Chapter 17 Copulas - Questions on coefficient of upper&lower tail dependence

[Bill SD]

Hi,
Having been through Chapter 17 on Copulas, have three basic questions on the coefficient of upper&lower tail dependence:

Question 1 - The definitions of the coefficient of upper and lower tail dependence include that: (i) the coefficient of the upper tail is the limiting value as we move further into the upper tail (from below) and (ii) the coefficient of the lower tail is the limiting value as we move further into the lower tail (from above). Why is it important to know if we're approaching the limit from above or below -surely the main thing that matters is the limit itself (which is 1 for upper tail and 0 for lower tail)?

Q2: Page 38 (2019 version of notes) includes several questions (a to g) asking to identify an appropriate copula for seven different types of data.
What are the factors that can help identify the appropriate copula - only the coefficients of the upper and lower tail dependence and the general purpose of the copula (ie. the 'independence' copula is obv suitable for data showing independence )?

Q3: Do the solutions (on the same page as the questions about identifying appropriate copulas) contain the exhaustive list of possible answers or only a sample of answers (and other correct answers exist but just aren't written here). [If the former, why isn't the Co-monotonic copula a good option for (question g) data with both upper and lower tail dependence?]

Thanks in advance

 

[Andrew Martin]

Hello

Question 1

When defining the probabilities in these limits, u itself must be between 0 and 1 (it is an input to the inverse of the CDF and the CDF has range from 0 to 1). So it makes sense to define the limits from the appropriate side when taking the limits to these values.

Question 2

In terms of what is in CS2, the characteristics discussed in the Core Reading about copula selection are association and the coefficients of tail dependence. I guess in general they could, for example, present some characteristics of a data set and expect you to use your knowledge of what a scatter of u and v would look like for different copulas to suggest a sensible choice (or even just present a u and v scatter and ask what type of copula it may come from).

Question 3

In general this looks to be a sample of answers. For example, for (c), I think we could use the Gaussian copula for positive association also. For (g), you're right in saying that the co-monotonic has upper and lower tail dependence. So, you're not necessarily wrong. However, for any real-world scenario, I think it is unlikely that the co-monotonic or counter-monotonic copulas would be appropriate. They have perfect positive or negative association and this is unlikely to be observed in real life unless, for example, one variable is just a monotonic function of the other. So, depending on the context of the question, I'd probably be a bit more cautious about suggesting these two for a given data set.

Hope this helps!

Andy

 

[CapitalActuary]

This is perhaps outside the scope of your exam, but I thought it might be useful add a few factors which inform how people generally pick a copula in practice.

The choice of copula in the 'real-world' depends on lots of factors, all of which overlap and intertwine:

• availability of data
• if the data is good enough, the statistical fit of copulas to the data
• related to the above, how well the model will back-test or be consistent with stress and scenario testing if a certain copula is picked
• expertise of the individuals involved
• how easy it is to explain what the copula is doing
• preferences of various stakeholders involved (e.g. model builders, model validators, regulators)
• purpose of the model, e.g. if you're just calculating tail VaR who really cares about the shape of the joint distribution overall, as long as the copula is doing what it needs to in the tail you care about
• ease of implementation
• availability of practitioner and/or academic research papers on use of that copula
• what ever was used last time / what everyone is used to / whatever is 'industry standard' (or perhaps some copula has become particularly 'in-vogue'!)
• probably other factors that don't come to mind right now

These factors apply both to the choice of copula distribution (e.g. Gaussian, Student's t, Frank, Clayton, Gumbel, some generalised version of one of these or a Vine copula involving them) and to the parameters that feed into the copula (like correlations, degrees of freedom, theta parameter(s)...).

I've designed, implemented, and validated capital models for by general insurance companies, and now I work in investment risk for a fund manager, where part of my job involves building risk models. Happy to elaborate on any of the above points, but again I do suspect the grizzly real-world details of how people build these models might be beyond what is expected by examiners.

 

[Bill SD]

This is perhaps outside the scope of your exam, but I thought it might be useful add a few factors which inform how people generally pick a copula in practice.

The choice of copula in the 'real-world' depends on lots of factors, all of which overlap and intertwine:

• availability of data
• if the data is good enough, the statistical fit of copulas to the data
• related to the above, how well the model will back-test or be consistent with stress and scenario testing if a certain copula is picked
• how easy it is to explain what the copula is doing...
• Happy to elaborate on any of the above points, but again I do suspect the grizzly real-world details of how people build these models might be beyond what is expected by examiners.

Thanks very much Andy and 'CapitalActuary' for extremely helpful answers and offer to elaborate.

Interested to know what you mean by the 'the statistical fit of copulas to the data' (second bullet quoted above). And also with 'explaining what the copula is doing', do you mean explaining how the copula is a good fit for the task or explaining to a non-actuary how a gaussian/gumbel/frank (etc) copula generally works.

 

[CapitalActuary]

Thanks very much Andy and 'CapitalActuary' for extremely helpful answers and offer to elaborate.

Interested to know what you mean by the 'the statistical fit of copulas to the data' (second bullet quoted above). And also with 'explaining what the copula is doing', do you mean explaining how the copula is a good fit for the task or explaining to a non-actuary how a gaussian/gumbel/frank (etc) copula generally works.

Statistical fitting
A copula can be fit to data just like any other CDF of a random variable. E.g. given a vector of observations x_1, x_2, ..., x_n you can fit a normal distribution to those values by using maximum likelihood estimation (MLE). You can do the same thing with a copula - using MLE or other methods - but instead of a vector you'll have multivariate observations like (x_1, y_1), (x_2, y_2), ..., (x_n, y_n) and you want to fit marginal distributions and a copula to these two vectors to describe the joint distribution of (X, Y).

Once you've done a fit, you can apply goodness of fit tests like AIC or Chi-squared like you'll be familiar with on 'usual' distributions from the actuarial exams. (Perhaps they cover fitting copulas as well, I'm not sure as I don't remember copulas being in the exams when I took them.)

If you see one type of copula fits another much better according to your goodness of fit tests, you might pick this one. That's all I was saying in this bullet point.

Explaining what the copula is doing
I meant both explaining why a copula is fit for purpose and also how to explain it to people who aren't familiar with them.

E.g. if you're just saying 'we want our variables to be correlated in the model' most people can understand that, and you just say 'ok we use something called a Gaussian copula for this - it's probably the most commonly used copula in finance' and management say "righto sounds good, I've heard of correlation before, carry on".

Explaining something like tail dependence is harder, e.g. if using a t copula to tweak the level of tail dependence. Then someone in senior management says "but I don't think there is tail dependence between profits, only between losses, and I also think the level of tail dependence is different between different classes of business" . So you say yeah but 1) we don't look at that end of the distribution in our capital model, 2) the t copula is symmetric so can't do tail dependence at one end of the distribution but not the other, and 3) the t copula only has 1 degrees of freedom parameter that we can't vary between different pairs of marginals.

Then you suggest: let's use a vine copula which lets us stitch together lots of Gumbels in order to achieve heavy asymmetric tail dependency, but then the regulator says "what on earth is this, we've not seen it before, please explain it to us and why the statistics we calculate on it look different to a t copula". So you say never mind, let's just use a grouped T instead which is a compromise because although its still symmetric we can tweak the tail dependency within groups and it's still basically like a T copula that the regulator is familiar with.

 

[CapitalActuary]

I should again re-iterate: I have no idea whether you would get any marks in CS2 for any of these sorts of points (I almost suspect not!). Just trying to give some background to how this stuff plays out in the workplace.