Chapter 18: Copulas

Question 1

The properties of a copula IOV

Answer 1

1. Increasing function of each of the individual variables in it
2. One CDF not 1 but the rest are then the copula should be equal to the CDF of the remaining variable
3. Valid probability value produced for any valid combination of variables

Question 2

Axioms for a good measure of concordance CUIS CCC

Answer 2

Completeness of domain: M defined for all value of X and Y
Unit range: M in (-1,1)
Independence: If so then M = 0
Symmetry: Mxy = Myx
Coherence: C1 > C2 then M1 > M2
Consistency: X = -Z then Mxy = -Mzy
Convergence: If c tends to C then m tends to M

Question 3

Merits of an explicit copula CITE SA

Answer 3

o Closed form function
o Integration avoided – closed form functions
o Tail dependency of each copula assessed
o Easy to use
o Small number of parameters involved
o Application to Heterogeneous variables is limited

Question 4

Applications of Gumbel copula UC

Answer 4

o Only upper tail dependence – no lower tail

o Credit loss portfolio modelling

Question 5

Application of a Frank copula NJE

Answer 5

o No tail dependency or symmetric form
o Joint survivor modelling
o Exchange rate movement modelling

Question 6

Application of Clayton copula LUI

Answer 6

o Lower tail dependency is possible
o Upper tail dependency not there
o Investment returns on a portfolio

Question 7

Merits of Gaussian copula CATS

Answer 7

• Closed form function does not exist
• Associations from -1 to 1 allowed for, hence very comprehensive
• Tail dependency lacks
• Single parameter definition – correlation derived between variables

Question 8

Benefits of a copula JEI

Answer 8

• Joint distribution does not have to be modelled
• Explicit relationship between interrelated factors shown
• Invariance - marginal distribution can be independently adjusted

Question 9

Types of copulas FIE

Answer 9

• Fundamental – copulas showing the basic dependencies that variables can display
• Implicit
• Explicit – copulas expressed as closed form functions

Question 10

Choosing and fitting a copula CUP MP

Answer 10

• Choosing
o Concordance
o Upper, lower tail lower tail dependencies
o Patterns of dependence
• Fitting
o MLE
o Parameterisation with rank correlation (Kendall’s tau)