# Copulas....

Discussion in 'SP9' started by Actuary_07, Mar 24, 2022.

1. ### Actuary_07Keen member

Hi,

This might be a very basic question - but I am not able to relate alpha and Kendal's tau - how do we calculate one from the other?
(this is in context of Q1 part 10 of September 2021 exam paper)

Thank you so much!

2. ### Anna BishopActEd TutorStaff Member

Hi Actuary_07

Good question. The Archimedean copulas have a Kendall's tau that is an exact function of the parameters. I'm pointing you to Sweeting p215, Table 10.2, or to the Course Notes, Mod 18, p37 (not to cop out of answering your query, but so you know where to locate this info if it comes up in the exam).

For example, for the Gumbel copula, tau = 1 - (1/alpha).

If the sample value of Kendall's tau is 0.5 say, then we would equate this to the formula above (think method of moments type idea):

1 - (1/alpha) = 0.5
1/alpha = 0.5
alpha = 2

Hope this helps!
Anna

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3. ### Actuary_07Keen member

Thank you soooo much Anna! This is super helpful! You are a life saviour (for SP9 , not to sound too dramatic!)

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4. ### Actuary_07Keen member

Hi!
another question related to copulas
I am not able to fully grasp the concept of lower tail dependency vs upper tail dependency
Thank you so much!

5. ### GottaStudyHardMember

I believe lower and upper tail dependency describes the association of the different risk factors as their values increase and as their values decrease. For example, if we look at a credit portfolio, the losses associated with a credit portfolio tend to be heavily associated during economically stressed periods, thus we use a Clayton copula to model the credit portfolios, since the association increases at extremely low values and the clayton copula has lower tail dependency (the Gumbel copula can also be used if the losses are recorded as positive values, and the Gumbel copula has upper tail dependency).

The lower/upper tail dependency structures help us to choose the appropriate copula to model the portfolio of risks. It also helps us to calibrate said copula, since the upper/lower tail dependence coefficients are linked to the alpha parameter used in archimedean copulas.

I hope this helps.

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6. ### CapitalActuaryVery Active Member

For upper/lower tail dependence measures and 'coefficients', see sections 11.6 and 11.8 here: https://www.actuaries.org.uk/system/files/documents/pdf/sm20100510.pdf
More generally for understanding copulas and their practical implications, read the entire paper, which is excellent. It's clearly explained and accessible even to people with no prior knowledge of such things.

7. ### Anna BishopActEd TutorStaff Member

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 CLayton 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 GUmbel 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

8. ### Actuary_07Keen member

It helps tons Anna! Thank you so much!
Also, does it mean that in the example 2, if I model losses as negative values, then they would exhibit strong negative tail dependency (with a coefficient of lower tail dependency close to 1) and so Clayton copula would be a good fit here?

9. ### Anna BishopActEd TutorStaff Member

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