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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
up vote 6 down vote accepted

In practice, I would begin with the recovery assumption. In the case of Greece, dealers are probably already quoting recovery swaps, allowing you to set this parameter directly. In general, you have to be willing to make assumptions based on history or on conversations with bankruptcy experts.

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

$$ \exp\left( -\int_0^T h(s) ds \right) $$

Note in particular that the "yield" has nothing to do with these calculations.

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