STOCHASTIC

172 BASIL A. KALYMON
more general position compared to the extremes of the Weingartner [6] discounting
method which uses the interest rate the firm will be paying on issued bonds, and the
Pye [4] discounting scheme in which the precise realized single-period market rate is
used. The discount factor as used in our model reflects the time preferences of the
decision maker, recognizing the imperfections of the market which do not allow him
complete flexibility in borrowing and lending. Such a discount factor would be affected
by the market interest rate in as much as the latter contains information on future
inflation, but would be principally determined by internal considerations. Note that
the importance of this factor is deemphasized in the model in that it is used merely to
compare different streams of interest payments and issuing costs and not to evaluate
the utility of repayment of the principal. As noted above in this regard, the model
assumes that the firm has already decided on the appropriate funding requirement in
accordance with its investment opportunities. Current practice employs a single discount
rate representing either a cost of capital or some type of single-period or multiperiod
market rate.
There has been little previous work in bond refunding as a multiperiod decision
process. Weingartner [6] is the only one to explicitly formulate the problem as a
multiperiod process. His model is completely deterministic, and assumes that a level
debt will be maintained, with only a single bond outstanding at a given time. Pye has
studied the use of a Markov process to represent interest rate fluctuation in [5], and
has used such a representation in [4] to determine the value of the call option by assuming
that a single refunding decision will be made in the most favorable period over the
term of the issued bond. Also, Bierman in [1] has proposed using a Markovian representation
of interest fluctuation to improve short-run timing when refunding a bond,
but does not consider the full multiperiod implications. References for work on the
term structure and fluctuations of interest rates may be found in Pye [5]. A summary
of current practice, (using essentially single decision approaches) as well as a discussion
of appropriate discounting rates to be used, can be found in Bowlin [2].
The general model to be discussed is more precisely defined in §1, and the existence
of optimal policies of special structure is proved in §2 under differing sets of assumptions.
Calculation of optimal debt size after a decision to refund has been made is
discussed in §3. In §4, simple necessary conditions for refunding are given for two level
debt models.
1. Model Formulation
In this section, the basic model to be studied is formally defined. Let Di represent
the net requirement for debt financing in period i, for i = 1, 2, • • • , n, with i = n
representing the chronologically first period of the n-period planning horizon. The
Di'a are assumed to be known. For additional clarification we stress that the financing
requirement as used in our model represents the exogenously determined amount of
debt financing which the firm envisions requiring. It is based on projected cash flows
aggregated over all the major projects/investments of the firm. (See Footnote 1.)
Let U , Xi and r, represent, respectively, the time to maturity, the size, and the coupon
rate of the bond outstanding in period i. Let p,,m represent the rate of interest that
must be paid on a bond issued in period i with time to maturity (or term) m. Defining
Pi by pi = pi,*>, we shall define
(1) Pi. = Vi(m, in),
where Vi(m, p,) is an increasing function in pt. Thus !\(m, p.) represents the term-
564 PART V. DYNAMIC MODELS