Credit Risk Guinea Pigs

Those of us taking ST9 for the first time offered always risked becoming actuarial lab rats. I feel like one in relation to the credit risk section (ST9-17 section in Acted course notes). Am I the only one feeling like this?

Take for example section 9.7 of McNeil “Copula Models” page 440-448 which is on the ST9 reading list.

This starts in Section 9.7.1 with a reference to Lemma 9.35 relating to conditionally independent doubly stochastic random times with (Ft)-adapted hazard-rate processes.

Lemma 9.35 comes from Section 9.6 which is not included in the required readings. To begin to understand what it's talking about I needed to go back to Section 9.2, “Mathematical Tools”. This section is not included in the readings. It starts with the statement :

“In our analysis we inevitably have to use basic notions from the theory of stochastic
processes, such as filtrations, stopping times or basic martingale theory. These issues
are covered in many standard textbooks on mathematical finance and probability
theory. For our purposes the technical level of Williams (1991) is sufficient.”

This means a fundamental understanding of measure-theoretic probability and stochastic processes to the level of Williams is effectively a pre-requisite for really understanding Section 9.2, which is in turn necessary for understanding pages 440-448. Is this the level of math now required for actuarial?

To understand somewhat Lemma 9.35 I needed to go through each of the following, not included in the “required” readings:

• Section 9.2.1 “Random Times and Hazard Rates”, Definition 9.1 “Cumulative hazard function and hazard rate”, Definition 9.6 “Cumulative hazard and hazard-rate processes”
• Section 9.2.3 “Doubly stochastic Random Times” at least Definition 9.11 “doubly stochastic random time” and Lemma 9.12,
• Section 9.6.2 “Conditionally independent default times” including Definition 9.32, and Lemma 9.33 (which extends Lemma 9.12)

However, not having yet studied measure-theoretic probability and martingales to the level of Williams (or comparably Chung, Billingsley, or Resnick), I'm concerned that my understanding may be superficial as these sections refer extensively to formal probability spaces, filtrations, filtered process, adaptedness, sigma-algebras and sub-sigma-algebras

The core readings for “Copula Models” pages 440-448 also include reference to the following concepts:

• On page 442, survival copulas - Pages 195/196 of McNeil, explain them but aren't included in the required readings for ST9, for example the Copula readings in ST9-12.
  
• On page 443 under “Static version”, the static threshold models of section 8.3. Section 8.3.2 discusses these models.
  
• On page 443 under “On calibration”, risk-neutrality and statistical procedures of Section 8.6; these are explained in sections 9.3 and 8.6.
  
• On page 446, in the one-factor Gauss copula model example, copula exchangeability. Definition 5.14 and pages 197/198 give the necessary background.
  
• Example 9.44 on page 446, LT-Archimedean copulas. Section 5.4.2 pages 222-224 exaplain them. The explanation relies on the Laplace-Stieltjes transform, so an understanding of Riemann-Stieltjes calculus and integration is necessary to background to really understand this section.
  
• Near the bottom of page 446, copula radial symmetry. Definition 5.13 explains this but it also refers to elliptical distributions, so it is necessary to refer to Section 3.3 pages 89-103 explaining spherical and elliptical distributions.
  
• In Example 9.44, default correlations; page 437 is required background here. Again we need the Laplace-Stieltjes transform but it also uses LT-Archimedean copulas with p-factor structure, meaning Section 5.4.3 is requried additional reading here.

None of the additional readings and mathematical background mentioned above are included in the core readings, yet all of them are needed in order to understand Section 9.7. There are some similar issues with some other sections of required mathematical reading for ST9-17, although to a slightly lesser extent.

I have two main concerns with the credit risk section:

• The required mathematical background is very extensive. The examples above make clear that Sections of McNeil included in the examinable credit risk readings require measure-theoretic probability, martingales and Riemann-Stieltjes level integration as mathematical background. Arguably the chapter 8 required readings assume understanding of stochastic calculus including Ito and Lebesgue calculus. Is the Institute of Actuaries really looking for ST9 candidates to have this level of math background?
  
• McNeil as a text is not very amenable to the approach of picking and choosing sections because, as should be clear from the examples above, each Section builds on many of the previous sections, like many math-oriented texts. Inadvertently or otherwise, many more sections of McNeil are really “required” reading than it seems initially.

The candidate is left with the following choices for these McNeil readings:

(a) learn all the sections omitted from the core reading but necessary to make sense of the core reading, with the result that much material is added to your required learning, or

(b) learn only the sections in the core reading, with the result that you may be able to write down the words describing the concepts but you won't know what they actually mean.

I also note that in the other heavily quantitative sections of the course, Acted course notes have helpfully summarised McNeil and filled in many of the missing details in McNeil. In contrast the Acted course notes include very little on the heavily quantitative aspects of credit risk in Chapters 8 and 9 (apart from KMV/Merton), such as all those mentioned above, and pages 344-345 and particularly 352-367. I'm wondering about the reasons for this omission. Was it considered not worthwhile, taking a view that detailed questions on the Mixture Model Approach and Copula Models for credit risk are unlikely to be examined? Or was it omitted for other practical reasons?

Finally I note that for the heavily quantitative credit risk sections such as pages 352-367 and pages 440-448, there are no examples in either Assignment 4, the Q&A bank Part 4, or the sample ST9 exam relating to this material (however, learning these sections from McNeil soaks up a huge amount of time in an already-large course). This means it's very difficult to understand what sorts of quantitative exam questions could arise in relation to the heavily quantitative credit risk material including mixture models and copula models. However it wouldn't really be fair not to examine them would it?