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In Trading Credit Curves Part I by JP Morgan we have that each point on a credit (CDS) curve represents:

$$PV(\text{Fee Leg}) = PV(\text{Contingent Leg})$$

which is

$$S_n \sum_{i=1}^{n}\Delta_i PS_i DF_i + \text{Accrual on Default} = (1-R)\sum_{i=1}^{n}(Ps(i-1)-Psi)DF_i$$

where the accrual on Default is $S_n \sum_{i=1}^{n}\frac{\Delta i}{2}(Ps(i-1)-Psi)DF_i$

where $S_n$ is the spread for protection to period n, $\Delta_i$ is the length of time period i in years, $PSi$ is the probability of survival to time t, $DFi$ is the risk free discount factor to time i, $R$ is the recovery rate on default

I cannot understand why the accrual on default bit is there and i cannot see how it has been derived and the reasoning behind it. I really dont see why you dont just sum to time n when there is a default and discount that? I dont understand why we need the $\Delta_i$ in the first term on the LHS as it seems superfluous.

I suppose really I dont understand the LHS of the equation derivation at all.

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The formula for the accrual on default $$ S_n \sum_{i=1}^n \frac{\Delta_i}{2}(Ps(i-1)-Ps(i))DF_i $$ is just an approximation that says conditional on default occurring within period $i$ (probability of $Ps(i-1)-Ps(i)$), defaults occurs on average in the middle of the period, thus the $\frac{\Delta_i}{2}$ average accrual time from beginning of period to default.

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  • $\begingroup$ Ok thanks, yeah i understand this now, but I dont see why we need the $\Delta_i$ in the LHS??? $\endgroup$ – Permian Feb 8 '18 at 9:24
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    $\begingroup$ $S_n \Delta_i$ is the fixed leg coupon paid on a full period ($S_n$ is a rate, not an amount). $\endgroup$ – Antoine Conze Feb 8 '18 at 9:31
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The accrual on default is like the accrued interest on a bond. A credit default swap can be looked as a synthetic bond. As such, with each passing day, interest is earned to the seller of protection (similar to a holder of a bond). The accrual is due to the seller of protection (holder of the bond) but has not been paid since interest is paid on a periodic basis but earned over the entire holding period.

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  • $\begingroup$ The big difference is that on credit event, the interest accrued on a bond is wiped out (just the notional repayment is accelerated), but on the credit default swap, the running spread accrues until the day of the credit event. $\endgroup$ – Dimitri Vulis Sep 17 at 22:01

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