Inferring Default Probabilities from Bond and Credit Prices
Summary
The document explains how to estimate default risk from market prices when a bond’s payoff depends on whether the issuer defaults. A simple comparison with a default-free bond can solve for default probability if recovery is assumed. The answer recommends setting recovery first, using market quotes such as recovery swaps when available or relying on historical evidence and expert judgment otherwise.
For practical pricing, it describes fitting a hazard rate so the expected cash flows of a bond or credit default swap match its market price. Rather than use one constant rate, the approach can fit a piecewise constant hazard curve across several maturities using observed debt prices. Cumulative default probability is then derived from the integrated hazard rate. The response cautions that bond yield alone is not the right input for this calculation. The estimates depend on the recovery assumption, the cash-flow model, and the quality and range of available market prices; the document does not give a worked calibration or discuss model uncertainty.
Key ideas
- A simple bond price comparison can imply default probability only after specifying a recovery rate.
- Recovery assumptions may come from traded recovery instruments, historical experience, or expert judgment.
- A hazard rate can be calibrated by matching modeled expected cash flows to observed bond or credit default swap prices.
- A piecewise hazard curve can be fitted across multiple maturities to represent changing default risk over time.
- Default probability through a horizon follows from integrating the hazard rate, rather than reading it directly from yield.
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Full text
# How to estimate probability of default from bond prices?
# 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?
## Answer by Brian B (score 9, accepted)
https://quant.stackexchange.com/a/2072
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.