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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

of default.-2-

In a seminal paper, Black and Scholes [1] 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.

Scholes model.

Merton [5] clarified and extended the Black-

While options are highly specialized and relatively unim-

portant financial instruments, both Black and Scholes [1] 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 [7] 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