I recently read chapter 14 of Gregory's *The xVA Challenge*.
He defines CVA as (formula 14.2)
$$
CVA = -LGD \cdot \sum\_{i = 1}^m EE(t\_i) \cdot PD(t\_{i-1}, t\_i),
$$
where $LGD$ is the Loss Given Default, $EE$ is the expected exposure and $PD(t\_{i-1}, t\_i)$ is the probability of default between times $t\_{i-1}$ and $t\_i$.

According to this definition CVA is always non-positive. But in equation (15.1) Gregory writes that
$$
\text{Risky value} = \text{Risk-free value} - CVA.
$$
For me this looks wrong: as the risky value should be less than the risk-free value and CVA is negative, there should be a plus sign in the equation above.
I already checked the [errata of this book](https://cvacentral.com/books/credit-value-adjustment/errata/) but this issue is not mentioned.

To illustrate this, assume that we have we have entered into a (plain vanilla) interest rate swap with a counterparty several months ago. The current risk-free value (i.e. the value without considering default risk) is given by EUR 100M.
Assume also that the CVA is EUR -10M. According to my understanding the value of the swap should than be EUR 90M = 100M - 10M.

What do you guys think?

**References**

Gregory, Jon. The xVA Challenge: counterparty credit risk, funding, collateral and capital. John Wiley & Sons, 2015