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# Calculation of adjusted exposure

#### liewpw05

##### New Member
Dear David,

As i was going through Michael Ong's Chapter 4 on expected loss, i have come across the following formulae
Adjusted exposure on default = Outstanding + Usage Given Default * Commitments
However in Table 4.2, Adjusted exposure = Outstanding + Usage Given Default *(Outstanding-Commitments).I am not sure which is correct.

A second related question is the definition of Commitments and the drawn & undrawn portions of the Commitment.The reading goes on to say the drawn portion of the commitment is considered as outstanding. could you also help to clarify this.

As to my understanding there is two parts to the commitment :

Risky Asset : Usage Given Default * Commitment
Riskless Asset : (1-Usage Given Default)*Commitment

Thanks

Regards
Peggy

Peggy

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi Peggy

They are related, your second leads to your first. In this framework, the cash can only be in one of two places: the bank has loaned in to the borrower (outstanding; the borrower has it) or the bank has yet to loan but has promised the cash is available to be borrowed (committed). At any given time, everything is covered by OS + COM, and therefore, Ong's correct is: AEAD = OS + (UGD)(COM).

In the example, it is maybe a bit of semantic confusion: the facility starts with $10 COM at time zero. At that point, prior to the question, the AED = (EGD)(COM). Then, the borrows draws down$5 MM. Now there is $5 MM OS (to your second, the cash leaves the bank, it is now outstanding) and the commitment has$5 MM remaining. So, it is really the commitment at the point in time, for the relevant quantity is $5 OS +$5 COM, not $5 OS +$10 COM.

another way to look at this is that the adjusted exposure is giving a multiple of 1.0 to outstanding because it has "left the bank," borrower has it, it is all exposure. And a multiple of UGD to any PROMISED (but not yet left the bank) Exposures because it has not left the bank, so:

(1)(OS) + (UGD)(COM) = AE, where OS + COM must cover the entire asset that either has been lent or may be lent due to promise (commitment).

Why go to this mathematical trouble? To parameterize the only random piece here. At any given time, the OS is known and the "remaining commitment" (see how that would clarify the question, if 4.2 said remaining commitment = \$5?) is known also. But the unknown (random) piece is "how much of the promise will be drawn" (= UGD * remaining commitment), so UGD can be modeled as a random variable. You can see by his language he suggests it can be modeled as a call option.

yes, your final eqs are correct. it is just a way to slice the promise into two pieces so the one piece counts as adjusted exposure.

Hope that helps! David

#### liewpw05

##### New Member
David,

That was a great help as the same example is extended to the next chapter on portfolio effects on expected and unexpected loss.

By the way, while reading the Chapters 5 & 6 of Ong, i noticed that a few AIMS require the candidate to "derive mathematically a certain formulae " my first impression is to just skip this since it will not be tested in the FRM multiple choice questions. What's your thoughts on this?

Thanks a lot

Regards,
Peggy

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi Peggy,

That's observant of you. I totally agree. The verb "to derive" was nowhere in last year's exam; it shows up in Credit and once in Culp commodity. My current hunch is that they are mistaken AIMs. I don't see how they can become test questions, so i'd not worry greatly about these (only b/c "to calculate" UL, etc is already a separate AIM, so they are not "to calculate")...I have a batch of problematic AIMs going to GARP (b/c this has troubled me too in prepping the credit risk; you probably noticed there are many grammatically flawed AIMs and some cuttoffs and some are vaguish...oh and there are maybe a half dozen redundancies...not a fine performance!), i'll let you know when i get f/back

David

#### liewpw05

##### New Member
Hi David,

In the Ong's reading on Loan Portfolio and Expected Loss, the calculation of the adjusted exposure includes the term UGD*Commitment where UGD= Usage given default or % drawdown.

Ong says " In the event of default,an unrestricted committment has the same risk exposure as a term loan of UGD*Commitment"

Does this mean under the normal course of doing business, the firm has access to 100% of the unrestricted commitment but in the event of financial distress, this ability is restricted to UGD% as a means for the bank to mitigate against credit losses?I am inferring this from the reading but not sure if the understanding is correct.

Thanks

Regards
Peggy

#### liewpw05

##### New Member
Hi again David,

I think my observation above on the UGD may be flawed. Further down the reading, i noticed that the UGD is actually an option given to the borrower by the bank in exchange for a option premium for the privilege to draw on the unused commitment in the event of financial distress. The UGD is hence not known in advance and has to be estimated from historical data. What's your thoughts on this?

Regards,
Peggy

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi Peggy,

I agree with your second point but I think your first is pretty close, too. About: â€œIn the event of default, an unrestricted committment has the same risk exposure as a term loan of UGD*Commitmentâ€ I think it is not about the maxmium possible exposure, which ex ante default, is the entire commitment (100% * commitment) but rather the expected "risky" exposure. So, elsewhere he make two points to support the idea that the risky asset (the adjusted exposure) is less than the full COM + OS. One, he says empirically we just observe the COM is not fully drawn at default. Two, he says we should analytically assume that the banks convenants are triggered to the banks protection (e.g., which to your point #1 could absolutely include the bank's option to reduce the committment; a sort of bank's option compounded on the borrower's option, if you will). So, unused committment hasn't left the bank's vault yet, so UGD*Unused Commitment is an *estimation* of what the borrower will draw down and just, probabilistically, it won't be 100%.

But I take your second to be indeed the theme of Ong's reading: "UGD is actually an option given to the borrower by the bank in exchange for a option premium" i.e., borrower pays the premium (fee) for the right but not the obligation to drawn down the commitment (and i can totally see GARP testing it this way!). And casting it in optionality terms highlights that it "is hence not known in advance" (in fact, I first thought, why not use an option pricing model to estimate the UGD, but I bet the variables around UGD are just too squishy). He says the UGD is mapped to credit rating; I didn't know that and his data looks pretty weak...so, yea, these are estimate where I bet some banks have better internal methods based on their own experience.

For me, where Ong is consistent with other readings is that, of all the params (EL, PD, UL), the most difficult to either parameterize (represent by a distribution ) or estimate based on historical are LGD and UGD.

David