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"
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 $$DVA_B(t,T) = \mathbb Q(t<\tau_B<T)\cdot E[LGD_B\cdot D(t,\tau_B )\cdot[NPV_B (\tau_B )]^+]$$
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?