CVA/CDVA - Worsened Credit Quality implies profit?

Question

In the book Counterparty Credit Risk, Collateral and Funding by Brigo et al I found the following:

• credit quality of investor WORSENS $\Rightarrow$ books POSITIVE MARK TO MKT
  
• credit quality of investor IMPROVES $\Rightarrow$ books NEGATIVE MARK TO MKT

"Citigroup in its press release on the first quarter revenues of 2009 reported a positive mark to market due to its worsened credit quality: “Revenues also included [...] a net 2.5$ billion positive CVA on derivative positions, excluding monolines, mainly due to the widening of Citi’s CDS spreads"

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Now, when trying to price some cashflows $\Pi_B(t,T)$ from the view of a bank (B) to some counterparty (C) we have the following pricing formula (including the CVA and DVA adjustment): $$E[\Pi_B^D(t,T)] = E[\Pi_B(t,T)] - CVA_B(t,T) + DVA_B(t,T)$$ where $$CVA_B(t,T) = \mathbb Q(t<\tau_C<T)\cdot E[LGD_C\cdot D(t,\tau_C )\cdot[-NPV_B (\tau_C )]^+]$$ and $$DVA_B(t,T) = \mathbb Q(t<\tau_B<T)\cdot E[LGD_B\cdot D(t,\tau_B )\cdot[NPV_B (\tau_B )]^+]$$

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Now I am wondering why exactly Citi group is having a positive MtM. If its credit spreads widen, then its default probability $\mathbb Q(t<\tau_B<T)$ increases. Hences the $DVA$ term increases and subsequently the price $E[\Pi_B^D(t,T)]$ which the bank would have to pay increases as well. So the bank is making a loss. I am sure there is a logical error somewhere, but where?

Answer 1

As I see it, the term $\Pi_B(t, T)$ is the value of the derivatives already owned by the bank. So, it's not some price they need to pay but an asset on the balance sheet. This increase in asset value leads to a profit.

Balance sheet Example

Imagine the balance sheet of OTC Subsidiary with rating A:

Assets                    | Liabilities
----------------------------------------------------
OTC derivative + CVA   100| Total Liabilities    100

Now OTC Subsidiary gets downgraded to rating B increasing the CVA by 10:

Assets                    | Liabilities
----------------------------------------------------
OTC derivative + CVA   110| Total Liabilities    100

A balance should balance and liabilities haven't changed (extra collateral wouldn't leave the balance sheet). Hence something should be added on the right side. I believe that how you want to name this quantitiy is mostly an accounting matter but one can choose to put this into the P/L. Putting it in the P/L might not be a really nice solution but it's what often used. Two discussions on this topic I found interesting can be found in Tracy Alloway in Banks face profits hit as fog and OTC Derivatives, Bilateral Trading and Central Clearing by David Murphy, Chapter 3.

Answer 2

The rules relating to mark-to-market accounting have always been, in my opinion, ridiculous.

Citigroup have to mark their liabilities to a fair value, and in this case, where it is their own debt, part of the pricing require that they consider the potential of their own default. The more likely it becomes, that the bank defaults the less the banks swaps is worth to the investors who hold them. The mark-to-market guidelines says Citigroup then should reduce the value of the liabilities on their books, they are allowed to book this difference as a gain/profit.

It is ridiculous because a bank spinning out of control and heading directly into bankruptcy are able to book huge paper profits while it at the same time is melting down.

I do not think you are missing something, except knowing about this one specific mark-to-market accounting practice/guideline.