Fitting copulas:
Some tips
Andreas Tsanakas, Cass Business School
Staple Inn, 16/11/06
Tip #1: Look at your data
(1)
 Usually there are some data, even if they are not enough to run a
formal Maximum Likelihood Estimation process
 Plot what you have and look at some heuristics

They may help you decide with model choice and sensible parameter
ranges
 Example



Plot ranks
Visually test for tail-dependence
Visually test for skewness
Tip #1: Look at your data
(2)
 Sample ranks are a data set on which a copula may be fitted

So work with those
 There are dependence measures that relate only to these ranks



Spearman’s rank correlation
Kendall tau
Blomqvist beta
 The estimates of these tend to be more stable than of the usual
correlation
 Can use directly to parameterise some models

Kendall’s tau works well with elliptical (Gaussian, t) and Archimedean
(Gumbel, Clayton) copulas
Rank plots
 Dependence patterns do not get distorted by marginals
‘Real world’
‘Copula world’
1
4
R2 = 0.64
3.5
0.75
3
R2 = 0.05
U2
X2
2.5
2
0.5
1.5
1
0.25
0.5
0
0
0
500
1000
X1
1500
2000
2500
0
0.25
0.5
U1
0.75
1
Example copulas: Gaussian (ρ=0.5)
 Has no tail dependence
Example copulas: t
 Does have tail dependence, same in both tails
Example copulas: Asymmetric t
 Has tail dependence and skewness
Tail correlations
 A local dependence metric that is sensitive to asymptotic tail
dependence (based on Schmidt and Schmid, 2006)
Cp, p   p 2
βp  
p  p2
0.45
Tail-Blomqvist correlation
0.4
0.35
Left tail p=0
0.3
Right tail p=1
0.25
0.2
Gaussian
T (dof=3)
Asym. T
0.15
0.1
0.05
0
-0.05
0
0.1
0.2
0.3
0.4
0.5
0.6
Confidence level
0.7
0.8
0.9
1
Compare 3 copulas
- Gaussian
- t
- Asymmetric t
Plot using 20,000 Samples
Very unstable!
Skew-rank correlations
 Try to identify skewness from a small sample (of 20)
 Generalisation of rank correlation with a 3rd moment adjustment
0.35
Gaussian
T (dof=3)
Asym. T
Skew Rank Correlation
0.3
The sensitivity to the 3rd
joint moments of the
sample ranks increases
with coefficient “a”.
a=0 gives the usual rank
correlation.
0.25
0.2
0.15
0.1
0.05
0
-0.05
-0.1
0
2
4
6
8
10
12
Coefficient a
14
16
18
20
Quadrant correlations
Can write (rank) correlation as sum of correlations in each quadrant
(Smith, 2002)
1
Gaussian copula
0.9
0.8
0.7
Rank(Y)

0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
Rank(X)
0.6
0.7
0.8
0.9
1
Quadrant correlations
Can write (rank) correlation as sum of correlations in each quadrant
(Smith, 2002)
1
Gaussian copula
0.9
0.8
Q2
Q1
0.7
Rank(Y)

0.6
0.5
0.4
Q3
0.3
Q4
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
Rank(X)
0.6
0.7
0.8
0.9
1
Quadrant correlations
Can write (rank) correlation as sum of correlations in each quadrant
(Smith, 2002)
1
t copula
0.9
0.8
0.7
Rank(Y)

0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
Rank(X)
0.6
0.7
0.8
0.9
1
Quadrant correlations
Can write (rank) correlation as sum of correlations in each quadrant
(Smith, 2002)
1
Asymmetric t
copula
0.9
0.8
0.7
Rank(Y)

0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
Rank(X)
0.6
0.7
0.8
0.9
1
Quadrant correlations
Plot the % breakdown of aggregate rank correlation to quadrants
% Breakdown of correlation to quadrants

5
Gaussian
T (dof=3)
Asym. T
4
3
2
1
0
-1
-2
-3
-4
0
1
2
Quadrant
3
4
5
Tip #2: Use judgement expertly
 There are 3 ways of using expert judgement in model calibration
 Method 1


Ask underwriters and other experts for sources of correlation
Make up some numbers
 Method 2



Ask questions that experts can answer meaningfully
Identify drivers of dependence
Associate those with your copula models
 Method 3


Adopt a full formal Bayesian framework
Not quite there yet!
Tip #3: Understand your models
 Copula models can often be expressed via factors (drivers)


So not as ad-hoc as you may think
Helps you chose an appropriate structure
 Gaussian copula: n (or less!) additive factors determine correlation
structure
 t-copula: same as Gaussian with one additional multiplicative factor
driving tail dependence, even for otherwise uncorrelated risks
 Archimedean copulas (e.g. Gumbel, Clayton): there is one factor,
conditional upon which all risks are independent
 p-factor Archimedean copulas: as above, but can use more than
one factor
Tip #4: Keep it simple
 Modern DFA software offers you a wealth possibilities
 But you may be tempted to overparameterise your model
 A small number of parameters (drivers) will give you a better
chance of making meaningful (interpretable) choices
 You may even be able to do a bit of estimation
Tip #5: Use other models for reference
 Suppose you have a peril model which you believe in

But it doesn’t cover all your risks
 Maybe the dependence structure that it implies between classes
can be used as a proxy for risks that aren’t covered?
 Fit a copula to the peril model simulations

Loads of (pseudo-)data!
 Alternatively suppose you have some particular pairs of risks for
which you have more data

This may help you get a feeling for sensible parameter ranges
Tip #6: Work backwards
 See what diversification credits different copula choices imply
 Do they make sense?
 The argument is of course circular, but it may help involve people
in the thinking
Tip #7: Take care of your data
 You may not have enough data today
 If you collect and maintain your data you’ll have more tomorrow
Tip #8: Adopt a positive attitude
 It is no good complaining about fitting a copula ‘being impossible’


It is difficult but not impossible in principle
Insisting on the difficulty does not make the problem of dependencies
go away
 There are some things you can do


Though they still may not work
Shouldn’t you be used to this?
Literature I used for this talk

On most things:
 McNeil, Frey and Embrechts (2005), Quantitative Risk Management,
Princeton University Press.

On quadrant correlations and normal mixtures
 Smith (2002), ‘Dependent Tails’, 2002 GIRO Convention,
http://www.actuaries.org.uk/Display_Page.cgi?url=/giro2002/index.xml

A technical paper from which I pinched an idea or two:
 Schmidt and Schmid (2006) ‘Nonparametric Inference on Multivariate Versions
of Blomqvist's Beta and Related Measures of Tail Dependence’, http://www.unikoeln.de/wiso-fak/wisostatsem/autoren/schmidt/publication.html
Sample literature on fitting copulas
(yes it exists)





Genest and Rivest (1993), ‘Statistical inference procedures for bivariate
Archimedean copulas,’ J. Amer. Statist. Assoc. 88, 1034-1043.
Joe (1997), Multivariate Models and Dependence Concepts, Chapman &
Hall.
McNeil, Frey and Embrechts (2005), Quantitative Risk Management,
Princeton University Press.
Denuit, Purcaru and Van Bellegem (2006), 'Bivariate archimedean copula
modelling for censored data in nonlife insurance'. Journal of Actuarial
Practice 13, 5-32.
Chen, Fan and Tsyrennikov (2006), ‘Efficient estimation of semiparametric
multivariate copula models. J. Am. Stat. Assoc. 101, 1228-1240.
Appendix - dependence measures






Consider risks X and Y, with cdfs F and G.
Let U=F(X), V=G(Y)
Let X’=X-E[X], Y’=Y-E[Y], U’=U-E[U], V’=V-E[V]
Assume sample of size n
x={x1,…,xn } is the sample from random variable X etc
u={u1,…,un } are the normalised sample ranks of X, i.e. numbers
1/n, 2/n,…,1, but ordered in the same way as the elements of x.
 Same for y, v.
Appendix - dependence measures

Pearson correlation coefficient
ρX, Y  

EX' Y'
  
E X'2 E Y'2
Spearman correlation coefficient
ρS X, Y   ρU, V 

Spearman correlation for the Gaussian copula
6
1 
ρS X, Y   arcsin r   ρX, Y 
π
2 
where r is the Pearson correlation of the underlying normal
distribution
Appendix - dependence measures
 Kendall correlation coefficient (population version)


  

 
~
~
~
~
τX, Y  P X  X Y  Y  0  P X  X Y  Y  0
~ ~
where ( X, Y ) is an independent copy of (X,Y)
 Kendall correlation coefficient (sample version)
n
τ̂X, Y    
 2
1
 signx
1i jn
j
 x i y j  y i 
 Kendall correlation coefficient elliptical copulas (incl. Gaussian, t)
τX, Y  
2
arcsinr 
π
where r is the Pearson correlation of the underlying elliptical
distribution
Appendix - dependence measures


Kendall correlation coefficient of the Pareto (flipped Clayton) copula


Copula function: Cu, v   u  v  1  1  u

Kendall’s τ:
θ
θ
θ
θ2
Kendall correlation coefficient of the Gumbel copula


Copula function: Cu, v   exp   ln u   ln v 

Kendall’s τ:
1
1
θ
θ

 1  v   1

θ 1/ θ

1/ θ
Appendix - dependence measures
 “Tail-Blomqvist” correlation coefficient (Schmidt and Schmid, 2006)
Cp, p   p 2
βX, Y; p  
p  p2
where 0<p<1 and C is the copula of (X, Y)
 For limβX, Y;p  λU  0 we have asymptotic upper tail-dependence
p1
βX, Y;p  λL  0 we have asymptotic lower tail-dependence
 For lim
p0
 For sample version use the empirical copula:
Ĉp, p   p2
β̂X, Y; p  
p  p2
 i j  # (uk , v k ) : uk  i / n, v k  j / n
Ĉn  ,  
n
n n
Appendix - dependence measures
 “Skew-rank correlation”
ρS X, Y; a 
    
EX'  EY'   aE| X' |  E| Y' |   E| X' |  E| Y' | 
EX' Y'  a E X' 2 Y'  E X' Y' 2
2 1/ 2
2 1/ 2
3 2/3
3 1/ 3
3 1/ 3
 Quadrant rank correlation (Smith, 2002)
ρS X, Y   ρS  X, Y   ρS  X, Y   ρS X, Y   ρS  X, Y 
ρS  X, Y  
EU' V ' IU'  0, V '  0,
  
E U'2 E V '2
ρS  X, Y  
EU' V ' IU'  0, V '  0,
etc, where I{A} is the indicator function of set A
  
E U'2 E V '2
3 2/3
