At a very simple level, consider two items of interest X and Y.
- Upper tail dependency is where a high value of X is associated with a high value of Y.
- Lower tail dependency is where a low value of X is associated with a low value of Y.
The coefficients of upper and lower tail dependency measure the degree of this dependency. They range between 0 (no dependency) and 1 (very strong dependency).
Example 1:
X = return on BP shares, Y = return on Shell shares
I'd expect to see strong lower tail dependency (ie coefficient of lower tail dependency close to 1).
This is because when BP shares yield very low returns, it is likely that Shell shares also yield very low returns, eg due to a factor that adversely affects the whole oil industry such as a ban on fossil fuels.
The C
Layton copula may be sensible here as it exhibits
Lower tail dependency.
Example 2:
X = credit default loss on corporate bond 1, Y = credit default loss on corporate bond 2
I am assuming that losses are measured as a positive quantity.
I'd expect to see strong upper tail dependency (ie coefficient of upper tail dependency close to 1).
This is because when one corporate bond defaults, it is likely that other corporate bonds will also default, eg due to a severe recession and many companies struggling.
The G
Umbel copula may be sensible here as it exhibits
Upper tail dependency.
A similar example to Example 2 could be X = losses on motor insurance portfolio, Y = losses on household insurance portfolio.
May expect to see upper tail dependency, due to an extremely bad winter leading to lots of motor accidents (high losses on motor) as well as lots of household claims (high losses on household).
It's worth now looking to make sure you have noted down the formulae for the coefficients of upper and lower tail dependency and can calculate these for simple copulas, eg independence, min, max copulas, and, if you are feeling brave, for the Clayton and Gumbel.
Does this help?
Anna
, not to sound too dramatic!)