# Robust way to calculate credit risky PV from CDS spreads

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.