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328 THE JOURNAL OF BUSINESS
with the portfolio model itself. This is because a wealthier investor (one whose wealth
exceeds d) with the same preferences and probability beliefs as a poorer one is, by
(21), a possible lender to the poorer one whose wealth is less than d. But the model
assumes that lending is safe, that is, that all loans are repaid with probability 1
while, as we have seen, the poorer investor may not be able to repay.
Case III.—A borrowing arrangement that is consistent with the assumed risklessness
of lending is one which permits borrowing to the extent that ability to repay,
that is, solvency, is guaranteed. Thus, a reasonable constraint on borrowing and
short sales, with considerable intuitive appeal as well, is given by
Pr {xi+1 > 0} = 1 j = 1,. . . , J - 1 . (23)
When (23) is substituted for (17), the solution to (10) is the same as when (17) is
used; that is, it is given by (14), (15), and (16). Thus, myopia is not optimal in this
case either.
V. THE GENERAL CASE
It is readily verified that the conclusions of Sections III and JV are not changed
if the number of risky investment opportunities is arbitrary. Moreover, the conclusions
hold for all of the functions (5), (6), and (7) whenever /x > 0, both with
no borrowing and in each of Cases I—III. Finally, when r, ^ \,j = 1, . . . , / — 1,
partial myopia, as defined by Mossin, is not optimal either in any of the preceding
cases. It should be noted that a solution need not exist in Case III unless the "noeasy-money
condition" holds. 4 A generalization of this condition (for the case when
yields are serially correlated) is given in Section VI. In the most general version of
Case III, the set ZJ{XJ) in (4) is given by those z;- which satisfy
za > 0 iiSj (24)
and (23).
When M = 0 in (6) and (7), [(S) is of no interest when /x < 0], complete myopia is
optimal in both Cases I and III but not in Case II, as is easily shown. The Mossin-
Leland conclusions concerning complete myopia when n = ti = . . . = »v_i = 1 and
partial myopia do not apply in (6) and (7) when y. < 0 either, except in Case III,
as we shall demonstrate below. In doing so, we shall also show that, when ^/0,
(5), (6), and (7) imply that the optimal investment policies at decision points 1,
. . . , J — I are never myopic in the presence of explicit borrowing limits of any
kind, with one exception.
When a solution to the portfolio problem at decision point / — 1 exists in the
presence of constraints (24) only, the optimal lending strategy ZI,J-AXJ-\) has the
form 2i,/_i(*/_i) = (1 — \AJ^I)XJ-I — Aj.-\n in the case of (6) (X = 1) and (7) and
the form Si,j_l(a;y_i) = Xj-i — Bj_, in the case of (5), where Aj-\ and Bj_i are
constants, generally positive, 6 which depend on r j—\.
Let us consider (6) and (7) when \,p > 0. Since Aj-i, and hence, — $I,J-I(#J-0I
4 Nils Hakansson, "Optimal Investment and Consumption Strategies under Risk for a Class of Utility
Functions," Econometrica 38 (September 1970): 587-607.
6 Nonpositive A}_x and B,_i imply that total short sales exceed or equal total long investments.
8 Nils Hakansson, "Risk Disposition and the Separation Property in Portfolio Selection," Journal of
Financial and Quantitative Analysis 4 (December 1969): 401-16.
3. MYOPIC PORTFOLIO POLICIES 405