STOCHASTIC

A DYNAMIC MODEL FOR BOND PORTFOLIO MANAGEMENT 143
The cash flow constraint for the start of period 3, and any additional periods in a
larger model, is analogous to that of period 2:
- S* yi k hi,2(ei) - X* (1 + ?*.s(ei, e»))**,»(«i, e8) - £* 8/a*(ei)6j*(ei)
(2.3) — 2* (1 + ff2.s( e i > e2))*j,»(ei, e>) + ]C* &»*(ei, «s) = /»(ei, e2),
V(ei, e.) e EiX E2.
The number of terms in this constraint is much larger because it is necessary to account
separately for bonds purchased at different points in time. For example, bonds sold
at the start of period 3 could have been purchased at the start of either period 1 or 2
and the gain or loss which results depends upon this distinction. As with constraint
(2.2), it is necessary to have one constraint of type (2.3) for each uncertain event sequence
ei, e2.
The sale of securities is constrained by inventory balance equations which require
that the amount of a security class sold at the start of period m must be less than or
equal to the amount of that security held during the previous period. The difference
between the amount previously held and that which is sold is the amount to be held
during the next period:
(2.4) -6!* + s{,2(ei) + hU(ei) = 0,
(2.5) —h\,t(ei) + Si,i(ei, e2) + Ai,»(ei, e2) = 0,
(2.6) -b,*(e,) + *2\»(ei, e») + Aj\»(ei, e2) =0, Vk £ K and (e,, e2) ^,Xft.
It is important to point out that this formulation of the problem includes security
classes that mature before the time horizon of the model. This is accomplished by
setting the hold variable for a matured security to zero (actually dropping the variable
from the model). This has the effect, through the inventory balance constraints, of
forcing the "sale" of the security at the time the security matures. The value of the
gain coefficient reflects the fact that the security matures at par with no transactions
cost.
Theoretically, we would like to solve this problem with the objective of maximizing
preference for terminal assets. This function would be difficult to specify though, since
it involves preference for assets over time. In addition, problems of realistic size would
be hard or even impossible to solve numerically because such objective functions are
nonlinear. Therefore, in lieu of a nonlinear preference function, we have added a set
of constraints which limit the realized net capital loss per year. This approach is
similar to the "safety first principle" which some, e.g., Telser [13], have used in portfolio
models. In our model, a limit on downside risk is expressed as an upper limit on
capital losses. These constraints lead to some hedging behavior as the possibility of an
increase in rates causes some short maturities to be held to avoid losses on sale.
Loss constraints are particularly appropriate for banks, in part because of a general
aversion to capital losses, but also because of capital adequacy and tax considerations.
Measures of adequate bank capital, such as that of the Federal Reserve Board of
Governors, relate the amount of capital required to the amount of "risk" in the bank's
assets. Thus, a bank's capital position affects its willingness to hold assets with capital
loss potential. Tax regulations are also important, because capital losses can be offset
against taxable income to reduce the size of the after-tax loss by roughly 50 %. As a
result, the amount of taxable income, which is sometimes relatively small in commercial
banks, imposes an upper limit on losses a bank is willing to absorb.
1. TWO-PERIOD CONSUMPTION MODELS AND PORTFOLIO REVISION 491