CVA using difference between 2 counterparty's spreads

Question

The approximation to calculate CVA as a spread is $CVA = Spread * Expected$ $Exposure$. I assume this means the counterparty's spread over a proxy for the risk free rate such as LIBOR or OIS. Is this an incorrect understanding?

I came across an FRM exam practice question created by GARP (the entity that runs the exam) which gave a scenario where both the bank and its counterparty experienced spread widenings due to rating downgrades. Where the banks spread went from 20bps to 150bps and its counterparty's spread went from 130bps to 170bps. The CVA a was given by (170-150)*Expected Exposure and was a reduction from the previous CVA of (130-20)*Expected Exposure. (Note, there is an assumption to ignore DVA)

This doesn't make sense to me. The counterparty to the bank suffered a downgrade and a resulting spread widening yet the CVA charge decreases?

I can't share the exact question, but here is essentially what it was

This was the question:

Foo Company is a frequent user of swaps with Acme Bank. Acme bank was recently downgraded from a rating of AA- to A+ and at the same time, Foo Company was downgraded from A to A-. As a result of the downgrades, the credit spread for Acme bank increased from 20bps to 150bps, and the credit spread for Foo Company increased from 130bps to 170 bps. The question asks what realistic changes the counterparties can request to their CVA charge, given a set of multiple choice.

The given answer was:

Because Foo Company has a lower credit rating than the bank, it would typically pay a CVA charge which would be a function of the relative credit spreads between the two entities. As the the credit spread between the two has narrowed after the downgrade to only 20bps, Foo Company can request a reduction in their CVA charge.

Answer 1

This is an oft-debated topic among CVA/DVA professionals at banks. The key question, as pointed out by one of the comments, is whether a bank can derive some type of benefit from the increase in its own credit spread (and thereby make less of a CVA charge on the proposed transaction). The two sides to the argument are (a) on the one hand, surely by symmetry there should be a zero CVA charge on a transaction between two counterparties of identical credit (assuming a transaction that is itself symmetric). This side would argue for the 20bp charge since both spreads should be taken into account (b) on the other hand, the only way that a counterparty can benefit from its own credit spread is by defaulting, which is not worth paying for. Hence one should price simply off the other side's credit spread.

Personally I prefer the former approach from a theoretical standpoint, but as I said there is considerable variation among banks as to the approach taken. Overall I would concur that a counterparty should request the reduction in CVA charge-after all, they have increased exposure to the bank defaulting.

Answer 2

What GARP considered for the question is a first-to-default CVA, meaning that the bank looks at what it expects to lose if its counterparty defaults before it does. In this case the formula would be: $\mathrm{CVA}=\left(Spread_{Counterparty} - Spread_{Bank} \right) \times EE$

In a sense, the bank "does not care" about what happens after it defaults. So, all other things bein equal, the likelier the bank's default, the lesser its CVA with the counterparty.