Two contingent payment legs
The protection buyer pays a running coupon while protection remains active and receives a contractual settlement after a qualifying credit event. A par spread balances the model value of those legs at inception; standardized contracts can instead combine a fixed coupon and an upfront payment.
A simple discrete model
Let notional be (N), payment dates (t_i), accrual fractions (alpha_i), discount factors (D_i), survival probabilities (Q_i=P( au>t_i)), recovery assumption (R), and annual coupon rate (c) in decimal units. Set (Q_0=1).
$$PV_{premium} \approx N c \sum_{i=1}^{n} D_i Q_i \alpha_i$$
$$PV_{protection} \approx N(1-R)\sum_{i=1}^{n}D_i(Q_{i-1}-Q_i)$$
This approximation puts default settlement at period endpoints and omits accrued premium on default. A production valuation must handle exact timing, accrual, calendars and the applicable contractual/model conventions.
Ignoring upfront and these refinements, the par spread is:
$$s_{par} \approx \frac{(1-R)\sum_iD_i(Q_{i-1}-Q_i)}{\sum_iD_iQ_i\alpha_i}$$
Hazard intensity and survival
A continuous-time hazard intensity is a conditional default rate per unit time, not itself a finite-period default probability. Under constant intensity (lambda):
$$Q(T)=e^{-\lambda T},\qquad P(\tau\leq T)=1-e^{-\lambda T}$$
The rough credit triangle (s\approx\lambda(1-R)) estimates intensity from spread under simplifying assumptions. It does not make (s/(1-R)) the unconditional probability of default over every horizon. A non-flat calibrated curve uses an integrated intensity and market/model assumptions.
Fixed coupon, upfront and mark-to-market
For the protection buyer, model value is protection-leg PV minus premium-leg PV before any upfront settlement. If the agreed coupon is below the market par spread, an upfront payment from the buyer can balance the contract. If it is above par spread, the direction can reverse. Accrued premium must be treated separately.
For an existing fixed-coupon contract, a widening market spread generally benefits the protection buyer and hurts the seller, holding other inputs fixed. A first-order estimate relates the spread difference to the risky annuity; use decimal/basis-point units consistently. It is not obtained by changing the existing contractual coupon.
Example: protection sold at150bp and offset with matching protection bought at100bp has a50bp annual coupon difference while the relevant contracts survive. Counterparty, collateral, default, maturity and settlement differences can prevent a perfect lock-in.
Recovery and basis
Recovery is an input to implied-curve calibration, and an actual auction settlement value is a separate realized contract outcome. A40% assumption is used for some senior-unsecured standard-converter cases; it is neither universal nor a forecast guaranteed to occur.
The CDS–bond basis compares a CDS spread with a specified bond-spread measure. Funding, collateral, liquidity, delivery options and counterparties matter; a quoted difference is not automatically risk-free arbitrage. Older LIBOR examples are historical and should not be silently reused with current curves or documentation.
Takeaway
Keep coupon, par spread, upfront, probability, intensity and risky annuity distinct. State the model assumptions and units before interpreting any number.
Reference: ISDA standard CDS converter specification (historical2009 conventions); ISDA CDS model resources.