Risky Coupons

In general, all cash flows are risky. Although U.S. Treasury securities are secured by the government's power to tax, that does not make them riskless. Nevertheless, we typically refer to a U.S. Treasury rate as the riskless rate and will continue to do so, as is the convention. Corporate bonds, on the other hand, carry no such guarantees and, to a lesser extent, neither do state nor municipal bonds. As such, these bonds are priced taking into account the perceived likelihood that the issuer may default on its obligation to make payments to creditors. We are now faced with the challenge of pricing bonds that are subject to credit risk.

Here is the intuition on how these bonds are priced: Assume that at each coupon date there is a small likelihood δ that the firm will default and the coupon (plus any subsequent coupons) will not be paid. Assume as well that the likelihood of default in any given period does not depend on the likelihood of default in any other period (default events are independent). Then, for an n-period bond, we can write the expected price as a discounted present value of expected coupons, that is, as:

Thus, is the likelihood that the coupon will be paid in period one, is the likelihood that the coupon is paid in periods one and two and finally, is the likelihood that the coupon is paid in period n and all previous periods. We can solve this power series using the techniques outlined previously to get:

Compare this to our earlier solution:

Admittedly, allowing for default makes pricing more complicated. Upon examination, it is easy to see, though, that setting the default likelihood to zero makes this equal to the standard pricing formula for an n-period bond with riskless coupons. In general, however, if default probabilities are nonzero, then the bond's value clearly declines with δ. In fact, as long as the default probability is greater than zero, then the price with risk is strictly less than the price of riskless bonds. Bondholders therefore pay a discount for risk bonds (alternatively, they pay a premium for riskless bonds).