Chapter 14 - Copulas

Question 1

General definintion of Copula

Answer 1

A copula expresses a multivariate cumulative distribution function in terms of the individual marginal cumulative distributions

Marginal risk distribution + Copula = Joint risk distribution

Question 2

Basic properties of copulas

Answer 2

An increasing function
F(x)>F(x) if x>x

C(1,1,1,ui,11) = ui
If we integrate out we get marginal distribution of i

For all (a1,a2,…,aN) and (b1,b2,…,bN) with ai<=bN)

Question 3

Axioms for good measure of concordance

Answer 3

Definition: Concordance does not imply that two variables directly influence each other (ie dependent) - rather, may be third variable that they are both dependent on (but not each other directly)

Completeness of domain [defined for all X and Y]
Symmetry [Mx,y = My,x]
Coherence [if Cx,y(u1,u2) >= Cw,z(u1,u2) for all u, then Mx,y >= Mw,z]
Unit range [ -1<= 1]
Independence [Mx,y = 0 if X and Y independent]
Consistency [if X=-Z, then Mx,y = -Mz,y]
Converegnce

Question 4

Clayton, Gumbel, Gaussian copula dependency structures

Answer 4

Clayton-no upper; non-zero lower
Equity sector returns

Gumbel-no lower; upper tail dependency
Credit portfolio

Gaussian-no lower/upper