STOCHASTIC

A DYNAMIC MODEL FOR BOND PORTFOLIO MANAGEMENT 141
The structure of the portfolio model presented here assumes that decisions are made
at the start of each period. At the beginning of the process, the manager starts with a
known portfolio and he faces a known set of interest rates. For ease of exposition it is
assumed that the initial portfolio is cash, but this assumption can be easily relaxed.
He can invest in any of a finite number of asset categories which can represent maturity
groups and/or types of bonds, such as U. S. Governments and municipals.
The results of this initial investment decision are subject to some uncertainty,
represented by a random "event" which occurs during the first period. An event is
defined by a set of interest rates and an exogenous cash flow. For example, one event
might represent a tightening of credit conditions in which interest rates increase and
the portfolio size has to be decreased to finance a rise in loans. It is assumed that there
are a finite number of such events which have a discrete probability distribution.
In addition, the portfolio manager knows the probability of each event, or it is appropriate
for him to behave as it he did. Because of this assumption, the paper treats
risk and uncertainty synonymously.
At the start of the second period, another set of investment decisions is made, taking
into account the initial period decisions and the random event which occurred. A
second random event determines the outcome of these decisions and is then followed
by a third-period decision. This process in general continues for n decision periods
with the rath random event determining the horizon value of the portfolio.
The mathematical formulation of the problem can be specified for a large number of
time periods, but for expositional convenience the formulation below has been limited
to three periods. These are a sufficient number to illustrate how the problem size can
be extended.
In the three-period formulation, bonds can be purchased at the start of periods 1, 2,
and 3 denoted by the subscript n. They can be sold or held at the start of any succeeding
period m, where subscript m = 2, 3. Categories of bonds are denoted by superscript
k. The random events which can occur in periods 1 and 2 are identified by
parenthetical terms ei and ei, respectively.
The decision variables of the model are buy (6n*), sell (s„,m), and hold (h„,„), where
each variable is expressed in dollars of initial purchase price and is constrained to be
greater than or equal to zero. Each of these decision variables is conditional upon events
which precede the time of the decision. For example, a sequence of decisions might be:
hi* = $1,000, h k i,i(ei) = $1,000, and s{,a(«i , ei) = $1,000. In this sequence $1,000 was
invested in type k bonds at the start of period 1. After random event ei occurred, the
bonds were held in period 2. Then after event ei the bonds were sold at the start of
period 3. The cash flow from the sale might be more or less than $1,000 depending
upon whether the sale price was more or less than $1,000. (The definition of the decision
variables and parameters is summarized in Table I.)
In addition to the exogenous cash flows associated with random events, there are
endogenous cash flows associated with the portfolio decisions. There is an income yield
(yn k ) stemming from the semiannual coupon interest on the bonds and a capital gain
or loss (ffn.m) which occurs when bonds are sold. Both are expressed as a percent of
initial purchase price. It is assumed that taxes are paid when income and/or gains are
received, so that the y and g coefficients are defined as after-tax. Transaction costs are
taken into account by adjusting the gain coefficient for the broker's commission, i.e.,
bonds are purchased at the "asked" price and sold at the "bid" price.
The model is designed on a cash accounting rather than on the accrual basis normally
used by banks for reporting purposes. On balance, this is a more accurate method be-
1. TWO-PERIOD CONSUMPTION MODELS AND PORTFOLIO REVISION 489