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:
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.
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.
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.
