With the fiscal situation of governments as shaky as a fat lady in high heels, calculating bond default probabilities is becoming a timely topic.

How do you determine the default probability of a bond? Trying to calculate the probability from fundamentals is a murky task. An alternative is to let the market do at least part of the calculation for you. If a liquid market exists for a bond then its price should tell you something about its default probability. The lower the price of the bond the higher the default probability.

Extracting the implied default probability from the price can be done using a little elementary probability theory and some simple logic. The details of the process are a little tedious however. For anyone interested in how it's done for coupon bonds, you can find the details here.

To try and simplify the picture a little I'm going to look at only zero coupon bonds and leave out a lot of the mathematical details. The result is a simple formula for default probability that can be used as a first approximation for coupon bonds also (emphasis on first approximation).

A zero coupon bond is the promise of a single cash payment at a specific date in the future. The size of the payment is the face value of the bond (also called the par value) and the payment date is called the maturity date.

To get at the default probability you should begin by asking how much you would be willing to pay for such a bond. The question is equivalent to asking how much a dollar in the future is worth to you right now. (What follows involves a little math with some terminology thrown in. If you're only interested in the default probability formula you can skip down to the end.)

Let \(d(T)\) be the amount you are willing to pay right now for a dollar that you will receive a time \(T\) from now. \(d(T)\) is called a discount factor and its value will depend on things like the inflation rate, the supply of dollars, and the return on other investment alternatives. It also depends on the probability of actually receiving the promised payment.

If \(P\) is the price of the bond and \(F\) is the face value then the following simple relationship should hold:

\[P = Fd(T)\]

In a market of bonds with the same duration and face value, it is \(d(T)\) that determines price differences. Some bond will have the highest price and therefore the highest \(d(T)\). This is the bond with the lowest default risk. You can assume the default risk on this bond is zero or at least negligible. If you buy the bond then you can be fairly certain of receiving payment of \(F\).

For any of the lower priced bonds, there is some probability that the full face value will not be paid. This is the default probability. In the event of a default, only a fraction of the face value will be paid.

You can find the \(d(T)\) for a risky bond by turning it into an equivalent risk free bond. You do this by using the payment's expectation value (average) in place of the \(F\) value for the risk free bond. The price formula in this case is then

\[P = F((1-p) + pR)d_0(T)\]

In this equation \(p\) is the default probability, \(R\) is the fraction of \(F\) that is paid on default (called the recovery rate), and \(d_0(T)\) is the discount factor for the risk free bond. The discount factor for the risky bond is therefore

\[d(T) = ((1-p) + pR)d_0(T)\]

If you solve this equation for the default probability you get:

\[p = \frac{1 - \alpha}{1 - R}\]

where \(\alpha\) is the ratio of the risky to the risk free discount factors or equivalently the ratio of the price of the risky bond to the price of the risk free bond.

\[\alpha = \frac{\mbox{Price of risky bond}}{\mbox{Price of risk free bond}}\]

This is a very simple formula for the default probability that you can also use as a first approximation for coupon bonds. When \(\alpha=R\) the probability of default is certain and when \(\alpha=1\) there is no chance of default.

The formula can be simplified somewhat by assuming \(R=0.5\) which is a common value for the recovery rate. In this case

\[p = 2(1 - \alpha)\]

For a more accurate calculation for coupon bonds see the Quantwolf Bond Default Probability Calculator.

© 2010-2012 Stefan Hollos and Richard Hollos

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