Credit VaR-2 exmaples with 2 different methods!!

[shanlane]

Hello,

As I am going over my notes again I noticed that there are two different ways of getting a value for CVaR from Stulz and from Allen (to be fair, Stulz never uses the term Credit VaR, just says "a typical VaR calculation..."). I just noticed that they are using the EXACT same transition matrix, credit spreads probabilities, etc.

Allen uses worst case loss - EL, while Stulz uses the current bond price and where it could end up with a certain level of confidence. In other words:

Stulz: Var = (Current bond price) - (where bond could be with certain amount of confidence)

Allen: Var = [Theoretcial price (mean of the distribution)] - [where bond could end up with a certain amount of confidence].

Could you please explain why these are different? Is there one way that we are actually supposed to be doing this?

Thanks,
Shannon

 

[David Harper CFA FRM]

Hi Shannon,

Can i trouble you to point me to the location of the Stulz reference, I am not familiar? In any case:

1. Strictly, either is okay because: this is the same absolute/relative VaR difference that arises in Market/OpRisk. Your first version (Stulz) is an "absolute CVaR" because it is a loss from current (initial value) and, therefore, implicitly includes the drift (expected loss). Your second is a relative VaR because it is a loss measured from the future expected value and excludes the EL. (the difference between market risk and credit risk VaR is: in market risk, the drift is positive [exp return] but in credit risk the drift is negative [EL]! In market risk, absolute VaR < relative VaR, but in credit risk, absolute VaR > relative VaR). Strictly, either is okay and the burden is to be specific with respect to the timframe (versus current or versus future?) and treatment of expected loss (EL)
2. In practical (FRM) terms, you can pretty much ignore the Stulz and rely on L. Allen: GARP has generally used CVaR (confidence) = WCL(confidence) - EL = UL(confidence). Why? Consistent with Basel, EL is assumed to be provisioned in income-statement-based charge offs, while CVaR calibarates economic/regulatory capital against UL.

Thanks,

 

[shanlane]

First of all, thank you for your help and your insights. This has been a really tough process.

These are both in the 2012 readings. Allen Ch 4 "Exending the Var Approach..." and Stulz Ch 18 "Crdit risks and credit derivatives". Both of them are in the section that describes CrdeitMetrics.

It seems like it is more than just absolute vs relative because Allen does not seem to use the market price at all. She just uses the theoretical price of a bond if it does not migrate (I guess this could be interpreted as the market price, but it really isnt), where Stulz uses a current market price. To make things worse, she (Allen) does not seem to interpolate correctly either. She gives a WCL price that is between the prices of the B and BB rated bonds even though it should be between the B and CCC ratings because of the cumulative density function that is given and the confidence level.

If you have anything else to add I would love to hear it. Very frustrating .

Thanks!

Shannon

 

[David Harper CFA FRM]

Hi Shannon,

I did not realize you were referring to their CreditMetrics exegeses. Exam-wise, to be really candid, I might not go to deep with their CM exegeses (the testability has been shallow or nil). But ...

They are both sourcing the same CM technical document. L. Allen doesn't interpolate, she makes a simplifying assumption that the future values are normally distributed ("assuming loan values were normally distributed"), in doing so, she's not using the actual CDF. Stulz plucks a "proper" 95th quantile from the sort (versus her 99% based on a normal deviate of 2.33). Stulz is superficial wrt current bond price ($108) but, IMO, these are superficial interpretative matters. In common they have:

future mean value = $107.09 is net of expected loss
future WCL (95%, Stulz) = 102.02; not really a worst case "loss", but a "worst case future VALUE"
future WCL (99%, Allen but per normal substitution) = $100.12

both imply a credit VaR is the difference between the future mean (107.09) and 102.02 @ 95% or ~100.12 @ 99%, which EXCLUDES expected loss (future negative drift).
i.e., Allen's diagram with UL (@ 99%) = EC (@ 99%) = 107.09 - 100.12 = $6.97 is consistent with GARP's default (above). L. Allen's Figure 4.3 strike me as both consistent with our overall idea and maybe all we really care about here (although she used a normal deviate, which doesn't really match her diagram but she advertises this difference).
... Stulz may casually reference the loss INCLUDING EL (108.00 - 102.02) but it's just a soft reference to an (valid, but un-named) absolute CVaR

I think it's more important, and made more sophisticated, by the fact that CM is a method that is trying to simultaneously incorporate credit deterioration and credit default (our vanilla CVaR approaches are often default-only models). This is a frankly more testable idea that i'd regret you to miss due to the weeds.

Thanks,

 

[shanlane]

Fantastic.

I know that this is getting alittle deep into CM, but I figured that I should look at it because it is really the only complete example of credit VaR anywhere in the curriculum.

Thanks!

Shannon

 

[David Harper CFA FRM]

Fair point, although not to disagree but rather to try and be exam-helpful, can I just remind:

• Ong's credit risk unexpected loss (UL) is really a CVaR (at low confidence corresponding to one standard deviation)
• The exam's request of CVaR has generally been of a much simpler sort so we don't want to lose sight of the simple case, for example, what is the 99% CVaR of a $100 bond with an PD (EDF) of 3.0%? answer = WCL(@99%) - EL = $100 - $3 = UL.
  
  I hope that's additive, thanks!

 

[shanlane]

That makes sense.

Thanks!
SHannon