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# CVA increase/decrease with Credit spread

#### Kavita.bhangdia

##### Active Member
Hi,
• Gregory ( chapter 12) says that CVA first increases with increase in credit spread but then dips..( table 12.1).please can you explain why does a CVA dips beyond a point? it should be a monotonically increasing function and then flatten out beyond a point. Why the decrease?
• Gregory ( in chapter 12) again says that: CVA is a linear combination of EE, and netting changes only the exposure and has no impact on recovery values, discount factors or default probabilities. The above sentence is not making any sense to me. CVA is a function of recovery value, EE and default probabilities.. So why does it say "no impact" on recovery values etc..
• According to Gregory chapter 12

Risky value = risk free value - CVA.

Okay I am completely missing the point here..shouldn't it be

Risky value = risk free value + CVA?

where am I going wrong?
Thanks,
Kavita

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#### Nicole Seaman

##### Director of FRM Operations
Staff member
Subscriber
@Kavita.bhangdia

Please notice that I have consolidated the three separate threads that you created into this one thread since they were all regarding Gregory, Chapter 12 CVA. This helps to keep our forum organized, and it helps other members to quickly find the answers to their questions when they use the search function instead of having to look through three separate threads. Thank you,

Nicole

#### QuantMan2318

##### Well-Known Member
Subscriber
Ha, the Great Gregory, the master of Counterparty Risk The point we have to understand here is that the Counter party risk deals with our receivables, the obligations of an opposite party to a financial instrument to our organization, therefore, the CVA is an adjustment to reduce the value of our receivables; An Accountant would call it an analogy to the provision for bad and doubtful debts, hence the reduction from the transaction value. (Risky value = Risk free value-CVA). I think the Risk free valuation adopted would be more. Those who work at the appropriate desk may explain this more succinctly.

Next let us see the Credit Spread impact, as the credit spread increases, we can see that both the hazard rates and the marginal default probability changes as a function of the exponent all of which goes into the computation of the CVA as well, we know that the graph of the negative exponent will curve up and then down (concave) and thus will increase and will finally dip. (I am simplifying here)

The formula for spread Marginal default probability is exp^[-spread/(1-R)*t-1]. I think this directly influences the CVA vis a vis the spread.

Another way to see it is the fact that CVA is an adjustment that is made that is roughly equal to the value of protection from a Credit Default Swap. Spread is nothing but the above value divided by the Risky Annuity.
When we simplify the above formula we get Spread = hazard rate * (1-R), therefore, for spread to increase, the hazard rate increases and when hazard rate rises, the marginal default probability rises as function of h*exp(-h*t) which leads to a upwards and then dipping curve. I think Quants (I am not a Quant though my profile name has it) will explain this property better as well.

As far as your points regarding Netting is concerned, I think you are confusing incremental CVA with CVA. CVA is used for standalone transactions. We cannot use it for a Netted set of transactions without making some adjustments. Gregory says that when we compute CVA for a Netted set of transactions is less or equal to the sum of the individual CVAs therefore how do we find what is the CVA contributed to each transaction in the Netted set? presto : find incremental CVA as the difference between CVA(NS+i) - CVA(NS), again we know that Netting affects only the Expected Exposures. It has no effect on the Marginal Probability or Recovery Values. Therefore the equivalent to the above formula is the formula for CVA replacing EE with incremental EE
Incremental CVA = (1-R)*sum(DF * EE_incremental*Marginal Prob)

Hope this helps somewhat

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#### ami44

##### Well-Known Member
Subscriber
• Gregory ( chapter 12) says that CVA first increases with increase in credit spread but then dips..( table 12.1).please can you explain why does a CVA dips beyond a point?

For most instruments will the EE(t) increase with growing t. If you have very high PDs the probability of defaulting at an early time rises, so that there there is not enough time to develop an exposure.
Imagine a swap that is worth zero at inception. For very high hazard rates, a default will occur only a few days after inception. At that time the swap will still not be much more worth than zero. Which means you did not loose much. Since the CVA is your expected loss, it's low in that case.

• Gregory ( in chapter 12) again says that: CVA is a linear combination of EE, and netting changes only the exposure and has no impact on recovery values, discount factors or default probabilities. The above sentence is not making any sense to me. CVA is a function of recovery value, EE and default probabilities.. So why does it say "no impact" on recovery values etc..

CVA is a function of recovery value, EE and PD. Netting has an impact on the EE, because exposures from different positions in a portfolio can set each off.

But recovery value and the probability of default of a counterparty are not effected by the fact if you can net exposures or not.

• According to Gregory chapter 12

Risky value = risk free value - CVA.
Okay I am completely missing the point here..shouldn't it be

Risky value = risk free value + CVA?
where am I going wrong?

That is really just a question how you define the sign of the CVA, isn't it?

Did that help?

#### QuantMan2318

##### Well-Known Member
Subscriber
That is really just a question how you define the sign of the CVA, isn't it?

Did that help?

Hi, @ami44, I have read a couple of your posts. I consider you as one who has both practical and theoretical knowledge of these things. Anyway I have a query here, originally CVA was defined that way right? Risky value = Risk free - CVA as there was no negative sign in the formula for CVA and it was therefore considered to be a mostly positive quantity adjusted as a loss to the Risk free value and hence the negative sign.

However, when CVA is negative as per the original formula (without the negative sign), we may add that to the Risk free value and I thought that Risk free value formula kept the concept clear by making two negatives as positive (-CVA is an income). Now if we change the formula for CVA by having a minus sign and putting Risky value as Risk free + CVA won't it confuse itself with the DVA? as it is DVA that is added to the Risk free value

I also wanted to write about the CVA increasing and then dipping as the spread increase in layman terms; Your explanation for this was precise. I think we may also view it as the value of protection received from the CDS as I mentioned above and that value falls when the counterparty is close to default right?

#### ami44

##### Well-Known Member
Subscriber
Hi Quantman2318,

CVA=protection from CDS
I do not have the Gregory text in front of me, so I apologize if I'm missimg something.
You might be right with your explanation, but I don't understand it. To get protection for the loss from a derivative you would need a cds with a varying Nominal, since the exposure varies with time. Also I don't understand your statement, that the value of a cds would fall if the counterparty is close to default. If I have a normal cds with a fixed nominal, than the value of that cds will always rise with increasing hazard rates or default probabilities.
Or what am I missing here?

Sign of CVA
Traditionally CVA and DVA are both defined as positive. That means CVA is the amount of loss that you expect, and as that has to be subtracted from the risk free value. Accordingly the DVA has to be added.
If you wanted, you could also define the CVA as negative since its a loss and the DVA positive because its a gain. The formula would change to
Risky value = risk free value + CVA + DVA
But to avoid confusion I would not do that though, better stick to the convention.

#### QuantMan2318

##### Well-Known Member
Subscriber
Hi Quantman2318,

CVA=protection from CDS
I do not have the Gregory text in front of me, so I apologize if I'm missimg something.
You might be right with your explanation, but I don't understand it. To get protection for the loss from a derivative you would need a cds with a varying Nominal, since the exposure varies with time. Also I don't understand your statement, that the value of a cds would fall if the counterparty is close to default. If I have a normal cds with a fixed nominal, than the value of that cds will always rise with increasing hazard rates or default probabilities.
Or what am I missing here?

Well, that is based on my own interpretation of the Gregory text. I will explain like this, you see, I worked as a controller for a factory, therefore we speak in terms of costs, CVA is an adjustment for the loss of your counterparty receivable, therefore the closest measure of that loss is the value of a CDS on that counterparty (akin to Opportunity costs) Gregory seems to toe this line of thinking in his text where he says that value of receiving protection from a CDS is (1-R)integral(Risk free discount)*Marginal Prob. Look at the CVA formula (1-R)sum(Risk free discount)*Marginal Prob*Expected Exposure and also look at his rough value of CVA, credit spread*EPE where EPE is EE averaged and what is spread? the above formula for value of CDS/Risky Anuity

Too similar to be ignored right? anyway, I didn't delve much into it in the first explanation given by me above (I restricted myself whatever I have read on CVA ) and again this is just a rough intuition to understand the derivation of various formulae for spreads, hazard and CVA that he has given

And you are right, the value of a CDS increases as the counterparty edges towards default, but I looked at it this way, you take a named CDS from a third party and the value of that CDS rises as the named counterparty edges towards default, however now there is no third party financial institution selling the CDS. You are your own third party and what do you do? your CVA becomes worthless as the counterparty comes more and more to default.

I am sorry @ami44 Please forgive me if I am wrong. That's why I have asked people like you to add to my explain Last edited:

#### brian.field

##### Well-Known Member
Subscriber
We have some smart people on this forum! @QuantMan2318 @ami44 - I am looking at you two! Subscriber

#### ami44

##### Well-Known Member
Subscriber
Quantman2318,

interesting view, never thought about it like that, but now that you said it, it makes sense. The value of the protection of a CDS is the CVA of a constant expected exposure of the amount of the CDS Nominal.
The protection of the CDS is worth the annuity times the CDS Spread (if spread is current market rate).
From that you get easily to the approximate formula CVA(running spread) = Spread(CDS) * EPE

But this all works only, if you approximate the calculation of the CVA by using a constant exposure (given by the EPE). In reality the expected exposure EE is not constant, but varies from current date to maturity. That is why the value of a CDS will always increase with increasing CDS Spread while the CVA of a derivate first increases, but than with very high spreads might fall again a little.
Your approximation CVA(running spread) = Spread(CDS) * EPE will not show that behaviour, but thats because its an approximation, the exact calculation does.

The Probability of default of some fictious protection seller does not seem relevant to me here.
To summarize: you can interpret the value of the protection of a CDS as a CVA on a constant expected exposure, but the other way around is not true. In general a CVA can not exactly be represented by a value of a CDS. It can be used as an approximation though.

Thank you Quantman2318 for this interesting thoughts. I learned something from it.
Also brian, thanks for your kind compliment.

#### maga

##### New Member
Hi David, Nicole. Am I right saying that DVA will decrease when bank's creditworthiness increases (due to for eg better quality of collateral posted, lower PD, narrower credit spread etc)? Likewise, CVA will decrease when counterparty's credit quality increases? Both CVA and DVA will increase as credit quality of counterparty and bank, respectively, decrease? Just wanted to check my understanding. Many thanks, Indre

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
@maga

If we use Gregory's (2nd edition) notation, then the essential relationship is: Risky value = RF value - bCVA = Rf value - (CVA - DVA). It's important to be really careful who is who, but I'll assume "we are the Bank" and we are calculating the Risky value for ourselves such that our credit exposure is the Counterparty. In the unilateral case, of course, an increase in CVA is greater risk to us, and therefore lower a value. In the bilateral case:
• If our own (we the bank) "creditworthiness increases (due to for eg better quality of collateral posted, lower PD, narrower credit spread etc)?" this is a decrease in DVA (aka, we are less risky to the counterparty) and--somewhat counterintutively--this decreases the value of the position from our perspective. This can be seen by the formula above: if our own creditworthiness increases, this is is a reduction in ↓DVA which reduces the risky value where ↓Risky value = Rf value - (CVA - DVA) = Rf value - CVA + ↓DVA. Similarly, a deterioration in our own credit quality increases the DVA and increases the value of the position to us (a higher DVA is added). Because ↑Risky value = Rf value - (CVA - DVA) = Rf value - CVA + ↑DVA. See https://www.bionicturtle.com/forum/...rparty-risk-cva-topic-review.6184/#post-45741
• Re: "CVA will decrease when counterparty's credit quality increases:" Yes correct, if the Counterparty's credit quality increases --> decrease in PD --> lower CVA and the Risky value increases (as a lower CVA is subtracted)
• Re: "Both CVA and DVA will increase as credit quality of counterparty and bank, respectively, decrease?" Yes true, because credit quality decrease implies higher PD and CVA is a function of PD. However, as above, note these ultimately have offsetting effects. (which should be somewhat intutive). In terms of risky value, this describes higher CVA and higher DVA, so that they are offsetting directionally in the Risky Value. I hope that helps!

#### Maiyaa

##### New Member
Well, that is based on my own interpretation of the Gregory text. I will explain like this, you see, I worked as a controller for a factory, therefore we speak in terms of costs, CVA is an adjustment for the loss of your counterparty receivable, therefore the closest measure of that loss is the value of a CDS on that counterparty (akin to Opportunity costs) Gregory seems to toe this line of thinking in his text where he says that value of receiving protection from a CDS is (1-R)integral(Risk free discount)*Marginal Prob. Look at the CVA formula (1-R)sum(Risk free discount)*Marginal Prob*Expected Exposure and also look at his rough value of CVA, credit spread*EPE where EPE is EE averaged and what is spread? the above formula for value of CDS/Risky Anuity

Too similar to be ignored right? anyway, I didn't delve much into it in the first explanation given by me above (I restricted myself whatever I have read on CVA ) and again this is just a rough intuition to understand the derivation of various formulae for spreads, hazard and CVA that he has given

And you are right, the value of a CDS increases as the counterparty edges towards default, but I looked at it this way, you take a named CDS from a third party and the value of that CDS rises as the named counterparty edges towards default, however now there is no third party financial institution selling the CDS. You are your own third party and what do you do? your CVA becomes worthless as the counterparty comes more and more to default.

I am sorry @ami44 Please forgive me if I am wrong. That's why I have asked people like you to add to my explain "You are your own third party and what do you do? your CVA becomes worthless as the counterparty comes more and more to default."--- CVA will decrease in magnitude or become worthless ? interesting thread.

#### Sujatha sundarji

##### New Member
Am still not able to follow this credit value adjustment topic. Can someone explain me why cva is added to the risk free value

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
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, Last edited: #### Sujatha sundarji ##### New Member #### nansverma ##### Member Thanks for the detailed explanation. One comment around marginal default probability- I thought it is defined as {exp(-h*t_i) - exp(-h*t_(i-1))} , i.e. difference of cumulative probabilities for two times instead of just the differential of cumulative default probability. Ha, the Great Gregory, the master of Counterparty Risk The point we have to understand here is that the Counter party risk deals with our receivables, the obligations of an opposite party to a financial instrument to our organization, therefore, the CVA is an adjustment to reduce the value of our receivables; An Accountant would call it an analogy to the provision for bad and doubtful debts, hence the reduction from the transaction value. (Risky value = Risk free value-CVA). I think the Risk free valuation adopted would be more. Those who work at the appropriate desk may explain this more succinctly. Next let us see the Credit Spread impact, as the credit spread increases, we can see that both the hazard rates and the marginal default probability changes as a function of the exponent all of which goes into the computation of the CVA as well, we know that the graph of the negative exponent will curve up and then down (concave) and thus will increase and will finally dip. (I am simplifying here) The formula for spread Marginal default probability is exp^[-spread/(1-R)*t-1]. I think this directly influences the CVA vis a vis the spread. Another way to see it is the fact that CVA is an adjustment that is made that is roughly equal to the value of protection from a Credit Default Swap. Spread is nothing but the above value divided by the Risky Annuity. When we simplify the above formula we get Spread = hazard rate * (1-R), therefore, for spread to increase, the hazard rate increases and when hazard rate rises, the marginal default probability rises as function of h*exp(-h*t) which leads to a upwards and then dipping curve. I think Quants (I am not a Quant though my profile name has it) will explain this property better as well. As far as your points regarding Netting is concerned, I think you are confusing incremental CVA with CVA. CVA is used for standalone transactions. We cannot use it for a Netted set of transactions without making some adjustments. Gregory says that when we compute CVA for a Netted set of transactions is less or equal to the sum of the individual CVAs therefore how do we find what is the CVA contributed to each transaction in the Netted set? presto : find incremental CVA as the difference between CVA(NS+i) - CVA(NS), again we know that Netting affects only the Expected Exposures. It has no effect on the Marginal Probability or Recovery Values. Therefore the equivalent to the above formula is the formula for CVA replacing EE with incremental EE Incremental CVA = (1-R)*sum(DF * EE_incremental*Marginal Prob) Hope this helps somewhat #### David Harper CFA FRM ##### David Harper CFA FRM Staff member Subscriber @nansverma Cumulative PD = 1 - exp(-λ*t) where λ is hazard rate. The unconditional PD (aka, joint) is the difference between cumulative PD: Unconditional PD = 1 - exp(-λ*t2) - [1 - exp(-λ*t1)] = exp(-λ*t1) - exp(-λ*t2), is maybe what you are thinking of. Say λ = 10%, to be dramatic. The unconditional PD (during third year) = exp(-λ*3) - exp(-λ*4) = 7.05%. I realize some call this a marginal PD, but I'm not crazy about that term b/c Malz defines marginal PD as the instantaneous rate of change, λ*exp(-λ*t), in this example which goes from 0.10*exp(-0.30) = 7.41% (at beginning of 3rd year) to 0.10*exp(-0.40) = 6.70% (at end of third year), and is a variable that seems to be far less used. Thanks, #### nc27 ##### Member Subscriber Hi @David Harper CFA FRM , I have two doubts that bothered me for a bit with respect to this topic.... say I have a derivative instruments portfolio (and netting condition as per my ISDA) composed of european options and irs swaps..... say that my exposure at the time of valuation represents an asset (+100$ in total, but then, individually, some intruments represent liabilities and others represent assets). Now, I am a risky company (rated bbb) and my financial counterparty is well rated (aa+) so that we will observe differences in cds spreads.

1) First doubt, is it possible to get a negative BCVA on the portfolio (DVA > CVA) ?

2) Second doubt, how can we interpret, from an accounting perspective, the fact that the bilateral adjusment might increase the company asset value:

-> for example if my bilateral adjustment on the portfolio is -10$then I will get risky value = risk free value - BCVA =$100 - (- $10 ) =$ 110.

If my above reasoning is consistent, how companies take into account those bilateral adjustments? (as required by IFRS 13)
the thing is, it would implies that a more risky companie could take advantage of a lower credit rating as the value of its assets might be artificially inflated.

I really hope you can help me on this one,
Thanks you!

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