## Market-Based Estimation of Implied Corporate Credit Risk

Corporations that raise capital in the debt markets need to pay a spread over risk-free instruments to compensate the investor for the increased *default risk* that they assume when buying bonds issued by the said corporation. Thus the buyer of such "credit-risky" bonds need to be able to estimate the likelihood of default and compare this to the default rates implied by the current trading spread of the bond. Sure, credit rating agencies do dish out ratings such as AA, AAA, BBB etc but these are far too broad and not that useful here.

There are more exotics techniques but lets first examine a simplistic back-of-the-envelope approach to calculate implied credit risk. Assume *p* is the **probability of default** of the corporation, and* r* is the **recovery rate** if/when a default happens.

Assume a corporate bond is currently yielding 12% and government bonds with the same term are yielding 10%. Further suppose that if the corporate bond defaulted the recovery rate would be 50%.

By definition, the (US) government bond is default-free and thus returns $1.1 annually for every $1 of face value. During this same time period, an investor holding a corporate bond could have one of two things happen. The issuer could default, with probability *p*, or it might not with probability (1-*p*). The payouts associated with these events are, respectively, $1.12 and $0.5 per dollar of face value. Generalising this it it would look as follows:

The *efficient-market hypothesis* implies that all available information is used in pricing the instrument thus the corporate bond price/yield already has the credit risk probability embedded in it. Thus an investor should be indifferent to investing in the government bond or the corporate bond. We can therefore look at the expected value of both instruments after one period and solve for the implied credit risk.

1.1 = (1-

*p*)1.12 + 0.5

*p*

=> 1.1 = 1.12 - 1.12

*p*+ 0.5

*p*

=> 0 = 0.02 - 0.62

*p*

=>

*p*= .02/.62 = 3.226%

In other words, this crude approach indicates that there is a 3.226% chance of default in the next 12 months for the said corporation.

As an aside, because only two events can happen, and we look only at distinct time epochs, this approach is effectively a *one-period, discrete-time binomial tree model*. Of course it is flawed in many ways (the real-world is continuous not discrete), but it's a good enough quick guess.

*12 Feb 2011*
*Damien Wintour*