# How to estimate probability of default from bond prices?

How do you use bond prices/yields to infer probabilities of default? I would think of it as follows:

Create a relationship between default free (e.g., Germany) and defaultable (e.g., Greece) bond prices and solve for $p$:

$$\begin{eqnarray} \tilde{B}(0,T) = B(0,T)rp + B(0,T)(1-p) \\ \frac{1}{1+\tilde{y}} = \frac{r}{1+y}p + \frac{1}{1+y}(1-p) \\ \left(\frac{1+y}{1+\tilde{y}} - 1\right)\frac{1}{r-1} = p, \end{eqnarray}$$ where $\tilde{B}(0,T)$ and $\tilde{y}$ are the defaultable bond price and associated yield respectively, $r$ is the recovery rate and $p$ is the probability of default.

A) Is this something you would actually use in practice?

B) How do you go about making an assumption on the recovery rate?

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Wow, also if someone could mention how to embed latex, that would be phenomenal. I apologize. –  SpeedBoots Sep 30 '11 at 17:05
@ SpeedBoots: to get your LaTeX rendered properly, you can just wrap an in-line expression in single dollar signs and a separate equation in double dollar signs. –  olaker Sep 30 '11 at 17:24
@olaker: Thanks! –  SpeedBoots Sep 30 '11 at 17:29

Once I have the recovery assumption, I can take any instrument, CDS or bond, and solve for the hazard rate $h$ that makes its sum of expected cashflows agree with the market price. once I have the hazard rate, the probability of defaulting before time $T$ is simply $e^{-hT}$.
What I actually do, though, is choose a set of anchor times $t_i$ for step-function $h(t)$ and simultaneously fit it, as best I can, to all observable debt instrument prices. Usually $\vec{t}=\{0.5, 1, 2, 3, 5, 7, 10\}$. The probability of defaulting before time $T$ is now
$$\exp\left( -\int_0^T h(s) ds \right)$$