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Suppose the credit risky present value of some future cash flow at time $T$ is to be calculated, and there are observable (market standard) CDS spreads on the obligor.

Now, I think that one could bootstrap the implied survival probability $S_T$ to time $T$ and use the formula for the PV factor, with $R$ being the assumed recovery rate and $DF_T$ the risk free discount factor:

$$ PV = DF_T[S_T + (1-S_T)R] $$

Alternatively, the CDS spread being a measure of compensation for credit risk, I thought of using an interpolated spread $x_T$ and calculating the PV of the premium leg via $L=\sum_{t_i} DF_{t_i}x_Td_{t_i}S_{t_i}$, with $t_i$ being the quarterly coupon payment dates, $d_{t_i}$ the daycount fraction from $t_{i-1}$ to $t_i$, and $S_{t_i}$ being the cumulative survival probability to $t_i$. This of course uses the simplified assumption of default occuring only at $t_i$ and hence I disregard accrued coupons for defaults in $(t_{i-1}, t_i)$ .

It does not seem as if $1-L = PV$ generally holds. Hence, I was wondering if there is any relation between both approaches and what the correct way to obtain a credit risky PV would be.

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