Capital Allocation Risk Measures

[obri600]

Hi there

Can anyone help me with the following.

1. What are the implications of a risk measure/capital allocation being coherent? Why is this desirable (other than having the properties listed in the core reading)?

2. In the game theory method why does n lines of business give rise to 2^n scenarios? For example, for 2 lines of business what are the four scenarios?

3. How do the co-measure variants of risk measures differ from the underlying risk measures? For example, what is the difference between the VaR and the Co-VaR?

Thanks

 

[Exam_Machine]

Maybe I can help with question 3 from your salvo

Hey,

All good questions, that I hope do not pop up in the exam!

As i understand it, the basic idea behind comeasures is looking at an overall risk measure for the whole business, then trying to find out what each of the lines of business contribute to that measure. Then what you do is allocate capital in those implied proportions to each line of business.

Hopefully you can see how this is automatically additive, as you are splitting down the risk measure for the portfolio into its constituent parts. Also, the comeasures are based on conditional expectations, where generally the "condition" relates to the overall portfolio risk measure. Expectation has the nice property of additivity as well.

Example: for Value at risk (VaR) and its comeasure CoVaR.
Say the risk measure for the portfolio is capital at the Value at Risk at say the 99.5th percentile (or one in 200). If we "zoom in" and look at that simulation that gives rise to that 99.5th percentile and break it down to look at what the corresponding capital requirements are for each line of business, we can see what the proportional contribution of each class is to the 99.5th portfolio outcome. In practice, we'll look at a narrow band around the 99.5th percentile as we're not likely to see much at that exact 99.5th percentile point (P(X= anything) = 0 for continous distributions etc).

This is why in the formula in the notes, we have that conditional expectation expression for CoVaR, where the conditional part is essentially saying that we are looking at the specific percentile for the whole portfolio, and averaging the contributions of line of business i (Xi in the notes) to that overall portfolio level percentile . That expectation then becomes the comeasure.

Hope that answers your question? (at least the third one)

 

[Duncan Brydon]

(i) Artzner (and others) drew up a list of axioms that they intended to be self-evidently desirable in an attempt to define a “good” risk measure and called measures that satisfied them “coherent”. It’s a matter of debate whether coherence is desirable.

Sub-additivity might be considered the main property of coherence. It says that diversification does not increase the amount of risk. An implication of this is that use of a sub-additive risk measure avoids manipulation of the regulatory system. If the measure used was not sub-additive, insurers might deliberately break up into subsidiaries to (reduce their total measured risk and so) reduce their capital requirements.

You might also be interested in:
http://www.acted.co.uk/forums/showthread.php?t=4283

(ii) There are n! scenarios but 2^n marginal scenarios.

(iii) Thanks to Exam_Machine for answering this. For Co-VaR, we would in practice look at all simulations in a small range around the percentile in question as Exam_Machine says. This is in order to get a reasonable sample size so that our results are credible.

I hope this is helpful.

 

[obri600]

Thanks for your replies!