1. Fitting copulas: Some tips Andreas Tsanakas, Cass Business School Staple Inn, 16/11/06

2. 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

3. 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

4. Rank plots Dependence patterns do not get distorted by marginals
‘Real world’ ‘Copula world’

5. Example copulas: Gaussian (ρ=0.5)
Has no tail dependence

6. Example copulas: t
Does have tail dependence, same in both tails

7. Example copulas: Asymmetric t
Has tail dependence and skewness

8. Tail correlations
A local dependence metric that is sensitive to asymptotic tail dependence (based on Schmidt and Schmid, 2006)
Left tail p=0
Right tail p=1
Compare 3 copulas
- Gaussian
- t
- Asymmetric t Plot using 20,000 Samples
Very unstable!

9. Skew-rank correlations
Try to identify skewness from a small sample (of 20)
Generalisation of rank correlation with a 3rd moment adjustment
The sensitivity to the 3rd joint moments of the sample ranks increases with coefficient “a”.
a=0 gives the usual rank correlation.

10. Quadrant correlations
Can write (rank) correlation as sum of correlations in each quadrant (Smith, 2002)
Gaussian copula

11. Quadrant correlations
Can write (rank) correlation as sum of correlations in each quadrant (Smith, 2002)
Gaussian copula

12. Quadrant correlations
Can write (rank) correlation as sum of correlations in each quadrant (Smith, 2002)
t copula

13. Quadrant correlations
Can write (rank) correlation as sum of correlations in each quadrant (Smith, 2002)
Asymmetric t copula

14. Quadrant correlations
Plot the % breakdown of aggregate rank correlation to quadrants

15. 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!

16. 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

17. 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

18. 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

19. 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

20. 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

21. 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?

22. 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,
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’,

23. Sample literature on fitting copulas (yes it exists)
Genest and Rivest (1993), ‘Statistical inference procedures for bivariate Archimedean copulas,’ J. Amer. Statist. Assoc. 88,
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,

24. 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.

25. Appendix - dependence measures
Pearson correlation coefficient
Spearman correlation coefficient
Spearman correlation for the Gaussian copula
where r is the Pearson correlation of the underlying normal distribution

26. Appendix - dependence measures
Kendall correlation coefficient (population version)
where is an independent copy of (X,Y)
Kendall correlation coefficient (sample version)
Kendall correlation coefficient elliptical copulas (incl. Gaussian, t)
where r is the Pearson correlation of the underlying elliptical distribution

27. Appendix - dependence measures
Kendall correlation coefficient of the Pareto (flipped Clayton) copula
Copula function:
Kendall’s τ:
Kendall correlation coefficient of the Gumbel copula

28. Appendix - dependence measures
“Tail-Blomqvist” correlation coefficient (Schmidt and Schmid, 2006)
where 0<p<1 and C is the copula of (X, Y)
For we have asymptotic upper tail-dependence
For we have asymptotic lower tail-dependence
For sample version use the empirical copula:

29. Appendix - dependence measures
“Skew-rank correlation”
Quadrant rank correlation (Smith, 2002)
etc, where I{A} is the indicator function of set A