More questions on Chapter 7

[Cheng]

Hi all!

I have more questions on Chapter 7 and hopefully someone can shed some light.

1) Section 1.1 says that capital allocation method should:
a) be consistent with the modelled capital for the whole business
(what does 'be consistent with the modelled capital' mean? can you provide an example of being consistent and inconsistent with the modelled capital?)
b) reflect the risk of individual business segments relative to one another and the business as a whole
c) reflect the interrelationship between risks and different business segments
(is this suggesting that the capital allocation method should allow for correlation and diversification benefit? how is c) different from b) ?)

2) Since expected policyholder default gives the expected loss beyond a threshold, why does the formula in computing EPD includes P(X>=b)?

i.e. formula given: EPD = P(X>=b)* E[(X-b)|X>=b], shouldn't it be
EPD = E[(X-b)|X>=b]?

3) I don't really understand co-measures in allocating capital, e.g. how it differs from other methods or what it implies. The notes seem to suggest that it's done using simulations while other allocation methods is based on mathematical formula..

a) Also, is co-measures a subset of proportional method where the risk measures are based upon conditional expectations? Hence, why is co-variance E[(Xi-ui)(X-u)] a co-measure when it's not based on conditional expectations? Is co-variance different from covariance?

b) how is co-TVaR differ from T-VaR and co X T-VaR differ from X T-VaR?

c) the formula for co-EPD seems to be rather different as compared to EPD..

Hope someone can help me with this. Material in this chapter is making me confused and I'm getting worried since the 'specimen Sept 2013 paper' has a big question on capital, according to the other post.

Thanks in advance!

 

[td290]

A quick glance through some recent SA3 posts will reveal that you're in good company in feeling confused/ranting about this chapter. I'm afraid large sections appear to have been plagiarised from other sources by someone who doesn't really have a thorough understanding of the subject themselves so the Core Reading is riddled with errors and doesn't always make a lot of sense. To answer your questions:

1) Some of this is pretty vague. Reasonable interpretations might be as follows:

a) The allocations for the subportfolios should add up to the total capital for the whole portfolio. The methodology/reasoning behind the allocation should draw on the methodology/reasoning behind the capital setting.

b) Riskier subportfolios should be allocated capital in proportion to the risk they expose the business to. The allocations should also be in proportion with the total capital for the whole portfolio.

c) Yes, this is about correlations. b) is saying the allocations should reflect to some extent the level of stand-alone risk of a subportfolio and c) is saying that they should also take into account correlations between subportfolios.

2) No. EPD is the expected loss to policyholders caused by the company defaulting. The policyholders only make losses if the company defaults, otherwise their loss is zero. That's why we multiply by P(X>=b).

3)a) Simulations are being used as an approximation to a theoretical model in just the same way here are they are normally, e.g. you could specify a collective risk (i.e. frequency-severity) model in which the frequencies are Poisson and the severities are lognormal. It would be very difficult to calculate the distribution of the aggregate loss in closed form so we approximate with a simulation-based model. Likewise the allocation methods are all theoretical ideas that, in most real-world scenarios, are very difficult to calculate exactly, so we approximate using simulation-based models. The co-measures are no different in this respect. Co-measures certainly overlap with other methods, e.g. Co-VaR, Co-TVaR and Co-XTVaR all give identical results to their Aumann-Shapley equivalents and differ only in the reasoning behind the approaches. Covariance is obviously a well-known statistical measure. As far as I know Co-EPD has no counterpart in any other system of allocating capital. Co-measures do not in general give the same answers as proportional methods.

b) Co-TVaR and Co-XTVaR are allocation methods and TVaR and XTVaR are risk measures. The risk measures set overall capital for the whole portfolio and the allocation methods allocate it to subportfolios.

c) See point 5 of my post: http://www.acted.co.uk/forums/showthread.php?t=8133.

 

[Sherwin]

For question(2), take an example.

X~U(0,10). Let b=9. What is the EPD now?

Your opinion seems to be 0.5, but the correct answer should be 0.5*Pr(X>9)=0.5*0.1=0.05.