After some googling, I have made some progress but not enough to come to a conclusion, so here we go:
Given that the CDS spread of a counterparty is 100bp (flat across time) and that the risk free interest rate is 0% (also flat), what is the annual implied probability of default, assuming that as the counterparty defaults, we have paid half of our annual spread? The maturity of the contract is 5 years and the expected recovery rate is 40%.
The present value of the CDS contract for us (the protection buyer) is:
$P(d)*Protection\ leg + (1-P(d))*Premium\ leg=0$,
$Protection\ leg= (1-R)*Notional-0.5*Spread*Notional$ and
$Premium\ leg= Spread*Notional$.
From this I solved that
In this case, I'm assuming this is the hazard rate $\lambda$, which is constant since the CDS term structure is flat. Now, following Hull, we can use the formula
to obtain the (approximate) implied probability of default happening during the time period $(0,t)$.
Now, what if I want to obtain the probability of default happening during, for example, $(1,2)$ or $(2,3)$? How do I properly condition the probabilities?