Copulas

Question 1

Drawbacks of using normal dist in multivariate modelling

Answer 1

same as univariate case: underestimates large negative return prob.
in multi case -> benefits of portfolio diversification likely exaggerated

Question 2

Threshold correlations

Answer 2

conventional correlation, but computed only on a selected subset of the data (e.g. conditional on both return series being below their p-th percentile if p < 0.5 and above if p>0.5). -> tells us about dependence across asset returns conditional on both returns being either very negative or very positive

Question 3

Correlations in the tails

Answer 3

Under normal -> threshold correlations go to zero for p -> 0 or p->1.

Question 4

Student t approach

Answer 4

replace normal with student. -> two variables have same tail thickness. however: TCs will always be symmetric, which is a constraint

Question 5

Copula approach

Answer 5

take different univariate (“marginal”) distributions and link these marginals across assets using copulas -> generate valid multivariate density (use absed on Sklar’s theorem: for general class of multivariate CDFs, there exists UNIQUE copula function G linking the marginals to form joint distribution)

Question 6

Steps for building copulas

Answer 6

1) Build and estimate n potentially different marginal distribution models using standard methods
2) decide on copula PDF and estimate it using probability outputs u_i from marginals as the data

Question 7

Common types of copulas

Answer 7

1) normal: can build nonnormal distributions (using nonnormal marginals). cannot generate asymmetric or dicontinuous threshold correlations though. extreme u’s: TCs still go to zero
2) Solved by t-Student copulas:

Question 8

TCs with Copulas pros ans cons

Answer 8

Pros: flexibility, attention to fit correlations in the tails

Cons: cannot nest any other key econometric framework, unclear how to pick copulas, no economic intuition for choice of copulas, unclear how to condition copulas on time-varying information flows