# Interpolation on CDS rates

I am just wondering if there is any way we could calculate a CDS Spread (not harzard rate) on a CDS curve. Most of the papers that I have come across so far discuss about interpolating the hazard rates using numerical root-finding algorithms by setting the break-even spread to 0. However can I assume that the hazard rates derived this way will give me a "proper" interpolation on the CDS spread? i.e., say if I have the 3 month and 6 month CDS spread, and I need to price the spread for a 4 month CDS, would it be OK if I simply assume the risky PV for the first 3 months is already 0 given the hazard rates that I have derived and just working on figuring out the final one month spread? Is there a direct, nice formula to compute for such a spread given hazard rate using piecewise constant hazard rate assumption between 3 month and 4 month?

Thanks!

The only criterium your interpolation must obey is the absence of arbitrage. Note that, assuming that $spread(3M) < spread(6M)$, $spread(4M)$ can take any value between $spread(3M)$ and $spread(6M)$ without creating an arbitrage opportunity (actually it can be even slightly less than $spread(3M)$ or slightly higher than $spread(6M)$) without creating an arbitrage opportunity).