GARP.FRM.PQ.P2 Credit VaR - About diversification in IRB approach~

[no_ming]

Hi, Mr. Harper, refer to the notes book 3 of IRB approach, I am so confused about the meaning of portfolio invariant, does it mean that IRB approach is not considering correlation between assets in the loan portfolio of bank and therefore no diversification effect is benefit within in the loan portfolio?

For the asset correlation at the bottom of page, is "R: Asset correlation to the single systematic risk factor" , is it equal to the correlation mentioned above in the portfolio invariant?

Thanks a lot

 

[David Harper CFA FRM]

Hi @no_ming

There is a problem with this statement: "does it mean that IRB approach is not considering correlation between assets in the loan portfolio of bank and therefore no diversification effect is benefit within in the loan portfolio?"
The IRB function uses the asymptotic single risk factor (ASRF) model, which qualifies the model as portfolio invariant which does assume diversification but does not consider direct correlation between an exposure and other loans in the portfolio. Although an imperfect analogy, I think of ASRF as analogous the CAPM in this respect: both assume the portfolio is well-diversified such that idiosyncratic risks are virtually eliminated (so notice the diversification assumption is very strong) and the only relevant single factor that matters is systemic risk factor. The beta, β, in CAPM (which is equal to cross-volatility multiplied by correlation, so is a function of the security's correlation with the market's excess return) is analogous to the correlation, R, in the ASRF-based Basel IRB function, where R is the correlation to the single systematic risk function. Of course the math is different, but at the end of the day, in both cases higher correlation implies higher risk (higher exposure).

Here is the most helpful paper I've ever read on the topic (previously assigned in the FRM): http://trtl.bz/bis-irb-risk-weight-function

And here is the relevant portion, I hope this is helpful!

The Basel risk weight functions used for the derivation of supervisory capital charges for Unexpected Losses (UL) are based on a specific model developed by the Basel Committee on Banking Supervision (cf. Gordy, 2003). The model specification was subject to an important restriction in order to fit supervisory needs:

The model should be portfolio invariant, i.e. the capital required for any given loan should only depend on the risk of that loan and must not depend on the portfolio it is added to. This characteristic has been deemed vital in order to make the new IRB framework applicable to a wider range of countries and institutions. Taking into account the actual portfolio composition when determining capital for each loan - as is done in more advanced credit portfolio models - would have been a too complex task for most banks and supervisors alike. The desire for portfolio invariance, however, makes recognition of institution-specific diversification effects within the framework difficult: diversification effects would depend on how well a new loan fits into an existing portfolio. As a result the Revised Framework was calibrated to well diversified banks. Where a bank deviates from this ideal it is expected to address this under Pillar 2 of the framework. If a bank failed at this, supervisors would have to take action under the supervisory review process (pillar 2).

In the context of regulatory capital allocation, portfolio invariant allocation schemes are also called ratings-based. This notion stems from the fact that, by portfolio invariance, obligor specific attributes like probability of default, loss given default and exposure at default suffice to determine the capital charges of credit instruments. If banks apply such a model type they use exactly the same risk parameters for EL and UL, namely PD, LGD and EAD.

4.1. The ASRF framework
In the specification process of the Basel II model, it turned out that portfolio invariance of the capital requirements is a property with a strong influence on the structure of the portfolio model. It can be shown that essentially only so-called Asymptotic Single Risk Factor (ASRF) models are portfolio invariant (Gordy, 2003). ASRF models are derived from “ordinary” credit portfolio models by the law of large numbers. When a portfolio consists of a large number of relatively small exposures, idiosyncratic risks associated with individual exposures tend to cancel out one-another and only systematic risks that affect many exposures have a material effect on portfolio losses. In the ASRF model, all systematic (or system-wide) risks, that affect all borrowers to a certain degree, like industry or regional risks, are modelled with only one (the “single”) systematic risk factor.

 

[no_ming]

Hi @no_ming

There is a problem with this statement: "does it mean that IRB approach is not considering correlation between assets in the loan portfolio of bank and therefore no diversification effect is benefit within in the loan portfolio?"
The IRB function uses the asymptotic single risk factor (ASRF) model, which qualifies the model as portfolio invariant which does assume diversification but does not consider direct correlation between an exposure and other loans in the portfolio. Although an imperfect analogy, I think of ASRF as analogous the CAPM in this respect: both assume the portfolio is well-diversified such that idiosyncratic risks are virtually eliminated (so notice the diversification assumption is very strong) and the only relevant single factor that matters is systemic risk factor. The beta, β, in CAPM (which is equal to cross-volatility multiplied by correlation, so is a function of the security's correlation with the market's excess return) is analogous to the correlation, R, in the ASRF-based Basel IRB function, where R is the correlation to the single systematic risk function. Of course the math is different, but at the end of the day, in both cases higher correlation implies higher risk (higher exposure).

Here is the most helpful paper I've ever read on the topic (previously assigned in the FRM): http://trtl.bz/bis-irb-risk-weight-function

And here is the relevant portion, I hope this is helpful!

Hi, Mr. Harper, for the practice exam 2016 question below which related to my question before, should (b) be the alternative answer for the diversification benefit besides from (a)?

 

[David Harper CFA FRM]

Hi @no_ming

Excellent question! Strictly speaking, in my opinion, the answer is only (a) and (b) is not valid; i.e., this Q&A is correctly specified. However, it's a challenging question, and you could argue it is borderline unfair. But the question writer knows what (s)he is doing. Because it says "Which of the following B2 approaches allow a bank to explicitly recognize diversification benefits?" The question tests a real understanding here, it its virtue: (a) is the best answer.

By our discussion above, you are right that (b) is tempting because the IRB depends on the portfolio invariance of ASRF model and, specifically, the IRB does assume the bank's portfolio is diversified (such that idiosyncratic risk is eliminated). Therefore, the IRB does grant the bank the benefit of recognizing diversification benefits. This is built-in and automatic by virtual of the correlation, ρ, assumption, which alone defines a position's systematic exposure (to the common macroecomic factor). So, does the IRB recognize the benefits of diversification? Yes, by definition and automatically. On the other hand, does the bank which uses the IRB have a method for explicitly calibrating their own diversification benefit, in the IRB model? No. Rather, the concentration risk (ie, lack of diversification) can be handled as an adjustment to the Pillar I minimum. So, in my opinion, this question succeeds by the inclusion of the word "explicitly." I hope that helps,

 

[patriciar]

Hi David,
I am not sure where to post my doubt as I haven't found a thread that answers my question:
In Basel II it is true that the capital for credit risk equals EAD*LGD*MA*(WCDR-PD), which essentially represents VaR(worst case)-ExpectedLosses, and where Capital is posted for UnexpectedLosses, So we could say, for credit risk, UL(capital)=VaR-EL, right?
On the other hand, in the credit Risk book we calculate the CreditVar as UnexpectedLoss-ExpectedLoss (CVaR=ULat a confidence level-EL).

What is happening here? I Think I should be missing something... cause these calculations are using the same factors but in a different way, right?

Thanks In advance

 

[David Harper CFA FRM]

Hi @patriciar I'll use a binomial to illustrate (this is a Malz example), let's assume $1.0 billion portfolio that contains 50 identical positions ($20.0 million per exposure) where each has a default probability of 2.0% (i.i.d. binomial). In credit risk, we can now refer to two points on the distribution: the 99.0% loss quantile which here is $80.0 million = 4 defaults * $20.0 million where BINOM.INV(50, 2%, 0.990) = 4 defaults is the worst expected number with 99.0% confidence; and the expected loss (EL) of $1.0 billion * 2.0% = $20.0 million.

The unexpected loss, UL(α) = loss quantile(α) - EL; in this case, UL(0.99) = $80 - $20 = $60. Typically the credit VaR (CVaR) is also defined as the UL such that CVaR = UL = quantile - EL. So you are correct about Basel's definition; e.g., see https://forum.bionicturtle.com/thre...k-capital-under-basel-ii-hull.8480/post-75611 including ....

For me, the key is to identify the ultimate, relevant distribution, which can be complexly or simply generated. Once specified, there is the sidebar definitional question of whether CVaR = UL or CVaR = EL + UL; our default in the FRM is to follow Malz et al and define CVaR = UL (i.e., net of expected loss). In the case of a single bond with PD = 2.0%, the EL is 2.0%, but the 0.95 quantile is awkwardly zero. That is, the distribution is zero loss until 0.98 then the 0.02 tail is entire loss; where does the 0.95 fall? It falls in the zero, because it's an odd distribution. So the 95.0% CVaR = 0.95 quantile - EL = zero - 2%. There's an infinite variety of distributions, but once we identify the distribution, it's merely an issue of identifying the 0.9X quantile and the EL. I hope that helps!

However, two things:

• Some authors do define CVaR(α) = UL(α) + EL; in this example, CVaR(0.99) = $60 + 20 = $80 such that this version of CVaR = loss quantile. (Basel itself has done this in the past, actually)
• But we would not say that CVaR(α) = UL(α) - EL. In our example, that would be CVaR(0.99) = 60 - 20 = 40 which is nobody's CVaR in this example. The EL and UL are definitive (unambiguous). We have to be a bit flexible to the fact that some define CVaR to include EL but CVaR = UL is our default. Hope that's helpful,