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# Gregory: CVA

#### MSharky

##### New Member
Subscriber

Why is it that CVA is lower for upward-sloping credit spread curves and higher for downward-sloping credit spread curves? Intuitively, as the CDS spreads increase, credit quality of the counter party worsens and so we should charge more to compensate for this increased risk (CVA increases).

#### ShaktiRathore

##### Well-Known Member
Subscriber
Hi,
I think its from the perspective of the counter party.
CVA(credit value adjustment) for firm B=EA*sA-EB*sB ...1)
sA=PD(A)*LGD(A) and E(A): exposure faced by B(firm); E(B): exposure faced by A where A is the counter party
CVA for counter party=EB*sB-EA*sA ..2)
As the credit spread increases the chance/probability(PD(A)) that the counter party shall default increases or that the value of sA increases as the PD(A) increases,assuming everything else remains constant as sA increases ,CVA for counter party decreases the as is evident from formula 2 above.
thanks

#### Kavita.bhangdia

##### Active Member
Hi David,

Gregory makes this statements about CVA:

2. CVA is LOWER for upwards sloping credit curve..WHY?
In my view it should be higher for upward sloping curve because we expect the spreads to increase and hence CVA should increase..

Thanks,
Kavita

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#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @Kavita.bhangdia I agree with you, I am similarly confused by this Gregory argument. (I will email him). First, he writes something very intuitive: "Let us first review the impact of increasing the credit spread of the counterparty in Table 12.1. The increase in credit spread [dh: credit curve = plot of credit spread versus maturity] clearly increases the CVA, but this effect is not linear since default probabilities are bounded by 100%."

Then he writes what I copied below, to your point. I checked his underlying XLS, they did not clarify for me, as they contain neither the table 12.3 nor a side-by-side comparison.

Here is my guess: notice he anchors all of the curves on a terminal spread of 500 bps; in this way, it sounds like his downward-sloping credit curve simply has higher overall starting spreads (?!). This would not be a highly intuitive comparison, in my humble opinion. Gregory (emphasis mine)
"Next, we look at the impact of changes in shape of the credit curve. In Chapter 10 (e.g., Figure 10.8), we considered upwards-sloping, flat and inverted credit curves all of which assumed a terminal 5-year credit spread of 500 bps. We discussed how, whilst they gave cumulative default probabilities that were approximately the same, the marginal default probabilities differed substantially. For a flat curve, default probability is approximately equally spaced whilst for an upwards (downwards)-sloping curve, defaults are back (front) loaded. We show the impact of curve shape on the CVA in Table 12.2. Even though the spread at the maturity of the swap (5Y) is the same in all cases, there are quite different results for the different curve shapes. Indeed, going from an upwards- to a downwards-sloping curve increases the CVA by 11%. We note that for EE profiles that are monotonic, such as forward contracts and cross-currency swaps, this impact is typically stronger (for example, for the case represented in Figure 12.3 the corresponding increase is 40% 15). This illustrates why we emphasised the shape of the credit curve as being an important part of the mapping process (Section 10.1.5)."-- Gregory, Jon (2012-09-07). Counterparty Credit Risk and Credit Value Adjustment: A Continuing Challenge for Global Financial Markets (The Wiley Finance Series) (Kindle Locations 6830-6840). John Wiley and Sons. Kindle Edition.

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#### Unusualskill

##### Member
Hi @David Harper CFA FRM ,

May I know why credit value adjustment will be lower for an upward-sloping credit spread curve compared to downward sloping credit spread curve? Thank you!

#### Nicole Seaman

##### Director of FRM Operations
Staff member
Subscriber
Hi @David Harper CFA FRM ,

May I know why credit value adjustment will be lower for an upward-sloping credit spread curve compared to downward sloping credit spread curve? Thank you!
Hello @Unusualskill

I moved your question here to this thread, which discusses the same question. A quick search of the forum brought up this thread for me.

Thank you,

Nicole

#### georgi

##### New Member
Hi @David Harper CFA FRM ,

May I know why credit value adjustment will be lower for an upward-sloping credit spread curve compared to downward sloping credit spread curve? Thank you!

I think CVA should be lower for an upward-sloping credit spread curve in the first periods of the life of the transaction (where credit spreads on the downward slope curve are still above the upward slope curve) -> in this case marginal default probabilieties would be higher using the downward slope curve and given that CVA is LGDxEExPD(t,t-1) the CVA from downward sloping curve would be higher.

At some point of time both curve will intersect and by then I would expect that the CVA estimated with the upward sloping curve will be higher.

#### nansverma

##### Member
Hi David, I have a couple of questions regarding CVA -
1. Risky Value = risk-free value - CVA. Is CVA here positive or negative ? My understanding is that its positive here as I would pay less for something risky than its risk-free value. Please clarify as Gregory notes shows it as negative later (which is fine as stand-alone value).
2. In CVA formula, expected exposure is discounted. Is it discounted back to MPOR (Margin period of risk)? My question is discounted as of which day ?
3. Why is marginal default probability used in CVA calculation ? If counterparty defaults before t_i, then payment at t_i is also at risk. Should we not be using cumulative default probabilities ?
4. Also from 3. if EE is discounted to MPOR time-frame, then why not consider default probability till MPOR ?
5. Page 143 in study notes - "Incremental EE can be negative, due to beneficial netting effects, which will lead to CVA being negative". Is this a typo ? I would think CVA is positive if EE is negative.

Thank you for your help and sorry for having too questions at one time,

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @nansverma you've asked a 5/6-part question, so I'll really just have to chip away at it as my time permits ... hopefully some others will chime in ...

1. Yes, you are correct. In your version (Gregory switched expressions when he updated his editions, but the substance is always the same, so we don't want to get too attached to +/- sign), CVA is a positive that reduces from the riskfree value to a lesser risky value: you are willing to pay less for something as the exposure to default increases. Further, you are showing the unilateral form which can be viewed as a special case of the bilateral form given by: Risky value = RF value - bCVA = Rf value - (CVA - DVA). Please see https://www.bionicturtle.com/forum/threads/cva-increase-decrease-with-credit-spread.9378/post-74255
Hi @Sujatha sundarji It's not, CVA is subtracted. The key unilateral relationship is given by Gregory's formula 14.1:
• Risky value = Riskfree value - CVA.
Intuitively, say you have a derivative contract with me (as your counterparty) which has a value of \$100.00 when you view me as posing zero counterparty risk to you. Now change the assumption and assume you become worried that I will pay the obligation (if you are in the money). The contract now has less value to you; its value is reduced by the CVA, which is the price of my counterparty risk to you.

As I explain above, we then generalize to the bilateral reality: you may be a counterparty risk to me, which from your perspective is DVA. And the formula generalizes to:
• Risky value = RF value - bCVA = Rf value - (CVA - DVA)
Now if we are equally risky in terms of counterparty risk, that is if CVA = DVA, then bCVA = 0 and then Risky value = Risk-free value because our respective risks cancel each other out. But if you are riskier than me, that if DVA > CVA, then bilateral CVA (bCVA) is positive and would be added. (More detail immediately above). I hope that's helpful,

2. The EE is discounted back to the present value (e.g., Year 0). CVA is a PV that offsets a M2M value.

3. The PDs in CVA are unconditional, not cumulative; the reason is related to above, this is a PV and unconditional PD is from the perspective of today. This is a mistake that GARP made until I explained to them why it needs to be an unconditional PD. Please see discussion at https://www.bionicturtle.com/forum/threads/garp-2017-p2-76.10344/ . Related, I responded to your question here https://www.bionicturtle.com/forum/threads/cva-increase-decrease-with-credit-spread.9378/post-74951

4. Not sure what you mean

5. Yes, you are correct! We picked that up from Gregory's 2nd edition, although he actually meant the same thing (in my interpretation). Please note:
• [2nd edition Gregory] "Incremental EE can be negative, due to beneficial netting effects, which will lead to a CVA being negative and, in such a case, it would be possible to transact at a loss due to the overall gain from CVA." -- Gregory, Jon. Counterparty Credit Risk and Credit Value Adjustment: A Continuing Challenge for Global Financial Markets (The Wiley Finance Series) (Kindle Locations 6896-6897). John Wiley and Sons. Kindle Edition.
• [3rd edition Gregory] "Incremental EE can be negative, due to beneficial netting effects, which will lead to a CVA being positive and, in such a case, it would be a benefit and not a cost." -- Gregory, Jon. The xVA Challenge: Counterparty Credit Risk, Funding, Collateral, and Capital (The Wiley Finance Series) (p. 318). Wiley. Kindle Edition.

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#### nansverma

##### Member
Thank you David. I looked at the other thread and didn't get a clear answer. @Nicole Seaman , I put it here as the other thread discusses CVA as function of credit spread and I thought my questions were different.