STOCHASTIC

150 STEPHEN P. BRADLEY AND DWIGHT B. CRANE
bles. Hence, if one profitable ray is generated at an iteration all new rays generated,
profitable or not, are in fact added to the restricted master as columns. As the restricted
master is augmented by more and more columns, those columns not in the current
basis are retained provided storage limitations permit. As storage limitations become
binding, those columns that price out most negatively are dropped.
5. Conclusions
The three-period three-event problem used to illustrate the model structure is
perhaps misleadingly small for illustrating the requirements of a realistic bond portfolio
problem. Let us consider a four-period model covering two years where the periods
are 3 months, 3 months, 6 months, and 1 year. Variable length periods are chosen to
reflect our greater uncertainty about events more distant in the future. Let the security
classes represent maturity categoriesof U.S. Government bills, notes, and bonds, e.g.,
3 months, 6 months, 1 year, 2, 3, 5,10, and 20 years. Buy, sell, and hold decisions must
be made for each of these security classes at the start of each period conditional on the
preceding sequence of uncertain events. After the decisions in each period a random
event occurs which determines the set of interest rates (i.e., the Government yield
curve) and the exogenous cash flow for the subsequent decision.
The number of random events in each time period depends on the degree of detail
desired and the resulting problem size. For the four-period model under consideration
we take a five-point approximation to the distribution of interest rates and exogenous
cash flow in each of the three-month periods and a three-point approximation in the
six-month and one-year periods. Our motivation is that over the first six months some
reasonable forecasting of interest rates is currently possible; however, beyond that
point merely assigning probabilities to the three events corresponding to rising, unchanging,
and falling rates is difficult. Since a tightening of credit conditions is normally
associated with an increase in rates, an exogenous cash outflow is assumed to occur
with a rate increase, representing a need for funds in other parts of the institution.
Similarly, no exogenous cash flow is assumed to occur with constant rates and a cash
inflow is associated with falling rates. The resulting general formulation has 2,421 constraints
and 5,328 variables which is clearly a large problem. By applying decomposition
the resulting linear programming restricted master has 181 constraints and 248
subproblems each of which can be solved very quickly. The number of decision variables
will of course remain the same; however, a large number of these will be zero.
With a model of this size the data requirements are rather extensive. This data
estimation problem stems from the need for portfolio managers to specify for each
event sequence both the interest rate structure and its associated probability. We have
simplified this effort somewhat in our model by requiring only three future interest rate
estimates for each future yield curve: the one-year rate, the twenty-year rate, and the
highest rate of the forecast yield curve. Our research indicates the remainder of the
yield curve can be satisfactorily approximated using the following equation: 2
.ft cl
y = at e ,
where
y = yield to maturity,
( = time to maturity,
a,b,c = parameters estimated from the three points on the yield curve.
1 An equation of this type was suggested to the authors by Kenneth E. Gray.
498 PART V. DYNAMIC MODELS