Download Copula-based top-down approaches in financial risk aggregation

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 Cech
University of Applied Sciences of bfi Vienna
Abstract
This article presents the concept of a copula-based top-down approach in the field of financial risk aggregation. Selected copulas
and their properties are presented. Copula parameter estimation and
goodness-of-fit tests are explained and algorithms for the simulation
of copulas and meta-distributions are provided. Further, the dependence structure between interest rate and credit risk factor changes
that are computed from sovereign and corporate bond indices is examined. No clear pattern of the dependence structure can be observed
as it varies substantially with the duration and the rating of the obligors. This could indicate that top-down approaches are too simplistic
to be implemented in practice. However, the results also suggest that
copula-based approaches for the data sample at hand seem preferable
to the assumption of a multivariate Gaussian distribution as none of
the marginal distributions examined are normally distributed and as
the Gaussian copula’s fit in terms of the AIC is worse than that of
other copulas. Further, the Gaussian copula seems to underestimate
the probability of joint strong risk factor changes for the data sample
at hand.
3
Contents
1 Introduction
5
2 Bottom-up and top-down approaches
7
3 Copula-based approaches
3.1 Introduction to copulas . . . . . . . . . . . . . . . . . . .
3.2 Modelling the marginal distributions . . . . . . . . . . .
3.3 Presentation of selected copulas . . . . . . . . . . . . . .
3.3.1 Gaussian copula . . . . . . . . . . . . . . . . . . .
3.3.2 Student t copula . . . . . . . . . . . . . . . . . .
3.3.3 BB1 copula . . . . . . . . . . . . . . . . . . . . .
3.3.4 Clayton copula . . . . . . . . . . . . . . . . . . .
3.3.5 Gumbel copula . . . . . . . . . . . . . . . . . . .
3.3.6 Frank copula . . . . . . . . . . . . . . . . . . . .
3.4 Copula-parameter estimation . . . . . . . . . . . . . . .
3.4.1 Maximum likelihood estimation . . . . . . . . . .
3.4.2 Parameter estimation using correlation measures .
3.4.3 Empirical copulas . . . . . . . . . . . . . . . . . .
3.5 Goodness-of-fit tests . . . . . . . . . . . . . . . . . . . .
3.6 Simulation of selected meta-distributions . . . . . . . . .
3.6.1 Simulation of meta-Gaussian distributions . . . .
3.6.2 Simulation of meta-Student t distributions . . . .
3.6.3 Simulation of bivariate meta-BB1 distributions . .
3.6.4 Simulation of bivariate meta-Clayton distributions
3.6.5 Simulation of bivariate meta-Gumbel distributions
3.6.6 Simulation of bivariate meta-Frank distributions .
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4 Implementation of a top-down approach and empirical evidence
46
5 Conclusion
61
Appendix
Appendix A: Rank-based correlation measures . . . . . . . . . . . .
Appendix B: GoF test – probability integral transform . . . . . . .
Appendix C: Empirical results for non-autocorrelation-adjusted data
61
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69
List of tables and figures
71
References
73
4
1
Introduction
According to the Basel Committee on Banking Supervision [54], risk aggregation refers to the development of quantitative risk measures that incorporate
multiple types or sources of risk. Amongst these types of risks are e.g. credit
risk, market risk (interest rate risk, stock price risk, etc.), insurance risk (life
and property and casualty insurance), operational risk, liquidity risk, asset
liability management (ALM) risk, business risk, etc. These quantitative risk
measures, defined over a specific time horizon, may then be used to estimate
the economic capital that is needed to absorb unexpectedly high potential
losses. 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 institution aspires, as the probability of default (i.e. the probability that the
economic capital cannot absorb the realised losses) is related to the confidence level of the risk measure.1
Such reasoning also forms the basis of the new Basel II regulatory framework (Basel Committee on Banking Supervision [55] and European Parliament and Council [56]), where banks are required to hold at least the minimum regulatory capital as a buffer against credit risk (regarded as the main
source of banking risks), whose magnitude does not only depend on the size
of the exposure but also on the riskiness of the credit portfolio.2 The one-year
survival 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 trading
book and for operational risk. The Basel II regulatory framework, however,
does not account for diversification effects between risk types (credit, market and operational risk), as the minimum regulatory capital requirements
for each risk-class are simply added to obtain the total minimum regulatory
1
The economic capital to be held in this context is then defined as the value-at-risk
with confidence level α which implies that a default probability of (1 − α) is conjectured.
For critical remarks see e.g. Pézier [57]. While the value-at-risk has been criticised as
it does not display the desirable feature of sub-additivity (see e.g. Arztner et al. [7]), it
is widely used in practice. Alternative risk measure like e.g. the expected shortfall (also
termed conditional value-at-risk, CVaR) are sub-additive and could easily be obtained by
the methodology presented in this paper. However, there is no direct linkage between the
expected shortfall and a financial institution’s probability of default.
2
The specific formulae to compute the regulatory minimum capital requirement for
credit 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]).
5
capital. This conservative approach implicitly assumes perfect positive correlation between the risk types.
The prudential rules of the Basel II accord should however not be mistaken as a guideline on how to allocate economic capital efficiently. Rather,
institution-internal models that go beyond the minimum regulatory requirements of Basel II (‘pillar 1’) are used in practice.3 Risk aggregation models
that quantify the diversification effects seem to be one necessary foundation
for the efficient allocation of economic capital. Several approaches to risk
aggregation have been proposed in the literature. Section 2 gives an introduction to top-down approaches as opposed to bottom-up approaches in the
context of risk aggregation and reviews existing literature.
Copula-based approaches, presented in section 3, seem adequate and
preferable to the widely employed assumption of multivariate Gaussian distributions4 of risk factor changes, if the risk factor changes are not normally
distributed.5 Section 3.1 gives an introduction to copula-based approaches
in the context of top-down risk aggregation. Various parametric distribution
functions that are used to model marginal distributions in the context of risk
aggregation are shortly mentioned in section 3.2. Section 3.3 presents some
selected bivariate and two multi-dimensional copulas in detail and compares
their properties. Specific equations for copula functions and copula densities
are also provided in this section. Different approaches to copula parameter
estimation are presented in section 3.4, and goodness-of-fit tests are presented
in section 3.5. Finally, section 3.6 provides algorithms for the simulation of
the presented copulas.
Section 4 first shortly addresses the results of two recent studies on the
implementation of a top-down risk aggregation model. These studies, using institution-internal data, find that the risk-factor changes seem to be
only slightly correlated. In the remainder of section 4, daily market data
(bond index returns) are used to examine the dependence structure between
interest rate risk and credit risk. The empirical results provide a very heterogeneous picture of the dependence structure between these two risk factor
3
Efforts on risk management system that go beyond the minimum capital requirements
of ‘pillar 1’ are also regulatorily required in ‘pillar 2’ of the Basel II accord.
4
or other widely used assumptions on joint distributions that jointly model the marginal
distributions and their dependence structure like e.g. the multivariate Student t or Weibull
distributions and the highly flexible multivariate generalized hyperbolic distribution (see
e.g. McNeil et al. [48], section 3.2.3).
5
Not everybody agrees on this statement, see e.g. Mikosch [51].
6
changes, depending on the maturity bands examined and the credit quality.
The goodness-of-fit of six copulas and empirical evidence of positive tail dependence 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 risk
types (for a review article see e.g. Saita [61] or Alexander [4]). These approaches may broadly be classified into bottom-up and top-down approaches.6
Bottom-up approaches try to model the distribution of various risk factors
and their impact on risk types, such as credit risk, market risk, etc. One
prominent example of a bottom-up approach is Credit Metrics, a credit risk
model which derives the profit and loss distribution of a credit portfolio from
the asset values of the obligors, which are modelled as linear combinations of
correlated industry index returns (see e.g. Crouhy et al. [14]). A bottom-up
approach in the context of risk aggregation would estimate the impact of
these risk factors (industry index returns) and, if necessary, additional risk
factors (such as the interest rate term structure, credit spreads, etc.) on the
profits and losses of other lines of business (e.g. the market portfolio) and
model the joint profit and loss distribution on that basis. Hence, the dependence between risk types (profits and losses of different lines of business) is
modelled indirectly: a joint distribution of risk factor changes is estimated
and the impact of these risk factor changes on the diverse financial portfolios’ profits and losses, defined as (generally non-linear) functions of the risk
factor changes, is modelled.
Top-down approaches, on the contrary, do not try to identify common
single risk factors that influence different types of risk, but rather start from
aggregated data, e.g. the profits and losses of different lines of business, such
as the returns of the credit portfolio or the market portfolio. Operational risk
in this context would be modelled as a portfolio of risk exposures with nonpositive profits and losses or returns. Empirical panel-data of (or assumptions
on) the profits and losses or returns of these portfolios allow to estimate a
joint distribution of the total returns, or the ‘total risk’. The single components that constitute the financial portfolios (or, alternatively expressed,
the single risk factors that influence the portfolio profits and losses) are not
6
See e.g. Cech and Jeckle [12]. The approaches are also referred to as base-level and
top-level aggregation, see e.g. Aas et al. [1].
7
Bottom-up approach
Top-down approach
Distribution of economic risk factors, e.g.
Distribution of portfolio returns, e.g.
•
•
•
•
•
•
•
•
•
•
interest rate term structure
credit spread term structure
equity returns
GDP growth
etc.
market portfolio returns
credit portfolio returns
insurance portfolio returns
losses due to operational risk
etc.
and dependence structure (copula)
and dependence structure (copula)
Profit and loss functions (domain: economic risk factors), e.g.
Joint profits and losses / returns.
•
•
•
•
•
market portfolio returns
credit portfolio returns
insurance portfolio returns
losses due to operational risk
etc.
Resulting in joint profits and losses / returns.
Figure 1: Bottom-up and top-down approaches.
addressed in this approach. Figure 1 schematically depicts the bottom-up
and the top-down approach.
In both approaches a common time horizon for the estimation of risk factor changes has to be found. Ideally, the time horizon would correspond to
the internal capital allocation cycle which conventionally is one year. Generally, the profit and losses or returns of credit and insurance risk are also
measured at least at this frequency and risk measures are estimated for a
one-year horizon. Market portfolio profits and losses and associated risk
measures are often measured and estimated on a daily basis, as the average
holding period of instruments in the market portfolio is generally short-term
and also because of regulatory directives. If one assumes that the market
portfolio profits and losses are normally distributed and i.i.d.7 , one can easily
compute one-year risk measures for the market portfolio by using the ‘squareroot-time’ formula and estimate the one-year unexpected loss8 as a multiple
of the one-day unexpected loss. The one-year unexpected
loss is computed
√
by 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 unexpected loss minus the one-year expected profit. This approach however
ignores the usual pre-defined market risk management intervention policies
like stop-loss limits, etc. as the risk measures are computed on the basis of
7
Independent and identically distributed, i.e. the returns are not autocorrelated.
I.e. the negative value of the one-sided confidence interval lower bound for a deviation
from the expected profit.
8
8
the current portfolio composition. I.e. the risk measures are computed for
a buy-and-hold portfolio, which leads to an overestimation (upward-bias9 ).
On the other hand side, non-normality of daily market risk factor changes
has been widely documented. The univariate risk factor changes are often
leptokurtic and left-skewed; furthermore the probability of joint extremely
negative returns is higher than implied by a multivariate normal distribution
(see e.g. Fortin and Kuzmics [27]). This again leads to an underestimation
of the risk measures (downward-bias) if normality of the market risk factor
changes is falsely assumed.
Aas et al. [1] in their model incorporate risk management intervention policies by simulating daily market portfolio returns under predefined
stop-loss policies and limits, using a constant conditional correlation (CCC)
GARCH(1,1) model to account for volatility clustering and leptokurtic return distributions. The distribution of 1-year market portfolio returns that
are obtained from the simulations of successive one-day returns are then used
to 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 sufficiently large data-set of e.g. credit or insurance portfolio profits and losses
on an annual basis so that a model for aggregated risk can be calibrated.
Rosenberg and Schuermann [59] overcome the problem of short time-series
by estimating linear regression functions to explain market and credit risk as
functions of macroeconomic risk factors, using panel-data for quarterly returns of the market and credit portfolio returns of a set of large banks. The
regression functions are calibrated using 9 years of historical data. Assuming constant regression parameters, market and credit portfolio returns and
their dependence structure are simulated by 29 years of historical quarterly
macroeconomic risk factor data as regressors (operational risk is modelled
separately).
If one wants to avoid the model risk associated with both models presented above, institution-internal time series of profits and losses or returns
for the different lines of business may be used to estimate a risk-aggregation
model. This again results in a very small data sample for the calibration of
the model if annual data is used as generally there are no long time series of
e.g. credit portfolio profits and losses available. Using monthly institutioninternal data seems to be a promising compromise.
9
Hickman et al. [38] show that risk management intervention policies can substantially
reduce the risk.
9
In both top-down and bottom-up approaches, the task of estimating joint
distributions (joint economic risk factors changes in the context of bottom-up
approaches and joint portfolio profits and losses in the context of top-down
approaches), may be decomposed into (a) the estimation of the marginal distributions (univariate risk factor changes or portfolio profits and losses) and
(b) the estimation of the dependence structure, if a copula-based approach
is used. Copulas may be thought of as a more flexible version of correlation matrices that are widely used in risk management models that assume
joint 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 and
Lee [69] (insurance, market, credit, ALM and operational risk), Dimakos and
Aas [20] (market, credit and operational risk), Rosenberg und Schuermann
[59] (market, credit and operational risk) and Tang and Valdez [67] (different types of insurance risk). While Kuritzkes et al. [47] in their simplifying
approach assume a joint normal distribution of the risk factor changes10 ,
the latter four articles describe a copula-based approach to aggregate the
risk of financial portfolios. Ward and Lee [69] and Dimakos and Aas [20]
use a Gaussian copula to combine the marginal distributions. The latter
study only models pairwise dependence between credit and market risk and
credit and operational risk without specifically modelling the dependence between market and operational risk. Rosenberg and Schuermann [59] estimate
the marginal distributions’ parameters and their correlation measures using
market data and values that were reported in other studies and regulatory
reports.11 The marginal distributions are combined by a Gaussian and a Student t copula to point out the effects of positive tail dependence (see section
3). They report that the choice of the copula (Gaussian or Student t) has a
more modest effect on risk than has the business mix (the weights assigned
to a bank’s financial portfolios). Tang and Valdez [67] use semi-annual data
for loss ratios for the aggregate Australian insurance industry from 1992 to
2002 to calibrate a model that aggregates risks of different lines of insurance
business (motor, household, fire and industrial special risks, liability, and
compulsary third party insurance). The marginal distributions are modelled
10
Hall [35] points out that the economic capital may be severely underestimated if a joint
Gaussian distribution is assumed while indeed the marginal distributions are non-normal.
11
The calibration of the marginal market and credit portfolio return distributions is
done for data that was obtained by a simulation in a bottom-up manner. The aggregation
of these risks is done in a top-down manner, where the correlation matrix reported in
Kuritzkes et al. [47] is used.
10
as gamma, log-normal and pareto distributions and the consequences of assuming 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 Pézier [5] and Aas et al. [1].
Medova and Smith [49] use Monte Carlo simulations to allow for a varying
exposure of the credit portfolio (employing a structural credit risk model).
Alexander and Pézier [5] estimate multiple linear regression models, regressing the profits and losses of 8 business units 12 on 6 risk factors 13 . Pearson’s
correlation coefficient is used as dependence measure. To account for tail
dependence (a higher probability of joint extreme events as compared to the
Gaussian distribution/copula; see section 3) the authors suggest to use the
tail correlations rather than the usual overall correlations. Aas et al. [1] use
a bottom-up approach to aggregate market, credit and ownership risk. For
aggregating (additionally) operational and business risk, they use a top-down
approach (employing a Gaussian copula).
3
3.1
Copula-based approaches
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 concept
for modelling dependence structures of joint distributions was independently
proposed by Höffding [39] some twenty years earlier). Recent textbooks on
copulas are e.g. Joe [42] and Nelsen [52], [53].
Copulas are functions that combine or couple (univariate) marginal distributions to a multivariate joint distribution. Sklar’s theorem (using a slightly
different notation in the original article) states that a n-dimensional joint distribution function F (x) evaluated at x = (x1 , x2 , . . . , xn ) may be expressed
in terms of the joint distribution’s copula C and its marginal distributions
F1 , F2 , . . . , Fn as
12
The business units are: Corporate finance, Trading and sales, Retail banking, Commercial banking, Payment and settlement, Agency and custody, Asset management, and
Retail brokerage.
13
Risk factors: 1Y treasury rate, 10Y - 1Y treasury rate (slope), implied interest rate
volatility, S&P 500 index, S&P 500 implied volatility, 10Y credit spread.
11
x ∈ Rn .
F (x) = C (F1 (x1 ), F2 (x2 ), . . . , Fn (xn )) ,
(1)
The copula function C is itself a multivariate distribution with uniform
marginal distributions on the interval U1 = [0, 1], C : Un1 → U1 . Reformulating formula 1 yields
C(u) = F F1−1 (u1 ), F2 (u2 )−1 , . . . , Fn (un )−1 ,
u ∈ Un1 ,
(2)
where u = (u1 , u2 , . . . , un ) = (F1 (x1 ), F2 (x2 ), . . . , Fn (xn )) are the respective univariate marginal distributions.
Thus, a copula-based approach allows a decomposition of a joint distribution into its marginal distributions and its copula. On the other hand
marginal distributions may be combined to a joint distribution assuming a
specific copula. The crucial point in using a copula-based approach is that
it allows for a separate modelling of
• the marginal distributions (i.e. the univariate profit and loss or return
distributions) and
• the dependence structure (the copula).
Figure 2 displays an example for the combination of two marginal distributions to a joint bivariate distribution. Assume that a financial institution holds two portfolios, a market portfolio and a credit portfolio. The
market portfolio’s annual return distribution is modelled as random variable
rM ∼ 0.1+0.15·t5 , where t5 is a Student’s t distributed random variable with
ν = 5 degrees of freedom. The credit portfolio’s annual return distribution is
modelled as rC ∼ ln(1.05 · B(30, 1.1)), where B(30, 1.1) is a beta distributed
random variable. Both portfolios exhibit ‘fat tails’, assigning a higher probability to extreme events than a normal distribution. The returns of the credit
portfolio are heavily left-skewed, assigning a higher probability to extreme
losses than to extreme gains. Table 1 displays mean and median values, the
standard deviation, skewness, and excess-kurtosis of the two return distributions.14 The data shows that both return distributions are non-normal. In a
14
The moments of the two distributions displayed in table 1 are the sample estimates of
1,000,000 simulated returns, using Monte Carlo simulation. Using this simulated values
for the depiction of the credit portfolio returns’ density in figure 2 as kernel smoothed
densities with Gaussian kernels, this explains the small right tail above the 0.05-threshold
displayed in figure 2. The simulated credit portfolio returns will not take on a value of
greater than 0.5, as the beta distribution is defined on the [0,1]-interval. (For a primer
on kernel smoothed densities see e.g. Scott and Sain [64]; on beta distributions, see e.g.
Johnson et al. [44], Chapter 25.)
12
density
2
1
-0.2
0
0.2
rM = F-1
(u )
M M
0.4
1
-0.1
-0.05
0
0.05
0.5
uM = FM(rM)
1
0.5
0
1
0
0.5
uC = FC(rC)
1
0.75
0.5
0.25
0.25
0.5
uM
0.75
joint distribution
marginal distributions
and copula
0
10
rC = F-1
(u )
C C
0.5
0
20
0
0.6
density
density
0
-0.4
uC
density
3
Figure 2: Example of the combination of a market and a credit portfolio
marginal return distributions to a joint returns distribution using a copulabased approach.
13
mean
median
standard deviation
skewness
excess kurtosis
rM
0.1000
0.1000
0.1930
0
4.8599
rC
0.0122
0.0225
0.0349
-1.9172
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 combined
to a joint distribution, as shown in figure 2.
Apart from the ability to combine arbitrary marginal distributions to a
joint distribution, copula-based approaches allow for a specific modelling of
the dependence structure, i.e. the copula.
One frequently observed empirical evidence is that extreme joint market movements are more frequently observed than implied by a multivariate
Gaussian distribution that is often used in market risk models.15 This empirical evidence is sometimes referred to as ‘correlation-breakdown’.
Copula-based approaches allow for a flexible modelling of the probability
of joint extreme observations (unconditional on the marginal distributions).
For example, a Student t copula assigns a higher probability to joint extreme
observations than does a Gaussian copula. This higher probability of joint
extreme observations as compared to the Gaussian copula is referred to as
positive tail dependence.
As an example, figure 3 shows scatter plots of two jointly distributed
standard normal random variables. These standard normal marginal distributions are combined by a Gaussian copula, a Student t copula, a Clayton
copula, and a Gumbel copula, respectively. The resulting joint distributions
(for arbitrary marginal distributions) are referred to as meta-Gaussian, metaStudent t, meta-Clayton and meta-Gumbel distributions. The top row shows
scatter plots of simulated joint distributions that have a correlation (in terms
of Spearman’s rho 16 ) of 0.4. The bottom row shows corresponding scatter
15
In passing, note that the multivariate Gaussian distribution in ‘copula-based approaches terms’ is a set of univariate Gaussian marginal distributions that are combined
by a Gaussian copula.
16
In copula-based approaches, rank-based correlation measures such as Spearman’s rho
and Kendall’s tau are preferable to the widely known Pearson correlation measure that is
14
ρS = 0.4:
(meta−)Gaussian
meta−Student t
meta−Clayton
meta−Gumbel
3
3
3
3
0
0
0
0
−3
−3
−3
−3
−3
ρS = 0.8:
0
3
−3
(meta−)Gaussian
0
3
−3
meta−Student t
0
3
−3
meta−Clayton
3
3
3
3
0
0
0
0
−3
−3
−3
−3
−3
0
3
−3
0
3
−3
0
3
0
3
meta−Gumbel
−3
0
3
Figure 3: Simulation scatter plots of bivariate meta-Gaussian, meta-Student
t, meta-Clayton and meta-Gumbel distributions. The top row shows scatter
plots of joint distributions with a Spearman’s rho correlation measure of
approximately 0.4, the bottom row shows scatter plots of joint distributions
with a Spearman’s rho of approximately 0.8. Both marginal are standard
normally distributed.
plots for joint distributions with a correlation of 0.8.
It can be seen that for identical marginal distributions and Spearman’s
rho the Student t copula assigns a higher probability to joint extreme events
than does the Gaussian copula. Assigning an equal probability to joint extreme positive deviations and to joint extreme negative deviations from the
median value, the Student t copula displays symmetric tail dependence.
Asymmetric tail dependence is prevalent if the probability of joint extreme negative realisations differs from that of joint extreme positive realisations. In figure 3 it can be seen that the Clayton copula assigns a higher
probability to joint extreme negative events than to joint extreme positive
events. The Clayton copula is said to display lower tail dependence, while
it displays zero upper tail dependence. The converse can be said about the
Gumbel copula (displaying upper but zero lower tail dependence). Table 2
used in the context of multivariate normal distributions. A short note on Spearman’s rho
and Kendall’s tau is given in Appendix A.
15
tail dep. Gaussian
lower
no
upper
no
symmetric
yes
Student t
yes
yes
yes
copula:
BB1 Clayton
yes
yes
yes
no
no
no
Gumbel
no
yes
no
Frank
no
no
yes
Table 2: Summary of which bivariate copulas display lower and upper tail
dependence and whether the positive tail dependence is symmetric.
gives an overview of which of the copulas presented in this article display
upper or lower tail dependence.17 In section 3.3 we will give a formal definition of upper and lower tail dependence and provide explicit formulas for
the magnitude of the tail dependence.
Some copulas allow to model both positive and negative dependence in
their ‘standard’ versions by assigning appropriate copula-parameters. Amongst
these copulas are e.g. the Gaussian, Student t and Frank copula. Figure 4
displays 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 special cases, the Clayton and Gumbel copula in their ‘standard’ version allow to model positive dependence only.18 Copula rotation allows to transform copulas such that they may be used to model negative dependence
also. Further, copula rotation allows to transform (bivariate) copulas depending on whether and/or where the empirical data at hand requires the
copula to display lower, upper or zero tail dependence. Denoting a bivariate copula density as c(u1 , u2 ), the so-called survival copula’s density
is 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 upper
tail dependence. In order to model discordance with e.g. a BB1 copula,
17
The flexible BB1 copula may also display either zero upper or zero lower tail dependence or symmetric tail dependence, depending on the parameterisation. In specific cases
the Gaussian and Student t copula may display also positive and no tail dependence,
respectively. See e.g. table 8 in section 3.3.
18
In fact, the Clayton copula may also be used in its standard version to model negative
dependence if the copula parameter θ ∈ [−1, 0). Such a parameterisation is not further
considered in the present article.
19
While the bivariate copula C(u1 , u2 ) returns the probability that both uniformly distributed marginal distributions take on values less than or equal to u1 and u2 , the survival
copula C −− (u1 , u2 ) returns the probability that both marginal distributions take on values
greater than u1 and u2 , respectively.
16
S
S
Frank copula, ρ = 0.4
15
3
4
10
2
2
density
6
0
1
5
0
1
0.5
u1
0.5
u1
u2
1
0
1
0.5
1
0 0
Gaussian copula, ρS = −0.4
0.5
1
0 0
0.5
u1
u2
Student t copula, ν = 3, ρS = −0.4
3
4
10
2
0
1
density
15
2
5
0
1
0.5
u1
1
0 0
0.5
u2
1
0 0
0.5
u2
Frank copula, ρS = −0.4
6
density
density
S
Student t copula, ν = 3, ρ = 0.4
density
density
Gaussian copula, ρ = 0.4
1
0
1
0.5
u1
1
0 0
0.5
u2
0.5
u1
1
0 0
0.5
u2
Figure 4: Densities of bivariate Gaussian, Student t and Frank copulas.
These copulas allow to model both concordance and discordance. The copulas in the top row display a Spearman’s rho of approximately 0.4 (copula
parameters: 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 of
approximately -0.4 (copula parameters: Gaussian: ρ = −0.42, Student t:
ν = 3 and ρ = −0.43, Frank: θ = −2.61). The densities are computed on
the interval [0.01, 0.99]2 .
17
+−
1.2
0.2 0.4 0.6 0.8
u1
u2
0.5
0.2 0.4 0.6 0.8
u1
0.6
0.4
1.2
1
1
0.8
2
0.2
32
1.2
u2
2
1.2
0.4
2
0.2 0.4 0.6 0.8
u1
0.6
0.2
2
2
2
0.8
u
u2
0.5
1
0.2
Gumbel C
1.2
0.4
0.5
1.2
0.2
0.6
1
0.4
−−
Gumbel C
1
0.6
0.8
0.2 0.4 0.6 0.8
u1
+−
Gumbel C
0.5
1
23
2
2
1.
1.2
1.2
1
0.2
0.5
1
0.4
0.2 0.4 0.6 0.8
u1
−+
3
0.5
1
0.5
++
0.6
3
0.2 0.4 0.6 0.8
u1
Gumbel C
0.8
0.2
u2
3
0.2 0.4 0.6 0.8
u1
0.4
1.2
2
3
2
0.5
0.8
0.6
1.2
1.2
0.5
u2
2
0.4
1
0.2
0.5
2
1.2
u
1.2
0.2
0.6
1
1
1 0.5
1
0.5
0.4
0.8
0.5 1
u2
0.6
2
0.8
Clayton C
0.5
1.
2
1.2
0.8
−−
Clayton C
3
0.5
−+
Clayton C
3
++
Clayton C
1
0.5
0.2 0.4 0.6 0.8
u1
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 approximately
0.4 (copula parameters: Clayton: θ = 0.76, Gumbel: θ = 1.38). The densities
are 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 of
Clayton and Gumbel copulas’ densities and of the rotated versions’ densities.
To present the consequences of assumptions on the copula in the context
of economic capital estimation, let us return to our simplified example, where
we assumed that a bank holds only two portfolios, a market portfolio with
annual returns rM ∼ 0.1 + 0.15 · t5 and a credit portfolio with annual returns
rC ∼ ln(1.05 · B(30, 1.1)). The correlation in terms of Spearman’s rho is
ρS = 0.4. Let us further assume that equal weights are assigned to these
portfolios, 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 credit
portfolio in year t, respectively. Table 3 shows several quantiles of total return distributions.20 The quantiles correspond to average one-year default
probabilities of Moody’s ratings from 1920 to 2004 reported in Hamilton et
al. [36], p.35. A bank that aspires a rating of e.g. ‘Ba’ has to hold enough
economic capital such that the total losses exceed the economic capital with
a probability of no more than 1.31%. The quantiles were computed under
20
The quantiles are obtained by a Monte Carlo simulation with 1,000,000 simulations
using antithetic sampling.
18
quantile
meta-Gaussian
meta-Student t
meta-Clayton
correlation=1
0.0432 (B)
-0.1202
-0.1214
-0.1265
-0.1406
0.0131 (Ba)
-0.2016
-0.2083
-0.2183
-0.2351
0.0030 (Baa)
-0.3153
-0.3363
-0.3482
-0.3695
0.0006 (Aa)
-0.4661
-0.5092
-0.5169
-0.5427
Table 3: Quantiles of the total return distribution, corresponding to average
Moody’s rating 1-year default probabilities, if meta-Gaussian, meta-Student
t 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 degrees
of freedom and a Clayton copula. Additionally, in order to demonstrate the
diversification effect, the quantiles were computed under the assumption of
perfect positive correlation. As can be seen in table 3, the effect of positive tail dependence (Student t copula and Clayton copula) increases, as the
quantile decreases. In our simplistic example the economic capital to be held
under the assumption of a Student t (Clayton) copula exceeds the economic
capital 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, we
shall shortly address how the marginal distributions may be modelled in the
following subsection.
3.2
Modelling the marginal distributions
In the context of top-down risk aggregation models, the following parametric
distribution functions are widely used to model the marginal distributions:
• Market portfolio returns
– Generalized hyperbolic (GH) distribution21 , or one if its special
cases such as the
– Normal inverse Gaussian (NIG) distribution or
– Student t and Gaussian distributions.
• Credit portfolio returns
– Beta distribution
21
See 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-smoothed
empirical distribution functions are widely employed.22
3.3
Presentation of selected copulas
This section presents some selected copulas from the family of elliptical and
Archimedean 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 uniformly distributed marginal distributions jointly take on value less than or
equal to u1 and u2 , respectively, are presented in table 4. Bivariate copula
densities that are needed in the context of parameter estimation and for the
depiction of data are presented in table 5.
The Gaussian and Student t copulas in their ‘standard versions’ allow for
a higher flexibility than the Archimedean copulas by enabling a modelling of
pairwise correlations that form the elements of the copula parameter matrix
P (‘capital Greek letter rho’).
22
For a primer on kernel smoothing, see e.g. Scott and Sain [64]. More detailed information can be found in Wand and Jones [68] and Silverman [65]
20
copula
parameters θ
Gaussian
ρ ∈ [−1, 1]
copula function C(u1 , u2 ; θ)
Φρ (Φ−1 (u1 ), Φ−1 (u2 )) =
R Φ−1 (u1 ) R Φ−1 (u2 )
−∞
−∞
√1
1−ρ2
2π
exp
2ρst−s2 −t2
2(1−ρ2 )
dsdt
or equivalently (see Roncalli [58])
R u1
0
Φ
Φ−1 (u2 )−ρΦ−1 (s)
√
1−ρ2
ds
where Φρ is the bivariate standard normal distribution
function with parameter ρ, and Φ−1 is the functional
inverse of the univariate standard normal c.d.f. Φ.
Student t
ν ∈ (0, ∞)
ρ ∈ [−1, 1]
−1
tν,ρ (t−1
ν (u1 ), tν (u2 )) =
R t−1
R t−1
ν (u1 )
ν (u2 )
−∞
−∞
√1
1−ρ2
2π
ν+2
s2 +t2 −2ρst − 2
ν(1−ρ2 )
1+
dsdt
or equivalently (see Roncalli [58])
R u1
0
q
tν+1
−1
t−1
ν+1
ν (u
√2 )−ρtν (s)
−1
2
ν+tν (s)
1−ρ2
ds
where tν,ρ is the bivariate Student t distribution and
t−1
ν is the functional inverse of the univariate
Student t c.d.f with ν degrees of freedom tν (.).
BB1
δ ∈ [1, ∞), θ ∈ (0, ∞)
Clayton
θ ∈ (0, ∞)
1+
Frank
θ ∈ [1, ∞)
θ ∈ (−∞, ∞)\0
u−θ
1 −1
δ 1δ
+ u−θ
2 −1
h
θ
θ
θ
exp − (− ln u1 ) + (− ln u2 )
− 1θ
!− θ1
− 1
−θ
u−θ
1 + u2 − 1
Gumbel
δ
ln 1 +
(e−θu1 −1)(e−θu2 −1)
e−θ −1
Table 4: Selected bivariate copula functions.
21
i1 θ
copula
Gaussian
probability density function c(u1 , u2 ; θ) =
√1
1−ρ2
exp
2ρy1 y2 −y12 −y22
2(1−ρ2 )
+
y12 +y22
2
∂ 2 C(u1 ,u2 ;θ)
∂u1 ∂u2
,
where y1 = Φ−1 (u1 ), y2 = Φ−1 (u2 ) and Φ−1 (.) is the
functional inverse of the standard normal c.d.f. Φ(.).
−1
Student t fν,ρ t−1
ν (u1 ), tν (u2 )
1
1
,
−1
fν (t−1
ν (u1 )) fν (tν (u2 ))
where fν,ρ is the p.d.f of the standard Student t distribution
function with ν degrees of freedom and correlation matrix ρ,
fν is the p.d.f of the univariate standard Student t distribution
and t−1
ν is the functional inverse of the univariate Student t
c.d.f with ν degrees of freedom t−1
ν (.).
BB1
δ−1 −θ
δ−1 −θ−1 −θ−1
u−θ
u2 − 1
u1
u2
·
1 −
h 1
i
1
2
1
1
· (1 + θ)a− θ −2 b δ −2 + (δθ − θ)a− θ −1 b δ −2 ,
h
δ
δ i
1
where a = 1 + b δ and b = u−θ
+ u−θ
.
1 −1
2 −1
Clayton
−θ
(1 + θ)u−θ−1
u2−θ−1 u−θ
1
1 + u2 − 1
Gumbel
exp(a) (− ln u1 )
Frank
− θ1 −2
h
i
2
1
b θ −2 + (θ − 1)b θ −2 ,
1
where a = −b θ , and b = (− ln u1 )θ + (− ln u2 )θ .
θ−1
(− ln u2 )θ−1
u1 u 2
θηe−θ(u1 +u2 )
2
[η−(1−e−θu1 )(1−e−θu2 )]
where η = 1 − e−θ .
,
Table 5: Probability density functions of selected bivariate copulas.
22
copula
Gaussian
parameters θ:
P
Student t
parameters θ:
ν, P
copula function C(u; θ)
R Φ−1 (u1 )
−∞
...
R Φ−1 (un )
−∞
√
1
(2π)n |P|
exp − 21 x0 P−1 x dx
−1
where Φ (.) is the functional inverse of the univariate standard
normal c.d.f. Φ(.).
R t−1
ν (u1 )
−∞
...
R t−1
ν (un )
−∞
Γ( ν+n
2 )
√
1+
Γ( ν2 ) (πν)n |P|
x0 P−1 x
ν
− ν+n
2
dx,
where t−1
ν is the functional inverse of the univariate Student t c.d.f.
with ν degrees of freedom tν (.) and Γ(.) is the Gamma function.
Table 6: n-dimensional Gaussian and Student t copula functions.
1 ρ1,2 ρ1,3 · · · ρ1,d
 ρ
1 ρ2,3 · · · ρ2,d
 1,2

..

.
P =  ρ1,3 ρ2,3 1

..
.
 ..
.
. . ..
 .
.
ρ1,d ρ2,d · · · · · · 1










(3)
Besides the copula parameter P, the Student t copula has an additional
scalar parameter ν, the degrees of freedom. These can, however, not be used
to explicitly model pairwise dependencies. Rather, the copula parameter ν,
being a scalar, affects all pairwise dependencies in the same manner. Table
6 and 7 provide the n-dimensional copula functions of Gaussian and Student
t copulas and their densities, respectively.
If the dependence of more than two dependent variables is to be modelled, the Archimedean copulas’ flexibility seems very restricted as either
only one (Clayton, Gumbel, Frank copulas) or only two (BB1 copula) scalar
parameters are used to parameterise the joint multidimensional dependence
structure.23 . This lack of flexibility can however be overcome by using hierarchical Archimedean copulas that are e.g. presented in Savu and Trede
[62]. A hierarchical copula joins two (or more) bivariate (or higher dimensional) Archimedean copulas by another Archimedean copula. The structure
of this approach is depicted in figure 6. If in the context of risk aggregation we want to combine the returns of, say, four financial portfolios we first
23
Formulas for n-dimensional Archimedean copulas can be found e.g. in Cherubini et
al. [13], pp.147ff.
23
copula
probability density function c(u; θ) =
Gaussian
φP (Φ−1 (u1 ), . . . , Φ−1 (un ))
∂ n C(u;θ)
∂u1 ...∂un
Qn
1
i=1 φ(Φ−1 (ui )) ,
where φP (.) is the p.d.f. of the multivariate standard normal distribution
with correlation matrix P, φ(.) is the p.d.f. of the univariate standard
normal distribution, and Φ−1 (.) is the functional inverse of the univariate
standard normal c.d.f. Φ(.).
Qn
−1
Student t fν,P t−1
ν (u1 ), . . . , tν (un )
1
,
i=1 fν (t−1
ν (ui ))
where fν,P is the p.d.f of the standard Student t distribution
function with ν degrees of freedom and correlation matrix P,
fν is the p.d.f of the univariate standard Student t distribution
and t−1
ν is the functional inverse of the univariate Student t
c.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 portfolio 3 and 4, respectively. These two copulas are then combined by a third
copula. Parameter estimation is done in the same manner as for the other
copulas (see subsection 3.4). For the simulation of hierarchical copulas, the
conditional inversion method has to be used (see Savu and Trede [62], p.10f).
The concept of positive upper and lower tail dependence of bivariate
copulas has already been introduced in section 3.1. Loosely speaking, lower
tail dependence λL describes the conditional probability that one of the two
random variables takes values below a very small value, given that also the
other random variable takes very small values. Upper tail dependence λU
can be described analogously. Formally,
C(α, α)
and
α→0
α→0
α
1 − 2α + C(α, α)
= lim− P (u1 > α|u2 > α) = lim−
,
α→1
α→1
1−α
λL =
λU
lim+ P (u1 ≤ α|u2 ≤ α) = lim+
(4)
(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 the
selected copulas are presented in table 8.
The concept of copula rotation has also been introduced already (see
figure 5 on p.18). Copulas may be rotated, depending on whether and/or
where the empirical data at hand requires the copula to display positive,
24
copula 3:
C3(C1(U1, U2), C2(U3, U4) ) ɽ [0,1]
copula 1:
C1(U1, U2) ɽ [0,1]
marginal distribution 1:
U1 = F1(X1) ɽ [0,1]
copula 2:
C2(U3, U4) ɽ [0,1]
marginal distribution 2:
U2 = F2(X2) ɽ [0,1]
marginal distribution 3:
U3 = F3(X3) ɽ [0,1]
marginal distribution 4:
U4 = F4(X4) ɽ [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
√
q
λL = λU = 2tν+1 − ν + 1 1−ρ
1+ρ
where tν+1 is the univariate Student t c.d.f with ν + 1 degrees of freedom
1
1
BB1
λL = 2− δθ
Clayton
λL = 2− θ
λU = 0
Gumbel
λL = 0
λU = 2 − 2 θ
Frank
λU = 2 − 2 δ
1
1
λL = λU = 0
Table 8: Lower and upper tail dependence, λL and λU , of selected bivariate
copulas.
25
negative or zero tail dependence. Let us define the vector ū = (ū1 , ū2 ),
where ūi = 1 − ui .24 Then the following observations are true
• ū1 and ū2 have copula C −− (u1 , u2 ) = u1 +u2 −1+C(1−u1 , 1−u2 ) with
density c−− (u1 , u2 ) = c(1 − u1 , 1 − u2 ). C −− is referred to as survival
copula.
• ū1 and u2 have copula C −+ (u1 , u2 ) = u2 − C(1 − u1 , u2 ) with density
c−+ (u1 , u2 ) = c(1 − u1 , u2 ).
• u1 and ū2 have copula C +− (u1 , u2 ) = u1 − C(u1 , 1 − u2 ) with density
c+− (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 dependence to model lower tail dependence, the corresponding C −− copula has to
be employed. If we want to use a copula C which is only suited to describe
positive dependence to model negative dependence, C −+ or C +− have to be
employed.
In the sub-sections below, the densities of selected bivariate copulas are
more closely regarded.
3.3.1
Gaussian copula
The Gaussian copula is the most widely used copula. It is the copula that
is implied by a multivariate Gaussian distribution (normal distribution). A
multivariate Gaussian distribution is a set of normally distributed marginal
distributions that are combined by a Gaussian copula. If other than normal marginal distributions are combined by a Gaussian copula, the resulting
joint distribution is referred to as meta-Gaussian distribution. Figure 2 on
p.13 contains an example of a meta-Gaussian distribution. Figure 7 displays
surface plots of Gaussian copula densities with a Spearman’s rho of 0.4 (top
left) and 0.8 (bottom left). The bivariate copula density goes to infinity at
u = (0, 0), and u = (1, 1) for ρ > 0 and at u = (0, 1) and u = (1, 0) for
ρ < 0. On the right hand side, corresponding (meta-) Gaussian distribution
densities with standard normal marginal distributions are displayed.
24
Note that Ū = 1 − U is uniformly distributed on the unit interval if U is uniformly
distributed on the unit interval.
26
We shall use the Gaussian copula as benchmark to which we compare the
other copulas.
3.3.2
Student t copula
The Student t copula is the copula that is implied by a multivariate Student t distribution (Student t marginal distributions combined by a Student
t copula). Like the Gaussian copula, the Student t copula has the parameter 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. The
higher ν, the higher the positive tail dependence (see table 8). Figure 8
displays surface plots of Student t copula densities with a Spearman’s rho
of 0.4 (top left) and 0.8 (bottom left). The bivariate copula density goes to
infinity at u = (0, 0), u = (0, 1), u = (1, 0), and u = (1, 1). On the right
hand side, corresponding contour plots of meta-Student t distribution densities with standard normal marginal distributions are displayed. Additionally,
contours of a Gaussian distribution with identical marginal distributions and
Spearman’s rho are plotted in light grey for comparison. It can be seen that
the Student t copula assigns a higher density to events near all four corners
than the Gaussian copula does. Differences between the Student t copula
and meta-distribution’s densities to those of the Gaussian copula with identical Spearman’s rho are summarised in the contour plots at the bottom of
figure 8, where grey shaded areas indicate that the densities of the Student
t copula or meta-distribution exceed that of the Gaussian copula.
As the degrees of freedom of a Student t copula increase, the copula
approaches a Gaussian copula. The Gaussian copula can be regarded as
a limiting case of the Student t copula, where ν → ∞. More in-depth
information 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 positively correlated bivariate dependence structures (copula parameters δ and
θ). Figure 9 displays contour plots of BB1 copula densities with an identical
Spearman’s rho of 0.4. The plot on the very left and on the very right hand
side are limiting cases of the BB1 copula. The very left BB1 copula has the
copula parameter δ = 1. This special case of the BB1 copula is called a
Clayton copula, and the BB1 copula parameter θ corresponds to the Clayton
copula parameter θ. The very right BB1 copula has the parameter θ tends
27
Gaussian copula, ρS = 0.4
(meta−) Gaussian distribution, ρS = 0.4
0.2
0.15
4
density
density
6
2
0.1
0.05
0
1
0.8
0.6
0.4
0.2
u2
0
0
3
0
0.2
0.4
u1
0.6
0.8
2
S
−2
3
1
(meta−) Gaussian distribution, ρ = 0.8
6
0.2
0.15
4
density
density
−3
2
1
−1 0
m ∼ N(0,1)
S
Gaussian copula, ρ = 0.8
2
0
1
0.8
0.6
0.4
0.2
u2
0
1
0
−1
−2
−3
m2 ∼ N(0,1)
1
0.1
0.05
0
3
0
0.2
0.4
u1
0.6
0.8
1
2
1
0
−1
−2
−3
m2 ∼ N(0,1)
−3
−2
2
1
−1 0
m ∼ N(0,1)
3
1
Figure 7: Densities of bivariate Gaussian copulas (left hand side) with a
Spearman’s rho of 0.4 (copula-parameter ρGaussian = 0.42) and 0.8 (copulaparameter ρGaussian = 0.81) evaluated on the interval [0.0001, 0.999]2 and
corresponding meta-distributions with standard normal marginal distributions (right hand side).
28
S
Student t copula, ν=3, ρ = 0.4
meta−Student t dist., ν=3, ρS = 0.4
3
2
2
0
1
m ∼ N(0,1)
density
5
0.8
0.6
0.2
0
0.8
0.6
0.4
0.2
0
1
−3
−3 −2 −1 0 1 2
m1 ∼ N(0,1)
u
1
S
meta−Student t dist., ρ = 0.8
2
0
2
m ∼ N(0,1)
5
0.6
−2
0.4
0.2
0
0
u
S
−2
S
ρ = 0.4
S
ρ = 0.8
0.5
u1
1
0.5
0
ρ = 0.8
3
m2 ∼ N(0,1)
0.5
0
0.5
u1
1
0
2
m1 ∼ N(0,1)
S
ρ = 0.4
1
u2
1
0
1
u1
2
0
0.8
0.6
0.4
0.2
3
0
0
2
0.8
m ∼ N(0,1)
density
3
S
Student t copula, ν=3, ρ = 0.8
u2
0
−1
−2
0.4
u2
0
1
1
−3
−3
0
m1 ∼ N(0,1)
3
−3
−3
0
m1 ∼ N(0,1)
3
Figure 8: Densities of bivariate Student t copulas with ν = 3 degrees of
freedom (left hand side) with a Spearman’s rho of 0.4 (copula-parameter
ρStudent t = 0.43) and 0.8 (copula-parameter ρStudent t = 0.83), evaluated on
the interval [0.0001, 0.9999]2 . Corresponding contour plots (contours at the
0.02, 0.05, 0.1, 0.2 and 0.3 level) of meta-Student t distributions with standard normal marginal distributions are plotted on the right hand side. Additionally, contours of a Gaussian meta-distribution with identical Spearman’s
rho and marginal distributions are plotted in light grey. The graphs in the
bottom row indicate in which areas the densities of the Student t copula or
meta distribution exceed that of a Gaussian copula or meta distribution with
identical Spearman’s rho (grey-shaded areas).
29
1
2
0.5
1
1.2
5
0.
1.2
1
1
0.5
u
2
1
1.21
1.2
1
1
u2
0.5
1 0.5
1
1.2
1 0.5
u2
1
1 1.2
1
0.5
10.5
u
3
0.2 0.4 0.6 0.8
u1
0.8
0.6
0.4
0.2
2
0.2 0.4 0.6 0.8
u1
2
3
23
23
5
0.
0.2 0.4 0.6 0.8
u1
0.8
0.6
0.4
0.2
3
2
1.2
0.8
0.6
0.4
0.2
BB1 (δ = 1.38, θ = 0+)
BB1 (δ = 1.2, θ = 0.2)
2
0.8
0.6
0.4
0.2
BB1 (δ = 1.1, θ = 0.48)
1.2
2
BB1 (δ = 1, θ = 0.76)
0.2 0.4 0.6 0.8
u1
Figure 9: Densities of bivariate BB1 copulas with different parameterisation.
All copulas have a a Spearman’s rho of approximately 0.4. The densities are
evaluated 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 parameter θ.
The next two sub-sections take a closer look on these two special cases of
the 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 dependence. These properties can be verified regarding the Clayton copula
density plots displayed in figure 10 on the left hand side. The top copula has
a 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 distributions
with standard normal marginal distributions are displayed on the right hand
side. The contour plots on the bottom of figure 10 show that the Clayton
copula assigns a higher probability to joint extremely negative realisations
as compared to the Gaussian copula, while it assigns a lower probability to
joint extremely positive realisations.
3.3.5
Gumbel copula
Figure 11 displays the densities of a survival Gumbel copula with a Spearman’s rho of 0.4 (top) and 0.8 (bottom). Like the Clayton copula, the survival
Gumbel copula displays lower tails dependence and no upper tail dependence.
The ‘tear shaped’ corresponding contour plots of meta-survival Gumbel distributions with standard normal marginal distributions are displayed on the
right hand side. The contour plots on the bottom of figure 11 show that
the survival Gumbel copula assigns a higher probability to joint extremely
negative realisations as compared to the Gaussian copula, while it assigns a
30
S
Clayton copula, ρ = 0.4
meta−Clayton dist., ρS = 0.4
3
2
2
0
1
m ∼ N(0,1)
density
5
0.8
0.6
0.2
0
0.8
0.6
0.4
0.2
0
1
−3
−3 −2 −1 0 1 2
m1 ∼ N(0,1)
u
1
S
meta−Clayton dist., ρ = 0.8
3
2
2
m ∼ N(0,1)
5
0.6
0.2
0
0
1
−3
−3 −2 −1 0 1 2
m1 ∼ N(0,1)
u1
S
S
ρ = 0.4
S
ρ = 0.8
1
0.5
0
ρ = 0.8
3
m2 ∼ N(0,1)
0.5
0
0.5
u1
1
3
S
ρ = 0.4
1
u2
1
0.5
u1
0.8
0.6
0.4
0.2
2
0
0
−1
−2
0.4
u
0
1
3
0
0
2
0.8
m ∼ N(0,1)
density
3
S
Clayton copula, ρ = 0.8
u2
0
−1
−2
0.4
u2
0
1
1
−3
−3
0
m1 ∼ N(0,1)
3
−3
−3
0
m1 ∼ N(0,1)
3
Figure 10: Densities of bivariate Clayton copulas (left hand side) with a
Spearman’s rho of 0.4 (copula-parameter θClayton = 0.76) and 0.8 (copulaparameter θClayton = 3.2), evaluated on the interval [0.0001, 0.9999]2 . Corresponding contour plots (contours at the 0.02, 0.05, 0.1, 0.2 and 0.3 level)
of meta-Clayton distributions with standard normal marginal distributions
are plotted on the right hand side. Additionally, contours of a Gaussian
meta-distribution with identical Spearman’s rho and marginal distributions
are plotted in light grey. The graphs in the bottom row indicate in which
areas the densities of the Clayton copula or meta distribution exceed that
of a Gaussian copula or meta distribution with identical Spearman’s rho
(grey-shaded areas).
31
S
surv. Gumbel copula, ρ = 0.4
meta−surv. Gumbel dist., ρS = 0.4
3
2
2
0
1
m ∼ N(0,1)
density
5
0.8
0.6
0.2
0
0.8
0.6
0.4
0.2
0
1
1
S
3
S
meta−surv. Gumbel dist., ρ = 0.8
3
2
2
m ∼ N(0,1)
5
0.6
0.2
0
0
1
−3
−3 −2 −1 0 1 2
m1 ∼ N(0,1)
u1
S
S
ρ = 0.4
S
ρ = 0.8
1
0.5
0
ρ = 0.8
3
m2 ∼ N(0,1)
0.5
0
0.5
u1
1
3
S
ρ = 0.4
1
u2
1
0.5
u1
0.8
0.6
0.4
0.2
2
0
0
−1
−2
0.4
u
0
1
3
0
0
2
0.8
m ∼ N(0,1)
density
−3
−3 −2 −1 0 1 2
m1 ∼ N(0,1)
u
surv. Gumbel copula, ρ = 0.8
u2
0
−1
−2
0.4
u2
0
1
1
−3
−3
0
m1 ∼ N(0,1)
3
−3
−3
0
m1 ∼ N(0,1)
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 0.3
level) of meta-survival Gumbel distributions with standard normal marginal
distributions are plotted on the right hand side. Additionally, contours of a
Gaussian meta-distribution with identical Spearman’s rho and marginal distributions are plotted in light grey. The graphs in the bottom row indicate
in which areas the densities of the survival Gumbel copula or meta distribution exceed that of a Gaussian copula or meta distribution with identical
Spearman’s rho (grey-shaded areas).
32
ρS = 0.8
0.8
u2
0.6
u2
0.6
m2 ∼ N(0,1)
0.8
ρS = 0.4
0.4
0.4
0.2
0.2
0.2 0.4 0.6 0.8
u1
ρS = 0.8
3
3
2
2
m2 ∼ N(0,1)
ρS = 0.4
1
0
−1
−2
0.2 0.4 0.6 0.8
u1
−3
−3 −2 −1 0 1 2
m1 ∼ N(0,1)
1
0
−1
−2
3
−3
−3 −2 −1 0 1 2
m1 ∼ N(0,1)
3
Figure 12: Contour plots of the log differences of a Clayton and a survival
Gumbel copula density (left two plots) and of corresponding log differences
of 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 metadistribution 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 true
for the ‘standard’ Gumbel copula C ++ ).
More closely examining the differences between a Clayton and a survival
Gumbel copula, figure 12 displays plots of the log-differences of a Clayton
and a Gumbel copula’s densities, ln cClayton (u1 , u2 ) − ln csurv. Gumbel (u1 , u2 )
and of meta distribution densities with standard normal marginal distributions, with a Spearman’s rho of 0.4 and 0.8, respectively. Grey-shaded
areas indicate that the Clayton copula’s or meta-distribution’s density exceeds that of a survival Gumbel copula. It can be seen that the Clayton
copula assigns a higher probability to joint extremely negative events, while
the survival Gumbel copula assigns a higher probability to joint extremely
positive 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 dependence 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 and
λL = 0.69.
33
S
Frank copula, ρ = 0.4
meta−Frank dist., ρS = 0.4
3
2
2
0
1
m ∼ N(0,1)
density
5
0.8
0.6
0.2
0
0.8
0.6
0.4
0.2
0
1
−3
−3 −2 −1 0 1 2
m1 ∼ N(0,1)
u
1
S
meta−Frank dist., ρ = 0.8
3
2
2
m ∼ N(0,1)
5
0.6
0.2
0
0
1
−3
−3 −2 −1 0 1 2
m1 ∼ N(0,1)
u1
S
S
ρ = 0.4
S
ρ = 0.8
1
0.5
0
ρ = 0.8
3
m2 ∼ N(0,1)
0.5
0
0.5
u1
1
3
S
ρ = 0.4
1
u2
1
0.5
u1
0.8
0.6
0.4
0.2
2
0
0
−1
−2
0.4
u
0
1
3
0
0
2
0.8
m ∼ N(0,1)
density
3
S
Frank copula, ρ = 0.8
u2
0
−1
−2
0.4
u2
0
1
1
−3
−3
0
m1 ∼ N(0,1)
3
−3
−3
0
m1 ∼ N(0,1)
3
Figure 13: Densities of bivariate Frank copulas (left hand side) with a Spearman’s rho of 0.4 (copula-parameter θF rank = 2.61) and 0.8 (copula-parameter
θF rank = 7.9), evaluated on the interval [0.0001, 0.9999]2 . Corresponding contour plots (contours at the 0.02, 0.05, 0.1, 0.2 and 0.3 level) of meta-Frank
distributions with standard normal marginal distributions are plotted on the
right hand side. Additionally, contours of a Gaussian meta-distribution with
identical Spearman’s rho and marginal distributions are plotted in light grey.
The graphs in the bottom row indicate in which areas the densities of the
Frank copula or meta distribution exceed that of a Gaussian copula or meta
distribution 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 tail
dependence. 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 provided
in figure 13. It can be seen that the Frank copula assigns a lower probability
to joint extremely negative or extremely positive realisations as compared to
the Gaussian copula.
3.4
Copula-parameter estimation
This section presents some widely used methods to estimate copula parameters from empirical data. The most commonly used method is to estimate
the parameters with maximum likelihood (MLE). This method is presented
in subsection 3.4.1 below. An alternative method is to estimate the copula
parameters 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 the
non-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 from
a multivariate distribution (our empirical observations), x̂1 , . . . , x̂N , where
x̂j = (x̂j,1 , . . . , x̂j,n ), j ∈ {1, . . . , N }.
Notice that assuming appropriate parametric models for the marginal
distributions F1 , . . . , Fn with parameters α1 , . . . , αn and for the copula C
with 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
Y
fi (xi ; αi ),
(6)
i=1
where c is the copula density and f1 , . . . , fn are the densities of the
marginal distributions. The above equation follows from equation 1 on p.12
Q
(Sklar’s theorem), as f (x) = ∂ n F (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 the
copula, θ, may be estimated from the empirical data with a MLE as
35
arg max
N
X
α1 ,...,αn ,θ
n
Y
ln c (F1 (x̂j,1 ; α1 ), . . . , Fn (x̂j,n ; αn ); θ)
j=1
!
fi (x̂j,i ; αi )
(7)
i=1
Such an approach is, however, computationally intensive because the
marginal distributions’ parameters and the copula parameters have to be
jointly estimated.
As demonstrated by Joe and Xu [43], the parameters of the meta-distribution
may be estimated in two steps by first estimating the marginal distributions’
parameters and then estimating the copula parameters. This approach is
referred to as the IFM (inference function for margins) method.
In the IFM method the marginal distributions’ parameters are estimated
with a MLE as
α̂i = arg max
α
i
N
X
ln fi (x̂j,i ; αi )
∀i ∈ {1, . . . , n}
(8)
j=1
Using the obtained parameter estimates for the marginal distributions α̂i ,
the copula parameters θ are estimated in a second step as
θ̂ = arg max
θ
N
X
ln c (F1 (x̂j,1 ; α̂1 ), . . . , Fn (x̂j,n ; α̂n ); θ)
n
Y
!
fi (x̂j,i ; α̂i )
(9)
i=1
j=1
Joe and Xu [43] show that the IFM method is highly efficient compared
to the parameter estimation presented in equation 7 where all parameters
are estimated simultaneously.
Both methods presented above are based on assumptions on the parametric distribution functions of the marginal distributions F1 , . . . , Fn . If any of
these assumptions are wrong, also the copula parameter estimates are biased.
The pseudo-log-likelihood method (also referred to as CML – canonical
maximum likelihood – or semiparametric method) overcomes this problem
by transforming the empirical observations x̂1 , . . . , x̂N into so-called pseudoobservations û1 , . . . , ûN , without assuming any specific functional form of
the marginal distributions. This method was first presented by Genest and
Rivest [29]. The pseudo-observations are computed as
36
ûj,i =
N
1 X
1x̂ ≤x̂
N + 1 k=1 k,i j,i
∀i ∈ {1, . . . , n}, j ∈ {1, . . . , N }
(10)
where 1x̂k,i ≤x̂j,i is an indicator function that takes a value of 1 if x̂k,i ≤ x̂j,i
and a value of 0 otherwise.25 The pseudo-observations are uniformly distributed between 0 and 1, ûi,n ∼ U (0, 1).
Using the pseudo-observations, the copula parameters are estimated with
a MLE as
θ̂ = arg max
θ
N
X
ln (c (ûj,1 , ûj,2 , . . . , ûj,n ; θ))
(11)
j=1
Scaillet and Fermanian [63] who conduct a Monte Carlo study to assess
the impact of misspecified marginal distributions suggest that ‘if one has any
doubt about the correct modeling of the margins, there is probably little to
loose but lots to gain from shifting towards a semiparametric approach’, i.e.
they suggest to generally use the pseudo-log-likelihood method rather than
the IFM method.
3.4.2
Parameter estimation using correlation measures
An alternative method to estimate the parameters of bivariate one-parametric
copulas is to compute rank-based correlation measures such as Spearman’s
rho ρS or Kendall’s tau τ K from the empirical data and to infer the copula
parameter from these correlation measures.
For some copulas a simple functional relationship exists between either
ρS and/or τ K and the copula parameter, hence the copula parameter may
easily be computed from the estimate of one of the two correlation measures
(see table 926 ; for information on ρS and τ K see Appendix A). If the Student
t copula parameter ρ is estimated from Kendall’s tau, additionally the parameter ν has to be estimated, using MLE.
25
Note that the sum of the indicator functions is divided by N +1 as also done in Demarta
and McNeil [18], p.8. This keeps the pseudo-observations away from the boundaries of the
unit cube where the density of many copulas take infinite values. Alternatively, the sum
of the indicator functions is divided by N by other studies.
26
The functional relationships are documented in e.g. Cherubini et al. [13], McNeil et
al. [48] and Nelsen [53].
37
Gaussian
ρS
6
π
arcsin ρ2
τK
2
π
arcsin ρ
copulas
Student t Clayton Gumbel
Frank
1−
2
π
arcsin ρ
θ
θ+2
1−
1
θ
12
θ
(D1 (θ) − D2 (θ))
1 − 4θ (1 − D1 (θ))
Table 9: Functional relationships between copula parameters and the correlation measures Spearman’s
rho ρS and Kendall’s tau τ K . Dk is the Debye
R
k
x
function Dk (x) = xkk 0 ett−1 dt.
The advantage of this approach is that it is computationally very fast.
However, Genest et al. [28] show that (for a bivariate Clayton copula) the
pseudo-log-likelihood approach presented in the above subsection outperforms the method using correlation coefficients. Still, the approach seems
useful as it allows to estimate starting values for numerical parameter estimations that are based on a MLE, speeding up the copula parameter estimation.
3.4.3
Empirical copulas
Empirical copulas may be used alternatively to the parametric copulas presented above. They were introduced by Deheuvels [16], [17] and are computed
from empirical pseudo-observations (see equation 10) as
Ĉ(u) =
N Y
n
1 X
1û ≤u .
N j=1 i=1 j,i i
(12)
The empirical copula asymptotically converges to the true copula for
N → ∞. It may be used for Monte Carlo simulations or for a visualisation of
the goodness-of-fit of some parametric copula, by comparing a parameterised
copula to the empirical copula. However, as pointed out by Scaillet and Fermanian [63], the empirical copula is not differentiable and graphical visualisations 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 information criterion AIC (Akaike [3]). This measure takes into account that
27
See e.g. Wand and Jones [68]and Silverman [65].
38
the likelihood increases with the number of copula parameters by adjusting
the measure accordingly. AIC is defined as
AIC = −2
N
X
ln c(û1,j , û2,j ; θ̂) + 2m
(13)
j=1
where m is the number of estimated (scalar) copula parameters and the
P
term N
j=1 ln c(û1,j , û2,j ; θ̂) is the log-likelihood. Hence, for a bivariate
Gaussian copula m = 1, while for a trivariate copula m = 3 and for a
four-dimensional Gaussian copula m = 6 (for the number of scalar parameters, i.e. the number of different elements of the Gaussian copula parameter
matrix P, see e.g. equation 3 on p.23). The lower the AIC the better the
goodness-of-fit.
Concerning statistical tests that explicitly test whether a parameterised
copula at hand is indeed the true copula, there does not yet exist one standard test. One widely used test is based on Rosenblatt’s [60] probability
integral transform. This test is explained in detail in Appendix B, where also
explicit formulas for computing tests statistics for selected bivariate copulas
and trivariate Gaussian and Student t copulas are provided. Statistical tests
based on the probability integral transform suffer from the fact that they test
for the whole joint distribution (i.e. the copula and the marginal distributions) while the focus should indeed be on the copula.28 A small simulation
study in Cech and Fortin [10], p.31, shows that the power of such tests is very
poor and that wrong hypotheses cannot be accurately rejected if the sample
size is small (N = 50). A small sample size is to be expected, however, when
monthly or less frequently measured data are used to calibrate the copulas
in a risk aggregation context.
An alternative test that, loosely speaking, examines the null hypothesis
of the Gaussian copula being the true copula against the Student t copula
can be conducted with a likelihood ratio test, where the Gaussian copula is
regarded 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 }
28
(14)
Note that if the sample size N is small the pseudo-observations ûi obtained by equation
10 may differ considerably from the uniformised observations Fi−1 (x̂i ), where Fi−1 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 estimated
under H0 . In our case k = 1 since the copula parameter ν is not estimated
under H0 . Note that 0 ≤ λ ≤ 1.
Research on statistical tests examining the goodness-of-fit of copulas is
still ongoing. Tests other than the ones presented above have been proposed
by 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-Gaussian
and meta-Student t distributions, and bivariate meta-BB1 (and its two special cases, meta-Clayton and meta-Gumbel) and meta-Frank distributions
are provided in subsections 3.6.1 to 3.6.6.
Before presenting the specific algorithms, we shortly remind the reader
on how to simulate dependent multivariate standard normal Gaussian distributions using the Cholesky factorization and show how antithetic sampling
is done.
For the simulation of meta-Gaussian and meta-Student t distributions,
dependent multivariate standard normal Gaussian distributions have to be
simulated (see subsections 3.6.1 and 3.6.2 below). This can be achieved, using
a Cholesky factorization of the correlation matrix P (i.e. the Gaussian and
Student t copula parameter P, see section 3.3). The Cholesky factorization
A is a lower triangular matrix that is linked to the matrix P by
AA0 = P.
(15)
Many statistical software packages provide functions on the computation
of A, detailed information on algorithms is given in Glassermann [32], pp.71ff.
A n-dimensional scenario of dependent multivariate standard normally
distributed random variables x = (x1 , . . . , xn ) ∼ N (0, 1) is simulated from independently distributed standard normal random variables z = (z1 , . . . , zn ) ∼
N (0, 1) i.i.d. and the matrix A as
29
In equation 14, the numerator is the likelihood of a Gaussian copula and the denominator is the likelihood of a Student t copula. The p-value when rejecting H0 is computed
as 1 − χ21 (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 we
may also define
x1 = z1
x2 = ρ · z1 +
q
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 standard error of the estimates that also ensures that simulated standard normally distributed random variables (∼ N (0, 1)) and uniformly distributed
random variables on the unit interval (∼ U (0, 1)) are not skewed (see e.g.
Glasserman [32], p.205ff). If m independent standard normally distributed
random 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 variables 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 corresponding meta-distributions) varies substantially, depending on the simulated copula. Table 10 shows the average computing time for a simulation of
100,000 scenarios of bivariate Gaussian, Student t30 , Clayton, Gumbel, Frank
30
Table 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 time
hardly differs (on average it increases by 0.5 milliseconds for ν = 100 ).
41
copulas
seconds
relative
Gaussian
0.072
1.00
Student t
1.002
13.95
Clayton
0.185
2.58
Gumbel
71.6
996
Frank BB1
0.078 85.3
1.09 1,187
Table 10: Computing time in seconds for the simulation of 100,000 scenarios of bivariate copulas and computing time relative to that for a Gaussian
copula.
and BB1 copulas, computing 100 simulations each. The simulations were
done on a ‘standard’ personal computer (3.5 GHz processor, 1 GB RAM),
using the software ‘Matlab’, version 7.2 in a MS-Windows environment. The
standard deviation of the computing time per simulation of the same copula
is very low (less than 3% in terms of the average computing time). It can be
seen that the Student t copula takes about 13 times longer to be simulated
than the Gaussian copula (using equation 17 to simulate bivariate dependent
standard normal random variables). This is mainly because the computation
of the univariate Student t distribution takes longer than that of a standard
normal distribution (see subsections 3.6.1 and 3.6.2 below).31 Simulations of
a 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 3.6.3
and 3.6.5 below). While the computing time for these two copulas may, of
course, be reduced using faster numerical methods32 , it seems clear that the
simulation of the Gumbel and BB1 copulas will always take longer than the
simulation of the other copulas presented in this article.
Concerning the simulations of trivariate, 4-dimensional and 5-dimensional
Gaussian and Student t copulas, table 11 displays the average computing time
and the computing time relative to that for a bivariate Gaussian copula (using equation 15 and 16 and the Matlab-function chol to simulate bivariate
dependent standard normal random variables). The simulation of a Student
t 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 n
increases.
In the subsections below algorithms for the simulation of multivariate
meta-Gaussian and meta-Student t distributions, and bivariate meta-BB1
(and the two special cases, meta-Clayton and meta Gumbel) and meta-Frank
31
The computation of 100,000 values of a Student t distribution on average takes about
15 times as long as does the computation of a standard normal distribution function, using
Matlab (functions normcdf and tcdf).
32
For the simulation above, the Matlab-function fminbnd was used.
42
copulas
dimensions
seconds
relative
Gaussian
3
4
5
0.115 0.161 0.212
1.60 2.24 2.95
Student t
3
4
5
1.430 1.909 2.393
19.90 26.57 33.31
Table 11: Computing time in seconds for the simulation of 100,000 scenarios
of n-dimensional Gaussian and Student t copulas and computing time relative
to that for a bivariate Gaussian copula.
distributions are provided. More information on the simulation of copulas
and 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 simulated
realisations. 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 random
variables zj into dependent random variable vectors xj according to
the copula parameter matrix P.
3. Transform the standard normally distributed variables xj into variables
that 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 normal
distribution function.
4. Compute the simulated joint realisations of the meta-Gaussian distribution aj as aj,i = Fj−1 (uj,i )∀j ∈ {1, . . . , n}, i ∈ {1, . . . , m}, where
Fj−1 is the functional inverse (the quantile function) of the j th marginal
distribution function Fj .
3.6.2
Simulation of meta-Student t distributions
Simulation of a n-dimensional meta-Student t distribution with m simulated
realisations. 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
33
See 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 be
computed by simulating ν standard normal variable vectors s̃k , k ∈
P
{1, . . . , ν}, and defining si = νk=1 s̃2k,i ∀i ∈ {1, . . . , m}.
2. Use a Cholesky factorization to transform the independent random
variables zj into dependent random variable vectors xj according to
the copula parameter matrix P.
3. Transform the standard normally distributed variables xj into variables
that
arequniformly
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 distribution aj as aj,i = Fj−1 (uj,i )∀j ∈ {1, . . . , n}, i ∈ {1, . . . , m}, where
Fj−1 is the functional inverse (the quantile function) of the j th marginal
distribution function Fj .
3.6.3
Simulation of bivariate meta-BB1 distributions
Simulation of a 2-dimensional (bivariate) meta-BB1 distribution with m simulated realisations. Copula parameters: δ, θ.
1. Simulate 2 independent uniformly distributed random variable vectors
s, u1 ∼ U (0, 1).
2. Use numerical methods to to find the elements of the vector u2 such
that
!
u2,i = 1 +
·
u−θ
1,i
δ
−1
δ
u−θ
1,i − 1
+
u−θ
2,i
−1/θ−1
δ 1/δ
+ u−θ
2,i − 1
δ 1/δ−1 −1
·
δ−1
u−θ
1,i − 1
u−θ−1
1,i
∀i ∈ {1, . . . , m}.
3. Compute the simulated joint realisations of the meta-BB1 distribution
aj as aj,i = Fj−1 (uj,i )∀j ∈ {1, 2}, i ∈ {1, . . . , m}, where Fj−1 is the functional inverse (the quantile function) of the j th marginal distribution
function Fj .
3.6.4
Simulation of bivariate meta-Clayton distributions
Simulation of a 2-dimensional (bivariate) meta-Clayton distribution with m
simulated realisations. Copula parameter: θ.
44
1. Simulate 2 independent uniformly distributed random variable vectors
s, u1 ∼ U (0, 1).
2. Compute the dependent uniformly distributed random variable vector
u2 ∼ U (0, 1) as
u2,i =
u−θ
1,i
− θ
si θ+1
− 1
θ
−1 +1
∀i ∈ {1, . . . , m}.
3. Compute the simulated joint realisations of the meta-Clayton distribution aj as aj,i = Fj−1 (uj,i )∀j ∈ {1, 2}, i ∈ {1, . . . , m}, where Fj−1 is the
functional inverse (the quantile function) of the j th marginal distribution function Fj .
3.6.5
Simulation of bivariate meta-Gumbel distributions
Simulation of a 2-dimensional (bivariate) meta-Gumbel distribution with m
simulated realisations. Copula parameter: θ.
1. Simulate 2 independent uniformly distributed random variable vectors
s, t1 ∼ U (0, 1).
2. Usenumerical
methods to find the elements of the vector t2 such that
∀i ∈ {1, . . . , m}.
t2,i 1 − ln θt2,i = si
3. Compute the dependent uniformly distributed random variable vectors
uj ∼ U (0, 1), j ∈ {1,2} as
1/θ
u1,i = exp t1,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 distribution aj as aj,i = Fj−1 (uj,i )∀j ∈ {1, 2}, i ∈ {1, . . . , m}, where Fj−1 is the
functional inverse (the quantile function) of the j th marginal distribution function Fj .
3.6.6
Simulation of bivariate meta-Frank distributions
Simulation of a 2-dimensional (bivariate) meta-Frank distribution with m
simulated realisations. Copula parameter: θ.
1. Simulate 2 independent uniformly distributed random variable vectors
s, u1 ∼ U (0, 1).
45
2. Compute the dependent uniformly distributed random variable vector
u2 ∼ U (0, 1)as
si (1−e−θ )
∀i ∈ {1, . . . , m}.
u2,i = − 1θ ln 1 + s e−θu1,i −1 −e−θu1,i
)
i(
3. Compute the simulated joint realisations of the meta-Frank distribution aj as aj,i = Fj−1 (uj,i )∀j ∈ {1, 2}, i ∈ {1, . . . , m}, where Fj−1 is the
functional inverse (the quantile function) of the j th marginal distribution function Fj .
4
Implementation of a top-down approach and
empirical evidence
Two recent articles by Cech and Fortin [10] and [11] have dealt with the estimation of copulas in a top-down risk aggregation approach using monthly
institution-internal profit and loss data. The sample size in both articles is
less than 50. One of the articles, [11], examines the dependence of the profits
and losses from a market, a credit and a hedge fund portfolio. Only weak
correlation in terms of Spearman’s rho is observed (−0.15 ≤ ρS ≤ 0.15),
and (accounting for the possibility of time delays in reporting returns for
the credit portfolio) there is no indication that lagged profits and losses of
the credit portfolio are highly correlated with the other portfolios’ profits
and losses. Several bivariate copulas (the same as those presented in this
article) and the trivariate Gaussian and Student t copulas are estimated. A
goodness-of-fit test based on the probability-integral transform cannot reject
the null hypothesis of the parameterised copula being the true copula for any
of the calibrated bi- and trivariate copulas. This surprising result is probably due to the small sample size, as also demonstrated in a small simultation
study in [10]. The other article, [10], examines the dependence structure
between observed market portfolio profits and losses and simulated credit
portfolio profits and losses, where more than 40,000 simulation paths are
available. The credit profit and loss paths are simulated by an institutioninternal model. Correlation measures and copula parameters for Gaussian,
Student t and Frank copulas are estimated for each of the more than 40,000
joint observations of the empirical market portfolio and one simulated credit
portfolio profits and loss path. The sample distribution of correlation measures 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 in
the 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, the
Frank copula on average displays a worse fit than the Gaussian and Student
t copula. Again, the null hypothesis of the copula at hand being the true
copula can be rejected only very rarely. While the probability of a type one
error when rejecting this null hypothesis tends to be highest for the Student
t copula (indicating that the Student t copula’s fit may be better than that
of the Gaussian copula), a likelihood ratio test shows that the hypothesis of
a Gaussian copula can be rejected in favour of a Student t copula in only
30.6%, 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 and
Student t copula are ambiguous.
In the present section we try to find further empirical evidence for the dependence structure of different types of risk, using daily market data rather
than monthly institution-internal data. The higher sample size allows for
an examination of the existence of non-zero tail dependence and promises a
higher power of the goodness-of-fit tests.
The data base includes daily sovereign and corporate bond indices from
the iboxx e index family for Euro denominated bonds34 for the period from
January 31st , 2000 until September 15th , 2006. For both sovereign and corporate bond indices, sub-indices with specific maturity bands are considered.
These maturity bands are
• all maturities
• 1Y to 3Y maturities
• 3Y to 5Y maturities
• 5Y to 7Y maturities
• 7Y to 10Y maturities
For corporate bond indices constituents are further grouped according to
their rating. These ratings are
• all ratings
• AAA-rated
• AA-rated
34
More 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 the
interest rate risk factor changes at day t as the log-returns of the sovereign
bond 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 excess
returns of corporate bonds over sovereign bonds for a given maturity band
and 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 subtracting the sovereign bond indices’ log-returns such that the resulting risk factor
changes cj,k,t only display the change in the value of the corporate bond
indices that is due to a change in the credit quality of the constituents (assuming a similar composition of the sovereign and corporate bond indices
with identical maturity band concerning duration, and assuming a constant
risk-appetite of the market participants).
From these riskless returns and excess returns, we construct data pairs
that consist of (ij,t , cj,k,t ) for maturity bands j and rating classes k, resulting in 25 bivariate empirical sample pairs. Scatter plots of these pairs are
displayed in figure 14. They indicate that many of the risk factor changes
are negatively correlated. Hence, negative interest rate risk factor changes
that are caused by an upward shift of the interest rate term structure tend to
occur simultaneously with positive credit risk factor changes that are caused
by an amelioration of the obligors’ credit quality. This finding is not surprising and is consistent with business cycle developments, where in times of
contraction 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 nonGaussian. Using a Jarque-Bera test (Jarque and Bera [41]), the null hypothesis 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
0.01
0
−0.01
0.01
0
−0.01
7−10Y, all ratings
credit risk
0
5−7Y, all ratings
credit risk
−0.01
0.01
3−5Y, all ratings
credit risk
0
credit risk
0.01
1−3Y, all ratings
0.01
0
−0.01
−0.01 0 0.01
interest rate risk
5−7Y, AAA
interest rate risk
7−10Y, AAA
0.01
0
−0.01
0.01
0
−0.01
0.01
0
−0.01
0.01
0
−0.01
credit risk
−0.01 0 0.01
interest rate risk
3−5Y, AAA
credit risk
−0.01 0 0.01
interest rate risk
1−3Y, AAA
credit risk
−0.01 0 0.01
interest rate risk
all mat., AAA
credit risk
−0.01 0 0.01
0.01
0
−0.01
−0.01 0 0.01
interest rate risk
5−7Y, AA
interest rate risk
7−10Y, AA
0.01
0
−0.01
0.01
0
−0.01
0.01
0
−0.01
0.01
0
−0.01
credit risk
−0.01 0 0.01
interest rate risk
3−5Y, AA
credit risk
−0.01 0 0.01
interest rate risk
1−3Y, AA
credit risk
−0.01 0 0.01
interest rate risk
all mat., AA
credit risk
−0.01 0 0.01
0.01
0
−0.01
−0.01 0 0.01
interest rate risk
5−7Y, A
interest rate risk
7−10Y, A
0.01
0
−0.01
0.01
0
−0.01
0.01
0
−0.01
0.01
0
−0.01
credit risk
−0.01 0 0.01
interest rate risk
3−5Y, A
credit risk
−0.01 0 0.01
interest rate risk
1−3Y, A
credit risk
−0.01 0 0.01
interest rate risk
all mat., A
credit risk
−0.01 0 0.01
0.01
0
−0.01
−0.01 0 0.01
−0.01 0 0.01
interest rate risk
3−5Y, BBB
interest rate risk
5−7Y, BBB
interest rate risk
7−10Y, BBB
0.01
0
−0.01
0.01
0
−0.01
0.01
0
−0.01
0.01
0
−0.01
credit risk
−0.01 0 0.01
interest rate risk
1−3Y, BBB
credit risk
−0.01 0 0.01
interest rate risk
all mat., BBB
credit risk
−0.01 0 0.01
credit risk
credit risk
credit risk
credit risk
credit risk
credit risk
all mat., all ratings
0.01
0
−0.01
−0.01 0 0.01
−0.01 0 0.01
−0.01 0 0.01
−0.01 0 0.01
−0.01 0 0.01
interest rate risk
interest rate risk
interest rate risk
interest rate risk
interest rate risk
Figure 14: Scatter plots of interest rate and credit risk factor changes.
49
to the assumption of a multivariate Gaussian distribution for the data sample
at hand.
For many of the marginal distributions (however for only one of the five
sovereign bond index return time series), autocorrelation is detected. To
adjust the data for autocorrelation (remember that we assume that the empirical pseudo-observations û are i.i.d.), an AR(2)-model is fitted where the
observed risk factor changes are modelled as
rt = β1 + β2 rt−1 + β3 rt−2 + t ,
(22)
where rt is either ij,t or cj,k,t .
The coefficients and their statistical significance are estimated using the
standard 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 significantly different from 0 at the 5% significance level, models of the type
rt = β1 + β2 rt−1 + t and rt = β1 + β3 rt−2 + t , respectively, are estimated. If
both β2 and β3 turn out not to be statistically significantly different from 0,
rt = β1 + t is modelled, where β̂1 ≡ r̄. For 21 out of the 25 credit risk factor
changes, AR(2) models are fitted. For only one of the interest rate risk factor
changes, autocorrelation is detected. Here, an AR(1)-model is fitted.
Being interested only in the innovations that cannot be explained by
lagged values of the observed returns, r̃t ≡ ˆt are used as the empirical return
observations in what follows. For the purpose of comparison, Spearman’s rho
and AIC goodness-of-fit measures of copulas are also computed for the original (unadjusted) observations. Results are presented in Appendix C. They
do not differ strongly from the results obtained from the autocorrelationadjusted observations. The 25 autocorrelation-adjusted data pairs include
1,727 observations each.
Table 12 shows sample estimates of Spearman’s rho for the 25 bivariate
empirical samples and reports, whether they are statistically significantly different from 0. In all but 3 samples, the Spearman’s rho correlation measure
is negative (ρ̂S < 0). In most of the cases ρS is found to be statistically significantly different from 0 at the 1% significance level (employing equation
35
Generally, in the context of estimating AR models, alternative estimation methods
are preferred to the OLS-method, as the estimated standard errors of the coefficients are
downward biased. However, the estimates of the coefficients are consistent and the bias
reduces as the sample size increases.
50
ρ̂S
rating
all
all
-0.53∗∗∗
AAA -0.78∗∗∗
AA
-0.26∗∗∗
A
-0.43∗∗∗
BBB -0.56∗∗∗
mean -0 .51
maturity bands
1Y-3Y
3Y-5Y
5Y-7Y
∗
∗∗∗
-0.04
-0.25
-0.28∗∗∗
-0.08∗∗∗ -0.28∗∗∗
0.00
∗∗∗
∗∗
0.10
-0.05
-0.13∗∗∗
∗∗∗
∗∗∗
0.09
-0.20
-0.22∗∗∗
-0.21∗∗∗ -0.23∗∗∗ -0.28∗∗∗
-0 .03
-0 .20
-0 .18
7Y-10Y
-0.35∗∗∗
-0.08∗∗∗
-0.29∗∗∗
-0.29∗∗∗
-0.30∗∗∗
-0 .26
mean
-0 .29
-0 .24
-0 .12
-0 .21
-0 .32
statistically 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 copulas are fitted to the data, using the pseudo-log-likelihood method (see section
3.4.1). The computing time for the copula estimation procedure varies considerably, depending on the copula. While the parameter estimation for the
BB1 copula takes about as long as for the Gaussian copula, the parameter estimation for the Clayton and Gumbel copulas takes about 20% of the
time and that for the Frank copula takes about 2% of the time. Parameters
estimation for a Student t copula takes about 150 times as long as for the
Gaussian copula.
For the BB1, Clayton and Gumbel copulas also the rotated versions
are fitted. Results are reported only for the rotated version with the best
goodness-of-fit in terms of the AIC. The average AIC that is obtained by
the copulas is displayed in figure 15. One can see that the Student t copula 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, the
Gaussian and the Clayton and Gumbel copulas, respectively. The Clayton
copula’s fit is inferior to that of all other copulas presented. The Frank copula
that assigns a low probability to joint extreme events yields a substantially
worse goodness-of-fit than the Student t and BB1 copulas.
Table 13 displays the AIC goodness-of-fit measures in detail. The asterisks beside the AIC measures indicate whether the null hypothesis of the
specific copula being the true copula can be rejected at the conventional statistical significance levels, employing the goodness-of-fit test based on the
probability integral transform presented in section 3.5. If a rotated copula is
51
0
−50
AIC
−100
−150
−200
−250
−300
Gaussian Student t
BB1
Clayton
Gumbel
Frank
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 page
panel A: all
maturities
Gauss
Student t
BB1
Clayton
Gumbel
Frank
ratings
all
-608.35∗∗∗
-694.07
-674.38(−+)
-543.63∗∗∗
(+−)
-647.39∗∗
(−+)
-605.81
1Y-3Y
-2.76∗∗∗
-84.08
-22.42∗∗
(+−)
-12.81∗∗∗
(−+)
-21.96∗∗
(+−)
-1.26∗∗∗
3Y-5Y
-123.15∗
-154.06
-144.17(+−)
-105.96(+−)
-126.87(+−)
-120.18∗
panel B: rating AAA
maturities
all
Gauss
-1652.81∗∗∗
Student t -1881.20
BB1
-1829.01∗∗∗
(−+)
Clayton -1493.76∗∗∗
(+−)
Gumbel -1782.38∗∗∗
(−+)
Frank
-1686.72∗∗
1Y-3Y
-7.74∗∗∗
-139.45
-33.91∗∗∗
(+−)
-23.82∗∗∗
(−+)
-33.37∗∗∗
(+−)
-10.19∗∗∗
3Y-5Y
-144.89∗∗∗
-213.94
-189.53∗∗
(+−)
-163.43∗∗∗
(−+)
-187.24∗∗
(+−)
-151.22∗∗∗
52
5Y-7Y
-151.28∗
-179.91
-172.57(−+)
-134.56(+−)
-160.22(−+)
-150.32∗
7Y-10Y
-235.98∗∗
-255.60
-249.54(−+)
-190.52∗∗
(−+)
-223.94(+−)
-227.03
5Y-7Y
7Y-10Y
0.26∗∗∗
-11.77∗∗∗
-27.49
-87.26
∗∗
-8.55(−+)
-35.35∗∗∗
(−+)
∗∗∗
-3.36(+−)
-19.22∗∗∗
(−+)
∗∗∗
-7.31∗∗∗
-30.47
(−+)
(−+)
1.96∗∗∗
-10.42∗∗∗
continued on next page
Table 13, continued from last page
panel C: rating AA
maturities
all
Gauss
-113.08∗∗∗
-223.39
Student t
-164.21∗∗
BB1
(−+)
-121.25∗∗∗
Clayton
(+−)
Gumbel
-158.63∗∗
(−+)
-127.39∗∗∗
Frank
1Y-3Y
-18.41∗∗∗
-131.05∗∗
-52.14∗∗∗
(−−)
-31.97∗∗∗
-51.17∗∗∗
(−−)
-19.17∗∗∗
3Y-5Y
-4.27∗∗∗
-55.76
-20.32∗(+−)
-13.11∗∗
(−+)
-20.13∗∗
(+−)
-3.51∗∗∗
5Y-7Y
-26.95∗∗
-46.11
-39.88(+−)
-25.09∗(+−)
-35.10(+−)
-27.55∗
7Y-10Y
-162.79
-195.47
-183.98(−+)
-138.67(+−)
-167.88(−+)
-156.83
panel D: rating A
all
maturities
Gauss
-363.34∗∗∗
Student t
-425.79
BB1
-410.88(−+)
-329.33∗∗
Clayton
(+−)
Gumbel
-393.44∗(−+)
Frank
-365.18
1Y-3Y
-16.53∗∗∗
-79.22∗
-35.59∗∗∗
(−−)
-22.21∗∗∗
(−−)
-32.02∗∗∗
-13.29∗∗∗
3Y-5Y
-75.69∗∗
-120.32
-101.45(+−)
-70.04∗∗
(+−)
-89.14(+−)
-75.80∗
5Y-7Y
-86.45∗∗
-121.34
-109.41(+−)
-81.44∗(+−)
-98.65(−+)
-85.10∗
7Y-10Y
-156.10∗∗
-171.37
-166.57(−+)
-129.20∗∗
(+−)
-153.61∗(−+)
-156.08∗∗
panel E: rating BBB
all
maturities
Gauss
-641.81∗∗∗
-725.40∗
Student t
BB1
-665.39∗∗
(−+)
-556.77∗∗∗
Clayton
(+−)
Gumbel
-669.99∗∗
(−+)
Frank
-691.04
1Y-3Y
-77.73∗∗∗
-177.40
-122.06∗∗∗
(+−)
-93.54∗∗∗
(−+)
-119.07∗∗∗
(+−)
-86.83∗∗∗
3Y-5Y
-88.20∗
-115.47
-103.58(+−)
-80.65(+−)
-94.87(−+)
-94.39
5Y-7Y
-152.08
-174.04
-171.14(−+)
-134.42(+−)
-157.95(−+)
-142.18
7Y-10Y
-174.90
-182.53
-179.04(−+)
-137.18(−+)
-160.32(+−)
-168.24
Table 13: AIC goodness-of-fit measures obtained by copulas fitted to the 25 bivariate observation. The subscripts
(−+) , (+−) and (−−) indicate that a rotated copula was
used. Asterisks indicate that the null hypothesis of the
specific copula being the true copula can be rejected at
the ∗ 10%, ∗∗ 5% or ∗∗∗ 1% significance level.
53
Gaussian
Student t
BB1
Clayton
Gumbel
Frank
significance level
5%
19
1
11
17
13
11
10%
22
3
12
19
15
17
1%
14
0
6
6
7
10
Table 14: Number of times that the null hypothesis of the copula at hand
being 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 best
fit in all 25 cases and that in all cases the two-parametric Student t and BB1
copulas achieve better results than their special cases (the Gaussian copula
and the Clayton and Gumbel copula, respectively). The Frank copula for
all data samples achieves a worse goodness-of-fit than the Student t or BB1
copula.
The Student t copula is also the copula for which the null hypothesis of
the 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 the
true copula is rejected in most of the cases. As far as the goodness-of fit of
the BB1 copula is concerned, the null hypothesis of the BB1 copula being
the true copula is rejected more often than the analogous null hypothesis for
the Student t copula.
Conducting additionally a likelihood ratio test, the null hypothesis of the
true copula being the Gaussian copula can be rejected in all of the cases in
favour of the Student t copula at the 1% (and even at the 0.2%) significance
level. This result is not surprising when the parameter distribution of the
Student t copula parameter estimate ν̂, displayed in figure 16, is regarded (as
the Gaussian copula corresponds to a Student t copula with ν → ∞). The
highest value for ν̂ is 12.09, the lowest is 2.83. The mean (median) value is
5.68 (5.17).
The results displayed in table 13 suggest that ‘positive tail dependence’36
36
As the risk factor changes for the data sample at hand are generally negatively correlated, 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 ρ̂S < 0. This definition differs
from the ‘official’ definition of lower and upper tail dependence given in equations 4 and
54
parameter dist. density
0.2
0.15
0.1
0.05
0
0
5
10
15
ν estimate
Figure 16: Gaussian kernel smoothed density of the parameter distribution
of the Student t copula parameter ν.
could be prevalent as the Student t and BB1 copulas that display positive
tail dependence in all cases have a better goodness-of-fit than the Gaussian
and Frank copula that do not. The data sample at hand is large enough
so that we may examine the potential existence of tail dependence in more
detail. To do so, we introduce the concept of corner dependence, which is an
empirical counterpart to the measures of positive tail dependence presented
in section 3.
For positively correlated pairs in terms of Spearman’s rho, ρ̂S > 0, we
define the empirical corner dependence as
= P (û1 ≤ α|û2 ≤ α) = P (û2 ≤ α|û1 ≤ α)
λ̂empirical
L,α
λ̂empirical
U,α
(23)
= P (û1 > 1 − α|û2 > 1 − α) = P (û2 > 1 − α|û1 > 1 − α),(24)
where û1 and û2 are the pseudo-observations (equation 10 on p.37).
For negatively correlated pairs in terms of Spearman’s rho, ρ̂S < 0, we
define the empirical corner dependence as
λ̂empirical
= P (û1 > 1 − α|û2 ≤ α) = P (û2 ≤ α|û1 > 1 − α)
L,α
(25)
λ̂empirical
= P (û1 ≤ α|û2 > 1 − α) = P (û2 > 1 − α|û1 ≤ α).
U,α
(26)
I.e. to compute e.g. the empirical corner dependence λ̂empirical
for posL,α
itively correlated pairs, we first identify the observation pairs for which
û2 ≤ α. λ̂empirical
is the fraction of these pairs for which û1 ≤ α.
L,α
5.
55
Further, for the Gaussian, Student t and Frank copula (symmetric copulas), we define the corner dependence that is implied by a specific copula for
ρ̂S > 0 as
C(α, α; θ̂)
λ̂implied
= λ̂implied
=
,
(27)
L,α
U,α
α
while for ρ̂S < 0 we define
λ̂implied
= λ̂implied
=
L,α
U,α
α − C(1 − α, α; θ̂)
.
α
(28)
For the BB1 copula (asymmetric copula) and its rotated versions we define
λ̂implied
L,α
λ̂implied
U,α
C ++ (α, α; θ̂)
=
α
2α − 1 + C ++ (1 − α, 1 − α; θ̂)
=
α
(29)
(30)
for C ++ and C −+ copulas, and
2α − 1 + C ++ (1 − α, 1 − α; θ̂)
α
++
C (α, α; θ̂)
=
α
=
λ̂implied
L,α
(31)
λ̂implied
U,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 the
implied corner dependence exceeds both the lower and upper empirical corner dependence and in how many cases it is below for α ∈ {0.05, 0.1}. The
most evident observation is that the Frank and the Gaussian copula tend to
underestimate the empirical corner dependence in all/most of the cases. The
Student t copula overestimates the corner dependence in a lot more cases for
α = 0.05 than it does for α = 0.1.
The corner dependence that is implied by the Student t and BB1 copulas
is more closely examined in table 16, where the number of cases where the
lower and upper empirical corner dependence is over- or underestimated is reported. The Student t copula more frequently overestimates both lower and
upper corner dependence than it underestimates these measures for α = 0.05.
56
all maturities
0.5
1Y−3Y
3Y−5Y
0.2
all ratings
0.1
0
e
G t B
F
0.5
AAA
0
e
G t B
F
0
0.2
0.1
0
e
e
G t B
G t B
F
F
0.4
AA
0.2
0
e
G t B
F
0.2
0.1
0
e
G t B
F
0.2
0.1
0
e
G t B
F
0.2
0.1
0
0.4
0.2
0.2
0.1
0
e
G t B
F
0
0.2
0.2
0.1
0.1
0
e
G t B
7Y−10Y
5Y−7Y
F
0
e
e
e
G t B
G t B
G t B
F
F
F
0.2
0.1
0
0.2
0.1
0
0.2
0.1
0
e
G t B
F
e
G t B
F
e
G t B
F
e
G t B
F
e
G t B
F
0.4
A
0.2
0
e
G t B
F
0.2
0.1
0
e
G t B
F
0.2
0.1
0
e
G t B
F
0.2
0.1
0
e
G t B
F
0.2
0.1
0
0.5
BBB
0
e
G t B
F
0.2
0.1
0
e
G t B
F
0.2
0.1
0
e
G t B
F
0.2
0.1
0
e
G t B
F
0.2
0.1
0
Figure 17: 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
emp
emp
implied
α = 0.1 λ̂implied < inf[λ̂emp
> sup[λ̂emp
L , λ̂U ] λ̂
L , λ̂U ]
22
0
Gaussian
4
5
Student t
Frank
25
0
emp
emp
implied
α = 0.05 λ̂implied < inf[λ̂emp
> sup[λ̂emp
L , λ̂U ] λ̂
L , λ̂U ]
Gaussian
20
0
Student t
1
12
Frank
24
0
Table 15: Number of cases where the implied corner dependence are below/exceed the empirical corner dependence for symmetric copulas.
57
α = 0.1 λ̂impl
< λ̂emp
L
L
Student t
10
BB1
14
λ̂impl
> λ̂emp
L
L
15
11
λ̂impl
< λ̂emp
U
U
14
15
λ̂impl
> λ̂emp
U
U
11
10
< λ̂emp
α = 0.05 λ̂impl
L
L
Student t
6
BB1
14
λ̂impl
> λ̂emp
L
L
19
11
λ̂impl
< λ̂emp
U
U
8
10
λ̂impl
> λ̂emp
U
U
17
15
Table 16: Number of cases where the implied upper and lower corner dependence of the Student t and BB1 copula are below/exceed the empirical
corner dependence.
No such behaviour can be observed for the BB1 copula.
Table 17 in detail shows the lower and upper empirical corner dependence
λ̂empirical
and λ̂empirical
for all 25 observation pairs for α ∈ {0.05, 0.1}, as well
L,α
U,α
as their mean and median values (on the very right hand side). The figures
reported as ‘abs. diff.’ are the absolute values of the log-differences of the
|. They indicate
−ln λ̂empirical
lower and upper corner dependence, | ln λ̂empirical
U,α
L,α
by how much the lower and upper corner dependence differ from each other.
The results suggest that lower and upper corner dependence differ considerably for the data sample at hand. This is more pronounced for α = 0.05 than
it 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 the
empirical corner dependence, the values in table 17 reported for ‘Diff. Ga
λL ’, ‘diff. Ga λU ’, ‘diff. t λL ’, ‘diff. t λU ’, ‘diff. B λL ’ and ‘diff. B λU ’ are
the log-differences of the implied and the empirical lower and upper corner
, for
− ln λ̂empirical
dependence measures, ln λ̂implied
− ln λ̂empirical
and ln λ̂implied
L,α
L,α
L,α
L,α
the Gaussian, Student t and BB1 copula, respectively. Negative values indicate that the implied corner dependence is lower than the empirical corner
dependence, which means that the parameterised copula underestimates the
probability of joint excessive observations. The values presented in table 17
show that the Gaussian copula seems to systematically underestimate the
probability of joint excessive events while the Student t copula overestimates
them; in addition these deviations are in absolute terms higher for α = 0.05
than they are for α = 0.1. However the corner dependence implied by the
Student t copula does not deviate as much from the empirical corner dependence as does the corner dependence implied by the Gaussian copula. For
the BB1 copula, the deviations for α = 0.05 are less pronounced than they
58
Table 17: Lower and upper empirical corner dependence and comparison
with the implied corner dependence of a Gaussian and a Student t copula.
α = 0.1
α = 0.05
α = 0.01
59
0.56
Diff t λ U
0.64
-1.41
-0.06
0.96
0.06
0.69
-2.11
0.18
0.41
0.28
Diff t λ L
0.12
0.12
0.79
0.79
-0.22
0.06
0.00
-0.22
0.06
all
all 1Y-3Y 3Y-5Y
0.40 0.19 0.24
0.43 0.17 0.26
0.08 0.06 0.07
-0.11 -0.46 -0.17
-0.19 -0.40 -0.24
0.05 -0.03 0.02
-0.04 0.04 -0.05
-0.05 -0.14 0.01
0.03 -0.31 -0.10
0.29 0.17 0.16
0.41 0.16 0.14
0.34 0.07 0.15
-0.06 -1.03 -0.23
-0.40 -0.96 -0.08
0.23 -0.18 0.16
-0.10 -0.11 0.31
0.08 -0.40 0.18
0.02 -0.83 0.20
Diff Ga λ U -0.13
abs. diff.
Diff Ga λ L
λL
λU
abs. diff.
Diff Ga λ L
Diff Ga λ U
Diff t λ L
Diff t λ U
Diff B λ L
Diff B λ U
λL
λU
abs. diff.
Diff Ga λ L
Diff Ga λ U
Diff t λ L
Diff t λ U
Diff B λ L
Diff B λ U
λL
λU
rating
maturity
0.83
-0.27
-0.09
0.06
1.10
-1.19
0.18
0.89 0.28
1.13
0.22 -0.14
0.06 0.41
0.85
0.71
0.99
0.99
-1.27
0.06
0.00
-1.27
0.06
0.43
1.12
-0.81
0.12
0.69
-0.12
0.06
-0.66
0.03
-2.24
0.12
0.69
-1.55
0.06
-0.01 1.26
0.68
-1.87 -0.27
0.12 0.06
0.69
-1.18
1.00
0.31
-1.06
0.06
0.69
-1.75
0.12
0.47
0.47
-1.36
0.06
0.00
-1.36
0.06
AA
all 1Y-3Y 3Y-5Y
0.26 0.25 0.17
0.31 0.21 0.17
0.20 0.18 0.03
-0.27 -0.60 -0.38
-0.47 -0.43 -0.34
0.11 -0.11 0.00
-0.10 0.07 0.03
-0.19 -0.22 -0.07
-0.11 -0.35 -0.30
0.20 0.16 0.15
0.22 0.08 0.09
0.11 0.69 0.49
-0.46 -0.75 -0.86
-0.57 -0.06 -0.37
0.22 0.16 -0.12
0.11 0.85 0.36
-0.31 -0.03 -0.25
0.09 0.06 -0.31
0.06 0.00
AAA
all 1Y-3Y 3Y-5Y 5Y-7Y 7Y-10Y
0.58 0.19 0.28 0.13
0.21
0.64 0.15 0.19 0.14
0.16
0.11 0.28 0.40 0.04
0.25
-0.05 -0.43 -0.30 -0.19 -0.48
-0.15 -0.15 0.09 -0.24 -0.23
0.07 0.11 -0.03 0.07 -0.08
-0.03 0.39 0.37 0.03
0.17
-0.01 -0.06 0.06 -0.08 -0.31
0.02 -0.07 0.06 -0.08
0.00
0.51 0.16 0.21 0.12
0.14
0.58 0.13 0.15 0.09
0.14
0.13 0.24 0.33 0.22
0.00
-0.06 -0.88 -0.42 -0.71 -0.67
-0.19 -0.64 -0.09 -0.49 -0.67
0.14 0.13 0.10 -0.13
0.10
0.01 0.37 0.42 0.09
0.10
0.01 -0.18 0.25 -0.52 -0.38
0.09 -0.51 -0.12 -0.15 -0.19
0.00 0.18
5Y-7Y 7Y-10Y
0.19
0.27
0.26
0.28
0.31
0.02
0.10 -0.10
-0.21 -0.13
0.28
0.02
-0.03
0.00
0.19
0.03
0.02
0.02
0.17
0.15
0.22
0.20
0.24
0.27
-0.22
0.11
-0.46 -0.15
0.13
0.37
-0.10
0.10
-0.04
0.39
0.02
0.20
0.24
0.24
-0.94
0.06
0.00
-0.94
0.06
0.87 0.69
0.87 0.69
-0.04 -0.15
0.06 0.12
0.00 0.00
-0.04 -0.15
0.00
0.00
0.80
0.80
-0.49
0.06
0.00
-0.49
0.06
A
all 1Y-3Y 3Y-5Y
0.32 0.22 0.22
0.36 0.20 0.24
0.12 0.06 0.10
-0.10 -0.47 -0.20
-0.22 -0.41 -0.30
0.08 -0.10 0.07
-0.04 -0.05 -0.03
-0.04 -0.24 0.01
0.03 -0.26 -0.15
0.27 0.19 0.19
0.33 0.13 0.14
0.20 0.37 0.29
-0.25 -0.90 -0.54
-0.44 -0.53 -0.25
0.10 -0.18 -0.02
-0.09 0.19 0.27
-0.11 -0.42 -0.08
0.04 -0.26 0.03
0.06 0.12
5Y-7Y 7Y-10Y
0.17
0.28
0.15
0.30
0.15
0.04
-0.16 -0.27
-0.01 -0.31
0.06 -0.09
0.21 -0.13
0.03 -0.16
0.10 -0.10
0.15
0.17
0.08
0.22
0.62
0.24
-0.60 -0.19
0.02 -0.43
-0.16
0.16
0.46 -0.07
-0.17
0.04
0.22
0.03
0.73
0.04
-0.42
0.06
0.69
-1.12
0.12
0.68 0.97
0.97
-0.07 0.32
0.06 0.12
0.00
0.32
0.43
-1.17
0.00
0.12
0.00
0.00
0.05
0.05
-0.78
0.12
0.00
-0.78
0.12
0.36
-0.74
0.58
0.60
0.54
0.01 -0.60
0.06
0.00
BBB
all 1Y-3Y 3Y-5Y 5Y-7Y 7Y-10Y mean median
0.35 0.27 0.20 0.23
0.25
0.42 0.25 0.24 0.27
0.26
0.20 0.09 0.16 0.17
0.02 0.13
0.11
0.04 -0.43 -0.10 -0.06 -0.12 -0.22
-0.17
-0.16 -0.34 -0.26 -0.23 -0.14 -0.23
-0.23
0.20 -0.04 0.11 0.09 -0.01 0.04
0.05
0.00 0.04 -0.05 -0.07 -0.04 0.03
-0.03
0.09 -0.06 0.06 0.02
0.01 -0.04
-0.04
0.07 -0.29 -0.09 -0.02
0.00 -0.08
-0.04
0.19 0.22 0.14 0.15
0.10
0.37 0.22 0.15 0.23
0.21
0.69 0.00 0.08 0.43
0.69 0.31
0.24
0.42 -0.70 -0.20 -0.08
0.35 -0.36
-0.42
-0.28 -0.70 -0.28 -0.51 -0.34 -0.39
-0.40
0.70 0.00 0.22 0.23
0.56 0.14
0.13
0.00 0.00 0.14 -0.20 -0.14 0.12
0.09
0.54 -0.01 0.19 0.11
0.61 0.00
-0.03
0.13 -0.60 0.04 -0.05
0.01 -0.07
0.02
0.00 0.12
5Y-7Y 7Y-10Y
0.19
0.25
0.23
0.26
0.17
0.02
-0.05 -0.15
-0.22 -0.17
0.18 -0.01
0.01 -0.03
0.11 -0.06
-0.04
0.03
0.17
0.09
0.20
0.17
0.13
0.63
-0.43
0.42
-0.55 -0.21
0.02
0.70
-0.10
0.07
-0.05
0.61
-0.23
0.24
are for the other two copulas, so one might infer that the BB1 copula’s fit
in that region is better than that of the Student t copula. Still, remember
that the Student t copula’s goodness-of-fit (indeed for the whole unit cube
region) is superior in all cases to that of the BB1 copula, allowing for a flexible modelling of asymmetric corner dependence. This suggests that either
this asymmetry is of minor importance as far as copula parameterisation for
the data sample at hand is considered or that other asymmetric copulas than
the BB1 copula should be employed.
Summarising, the following may be stated. The dependence structure
between daily interest rate and credit risk factor changes seems to be very
heterogeneous in an unsystematic way, depending on the rating of the obligors (credit risk) and the time until maturity of the financial instruments.
These findings indicate that the top-down approach presented in this article
might be overly simplistic. However, the data also suggest that copula-based
approaches in a risk aggregation context (e.g. in a bottom-up approach)
seem preferable to approaches based on a multivariate Gaussian distribution
as (a) the marginal dsitributions are not normally distributed and (b) the
Gaussian copula fits the data worse than other copulas including the Student t copula. The Student t copula achieves the best goodness-of-fit (in
terms of the AIC). A likelihood ratio test rejects the null hypothesis of a
Gaussian copula in favour of a Student t copula in all cases considered. The
BB1 copula also yields good results (in terms of the AIC), however the null
hypothesis of the BB1 copula being the true copula is rejected quite often
(as compared to the Student t copula). The Frank copula yields inferior
results for the data sample at hand. Concerning the computing time needed
for the estimation and simulation of copulas it is found that interestingly for
Student t copulas parameter estimation takes very long while simulation is
comparably 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 in
the field of financial risk aggregation. After reviewing recent literature on
the subject in section 2, copula-based approaches are presented in section 3.
Here, the Gaussian, Student t, BB1, Clayton, Gumbel, and Frank copula are
presented and their properties are examined. Specific equations for the copula functions and their densities are provided. In addition, section 3 shows
how copula parameters are estimated and presents goodness-of-fit measures
and tests. Algorithms for the simulation of copulas and meta-distributions
are also provided.
Section 4 examines the dependence structure between interest rate and
credit risk factor changes that are computed from sovereign and corporate
bond indices. No clear pattern can be observed as the dependence structure
varies 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-down
approach 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 assumption
of a multivariate Gaussian distribution: none of the marginal distributions
examined are normally distributed. Also, the Gaussian copula’s fit in terms
of 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 measures
are used rather than the linear correlation coefficient (Pearson’s rho), that is
used in the context of multivariate Gaussian distributions. Two widely used
rank-based correlation measures are
• Spearman’s rho ρS and
• Kendall’s tau τ K .
The sample estimate of Spearman’s rank correlation coefficient (Spearman’s rho) is calculated as
2
6 N
i=1 di
ρ̂ = 1 −
N (N 2 − 1)
P
S
(33)
where N is the sample size, di = R(x1,i )−R(x2,i ) with R(.) the ranks (in case
of 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 correlation
coefficient of the obervation’s ranks or, equivalently, to Pearson’s correlation
coefficient of the pseudo-observations (see equation 10 on p.37), i.e.
ρ̂S =
cov(U1 , U2 )
σU1 σU2
(34)
The null hypothesis of a correlation coefficient of zero is tested by employing the formula
√
ρ̂ N − 2
(35)
t= √
1 − ρ̂2
This test is employed for significance testing of both Spearman’s rank correlation coefficient and Pearson’s correlation coefficient (see Glasser and Winter
[31] and Kendall et al. [46]). The test statistic t is Student t distributed with
ν = N − 2 degrees of freedom. Thus, the corresponding p-value is computed
as 2 (1 − tν=N −2 (|t|)), where tν is the Student t distribution function with ν
degrees of freedom.
37
see eg. Hartung et al. [37], p.554.
62
An alternative test for testing the independence of two jointly observed
samples is based on the Hotelling-Pabst statistic (Hotelling and Pabst [40])
P
2
D= N
i=1 di . The test statistic T is defined as
T = q
PN
d2 − (N 3 − N )
= q i=1 i
Var(D)
(N − 1)(N + 1)2 N 2
D − E(D)
6
(36)
if there are no ties (see e.g. Hartung et al. [37], p.556f). For N > 30, T
is approximately standard normally distributed, so the p-value for rejecting
the 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 type
H0 : ρS ≥ ρ0
against the alternative hypotheses
H1 : 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 correlation coefficient (rather than for Pearson’s correlation coefficient) according
to findings by David and Mallows [15], Fieller et al. [23], Fieller and Pearson
[24] is used
t= √
√
1
(z − ζ) N − 3,
1.06
(37)
where
z = arctanh(ρ̂i,j ) =
1 1 + ρ̂i,j
ln
2 1 − ρ̂i,j
and
ζ = arctanh(ρ0 ) +
ρ0
1 1 + ρ0
ρ0
= ln
+
.
2(N − 1)
2 1 − 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 standard
Gaussian c.d.f. at significance level α. The p-values for the probability of a
type 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’s
tau, Kendall [45]) is calculated as
τ̂ K =
4P
−1
N (N − 1)
63
(38)
1
ρS
0.5
0
−0.5
−1
−1
−0.5
0
0.5
1
τK
Figure 18: Bounds of Spearman’s rho ρS as a function of Kendall’s tau τ K
for a given dependence structure.
where
P =
N
−1
X
N
X
1x̃2,i <x̃2,j
i=1 j=i+1
where x̃2,i are the re-arranged observations of x2,i , when the joint observations are ordered according to the values of the observations of the first data
sample x1,i . 2P can also be interpreted as the number of concordant pairs
minus the number of discordant pairs.38
As shown by Durbin and Stuart [21], there exists a functional relationship
between Spearman’s rho ρS and Kendall’s tau τ K
(
− 12
+
3 K
τ − 21
2
τ K + 12 (τ K )2
≤ ρS ≤
≤ ρs ≤
1
+ τ K − 12 (τ K )2
2
3 K
τ + 12
2
if τ K ≥ 0
if τ K < 0
(39)
Figure 18 depicts these bounds graphically.
38
Internet-references on the computation of τ̂ K are e.g.
http://en.wikipedia.org/wiki/Kendall’s tau and
http://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 test
is 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 function F (x1 , . . . , xn ). Let Fi (xi ) = P (Xi ≤ xi ) be the distribution function
of 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 probabilities 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 independently
distributed 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 the
conditional distribution of Ui , given the values U1 , . . . , Ui−1 , is
∂ i−1 Ci (u1 , . . . , ui )
Ci (ui |u1 , . . . , ui−1 ) =
∂u1 · · · ∂ui−1
,
∂ i−1 Ci−1 (u1 , . . . , ui−1 )
,
∂u1 · · · ∂ui−1
i = 2, . . . , n.
Using the conditional distributions Ci , the transformed variables Zi can thus
written 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
39
see e.g Rosenblatt [60]. These tests do in fact test for the whole joint distribution, not
just for the copula. Test results may thus, in general, be affected by the assumption on
the marginals.
65
Sn =
Pn
−1
2
i=1 (Φ (Zi ))
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 table 18.
For rotated bivariate copulas, S2 is
• C
−−
• C
−+
−1
2
−1
2
2
: S2 = (Φ (u1 )) + Φ
: S2 = (Φ (u1 )) + Φ
−1
−1
1−
∂C(1−u1 ,1−u2 ;θ̂)
∂(1−u1 )
∂C(1−u1 ,u2 ;θ̂)
∂(1−u1 )
• C +− : S2 = (Φ−1 (u1 )) + Φ−1 1 −
2
2
∂C(u1 ,1−u2 ;θ̂)
∂(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 ) =
=
∂ 2 C(u1 ,u2 ,v3 )
∂u1 ∂u2
∂ 2 C(u1 ,u2 )
∂u1 ∂u2
=
∂ 2 C(u1 ,u2 ,v3 )
∂u1 ∂u2
c(u1 , u2 )
with c(u1 , u2 ) the bivariate copula’s density. Z1 and Z2 are defined as in
the bivariate case. Explicit formulas for S3 , for the trivariate Gaussian and
Student t copulas, are provided in table 19.
If one wants to test whether a parameterised copula is indeed the true
copula, values of S2 or S3 are computed from the N tuples of empirical
66
copula
Gaussian
2
S = (Φ−1 (u1 )) + Φ−1
∂C(u1 ,u2 ;θ̂)
∂u1
2
2
S = (Φ−1 (u1 )) +
2


(Φ−1 (u1 ))
−
exp
+x2 −2ρ̂xΦ−1 (u1 )
2(1−ρ̂2 )


 R Φ−1 (u2 )



 −1  −∞
+ Φ 






√
2π
2
!


dx 

 ,



1−ρ̂2
φ(Φ−1 (u1 ))
where φ(.) is the p.d.f. of the univariate standard normal distribution and
Φ−1 (.) is the functional inverse of the univariate standard normal c.d.f.
2
Student t S = (Φ−1 (u1 )) +






 −1 
+ Φ 




R t−1 (u2 )
ν̂
−∞
Γ ν̂+2
2
√
ν̂
Γ 2 π ν̂ 1−ρ̂2
(
( )
)
1+
2
t−1 (u1 ) +x2 −2ρ̂xt−1 (u1 )
ν̂
ν̂
(1−ρ̂2 )ν̂
(
)
2
!− ν̂+2
2
dx 


 ,


fν̂ (t−1
(u1 ))
ν̂
where fν is the p.d.f of the univariate standard Student t distribution and
t−1
ν is the functional inverse of the univariate Student t c.d.f with ν d.o.f.
BB1
2
1
1
θ̂
δ̂−1 −θ̂−1
S = (Φ−1 (u1 )) + Φ−1 a− θ̂ −1 b δ̂ −1 (u−
u1
1 − 1)
1
where a = 1 + b δ̂
−1
and b =
2
θ̂
u−
1 −1
θ̂−1
u−
1
−1
δ̂
θ̂
u−
1
Clayton
S = (Φ (u1 )) + Φ
Gumbel
S = (Φ−1 (u1 )) + Φ−1 exp −b θ̂
2
h
1
2
,
δ̂ −θ̂
+ u2 − 1
.
+
u2−θ̂
− 1 −1
−1
!!2
θ̂
(− ln u1 )θ̂−1 1 −1
b θ̂
u1
2
,
i
where b = (− ln u1 )θ̂ + (− ln u2 )θ̂ .
Frank
2
S = (Φ−1 (u1 )) + Φ−1
e−θ̂u1 (e−θ̂u2 −1)
2
e−θ̂ +(e−θ̂u1 −1)(e−θ̂u2 −1)−1
Table 18: Equations for S2 required for the goodness-of-fit test for selected
bivariate copulas.
67
68
2
(u1 ) +

Φ
−1

−∞
R Φ−1 (u2 )

−∞
R Φ−1 (u3 )
exp − 1
2
−
√
1−ρ̂2
1,2
(
φd=3,P̂ (Φ−1 (u1 ),Φ−1 (u2 ),x)dx


 +



dx 
2
φ(Φ−1 (u1 ))·φ(Φ−1 (u2 ))·c(u1 ,u2 ;ρ̂)
φ(Φ−1 (u1 ))·φ(Φ−1 (u2 ))·c(u1 ,u2 ;ρ̂)
)
φ(Φ−1 (u1 ))
2π
+x2 −2ρ̂1,2 xΦ−1 (u1 )
2(1−ρ̂2 )
1,2
2
(Φ−1 (u1 ))
Φ
−∞
R Φ−1 (u3 )
2
, or, explicitly,
 ,
dx 
!!2
2 −1
2 −1
−1
−1
−1
−1
ρ̂
+ Φ−1 (u2 )
ρ̂
+x2 ρ̂
+2Φ−1 (u1 )Φ−1 (u2 )ρ̂
+2Φ−1 (u1 )xρ̂
+2Φ−1 (u2 )xρ̂
1,1
2,2
3,3
1,2
1,3
2,3
(2π)3/2 |ρ̂|1/2
(Φ−1 (u1 ))
+
−1
!!2

Φ−1


( ν̂2 )(πν̂)3/2 |ρ̂|1/2
Γ
( ν̂+3
2 )
Γ
1+
( ν̂2 )πν̂
Γ
+
fν̂ (t−1
(u1 ))
ν̂
)
(
fν̂ t−1
(u1 )
ν̂
(
Φ−1
fd=3,ν̂,P̂ (t−1
(u1 ),t−1
(u2 ),x)dx
ν̂
ν̂
− ν̂+2
2

 +

dx 
2
−1
−1
fν̂ (tν̂
(u1 ))·fν̂ (tν̂
(u2 ))·c(u1 ,u2 ;ρ̂)
ν̂
−∞
R t−1 (u3 )
−1
+x2 −2ρ̂1,2 xt
(u1 )
ν̂
(1−ρ̂2 )ν̂
1,2
2
(t−1 (u1 ))
1+ ν̂
!!2
!!2
)
(
−1
·fν̂ tν̂
(u2 )
)·c(u1 ,u2 ;ρ̂)
− ν̂+3
2

 ,

dx 
2
, or, explicitly,
2 −1
2 −1
−1
−1
−1
−1
−1
−1
−1
−1
−1
ρ̂
+ t
(u2 )
ρ̂
+x2 ρ̂
+2t
(u1 )t
(u2 )ρ̂
+2t
(u1 )xρ̂
+2t
(u2 )xρ̂
1,1
2,2
3,3
1,2
1,3
2,3
ν̂
ν̂
ν̂
ν̂
ν̂
ν̂
(u1 ))
(t−1
ν̂
1−ρ̂2
1,2
( ν̂+2
2 )
√
Γ
fν̂ (t−1
(u1 ))
ν̂
fd=2,ν̂,P̂ (t−1
(u1 ),x)dx
ν̂
where fd=i,ν,P is the p.d.f of the i-dimensional standard Student t distribution function with ν degrees of freedom and correlation matrix P,
fν is the p.d.f of the univariate standard Student t distribution, t−1
ν is the functional inverse of the univariate Student t c.d.f with ν degrees
and c(u1 , u2 ; ρ̂) is the bivariate Student t copula density.
ν̂
−∞
R t−1 (u3 )
ν̂
−∞
R t−1 (u2 )
ν̂
−∞
R t−1 (u2 )

2 


(u1 ) + Φ−1 






+ Φ−1 



S= Φ
−1
2
S = Φ−1 (u1 ) +
where φd=i,P (.) is the p.d.f. of the i-dimensional standard normal distribution with correlation matrix P, φ(.) is the p.d.f. of the univariate
standard normal distribution, Φ−1 (.) is the functional inverse of the univariate standard normal c.d.f. and c(u1 , u2 ; ρ̂) is the bivariate
Gaussian copula density.


+ Φ−1 

exp
φ(Φ−1 (u1 ))
!!2
c(u1 ,u2 ;θ̂)
∂ 2 C(u1 ,u2 ,u3 ;θ̂)
∂u1 ∂u2
φd=2,P̂ (Φ−1 (u1 ),x)dx
Φ
−1
Table 19: Equations for S that is needed for the goodness-of-fit test for trivariate Gaussian and Student t copulas.
For the bivariate copula density functions c(u1 , u2 ; θ̂), refer to Table 5 on p.22.
Student t
S= Φ
Gaussian
−1
2
2 1 ,u2 ;θ̂)
(u1 ) + Φ−1 ∂C(u∂u
+
1

 R Φ−1 (u2 )



 −∞
2
−1
−1 

S = Φ (u1 ) + Φ 




S= Φ
copula
−1
ρ̂S
rating
all
AAA
AA
A
BBB
all
-0.52∗∗∗
-0.78∗∗∗
-0.26∗∗∗
-0.41∗∗∗
-0.55∗∗∗
maturity bands
1Y-3Y 3Y-5Y 5Y-7Y
-0.04∗
-0.24∗∗∗ -0.27∗∗∗
-0.06∗∗∗ -0.29∗∗∗ 0.00
0.12∗∗∗ -0.05∗∗ -0.13∗∗∗
0.09∗∗∗ -0.21∗∗∗ -0.20∗∗∗
-0.21∗∗∗ -0.22∗∗∗ -0.26∗∗∗
7Y-10Y
-0.32∗∗∗
-0.08∗∗∗
-0.29∗∗∗
-0.27∗∗∗
-0.29∗∗∗
Table 20: Spearman’s rho for the 25 bivariate unadjusted observation pairs.
pseudo-observations û1 , . . . , ûN according to the equations presented in tables 18 and 19. Subsequently one tests whether the N values obtained for
Sn are from a chi-square distribution with n degrees of freedom using e.g. a
Kolmogorov-Smirnov40 or an Anderson-Darling41 goodness-of-fit test.
Appendix C: Empirical results for non-autocorrelationadjusted data
The tables below present sample estimates of Spearman’s rho (table 20 and
AIC goodness-of-fit measures (table 21) that are obtained by various copulas for the original, i.e. unadjusted data sample presented in section 4. The
tables correspond to tables 12 and 13. The unadjusted data sample displays
autocorrelation so the results reported in tables 20 and 21 below are biased.
Table 21, continued on next page
panel A: all
maturities
Gauss
Student t
BB1
Clayton
Gumbel
Frank
ratings
all
-578.50∗∗∗
-665.11
-646.94(−+)
-529.69∗∗∗
(+−)
-626.18∗(−+)
-580.01∗
1Y-3Y
-3.34∗∗∗
-86.22∗
-24.32∗∗
(+−)
-13.98∗∗∗
(−+)
-23.85∗∗
(+−)
-1.06∗∗∗
3Y-5Y
-115.85∗∗
-157.93
-143.01∗(+−)
-113.75∗∗
(+−)
-131.58∗(−+)
-111.00∗∗
5Y-7Y
-138.15∗
-170.00
-161.85(−+)
-128.68∗(+−)
-152.24(−+)
-136.35
7Y-10Y
-204.69∗
-227.30
-222.81(−+)
-169.95∗(+−)
-202.20(−+)
-189.41
Continued on next page
40
41
See e.g. Hartung et al. [37], p.183.
See Anderson Darling [6] or e.g. Giles [30].
69
Table 21, continued from last page
panel B: rating AAA
maturities
all
Gauss
-1655.84∗∗∗
Student t -1899.27
-1840.76∗∗∗
BB1
(−+)
Clayton -1497.27∗∗∗
(+−)
Gumbel -1792.67∗∗∗
(−+)
-1700.96∗∗
Frank
1Y-3Y
-4.53∗∗∗
-142.54∗∗
-31.24∗∗∗
(+−)
-20.07∗∗∗
(−+)
-31.07∗∗∗
(+−)
-5.61∗∗∗
3Y-5Y
-154.37∗∗∗
-221.96
-197.71∗∗
(+−)
-169.97∗∗
(−+)
∗∗
-194.56(+−)
-157.71∗∗
5Y-7Y
0.62∗∗∗
-29.12
-8.12∗∗
(+−)
-2.79∗∗
(+−)
∗∗
-7.22(+−)
1.99∗∗∗
7Y-10Y
-13.19∗∗∗
-99.05
-39.40∗∗∗
(−+)
-23.06∗∗∗
(−+)
∗∗∗
-33.27(+−)
-10.49∗∗∗
panel C: rating AA
all
maturities
Gauss
-112.26∗∗∗
-219.51
Student t
-162.90∗(−+)
BB1
Clayton
-119.97∗∗∗
(+−)
-157.83∗∗
Gumbel
(−+)
Frank
-127.87∗∗
1Y-3Y
-24.04∗∗∗
-136.33∗∗∗
-57.82∗∗∗
(−−)
-36.02∗∗∗
-56.41∗∗∗
(−−)
-28.20∗∗∗
3Y-5Y
-3.85∗∗
-56.53
-19.22∗(+−)
-12.63∗∗
(−+)
-19.03∗(+−)
-3.17∗∗
5Y-7Y
-29.66∗∗
-51.47
-42.47∗(+−)
-29.14∗∗
(−+)
-38.50∗(+−)
-31.09∗∗
7Y-10Y
-172.20∗
-206.99
-196.14(−+)
-145.43∗∗
(−+)
-175.89(+−)
-163.28
panel D: rating A
all
maturities
Gauss
-339.51∗∗
Student t
-398.33
-384.77(−+)
BB1
Clayton
-307.51∗∗
(+−)
-368.31(−+)
Gumbel
Frank
-340.09
1Y-3Y
-15.36∗∗∗
-76.69∗
-33.03∗∗∗
(−−)
-19.71∗∗∗
(−−)
-30.39∗∗∗
-14.04∗∗∗
3Y-5Y
-77.23∗
-115.28
-99.13(+−)
-68.46∗(+−)
-87.59∗(+−)
-77.47∗
5Y-7Y
-77.13∗∗
-115.31
-101.70∗(+−)
-75.76∗∗
(+−)
-91.44∗(−+)
-75.06∗∗
7Y-10Y
-135.76∗
-148.64
-148.14(−+)
-117.98(+−)
-138.69(−+)
-129.90
panel E: rating BBB
maturities
all
Gauss
-605.70∗∗∗
Student t
-695.43∗
BB1
-669.98∗∗∗
(−+)
Clayton
-551.89∗∗∗
(+−)
Gumbel
-654.12∗∗∗
(−+)
Frank
-655.19∗∗∗
1Y-3Y
-73.60∗∗∗
-183.05
-120.23∗∗
(+−)
-89.30∗∗∗
(−+)
-116.88∗∗
(+−)
-79.97∗∗∗
3Y-5Y
-88.13∗∗
-115.04
-104.12∗(+−)
-85.93∗(+−)
-100.02(−+)
-92.57∗
5Y-7Y
-137.27
-159.41
-159.48(−+)
-133.25(+−)
-152.35(−+)
-124.47
7Y-10Y
-157.93
-171.81
-166.87(−+)
-132.78(+−)
-153.61(−+)
-152.23
Continued on next page
70
Table 21, continued from last page
Table 21: AIC goodness-of-fit measures obtained by copulas fitted to the 25 bivariate unadjusted observation.
The subscripts (−+) , (+−) and (−−) indicate that a rotated copula was used. Asterisks indicate that the null
hypothesis of the specific copula being the true copula
can be rejected at the ∗ 10%, ∗∗ 5% or ∗∗∗ 1% significance
level.
List of Tables
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Sample moments marginal distributions, introductory example
Summary table lower, upper and symmetric tail dependence. .
Quantiles of total return distribution, introductory example .
Selected bivariate copula functions . . . . . . . . . . . . . . .
Probability density functions of selected bivariate copulas . . .
n-dimensional Gaussian and Student t copula functions . . . .
n-dimensional Gaussian and Student t copula density functions
Lower and upper tail dependence, λL and λU , of selected bivariate copulas . . . . . . . . . . . . . . . . . . . . . . . . . .
Functional relationships between copula parameters and correlation measures. . . . . . . . . . . . . . . . . . . . . . . . . .
Computing time for selected bivariate copulas . . . . . . . . .
Computing time for n-dimensional Gaussian and Student t
copulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Spearman’s rho for the 25 bivariate observation pairs . . . . .
AIC goodness-of-fit measures obtained by bivariate copulas .
Rejection of the null hypothesis that the copula at hand is the
true copula . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Comparison of implied and empirical corner dependence for
symmetric copulas . . . . . . . . . . . . . . . . . . . . . . . .
Comparison of implied and empirical upper and lower corner
dependence for Student t and BB1 copulas . . . . . . . . . . .
Lower and upper empirical corner dependence 1 . . . . . . . .
GoF-test: S2 for selected bivariate copulas . . . . . . . . . . .
GoF-test: S3 for tivariate Gaussian and Student t copulas . .
Spearman’s rho for unadjusted observations . . . . . . . . . .
71
14
16
19
21
22
23
24
25
38
42
43
51
53
54
57
58
59
67
68
69
21
AIC goodness-of-fit measures for the unadjusted observations
71
List of Figures
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
Bottom-up and top-down approaches . . . . . . . . . . . . . .
Introductory example for copula-based approaches . . . . . . .
Scatter plots of bivariate meta-distributions . . . . . . . . . .
Densities of bivariate Gaussian, Student t and Frank copulas .
Densities of rotated Clayton and Gumbel copulas . . . . . . .
Structure of a four-dimensional hierarchical Archimedean copula
Densities of bivariate Gaussian copulas and meta-Gaussian
distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Densities of bivariate Student t copulas and meta-Student t
distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Densities of bivariate BB1 copulas . . . . . . . . . . . . . . . .
Densities of bivariate Clayton copulas and meta-Clayton distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Densities of bivariate Gumbel copulas and meta-Gumbel distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Log differences of a Clayton and a survival Gumbel copulas’
densities and of corresponding meta-distributions’ densities . .
Densities of bivariate Frank copulas and meta-Frank distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Scatter plots of interest rate and credit risk factor changes . .
Mean AIC of selected copulas . . . . . . . . . . . . . . . . . .
Parameter distribution of the Student t copula parameter ν .
Empirical and implied lower and upper corner dependence . .
Bounds of Spearman’s rho ρS as a function of Kendall’s tau τ K
72
8
13
15
17
18
25
28
29
30
31
32
33
34
49
52
55
57
64
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