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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??? $\endgroup$
    – Trajan
    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). $\endgroup$ 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. $\endgroup$ Sep 17 '19 at 22:01
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Simply speaking, as mentioned by Antoine, the accrual arises because default may happen between two payment dates and the accrued payment should be paid. $\Delta_i$ is the year fraction. Since $S_n$ is quoted as an annual rate, $S_n\Delta_i$ is the payment amount per $1 notional.

However, in the formula you mentioned, default is modeled at the same frequency as coupon payments. More generally, we can model default at a different frequency in a more granular way.

As a simplified example, let $t=1,\ldots,T$ be possible default dates, such as months and $d$ be the period between two payment dates ($d=3$ for quarterly payments, in this case $\Delta\approx0.25$ depending on day count convention). Let $\tau$ be the random default time with $1\leq\tau\leq T$ and $\tau\in\mathbb{Z}$. We assume that $T$ is a multiple of $\Delta$. Then the formula becomes $$ \frac{S_n}{d}\sum_{j=1}^{T/d}\mathbb{E}_0^Q\left[\tilde{DF}_{jd}I(\tau>jd)\right]+\frac{S_n}{d}\sum_{i=1}^T\mathbb{E}_0^Q\left[\tilde{DF}_i\left(\frac{i}{d}-\left\lfloor\frac{i}{d}\right\rfloor\right)I(\tau=i)\right] = (1-R)\sum_{i=1}^{T}\mathbb{E}_0^Q\left[\tilde{DF}_iI(\tau=i)\right] $$ where $\mathbb{E}_0^Q$ denotes conditional expectation at time 0 under the risk-neutral measure, $I(A)$ is an indicator function of event $A$, and $\lfloor x\rfloor$ is the largest integer that is less than or equal to $x$.

Here, $jd$ runs over all payment dates while $i$ runs over all possible default dates. The accrual on default arises due to mismatch between values of $jd$ and those of $i$. For example, if default happens at $i=4$, then one month has passed after the last payment date, the accrual is the payment corresponding to this 1 month period.

Finally, we assume independence between default and discount factor under the risk-neutral measure and define $$ DF_{t}=\mathbb{E}_0^Q[\tilde{DF_t}],\quad SP_{t}=P^Q(\tau>t)=\mathbb{E}_0^Q[I(\tau>t)] $$ then $\mathbb{E}_0^Q[I(\tau=t)]=SP_{t-1}-SP_{t}$. We can rewrite the above equation as $$ \frac{S_n}{d}\sum_{j=1}^{T/d}DF_{jd}SP_{jd}+\frac{S_n}{d}\sum_{i=1}^TDF_i\left(\frac{i}{d}-\left\lfloor\frac{i}{d}\right\rfloor\right)(SP_{i-1}-SP_i) = (1-R)\sum_{i=1}^{T}DF_i(SP_{i-1}-SP_i) $$

Now to conform to the equation you mentioned, we have to rewrite the summations of $i$ using $j$ with step $d$. Let $\tau^d$ be the random default time at payment dates, defined as $\tau^d=jd$ if $(j-1)d<\tau\leq jd$. For example, $\tau^d=3$ means default happens between time $0$ (excluded) and time $3$ (included). Then, we have $$ P^Q(\tau^d=jd)=\sum_{k=(j-1)d+1}^{jd}(SP_{k-1}-SP_{k})=SP_{(j-1)d}-SP_{jd} $$ To add discount factor, we need to make the following assumption: $$ DF_{(j-1)d+1}\approx\cdots\approx DF_{jd-1}\approx DF_{jd} $$ Then, the RHS becomes $$ (1-R)\sum_{j=1}^{T/d}DF_{jd}(SP_{(j-1)d}-SP_{jd}) $$ To rewrite the second term on the LHS, we assume $$ \sum_{i=1}^TDF_i\left(\frac{i}{d}-\left\lfloor\frac{i}{d}\right\rfloor\right)(SP_{i-1}-SP_i)\approx\sum_{j=1}^{T/d}DF_{jd}\frac{1}{2}(SP_{(j-1)d}-SP_{jd}) $$ which combines the assumption on the discount factor and an additional assumption that $x-\lfloor x\rfloor$ is on average around $\frac{1}{2}$.

Finally set $\Delta=1/d$, we arrive at the equation you mentioned, after re-defining the frequency.

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