STOCHASTIC

ON OPTIMAL MYOPIC PORTFOLIO POLICIES 329
may be arbitrarily large, any finite borrowing limit has a chance to be binding.
Consequently, for fj-i{x) to be a positive linear transformation oifj(x) in the presence
of a borrowing limit, it must be a positive linear transformation of //(*) whether
the borrowing limit is binding or not. From Section IV, it is apparent that, when
Mj — 2 and (17) holds, a necessary and sufficient condition for/^_i(*) to be a positive
linear transformation of fj(x) is that z*,j_i(a5j_i) has the form Z*,J_I(KJ_I) =
a2,j-i(\xj-i + M), where ai,j-, is a nonnegative constant. When there is more than
one risky asset and (24) holds, this condition generalizes to
Z*J--I(^-I) = ai.j-iO^xj-i + M) i = 2, . . . , Mj_i , (25)
where the a,-,j_i are constants, nonnegative only for i (E Sj^i. 7 It is now clear
that the optimal investment strategy z*_i will have the form (25) if and only if
(1) the borrowing limit is not binding or (2) the borrowing limit has the form
M j-i
—*i.j-i(= J2 * 1, X, M > 0 . (26)
The latter assertion follows from (2) and the fact that this form of the borrowing
limit does give the solution (25) for any Fj-lt as is easily verified; moreover, only
(26) is capable of giving a solution of form (25) when the borrowing limit is binding.
Since knowledge of whether any given borrowing limit is binding or not requires
knowledge of Fj-i, it follows that fj-i(x) is myopic in the presence of a borrowing
limit only if this limit has the form (26). By induction, fi(x), . . . ,fj-i(x) are then
myopic in the case of (6) and (7) for X,yu > 0 if and only if the borrowing limit in
period j is given by
(XCy - \)XJ + Cm \Cj> 1, X, » > 0, j = 1, . . . , / - 1 . (27)
When ii < 0 or X < 0 in (6) and (7), any borrowing limit would, to be consistent
with myopia, again have to have the form (27). But when M < 0, we must have
X > 0 and vice versa so that (27) cannot be nonnegative for all Xj > 0 for which
borrowing may be desired, a basic requirement of any "true" borrowing limit. The
situation in the case of function (5) is analogous. As a result, fi(xi), . . . ,/j_i(a;/_i)
can never be myopic for (5), (6), and (7) when X < 0 or //. < 0 in the presence of
a borrowing limit.
Turning now to the solvency constraint (23), we obtain whenever a solution exists
for (6) and (7) that the greatest lower bound on b such that Pr[xj < b] > 0, for
any decision at decision point / — 1 which satisfies (24), is KJ-I(\XJ-I + y.) +
fSi.J-iCffj-i), where Kj_t is a constant which depends on Fj-_i. Since //(—MA) = °°
for X > 0, we obtain, letting xj = KJ-I(\XJ.I + M) + ^i,/-i(*y-i), that \xj +
M > 0, which implies, since X and /* cannot both be negative, xj > 0 when
M < 0. Thus the solvency constraint (23) is not binding when n < 0 but may
be when fi > 0. Consequently, the induced utility functions fi(x), . . . ,fj-i{x) are
myopic for the class (6) and (7) when n < 0 in the presence of (24) and the solvency
constraint (23).
7 Ibid.
406 PART IV. DYNAMIC MODELS REDUCIBLE TO STATIC MODELS