Risk Aggregation:
Copula Approach
Ken Seng Tan,
Ph.D., ASA, CERA
Canada Research Chair in
Quantitative Risk Management
April 18-19, 2009
kstan@uwaterloo.ca
Introduction


The goal of integrated risk management in a
financial institution is to both measure and
manage risk and capital across a diverse
range of activities in the banking, securities,
and insurance sectors
This requires an approach for aggregating
different risk types, and hence risk
distributions

a problem found in many applications in finance
including risk management and portfolio choice.
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Risk Aggregation
Risk Driver
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Topics
1.
2.
3.
4.
5.
Measures of Association
Copulas
Which Copula to Use?
Applications
Concluding Remarks
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Topic I

Measures of Association

Comovements (or dependence) between variables
 Pearson correlation








Its potential pitfalls
Comonotonic risks
Rank correlations
Tail dependence
Copulas
Which Copula to Use?
Applications
Concluding Remarks
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Pearson Correlation of
Coefficient: ρ(X,Y)


Most common measure of dependence
Definition:
 ( X ,Y ) 

Cov( X , Y )
E ( XY )  E ( X ) E (Y )

Var ( X )Var (Y )
Var ( X )Var (Y )
Properties:



-1 ≤ ρ(X,Y) ≤ 1
If X & Y are independent, then ρ(X,Y) = 0.
If |ρ(X,Y)| = 1, then X and Y are said to be perfectly linearly
dependent
 X = aY + b, for nonzero a
 linear correlation coefficient

Invariant under strictly increasing linear transformations:
  aX  b, cY  d     X , Y  ,
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a  0, c  0
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Is Pearson Correlation a Good
Measure of Dependence?

Var(X) and Var(Y) must be finite


Possible values of correlation depend on the
marginal (and joint) distribution of X and Y



Problems with heavy-tailed distributions
All values between -1 and 1 are not necessary attainable
Perfectly positively (negatively) dependent risks do
not necessarily have a Pearson correlation of 1 (-1)
Correlation is not invariant under non-linear
transformations of risks
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Example:
Attainable Correlations



Suppose X ~ N(0,1), Y ~ N(0,σ2)
For a given ρ(X ,Y), what can you say about ρ(eX , eY)?
 nonlinear transformation
 eX and eY are lognormally distributed
Now assume σ = 4:
 If ρ(X ,Y) = 1


If ρ(X ,Y) = -1


ρ(eX , eY) = 0.01372 = ρ(eZ , eσZ) for Z ~ N(0,1),
ρ(eX , eY) = ρ(eZ , e-σZ) = -0.00025
This implies for -1≤ ρ(X ,Y) ≤ 1
 -0.00025 ≤ ρ (eX , eY) ≤ 0.01372

Implications?
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What can we conclude from the
last example?

Pearson correlation is an effective way to
represent comovements between variables if
they are linked by linear relationships, but it
may be severely flawed in the presence of
non-linear links

Need better measures of dependence!
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Comonotonic risks

Comonotonicity is an extension of the concept of perfect correlation
to random variables with non-linear relations.

Two risks X and Y are comonotonic if there exists a r.v. Z and
increasing functions u and v such that X = u(Z) and Y = v(Z)
X and Y are countermonotonic if u increasing and v decreasing, or
vice versa.


Example I: Last example with X ~ N(0,1) and Y ~ N(0,σ2)
 eX & eY are comonotoic when ρ(X,Y) = 1
 eX & eY are countermonotoic when ρ(X,Y) = -1

Example II: ceding company’s risk and reinsurer’s risk

Not perfectly (linearly) dependent but they are comonotonic
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Rank Correlations

Non-parametric (or distribution-free) measures
of association by looking at the ranks of the data




Only need to know the ordering (or ranks) of the
sample for each variable and not its actual numerical
value
Does not depend on marginal distributions
Invariant under strictly monotone transforms
Two variants of rank correlation:


Kendall’s Tau (ρτ)
Spearman’s Rho (ρS)
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Properties of Rank Correlations

1    X , Y   1 and  1  S ( X , Y )  1
  X , Y   S  X , Y 
If X & Y are comonotonic
If X & Y are countermonotonic
If X & Y are independent

1
1
0
1
1
0
Rank correlation measures the degree of monotonic dependence
between X and Y, whereas linear correlation measures the
degree of linear dependence

rank correlations are alternatives to the linear correlation coefficient
as a measure of dependence for nonelliptical distributions
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Coefficients of Tail Dependence

In risk management, we are often concerned with
extreme values, particularly their dependence in the tails

The concept of tail dependence relates to the amount of
dependence in the upper-right-quadrant or lower-leftquadrant tail of a bivariate distribution



Provide measures of extremal dependence
A measure of joint downside risk or joint upside potential
A bivariate distribution can have either




upper tail dependence, or
lower tail dependence, or
both, or
none (i.e. tail independence)
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Simulated Samples of Some
Bivariate Distributions
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Topic II

Measures of Association

Copulas

What is a copula?






Key results
Some examples of copulas
Other properties of copulas
Which Copula to Use?
Applications
Concluding Remarks
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Basic Copula Primer



Copulas provide important theoretical insights and practical
applications in multivariate modeling
The key idea of the copula approach is that a joint distribution
can be factored into the marginals and a dependence function
called a copula.
 The dependence structure is entirely determined by the copula
Using a copula, marginal risks that are initially estimated
separately can then be combined in a joint risk (or aggregate)
distribution that preserves the original characteristics of the
marginals.


facilitate a bottom-up approach to multivariate model building;
Given marginal distributions, the joint distribution is completely
determined by its copula.
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Basic Copula Primer (cont’d)

This implies that the multivariate modeling can be
decomposed into two steps:




Define the appropriate marginals and
Choose the appropriate copula
The separation of marginal and dependence is also useful
from a practical (or calibration) point of view;
Copulas express dependence on a quantile scale
 allow us to define a number of useful alternative
dependence measures
 useful for describing the dependence of extreme
outcomes
 the concept of quantile is also natural in risk
management (e.g. VaR)
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What is a Copula?

A copula C is a
multivariate uniform
distribution function (d.f.)
with standard uniform
marginals

We focus on bivariate case

Bivariate copula:

Connection between
(bivariate) d.f., marginals and
copula function:
FX ,Y  x, y   C  FX  x  , FY  y  
FX ,Y  x, y 
 Pr  X  x, Y  y 
C(u,v) = Pr( U ≤ u, V ≤ v )
where U, V ~ Uniform(0,1)
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 Pr  FX  X   FX  x  , FY Y   FY  y  
 Pr U  FX  x  ,V  FY  y  
 C  FX  x  , FY  y  
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Does such a copula always exist?
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Mathematical Foundation:
Sklar’s Theorem (1959)

Suppose X and Y are r.v. with continuous d.f. FX & FY.

If C is any copula, then
C  FX  x  , FY  y  
is a joint d.f. with marginals FX & FY.

Conversely, if FX,Y(x,y) is a joint d.f. with marginals FX & FY , then
there exists a unique copula C such that
 
    
FX ,Y x, y  C FX x , FY y

Key result: Decomposition of multivariate d.f.

Marginal information is embedded in FX & FY and the dependence
structure is captured by the copula C(·,·)
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Examples of Copula:
FX ,Y  x, y   C  FX  x  , FY  y  
 Independence copula:
C  u, v   u  v
C  FX  x  , FY  y    FX  x   FY  y   FX ,Y  x, y 
 Gaussian copula (or Normal copula) with correlation  :
C  CGa (u, v)   2 ( 1 (u ),  1 (v);  )
where
 2 (, ,  ) is the std bivariate normal d.f. with correlation 
and  1   is the inverse of the standard normal d.f.
 Student t copula with degree of freedom  and correlation :
C  Cvt ,   u, v   Cvt ,   tv1  u  , tv1  v  
where tv1   : inverse of the univariate Student t d.f. with v degrees of freedom
 Gumbel copula, Clayton copula, Frank copula, etc ...
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Copulas: Gaussian vs Student t
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Other Properties of Copulas
Flexibility
1.

Useful when “off-the-shelf” multivariate distributions
inadequately characterize the joint risk distribution
Easy to simulate
Invariant property
Dependence measures
2.
3.
4.

Offers important insights to modeling dependence
via
a) rank correlations
b) tail dependence
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(1) Flexibility:

The power of copula lies in its flexibility in creating
multivariate d.f. via arbitrary marginals


FX ,Y  x, y   C  FX  x  , FY  y  
Useful when “off-the-shelf” multivariate distributions inadequately
characterize the joint risk distribution
Example:

In credit risk modeling, the
default time may be
modeled as


The recovery rate may be
modeled as


X1 ~ Inverse Gaussian
X2 ~ Beta
Interested in the joint
distribution of X1 and X2
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see Jouanin, Riboulet and Roncalli (2004)
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2) Easy to simulate

Offers Monte Carlo risk studies




risk measures
economic capital
stress testing …
Simulated samples:

Gaussian copula

ρ = 0.7

Gumbel copula:

θ = 2.0
Clayton copula:
 θ = 2.2

t-copula:


v = 4 ρ =0.71
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3) Invariant Property:




Let C be a copula for X & Y,
If g(.) and h(.) are strictly increasing functions
Then C is also the copula for g(X) and h(Y)
This is due to the fact that copula relates the quantiles of the
two distributions rather than the original variables
Example:



 y 
Invariant under strictly increasing transformations of the
marginals


FX ,Y  x, y   C  FX  x  , FY
Consider two standard normals X & Y and let their dependence
be represented by the Gaussian copula.
Under increasing transforms, eX & eY still have the Gaussian
copula
Useful with confidentiality of banks’ or insurers’ data.

Copulas can be estimated even if data is transformed
appropriately.
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4a) Kendall’s Tau and Spearman’s
Rho via Copula
 Both rank correlations depend only on the (unique) copula:
1 1
 ( X , Y )  4   C (u, v) dC (u, v)  1
0 0
1 1
 S ( X , Y )  12   C (u, v)  uv dudv
0 0
 Invariant under monotonic transformation
 Gaussian copula:
2
6

  X , Y   arcsin  &  S  X , Y   arcsin


2
2
 t copula:   X , Y   arcsin  (independent of the d.f.)

 Useful for fitting copulas to data
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4b) Tail Dependence via Copula

Recall that tail dependence relates to the magnitude of
dependence in the upper-right-quadrant or lower-left-quadrant tail
of a bivariate distribution


the joint exceedance (tail) probabilities at high (and low) quantiles
examine tail dependence either for a fixed quantile or asymptotically.
 Joint Exceedance Probability (for Upper Tail Dependence)

1
Y
 Pr Y  F
  X  F  
1
X
1  2  C  ,  

for quantile  close to 1
1
 Joint Exceedance Probability (for Lower Tail Dependence)

1
Y
 Pr Y  F
  X  F   
1
X
C  ,  

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for quantile  close to 0
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Comparison of Tail dependence: Gaussian
vs t copulas (std normal marginals)
 copula parameters:  =0.7,  =3
 quantiles lines (vertical and horizontal): 0.5% and 99.5%
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Joint Exceedance Probabilities at
High Quantitles

Joint exceedance probabilities are given for Normal copula
For t-copula, we report the ratio of the joint exceedance
probabilities of t-copula to normal-copula

From Table 5.2 of McNeil, Frey and Embrechts (2005)

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Joint 99% (or equivalently 1%) Exceedance
Probabilities in High Dimensions
Consider daily returns on five stocks with constant ρ = 0.5.
 Impact on the choice of copula?

Prob. on any day all returns
are below 1% quantile
Gaussian
t (4 d.f.)
How often does such an
event happen on average?
7.48 x 10-5
once every 53.1 years
(7.48 x 10-5) x 7.68
once every 6.9 years
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Asymptotic Tail Dependence
Limiting probability
Asymptotic upper tail dependence is obtained by taking the limit
α-quantile  1
Asymptotic lower tail dependence is obtained by taking the limit
α-quantile  0
limiting probability > 0 implies tail dependence




Gaussian
Asymptotic tail independence (ρ < 1)
t
Asymptotic tail dependence (ρ > -1)
Gumbel
Asymptotic upper tail dependence (θ > 1)
Clayton
Asymptotic lower tail dependence (θ > 0)
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Simulated Copulas with Standard
Normal Marginals


In all cases, linear
correlation is around
0.7
Gumbel copula:


Clayton copula:


θ = 2.2
t copula:

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θ = 2.0
v = 4 ρ =0.71
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Topic III

Measures of Association
Copulas

Which Copula to Use?

Applications
Concluding Remarks


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Which Copula to Use?


Given observed data set: { (x1,y1), …, (xT,yT) } how
do we select a copula that reflects the underlying
characteristics of the data?
Parameter estimation








Goodness-of-fit test

Model selection

One-step approach
Two-step approach
Model validation


Kolmogorov-Smirnov test
Anderson-Darling test …


Examine tail dependence
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Principle of parsimony
Akaike’s Information Criterion (AIC)
Schwartz Bayesian Criterion (SBC)
Klugman, Panjer and Willmot (2008)
Loss Models: From Data to Decisions.
Venter (2002) “Tails of Copulas”
Genest, Remillard and Beaudoinc (in
press): “Goodness-of-fit tests for
copulas: A review and a power study”
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Parameter Estimation:
One-Step Approach
FX ,Y ( x, y)  C  FX  x  , FY  y  
# of parameters: nC


nX

nY
Direct Maximum Likelihood (ML) method


Estimate jointly the marginals and the copula function
using the method of ML
nC + nX + nY dimensions optimization problem
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Parameter Estimation:
Two-Step Approach

Inference-functions for Margins (IFM) method
 Step 1:

for each risk factor, independently determine parametric form of
marginal, say, using method of ML
nX parameters for 1st factor and nY parameters for 2nd factor
Step 2:



given marginals, determine copula using method of ML

nC dimensions optimization problem

Pseudo-likelihood method/Semi-parametric Approach
 Similar to IFM except that the marginals are the empirical cdf

Rank-correlation-based Method of Moments
 Calibrating copula by matching to the empirical rank correlations,
independent of marginals
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Topic IV

Measures of Association
Copulas
Choosing the Right Copula

Applications

Concluding Remarks


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Frees, Carriere, and Valdez (1996):
“Annuity Valuation with Dependent Mortality”



Gompertz marginals (for both males and females)
and Frank's copula are calibrated to the joint lives
data from a large Canadian insurer.
The estimation results show strong positive
dependence between joint lives with real economic
significance.
The study shows a reduction of approximately 5% in
annuity values when dependent mortality models
are used, compared to the standard models that
assume independence.
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Klugman and Parsa (1999): “Fitting Bivariate
Loss Distributions with Copulas”




Calibrate Frank’s copula to the joint distribution of loss and
allocated loss adjustment expense (ALAE) for a liability line using
1,500 claims supplied by Insurance Services Office.
Marginals:
 Examine a number of severity distributions
 Loss data: 2-parameter inverse paralogistic distribution
 ALAE: 3-parameter inverse Burr distribution
Discuss ML inference for copulas and bivariate goodness-of-fit
tests
Frees and Valdez (1997) “Understanding relationships using
copulas”
 Using similar data, they adopt Pareto marginals for both
distributions and consider Frank’s copula and Gumbel copula
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Kole, Koedijk and Verbeek (2007):
“Selecting Copulas for Risk Management”



They show the importance of selecting an accurate copula for
risk management.
They extend standard goodness-of-fit tests to copulas.
Using a portfolio consisting of stocks, bonds and real estate,
these tests provide clear evidence in favor of the Student's t
copula, and reject both Gaussian copula and Gumbel copula.
 Gaussian copula underestimates the probability of joint extreme
downward movements, while the Gumbel copula overestimates
this risk.
 Gaussian copula is too optimistic on diversification benefits, while
the Gumbel copula is too pessimistic.
 These differences are significant.
 They also conclude that both dependence in the center and
dependence in the tails are important
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Rosenberg and Schuermann (2006):
“A general approach to integrated risk management with
skewed, fat-tailed risks”



A comprehensive study of banks’ returns driven by
credit , market, and operational risks
They propose a copula-based methodology to
integrate a bank’s distributions of credit, market, and
operational risk-driven returns.
Their empirical analysis uses information from
regulatory reports, market data, and vendor data


most of them are publicly available, industry-wide data
They examine the sensitivity of risk estimates to
business mix, dependence structure, risk measure,
and estimation method
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Fig 2 of Rosenberg
and Schuermann (2006)
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Rosenberg and Schuermann (2006)
(cont’d)

Their findings:




Given a risk type, total risk is more sensitive to
differences in business mix or risk weights than to
differences in inter-risk correlations
The choice of copula (normal versus t ) has a
modest effect on total risk
Assuming perfect correlation overestimates risk
by more than 40%.
Assuming joint normality of the risks,
underestimates risk by a similar amount
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Concluding Remarks

In this presentation,




we discussed various dependence measures,
highlighted pitfalls with the commonly used linear
correlation;
we introduced copula, particularly its role in modeling
dependence and joint risk distributions;
we reviewed various ways of calibrating copula to
empirical data;
we also examined some of its applications in
insurance, finance, and risk management,
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Concluding Remarks (cont’d)

A quote from Embrechts (2008) “Copulas: A personal view” :


Nevertheless copula has some obvious advantages:



the separation of marginals and dependence modeling is appealing, particularly
for problems with a large number of risk drivers
it can still be a powerful tool, providing a simple way of coupling marginal d.f.
while inducing dependence
Tail dependence is important, especially for risk management
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“… the question “which copula to use?” has no obvious answer. There definitely
are many problems out there for which copula modeling is very useful. … Copula
theory does not yield a magic trick to pull the model out of a hat.”
“One of my probability friends, at the height of the copula craze to credit risk
pricing, told me that “The Gauss–copula is the worst invention ever for credit risk
management.” ” Embrechts (2008)
Numerous studies have supported the use of the t-copula, as opposed to the
Gaussian copula
“All models are wrong but some are useful” George E.P. Box
kstan@uwaterloo.ca
SOA CERA - EPP
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References
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P. Embrechts (2008) “Copulas: A personal view” www.math.ethz.ch/~embrechts/
A. McNeil, R. Frey, P. Embrechts (2005) “Quantitative Risk Management” Princeton
University Press.
J. Yan (2007) “Enjoy the Joy of Copulas: With a Package copula”. Journal of Statistical
Software vol. 21 issue #4.
Copula R package (freeware) cran.r-project.org
C. Genest, B. Remillard, and D. Beaudoin “Goodness-of-fit tests for copulas: A review and
a power study” forthcoming in Insurance, Mathematics and Economics.
E.W. Frees and E.A. Valdez (1997) “Understanding relationships using copulas” North
American Actuarial Journal 2(1):1-25
E.W. Frees, J. Carriere, and E.A. Valdez (1996) “Annuity Valuation with Dependent
Mortality.” Journal of Risk and Insurance, 63(2):229-261.
J-F Jouanin, G. Riboulet and T. Roncalli (2004) “Financial Applications of Copula
Functions” in Risk Measures for the 21st Century editor G. Szego.
E. Kole, K. Koedijk, and M. Verbeek (2007) “Selecting copulas for risk management”, J of
Banking & Finance 31:2405-2423.
S.A. Klugman, H.H. Panjer, and G.E. Willmot (2008) Loss Models: From Data to
Decisions. 3rd edition. Wiley.
S.A. Klugman and R. Parsa (1999) “Fitting bivariate loss distributions with copulas”
Insurance: Mathematics and Economics 24:139-148.
J.V. Rosenberg and T. Schuermann (2006) “A general approach to integrated risk
management with skewed, fat-tailed risks”, J of Financial Economics 79:569-614
G. Venter (2002) “Tails of Copulas”. Proceedings of the Casualty Actuarial Society,
LXXXIX 2:68–113.
kstan@uwaterloo.ca SOA CERA - EPP
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