Dividing CVA by Duration --> gives Credit Spread

[rajeshtr]

Hi David, I am not able to follow how the CVA/Notional is converted into Credit Spread just by dividing by Duration. is there any Conceptual reason or logical way of thinking through this?

 

[David Harper CFA FRM]

Hi @rajeshtr I haven't looked at that closely but it actually does offer a bit of intuition because I see that earlier Gregory suggests we can also (additionally? alternatively?) divide by the "risky annuity" the following (emphasis mine):

"A simple calculation would involve dividing the CVA by the risky annuity [footnote 8: The risky annuity represents the value of receiving a unit amount in each period as long as the counterparty does not default] value for the maturity in question. For the previous calculation, a risky annuity of 3.65 would be obtained using the simple formula described in Appendix 10B (the accurate result for an interval of 0.25 years is 3.59). From the result above, we would therefore obtain the CVA as a spread, being 0.253%/ 3.65 × 10,000 = 6.92 bps (per annum)." -- Gregory, Jon. Counterparty Credit Risk and Credit Value Adjustment: A Continuing Challenge for Global Financial Markets (The Wiley Finance Series) (Kindle Locations 6689-6693). John Wiley and Sons. Kindle Edition.

I don't see the calculation for the "risky duration" but at least directionally it makes sense to me given the reference to risky annuity (which connotes coupons with counterparty risk): this is a way to "plus up" the CVA if it were paid (or charged) over time rather than up-front. Imagine I am your risky counterparty and the MtM riskfree derivative value is $100.00 but I am risky and we have a 10-year contract. You price my up-front counterparty risk at $10.00 so the adjusted value = $100.00 - $10.00 upfront CVA = $90.00. But let's change the deal so that you charge me the CVA over the ten years. But I am risky, is the point of the CVA! So you want more than $10/10 = $1.00 per year because I can default on year 1, year 2 .... year 10. I can't see the risky duration calculation (yet) but directionally, like any duration (on a coupon bond), it's less than the maturity, so it makes sense that you'd divide the upfront CVA by less than the full maturity. (it has the same effect as assigning a higher discount rate to the per annum CVA charge, obviously). I hope that is some directional intuition, sorry I don't have a fix on the "risky duration" (if that's even different than mod/mac duration).

btw, I do see your question at https://forum.bionicturtle.com/threads/cds-bond-basis-factors-confusing-impact.10284/ and I definitely have thoughts, but I will get to that tomorrow, if that's okay. Thanks!