copula-based top-down approaches in financial risk aggregation · dence structure between interest...

Number 32

Working Paper Series by the University of Applied Sciences of bfi Vienna

Copula-based top-down approaches in financial risk aggregation

December 2006

Christian CechUniversity of Applied Sciences of bfi Vienna

Abstract

This article presents the concept of a copula-based top-down ap-proach in the field of financial risk aggregation. Selected copulasand their properties are presented. Copula parameter estimation andgoodness-of-fit tests are explained and algorithms for the simulationof copulas and meta-distributions are provided. Further, the depen-dence structure between interest rate and credit risk factor changesthat are computed from sovereign and corporate bond indices is ex-amined. No clear pattern of the dependence structure can be observedas it varies substantially with the duration and the rating of the oblig-ors. This could indicate that top-down approaches are too simplisticto be implemented in practice. However, the results also suggest thatcopula-based approaches for the data sample at hand seem preferableto the assumption of a multivariate Gaussian distribution as none ofthe marginal distributions examined are normally distributed and asthe Gaussian copula’s fit in terms of the AIC is worse than that ofother copulas. Further, the Gaussian copula seems to underestimatethe probability of joint strong risk factor changes for the data sampleat hand.

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Contents

1 Introduction 5

2 Bottom-up and top-down approaches 7

3 Copula-based approaches 113.1 Introduction to copulas . . . . . . . . . . . . . . . . . . . . . . 113.2 Modelling the marginal distributions . . . . . . . . . . . . . . 193.3 Presentation of selected copulas . . . . . . . . . . . . . . . . . 20

3.3.1 Gaussian copula . . . . . . . . . . . . . . . . . . . . . . 263.3.2 Student t copula . . . . . . . . . . . . . . . . . . . . . 273.3.3 BB1 copula . . . . . . . . . . . . . . . . . . . . . . . . 273.3.4 Clayton copula . . . . . . . . . . . . . . . . . . . . . . 303.3.5 Gumbel copula . . . . . . . . . . . . . . . . . . . . . . 303.3.6 Frank copula . . . . . . . . . . . . . . . . . . . . . . . 35

3.4 Copula-parameter estimation . . . . . . . . . . . . . . . . . . 353.4.1 Maximum likelihood estimation . . . . . . . . . . . . . 353.4.2 Parameter estimation using correlation measures . . . . 373.4.3 Empirical copulas . . . . . . . . . . . . . . . . . . . . . 38

3.5 Goodness-of-fit tests . . . . . . . . . . . . . . . . . . . . . . . 383.6 Simulation of selected meta-distributions . . . . . . . . . . . . 40

3.6.1 Simulation of meta-Gaussian distributions . . . . . . . 433.6.2 Simulation of meta-Student t distributions . . . . . . . 433.6.3 Simulation of bivariate meta-BB1 distributions . . . . . 443.6.4 Simulation of bivariate meta-Clayton distributions . . . 443.6.5 Simulation of bivariate meta-Gumbel distributions . . . 453.6.6 Simulation of bivariate meta-Frank distributions . . . . 45

4 Implementation of a top-down approach and empirical evi-dence 46

5 Conclusion 61

Appendix 61Appendix A: Rank-based correlation measures . . . . . . . . . . . . 62Appendix B: GoF test – probability integral transform . . . . . . . 64Appendix C: Empirical results for non-autocorrelation-adjusted data 69

List of tables and figures 71

References 73

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1 Introduction

According to the Basel Committee on Banking Supervision [54], risk aggrega-tion refers to the development of quantitative risk measures that incorporatemultiple types or sources of risk. Amongst these types of risks are e.g. creditrisk, market risk (interest rate risk, stock price risk, etc.), insurance risk (lifeand property and casualty insurance), operational risk, liquidity risk, assetliability management (ALM) risk, business risk, etc. These quantitative riskmeasures, defined over a specific time horizon, may then be used to estimatethe economic capital that is needed to absorb unexpectedly high potentiallosses. Apart from the properties of the types of risks and the time horizon,the amount of economic capital depends on the rating that a financial in-stitution aspires, as the probability of default (i.e. the probability that theeconomic capital cannot absorb the realised losses) is related to the confi-dence level of the risk measure.1

Such reasoning also forms the basis of the new Basel II regulatory frame-work (Basel Committee on Banking Supervision [55] and European Parlia-ment and Council [56]), where banks are required to hold at least the mini-mum regulatory capital as a buffer against credit risk (regarded as the mainsource of banking risks), whose magnitude does not only depend on the sizeof the exposure but also on the riskiness of the credit portfolio.2 The one-yearsurvival probability (of the financial institution) is targeted at 99.9% per year(i.e. the expected probability of default is no more than 0.1%). Additionally,banks have to hold minimum regulatory capital for market risk in the tradingbook and for operational risk. The Basel II regulatory framework, however,does not account for diversification effects between risk types (credit, mar-ket and operational risk), as the minimum regulatory capital requirementsfor each risk-class are simply added to obtain the total minimum regulatory

1The economic capital to be held in this context is then defined as the value-at-riskwith confidence level α which implies that a default probability of (1− α) is conjectured.For critical remarks see e.g. Pezier [57]. While the value-at-risk has been criticised asit does not display the desirable feature of sub-additivity (see e.g. Arztner et al. [7]), itis widely used in practice. Alternative risk measure like e.g. the expected shortfall (alsotermed conditional value-at-risk, CVaR) are sub-additive and could easily be obtained bythe methodology presented in this paper. However, there is no direct linkage between theexpected shortfall and a financial institution’s probability of default.

2The specific formulae to compute the regulatory minimum capital requirement forcredit risk in the Basel II accord are derived on the basis of a structural model (Merton[50] model), where it is assumed that the credit portfolio is asymptotically fine-grained,i.e. it is assumed that idiosyncratic risk is diversified away completely (see e.g. Finger [25]and Gordy [33]).

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capital. This conservative approach implicitly assumes perfect positive cor-relation between the risk types.

The prudential rules of the Basel II accord should however not be mis-taken as a guideline on how to allocate economic capital efficiently. Rather,institution-internal models that go beyond the minimum regulatory require-ments of Basel II (‘pillar 1’) are used in practice.3 Risk aggregation modelsthat quantify the diversification effects seem to be one necessary foundationfor the efficient allocation of economic capital. Several approaches to riskaggregation have been proposed in the literature. Section 2 gives an intro-duction to top-down approaches as opposed to bottom-up approaches in thecontext of risk aggregation and reviews existing literature.

Copula-based approaches, presented in section 3, seem adequate andpreferable to the widely employed assumption of multivariate Gaussian dis-tributions4 of risk factor changes, if the risk factor changes are not normallydistributed.5 Section 3.1 gives an introduction to copula-based approachesin the context of top-down risk aggregation. Various parametric distributionfunctions that are used to model marginal distributions in the context of riskaggregation are shortly mentioned in section 3.2. Section 3.3 presents someselected bivariate and two multi-dimensional copulas in detail and comparestheir properties. Specific equations for copula functions and copula densitiesare also provided in this section. Different approaches to copula parameterestimation are presented in section 3.4, and goodness-of-fit tests are presentedin section 3.5. Finally, section 3.6 provides algorithms for the simulation ofthe presented copulas.

Section 4 first shortly addresses the results of two recent studies on theimplementation of a top-down risk aggregation model. These studies, us-ing institution-internal data, find that the risk-factor changes seem to beonly slightly correlated. In the remainder of section 4, daily market data(bond index returns) are used to examine the dependence structure betweeninterest rate risk and credit risk. The empirical results provide a very het-erogeneous picture of the dependence structure between these two risk factor

3Efforts on risk management system that go beyond the minimum capital requirementsof ‘pillar 1’ are also regulatorily required in ‘pillar 2’ of the Basel II accord.

4or other widely used assumptions on joint distributions that jointly model the marginaldistributions and their dependence structure like e.g. the multivariate Student t or Weibulldistributions and the highly flexible multivariate generalized hyperbolic distribution (seee.g. McNeil et al. [48], section 3.2.3).

5Not everybody agrees on this statement, see e.g. Mikosch [51].

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changes, depending on the maturity bands examined and the credit quality.The goodness-of-fit of six copulas and empirical evidence of positive tail de-pendence is examined for 25 data pairs with a sample size of N = 1, 727 each.

Section 5 concludes.

2 Bottom-up and top-down approaches

In general one can employ different approaches to aggregate different risktypes (for a review article see e.g. Saita [61] or Alexander [4]). These ap-proaches may broadly be classified into bottom-up and top-down approaches.6

Bottom-up approaches try to model the distribution of various risk factorsand their impact on risk types, such as credit risk, market risk, etc. Oneprominent example of a bottom-up approach is Credit Metrics, a credit riskmodel which derives the profit and loss distribution of a credit portfolio fromthe asset values of the obligors, which are modelled as linear combinations ofcorrelated industry index returns (see e.g. Crouhy et al. [14]). A bottom-upapproach in the context of risk aggregation would estimate the impact ofthese risk factors (industry index returns) and, if necessary, additional riskfactors (such as the interest rate term structure, credit spreads, etc.) on theprofits and losses of other lines of business (e.g. the market portfolio) andmodel the joint profit and loss distribution on that basis. Hence, the depen-dence between risk types (profits and losses of different lines of business) ismodelled indirectly: a joint distribution of risk factor changes is estimatedand the impact of these risk factor changes on the diverse financial portfo-lios’ profits and losses, defined as (generally non-linear) functions of the riskfactor changes, is modelled.

Top-down approaches, on the contrary, do not try to identify commonsingle risk factors that influence different types of risk, but rather start fromaggregated data, e.g. the profits and losses of different lines of business, suchas the returns of the credit portfolio or the market portfolio. Operational riskin this context would be modelled as a portfolio of risk exposures with non-positive profits and losses or returns. Empirical panel-data of (or assumptionson) the profits and losses or returns of these portfolios allow to estimate ajoint distribution of the total returns, or the ‘total risk’. The single com-ponents that constitute the financial portfolios (or, alternatively expressed,the single risk factors that influence the portfolio profits and losses) are not

6See e.g. Cech and Jeckle [12]. The approaches are also referred to as base-level andtop-level aggregation, see e.g. Aas et al. [1].

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Bottom-up approach Distribution of economic risk factors, e.g.

• interest rate term structure • credit spread term structure • equity returns • GDP growth • etc.

and dependence structure (copula) Profit and loss functions (domain: economic risk factors), e.g.

• market portfolio returns • credit portfolio returns • insurance portfolio returns • losses due to operational risk • etc.

Resulting in joint profits and losses / returns.

Top-down approach Distribution of portfolio returns, e.g.

• market portfolio returns • credit portfolio returns • insurance portfolio returns • losses due to operational risk • etc.

and dependence structure (copula) Joint profits and losses / returns.

Figure 1: Bottom-up and top-down approaches.

addressed in this approach. Figure 1 schematically depicts the bottom-upand the top-down approach.

In both approaches a common time horizon for the estimation of risk fac-tor changes has to be found. Ideally, the time horizon would correspond tothe internal capital allocation cycle which conventionally is one year. Gen-erally, the profit and losses or returns of credit and insurance risk are alsomeasured at least at this frequency and risk measures are estimated for aone-year horizon. Market portfolio profits and losses and associated riskmeasures are often measured and estimated on a daily basis, as the averageholding period of instruments in the market portfolio is generally short-termand also because of regulatory directives. If one assumes that the marketportfolio profits and losses are normally distributed and i.i.d.7, one can easilycompute one-year risk measures for the market portfolio by using the ‘square-root-time’ formula and estimate the one-year unexpected loss8 as a multipleof the one-day unexpected loss. The one-year unexpected loss is computedby multiplying the one-day unexpected loss by

√260 ≈ 16 (assuming 260

trading days per annum) and the one-year value at risk is the one-year un-expected loss minus the one-year expected profit. This approach howeverignores the usual pre-defined market risk management intervention policieslike stop-loss limits, etc. as the risk measures are computed on the basis of

7Independent and identically distributed, i.e. the returns are not autocorrelated.8I.e. the negative value of the one-sided confidence interval lower bound for a deviation

from the expected profit.

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the current portfolio composition. I.e. the risk measures are computed fora buy-and-hold portfolio, which leads to an overestimation (upward-bias9).On the other hand side, non-normality of daily market risk factor changeshas been widely documented. The univariate risk factor changes are oftenleptokurtic and left-skewed; furthermore the probability of joint extremelynegative returns is higher than implied by a multivariate normal distribution(see e.g. Fortin and Kuzmics [27]). This again leads to an underestimationof the risk measures (downward-bias) if normality of the market risk factorchanges is falsely assumed.

Aas et al. [1] in their model incorporate risk management interven-tion policies by simulating daily market portfolio returns under predefinedstop-loss policies and limits, using a constant conditional correlation (CCC)GARCH(1,1) model to account for volatility clustering and leptokurtic re-turn distributions. The distribution of 1-year market portfolio returns thatare obtained from the simulations of successive one-day returns are then usedto model the one-year market risk and its correlation with other risk types.This very promising approach is, however, only useful if there exists a suffi-ciently large data-set of e.g. credit or insurance portfolio profits and losseson an annual basis so that a model for aggregated risk can be calibrated.

Rosenberg and Schuermann [59] overcome the problem of short time-seriesby estimating linear regression functions to explain market and credit risk asfunctions of macroeconomic risk factors, using panel-data for quarterly re-turns of the market and credit portfolio returns of a set of large banks. Theregression functions are calibrated using 9 years of historical data. Assum-ing constant regression parameters, market and credit portfolio returns andtheir dependence structure are simulated by 29 years of historical quarterlymacroeconomic risk factor data as regressors (operational risk is modelledseparately).

If one wants to avoid the model risk associated with both models pre-sented above, institution-internal time series of profits and losses or returnsfor the different lines of business may be used to estimate a risk-aggregationmodel. This again results in a very small data sample for the calibration ofthe model if annual data is used as generally there are no long time series ofe.g. credit portfolio profits and losses available. Using monthly institution-internal data seems to be a promising compromise.

9Hickman et al. [38] show that risk management intervention policies can substantiallyreduce the risk.

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In both top-down and bottom-up approaches, the task of estimating jointdistributions (joint economic risk factors changes in the context of bottom-upapproaches and joint portfolio profits and losses in the context of top-downapproaches), may be decomposed into (a) the estimation of the marginal dis-tributions (univariate risk factor changes or portfolio profits and losses) and(b) the estimation of the dependence structure, if a copula-based approachis used. Copulas may be thought of as a more flexible version of correla-tion matrices that are widely used in risk management models that assumejoint normality. Copula-based approaches are discussed in detail in section 3.

Work on top-down approaches has been done by Kuritzkes et al. [47](insurance, market, credit, ALM, operational and business risk), Ward andLee [69] (insurance, market, credit, ALM and operational risk), Dimakos andAas [20] (market, credit and operational risk), Rosenberg und Schuermann[59] (market, credit and operational risk) and Tang and Valdez [67] (differ-ent types of insurance risk). While Kuritzkes et al. [47] in their simplifyingapproach assume a joint normal distribution of the risk factor changes10,the latter four articles describe a copula-based approach to aggregate therisk of financial portfolios. Ward and Lee [69] and Dimakos and Aas [20]use a Gaussian copula to combine the marginal distributions. The latterstudy only models pairwise dependence between credit and market risk andcredit and operational risk without specifically modelling the dependence be-tween market and operational risk. Rosenberg and Schuermann [59] estimatethe marginal distributions’ parameters and their correlation measures usingmarket data and values that were reported in other studies and regulatoryreports.11 The marginal distributions are combined by a Gaussian and a Stu-dent t copula to point out the effects of positive tail dependence (see section3). They report that the choice of the copula (Gaussian or Student t) has amore modest effect on risk than has the business mix (the weights assignedto a bank’s financial portfolios). Tang and Valdez [67] use semi-annual datafor loss ratios for the aggregate Australian insurance industry from 1992 to2002 to calibrate a model that aggregates risks of different lines of insurancebusiness (motor, household, fire and industrial special risks, liability, andcompulsary third party insurance). The marginal distributions are modelled

10Hall [35] points out that the economic capital may be severely underestimated if a jointGaussian distribution is assumed while indeed the marginal distributions are non-normal.

11The calibration of the marginal market and credit portfolio return distributions isdone for data that was obtained by a simulation in a bottom-up manner. The aggregationof these risks is done in a top-down manner, where the correlation matrix reported inKuritzkes et al. [47] is used.

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as gamma, log-normal and pareto distributions and the consequences of as-suming Gaussian, Student t and Cauchy copulas are addressed.

Work on bottom-up approaches has been done by Medova and Smith[49] (market and credit risk), Alexander and Pezier [5] and Aas et al. [1].Medova and Smith [49] use Monte Carlo simulations to allow for a varyingexposure of the credit portfolio (employing a structural credit risk model).Alexander and Pezier [5] estimate multiple linear regression models, regress-ing the profits and losses of 8 business units 12 on 6 risk factors 13. Pearson’scorrelation coefficient is used as dependence measure. To account for taildependence (a higher probability of joint extreme events as compared to theGaussian distribution/copula; see section 3) the authors suggest to use thetail correlations rather than the usual overall correlations. Aas et al. [1] usea bottom-up approach to aggregate market, credit and ownership risk. Foraggregating (additionally) operational and business risk, they use a top-downapproach (employing a Gaussian copula).

3 Copula-based approaches

3.1 Introduction to copulas

Copula-based approaches are a rather new methodology in risk management.The term copula was introduced by Sklar [66] in 1959 (a similar conceptfor modelling dependence structures of joint distributions was independentlyproposed by Hoffding [39] some twenty years earlier). Recent textbooks oncopulas are e.g. Joe [42] and Nelsen [52], [53].

Copulas are functions that combine or couple (univariate) marginal distri-butions to a multivariate joint distribution. Sklar’s theorem (using a slightlydifferent notation in the original article) states that a n-dimensional joint dis-tribution function F (x) evaluated at x = (x1, x2, . . . , xn) may be expressedin terms of the joint distribution’s copula C and its marginal distributionsF1, F2, . . . , Fn as

12The business units are: Corporate finance, Trading and sales, Retail banking, Com-mercial banking, Payment and settlement, Agency and custody, Asset management, andRetail brokerage.

13Risk factors: 1Y treasury rate, 10Y - 1Y treasury rate (slope), implied interest ratevolatility, S&P 500 index, S&P 500 implied volatility, 10Y credit spread.

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F (x) = C (F1(x1), F2(x2), . . . , Fn(xn)) , x ∈ Rn. (1)

The copula function C is itself a multivariate distribution with uniformmarginal distributions on the interval U1 = [0, 1], C : Un

1 → U1. Reformu-lating formula 1 yields

C(u) = F(F−1

1 (u1), F2(u2)−1, . . . , Fn(un)−1

), u ∈ Un

1 , (2)

where u = (u1, u2, . . . , un) = (F1(x1), F2(x2), . . . , Fn(xn)) are the respec-tive univariate marginal distributions.

Thus, a copula-based approach allows a decomposition of a joint distri-bution into its marginal distributions and its copula. On the other handmarginal distributions may be combined to a joint distribution assuming aspecific copula. The crucial point in using a copula-based approach is thatit allows for a separate modelling of

• the marginal distributions (i.e. the univariate profit and loss or returndistributions) and

• the dependence structure (the copula).

Figure 2 displays an example for the combination of two marginal dis-tributions to a joint bivariate distribution. Assume that a financial insti-tution holds two portfolios, a market portfolio and a credit portfolio. Themarket portfolio’s annual return distribution is modelled as random variablerM ∼ 0.1+0.15·t5, where t5 is a Student’s t distributed random variable withν = 5 degrees of freedom. The credit portfolio’s annual return distribution ismodelled as rC ∼ ln(1.05 ·B(30, 1.1)), where B(30, 1.1) is a beta distributedrandom variable. Both portfolios exhibit ‘fat tails’, assigning a higher proba-bility to extreme events than a normal distribution. The returns of the creditportfolio are heavily left-skewed, assigning a higher probability to extremelosses than to extreme gains. Table 1 displays mean and median values, thestandard deviation, skewness, and excess-kurtosis of the two return distribu-tions.14 The data shows that both return distributions are non-normal. In a

14The moments of the two distributions displayed in table 1 are the sample estimates of1,000,000 simulated returns, using Monte Carlo simulation. Using this simulated valuesfor the depiction of the credit portfolio returns’ density in figure 2 as kernel smootheddensities with Gaussian kernels, this explains the small right tail above the 0.05-thresholddisplayed in figure 2. The simulated credit portfolio returns will not take on a value ofgreater than 0.5, as the beta distribution is defined on the [0,1]-interval. (For a primeron kernel smoothed densities see e.g. Scott and Sain [64]; on beta distributions, see e.g.Johnson et al. [44], Chapter 25.)

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-0.1 -0.05 0 0.050

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20

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rC = FC-1(uC)

0.25 0.5 0.75

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mar

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-0.4 -0.2 0 0.2 0.4 0.60

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rM = FM-1(uM)

Figure 2: Example of the combination of a market and a credit portfoliomarginal return distributions to a joint returns distribution using a copula-based approach.

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rM rC

mean 0.1000 0.0122median 0.1000 0.0225standard deviation 0.1930 0.0349skewness 0 -1.9172excess kurtosis 4.8599 5.6499

Table 1: Sample moments for rM ∼ 0.1+0.15·t5 and rC ∼ ln(1.05·B(30, 1.1)).

copula-based approach these marginal distributions may easily be combinedto a joint distribution, as shown in figure 2.

Apart from the ability to combine arbitrary marginal distributions to ajoint distribution, copula-based approaches allow for a specific modelling ofthe dependence structure, i.e. the copula.

One frequently observed empirical evidence is that extreme joint mar-ket movements are more frequently observed than implied by a multivariateGaussian distribution that is often used in market risk models.15 This em-pirical evidence is sometimes referred to as ‘correlation-breakdown’.

Copula-based approaches allow for a flexible modelling of the probabilityof joint extreme observations (unconditional on the marginal distributions).For example, a Student t copula assigns a higher probability to joint extremeobservations than does a Gaussian copula. This higher probability of jointextreme observations as compared to the Gaussian copula is referred to aspositive tail dependence.

As an example, figure 3 shows scatter plots of two jointly distributedstandard normal random variables. These standard normal marginal distri-butions are combined by a Gaussian copula, a Student t copula, a Claytoncopula, and a Gumbel copula, respectively. The resulting joint distributions(for arbitrary marginal distributions) are referred to as meta-Gaussian, meta-Student t, meta-Clayton and meta-Gumbel distributions. The top row showsscatter plots of simulated joint distributions that have a correlation (in termsof Spearman’s rho16) of 0.4. The bottom row shows corresponding scatter

15In passing, note that the multivariate Gaussian distribution in ‘copula-based ap-proaches terms’ is a set of univariate Gaussian marginal distributions that are combinedby a Gaussian copula.

16In copula-based approaches, rank-based correlation measures such as Spearman’s rhoand Kendall’s tau are preferable to the widely known Pearson correlation measure that is

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−3 0 3

−3

0

3

(meta−)Gaussian

−3 0 3

−3

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meta−Student t

−3 0 3

−3

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meta−Clayton

−3 0 3

−3

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meta−Gumbel

−3 0 3

−3

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(meta−)Gaussian

−3 0 3

−3

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meta−Student t

−3 0 3

−3

0

3

meta−Clayton

−3 0 3

−3

0

3

meta−Gumbel

ρS = 0.4:

ρS = 0.8:

Figure 3: Simulation scatter plots of bivariate meta-Gaussian, meta-Studentt, meta-Clayton and meta-Gumbel distributions. The top row shows scatterplots of joint distributions with a Spearman’s rho correlation measure ofapproximately 0.4, the bottom row shows scatter plots of joint distributionswith a Spearman’s rho of approximately 0.8. Both marginal are standardnormally distributed.

plots for joint distributions with a correlation of 0.8.

It can be seen that for identical marginal distributions and Spearman’srho the Student t copula assigns a higher probability to joint extreme eventsthan does the Gaussian copula. Assigning an equal probability to joint ex-treme positive deviations and to joint extreme negative deviations from themedian value, the Student t copula displays symmetric tail dependence.

Asymmetric tail dependence is prevalent if the probability of joint ex-treme negative realisations differs from that of joint extreme positive reali-sations. In figure 3 it can be seen that the Clayton copula assigns a higherprobability to joint extreme negative events than to joint extreme positiveevents. The Clayton copula is said to display lower tail dependence, whileit displays zero upper tail dependence. The converse can be said about theGumbel copula (displaying upper but zero lower tail dependence). Table 2

used in the context of multivariate normal distributions. A short note on Spearman’s rhoand Kendall’s tau is given in Appendix A.

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copula:tail dep. Gaussian Student t BB1 Clayton Gumbel Frank

lower no yes yes yes no noupper no yes yes no yes no

symmetric yes yes no no no yes

Table 2: Summary of which bivariate copulas display lower and upper taildependence and whether the positive tail dependence is symmetric.

gives an overview of which of the copulas presented in this article displayupper or lower tail dependence.17 In section 3.3 we will give a formal defi-nition of upper and lower tail dependence and provide explicit formulas forthe magnitude of the tail dependence.

Some copulas allow to model both positive and negative dependence intheir ‘standard’ versions by assigning appropriate copula-parameters. Amongstthese copulas are e.g. the Gaussian, Student t and Frank copula. Figure 4displays the bivariate densities of these 3 copulas for a Spearman’s row of 0.4(top row) and for a Spearman’s rho of -0.4 (bottom row).

Other (bivariate) copulas like e.g. the BB1 copula and its two spe-cial cases, the Clayton and Gumbel copula in their ‘standard’ version al-low to model positive dependence only.18 Copula rotation allows to trans-form copulas such that they may be used to model negative dependencealso. Further, copula rotation allows to transform (bivariate) copulas de-pending on whether and/or where the empirical data at hand requires thecopula to display lower, upper or zero tail dependence. Denoting a bi-variate copula density as c(u1, u2), the so-called survival copula’s densityis c−−(u1, u2) = c(1 − u1, 1 − u2).

19 In the case of e.g. the Gumbel copula,the survival copula is used to model lower tail dependence and no uppertail dependence. In order to model discordance with e.g. a BB1 copula,

17The flexible BB1 copula may also display either zero upper or zero lower tail depen-dence or symmetric tail dependence, depending on the parameterisation. In specific casesthe Gaussian and Student t copula may display also positive and no tail dependence,respectively. See e.g. table 8 in section 3.3.

18In fact, the Clayton copula may also be used in its standard version to model negativedependence if the copula parameter θ ∈ [−1, 0). Such a parameterisation is not furtherconsidered in the present article.

19While the bivariate copula C(u1, u2) returns the probability that both uniformly dis-tributed marginal distributions take on values less than or equal to u1 and u2, the survivalcopula C−−(u1, u2) returns the probability that both marginal distributions take on valuesgreater than u1 and u2, respectively.

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00.5

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Gaussian copula, ρS = 0.4

u1

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u1

dens

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Figure 4: Densities of bivariate Gaussian, Student t and Frank copulas.These copulas allow to model both concordance and discordance. The cop-ulas in the top row display a Spearman’s rho of approximately 0.4 (copulaparameters: Gaussian: ρ = 0.42, Student t: ν = 3 and ρ = 0.43, Frank:θ = 2.61). The copulas in the bottom row display a Spearman’s rho ofapproximately -0.4 (copula parameters: Gaussian: ρ = −0.42, Student t:ν = 3 and ρ = −0.43, Frank: θ = −2.61). The densities are computed onthe interval [0.01, 0.99]2.

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0.5

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1

11.2

1.2

23

Clayton C++

u1

u 2

0.2 0.4 0.6 0.8

0.2

0.4

0.6

0.8

0.5

0.51

1

1.2

1.2

23

Clayton C−+

u1

u 2

0.2 0.4 0.6 0.8

0.2

0.4

0.6

0.8

0.5

0.51

11.2

1.2

2

3

Clayton C+−

u1

u 2

0.2 0.4 0.6 0.8

0.2

0.4

0.6

0.80.5

0.511

1.2

1.2

23

Clayton C−−

u1

u 2

0.2 0.4 0.6 0.8

0.2

0.4

0.6

0.8

0.5

0.5

1

1

1.2

1.2

2

2

3

Gumbel C++

u1

u 2

0.2 0.4 0.6 0.8

0.2

0.4

0.6

0.8

0.5

0.51

1

1.2

1.2

2

2

3

Gumbel C−+

u1

u 2

0.2 0.4 0.6 0.8

0.2

0.4

0.6

0.8

0.5

0.51

1

1.2

1.2

2

2

3

Gumbel C+−

u1

u 2

0.2 0.4 0.6 0.8

0.2

0.4

0.6

0.8

0.5

0.5

1

11.

2

1.2

2

2

3

Gumbel C−−

u1

u 2

0.2 0.4 0.6 0.8

0.2

0.4

0.6

0.8

Figure 5: Contour plots of the densities of a Clayton (top row) and a Gumbel(bottom row) copula C, and of their rotated versions C−+, C+− and C−−.Spearman’s rho for both copulas in their standard version is approximately0.4 (copula parameters: Clayton: θ = 0.76, Gumbel: θ = 1.38). The densitiesare computed on the interval [0.01, 0.99]2.

the rotated versions C−+ or C+− with densities c−+(u1, u2) = c(1 − u1, u2)and c+−(u1, u2) = c(u1, 1 − u2) are used. Figure 5 displays contour plots ofClayton and Gumbel copulas’ densities and of the rotated versions’ densities.

To present the consequences of assumptions on the copula in the contextof economic capital estimation, let us return to our simplified example, wherewe assumed that a bank holds only two portfolios, a market portfolio withannual returns rM ∼ 0.1+0.15 · t5 and a credit portfolio with annual returnsrC ∼ ln(1.05 · B(30, 1.1)). The correlation in terms of Spearman’s rho isρS = 0.4. Let us further assume that equal weights are assigned to theseportfolios, such that the bank’s total return in year t is rt = 0.5rM,t +0.5rC,t,where rM,t and rC,t are the realised returns of the market and the creditportfolio in year t, respectively. Table 3 shows several quantiles of total re-turn distributions.20 The quantiles correspond to average one-year defaultprobabilities of Moody’s ratings from 1920 to 2004 reported in Hamilton etal. [36], p.35. A bank that aspires a rating of e.g. ‘Ba’ has to hold enougheconomic capital such that the total losses exceed the economic capital witha probability of no more than 1.31%. The quantiles were computed under

20The quantiles are obtained by a Monte Carlo simulation with 1,000,000 simulationsusing antithetic sampling.

18

quantile 0.0432 (B) 0.0131 (Ba) 0.0030 (Baa) 0.0006 (Aa)meta-Gaussian -0.1202 -0.2016 -0.3153 -0.4661meta-Student t -0.1214 -0.2083 -0.3363 -0.5092meta-Clayton -0.1265 -0.2183 -0.3482 -0.5169correlation=1 -0.1406 -0.2351 -0.3695 -0.5427

Table 3: Quantiles of the total return distribution, corresponding to averageMoody’s rating 1-year default probabilities, if meta-Gaussian, meta-Studentt and meta-Clayton distributions with a Spearman’s rho of 0.4 are assumed.

the assumption of a Gaussian copula, a Student t copula with ν = 3 degreesof freedom and a Clayton copula. Additionally, in order to demonstrate thediversification effect, the quantiles were computed under the assumption ofperfect positive correlation. As can be seen in table 3, the effect of posi-tive tail dependence (Student t copula and Clayton copula) increases, as thequantile decreases. In our simplistic example the economic capital to be heldunder the assumption of a Student t (Clayton) copula exceeds the economiccapital under the assumption of a Gaussian copula by 0.94% (4.92%), if a‘B’-rating is aspired, and by 7.10% (8.34%), if a ‘Aa’-rating is aspired.

Before presenting some selected copulas in detail in subsection 3.3, weshall shortly address how the marginal distributions may be modelled in thefollowing subsection.

3.2 Modelling the marginal distributions

In the context of top-down risk aggregation models, the following parametricdistribution functions are widely used to model the marginal distributions:

• Market portfolio returns

– Generalized hyperbolic (GH) distribution21, or one if its specialcases such as the

– Normal inverse Gaussian (NIG) distribution or

– Student t and Gaussian distributions.

• Credit portfolio returns

– Beta distribution

21See e.g. Aas and Haff [2]

19

– Weibull distribution

• Insurance portfolio returns and operational risk

– Pareto distribution

– log-normal distribution

– Gamma distribution

Alternatively nonparametric approaches like e.g. the use of kernel-smoothedempirical distribution functions are widely employed.22

3.3 Presentation of selected copulas

This section presents some selected copulas from the family of elliptical andArchimedean copulas. These are

• Elliptical copulas

– Gaussian copula

– Student t copula

• Archimedean copulas

– BB1 copula and its two special cases, the

– Clayton copula and the

– Gumbel copula.

– Frank copula

Bivariate copula functions C(u1, u2), i.e. the probability that both uni-formly distributed marginal distributions jointly take on value less than orequal to u1 and u2, respectively, are presented in table 4. Bivariate copuladensities that are needed in the context of parameter estimation and for thedepiction of data are presented in table 5.

The Gaussian and Student t copulas in their ‘standard versions’ allow fora higher flexibility than the Archimedean copulas by enabling a modelling ofpairwise correlations that form the elements of the copula parameter matrixP (‘capital Greek letter rho’).

22For a primer on kernel smoothing, see e.g. Scott and Sain [64]. More detailed infor-mation can be found in Wand and Jones [68] and Silverman [65]

20

copula parameters θ copula function C(u1, u2; θ)

Gaussian ρ ∈ [−1, 1] Φρ (Φ−1(u1), Φ−1(u2)) =

∫ Φ−1(u1)−∞

∫ Φ−1(u2)−∞

1

2π√

1−ρ2exp

(2ρst−s2−t2

2(1−ρ2)

)dsdt

or equivalently (see Roncalli [58])

∫ u10 Φ

(Φ−1(u2)−ρΦ−1(s)√

1−ρ2

)ds

where Φρ is the bivariate standard normal distributionfunction with parameter ρ, and Φ−1 is the functionalinverse of the univariate standard normal c.d.f. Φ.

Student t ν ∈ (0,∞) tν,ρ (t−1ν (u1), t

−1ν (u2)) =

ρ ∈ [−1, 1]∫ t−1

ν (u1)−∞

∫ t−1ν (u2)−∞

1

2π√

1−ρ2

(1 + s2+t2−2ρst

ν(1−ρ2)

)− ν+22 dsdt

or equivalently (see Roncalli [58])

∫ u10 tν+1

(√ν+1

ν+t−1ν (s)2

t−1ν (u2)−ρt−1

ν (s)√1−ρ2

)ds

where tν,ρ is the bivariate Student t distribution andt−1ν is the functional inverse of the univariate

Student t c.d.f with ν degrees of freedom tν(.).

BB1 δ ∈ [1,∞), θ ∈ (0,∞)

(1 +

[(u−θ

1 − 1)δ

+(u−θ

2 − 1)δ] 1

δ

)− 1θ

Clayton θ ∈ (0,∞)(u−θ

1 + u−θ2 − 1

)− 1θ

Gumbel θ ∈ [1,∞) exp(−[(− ln u1)

θ + (− ln u2)θ] 1

θ

)

Frank θ ∈ (−∞,∞)\0 −1θln(1 +

(e−θu1−1)(e−θu2−1)e−θ−1

)Table 4: Selected bivariate copula functions.

21

copula probability density function c(u1, u2; θ) = ∂2C(u1,u2;θ)∂u1∂u2

Gaussian 1√1−ρ2

exp(

2ρy1y2−y21−y2

22(1−ρ2) + y2

1+y22

2

),

where y1 = Φ−1(u1), y2 = Φ−1(u2) and Φ−1(.) is thefunctional inverse of the standard normal c.d.f. Φ(.).

Student t fν,ρ

(t−1ν (u1), t−1

ν (u2))

1fν(t−1

ν (u1))1

fν(t−1ν (u2))

,

where fν,ρ is the p.d.f of the standard Student t distributionfunction with ν degrees of freedom and correlation matrix ρ,fν is the p.d.f of the univariate standard Student t distributionand t−1

ν is the functional inverse of the univariate Student tc.d.f with ν degrees of freedom t−1

ν (.).

BB1(u−θ

1 − 1)δ−1 (

u−θ2 − 1

)δ−1u−θ−1

1 u−θ−12 ·

·[(1 + θ)a−

1θ−2b

2δ−2 + (δθ − θ)a−

1θ−1b

1δ−2],

where a =(1 + b

1δ

)and b =

[(u−θ

1 − 1)δ

+(u−θ

2 − 1)δ]

.

Clayton (1 + θ)u−θ−11 u−θ−1

2

(u−θ

1 + u−θ2 − 1

)− 1θ−2

Gumbel exp(a) (− ln u1)θ−1(− ln u2)

θ−1

u1u2

[b

2θ−2 + (θ − 1)b

1θ−2],

where a = −b1θ , and b =

[(− lnu1)θ + (− lnu2)θ

].

Frank θηe−θ(u1+u2)

[η−(1−e−θu1)(1−e−θu2)]2,

where η = 1− e−θ.

Table 5: Probability density functions of selected bivariate copulas.

22

copula copula function C(u; θ)

Gaussian∫ Φ−1(u1)−∞ . . .

∫ Φ−1(un)−∞

1√(2π)n|P|

exp(−1

2x′P−1x

)dx

parameters θ: where Φ−1(.) is the functional inverse of the univariate standardP normal c.d.f. Φ(.).

Student t∫ t−1

ν (u1)−∞ . . .

∫ t−1ν (un)−∞

Γ( ν+n2 )

Γ( ν2 )√

(πν)n|P|

(1 + x′P−1x

ν

)− ν+n2 dx,

parameters θ: where t−1ν is the functional inverse of the univariate Student t c.d.f.

ν,P with ν degrees of freedom tν(.) and Γ(.) is the Gamma function.

Table 6: n-dimensional Gaussian and Student t copula functions.

P =

1 ρ1,2 ρ1,3 · · · ρ1,d

ρ1,2 1 ρ2,3 · · · ρ2,d

ρ1,3 ρ2,3 1...

......

. . ....

ρ1,d ρ2,d · · · · · · 1

(3)

Besides the copula parameter P, the Student t copula has an additionalscalar parameter ν, the degrees of freedom. These can, however, not be usedto explicitly model pairwise dependencies. Rather, the copula parameter ν,being a scalar, affects all pairwise dependencies in the same manner. Table6 and 7 provide the n-dimensional copula functions of Gaussian and Studentt copulas and their densities, respectively.

If the dependence of more than two dependent variables is to be mod-elled, the Archimedean copulas’ flexibility seems very restricted as eitheronly one (Clayton, Gumbel, Frank copulas) or only two (BB1 copula) scalarparameters are used to parameterise the joint multidimensional dependencestructure.23. This lack of flexibility can however be overcome by using hi-erarchical Archimedean copulas that are e.g. presented in Savu and Trede[62]. A hierarchical copula joins two (or more) bivariate (or higher dimen-sional) Archimedean copulas by another Archimedean copula. The structureof this approach is depicted in figure 6. If in the context of risk aggrega-tion we want to combine the returns of, say, four financial portfolios we first

23Formulas for n-dimensional Archimedean copulas can be found e.g. in Cherubini etal. [13], pp.147ff.

23

copula probability density function c(u; θ) = ∂nC(u;θ)∂u1...∂un

Gaussian φP (Φ−1(u1), . . . , Φ−1(un))

∏ni=1

1φ(Φ−1(ui))

,

where φP(.) is the p.d.f. of the multivariate standard normal distributionwith correlation matrix P, φ(.) is the p.d.f. of the univariate standardnormal distribution, and Φ−1(.) is the functional inverse of the univariatestandard normal c.d.f. Φ(.).

Student t fν,P

(t−1ν (u1), . . . , t−1

ν (un))∏n

i=11

fν(t−1ν (ui))

,

where fν,P is the p.d.f of the standard Student t distributionfunction with ν degrees of freedom and correlation matrix P,fν is the p.d.f of the univariate standard Student t distributionand t−1

ν is the functional inverse of the univariate Student tc.d.f with ν degrees of freedom t−1

ν (.).

Table 7: n-dimensional Gaussian and Student t copula density functions.

calibrate two copulas that combine the returns of portfolio 1 and 2, and port-folio 3 and 4, respectively. These two copulas are then combined by a thirdcopula. Parameter estimation is done in the same manner as for the othercopulas (see subsection 3.4). For the simulation of hierarchical copulas, theconditional inversion method has to be used (see Savu and Trede [62], p.10f).

The concept of positive upper and lower tail dependence of bivariatecopulas has already been introduced in section 3.1. Loosely speaking, lowertail dependence λL describes the conditional probability that one of the tworandom variables takes values below a very small value, given that also theother random variable takes very small values. Upper tail dependence λU

can be described analogously. Formally,

λL = limα→0+

P (u1 ≤ α|u2 ≤ α) = limα→0+

C(α, α)

αand (4)

λU = limα→1−

P (u1 > α|u2 > α) = limα→1−

1− 2α + C(α, α)

1− α, (5)

provided the limit exists with λL, λU ∈ [0, 1]. For symmetric copulas λL =λU . Formulas for the magnitude of lower and upper tail dependence for theselected copulas are presented in table 8.

The concept of copula rotation has also been introduced already (seefigure 5 on p.18). Copulas may be rotated, depending on whether and/orwhere the empirical data at hand requires the copula to display positive,

24

marginal distribution 1:U1 = F1(X1) [0,1]




copula 1:C1(U1, U2) [0,1]

copula 2:C2(U3, U4) [0,1]

copula 3:C3(C1(U1, U2), C2(U3, U4) ) [0,1]

Figure 6: Structure of a four-dimensional hierarchical Archimedean copula.

copula lower tail dependence λL upper tail dependence λU

Gaussian λL = λU = 0 (iff ρ < 1; λL = λU = 1 iff ρ = 1)

Student t λL = λU = 2tν+1

(−√

ν + 1√

1−ρ1+ρ

)where tν+1 is the univariate Student t c.d.f with ν + 1 degrees of freedom

BB1 λL = 2−1δθ λU = 2− 2

1δ

Clayton λL = 2−1θ λU = 0

Gumbel λL = 0 λU = 2− 21θ

Frank λL = λU = 0

Table 8: Lower and upper tail dependence, λL and λU , of selected bivariatecopulas.

25

negative or zero tail dependence. Let us define the vector u = (u1, u2),where ui = 1− ui.

24 Then the following observations are true

• u1 and u2 have copula C−−(u1, u2) = u1+u2−1+C(1−u1, 1−u2) withdensity c−−(u1, u2) = c(1 − u1, 1 − u2). C−− is referred to as survivalcopula.

• u1 and u2 have copula C−+(u1, u2) = u2 − C(1 − u1, u2) with densityc−+(u1, u2) = c(1− u1, u2).

• u1 and u2 have copula C+−(u1, u2) = u1 − C(u1, 1 − u2) with densityc+−(u1, u2) = c(u1, 1− u2).

If C(u1, u2) is symmetric, then c(u1, u2) = c−−(u1, u2) and c−+(u1, u2) =c+−(u1, u2).

If we want to use a copula C which is suited to describe upper tail depen-dence to model lower tail dependence, the corresponding C−− copula has tobe employed. If we want to use a copula C which is only suited to describepositive dependence to model negative dependence, C−+ or C+− have to beemployed.

In the sub-sections below, the densities of selected bivariate copulas aremore closely regarded.

3.3.1 Gaussian copula

The Gaussian copula is the most widely used copula. It is the copula thatis implied by a multivariate Gaussian distribution (normal distribution). Amultivariate Gaussian distribution is a set of normally distributed marginaldistributions that are combined by a Gaussian copula. If other than nor-mal marginal distributions are combined by a Gaussian copula, the resultingjoint distribution is referred to as meta-Gaussian distribution. Figure 2 onp.13 contains an example of a meta-Gaussian distribution. Figure 7 displayssurface plots of Gaussian copula densities with a Spearman’s rho of 0.4 (topleft) and 0.8 (bottom left). The bivariate copula density goes to infinity atu = (0, 0), and u = (1, 1) for ρ > 0 and at u = (0, 1) and u = (1, 0) forρ < 0. On the right hand side, corresponding (meta-) Gaussian distributiondensities with standard normal marginal distributions are displayed.

24Note that U = 1 − U is uniformly distributed on the unit interval if U is uniformlydistributed on the unit interval.

26

We shall use the Gaussian copula as benchmark to which we compare theother copulas.

3.3.2 Student t copula

The Student t copula is the copula that is implied by a multivariate Stu-dent t distribution (Student t marginal distributions combined by a Studentt copula). Like the Gaussian copula, the Student t copula has the param-eter ρ in the bivariate case (table 4) or P in higher dimensions (table 6).Additionally it has the (scalar) parameter ν, the degrees of freedom. Thehigher ν, the higher the positive tail dependence (see table 8). Figure 8displays surface plots of Student t copula densities with a Spearman’s rhoof 0.4 (top left) and 0.8 (bottom left). The bivariate copula density goes toinfinity at u = (0, 0), u = (0, 1), u = (1, 0), and u = (1, 1). On the righthand side, corresponding contour plots of meta-Student t distribution densi-ties with standard normal marginal distributions are displayed. Additionally,contours of a Gaussian distribution with identical marginal distributions andSpearman’s rho are plotted in light grey for comparison. It can be seen thatthe Student t copula assigns a higher density to events near all four cornersthan the Gaussian copula does. Differences between the Student t copulaand meta-distribution’s densities to those of the Gaussian copula with iden-tical Spearman’s rho are summarised in the contour plots at the bottom offigure 8, where grey shaded areas indicate that the densities of the Studentt copula or meta-distribution exceed that of the Gaussian copula.

As the degrees of freedom of a Student t copula increase, the copulaapproaches a Gaussian copula. The Gaussian copula can be regarded asa limiting case of the Student t copula, where ν → ∞. More in-depthinformation on Student t copulas can e.g. be found in Demarta and McNeil[18].

3.3.3 BB1 copula

The two-parametric BB1 copula allows for a high flexibility in modelling pos-itively correlated bivariate dependence structures (copula parameters δ andθ). Figure 9 displays contour plots of BB1 copula densities with an identicalSpearman’s rho of 0.4. The plot on the very left and on the very right handside are limiting cases of the BB1 copula. The very left BB1 copula has thecopula parameter δ = 1. This special case of the BB1 copula is called aClayton copula, and the BB1 copula parameter θ corresponds to the Claytoncopula parameter θ. The very right BB1 copula has the parameter θ tends

27

0 0.2 0.4 0.6 0.8 1

00.2

0.40.6

0.810

2

4

6

u1


u2

dens

ity

−3 −2 −1 0 1 2 3

−3−2

−10

12

30

0.05

0.1

0.15

0.2

m1 ∼ N(0,1)

(meta−) Gaussian distribution, ρS = 0.4

m2 ∼ N(0,1)

dens

ity

0 0.2 0.4 0.6 0.8 1

00.2

0.40.6

0.810

2

4

6

u1


u2

dens

ity

−3 −2 −1 0 1 2 3

−3−2

−10

12

30

0.05

0.1

0.15

0.2

m1 ∼ N(0,1)

(meta−) Gaussian distribution, ρS = 0.8

m2 ∼ N(0,1)

dens

ity

Figure 7: Densities of bivariate Gaussian copulas (left hand side) with aSpearman’s rho of 0.4 (copula-parameter ρGaussian = 0.42) and 0.8 (copula-parameter ρGaussian = 0.81) evaluated on the interval [0.0001, 0.999]2 andcorresponding meta-distributions with standard normal marginal distribu-tions (right hand side).

28

u1

u 2

ρS = 0.4

0 0.5 10

0.5

1

m1 ∼ N(0,1)

m2 ∼

N(0

,1)

ρS = 0.4

−3 0 3−3

0

3

u1

u 2

ρS = 0.8

0 0.5 10

0.5

1

m1 ∼ N(0,1)

m2 ∼

N(0

,1)

ρS = 0.8

−3 0 3−3

0

3

0 0.2 0.4 0.6 0.8 10

0.20.4

0.60.8

10

5

u1

Student t copula, ν=3, ρS = 0.4

u2

dens

ity

m1 ∼ N(0,1)

m2 ∼

N(0

,1)

meta−Student t dist., ν=3, ρS = 0.4

−3 −2 −1 0 1 2 3−3

−2

−1

0

1

2

3

0 0.2 0.4 0.6 0.8 10

0.20.4

0.60.8

10

5

u1

Student t copula, ν=3, ρS = 0.8

u2

dens

ity

m1 ∼ N(0,1)

m2 ∼

N(0

,1)

meta−Student t dist., ρS = 0.8

−2 0 2

−2

0

2

Figure 8: Densities of bivariate Student t copulas with ν = 3 degrees offreedom (left hand side) with a Spearman’s rho of 0.4 (copula-parameterρStudent t = 0.43) and 0.8 (copula-parameter ρStudent t = 0.83), evaluated onthe interval [0.0001, 0.9999]2. Corresponding contour plots (contours at the0.02, 0.05, 0.1, 0.2 and 0.3 level) of meta-Student t distributions with stan-dard normal marginal distributions are plotted on the right hand side. Addi-tionally, contours of a Gaussian meta-distribution with identical Spearman’srho and marginal distributions are plotted in light grey. The graphs in thebottom row indicate in which areas the densities of the Student t copula ormeta distribution exceed that of a Gaussian copula or meta distribution withidentical Spearman’s rho (grey-shaded areas).

29

0.5

0.5

0.5

11

1 1.2

1.2

23

BB1 (δ = 1, θ = 0.76)

u1

u 2

0.2 0.4 0.6 0.8

0.20.40.60.8

0.5

0.5

1

1

1

1

1.2

1.2

2

2

3

BB1 (δ = 1.1, θ = 0.48)

u1

u 2

0.2 0.4 0.6 0.8

0.20.40.60.8 0.

5

0.5

1

1

1

1

1.2

1.2

2

2

3

BB1 (δ = 1.38, θ = 0+)

u1

u 2

0.2 0.4 0.6 0.8

0.20.40.60.8

0.5

0.5

1

1

1

1

1.2

1.2

2

2

3

3

BB1 (δ = 1.2, θ = 0.2)

u1

u 2

0.2 0.4 0.6 0.8

0.20.40.60.8

Figure 9: Densities of bivariate BB1 copulas with different parameterisation.All copulas have a a Spearman’s rho of approximately 0.4. The densities areevaluated on the interval [0.01, 0.99]2.

towards zero. This special case of the BB1 copula is called a Gumbel copula,and the BB1 copula parameter δ corresponds to the Gumbel copula param-eter θ.

The next two sub-sections take a closer look on these two special cases ofthe BB1 copula and compare their densities to that of a Gaussian copula.

3.3.4 Clayton copula

The Clayton copula displays lower tail dependence and zero upper tail de-pendence. These properties can be verified regarding the Clayton copuladensity plots displayed in figure 10 on the left hand side. The top copula hasa Spearman’s rho of 0.4, the bottom copula has a Spearman’s rho of 0.8. The‘triangle-shaped’ corresponding contour plots of meta-Clayton distributionswith standard normal marginal distributions are displayed on the right handside. The contour plots on the bottom of figure 10 show that the Claytoncopula assigns a higher probability to joint extremely negative realisationsas compared to the Gaussian copula, while it assigns a lower probability tojoint extremely positive realisations.

3.3.5 Gumbel copula

Figure 11 displays the densities of a survival Gumbel copula with a Spear-man’s rho of 0.4 (top) and 0.8 (bottom). Like the Clayton copula, the survivalGumbel copula displays lower tails dependence and no upper tail dependence.The ‘tear shaped’ corresponding contour plots of meta-survival Gumbel dis-tributions with standard normal marginal distributions are displayed on theright hand side. The contour plots on the bottom of figure 11 show thatthe survival Gumbel copula assigns a higher probability to joint extremelynegative realisations as compared to the Gaussian copula, while it assigns a

30

u1

u 2

ρS = 0.4

0 0.5 10

0.5

1

m1 ∼ N(0,1)

m2 ∼

N(0

,1)

ρS = 0.4

−3 0 3−3

0

3

u1

u 2

ρS = 0.8

0 0.5 10

0.5

1

m1 ∼ N(0,1)

m2 ∼

N(0

,1)

ρS = 0.8

−3 0 3−3

0

3

m1 ∼ N(0,1)

m2 ∼

N(0

,1)

meta−Clayton dist., ρS = 0.8

−3 −2 −1 0 1 2 3−3

−2

−1

0

1

2

3

0 0.2 0.4 0.6 0.8 10

0.20.4

0.60.8

10

5

u1

Clayton copula, ρS = 0.8

u2

dens

ity

0 0.2 0.4 0.6 0.8 10

0.20.4

0.60.8

10

5

u1

Clayton copula, ρS = 0.4

u2

dens

ity

m1 ∼ N(0,1)

m2 ∼

N(0

,1)

meta−Clayton dist., ρS = 0.4

−3 −2 −1 0 1 2 3−3

−2

−1

0

1

2

3

Figure 10: Densities of bivariate Clayton copulas (left hand side) with aSpearman’s rho of 0.4 (copula-parameter θClayton = 0.76) and 0.8 (copula-parameter θClayton = 3.2), evaluated on the interval [0.0001, 0.9999]2. Cor-responding contour plots (contours at the 0.02, 0.05, 0.1, 0.2 and 0.3 level)of meta-Clayton distributions with standard normal marginal distributionsare plotted on the right hand side. Additionally, contours of a Gaussianmeta-distribution with identical Spearman’s rho and marginal distributionsare plotted in light grey. The graphs in the bottom row indicate in whichareas the densities of the Clayton copula or meta distribution exceed thatof a Gaussian copula or meta distribution with identical Spearman’s rho(grey-shaded areas).

31

u1

u 2

ρS = 0.4

0 0.5 10

0.5

1

m1 ∼ N(0,1)

m2 ∼

N(0

,1)

ρS = 0.4

−3 0 3−3

0

3

u1

u 2

ρS = 0.8

0 0.5 10

0.5

1

m1 ∼ N(0,1)

m2 ∼

N(0

,1)

ρS = 0.8

−3 0 3−3

0

3

m1 ∼ N(0,1)

m2 ∼

N(0

,1)

meta−surv. Gumbel dist., ρS = 0.8

−3 −2 −1 0 1 2 3−3

−2

−1

0

1

2

3

0 0.2 0.4 0.6 0.8 10

0.20.4

0.60.8

10

5

u1

surv. Gumbel copula, ρS = 0.8

u2

dens

ity

0 0.2 0.4 0.6 0.8 10

0.20.4

0.60.8

10

5

u1

surv. Gumbel copula, ρS = 0.4

u2

dens

ity

m1 ∼ N(0,1)

m2 ∼

N(0

,1)

meta−surv. Gumbel dist., ρS = 0.4

−3 −2 −1 0 1 2 3−3

−2

−1

0

1

2

3

Figure 11: Densities of bivariate survival Gumbel copulas (left hand side)with a Spearman’s rho of 0.4 (copula-parameter θGumbel = 1.38) and 0.8(copula-parameter θGumbel = 2.6), evaluated on the interval [0.0001, 0.9999]2.Corresponding contour plots (contours at the 0.02, 0.05, 0.1, 0.2 and 0.3level) of meta-survival Gumbel distributions with standard normal marginaldistributions are plotted on the right hand side. Additionally, contours of aGaussian meta-distribution with identical Spearman’s rho and marginal dis-tributions are plotted in light grey. The graphs in the bottom row indicatein which areas the densities of the survival Gumbel copula or meta distri-bution exceed that of a Gaussian copula or meta distribution with identicalSpearman’s rho (grey-shaded areas).

32

u1

u 2

ρS = 0.4

0.2 0.4 0.6 0.8

0.2

0.4

0.6

0.8

u1

u 2

ρS = 0.8

0.2 0.4 0.6 0.8

0.2

0.4

0.6

0.8

m1 ∼ N(0,1)

m2 ∼

N(0

,1)

ρS = 0.4

−3 −2 −1 0 1 2 3−3

−2

−1

0

1

2

3

m1 ∼ N(0,1)

m2 ∼

N(0

,1)

ρS = 0.8

−3 −2 −1 0 1 2 3−3

−2

−1

0

1

2

3

Figure 12: Contour plots of the log differences of a Clayton and a survivalGumbel copula density (left two plots) and of corresponding log differencesof meta-distribution densities with standard normal marginal distributions(right two plots), when both copulas display a Spearman’s rho of 0.4 and 0.8,respectively (copula parameters for ρS = 0.4: θClayton = 0.76, θsurv. Gumbel =1.38, copula parameters for ρS = 0.8: θClayton = 3.2, θsurv. Gumbel = 2.6).Grey-shaded areas indicate that the density of a Clayton copula or meta-distribution exceeds that of a survival Gumbel copula or meta-distribution(log-differences > 0). Contour lines are plotted at the -0.2, -0.1, -0.05, 0, 0.05,0.1 and 0.2 levels. The log differences are evaluated on the [0.0001, 0.9999]2

and [−3, 3]2 interval.

lower probability to joint extremely positive realisations (the converse is truefor the ‘standard’ Gumbel copula C++).

More closely examining the differences between a Clayton and a survivalGumbel copula, figure 12 displays plots of the log-differences of a Claytonand a Gumbel copula’s densities, ln cClayton(u1, u2) − ln csurv. Gumbel(u1, u2)and of meta distribution densities with standard normal marginal distri-butions, with a Spearman’s rho of 0.4 and 0.8, respectively. Grey-shadedareas indicate that the Clayton copula’s or meta-distribution’s density ex-ceeds that of a survival Gumbel copula. It can be seen that the Claytoncopula assigns a higher probability to joint extremely negative events, whilethe survival Gumbel copula assigns a higher probability to joint extremelypositive events. These findings are in line with the lower tail dependence λL

for these copulas (see table 8 on p.25). For the Clayton the lower tail de-pendence measures are λL = 0.40 and λL = 0.81 for ρS = 0.4 and ρS = 0.8,respectively, while for the survival Gumbel copula they are λL = 0.35 andλL = 0.69.

33

u1

u 2

ρS = 0.4

0 0.5 10

0.5

1

m1 ∼ N(0,1)

m2 ∼

N(0

,1)

ρS = 0.4

−3 0 3−3

0

3

u1

u 2

ρS = 0.8

0 0.5 10

0.5

1

m1 ∼ N(0,1)

m2 ∼

N(0

,1)

ρS = 0.8

−3 0 3−3

0

3

m1 ∼ N(0,1)

m2 ∼

N(0

,1)

meta−Frank dist., ρS = 0.8

−3 −2 −1 0 1 2 3−3

−2

−1

0

1

2

3

0 0.2 0.4 0.6 0.8 10

0.20.4

0.60.8

10

5

u1


u2

dens

ity

0 0.2 0.4 0.6 0.8 10

0.20.4

0.60.8

10

5

u1


u2

dens

ity

m1 ∼ N(0,1)

m2 ∼

N(0

,1)

meta−Frank dist., ρS = 0.4

−3 −2 −1 0 1 2 3−3

−2

−1

0

1

2

3

Figure 13: Densities of bivariate Frank copulas (left hand side) with a Spear-man’s rho of 0.4 (copula-parameter θFrank = 2.61) and 0.8 (copula-parameterθFrank = 7.9), evaluated on the interval [0.0001, 0.9999]2. Corresponding con-tour plots (contours at the 0.02, 0.05, 0.1, 0.2 and 0.3 level) of meta-Frankdistributions with standard normal marginal distributions are plotted on theright hand side. Additionally, contours of a Gaussian meta-distribution withidentical Spearman’s rho and marginal distributions are plotted in light grey.The graphs in the bottom row indicate in which areas the densities of theFrank copula or meta distribution exceed that of a Gaussian copula or metadistribution with identical Spearman’s rho (grey-shaded areas).

34

3.3.6 Frank copula

Like the Gaussian copula, the Frank copula does not display positive taildependence. Plots of Frank copula densities and meta-Frank distributions’densities for Frank copulas with a Spearman’s rho of 0.4 and 0.8 are providedin figure 13. It can be seen that the Frank copula assigns a lower probabilityto joint extremely negative or extremely positive realisations as compared tothe Gaussian copula.

3.4 Copula-parameter estimation

This section presents some widely used methods to estimate copula param-eters from empirical data. The most commonly used method is to estimatethe parameters with maximum likelihood (MLE). This method is presentedin subsection 3.4.1 below. An alternative method is to estimate the copulaparameters via correlation measures such as Spearman’s rho or Kendall’s tau.This method is presented in subsection 3.4.2. Finally, the computation of thenon-parametric empirical copula is shortly adressed in subsection 3.4.3.

3.4.1 Maximum likelihood estimation

Assume we observe N vectors of n-dimensional i.i.d. random variables froma multivariate distribution (our empirical observations), x1, . . . , xN , wherexj = (xj,1, . . . , xj,n), j ∈ {1, . . . , N}.

Notice that assuming appropriate parametric models for the marginaldistributions F1, . . . , Fn with parameters α1, . . . , αn and for the copula Cwith parameters θ, we may write the density of the multivariate distribution,f , as

f(x) = c (F1(x1; α1), F2(x2; α2), . . . , Fn(xn; αn); θ)n∏

i=1

fi(xi; αi), (6)

where c is the copula density and f1, . . . , fn are the densities of themarginal distributions. The above equation follows from equation 1 on p.12(Sklar’s theorem), as f(x) = ∂nF (x)/(

∏∂xi). See tables 5 and 7 in section

3.3 for the densities c of selected copulas.

The parameters of both the marginal distributions, α1, . . . , αn, and thecopula, θ, may be estimated from the empirical data with a MLE as

35

arg maxα1,...,αn,θ

N∑j=1

ln

(c (F1(xj,1; α1), . . . , Fn(xj,n; αn); θ)

n∏i=1

fi(xj,i; αi)

)(7)

Such an approach is, however, computationally intensive because themarginal distributions’ parameters and the copula parameters have to bejointly estimated.

As demonstrated by Joe and Xu [43], the parameters of the meta-distributionmay be estimated in two steps by first estimating the marginal distributions’parameters and then estimating the copula parameters. This approach isreferred to as the IFM (inference function for margins) method.

In the IFM method the marginal distributions’ parameters are estimatedwith a MLE as

αi = arg maxαi

N∑j=1

ln fi(xj,i; αi) ∀i ∈ {1, . . . , n} (8)

Using the obtained parameter estimates for the marginal distributions αi,the copula parameters θ are estimated in a second step as

θ = arg maxθ

N∑j=1

ln

(c (F1(xj,1; α1), . . . , Fn(xj,n; αn); θ)

n∏i=1

fi(xj,i; αi)

)(9)

Joe and Xu [43] show that the IFM method is highly efficient comparedto the parameter estimation presented in equation 7 where all parametersare estimated simultaneously.

Both methods presented above are based on assumptions on the paramet-ric distribution functions of the marginal distributions F1, . . . , Fn. If any ofthese assumptions are wrong, also the copula parameter estimates are biased.

The pseudo-log-likelihood method (also referred to as CML – canonicalmaximum likelihood – or semiparametric method) overcomes this problemby transforming the empirical observations x1, . . . , xN into so-called pseudo-observations u1, . . . , uN , without assuming any specific functional form ofthe marginal distributions. This method was first presented by Genest andRivest [29]. The pseudo-observations are computed as

36

uj,i =1

N + 1

N∑k=1

1xk,i≤xj,i∀i ∈ {1, . . . , n}, j ∈ {1, . . . , N} (10)

where 1xk,i≤xj,iis an indicator function that takes a value of 1 if xk,i ≤ xj,i

and a value of 0 otherwise.25 The pseudo-observations are uniformly dis-tributed between 0 and 1, ui,n ∼ U(0, 1).

Using the pseudo-observations, the copula parameters are estimated witha MLE as

θ = arg maxθ

N∑j=1

ln (c (uj,1, uj,2, . . . , uj,n; θ)) (11)

Scaillet and Fermanian [63] who conduct a Monte Carlo study to assessthe impact of misspecified marginal distributions suggest that ‘if one has anydoubt about the correct modeling of the margins, there is probably little toloose but lots to gain from shifting towards a semiparametric approach’, i.e.they suggest to generally use the pseudo-log-likelihood method rather thanthe IFM method.

3.4.2 Parameter estimation using correlation measures

An alternative method to estimate the parameters of bivariate one-parametriccopulas is to compute rank-based correlation measures such as Spearman’srho ρS or Kendall’s tau τK from the empirical data and to infer the copulaparameter from these correlation measures.

For some copulas a simple functional relationship exists between eitherρS and/or τK and the copula parameter, hence the copula parameter mayeasily be computed from the estimate of one of the two correlation measures(see table 926; for information on ρS and τK see Appendix A). If the Studentt copula parameter ρ is estimated from Kendall’s tau, additionally the pa-rameter ν has to be estimated, using MLE.

25Note that the sum of the indicator functions is divided by N+1 as also done in Demartaand McNeil [18], p.8. This keeps the pseudo-observations away from the boundaries of theunit cube where the density of many copulas take infinite values. Alternatively, the sumof the indicator functions is divided by N by other studies.

26The functional relationships are documented in e.g. Cherubini et al. [13], McNeil etal. [48] and Nelsen [53].

37

copulasGaussian Student t Clayton Gumbel Frank

ρS 6π

arcsin ρ2

1− 12θ

(D1(θ)−D2(θ))

τK 2π

arcsin ρ 2π

arcsin ρ θθ+2

1− 1θ

1− 4θ(1−D1(θ))

Table 9: Functional relationships between copula parameters and the corre-lation measures Spearman’s rho ρS and Kendall’s tau τK . Dk is the Debyefunction Dk(x) = k

xk

∫ x0

tk

et−1dt.

The advantage of this approach is that it is computationally very fast.However, Genest et al. [28] show that (for a bivariate Clayton copula) thepseudo-log-likelihood approach presented in the above subsection outper-forms the method using correlation coefficients. Still, the approach seemsuseful as it allows to estimate starting values for numerical parameter esti-mations that are based on a MLE, speeding up the copula parameter esti-mation.

3.4.3 Empirical copulas

Empirical copulas may be used alternatively to the parametric copulas pre-sented above. They were introduced by Deheuvels [16], [17] and are computedfrom empirical pseudo-observations (see equation 10) as

C(u) =1

N

N∑j=1

n∏i=1

1uj,i≤ui. (12)

The empirical copula asymptotically converges to the true copula forN →∞. It may be used for Monte Carlo simulations or for a visualisation ofthe goodness-of-fit of some parametric copula, by comparing a parameterisedcopula to the empirical copula. However, as pointed out by Scaillet and Fer-manian [63], the empirical copula is not differentiable and graphical visual-isations are hard to interpret, as the empirical copula is a ‘step-function’.They suggest to use kernel-smoothed representations of empirical copulas27

3.5 Goodness-of-fit tests

A widely used measure for the goodness-of-fit of a copula is the Akaike in-formation criterion AIC (Akaike [3]). This measure takes into account that

27See e.g. Wand and Jones [68]and Silverman [65].

38

the likelihood increases with the number of copula parameters by adjustingthe measure accordingly. AIC is defined as

AIC = −2N∑

j=1

ln(c(u1,j, u2,j; θ)

)+ 2m (13)

where m is the number of estimated (scalar) copula parameters and the

term∑N

j=1 ln(c(u1,j, u2,j; θ)

)is the log-likelihood. Hence, for a bivariate

Gaussian copula m = 1, while for a trivariate copula m = 3 and for afour-dimensional Gaussian copula m = 6 (for the number of scalar parame-ters, i.e. the number of different elements of the Gaussian copula parametermatrix P, see e.g. equation 3 on p.23). The lower the AIC the better thegoodness-of-fit.

Concerning statistical tests that explicitly test whether a parameterisedcopula at hand is indeed the true copula, there does not yet exist one stan-dard test. One widely used test is based on Rosenblatt’s [60] probabilityintegral transform. This test is explained in detail in Appendix B, where alsoexplicit formulas for computing tests statistics for selected bivariate copulasand trivariate Gaussian and Student t copulas are provided. Statistical testsbased on the probability integral transform suffer from the fact that they testfor the whole joint distribution (i.e. the copula and the marginal distribu-tions) while the focus should indeed be on the copula.28 A small simulationstudy in Cech and Fortin [10], p.31, shows that the power of such tests is verypoor and that wrong hypotheses cannot be accurately rejected if the samplesize is small (N = 50). A small sample size is to be expected, however, whenmonthly or less frequently measured data are used to calibrate the copulasin a risk aggregation context.

An alternative test that, loosely speaking, examines the null hypothesisof the Gaussian copula being the true copula against the Student t copulacan be conducted with a likelihood ratio test, where the Gaussian copula isregarded as a limiting case of the Student t copula with ν → ∞. Formally,H0 : θ ∈ Θ0 is tested against H1 : θ ∈ Θ1, where Θ0 ⊆ Θ1. Given H0 is true,the test statistic t = −2 ln λ, where

λ =sup{L(θ|x) : x ∈ Θ0}sup{L(θ|x) : x ∈ Θ1}

, (14)

28Note that if the sample size N is small the pseudo-observations ui obtained by equation10 may differ considerably from the uniformised observations F−1

i (xi), where F−1i are the

functional inverse of the true marginal distributions.

39

is asymptotically χ2k distributed as the sample size approaches infinity.29 The

degrees of freedom k are equal to the number of parameters not estimatedunder H0. In our case k = 1 since the copula parameter ν is not estimatedunder H0. Note that 0 ≤ λ ≤ 1.

Research on statistical tests examining the goodness-of-fit of copulas isstill ongoing. Tests other than the ones presented above have been proposedby e.g. Fermanian [22] and Berg and Bakken [8].

3.6 Simulation of selected meta-distributions

In this section algorithms for the simulation of multivariate meta-Gaussianand meta-Student t distributions, and bivariate meta-BB1 (and its two spe-cial cases, meta-Clayton and meta-Gumbel) and meta-Frank distributionsare provided in subsections 3.6.1 to 3.6.6.

Before presenting the specific algorithms, we shortly remind the readeron how to simulate dependent multivariate standard normal Gaussian distri-butions using the Cholesky factorization and show how antithetic samplingis done.

For the simulation of meta-Gaussian and meta-Student t distributions,dependent multivariate standard normal Gaussian distributions have to besimulated (see subsections 3.6.1 and 3.6.2 below). This can be achieved, usinga Cholesky factorization of the correlation matrix P (i.e. the Gaussian andStudent t copula parameter P, see section 3.3). The Cholesky factorizationA is a lower triangular matrix that is linked to the matrix P by

AA′ = P. (15)

Many statistical software packages provide functions on the computationof A, detailed information on algorithms is given in Glassermann [32], pp.71ff.

A n-dimensional scenario of dependent multivariate standard normallydistributed random variables x = (x1, . . . , xn) ∼ N(0, 1) is simulated from in-dependently distributed standard normal random variables z = (z1, . . . , zn) ∼N(0, 1) i.i.d. and the matrix A as

29In equation 14, the numerator is the likelihood of a Gaussian copula and the denomi-nator is the likelihood of a Student t copula. The p-value when rejecting H0 is computedas 1− χ2

1(t), where t = −2 ln λ is the test statistic.

40

x1 = z1

x2 = A1,2 · z1 + A2,2 · z2

...

xn = A1,n · z1 + A2,n · z2 + . . . + An,n · zn. (16)

where Ai,j is the (i, j)th element of A.

For a scenario of a dependent bivariate standard normal distribution wemay also define

x1 = z1

x2 = ρ · z1 +√

1− ρ2 · z2 (17)

where ρ is the (1, 2)th element of P, ρ ≡ ρ1,2, according to equation 3.

Antithetic sampling is one easily applicable method to reduce the stan-dard error of the estimates that also ensures that simulated standard nor-mally distributed random variables (∼ N(0, 1)) and uniformly distributedrandom variables on the unit interval (∼ U(0, 1)) are not skewed (see e.g.Glasserman [32], p.205ff). If m independent standard normally distributedrandom variables, z ∼ N(0, 1) i.i.d., are to be simulated we define

zi =

{rv ∼ N(0, 1) ∀i ≤ m/2

−zi−m/2 ∀i > m/2(18)

For the simulation of m independent uniformly distributed random vari-ables on the unit interval, t ∼ U(0, 1) i.i.d., we define

ti =

{rv ∼ U(0, 1) ∀i ≤ m/2

1− ti−m/2 ∀i > m/2(19)

The average computing time for the simulation of copulas (and thus cor-responding meta-distributions) varies substantially, depending on the simu-lated copula. Table 10 shows the average computing time for a simulation of100,000 scenarios of bivariate Gaussian, Student t30, Clayton, Gumbel, Frank

30Table 10 shows the average computing for a Student t copula with ν = 3. Additionally,simulations were done for a Student t copula with ν = 100. The average computing timehardly differs (on average it increases by 0.5 milliseconds for ν = 100 ).

41

copulas Gaussian Student t Clayton Gumbel Frank BB1seconds 0.072 1.002 0.185 71.6 0.078 85.3relative 1.00 13.95 2.58 996 1.09 1,187

Table 10: Computing time in seconds for the simulation of 100,000 scenar-ios of bivariate copulas and computing time relative to that for a Gaussiancopula.

and BB1 copulas, computing 100 simulations each. The simulations weredone on a ‘standard’ personal computer (3.5 GHz processor, 1 GB RAM),using the software ‘Matlab’, version 7.2 in a MS-Windows environment. Thestandard deviation of the computing time per simulation of the same copulais very low (less than 3% in terms of the average computing time). It can beseen that the Student t copula takes about 13 times longer to be simulatedthan the Gaussian copula (using equation 17 to simulate bivariate dependentstandard normal random variables). This is mainly because the computationof the univariate Student t distribution takes longer than that of a standardnormal distribution (see subsections 3.6.1 and 3.6.2 below).31 Simulations ofa Gumbel and BB1 copula take much longer than those for the other copulas.This is due to the necessity of using numerical methods (see subsections 3.6.3and 3.6.5 below). While the computing time for these two copulas may, ofcourse, be reduced using faster numerical methods32, it seems clear that thesimulation of the Gumbel and BB1 copulas will always take longer than thesimulation of the other copulas presented in this article.

Concerning the simulations of trivariate, 4-dimensional and 5-dimensionalGaussian and Student t copulas, table 11 displays the average computing timeand the computing time relative to that for a bivariate Gaussian copula (us-ing equation 15 and 16 and the Matlab-function chol to simulate bivariatedependent standard normal random variables). The simulation of a Studentt copula on average takes about 12 times as long as that of a Gaussian copula.As expected, the computation time increases as the number of dimensions nincreases.

In the subsections below algorithms for the simulation of multivariatemeta-Gaussian and meta-Student t distributions, and bivariate meta-BB1(and the two special cases, meta-Clayton and meta Gumbel) and meta-Frank

31The computation of 100,000 values of a Student t distribution on average takes about15 times as long as does the computation of a standard normal distribution function, usingMatlab (functions normcdf and tcdf).

32For the simulation above, the Matlab-function fminbnd was used.

42

copulas Gaussian Student tdimensions 3 4 5 3 4 5

seconds 0.115 0.161 0.212 1.430 1.909 2.393relative 1.60 2.24 2.95 19.90 26.57 33.31

Table 11: Computing time in seconds for the simulation of 100,000 scenariosof n-dimensional Gaussian and Student t copulas and computing time relativeto that for a bivariate Gaussian copula.

distributions are provided. More information on the simulation of copulasand meta-distributions can be found in e.g. Cherubini et al [13], p.181ff. 33

3.6.1 Simulation of meta-Gaussian distributions

Simulation of a n-dimensional meta-Gaussian distribution with m simulatedrealisations. Copula parameters: P (n× n matrix).

1. Simulate n independent standard normal random variable vectors zj ∼N(0, 1), j ∈ {1, . . . , n}.

2. Use a Cholesky factorization to transform the independent randomvariables zj into dependent random variable vectors xj according tothe copula parameter matrix P.

3. Transform the standard normally distributed variables xj into variablesthat are uniformly distributed between 0 and 1 uj by defining uj,i =Φ(xj,i)∀j ∈ {1, . . . , n}, i ∈ {1, . . . ,m}, where Φ is the standard normaldistribution function.

4. Compute the simulated joint realisations of the meta-Gaussian distri-bution aj as aj,i = F−1

j (uj,i)∀j ∈ {1, . . . , n}, i ∈ {1, . . . ,m}, whereF−1

j is the functional inverse (the quantile function) of the jth marginaldistribution function Fj.

3.6.2 Simulation of meta-Student t distributions

Simulation of a n-dimensional meta-Student t distribution with m simulatedrealisations. Copula parameters: P (n× n matrix) and ν.

1. Simulate n independent standard normal random variable vectors zj ∼N(0, 1), j ∈ {1, . . . , n} and one chi-square distributed random variable

33See Joe [42] and Cech and Fortin [9] for the BB1 copula.

43

vector with ν degrees of freedom s ∼ χ2ν .

If the degrees of freedom ν are an integer value, the vector s can becomputed by simulating ν standard normal variable vectors sk, k ∈{1, . . . , ν}, and defining si =

∑νk=1 s2

k,i∀i ∈ {1, . . . ,m}.

2. Use a Cholesky factorization to transform the independent randomvariables zj into dependent random variable vectors xj according tothe copula parameter matrix P.

3. Transform the standard normally distributed variables xj into variablesthat are uniformly distributed between 0 and 1 uj by defining uj,i =

tν(xj,i ·

√ν/si

)∀j ∈ {1, . . . , n}, i ∈ {1, . . . ,m}, where tν is the Student

t distribution function with ν degrees of freedom.

4. Compute the simulated joint realisations of the meta-Student t dis-tribution aj as aj,i = F−1

j (uj,i)∀j ∈ {1, . . . , n}, i ∈ {1, . . . ,m}, whereF−1

j is the functional inverse (the quantile function) of the jth marginaldistribution function Fj.

3.6.3 Simulation of bivariate meta-BB1 distributions

Simulation of a 2-dimensional (bivariate) meta-BB1 distribution with m sim-ulated realisations. Copula parameters: δ, θ.

1. Simulate 2 independent uniformly distributed random variable vectorss,u1 ∼ U(0, 1).

2. Use numerical methods to to find the elements of the vector u2 suchthat

u2,i =

(1 +

((u−θ

1,i − 1)δ

+(u−θ

2,i − 1)δ)1/δ

)−1/θ−1

·

·((

u−θ1,i − 1

)δ+(u−θ

2,i − 1)δ)1/δ−1 (

u−θ1,i − 1

)δ−1u−θ−1

1,i

∀i ∈ {1, . . . ,m}.

3. Compute the simulated joint realisations of the meta-BB1 distributionaj as aj,i = F−1

j (uj,i)∀j ∈ {1, 2}, i ∈ {1, . . . ,m}, where F−1j is the func-

tional inverse (the quantile function) of the jth marginal distributionfunction Fj.

3.6.4 Simulation of bivariate meta-Clayton distributions

Simulation of a 2-dimensional (bivariate) meta-Clayton distribution with msimulated realisations. Copula parameter: θ.

44


2. Compute the dependent uniformly distributed random variable vectoru2 ∼ U(0, 1) as

u2,i =(u−θ

1,i

(s− θ

θ+1

i − 1)

+ 1)− 1

θ

∀i ∈ {1, . . . ,m}.

3. Compute the simulated joint realisations of the meta-Clayton distribu-tion aj as aj,i = F−1

j (uj,i)∀j ∈ {1, 2}, i ∈ {1, . . . ,m}, where F−1j is the

functional inverse (the quantile function) of the jth marginal distribu-tion function Fj.

3.6.5 Simulation of bivariate meta-Gumbel distributions

Simulation of a 2-dimensional (bivariate) meta-Gumbel distribution with msimulated realisations. Copula parameter: θ.

1. Simulate 2 independent uniformly distributed random variable vectorss, t1 ∼ U(0, 1).

2. Use numerical methods to find the elements of the vector t2 such thatt2,i

(1− ln t2,i

θ

)= si ∀i ∈ {1, . . . ,m}.

3. Compute the dependent uniformly distributed random variable vectorsuj ∼ U(0, 1), j ∈ {1, 2} as

u1,i = exp(t1/θ1,i ln t2,i

)and

u2,i = exp((1− t1,i)

1/θ ln t2,i

)∀i ∈ {1, . . . ,m}.

4. Compute the simulated joint realisations of the meta-Gumbel distribu-tion aj as aj,i = F−1



3.6.6 Simulation of bivariate meta-Frank distributions

Simulation of a 2-dimensional (bivariate) meta-Frank distribution with msimulated realisations. Copula parameter: θ.


45

2. Compute the dependent uniformly distributed random variable vectoru2 ∼ U(0, 1) as

u2,i = −1θln(1 +

si(1−e−θ)si(e−θu1,i−1)−e−θu1,i

)∀i ∈ {1, . . . ,m}.

3. Compute the simulated joint realisations of the meta-Frank distribu-tion aj as aj,i = F−1



4 Implementation of a top-down approach and

empirical evidence

Two recent articles by Cech and Fortin [10] and [11] have dealt with the es-timation of copulas in a top-down risk aggregation approach using monthlyinstitution-internal profit and loss data. The sample size in both articles isless than 50. One of the articles, [11], examines the dependence of the profitsand losses from a market, a credit and a hedge fund portfolio. Only weakcorrelation in terms of Spearman’s rho is observed (−0.15 ≤ ρS ≤ 0.15),and (accounting for the possibility of time delays in reporting returns forthe credit portfolio) there is no indication that lagged profits and losses ofthe credit portfolio are highly correlated with the other portfolios’ profitsand losses. Several bivariate copulas (the same as those presented in thisarticle) and the trivariate Gaussian and Student t copulas are estimated. Agoodness-of-fit test based on the probability-integral transform cannot rejectthe null hypothesis of the parameterised copula being the true copula for anyof the calibrated bi- and trivariate copulas. This surprising result is proba-bly due to the small sample size, as also demonstrated in a small simultationstudy in [10]. The other article, [10], examines the dependence structurebetween observed market portfolio profits and losses and simulated creditportfolio profits and losses, where more than 40,000 simulation paths areavailable. The credit profit and loss paths are simulated by an institution-internal model. Correlation measures and copula parameters for Gaussian,Student t and Frank copulas are estimated for each of the more than 40,000joint observations of the empirical market portfolio and one simulated creditportfolio profits and loss path. The sample distribution of correlation mea-sures such as Spearman’s rho and Kendall’s tau is centered around zero,indicating uncorrelatedness of the two time series. In more than 99.602%(99.995%) of the cases, the observed Spearman’s rho (Kendall’s tau) is inthe interval [−0.2, 0.2], and the null hypothesis of ρS = 0 can be rejected in

46

only 0.5% of all cases at the 10% significance level. In terms of the AIC, theFrank copula on average displays a worse fit than the Gaussian and Studentt copula. Again, the null hypothesis of the copula at hand being the truecopula can be rejected only very rarely. While the probability of a type oneerror when rejecting this null hypothesis tends to be highest for the Studentt copula (indicating that the Student t copula’s fit may be better than thatof the Gaussian copula), a likelihood ratio test shows that the hypothesis ofa Gaussian copula can be rejected in favour of a Student t copula in only30.6%, 12.2% and 0.5% of the cases at the 10%, 5% and 1% significance level,respectively. Hence the results for the goodness-of-fit of the Gaussian andStudent t copula are ambiguous.

In the present section we try to find further empirical evidence for the de-pendence structure of different types of risk, using daily market data ratherthan monthly institution-internal data. The higher sample size allows foran examination of the existence of non-zero tail dependence and promises ahigher power of the goodness-of-fit tests.

The data base includes daily sovereign and corporate bond indices fromthe iboxx e index family for Euro denominated bonds34 for the period from

January 31st, 2000 until September 15th, 2006. For both sovereign and cor-porate bond indices, sub-indices with specific maturity bands are considered.These maturity bands are

• all maturities

• 1Y to 3Y maturities




For corporate bond indices constituents are further grouped according totheir rating. These ratings are

• all ratings

• AAA-rated

• AA-rated

34More information can be found at www.iboxx.com.

47

• A-rated

• BBB-rated

We are interested in the dependence structure of joint risk factor changes,specifically of joint interest rate and credit risk factor changes. We define theinterest rate risk factor changes at day t as the log-returns of the sovereignbond indices, ij,t, with maturity bands j

ij,t = ln Psov,j,t − ln Psov,j,t−1 ∀j ∈ {1, . . . , 5}. (20)

The credit risk factor changes at day t, cj,k,t, are defined as the excessreturns of corporate bonds over sovereign bonds for a given maturity bandand a given rating

cj,k,t = ln Pj,k,t − ln Pj,k,t−1 − ij,t ∀j ∈ {1, . . . , 5}, k ∈ {1, . . . , 5}. (21)

Hence, the corporate bond indices’ log-returns are adjusted by subtract-ing the sovereign bond indices’ log-returns such that the resulting risk factorchanges cj,k,t only display the change in the value of the corporate bondindices that is due to a change in the credit quality of the constituents (as-suming a similar composition of the sovereign and corporate bond indiceswith identical maturity band concerning duration, and assuming a constantrisk-appetite of the market participants).

From these riskless returns and excess returns, we construct data pairsthat consist of (ij,t, cj,k,t) for maturity bands j and rating classes k, result-ing in 25 bivariate empirical sample pairs. Scatter plots of these pairs aredisplayed in figure 14. They indicate that many of the risk factor changesare negatively correlated. Hence, negative interest rate risk factor changesthat are caused by an upward shift of the interest rate term structure tend tooccur simultaneously with positive credit risk factor changes that are causedby an amelioration of the obligors’ credit quality. This finding is not sur-prising and is consistent with business cycle developments, where in times ofcontraction the interest rates and the credit quality tend to decrease jointly,while in times of expansions they tend to increase jointly.

A closer look on the marginal distributions shows that they all are non-Gaussian. Using a Jarque-Bera test (Jarque and Bera [41]), the null hypoth-esis of normally distributed marginal distributions can be rejected at the 1%significance level in all cases (in fact they can be rejected even at the 0.012%significance level). Hence, a copula-based approach clearly seems preferable

48

−0.01 0 0.01

−0.01

0

0.01

interest rate risk

cred

it ris

k

all mat., all ratings

−0.01 0 0.01

−0.01

0

0.01

interest rate risk

cred

it ris

k

1−3Y, all ratings

−0.01 0 0.01

−0.01

0

0.01

interest rate risk

cred

it ris

k

3−5Y, all ratings

−0.01 0 0.01

−0.01

0

0.01

interest rate risk

cred

it ris

k

5−7Y, all ratings

−0.01 0 0.01

−0.01

0

0.01

interest rate risk

cred

it ris

k

7−10Y, all ratings

−0.01 0 0.01

−0.01

0

0.01

interest rate risk

cred

it ris

k

all mat., AAA

−0.01 0 0.01

−0.01

0

0.01

interest rate risk

cred

it ris

k

1−3Y, AAA

−0.01 0 0.01

−0.01

0

0.01

interest rate risk

cred

it ris

k

3−5Y, AAA

−0.01 0 0.01

−0.01

0

0.01

interest rate risk

cred

it ris

k

5−7Y, AAA

−0.01 0 0.01

−0.01

0

0.01

interest rate risk

cred

it ris

k

7−10Y, AAA

−0.01 0 0.01

−0.01

0

0.01

interest rate risk

cred

it ris

k

all mat., AA

−0.01 0 0.01

−0.01

0

0.01

interest rate risk

cred

it ris

k

1−3Y, AA

−0.01 0 0.01

−0.01

0

0.01

interest rate risk

cred

it ris

k

3−5Y, AA

−0.01 0 0.01

−0.01

0

0.01

interest rate risk

cred

it ris

k

5−7Y, AA

−0.01 0 0.01

−0.01

0

0.01

interest rate risk

cred

it ris

k

7−10Y, AA

−0.01 0 0.01

−0.01

0

0.01

interest rate risk

cred

it ris

k

all mat., A

−0.01 0 0.01

−0.01

0

0.01

interest rate risk

cred

it ris

k

1−3Y, A

−0.01 0 0.01

−0.01

0

0.01

interest rate risk

cred

it ris

k

3−5Y, A

−0.01 0 0.01

−0.01

0

0.01

interest rate risk

cred

it ris

k

5−7Y, A

−0.01 0 0.01

−0.01

0

0.01

interest rate risk

cred

it ris

k

7−10Y, A

−0.01 0 0.01

−0.01

0

0.01

interest rate risk

cred

it ris

k

all mat., BBB

−0.01 0 0.01

−0.01

0

0.01

interest rate risk

cred

it ris

k

1−3Y, BBB

−0.01 0 0.01

−0.01

0

0.01

interest rate risk

cred

it ris

k

3−5Y, BBB

−0.01 0 0.01

−0.01

0

0.01

interest rate risk

cred

it ris

k

5−7Y, BBB

−0.01 0 0.01

−0.01

0

0.01

interest rate risk

cred

it ris

k

7−10Y, BBB

Figure 14: Scatter plots of interest rate and credit risk factor changes.

49

to the assumption of a multivariate Gaussian distribution for the data sampleat hand.

For many of the marginal distributions (however for only one of the fivesovereign bond index return time series), autocorrelation is detected. Toadjust the data for autocorrelation (remember that we assume that the em-pirical pseudo-observations u are i.i.d.), an AR(2)-model is fitted where theobserved risk factor changes are modelled as

rt = β1 + β2rt−1 + β3rt−2 + εt, (22)

where rt is either ij,t or cj,k,t.

The coefficients and their statistical significance are estimated using thestandard OLS-estimates of a classical normal linear regression (see e.g. Greene[34]).35 If the estimates of either β2 or β3 turn out not to be statistically sig-nificantly different from 0 at the 5% significance level, models of the typert = β1 + β2rt−1 + εt and rt = β1 + β3rt−2 + εt, respectively, are estimated. Ifboth β2 and β3 turn out not to be statistically significantly different from 0,rt = β1 + εt is modelled, where β1 ≡ r. For 21 out of the 25 credit risk factorchanges, AR(2) models are fitted. For only one of the interest rate risk factorchanges, autocorrelation is detected. Here, an AR(1)-model is fitted.

Being interested only in the innovations that cannot be explained bylagged values of the observed returns, rt ≡ εt are used as the empirical returnobservations in what follows. For the purpose of comparison, Spearman’s rhoand AIC goodness-of-fit measures of copulas are also computed for the orig-inal (unadjusted) observations. Results are presented in Appendix C. Theydo not differ strongly from the results obtained from the autocorrelation-adjusted observations. The 25 autocorrelation-adjusted data pairs include1,727 observations each.

Table 12 shows sample estimates of Spearman’s rho for the 25 bivariateempirical samples and reports, whether they are statistically significantly dif-ferent from 0. In all but 3 samples, the Spearman’s rho correlation measureis negative (ρS < 0). In most of the cases ρS is found to be statistically sig-nificantly different from 0 at the 1% significance level (employing equation

35Generally, in the context of estimating AR models, alternative estimation methodsare preferred to the OLS-method, as the estimated standard errors of the coefficients aredownward biased. However, the estimates of the coefficients are consistent and the biasreduces as the sample size increases.

50

ρS maturity bandsrating all 1Y-3Y 3Y-5Y 5Y-7Y 7Y-10Y mean

all -0.53∗∗∗ -0.04∗ -0.25∗∗∗ -0.28∗∗∗ -0.35∗∗∗ -0 .29AAA -0.78∗∗∗ -0.08∗∗∗ -0.28∗∗∗ 0.00 -0.08∗∗∗ -0 .24AA -0.26∗∗∗ 0.10∗∗∗ -0.05∗∗ -0.13∗∗∗ -0.29∗∗∗ -0 .12A -0.43∗∗∗ 0.09∗∗∗ -0.20∗∗∗ -0.22∗∗∗ -0.29∗∗∗ -0 .21

BBB -0.56∗∗∗ -0.21∗∗∗ -0.23∗∗∗ -0.28∗∗∗ -0.30∗∗∗ -0 .32mean -0 .51 -0 .03 -0 .20 -0 .18 -0 .26statistically significantly different from 0 at the∗ 10%, ∗∗ 5%, ∗ ∗ ∗ 1% significance level.

Table 12: Spearman’s rho for the 25 bivariate observation pairs.

37 in the Appendix).

Bivariate Gaussian, Student t, BB1, Clayton, Gumbel, and Frank copu-las are fitted to the data, using the pseudo-log-likelihood method (see section3.4.1). The computing time for the copula estimation procedure varies con-siderably, depending on the copula. While the parameter estimation for theBB1 copula takes about as long as for the Gaussian copula, the parame-ter estimation for the Clayton and Gumbel copulas takes about 20% of thetime and that for the Frank copula takes about 2% of the time. Parametersestimation for a Student t copula takes about 150 times as long as for theGaussian copula.

For the BB1, Clayton and Gumbel copulas also the rotated versionsare fitted. Results are reported only for the rotated version with the bestgoodness-of-fit in terms of the AIC. The average AIC that is obtained bythe copulas is displayed in figure 15. One can see that the Student t cop-ula on average yields the best goodness-of-fit, followed by the BB1 copula.Both copulas yield a better goodness-of-fit than their restricted versions, theGaussian and the Clayton and Gumbel copulas, respectively. The Claytoncopula’s fit is inferior to that of all other copulas presented. The Frank copulathat assigns a low probability to joint extreme events yields a substantiallyworse goodness-of-fit than the Student t and BB1 copulas.

Table 13 displays the AIC goodness-of-fit measures in detail. The as-terisks beside the AIC measures indicate whether the null hypothesis of thespecific copula being the true copula can be rejected at the conventional sta-tistical significance levels, employing the goodness-of-fit test based on theprobability integral transform presented in section 3.5. If a rotated copula is

51

Gaussian Student t BB1 Clayton Gumbel Frank−300

−250

−200

−150

−100

−50

0

AIC

Figure 15: Mean AIC obtained by selected bivariate copulas.

parameterised, this is indicated by a subscript beside the AIC measure.

Table 13, continued on next pagepanel A: all ratingsmaturities all 1Y-3Y 3Y-5Y 5Y-7Y 7Y-10Y

Gauss -608.35∗∗∗ -2.76∗∗∗ -123.15∗ -151.28∗ -235.98∗∗

Student t -694.07 -84.08 -154.06 -179.91 -255.60BB1 -674.38(−+) -22.42∗∗(+−) -144.17(+−) -172.57(−+) -249.54(−+)

Clayton -543.63∗∗∗(+−) -12.81∗∗∗(−+) -105.96(+−) -134.56(+−) -190.52∗∗(−+)

Gumbel -647.39∗∗(−+) -21.96∗∗(+−) -126.87(+−) -160.22(−+) -223.94(+−)

Frank -605.81 -1.26∗∗∗ -120.18∗ -150.32∗ -227.03

panel B: rating AAAmaturities all 1Y-3Y 3Y-5Y 5Y-7Y 7Y-10Y

Gauss -1652.81∗∗∗ -7.74∗∗∗ -144.89∗∗∗ 0.26∗∗∗ -11.77∗∗∗

Student t -1881.20 -139.45 -213.94 -27.49 -87.26BB1 -1829.01∗∗∗(−+) -33.91∗∗∗(+−) -189.53∗∗(+−) -8.55∗∗(−+) -35.35∗∗∗(−+)

Clayton -1493.76∗∗∗(+−) -23.82∗∗∗(−+) -163.43∗∗∗(−+) -3.36∗∗∗(+−) -19.22∗∗∗(−+)

Gumbel -1782.38∗∗∗(−+) -33.37∗∗∗(+−) -187.24∗∗(+−) -7.31∗∗∗(−+) -30.47∗∗∗(−+)

Frank -1686.72∗∗ -10.19∗∗∗ -151.22∗∗∗ 1.96∗∗∗ -10.42∗∗∗

continued on next page

52

Table 13, continued from last page

panel C: rating AAmaturities all 1Y-3Y 3Y-5Y 5Y-7Y 7Y-10Y

Gauss -113.08∗∗∗ -18.41∗∗∗ -4.27∗∗∗ -26.95∗∗ -162.79Student t -223.39 -131.05∗∗ -55.76 -46.11 -195.47

BB1 -164.21∗∗(−+) -52.14∗∗∗(−−) -20.32∗(+−) -39.88(+−) -183.98(−+)

Clayton -121.25∗∗∗(+−) -31.97∗∗∗ -13.11∗∗(−+) -25.09∗(+−) -138.67(+−)

Gumbel -158.63∗∗(−+) -51.17∗∗∗(−−) -20.13∗∗(+−) -35.10(+−) -167.88(−+)

Frank -127.39∗∗∗ -19.17∗∗∗ -3.51∗∗∗ -27.55∗ -156.83

panel D: rating Amaturities all 1Y-3Y 3Y-5Y 5Y-7Y 7Y-10Y

Gauss -363.34∗∗∗ -16.53∗∗∗ -75.69∗∗ -86.45∗∗ -156.10∗∗

Student t -425.79 -79.22∗ -120.32 -121.34 -171.37BB1 -410.88(−+) -35.59∗∗∗(−−) -101.45(+−) -109.41(+−) -166.57(−+)

Clayton -329.33∗∗(+−) -22.21∗∗∗(−−) -70.04∗∗(+−) -81.44∗(+−) -129.20∗∗(+−)

Gumbel -393.44∗(−+) -32.02∗∗∗ -89.14(+−) -98.65(−+) -153.61∗(−+)

Frank -365.18 -13.29∗∗∗ -75.80∗ -85.10∗ -156.08∗∗

panel E: rating BBBmaturities all 1Y-3Y 3Y-5Y 5Y-7Y 7Y-10Y

Gauss -641.81∗∗∗ -77.73∗∗∗ -88.20∗ -152.08 -174.90Student t -725.40∗ -177.40 -115.47 -174.04 -182.53

BB1 -665.39∗∗(−+) -122.06∗∗∗(+−) -103.58(+−) -171.14(−+) -179.04(−+)

Clayton -556.77∗∗∗(+−) -93.54∗∗∗(−+) -80.65(+−) -134.42(+−) -137.18(−+)

Gumbel -669.99∗∗(−+) -119.07∗∗∗(+−) -94.87(−+) -157.95(−+) -160.32(+−)

Frank -691.04 -86.83∗∗∗ -94.39 -142.18 -168.24

Table 13: AIC goodness-of-fit measures obtained by cop-ulas fitted to the 25 bivariate observation. The subscripts

(−+), (+−) and (−−) indicate that a rotated copula wasused. Asterisks indicate that the null hypothesis of thespecific copula being the true copula can be rejected atthe ∗ 10%, ∗∗ 5% or ∗∗∗ 1% significance level.

53

significance level10% 5% 1%

Gaussian 22 19 14Student t 3 1 0

BB1 12 11 6Clayton 19 17 6Gumbel 15 13 7Frank 17 11 10

Table 14: Number of times that the null hypothesis of the copula at handbeing the true copula is rejected at the 10%-, 5%- and 1%-significance level.

We can observe that in terms of the AIC the Student t copula has the bestfit in all 25 cases and that in all cases the two-parametric Student t and BB1copulas achieve better results than their special cases (the Gaussian copulaand the Clayton and Gumbel copula, respectively). The Frank copula forall data samples achieves a worse goodness-of-fit than the Student t or BB1copula.

The Student t copula is also the copula for which the null hypothesis ofthe copula at hand being the true copula can be rejected in least of the cases(see also table 14). The null hypothesis of the Gaussian copula being thetrue copula is rejected in most of the cases. As far as the goodness-of fit ofthe BB1 copula is concerned, the null hypothesis of the BB1 copula beingthe true copula is rejected more often than the analogous null hypothesis forthe Student t copula.

Conducting additionally a likelihood ratio test, the null hypothesis of thetrue copula being the Gaussian copula can be rejected in all of the cases infavour of the Student t copula at the 1% (and even at the 0.2%) significancelevel. This result is not surprising when the parameter distribution of theStudent t copula parameter estimate ν, displayed in figure 16, is regarded (asthe Gaussian copula corresponds to a Student t copula with ν → ∞). Thehighest value for ν is 12.09, the lowest is 2.83. The mean (median) value is5.68 (5.17).

The results displayed in table 13 suggest that ‘positive tail dependence’36

36As the risk factor changes for the data sample at hand are generally negatively cor-related, we define ‘positive tail dependence’ in this section as λL = limα→0+ P (u1 >1−α|u2 ≤ α) and λU = limα→0+ P (u1 ≤ α|u2 > 1−α) for ρS < 0. This definition differsfrom the ‘official’ definition of lower and upper tail dependence given in equations 4 and

54

0 5 10 150

0.05

0.1

0.15

0.2

ν estimate

para

met

er d

ist.

dens

ity

Figure 16: Gaussian kernel smoothed density of the parameter distributionof the Student t copula parameter ν.

could be prevalent as the Student t and BB1 copulas that display positivetail dependence in all cases have a better goodness-of-fit than the Gaussianand Frank copula that do not. The data sample at hand is large enoughso that we may examine the potential existence of tail dependence in moredetail. To do so, we introduce the concept of corner dependence, which is anempirical counterpart to the measures of positive tail dependence presentedin section 3.

For positively correlated pairs in terms of Spearman’s rho, ρS > 0, wedefine the empirical corner dependence as

λempiricalL,α = P (u1 ≤ α|u2 ≤ α) = P (u2 ≤ α|u1 ≤ α) (23)

λempiricalU,α = P (u1 > 1− α|u2 > 1− α) = P (u2 > 1− α|u1 > 1− α),(24)

where u1 and u2 are the pseudo-observations (equation 10 on p.37).

For negatively correlated pairs in terms of Spearman’s rho, ρS < 0, wedefine the empirical corner dependence as

λempiricalL,α = P (u1 > 1− α|u2 ≤ α) = P (u2 ≤ α|u1 > 1− α) (25)

λempiricalU,α = P (u1 ≤ α|u2 > 1− α) = P (u2 > 1− α|u1 ≤ α). (26)

I.e. to compute e.g. the empirical corner dependence λempiricalL,α for pos-

itively correlated pairs, we first identify the observation pairs for whichu2 ≤ α. λempirical

L,α is the fraction of these pairs for which u1 ≤ α.

5.

55

Further, for the Gaussian, Student t and Frank copula (symmetric copu-las), we define the corner dependence that is implied by a specific copula forρS > 0 as

λimpliedL,α = λimplied

U,α =C(α, α; θ)

α, (27)

while for ρS < 0 we define

λimpliedL,α = λimplied

U,α =α− C(1− α, α; θ)

α. (28)

For the BB1 copula (asymmetric copula) and its rotated versions we define

λimpliedL,α =

C++(α, α; θ)

α(29)

λimpliedU,α =

2α− 1 + C++(1− α, 1− α; θ)

α(30)

for C++ and C−+ copulas, and

λimpliedL,α =

2α− 1 + C++(1− α, 1− α; θ)

α(31)

λimpliedU,α =

C++(α, α; θ)

α(32)

for C−− and C+− copulas.

Figure 17 shows the empirical lower and upper corner dependence (‘e’)and the implied corner dependence of the Gaussian (‘G’), Student t (‘t’), BB1(‘B’), and Frank (‘F’) copula for α = 0.1. For symmetric copulas (Gaussian,Student t and Frank) table 15 summarises in how many of the 25 cases theimplied corner dependence exceeds both the lower and upper empirical cor-ner dependence and in how many cases it is below for α ∈ {0.05, 0.1}. Themost evident observation is that the Frank and the Gaussian copula tend tounderestimate the empirical corner dependence in all/most of the cases. TheStudent t copula overestimates the corner dependence in a lot more cases forα = 0.05 than it does for α = 0.1.

The corner dependence that is implied by the Student t and BB1 copulasis more closely examined in table 16, where the number of cases where thelower and upper empirical corner dependence is over- or underestimated is re-ported. The Student t copula more frequently overestimates both lower andupper corner dependence than it underestimates these measures for α = 0.05.

56

e G t B F0

0.5

e G t B F0

0.1

0.2

e G t B F0

0.10.2

e G t B F0

0.10.2

e G t B F0

0.10.2

e G t B F0

0.5

e G t B F0

0.10.2

e G t B F0

0.2

0.4

e G t B F0

0.1

0.2

e G t B F0

0.10.2

e G t B F0

0.2

0.4

e G t B F0

0.10.2

e G t B F0

0.1

0.2

e G t B F0

0.1

0.2

e G t B F0

0.10.2

e G t B F0

0.2

0.4

e G t B F0

0.10.2

e G t B F0

0.10.2

e G t B F0

0.10.2

e G t B F0

0.10.2

e G t B F0

0.5

e G t B F0

0.10.2

e G t B F0

0.10.2

e G t B F0

0.10.2

e G t B F0

0.10.2

all ratings

AAA

AA

A

BBB

all maturities 1Y−3Y 3Y−5Y 5Y−7Y 7Y−10Y

Figure 17: Empirical lower and upper corner dependence (‘e’) and the impliedcorner dependence of the Gaussian (‘G’), Student t (‘t’), BB1 (‘B’), and Frank(‘F’) copula for α = 0.1

α = 0.1 λimplied < inf[λempL , λemp

U ] λimplied > sup[λempL , λemp

U ]Gaussian 22 0Student t 4 5

Frank 25 0

α = 0.05 λimplied < inf[λempL , λemp

U ] λimplied > sup[λempL , λemp

U ]Gaussian 20 0Student t 1 12

Frank 24 0

Table 15: Number of cases where the implied corner dependence are be-low/exceed the empirical corner dependence for symmetric copulas.

57

α = 0.1 λimplL < λemp

L λimplL > λemp

L λimplU < λemp

U λimplU > λemp

U

Student t 10 15 14 11BB1 14 11 15 10

α = 0.05 λimplL < λemp

L λimplL > λemp

L λimplU < λemp

U λimplU > λemp

U

Student t 6 19 8 17BB1 14 11 10 15

Table 16: Number of cases where the implied upper and lower corner de-pendence of the Student t and BB1 copula are below/exceed the empiricalcorner dependence.

No such behaviour can be observed for the BB1 copula.

Table 17 in detail shows the lower and upper empirical corner dependenceλempirical

L,α and λempiricalU,α for all 25 observation pairs for α ∈ {0.05, 0.1}, as well

as their mean and median values (on the very right hand side). The figuresreported as ‘abs. diff.’ are the absolute values of the log-differences of thelower and upper corner dependence, | ln λempirical

L,α − ln λempiricalU,α |. They indicate

by how much the lower and upper corner dependence differ from each other.The results suggest that lower and upper corner dependence differ consider-ably for the data sample at hand. This is more pronounced for α = 0.05 thanit is for α = 0.1. The Student t copula does not allow for such differences.

Focussing on the deviation of the implied corner dependence from theempirical corner dependence, the values in table 17 reported for ‘Diff. GaλL’, ‘diff. Ga λU ’, ‘diff. t λL’, ‘diff. t λU ’, ‘diff. B λL’ and ‘diff. B λU ’ arethe log-differences of the implied and the empirical lower and upper cornerdependence measures, ln λimplied

L,α − ln λempiricalL,α and ln λimplied

L,α − ln λempiricalL,α , for

the Gaussian, Student t and BB1 copula, respectively. Negative values indi-cate that the implied corner dependence is lower than the empirical cornerdependence, which means that the parameterised copula underestimates theprobability of joint excessive observations. The values presented in table 17show that the Gaussian copula seems to systematically underestimate theprobability of joint excessive events while the Student t copula overestimatesthem; in addition these deviations are in absolute terms higher for α = 0.05than they are for α = 0.1. However the corner dependence implied by theStudent t copula does not deviate as much from the empirical corner depen-dence as does the corner dependence implied by the Gaussian copula. Forthe BB1 copula, the deviations for α = 0.05 are less pronounced than they

58

ratin

gm

atur

ityal

l 1Y

-3Y

3Y-5

Y5Y

-7Y

7Y-1

0Yal

l 1Y

-3Y

3Y-5

Y5Y

-7Y

7Y-1

0Yal

l 1Y

-3Y

3Y-5

Y5Y

-7Y

7Y-1

0Yal

l 1Y

-3Y

3Y-5

Y5Y

-7Y

7Y-1

0Yal

l 1Y

-3Y

3Y-5

Y5Y

-7Y

7Y-1

0Ym

ean

med

ian

λL

0.40

0.19

0.24

0.19

0.27

0.58

0.19

0.28

0.13

0.21

0.26

0.25

0.17

0.17

0.28

0.32

0.22

0.22

0.19

0.25

0.35

0.27

0.20

0.23

0.25

λU

0.43

0.17

0.26

0.26

0.28

0.64

0.15

0.19

0.14

0.16

0.31

0.21

0.17

0.15

0.30

0.36

0.20

0.24

0.23

0.26

0.42

0.25

0.24

0.27

0.26

abs.

diff

.0.

080.

060.

070.

310.

020.

110.

280.

400.

040.

250.

200.

180.

030.

150.

040.

120.

060.

100.

170.

020.

200.

090.

160.

170.

020.

130.

11D

iff G

a λL

-0.1

1-0

.46

-0.1

70.

10-0

.10

-0.0

5-0

.43

-0.3

0-0

.19

-0.4

8-0

.27

-0.6

0-0

.38

-0.1

6-0

.27

-0.1

0-0

.47

-0.2

0-0

.05

-0.1

50.

04-0

.43

-0.1

0-0

.06

-0.1

2-0

.22

-0.1

7D

iff G

a λU

-0.1

9-0

.40

-0.2

4-0

.21

-0.1

3-0

.15

-0.1

50.

09-0

.24

-0.2

3-0

.47

-0.4

3-0

.34

-0.0

1-0

.31

-0.2

2-0

.41

-0.3

0-0

.22

-0.1

7-0

.16

-0.3

4-0

.26

-0.2

3-0

.14

-0.2

3-0

.23

Diff

t λ

L0.

05-0

.03

0.02

0.28

0.02

0.07

0.11

-0.0

30.

07-0

.08

0.11

-0.1

10.

000.

06-0

.09

0.08

-0.1

00.

070.

18-0

.01

0.20

-0.0

40.

110.

09-0

.01

0.04

0.05

Diff

t λ

U-0

.04

0.04

-0.0

5-0

.03

0.00

-0.0

30.

390.

370.

030.

17-0

.10

0.07

0.03

0.21

-0.1

3-0

.04

-0.0

5-0

.03

0.01

-0.0

30.

000.

04-0

.05

-0.0

7-0

.04

0.03

-0.0

3D

iff B

λL

-0.0

5-0

.14

0.01

0.19

0.03

-0.0

1-0

.06

0.06

-0.0

8-0

.31

-0.1

9-0

.22

-0.0

70.

03-0

.16

-0.0

4-0

.24

0.01

0.11

-0.0

60.

09-0

.06

0.06

0.02

0.01

-0.0

4-0

.04

Diff

B λ

U0.

03-0

.31

-0.1

00.

020.

020.

02-0

.07

0.06

-0.0

80.

00-0

.11

-0.3

5-0

.30

0.10

-0.1

00.

03-0

.26

-0.1

5-0

.04

0.03

0.07

-0.2

9-0

.09

-0.0

20.

00-0

.08

-0.0

4λ

L0.

290.

170.

160.

170.

150.

510.

160.

210.

120.

140.

200.

160.

150.

150.

170.

270.

190.

190.

170.

090.

190.

220.

140.

150.

10λ

U0.

410.

160.

140.

220.

200.

580.

130.

150.

090.

140.

220.

080.

090.

080.

220.

330.

130.

140.

200.

170.

370.

220.

150.

230.

21ab

s. di

ff.

0.34

0.07

0.15

0.24

0.27

0.13

0.24

0.33

0.22

0.00

0.11

0.69

0.49

0.62

0.24

0.20

0.37

0.29

0.13

0.63

0.69

0.00

0.08

0.43

0.69

0.31

0.24

Diff

Ga λ

L-0

.06

-1.0

3-0

.23

-0.2

20.

11-0

.06

-0.8

8-0

.42

-0.7

1-0

.67

-0.4

6-0

.75

-0.8

6-0

.60

-0.1

9-0

.25

-0.9

0-0

.54

-0.4

30.

420.

42-0

.70

-0.2

0-0

.08

0.35

-0.3

6-0

.42

Diff

Ga λ

U-0

.40

-0.9

6-0

.08

-0.4

6-0

.15

-0.1

9-0

.64

-0.0

9-0

.49

-0.6

7-0

.57

-0.0

6-0

.37

0.02

-0.4

3-0

.44

-0.5

3-0

.25

-0.5

5-0

.21

-0.2

8-0

.70

-0.2

8-0

.51

-0.3

4-0

.39

-0.4

0D

iff t

λL

0.23

-0.1

80.

160.

130.

370.

140.

130.

10-0

.13

0.10

0.22

0.16

-0.1

2-0

.16

0.16

0.10

-0.1

8-0

.02

0.02

0.70

0.70

0.00

0.22

0.23

0.56

0.14

0.13

Diff

t λ

U-0

.10

-0.1

10.

31-0

.10

0.10

0.01

0.37

0.42

0.09

0.10

0.11

0.85

0.36

0.46

-0.0

7-0

.09

0.19

0.27

-0.1

00.

070.

000.

000.

14-0

.20

-0.1

40.

120.

09D

iff B

λL

0.08

-0.4

00.

18-0

.04

0.39

0.01

-0.1

80.

25-0

.52

-0.3

8-0

.31

-0.0

3-0

.25

-0.1

70.

04-0

.11

-0.4

2-0

.08

-0.0

50.

610.

54-0

.01

0.19

0.11

0.61

0.00

-0.0

3D

iff B

λU

0.02

-0.8

30.

200.

020.

200.

09-0

.51

-0.1

2-0

.15

-0.1

90.

090.

06-0

.31

0.22

0.03

0.04

-0.2

60.

03-0

.23

0.24

0.13

-0.6

00.

04-0

.05

0.01

-0.0

70.

02

λL

0.12

0.12

0.06

0.18

0.00

0.18

0.06

0.06

0.06

0.06

0.00

0.12

0.06

0.06

0.06

0.12

0.00

0.06

0.12

0.00

0.12

0.12

0.00

0.12

0.00

λU

0.18

0.06

0.06

0.06

0.06

0.41

0.06

0.12

0.12

0.12

0.06

0.06

0.06

0.06

0.06

0.12

0.00

0.06

0.06

0.06

0.12

0.00

0.00

0.12

0.06

abs.

diff

.0.

410.

690.

001.

100.

850.

000.

690.

690.

690.

690.

000.

000.

000.

000.

000.

690.

000.

000.

36D

iff G

a λ

L0.

28-2

.11

-0.2

2-1

.19

0.71

-1.2

7-0

.12

-1.5

5-1

.18

-1.7

5-1

.36

-0.9

4-0

.04

-0.1

5-0

.49

-1.1

20.

32-1

.17

-0.7

8-0

.74

Diff

Ga

λU

-0.1

3-1

.41

-0.2

2-0

.09

0.22

-0.1

4-1

.27

-0.8

1-2

.24

-1.8

7-0

.27

-1.0

6-1

.36

-0.9

4-0

.04

-0.1

5-0

.49

-0.4

2-0

.07

0.32

-0.7

80.

01-0

.60

Diff

t λ

L0.

96-0

.06

0.79

-0.2

71.

130.

991.

120.

030.

680.

310.

470.

240.

870.

690.

800.

040.

970.

430.

050.

54D

iff t

λU

0.56

0.64

0.79

0.83

0.89

0.28

0.99

0.43

-0.6

6-0

.01

1.26

1.00

0.47

0.24

0.87

0.69

0.80

0.73

0.68

0.97

0.05

0.58

0.60

α = 0.1 α = 0.05 α = 0.01

BB

Bal

l A

AA

AA

A

Table 17: Lower and upper empirical corner dependence and comparisonwith the implied corner dependence of a Gaussian and a Student t copula.

59

are for the other two copulas, so one might infer that the BB1 copula’s fitin that region is better than that of the Student t copula. Still, rememberthat the Student t copula’s goodness-of-fit (indeed for the whole unit cuberegion) is superior in all cases to that of the BB1 copula, allowing for a flex-ible modelling of asymmetric corner dependence. This suggests that eitherthis asymmetry is of minor importance as far as copula parameterisation forthe data sample at hand is considered or that other asymmetric copulas thanthe BB1 copula should be employed.

Summarising, the following may be stated. The dependence structurebetween daily interest rate and credit risk factor changes seems to be veryheterogeneous in an unsystematic way, depending on the rating of the oblig-ors (credit risk) and the time until maturity of the financial instruments.These findings indicate that the top-down approach presented in this articlemight be overly simplistic. However, the data also suggest that copula-basedapproaches in a risk aggregation context (e.g. in a bottom-up approach)seem preferable to approaches based on a multivariate Gaussian distributionas (a) the marginal dsitributions are not normally distributed and (b) theGaussian copula fits the data worse than other copulas including the Stu-dent t copula. The Student t copula achieves the best goodness-of-fit (interms of the AIC). A likelihood ratio test rejects the null hypothesis of aGaussian copula in favour of a Student t copula in all cases considered. TheBB1 copula also yields good results (in terms of the AIC), however the nullhypothesis of the BB1 copula being the true copula is rejected quite often(as compared to the Student t copula). The Frank copula yields inferiorresults for the data sample at hand. Concerning the computing time neededfor the estimation and simulation of copulas it is found that interestingly forStudent t copulas parameter estimation takes very long while simulation iscomparably fast; the contrary can be said about the BB1 copula.

60

5 Conclusion

This article presents the concept of a copula-based top-down approach inthe field of financial risk aggregation. After reviewing recent literature onthe subject in section 2, copula-based approaches are presented in section 3.Here, the Gaussian, Student t, BB1, Clayton, Gumbel, and Frank copula arepresented and their properties are examined. Specific equations for the cop-ula functions and their densities are provided. In addition, section 3 showshow copula parameters are estimated and presents goodness-of-fit measuresand tests. Algorithms for the simulation of copulas and meta-distributionsare also provided.

Section 4 examines the dependence structure between interest rate andcredit risk factor changes that are computed from sovereign and corporatebond indices. No clear pattern can be observed as the dependence structurevaries substantially depending on the duration and the rating of the obligors.While on the one hand this may be taken as an indication of the top-downapproach presented in this article being too simplistic to be implemented,the results on the other hand suggest that copula-based approaches (e.g.in the context of bottom-up approaches) seem preferable to the assumptionof a multivariate Gaussian distribution: none of the marginal distributionsexamined are normally distributed. Also, the Gaussian copula’s fit in termsof the AIC is worse than that of the Student t and the BB1 copula.

61

Appendix

Appendix A: Rank-based correlation measures

In the context of copula-based approaches, rank-based correlation measuresare used rather than the linear correlation coefficient (Pearson’s rho), that isused in the context of multivariate Gaussian distributions. Two widely usedrank-based correlation measures are

• Spearman’s rho ρS and

• Kendall’s tau τK .

The sample estimate of Spearman’s rank correlation coefficient (Spear-man’s rho) is calculated as

ρS = 1− 6∑N

i=1 d2i

N(N2 − 1)(33)

where N is the sample size, di = R(x1,i)−R(x2,i) with R(.) the ranks (in caseof ties midranks) of the observed values x1,i and x2,i.

37 If there are no ties,Spearman’s rank correlation coefficient corresponds to Pearson’s correlationcoefficient of the obervation’s ranks or, equivalently, to Pearson’s correlationcoefficient of the pseudo-observations (see equation 10 on p.37), i.e.

ρS =cov(U1, U2)

σU1σU2

(34)

The null hypothesis of a correlation coefficient of zero is tested by em-ploying the formula

t =ρ√

N − 2√1− ρ2

(35)

This test is employed for significance testing of both Spearman’s rank correla-tion coefficient and Pearson’s correlation coefficient (see Glasser and Winter[31] and Kendall et al. [46]). The test statistic t is Student t distributed withν = N − 2 degrees of freedom. Thus, the corresponding p-value is computedas 2 (1− tν=N−2(|t|)), where tν is the Student t distribution function with νdegrees of freedom.

37see eg. Hartung et al. [37], p.554.

62

An alternative test for testing the independence of two jointly observedsamples is based on the Hotelling-Pabst statistic (Hotelling and Pabst [40])D =

∑Ni=1 d2

i . The test statistic T is defined as

T =D − E(D)√

Var(D)=

6∑N

i=1 d2i − (N3 −N)√

(N − 1)(N + 1)2N2(36)

if there are no ties (see e.g. Hartung et al. [37], p.556f). For N > 30, Tis approximately standard normally distributed, so the p-value for rejectingthe null hypothesis of independence can be approximated as 2 (1− Φ (|T |)),where Φ(.) is the univariate standard normal c.d.f.

To test null hypotheses of the typeH0 : ρS ≥ ρ0

against the alternative hypothesesH1 : ρS < ρ0,the following test statistic which is based on Fisher’s r to z transformation(Fisher [26]; for detailed instructions see e.g. Hartung et al. [37], p.548f.)and which is slightly modified to be applicable for Spearman’s rank correla-tion coefficient (rather than for Pearson’s correlation coefficient) accordingto findings by David and Mallows [15], Fieller et al. [23], Fieller and Pearson[24] is used

t =1√1.06

(z − ζ)√

N − 3, (37)

where

z = arctanh(ρi,j) =1

2ln

1 + ρi,j

1− ρi,j

and

ζ = arctanh(ρ0) +ρ0

2(N − 1)=

1

2ln

1 + ρ0

1− ρ0

+ρ0

2(N − 1).

The null hypothesis can be rejected at confidence level α if t < Φ−1(α),i.e. if the test statistic is smaller than the functional inverse of the standardGaussian c.d.f. at significance level α. The p-values for the probability of atype 1 error if the null hypothesis is rejected can thus be computed as Φ(t).

The sample estimate of Kendall’s rank correlation coefficient (Kendall’stau, Kendall [45]) is calculated as

τK =4P

N(N − 1)− 1 (38)

63

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1

ρS

τK

Figure 18: Bounds of Spearman’s rho ρS as a function of Kendall’s tau τK

for a given dependence structure.

where

P =N−1∑i=1

N∑j=i+1

1x2,i<x2,j

where x2,i are the re-arranged observations of x2,i, when the joint observa-tions are ordered according to the values of the observations of the first datasample x1,i. 2P can also be interpreted as the number of concordant pairsminus the number of discordant pairs.38

As shown by Durbin and Stuart [21], there exists a functional relationshipbetween Spearman’s rho ρS and Kendall’s tau τK

{32τK − 1

2≤ ρS ≤ 1

2+ τK − 1

2(τK)2 if τK ≥ 0

−12

+ τK + 12(τK)2 ≤ ρs ≤ 3

2τK + 1

2if τK < 0

(39)

Figure 18 depicts these bounds graphically.

38Internet-references on the computation of τK are e.g.http://en.wikipedia.org/wiki/Kendall’s tau andhttp://www.quantlet.com/mdstat/scripts/estat zko/ktau/estat/bpreview/006 kendallstau.html.

64

Appendix B: GoF test – probability integral transform

This goodness-of-fit uses the probability-integral transformation.39 The testis presented e.g in Dias [19], p.27f., where the following is taken from.

Let (X1, . . . , Xn)T be a random vector with continuous distribution func-tion F (x1, . . . , xn). Let Fi(xi) = P (Xi ≤ xi) be the distribution functionof the univariate margin Xi, i = 1, . . . , n and Fi|1,...,i−1(xi|x1, . . . , xi−1) =P (Xi ≤ xi|X1 = x1, . . . , Xi−1 = xi−1) for i = 2, . . . , n (conditional probabili-ties of Xi, given X1, . . . , Xi−1). Consider the n transformations

T (x1) = P (X1 ≤ x1) = F1(x1),

T (x2) = P (X2 ≤ x2|X1 = x1) = F2|1(x2|x1),

...

T (xn) = P (Xn ≤ xn|X1 = x1, . . . , Xn−1 = xn−1) = Fn|1,...,n−1(xn|x1, . . . , xn−1).

Then the Zi = T (Xi), i = 1, . . . , n, are uniformly and independentlydistributed on [0, 1]n. Suppose that C is a copula such that

F (x1, x2, . . . , xn) = C(F1(x1), F2(x2), . . . , Fn(xn)).

If Ci(u1, . . . , ui) denotes the joint i-marginal distribution

Ci(u1, . . . , ui) = C(u1, . . . , ui, 1, . . . , 1), i = 2, . . . , n− 1,

of (U1, . . . , Ui), with C1(u1) = u1, Cn(u1, . . . , un) = C(u1, . . . , un), then theconditional distribution of Ui, given the values U1, . . . , Ui−1, is

Ci(ui|u1, . . . , ui−1) =∂i−1Ci(u1, . . . , ui)

∂u1 · · · ∂ui−1

/∂i−1Ci−1(u1, . . . , ui−1)

∂u1 · · · ∂ui−1

, i = 2, . . . , n.

Using the conditional distributions Ci, the transformed variables Zi can thuswritten as

Zi = Ci(Fi(Xi)|F1(X1), . . . , Fi−1(Xi−1), i = 2, . . . , n,

and Z1 = F1(X1). If (F1(X1), . . . , Fn(Xn)) has distribution function C,then the Φ−1(Zi), i = 1, . . . , n are iid standard normally distributed and

39see e.g Rosenblatt [60]. These tests do in fact test for the whole joint distribution, notjust for the copula. Test results may thus, in general, be affected by the assumption onthe marginals.

65

Sn =∑n

i=1(Φ−1(Zi))

2 has a chi-square distribution with n degrees of freedom.

In particular for bivariate copulas (n = 2),

S2 = (Φ−1(Z1))2 + (Φ−1(Z2))

2,

where

Z1 = F1(X1) = u1, and

Z2 = C2(F2(X2)|F1(X1)) = C2(u2|u1) =∂C(u1, u2)

∂u1

.

Explicit formulas for S2 for selected bivariate copulas are provided in ta-ble 18.

For rotated bivariate copulas, S2 is

• C−−: S2 = (Φ−1(u1))2+(Φ−1

(1− ∂C(1−u1,1−u2;θ)

∂(1−u1)

))2

• C−+: S2 = (Φ−1(u1))2+(Φ−1

(∂C(1−u1,u2;θ)

∂(1−u1)

))2

• C+−: S2 = (Φ−1(u1))2+(Φ−1

(1− ∂C(u1,1−u2;θ)

∂(u1)

))2

For trivariate copulas (n = 3), we have

S3 = (Φ−1(Z1))2 + (Φ−1(Z2))

2 + (Φ−1(Z1))2,

where

Z3 = C3(F3(X3)|F1(X1), F2(X2)) = C3(u3|u1, u2) =

=

∂2C(u1,u2,v3)∂u1∂u2

∂2C(u1,u2)∂u1∂u2

=

∂2C(u1,u2,v3)∂u1∂u2

c(u1, u2)

with c(u1, u2) the bivariate copula’s density. Z1 and Z2 are defined as inthe bivariate case. Explicit formulas for S3, for the trivariate Gaussian andStudent t copulas, are provided in table 19.

If one wants to test whether a parameterised copula is indeed the truecopula, values of S2 or S3 are computed from the N tuples of empirical

66

copula S = (Φ−1(u1))2+(Φ−1

(∂C(u1,u2;θ)

∂u1

))2

Gaussian S = (Φ−1(u1))2+

+

Φ−1

∫ Φ−1(u2)

−∞

exp

(−(Φ−1(u1))

2+x2−2ρxΦ−1(u1)

2(1−ρ2)

)2π√

1−ρ2dx

φ(Φ−1(u1))

2

,

where φ(.) is the p.d.f. of the univariate standard normal distribution andΦ−1(.) is the functional inverse of the univariate standard normal c.d.f.

Student t S = (Φ−1(u1))2+

+

Φ−1

∫ t−1

ν(u2)

−∞

Γ( ν+22 )

Γ( ν2 )πν

√1−ρ2

(1+

(t−1ν

(u1))2+x2−2ρxt−1

ν(u1)

(1−ρ2)ν

)− ν+22

dx

fν(t−1ν

(u1))

2

,

where fν is the p.d.f of the univariate standard Student t distribution andt−1ν is the functional inverse of the univariate Student t c.d.f with ν d.o.f.

BB1 S = (Φ−1(u1))2+(Φ−1

(a−

1

θ−1b

1

δ−1(u−θ

1 − 1)δ−1u−θ−11

))2,

where a =(1 + b

1δ

)and b =

[(u−θ

1 − 1)δ

+(u−θ

2 − 1)δ].

Clayton S = (Φ−1(u1))2+

(Φ−1

(u−θ−1

1

(u−θ

1 + u−θ2 − 1

)− 1

θ−1))2

Gumbel S = (Φ−1(u1))2+(Φ−1

(exp

(−b

1

θ

)(− ln u1)θ−1

u1b

1

θ−1))2

,

where b =[(− lnu1)θ + (− lnu2)θ

].

Frank S = (Φ−1(u1))2+(Φ−1

(e−θu1(e−θu2−1)

e−θ+(e−θu1−1)(e−θu2−1)−1

))2

Table 18: Equations for S2 required for the goodness-of-fit test for selectedbivariate copulas.

67

copula

S=( Φ

−1(u

1)) 2 +

( Φ−

1( ∂

C(u

1,u

2;θ

)∂

u1

)) 2 +

( Φ−

1

( ∂2

C(u

1,u

2,u

3;θ

)∂

u1

∂u2

c(u

1,u

2;θ

)

)) 2

Gau

ssia

nS

=( Φ

−1(u

1)) 2 +

( Φ−

1

( ∫ Φ−

1(u

2)

−∞

φd=

2,P

(Φ−

1(u

1),

x)d

x

φ(Φ−

1(u

1))

)) 2+

( Φ−

1

( ∫ Φ−

1(u

3)

−∞

φd=

3,P

(Φ−

1(u

1),

Φ−

1(u

2),

x)d

x

φ(Φ−

1(u

1))·φ

(Φ−

1(u

2))·c

(u1,u

2;ρ

)

)) 2,or

,ex

plic

itly

,

S=( Φ

−1(u

1)) 2 +

Φ−

1

∫ Φ−1(u

2)

−∞

exp

( −(Φ−

1(u

1) )

2+

x2−

2ρ1

,2xΦ−

1(u

1)

2(1−

ρ2 1

,2)

)2

π√

1−

ρ2 1

,2

dx

φ(Φ−

1(u

1))

2

+

+

Φ−1

∫ Φ−1(u

3)

−∞

exp( −1 2

( (Φ−

1(u

1) )

2ρ−

11

,1+(Φ−

1(u

2) )

2ρ−

12

,2+

x2

ρ−

13

,3+

2Φ−

1(u

1)Φ−

1(u

2)ρ−

11

,2+

2Φ−

1(u

1)x

ρ−

11

,3+

2Φ−

1(u

2)x

ρ−

12

,3

))(2

π)3

/2|ρ|1

/2

dx

φ(Φ−

1(u

1))·φ

(Φ−

1(u

2))·c

(u1,u

2;ρ

)

2

,

whe

reφ

d=

i,P(.

)is

the

p.d.

f.of

the

i-di

men

sion

alst

anda

rdno

rmal

dist

ribu

tion

wit

hco

rrel

atio

nm

atri

xP

,φ(.

)is

the

p.d.

f.of

the

univ

aria

test

anda

rdno

rmal

dist

ribu

tion

,Φ−

1(.

)is

the

func

tion

alin

vers

eof

the

univ

aria

test

anda

rdno

rmal

c.d.

f.an

dc(

u1,u

2;ρ

)is

the

biva

riat

eG

auss

ian

copu

lade

nsity.

Stu

den

tt

S=( Φ

−1(u

1)) 2 +

( Φ−

1

( ∫ t−1

ν(u

2)

−∞

fd=

2,ν

,P(t−

1ν

(u1),

x)d

x

fν(t−

1ν

(u1) )

)) 2+

( Φ−

1

( ∫ t−1

ν(u

3)

−∞

fd=

3,ν

,P(t−

1ν

(u1),

t−1

ν(u

2),

x)d

x

fν(t−

1ν

(u1) )·f

ν(t−

1ν

(u2) )·c

(u1,u

2;ρ

)

)) 2,or

,ex

plic

itly

,

S=( Φ

−1(u

1)) 2 +

Φ−

1

∫ t−1 ν(u

2)

−∞

Γ(ν

+2

2)

Γ(ν 2

)πν√

1−

ρ2 1

,2

( 1+

(t−

1ν

(u1) )

2+

x2−

2ρ1

,2x

t−

1ν

(u1)

(1−

ρ2 1

,2)ν

) −ν+2

2

dx

fν(t−

1ν

(u1) )

2

+

+

Φ−

1

∫ t−1 ν(u

3)

−∞

Γ(ν

+3

2)

Γ(ν 2

)(π

ν)3

/2|ρ|1

/2

( 1+

(t−

1ν

(u1) )

2ρ−

11

,1+(t−

1ν

(u2) )

2ρ−

12

,2+

x2

ρ−

13

,3+

2t−

1ν

(u1)t−

1ν

(u2)ρ−

11

,2+

2t−

1ν

(u1)x

ρ−

11

,3+

2t−

1ν

(u2)x

ρ−

12

,3ν

) −ν+3

2

dx

fν(t−

1ν

(u1) )·f

ν(t−

1ν

(u2) )·c

(u1,u

2;ρ

)

2

,

whe

ref d

=i,

ν,P

isth

ep.

d.fof

the

i-di

men

sion

alst

anda

rdSt

uden

tt

dist

ribu

tion

func

tion

wit

hν

degr

ees

offr

eedo

man

dco

rrel

atio

nm

atri

xP

,f ν

isth

ep.

d.fof

the

univ

aria

test

anda

rdSt

uden

tt

dist

ribu

tion

,t−

1ν

isth

efu

ncti

onal

inve

rse

ofth

eun

ivar

iate

Stud

ent

tc.

d.fw

ith

νde

gree

san

dc(

u1,u

2;ρ

)is

the

biva

riat

eSt

uden

tt

copu

lade

nsity.

Tab

le19

:E

quat

ions

for

Sth

atis

nee

ded

for

the

goodnes

s-of

-fit

test

for

triv

aria

teG

auss

ian

and

Stu

den

tt

copula

s.For

the

biv

aria

teco

pula

den

sity

funct

ions

c(u

1,u

2;θ

),re

fer

toTab

le5

onp.2

2.

68

ρS maturity bandsrating all 1Y-3Y 3Y-5Y 5Y-7Y 7Y-10Y

all -0.52∗∗∗ -0.04∗ -0.24∗∗∗ -0.27∗∗∗ -0.32∗∗∗

AAA -0.78∗∗∗ -0.06∗∗∗ -0.29∗∗∗ 0.00 -0.08∗∗∗

AA -0.26∗∗∗ 0.12∗∗∗ -0.05∗∗ -0.13∗∗∗ -0.29∗∗∗

A -0.41∗∗∗ 0.09∗∗∗ -0.21∗∗∗ -0.20∗∗∗ -0.27∗∗∗

BBB -0.55∗∗∗ -0.21∗∗∗ -0.22∗∗∗ -0.26∗∗∗ -0.29∗∗∗

Table 20: Spearman’s rho for the 25 bivariate unadjusted observation pairs.

pseudo-observations u1, . . . , uN according to the equations presented in ta-bles 18 and 19. Subsequently one tests whether the N values obtained forSn are from a chi-square distribution with n degrees of freedom using e.g. aKolmogorov-Smirnov40 or an Anderson-Darling41 goodness-of-fit test.

Appendix C: Empirical results for non-autocorrelation-adjusted data

The tables below present sample estimates of Spearman’s rho (table 20 andAIC goodness-of-fit measures (table 21) that are obtained by various copu-las for the original, i.e. unadjusted data sample presented in section 4. Thetables correspond to tables 12 and 13. The unadjusted data sample displaysautocorrelation so the results reported in tables 20 and 21 below are biased.

Table 21, continued on next pagepanel A: all ratingsmaturities all 1Y-3Y 3Y-5Y 5Y-7Y 7Y-10Y

Gauss -578.50∗∗∗ -3.34∗∗∗ -115.85∗∗ -138.15∗ -204.69∗

Student t -665.11 -86.22∗ -157.93 -170.00 -227.30BB1 -646.94(−+) -24.32∗∗(+−) -143.01∗(+−) -161.85(−+) -222.81(−+)

Clayton -529.69∗∗∗(+−) -13.98∗∗∗(−+) -113.75∗∗(+−) -128.68∗(+−) -169.95∗(+−)

Gumbel -626.18∗(−+) -23.85∗∗(+−) -131.58∗(−+) -152.24(−+) -202.20(−+)

Frank -580.01∗ -1.06∗∗∗ -111.00∗∗ -136.35 -189.41

Continued on next page40See e.g. Hartung et al. [37], p.183.41See Anderson Darling [6] or e.g. Giles [30].

69

Table 21, continued from last pagepanel B: rating AAAmaturities all 1Y-3Y 3Y-5Y 5Y-7Y 7Y-10Y

Gauss -1655.84∗∗∗ -4.53∗∗∗ -154.37∗∗∗ 0.62∗∗∗ -13.19∗∗∗

Student t -1899.27 -142.54∗∗ -221.96 -29.12 -99.05BB1 -1840.76∗∗∗(−+) -31.24∗∗∗(+−) -197.71∗∗(+−) -8.12∗∗(+−) -39.40∗∗∗(−+)

Clayton -1497.27∗∗∗(+−) -20.07∗∗∗(−+) -169.97∗∗(−+) -2.79∗∗(+−) -23.06∗∗∗(−+)

Gumbel -1792.67∗∗∗(−+) -31.07∗∗∗(+−) -194.56∗∗(+−) -7.22∗∗(+−) -33.27∗∗∗(+−)

Frank -1700.96∗∗ -5.61∗∗∗ -157.71∗∗ 1.99∗∗∗ -10.49∗∗∗

panel C: rating AAmaturities all 1Y-3Y 3Y-5Y 5Y-7Y 7Y-10Y

Gauss -112.26∗∗∗ -24.04∗∗∗ -3.85∗∗ -29.66∗∗ -172.20∗

Student t -219.51 -136.33∗∗∗ -56.53 -51.47 -206.99BB1 -162.90∗(−+) -57.82∗∗∗(−−) -19.22∗(+−) -42.47∗(+−) -196.14(−+)

Clayton -119.97∗∗∗(+−) -36.02∗∗∗ -12.63∗∗(−+) -29.14∗∗(−+) -145.43∗∗(−+)

Gumbel -157.83∗∗(−+) -56.41∗∗∗(−−) -19.03∗(+−) -38.50∗(+−) -175.89(+−)

Frank -127.87∗∗ -28.20∗∗∗ -3.17∗∗ -31.09∗∗ -163.28

panel D: rating Amaturities all 1Y-3Y 3Y-5Y 5Y-7Y 7Y-10Y

Gauss -339.51∗∗ -15.36∗∗∗ -77.23∗ -77.13∗∗ -135.76∗

Student t -398.33 -76.69∗ -115.28 -115.31 -148.64BB1 -384.77(−+) -33.03∗∗∗(−−) -99.13(+−) -101.70∗(+−) -148.14(−+)

Clayton -307.51∗∗(+−) -19.71∗∗∗(−−) -68.46∗(+−) -75.76∗∗(+−) -117.98(+−)

Gumbel -368.31(−+) -30.39∗∗∗ -87.59∗(+−) -91.44∗(−+) -138.69(−+)

Frank -340.09 -14.04∗∗∗ -77.47∗ -75.06∗∗ -129.90

panel E: rating BBBmaturities all 1Y-3Y 3Y-5Y 5Y-7Y 7Y-10Y

Gauss -605.70∗∗∗ -73.60∗∗∗ -88.13∗∗ -137.27 -157.93Student t -695.43∗ -183.05 -115.04 -159.41 -171.81

BB1 -669.98∗∗∗(−+) -120.23∗∗(+−) -104.12∗(+−) -159.48(−+) -166.87(−+)

Clayton -551.89∗∗∗(+−) -89.30∗∗∗(−+) -85.93∗(+−) -133.25(+−) -132.78(+−)

Gumbel -654.12∗∗∗(−+) -116.88∗∗(+−) -100.02(−+) -152.35(−+) -153.61(−+)

Frank -655.19∗∗∗ -79.97∗∗∗ -92.57∗ -124.47 -152.23

Continued on next page

70

Table 21, continued from last pageTable 21: AIC goodness-of-fit measures obtained by cop-ulas fitted to the 25 bivariate unadjusted observation.The subscripts (−+), (+−) and (−−) indicate that a ro-tated copula was used. Asterisks indicate that the nullhypothesis of the specific copula being the true copulacan be rejected at the ∗ 10%, ∗∗ 5% or ∗∗∗ 1% significancelevel.

List of Tables

1 Sample moments marginal distributions, introductory example 142 Summary table lower, upper and symmetric tail dependence. . 163 Quantiles of total return distribution, introductory example . 194 Selected bivariate copula functions . . . . . . . . . . . . . . . 215 Probability density functions of selected bivariate copulas . . . 226 n-dimensional Gaussian and Student t copula functions . . . . 237 n-dimensional Gaussian and Student t copula density functions 248 Lower and upper tail dependence, λL and λU , of selected bi-

variate copulas . . . . . . . . . . . . . . . . . . . . . . . . . . 259 Functional relationships between copula parameters and cor-

relation measures. . . . . . . . . . . . . . . . . . . . . . . . . . 3810 Computing time for selected bivariate copulas . . . . . . . . . 4211 Computing time for n-dimensional Gaussian and Student t

copulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4312 Spearman’s rho for the 25 bivariate observation pairs . . . . . 5113 AIC goodness-of-fit measures obtained by bivariate copulas . 5314 Rejection of the null hypothesis that the copula at hand is the

true copula . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5415 Comparison of implied and empirical corner dependence for

symmetric copulas . . . . . . . . . . . . . . . . . . . . . . . . 5716 Comparison of implied and empirical upper and lower corner

dependence for Student t and BB1 copulas . . . . . . . . . . . 5817 Lower and upper empirical corner dependence 1 . . . . . . . . 5918 GoF-test: S2 for selected bivariate copulas . . . . . . . . . . . 6719 GoF-test: S3 for tivariate Gaussian and Student t copulas . . 6820 Spearman’s rho for unadjusted observations . . . . . . . . . . 69

71

21 AIC goodness-of-fit measures for the unadjusted observations 71

List of Figures

1 Bottom-up and top-down approaches . . . . . . . . . . . . . . 82 Introductory example for copula-based approaches . . . . . . . 133 Scatter plots of bivariate meta-distributions . . . . . . . . . . 154 Densities of bivariate Gaussian, Student t and Frank copulas . 175 Densities of rotated Clayton and Gumbel copulas . . . . . . . 186 Structure of a four-dimensional hierarchical Archimedean copula 257 Densities of bivariate Gaussian copulas and meta-Gaussian

distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288 Densities of bivariate Student t copulas and meta-Student t

distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299 Densities of bivariate BB1 copulas . . . . . . . . . . . . . . . . 3010 Densities of bivariate Clayton copulas and meta-Clayton dis-

tributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3111 Densities of bivariate Gumbel copulas and meta-Gumbel dis-

tributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3212 Log differences of a Clayton and a survival Gumbel copulas’

densities and of corresponding meta-distributions’ densities . . 3313 Densities of bivariate Frank copulas and meta-Frank distribu-

tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3414 Scatter plots of interest rate and credit risk factor changes . . 4915 Mean AIC of selected copulas . . . . . . . . . . . . . . . . . . 5216 Parameter distribution of the Student t copula parameter ν . 5517 Empirical and implied lower and upper corner dependence . . 5718 Bounds of Spearman’s rho ρS as a function of Kendall’s tau τK 64

72

References

[1] Kjersti Aas, Xeni K Dimakos, and Anders Øksendal. Risk capital ag-gregation. Journal of Risk, 2006.

[2] Kjersti Aas and Ingrid Hobæk Haff. The generalised hyperbolic skewstudents t-distribution. Journal of Financial Econometrics, 4(2), 2006.

[3] Hirotugu Akaike. A new look at the statistical model identification.IEEE Transactions on Automatic Control, 19(6):716–723, 1974.

[4] Carol Alexander. The Present and Future of Financial Risk Manage-ment. Journal of Financial Econometrics, 3(1):3–25, 2005.

[5] Carol Alexander and Jacques Pezier. On the Aggregation of Firm-WideMarket and Credit Risks. ISMA Centre Discussion Papers in Finance2003-13, University of Reading, 2003.

[6] T. W. Anderson and D. A. Darling. Asymptotic Theory of CertainGodness of Fit Criteria Based on Stochastic Processes. Annals of Math-ematical Statistics, 23:193–212, 1952.

[7] Philippe Arztner, Freddy Delbaen, Jean-Marc Eber, and David Heath.Coherent Measures of Risk. Mathematical Finance, 9(3), 2006.

[8] Daniel Berg and Henrik Bakken. A Goodness-of-fit Test for Copulaebased on the Probability Integral Transform. Statistical Research Re-port No. 10, University of Oslo, Department of Mathematics, ISSN 0806-3842, 2005.

[9] Christian Cech and Ines Fortin. Messung der Abhangigkeitsstrukturzwischen Markt- und Kreditrisiko. Wirtschaft und Management - Risiko-management, 2(3):65–85, 2005.

[10] Christian Cech and Ines Fortin. Investigating the dependence structurebetween market and credit portfolios’ profits and losses in a top-downapproach using institution-internal simulated data. Study by the Uni-versity of Applied Sciences bfi Vienna, 2006.

[11] Christian Cech and Ines Fortin. Modelling the dependence structure be-tween financial portfolios’ profits and losses. A top-down approach usinginstitution-internal data. Study by the University of Applied Sciencesbfi Vienna, unpublished, 2006.

73

[12] Christian Cech and Michael Jeckle. Aggregation von Markt- und Kred-itrisiko. Working paper series by the university of applied sciences bfiVienna, No. 15, 2005.

[13] U. Cherubini, E. Luciano, and W. Vecchiato. Copula methods in finance.Wiley Finance, 2004.

[14] Michel Crouhy, Dan Galai, and Robert Mark. A comparative analysis ofcurrent credit risk models. Journal of Banking and Finance, 24:59–117,2000.

[15] F. N. David and C. L. Mallows. The variance of spearman’s rho innormal samples. Biometrika, 48:19–28, 1961.

[16] Paul Deheuvels. La fonction de dependance empirique et ses proprietes,un test non parametrique d’independance. Bulletin de l’academie Royalede Belgique, Classe de Sciences, 65:274–292, 1979.

[17] Paul Deheuvels. A non-parametric test for independence. Publicationsde l’Institut de Statistique de l’Universite de Paris, 26:29–50, 1981.

[18] Stefano Demarta and Alexander J. McNeil. The t Copula and RelatedCopulas. working paper department of mathematics, ETHZ, 2004.

[19] Alexandra da Costa Dias. Copula Inference for Finance and Insurance.PhD thesis, Swiss Federals Institute of Technology (ETH) Zurich, 2004.

[20] Xeni K. Dimakos and Kjersti Aas. Integrated Risk Modelling. StatisticalModelling, 2004.

[21] J. Durbin and A. Stuart. Inversions of rank correlations. Journal of theRoyal Statistical Society, Series B, 2:303–309, 1951.

[22] Jean-David Fermanian. Goodness-of-fit tests for copulas. Journal ofMultivariate Analysis, 95:119–152, 2005.

[23] E. C. Fieller, H. O. Hartley, and E. S. Pearson. Tests for rank correlationcoefficients: I. Biometrika, 44:470–481, 1957.

[24] E. C. Fieller and E. S. Pearson. Tests for rank correlation coefficients:Ii. Biometrika, 48:29–40, 1961.

[25] Christopher C. Finger. The one-factor creditmetrics model in the newbasel capital accord. RiskMetrics Journal, 2(1):9–18, 2001.

74

[26] R. A. Fisher. Frequency distribution of the values of the correlationcoefficient in samples from an indefinitely large population. Biometrika,10:507–521, 1915.

[27] Ines Fortin and Christoph Kuzmics. Tail-dependence in Stock-ReturnPairs. International Journal of Intelligent Systems in Accounting, Fi-nance and Management, 11:89–107, 2002.

[28] Christian Genest, Kilani Ghoudi, and Louis-Paul Rivest. A semipara-metric estimation procedure of dependence parameters in multivariatefamilies of distributions. Biometrika, 82:543–552, 1995.

[29] Christian Genest and Louis-Paul Rivest. Statistical Inference Proceduresfor Bivariate Archimedian Copulas. Journal of the American StatisticalAssociation, 88:1034–1043, 1993.

[30] David E. A. Giles. A Saddlepoint Approximation to the DistributionFunction of the Anderson-Darling Tests Statistic. Econometrics workingpaper EWP 0005, ISBN 1485-6441, University of Columbia, 2000.

[31] G. J. Glasser and R. F. Winter. Critical values of the coefficient ofrank correlation for testing the hypothesis of independence. Biometrika,48:444–448, 1960.

[32] Paul Glasserman. Monte Carlo Methods in Financial Engineering.Springer, 2004.

[33] Michael B. Gordy. A risk-factor model foundation for ratings-based bankcapital rules. Journal of Financial Intermediation, 12(3):199–232, 2003.

[34] William H. Greene. Econometric analysis. Pentice Hall, 5th edition,2003.

[35] Christopher Hall. Economic capital: towards an integrated risk frame-work. Risk (October 2002), pages 33–38, 2002.

[36] D. Hamilton, P. Varma, S. Ou, and R. Cantor. Default and RecoveryRates of Corporate Bond Issuers 1920-2004. Moodys Investors Services,2005.

[37] Joachim Hartung, Barbel Elpelt, and Karl-Heinz Klosener. Statistik.Oldenburg, 13th edition, 2002.

[38] Andrew Hickman, Jim Rich, and Curtis Tange. A Consistent Frameworkfor Market and Credit Risk Management. The Risk Desk, 2(11), 2002.

75

[39] W. Hoffding. Maßstabinvariante Korrelationstheorie. PhD thesis, 1940.

[40] H Hotelling and M. R. Pabst. Rank correlation and tests of significanceinvolving no assumption of normality. Annals of Mathematical Statistics,7:29–43, 1936.

[41] C. M. Jarque and A. K. Bera. A test for normality of observations andregression residuals. Internat. Statist. Rev., 55:163–172, 1987.

[42] Harry Joe. Multivariate Models and Dependence Concepts. Chapmanand Hall, 1997.

[43] Harry Joe and J. Xu, James. The Estimation Method of Inference Func-tions for Margins for Multivariate Models. working paper, Departmentof Statistics, University of British Columbia, 1996.

[44] Norman Johnson, Samuel Kotz, and N. Balakrishnan. Continuous uni-variate distributions, volume 2. John Wiley and Sons, Inc., second edi-tion, 1995.

[45] M. G. Kendall. A New Measure of Rank Correlation. Biometrika, 30:81–89, 1938.

[46] M. G. Kendall, F. H. Kendall, and B. B. Smith. The distribution ofspearmans coefficient of rank correlation in a universe in which all rank-ings occur an equal number of times. Biometrika, 30:251–273, 1939.

[47] A. Kuritzkes, T. Schuermann, and S. Weiner. Risk Measurement, RiskManagement and Capital Adequacy in Financial Conglomerates. Whar-ton paper, 2003.

[48] Alexander J. McNeil and Embrechts Paul Frey, Rudiger and. Quanti-tative Risk Management. Concepts, Techniques and Tools. Princetonuniversity press, 2005.

[49] E.A. Medova and R.G. Smith. A framework to measure integrated risk.Quantitative Finance, 5:105–121, 2005.

[50] Robert C. Merton. On the pricing of corporate debt: the risk structureof interest rates. Journal of Finance, 29:449–470, 1974.

[51] Thomas Mikosch. Copulas: Tales and facts. Working paper, Universityof Copenhagen, 2005.

[52] R. Nelsen. An Introduction to Copulas. Springer, 1st edition, 1999.

76

[53] R. Nelsen. An Introduction to Copulas. Springer, 2nd edition, 2006.

[54] Basel Committee on Banking Supervision. Trends in risk integrationand aggregation. Bank for International Settlements, 2003.

[55] Basel Committee on Banking Supervision. International Convergenceof Capital Measurement and Capital Standards. Bank for InternationalSettlements, 2006.

[56] European Parliament and Council of the European Union. Capitalrequirements for credit institutions and investment firms. Directives2006/48/EC and 2006/49/EC. Official Journal of the European Union,2006.

[57] Jacques Pezier. Application-Based Financial Risk Aggregation Meth-ods. ISMA Centre Discussion Papers in Finance 2003-11, University ofReading, 2003.

[58] T. Roncalli. Gestion des Risques Multiples. Credit Lyonnais, workingpaper, 2002.

[59] J. Rosenberg and T. Schuermann. A General Approach to IntegratedRisk Management with Skewed, Fat-Tailed Risks. Federal Reserve Bankof New York, staff report no. 185, 2004.

[60] M. Rosenblatt. Remarks on a multivariate transformation. Annals ofMathematical Statistics, 23:470–472, 1952.

[61] Francesco Saita. Risk Capital Aggregation: the Risk Manager’s Perspec-tive. NewFin Research Center and IEMIF Universita Bocconi, workingpaper, 2004.

[62] Cornelia Savu and Mark Trede. Hierarchical Archimedean Copulas.working paper, Institute of Econometrics, University of Munster, 2006.

[63] Olivier Scaillet and Jean-David Fermanian. Some statistical pitfalls incopula modeling for financial applications. In Capital Formation, Gov-ernance and Banking, pages 57–72. Nova Science Publishers, 2005.

[64] David W. Scott and Stephan R. Sain. Multi-dimensional Density Esti-mation. In C.R. Rao and E.J. Wegman, editors, Handbook of Statistics—Vol 23: Data Mining and Computational Statistics. Elsevier, Amster-dam, 2004.

[65] B.W. Silverman. Density Estimation. Chapman and Hall, 1986.

77

[66] A. Sklar. Fonctions de Repartition a n dimension et leurs Marges. Pub-lications de l’Institut de Statistique de l’Universite de Paris, 8:229–331,1959.

[67] Andrew Tang and Emiliano A. Valdez. Economic Capital and the Ag-gregation of Risks using Copulas. School of Actuarial Studies and Uni-versity of New South Wales, 2005.

[68] M.P. Wand and M.C. Jones. Kernel Smoothing. Chapman and Hall,1995.

[69] Lisa S. Ward and David H. Lee. Practical Application of the Risk-Adjusted Return on Capital Framework. CAS Forum summer 2002,Dynamic Financial Analysis discussion paper, 2002.

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Working Papers und Studien der Fachhochschule des bfi Wien 2006 erschienene Titel Working Paper Series No 22 Thomas Wala: Steueroptimale Rechtsform. Didactic Series. Wien Mai 2006 Working Paper Series No 23 Thomas Wala: Planung und Budgetierung. Entwicklungsstand und Perspektiven. Didactic Series. Wien Mai 2006 Working Paper Series No 24 Thomas Wala: Verrechnungspreisproblematik in dezentralisierten Unternehmen. Didactic Series. Wien Mai 2006 Working Paper Series No 25 Felix Butschek: The Role of Women in Industrialization. Wien Mai 2006 Working Paper Series No 26 Thomas Wala: Anmerkungen zum Fachhochschul-Ranking der Zeitschrift INDUSTRIEMAGAZIN. Wien Mai 2006 Working Paper Series No 27 Thomas Wala / Nina Miklavc: Betreuung von Diplomarbeiten an Fachhochschulen. Didactic Series. Wien Juni 2006 Working Paper Series No 28 Grigori Feiguine: Auswirkungen der Globalisierung auf die Entwicklungsperspektiven der russischen Volkswirtschaft. Wien Juni 2006 Working Paper Series No 29 Barbara Cucka: Maßnahmen zur Ratingverbesserung. Empfehlungen von Wirtschaftstreuhändern. Eine ländervergleichende Untersuchung der Fachhochschule des bfi Wien GmbH in Kooperation mit der Fachhochschule beider Basel Nordwestschweiz. Wien Juli 2006 Working Paper Series No 30 Evamaria Schlattau: Wissensbilanzierung an Hochschulen. Ein Instrument des Hochschulmanagements. Wien Oktober 2006 Working Paper Series No 31 Susanne Wurm: The Development of Austrian Financial Institutions in Central, Eastern and South-Eastern Europe, Comparative European Economic History Studies. Wien November 2006 Studien Breinbauer, Andreas / Bech, Gabriele: „Gender Mainstreaming“. Chancen und Perspektiven für die Logistik- und Transportbranche in Österreich und insbesondere in Wien. Study. Vienna March 2006 Breinbauer, Andreas / Paul, Michael: Marktstudie Ukraine. Zusammenfassung von Forschungsergebnissen sowie Empfehlungen für einen Markteintritt. Study. Vienna July 2006 Breinbauer, Andreas / Kotratschek, Katharina: Markt-, Produkt- und KundInnenanforderungen an Transportlösungen. Abschlussbericht. Ableitung eines Empfehlungskataloges für den Wiener Hafen hinsichtlich der Wahrnehmung des Binnenschiffverkehrs auf der Donau und Definition der Widerstandsfunktion, inklusive Prognosemodellierung bezugnehmend auf die verladende Wirtschaft mit dem Schwerpunkt des Einzugsgebietes des Wiener Hafens. Wien August 2006

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