Hi,I am trying to determine the relationship between the spread of zero-coupon bond over the risk free and its recovery rate in the case of default. The purpose is to find the probability density function for the default of a bond and will be used to value a credit default swap on that bond.I start with the following variables and relationships:V = Value of bond including accrued interestA(t) = Accrued interest as a percentage of face valueF = Face valuer = risk free rates = spread over risk freeV = F*(1+A(t))Bond Price = V.exp(-(r+s).t)Bond Price = (1-p).V.exp(-r.t) + p(V*R).exp(-r.t)From this, we can find P(t), the probability function and then can find the pdf by taking the derivative w.r.t. t.P(t) = [1 – exp(-st)]/(1-R)In this relationship, as R goes to 1, the P goes to inf. The problem I see with this is that s is not independent from R. I want to be able to write s in terms of R (or R in terms of s) and derive a relationship that makes sense (one which results in a probability function with the range [0,1]. Alternatively, is there a way to estimate the relationship empirically?I am looking for a (hopefully) somewhat simple solution. Not in a QF program… I am in an undergrad business program with a few years of math background. Am trying to work with the Hull/White paper on CDS (found here) and need to find a pdf (q(t) on p.13) to then create a numerical model in excel. I appreciate anyone’s thoughts.

You only have a problem if R = 1 and s > 0. This makes sense because a bond that will lose nothing if it defaults is a risk-free bond, if it returns more than the risk-free rate you have an inconsistency. No probability of default can fix that.More generally, if exp(-st) < R then you can buy the bond today for less than its present value even assuming default. Obviously, no probability of default will make that consistent.It's good that your model gives you impossible probabilities for inconsistent situations. It gives you faith that it is correct, and it's less likely to get you into trouble.My other comment is that you're being sloppy about time. Only A is shown as an explicit function of time, but s, v, r, P and R can all depend on time. Your formulae for Bond Price may give different values for different values of t.