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%.
My try:
The present value of the CDS contract for us (the protection buyer) is:
$P(d)*Protection\ leg + (1-P(d))*Premium\ leg=0$,
where
$Protection\ leg= (1-R)*Notional-0.5*Spread*Notional$ and
$Premium\ leg= Spread*Notional$.
From this I solved that
$P(d)=\frac{Spread}{(1-R)+0.5*Spread}$.
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
$P(0,t)=1-e^{(-\lambda*t)}$
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?
Thank you!