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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

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If you provided a source for your definition of "credit exposure" and "wrong-way risk", we could probably give an answer more easily.

Credit exposure is not "conditional on default". It basically represents how much a counterparty owes you at a given time $t$.

When you compute the CVA, you usually assume that the correlation between counterparty credit quality and your credit exposure is 0. Wrong and right way risk are basically "corrections" of this assumption.

In case of "right-way" if the counterparty is having a hard time, our exposure to them is likely to be low: the correlation between counterparty credit and exposure is not 0.

For example, you agree with a coal mine to buy some of their coal every month over the next 10 years for a fixed price determined today. The trade is profitable if the price of coal goes up. If the price of coal goes up, the mine is likely to be successful. However if the coal goes down, the mine might shut down because operating costs are simply higher than the price of coal, but if the price of coal went down, your original trade was probably worth not much anymore.

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  • $\begingroup$ The CVA formula above "includes" wrong-way risk because the term is $EE(t_j|t_j =\tau_c)$ , as opposed to the "simple" CVA where the term would be $EE(t_j)$ (i.e. it assumes exposure and credit quality are uncorrelated). $\endgroup$
    – SRKX
    Sep 28, 2015 at 14:58
  • $\begingroup$ My question is what's the difference between the two. $\endgroup$
    – AfterWorkGuinness
    Sep 28, 2015 at 15:03
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    $\begingroup$ The difference is that one assumes exposure has correlation 0 with credit quality, while the other doesn't. $\endgroup$
    – SRKX
    Sep 28, 2015 at 15:31
  • $\begingroup$ What I don't get is how expected exposure is different from expected exposure at time to conditional on t being default time. In the author's definition of exposure above, he says it is conditional on default $\endgroup$
    – AfterWorkGuinness
    Sep 28, 2015 at 16:18
  • $\begingroup$ Is it what you understand from the statement "Credit exposure is the loss in the event that the counterparty defaults"? I think this is to be read as "if the counterparty was to default". (i.e expected exposure averages the exposure across all paths, not only defaulting -> which doesn't change CVA if correlation with credit quality is 0) $\endgroup$
    – SRKX
    Sep 28, 2015 at 16:25

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