Decomposing Carry and Roll for Upfront Credit Default Swaps

Summary

The document sets out a conventional decomposition of credit default swap position profit and loss into carry, roll down, and curve shift. It gives formulas for carry and roll down using the horizon, notional, quoted spread, maturity-specific spread change, and risky present value of a basis point. The question is how to adapt those terms when a CDS is quoted with a standardized coupon and an upfront payment rather than a par spread.

The author points out that the cited formulas come from the pre Big Bang CDS convention and asks whether upfront value should be amortized over the holding horizon, assuming zero interest rates, before calculating implied spread curve movement for zero profit and loss. No answer or supporting evidence is included, so the treatment of upfront cash flows remains unresolved. The formulas are presented as background definitions rather than a validated method for upfront CDS positions; accrued premium, discounting, and contract specifics may also matter in an implementation.

Key ideas

• CDS position profit and loss can be separated into carry, roll down, and curve shift.
• The stated conventional carry term scales horizon, notional, and CDS spread.
• The stated roll down term uses the spread difference across maturities and risky present value of a basis point.
• The document asks how upfront payments under standardized coupon conventions should affect carry and roll calculations.
• It does not provide an answer or establish that linear amortization is appropriate.

Tags

# Carry & Roll for Upfront CDS


# Carry & Roll for Upfront CDS












We can decompose the P&L for a CDS Position as follows:

P&L = P&L(Carry) + P&L(Roll-Down) + P&L(Curve Shift)

For a fixed horizon (h), Notional (N) and CDS Spread for a maturity T (S(T)) we can find the following definitions of the first two terms in several research papers:

P&L(Carry) = h * N * S

P&L(Roll-Down) = (S(T) - S(T-h))*Risky PV01(T-h) (as seen from the cds curve today)

with the usual definition of Risky PV01 in the papers.

With this, we can back out the implied curve Shift to make the above P&L equal to zero. However, all papers and books I found describe these methods in the pre big bang CDS world (with par spreads instead of upfront CDS).

Question: Do you know any literature on how to rewrite these formulas for the upfront CDS case? Example:

If (e.g. as a protection buyer with standardized fixed coupon of 1% and quoted spread of 0.8%) we receive an upfront payment of U, how would you modify the Carry and Roll formula? Obviously, there are rates effects as I can reinvest the upfront but that's not my point (so assume rates are zero). As the upfront amount is based on the full maturity we probably should not take the full amount into account for the horizon h. Should we, e.g., amortize linearly?

It would be then interesting to back out the implied curve shift for the quoted spread in order to have zero P&L in the upfront case (this obviously will differ from the implied shift in the non-upfront case).

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.