# sign of CVA (Credit Value Adjustment)

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 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

This was originally meant as a comment but was too long to be considered as such.

It's all a matter of convention but I would agree with you that there is a sign problem.

If you look at it from the perspective of the desk which makes the trade, CVA constitutes a cost (hence a negative amount) which, all other things equal, decreases the overall value of your portfolio once factored in.

To illustrate this, suppose you enter an uncollateralised long option trade with a counterparty. Further assume that this option has a positive payout in all states of the world, i.e. a positive premium (e.g. long call).

What happens then is that you are paying a certain amount of cash to the counterparty, in exchange for a promise of it delivering you the payout at the contract's expiry. You are thus exposed to the risk of that counterparty's defaulting on it's obligation, i.e. credit risk.

Say the option value is 100\$in the market ("risk-free" value, in the sense: not counterparty dependent). Using the above formula, you compute a CVA of -10\$. This number is negative since it is what you expect to lose should the counterparty default, i.e. what you won't get back on what it promised you at inception.

Assuming you pay the "risk-free" value for the option to the counterparty, your books accounting will most likely show the following new lines:

• Short 100\$cash (premium paid) • Long 100\$ worth of option ("risk-free" value of the deal)
• CVA cost of -10\$(charge for credit risk, dependent on the CTP) Such that you actually "lose" 10$ when the trade is made, due to the unprovisioned/unhedged credit risk you've just added to your books.

Knowing this, suppose you instead agree to pay to the counterparty a premium equal to "risk-free value + CVA", which is more in line with the real economic value of the trade from your perspective. You would then have:

• Short 90\$cash (credit risk adjusted premium) • Long 100\$ worth of option ("risk-free" value of the deal)
• CVA of -10\$(CVA charge of the deal) so that you end up being neutral once the deal is made. This is usually how this happens in practice since there exists a separate XVA desk. Put differently, the desk which makes the option trade (e.g. an equity derivatives desk) values that trade at its "risk-free" value. However, it needs to adjust the premium quoted/paid for credit risk, in the form of a credit value adjustment, which is computed by a dedicated XVA desk, who shall later be in charge of managing the credit risk transferred to it. You should ask more from your counterparty to compensate for the counterparty credit risk that you will assume by entering a trade with it. For an identical trade, the price that you would ask a very risky counterparty to pay should be greater than the one you would ask a very solid counterparty to pay. Regarding the sign it is a matter of preference, you can of course compute the CVA as a negative value and then substract it from the price, but the convention is to compute the CVA as a positive quantity, and add it to the price. • Thank you for your answer. This were true if by "value" the author meant "price which I charge the counterpart". But usually value is the market-value, no? – Cettt Nov 12 '19 at 15:24 • With the CVA, there is no unique price anymore, as the price will depend on the counterpart. So, price means the price for which you will agree to enter into the trade with the counterparty. – byouness Nov 12 '19 at 18:08 • Assume I entered into an interest rate swap with my counterparty and the risk-free value (or market value) is +100$ (which means that the counterparty would have to pay me 100\$if they want to terminate the contract) and the CVA is -10\$. Then I would argue that the value of that swap is only 90\\$. The price which I agreed to enter the swap is irrelevant, isn't it. – Cettt Nov 13 '19 at 8:03
• To the 100M you have to add 10M to account for the fact that your counterpart might default during the life of the swap. The 10 will become zero if your trade is with a (theoretical) counterpart that cannot default ever and it gets bigger the more likely to default your counterpart is. For an IRS usually you impact the CVA on the rate or margin instead of the market value. So, instead of paying Libor to receive 1% e.g. you would receive 1.02% or 1.03% etc. – byouness Nov 13 '19 at 8:53
• So if I trade with a counterparty who can default the value is 110. But if the counterparty cannot default the value is 100. By this logic, in order to increase value one can just look for counterparties which are guaranteed to default and only trade with them. – Cettt Nov 13 '19 at 9:32