STOCHASTIC

MANAGEMENT SCIENCE
Vol. 18, No. 3, November, 1971
Printed in U.S.A.
BOND REFUNDING WITH STOCHASTIC INTEREST
RATES*f
BASIL A. KALYMON
University of California, Los Angeles
The bond refunding problem is formulated as a multiperiod decision process in
which future interest rates are determined by a Markovian stochastic process. It is
assumed that a single bond is to be outstanding at a given time. Given the future requirements
for debt financing, the decision maker must decide whether to keep his
current bond or to refund by issuing a new bond at the current market interest rates.
Over a finite planning horizon, the structure of policies which minimize expected total
discounted costs is studied.
Introduction
The class of models considered in this paper is concerned with the bond refunding
problem formulated as a multiperiod decision process. The bond refunding problem to
be studied assumes that in each of a number of discrete future periods there is a net
requirement for financing which is to be met by having outstanding a single bond
the original term of which is optional. The bond is callable before maturity, and set-up
costs are charged for refunding and issuing a new bond. Given that interest rates
payable on new bonds vary with term to maturity and change through time, the decision
maker must decide in each period whether to keep his current bond (if unexpired),
or to refund and issue a new bond at current market rates. A penalty cost shall
be charged for not meeting fund requirements, and a bonus shall be allowed for any
excess funds during a given period. The objective shall be to minimize the expected
total discounted cost over a finite planning horizon.
Since only a single bond is allowed to be outstanding at any given time, the applicability
of the model is restricted to situations in which either the funding requirement is
unchanged from present levels, or the future requirement is constant but higher than
the currently outstanding bond or there is envisioned a slow increase in the funding
requirement but it is desirable to maintain only a single major debt outstanding at a
time. Thus, the penalty cost or bonus may be used to model the cost of minor shortterm
borrowings or deposits which result. As such, this penalty/bonus would reflect
the secondary motivation (besides interest payment savings) cited in bond refunding,
namely the readjustment of debt level (see [6]).
Over the finite horizon, a nonstationary per-period discount rate shall be assumed,
which might represent a single discount factor based on the long-run cost of capital
or vary with the expected single-period rates. Such a discounting scheme represents a
* Received February 1970; revised August 1970.
t This paper is based on a section of the author's Ph.D. dissertation [3] in the Department of
Administrative Sciences at Yale University and was partially supported by the National Science
Foundation. The author is indebted to Harvey M. Wagner for his guidance and encouragement
in this research.
1 For example, if an investment of $1 million is to be made in the first period which provides
no return for the first 3 periods, and provides a return of $250 thousand in each of the
next 7 periods, then the required financing for this project would be $1 million for each of the
first 3 periods, $750 thousand in period 4, and decreasing by $250 thousand in each period thereafter.
The net requirement would represent the total requirement over all projects of the given
firm.
3. MODELS OF OPTION STRATEGY 563