# Accrual in Default Derivation of Credit CDS Curve

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.

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.

• Ok thanks, yeah i understand this now, but I dont see why we need the $\Delta_i$ in the LHS??? Feb 8 '18 at 9:24
• $S_n \Delta_i$ is the fixed leg coupon paid on a full period ($S_n$ is a rate, not an amount). Feb 8 '18 at 9:31

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.

• 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. Sep 17 '19 at 22:01

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.