CVA

[Mish]

David,

Study Notes of Credit Risk - Q39.7.1 on pg 183.
If adjusted mid-market value to B is +11 million, shouldn't adjusted mid-market value to A is -11 million? The solution on page 187 reads 9 million.

Thanks,
Michelle

 

[David Harper CFA FRM]

Hi Michelle,

Yes, absolutely, my typo, the value to A should be -11 not +9. Apologies (answer given is correct, but that phrase is incorrect!). I have tagged this for edit-fix. Thank you!

 

[Hardlearner]

Hi David,

according to Hull: If a market participant expects to be a net PAYER of collateral the formula is:
f = fnd - CVA + DVA -CRA

Question 1:
If a market participant expects to be a net RECEIVER of collateral the formula is:
f = fnd - CVA + DVA +(!) CRA
Is this correct?

Question 2:
Regardless of payer / receiver the formula for calculation CRA is always:
CRA = fnd[r(c)] - fnd[r(f)]

Thank you in advance
Best regards

 

[David Harper CFA FRM]

Hi @Hardlearner

It's a new reading so I'm not (candidly) not fully seasoned myself with it, but I am pretty confident the answer to both questions is yes. I
*assume* Hull's CRA is the same FCA+FBA (funding cost and benefit), but regardless this looks to correspond to Gregory's:

Interest will typically be paid on cash collateral at the overnight indexed swap (OIS 5 ) rate (for example, EONIA in Europe, Fed Funds in the US). The logic behind this is that since collateral may only be held for short periods (due to the variation of exposure), then only a short-term interest rate can be paid. However, OIS is not necessarily the most appropriate collateral rate, especially for long-dated exposures where collateral may be held in substantial amounts for a long period. This may lead to a negative carry problem due to an institution funding the collateral posted at a rate significantly above LIBOR but receiving only the OIS rate (less than LIBOR) for the collateral posted. Sometimes, a collateral receiver may possibly agree to post a rate higher than OIS to compensate for this funding mismatch. Another reason for a collateral giver [david note: I think typo, he must mean "receiver" here] to pay a return in excess of OIS would be to incentivise the posting of cash over other more risky and volatile securities. -- Gregory 5.2.7

So f = f(nd) - CFA + DVA - CRA is true:

• if the market participant, who posts collateral, is paid (by collateral receiver) a collateral rate less than the rate required to fund the collateral ("economic rate"), positive CRA reduces (f); i.e., reduces the net value of the derivative to the market participant. This is Gregory's "negative carry problem due to an institution funding the collateral posted at a rate significantly above LIBOR but receiving only the OIS rate (less than LIBOR) for the collateral posted."
• if the market participant is paid more than the funding cost of the collateral (economic rate), negative CRA would appropriately increase the net value of (f). So f = ... - (-CRA) = ... +CRA makes sense.

Re: CRA = fnd[r(c)] - fnd[r(f)], yes this looks to work for both counterparties.

• If r(c) < r(f), then f(nd)[r(c)] > f(nd)[r(f)] ... lower discount rate increases present value ... and this is positive CRA, consistent with an decrease in overall value to the "market participant" who is paid a daily rate on the collateral, r(c), which is less than the "economic rate," in this case, r(f)
• Conversely, r(c) > r(f), then f(nd)[r(c)] < f(nd)[r(f)] ... and negative CRA contributes an increase in derivatives value. I hope that helps!

 

[Hardlearner]

Hi David,
thank you very much for your detailed answer! I'm very glad, that you had time to review my thoughts. Unfortunately I have the following Point:
You write:
"if the market participant is paid more than the funding cost of the collateral (economic rate), negative CRA would appropriately increase the net value of (f). So f = ... - (-CRA) = ... +CRA makes sense."

Yes, but to my mind this describes the position of the participant, who is payer of collateral (he posts collateral) in the transaction. When he receives more than the economic rate from the receiver of his collateral then CRA is negative and the value of his position increases, due to -(-CRA)=+CRA.

But I had the position of the RECEIVER of the collateral in my mind. The person who has a claim against the payer of collateral.

Can the value of his position calculated like this: f = fnd - CVA + DVA +(!) CRA ?

Case 1:
When the RECEIVER pays less than the economic rate, than CRA is positive and the value of his position increases.
Case 2:
When the RECEIVER pays more than the economic rate, than CRA is negative and the value of his position decreases.

So it seems the formula for the receiver of collataral (the person who has a claim..)

f = fnd - CVA + DVA +(!) CRA

is correct. Do you agree?

Thanks a lot!

 

[David Harper CFA FRM]

Hi @Hardlearner

I sort of agree with this because the collateral rate and the economic rate do not depend on our counterparty perspective (these rates are true for both, although only one of the counterparties has a funding/economic rate to pay):

Case 1:
When the RECEIVER pays less than the economic rate, than CRA is positive and the value of his position increases.
Case 2:
When the RECEIVER pays more than the economic rate, than CRA is negative and the value of his position decreases.

i.e., if (collateral) receiver pay less than economic rate, his/her value increases

But we can't then insert this one term (from the receiver perspective) into a formula which applies to the receiver's counterparty. As mentioned before, I *think* CRA = funding value adjustment (FVA) which parses into a funding cost and benefit (FBA and FBA), such that Hull basically maps to BCVA. If this assumption is true, then from the perspective of the market participant (collateral "giver" if you like):

• f = f(nd) - CVA + DVA - CRA has the corresponding value to the counterparty (the "receiver) of:
• f = (-) [f(nd) - CVA + DVA - CRA] = -f(nd) + CVA - DVA + CRA; to your point about CRA, however, this is awkward because CVA represents a "unilateral" price of the other's counterparty's risk and DVA represents the price of the institution's own perspective. So, if we want to switch our perspective from the "giver" (participant, institution) to the "receiver," my thought is we'd want to follow Gregory and expend the CRA into its compents. Let CRA = funding value adjustment (FVA) = funding cost adjustment (FVA) - Funding benefit adjustment (FBA)
  
• f = f(nd) - CVA + DVA - CRA = f(nd) - CVA + DVA - (FCA - FBA); and now this formula can be used for the receiver without disputing your two cases above

Please note I this is my tentative assumption (I haven't had time to confirm), but you can see that so far I basically agree with you, I just think it's a tricky matter of "which counterparty perspective?" ie.,

If we have an institution (I) and its counterparty (C), then the formula from the perspective of the institution is:

• f = f(nd) - CVA(I) + DVA(C) - CRA, which under identical model/parameter assumptions implies
• value to the counterparty = -f(nd) + CVA(I) - DVA(C) + CRA, but that's awkward mix of perspectives and because from the counterparty's perspective DVA(C) is their own CVA and CVA(I) is their own DVA, so the counterparty still uses:
  
• value (counterparty) = f(nd) - CVA + DVA - (FCA - FBA)

 

[Hardlearner]

Hi David,

thank you very much for your detailed explanation. I appreciate very much that you spend so much time to make things clear. Especially the reference to Gregory with CRA=FCA-FBA helps me to memorize that CRA is positive for the market participant who is collateral payer (he posts collateral) when the rate paid on collateral (his benefit) is less than the economic rate for that collateral (his costs, eg. funding costs in Gregory).

My very last question concerning that topic:
According to Hull p. 13: CRA:=fnd(rc) –fnd(rf)
Then he states: “With the assumptions in a) to e) above, CVA=DVA=0 and the value of the portfolio in equation (2) simplifies to fnd(rc).”

Economically I can get his conclusion, but when I put CRA:=fnd(rc) –fnd(rf) into equation (2)

f = fnd –CVA + DVA – CRA
f = fnd –CVA + DVA –[(fnd(rc)-fnd(rf)] and CVA=DVA=0
f = fnd – fnd(rc) + fnd(rf)

But according to Hull it simplifies to fnd(rc)!

What is my error?

Thanks a lot!

 

[David Harper CFA FRM]

Hi @Hardlearner

Sure thing, well, I think he's just showing the case where CRA = 0, and f = fnd = fnd(rc) = fnd(r); i.e., the original fnd is implicitly the risk-free rate (OIS, to the theme of the paper). The reason I assume that is:

To examine the possible impact of cash collateral consider an idealized situation where the following is true
a) There is a two-way zero-threshold collateral agreement. This means that, when the valueof the outstanding derivatives portfolio to one side is X, the other side is required to post max(X, 0) as collateral.
b) There is no minimum transfer amount and collateral is transferred continuously.
c) All collateral must be in the form of cash.
d) Collateral is always posted right up to the time of a default.
e) Transactions can be replaced at their mid-market value at the time of a default.
In this situation, the collateral is a perfect hedge for losses due to default so that CVA and DVA are both zero. Because the credit risk has been hedged, the collateral investment is risk-free and should earn the risk-free rate of interest. If the interest paid on cash collateral is the risk-free rate, the CRA is zero.

i.e., Under true assumptions (a) to (e), he seems to setup f = fnd - 0 + 0 - 0 = fnd, with rf=c, such that fnd = fnd(c) =fnd(rf)

• Notice that he introduces r(c) - r(f) by "Suppose next that the rate paid on cash collateral is different from the risk-free rate." ... a difference that warrants CRA, without this difference, CRA was not needed
• So, maybe I miss something but I think under true (a) to (e), rf = c, such that fnd = fnd(rf) = fnd(c); or

 

[Hardlearner]

Hi David,

thanks agin for your help and that you pulled me back on the right way:
Hulls statement "With the assumptions in a) to e) above, CVA=DVA=0 and the value of the portfolio in equation (2) simplifies to fnd(rc).” only belongs to the case where CRA=0 due to rf=rc.

Due to the fact, that the statement above follows the case where Hull describes rc<>rf I thought, that the statement belongs to this case. But this was my fault.

So thanks a lot for correcting me!