IN DEGREE PROJECT TECHNOLOGY,FIRST CYCLE, 15 CREDITS

, STOCKHOLM SWEDEN 2017

Exploring the Factors of the Credit Default Swap Spread in Different Business Sectors

KRISTOFER ENGMAN

BETTY ÅLANDER

KTH ROYAL INSTITUTE OF TECHNOLOGYSCHOOL OF ENGINEERING SCIENCES

Exploring the Factors of the Credit Default Swap Spread in Different Business Sectors

KRISTOFER ENGMAN BETTY ÅLANDER Degree Projects in Applied Mathematics and Industrial Economics Degree Programme in Industrial Engineering and Management KTH Royal Institute of Technology year 2017 Supervisor at SEB: Morten Karlsmark, Salla Franzén, Supervisors at KTH: Pierre Nyquist, Hans Lööf Examiner at KTH: Henrik Hult

TRITA-MAT-K 2017:05 ISRN-KTH/MAT/K--17/05--SE Royal Institute of Technology School of Engineering Sciences KTH SCI SE-100 44 Stockholm, Sweden URL: www.kth.se/sci

Abstract

In this study, we investigate the effect of market factors on credit default swap spreadsaggregated by specific business sectors. The market factors include commodity spotprices, foreign exchange spot prices, equity index prices and interest swap rates. Usinglinear regression modelling, we find that many of the factors are correlated to the creditdefault swap spreads. To examine the collective effect of the factors on the credit defaultswap spread, we produce linear models using best subsets regression.

The empirical results suggest that many of the factors are significant in explaining thecredit default swap. Our models show significance of regression on a 99% level, and mostvariables have correlations that are consistent with previous research. Notably, we findthat the factors show different levels of significance for each of the sectors. Based onthis investigation we conclude that there in fact exist relationships between the marketfactors and the credit default swap spread changes, and that these relationships arebusiness sector specific.

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Marknadsfaktorers inverkan pa spreaden for kreditswapparinom olika affarsomraden

Sammanfattning

I denna studie undersoker vi marknadsfaktorers inverkan pa spreaden for kreditswapparaggregerade med avseende pa utvalda affarsomraden. Marknadsfaktorerna som inklud-eras i studien ar avistapriser for ravaror, avistapriser for utlandska valutakurser, aktie-index priser och ranteswapkurser. Genom modellering med linjarregression finner vi attmanga av faktorerna pavisar korrelation med spreaden for kreditswappar. For att under-soka den gemensamma effekten som faktorerna har pa spreaden for kreditswappar skaparvi linjara modeller genom att testa alla mojliga permutationer av variablerna.

De empiriska resultaten antyder att manga av faktorerna uppvisar signifikans i sinforklarande formaga av spreaden for kreditswappar. Regressionsmodellerna pavisarsignifikans pa en 99%-niva och majoriteten av variablerna visar pa korrelationer somaterspeglar tidigare forskning inom omradet. I synnerhet ser vi att faktorerna visarolika signifikansnivaer for de olika affarsomradena. Darav dras slutsatsen att det finnsett samband mellan marknadsfaktorerna och spreaden for kreditswappar, samt att dessaar affarsomradesspecifika.

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Acknowledgements

We would like to thank our supervisors at the Royal Institute of Technology (KTH),Dr. Pierre Nyquist at the Department of Mathematics and Prof. Hans Loof at theDepartment of Industrial Economics and Management for the assistance before andduring the study.

We would also like to thank Skandinaviska Enskilda Banken AB (SEB), and especiallyDr. Morten Karlsmark and Dr. Salla Franzen, for the help with choosing a subject,providing us with data and for the valuable feedback.

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Contents

1 Introduction 61.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.2 Research Question . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3 Goal and Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.4 Scope and Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Theory 92.1 Counterparty Risk Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.1.1 Lending Risk vs. Counterparty Risk . . . . . . . . . . . . . . . . . 92.1.2 Pre-settlement Risk and Settlement Risk . . . . . . . . . . . . . . 92.1.3 Components of Counterparty Risk and Wrong-Way Risk . . . . . . 102.1.4 The Credit Default Swap . . . . . . . . . . . . . . . . . . . . . . . 112.1.5 Pricing Credit Default Swaps . . . . . . . . . . . . . . . . . . . . . 11

2.2 Mathematical Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.2.1 Multiple Linear Regression . . . . . . . . . . . . . . . . . . . . . . 122.2.2 Underlying Assumptions . . . . . . . . . . . . . . . . . . . . . . . . 132.2.3 Method of Least Squares . . . . . . . . . . . . . . . . . . . . . . . 132.2.4 t-test and F -test . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.2.5 Confidence Intervals of Estimated Regression Coefficients . . . . . 142.2.6 Heteroscedasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.2.7 Multicollinearity Diagnostics . . . . . . . . . . . . . . . . . . . . . 152.2.8 Basic Statistical Measures . . . . . . . . . . . . . . . . . . . . . . . 162.2.9 Model Evaluation Methods . . . . . . . . . . . . . . . . . . . . . . 172.2.10 Variable Selection Techniques . . . . . . . . . . . . . . . . . . . . . 192.2.11 Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.3 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3 Methodology 223.1 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.1.1 Response Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.1.2 Predictor Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.1.3 Data Structuring . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.2 Model Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.2.1 Structure of Regression Model . . . . . . . . . . . . . . . . . . . . 253.2.2 Residual Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.2.3 Variable Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.2.4 Reduced Model Summary . . . . . . . . . . . . . . . . . . . . . . . 26

4 Results 284.1 Residual Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4.1.1 Normal Q-Q Plot and Heteroscedasticity . . . . . . . . . . . . . . . 284.1.2 Multicollinearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

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4.2 Variable Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304.3 Model Choice and Resulting Models . . . . . . . . . . . . . . . . . . . . . 30

5 Discussion 325.1 Discussion of Residual Analysis . . . . . . . . . . . . . . . . . . . . . . . . 325.2 Discussion of Final Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

5.2.1 Commodities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335.2.2 Foreign Exchange Spot Rates . . . . . . . . . . . . . . . . . . . . . 345.2.3 Equity Indices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355.2.4 Interest Swap Rates . . . . . . . . . . . . . . . . . . . . . . . . . . 36

5.3 Evaluation of Chosen Methods . . . . . . . . . . . . . . . . . . . . . . . . 375.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

5.4.1 Further Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

References 39

6 Appendix 416.1 Table 5: Predictor Variables . . . . . . . . . . . . . . . . . . . . . . . . . . 416.2 Table 6: Variance Inflation Factors . . . . . . . . . . . . . . . . . . . . . . 416.3 Figure 3: Residual analyses after transformation . . . . . . . . . . . . . . 426.4 Figure 4: Residual analyses before transformation . . . . . . . . . . . . . 436.5 Figure 5: Correlation plot for the Basic Materials sector . . . . . . . . . . 446.6 Figure 6: Correlation plot for the Energy sector . . . . . . . . . . . . . . . 456.7 Figure 7: Correlation plot for the Financials sector . . . . . . . . . . . . . 466.8 Table 7: Models for the Basic Materials sector after best subsets regression 476.9 Table 8: Models for the Energy sector after best subsets regression . . . . 486.10 Table 9: Models for the Financials sector after best subsets regression . . 49

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1 Introduction

1.1 Background

At the turn of the century, credit derivatives became increasingly popular in the deriva-tives market. The market for credit derivatives grew from a total notional principal of$800 billion in 2000 to a high point of $50 trillion before the financial crisis 2007, afterwhich it decreased and subsequently stabilised [1]. Credit derivatives are an integral partof counterparty credit risk. This is the risk resulting from the failure of a counterpartyto fulfill its contractual obligations in a financial contract. Historically, counterpartyrisk was generally managed by performing trades with counterparties regarded as hav-ing solid finances [2]. Counterparty risk has always been integral for risk management,but until the financial crisis, its importance was obscured by the myth of the ”too bigto fail” institutions.

In light of the financial distress caused by the financial crisis, the resulting generaldecrease in credit quality on the market, as well as new regulations, the interest in coun-terparty risk skyrocketed [2]. A key method of mitigating counterparty risk is throughhedging with a credit derivative known as the credit default swap (CDS). Therefore, itis of great interest to determine the factors that affect the CDS spread for monitoringpurposes.

The most well-known models for pricing these kinds of derivatives are those of Black,Scholes, and Merton, known as the structured models [3, 4]. Such models use theunderlying assumption that a firm defaults when the value of the firm’s assets dropsbelow a certain threshold compared to its level of debt. Previous studies have proposedseveral models of the CDS spread empirically using linear regression with predictorsbased on economic theory. However, these studies were primarily concentrated on debt-related variables such as the firm’s leverage ratio. Since the value of a firm’s assetsalso play a central role for the occurance of firm default, we will in our study introduceasset-related variables in addition to proxies for the market condition to our models ofthe CDS spread.

1.2 Research Question

In this study, we formulate and analyse several models for the CDS spreads using mul-tiple linear regression. We regress the CDS spreads of firms in different sectors againstcommon market factors to see how well they correlate and to see if we can find a modelthat adequately describes the CDS spread. Thus, the research questions we want toanswer with this study are:

Is there a correlation between business sector specific credit default swap spreads andcommon market factors?

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and if a correlation is found,

Can the credit default swap spread be modelled, using these factors, with linear regressionmodelling?

Following the results of the study, we will also look at how the models compare toprevious research on the determinants of CDS spreads, and how they can be interpretedfrom a financial perspective.

1.3 Goal and Purpose

The purpose of this study is to test if our set of market factors can describe the CDSspread of specific business sectors, and to find a model using these factors. Because ofthe stochastic nature of the financial market, we expect the explanatory and predictiveability of the model to be moderate.

As stated in the Background, the CDS is a popular credit derivative, and followingthe financial crisis the value of the risk mitigation they provide has become all themore evident. The CDS is a relatively recent phenomenon. Empirical research on themodelling of these was still in its early stages in 2007 [5]. It would be of value todetermine novel models of CDS spreads that financial institutions could use to predicthow market factor changes affect the CDS spread.

CDS spreads are also used as an implicit measure of the probability of default. Oneuse of the probability of default is to monitor possible wrong-way risk. Since financialinstitutions perform transactions of large volumes, the credit risk associated with wrong-way risk is important to consider. An improved modelling of the probability of defaultthrough CDS spreads would therefore be valuable. By looking at the correlation ofmarket factors on the CDS spread, the model can assist in monitoring wrong-way riskfor companies in specific sectors.

Additionally, these models could be used in synergy with existing risk management mod-els when making investment decisions, providing a short-term market-based estimationof the expected loss of the transaction. If our models indicate that an investment with acounterparty can be considered as safe, the financial institution could increase the vol-ume of which they trade with said counterparty. This could be a profitable investmentopportunity.

1.4 Scope and Limitations

To limit the size of this study, we have posed some limitations on its scope, primarilyrelating to the data that will be analysed.

The study will include analysis of CDS spreads corresponding to firms in the followingbusiness sectors: Basic Materials, Energy and Financials. These were chosen based on

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the empirical assumption that different predictors should be significant for each of thesesectors, and based on the availability of data. To facilitate the calculations, we willconsolidate the firms by business sector when we analyse the spreads and thus look atthe CDS spread for a business sector as a whole instead of for a specific firm. This willbe explained further in later sections.

The predictor variables used for the study are: commodity spot prices, equity indices,foreign exchange spot rates, and interest swap rates. These were chosen to represent asample of the financial market, and because of the availability of data. In this study, wedid not account for possible endogeneity arising from the chosen predictors.

Furthermore, we have limited the span of time from which the data is collected. Thedata period is from 2008-01-02 to 2017-03-15. This study will not account for the effectof time series in the data. However, the choice of looking at the weekly relative changesin the data will reduce the effect of time series. This will be explained further in theMethodology section.

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2 Theory

In this section, we will present the theory that represents the backbone for the study’srelevance as well as its quantitative part. A presentation will firstly be made on the areaof counterparty risk, followed by a short introduction of the mathematical methods usedin the study. Finally, we will present a short literature review on the subject.

2.1 Counterparty Risk Theory

Counterparty credit risk, commonly referred to as counterparty risk, is the risk thata counterparty of a financial contract cannot fulfill its contractual obligations due toinsolvency. Counterparty risk is often likened to lending risk, however there is a sub-stantial difference in the risk environment between these transaction types. In the comingsubsections, counterparty risk as a concept will be presented, followed by methods formitigating said risk. If nothing else is specified, the theory in this section is collectedfrom the literature written by Jon Gregory, and John Hull [1, 2]

2.1.1 Lending Risk vs. Counterparty Risk

In lending risk, the exposure at risk at any time during the lending period can generallybe thought of as constant. This is because market variables such as interest rates willonly create moderate levels of uncertainty over the amount owed. Moreover, only thebondholder (the lender) will take on credit risk. If the bond defaults, the issuer of abond (the borrower) does not face a loss.

Similarly to lending risk, the cause of loss in counterparty risk is the counterparty be-ing unable or unwilling to meet the contractual obligations. In this case, however, theexposure during the contract period is uncertain. This is because the underlying inmost cases is a financial derivative, which generally exhibits substantial volatility. Fur-thermore, since a derivative can have a negative value, the value of the contract canbe both positive and negative. Thus, counterparty risk is bilateral, meaning that eachcounterparty holds risk against the other.

2.1.2 Pre-settlement Risk and Settlement Risk

Generally, we divide the counterparty risk of a transaction into two parts: pre-settlementrisk and settlement risk. The pre-settlement risk is the risk that a counterparty willdefault during the lifetime of the transaction, prior to the settlement period or expirationof the transaction, while settlement risk is the risk of default during the settlementperiod. Usually, counterparty risk refers to the pre-settlement risk, however, settlementrisk is also important to consider.

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Settlement risk is characterised by a very large potential exposure (value at risk in aninvestment), which could even amount to 100 % of the notional of the investment [2]. Theprobability of default of the counterparty during the settlement period, however, is verylow in most cases. Pre-settlement risk generally has a much lower potential exposure,but there is a substantially higher probability that the counterparty will default sincethe time horizon is considerably longer. The balance between settlement risk and pre-settlement risk depends on the nature of the underlying derivative that is traded. Inlater sections, counterparty risk refer to the pre-settlement risk.

2.1.3 Components of Counterparty Risk and Wrong-Way Risk

The main components of counterparty risk are the Expected Loss (EL), Probability ofDefault (PD), Exposure At Default (EAD), and Loss Given Default (LGD). They arerelated as follows:

EL = PD · EAD · LGD

The PD is the probability that the counterparty will default during the contract period,the EAD is the value of the underlying asset in an investment at the time of the defaultand the LGD is equal to one minus the recovery rate, the amount of the notional that willbe recovered at default [6]. The LGD is generally set at 60 %, but can vary dependingon for example the region in which the counterparty operates [1]. For more informa-tion on the estimation of these parameters, we refer to The Basel II Risk Parameters:Estimation, Validation, and Stress Testing by Engelmann & Rauhmeier [6].

The components are also related to capital requirements imposed on financial institutionsthat are exposed to credit risk by the Basel Accords. The capital requirement is defined[7]:

K = LGD · [Φ(f(PD,R))− PD] · C,

where R is a correlation coefficient for the asset, Φ is the standard normal cumulativedistribution, f(PD,R) is a function depending on PD and R, and C is the full maturityadjustment as a function of PD and the maturity of the asset.

Throughout the duration of a financial contract, the PD and the EAD will vary tosome extent. The PD may for example vary due to a change in credit quality of thefirm, and the EAD will change according to market conditions [2]. Generally thesecomponents are thought to be independent, such that the probability of the PD andEAD values simultaneously increasing is very low. However, this may not be the case,as was illustrated by the market events in 2007 and onwards [2]. The case when the PDand the EAD are increasing at the same time is known as wrong-way risk (WWR), whilean increase in EAD and decrease in PD, is known as right-way risk (RWR). Thus, anincrease in the EAD can be both beneficial and detrimental depending on the PD.

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A simple example of WWR is buying a put option on a stock where the underlying inquestion consists of assets that are highly correlated to those of the counterparty. Theput option’s value will increase if the stock goes down, in which case the counterparty’scredit quality will likely be deteriorating.

2.1.4 The Credit Default Swap

A natural step for firms in the financial market is to attempt to mitigate counterpartyrisk. This can be done through a number of methods such as netting, collateral, andhedging. Hedging is the process of neutralising the risk of a transaction by taking aposition that offsets the risk associated with the transaction, for example by using theunderlying of the transaction that is hedged. However, a wide range of derivatives canbe used for hedging, and one of note is the credit default swap (CDS).

The CDS is the most common type of credit derivative, and provides a way for companiesto trade credit risk in the same way that they trade market risk: through asset-backedderivatives [1]. It is a contract that provides an insurance for a transaction against therisk of the default of a specific firm. In the context of the CDS, the counterparty isknown as the reference entity, and the default of said counterparty is called a creditevent.

The buyer of a CDS gains the right to sell bonds issued by the reference entity fortheir face value when a credit event occurs, and the seller of a CDS is then obliged topurchase these bonds for their face value. The face value of the bonds that can be soldis known as the CDS’s notional principal. In return, the buyer of a CDS will makeperiodic payments, normally quarterly, until the end of the insurance period or until acredit event occurs. The total amount that the buyer of the CDS pays per year, as apercent of the notional, is called the CDS spread.

2.1.5 Pricing Credit Default Swaps

The most well-known mathematical approaches for pricing the CDS are structural andreduced form models.

The structured models are based on Fischer Black, Myron Scholes, and Robert Merton’sassumption that a firm defaults when the value of its assets falls below a certain level[3, 4]. The weakness of structured models is that some of the variables needed in themodels are difficult to estimate with an adequately high probability [8].

A later development is the concept of reduced models based on the research by for exam-ple Jarrow & Turnbull, where the credit risk instead is determined by the probability ofdefault modeled as a stochastic process (known as the hazard rate) with a set recoveryrate in case of default [9]. The reduced models of Darrel Duffie are also frequently usedin the literature [10].

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A third method of modelling credit risk is through empirical modelling [5, 8]. Empiricalmodelling is a form of ex post modelling, using historical data to determine a correlationor to predict expected future outcomes, commonly through mathematical modellingusing some form of regression analysis. A selection of these studies will be presented inthe later section named Literature Review.

In this study, empirical modelling through regression analysis will be performed usingthe Black-Scholes-Merton assumption of a correlation between the firm’s probability ofdefault and the value of its assets compared to its level of debt.

2.2 Mathematical Theory

This section will describe the mathematical theory that is used in the quantitative partof the study. If nothing else is specified, the theory in this section is collected from theliterature written by Douglas Montgomery et al., and Trevor Hastie et al. [11, 12].

2.2.1 Multiple Linear Regression

Regression analysis is a commonly used method within mathematical statistics for inves-tigating and modelling relationships between variables. Regression analysis has numer-ous applications in many different fields. The main goal is to find the best linear modelwith respect to a set of observations of the chosen variables. Ordinary least squares isthe most commonly used method to obtain this model.

Generally, the inferences made using a regression model are:

– Identifying the relative effects of the predictor variables

– Prediction and/or estimation

– Selection of an appropriate set of variables for the model

The multiple linear regression model is defined as the relationship between a responsevariable y, and multiple predictors {x1, x2, ..., xk}, and can be expressed as follows:

y = Xβ + ε,

where

y =

y1y2...yn

, X =

1 x11 x12 ... x1k1 x21 x22 ... x2k...

......

...1 xn1 xn2 ... xnk

, β =

β0β1...βk

and ε =

ε1ε2...εn

.The value k represents the number of predictors, and n represents the number of obser-vations, such that xij is the i:th observation of predictor xj , and yi is the i:th observation

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of the response variable. The vector β consists of the parameters βj , for j = 0, 1, . . . , k,and are unknown regressions coefficients that will be estimated by performing multiplelinear regression. Thus, we have p = k+1 coefficients: one for the intercept, β0, and onefor each of the k predictor variables, {β1, . . . , βk}. The vector ε is a vector containingthe error terms for all n observations.

2.2.2 Underlying Assumptions

To apply multiple linear regression, some underlying assumptions have to be made [11,13, 14]:

– The relationship between the response, y, and the predictors, {x1, x2, ..., xk}, islinear.

– The error term ε has zero mean, E(ε) = 0

– The model is homoscedastic, which means that the error term ε has constantvariance, V ar(ε) = σ2

– The errors are uncorrelated

– The errors are normally distributed

If the aforementioned assumptions are not fulfilled, the regression model may be ill-conditioned and might present misleading results.

2.2.3 Method of Least Squares

To estimate the regression coefficients β, we wish to find the vector of least squaresestimators, denoted β. This is done by finding the solution to the following optimisationproblem:

minβ

S(β) = ε′ε = (y −Xβ)′(y −Xβ),

where y are the observations of the response and X are the observations of the predictors.This has the optimal solution:

β = (X′X)−1X′y.

The derived least squares estimator β has the following properties:

– β is an unbiased estimator of β, E(β) = β.

– V ar(βj) = σ2(X ′X)−1jj , for j = 1, . . . , p.

– Cov(βi, βj) = σ2(X ′X)−1ij , where i 6= j.

– β is the best linear unbiased estimator of β. This follows from the Gauss-Markovtheorem.

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2.2.4 t-test and F -test

The t-test can be used to test the significance of a regression coefficient for a specificregression model. The hypotheses for testing the significance of the predictor xj corre-sponding to the regression coefficient βj are:

H0 : βj = 0, H1 : βj 6= 0.

If the null hypothesis, H0, is rejected, we can with a predetermined confidence levelconclude that the regression coefficient corresponding to predictor xj is non-zero andthat the predictor should be included in the model. This implies that the predictor issignificant for the regression model on the predetermined confidence level.

The t-test statistic for predictor xj is defined as:

tj =βj√

σ2(X ′X)−1jj

,

where σ2 is an estimation of the variance, commonly defined as:

σ2 =SSResn− p

= MSRes.

SSRes will be explained further in section 2.2.8.

The null hypothesis is rejected if:

|tj | > tα/2,n−k−1,

where α is the the chosen significance level for the t-test.

The t-test is equivalent to the partial F-test. The relationship between a t-test withv degrees of freedom, and a F -test with 1 degree of freedom in the numerator and vdegrees of freedom in the denominator is [11]:

t2j = Fj .

2.2.5 Confidence Intervals of Estimated Regression Coefficients

Another way to examine the hypotheses presented above is by constructing confidenceintervals. These are created for each regression coefficient with a predetermined sig-nificance level α. We can say that the true value of the coefficient will be within theconfidence interval with a confidence level of 1−α. If zero is not included in the confidence

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interval, then we can reject the null hypothesis, H0, with the predetermined confidencelevel. A confidence interval for the regression coefficient βj is defined as:

Iβj = βj ± tα/2,n−p√σ2(X ′X)−1jj .

2.2.6 Heteroscedasticity

In many cases, the assumption of constant variance of the error terms is not fulfilled,and this phenomenon is known as heteroscedasticity. To alleviate the effects of het-eroscedasticity, a transform is usually performed. If the transformation does not reducethe heteroscedasticity to an adequate level, it is also possible to perform a weighted leastsquares estimation, where the weights are the inverse of the variance, to reduce the errorterms to unity. Another possibility is to perform a robust regression model, for exampleusing White’s robust standard errors [15].

2.2.7 Multicollinearity Diagnostics

In multiple regression modelling, there are several phenomena that can cause degradationin the resulting model’s adequacy. An important assumption is that there is a linearrelationship between the response variable and the predictors. However, in some casesthere may also exist near-linear dependencies between the predictors, also known asmulticollinearity.

Generally, some multicollinearity is almost always present between predictors, and wecan assume that the inferences presented in section 2.2.1 can be made with only somedegree of uncertainty. However, inferences from models with severe multicollinearity mayproduce misleading results. With severe multicollinearity, our least squares estimatedcoefficients will have a very large variance, and the Euclidean distance between ourestimators and their true values can be very large.

To visualise the near-linear dependencies between the predictors, a correlation plot canbe useful. The correlations in this plot correspond to the correlations found when per-forming a simple linear regression where one variable is chosen as the response andanother as the sole predictor.

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Variance Inflation Factor

To identify multicollinearity, the variance inflation factors (VIF) can be examined. Thismethod examines the diagonal elements of the C = (X ′X)−1-matrix. The VIF valueassociated with predictor xj is denoted Cjj and is defined as:

V IFj = Cjj = (X ′X)−1jj = (1−R2j )−1.

Here, R2j is the coefficient of determination when we regress predictor xj against the

remaining predictors. The coefficient of determination will be explained further in sec-tion 2.2.9. An R2

j near unity implies that there exists a near-linear relationship betweenthe predictor xj and the remaining predictors. A V IFj value exceeding 10 implies thatpredictor xj is multicollinear, and many V IFj larger than 10 indicates strong multi-collinearity in the data set [11]. Since Cjj = (X ′X)−1jj , we can from the definition ofconfidence intervals in section 2.2.5 see that a high level of multicollinearity will result ina wide confidence interval for the regression coefficients, and thus decreases the usabilityof the model.

Condition Number

Another way of identifying multicollinearity is to perform an eigensystem analysis. Thismethod examines the eigenvalues of the matrix X ′X. If there exists a near-linear rela-tionship between the columns of X, the eigenvalues will be small. One way of quantifyingthe multicollinearity from the above phenomenon is through calculating the conditionnumber, which is the fraction of the largest and the smallest eigenvalues of the ma-trix:

κ =λmaxλmin

.

A κ that is less than 100 implies that there is no serious multicollinearity in the data,a κ between 100 and 1000 implies multicollinearity, and a κ larger than 1000 indicatessevere multicollinearity [11].

2.2.8 Basic Statistical Measures

To examine the explainability of a regression model, three measures are used: Regression,Residual, and Total Sum of Squares.

Regression Sum of Squares

The regression sum of squares measures the variation in the observed data and quantifiesthe amount of variability in the observations accounted for by the model. The regressionsum of squares is defined as:

SSR =

n∑i=1

(yi − y)2,

16

where yi is the fitted value of the response and y =1

n

∑ni=1 yi is the mean value of the

response.

Residual Sum of Squares

The residual sum of squares measures the variation in the error terms. It is an indicationof how much the data differs from the estimated regression model. The residual sum ofsquares is defined as:

SSRes =n∑i=1

(yi − yi)2.

Total Sum of Squares

The total sum of squares, which is the sum of the regression and residual sum of squares,measures the total variability in the observations and is defined as:

SST = SSR + SSRes =

n∑i=1

(yi − y)2.

2.2.9 Model Evaluation Methods

There exists multiple methods for comparing and evaluating different regression models.The measures used in this study are: the coefficient of determination (R2), adjustedR2, PRESS statistic, akaike information criterion, bayesian information criterion, andMallow’s Cp.

Coefficient of Determination

The coefficient of determination, commonly referred to as R2, is a measure of the pre-dictors’ ability to explain the variance in the response, and shows if the model replicatesthe observations adequately. Since the SST is always larger than or equal to SSR, themeasure is always within the interval 0 ≤ R2 ≤ 1. A value close to unity indicatesthat nearly all the variance in the response variable can be explained by the predictors.Consequently, models with large values of R2 are generally desired. However, by addingmore predictors, it is always possible to obtain a larger R2. Therefore the measure shouldbe used with carefulness. R2 is defined as:

R2 =SSRSST

= 1− SSResSST

.

17

Adjusted R2

Adjusted R2, commonly denoted R2Adj , is a measure that adjusts the R2 for its degrees

of freedom. Thus, the R2Adj only increases if the mean square error decreases, which

is not always the case when adding more predictors. This makes this statistic a moreattractive model selection measure than R2. R2

Adj is defined as:

R2Adj = 1− SSRes/(n− p)

SST /(n− 1).

PRESS Statistic

The PRESS statistic measures how well the regression model will do when predictingnew data points. This statistic is the sum of squares of the ordinary residuals adjustedfor the observation’s distance from the centroid of the x-space through the diagonalelements of the hat matrix H, where hii = x′i(X

′X)−1xi. If we are seeking a model withhigh predictive ability, the model with the smallest PRESS statistic should be chosen.The PRESS statistic is defined as:

PRESS =n∑i=1

( εi1− hii

)2.

Akaike Information Criterion and Bayesian Information Criterion

The akaike information criterion, or AIC, measures the quality of a model based on a setof data. It makes a trade-off between the model’s complexity and how well the model fitsthe data. Similarly, the bayesian information criterion, or BIC, measures the quality ofa model and penalises model complexity. However, the penalisation of model complexityfor BIC is higher than for the AIC measure. Hence, the AIC measure will prefer largermodels than the BIC measure. Small values of AIC and BIC are desirable.

For the ordinary least squares case, AIC and BIC are defined as:

AIC = −nln(SSRes

n

)+ 2p, BIC = −nln

(SSResn

)+ pln(n).

Mallows’s Cp Statistic

Mallows’s Cp measures the precision and bias of different models. Small values of Cpare desirable, since this indicates that the model has small variance and therefore ismore accurate when performing regression. The measure takes into account the issue ofoverfitting since it penalises models with a large number of predictors. Mallows’s Cp isdefined as:

Cp =SSRes(p)

σ2− n+ 2p.

18

2.2.10 Variable Selection Techniques

In situations where there are a large amount of candidate predictors for the regressionmodel, it is desirable to find the best subset of these variables. Methods that canfacilitate this selection of variables are known as best subsets regression methods. Thesefind the best possible subset of the candidate variables with respect to a chosen statisticalmeasure among those presented in section 2.2.9.

All Possible Regressions

All possible regressions, also known as the exhaustive subset selection method, is amethod that tests all the possible subsets of the candidate variables and finds the bestone according to one of the statistical measures mentioned above. It starts with the nullmodel, including only the intercept term β0, and successively adds predictors until alldifferent combinations for the models have been tested. Since we test all the possiblemodels, the result will represent the best subset according to the chosen statistic.

The drawback of using this method is that it requires a large amount of computationalpower to perform. In the case of k candidate predictors, the algorithm will perform 2k

estimations. For today’s computational power, it is possible to perform the exhaustivemethod for up to approximately 30 candidate predictors [11].

Forward Stepwise Selection

For models that have more than 30 candidate predictors, the forward stepwise selectionmethod can be used. Using this method, the assumption is made that there are zeropredictors in the model initially. Then, the method will add the predictor with the largestsimple correlation with the response variable. In other words, it is the predictor thatwill produce the largest F statistic. The algorithm will thereafter successively add thepredictor with the largest correlation with the response, when adjusting for the effect ofthe previously added predictors. This is done until the model contains a predeterminedamount of variables, or until all model sizes have been tested with respect to a statisticalmeasure [11].

For larger models, the exhaustive and the forward approach are generally used in unison.Firstly, one performs forward subset selection to find the 30 best predictors for themodel. Secondly, an exhaustive approach is performed on the predictors selected fromthe forward approach to find the best possible subset using one of the statistical measurespresented earlier [11].

2.2.11 Transformations

In the case of non-normality in the data used in the linear regression model, transfor-mations of the data can be performed. Sometimes, we can use experience or theoreticalmodels to determine the best transformation. In many cases, however, it needs to bedetermined analytically. For financial data, the logarithmic transform is commonly used.

19

The logarithmic transform is also useful since it decreases the level of heteroscedasticityfor data that is not normally distributed [15].

When using the logarithmic transformation, the model takes the form:

lny = lnβ0 + β1x1 + · · ·+ βkxk + lnε,

which corresponds to the following assumed relationship between the response and thepredictor variables:

y = β0eβ1x1+···+βkxkε.

2.3 Literature Review

Previous research on the subject of CDS spreads have examined a number of determi-nants of CDS spreads chosen on the basis of economic theory. This section will present aselection of the previous studies performed in the area, and some notable results.

Merton’s model and recent extensions predict a negative relationship between the risk-free rate and the bond spread. This is confirmed by studies performed by Longstaff& Schwartz, Duffee, and Skinner & Townend [16, 17, 18]. Empirical studies on creditrisk modelling have generally used zero-coupon rates extracted from government bondsas their proxy for the default-free interest rate, however, in 1998 financial markets havemoved away from using these. Today, the default-free rate is widely proxied by swap andrepo contracts, and this has also been done in more recent studies [8]. Notably, Blancoet al., Houweling & Vorst, and Hull et al. used the swap rate as a proxy for the risk-freerate, and found that it more closely matches the CDS market’s use of the risk-free ratethan using the treasury rate [8, 19, 20].

Aunon-Nerin et al. performed a thorough analysis of the economic theory on pricingmodels of the CDS and identified possibly interesting determinants for the spread, in-cluding credit rating, yield curves, stock prices, interest rates and leverage [21]. Theyfound that the identified factors provide a large explanability for the CDS spreads. No-tably, they confirmed the negative correlation between local interest rates and CDSspreads, and also found that US risk-free rates are significant in explaining CDS spreadseven in different countries. Furthermore, they showed a negative correlation betweenthe stock price and the CDS spread. The most important factor for the CDS spreadsuggested by their model is the credit risk rating of the company, which is expected fromthe risk-neutral pricing models [1].

Many studies have examined the same variables that were presented by Aunon-Nerin etal., and performed more thorough empirical modelling of default risk premia. Houweling& Vorst used reduced-form models to test their pricing performance [8]. Collin-Dufresneet al., Campbell & Taksler, Fabozzi et al., and Ericsson et al. performed linear regres-sion analysis on the relationship between credit spreads and key variables suggested byeconomic theory [22, 23, 24]. However, most of these looked at credit spreads between

20

corporate bond yields and a benchmark risk-free rate while our study examines creditspreads as proxied by CDS spreads. Using CDS spreads has the advantage of avoid-ing noise arising from an inadequate model of the risk-free yield curve, but should stillproduce similar results [24].

Fabozzi et al. also performed linear regression using similar variables to the above stud-ies, but used CDS spreads as the credit spread. They showed that there is a significantcorrelation between CDS spreads and the business sector in which the firm operates in[5].

In a later study by Longstaff et al. examined the lead-lag relationship between thestock returns and CDS spread changes, and found that the stock market leads the CDSmarket [25]. Norden & Weber found in their study that there is a significant negativecorrelation between stock returns and CDS spread changes, and also confirm the lead-lagrelationship presented by Longstaff et al. [26].

21

3 Methodology

In this section, we will describe the methodology that was used to answer the researchquestions of the study. A presentation will be made of the data used in the study,followed by the mathematical methodology used to obtain our results.

3.1 Data

This section will present the response and the predictor variables used in the study andhow they are structured for the analysis. The data used in this study was provided bySEB.

3.1.1 Response Variables

The response variables used in the study were five year CDS spreads for firms active in anumber of various business sectors and countries. The data is collected on a time horizonof approximately seven and a half years spanning between 2009-09-21 and 2017-03-14,and includes a total of approximately one million observations of the CDS spreads. Wechose to only look at investment grade companies, since they have the most liquid CDScontracts, leaving out the high yield companies. Investment grade companies have acredit rating of ”BBB” and up, and high yield companies have a credit rating below”BBB”. The weekly relative change in the CDS spreads was used in the analysis aspresented in section 3.1.3.

The CDS spreads were from firms in the following business sectors: Basic Materials,Energy and Financials. See Table 1 and Table 2 below for a more detailed presentationof the data included in each sector. For each sector, and for each viable observationdate, we calculated the arithmetic mean of the changes in CDS spreads of the firms tofind a sector specific CDS spread. On each data set of sector specific CDS spreads, weperformed a separate regression analysis using the same predictor variables.

Table 1: The geographical distribution of the companies included in the CDS data foreach sector

Region\Sector Basic Materials Energy Financials

Europe 21 8 72North America 1 1 13East Europe 0 2 5Latin America 0 1 0

Total 22 12 90

22

Table 2: The credit rating distribution of the companies included in the CDS data foreach sector. We only use the CDS spreads for investment grade companies.

Rating Basic Materials Energy Financials

AAA 0 0 0AA 0 2 17A 3 2 36BBB 6 5 17BB 8 1 9B 1 0 4CCC 1 0 0NA 3 2 7

Total 22 12 90

3.1.2 Predictor Variables

The predictors used in the model were daily quotes of a number of variables: commodityspot prices, FX spot rates, equity indices and interest swap rates. The data set consistedof roughly 89 000 observations of the variables during a period between 2008-01-02 and2017-03-15. For this study, the weekly relative change for these variables was used, aspresented in section 3.1.3. The predictors used in the study were:

1. Commodity Spot PricesThe chosen commodity spot prices were a selection of some of the most liquidcommodities on the market that were deemed to possibly be interesting for theselected sectors. The commodity spot prices are quoted in either US dollars orEuro, but primarily in US dollars. We looked at the spot prices for:

– Corn bushels

– Wheat bushels

– Troys of gold

– Barrels of Brent crude oil (ICE)

– Barrels of WTI crude oil (Nymex)

– Metric tonnes of aluminium

– Metric tonnes of copper

– Metric tonnes of lead

– Metric tonnes of nickel

– Metric tonnes of zinc

– MWh of electricity

– Pounds of coffee

– Pounds of sugar no. 11

2. Foreign Exchange (FX) Spot RatesAn FX spot rate is the rate for which two currencies are exchanged at a point intime. They are named as ’foreign/domestic’, and are quoted as the spot price of

23

the foreign currency in the domestic currency. For example, the EUR/USD FXspot is the price of one Euro in US dollars.

We looked at the FX spot rates for:

– EUR/USD

– GBP/USD

– USD/NOK

– USD/SEK

3. Equity IndicesAn equity index is a portfolio of stocks in a market, typically the ones with thelargest market share. They are intended to represent a proxy for the condition ofa localised stock market, and are quoted in the local currency.

The equity indices we looked at were:

– DAX index (30 most traded stocks on the Deutsche Borse, Germany)

– MSCI world index (Morgan Stanley Capital International World Index)

– OBX index (25 most traded stocks on the Oslo Stock Exchange, Norway)

– OMX index (30 most traded stock on the Stockholm Stock Exchange, Sweden)

– S&P 500 index (Standard & Poor’s index of the 500 largest American com-panies)

– Euro Stoxx 50 index (Stock index of the 50 largest and most liquid Eurozonecompanies)

4. Interest Swap RatesThe interest swap rates are zero-coupon rates calculated from swap quotes. Theseswap rates are an exchange of a fixed interest rate for a floating interest rate. Thefloating interest rate is the LIBOR, plus a risk premium. The interest swap rateis widely considered to be a proxy for the risk-free rate [8]. The maturity of theLIBOR used for each currency is written within a parenthesis. The data is quotedin percent.

We looked at the 1, 5, and 10 year swaps for:

– Danish krone (6 months)

– Euro (6 months)

– Norwegian krone (6 months)

– Swedish krona (3 months)

– US dollar (3 months)

24

In total, we had 38 predictors. However, because of missing data and high multicollinear-ity, we removed the observations of zinc, MSCI World index, Euro Stoxx 50 index, as wellas the 1y and 5y interest swap rates for all currencies. This reduced the total amountof predictors to 25. The final predictors are presented in Table 5 of the appendix.

3.1.3 Data Structuring

To examine the impact of the predictors on the response, we looked at the weekly changesin the observations. We used the relative change, such that we examined the percentchange in the variables per week instead of the absolute changes. To enable transform ofthe data, we added one to the relative change of every observation such that they werecentered around unity. For the set of n observations {Z1, Z2, . . . , Zi, Zi+1, . . . , Zn}, therelative change centered around unity is:

∆Zi =Zi+1 − Zi

Zi+ 1 =

Zi+1

Zi

Some previous studies used the absolute change for some variables, such as interest rates.However, in this study we chose to only look the relative changes.

After matching the dates of the response variables and the predictor variables, we hada total of 1 939 observations over a time span between 2009-09-21 and 2017-03-10. Forthese observations, we calculated the weekly relative change according to the aboveformula, resulting in a total of 388 usable observations. For each of these observations,we had one response variable and the above presented 25 predictors.

3.2 Model Analysis

Using the data presented in section 3.1, we performed a thorough regression analysis.The analysis was performed using the programming language R in the statistical softwareRStudio.

3.2.1 Structure of Regression Model

For the regression analysis, we assumed that the regression model was structured as amultiple linear regression model. To facilitate comparison with previous research, wealso performed a logarithmic transform of the response variable, such that our modelbecame:

y′sector = β′0 + β1x1 + β2x2 + ...+ β24x24 + β25x25 + ε′,

25

where y′sector represents the logarithmically transformed weekly changes in the CDSspread for the selected business sector, β′0 = ln(β0), and ε′ = ln(ε).

The predictors, {x1, x2, ..., x24, x25} are presented in Table 5 of the appendix. As pre-sented in the section 3.1.3, the data points used in the study were the weekly relativechanges for each observations of the variables.

Since we wanted to look at sector specific correlations, we created one model for each ofthe selected business sectors.

3.2.2 Residual Analysis

To inspect the normality assumption for the linear regression model, we performed aresidual analysis of the models. Firstly, we calculated the R-student residual of the re-gression model and plotted these against the estimated values for the response, y. Then,we produced a quantile-quantile normality plot (also known as the Q-Q plot). Thesetwo plots were subsequently inspected for any violations of the normality assumptionand for heteroscedasticity.

Furthermore, we tested the models for multicollinearity. This was done by calculatingthe models’ V IFj and condition number, κ. In order to facilitate the discussion of thesecorrelations, we produced a plot of the simple correlations for the predictors and theresponse.

3.2.3 Variable Selection

For each sector’s regression model we performed variable selection using best subsetselection regression.

Since we already reduced the amount of variables to less than 30 in section 3.1, theunified forward-exhaustive approach presented in section 2.2.10 was not necessary. Wetherefore immediately performed an exhaustive all possible regressions procedure to findthe optimal model from our set of 25 predictor variables. From the exhaustive selectionwe obtain three models, each being the optimal solution with respect to R2

adj , BIC orMallow’s Cp statistics respectively.

Thus, we obtained a total of nine models from the best subsets regression, three for eachsector.

3.2.4 Reduced Model Summary

Finally, we chose the three best models, one for each sector, based on the PRESS statistic.The PRESS statistic was used since it is a common model selection statistic along with

26

those presented in section 2.2.9. Since all the other statistics had already been utilized tofind the nine candidate models, we used the PRESS statistic for our final selection.

The final models were then analysed using the models’ coefficients as well as commonstatistics such as R2 and R2

adj . We also performed a t-test for the regression coefficientsto test their significance for the model, followed by an F -test to test the significance ofregression.

27

4 Results

In this section, we will present the results of the methodology presented in the previoussection. The final models will then be discussed in the following section.

4.1 Residual Analysis

Firstly, we will present the results from the residual analysis. The results generally indi-cate that the underlying assumptions of the linear regression analysis are not perfectlyfulfilled.

4.1.1 Normal Q-Q Plot and Heteroscedasticity

The results from the residual analysis are presented in Figure 1 below. Only the BasicMaterials sector’s residual analysis is presented since the results are similar for all thesectors. The Q-Q plots and residual plots for all sectors can be found in Figure 3, and theuntransformed residual plots in Figure 4 in appendix. The Normal Q-Q plots indicatea heavy-tailed distribution for all the sectors, and the residual plots show an indicationof heteroscedasticity.

−0.1 0.0 0.1 0.2

−0.

15−

0.05

0.05

0.10

0.15

0.20

Fitted values

Res

idua

ls

Residuals vs Fitted

−3 −2 −1 0 1 2 3

−4

−2

02

4

Theoretical Quantiles

Sta

ndar

dize

d re

sidu

als

Normal Q−Q

Basic Materials Sector

Figure 1: Residual plots for the Basic Materials sector.

28

4.1.2 Multicollinearity

The calculated Variance Inflation Factors (VIFs) are presented in Table 6 in the ap-pendix. Two of the 25 predictors used in the study have VIFs that are larger thanthe recommended limit of 10, which indicates that the data displays moderate multi-collinearity. The equity indices and the interest swap rates are the groups of predictorvariables containing the highest VIF values. In the commodities group, the crude oilspot prices show moderate multicollinearity. The obtained condition number of 161.99confirms that there is no serious problem with multicollinearity.

The correlation plot for the variables is presented below in Figure 2. Since the correlationplots for the different sectors are almost identical, we have here only presented the resultsfor the Basic Materials sector. The correlation plots for all the sectors can be found inFigure 5 to 7 in appendix, where the y variable is the CDS spread for each sector.

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

y X1.Corn

X2.Wheat

X3.Gold

X4.Brent.Crude.Oil

X5.Aluminium

X6.Copper

X7.Lead

X8.Nickel

X9.Electricity.MWh

X10.Coffee

X11.Sugar..11

X12.Nym

ex.Crude.Oil

X13.FX

.EUR.USD

X14.FX

.GBP

.USD

X15.FX

.USD

.NOK

X16.FX

.USD

.SEK

X17.DA

X.Index

X18.OBX

.Index

X19.OMX.Index

X20.SP

X.Index

X21.IR.DKK

.10y

X22.IR.EUR.10y

X23.IR.NOK.10y

X24.IR.SEK

.10y

X25.IR.USD

.10y

yX1.CornX2.Wheat

X3.GoldX4.Brent.Crude.Oil

X5.AluminiumX6.Copper

X7.LeadX8.Nickel

X9.Electricity.MWhX10.CoffeeX11.Sugar..11

X12.Nymex.Crude.OilX13.FX.EUR.USD

X14.FX.GBP.USDX15.FX.USD.NOK

X16.FX.USD.SEKX17.DAX.Index

X18.OBX.IndexX19.OMX.Index

X20.SPX.IndexX21.IR.DKK.10y

X22.IR.EUR.10yX23.IR.NOK.10y

X24.IR.SEK.10yX25.IR.USD.10y

Figure 2: Simple correlation matrix for the response y, here for the CDS spread of theBasic Materials sector, and the predictors.

29

4.2 Variable Selection

The resulting models after exhaustive selection with respect to the three statistical mea-sures R2

adj , BIC, and Mallow’s Cp are presented in Tables 7 to 9 in appendix, one for each

sector. The R2adj statistic chooses the largest models, and the BIC chooses the smallest.

The models for the Basic Materials sector had the highest R2adj , at approximately 47%,

while the Energy sector had the lowest at approximately 38%. All the models showsignificance of regression at the 99% confidence level.

4.3 Model Choice and Resulting Models

We then inspect the obtained models using the PRESS statistic, presented in Table 3below. The best PRESS statistics for each sector has been bolded. The correspondingmodel is used as the final model in the rest of the study.

Table 3: The PRESS-statistics for the models obtained by the variable selectiontechniques. The best PRESS statistics for each sector has been bolded.

Model\Sector Basic Materials Energy Financials

R2adj 0.6773 0.9989 0.8099

BIC 0.6807 0.9928 0.8142Mallow’s Cp 0.6679 0.9882 0.8028

The resulting models for each sector, including only the statistically significant predictorson a 95 % confidence level or above, thus are:

Basic Materials

y Basic Materials =β0 + β6 Copper + β7 Lead + β13 FX EUR/USD + β15 FX USD/NOK+

β17 DAX Index + β19 OMX Index + β22 IR EUR 10y + β25 IR USD 10y

Energy

y Energy =β3 Gold + β4 Brent Crude Oil + β6 Copper + β14 FX GBP/USD+

β15 FX USD/NOK + β17 DAX Index + β20 SPX Index

Financials

y Financials =β0 + β3 Gold + β6 Copper + β13 FX EUR/USD + β14 FX GBP/USD+

β17 DAX Index + β19 OMX Index + β20 SPX Index

Table 4 presents a more detailed view of the final models, including the regression co-efficients, statistical measures, and standard errors. These also include the coefficientsthat had a significance level below 95 %.

30

Table 4: Final models for each sector

Response variable: CDS Spread

Basic Materials Energy Financials

X3.Gold 0.336∗∗∗ 0.410∗∗∗

(0.125) (0.114)X4.Brent.Crude.Oil −0.171∗∗

(0.076)X6.Copper −0.352∗∗∗ −0.330∗∗∗ −0.199∗∗

(0.105) (0.110) (0.097)X7.Lead 0.144∗

(0.081)X9.Electricity.MWh −0.066

(0.044)X13.FX.EUR.USD −0.469∗ −1.495∗∗∗

(0.256) (0.236)X14.FX.GBP.USD 0.439∗ 0.451∗

(0.246) (0.237)X15.FX.USD.NOK 0.420∗ 0.760∗∗∗

(0.219) (0.210)X17.DAX.Index −0.555∗∗∗ −0.367∗∗ −0.732∗∗∗

(0.158) (0.148) (0.181)X19.OMX.Index −0.495∗∗∗ −0.385∗

(0.184) (0.207)X20.SPX.Index −0.901∗∗∗ −0.397∗

(0.228) (0.218)X22.IR.EUR.10y 0.106∗∗∗

(0.040)X25.IR.USD.10y −0.258∗∗∗

(0.072)Constant 1.458∗∗∗ 0.236 2.416∗∗∗

(0.481) (0.459) (0.240)

Observations 365 366 363R2 0.483 0.395 0.474Adjusted R2 0.472 0.383 0.462Residual Std. Error 0.042 (df = 356) 0.051 (df = 358) 0.046 (df = 354)F Statistic 41.603∗∗∗ (df = 8; 356) 33.420∗∗∗ (df = 7; 358) 39.935∗∗∗ (df = 8; 354)

Note: ∗p<0.1; ∗∗p<0.05; ∗∗∗p<0.01

31

5 Discussion

This section will discuss the results presented in section 4. It also contains an evaluationof our methods, as well as a discussion of the chosen methodology. In addition, arecommendation for further studies on the subject will be provided.

5.1 Discussion of Residual Analysis

The results presented in section 4.1 generally indicate that the underlying assumptionsof linear regression analysis are not perfectly fulfilled.

Figure 3 from section 4.1.1 indicates that the data for all the sectors’ CDS spreadshas a heavy-tailed distribution. The figure furthermore indicates that all sectors showsome degree of heteroscedasticity. The violations of the underlying assumptions mayaffect the level of certainty when making the inferences from the regression model. Thenormality assumption is for example needed when performing tests of significance forthe regression coefficients. However, we deem that the deviations from the underlyingassumptions are not severe enough to warrant a dismissal of the inferences made fromthe final models.

The condition number indicates that there is no severe issue with multicollinearty in ourdata set, and since it is nearly 100, the effects of multicollinearity are negligible. The VIFvalues are presented in Table 6 of the appendix. As noted earlier, some of the predictorvariables have VIF values larger than 10, indicating multicollinearity. Unsurprisingly, thefactors with the highest VIF values are the equity indices and interest swap rates. Thehigh values within these two groups follow from the fact that these have high correlationwith each other, which can be seen in the correlation plots presented in Figure 2, andFigures 5 to 7 in the appendix.

This correlation most likely arises from the fact that we live in a global and intercon-nected world, which makes economies integrated across borders of countries and con-tinents. Today, most companies are global actors, so their business activities have animpact on numerous local equity indices and interest rates and not only the ones wherethey are situated. As such, it is natural that our equity indices show high multicollinear-ity.

The same is true for the correlation between the interest swap rates. To create a stabletrading environment between the global economies, it is common that some currenciesare pegged to another currency. Therefore, they move in sync and should have a highcorrelation to one another. The correlation plots presented in Figure 2, and Figures 5to 7 in the appendix, suggests that the interest swap rates for the Danish krone and theEuro are especially correlated.

In addition to the above mentioned variables, the crude oil commodities showed highmulticollinearity. This is not surprising since the underlying is the same commodity,

32

only collected from different regions in the world with some small variations in quality.In the correlation plots, we can see that the simple correlation between these is positiveand close to unity.

5.2 Discussion of Final Models

In this section, we will analyse the final models, found in Table 4, from a mathematicaland financial point of view. Generally, all the models show significance of regression on aconfidence level above 99%, and have an R2

Adj between 38% and 47%. When including allpredictors chosen for the final models (independent of their significance level), we obtaina model size of 8, 7 and 8 variables for the Basic Materials, Energy and Financials sectorrespectively.

To discuss the final models, we analyse the predictors separately by type: commodities,FX spot rates, equity indices, and interest swap rates.

5.2.1 Commodities

As was presented in the section 1.1 of this study, we have based the research questions onthe assumptions of Black-Scholes-Merton’s structural model, where a firm defaults whenthe value of its asset fall below its level of debt. Thus, commodities sold or purchased by afirm should in theory affect its probability of default. If the price of a commodity sold bythe company increases, the company will increase its revenue. This should decrease thefirm’s probability of default and by extensions also lower its CDS spread. The oppositeoccurs if the price of the commodity instead decreases. Consequently, the CDS spreadwill most likely have a negative relationship to the prices of commodities sold and thereverse should be true for commodities being purchased.

Copper was a significant predictor for all sectors and showed a negative relationshipwith the CDS spread. One reason why we can see this relationship could be that whenthe economy is strong there is a high demand for copper since it is an important rawmaterial for most non-service related sectors. The opposite should be true when theeconomy is in recession.

For the Basic Materials sector, lead was significant in addition to copper. Our modelsuggests a positive correlation between the lead price and the CDS spread. Since lead isa Basic Material and therefore may be sold by firms in the sector, it would be reasonableto assume that the correlation should be negative. However, since some Basic Materialssuch as alloys use lead in their production, it might show a positive correlation due toit also being purchased by firms in this sector.

Gold is a liquid commodity that historically has had a high price when the economy hasbeen in recession, since many consider it a safe investment. Thus, it can be viewed as aproxy for the macroeconomic condition, having an inverse relation with the welfare of the

33

economy. Gold was significant for both the Energy sector and the Financial sector, witha positive relationship in both cases. This is not surprising since a recession in generalshould increase the probability of default, and thus CDS spread for companies. The priceof gold and the CDS spreads therefore move in the same direction. However, we expectedgold to also be significant for the Basic Materials sector for the same reason.

For the Energy sector, Brent crude oil was significant along with copper and gold. Manyof the companies in this sector are working with drilling, refinery, and distribution ofpetroleum based products, so the significance for this predictor was expected. Since wehave a negative relationship between the CDS spread and the Brent crude oil, our modelsuggests that the companies are oil-related and on the sell side. When the oil priceincreases, the companies should perform better and thus their probability of defaultdecreases, and also the CDS spread.

5.2.2 Foreign Exchange Spot Rates

To our knowledge, there are no previous studies investigating the relationship betweenforeign exchange spot rates and CDS spreads. The spot rates should, however, affectthe CDS spread since today’s global trade implies that a company’s result margins areaffected by the movements of currencies’ values.

For the Basic Materials sector, the EUR/USD FX and the USD/NOK FX spot rateswere significant. In the data set, only two firms were from countries where the Euro isnot the domestic currency, which could explain the negative correlation of the EUR/USDFX spot rate. If there is a recession in the European economy, the price of the Euro willfall, which in most cases would lead to an decrease in the EUR/USD FX spot rate. In arecession, the probability of default for firms increase, which implies an increase of theCDS spread.

Since the prices of raw materials are globally quoted in US dollars, the revenue of thecompanies in the Basic Materials sector is reliant on the exchange rate of their domesticcurrency against the US dollar. Thus, the correlation between the EUR/USD FX spotrate and the CDS spread for the Basic Materials sector could also suggest that thevolume of which firms purchase raw materials or services in US dollars for this sectoris larger than the volume of which firms sell raw materials. This could for example betrue if the products sold by the firm are semi-refined, such as alloys that were mentionedearlier. If the price of the Euro against the US dollar increases, the companies couldface a loss from the currency exchange when purchasing raw materials quoted in USdollars.

The USD/NOK FX showed a positive correlation to the CDS spread of the Basic Mate-rials sector. This correlation was expected, since oil is an important raw material for theBasic Material sector and Norway is a major actor in the oil industry. Norway’s largepresence in the oil industry causes the Norwegian krone to be negatively correlated to

34

the price of oil. The negative correlation between the currency swap and oil prices canalso be seen in the simple correlation plots.

For the Energy sector, the GBP/USD and USD/NOK FX spot rates are significant andpositively correlated to the CDS spread. Since Norway has a large energy industry, it isnot surprising that the USD/NOK FX spot rate is significant for the model. Further-more, as was stated in the above paragraph, Norway’s large presence in the oil industrycauses their economy to depend largely on the price of oil. This would explain thepositive correlation suggested by the model.

The significance of the GBP/USD FX spot rate was unexpected in the model, but hada much lower significance level than the USD/NOK FX spot rate. The GBP/USD FXspot rate can have been affected by the recent news of Great Britain leaving the EU,which lowered the yield expectations of the British pound. However, since it occurredduring the last year, it is unlikely that it would have that much of an impact on theresults.

For the Financials sector, the EUR/USD and the GBP/USD FX spot rates were signif-icant. The European and the American market are two of the largest economies in theworld by GDP. Hence, it is not surprising that the EUR/USD FX spot rate is significantfor the Financial sector. The negative relationship can partly be explained by the factthat most of the companies within the sector in our study are based in Europe. Asstated for the Basic Materials sector, the correlation might be due to the effect of theEuropean economy’s condition on the CDS spread.

Furthermore, the GBP/USD FX spot rate is significant for the Financial sector and hasa positive relationship with the CDS spread. As mentioned previously for the Energysector, this might follow from the effects in the wake of Great Britain leaving the EU, butis unlikely. However, this FX spot rate was notably less significant than the EUR/USDFX spot rate.

5.2.3 Equity Indices

The study conducted by Aunon-Nerin et al. showed that stock returns had a negativecorrelation with CDS spreads, and this is mirrored in our results [21]. Equity indices,which represent a portfolio of stocks, have a negative correlation with the CDS spread forall our models. This was expected, since increasing stock returns reflects that a companyis performing well, which should lower its probability of default and by extension its CDSspread. Analogously, if an equity index falls, it implies that the economy is performingworse, and companies will more likely default.

For all sectors, the DAX index was a significant predictor. Since our data is mainlycollected from Europe, and Germany is Europe’s largest economy, this was an expectedresult.

35

For the Basic Materials sector, the OMX index was also significant along with the DAXindex. Both the negative correlation and the significance are expected results, since theOMX index contains companies active on the European market. The correlation couldalso arise from that firms included in the OMX market purchase large quantities fromthe companies in this sector, since many of them are heavy industry companies thathave a large demand for basic materials.

For the Energy sector, the S&P index was significant along with DAX index. Amongthe chosen companies in the Energy sector, the only one based in the US was a highyield company and thus was excluded from the analysis. The US is the world’s largesteconomy, and as such it is natural that it would affect companies around the world. TheEnergy sector contains companies that both drill oil and produce electricity and sincethe S&P index contains a number of similar companies, they could correlate.

For the Financials sector the DAX index, OMX index, and S&P index were all significant.Many of the firms in this sector are included in the above indices, so this correlationwas expected. Furthermore, the financial institutions provide loans for many companiesrepresented in the indices, which means that their performance is correlated to the healthof these businesses.

5.2.4 Interest Swap Rates

Previous research suggests a negative correlation between the risk-free rate (as proxiedby the treasury rate) and the CDS spread [16, 17, 18, 21]. This is also supported byrisk-neutral pricing models for the CDS [24]. Since the interest swap rate is the currentlymost used proxy for the risk-free rate, they should show a similar correlation with theCDS spread [8, 19, 20]. However, contrary to previous research, we found that the risk-free rates as proxied by swap rates were not significant for every sector and not alwaysnegative.

The interest swap rates were only found to be significant for the Basic Materials sector.Our method selected the Euro and the US dollar interest swap rates as significant forthe Basic Materials sector, with a positive and negative relationship respectively.

However, while looking at the correlation matrix we see that all interest swap ratesexhibit a negative correlation to the CDS spread when looking at simple correlations.This indicates that our choice of variables produces an interference that changes the signof some interest swap rates’ correlation to the CDS spread.

As presented in Table 1, 21 of the 22 companies in the Basic Materials sector are fromEurope, and thus it is natural that the Euro interest swap rates will affect its CDS spreadof the Basic Materials sector. It is, however, interesting that the correlation is positiveand not negative. The US dollar interest swap rate produced a negative correlation, asis consistent with previous studies.

36

Our model suggests that interest swap rates are not significant for the Financial sector’sCDS spreads. This was unexpected, since the interest swap rate should affect the revenuein the Financial sector. Banks generally lend out money long-term, but themselvesborrow money short-term. The profit for the bank from the transaction is thus basedon the spread between these rates. Thus, it was expected that the included long-termswap rates would be significant for this sector.

5.3 Evaluation of Chosen Methods

In our present methodology, the underlying assumptions of linear regression analysisare not perfectly fulfilled, which is known to affect the final results. To improve theunderlying assumptions, a more analytical method of choosing our transform could beused, such as Box-Cox. This, however, may affect the coefficients of the final model andmake the model more difficult to interpret. To mitigate the effects of heteroscedasticity,a White’s robust standard errors regression model could be used. However, this was notconsidered in the study.

We used a time interval of a week to calculate the change in our variables. Due totime series, it might be more effective to look at monthly changes, but with our dataset this would have resulted in a very small set of observations. The choice of lookingat weekly changes has been made in previous literature, and as such lends credence toit. For example, the study performed by Blanco et al. uses the weekly changes, whichalso leads to a reduction of noise [19]. Daily changes were tested, but it produced alarger amount of time series interference than weekly changes and thus was rejected asan alternative method for our study.

To look at the CDS spread on a sector level, we calculated the arithmetic mean of theCDS spreads for the companies that were included in said sector. Since the marketshare of the companies may markedly differ, a natural choice would have been to lookat a weighted mean with respect to each company’s market share. It could also beargued that the sectoral division of the CDS spreads is too wide, since we for examplein the Basic Materials sector have included companies that produce both paper and rawmetals. The spot price of lead should reasonably not affect the paper business, but itcould affect the mining business.

5.4 Conclusion

In this study, we have investigated the correlation between common market factors andcredit default swap spreads for specific sectors and created a linear model of these usinglinear regression analysis.

Our empirical results show that there indeed exists correlation between the market fac-tors we have examined and the credit default swap spreads. These correlations are con-

37

sistent with previous research on the determination of factors for the pricing of creditdefault swaps. Furthermore, we have produced models of the credit default swap spreadfor three separate sectors using linear regression modelling. Three models were selectedfor each sector using best subsets regression and from these, a final model was selected us-ing the PRESS statistic. Each of the final models for the different sectors obtained usinglinear regression showed significance of regression on the 99 % confidence level.

From the obtained models, we found novel correlations between the examined variablesand the credit default swap that are of significance for modelling. In particular, we foundthat commodities showed significance for different sectors, which is consistent with theassumptions of the structural model proposed by Black-Scholes-Merton. Moreover, wefound that FX Spot prices also showed significance for the credit default swap spread.We shared many common results with previous studies, however there were some incon-sistencies.

5.4.1 Further Studies

From the results presented in section 4, it is evident that there are business sector specificfactors that affect CDS spreads. Thus, our study suggests the need for further studiesthat explore the determining factors for other business sectors.

However, since we have not considered the effects of time series on our data set, someimportant correlations might have been missed. A lead-lag analysis could also provideuseful insights.

Endogeneity and heteroscedasticity could have affected the results of our study as well.To mitigate heteroscedasticity, a regression using White’s robust standard errors couldbe useful. Since our sectoral division was very wide, a weighted average or a morethorough partitioning could provide different results. Therefore, further studies shouldalso investigate the influence of the above on the results.

38

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[17] G. R. Duffee. The relation between treasury yields and corporate bond yield spreads.Journal of Finance, 53(6):2225–2241, 1998.

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[20] J. Hull, M. Predescu, and A. White. The relationship between credit default swapspreads, bond yields, and credit rating announcements. Journal of Banking andFinance, 28(11):2789–2811, 2004.

[21] Daniel Aunon-Nerin, Didier Cossin, Tomas Hricko, and Zhijiang Huang. Exploringfor the determinants of credit risk in credit default swap transaction data: Is fixed-income markets’ information sufficient to evaluate credit risk? SSRN ElectronicJournal, 2002.

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6 Appendix

6.1 Table 5: Predictor Variables

Variable Commodities Variable FX Spot Rates Variable Equity Indices Variable Interest Swap Rates

x1 Corn x13 FX EUR/USD x17 DAX Index x21 IR DKK 10yx2 Wheat x14 FX GBP/USD x18 OBX Index x22 IR EUR 10yx3 Gold x15 FX USD/NOK x19 OMX Index x23 IR NOK 10yx4 Brent Crude Oil x16 FX USD/SEK x20 SPX Index x24 IR SEK 10yx5 Aluminium x25 IR USD 10yx6 Copperx7 Leadx8 Nickelx9 Electricityx10 Coffeex11 Sugar no. 11x12 Nymex Crude Oil

6.2 Table 6: Variance Inflation Factors

Variable Commodities Variable FX Spot Rates Variable Equity Indices Variable Interest Swap Rates

x1 2.1092 x13 4.6678 x17 5.3420 x21 11.9926x2 2.0350 x14 1.6934 x18 4.6016 x22 10.9826x3 1.4653 x15 4.5068 x19 5.3484 x23 3.3275x4 5.7004 x16 6.0818 x20 3.5231 x24 5.8169x5 2.0222 x25 3.1835x6 2.8834x7 2.2430x8 1.9909x9 1.1315x10 1.2398x11 1.2874x12 5.8670

41

6.3 Figure 3: Residual analyses after transformation

−0.1 0.0 0.1 0.2

−0.

15−

0.05

0.05

0.10

0.15

0.20

Fitted values

Res

idua

ls

Residuals vs Fitted

−3 −2 −1 0 1 2 3

−4

−2

02

4

Theoretical Quantiles

Sta

ndar

dize

d re

sidu

als

Normal Q−Q

Basic Materials Sector

(a) Residual analysis for the BasicMaterials sector

−0.1 0.0 0.1 0.2

−0.

2−

0.1

0.0

0.1

0.2

Fitted values

Res

idua

ls

Residuals vs Fitted

−3 −2 −1 0 1 2 3

−4

−2

02

46

Theoretical Quantiles

Sta

ndar

dize

d re

sidu

als

Normal Q−Q

Energy Sector

(b) Residual analysis for the Energy sector

−0.1 0.0 0.1 0.2

−0.

15−

0.05

0.00

0.05

0.10

0.15

Fitted values

Res

idua

ls

Residuals vs Fitted

−3 −2 −1 0 1 2 3

−3

−2

−1

01

23

Theoretical Quantiles

Sta

ndar

dize

d re

sidu

als

Normal Q−Q

Financials Sector

(c) Residual analysis for the Financialssector

42

6.4 Figure 4: Residual analyses before transformation

0.9 1.0 1.1 1.2

−0.

10.

00.

10.

2

Fitted values

Res

idua

ls

Residuals vs Fitted

−3 −2 −1 0 1 2 3

−4

−2

02

46

Theoretical Quantiles

Sta

ndar

dize

d re

sidu

als

Normal Q−Q

Basic Materials Sector

(a) Residual analysis for the BasicMaterials sector

0.9 1.0 1.1 1.2

−0.

10.

00.

10.

2

Fitted values

Res

idua

ls

Residuals vs Fitted

−3 −2 −1 0 1 2 3

−4

−2

02

46

Theoretical Quantiles

Sta

ndar

dize

d re

sidu

als

Normal Q−Q

Energy Sector

(b) Residual analysis for the Energy sector

0.9 1.0 1.1 1.2

−0.

15−

0.05

0.00

0.05

0.10

0.15

Fitted values

Res

idua

ls

Residuals vs Fitted

−3 −2 −1 0 1 2 3

−2

02

4

Theoretical Quantiles

Sta

ndar

dize

d re

sidu

als

Normal Q−Q

Financials Sector

(c) Residual analysis for the Financialssector

43

6.5 Figure 5: Correlation plot for the Basic Materials sector

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

y X1.Corn

X2.Wheat

X3.Gold

X4.Brent.Crude.Oil

X5.Aluminium

X6.Copper

X7.Lead

X8.Nickel

X9.Electricity.MWh

X10.Coffee

X11.Sugar..11

X12.Nym

ex.Crude.Oil

X13.FX

.EUR.USD

X14.FX

.GBP

.USD

X15.FX

.USD

.NOK

X16.FX

.USD

.SEK

X17.DA

X.Index

X18.OBX

.Index

X19.OMX.Index

X20.SP

X.Index

X21.IR.DKK

.10y

X22.IR.EUR.10y

X23.IR.NOK.10y

X24.IR.SEK

.10y

X25.IR.USD

.10y

yX1.CornX2.Wheat

X3.GoldX4.Brent.Crude.Oil

X5.AluminiumX6.Copper

X7.LeadX8.Nickel

X9.Electricity.MWhX10.CoffeeX11.Sugar..11

X12.Nymex.Crude.OilX13.FX.EUR.USD

X14.FX.GBP.USDX15.FX.USD.NOK

X16.FX.USD.SEKX17.DAX.Index

X18.OBX.IndexX19.OMX.Index

X20.SPX.IndexX21.IR.DKK.10y

X22.IR.EUR.10yX23.IR.NOK.10y

X24.IR.SEK.10yX25.IR.USD.10y

44

6.6 Figure 6: Correlation plot for the Energy sector

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

y X1.Corn

X2.Wheat

X3.Gold

X4.Brent.Crude.Oil

X5.Aluminium

X6.Copper

X7.Lead

X8.Nickel

X9.Electricity.MWh

X10.Coffee

X11.Sugar..11

X12.Nym

ex.Crude.Oil

X13.FX

.EUR.USD

X14.FX

.GBP

.USD

X15.FX

.USD

.NOK

X16.FX

.USD

.SEK

X17.DA

X.Index

X18.OBX

.Index

X19.OMX.Index

X20.SP

X.Index

X21.IR.DKK

.10y

X22.IR.EUR.10y

X23.IR.NOK.10y

X24.IR.SEK

.10y

X25.IR.USD

.10y

yX1.CornX2.Wheat

X3.GoldX4.Brent.Crude.Oil

X5.AluminiumX6.Copper

X7.LeadX8.Nickel

X9.Electricity.MWhX10.CoffeeX11.Sugar..11

X12.Nymex.Crude.OilX13.FX.EUR.USD

X14.FX.GBP.USDX15.FX.USD.NOK

X16.FX.USD.SEKX17.DAX.Index

X18.OBX.IndexX19.OMX.Index

X20.SPX.IndexX21.IR.DKK.10y

X22.IR.EUR.10yX23.IR.NOK.10y

X24.IR.SEK.10yX25.IR.USD.10y

45

6.7 Figure 7: Correlation plot for the Financials sector

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

y X1.Corn

X2.Wheat

X3.Gold

X4.Brent.Crude.Oil

X5.Aluminium

X6.Copper

X7.Lead

X8.Nickel

X9.Electricity.MWh

X10.Coffee

X11.Sugar..11

X12.Nym

ex.Crude.Oil

X13.FX

.EUR.USD

X14.FX

.GBP

.USD

X15.FX

.USD

.NOK

X16.FX

.USD

.SEK

X17.DA

X.Index

X18.OBX

.Index

X19.OMX.Index

X20.SP

X.Index

X21.IR.DKK

.10y

X22.IR.EUR.10y

X23.IR.NOK.10y

X24.IR.SEK

.10y

X25.IR.USD

.10y

yX1.CornX2.Wheat

X3.GoldX4.Brent.Crude.Oil

X5.AluminiumX6.Copper

X7.LeadX8.Nickel

X9.Electricity.MWhX10.CoffeeX11.Sugar..11

X12.Nymex.Crude.OilX13.FX.EUR.USD

X14.FX.GBP.USDX15.FX.USD.NOK

X16.FX.USD.SEKX17.DAX.Index

X18.OBX.IndexX19.OMX.Index

X20.SPX.IndexX21.IR.DKK.10y

X22.IR.EUR.10yX23.IR.NOK.10y

X24.IR.SEK.10yX25.IR.USD.10y

46

6.8 Table 7: Models for the Basic Materials sector after best subsetsregression

Response variable: CDS Spread

R2adj BIC Cp

X6.Copper −0.353∗∗∗ −0.248∗∗∗ −0.352∗∗∗

(0.116) (0.085) (0.105)X7.Lead 0.127 0.144∗

(0.082) (0.081)X8.Nickel 0.083

(0.073)X9.Electricity.MWh −0.048

(0.040)X13.FX.EUR.USD −0.553∗∗ −0.469∗

(0.276) (0.256)X14.FX.GBP.USD 0.236

(0.217)X15.FX.USD.NOK 0.445∗∗ 0.420∗

(0.225) (0.219)X16.FX.USD.SEK 0.643∗∗∗

(0.168)X17.DAX.Index −0.518∗∗∗ −0.623∗∗∗ −0.555∗∗∗

(0.168) (0.155) (0.158)X19.OMX.Index −0.429∗∗ −0.548∗∗∗ −0.495∗∗∗

(0.191) (0.183) (0.184)X20.SPX.Index −0.312

(0.208)X22.IR.EUR.10y 0.089∗∗ 0.106∗∗∗

(0.041) (0.040)X25.IR.USD.10y −0.230∗∗∗ −0.258∗∗∗

(0.075) (0.072)Constant 1.464∗∗∗ 0.777∗∗∗ 1.458∗∗∗

(0.520) (0.238) (0.481)

Observations 362 365 365R2 0.491 0.460 0.483Adjusted R2 0.473 0.454 0.472Residual Std. Error 0.042 (df = 349) 0.043 (df = 360) 0.042 (df = 356)F Statistic 28.035∗∗∗ (df = 12; 349) 76.717∗∗∗ (df = 4; 360) 41.603∗∗∗ (df = 8; 356)

Note: ∗p<0.1; ∗∗p<0.05; ∗∗∗p<0.01

47

6.9 Table 8: Models for the Energy sector after best subsets regres-sion

Response variable: CDS Spread

R2adj BIC Cp

X3.Gold 0.300∗∗ 0.336∗∗∗

(0.135) (0.125)X4.Brent.Crude.Oil −0.229∗∗∗ −0.171∗∗

(0.083) (0.076)X6.Copper −0.387∗∗∗ −0.286∗∗∗ −0.330∗∗∗

(0.131) (0.103) (0.110)X7.Lead 0.106

(0.100)X10.Coffee 0.071

(0.059)X13.FX.EUR.USD −0.762∗

(0.397)X14.FX.GBP.USD 0.564∗∗ 0.439∗

(0.262) (0.246)X15.FX.USD.NOK 0.752∗∗ 0.594∗∗∗ 0.760∗∗∗

(0.308) (0.183) (0.210)X16.FX.USD.SEK −0.545

(0.397)X17.DAX.Index −0.407∗∗ −0.439∗∗∗ −0.367∗∗

(0.157) (0.146) (0.148)X20.SPX.Index −0.885∗∗∗ −0.927∗∗∗ −0.901∗∗∗

(0.236) (0.229) (0.228)X21.IR.DKK.10y −0.238

(0.153)X22.IR.EUR.10y 0.189∗

(0.107)X25.IR.USD.10y −0.065

(0.095)Constant 1.541∗ 1.061∗∗∗ 0.236

(0.828) (0.287) (0.459)

Observations 366 367 366R2 0.411 0.371 0.395Adjusted R2 0.387 0.364 0.383Residual Std. Error 0.050 (df = 351) 0.051 (df = 362) 0.051 (df = 358)F Statistic 17.474∗∗∗ (df = 14; 351) 53.445∗∗∗ (df = 4; 362) 33.420∗∗∗ (df = 7; 358)

Note: ∗p<0.1; ∗∗p<0.05; ∗∗∗p<0.01

48

6.10 Table 9: Models for the Financials sector after best subsets re-gression

Response variable: CDS Spread

R2adj BIC Cp

X3.Gold 0.357∗∗∗ 0.305∗∗∗ 0.410∗∗∗

(0.120) (0.110) (0.114)X6.Copper −0.216∗∗ −0.199∗∗

(0.100) (0.097)X9.Electricity.MWh −0.057 −0.066

(0.044) (0.044)X11.Sugar..11 0.058

(0.050)X13.FX.EUR.USD −1.547∗∗∗ −1.281∗∗∗ −1.495∗∗∗

(0.241) (0.194) (0.236)X14.FX.GBP.USD 0.456∗ 0.451∗

(0.236) (0.237)X17.DAX.Index −0.712∗∗∗ −0.915∗∗∗ −0.732∗∗∗

(0.184) (0.134) (0.181)X19.OMX.Index −0.366∗ −0.385∗

(0.207) (0.207)X20.SPX.Index −0.325 −0.650∗∗∗ −0.397∗

(0.224) (0.203) (0.218)X22.IR.EUR.10y 0.051

(0.045)X25.IR.USD.10y −0.132

(0.083)Constant 2.437∗∗∗ 2.543∗∗∗ 2.416∗∗∗

(0.247) (0.210) (0.240)

Observations 363 369 363R2 0.480 0.456 0.474Adjusted R2 0.464 0.450 0.462Residual Std. Error 0.046 (df = 351) 0.047 (df = 364) 0.046 (df = 354)F Statistic 29.507∗∗∗ (df = 11; 351) 76.172∗∗∗ (df = 4; 364) 39.935∗∗∗ (df = 8; 354)

Note: ∗p<0.1; ∗∗p<0.05; ∗∗∗p<0.01

49

TRITA -MAT-K 2017:05

ISRN -KTH/MAT/K--17/05--SE

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