STOCHASTIC

142 STEPHEN P. BRADLEY AND DWIGHT B. CRANE
TABLE I
Definition of Variables
bn k (-) — amount of security class k purchased at beginning of period n; in dollars of initial purchase
price
Sn,m(-) = amount of security class k, which was purchased at the beginning of period n and sold
at the beginning of period m; in dollars of initial purchase price
^n,m(-) = amount of security class k which was purchased at the beginning of period n and held
(as opposed to sold) at the beginning of period m; in dollars of initial purchase price
9".m(-) = capital gain or loss on security class k purchased at the beginning of period n and sold
at the beginning of period m; in percent of initial purchase price
2/n*(-) = income yield from interest coupons on security class k, purchased at the beginning of
period n\ in percent of initial purchase price
/n(-) = incremental amount of funds either made available to or withdrawn from the portfolio
at the beginning of period n\ in dollars
L{-) = upper bound on the realized capital loss (after taxes) from sales during a year; in
dollars
(•) indicates that the variable is conditional on the sequence of uncertain events which
precede it
cause it takes into account when cash is actually available for reinvestment. It is not
completely accurate, however, because it ignores the tax effects of amortization of
bond premiums and accretion of discounts of municipal bonds which might be contained
in the starting portfolio. If this were judged to be a significant problem, it
potentially could be solved by setting up separate bond categories for each combination
of maturity and initial purchase price and then adjusting the cash flow and book
value.
The decisions of the model are constrained by a set of cash flow constraints which
express a limit on the funds available for the portfolio. We assume one security class
is a risk free asset, and hence we are always fully invested. At the start of the first
time period a flow of funds, /i , is made available for the portfolio. Given our simplifying
assumption that there are no bonds in the portfolio before the start of the model,
the first-period cash flow constraint simply limits the sum of all purchases to be equal
to/i:
(2.1) £>/ =/!.
Before the start of the second period an uncertain event occurs in which a new set of
interest rates obtains and there is an exogenous cash flow to or from the portfolio.
New investments in the portfolio at the start of the second period are limited by this
cash flow, plus the income received from bonds held in the portfolio during the previous
period and the cash generated from any sales at the start of the period 1
(2.2) -£*)/, V - E*(l + ffi.2(ei))*Ue,) + £*k*(ei) = Mei), Ve, £ £, .
Note that the second-period decision variables, Si,2(ei) and W(ei), and the gain or loss
resulting from security sales, gi.iiei), are conditional upon the event which occurs in
the first period. Since there are a number of such events, there must be one constraint
of type (2.2) for each event which can occur.
1 For expositional convenience, this and subsequent equations assume that interest is paid each
period. In the actual model, if the initial two periods were three months each and interest were
paid in six-month intervals, there would be no cash flows until the end of the second period.
490 PART V. DYNAMIC MODELS