et al. (2006) ana-

lyze EEX (i.e. Phelix) electricity spot prices on a hourly and daily base load basis for the pe-

riod from 2000 to 2004 as well as the prices of various futures contracts for the period from

2003 to 2004. Blöchlinger (2008) extends this period by analyzing historical Phelix spot price


15

data from 2001 to April 2007 and Phelix futures prices between 2003 and 2007. With regard

to Swissix based products, Giger (2008) analyzes spot prices for a period ranging from De-

cember 2006 to March 2008. For the Nordic power market, one of the pioneering markets

with regard to the world-wide deregulation process, Weron (2005) studies the hourly spot

prices of the Nordic Power exchange Nord Pool for a period from 1992 to 2004. In a cross-

section analysis, Meyer-Brandis and Tankov (2007) further compare electricity spot prices of

several power exchanges across Europe and the USA with historical data up to 2006. In the

following, the main results of these studies shall be discussed in a general setting. The most

fundamental characteristics of electricity spot prices can thereby be summarized under the

notions strong seasonalities, mean reverting behaviour and jumps, spikes as well as extreme

volatility.

2.2.1 Spot Price Characteristics

2.2.1.1 Seasonalities

Blöchlinger (2008, p.8) states that, compared to other commodities, the seasonal patterns of

electricity spot prices are among the most complicated. In order to see why electricity prices

exhibit seasonal fluctuations, we must first recognize that the demand for electricity itself

underlies fluctuations. Electricity demand (and thus electricity prices, as we will see later)

thereby typically exhibits three different types of seasonalities: within a single day, during the

week, and during the year. Simonsen et al. (2004, p.6) and Weron (2005, p.4) remark that

the amount of power demanded at various points in time depends to a large extent on the

level of human activity as well as weather and climate conditions. During the day, we can

generally observe a drastic increase in electricity demand between 5 a.m. and 8 a.m. when

people get up and business activities are initiated (cf. e.g. Borchert et al., 2006, p.52-53 or

Blöchlinger, 2008, p.8). On the other side, around 8 p.m. electricity demand declines rapidly,

as most human activities come to a halt. As shown by Blöchlinger (2008, p.8-10), the specific

hourly pattern differs largely with respect to the yearly seasons. While we can generally ob-

serve two peaks on a typical winter day, a less pronounced at noon and a more pronounced

around 7 p.m., we typically only see one peak at noon during a summer day. Clearly, short-

term electricity demand is strongly affected by temperature, with significant amounts of elec-

tricity being used for heating and air conditioning purposes (cf. Lemming, 2003, p.4). Fur-

thermore, electricity consumption is lower on Saturdays and Sundays, which is due to the

fact that many businesses do not operate during the weekend. Lastly, electricity demand

during the year seems to depend on the respective power market under analysis. For in-

stance, Simonsen et al. (2004, p.6) find that in some Nordic countries, especially when elec-

tricity is used country-wide for heating, electricity consumption is significantly higher during

the winter compared to the summer. Simonsen (2004, p.6) further notes that the opposite


16

may be true for other countries or regions. For instance, in California, the permanent use of

air conditioning systems during the hot summer months results in higher electricity consump-

tion than during mild winter months. For the spot power market of the EEX, Blöchlinger

(2008, p.8) does not find any clear evidence for either of these extreme cases.

Once we have accepted that electricity demand shows seasonal behavior, we must demon-

strate that the demand fluctuations translate into corresponding seasonalities in electricity

prices. This relation is for instance shown in an empirical context by Borchert et al. (2006,

p.52-53). Due to the highly inelastic demand, combined with the lack of sufficient and effi-

cient storing possibilities, market prices for power are to a large extent determined by the

power supply, which is based on the merit order of power generation technologies (Burger et

al., 2003, p.2). The merit order is established by stacking the various power plants of a power

system according to their marginal costs, creating a strictly increasing curve that returns the

marginal costs for any possible accumulated power load. To the left, the curve includes all

power plants with high fixed and low variable costs, such as for instance nuclear, coal and

run-of-river hydro power plants (He, 2007, p.57). Since these power plants are characterized

by very low marginal costs once they are running, they are usually called in first and conse-

quently form the base load. More to the right, the merit order includes power plants that have

high variable costs, such as gas or oil power plants. Hence, they are only operated in peak

hours, i.e. when demand is too high to just rely on the other power plants. Interrelating this

with the demand curve, we can note that a low demand leads to an intersection of supply

and demand curves at a low level, so that during these hours only the power plants with the

lowest marginal generation costs are in operation, resulting in a relatively low market clearing

price. The higher the load the more power plants with steadily increasing marginal costs are

employed in the generation process, leading to correspondingly higher electricity prices. Be-

sides demand, also power supply may be subject to climate conditions and seasonal varia-

tions, leading to further fluctuations in power prices. For instance, hydro storage power plants

highly depend on rainfall and snow melting, with precipitation following annual cycles (We-

ron, 2005, p.4; Lemming, 2003, p.4).

2.2.1.2 Mean Reversion

According to He (2007, p.59), there exists a strong consensus among researchers that elec-

tricity prices show a mean reverting behavior, as do most commodities. This means that

electricity prices show significant jumps and spikes in the short-term, but in the longer-term

they are always pulled back towards a long-term mean (cf. Borchert et al., 2006, p.45-55).

Geman and Roncoroni (2006, p.1227) further specify that commodities typically exhibit a

mean reverting behavior towards a price level that is characterized by the marginal costs of

production and that may be constant, periodic or periodic with a trend. Clearly, electricity


17

prices mean revert around a periodical trend driven by the above mentioned seasonalities,

outages of power plants as well as fluctuations in demand and supply due to changing

weather condition. According to Pilipovic (2007, p.24), the rate of mean reversion depends

on how fast the responsible events dissipate and on the ability of the supply side to react to

these events by rebalancing supply and demand. Burger et al. (2003, p.3) further state that

mean reversion is rather fast and usually takes place within days or weeks, at most. In the

long-run, however, the mean reverting level itself can be subject to permanent shifts, for in-

stance, initiated by changes in the availability and prices of energy sources or by sustainably

changing demand patterns (Borchert et al., 2006, p.55).

2.2.1.3 Jumps, Spikes and exceptionally high Volatility

Besides seasonal patterns and mean reverting behavior, electricity prices typically also exhi-

bit jumps. Note, however, that since prices do not persist on the level they initially jump up to,

it makes sense to rather use the notion „spike‟ (Blöchlinger, 2008, p.10). According to Geman

and Roncoroni (2006, p.1228), a spike can thereby be regarded as one or several subse-

quent upward jumps, which are directly followed by a significant down movement in the price.

The initial price jumps can be initiated by unexpected outages of power plants (Weber, 2005,

p.15) or sudden increases in demand due to extreme weather conditions (Geman and Ron-

coroni, 2006, p.1228). As a result, the demand curve will intersect the merit order further to

the right, indicating that power plants with higher marginal costs are being operated. Howev-

er, as these shocks in demand or supply are typically of a short-term nature, prices shortly

return to the normal level. Again, efficient storing possibilities could mitigate this process and

lead to smoother price movements. As this is not the case, the large price movements, which

are responsible for a significant part of the total variation in electricity prices, explain the ex-

tremely high volatilities in spot power markets (Weron, 2005, p.3). In fact, various studies, for

instance Pilipovic (1998, cited in Weber, 2005, p.15) have shown that electricity prices show

the highest volatilities among all traded commodities. Corresponding to He (2007, p.58), an-

nualized volatilities of 1000% are not uncommon to observe in hourly spot price data. He

(2007, p.59) further states that volatility is heteroskedastic as it tends to be high when prices

are at a high level. This corresponds to the observations of Weron (2005, p.3) and Simonsen

et al. (2004, p.11) that spikes are more likely to occur during peak hours and in months of

high demand for electricity.

2.2.2 Price Characteristics of Futures Contracts

On most power exchanges trading platforms for futures contracts were offered shortly after

trading in spot contracts was established (Weber, 2005, p.17). The intention of this was to

provide market participants with the ability to cope with the uncertainties in electricity spot

prices (cf. Blöchlinger, 2008). This is particularly important when considering the extreme


18

volatility of electricity spot prices, as shown in the precedent section. Weber (2005) reasons

that the prices of power futures contracts for electricity exhibit characteristics that are very

distinct to those of spot power contracts but similar to that of other financial contracts. Hence,

futures prices do not show strong seasonal patterns, have no mean reversion in general and

feature a much lower volatility than spot prices. According to Pilipovic (2007, p.26), the vola-

tilities of futures prices decrease with increasing expiration of the respective contracts. This is

a result of the fact that it is reasonable to expect that supply and demand are balanced in the

long run. Consequently, futures prices reflect the corresponding equilibrium price level, which

is relatively stable over time. Due to these reasons, and given the fact that they are storable,

futures and forward contracts are often regarded as the basic tradable assets within power

markets (Blöchlinger, 2008, p.1).

Dependence: Linear Correlation, Copulas and Measures of Association

19

3 DEPENDENCE: LINEAR CORRELATION, COPULAS AND MEAS-

URES OF ASSOCIATION

In risk management, it is of utmost importance to get an idea about how certain random va-

riables move together. Thus, whenever we are dealing with the issue of modeling dependen-

cies among random variables, methods such as correlation measures or copula models

come into play. However, unlike the concept of Pearson‟s linear correlation, copulas allow for

a far more complex and more flexible way to describe and model such dependence struc-

tures.

This section starts with the presentation of the theoretical background behind copula models.

Compared to other theoretical contributions discussing the issue of modeling dependence

structures, the first subsection may follow a somewhat unconventional but yet logic structure.

We commence with the definition of Pearson‟s linear correlation as a dependence measure

and immediately proceed to discuss the limitations which are inherent to this concept. Having

recognized the fact that Pearson‟s linear correlation can only be applied in a sensible way in

certain cases, we move on to cover the more general copula models, which help us to con-

ceive dependence at a deeper level. Finally, we will derive Spearman‟s rho and Kendall‟s tau

as alternative dependence measures, i.e. measures of association. Since they are consistent

with the afore mentioned concept of copulas, they represent improved and adequate alterna-

tives to the initially discussed linear correlation. Following the theoretical considerations

about dependence structures, the transition to the empirical part will be initiated by discuss-

ing two topics that are central when fitting copulas to data. Firstly, this includes the presenta-

tion of the main estimation methods and secondly, the evaluation of different copula model

specifications based on a variety of goodness of fit measures.

3.1 Theoretical Background of Copulas and Dependence

3.1.1 Pearson’s Linear Correlation

Measures of dependence are commonly used to represent the dependence structure of a

pair of random variables by a scalar (cf. Schmidt, 2007, p.21). Out of these, Pearson‟s linear

correlation is undeniably the most popular and most frequently applied measure in practice.

Hence, we will begin our analysis of stochastic dependence herewith.

Definition 3.1 (Pearson’s linear correlation) Let 𝑋1 and 𝑋2 be two vectors of random va-

riables with finite and nonzero variances, then Pearson‟s linear correlation coefficient is


20

𝜌 𝑋1 ,𝑋2 =

𝐶𝑜𝑣 𝑋1 ,𝑋2

𝑉𝑎𝑟 𝑋1 𝑉𝑎𝑟 𝑋2 (3.1)

with 𝐶𝑜𝑣 𝑋1 ,𝑋2 and 𝑉𝑎𝑟 𝑋𝑖 being the covariance of 𝑋1 and 𝑋2, respectively the variance of

𝑋𝑖 (cf. Embrechts et al., 2001, p.9).

Pearson‟s linear correlation is a measure of a specific kind of dependence, namely a linear

one. Thus, 𝑋1 and 𝑋2 being perfectly linearly dependent in the sense of 𝑋2 = 𝛼 + 𝛽𝑋1 with

𝛼 ∈ ℝ and 𝛽 ∈ ℝ\{0} is equivalent to 𝜌 𝑋1 ,𝑋2 = 1. To be more precise, 𝛽 > 0 implies a per-

fectly positive and 𝛽 < 0 implies a perfectly negative linear dependence (cf. Embrechts et al.,

2001, p.10; McNeil et al., 2005, p.202). It is exactly these values that form the bounds of the

possible range of values that can be taken by Pearson‟s linear correlation coefficient. In any

other case we can hence state that −1 < 𝜌 𝑋1 ,𝑋2 < 1. Particularly, in the case that 𝛽 = 0,

i.e. 𝑋1 and 𝑋2 are independent, it applies that 𝜌 𝑋1 ,𝑋2 = 0. It is, however, important to rec-

ognize that the inverse of this statement does not hold in general, i.e. a correlation of zero

does not per se imply independence. Furthermore, it holds for 𝛽1 ,𝛽2 > 0 that 𝜌 𝛼1 +

𝛽1𝑋1,𝛼2 + 𝛽2𝑋2 = 𝜌 𝑋1 ,𝑋2 . This directly implies that Pearson‟s linear correlation is invariant

under strictly increasing linear transformations. On the other side, it is not invariant under

nonlinear strictly increasing transformations of the form 𝑇:ℝ → ℝ , i.e. 𝜌 𝑇(𝑋1),𝑇(𝑋2) ≠

𝜌 𝑋1 ,𝑋2 . As we will see later, this is an important point when distinguishing Pearson‟s linear

correlation from other measures of dependence.

According to Embrechts et al. (2001, p.10), the reasons for making Peason‟s linear correla-

tion the first choice in many application lies in the fact that it is simple to calculate and that it

is a natural measure of dependence when working within the family of elliptical distributions,

such as for instance the multivariate normal or the multivariate t-distribution. Their name

comes from the fact that elliptical distributions are distributions which have a density that is

constant on ellipsoids, e.g. in the bivariate case the contour lines of the density surface form

ellipses (cf. Embrechts et al., 1999, p.2). Elliptical distributions are fully characterized by a

vector of means, a variance-covariance matrix and a characteristic generator function3. Later

on, it will become evident that means and variances are determined by the marginal distribu-

tions, so that in the case of elliptical distributions, copulas only depend on the correlation

matrix and the characteristic generator function. In this sense, the correlation matrix has a

natural parametric role in this class of distributions (McNeil et al., 2005, p.201). Thus, as long

as we have a multivariate normal distribution (or any other elliptical distribution), it is com-

pletely unproblematic to use a correlation matrix based on Pearson‟s linear correlation to get

an idea about the dependence structure of the underlying variables (Embrechts et al., 1999,

3 For a more elaborate presentation of elliptical distributions and their characteristics compare for instance Asche (2004, p.29),

Lindskog (2003, p.3) and Schmidt (2007, p.21).


21

p.2). However, as remarked by Lindskog (2000, p.1), empirical studies suggest that most

financial data is not adequately represented by a multivariate normal distribution, mainly due

to heavy tails and extreme events occurring more frequently than implied by a multivariate

normal distribution. As indicated by the specific spot price characteristics presented in Sec-

tion 2.2, the same may hold for electricity prices. Considering this, the elliptical world must be

left behind and Pearson‟s linear correlation can no longer be seen as a suitable measure of

dependence.

Following the argumentation of Schmidt (2007, pp.21-22), Embrechts et al. (1999, pp.1-7)

and McNeil et al. (2005, pp.201-206)), further limitations and pitfalls regarding the use of

Pearson‟s linear correlation as a dependence measure can be summarized as follows. To

start with, Pearson‟s linear correlation is only a scalar measure of dependence and as such it

is unable to capture the whole dependence structure of the underlying variables. Secondly, a

correlation coefficient of zero does not in general imply independence. Although there is an

equivalence between zero correlation and independence for normal distributions, this does

not even apply to t-distributions, although they belong to the same family of elliptical distribu-

tions. Thirdly, Pearson‟s linear correlation is invariant under strictly increasing linear trans-

formations, but not under more general nonlinear strictly increasing transformations. The

consequence of this is that for instance two logarithmically transformed random variables

may have a different correlation than the untransformed random variables. Fourthly, Pear-

son‟s linear correlation is only defined when the variances of the underlying variables are

finite. In this sense, Pearson‟s linear correlation is not an adequate measure of dependence

for very heavy-tailed variables. Fifthly, depending on the marginal distributions of the underly-

ing variables, it is not necessarily the case that all values in the interval [-1, 1] are attainable

by the linear correlation coefficient. Although it holds true if the variables follow an elliptical

distribution, we cannot claim that the same applies for other distributions. Hence we must

adhere to the statement that perfectly negatively dependent variables do not per se have a

correlation of minus one and perfectly positively dependent variables do not necessarily ex-

hibit a correlation of plus one. Furthermore, the interpretation that small correlations imply

weak dependence may be misleading. Lastly, it does not in general hold that a pair of ran-

dom variables with some marginal distributions and a specific linear correlation uniquely de-

termines the multivariate distribution. In other words, unless we restrict ourselves to elliptical

distributions, two bivariate distributions may have a completely different dependence struc-

ture despite having identical marginal distributions and an identical linear correlation. To

summarize the key result of the last two pitfalls, we may state that it can be quite dangerous

to conclude on the dependence structure solely based on information about the underlying

marginal distributions and linear correlations.


22

The previous remarks have shown us that the imprudent use of Pearson‟s linear correlation

can lead to severe misinterpretations, especially in the case of distributions outside of the

elliptical world. Or to say it in the words of Embrechts et al. (1999, p.1) “[Pearson‟s linear]

correlation is a minefield for the unwary.” This renders it necessary to introduce other tech-

niques to model dependence structures which avoid at least some of these problems. For

this reason, we will now turn the focus towards copulas and subsequently to some copula

related measures of association. They seem to be promising concepts in a broader range of

applications.

3.1.2 Copulas

According to Durante and Sempi (2009, pp.1-2), Trivedi and Zimmer (2007, p.3) and Quesa-

da-Molina et al. (2003, p.499), the history of copulas dates back as far as the 1940s and

1950s when, among others, Hoeffding (1940), Fréchet (1951) and Gumbel (1958) released

papers on copula related subjects. Nevertheless, it was not until 1959 when Sklar (1959) first

made use of the term „copula‟. Beyond that, by proving the theorem that is now carrying his

name, Sklar achieved the deepest result in the context of copulas (Durante and Sempi, 2009,

pp.1-2). At that time, however, the use of copulas merely encompassed the construction of

the theory of probabilistic metric spaces (Quesada-Molina, 2003, p.499). It is thus not sur-

prising, that Nelson (2006, p.1) refers to copulas as being a rather modern phenomenon,

stating that they have only recently been rediscovered. For one part, this is due to several

conferences devoted to copulas that were being held in the 1990s and early 2000s which

have strongly contributed to the further development of the field. The publications by Joe

(1997) and Nelson (1999) emerged as standard references of copula theory, further increas-

ing the popularity of copulas. Moreover, copula models started to attract the interest by re-

searchers of various applied sciences, most notably finance. Charpentier et al. (2006, p.1) go

as far as saying that copulas have become a standard tool in finance, with applications com-

prising for instance credit derivatives, option pricing and risk management (cf. also Cherubini

et al., 2004). Due to these developments, the copula related body of literature has seen a

tremendous increase over the past twenty years.

3.1.2.1 Preliminaries

Before we can enter into the deeper mathematics of copulas, we need to introduce some

basic notions and concepts which turn out to be central in the context of copulas. Firstly, and

although it is assumed that the reader is familiar with basic statistics, we will shortly recapitu-

late the definition of the distribution function of a continuous random variable, with the inten-

tion to clarify the notation used in the subsequent remarks (cf. Asche, 2004, pp.6-10).


23

Definition 3.2 (Cumulative distribution function) The cumulative distribution function (cdf),

often just referred to as distribution function, denotes the probability that the random variable

𝑋 takes at most a value 𝑥, i.e. 𝐹 𝑥 = 𝑃 𝑋 ≤ 𝑥 .

Whenever we are working with a set of random variables 𝑿 = 𝑋1 ,… ,𝑋𝑛 , as it is inevitably the

case when we intend to analyze dependence structures, we have to further distinguish be-

tween multivariate and marginal distribution functions. A multivariate or joint cdf indicates the

probability that each of the random variables takes at most a certain value, i.e. 𝐹 𝑥1 ,… , 𝑥𝑛 =

𝑃 𝑋1 ≤ 𝑥1 ,… ,𝑋𝑛 ≤ 𝑥𝑛 . By contrast, the 𝑖-th marginal cdf defines the distribution of a single

component 𝑋𝑖 of 𝑿 independent of the distribution of the other components, i.e. 𝐹𝑖 𝑥𝑖 =

𝑃 𝑋𝑖 ≤ 𝑥𝑖 . The relation between joint and marginal cdf (in the bivariate case) is given by the

following limit (for the case of the first component):

𝐹1 𝑥1 = lim𝑥2→∞

𝐹 𝑥1 ,𝑥2 (3.2)

Basically, probabilities are defined as an integral of the probability density function (pdf), i.e.

𝑃 𝑎 ≤ 𝑋 ≤ 𝑏 = 𝑓 𝑥 𝑏

𝑎𝑑𝑥. As for the cdfs, it is possible to express multivariate and marginal

densities. The relation between the cdf and the pdf is given by

𝐹 𝑥1 ,… , 𝑥𝑛 = 𝑃 𝑋1 ≤ 𝑥1 ,… ,𝑋𝑛 ≤ 𝑥𝑛 = ⋯ 𝑓 𝑥1 ,… , 𝑥𝑛

𝑥𝑛

−∞

𝑥1

−∞

𝑑𝑥1 ⋯𝑑𝑥𝑛 (3.3)

and, in the case of continuously differentiable cdfs, by

𝑑𝐹(𝑥)

𝑑𝑥= 𝑓(𝑥) (3.4)

Next, let us have a look at the notion of the so called generalized inverse, as it is stated by

McNeil et al. (2005, pp.494-495).

Definition 3.3 (Generalized inverse) Let 𝑇 be an increasing function such that 𝑦 > 𝑥 ⇒

𝑇(𝑦) ≥ 𝑇 𝑥 for all pairs and 𝑇 𝑦 > 𝑇 𝑥 for some pair, then the (left-continuous) genera-

lized inverse of 𝑇 is defined as 𝑇← 𝑦 = inf {𝑥 ∶ 𝑇(𝑥 ≥ 𝑦)}

Applying the idea of the generalized inverse to distribution functions leads to the notion of the

quantile function and the following proposition (cf. McNeil et al., 2005, pp.185-186):

Proposition 3.4 (Quantile and probability transformation) If 𝐺 denotes a distribution function

and 𝐺← is its generalized inverse, then the quantile transformation states that 𝑃 𝐺←(𝑈) ≤

𝑥 = 𝐺(𝑥) given that 𝑈 ~ 𝑈(0, 1). Conversely, the probability transformation states that if 𝑌

has a distribution function 𝐺, then 𝐺 𝑌 ~ 𝑈(0, 1).

In fact, the quantile and the probability transformation represent two sides of the same coin.

The probability transformation represents the usual way of reading a cdf, i.e. applying a


24

probability transformation on a random variable that follows a certain distribution provides us

with the corresponding probability information. By contrast, the quantile transformation is of

central importance in stochastic simulation. It basically states that by generating uniformly

distributed random variables 𝑈 and applying the inverse of a distribution function 𝐺 on them,

we can generate random variables of the desired cdf 𝐺.

3.1.2.2 Definition of Copula, Sklar’s Theorem and Basic Properties

The following remark will provide us with the standard operational definition of a copula

based on McNeil et al. (2005, p.185) and Trivedi & Zimmer (2007, pp.9-10)4.

Definition 3.5 (Copula) A copula 𝐶 𝒖 = 𝐶(𝑢1 ,… , 𝑢𝑑) in 𝑑 dimensions is a 𝑑 -dimensional

distribution function on [0,1]𝑑 whose marginals are uniformly distributed. Thus, a copula is a

mapping of the unit hypercube into the unit interval, i.e. 𝐶: [0,1]𝑑 → [0,1].

Since copulas represent multivariate distributions, copulas have properties analogous to

those of any other joint cdf. Following McNeil (2005, p.185) three properties must hold for a

function in order to be qualified as a copula: Firstly, 𝐶(𝑢1 ,… ,𝑢𝑑) is an increasing function in

each component 𝑢𝑖 . Secondly, by setting all the components 𝑢𝑗 = 1 with 𝑗 ≠ 𝑖 we obtain the

marginal component 𝑢𝑖 , i.e. 𝐶 1,… ,1,𝑢𝑖 , 1,… ,1 = 𝑢𝑖 . Or, to put it in the words of Trivedi and

Zimmer (2007, p.10), given that we know 𝑑 − 1 of the random variables with marginal proba-

bility one, then the joint probability of the 𝑑 outcomes corresponds to the probability of the

remaining outcome. Clearly, this property can be seen as the requirement of marginal distri-

butions that are uniform. Thirdly, for a random vector (𝑈1 ,… ,𝑈𝑑)′ having a distribution func-

tion 𝐶 and for values 𝑎1 ,… ,𝑎𝑑 and 𝑏1 ,… , 𝑏𝑑 ∈ [0,1]𝑑 with 𝑎𝑖 ≤ 𝑏𝑖 the probability 𝑃 𝑎1 ≤

𝑈1 ≤ 𝑏1 ,… , 𝑎𝑑 ≤ 𝑈𝑑 ≤ 𝑏𝑑 has to be non-negative. This last property is often referred to as

„rectangle inequality‟. Together, the three properties characterize a copula and hence provide

an alternative way to define a copula. As a result, we can state that any function which fulfills

these properties represents a copula. As a concluding remark, it is important to note that

other papers use slightly distinct expressions for these properties. For instance, according to

Asche (2004, p.12) and Embrechts (2001, p.3), 𝐶 must be grounded and 𝑑-increasing. Trive-

di and Zimmer (2007, p.10) further elaborate the term „grounded‟ as 𝐶 𝑢1,… ,𝑢𝑑 = 0 if 𝑢𝑖 = 0

for any 𝑖 ≤ 𝑑. Hence, if the marginal probability of one outcome is zero, the joint probability of

all outcomes is zero, too. Moreover, the above mentioned rectangle inequality is equivalent

to the expression that 𝐶 is 𝑑 -increasing. In the bivariate case, this property is often

represented by 𝐶 𝑢1,2,𝑢2,2 − 𝐶 𝑢1,2 ,𝑢2,1 − 𝐶 𝑢1,1,𝑢2,2 + 𝐶(𝑢1,1,𝑢2,1) ≥ 0 for two marginals

𝑢1 and 𝑢2 with two values each and 𝑢1,1 ≤ 𝑢1,2 and 𝑢2,1 ≤ 𝑢2,2.

4 For a more abstract definition that interprets copulas as a subset of general multivariate distributions see for instance Em-

brechts et al. (2001), Asche (2004) or Nelson (2006).


25

But let us now have a closer look at what copulas actually are. According to Nelson (2006,

p.2), the latin word „copula‟ can be translated as a link, tie or bond. And this is exactly what a

copula is. A copula couples a multivariate distribution to its univariate marginal distributions.

This relation between marginal and multivariate cdfs is covered in Sklar‟s theorem. It is a key

result in the context of the application of copulas and shall now be discussed.

Theorem 3.6 (Sklar, 1959) Given a multivariate distribution function 𝐹 with marginal distribu-

tions 𝐹1 ,… ,𝐹𝑑 , then there exists a copula 𝐶 such that for all 𝑥1 ,… , 𝑥𝑑 ∈ [−∞,∞]

𝐹 𝑥1 ,… , 𝑥𝑑 = 𝐶 𝐹1 𝑥1 ,… ,𝐹𝑑 𝑥𝑑 (3.5)

As stated by Embrechts et al. (2001, p.4), the elegance of Sklar‟s theorem lies in the way it

shows us how the univariate marginal distributions and the multivariate dependence struc-

ture, represented by the copula function, can be separated. Basically, Sklar‟s theorem can be

interpreted in two ways. On one side, if 𝐶 is a copula and 𝐹1 ,… ,𝐹𝑑 represent univariate distri-

bution functions, then the multivariate distribution function 𝐹 is defined as in the formula

above. A joint distribution function can thus be generated by coupling the marginals with a

copula. On the other side, a copula can be extracted from a joint distribution function and the

corresponding marginal distributions. Sklar‟s theorem hence also shows that all multivariate

distribution functions contain a copula. In particular, the copula function 𝐶 is unique if the

marginals are continuous. To see this second interpretation more clearly, we can rewrite

Sklar‟s theorem by applying the concept of the generalized inverse 𝑥𝑖 = 𝐹𝑖← 𝑢𝑖 on the left-

hand side and 𝐹 𝑥𝑖 = 𝑢𝑖 on the right-hand side (cf. McNeil et al., 2005, p.187):

𝐶 𝑢1,… ,𝑢𝑑 = 𝐹 𝐹1← 𝑢1 ,… ,𝐹𝑑

← 𝑢𝑑 (3.6)

The expressions (3.5) and (3.6) are essential in the application of copulas (cf. Schmidt, 2007,

p.7 and McNeil et al., 2005, p.187). The importance of the second approach is rather theoret-

ical and lies in the extraction of a copula from a multivariate distribution function. Later on, we

will see that the derivation of the Gaussian and the t-copula are examples of this approach.

By contrast, the first way of interpreting Sklar‟s theorem is the starting point of many empiri-

cal applications (cf. Trivedi and Zimmer, 2007, pp.10-12). This approach allows us to con-

clude on the joint distribution function by separately specifying the marginal distribution for

each random variable and the copula function. This makes the estimation of the joint cdf very

flexible, for instance letting the marginal distributions stem from different families. In particu-

lar, we can express (3.6) as

𝐹 𝑥1 ,… , 𝑥𝑑 ;𝜃 = 𝐶 𝐹1 𝑥1 ,… ,𝐹𝑑 𝑥𝑑 ;𝜃 (3.7)

where 𝜃 represents the parameter vector of the copula, characterizing the dependence be-

tween the marginal distributions. Note, in the case of independence the copula is simply the


26

product of the marginals, reducing the problem to the rather trivial task of estimating the indi-

vidual marginal distributions.

The definition of copulas implies that they are cumulative distribution functions. For certain

applications, however, it may be more convenient to dispose of copula densities. On one

side, this may involve the illustration of the dependence structure through plotting the pdf,

which is in many cases more intuitive than plotting the cdf (Schmidt, 2007, p.8). More impor-

tantly, copula densities are required whenever we intend to fit copulas to a data set using a

maximum likelihood approach. Although not all copula functions have densities, all the para-

metric copulas discussed throughout this section do so. Following McNeil et al. (2005,

p.197), we may characterize the copula density 𝑐 of a copula 𝐶 (given differentiability) as

𝑐 𝑢1 ,… ,𝑢𝑑 =

𝜕𝑑𝐶 𝑢1 ,… ,𝑢𝑑

𝜕𝑢1 ⋯𝜕𝑢𝑑 (3.8)

When discussing Pearson‟s linear correlation in the precedent subsection, we realized that it

is only invariant under strictly increasing linear transformations. Copulas, on the contrary,

possess the superior property that they are invariant under any strictly increasing, i.e. mono-

tonic transformation of the marginal distributions. As a consequence, the dependence struc-

ture of the respective random variables will remain unchanged after the transformation. For

instance, logarithmically transformed random variables will still have the same dependence

structure, expressed by the copula, as the untransformed variables. The following proposi-

tion, whose proof can be found in McNeil et al. (2005, p.188), formalizes this property.

Proposition 3.7 (Invariance of copulas) Let 𝑋1 ,… ,𝑋𝑑 be random variables with continuous

marginal distributions and copula 𝐶. By referring to 𝑇1 ,… ,𝑇𝑑 as strictly increasing functions,

the transformed random variables 𝑇1 𝑋1 ,… ,𝑇𝑑 𝑋𝑑 will have the same copula 𝐶.

The Fréchet-Hoeffding bounds, which will be discussed next based on McNeil et al. (2005,

pp.188-190), Schmidt (2007, pp.10-12) and Trivedi and Zimmer (2007, pp.9-14), constitute

another important result in the context of copulas.

Theorem 3.8 (Fréchet-Hoeffding bounds) The Fréchet-Hoeffding bounds represent universal

bounds in that sense that any cumulative distribution function, and hence every copula, is

bounded by the lower and upper bounds

𝑚𝑎𝑥 𝑢𝑖 + 1− 𝑑

𝑑

𝑖=1

, 0 ≤ 𝐶(𝒖) ≤ 𝑚𝑖𝑛 𝑢1 ,… ,𝑢𝑑 (3.9)

According to Trivedi and Zimmer (2007, pp.12-13), the practical relevance of the Fréchet-

Hoeffding bounds becomes evident when we intend to select a reasonable copula. It may

thereby be sensible to choose a copula that covers the whole space between the lower and


27

the upper bound. Furthermore, if the copula parameter 𝜃 reaches its upper (lower) limit within

the permissible range, the copula should converge to the upper (lower) Fréchet-Hoeffding

bound. However, depending on the parametric form of a certain copula, not the full range of

dependence structures is attainable. This makes the application of certain copulas more or

less reasonable depending on what data set is analyzed.

Figure 3 Fréchet-Hoeffding bounds for C(u,v). The upper Fréchet-Hoeffding bound is

represented by the front surface of the pyramid-shaped body, while the surface

spanned by the bottom and rear side corresponds to the lower bound. Source:

Schmidt, 2007, p.11.

It is important to note that the Fréchet-Hoeffding bounds allow for specific interpretations with

regard to dependence. Particularly, in the bivariate case, the Fréchet-Hoeffding bounds are

copula functions themselves5. These copulas, known as comonotonicity and countermonoto-

nicity copula, together with the independence copula constitute the class of the fundamental

copulas. As their name suggests, these copulas represent some fundamental dependence

structures. Before elaborating on these copulas, the illustration of the Fréchet-Hoeffding

bounds for the two dimensional case (cf. Figure 3) shall be commented. The surface

spanned by the bottom and rear side of the pyramid represents the lower bound, while the

surface given by the front side equals the upper bound. In accordance with Theorem 3.8,

every copula must lie within the interior of this pyramid.

5 It should be particularized that the lower Fréchet-Hoeffding bound is not a copula for d ≥ 3, while the upper bound actually

satisfies the definition of a d-dimensional copula function for all d ≥ 2.


28

3.1.2.3 Fundamental Copulas

The independence copula

𝐶Π 𝑢1,… , 𝑢𝑑 = 𝑢𝑖

𝑑

𝑖=1

(3.10)

refers to a dependence structure where there is no dependence between the random va-

riables. Sklar‟s theorem directly implies that random variables are independent if and only if

the independence copula describes their dependence structure (McNeil et al., 2005, p.189).

According to Trivedi and Zimmer (2007, p.15), the importance of the independence copula

lies in its function as a benchmark for independence.

The comonotonicity copula

𝐶𝑀 𝑢1 ,… ,𝑢𝑑 = 𝑚𝑖𝑛 𝑢1,… ,𝑢𝑑 (3.11)

is the upper Fréchet-Hoeffding bound and relates to the case of perfect positive dependence.

Following McNeil et al. (2005, p.190), a number of random variables is referred to as perfect-

ly positively dependent if they are almost surely strictly increasing functions of each other, i.e.

𝑋𝑖 = 𝑇𝑖 𝑋1 for 𝑖 = 2,… ,𝑑.

The countermonotonicity copula

𝐶𝑊(𝑢1 ,𝑢2) = 𝑚𝑎𝑥 𝑢1 + 𝑢2 − 1, 0 (3.12)

is the two-dimensional lower Fréchet-Hoeffding bound and describes the other extreme,

namely perfect negative dependence. Two random variables are perfectly negatively depen-

dent if one random variable is almost surely a strictly decreasing function of the other. For-

mally, it holds 𝑋2 = 𝑇 𝑋1 with 𝑇 being a strictly decreasing function.

3.1.2.4 Elliptical and Archimedean Copulas

Obviously, there exist many functions that fulfill the definition of a copula and the body of

copula literature is characterized by a correspondingly vast number of different copulas. In

this subsection, some important parametric copula families will be presented in more detail.

Together, they represent a broad spectrum of dependence structures, allowing for the recon-

struction of many characteristics of empirical data. Consequently, these copulas are not only

popular in the literature but also frequently applied in empirical studies (cf. Trivedi and Zim-

mer, 2007, p.15). Moreover, as expressed in equation (3.7), each copula family is deter-

mined by a single parameter or a vector thereof. More precisely, all copulas taken into con-

sideration are, in the bivariate case, characterized by a single parameter, except for the t-


29

copula, which is subject to an additional second parameter6. Durante and Sempi (2009, p.14)

list a number of requirements that should preferably be fulfilled by a copula family. In particu-

lar, they should allow for a probabilistic interpretation, represent a flexible and wide range of

dependence and be easy to handle. With respect to the range of dependence Trivedi and

Zimmer (2007, p.13) further state that a copula family should comprise the independence,

comonotonicity and countermonotonicity copula in order to be qualified as comprehensive.

Table 3 thereby reveals for each copula family under consideration firstly, what permissible

parameter ranges are and secondly, for what parameter values the three fundamental copu-

las are obtained, provided that they are attainable. Since we restrict ourselves to some basic

copula families (i.e. Gaussian, t-, Gumbel, Clayton and Frank copula), the third of the above

mentioned requirements holds as well.

Table 3 Copula families, generator functions and permissible parameter ranges. The first column provides the generator

functions for the Archimedean copula families The second column exhibits the permissible parameter range for the Gaussian

(𝐶ρ𝐺𝑎 ), t- (𝐶ρ ,ν

t ), Gumbel (𝐶𝜃𝐺𝑢 ), Clayton (𝐶𝜃

𝐶𝑙 ) and Frank (𝐶𝜃𝐹𝑟 ) copula families. Furthermore, the table indicates for which para-

meter values the countermonotonicity (𝐶W ), the independence (𝐶Π ) and the comonotonicity (𝐶M) copulas are obtained. Note

that the entries for the Gaussian and t-copula as well as for the countermonotonicity copula uniquely refer to the bivariate

case. Based on: McNeil et al., 2005, p.220; Cherubini et al., 2004, pp.112-128.

Copula

family

Generator function

𝝍 𝒕

Permissible parameter

range 𝑪𝐖 𝑪𝚷 𝑪𝐌

𝐶ρ𝐺𝑎 N/A 𝜌 ∈ [−1,1] ρ = −1 ρ = 0 ρ = 1

𝐶ρ,νt N/A 𝜌 ∈ [−1,1] ρ = −1 ρ ≠ 0 ρ = 1

𝐶𝜃𝐺𝑢 −ln t

𝜃 𝜃 ∈ [1,∞) Not attainable 𝜃 = 1 𝜃 → ∞

𝐶𝜃𝐶𝑙

1

𝜃 𝑡−𝜃 − 1 𝜃 ∈ [−1,∞) \ {0} 𝜃 = −1 𝜃 → 0 𝜃 → ∞

𝐶𝜃𝐹𝑟 −ln

𝑒−𝜃𝑡 − 1

𝑒−𝜃 − 1 𝜃 ∈ ℝ \ {0} 𝜃 → −∞ 𝜃 → 0 𝜃 → ∞

In the section about Sklar‟s theorem, it was mentioned that it is possible to extract copulas

from multivariate distributions. The copulas within the elliptical class originate exactly from

that approach, representing copulas inherent to multivariate elliptical distributions. In particu-

lar, the Gaussian and the t-copula are derived from the multivariate normal respectively t-

distribution by applying expression (3.6).

6 In the following, the copula functions are represented in their d-variate form. In the subsequent analysis, however, we will

investigate the dependence structure of return series of electricity prices in a pair wise manner. Consequently, we will then

restrict ourselves to the bivariate form of the individual copula families.


30

The Gaussian copula is given by

𝐶Σ𝐺𝑎 𝑢1,… ,𝑢𝑑 = ΦΣ Φ

−1 u1 ,… ,Φ−1 ud (3.13)

where ΦΣ denotes the cdf of a 𝑑-variate standard normal distribution with correlation matrix Σ

and Φ is the univariate standard normal distribution (cf. McNeil et al., 2005, p.191). The

Gaussian copula covers dependence structures between perfect positive dependence, inde-

pendence and perfect negative dependence, with the strength of dependence being deter-

mined by Σ. In the bivariate case, it is basically sufficient to know the linear correlation coeffi-

cient 𝜌 between the two random variables, which constitutes the only parameter, as

Σ =

1 ρρ 1

(3.14)

This reinforces the previous statement that Pearson‟s linear correlation fully describes the

dependence structure of normal distributions respectively elliptical distributions in general (cf.

Schmidt, 2007, p.14). The parametric role of Σ can be ascribed to the fact that through the

standardization procedure of the marginal distributions the random variables 𝒀~𝑁 𝝁,Ω are

transformed to 𝑿~𝑁 𝟎, Σ by strictly increasing transformations. As we know from Proposition

3.7, such transformations leave copulas unaffected. Cherubini et al. (2004, p.114) point out

that the Gaussian copula results in a multivariate normal distribution only if it is combined

with normal marginal distributions. This does not imply, however, that, once it has been ex-

tracted, the Gaussian copula cannot be applied to some arbitrary marginal distributions. Ra-

ther, the resulting joint distribution would be non-normal, further enlarging the range of multi-

variate distributions that can be modeled. Following McNeil et al. (2005, p.193), these multi-

variate (non-normal) distributions are referred to as meta-Gaussian. For instance, prominent

credit risk models use exponential marginal cdfs together with the Gaussian copula to model

companies‟ default times (McNeil et al., 2005, p.193). In the same way as the meta-Gaussian

distribution refers to the Gaussian copula, the notation of a meta-distribution can be ex-

tended to other copulas. The fact, that we can construct a meta-Gaussian, meta-t-, meta-

Gumbel etc. distribution with the same marginals and the same correlation again clarifies the

limitations of Pearson‟s linear correlation as a single measure of dependence whenever we

do not combine the marginals of an elliptical distribution with the corresponding copula.

The t-copula is represented as

𝐶Σ ,ν𝑡 𝑢1,… ,𝑢𝑑 = tΣ ,ν tν

−1 u1 ,… , tν−1 ud (3.15)

with tΣ ,ν and tν describing the cdf of the 𝑑-variate respectively univariate t-distribution with ν

degrees of freedom (cf. McNeil et al., 2005, p.191). Analogous to the Gaussian copula, the t-

copula is extracted from a multivariate t-distribution, provided that the marginals are also t-

distributed. According to Trivedi and Zimmer (2007, p.17), the t-copula converges to the


31

Gaussian copula as ν approaches infinity. Σ again corresponds to the correlation matrix and,

analogously to the Gaussian copula, the t-copula is parameterized by the linear correlation

coefficient in the bivariate case. As for all elliptical distributions, with the exception of the

normal distribution, zero correlation in the components does not imply independence. Hence,

while comonotonicity and countermonotonicity can be achieved in the same way as in the

case of the Gaussian copula, it is not possible to obtain the independence copula with ρ = 0

as long as ν < ∞.

Figure 4 compares the copula densities of a Gaussian copula and a t-copula for ρ = 0.3 and

ν = 2. Firstly, we can observe that both copulas are symmetric, with the lower left quadrant

being equally pronounced as the upper right quadrant. Secondly, despite being quite similar

in the center, the behavior at the four corner points differs substantially. In particular, the t-

copula features lower and upper tail dependence, while the Gaussian copula does not show

any tail dependence for ρ ≠ ± 1 (cf. McNeil et al., 2005, pp.190-195). Tail dependence here-

by refers to the occurrence of joint extremal events in the sense that there is a tendency for

𝑋2 to take extreme values when 𝑋1 takes extreme values and vice versa. Data exhibiting joint

extremal events can hence be more accurately modeled via a t-copula than via a Gaussian

copula. Subsection 3.1.3.3 will further elaborate on the topic of tail dependence.

Figure 4 Gaussian and t-copula densities. (a) illustrates the Gaussian copula density,

which is characterized by its symmetry and the absence of tail dependence. (b) shows

the t-copula density, which is also symmetric but shows both upper and lower tail de-

pendence. Source: Schmidt, 2007, p.14.

Elliptical copulas are easily parameterized by the linear correlation matrix, but they are not

without drawbacks. To begin with, empirical data often does not follow a joint elliptical distri-

bution. This can be partially compensated by combining arbitrary marginals with e.g. a Gaus-

sian copula, creating a meta-Gaussian distribution. Embrechts et al. (2001, p.24) further

state that we are in no way restricted to stay within a single distributional family with regard to

the marginals. By using different marginal distributions for the individual components further

flexibility in the modelling of multivariate distributions is gained. It is hence not surprising,

that, as pointed out by Asche (2004, p.20), many applications in finance achieve a good re-

presentation of the empirical dependence structure by simply using a Gaussian or t-copula.


32

What remains unsolved, however, is that the elliptical copulas have radial symmetry, so that

asymmetric dependence structures cannot be modeled with these copulas (cf. Embrechts et

al., 2001, p.15). Yet, we may have empirical data which exhibits stronger tail dependence

either to the up- or downside. The Archimedean copulas, a second class of copulas beside

the elliptical ones, thereby have fundamentally different features. They describe specific de-

pendence structures often found in empirical data, such as the cases of upper or lower tail

dependence. Furthermore, Archimedean copulas have simple closed forms (cf. Gartner,

2007, p.39). Contrary to the elliptical copulas, Archimedean copulas are not extracted from

multivariate distributions but rather originate from mathematical construction. This is also why

McNeil et al. (2005, p.190) refer to them as explicit copulas in contrast to the elliptical, implicit

copulas.

Before having a closer look at some popular Archimedean copula families, we will first ex-

amine the general definition of an Archimedean copula.

Definition 3.9 (Archimedean copulas) Given a continuous, strictly decreasing, convex func-

tion 𝜓 from [0,1] to [0,∞] with 𝜓 1 = 0 and its pseudo-inverse 𝜓 −1 : 0,∞ → [0,1], a copula

which fulfils

𝐶 𝑢1 ,… ,𝑢𝑑 = 𝜓 𝜓 −1 𝑢1 + 𝜓 −1 𝑢2 +⋯+ 𝜓 −1 𝑢𝑑 (3.16)

is called an Archimedean copula (cf. Embrechts et al., 2001, p.31; Durante and Sempi, 2009,

p.15 and Cherubini et al., 2004, p.121).

The function 𝜓 is denoted as the generator function. Note that the respective Archimedean

copula is only strict if 𝜓 0 = ∞, in which case 𝜓 −1 corresponds to the usual inverse 𝜓−1.

The generator functions that lead to the following copula families are provided in Table 3.

The Gumbel copula

𝐶𝜃𝐺𝑢 𝑢1,… ,𝑢𝑑 = exp − −ln𝑢1

𝜃 +⋯+ −ln𝑢𝑑 𝜃

1𝜃 (3.17)

covers dependence structures between independence and perfect positive dependence,

hence it is not comprehensive. As it becomes evident from Figure 5 (a), the Gumbel copula

exhibits strong upper but only weak lower tail dependence. As such, it may propose an ap-

propriate model for the joint distribution of random variables where the outcomes are strongly

correlated at high values and to a less extent at low values (Trivedi and Zimmer, 2007, p.19).


33

The Clayton copula

𝐶𝜃𝐶𝑙 𝑢1,… ,𝑢𝑑 = 𝑢1

−𝜃 +⋯+ 𝑢𝑑−𝜃 − 𝑑 + 1

−1𝜃 (3.18)

is only strict if 𝜃 ∈ (0,∞). In this case, the Clayton copula is not comprehensive, as it does

not cover the case of countermonotonicity (in the bivariate case). By letting 𝜃 ∈ −1,∞ \ {0}

this can be solved, however, at the cost of the Clayton copula losing its property of being

strict. Furthermore, this implies that the Clayton copula is not given by the expression shown

above, but by the maximum of this term with zero (cf. McNeil et al., 2005, p.220). The Clay-

ton copula can be used to model strong lower tail dependence while holding upper tail de-

pendence relatively low (cf. Figure 5 (b)). This may represent an adequate model for applica-

tions in finance, where we can observe a strong correlation across the components in down

markets.

The Frank copula

𝐶𝜃𝐹𝑟 𝑢1 ,… ,𝑢𝑑 = −

1

𝜃ln 1 +

exp −𝜃𝑢1 − 1 ∙ … ∙ exp −𝜃𝑢𝑑 − 1

exp −𝜃 − 1 d−1 (3.19)

is comprehensive as it interpolates between perfect negative and perfect positive depen-

dence, at least in the bivariate case where the countermonotonicity copula is attainable. Ac-

cording to Embrechts et al. (2001, p.32), the Frank copula is the only Archimedean family

showing radial symmetry, comparable to the Gaussian or t-copula. As visible from Figure 5

(c), the Frank copula exhibits strongest dependence in the center, while having no tail de-

pendence. According to Trivedi and Zimmer (2007, p.19), the Frank copula is often used in

empirical studies.

Figure 5 Gumbel, Clayton and Frank copula densities. (a) illustrates the density of the Gumbel copula with its upper

tail dependence. (b) plots the density of the counterpart to the Gumbel copula, namely the Clayton copula. It is cha-

racterized by lower tail dependence. (c) visualizes the Frank copula density which, similar to the Gaussian copula

density, is symmetric and does not exhibit any tail dependence. Source: Schmidt, 2007, p.18.


34

3.1.3 Measures of Association

In Section 3.1.1 we discussed Pearson‟s linear correlation as a first, well-known measure of

dependence. It proved to be a concept which is subject to several limitations. The fact that

Pearson‟s linear correlation is not scale-invariant is one of these drawbacks. This means

that, although it is under linear transformations, it is not invariant under strictly increasing

transformations in general. Yet, there exists a variety of more appropriate, scale-invariant

ways to measure dependence. Moreover, these measures are general enough to be reason-

able for any dependence structure, not only linear dependence. Some authors refer to them

as measures of dependence (cf. for instance Schmidt, 2007, p.21). We will follow the termi-

nology of Nelson (2006, p.157) and denote them as measures of association. This is clearly

a more general term, with two random variables being referred to as „associated‟ whenever

they are not independent (Cherubini et al., 2004, p.95). As opposed to this, Cherubini et al.

(2004, p.38) point out that dependence, in a strict sense, only refers to positive comove-

ments of random variables.

The most popular scale-invariant measures of association are probably the rank correlations

Kendall‟s tau and Spearman‟s rho7. They provide reasonable alternatives to Pearson‟s linear

correlation coefficient and are also applicable to non-elliptical distributions where the latter

proves to be inappropriate as demonstrated in Section 3.1.1. As their name suggests, the

measures of rank correlation are not based on the data itself but rather on the ranks of it.

Moreover, both coefficients measure a specific form of dependence known under the notion

of „concordance‟. We will hence first provide a definition of this property based on Nelson

(2006, pp.157-158) and McNeil et al. (2005, p.206).

Definition 3.10 (Concordance, discordance) Two observations 𝑥1 ,𝑥2 and (𝑥 1, 𝑥 2) ∈ ℝ2 of

the random vectors 𝑋1 and 𝑋2 are called concordant if 𝑥1 − 𝑥 1 𝑥2 − 𝑥 2 > 0 and discordant

if 𝑥1 − 𝑥 1 𝑥2 − 𝑥 2 < 0.

In other words, a pair of random vectors is known to be concordant if large (small) values in

one variable have a tendence to be associated with large (small) values in the other. Con-

versely, for a discordant pair of random variables large (small) values tend to be associated

with small (large) values.

We will now take a closer look at what properties must be fulfilled by a measure of associa-

tion in order to be known as a measure of concordance (cf. Embrechts et al., 2001, p.12 and

Cherubini et al., 2004, p.96).

7 Note that Spearman‟s rho and Kendall‟s tau are not the only enhanced dependence measures. Further measures such as

Blomqvist‟s beta, Gini‟s gamma and Schweizer and Wolff‟s sigma are covered in detail in Nelson (2006).


35

Definition 3.11 (Measure of concordance) A measure of association 𝜅𝑋1 ,𝑋2= 𝜅𝑋2 ,𝑋1

= 𝜅𝐶 is

known to be a measure of concordance if, for two vectors of continuous random variables 𝑋1

and 𝑋2 with copula 𝐶, the following properties apply:

1. Completeness: 𝜅𝑋1 ,𝑋2 is defined for every pair 𝑋1 and 𝑋2

2. Normalization: 𝜅𝑋1 ,𝑋2 ∈ −1, 1

3. Symmetry: 𝜅𝑋1 ,𝑋2= 𝜅𝑋2 ,𝑋1

4. If 𝑋1 and 𝑋2 are independent, then 𝜅𝑋1 ,𝑋2= 0

5. 𝜅−𝑋1 ,𝑋2= 𝜅𝑋1 ,−𝑋2

= −𝜅𝑋1 ,𝑋2

6. If 𝜅𝑋1 ,𝑋2= 𝐶 ≺ 𝐶 , then 𝜅𝐶 ≤ 𝜅𝐶

7. 𝜅𝑋1 ,𝑋2 converges if the copula converges, i.e. lim𝑛→∞ 𝜅𝐶𝑙𝑖𝑚 = 𝜅𝐶 for 𝑋1 and 𝑋2being conti-

nuous random vectors with the corresponding copula 𝐶𝑙𝑖𝑚 converging to 𝐶

Nelson (2006) provides a proof showing that Kendall‟s tau and Spearman‟s rho fulfill the

properties of a concordance measure. Hence, all properties of Definition 3.11 also apply to

Kendall‟s tau and Spearman‟s rho. For instance, the rank correlations take values in the in-

terval [-1, 1]. It is hereby important to note that for continuous random variables any value

between minus one and one is attainable for given marginals, which is a major improvement

compared to Pearson‟s linear correlation coefficient. Furthermore, the value zero is obtained

if the variables are independent, while the opposite must not necessarily hold. Moreover, all

concordance measures are bounded by the case of comonotonicity and countermonotonicity,

i.e. 𝜅𝑋1 ,𝑋2= 1 ⇔ 𝐶 = 𝐶𝑀 and 𝜅𝑋1 ,𝑋2

= −1 ⇔ 𝐶 = 𝐶𝑊 (Cherubini et al., 2004, p.96 and Em-

brechts et al., 2001, p.15). These properties underline the conceptual proximity of concor-

dance measures to copula models. As it will become evident later, we can further state that

these measures of concordance depend only on the copula but not on the marginals of a

multivariate distribution (McNeil et al., 2005, p.206). Clearly, this represents another distinc-

tion from Pearson‟s linear correlation. Despite all these improvements compared to Pear-

son‟s linear correlation, Kendall‟s tau and Spearman‟s rho are still subject to some limita-

tions. Both measures are scalar measures and consequently incapable of characterizing the

entire dependence structure as it is possible with copulas. For instance, they do not reveal

whether the dependence structure is symmetric or asymmetric with respect to the joint tails.

Moreover, it is still a fallacy to assume that the marginal distributions in connection with a

rank correlation matrix fully determine the joint distribution of the underlying random va-

riables.


36

3.1.3.1 Kendall’s Tau

In its population version, Kendall‟s tau is represented by the probability of concordance mi-

nus the probability of discordance, as expressed by the following definition (cf. McNeil et al.,

2005, p.207).

Definition 3.12 (Kendall’s tau) For two independent and identically distributed pairs of ran-

dom variables 𝑋1 ,𝑋2 and 𝑋 1 ,𝑋 2 Kendall‟s tau is given by

𝜌𝜏 𝑋1 ,𝑋2 = 𝑃 𝑋1 − 𝑋 1 𝑋2 − 𝑋 2 > 0 − 𝑃 𝑋1 − 𝑋 1 𝑋2 − 𝑋 2 < 0 (3.20)

An alternative definition makes use of the expectation operator, expressing Kendall‟s tau as

𝜌𝜏 𝑋1 ,𝑋2 = 𝐸 sign 𝑋1 − 𝑋 1 𝑋2 − 𝑋 2 (3.21)

Following McNeil et al. (2005, p.229), the empirical or sample analogue of this theoretical

expression is given by the following standard estimator of Kendall‟s tau

𝑟𝜏 𝑋𝑖 ,𝑋𝑗 =

𝑛2 −1

sign 𝑋𝑡 ,𝑖 − 𝑋𝑠,𝑖 𝑋𝑡 ,𝑗 − 𝑋𝑠,𝑗

1≤𝑡<𝑠≤𝑛

(3.22)

where 𝑋𝑡 ,𝑖 and 𝑋𝑡 ,𝑗 refer to the 𝑡-th observations of two random vectors with 𝑛 observations.

To put it simple, the sample version of Kendall‟s tau is the difference of the number of con-

cordant pairs and the number of discordant pairs divided by the number of total pairs of ob-

servations. The calculation of this estimator is, however, computationally intensive for large

𝑛, as it comprises the evaluation of every pair of observations. This motivates the use of

Spearman‟s rho, which is less time-consuming in its computation while principally conveying

the same information with respect to rank correlation.

In terms of copulas, Kendall‟s tau can be represented as follows, indicating that it does in-

deed only depend on the respective copula function:

𝜌𝜏 𝑋1 ,𝑋2 = 4 𝐶 𝑢1 ,𝑢2 𝑑𝐶 𝑢1 ,𝑢2 − 1

1

0

1

0

(3.23)

3.1.3.2 Spearman’s Rho

Analogously to Definition 3.12 (Kendall‟s tau), the population version of Spearman‟s rho can

be defined using the notion of concordance and discordance (cf. Nelson, 2006, p.167).

Definition 3.13 (Spearman’s rho) Let 𝑋1 ,𝑋2 , 𝑋 1 ,𝑋 2 and 𝑋 1 ,𝑋 2 stand for three indepen-

dent and identically distributed pairs of random vectors, all with a joint distribution function H.

Spearman‟s rho is then defined in proportion to the probability of concordance of 𝑋1 ,𝑋2 and

𝑋 1 ,𝑋 2 minus the corresponding probability of discordance.


37

𝜌𝑠 𝑋1 ,𝑋2 = 3 𝑃 𝑋1 − 𝑋 1 𝑋2 − 𝑋 2 > 0 − 𝑃 𝑋1 − 𝑋 1 𝑋2 − 𝑋 2 < 0 (3.24)

McNeil et al. (2005, p.207) and Schmidt (2007, p.23) present another definition that is closely

related to the quantile respectively probability transformation (cf. Proposition 3.4) and hence

more intuitive in the context of copulas. Basically, applying the respective cdfs to the random

vectors results in the data being represented, up to a multiplicative factor, in terms of ranks.

Spearman‟s rho can then be interpreted as the linear correlation of the probability trans-

formed variables (Embrechts et al., 2001, p.6). Formally, if we let 𝑋1 and 𝑋2 be two random

variables with marginal distribution functions 𝐹1 and 𝐹2, then Spearman‟s rho is defined as

𝜌𝑠 𝑋1 ,𝑋2 = 𝜌 𝐹1 𝑋1 ,𝐹2(𝑋2) , with 𝜌 being the linear correlation coefficient (cf. Definition

3.1).

According to McNeil et al. (2005, p.229), the unbiased estimator of Spearman‟s rho is given

by

𝑟𝑡 𝑋𝑖 ,𝑋𝑗 =

12

𝑛 𝑛2 − 1 rank 𝑋𝑡 ,𝑖 −

1

2 𝑛 + 1 rank 𝑋𝑡 ,𝑗 −

1

2 𝑛 + 1

𝑛

𝑡=1

(3.25)

where rank 𝑋𝑡 ,𝑖 denotes the rank of the 𝑡-th observation 𝑋𝑡 ,𝑖 among all 𝑛 observations.

Following Nelson (2006, p.167), it is further possible to express Spearman‟s rho in terms of

copulas. The following relation hereby holds:

𝜌𝑠 𝑋1 ,𝑋2 = 12 𝐶 𝑢1,𝑢2 𝑑𝑢1𝑑𝑢2

1

0

1

0

− 3 (3.26)

Although Kendall‟s tau and Spearman‟s rho both measure a dependence known as concor-

dance, they may yield quite different values for one and the same pair of random variables.

However, the following functional relationship always holds between the two rank correla-

tions (cf. Cherubini et al., 2004, p.103).

3

2𝜌𝜏 −

1

2≤ 𝜌𝑠 ≤

1

2+ 𝜌𝜏 −

1

2𝜌𝜏

2 for 𝜌𝜏 ≥ 0 (3.27)

1

2𝜌𝜏

2 + 𝜌𝜏 −1

2≤ 𝜌𝑠 ≤

3

2𝜌𝜏 +

1

2for 𝜌𝜏 < 0 (3.28)

The attainable region is illustrated in Figure 6, where also the estimated rank correlations of

the various return series analyzed in Section 4 are depicted. Note that the interval of attaina-

ble values for one measure is largest if the other measure takes a value of zero. In contrast,

if one measure is plus or minus one, the other measure will have the same value.


38

Figure 6 Relation between Kendall‟s tau and Spearman‟s rho. The

region of attainable combinations between Kendall‟s tau and Spear-

man‟s rho is given by the area bounded by the two bold lines that go

through (-1, -1) to (0, -0.5) to (1, 1) to (0, 0.5). The points marked

with an x show the combinations of 𝛒𝛕 and 𝛒𝐬 of the various return

series analyzed in Section 4.

From the definitions of Kendall‟s tau and Spearman‟s rho incorporating the copula function,

we can observe that both measures are increasing functions of the value of the respective

copula (cf. Embrechts et al., 2001, p.15). Moreover, the relation between the parameter of

the various copulas under consideration and Kendall‟s tau on one side and Spearman‟s rho

on the other side are depicted in Table 4. These relations provide one possible way of fitting

copulas to data, as we will see in Section 3.2.1.4.

-1

-0.5

0

0.5

1

-1 -0.5 0 0.5 1

Spe

arm

an's

rh

o

Kendall's tau

Table 4 Relation between the copula parameter and 𝝆𝝉, 𝝆𝒔, 𝝀𝒖 and 𝝀𝒍. Note that for Spearman‟s rho analytic expressions do

not exist for some copula families, i.e. it must be computed numerically. Also note that 𝐷1 𝜃 denotes the Debye function

𝐷1 𝜃 = 𝜃−1 𝑡/ exp 𝑡 − 1 dt𝜃

0. Based on: McNeil et al., 2005, p.222; Cherubini et al., 2004, p.126; Asche, 2004, p.49;

Jondeau, 2006, pp.246-252.

Copula

family Kendall’s tau 𝜌𝜏

Spearman’s rho

𝜌𝑠 Upper tail dependence 𝜆𝑢 Lower tail dependence 𝜆𝑙

𝐶ρ𝐺𝑎

2

𝜋arcsin 𝜌

6

𝜋arcsin

𝜌

2

1,0, 𝜌 = 1𝜌 < 1

1,0, 𝜌 = 1𝜌 < 1

𝐶ρ,νt

2

𝜋arcsin 𝜌

No closed form

expression 2𝑡𝑣+1 −

𝑣 + 1 1− 𝜌

1 + 𝜌 2𝑡𝑣+1 −

𝑣 + 1 1− 𝜌

1 + 𝜌

𝐶𝜃𝐺𝑢 1 − 𝜃−1

No closed form

expression 2− 21/𝜃 0

𝐶𝜃𝐶𝑙

𝜃

𝜃 + 2 N/A 0 2

−1/𝜃 ,0,

𝜃 > 0𝜃 ≤ 0

𝐶𝜃𝐹𝑟 1 −

12 𝐷1 𝜃 −𝐷2 𝜃

𝜃 1−

4 1− 𝐷1 𝜃

𝜃 0 0


39

3.1.3.3 Tail Dependence

Tail dependence refers to the dependence in the upper and lower tail of a bivariate distribu-

tion. The coefficient of tail dependence is hence a scalar measure of extremal dependence.

The following definition is based on limiting conditional probabilities of quantile exceedances

(cf. McNeil et al., 2005, pp.208-209). Basically, it refers to the probability that 𝑋2 exceeds its

𝑞-quantile conditional on 𝑋2 exceeding its 𝑞-quantile, considering the limiting cases of 𝑞.

Definition 3.14 (Coefficient of upper and lower tail dependence) The coefficients of upper

and lower tail dependence of two random variables 𝑋1 and 𝑋2 with cumulative distribution

functions 𝐹1 and 𝐹2 are given by

𝜆𝑢 𝑋1 ,𝑋2 = lim𝑞→1−

𝑃 𝑋2 > 𝐹2←(𝑞)|𝑋1 > 𝐹1

←(𝑞) (3.29)

𝜆𝑙 𝑋1,𝑋2 = lim𝑞→0+

𝑃 𝑋2 ≤ 𝐹2←(𝑞)|𝑋1 ≤ 𝐹1

←(𝑞) (3.30)

𝜆𝑢 and 𝜆𝑙 take values in the interval [0,1], where 𝜆𝑢 𝜆𝑙 ∈ (0,1] means that 𝑋1 and 𝑋2 show

extremal dependence in the upper (lower) tail while 𝜆𝑢 𝜆𝑙 = 0 indicates that the random va-

riables are asymptotically independent in the upper (lower) tail.

Like the rank correlations, the coefficient of tail dependence can be expressed in terms of

copulas, provided that 𝐹1 and 𝐹2 are continuous. Moreover, the following two formulas indi-

cate that the measure of tail dependence is fully characterized by the copula function (Che-

rubini et al., 2004, p.43).

𝜆𝑢 𝑋1,𝑋2 = lim

𝑞→1−

1 − 2𝑞 + 𝐶 𝑞, 𝑞

1 − 𝑞 (3.31)

𝜆𝑙 𝑋1,𝑋2 = lim

𝑞→0+

𝐶 𝑞, 𝑞

𝑞 (3.32)

The relation between the upper and lower tail dependence measures and the copula para-

meters is presented in Table 4. Basically, these relations show exactly what has already

been mentioned in Section 3.1.2.4 when the various copula families were discussed. The

Gaussian copula hence shows neither upper nor lower tail dependence, except if the random

vectors are perfectly correlated. The t-copula on the contrary exhibits upper and lower tail

dependence provided that 𝜌 > −1 (cf. Schmidt, 2007, p.27). The Gumbel copula has a non-

zero upper tail dependence coefficient and a lower coefficient of tail dependence of zero,

while it is vice versa for the Clayton copula. Finally, the Frank copula, similar to the Gaussian

copula, is asymptotically independent in both tails.


40

3.2 Fitting Copulas to Data

3.2.1 Estimation of Copulas

The starting point in the process of fitting a copula to a given data set is to assume that the

dependence structure can be adequately represented by a parametric copula family

𝐶𝜃 ∙ ;𝜃 ,𝜃 ∈ Θ, where Θ denotes the range of permissible values for the parameters. In this

sense, prior to the actual parameter estimation, we must first decide what copula family or

families may provide for an accurate representation of the dependence structure. Once this

has been decided, the problem of estimating a copula basically amounts to finding a value

for 𝜃 given a specific parametric copula family 𝐶𝜃 . However, we have to take into account

that the dependence structure of a set of random vectors 𝑿 also depends on the characteri-

zation of the marginals. Following Kpanzou (2008, p.23), we can represent the multivariate

cdf as

𝐹 𝒙;𝜶;𝜃 = 𝐶 𝐹1 𝑥1;𝛼1 ,… ,𝐹𝑑 𝑥𝑑 ;𝛼𝑑 ;𝜃 (3.33)

where 𝐶 denotes the copula function with parameter vector 𝜃 and 𝐹1 ,… ,𝐹𝑑 refer to the mar-

ginal distributions that are parameterized by 𝛼1 ,… ,𝛼𝑑 .

According to Fermanian and Scaillet (2003, p.3), estimating a multivariate distribution via

copulas has the advantage that the estimation process is divided into two separate problems.

Firstly, we must characterize the marginal distribution of each random variable by estimating

𝜶 and secondly, the dependence structure for these marginal distributions is derived by esti-

mating 𝜃. According to Karlqvist (2008, p.4) and Fermanian and Scaillet (2003, p.3) two main

estimation methods exist: fully parametric and semi-parametric. With regard to the fully pa-

rametric methods, Trivedi and Zimmer (2007, p.53) further differentiate between a full maxi-

mum likelihood approach (FML), which estimates all parameters simultaneously, and a two

step maximum likelihood method, where the marginals are estimated in a separate first step

prior to estimating the copula parameters. The latter is subsequently referred to as the infe-

rence method for margins (IFM). Cherubini et al. (2003, p.153) state that a main problem in

the attempt to fitting both the marginals and the copula function is that there exists a vast

number of combinations for the choice of the marginals and the copula family. By contrast,

the semi-parametric canonical maximum likelihood method (CML) simplifies the estimation

process as it makes use of the empirical distribution function to circumvent the problem of

estimating the marginal distributions parametrically. McNeil et al. (2005, p.229) further sug-

gest a method-of-moments procedure using estimates of the rank correlations Kendall‟s tau

and Spearman‟s rho to estimate 𝜃. Finally, the empirical copula provides a way to character-

ize the dependence structure in a completely nonparametric way.


41

3.2.1.1 Full Maximum Likelihood Approach (FML)

Using the canonical representation (cf. Cherubini et al., 2004, p.154), we can directly rewrite

expression (3.33) as

𝑓 𝒙;𝜶;𝜃 = 𝑐 𝐹1 𝑥1;𝛼1 ,… ,𝐹𝑑 𝑥𝑑 ;𝛼𝑑 ;𝜃 ∙ 𝑓𝑗 𝑥𝑗 ;𝛼𝑗

𝑑

𝑗=1

(3.34)

where 𝑐 denotes the copula density as shown in (3.8), while 𝑓 and 𝑓𝑗 refer to the multivariate

respectively the 𝑗-th univariate pdf of the random variables 𝑿. Taking the natural logarithm

and summing up for all observations in the sample 𝑿 = 𝑥1,𝑖 ,… , 𝑥𝑑 ,𝑖 𝑖=1

𝑛 the log likelihood

function is given by

ln 𝐿 𝜶;𝜃 = ln 𝑐 𝐹1 𝑥1,𝑖 ;𝛼1 ,… ,𝐹𝑑 𝑥𝑑 ,𝑖 ;𝛼𝑑 ;𝜃 +

𝑛

𝑖=1

ln𝑓𝑗 𝑥𝑗 ,𝑖 ;𝛼𝑗

𝑑

𝑗=1

𝑛

𝑖=1

(3.35)

The FML estimator is then obtained through the maximization of (3.35) with respect to both

the parameters of the marginals and the copula parameter, i.e.

𝜶 ,𝜃 𝐹𝑀𝐿

′= argmax

𝜶,𝜃 ln 𝐿 𝜶;𝜃 (3.36)

which represents the solution of

∂ln 𝐿

∂𝛼1,⋯ ,

∂ln 𝐿

∂𝛼𝑑,∂ln 𝐿

∂𝜃 = 𝟎′ (3.37)

Assuming that the usual regularity conditions hold, this maximum likelihood estimator is con-

sistent and asymptotically efficient as well as asymptotically normally distributed with the co-

variance matrix given by the inverse of Fisher‟s information matrix (cf. Cherubini et al., 2004,

p.154).

3.2.1.2 Inference Method for Margins (IFM)

The log likelihood function presented in equation (3.35) can actually be separated in two dis-

tinct terms

ln 𝐿𝑗 𝜶 = ln 𝑓𝑗 𝑥𝑗 ,𝑖 ;𝛼𝑗

𝑑

𝑗=1

𝑛

𝑖=1

(3.38)

and

ln 𝐿𝑐 𝜶 ;𝜃 = ln 𝑐 𝐹1 𝑥1,𝑖 ;𝛼 1 ,… ,𝐹𝑑 𝑥𝑑 ,𝑖 ;𝛼 𝑑 ;𝜃

𝑛

𝑖=1

(3.39)

Contrary to the FML approach, where all parameters are estimated in a single step, the IFM

estimator is derived in two steps:


42

1. In a first step, the parameters 𝜶 of the marginal distributions are estimated as

𝜶 = argmax𝜶

ln 𝐿𝑗 𝜶 (3.40)

2. In a second step, given the estimates 𝜶 derived in the first step, the copula parameter 𝜃 is

estimated as

𝜃 = argmax𝜃

ln 𝐿𝑐 𝜶 ;𝜃 (3.41)

The IFM estimator 𝜶 ,𝜃 𝐼𝐹𝑀

′ then represents the solution of (cf. Kpanzou, 2008, p.24)

∂ln 𝐿1

∂𝛼1,⋯ ,

∂ln 𝐿𝑑∂𝛼𝑑

,∂ln 𝐿𝑐∂𝜃

= 𝟎′ (3.42)

The different expressions in (3.37) and (3.42) show that the FML and IFM estimators are not

equivalent in general. Trivedi and Zimmer (2007, p.59) further point out that the IFM estima-

tor is less efficient than the FML estimator. According to Durrlemann et al. (2000, p.3), the

IFM estimator is nevertheless consistent and asymptotically normally distributed with the co-

variance matrix given by the inverse of Godambe‟s information matrix.

From a computational perspective, however, we can state that the IFM method is more effi-

cient than the FML approach. This is mainly an implication of separating the estimation pro-

cedure into two distinct steps, resulting in a reduction of possible combinations of marginals

and copula choice. Thus, the IFM method provides an attractive alternative to the FML me-

thod, especially for higher dimensions where the estimation procedure is computationally

very intensive (cf. Trivedi and Zimmer, 2007, p.58). Cherubini et al. (2004, p.158) further

state that the IFM method can also be of an ancillary use to the estimation via the FML me-

thod by providing starting values for the estimation via the latter.

It is important to see that the correct specification of the marginal distributions is of utmost

importance in the context of both the FML and IFM approach, as wrong assumptions about

the shape of the marginals may have a large impact on the estimation of the copula parame-

ter. The canonical maximum likelihood approach circumvents this problem by not imposing

any specific parametric form on the distribution of the marginals. Furthermore, the computa-

tional effort is further reduced compared to the previously mentioned methods, as the copula

parameter is the only one to be estimated.

3.2.1.3 Canonical Maximum Likelihood Approach (CML)

Basically, the CML method represents an altered, semi-parametric form of the IFM. Following

Cherubini et al. (2004, p.160), the two steps of the estimation process are now given as fol-

lows:


43

1. In a first step, the marginals are estimated in a nonparametric way using the empirical dis-

tribution function, i.e. 𝐹 𝑗 𝑥𝑗 ,𝑖 for 𝑗 = 1,… ,𝑑. Basically, the data 𝑥1,𝑖 ,… , 𝑥𝑑 ,𝑖 is transformed

into uniform variates 𝑢 1,𝑖 ,… , 𝑢 𝑑 ,𝑖 without specifying the marginals.

2. In a second step, the copula parameter 𝜃 is estimated as

𝜃 = argmax

𝜃 ln 𝑐 𝐹 1 𝑥1,𝑖 ,… ,𝐹 𝑑 𝑥𝑑 ,𝑖 ; 𝜃

𝑛

𝑖=1

(3.43)

3.2.1.4 Calibration with Kendall’s tau and Spearman’s rho

Durrlemann et al. (2000, p.7) and McNeil et al. (2005, p.229) further suggest an estimation

method where the copula parameter is chosen in such a way that they fit either of the two

rank correlation coefficients presented in Section 3.1.3. For this purpose, we must in a first

step calculate the empirical estimates of Kendall‟s tau or Spearman‟s rho according to equa-

tions (3.22) or (3.25), respectively. Subsequently, we can make use of the functional rela-

tionships between either Kendall‟s tau and 𝜃 or Spearman‟s rho and 𝜃 (cf. Table 4) in order

to calculate the copula parameter. It must be noted, however, that analytical solutions are

only available in some cases. In all other cases, a numerical derivation is inevitable.

For instance, the single parameter 𝜌 of the Gaussian copula can be calibrated either as

𝜌 = sin 𝜋𝑟𝜏2 (3.44)

or

𝜌 = 2 sin 𝜋𝑟𝑠6 (3.45)

With regard to the t-copula, the copula parameter 𝜌 can be likewise calibrated using expres-

sion (3.44). By contrast, the application of equation (3.45) would result in an error, as the

relation between 𝜌𝑠 and 𝜌 is not given in closed form (McNeil et al., 2005, p.230). Moreover,

we have to recognize that only 𝜌 can be calibrated using Kendall‟s tau, while the estimation

of the second t-copula parameter 𝜈 requires the use of a maximum likelihood approach.

3.2.1.5 Nonparametric Method

Another way to express the dependence structure of a set of random variables is to derive

the empirical copula, which was first introduced by Deheuvels (1979). Unlike the copulas

presented in Section 3.1.2.4, the empirical copula is of a nonparametric nature. Following

Fermanian (2003, p.6) and Genest et al. (2007, p.201), we write the empirical copula in the

𝑑-variate case as


44

𝐶𝑛 𝑢1 ,… ,𝑢𝑑 =1

𝑛 𝟏 𝐹𝑘 𝑥𝑖,𝑘 ≤ 𝑢𝑘

𝑑

𝑘=1

𝑛

𝑖=1

(3.46)

where 𝐹𝑘 denotes the empirical marginal cdf and 𝟏 is an indicator function taking the value 1

if the condition is satisfied and 0 otherwise.

The empirical copula is important in the context of copula selection, as it is an integral part of

many goodness of fit tests. The reasoning behind this is that we should choose the parame-

tric copula that is closest to the empirical copula.

3.2.2 Goodness of Fit Tests for Copulas

Once the parameters of a set of copulas have been estimated, the next step in the process

of fitting copulas to empirical data consists of selecting the right copula out of all the copulas

under consideration. In this context, it must be noted that no single test exists that allows us

to find the copula that best fits to the data under analysis. Rather, literature about copulas is

characterized by a variety of criteria and goodness of fit measures.

Given that we have followed a maximum likelihood approach in order to estimate the para-

meters of a set of copulas, a first obvious criterion is to compare the maximized value of the

log likelihood function ln 𝐿 𝜶 ;𝜃 8. Clearly, the model specification with the highest maximum

log likelihood value implies the best fit relative to all other model specifications. However, as

pointed out by Gartner (2007, p.82), simply comparing the maximum log likelihood values

may be problematic due to several reasons. Firstly, it is only sensible to compare log likelih-

ood values if the number of estimated parameters is the same across all maximum likelihood

functions. The reason for this lies in the fact that the addition of further parameters to a spe-

cific maximum likelihood function generally leads to a monotonic increase in the maximized

log likelihood value. Secondly, maximum log likelihood values only provide a useful criterion

if in the estimation process the same data is used for each model specification. Expanding or

reducing the data set arbitrarily may have a great impact on the maximized log likelihood

value. Finally, log likelihood values can only be compared if the same estimation method has

been applied with respect to all model specifications. Hence it is not advisable to compare,

for instance, the maximum log likelihood value of a Gaussian copula estimated with the FML

method with that of a t-copula estimated with the IFM or CML method. The last two issues

are not problematic once we are aware of them. However, with regard to the first issue, the

Akaike information criterion (AIC)

𝐴𝐼𝐶 = −2 ln 𝐿 𝜶 ;𝜃 + 2𝑘 (3.47)

8 Note that we use the notation ln 𝐿 𝜶 ;𝜃 for the maximized log likelihood value independently of whether it refers to (3.35),

(3.39) or (3.43) of the FML, IFM or CML method, respectively.


45

may present a more sophisticated alternative (cf. Chollete, 2008, p.61). Hereby, 𝑘 refers to

the number of estimated parameters and implies a penalty for overfitting the model. Never-

theless, Gartner (2007, p.85) states that the AIC suffers from the issue of favoring model

specifications with a larger number of parameters the higher the number of observations in

the sample. In this sense, the Bayesian information criterion (BIC)

𝐵𝐼𝐶 = −2ln 𝐿 𝜶 ;𝜃 + 𝑘 ln 𝑛 (3.48)

provides a further improvement, as it makes the penalty term depend on the number of ob-

servations 𝑛. Note that for both the AIC and BIC the model specification with the lowest value

implies the best fit.

According to Fischer et al. (2007, p.11), criteria based on the maximum log likelihood value

may lead to misleading conclusions. A second group of criteria measures the distance be-

tween the empirical copula (cf. (3.46)) and a set of hypothesized parametric copulas. Basi-

cally, the idea is to test whether a hypothesized distribution provides an adequate fit to the

observed data (Kole et al., 2006, p.6). In that case, the empirical cdf 𝐶𝑛 will almost surely

converge to the hypothesized cdf 𝐶𝜃 . Thus, a lower distance will imply a better fit to the ac-

tual data. The Kolmogorov-Smirnov distance

𝐷𝐾𝑆 = max𝑖 ,𝑗 𝐶𝑛 𝑢1,𝑖 ,𝑢2,𝑗 − 𝐶𝜃 𝑢1,𝑖 ,𝑢2,𝑗 (3.49)

provides a first choice for such a measure of distance. However, as 𝐷𝐾𝑆 puts large weight on

deviations in the centre of a distribution, the Anderson-Darling distance measure

𝐷𝐴𝐷 = max

𝑖 ,𝑗

𝐶𝑛 𝑢1,𝑖 ,𝑢2,𝑗 − 𝐶𝜃 𝑢1,𝑖 ,𝑢2,𝑗

𝐶𝜃 𝑢1,𝑖 ,𝑢2,𝑗 ∙ 1− 𝐶𝜃 𝑢1,𝑖 ,𝑢2,𝑗

(3.50)

may prove to be a better alternative, especially if the tails of a distribution are of increased

importance (cf. Kole et al., 2006, p.7). That is to say, compared to the Kolmogorov-Smirnov

test statistic, 𝐷𝐴𝐷 puts more weight on deviations in the tails. Note that the above formulas for

both 𝐷𝐾𝑆 and 𝐷𝐴𝐷 use the absolute difference between the empirical and the hypothesized

copula and not the squared deviation as it was originally proposed. Following Fischer (2003,

p.4), it may also be sensible to calculate an integrated version of the Anderson-Darling test

statistic to reduce the impact of outliers:

𝐷𝐼𝐴𝐷 = 𝐶𝑛 𝑢1,𝑖 ,𝑢2,𝑗 − 𝐶𝜃 𝑢1,𝑖 ,𝑢2,𝑗

2

𝐶𝜃 𝑢1,𝑖 ,𝑢2,𝑗 ∙ 1− 𝐶𝜃 𝑢1,𝑖 ,𝑢2,𝑗

𝑛

𝑗=1

𝑛

𝑖=1

(3.51)

According to Junker (2003, p.78), 𝐷𝐼𝐴𝐷 is more robust and provides a more complete picture

of the goodness of fit as, contrary to 𝐷𝐾𝑆 and 𝐷𝐴𝐷 , it does not solely rely on the largest devia-


46

tion in the sample. A further measure of distance is given by the Cramer-von-Mises statistic

(cf. for instance Omelka et al., 2009, p.9)

𝐷𝐶𝑣𝑀 = 𝐶𝑛 𝑢1,𝑖 ,𝑢2,𝑗 − 𝐶𝜃 𝑢1,𝑖 ,𝑢2,𝑗

2𝑛

𝑗=1

𝑛

𝑖=1

(3.52)

Alternatively, many studies about copulas employ the square root of 𝐷𝐶𝑣𝑀 as a distance

measure known under the notion L2-norm (cf. for instance Karlqvist, 2008, p.9 and Fischer,

2003, p.4).

It is important to see that the distributions of the test statistics are in general nonstandard

under the null hypothesis. Additionally, Kole et al. (2006, p.7) state that the estimation of the

parameters of the hypothesized distribution often takes place based on the same data set,

making it inevitable to run simulations in order to properly evaluate the test statistics.

Applying Copulas to the Electric Power Industry

47

4 APPLYING COPULAS TO THE ELECTRIC POWER INDUSTRY

In this section, the previously presented concepts of stochastic dependence are applied to

the electric power industry in order to analyze the dependence structure between various

pairs of return series of electricity prices. In particular, the analysis consists of a Phelix and

Swissix spot analysis, a Phelix Year Futures analysis and a further analysis involving syn-

thetic time series for Phelix futures contracts with various delivery periods and times to deli-

very. The main focus hereby lies on Phelix products, i.e. contracts involving the delivery of

electricity within the market area of Germany/Austria traded on a day-ahead basis at the EEX

(cf. Section 2.1.3). As it is evident from Table 2, Phelix spot and futures contracts play a cen-

tral role within the trading activities at the EEX. The inclusion of Swissix products into the

spot analysis further allows concluding on the dependence structure between the Ger-

man/Austrian and the Swiss market area. The Swissix is thereby not only closely related to

the Phelix with respect to its price movements, it is also constructed in an analogous way,

making comparisons between the two indexes straightforward. With respect to the futures

analysis, we must note that all futures products have a maturity in the sense that any specific

contract ceases to exist after a limited time of trading when it enters the delivery period (cf.

Table 1). The second part of the analysis will hence use data of Phelix Year Futures con-

tracts, as they allow for the longest range of observations. If we instead want to analyze the

interdependence between futures contracts with various delivery periods and times to deli-

very (and between spot and futures prices), we will have to calculate synthetic return series

of electricity with delivery one month, one quarter and one year ahead, two months, two

quarters and two years ahead, etc. The third part of the analysis addresses these kinds of

dependencies.

Within the spot analysis, the focus lies on the investigation of the dependence structure be-

tween Phelix and Swissix products on one side (cf. Section 4.2.3) and between base and

peak load contracts on the other side (Section 4.2.4). With respect to the Phelix Year Futures

contracts, the dependence structure between base and peak load products (Section 4.3.3)

and between 2010 and 2011 contracts (Section 4.3.4) are analyzed. The last part of the

analysis finally deals with the dependence structure between synthetic futures contracts of

the same delivery period but different time to delivery (Section 4.4.3), between synthetic re-

turn series of futures contracts with different delivery periods (Section 4.4.4) as well as be-

tween a spot return series and several synthetic futures contracts (Section 4.4.5).


48

4.1 General Remarks about the Estimation Procedure

After describing the composition of the data set and presenting the descriptive statistics for

the prices and log returns of the data set‟s various time series, the following steps are ap-

plied in all subsequent analyses:

1. Estimation of the marginal distributions9

In Section 3, we have learnt that the representation of a multivariate distribution via copulas

has the advantage that the marginal distributions of the individual components and the de-

pendence structure between these components can be modeled separately. In accordance

with this, our first action is to address the issue of how the marginals can be adequately cha-

racterized both parametrically and non-parametrically. It must be noted that, with regard to

the parametric way of characterizing the marginals, the usual starting point is to assume a

normal distribution for the marginal behavior of return data. However, the Jarque-Bera test

statistics (cf. for instance Table 5) will reveal that the return series of the various data sets

under examination reject the null hypothesis of a normal distribution with high confidence.

The Kolmogorov-Smirnov test statistics of fitting a normal distribution to the return series will

in general confirm this observation. The reason for the distinct behavior can mainly be as-

cribed to kurtosis taking considerably higher values than implied by a normal distribution. To

account for this, a first step may be to model the marginal cdfs by a t-distribution, which

shows fatter tails compared to a normal distribution. However, the descriptive statistics will

also reveal that most return series exhibit values for skewness that are considerably distinct

from zero. Considering that the t-distribution is a symmetric distribution and as such unable

to capture positive or negative skewness, the parametric model for the marginals can further

be enhanced by relying on a non-central (or skewed) t-distribution, as for instance proposed

by Cherubini et al. (2004, p.159). In particular, the applied model is characterized by four

parameters, namely a location (lo), scale (sc), shape and degree of freedom (df) parameter10.

Furthermore, Kolmogorov-Smirnov test statistics will confirm that the non-parametric charac-

terization of the marginals via the empirical distribution function provides an adequate alter-

native for the parametric representation of the true distribution for all return series under con-

sideration.

2. Probability transformation of the observed random variables11

Having recognized that we can rely on the non-central t-distribution to parametrically charac-

terize the marginal distributions, the observed random variables of the various return series

9 The respective results (cf. for instance Table 6 and Figure A2) were obtained through calculations with the statistical software

R by applying Code 1 and Code 2 presented in Appendix C. 10 For more details on the concrete model the reader is referred to the corresponding software documentation. 11

cf. Code 3


49

are transformed for the purpose of subsequently being able to estimate the copula parame-

ters. In particular, we will transform the 𝑛 random variables of the 𝑗-th return series 𝑥𝑗 ,1 ,… , 𝑥𝑗 ,𝑛

into corresponding uniformly distributed random variables 𝑢𝑗 ,1nct,… ,𝑢𝑗 ,𝑛

nct based on the probabil-

ity transformation 𝐹𝑗nct 𝑥𝑗 ,𝑖 ;𝛼 𝑗

nct = 𝑢𝑗 ,𝑖nct where 𝐹𝑗

nct corresponds to the 𝑗-th marginal cdf of the

non-central t-distribution with the respective parameters 𝛼 𝑗nct as estimated in the first step.

Based on the insight that the marginals can also be modelled non-parametrically, we calcu-

late a second set of uniformly distributed random variables based on the empirical distribu-

tion function. In particular, the 𝑛 random variables of the 𝑗-th return series 𝑥𝑗 ,1 ,… , 𝑥𝑗 ,𝑛 are now

transformed based on the probability transformation 𝐹 𝑗 𝑥𝑗 ,𝑖 = 𝑢 𝑗 ,𝑖 , where 𝐹 𝑗 denotes the em-

pirical distribution function of the 𝑗-th component.

3. Estimation of the copula parameters12

Once the work with the univariate data is accomplished, the focus shifts towards the task of

modeling the bivariate dependence structures. For this purpose, Gaussian, t-, Gumbel, Clay-

ton and Frank copulas are fitted to the various pairs of return series under consideration. On

one side, the estimation of the respective copula parameters is based on the first set of the

probability transformed random variables, i.e. 𝑢𝑗 ,𝑖nct. This basically corresponds to passing the

second step of the IFM method, solving expression (3.41). Additionally, the copula parame-

ters are estimated based on the second set of the probability transformed random variables

𝑢 𝑗 ,𝑖 . In fact, the estimation procedure then corresponds to the CML method and implies solv-

ing expression (3.43). As a third alternative, it would also be possible to calculate the copula

parameters based on the estimates of Kendall‟s tau and Spearman‟s rho as proposed in

Section 3.2.1.4. Calculations for the Gaussian, Gumbel and Clayton copula have revealed,

however, that the resulting copula parameter estimates are rather distinct from the ones

based on the two maximum likelihood approaches. Furthermore, for the t-copula (probably

the most important copula in the present analysis) the second parameter cannot be esti-

mated based on the rank correlations. As a consequence, we will subsequently solely focus

on the estimation of the copula parameters via the IFM and CML method.

4. Goodness of fit testing13

Beside the parameter estimates, the subsequent sub-analyses also provide values for the

test statistics presented in Section 3.2.2 in order to evaluate the goodness of fit of the various

copula model specifications. The goodness of fit measures within the first group (ln 𝐿 𝜶 ;𝜃 ,

AIC and BIC) are thereby closely related to the maximum likelihood procedure applied within

the context of estimating the copula parameters. Consequently, the corresponding values are

12

cf. Code 4 13 cf. Code 5


50

sometimes directly provided by statistical software, together with the parameter estimates.

Alternatively, and as done in our case, ln 𝐿 𝜶 ;𝜃 can be calculated as the sum of the natural

logarithms of the copula densities evaluated at each pair of observations using the respective

copula parameter estimates. By contrast, the second group of goodness of fit measures

(𝐷𝐾𝑆, 𝐷𝐴𝐷 , 𝐷𝐼𝐴𝐷 and 𝐷𝐶𝑣𝑀) is based on the distance between the empirical copula (cf. expres-

sion 3.46) and the cdf of the various fitted copula families. The values for the cdf are thereby

obtained analogously to the pdf, i.e. as a function of each pair of observations using the re-

spective copula parameter estimates.

5. Generating a bivariate random sample of returns14

The parameter estimates of the marginal distributions can be used in conjunction with a spe-

cific copula model to generate a bivariate sample of random returns (cf. for instance Figure

A3). Clearly, this procedure can be seen as a main result of the analysis, as it allows us to

reproduce the dependence structure for an arbitrary amount of randomly generated vari-

ables. Moreover, when compared with the empirical point cloud of the historical returns, the

respective point clouds may present a further way to decide on the goodness of fit of the var-

ious copula model specifications. Especially in the cases where the goodness of fit measures

are contradictory, these figures may provide additional insight.

4.2 Phelix and Swissix Spot Analysis

4.2.1 Data Set and Descriptive Statistics

The spot data set includes the price respectively return series of various Phelix and Swissix

spot contracts. In the subsequent analysis, we will refer to these series by making use of the

following notions:

- Phelix Hourly (PhH): Historical time series of hourly prices of electricity traded on a

day-ahead basis with delivery in the German/Austrian market area on the respective

hour of the next day.

- Swissix Hourly (SwH): Historical time series of hourly prices of electricity traded on a

day-ahead basis with delivery in the Swiss market area on the respective hour of the

next day.

- Phelix Day Base (PhB): Historical time series of daily prices of electricity traded on a

day-ahead basis with delivery in the German/Austrian market area during the base

hours of the next day. Corresponds to the daily calculated arithmetic average of the

auction prices of the hours 1 to 24 of the Phelix Hourly.

14 cf. Code 7


51

- Phelix Day Peak (PhP): Historical time series of daily prices of electricity traded on a

day-ahead basis with delivery in the German/Austrian market area during the peak

hours of the next day. Corresponds to the daily calculated arithmetic average of the

auction prices of the hours 9 to 20 of the Phelix Hourly.

- Swissix Day Base (SwB): Historical time series of daily prices of electricity traded on

a day-ahead basis with delivery in the Swiss market area during the base hours of the

next day. Corresponds to the daily calculated arithmetic average of the auction prices

of the hours 1 to 24 of the Phelix Hourly.

- Swissix Day Peak (SwP): Historical time series of daily prices of electricity traded on

a day-ahead basis with delivery in the Swiss market area during the peak hours of the

next day. Corresponds to the daily calculated arithmetic average of the auction prices

of the hours 9 to 20 of the Phelix Hourly.

The sample period of the Phelix and Swissix spot data set covers three years, starting 1

January 2007 and ending 31 December 2009. The main reason for choosing 1 January 2007

as the starting point of the analysis is the fact that Swissix based products have only been

traded since mid-December 2006. For consistency reasons, the sample period has been

decided to be the same for the other time series as well. Obviously though, the number of

observations differs for daily and hourly data. Base and peak spot prices constitute daily av-

erages of the corresponding hourly prices, resulting in a ratio of 24 to 1 with regard to the

number of observations.

Due to the specific characteristics of electricity spot prices, some adjustments in the data set

are indispensable. Firstly, we have seen in Section 2.2.1.1 that electricity spot prices gener-

ally show strong seasonal patterns. In order to gain insight into the seasonalities of the hourly

and daily spot products under analysis, Figure 7 visualizes the variations in the average spot

price of the Phelix and Swissix time series during (a) the hours of the day, (b) the days of the

week and (c) the months of the year. Firstly, and despite the different time period under con-

sideration15, the observation of Blöchlinger (2008, p.7), stating that electricity prices show a

steady increase from 5 a.m. onwards and do not continually decrease before 8 p.m., is as-

serted by Figure 7 (a). Furthermore, we can observe that the average prices of the peak load

hours H9 to H20 clearly lie above those of the off peak hours H21 to H8. The Phelix and

Swissix data thereby shows a similar pattern, with the price averages of the Swissix being

constantly above the Phelix counterparts. It is noticeable, however, that the difference is lar-

ger at certain hours, especially during the afternoon and during the night.

15

Blöchinger (2008, p.7) analyzes hourly and daily Phelix data for a period ranging from 1 January 2001 to 30 April 2007.


52

Figure 7 Seasonalities during the day, week and year. Figure (a) visualizes the hourly averages of

the Phelix Hourly and Swissix Hourly time series. Note that the hourly price average is thereby cal-

culated over the entire sample. A further differentiation through the separate calculation of the aver-

ages of certain days of the week and during specific months would probably allow for further inter-

pretations. Figure (b) depicts the averages of the Phelix Day Base and the Swissix Day Base for

each day of the week during the entire sample period. Figure (c) finally shows the Phelix Day Base

and Swissix Day Base averages for the various months of the year. All prices are in EUR/MWh.

0.00

10.00

20.00

30.00

40.00

50.00

60.00

70.00

80.00

H1 H2 H3 H4 H5 H6 H7 H8 H9 H10H11H12H13H14H15H16H17H18H19H20H21H22H23H24

(a) Seasonalities during the day

Phelix Swissix

0.00

10.00

20.00

30.00

40.00

50.00

60.00

70.00

Mon Tue Wed Thu Fri Sat Sun

(b) Seasonalities during the week

Phelix Swissix

0.00

10.00

20.00

30.00

40.00

50.00

60.00

70.00

80.00

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

(c) Seasonalities during the year

Phelix Swissix


53

Figure 7 (b) reinforces the fact that electricity prices are on average lower on Saturdays and

Sundays, while they are relatively constant during the other days of the week. With respect to

the seasonal patterns during the year, Figure 7 (c) reveals that Phelix prices are above aver-

age between September and November. Apart from this, no clear seasonal pattern is ob-

servable for the Phelix. The Swissix on the other side evidently shows higher price averages

during the winter months relative to the summer months. This contrasts with the observation

of Giger (2008, p.16) stating that the seasonal variation during the year is relatively low in

Switzerland. These seasonalities during the day, during the week and, to a less extent, dur-

ing the year have a substantial impact on the variability in electricity prices. In order to adjust

for a fair amount of these non-random effects, the returns of the daily time series are not

simply calculated from day to day, but rather from Monday to Monday in the following week,

Tuesday to Tuesday in the following week and so on. Similarly, for hourly data the return is

calculated from, say, Monday 1 a.m. to Monday 1 a.m. in the following week instead of Mon-

day 1 a.m. to Monday 2 a.m. More generally, with 𝑌𝑡 denoting the electricity spot price ob-

served at time t, the following deseasonalized expression of the log return is applied:

𝑋𝑡 = ln

𝑌𝑡𝑌𝑡−𝑝

(4.1)

where 𝑝 = 7 for daily spot data and 𝑝 = 168 for hourly spot data. This simple but effective

procedure allows us to adjust for a major amount of the hourly and daily variation caused by

seasonal effects. Small incorrections originate from this method not distinguishing between

holidays and workdays, thus ignoring the fact that holidays show load and price patterns

similar to Sundays.

Figure 8 Autocorrelation function for daily log returns of spot products. Figure (a) presents the autocorrelation

function of the Phelix Day Base, Figure (b) illustrates the autocorrelation function of the Swissix Day Base. Note

that the return series without deseasonalization (i.e. p = 1 in expression (4.1)) show strong spikes every seventh

lag. The autocorrelation function of the return series with deseasonalization (i.e. p = 7 in expression (4.1)) on the

other side is much less pronounced.

-0.5

0

0.5

1

0 10 20 30 40 50 60 70 80 90 100

(a) Autocorrelations of the Phelix Day Base

Phelix Day Base (without deseasonalization)

Phelix Day Base (with deseasonalization)

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 10 20 30 40 50 60 70 80 90 100

(b) Autocorrelations of the Swissix Day Base

Swissix Day Base (without deseasonalization)

Swissix Day Base (with deseasonalization)


54

The analysis of the autocorrelation function of the Phelix Day Base respectively Swissix Day

Base log returns (without deseasonalization) provides additional evidence for the existence

of seasonalities during the week (cf. Figure 8). However, Figure 8 also reveals that, when

using the above proposed method of deseasonalizing the returns, the autocorrelations of the

log returns are much less pronounced.

Secondly, prices equal to zero are not uncommon in the spot segment of the electric power

industry and some products even allow for negative prices, for instance those based on the

Phelix (cf. Table 1). The above expression to derive the returns is, however, incompatible

with strictly non-positive values, for which the natural logarithm is not defined. For the subse-

quent analysis, we will hence eliminate all data points from the return vectors where a price

below two Euros appears either in the nominator or denominator of the logarithmic function

when calculating the returns. Consequently, the series are filtered of spikes to the downside,

involving only the random Wiener process components.

A further remark concerns the invariance of copulas and the rank correlation coefficients with

respect to strictly increasing transformations of the marginals (cf. Proposition 3.7). Basically,

the relation between the log return (4.1) and the simple gross return

1 + 𝑅𝑡 =

𝑌𝑡𝑌𝑡−𝑝

(4.2)

is of a logarithmic nature (cf. Ruppert, 2004, p.77). Since the logarithm is a monotonically

increasing transformation, the results should not depend on whether we use log returns or

gross returns, i.e. the dependence structure should be the same. A similar argumentation

holds for the simple net return Rt, with the only difference to the simple gross return being an

absolute term equal to one. The invariance is most easily demonstrated by calculating the

ranks of the individual observations under all three return specifications. The fact that we

obtain the same ranks for each individual observation implies the same values for the rank

correlations Kendall‟s tau and Spearman‟s rho. This in turn implies that the copula parame-

ters are the same for all three return specifications, given that the relations in Table 4 hold.

Moreover, it is important to note that the log return shown in expression (4.1) corresponds to

the difference in the logarithmic prices, i.e. 𝑋𝑡 = ln 𝑌𝑡 − ln 𝑌𝑡−𝑝 . However, the use of the

absolute price difference 𝑌𝑡 − 𝑌𝑡−𝑝 leads to totally different ranks and consequently also to a

different dependence structure, as it does not constitute a monotonically increasing transfor-

mation of the log return. The same applies to the analysis of the price 𝑌𝑡 itself or its logarithm

ln 𝑌𝑡 .


55

Table 5 Descriptive statistics of the Phelix and Swissix spot data set. Panel A represents the

statistics for the prices, Panel B for the log returns of the various time series. All prices are in

EUR/MWh.

Panel A: Prices

Phelix

Hourly

Swissix

Hourly

Phelix Day

Base

Phelix Day

Peak

Swissix

Day Base

Swissix

Day Peak

# Obs. 26301 26301 1096 1096 1096 1096

Min -500.02 0.06 -35.57 6.76 16.45 17.99

Max 821.90 553.88 158.97 248.38 179.90 265.84

Mean 47.55 56.11 47.55 58.35 56.11 67.29

Median 41.59 50.85 41.44 49.74 52.94 63.95

St. Dev. 29.54 30.58 21.32 28.27 23.53 30.20

Variance 872.73 934.94 454.34 799.46 553.86 912.14

Skewness 2.71 1.87 0.94 1.49 0.65 1.13

Kurtosis 47.62 14.83 1.33 4.20 0.53 3.42

JB 2517215 256356 244 1211 90 769

P-value 0.000 0.000 0.000 0.000 0.000 0.000

Panel B: Log returns

Phelix

Hourly

Swissix

Hourly

Phelix Day

Base

Phelix Day

Peak

Swissix

Day Base

Swissix

Day Peak

# Obs. 25645 25645 1086 1086 1086 1086

Min -3.80 -3.01 -2.08 -2.38 -1.03 -1.14

Max 3.07 2.12 1.86 2.04 0.96 1.29

Mean 0.0000 -0.0011 0.0006 0.0010 -0.0016 -0.0014

Median 0.0036 0.0008 0.0035 0.0082 -0.0009 0.0041

St. Dev. 0.3996 0.2841 0.2923 0.3033 0.2014 0.2223

Variance 0.1596 0.0807 0.0854 0.0920 0.0406 0.0494

Skewness -0.2778 -0.2958 -0.5277 -0.4864 -0.1478 -0.0488

Kurtosis 10.22 8.76 6.64 8.45 2.68 4.03

JB 112044 82396 2047 3277 328 734

P-value 0.000 0.000 0.000 0.000 0.000 0.000

Table 5 represents the descriptive statistics of the Phelix and Swissix time series under con-

sideration. The figures in Panel A hereby refer to the descriptive statistics of the historically

observed prices without any adjustments. The time series consist of 26301 (hourly data) re-

spectively 1096 (daily data) observations. The PhH series (and logically also the PhB) has a

mean of 47.55 EUR/MWh, while the mean of the SwH (and hence of the SwB) equals 56.11

EUR/MWh. By contrast, the PhP and SwP series exhibit a mean of 58.35 EUR/MWh and

67.29 EUR/MWh, respectively. These figures indicate on one side that the prices of the

Swissix series lie consistently above those of the Phelix series, on average close to 20%.

According to CKW (2009, p.34), these differences can be ascribed to the lower market liquid-

ity as well as the costs of import due to limitations in the cross-border transmission between

Switzerland and Germany. Furthermore, the statistics reveal that the daily peak load series

are above the daily base load ones. This is an obvious implication of the fact that PhP and


56

SwP are the arithmetic average of the peak hours, which are generally higher than the off-

peak hours due to the typically observed seasonalities. Evidently, the maximum values of the

daily series lie three to four times above the reported mean values. For PhH and SwH, this

multiple even rises above ten, with the corresponding maximum prices reaching stunning

821.90 EUR/MWh and 553.88 EUR/MWh, respectively. Regarding the minimum prices, note

that the Phelix based products can attain prices below zero, as shown by PhH with -500.02

EUR/MWh and PhB with -35.57 EUR/MWh. SwH and its averages, on the other side, are

bounded by a minimum price equal to zero. The values for the standard deviation and va-

riance reveal that the prices of the hourly series compared with the daily averages and the

daily peak load series compared with the daily base load ones exhibit a higher volatility. Fur-

thermore, the Swissix series show in general values for the volatility that are slightly higher

than those of the corresponding Phelix series. The statistics for skewness and (excess) kur-

tosis indicate towards a positively skewed, leptokurtic distribution of the prices. This observa-

tion is asserted by the median values, which are consistently below the corresponding mean

values. A possible explanation for the longer right tail may lie in the existence of relatively

few, extremely high observations due to the occurrence of spikes. This would also explain

why skewness and kurtosis are higher for the hourly series compared to the daily series,

where extreme prices are averaged out to some extent in the latter. Further note that skew-

ness and kurtosis are lower for Swissix than for Phelix contracts and higher for peak load

than for base load contracts. The Jarque-Bera test statistics strongly reject the null hypothe-

sis of a normal distribution for all time series under consideration, with the rounded p-values

being 0.000. This observation is not unexpected, as we usually assume that not the prices

but rather the returns follow a normal distribution, at least approximately. This would in turn

imply that the prices underlie a lognormal distribution, which is characterized by a longer right

tail, similar to the above observation.

Panel B represents the descriptive statistics of the log returns calculated in accordance with

equation (4.1) and applying the above mentioned adjustments. The number of observations

has thereby shrunk imperceptibly to 25645 (hourly data) respectively 1086 (daily data). The

mean returns are slightly positive (negative) for the Phelix (Swissix) contracts and overall

very close to zero. The maximum returns lie between 0.96 (for SwB) and 3.07 (for PhH), the

minimum returns between -1.03 (for SwB) and -3.80 (for PhH). The returns are hence simi-

larly pronounced to the up as to the down side. With regard to the standard deviation of the

log returns, we can state that the values of the two daily Phelix indexes are close to 0.3,

those for Swissix are around 0.2. This amounts to an annualized standard deviation of rough-

ly 580% and 380%, respectively. Contrary to the prices, the returns are hence less volatile

for the Swissix contracts as compared with the Phelix ones. However, it still holds that the

volatility is higher for the hourly series compared to the daily series. The figures for (excess)


57

kurtosis again hint towards fat tails due to extreme observation in both directions. Kurtosis is

hereby again higher for hourly (vs. daily), peak load (vs. base load) and Phelix (vs. Swissix)

return series. Contrary to the above observation, skewness is negative for all return series,

indicating at a longer left tail. The test statistics of the Jarque-Bera test for normality lead to

similar results as above. Unexpectedly though, for the daily series the rejection of the null

hypothesis is even stronger than for the corresponding time series of prices.

Figure A1 (cf. Appendix A) illustrates the historical price and return movements of the various

spot series under analysis, asserting many of the above observations.

4.2.2 Estimation of the Marginals

Table 6 presents the parameter estimates and Kolmogorov-Smirnov test statistics (including

p-values) for various specifications of the marginal distributions of the return series of the

spot data set. Column 1 thereby shows the parameter estimates of fitting a normal distribu-

tion to the data. Basically, these estimates correspond to the values of the mean and stan-

dard deviation listed in Table 5. Similar to the Jarque-Bera test statistics, the Kolmogorov-

Smirnov test rejects the null hypothesis of a normal distribution for all return series with a

confidence level of 99.9% or more. Column 3 exhibits the estimated parameters of the non-

central t-distribution. The corresponding Kolmogorov-Smirnov test statistics reveal that the

underlying null distribution cannot be rejected for any of the return series with a confidence

level of 90% or more, as all p-values are above 0.102. Further note that the p-values of the

daily return series are considerably higher than those of the hourly data, taking values be-

tween 0.805 and 0.985. The non-central t-distribution hence provides a reasonable fit for the

marginal distributions of the spot return series. Additionally, the present analysis involves the

empirical marginal distribution function. Clearly, as the latter represents a non-parametric

estimation method, Column 5 does not provide any parameter estimates. It may nevertheless

be important to employ a Kolmogorov-Smirnov test in order to identify any misspecifications.

All test statistics thereby have a rounded p-value of 1.000, indicating that the empirical distri-

bution function provides an adequate fit. Figure A2 (cf. Appendix A) further illustrates the

histogram, the density of a fitted normal and non-central t-distribution as well as the empirical

density function for all return series of the Phelix and Swissix spot data set. Evidently, the

fitted normal distribution only provides a very crude approximation of the true distribution.

The non-central t-distribution on the other hand is in all cases nearly identical to the empirical

distribution.


58

Table 6 Marginal parameter estimates for the Phelix and Swissix spot data set. Besides the estimated parameters of fitting a

normal and a non-central t- distribution to the various return series of the data set, the table also provides Kolmogorov-

Smirnov test statistics (with p-values) for the normal, non-central t- and empirical distribution.

Series Normal distribution Non-central t-distribution Empirical distribution

𝛼 𝐷𝐾𝑆 𝛼 𝐷𝐾𝑆 𝛼 𝐷𝐾𝑆

Phelix Hourly μ = 0.0000

σ = 0.3996

0.1111

(0.000)

lo = 0.0118

sc = 0.1908

sh = -0.0448

df = 2.0989

0.0060

(0.322)

N/A 0.0010

(1.000)

Swissix Hourly μ = -0.0011

σ = 0.2841

0.0980

(0.000)

lo = 0.0129

sc = 0.1536

sh = -0.0772

df = 2.3712

0.0076

(0.102)

N/A 0.0005

(1.000)

Phelix Day

Base

μ = 0.0006

σ = 0.2923

0.0740

(0.000)

lo = 0.0214

sc = 0.1937

sh = -0.0941

df = 3.3749

0.0150

(0.968)

N/A 0.0009

(1.000)

Phelix Day

Peak

μ = 0.0010

σ = 0.3033

0.0911

(0.000)

lo = 0.0219

sc = 0.1822

sh = -0.0991

df = 2.8236

0.0139

(0.985)

N/A 0.0009

(1.000)

Swissix Day

Base

μ = -0.0016

σ = 0.2014

0.0608

(0.001)

lo = 0.0227

sc = 0.1454

sh = -0.1683

df = 3.8001

0.0166

(0.926)

N/A 0.0009

(1.000)

Swissix Day

Peak

μ = -0.0014

σ = 0.2223

0.0799

(0.000)

lo = 0.0169

sc = 0.1392

sh = -0.1219

df = 2.8386

0.0195

(0.805)

N/A 0.0018

(1.000)

4.2.3 Phelix vs. Swissix

The values for Pearson‟s linear correlation, Kendall‟s tau and Spearman‟s rho between the

Phelix Day Base and Swissix Day Base return series are 0.612, 0.427 and 0.584, respective-

ly. For the relation between the Phelix Day Peak and Swissix Day Peak, the corresponding

values amount to 0.671, 0.445 and 0.608. Furthermore, the Phelix Hourly and Swissix Hourly

return series have correlation coefficients of 0.433, 0.310 and 0.435. These figures indicate

towards the conclusion that a moderately high dependence exists between PhB and SwB as

well as between PhP and SwP, with the dependence of the latter being slightly more pro-

nounced. By contrast, the strength of dependence between PhH and SwH is considerably

lower.

In the following, we will apply copula models to the three above mentioned combinations of

Phelix and Swissix spot return series in order to analyze the respective dependence struc-

tures more closely. The corresponding results are presented in Table 7, revealing both the


59

copula parameter estimates and the various goodness of fit measures for the Gaussian, t-,

Gumbel, Clayton and Frank copula families. Note that Table 7 thereby comprises the figures

based on the historical returns being transformed via the cdf of a fitted non-central t-

distribution. By contrast, Table B1 (cf. Appendix B) presents the corresponding results for a

probability transformation based on the empirical distribution function.

Panel A of Table 7 represents the parameter estimates for the various copula families under

consideration with regard to the dependence structure between the Phelix Day Base and the

Swissix Day Base return series. The Gaussian copula has an estimated parameter value of

0.6066, which is somewhere between Pearson‟s and Spearman‟s correlation coefficient

mentioned above. The t-copula has a similar value for the first parameter 𝜌 (0.6189) and a

relatively low value for ν (3.8437). This indicates that the estimated t-copula is rather distinct

from the Gaussian copula, as the t-copula converges to the Gaussian copula for ν → ∞. The

Table 7 Copula parameter estimates (Phelix vs. Swissix) for the Phelix and Swissix spot return series based on non-central

t-distributed marginals. Besides the parameter estimates of fitting a Gaussian, t-, Gumbel, Clayton and Frank copula to the

various pairs of return series, the table also provides a variety of corresponding goodness of fit measures.

Panel A: Phelix Day Base vs. Swissix Day Base

𝜃 ln 𝐿 𝜶 ;𝜃 𝐴𝐼𝐶 𝐵𝐼𝐶 𝐷𝐾𝑆 𝐷𝐴𝐷 𝐷𝐼𝐴𝐷 𝐷𝐶𝑣𝑀

𝐶ρ𝐺𝑎 0.6066 249.17 -496.35 -491.36 0.0248 0.0716 0.4819 0.0648

𝐶ρ,𝑣t 0.6189, 3.8437 283.89 -563.78 -553.80 0.0193 0.0709 0.3603 0.0468

𝐶𝜃𝐺𝑢 1.7080 261.71 -521.43 -516.44 0.0246 0.1290 0.9115 0.0611

𝐶𝜃𝐶𝑙 0.9917 197.47 -392.93 -387.94 0.0593 0.1228 2.8305 0.5704

𝐶𝜃𝐹𝑟 4.6813 241.14 -480.29 -475.30 0.0252 0.1858 1.3894 0.1001

Panel B: Phelix Day Peak vs. Swissix Day Peak


𝐶ρ𝐺𝑎 0.6487 296.52 -591.04 -586.05 0.0296 0.0769 0.7501 0.1051

𝐶ρ,𝑣t 0.6439, 3.5728 327.27 -650.53 -640.55 0.0258 0.0576 0.5248 0.0823

𝐶𝜃𝐺𝑢 1.8014 311.21 -620.43 -615.44 0.0187 0.1385 1.0770 0.0548

𝐶𝜃𝐶𝑙 1.1006 226.03 -450.07 -445.08 0.0644 0.1381 3.1058 0.6452

𝐶𝜃𝐹𝑟 4.9713 267.17 -532.35 -527.36 0.0339 0.2237 2.2904 0.1867

Panel C: Phelix Hourly vs. Swissix Hourly


𝐶ρ𝐺𝑎 0.4487 2883.33 -5764.66 -5759.67 0.1738 0.3482 375.41 70.97

𝐶ρ,𝑣t 0.4671, 3.2290 3979.36 -7954.73 -7944.74 0.1748 0.3501 388.54 70.72

𝐶𝜃𝐺𝑢 1.4355 3278.92 -6555.84 -6550.85 0.1735 0.3476 368.03 67.90

𝐶𝜃𝐶𝑙 0.6650 2538.03 -5074.06 -5069.07 0.1759 0.3525 497.53 95.21

𝐶𝜃𝐹𝑟 3.1742 2924.29 -5846.57 -5841.58 0.1746 0.3497 308.66 58.79


60

parameter estimate of the Gumbel copula equals 1.7080 and hence lies between indepen-

dence and comonotonicity. The same applies to the Clayton copula, with the estimated pa-

rameter value being close to one (𝜃 = 0.9917). Finally, the Frank copula has a parameter

value equal to 4.6813. All goodness of fit measures hint towards the conclusion that the t-

copula best represents the dependence structure between the PhB and the SwB return se-

ries. With respect to the alternative estimation via the CML method, Table B1 shows that the

estimated copula parameters are in proximity of the above mentioned. Furthermore, all

goodness of fit measures likewise favor the t-copula. Figure A3 (a) illustrates the empirical

point cloud of the PhB and SwB historical returns and provides a comparison with bivariate

samples of randomly generated returns of equal size in accordance with the non-central t-

distributed marginals and the various copula model specifications. The point clouds for the

Gumbel (d) and the Clayton (e) copula model hereby reveal that the asymmetric nature of

these model specifications with respect to tail dependence is not reconcilable with the rather

symmetric dependence structure of the historical PhB and SwB returns. Furthermore, the

symmetric Gaussian (b) and Frank (f) copula are too spread out towards both joint tails as a

result of the inexistent tail dependence. The point cloud of the t-copula (c), on the other side,

seems to provide a rather good fit in comparison with the empirical point cloud.

With respect to the dependence structure between the Phelix Day Peak and the Swissix Day

Peak return series, Panel B reveals that all copula parameters (except ν) are a fair amount

higher than their counterparts depicted in Panel A. This observation allows us to interpret

that the dependence among the peak products is higher than among the base products. Re-

garding the various goodness of fit measures, we notice that all test statistics suggest that

the t-copula provides the best fit, except 𝐷𝐾𝑆 and 𝐷𝐶𝑣𝑀, both of which prefer the Gumbel cop-

ula followed by the t-copula. The results for the CML method presented in Table B1 by and

large correspond to those mentioned above. The only distinction lies in the observation that

𝐷𝐶𝑣𝑀 also favors the t-copula. Basically, the point clouds presented in Figure A4 lie in accor-

dance with these results, i.e. the bivariate return sample generated from the t-copula (c) is

able to capture the empirical dependence structure (a) relatively well. However, the latter

also reveals a slightly more spread out lower tail, which would explain why two of the dis-

tance measures tend towards the Gumbel copula model.

Panel C finally lists the copula parameters for the potential copula models of the dependence

structure between the Phelix Hourly and the Swissix Hourly return series. As a first observa-

tion, we can state that all parameter values are considerably below the ones presented in

Panel A and B, indicating that the dependence structure among the Swissix and the Phelix is

less pronounced for hourly data than it is for the daily averages. Unfortunately, the goodness

of fit measures do not allow for a clear interpretation, as ln 𝐿 𝜶 ;𝜃 , AIC and BIC point to-

wards the t-copula model, while 𝐷𝐾𝑆 and 𝐷𝐴𝐷 favor the model based on the Gumbel copula


61

and 𝐷𝐼𝐴𝐷 as well as 𝐷𝐶𝑣𝑀 indicate at the Frank copula. These observations are confirmed by

Table B1, as all goodness of fit measures imply the same copula models as the ones above.

Figure A5, however, clearly shows that the t-copula based point cloud (c) seems to best fit

the empirical return sample (a)16. In other words, the Gumbel (d) and Clayton (e) copula

again appear too asymmetric with respect to tail dependence while the Gaussian (b) and

Frank (f) copula lack tail dependence.

4.2.4 Base vs. Peak

In the second part of the spot analysis we shall move the focus of the analysis away from the

dependence between various contracts of different (i.e. German/Austrian and Swiss) market

areas and investigate the dependence structure between base and peak contracts of the

same market area. The values for the three correlation coefficients (Pearson, Kendall,

Spearman) are 0.951, 0.824 and 0.953 for the dependence between the Phelix Day Base

and Phelix Day Peak return series, respectively 0.954, 0.802 and 0.944 for the Swissix Day

Base and Phelix Day Peak series. These figures allow us to conclude that the strength of the

dependence between return series that differ only with respect to their load profile is very

high. In order to further analyze those dependence structures, Table 8 first presents the cop-

ula parameter estimation results for PhB and PhP and subsequently for SwB and SwP, both

based on non-central t-distributed marginals. Table B2 further lists the corresponding results

for the copula parameters being estimated via the CML method.

The copula parameter estimates for the Phelix Day Base and Phelix Day Peak return series

presented in Panel A of Figure 8 reveal values that are distinctly away from the independ-

ence copula. With the exception of 𝐷𝐴𝐷 , all goodness of fit measures hint towards the t-

copula as the best fit copula. According to the CML based estimation of the copula parame-

ters (cf. Table B2), 𝐷𝐴𝐷 is in line with the other distance measures, providing further evidence

for the t-copula model. Figure A6 confirms these results, as the random sample based on the

t-copula (c) provides for a rather accurate representation of the empirical returns (a). The

asymmetric structure of the Gumbel (d) and Clayton (e) samples with regard to tail depend-

ence is clearly not existent in the empirical sample. Likewise, the lack of tail dependence, as

it is observable in the case of the Frank copula (f), does not conform to the empirical de-

pendence structure. Although principally also characterized by no upper or lower tail de-

pendence, the Gaussian copula based sample (b) provides a much better fit, although still

not as good as the t-copula one.

16

Note that the copula based randomly generated samples only comprise 1000 data points compared to the 25645 observa-

tions of the empirical point cloud due to computational reasons.


62

Table 8 Copula parameter estimates (Base vs. Peak) for the Phelix and Swissix spot return series based on non-central t-

distributed marginals. Besides the parameter estimates of fitting a Gaussian, t-, Gumbel, Clayton and Frank copula to the


Panel A: Phelix Day Base vs. Phelix Day Peak


𝐶ρ𝐺𝑎 0.9538 1306.44 -2610.87 -2605.88 0.0172 0.0414 0.2579 0.0367

𝐶ρ,𝑣t 0.9597, 4.2817 1394.16 -2784.33 -2774.35 0.0162 0.0463 0.2276 0.0285

𝐶𝜃𝐺𝑢 5.0974 1309.78 -2617.57 -2612.58 0.0206 0.0703 0.5402 0.0582

𝐶𝜃𝐶𝑙 5.6717 1119.66 -2237.33 -2232.34 0.0537 0.1286 2.6121 0.4645

𝐶𝜃𝐹𝑟 20.687 1285.31 -2568.61 -2563.62 0.0202 0.2221 1.1957 0.0656

Panel B: Swissix Day Base vs. Swissix Day Peak


𝐶ρ𝐺𝑎 0.9520 1285.47 -2568.95 -2563.96 0.0186 0.0687 0.3747 0.0461

𝐶ρ,𝑣t 0.9518, 5.3339 1304.29 -2604.59 -2594.61 0.0189 0.0626 0.3568 0.0457

𝐶𝜃𝐺𝑢 4.7398 1245.17 -2488.34 -2483.35 0.0204 0.1083 0.7663 0.0680

𝐶𝜃𝐶𝑙 5.1665 1062.73 -2123.46 -2118.47 0.0571 0.1337 2.7662 0.4766

𝐶𝜃𝐹𝑟 17.9959 1170.14 -2338.27 -2333.28 0.0228 0.1879 2.0398 0.0992

With respect to the analysis of the dependence structure between the Swissix Day Base and

Swissix Day Peak return series, we observe that all copula parameters are slightly below

those presented in Panel A except the t-copula‟s second parameter 𝜈. Again, all but one

goodness of fit test statistics are in favour of the t-copula. While Table 8 sees a diverging

result for 𝐷𝐾𝑆, Table B2 lists 𝐷𝐴𝐷 as a deviator. These little inconsistencies may be a result of

the way 𝐷𝐾𝑆 and 𝐷𝐴𝐷 are structured as distance measures, making them vulnerable to out-

liers. The samples of the various copula model specifications illustrated in Figure A7 are

much in line with those presented before. The t-copula (c) again implies an adequately accu-

rate fit, directly followed by the Gaussian copula (b). The sample point clouds of the other

three copula families, on the contrary, differ too much with regard to the upper, the lower or

both joint tails.

4.3 Phelix Year Futures Analysis


With Phelix Month Futures contracts being traded for the current and the next nine months,

the maximum sample period that can be covered by a single monthly contract equals ten

months. Phelix Quarter Futures are available for the next eleven quarters, resulting in a

sample period of eleven quarters. Although this period is close to the sample period under

consideration in this analysis (i.e. three years), it must be noted that the trading volume is


63

rather low at the time a certain contract is first put up for trading, and that the number of con-

tracts traded increases constantly with time. In fact, some contracts are not even traded, or

at least not very actively, during the first months of trading. In order to cover a period of three

years of reliable data, we must hence rely on Phelix Year Futures contracts. The Phelix Year

Futures contracts with delivery in 2010 and 2011 are particularly interesting, as they are al-

ready in their fourth respectively third year of trading, implying a trading volume that is rea-

sonably high. By contrast, contracts with a later time of delivery have been traded for a

shorter period while contracts with an earlier time of delivery fail to provide data for the year

2009. As a consequence of these considerations, the following contracts are included in the

data set:

- Phelix Jan 2010 Base (10B): Historical time series of daily prices (except weekends

and statutory holidays) of electricity traded with delivery within the German/Austrian

market area during the base hours of each day (including Saturdays and Sundays) of

the year 2010.

- Phelix Jan 2010 Peak (10P): Historical time series of daily prices (except weekends


market area during the peak hours of each day (except Saturdays and Sundays) of

the year 2010.

- Phelix Jan 2011 Base (11B): Historical time series of daily prices (except weekends


market area during the base hours of each day (including Saturdays and Sundays) of

the year 2011.

- Phelix Jan 2011 Peak (11P): Historical time series of daily prices (except weekends


market area during the peak hours of each day (except Saturdays and Sundays) of

the year 2011.

As already mentioned above, the sample period again encompasses three years, ranging

from 1 January 2007 to 31 December 2009. It is important to note that futures prices, al-

though calculated on a daily basis, are only available for weekdays, as the respective con-

tracts are not traded during weekends and statutory holidays (cf. Table 1). This results in a

reduced number of observations in comparison with the above spot contracts, for which trad-

ing takes place each single day of the week during the entire year. Section 2.2 has shown

that the prices of power futures contracts exhibit no seasonal effects comparable to those of

electricity spot prices. In order to get the log returns, it is hence possible to use the usual

formula, i.e. 𝑝 = 1 in expression (4.1). Furthermore, we can observe that all prices are sub-

stantially above two Euros, making a filter such as the one applied in Section 4.2.1 unneces-


64

sary. However, it has been found that log returns equal to zero significantly bias the subse-

quent analysis. The Phelix Year Futures series are hence filtered accordingly in order to

avoid these problems.

Table 9 Descriptive statistics of the Phelix Year Futures data set. Panel A

represents the statistics for the prices, Panel B for the log returns of the

various time series. All prices are in EUR/MWh.

Panel A: Prices

Phelix Jan

2010 Base

Phelix Jan

2010 Peak

Phelix Jan

2011 Base

Phelix Jan

2011 Peak

# Obs. 753 753 753 753

Min 42.65 58.13 45.55 66.59

Max 89.00 127.50 89.67 128.58

Mean 57.78 82.80 59.72 86.16

Median 54.53 79.62 55.45 81.00

St. Dev. 9.95 14.56 9.07 12.97

Variance 98.96 211.98 82.22 168.12

Skewness 1.11 1.00 1.43 1.49

Kurtosis 0.52 0.54 1.12 1.29

JB 162 136 296 330

P-value 0.000 0.000 0.000 0.000

Panel B: Log returns

Phelix Jan

2010 Base

Phelix Jan

2010 Peak

Phelix Jan

2011 Base

Phelix Jan

2011 Peak

# Obs. 573 573 573 573

Min -0.0634 -0.0654 -0.0643 -0.0619

Max 0.0693 0.0482 0.0732 0.0597

Mean -0.0004 -0.0005 -0.0002 -0.0001

Median -0.0002 -0.0002 0.0005 0.0005

St. Dev. 0.0138 0.0118 0.0122 0.0103

Variance 0.0002 0.0001 0.0001 0.0001

Skewness 0.0027 -0.1537 0.1683 -0.1465

Kurtosis 3.07 2.88 4.18 4.77

JB 225 200 419 545

P-value 0.000 0.000 0.000 0.000

The descriptive statistics of the Phelix Year Futures series are provided in Table 9. The fig-

ures in Panel A refer to the descriptive statistics of the historically observed prices without

any adjustments. Each of the time series contains 753 observations. The 10B and 11B series

exhibit a mean of 57.78 EUR/MWh and 59.72 EUR/MWh, the 10P and 11P series even

reach 82.80 EUR/MWh and 86.16 EUR/MWh, respectively. The mean values of the 2011

contracts thereby lie above those of the 2010 contracts. Obviously, these values are consi-

derably higher than the spot counterparts described in Section 4.2.1. It must be noted how-

ever, that the Phelix Year Futures prices have seen a tremendous decline over the second


65

half of 2008 following a period of continuous appreciation (cf. Figure A8). The minimum and

maximum values of the Phelix Year Futures series are between 42.65 EUR/MWh (10B) and

66.59 EUR/MWh (11P), respectively between 89.00 EUR/MWh (10B) and 128.58 EUR/MWh

(11P). The minimum values thereby lie within 25% of the respective means and the maxi-

mum values lie within 50% thereof. Consequently, extreme observations are less extreme

than in the case of the spot data set, where the minimum and maximum values represent a

multiple of the respective mean. This is confirmed by the values of the (excess) kurtosis,

showing that a leptokurtic feature is evident but less pronounced than for the spot series.

Moreover, positive skewness again indicates towards price distributions with longer right

tails. Finally, the Jarque-Bera test statistics once more imply that the underlying normal dis-

tribution hypothesis must be rejected for all four time series. Interestingly, the rejection is

stronger for 2011 contracts than for 2010 contracts due to the more pronounced skewness

and kurtosis.

Panel B represents the descriptive statistics of the log returns. As a result of the above men-

tioned deletion of zero return observations, the number of observations has significantly de-

creased to 573. The mean returns are again close to zero and slightly negative for all series.

The minimum returns are all around -0.06, while the maximum returns vary between 0.05

and 0.07. Moreover, the standard deviations of the return series are close to 0.01. On an

annual basis, this amounts to between 16% (11P) and 22% (10B). These values clearly allow

us to say that, contrary to Phelix and Swissix spot products, Phelix Year Futures contracts

show characteristics similar to other financial assets. Positive (excess) kurtosis is again

present in the return series and more pronounced than for the prices themselves. With re-

spect to skewness, no clear pattern is visible, as it is negative for 10P and 11P, positive for

11B and near zero for 10B. The Jarque-Bera test statistics are consistently above those pre-

sented in Panel A.


Analogous to Section 4.2.2, this subsection presents the results of parametrically fitting a

non-central t-distribution and non-parametrically fitting an empirical distribution function to the

return series of the Phelix Year Futures data set.

The p-values of the Kolmogorov-Smirnov test statistics presented in the second column of

Table 10 suggest that the normal distribution does not provide an adequate representation of

the marginals of the Phelix Year Futures data set, although some of the p-values lie above

the Jarque-Bera counterparts presented in Table 9. By contrast, the non-central t-distribution

seems to provide an adequate fit with p-values being between 0.632 and 0.952. Further-

more, the Kolmogorov-Smirnov test statistics of the empirical distribution indicate a perfect fit

for all return series of the data set. At this point, it is important to emphasize the importance


66

of the decision to omit all observations with a zero return. In fact, not doing so would result in

some of the return series not being properly represented by the empirical distribution function

with p-values being close to zero. For a graphical representation of the various fitted distribu-

tions also compare Figure A9.

Table 10 Marginal parameter estimates for the Phelix Year Futures data set. Besides the estimated parameters of fitting a

normal and a non-central t- distribution to the various return series of the data set, the table also provides Kolmogorov-

Smirnov test statistics (with p-values) for the normal, non-central t- and empirical distribution.



Phelix Jan

2010 Base

μ = -0.0004

σ = 0.0138

0.0867

(0.000)

lo = 0.0023

sc = 0.0092

sh = -0.2941

df = 2.9942

0.0271

(0.793)

N/A 0.0035

(1.000)

Phelix Jan

2010 Peak

μ = -0.0005

σ = 0.0118

0.0770

(0.002)

lo = 0.0014

sc = 0.0084

sh = -0.2342

df = 3.5362

0.0312

(0.632)

N/A 0.0017

(1.000)

Phelix Jan

2011 Base

μ = -0.0002

σ = 0.0122

0.0879

(0.000)

lo = 0.0023

sc = 0.0081

sh = -0.3130

df = 2.9093

0.0270

(0.799)

N/A 0.0035

(1.000)

Phelix Jan

2011 Peak

μ = -0.0001

σ = 0.0103

0.0655

(0.015)

lo = 0.0020

sc = 0.0077

sh = -0.2989

df = 4.1317

0.0216

(0.952)

N/A 0.0052

(1.000)

4.3.3 Base vs. Peak

This part of the Phelix Year Futures analysis comprises the investigation of the dependence

structure, firstly, between the Phelix Year 2010 Base and Phelix Year 2010 Peak return se-

ries and, secondly, between the Phelix Year 2011 Base and Phelix Year 2011 Peak series.

The corresponding values of the correlation coefficients by Pearson, Kendall and Spearman

are 0.927, 0.755, 0.910, respectively 0.883, 0.668, 0.844. Similar to the dependence be-

tween base and peak load return series within the spot data set, these values hint towards a

strong dependence also between otherwise identical Futures base and peak load contracts.

Table 11 (for non-central t-distributed marginals) and Table B3 (for empirically distributed

marginals) present the results of a more elaborate analysis of the dependence structure us-

ing the various copula models under consideration.


67

Panel A of Table 11 reveals that the copula parameter estimates for analyzing the depend-

ence structure between the Phelix Year 2010 Base and Phelix Year 2010 Peak return series

are again relatively high and closer to the case of comonotonicity than to independence. The

goodness of fit measures do not contradict each other and clearly indicate at a t-copula

based dependence structure. By contrast, Panel A of Table B3 reveals a distinct conclusion

for 𝐷𝐴𝐷 , which is probably again due to outliers. Figure A10 provides further evidence in fa-

vour of the t-copula, as it is the only copula model specification that accurately represents the

joint tails.

With respect to the dependence structure between the Phelix Year 2011 Base and Phelix

Year 2011 Peak, Panel B of Table 11 as well as Panel B of Table B3 unambiguously hint

towards the t-copula. Again, this conclusion is supported by the point clouds presented in

Figure A11.

4.3.4 2010 vs. 2011

Apart from the analysis of the dependence structure between return series of Phelix Year

Futures contracts with different load profiles, the subsequent analysis concentrates on the

dependence between contracts with a different time to delivery. Firstly, we investigate the

dependence structure between the Phelix Year 2010 Base and Phelix Year 2011 Base return

series. Pearson‟s, Kendall‟s and Spearman‟s correlation coefficients take values of 0.942,

Table 11 Copula parameter estimates (Base vs. Peak) for the Phelix Year Futures return series based on non-central t-



Panel A: Phelix Year 2010 Base vs. Phelix Year 2010 Peak


𝐶ρ𝐺𝑎 0.9275 563.97 -1125.95 -1121.60 0.0356 0.0974 0.6654 0.0845

𝐶ρ,𝑣t 0.9292, 4.0380 584.10 -1164.20 -1155.50 0.0333 0.0933 0.6166 0.0776

𝐶𝜃𝐺𝑢 3.9369 552.22 -1102.44 -1098.09 0.0382 0.1390 1.0737 0.1013

𝐶𝜃𝐶𝑙 4.1328 468.66 -935.31 -930.96 0.0791 0.1755 1.7012 0.3045

𝐶𝜃𝐹𝑟 14.5971 520.36 -1038.72 -1034.37 0.0389 0.1615 1.6089 0.1669

Panel B: Phelix Year 2011 Base vs. Phelix Year 2011 Peak


𝐶ρ𝐺𝑎 0.8677 400.49 -798.99 -794.63 0.0272 0.1064 0.6175 0.0713

𝐶ρ,𝑣t 0.8695, 4.0893 420.10 -836.20 -827.50 0.0244 0.0869 0.5258 0.0619

𝐶𝜃𝐺𝑢 2.8308 380.04 -758.07 -753.72 0.0359 0.1712 1.4534 0.1104

𝐶𝜃𝐶𝑙 2.9700 356.73 -711.46 -707.11 0.0543 0.1191 1.2207 0.2222

𝐶𝜃𝐹𝑟 10.1263 367.32 -732.65 -728.30 0.0356 0.2097 1.9666 0.1666


68

0.772 and 0.921, respectively. Secondly, the analysis takes into account the relation between

the returns of Phelix Year 2010 Peak and Phelix Year 2011 Peak contracts. The correspond-

ing correlation coefficients are given as 0.892, 0.698 and 0.875. The 2010 and 2011 con-

tracts are consequently highly correlated, whereby the dependence is more pronounced for

base load contracts than for peak load ones.

Table 12 Copula parameter estimates (2010 vs. 2011) for the Phelix Year Futures return series based on non-central t-



Panel A: Phelix Year 2010 Base vs. Phelix Year 2011 Base


𝐶ρ𝐺𝑎 0.9369 602.18 -1202.37 -1198.02 0.0351 0.0804 0.6240 0.0764

𝐶ρ,𝑣t 0.9372, 3.4593 624.81 -1245.62 -1236.92 0.0322 0.0883 0.5684 0.0677

𝐶𝜃𝐺𝑢 4.1738 584.89 -1167.79 -1163.44 0.0337 0.1225 1.0415 0.0904

𝐶𝜃𝐶𝑙 4.7373 520.10 -1038.20 -1033.85 0.0614 0.1428 1.5627 0.2710

𝐶𝜃𝐹𝑟 15.6139 551.91 -1101.82 -1097.47 0.0373 0.1768 1.7129 0.1628

Panel B: Phelix Year 2010 Peak vs. Phelix Year 2011 Peak


𝐶ρ𝐺𝑎 0.8887 446.93 -891.87 -887.52 0.0296 0.0704 0.3727 0.0541

𝐶ρ,𝑣t 0.8888, 13.8778 448.92 -893.85 -885.14 0.0298 0.0659 0.3584 0.0523

𝐶𝜃𝐺𝑢 2.9922 411.23 -820.45 -816.10 0.0286 0.1332 0.9505 0.0781

𝐶𝜃𝐶𝑙 2.9961 366.29 -730.59 -726.23 0.0715 0.1534 1.6163 0.3297

𝐶𝜃𝐹𝑟 11.1110 408.77 -815.54 -811.19 0.0336 0.1656 1.3933 0.1201

In accordance with previous results, the various copula parameter estimates for the Phelix

Year 2010 Base and Phelix Year 2011 Base return series presented in Panel A of Table 12

are decisively above the values that would correspond to the case of independence. More-

over, in Panel A of both Table 12 and Table B4, 𝐷𝐴𝐷 is the only distance measure that does

not point towards the t-copula. Consequently, we must again assume that this is a result of

some outliers influencing the corresponding test statistic. Figure A12 further confirms that the

t-copula provides an adequate representation of the empirical dependence structure of the

returns.

The results for the dependence structure between the Phelix Year 2010 Peak and Phelix

Year 2011 Peak return series (cf. Panel B of Table 12 and Table B4) go into the same direc-

tions as those for the base load contracts. It must be noted, though, that the parameter esti-

mates are significantly lower. Only the second parameter 𝜈 of the t-copula is rather high, put-

ting it closer to the corresponding Gaussian copula. Again, for non-central t-distributed mar-


69

ginals, the majority of the goodness of fit measures decides in favor of the t-copula, with only

BIC and 𝐷𝐾𝑆 hinting towards a representation of the dependence structure via the Gaussian,

respectively via the Gumbel copula. The respective results for the CML method present even

more evidence in favour of the t-copula, as all test statistics unanimously prefer the t-copula

to the other model specifications. Figure A13 confirms that the t-copula based model pro-

vides an appropriate representation of the dependence structure. Due to the relatively high

second parameter 𝜐, however, the difference towards the Gaussian copula model is rather

small.

4.4 Further Analysis involving various Phelix Futures Contracts


The data sets of the first and second part of the empirical analysis are directly attained as

historical price data, as each time series of prices corresponds to a single contract traded at

the EEX. By contrast, the return series of the present analysis are constructed of various

Phelix futures contracts with different delivery periods and times of delivery. In particular, the

following synthetic time series are considered:

- 1 Month ahead (1M): Synthetic time series of daily prices (except weekends and sta-

tutory holidays) of electricity traded with delivery within the German/Austrian market

area during the base hours of each day of the next month.

- 2 Months ahead (2M): Synthetic time series of daily prices (except weekends and sta-

tutory holidays) of electricity traded with delivery within the German/Austrian market

area during the base hours of each day of the month after the next.

- 1 Quarter ahead (1Q): Synthetic time series of daily prices (except weekends and

statutory holidays) of electricity traded with delivery within the German/Austrian mar-

ket area during the base hours of each day of the next quarter.

- 2 Quarters ahead (2Q): Synthetic time series of daily prices (except weekends and

statutory holidays) of electricity traded with delivery within the German/Austrian mar-

ket area during the base hours of each day of the quarter after the next.

- 1 Year ahead (1Y): Synthetic time series of daily prices (except weekends and statu-

tory holidays) of electricity traded with delivery within the German/Austrian market

area during the base hours of each day of the next year.

- 2 Years ahead (2Y): Synthetic time series of daily prices (except weekends and statu-

tory holidays) of electricity traded with delivery within the German/Austrian market

area during the base hours of each day of the year after the next.


70

- Spot (Sp): Historical time series of daily prices (except weekends and statutory holi-

days) of electricity traded on a day-ahead basis with delivery in the German/Austrian

market area during the base hours of the next day. Corresponds to the prices of the

Phelix Day Base during weekdays.

The sample period of the present data set again covers the three years from 1 January 2007

to 31 December 2009. Furthermore, the returns are calculated similar to Section 4.3.1 using

𝑝 = 1 in expression (4.1). However, the procedure differs in such a way that we always use

the monthly, quarterly or yearly contract one or two delivery periods ahead. For instance, if

we want to calculate the 1 Month (1 Quarter, 1 Year) ahead return series, on 2 January 2007

𝑌𝑡 is given by the price of the Phelix Feb 2007 Month (Phelix Apr 2007 Quarter, Phelix Jan

2008 Year) Base contract and on 2 April 2007 𝑌𝑡 equals the price of the Phelix May 2007

Month (Phelix Jul 2007 Quarter, Phelix Jan 2008 Year) Base contract17. This basically means

that once a period equal to the delivery period under consideration has elapsed, the next

contract of the same delivery period but with a later time of delivery is used. However, any

change of the contract may involve a significant jump in the time series due to the possibility

of the products having a different mean (cf. Blöchlinger, 2008, p.23). Herein, this problem is

solved by applying an overlap of one day every time a specific contract approaches its last

trading day and a new contract comes into play. With this, we can avoid to calculate a return

across two distinct contracts. A further difficulty arises from the fact that trading for quarterly

and yearly contracts is ceased three days before the beginning of the delivery period (cf.

Table 1). For instance, the above mentioned Phelix Apr 2007 Quarter Base contract is only

traded until 28 March 2007. We can hence calculate the last return with this contract on 28

March 2007 using the prices on 27 and 28 March 2007. The next return in the synthetic re-

turn series is then calculated using the prices of the Phelix Jul 2007 Quarter Base contract

on 28 and 29 March 2007. The fact that we have two different prices of two different con-

tracts for 28 March 2007 corresponds to what was meant with the above mentioned overlap

of one day. Note that monthly contracts, on the other side, are also traded during the delivery

month. As a final adjustment and analogously to Section 4.3.1, we omit all data points with a

log return equal to zero in order to avoid any misspecifications of the marginals.

17 With respect to the notations used at the EEX for the various monthly, quarterly and yearly contracts, it is important to note

that the name of a contract is given by the month in which the delivery period begins. For instance for 2007, the Phelix Month

Futures are called Phelix Jan 2007, Phelix Feb 2007, and so on until Phelix Dec 2007, the Phelix Quarter Futures are named

Phelix Jan 2007, Phelix Apr 2007, Phelix Jul 2007 and Phelix Oct 2007 and, lastly, the only Phelix Year Futures contract for

2007 is named Phelix Jan 2007.


71

Table 13 Descriptive statistics of the synthetic time series.

1 Month

ahead

2 Months

ahead

1 Quarter

ahead

2 Quarters

ahead

1 Year

ahead

2 Years

ahead

Spot

# Obs. 679 679 679 679 679 679 679

Min -0.1461 -0.1423 -0.0558 -0.0615 -0.0591 -0.0634 -1.1194

Max 0.1298 0.0780 0.0635 0.0745 0.0651 0.0693 1.8633

Mean -0.0034 -0.0020 -0.0014 -0.0009 -0.0004 0.0000 0.0166

Median -0.0026 -0.0016 -0.0015 -0.0008 -0.0006 0.0007 0.0086

St. Dev. 0.0254 0.0212 0.0168 0.0151 0.0136 0.0122 0.2655

Variance 0.0006 0.0004 0.0003 0.0002 0.0002 0.0001 0.0705

Skewness -0.2239 -0.1992 -0.0009 -0.0027 -0.0585 -0.0553 0.4329

Kurtosis 3.5813 3.9002 0.7836 2.3036 2.5313 3.8913 5.0325

JB 369 435 17 150 182 429 738

P-value 0.000 0.000 0.000 0.000 0.000 0.000 0.000

Table 13 shows the descriptive statistics of the synthetic return series. The number of obser-

vations is 679, which is above the one used in the Phelix Year Futures analysis. The reason

for this lies in the fact that fewer observations have to be omitted due to a zero return. The

mean returns are once more very close to zero and negative for nearly all synthetic futures

series. Note that the mean return for the Spot series (0.0166) is substantially above the cor-

responding Phelix Day Base mean return (0.0006) presented in Section 4.2.1, although the

only difference lies in the number of observations, with the Spot series taking into account

solely the returns during weekdays. The maximum and minimum returns are obviously most

pronounced for the Spot series. Furthermore, we can observe that the monthly return series

show higher extreme values than the quarterly or yearly series. Similar results also hold for

the standard deviation, which is higher for the Spot series than for the monthly return series,

for which it is in turn higher than for the quarterly or yearly series. Skewness is negative for

all synthetic futures series and (excess) kurtosis is in most cases well pronounced. Further

note that the Spot series, in contrast to the Phelix Day Base return series, shows positive

skewness. The Jarque-Bera test statistics reject the null hypothesis of a normal distribution in

all cases.


Analogous to Section 4.2.2 and 4.3.2, this subsection presents the results of parametrically

fitting a non-central t-distribution and non-parametrically fitting an empirical distribution func-

tion to the synthetic return series of the data set.

The p-values of the Kolmogorov-Smirnov test statistics listed in the second column of Table

14 are less unambiguous than the ones presented in the previous two analyses. While a re-

jection of the normal distribution null hypothesis is evident for the 2Q, 1Y, 2Y and Sp return

series, the hypothesis can only be rejected for the 1M, 2M and 1Q series with a significance


72

level of 0.023, 0.068 and 0.216, respectively. Nevertheless, the test statistics of fitting a non-

central t-distribution reveal a better fit in all cases, with p-values being above 0.601 Figure

A15 further confirms that the non-central t-distribution provides a substantial improvement in

approaching the empirical density function compared to the normal distribution. Finally, the

Kolmogorov-Smirnov test statistics for the empirical distribution again imply a perfect fit.

Table 14 Marginal parameter estimates for the synthetic return series. Besides the estimated parameters of fitting a normal

and a non-central t- distribution to the various return series of the data set, the table also provides Kolmogorov-Smirnov test

statistics (with p-values) for the normal, non-central t- and empirical distribution.



1 Month ahead μ = -0.0034

σ = 0.0254

0.0573

(0.023)

lo = 0.0025

sc = 0.0198

sh = -0.3336

df = 4.6548

0.0179

(0.982)

N/A 0.0029

(1.000)

2 Months

ahead

μ = -0.0020

σ = 0.0212

0.0499

(0.068)

lo = 0.0013

sc = 0.0164

sh = -0.2213

df = 4.7189

0.0178

(0.982)

N/A 0.0029

(1.000)

1 Quarter

ahead

μ = -0.0014

σ = 0.0168

0.0405

(0.216)

lo = -0.0006

sc = 0.0144

sh = -0.0688

df = 7.3778

0.0203

(0.943)

N/A 0.0029

(1.000)

2 Quarters

ahead

μ = -0.0009

σ = 0.0151

0.0656

(0.006)

lo = 0.0024

sc = 0.0118

sh = -0.3145

df = 4.2016

0.0241

(0.825)

N/A 0.0029

(1.000)

1 Year ahead μ = -0.0004

σ = 0.0136

0.0731

(0.001)

lo = 0.0028

sc = 0.0101

sh = -0.3322

df = 3.8141

0.025

(0.790)

N/A 0.0029

(1.000)

2 Years ahead μ = -0.0000

σ = 0.0122

0.0982

(0.000)

lo = 0.0024

sc = 0.0079

sh = -0.3014

df = 2.8669

0.0294

(0.601)

N/A 0.0044

(1.000)

Spot μ = 0.0166

σ = 0.2655

0.0659

(0.006)

lo = -0.0066

sc = 0.1916

sh = 0.1183

df = 4.0251

0.0218

(0.902)

N/A 0.0015

(1.000)

Before proceeding with the analyses of the dependence structure between return series of

synthetic futures contracts with different time to delivery, between return series of synthetic

futures contracts with different delivery period and, finally, between the Spot and the syn-

thetic futures return series, Table 15 presents the correlation matrices (in accordance with

Pearson, Kendall and Spearman) for all return series within the current data set.


73

There are three main conclusions with regard to the correlation matrices: Firstly, return series

of futures contracts with the same delivery period but different time to delivery are highly cor-

related. Secondly, the correlation matrices confirm the observation that correlations tend to

increase with decreasing spread in the time to delivery of the synthetic futures contracts (cf.

Blöchlinger, 2008, p.32). In this sense, the correlation between 1M and 1Q as well as be-

tween 1Q and 1Y is higher than the correlation between 1M and 1Y. A last important conclu-

sion conveys that the correlation between the spot return series and the synthetic futures

return series is negligible as it is close to zero.

Table 15 Correlation matrices for the synthetic return series. Provided are the matrices in

accordance with Pearson‟s linear correlation and Kendall‟s respectively Spearman‟s rank

correlation coefficient.

Panel A: Pearson‟s linear correlation

1M 2M 1Q 2Q 1Y 2Y Sp

1M 1.000 0.841 0.746 0.528 0.457 0.380 0.006

2M 1.000 0.871 0.677 0.606 0.525 -0.023

1Q 1.000 0.822 0.755 0.673 -0.008

2Q 1.000 0.872 0.830 0.025

1Y 1.000 0.950 -0.010

2Y 1.000 -0.002

Sp 1.000

Panel B: Kendall‟s rank correlation


1M 1.000 0.675 0.619 0.419 0.358 0.306 0.035

2M 1.000 0.747 0.534 0.480 0.417 0.016

1Q 1.000 0.643 0.579 0.504 0.014

2Q 1.000 0.714 0.645 0.000

1Y 1.000 0.796 0.004

2Y 1.000 0.009

Sp 1.000

Panel C: Spearman‟s rank correlation


1M 1.000 0.853 0.792 0.587 0.512 0.445 0.051

2M 1.000 0.903 0.718 0.661 0.588 0.022

1Q 1.000 0.829 0.766 0.690 0.022

2Q 1.000 0.878 0.825 0.002

1Y 1.000 0.939 0.005

2Y 1.000 0.014

Sp 1.000


74

4.4.3 Different Time to Delivery

Table 16 and Table B5 present the results of fitting the Gaussian, t-, Gumbel, Clayton and

Frank copula families to the various pairs of return series of synthetic futures contracts with

the same delivery period but different time to delivery. In particular, this includes the depend-

ence between the return series 1 Month ahead vs. 2 Months ahead, 1 Quarter ahead vs. 2

Quarters ahead, and, finally 1 Year ahead vs. 2 Years ahead.

For all three combinations of return series, the copula parameter estimates reveal a high

dependence. Note that the strongest dependence is inherent to the dependence structure of

the return series 1 Year ahead vs. 2 Years ahead. The various goodness of fit measures

clearly indicate towards a best representation of the dependence structure via the t-copula.

Deviations in the individual test statistics thereby solely originate from 𝐷𝐾𝑆 and 𝐷𝐴𝐷 . Probably,

the corresponding values are again influenced by outliers. Figures A16, A17 and A18 further

Table 16 Copula parameter estimates (different time to delivery) for the synthetic return series based on non-central t-


various pairs of return series, the table also provides a variety of corresponding goodness of fit.

Panel A: 1 Month ahead vs. 2 Months ahead


𝐶ρ𝐺𝑎 0.8531 441.73 -881.45 -876.93 0.0211 0.0812 0.3006 0.0274

𝐶ρ,𝑣t 0.8640, 5.4514 465.34 -926.68 -924.16 0.0170 0.0908 0.2611 0.0209

𝐶𝜃𝐺𝑢 2.7496 426.21 -850.42 -845.90 0.0335 0.1206 0.7833 0.0615

𝐶𝜃𝐶𝑙 2.5149 358.88 -715.76 -711.24 0.0648 0.1333 2.2406 0.3592

𝐶𝜃𝐹𝑟 10.4483 445.03 -888.06 -883.54 0.0300 0.1087 0.7703 0.0624

Panel B: 1 Quarter ahead vs. 2 Quarters ahead


𝐶ρ𝐺𝑎 0.8325 400.96 -799.91 -795.39 0.0270 0.0791 0.3322 0.0407

𝐶ρ,𝑣t 0.8384, 5.7519 414.27 -824.54 -822.02 0.0252 0.0655 0.3037 0.0374

𝐶𝜃𝐺𝑢 2.5440 381.30 -760.60 -756.08 0.0292 0.1446 0.8003 0.0580

𝐶𝜃𝐶𝑙 2.2376 324.46 -646.92 -642.40 0.0625 0.1303 2.4605 0.4279

𝐶𝜃𝐹𝑟 9.2792 392.50 -783.00 -778.48 0.0272 0.2373 1.1158 0.0544

Panel C: 1 Year ahead vs. 2 Years ahead


𝐶ρ𝐺𝑎 0.9471 771.91 -1541.83 -1537.31 0.0304 0.1028 0.5556 0.0539

𝐶ρ,𝑣t 0.9476, 3.3821 799.78 -1595.56 -1593.04 0.0305 0.0895 0.5140 0.0516

𝐶𝜃𝐺𝑢 4.5567 749.22 -1496.44 -1491.92 0.0334 0.1466 0.9568 0.0702

𝐶𝜃𝐶𝑙 5.3210 669.41 -1336.82 -1332.30 0.0512 0.1176 1.7037 0.2433

𝐶𝜃𝐹𝑟 17.4347 713.62 -1425.24 -1420.72 0.0349 0.2031 1.7268 0.1053


75

confirm the conclusion that the t-copula provides the best fit to the empirical dependence

structure.

4.4.4 Different Delivery Period

The copula parameter estimates and test statistics for the various synthetic futures contracts

with different delivery period are provided in Table 17 and Table B6. In particular, the analy-

sis includes the dependence between the return series 1 Month ahead vs. 1 Quarter ahead,

1 Month ahead vs. 1 Year ahead, and, finally 1 Quarter ahead vs. 1 Year ahead.

Note that, according to the estimated copula parameters, the dependence is stronger for 1M

vs. 1Q and 1Q vs. 1Y than for 1M vs. 1Y. This coincides with what was mentioned in the con-

text of the previously presented correlation matrices. Unfortunately, the goodness of fit

measures do not allow for an unambiguous conclusion. While there exists some evidence in

Table 17 Copula parameter estimates (different delivery period) for the synthetic return series based on non-central t-



Panel A: 1 Month ahead vs. 1 Quarter ahead


𝐶ρ𝐺𝑎 0.7795 317.57 -633.15 -628.63 0.0372 0.0813 0.5786 0.0769

𝐶ρ,𝑣t 0.8101, 4.3404 354.09 -704.17 -701.65 0.0298 0.0759 0.3653 0.0391

𝐶𝜃𝐺𝑢 2.3224 317.77 -633.54 -629.02 0.0363 0.1301 1.0728 0.1034

𝐶𝜃𝐶𝑙 1.8536 257.86 -513.72 -509.20 0.0722 0.1455 2.6837 0.4504

𝐶𝜃𝐹𝑟 8.7277 355.86 -709.71 -705.19 0.0278 0.1025 0.5954 0.0554

Panel B: 1 Month ahead vs. 1 Year ahead


𝐶ρ𝐺𝑎 0.4980 96.76 -191.51 -186.99 0.0312 0.0719 0.2901 0.0323

𝐶ρ,𝑣t 0.5055, 19.4615 98.49 -192.99 -190.47 0.0307 0.0756 0.2784 0.0288

𝐶𝜃𝐺𝑢 1.4331 80.29 -158.58 -154.06 0.0334 0.1291 0.9125 0.0768

𝐶𝜃𝐶𝑙 0.6938 75.27 -148.54 -144.02 0.0539 0.1102 1.4259 0.2179

𝐶𝜃𝐹𝑟 3.6393 103.77 -205.54 -201.02 0.0236 0.0929 0.3094 0.0287

Panel C: 1 Quarter ahead vs. 1 Year ahead


𝐶ρ𝐺𝑎 0.7687 303.45 -604.90 -600.38 0.0315 0.0934 0.4270 0.0488

𝐶ρ,𝑣t 0.7755, 6.5179 312.18 -620.36 -617.84 0.0321 0.0735 0.3831 0.0443

𝐶𝜃𝐺𝑢 2.1543 283.16 -564.31 -559.79 0.0359 0.1643 0.9908 0.0637

𝐶𝜃𝐶𝑙 1.7326 247.59 -493.17 -488.65 0.0634 0.1372 2.4934 0.4265

𝐶𝜃𝐹𝑟 7.4584 300.61 -599.21 -594.69 0.0298 0.2013 1.1740 0.0623


76

favor of the t-copula, many test statistics also clearly hint towards the Frank copula. In fact,

one may be able to recognize an empirical dependence structure in Figure A19, A20 and

A21 that is somewhere between the t-copula and the Frank copula, in particular with regard

to the dependence in both joint tails.

4.4.5 Spot vs. Futures

Table 18 and Table B7 finally exhibit the results of fitting the copula models to the various

pairs of combining the Spot return series with the synthetic futures return series. In particular,

this includes the dependence between the return series Spot vs. 1 Month ahead, Spot vs. 1

Quarter ahead, and, finally Spot vs. 1 Year ahead.

Table 18 Copula parameter estimates (Spot vs. Futures) for the synthetic return series based on non-central t-distributed

marginals. Besides the parameter estimates of fitting a Gaussian, t-, Gumbel, Clayton and Frank copula to the various pairs

of return series, the table also provides a variety of corresponding goodness of fit.

Panel A: Spot vs. 1 Month ahead


𝐶ρ𝐺𝑎 0.0409 0.57 0.87 5.39 0.0320 0.1313 0.9085 0.0584

𝐶ρ,𝑣t 0.0565, 6.9227 6.02 -8.04 -5.52 0.0318 0.1039 0.5925 0.0525

𝐶𝜃𝐺𝑢 1.0252 0.73 0.54 5.06 0.0322 0.1387 1.0010 0.0606

𝐶𝜃𝐶𝑙 0.1075 3.18 -4.35 0.17 0.0336 0.0878 0.5406 0.0552

𝐶𝜃𝐹𝑟 0.3301 0.97 0.06 4.58 0.0335 0.1282 0.8688 0.0597

𝐶Π N/A N/A N/A N/A 0.0295 0.1528 1.2557 0.0837

Panel B: Spot vs. 1 Quarter ahead


𝐶ρ𝐺𝑎 0.0058 0.01 1.98 6.50 0.0295 0.1093 0.8554 0.0559

𝐶ρ,𝑣t 0.0145, 12.7820 1.49 1.02 3.55 0.0279 0.0929 0.6538 0.0500

𝐶𝜃𝐺𝑢 1.0000 0.00 2.00 6.52 0.0292 0.1121 0.8952 0.0570

𝐶𝜃𝐶𝑙 0.0760 1.74 -1.48 3.04 0.0314 0.0804 0.5262 0.0542

𝐶𝜃𝐹𝑟 0.1316 0.15 1.69 6.21 0.0301 0.1048 0.7722 0.0546

𝐶Π N/A N/A N/A N/A 0.0292 0.1121 0.8952 0.0682

Panel C: Spot vs. 1 Year ahead


𝐶ρ𝐺𝑎 -0.0027 0.00 1.99 6.52 0.0318 0.0794 0.4478 0.0429

𝐶ρ,𝑣t -0.0027, 3.0000 0.00 3.99 6.52 0.0318 0.0794 0.4478 0.0429

𝐶𝜃𝐺𝑢 1.0000 0.00 2.00 6.52 0.0321 0.0782 0.4392 0.0429

𝐶𝜃𝐶𝑙 0.0376 0.43 1.15 5.67 0.0348 0.0701 0.3524 0.0444

𝐶𝜃𝐹𝑟 0.0277 0.01 1.99 6.51 0.0328 0.0765 0.4276 0.0431

𝐶Π N/A N/A N/A N/A 0.0321 0.0782 0.4392 0.0509


77

In all cases, the estimated copula parameters are close to the parameter values applying to

the case of independence (cf. Table 3). As a consequence, the above tables also comprise

the measures of distance with respect to the independence copula. The observation that

none of the copula models is clearly favored by the various test statistics may lead us to the

conclusion that the respective dependence structures cannot be modelled adequately by the

copula models under consideration. Clearly though, as the pairs of returns are close to inde-

pendence, the dependence structures are not of great importance in practical work.

4.5 Summary of the Results across all Parts of the Analysis

The results of the above analyses can be summarized as follows:

- The dependence structures of the daily returns of Phelix vs. Swissix spot contracts

are best described via the t-copula. The dependence of the peak load contracts is

thereby slightly more pronounced than that of the base load contracts.

- The dependence structure of the hourly returns of Phelix vs. Swissix spot contracts is

not unambiguously represented by a t-copula model, although the randomly generat-

ed return samples may indicate so.

- The dependence structures of the returns of base vs. peak load spot contracts may

be represented by t-copula models. The strength of dependence is approximately

equally high for the Swissix contracts than for the Phelix contracts.

- The dependence structures of the returns of base vs. peak load Phelix Year Futures

contracts correspond to t-copula models. The dependence is thereby stronger for the

2010 contracts than for the 2011 ones.

- The dependence structures of the returns of 2010 vs. 2011 Phelix Year Futures con-

tracts can be characterized by a t-copula. The dependence of the base load contracts

is more pronounced than the dependence of the peak load contracts.

- The dependence structures of the returns of the synthetic futures contracts with dif-

ferent time to delivery are best represented by t-copula models. The strength of de-

pendence is thereby highest for the series with a yearly delivery period.

- The dependence structures of the returns of the synthetic futures contracts with dif-

ferent delivery period are not clearly represented by any of the copula models, al-

though they are located somewhere between the t- and the Frank copula. The de-

pendence is, however, stronger for the combinations implying a lower spread with re-

gard to the different delivery period of the contracts.

- The dependence structures of the returns of Spot vs. synthetic futures contracts is

close to independence.


78

With regard to the representation of the various dependence structures via copula models,

we must first state that the results based on the marginals being parametrically estimated in

general coincide with the results based on the CML method. Furthermore, the analyses pro-

vide us with the conclusion that the t-copula provides an appropriate model in the majority of

the analyzed cases. In particular, this holds for the following pairs of return series: PhB vs.

SwB, PhP vs. SwP, PhB vs. PhP, SwB vs. SwP, 10B vs. 10P, 11B vs. 11P, 10B vs. 11B,

10P vs. 11P, 1M vs. 2M, 1Q vs. 2Q and 1Y vs. 2Y. Basically, this has been confirmed by the

fact that all or most goodness of fit measures point into the direction of the t-copula. Note that

in some cases there exist some contradictions within the various test statistics. In many of

these cases, however, this is mainly due to either 𝐷𝐾𝑆 or 𝐷𝐴𝐷 being heavily influenced by

outliers. By contrast, the goodness of fit measures of the following pairs of return series do

not clearly point towards any specific copula model: PhH vs. SwH, 1M vs. 1Q, 1M vs. 1Y, 1Q

vs. 1Y, Sp vs. 1M, Sp vs. 1Q and Sp vs. 1Y.

The above analyses have further shown that there exists a broad range of dependence

structures with various degrees of strength. Based on the correlation parameter of the t-

copula models, the various dependencies can be ordered as follows: PhB vs. PhP (0.9597),

SwB vs. SwP (0.9518), 1Y vs. 2Y (0.9476), 10B vs. 11B (0.9372), 10B vs. 10P (0.9292), 10P

vs. 11P (0.8888), 11B vs. 11P (0.8695), 1M vs. 2M (0.8640), 1Q vs. 2Q (0.8384). 1M vs. 1Q

(0.8101), 1Q vs. 1Y (0.7755), PhB vs. SwB (0.6189), PhP vs. SwP (0.6439), 1M vs. 1Y

(0.5055), PhH vs. SwH (0.4671), Sp vs. 1M (0.0565), Sp vs. 1Q (0.0145), Sp vs. 1Y (-

0.0027).

Finally, several conclusions can be drawn with regard to the different correlation coefficients

applied throughout the various analyses. On one side, we can observe that Spearman‟s rho

is overall rather close to Pearson‟s linear correlation coefficient. In particular, the largest dif-

ference between the estimates of 𝜌 and 𝜌𝑠 amounts to 0.062 respectively -0.065. Further-

more, Kendall‟s tau is in all cases substantially below Spearman‟s rho. Figure 6 confirms the

observation that the estimates of 𝜌𝜏 lie consistently below those of 𝜌𝑠, as all points lie above

the 45° line.

Conclusion

79

5 CONCLUSION

The origin of the present thesis lies in the deregulation of electricity markets, which has sub-

stantially changed the sphere of activity of companies within the electric power industry. With

prices now being determined by the forces of supply and demand, electricity spot prices ex-

hibit some unique characteristics such as seasonal patterns, mean reversion, spikes and

exceptionally high volatility. Factors like the non-storability of electricity, the requirement of

instantaneous equilibrium of power supply and demand, the inelastic and seasonal nature of

electricity demand and the merit order of electricity supply hereby play a major role in ex-

plaining these features. The resulting price uncertainty makes the application of adequate

financial risk management tools a necessity. While Pearson‟s linear correlation coefficient is

often employed as a measure of dependence in empirical applications, copula models pro-

vide a much more sophisticated and flexible way to characterize a dependence structure.

Copula models thereby render it possible to separate the characterization of the dependence

structure from the characterization of the marginals, as the copula represents the function

that puts marginal distributions in relation to their joint distribution. Furthermore, copula mod-

els are sensible under any observable return specification. In particular, copula models are

able to take into account various types of tail dependence, allowing for the accurate repre-

sentation of joint extreme events in the respective models. The present thesis has thereby

presented the Gaussian and t-copula as examples for the elliptical copula class as well as

the Gumbel, Clayton and Frank copulas, which belong to the class of Archimedean copulas.

The Gumbel, Clayton and t-copula are insofar of central importance, as they allow for upper,

lower and symmetric tail dependence, respectively. The thesis has further discussed several

estimation methods and goodness of fit measures that are frequently applied in the context

of fitting copulas to a given data set. The two step inference method for margins is often

used, exploiting the fact that the marginals and the dependence structure can be estimated

separately. The canonical maximum likelihood method represents a semi-parametric version

thereof, replacing the parametric estimation of the marginals by the empirical distribution

function.

In the empirical part of the thesis, copula models were employed in order to analyze the de-

pendence structure between Phelix and Swissix spot return series, Phelix Year Futures re-

turn series and synthetic return series of various futures contracts based on the Phelix. In

particular, Gaussian, t-, Gumbel, Clayton and Frank copula families were fitted via the infer-

ence method for margins (based on the marginals following a non-central t-distribution) and

via the canonical maximum likelihood method. The goodness of fit measures of the resulting

copula models thereby revealed that the t-copula represents in most cases the best way of

characterizing the various dependence structures. This observation has further been con-

Conclusion

80

firmed by the point clouds of randomly generated return samples that were constructed in

accordance with the marginal and copula model specifications. In particular, the t-copula is

able to explain the dependence structure of the return movements between Phelix and Swis-

six contracts differing only with respect to the market area of delivery, between base and

peak load spot contracts, between Phelix Year Futures contracts with different load profiles,

between Phelix Year Futures contracts with different times of delivery and, finally, between

synthetic futures contracts with the same delivery period but different time to delivery.

The present analysis can be extended in various directions by future research. On one side,

the analysis has only considered products traded at the EEX. The European market for elec-

tricity, however, is characterized by a variety of power exchanges and by a likewise range of

different tradable products. Evidently, it would be an interesting task to investigate the vari-

ous dependence structures of the return series of these products. But even in the area of the

EEX, the possibilities are far from being exhausted. For instance, the analysis is extendable

by considering the dependence structures between return series of intraday traded contracts,

contracts with delivery within the French market area or options contracts.

The cooperation between the EEX AG and the Powernext SA has demonstrated that the

European electricity market is in a phase of integration. Time will show whether this will bring

us closer towards the goal of an integrated Europe-wide market for electricity and correspon-

dingly a single price for electricity. If this was the case, the dependence structure of the re-

turns of various comparable power products examined for different time periods should expe-

rience a strengthening. Obviously, this would suggest time-variation with respect to the biva-

riate return observations, which would in turn imply the use of copula functions that allow for

time-variation in the parameters.

But also with respect to the application of copula models there is still room for future re-

search. While the present analysis involved the basic copula families often employed in em-

pirical studies, literature about copulas offers a variety of other copula functions that may

present an even better fit than the t-copula. Another possibility would be to consider combi-

nations of copulas. This might present a promising alternative, especially in the cases where

the goodness of fit tests indicate at two or more distinct copula families.

Appendix A: Additional Figures

81

APPENDIX A: ADDITIONAL FIGURES

0.00

50.00

100.00

150.00

200.00

250.00

300.00

350.00

01.01.2007 01.01.2008 01.01.2009

(a) Phelix Hourly (prices)

-3.00

-2.00

-1.00

0.00

1.00

2.00

3.00

4.00

01.01.2007 01.01.2008 01.01.2009

(b) Phelix Hourly (log returns)

0.00

50.00

100.00

150.00

01.01.2007 01.01.2008 01.01.2009

(c) Swissix Hourly (prices)

-2.00

-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

01.01.2007 01.01.2008 01.01.2009

(d) Swissix Hourly (log returns)

-50.00

0.00

50.00

100.00

150.00

200.00

01.01.2007 01.01.2008 01.01.2009

(e) Phelix Day Base (prices)

-3.00

-2.00

-1.00

0.00

1.00

2.00

3.00

01.01.2007 01.01.2008 01.01.2009

(f) Phelix Day Base (log returns)

0.00

50.00

100.00

150.00

200.00

250.00

300.00

01.01.2007 01.01.2008 01.01.2009

(g) Phelix Day Peak (prices)

-3.00

-2.00

-1.00

0.00

1.00

2.00

3.00

01.01.2007 01.01.2008 01.01.2009

(h) Phelix Day Peak (log returns)


82

Figure A1 Historical movement of the prices and log returns of the various time series of the Phelix and Swissix spot data set.

0.00

50.00

100.00

150.00

200.00

01.01.2007 01.01.2008 01.01.2009

(i) Swissix Day Base (prices)

-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

01.01.2007 01.01.2008 01.01.2009

(j) Swissix Day Base (log returns)

0.00

50.00

100.00

150.00

200.00

250.00

300.00

01.01.2007 01.01.2008 01.01.2009

(k) Swissix Day Peak (prices)

-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

01.01.2007 01.01.2008 01.01.2009

(l) Swissix Day Peak (log returns)


83

Figure A2 Histogram of the various return series of the Phelix and Swissix spot data set. Furthermore, the blue

curve refers to the density of the empirical, the green curve to that of a normal and the red curve to that of a

non-central t-distribution.


84

Figure A3 Phelix Day Base vs. Swissix Day Base point clouds. Figure (a) illustrates the empirical returns, Figures (b) to (f)

represent a sample with 1086 simulated points based on the parameters of the respective non-central t-distributed marginals

and various copula parameter estimates. The vertical and horizontal lines refer to the 0.02 and 0.98 quantiles.


85

Figure A4 Phelix Day Peak vs. Swissix Day Peak point clouds. Figure (a) illustrates the empirical returns, Figures (b) to (f)




86

Figure A5 Phelix Hourly vs. Swissix Hourly point clouds. Figure (a) illustrates the empirical returns, Figures (b) to (f) represent a

sample with 1000 simulated points based on the parameters of the respective non-central t-distributed marginals and various

copula parameter estimates. The vertical and horizontal lines refer to the 0.01 and 0.99 quantiles.


87

Figure A6 Phelix Day Base vs. Phelix Day Peak point clouds. Figure (a) illustrates the empirical returns, Figures (b) to (f)




88

Figure A7 Swissix Day Base vs. Swissix Day Peak point clouds. Figure (a) illustrates the empirical returns, Figures (b) to (f)




89

Figure A8 Historical movement of the prices and log returns of the various time series of the Phelix Year Futures data set.

0.00

50.00

100.00

02.01.2007 02.01.2008 02.01.2009

(a) Phelix Jan 2010 Base (prices)

-0.08

-0.06

-0.04

-0.02

0.00

0.02

0.04

0.06

0.08

02.01.2007 02.01.2008 02.01.2009

(b) Phelix Jan 2010 Base (log returns)

0.00

50.00

100.00

150.00

02.01.2007 02.01.2008 02.01.2009

(c) Phelix Jan 2010 Peak (prices)

-0.08

-0.06

-0.04

-0.02

0.00

0.02

0.04

0.06

02.01.2007 02.01.2008 02.01.2009

(d) Phelix Jan 2010 Peak (log returns)

0.00

50.00

100.00

02.01.2007 02.01.2008 02.01.2009

(e) Phelix Jan 2011 Base (prices)

-0.08-0.06-0.04-0.020.000.020.040.060.080.10

02.01.2007 02.01.2008 02.01.2009

(f) Phelix Jan 2011 Base (log returns)

0.00

50.00

100.00

150.00

02.01.2007 02.01.2008 02.01.2009

(g) Phelix Jan 2011 Peak (prices)

-0.08

-0.06

-0.04

-0.02

0.00

0.02

0.04

0.06

0.08

02.01.2007 02.01.2008 02.01.2009

(h) Phelix Jan 2011 Peak (log returns)


90

Figure A9 Histogram of the various Phelix Year Futures return series. Furthermore, the blue curve refers to the

density of the empirical, the green curve to that of a normal and the red curve to that of a non-central t-

distribution.


91

Figure A10 Phelix Jan 2010 Base vs. Phelix Jan 2010 Peak point clouds. Figure (a) illustrates the empirical returns, Figures (b)

to (f) represent a sample with 573 simulated points based on the parameters of the respective non-central t-distributed margin-

als and various copula parameter estimates. The vertical and horizontal lines refer to the 0.02 and 0.98 quantiles.


92

Figure A11 Phelix Jan 2011 Base vs. Phelix Jan 2011 Peak point clouds. Figure (a) illustrates the empirical returns, Figures (b)




93

Figure A12 Phelix Jan 2010 Base vs. Phelix Jan 2011 Base point clouds. Figure (a) illustrates the empirical returns, Figures (b)




94

Figure A13 Phelix Jan 2010 Peak vs. Phelix Jan 2011 Peak point clouds. Figure (a) illustrates the empirical returns, Figures (b)




95

Figure A14 Historical movement of the synthetic return series.

-0.20

-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

08.01.2007 08.01.2008 08.01.2009

(a) 1 Month ahead

-0.20

-0.15

-0.10

-0.05

0.00

0.05

0.10

08.01.2007 08.01.2008 08.01.2009

(b) 2 Months ahead

-0.08

-0.06

-0.04

-0.02

0.00

0.02

0.04

0.06

0.08

08.01.2007 08.01.2008 08.01.2009

(c) 1 Quarter ahead

-0.08-0.06-0.04-0.020.000.020.040.060.080.10

08.01.2007 08.01.2008 08.01.2009

(d) 2 Quarters ahead

-0.08

-0.06

-0.04

-0.02

0.00

0.02

0.04

0.06

0.08

08.01.2007 08.01.2008 08.01.2009

(e) 1 Year ahead

-0.08

-0.06

-0.04

-0.02

0.00

0.02

0.04

0.06

0.08

08.01.2007 08.01.2008 08.01.2009

(f) 2 Years ahead

-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

2.00

2.50

08.01.2007 08.01.2008 08.01.2009

(g) Spot


96


97

Figure A15 Histogram of the various synthetic return series. Furthermore, the blue curve refers to the density

of the empirical, the green curve to that of a normal and the red curve to that of a non-central t-distribution.


98

Figure A16 1 Month ahead vs. 2 Months ahead point clouds. Figure (a) illustrates the empirical returns, Figures (b) to (f)

represent a sample with 679 simulated points based on the parameters of the respective non-central t-distributed marginals and

various copula parameter estimates. The vertical and horizontal lines refer to the 0.02 and 0.98 quantiles.


99

Figure A17 1 Quarter ahead vs. 2 Quarters ahead point clouds. Figure (a) illustrates the empirical returns, Figures (b) to (f)




100

Figure A18 1 Year ahead vs. 2 Years ahead point clouds. Figure (a) illustrates the empirical returns, Figures (b) to (f) represent

a sample with 679 simulated points based on the parameters of the respective non-central t-distributed marginals and various



101

Figure A19 1 Month ahead vs. 1 Quarter ahead point clouds. Figure (a) illustrates the empirical returns, Figures (b) to (f)




102

Figure A20 1 Month ahead vs. 1 Year ahead point clouds. Figure (a) illustrates the empirical returns, Figures (b) to (f) represent

a sample with 679 simulated points based on the parameters of the respective non-central t-distributed marginals and various



103

Figure A21 1 Quarter ahead vs. 1 Year ahead point clouds. Figure (a) illustrates the empirical returns, Figures (b) to (f)




104

Figure A22 Spot vs. 1 Month ahead point clouds. Figure (a) illustrates the empirical returns, Figures (b) to (f) represent a sam-

ple with 679 simulated points based on the parameters of the respective non-central t-distributed marginals and various copula

parameter estimates. The vertical and horizontal lines refer to the 0.02 and 0.98 quantiles.


105

Figure A23 Spot vs. 1 Quarter ahead point clouds. Figure (a) illustrates the empirical returns, Figures (b) to (f) represent a

sample with 679 simulated points based on the parameters of the respective non-central t-distributed marginals and various



106

Figure A24 Spot vs. 1 Year ahead point clouds. Figure (a) illustrates the empirical returns, Figures (b) to (f) represent a sample

with 679 simulated points based on the parameters of the respective non-central t-distributed marginals and various copula

parameter estimates. The vertical and horizontal lines refer to the 0.02 and 0.98 quantiles.

Appendix B: Additional Tables

107

APPENDIX B: ADDITIONAL TABLES

Table B1 Copula parameter estimates (Phelix vs. Swissix) for the Phelix and Swissix spot return series based on empirically



Panel A: Phelix Day Base vs. Swissix Day Base


𝐶ρ𝐺𝑎 0.5983 240.61 -479.22 -474.23 0.0165 0.0610 0.2748 0.0421

𝐶ρ,𝑣t 0.6234, 4.7234 270.28 -536.55 -526.57 0.0145 0.0383 0.1598 0.0224

𝐶𝜃𝐺𝑢 1.6143 224.21 -446.42 -441.43 0.0202 0.1278 1.1795 0.0894

𝐶𝜃𝐶𝑙 1.0276 203.11 -404.22 -399.23 0.0488 0.0998 2.4392 0.4863

𝐶𝜃𝐹𝑟 4.6685 240.61 -479.23 -474.24 0.0187 0.1968 1.0256 0.0755

Panel B: Phelix Day Peak vs. Swissix Day Peak


𝐶ρ𝐺𝑎 0.6401 286.31 -570.62 -565.63 0.0191 0.0641 0.4550 0.0578

𝐶ρ,𝑣t 0.6507, 5.0039 309.48 -614.96 -604.98 0.0190 0.0514 0.3162 0.0456

𝐶𝜃𝐺𝑢 1.6945 267.34 -532.68 -527.69 0.0181 0.1343 1.1201 0.0497

𝐶𝜃𝐶𝑙 1.1152 227.15 -452.30 -447.31 0.0541 0.1135 2.7689 0.5517

𝐶𝜃𝐹𝑟 4.9385 265.24 -528.47 -523.48 0.0266 0.2063 1.7490 0.1323

Panel C: Phelix Hourly vs. Swissix Hourly


𝐶ρ𝐺𝑎 0.4491 2888.72 -5775.45 -5770.45 0.1715 0.3435 376.8312 70.6286

𝐶ρ,𝑣t 0.4694, 3.4668 3905.46 -7806.91 -7796.93 0.1723 0.3452 386.2600 69.9505

𝐶𝜃𝐺𝑢 1.4313 3219.72 -6437.43 -6432.44 0.1712 0.3429 372.5204 68.4480

𝐶𝜃𝐶𝑙 0.6577 2513.68 -5025.36 -5020.37 0.1736 0.3477 501.5635 95.1862

𝐶𝜃𝐹𝑟 3.1599 2909.67 -5817.35 -5812.36 0.1722 0.3450 311.5587 58.8950


108

Table B2 Copula parameter estimates (Base vs. Peak) for the Phelix and Swissix spot return series based on empirically



Panel A: Phelix Day Base vs. Phelix Day Peak


𝐶ρ𝐺𝑎 0.9557 1328.79 -2655.59 -2650.60 0.0149 0.0314 0.0995 0.0171

𝐶ρ,𝑣t 0.9598, 4.4030 1419.36 -2834.73 -2824.75 0.0127 0.0310 0.0972 0.0139

𝐶𝜃𝐺𝑢 5.0741 1332.10 -2662.20 -2657.21 0.0194 0.0490 0.3853 0.0480

𝐶𝜃𝐶𝑙 5.7034 1120.60 -2239.20 -2234.21 0.0474 0.1144 2.3980 0.4135

𝐶𝜃𝐹𝑟 20.5918 1279.15 -2556.30 -2551.31 0.0167 0.1550 0.8700 0.0471

Panel B: Swissix Day Base vs. Swissix Day Peak


𝐶ρ𝐺𝑎 0.9539 1307.58 -2613.16 -2608.17 0.0103 0.0249 0.0673 0.0102

𝐶ρ,𝑣t 0.9518, 5.1133 1329.92 -2655.85 -2645.87 0.0101 0.0258 0.0561 0.0088

𝐶𝜃𝐺𝑢 4.7295 1267.72 -2533.43 -2528.44 0.0151 0.0637 0.3941 0.0398

𝐶𝜃𝐶𝑙 5.2095 1066.18 -2130.35 -2125.36 0.0453 0.1098 2.3468 0.3924

𝐶𝜃𝐹𝑟 17.9381 1163.53 -2325.05 -2320.06 0.0193 0.1544 1.3223 0.0612

Table B3 Copula parameter estimates (Base vs. Peak) for the Phelix Year Futures return series based on empirically distrib-

uted marginals. Besides the parameter estimates of fitting a Gaussian, t-, Gumbel, Clayton and Frank copula to the various

pairs of return series, the table also provides a variety of corresponding goodness of fit measures.

Panel A: Phelix Year 2010 Base vs. Phelix Year 2010 Peak


𝐶ρ𝐺𝑎 0.9301 574.12 -1146.24 -1141.88 0.0253 0.0674 0.1815 0.0286

𝐶ρ,𝑣t 0.9258, 4.0829 593.57 -1183.15 -1174.45 0.0240 0.0742 0.1596 0.0230

𝐶𝜃𝐺𝑢 3.8443 565.81 -1129.63 -1125.28 0.0284 0.0828 0.5134 0.0649

𝐶𝜃𝐶𝑙 3.9697 447.96 -893.91 -889.56 0.0565 0.1310 1.2794 0.2134

𝐶𝜃𝐹𝑟 14.3083 506.59 -1011.19 -1006.83 0.0297 0.1010 0.7304 0.0770

Panel B: Phelix Year 2011 Base vs. Phelix Year 2011 Peak


𝐶ρ𝐺𝑎 0.8770 420.09 -838.17 -833.82 0.0251 0.0509 0.1927 0.0344

𝐶ρ,𝑣t 0.8679, 3.8633 444.93 -885.86 -877.16 0.0230 0.0468 0.1417 0.0238

𝐶𝜃𝐺𝑢 2.8250 405.89 -809.78 -805.43 0.0283 0.1086 0.8575 0.0879

𝐶𝜃𝐶𝑙 2.9669 354.06 -706.12 -701.77 0.0436 0.1094 0.8290 0.1432

𝐶𝜃𝐹𝑟 9.9987 360.43 -718.86 -714.51 0.0301 0.1703 1.0986 0.0858


109

Table B4 Copula parameter estimates (2010 vs. 2011) for the Phelix Year Futures return series based on empirically distrib-


pairs of return series, the table also provides a variety of corresponding goodness of fit measures.

Panel A: Phelix Year 2010 Base vs. Phelix Year 2011 Base


𝐶ρ𝐺𝑎 0.9402 617.12 -1232.23 -1227.88 0.0197 0.0419 0.1111 0.0200

𝐶ρ,𝑣t 0.9361, 3.9362 640.09 -1276.18 -1267.48 0.0172 0.0472 0.0908 0.0148

𝐶𝜃𝐺𝑢 4.1309 606.03 -1210.05 -1205.70 0.0281 0.0765 0.4153 0.0525

𝐶𝜃𝐶𝑙 4.5059 500.90 -999.80 -995.45 0.0472 0.1145 1.0956 0.1777

𝐶𝜃𝐹𝑟 15.3369 538.25 -1074.49 -1070.14 0.0225 0.1349 0.7410 0.0644

Panel B: Phelix Year 2010 Peak vs. Phelix Year 2011 Peak


𝐶ρ𝐺𝑎 0.8985 471.79 -941.59 -937.23 0.0188 0.0387 0.0813 0.0137

𝐶ρ,𝑣t 0.8891, 10.3672 477.43 -950.86 -942.15 0.0162 0.0334 0.0566 0.0081

𝐶𝜃𝐺𝑢 3.0023 440.30 -878.60 -874.25 0.0213 0.0923 0.5227 0.0489

𝐶𝜃𝐶𝑙 3.0912 372.66 -743.32 -738.97 0.0476 0.1112 1.1710 0.2200

𝐶𝜃𝐹𝑟 11.0555 405.15 -808.30 -803.95 0.0225 0.1791 0.7904 0.0464


110

Table B5 Copula parameter estimates (different time to delivery) for the synthetic return series based on empirically distrib-


pairs of return series, the table also provides a variety of corresponding goodness of fit.

Panel A: 1 Month ahead vs. 2 Months ahead


𝐶ρ𝐺𝑎 0.8203 379.46 -756.93 -752.41 0.0298 0.0630 0.4440 0.0613

𝐶ρ,𝑣t 0.8613, 5.4803 444.03 -884.06 -881.54 0.0187 0.0803 0.1877 0.0160

𝐶𝜃𝐺𝑢 2.3879 343.25 -684.49 -679.97 0.0386 0.1204 1.4762 0.1511

𝐶𝜃𝐶𝑙 2.4813 349.54 -697.07 -692.55 0.0666 0.1364 2.1749 0.3577

𝐶𝜃𝐹𝑟 10.3944 442.87 -883.75 -879.23 0.0203 0.0950 0.4544 0.0349

Panel B: 1 Quarter ahead vs. 2 Quarters ahead


𝐶ρ𝐺𝑎 0.7932 336.76 -671.51 -666.99 0.0348 0.0696 0.3841 0.0589

𝐶ρ,𝑣t 0.8368, 6.3693 396.97 -789.94 -787.42 0.0232 0.0580 0.1354 0.0176

𝐶𝜃𝐺𝑢 2.2296 305.94 -609.89 -605.36 0.0358 0.1277 1.4551 0.1425

𝐶𝜃𝐶𝑙 2.2295 320.79 -639.57 -635.05 0.0601 0.1202 2.1053 0.3578

𝐶𝜃𝐹𝑟 9.1794 388.43 -774.87 -770.35 0.0221 0.1700 0.6041 0.0287

Panel C: 1 Year ahead vs. 2 Years ahead


𝐶ρ𝐺𝑎 0.9512 798.55 -1595.10 -1590.58 0.0138 0.0331 0.0760 0.0105

𝐶ρ,𝑣t 0.9480, 3.5795 825.88 -1647.76 -1645.24 0.0124 0.0359 0.0651 0.0088

𝐶𝜃𝐺𝑢 4.5625 777.59 -1553.18 -1548.66 0.0179 0.0728 0.3710 0.0353

𝐶𝜃𝐶𝑙 5.3011 667.47 -1332.94 -1328.42 0.0394 0.1104 1.1845 0.1676

𝐶𝜃𝐹𝑟 17.3975 708.60 -1415.21 -1410.69 0.0214 0.1486 0.8121 0.0435


111

Table B6 Copula parameter estimates (different delivery period) for the synthetic return series based on non-central t-



Panel A: 1 Month ahead vs. 1 Quarter ahead


𝐶ρ𝐺𝑎 0.7415 271.02 -540.04 -535.51 0.0403 0.0825 0.9433 0.1334

𝐶ρ,𝑣t 0.8109, 5.0708 338.44 -672.88 -670.36 0.0234 0.0718 0.2760 0.0262

𝐶𝜃𝐺𝑢 2.0444 250.91 -499.82 -495.30 0.0462 0.1416 2.1556 0.2331

𝐶𝜃𝐶𝑙 1.8975 261.32 -520.64 -516.12 0.0595 0.1212 2.3485 0.3812

𝐶𝜃𝐹𝑟 8.6433 352.56 -703.13 -698.61 0.0268 0.0782 0.3466 0.0345

Panel B: 1 Month ahead vs. 1 Year ahead


𝐶ρ𝐺𝑎 0.4786 88.37 -174.73 -170.21 0.0233 0.0595 0.2737 0.0367

𝐶ρ,𝑣t 0.4900, 20.8874 91.42 -178.85 -176.33 0.0211 0.0660 0.2391 0.0296

𝐶𝜃𝐺𝑢 1.3505 61.13 -120.26 -115.74 0.0386 0.1042 1.5362 0.1502

𝐶𝜃𝐶𝑙 0.6987 74.81 -147.61 -143.09 0.0503 0.1013 1.3358 0.2038

𝐶𝜃𝐹𝑟 3.6265 103.43 -204.86 -200.34 0.0185 0.0563 0.1432 0.0145

Panel C: 1 Quarter ahead vs. 1 Year ahead


𝐶ρ𝐺𝑎 0.7359 264.85 -527.70 -523.18 0.0342 0.0684 0.3932 0.0588

𝐶ρ,𝑣t 0.7726, 7.3057 297.91 -591.82 -589.30 0.0254 0.0508 0.2055 0.0287

𝐶𝜃𝐺𝑢 1.9423 229.57 -457.13 -452.61 0.0371 0.1409 1.5987 0.1424

𝐶𝜃𝐶𝑙 1.7389 246.93 -491.85 -487.33 0.0620 0.1239 2.1580 0.3709

𝐶𝜃𝐹𝑟 7.3941 298.78 -595.57 -591.05 0.0278 0.1816 0.6801 0.0385


112

Table B7 Copula parameter estimates (Spot vs. Futures) for the synthetic return series based on empirically distributed

marginals. Besides the parameter estimates of fitting a Gaussian, t-, Gumbel, Clayton and Frank copula to the various pairs

of return series, the table also provides a variety of corresponding goodness of fit.

Panel A: Spot vs. 1 Month ahead


𝐶ρ𝐺𝑎 0.0269 0.25 1.51 6.03 0.0268 0.1396 0.8242 0.0432

𝐶ρ,𝑣t 0.0490, 22.0213 2.48 -0.96 1.56 0.0265 0.1178 0.6072 0.0394

𝐶𝜃𝐺𝑢 1.0032 0.02 1.96 6.49 0.0294 0.1520 1.0144 0.0498

𝐶𝜃𝐶𝑙 0.1104 3.22 -4.45 0.07 0.0273 0.0823 0.4273 0.0395

𝐶𝜃𝐹𝑟 0.3266 0.95 0.10 4.62 0.0272 0.1280 0.7057 0.0428

𝐶Π NA NA NA NA 0.0299 0.1539 1.0511 0.0625

Panel B: Spot vs. 1 Quarter ahead


𝐶ρ𝐺𝑎 -0.0109 0.04 1.92 6.44 0.0255 0.1306 0.8025 0.0373

𝐶ρ,𝑣t 0.0093, 45.4897 0.85 2.31 4.83 0.0227 0.1159 0.6215 0.0331

𝐶𝜃𝐺𝑢 1.0000 0.00 2.00 6.52 0.0241 0.1250 0.7243 0.0352

𝐶𝜃𝐶𝑙 0.0827 1.98 -1.97 2.55 0.0218 0.0744 0.3934 0.0361

𝐶𝜃𝐹𝑟 0.1411 0.18 1.64 6.16 0.0216 0.1139 0.6084 0.0346

𝐶Π NA NA NA NA 0.0241 0.1250 0.7243 0.0426

Panel C: Spot vs. 1 Year ahead


𝐶ρ𝐺𝑎 -0.0022 0.00 2.00 6.52 0.0169 0.0784 0.2597 0.0166

𝐶ρ,𝑣t -0.0019, 0.0000 0.00 4.00 6.52 0.0170 0.0782 0.2590 0.0166

𝐶𝜃𝐺𝑢 1.0000 0.00 2.00 6.52 0.0173 0.0774 0.2543 0.0166

𝐶𝜃𝐶𝑙 0.0388 0.45 1.10 5.62 0.0211 0.0607 0.1923 0.0185

𝐶𝜃𝐹𝑟 0.0320 0.01 1.98 6.50 0.0182 0.0751 0.2438 0.0169

𝐶Π NA NA NA NA 0.0173 0.0774 0.2543 0.0205

Appendix C: Codes for Matlab and R

113

APPENDIX C: CODES FOR MATLAB AND R

On the one hand, this thesis uses R, a programming language and computing environment

with a focus on the implementation of statistical techniques (cf. www.r-project.org). R has the

advantage of being highly extensible via so called packages and has evolved to a standard

tool among statisticians. Those packages that are not part of the initial configuration are

loaded through the command library(„package name‟). For the following codes to work, it is

hence required that the respective packages have been downloaded beforehand.

On the other hand, this thesis also makes use of Matlab as a further numerical computing

environment (cf. www.mathworks.com). Note that the standard edition of Matlab can be ex-

tended by various toolboxes. The following Matlab codes thereby require the installation of

the Statistics toolbox.

Code 1 [R] Graph with histogram, fitted normal, non-central (skewed) t- and empirical distribution function for the marginals (cf.

for instance Figure A2).

library(sn)

X = read.table("G://1_SpotAnalysis/2_Returns&Marginals/PhHSwH_log.txt")

h = hist(X[,1], breaks=20, prob=TRUE, col="gray85", main="(a) Phelix Hourly", xlab="", ylab="")

xfit = seq(min(X[,1]), max(X[,1]), length=40)

yfit = dnorm(xfit, mean=mean(X[,1]), sd=sd(X[,1]))

lines(xfit, yfit, col="green3", lwd=2)

fit1 = st.mle(y=X[,1])

yfit = dst(xfit, fit1$dp[1], fit1$dp[2], fit1$dp[3], fit1$dp[4])

lines(xfit, yfit, col="red", lwd=2)

lines(density(X[,1]), col="blue", lwd=2)

h = hist(X[,2], breaks=20, prob=TRUE, col="gray85", main="(b) Swissix Hourly", xlab="", ylab="")

xfit = seq(min(X[,2]), max(X[,2]), length=40)

yfit = dnorm(xfit, mean=mean(X[,2]), sd=sd(X[,2]))

lines(xfit, yfit, col="green3", lwd=2)


yfit = dst(xfit, fit2$dp[1], fit2$dp[2], fit2$dp[3], fit2$dp[4])

lines(xfit, yfit, col="red", lwd=2)

lines(density(X[,2]), col="blue", lwd=2)

Code 2 [R] Parameter estimates and Kolmogorov-Smirnov test statistics (incl. p-values) of fitting a normal, non-central (skewed)

t- and empirical distribution function to the marginals (cf. for instance Table 6).

library(sn)

library(QRMlib)

X = read.table("G://1_SpotAnalysis/PhHSwH_log.txt")


114

ks.test(X[,1],"pnorm", mean=mean(X[,1]), sd=sd(X[,1]))

ks.test(X[,2],"pnorm", mean=mean(X[,2]), sd=sd(X[,2]))



ks.test(X[,1],"pst", fit1$dp[1], fit1$dp[2], fit1$dp[3], fit1$dp[4])

ks.test(X[,2],"pst", fit2$dp[1], fit2$dp[2], fit2$dp[3], fit2$dp[4])

ks.test(X[,1],"edf")

ks.test(X[,2],"edf")

Code 3 [R] Transforming the observed log returns into uniformly distributed variables based on the fitted non-central (skewed) t-

distribution and the empirical distribution function.

library(sn)

library(QRMlib)

X = read.table("G://1_SpotAnalysis/PhHSwH_log.txt")



U_skewt1 = pst(X[,1], fit1$dp[1], fit1$dp[2], fit1$dp[3], fit1$dp[4])

U_skewt2 = pst(X[,2], fit2$dp[1], fit2$dp[2], fit2$dp[3], fit2$dp[4])

U_skewt = cbind(U_skewt1, U_skewt2)

write.table(U_skewt, file = "G://1_SpotAnalysis/PhHSwH_Mskewt.txt")

U_empirical1 = edf(X[,1])

U_empirical2 = edf(X[,2])

U_empirical = cbind(U_empirical1, U_empirical2)

write.table(U_empirical, file = "G://1_SpotAnalysis/PhHSwH_Mempirical.txt")

Code 4 [Matlab] Parameter estimates of the Gaussian, t-, Gumbel, Clayton and Frank Copulas with respect to the uniformly

distributed variables obtained from Code 3 (Note that the .txt files must first be transformed into .xlsx files).

X = xlsread('G://1_SpotAnalysis/PhHSwH_Mskewt.xlsx')

rhoGauss = copulafit('Gauss', X)

[rhot, nu] = copulafit('t', X)

thetaGumbel = copulafit('Gumbel', X)

thetaClayton = copulafit('Clayton', X)

thetaFrank = copulafit('Frank', X)

X = xlsread('G://1_SpotAnalysis/PhHSwH_Mempirical.xlsx')

rhoGauss = copulafit('Gauss', X)

[rhot, nu] = copulafit('t', X)

thetaGumbel = copulafit('Gumbel', X)

thetaClayton = copulafit('Clayton', X)

thetaFrank = copulafit('Frank', X)


115

Code 5 [Matlab] Pdf and cdf values based on the parameter estimates obtained from Code 4.

PDFGauss = copulapdf('Gauss', X, rhoGauss)

PDFt = copulapdf('t', X, rhot(2), nu)

PDFGumbel = copulapdf('Gumbel', X, thetaGumbel)

PDFClayton = copulapdf('Clayton', X, thetaClayton)

PDFFrank = copulapdf('Frank', X, thetaFrank)

xlswrite('G://1_SpotAnalysis/4_GoodnessofFit/PDFCDF/PhHSwHPDFMskewt.xlsx',[PDFGauss PDFt PDFGumbel PDFClayton

PDFFrank])

CDFGauss = copulacdf('Gauss', X, rhoGauss)

CDFt = copulacdf('t', X, rhot(2), nu)

CDFGumbel = copulacdf('Gumbel', X, thetaGumbel)

CDFClayton = copulacdf('Clayton', X, thetaClayton)

CDFFrank = copulacdf('Frank', X, thetaFrank)

xlswrite('G:// 1_SpotAnalysis/4_GoodnessofFit/PDFCDF/PhHSwH CDFMskewt.xlsx',[CDFGauss CDFt CDFGumbel CDFClay-

ton CDFFrank])

Code 6 [Matlab] Pearson‟s linear, Kendall‟s and Spearman‟s rank correlation coefficient.

A = textread('G://1_SpotAnalysis/PhHSwH_log.txt')

corr(A, 'Type', 'Pearson')

corr(A, 'Type', 'Kendall')

corr(A, 'Type', 'Spearman')

Code 7 [R] Empirical and randomly generated bivariate point clouds (based on marginals following a non-central t-distribution

given the estimated parameters of the various copula families) with 0.02 and 0.98-quantiles.

library(sn)

library(copula)

library(fields)

X = read.table("G://1_SpotAnalysis/2_Returns&Marginals/PhBSwB_log.txt")



plot(X, main="(a) Phelix Day Base vs. Swissix Day Base – Empirical", xlab="Phelix Day Base", ylab="Swissix Day Base")

xhi=qst(0.98,location=fit1$dp[1], scale=fit1$dp[2], shape=fit1$dp[3], df= fit1$dp[4])

xlo=qst(0.02,location=fit1$dp[1], scale=fit1$dp[2], shape=fit1$dp[3], df= fit1$dp[4])

yhi=qst(0.98,location=fit2$dp[1], scale=fit2$dp[2], shape=fit2$dp[3], df= fit2$dp[4])

ylo=qst(0.02,location=fit2$dp[1], scale=fit2$dp[2], shape=fit2$dp[3], df= fit2$dp[4])

xline(xhi); xline(xlo); yline(yhi); yline(ylo)

x=rmvdc(mvdc(normalCopula(param=0.6066), c("st","st"), paramMargins=list(list(location=fit1$dp[1], scale=fit1$dp[2],

shape=fit1$dp[3], df= fit1$dp[4]), list(location=fit2$dp[1],scale=fit2$dp[2], shape=fit2$dp[3], df=fit2$dp[4]))), length(X[,1]))

plot(x, main="(b) Phelix Day Base vs. Swissix Day Base – Gaussian copula", xlab="Phelix Day Base", ylab="Swissix Day Base")





116



x=rmvdc(mvdc(tCopula(param=0.6189, df=3.8437), c("st","st"), paramMargins=list(list(location=fit1$dp[1], scale=fit1$dp[2],


plot(x, main="(c) Phelix Day Base vs. Swissix Day Base – t copula", xlab="Phelix Day Base", ylab="Swissix Day Base")






x=rmvdc(mvdc(gumbelCopula(param=1.7080), c("st","st"), paramMargins=list(list(location=fit1$dp[1], scale=fit1$dp[2],


plot(x, main="(d) Phelix Day Base vs. Swissix Day Base – Gumbel copula", xlab="Phelix Day Base", ylab="Swissix Day Base")






x=rmvdc(mvdc(claytonCopula(param=0.9917), c("st","st"), paramMargins=list(list(location=fit1$dp[1], scale=fit1$dp[2],


plot(x, main="(e) Phelix Day Base vs. Swissix Day Base – Clayton copula", xlab="Phelix Day Base", ylab="Swissix Day Base")






x=rmvdc(mvdc(frankCopula(param=4.6813), c("st","st"), paramMargins=list(list(location=fit1$dp[1], scale=fit1$dp[2],


plot(x, main="(f) Phelix Day Base vs. Swissix Day Base – Frank copula", xlab="Phelix Day Base", ylab="Swissix Day Base")






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Declaration of Authorship

122

DECLARATION OF AUTHORSHIP

I hereby declare

- that I have written this thesis without any help from others and without the use of

documents and aids other than those stated above,

- that I have mentioned all used sources and that I have cited them correctly according

to established academic citation rules.

St. Gallen, 16 August 2010

Pascal Fischbach

copula-models in the electric power industry

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