Why CDS Risky Duration Includes Half a Coupon at Default
Summary
The document explains the half-period term in a common credit default swap spread formula. The denominator includes both premium payments made when the contract survives to a coupon date and premium accrued if default occurs between coupon dates. Because the protection buyer owes accrued premium through the default date, the default time within the period affects the expected premium leg.
The formula’s half-coupon adjustment follows from assuming default is equally likely at any time during a coupon period. Under that simplifying assumption, the expected elapsed portion is half the period, so the accrued premium is approximated as half a coupon. The answers also distinguish CDS treatment from a cash bond, where accrued coupon is generally lost on default. This is a convention-based approximation; actual accrued premium depends on the default date and contract terms.
Key ideas
- The CDS premium leg can include an accrued payment when default occurs between coupon dates.
- The half-period adjustment assumes default is uniformly likely throughout a coupon period.
- The denominator’s survival component represents scheduled premiums, while its default component represents accrued premiums.
- A cash bond’s accrued coupon treatment on default differs from the CDS accrued-premium obligation.
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# The factor of 1/2 used in CDS spread derivation
# The factor of 1/2 used in CDS spread derivation
Looking at the CDS spread formula, most of the variables are intuitive but only “2” in the equation that I’m stuck with.
$$S = \frac{(1-R)\sum D(t_i )(q_{i-1} - q_i)}{\sum D(t_i)q(t_i)d_i + D(t_i)(q_{i-1} - q_i){d_i\over2}}$$
Where:
- D(t)= discount factor for date t
- q(t)=survival probability at time t
- S=annual premium
- d=accrual days
Q: what does “2” represent here? Why does it have to be divided by 2?
P.S. I’ve also read this post but still cannot understand.
## Answer by siou0107 (score 2, accepted)
https://quant.stackexchange.com/a/51527
This adjustment takes into account accrued premium at default. Upon default in period $\left(t_{i - 1}, t_i\right]$, the protection buyer owes the protection seller $S \times d\left(t_{i - 1}, \tau\right)$, where $\tau$ is the default time.
Basically, the default can occur at any time between two coupon dates, but it is reasonable to assume it happens midway (on average). With this assumption, the protection buyer thus owes half the coupon payment to the protection seller upon default.
Thus, in your denominator, which practitioners call the risky duration of the CDS, the first part correspond to the coupons paid upon survival and the second part corresponds to the coupon paid upon default.
## Answer by Dimitri Vulis (score 1)
https://quant.stackexchange.com/a/51529
One important difference between a cash bond and a CDS is that when a cash bond defaults, all the accrued coupon is wiped out, but the notional payment is accelerated. For this reason, the exact date of the default is not very important. The bond holder gets recovery on the principal, but no recovery on the accrued coupon, so you don't have the similar term for a bond, and don't really care how much the accrued coupon had been.
But in the CDS, the running spread continues accruing until the date of the default. For this reason, it is important to determine not only the fact that a default has occurred, but also the exact date when it occurred. CDS inherited this convention from old-fashioned letters of credit (LOC). If you assume that a default is equally likely to occur on any day within the coupon period, then the expectation is that it will happen exactly half-way into the period. This is where the half comes from.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.