To be presented at the American Finance Association Meetings, New York, December 1973.
The value of a particular issue of corporate debt depends esentially
on three items:
(1) the required rate of return on riskless (in terms of
default) debt (e.&., government bonds or very high-grade corporate bonds);
(2) the various provisions and restrictions contained in the indenture (e.g.,
maturity date, coupon rate, call terms, seniority in the event of default,
sinking fund, etc.); (3) the probability that the firm will be unable to
satisfy some or all of the indenture requiremerits (i.e., the probability of
While a number of theories and empirical studies has been published
on the term structure of interest rates (item 1), there has been no systematic
development of a theory for pricing bonds when there is a significant proba-
bility of default.
The purpose of this paper is to present such a theory
which might be called a theory of the risk structure of interest rates.
use of the term “risk” is restricted to the possible gains or losses to bond-
holders as a result of (unanticipated) changes in the probability of default
and does not include the gains or losses inherent to all bonds caused by
(unanticipated) changed in interest rates in general.
Throughout most of
the analysis, a given term structure is assumed and hence, the price differ-
entials among bonas will be solely caused by differences in the probability
In a seminal paper, Black and Scholes  present a complete
general equilibrium theory of option pricing which is particularly attract-
ive because the final formula is a function of “observable” variables.
Therefore, the model is subject to direct empirical tests which they [21
performed with some success.
Merton  clarified and extended the Black-
While options are highly specialized and relatively unim-
portant financial instruments, both Black and Scholes  and Merton [5, 6]
recognized that the same basic approach could be applied in developing a
pricing theory for corporate liabilities in general.
In Section II of the paper, the basic equation for the pricing
of financial instruments is developed along Black-Scholes lines.
Section III, the model is applied to the simplest form of corporate debt,
the discount bond where no coupon payments are made, and a formula for com-
puting the risk structure of interest rates is presented. In Section IV, com-
parative statics are used to develop graphs of the risk structure, and the
question of whether the term premium is an adequate measure of the risk of
a bond is answered.
In Section V, the validity in the presence of bank-
ruptcy of the famous Modigliani-Miller theorem  is proven, and the re-
quired return on debt as a function of the debt-to-equity ratio is deduced.
In Section VI, the analysis is extended to include coupon and callable