# CVA as a running spread - risk annuity calculation in the Monte Carlo framework

I have simulated future term structures in the one-factor Hull-White model and calculated the CVA of a particular trade (let's say, now I have it in absolute value, in dollars). However, I want to represent this CVA value as a running spread. As far as I understood from the Gregory's book (2011) "Counterparty credit risk - The new challenge for global financial markets", to get the annual spread value I have to divide my absolute CVA value by the CDS risky annuity (and then also by the par value to get the value in %). This is something where I am stuck.

The formula of the risk annuity given in the book is (1-exp(-(r+h)(T-t)))/(r+h) where r is the constant continiously compounded interest rate and h is the hazard rate. For my case T-t = 12 years, and the yearly CDS spread is 300 bps. However, in my example r is not constant (I have the upward-sloping initial interest rate term structure).

Can anyone advice me how to estimate CVA as a running spread in this case? If u also can offer a good paper as a reference, I would also be very grateful.

Thanks in advance.

## 1 Answer

This only produces an approximation. As per Gregory (page 256)

However, adding a spread to a contract such as a swap, the problem is non-linear since the spread itself will have an impact on the CVA. The correct value should be calculated recursively (since the spread is risky too) until the risky MTM of the contract is zero.

He points to a simpler solution that doesn't require a recursive solution in Vrins and Gregory (2011)

In regards to what value of r to use for an upward sloping curve, I do believe (but am not certain) you want a risk free rate (such as OIS) for the same maturity as T-t.

• Thank you very much for your answer, unfortunately I can't evaluate it due to reputation constraints – QuackQuack Oct 4 '15 at 20:57
• You're welcome. Now that you've selected an answer, you have sufficient points to up vote :) – AfterWorkGuinness Oct 4 '15 at 21:52