Credit Exposure

[Stuti]

Hi, it is mentioned in the Credit Exposure chapter that Exposure is proportional to T^(1/2). How do we know this result?

 

[David Harper CFA FRM]

Hi @Stuti

It's the same square root rule that let's us scale volatility/VaR per sqrt(ΔT) but therefore importantly it is conditional on the same i.i.d. assumption, namely that the returns of the credit risk factor are independent. Further, this is the simplest possible credit exposure profile; i.e., a single i.i.d. credit risk factor. For example, in a swap, sqrt(ΔT) tends to characterize the early diffusion effect (although, even here, we know the interest rate risk factor is not i.i.d. as mean reversion is not auto independence) but the later dates are characterized by "amortization effect" which is separate. So, exposure proportional to sqrt(ΔT) is very specific to only the i.i.d. factor(s) involved. I hope that helps!

 

[nansverma]

One more thing : Gregory mentions exposure being proportional to sqrt (T). But isnt that the case when we talk about volatility or VaR(PFE) ? If we are just talking about expected exposure, then that would be proportional to just T ?

 

[Nicole Seaman]

One more thing : Gregory mentions exposure being proportional to sqrt (T). But isnt that the case when we talk about volatility or VaR(PFE) ? If we are just talking about expected exposure, then that would be proportional to just T ?

Hello @nansverma

Please note that I moved your question here to this thread that already discusses this. There is over a decade of FRM discussions in the forum so it is helpful if you use the search or tag function to see if your questions have already been answered. This saves you time so you don't have to wait for an answer. It also saves David time so he isn't answering the same questions more than once. As the exam gets closer, the forum will be very busy and we want to make sure everyone gets the answers to their questions If David's answer above doesn't answer your question, feel free to let us know and someone can elaborate further.

Thank you,

Nicole

 

[nansverma]

Thank you Nicole. I will keep in mind for future. Since my question was from the part just few pages after my initial question in the study notes, I ended up asking in the same thread. Having said that, yes my question is kind of answered as David's answer is what I expected to be the reason. In the study notes, sqrt(T) rule follows right after expected exposure values, so I was confused expectation is not proportional to sqrt(T).

 

[nansverma]

Hi David, In gregory notes about expected exposure etc. - In Fig 7.14, EFV of a payer swap is always negative (seems like return %). First, I wanted to confirm EFV is just the expected value of all possible values , which is equivalent to fair value of the swap, is that correct ? If yes, is it correct that the fair value is always negative for all the future time (as calculated today) ? Thank you

 

[David Harper CFA FRM]

Hi @nansverma In Gregory's Figure 7.14 (where the lines are not very well distinguished), I think for this illustrated payer (i.e., pay fixed, receive floating) interest rate swap the EE is the upper, positive line, the EFV is the middle (less positive) and the NEE is the negative line. Due to the importance of the "expected future value," I have copied it from Gregory below. But you are basically correct to say EFV is "the expected value of all possible values , which is equivalent to fair value [future M2M] of the swap." Realistically, it will be the average of Monte Carlo simulated future values. In the case of this swap, it is heavily influenced by the forward interest rate curve. However, it is not necessarily negative or positive because, if both counterparties use the same simulation (assume they are at least roughly similar), a positive EFV for one counterparty will be a negative EFV for the other!

This is unlike the EE and NEE, which are not thusly "symmetrical." Recall the essential counterparty formula: Exposure = max(value, 0) such that negative exposure = min(value, 0). EE and PFE must be positive because they are, respectively, the average and conditional quantile of only positive values (the exposure part of the distribution); similarly, NEE must be negative. I hope that's helpful!

Gregory (emphasis mine):

7.2.1 Expected future value: This component represents the forward or expected value of the netting set at some point in the future. As mentioned above, due to the relatively long time horizons involved in measuring counterparty risk, the expected value can be an important component, whereas for market risk VAR assessment (involving only a time horizon of ten days), it is not. Expected future value (EFV) represents the expected (average) of the future value calculated with some probability measure in mind (to be discussed later). EFV may vary significantly from current value for a number of reasons:

• Cashflow differential. Cashflows in derivatives transactions may be rather asymmetric. For example, early in the lifetime of an interest rate swap, the fixed cashflows will typically exceed the floating ones, assuming the underlying yield curve is upwards-sloping as is most common. Another example is a cross-currency swap where the payments may differ by several per cent annually due to a differential between the associated interest rates. The result of asymmetric cashflows is that a party may expect a transaction in the future to have a value significantly above (below) the current one due to paying out (receiving) net cashflows. Note that this can also apply to transactions maturing due to final payments (e.g. cross-currency swaps).
• Forward rates. Forward rates can differ significantly from current spot variables. This difference introduces an implied drift (trend) in the future evolution of the underlying variables in question (assuming one believes this is the correct drift to use, as discussed in more detail in Section 10.4). Drifts in market variables will lead to a higher or lower future value for a given netting set, even before the impact of volatility. Note that this point is related to the point above on cashflow differential, since some or all of this is a result of forward rates being different from spot rates. Asymmetric collateral agreements. If collateral agreements are asymmetric (such as a one-way collateral posting) then the future value may be expected to be higher or lower reflecting respectively unfavourable or favourable collateral terms. More discussion on the impact of collateral terms is given in Chapter 11." -- Gregory, Jon. The xVA Challenge: Counterparty Credit Risk, Funding, Collateral, and Capital (The Wiley Finance Series) (pp. 114-115). Wiley. Kindle Edition.

 

[nansverma]

Hi @nansverma In Gregory's Figure 7.14 (where the lines are not very well distinguished), I think for this illustrated payer (i.e., pay fixed, receive floating) interest rate swap the EE is the upper, positive line, the EFV is the middle (less positive) and the NEE is the negative line. Due to the importance of the "expected future value," I have copied it from Gregory below. But you are basically correct to say EFV is "the expected value of all possible values , which is equivalent to fair value [future M2M] of the swap." Realistically, it will be the average of Monte Carlo simulated future values. In the case of this swap, it is heavily influenced by the forward interest rate curve. However, it is not necessarily negative or positive because, if both counterparties use the same simulation (assume they are at least roughly similar), a positive EFV for one counterparty will be a negative EFV for the other!

This is unlike the EE and NEE, which are not thusly "symmetrical." Recall the essential counterparty formula: Exposure = max(value, 0) such that negative exposure = min(value, 0). EE and PFE must be positive because they are, respectively, the average and conditional quantile of only positive values (the exposure part of the distribution); similarly, NEE must be negative. I hope that's helpful!

Gregory (emphasis mine):

Thank you once again !