# Default Probability Implied in Bond Prices?

Say I am trying to find the probability of default on JP Morgan implied by the price of their fixed income assets. Can this be done? Are there any pitfalls to this approach? I have heard of this approach being used, but haven't been able to find many clear implementations in the literature. How can I estimate JP Morgan's default probability using their quoted bond prices?

## 3 Answers

Assume :

• $$R$$ a recovery rate,
• a continuous payment
• a flat intensity $$\lambda$$ i.e $$\mathbb{P}(\tau>t)=e^{-\lambda t}$$
• a flat discount rate $$r$$

## With bonds prices

Assuming

• JPM bond pays a coupon rate of $$\kappa$$
• the risk free bond (being US bonds) pays a coupon rate of $$\kappa^{risk~free}$$

you have :

$$\text{PV}(\text{Bond}_{JPM}) = \int_{0}^T \kappa e^{-(r+\lambda) t} dt + R\int_{0}^T\lambda e^{-(r+\lambda)t}dt + e^{(r+\lambda)T}$$

where as

$$\text{PV}(\text{Bond}_{risk~free})=\int_{0}^T \kappa^{risk~free} e^{-rt}dt + e^{-rT}$$

with this, you can find some $$r$$ and $$\lambda$$ solving above equations.

## With CDS

If you have the CDS of maturity $$T$$ on JPM, as a quoted spread $$s$$, you will have the following $$\lambda \approx \frac{s}{1-R}$$

## Be careful

As mentionned in other answers, recovery is the key.

One more thing that must be considered is the expected recovery rate. A model that ignores this rate is not tied to the real world. To estimate the probability of default, you would need to find the rate that needs to be applied to each time step/payment such that risk free discounting of payments yields the price of the bond. Specifically, Price = $\sum{P((}$No default in period $)* PV($payment at risk-free rate$))$. By taking the spread over the risk-free rate, recovery (inherent in the price of the bond) is ignored. You can get an estimate of the probability of default by dividing the spread by $1-recovery rate$. If there is are no coupon payments, these should be the same.

I think it depends on your goals and how sophisticated you wish to be. At the lowest level, one can just take the spread of JPM over some relatively risk free rate (Treasurys or swaps) and declare that is the probability of default. Others (e.g. Elton, Gruber, et al in Explaining the Rate Spread on Corporate Bonds) try to measure the components. While there is a default component, for bonds there is also a tax premium and a risk premium. They try to measure those components and separate out the remainder as the probability of default if I recall correctly. If I were asked to do that project, I’d say that much of the spread movement is due to systemic factors such as macroeconomics that do not reflect JPM risk, and instead use some kind of Stein estimator approach combining the market spread with fundamental factors, maybe a Merton model input, and priors about too big to fail. I’m sure others would see this differently. Oh, and upon re-reading this of course you would have to make some recovery rate assumptions if you are using a spread model cause not all of the face value is lost upon default. There was actually a very similar question posted a while ago on this you can look up.