Credit exposure defines the loss in the event of a counterparty defaulting, and expected exposure is the average of all credit exposures.
BUT
When adjusting the CVA calculation to account for wrong-way-risk we replace expected exposure with expect exposure at time T* conditional on this being the counterparty default time.
If exposure is already conditional on default, how is this any different?
EDIT
Here are the definitions of credit exposure and wrong way risk from the John Gregory's book as requested:
Wrong-way risk is used to indicate an unfavorable dependence between exposure and counterparty credit quality
Credit exposure is the loss in the event that the counterparty defaults. Credit exposure is characterized by the fact that a positive value of a financial instrument corresponds to a claim on a defaulted counterparty
To characterize exposure we need to answer two questions:
- What is the current exposure (the maximum loss if the counterparty defaults now)?
- What is the exposure in the future (what could be the loss if the counterparty defaults at some point in the future)? This second question is far more complex to answer
Source text in question (emphasis mine):
The presence of wrong-way risk will (unsurprisingly) increase CVA. However, the magnitude of this increase will be hard to quantify, as we shall show in some examples. Wrong-way risk also prevents one from using the (relatively) simple formulas used for CVA in Chapter 12.We can still use the same CVA expression as long as we calculate the exposure conditional $$CVA \approx (1 - Recovery) \sum\limits_{j=1}^MDF(t_{j})EE(t_{j}| t_{j} = \tau_{c})PD(t_{j-1},t_{j})$$
where $EE(t_{j}| t_{j} = \tau_{c})$ represents the expected exposure at time $t_{j}$ conditional on this being the counterparty default time ($\tau_{c}$). This replaces the previous exposure, which was unconditional. As long as we use the conditional exposure, everything is correct.upon default of the counterparty. Returning to equation (12.2), we simply rewrite the expression as